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A supernova remnant coincident with the slow X-ray pulsar AX J1845–0258 B. M. Gaensler11affiliation: Center for Space Research, Massachusetts Institute of Technology, 70 Vassar Street, Cambridge, MA 02139; bmg@space.mit.edu 22affiliation: Hubble Fellow , E. V. Gotthelf33affiliation: Columbia Astrophysics Laboratory, Columbia University, 550 West 120th Street, New York, NY 10027; evg@astro.columbia.edu and G. Vasisht44affiliation: Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109; gv@astro.caltech.edu Abstract We report on Very Large Array observations in the direction of the recently-discovered slow X-ray pulsar AX J1845–0258. In the resulting images, we find a $5^{\prime}$ shell of radio emission; the shell is linearly polarized with a non-thermal spectral index. We class this source as a previously unidentified, young ($<8000$ yr), supernova remnant (SNR), G29.6+0.1, which we propose is physically associated with AX J1845–0258. The young age of G29.6+0.1 is then consistent with the interpretation that anomalous X-ray pulsars (AXPs) are isolated, highly magnetized neutron stars (“magnetars”). Three of the six known AXPs can now be associated with SNRs; we conclude that AXPs are young ($\lesssim$10 000 yr) objects, and that they are produced in at least 5% of core-collapse supernovae. ISM: individual (G29.6+0.1) – ISM: supernova remnants – pulsars: individual (AX J1845–0258) – radio continuum: ISM – stars: neutron – X-rays: stars \submitted (To appear in The Astrophysical Journal Letters) 1 Introduction It is becoming increasingly apparent that isolated neutron stars come in many flavors besides traditional radio pulsars. In recent years, the neutron star zoo has widened to include $\sim$10 radio-quiet neutron stars (bj98 ), six anomalous X-ray pulsars (AXPs; mer99 ) and four soft $\gamma$-ray repeaters (SGRs; kou99 ). There is much uncertainty and debate as to the nature of these sources; one way towards characterizing their properties is through associations with supernova remnants (SNRs). An association with a SNR gives an independent estimate of a neutron star’s age and distance, while the position of the neutron star with respect to the SNR’s center can be used to estimate the transverse space velocity of the compact object. A case in point are the AXPs. Some authors propose that the AXPs are accreting systems (van Paradijs (Taam); ms95 ; Ghosh (Angelini)), while others argue that AXPs are “magnetars”, isolated neutron stars with very strong magnetic fields, $B\gtrsim 10^{14}$ G (td96b ; hh97 ; mel99 ). However, the association of the AXP 1E 1841–045 with the very young ($\lesssim$2 kyr) SNR G27.4+0.0 (vg97 ) makes the case that 1E 1841–045 is a young object. Assuming that the pulsar was born spinning quickly, it is difficult to see how accretion could have slowed it down to its current period in such a short time. This result thus favors the magnetar model for 1E 1841–045, and indeed the magnetic field inferred from its period and period derivative, and assuming standard pulsar spin-down, is $B\approx 8\times 10^{14}$ G. AX J1845–0258 (also called AX J1844.8–0258) is a 6.97 sec X-ray pulsar, found serendipitously in an ASCA observation of the (presumably unassociated) SNR G29.7–0.3 (gv98 , hereafter GV98; tkk+98 , hereafter T98). The long pulse period, low Galactic latitude and soft spectrum of AX J1845–0258 led GV98 and T98 to independently propose that this source is an AXP (a conclusion which still needs to be confirmed through measurement of a period derivative). The high hydrogen column density inferred from photoelectric absorption ($N_{H}\approx 10^{23}$ cm${}^{-2}$) suggests that AX J1845–0258 is distant; T98 put it in the Scutum arm, with a consequent distance of 8.5 kpc, while GV98 nominate 15 kpc. Because AX J1845–0258 was discovered at the very edge of the ASCA GIS field-of-view, its position from these observations could only be crudely estimated, with an uncertainty of $\sim 3^{\prime}$. A subsequent (1999 March) 50 ks on-axis ASCA observation has since been carried out (vgtg99 ). No pulsations are seen in these data, but a faint point source, AX J184453.3–025642, is detected within the error circle for AX J1845–0258. Vasisht et al. (1999) determine an accurate position for AX J184453.3–025642, and argue that it corresponds to AX J1845–0258 in a quiescent state. Significant variations in the flux density of AX J1845–0258 were also reported by T98. The region containing AX J1845–0258 has been surveyed at 1.4 GHz as part of the NVSS (ccg+98 ). An image from this survey shows a $\sim 5^{\prime}$ shell near the position of the pulsar. We here report on multi-frequency polarimetric observations of this radio shell, at substantially higher sensitivity and spatial resolution than offered by the NVSS. Our observations and analysis are described in §2, and the resulting images are presented in §3. In §4 we argue that the radio shell coincident with AX J1845–0258 is a new SNR, and consider the likelihood of an association between the two sources. 2 Observations and Data Reduction Radio observations of the field of AX J1845–0258 were made with the D-configuration of the Very Large Array (VLA) on 1999 March 26. The total observing time was 6 hr, of which 4.5 hr was spent observing in the 5 GHz band, and the remainder in the 8 GHz band. 5 GHz observations consisted of a 100 MHz bandwidth centered on 4.860 GHz; 8 GHz observations were similar, but centered on 8.460 GHz. Amplitudes were calibrated by observations of 3C 286, assuming flux densities of 7.5 and 5.2 Jy at 5 GHz and 8 GHz respectively. Antenna gains and instrumental polarization were calibrated using regular observations of MRC 1801+010. Four Stokes parameters (RR, LL, RL, LR) were recorded in all observations. To cover the entire region of interest, observations were carried out in a mosaic of 2 (3) pointings at 5 (8) GHz. Data were edited and calibrated in the MIRIAD package. In total intensity (Stokes $I$), mosaic images of the field were formed using uniform weighting and maximum entropy deconvolution. The resulting images were then corrected for both the mean primary beam response of the VLA antennas and the mosaic pattern. The resolution and noise in the final images are given in Table 1. Images of the region were also formed in Stokes $Q$, $U$ and $V$. These images were made using natural weighting to give maximum sensitivity, and then deconvolved using a joint maximum entropy technique (Sault (Bock)). At each of 5 and 8 GHz, a linear polarization image $L$ was formed from $Q$ and $U$. Each $L$ image was clipped where the polarized emission or the total intensity was less than 5$\sigma$. In order to determine a spectral index from these data, it is important to ensure that the images contain the same spatial scales. We thus spatially filtered each total intensity image (see gbm+98 ), removing structure on scales larger than $5\arcmin$ and smoothing each image to a resolution of $15\arcsec$. The spatial distribution of spectral index was then determined using the method of “T–T” (temperature-temperature) plots (tpkp62 ; gbm+98 ). 3 Results Total intensity images of the region are shown in Figure 1. At both 5 and 8 GHz, a distinct shell of emission is seen, which we designate G29.6+0.1; observed properties are given in Table 1. The shell is clumpy, with a particularly bright clump on its eastern edge. In the east the shell is quite thick (up to 50% of the radius), while the north-western rim is brighter and narrower. Two point sources can be seen within the shell interior. At 5 GHz, the shell appears to be sitting upon a plateau of diffuse extended emission; this emission is resolved out at 8 GHz. Significant linear polarization at 5 GHz is seen from much of the shell, particularly in the two brightest parts of the shell on the eastern and western edges. Where detected, the fractional polarization is 2–20%. At 8 GHz, linear polarization is seen only from these two regions, with fractional polarization 5–40%. No emission was detected in Stokes $V$, except for instrumental effects associated with the offset of the VLA primary beam between left- and right-circular polarization. Meaningful T–T plots were obtained for three regions of the SNR, as marked in Figure 1; the spectral index, $\alpha$ ($S_{\nu}\propto\nu^{\alpha}$), for each region is marked. There appear to be distinct variations in spectral index around the shell, but all three determinations fall in the range $-0.7\lesssim\alpha\lesssim-0.4$. Two point sources are visible within the field. The more northerly of the two is at $18^{\rm h}44^{\rm m}55\farcs 11$, $-02\arcdeg 55\arcmin 36\farcs 9$ (J2000), with $S_{\rm 5\,GHz}=0.8\pm 0.1$ mJy and $\alpha=+0.5\pm 0.3$, while the other is at $18^{\rm h}44^{\rm m}50\farcs 59$, $-02\arcdeg 57\arcmin 58\farcs 5$ (J2000) with $S_{\rm 5\,GHz}=2.0\pm 0.3$ mJy and $\alpha=-0.4\pm 0.1$. Positional uncertainties for both sources are $\approx 0\farcs 3$ in each coordinate. No emission is detected from either source in Stokes $Q$, $U$ or $V$. 4 Discussion The source G29.6+0.1 is significantly linearly polarized and has a non-thermal spectrum. Furthermore, the source has a distinct shell morphology, and shows no significant counterpart in 60 $\mu$m IRAS data. These are all the characteristic properties of supernova remnants (e.g. wg96 ), and we thus classify G29.6+0.1 as a previously unidentified SNR. 4.1 Physical Properties of G29.6+0.1 Distances to SNRs are notoriously difficult to determine. The purported $\Sigma-D$ relation has extremely large uncertainties, and this source is most likely too faint to show H i absorption. So while we cannot determine a distance to G29.6+0.1 directly, we can attempt to estimate its distance by associating it with other objects in the region. Indeed hydrogen recombination lines from extended thermal material have been detected from the direction of G29.6+0.1 (Lockman (Pisano)), at systemic velocities of $+42$ and $+99$ km s${}^{-1}$. Adopting a standard model for Galactic rotation (Fich (Blitz)), these velocities correspond to possible distances of 3, 6, 9 or 12 kpc, a result which is not particularly constraining. Nevertheless, G29.6+0.1 is of sufficiently small angular size that we can put an upper limit on its age simply by assuming that it is located within the Galaxy. At a maximum distance of 20 kpc, the SNR is $27.5\pm 1.5$ pc across. For a uniform ambient medium of density $n_{0}$ cm${}^{-3}$, the SNR has then swept up $(260\pm 40)n_{0}$ $M_{\sun}$ from the ISM which, for typical ejected masses and ambient densities, corresponds to a SNR which has almost completed the transition from free expansion to the adiabatic (Sedov-Taylor) phase (see e.g. dj96 ). Thus expansion in the adiabatic phase acts as an upper limit, and we can derive a maximum age for G29.6+0.1 of $(13\pm 4)\left(n_{0}/E_{51}\right)^{1/2}$ kyr, where $E_{51}$ is the kinetic energy of the explosion in units of $10^{51}$ erg. For a typical value $n_{0}/E_{51}=0.2$ (Frail (Goss)), we find that the age of the SNR must be less than 8 kyr. For distances nearer than 20 kpc, the SNR is even younger. For example, at a distance of 10 kpc, the SNR has swept up sufficiently little material from the ISM that it is still freely expanding, and an expansion velocity 5000 km s${}^{-1}$ then corresponds to an age 1.4 kyr. 4.2 An association with AX J1845–0258? G29.6+0.1 is a young SNR in the vicinity of a slow X-ray pulsar. If the two can be shown to be associated, and if we assume that AX J1845–0258 was born spinning rapidly, then the youth of the system argues that AX J1845–0258 has slowed down to its current period via electromagnetic braking rather than accretion torque, and that it is thus best interpreted as a magnetar (cf. vg97 ). Indeed if one assumes that the source has slowed down through the former process, its inferred dipole magnetic field is $\sim 9t_{3}^{-1/2}\times 10^{14}$ G, for an age $t_{3}$ kyr. For ages in the range 1.4–8 kyr (§4.1 above), this results in a field in the range $(3-8)\times 10^{14}$ G, typical of other sources claimed to be magnetars. But are the two sources associated? Associations between neutron stars and SNRs are judged on various criteria, including agreements in distance and in age, positional coincidence, and evidence for interaction. Age and distance are the most fundamental of these, but unfortunately existing data on AX J1845–0258 provide no constraints on an age, and suggest only a very approximate distance of $\sim$10 kpc (GV98; T98). The source AX J184453.3–025642 (vgtg99 ) is located well within the confines of G29.6+0.1, less than $40\arcsec$ from the center of the remnant (see Figure 1). Vasisht et al. (1999) argue that AX J1845–0258 and AX J184453.3–025642 are the same source; if we assume that this source is associated with the SNR and was born at the remnant’s center, then we can infer an upper limit on its transverse velocity of $1900d_{10}/t_{3}$ km s${}^{-1}$, where the distance to the system is $10d_{10}$ kpc. In §4.1 we estimated $d_{10}/t_{3}\sim 0.3-0.7$, and so the velocity inferred falls comfortably within the range seen for the radio pulsar population (e.g. ll94 ; cc98 ) Alternatively, if we assume a transverse velocity of $400v_{400}$ km s${}^{-1}$, we can infer an age for the system of $<5d_{10}/v_{400}$ kyr, consistent with the determinations above. There is no obvious radio counterpart to the X-ray pulsar — both radio point sources in the region are outside all of the X-ray error circles. At the position of AX J184453.3–025642, we set a 5$\sigma$ upper limit of 1 mJy on the 5 GHz flux density of any point source. We also need to consider the possibility that the positional alignment of AX J184453.3–025642 and G29.6+0.1 is simply by chance. The region is a complex part of the Galactic Plane — there are 15 catalogued SNRs within 5°— and it seems reasonable in such a region that unassociated SNRs and neutron stars could lie along the same line of sight (gj95c ). Many young radio pulsars have no associated SNR (Braun (Goss)), so there is no reason to demand that even a young neutron star be associated with a SNR. The first quadrant of the Galaxy is not well-surveyed for SNRs, so we estimate the likelihood of a chance association by considering the fourth quadrant, which has been thoroughly surveyed for SNRs by Whiteoak & Green (1996). In a representative region of the sky defined by $320^{\circ}\leq l\leq 355^{\circ}$ and $|b|\leq 1.5^{\circ}$, we find 44 SNRs in their catalogue. Thus for the $\sim$10 radio-quiet neutron stars, AXPs and SGRs at comparable longitudes and latitudes, there is a probability $1.6\times 10^{-3}$ that at least one will lie within $40\arcsec$ of the center of a known SNR by chance. Of course in the present case we have carried out a targeted search towards a given position, and so the probability of spatial coincidence is somewhat higher than for a survey; nevertheless, we regard it unlikely that AX J184453.3–025642 should lie so close to the center of an unrelated SNR, and hence propose that the pulsar and the SNR are genuinely associated. There is good evidence that magnetars power radio synchrotron nebulae through the injection of relativistic particles into their environment (kfk+94 ; Frail (Kulkarni)). The two such sources known are filled-center nebulae with spectral indices $\alpha\sim-0.7$, and in one case the neutron star is substantially offset from the core of its associated nebula (hkc+99 ). In Figure 1, the clump of emission with peak at $18^{\rm h}44^{\rm m}56^{\rm s}$, $-02\arcdeg 57\arcmin$ (J2000) has such properties, and one can speculate that it corresponds to such a source. Alternatively, compact steep-spectrum features are seen in other shell SNRs, and may be indicative of deceleration of the shock in regions where it is expanding into a dense ambient medium (dbwg91 ; gbm+98 ). 5 Conclusions Radio observations of the field of the slow X-ray pulsar AX J1845–0258 reveal a linearly polarized non-thermal shell, G29.6+0.1, which we classify as a previously undiscovered supernova remnant. We infer that G29.6+0.1 is young, with an upper limit on its age of 8000 yr. The proposed quiescent counterpart of AX J1845–0258, AX J184453.3–025642, is almost at the center of G29.6+0.1, from which we argue that the pulsar and SNR were created in the same supernova explosion. The young age of the system provides further evidence that anomalous X-ray pulsars are isolated magnetars rather than accreting systems, although we caution that the apparent flux variability of AX J1845–0258 raises questions over both its classification as an AXP and its positional coincidence with G29.6+0.1. Future X-ray measurements should be able to clarify the situation. There are now six known AXPs, three of which have been associated with SNRs. In every case the pulsar is at or near the geometric center of its SNR. This result is certainly consistent with AXPs being young, isolated neutron stars, as argued by the magnetar hypothesis. If one considers the radio pulsar population, the fraction of pulsars younger than a given age which can be convincingly associated with SNRs drops as the age threshold increases. The age below which 50% of pulsars have good SNR associations is $\sim$20 kyr, and for several of these the pulsar is significantly offset from the center of its SNR (e.g. fk91 ; fggd96 ). Thus if the SNRs associated with both AXPs and radio pulsars come from similar explosions and evolve into similar environments, this seems good evidence that AXPs are considerably younger than 20 kyr. Indeed all of the three SNRs associated with AXPs have ages $<$10 kyr (gv97 ; pof+98 ; §4.1 of this paper). While the sample of AXPs is no doubt incomplete, this implies a Galactic birth-rate for AXPs of $>$0.6 kyr${}^{-1}$. This corresponds to $(5\pm 2)$% of core-collapse supernovae (ctt+97 ), or 3%–20% of the radio pulsar population (lml+98 ; bj98 ). There is mounting evidence that soft $\gamma$-ray repeaters (SGRs) are also magnetars (ksh+99 ). However of the four known SGRs, two (0526–66 and 1627–41) are on the edge of young SNRs (cdt+82 ; Smith (Bradt)), a third (1900+14) is on the edge of an old SNR (vkfg94 ), and the fourth (1806–20) has no associated SNR blast wave (kfk+94 ). This suggests that SGRs represent an older, or higher velocity, manifestation of magnetars than do AXPs. B.M.G. thanks Bob Sault for advice on calibration. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. B.M.G. acknowledges the support of NASA through Hubble Fellowship grant HF-01107.01-98A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA under contract NAS 5–26555. E.V.G. & G.V.’s research is supported by NASA LTSA grant NAG5–22250 References Braun (Goss) Braun, R., Goss, W. M., & Lyne, A. G. 1989, ApJ, 340, 355. 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From yeV to TeV: Search for the Neutron Electric Dipole Moment D. H. Beck\fromins:uiuc    D. Budker\fromins:berkeley\fromins:LBNL    and B. K.  Park\fromins:berkeley  for the nEDM Collaboration\fromins:nEDM ins:uiucins:uiucins:berkeleyins:berkeleyins:LBNLins:LBNLins:nEDMins:nEDM Abstract The existence of electric dipole moments (EDM) for fundamental particles signals time-reversal symmetry (T) violation accompanied by violation of parity (P); only upper limits have been established to date. Time-reversal violation in turn implies CP violation under the assumption that CPT is a good symmetry. The neutron is an attractive system for an EDM search, both because it is neutral and because a neutron EDM would be relatively easier to interpret than the comparable quantity for a nucleus or even an atom. We introduce briefly the key experimental requirements for such search and describe some aspects of the neutron EDM experiment planned for the Spallation Neutron Source at the U.S. Oak Ridge National Laboratory. \instlist Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL, USA Department of Physics, University of California Berkeley, Berkeley, CA 94720-7300, USA Nuclear Science Division, E. O. Lawrence National Laboratory, Berkeley, CA 94720, USA http://www.phy.ornl.gov/nedm/ \PACSes\PACSit14.20.DhNeutron properties 1 Introduction Searches for electric dipole moments (EDM) of fundamental particles have a history that dates back to the 1950’s. In this era, the violation of parity was first postulated and then observed in weak-interaction experiments of Wu, et al. [1], marking the beginning of the demise of discrete symmetries in general (so far, with the exception of CPT invariance). The first EDM search for the neutron was done early in the decade by Smith, Purcell and Ramsey as search for parity violation in a scattering experiment at Oak Ridge. Although the measurements were completed by 1951, the paper was not published until 1957 [2], after the publication of the Lee and Yang paper [3] on the weak interaction, because the upper limit determined, $\left|d_{n}\right|\leq 3.9\times 10^{-20}$ e$\cdot$cm (all 90% C.L.), was not thought to be interesting at the time. In fact, upper limits on EDM have continued to limit theory in important ways over the intervening 60 years. The CP violation in the standard model generates very small EDM, typically coming in only at the level of a third order loop [4]. Therefore, in the search for new physics, EDM are attractive because the standard model ‘background’ is several orders of magnitude below currently observed limits. There are two prominent threads that argue for continued searches. First, in 1967 Sakharov argued [5] that any process that could generate the matter-antimatter asymmetry observed in the universe must involve CP violation; the aforementioned CP violation in the standard model is much too small to account for the observed asymmetry [6]. Second, in the currently popular models of physics beyond the standard model, e.g., SUSY, there is much more opportunity for the necessary complex couplings (analogous to those in the CKM matrix) in the SUSY breaking mechanisms of our low-energy world. The current set of EDM upper limits has significantly restricted the space of possible SUSY models [7]. With these experiments, we are therefore exploring aspects of physics beyond the standard model that pertain to the TeV-scale energies, and in ways that complement the direct searches at the LHC. 2 EDM Experiments Most EDM experiments look for precession of a fundamental particle’s angular momentum in an external electric field. The interaction energy of the electric dipole in the electric field combines with that of the T-allowed magnetic moment in an external magnetic field in a way that either increases or decreases the normal precession frequency. These effects are very small; the interaction energy at the sensitivity limit of our proposed Oak Ridge experiment is about $10^{-23}$ eV or 10 yocto-electron volts (yeV); about ten orders of magnitude smaller than the corresponding magnetic energy. We next survey a few of the key experiments and techniques in the field. A landmark EDM experiment was carried out at Berkeley by Eugene Commins’ group. Taking advantage of the relativistic enhancement of atomic EDM in heavy atoms [8], Commins used an atomic beam of polarized (paramagnetic) thallium to measure an EDM which is mostly sensitive to an EDM of the electron. It is essentially a variant of the Ramsey separated oscillatory field method [9] wherein the magnetic moments of optically polarized atoms are flipped by a first $\pi/2$ pulse, precess freely in a region where the magnetic and electric fields are either parallel or anti-parallel, and then are analyzed by applying a second, synchronous $\pi/2$ pulse. The resulting polarization is then probed by the same laser used to produce the initial polarization. The Commins group measured an upper limit $\left|d_{e}\right|\leq 1.5\times 10^{-27}$ e$\cdot$cm [10]. Essentially the same limit of $\left|d_{e}\right|\leq 1.05\times 10^{-27}$ e$\cdot$cm [11] has been obtained recently by the Hinds group at Imperial College in an experiment otherwise similar, but taking the advantage of the large internal electric field in the polar YbF molecule. The record for the smallest EDM upper limit is held by the Hg experiment in Seattle. In this diamagnetic atom, the atomic EDM is most sensitive to the intrinsic nuclear EDM, and there is enhancement from the finite size of the nucleus. The experiment uses a pair of cells with the same nominal magnetic field, but opposite electric fields, and a second pair of cells with no electric field acting as a co-magnetometer. The atoms are first optically pumped with a circularly polarized laser beam propagating perpendicular to the magnetic field and modulated at the Larmor frequency. After the atoms are polarized, the light polarization is switched to linear, and a linear polarizer analyzes optical rotation in the transmitted light induced by the precessing atoms. The limit on the ${}^{199}$Hg EDM is $\left|d_{Hg}\right|\leq 2.6\times 10^{-29}$ e$\cdot$cm [12]. The current limit on the neutron EDM is held by the ILL experiment. It utilizes ultra-cold neutrons (UCN) with energies $\raisebox{-2.58pt}{$\,\stackrel{{\scriptstyle\raisebox{-0.86pt}{$\textstyle<$}% }}{{\sim}}\,$}110$ neV, which can be trapped by the Fermi potentials of certain materials like SiO${}_{2}$ and diamond-like carbon (the UCN-storage possibility was first envisioned by Zel’dovich in 1959 [13], see also Ref. [14]). The UCN are produced by slowing cold neutrons via scattering from a receding turbine blade. After the neutrons are polarized by transmission through a magnetized foil, they precess in a cell with parallel or anti-parallel $\vec{E}$ and $\vec{B}$ fields and are analyzed in a Ramsey separated oscillatory field experiment as described above. The limit set in this experiment is $\left|d_{n}\right|\leq 2.9\times 10^{-26}$ e$\cdot$cm [15]. New n-EDM experiments are currently being developed for neutron facilities at ILL, PSI, Munich, TRIUMF, and the Oak Ridge Spallation Neutron Source. All primarily seek to increase the UCN density significantly above the $1$ cm${}^{-3}$ in the first ILL experiment. Additional co-magnetometry is also considered to be an important improvement to help circumvent the geometric-phase systematic uncertainty [16, 17] that ultimately limited the ILL experiment described above. 3 Neutron EDM experiment at the SNS 3.1 General description The sensitivity limit of all measurements of electric dipole moments is, up to a numerical factor {eqnletter} δd ∼1E 1N τT, where $E$ is the electric field, $\tau$ is the coherence time and $T$ the overall measurement time. The obvious factors to attack are therefore $E$, $N$ and $\tau$. In order to increase $N$, this experiment (see Fig. 1) uses the downscattering of neutrons by resonant creation of the phonon Landau-Feynman excitations in superfluid ${}^{4}$He at $T=0.3-0.5$ K. Cold neutrons with wavelengths of 8.9 Å, produced either by a graphite monochromator or a chopper, enter the experiment from a standard polarizing guide. The wavelength corresponds to the crossing of the dispersion curves of the neutrons and superfluid elementary excitations, enabling the neutrons to efficiently give up their energy and momentum to the phonons/rotons [18], which are eventually absorbed by the cold walls of the container. In fact, thus scattered neutrons are at a temperature much below that of the bath (the upscattering rate is small) and have an energy below that of the Fermi potential associated with the polystyrene coated acrylic container for the superfluid. In this manner, neutron densities of order 100 cm${}^{-3}$ can be produced. However, even this large number of polarized neutrons, trapped in two three-liter cells of superfluid helium (with opposing electric fields), is neither sufficient for direct detection using SQUID or atomic magnetometers, nor is it amenable to the Ramsey technique. Thus, a second major point of departure for this experiment is the introduction of polarized ${}^{3}$He (with a magnetic moment about 10% larger than that of the neutron) into the superfluid to act as the ‘detector’ [19]. In this case, we produce polarized ${}^{3}$He atoms using an atomic beam source (ABS). The neutron-capture reaction on ${}^{3}$He {eqnletter} n + ${}^{3}$He →p + t + 764 keV is highly spin dependent, with a cross section of megabarns when the spins are anti-parallel and effectively zero with spins parallel. Therefore, by applying a single $\pi/2$ pulse, flipping both the neutron and ${}^{3}$He spins, the difference in their precession rates can be measured by observing the scintillation light produced in the liquid helium by the capture products. Because of the large cross section, the optimal density of ${}^{3}$He is low, about $10^{12}$ cm${}^{-3}$ corresponding to a ${}^{3}$He/${}^{4}$He fraction of about $5\times 10^{-11}$, or about four orders of magnitude below the natural concentration. Even at this low density, however, we can use the overall precession rate of the polarized ${}^{3}$He, measured by SQUID, to determine the magnetic field. Aside from a small gravitational offset (because of the lower neutron temperature and larger ${}^{3}$He mass), the neutrons and ${}^{3}$He atoms sample exactly the same space inside the superfluid-filled cells, thus an effective co-magnetometer is built-in. To maximize the effectiveness of the co-magnetometer, a spin-dressing technique has been developed and experimentally validated [20, 21], in which the gyromagnetic ratios of the neutrons and ${}^{3}$He are rendered effectively the same by applying off-resonant radio-frequency fields. Maximum sensitivity results when measuring for approximately the neutron lifetime, with the capture lifetime tuned (by adjusting the ${}^{3}$He density) to about the same value. Because the ${}^{3}$He will eventually be depolarized by wall collisions and field gradients, we must have a technique for removing it from the system and supplying a new charge of highly polarized ${}^{3}$He. For this purpose, we will again use the phonons in the superfluid, this time produced by a heater. These phonons scatter the ${}^{3}$He toward the cold end of the region (see Refs. [22, 23] and references therein) where the heater has produced a temperature gradient. After dumping the volume containing the concentrated, somewhat depolarized ${}^{3}$He, refilling it with pure ${}^{4}$He superfluid, a new set of highly polarized ${}^{3}$He atoms is injected into the experiment from the ABS. With a design value for the electric field of 50 kV/cm and a 300-day live-time, this experiment is expected to reach the level of about $8\times 10^{-28}$ e$\cdot$cm. The largest systematic uncertainties are expected to be from the geometric-phase effect ($\sim 2\times 10^{-28}$ e$\cdot$cm, limited by the uniformity of the $B_{0}$ holding field), with contributions from an effective magnetic field coming from the neutron scattering from polarized ${}^{3}$He, as well as from leakage currents at the $\sim 1\times 10^{-28}$ e$\cdot$cm level. 3.2 Field monitoring In order to control systematic uncertainties due to the motional magnetic field, it is essential to maintain stable, homogeneous electric field over the measurement cycle, as well as apply accurate electric-field reversal in order to reduce systematic effects quadratic in electric field. An accurate monitoring of electric field is necessary to ensure this has been achieved. The Kerr effect in superfluid helium, already present in the experiment, provides a useful non-contact method of monitoring the electric field, especially given the harsh environment of the SNS nEDM experiment. The applied electric field causes the helium medium to become birefringent, which is then detected by laser polarimetry. The Kerr constant of superfluid helium has been measured to demonstrate the feasibility of the technique [24], and in order to minimize effect of spurious birefringence from optical windows, we have developed a double-pass cancellation scheme and demonstrated the scheme in a proof-of-concept setup [25]. We project sensitivity of $\delta E/E\approx 1\%$. Non-linear magneto-optical rotation (NMOR) magnetometers can be used to monitor the magnetic field and ensure that the geometric phase effect from the magnetic field inhomogeneity is below the level specified above [26]. 4 Conclusion The EDM of elementary particles continue to provide important constraints on physics beyond the standard model. Because they are sensitive to CP-violating couplings, they are complementary to the direct searches at the LHC. 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A Non-Binary Associative Memory with Exponential Pattern Retrieval Capacity and Iterative Learning Amir Hesam Salavati${}^{\dagger}$, K. Raj Kumar${}^{\ddagger}$, and Amin Shokrollahi${}^{\dagger}$ $\dagger$:Laboratoire d’algorithmique (ALGO) Ecole Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland E-mail: {hesam.salavati,amin.shokrollahi}@epfl.ch $\ddagger$:Qualcomm Research India Bangalore - 560066, India E-mail: kumarraj@qti.qualcomm.com Abstract We consider the problem of neural association for a network of non-binary neurons. Here, the task is to first memorize a set of patterns using a network of neurons whose states assume values from a finite number of integer levels. Later, the same network should be able to recall previously memorized patterns from their noisy versions. Prior work in this area consider storing a finite number of purely random patterns, and have shown that the pattern retrieval capacities (maximum number of patterns that can be memorized) scale only linearly with the number of neurons in the network. In our formulation of the problem, we concentrate on exploiting redundancy and internal structure of the patterns in order to improve the pattern retrieval capacity. Our first result shows that if the given patterns have a suitable linear-algebraic structure, i.e. comprise a sub-space of the set of all possible patterns, then the pattern retrieval capacity is in fact exponential in terms of the number of neurons. The second result extends the previous finding to cases where the patterns have weak minor components, i.e. the smallest eigenvalues of the correlation matrix tend toward zero. We will use these minor components (or the basis vectors of the pattern null space) to both increase the pattern retrieval capacity and error correction capabilities. An iterative algorithm is proposed for the learning phase, and two simple neural update algorithms are presented for the recall phase. Using analytical results and simulations, we show that the proposed methods can tolerate a fair amount of errors in the input while being able to memorize an exponentially large number of patterns. Neural associative memory, Error correcting codes, message passing, stochastic learning, dual-space method I Introduction Neural associative memory is a particular class of neural networks capable of memorizing (learning) a set of patterns and recalling them later in presence of noise, i.e. retrieve the correct memorized pattern from a given noisy version. Starting from the seminal work of Hopfield in 1982 [1], various artificial neural networks have been designed to mimic the task of the neuronal associative memory (see for instance [2], [3], [4], [5], [6]). In essence, the neural associative memory problem is very similar to the one faced in communication systems where the goal is to reliably and efficiently retrieve a set of patterns (so called codewords) form noisy versions. More interestingly, the techniques used to implement an artificial neural associative memory looks very similar to some of the methods used in graph-based modern codes to decode information. This makes the pattern retrieval phase in neural associative memories very similar to iterative decoding techniques in modern coding theory. However, despite the similarity in the task and techniques employed in both problems, there is a huge gap in terms of efficiency. Using binary codewords of length $n$, one can construct codes that are capable of reliably transmitting $2^{rn}$ codewords over a noisy channel, where $0<r<1$ is the code rate [7]. The optimal $r$ (i.e. the largest possible value that permits the almost sure recovery of transmitted codewords from the corrupted received versions) depends on the noise characteristics of the channel and is known as the Shannon capacity [8]. In fact, the Shannon capacity is achievable in certain cases, for example by LDPC codes over AWGN channels. In current neural associative memories, however, with a network of size $n$ one can only memorize $O(n)$ binary patterns of length $n$ [9], [2]. To be fair, it must be mentioned that these networks are designed such that they are able to memorize any possible set of randomly chosen patterns (with size $O(n)$ of course) (e.g., [1], [2], [3], [4]). Therefore, although humans cannot memorize random patterns, these methods provide artificial neural associative memories with a pleasant sense of generality. However, this generality severely restricts the efficiency of the network since even if the input patterns have some internal redundancy or structure, current neural associative memories could not exploit this redundancy in order to increase the number of memorizable patterns or improve error correction during the recall phase. In fact, concentrating on redundancies within patterns is a fairly new viewpoint. This point of view is in harmony to coding techniques where one designs codewords with certain degree of redundancy and then use this redundancy to correct corrupted signals at the receiver’s side. In this paper, we focus on bridging the performance gap between the coding techniques and neural associative memories. Our proposed neural network exploits the inherent structure of the input patterns in order to increase the pattern retrieval capacity from $O(n)$ to $O(a^{n})$ with $a>1$. More specifically, the proposed neural network is capable of learning and reliably recalling given patterns when they come from a subspace with dimension $k<n$ of all possible $n$-dimensional patterns. Note that although the proposed model does not have the versatility of traditional associative memories to handle any set of inputs, such as the Hopfield network [1], it enables us to boost the capacity by a great extent in cases where there is some input redundancy. In contrast, traditional associative memories will still have linear pattern retrieval capacity even if the patterns good linear algebraic structures. In [10], we presented some preliminary results in which two efficient recall algorithms were proposed for the case where the neural graph had the structure of an expander [11]. Here, we extend the previous results to general sparse neural graphs as well as proposing a simple learning algorithm to capture the internal structure of the patterns (which will be used later in the recall phase). The remainder of this paper is organized as follows: In Section II, we will discuss the neural model used in this paper and formally define the associative memory problem. We explain the proposed learning algorithm in Section III. Sections IV and V are respectively dedicated to the recall algorithm and analytically investigating its performance in retrieving corrupted patterns. In Section VI we address the pattern retrieval capacity and show that it is exponential in $n$. Simulation results are discussed in Section VII. Section VIII concludes the paper and discusses future research topics. Finally, the Appendices contain some extra remarks as well as the proofs for certain lemmas and theorems. II Problem Formulation and the Neural Model II-A The Model In the proposed model, we work with neurons whose states are integers from a finite set of non-negative values $\mathcal{Q}=\{0,1,\dots,Q-1\}$. A natural way of interpreting this model is to think of the integer states as the short-term firing rate of neurons (possibly quantized). In other words, the state of a neuron in this model indicates the number of spikes fired by the neuron in a fixed short time interval. Like in other neural networks, neurons can only perform simple operations. We consider neurons that can do linear summation over the input and possibly apply a non-linear function (such as thresholding) to produce the output. More specifically, neuron $x$ updates its state based on the states of its neighbors $\{s_{i}\}_{i=1}^{n}$ as follows: 1. It computes the weighted sum $h=\sum_{i=1}^{n}w_{i}s_{i},$ where $w_{i}$ denotes the weight of the input link from the $i^{th}$ neighbor. 2. It updates its state as $x=f(h),$ where $f:\mathbb{R}\rightarrow\mathcal{Q}$ is a possibly non-linear function from the field of real numbers $\mathbb{R}$ to $\mathcal{Q}$. We will refer to these two as ”neural operations” in the sequel. II-B The Problem The neural associative memory problem consists of two parts: learning and pattern retrieval. II-B1 The learning phase We assume to be given $C$ vectors of length $n$ with integer-valued entries belonging to $\mathcal{Q}$. Furthermore, we assume these patterns belong to a subspace of $\mathcal{Q}^{n}$ with dimension $k\leq n$. Let $\mathcal{X}_{C\times n}$ be the matrix that contains the set of patterns in its rows. Note that if $k=n$, then we are back to the original associative memory problem. However, our focus will beon the case where $k<n$, which will be shown to yield much larger pattern retrieval capacities. Let us denote the model specification by a triplet $(\mathcal{Q},n,k)$. The learning phase then comprises a set of steps to determine the connectivity of the neural graph (i.e. finding a set of weights) as a function of the training patterns in $\mathcal{X}$ such that these patterns are stable states of the recall process. More specifically, in the learning phase we would like to memorize the patterns in $\mathcal{X}$ by finding a set of non-zero vectors $w_{1},\dots,w_{m}\in\mathbb{R}^{n}$ that are orthogonal to the set of given patterns. Remark here that such vectors exist (for instance the basis of the null-space). Our interest is to come up with a neural scheme to determine these vectors. Therefore, the inherent structure of the patterns are captured in the obtained null-space vectors, denoted by the matrix $W\in\mathbb{R}^{m\times n}$, whose $i^{\mbox{th}}$ row is $w_{i}$. This matrix can be interpreted as the adjacency matrix of a bipartite graph which represents our neural network. The graph is comprised on pattern and constraint neurons (nodes). Pattern neurons, as they name suggest, correspond to the states of the patterns we would like to learn or recall. The constrain neurons, on the other hand, should verify if the current pattern belongs to the database $\mathcal{X}$. If not, they should send proper feedback messages to the pattern neurons in order to help them converge to the correct pattern in the dataset. The overall network model is shown in Figure 1. II-B2 The recall phase In the recall phase, the neural network should retrieve the correct memorized pattern from a possibly corrupted version. In this case, the states of the pattern neurons $x_{1},x_{2},\dots,x_{n}$ are initialized with the given (noisy) input pattern. Here, we assume that the noise is integer valued and additive111It must be mentioned that neural states below $0$ and above $Q-1$ will be clipped to $0$ and $Q-1$, respectively. This is biologically justified as the firing rate of neurons can not exceed an upper bound and of course can not be less than zero.. Therefore, assuming the input to the network is a corrupted version of pattern $x^{\mu}$, the state of the pattern nodes are $x=x^{\mu}+z$, where $z$ is the noise. Now the neural network should use the given states together with the fact that $Wx^{\mu}=0$ to retrieve pattern $x^{\mu}$, i.e. it should estimate $z$ from $Wx=Wz$ and return $x^{\mu}=x-z$. Any algorithm designed for this purpose should be simple enough to be implemented by neurons. Therefore, our objective is to find a simple algorithm capable of eliminating noise using only neural operations. II-C Related Works Designing a neural associative memory has been an active area of research for the past three decades. Hopfield was the first to design an artificial neural associative memory in his seminal work in 1982 [1]. The so-called Hopfield network is inspired by Hebbian learning [12] and is composed of binary-valued ($\pm 1$) neurons, which together are able to memorize a certain number of patterns. In our terminology, the Hopfield network corresponds to a $(\{-1,1\},n,n)$ neural model. The pattern retrieval capacity of a Hopfield network of $n$ neurons was derived later by Amit et al. [13] and shown to be $0.13n$, under vanishing bit error probability requirement. Later, McEliece et al. [9] proved that under the requirement of vanishing pattern error probability, the capacity of Hopfield networks is $n/(2\log(n)))=O(n/\log(n))$. In addition to neural networks with online learning capability, offline methods have also been used to design neural associative memories. For instance, in [2] the authors assume the complete set of pattern is given in advance and calculate the weight matrix using the pseudo-inverse rule [14] offline. In return, this approach helps them improve the capacity of a Hopfield network to $n/2$, under vanishing pattern error probability condition, while being able to correct one bit of error in the recall phase. Although this is a significant improvement to the $n/\log(n)$ scaling of the pattern retrieval capacity in [9], it comes at the price of much higher computational complexity and the lack of gradual learning ability. While the connectivity graph of a Hopfield network is a complete graph, Komlos and Paturi [15] extended the work of McEliece to sparse neural graphs. Their results are of particular interest as physiological data is also in favor of sparsely interconnected neural networks. They have considered a network in which each neuron is connected to $d$ other neurons, i.e., a $d$-regular network. Assuming that the network graph satisfies certain connectivity measures, they prove that it is possible to store a linear number of random patterns (in terms of $d$) with vanishing bit error probability or $C=O(d/\log n)$ random patterns with vanishing pattern error probability. Furthermore, they show that in spite of the capacity reduction, the error correction capability remains the same as the network can still tolerate a number of errors which is linear in $n$. It is also known that the capacity of neural associative memories could be enhanced if the patterns are of low-activity nature, in the sense that at any time instant many of the neurons are silent [14]. However, even these schemes fail when required to correct a fair amount of erroneous bits as the information retrieval is not better compared to that of normal networks. Extension of associative memories to non-binary neural models has also been explored in the past. Hopfield addressed the case of continuous neurons and showed that similar to the binary case, neurons with states between $-1$ and $1$ can memorize a set of random patterns, albeit with less capacity [16]. Prados and Kak considered a digital version of non-binary neural networks in which neural states could assume integer (positive and negative) values [17]. They show that the storage capacity of such networks are in general larger than their binary peers. However, the capacity would still be less than $n$ in the sense that the proposed neural network can not have more than $n$ patterns that are stable states of the network, let alone being able to retrieve the correct pattern from corrupted input queries. In [3] the authors investigated a multi-state complex-valued neural associative memory for which the estimated capacity is $C<0.15n$. Under the same model but using a different learning method, Muezzinoglu et al. [4] showed that the capacity can be increased to $C=n$. However the complexity of the weight computation mechanism is prohibitive. To overcome this drawback, a Modified Gradient Descent learning Rule (MGDR) was devised in [18]. In our terminology, all these models are $(\{e^{2\pi js/k}|0\leq s\leq k-1\},n,n)$ neural associative memories. Given that even very complex offline learning methods can not improve the capacity of binary or multi-sate neural associative memories, a group of recent works has made considerable efforts to exploit the inherent structure of the patterns in order to increase capacity and improve error correction capabilities. Such methods focus merely on memorizing those patterns that have some sort of inherent redundancy. As a result, they differ from previous methods in which the network was deigned to be able to memorize any random set of patterns. Pioneering this approach, Berrou and Gripon [19] achieved considerable improvements in the pattern retrieval capacity of Hopfield networks, by utilizing Walsh-Hadamard sequences. Walsh-Hadamard sequences are a particular type of low correlation sequences and were initially used in CDMA communications to overcome the effect of noise. The only slight downside to the proposed method is the use of a decoder based on the winner-take-all approach which requires a separate neural stage, increasing the complexity of the overall method. Using low correlation sequences has also been considered in [5], where the authors introduced two novel mechanisms of neural association that employ binary neurons to memorize patterns belonging to another type of low correlation sequences, called Gold family [20]. The network itself is very similar to that of Hopfield, with a slightly modified weighting rule. Therefore, similar to a Hopfield network, the complexity of the learning phase is small. However, the authors failed to increase the pattern retrieval capacity beyond $n$ and it was shown that the pattern retrieval capacity of the proposed model is $C=n$, while being able to correct a fair number of erroneous input bits. Later, Gripon and Berrou came up with a different approach based on neural cliques, which increased the pattern retrieval capacity to $O(n^{2})$ [6]. Their method is based on dividing a neural network of size $n$ into $c$ clusters of size $n/c$ each. Then, the messages are chosen such that only one neuron in each cluster is active for a given message. Therefore, one can think of messages as a random vector of length $c\log(n/c)$, where the $\log(n/c)$ part specifies the index of the active neuron in a given cluster. The authors also provide a learning algorithm, similar to that of Hopfield, to learn the pair-wise correlations within the patterns. Using this technique and exploiting the fact that the resulting patterns are very sparse, they could boost the capacity to $O(n^{2})$ while maintaining the computational simplicity of Hopfield networks. In contrast to the pairwise correlation of the Hopfield model, Peretto et al. [21] deployed higher order neural models: the models in which the state of the neurons not only depends on the state of their neighbors, but also on the correlation among them. Under this model, they showed that the storage capacity of a higher-order Hopfield network can be improved to $C=O(n^{p-2})$, where $p$ is the degree of correlation considered. The main drawback of this model is the huge computational complexity required in the learning phase, as one has to keep track of $O(n^{p-2})$ neural links and their weights during the learning period. Recently, the present authors introduced a novel model inspired by modern coding techniques in which a neural bipartite graph is used to memorize the patterns that belong to a subspace [10]. The proposed model can be also thought of as a way to capture higher order correlations in given patterns while keeping the computational complexity to a minimal level (since instead of $O(n^{p-2})$ weights one needs to only keep track of $O(n^{2})$ of them). Under the assumptions that the bipartite graph is known, sparse, and expander, the proposed algorithm increased the pattern retrieval capacity to $C=O(a^{n})$, for some $a>1$, closing the gap between the pattern retrieval capacities achieved in neural networks and that of coding techniques. For completeness, this approach is presented in the appendix (along with the detailed proofs). The main drawbacks in the proposed approach were the lack of a learning algorithm as well as the expansion assumption on the neural graph. In this paper, we focus on extending the results described in [10] in several directions: first, we will suggest an iterative learning algorithm, to find the neural connectivity matrix from the patterns in the training set. Secondly, we provide an analysis of the proposed error correcting algorithm in the recall phase and investigate its performance as a function of input noise and network model. Finally, we discuss some variants of the error correcting method which achieve better performance in practice. It is worth mentioning that an extension of this approach to a multi-level neural network is considered in [22]. There, the novel structure enables better error correction. However, the learning algorithm lacks the ability to learn the patterns one by one and requires the patterns to be presented all at the same time in the form of a big matrix. In [23] we have further extended this approach to a modular single-layer architecture with online learning capabilities. The modular structure makes the recall algorithm much more efficient while the online learning enables the network to learn gradually from examples. The learning algorithm proposed in this paper is also virtually the same as the one we proposed in [23], giving it the advantage of Another important point to note is that learning linear constraints by a neural network is hardly a new topic as one can learn a matrix orthogonal to a set of patterns in the training set (i.e., $Wx^{\mu}=0$) using simple neural learning rules (we refer the interested readers to [24] and [25]). However, to the best of our knowledge, finding such a matrix subject to the sparsity constraints has not been investigated before. This problem can also be regarded as an instance of compressed sensing [26], in which the measurement matrix is given by the big patterns matrix $\mathcal{X}_{C\times n}$ and the set of measurements are the constraints we look to satisfy, denoted by the tall vector $b$, which for simplicity reasons we assume to be all zero. Thus, we are interested in finding a sparse vector $w$ such that $\mathcal{X}w=0$. Nevertheless, many decoders proposed in this area are very complicated and cannot be implemented by a neural network using simple neuron operations. Some exceptions are [27] and [28] which are closely related to the learning algorithm proposed in this paper. II-D Solution Overview Before going through the details of the algorithms, let us give an overview of the proposed solution. To learn the set of given patterns, we have adopted the neural learning algorithm proposed in [29] and modified it to favor sparse solutions. In each iteration of the algorithm, a random pattern from the data set is picked and the neural weights corresponding to constraint neurons are adjusted is such a way that the projection of the pattern along the current weight vectors is reduced, while trying to make the weights sparse as well. In the recall phase, we exploit the fact that the learned neural graph is sparse and orthogonal to the set of patterns. Therefore, when a query is given, if it is not orthogonal to the connectivity matrix of the weighted neural graph, it is noisy. We will use the sparsity of the neural graph to eliminate this noise using a simple iterative algorithm. In each iteration, there is a set of violated constraint neurons, i.e. those that receive a non-zero sum over their input links. These nodes will send feedback to their corresponding neighbors among the pattern neurons, where the feedback is the sign of the received input-sum. At this point, the pattern nodes that receive feedback from a majority of their neighbors update their state according to the sign of the sum of received messages. This process continues until noise is eliminated completely or a failure is declared. In short, we propose a neural network with online learning capabilities which uses only neural operations to memorize an exponential number of patterns. III Learning Phase Since the patterns are assumed to be coming from a subspace in the $n$-dimensional space, we adapt the algorithm proposed by Oja and Karhunen [29] to learn the null-space basis of the subspace defined by the patterns. In fact, a very similar algorithm is also used in [24] for the same purpose. However, since we need the basis vectors to be sparse (due to requirements of the algorithm used in the recall phase), we add an additional term to penalize non-sparse solutions during the learning phase. Another difference with the proposed method and that of [24] is that the learning algorithm proposed in [24] yields dual vectors that form an orthogonal set. Although one can easily extend our suggested method to such a case as well, we find this requirement unnecessary in our case. This gives us the additional advantage to make the algorithm parallel and adaptive. Parallel in the sense that we can design an algorithm to learn one constraint and repeat it several times in order to find all constraints with high probability. And adaptive in the sense that we can determine the number of constraints on-the-go, i.e. start by learning just a few constraints. If needed (for instance due to bad performance in the recall phase), the network can easily learn additional constraints. This increases the flexibility of the algorithm and provides a nice trade-off between the time spent on learning and the performance in the recall phase. Both these points make an approach biologically realistic. It should be mentioned that the core of our learning algorithm here is virtually the same as the one we proposed in [23]. III-A Overview of the proposed algorithm The problem to find one sparse constraint vector $w$ is given by equations (1), (2), in which pattern $\mu$ is denoted by $x^{\mu}$. {subequations} $$\min\sum_{\mu=1}^{C}s|x^{\mu}\cdot w|^{2}+\eta g(w).$$ (1) subject to: $$\|w\|_{2}=1$$ (2) In the above problem, $\cdot$ is the inner-product, $\|.\|_{2}$ represent the $\ell_{2}$ vector norm, $g(w)$ a penalty function to encourage sparsity and $\eta$ is a positive constant. There are various ways to choose $g(w)$. For instance one can pick $g(w)$ to be $\|.\|_{1}$, which leads to $\ell_{1}$-norm penalty and is widely used in compressed sensing applications [27], [28]. Here, we will use a different penalty function, as explained later. To form the basis for the null space of the patterns, we need $m=n-k$ vectors, which we can obtain by solving the above problem several times, each time from a random initial point222It must be mentioned that in order to have exactly $m=n-k$ linearly independent vectors, we should pay some additional attention when repeating the proposed method several time. This issue is addressed later in the paper.. As for the sparsity penalty term $g(w)$ in this problem, in this paper we consider the function $$g(w)=\sum_{i=1}^{n}\tanh(\sigma w_{i}^{2}),$$ where $\sigma$ is chosen appropriately. Intuitively, $\tanh(\sigma w_{i}^{2})$ approximates $|\hbox{sign}(w_{i})|$ in $\ell_{0}$-norm. Therefore, the larger $\sigma$ is, the closer $g(w)$ will be to $\|.\|_{0}$. By calculating the derivative of the objective function, and by considering the update due to each randomly picked pattern $x$, we will get the following iterative algorithm: {subequations} $$y(t)=x(t)\cdot w(t)$$ (3) $$\tilde{w}(t+1)=w(t)-\alpha_{t}\left(2y(t)x(t)+\eta\Gamma(w(t))\right)$$ (4) $$w(t+1)=\frac{\tilde{w}(t+1)}{\|\tilde{w}(t+1)\|_{2}}$$ (5) In the above equations, $t$ is the iteration number, $x(t)$ is the sample pattern chosen at iteration $t$ uniformly at random from the patterns in the training set $\mathcal{X}$, and $\alpha_{t}$ is a small positive constant. Finally, $\Gamma(w):\mathcal{R}^{n}\rightarrow\mathcal{R}^{n}=\nabla g(w)$ is the gradient of the penalty term for non-sparse solutions. This function has the interesting property that for very small values of $w_{i}(t)$, $\Gamma(w_{i}(t))\simeq 2\sigma w_{i}(t)$. To see why, consider the $i^{th}$ entry of the function $\Gamma(w(t)))$ $$\Gamma_{i}(w(t))=\partial g(w(t))/\partial w_{i}(t)=2\sigma_{t}w_{i}(t)(1-% \tanh^{2}(\sigma w_{i}(t)^{2}))$$ It is easy to see that $\Gamma_{i}(w(t))\simeq 2\sigma w_{i}(t)$ for relatively small $w_{i}(t)$’s. And for larger values of $w_{i}(t)$, we get $\Gamma_{i}(w(t))\simeq 0$ (see Figure 2). Therefore, by proper choice of $\eta$ and $\sigma$, equation (4) suppresses small entries of $w(t)$ by pushing them towards zero, thus, favoring sparser results. To simplify the analysis, with some abuse of notation, we approximate the function $\Gamma(w^{(\ell)}(t))$ with the following function: $$\Gamma_{i}(w^{(\ell)}(t))=\left\{\begin{array}[]{ll}w_{i}^{(\ell)}(t)&\mbox{if% $|w_{i}^{(\ell)}(t)|\leq\theta_{t}$};\\ 0&\mbox{otherwise},\end{array}\right.$$ (6) where $\theta_{t}$ is a small positive threshold. Following the same approach as [29] and assuming $\alpha_{t}$ to be small enough such that equation (5) can be expanded as powers of $\alpha_{t}$, we can approximate equation (III-A) with the following simpler version: {subequations} $$y(t)=x(t)\cdot w(t)$$ (7) $$w(t+1)=w(t)-\alpha_{t}\left(y(t)\left(x(t)-\frac{y(t)w(t)}{\|w(t)\|_{2}^{2}}% \right)+\eta\Gamma(w(t))\right)$$ (8) In the above approximation, we also omitted the term $\alpha_{t}\eta\left(w(t)\cdot\Gamma(w(t))\right)w(t)$ since $w(t)\cdot\Gamma(w(t))$ would be negligible, specially as $\theta_{t}$ in equation (6) becomes smaller. The overall learning algorithm for one constraint node is given by Algorithm 1. In words, in Algorithm 1 $y(t)$ is the projection of $x(t)$ on the basis vector $w(t)$. If for a given data vector $x(t)$, $y(t)$ is equal to zero, namely, the data is orthogonal to the current weight vector $w(t)$, then according to equation (8) the weight vector will not be updated. However, if the data vector $x(t)$ has some projection over $w(t)$ then the weight vector is updated towards the direction to reduce this projection. Since we are interested in finding $m$ basis vectors, we have to do the above procedure at least $m$ times in parallel.333In practice, we may have to repeat this process more than $m$ times to ensure the existence of a set of $m$ linearly independent vectors. However, our experimental results suggest that most of the time, repeating $m$ times would be sufficient. Remark 1. Although we are interested in finding a sparse graph, note that too much sparseness is not desired. This is because we are going to use the feedback sent by the constraint nodes to eliminate input noise at pattern nodes during the recall phase. Now if the graph is too sparse, the number of feedback messages received by each pattern node is too small to be relied upon. Therefore, we must adjust the penalty coefficient $\eta$ such that resulting neural graph is sufficiently sparse. In the section on experimental results, we compare the error correction performance for different choices of $\eta$. III-B Convergence analysis In order to prove that Algorithm 1 converges to the proper solution, we use results from statistical learning. More specifically, we benefit from the convergence of Stochastic Gradient Descent (SGD) algorithms [30]. To prove the convergence, let $E(w)=\sum_{\mu}|x^{\mu}\cdot w|^{2}$ be the cost function we would like to minimize. Furthermore, let $A=\mathbb{E}\{xx^{T}|x\in\mathcal{X}\}$ be the corelation matrix for the patterns in the training set. Therefore, due to uniformity assumption for the patterns in the training set, one can rewrite $E(w)=w^{T}Aw$. Finally, denote $A_{\mu}=x^{\mu}(x^{\mu})^{T}$. Now consider the following assumptions: A1. $\|A\|_{2}\leq\Upsilon<\infty$ and $\sup_{\mu}\|A_{\mu}\|_{2}=\|x^{\mu}\|^{2}\leq\zeta<\infty$. A2. $\alpha_{t}>0$, $\sum\alpha_{t}\rightarrow\infty$ and $\sum\alpha_{t}^{2}<\infty$, where $\alpha_{t}$ is the small learning rate defined in III-A. The following lemma proves the convergence of Algorithm 1 to a local minimum $w^{*}$. Lemma 1. Let assumptions A1 and A2 hold. Then, Algorithm 1 converges to a local minimum $w^{*}$ for which $\nabla E(w^{*})=0$. Proof. To prove the lemma, we use the convergence results in [30] and show that the required assumptions to ensure convergence holds for the proposed algorithm. For simplicity, these assumptions are listed here: 1. The cost function $E(w)$ is three-times differentiable with continuous derivatives. It is also bounded from below. 2. The usual conditions on the learning rates are fulfilled, i.e. $\sum\alpha_{t}=\infty$ and $\sum\alpha_{t}^{2}<\infty$. 3. The second moment of the update term should not grow more than linearly with size of the weight vector. In other words, $$E(w)\leq a+b\|w\|_{2}^{2}$$ for some constants $a$ and $b$. 4. When the norm of the weight vector $w$ is larger than a certain horizon $D$, the opposite of the gradient $-\nabla E(W)$ points towards the origin. Or in other words: $$\inf{\|w\|_{2}>D}w\cdot\nabla E(w)>0$$ 5. When the norm of the weight vector is smaller than a second horizon $F$, with $F>D$, then the norm of the update term $\left(2y(t)x(t)+\eta\Gamma(w(t))\right)$ is bounded regardless of $x(t)$. This is usually a mild requirement: $$\forall x(t)\in\mathcal{X},\ \sup_{\|w\|_{2}\leq F}\|\left(2y(t)x(t)+\eta% \Gamma(w(t))\right)\|_{2}\leq K_{0}$$ To start, assumption $1$ holds trivially as the cost function is three-times differentiable, with continuous derivatives. Furthermore, $E(w)\geq 0$. Assumption $2$ holds because of our choice of the step size $\alpha_{t}$, as mentioned in the lemma description. Assumption $3$ ensures that the vector $w$ could not escape by becoming larger and larger. Due to the constraint $\|w\|_{2}=1$, this assumption holds as well. Assumption $4$ holds as well because: $$\displaystyle\mathbb{E}_{\mu}\left(2A_{\mu}w+\eta\Gamma(w)\right)^{2}$$ $$\displaystyle=$$ $$\displaystyle 4w^{T}\mathbb{E}_{\mu}(A_{\mu}^{2})w+\eta^{2}\|\Gamma(w)\|_{2}^{2}$$ (9) $$\displaystyle+$$ $$\displaystyle 4\eta w^{T}\mathbb{E}_{\mu}(A_{\mu})\Gamma(w)$$ $$\displaystyle\leq$$ $$\displaystyle 4\|w\|_{2}^{2}\zeta^{2}+\eta^{2}\|w\|_{2}^{2}+4\eta\Upsilon\|w\|% _{2}^{2}$$ $$\displaystyle=$$ $$\displaystyle\|w\|_{2}^{2}(4\zeta^{2}+4\eta\Upsilon+\eta^{2})$$ Finally, assumption $5$ holds because: $$\displaystyle\|2A_{\mu}w+\eta\Gamma(w)\|_{2}^{2}$$ $$\displaystyle=$$ $$\displaystyle 4w^{T}A_{\mu}^{2}w+\eta^{2}\|\Gamma(w)\|_{2}^{2}$$ (10) $$\displaystyle+$$ $$\displaystyle 4\eta w^{T}A_{\mu}\Gamma(w)$$ $$\displaystyle\leq$$ $$\displaystyle\|w\|_{2}^{2}(4\zeta^{2}+4\eta\zeta+\eta^{2})$$ Therefore, $\exists F>D$ such that as long as $\|w\|_{2}^{2}<F$: $$\sup_{\|w\|_{2}^{2}<E}\|2A_{\mu}w+\eta\Gamma(w)\|_{2}^{2}\leq(2\zeta+\eta)^{2}% F=\hbox{constant}$$ (11) Since all necessary assumptions hold for the learning algorithm 1, it converges to a local minimum where $\nabla E(w^{*})=0$. ∎ Next, we prove the desired result, i.e. the fact that at the local minimum, the resulting weight vector is orthogonal to the patterns, i.e. $Aw=0$. Theorem 2. In the local minimum where $\nabla E(w^{*})=0$, the optimal vector $w^{*}$ is orthogonal to the patterns in the training set. Proof. Since $\nabla E(w^{*})=2Aw^{*}+\eta\Gamma(w^{*})=0$, we have: $$w^{*}\cdot\nabla E(w^{*})=2(w^{*})^{T}Aw^{*}+\eta w^{*}\cdot\Gamma(w^{*})$$ (12) The first term is always greater than or equal to zero. Now as for the second term, we have that $|\Gamma(w_{i})|\leq|w_{i}|$ and $\hbox{sign}(w_{i})=\hbox{sign}(\Gamma(w_{i}))$, where $w_{i}$ is the $i^{th}$ entry of $w$. Therefore, $0\leq w^{*}\cdot\Gamma(w^{*})\leq\|w^{*}\|_{2}^{2}$. Therefore, both terms on the right hand side of (12) are greater than or equal to zero. And since the left hand side is known to be equal to zero, we conclude that $(w^{*})^{T}Aw^{*}=0$ and $\Gamma(w^{*})=0$. The former means $(w^{*})^{T}Aw^{*}=\sum_{\mu}(w^{*}\cdot x^{\mu})^{2}=0$. Therefore, we must have $w^{*}\cdot x^{\mu}=0$, for all $\mu=1,\dots,C$. This simply means that the vector $w^{*}$ is orthogonal to all the patterns in the training set. ∎ Remark 2. Note that the above theorem only proves that the obtained vector is orthogonal to the data set and says nothing about its degree of sparsity. The reason is that there is no guarantee that the dual basis of a subspace be sparse. The introduction of the penalty function $g(w)$ in problem (III-A) only encourages sparsity by suppressing the small entries of $w$, i.e. shifting them towards zero if they are really small or leaving them intact if they are rather large. And from the fact that $\Gamma(w^{*})=0$, we know this is true as the entries in $w^{*}$ are either large or zero, i.e. there are no small entries. Our experimental results in section VII show that in fact this strategy works perfectly and the learning algorithm results in sparse solutions. III-C Avoiding the all-zero solution Although in problem (III-A) we have the constraint $\|w\|_{2}=1$ to make sure that the algorithm does not converge to the trivial solution $w=0$, due to approximations we made when developing the optimization algorithm, we should make sure to choose the parameters such that the all-zero solution is still avoided. To this end, denote $w^{\prime}(t)=w(t)-\alpha_{t}y(t)\left(x(t)-\frac{y(t)w(t)}{\|w(t)\|_{2}^{2}}\right)$ and consider the following inequalities: $$\displaystyle\|w(t+1)\|_{2}^{2}$$ $$\displaystyle=$$ $$\displaystyle\|w(t)-\alpha_{t}y(t)\left(x(t)-\frac{y(t)w(t)}{\|w(t)\|_{2}^{2}}\right)$$ $$\displaystyle-$$ $$\displaystyle\alpha_{t}\eta\Gamma(w(t))\|_{2}^{2}$$ $$\displaystyle=$$ $$\displaystyle\|w^{\prime}(t)\|^{2}+\alpha_{t}^{2}\eta^{2}\|\Gamma(w(t))\|^{2}$$ $$\displaystyle-$$ $$\displaystyle 2\alpha_{t}\eta\Gamma(w(t))\cdot w^{\prime}(t)$$ $$\displaystyle\geq$$ $$\displaystyle\|w^{\prime}(t)\|_{2}^{2}-2\alpha_{t}\eta\Gamma(w(t))\cdot w^{% \prime}(t)$$ Now in order to have $\|w(t+1)\|_{2}^{2}>0$, we must have that $2\alpha_{t}\eta|\Gamma(w(t))^{T}w^{\prime}(t)|<\|w^{\prime}(t)\|_{2}^{2}$. Given that, $|\Gamma(w(t))\cdot w^{\prime}(t)|\leq\|w^{\prime}(t)\|_{2}\|\Gamma(w(t))\|_{2}$, it is therefore sufficient to have $2\alpha_{t}\eta\|\Gamma(w(t))\|_{2}<\|w^{\prime}(t)\|_{2}$. On the other hand, we have: $$\displaystyle\|w^{\prime}(t)\|_{2}^{2}$$ $$\displaystyle=$$ $$\displaystyle\|w(t)\|_{2}^{2}+\alpha_{t}^{2}y(t)^{2}\|x(t)-\frac{y(t)w(t)}{\|w% (t)\|_{2}^{2}}\|_{2}^{2}$$ (14) $$\displaystyle\geq$$ $$\displaystyle\|w(t)\|_{2}^{2}$$ As a result, in order to have $\|w(t+1)\|_{2}^{2}>0$, it is sufficient to have $2\alpha_{t}\eta\|\Gamma(w(t))\|_{2}<\|w(t)\|_{2}$. Finally, since we have $|\Gamma(w(t))|\leq|w(t)|$ (entry-wise), we know that $\|\Gamma(w(t))\|_{2}\leq\|w(t)\|_{2}$. Therefore, having $2\alpha_{t}\eta<1\leq\|w(t)\|_{2}/\|\Gamma(w(t))\|_{2}$ ensures $\|w(t)\|_{2}>0$. Remark 3. Interestingly, the above choice for the function $w-\eta\Gamma(w)$ looks very similar to the soft thresholding function (15) introduced in [27] to perform iterative compressed sensing. The authors show that their choice of the sparsity function is very competitive in the sense that one can not get much better results by choosing other thresholding functions. However, one main difference between their work and that of ours is that we enforce the sparsity as a penalty in equation (4) while they apply the soft thresholding function in equation (15) to the whole $w$, i.e. if the updated value of $w$ is larger than a threshold, it is left intact while it will be put to zero otherwise. $$f_{t}(x)=\left\{\begin{array}[]{ll}x-\theta_{t}&\mbox{if $x>\theta_{t}$};\\ x+\theta_{t}&\mbox{if $x<-\theta_{t}$}\\ 0&\mbox{otherwise}.\end{array}\right.$$ (15) where $\theta_{t}$ is the threshold at iteration $t$ and tends to zero as $t$ grows. III-D Making the Algorithm Parallel In order to find $m$ constraints, we need to repeat Algorithm 1 several times. Fortunately, we can repeat this process in parallel, which speeds up the algorithm and is more meaningful from a biological point of view as each constraint neuron can act independently of other neighbors. Although doing the algorithm in parallel may result in linearly dependent constraints once in a while, our experimental results show that starting from different random initial points, the algorithm converges to different distinct constraints most of the time. And the chance of getting redundant constraints reduces if we start from a sparse random initial point. Besides, as long as we have enough distinct constraints, the recall algorithm in the next section can start eliminating noise and there is no need to learn all the distinct basis vectors of the null space defined by the training patterns (albeit the performance improves as we learn more and more linearly independent constraints). Therefore, we will use the parallel version to have a faster algorithm in the end. IV Recall Phase In the recall phase, we are going to design an iterative algorithm that corresponds to message passing on a graph. The algorithm exploits the fact that our learning algorithm resulted in the connectivity matrix of the neural graph which is sparse and orthogonal to the memorized patterns. Therefore, given a noisy version of the learned patterns, we can use the feedback from the constraint neurons in Fig. 1 to eliminate noise. More specifically, the linear input sums to the constraint neurons are given by the elements of the vector $W(x^{\mu}+z)=Wx^{\mu}+Wz=Wz$, with $z$ being the integer-valued input noise (biologically speaking, the noise can be interpreted as a neuron skipping some spikes or firing more spikes than it should). Based on observing the elements of $Wz$, each constraint neuron feeds back a message (containing info about $z$) to its neighboring pattern neurons. Based on this feedback, and exploiting the fact that $W$ is sparse, the pattern neurons update their states in order to reduce the noise $z$. It must also be mentioned that we initially assume assymetric neural weights during the recall phase. More specifically, we assume the backward weight from constraint neuron $i$ to pattern neuron $j$, denoted by $W^{b}_{ij}$ be equal to the sign of the weight from pattern neuron $i$ to constraint neuron $j$, i.e. $W^{b}_{ij}=\hbox{sign}(W_{ij})$, where sign(x) is equal to $+1$, $0$ or $-1$ if $x>0$, $x=0$ or $x<0$, respectively. This assumption simplifies the error correction analysis. Later in section IV-B, we are going to consider another version of the algorithm which works with symmetric weights, i.e. $W^{b}_{ij}=W_{ij}$, and compare the performance of all suggested algorithms together in section VII. IV-A The Recall Algorithms The proposed algorithm for the recall phase comprises a series of forward and backward iterations. Two different methods are suggested in this paper, which slightly differ from each other in the way pattern neurons are updated. The first one is based on the Winner-Take-All approach (WTA) and is given by Algorithm 2. In this version, only the pattern node that receives the highest amount of normalized feedback updates its state while the other pattern neurons maintain their current states. The normalization is done with respect to the degree of each pattern neuron, i.e. the number of edges connected to each pattern neuron in the neural graph. The winner-take-all circuitry can be easily added to the neural model shown in Figure 1 using any of the classic WTA methods [14]. The second approach, given by Algorithm 3, is much simpler: in every iteration, each pattern neuron decides locally whether or not to update its current state. More specifically, if the amount of feedback received by a pattern neuron exceeds a threshold, the neuron updates its state; otherwise, it remains unchanged.444Note that in order to maintain the current value of a neuron in case no input feedback is received, we can add self-loops to pattern neurons in Figure 1. These self-loops are not shown in the figure for clarity. In both algorithms, the quantity $g^{(2)}_{j}$ can be interpreted as the number of feedback messages received by pattern neuron $x_{j}$ from the constraint neurons. On the other hand, the sign of $g^{(1)}_{j}$ provides an indication of the sign of the noise that affects $x_{j}$, and $|g^{(1)}_{j}|$ indicates the confidence level in the decision regarding the sign of the noise. It is worthwhile mentioning that the Majority-Voting decoding algorithm is very similar to the Bit-Flipping algorithm of Sipser and Spielman to decode LDPC codes [31] and a similar approach in [32] for compressive sensing methods. Remark 4. To give the reader some insight about why the neural graph should be sparse in order for the above algorithms to work, consider the backward iteration of both algorithms: it is based on counting the fraction of received input feedback messages from the neighbors of a pattern neuron. In the extreme case, if the neural graph is complete, then a single noisy pattern neuron results in the violation of all constraint neurons in the forward iteration. As a result, in the backward iteration all the pattern neurons receive feedback from their neighbors and it is impossible to tell which of the pattern neuron is the noisy one. However, if the graph is sparse, a single noisy pattern neuron only makes some of the constraints unsatisfied. Consequently, in the recall phase only the nodes which share the neighborhood of the noisy node receive input feedbacks. And the fraction of the received feedbacks would be much larger for the original noisy node. Therefore, by merely looking at the fraction of received feedback from the constraint neurons, one can identify the noisy pattern neuron with high probability as long as the graph is sparse and the input noise is reasonable bounded. IV-B Some Practical Modifications Although algorithm 3 is fairly simple and practical, each pattern neuron still needs two types of information: the number of received feedbacks and the net input sum. Although one can think of simple neural architectures to obtain the necessary information, we can modify the recall algorithm to make it more practical and simpler. The trick is to replace the degree of each node $x_{j}$ with the $\ell_{1}$-norm of the outgoing weights. In other words, instead of using $\|w_{j}\|_{0}=d_{j}$, we use $\|w_{j}\|_{1}$. Furthermore, we assume symmetric weights, i.e $W^{b}_{ij}=W_{ij}$. Interestingly, in some of our experimental results corresponding to denser graphs, this approach performs much better, as will be illustrated in section VII. One possible reason behind this improvement might be the fact that using the $\ell_{1}$-norm instead of the $\ell_{0}$-norm in 3 will result in better differentiation between two vectors that have the same number of non-zero elements, i.e. have equal $\ell_{0}$-norms, but differ from each other in the magnitude of the element, i.e. their $\ell_{1}$-norms differ. Therefore, the network may use this additional information in order to identify the noisy nodes in each update of the recall algorithm. V Performance Analysis In order to obtain analytical estimates on the recall probability of error, we assume that the connectivity graph $W$ is sparse. With respect to this graph, we define the pattern and constraint degree distributions as follows. Definition 1. For the bipartite graph $W$, let $\lambda_{i}$ ($\rho_{j}$) denote the fraction of edges that are adjacent to pattern (constraint) nodes of degree $i$ ($j$). We call $\{\lambda_{1},\dots,\lambda_{m}\}$ and $\{\rho_{1},\dots,\rho_{n}\}$ the pattern and constraint degree distribution form the edge perspective, respectively. Furthermore, it is convenient to define the degree distribution polynomials as $$\lambda(z)=\sum_{i}\lambda_{i}z^{i-1}\hbox{ and }\rho(z)=\sum_{i}\rho_{i}z^{i-% 1}.$$ The degree distributions are determined after the learning phase is finished and in this section we assume they are given. Furthermore, we consider an ensemble of random neural graphs with a given degree distribution and investigate the average performance of the recall algorithms over this ensemble. Here, the word ”ensemble” refers to the fact that we assume having a number of random neural graphs with the given degree distributions and do the analysis for the average scenario. To simplify analysis, we assume that the noise entries are $\pm 1$. However, the proposed recall algorithms can work with any integer-valued noise and our experimental results suggest that this assumption is not necessary in practice. Finally, we assume that the errors do not cancel each other out in the constraint neurons (as long as the number of errors is fairly bounded). This is in fact a realistic assumption because the neural graph is weighted, with weights belonging to the real field, and the noise values are integers. Thus, the probability that the weighted sum of some integers be equal to zero is negligible. We do the analysis only for the Majority-Voting algorithms since if we choose the Majority-Voting update threshold $\varphi=1$, roughly speaking, we will have the winner-take-all algorithm.555It must be mentioned that choosing $\varphi=1$ does not yield the WTA algorithm exactly because in the original WTA, only one node is updated in each round. However, in this version with $\varphi=1$, all nodes that receive feedback from all their neighbors are updated. Nevertheless, the performance of the both algorithms is rather similar. As mentioned earlier, in this paper we will perform the analysis for general sparse bipartite graphs. However, restricting ourselves to a particular type of sparse graphs known as ”expander” allows us to prove stronger results on the recall error probabilities. More details can be found in Appendix C and in [10]. However, since it is very difficult, if not impossible in certain cases, to make a graph expander during an iterative learning method, we focus on the more general case of sparse neural graphs. To start the analysis, let $\mathcal{E}_{t}$ denote the set of erroneous pattern nodes at iteration $t$, and $\mathcal{N}(\mathcal{E}_{t})$ be the set of constraint nodes that are connected to the nodes in $\mathcal{E}_{t}$, i.e. these are the constraint nodes that have at least one neighbor in $\mathcal{E}_{t}$. In addition, let $\mathcal{N}^{c}(\mathcal{E}_{t})$ denote the (complimentary) set of constraint neurons that do not have any connection to any node in $\mathcal{E}_{t}$. Denote also the average neighborhood size of $\mathcal{E}_{t}$ by $S_{t}=\mathbb{E}(|\mathcal{N}(\mathcal{E}_{t})|)$. Finally, let $\mathcal{C}_{t}$ be the set of correct pattern nodes. Based on the error correcting algorithm and the above notations, in a given iteration two types of error events are possible: 1. Type-1 error event: A node $x\in\mathcal{C}_{t}$ decides to update its value. The probability of this phenomenon is denoted by $P_{e_{1}}(t)$. 2. Type-2 error event: A node $x\in\mathcal{E}_{t}$ updates its value in the wrong direction. Let $P_{e_{2}}(t)$ denote the probability of error for this type. We start the analysis by finding explicit expressions and upper bounds on the average of $P_{e_{1}}(t)$ and $P_{e_{2}}(t)$ over all nodes as a function $S_{t}$. We then find an exact relationship for $S_{t}$ as a function of $|\mathcal{E}_{t}|$, which will provide us with the required expressions on the average bit error probability as a function of the number of noisy input symbols, $|\mathcal{E}_{0}|$. Having found the average bit error probability, we can easily bound the block error probability for the recall algorithm. V-A Error probability - type 1 To begin, let $P^{x}_{1}(t)$ be the probability that a node $x\in\mathcal{C}_{t}$ with degree $d_{x}$ updates its state. We have: $$P^{x}_{1}(t)=\hbox{Pr}\{\frac{|\mathcal{N}(\mathcal{E}_{t})\cap\mathcal{N}(x)|% }{d_{x}}\geq\varphi\}$$ (16) where $\mathcal{N}(x)$ is the neighborhood of $x$. Assuming random construction of the graph and relatively large graph sizes, one can approximate $P^{x}_{1}(t)$ by $$P_{1}^{x}(t)\approx\sum_{i=\lceil\varphi d_{x}\rceil}^{d_{x}}{d_{x}\choose i}% \left(\frac{S_{t}}{m}\right)^{i}\left(1-\frac{S_{t}}{m}\right)^{d_{x}-i}.$$ (17) In the above equation, $S_{t}/m$ represents the probabaility of having one of the $d_{x}$ edges connected to the $S_{t}$ constraint neurons that are neighbors of the erroneous pattern neurons. As a result of the above equations, we have: $$P_{e_{1}}(t)=\mathbb{E}_{d_{x}}(P^{x}_{1}(t)),$$ (18) where $\mathbb{E}_{d_{x}}$ denote the expectation over the degree distribution $\{\lambda_{1},\dots,\lambda_{m}\}$. Note that if $\varphi=1$, the above equation simplifies to $$P_{e_{1}}(t)=\lambda\left(\frac{S_{t}}{m}\right)$$ V-B Error probability - type 2 A node $x\in\mathcal{E}_{t}$ makes a wrong decision if the net input sum it receives has a different sign than the sign of noise it experiences. Instead of finding an exact relation, we bound this probability by the probability that the neuron $x$ shares at least half of its neighbors with other neurons, i.e. $P_{e_{2}}(t)\leq\hbox{Pr}\{\frac{|\mathcal{N}(\mathcal{E^{*}}_{t})\cap\mathcal% {N}(x)|}{d_{x}}\geq 1/2\}$, where $\mathcal{E^{*}}_{t}=\mathcal{E}_{t}\setminus x$. Letting $P^{x}_{2}(t)=\hbox{Pr}\{\frac{|\mathcal{N}(\mathcal{E^{*}}_{t})\cap\mathcal{N}% (x)|}{d_{x}}\geq 1/2|\hbox{deg}(x)=d_{x}\}$, we will have: $$P^{x}_{2}(t)=\sum_{i=\lceil d_{x}/2\rceil}^{d_{x}}{d_{x}\choose i}\left(\frac{% S^{*}_{t}}{m}\right)^{i}\left(1-\frac{S^{*}_{t}}{m}\right)^{d_{x}-i}$$ (19) where $S^{*}_{t}=\mathbb{E}(|\mathcal{N}(\mathcal{E}^{*}_{t})|)$ Therefore, we will have: $$P_{e_{2}}(t)\leq\mathbb{E}_{d_{x}}(P^{x}_{2}(t))$$ (20) Combining equations (18) and (20), the bit error probability at iteration $t$ would be $$\displaystyle P_{b}(t+1)$$ $$\displaystyle=$$ $$\displaystyle\hbox{Pr$\{x\in\mathcal{C}_{t}\}$}P_{e_{1}}(t)+\hbox{Pr$\{x\in% \mathcal{E}_{t}\}$}P_{e_{2}}(t)$$ (21) $$\displaystyle=$$ $$\displaystyle\frac{n-|\mathcal{E}_{t}|}{n}P_{e_{1}}(t)+\frac{|\mathcal{E}_{t}|% }{n}P_{e_{2}}(t)$$ And finally, the average block error rate is given by the probability that at least one pattern node $x$ is in error. Therefore: $$P_{e}(t)=1-(1-P_{b}(t))^{n}$$ (22) Equation (22) gives the probability of making a mistake in iteration $t$. Therefore, we can bound the overall probability of error, $P_{E}$, by setting $P_{E}=\lim_{t\rightarrow\infty}P_{e}(t)$. To this end, we have to recursively update $P_{b}(t)$ in equation (21) and using $|\mathcal{E}_{t+1}|\approx nP_{b}(t+1)$. However, since we have assumed that the noise values are $\pm 1$, we can provide an upper bound on the total probability of error by considering $$\displaystyle P_{E}$$ $$\displaystyle\leq$$ $$\displaystyle P_{e}(1)$$ (23) In other words, we assume that the recall algorithms either correct the input error in the first iteration or an error is declared. Obviously, this bound is not tight as in practice and one might be able to correct errors in later iterations. In fact simulation results confirm this expectation. However, this approach provides a nice analytical upper bound since it only depends on the initial number of noisy nodes. As the initial number of noisy nodes grow, the above bound becomes tight. Thus, in summary we have: $$P_{E}\leq 1-(1-\frac{n-|\mathcal{E}_{0}|}{n}\bar{P}_{1}^{x}-\frac{|\mathcal{E}% _{0}|}{n}\bar{P}_{2}^{x})^{n}$$ (24) where $\bar{P}_{i}^{x}=\mathbb{E}_{d_{x}}\{P_{i}^{x}\}$ and $|\mathcal{E}_{0}|$ is the number of noisy nodes in the input pattern initially. Remark 5. One might hope to further simplify the above inequalities by finding closed form approximation of equations (17) and (19). However, as one expects, this approach leads to very loose and trivial bounds in many cases. Therefore, in our experiments shown in section VII we compare simulation results to the theoretical bound derived using equations (17) and (19). Now, what remains to do is to find an expression for $S_{t}$ and $S^{*}_{t}$ as a function of $|\mathcal{E}_{t}|$. The following lemma will provide us with the required relationship. Lemma 3. The average neighborhood size $S_{t}$ in iteration $t$ is given by: $$S_{t}=m\left(1-(1-\frac{\bar{d}}{m})^{|\mathcal{E}_{t}|}\right)$$ (25) where $\bar{d}$ is the average degree for pattern nodes. Proof. The proof is given in Appendix A. ∎ VI Pattern Retrieval Capacity It is interesting to see that, except for its obvious influence on the learning time, the number of patterns $C$ does not have any effect in the learning or recall algorithm. As long as the patterns come from a subspace, the learning algorithm will yield a matrix which is orthogonal to all of the patterns in the training set. And in the recall phase, all we deal with is $Wz$, with $z$ being the noise which is independent of the patterns. Therefore, in order to show that the pattern retrieval capacity is exponential with $n$, all we need to show is that there exists a ”valid” training set $\mathcal{X}$ with $C$ patterns of length $n$ for which $C\propto a^{rn}$, for some $a>1$ and $0<r$. By valid we mean that the patterns should come from a subspace with dimension $k<n$ and the entries in the patterns should be non-negative integers. The next theorem proves the desired result. Theorem 4. Let $\mathcal{X}$ be a $C\times n$ matrix, formed by $C$ vectors of length $n$ with non-negative integers entries between $0$ and $Q-1$. Furthermore, let $k=rn$ for some $0<r<1$. Then, there exists a set of such vectors for which $C=a^{rn}$, with $a>1$, and $\hbox{rank}(\mathcal{X})=k<n$. Proof. The proof is based on construction: we construct a data set $\mathcal{X}$ with the required properties. To start, consider a matrix $G\in\mathbb{R}^{k\times n}$ with rank $k$ and $k=rn$, with $0<r<1$. Let the entries of $G$ be non-negative integers, between $0$ and $\gamma-1$, with $\gamma\geq 2$. We start constructing the patterns in the data set as follows: consider a set of random vectors $u^{\mu}\in\mathbb{R}^{k}$, $\mu=1,\dots,C$, with integer-valued entries between $0$ and $\upsilon-1$, where $\upsilon\geq 2$. We set the pattern $x^{\mu}\in\mathcal{X}$ to be $x^{\mu}=u^{\mu}\cdot G$, if all the entries of $x^{\mu}$ are between $0$ and $Q-1$. Obviously, since both $u^{\mu}$ and $G$ have only non-negative entries, all entries in $x^{\mu}$ are non-negative. Therefore, it is the $Q-1$ upper bound that we have to worry about. The $j^{th}$ entry in $x^{\mu}$ is equal to $x_{j}^{\mu}=u^{\mu}\cdot G_{j}$, where $G_{j}$ is the $j^{th}$ column of $G$. Suppose $G_{j}$ has $d_{j}$ non-zero elements. Then, we have: $$x_{j}^{\mu}=u^{\mu}\cdot G_{j}\leq d_{j}(\gamma-1)(\upsilon-1)$$ Therefore, denoting $d^{*}=\max_{j}d_{j}$, we could choose $\gamma$, $\upsilon$ and $d^{*}$ such that $$Q-1\geq d^{*}(\gamma-1)(\upsilon-1)$$ (26) to ensure all entries of $x^{\mu}$ are less than $Q$. As a result, since there are $\upsilon^{k}$ vectors $u$ with integer entries between $0$ and $\upsilon-1$, we will have $\upsilon^{k}=\upsilon^{rn}$ patterns forming $\mathcal{X}$. Which means $C=\upsilon^{rn}$, which would be an exponential number in $n$ if $\upsilon\geq 2$. ∎ As an example, if $G$ can be selected to be a sparse $200\times 400$ matrix with $0/1$ entries (i.e. $\gamma=2$) and $d^{*}=10$, and $u$ is also chosen to be a vector with $0/1$ elements (i.e. $\upsilon=2$), then it is sufficient to choose $Q\geq 11$ to have a pattern retrieval capacity of $C=2^{rn}$. Remark 6. Note that inequality (26) was obtained for the worst-case scenario and in fact is very loose. Therefore, even if it does not hold, we will still be able to memorize a very large number of patterns since a big portion of the generated vectors $x^{\mu}$ will have entries less than $Q$. These vectors correspond to the message vectors $u^{\mu}$ that are ”sparse” as well, i.e. do not have all entries greater than zero. The number of such vectors is a polynomial in $n$, the degree of which depends on the number of non-zero entries in $u^{\mu}$. VII Simulation Results VII-A Simulation Scenario We have simulated the proposed learning and recall algorithms for three different network sizes $n=200,400,800$, with $k=n/2$ for all cases. For each case, we considered a few different setups with different values for $\alpha$, $\eta$, and $\theta$ in the learning algorithm 1, and different $\varphi$ for the Majority-Voting recall algorithm 3. For brevity, we do not report all the results for various combinations but present only a selection of them to give insight on the performance of the proposed algorithms. In all cases, we generated $50$ random training sets using the approach explained in the proof of theorem 4, i.e. we generated a generator matrix $G$ at random with $0/1$ entries and $d^{*}=10$. We also used $0/1$ generating message words $u$ and put $Q=11$ to ensure the validity of the generated training set. However, since in this setup we will have $2^{k}$ patterns to memorize, doing a simulation over all of them would take a lot of time. Therefore, we have selected a random sample sub-set $\mathcal{X}$ each time with size $C=10^{5}$ for each of the $50$ generated sets and used these subsets as the training set. For each setup, we performed the learning algorithm and then investigated the average sparsity of the learned constraints over the ensemble of $50$ instances. As explained earlier, all the constraints for each network were learned in parallel, i.e. to obtain $m=n-k$ constraints, we executed Algorithm 1 from random initial points $m$ time. As for the recall algorithms, the error correcting performance was assessed for each set-up, averaged over the ensemble of $50$ instances. The empirical results are compared to the theoretical bounds derived in Section V as well. VII-B Learning Phase Results In the learning algorithm, we pick a pattern from the training set each time and adjust the weights according to Algorithm 1. Once we have gone over all the patterns, we repeat this operation several times to make sure that update for one pattern does not adversely affect the other learned patterns. Let $t$ be the iteration number of the learning algorithm, i.e. the number of times we have gone over the training set so far. Then we set $\alpha_{t}\propto\alpha_{0}/t$ to ensure the conditions of Theorem 1 is satisfied. Interestingly, all of the constraints converged in at most two learning iterations for all different setups. Therefore, the learning is very fast in this case. Figure 3 illustrates the percentage of pattern nodes with the specified sparsity measure defined as $\varrho=\kappa/n$, where $\kappa$ is the number of non-zero elements. From the figure we notice two trends. The first is the effect of sparsity threshold, which as it is increased, the network becomes sparser. The second one is the effect of network size, which as it grows, the connections become sparser. VII-C Recall Phase Results For the recall phase, in each trial we pick a pattern randomly from the training set, corrupt a given number of its symbols with $\pm 1$ noise and use the suggested algorithm to correct the errors. A pattern error is declared if the output does not match the correct pattern. We compare the performance of the two recall algorithms: Winner-Take-All (WTA) and Majority-Voting (MV). Table I shows the simulation parameters in the recall phase for all scenarios (unless specified otherwise). Figure 4 illustrates the effect of the sparsity threshold $\theta$ on the performance of the error correcting algorithm in the recall phase. Here, we have $n=400$ and $k=200$. Two different sparsity thresholds are compared together, namely $\theta_{t}\propto 0.031/t$ and $\theta_{t}\propto 0.021/t$. Clearly, as network becomes sparser, i.e. $\theta$ increases, the performance of both recall algorithms improve. In Figure 5 we have investigated the effect of network size on the performance of recall algorithms by comparing the pattern error rates for two different network size, namely $n=800$ and $n=400$ with $k=n/2$ in both cases. As obvious from the figure, the performance improves to a great extent when we have a larger network. This is partially because of the fact that in larger networks, the connections are relatively sparser as well. Figure 6 compares the results obtained in simulation with the upper bound derived in Section V. Note that as expected, the bound is quite loose since in deriving inequality (22) we only considered the first iteration of the algorithm. We have also investigated the tightness of the bound given in equation (23) with simulation results. To this end, we compare $P_{e}(1)$ and $\lim_{t\rightarrow\infty}P_{e}(t)$ in our simulations for the case of $\pm 1$ noise. Figure 7 illustrates the result and it is evident that allowing the recall algorithm to iterate improves the final probability of error to a great extent. Finally, we investigate the performance of the modified more practical version of the Majority-Voting algorithm, which was explained in Section IV-B. Figure 8 compares the performance of the WTA and original MV algorithms with the modified version of MV algorithm for a network with size $n=200$, $k=100$ and learning parameters $\alpha_{t}\propto 0.45/t$, $\eta=0.45$ and $\theta_{t}\propto 0.015/t$. The neural graph of this particular example is rather dense, because of small $n$ and sparsity threshold $\theta$. Therefore, here the modified version of the Majority-Voting algorithm performs better because of the extra information provided by the $\ell_{1}$-norm (than the $\ell_{0}$-norm in the original version of the Majority-Voting algorithm). However, note that we did not observe this trend for the other simulation scenarios where the neural graph was sparser. VIII Conclusions and Future Works In this paper, we proposed a neural associative memory which is capable of exploiting inherent redundancy in input patterns to enjoy an exponentially large pattern retrieval capacity. Furthermore, the proposed method uses simple iterative algorithms for both learning and recall phases which makes gradual learning possible and maintain rather good recall performances. The convergence of the proposed learning algorithm was proved using techniques from stochastic approximation. We also analytically investigated the performance of the recall algorithm by deriving an upper bound on the probability of recall error as a function of input noise. Our simulation results confirms the consistency of the theoretical results with those obtained in practice, for different network sizes and learning/recall parameters. Improving the error correction capabilities of the proposed network is definitely a subject of our future research. We have already started investigating this issue and proposed a different network structure which reduces the error correction probability by a factor of $10$ in many cases [22]. We are working on different structures to obtain even more robust recall algorithms. Extending this method to capture other sorts of redundancy, i.e. other than belonging to a subspace, will be another topic which we would like to explore in future. Finally, considering some practical modifications to the learning and recall algorithms is of great interest. One good example is simultaneous learn and recall capability, i.e. to have a network which learns a subset of the patterns in the subspace and move immediately to the recall phase. Now during the recall phase, if the network is given a noisy version of the patterns previously memorized, it eliminates the noise using the algorithms described in this paper. However, if it is a new pattern, i.e. one that we have not learned yet, the network adjusts the weights in order to learn this pattern as well. Such model is of practical interest and closer to real-world neuronal networks. Therefore, it would be interesting to design a network with this capability while maintaining good error correcting capabilities and large pattern retrieval capacities. Acknowledgment The authors would like to thank Prof. Wulfram Gerstner and his lab members, as well as Mr. Amin Karbasi for their helpful comments and discussions. This work was supported by Grant 228021-ECCSciEng of the European Research Council. Appendix A Average neighborhood size In this appendix, we find an expression for the average neighborhood size for erroneous nodes, $S_{t}=\mathbb{E}(|\mathcal{N}(\mathcal{E}_{t})|)$. Towards this end, we assume the following procedure for constructing a right-irregular bipartite graph: • In each iteration, we pick a variable node $x$ with a degree randomly determined according to the given degree distribution. • Based on the given degree $d_{x}$, we pick $d_{x}$ constraint nodes uniformly at random with replacement and connect $x$ to the constraint node. • We repeat this process $n$ times, until all variable nodes are connected. Note that the assumption that we do the process with replacement is made to simplify the analysis. This assumption becomes more exact as $n$ grows. Having the above procedure in mind, we will find an expression for the average number of constraint nodes in each construction round. More specifically, we will find the average number of constraint nodes connected to $i$ pattern nodes at round $i$ of construction. This relationship will in turn yields the average neighborhood size of $|\mathcal{E}_{t}|$ erroneous nodes in iteration $t$ of error correction algorithm described in section IV. With some abuse of notations, let $S_{e}$ denote the number of constraint nodes connected to pattern nodes in round $e$ of construction procedure mentioned above. We write $S_{e}$ recursively in terms of $e$ as follows: $$\displaystyle S_{e+1}$$ $$\displaystyle=$$ $$\displaystyle\mathbb{E}_{d_{x}}\{\sum_{j=0}^{d_{x}}{d_{x}\choose j}\left(\frac% {S_{e}}{m}\right)^{d_{x}-j}\left(1-\frac{S_{e}}{m}\right)^{j}(S_{e}+j)\}$$ (27) $$\displaystyle=$$ $$\displaystyle\mathbb{E}_{d_{x}}\{S_{e}+d_{x}(1-S_{e}/m)\}$$ $$\displaystyle=$$ $$\displaystyle S_{e}+\bar{d}(1-S_{e}/m)$$ Where $\bar{d}=\mathbb{E}_{d_{x}}\{d_{x}\}$ is the average degree of the pattern nodes. In words, the first line calculates the average growth of the neighborhood when a new variable node is added to the graph. The proceeding equalities directly follows from relationship on binomial sums. Noting that $S_{1}=\bar{d}$, one obtains: $$\displaystyle S_{t}=m\left(1-(1-\frac{\bar{d}}{m})^{|\mathcal{E}_{t}|}\right)$$ (28) In order to verify the correctness of the above analysis, we have performed some simulations for different network sizes and degree distributions obtained from the graphs returned by the learning algorithm. We generated $100$ random graphs and calculated the average neighborhood size in each iteration over these graphs. Furthermore, two different network sizes were considered $n=100,200$ and $m=n/2$ in all cases, where $n$ and $m$ are the number of pattern and constraint nodes, respectively. The result for $n=100,m=50$ is shown in Figure 9, where the average neighborhood size in each iteration is illustrated and compared with theoretical estimations given by equation (28). Figure 10 shows similar results for $n=200$, $m=100$. In the figure,the dashed line shows the average neighborhood size over these graphs. The solid line corresponds to theoretical estimations. It is obvious that the theoretical value is an exact approximation of the simulation results. Appendix B Expander Graphs This section contains the definitions and the necessary background on expander graphs. Definition 2. A regular $(d_{p},d_{c},n,m)$ bipartite graph $W$ is a bipartite graph between $n$ pattern nodes of degree $d_{p}$ and $m$ constraint nodes of degree $d_{c}$. Definition 3. An $(\alpha n,\beta d_{p})$-expander is a $(d_{p},d_{c},n,m)$ bipartite graph such that for any subset $\mathcal{P}$ of pattern nodes with $|\mathcal{P}|<\alpha n$ we have $|\mathcal{N}(\mathcal{P})|>\beta d_{p}|\mathcal{P}|$ where $\mathcal{N}(\mathcal{P})$ is the set of neighbors of $\mathcal{P}$ among the constraint nodes. The following result from [31] shows the existence of families of expander graphs with parameter values that are relevant to us. Theorem 5. [31] Let $W$ be a randomly chosen $(d_{p},d_{c})-$regular bipartite graph between $n$ $d_{p}-$regular vertices and $m=(d_{p}/d_{c})$ $d_{c}-$regular vertices. Then for all $0<\alpha<1$, with high probability, all sets of $\alpha n$ $d_{p}-$regular vertices in $W$ have at least $$n\left(\frac{d_{p}}{d_{c}}(1-(1-\alpha)^{d_{c}})-\sqrt{\frac{2d_{c}\alpha h(% \alpha)}{\log_{2}e}}\right)$$ neighbors, where $h(\cdot)$ is the binary entropy function. The following result from [33] shows the existence of families of expander graphs with parameter values that are relevant to us. Theorem 6. Let $d_{c}$, $d_{p}$, $m$, $n$ be integers, and let $\beta<1-1/d_{p}$. There exists a small $\alpha>0$ such that if $W$ is a $(d_{p},d_{c},n,m)$ bipartite graph chosen uniformly at random from the ensemble of such bipartite graphs, then $W$ is an $(\alpha n,\beta d_{p})$-expander with probability $1-o(1)$, where $o(1)$ is a term going to zero as $n$ goes to infinity. Appendix C Analysis of the Recall Algorithms for Expander Graphs C-A Analysis of the Winner-Take-All Algorithm We prove the error correction capability of the winner-take-all algorithm in two steps: first we show that in each iteration, only pattern neurons that are corrupted by noise will be chosen by the winner-take-all strategy to update their state. Then, we prove that the update is in the right direction, i.e. toward removing noise from the neurons. Lemma 7. If the constraint matrix $W$ is an $(\alpha n,\beta d_{p})$ expander, with $\beta>1/2$, and the original number of erroneous neurons are less than or equal to $2$, then in each iteration of the winner-take-all algorithm only the corrupted pattern nodes update their value and the other nodes remain intact. For $\beta=3/4$, the algorithm will always pick the correct node if we have two or fewer erroneous nodes. Proof. If we have only one node $x_{i}$ in error, it is obvious that the corresponding node will always be the winner of the winner-take-all algorithm unless there exists another node that has the same set of neighbors as $x_{i}$. However, this is impossible as because of the expansion properties, the neighborhood of these two nodes must have at least $2\beta d_{p}$ members which for $\beta>1/2$ is strictly greater than $d_{p}$. As a result, no two nodes can have the same neighborhood and the winner will always be the correct node. In the case where there are two erroneous nodes, say $x_{i}$ and $x_{j}$, let $\mathcal{E}$ be the set $\{x_{i},x_{j}\}$ and $\mathcal{N}(\mathcal{E})$ be the corresponding neighborhood on the constraint nodes side. Furthermore, assume $x_{i}$ and $x_{j}$ share $d_{p^{\prime}}$ of their neighbors so that $|\mathcal{N}(\mathcal{E})|=2d_{p}-d_{p^{\prime}}$. Now because of the expansion properties: $$|\mathcal{N}(\mathcal{E})|=2d_{p}-d_{p^{\prime}}>2\beta d_{p}\Rightarrow d_{p^% {\prime}}<2(1-\beta)d_{p}.$$ Now we have to show that there are no nodes other than $x_{i}$ and $x_{j}$ that can be the winner of the winner-take-all algorithm. To this end, note that only those nodes that are connected to $N(\mathcal{E})$ will receive some feedback and can hope to be the winner of the process. So let’s consider such a node $x_{\ell}$ that is connected to $d_{p_{\ell}}$ of the nodes in $N(\mathcal{E})$. Let $\mathcal{E}^{\prime}$ be $\mathcal{E}\cup\{x_{\ell}\}$ and $N(\mathcal{E}^{\prime})$ be the corresponding neighborhood. Because of the expansion properties we have $|N(\mathcal{E}^{\prime})|=d_{p}-d_{p_{\ell}}+|N(\mathcal{E})|>3\beta d_{p}$. Thus: $$\displaystyle d_{p_{\ell}}$$ $$\displaystyle<$$ $$\displaystyle d_{p}+|N(\mathcal{E})|-3\beta d_{p}=3d_{p}(1-\beta)-d_{p^{\prime% }}.$$ Now, note that the nodes $x_{i}$ and $x_{j}$ will receive some feedback from $2d_{p}-d_{p^{\prime}}$ edges because we assume there is no noise cancellation due to the fact that neural weights are real-valued and noise entries are integers. Since $2d_{p}-d_{p^{\prime}}>3d_{p}(1-\beta)-d_{p^{\prime}}$ for $\beta>1/2$, we conclude that $d_{p}-d_{p^{\prime}}>d_{p_{\ell}}$ which proves that no node outside $\mathcal{E}$ can be picked during the winner-take-all algorithm as long as $|\mathcal{E}|\leq 2$ for $\beta>1/2$. ∎ In the next lemma, we show that the state of erroneous neurons is updated in the direction of reducing the noise. Lemma 8. If the constraint matrix $W$ is an $(\alpha n,\beta d_{p})$ expander, with $\beta>3/4$, and the original number of erroneous neurons is less than or equal $e_{\min}=2$, then in each iteration of the winner-take-all algorithm the winner is updated toward reducing the noise. Proof. When there is only one erroneous node, it is obvious that all its neighbors agree on the direction of update and the node reduces the amount of noise by one unit. If there are two nodes $x_{i}$ and $x_{j}$ in error, since the number of their shared neighbors is less than $2(1-\beta)d_{p}$ (as we proved in the last lemma), then more than half of their neighbors would be unique if $\beta\geq 3/4$. These unique neighbors agree on the direction of update. Therefore, whoever the winner is will be updated to reduce the amount of noise by one unit. ∎ The following theorem sums up the results of the previous lemmas to show that the winner-take-all algorithm is guaranteed to perform error correction. Theorem 7. If the constraint matrix $W$ is an $(\alpha n,\beta d_{p})$ expander, with $\beta\geq 3/4$, then the winner-take-all algorithm is guaranteed to correct at least $e_{\min}=2$ positions in error, irrespective of the magnitudes of the errors. Proof. The proof is immediate from Lemmas 7 and 8. ∎ C-B Analysis of the Majority Algorithm Roughly speaking, one would expect the Majority-Voting algorithm to be sub-optimal in comparison to the winner-take-all strategy, since the pattern neurons need to make independent decisions, and are not allowed to cooperate amongst themselves. In this subsection, we show that despite this restriction, the Majority-Voting algorithm is capable of error correction; the sub-optimality in comparison to the winner-take-all algorithm can be quantified in terms of a larger expansion factor $\beta$ being required for the graph. Theorem 8. If the constraint matrix $W$ is an $(\alpha n,\beta d_{p})$ expander with $\beta>\frac{4}{5}$, then the Majority-Voting algorithm with $\varphi=\frac{3}{5}$ is guaranteed to correct at least two positions in error, irrespective of the magnitudes of the errors. Proof. As in the proof for the winner-take-all case, we will show our result in two steps: first, by showing that for a suitable choice of the Majority-Voting threshold $\varphi$, that only the positions in error are updated in each iteration, and that this update is towards reducing the effect of the noise. Case 1 First consider the case that only one pattern node $x_{i}$ is in error. Let $x_{j}$ be any other pattern node, for some $j\neq i$. Let $x_{i}$ and $x_{j}$ have $d_{p^{\prime}}$ neighbors in common. As argued in the proof of Lemma 7, we have that $$d_{p^{\prime}}<2d_{p}(1-\beta).$$ (29) Hence for $\beta=\frac{4}{5}$, $x_{i}$ receives non-zero feedback from at least $\frac{3}{5}d_{p}$ constraint nodes, while $x_{j}$ receives non-zero feedback from at most $\frac{2}{5}d_{p}$ constraint nodes. In this case, it is clear that setting $\varphi=\frac{3}{5}$ will guarantee that only the node in error will be updated, and that the direction of this update is towards reducing the noise. Case 2 Now suppose that two distinct nodes $x_{i}$ and $x_{j}$ are in error. Let $\mathcal{E}=\{x_{i},x_{j}\}$, and let $x_{i}$ and $x_{j}$ share $d_{p^{\prime}}$ common neighbors. If the noise corrupting these two pattern nodes, denoted by $z_{i}$ and $z_{j}$, are such that $\hbox{sign}(z_{i})=\hbox{sign}(z_{j})$, then both $x_{i}$ and $x_{j}$ receive $-\hbox{sign}(z_{i})$ along all $d_{p}$ edges that they are connected to during the backward iteration. Now suppose that $\hbox{sign}(z_{i})\neq\hbox{sign}(z_{j})$. Then $x_{i}$ ($x_{j}$) receives correct feedback from at least the $d_{p}-d_{p^{\prime}}$ edges in $\mathcal{N}(\{x_{i}\})\backslash\mathcal{E}$ (resp. $\mathcal{N}(\{x_{j}\})\backslash\mathcal{E}$) during the backward iteration. Therefore, if $d_{p^{\prime}}<d_{p}/2$, the direction of update would be also correct and the feedback will reduce noise during the update. And from equation (29) we know that for $\beta=4/5$, $d_{p^{\prime}}\leq 2d_{p}/5<d_{p}/2$. Therefore, the two noisy nodes will be updated towards the correct direction. Let us now examine what happens to a node $x_{\ell}$ that is different from the two erroneous nodes $x_{i},x_{j}$. Suppose that $x_{\ell}$ is connected to $d_{p_{\ell}}$ nodes in $\mathcal{N}(\mathcal{E})$. From the proof of Lemma 7, we know that $$\displaystyle d_{p_{\ell}}$$ $$\displaystyle<$$ $$\displaystyle 3d_{p}(1-\beta)-d_{p^{\prime}}$$ $$\displaystyle\leq$$ $$\displaystyle 3d_{p}(1-\beta).$$ Hence $x_{\ell}$ receives at most $3d_{p}(1-\beta)$ non-zero messages during the backward iteration. For $\beta>\frac{4}{5}$, we have that $d_{p}-2d_{p}(1-\beta)>3d_{p}(1-\beta)$. Hence by setting $\beta=\frac{4}{5}$ and $\varphi=[d_{p}-2d_{p}(1-\beta)]/d_{p}=\frac{3}{5}$, it is clear from the above discussion that we have ensured the following in the case of two erroneous pattern nodes: • The noisy pattern nodes are updated towards the direction of reducing noise. • No pattern node other than the erroneous pattern nodes is updated. ∎ C-C Minimum Distance of Patterns Next, we present a sufficient condition such that the minimum Hamming distance666Two (possibly non-binary) $n-$length vectors $x$ and $y$ are said to be at a Hamming distance $d$ from each other if they are coordinate-wise equal to each other on all but $d$ coordinates. between these exponential number of patterns is not too small. In order to prove such a result, we will exploit the expansion properties of the bipartite graph $W$; our sufficient condition will be in terms of a lower bound on the parameters of the expander graph. Theorem 9. Let $W$ be a $(d_{p},d_{c},n,m)-$regular bipartite graph, that is an $(\alpha n,\beta d_{p})$ expander. Let $\mathcal{X}$ be the set of patterns corresponding to the expander weight matrix $W$. If $$\beta>\frac{1}{2}+\frac{1}{4d_{p}},$$ then the minimum distance between the patterns is at least $\lfloor\alpha n\rfloor+1$. Proof. Let $d$ be less than $\alpha n$, and $W_{i}$ denote the $i^{th}$ column of $W$. If two patterns are at Hamming distance $d$ from each other, then there exist non-zero integers $c_{1},c_{2},\dots,c_{d}$ such that $$c_{1}W_{i_{1}}+c_{2}W_{i_{2}}+\cdots+c_{d}W_{i_{d}}=0,$$ (30) where $i_{1},\dots,i_{d}$ are distinct integers between $1$ and $n$. Let $\mathcal{P}$ denote any set of pattern nodes of the graph represented by $W$, with $|\mathcal{P}|=d$. As in [32], we divide $\mathcal{N}(\mathcal{P})$ into two disjoint sets: $\mathcal{N}_{unique}(\mathcal{P})$ is the set of nodes in $\mathcal{N}(\mathcal{P})$ that are connected to only one edge emanating from $\mathcal{P}$, and $\mathcal{N}_{shared}(\mathcal{P})$ comprises the remaining nodes of $\mathcal{N}(\mathcal{P})$ that are connected to more than one edge emanating from $\mathcal{P}$. If we show that $|\mathcal{N}_{unique}(\mathcal{P})|>0$ for all $\mathcal{P}$ with $|\mathcal{P}|=d$, then (30) cannot hold, allowing us to conclude that no two patterns with distance $d$ exist. Using the arguments in [32, Lemma 1], we obtain that $$|\mathcal{N}_{unique}(\mathcal{P})|>2d_{p}|\mathcal{P}|\left(\beta-\frac{1}{2}% \right).$$ Hence no two patterns with distance $d$ exist if $$2d_{p}d\left(\beta-\frac{1}{2}\right)>1\Leftrightarrow\beta>\frac{1}{2}+\frac{% 1}{2d_{p}d}.$$ By choosing $\beta>\frac{1}{2}+\frac{1}{4d_{p}}$, we can hence ensure that the minimum distance between patterns is at least $\lfloor\alpha n\rfloor+1$. ∎ C-D Choice of Parameters In order to put together the results of the previous two subsections and obtain a neural associative scheme that stores an exponential number of patterns and is capable of error correction, we need to carefully choose the various relevant parameters. 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Interventional Few-Shot Learning Zhongqi Yue1,3  Hanwang Zhang1  Qianru Sun2  Xian-Sheng Hua3 1Nanyang Technological University, 2Singapore Management University, 3Alibaba Group yuez0003@ntu.edu.sg, hanwangzhang@ntu.edu.sg,  qianrusun@smu.edu.sg,  xiansheng.hxs@alibaba-inc.com Abstract We uncover an ever-overlooked deficiency in the prevailing Few-Shot Learning (FSL) methods: the pre-trained knowledge is indeed a confounder that limits the performance. This finding is rooted from our causal assumption: a Structural Causal Model (SCM) for the causalities among the pre-trained knowledge, sample features, and labels. Thanks to it, we propose a novel FSL paradigm: Interventional Few-Shot Learning (IFSL). Specifically, we develop three effective IFSL algorithmic implementations based on the backdoor adjustment, which is essentially a causal intervention towards the SCM of many-shot learning: the upper-bound of FSL in a causal view. It is worth noting that the contribution of IFSL is orthogonal to existing fine-tuning and meta-learning based FSL methods, hence IFSL can improve all of them, achieving a new 1-/5-shot state-of-the-art on miniImageNet, tieredImageNet, and cross-domain CUB. Code is released at https://github.com/yue-zhongqi/ifsl. 1 Introduction Few-Shot Learning (FSL) — the task of training a model using very few samples — is nothing short of a panacea for any scenario that requires fast model adaptation to new tasks [69], such as minimizing the need for expensive trials in reinforcement learning [31] and saving computation resource for light-weight neural networks [28, 26]. Although we knew that, more than a decade ago, the crux of FSL is to imitate the human ability of transferring prior knowledge to new tasks [19], not until the recent advances in pre-training techniques, had we yet reached a consensus on “what & how to transfer”: a powerful neural network $\Omega$ pre-trained on a large dataset $\mathcal{D}$. In fact, the prior knowledge learned from pre-training prospers today’s deep learning era, e.g., $\mathcal{D}$ = ImageNet, $\Omega$ = ResNet in visual recognition [25, 24]; $\mathcal{D}$ = Wikipedia, $\Omega$ = BERT in natural language processing [65, 17]. In the context of pre-trained knowledge, we denote the original FSL training set as support set $\mathcal{S}$ and the test set as query set $\mathcal{Q}$, where the classes in $(\mathcal{S},\mathcal{Q})$ are unseen (or new) in $\mathcal{D}$. Then, we can use $\Omega$ as a backbone (fixed or partially trainable) for extracting sample representations $\mathbf{x}$, and thus FSL can be achieved simply by fine-tuning the target model on $\mathcal{S}$ and test it on $\mathcal{Q}$ [13, 18]. However, the fine-tuning only exploits the $\mathcal{D}$’s knowledge on “what to transfer”, but neglects “how to transfer”. Fortunately, the latter can be addressed by applying a post-pre-training and pre-fine-tuning strategy: meta-learning [55]. Different from fine-tuning whose goal is the “model” trained on $\mathcal{S}$ and tested on $\mathcal{Q}$, meta-learning aims to learn the “meta-model” — a learning behavior — trained on many learning episodes $\{(\mathcal{S}_{i},\mathcal{Q}_{i})\}$ sampled from $\mathcal{D}$ and tested on the target task $(\mathcal{S},\mathcal{Q})$. In particular, the behavior can be parameterized by $\phi$ using model parameter generator [49, 21] or initialization [20]. After meta-learning, we denote $\Omega_{\phi}$ as the new model starting point for the subsequent fine-tuning on target task $(\mathcal{S},\mathcal{Q})$. Figure 1 illustrates the relationships among the above discussed FSL paradigms. It is arguably a common sense that the stronger the pre-trained $\Omega$ is, the better the downstream model will be. However, we surprisingly find that this may not be always the case in FSL. As shown in Figure 1(a), we can see a paradox: though stronger $\Omega$ improves the performance on average, it indeed degrades that of samples in $\mathcal{Q}$ dissimilar to $\mathcal{S}$. To illustrate the “dissimilar”, we show a 5-shot learning example in Figure 1(b), where the prior knowledge on “green grass” and “yellow grass” is misleading. For example, the “Lion” samples in $\mathcal{Q}$ have “yellow grass”, hence they are misclassified as “Dog” whose $\mathcal{S}$ has major “yellow grass”. If we use stronger $\Omega$, the seen old knowledge (“grass” & “color”) will be more robust than the unseen new knowledge (“Lion” & “Dog”), and thus the old becomes even more misleading. We believe that such a paradox reveals an unknown systematic deficiency in FSL, which has been however hidden for years by our gold-standard “fair” accuracy, averaged over all the random $(\mathcal{S},\mathcal{Q})$ test trials, regardless of the similarity between $\mathcal{S}$ and $\mathcal{Q}$ (cf. Figure 1(a)). Though Figure 2 only illustrates the fine-tune FSL paradigm, the deficiency is expected in the meta-learning paradigm, as fine-tune is also used in each meta-train episode (Figure 1). We will analyze them thoroughly in Section 5. In this paper, we first point out that the cause of the deficiency: pre-training can do evil in FSL, and then propose a novel FSL paradigm: Interventional Few-Shot Learning (IFSL), to counter the evil. Our theory is based on the assumption of the causalities among the pre-trained knowledge, few-shot samples, and class labels. Specifically, our contributions are summarized as follows. • We begin with a Structural Causal Model (SCM) assumption in Section 2.2, which shows that the pre-trained knowledge is essentially a confounder that causes spurious correlations between the sample features and class labels in support set. As an intuitive example in Figure 1(b), even though the “grass” feature is not the cause of the “Lion” label, the prior knowledge on “grass” still confounds the classifier to learn a correlation between them. • In Section 2.3, we illustrate a causal justification of why the proposed IFSL fundamentally works better: it is essentially a causal approximation to many-shot learning. This motivates us to develop three effective implementations of IFSL using the backdoor adjustment [46] in Section 3. • Thanks to the causal intervention, IFSL is naturally orthogonal to the downstream fine-tuning and meta-learning based FSL methods [20, 66, 29]. In Section 5.2, IFSL improves all baselines by a considerable margin, achieving new 1-/5-shot state-of-the-arts: 73.51%/83.21% on miniImageNet [66], 83.07%/88.69% on tieredImageNet [52], and 50.71%/64.43% on cross-domain CUB [70]. • We further diagnose the detailed performances of FSL methods across different similarities between $\mathcal{S}$ and $\mathcal{Q}$. We find that IFSL outperforms all baselines in every inch. 2 Problem Formulations 2.1 Few-Shot Learning We are interested in a prototypical FSL: train a $K$-way classifier on an $N$-shot support set $\mathcal{S}$, where $N$ is a small number of training samples per class (e.g., $N$=1 or 5); then test the classifier on a query set $\mathcal{Q}$. As illustrated in Figure 1, we have the following two paradigms to train the classifier $P(y|\mathbf{x};\theta)$, predicting the class $y\in\{1,...,K\}$ of a sample $\mathbf{x}$: Fine-Tuning. We consider the prior knowledge as the sample feature representation $\mathbf{x}$, encoded by the pre-trained network $\Omega$ on dataset $\mathcal{D}$. In particular, we refer $\mathbf{x}$ to the output of the frozen sub-part of $\Omega$ and the rest trainable sub-part of $\Omega$ (if any) can be absorbed into $\theta$. We train the classifier $P(y|\mathbf{x};\theta)$ on the support set $\mathcal{S}$, and then evaluate it on the query set $\mathcal{Q}$ in a standard supervised way. Meta-Learning. Yet, $\Omega$ only carries prior knowledge in a way of “representation”. If the dataset $\mathcal{D}$ can be re-organized as the training episodes $\{(\mathcal{S}_{i},\mathcal{Q}_{i})\}$, each of which can be treated as a “sandbox” that has the same $N$-shot-$K$-way setting as the target $(\mathcal{S},\mathcal{Q})$. Then, we can model the “learning behavior” from $\mathcal{D}$ parameterized as $\phi$, which can be learned by the above fine-tuning paradigm for each $(\mathcal{S}_{i},\mathcal{Q}_{i})$. Formally, we denote $P_{\phi}(y|\mathbf{x};\theta)$ as the enhanced classifier equipped with the learned behavior. For example, $\phi$ can be the classifier weight generator [21], distance kernel function in $k$-NN [66], or even $\theta$’s initialization [20]. Considering $L_{\phi}(\mathcal{S}_{i},\mathcal{Q}_{i};\theta)$ as the loss function of $P_{\phi}(y|\mathbf{x};\theta)$ trained on $\mathcal{S}_{i}$ and tested on $\mathcal{Q}_{i}$, we can have $\phi\leftarrow\arg\min_{(\phi,\theta)}\mathbb{E}_{i}\left[L_{\phi}(\mathcal{S}% _{i},\mathcal{Q}_{i};\theta)\right]$, and then we fix the optimized $\phi$ and fine-tune for $\theta$ on $\mathcal{S}$ and test on $\mathcal{Q}$. Please refer to Appendix 5 for the details of various fine-tuning and meta-learning settings. 2.2 Structural Causal Model From the above discussion, we can see that $(\phi,\theta)$ in meta-learning and $\theta$ in fine-tuning are both dependent on the pre-training. Such “dependency” can be formalized with a Structural Causal Model (SCM) [46] proposed in Figure 2(a), where the nodes denote the abstract data variables and the directed edges denote the (functional) causality, e.g., $X\rightarrow Y$ denotes that $X$ is the cause and $Y$ is the effect. Now we introduce the graph and the rationale behind its construction at a high-level. Please see Section 3 for the detailed functional implementations. $\boldsymbol{D\to X}$. We denote $X$ as the feature representation and $D$ as the pre-trained knowledge, e.g., the dataset $\mathcal{D}$ and its induced model $\Omega$. This link assumes that the feature $X$ is extracted by using $\Omega$. $\boldsymbol{D\rightarrow C\leftarrow X}$. We denote $C$ as the transformed representation of $X$ in the low-dimensional manifold, whose base is inherited from $D$. This assumption can be rationalized as follows. 1) $D\rightarrow C$: a set of data points are usually embedded in a low-dimensional manifold. This finding can be dated back to the long history of dimensionality reduction [63, 53]. Nowadays, there are theoretical [3, 10] and empirical [82, 76] evidences showing that disentangled semantic manifolds emerge during training deep networks. 2) $X\rightarrow C$: features can be represented using (or projected onto) the manifold base linearly [64, 11] or non-linearly [8]. In particular, as later discussed in Section 3, we explicitly implement the base as feature dimensions (Figure 2(b)) and class-specific mean features (Figure 2(c)). $\boldsymbol{X\rightarrow Y\leftarrow C}$. We denote $Y$ as the classification effect (e.g., logits), which is determined by $X$ via two ways: 1) the direct $X\rightarrow Y$ and 2) the mediation $X\rightarrow C\rightarrow Y$. In particular, the first way can be removed if $X$ can be fully represented by $C$ (e.g., feature-wise adjustment in Section 3). The second way is inevitable even if the classifier does not take $C$ as an explicit input, because any $X$ can be inherently represented by $C$. To illustrate, suppose that $X$ is a linear combination of two base vectors plus a noise residual: $\mathbf{x}=c_{1}\mathbf{b}_{1}+c_{2}\mathbf{b}_{2}+\mathbf{e}$, any classifier $f(\mathbf{x})$ = $f(c_{1}\mathbf{b}_{1}+c_{2}\mathbf{b}_{2}+\mathbf{e})$ will implicitly exploit the $C$ representation in terms of $\mathbf{b}_{1}$ and $\mathbf{b}_{2}$. In fact, this assumption also fundamentally validates unsupervised representation learning [7]. To see this, if $C\not\rightarrow Y$ in Figure 2(a), uncovering the latent knowledge representation from $P(Y|X)$ would be impossible, because the only path left that transfers knowledge from $D$ to $Y$: $D\rightarrow X\rightarrow Y$, is cut off by conditioning on $X$: $D\not\rightarrow X\rightarrow Y$. An ideal FSL model should capture the true causality between $X$ and $Y$ to generalize to unseen samples. For example, as illustrated in Figure 1(b), we expect that the “Lion” prediction is caused by the “lion” feature per se, but not the background “grass”. However, from the SCM in Figure 2(a), the conventional correlation $P(Y|X)$ fails to do so, because the increased likelihood of $Y$ given $X$ is not only due to “X causes Y” via $X\rightarrow Y$ and $X\rightarrow C\rightarrow Y$, but also the spurious correlation via 1) $D\rightarrow X$, e.g., the “grass” knowledge generates the “grass” feature, and 2) $D\rightarrow C\rightarrow Y$, e.g., the “grass” knowledge generates the “grass” semantic, which provides useful context for “Lion” label. Therefore, to pursue the true causality between $X$ and $Y$, we need to use the causal intervention $P(Y|do(X))$ [48] instead of the likelihood $P(Y|X)$ for the FSL objective. 2.3 Causal Intervention via Backdoor Adjustment By now, an astute reader may notice that the causal graph in Figure 2(a) is also valid for Many-Shot Learning (MSL), i.e., conventional learning based on pre-training. Compared to FSL, the $P(Y|X)$ estimation of MSL is much more robust. For example, on miniImageNet, a 5-way-550-shot fine-tuned classifier can achieve 95% accuracy, while a 5-way-5-shot one only obtains 79%. We used to blame FSL for insufficient data by the law of large numbers in point estimation [16]. However, it does not answer why MSL converges to the true causal effects as the number of samples increases infinitely. In other words, why $P(Y|do(X))\approx P(Y|X)$ in MSL while $P(Y|do(X))\not\approx P(Y|X)$ in FSL? To answer the question, we need to incorporate the endogenous feature sampling $\mathbf{x}\sim P(X|I)$ into the estimation of $P(Y|X)$, where $I$ denotes the sample ID. We have $P(Y|X=\mathbf{x}_{i})\coloneqq\mathbb{E}_{\mathbf{x}\sim P(X|I)}P(Y|X=\mathbf{% x},I=i)=P(Y|I)$, i.e., we can use $P(Y|I)$ to estimate $P(Y|X)$. In Figure 3(a), the causal relation between $I$ and $X$ is purely $I\to X$, i.e., $X\to I$ does not exist, because tracing the $X$’s ID out of many-shot samples is like to find a needle in a haystack, given the nature that a DNN feature is an abstract and diversity-reduced representation of many samples [23]. However, as shown in Figure 3(b), $X\rightarrow I$ persists in FSL, because it is much easier for a model to “guess” the correspondence, e.g., the 1-shot extreme case that has a trivial 1-to-1 mapping for $X\leftrightarrow I$. Therefore, as we formally show in Appendix 1, the key causal difference between MSL and FSL is: MSL essentially makes $I$ an instrumental variable [1] that achieves $P(Y|X)\coloneqq P(Y|I)\approx P(Y|do(X))$. Intuitively, we can see that all the causalities between $I$ and $D$ in MSL are all blocked by colliders111In causal graph, the junction $A\rightarrow B\leftarrow C$ is called a “collider”, making $A$ and $C$ independent even though $A$ and $C$ are linked via $B$ [46]. For example, $A$ = “Quality”, $C$ = “Luck”, and $B$ = “Paper Acceptance”., making $I$ and $D$ independent. So, the feature $X$ is essentially “intervened” by $I$, no longer dictated by $D$, e.g., neither “yellow grass” nor “green grass” dominates “Lion” in Figure 1(b), mimicking the casual intervention by controlling the use of pre-trained knowledge. In this paper, we propose to use the backdoor adjustment [46] to achieve $P(Y|do(X))$ without the need for many-shot, which certainly undermines the definition of FSL. The backdoor adjustment assumes that we can observe and stratify the confounder, i.e., $D=\{d\}$, where each $d$ is a stratification of the pre-trained knowledge. Formally, as shown in Appendix 2, the backdoor adjustment for the graph in Figure 2(a) is: $$P\left(Y|do(X=\boldsymbol{x})\right)=\sum_{d}P\left(Y|X=\boldsymbol{x},D=d,C=g% (\mathbf{x},d)\right)P(D=d),$$ (1) where $g$ is a function defined later. However, it is not trivial to instantiate $d$, especially when $D$ is a 3rd-party delivered pre-trained network where the dataset is unobserved [22]. Next, we will offer three practical implementations of Eq. (1) for Interventional FSL. 3 Interventional Few-Shot Learning Our implementation idea is inspired from the two inherent properties of any pre-trained DNN. First, each feature dimension carries a semantic meaning, e.g., every channel in convolutional neural network is well-known to encode visual concepts [82, 76]. So, each feature dimension represents a piece of knowledge. Second, most prevailing pre-trained models use a classification task as the objective, such as the 1,000-way classifier of ResNet [25] and the token predictor of BERT [17]. Therefore, the classifier can be considered as the distilled knowledge, which has been already widely adopted in literature [26]. Next, we will detail the proposed Interventional FSL (IFSL) by providing three different implementations222We assume that the combinations of the feature dimensions or classes are linear, otherwise the adjustment requires prohibitive $\mathcal{O}(2^{n})$ sampling. We will relax this assumption in future work. for $g(\mathbf{x},d)$, $P(Y|X,D,C)$, and $P(D)$ in Eq. (1). In particular, the exact forms of $P(Y|\cdot)$ across different classifiers are given in Appendix 5. Feature-wise Adjustment. Suppose that $\mathcal{F}$ is the index set of the feature dimensions of $\mathbf{x}$, e.g., from the last-layer of the pre-trained network $\Omega$. We divide $\mathcal{F}$ into $n$ equal-size disjoint subsets, e.g., the output feature dimension of ResNet-10 is 512, if $n=8$, the $i$-th set will be a feature dimension index set of size 512/8 = 64, i.e., $\mathcal{F}_{i}=\{64(i-1)+1,...,64i\}$. The stratum set of pre-trained knowledge is defined as $D:=\{d_{1},\ldots,d_{n}\}$, where each $d_{i}=\mathcal{F}_{i}$. (i) $g(\mathbf{x},d_{i})\coloneqq\{k|k\in\mathcal{F}_{i}\cap\mathcal{I}_{t}\}$, where $\mathcal{I}_{t}$ is an index set whose corresponding absolute values in $\mathbf{x}$ are larger than the threshold $t$. The reason is simple: if a feature dimension is inactive in $\mathbf{x}$, its corresponding adjustment can be omitted. We set $t$=1e-3 in this paper. (ii) $P(Y|X,D,C)=P(Y|[\mathbf{x}]_{c})$, where $c=g(\mathbf{x},d_{i})$ is implemented as the index set defined above, $[\mathbf{x}]_{c}=\{x_{k}\}_{k\in c}$ is a feature selector which selects the dimensions of $\mathbf{x}$ according to the index set $c$. The classifier takes the adjusted feature $[\mathbf{x}]_{c}$ as input. Note that $d$ is already absorbed in $c$, so $[\mathbf{x}]_{c}$ is essentially a function of $(X,D,C)$. (iii) $P(d_{i})=1/n$, where we assume a uniform prior for the adjusted features. (iv) The overall feature-wise adjustment is: $$P(Y|do(X=\boldsymbol{x}))=\frac{1}{n}\sum_{i=1}^{n}P(Y|[\mathbf{x}]_{c}),~{}~{% }~{}\textrm{where}~{}~{}c={\{k|k\in\mathcal{F}_{i}~{}\cap~{}\mathcal{I}_{t}\}}.$$ (2) It is worth noting that the feature-wise adjustment is always applicable, as we can always have the feature representation $\mathbf{x}$ from the pre-trained network. Interestingly, our feature-wise adjustment sheds some light on the theoretical justifications for the multi-head trick in transformers [65]. We will explore this in future work. Class-wise Adjustment. Suppose that there are $m$ pre-training classes, denoted as $\mathcal{A}=\{a_{1},\ldots a_{m}\}$. In class-wise adjustment, each stratum of pre-trained knowledge is defined as a pre-training class, i.e., $D:=\{d_{1},\ldots,d_{m}\}$ and each $d_{i}=a_{i}$. (i) $g(\mathbf{x},d_{i})\coloneqq P(a_{i}|\mathbf{x})\mathbf{\bar{x}}_{i}$, where $P(a_{i}|\mathbf{x})$ is the pre-trained classifier’s probability output that $\mathbf{x}$ belongs to class $a_{i}$, and $\mathbf{\bar{x}}_{i}$ is the mean feature of pre-training samples from class $a_{i}$. Note that unlike feature-wise adjustment where $c$ is an index set, here $c=g(\mathbf{x},d_{i})$ is implemented as a real vector. (ii) $P(Y|X,D,C)=P(Y|\mathbf{x}\oplus g(\mathbf{x},d_{i}))$, where $\oplus$ denotes vector concatenation. (iii) $P(d_{i})=1/m$, where we assume a uniform prior of each class. (iv) The overall class-wise adjustment is: $$P(Y|do(X=\mathbf{x}))=\frac{1}{m}\sum_{i=1}^{m}P(Y|\mathbf{x}\oplus P\left(a_{% i}|\mathbf{x})\mathbf{\bar{x}}_{i}\right)\approx P(Y|\mathbf{x}\oplus\frac{1}{% m}\sum_{i=1}^{m}P\left(a_{i}|\mathbf{x})\mathbf{\bar{x}}_{i}\right),$$ (3) where we adopt the Normalized Weighted Geometric Mean (NWGM) [71] approximation to move the outer sum $\sum P$ into the inner $P(\sum)$. This greatly reduces the network forward-pass consumption as $m$ is usually large in pre-training dataset. Please refer to Appendix 3 for the detailed derivation. Combined Adjustment. We can combine feature-wise and class-wise adjustment to make the stratification in backdoor adjustment much more fine-grained. Our combination is simple: applying feature-wise adjustment after class-wise adjustment. Thus, we have: $$P(Y|do(X=\mathbf{x}))\approx\frac{1}{n}\sum_{i=1}^{n}P(Y|[\mathbf{x}]_{c}% \oplus\frac{1}{m}\sum_{j=1}^{m}[P(a_{j}|\mathbf{x})\mathbf{\bar{x}}_{j}]_{c}),% ~{}\textrm{where}~{}c={\{k|k\in\mathcal{F}_{i}~{}\cap~{}\mathcal{I}_{t}\}}.$$ (4) 4 Related Work Few-Shot Learning. FSL has a wide spectrum of methods, including fine-tuning [13, 18], optimizing model initialization [20, 42], generating model parameters [54, 36], learning a feature space for a better separation of sample categories [66, 77], feature transfer [58, 43], and transductive learning that additionally uses query set data [18, 29, 27]. Thanks to them, the classification accuracy has been drastically increased [29, 77, 73, 37]. However, accuracy as a single number cannot explain the paradoxical phenomenon in Figure 2. Our work offers an answer from a causal standpoint by showing that pre-training is a confounder. We not only further improve the accuracy of various FSL methods, but also explain the reason behind the improvements. In fact, the perspective offered by our work can benefit all the tasks that involve pre-training—any downstream task can be seen as FSL compared to the large-scale pre-training data. Negative Transfer. The above phenomenon is also known as the negative transfer, where learning in source domain contributes negatively to the performance in target domain [44]. Many research works have being focused on when and how to conduct this transfer learning [30, 4, 81]. Yosinski et al. [74] split ImageNet according to man-made objects and natural objects as a test bed for feature transferability. They resemble the $\mathcal{S}\not\sim\mathcal{Q}$ settings used in Figure 1(a). Other work also revealed that using deeper backbone might lead to degraded performance when the domain gap between training and test is large [33]. Some similar findings are reported in the few-shot setting [50] and NLP tasks [62]. Unfortunately, they didn’t provide a theoretical explanation why it happens. Causal Inference. Our work aims to deal with the pre-training confounder in FSL based on causal inference [48]. Causal inference was recently introduced to machine learning [40, 9] and has been applied to various fields in computer vision, including image classification [12, 38], imitation learning [15], long-tailed recognition [60] and semantic segmentation [78]. We are the first to approach FSL from a causal perspective. We would like to highlight that data-augmentation based FSL can also be considered as approximated intervention. These methods learn to generate additional support samples with image deformation [14, 79] or generative models [2, 80]. This can be view as physical interventions on the image features. Regarding the causal relation between image $X$ and label $Y$, some works adopted anti-causal learning [41], i.e., $Y\to X$, where the assumption is that labels $Y$ are disentangled enough to be treated as Independent Mechanism (IM) [45, 59], which generates observed images $X$ through $Y\to X$. However, our work targets at the more general case where labels can be entangled (e.g.“lion” and “dog” share the semantic “soft fur”) and the IM assumption may not hold. Therefore, we use causal prediction $X\to Y$ as it is essentially a reasoning process, where the IM is captured by $D$, which is engineered to be disentangled through CNN (e.g., the conv-operations are applied independently). In this way, $D$ generates visual features through $D\to X$ and emulates human’s naming process through $D\to Y$ (e.g., “fur”, “four-legged”$\to$ “meerkat”). In fact, the causal direction $X\to Y$ (NOT anti-causal $Y\to X$) has been empirically justified in complex CV tasks [32, 72, 67, 60, 61]. 5 Experiments 5.1 Datasets and Settings Datasets. We conducted experiments on benchmark datasets in FSL literature: 1) miniImageNet [66] containing 600 images per class over 100 classes. We followed the split proposed in [51]: 64/16/20 classes for train/val/test. 2) tieredImageNet [52] is much larger compared to miniImageNet with 608 classes and each class around 1,300 samples. These classes were grouped into 34 higher-level concepts and then partitioned into 20/6/8 disjoint sets for train/val/test to achieve larger domain difference between training and testing. 3) Caltech-UCSD Birds-200-2011 (CUB) [70] for cross-domain evaluation. It contains 200 classes and each class has around 60 samples. The models used for CUB test were trained on the miniImageNet. Training and evaluation settings on miniImageNet and tieredImageNet are included in Appendix 5. Implementation Details. We pre-trained the 10-layer ResNet (ResNet-10) [25] and the WideResNet (WRN-28-10) [75] as our backbones. Our proposed IFSL supports both fine-tuning and meta-learning. For fine-tuning, we applied average pooling on the last residual block and used the pooled features to train classifiers. For meta-learning, we deployed 5 representative methods that cover a large spectrum of meta-learning based FSL: 1) model initialization: MAML [20], 2) weight generator: LEO [54], transductive learning: SIB [29], 4) metric learning: MatchingNet (MN) [66], and 5) feature transfer: MTL [58]. For both fine-tuning and meta-learning, our IFSL aims to the learn classifier $P(Y|do(X))$ instead of the conventional $P(Y|X)$. Detailed implementations are given in Appendix 5. Evaluation Metrics. Our evaluation is based on the following metrics: 1) Conventional accuracy (Acc) is the average classification accuracy commonly used in FSL [20, 66, 58]. 2) Hardness-specific Acc. For each query, we define a hardness that measures its semantic dissimilarity to the support set, and accuracy is then computed at different levels of query hardness. Specifically, query hardness is computed by $h=\log\left((1-s)/s\right)$ and $s=exp{\langle\mathbf{r}^{+},\mathbf{p}_{c=gt}^{+}\rangle}/{\sum\nolimits_{c}% exp\langle\mathbf{r}^{+},\mathbf{p}_{c}^{+}\rangle}$, where $\langle\cdot\rangle$ is the cosine similarity, $(\cdot)^{+}$ represents the ReLU activation function, $\mathbf{r}$ denotes the $\Omega$ prediction logits of query, $\mathbf{p}_{c}$ denotes the average prediction logits of class $c$ in the support set and $gt$ is the ground-truth of query. Using Hardness-specific Acc is similar to evaluating the hardness of FSL tasks [18], while ours is query-sample-specific and hence is more fine-grained. Later, we will show its effectiveness to unveil the spurious effects in FSL. 3) Feature localization accuracy (CAM-Acc) quantifies if a model “pays attention” to the actual object when making prediction. It is defined as the percentage of pixels inside the object bounding box by using Grad-CAM [56] score larger than $0.9$. Compared to Acc that shows if the prediction is correct, CAM-Acc reveals whether the prediction is based on the correct visual cues. 5.2 Results and Analysis Conventional Acc. 1) From Table 5.1, we observe that IFSL consistently improves fine-tuning and meta-learning in all settings, which suggests that IFSL is agnostic to methods, datasets, and backbones. 2) In particular, the improvements are typically larger on 1-shot than 5-shot. For example, in fine-tuning, the average performance gain is 1.15% on 5-shot and 3.58% on 1-shot. The results support our analysis in Section 2.3 that FSL models are more prone to bias in lower-shot settings. 3) Regarding the average improvements on fine-tuning vs. meta-learning (e.g.$k$-NN and MN), we observe that IFSL improves more on fine-tuning in most cases. We conjecture that this is because meta-learning is an implicit form of intervention, where randomly sampled meta-training episodes effectively stratify the pre-trained knowledge. This suggests that meta-learning is fundamentally superior over fine-tuning due to increased robustness against confounders. We will investigate this potential theory in future work. 4) Additionally we see that the improvements on miniImageNet are usually larger than that on tieredImageNet. A possible reason is the much larger training set for tieredImageNet: it substantially increases the breadth of the pre-trained knowledge and the resulting models explain query samples much better. 5) According to Table 5.1 and Table 5.1, it is clear that our $k$-NN+IFSL outperforms IdeMe-Net [14] using the same pre-trained ResNet-10. This shows that using data augmentation — a method of physical data intervention as in IdeMe-Net [14] is inferior to our causal intervention in IFSL. 6) Overall, our IFSL achieves the new state-of-the-art on both datasets. Note that IFSL is flexible to be plugged into different baselines. Hardness-specific Acc. 1) Figure 5() shows the plot of Hardness-specific Acc of fine-tuning. We notice that when query becomes harder, ResNet-10 (blue curves) becomes superior to WRN-28-10 (red curves). This tendency is consistent with Figure 1(a) illustrating the effect of the confounding bias caused by pre-training. 2) Intriguingly, in Figure 5(), we notice that this tendency is reversed for meta-learning, i.e., deeper backbone always performs better. The improved performance of deeper backbone on hard queries suggests that meta-learning should have some functions to remove the confounding bias. This evidence will inspire us to provide a causal view of meta-learning in future work. 3) Overall, Figure 5 shows that using IFSL futher improves fine-tuning and meta-learning consistently across all hardness, validating the effectiveness of the proposed causal intervention. CAM-Acc & Visualization. In Figure 6, we compare +IFSL to baseline linear classifier on the left and to baseline MAML [20] on the right, and summarize CAM-Acc results in the upper-right table. From the visualization, we see that using IFSL let the model pay more attention to the objects. However, notice that all models failed in the categories colored as red. A possible reason behind the failures is the extremely small size of the object — models have to resort to context for prediction. From the numbers, we can see our improvements for 1-shot are larger than that for 5-shot, consistent with our findings using other evaluation metrics. These results suggest that IFSL helps models use the correct visual semantics for prediction by removing the confounding bias. Cross-Domain Generalization Ability. In Table 5.1, we show the testing results on CUB using the models trained on the miniImageNet. The setting is challenging due to the big domain gap between the two datasets. We chose linear classifier as it outperforms cosine and $k$-NN in cross-domain setting and compared with transductive method — SIB. The results clearly show that IFSL works well in this setting and brings consistent improvements, with the average 1.94% of Acc. In addition, we can see that applying IFSL brings larger improvements to the inductive linear classifier than to the transductive SIB. It is possibly because transductive methods involve unlabeled query data and performs better than inductive methods with the additional information. Nonetheless we observe that IFSL can further improve SIB in cross-domain (Table 5.1) and single-domain (Table 5.1) generalization. 6 Conclusions We presented a novel casual framework: Interventional Few-Shot Learning (IFSL), to address an overlooked deficiency in recent FSL methods: the pre-training is a confounder hurting the performance. Specifically, we proposed a structural causal model of the causalities in the process of FSL and then developed three practical implementations based on the backdoor adjustment. To better illustrate the deficiency, we diagnosed the classification accuracy comprehensively across query hardness, and showed that IFSL improves all the baselines across all the hardness. It is worth highlighting that the contribution of IFSL is not only about improving the performance of FSL, but also offering a causal explanation why IFSL works well: it is a causal approximation to many-shot learning. We believe that IFSL may shed light on exploring the new boundary of FSL, even though FSL is well-known to be ill-posed due to insufficient data. To upgrade IFSL, we will seek other observational intervention algorithms for better performance, and devise counterfactual reasoning for more general few-shot settings such as domain transfer. 7 Broader Impact The proposed method aims to improve the Few-Shot Learning task. 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Supplementary Material for Interventional Few-Shot Learning This supplementary material is organized as follows: • Section A.1 details our analysis in Section 2.3 by showing many-shot learning converges to true causal effect through instrumental variable (IV); • Section A.2 gives the derivation for the backdoor adjustment formula in Eq. (1); • Section A.3 presents the detailed derivation for the NWGM approximation used in Eq. (3) and (4); • Section A.4 includes the algorithms for adding IFSL to fine-tuning and meta-learning; • Section A.5 shows the implementation details for pre-training (Section A.5.1), fine-tuning (Section A.5.2) and meta-learning (Section A.5.3); • Section A.6 includes additional experimental results on Conventional Acc (Section A.6.1), Hardness-Specific Acc (Section A.6.2), CAM-Acc (Section A.6.3) and cross-domain evaluation (Section A.6.4). A.1 Instrumental Variable In this section, we will show that in our causal graph for many-shot learning, the sampling ID $I$ is essentially an instrumental variable for $X\rightarrow Y$ that achieves $P(Y|I)\approx P(Y|do(X))$. Before introducing instrumental variable, we first formally define d-separation [46], which gives a criterion to study the dependencies between nodes (data variables) in any structural causal model. d-separation. A set of nodes $Z$ blocks a path $p$ if and only if 1) $p$ contains a chain $A\rightarrow B\rightarrow C$ or a fork $A\leftarrow B\rightarrow C$ and the middle node $B$ is in $Z$; 2) $p$ contains a collider $A\rightarrow B\leftarrow C$ such that the middle node $B$ and its descendants are not in $Z$. If conditioning on $Z$ blocks every path between $X$ and $Y$, we say $X$ and $Y$ are d-separated conditional on $Z$, i.e., $X$ and $Y$ are independent given $Z$ ($X\mathchoice{\mathrel{\hbox to 0.0pt{$\displaystyle\perp$}\mskip 2.0mu {% \displaystyle\perp}}}{\mathrel{\hbox to 0.0pt{$\textstyle\perp$}\mskip 2.0mu {% \textstyle\perp}}}{\mathrel{\hbox to 0.0pt{$\scriptstyle\perp$}\mskip 2.0mu {% \scriptstyle\perp}}}{\mathrel{\hbox to 0.0pt{$\scriptscriptstyle\perp$}\mskip 2% .0mu {\scriptscriptstyle\perp}}}Y|Z$). Instrumental Variable. For a structual causal model $\mathcal{G}$, a variable Z is an instrumental variable (IV) to $X\rightarrow Y$ by satisfying the graphical criteria [48]: 1) $(Z\mathchoice{\mathrel{\hbox to 0.0pt{$\displaystyle\perp$}\mskip 2.0mu {% \displaystyle\perp}}}{\mathrel{\hbox to 0.0pt{$\textstyle\perp$}\mskip 2.0mu {% \textstyle\perp}}}{\mathrel{\hbox to 0.0pt{$\scriptstyle\perp$}\mskip 2.0mu {% \scriptstyle\perp}}}{\mathrel{\hbox to 0.0pt{$\scriptscriptstyle\perp$}\mskip 2% .0mu {\scriptscriptstyle\perp}}}Y)_{\mathcal{G}_{\overline{X}}}~{}$; 2) $(Z\not\mathchoice{\mathrel{\hbox to 0.0pt{$\displaystyle\perp$}\mskip 2.0mu {% \displaystyle\perp}}}{\mathrel{\hbox to 0.0pt{$\textstyle\perp$}\mskip 2.0mu {% \textstyle\perp}}}{\mathrel{\hbox to 0.0pt{$\scriptstyle\perp$}\mskip 2.0mu {% \scriptstyle\perp}}}{\mathrel{\hbox to 0.0pt{$\scriptscriptstyle\perp$}\mskip 2% .0mu {\scriptscriptstyle\perp}}}X)_{\mathcal{G}}~{}$, where $\mathcal{G}_{\overline{X}}$ is the manipulated graph where all incoming arrows to node $X$ are deleted. For the SCM of many-shot learning in Figure 4(a), it is easy to see that $I$ satisfies both criteria and therefore it is an IV for $X\rightarrow Y$. However, in the few-shot SCM in Figure 4(b), the paths $I\leftarrow X\leftarrow D\rightarrow C\rightarrow Y$ and $I\leftarrow X\rightarrow C\rightarrow Y$ are not blocked in $\mathcal{G}_{\overline{X}}$, which means the first criterion is not met $(I\not\mathchoice{\mathrel{\hbox to 0.0pt{$\displaystyle\perp$}\mskip 2.0mu {% \displaystyle\perp}}}{\mathrel{\hbox to 0.0pt{$\textstyle\perp$}\mskip 2.0mu {% \textstyle\perp}}}{\mathrel{\hbox to 0.0pt{$\scriptstyle\perp$}\mskip 2.0mu {% \scriptstyle\perp}}}{\mathrel{\hbox to 0.0pt{$\scriptscriptstyle\perp$}\mskip 2% .0mu {\scriptscriptstyle\perp}}}Y)_{\mathcal{G}_{\overline{X}}}~{}$ and $I$ is not an instrumental variable in the few-shot learning case. Instrumental variable can help find the true causal effect even in the presence of confounder. This is due to the collider junction that makes the IV and confounder independent ($I\mathchoice{\mathrel{\hbox to 0.0pt{$\displaystyle\perp$}\mskip 2.0mu {% \displaystyle\perp}}}{\mathrel{\hbox to 0.0pt{$\textstyle\perp$}\mskip 2.0mu {% \textstyle\perp}}}{\mathrel{\hbox to 0.0pt{$\scriptstyle\perp$}\mskip 2.0mu {% \scriptstyle\perp}}}{\mathrel{\hbox to 0.0pt{$\scriptscriptstyle\perp$}\mskip 2% .0mu {\scriptscriptstyle\perp}}}D$ in Figure 4(a)). To see this, we will first consider a simplified case of Figure 4(a) where each causal link represents a linear relationship and we aim to find the true causal effect from $X\rightarrow Y$ through linear regression. Without loss of generality, let $I,X,Y$ take the value of real number. Denote $r_{IX},r_{XY},$ and $r_{IY}$ as the slope of regression line between $I$ and $X$, $X$ and $Y$, $I$ and $Y$ respectively. Notice that $r_{XY}$ is spurious as it is contaminated by the backdoor path $X\leftarrow D\rightarrow C\rightarrow Y$. However, since the path $I\rightarrow X\leftarrow D\rightarrow C\rightarrow Y$ is blocked due to collider at $X$, $r_{IY}$ is free from confounding bias. Therefore $r_{IY}/r_{IX}$ gives the true causal effect from $X\rightarrow Y$. Similarly, in the classification case of many-shot learning, a classifier is trained to maximize the conditional probability on the IV $P(Y|I)$. As the ID-sample matching $I\rightarrow X$ is deterministic, the classifier eventually learns to predict based on the true causal relationship $X\rightarrow Y$. Yet in the complex case of image classification, it is unreasonable to assume linear relationships between variables. In the nonlinear case, it is shown in [6] that observations on IV provide a bound for the true causal effect. This means that learning based on $P(Y|I)$ provides an approximation to the true causal effect, i.e.$P(Y|I)\approx P(Y|do(X))$. A.2 Derivation of Backdoor Adjustment for the Proposed Causal Graph We will show the derivation of the backdoor adjustment for the causal graph in Figure 3(a) using the three rules of do-calculus [47]. For a causal directed acyclic graph $\mathcal{G}$, let $X,Y,Z$ and $W$ be arbitrary disjoint sets of nodes. We use $\mathcal{G}_{\overline{X}}$ to denote the manipulated graph where all incoming arrows to node $X$ are deleted. Similarly $\mathcal{G}_{\underline{X}}$ represents the graph where outgoing arrows from node $X$ are deleted. We use lower case $x,y,z$ and $w$ for specific values taken by each set of nodes: $X=x,Y=y,Z=z$ and $W=w$. For any interventional distribution compatible with $\mathcal{G}$, we have the following three rules: Rule 1 Insertion/deletion of observations: $$P(y|do(x),z,w)=P(y|do(x),w),\mathrm{if}(Y\mathchoice{\mathrel{\hbox to 0.0pt{$% \displaystyle\perp$}\mskip 2.0mu {\displaystyle\perp}}}{\mathrel{\hbox to 0.0% pt{$\textstyle\perp$}\mskip 2.0mu {\textstyle\perp}}}{\mathrel{\hbox to 0.0pt{% $\scriptstyle\perp$}\mskip 2.0mu {\scriptstyle\perp}}}{\mathrel{\hbox to 0.0pt% {$\scriptscriptstyle\perp$}\mskip 2.0mu {\scriptscriptstyle\perp}}}Z|X,W)_{% \mathcal{G}_{\overline{X}}}$$ (A5) Rule 2 Action/observation exchange: $$P(y|do(x),do(z),w)=P(y|do(x),z,w),\mathrm{if}(Y\mathchoice{\mathrel{\hbox to 0% .0pt{$\displaystyle\perp$}\mskip 2.0mu {\displaystyle\perp}}}{\mathrel{\hbox t% o 0.0pt{$\textstyle\perp$}\mskip 2.0mu {\textstyle\perp}}}{\mathrel{\hbox to 0% .0pt{$\scriptstyle\perp$}\mskip 2.0mu {\scriptstyle\perp}}}{\mathrel{\hbox to % 0.0pt{$\scriptscriptstyle\perp$}\mskip 2.0mu {\scriptscriptstyle\perp}}}Z|X,W)% _{\mathcal{G}_{\overline{X}\underline{Z}}}$$ (A6) Rule 3 Insertion/deletion of actions: $$P(y|do(x),do(z),w)=P(y|do(x),w),\mathrm{if}(Y\mathchoice{\mathrel{\hbox to 0.0% pt{$\displaystyle\perp$}\mskip 2.0mu {\displaystyle\perp}}}{\mathrel{\hbox to % 0.0pt{$\textstyle\perp$}\mskip 2.0mu {\textstyle\perp}}}{\mathrel{\hbox to 0.0% pt{$\scriptstyle\perp$}\mskip 2.0mu {\scriptstyle\perp}}}{\mathrel{\hbox to 0.% 0pt{$\scriptscriptstyle\perp$}\mskip 2.0mu {\scriptscriptstyle\perp}}}Z|X,W)_{% \mathcal{G}_{\overline{X}\overline{Z(W)}}},$$ (A7) where $Z(W)$ is the set of nodes in $Z$ that are not ancestors of any $W$-node in $\mathcal{G}_{\overline{X}}$. In our causal graph, the desired interventional distribution $P(Y|do(X=\mathbf{x}))$ can be derived by: $$\displaystyle P(Y|do(\mathbf{x}))$$ $$\displaystyle=\sum_{d}P(Y|do(X=\mathbf{x}),D=d)P(D=d|do(X=\mathbf{x}))$$ (A8) $$\displaystyle=\sum_{d}P(Y|do(X=\mathbf{x}),D=d)P(D=d)$$ (A9) $$\displaystyle=\sum_{d}P(Y|X=\mathbf{x},D=d)P(D=d)$$ (A10) $$\displaystyle=\sum_{d}\sum_{c}P(Y|X=\mathbf{x},D=d,C=c)P(C=c|X=\mathbf{x},D=d)% P(D=d)$$ (A11) $$\displaystyle=\sum_{d}P(Y|X=\mathbf{x},D=d,C=g(\mathbf{x},d))P(D=d),$$ (A12) where Eq. (A8) and Eq. (A11) follow the law of total probability; Eq. (A9) uses Rule 3 given $D\mathchoice{\mathrel{\hbox to 0.0pt{$\displaystyle\perp$}\mskip 2.0mu {% \displaystyle\perp}}}{\mathrel{\hbox to 0.0pt{$\textstyle\perp$}\mskip 2.0mu {% \textstyle\perp}}}{\mathrel{\hbox to 0.0pt{$\scriptstyle\perp$}\mskip 2.0mu {% \scriptstyle\perp}}}{\mathrel{\hbox to 0.0pt{$\scriptscriptstyle\perp$}\mskip 2% .0mu {\scriptscriptstyle\perp}}}X$ in $\mathcal{G}_{\overline{X}}$; Eq. (A10) uses Rule 2 to change the intervention term to observation as $(Y\mathchoice{\mathrel{\hbox to 0.0pt{$\displaystyle\perp$}\mskip 2.0mu {% \displaystyle\perp}}}{\mathrel{\hbox to 0.0pt{$\textstyle\perp$}\mskip 2.0mu {% \textstyle\perp}}}{\mathrel{\hbox to 0.0pt{$\scriptstyle\perp$}\mskip 2.0mu {% \scriptstyle\perp}}}{\mathrel{\hbox to 0.0pt{$\scriptscriptstyle\perp$}\mskip 2% .0mu {\scriptscriptstyle\perp}}}X|D)$ in $\mathcal{G}_{\underline{X}}$; Eq. (A12) is because in our causal graph, $C$ takes a deterministic value given by function $g(\mathbf{x},d)$. This reduces summation over all values of $C$ in Eq. (A11) to a single probability measure in Eq. (A12). A.3 Derivation of NWGM Approximation We will show the derivation of NWGM approximation used in Eq. (3) and (4). In a $K$-way FSL problem, let $f(\cdot)$ be a classifier function that calculates logits for $K$ classes and $\sigma$ be the softmax function over $K$ classes. The approximation effectively moves the outer expectation inside the classifier function: $\operatorname{\mathbb{E}}\left[\sigma(f(\cdot))\right]\approx\sigma(f(% \operatorname{\mathbb{E}}[\cdot]))$. We will first show the derivation for moving the expectation inside softmax function, i.e., $\operatorname{\mathbb{E}}[\sigma\left(f(\cdot)\right)]\approx\sigma\left(% \operatorname{\mathbb{E}}[f(\cdot)]\right)$. Without loss of generality, the backdoor adjustment formula in Eq. (3) and Eq. (4) can be written as: $$P(Y=y|do(X=\mathbf{x}))=\sum_{d\in D}\sigma(f_{y}(\mathbf{x}\oplus\mathbf{c}))% P(d),$$ (A13) where $D$ represents the set of stratifications, $f_{y}$ is the classifier logit for class $y$, $\mathbf{c}=g(\mathbf{x},d)$ is the feature concatenated to $\mathbf{x}$ in Eq. (3) and (4) and $P(d)$ is the prior for each stratificaction. It is shown in [5] that Eq. (A13) can be approximated by the Normalized Weighted Geometric Mean (NWGM) as: $$\displaystyle\sum_{d\in D}\sigma(f_{y}(\mathbf{x}\oplus\mathbf{c}))P(d)$$ $$\displaystyle\approx NWGM_{d\in D}(\sigma(f_{y}(\mathbf{x}\oplus\mathbf{c})))$$ (A14) $$\displaystyle=\frac{\prod_{d}[exp(f_{y}(\mathbf{x}\oplus\mathbf{c}))]^{P(d)}}{% \sum_{i=1}^{K}\prod_{d}[exp(f_{i}(\mathbf{x}\oplus\mathbf{c}))]^{P(d)}}$$ (A15) $$\displaystyle=\frac{exp(\sum_{d}f_{y}(\mathbf{x}\oplus\mathbf{c})P(d))}{\sum_{% i=1}^{K}exp(\sum_{d}f_{i}(\mathbf{x}\oplus\mathbf{c})P(d))}$$ (A16) $$\displaystyle=\sigma\left(\operatorname{\mathbb{E}}_{d}[f_{y}(\mathbf{x}\oplus% \mathbf{c})]\right),$$ (A17) where Eq. (A14) follows [5], Eq. (A15) follows the definition of NWGM, Eq. (A16) is because $exp(a)^{b}=exp(ab)$. Next we will show the derivation for linear, cosine and $k$-NN classifier to further move the expectation inside the classifier function, i.e., $\sigma(\operatorname{\mathbb{E}}[f(\cdot)])=\sigma(f(\operatorname{\mathbb{E}}% [\cdot]))$. For the linear classifier, $f(\mathbf{x}\oplus\mathbf{c})=\mathbf{W}_{1}\mathbf{x}+\mathbf{W}_{2}\mathbf{c}$, where $\mathbf{W}_{1},\mathbf{W}_{2}\in\mathbb{R}^{K\times N}$ denote the learnable weight, $N$ is the feature dimension, which is the same for $\mathbf{x}$ and $\mathbf{c}$ in Eq. (3) and (4). The bias term is dropped as it does not impact our analysis. Now the expectation can be further moved inside the classifier function through: $$\displaystyle\sum_{d}f(\mathbf{x}\oplus\mathbf{c}))P(d)$$ $$\displaystyle=\sum_{d}(\mathbf{W}_{1}\mathbf{x}+\mathbf{W}_{2}\mathbf{c})P(d)$$ (A18) $$\displaystyle=\mathbf{W}_{1}\mathbf{x}+\sum_{d}\mathbf{W}_{2}\mathbf{c}P(d)$$ (A19) $$\displaystyle=f(\mathbf{x}\oplus\sum_{d}\mathbf{c}P(d)),$$ (A20) where Eq. (A19) is because the feature vector $\mathbf{x}$ is the same for all $d$ and $\operatorname{\mathbb{E}}_{d}[\mathbf{x}]=\mathbf{x}$. For the cosine classifier, $f(\mathbf{x}\oplus\mathbf{c})=(\mathbf{W}_{1}\mathbf{x}+\mathbf{W}_{2}\mathbf{% c})/\left\lVert\mathbf{x}\oplus\mathbf{c}\right\rVert\left\lVert\mathbf{W}\right\rVert$, where $\mathbf{W}\in\mathbb{R}^{K\times 2N}$ is the concatenation of $\mathbf{W}_{1}$ and $\mathbf{W}_{2}$. In the special case where $\mathbf{x}$ and $\mathbf{c}$ are unit vector, $\left\lVert\mathbf{x}\oplus\mathbf{c}\right\rVert$ is $\sqrt{2}$ and the cosine classifier function reduces to a linear combination of terms involving only $\mathbf{x}$ and only $\mathbf{c}$. From there, the analysis for linear classifier follows and we have $\sigma(\operatorname{\mathbb{E}}f(\cdot))=\sigma(f(\operatorname{\mathbb{E}}% \cdot))$ for cosine classifier. In the general case where $\mathbf{x}$ and $\mathbf{c}$ are not unit vector, moving the expectation inside cosine classifier function is an approximation $\sigma(\operatorname{\mathbb{E}}[f(\cdot)])\approx\sigma(f(\operatorname{% \mathbb{E}}[\cdot]))$. For the $k$-NN classifier, our implementation calculates class centroids using the mean feature of the $K$ support sets and then uses the nearest centroid for prediction ($1$-NN). Specifically, let $\mathbf{x}$ be a feature vector and $\mathbf{x^{\prime}}$ be the $i$th class centroid, $i\in\{1,\ldots,K\}$. The logit for class $i$ is given by $f_{i}(\mathbf{x})=-\left\lVert\mathbf{x}-\mathbf{x^{\prime}}\right\rVert^{2}$. It is shown in [57] that $k$-NN classifier that uses squared Euclidean distance to generate logits is equivalent to a linear classifier with a particular parameterization. Therefore, our analysis on linear classifier follows for $k$-NN. In summary, the derivation of $\operatorname{\mathbb{E}}[\sigma(f(\cdot))]\approx\sigma(f(\operatorname{% \mathbb{E}}[\cdot]))$ is a two-stage process where we first move the expectation inside the softmax function and then further move it inside the classifier function. A.4 Algorithms for Fine-tuning and Meta-Learning with IFSL In this section, we will briefly revisit the settings of fine-tuning and meta-learning and introduce how to integrate IFSL into them. In fine-tuning, the goal is to train a classifier $\theta$ conditioned on the current support set $\mathcal{S}=\{(\mathbf{x}_{i},y_{i})\}_{i=1}^{n_{s}}$, where $\mathbf{x}_{i}$ is the feature generated by $\Omega$ for $i$th sample, $y_{i}$ is the ground-truth label for $i$th sample and $n_{s}$ is the support set size. This is achieved by first predicting the support label $\hat{y}$ using the classifier $P(y|\mathbf{x};\theta)$. Then with the predicted label $\hat{y}$ and ground-truth label $y$, one can calculate a loss $\mathcal{L}(\hat{y},y)$ (usually cross-entropy loss) to update the classifier parameter, e.g.through stochastic gradient descent. Adding IFSL to fine-tuning is simple: 1) Pick an adjustment strategy introduced in Section 3. Each implementation defines the set of pre-trained knowledge stratifications $D$, function form of $g(X,D)$, function form of $P(Y|X,D,C)$ and the prior $P(D)$; 2) The classifier prediction is now based on $P(Y|do(X);\theta)$. The process of fine-tuning with IFSL is summarized in Algorithm 1. Note that for the non-parametric $k$-NN classifier, the fine-tuning process is not applicable. When adding IFSL to $k$-NN, each sample is represented by the adjusted feature instead of original feature $\mathbf{x}$. Please refer to the classifier inputs in Eq. (2), (3) and (4) for the exact form of adjusted feature. In meta-learning, the goal is to learn the additional “learning behavior” parameterized by $\phi$ using training episodes $\{(\mathcal{S}_{i},\mathcal{Q}_{i})\}$ sampled from training dataset $\mathcal{D}$. The classifier in meta-learning makes predictions by additionally conditioning on the learning behavior, written as $P_{\phi}(y|\mathbf{x};\theta)$. Within each episode, $\theta$ is first fine-tuned on the support set $\mathcal{S}_{i}$. Then the fine-tuned model is tested on the query set $\mathcal{Q}_{i}$ to obtain the loss $\mathcal{L}_{\phi}(\mathcal{S}_{i},\mathcal{Q}_{i})$ (e.g.using cross-entropy loss). Finally the loss is used to update $\phi$ using an optimizer. It is also easy to integrate IFSL into meta-learning by only changing the classifier from $P_{\phi}(y|\mathbf{x};\theta)$ to $P_{\phi}(y|do(\mathbf{x});\theta)$. The flow of meta-learning with IFSL is presented in Algorithm 2. Firstly notice that the initialization of $\theta$ in each task may depend on $\phi$ or $\mathcal{S}_{i}$. For example, in MAML [20] $\phi$ essentially defines an initialization of model parameters, and in LEO [54] the initial classifier parameter is generated conditioned on $\phi$ and $\mathcal{S}_{i}$. Secondly, although the fine-tuning of $\theta$ largely follows Algorithm 1, some meta-learning methods additionally utilize meta-knowledge $\phi$. For example, in SIB the gradients for updating $\theta$ are predicted by $\phi$ using unlabelled query features instead of calculated from $\mathcal{L}(\hat{y},y)$ as in Algorithm 1. A.5 Implementation Details A.5.1 Pre-training Prior to fine-tuning or meta-learning, we pre-trained a deep neural network (DNN) as feature extractor on the train split of a dataset. We use ResNet-10[25] or WRN-28-10[75] as feature extractor backbone. This section will present the architecture and exact training procedure for our backbones. Network Architecture. The architecture of our ResNet-10 and WRN-28-10 backbone is shown in Figure A1. Specifically, each convolutional layer is described as “$n\times n$ conv, $p$”, where $n$ is the kernel size and $p$ is the number of output channels. Convolutional layers with “$/2$” have a stride of 2 and are used to perform downsampling. The solid curved lines represent identity shortcuts, and the dotted lines are projection shortcuts implemented by $1\times 1$ convolutions. The batch normalization and ReLU layers are omitted in Figure A1 to highlight the key structure of the two backbones. Pre-training Procedure. The networks are trained from scratch with stochastic gradient descent in a fully-supervised manner, i.e., minimizing cross-entropy loss on the train split of a dataset. Specifically the training is conducted on 90 epochs with early stopping using validation accuracy. We used batch size of 256 and image size of $84\times 84$. For data augmentation, a random patch is sampled from an image, resized to $84\times 84$ and randomly flipped along horizontal axis before used for training. The initial learning rate is set to 0.1 and it is scaled down by factor of 10 every 30 epochs. A.5.2 Fine-Tuning We consider linear, cosine and $k$-NN classifier for our fine-tuning experiments. In a $K$-way FSL problem, the detailed implementations for the classifier function $f(\mathbf{x})$ are: Linear. $f(\mathbf{x})=\mathbf{W}\mathbf{x}+\mathbf{b}$, where $\mathbf{x}$ is the input feature, $\mathbf{W}\in\mathbb{R}^{K\times N}$ is the learnable weight parameter, $N$ is the feature dimension and $\mathbf{b}\in\mathbb{R}^{K}$ is the learnable bias parameter. Cosine. $f(\mathbf{x})=\mathbf{W}\mathbf{x}/\left\lVert\mathbf{W}\right\rVert\left% \lVert\mathbf{x}\right\rVert$, where $\mathbf{W}\in\mathbb{R}^{K\times N}$ is the learnable weight parameter. We implemented cosine classifier without using the bias term. $k$-NN. Our implementation of $k$-NN is similar to [57, 68]. For each of the $K$ classes, we first calculated the average support set feature (centroid) denoted as $\mathbf{x}_{i},i\in\{1,\ldots,K\}$. The classifier output for class $i$ is then given by $f_{i}(\mathbf{x})=-\left\lVert\mathbf{x}-\mathbf{x}_{i}\right\rVert^{2}$. Notice that the prediction given by this classifier will be the nearest centroid. We froze the backbone and used the average pooling layer output of $\Omega$ to learn the classifier. The output logits from classifier functions are normalized using softmax to generate probability output $P(y|\mathbf{x})$. For linear and cosine classifier, we followed [13] and trained the classifier for 100 iteration with a batch size of 4. For fine-tuning baseline, we set the learning rate as $1\times 10^{-2}$ and weight decay as $1\times 10^{-3}$. For IFSL, we set the learning rate as $5\times 10^{-3}$ and weight decay as $1\times 10^{-3}$. $k$-NN classifier is non-parametric and can be initialized directly from support set. A.5.3 Meta-Learning MAML. MAML [20] aims to learn an initialization of network parameters such that it can be fine-tuned within a few steps to solve a variety of few-shot classification tasks. When using pre-trained network with MAML, it has been shown that learning initialization of the backbone can lead to unsatisfactory performance [13, 58]. Therefore in our experiment, we froze the backbone and appended a 2-layer MLP with ReLU activation in between the hidden layers and a linear classifier after the average pooling layer of $\Omega$. The hidden dimension of the layers in MLP is the same as output dimension of $\Omega$ (512 for ResNet-10 and 640 for WRN-28-10). The initialization of MLP and the linear classifier is meta-learnt using MAML. For hyper-parameters, we set the inner loop learning rate $\alpha=0.01$, the outer loop learning rate $\beta=0.01$ and the number of adaptation steps as $20$. For IFSL, we adopted the same hyper-parameter setting and set $n$=8 for feature-wise and combined adjustment. Implementation-wise, we adopted the released code111https://github.com/wyharveychen/CloserLookFewShot from [13] and performed experiments on MAML without using first-order approximation. Following the implementation in [13], the model was trained on 10,000 randomly sampled tasks with model selection using validation accuracy. We used 2,000 randomly sampled tasks for validation and testing. MTL. MTL [58] learns scaling and shifting parameters at each convolutional layer of the backbone. We used the MTL implementation released by the author222https://github.com/yaoyao-liu/meta-transfer-learning which adopts linear classifier. We integrated our ResNet-10 and WRN-28-10 backbones into the released code. The learning rate for scaling and shifting weights $\phi_{SS}$ and initial classifier parameters was set to $1\times 10^{-4}$ uniformly. We set the inner loop learning rate for classifier as $1\times 10^{-2}$ and the inner loop update step as 100. For IFSL, we adopted the same hyper-parameter setting and set $n$=8 for feature-wise and combined adjustment. We trained the MTL model on 10,000 randomly sampled tasks with model selection using validation accuracy and used 2,000 randomly sampled tasks for validation and testing. We used 3 RTX 2080 Ti for MTL experiments on WRN-28-10 backbone. LEO. LEO [54] learns to generate classifier parameters conditioned on support set and the generated parameters are further fine-tuned within each FSL task. Our experiments were conducted on the released code of LEO333https://github.com/deepmind/leo using linear classifier. Following author’s implementation, we saved the center cropped features from our pre-trained backbones and used the saved features to train LEO. For baseline, we used the hyper-parameter settings released by the author. For IFSL, we set $n$=8 for feature-wise and combined adjustment and halved the outer loop learning rate compared to baseline. The model was trained up to 100,000 randomly sampled tasks from training split with early stopping using validation accuracy. We used 2,000 randomly sampled tasks for validation and testing. Matching Net. Matching Net (MN) [66] is a metric-based method that learns a distance kernel function for $k$-NN. We used the Matching Net implementation in [13]. The implementation follows the setup in [66] and uses LSTM-based fully conditional embedding. We set the learning rate as 0.01 uniformly. For IFSL, we used $n$=16 for feature-wise and combined adjustment. The model was trained using 10,000 randomly sampled tasks with model selection using validation accuracy. We used 2,000 randomly sampled tasks for validation and testing. SIB. SIB [29] initializes classifier from support set and generates gradients conditioned on unlabelled query set features to update classifier parameters. We followed the SIB implementation released by the author444https://github.com/hushell/sib_meta_learn which uses cosine classifier. In the transductive setting, the query set size is set to 15. In the inductive setting, we used only 1 query sample randomly selected from the $K$ classes in each episode. In terms of hyper-parameter settings, we took 3 synthetic gradient steps ($K=3$) for all our experiments. For baseline, the learning rate for SIB network and classifier was set to $1\times 10^{-3}$ following author’s implementation. For IFSL, we set the learning rate to $5\times 10^{-4}$ and used $n$=4 for feature-wise and combined adjustment. In both transductive and inductive settings, we meta-trained SIB using 50,000 randomly sampled tasks with model selection using validation accuracy. We used 2,000 randomly sampled tasks for validation and testing. A.6 Additional Results In this section, we include additional results on 1) Conventional Acc in Table A.6.1 supplementary to Table 1; 2) Hardness-Specific Acc in Figure A2 for miniImageNet and Figure A3 for tieredImageNet, supplementary to Figure 5; 3) CAM-Acc in Table A.6.3 supplementary to Figure 6; 4) Cross-Domain Evaluation in Table A.6.4 supplementary to Table 3. A.6.1 Conventional Acc A.6.2 Hardness-Specific Acc A.6.3 CAM-Acc A.6.4 Cross-Domain Evaluation

A short review and primer on using video for psychophysiological observations in human-computer interaction applications Teppo Valtonen 1Quantified Employee unit, Finnish Institute of Occupational Health, , POBox 40, 00250, Helsinki, Finland 1teppo. valtonen @ttl. fi Abstract The application of psychophysiological measures in human-computer interaction is a growing field with significant potential for future smart personalised systems. Working in this emerging field requires comprehension of an array of physiological signals and analysis techniques. An important aspect in measuring psychophysiological variables in real-world settings is the invasiveness of the measurement setup. Video is a signal which can be captured from a distance without interrupting the subject. Furthermore, the advancements in camera technologies enable detecting a growing variety of psychophysiological phenomena from a video signal with an increasing accuracy. This paper aims to serve as a primer for the novice, enabling rapid familiarisation with the latest core concepts. We put special emphasis on everyday human-computer interface applications to distinguish from the more common clinical or sports uses of psychophysiology. This paper is an extract from a comprehensive review of the entire field of ambulatory psychophysiology, including 12 similar chapters, plus application guidelines and systematic review. Thus any citation should be made using the following reference: B. Cowley, M. Filetti, K. Lukander, J. Torniainen, A. Henelius, L. Ahonen, O. Barral, I. Kosunen, T. Valtonen, M. Huotilainen, N. Ravaja, G. Jacucci. The Psychophysiology Primer: a guide to methods and a broad review with a focus on human-computer interaction. Foundations and Trends in Human-Computer Interaction, vol. 9, no. 3-4, pp. 150–307, 2016. Keywords:video, psychophysiology, human-computer interaction, primer, review, chapter 1 Introduction Advances in camera technologies make video an attractive possibility for measuring a variety of physiological phenomena, especially in environments such as the workplace, where contextual factors (e.g., ambient light) can be accounted for. Ever smaller and more accurate camera systems enable unobtrusive observations with sufficiently high precision for many interesting applications, and as long as there is a line of sight between the camera and the object of interest, a video signal may reveal, for example, a person’s cognitive state without interrupting the work that is being done. Here we consider digital video-based systems aimed at assessing some aspect of physiology or behaviour from a distance, in order to augment human–computer interaction. We exclude the ocular system from discussion, since it is dealt with in Cowley et al. (2016). 2 Background Typically, a video signal comprises measurements of the intensity of electromagnetic radiation in the spectra of visible (wavelengths of about 390 to 700 nm) and infrared (wavelengths from about 700 to 1000 nm) light, on a plane (for example, the image sensor in digital cameras). Changes in intensity arise mainly from a change in the original light source or the various points of reflection along the path of the ray of light from the source to the sensor. Accordingly, any movement within the measurement space (for instance, the eyebrows rising when the subject is surprised or expansion of the lungs when one is inhaling) or changes in the reflective properties of the reflection points (such as a change in skin colour due to increased blood flow) may be detected via the sensor. There is great variety in the video technologies and systems available today. High-speed cameras A typical video camera captures 24 to 30 frames per second, depending on the encoding. While this is sufficient to make a video stream seem smooth for the human visual system, systems with higher frame rates have been developed too. One of the fastest methods, known as compressed ultra-fast photography (CUP), can capture non-repetitive time-evolving events at up to 1011 frames per second (Gao et al., 2014). For many physiological phenomena, a frame rate on the magnitude of 100 frames per second is adequate, and 200 frames a second may already allow, for example, the use of video-based photoplethysmography in clinical settings (Sun et al., 2012a). Webcams The first system to feature a video camera that streamed an image in real time through a computer network came about in 1991 (Stafford-Fraser, 1995). The camera was pointed at a coffee pot in the Cambridge University Computer Lab. Since then, video cameras have become a basic feature in laptop computers and the screens of desktop computers, and they have been used mainly for video calls. Cameras in hand-held devices While cameras forming part of traditional computers are widespread, probably the most ubiquitous camera systems today are those embedded in hand-held devices, since almost all modern mobile phones and tablet computers have one or more cameras on their faces. The main camera typically points away from the user and is intended for photography. There is often another, however, intended for video calls and points in the same direction as the screen. In addition to conveying a video image to a caller, the front-facing camera can be used to detect, for example, whether or not there is a face in front of the screen. Once a face is detected, it may reveal various attributes of the user, such as emotional engagement as assessed from facial expressions (Kang et al., 2008). Also, however, as their name suggests, hand-held devices are often held in the user’s hand. It has been demonstrated that the optical sensor of a mobile phone can detect, for example, the following elements from touch: breathing rate, heart rate, blood oxygen saturation, and even atrial fibrillation or blood loss (Scully et al., 2012). 3D camera systems Whereas a single-sensor camera system typically is limited to collecting emitted visual information on a two-dimensional plane, adding sensors and possibly projectors to the system may enable the observation of three-dimensional structures. A system with two appropriately placed cameras (i.e., a stereo camera system) functions in the same way as human binocular vision and can produce three-dimensional images. In another commonly used method, the system utilises projections of structured light patterns onto a three-dimensional surface, whereby the sensor’s detection of distortions in the patterns received may reveal the precise three-dimensional co-ordinates of the surface (Ma et al., 2009). A third method, one that is quite new, uses a time-of-flight (ToF) camera system, in which distances from the source of a light pulse to a camera via each point in the visual field can be resolved from the time of flight of each light pulse on the basis of the known speed of light (Gokturk et al., 2004). Systems of this type have been used, for example, to monitor sleep (Lee et al., 2015). 3 Methods With the above foundations laid, we now describe methods that have been used to extract information about human psychophysiology from a video signal. While there are diverse methods, we concentrate on three main categories here: light intensity analysis, 2D morphological analysis, and 3D morphological analysis. All of these areas are showing rapid development, and some solutions are still experimental. For practical methods, therefore, more research might be required. Light intensity analysis is the basis for video signal analysis and enables most higher-level interpretations. Even on its own, however, simply detecting changes in the intensity of the light in a fixed area in a video image may illuminate interesting psychophysiological variables. For example, while a plethysmograph reveals changes in volume in a body, typically due to changes in the amount of blood or air contained in that part of the body, photoplethysmography is an optical technique that can be used to detect variations in the intensity of light reflected from tissue that arise from changes in blood flow (Allen, 2007). For an optimal result, various light intensity parameters should be considered. These depend on the application. For example, pulses in line with heart rate seem the most apparent in the green colour channel of a colour camera feed (Sun et al., 2012b). 2D morphological analysis is the analysis of interesting areas or shapes in a 2D image, and it is based on the detection of edges between areas that differ in light intensity. For HCI purposes, the most interesting part of the body is the human face. Face detection and recognition are established research topics, and there are free tools available for these (for example, OpenCV FaceRecognizer111 See http://docs.opencv.org/2.4/modules/contrib/doc/facerec/facerec_tutorial.html0.). Samal and Iyengar (1992) provide a good description of the process, from face detection all the way to the analysis of facial expressions and the classification of faces. More recently, Zhao et al. (2003) undertook an extensive review of face recognition. In a recent, thorough review, Martinez and Valstar (2016) concentrates on automatic recognition of facial expressions. 3D morphological analysis is a broad category of analysis methods that rely on different optical sensor systems producing data on 3D structures within a sensor’s field of view. For example, in addition to using intensity analysis, one can collect plethysmographic data from a distance by measuring the movement of a body in three-dimensional space with an optical sensor. Even consumer-grade 3D sensors used in gaming may be utilised to measure heart and respiration rate (Bernacchia et al., 2014). 4 Applications Novel video technologies and methods of signal processing give rise to interesting applications for observing psychophysiological phenomena. Here we describe two of the most interesting video-based applications for HCI: photoplethysmography and the recognition of facial expressions. Plethysmographic data can provide basic information on psychophysiology. More thorough description is given, for example, in Cowley et al. (2016). Here we consider using video cameras for PPG, an optical technique that can be utilised to detect changes in blood flow (Allen, 2007). For example, Sun and colleagues extracted a PPG signal from a video and analysed pulse rate variability (a possible surrogate measure for HRV; see, for example, Gil et al. (2010)) from the palm of the subject’s hand, using a monochrome CMOS camera running at 200 frames per second in 10-bit greyscale (Sun et al., 2012a). Tarassenko et al. (2014) measured both heart and respiration rate in a clinical set-up from a five-megapixel face video with eight bits per pixel, recorded at 12 frames per second. For broader applicability, even low-cost webcams have been demonstrated to function as photoplethysmographic sensors. In another of their studies, Sun et al. (2012b) compared a high-performance camera and a low-cost webcam in normal office lighting. They fixed a high-speed colour CMOS camera and a colour webcam in front of the face, along with a gold-standard pulse oximetry contact sensor on the index finger of the user’s left hand. To ensure that they were measuring light reflected from the skin, the authors manually determined the region of interest (ROI) in each frame of the video signals. For HR detection, they used the green colour channel to analyse changes in the average intensity of the pixels within the ROI, since, especially in the webcam signal, pulsations were most apparent in this particular channel. They concluded that both imaging PPG systems can successfully measure important physiological variables (in their case, HR). Mental state is high-level information, and identifying and conveying that information is gaining burgeoning interest in HCI research. Knowledge of the user’s mental state could augment not only user interfaces but also remote collaboration, telecommuting, and video conferencing. Facial expressions are an obvious signal as to the mental state, and the recognition of facial expressions is natural (and automatic) for humans; it is an important part of our communication. Both voluntary and involuntary facial actions convey, in particular, emotional information that is otherwise difficult to express – and difficult to conceal in face-to-face interaction (Ekman, 2003). Facial expressions can be categorised as reflecting six canonical emotions in addition to a neutral expression: anger, disgust, fear, happiness, surprise, and sadness (Dalgleish and Power, 1999). For an excellent review of automatic facial expression recognition, we recommend the work of Martinez and Valstar (2016). Traditionally, the process has begun with databases of portraits of actors mimicking the emotions (for example, as shown in Figure 1). A neutral expression is used as a reference for training the algorithm. The core challenges with such a category-based approach are that the emotions each appear relatively rarely and that some expressions may differ in meaning on the basis of context. For example, someone might smile when embarrassed, not just when happy. Another method involves looking at the basic units of muscle activity in the human face, termed ‘action units’, in keeping with the Facial Action Coding System (FACS; see (Ekman and Friesen, 1976)). With the FACS approach, interpretation of mental state can be done at a later stage in the analysis pipeline, with the aid of additional information on the context. A third approach is to represent the mental state on two or more dimensions of affect, such as continua for arousal (ranging from relaxed to aroused) and valence (from pleasant to unpleasant). However, neither action units nor values on the affective dimensions are always detected with the current methods, especially in real-world settings. 5 Conclusion The recent progress in video and signal processing methods renders video an interesting alternative to many traditional means of obtaining psychophysiological measurements, in areas such as plethysmography. In addition, video may enable new HCI applications, such as the remote and automatic identification of the mental state. Many of these methods are still in the early phases of development and require more research before they can achieve greater feasibility; however, there is already inexpensive hardware available, and, for many areas of study, excellent-quality free, open-source software tools and libraries exist. It is clear that the use of video in HCI is only just beginning. References Allen [2007] John Allen. Photoplethysmography and its application in clinical physiological measurement. Physiological measurement, 28(3):R1, 2007. Bernacchia et al. [2014] Natascia Bernacchia, Lorenzo Scalise, Luigi Casacanditella, Ilaria Ercoli, Paolo Marchionni, and Enrico Primo Tomasini. Non contact measurement of heart and respiration rates based on kinect™. In Medical Measurements and Applications (MeMeA), 2014 IEEE International Symposium on, pages 1–5. IEEE, 2014. Cowley et al. 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A simple numerical method of second and third orders convergence for solving a fully third order nonlinear boundary value problem Dang Quang A${}^{\text{a}}$, Dang Quang Long${}^{\text{b}}$ ${}^{\text{a}}$ Center for Informatics and Computing, VAST 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam Email: dangquanga@cic.vast.vn ${}^{\text{b}}$ Institute of Information Technology, VAST, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam Email: dqlong88@gmail.com ( ; ) Abstract In this paper we consider a fully third order nonlinear boundary value problem which is of great interest of many researchers. First we establish the existence, uniqueness of solution. Next, we propose simple iterative methods on both continuous and discrete levels. We prove that the discrete methods are of second order and third accuracy due to the use of appropriate formulas for numerical integration and obtain estimate for total error. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the iterative method. Keywords: Third order nonlinear equation; Existence and uniqueness of solution; Iterative method; Third order accuracy; Total error AMS Subject Classification: 34B15, 65L10 1 Introduction Boundary value problems (BVPs) for third order nonlinear differential equations appear in many applied fields, such as flexibility mechanics, chemical engineering, heat conduction and so on. A lot of works are devoted to the qualitative aspects of the problems (see e.g. [6, 7, 22, 23, 33, 35, 37]). There are also many methods concerning the solution of third order BVPs including analytical methods [1, 28, 32] and numerical methods by using interpolation polynomials [3], quartic splines [21], [31], quintic splines[26], Non-polynomial splines [24], [25], [34], and wavelet [19]. The majority of the mentioned above numerical methods are devoted to linear equations or special nonlinear third order differential equations. In this paper we consider the following BVP $$\begin{split}\displaystyle u^{(3)}(t)&\displaystyle=f(t,u(t),u^{\prime}(t),u^{% \prime\prime}(t)),\quad 0<t<1,\\ \displaystyle u(0)&\displaystyle=c_{1},u^{\prime}(0)=c_{2},u^{\prime}(1)=c_{3}% .\end{split}$$ (1) Some authors studied the existence and positivity of solution for this problem, for example, by using the lower and upper solutions method and fixed point theorem on cones, in [36] Yao and Feng established the existence of solution and positive solution for the case $f=f(t,u(t))$, in [20] Feng and Liu obtained existence results by the use of the lower and upper solutions method and a new maximum principle for the case $f=f(t,u(t),u^{\prime}(t))$. It should be emphasized that the results of these two works are pure existence but not methods for finding solutions. Many researchers are interested in numerical solution of the problem (1) without attention to qualitative aspects of it or refer to the book [2]. Below we mention some works devoted to solution methods for the problem (1). Namely, Al Said et al. [4] have solved a third order two point BVP using cubic splines. Noor et al. [29] generated second order method based on quartic splines. Other authors [8, 26] generated finite difference using fourth degree B-spline and quintic polynomial spline for this problem subject to other boundary conditions. El-Danaf [10] constructed a new spline method based on quartic nonpolynomial spline functions that has a polynomial part and a trigonometric part to develop numerical methods for a linear differential equation with the boundary conditions as in (1). Recently, in 2016 Pandey [30] solved the problem for the case $f=f(t,u)$ by the use of quartic polynomial splines. The convergence of the method at least $O(h^{2})$ for the linear case $f=f(t)$ was proved. In the next year this author in [31] proposed two difference schemes for the general case $f=f(t,u(t),u^{\prime}(t),u^{\prime\prime}(t))$ and also established the second order accuracy for the linear case. In the beginning of 2019 Chaurasia et al. [9] use exponential amalgamation of cubic spline functions to form a novel numerical method of second-order accuracy. It should be emphasized that all mentioned above authors only draw attention to the construction of discrete analog of the problem (1) and estimate the error of the obtained solution assuming that the nonlinear system of algebraic equations can be solved by known iterative methods. Thus, they did not take into account the errors arisen in the last iterative methods. Motivated by these facts, in this paper we propose a completely different method, specifically, an iterative method on both continuous and discrete levels for the problem (1). We give an analysis of total error of the solution actually obtained. This error includes the error of the iterative method on continuous level and the error arisen in numerical realization of this iterative method. The obtained total error estimate suggests to choose suitable grid size for discretization if desiring to get approximate solution with a given accuracy. In order to justify the total error estimate, first we establish some results on existence, uniqueness of solution. These results are obtained by the method developed in [11]-[18]. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the iterative method. 2 Existence results For simplicity of presentation we consider the problem (1) with homogeneous boundary conditions, i.e., the problem $$\begin{split}\displaystyle u^{(3)}(t)&\displaystyle=f(t,u(t),u^{\prime}(t),u^{% \prime\prime}(t)),\quad 0<t<1,\\ \displaystyle u(0)&\displaystyle=0,u^{\prime}(0)=0,u^{\prime}(1)=0.\end{split}$$ (2) To investigate this problem we associate it with an operator equation as follows. For functions $\varphi(x)\in C[0,1]$ consider the nonlinear operator $A$ defined by $$(A\varphi)(t)=f(t,u(t),u^{\prime}(t),u^{\prime\prime}(t)),$$ (3) where $u(t)$ is the solution of the problem $$\begin{split}\displaystyle u^{\prime\prime\prime}(t)&\displaystyle=\varphi(t),% \quad 0<t<1\\ \displaystyle u(0)&\displaystyle=0,u^{\prime}(0)=0,u^{\prime}(1)=0.\end{split}$$ (4) Proposition 2.1 If the function $\varphi(x)$ is a fixed point of the operator $A$, i.e., $\varphi(t)$ is a solution of the operator equation $\varphi=A\varphi$ , then the function $u(t)$ determined from the BVP (4) solves the problem (2). Conversely, if $u(t)$ is a solution of the BVP (2) then the function $\varphi(t)=f(t,u(t),u^{\prime}(t),u^{\prime\prime}(t))$ is a fixed point of the operator $A$ defined above by (3), (4). Thus, the problem (2) is reduced to the fixed point problem for $A$. Now, we study the properties of $A$. For this purpose, notice that the problem (2) has a unique solution representable in the form $$u(t)=\int_{0}^{1}G_{0}(t,s)\varphi(s)ds,\quad 0<t<1,$$ (5) where $G_{0}(t,s)$ is the Green function of the problem (4) $$\displaystyle G_{0}(t,s)=\left\{\begin{array}[]{ll}\dfrac{s}{2}(t^{2}-2t+s),% \quad 0\leq s\leq t\leq 1,\\ \,\,\dfrac{t^{2}}{2}(s-1),\quad 0\leq t\leq s\leq 1.\\ \end{array}\right.$$ Differentiating both sides of (5) gives $$\displaystyle u^{\prime}(t)$$ $$\displaystyle=\int_{0}^{1}G_{1}(t,s)\varphi(s)ds,$$ (6) $$\displaystyle u^{\prime\prime}(t)$$ $$\displaystyle=\int_{0}^{1}G_{2}(t,s)\varphi(s)ds,$$ (7) where $$G_{1}(t,s)=\left\{\begin{array}[]{ll}s(t-1),\quad 0\leq s\leq t\leq 1,\\ t(s-1),\quad 0\leq t\leq s\leq 1,\\ \end{array}\right.$$ $$G_{2}(t,s)=\left\{\begin{array}[]{ll}s,&\quad 0\leq s\leq t\leq 1,\\ s-1,&\quad 0\leq t\leq s\leq 1.\\ \end{array}\right.$$ (8) It is easily seen that $G_{0}(t,s)\leq 0,\;G_{1}(t,s)\leq 0$ in $Q=[0,1]^{2}$ and $$\displaystyle M_{0}=\max_{0\leq t\leq 1}\int_{0}^{1}|G(t,s)|\ ds=\dfrac{1}{12}% ,\quad M_{1}=\max_{0\leq t\leq 1}\int_{0}^{1}|G_{1}(t,s)|\ ds=\dfrac{1}{8},$$ (9) $$\displaystyle M_{2}=\max_{0\leq t\leq 1}\int_{0}^{1}|G_{2}(t,s)|\ ds=\dfrac{1}% {2}.$$ Next, for each fixed real number $M>0$ introduce the domain $$\mathcal{D}_{M}=\{(t,x,y,z)|\ 0\leq t\leq 1,\,\,|x|\leq M_{0}M,\,\,|y|\leq M_{% 1}M,\,\,|z|\leq M_{2}M\},$$ and as usual, by $B[O,M]$ we denote the closed ball of radius $M$ centered at $0$ in the space of continuous in $[0,1]$ functions, namely, $B[O,M]=\{\varphi\in C[0,1]|\ \|\varphi\|\leq M\},$ where $\|\varphi\|=\max_{0\leq t\leq 1}|\varphi(t)|.$ By the analogous techniques as in [11]-[18] we have proved the following results. Theorem 2.2 (Existence of solutions) Suppose that there exists a number $M>0$ such that the function $f(t,x,y,z)$ is continuous and bounded by $M$ in the domain $\mathcal{D}_{M}$, i.e., $$|f(t,x,y,z)|\leq M$$ for any $(t,x,y,z)\in\mathcal{D}_{M}.$ Then, the problem (1) has a solution $u(t)$ satisfying $$|u(t)|\leq M_{0}M,\;|u^{\prime}(t)|\leq M_{1}M,\;|u^{\prime\prime}(t)|\leq M_{% 2}M\text{ for any }0\leq t\leq 1.$$ Theorem 2.3 (Existence and uniqueness of solution) Assume that there exist numbers $M,L_{0},L_{1}$, $L_{2}\geq 0$ such that $$|f(t,x,y,z)|\leq M,$$ $$\displaystyle|f(t,x_{2},y_{2},z_{2})-f(t,x_{1},y_{1},z_{1})|\leq L_{0}|x_{2}-x% _{1}|+L_{1}|y_{2}-y_{1}|+L_{2}|z_{2}-z_{1}|$$ (10) for any $(t,x,y,z),(t,x_{i},y_{i},z_{i})\in\mathcal{D}_{M}\ (i=1,2)$ and $$q:=L_{0}M_{0}+L_{1}M_{1}+L_{2}M_{2}<1.$$ Then, the problem (2) has a unique solution $u(t)$ such that $|u(t)|\leq M_{0}M,$ $|u^{\prime}(t)|\leq M_{1}M,\,\,|u^{\prime\prime}(t)|\leq M_{2}M$ for any $0\leq t\leq 1$. Remark. The problem (1) for $u(t)$ with non-homogeneous boundary conditions can be reduced to the problem with homogeneous for function $v(t)$ if setting $u(t)=v(t)+P_{2}(t)$, where $P_{2}(t)$ is the second degree polynomial satisfying the boundary conditions $P_{2}(0)=c_{1},P^{\prime}_{2}(0)=c_{2},P_{2}(1)=c_{3}$. 3 Iterative method on continuous level Consider the following iterative method for solving the problem (2): 1. Given $$\varphi_{0}(t)=f(t,0,0,0).$$ (11) 2. Knowing $\varphi_{k}(t)$ $(k=0,1,...)$ compute $$\begin{split}\displaystyle u_{k}(t)&\displaystyle=\int_{0}^{1}G_{0}(t,s)% \varphi_{k}(s)ds,\\ \displaystyle y_{k}(t)&\displaystyle=\int_{0}^{1}G_{1}(t,s)\varphi_{k}(s)ds,\\ \displaystyle z_{k}(t)&\displaystyle=\int_{0}^{1}G_{2}(t,s)\varphi_{k}(s)ds,% \end{split}$$ (12) 3. Update $$\varphi_{k+1}(t)=f(t,u_{k}(t),y_{k}(t),z_{k}(t)).$$ (13) Set $$p_{k}=\dfrac{q^{k}}{1-q}\|\varphi_{1}-\varphi_{0}\|.$$ Theorem 3.1 (Convergence) Under the assumptions of Theorem 2.3 the above iterative method converges and there hold the estimates $$\|u_{k}-u\|\leq M_{0}p_{k},\quad\|u^{\prime}_{k}-u^{\prime}\|\leq M_{1}p_{k},% \quad\|u^{\prime\prime}_{k}-u^{\prime\prime}\|\leq M_{2}p_{k},$$ where $u$ is the exact solution of the problem (2) and $M_{0},M_{1},M_{2}$ are given by (9). This theorem follows straightforward from the convergence of the successive approximation method for finding fixed point of the operator $A$ and the representations (5)-(7) and (12). 4 Discrete iterative method 1 To numerically realize the above iterative method we construct the corresponding discrete iterative methods. For this purpose cover the interval $[0,1]$ by the uniform grid $\bar{\omega}_{h}=\{t_{i}=ih,\;h=1/N,i=0,1,...,N\}$ and denote by $\Phi_{k}(t),U_{k}(t),Y_{k}(t),Z_{k}(t)$ the grid functions, which are defined on the grid $\bar{\omega}_{h}$ and approximate the functions $\varphi_{k}(t),u_{k}(t),y_{k}(t),z_{k}(t)$ on this grid, respectively. First, consider the following discrete iterative method, named Method 1: 1. Given $$\Phi_{0}(t_{i})=f(t_{i},0,0,0),\ i=0,...,N.$$ (14) 2. Knowing $\Phi_{k}(t_{i}),\;k=0,1,...;\;i=0,...,N,$ compute approximately the definite integrals (12) by trapezium formulas $$\begin{split}\displaystyle U_{k}(t_{i})&\displaystyle=\sum_{j=0}^{N}h\rho_{j}G% _{0}(t_{i},t_{j})\Phi_{k}(t_{j}),\\ \displaystyle Y_{k}(t_{i})&\displaystyle=\sum_{j=0}^{N}h\rho_{j}G_{1}(t_{i},t_% {j})\Phi_{k}(t_{j}),\\ \displaystyle Z_{k}(t_{i})&\displaystyle=\sum_{j=0}^{N}h\rho_{j}G_{2}^{*}(t_{i% },t_{j})\Phi_{k}(t_{j}),\;i=0,...,N,\end{split}$$ (15) where $\rho_{j}$ are the weights of the trapezium formula $$\rho_{j}=\begin{cases}1/2,\;j=0,N\\ 1,\;j=1,2,...,N-1\end{cases}$$ and $$G_{2}^{*}(t,s)=\begin{cases}s,&0\leq s<t\leq 1,\\ s-1/2,&s=t,\\ s-1,&0\leq t<s\leq 1.\end{cases}$$ (16) 3. Update $$\Phi_{k+1}(t_{i})=f(t_{i},U_{k}(t_{i}),Y_{k}(t_{i}),Z_{k}(t_{i})).$$ (17) In order to get the error estimates for the numerical approximate solution for $u(t)$ and its derivatives on the grid we need some following auxiliary results. Proposition 4.1 Assume that the function $f(t,x,y,z)$ has all continuous partial derivatives up to second order in the domain $\mathcal{D}_{M}$. Then for the functions $u_{k}(t),y_{k}(t),z_{k}(t),k=0,1,...$, constructed by the iterative method (11)-(13) there hold $z_{k}(t)\in C^{3}[0,1],\;y_{k}(t)\in C^{4}[0,1],\;u_{k}(t)\in C^{5}[0,1].$ Proof 4.2 We prove the proposition by induction. For $k=0,$ by the assumption on the function $f$ we have $\varphi_{0}(t)\in C^{2}[0,1]$ since $\varphi_{0}(t)=f(t,0,0,0)$. Taking into account the expression (8) of the function $G_{2}(t,s)$ we have $$z_{0}(t)=\int_{0}^{1}G_{2}(t,s)\varphi_{0}(s)ds=\int_{0}^{t}s\varphi_{0}(s)ds-% \int_{t}^{1}(s-1)\varphi_{0}(s)ds.$$ It is easy to see that $z_{0}^{\prime}(t)=\varphi_{0}(t)$. Therefore, $z_{0}(t)\in C^{3}[0,1]$. It implies $y_{0}(t)\in C^{4}[0,1],\;u_{0}(t)\in C^{5}[0,1]$. Now suppose $z_{k}(t)\in C^{3}[0,1],\;y_{k}(t)\in C^{4}[0,1],\;u_{k}(t)\in C^{5}[0,1].$ Then, because $\varphi_{k+1}(t)=f(t,u_{k}(t),y_{k}(t),z_{k}(t))$ and the function $f$ by the assumption has continuous derivative in all variables up to order 2, it follows that $\varphi_{k+1}(t)\in C^{2}[0,1]$. Repeating the same argument as for $\varphi_{0}(t)$ above we obtain that $z_{k+1}(t)\in C^{3}[0,1],\;y_{k+1}(t)\in C^{4}[0,1],\;u_{k+1}(t)\in C^{5}[0,1].$ Thus, the proposition is proved. Proposition 4.3 For any function $\varphi(t)\in C^{2}[0,1]$ there hold the estimates $$\int_{0}^{1}G_{n}(t_{i},s)\varphi(s)ds=\sum_{j=0}^{N}h\rho_{j}G_{n}(t_{i},t_{j% })\varphi(t_{j})+O(h^{2}),\quad(n=0,1)$$ (18) $$\int_{0}^{1}G_{2}(t_{i},s)\varphi(s)ds=\sum_{j=0}^{N}h\rho_{j}G_{2}^{*}(t_{i},% t_{j})\varphi(t_{j})+O(h^{2}).$$ (19) Proof 4.4 In the case $n=0,1$, since the functions $G_{n}(t_{i},s)$ are continuous at $s=t_{i}$ and are polynomials in $s$ in the intervals $[0,t_{i}]$ and $[t_{i},1]$ we have $$\displaystyle\int_{0}^{1}G_{n}(t_{i},s)\varphi(s)ds=\int_{0}^{t_{i}}G_{n}(t_{i% },s)\varphi(s)ds+\int_{t_{i}}^{1}G_{n}(t_{i},s)\varphi(s)ds$$ $$\displaystyle=h\big{(}\tfrac{1}{2}G_{n}(t_{i},t_{0})\varphi(t_{0})+G_{n}(t_{i}% ,t_{1})\varphi(t_{1})+...+G_{n}(t_{i},t_{i-1})\varphi(t_{i-1})+\tfrac{1}{2}G_{% 2}(t_{i},t_{i})\varphi(t_{i})\big{)}$$ $$\displaystyle+h\big{(}\tfrac{1}{2}G_{n}(t_{i},t_{i})\varphi(t_{i})+G_{n}(t_{i}% ,t_{i+1})\varphi(t_{i+1})+...+G_{n}(t_{i},t_{N-1})\varphi(t_{N-1})$$ $$\displaystyle+\tfrac{1}{2}G_{n}(t_{i},t_{N})\varphi(t_{N})\big{)}+O(h^{2})$$ $$\displaystyle=\sum_{j=0}^{N}h\rho_{j}G_{n}(t_{i},t_{j})\varphi(t_{j})+O(h^{2})% \quad(n=0,1).$$ Thus, the estimate (18) is established. The estimate (19) is obtained using the following result, which is easily proved. Lemma 4.5 Let $p(t)$ be a function having continuous derivatives up to second order in the interval $[0,1]$ except for the point $t_{i},\ 0<t_{i}<1$, where it has a jump. Denote $\lim_{t\rightarrow{t_{i}-0}}p(t)=p_{i}^{-}$, $\lim_{t\rightarrow{t_{i}+0}}p(t)=p_{i}^{+},$ $p_{i}=\tfrac{1}{2}(p_{i}^{-}+p_{i}^{+})$. Then $$\int_{0}^{1}p(t)dt=\sum_{j=0}^{N}h\rho_{j}p(j)+O(h^{2}),$$ (20) where $p_{j}=p(t_{j}),j\neq i.$ Proposition 4.6 Under the assumption of Proposition 4.1 for any $k=0,1,...$ there hold the estimates $$\|\Phi_{k}-\varphi_{k}\|=O(h^{2}),$$ (21) $$\begin{split}\displaystyle\|U_{k}-u_{k}\|&\displaystyle=O(h^{2}),\;\|Y_{k}-y_{% k}\|=O(h^{2}),\;\|Z_{k}-z_{k}\|=O(h^{2}).\end{split}$$ (22) where $\|.\|_{C(\bar{\omega}_{h})}$ is the max-norm of function on the grid $\bar{\omega}_{h}$. Proof 4.7 We prove the proposition by induction. For $k=0$ we have immediately $\|\Phi_{0}-\varphi_{0}\|=0$. Next, by the first equation in (12) and Proposition 4.3 we have $$u_{0}(t_{i})=\int_{0}^{1}G_{0}(t_{i},s)\varphi_{0}(s)ds=\sum_{j=0}^{N}h\rho_{j% }G_{0}(t_{i},t_{j})\varphi_{0}(t_{j})+O(h^{2})$$ (23) for any $i=0,...,N$. On the other hand, in view of the first equation in (15) we have $$U_{0}(t_{i})=\sum_{j=0}^{N}h\rho_{j}G_{0}(t_{i},t_{j})\varphi_{0}(t_{j}).$$ (24) Therefore, $|U_{0}(t_{i})-u_{0}(t_{i})|=O(h^{2})$. Consequently, $\|U_{0}-u_{0}\|=O(h^{2})$. Similarly, we have $$\|Y_{0}-y_{0}\|=O(h^{2}),\;\|Z_{0}-z_{0}\|=O(h^{2}).$$ (25) Now suppose that (21) and (22) are valid for $k\geq 0$. We shall show that these estimates are valid for $k+1$. Indeed, by the Lipshitz condition of the function $f$ and the estimates (22) it is easy to obtain the estimate $$\|\Phi_{k+1}-\varphi_{k+1}\|=O(h^{2})$$ (26) Now from the first equation in (12) by Proposition 4.3 we have $$u_{k+1}(t_{i})=\int_{0}^{1}G_{0}(t_{i},s)\varphi_{k+1}(s)ds=\sum_{j=0}^{N}h% \rho_{j}G_{0}(t_{i},t_{j})\varphi_{k+1}(t_{j})+O(h^{2})$$ On the other hand by the first formula in (15) we have $$U_{k+1}(t_{i})=\sum_{j=0}^{N}h\rho_{j}G_{0}(t_{i},t_{j})\Phi_{k+1}(t_{j}).$$ From the above equalities, having in mind the estimate (26) we obtain the estimate $$\|U_{k+1}-u_{k+1}\|=O(h^{2}).$$ Similarly, we obtain $$\|Y_{k+1}-y_{k+1}\|=O(h^{2}),\;\|Z_{k+1}-z_{k+1}\|=O(h^{2}).$$ Thus, by induction we have proved the proposition. Now combining Proposition 4.6 and Theorem 3.1 results in the following theorem. Theorem 4.8 For the approximate solution of the problem (2) obtained by the discrete iterative method on the uniform grid with gridsize $h$ there hold the estimates $$\begin{split}\displaystyle\|U_{k}-u\|&\displaystyle\leq\left(M_{0}+\frac{1}{r}% \right)p_{k}d+O(h^{2}),\;\|Y_{k}-u^{\prime}\|\leq M_{1}p_{k}d+O(h^{2}),\\ \displaystyle\|Z_{k}-u^{\prime\prime}\|&\displaystyle\leq M_{2}p_{k}d+O(h^{2})% .\end{split}$$ Remark 1. We perform the discrete iterative process (14)-(17) until $\|\Phi_{k+1}-\Phi_{k}\|\leq TOL$, where $TOL$ is a given tolerance. From Theorem 4.8 it is seen that the accuracy of the discrete approximate solution depends on both the number $q$ defined in Theorem 2.3, which determines the number of iterations of the continuous iterative method and the gridsize $h$. The number $q$ presents the nature of the BVP, therefore, it is necessary to choose appropriate $h$ consistent with $q$ because the choice of very small $h$ does not increase the accuracy of the approximate discrete solution. Below, in examples we shall see this fact. Remark 2. As mentioned in the Introduction, in 2016 Pandey [30] discretized the problem (1) by quartic splines and proved the second order convergence only for the linear case (when $f=f(x)$). Next year, in [31] he constructed two difference schemes for the problem and also proved the second order convergence for the linear case. The obtained system of difference equations are solve iteratively by the Gauss-Seidel or Newton-Raphson method. The error arising in these iterative methods are not considered together with the error of the discretization. 5 Discrete iterative method 2 Consider another discrete iterative method, named Method 2. The steps of this method are the same of Method 1 with essential difference in Step 2 and now the number of grid points is even, $N=2n$. Namely, 2’. Knowing $\Phi_{k}(t_{i}),\;k=0,1,...;\;i=0,...,N,$ compute approximately the definite integrals (12) by the modified Simpson formulas $$\begin{split}\displaystyle U_{k}(t_{i})&\displaystyle=F(G_{0}(t_{i},.)\Phi_{k}% (.)),\\ \displaystyle Y_{k}(t_{i})&\displaystyle=F(G_{1}(t_{i},.)\Phi_{k}(.)),\\ \displaystyle Z_{k}(t_{i})&\displaystyle=F(G_{2}^{*}(t_{i},.)\Phi_{k}(.)),\end% {split}$$ (27) where $$F(G_{l}(t_{i},.)\Phi_{k}(.))=\begin{cases}\sum_{j=0}^{N}h\rho_{j}G_{l}(t_{i},t% _{j})\Phi_{k}(t_{j})\;\text{ if }i\text{ is even }\\ \sum_{j=0}^{N}h\rho_{j}G_{l}(t_{i},t_{j})\Phi_{k}(t_{j})+\dfrac{h}{6}\Big{(}G_% {l}(t_{i},t_{i-1})\Phi_{k}(t_{i-1})-2G_{l}(t_{i},t_{i})\Phi_{k}(t_{i})\\ \quad+G_{l}(t_{i},t_{i+1})\Phi_{k}(t_{i+1})\Big{)}\;\text{ if }i\text{ is odd % },\\ l=0,1;\;i=0,1,2,...,N.\end{cases}$$ (28) $\rho_{j}$ are the weights of the Simpson formula $$\rho_{j}=\begin{cases}1/3,\;j=0,N\\ 4/3,\;j=1,3,...,N-1\\ 2/3,\;j=2,4,...,N-2,\end{cases}$$ $F(G_{2}^{*}(t_{i},.)\Phi_{k}(.))$ is calculated in the same way as $F(G_{l}(t_{i},.)\Phi_{k}(.))$ above, where $G_{l}$ is replaced by $G_{2}^{*}$ defined by the formula (16). Proposition 5.1 Assume that the function $f(t,x,y,z)$ has all continuous partial derivatives up to fourth order in the domain $\mathcal{D}_{M}$. Then for the functions $u_{k}(t),y_{k}(t),z_{k}(t),\varphi_{k+1}(t)$, $k=0,1,...$, constructed by the iterative method (11)-(13) there hold $z_{k}(t)\in C^{5}[0,1],\;y_{k}(t)\in C^{6}[0,1],\;u_{k}(t)\in C^{7}[0,1],% \varphi_{k+1}(t)\in C^{4}[0,1].$ Proposition 5.2 For any function $\varphi(t)\in C^{4}[0,1]$ there hold the estimates $$\int_{0}^{1}G_{l}(t_{i},s)\varphi(s)ds=F(G_{l}(t_{i},.)\varphi(.))+O(h^{3}),% \quad(l=0,1)$$ (29) $$\int_{0}^{1}G_{2}(t_{i},s)\varphi(s)ds=F(G_{2}^{*}(t_{i},.)\varphi(.))+O(h^{3}).$$ (30) Proof 5.3 Recall that the interval $[0,1]$ is divided into $N=2n$ by the points $t_{i}=ih,h=1/N$. In each subinterval $[0,t_{i}]$ and $[t_{i},1$ the functions $G_{l}(t_{i},s)$ are continuous as polynomials. Therefore, if $i$ is even number, $i=2m$ then we represent $$\int_{0}^{1}G_{l}(t_{i},s)\varphi(s)ds=\int_{0}^{t_{2m}}\;+\int_{t_{2m}}^{1}.$$ Applying the Simpson formula to the integrals in the right-hand side we obtain $$\int_{0}^{1}G_{l}(t_{i},s)\varphi(s)ds=F(G_{l}(t_{i},.)\varphi(.))+O(h^{4})$$ because by assumption $\varphi(t)\in C^{4}[0,1]$. Now consider the case when $i$ is odd number, $i=2m+1$. In this case we represent $$I=\int_{0}^{1}G_{l}(t_{i},s)\varphi(s)ds=\int_{0}^{t_{2m}}\;+\int_{t_{2m}}^{t_% {2m+1}}+\int_{t_{2m+1}}^{t_{2m+2}}+\int_{t_{2m+2}}^{1}.$$ (31) For simplicity we denote $$f_{j}=G_{l}(t_{i},s_{j})\varphi(s_{j})$$ Applying the Simpson formula to the first and the fourth integrals in the right-hand side (31) and the trapezium formula to the second and the third integrals there, we obtain $$\displaystyle I$$ $$\displaystyle=\dfrac{h}{3}[f_{0}+f_{2m}+4(f_{1}+f_{3}+...+f_{2m-1})+2(f_{2}+f_% {4}+...+f_{2m-2})]+O(h^{4})$$ $$\displaystyle+\dfrac{h}{2}(f_{2m}+f_{2m+1})+O(h^{3})+\dfrac{h}{2}(f_{2m+1}+f_{% 2m+2})+O(h^{3})$$ $$\displaystyle+\dfrac{h}{3}[f_{2m+2}+f_{2n}+4(f_{2m+3}+f_{2m+5}+...+f_{2n-1})+2% (f_{2m+4}+f_{2m+6}+...+f_{2n-2})]+O(h^{4})$$ $$\displaystyle=\dfrac{h}{3}[f_{0}+f_{2n}+4(f_{1}+f_{3}+...+f_{2n-1})+2(f_{2}+f_% {4}+...+f_{2n-2})]$$ $$\displaystyle+\dfrac{h}{6}(f_{2m}-2f_{2m+1}+f_{2m+2})+O(h^{3})$$ $$\displaystyle=F(G_{l}(t_{i},.)\varphi(.))+O(h^{3})$$ Thus, in the both cases of $i$, even or odd, we have the estimate (29). The estimate (30) is obtained analogously as (29) if taking into account that $$2G_{2}^{*}(t_{i},t_{i})=G_{2}^{-}(t_{i},t_{i})+G_{2}^{+}(t_{i},t_{i}),$$ where $$\displaystyle G_{2}^{\pm}(t_{i},t_{i})=\lim_{s\rightarrow t_{i}\pm 0}G_{2}(t_{% i},s)$$ Theorem 5.4 Under the assumptions of Proposition 5.1, for the approximate solution of the problem (2) obtained by the discrete iterative method 2 on the uniform grid with gridsize $h$ there hold the estimates $$\begin{split}\displaystyle\|U_{k}-u\|&\displaystyle\leq\left(M_{0}+\frac{1}{r}% \right)p_{k}d+O(h^{3}),\;\|Y_{k}-u^{\prime}\|\leq M_{1}p_{k}d+O(h^{3}),\\ \displaystyle\|Z_{k}-u^{\prime\prime}\|&\displaystyle\leq M_{2}p_{k}d+O(h^{3})% .\end{split}$$ 6 Examples Consider some examples for confirming the validity of the obtained theoretical results and the efficiency of the proposed iterative method. Example 1. (Problem 2 in [30]) Consider the problem $$\displaystyle\begin{split}\displaystyle u^{\prime\prime\prime}(x)&% \displaystyle=x^{4}u(x)-u^{2}(x)+f(x),\;0<x<1,\\ \displaystyle u(0)&\displaystyle=0,\;u^{\prime}(0)=-1,\;u^{\prime}(1)=\sin(1),% \end{split}$$ where $f(x)$ is calculated so that the exact solution of the problem is $$u^{*}(x)=(x-1)\sin(x).$$ It is easy to verify that with $M=7$ all conditions of Theorem 2.3 are satisfied, so the problem has a unique solution. The results of the numerical experiments with two different tolerances are given in Tables 1- 3. In the above tables $N$ is the number of grid points, $K$ is the number of iterations, $Error_{trap},\;Error_{Simp}$ are errors $\|U_{K}-u^{*}\|$ in the cases of using Method 1 and Method 2, respectively,$Order$ is the order of convergence calculated by the formula $$Order=\log_{2}\frac{\|U^{N/2}_{K}-u^{*}\|}{\|U^{N}_{K}-u^{*}\|}.$$ In the above formula the superscripts $N/2$ and $N$ of $U_{K}$ mean that $U_{K}$ is computed on the grid with the corresponding number of grid points. From the tables we observe that for each tolerance the number of iterations is constant and the errors of the approximate solution decrease with the rate (or order) close to 2 for Method 1 and close to 3 for Method 2 until they cannot improved. This can be explained as follows. Since the total error of the actual approximate solution consists of two terms: the error of the iterative method on continuous level and the error of numerical integration at each iteration, when these errors are balanced, the further increase of number of grid points $N$(or equivalently, the decrease of grid size $h$) cannot in general improve the accuracy of approximate solution. Notice that in [30] the author used Newton-Raphson iteration method to solve nonlinear system of equations arisen after discretization of the differential problem. Iteration process is continued until the maximum difference between two successive iterations , i.e., $\|U_{k+1}-U_{k}\|$ is less than $10^{-10}$. The number of iterations for achieving this tolerance is not reported. The accuracy for some different $N$ is (see [30, Table 2]) From the tables of our results and of Pandey it is clear that our method gives much better accuracy. Example 2. (Problem 2 in [31]) Consider the problem $$\displaystyle\begin{split}\displaystyle u^{\prime\prime\prime}(x)&% \displaystyle=-xu^{\prime\prime}(x)-6x^{2}+3x-6,\;0<x<1,\\ \displaystyle u(0)&\displaystyle=0,\;u^{\prime}(0)=0,\;u^{\prime}(1)=0.\end{split}$$ It is easy to verify that with $M=9$ all conditions of Theorem 2.3 are satisfied, so the problem has a unique solution. This solution is $u(x)=x^{2}(\frac{3}{2}-x)$. The results of the numerical experiments with different tolerances are given in Tables 5, 6 and 7. Notice that [31] the author used Gauss-Seidel iteration method to solve linear system of equations arisen after discretization of the differential problem. Iteration process is continued until the maximum difference between two successive iterations , i.e., $\|U_{k+1}-U_{k}\|$ is less than $10^{-10}$. The results for some different $N$ are From the tables of our results and of Pandey it is clear that our method gives better accuracy and requires less computational work. Example 3. Consider the problem $$\displaystyle\begin{split}\displaystyle u^{\prime\prime\prime}(x)&% \displaystyle=(u(x))^{2}+u^{\prime}(x)-e^{2x},\;0<x<1,\\ \displaystyle u(0)&\displaystyle=1,\;u^{\prime}(0)=1,\;u^{\prime}(1)=e.\end{split}$$ It is easy to verify that with $M=10$ all conditions of Theorem 2.3 are satisfied, so the problem has a unique solution. This solution is $u(x)=e^{x}$. The results of the numerical experiments with different tolerances are given in Tables 9 and 10. Example 4. Consider the problem for fully third order differential equation $$\displaystyle\begin{split}\displaystyle u^{\prime\prime\prime}(x)&% \displaystyle=-e^{u(x)}-e^{u^{\prime}(x)}-\frac{1}{10}(u^{\prime\prime}(x))^{2% },\;0<x<1,\\ \displaystyle u(0)&\displaystyle=0,\;u^{\prime}(0)=0,\;u^{\prime}(1)=0.\end{split}$$ It is easy to verify that with $M=3$ all conditions of Theorem 2.3 are satisfied, so the problem has a unique solution. The numerical solution of the problem is depicted in Figure 1. 7 Conclusion In this paper we established the existence and uniqueness of solution for a boundary value problem for fully third order differential equation. Next, for finding this solution we proposed an iterative method at both continuous and discrete levels. The numerical realization of the discrete iterative method is very simple. It is based on popular rules for numerical integration. One of the important results is that we obtained an estimate for total error of the approximate solution which is actually obtained. This total error depends on the number of iterations performed and the discretization parameter. The validity of the theoretical results and the efficiency of the iterative method are illustrated in examples. 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The questionable impact of population-wide public testing in reducing SARS-CoV-2 infection prevalence in the Slovak Republic Jozef Černák111jozefcernak@gmail.com Abstract Mina and Andersen, authors of the Perspectives in Science: "COVID-19 Testing: One Size Does Not Fit All" have referred to results and adopted conclusions from recently published governmental report Pavelka et al. “The effectiveness of population wide, rapid antigen test based screening in reducing SARS-CoV-2 infection prevalence in Slovakia” without critical consideration, and rigorous verification. We demonstrate that the authors refer to conclusions that are not supported by experimental data. Further, there is a lack of objective, independent information and studies regarding the widespread, public testing program currently in force in the Slovak Republic. We offer an alternative explanation of observed data as they have been provided by the Slovak Republic government to fill this information gap. We also provide explanations and conclusions that more accurately describe viral spread dynamics. Drawing from available public data and our simple but rigorous analysis, we show that it is not possible to make clear conclusions about any positive impact of the public testing program in the Slovak Republic. In particular, it is not possible to conclude that this testing program forces the curve down for the SARS-CoV-2 virus outbreak. We think that Pavelka et al. did not consider many fundamental phenomena in their proposed computer simulations and data analysis - in particular: the complexity of SARS-CoV-2 virus spread. In complex spatio-temporal dynamical systems, small spatio-temporal fluctuations can dramatically change the dynamics of virus spreading on large scales. Department of Nuclear and Sub-Nuclear Physics, Faculty of Science, Institute of Physics, Pavol Jozef Šafárik University in Košice, Košice, Slovak Republic INTRODUCTION: Mina and Andresen in the paper [1] refer to mathematical models that incorporate relevant variation in viral loads and test accuracy [2]. On that basis, they suggest that - with frequent, large-scale sampling of a population - detection of herd effects was possible. The authors [1] referred to the public testing currently in effect in the Slovak Republic [3]. Unfortunately, they have adopted research conclusions that are not supported by appropriate mathematical models, which require additional variables and factors. Further, the authors of the initial report [3] have not rigorously compared the results with actual measurement data from the Slovak Republic over a longer term window, i.e. a few weeks before and after public -wide testing (31. October - 1. November 2020). Our independent and simple analysis of WHO available global data from Slovak Republic (S1) shows quit the opposite effect and throws into doubt the conclusions the authors [3]. RESULTS: We have analyzed the scaling properties [4] of daily count $i(t)$ as well as cumulative count $I(t)$ of infected cases where $t$ is time in days during the first and second SARS-CoV-2 virus waves. Our results Figure 1 show scaling properties of $i(t)\sim t^{\pm\beta}$, $I(t)\sim t^{\alpha}$ where $\alpha$ and $\beta$ are scaling exponents. Double logarithmic scales Figure 1 are much more suitable to demonstrate scaling properties and to identify significant changes of virus spread dynamics, for example to recognize outbreak waves as well as the dynamics of outbreak growth and decay during a time of the wave. A power law decay of daily count of infected cases $i(t)\sim t^{-\beta}$ shows that a decay of outbreak follows a slow dynamics and can take a long time depending on both an exponent $\beta$ and a number of daily infected cases $N$ in tipping point of daily count of infected cases $i(t)$ (in a preparation to publish). We have analyzed only one component of mobility Figure 2 (S2): retail and recreation, that carries important information about the effectiveness of public policy measures i.e. demonstrating that these measures decrease average mobility and therefore the average number of daily personal contacts. In Figure 1 we can identify in these neighbor countries a common tipping point of daily count of infected cases $i(t)$ on 1. November 2020. We compare a temporal evolution of a retail mobility Figure 2 (A) and rescaled daily count of infected cases Figure 2 (B) in Czech Republic and Slovak Republic). You can see common features of retail mobility and daily count of infected cases before the tipping point and quit different features of mobility as well as daily count of infected cases subsequent the tipping point. Retail mobility Figure 2 (A) and daily count of infected cases Figure 2 (B) clearly demonstrate that, if countries applied similar public policy measures to decrease mobility, that the dynamic of virus spread has similarly decayed in both countries. After public-wide testing in the Slovak Republic (31. October-1. November 2020), mobility dynamics Figure 2 (A) as well as rescaled daily count of infected cases Figure 2 (B) dramatically changed in the Slovak Republic. The rescaled daily count of infected cases in Slovak Republic shows a much more higher daily count of infected cases as when both countries shared similar public policy measures to control low mobility. DISCUSSION: Our criticism is focused on the work of Pavelka et al. [3]. We think that Pavelka et al. [3] made several conceptual mistakes during their analysis of available data and their assumptions regarding governmental public policy measures. Most egregiously, they did not discuss the potential effects of false negative results in that report [3]. SD Biosensor claims that a combined negative agreement with PCR tests is $99.7\%$ (see the company data sheet regarding tests results in Switzerland). Based on data provided by the authors [3] and SD Biosensors, we estimate $15600$ false negative tests (i.e. the infected cases that were falsely evaluated as negative cases). Shortly after the public testing phase, the Slovak Republic government permitted the free movement of tested persons Figure 2 (A), while it has drastically restricted the free movement of healthy persons that opted to not participate in the public testing program. This increase in mobility of tested population (Figure 2 (A)) - including persons who have false negative test results - would logically suggest an uncontrolled increase of infection in all regions of the Slovak Republic within the following 7- 14 days after testing. This has now been confirmed by publicly available data Figure 1 (B) and Figure 2 (B). We note that it is necessary to consider the long incubation period of SARS-COV-2 virus and the average time when first syndromes could occur [5]. The authors [3] have not discussed the important impacts of other measures that were applied before public testing began, for example a decrease in mobility - very similar to that experienced in Czech Republic and Slovak Republic Figure 2 (A) who share the same tipping point of daily count of infected cases at 1. November 2020 Figures 1 (A), (B) and 2 (B). We note that, at this tipping point, the reproductive number has been $R<1$ and public testing program in the Slovak Republic had been started. Importantly, the author’s computer simulations [3] did not take into account the influx of new infected cases from abroad due to periodic - and massive - migration of work forces between the Slovak Republic, Czech Republic and other countries. The history of the SARS-CoV-2 outbreak in the Czech Republic and Slovak Republic - prior to the Slovak’s Republic testing program Figures 1 (A), (B), and 2 (B) - show that these countries were strongly coupled, with similar daily counts of infections and similarly decreasing infection trends due to low-mobility and other important public-policy measures taken in the Slovak Republic and neighboring countries. Subsequent to the "tipping point", the decreasing trend in daily infections in the Slovak Republic virtually stopped within a few days. Daily count of infected cases Figure 2 (B) started again to increase. This is in contrast to the situation in the Czech Republic Figures 1 (A), (B) and 2 (B). This directly demonstrates that the public testing program has not had any positive effect on daily infection rates. We show in Figure 2 (B), that public testing - in an environment where tests are not precise and there is a relative high mobility of tested persons (many with false negative test results) - can initiate new outbreaks. Our interpretation of available data is entirely contrary to the interpretations and conclusions as presented in [3] and uncritically adopted by other authors [1]. We are confident that our conclusions are well supported by other authors who have investigated the SARS outbreak and mathematically investigated the impacts of quarantine and other public-policy measures in the past [6]. These authors concluded that quarantine appears to have formed the most effective basis for control in several countries and should be equally effective on a smaller scale, likely contributing to the prevention of major outbreaks in other countries. On the other hand, in the absence of such effective measures, SARS has the potential to spread very widely. Considerable effort will be necessary to implement such measures in those settings where transmission is ongoing, but such efforts are essential to quell local outbreaks and reduce the risk of further global dissemination [6]. We think that in the context of large scale populations, it is very difficult to control the effectiveness of wide public quarantine (personal remark: i.e. without drastic violation of human rights) due to the complexity of virus spread as well as of personal contact interactions [4]. CONCLUSIONS: We believe that a detailed and correct analysis of SARS-CoV-2 virus spread in the Czech and Slovak Republics is very important and could be useful for a better understanding of dynamic of SARS-CoV-2 outbreak. Both the Czech Republic and Slovak Republic have been successful in stopping the first wave of SARS-CoV-2 outbreak Figure 1. On the other hand, the countries have not be able to smoothly manage the second wave. The current approaches to manage the outbreak in these countries are quite different. In the case of the Czech Republic, the main tools are to limit mobility and increase testing, while the Slovak Republic engages in a model of very intensive and frequent testing virtually everywhere [2, 3] with a relative high mobility allowed in tested populations. Since the tipping point (1. November 2020), the data does not support any positive impact of this approach in the Slovak Republic Figure 2 (B). We think that this is due to the complexity of virus spread, rapid and uncoordinated shifts in public policy, non-optimal communication with citizens and a very low effectiveness of quarantine control on large scales [6] Acknowledgment We thank Geir Helgesen for valuable discussion and Ben Dowling for reading the manuscript. Supplementary materials References [1] M. J. Mina and K. G. Andersen, Science, DOI: 10.1126/science.abe9187 (2020). [2] D. B. Larremore et al., Sci. Adv. DOI: 10.1 126/sciadv.abd5393 (2020). [3] M. Pavelka et al., CMMID Repository, 11 November 2020; https://bit.ly/36BkxV5. [4] M. Tizzoni et al. Sci Rep 5, 15111 (2015) DOI: 10.1038/srep15111. [5] B. Hu, H. Guo, P. Zhou, et al., Nat Rev Microbiol (2020). DOI: 10.1038/s41579-020-00459-7 [6] M. Lipsitch at.al., Science 300, 1966 (2003).

hep-ph/0107256 SNS-PH/01-10 November 19, 2020 SN1a data and the CMB of Modified Curvature at Short and Large Distances Mar Bastero-Gil and Laura Mersini Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7, I-56126 Pisa, Italy The SN1a data, although inconclusive, when combined with other observations makes a strong case that our universe is presently dominated by dark energy. We investigate the possibility that large distance modifications of the curvature of the universe would perhaps offer an alternative explanation of the observation. Our calculations indicate that a universe made up of no dark energy but instead, with a modified curvature at large scales, is not scale-invariant, therefore quite likely it is ruled out by the CMB observations. The sensitivity of the CMB spectrum is checked for the whole range of mode modifications of large or short distance physics. The spectrum is robust against modifications of short-distance physics and the UV cutoff when: the initial state is the adiabatic vacuum, and the inflationary background space is de Sitter. E-mails: bastero@cibs.sns.it, mersini@cibs.sns.it 1 Introduction Based on the theoretical cosmological models of inflation, the interpretation of the current astrophysical observations such as the SN1a data [1], suggest that our universe contains a large amount of dark energy [2]. However, alternative models, free of dark energy, which may fit in the allowed range of parameters suggested by observation, are not excluded. In this paper we investigate claims to a possibly different interpretation of the SN1A data, for these alternative cosmological models: a FRW universe with no dark energy but with a modified curvature at large enough distances. The hope then is that either the Friedman equation for the expansion is modified, or that the light from SN1a that reaches us, while passing through these regions of different curvature, would be deflected, thereby ‘‘appearing’’ to have the same effect as an accelerating universe111We thank A. Riotto for bringing this idea to our attention.${}^{,}$222See however Ref. [3] for constraints on models with spatial variations of the vacuum energy density.. We examine metric perturbations in this modified background geometry (traced back at the time of inflation). Metric perturbations are responsible for the generation of the large scale structure and temperature anisotropies of the CMB. The inflaton field (in 4 dimensions), through the Friedmann equation, determines the expansion rate $H$ for the curvature of the background geometry. The metric perturbations satisfy a Klein-Gordon scalar field equation, minimally coupled to gravity [4]. The scalar field has a generalized mass squared $\Omega_{n}(\eta)^{2}$ that receives the contributions of two terms: the field frequency squared and the field coupling to the background curvature term. The coupling of the field to the curvature results in a modified propagation at long wavelengths since the curvature of the universe is modified at large distances compared to the intermediate scales. Examples of modified gravity can be found in [5, 6, 7, 8]. Then, the modified propagation of wavelengths of the same scale as the background curvature deviation scale can be attributed to a nonlinear dispersed frequency of the field at those wavelengths, for as long as the generalized mass squared, $\Omega_{n}(\eta)^{2}$ in the field equation, remains the same. This equivalence noticed in [9] is very useful for calculating the effects of modified large distance curvature in observations. Our model consists of a (2-parameter) family of nonlinear dispersion relations for the generalized frequency of the field, that take account of the modification of the curvature at large distances. The family of dispersion functions is nearly linear for most of the range $k<M_{P}$, except a nonlinear deviation centered around some low value of momenta $k_{0}$. It is this deviation bump that reflects the modifications of the generalized frequency of the field at low momenta $k_{0}$ due to the modification of the curvature at large distances $k^{-1}_{0}$. The dispersion function introduced in Sect. 2, although nonlinear in the transplanckian regime, it is nevertheless a smooth function there, asymptotically approaching a constant value at time-infinity, thus having a well defined initial vacuum state [10]. The analytical calculation of the CMB spectrum is based on the Bogoliubov coefficient method. The details of the exact solutions for this class of dispersion functions [10] are given in the Appendix. In Sect. 3 we check the sensitivity of CMB spectrum to the bump parameters $k_{0}$ and $B$ (scale location and amplitude) that control the deviation behavior from a linear frequency dispersion at low values of the momenta; i.e., the allowed range of curvature modifications at very large or very short distances that may agree with observation. We use CMBFAST in Sect. 3 to plot the spectrum, by replacing the standard primordial power spectrum $\delta_{H}^{0}(k)$ with that derived analytically in Sect. 2 for the model considered. We comment and summarize the results in Section 4. It is shown that the CMB spectrum is sensitive only to the choice of the initial vacuum state and the departure from linearity in the low momenta regime. However, for an adiabatic initial vacuum state, the CMB spectrum of a de Sitter expansion does not depend in the details of nonlinearity in the transplanckian regime [11, 12, 13, 9, 14, 15] . 2 The Model The generalized Friedmann-Lemaitre-Robertson-Walker (FLRW) line-element in the presence of scalar and tensor perturbations, takes the form [16] $$\displaystyle ds^{2}$$ $$\displaystyle=$$ $$\displaystyle a^{2}(\eta)\left\{-d\eta^{2}+\left[\delta_{ij}+h(\eta,{\bf n})Q% \delta_{ij}\right.\right.$$ (1) $$\displaystyle\left.\left.+h_{l}(\eta,{\bf n})\frac{Q_{ij}}{n^{2}}+h_{gw}(\eta,% {\bf n})Q_{ij}\right]dx^{i}dx^{j}\right\}\,,$$ where $\eta$ is the conformal time and $a(\eta)$ the scale factor. The dimensionless quantity ${\bf n}$ is the comoving wavevector, related to the physical vector ${\bf k}$ by ${\bf k}={\bf n}/a(\eta)$. The function $(h,\,h_{l})$ and $h_{gw}$ represent the scalar and tensor perturbations respectively. The power spectrum of the perturbations can be computed once we solve the equations in the scalar and tensor sector. The equation for the metric perturbations corresponds to a Klein-Gordon equation of a minimally coupled scalar field, $\mu_{n}$, in a time dependent background333We refer the reader for the details of the procedure to Refs. [17] and related references [4]. $$\mu_{n}^{\prime\prime}+\Omega_{n}(\eta)^{2}\mu_{n}=0\,,$$ (2) where the prime denotes derivative with respect to conformal time $\eta$, and the generalised comoving frequency is444Note that from here on we use the symbol $a$ instead of $a(\eta)$ for the scale factor. $$\Omega_{n}(\eta)^{2}=n^{2}-\frac{a^{\prime\prime}}{a}=a^{2}k^{2}-\frac{a^{% \prime\prime}}{a}\,.$$ (3) The dynamics of the scale factor is determined by the evolution of the background inflaton field $\phi$, with potential $V(\phi)$, and the Friedmann equation. There are mechanisms that may produce different scale factors by modifying gravity at large (e.g. [5, 6, 7, 8]) or short distances ([13]).The present large distance modification scales can be traced back in time and would correspond to deviations in the primordial scale factor and spectrum. We can denote this “distance-dependent” scale-factor by ${\cal A}$. The coupling of the field to this background curvature results in a modified propagation of the field at long wavelengths. Therefore, modifications of the scale factor or curvature ($\mathcal{A}$) of the universe at large scales can be attributed to a dispersed effective frequency $(n_{eff})$, such that the generalized comoving frequency Eq. (3) remains the same, in the following manner $$\Omega_{n}(\eta)^{2}=n^{2}-\frac{\mathcal{A}^{\prime\prime}}{\mathcal{A}}=n_{% eff}^{2}-\,\frac{a^{\prime\prime}}{a}\,.$$ (4) $n_{eff}$ denotes the dispersed comoving frequency of the field due to absorbing the modification terms to the curvature, ${\cal A}^{\prime\prime}/{\cal A}$. Therefore, the dispersion function for the generalized frequency results from the modified curvature at very large and very short distances. It deviates from linearity at small momentum $k$ and asymptotically approaches a constant value in the transplanckian regime. The dispersion relation for the generalized comoving frequency $\Omega_{n}(\eta)$ is simply555From here on, we absorb the term $a^{\prime\prime}/a$ of Eq. (4) into the definition of the dispersion function $F[k]$. [10]: $\Omega_{n}(\eta)=a(\eta)F[n/a(\eta)]$. The 2-parameter family of dispersion functions $F[k]$ of our model (see Fig. 1) is: $$\displaystyle F[k]^{2}$$ $$\displaystyle=$$ $$\displaystyle(k^{2}-k_{1}^{2})V_{0}(x,x_{0})+k^{2}~{}V_{1}(x-x_{0})+k_{1}^{2}\,,$$ (5) $$\displaystyle V_{0}(x,x_{0})$$ $$\displaystyle=$$ $$\displaystyle\frac{C}{1+e^{x}}+\frac{E~{}e^{x}}{(1+e^{x})(1+e^{(x-x_{0})})}\,,$$ (6) $$\displaystyle V_{1}(x-x_{0})$$ $$\displaystyle=$$ $$\displaystyle-B\frac{e^{x}}{(1+e^{(x-x_{0})})^{2}}\,,$$ (7) where the dimensionless wavevector is $x=k/k_{C}$, $k_{C}=M_{P}$ is the cutoff scale, $k_{0}\ll k_{C}$, (i.e. $x_{0}\ll\ 1)$ is the value at which we deviate from linearity at low momentum, the deviation amplitude is controlled by the parameter $B$, and the constant parameter $k_{1}<k_{C}$ is the asymptotic value of the frequency in the transplanckian regime ($k\rightarrow\infty$). $C,E,B,x_{0}$ are dimensionless parameters. As already discussed in Refs. [11, 12, 13], Eq. (2) represents particle production in a time-dependent background [18, 19]. We will follow the method of Bogoliubov transformation to calculate the spectrum. The frequency $\Omega_{n}(\eta)^{2}$ (which is the same as a ‘time-dependent mass squared‘ term) goes asymptotically to constant values at late and early times. Therefore the initial and final vacuum states are well defined. At early times the wavefunction should behave as a plane wave: $$\mu_{n}\rightarrow_{\eta\rightarrow-\infty}\frac{1}{\sqrt{2\Omega_{n}^{in}}}e^% {-i\Omega^{in}_{n}\eta}\,.$$ (8) But at late times one has a squeezed state due to the curved background that mixes positive and negative frequencies. The evolution of the mode function $\mu_{n}$ at late times fixes the Bogoliubov coefficients $\alpha_{n}$ and $\beta_{n}$, $$\mu_{n}\rightarrow_{\eta\rightarrow+\infty}\frac{\alpha_{n}}{\sqrt{2\Omega^{% out}_{n}}}e^{-i\Omega^{out}_{n}\eta}+\frac{\beta_{n}}{\sqrt{2\Omega^{out}_{n}}% }e^{+i\Omega^{out}_{n}\eta}\,.$$ (9) with the normalization condition: $$|\alpha_{n}|^{2}-|\beta_{n}|^{2}=1\,.$$ (10) In the above expressions, $\Omega^{in}_{n}$ and $\Omega^{out}_{n}$ denote the asymptotic values of $\Omega_{n}(\eta)$ when $\eta\rightarrow\mp\infty$. Details of the exact solution for Eq. (2) with the dispersed frequency given by Eqs. (5-7) are given in the Appendix. The final expression for the Bogoliubov coefficient $|\beta_{n}|^{2}$ is: $$|\beta_{n}|^{2}=\frac{\sinh^{2}(2\pi\hat{\Omega}_{-})+\Gamma(k_{0},B)}{\sinh^{% 2}(2\pi\hat{\Omega}_{+})-\sinh^{2}(2\pi\hat{\Omega}_{-})}\,,$$ (11) where $\hat{\Omega}_{i}=\Omega_{i}/n$, $\hat{\Omega}_{\pm}=(\hat{\Omega}^{out}\pm\hat{\Omega}^{in})/2$, and the deviation function $\Gamma(k_{0},B)$ that contains the departure from thermality in the spectrum is $$\Gamma(k_{0},B)=\cosh^{2}(\frac{\pi}{2}\sqrt{4Be^{-x_{0}}-1})\,.$$ (12) When $B=0$, for $\Omega^{in}>\Omega^{out}$, then it is clear from Eq. (11) that the spectrum of created particles is nearly thermal to high accuracy, $$\left|\frac{\beta_{n}}{\alpha_{n}}\right|^{2}\simeq e^{-4\pi\sqrt{C}}\,.$$ (13) The function $\Gamma(k_{0},B)$ represents the $deviation$ of the spectrum from thermal behavior due to the non-linearities at low momentum. Therefore, the amplitude of the power spectrum, $\delta_{H}(k)$, will be modified by $\Gamma(k_{0},B)$ due to the non-linear dispersion function introduced at around $x_{0}<1$. In de Sitter space, the Bogoliubov coefficients would not depend on $k$ except their dependence in the bump parameters through the deviation function $\Gamma(k_{0},B)$. This function represents the departure from thermality in the particle creation number, $|\beta_{n}|^{2}$ and it confirms B. L. Hu idea [20] that near thermal radiance can be characterized by departure from exponential scaling. It is straightforward to derive the CMB power spectrum, $P(n)$, analytically from (the exact solution for) the Bogoliubov coefficients $\alpha_{n},\beta_{n}$ [15] $$P(n)=\frac{n^{3}}{2\pi^{2}}|\frac{\mu_{n}}{a}|^{2}\simeq|\beta_{n}+\alpha_{n}|% ^{2}\,.$$ (14) The deviation of the spectrum from scale-invariance in this class of models depends on the parameters of large-distance curvature modifications, namely: the scale of modified long wavelength modes, $k_{0}^{-1}$, and the deviation amplitude $B$. The expression for the Bogoliubov coefficient and Eq. (13) indicate that: for a well-defined initial vacuum state666The field is in an initial Bunch-Davies vacuum., the spectrum is insensitive to the nonlinear dispersion relation in the transplanckian regime (modifications of short-distance physics). The unusual CMB spectrum plotted in the next Section with CMBFAST, demonstrates that modifications of the large scale curvature of the universe produce a tilt due to the departure from scale-invariance, and therefore conflict with the observed CMBR spectrum. In general the tilt is enhanced for modifications at superhorizon scales ($k_{0}\leq H_{0}$) because it is the low energy modes that dominate the spectrum in the Bogoliubov coefficient. Although departure from scale invariance is smaller at the last scattering horizon scale, $H_{LS}$, the range of deviation parameters is constrained by the amplitude of the first peak. The deviation introduced to the spectral index, $n_{s}$ from higher energy modes (wavelengths shorter than the last scattering horizon $k_{0}>H_{LS}$) becomes negligible because high energy modes do not contribute significantly to the spectrum. However, the shorter wavelengths would correspond to the intermediate FRW regime rather than the large distance scales, a regime which is scrutinized by direct observation. 3 CMB Spectrum Recent Boomerang and MAXIMA-1 CMB experiments [21, 22] have, to high accuracy, constrained the cosmological parameters, derived from the family of inflationary adiabatic models, to: total energy density $\Omega_{tot}=0.90\pm 0.15$ and spectral index $n_{s}=0.99\pm 0.09$ at a 95% confidence level [23]. The current data favors a universe with dark energy density $\Omega_{\Lambda}=0.7$. In this part, we explore the cosmological consequences of the alternative model that was given in Section 2 (Fig. 1). CMB is the most difficult test of precision cosmology that these models should pass. This model contains no dark energy, $\Omega_{\Lambda}=0$, however it describes a universe which at large distances has a modified curvature from the metric of the FRW universe at intermediate scale. In Fig.2 we show the CMB power spectra obtained for different representative values of the deviation parameters $k_{0}$ and $B$ in the dispersion function Eqs. (5-7). The conventional parameters that go in the input of CMBFAST are: $(\Omega_{tot},\Omega_{b},\Omega_{c},\Omega_{\Lambda}$), which stand for total energy density, baryonic, cold dark matter and the cosmological constant energy density respectively; and $n_{s}$ which is the scalar spectral index. We modified the power spectrum amplitude $\delta_{H}^{0}(k)$ in the POWERSFLAT subroutine of CMBFAST, in order to contain the deviation from the thermal spectrum (for the exact calculation reported in Section 2). The modified perturbation amplitude $\delta_{H}^{2}(k)$ is expressed in terms of $\delta_{H}^{0},\,k_{0},\,B$, where $\delta_{H}^{0}$ is the unmodified amplitude of the scale-invariant power spectrum, $k_{0}$ corresponds to the location-scale where the curvature is modified, and B measures the amplitude of deviation in the curvature at scale $k_{0}$. The values of the conventional parameters were taken to be (1,0.03, 0.97,0) for all the deviation plots ($II-V$), but the deviation parameters in the 4 plots below in Fig. 2 are in respective order: I (solid line): ($$k_{0}=0$$, $$B=0$$, $$\Omega_{\Lambda}=0.7$$) II (long-dashed line): ($$k_{0}=10^{-6}$$ hMpc$${}^{-1}$$, $$B=2$$) III (dashed line): ($$k_{0}=0.05$$ hMpc$${}^{-1}$$, $$B=2$$) IV (dot-dashed line ): ($$k_{0}=5$$ hMpc$${}^{-1}$$, $$B=2$$) V (dotted line ): ($$k_{0}=0.05$$ hMpc$${}^{-1}$$, $$B=2.5$$) All plots were normalized to COBE. Shown for comparison is also plot $I$ corresponding to the conventional CMB spectrum with $\Omega_{\Lambda}=0.7$. As it can be seen from the plots in Fig. 2, there are distinct features of the CMB spectra corresponding to the dispersion function in comparison to the standard spectrum obtained for ($\Lambda$)CDM models. There is an overall tilt produced in the spectrum which signals departure from the scale invariance. This tilt is a function of the amplitude and scale of the modification, $k_{0},B$, introduced in Sect.2 (Eq.11), such that it increases for low values of the deviation momentum scale $k_{0}$ and large deviation amplitude $B$. Let us consider the 3 regimes into which the curvature modifications can be introduced: (1) Modifications at superHubble scales ($k_{0}<H_{0}$). The departure from scale invariance is the strongest because the low energy modes dominate the spectrum, (II) in Fig. 2. Models predicting curvature modifications in regime (1) quite likely are ruled out due to a strongly tilted spectrum. (2) Modifications in the distance range between the current horizon $H_{0}$ and last scattering horizon scale $H_{LS}$ ($H_{0}\leq k_{0}\leq H_{LS}$). For this range, the tilt is less pronounced then in regime (1). The main constraint comes from the tilt and it tightly limits the amplitude of deviation in the first peak. For modifications around the last scattering horizon scale, $k_{0}\approx H_{LS}$ the departure from scale invariance is vanishing, therefore the constraints are relaxed. However, even in this case the parameter B is tightly constrained to deviation by less than 10%, in order for the amplitude of the first peak $A_{1}$ to be in the allowed range of 4500-5500 $\mu K^{2}$ [21, 22]. In Fig. 2 we show the CMB spectra for these tuned values of $k_{0},\,B$; for comparison we also plotted the CMB for a value of $B=2.5$, which is outside the allowed range. (3) Modifications at distances shorter than the last scattering horizon ($k_{0}>H_{LS}$). As we approach higher energy modes, the effect of the modification in the tilt of the spectrum is suppressed, therefore the departure from the conventional spectrum is vanishing. Nevertheless, these length scales do not correspond to large distances anymore, instead they are in the intermediate regime of FRW Universe. Thus the possibility of curvature modifications at such scales (galactic and intergalactic) is ruled out by direct observation up to very short distances (less than 1 mm). Clearly, there is no tilt or departure from the conventional CMB produced in the limit of modifications of very short distance physics, (very high momenta $k_{0}\rightarrow\infty$, i.e., transplanckian regime). The claim of the model was to ‘‘offer an alternative explanation’’ to the SN1a data, namely: either the conventional Friedmann equation is modified or; the light of the SN1a passing through regions of modified curvature would get deflected, and therefore when received by us would appear as if indicating an accelerating universe. Although this alternative approach to the SN1a data might be theoretically appealing, we conclude that the CMB data tightly constrains it and makes it unlikely to bear any resemblance to reality. The method used in this work, can also be adopted to check if higher dimensional models that predict modified gravity at large scales and modified equations for the perturbations 777These models naturally modify the curvature around horizon and Planck length-scales due to the higher dimensional gravity effects that switch on at very large or very short distances, but nevertheless with contributions from higher graviton excitations suppressed [24]. [5, 6, 7, 8] satisfy the CMB constraints. 4 Summary In this work we investigated claims that a modified large-distance curvature may offer an alternative explanation for the SN1a data. To check these claims, we studied the sensitivity of CMB spectrum to the whole range of modes, $0\leq k\leq\infty$, when short and large distance regimes are modified. In [9] it was noticed that a modified curvature of the universe at large distance (when traced back at the time of inflation888It should be noted that in the case of higher dimensional multigravity [5, 6, 7, 8, 24], it is not clear how the metric perturbation equations are modified. ) gives rise to a dispersed frequency for the cosmic perturbations. The field is minimally coupled to the curvature thus its propagation feels the modifications in the background geometry. We adopted the method of Ref. [9] in order to find out the effects of curvature deviations on the current astrophysical observables. The role of a modified curvature of the universe at large distances on the inflationary metric perturbations was analytically described by a family of dispersion relations. The modification modulates the generalized frequencies of the inflationary perturbation modes at small values of the momenta $k$ by departing from linearity around some certain small momenta $k=k_{0}$ ($k_{0}\leq M_{P}$) with a deviation amplitude $B$. The nonlinear feature of the dispersion relations, at small momenta $k_{0}$ and in the transplanckian regime, tracks the curvature deviations at large and short distances, from the conventional FRW universe of intermediate scales. One of the parameters ($k_{1}^{2}<k_{C}^{2}$), in this class of dispersion functions was constrained in order to satisfy the Starobinsky bound for negligible backreaction [14]. The analytical expression for the CMBR spectrum (Sect. 2), as well as the CMBFAST plots of this class of models, deviate from the black body scale invariant spectrum. The deviation function $\Gamma(k_{0},B)$, given in Sect. 2 and the Appendix, which measures departure from the scale-invariant spectrum (deviation from thermality in the the Bogoliubov coefficient), depends on two free parameters, the scale $k_{0}$ and the amplitude of the curvature modifications $B$. The tilt produced in the spectrum due to $\Gamma(k_{0},B)$ is present for all modification scales $k_{0}\leq M_{P}$ (these values of the physical momenta correspond to the time of inflation). The tilt is less pronounced for scale modifications corresponding to length scales less or equal to the horizon of the last scattering surface, and in this case, the main constraint comes from the modifications to the amplitude of the first acoustic peak and the fact that curvature modifications in the intermediate FRW Universe scales are under direct observation. It remains interesting to answer why the only curvature modifications that for a small range of $k_{0}$ and $B$ can reconcile with the conventional CMB spectrum are allowed only around the last scattering $H_{LS}$ scales. The scale and amplitude of the deviations from the conventional spectrum, are severely constrained from the observed CMB spectrum to be within $10\%$ of the scale and amplitude of the first peak. Although it is counterintuitive, since large distance would correspond to low energy theories, our results indicate that any modifications in the large scale curvature of the universe, is tightly constrained from CMB data, to a very small range of deviations from the curvature of the intermediate FRW universe. Perhaps, there is a natural way that would explain such a universe with an FRW spacetime at intermediate and very large distances but with small curvature deviations around $H_{LS}$, without the need to appeal to fine-tuning. But if not, then theoretical cosmological models would have to account for the negative pressure dark energy of the universe. The analysis of the sensitivity of the CMB spectrum for the whole range of modes in a de Sitter background space, with modifications in the short and large distance physics, reveal: the spectrum is insensitive to the details of short-distance physics and the cuttoff scale $k_{C}$ (the transplanckian regime) only for an initial adiabatic vacuum state; the scale-invariance of the spectrum and the amplitude of the first acoustic peak are very sensitive to modifications of large distance physics (low momentum modes); the spectrum is also highly sensitive to the choice of the initial conditions999It has been argued by many authors [9, 14, 12] that the adiabatic vacuum is the right choice for the initial conditions. Even for the same dispersion model, a different choice for the initial vacuum state will clearly result in a different particle spectrum, therefore one has to be careful to distinguish if the features observed in the CMB spectrum are signatures of new physics or only of the choice of initial conditions.. In our class of dispersion models, the initial vacuum state is well-defined since the background ($\Omega_{n}(\eta)^{2}$) goes asymptotically flat at early times (Bunch-Davis vacuum [25]). The CMB spectrum for this class of models is indeed insensitive to short distance modifications, as it can be checked by taking the limit when large scale modification parameters ${k_{0},B}$ go to zero, in which case the conventional scale-invariant spectrum is recovered. Therefore all the features observed in Fig. 2, are due to large-scale curvature modifications only. Acknowledgment: We are very grateful to S.Dodelson for his help with the CMBFAST. We want to thank A. Riotto, R. Kolb, L. Parker, A. Kempf, P. Frampton, G. Siegl, I. Kogan, for beneficial discussions and comments. We also would like to thank P. Kanti for useful discussions in the early stages of this work. We acknowledge Lloyd Knox for making CMBFAST program available. 5 Appendix The family of dispersion functions we used in Sect. 2 to model the deviation of the curvature at large and short distances is given by: $$\displaystyle F[k]^{2}$$ $$\displaystyle=$$ $$\displaystyle(k^{2}-k_{1}^{2})V_{0}(x,x_{0})+k^{2}~{}V_{1}(x-x_{0})+k_{1}^{2}\,,$$ (15) $$\displaystyle V_{0}(x,x_{0})$$ $$\displaystyle=$$ $$\displaystyle\frac{C}{1+e^{x}}+\frac{E~{}e^{x}}{(1+e^{x})(1+e^{(x-x_{0})})}\,,$$ (16) $$\displaystyle V_{1}(x-x_{0})$$ $$\displaystyle=$$ $$\displaystyle-B\frac{e^{x}}{(1+e^{(x-x_{0})})^{2}}\,,$$ (17) where101010The momentum $k$ has been shifted by $k_{C}$ such that $\Omega_{n}\approx x$ for small positive values, $x\ll 1$. $x=k/k_{C}$, $x_{0}=k_{0}/k_{C}$; $k_{C}=M_{P}$ is the cutoff scale, $k_{0}\ll k_{C}$ ($x_{0}\ll 1$) is the value at which we deviate from linearity at low momentum, and the amplitude of the “bump/deviation” is controlled by $B$ (see Fig. 1). The parameter $k_{1}=n_{1}/a(\eta)$ gives the asymptotic constant value at initial time for the frequency in the limit ($k\gg k_{C}$), i.e., in the transplanckian regime. On the other hand, in order to ensure the linear behavior at very low values of the momenta, $x\ll 1$, we impose the following constraints for any value of the deviation parameters $x_{0}$ and B: $$\displaystyle V_{0}(x\ll 1,x_{0})$$ $$\displaystyle\simeq$$ $$\displaystyle 1\,,\;\;\;\;\;V_{0}(x\ll 1,x_{0})^{\prime\prime}\simeq 0\,,$$ (18) $$\displaystyle V_{1}(x\ll 1,x_{0})$$ $$\displaystyle\simeq$$ $$\displaystyle 0\,,$$ (19) where prime denotes derivative with respect to the physical momentum $k$. The generalised comoving frequency $\Omega_{n}(\eta)$ is then given by: $$\displaystyle\Omega_{n}(\eta)^{2}$$ $$\displaystyle=$$ $$\displaystyle a(\eta)^{2}F[n/a(\eta)]^{2}$$ (20) $$\displaystyle=$$ $$\displaystyle\!\!\!\!(n^{2}-n_{1}^{2})\left[\frac{C}{1+e^{x}}+\frac{E~{}e^{x}}% {(1+e^{x})(1+e^{(x-x_{0})})}\right]-n^{2}\left[\frac{B~{}e^{x}}{(1+e^{(x-x_{0}% )})^{2}}\right]+n_{1}^{2}\,,$$ with $n=a(\eta)k$, $a(\eta)=-|\eta_{C}|/\eta$ during de Sitter inflation ($|\eta_{C}|=1/H(\eta_{C})$), and $$x=\frac{k}{k_{C}}=-\frac{n}{|\eta_{C}|k_{C}}\eta\,.$$ (21) The generalised frequency $\Omega_{n}(\eta)$ goes to constant values at $\eta\rightarrow\pm\infty$, such that: $$\displaystyle\Omega_{n}(\eta)$$ $$\displaystyle\rightarrow_{\eta\rightarrow-\infty}$$ $$\displaystyle\Omega_{n}^{in}=n_{1}\,,$$ (22) $$\displaystyle\Omega_{n}(\eta)$$ $$\displaystyle\rightarrow_{\eta\rightarrow+\infty}$$ $$\displaystyle\Omega_{n}^{out}=\sqrt{n_{1}^{2}+C(n^{2}-n_{1}^{2})}\,,$$ (23) with $\Omega_{n}^{out}\simeq\sqrt{C}n$ when $n\gg n_{1}$. Under the change of variables $\eta\rightarrow u=exp(dn\eta)$, where $d=1/(|\eta_{C}|k_{C})$, the scalar wave equation (2) for the mode $\mu_{n}$ becomes: $$\left[\partial^{2}_{u}+\frac{1}{u}\partial_{u}+V(u)\right]\mu_{n}=0\,,$$ (24) where: $$V(u)=\hat{D}+\frac{\hat{C}}{u(1+u)}+\frac{\hat{E}}{u(u+1)(\gamma_{0}+u)}-\frac% {\hat{B}}{u(u+\gamma_{0})^{2}}\,,$$ (25) and, $$\displaystyle\hat{C}=C[(n^{2}-n^{2}_{1})/(n~{}d)^{2}]\,,\;\;\;\hat{E}=E[(n^{2}% -n^{2}_{1})/(n~{}d)^{2}]\,,$$ $$\displaystyle\hat{D}=(n_{1}/n~{}d)^{2}\,,$$ $$\displaystyle\hat{B}=B/d^{2}\,,$$ and $\gamma_{0}=exp(-k_{0}/k_{C})$. Eq. (24) is exactly solvable in terms of the Riemann generalised hypergeometric functions [26] with the constraint $\hat{E}=\hat{C}/(1-\gamma_{0})$, $$\mu_{n}\propto P\left(\begin{array}[]{cccc}0&\infty&-\gamma_{0}&\\ a&c&b&u\\ a^{*}&c^{*}&b^{*}&\end{array}\right)\,.$$ (26) As explained in Section 2, because of the asymptotic behavior of $\Omega_{n}(\eta)$, the initial and final vacua are well defined and the mode functions $\mu_{n}$ behave as plane waves in the asymptotic limits $\eta\rightarrow\mp\infty$. The exact solution which matches this asymptotic behavior is then given by: $$\displaystyle\mu^{in}(\eta)=N^{in}(u)^{a}(u+\gamma_{0})^{b}~{}_{2}F_{1}[a+b+c,% a+b+c^{*},1+a-a^{*},-\frac{u}{\gamma_{0}}]\,,$$ (27) where $N^{in}$ is a normalization constant, and $$\displaystyle a$$ $$\displaystyle=$$ $$\displaystyle-i\hat{\Omega}^{in}=-i\sqrt{\hat{D}}\,,$$ (28) $$\displaystyle b$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{2}\left(1+\sqrt{1-4\hat{B}e^{-x_{0}}}\right)\,,$$ (29) $$\displaystyle c$$ $$\displaystyle=$$ $$\displaystyle-i\hat{\Omega}^{out}=-i\sqrt{\hat{D}+\hat{C}}\,.$$ (30) At late times the solution becomes a squeezed state by mixing of positive and negative frequencies: $$\displaystyle\mu^{out}_{n}(\eta)$$ $$\displaystyle=$$ $$\displaystyle N^{out}~{}(u)^{a}(u+\gamma_{0})^{b}\times$$ $$\displaystyle\!\!\!\!\!\!\left(\frac{\Gamma(1+a-a^{*})\Gamma(c^{*}-c)}{\Gamma(% a+b+c^{*})\Gamma(1-a^{*}-b-c)}~{}_{2}F_{1}[a+b+c,a^{*}+b+c,1+c-c^{*},-\frac{% \gamma_{0}}{u}]\right.$$ $$\displaystyle+$$ $$\displaystyle\!\!\!\!\!\!\left.\frac{\Gamma(1+a-a^{*})\Gamma(c^{*}-c)}{\Gamma(% a+b+c)\Gamma(1-a^{*}-b-c^{*})}~{}_{2}F_{1}[a+b+c^{*},a^{*}+b+c^{*},1+c^{*}-c,-% \frac{\gamma_{0}}{u}]\right)\,,$$ $$\displaystyle\mu^{out}_{n}$$ $$\displaystyle\rightarrow_{\eta\rightarrow+\infty}$$ $$\displaystyle\frac{\alpha_{n}}{\sqrt{2\Omega^{out}_{n}}}e^{-i\Omega^{out}_{n}% \eta}+\frac{\beta_{n}}{\sqrt{2\Omega^{out}_{n}}}e^{+i\Omega^{out}_{n}\eta}\,,$$ (31) with $|\beta_{n}|^{2}$ being the Bogoliubov coefficient equal to the particle creation number per mode $n$ and $\hat{\Omega}_{i}=\Omega_{i}/n$. Using the linear transformation properties of hypergeometric functions [26], we find that $$\left|\frac{\beta_{n}}{\alpha_{n}}\right|^{2}=\frac{\sinh^{2}(2\pi\hat{\Omega}% _{-})+\Gamma(k_{0},\hat{B})}{\sinh^{2}(2\pi\hat{\Omega}_{+})+\Gamma(k_{0},\hat% {B})}\,,$$ (32) where, $$\displaystyle\hat{\Omega}_{\pm}=\frac{\hat{\Omega}^{out}\pm\hat{\Omega}^{in}}{% 2}\,,\;\;\;\;\hat{\Omega}^{(}i)=\frac{\Omega^{i}}{nb}\,,$$ (33) and the deviation function $\Gamma(k_{0},\hat{B})$ is $$\Gamma(k_{0},\hat{B})=\cosh^{2}(\frac{\pi}{2}\sqrt{4\hat{B}e^{-x_{0}}-1})\,.$$ (34) When $B=0$ and $\Omega^{in}>\Omega^{out}$, then it is clear from Eq. (32) that the spectrum of created particles is nearly thermal to high accuracy, $$\left|\frac{\beta_{n}}{\alpha_{n}}\right|^{2}\simeq e^{-4\pi\sqrt{C}}\,,$$ (35) as expected in de Sitter expansion. 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Structure analysis of interstellar clouds: II. Applying the $\Delta$-variance method to interstellar turbulence V. Ossenkopf 11. Physikalisches Institut der Universität zu Köln, Zülpicher Straße 77, 50937 Köln, Germany 12SRON Netherlands Institute for Space Research, P.O. Box 800, 9700 AV Groningen, Netherlands 23Kapteyn Astronomical Institute, University of Groningen, PO box 800, 9700 AV Groningen, Netherlands 3    M. Krips 11. Physikalisches Institut der Universität zu Köln, Zülpicher Straße 77, 50937 Köln, Germany 14Harvard-Smithsonian Center for Astrophysics, SMA project, 60 Garden Street, MS 78 Cambridge, MA 02138, USA4    J. Stutzki 11. Physikalisches Institut der Universität zu Köln, Zülpicher Straße 77, 50937 Köln, Germany 1 (Received: December 23, 2002; accepted February 22, 2008) Key Words.: Methods: data analysis – Methods: statistical – ISM: clouds – ISM: structure Abstract Context: The $\Delta$-variance analysis is an efficient tool for measuring the structural scaling behaviour of interstellar turbulence in astronomical maps. It has been applied both to simulations of interstellar turbulence and to observed molecular cloud maps. In paper I we proposed essential improvements to the $\Delta$-variance analysis and tested them on artificial structures with known characteristics. Aims: In this paper we apply the improved $\Delta$-variance analysis to simulations of interstellar turbulence and observations of molecular clouds. We tested the new capabilities in practical use and studied properties of interstellar turbulence that could not have been addressed before. Methods: We selected three example data sets that profit in particular from the improved $\Delta$-variance method: i) a hydrodynamic turbulence simulation with prominent density and velocity structures, ii) an observed intensity map of $\rho$ Oph with irregular boundaries and variable uncertainties of the different data points, and iii) a map of the turbulent velocity structure in the Polaris Flare affected by the intensity dependence on the centroid velocity determination. Results: The tests confirm the extended capabilities of the improved $\Delta$-variance analysis. Prominent spatial scales were accurately identified and artifacts from a variable reliability of the data were removed. The analysis of the hydrodynamic simulations showed that the injection of a turbulent velocity structure creates the most prominent density structures are produced on a scale somewhat below the injection scale. The new analysis of a $\rho$ Oph continuum map reveals an intermediate stage in the molecular cloud evolution showing both signatures of the typical molecular cloud scaling behaviour and the formation of condensed cores. When analysing the velocity structure of the Polaris Flare we show that a universal power law connects scales from 0.03 pc to 3 pc. However, a plateau in the $\Delta$-variance spectrum around 5 pc indicates that the visible large-scale velocity gradient is not converted directly into a turbulent cascade here. It is obvious that, for any turbulent structure, effects of low-number statistics become important on the driving scale. Conclusions: 1 Introduction Observations of interstellar clouds show a complex, filamentary structure which can be attributed to turbulence in the interstellar medium (Franco & Carramiñana, 1999; Ossenkopf et al., 2000; Mac Low & Klessen, 2004). To understand the processes governing the structure and evolution of the clouds, turbulence models have to be constructed and compared to observational data. Their parameters and implementational details need to be adjusted to fit the observed behaviour. Due to the random nature of turbulence, simulations will never provide an exact reproduction of the observed data sets but will only reproduce general statistical properties like scaling relations. The most common scaling relation to characterise turbulent structures is the power spectrum of fluctuations, both in density and in velocity. Here, we encourage using another quantity, the $\Delta$-variance spectrum, introduced by Stutzki et al. (1998), a wavelet-based method to measure the relative amount of structural variation as a function of the size scale. Due to the mutual relations between the $\Delta$-variance spectrum and the radially averaged power-spectrum, the $\Delta$-variance analysis can be considered as a very robust method of evaluating the power spectrum of a structure. The advantages of the $\Delta$-variance method result from the smooth wavelet filter shape, which provides a robust way for an angular average independent of gridding effects, and from the insensitivity to edge effects as discussed by Bensch et al. (2001). In parallel to the structure scaling analysis, clump decomposition algorithms like GAUSSCLUMP (Stutzki & Güsten, 1990) found that the clump mass spectrum $dN/dM\propto M^{-\gamma}$ of sufficiently large molecular cloud data sets also tend to follow power laws over many orders of magnitude with the spectral index $\gamma$ in a relatively narrow range between 1.7 to 1.9 (Kramer et al., 1998; Heithausen et al., 1998). Stutzki et al. (1998) demonstrated that a clump ensemble with such a mass spectrum and with a power-law mass-size scaling relation results in a cloud image with the $\Delta$-variance scaling index determined by the number-mass and the mass-size spectral indices. However, it was questioned by e.g. Vázquez-Semadeni et al. (1997); Ballesteros-Paredes & Mac Low (2002) whether the observed mass-size relation reflects true properties of the underlying structure. It may rather represent an observational artifact. From these results it is obvious that there is a strong interest in studying the spatial scaling behaviour of observed maps of the interstellar medium via the $\Delta$-variance analysis. In paper I we have proposed several essential improvements to the original $\Delta$-variance method. We have investigated the use of different wavelets and calibrated their spatial resolution. Unfortunately, it turns out that it is not possible to define a single optimum wavelet for all purposes because different wavelets exhibit a different power in the detection of the characteristic structures. A good compromise is given by the Mexican-hat filter with a diameter ratio of 1.5. When the main focus lies on the measurement of the spectral index, the French-hat filter with a diameter ratio of about 2.3 is also suitable. We also introduced a significance function to weight the different data points. This allows us to analyse observed data where the signal-to-noise ratio is not uniform across the mapped area, but spatially varying. This should permit to distinguish the influence of variable noise from actual small-scale structure in the maps. The need for such a treatment became very obvious when Ossenkopf & Mac Low (2002) used the $\Delta$-variance analysis to characterise the velocity structure detected in molecular line observations of the Polaris Flare taken by Falgarone et al. (1998), Bensch et al. (2001), and Heithausen & Thaddeus (1990). Comparing the $\Delta$-variance analysis with the size-linewidth relation, we found that the $\Delta$-variance of the centroid velocity maps produced wrong results on scales where the maps do not show any noticeable emission. With the introduction of the significance function, the new $\Delta$-variance analysis should be able to also reliably analyse such data sets. Moreover, the use a weighting function also allows us to use computational methods like the fast Fourier transform to obtain the $\Delta$-variance spectrum even for maps with irregular boundaries or maps which are only sparsely filled by significant values. In paper I we tested the properties of the improved $\Delta$-variance analysis using simple artificial data sets. In this paper we will apply it to more realistic data sets, either simulations of interstellar turbulence or to actual observational data. In Sect. 2 we recapitulate the formalism of the new $\Delta$-variance analysis and summarise the results that we obtained from the application to the test data sets. In Sect. 3 we apply the analysis to a hydrodynamic simulation, to the $\rho$-Oph dust continuum map by Motte et al. (1998) and to the centroid velocity map for the Polaris Flare. We discuss the conclusions on their structure in Sect. 4. 2 The improved $\Delta$-variance method In this section we summarise the main properties of the improved $\Delta$-variance analysis proposed in paper I, focusing on the new points, not covered in the original $\Delta$-variance definition by Stutzki et al. (1998). The $\Delta$-variance measures the amount of structure on a given scale $l$ in a map $f(\@vec{r})$ by filtering the map with a spherically symmetric wavelet of size $l$ and computing the variance of the thus filtered map: $$\sigma_{\Delta}^{2}(l)=\left\langle\left(f(\@vec{r})*{\bigodot}_{l}(\@vec{r})% \right)^{2}\right\rangle_{\@vec{r}}$$ (1) where, the average is taken over the area of the map, the symbol $*$ stands for a convolution, and $\bigodot_{l}$ describes the filter function composed of positive inner “core” and a negative annulus, both normalised to integral values of unity $${\bigodot}_{l}(\@vec{r})=\bigodot_{l,{\rm core}}(\@vec{r})-\bigodot_{l,{\rm ann% }}(\@vec{r})\;.$$ (2) We have studied the “French-hat” filter with constant values in both parts: $$\displaystyle\bigodot_{l,{\rm core}}(\@vec{r})$$ $$\displaystyle=$$ $$\displaystyle{4\over\pi l^{2}}\left\{\begin{array}[]{ll}1&:|\@vec{r}|\leq l/2% \\ 0&:|\@vec{r}|>l/2\end{array}\right.$$ $$\displaystyle\bigodot_{l,{\rm ann}}(\@vec{r})$$ $$\displaystyle=$$ $$\displaystyle{4\over\pi l^{2}}\left\{\begin{array}[]{ll}{1/(v^{2}-1)}&:l/2<|% \@vec{r}|\leq v\times l/2\\ 0&:|\@vec{r}|\leq l/2,|\@vec{r}|>v\times l/2\end{array}\right.$$ (3) and a “Mexican hat” consisting of two Gaussian functions: $$\displaystyle\bigodot_{l,{\rm core}}(\@vec{r})$$ $$\displaystyle=$$ $$\displaystyle{4\over\pi l^{2}}\exp\left(\@vec{r}^{2}\over(l/2)^{2}\right)$$ (4) $$\displaystyle\bigodot_{l,{\rm ann}}(\@vec{r})$$ $$\displaystyle=$$ $$\displaystyle{4\over\pi l^{2}(v^{2}-1)}\left[\exp\left(\@vec{r}^{2}\over(vl/2)% ^{2}\right)-\exp\left(\@vec{r}^{2}\over(l/2)^{2}\right)\right]$$ where $l$ is the core diameter and $v$ is the diameter ratio between the annulus and the core of the filter. Plotting the $\Delta$-variance as a function of the filter size $l$ then provides a spectrum showing the relative amount of structure in a given map as a function of the structure size. The effective filter size, given by the average distance of points in the core and the annulus, deviates from the core diameter $l$ as $${l_{\rm eff}\over l}=\left\{\displaystyle 0.29v+0.26\quad{\rm for\;the\;French% \;hat}\hfill\atop\displaystyle 0.41v+0.46\quad{\rm for\;the\;Mexican\;hat.}% \hfill\right.$$ (5) Thus structures with a particular size should show up as prominent peaks in the $\Delta$-variance spectrum on a scale $l_{\rm eff}$ corresponding to that size. Test with artificial data sets in paper I have shown, however, that the peak positions always falls 10-20 % below the maximum structure size. Taking this systematic offset into account we can nevertheless reliably calibrate the spatial resolution of the $\Delta$-variance analysis. A major improvement of the new $\Delta$-variance algorithm was the introduction of a weighting function to the data $w_{\rm data}(\@vec{r})$. This simultaneously solved the problems of the edge treatment of finite maps and the analysis of data with a variable uncertainty across the map. The weight function varies between 0 and 1, representing the reliability of the individual data points, and it extends beyond the original map size, padding it with zeros at the boundaries. Instead of the original map $f(\@vec{r})$, an extended map, $f_{\rm padded}(\@vec{r})=f(\@vec{r})\times w_{\rm data}(\@vec{r})$ inside the original data area, $f_{\rm padded}(\@vec{r})=0$ outside, is analysed. This padded map can be periodically continued without wrap-around effects, so that the filter convolution can be efficiently computed in Fourier space involving a fast Fourier transform and a map multiplication. To avoid that data points within the padded area or with a low weighting are counted like normal zero-value data, but disregarded in the computation of the variance, the filter has to be re-normalised at each position in the map in such a way that the integral weights of core and annulus remain unity when excluding the padded points and when taking the weighting of the normal data points into account. Instead of one convolution (Eq. 1), one has to compute four convolutions $$\displaystyle G_{l,{\rm core}}(\@vec{r})$$ $$\displaystyle=$$ $$\displaystyle f_{\rm padded}(\@vec{r})*\bigodot_{l,{\rm core}}(\@vec{r^{\prime% }})$$ $$\displaystyle G_{l,{\rm ann}}(\@vec{r})$$ $$\displaystyle=$$ $$\displaystyle f_{\rm padded}(\@vec{r})*\bigodot_{l,{\rm ann}}(\@vec{r^{\prime}})$$ $$\displaystyle W_{l,{\rm core}}(\@vec{r})$$ $$\displaystyle=$$ $$\displaystyle w(\@vec{r})*\bigodot_{l,{\rm core}}(\@vec{r^{\prime}})$$ $$\displaystyle W_{l,{\rm ann}}(\@vec{r})$$ $$\displaystyle=$$ $$\displaystyle w(\@vec{r})*\bigodot_{l,{\rm ann}}(\@vec{r^{\prime}})$$ (6) and combine the results while re-normalising with the effective filter weight for the valid data $$F_{\rm l}(\@vec{r})={G_{\rm{\it l},core}(\@vec{r})\over W_{\rm{\it l},core}(% \@vec{r})}-{G_{\rm{\it l},ann}(\@vec{r})\over W_{\rm{\it l},ann}(\@vec{r})}\;.$$ (7) From the actual filter weight computed for each point in the map we can derive a significance function as the product of both normalisation factors $$W_{\rm{\it l},tot}(\@vec{r})=W_{\rm{\it l},core}(\@vec{r})W_{\rm{\it l},ann}(% \@vec{r})\;.$$ (8) This provides the actual significance of the data points in the convolved map which is used when computing the $\Delta$-variance of the whole map $$\sigma_{\Delta}^{2}(l)={\sum_{\rm map}(F_{\rm l}(\@vec{r})-\langle F_{\rm l}% \rangle)^{2}W_{\rm{\it l},tot}(\@vec{r})\over\sum_{\rm map}W_{\rm{\it l},tot}(% \@vec{r})}\;.$$ (9) With this generalised concept, the $\Delta$-variance analysis can be applied to arbitrary data sets. They must be projected onto some regular grid but they do not need to contain regular boundaries as the corresponding “empty” grid points can be marked with a zero significance. Varying noise or other changes in the data reliability can be expressed in the significance function $w_{\rm data}(\@vec{r})$. This applies e.g. to maps where not all points are observed with the same integration time so that they show a different noise level. In paper I we used a weighting function given by the inverse noise RMS and in Sect. 3.2 we will study the impact of the selection of the weighting function for observed data. The only remaining requirement for the applicability of the $\Delta$-variance analysis is a sufficiently large spatial dynamic range in the data. Bensch et al. (2001) had shown that a map has to contain at least 30 pixels in each direction to obtain reasonable error bars of the $\Delta$-variance spectrum. Numerical tests with noisy data in paper I showed that this critical size needs to be extended by a factor of about one over the average data significance in the case of data with a variable reliability. 3 Applications 3.1 Hydrodynamic simulations In the papers by Mac Low & Ossenkopf (2000), Ossenkopf & Mac Low (2002), and Ossenkopf et al. (2001) we have demonstrated the general applicability of the $\Delta$-variance analysis to extract characteristic structure sizes and scaling laws from (magneto-)hydrodynamic simulations performed with a variety of codes. Here, we need to test whether the $\Delta$-variance with adapted filter functions improves the sensitivity of this method. The weighting function is irrelevant in this case because the data do not suffer from noise or another cause of variable reliability across the data set. We have applied the analysis to a variety of simulations presented in Ossenkopf et al. (2001), but we present the results here only for a single model, the first inertial stage of the small-scale driven hydrodynamic turbulence computed by smooth-particle hydrodynamics (SPH), S02 at $t=0$. In this simulation the velocity field is driven by a Gaussian field of random fluctuations within a finite wavenumber range, $k=7\dots 8$. This means, that the driving process introduces characteristic variations into the velocity structure with the same scale length as used in the artificial sine wave field used in paper I, but with wavenumbers between 7 and 8. Thus the $\Delta$-variance spectrum should measure a peak variation for scales of $1/(\sqrt{2}k)$, i.e. between 0.088 and 0.101 of the size of the whole data cube. Selecting a model which is driven on small scales guarantees that we can identify a clear peak for these structures leaving enough dynamic range on smaller and larger scales. We select the initial stage of fully evolved turbulence in the simulation to make sure that the turbulent driving is the only process creating structures in the data set, avoiding effects of self-gravity. In this way we have a data set which is best suited to test the structure recognition by the $\Delta$-variance analysis, as it should directly detect the scales of the driving process in the velocity structure and determine the scaling of the turbulent cascade on lower scales. Results for the other simulations did not provide any fundamentally different results, but are less clear to interpret because either the driving scale is closer to the edge of the dynamic range or the contained structure is less well known. Figure 1 shows the $\Delta$-variance spectra of the $z$-projection of the $z$ component of the velocity field of the turbulence simulation computed with four different filter functions. We find a clear shift between the peak positions measured by the different filters. The Mexican-hat filter gives systematically larger lags for the peak than the French hat and the lag of the peak grows with growing annulus-to-core diameter ratio, $v$. The minimum peak lag, given by the French hat with $v=1.5$, falls at 0.084, the maximum lag, given by the Mexican hat with $v=3$, at 0.11. This behaviour is consistent with the results obtained for the simple sine wave field in paper I. It can be understood as a result of the variable width and the shape of the filters in Fourier space. The broadening of the filter function reduces the contrast which leads in a spectrum with a steep slope on small scales and a shallow slope on large scales to an effective shift of the peak position. The steep decay of the French-hat filter function for large lags leads to a somewhat lower peak position, but is always accompanied by side lobes of the Bessel function visible as artificial secondary peaks at large lags. For the turbulence simulations these secondary peaks are not as pronounced as for the sine wave field, but they are also visible in Fig. 1. For the diameter ratios $v$ of about 1.5 for the Mexican hat and 2.3 for the French hat, deduced as optimum values in paper I, the peak position falls at about 0.088 in both cases, a value slightly lower than the expected average structure size. The different width of the peak has a direct impact on the slope of the turbulent structures measured at small lags. With the broad peaks produced by the Mexican-hat filter, the slope is affected down to relatively small scales. As the filter diameter ratio $v$ constrains the minimum scale which can be resolved, a clear power law becomes only visible for the French-hat filter with $v=1.5$. Small diameter ratios are always favourable with respect to the dynamic range which can be covered in the $\Delta$-variance analysis because of the minimum filter size and the constraint that the overall filter must always remain small compared to the analysed map. For the French hat with $v=3.0$ and the Mexican hat with $v=1.5$ the remaining dynamic range is just marginally sufficient to reliably determine the spectral index. We find slopes between 2.8 and 3.1 corresponding to power spectral indices $\zeta=4.8\dots 5.1$. This is much higher than the Kolmogorov index of $\zeta=3.67$ indicating that the scaling on small scales is not determined by a self-similar turbulent cascade, but by the numerical viscosity in these simulations damping small scale structures. For simulations driven on large scales resulting in a larger dynamic range below the peak, we find a limited inertial range with a slope of about two, corresponding to $\zeta\approx 4$ which is consistent with a cascade of Burger’s turbulence (Ossenkopf & Mac Low, 2002; Ossenkopf et al., 2001, see). In a second step we investigate how the turbulent velocity scaling translates into the creation of turbulent density enhancements. Figure 2 shows the $\Delta$-variance spectra of the density structure seen in the same step of the simulation using the same filters as applied to the velocity structure. We find the same systematic deviations between the results seen by different filters but a generic shift of the peak position to shorter lags by a factor 0.75-0.8 with respect to the peaks of the velocity structure. The slope of the turbulent density structure on small scales is shallower by $1.0\pm 0.05$. For the mutual comparison between density and velocity structure, the selection of the filter is thus irrelevant as long as the same filter is applied in both cases. We see that injesting energy on a particular scale does not create density enhancements on that scale, but rather on a scale smaller by a factor 0.75-0.8. This shift has not been noticed before by Mac Low & Ossenkopf (2000) as only the systematic tests of the filter functions provided enough sensitivity with respect to a reliable scale detection. It seems that the turbulent cascade builds up density fluctuations on all scales below the driving scale, but that those density enhancements act themselves as points of an efficient energy conversion between the scales creating new density structures so that dominant density scale falls somewhat below the initial scale. Finally we need to address the significance of the measured structure size and the scaling indices with respect to the random fluctuations always present in turbulence simulations and with respect to the relation between the three-dimensional (3-D) structure and the two-dimensional (2-D) projections which we can measure in astronomical observations. For turbulent density structures this comparison has been done by Mac Low & Ossenkopf (2000) using the “traditional´´ $\Delta$-variance filter with the diameter ratio $v=3.0$. We will repeat it here for the velocity structure as the direct carrier of turbulent energy using one of more sensitive filter functions. Figure 3 shows the $\Delta$-variance spectra of the $z$-component of the hydrodynamic simulation introduced above, computed with a French-hat filter with $v=2.3$. The three broken lines show the $\Delta$-variance spectra computed for the three different orthogonal projections of the velocity data cube. The solid curve shows the spectrum computed for the 3-D structure but rescale as if computed in 2-D by a factor proportional to the lag, to compensate for the different exponents of the $\Delta$-variance spectra depending on the dimensionality of the considered space, and shifted by a factor $\pi/4$, to account for the average reduction of a random structure size when projected from 3-D onto a plane. The solid line, thus represents our best knowledge on the velocity structure actually present in the simulations, while the broken lines represent possible observer’s views onto that structure. We find a considerable variation between the $\Delta$-variance spectra seen in the different directions, giving a feeling for the statistical uncertainty when measuring the scaling in turbulent simulations111The computational uncertainty given by the finite size of any data set was discussed in detail by Bensch et al. (2001).. Nevertheless, both the peak position and the exponent at small lags agree between all three curves. Compared to the full 3-D structure, there is, however, a systematic shift of the peak to somewhat larger lags and a slight reduction of the slope at small lags. For the 3-D structure the peak is seen at a lag of about 0.075, which is about 20 % smaller than the expected scale for the maximum variation, while the peak for the projections falls between 0.081 and 0.092, i.e. only 10 % below the expected scale. In the 2-D projections, the peak is always broader and the slope at small lags is always somewhat shallower, similar to the impact of broader filter functions. It is important to notice, that the corresponding plots for the turbulent density structure, showing a shallower scaling at small lags, exhibit a very good match between the $\Delta$-variance spectra computed in 2-D and in 3-D, i.e. neither a shift of the peak nor different slopes at small lags. This seems to indicate that the turbulent velocity cascade does not behave fully isotropic. A similar effect would be expected for grid-based hydrodynamic simulations where the dissipation is strong between neighbouring cells in the $x$-, $y$-, and $z$-directions. However, the simulations studied here used an SPH code which should not show any intrinsic anisotropy. The cause for the anisotropy of the turbulent velocity scaling is thus so far unknown. 3.2 Maps with variable noise Maps with variable noise are obtained e.g. in observations with detector arrays showing a pixel-to-pixel variation in the sensitivity. They are produced in single-pixel observations when a drift in the receiver sensitivity or the atmospheric conditions changes the noise in the data during the measurement and they result from mosaicing observations with variable integration times for different regions of the field. All these cases can be analysed in terms of the improved $\Delta$-variance as long as the spatial distribution of the noise across the map is well known so that a corresponding significance map can be defined which is used to weight the different points in the $\Delta$-variance analysis. Then the $\Delta$-variance spectrum is able to distinguish between small-scale noise contributions in regions with a high noise level and real small-scale structures in regions with a low noise level. As a challenging example of a data set where a variable noise is produced by the observation of different points of the map with varying integration time we use the 1.3 mm continuum map of $\rho$ Oph obtained by Motte et al. (1998). It is the result of a mosaicing observation where an efficient use of the array receiver is given by several observations of the source with different orientations of the array. The combination of these observations then results in a poorer coverage of the outer regions of the source compared to the central regions in terms of the total integration time spent on each point. If the source is covered in total with $N_{\rm tot}$ observations, we can characterise the integration time at each point by the number of coverages including this point $1\leq N\leq N_{\rm max}$, where $N_{\rm max}\leq N_{\rm tot}$. As the noise at each point is inversely proportional to the square root of the integration time, we can use the value of $\sqrt{N}$ as a measure for the data reliability across the map. In addition to the variable noise the map has highly irregular boundaries. Figure 4 shows the intensity map. In contrast to the original publication we show the intensity with equidistant contours on a linear scale because the linear-scale presentation gives a better feeling for the structure that is measurable by means of a statistical method like the $\Delta$-variance analysis. To emphasise the irregular noise behaviour we plot one contour at 20 mJy/15${}^{\prime\prime}$-beam, which is below the noise level in the outer parts of the map and above the noise level in the inner parts. Consequently, this contour shows partly real structure and partly artificial structure from the noise. Figure 5 contains the corresponding map of significance values defined as the square root of the number of integrations at each point, $\sqrt{N}$, thus measuring the inverse noise RMS. To demonstrate the influence of the significance weighting we show in Fig. 6 the $\Delta$-variance spectra computed for the $\rho$ Oph map using three different weighting functions. The lower spectrum is generated when the weights are ignored, i.e. simply set to unity at all valid data points. The upper graph is produced when the full weighting function from Fig. 5 is used. The intermediate curve follows when we introduce and upper limit to the weighting function motivated by the idea that above a certain significance limit a further reduction of the noise level does not improve the structure characterisation any more. All computations use the filter truncation outside of the area where data have been taken. The irregular boundaries make it impossible to construct a useful periodic continuation here. The $\Delta$-variance spectrum computed without weights seems to indicate a wide range of scales with a power-law behaviour from about 0.2${}^{\prime}$ to 2${}^{\prime}$ whereas the $\Delta$-variance spectrum computed with the full weights shows a steepening starting at 0.5-0.7${}^{\prime}$. Bensch et al. (2001) have shown, however, that such a behaviour is exactly to be expected from the finite beam of the observations. The data are given at a resolution of 15${}^{\prime\prime}$ and the corresponding beam smearing is known to steepen the $\Delta$-variance spectrum up to scales of about one arcminute. This steepening can be modelled theoretically using the analytic expressions for the beam convolution from Bensch et al.. In fact, the $\Delta$-variance spectrum computed from the $\rho$ Oph map with full weights can be fitted by a single power-law structure with $\alpha=0.68$ from $\la 0.2^{\prime}$ up to about 2${}^{\prime}$ and a convolution function of a 15${}^{\prime\prime}$ HPBW beam (see Fig. 7). The fitted exponent of 0.68 falls into the range measured in molecular line observations of molecular clouds covering exponents between 0.5 and 1.3 (Bensch et al., 2001; Elmegreen & Scalo, 2004; Falgarone et al., 2004). In contrast, the two lower curves cannot be fitted in the same way. These spectra would lead to the conclusion of a surplus of small-scale structure relative to a power-law scaling relation. Such a relative surplus of structure on small scales is hard to explain as it would require additional driving processes on these scales overcompensating the known dissipation of turbulence by ambipolar diffusion and molecular viscosity (Klessen et al., 2000). Gravitational collapse is not able to create these structures; it always affects the whole $\Delta$-variance spectrum, not only the small-scale tail (Ossenkopf et al., 2001). A surplus of small-scale structure is also in contrast to the analysis of Motte et al. (1998) who found a relative lack of small structures in terms of a flatter clump mass spectrum for small clumps. Thus we conclude from the scaling behaviour that the full weighting of intensity maps by their inverse noise RMS results in the most reliable $\Delta$-variance spectra. With this weighting the $\Delta$-variance analysis is able to distinguish insignificant small-scale structure, dominating the lowest contour in Fig. 4, from significant structures which are intuitively better presented by the contours chosen in the original plot by Motte et al.. The increase of the absolute value of the $\Delta$-variance at large lags when using the weighting function is explained by the relative increase of the contribution of the bright cores in the map when virtually reducing the map size by weighting the outer parts by lower significance values. To get a feeling for the reliability of the different points in the $\Delta$-variance spectrum we plot in Fig. 7 the $\Delta$-variance spectrum including the error bars. The error bars arise from the statistical uncertainty of the measurement of the average variance in a filtered map (Bensch et al. 2001). Due to the lower number of statistically independent points in maps convolved with a larger filter, the $\Delta$-variance is most uncertain at the largest lags. In spite of the large error bars, the general scaling behaviour can be accurately traced. The solid line shows the fit to the data using a power-law description of the structure scaling and the convolution by a 15${}^{\prime\prime}$ beam. In the $\Delta$-variance spectrum one can clearly see that the dominating structure has a scale of about 2.2${}^{\prime}$, i.e. 0.1 pc. This corresponds to the typical size of the cores identified by Motte et al.. The contribution of significant structures is continued up to 7-9${}^{\prime}$, i.e. 0.3-0.4 pc. This scale agrees with the size of the largest identified core. Above about 9${}^{\prime}$ the $\Delta$-variance spectrum decays with $\alpha=-2$ indicating a lack of further correlated structure on larger scales. It is not clear whether the subtraction of large-scale emission unavoidable in the used observing mode has removed some large-scale correlation which is present in the cloud but cannot be detected from the map. At small lags the spectrum indicates no separation between the scales where “cores” and “clumps” (condensations) were defined by Motte et al.. The fit in Fig. 7 shows that on scales from 0.2${}^{\prime}$ to 2${}^{\prime}$, i.e. within a dynamic range of a factor ten, the spectrum is described by a single power-law smoothed by the observational beam. The $\Delta$-variance spectrum suggests that the same processes drive the formation of the somewhat larger “cores” and the somewhat smaller condensations. The break in the spectrum at 0.4${}^{\prime}$, that seems to suggest a change in the scaling law of the observed structure, is only produced by the beam smearing and is quantitatively in agreement with a continuation of the power law observed on larger scales down to at least 0.2${}^{\prime}$. The underlying structure can be described by a perfect a power law in contradiction to the clump mass spectrum studied by Motte et al. who found a significant turn-down at a mass of about 0.5 $M_{\sun}$. This difference is even more intriguing because of the opposite situation in molecular line studies of the Polaris Flare where the clump mass spectrum shows a perfect power law (Heithausen et al., 1998) but the $\Delta$-variance spectrum shows a steepening towards small scales (Bensch et al. 2001, see also Ossenkopf et al. 2000). From the theoretical modelling of the translation of a clump spectrum into a corresponding power spectral index of an fBm by Stutzki et al. (1998), we would expect a fixed relation between the measured clump mass spectrum and the corresponding $\Delta$-variance spectrum in both cases. However, our examples violate this relation. The mass-size relation of the clumps may be affected by optical depth effects and a large part of the observationally identified clumps may result from the superposition of different structures along the line of sight, not well separated in velocity space (Ballesteros-Paredes & Mac Low, 2002; Ossenkopf, 2003). Further systematic studies are necessary to understand the actual physical processes interrelating the structure size spectra and the clump mass spectra. We can compare the new results with the outcome of previous $\Delta$-variance analyses, because all previous conclusions on the slopes of the $\Delta$-variance spectra remain valid. The new $\Delta$-variance method has improved our ability to precisely detect prominent scales, it has calibrated the absolute scales and it increased the statistical significance of the spectra by taking variable data reliability and edge effects into account, but none of these points should significantly affect the general scaling behaviour measured in our previous papers. The $\rho$ Oph map shows a behaviour which is intermediate between that observed e.g. by Bensch et al. (2001) in molecular molecular lines, where we find a power-law $\Delta$-variance spectrum on small scales and a dominance of large-scale structure, and the spectrum measured for the 1.3 mm continuum map of Serpens (Testi & Sargent, 1998) analysed by Ossenkopf et al. (2001), where small cores dominate the spectrum resulting in a steep decay on large scales. In the $\rho$-Oph map we find both effects in one spectrum. The dense cores represent the dominating size scale but we can clearly resolve the scaling of significant structure on smaller scales. Taking all facts from the scaling behaviour and the clump size detection together we conclude that an appropriate characterisation of the map in terms of a $\Delta$-variance spectrum is only obtained when weighting the intensity maps by the inverse noise RMS. Otherwise a variable noise in the spectra always tends to mimic small-scale structure which might be taken for real. 3.3 Velocity centroid maps The situation is more difficult in the analysis of maps representing other quantities than intensities. Then the weighting function given by the inverse noise RMS of the observation is not necessarily a good measure for the significance of the data. We study one such example here. Investigating the velocity structure in the Polaris Flare molecular cloud, Ossenkopf & Mac Low (2002) applied the ordinary $\Delta$-variance analysis to maps of centroid velocities in CO line data. The scaling behaviour of the velocity field measured in terms of the $\Delta$-variance spectrum was compared to model simulations of interstellar turbulence. The data sets were given by three nested CO maps taken with different telescopes at different resolutions. The maps taken at high resolutions with the IRAM 30 m telescope (Falgarone et al., 1998) and the KOSMA 3 m telescope (Bensch et al. 2001) only covered regions with sufficiently bright emission so that a reliable determination of the line centroid velocities was possible at all points. The centroid $\Delta$-variance spectra derived for these two maps showed a continuous power-law spectrum with a slight steepening towards the smallest lags. In contrast, the map on the largest scale taken with the CfA 1.2 m telescope (Heithausen & Thaddeus, 1990) contains many data points where no emission above the noise limit was detected. Moreover, it was difficult to obtain a reliable determination of the centroid velocities in regions where the line intensities only exceeded the noise RMS by a factor of a few (see Ossenkopf & Mac Low 2002). The resulting $\Delta$-variance spectrum did not show a continuation of the power-law behaviour from the two maps on smaller scales, but turned essentially flat. This is in contradiction to an eye inspection of the centroid map plotted in Fig. 8 showing a large-scale velocity gradient which should appear as well as large-scale structure in the $\Delta$-variance spectrum. The corresponding map of line integrated intensities, plotted in Fig. 9, shows that the map contains large regions without emission. When the ordinary $\Delta$-variance counts their centroid velocity with the same weight as that from points in the actual molecular cloud the “empty regions” statistically hide the variations in the regions with significant values. The virtual lack of large-scale velocity variations in the $\Delta$-variance spectrum is thus due to the missing significance weighting. The knowledge on the reliability of the centroid velocities coming from the corresponding line intensities has to be taken into account using the improved $\Delta$-variance analysis. Unfortunately, it is not obvious how the significance of the centroid velocities is related to the line intensities. We have tested weighting functions based on three assumptions: i) The significance of the centroid velocities is determined by the integrated line intensities at each point. ii) The zero value of the weighting function corresponds to a minimum intensity of 0.65 K km/s which means that at least 10 velocity channels show an intensity above the noise RMS. iii) Integrated intensities above some limit do not further increase the significance of the centroid velocities. The weighting function is unity at all points with higher intensities. Figure 10 shows the resulting $\Delta$-variance spectra of the centroid map when weighting functions with different upper intensity thresholds are applied. For an easy comparison to the previous results we also include the graph obtained without any weighting. The introduction of the weighting function results in a completely different $\Delta$-variance spectrum at large lags. Whereas the previous computation showed a flat spectrum we find now a strong increase of the spectrum above $3{}^{\circ}$. Unfortunately, the results only give rough guidelines for the selection of the optimum weighting function. The best continuation of the $\Delta$-variance spectra from the smaller scales provided by the KOSMA map is obtained when the upper intensity limit corresponds to four times the lower limit. The two curves only using a narrow dynamic intensity range for the weighting function are still heavily influenced by observational noise visible as increased $\Delta$-variance values at small lags. On the other hand it is not clear whether the complete suppression of the noise effect in the two curves with the highest upper limits is realistic, so that we conclude that in this example the significance of the centroid velocities does not further increase for integrated intensities above 2.6 K km/s. Nevertheless, a final answer to this question still has to come from a theoretical model for the quantitative impact of the observational noise on the noise in centroid velocities. In spite of the uncertainty of the noise contribution at small lags, we can draw new essential conclusions on the velocity structure of the Polaris Flare from the $\Delta$-variance spectrum. The power-law scaling behaviour of the velocity structure detected previously only for smaller sizes is now continued up to scales of 1.3${}^{\circ}$, i.e. 3-4 pc. A single power law with an exponent $\alpha\approx 0.9$ can be used to cover the length scale range of about a factor 100. The $\Delta$-variance spectrum also shows a plateau around 2${}^{\circ}$, i.e. 5 pc, indicating the relative deficiency of motions on that scale. Above $3{}^{\circ}$ the $\Delta$-variance spectrum rises again tracing the global velocity gradient visible in Fig. 8. This behaviour indicates that the large-scale gradient is not converted directly by shear motions into the turbulent cascade but that the turbulent cascade starts at somewhat smaller scales. This points towards a shock producing the large-scale gradient which is only converted into turbulent energy on the scale of previously existing density fluctuations in the cloud and excludes Galactic rotation as the main driving force. 4 Conclusions In paper I we have proposed two essential improvements of the $\Delta$-variance analysis. Here, we have tested their actual impact when applied to data sets characterising interstellar turbulence. The first improvement was the introduction of a weighting function for each pixel in the map. This allows us to study data sets with a variable data reliability across the map and to simultaneously solve boundary problems even for maps with irregular boundaries. Maps with a variable data reliability are eventually obtained in most observations, either due to a local or a temporal variability of the detector sensitivity or the atmosphere or due to different integration times spent for different points of a map. By applying the improved $\Delta$-variance analysis to observed data we find that only the use of a significance function to weight the different data points allows us to distinguish the influence of variable noise from actual small-scale structure in the maps. In the analysis of intensity maps the weighting function is best provided by the inverse RMS in the data points. The situation is more complex for derived quantities, like centroid velocities, without a simple analytic relation between the uncertainty of the quantity and the observational noise. Here, in general two thresholds can be defined – a lower threshold below which all data have to be ignored and an upper threshold above which the significance of the data is not further improved by decreasing the noise. For the centroid velocities this means that the integrated line intensities between the two boundaries may serve as weighting function. The second improvement of the $\Delta$-variance analysis is its optimisation with respect to the shape of the wavelet used to filter the observed maps. The application of different filters in the analysis of hydrodynamic simulations confirmed the result from paper I, that a Mexican-hat filter with a diameter ratio $v=1.5$ is well suited to resolve prominent structure scales and to measure the slope of the turbulent cascade, however, it turned out that the impact of the detailed shape of the $\Delta$-variance filter is less significant for realistic data than for the artificial test data used in paper I. The turbulent structure is well resolved for a wide set of filters as long as one consistent filter shape is used throughout the full analysis of a data set. Comparing the density and velocity structure of the simulations shows a small but significant shift between the scale of the most prominent velocity structures, created by the energy injection, and the most prominent density structures produced by the velocity field. Applying the new method to the example of the dust emission map of $\rho$ Oph by Motte et al. (1998) shows that the spatial scaling behaviour there can be described perfectly by a power law interconnecting the range of small clumps and more massive cores. The method can reproduce the size of the dominant cores and we find no indication for large-scale correlation between the clumps and cores in the data. The $\Delta$-variance spectrum shows no break in the scaling behaviour between cores and condensations in contrast to the mass spectrum derived by Motte et al. (1998). The reason for the different behaviour of the two measures has to be topic of a future investigation. In the example of the analysis of the velocity structure in the Polaris Flare we show that the power-law scaling behaviour established by Ossenkopf & Mac Low (2002) for the small scales is continued to large scales. However, a plateau in the $\Delta$-variance spectrum around 5 pc indicates that the existing large-scale velocity gradient is not converted directly into a turbulent cascade. A possible explanation for this behaviour is the existence of a shock producing the large-scale gradient which is only converted into turbulent energy on the scale of the individual density fluctuations in the cloud. This scenario would be consistent with the affiliation of the molecular cloud to a large H I supershell by Meyerdierks et al. (1991). Combining the results from the turbulence simulation with the analysis of the Polaris Flare velocity structure indicates that a large-scale velocity field does not automatically produce density structures on those scales, but that a full turbulence cascade covering density and velocity fluctuations evolves predominantly at seeds of primordially existing density fluctuations, which may have been produced by previous velocity fields on larger scales. When interpreting turbulent structures in interstellar clouds it has to be taken into account that close to the scale of the energy injection a statistical analysis of the turbulent cascade is always affected by low number statistics as few density “seeds” may dominate the shape of the scaling relations there. A reliable statistics is only given on smaller scales. Acknowledgements. We thank R. Klessen for providing us with the numerical data of the SPH simulations used in Sect. 3.1. We thank Ph. André for the data of the $\rho$ Oph continuum observations used in Sect. 3.2. We thank F. Bensch for useful discussions and J. Ballesteros-Paredes for carefully refereeing this paper suggesting significant improvements. This work has been supported by the Deutsche Forschungsgemeinschaft through grant 494B. It has made use of NASA’s Astrophysics Data System Abstract Service. References Ballesteros-Paredes & Mac Low (2002) Ballesteros-Paredes J., Mac Low, M.-M. 2002, ApJ 570, 734B Bensch et al. (2001) Bensch F., Stutzki J., & Ossenkopf V. 2001, A&A 366, 636 Décamp & Le Bourlot (2002) Décamp N., Le Bourlot J. 2002, A&A 389, 1055 Elmegreen & Scalo (2004) Elmegreen B.G., Scalo J. 2004, ARAA 42, 211 Falgarone et al. (1998) Falgarone E., Panis J.-F., Heithausen A. et al. 1998, A&A 331, 669 Falgarone et al. (2004) Falgarone E., Hily-Blant P., Levrier F. 2004, Ap&SS 292, 89 Franco & Carramiñana (1999) Franco J., Carramiñana A. 1999, Interstellar Turbulence, Cambridge Univ. 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E${}_{6}$ inspired SUSY models with Custodial Symmetry R. Nevzorov${}^{*}$ Alikhanov Institute for Theoretical and Experimental Physics, Moscow, 117218, Russia ${}^{*}$E-mail: nevzorov@itep.ru Abstract The breakdown of $E_{6}$ within the supersymmetric (SUSY) Grand Unified Theories (GUTs) can result in SUSY extensions of the standard model (SM) based on the SM gauge group together with extra $U(1)$ gauge symmetry under which right–handed neutrinos have zero charge. In these $U(1)_{N}$ extensions of the minimal supersymmetric standard model (MSSM) a single discrete $\tilde{Z}^{H}_{2}$ symmetry may be used to suppress the most dangerous operators, that give rise to proton decay as well as non–diagonal flavour transitions at low energies. The SUSY models under consideration involves $Z^{\prime}$ and extra exotic matter beyond the MSSM. We discuss leptogenesis within this SUSY model and argue that the extra exotic states may lead to the non–standard Higgs decays. keywords: Grand Unified Theories; Supersymmetry; Leptogenesis; Higgs boson. \bodymatter 1 Introduction Supersymmetric (SUSY) extensions of the standard model (SM) allows one to embed SM into Grand Unified Theories (GUTs) based on simple gauge groups such as $SU(5)$, $SO(10)$ or $E_{6}$. Indeed, it was found that the electroweak (EW) and strong gauge couplings extrapolated to high energies using the renormalisation group equation (RGE) evolution converge to a common value at some high energy scale in the framework of the minimal SUSY standard model (MSSM) [1, 2, 3, 4]. The incorporation of the SM gauge interactions within GUTs permits, in particular, to explain the peculiar assignment of $U(1)_{Y}$ charges postulated in the SM. It is well known that each family of quarks and leptons fills in complete 16 dimensional spinor representation of $SO(10)$ that also predicts the existence of right–handed neutrino, allowing it to be used for both the see–saw mechanism and leptogenesis. In $N=1$ SUSY GUT based on $E_{6}$ the complete fundamental $27$ representation, that decomposes under $SO(10)\times U(1)_{\psi}$ subgroup as $$27\to\left(16,\,\frac{1}{\sqrt{24}}\right)\oplus\left(10,\,-\frac{2}{\sqrt{24}% }\right)\oplus\left(1,\,\frac{4}{\sqrt{24}}\right)\,,$$ (1) contains Higgs doublet. It is assigned to $\left(10,\,-\frac{2}{\sqrt{24}}\right)$. The SM gauge bosons belong to the adjoint representation of $E_{6}$, i.e. $78$–plet. In $N=2$ SUSY GUT based on the $E_{8}$ gauge group all SM particles belong to $248$ dimensional representation of $E_{8}$ that decomposes under its $E_{6}$ subgroup as follows $$248\to 78\oplus\,3\times 27\,\oplus\,3\times\overline{27}\,\oplus\,8\times 1\,.$$ (2) The local version of SUSY (supergravity) results in a partial unification of the SM gauge interactions with gravity. However supergravity (SUGRA) is a non–renormalisable theory and has to be considered as an effective low energy limit of some renormalisable or even finite theory. Currently, the best candidate for such an underlying theory, i.e. hypothetical single framework that explains and links together all physical aspects of the universe, is ten–dimensional heterotic superstring theory based on $E_{8}\times E^{\prime}_{8}$ [5]. Compactification of extra dimensions leads to an effective supergravity and results in the breakdown of $E_{8}$ to $E_{6}$ or its subgroups in the observable sector. The remaining $E^{\prime}_{8}$ plays the role of a hidden sector which gives rise to spontaneous breakdown of SUGRA. 2 The $U(1)_{N}$ extensions of the MSSM In orbifold SUSY GUTs the $E_{6}$ gauge group can be broken down to $SU(3)_{C}\times SU(2)_{W}\times U(1)_{Y}\times U(1)_{\chi}\times U(1)_{\psi}$, where the $U(1)_{\psi}$ and $U(1)_{\chi}$ symmetries are defined by: $E_{6}\to SO(10)\times U(1)_{\psi}$, $SO(10)\to SU(5)\times U(1)_{\chi}$ [6]. In order to ensure anomaly cancellation in this case the particle content below the GUT scale $M_{X}$ should be extended to include three $27$–plets. Each $27$–plet, referred to as $27_{i}$ with $i=1,2,3$, includes one generation of ordinary matter, a SM singlet field $S_{i}$ (see last term in Eq. (1)), that carries non–zero $U(1)_{\psi}$ charge, as well as Higgs–like doublets ($H^{u}_{i}$ and $H^{d}_{i}$) and charged $\pm 1/3$ exotic quarks ($D_{i}$ and $\bar{D}_{i}$) which are associated with $\left(10,\,-\frac{2}{\sqrt{24}}\right)$ in Eq. (1). In addition the splitting of bulk 27–plets can give rise to a set of $M_{l}$ and $\overline{M}_{l}$ supermultiplets with opposite quantum numbers. The presence of exotic matter in the $E_{6}$ inspired SUSY models generically leads to rapid proton decay and non–diagonal flavour transitions at low energies. To suppress flavour changing processes as well as the most dangerous baryon and lepton number violating operators one can impose a single discrete $\tilde{Z}^{H}_{2}$ symmetry. All states from complete $27_{i}$–plets are odd whereas all supermultiplets $M_{l}$ are even under this $\tilde{Z}^{H}_{2}$ symmetry. Because $M_{l}$ can be used for the breakdown of gauge symmetry this set of supermultiplets should contain $H_{u}$, $H_{d}$, $S$ and $N^{c}_{H}$. Since superfield $N^{c}_{H}$ has the same $U(1)_{\psi}$ and $U(1)_{\chi}$ charges as the right–handed neutrino the large vacuum expectation values (VEVs) of $N^{c}_{H}$ and $\overline{N}_{H}^{c}$ break $U(1)_{\psi}\times U(1)_{\chi}$ down to $U(1)_{N}$ generating large Majorana masses for the right–handed neutrinos. Only in this $E_{6}$ inspired $U(1)$ extension of the MSSM, i.e. in the so–called Exceptional Supersymmetric Standard Model (E${}_{6}$SSM)[7, 8], the right–handed neutrinos may be superheavy, shedding light on the origin of the mass hierarchy in the lepton sector and providing a mechanism for the generation of lepton and baryon asymmetry of the universe [9]. Different phenomenological implications of the several variants of the E${}_{6}$SSM were considered in Refs. 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21. In particular, the renormalisation group (RG) flow of the gauge and Yukawa couplings as well as the theoretical upper bound on the lightest Higgs boson mass were examined in the vicinity of the quasi–fixed point [21] that appears as a result of the intersection of the invariant and quasi–fixed lines [22, 23]. Within the constrained version of the E${}_{6}$SSM and its modifications the particle spectrum, the corresponding collider signatures and the implications for dark matter were analysed in Refs. 24, 25, 26, 27, 28, 29. Here we assume that $U(1)_{\psi}\times U(1)_{\chi}$ symmetry is broken down to $U(1)_{N}\times Z_{2}^{M}$, where $Z_{2}^{M}=(-1)^{3(B-L)}$ is a matter parity. This can occur because $Z_{2}^{M}$ is a discrete subgroup of $U(1)_{\psi}$ and $U(1)_{\chi}$. In the simplest case the set of the $Z^{H}_{2}$–even supermultiplets $M_{l}$ should also include a lepton $SU(2)_{W}$ doublet $L_{4}$ to allow the lightest exotic quarks to decay [6]. The supermultiplets $\overline{M}_{l}$ can be either even or odd under the $\tilde{Z}^{H}_{2}$ symmetry. The simplest scenario imply that $\overline{S}$, $\overline{H}_{u}$ and $\overline{H}_{d}$ are odd whereas $\overline{L}_{4}$ is even under $\tilde{Z}^{H}_{2}$. It is expected that the $\tilde{Z}^{H}_{2}$-odd supermultiplets $\overline{S}$, $\overline{H}_{u}$ and $\overline{H}_{d}$ get combined with the superposition of the appropriate components from $27_{i}$ forming vectorlike states with masses of order of $M_{X}$. At the same time the supermultiplets $L_{4}$ and $\overline{L}_{4}$ should form TeV scale vectorlike states to render the lightest exotic quarks unstable. The most general renormalisable superpotential which is allowed by the $\tilde{Z}^{H}_{2}$, $Z_{2}^{M}$ and $SU(3)\times SU(2)_{W}\times U(1)_{Y}\times U(1)_{N}$ symmetries can be written as $$\begin{array}[]{c}W=\lambda S(H_{u}H_{d})+\lambda_{\alpha\beta}S(H^{d}_{\alpha% }H^{u}_{\beta})+\kappa_{ij}S(D_{i}\overline{D}_{j})+\tilde{f}_{\alpha\beta}S_{% \alpha}(H^{d}_{\beta}H_{u})\\ +f_{\alpha\beta}S_{\alpha}(H_{d}H^{u}_{\beta})+h^{E}_{i\alpha}e^{c}_{i}(H^{d}_% {\alpha}L_{4})+\mu_{L}L_{4}\overline{L}_{4}+W_{N}\\ +g^{D}_{ij}(Q_{i}L_{4})\overline{D}_{j}+W_{\rm MSSM}(\mu=0)\,,\end{array}$$ (3) where $\alpha,\beta=1,2$ and $i,j=1,2,3$ while $W_{\rm MSSM}(\mu=0)$ is the MSSM superpotential with the bilinear mass parameter $\mu$ set to zero and $$W_{N}=\frac{1}{2}M_{ij}N_{i}^{c}N_{j}^{c}+\tilde{h}_{ij}N_{i}^{c}(H_{u}L_{j})+% h_{i\alpha}N_{i}^{c}(H^{u}_{\alpha}L_{4})\,.$$ (4) In Eqs. (3) and (4) $e_{i}^{c}$ and $N^{c}_{i}$ are the right-handed charged leptons and neutrinos whereas $Q_{i}$ and $L_{j}$ are the left-handed quark and lepton doublets respectively. The second last term in Eq. (3) ensures that the lightest exotic quarks decay within a reasonable time when the couplings $g^{D}_{ij}$ are sufficiently large and the components of the supermultiplets $L_{4}$ and $\overline{L}_{4}$ have masses of the order of a few TeV. Since in this case extra matter beyond the MSSM fill in complete $SU(5)$ representations the gauge coupling unification in the SUSY model under consideration can be achieved for any phenomenologically acceptable value of $\alpha_{3}(M_{Z})$, consistent with its central measured low energy value [11, 6]. The $Z^{H}_{2}$–even supermultiplets $H_{u}$, $H_{d}$ and $S$ gain non–zero VEVs, i.e. $\langle H_{d}\rangle=v_{1}/\sqrt{2}$, $\langle H_{u}\rangle=v_{2}/\sqrt{2}$ and $\langle S\rangle=s/\sqrt{2}$, which are much smaller than the VEVs of $N^{c}_{H}$ and $\overline{N}_{H}^{c}$. In phenomenologically viable scenarios the SM singlet superfield $S$ has to acquire VEV which is much larger than $1\,\mbox{TeV}$ breaking $U(1)_{N}$ gauge symmetry and inducing sufficiently large masses of $Z^{\prime}$ boson and exotic fermion states. The neutral components of $H_{u}$ and $H_{d}$ develop VEVs, so that $v=\sqrt{v_{1}^{2}+v_{2}^{2}}\simeq 246\,\mbox{GeV}$. These VEVs trigger the breakdown of the $SU(2)_{W}\times U(1)_{Y}$ symmetry down to $U(1)_{em}$ associated with electromagnetism and give rise to the masses of ordinary quarks and leptons. In the framework of the E${}_{6}$SSM the Higgs sector was explored in Ref. 7. When CP-invariance is preserved the Higgs spectrum contains three CP-even, one CP-odd and two charged states. The SM singlet dominated CP-even state and the $Z^{\prime}$ gauge boson are almost degenerate. If $\lambda<g^{\prime}_{1}$, where $g^{\prime}_{1}$ is the $U(1)_{N}$ gauge coupling, the SM singlet dominated Higgs boson is the heaviest CP-even state. In this case the rest of the Higgs spectrum is basically indistinguishable from the one in the MSSM. When $\lambda\gtrsim g^{\prime}_{1}$ the Higgs spectrum has a very hierarchical structure, which is similar to the one in the NMSSM with the approximate PQ symmetry [30]. As a consequence the mass matrix of the CP–even Higgs sector can be diagonalised using the perturbation theory [31, 32, 33]. If $\lambda\gtrsim g^{\prime}_{1}$ the MSSM–like CP-even, CP-odd and charged states have almost the same masses and lie beyond the TeV range. For the analysis of the phenomenological implications of the SUSY models discussed above it is convenient to introduce the $Z_{2}^{E}$ symmetry, which can be defined such that $\tilde{Z}^{H}_{2}=Z_{2}^{M}\times Z_{2}^{E}$. The supermultiplets $S_{\alpha}$, $H^{u}_{\alpha}$, $H^{d}_{\alpha}$, $D_{i}$, $\bar{D}_{i}$, $L_{4}$ and $\overline{L}_{4}$ are odd under the $Z_{2}^{E}$ symmetry. The components of all other supermultiplets are $Z_{2}^{E}$ even. Because the Lagrangian is invariant under both $Z_{2}^{M}$ and $\tilde{Z}^{H}_{2}$ symmetries, the $Z_{2}^{E}$ symmetry is also conserved. This implies that in collider experiments the exotic particles, which are odd under the $Z_{2}^{E}$ symmetry, can only be created in pairs and the lightest exotic state has to be absolutely stable. Using the method proposed in Ref. 34 it was argued that the masses of the lightest exotic fermions, which are predominantly linear superpositions of the fermion components of the superfields $S_{\alpha}$, do not exceed $60-65\,\mbox{GeV}$ [13]. Thus these states tend to be the lightest exotic particles in the spectrum. Moreover the lightest exotic fermion is also the lightest SUSY particle (LSP). Although the couplings of the corresponding states to the SM gauge bosons and fermions are quite small the lightest exotic state could account for all or some of the observed cold dark matter density if it had a mass close to half the $Z$ mass. However in this case the SM–like Higgs boson would decay almost 100% of the time into the fermion components of $S_{\alpha}$. All other branching ratios would be strongly suppressed. Basically such scenario has been already ruled out by the LHC experiments. On the other hand if the lightest exotic fermions are substantially lighter than $M_{Z}$ the annihilation cross section for $\mbox{LSP}+\mbox{LSP}\to\mbox{SM particles}$ becomes too small leading to a relic density that is much larger than its measured value. The simplest phenomenologically viable scenarios imply that the fermion components of $S_{\alpha}$ are significantly lighter than $1\,\mbox{eV}$111The presence of very light neutral fermions in the particle spectrum might have interesting implications for neutrino physics [35].. In this scenario the lightest SUSY particles form hot dark matter in the Universe. When the masses of the fermion components of $S_{\alpha}$ are considerably smaller than $1\,\mbox{eV}$ these states give only a very minor contribution to the dark matter density. At the same time the invariance of the Lagrangian under the $Z_{2}^{M}$ symmetry ensures that the $R$-parity is also conserved and the lightest ordinary neutralino is stable. In this case the lightest ordinary neutralino may account for all or some of the observed cold dark matter density. The scenarios discussed above are realised if $\tilde{f}_{\alpha\beta}\sim f_{\alpha\beta}\lesssim 10^{-6}$. When the Yukawa couplings of the superfields $S_{\alpha}$ are very small the terms $\tilde{f}_{\alpha\beta}S_{\alpha}(H^{d}_{\beta}H_{u})$ and $f_{\alpha\beta}S_{\alpha}(H_{d}H^{u}_{\beta})$ in the superpotential (3) can be ignored. In this limit the low–energy effective Lagrangian possesses an approximate global $U(1)_{E}$ symmetry below the scale $M_{1}$ where $M_{1}$ is the mass of the lightest right–handed neutrinos. The $U(1)_{E}$ charges of the exotic matter fields are summarised in Table 2. Both $U(1)_{B-L}$ and $U(1)_{E}$ symmetries are explicitly broken because of the interactions of matter supermultiplets with $N_{i}^{c}$ in $W_{N}$. As a consequence the decays of the lightest right–handed neutrino/sneutrino induce simultaneously $U(1)_{E}$ and $U(1)_{B-L}$ asymmetries. These asymmetries would not be washed out in the limit $\tilde{f}_{\alpha\beta},f_{\alpha\beta}\to 0$. Moreover the sufficiently small values of the $U(1)_{E}$ violating Yukawa couplings, i.e $\tilde{f}_{\alpha\beta},f_{\alpha\beta}\lesssim 10^{-7}$, should not erase the induced $U(1)_{E}$ asymmetry [36, 37, 38]. The non-zero values of $\tilde{f}_{\alpha\beta}$ and $f_{\alpha\beta}$ break the $U(1)_{E}$ symmetry and the lightest exotic state that carries the $U(1)_{E}$ charge becomes unstable. It decays into the fermion components of $S_{\alpha}$ so that the generated $U(1)_{E}$ asymmetry gets converted into the hot dark matter density. 3 Generation of baryon asymmetry A potential drawback of supersymmetric thermal leptogenesis is the lower bound on $M_{1}$. Indeed, it was shown that the appropriate amount of the baryon asymmetry in the SM and MSSM can be induced only if $M_{1}$ is larger than $10^{9}\,\mbox{GeV}$ [39, 40]. In the framework of supergravity this lower bound on $M_{1}$ leads to the gravitino problem [41, 42]. After inflation the universe thermalizes with a reheat temperature $T_{R}$. If $T_{R}>M_{1}$, the right-handed neutrinos are produced by thermal scattering and thermal leptogenesis could take place. At the same time when $T_{R}\gtrsim 10^{9}\,\mbox{GeV}$ such a high reheating temperature results in an overproduction of gravitinos which tend to decay during or after Big Bang Nucleosynthesis (BBN) destroying the agreement between the predicted and observed light element abundances. It was argued that the gravitino density becomes low enough if $T_{R}\lesssim 10^{6-7}\,\mbox{GeV}$ [43, 44, 45]. In order to avoid the gravitino problem we fix $M_{1}\simeq 10^{6}\,\mbox{GeV}$. We also assume that two other right-handed neutrino states have masses $M_{2,3}\sim 10^{6-7}\,\mbox{GeV}$. For so low $M_{i}$ the absolute values of the Yukawa couplings $|\tilde{h}_{ij}|$ should be rather small to reproduce the left–handed neutrino mass scale $m_{\nu}\lesssim 0.1\,\mbox{eV}$, i.e. $|\tilde{h}_{ij}|^{2}\ll 10^{-8}$. So small Yukawa couplings can be ignored in the leading approximation. Then only the new channels of the decays of the lightest right–handed neutrino $N_{1}$ and its superpartner $\widetilde{N}_{1}$, i.e. $$N_{1}\to L_{4}+H^{u}_{\alpha},\quad N_{1}\to\widetilde{L}_{4}+\widetilde{H}^{u% }_{\alpha},\quad\widetilde{N}^{*}_{1}\to L_{4}+\widetilde{H}^{u}_{\alpha},% \quad\widetilde{N}_{1}\to\widetilde{L}_{4}+H^{u}_{\alpha},$$ (5) give rise to the generation of lepton asymmetry. This process is controlled by the CP (decay) asymmetries associated with the decays of $N_{1}$, i.e. $$\varepsilon^{\alpha}_{1,\,\ell_{4}}=\frac{\Gamma^{\alpha}_{N_{1}\ell_{4}}-% \Gamma^{\alpha}_{N_{1}\bar{\ell}_{4}}}{\sum_{\beta}\left(\Gamma^{\beta}_{N_{1}% \ell_{4}}+\Gamma^{\beta}_{N_{1}\bar{\ell}_{4}}\right)}\,,\qquad\varepsilon^{% \alpha}_{1,\,\widetilde{\ell}_{4}}=\frac{\Gamma^{\alpha}_{N_{1}\widetilde{\ell% }_{4}}-\Gamma^{\alpha}_{N_{1}\widetilde{\ell}^{*}_{4}}}{\sum_{\beta}\left(% \Gamma^{\beta}_{N_{1}\widetilde{\ell}_{4}}+\Gamma^{\beta}_{N_{1}\widetilde{% \ell}^{*}_{4}}\right)}\,,$$ (6) and $\widetilde{N}_{1}$, i.e. $$\varepsilon^{\alpha}_{\widetilde{1},\,\ell_{4}}=\frac{\Gamma^{\alpha}_{% \widetilde{N}_{1}^{*}\ell_{4}}-\Gamma^{\alpha}_{\widetilde{N}_{1}\bar{\ell}_{4% }}}{\sum_{\beta}\left(\Gamma^{\beta}_{\widetilde{N}_{1}^{*}\ell_{4}}+\Gamma^{% \beta}_{\widetilde{N}_{1}\bar{\ell}_{4}}\right)}\,,\qquad\varepsilon^{\alpha}_% {\widetilde{1},\,\widetilde{\ell}_{4}}=\frac{\Gamma^{\alpha}_{\widetilde{N}_{1% }\widetilde{\ell}_{4}}-\Gamma^{\alpha}_{\widetilde{N}_{1}^{*}\widetilde{\ell}^% {*}_{4}}}{\sum_{\beta}\left(\Gamma^{\beta}_{\widetilde{N}_{1}\widetilde{\ell}_% {4}}+\Gamma^{\beta}_{\widetilde{N}_{1}^{*}\widetilde{\ell}^{*}_{4}}\right)}\,.$$ (7) In Eqs. (6) and (7) the superscripts $\alpha$ and $\beta$ represent the components of the supermultiplets $H^{u}_{\alpha}$ and $H^{u}_{\beta}$ in the final state. At the tree level the partial decay widths associated with the new channels (5) are given by $$\Gamma^{\alpha}_{N_{1}\ell_{4}}+\Gamma^{\alpha}_{N_{1}\bar{\ell}_{4}}=\Gamma^{% \alpha}_{N_{1}\widetilde{\ell}_{4}}+\Gamma^{\alpha}_{N_{1}\widetilde{\ell}^{*}% _{4}}=\Gamma^{\alpha}_{\widetilde{N}_{1}^{*}\ell_{4}}=\Gamma^{\alpha}_{% \widetilde{N}_{1}\bar{\ell}_{4}}=\Gamma^{\alpha}_{\widetilde{N}_{1}\widetilde{% \ell}_{4}}=\Gamma^{\alpha}_{\widetilde{N}_{1}^{*}\widetilde{\ell}_{4}^{*}}=% \frac{|h_{1\alpha}|^{2}}{8\pi}M_{1}$$ (8) and all decay asymmetries (6) and (7) vanish. The non–zero values of the CP asymmetries arise after the inclusion of one–loop vertex and self–energy corrections to the decay amplitudes of $N_{1}$ and $\widetilde{N}_{1}$. In this context it is worth noting that the supermultiplets $H^{u}_{\alpha}$ can be redefined so that only one doublet $H^{u}_{1}$ interacts with $L_{4}$ and $N^{c}_{1}$. Therefore without loss of generality $h_{12}$ in $W_{N}$ may be set to zero. In this limit $\varepsilon^{2}_{1,\,\ell_{4}}=\varepsilon^{2}_{1,\,\widetilde{\ell}_{4}}=% \varepsilon^{2}_{\widetilde{1},\,\ell_{4}}=\varepsilon^{2}_{\widetilde{1},\,% \widetilde{\ell}_{4}}=0$. When SUSY breaking scale is negligibly small as compared with $M_{1}$, $h_{j1}=|h_{j1}|e^{i\varphi_{j1}}$ and $M_{j}$ are real the non–zero asymmetries are given by $$\varepsilon^{1}_{1,\,\ell_{4}}=\varepsilon^{1}_{1,\,\widetilde{\ell}_{4}}=% \varepsilon^{1}_{\widetilde{1},\,\ell_{4}}=\varepsilon^{1}_{\widetilde{1},\,% \widetilde{\ell}_{4}}=\frac{1}{8\pi}\Biggl{[}\sum_{j=2,3}|h_{j1}|^{2}f\left(% \frac{M^{2}_{j}}{M_{1}^{2}}\right)\sin 2\Delta\varphi_{j1}\Biggr{]}\,,$$ (9) where $\Delta\varphi_{j1}=\varphi_{j1}-\varphi_{11}$ and $$f(z)=f^{V}(z)+f^{S}(z)\,,\qquad f^{S}(z)=\dfrac{2\sqrt{z}}{1-z}\,,\qquad f^{V}% (z)=-\sqrt{z}\,\ln\left(\dfrac{1+z}{z}\right)\,.$$ (10) Because the Yukawa couplings of the superfields $N^{c}_{i}$ to the supermultiplets $H^{u}_{\alpha}$ and $L_{4}$ violate both $U(1)_{E}$ and $U(1)_{B-L}$ the decay channels of the lightest right–handed neutrino/sneutrino (5) induce simultaneously $U(1)_{B-L}$ and $U(1)_{E}$ asymmetries. These asymmetries are determined by the same set of the CP asymmetries (9). The evolution of the $U(1)_{B-L}$ and $U(1)_{E}$ asymmetries are described by the system of Boltzmann equations. The generated baryon asymmetry can be estimated as follows $$Y_{\Delta B}\sim 10^{-3}\varepsilon^{1}_{1,\,\ell_{4}}\eta\,,$$ (11) where $Y_{\Delta B}$ is the baryon asymmetry relative to the entropy density, i.e. $$Y_{\Delta B}=\dfrac{n_{B}-n_{\bar{B}}}{s}\biggl{|}_{0}=(8.75\pm 0.23)\times 10% ^{-11}\,.$$ (12) In Eq. (11) $\eta$ is an efficiency factor. It varies from 0 to 1. In the strong washout scenario $\eta$ is given by $$\eta\simeq H(T=M_{1})/\Gamma_{1}\,,$$ (13) where $H$ is the Hubble expansion rate $$H=1.66g_{*}^{1/2}\dfrac{T^{2}}{M_{Pl}}\,,$$ (14) $g_{*}=n_{b}+\dfrac{7}{8}\,n_{f}$ is the number of relativistic degrees of freedom and $$\Gamma_{1}=\Gamma^{1}_{N_{1}\ell_{4}}+\Gamma^{1}_{N_{1}\bar{\ell}_{4}}=\dfrac{% |h_{11}|^{2}}{8\pi}\,M_{1}\,.$$ (15) As follows from Eq. (9) the values of the CP asymmetries are determined by the CP–violating phases $\Delta\varphi_{j1}$ and the absolute values of the Yukawa couplings $|h_{21}|$ and $|h_{31}|$ but do not depend on $|h_{11}|$. To simplify our analysis we fix $|h_{31}|=0$ and $(M_{2}/M_{1})=10$. At the same time the efficiency factor $\eta$ is set by the lightest right–handed neutrino mass $M_{1}$ and $|h_{11}|$. We restrict our consideration here by the values of $|h_{11}|^{2}\gg|\tilde{h}_{ik}|^{2}$, i.e. $|h_{11}|^{2}\gtrsim 10^{-8}$. For $\Delta\varphi_{21}=\pi/4$ we find $$\log|\eta|\simeq-2\log|h_{11}|-10.2\,,\qquad\qquad\log|\omega|\simeq-2\log|h_{% 11}|+2\log|h_{21}|-12.1\,,$$ (16) where $\omega=\varepsilon^{1}_{1,\,\ell_{4}}\eta$. Eq. (16) indicates that $\eta$ varies from $10^{-2}$ to $10^{-4}$ when $|h_{11}|$ increases from $10^{-4}$ to $10^{-3}$. The dependence of $|\omega|$, that determines the generated baryon asymmetry (11), on $|h_{21}|$ and $|h_{11}|$ is explored in Fig. 1. This figure illustrates that for $\Delta\varphi_{21}=\pi/4$ and $|h_{21}|\sim 0.1$ the phenomenologically acceptable baryon density, corresponding to $\omega\sim 10^{-7}-10^{-6}$, can be obtained if $|h_{11}|$ varies between $10^{-4}$ and $10^{-3}$. If $\tilde{f}_{\alpha\beta},f_{\alpha\beta}\lesssim 10^{-7}$ the induced dark matter and baryon number densities should be of the same order of magnitude. 4 Exotic Higgs decays As it was mentioned before the lightest and second lightest exotic states ($\chi^{0}_{1}$ and $\chi^{0}_{2}$) are mostly linear superpositions of the fermion components of the superfields $S_{\alpha}$. In the simplest phenomenologically viable scenarios $\chi^{0}_{1}$ should have mass $m_{\chi_{1}}\ll 1\,\mbox{eV}$. At the same time $\chi^{0}_{2}$ can be considerably heavier if some of the Yukawa couplings $\tilde{f}_{\alpha\beta}$ and $f_{\alpha\beta}$ are much larger than $10^{-6}-10^{-5}$. Although $\chi^{0}_{1}$ and $\chi^{0}_{2}$ tend to be rather light their couplings to the $Z$–boson and other SM particles can be negligibly small because these states are predominantly the fermion components of the SM singlet superfields $S_{\alpha}$ [15]. As a result any possible signal, which $\chi^{0}_{1}$ and $\chi^{0}_{2}$ could give rise to at former and present collider experiments, would be extremely suppressed and such states could escape their experimental detection. The couplings of the lightest Higgs boson $h_{1}$ to $\chi^{0}_{1}$ and $\chi^{0}_{2}$ are determined by their masses [13]. Since $\chi^{0}_{1}$ is extremely light it does not affect Higgs phenomenology. The coupling of the SM–like Higgs state $h_{1}$ to the second lightest exotic particle $X^{h}_{22}\simeq|m_{\chi_{2}}|/v$ [13]. This coupling gives rise to the decays of $h_{1}$ into $\chi^{0}_{2}$ pairs with partial width given by $$\Gamma(h_{1}\to\chi^{0}_{2}\chi^{0}_{2})=\frac{(X^{h}_{22})^{2}m_{h_{1}}}{4\pi% }\left(1-4\frac{|m_{\chi_{2}}|^{2}}{m^{2}_{h_{1}}}\right)^{3/2}\,,$$ (17) where $m_{h_{1}}$ is the lightest Higgs boson mass. From Eq. (17) it follows that the partial decay width of the non–standard Higgs decays depend rather strongly on $m_{\chi_{2}}$. The branching ratio of $h_{1}\to\chi^{0}_{2}\chi^{0}_{2}$, can be substantial if the second lightest exotic fermion has a mass of order of the $b$–quark mass $m_{b}$. To avoid the suppression of the branching ratios for Higgs decays into SM particles we restrict our consideration to the GeV scale masses of the second lightest exotic particle. After being produced $\chi^{0}_{2}$ sequentially decay into $\chi^{0}_{1}$ and fermion–antifermion pair via virtual $Z$. Thus the exotic decays of the SM–like Higgs discussed above results in two fermion–antifermion pairs and missing energy in the final state. Nevertheless due to the small coupling of the lightest and second lightest exotic fermions to the $Z$–boson $\chi^{0}_{2}$ tends to live longer than $10^{-8}\,\mbox{sec}$. Therefore it typically decays outside the detectors and can not be observed at the LHC directly. 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A weak randomness notion for probability measures André Nies Department of Computer Science, The University of Auckland, Private Bag 92019, Auckland, New Zealand, andre@cs.auckland.ac.nz Frank Stephan Department of Mathematics and Department of Computer Science, National University of Singapore, 10 Lower Kent Ridge Road, Block S17, Singapore 119076, Republic of Singapore, fstephan@comp.nus.edu.sg (January 14, 2021) Abstract We study probability measures on Cantor space, thinking of them as statistical superpositions of bit sequences. We say that a measure $\mu$ on the space of infinite bit sequences is Martin-Löf absolutely continuous if the non-Martin-Löf random bit sequences form a $\mu$ null set. We analyse this notion as a weak randomness notion for measures. We begin with examples and a robustness property related to Solovay test. Then we study the growth of initial segment complexity for measures (defined as a $\mu$-average over the complexity of strings of the same length) and relate it to our weak randomness property. We introduce K-triviality for measures. We seek an appropriate effective version of the Shannon-McMillan-Breiman theorem where the trajectories are replaced by measures. keywords: algorithmic randomness, probability measure on Cantor space, Kolmogorov complexity, statistical superposition, quantum states \subjclass 03D32,68Q30 1 Introduction The theory of algorithmic randomness is usually developed for bit sequences. A central randomness notion based on algorithmic tests is the one due to Martin-Löf [10]. Let $\{0,1\}^{{\mathbb{N}}}$ denote the topological space of infinite bit sequences. A probability measure $\mu$ on $\{0,1\}^{{\mathbb{N}}}$ can be seen as a statistical superposition of bit sequences. The bit sequences $Z$ form an extreme case: the corresponding measure $\mu$ is the Dirac measure $\delta_{Z}$, i.e., $\mu$ is concentrated on $\{Z\}$. The opposite extreme is the uniform measure $\mathbf{\lambda}$ which independently gives each bit value the probability $1/2$. The uniform measure represents the maximum disorder as no bit sequence is preferred over any other. We introduce an algorithmic randomness notion for probability measures. The notion is weak enough so that $\mathbf{\lambda}$ qualifies as random. Thus, randomness in this weak sense is compatible with, from another point of view, being computable. Nothing changes for bit sequences: $\delta_{Z}$ is random iff $Z$ is Martin-Löf random. Recall that a measure $\mu$ on $\{0,1\}^{{\mathbb{N}}}$ is called absolutely continuous if each $\mathbf{\lambda}$-null set is a $\mu$-null set. Our notion is a weakening of absolute continuity, requiring the $\mathbf{\lambda}$-null set in the hypothesis to be effective in the sense of Martin-Löf. Given that there is a universal Martin-Löf test, and hence a largest effective null set, all we have to require is that $\mu(\mathcal{C})=0$ where $\mathcal{C}$ is the class of bit sequences that are not Martin-Löf  random. We provide a brief overview of the paper. Some background and formal definitions will be given in Section 2. In Section 3 we extend the well-known equivalence of Martin-Löf test and Solovay tests to measures. In Section 4 we show that our weak notion of randomness is implied by a notion of Martin-Löf randomness for measures that is obtained when viewing the measures as points in a canonical computable probability space, in the sense of [7]. That stronger randomness notion forces the measure to be atomless. Randomness of infinite bit sequences is linked to the descriptive complexity of their initial segments via the Levin-Schnorr theorem, which intuitively says that randomness of $Z$ means incompressibility, up to the same constant $b$, of all the initial segments $x$. Formally, one requires that $K(x)\geq|x|-b$ for each initial segment $x$ of $Z$, where $K(x)$ is the prefix free version of Kolmogorov complexity of a string $x$. The “$n$-th initial segment” of a measure $\mu$ is given by its values $\mu[x]$ for all strings $x$ of length $n$, where $[x]$ denotes the set of infinite sequences extending $x$. It is natural to define the initial segment complexity $K(\mu\!\upharpoonright_{n})$ of this initial segment as the $\mu$-average of the individual complexities of those strings. With this definition, in Section 5 we show that both implications of the analog of the Levin-Schnorr theorem fail. However, we also show in Proposition 7.4 that for measures that are random in our weak sense, $K(\mu\!\upharpoonright_{n})/n$ converges to $1$. Thus, such measures have effective dimension 1; see, for example, Downey and Hirschfeldt [4, Section 12.3]. Opposite to random bit sequences are the $K$-trivial sequences, where the initial segment complexity grows no faster than that of a computable set; for background see e.g. Nies [14, Section 5.3]. In Section 6 we extend this notion to statistical superpositions of bit sequences: we introduce $K$-trivial measures and show that they all have countable support. The Shannon-McMillan-Breiman Theorem from the 1950s (see [19], where it is called the Entropy Theorem) says informally that for an ergodic measure $\rho$ on $\{0,1\}^{{\mathbb{N}}}$, outside a null set every bit sequence $Z$ reflects the entropy of the measure by the limiting weighted information content on its sufficiently large initial segments. In the final Section 7 we study what happens when $Z$ is replaced by a measure $\mu$ that is Martin-Löf a.c. with respect to $\rho$ and we take the $\mu$-average of the information contents at the same length. We note that our research was partly motivated by a recent attempt to define Martin-Löf randomness for quantum states corresponding to infinitely many qubits [15]. Using the terminology of [15], probability measures correspond to the quantum states $\rho$ where the matrix $\rho\!\upharpoonright_{M_{n}}$ is diagonal for each $n$, where $M_{n}$ is the algebra of $2^{m}\times 2^{m}$ complex matrices. For general background on recursion theory and algorithmic randomness we refer the readers to the textbooks of Calude [1], Downey and Hirschfeldt [4], Li and Vitányi [9], Nies [14], Odifreddi [16, 17], and Soare [20]. There are also lecture notes on recursion theory available online [21]. 2 Measures and Randomness We begin with some background on randomness for bit sequences. We use standard notation: letters $Z,X,\ldots$ denote elements of the space of infinite bit sequences $\{0,1\}^{{\mathbb{N}}}$, $\sigma,\tau$ denote finite bit strings, and $[\sigma]=\{Z\colon\,Z\succ\sigma\}$ is the set of infinite bit strings extending $\sigma$. $Z\!\upharpoonright_{n}$ denotes the string consisting of the first $n$ bits of $Z$. For quantities $r,s$ depending on the same parameters, we write $r\leq^{+}s$ for $r\leq s+O(1)$. A subset $G$ of $\{0,1\}^{{\mathbb{N}}}$ is called effectively open if $G=\bigcup_{i}[\sigma_{i}]$ for a computable sequences ${\left\langle{\sigma_{i}}\right\rangle}_{i\in{\mathbb{N}}}$ of strings. Definition 2.1. A Martin-Löf test (ML-test, for short) is a sequence ${\left\langle{G_{m}}\right\rangle}$ of uniformly effectively open sets such that $\mathbf{\lambda}G_{m}\leq 2^{-m}$ for each $m$. A bit sequence $Z$ fails the test if $Z\in\bigcap_{m}G_{m}$, otherwise it passes the test. $Z$ is Martin-Löf random (ML-random) if $Z$ passes each ML-test. Let $K(x)$ denote the prefix free version of descriptive (i.e., Kolmogorov) complexity of a bit string $x$. Theorem 2.2 (Levin [8], Schnorr [18]). $Z$ is ML-random $\Leftrightarrow$ $\exists b\forall nK(Z\!\upharpoonright_{n})\geq n-b$. Using the notation of [14, Ch. 3], let $\mathcal{R}_{b}$ denote the set of bit sequences $Z$ such that $K(Z\!\upharpoonright_{n})<n-b$ for some $n$. It is easy to see that ${\left\langle{\mathcal{R}_{b}}\right\rangle}_{b\in{\mathbb{N}}}$ forms a Martin-Löf test. The Levin-Schnorr theorem says that this test is universal: $Z$ is ML-random iff it passes the test. Unless otherwise stated, all measures will be probability measures. We use the letters $\mu,\nu,\rho$ for probability measures; $\mathbf{\lambda}$ denotes the uniform measure. So $\mathbf{\lambda}[\sigma]=2^{-|\sigma|}$. We now provide the formal definition of our weak randomness notion for measures. Definition 2.3. A measure $\mu$ is called Martin-Löf absolutely continuous (Martin-Löf  a.c., for short) if $\inf_{m}\mu(G_{m})=0$ for each Martin-Löf test ${\left\langle{G_{m}}\right\rangle}_{m\in{\mathbb{N}}}$. If $\inf_{m}\mu(G_{m})=0$ we say that $\mu$ passes the test. If $\inf_{m}\mu(G_{m})\geq\delta$ where $\delta>0$ we say $\mu$ fails the test at level $\delta$. In the definition it suffices to consider Martin-Löftests ${\left\langle{G_{m}}\right\rangle}$ such that $G_{m}\supseteq G_{m+1}$ for each $m$, because we can replace ${\left\langle{G_{m}}\right\rangle}$ by the Martin-Löf test $\hat{G}_{m}=\bigcup_{k>m}G_{k}$, and of course $\inf_{m}\mu(\hat{G}_{m})=0$ implies $\inf_{m}\mu(G_{m})=0$. So we can change the definition above, replacing the condition $\inf_{m}G_{m}=0$ by the only apparently stronger condition $\lim_{m}G_{m}=0$. The intersection of a universal ML-test consists of the non-Martin-Löf  random sequences. Since such a test exists, we have: Fact 2.4. A measure $\mu$ is Martin-Löf a.c.  iff the sequences which are not Martin-Löf random form a $\mu$-null set. We already mentioned the diametrically opposite examples, which give an impression of the range of the notion. Example 2.5. The uniform measure $\mathbf{\lambda}$ is a Martin-Löf a.c. measure. A Dirac measure $\delta_{Z}$ is Martin-Löf a.c.  iff $Z$ is Martin-Löf  random. Each ML-random sequence $Z$ satisfies the law of large numbers $\lim_{n}|\{i<n\colon Z(i)=1\}|=1/2$; see e.g. [14, Prop. 3.2.19]. So if $\mu$ is Martin-Löf a.c., then $\mu$ almost surely, $Z$ satisfies the law of large numbers. Thus, for $p\neq 1/2$, a Bernoulli measure on $\{0,1\}^{{\mathbb{N}}}$, that independently gives probability $p$ to a $0$ in each position, is not a Martin-Löf a.c. measure. For a measure $\nu$ and string $\sigma$ with $\nu[\sigma]>0$ let $\nu_{\sigma}$ be the localisation: $\nu_{\sigma}(A)=2^{-|\sigma|}\nu(A\cap[\sigma])$. Clearly if $\nu$ is Martin-Löf a.c.  then so is $\nu_{\sigma}$. A set $S$ of probability measures is called convex if $\mu_{i}\in S$ for $i\leq k$ implies that the convex combination $\mu=\sum_{i}\alpha_{i}\mu_{i}$ is in $S$, where the $\alpha_{i}$ are reals in $[0,1]$ and $\sum_{i}\alpha_{i}=1$. The extreme points of $S$ are the ones that can only be written as convex combinations of length 1 of elements of $S$. Proposition 2.6. The Martin-Löf a.c. probability measures form a convex set. Its extreme points are the Martin-Löf a.c. Dirac measures, i.e. the measures $\delta_{Z}$ where $Z$ is a ML-random bit sequence. Proof 2.7. Convexity is trivial: let $\mu=\sum_{i}\alpha_{i}\mu_{i}$ as above where the $\mu_{i}$ are Martin-Löf a.c. measures. Suppose ${\left\langle{G_{m}}\right\rangle}$ is a Martin-Löf test. Then $\lim_{m}\mu_{i}(G_{m})=0$ for each $i$, and hence $\lim_{m}\mu(G_{m})=0$. If $\mu$ is a Dirac measure then it is an extreme point of the Martin-Löf a.c. measures. Conversely, if $\mu$ is not Dirac there is a least number $t$ such that the decomposition $\mu=\sum_{|\sigma|=t,\mu[\sigma]>0}\mu[\sigma]\cdot\mu_{\sigma}$ is nontrivial. Hence $\mu$ is not an extreme point. 3 Solovay tests Recall that a Solovay test is a sequence ${\left\langle{S_{k}}\right\rangle}_{k\in{\mathbb{N}}}$ of uniformly $\Sigma^{0}_{1}$ sets such that $\sum_{k}S_{k}<\infty$. A bit sequence $Z$ passes such a test if $Z\not\in S_{k}$ for almost every $k$. (Each ML-test is a Solovay test, but the passing condition is stronger for Solovay tests). A basic fact from the theory of algorithmic randomness (e.g. [14, 3.2.9]) states that $Z$ is ML-random iff $Z$ passes each Solovay test. We say that a measure $\mu$ passes a Solovay test ${\left\langle{S_{k}}\right\rangle}_{k\in{\mathbb{N}}}$ if $\lim_{k}\mu(S_{k})=0$. The following characterises the Martin-Löf a.c. measures with countable support. Fact 3.1. Let $\mu=\sum_{k}c_{k}\delta_{Z_{k}}$ where $\forall k\,[0<c_{k}\leq 1]$ and $\sum_{k}c_{k}=1$. Then $\mu$ is Martin-Löf a.c. iff all $Z_{k}$ are Martin-Löf random. Proof 3.2. The implication from left to right is immediate. For the converse implication, given a Martin-Löf test ${\left\langle{G_{m}}\right\rangle}$, note that the $Z_{k}$ pass this test as a Solovay test. Hence for each $r$, there is $M$ such that $Z_{k}\not\in G_{m}$ for each $k\leq r$ and $m\geq M$. This implies that $\mu(G_{m})\leq\sum_{k>r}c_{k}$. So $\lim_{m}\mu(G_{m})=0$. The fact that passing all Martin-Löf tests is equivalent to passing all Solovay tests generalises to measures. We use the following variant for measures of a result by Tejas Bhojraj (in preparation) that he proved in the setting of quantum states. Proposition 3.3. A measure $\mu$ is Martin-Löf a.c. iff $\mu$ passes each Solovay test. Proof 3.4. Each Martin-Löf test is a Solovay test. So the implication from right to left is immediate. For the implication from left to right, suppose that ${\left\langle{S_{k}}\right\rangle}$ is a Solovay test that $\mu$ fails. So there is $\delta>0$ such that $\mu(S_{k})>\delta>0$ for infinitely many $k$. We will define a Martin-Löf test ${\left\langle{G_{m}}\right\rangle}_{m\in{\mathbb{N}}}$ that $\mu$ fails at level $\delta/2$, i.e. $\inf_{m}\mu(G_{m})\geq\delta/2$. Let $S_{k,t}$ denote the clopen set given by the strings $\sigma$ of length $t$ such that $[\sigma]\subseteq S_{k}$. First we use a minor modification of the proof that the two test notions are equivalent for bit sequences (e.g. [14, Prop. 3.2.19]): Let $G_{m,t}$ be the clopen set generated by strings $\sigma$ of length $t$ such that $[\sigma]\subseteq S_{k,t}$ for $\delta 2^{m-1}$ many $k\leq t$. Similar to the proof of [14, Prop. 3.2.19] one shows that $\mathbf{\lambda}G_{m,t}\leq 2^{-m}/\delta$. Let $G_{m}=\bigcup_{t}G_{m,t}$ and note that $G_{m}$ is effectively open uniformly in $m$. So, fixing a $k\in{\mathbb{N}}$ such that $k>\log_{2}(1/\delta)$, ${\left\langle{G_{k\cdot m}}\right\rangle}_{m\in{\mathbb{N}}}$ forms a Martin-Löf test. Given $m$, we pick $t\in{\mathbb{N}}$ sufficiently large so that for some set $M\subseteq\{0,\ldots,t-1\}$ of size $2^{m}$ we have $\mu(S_{k,t})>\delta$ for each $k\in M$. We show that $\mu(G_{m,t})>\delta/2$. Let $\sigma$ range over strings of length $t$. If $[\sigma]\not\subseteq G_{m,t}$ then $\sum_{k\leq t}\mu([\sigma]\cap S_{k,t})\leq 2^{m-1}\delta\mu[\sigma]$ by definition of $G_{m}$. Therefore $$\sum_{[\sigma]\not\subseteq G_{m,t}}\sum_{k\leq t}\mu([\sigma]\cap S_{k,t})% \leq 2^{m-1}\delta.$$ Since $2^{m}\delta\leq\sum_{k\in M}\mu(S_{k,t})$, this implies $\sum_{[\sigma]\subseteq G_{m}}\sum_{k\in M}\mu[\sigma]>2^{m-1}\delta$. Since $|M|=2^{m}$ this shows $\mu G_{m,t}>\delta/2$ as required. 4 Full Martin-Löf  randomness of measures Let $\mathcal{M}(\{0,1\}^{{\mathbb{N}}})$ be the space of probability measures on Cantor space (which is canonically a compact topological space). A probability measure $\mathbb{P}$ on this space has been introduced implicitly in Mauldin and Monticino [11]. Culver’s thesis [3] shows that this measure is computable. So the framework of [7] yields a definition of Martin-Löf randomness for points in the space $\mathcal{M}(\{0,1\}^{{\mathbb{N}}})$. To define $\mathbb{P}$, first let $\mathcal{R}$ be the closed set of representations of probability measures; namely, $\mathcal{R}$ consists of the functions $X\colon\{0,1\}^{*}\to[0,1]$ such that $X_{\emptyset}=1$ and $X_{\sigma}=X_{\sigma 0}+X_{\sigma 1}$ for each string $\sigma$. $\mathbb{P}$ is the unique measure on $\mathcal{R}$ such that for each string $\sigma$ and $r,s\in[0,1]$, we have $P(X_{\sigma 0}\leq r\mid X_{\sigma}=s)=\min(1,r/s)$. Intuitively, we choose $X_{\sigma 0}$ at random w.r.t. the uniformly distribution on the interval $[0,X_{\sigma}]$, and the choices made at different strings are independent. Proposition 4.1. Every probability measure $\mu$ that is Martin-Löf random wrt to $\mathbb{P}$ is Martin-Löf absolutely continuous. For the duration of this proof let $\mu$ range over ${\mathcal{M}(\{0,1\}^{{\mathbb{N}}})}$. For an open set $G\subseteq\{0,1\}^{{\mathbb{N}}}$, let $r_{G}=\int\mu(G)d\mathbb{P}(\mu)$. Our proof of Prop. 4.1 is based on two facts. For open $G\subseteq\{0,1\}^{{\mathbb{N}}}$, let $r_{G}=\int\mu(G)d\mathbb{P}(\mu)$. Fact 4.2. $r_{G}=\mathbf{\lambda}(G)$. Proof 4.3. Clearly, for each $n$ we have $$\sum_{|\sigma|=n}r_{[\sigma]}=\int\sum_{|\sigma|=n}\mu([\sigma])d\mathbb{P}(% \mu)=1.$$ Furthermore, $r_{\sigma}=r_{\eta}$ whenever $|\sigma|=|\eta|=n$ because there is a $\mathbb{P}$-preserving transformation $T$ of $\mathcal{M}(\{0,1\}^{{\mathbb{N}}})$ such that $\mu([\sigma])=T(\mu)([\eta])$. Therefore $r_{[\sigma]}=2^{-|\sigma|}$. If $\sigma,\eta$ are incompatible then $r_{[\sigma]\cup[\eta]}=r_{[\sigma]}+r_{[\eta]}$. Now it suffices to write $G=\bigcup_{i}[\sigma_{i}]$ where the strings $\sigma_{i}$ are incompatible, so that $\mathbf{\lambda}G=\sum_{i}2^{-|\sigma_{i}|}$. Fact 4.4. Let $\mu\in\mathcal{M}(\{0,1\}^{{\mathbb{N}}})$ and let ${\left\langle{G_{m}}\right\rangle}_{m\in{\mathbb{N}}}$ be a ML-test such that there is $\delta\in{\mathbb{Q}}^{+}$ with $\forall m\,\mu(G_{m})>\delta$. Then $\mu$ is not ML-random w.r.t. $\mathbb{P}$. Proof 4.5. Observe that by the foregoing fact $$\delta\cdot\mathbb{P}(\{\mu\colon\mu(G_{m})\geq\delta\})\leq\int\mu(G_{m})d% \mathbb{P}(\mu)=\mathbf{\lambda}(G_{m})\leq 2^{-m}.$$ Let $\mathcal{G}_{m}=\{\mu\colon\mu(G_{m})>\delta\}$ which is uniformly effectively open in the space of measures $\mathcal{M}(\{0,1\}^{{\mathbb{N}}})$. Fix $k$ such that $2^{-k}\leq\delta$; then ${\left\langle{\mathcal{G}_{m+k}}\right\rangle}_{m\in{\mathbb{N}}}$ is a ML-test w.r.t. $\mathbb{P}$ that succeeds on $\mu$. Culver [3] shows that each measure $\mu$ that is Martin-Löf random w.r.t.  $\mathbb{P}$ is non-atomic. So because of the measures $\delta_{Z}$ for Martin-Löf random bit sequences $Z$, the converse of Prop. 4.1 fails: not every Martin-Löf a.c. measure is Martin-Löf random with respect to $\mathbb{P}$. 5 Initial segment complexity of a measure $\mu$ Let $K(\mu\!\upharpoonright_{n})=\sum_{|x|=n}K(x)\mu[x]$ be the $\mu$-average of all the $K(x)$ over all strings $x$ of length $n$. In a similar way we define $C(\mu\!\upharpoonright_{n})$. Note that for a Dirac measure $\delta_{Z}$, we have $K(\delta_{Z}\!\upharpoonright_{n})=K(Z\!\upharpoonright_{n})$. In this section we use standard inequalities such as $C(x)\leq^{+}K(x)$, $K(x)\leq^{+}|x|+2\log|x|$ and $K(0^{n})\leq^{+}2\log n$. We also use that for each $r$ there are at most $2^{r}-1$ strings such that $C(x)<r$. See e.g. [14, Ch. 2]. Recall that $\mathbf{\lambda}$ denotes the uniform measure on $\{0,1\}^{{\mathbb{N}}}$. Fact 5.1. (a) $C(\mathbf{\lambda}\!\upharpoonright_{n})\geq^{+}n$. (b) $K(\mathbf{\lambda}\!\upharpoonright_{n})\geq^{+}n+K(n)$. Proof 5.2. Chaitin [2] showed that there is a constant $c$ such that, for all $d$, there are at most $2^{n+c-d}$ strings $x\in\{0,1\}^{n}$ with $C(x)\leq n-d$. Similarly, among the strings of length $n$, there are at most $2^{n+c-d}$ strings with $K(x)\leq n+K(n)-d$. In other words, the fraction of strings of length $n$ where, for $(a)$, $C(x)\leq n-d$, and, for $(b)$, $K(x)\leq n+K(n)-d$, respectively, is in each case at most $2^{c-d}$. Now for each $d$, from the estimated lower bound $n$ and $n+K(n)$, respectively, one subtracts the fraction of the strings of length $n$ for which the Kolmogorov complexity is at least $d$ below the average in order to correct the lower bound. For, if the complexity of a string is $r$ below the lower bound, it has to be considered $r$ times, for $d=1,\ldots,r$. Let $c_{d}$ is the fraction of strings of length $n$ with $C(x)\leq n-d$ and $k_{d}$ is the fraction of strings with $K(x)\leq n+K(n)-d$. Then $$C(\mathbf{\lambda}\!\upharpoonright_{n})\geq n-\sum_{d\geq 0}c_{d}\mbox{ and }% K(\mathbf{\lambda}\!\upharpoonright_{n})\geq n+K(n)-\sum_{d\geq 0}k_{d}.$$ Using Chaitin’s bounds gives then the corrected estimates on the averages of $$C(\mathbf{\lambda}\!\upharpoonright_{n})\geq n-\sum_{d\geq 0}2^{c-d}\mbox{ and% }K(\mathbf{\lambda}\!\upharpoonright_{n})\geq n+K(n)-\sum_{d\geq 0}2^{c-d}.$$ Now one uses that $\sum_{d\geq 0}2^{c-d}\leq 2^{c+1}$ and that $2^{c+1}$ is a constant independent of $n$ and only dependent on the universal machine in order to get that $C(\mathbf{\lambda}\!\upharpoonright_{n})$ $\geq^{+}$ $n$ and $K(\mathbf{\lambda}\!\upharpoonright_{n})$ $\geq^{+}$ $n+K(n)$. We say that a measure $\mu$ has complex initial segments if $K(\mu\!\upharpoonright_{n})\geq^{+}n$. We will show that both implications of the analog of the Levin-Schnorr Theorem 2.2 fail for measures. One implication, that a Martin-Löf a.c. measure cannot have initial segment complexity growing slower than $n-O(1)$, is disproved by a simple example of a measure with countable support. Example 5.3. There is a Martin-Löf a.c. measure $\mu$ such that $K(\mu\!\upharpoonright_{n})\leq^{+}n-\log n$. Proof 5.4. We let $\mu=\sum c_{k}\delta_{Z_{k}}$ where $Z_{k}$ is Martin-Löf random and $0^{n_{k}}\prec Z_{k}$ for a sequence ${\left\langle{c_{k}}\right\rangle}$ of reals in $[0,1]$ that add up to $1$, and a sufficiently fast growing sequence $n_{k}$. Then $\mu$ is Martin-Löf a.c. by Fact 3.1. For $n$ such that $n_{k}\leq n<n_{k+1}$ we have $$\displaystyle K(\mu\!\upharpoonright_{n})$$ $$\displaystyle\leq^{+}$$ $$\displaystyle(\sum_{l=0}^{k}c_{l})\cdot(n+2\log n)+\sum_{l=k+1}^{\infty}c_{l}% \cdot 2\log n$$ $$\displaystyle\leq^{+}$$ $$\displaystyle(1-c_{k+1})n+2\log n.$$ Hence, to achieve $K(\mu\!\upharpoonright_{n})\leq^{+}n-\log n$ it suffices to ensure that $3\log(n_{k+1})<c_{k+1}n_{k}$. For instance, we can let $c_{k}=1/((k+1)(k+2))$ and $n_{k}=2^{k+4}$. To falsify the converse implication, we need to provide a measure $\mu$ such that $K(\mu\!\upharpoonright_{n})\geq^{+}n$ yet $\mu$ is not a Martin-Löf a.c. measure. This will be immediate from the following fact on the growth of initial segment complexity for bit sequences. Theorem 5.5. There are a Martin-Löf random $X$ and a not Martin-Löf random $Y$ such that, for all $n$, $K(X\upharpoonright n)+K(Y\upharpoonright n)\geq^{+}2n$. Proof 5.6. Let $X$ be a low Martin-Löf random set (i.e., $X^{\prime}\equiv_{T}\emptyset^{\prime}$). We claim that there is a strictly increasing function $f$ such that the complement of the range of $f$ is a recursively enumerable set $E$, and $K(X\upharpoonright m)\geq m+3n$ for all $m\geq f(n)$. To see this, recall that $\lim_{n}K(X\!\upharpoonright_{n})-n=\infty$. Since $X$ is low there is a computable function $p$ such that for all $n$, $\lim_{s}p(n,s)$ is the maximal $m$ such that $K(X\upharpoonright m)\leq m+3n$. Define $f(n,s)$ for $n\leq s$ as follows. $f(0,0)=0$; for $s>0$ let $n$ be least such that $p(n,s-1)\neq p(n,s)$ or $n=s$. Let $f(n,s)=r$ be larger than all previous values assigned by $f$ to a pair, and let $f(m,s)=r+m-n$ for $n<m\leq s$. Let $f(m,s)=f(m,s-1)$ for $m<n$. Then $f(n)=\lim_{s}f(n,s)$ is a function as required, which verifies the claim. Now let $g(n)=\max\{m:f(m)\leq n\}$ (with the convention that $\max(\emptyset)=0$). Since $g$ is unbounded, by a result of Miller and Yu [13, Cor. 3.2] there is a Martin-Löf random $Z$ such that there exist infinitely many $n$ with $K(Z\upharpoonright n)\leq n+g(n)/2$. Let $Y=\{n+f(n):n\in Z\}$. Note that $K(Z\upharpoonright n)\leq K(Y\upharpoonright n)+g(n)+K(g(n))$, as one can enumerate the set $E$ until there are, up to $n$, only $g(n)$ many places not enumerated and then one can reconstruct $Z\upharpoonright n$ from $Y\upharpoonright n$ and $g(n)$ and the last $g(n)$ bits of $Z$. As $Z$ is Martin-Löf random, $K(Z\upharpoonright n)\geq^{+}n$ and so, $K(Y\upharpoonright n)\geq^{+}n-g(n)-K(g(n))\geq^{+}n-2g(n)$. The definitions of $X,f,g$ give $K(X\upharpoonright n)\geq n+3g(n)$. This shows that $K(X\upharpoonright n)+K(Y\upharpoonright n)\geq 2n$ for almost all $n$. However, the set $Y$ is not Martin-Löf random, as there are infinitely many $n$ with $K(Z\upharpoonright n)\leq^{+}n+g(n)/2$. Now $Y\upharpoonright n+g(n)$ can be computed from $Z\upharpoonright n$ and $g(n)$, as one needs only to enumerate $E$ until the $g(n)$ nonelements of $E$ below $n$ are found and they allow to see where the zeroes have to be inserted into the string $Z\upharpoonright n$ in order to obtain $Y\upharpoonright n+g(n)$. Note furthermore, that $K(g(n))\leq g(n)/4$ for almost all $n$ and thus $K(Y\upharpoonright n+g(n))\leq^{+}n+3/4\cdot g(n)$ for infinitely many $n$, so $Y$ cannot be Martin-Löf random. Corollary 5.7. There is a measure $\mu$ with complex initial segments which is not Martin-Löf a.c. Proof 5.8. The measure $\mu=(\delta_{X}+\delta_{Y})/2$ has only two equal-weighted atoms and one of these atoms is not Martin-Löf random. So every component of a universal Martin-Löf test has at least $\mu$-measure $1/2$. On the other hand, $K(\mu\upharpoonright n)\geq n$ for almost all $n$ by the preceding result. A bit sequence $Z\in\{0,1\}^{{\mathbb{N}}}$ is called strongly Chaitin random if there is $d$ such that $K(Z\!\upharpoonright_{n})\geq n+K(n)-d$ for infinitely many $n$. (This is equivalent to 2-randomness by [12]; for detail see e.g. [14, 8.1.14] or [4]). We may extend this notion to measures. Each such measure is Martin-Löf a.c.: Theorem 5.9. Suppose that $\mu$ is a measure such that $K(\mu\!\upharpoonright_{n})\geq n+K(n)-r$ for infinitely many $n$. Then $\mu$ is a Martin-Löf a.c. measure. Proof 5.10. Suppose that $\mu$ is not a Martin-Löf a.c. measure. So there is a Martin-Löf test ${\left\langle{G_{d}}\right\rangle}_{d\in{\mathbb{N}}}$ and $\epsilon>0$ such that $\mu(G_{d})>\epsilon$ for each $d$. We view $G_{d}$ as given by an enumeration of strings, uniformly in $d$; thus $G_{d}=\bigcup_{i}[\sigma_{i}]$ for an effectively computable sequences ${\left\langle{\sigma_{i}}\right\rangle}$. Let $G_{d}^{\leq n}$ denote the clopen set generated by the strings in this enumeration of length at most $n$. (Note that this set is not effectively given as a clopen set, but we effectively have a description of it as a $\Sigma^{0}_{1}$ set). Let $c$ be a constant such that $K(x)\leq n+K(n)+c$ for each $x$ of length $n$. If $x$ is a string of length $n$ such that $[x]\subseteq G_{d}^{\leq n}$ then $$K(x\mid n,d)\leq^{+}n-d.$$ To see this let $M$ be the fixed machine that, on a pair of auxiliary inputs $n,d$, waits for $[x]\subseteq G_{d}^{\leq n}$ and once that happens provides a description of length $n-d$ for $x$ (so the descriptions for different $x$ are prefix free). It follows that for such $x$ (and after increasing $c$ if necessary) $$K(x)\leq n+K(n)-d+2\log d+c.$$ For each $d,n$, letting $x$ range over strings of length $n$, we have $$\displaystyle K(\mu\!\upharpoonright_{n})$$ $$\displaystyle=$$ $$\displaystyle\sum K(x)\mu[x]$$ $$\displaystyle=$$ $$\displaystyle\sum_{[x]\subseteq G_{d}^{\leq n}}K(x)\mu[x]+\sum_{[x]\not% \subseteq G_{d}^{\leq n}}K(x)\mu[x]$$ The first summand is bounded above by $\mu(G_{d}^{\leq n})(n+K(n)-d+2\log d+c)$, the second by $(1-\mu(G_{d}^{\leq n}))(n+K(n)+c)$. We obtain $K(\mu\!\upharpoonright_{n})\leq n+K(n)+c-\mu(G_{d}^{\leq n})d/2$. Now for each $d$, for sufficiently large $n$ we have $\mu(G_{d}^{\leq n})>\epsilon$. So given $r$ let $d=2r/\epsilon$; then for large enough $n$ we have $K(\mu\!\upharpoonright_{n})\leq n+K(n)+c-r$. So $\mu$ is not strongly Chaitin random. We ask whether a known fact for bit sequences lifts to measures. Question 5.11. Is every strongly Chaitin random measure Martin-Löf a.c.  relative to the halting problem ${\emptyset^{\prime}}$? 6 $K$-triviality for measures Definition 6.1. A measure $\mu$ is called $K$-trivial if $K(\mu\!\upharpoonright_{n})\leq^{+}K(n)$ for each $n$. For Dirac measures $\delta_{A}$ this is the same as saying that $A$ is $K$-trivial in the usual sense. More generally, any finite convex combination of such Dirac measures is $K$-trivial. Proposition 6.2. Suppose $\mu$ is $K$-trivial. Then $\mu$ is supported by its set of atoms. Thus, if $\mu$ is $K$-trivial for constant $b$ then $\mu$ has the form $\sum_{r<N}\alpha_{r}\delta_{A_{r}}$ where $N\leq\infty$ and each $\alpha_{r}$ is positive and $\sum_{r<N}\alpha_{r}=1$. Clearly each $A_{r}$ is $K$-trivial for constant $b/\alpha_{r}$. Proof 6.3. Assume for a contradiction that $\mu$ gives a measure of $\epsilon>0$ to the set of its non-atoms. Note that there is a constant $b$ such that $K(x)\geq K(|x|)-b$ for each $x$. Fix $c$ arbitrary with the goal of showing that $K(\mu\!\upharpoonright_{n})\geq K(n)-b+\epsilon c/2$ for large enough $n$. There is $d$ (in fact $d=O(2^{c})$) such that for each $n$ there are at most $d$ strings $x$ of length $n$ with $K(x)\leq K(n)+c$ (see e.g. [14, 2.2.26]). Let $S_{n}=\{x\colon\,|x|=n\land\mu[x]\leq\epsilon/2d\}$. By hypothesis we have $\mu[S_{n}]^{\prec}\geq\epsilon$ for large enough $n$. Therefore by choice of $d$ we have $\mu[S_{n}\cap\{x\colon\,K(x)>|x|+c\}]^{\prec}\geq\epsilon/2$. Now we can give a lower bound for the $\mu$ average of $K(x)$ over all strings $x$ of length $n$: $$\sum_{|x|=n}K(x)\mu[x]\geq(1-\epsilon/2)(K(n)-b)+(\epsilon/2)(K(n)+c)\geq K(n)% -b+\epsilon c/2,$$ as required. Notice that we only used the weaker hypothesis that $\liminf_{n}[K(\mu\!\upharpoonright_{n})-K(n)]$ is finite. It would be interesting to characterise the countable convex combinations of $K$-trivials that yield $K$-trivial measures. The following is easily checked. Fact 6.4. Suppose that $A_{r}$ is $K$-trivial with constant $b_{r}$, and $\sum_{r}\alpha_{r}b_{r}\leq c<\infty$ where each $\alpha_{r}$ is positive and $\sum\alpha_{r}=1$. Then $\mu=\sum_{r}\alpha_{r}\delta_{A_{r}}$ is $K$-trivial with constant $c$. For instance, we can build a computable $K$-trivial measure with infinitely many atoms as follows. Let $A_{r}=0^{r+1}1^{\infty}$, so that $K(A_{r}\!\upharpoonright_{n})\leq^{+}K(n)+2\log r$. Let $\mu=\sum 2^{-r+1}A_{r}$. By the above fact $\mu$ is $K$-trivial. If we vary the construction by letting $A_{r}=0^{r+1}1B$ where $B$ is $K$-trivial but non-recursive, we obtain a $K$-trivial $\mu$ with infinitely many atoms, and none of them recursive. On the other hand, the following example shows that not every infinite convex combination of $K$-trivial Dirac measures yields a $K$-trivial measure. Let $\mu=\sum_{k}\alpha_{k}\delta_{A_{k}}$, where $A_{k}=\{\ell:\ell\in\Omega\wedge\ell<k\}$, and $\alpha_{k}=(k+1)^{-1/2}-(k+2)^{-1/2}$. All sets $A_{k}$ are finite and thus $K$-trivial. Furthermore, the sum of all $\alpha_{k}$ is $1$. We have $K(\mu\!\upharpoonright_{n})=\sum_{x\in\{0,1\}^{n}}K(x)\mu(x)\geq(\sum_{m\geq n% }\alpha_{m})\cdot K(\Omega\upharpoonright n)\geq(n+2)^{-1/2}\cdot(n+2)=\sqrt{n% +2}$ for almost all $n$ and thus the average grows faster than $K(n)+c$. So the measure is not $K$-trivial. In a sense, an atomless measure can come arbitrarily close to being $K$-trivial. Proposition 6.5. For each nondecreasing unbounded function $f$ which is computably approximable from above there is a non-atomic measure $\mu$ such that $K(\mu\upharpoonright n)\leq^{+}K(n)+f(n)$. Proof 6.6. There is a recursively enumerable set $A$ such that, for all $n$, $A$ has up to $n$ and up to a constant $f(n)/2$ non-elements. One let $\mu$ be the measure such that $\mu(x)=2^{-m}$ in the case that all ones in $x$ are not in $A$ and $\mu(x)=0$ otherwise, here $m$ is the number of non-elements of $A$ below $|x|$. One can see that when $\mu(x)=2^{-m}$ then $x$ can be computed from $|x|$ and the string $b_{0}b_{1}\ldots b_{m-1}$ which describes the bits at the non-elements of $A$. Thus $K(x)\leq^{+}K(|x|)+K(b_{0}b_{1}\ldots b_{m-1})\leq^{+}K(|x|)+2m$. It follows that $K(\mu\upharpoonright n)\leq^{+}K(n)+f(n)$, as the $\mu$-average of strings $x\in\{0,1\}^{n}$ with $K(x)\leq^{+}K(n)+f(n)$ is at most $K(n)+f(n)$ plus a constant. 7 Towards effective Shannon-McMillan-Breiman for measures We review some notions from the field of symbolic dynamics, a mathematical area closely related to Shannon information theory. It is useful to admit alphabets other than the binary one. Let $\mathbb{A}^{\infty}$ denote the topological space of one-sided infinite sequences of symbols in an alphabet $\mathbb{A}$. Randomness notions etc. carry over from the case of $\mathbb{A}=\{0,1\}$. A dynamics on $\mathbb{A}^{\infty}$ is given by the shift operator $T$, which erases the first symbol of a sequence. A measure $\rho$ on $\mathbb{A}^{\infty}$ is called shift invariant if $\rho(G)=\rho(T^{-1}(G))$ for each open (and hence each measurable) set $G$. The empirical entropy of a measure $\rho$ along $Z\in\mathbb{A}^{\infty}$ is given by the sequence of random variables $$h^{\rho}_{n}(Z)=-\frac{1}{n}\log_{|\mathbb{A}|}\rho[Z\!\upharpoonright_{n}].$$ A shift invariant measure $\rho$ on $\mathbb{A}^{\infty}$ is called ergodic if every $\rho$ integrable function $f$ with $f\circ T=f$ is constant $\rho$-almost surely. An equivalent condition that is easier to check is the following: for $u,v\in\mathbb{A}^{*}$, $$\lim_{N}\frac{1}{N}\sum_{k=0}^{n-1}\rho([u]\cap T^{-k}[v])=\rho[u]\rho[v].$$ For ergodic $\rho$, the entropy $H(\rho)$ is defined as $\lim_{n}H_{n}(\rho)$, where $$H_{n}(\rho)=-\frac{1}{n}\sum_{|w|=n}\rho[w]\log\rho[w].$$ Thus, $H_{n}(\rho)=\mathbb{E}_{\rho}h^{\rho}_{n}$ is the expected value with respect to $\rho$. One notes that $H_{n+1}(\rho)\leq H_{n}(\rho)\leq 1$ so that the limit exists. A well-known result from the 1950s due to Shannon, McMillan and Breiman (see, e.g., [19]) states that for an ergodic measure $\rho$, for $\rho$-a.e. $Z$ the empirical entropy along $Z$ converges to the entropy of the measure. Theorem 7.1 (SMB theorem). Let $\rho$ be an ergodic measure on the space $\mathbb{A}^{\infty}$. For $\rho$-a.e.  $Z$ we have $\lim_{n}h^{\rho}_{n}(Z)=H(\rho)$. A measure $\rho$ on $\mathbb{A}^{\infty}$ is called computable if the real $\rho[x]$ is computable, uniformly in $x\in\mathbb{A}^{*}$. For such a measure we can define Martin-Löf tests and Martin-Löf randomness with respect to $\rho$ ($\rho$-ML randomness for short) as above; in Definition 2.1 we simply replace the uniform measure $\mathbf{\lambda}$ by $\rho$. The basic theory of $\rho$-ML randomness is developed in a way similar to the uniform case. In particular, there is a universal $\rho$-ML test. Recalling Fact 2.4, we say that a measure $\mu$ is Martin-Löf a.c. with respect to $\rho$ if $\mu(\mathcal{C})=0$ where $\mathcal{C}$ is the class of sequences in $\mathbb{A}^{\infty}$ that are not ML-random with respect to $\rho$. If a computable measure $\rho$ is shift invariant, then $\lim_{n}h^{\rho}_{n}(Z)$ exists for each $\rho$-ML-random $Z$ by a result of Hochman [5]. Hoyrup [6, Thm. 1.2] gave an alternative proof for ergodic $\rho$, and also showed that in that case we have $\lim_{n}h^{\rho}_{n}(Z)=H(\rho)$ for each $\rho$-ML random $Z$. We extend this result to measures $\mu$ that are Martin-Löf a.c. with respect to $\rho$. We require as a hypothesis that the $h_{n}^{\rho}$ are uniformly bounded. This holds e.g. for Bernoulli measures and the measures given by a Markov process. On the other hand, using a renewal process it is not hard to construct an ergodic computable measure $\rho$ where this hypothesis fails. For instance, over the binary alphabet let $\rho$ be the shift invariant measure such that $\rho[10^{k}1]=2^{-k-1}$ for each $k\in{\mathbb{N}}$. Theorem 7.2. Let $\rho$ be a computable ergodic measure on the space $\mathbb{A}^{\infty}$ such that for some constant $D$, each $h_{n}^{\rho}$ is bounded above by $D$. Suppose the measure $\mu$ is Martin-Löf a.c.  with respect to $\rho$. Then $\lim_{n}E_{\mu}h^{\rho}_{n}=H(\rho)$. Proof 7.3. For each pair of rationals $s<t$ and each $k\in{\mathbb{N}}$, let $U^{s,t}_{k}$ be the effectively open set of those $Z\in\mathbb{A}^{\infty}$ such that the sequence $h_{n}^{\rho}(Z)$ has $k$ upcrossings of $[s,t]$. By Hochman [5, Thm. 1.3 ] we have $\rho(U^{s,t}_{k})<c\cdot\alpha^{k}$ for positive constants $\alpha<1$ and $c$ that depend only on $s,t$. Therefore a subsequence of $U^{s,t}_{k}$ is a Martin-Löf test w.r.t. $\rho$, and hence $\mu(\bigcap_{k}U^{s,t}_{k})=0$. Since $s<t$ are arbitrary and $h^{\rho}_{n}$ is bounded by a fixed constant $D$ this implies that $h_{n}^{\rho}(Z)$ converges to a value $g(Z)$, $\mu$-almost surely. By Hoyrup’s result, $\lim_{n}h^{\rho}_{n}(Z)=H(\rho)$ for each $\rho$-ML random $Z$. Since the sequences that are not ML-random w.r.t. $\rho$ form a null set w.r.t. $\mu$, we infer that $g(Z)=H(\rho)$ for $\mu$-a.e. $Z$. The Dominated Convergence Theorem now shows that $\lim_{n}\mathbb{E}_{\mu}h_{n}^{\rho}=E_{\mu}g=H(\rho)$, as required. Proposition 7.4. Let $\rho$ be a computable ergodic invariant measure, and suppose $\mu$ is a Martin-Löf a.c. measure with respect to $\rho$. Then $\lim_{n}\frac{1}{n}K(\mu\!\upharpoonright_{n})=\lim_{n}\frac{1}{n}C(\mu\!% \upharpoonright_{n})=H(\rho)$. Proof 7.5. We can use $K$ and $C$ interchangeably because $C(x)\leq^{+}K(x)\leq^{+}C(x)+K(C(x))$ [14, 2.4.1]. In analogy to the functions $h_{n}$, let $k_{n}(Z)=K(Z\!\upharpoonright_{n})/n$. The argument for $K$, say, is very similar to the one in the theorem above, replacing the functions $h_{n}$ by $k_{n}$. Note that $k_{n}$ is bounded above by a constant because $K(x)\leq^{+}|x|+2\log|x|$. With the $U^{s,t}_{k}$ defined analogously, $\rho(U^{s,t}_{k})<c\cdot\alpha^{k}$ for some positive constants $c,\alpha$ with $\alpha<1$ by Hochman [5, Thm. 1.4 ]. Hoyrup’s result [6, Thm. 1.2] states that $\lim_{n}k_{n}(Z)=H(\rho)$ for each $\rho$-ML random $Z$. Now we can apply the Dominated Convergence Theorem as before. Acknowledgment. A. Nies was supported in part by the Marsden Fund of the Royal Society of New Zealand, UoA 13-184. F. Stephan is supported in part by the Singapore Ministry of Education Academic Research Fund Tier 2 grant MOE2016-T2-1-019 / R146-000-234-112. A preliminary version was summarised in the Logic Blog 2018 on the first author’s homepage at https://www.cs.auckland.ac.nz/~nies/. References [1] Cristian S. Calude. Information and Randomness - An Algorithmic Perspective. Second Edition, Springer, Heidelberg, 2002. [2] Gregory J. Chaitin. A theory of program size formally identical to information theory. Journal of the Association for Computing Machinery, 22:329-340, 1975. [3] Quinn Culver. Topics in Algorithmic Randomness and Effective Probability. PhD thesis, University of Notre Dame, 2015. [4] Rodney G. Downey and Denis Hirschfeldt. Algorithmic Randomness and Complexity. Springer-Verlag, Berlin, 2010. 855 pages. [5] Michael Hochman. Upcrossing inequalities for stationary sequences and applications. Annals of Probability, 37(6):2135–2149, 2009. [6] Mathieu Hoyrup. The dimension of ergodic random sequences. In Christoph Dürr and Thomas Wilke, editors, STACS 2012, pages 567–576, 2012. [7] Mathie Hoyrup and Cristóbal Rojas. Computability of probability measures and Martin-Löf randomness over metric spaces. Information and Computation, 207(7):830–847, 2009. [8] Leonid A. Levin. The concept of a random sequence. Doklady Akademii Nauk SSSR, 212:548–550, 1973. [9] Ming Li and Paul Vitányi. An Introduction to Kolmogorov Complexity and its Applications. Texts in Computer Science. Springer, New York, third edition, 2008. [10] Per Martin-Löf. The definition of random sequences. Information and Control, 9:602–619, 1966. [11] Daniel Mauldin and Michael Monticino. Randomly generated distributions. Israel Journal of Mathematics, 91(1-3):215–237, 1995. [12] Joeseph S. Miller. The $K$-degrees, low for $K$-degrees, and weakly low for $K$ sets. Notre Dame Journal of Formal Logic, 50(4):381–391, 2009. [13] Joseph S. Miller and Liang Yu. Oscillation in the initial segment complexity of random reals. Advances in Mathematics, 226(6):4816–4840, 2011. [14] André Nies. Computability and Randomness, volume 51 of Oxford Logic Guides. Oxford University Press, Oxford, 2009. 444 pages. Paperback version 2011. [15] André Nies and Volkher Scholz. Martin-Löf random quantum states. Technical Report on http://www.arxiv.org no 1709.08422, 2017, new version 2018. [16] Piergiorgio Odifreddi. Classical Recursion Theory. North-Holland, Amsterdam, 1989. [17] Piergiorgio Odifreddi. Classical Recursion Theory, Volume II. Elsevier, Amsterdam, 1999. [18] Claus-Peter Schnorr. Process complexity and effective random tests. Journal on Computer and System Sciences, 7:376–388, 1973. [19] Paul C. Shields. The Ergodic Theory of Discrete Sample Paths. Graduate Studies in Mathematics 13. American Mathematical Society, 1996. [20] Robert I. Soare. Recursively Enumerable Sets and Degrees. A Study of Computable Functions and Computably Generated Sets. Springer, Heidelberg, 1987. [21] Frank Stephan. Recrsuion Theory. Lecture notes, School of Computing, National University of Singapore, Technical Report TR10/12, 2012.

Bayesian inference and prediction for mean-mixtures of normal distributions 111 Pankaj Bhagwat${}^{a}$ & Éric Marchand a Université de Sherbrooke, Département de mathématiques, Sherbrooke Qc, CANADA, J1K 2R1  (e-mails: pankaj.uttam.bhagwat@usherbrooke.ca; eric.marchand@usherbrooke.ca) Abstract We study frequentist risk properties of predictive density estimators for mean mixtures of multivariate normal distributions, involving an unknown location parameter $\theta\in\mathbb{R}^{d}$, and which include multivariate skew normal distributions. We provide explicit representations for Bayesian posterior and predictive densities, including the benchmark minimum risk equivariant (MRE) density, which is minimax and generalized Bayes with respect to an improper uniform density for $\theta$. For four dimensions or more, we obtain Bayesian densities that improve uniformly on the MRE density under Kullback-Leibler loss. We also provide plug-in type improvements, investigate implications for certain type of parametric restrictions on $\theta$, and illustrate and comment the findings based on numerical evaluations. Keywords and phrases: Bayes predictive density; Dominance; Kullback-Leibler loss; Minimax; Minimum risk equivariant; Mean mixtures; Multivariate Normal, Skew-normal distribution. 1 Introduction The findings of this paper relate to predictive density estimation for mean mixture of normal distributions. The modelling of data via mixing multivariate normal distributions has found many applications and lead to methodological challenges for statistical inference. These include finite mixtures, as well as continuous mixing on the mean and/or the variance. Whereas, scale or variance mixtures of multivariate normal distributions compose a quite interesting subclass of spherically symmetric distributions, modelling asymmetry requires mixing on the mean and prominent examples are generated via variance-mean mixtures (e.g., [7]), as well as mean mixtures of multivariate normal distributions (e.g., [1, 5]) and references therein). Moreover, such mean mixtures, which are the subject of study here, generate or are connected to multivariate skew-normal distributions (e.g., [6]) which have garnered much interest over the years. The development of shrinkage estimation techniques, namely since Stein’s inadmissibility finding ([26]) concerning the maximum likelihood or best location equivariant estimator under squared error loss in three dimensions or more, has had a profound impact on statistical theory, thinking, methods, and practice. Examples include developments on sparsity and regularization methods, empirical Bayes estimation, adpative inference, small area estimation, non-parametric function estimation, and predictive density estimation. Cast in a decision-theoretic framework, Stein’s original result has been expanded in many diverse ways, namely to other distributions or probability models, and namely for spherically symmetric and elliptically symmetric distributions (see for instance, [11]). There have been fewer findings for multivariate skew-normal or mean mixtures of normal distributions, but the recent work of Kubokawa et al. [18] establishes point estimation minimax improvements of the best location equivariant estimator under quadratic loss, when the dimension of the location parameter is greater than or equal to four, and with underlying known perturbation parameter. Predictive density estimation has garnered much interest over the past twenty years or so, and addresses fundamental issues in statistical predictive analysis. Decision-theoretic links between shrinkage point estimation and shirnkage predictive density estimation for normal models have surfaced (e.g., [14], [13]) and stimulated much activity (see for instance [12]), including findings for restricted parameter spaces (e.g., [10]). The main objective of this work is thus to explore the problem of predictive density estimation for mean mixtures of normal (MMN) distributions. A secondary objective is to provide novel representations for Bayesian posterior distributions and predictive densities, which have been found to be lacking in the literature. Following early findings of Komaki (e.g., [14]) on the predictive density estimation problem for multivariate normal models under Kullback-Leibler loss, George, Liang and Xu in [13] exhibited further parallels with the point estimation problem for normal distribution under quadratic loss. They provide sufficient conditions on marginal distributions and prior distributions to get improved shrinkage predictive density estimators when the dimension is greater than or equal to three. Thus, motivated by these connections, it is interesting to investigate whether such shrinkage plays any role in the predictive density estimation problem for mean-mixture of multivariate normal models and we focus on frequentist risk efficiency of predictive density estimators under Kullback-Leibler loss. Our contribution here consists in identifying classes of plug-in type predictive densities and of Bayes predictive densities which are minimax and dominate the benchmark minimum equivariant estimator (MRE) for the case when the dimension of the location parameter is greater than or equal to four. The organization of this manuscript is as follows. Section 2.1 contains several introductory definitions, properties and examples of MMN models, including a useful canonical form which subdivides the MMN distributed vector into $d$ independent components, one of which a univariate MMN distribution and the others as normal distributions. Section 2.2 focuses on the predictive estimation framework with a KL loss decomposition, and an initial representation for the MRE density accompanied by various examples. Section 2.3 expands on the calculation of minimax risk and a representation in terms of the entropy of a univariate distribution. Section 3 is devoted to Bayesian posterior and predictive analysis with several novel representations. Sections 4.1 and 4.2, namely Theorems 4.4, Theorem 4.5 and Corollary 4.1, contain the main dominance findings, with plug-in type and Bayesian improvements. In both cases, the main technique employed rests upon the canonical transformation presented in Section 2.1 and permits to split up the KL risk as the addition of two parts, one of which can be operated on using known normal model prediction analysis findings. Section 4.3 deals with parametric restrictions and further applications of Theorems 4.4 and 4.5. Finally, some further illustrations are provided in Section 5. 2 Preliminary results and definitions Here are some details, properties and definitions on mean mixture of normal distributions, its canonical form, and predictive density estimation. In the following, we will denote $\phi_{d}(z;\Sigma)$ the probability density function (pdf) of a $N_{d}(0,\Sigma)$ distribution evaluated at $z\in\mathbb{R}^{d}$ and for positive definite $\Sigma$. When $\Sigma=I_{d}$, we may simplify the writing to $\phi_{d}(z)$, and then for $d=1$ to $\phi(z)$. We will denote $\Phi$ the cdf of a $N(0,1)$ distribution. 2.1 The model The distributions of interest are mean-mixtures of multivariate normal distributions, both for our observables and densities to be estimated by a predictive density estimator. Such distributions connect to multivariate skew-normal distributions and have been the object of interest in recent work with studies of stochastic properties (e.g., [1], [5]), and shrinkage estimation about its location parameter ([18]). Definition 2.1. A random vector $X\in\mathbb{R}^{d}$ is said to have a mean-mixture of normal distributions (MMN), denoted as $X\sim MMN_{d}(\theta,a,\Sigma,\mathcal{L})$, if it admits the representation $$X|V=v\sim N_{d}(\theta+va\,,\Sigma)\,,\,V\sim\mathcal{L}\,,$$ (2.1) where $\theta\in\mathbb{R}^{d}$ is a location parameter, $a\in\mathbb{R}^{d}-\{0\}$ is a known perturbation vector, $\Sigma$ is a known positive definite covariance matrix, and $V$ is a scalar random variable with cdf $\mathcal{L}$. Alternatively, the random vector $X$ has stochastic representation $$\displaystyle X=\theta+\Sigma^{1/2}Z+Va\,,$$ (2.2) where $Z\sim N_{d}(0,I_{d})$ and $V\sim\mathcal{L}$ on $\mathbb{R}$, and its probability density function can be expressed as: $$\displaystyle p(x|\theta)$$ $$\displaystyle=\mathbb{E}^{V}\left\{\phi_{d}\left(x-\theta-Va,\Sigma\right)\right\}$$ $$\displaystyle=\phi_{d}\left(x-\theta,\Sigma\right)\,\mathbb{E}^{V}\left(e^{-\frac{V^{2}}{2}a^{T}\Sigma^{-1}a}e^{V\,(x-\theta)^{T}\Sigma^{-1}a}\right)\,.$$ (2.3) Thus, we note that the density function of MMN random vector can be decomposed in two parts: one symmetrical density $\phi_{d}(\cdot)$ and the other part which is a function of the projection of $(x-\theta)$ in the direction of $\Sigma^{-1}a$. Moreover, this construction isolates the asymmetry in the direction $\Sigma^{-1}a$ and the scale is controlled by the random variable $V$. Remark 2.1. It is easy to see that the family of MMN distributions is closed under linear combinations of independent components. Specifically, if $X_{i}|\theta\sim MMN_{d}(\theta,a,\Sigma_{i},\mathcal{L}_{i})$, $i=1,\ldots,n$, are independently distributed, then $\sum_{i=1}^{n}b_{i}X_{i}|\theta\sim MMN_{d}((\sum_{i=1}^{n}b_{i})\,\theta,a,\sum_{i=1}^{n}b_{i}^{2}\,\Sigma_{i},\mathcal{L}_{0})$ with $\mathcal{L}_{0}$ the cdf of the mixing variable $V_{0}=^{d}\sum_{i=1}^{n}b_{i}V_{i}$. Namely, for the identically distributed case with $\Sigma_{i}=\Sigma$ and the sample mean with $b_{i}=1/n$, we obtain that $$\bar{X}|\theta\,\sim\,MMN_{d}(\theta,a,\Sigma/n,\mathcal{L}_{0})\,,\hbox{ with }\mathcal{L}_{0}\,\hbox{ the cdf of }\bar{V}\,.$$ It thus follows, as observed in [18], that findings applicable for a single MMN distributed observable $X$ can be extended to the random sample case. We now turn our attention to a fundamental decomposition, or canonical form, ([1]) for MMN distributions which will be most useful. Lemma 2.1. For a random vector $X\sim MMN_{d}(\theta,a,\Sigma,\mathcal{L})$ as in (2.1), there exists an orthogonal matrix $H$ such that the first row of $H$ is proportional to $a^{\top}\,\Sigma^{-1/2}$ and $Z=H\Sigma^{-1/2}X$ has a $MMN_{d}(H\Sigma^{-1/2}\theta,a_{0},I_{d},\mathcal{L})$ distribution with $a_{0}=(\sqrt{a^{T}\Sigma^{-1}a},0,\dots,0)^{T}$. Such a $Z$ may be referred to as a canonical form and is comprised of $d$ independent components. Moreover $Z-H\Sigma^{-1/2}\theta$ has $d-1$ components which are $N(0,1)$ distributed and another distributed as $MMN_{1}(0,a_{0},1,\mathcal{L})\,$. Such a canonical form construction is not unique and depends on the choice of $H$. As already mentioned, the family of MMN distributions contains many interesting distributions and we refer to the above-mentioned references for various properties. We expand here with illustrations, which will also inform us for our predictive density problem and related Bayesian posterior analysis. A prominent example is the multivariate skew normal distribution due to Azzalini and Dalla Valle [6]. If we consider $V\sim TRN(0,1),$ the standard truncated normal distribution on $R_{+}$ in (2.1), we get the multivariate skew-normal family of distributions with densities $$p(x|\theta)\,=\,2\phi_{d}\left(x-\theta;\Sigma+aa^{T}\right)\Phi\left(\frac{(x-\theta)^{\top}\Sigma^{-1}a}{\sqrt{1+a^{\top}\Sigma^{-1}a}}\right).$$ (2.4) We denote this as $X\sim SN_{d}(\theta,a,\Sigma)$. Here, we note that $V\sim\sqrt{\chi^{2}_{1}}$, i.e. the square root of a Chi-square distribution with $k=1$ degrees of freedom. Various other choices of the mixing density have appeared in the literature (e.g., [5]), namely cases where $V\sim\sqrt{\chi^{2}_{k}}$ or $V$ is Gamma distributed. Here is a general result containing such cases as well as many others. Theorem 2.1. For a mixing density of the form $$\displaystyle\ell(v)=h(v)\,e^{-vc_{2}}\,e^{-\frac{v^{2}}{2}c_{1}}\,\mathbb{I}_{(0,\infty)}(v)\,,$$ (2.5) with $c_{1}>0,c_{2}\in\mathbb{R}$ or $c_{1}=0,c_{2}\geq 0$, the corresponding pdf of $X$ in (2.1) is given by $$\displaystyle p(x|\theta)$$ $$\displaystyle=\frac{1}{c_{1}^{\prime}}\,\phi_{d}\left(x-\theta,\Sigma\right)\,\frac{\mathbb{E}\left[\left.h\left\{\frac{1}{c_{1}^{\prime}}\left(Z+\frac{c_{2}^{\prime}}{c_{1}^{\prime}}\right)\right\}\right|Z+\frac{c_{2}^{\prime}}{c_{1}^{\prime}}\geq 0\right]}{R\left(\frac{c_{2}^{\prime}}{c_{1}^{\prime}}\right)},$$ (2.6) with $Z\sim N(0,1)$, $c_{1}^{\prime}=\left(c_{1}+a^{\top}\Sigma^{-1}a\right)^{1/2}$, $c_{2}^{\prime}=(x-\theta)^{\top}\Sigma^{-1}a-c_{2}\,$, and $R(\cdot)$ the reverse Mill’s ratio given by $R(t)=\phi(t)/\Phi(t),t\in\mathbb{R}$. Proof. The result follows from (2.3) as $$\displaystyle\mathbb{E}^{V}\left(e^{-\frac{V^{2}}{2}a^{T}\Sigma^{-1}a}e^{V(x-\mu)^{T}\Sigma^{-1}a}\right)$$ $$\displaystyle=$$ $$\displaystyle\int\limits_{0}^{\infty}\,e^{-\frac{v^{2}}{2}(c_{1}^{\prime})^{2}}e^{vc_{2}^{\prime}}\,h(v)\,dv$$ $$\displaystyle=$$ $$\displaystyle\frac{\sqrt{2\pi}}{c_{1}^{\prime}}\,\,e^{\frac{c_{2}^{\prime 2}}{2c_{1}^{\prime 2}}}\int\limits_{0}^{\infty}h(v)\,\frac{c_{1}^{\prime}}{\sqrt{2\pi}}\;e^{-\frac{c_{1}^{\prime 2}\left(v-\frac{c_{2}^{\prime}}{(c_{1}^{\prime})^{2}}\right)^{2}}{2}}\,dv$$ $$\displaystyle=$$ $$\displaystyle\frac{\sqrt{2\pi}}{c_{1}^{\prime}}\;e^{\frac{c_{2}^{\prime 2}}{2c_{1}^{\prime 2}}}\;\mathbb{E}\left\{\left.h\left(\frac{Z}{c_{1}^{\prime}}+\frac{c_{2}^{\prime}}{c_{1}^{\prime 2}}\right)\right|Z+\frac{c_{2}^{\prime}}{c_{1}^{\prime}}\geq 0\right\}\,\Phi\left(\frac{c_{2}^{\prime}}{c_{1}^{\prime}}\right)\,.\qed$$ We point out that the above Theorem applies for $c_{1}=c_{2}=0$ and thus covers all absolutely continuous distributions on $\mathbb{R}_{+}$. Here are nevertheless specific examples of Theorem 2.1 and model density (2.6). Example 2.1. (A) Gamma mixing with $h(v)\,=\,\frac{v^{\alpha-1}}{\Gamma(\alpha)\beta^{\alpha}}$. Theorem 2.1 applies with $c_{1}=0$ and $c_{2}=1/\beta$, and the model density is given by (2.6) with the above $h$, $c_{1}^{\prime}\,=\,\left(a^{\top}\Sigma^{-1}a\right)^{1/2}$ and $c_{2}^{\prime}\,=\,(x-\theta)^{\top}\Sigma^{-1}a-(1/\beta)$. The density was studied in [1, 2]. The exponential case with $\alpha=1$ simplifies with $$\noindent p(x|\theta)=\frac{1}{\beta c_{1}^{\prime}}\,\,\frac{\phi_{d}\left(x-\theta;\Sigma\right)}{R\left(\frac{c_{2}^{\prime}}{c_{1}^{\prime}}\right)}\,.$$ (2.7) More generally for positive integer $\alpha$, the density’s expression brings into play the $(\alpha-1)^{\hbox{th}}$ lower-truncated moment of a normal distribution. For instance, with $\mathbb{E}\left\{(Z+\Delta)|Z+\Delta\geq 0\right\}\,=\,\Delta\,+\,R(\Delta),$ we obtain for the case $\alpha=2$ the model density: $$p(x|\theta)\,=\,\frac{\phi_{d}(x-\theta,\Sigma)}{(c_{1}^{\prime}\,\beta)^{2}}\,\left\{\frac{c_{2}^{\prime}/c_{1}^{\prime}}{R(c_{2}^{\prime}/c_{1}^{\prime})}\,+\,1\right\}\,,$$ with the above $c_{1}^{\prime}$ and $c_{2}^{\prime}$. (B) $\sqrt{\chi_{k}^{2}}$ mixing with $h(v)\,=\,\frac{(\frac{1}{2})^{k/2-1}}{\Gamma(k/2)}\,v^{k-1}$, $c_{1}=1$, $c_{2}=0$, and $k>0$. The corresponding model density is given by (2.6) with the above $h$, $c_{1}^{\prime}=\left(1+a^{\top}\Sigma^{-1}a\right)^{1/2}$, and $c_{2}^{\prime}\,=\,(x-\theta)^{\top}\Sigma^{-1}a$. The density was given in [5] and, as previously noted, the case $k=1$ reduces to the skew-normal case in (2.4). As in Example (A) for positive integer $k$, the density’s expression involves a lower-truncated moment of a normal distribution. (C) Kummer type II mixing with $c_{2}=c/\sigma$, $c_{1}=0$, $h(v)=\frac{\sigma^{b}}{\Gamma(a)\,\psi(a,1-b,c)}\,\frac{v^{a-1}}{(v+\sigma)^{a+b}}$ with $a,c,\sigma>0$, $b\in\mathbb{R}$, and $\psi$ the confluent hypergeometric function of type II defined for $\gamma_{1},\gamma_{3}>0$ and $\gamma_{2}\in\mathbb{R}$ as $\psi(\gamma_{1},\gamma_{2},\gamma_{3})\,=\,\frac{1}{\Gamma(\gamma_{1})}\,\int_{\mathbb{R}_{+}}t^{\gamma_{1}-1}(1+t)^{\gamma_{2}-\gamma_{1}-1}\,e^{-\gamma_{3}t}\,dt$. This class of densities includes for $b=-a$ the Gamma densities in (A), as well as Beta type II densities for $c=0$ and $b>0$ The resulting mean-mixture density is given by (2.6) and involves interesting expectations of the form $\mathbb{E}\left(\frac{W^{a-1}}{(W+\sigma)^{a+b}}|W\geq 0\right)$ where $W\sim N(\Delta,1)$ with $\Delta=c_{2}^{\prime}/c_{1}^{\prime}$. 2.2 The prediction problem Consider $X|\theta\sim MMN_{d}(\theta,a,\Sigma_{X},\mathcal{L}_{1})$ and $Y|\theta\sim MMN_{d}(\theta,a,\Sigma_{Y},\mathcal{L}_{2}),$ independently distributed as in Definition 2.1, i.e. $$X|\theta,V_{1}\sim N_{d}(\theta+V_{1}\;a,\Sigma_{X})\,,\,Y|\theta,V_{2}\sim N_{d}(\theta+V_{2}\;a,\Sigma_{Y})\,,\hbox{ with }V_{1}\sim\mathcal{L}_{1}\,,\,V_{2}\sim\mathcal{L}_{2}\,.$$ (2.8) Let $p(x|\theta)$ and $q(y|\theta)$ denote the conditional densities of $X$ and $Y$ given $\theta$, respectively. Based on observing $X=x$, we consider the problem of finding a suitable predictive density estimator $\hat{q}(y;x)$ for $q(y|\theta)\,,y\in\mathbb{R}^{d}\,.$ The ubiquitous Kullack-Leibler (KL) divergence between two Lebesgue densities $f$ and $g$ on $\mathbb{R}^{m}$, defined as $$\rho(f,g)\,=\,\int_{\mathbb{R}^{m}}f(t)\,\log\frac{f(t)}{g(t)}\,dt\,,$$ is the basis of Kullback-Leibler loss given by $$L(\theta,\hat{q})\,=\,\rho(q_{\theta},\hat{q})\,.$$ (2.9) We will make use of Lemma 2.1’s canonical form as in (2.1) to transform a mean mixture of normal distributions vector into two independent components and to capitalize on the corresponding simplification for KL divergence which is as follows. Lemma 2.2. Let $T=(T_{(1)},T_{(2)})\in\mathbb{R}^{m}$ and $U=(U_{(1)},U_{(2)})\in\mathbb{R}^{m}$ be random vectors subdivided into independently distributed components $T_{(i)}$ and $U_{(i)}$ of dimensions $m_{i}$ for $i=1,2$ with $m_{1}+m_{2}=m$. Denote $f$ and $g$ the densities of $T$ and $U$, respectively, and $f_{1},f_{2},g_{1},g_{2}$ the densities of $T_{(1)},T_{(2)},U_{(1)},U_{(2)}$, respectively. Then, we have $$\rho(f,g)\,=\,\rho(f_{1},g_{1})\,+\,\rho(f_{2},g_{2})\,.$$ (2.10) Proof. By independence, we have $$\rho(f,g)\,=\,\mathbb{E}^{T}\left\{\log\left(\frac{f_{1}(T_{1})f_{2}(T_{2})}{g_{1}(T_{1})g_{2}(T_{2})}\right)\right\}\,=\,\mathbb{E}^{T}\left\{\log\left(\frac{f_{1}(T_{1})}{g_{1}(T_{1})}\right)\right\}\,+\,\mathbb{E}^{T}\left\{\log\left(\frac{f_{2}(T_{2})}{g_{2}(T_{2})}\right)\right\}\,,$$ which is (2.10). ∎ We evaluate the performance of the density estimators using KL loss in (2.9), and the associated KL risk function $$\displaystyle R_{KL}(\theta,\hat{q})=\int_{\mathbb{R}^{d}}\{\int_{\mathbb{R}^{d}}q(y|\theta)\;\text{log}\;\frac{q(y|\theta)}{\hat{q}(y;x)}\;dy\}\;p(x|\theta)\;dx.$$ (2.11) For a prior density $\pi$ for $\theta$ with respect to a $\sigma-$finite measure $\nu$, it is known (e.g., [3, 4]) that the Bayes predictive density is given by $$\displaystyle\hat{q}_{\pi}(y;x)=\int_{\mathbb{R}^{d}}q(y|\theta)\;p(x|\theta)\;\pi(\theta)\;d\nu(\theta).$$ (2.12) A benchmark predictive density estimator for $q(y|\theta),y\in\mathbb{R}^{d}$, is given by the Bayes predictive density estimator $\hat{q}_{U}(y;X),y\in\mathbb{R}^{d}$, with respect to the uniform prior density on $\mathbb{R}^{d}$. It is known to be the minimum risk equivariant (MRE) predictive density estimator under changes of location, as well as minimax. In [16], a representation, which applies to both integrated squared-error loss and KL loss, for $\hat{q}_{U}$ is provided. For our prediction problem, the following result makes use of this representation and summarizes the above optimality properties. Lemma 2.3. The MRE predictive density estimator of the density of $Y$ relative to model (2.8) under KL loss, is given by the Bayes predictive density $\hat{q}_{U}$ under prior $\pi_{U}(\theta)=1\,$. Furthermore, we have $$\displaystyle\hat{q}_{U}(\cdot;X)\sim MMN_{d}(X,a,\Sigma_{X}+\Sigma_{Y},\mathcal{L}_{3})\,,$$ (2.13) where $\mathcal{L}_{3}$ is the cdf of $V_{3}=V_{2}-V_{1}$. Finally, $\hat{q}_{U}(y;X)$ is minimax under KL loss. Proof. The MRE and minimax properties are given in [24] and [20], respectively. For a location family prediction problem with $X\sim p(x-\theta)$ and $Y\sim q(y-\theta)$ independently distributed, it is shown in [16] that $$\hat{q}_{U}(y;X)\,=\,q*\bar{p}(y-x)\,,\hbox{ with }\bar{p}(t)=p(-t)\,,$$ i.e., the convolution of $q$ and the additive inverse of $p$ followed by a change of location equal to $x$. For model (2.1), the above convolution $q*\bar{p}$ is given by the density of $Y-X$ in model (2.1) with $\theta=0$, and the result follows since $$Y-X|V_{1},V_{2}\sim N_{d}((V_{2}-V_{1})\,a,\Sigma_{X}+\Sigma_{Y})\,.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\qed$$ Here, we can see that the MRE density estimator also belongs to the class of MMN distributions with same perturbation parameter $a$ and location parameter $x$. As well, the distribution of the difference $V_{2}-V_{1}$ plays a key role in Theorem 2.3’s representation of the MRE predictive density, and as illustrated in the next subsection of examples. 2.3 Minimax risk and entropy The Kullback-Leibler risk expressions brings into play the entropy associated with MMN distributions. Such a measure is not easily manipulated into a closed form (see for instance [9] for the study of entropy for skewed-normal distributions), but they can be expressed in terms of the entropy of a univariate MMN distribution, as illustrated with the following expansion of the constant and minimax risk of the MRE density $\hat{q}_{U}$ in the context of model (2.8). For a Lebesgue density on $\mathbb{R}^{d}$, defined as $$H(f)\,=\,-\int_{\mathbb{R}^{d}}f(t)\,\log f(t)\,dt\,,$$ we will make use of the following well-known and easily established properties. Lemma 2.4. (a) For $T\in\mathbb{R}^{d}$ with density $f$ and $U=\psi(T)\sim g$ with $\psi:\mathbb{R}^{d}\to\mathbb{R}^{d}$ invertible with inverse Jacobian $J_{\psi}$, we have $H(g)\,=\,-\mathbb{E}\log|J_{\psi}|+H(f)\,;$ (b) Let $T=(T_{(1)},T_{(2)})\sim f$ be a random vector with independently distributed components $T_{(1)}\sim f_{1}$ on $\mathbb{R}^{m_{1}}$ and $T_{(2)}\sim f_{2}$ on $\mathbb{R}^{m_{2}}$. Then (as in Lemma 2.2), we have $H(f)\,=\,H(f_{1})\,+\,H(f_{2})\,$. As implied by part (a) of the above lemma, the entropy $H(f_{\mu})$ is constant as a function of $\mu$ for location family densities $f_{\mu}(t)=f_{0}(t-\mu)$, as is the case for $MMN_{d}(\mu,b,\Sigma,\mathcal{L})$ densities. Now, we have the following dimension reduction decomposition for the entropy $H_{d}(b,\Sigma,\mathcal{L})$ of a $MMN_{d}(0,b,\Sigma,\mathcal{L})$ density. Lemma 2.5. We have for $d\geq 2$: $$H_{d}(b,\Sigma,\mathcal{L})\,=\,H_{1}(\sqrt{b^{\top}\Sigma^{-1}b},1,\mathcal{L})\,+\,\frac{d-1}{2}\,\{1+\log(2\pi)\}\,+\,\frac{1}{2}\log|\Sigma|\,.$$ Proof. Let $X\sim MMN_{d}(0,b,\Sigma,\mathcal{L})$, which has entropy $H_{d}(b,\Sigma,\mathcal{L})$, and set $Z=H\,\Sigma^{-1/2}\,X\sim f_{Z}$ with $H$ orthogonal having first row $\frac{b^{\top}\Sigma^{-1/2}}{\sqrt{(b^{\top}\Sigma^{-1}b)}}$. It follows from part (a) of Lemma 2.4 that $H(f_{Z})=-\frac{1}{2}\,\log|\Sigma|+H_{d}(b,\Sigma,\mathcal{L})$. From Lemma 2.1, we have $Z=(Z_{1},Z_{(2)})^{\top}$ with $Z_{1}\sim MMN_{1}(0,\sqrt{(b^{\top}\Sigma^{-1}b)},1,\mathcal{L})$ and $Z_{2}\sim N_{d-1}(0,I_{d-1})$ independently distributed, and the result follows from part (b) of Lemma 2.4 and a straightforward evaluation of the entropy $H(\phi_{d-1})$. ∎ With the above, we conclude with an expression for the constant and minimax risk. Theorem 2.2. In the context of model (2.8), the Kullback-Leibler risk of the MRE density $\hat{q}_{U}$ is given by $$R_{KL}(\theta,\hat{q}_{U})\,=\,H_{1}(\sqrt{a^{\top}\Sigma_{S}^{-1}a},1,\mathcal{L}_{3})\,-\,H_{1}(\sqrt{a^{\top}\Sigma_{Y}^{-1}a},1,\mathcal{L}_{2})+\frac{1}{2}\log\frac{\Sigma_{S}}{\Sigma_{Y}}\,,$$ (2.14) with $\Sigma_{S}\,=\,\Sigma_{X}+\Sigma_{Y}$. Proof. We have for $\theta\in\mathbb{R}^{d}\,$ $$\displaystyle R_{KL}(\theta,\hat{q}_{U})\,$$ $$\displaystyle=$$ $$\displaystyle\,\mathbb{E}_{\theta}\{\log q_{\theta}(Y)\,-\,\log\hat{q}_{U}(Y;X)\}$$ $$\displaystyle=$$ $$\displaystyle\,H(\hat{q}_{U})\,-\,H(q_{0})\,$$ $$\displaystyle=$$ $$\displaystyle\,H_{d}(a,\Sigma_{S},\mathcal{L}_{3})\,-\,H_{d}(a,\Sigma_{Y},\mathcal{L}_{2})\,,$$ by the independence of $X$ and $Y$, the constancy of location family density $q_{\theta}$, and since $Y-X|\theta\sim MMN_{d}(0,a,\Sigma_{S},\mathcal{L}_{3})$. The result then follows from Lemma 2.5. ∎ The particular case with $\Sigma_{X}=\sigma^{2}_{X}I_{d}$ and $\Sigma_{Y}=\sigma^{2}_{Y}I_{d}$ follows directly from (2.14) and yields $$R_{KL}(\theta,\hat{q}_{U})\,=\,H_{1}(\frac{\|a\|}{\sigma_{S}},1,\mathcal{L}_{3})\,-\,H_{1}(\frac{\|a\|}{\sigma_{Y}},1,\mathcal{L}_{2})+\frac{d}{2}\log\frac{\sigma^{2}_{S}}{\sigma^{2}_{Y}}\,,$$ (2.15) 2.4 Minimum risk predictive density: Examples Theorem 2.1 tells us that the minimum risk predictive density is given by $\hat{q}_{U}(\cdot;X)\sim\hbox{MMN}_{d}(X,a,\Sigma_{X}+\Sigma_{Y},\mathcal{L}_{3})$ with $\mathcal{L}_{3}$ the cdf of $V_{2}-V_{1}$. The result is quite general and can be viewed as an extension of the multivariate normal case with $a=0$ and $\hat{q}_{U}(\cdot;X)\sim\hbox{N}_{d}(X,\Sigma_{X}+\Sigma_{Y})$. Here are some interesting examples. When continuous, the mixing distributions can be taken to have a scale parameter equal to one without loss of generality, since a multiple can be integrated into the shape vector $a$. (A) For the case of degenerate $V_{2}$ with $\mathbb{P}(V_{2}=v_{2})=1$, i.e., when the distribution of $Y|\theta$ is normal with $Y\sim N_{d}(\theta+av_{2},\Sigma_{Y})$, the minimum risk equivariant predictive density reduces to $\hat{q}_{U}(\cdot;X)\sim MMN_{d}(X+av_{2},-a,\Sigma_{X}+\Sigma_{Y},\mathcal{L}_{1})$. (B) For the case of degenerate $V_{1}$ with $\mathbb{P}(V_{1}=v_{1})=1$, i.e., when the distribution of $X$ is normal with $X|\theta\sim N_{d}(\theta+av_{1},\Sigma_{Y})$, the minimum risk equivariant predictive density reduces to $\hat{q}_{U}(\cdot;X)\sim MMN_{d}(X-av_{1},a,\Sigma_{X}+\Sigma_{Y},\mathcal{L}_{2})$. (C) We consider in this example $V_{1},V_{2}$ i.i.d. exponentially distributed with densities $f(t)=e^{-t}\,\mathbb{I}_{(0,\infty)}(t)$, as well as $\Sigma_{X}=\sigma^{2}_{X}\,I_{d}$ and $\Sigma_{Y}=\sigma^{2}_{Y}\,I_{d}$. Here the distribution of $V_{3}$ is Laplace or double-exponential with density $\frac{1}{2}\,e^{-|v_{3}|}$ on $\mathbb{R}$. Therefore, from Theorem 2.1, we have $$\displaystyle\hat{q}_{U}(y;x)\,$$ $$\displaystyle=$$ $$\displaystyle\,\int_{\mathbb{R}}\frac{1}{2}\,e^{-|v_{3}|}\frac{1}{\sigma_{S}^{d}}\,\phi_{d}(\frac{y-x-av_{3}}{\sigma_{S}})\,dv_{3}\,,$$ $$\displaystyle=$$ $$\displaystyle\phi_{d}\left(y-x;\sigma^{2}_{S}I_{d}\right)\,\int_{\mathbb{R}_{+}}e^{-(v_{3}^{2}\frac{\|a\|^{2}}{2\sigma^{2}_{S}}+v_{3})}\;\cosh\left(v_{3}(\frac{(y-x)^{\top}a}{\sigma^{2}_{S}}\right)\,dv_{3}$$ with $\sigma_{S}=(\sigma^{2}_{X}+\sigma^{2}_{Y})^{1/2}$. By making use of Lemma 5.9 in the Appendix with $A=\frac{\|a\|^{2}}{\sigma^{2}_{S}}$, $B=-1\,\pm\,\frac{(y-x)^{\top}a}{\sigma^{2}_{S}}$, and $c=0$, we obtain (for $a\neq 0$) $$\begin{split}\hat{q}_{U}(y;x)\,=&\,\sqrt{\frac{\pi\sigma^{2}_{S}}{2\|a\|^{2}}}\,\phi_{d}(y-x;\sigma^{2}_{S}I_{d})\,e^{\frac{\sigma^{2}_{S}}{2\|a\|^{2}}\,+\,\frac{\{(y-x)^{\top}a\}^{2}}{2\sigma^{2}_{S}\|a\|^{2}}}\\ \,&\times\,\left[\left\{e^{-\frac{(y-x)^{\top}a}{\|a\|^{2}}}\,\Phi\left(\frac{\sigma_{S}}{\|a\|}(\frac{(y-x)^{\top}a}{\sigma^{2}_{S}}-1)\right)\right\}+\left\{e^{\frac{(y-x)^{\top}a}{\|a\|^{2}}}\,\Phi\left(-\frac{\sigma_{S}}{\|a\|}(\frac{(y-x)^{\top}a}{\sigma^{2}_{S}}+1)\right)\right\}\right]\,.\end{split}$$ (D) Consider $V_{1},V_{2}$ i.i.d. truncated normal distributed $\hbox{TN}(0,1)$ (or equivalently as $\sqrt{\chi^{2}_{1}}$) for which $X$ and $Y$ are i.i.d. as multivariate skew-normal as in (2.4). A straightforward calculation yields the density $$g_{V_{3}}(t)\,=\,2\sqrt{2}\;\phi(\frac{t}{\sqrt{2}})\;\Phi(-\frac{|t|}{\sqrt{2}})\,\,\mathbb{I}_{\mathbb{R}}(t)\,,$$ for $V_{3}=^{d}V_{1}-V_{2}$. It follows from Theorem 2.1, for $\Sigma_{X}=\sigma^{2}_{X}\,I_{d}$ and $\Sigma_{Y}=\sigma^{2}_{Y}\,I_{d}$, denoting $\sigma_{S}=(\sigma^{2}_{X}+\sigma^{2}_{Y})^{1/2}$, that $$\displaystyle\hat{q}_{U}(y;x)\,$$ $$\displaystyle=$$ $$\displaystyle\,\int_{\mathbb{R}}\,2\sqrt{2}\;\phi(\frac{t}{\sqrt{2}})\;\Phi(-\frac{t}{\sqrt{2}})\,\phi_{d}(y-x-at;\sigma_{S}^{2}I_{d})\,dt\,,$$ $$\displaystyle=$$ $$\displaystyle\frac{2}{\sqrt{\pi}}\,\phi_{d}(y-x;\sigma_{S}^{2}I_{d})\,\int_{\mathbb{R}_{+}}\Phi(-\frac{t}{\sqrt{2}})\,e^{-\frac{t^{2}}{2}(\frac{1}{2}+\frac{a^{\top}a}{\sigma_{S}^{2}})}\left\{e^{\frac{(y-x)^{\top}at}{\sigma^{2}_{S}}}+e^{-\frac{(y-x)^{\top}at}{\sigma^{2}_{S}}}\right\}dt\,.$$ Now, by making use of Lemma 5.9 with $c=-\frac{\sqrt{2}}{2}$, $A=\frac{2\sigma_{S}^{2}}{\sigma_{S}^{2}+2a^{\top}a}$, and $B=\pm\frac{(y-x)^{\top}a}{\sigma^{2}_{S}}$, collecting terms, and setting $f_{k}=\sqrt{\sigma^{2}_{S}+ka^{\top}a}$, we obtain the minimum risk equivariant predictive density $$\displaystyle\hat{q}_{U}(y;x)\,$$ $$\displaystyle=$$ $$\displaystyle\,\frac{4\sigma_{S}}{f_{1}}\,\phi_{d}\left(y-x;\sigma^{2}_{S}(I_{d}+\frac{aa^{\top}}{f_{1}^{2}})\right)\,$$ $$\displaystyle\!\times$$ $$\displaystyle\!\!\!\left\{\Phi_{2}\left(-\frac{(y-x)^{\top}a}{f_{1}f_{2}},\frac{\sqrt{2}(y-x)^{\top}a}{\sigma_{S}f_{2}};\frac{-\sigma_{S}}{\sqrt{2}\,f_{2}}\right)\,+\,\Phi_{2}\left(\frac{(y-x)^{\top}a}{f_{1}f_{2}},-\frac{\sqrt{2}(y-x)^{\top}a}{\sigma_{S}f_{2}};\frac{-\sigma_{S}}{\sqrt{2}f_{2}}\right)\right\}\,,$$ where $\Phi_{2}(z_{1},z_{2};\rho)$ the cdf evaluated at $z_{1},z_{2}\in\mathbb{R}$ of a bivariate normal distributions with means equal to $0$, variances equal to $1$ and covariance equal to $\rho$. In the evaluation above, we made use of the identities $(I-\frac{aa^{\top}}{f_{2}^{2}})^{-1}\,=\,I+\frac{aa^{\top}}{f_{1}^{2}}$ and $|I+\frac{aa^{\top}}{f_{1}^{2}}|\,=\,1+\frac{a^{\top}a}{f_{1}^{2}}\,$, which is a special case of the Sherman-Morrison formula for the matrix inversion of $A+b_{1}b_{2}^{\top}$ with $A$ being a square matrix and $b_{1}$ and $b_{2}$ vectors of the same dimension. 3 Bayes posterior analysis and predictive densities In this section, we expand on and document representations for Bayesian posterior and predictive densities for mean mixture of normal distributions. 3.1 Posterior densities Bayesian posterior analysis of MMN models relate to the general form $$X|K,\theta\sim f_{\theta,K}\,,\,K\sim g\,,\,\hbox{ and }\theta\sim\pi\,,$$ (3.16) with observable $X\in\mathbb{R}^{d}$, density $g$ of $K$ free of $\theta$, and $\pi$ prior density for $\theta\in\mathbb{R}^{d}$. Such a set-up leads to the following intermediate result, taken from [21]. Lemma 3.6. For model (3.16), the posterior distribution of $U=^{d}\theta|x$ admits the representation $$U|K^{\prime}\sim\pi_{k^{\prime},x}\hbox{ with }K^{\prime}\sim g_{\pi,x}\,,$$ (3.17) $\pi_{k^{\prime},x}$ being the posterior density of $\theta$ as if $K=k^{\prime}$ had been observed, and $g_{\pi,x}(k^{\prime})\propto g(k^{\prime})\,m_{\pi,k^{\prime}}(x)$ with $m_{\pi,k^{\prime}}$ being the marginal density of $X$ as if $K=k^{\prime}$ had been observed. We now apply the above to MMN distributions as in Definition 2.1. Example 3.2. We apply Lemma 3.6 to $X|\theta\sim MMN_{d}(\theta,a,\Sigma,\mathcal{L})$ and the prior $\theta\sim N_{d}(\mu,\Delta)$ with $\Sigma,\Delta>0$. The above fits into model (3.16) with $g$ taken to be the density of the mixing parameter $K=V\sim\mathcal{L}$, and $f_{\theta,k}$ the $N_{d}(\theta+ka,\Sigma)$ density. Conditional on $K=k^{\prime}$, standard Bayesian analysis for the normal model tells us that $$\theta|k^{\prime},x\sim N_{d}\left((I-P)x+P\mu-k^{\prime}a,(I-P)\,\Sigma\right)\,,\,\hbox{ and }X|k^{\prime}\sim N_{d}(\mu+k^{\prime}a,\Sigma+\Delta)\,,$$ (3.18) with $P=\Sigma\,(\Sigma+\Delta)^{-1}$, which yields the densities $\pi_{k^{\prime},x}$ and $m_{\pi,k^{\prime}}$ of Lemma 3.6. Then from Lemma 3.6, we infer that $$\theta|x\sim MMN_{d}\left((I-P)x+P\mu,\,a^{*}=-a\,,\,(I-P)\,\Sigma\,,\mathcal{L}^{*}\right)\,,$$ (3.19) where the distribution $\mathcal{L}^{*}$ has density $$g_{\pi,x}(k^{\prime})\,\propto g(k^{\prime})\,e^{-\frac{A}{2}k^{\prime 2}+Bk^{\prime}}\,,\,\hbox{ with }A=a^{\top}(\Sigma+\Delta)^{-1}a\,\hbox{ and }B=(x-\mu)^{\top}(\Sigma+\Delta)^{-1}a\,.$$ (3.20) Furthermore, it follows immediately that $$\mathbb{E}(\theta|x)\,=\,(I-P)x+P\mu\,-\,P\,a\,\mathbb{E}(K^{\prime})\,,\,\hbox{ with }K^{\prime}\sim g_{\pi,x}\,.$$ (3.21) Remark 3.2. For the improper prior density $\pi(\theta)\,=\,1$, one obtains $\theta|x\sim MMN_{d}(x,-a,\Sigma,\mathcal{L})$ by a direct calculation. It can also be inferred from the above Example with $\Delta=\tau^{2}I_{d}$ and $\tau^{2}\to\infty$. Example 3.3. It is interesting to further study the above posterior distributions for the particular cases where the mixing density (i.e., $V$ or $K$) of the MMN model is of the form $$g(k)\propto e^{-c_{1}k^{2}/2-c_{2}k}\;\mathbb{I}_{(0,\infty)}(k),$$ (3.22) with $c_{1}>0,c_{2}\in\mathbb{R}$ or $c_{1}=0,c_{2}>0$. Several of these distributions were presented in Example 2.1, but we recall that the cases $c_{1}>0$ for instance, which correspond to truncated normal distributions on $(0,\infty)$, lead to skew-normal densities (2.4) for $c_{2}=0$. In the following, denote $\hbox{TN}\left(a,b;(0,\infty)\right)$ as a truncated normal distribution on $(0,\infty)$ with shape parameter $a\in\mathbb{R}$, scale parameter $b>0$, density $\frac{1}{b}\,\frac{\phi((y-a)/b)}{\Phi(a/b)}\,\mathbb{I}_{(0,\infty)}(y)$, and expectation $a\,+\,bR(a/b)$, with the reverse Mill’s ratio $R(\cdot)$. Now, it is easily seen for cases where $K\sim g$ as in (3.22) that $$\displaystyle g_{\pi,x}(k^{\prime})\,$$ $$\displaystyle\propto$$ $$\displaystyle e^{-(c_{1}+A)k^{\prime 2}/2\,+\,(B-c_{2})k^{\prime}}\,\mathbb{I}_{(0,\infty)}(k^{\prime})$$ $$\displaystyle\propto$$ $$\displaystyle\phi\left(\sqrt{A+c_{1}}\,k^{\prime}\,-\,\frac{(B-c_{2})}{\sqrt{A+c_{1}}}\right)\mathbb{I}_{(0,\infty)}(k^{\prime})\,,$$ which is the density of a $\hbox{TN}\left(\frac{B-c_{2}}{A+c_{1}},\frac{1}{\sqrt{A+c_{1}}};(0,\infty)\right)$ distribution. Hence, the above, which yields the density associated with $\mathcal{L}$, provides a complete description of the posterior distribution in (3.19) for all considered cases of mixing density (3.22). Analogously, the corresponding expectation $\mathbb{E}(K^{\prime})\,=\,\frac{B-c_{2}}{A+c_{1}}\,+\frac{1}{\sqrt{A+c_{1}}}\,R(\frac{B-c_{2}}{\sqrt{A+c_{1}}})$ provides an explicit expression for the posterior expectation $\mathbb{E}(\theta|x)$ in (3.21). 3.2 Predictive densities We now continue the above posterior analysis by focussing on the Bayes predictive density (i.e., the conditional density of $Y$ given $X=x$) for MMN distributions and a normally distributed prior for the unknown location parameter. In doing so, the following extension come into play. Definition 3.2. A random vector $Z\in\mathbb{R}^{d}$ is said to have a mean mixture of normal distribution with two directions, denoted as $Z\sim MMN_{d}(\theta,a_{1},a_{2},\Sigma,\mathcal{L})$, if it admits the representation $$\displaystyle Z|V_{1},V_{2}$$ $$\displaystyle\sim N_{d}\left(\theta+a_{1}W_{1}+a_{2}W_{2}\,,\Sigma\right)\hbox{ with }(W_{1},W_{2})\sim\mathcal{L},$$ where $\theta\in\mathbb{R}^{d}$ is a location parameter, $a_{1},a_{2}\in\mathbb{R}^{d}$ are known perturbation vectors, $\Sigma$ is a known positive definite covariance matrix, and $W_{1},W_{2}$ are scalar random variable with joint cdf $\mathcal{L}$. We make use of the following intermediate result provided in [21] and applicable to mixture models of the form: $$X|K,\theta\sim f_{\theta,K}\hbox{ with }K\sim g\,;Y|J,\theta\sim f_{\theta,J}\hbox{ with }J\sim h,\hbox{ and }\theta\sim\pi.$$ (3.23) In the above set-up, $X\in\mathbb{R}^{d}$ is observable, the mixing variables $K$ and $J$ are independently distributed with distributions free of $\theta$, the variables $X$ and $Y$ are conditionally independent on $\theta$, and $\pi$ is a prior density for $\theta\in\mathbb{R}^{d}$ with respect to a $\sigma-$finite measure $\nu$. Lemma 3.7. For model (3.23), setting $\pi_{k^{\prime},x}$ and $g_{\pi,x}$ as in Lemma 3.6, the Bayes predictive density of $Y$ admits the mixture representation $$Y|J^{\prime},K^{\prime}\sim q_{\pi}(\cdot|J^{\prime},K^{\prime}),\hbox{ with }J^{\prime}\sim h,K^{\prime}\sim g_{\pi,x}\hbox{ independent },$$ and $q_{\pi}(y|j^{\prime},k^{\prime})\,=\,\int_{\mathbb{R}^{d}}q_{\theta,j^{\prime}}(y)\,\pi_{k^{\prime},x}(\theta)\,d\nu(\theta)$, which can be interpreted as the Bayes predictive density for $Y$ as if $Y\sim q_{\theta,j^{\prime}}$ and $K=k^{\prime}$ had been observed. Applied to mean mixture of multivariate normal distributions with a normal distributed prior, we obtain the following presented as a theorem. Theorem 3.3. (a) For $X|\theta\sim MMN_{d}(\theta,a_{X},\sigma_{X}^{2}I_{d},\mathcal{L}_{1})$ and $Y|\theta\sim MMN_{d}(\theta,a_{Y},\sigma_{Y}^{2}I_{d},\mathcal{L}_{2})$ independent with prior $\theta\sim N_{d}(\mu,\tau^{2}I_{d})$, the Bayes predictive density for $Y$ is that of a $$MMN_{d}\left(\omega x+(1-\omega)\mu,-\omega a_{X},a_{Y},(\omega\sigma_{X}^{2}+\sigma_{Y}^{2})I_{d},\mathcal{L}\right)\,$$ distribution, with $\mathcal{L}$ the joint cdf of $(K^{\prime},J^{\prime})$ with independently distributed $K^{\prime}\sim g_{\pi,x}$ as in (3.20) and $J^{\prime}\sim\mathcal{L}_{2}$, with $\omega=\tau^{2}/(\tau^{2}+\sigma_{X}^{2})$, $A=\|a_{X}\|^{2}/(\sigma_{X}^{2}+\tau^{2})$, and $B=\{(x-\mu)^{\top}a_{X}\}/(\sigma_{X}^{2}+\tau^{2}))$. (b) Moreover, whenever $a_{Y}=ca_{X}$ for $a_{X}\neq 0$ and a fixed $c\in\mathbb{R}$, the above predictive distribution is $MMN_{d}\,\left(\omega x+(1-\omega)\mu,a_{X},(\omega\sigma_{X}^{2}+\sigma_{Y}^{2})I_{d},\mathcal{L}_{3}\right)$, with $\mathcal{L}_{3}$ the cdf of $cJ^{\prime}-\omega K^{\prime}$, and $(J^{\prime},K^{\prime})$ distributed as above. Finally, for $a_{X}=0$, i.e., for $X|\theta\sim N_{d}(\theta,\sigma_{X}^{2}I_{d})$, the predictive distribution is $MMN_{d}(\omega x+(1-\omega)\mu,a_{Y},(\omega\sigma_{X}^{2}+\sigma_{Y}^{2})I_{d},\mathcal{L}_{2})$ Proof. Part (b) follows immediately from part (a). For part (a), consider $X^{\prime}=X-K^{\prime}a_{X}$ and $Y^{\prime}=Y-J^{\prime}a_{Y}$. The result then follows from Lemma 3.7 with the familiar predictive density estimation result: $$Y^{\prime}|J^{\prime},K^{\prime},X^{\prime}\sim N_{d}\left(\omega X^{\prime}+(1-\omega)\mu,(\omega\sigma_{X}^{2}+\sigma_{Y}^{2})I_{d}\right),$$ implying $$q_{\pi}(\cdot|J^{\prime},K^{\prime})\,\sim\,N_{d}\left(\omega x+(1-\omega)\mu-\omega a_{X}K^{\prime}+a_{Y}J^{\prime},(\omega\sigma_{X}^{2}+\sigma_{Y}^{2})I_{d}\right)\,,$$ matching Definition 3.2 with $(W_{1},W_{2})=^{d}(K^{\prime},J^{\prime})$. ∎ Remark 3.3. We point out that the minimum risk predictive density matches the density in (b) with $\tau^{2}=\infty$, i.e., $\omega=1$. 4 Dominance Results In this section, we first provide KL risk improvements on the MRE predictive density $\hat{q}_{U}$ for estimating the density of $Y|\theta\sim MMN_{d}(\theta,a,\sigma^{2}_{Y}I_{d},\mathcal{L}_{2})$ based on $X|\theta\sim MMN_{d}(\theta,a,\sigma^{2}_{X}I_{d},\mathcal{L}_{1})$ with $d\geq 4$. Such improvements are necessarily minimax as a consequence of Theorem 2.3. Our findings cover two types of improvements: (i) plug-in type (Section 4.1), and (ii) Bayesian improvements (Section 4.2). Furthermore, we provide analogue results for certain type of restricted parameter spaces which are also applicable for $d=2,3$. Examples will be provided in Section 5. The restriction to covariance matrices that are multiple of identity is justified by convenience and the fact that there is no loss of generality in doing so. Remark 4.4. Predictive density estimates are intrinsic by nature which implies that the developments of this section, presented for $\Sigma_{X}=\sigma^{2}_{X}I_{d}$ and $\Sigma_{Y}=\sigma^{2}_{Y}I_{d}$ in model (2.1) with known $\sigma^{2}_{X}$ and $\sigma^{2}_{Y}$, apply as well for $\Sigma_{Y}=c\Sigma_{X}$ with known $\Sigma_{X},\Sigma_{Y}$, and $c=\sigma^{2}_{Y}/\sigma^{2}_{X}$. Indeed, one can consider $X^{\prime}=\Sigma_{X}^{-1/2}X$ for which $X|\theta\sim MMN_{d}(\Sigma_{X}^{-1/2}\theta,\Sigma_{X}^{-1/2}a,I_{d},\mathcal{L}_{1})$ to estimate the density of $Y^{\prime}=\Sigma_{X}^{-1/2}Y$, for which $Y^{\prime}|\theta\sim MMN_{d}(\Sigma_{X}^{-1/2}\theta,\Sigma_{X}^{-1/2}a,cI_{d},\mathcal{L}_{2})$. In doing so, one produces a predictive density estimator $q_{1}(y^{\prime})\,=\,\hat{q}(y^{\prime};x^{\prime}),y^{\prime}\in\mathbb{R}^{d}$, for the density $q_{Y^{\prime}}$ of $Y^{\prime}$, which equates to $q_{2}(y)\,=\,\hat{q}(\Sigma_{X}^{-1/2}y;\Sigma_{X}^{-1/2}x)\,|\Sigma_{X}^{-1/2}|$; $y\in\mathbb{R}^{d}$; as a predictive density estimator of the density $q_{Y}$ of $Y$. Moreover, the Kullback-Leibler $\rho(q_{Y^{\prime}},q_{1})$ and $\rho(q_{Y},q_{2})$ are equal, i.e. $$\int_{\mathbb{R}^{d}}q_{Y^{\prime}}(t)\,\log\frac{q_{Y^{\prime}}(t)}{q_{1}(t)}\,dt\,=\,\int_{\mathbb{R}^{d}}q_{Y}(t)\,\log\frac{q_{Y}(t)}{q_{2}(t)}\,dt\,,$$ as seen with the change of variables $t\to\Sigma_{X}^{-1/2}t$. 4.1 Plug-in type improvements In the normal case with $X|\theta\sim N_{d}\left(\theta,\sigma^{2}_{X}I_{d}\right)$ and $Y|\theta\sim N_{d}\left(\theta,\sigma^{2}_{Y}I_{d}\right)$ independently distributed, the MRE predictive density $\hat{q}_{U}(\cdot;X)\sim N_{d}\left(X,(\sigma^{2}_{X}+\sigma^{2}_{Y})I_{d}\right)$ is inadmissible for $d\geq 3$ and can be improved by plug-in type densities of the form $q_{\hat{\theta}}(\cdot;X)\sim N_{d}\left(\hat{\theta}(X),(\sigma^{2}_{X}+\sigma^{2}_{Y})I_{d}\right)$. Indeed, the KL risk performance of $q_{\hat{\theta}}$ relates directly to the “dual” point estimation risk of $\hat{\theta}(X)$ for estimating $\theta$ under squared error loss $\|\hat{\theta}-\theta\|^{2}$, with $q_{\hat{\theta}}(\cdot;X)$ dominating $\hat{q}_{U}(\cdot;X)$ if and only if $\hat{\theta}(X)$ dominates $X$ ([10]). For MMN distributions, such a duality does not deploy itself in the same way, but does so after transformation of $(X,Y)$ to a canonical form and through the intrinsic nature of predictive densities. The following result exhibits this and is applicable to $d\geq 4$. Theorem 4.4. Consider $X,Y$ distributed as in model (2.8) with $a\neq 0,d\geq 4,\theta\in\mathbb{R}^{d},\Sigma_{X}=\sigma^{2}_{X}I_{d},\hbox{ and }\Sigma_{Y}=\sigma^{2}_{Y}I_{d}$, and the problem of obtaining a predictive density estimator $\hat{q}(y;X)$, $y\in\mathbb{R}^{d}$, for the density of $Y$. Let $H=\begin{pmatrix}{h_{1}^{\top}}\\ {H_{2}}\end{pmatrix}$ be an $d\times d$ orthogonal matrix such that $h_{1}=\frac{a}{\|a\|}$. Define the densities $$q_{1}(\cdot;X)\sim MMN_{1}\left(h_{1}^{\top}X,\|a\|,(\sigma^{2}_{X}+\sigma^{2}_{Y}),\mathcal{L}_{3}\right)\hbox{ and }q_{2,\hat{\zeta}_{2}}(\cdot;X)\sim N_{d-1}\left(\hat{\zeta}_{2}(H_{2}X),(\sigma^{2}_{X}+\sigma^{2}_{Y})I_{d-1}\right)\,.$$ Then, the predictive density $q_{H,\hat{\zeta}_{2}}(y;X)\,=\,q_{1}(h_{1}^{\top}y;X)\,\times\,q_{2,\hat{\zeta}_{2}}(H_{2}y;X)$, $y\in\mathbb{R}^{d}$, dominates $\hat{q}_{U}$ under KL loss if and only if $\hat{\zeta}_{2}(Z_{2})$ dominates $Z_{2}$ as an estimator of $\zeta_{2}\in\mathbb{R}^{d-1}$ under squared error loss $\|\hat{\zeta}_{2}-\zeta_{(2)}\|^{2}$ and for the model $Z_{2}|\zeta_{2}\sim N_{d-1}\left(\zeta_{(2)},\sigma^{2}_{X}\,I_{d-1}\right)$. Proof. Set $$X^{\prime}\,=\,HX=\begin{pmatrix}{X_{1}^{\prime}}\\ {X_{(2)}^{\prime}}\end{pmatrix}\,,\,Y^{\prime}\,=\,HY=\begin{pmatrix}{Y_{1}^{\prime}}\\ {Y_{(2)}^{\prime}}\end{pmatrix}\,,\hbox{ and }\zeta\,=\,H\theta=\begin{pmatrix}{\zeta_{1}}\\ {\zeta_{(2)}}\end{pmatrix}\,,$$ (4.24) with $X_{1}^{\prime}\,=\,h_{1}^{\top}X$, $X_{(2)}^{\prime}=H_{2}X$, $Y_{1}^{\prime}\,=\,h_{1}^{\top}Y$, $X_{(2)}^{\prime}=H_{2}X$, $\zeta_{1}\,=\,h_{1}^{\top}\theta$, and $\zeta_{(2)}=H_{2}\theta$. From Lemma 2.1, we have that $X_{1}^{\prime}$, $X_{(2)}^{\prime}$, $Y_{1}^{\prime}$, and $Y_{(2)}^{\prime}$ are independently distributed with $X_{1}^{\prime}\sim MMN_{1}\left(\zeta_{1},\|a\|,\sigma^{2}_{X},\mathcal{L}_{1}\right)$, $Y_{1}^{\prime}\sim MMN_{1}\left(\zeta_{1},\|a\|,\sigma^{2}_{Y},\mathcal{L}_{2}\right)$, $X_{(2)}^{\prime}\sim N_{d-1}(\zeta_{(2)},\sigma^{2}_{X}I_{d-1})$, and $Y_{(2)}^{\prime}\sim N_{d-1}(\zeta_{(2)},\sigma^{2}_{Y}I_{d-1})$. Now consider the class of predictive densities of the form $$q_{\hat{\zeta_{2}}}(y^{\prime};X^{\prime})\,=\,q_{1}(y_{1}^{\prime};X_{1}^{\prime})\times q_{2,\hat{\zeta_{2}}}(y_{2}^{\prime};X_{2}^{\prime})\,,y^{\prime}=(y_{1}^{\prime},y_{(2)}^{\prime})\in\mathbb{R}^{d},$$ (4.25) for estimating the density of $Y^{\prime}$. As in Remark 4.4, the Kullback-Leibler risk performance of $q_{H,\hat{\zeta}_{2}}(\cdot;X)$ for estimating the density of $Y$ is equivalent to the Kullback-Leibler risk performance of $q_{\hat{\zeta_{2}}}(\cdot;X^{\prime})$ for estimating the density of $Y^{\prime}$. Furthermore, observe that the MRE density estimator $\hat{q}_{U}$ equates to density $q_{\hat{\zeta}_{2,0}}(\cdot;X^{\prime})$ with $\hat{\zeta}_{2,0}(Y_{2}^{\prime})\,=Y_{2}^{\prime}$. It thus follows, with the independence of the components of $Y^{\prime}$ and $X^{\prime}$, Lemma 2.2, and setting $Z_{2}=X_{(2)}$ that $$\displaystyle R_{KL}(\theta,\hat{q}_{U})-R_{KL}(\theta,q_{H,\hat{\zeta}_{2}})$$ $$\displaystyle=$$ $$\displaystyle R_{KL}(\theta,q_{\hat{\zeta}_{2,0}})\,-R_{KL}(\theta,q_{\hat{\zeta}_{2}})$$ (4.26) $$\displaystyle=$$ $$\displaystyle\mathbb{E}\log\left(\frac{q_{1}(Y_{1}^{\prime};X_{1}^{\prime})}{q_{1}(Y_{1}^{\prime};X_{1}^{\prime})}\right)\,+\,\mathbb{E}\log\left(\frac{q_{2,\hat{\zeta}_{2}}(Y_{2}^{\prime};X_{2}^{\prime})}{q_{2,\hat{\zeta}_{2,0}}(Y_{2}^{\prime};X_{2}^{\prime})}\right)$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{2(\sigma^{2}_{X}+\sigma^{2}_{Y})}\left(\mathbb{E}\;||\hat{\zeta}_{2}(Z_{2})-\zeta_{2}||^{2}-\mathbb{E}\;||Z_{2}-\zeta_{2}||^{2}\right)\,,$$ which yields the result. ∎ The above dominance finding is quite general with respect to the specifications of $a,\mathcal{L}_{1}$, and $\mathcal{L}_{2}$ of model (2.8). Furthermore, observe by examining (4.26) that the risk difference depends on $\theta$ only through $\zeta_{(2)}=H_{2}\theta$ and this for any choice of $H_{2}$. More strikingly as seen with (4.26), the risk difference does not depend on the mixing distributions $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ and can be simply described by a quadratic risk difference of point estimators which arise in a $(d-1)$ variate normal distribution problem. An illustration of Theorem 4.4 will be presented in Section 5. 4.2 Bayesian improvements We now focus on Bayesian predictive densities that dominate $\hat{q}_{U}$. In doing so, we work with canonical forms as in Lemma 2.1, apply the partitioning argument of Lemma 2.2, and take advantage of known results for prediction in $(d-1)$ multivariate normal models. We consider a class of improper priors on $\theta$ which is the product measure of a (improper) uniform density over the linear subspace spanned by $a$ and a second component of the prior ($\pi_{0}$) supported on the subspace orthogonal to $a$. The measure of this nature splits resulting Bayes predictive densities into independent parts and leads to a decomposition the KL risk in two additive parts. Hence, the dominance result is obtained by dominating the part of the KL risk corresponding to the orthogonal space to $a$, where transformed variables are $N_{d-1}$ distributed and where we can capitalize on known results. Namely, the superharmonicity of $\pi_{0}$, or its associated marginal density or its associated square root marginal density, will suffice for dominance and minimaxity. Theorem 4.5. Consider $X,Y$ distributed as in model (2.8) with $\Sigma_{X}=\sigma^{2}_{X}I_{d}$, $\Sigma_{Y}=\sigma^{2}_{Y}I_{d}$, and $d\geq 2$. Let $H=\begin{pmatrix}{h_{1}^{\top}}\\ {H_{2}}\end{pmatrix}$ be an $d\times d$ orthogonal matrix such that $h_{1}=\frac{a}{\|a\|}$. Let $X^{\prime},Y^{\prime}$, and $\zeta$ be defined as in (4.24) and consider prior densities of the form $$\pi(\theta)=\pi_{0}\left(\zeta_{(2)}\right).$$ (4.27) (a) Then, the Bayes predictive density for $Y$ is given by $$\hat{q}_{\pi}(y;X)\,=\,\hat{q}^{\prime}_{\pi}(Hy;X^{\prime})\,,y\in\mathbb{R}^{d},$$ (4.28) with $\hat{q}^{\prime}_{\pi}(\cdot;x^{\prime})$ the Bayes predictive density for $Y^{\prime}$ based on $X^{\prime}$, given by $$\hat{q}^{\prime}_{\pi}(y^{\prime};X^{\prime})\,=\,\hat{q}_{U}(y_{1}^{\prime};X_{1}^{\prime})\times\,\hat{q}^{\prime}_{\pi_{0}}(y_{(2)}^{\prime};X^{\prime}_{(2)})\,,$$ (4.29) with: (i) $\hat{q}_{U}(\cdot;X_{1}^{\prime})$ the MRE density, given in Theorem 2.3, of $Y_{1}^{\prime}\sim MMN_{1}(\zeta_{1},\|a\|,\sigma^{2}_{Y},\mathcal{L}_{2})$ based on $X_{1}^{\prime}\sim MMN_{1}(\zeta_{1},\|a\|,\sigma^{2}_{X},\mathcal{L}_{1})$, and (ii) $\hat{q}^{\prime}_{\pi_{0}}(\cdot;X_{2}^{\prime})$ the Bayes predictive density for $Y^{\prime}_{(2)}\sim N_{d-1}(\zeta_{(2)},\sigma^{2}_{Y}I_{d-1})$ based on $X^{\prime}_{(2)}\sim N_{d-1}\left(\zeta_{(2)},\sigma^{2}_{X}I_{d-1}\right)$ and for prior density $\pi_{0}(\zeta_{(2)})$ for $\zeta_{(2)}$; (b) If $d\geq 4$, then $\hat{q}_{\pi}$ given in (4.28) dominates the MRE $\hat{q}_{U}$, and is therefore minimax, if and only if $\hat{q}^{\prime}_{\pi_{0}}(\cdot;X_{2}^{\prime})$ dominates the MRE density for $Y^{\prime}_{(2)}$ based on $X^{\prime}_{(2)}$ given by a $N_{d-1}(X^{\prime}_{(2)},(\sigma^{2}_{X}+\sigma^{2}_{Y})I_{d-1})$ density. Proof. (a) Eq. (4.28) follows from the transformation of variables under the orthogonal matrix $H$. Note that the distribution of the transformed variables is $$\displaystyle\>\>X^{{}^{\prime}}\sim MMN_{d}(\zeta,a_{0},\sigma^{2}_{X}I_{d},\mathcal{L}_{1})$$ $$\displaystyle\hbox{and }\;\;Y^{{}^{\prime}}\sim MMN_{d}(\zeta,a_{0},\sigma^{2}_{Y}I_{d},\mathcal{L}_{2}),$$ where $a_{0}=\left(\frac{a}{\|a\|},0,\ldots,0\right)^{\top}$. The prior of the form (4.27) induces an improper uniform measure on $\zeta_{1}$ and independent $\pi_{0}(\zeta_{(2)})$ on $\zeta_{(2)}$. Along with the conditional independence of $Y^{{}^{\prime}}_{1}$ and $Y^{{}^{\prime}}_{(2)}$ given $\zeta$, we get the Bayes predictive density as (4.29). (b) Observe that the MRE density estimator $\hat{q}_{U}(\cdot;X)$ corresponds to $\pi_{0}(\theta)=1$, i.e., the improper uniform density on $\zeta_{(2)}\in\mathbb{R}^{d-1}$. By virtue of Lemma 2.2, the KL risk difference between $\hat{q}_{U}(\cdot;X)$ and $\hat{q}_{\pi}(\cdot;X)$ is then expressed as $$\displaystyle R_{KL}(\theta,\hat{q}_{U})-R_{KL}(\theta,\hat{q}_{\pi})$$ $$\displaystyle=$$ $$\displaystyle\mathbb{E}\;\text{log}\;\hat{q}_{\pi}(Y;X)-\mathbb{E}\;\text{log}\;\hat{q}_{U}(Y;X)$$ $$\displaystyle=$$ $$\displaystyle\mathbb{E}\;\text{log}\;\hat{q}^{\prime}_{\pi_{0}}(Y_{(2)}^{\prime};X^{\prime}_{(2)})-\mathbb{E}\;\text{log}\;\hat{q}^{\prime}_{U}(Y_{(2)}^{\prime};X^{\prime}_{(2)})$$ $$\displaystyle=$$ $$\displaystyle R_{KL}(\zeta_{(2)},\hat{q}^{\prime}_{U})-R_{KL}(\zeta_{(2)},\hat{q}^{\prime}_{\pi_{0}}),$$ and part $\bf(b)$ follows. ∎ Remark 4.5. Theorem 4.5’s dominance finding in part (b) is unified with respect to the model settings $a$, $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$, as well as the dimension $d\geq 4,\sigma_{X}^{2}$, and $\sigma_{Y}^{2}$. Furthermore, as seen in the lines of the proof, the difference in risks between the predictive densities $\hat{q}_{U}$ and $\hat{q}_{\pi}$: (i) does not depend on the mixing $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$, and (ii) depends on $\theta$ only through $\zeta_{(2)}=H_{2}\theta$. Starting with [14], continuing namely with [13], several Bayesian predictive densities $\hat{q}^{\prime}_{\pi_{0}}(\cdot;X_{2}^{\prime})$ have been shown to satisfy the dominance condition in part (b) of the above Theorem. Such choices lead to dominating predictive densities of $\hat{q}_{U}$. In [13], analogously to the quadratic risk estimation problem with multivariate normal observables (e.g., [27, 11]), sufficient conditions for minimaxity are conveniently expressed in terms of the marginal density of $Z\sim N_{d-1}(\zeta_{(2)},\sigma^{2}I_{d-1})$ associated with density $\pi_{0}$ and given by $$m_{\pi_{0}}(z,\sigma^{2})=\int_{\mathbb{R}^{d-1}}\phi_{d-1}(z-\zeta_{(2)},\sigma^{2}I_{d-1})\,\pi_{0}(\zeta_{(2)})\,d\zeta_{(2)}\,.$$ The superharmonicity of either $\pi_{0}$, $m_{\pi_{0}}(z,\sigma^{2})$ for $z\in\mathbb{R}^{d-1}$, for various values of $\sigma^{2}$, or as well of $\sqrt{m_{\pi_{0}}(z,\sigma^{2})}$, each lead to sufficient conditions for minimaxity. We recall here that the superharmonicity of $h:\mathbb{R}^{d-1}\to\mathbb{R}$ holds whenever the Laplacian $\Delta^{2}h(t)=\sum_{i=1}^{d-1}\frac{\partial^{2}h(t)}{\partial t_{i}^{2}}$ exists with $\Delta^{2}h(t)\leq 0$ for $t\in\mathbb{R}^{d-1}$. Corollary 4.1. Consider the prediction context of Theorem 4.5 and a prior density $\pi_{0}$ as in (4.27) other than the uniform density. Suppose that $m_{\pi_{0}}(z,\sigma^{2}_{X})$ is finite for all $z\in\mathbb{R}^{d-1}$ and that $d\geq 4$. Then, the following conditions are each sufficient for $\hat{q}_{\pi}(\cdot;X)$ given in (4.28) with prior density as in (4.27) to dominate the MRE density $\hat{q}_{U}$: (i) $\Delta^{2}\,m_{\pi_{0}}(z,\sigma^{2})\leq 0$, $z\in\mathbb{R}^{d-1}$, for $\frac{\sigma^{2}_{X}\sigma^{2}_{Y}}{\sigma^{2}_{X}+\sigma^{2}_{Y}}<\sigma^{2}<\sigma^{2}_{X}\,$, with strict inequality on a set of positive Lebesgue measure on $\mathbb{R}^{d-1}$ for at least one $\sigma^{2}$; (ii) $\Delta^{2}\sqrt{m_{\pi_{0}}(z,\sigma^{2})}\leq 0$, $z\in\mathbb{R}^{d-1}$, for $\frac{\sigma^{2}_{X}\sigma^{2}_{y}}{\sigma^{2}_{X}+\sigma^{2}_{Y}}<\sigma^{2}<\sigma^{2}_{X}\,$, with strict inequality on a set of positive Lebesgue measure on $\mathbb{R}^{d-1}$ for at least one $\sigma^{2}$; (iii) The prior $\pi_{0}$ is such that $\Delta^{2}\pi_{0}(\zeta_{(2)})\leq 0$ a.e. Proof. The results follow from part (b) of Theorem 4.5 and Theorem 1 - Corollary 2 in [13]. ∎ Choices of the prior density $\pi_{0}$ satisfying the conditions of Corollary 4.1 thus rest upon analyses for the normal case which are plentiful. In particular, several examples of $\pi_{0}$, and the resulting predictive density $\hat{q}_{\pi_{0}}^{\prime}$, are provided in [13]. These provide explicit representations of minimax predictive densities $\hat{q}_{\pi}$ given in (4.28). A detailed example is presented in Section 5. The orthogonality decomposition used in this Section leads to a further interesting representation which generalizes the one obtained in the multivariate normal case, and for which we now expand upon. For the multivariate normal case, referring to Theorem 4.5’s decomposition, with $X_{(2)}^{\prime}\sim N_{d-1}(\zeta_{(2)},\sigma^{2}_{X}I_{d-1})$ independent of $Y_{(2)}^{\prime}\sim N_{d-1}(\zeta_{(2)},\sigma^{2}_{Y}\,I_{d-1})$, a well-known representation of the Bayes predictive density associated with prior density $\pi_{0}$ for $\zeta_{(2)}$, given by [13], is $$\hat{q}_{\pi_{0}}^{\prime}(y_{(2)}^{\prime};x_{(2)}^{\prime})=\hat{q}_{U}^{\prime}(y_{(2)}^{\prime};x_{(2)}^{\prime})\times\frac{m_{\pi_{0}}(w_{(2)}^{\prime};\sigma^{2}_{W})}{m_{\pi_{0}}(x_{(2)}^{\prime},\sigma^{2}_{X})}\,,$$ (4.30) with $w_{(2)}^{\prime}=\frac{\sigma^{2}_{X}y_{(2)}^{\prime}+\sigma^{2}_{Y}x_{(2)}^{\prime}}{\sigma^{2}_{X}+\sigma^{2}_{Y}}$ and $\sigma^{2}_{W}=\frac{\sigma^{2}_{X}\sigma^{2}_{Y}}{\sigma^{2}_{X}+\sigma^{2}_{Y}},$ and where $\hat{q}_{U}^{\prime}(\cdot;X_{(2)}^{\prime})$ is the MRE predictive density of the density of $Y_{(2)}^{\prime}$ based on $X_{(2)}^{\prime}$, and given by a $N_{d-1}(x_{(2)}^{\prime},(\sigma_{X}^{2}+\sigma_{Y}^{2})\,I_{d-1})$ density. For the MMN case, we now have the following. Lemma 4.8. For a prior $\pi_{0}$ and $H$ in Theorem 4.5, the corresponding Bayes predictive density $\hat{q}_{\pi}$ admits the representation $$\hat{q}_{\pi}(y;x)=\hat{q}_{U}(y;x)\times\frac{m_{\pi_{0}}(H_{2}\,w,\sigma^{2}_{W})}{m_{\pi_{0}}(H_{2}\,x,\sigma^{2}_{X})}\,,$$ (4.31) with $w=\frac{\sigma^{2}_{X}y+\sigma^{2}_{Y}x}{\sigma^{2}_{X}+\sigma^{2}_{Y}}$. Proof. Using the set-up of Theorem 4.5, and expressions (4.28) and (4.29), the MRE predictive density is obtained as $$\hat{q}_{U}(y;X)\,=\,\hat{q}_{U}(y_{1}^{\prime},X_{1}^{\prime})\times\hat{q}_{U}^{\prime}(y_{(2)}^{\prime};X_{(2)}^{\prime})\,,y\in\mathbb{R}^{d}.$$ Therefore, from (4.28) and (4.29) again, as well as from 4.30, we obtain $$\hat{q}_{\pi}(y;X)\,=\,\hat{q}_{U}^{\prime}(h_{1}^{\top}y;X_{1}^{\prime})\times\hat{q}_{U}^{\prime}(H_{2}y;X_{2}^{\prime})\times\frac{m_{\pi_{0}}(w_{(2)}^{\prime};\sigma^{2}_{W})}{m_{\pi_{0}}(x_{(2)}^{\prime},\sigma^{2}_{X})}\,,$$ which yields the result. ∎ To conclude describing the dominance findings of this section and of Section 4.1, we point out that the plug-in type improvements of Theorem 4.5 and the Bayesian dominance results of Theorem 4.5 and Corollary 4.1 are applicable regardless of the choice of the orthogonal completion $H_{2}$ of $H$, thus adding to choices of $\pi_{0}$ leading to minimaxity. Furthermore, the above developments are unified and the findings are applicable for all MMN models (2.8) with $\Sigma_{X}=\sigma^{2}_{X}I_{d}$ and $\Sigma_{Y}=\sigma^{2}_{Y}I_{d}$, as well as for $\Sigma_{Y}=c\Sigma_{X}$ as justified in Remark 4.4. Remark 4.6. A particular appealing choice of $H_{2}$, which will be further explored below in Sections 4.3 and 5, is the such that $H_{2}^{\top}H_{2}\,=\,I_{d}-\frac{\;\;aa^{\top}}{a^{\top}a}$ in which case $$\|\zeta_{(2)}\|^{2}\,=\,\theta^{\top}\left(I_{d}-\frac{\;\;aa^{\top}}{a^{\top}a}\right)\theta\,,$$ (4.32) and spherically symmetric densities $\pi_{2}(\zeta_{2})\,=\,g\left(\|\zeta_{2}\|^{2}\right)$ lead to prior densities in (4.27) of the form $$\pi(\theta)\,=\,g\left\{\theta^{\top}\left(I_{d}-\frac{\;\;aa^{\top}}{a^{\top}a}\right)\theta\right\}\,=\,g\left(\left\|\theta-\frac{a^{\top}\theta}{a^{\top}a}\,a\right\|^{2}\right).$$ (4.33) Such densities do not depend on $\|a\|$ and have contours given by hypersurfaces of cylinders with axis given by $a$ (or $h_{1}=\frac{a}{\|a\|}$). Here is an example of three contours for $d=3$ and $a=(1,1,1)^{\top}$. 4.3 Restricted parameter spaces Theorem 4.5’s decomposition also leads to implications when there exists parametric restrictions on $\zeta_{(2)}\,=\,H\theta$. Statistical models where parametric restrictions are present appear naturally in a great variety of contexts, and there is a large literature on related inferential problems, namely for a decision-theoretic approach (e.g., [23, 28]). Questions of predictive analysis under parametric restrictions are also of interest with findings obtained in [22, 17, 10]. Namely, for normal models, specifically model (2.8) with $a=0$, $\Sigma_{X}=\sigma^{2}_{X}I_{d}$, $\Sigma_{Y}=\sigma^{2}_{Y}I_{d}$ with $\theta$ constrained to a convex set $C_{0}$ with non-empty interior, [10] showed that the Bayes predictive density associated with the uniform prior for $\theta$ on $C_{0}$ dominates the MRE predictive density under Kullback-Leibler loss. The next results extends this finding to MMN models. Theorem 4.6. Consider $X,Y$ distributed as in model (2.8) with $\Sigma_{X}=\sigma^{2}_{X}I_{d}$, $\Sigma_{Y}=\sigma^{2}_{Y}I_{d}$, and $d\geq 2$. Let $C\subset\mathbb{R}^{d-1}$ be a convex set with non-empty interior, and let $\pi_{C}(\theta)=\pi_{0,U}(\zeta_{(2)})=I_{C}(\zeta_{(2)})\,.$ Then $\hat{q}_{\pi_{C}}(\cdot;X)$ dominates $\hat{q}_{U}(\cdot;X)$ under KL risk and the restriction $\theta\in\{\theta\in\mathbb{R}^{d}\,:\,H_{2}\theta\in C\}$. Proof. As in Theorem 4.5 and the given proof, we infer that $\hat{q}_{\pi}$ given in (4.28) with prior density $\pi(\theta)\,=\,\pi_{0}(\zeta_{(2)})$ for $\zeta_{(2)}=H_{2}\theta$ dominates $\hat{q}_{U}$ if and only if $\hat{q}^{\prime}_{\pi_{0}}(\cdot;X_{2}^{\prime})$ dominates the MRE density for $Y_{(2)}^{\prime}\sim N_{d-1}(\zeta_{2},\sigma^{2}_{Y}I_{d-1})$. 222Said otherwise, part (b) of Theorem 4.5 could have been stated for $d\geq 2$, but this would lead to knowingly vacuous conditions in the absence of a parametric restriction. But, since this latter dominance holds precisely for density $\pi=\pi_{C}$ for the uniform density choice $\pi_{0}=\pi_{0,U}$ as shown in [10], the result follows. ∎ The setting of $C$ above is quite general and interesting examples includes balls and cones. As earlier, the finding is unified and general to the MMN models. Here are two applications of Theorem 4.6. Example 4.4. Suppose $d=2$, $a=(1,1)^{\top}$, and the parametric restriction $\underline{c}\leq\theta_{1}-\theta_{2}\leq\bar{c}$, with $C=(\underline{c},\bar{c})$ a strict subset of $\mathbb{R}$. The MRE density $\hat{q}_{U}(\cdot;X)$ is that of $MMN_{2}(X,a,(\sigma^{2}_{X}+\sigma^{2}_{Y})I_{2},\mathcal{L}_{3})$ distribution. In the context of Theorem 4.6, we have $\zeta_{(2)}\,=\,\frac{\theta_{1}-\theta_{2}}{\sqrt{2}}$ and the prior density $\pi_{C}(\theta)=I_{C}(\theta_{1}-\theta_{2})$. Theorem 4.5 tells us that the Bayes predictive density $\hat{q}_{\pi_{C}}$ dominates the MRE $\hat{q}_{U}$ with respect to KL loss and under the given parametric restriction. 333In Example 4.4, for the compact interval case say without loss of generality $\underline{c}=-m$ and $\bar{c}=m$, there exists a much larger class of dominating predictive densities obtained by replacing the uniform density for $\zeta_{(2)}$ by an even density $\pi_{0}$ supported on $(-m,m)$ that is increasing and logconcave on $(0,m)$. This is established as in Theorem 4.6 and making use of Theorem 3.2 in [10], which exploits a related point estimation finding in [19]. An explicit expression for $\hat{q}_{\pi_{C}}$ is available from Lemma 4.8 with $\pi_{0}$ the uniform $U(\frac{\underline{c}}{\sqrt{2}},\frac{\bar{c}}{\sqrt{2}})$ density for $\zeta_{(2)}$. As evaluated in [17], we obtain $$\displaystyle\left(\frac{\sqrt{2}}{\bar{c}-\underline{c}}\right)m_{\pi_{0}}(z,\sigma^{2})\,$$ $$\displaystyle=$$ $$\displaystyle\,\int_{\underline{c}/\sqrt{2}}^{\bar{c}/\sqrt{2}}\phi\left(z-\zeta_{(2)},\sigma^{2}\right)\,d\zeta_{(2)}$$ $$\displaystyle=$$ $$\displaystyle\Phi\left(\frac{z+\bar{c}/\sqrt{2}}{\sigma}\right)\,-\,\Phi\left(\frac{z+\underline{c}/\sqrt{2}}{\sigma}\right)\,,$$ and (4.31) then yields $$\hat{q}_{\pi_{C}}(y;x)\,=\,\hat{q}_{U}(y;x)\,\;\frac{\Phi\left(\frac{w+\bar{c}/\sqrt{2}}{\sigma_{W}}\right)\,-\,\Phi\left(\frac{w+\underline{c}/\sqrt{2}}{\sigma_{W}}\right)}{\Phi\left(\frac{x+\bar{c}/\sqrt{2}}{\sigma_{X}}\right)\,-\,\Phi\left(\frac{x+\underline{c}/\sqrt{2}}{\sigma_{X}}\right)}\,,y\in\mathbb{R},$$ with $w\,=\,\frac{\sigma^{2}_{X}y\,+\,\sigma^{2}_{Y}x}{\sigma^{2}_{X}+\sigma^{2}_{Y}}$, $\sigma^{2}_{W}\,=\,\frac{\sigma^{2}_{X}\sigma^{2}_{Y}}{\sigma^{2}_{X}+\sigma^{2}_{Y}}$, and $\hat{q}_{U}$ the MRE density which is that of a $MMN_{1}(x,a,(\sigma^{2}_{X}+\sigma^{2}_{Y}),\mathcal{L}_{3})$ distribution. Example 4.5. Theorem 4.6 applies for $\theta$ restricted to a cylinder of radius, say $m$, with the axis along the direction $a$, i.e., $$C_{m}=\left\{\theta\in\mathbb{R}^{d}:\left\|\theta-\frac{a^{\top}\theta}{a^{\top}a}a\right\|\leq m\right\};$$ examples of which are drawn in Figure 1. The dominating predictive density $\hat{q}_{\pi_{C_{m}}}$ is Bayes with respect to the uniform prior density on $C_{m}$, which corresponds to (4.33) with $g(t)\,=\,I_{(0,m)}(t)$. An explicit expression for $\hat{q}_{\pi_{C_{m}}}$ can be derived from Lemma 4.8 with $\pi_{0}$ the uniform density on the ball $B_{m}=\{t\in\mathbb{R}^{d-1}:\|t\|\leq m\}$ and marginal density $$\displaystyle m_{\pi_{0}}(z,\sigma^{2})\,$$ $$\displaystyle=$$ $$\displaystyle\,\int_{B_{m}}\phi_{d-1}\left(z-\zeta_{(2)},\sigma^{2}I_{d-1}\right)\,d\zeta_{(2)}$$ $$\displaystyle=$$ $$\displaystyle F_{d-1,\frac{\|z\|^{2}}{\sigma^{2}}}(\frac{m^{2}}{\sigma^{2}})\,,$$ with $F_{\nu,\lambda}$ the cdf of a $\chi_{\nu}^{2}(\lambda)$ distribution. From (4.31), we thus obtain $$\hat{q}_{\pi_{C_{m}}}(y;x)\,=\,\hat{q}_{U}(y;x)\,\left(\frac{F_{d-1,\frac{\|H_{2}w|^{2}}{\sigma^{2}_{W}}}(\frac{m^{2}}{\sigma^{2}_{W}})}{F_{d-1,\frac{\|H_{2}x|^{2}}{\sigma^{2}_{X}}}(\frac{m^{2}}{\sigma^{2}_{X}})}\right),\,y\in\mathbb{R}^{d},$$ with $\|H_{2}t\|^{2}\,=\,t^{\top}\left(I-\frac{\;\;aa^{\top}}{a^{\top}a}\right)t\,$, for $t\in\mathbb{R}^{d}$, $w\,=\,\frac{\sigma^{2}_{X}y\,+\,\sigma^{2}_{Y}x}{\sigma^{2}_{X}+\sigma^{2}_{Y}}$, $\sigma^{2}_{W}\,=\,\frac{\sigma^{2}_{X}\sigma^{2}_{Y}}{\sigma^{2}_{X}+\sigma^{2}_{Y}}$, and $\hat{q}_{U}$ the MRE density which is that of a $MMN_{d}(x,a,(\sigma^{2}_{X}+\sigma^{2}_{Y})I_{d},\mathcal{L}_{3})$ distribution. 5 Illustrations We provide here illustrations of Theorems 4.4 and 4.5 accompanied by numerical comparisons and various observations. Example 5.6. (A Bayesian minimax predictive density) In the context of Theorem 4.5, consider $H_{2}$ as in Remark 4.6 combined with the harmonic prior density for $\zeta_{(2)}\in\mathbb{R}^{d-1}$ given by $\pi_{0}(\zeta_{(2)})=\|\zeta_{(2)}\|^{-(d-3)}$ and which generates via (4.33) an “adjusted” harmonic prior density on $\theta$ given by $$\pi_{H}(\theta)=\left\|\theta-\frac{a^{\top}\theta}{a^{\top}a}a\right\|^{-(d-3)}\,.$$ (5.34) Thus, the prior density is the product measure on $\mathbb{R}^{d}$ with uniform prior on the linear subspace spanned by $a$ and the above harmonic measure on the $(d-1)-$dimensional chosen subspace orthogonal to $a$. Since $\pi_{0}$ is superharmonic on $\mathbb{R}^{d-1}$ for $d\geq 4$, it follows from Corollary 4.1 that the Bayes predictive density $\hat{q}_{\pi_{H}}(\cdot;X)$ given in (4.28), as well as in (5.36) below, dominates the MRE density $\hat{q}_{U}$ and is consequently minimax. An explicit expression for $\hat{q}_{\pi_{H}}$ is available from Lemma 4.31 with marginal density $$\displaystyle m_{\pi_{0}}(z,\sigma^{2})=\int_{\mathbb{R}^{d-1}}\phi_{d-1}(z-\zeta_{(2)},\sigma^{2}I_{d-1})\,\frac{1}{||\zeta_{(2)}||^{(d-3)}}\,d\zeta_{(2)}=\sigma^{3-d}\,\mathbb{E}\,T^{\frac{(3-d)}{2}},$$ where $T\sim\chi^{2}_{d-1}\left(\frac{||z||^{2}}{\sigma^{2}}\right)$. In particular for odd $d\geq 5$, as shown in the Appendix, one may obtain $$m_{\pi_{0}}(z,\sigma^{2})=\left(||z||^{2}\right)^{\frac{3-d}{2}}\left(1-e^{-\frac{||z||^{2}}{2\sigma^{2}}}\sum\limits_{k=0}^{\frac{d-5}{2}}\left(\frac{||z||^{2}}{2\sigma^{2}}\right)^{k}\frac{1}{k!}\right)=r(||z||^{2},\sigma^{2})\,\hbox{ (say) },$$ (5.35) which relates to known results on the inverse moments of a chi-square variable with even degrees of freedom (e.g., [8]), as well a closed form for an incomplete gamma function which intervenes in Komaki’s [14] representation of $m_{\pi_{0}}$. From (4.31) and the above, we thus have $$\hat{q}_{\pi_{H}}(y;x)\,=\hat{q}_{U}(y;x)\,\frac{r\left(\left\|w-\frac{a^{\top}w}{a^{\top}a}a\right\|^{2},\sigma^{2}_{W}\right)}{r\left(\left\|x-\frac{a^{\top}x}{a^{\top}a}a\right\|^{2},\sigma^{2}_{X}\right)}\,,y\in\mathbb{R}^{d}\,,$$ (5.36) where $w$ and $\sigma^{2}_{W}$ are as given in Lemma 4.8. Risk differences between $\hat{q}_{U}$ and $\hat{q}_{\pi_{H}}$ are plotted in Figure 1(a) and Figure 1(b) as a function of $\|\zeta_{(2)}\|^{2}$, or equivalently as a function of $$\displaystyle t=\frac{\|\zeta_{(2)}\|^{2}}{d-1}=\frac{1}{d-1}\left\|\theta-\frac{a^{\top}\theta}{a^{\top}a}a\right\|^{2}\,,$$ i.e., in terms of the average squared component of $\zeta_{(2)}$. The actual risks depend on the underlying mixing distributions $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$, but not the risk differences as previously observed in Remark 4.5. Observe as well that $t$ is independent of $\|a\|$ and only depends on the direction $a/\|a\|$. Figure 1(a) has $\sigma^{2}_{X}=1,\sigma^{2}_{Y}=2$ and varying $d$, while Figure 1(b) has fixed $d=5,\sigma^{2}_{X}=1$ with $\sigma^{2}_{Y}=c\sigma^{2}_{X}$ and varying $c$. As seen with Figure 1(a), the improvement in KL risk vanishes at $t\to\infty$, but gains in prominence with increasing $d$, and with the proximity of $\theta$ to the linear subspace spanned by $a$. As seen with Figure 1(b), the KL risk difference loses in prominence with larger $c$ which is consistent with the fact that MRE density gains in reliability when the variance $\sigma^{2}_{X}$ of the observable decreases. Frequentist risk ratios between $\hat{q}_{U}$ and $\hat{q}_{\pi_{H}}$ are plotted in Figure 1(c) for $\sigma^{2}_{X}=1,\sigma^{2}_{Y}=2$ and varying $d$. These ratios depend additionally on the mixing distributions $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ and they are set here with $\sqrt{\chi^{2}_{1}}$ mixing (Example 2.1 (B)), i.e., $X|\theta$ and $Y|\theta$ have skew-normal distributions with densities given in (2.4) and MRE density expanded upon in part (D) of Section 2.4. We further set $a=\boldsymbol{1}_{d}=(1,\ldots,1)^{\top}$, in which case the harmonic prior density on $\theta$ in (5.34) reduces to $\pi_{0}(\theta)=||\theta-\bar{\theta}\boldsymbol{1}_{d}||^{-(d-3)}$ with $\bar{\theta}=\frac{1}{d}\sum\limits_{i=1}^{d}\theta_{i}$. With the above settings, the constant (and minimax) risk of the MRE density can be computed from (2.15). For instance, we obtain $R(\theta,\hat{q}_{U})\approx 1.0954$ for $d=5$, $\approx 1.5187$ for $d=7$ and $\approx 1.9403$ for $d=9$. These are close to linear with the term $\frac{d}{2}\log\frac{\sigma^{2}_{S}}{\sigma^{2}_{Y}}=\frac{d}{2}\log\frac{3}{2}$ ($\approx 1.0137$ for $d=5$, $\approx 1.4191$ for $d=7$ and $\approx 1.8246$ for $d=9$), representing the MRE risk for the normal case with $a=0$, being dominant in (2.15). As seen in Figure 1(c), where the risk ratios are plotted with respect to $t=\frac{1}{d-1}\,||\theta-\bar{\theta}\boldsymbol{1}_{d}||^{2}$, the gains increase in $d$ and with the closeness of the $\theta_{i}$’s to $\bar{\theta}$. Example 5.7. (Plug-in type improved predictive density) In the context of Theorem 4.4, consider plug-in type predictive densities $q_{H,\hat{\zeta}_{2}}(y;X)\,=\,q_{1}(h_{1}^{\top}y;X)\,\times\,q_{2,\hat{\zeta}_{2}}(H_{2}y;X)$ with the choice of the James-Stein estimator $\hat{\zeta}_{2}(Z_{2})=\left(1-\frac{(d-3)\sigma^{2}_{X}}{||Z_{2}||^{2}}\right)\,Z_{2}$ leading to the dominance of $q_{H,\hat{\zeta}_{2}}$ over $\hat{q}_{U}$ for $d\geq 4$. Both the dominating predictive density $q_{H,\hat{\zeta}_{2}}$ and the actual difference in risks do depend on the choice of $H_{2}$, but the KL risk difference, as given in (4.26) and mentioned at the end of Section 4.1, is independent of the underlying mixing distributions and will thus coincide with the corresponding difference stemming for $d-1$ dimensional normal models and which have appeared many times in the literature. The difference in risks will be a function of $\zeta_{(2)}=H_{2}\theta$ in general, and more precisely as a function of $\|\zeta_{(2)}\|^{2}$ in this case given that the James-Stein estimator is equivariant with respect to orthogonal transformations. It is thus more interesting to look at the ratio of Kullback-Leibler risks and such ratios are presented in Figure 1(d) with the same settings as in Example 5.6, i.e., multivariate skew-normal models with $\sqrt{\chi^{2}_{1}}$ mixing, $\sigma^{2}_{X}=1,\sigma^{2}_{Y}=2$, and $a=(1,\cdots,1)^{T}.$ Again here, the risk ratios are plotted with respect to $t=\frac{1}{d-1}\,||\theta-\bar{\theta}\boldsymbol{1}_{d}||^{2}$, the gains increase in $d$ and with the closeness of the $\theta_{i}$’s to $\bar{\theta}$. Concluding remarks In this work, we have addressed the problem of determining efficient predictive densities under Kullback-Leibler frequentist risk for multivariate skew-normal distributions and, more generally, for mean mixtures of multivariate normal (MMN) distributions, and provided Bayesian and plug-in type predictive densities which dominate the MRE density, and are minimax in four dimensions or more. In doing so, we have made use of a canonical transformation which leads to the decomposition of the Kullback-Leibler risk for the predictive densities being considered into two additive parts, one of which matching that of the MRE and minimax density, the other relating to a normal model and permitting improvement in view of shrinkage predictive density estimation results for such models. Further implications are provided for certain type of parametric restrictions. In addition, motivated by the relative paucity of analytical representations for Bayesian posterior and predictive densities, we have contributed such explicit representations. This work represents, to the best of our knowledge, a first foray of the study of predictive density estimation for MMN distributions. The findings are thus novel and they are also unified. The canonical transformation technique may well find further applications in predictive analysis, such as for mean-variance mixture of normal distributions. Extensions to other choices of loss (e.g., $\alpha$-divergence) and to unknown covariance structures would be most interesting to investigate as well. Acknowledgements Éric Marchand’s research is supported in part by the Natural Sciences and Engineering Research Council of Canada. Pankaj Bhagwat is grateful to the ISM (Institut des sciences mathématiques) for financial support. Thanks to Jean-Philippe Burelle for useful discussions on geometric representations related to prior density (4.33). Appendix Lemma 5.9. For all $B,c\in\mathbb{R}$, $A\in\mathbb{R}_{+}$, we have $$\int_{0}^{\infty}\Phi(ct)\,e^{-\frac{t^{2}}{2A}+Bt}\,dt\,=\,e^{\frac{AB^{2}}{2}}\,\sqrt{2\pi A}\,\,\Phi_{2}(\frac{cAB}{\sqrt{1+c^{2}A}},B\sqrt{A};\frac{c\sqrt{A}}{\sqrt{1+c^{2}A}})\,.$$ (5.37) Proof. We have $$\displaystyle e^{\frac{-AB^{2}}{2}}\,(2\pi A)^{-1/2}\,\int_{0}^{\infty}\Phi(ct)\,e^{-\frac{t^{2}}{2A}+Bt}\,dt\,\,$$ $$\displaystyle=$$ $$\displaystyle\int_{0}^{\infty}\Phi(ct)\,\frac{1}{\sqrt{A}}\,\phi(\frac{t-AB}{\sqrt{A}})\,dt$$ $$\displaystyle=$$ $$\displaystyle\mathbb{P}\left(U-cT\leq 0,-T\leq 0\right)\,,$$ with $U,T$ independently distributed as $N(0,1)$ and $N(\theta_{T}=AB,\sigma^{2}_{T}=A)$, respectively. The result follows since $$(U-cT,-T)^{\top}\sim N_{2}\left(\left(\begin{array}[]{r}-cAB\\ AB\end{array}\right),\left[\begin{array}[]{rr}1+c^{2}A&cA\\ cA&A\end{array}\right]\right)\,.\;\qed$$ Proof of (5.35). With the standard representation $T|K\sim\chi^{2}_{d-1+2K}$ with $K\sim Poisson\left(\frac{||z||^{2}}{2\sigma^{2}}\right)$, we have $$\displaystyle\mathbb{E}\,T^{\,(3-d)/2}$$ $$\displaystyle=$$ $$\displaystyle\sum\limits_{k=0}^{\infty}e^{-\frac{||z||^{2}}{2\sigma^{2}}}\frac{1}{k!}\left(\frac{||z||^{2}}{2\sigma^{2}}\right)^{k}\mathbf{E}\left(\chi^{2}_{d-1+2k}\right)^{\frac{(3-d)}{2}}$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{2^{\frac{d-3}{2}}}e^{-\frac{||z||^{2}}{2\sigma^{2}}}\sum\limits_{k=0}^{\infty}\left(\frac{||z||^{2}}{2\sigma^{2}}\right)^{k}\frac{1}{\Gamma(\frac{d-1}{2}+k)}$$ $$\displaystyle=$$ $$\displaystyle e^{-\frac{||z||^{2}}{2\sigma^{2}}}\left(\frac{||z||^{2}}{2\sigma^{2}}\right)^{-\frac{d-3}{2}}\sum\limits_{k=\frac{d-3}{2}}^{\infty}\left(\frac{||z||^{2}}{2\sigma^{2}}\right)^{k}\frac{1}{k!}\,,$$ which yields (5.35). ∎ References [1] Abdi, M., Madadi, M., Balakrishnan, N. & Jamalizadeh, A. (2021). 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A Good Practice Towards Top Performance of Face Recognition: Transferred Deep Feature Fusion Lin Xiong${}^{1}$${}^{*}$${}^{\dagger}$, Jayashree Karlekar${}^{1}$${}^{*}$, Jian Zhao${}^{2}$${}^{*}$${}^{\dagger}$, Jiashi Feng${}^{2}$,  , Sugiri  Pranata${}^{1}$, and Shengmei  Shen${}^{1}$ ${}^{1}$L. Xiong, J. Karlekar, S. Pranata and S.M. Shen are with Panasonic R&D Center Singapore, Singapore (lin.xiong@sg.panasonic.com; karlekar.jayashree@sg.panasonic.com; shengmei.shen@sg.panasonic.com).${}^{2}$J. Zhao and J.S. Feng are with Department of Electrical and Computer Engineering, National University of Singapore, Singapore (zhaojian90@u.nus.edu; elefjia@nus.edu.sg). J. Zhao was an intern at Panasonic R&D Center Singapore during this work.${}^{*}$ L. Xiong, J. Zhao and J. Karlekar make an equal contribution.${}^{\dagger}$ L. Xiong and J. Zhao are the corresponding author. Abstract Unconstrained face recognition performance evaluations have traditionally focused on Labeled Faces in the Wild (LFW) dataset for imagery and the YouTubeFaces (YTF) dataset for videos in the last couple of years. Spectacular progress in this field has resulted in a saturation on verification and identification accuracies for those benchmark datasets. In this paper, we propose a unified learning framework named transferred deep feature fusion targeting at the new IARPA Janus Bechmark A (IJB-A) face recognition dataset released by NIST face challenge. The IJB-A dataset includes real-world unconstrained faces from 500 subjects with full pose and illumination variations which are much harder than the LFW and YTF datasets. Inspired by transfer learning, we train two advanced deep convolutional neural networks (DCNN) with two different large datasets in source domain, respectively. By exploring the complementarity of two distinct DCNNs, deep feature fusion is utilized after feature extraction in target domain. Then, template specific linear SVMs is adopted to enhance the discrimination of framework. Finally, multiple matching scores corresponding different templates are merged as the final results. This simple unified framework outperforms the state-of-the-art by a wide margin on IJB-A dataset. Based on the proposed approach, we have submitted our IJB-A results to National Institute of Standards and Technology (NIST) for official evaluation. Face Recognition, Deep Convolutional Neural Network, Feature Fusion, Model Ensemble, SVMs. I Introduction Face recognition performance using features of Deep Convolutional Neural Network (DCNN) have been dramatically improved in recent years. Many state-of-the-art algorithms claim very close [7],[12] or even have surpassed [13], [22],[27] human performance on Labeled Faces in the Wild (LFW) dataset. The saturation in recognition accuracy for current benchmark dataset has come. In order to push the development of frontier in regarding to unconstrained face recognition, a new face dataset template-based IJB-A is introduced recently [20], whose setting and solutions are aligned better with the requirements of real applications. The IJB-A dataset is created to provide the latest and most challenging dataset for both verification and identification as shown is Fig.1. Unlike LFW and YTF, this dataset includes both image and video of subjects manually annotated with facial bounding boxes to avoid the near frontal condition, along with protocols for evaluation of both verification and identification. Those protocols significantly deviate from standard protocols for many face recognition algorithms [28],[29]. Moreover, the concept of template is introduced, simultaneously. A template refers to a collection of all media (images and/or video frames) of an interested face captured under different conditions that can be utilized as a combined single representation for matching task. The template-based setting reflects many real-world biometric scenarios, where capturing a subject’s facial appearance is possible more than once under different acquisition ways. In other words, this new IJB-A face recognition task requires to deal with a more challenging set-to-set matching problem successfully regardless of face capture settings (illumination, sensor, resolution) or subject conditions (facial pose, expression, occlusion). Our contributions can be summarized as following aspects: 1. A unified learning framework named transferred deep feature fusion is proposed for face verification and identification. 2. Two latest DCNN models are trained in source domain with two different large datasets in order to take full advantage of complementary between models and datasets. 3. Two-stage fusion are designed, one for features and another for similarity scores. 4. One-vs-rest template specific linear SVMs with chosen negative set is trained in target domain. In this paper, we propose a unified learning framework named transferred deep feature fusion. It can effectively integrate superiority of each module and outperform the state-of-the-art on IJB-A dataset. Inspired by transfer learning [1], facial feature encoding model of subjects are trained offline in a source domain, and this feature encoding model is transferred to a specific target domain where limited available faces of new subjects can be encoded. Specifically, in order to capture the intrinsic discrimination of subjects and enhance the generalization capability of face recognition models, we deploy two advanced deep convolutional neural networks (DCNN) with distinct architectures to learn the representation of faces on two different large datasets (each one has no overlap with IJB-A dataset) in source domain. These two DCNN models provide distinct feature representations which can better characterize the data distribution from different perspectives. The complementary between two distinct models is beneficial for feature representation [17]. Thus, representing a face from different perspectives could effectively decrease ambiguity among subjects and enhance the generalization performance of face recognition especially on extremely large number of subjects. After offline training procedure, those two DCNN models are transferred to target domain where templates of IJB-A dataset as inputs are performed feature extraction with shared weigths and biases, respectively. Then, features from two DCNN models are combined in order to obtain more discriminative representation. Finally, template specific linear SVMs are trained on fused features for classification. Furthermore, for set-to-set matching problem, multiple matching scores are merged into a single one [43],[45],[33] for each template pair as the final results. Comprehensive evaluations on IJB-A public dataset well demonstrate the significant superiority of the proposed learning framework over other state-of-the-art methods. Based on the proposed approach, we have submitted our IJB-A results to NIST for official evaluation. This paper is organized as follows. We review the related work in Section II. Section III shows the details of transferred deep feature fusion. In Section IV, a comprehensive evaluation on IJB-A dataset is shown. Finally, the conclusion remarks and the future work are presented in Section V. II Related Work Recently, all the top performing methods for face recognition on LFW and YTF are all based on DCNN architectures. Such as the VGG-Face model [14], as a typical application of the VGG-16 convolutional network architecture [8] trained on a reasonably and publicly large face dataset of 2.6M images of 2622 subjects, provides state-of-the-art performance. This dataset is called as VGG-Face data for convenience in the following section. FaceNet [22] utilizes the DCNN with inception module [18] for unconstrained face recognition. This network is trained using a private huge dataset of over 200M images and 8M subjects. DeepFace [7] deploys a DCNN coupled with 3D alignment, where facial pose is normalized by warping facial landmarks to a canonical position prior to encoding face images. DeepID2+ [12] and DeepID3 [13] extend the FaceNet model by including joint Bayesian metric learning [3] and multi-task learning. More better unconstrained face recognition performance is provided by them. Moreover, DeepFace is trained using a private dataset of 4.4M images and 4,030 subjects. DeepID2+ and DeepID3 are trained also using a private dataset of 202,595 images and 10,117 subjects with 25 networks and 50 networks, respectively. The idea of multiple model ensemble is involved. Moreover, many approaches use metric learning in the form of triplet loss similarity or joint Bayesian for the final loss to learn an optimal embedding for face recognition [22],[14],[27]. Thus, a recent study [16] concludes that multiple networks ensemble and metric learning are crucial for improvement on LFW. With the advent of IJB-A dataset introduced by NIST in 2015, the task of template-based unconstrained face recognition has attracted extensive attention. So far as we known, most algorithms for this challenging problem are also based on DCNN architecture as top performing methods did on LFW and YTF. Chen et al. [27] achieve good performance by extracting feature representations via a DCNN trained on public dataset which includes 490,356 images and 10,548 subjects. And then, those features as inputs are applied to learn metric matrix in order to project the feature vector into a low-dimensional space, meanwhile, maximizing the between-class variation and minimizing within-class variation via joint Bayesian metric learning. B-CNN [30] applies the bilinear CNN architecture to face identification. Deep Multi-pose [44] utilizes five pose specialized sub-networks with 3D pose rendering to encode multiple pose-specific features. Sensitivity of the recognition system to pose variations is reduced since an ensemble of pose-specific deep features is adopted. Pooling faces [45] aligns faces in 3D and bins them according to head pose and image quality. Pose-Aware Models (PAMs) [43] handles pose variability by learning Pose-Aware Models for frontal, half-profile and full-profile poses in order to improve face recognition performance in wild. Masi et al. [33] even question whether need to collect millions of faces or not for effective face recognition. Thus, a far more accessible means of increasing training data sizes is proposed. Pose, 3D shape and expression are utilized to synthesize more faces from CASIA-WebFace dataset [9]. Triplet Probabilistic Embedding (TPE) [42] couples a DCNN-based approach with a low-dimensional discriminative embedding learned using triplet probability constraints to solve the unconstrained face verification problem. TPE obtains better performance than previous algorithms on IJB-A dataset. Template Adaptation (TA) [34] proposes the idea of template adaptation which is a form of transfer learning to the set of media in a template. Combining DCNN features with template adaptation, it obtains better performance than TPE on IJB-A task. Ranjan et al. propose an all-in-one method [46] employed a multi-task learning framework that regularizes the shared parameters of CNN and builds a synergy among different domains and tasks. Until recently, Yang et al. propose Neural Aggregation Network (NAN) [47] which produces a compact and fixed-dimension feature representation. It adaptively aggregates the features to form a single feature inside the convex hull spanned by them. What’s more interesting is that NAN learns to advocate high-quality face images while repelling low-quality ones such as blurred, occluded and improperly exposed faces. Thus, the face recognition performance on IJB-A dataset is pushed to reach an unprecedented height. Just a few days ago, Ranjan et al. [49] add an $L_{2}$-constraint to the feature descriptors which restricts them to lie on a hypersphere of a fixed radius. Therefore, minimizing the softmax loss is equivalent to maximizing the cosine similarity for the positive pairs and minimizing it for the negative pairs. In this way, the verification performance on IJB-A dataset is refreshed again. In the current work, we also follow the similar way–DCNN model should be a good baseline. By virtue of the complementary between different DCNN architectures and datasets, we can obtain a more general feature representation model via ensemble strategy. Intrinsic discrimination of subjects is also important for face recognition, inspired by transfer learning, template specific linear one-vs-rest SVMs are trained in target domain. It shares similar idea as TA [34] while different negative set is chosen. Similar to [43],[45],[33], multiple matching scores are merged into a single one for set-to-set matching whereas an easier way is adopted. Last, we also deploy TPE to further enhance performance of face recognition. More detailed information about our learning framework can be found in the next section part. III Transferred Deep Feature Fusion It is necessary that DCNN architectures are trained on tremendous dataset. However, IJB-A datasets contains 500 subjects with 5,396 images and 2,042 videos sampled to 20,412 frames in total. This is obviously inadequate. Unlike [33] where training data is increased by synthesizing faces based on pose, 3D shape and expression variations, inspired by domain adaptation, we need other huge labeled face datasets in source domain to train DCNN model. It is different from replacing the final entropy loss layer for a new task and fine-tuning the DCNN model on this new objective using data from the target domain [11]. We focus on training DCNN model and the one-vs-rest linear SVMs in source domain and target domain, separately. Last, one-shot-similarity (OSS) [2] is utilized to calculate similarity scores and we fuse those multiple matching scores into a single one for final performance evaluation. As shown in Fig.2, our learning framework consists three components: two distinct DCNN models are trained with two different large face datasets in source domain illustrated in middle component, respectively. In target domain, the new unseen data as inputs are fed into those two DCNN architectures with the shared weights and biases learned from source domain for feature extraction, respectively. Then, all features are combined in the first fusion stage. Template specific one-vs-rest SVMs are trained on those fused features in order to boost the intrinsic discrimination of subjects. Last but not least, multiple matching scores computed by OSS is weighted to one final score for verification and identification in the second fusion stage of upper and lower components, respectively. The detailed of each components of our learning framework are presented in the following subsections. III-A Deep feature learning in source domain In this part, we discuss detailedly two DCNN models and two extra huge datasets for training in source domain. Since Network-in-Network (NIN) [6] has been proposed, the depth of DCNN is refreshed again and again. Recent works [15],[40],[48] have shown that convolutional networks with small filters can be substantially deeper, more accurate, and efficient to train if they contain shorter connections between layers close to the input and those close to the output. The bypassing paths are presumed to be the key factor that eases the training of these very deep networks. This point is further supported by ResNets [32], in which pure identity mappings are used as bypassing paths. ResNets have achieved impressive, record-breaking performance on ImageNet [25]. Until recently, Xie et al. [39] reconstruct the building block of ResNets with aggregating a set of transformations. This simple design results in a homogeneous, multi-branch architecture that has only a few hyper-parameters to set. A new dimension called cardinality is proposed, which as an essential factor in addition to the dimension of depth and width. Thus, it is codenamed ResNext. A typical block of ResNext is shown in Fig.3. Considering the balance between performance and efficiency, we choose ResNext 50 as the first DCNN model. For public large face dataset, the VGG-Face should be a better choice for ResNext 50. The original VGG-Face dataset includes 2,109,307 available images and 2,614 subjects. First, we utilize ground-truth bounding box given by dataset to crop and resize face images from the original ones. Each face image is 144$\times$144. An off-the-shelf CNN model pre-trained on CASIA-WebFace is deployed to do noisy data cleaning. Moreover, the overlap subject with IJB-A dataset should be removed. Finally, we obtain 1,648,187 images and 2,613 subjects in total. For partition of training and validation parts, we refer to ImageNet. 90% of the total images (1,483,368) are served as training data. 5% of the total images (82,410) are viewed as validation data. Our implementation for VGG-Face on ResNext 50 is implemented by MXNet [26]. The image is resized from 144$\times$144 to 480$\times$480 for data augmentation. A 224$\times$224 crop is randomly sampled from 480$\times$480 or its horizontal flip, with the per-pixel mean substracted. The standard color augmentation [4] is used. We adopt batch normalization (BN) [19] right after each convolution and before ReLU. We initialize the weights as in [21] and train ResNext 50 from scratch. NAG with a mini-batch size of 256 is utilized on our GPU cluster machine. The learning rate starts from 0.1 and is divided by 10 every 30 epoch and the model is trained for up to 125 epoch. The weight decay is 0.0001 and the momentum is 0.9. The cardinality is 32. The training and validation curves are shown in Fig.4. Finally, we obtain the validation performance 95.63% at top1 and 97.00% at top 5, respectively. Inspired by NIN, an orthogonal approach to making networks deeper (e.g., with the help of skip connections) is to increase the network width. The GoogLeNet [18] uses an ”Inception module” which concatenates features maps produced by filers of different sizes. Different from ResNext which enhances representational power of network via extremely deep architecture, GoogLeNet depends on wider structure to boost capacity of network. Along with the BN emergence, training DCNN becomes easier than before. Thus, GoogLeNet-BN is our second DCNN model. To train GoogLeNet-BN on a much bigger dataset with large number of subjects. Data preprocessing is done as following steps. We use OpenCV to detect face and utilize bounding box to crop and resize face images. Each image is 256$\times$256. There are 582,405 images can not be detected, so we delete them. The overlap subject with IJB-A dataset should be removed. Considering the data distribution, we only keep those identities which have 40-500 images. Finally, we obtain 4,356,052 images and 53,317 subjects in total. Our implementation for our face data on GoogLeNet-BN is implemented by caffe [10]. A 224$\times$224 crop is randomly sampled from 256$\times$256 or its horizontal flip. We initialize the weights as in [21] and train GoogLeNet from scratch. SGD with a mini-batch size of 256 is utilized on our GPU cluster machine. The learning rate starts from 0.1 and exp policy is adopted. The weight decay is 0.0001 and the momentum is 0.9. The model are trained for up to 60$\times$$10^{4}$ iterations. We stop training procedure when the error is not decreasing. III-B Template-based unconstrained face recognition After finish training procedure of two DCNN models in source domain. Weights and biases of ResNext 50 and GoogLeNet-BN are shared into target domain. Each face image or frame of video from target domain is viewed as input to feed into those two models, respectively. For ResNext 50, the penultimate global average pooling layer is served as feature extraction layer. It has 2,048 output size. Thus, the feature dimension is 2,048. Given an image or frame ${{\mathbf{x}}_{i}}\in{\mathbb{R}^{d}}$ from a mini-batch of size $M$, where ${d}$ is the dimension of image or frame. ${f_{R}}\left({{{\mathbf{x}}_{i}}}\right)\in{\mathbb{R}^{{d_{1}}}}$ denotes the feature from ResNext 50, where ${d_{1}}<d$ and ${d_{1}}=2048$. Similarly, for GoogLeNet-BN, 7$\times$7 average pooling layer is treated as feature extraction layer. The channel size is 1,024. So, the feature dimension is 1,024. Let ${\text{ }}{f_{G}}\left({{{\mathbf{x}}_{i}}}\right)\in{\mathbb{R}^{{d_{2}}}}$ is the feature from GoogLenet-BN, where ${d_{2}}=1024$. In the first-stage fusion, ${f_{R}}\left({{{\mathbf{x}}_{i}}}\right)$ and ${f_{G}}\left({{{\mathbf{x}}_{i}}}\right)$ are concatenated into ${\text{ }}{f_{F}}\left({{{\mathbf{x}}_{i}}}\right)\in{\mathbb{R}^{{d_{3}}}}$, where ${d_{3}}=3072$. Finally, each feature is normalized to unit via $L{}_{2}$ norm for the next procedure. After feature fusion, in order to train a more discriminative model in target domain, template specific one-vs-rest SVMs play an important role. Specifically, the weights and biases terms for template specific SVMs are learned by optimizing the following ${L_{2}}$-regularized ${L_{2}}$-loss objective function: $$\displaystyle\mathop{\min}\limits_{\mathbf{w}}\frac{1}{2}{{\mathbf{w}}^{T}}{% \mathbf{w}}+{\lambda_{+}}\sum\limits_{i=1}^{{N_{+}}}{\max{{\left[{0,1-{y_{i}}{% {\mathbf{w}}^{T}}{f_{F}}\left({{{\mathbf{x}}_{i}}}\right)}\right]}^{2}}}$$ (1) $$\displaystyle+{\lambda_{-}}\sum\limits_{i=1}^{{N_{-}}}{\max{{\left[{0,1-{y_{j}% }{{\mathbf{w}}^{T}}{f_{F}}\left({{{\mathbf{x}}_{j}}}\right)}\right]}^{2}}}$$ where ${\mathbf{w}}$ denote the weights including bias term, ${y_{i}}\in\left\{{-1,1}\right\}$ denotes the label indicating whether the current sample being negative or possible, ${{N_{+}}}$ indicates the number of positive samples, ${{N_{-}}}$ is the number of negative ones, ${N_{-}}\gg{N_{+}}$. Moreover, the constraint for negative samples ${\lambda_{-}}=C\frac{{{N_{+}}+{N_{-}}}}{{2{N_{-}}}}$, the constraint for positive samples ${\lambda_{+}}=C\frac{{{N_{+}}+{N_{-}}}}{{2{N_{+}}}}$, where $C$ is a trade-off factor. A template includes images or/and frames of video. For the feature of video frame, we compute the average media encodings. Let $t_{j}^{V}$ denotes average media encoding of video $j$. $$t_{j}^{V}=\frac{1}{{N_{j}^{V}}}\sum\limits_{i=1}^{N_{j}^{V}}{{f_{F}}\left({{{% \mathbf{x}}_{i}}}\right)}$$ (2) where $N_{j}^{V}$ is the number of frame in video $j$, ${{{\mathbf{x}}_{i}}}$ denotes $i$ frame of video $j$. In other words, all features of video frames are aggregate one feature. Thus, the deep facial representations for the $a$th template can be expressed as $${T_{a}}=\left\{{t_{i}^{I},...,t_{{N_{a}}}^{V}}\right\}$$ (3) where ${t_{i}^{I}}$ denotes $i$th image, ${{N_{a}}}$ express the number of image and video. All media encoding need to perform unit normalization. For verification (a.k.a 1:1 compare), the positive sample of template specific SVM is probe template, the large-scale negative samples include the whole training set. For identification (a.k.a 1:N search), the probe template specific SVMs adopt the whole training set as the large-scale negative samples; whereas for gallery template specific SVM, we adopt other gallery templates and the whole training set as large-scale negative samples. Based on One shot similarity (OSS), we compute similarity between two features $p$ and $q$ via $s\left({p,q}\right)=\frac{1}{2}\mathcal{P}\left(q\right)+\frac{1}{2}\mathcal{Q% }\left(p\right)$ where $\mathcal{P}\left(q\right)$ denotes the trained probe template specific SVM model and $\mathcal{Q}\left(p\right)$ indicates the trained gallery template specific SVM model. One template exists many features as Eqn.3, the resulting multiple matching scores should be ensembled into a single one for each template pair in second-stage fusion. $$s\left({{T_{a}},{T_{b}}}\right)=\frac{{\sum\limits_{{t_{i}}\in{T_{a}},{t_{j}}% \in{T_{b}}}{s\left({{t_{i}},{t_{j}}}\right){e^{\beta{\text{ }}s\left({{t_{i}},% {t_{j}}}\right)}}}}}{{\sum\limits_{{t_{i}}\in{T_{a}},{t_{j}}\in{T_{b}}}{{e^{% \beta{\text{ }}s\left({{t_{i}},{t_{j}}}\right)}}}}}$$ (4) where $\beta=0$ is enough in our following experiments. IV Experiments and analysis In this section, we describe the results for evaluation of the experimental system on the IJB-A verification and identification protocols. The IJB-A dataset contains face images and video frames captured from unconstrained settings which are aligned better with the requirements of real applications. There are 500 subjects with 5,396 images and 2,042 videos sampled to 20,412 frames in total. Full pose variation and wide variations in imaging conditions are the main features of IJB-A dataset, which makes the face recognition very challenging. In our experiments, we just utilize the ground-truth bounding box to crop face image from the original one and resize to 224$\times$224 for each image or frame. We do not use any off-the-shelf pre-trained DCNN model to clean data. We also do not deploy any face detector and do not perform any face alignment procedure. A remarkable feature of this dataset is that the concept of template is introduced. Each training an testing sample in called a template which comprises a mixture of static images and sampled video frames. Each static image or a frame of video corresponds with a media. On average, each subject has 11.4 images and 4.2 videos. There are 10 training and testing splits. Each of them contains 333 and 167 subjects, respectively. In TableI, we list the performance of state-of-the-art algorithms on IJB-A dataset. Our performance achieves the best of them for both verification and identification protocols. When we use the TPE to learn a discriminative mapping space while keep the original feature dimension using the training splits of IJB-A. It slightly improves the performance and achieves the new record TAR of 0.921 @ FAR = 0.001, TAR of 0.961 @ FAR = 0.01 and TAR of 0.989 @ FAR = 0.1 for verification. Our method performs significantly better than state-of-the-other algorithms in other indicators as well. These results clearly suggests the effectiveness of our proposed learning framework. In [49], the author reports the results for a very low FAR of 0.0001. Thus, in TableII, we also report the performance @ FAR = 0.0001 for verification protocol, our results still slightly better than $L_{2}$-softmax, even TPE is added. We illustrate the identification results for IJB-A split1 on close protocol in Fig.5. The first column shows the query images from probe templates. The remaining 5 columns show the corresponding top-5 queried gallery templates. For each template, we provide Template ID, Subject ID and similarity score. For all five rows, our approach can successfully find the subjects in rank 1. Finally, we visualize the verification results in Fig.6 and Fig.7 for IJB-A split1 to gain insight into template based unconstrained face recognition. After computing the similarities for all pairs of probe and reference templates, we sort the resulting list. Each row represents a probe and reference template pair. The original templates within IJB-A contain from one to dozens of media. Up to eight individual media are shown with the last space showing a mosaic of the remaining media in the template. Between the templates are the Template IDs for probe and reference as well as the best mated and best non-mated similarity. Fig.6 (a) shows the highest mated similarities. In the thirty highest scoring correct matches, we note that every reference template contains dozens of media. The probe templates also contain dozens of media that matches well. Fig.6 (b) shows the lowest mated template pairs, representing failed matching. The thirty lowest mated results from single-media reference templates are under extremely challenging unconstrained conditions. There extremely difficult cases cannot be solved even using our proposed approach. Fig.7 (a) showing the best non-mated similarities shows the most certain nonmates, again often involving large templates with enough guidance from the relevant and historical information. Fig.7 (b) showing the worst non-mated pairs highlights the unstable errors involving single-media reference templates representing impostors in challenging orientation. V Conclusion In this paper, we propose a unified learning framework named transferred deep feature fusion. It can effectively integrate superiority of each module and outperform the state-of-the-art on IJB-A dataset. Inspired by transfer learning, facial feature encoding model of subjects are trained offline in a source domain, and this feature encoding model is transferred to a specific target domain where limited available faces of new subjects can be encoded. Specifically, in order to capture the intrinsic discrimination of subjects and enhance the generalization capability of face recognition models, we deploy two advanced deep convolutional neural networks (DCNN) with distinct architectures to learn the representation of faces on two different large datasets (each one has no overlap with IJB-A dataset) in source domain. These two DCNN models provide distinct feature representations which can better characterize the data distribution from different perspectives. The complementary between two distinct models is beneficial for feature representation. Thus, representing a face from different perspectives could effectively decrease ambiguity among subjects and enhance the generalization performance of face recognition especially on extremely large number of subjects. After offline training procedure, those two DCNN models are transferred to target domain where templates of IJB-A dataset as inputs are performed feature extraction with shared weigths and biases, respectively. Then, two-stage fusion is designed, features from two DCNN models are combined in order to obtain more discriminative representation in first-stage. Finally, template specific linear SVMs are trained on fused features for classification. Furthermore, for set-to-set matching problem, multiple matching scores are merged into a single one for each template pair as the final results in the second-stage of fusion. 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Coassociative Lie algebras D.-G. Wang, J.J. Zhang and G. Zhuang Wang: School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, P.R.China dgwang@mail.qfnu.edu.cn, dingguo95@126.com Zhang: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195, USA zhang@math.washington.edu Zhuang: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195, USA gzhuang@math.washington.edu Abstract. A coassociative Lie algebra is a Lie algebra equipped with a coassociative coalgebra structure satisfying a compatibility condition. The enveloping algebra of a coassociative Lie algebra can be viewed as a coalgebraic deformation of the usual universal enveloping algebra of a Lie algebra. This new enveloping algebra provides interesting examples of noncommutative and noncocommutative Hopf algebras and leads to a classification of connected Hopf algebras of Gelfand-Kirillov dimension four in [WZZ]. Key words and phrases:Lie algebra, coalgebra, coassociative Lie algebra, universal enveloping algebra, Hopf algebra, Gelfand-Kirillov dimension 2000 Mathematics Subject Classification: Primary 16A24, 16W30, 57T05 Contents 0 Introduction 1 Definitions 2 Results on enveloping algebras 3 Elementary properties of coassociative Lie algebras 4 Examples Dedicated to Kenny Brown and Toby Stafford on the occasion of their 60th birthdays. 0. Introduction We introduce the notion of a coassociative Lie algebra which generalizes in an obvious way both a Lie algebra and a coassociative coalgebra without counit. The enveloping algebra of a coassociative Lie algebra is a bialgebra that is a generalization of the usual universal enveloping algebra of a Lie algebra. The enveloping algebra of a coassociative Lie algebra should be considered as a coalgebraic deformation of the usual universal enveloping algebra on one hand, and potentially as an algebraic deformation of the coordinate ring (regular functions) of certain algebraic groups or semigroups on the other. Let $k$ be a base field that is algebraically closed and everything is over $k$. Let ${\mathfrak{g}}$ denote an ordinary Lie algebra and $L$ a coassociative Lie algebra. Let $U({\mathfrak{g}})$ (respectively, $U(L)$) denote the enveloping algebra of ${\mathfrak{g}}$ (respectively, $L$). It is well-known that $U({\mathfrak{g}})$ is a Hopf algebra. In contrast, $U(L)$ is not a Hopf algebra in general. Theorem 0.1 (Theorem 2.5). The enveloping algebra $U(L)$ of a coassociative Lie algebra $L$ is a Hopf algebra if and only if $L$ is locally conilpotent. Most of unexplained terms will be defined in Sections 1 and 2. Starting with conilpotent coalgebras or nilpotnent Lie algebras, one can construct families of nontrivial coassociative Lie algebras based on them. For instance, we give explicit examples in Example 4.4 (based on strictly upper triangular matrix coalgebras) and Example 4.3 (based on Heisenberg Lie algebras). However, it is quite unsatisfactory that there is no nontrivial Lie structure based on cosemisimple coalgebras [Proposition 3.6] and no nontrivial coalgebra structure based on the semisimple Lie algebra $sl_{2}$ [Theorem 3.9]. It would be interesting if a generalized version of coassociative Lie structure can be constructed on cosemisimple coalgebras and/or semisimple Lie algebras. Coassociative Lie algebras are helpful in the classification of connected Hopf algebras of low Gelfand-Kirillov dimension (denoted by $\operatorname{GKdim}$). Definitions and basic properties of GK-dimension can be found in the first three chapters of [KL]. Our reference book for Hopf algebras is [Mo]. Let $P(H)$ be the subspace of all primitive elements in $H$, and $p(H)$ denote the dimension of $P(H)$. If $\operatorname{GKdim}H\leq 2$, then it is well-known that $H\cong U({\mathfrak{g}})$ for a Lie algebra ${\mathfrak{g}}$ of dimension $p(H)$. If $\operatorname{GKdim}H=3$, then it follows from a result of the third-named author [Zh2] that $H$ is isomorphic to either (i) $U({\mathfrak{g}})$ for a Lie algebra ${\mathfrak{g}}$ of dimension $3$, or (ii) $U(L)$ for a coassociative Lie algebra $L$ of dimension $3$. In [WZZ], we give a classification in GK-dimension 4. Theorem 0.2. [WZZ, Theorem 0.3 and Remark 0.4] Suppose that $k$ is of characteristic zero. Let $H$ be a connected Hopf algebra of GK-dimension four. Then one of the following occurs. (a) If $p(H)=4$, then $H\cong U({\mathfrak{g}})$ for a Lie algebra ${\mathfrak{g}}$ of dimension $4$. (b) If $p(H)=3$, then $H\cong U(L)$ for an anti-cocommutative coassociative Lie algebra $L$ of dimension $4$. (c) If $p(H)=2$, then $H$ is not isomorphic to either $U({\mathfrak{g}})$ or $U(L)$ as Hopf algebras, for any Lie algebra ${\mathfrak{g}}$ or any coassociative Lie algebra $L$. Such a Hopf algebra is isomorphic to one of the four families of Hopf algebras explicitly constructed in [WZZ, Section 4]. The Hopf algebras in Theorem 0.2(b,c) are completely classified in [WZZ]. The proof of Theorem 0.2 is heavily dependent on the study of coassociative Lie algebras. A result of Milnor-Moore-Cartier-Kostant [Mo, Theorem 5.6.5] states that any cocommutative connected Hopf algebra over a field of characteristic zero is isomorphic to $U({\mathfrak{g}})$ for some Lie algebra ${\mathfrak{g}}$. This applies to case (a) of Theorem 0.2. However, the Hopf algebras in Theorem 0.2(b,c) are not cocommutative, hence not isomorphic to $U({\mathfrak{g}})$ for a usual Lie algebra ${\mathfrak{g}}$. A generalization of Theorem 0.2(b) states that if $p(H)=\operatorname{GKdim}H-1$, then $H$ is isomorphic to $U(L)$ for some coassociative Lie algebra $L$ [WZZ, Theorem 0.5]. The Hopf algeba $U(\mathfrak{g})$ is always involutory. In general $U(L)$ is not involutory. By using the Calabi-Yau property of the enveloping algebra of unimodular Lie algebras, we prove the following result. Theorem 0.3 (Theorem 2.7). Let ${\mathfrak{g}}$ be a finite dimensional unimodular Lie algebra. Suppose that $L:=({\mathfrak{g}},\delta)$ is a conilpotent coassociative Lie algebra. Then $U(L)$ is involutory. Coassociative Lie algebras are not understood fully, and a lot of basic questions are unsolved. For example, Question 0.4. Let $n(L)$ be the nilpotency of $L$ and $con(L)$ be the conilpotency of $L$. If $n(L)+con(L)<\infty$, then is there the bound for the number $n(L)+con(L)-\dim L$? Preliminary analysis of known examples shows that if $n(L)+con(L)<\infty$, then $$n(L)+con(L)-\dim L\leq 1.$$ As one may guess, this paper grows out of the study of connected Hopf algebras [WZZ]. By Theorem 0.2(b), certain classes of connected Hopf algebras are the enveloping algebras of coassociative Lie algebras, which are noncommutative and noncocommutative. This construction is very different from the various classical constructions of quantum groups which we are familiar with. One way of thinking of a coassociative Lie algebra is that it has an additional coalgebra structure on a Lie algebra. Our work [WZZ] suggests that certain “nilpotent” quantum groups could be represented by a Lie algebra with an extra coalgebra structure. This idea might be worth pursuing further. 1. Definitions In this section, we start with some definitions and give some easy examples of coassociative Lie algebras. We would like to thank Milen Yakimov for suggesting the name “coassociative Lie algebra”. Definition 1.1. A Lie algebra $(L,[\;,\;])$ together with a coproduct $\delta:L\to L\otimes L$ is called a coassociative Lie algebra if (a) $(L,\delta)$ is a coassociative coalgebra without counit, and (b) two operations $\delta$ and $[\;,\;]$ satisfy the following compatibility condition (E1.1.1) $$\delta([a,b])=b_{1}\otimes[a,b_{2}]+[a,b_{1}]\otimes b_{2}+[a_{1},b]\otimes a_% {2}+a_{1}\otimes[a_{2},b]+[\delta(a),\delta(b)]$$ for all $a,b\in L$. Here $\delta(x)=x_{1}\otimes x_{2}$, which is basically the Sweedler’s notation with the summation indicator omitted. Definition 1.1 is not complete without the following remark. Remark 1.2. For a general Lie algebra $L$, the product $L\otimes L$ is not a Lie algebra in any natural way. To make sense of $[\delta(a),\delta(b)]$ in (E1.1.1), we need to embed $L$ into the usual universal enveloping algebra $U(L)$. Then $L\otimes L$ can be naturally identified with a subspace of $U(L)\otimes U(L)$. Under this identification, $[f,g]$ is defined as the commutator of $f$ and $g$ in the associative algebra $U(L)\otimes U(L)$ for all elements $f,g\in L\otimes L$. The equation (E1.1.1) implies that the element $[\delta(a),\delta(b)]\in U(L)\otimes U(L)$ is actually in the subspace $L\otimes L$, namely, (E1.2.1) $$[\delta(a),\delta(b)]\in L\otimes L.$$ Inside $U(L)\otimes U(L)$, the equation (E1.1.1) can be written as (E1.2.2) $$\delta([a,b])=[a\otimes 1+1\otimes a,\delta(b)]+[\delta(a),b\otimes 1+1\otimes b% ]+[\delta(a),\delta(b)]$$ for all $a,b\in L$. Let $L^{1}$ denote the Lie algebra extension $L\oplus k1$ where $1$ is in the center (i.e., $[1,L^{1}]=0$). Define $\Delta:L^{1}\to L^{1}\otimes L^{1}$ by (E1.2.3) $$\Delta(1)=1\otimes 1,\quad\Delta(a)=a\otimes 1+1\otimes a+\delta(a)$$ for all $a\in L$. By embedding $L^{1}=L\oplus k1$ into $U(L)$ naturally, we can define $[f,g]$ for all elements in $f,g\in L^{1}$. For any $k$-space $V$, let $V^{\otimes 2}$ denote $V\otimes V$. Lemma 1.3. Let $L$ be a Lie algebra, and let $\delta:L\to L^{\otimes 2}$ be any linear map. Then $L$ is a coassociative Lie algebra if and only if $(L^{1},\Delta)$ is a counital coalgebra and the equation (E1.3.1) $$\Delta([a,b])=[\Delta(a),\Delta(b)]$$ holds in $U(L)^{\otimes 2}$ for all $a,b\in L^{1}$. Proof. A direct calculation shows that $\Delta$ is coassociative if and only if $\delta$ is coassociative. It is also easy to see that (E1.1.1) is equivalent to (E1.3.1). ∎ It follows from (E1.2.2) that the kernel $\ker\delta$ is a Lie subalgebra of $L$. Let $$\Phi(a,b):=\delta([a,b])-(b_{1}\otimes[a,b_{2}]+[a,b_{1}]\otimes b_{2}+[a_{1},% b]\otimes a_{2}+a_{1}\otimes[a_{2},b]).$$ Then (E1.1.1) becomes (E1.3.2) $$\Phi(a,b)=[\delta(a),\delta(b)].$$ Let $L$ be a coassociative Lie algebra. If $\delta=0$, it is an ordinary Lie algebra. If $[\;,\;]=0$, then $L$ is simply a coassociative coalgebra without counit. Let $\operatorname{{\sf CoLieAlg}}$ denote the category of coassociative Lie algebras, $\operatorname{{\sf LieAlg}}$ denote the category of Lie algebras and $\operatorname{{\sf CoAlg}}$ denote the category of coalgebras without counit. Then both $\operatorname{{\sf LieAlg}}$ and $\operatorname{{\sf CoAlg}}$ are full subcategories of $\operatorname{{\sf CoLieAlg}}$. Now we give some simple examples of coassociative Lie algebras with nontrivial Lie bracket $[\;,\;]$ and nontrivial coproduct $\delta$, in which both sides of (E1.1.1) are trivially zero. More complicated examples will be given in later sections. Example 1.4. Let $L=kx\oplus ky$ with $[x,y]=y$ and $\delta(x)=\lambda y\otimes y$ for any $\lambda\in k$ and $\delta(y)=0$. Then $L$ is a coassociative Lie algebra. It is clear that $(L,[\;,\;])$ is a Lie algebra and $(L,\delta)$ is a coalgebra. Note that (E1.1.1) is trivial when $a=b$. So it suffices to check (E1.1.1) for $a=x$ and $b=y$, in which case both sides of (E1.1.1) are zero. It can be verified that, up to isomorphism, this is the unique coassociative Lie algebra with an underlying non-abelian Lie algebra of dimension 2. Here is a slightly different situation. Lemma 1.5. Let $(L,[\;,\;])$ be a Lie algebra and $(L,\delta)$ be a coalgebra. Suppose that $\delta(L)\subset Z\otimes Z$ where $Z$ is the center of $(L,[\;,\;])$ and that $\delta([L,L])=0$. Then $L$ is a coassociative Lie algebra. Proof. The hypotheses imply that both sides of (E1.1.1) are zero. ∎ Example 1.6. Suppose that $L$ is a Lie algebra containing a Lie ideal $W$ such that both $W$ and $L/W$ are abelian. For example, $L$ is a Lie algebra of nilpotency 2. Let $\delta:L\xrightarrow{\pi}L/W\to W^{\otimes 2}$ be a $k$-linear map where $\pi$ is the quotient map. A direct calculation shows that $(\delta\otimes 1)\delta=(1\otimes\delta)\delta=0$. Thus $(L,\delta)$ is coassociative. By the definition of $\delta$, the hypotheses of Lemma 1.5 hold; whence $L$ is a coassociative Lie algebra. Below are two special cases. (a) Let $L$ be the 3-dimensional Heisenberg Lie algebra with a basis $\{x,y,z\}$ such that $[x,y]=z$ and $z$ is central. Define $\delta(x)=z\otimes z$ and $\delta(y)=\lambda z\otimes z$, for some $\lambda\in k$, and $\delta(z)=0$. By the above paragraph, $L$ is a coassociative Lie algebra. (b) Let $W_{1}$ be any Lie algebra and $W$ be a vector space. Let $\phi:W_{1}\to W$ be any $k$-linear map. Define a Lie bracket on $L:=W_{1}\oplus W$ by $$[w_{1}+w,t_{1}+t]=\phi([w_{1},t_{1}]_{W_{1}})$$ for all $w_{1},t_{1}\in W_{1}$ and $w,t\in W$. Then $W$ is a Lie ideal of $L$ such that both $L/W$ and $W$ are abelian. Then every $k$-linear map $\delta:L\xrightarrow{\pi}W_{1}\to W\otimes W$ defines a coassociative Lie algebra $L$. Next, we give an example of “almost coassociative Lie algebra”, which is dependent on the embedding of $L$ into an associative algebra. If $A$ is any associative algebra, define $[\;,\;]_{A}$ to be the commutator of $A$. Example 1.7. Let $H$ be a bialgebra and let $H_{+}:=\ker\epsilon$ where $\epsilon$ is the counit of $H$. Let $L$ be a Lie subalgebra of $(H_{+},[\;,\;]_{H})$ (for example, $L=H_{+}$). Define $\delta(a)=\Delta(a)-1\otimes a-a\otimes 1$ for all $a\in H$. Then $(H_{+},\delta)$ is a coalgebra. Suppose that $\delta(L)\subset L^{\otimes 2}$. Then $(L,\delta)$ is a coalgebra. However, $(L,[\;,\;]_{H},\delta)$ is generally not a coassociative Lie algebra. Let $H$ be the 4-dimensional Taft Hopf algebra: $$k\langle g,x\rangle/(xg+gx=0,\;g^{2}=1,\;x^{2}=0)$$ with $\Delta(g)=g\otimes g$, $\Delta(x)=x\otimes 1+g\otimes x$, and $\epsilon(g)=1$, $\epsilon(x)=0$. Let $L$ be $H_{+}=kx\oplus k(g-1)\oplus k(gx)$. Let $y=g-1$ and $z=gx$. Then $[x,y]_{H}=-2z$, $[x,z]_{H}=0$, $[y,z]_{H}=2x$, $\delta(x)=y\otimes x$, $\delta(y)=y\otimes y$, and $\delta(z)=z\otimes y$. Hence $$[\delta(x),\delta(z)]_{H^{\otimes 2}}=[y\otimes x,z\otimes y]_{H^{\otimes 2}}=% yz\otimes xy-zy\otimes yx=:(*).$$ Using the facts $[x,y]_{H}=-2z$ and $[y,z]_{H}=2x$, we have $$(*)=yz\otimes xy-(yz-2x)\otimes(xy+2z)=2yz\otimes z+2x\otimes xy+4x\otimes z,$$ which is not in $L\otimes L$ if we embed $L$ into $U(L)$. Hence (E1.1.1) does not hold for $(L,[\;,\;]_{H},\delta)$, and consequently, $L$ is not a coassociative Lie algebra. After identifying $yz$ with $-z$, and $xy$ with $-x-z$ in $H$, (E1.1.1) does hold in $H\otimes H$. The enveloping algebra of a coassociative Lie algebra is defined as follows. Definition 1.8. Let $L$ be a coassociative Lie algebra. The enveloping algebra of $L$, denoted by $U(L)$, is defined to be a bialgebra, whose algebra structure equals that of the enveloping algebra of the underlying Lie algebra $L$, namely, $$U(L)=k\langle L\rangle/(ab-ba=[a,b],\forall\;a,b\in L),$$ and whose coalgebra structure is determined by $$\Delta(a)=a\otimes 1+1\otimes a+\delta(a),\quad\epsilon(a)=0$$ for all $a\in L$. By (E1.3.1) it is easy to see that $U(L)$ is a bialgebra. We will also use $U(L,\delta)$ to denote $U(L)$ if we want to emphasize the coproduct $\delta$. It is clear that the assignment $L\to U(L)$ defines a functor from $\operatorname{{\sf CoLieAlg}}$ to ${\sf{BiAlg}}$ where ${\sf{BiAlg}}$ is the category of bialgebras. Example 1.9. If $\dim L=1$, then there are exactly two coassociative Lie algebra structures on $L$, up to isomorphism. One is determined by $\delta=0$. In this case, the enveloping algebra is $U(L)=k[x]$ with $x$ being a primitive element. Consequently, $U(L)$ is a Hopf algebra. The other is determined by $\delta(x)=x\otimes x$. Here, $U(L)=k[g]$ where $g=1+x$ and $g$ is a group-like element in $U(L)$. In this case, $U(L)$ is not a Hopf algebra because the group-like element $g$ is not invertible in $U(L)$. Let $({\mathfrak{g}},\delta)$ be a coassociative Lie algebra with underlying Lie algebra ${\mathfrak{g}}$. Then Poincaré-Birkhoff-Witt (PBW) theorem holds for $U({\mathfrak{g}},\delta)$, since, algebraically it is the usual enveloping algebra $U({\mathfrak{g}})$. The difference between $U({\mathfrak{g}},\delta)$ and $U({\mathfrak{g}})$ is their coalgebra structures. For many examples of $({\mathfrak{g}},\delta)$, one can construct explicitly a family of bialgebras $B(q)$ dependent on $({\mathfrak{g}},\delta)$, where $q\in k$, such that $B(1)=U({\mathfrak{g}},\delta)$ and $B(0)=U({\mathfrak{g}})$. Hence $U({\mathfrak{g}},\delta)$ can be considered as a coalgebraic deformation of $U({\mathfrak{g}})$. However, we will not pursue this topic further. Since $U(L)$ is generated by $L$ as an algebra, $U(L)$ is cocommutative if and only if the underlying coalgebra $L$ is cocommutative. Similarly, $U(L)$ is commutative if and only if the underlying Lie algebra $L$ is abelian. Let $\delta^{n}=(\delta\otimes 1^{\otimes n-1})(\delta\otimes 1^{\otimes n-2})% \cdots(\delta\otimes 1)\delta$. Here is a list of definitions. Definition 1.10. Let $L_{1},L_{2},L$ be coassociative Lie algebras. (a) We say that $L_{1}$ and $L_{2}$ are quasi-equivalent if $U(L_{1})$ is isomorphic to $U(L_{2})$ as bialgebras. (b) A Lie algebra ${\mathfrak{g}}$ is called rigid if every compatible $\delta$-structure on ${\mathfrak{g}}$ is zero. (c) A coalgebra $C$ is called rigid if every compatible Lie-structure on $C$ is trivial. (d) A coalgebra $(C,\delta)$ is called anti-cocommutative if $\tau\delta=-\delta$, where the flip $\tau:C^{\otimes 2}\to C^{\otimes 2}$ is defined by $\tau(a\otimes b)=b\otimes a$. (e) The nilpotency of $L$, denoted by $n(L)$, is defined to be the nilpotency of the underlying Lie algebra $L$. (f) An element $x\in L$ is called conilpotent if $\delta^{n}(x)=0$ for some $n>0$. We say $L$ is locally conilpotent if every element in $L$ is conilpotent. (g) We call $L$ $n$-conilpotent if $\delta^{n}(L)=0$. The smallest such $n$, denoted by $con(L)$, is called conilpotency of $L$. 2. Results on enveloping algebras In this section we study some properties of the enveloping algebras $U(L)$. Let $B$ be a bialgebra with coproduct $\Delta$. Define $\delta_{B}:B\to B^{\otimes 2}$ by $$\delta_{B}(x)=\Delta(x)-x\otimes 1-1\otimes x$$ for all $x\in B$. Definition 2.1. A subspace $V$ in a bialgebra $B$ is called a $\delta$-space of $B$ if (a) $V$ is a Lie subalgebra of $(B,[\;,\;]_{B})$, (b) $\epsilon(V)=0$, (c) $\delta_{B}(V)\subset V^{\otimes 2}$ inside $B^{\otimes 2}$, and (d) $B$ is an ${\mathbb{N}}$-filtered algebra with an exhaustive filtration defined by $F_{n}(B):=(k1+V)^{n}$, for $n\geq 0$, such that the associated graded ring $gr_{F}B$ is isomorphic to the commutative polynomial ring $k[V]$. Remark 2.2. If $L$ is coassociative Lie algebra, then $L$ is $\delta$-space of $U(L)$. In general, a $\delta$-space of $U(L)$ is not unique. See Corollary 2.6. Lemma 2.3. If $V$ is a $\delta$-space of $B$, then $(V,[\;,\;]_{B},\delta_{B})$ is a coassociative Lie algebra and $B\cong U(V)$ as bialgebras. Proof. Let $U$ be the usual enveloping algebra of the Lie algebra $(V,[\;,\;]_{B})$. Then there is an algebra homomorphism $\phi:U\rightarrow B$ such that $\phi\mid_{V}=Id_{V}$. It follows from Definition 2.1(c) that $B$ is generated by $V$ and that the set $\{v_{1}^{n_{1}}\cdots v_{d}^{n_{d}}\mid n_{i}\geq 0\}$ is a $k$-linear basis of $B$ where $\{v_{1},\cdots,v_{j},\cdots\}$ is a $k$-linear basis of $V$. Since $\{v_{1}^{n_{1}}\cdots v_{d}^{n_{d}}\mid n_{i}\geq 0\}$ is also a $k$-linear basis of $U$ by PBW theorem, $\phi$ is an isomorphism of algebras. Note that $B$ is a bialgebra and generated by $V$ as an algebra, one can defined a canonical bialgebra structure $\Delta_{U}$ on $U$ via $\phi$ such that $\phi$ is an isomorphism of bialgebras. Let $\delta_{U}(v)=\Delta_{U}(v)-v\otimes 1-1\otimes v$ for $v\in V$. Since $\delta_{B}(x)=\Delta_{B}(x)-x\otimes 1-1\otimes x$, $\delta_{U}(v)=\delta_{B}(v)$ for all $v\in V$ (we are identifying the subspace $V\subset B$ with the subspace $V\subset U$ via the map $\phi\mid_{V}=Id_{V}$). By Definition 2.1(b), one sees easily that $\delta_{U}(v)\in V^{\otimes 2}$ for all $v\in V$. Since $U$ is a bialgebra (via the map $\phi$), (E1.3.1) holds. Now by Lemma 1.3 and Definition 1.8, $(V,[\;,\;]_{U},\delta_{U})$ is a coassociative Lie algebra with enveloping algebra $U$. Since $(V,[\;,\;]_{B},\delta_{B})=(V,[\;,\;]_{U},\delta_{U})$ by construction and $U\cong B$ as bialgebras, the results follow. ∎ Let $(L,\delta)$ be a coalgebra (without counit). Let $L^{1}=k1\oplus L$, and $\Delta:L^{1}\to(L^{1})^{\otimes 2}$ be defined as (E1.2.3). Moreover, let $\epsilon:L^{1}\to k$ be defined by $\epsilon(1)=1,\epsilon(x)=0$ for all $x\in L$. Then the assignment $(L,\delta)\to(L^{1},\Delta,\epsilon)$ defines a functor from $\operatorname{{\sf CoAlg}}$ to $\operatorname{{\sf CouAlg}}$, where $\operatorname{{\sf CouAlg}}$ is the category of counital coassociative coalgebras. The following lemma is easy. Lemma 2.4. Let $(L,\delta)$ be a coalgebra. Then $(L,\delta)$ is locally conilpotent if and only if $(L^{1},\Delta,\epsilon)$ is a connected counital coalgebra. Proof. Since $L$ is a sum of its finite dimensional subcoalgebras, we can assume without loss of generality that $L$ is finite dimensional. Note that $L$ can be identified with the quotient coalgebra $L^{1}/k1$. By taking the $k$-linear dual, $L^{*}$ becomes a subalgebra (without a unit) of $(L^{1})^{*}$. In fact, $L^{*}$ is a maximal ideal of $(L^{1})^{*}$ of codimension $1$. Now the lemma is equivalent to the statement that $L^{*}$ is a nilpotent ideal if and only if $L^{*}$ is the unique maximal ideal of $(L^{1})^{*}$, which is an easy ring-theoretical fact. ∎ Now we are ready to prove Theorem 0.1. Theorem 2.5. Let $L$ be a coassociative Lie algebra. Then the following are equivalent: (a) $L$ is locally conilpotent; (b) $U(L)$ is a connected Hopf algebra; and (c) $U(L)$ is a Hopf algebra. Proof. (a) $\Rightarrow$ (b). Since $(L,\delta)$ is locally conilpotent, $(L^{1},\Delta,\epsilon)$ is a connected coalgebra by Lemma 2.4. Since $U(L)$ is generated by $L^{1}$ as an algebra, $U(L)$ is connected as a coalgebra. It follows from [Mo, Lemma 5.2.10] that a connected bialgebra is automatically a Hopf algebra. (b) $\Rightarrow$ (c). This is clear. (c) $\Rightarrow$ (a). We proceed by contradiction. Suppose that $U(L)$ is a Hopf algebra, but $(L,\delta)$ is not locally conilpotent. Then $(L^{1},\Delta)$ is not connected, whence its coradical is strictly larger that $k1$. Pick a simple subcoalgebra with counit of $L^{1}$, say $C$, which not equal to $k1$. Since $k$ is algebraically closed, $C$ is isomorphic to a matrix coalgebra $\bigoplus_{i,j=1,\ldots,n}kx_{ij}$ with $$\Delta(x_{ij})=\sum_{s=1}^{n}x_{is}\otimes x_{sj},\quad{\text{and}}\quad% \epsilon(x_{ij})=\delta_{ij}$$ for all $1\leq i,j\leq n$. Here $\delta_{ij}$ is the Kronecker delta. Let $y_{ij}=x_{ij}-\delta_{ij}$, for all $i,j$. Note that $L=\ker(\epsilon:L^{1}\to k)$. Then $\bigoplus_{i,j=1,\ldots,n}ky_{ij}\subset L$ is a simple subcoalgebra of $L$ such that $\delta(y_{ij})=\sum_{s=1}^{n}y_{is}\otimes y_{sj}$ for all $1\leq i,j\leq n$. Let $z_{ij}=S(x_{ij})$ for all $1\leq i,j\leq n$. Let $X$ be the matrix $(x_{ij})_{n\times n}$ and $Z$ be the matrix $(z_{ij})_{n\times n}$. Then the antipode axiom implies that $XZ=ZX=I_{n}$ where $I_{n}$ is the identity $n\times n$-matrix. Note that $L$ is a $\delta$-space of $U(L)$, and hence $U(L)$ has a filtration defined by $F_{n}=(L^{1})^{n}$ such that $\operatorname{gr}_{F}U(L)$ is isomorphic to the commutative polynomial ring $k[L]$. One can extend this filtration naturally from $U(L)$ to the matrix algebra $M_{n}(U(L))$ such that $\operatorname{gr}_{F}(M_{n}(U(L))\cong M_{n}(k[L])$. Let $\operatorname{gr}$ also denote the leading terms of elements in $\operatorname{gr}_{F}(M_{n}(U(L))$. Then the equation $XZ=I_{n}$ implies that $\operatorname{gr}(X)\operatorname{gr}(Z)=0$. Note that $\operatorname{gr}(X)=(y_{ij})\in M_{n}(k[L])$ and thus $\det\operatorname{gr}(X)=\det(y_{ij})$, which is nonzero in the commutative polynomial subring $k[y_{ij}]\subset k[L]$. Consequently, the equation $\operatorname{gr}(X)\operatorname{gr}(Z)=0$ implies that $\operatorname{gr}(Z)=0$. Hence $Z=0$, yielding a contradiction. The assertion follows. ∎ Corollary 2.6. If ${\text{char}}\;k=0$, then every locally conilpotent cocommutative coassociative Lie algebra is quasi-equivalent to a Lie algebra. Proof. Let $L$ be any locally conilpotent cocommutative coassociative Lie algebra. Then $U(L)$ is a connected cocommutative Hopf algebra by Theorem 2.5. By Milnor-Moore-Cartier-Kostant Theorem [Mo, Theorem 5.6.5], $U(L)$ is isomorphic to $U({\mathfrak{g}})$ for some Lie algebra ${\mathfrak{g}}$. The assertion follows. ∎ Recall that a Lie algebra ${\mathfrak{g}}$ is called unimodular if $ad(x)$ has zero trace for all $x\in{\mathfrak{g}}$, where $ad(x)\in\operatorname{End}_{k}(\mathfrak{g})$ is the $k$-linear map sending $y\in\mathfrak{g}$ to $[x,y]$. Combining results of Koszul [Ko] and Yekutieli [Ye, Theorem A] (also see [BZ, Proposition 6.3] and [HVZ, Theorem 5.3 and Lemma 4.1]), ${\mathfrak{g}}$ is unimodular if and only if $U({\mathfrak{g}})[d]$ is the rigid dualizing complex over $U({\mathfrak{g}})$ if $d:=\dim{\mathfrak{g}}$ is finite. By [BZ, Proposition 6.3] and [HVZ, Theorem 5.3], ${\mathfrak{g}}$ is unimodular if and only if the Hopf algebra $U({\mathfrak{g}})$ is unimodular in the sense of [LWZ], if and only if $U({\mathfrak{g}})$ is Calabi-Yau, and if and only if the homological integral of $U({\mathfrak{g}})$ (as defined in [LWZ]) is trivial. It is well-known that all Heisenberg Lie algebras are unimodular, and that the 2-dimensional non-abelian Lie algebra is not. Next we verify Theorem 0.3. Theorem 2.7. Let ${\mathfrak{g}}$ be a finite dimensional unimodular Lie algebra. Suppose $({\mathfrak{g}},\delta)$ is a coassociative Lie algebra such that $\delta$ is conilpotent. Then the Hopf algebra $U({\mathfrak{g}},\delta)$ is involutory. Proof. Let $H$ (respectively $K$) denote the Hopf algebra $U({\mathfrak{g}},\delta)$ (respectively, $U({\mathfrak{g}})$). Let $\mu_{H}$ and $\mu_{K}$ denote the Nakayama automorphisms of $H$ and $K$ respectively. Since the Nakayama automorphism is defined uniquely up to an inner automorphism, and the units in $H$ are those elements in $k^{\times}$, the Nakayama automorphism of $H$ (and of $K$) is unique. Since $H=K$ as algebras by Definition 1.8, we have $\mu_{H}=\mu_{K}$. Since ${\mathfrak{g}}$ is unimodular, $\mu_{K}=Id_{K}$ by [BZ, Proposition 6.3(c)]. As a consequence $\mu_{H}=Id_{H}$. Since ${\mathfrak{g}}$ is unimodular, the homological integral of $K$, denoted by $\int^{l}_{K}$, is trivial. Consequently, $\int^{l}_{K}$ equals the trivial module $K/({\mathfrak{g}})$ where $({\mathfrak{g}})$ is the ideal of $K$ generated by subspace ${\mathfrak{g}}$. The homological integral is only dependent on the algebra structure of the Hopf algebra, so we have that $\int^{l}_{H}=H/({\mathfrak{g}})$. This implies that $\int^{l}_{H}$ is trivial. Therefore the left winding automorphism associated to $\int^{l}_{H}$, denoted by $\Xi^{l}_{\int^{l}}$, is the identity map of $H$. Combining the above with [BZ, Theorem 0.3], we have that $$Id_{H}=\mu_{H}=S_{H}^{2}\circ\Xi^{l}_{\int^{l}}=S_{H}^{2}\circ Id_{H}=S_{H}^{2}$$ where $S_{H}$ is the antipode of $H$. Hence $S_{H}^{2}=Id_{H}$, and $H$ is involutory. ∎ Example 4.2 shows that $U({\mathfrak{g}},\delta)$ may not be involutory if ${\mathfrak{g}}$ is not unimodular. This is another way of showing that $U({\mathfrak{g}},\delta)$ is not isomorphic to $U({\mathfrak{g}}^{\prime})$ for any Lie algebra ${\mathfrak{g}}^{\prime}$. For the rest of this section we assume that ${\text{char}}\;k\neq 2$. Lemma 2.8. Let $L$ be an anti-cocommutative coalgebra. Then (a) $con(L)\leq 2$, and as a consequence, $L$ is conilpotent; and (b) $\delta(L)\subset(\ker\delta)^{\otimes 2}$. Proof. (a) A standard calculation by using Sweedler’s notation shows that $$(1\otimes\delta)\tau\delta=(\tau\otimes 1)(1\otimes\tau)(1\otimes\delta)\delta.$$ Since $\tau\delta=-\delta$ by assumption, the left-hand side of the equation becomes $-(1\otimes\delta)\delta$ while the right-hand side is $(1\otimes\delta)\delta$. As a consequence, $(1\otimes\delta)\delta=0$. (b) For any $x\in L$, write $\delta(x)=\sum_{i=1}^{n}x_{i}\otimes y_{i}$ for a minimal integer $n$. Then $\{x_{i}\}_{i=1}^{n}$ is linearly independent. Since $con(L)\leq 2$, $$0=(1\otimes\delta)\delta(x)=\sum_{i=1}^{n}x_{i}\otimes\delta(y_{i}).$$ Since $\{x_{i}\}_{i=1}^{n}$ is linearly independent, $\delta(y_{i})=0$ for all $i$. This means that $\delta(L)\subset L\otimes\ker\delta$. Similarly, $\delta(L)\subset\ker\delta\otimes L$. The assertion follows. ∎ Lemma 2.8 also implies a nice fact about the form of the antipode. Proposition 2.9. Suppose that $L$ is anti-cocommutative and that $U(L)$ is involutory. Then $S(x)=-x$ for all $x\in L$. Proof. It follows from Definition 1.1 that the kernel of $\delta$, denoted by $K$, is a Lie subalgebra of $L$. If $x\in K$, then $\Delta(x)=x\otimes 1+1\otimes x$. The antipode axiom implies that $S(x)=-x$. If $x\in L\setminus K$, it follows from Lemma 2.8(b) that $$\delta(x)=\sum_{i<j}a_{ij}(x_{i}\otimes x_{j}-x_{j}\otimes x_{i}),$$ for some $x_{i}\in K$ and some $a_{ij}\in k$. Hence $$\Delta(x)=x\otimes 1+1\otimes x+\sum_{i<j}a_{ij}(x_{i}\otimes x_{j}-x_{j}% \otimes x_{i}).$$ Applying the antipode axiom, and using the fact that $S(x_{i})=-x_{i}$, we have that $$0=S(x)+x+\sum_{i<j}a_{ij}(S(x_{i})x_{j}-S(x_{j})x_{i})=S(x)+x+\sum_{i<j}a_{ij}% (-x_{i}x_{j}+x_{j}x_{i}).$$ So $S(x)=-x-Y$ where $Y=\sum_{i<j}a_{ij}(-x_{i}x_{j}+x_{j}x_{i})\in K$. Applying $S$ to $S(x)=-x-Y$, and using the hypothesis that $S^{2}=Id$, we have $x=-S(x)+Y$. Thus $Y=0$, and $S(x)=-x$. ∎ 3. Elementary properties of coassociative Lie algebras In this section some elementary properties of coassociative Lie algebras are discussed. First we need some lemmas that will simplify computations. Lemma 3.1. Let $L$ be a coassociative Lie algebra. Let $\{x_{i}\}_{i\in I}$ be a totally ordered $k$-linear basis of $L$. For any $a,b\in L$, write $\delta(a)=\sum_{i}x_{i}\otimes a_{i}$ and $\delta(b)=\sum_{i}x_{i}\otimes b_{i}$. Then (a) $[a_{i},b_{i}]=0$ for all $i$; and (b) $[a_{i},b_{j}]+[a_{j},b_{i}]=0$ for all $i,j$. If $\delta(L)\subset L^{\prime}\otimes L^{\prime\prime}$ for some subspaces $L^{\prime}$ and $L^{\prime\prime}$ of $L$, then (c) $[\delta(a),\delta(b)]\in[L^{\prime},L^{\prime}]\otimes[L^{\prime\prime},L^{% \prime\prime}]$; (d) $\Phi(a,b)\in[L^{\prime},L^{\prime}]\otimes[L^{\prime\prime},L^{\prime\prime}]$; and (e) $\delta([L,L])\subset[L^{\prime},L]\otimes L^{\prime\prime}+L^{\prime}\otimes[L% ^{\prime\prime},L]+[L^{\prime},L^{\prime}]\otimes[L^{\prime\prime},L^{\prime% \prime}]$. Proof. We compute $[\delta(a),\delta(b)]$ in $U(L)^{\otimes 2}$ as follows $$\displaystyle\;[\delta(a),\delta(b)]$$ $$\displaystyle=\sum_{i,j}x_{i}x_{j}\otimes a_{i}b_{j}-\sum_{i,j}x_{i}x_{j}% \otimes b_{i}a_{j}$$ $$\displaystyle=\sum_{i<j}x_{i}x_{j}\otimes a_{i}b_{j}+\sum_{j>i}x_{j}x_{i}% \otimes a_{j}b_{i}+\sum_{i}x_{i}^{2}\otimes a_{i}b_{i}$$ $$\displaystyle\quad-\sum_{i<j}x_{i}x_{j}\otimes b_{i}a_{j}-\sum_{j>i}x_{j}x_{i}% \otimes b_{j}a_{i}-\sum_{i}x_{i}^{2}\otimes b_{i}a_{i}$$ $$\displaystyle=\sum_{i<j}x_{i}x_{j}\otimes(a_{i}b_{j}-b_{i}a_{j})+\sum_{i}x_{i}% ^{2}\otimes[a_{i},b_{i}]$$ $$\displaystyle\quad+\sum_{j>i}(x_{i}x_{j}+[x_{j},x_{i}])\otimes a_{j}b_{i}-\sum% _{j>i}(x_{i}x_{j}+[x_{j},x_{i}])\otimes b_{j}a_{i}$$ $$\displaystyle=\sum_{i<j}x_{i}x_{j}\otimes([a_{i},b_{j}]+[a_{j},b_{i}])+\sum_{i% }x_{i}^{2}\otimes[a_{i},b_{i}]$$ $$\displaystyle\quad\qquad\qquad+\sum_{i<j}[x_{i},x_{j}]\otimes(b_{j}a_{i}-a_{j}% b_{i}).$$ Since $\{x_{i}x_{j}\}_{i\leq j}$ are linearly independent in $U(L)/L$ and $[x_{i},x_{j}]\in L$ for all $i,j$, we have $[a_{i},b_{j}]+[a_{j},b_{i}]=0$, $[a_{i},b_{i}]=0$. Parts (a) and (b) follow. Now assume that $\delta(L)\subset L^{\prime}\otimes L^{\prime\prime}$. By the above computation and parts (a,b), $[\delta(a),\delta(b)]=\sum_{i<j}[x_{i},x_{j}]\otimes(b_{j}a_{i}-a_{j}b_{i})$, which is in $[L^{\prime},L^{\prime}]\otimes U(L)$, as we can assume $x_{i}\in L^{\prime}$ whenever $a_{i}$ (or $b_{i}$) is nonzero. By symmetry, $[\delta(a),\delta(b)]\in U(L)\otimes[L^{\prime\prime},L^{\prime\prime}]$. Hence $[\delta(a),\delta(b)]\in[L^{\prime},L^{\prime}]\otimes[L^{\prime\prime},L^{% \prime\prime}]$. This is part (c). Part (d) follows from the equation $\Phi(a,b)=[\delta(a),\delta(b)]$. Part (e) follows from part (c) and (E1.1.1). ∎ If $V$ and $W$ are subspaces of a vector space $A$, let $V/W$ denote $V/(V\cap W)$. Definition 3.2. (a) A subspace $V$ of a Lie algebra $L$ is said to have small centralizer if $(\ker ad(x))\cap V$ has dimension 1 for all $x\in V\setminus\{0\}$. (b) Let $Z$ be a Lie ideal of $L$. A subspace $V\subset L$ is said to have small centralizer modulo $Z$ if the quotient space $V/Z$ in $L/Z$ has small centralizer. Proposition 3.3. Let $L$ be a coassociative Lie algebra and $Z$ be a Lie ideal of $L$. Suppose that $L^{\prime}$ and $L^{\prime\prime}$ are two subspaces of $L$ such that (a) $[L^{\prime},Z]=[L^{\prime\prime},Z]=0$, (b) $L^{\prime}$ and $L^{\prime\prime}$ have small centralizers modulo $Z$, and (c) $\delta(L)\subset L^{\prime\prime}\otimes L^{\prime}+(Z\otimes L+L\otimes Z)$. Then $\dim\delta(L)/(Z\otimes L+L\otimes Z)\leq 1.$ Remark 3.4. Since $Z$ is a Lie ideal of $L$, we have that $L/Z$ is a quotient Lie algebra, but $L/Z$ may not be a quotient of the coassociative Lie algebra $L$. Proof of Proposition 3.3. Let $W=(Z\otimes L+L\otimes Z)\cap\delta(L)$. Without loss of generality we may assume that $\delta(L)\neq W$. Let $a$ and $b$ be any two elements in $L$. The assertion is equivalent to Claim: $\delta(a)$ and $\delta(b)$ are linearly dependent in $\delta(L)/W$. Proof of the Claim: If $\delta(a)$ or $\delta(b)$ is in $W$, the claim is obvious, so we assume during the proof that $\delta(a)$ and $\delta(b)$ are not in $W$. In the rest of the proof, we will pick a $k$-linear basis $\{z_{j}\}_{j\geq 1}$ of $Z$, extend it to a basis $\{x_{i}\}_{i\geq 1}\cup\{z_{j}\}_{j\geq 1}$ of $L^{\prime\prime}+Z$ where $x_{i}\in L^{\prime\prime}\setminus Z$, then extend it to a basis $\{x_{i}\}_{i\geq 1}\cup\{z_{j}\}_{j\geq 1}\cup\{l_{s}\}_{s\geq 1}$ of the whole space $L$ where $l_{s}\in L\setminus(L^{\prime\prime}+Z)$. For simplicity, we use integers to index the basis elements. Since $\delta(L)\subset L^{\prime\prime}\otimes L^{\prime}+(Z\otimes L+L\otimes Z)$, for any $a\in L$, $$\delta(a)=\sum_{i\geq 1}x_{i}\otimes a_{i}+\sum_{j\geq 1}z_{j}\otimes a^{% \prime}_{j}+\sum_{s\geq 1}l_{s}\otimes a^{\prime\prime}_{s}$$ where $a_{i}\in L^{\prime}+Z$, $a^{\prime}_{j}\in L$ and $a^{\prime\prime}_{s}\in Z$. Case 1: Suppose that $\delta(a),\delta(b)\in u\otimes L^{\prime}+(Z\otimes L+L\otimes Z)$ for some $u\in L^{\prime\prime}\setminus Z$. In this case we can choose $x_{1}=u$. By the choice of the $k$-linear basis, we can write $$\delta(a)=u\otimes a_{1}+\sum_{i\geq 2}x_{i}\otimes a_{i}+\sum_{j\geq 1}z_{j}% \otimes a^{\prime}_{j}+\sum_{s\geq 1}l_{s}\otimes a^{\prime\prime}_{s},$$ where $a_{1}\in L^{\prime}+Z$, $a_{i}\in Z$ for all $i\geq 2$, $a^{\prime}_{j}\in L$ and $a^{\prime\prime}_{s}\in Z$ for all $j,s$. Similarly, $$\delta(b)=u\otimes b_{1}+\sum_{i\geq 2}x_{i}\otimes b_{i}+\sum_{j\geq 1}z_{j}% \otimes b^{\prime}_{j}+\sum_{k\geq 1}l_{s}\otimes b^{\prime\prime}_{s}$$ where $b_{1}\in L^{\prime}+Z$, $b_{i}\in Z$ for all $i\geq 2$, and $b^{\prime}_{j}\in L$ and $b^{\prime\prime}_{s}\in Z$ for all $j,s$. By Lemma 3.1(a), $[a_{1},b_{1}]=0$. Since $L^{\prime}$ has small centralizer modulo $Z$, $b_{1}\in ka_{1}+Z$. Thus $\delta(a)$ and $\delta(b)$ are linearly dependent in $\delta(L)/W$. Case 2: Suppose that $\delta(a),\delta(b)\in L^{\prime\prime}\otimes u+(Z\otimes L+L\otimes Z)$ for some $u\in L^{\prime}\setminus Z$. Case 2 is equivalent to Case 1 by symmetry. Hence the claim follows by Case 1. Case 3: Suppose that $\delta(a)\in u\otimes L^{\prime}+(Z\otimes L+L\otimes Z)$ for some $u\in L^{\prime\prime}\setminus Z$. In this case, we can choose $x_{1}=u$, and $\delta(a)$ can be written as in Case 1. In particular, $\delta(a)\in L^{\prime\prime}\otimes a_{1}+(Z\otimes L+L\otimes Z)$. Write $$\delta(b)=u\otimes b_{1}+\sum_{i\geq 2}x_{i}\otimes b_{i}+\sum_{j\geq 1}z_{j}% \otimes b^{\prime}_{j}+\sum_{s\geq 1}l_{s}\otimes b^{\prime\prime}_{s}$$ where $b_{i}\in L^{\prime}+Z$ for all $i\geq 1$, and $b^{\prime}_{j}\in L$, $b^{\prime\prime}_{s}\in Z$ for all $j,s$. By Lemma 3.1(a), $[a_{1},b_{1}]=0$. Since $L^{\prime}$ has small centralizer modulo $Z$, $b_{1}=\lambda a_{1}+z$ for some $\lambda\in k$ and $z\in Z$. Replacing $b_{1}$ by $b_{1}-\lambda a_{1}$, we may assume that $b_{1}\in Z$. By Lemma 3.1(b), for every $i\geq 2$, $[a_{1},b_{i}]=-[a_{i},b_{1}]\in Z$ since $b_{1}$ is in $Z$. Since $L^{\prime\prime}$ has small centralizer modulo $Z$, $b_{i}=\lambda_{i}a_{1}+z_{i}$, for $\lambda_{i}\in k$ and $z_{i}\in Z$ for all $i\geq 2$. Thus $$\delta(b)=u\otimes b_{1}+\sum_{i\geq 2}x_{i}\otimes(\lambda_{i}a_{1}+z_{i})+% \sum_{j\geq 1}z_{j}\otimes b^{\prime}_{j}+\sum_{s\geq 1}l_{s}\otimes b^{\prime% \prime}_{s}$$ which is in $L^{\prime\prime}\otimes a_{1}+(Z\otimes L+L\otimes Z)$. Therefore both $\delta(a)$ and $\delta(b)$ are in $L^{\prime\prime}\otimes a_{1}+(Z\otimes L+L\otimes Z)$. The claim follows from Case 2. Case 4: Suppose that either $\delta(a)$ or $\delta(b)$ is in $L^{\prime\prime}\otimes u+(Z\otimes L+L\otimes Z)$ for some $u\in L^{\prime}\setminus Z$. The claim follows by symmetry and Case 3. Case 5 [the general case]: By the choice of $k$-linear basis, we can write $$\displaystyle\delta(a)$$ $$\displaystyle=\sum_{i\geq 1}x_{i}\otimes a_{i}+\sum_{j\geq 1}z_{j}\otimes a^{% \prime}_{j}+\sum_{s\geq 1}l_{s}\otimes a^{\prime\prime}_{s},$$ $$\displaystyle\delta(b)$$ $$\displaystyle=\sum_{i\geq 1}x_{i}\otimes b_{i}+\sum_{j\geq 1}z_{j}\otimes b^{% \prime}_{j}+\sum_{s\geq 1}l_{s}\otimes b^{\prime\prime}_{s},$$ where $a_{i},b_{i}\in L^{\prime}+Z$, $a^{\prime}_{j},b^{\prime}_{j}\in L$ and $a^{\prime\prime}_{s},b^{\prime\prime}_{s}\in Z$ for all $i,j,s$. Without loss of generality, we may assume that $a_{1}\in L^{\prime}\setminus Z$. By Lemma 3.1(a), $[a_{1},b_{1}]=0$. Since $L^{\prime}$ has small centralizer modulo $Z$, $b_{1}=\lambda a_{1}+z$ for some $\lambda\in k$ and where $z\in Z$. Replacing $b_{1}$ by $b_{1}-\lambda a_{1}$, we may assume that $b_{1}\in Z$. By Lemma 3.1(b), for every $i\geq 2$, $[a_{1},b_{i}]=-[a_{i},b_{1}]\in Z$ since $b_{1}$ are in the Lie ideal $Z$. Since $L^{\prime}$ has small centralizer modulo $Z$, $b_{i}=\lambda_{i}a_{1}+z_{i}$, for $\lambda_{i}\in k$ and $z_{i}\in Z$ for all $i\geq 2$. Together with the fact $b_{1}\in Z$, we have that $\delta(b)\in L^{\prime\prime}\otimes a_{1}+(Z\otimes L+L\otimes Z)$. The claim now follows from Case 4. ∎ For a subset $S\subset L$, the centralizer of $S$ in $L$ is defined to be $$C_{S}(L)=\{y\in L\mid[x,y]=0,\forall\;x\in S\}.$$ Lemma 3.5. If there is an element $a\in L$ such that $\delta(a)=x\otimes y\neq 0$, then $\delta(L)\subset C_{\{x\}}(L)\otimes C_{\{y\}}(L)$. Proof. By symmetry, it suffices to show that $\delta(L)\subset L\otimes C_{\{y\}}(L)$. Pick a basis $\{x_{i}\}$ of $L$ such that $x_{1}=x$. Then $\delta(a)=x_{1}\otimes y$. For any $b\in L$, write $\delta(b)=\sum_{i}x_{i}\otimes b_{i}$. By Lemma 3.1(a), $[y,b_{1}]=0$. For any $i\geq 2$, by Lemma 3.1(b), $[y,b_{i}]=-[0,b_{1}]=0$. The assertion follows. ∎ Proposition 3.6. Let $C$ be the coradical of a coassociative Lie algebra $L$. Then $[C,C]=0$. As a consequence, cosemisimple coalgebras are rigid. Proof. Since $k$ is algebraically closed, $C=\bigoplus_{i}M_{n_{i}}(k)$ for a set of positive integers $\{n_{i}\}_{i\in I}$. Let $x,y\in C$; we need to show $[x,y]=0$. By linearity, we may assume that $x$ and $y$ are some basis elements in $C$. We need to consider two cases. Case 1: We have that $x$ and $y$ are in the same matrix subcoalgebra, say $M_{n}(k)$. If $n=1$, $x=y$. The assertion is trivial. Now assume that $n>1$. Then we may assume that $x=x_{ij}$ and $y=x_{kl}$ for some $i,j,k,l$. Consider $\delta(x_{1j})=\sum_{s}x_{1s}\otimes x_{sj}$ and $\delta(x_{2l})=\sum_{t}x_{2t}\otimes x_{tl}$. By Lemma 3.1(b), $[x_{sj},x_{tl}]=0$ for all $s,t$. The assertion follows. Case 2: We have that $x$ and $y$ are in different matrix subcoalgebras. Then we may assume that $x=x_{ij}\in M_{n_{1}}(k)$ and $y=y_{kl}\in M_{n_{2}}(k)$. Consider $\delta(x_{1j})=\sum x_{1s}\otimes x_{sj}$ and $\delta(y_{1l})=\sum y_{1t}\otimes y_{tl}$. By Lemma 3.1(b), $[x_{sj},y_{tl}]=0$ for all $s,t$. The assertion follows. ∎ The following lemma is also true. The proof is omitted since it is straightforward and somewhat similar to the proof of Proposition 3.6. Lemma 3.7. If $C_{1}$ and $C_{2}$ are subcoalgebras of a coassociative Lie algebra such that $C_{1}\cap C_{2}=\{0\}$, then $[\delta(C_{1}),\delta(C_{2})]=0$. Lemma 3.8. Let $L$ be a coalgebra. Then we have the following statements. (a) For every $x\in L$, write $\delta(x)=\sum_{i=1}^{n}w_{i}\otimes v_{i}$ for a minimal $n$. Then $\sum_{i}k\delta(v_{i})\subset L\otimes V_{x}$ and $\sum_{i}k\delta(w_{i})\subset W_{x}\otimes L$ for some subspaces $V_{x}\subset\sum_{i}kv_{i}$ and $W_{x}\subset\sum_{i}kw_{i}$ of dimension no more than $\dim\delta(L)$. (b) If $\delta(L)$ is 1-dimensional and $L$ is not 2-conilpotent, then $\delta(L)$ has a basis element of the form $T\otimes T$ for some $0\neq T\in L$. Proof. (a) Let $\{y_{t}\}_{t=1}^{m}$ be a basis of $\sum_{s}k\delta(v_{s})$ for some $m\leq\dim\delta(L)$. Then there are elements $a_{1},\ldots,a_{m}\in L$ such that $$\sum_{i}\delta(w_{i})\otimes v_{i}=\sum_{i}w_{i}\otimes\delta(v_{i})=\sum_{t=1% }^{m}a_{t}\otimes y_{t}\in(\sum_{t=1}^{m}ka_{t})\otimes L\otimes L.$$ This implies that $\delta(w_{i})\in(\sum_{t}ka_{t})\otimes L$ for each $i$. Since $\{y_{t}\}_{t=1}^{m}$ is a basis of $\sum_{s}k\delta(v_{s})$, the equation $$\sum_{i}w_{i}\otimes\delta(v_{i})=\sum_{t=1}^{m}a_{t}\otimes y_{t}$$ implies that $\sum_{t}ka_{t}\subset\sum_{i}kw_{i}$. Therefore the second assertion follows by taking $W_{x}=\sum_{t}ka_{t}$. The first assertion is similar. (b) Pick $0\neq\Omega\in\delta(L)$, and let $\{x_{i}\}$ be a finite set of linearly independent elements in $L$ such that $\Omega=\sum_{i,j}a_{ij}x_{i}\otimes x_{j}$. Since $\delta(L)$ is 1-dimensional, $\delta(x_{i})=b_{i}\Omega$ for some $b_{i}\in k$. Pick $x\in L$ such that $\delta(x)=\Omega$ and $(\delta\otimes 1)\delta(x)\neq 0$ (since $L$ is not $2$-conilpotent). Then $$(\delta\otimes 1)\delta(x)=\sum_{i,j}a_{i,j}b_{i}\Omega\otimes x_{j}=\Omega% \otimes(\sum_{i,j}a_{ij}b_{i}x_{j})=\Omega\otimes T$$ where $T:=\sum_{i,j}a_{ij}b_{i}x_{j}\in L$, and $$(1\otimes\delta)\delta(x)=\sum_{i,j}a_{ij}x_{i}\otimes b_{j}\Omega=(\sum_{i,j}% a_{ij}b_{j}x_{i})\otimes\Omega=S\otimes\Omega$$ where $S:=\sum_{i,j}a_{ij}b_{j}x_{i}$. By coassociativity, $\Omega\otimes T=S\otimes\Omega$. This implies that Hence $\Omega=c^{\prime}T\otimes T$ and $S=c^{\prime\prime}T$ for some $c^{\prime},c^{\prime\prime}\in k^{\times}$. Since $k$ is algebraically closed, we can choose $c^{\prime}=1$ by a scalar change of $T$. ∎ Now we prove the main result of this section. Let $$\left\{e:=\begin{pmatrix}0&1\\ 0&0\end{pmatrix},\quad f:=\begin{pmatrix}0&0\\ 1&0\end{pmatrix},\quad h:=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}\right\}$$ be a standard $k$-basis of $sl_{2}$. Theorem 3.9. The simple Lie algebra $sl_{2}$ is rigid. Proof. By using the standard basis of $sl_{2}$, it is straightforward to check that $sl_{2}$ has small centralizers (details are omitted). Let $L=(sl_{2},\delta)$ be a coassociative Lie algebra. We need to show that $\delta=0$. By Proposition 3.3 for $L^{\prime}=L^{\prime\prime}=L=sl_{2}$ and $Z=0$, $\dim\delta(L)\leq 1$. To avoid the triviality, we assume that $\dim\delta(L)=1$ and let $\Omega\in\delta(L)$ be a nonzero element. If $L$ is not 2-conilpotent, then Lemma 3.8 says that $\Omega=T\otimes T$ where $T=t_{1}e+t_{2}f+t_{3}h$ for some $t_{1},t_{2},t_{3}\in k$. Since $\Omega\neq 0$, not all $t_{i}$ are zero. Suppose $\delta(e)=a\Omega$, $\delta(f)=b\Omega$ and $\delta(h)=c\Omega$ for some $a,b,c\in k$. By (E1.1.1), we have $$\displaystyle 2b\Omega$$ $$\displaystyle=\delta(2f)=\delta([f,h])$$ $$\displaystyle=[f\otimes 1+1\otimes f,c\Omega]+[b\Omega,h\otimes 1+1\otimes h]$$ $$\displaystyle=\{-2bt_{1}e+(2ct_{3}+2bt_{2})f-ct_{1}h\}\otimes T$$ $$\displaystyle\qquad\qquad\qquad+T\otimes\{-2bt_{1}e+(2ct_{3}+2bt_{2})f-ct_{1}h\}.$$ Hence $b(t_{1},t_{2},t_{3})=(-2bt_{1},2bt_{2}+2ct_{3},-ct_{1})$, or $$M\begin{pmatrix}t_{1}\\ t_{2}\\ t_{3}\end{pmatrix}=0\quad{\text{where}}\quad M=\begin{pmatrix}b&0&0\\ 0&b&2c\\ c&0&b\end{pmatrix}.$$ Since not all $t_{1},t_{2},t_{3}$ are zero, the determinant of the matrix $M$, which is $b^{3}$, is zero. Hence $b=0$. Since $e$ and $f$ plays a very similar role, by symmetry, $a=0$. By (E1.1.1) and the fact $\delta(e)=\delta(f)=0$, we have that $\delta(h)=\delta([e,f])=0$. Thus $c=0$, whence $\delta=0$, yielding a contradiction. Therefore $\delta$ is 2-conilpotent. Since $\delta$ is 2-conilpotent, by Theorem 2.5, $U(sl_{2},\delta)$ is a connected Hopf algebra of GK-dimension 3. By comparing with the list in the classification of connected Hopf algebras of GK-dimension 3 [Zh2, Theorem 1.3], $U(sl_{2},\delta)$ must be isomorphic to $U(sl_{2})$. Since $U(sl_{2})$ is cocommutative, $(sl_{2},\delta)$ must be cocommutative. Hence $$\displaystyle\Omega$$ $$\displaystyle=a_{11}e\otimes e+a_{12}(e\otimes f+f\otimes e)+a_{13}(e\otimes h% +h\otimes e)$$ $$\displaystyle\qquad\qquad+a_{22}f\otimes f+a_{23}(f\otimes h+h\otimes f)+a_{33% }h\otimes h\neq 0.$$ Since $\delta(L)$ is 1-dimensional, the kernel $L_{0}=\ker(\delta)$ is a 2-dimensional Lie subalgebra of $sl_{2}$. By an elementary computation, it is easy to verify that any 2-dimensional Lie subalgebra of $sl_{2}$ is either (i) $kf+kh$, or (ii) $ke+kh$, or (iii) $k(e+ah)+k(-4af+h)$ for some $a\in k^{\times}$. In the first case, $(\delta\otimes 1)\Omega=a_{11}\delta(e)\otimes e+a_{12}\delta(e)\otimes f+a_{1% 3}\delta(e)\otimes h$ and without loss of generality, we can assume that $\delta(e)=\Omega$. Since $(sl_{2},\delta)$ is 2-conilpotent, $(\delta\otimes 1)(\Omega)=0$, which implies that $a_{11}=a_{12}=a_{13}=0$. It is easy to see that $$\displaystyle-\Phi(e,h)$$ $$\displaystyle=-\delta(-2e)+[\delta(e),h\otimes 1+1\otimes h]$$ $$\displaystyle=2\Omega+4a_{22}f\otimes f+2a_{23}(f\otimes h+h\otimes f).$$ The equations $\Phi(e,h)=[\delta(e),\delta(h)]=[\delta(e),0]=0$ imply that $a_{22}=a_{23}=a_{33}=0$. Hence $\Omega=0$, a contradiction. The assertion follows. The second case is similar. The final case is when $e+ah,h-4af\in\ker(\delta)$ for some $a\in k^{\times}$. Let $$\displaystyle e^{\prime}$$ $$\displaystyle=\begin{pmatrix}1&0\\ -2a&1\end{pmatrix}\begin{pmatrix}0&1\\ 0&0\end{pmatrix}\begin{pmatrix}1&0\\ -2a&1\end{pmatrix}^{-1}=\begin{pmatrix}2a&1\\ -4a^{2}&-2a\end{pmatrix}$$ $$\displaystyle h^{\prime}$$ $$\displaystyle=\begin{pmatrix}1&0\\ -2a&1\end{pmatrix}\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}\begin{pmatrix}1&0\\ -2a&1\end{pmatrix}^{-1}=\begin{pmatrix}1&0\\ -4a&-1\end{pmatrix}$$ $$\displaystyle f^{\prime}$$ $$\displaystyle=\begin{pmatrix}1&0\\ -2a&1\end{pmatrix}\begin{pmatrix}0&0\\ 1&0\end{pmatrix}\begin{pmatrix}1&0\\ -2a&1\end{pmatrix}^{-1}=\begin{pmatrix}0&0\\ 1&0\end{pmatrix}=f.$$ Then $\{e^{\prime},f^{\prime},h^{\prime}\}$ is a new standard basis of $sl_{2}$. It is clear that $e^{\prime}=e+2ah-4a^{2}f\in\ker(\delta)$ and $h^{\prime}=h-4af\in\ker(\delta)$. Thus it is equivalent to the second case. Combining all these cases, the assertion follows. ∎ Similar to Theorem 3.9 we show the following. Theorem 3.10. Write $gl_{2}=sl_{2}\oplus kz$ where $kz$ is the center of $gl_{2}$. If $(gl_{2},\delta)$ is a coassociative Lie algebra, then $\delta\mid_{sl_{2}}=0$ and $\delta(z)=az\otimes z$ for some scalar $a\in k$. Sketch of Proof. Some tedious computations are omitted in the following proof. First of all, the $\delta$ given in the theorem gives rise to a coassociative Lie algebra structure on $gl_{2}$. Now we assume that $(gl_{2},\delta)$ is a coassociative Lie algebra. Applying Proposition 3.3 to $(L,L^{\prime},L^{\prime\prime},Z)=(gl_{2},sl_{2},sl_{2},kz)$, we obtain that $$\dim(\delta(gl_{2})/(z\otimes gl_{2}+gl_{2}\otimes z))\leq 1.$$ Therefore there is an element $\Omega\in sl_{2}\otimes sl_{2}$ such that, for every $x\in gl_{2}$, (E3.10.1) $$\delta(x)=\sigma(x)\otimes z+z\otimes\tau(x)+\lambda(x)\Omega$$ for some $\sigma(x)\in sl_{2},\tau(x)\in gl_{2}$ and $\lambda(x)\in k$. Both $\sigma(x)$ and $\tau(x)$ are uniquely determined by (E3.10.1). If $\Omega\neq 0$, then $\lambda(x)$ is also uniquely determined by (E3.10.1). Setting $x=z$ in (E3.10.1), we have (E3.10.2) $$\delta(z)=\sigma(z)\otimes z+z\otimes\tau(z)+\lambda(z)\Omega.$$ Equation (E1.2.2) for $(a,b)=(z,x)$ implies that $$\displaystyle 0=$$ $$\displaystyle[\sigma(z),x]\otimes z+z\otimes[\tau(z),x]+[\lambda(z)\Omega,x% \otimes 1+1\otimes x]$$ $$\displaystyle+[\sigma(z),\sigma(x)]\otimes z^{2}+z^{2}\otimes[\tau(z),\tau(x)]$$ $$\displaystyle+[\lambda(z)\Omega,\sigma(x)\otimes z]+[\lambda(z)\Omega,z\otimes% \tau(x)]$$ $$\displaystyle+[\sigma(z)\otimes z,\lambda(x)\Omega]+[z\otimes\tau(z),\lambda(x% )\Omega].$$ Since the terms in the above equation live in different $k$-subspaces of $U(gl_{2})\otimes U(gl_{2})$, we have $[\sigma(z),x]=0=[\tau(z),x]$ for all $x\in sl_{2}$. Thus $\sigma(z)=0$ and $\tau(z)\in kz$. In this case, (E1.2.2) becomes, for every $x\in gl_{2}$, (E3.10.3) $$0=\lambda(z)([\Omega,x\otimes 1+1\otimes x]+[\Omega,\sigma(x)\otimes z]+[% \Omega,z\otimes\tau(x)]).$$ First we claim that $\lambda(z)\Omega=0$. If not, we may assume that $\Omega\neq 0$ and $\lambda(z)=1$. Using the fact that terms live in different $k$-subspaces of $U(gl_{2})\otimes U(gl_{2})$, (E3.10.3) implies that $$[\Omega,x\otimes 1+1\otimes x]=[\Omega,\sigma(x)\otimes z]=[\Omega,z\otimes% \tau(x)]=0$$ for all $x\in gl_{2}$. A computation shows that the first equation implies that $\Omega=c(h\otimes h+2(e\otimes f+f\otimes e))$ for some $0\neq c\in k$. The second and third equations imply that $\sigma(x)=0$ and $\tau(x)\in kz$. Going back to (E3.10.1), we have, for every $x\in gl_{2}$, $$\delta(x)=\phi(x)z\otimes z+\lambda(x)\Omega$$ for some $\phi(x)\in k$. For any $x,y\in sl_{2}$, (E1.2.2) says that $$\delta([x,y])=[x\otimes 1+1\otimes x,\lambda(y)\Omega]+[\lambda(x)\Omega,y% \otimes 1+1\otimes y]=0.$$ Since $sl_{2}=[sl_{2},sl_{2}]$, $\delta\mid_{sl_{2}}=0$. The coassociativity on $z$ shows that $\Omega\otimes z=0$, a contradiction. Therefore we proved our claim. For the rest of the proof, we have $\lambda(z)\Omega=0$ and $\delta(z)=\sigma(z)\otimes z+z\otimes\tau(z)$. Then $kz$ is an ideal of the coassociative Lie algebra $(gl_{2},\delta)$ and $(gl_{2}/kz,\overline{\delta})\cong(sl_{2},\overline{\delta})$ is a quotient coassociative Lie algebra where $\overline{\delta}$ is the induced coproduct. By Theorem 3.9, $\overline{\delta}=0$. This means that $\Omega=0$. For each $x\in gl_{2}$, write $$\delta(x)=\sigma_{1}(x)\otimes z+z\otimes\sigma_{2}(x)+\nu(x)z\otimes z$$ for some $\sigma_{1}(x),\sigma(x)\in sl_{2}$ and some $\nu(x)\in k$. By (E1.2.2), we have $$\displaystyle\delta([x,y])=$$ $$\displaystyle[\sigma_{1}(x),y]\otimes z+z\otimes[\sigma_{2}(x),y]+[x,\sigma_{1% }(y)]\otimes z+z\otimes[x,\sigma_{2}(y)]$$ $$\displaystyle+[\sigma_{1}(x),\sigma_{1}(y)]\otimes z^{2}+z^{2}\otimes[\sigma_{% 2}(x),\sigma_{2}(y)]$$ for all $x,y\in gl_{2}$. Setting $y=z$, we have $[x,\sigma_{i}(z)]=0$ for all $x\in sl_{2}$. This implies that $\sigma_{i}(z)=0$ for $i=1,2$. Setting $x,y\in sl_{2}$, we have that $$[\sigma_{1}(x),\sigma_{1}(y)]=0=[\sigma_{2}(x),\sigma_{2}(y)]$$ and that $\delta(sl_{2})\subset sl_{2}\otimes z+z\otimes sl_{2}$. Since $sl_{2}$ has small centralizers, $\dim\sigma_{1}(sl_{2})\leq 1$ and $\dim\sigma_{2}(sl_{2})\leq 1$. Combining above facts, there exist $w_{1},w_{2}\in sl_{2}\setminus\{0\}$, $c\in k$ and linear maps $\phi_{1},\phi_{2}:sl_{2}\to k$, such that $$\displaystyle\delta(z)$$ $$\displaystyle=cz\otimes z,$$ $$\displaystyle\delta(x)$$ $$\displaystyle=\phi_{1}(x)w_{1}\otimes z+\phi_{2}(x)z\otimes w_{2}$$ for all $x\in sl_{2}$. Let $\{e,f,h\}$ be the standard basis of $sl_{2}$, and write $w_{1}=ae+bf+ch\neq 0$ for some $a,b,c\in k$. A calculation using explicit Lie product of the elements $e,f$, and $h$ shows that (E1.2.2) implies that $\phi_{1}=0$. By symmetry, $\phi_{2}=0$. Thus the assertion follows. ∎ 4. Examples We present several families of coassociative Lie algebras in this section. One-dimensional ones are listed in Example 1.9. Here is the 2-dimensional case. Example 4.1. If $\dim L=2$, then there are two Lie algebra structures on $L$, up to isomorphism. Namely, $L$ is either abelian or non-abelian. If $L$ is abelian, then the classification of coassociative Lie algebra structures on $L$ is equivalent to the classification of coalgebra structures on $L$. It is easy to show that $\delta$-structure in $L$ is isomorphic to one of the following: (4.1.1) $\delta=0$; (4.1.2) $(L,\delta)$ is cosemisimple; (4.1.3) $L=kx_{1}\oplus kx_{2}$ and $\delta(x_{1})=x_{1}\otimes x_{1}$, $\delta(x_{2})=0$; (4.1.4) $L=kx_{1}\oplus kx_{2}$ and $\delta(x_{1})=x_{1}\otimes x_{1}$, $\delta(x_{2})=x_{1}\otimes x_{2}+x_{2}\otimes x_{1}$; (4.1.5) $L=kx_{1}\oplus kx_{2}$ and $\delta(x_{1})=0$, $\delta(x_{2})=x_{1}\otimes x_{1}$. If $L$ is non-abelian, then $L$ has a basis $\{x_{1},x_{2}\}$ such that $[x_{1},x_{2}]=x_{2}$. We have two cases. Case 1: $\delta(x_{2})=0$. We are only interested in nonzero $\delta$-structures. Write $\delta(x_{1})=\sum_{i,j}a_{ij}x_{i}\otimes x_{j}\neq 0$. In this case $$\Phi(x_{1},x_{2})=-[\delta(x_{1}),x_{2}\otimes 1+1\otimes x_{2}]=-(a_{11}(x_{2% }\otimes x_{1}+x_{1}\otimes x_{2})+(a_{12}+a_{21})x_{2}\otimes x_{2}).$$ By (E1.3.2), $\Phi(x_{1},x_{2})=[\delta(x_{1}),\delta(x_{2})]=0$. Hence $a_{11}=0$ and $a_{12}+a_{21}=0$. Let $a=a_{12}$ and $b=a_{22}$. We have that $\delta(x_{1})=ax_{1}\otimes x_{2}-ax_{2}\otimes x_{1}+bx_{2}\otimes x_{2}$. Now coassociativity of $\delta$ shows that $a=0$. Thus $\delta(x_{2})=0$ and $\delta(x_{1})=bx_{2}\otimes x_{2}$. Up to a base change, we may assume $b=1$. Hence this is Example 1.4. Case 2: $\delta(x_{2})\neq 0$. Since $L$ is 2-dimensional and non-nilpotent, it has small centralizers. By Lemma 3.3, $\delta(L)$ is 1-dimensional. Since $\delta(x_{2})\neq 0$, by replacing $x_{1}$ by $x_{1}-ax_{2}$ for some suitable $a\in k$, we have $\delta(x_{1})=0$. Write $\delta(x_{2})=\sum_{i,j}b_{ij}x_{i}\otimes x_{j}\neq 0$. In this case, $$\displaystyle 0=[\delta(x_{1}),\delta(x_{2})]$$ $$\displaystyle=\Phi(x_{1},x_{2})=\delta(x_{2})-[x_{1}\otimes 1+1\otimes x_{1},% \delta(x_{2})]$$ $$\displaystyle=b_{11}x_{1}\otimes x_{1}+b_{12}x_{1}\otimes x_{2}+b_{21}x_{2}% \otimes x_{1}+b_{22}x_{2}\otimes x_{2}$$ $$\displaystyle\quad-(b_{12}x_{1}\otimes x_{2}+b_{21}x_{2}\otimes x_{1}+2b_{22}x% _{2}\otimes x_{2})$$ $$\displaystyle=b_{11}x_{1}\otimes x_{1}-b_{22}x_{2}\otimes x_{2}.$$ Hence $b_{11}=b_{22}=0$ and $\delta(x_{2})=b_{12}x_{1}\otimes x_{2}+b_{21}x_{2}\otimes x_{1}$. The coassociativity of $\delta$ implies that $b_{12}=b_{21}=0$. Therefore $\delta=0$ in this case, yielding a contradiction. Combining these two cases, the only nonzero $\delta$-structure on the 2-dimensional non-abelian Lie algebra is the one in Example 1.4, up to isomorphisms. One nice fact in 2-dimensional case is that $\delta$ is always cocommutative. If $\delta$ is conilpotent, then $L$ is quasi-equivalent to a Lie algebra by Corollary 2.6. Next, we consider some higher-dimensional examples. Example 4.2. Let ${\mathfrak{g}}$ be a 3-dimensional Lie algebra with a $k$-linear basis $\{x,y,z\}$ such that its Lie structure is determined by $$[x,y]=y,\quad[z,y]=0,\quad[z,x]=-z+\lambda y,$$ for any $\lambda\in k$. Let $L=({\mathfrak{g}},\delta)$ where the coproduct $\delta$ is determined by $$\delta(x)=\delta(y)=0,\quad\delta(z)=x\otimes y-y\otimes x.$$ It is routine to check that $L$ is a coassociative Lie algebra (using Definition 1.1). It is obvious that $\delta$ is conilpotent and anti-cocommutative. Let $H$ be the enveloping algebra $U(L)$. It follows from the antipode axiom that $S(x)=-x$, $S(y)=-y$, and $S(z)=-z+y$. Hence $S^{2}(z)=z-2y$ and $H$ is not involutory. By Theorem 2.7, ${\mathfrak{g}}$ is not unimodular, which can also be verified directly. Let ${\mathfrak{h}}_{2n+1}$ be the $(2n+1)$-dimensional Heisenberg Lie algebra with a standard basis $\{x_{1},\cdots,x_{n},y_{1},\cdots,y_{n},z\}$. Here $[x_{i},y_{i}]=z$ for all $i$, and all other brackets are zero. Let $A=(a_{ij})$, $B=(b_{ij})$, $C=(c_{ij})$ and $D=(d_{ij})$ denote $n\times n$-matrices over $k$, and let $E=(e_{i})$ be an $n$-column vector over $k$. Example 4.3. Each of the following $\delta$ defines a coassociative coalgebra structure on ${\mathfrak{h}}_{2n+1}$ such that $({\mathfrak{h}}_{2n+1},\delta)$ is a coassociative Lie algebra. (a) For every $i$, $\delta(x_{i})=\delta(z)=0$, and $$\delta(y_{i})=\sum_{j}(a_{ij}x_{j}+b_{ij}y_{j})\otimes z+\sum_{j}z\otimes(c_{% ij}x_{j}-b_{ij}y_{j})+e_{i}z\otimes z,$$ where the coefficient matrices $A=(a_{ij})$, $B=(b_{ij})$, $C=(c_{ij})$ and $E=(e_{i})$ satisfy (i) $BA=B^{2}=BC=0$; (ii) $BE=0$; (iii) $A+A^{\tau}+C+C^{\tau}=0$; (iv) $AB^{\tau}=BA^{\tau}$; and (v) $CB^{\tau}=BC^{\tau}$. (b) For every $i$, $\delta(x_{i})=e_{i}z\otimes z$, $\delta(z)=0$, and $\delta(y_{i})=\sum b_{ij}(x_{j}\otimes z+z\otimes x_{j})$, where the coefficient matrix $B=(b_{ij})$ satisfies $B=B^{\tau}$. (c) For every $i$, $\delta(x_{i})=0$, $\delta(z)=z\otimes z$, and $$\delta(y_{i})=\sum_{j}(a_{ij}x+b_{ij}y_{j})\otimes z+\sum_{j}z\otimes(-a_{ji}x% _{j}+(\delta_{ij}-b_{ij})y_{j}),$$ where the coefficient matrices $A=(a_{ij})$ and $B=(b_{ij})$ satisfy the conditions $B^{2}=B$; $BA=A$; and $BA^{\tau}=0$. Proof. (a) Easy computations show that $$\displaystyle(\delta\otimes 1)$$ $$\displaystyle\delta(y_{i})$$ $$\displaystyle=\sum_{j}b_{ij}\delta(y_{j})\otimes z$$ $$\displaystyle=\sum_{j,s}(b_{ij}a_{js}x_{s}+b_{ij}b_{js}y_{s})\otimes z\otimes z% +\sum_{j,s}z\otimes(b_{ij}c_{js}x_{s}-b_{ij}b_{js}y_{s})\otimes z$$ $$\displaystyle\qquad\qquad+\sum_{j}b_{ij}e_{j}z\otimes z\otimes z,\;\quad{\text% {and}}$$ $$\displaystyle(1\otimes\delta)$$ $$\displaystyle\delta(y_{i})$$ $$\displaystyle=\sum_{j}z\otimes(-b_{ij})\delta(y_{j})$$ $$\displaystyle=\sum_{j,s}z\otimes(-b_{ij}a_{js}x_{s}-b_{ij}b_{js}y_{s})\otimes z% +\sum_{j,s}z\otimes z\otimes(-b_{ij}c_{js}x_{s}+b_{ij}b_{js}y_{s})$$ $$\displaystyle\qquad\qquad-\sum_{j}b_{ij}e_{j}z\otimes z\otimes z.$$ Coassociativity is equivalent to equations $BA=B^{2}=BC=0$ and $BE=0$. To check the condition (E1.1.1) we note that (E1.1.1) is trivial when $(a,b)=(z,z),(x_{i},z),(y_{i},z)$, and $(x_{i},x_{j})$. It suffices to verify (E1.1.1) for $(a,b)=(x_{i},y_{j})$ and $(a,b)=(y_{i},y_{j})$ for all $1\leq i,j\leq n$. If $(a,b)=(x_{i},y_{j})$, we have that LHS of (E1.1.1) $$\displaystyle=\delta([x_{i},y_{j}])=\delta(\delta_{ij}z)=0,\;{\text{and}}$$ RHS of (E1.1.1) $$\displaystyle=[x_{i}\otimes 1+1\otimes x_{i},\delta(y_{j})]=b_{ji}z\otimes z-b% _{ji}z\otimes z=0.$$ Hence (E1.1.1) holds for $(a,b)=(x_{i},y_{j})$. If $(a,b)=(y_{i},y_{j})$, we have that LHS of (E1.1.1) $$\displaystyle=\delta([y_{i},y_{j}])=\delta(0)=0,\;{\text{and}}$$ RHS of (E1.1.1) $$\displaystyle=[y_{i}\otimes 1,\delta(y_{j})]+[1\otimes y_{i},\delta(y_{j})]$$ $$\displaystyle\quad+[\delta(y_{i}),y_{j}\otimes 1]+[\delta(y_{i}),1\otimes y_{j% }]+[\delta(y_{i}),\delta(y_{j})]$$ $$\displaystyle=a_{ji}z\otimes z+c_{ji}z\otimes z$$ $$\displaystyle\quad+a_{ij}z\otimes z+c_{ij}z\otimes z$$ $$\displaystyle\quad+(\sum_{s}a_{is}b_{js}-b_{is}a_{js})z\otimes z^{2}+(\sum_{s}% -c_{is}b_{js}+b_{is}c_{js})z^{2}\otimes z.$$ Hence (E1.1.1) holds if and only if $A+A^{\tau}+C+C^{\tau}=0$, $AB^{\tau}=BA^{\tau}$ and $CB^{\tau}=BC^{\tau}$. This completes the proof of (a). The proofs of (b) and (c) are similar and therefore omitted. ∎ We consider one last example. Let $U_{n}$ be the strictly upper triangular $n\times n$-matrix coalgebra, namely, it is the coalgebra with basis $\{x_{ij}\}_{1\leq i<j\leq n}$ such that $$\delta(x_{ij})=\sum_{i<s<j}x_{is}\otimes x_{sj}\qquad{\text{for all $1\leq i,j% \leq n$}}.$$ In the following proposition, let $E=(e_{i}),F=(f_{i})$ and $G=(g_{i})$ be three arbitrary vectors in $k^{n-1}$. For $1\leq i<j\leq n$, define $$a_{ij}=g_{i}+g_{i+1}+\cdots+g_{j-1}.$$ It follows from the definition that $a_{is}+a_{sj}=a_{ij}$ for all $1\leq i<s<j\leq n$. Example 4.4. Let $n\geq 3$. Then the following anti-commutative $k$-bilinear map $[\;,\;]:U_{n}^{\otimes 2}\to U_{n}$ defines a Lie algebra structure on $U_{n}$ such that $(U_{n},[\;,\;])$ is a coassociative Lie algebra. (E4.4.1) $$\displaystyle\;[x_{1n},x_{1n}]$$ $$\displaystyle=0,$$ (E4.4.2) $$\displaystyle[x_{st},x_{ij}]$$ $$\displaystyle=0\qquad\qquad\qquad{\text{ if $(s,t)\neq(1,n)$ and $(i,j)\neq(1,% n)$}},$$ (E4.4.3) $$\displaystyle[x_{1n},x_{ij}]$$ $$\displaystyle=a_{ij}x_{ij}\qquad\qquad{\text{ if $(i,j)\neq(1,n),(1,n-1),(2,n)% $}},$$ (E4.4.4) $$\displaystyle[x_{1n},x_{1n-1}]$$ $$\displaystyle=a_{1n-1}x_{1n-1}+\sum_{i=1}^{n-1}e_{i}x_{ii+1},$$ (E4.4.5) $$\displaystyle[x_{1n},x_{2n}]$$ $$\displaystyle=a_{2n}x_{2n}+\sum_{i=1}^{n-1}f_{i}x_{ii+1}.$$ Proof. First we prove that $(U_{n},[\;,\;])$ is a Lie algebra. Let $K=\bigoplus_{(i,j)\neq(1,n)}kx_{ij}\subset L$. By definition, we have $[K,K]=0$ and $[L,K]=[K,L]\subset K$. Since we define $[\;,\;]$ to be anti-commutative, it suffices to show the Jacobi identity $$[a,[b,c]]=[b,[a,c]]+[[a,b],c]$$ for all $a,b,c\in U_{n}$. If $a,b,c\in K$, then the Jacobi identity is trivially true. If $a=b=x_{1n}$ and $c\in K$, the Jacobi identity is also true since it is true for all $a=b$. The Jacobi identity is stable under permutation and thus the remaining case to consider is when $a=x_{1n}$ and $b,c\in K$. In this case, $$\displaystyle\;[a,[b,c]]$$ $$\displaystyle=[a,0]=0,$$ $$\displaystyle[b,[a,c]]+[[a,b],c]$$ $$\displaystyle\in[K,K]+[K,K]=\{0\}.$$ Hence the Jacobi identity holds and $(U_{n},[\;,\;])$ is a Lie algebra. To prove $(U_{n},[\;,\;])$ is a coassociative Lie algebra, we need to verify (E1.1.1). By linearity, it suffices to check (E1.1.1) for cases listed in (E4.4.1)-(E4.4.5). Case 1: If $(a,b)=(x_{1n},x_{1n})$, (E1.1.1) is automatic (for any $a=b$). Case 2: Suppose that $(a,b)=(x_{st},x_{ij})$ for $(s,t)\neq(1,n)$ and $(i,j)\neq(1,n)$. Since $\delta(L)\subset K\otimes K$ and $[K,K]=0$, both sides of (E1.1.1) are zero. Case 3: Suppose that $(a,b)=(x_{1n},x_{ij})$ for $(i,j)\neq(1,n)$. Using the fact $[K,K]=0$, we have LHS of (E1.1.1) $$\displaystyle=\delta([x_{1n},x_{ij}])=a_{ij}\ \sum_{i<s<j}x_{is}\otimes x_{sj},$$ RHS of (E1.1.1) $$\displaystyle=[x_{1n}\otimes 1+1\otimes x_{1n},\sum_{i<s<j}x_{is}\otimes x_{sj}]$$ $$\displaystyle=(a_{is}+a_{sj})\sum_{i<s<j}x_{is}\otimes x_{sj}=a_{ij}\sum_{i<s<% j}x_{is}\otimes x_{sj}.$$ Hence (E1.1.1) holds. This takes care of cases in (E4.4.3)-(E4.4.5). Combining all the above cases we have checked (E1.1.1). Therefore $(U_{n},[\;,\;])$ is a coassociative Lie algebra. ∎ Acknowledgments The authors thank Chelsea Walton for reading an earlier version of this paper and for her useful comments and thank Milen Yakimov for suggesting them the name “coassociative Lie algebra” for the main object studied in this paper. The authors also thank the referee for his/her valuable comments and suggestions. Part of the research was completed when J.J. 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Zhang, Homological integral of Hopf algebras, Trans. Amer. Math. Soc. 359 (2007), no. 10, 4945–4975. [Mo] S. Montgomery, Hopf Algebras and their Actions on Rings, CBMS Regional Conference Series in Mathematics, 82, Providence, RI, 1993. [WZZ] D.-G. Wang, J.J. Zhang and G.-B. Zhuang, Classification of connected Hopf algebras of GKdimension four, preprint, arXiv:1302.2270 (2013). [Ye] A. Yekutieli, The rigid dualizing complex of a universal enveloping algebra, J. Pure Appl. Algebra, 150 (2000), 85-93. [Zh1] G. Zhuang, Existence of Hopf subalgebras of GK-dimension two, J. Pure Appl. Algebra 215 (2011), 2912-2922. [Zh2] G. Zhuang, Properties of pointed and connected Hopf algebras of finite Gelfand-Kirillov dimension, J. Lond. Math. Soc. (to appear).

Chiral quantum optics in photonic sawtooth lattices Eduardo Sánchez-Burillo Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany    Chao Wan Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany Fakultät Physik at Ludwig-Maximilians-Universität, Schellingstraße 4, 80799 München, Germany    David Zueco Instituto de Ciencia de Materiales de Aragón and Departamento de Física de la Materia Condensada, CSIC-Universidad de Zaragoza, Calle Pedro Cerbuna 12, 50009 Zaragoza, Spain Fundación ARAID, Paseo María Agustín 36, 50004 Zaragoza, Spain    Alejandro González-Tudela Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, Madrid 28006, Spain (December 1, 2020) Abstract Chiral quantum optics has become a burgeoning field due to its potential applications in quantum networks or quantum simulation of many-body physics. Current implementations are based on the interplay between local polarization and propagation direction of light in nanophotonic structures. In this manuscript, we propose an alternative platform based on coupling quantum emitters to a photonic sawtooth lattice, a one-dimensional model with an effective flux per plaquette introduced by complex tunnelings. We study the dynamics emerging from such structured photonic bath and find the conditions to obtain quasi-perfect directional emission when the emitters are resonant with the band. In addition, we find that the photons in this bath can also mediate complex emitter-emitter interactions tunable in range and phase when the emitters transition frequencies lie within a band-gap. Since these effects do not rely on polarization they can be observed in platforms beyond nanophotonics such as matter-waves or circuit QED ones, of which we discuss a possible implementation. Introduction.- Designing robust non-reciprocal optical devices at the classical and quantum level has remained a challenge for many years with fundamental and practical implications (see, e.g., Refs. Jalas et al. (2013); Sayrin et al. (2015); Sounas and Alù (2017); Caloz et al. (2018), and references therein). On the classical level, the search was mainly focused on systems breaking Lorentz-reciprocity such as magneto-optical materials. On the quantum side, nanophotonic systems have recently emerged as a powerful candidate due to the link between the local polarization and propagation direction that appears thanks to the subwavelength light confinement Bliokh et al. (2015); Mechelen and Jacob (2016). Using this connection and the intrinsic polarization of classical and quantum emitters (nanoparticles, atoms, or quantum dots) many experiments have shown chiral light-matter coupling in waveguides Mitsch et al. (2014); Petersen et al. (2014); Scheucher et al. (2016); Söllner et al. (2015); Coles et al. (2016), and harnessed it to achieve, e.g., optical isolation at the single-photon level Sayrin et al. (2015). Apart from these experiments, chiral quantum optical systems Lodahl et al. (2017) have also been proposed to build spin-photon interfaces Mahmoodian et al. (2016) and directional amplifiers Metelmann and Clerk (2015); Malz et al. (2018) in quantum networks, to engineer novel many-body spin or photonic states Ramos et al. (2014, 2016); Vermersch et al. (2016); Pichler et al. (2015); Guimond et al. (2016), to emit non-classical light Mahmoodian et al. (2018); Downing et al. (2019), or to induce exotic self-organization patterns Eldredge et al. (2016), among other phenomena. All these exciting prospects have triggered the interest to implement these chiral couplings also beyond the optical regime. There are theoretical proposals to do so using complex non-local light-matter interactions Ramos et al. (2014, 2016); Vermersch et al. (2016) or with moving atoms Calajó and Rabl (2017), but their experimental realization remains so far elusive. In this manuscript, we explore an alternative in which quantum emitters (QEs) couple locally to a photonic bath which inherently breaks time-reserval symmetry. In particular, we use a minimal one-dimensional model named as photonic sawtooth lattice (see Fig. 1(a)), also labelled as $\Delta$ chain Nakamura and Kubo (1996) or triangle lattice Hyrkäs et al. (2013); Flach et al. (2014) in the literature. This is a one-dimensional model with closed loops, which allows for a complex coupling ($\phi\neq 0$ in Fig. 1) between the degrees of freedom in the photonic lattice vertices defining an effective magnetic flux per loop 111If there are no loops, complex couplings have nonphysical implications.. We first propose an implementation to simulate this model based on circuit QED technologies Astafiev et al. (2010); Hoi et al. (2011, 2013a); van Loo et al. (2013); Hoi et al. (2013b); Liu and Houck (2017); Mirhosseini et al. (2018); Roushan et al. (2016); Gu et al. (2017). We then predict that one can indeed obtain quasi-perfect directional emission Lodahl et al. (2017) when the QEs are resonant with the bands appearing in these systems. In our proposal, rather than selecting a given momentum using the destructive interference induced by non local couplings Ramos et al. (2014, 2016); Vermersch et al. (2016), it is the asymmetric nature of the “sawtooth” band structure the one responsible of the chirality of the emission. In addition to that, the properties of the bath lead to other phenomena such as the emergence of a sublattice-dependent directional emission or QE dipole interactions tunable in range an phase when the QEs frequencies lie in a band-gap. Model and implementation.- The Hamiltonian of the full system (photons and QEs) is ($\hbar=1$): $$H=H_{\text{ST}}+\Delta\sum_{j=1}^{M}\sigma_{j}^{\dagger}\sigma_{j}+H_{\text{% int}},$$ (1) where $\Delta$ is the transition frequency of each QE described as a two-level system with ladder operators $\sigma_{j}^{(\dagger)}$ for the $j$-th QE, $M$ is the number of QEs, $H_{\text{int}}$ is the interaction Hamiltonian, and $H_{\text{ST}}$ is the Hamiltonian of the sawtooth lattice. The latter reads: $$\displaystyle H_{\text{ST}}=$$ $$\displaystyle\omega_{B}\sum_{n=1}^{N}(a_{n}^{\dagger}a_{n}+b_{n}^{\dagger}b_{n% })-J_{AA}\sum_{n=1}^{N}(a_{n}^{\dagger}a_{n+1}+\text{H.c.})$$ $$\displaystyle-J_{AB}\sum_{n=1}^{N}(a_{n}^{\dagger}b_{n}+e^{-i\phi}a_{n+1}^{% \dagger}b_{n}+\text{H.c.}),$$ (2) where $a_{n}$ and $b_{n}$ are the annihilation operators of the $n$-th $a$ and $b$ modes (notice that the photonic bath is bipartite and must be described by two sublattices $A$ and $B$, see Fig. 1(a)), $N$ is the number of modes in each sublattice, $\omega_{B}$ is the bare energy of each resonator that we take as the energy reference, i.e., $\omega_{B}\equiv 0$ , $J_{AA}$ is the nearest-neighbour coupling between the $A$ lattice sites, $J_{AB}$ stands for the hopping strength between the $a$ and $b$ modes, and $\phi$ is the phase difference in each loop (see Fig. 1(a)). As the physics of Eq. (Chiral quantum optics in photonic sawtooth lattices) does not rely on polarization, it can be obtained in platforms beyond optical ones Mitsch et al. (2014); Petersen et al. (2014); Scheucher et al. (2016); Söllner et al. (2015); Coles et al. (2016) such as cold atoms in state-dependent lattices de Vega et al. (2008); Navarrete-Benlloch et al. (2011); Krinner et al. (2018), where such complex loops have already been engineered Aidelsburger et al. (2013), or circuit QED platforms Astafiev et al. (2010); Hoi et al. (2011, 2013a); van Loo et al. (2013); Hoi et al. (2013b). In Fig. 1(b) we show a way on how to engineer such looped geometries and complex phases using superconducting qubits as implemented for a single loop in Ref. Roushan et al. (2016). For the qubit-resonator coupling, we assume that the qubit is capacitively coupled to an LC resonator as shown in Fig. 1(b), whereas the resonators are inductively coupled via a SQUID-type loop based on Josephson junctions Peropadre et al. (2013); Chen et al. (2014). These Josephson junctions can be controlled via an external flux $\phi_{\rm ext}(t)$ entering the loop leading to the time-dependent coupling Hamiltonian between two resonators: $$H_{\text{aux}}=\sum_{i=1,2}\omega_{i}a_{i}^{\dagger}a_{i}+J(t)(a_{1}^{\dagger}% +a_{1})(a_{2}^{\dagger}+a_{2})$$ (3) Then, setting $\omega_{1}=\omega$ and $\omega_{2}=\omega+\delta$, taking $J(t)=J\cos(\delta t+\phi)$, and assuming $J,\delta\ll\omega$, we arrive to the effective Hamiltonian we want to simulate, cf. Eq. (Chiral quantum optics in photonic sawtooth lattices). For the experimentally implemented loop of Ref. Roushan et al. (2016), couplings of the order of $4.1$ MHz (i.e. $J/\omega\sim 10^{-3}$) have been measured. Thus, concatenating several of these cells and placing selectively the qubits, as done in Refs. Liu and Houck (2017); Mirhosseini et al. (2018) for simple coupled cavity arrays, one will be able to explore the phenomena predicted in this manuscript. For the interaction term $H_{\text{int}}$ (last term of Eq. (1)), we consider point-like and dipole-field coupling under the rotating-wave approximation, which is valid when the coupling strength is small with respect to the other energy scales of the system Cohen-Tannoudji et al. (1992): $$H_{\text{int}}=g\left(\sum_{j=1}^{M_{a}}\sigma_{j}^{+}a_{x_{j}}+\sum_{j=M_{a}+% 1}^{M}\sigma_{j}^{+}b_{x_{j}}\right)+\text{H.c.}$$ (4) Here $g$ is the coupling constant, $M_{a}$ is the number of qubits coupled to $A$ (so $M_{b}\equiv M-M_{a}$ are coupled to $B$), and $x_{j}$ is the position of the $j$-th qubit. Since we are interested in predictions in the thermodynamic limit, that is, when $N\rightarrow\infty$, we can take periodic boundary conditions for the bath and introduce the following plane wave modes $\hat{a}_{k}/\hat{b}_{k}\equiv 1/\sqrt{N}\sum_{n=1}^{N}e^{-ikn}a_{n}/b_{n}$, in terms of which $H_{\text{ST}}$ (Eq. (Chiral quantum optics in photonic sawtooth lattices)) reads: $$H_{\text{ST}}=\sum_{k}\left(\hat{a}_{k}^{\dagger}\;\hat{b}_{k}^{\dagger}\right% )h_{\text{ST}}(k)\left(\begin{array}[]{c}\hat{a}_{k}\\ \hat{b}_{k}\end{array}\right),$$ (5) with $h_{\text{ST}}(k)$ $$h_{\text{ST}}(k)=\left(\begin{array}[]{cc}-2J_{AA}\cos k&f(k,\phi)\\ f^{*}(k,\phi)&0\end{array}\right),$$ (6) where $f(k,\phi)=-J_{AB}(1+e^{-i(k+\phi)})$. Notice that we introduce the $\hat{\cdot}$ notation to distinguish the operators in real/momentum space. We can diagonalize $h_{\text{ST}}(k)$ such that $H_{\text{ST}}=\sum_{k}(\omega_{u}(k)\,\hat{u}_{k}^{\dagger}\hat{u}_{k}+\omega_% {l}(k)\,\hat{l}_{k}^{\dagger}\hat{l}_{k})$, where $\hat{u}_{k}$ and $\hat{l}_{k}$ are related to $\hat{a}_{k}$ and $\hat{b}_{k}$ by means of a unitary transformation: $$\left(\begin{array}[]{c}\hat{u}_{k}\\ \hat{l}_{k}\end{array}\right)=\left(\begin{array}[]{cc}\cos(\theta_{k})e^{-i% \varphi_{k}}&-\sin(\theta_{k})\\ \sin(\theta_{k})e^{-i\varphi_{k}}&\cos(\theta_{k})\end{array}\right)\left(% \begin{array}[]{c}\hat{a}_{k}\\ \hat{b}_{k}\end{array}\right),$$ (7) where the particular form of $\theta_{k}$ and $\varphi_{k}$ is shown in the Sup. Material Sup . The bands of the model, $\omega_{u/l}(k)$, read $$\displaystyle\omega_{u/l}(k)=-J_{AA}\cos k\pm\sqrt{J_{AA}^{2}\cos^{2}k+4J_{AB}% ^{2}\cos^{2}\left((k+\phi)/2\right)}.$$ (8) A celebrated feature of the sawtooth lattice is the appearance of flat bands, setting $J_{AB}/J_{AA}=\sqrt{2}$ and $\phi=0$ Leykam et al. (2018). Here, we are however interested in the implications of a nontrivial phase, $\phi\neq 0$, which leads nonsymmetric band structure in $k$-space (see Fig. 1(c)), due to the explicit breaking of time reversal ($H_{\text{ST}}\neq H_{\text{ST}}^{*}$, which implies $h_{\text{ST}}(k)\neq h_{\text{ST}}(-k)$). This cannot happen in a standard 1D photonic system (without loops) since one can get rid of the phase $\phi$ by means of local transformations of the bosonic operators. Apart from the $k$-asymmetry, there appears an extra band-gap between both bands for most values of $\phi$, except for $\phi=\pm\pi/2$ (see again Fig. 1(c)) where the two bands touch at single $k$-points $k=\pm\pi/2$. These type of singular bandgaps lead to exotic QE dynamics and interactions in higher dimensions González-Tudela and Cirac (2018); Perczel and Lukin (2018); González-Tudela and Cirac (2018). This will not be the case for the bath considered in this manuscript since the coupling strength to the upper/lower band $\omega_{l/u}(k)$, defined by the functions $\theta_{k},\varphi_{k}$, turn this point into a trivial band-crossing, as we explain in Sup. Material Sup . QE resonant with the band: Directional emission.- We first study the spontaneous decay of a single qubit ($M=1$ in (1) and (4)) when the QE transition frequency lies within the bands, i.e., $\Delta\in\omega_{u/l}(k)$. The state at time $t$ reads $|\Psi_{D}(t)\rangle=e^{-iHt}\sigma^{\dagger}\left|\mathrm{vac}\right\rangle$, where $\left|\mathrm{vac}\right\rangle$ is the global vacuum state and $D$ stands for the sublattice the QE is coupled to: $D=A,B$. As the number of excitations $\mathcal{N}\equiv\sum_{n=1}^{N}(a_{n}^{\dagger}a_{n}+b_{n}^{\dagger}b_{n})+% \sigma^{\dagger}\sigma$ is a conserved quantity under the rotating-wave approximation (4), the state $|\Psi_{D}(t)\rangle$ can be spanned in the single-excitation subspace as $$|\Psi_{D}(t)\rangle=\left(c_{e}^{D}(t)\sigma^{+}+\sum_{k}\left(c_{u}^{D}(k,t)% \hat{u}_{k}^{\dagger}+c_{l}^{D}(k,t)\hat{l}_{k}^{\dagger}\right)\right)|0\rangle.$$ (9) This allows us to calculate exactly the dynamics either numerically or semi-analytically using the resolvent operator approach Cohen-Tannoudji et al. (1992). In the latter, the QE probablity amplitude, $c_{e}^{D}(t)$, is obtained as the inverse Laplace transform of the QE Green Function operator $c_{e}^{D}(t)={\mathcal{L}}^{-1}[1/(s+\Sigma_{e}^{D}(z))]$ where the self-energy reads Sup : $$\Sigma_{e}^{D}(z)=\sum_{k}\sum_{\alpha=u,l}\frac{|\langle 0|\alpha_{k}H_{\text% {int}}\sigma^{+}|0\rangle|^{2}}{z-\omega_{\alpha}(k)}\,.$$ (10) Within the Markovian approximation the dependence on $z$ of the self-energy is neglected and replaced by $z=\Delta+i0^{+}$ when doing the inverse Laplace, yielding $c_{e}^{D}(t)\simeq e^{-i\Sigma_{e}^{D}(\Delta+i0^{+})t}$, where $\Sigma_{e}^{D}(\Delta+i0^{+})=\delta\omega_{D}-i\gamma_{D}/2$. Therefore, $\delta\omega_{D}$ and $\gamma_{D}$ correspond to the renormalization of the excited state frequency and linewidth, respectively. In Fig. 2 we plot these functions, $\delta\omega_{D}$ and $\gamma_{D}$, for $\phi=\pi/3$, marking the four band limits with vertical lines. The main result here is that both the decay rate and Lamb-shift depends on the sublattice the QE is coupled, which is a consequence of sublattice symmetry breaking of the bath for any $J_{AB}\neq 0$. When $\phi\neq\pi/2$, the latter also leads to the cancellation of the standard 1D band-edge divergence of $\gamma_{A}$ when $\Delta$ matches the the upper [lower] band-edge of $\omega_{l}(k)$ $[\omega_{u}(k)]$ for $\phi\in[0,\pi/2)$ [$(\pi/2,\pi]$]. Similar behaviour was also found in two-dimensional photonic crystals without sublattice symmetry González-Tudela and Galve (2018). When calculating the exact QE dynamics (not shown), we find band-edge related dynamics such as fractional and power-law decays in the long-time limit when the atomic frequencies are close to the band edges, which are very similar to the ones appearing in other band-gap photonic materials Khalfin (1958); Bykov (1975); Fonda et al. (1978); Hack (1982); Onley and Kumar (1992); John and Quang (1994); Gaveau and Schulmann (1995); Garmon et al. (2013); Redchenko and Yudson (2014); Lombardo et al. (2014); Sánchez-Burillo et al. (2016); González-Tudela and Cirac (2018); González-Tudela and Galve (2018). Thus, for this manuscript, we focus on the bath emission dynamics which indeed displays very distinctive features from other photonic baths. As a first evidence of that, we plot in Fig. 3(a) a snapshot of the photon population in real space for a situation where the emission is highly directional, which corresponds to a QE coupled to the $B$ sublattice, with parameters $J_{AB}=0.2J_{AA}$, $\phi=1.5$, and $\Delta=-0.5J_{AA}$. We want to note that even though the bath breaks $\pm k$ symmetry for any $\phi\neq 0$, we numerically find that the degree of directional emission depends strongly on other parameters, especially $\Delta$. Thus, having a bath with $\phi\neq 0$ is therefore a necessary but not sufficient condition for efficient directional emission. Let us further understand the origin and possibilities of the directional emission in this system by considering that the QE is resonant with $\omega_{l}(k)$, and taking the limit when $g$ is small enough such that we are in the Markov regime. In this limit, the QE dynamics is dominated by the resonant $k$-modes defined by $\omega_{l}(k_{R/L})=\Delta$, where $k_{R/L}$ correspond to the momenta of the right- and left-moving photons, respectively (see Fig. 1(b)). Furthermore, within each direction the excitations split between the ones propagating in the $A$ or $B$ sublattices. Thus, the decay rate $\gamma_{D}$ introduced as the imaginary part of the self-energy (Eq. (10)) can be separated into four different contributions: $$\displaystyle\gamma_{D}=\Gamma^{D}_{a}(k_{R})+\Gamma^{D}_{a}(k_{L})+\Gamma^{D}% _{b}(k_{R})+\Gamma^{D}_{b}(k_{L})\,,$$ (11) where $\Gamma^{D}_{\alpha}(q)$ denotes the decay rate into the $\alpha$-sublattice at momentum $q$ for a QE coupled to the $D$ sublattice, reading: $$\displaystyle\Gamma^{A(B)}_{a(b)}(q)$$ $$\displaystyle=\frac{|\sin(\theta_{q})|^{4}\left(|\cos(\theta_{q})|^{4}\right)}% {|v_{l}(q)|}$$ (12) $$\displaystyle\Gamma^{A}_{b}(q)$$ $$\displaystyle=\frac{|\sin(\theta_{q})|^{2}|\cos(\theta_{q})|^{2}}{|v_{l}(q)|}=% \Gamma^{B}_{a}(q)$$ (13) where $v_{l}(k)$ is the group velocity of the modes in the lower band, $v_{l}(k)=\partial_{k}\omega_{l}(k)$. With these functions we can define a global directionality ratio: $$\displaystyle R_{L/R}^{D}=\frac{\sum_{\alpha}\Gamma^{D}_{\alpha}(k_{R/L})}{% \sum_{\alpha}\left(\Gamma^{D}_{\alpha}(k_{R})+\Gamma^{D}_{\alpha}(k_{L})\right)}$$ (14) with $\alpha=a,b$, that tell us the ratio of light emitted in the left/right side in both sublattices, and a local one which distinguishes between sublattices $R^{D}_{R/L,a/b}$ with the same expressions but without the summation in $\alpha$. In Fig. 3(b), we plot $R^{B}_{L}$ as a function of $\Delta$ and $\phi>0$ for a tunneling $J_{AB}=0.2J_{AA}$. Note that the role of the $L/R$ is reversed by switching the sign of $\phi$. There, we observe that we can find non-reciprocal emission, that is, $R_{\alpha}^{D}>1/2$ for any $\phi\neq 0$. However, in order to find $R^{B}_{L}\approx 1$ one has to go to the limit where $J_{AB}/J_{AA}\ll 1$, $\phi\rightarrow\pi/2$, and $\Delta\to 0^{-}$ where the slope around $k=\pm\pi/2$ is very different yielding a highly asymmetric density of states for the left/right moving modes. The left/right character of the emission can be switched with the sign of $\phi$. When the QE couples to the $A$ sublattice instead, the global $R^{A}_{L/R}\approx 1/2$, however, locally in each sublattice can be made very directional, $R_{L,b}^{A},R_{R,a}^{A}\approx 1$ (see Sup ). This is possible because in that case $\theta_{k_{R/L}}$ is such that $\Gamma_{a}^{A}(k)$ is drastically much larger for $k_{R}$, compensating the density of states. To our knowledge, this sublattice-dependent chirality does not appear in other photonic reservoirs considered in the literature. QEs outside of the band: Tunable complex interactions.- We focus now on the regime where $\Delta\notin\omega_{l/u}(k)$, such that the physics is dominated by the bound states (BSs) John (1984, 1987); Tong et al. (2010a, b); Longo et al. (2010, 2011); Yang et al. (2013); Lü et al. (2013); Sánchez-Burillo et al. (2014, 2016); Calajó et al. (2016a, b); Shi et al. (2016a, 2018a); Bello et al. (2018). In the single-excitation subspace, the BS wavefunction of a single emitter coupled to the $D$ sublattice reads: $$\left|\Psi_{m}^{D}\right\rangle=\sum_{n}(c_{m,a}^{D}(n)a_{n}^{\dagger}+c_{m,b}% ^{D}(n)b_{n}^{\dagger})\left|0\right\rangle+c_{m,e}^{D}\sigma^{\dagger}\left|0% \right\rangle.$$ (15) where $m=-1,0,1$ denotes the different BSs that can appear in the upper/middle/lower band-gap, respectively. Their wavefunction, $c_{n,a/b}^{D}(n)$, and energy can be found from the secular equation: $H\left|\Psi_{m}^{D}\right\rangle=E_{m}^{D}\left|\Psi_{m}^{D}\right\rangle$ with $E_{m}^{D}\notin\omega_{l,u}(k)$ (see SM  Sup ). Using this equation, we can prove that there always exists one bound state $\left|\Psi_{\mp 1}^{D}\right\rangle$ below [above] $\omega_{l[u]}(k)$, because the self-energy always diverges at these band-edges Shi et al. (2016b); Calajó et al. (2016c), such that the interaction with the bath is able to push one state out of the band. However, in the middle band-gap that appears when $\phi\neq\pm\pi/2$, an extra BS $\left|\Psi_{0}^{D}\right\rangle$ emerges if and only if (i) $\Delta>0$, $|\phi|<\pi/2$ or (ii) $\Delta<0$ and $\phi\in(-\pi,-\pi/2)\cup(\pi/2,\pi)$ for $D=B$ or $A$, respectively Sup . The underlying reason of this condition is the finite value of the self-energy in one of the band-edges, as shown in Fig. 2(a), which defines a critical detuning for the existence of the BS. To illustrate the main features of these BSs, we plot in Fig. 4 their wavefunction coefficients both in momentum and real space of an instance of a BS; concretely, we choose the interband one $\left|\Psi_{0}^{D}\right\rangle$. i) Contrarily to what happens in emission, the absolute value of the wavefunction $|c^{D}_{m,\alpha}(n)|$ is always symmetrically distributed around the QE no matter the band-gap or parameters considered (see Fig. 4(b)); ii) as it occurs with other photonic lattices John (1984, 1987); Tong et al. (2010a, b); Longo et al. (2010, 2011); Yang et al. (2013); Lü et al. (2013); Sánchez-Burillo et al. (2014, 2016); Calajó et al. (2016a, b); Shi et al. (2016a, 2018a); Bello et al. (2018), the BS are exponentially localized around the emitter with a localization length which can be tuned: the closer the $E^{D}_{m}$ lies to one of the band-edges, the larger is the localization length; iii) the main distinctive feature is that the BSs acquire a tunable complex phase $c^{D}_{m,\alpha}(n)\propto e^{i\varphi^{D}_{m}n}$. In the small coupling limit, this phase $\varphi^{D}_{m}$ matches the position of the band edge closest to $\Delta$. The positions of these band edges change with both the ratio $J_{AB}/J_{AA}$ and $\phi$. This dependence is seen in Fig. 1(c); e.g., the minimum of the upper band runs from 0 to $\pi/2$, so the phase of the interband bound state can be tuned in this range provided $\Delta$ tends to this band edge. In the example of Fig. 4, the momentum of the closest band edge occurs at $k_{\mathrm{edge}}\simeq\pi/3$, so the coefficients in momentum space $|c_{0,\alpha}^{B}(k)|^{2}$ are distributed around $\pi/3$ (panel (a)) and their real and imaginary parts in real space for the $b$ modes, $\text{Re}(c_{0,b}^{B}(n))$ and $\text{Im}(c_{0,b}^{B}(n))$ have periodicity $2\pi/(\pi/3)=6$ (panel (b)). One of the main interests of these BS is that when many emitters couple to the bath, they can mediate interactions between QEs which can be harnessed to simulate spin models with tunable interactions. In the Markovian approximation, an effective Hamiltonian for the qubits can be derived Douglas et al. (2015); González-Tudela et al. (2015): $$\displaystyle H_{\text{qb}}=\sum_{i<j}(J^{D_{i}D_{j}}_{ij}\sigma_{i}^{\dagger}% \sigma_{j}+\text{H.c.})\,,$$ (16) where $J^{D_{i}D_{j}}_{ij}$ inherits the shape of the BS wavefunction with energy $\Delta$, i.e., $J_{ij}^{AA/BB}\propto c_{a/b}^{A/B}(r_{ij})$ and $J^{AB}_{ij}\propto c^{A}_{b}(r_{ij})$ (see SM Sup ). When $\Delta$ lies in a band-gap, one can then control not only the effective range of the interactions, but, as discussed in the previous paragraph and in Fig. 4, also its phase. In particular, the physical phase of the photonic lattice $\phi$ is inherited by the effective spin-spin interactions (see SM Sup ). We finally want to note that even richer many-body dynamics will appear in the non-perturbative regime replacing spins by polaritons Shi et al. (2018b). Conclusions.- We have studied the dynamics and interactions of quantum emitters coupled to a minimal one-dimensional model breaking time-reversal symmetry, i.e., the photonic sawtooth lattice. We have found that when the emitters are resonant with the band they decay in an asymmetric fashion into left/right moving modes. Optimizing the parameters we identified regimes of quasi-perfect directionality, or more exotic ones in which the emitter decays in both directions but to a different sublattice in each of the them. Thus, these systems can be an alternative way of exploring chiral quantum optics without the need of using polarization or moving emitters. In addition to that, when the emitter frequency lies in a band-gap we have found the emergence of bound states whose not only their spatial range, but also their complex phase can be tuned through the system parameters. Since these bound states ultimately mediate interactions between emitters when many of them couple to the bath, our proposed setup provides access to the simulation of a large class of spin models with complex interactions. Furthermore, we discussed a particular implementation where to observe such phenomenology based on state-of-the-art superconducting technologies. Acknowledgments.- ESB acknowledges ERC Advanced Grant QUENOCOBA under the EU Horizon 2020 program (grant agreement 742102). AGT and DZ acknowledge support from CSIC Research Platform PTI-001. AGT acknowledges funding from the national project PGC2018-094792-B-I00 from Ministerio de Ciencia e Innovación. Supplemental Material: Chiral quantum optics in photonic sawtooth lattices In this Supplementary Material, we give more details of the: i) diagonalization of the bath Hamiltonian in Section SM1; ii) calculation of the single emitter self-energy in Section SM2; iii) the structure of the qubit-band couplings in Section SM3; iv) the absence of a nonanalyticity in the bands for $\phi=\pi/2$; v) more details in the sublattice directional behaviour in Section SM5; vi) the features of the bound states energies and wavefunctions in Section SM6; vii) the connection between the bound states and the two emitter self-energy in Section SM7. SM1 Diagonalization of the sawtooth lattice In this Section, we give some details on the diagonalization of the sawtooth Hamiltonian (Eq. (Chiral quantum optics in photonic sawtooth lattices)). The bosonic operators which diagonalize the model, $u_{k}/l_{k}$ (see Eq. (7)) are related to $a_{k}$ and $b_{k}$ by means of a unitary transformation $P_{k}$ (Eq. (7)). The latter reads $$\displaystyle P_{k}$$ $$\displaystyle=\left(\begin{array}[]{cc}\cos(\theta_{k})e^{i\varphi_{k}}&\sin(% \theta_{k})e^{i\varphi_{k}}\\ -\sin(\theta_{k})&\cos(\theta_{k})\end{array}\right)$$ $$\displaystyle=\left(\begin{array}[]{cc}N_{u}(k)f^{*}(k,\phi)&N_{l}(k)f^{*}(k,% \phi)\\ N_{u}(k)(\omega_{u}(k)+2J_{AA}\cos k)&N_{l}(k)(\omega_{l}(k)+2J_{AA}\cos k)% \end{array}\right)$$ (SM1) where $N_{u/l}(k)$ is a normalization factor $$N_{u/l}(k)=\frac{1}{\sqrt{|f(k,\phi)|^{2}+(\omega_{u/l}(k)+2J_{AA}\cos k)^{2}}}.$$ (SM2) SM2 Single-qubit self-energy To compute $c_{e}^{D}(t)$, we use the resolvent operator method Cohen-Tannoudji et al. (1992), which tell us that the probability amplitude can be computed as: $$c_{e}^{D}(t)=-\frac{1}{2\pi i}\int_{-\infty}^{\infty}dE\,G_{e}^{D}(E+i0^{+})e^% {-iEt},$$ (SM3) where $G_{e}^{D}(z)$ is the single-qubit Green function when it is coupled to the sublattice $D$: $$G_{e}^{D}(z)=\frac{1}{z-\Delta-\Sigma_{e}^{D}(z)},$$ (SM4) being $\Sigma_{e}^{D}(z)$ the so-called self-energy. In this Section, we derive the expressions for the single-qubit self-energy when the qubit is locally coupled to $A$ or $B$. The $\Sigma_{e}^{D}(z)$ of our two band model reads: $$\Sigma_{e}^{D}(z)=\sum_{k}\sum_{\alpha=u,l}\frac{|\langle 0|\alpha_{k}H_{\text% {int}}\sigma^{+}|0\rangle|^{2}}{z-\omega_{\alpha}(k)}.$$ (SM5) Considering $H_{\text{int}}$ (Eq. (4) for a single qubit) coupled to $A$ or $B$ and taking into account the relation between $(u_{k},l_{k})$ and $(a_{k},b_{k})$ (see Eqs. (7) and (SM1)) and the expressions for $\omega_{u/l}(k)$ (see Eq. (8)): $$\displaystyle\Sigma_{e}^{A}(z)=\frac{g^{2}}{2\pi}\int_{-\pi}^{\pi}dk\frac{z}{z% ^{2}+2zJ_{AA}\cos k-|f(k,\phi)|^{2}},$$ (SM6) $$\displaystyle\Sigma_{e}^{B}(z)=\frac{g^{2}}{2\pi}\int_{-\pi}^{\pi}dk\frac{z+2J% _{AA}\cos k}{z^{2}+2zJ_{AA}\cos k-|f(k,\phi)|^{2}}.$$ (SM7) We take here the thermodynamic limit: $N\to\infty$. One can solve these integrals by means of the change of variable $y\equiv e^{ik}$. The integration domain is now the unit circle in the complex plane: $$\displaystyle\Sigma_{e}^{A}(z)=\frac{g^{2}}{2\pi i}\oint dy\frac{zJ_{AA}}{(zJ_% {AA}-J_{AB}^{2}e^{i\phi})(y-y_{+})(y-y_{-})},$$ (SM8) $$\displaystyle\Sigma_{e}^{B}(z)=\frac{g^{2}}{2\pi i}\oint dy\frac{J_{AA}y^{2}+% zy+J_{AA}}{(zJ_{AA}-J_{AB}^{2}e^{i\phi})\,y(y-y_{+})(y-y_{-})},$$ (SM9) where $y_{\pm}$ are: $$y_{\pm}=\frac{2J_{AB}^{2}-z^{2}\pm\sqrt{(2J_{AB}^{2}-z^{2})^{2}-4(z^{2}J_{AA}^% {2}+J_{AB}^{4}-2zJ_{AA}J_{AB}^{2}\cos\phi)}}{2(zJ_{AA}-J_{AB}^{2}e^{i\phi})}.$$ (SM10) We define $y_{\text{min/max}}$ as the minimum/maximum of $\{y_{-},y_{+}\}$ with respect to the absolute values $|y_{\pm}|$. Applying the Cauchy’s residue theorem and taking into account that $(|y_{+}|-1)(|y_{-}|-1)<1$ for all $z\in\mathbb{C}$ with $\text{Im}(z)\neq 0$: $$\displaystyle\Sigma_{e}^{A}(z)=\frac{g^{2}z\,\text{sign}(|y_{-}|-|y_{+}|)}{(zJ% _{AA}-J_{AB}^{2}e^{i\phi})(y_{+}-y_{-})},$$ (SM11) $$\displaystyle\Sigma_{e}^{B}(z)=\frac{g^{2}J_{AA}}{zJ_{AA}-J_{AB}^{2}e^{i\phi}}% \left(\frac{1}{y_{+}y_{-}}+\frac{y_{\text{min}}^{2}+(z/J_{AA})y_{\text{min}}+1% }{y_{\text{min}}(y_{\text{min}}-y_{\text{max}})}\right).$$ (SM12) E.g. if we consider that $\Delta$ is embedded in the lower band, it is straightforward to derive Eq. (11) from Eqs. (SM11) and (SM12). SM3 Qubit-band couplings Here, we write down the qubit-band coupling for both bands. Let us consider the interaction Hamiltonian $H_{\text{int}}$ (Eq. (4)) for a single qubit. For the sake of simplicity, the qubit will be coupled to $A$. We write $H_{\text{int}}^{A}$ in terms of $u_{k}$ and $l_{k}$ (see Eq. (7)): $$\displaystyle H_{\text{int}}^{A}=\frac{g}{\sqrt{N}}\sigma^{+}\sum_{k}e^{ikx_{0% }}(\cos(\theta_{k})e^{i\varphi_{k}}\hat{u}_{k}+\sin(\theta_{k})e^{i\phi_{k}}% \hat{l}_{k})+\text{H.c.}$$ (SM13) where $x_{0}$ is the position of the qubit and $\cos/\sin(\theta_{k})$ are the matrix elements of the unitary transformation $P_{k}$ (see Eq. (SM1)). The latter determines the coupling strength to each band: $G_{u,A}(k)=|\cos(\theta_{k})|^{2}$ and $G_{l,A}(k)=|\sin(\theta_{k})|^{2}$, up to the density of states, which is given by $1/|\partial\omega_{u/l}(k)|$. SM4 Trivial crossing point The existence of a nonanalytical point in the band structure can cause exotic behaviour both in the dynamics of the emitters and in the effective interactions González-Tudela and Cirac (2018). We explain in this section why the apparent kink in the bands at $k=\pm\pi/2$ when $\phi=\pm\pi/2$ (see Fig. 1(c) for $\phi=\pi/2$) is actually a trivial crossing point. We first plot both the real and imaginary parts of the self-energy $\Sigma_{e}^{D}(z)$, Eqs. (SM11) and (SM12), as a function of $\Delta$ when $\phi=\pi/2$ in Fig. SM1. As seen, both are smooth functions of $\Delta$, which indicates that the couped qubit does not feel a nonanalyticity in the dispersion relations $\omega_{u/l}(k)$. We confirm this by studying how the QE couples to the bands for $\phi=\pm\pi/2$. We define two new bands $\omega_{\pm}(k)$, together with the corresponding couplings $G_{\pm,A}(k)$ (see Sect. SM3): $$\displaystyle\omega_{\pm}(k)\equiv$$ $$\displaystyle\left\{\begin{array}[]{c}\omega_{u/l}(k)\quad\text{if}\;k<\pi/2,% \\ \omega_{l/u}(k)\quad\text{if}\;k>\pi/2.\end{array}\right.$$ (SM14) $$\displaystyle G_{\pm,A}(k)\equiv$$ $$\displaystyle\left\{\begin{array}[]{c}G_{u/l,A}(k)\quad\text{if}\;k<\pi/2,\\ G_{l/u,A}(k)\quad\text{if}\;k>\pi/2.\end{array}\right.$$ (SM15) We plot both $\omega_{\pm}(k)$ and $G_{\pm,A}(k)$ in Fig. SM2 for $\phi=\pi/2$. As seen, these bands $\omega_{\pm}(k)$ do not have any kink and the QE couples smoothly to both of them. In conclusion, the apparent nonanalytical behavior is actually an artifact of the definition of the bands. SM5 Sublattice-dependent directional emission We showed in the main text that a qubit coupled to $B$ emits left-moving photons when $\Delta$ is close to the upper limit of $\omega_{l}(k)$, $\phi\to\pi/2$, and $J_{AB}\ll J_{AA}$. (see Fig. 3). If it is coupled to $A$ instead, the behaviour changes. We show in Fig. SM3 the directionality ratios in each sublattice when the qubit is coupled to $A$, $R_{L,a/b}^{A}$, as a function of $\Delta$ and $\phi$ for $J_{AA}=1$ and $J_{AB}=0.2$. The qubit still decays into left-moving $b$ modes when $\Delta$ is close the upper band limit of $\omega_{l}(k)$ and $\phi\to\pi/2$, whereas it emits right-moving $a$ photons under the same conditions (notice that $R_{L,b}^{A}$ tends to $1$ in this regime, whilst $R_{L,a}^{A}\to 0$). This is possible because, in this regime, the matrix elements of $P_{k}$ are such that the numerator of the couplings $\Gamma_{a,b}^{A}(k)$ (see Eqs. (12)-) compensates the sharp difference between the density of states for $k_{L}$ and $k_{R}$. Actually, the global directionality ratio $\eqref{eq:directionality_ratio}$ is totally symmetric: $R_{L}^{A}=R_{R}^{A}=1/2$, which is straightforwardly derived from the expressions (12), (SM5), (SM1), and (8). We do not know whether this can be related to any symmetry of the model and we leave this potential connection for future projects. We show an instance of emission into opposite directions in Fig. SM4. SM6 Bound states In this section, we discuss the existence conditions of the bound states (BSs) and we compute their wavefunctions in real space. As mentioned in the main text, we have to impose the eigenvalue equation $H\left|\Psi_{m}^{D}\right\rangle=E_{m}^{D}\left|\Psi_{m}^{D}\right\rangle$ with the energy $E_{m}^{D}$ outside of the bands. This can be mapped into finding the roots of the following function: $$F_{D}(E)\equiv E-\Delta-\Sigma_{e}^{D}(E),$$ (SM16) with $E\notin\omega_{l,u}(k)$ Shi et al. (2016a). It can be easily proved from Eq. (10) that $\Sigma_{e}^{D}(E)$ is a decreasing function, so $F_{D}(E)$ is an increasing function. Besides, $\lim_{E\to\pm\infty}F_{D}(E)=\pm\infty$. Then, according to the behaviour of $\Sigma_{e}^{D}(E)$ in the band edges, we can figure out whether there exist or not a bound state in each of the band gaps: • A BS exists with $E_{\text{bs}}<\omega_{l}(k)$ ($E_{\text{bs}}>\omega_{l}(k)$) for all the values of the parameters if and only if $F_{D}(E)>(<)0$ when $E$ tends to the minimum of $\omega_{l}(k)$ (maximum of $\omega_{u}(k)$). We plot $\Sigma_{e}^{D}(E)$ for $E$ outside of the band in Fig. 2 and show that it diverges in the the lowest/highest energy band-edge, which guarantees the aforementioned conditions, so the existence of two BS below $\omega_{l}(k)$ and over $\omega_{u}(k)$, which we label as $\left|\Psi_{-1}^{D}\right\rangle$ and $\left|\Psi_{+1}^{D}\right\rangle$, respectively. • The situation is different in the middle band-gap. In Fig. 2(a), we observe when the QE is coupled to the B lattice, the self-energy diverges in both the upper/lower middle band-edges. Thus, a middle BS, $\left|\Psi_{0}^{B}\right\rangle$, always exists. On the other hand, if the emitter is coupled to $A$, the state $\left|\Psi_{0}^{A}\right\rangle$ exists if $\Delta>0$ when $|\phi|<\pi/2$ because the self-energy $\Sigma_{e}^{A}(E)$ vanishes when $E$ tends to the maximum of $\omega_{l}(k)$. When $\phi\in(-\pi,-\pi/2)$ or $\phi\in(\pi/2,\pi)$, the existence condition is $\Delta<0$ (not shown). Concerning the wavefunctions, if $D=A$, the coefficients read $$\displaystyle c_{m,a}^{A}(n)=\frac{gc_{e}^{A}}{2\pi}\int_{-\pi}^{\pi}dk\,e^{ikn}$$ $$\displaystyle\left(\frac{|(P_{k})_{11}|^{2}}{E_{m}^{D}-\omega_{u}(k)}\right.$$ $$\displaystyle+\left.\frac{|(P_{k})_{12}|^{2}}{E_{m}^{A}-\omega_{l}(k)}\right),$$ (SM17) $$\displaystyle c_{m,b}^{A}(n)=\frac{gc_{e}^{A}}{2\pi}\int_{-\pi}^{\pi}dk\,e^{ikn}$$ $$\displaystyle\left(\frac{(P_{k})_{21}(P_{k})_{11}^{*}}{E_{m}^{A}-\omega_{u}(k)% }\right.$$ $$\displaystyle+\left.\frac{(P_{k})_{22}(P_{k})_{12}^{*}}{E_{m}^{A}-\omega_{l}(k% )}\right),$$ (SM18) where $E_{m}^{A}$ is the energy of $\left|\Psi_{\text{m,bs}}^{A}\right\rangle$ and $c_{e}^{A}$ is obtained imposing the normalization condition. Doing the maths: $$\displaystyle c_{m,a}^{A}(n)=\frac{gc_{e}}{2\pi}\int_{-\pi}^{\pi}dk\frac{e^{% ikn}\,E_{\text{m}}^{A}}{(E_{\text{m}}^{A})^{2}+J_{AA}2E_{\text{m}}^{A}\cos k-|% f(k,\phi)|^{2}},$$ (SM19) $$\displaystyle c_{m,b}^{A}(n)=\frac{-gc_{e}}{2\pi}\int_{-\pi}^{\pi}dk\frac{e^{% ikn}f^{*}(k,\phi)}{(E_{\text{m}}^{A})^{2}+2J_{AA}E_{\text{m}}^{A}\cos k-|f(k,% \phi)|^{2}}.$$ (SM20) If the qubit is instead coupled to $B$: $$\displaystyle c_{m,a}^{B}(n)=\frac{gc_{e}^{B}}{2\pi}\int_{-\pi}^{\pi}dk\,e^{ikn}$$ $$\displaystyle\left(\frac{(P_{k})_{11}(P_{k})_{21}^{*}}{E_{m}^{D}-\omega_{u}(k)% }\right.$$ $$\displaystyle+\left.\frac{(P_{k})_{12}(P_{k})_{22}^{*}}{E_{m}^{B}-\omega_{l}(k% )}\right),$$ (SM21) $$\displaystyle c_{m,b}^{B}(n)=\frac{gc_{e}^{B}}{2\pi}\int_{-\pi}^{\pi}dk\,e^{ikn}$$ $$\displaystyle\left(\frac{|(P_{k})_{21})|^{2}}{E_{m}^{B}-\omega_{u}(k)}\right.$$ $$\displaystyle\left.+\frac{|(P_{k})_{22})|^{2}}{E_{m}^{B}-\omega_{l}(k)}\right),$$ (SM22) which becomes: $$\displaystyle c_{m,a}^{B}(n)=\frac{-gc_{e}^{B}}{2\pi}\int_{-\pi}^{\pi}dk\frac{% e^{ikn}f(k,\phi)}{(E_{\text{m}}^{B})^{2}+2J_{AA}E_{\text{m}}^{B}\cos k-|f(k,% \phi)|^{2}},$$ (SM23) $$\displaystyle c_{m,b}^{B}(n)=\frac{gc_{e}^{B}}{2\pi}\int_{-\pi}^{\pi}dk\frac{e% ^{ikn}(E_{\text{m}}^{B}+2\cos k)}{(E_{\text{m}}^{B})^{2}+2J_{AA}E_{\text{m}}^{% B}\cos k-|f(k,\phi)|^{2}}.$$ (SM24) Notice that all these expressions look similar to $\Sigma_{e}^{D}(z)$ (see Eqs. (SM6) and (SM7)), so we can calculate the coefficients in terms of complex integrals (Eqs. (SM8) and (SM9)). The change of variable is still $y=e^{ik}$ if $n\geq 1$, but $y=e^{-ik}$ if $n\leq-1$. In the first case, the poles of the integral are $y_{\pm}$, while in the second are their complex conjugates $y_{\pm}^{*}$. SM7 Two-qubit self-energy We show here the expressions for the collective self-energy $\Sigma_{c}^{D_{12}}$ (Eq. (16)). The computation is totally analogous to the single-qubit self-energy (see App. SM2). They read $$\displaystyle\Sigma_{c}^{AA}(z;r_{12})=\frac{g^{2}}{2\pi}\int_{-\pi}^{\pi}dk% \frac{e^{ikr_{12}}\,z}{z^{2}+2zJ_{AA}\cos k-|f(k,\phi)|^{2}},$$ (SM25) $$\displaystyle\Sigma_{c}^{BB}(z;r_{12})=\frac{g^{2}}{2\pi}\int_{-\pi}^{\pi}dk% \frac{e^{ikr_{12}}(z+2J_{AA}\cos k)}{z^{2}+2zJ_{AA}\cos k-|f(k,\phi)|^{2}},$$ (SM26) $$\displaystyle\Sigma_{c}^{AB}(z;r_{12})=-\frac{g^{2}}{2\pi}\int_{-\pi}^{\pi}dk% \frac{e^{ikr_{12}}f^{*}(k,\phi)}{z^{2}+2zJ_{AA}\cos k-|f(k,\phi)|^{2}},$$ (SM27) where $r_{12}=x_{2}-x_{1}$ is the relative position of the qubits. Notice that $\Sigma_{c}^{AA}(z;r_{12})$, $\Sigma_{c}^{BB}(z;r_{12})$, and $\Sigma_{c}^{AB}(z;r_{12})$ are proportional to the bound-state coefficients $c_{m,a}^{A}(r_{12})$, $c_{m,b}^{B}(r_{12})$, and $c_{m,b}^{A}(r_{12})$ respectively, by changing the bound-state energies $E_{m}^{D}$ by $z$ (see Eqs. (SM19), (SM24), and (SM20)). It is here where it becomes evident that the effective interactions are mediated by the bound states. 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High Resolution Polarimetry of the Inner Galaxy Bryan M. Gaensler${}^{1}$, J. M. Dickey${}^{2}$, N. M. McClure-Griffiths${}^{3}$, N. S. Bizunok${}^{4}$, A. J. Green${}^{5}$ ${}^{1}$Harvard-Smithsonian Center for Astrophysics, Cambridge MA 02138, USA ${}^{2}$University of Minnesota, Minnesota MN 55455, USA ${}^{3}$Australia Telescope National Facility, Epping NSW 1710, Australia ${}^{4}$Boston University, Boston MA 02215, USA ${}^{5}$University of Sydney, NSW 2006, Australia Abstract We present our results from the Southern Galactic Plane Survey, an effort to map the fourth quadrant of the Milky Way in linear polarization at a frequency of 1.4 GHz and at a resolution of 1–2 arcmin. These data are a powerful probe of both the turbulence and large-scale structure of magneto-ionic gas, and have revealed a variety of new features in the interstellar medium. December 6, 2020 Introduction The Milky Way was the first celestial radio source discovered, and was subsequently one of the first sources to be detected in linear polarization. There are two sources of this polarization: discrete objects such as supernova remnants (SNRs), and a diffuse polarized background produced by the relativistic component of the interstellar medium (ISM). All of this emission undergoes Faraday rotation as it propagates towards us, either in the source itself or in intervening material. With sufficiently high angular and frequency resolution, we can use the properties of this polarized emission to map out the distribution of ionized gas and magnetic fields in individual sources and in the ambient ISM. Only recently have instruments and techniques advanced to a point where such studies are feasible gld+99 ; hkd00 ; wdj+93 . Motivated by the spectacular single-dish polarization surveys of Duncan et al dhjs97 ; drrf99 , we have made polarimetric images of the entire fourth quadrant of the Galaxy with the Australia Telescope Compact Array (ATCA). These data have been taken as part of the Southern Galactic Plane Survey (SGPS; mgd+01 ). While the primary focus of the SGPS is to study the Galactic distribution of H i, the ATCA simultaneously receives full polarimetric continuum data, which have allowed us to map out the distribution of linearly polarized emission in the survey region. While the full survey has now been completed, a detailed analysis has only been carried out on a 28-deg${}^{2}$ test region, covering the range $325\hbox{$.\!\!^{\circ}$}5<l<332\hbox{$.\!\!^{\circ}$}5$, $-0\hbox{$.\!\!^{\circ}$}5<b<+3\hbox{$.\!\!^{\circ}$}5$. We here summarize the main results of this analysis; this work is described in more detail by Gaensler et al gdm+00 . Observations and Reduction The ATCA is a 6-element synthesis telescope, located near Narrabri, NSW, Australia. Observations of the test region of the SGPS were carried out in nine observing runs in 1997 and 1998, and comprised 190 separate telescope pointings (see mgd+01 for details). Data were recorded in nine spectral channels spread across 96 MHz of bandwidth and centered at a frequency of 1384 MHz. The two sources MRC B1438–481 and MRC B1613–586 were observed over a wide range in parallactic angle in order to solve for the instrumental polarization characteristics of each antenna shb96 . For each spectral channel, images of the field in Stokes $I$, $Q$, $U$ and $V$ were deconvolved jointly using the maximum entropy algorithm PMOSMEM sbd99 and then smoothed to a resolution of $\sim$1 arcmin. The final sensitivity in each image is $\stackrel{{\scriptstyle<}}{{{}_{\sim}}}0.5$ mJy beam${}^{-1}$. Images of linearly polarized intensity, $L=(Q^{2}+U^{2})^{1/2}$, linearly polarized position angle, $\Theta=\frac{1}{2}\tan^{-1}(U/Q)$, and uncertainty in position angle, $\Delta\Theta=\sigma_{Q,U}/2L$, were then formed from each pair of $Q$ and $U$ images. The nine $L$ maps (one per spectral channel) were then averaged together to make a final image of $L$ for the entire test region, while the nine $\Theta$ and $\Delta\Theta$ maps were used to derive an image of the rotation measure (RM) over the field. Results In Figure 1 we show images of $I$ and $L$ from both our 1.4-GHz ATCA observations and from the 2.4-GHz Parkes survey of Duncan et al dhjs97 (resolution $10\hbox{$.\mskip-4.0mu ^{\prime}$}4$). The total intensity images show the presence of SNRs and H ii regions (see mgd+01 for further discussion). Although the interferometric ATCA observations are not sensitive to the diffuse emission seen by Parkes, it is clear that the same features are present in both data-sets. At first glance it seems that the $L$ images have very little in common with the Stokes $I$ emission. In particular, the ATCA $L$ image is dominated by diffuse polarization spread all over the field of view, composed of discrete patches separated by narrow “canals” of reduced polarization. While none of this emission is correlated with total intensity, there does seem to be a good match between the brightest polarized regions of the ATCA and Parkes data, despite the differing frequencies and resolutions of these data-sets. Using the images of $\Theta$ and $\Delta\Theta$, we can determine the variation of polarization position angle with frequency wherever we detect polarized emission. The resulting RMs are generally small and negative, with a mean RM for the entire field of $-12.9\pm 0.1$ rad m${}^{-2}$; 50% of the RMs have magnitudes smaller than $\pm$25 rad m${}^{-2}$ and 98% are smaller than $\pm$100 rad m${}^{-2}$. The ATCA $L$ image reveals two large voids of reduced polarization, each elliptical and several degrees in extent. One void is centered on $(l,b)=(332\hbox{$.\!\!^{\circ}$}4,+1\hbox{$.\!\!^{\circ}$}4)$ (“void 1”) and the other on $(328\hbox{$.\!\!^{\circ}$}2,-0\hbox{$.\!\!^{\circ}$}5)$ (“void 2”); both voids are also seen in the 2.4-GHz Parkes polarization map. The RMs around the edges of these voids range up to $\pm 400$ rad m${}^{-2}$, in distinction to the low RMs seen over the rest of the field. A careful examination shows one marked correspondence between the ATCA Stokes $I$ and $L$ images: at $(326.3,+0.8)$, the bright H ii region RCW 94 shows reduced polarization towards its interior, and is further surrounded by a halo in which no polarization at all is seen. This is shown in more detail in Figure 2. Finally, of the numerous unresolved sources distributed across the field, 21 of these sources show detectable linear polarization. The RMs for these sources fall in the range –1400 rad m${}^{-2}$ to +200 rad m${}^{-2}$. Discussion Diffuse Emission We first note that the incomplete $u-v$ coverage of an interferometer affects images of polarization in complicated ways. While it is physically required that $L\leq I$, and we generally expect that structures seen in $L$ might correspond to similar structures in $I$, neither situation will be generally observed in interferometric data. This is because an interferometer can not detect structures larger than a certain size, corresponding to the closest spacings between its antenna elements (in the case of the ATCA, this maximum scale of $\sim 35^{\prime}$). A source larger than this maximum scale will not be seen in Stokes I; if it is also a uniformly polarized source, it will not be detected in polarization either. However, magnetic field structure within the source, plus variations in the Faraday rotation along different lines-of-sight, can introduce power in Stokes $Q$ and $U$ on smaller scales, to which the interferometer is sensitive. We thus can often observe complicated structures in polarization which have no counterpart in total intensity gldt98 ; gld+99 ; hkd00 ; wdj+93 . Clearly such an effect is occurring here, and is producing virtually all the linear polarization seen in Figure 1. We can crudely divide up the diffuse polarization we see into two components. The brightest polarization seen with the ATCA matches well the bright polarized structures seen with Parkes. Since the amount of Faraday-induced polarization is very strongly dependent on both resolution and frequency, the fact that two such disparate data-sets show similar structures implies that these bright polarized structures are intrinsic to the emitting regions. By comparing the RMs observed for this emission to those observed for pulsars in this part of the sky, we can conclude that the distance to this emission is in the range 1.3–4.5 kpc. The depolarizing effects of RCW 94 (discussed further below) imply that the polarized emission is $>$3 kpc distant, while the lack of depolarization against other H ii regions gives an upper limit of $6.5$ kpc. Dickey dic97 has made H i absorption measurements against this emission to derive a lower limit on its distance of 2 kpc. Taking into account all these constraints, we argue that the mean distance to the source of polarized emission is $3.5\pm 1.0$ kpc, corresponding to the Crux spiral arm of our Galaxy. The rest of the ATCA field is filled with fainter diffuse polarization, which does not have any counterpart in the Parkes data. This emission is best explained as being due to Faraday rotation in foreground material. The RMs measured for this emission imply that they are caused by foreground clouds of RM $\sim 5$ rad m${}^{-2}$, consistent with the conclusions made by Wieringa et al wdj+93 . Voids in Polarization To the best of our knowledge, voids in polarization such as those described here have not been previously reported. There are two possible explanations to account for these structures: either they represent regions where the level of intrinsic polarization is low, or they are the result of propagation through a foreground object, whose properties have depolarized the emission at both 1.4 and 2.4 GHz. If the voids are intrinsic to the emitting regions, then the distance of 3.5 kpc inferred above implies that they are hundreds of parsecs across — it is hard to see what could produce such uniformly low polarized intensity across such large regions. We thus think it unlikely that the voids are intrinsic to the emitting regions. We thus favor the possibility that the voids are caused by depolarizing effects in foreground material. We have considered in detail the various ways in which foreground Faraday rotation can produce the observed structure, and can rule out bandwidth and gradient depolarization as possible mechanisms (see gdm+00 for details). The only remaining possibility is that depolarization in the voids is due to beam depolarization, in which the RM varies randomly on small scales. We have developed a detailed model for “void 1” to confirm this. We consider void 1 to be a caused by a sphere of uniform electron density $n_{e}$ cm${}^{-3}$, centered on $(332\hbox{$.\!\!^{\circ}$}5,+1\hbox{$.\!\!^{\circ}$}2)$ with a radius of $1\hbox{$.\!\!^{\circ}$}4$ and at a distance to us of $d$ kpc. Within the sphere, we suppose that there are random and ordered components to the magnetic field, and that these two components have identical amplitudes $B$ $\mu$G. The ordered component is uniformly oriented at an angle $\theta$ to the line of sight. We assume that the random component is coherent within individual cells of size $l$ pc, but that the orientation from cell to cell is random. Uniformly polarized rays which propagate through a different series of cells will experience differing levels of Faraday rotation, resulting in beam depolarization when averaged over many different paths. By calculating the properties of the polarized signal which emerges after propagating through this source, we find that we can account for the observed properties of void 1 if $n_{e}\sim 20$ cm${}^{-3}$, $B\sim 5$ $\mu$G, $\theta\stackrel{{\scriptstyle>}}{{{}_{\sim}}}80^{\circ}$, $d\sim 300$ pc and $l\sim 0.2$ pc (see gdm+00 for details). These properties are consistent with those of an H ii region of comparatively low emission measure. Indeed Figure 3 demonstrates that H$\alpha$ emission fills void 1, its morphology and perimeter matching exactly to that of the void. It is interesting to note that the O9V star HD 144695 is very close to the projected center of void 1, and is at a distance of $300\pm 160$ pc. The radius of the Strömgren sphere which this star would produce is consistent with the extent of the void. It is thus reasonable to propose that the star is embedded in and powers the surrounding ionized bubble. Two properties of the voids which our simple model cannot account for are the requirement that the uniform component of the magnetic field be largely oriented in the plane of the sky, but that we generally observe coherent regions of large RM (of the order of a few hundred rad m${}^{-2}$) around the edges of the voids. We suggest that both these results can be explained by the field geometry which arises during the expansion phase of an H ii region as it interacts with surrounding material. This produces a magnetic field perpendicular to the line of sight over most of the void, but which is parallel to the line of sight (and can thus potentially produce high RMs) around the perimeter. Depolarization seen towards RCW 94 The reduced polarization seen coincident with RCW 94 in Figure 2 presumably results from beam depolarization, just as for the H ii region argued to produce void 1. However, the effects of beam depolarization are expected to be weakest around the edges of the source, and thus cannot account for the halo of complete depolarization surrounding RCW 94. We rather account for this depolarization halo by requiring the electron density to be approximately constant across RCW 94, but to fall off rapidly beyond the boundaries of the source. This produces a sharp gradient in RM around the edges of the source, resulting in complete depolarization. The presence of significant CO emission at the same position and systemic velocity as for RCW 94 bact89 suggests that the H ii region is interacting with a molecular cloud. This possibility is supported by H i observations of the region, which show that RCW 94 is embedded in a shell of H i emission, which is further surrounded by a ring of decreased H i emission mdg+00b . McClure-Griffiths et al mdg+00b ; mgd+01 argue that this structure in H i confirms that RCW 94 is embedded in a molecular cloud, the shell of emission resulting from H${}_{2}$ molecules dissociated by the H ii region, and the surrounding region of reduced H i corresponding to regions of undisturbed molecular material. Simulations of H ii regions evolving within molecular clouds (rtf95 and references therein) show that for certain forms of the density profile within the parent cloud, the shock driven into the cloud by the embedded expanding H ii region can produce a halo of partially ionized material around the latter’s perimeter, which would produce the fall-off in $n_{e}$ required to produce the depolarization halo observed. Point Sources With the exception of one source known to be a pulsar, the polarized point sources in our field are presumably extragalactic, and their RMs thus probe the entire line-of-sight through the Galaxy. When combined with information from pulsar RMs, we can use these data to constrain the geometry of the overall Galactic magnetic field. So far we have compared the RMs in our test region to those expected for a bisymmetric spiral configuration, and have found that pitch angles in the lower end of the range allowed by pulsars ($p\sim-4.5^{\circ}$) are favored dom+01 . We are in the process of carrying out a more detailed study using the RMs of 163 background sources from the entire SGPS, in which we are comparing these measurements to the distributions expected for a wider variety of geometries and model parameters (Bizunok et al, in preparation). Conclusions The ATCA’s sensitivity, spatial resolution and spectral flexibility have allowed us to study linear polarization and Faraday rotation from the inner Galaxy in an unprecedented detail. Even though the test region we have considered covers less than 7% of the full survey, we have been able to identify a variety of distinct polarimetric phenomena, and have used these to map out both global and turbulent structures in the magneto-ionized ISM. We anticipate that our analysis of the full SGPS will result in a comprehensive study of magnetic fields and turbulence in the inner Galaxy. Acknowledgements. The Australia Telescope is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO. H$\alpha$ data were taken from the Southern H-Alpha Sky Survey Atlas (SHASSA), which is supported by the National Science Foundation. B.M.G. is supported by a Clay Fellowship awarded by the Harvard-Smithsonian Center for Astrophysics, while J.M.D. acknowledges the support of NSF grant AST-9732695 to the University of Minnesota. References (1) Bronfman, L., Alvarez, H., Cohen, R. S., and Thaddeus, P., 1989, ApJS, 71, 481 (2) Dickey, J. M., 1997, ApJ, 488, 258 (3) Dickey, J. M., Onken, C. A., McClure-Griffiths, N. M., Gaensler, B. M., Green, A. J., Haynes, R. F., and Wieringa, M. H., in Radio Polarization: A New Probe of the Galaxy, edited by T. L. 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First X-ray observations of Low-Power Compact Steep Spectrum Sources M.  Kunert-Bajraszewska${}^{1}$, A. Labiano,${}^{2,3}$, A. Siemiginowska${}^{3}$, M. Guainazzi,${}^{2}$ ${}^{1}$ Toruń Centre for Astronomy, Faculty of Physics, Astronomy and Informatics, NCU, Grudziacka 5, 87-100 Toruń, Poland ${}^{2}$European Space Astronomy Centre of ESA, PO Box 78, Villanueva de la Cañada, 28691, Madrid, Spain ${}^{3}$Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA () Abstract We report first X-ray Chandra observations of a sample of seven low luminosity compact (LLC) sources. They belong to a class of young compact steep spectrum (CSS) radio sources. Four of them have been detected, the other three have upper limit estimations for X-ray flux, one CSS galaxy is associated with an X-ray cluster. We have used the new observations together with the observational data for known strong CSS and gigahertz-peaked spectrum (GPS) objects and large scale FR Is and FR IIs to study the relation between morphology, X-ray properties and excitation modes in radio-loud AGNs. We found that: (1) The low power objects fit well to the already established X-ray - radio luminosity correlation for AGNs and occupy the space among, weaker in the X-rays, FR I objects. (2) The high excitation galaxies (HEG) and low excitation galaxies (LEG) occupy distinct locus in the radio/X-ray luminosity plane, notwithstanding their evolutionary stage. This is in agreement with the postulated different origin of the X-ray emission in these two group of objects. (3) We have tested the AGN evolution models by comparing the radio/X-ray luminosity ratio with the size of the sources, and indirectly, with their age. We conclude that the division for two different X-ray emission modes, namely originate in the base of the relativistic jet (FR Is) or in the accretion disk (FR IIs) is already present among the younger compact AGNs. (4) Finally, we found that the CSS sources are less obscured than the more compact GPSs in X-rays. However, the anti-correlation between X-ray column density and radio size does not hold for the whole sample of GPS and CSS objects. keywords: galaxies-active, galaxies-evolution, X-rays-galaxies ††pagerange: First X-ray observations of Low-Power Compact Steep Spectrum Sources–LABEL:lastpage††pubyear: 2013 1 Introduction We still know little about how radio galaxies are born and how they subsequently evolve, but it is generally accepted that the GHz Peaked Spectrum (GPS) and Compact Steep Spectrum (CSS) radio sources are young, smaller versions of the large-scale powerful radio sources (O’Dea et al., 1991; Fanti et al., 1990, 1995; Readhead et al., 1996a; O’Dea & Baum, 1997). Recently, the High Frequency Peakers have been added to the sequence, as possible progenitors of GPS sources (e.g., Orienti et al., 2007, and references therein). The GPS and CSS sources are powerful but compact radio sources whose spectra are generally simple and convex with peaks near 1 GHz and 100 MHz respectively. The GPS sources are contained within the extent of the optical narrow emission line region ($\lesssim 1$ kpc) while the CSS sources are contained within the host galaxy ($\lesssim 15$ kpc, see O’Dea, 1998, for a review). In the general scenario of the evolution of powerful radio-loud AGNs, GPS sources evolve into CSS sources and these into supergalactic-size FR I or FR II objects (Fanaroff &Riley, 1974). The dynamic evolution of the double-lobed radio sources, characterized by the total extent of the source, advance speed of the hotspots and the dependence of the density distribution of the interstellar and intergalactic medium along the way of the propagating jets and lobes, predicts the increase of the radio power with the linear size of the source in the GPS and CSS phase until they reach the 1-3 kpc size. Then the larger CSS objects should start to slowly decrease their luminosity but the sharp radio power decrease is visible only in the FR I and FR II phase of evolution (Begelman & Cioffi, 1989; De Young, 1993; Carvalho, 1994; Fanti et al., 1995; Begelman, 1996; De Young, 1997; Kaiser & Alexander, 1997b; Carvalho, 1998; Snellen et al., 2000; Kino & Kawakatu, 2005; Kawakatu & Kino, 2006; Kaiser & Best, 2007; Kawakatu et al., 2009a). Finally, after the cut off of the material supply to the central engine of the galaxy, the sources begin their fading phase. They can come back on the main evolutionary sequence after the re-ignition of the radio activity (e.g., Kozieł-Wierzbowska et al., 2012; Konar et al., 2012). However, population studies have drawn attention to the existence of far too many compact sources compared to the number of large-scale objects (O’Dea & Baum, 1997). It has been proposed then that some of the young radio-loud AGNs, namely the GPS and CSS sources, can be short-lived objects (Reynolds & Begelman, 1997; Czerny et al., 2009; Kawakatu et al., 2009b; Kunert-Bajraszewska et al., 2010) and that not one but a few evolutionary paths exist (Marecki et al., 2003; Kunert-Bajraszewska et al., 2010; An & Baan, 2012). Detection of several candidates for dying compact sources (Giroletti et al., 2005; Kunert-Bajraszewska et al., 2006, 2010; Orienti et al., 2010) supports this view. The determining factors for the further evolution of compact radio objects could occur at sub-galactic (or even nuclear) scales, or they could be related to the radio jet-ISM interactions and evolution. Our previous studies suggest that the evolutionary track could be related to the interaction, strength of the radio source, and excitation levels of the ionized gas (Kunert-Bajraszewska et al., 2010; Kunert-Bajraszewska & Labiano, 2010), instead of the radio morphology of the young radio source. The characteristics (size, radio power and young age) of GPS and CSS sources make them excellent probes of interaction (and therefore evolution) of radio sources. Furthermore, they have not completely broken through the ISM, so these interactions are expected to be more important than in the larger sources. Observations of UV, HI and, especially, of the ionized gas in GPS and CSS sources suggest the presence of such interactions (Labiano, 2008a; Labiano et al., 2008b; Holt et al., 2006; Labiano et al., 2005; Axon et al., 2000; de Vries et al., 1999, 1997; Gelderman & Whittle, 1994). Additional clues on the evolution of compact GPS and CSS sources may come from the X-ray band, but still little is known about the nature of the X-ray emission in these young sources. Theoretical models predict strong X-ray emission from young radio sources, due to the recent triggering of the nuclear activity and/or the expansion through the ISM (e.g., Siemiginowska et al., 2012; Siemiginowska, 2009, and references therein). The Chandra and XMM-Newton observations of GPS and CSS objects made so far have focused on sources with high radio emission (e.g., Siemiginowska et al., 2008; Vink et al., 2006; Guainazzi et al., 2004, 2006; Tengstrand et al., 2009). These sources, when included in the L${}_{2-10}$ keV versus L${}_{5GHz}$ diagram, group in the region occupied by powerful FR II sources (Tengstrand et al., 2009). Therefore, the location of GPS and CSS sources in the radio to X-ray luminosity diagram is consistent with them being powered by accretion, and therefore evolving onto a track of constant X-ray, accretion-driven luminosity to FR IIs, as well as with the correlation between radio and X-ray luminosity observed in FR Is, which would point to a common origin for the emission in these two bands. In this paper, we present the first X-ray observations of low power radio sources, starting to fill the gap in the L${}_{2-10}$ keV versus L${}_{5GHz}$ diagram, and shedding some light on the origin of high-energy emission of young radio sources and their evolution. 2 The sample The current sample consists of 7 sources (0810+077, 0907+049, 0942+355, 1321+045, 1542-390, 1558+536, 1624+049) taken from the Low Luminosity Compact sources sample (LLC, Kunert-Bajraszewska & Thomasson, 2009; Kunert-Bajraszewska et al., 2010). The LLC consists of 44 nearby (z$<$0.9) sources, selected from the final release of FIRST (White et al., 1997), and the GB6 (Gregory et al., 1996) and SDSS surveys, and observed with MERLIN at L-band and C-band. The main selection criterion for the LLC was the luminosity limit: L${}_{5GHz}<5\times 10^{42}$ erg s${}^{-1}$. The radio and optical properties of the LLC were discussed and analyzed in Kunert-Bajraszewska et al. (2010) and Kunert-Bajraszewska & Labiano (2010) respectively. The 7 current sources form the so called pilot sample and were selected to represent different stages of the radio source evolution within the ISM: weak or undetected radio core and strong lobes or breaking up radio lobes with bright radio core, and linear sizes ranging from 2 to 17 kpc . 3 X-ray observations and data reduction The sample was observed using Chandra ACIS-S3 with 1/8 subarray and standard pointings, with exposure times of $\sim$ 9500 seconds (see Table 1). The Chandra data were reduced using CIAO 4.5 (Fruscione et al., 2006) with the calibration files from CALDB 4.4.5. All our sources are contained within the FWHM of the PSF. We used a circular extraction region for each source, with radius $2\arcsec$, which also contains all the radio emission. The background regions consist of four circular regions of radius 10$\arcsec$ around the source. The CIAO default tools were used to extract the spectra and associated rmf and arf files. The total counts detected for each source are listed in Table 1. We used Sherpa (Freeman et al., 2001) to fit the spectra, using an absorbed power-law in the 0.5-7 keV energy range: $$\hfill N(E)=e^{-N_{H}^{Gal}\sigma(E)}\times e^{-N_{H}^{z_{obs}}\sigma[E(1+z_{% obs})]}\times AE^{-\Gamma}\hfill$$ (1) where N(E) is in photons cm${}^{-2}$ s${}^{-1}$, A is the normalization at 1 keV, $\Gamma$ is the photon index of the power law, $\sigma$(E) and $\sigma$[E(1+ z${}_{obs}$)] are the absorption cross-sections (Morrison & McCammon, 1983; Wilms et al., 2000), and N${}_{H}^{Gal}$ and N${}_{H}^{z_{obs}}$ are the column densities of the Milky Way (Kalberla et al., 2005; Dickey & Lockman, 1990) and the source. The Galactic absorption was kept constant during fitting. The second absorption component is assumed to be intrinsic to the quasar and located at the redshift of the source. The model was applied to all sources. However, 0907+049, 1558+536 and 1624+049 do not have enough counts to produce a reasonable fit. The results are summarized in Table 1. We use H${}_{0}$=71, $\Omega_{M}=0.27,\Omega_{\Lambda}=0.73$ (Spergel et al., 2003) throughout the paper. 4 Discussion 4.1 The X-ray and radio morphology We have observed a pilot sample of Low Luminosity Compact (LLC) sources (7 out of 44 objects) with Chandra. Four of them have been detected, the other three have upper limit estimations for X-ray flux (see Table 1). One of the objects, 1321+045, appeared to be associated with an X-ray cluster and has been discussed in a separate paper (Kunert-Bajraszewska et al., 2013). The Chandra ACIS-S images of two of the sources discussed here with the largest number of X-ray photons are shown in Fig. 1. We also overlayed the radio MERLIN 1.6 GHz contours on the X-ray emission with the indications of radio components. 0810+077 is a quasar classified as LEG. Its radio morphology consist of three components: the weak central one (C) and two jets/lobes (E and W). The optical counterpart is coincident with the component C and we suggested this could be a radio core (Kunert-Bajraszewska et al., 2010). However, this is based on observations at only one radio frequency so it should be treated as tentative. The brightest part of the X-ray emission lies between the components C and W. The potential offset between the centroid of the X-ray emission and component C or W can be consistent with the astrometric uncertainty of Chandra. 0942+355 is a galaxy classified as HEG, larger than 0810+077 and with more complex radio structure. Its 1.6 GHz asymmetric radio morphology consist of three components: weak radio core (C) and two lobes (E and W). There are also 5 GHz observations of this source (Kunert-Bajraszewska et al., 2010) showing only emission from the south-eastern radio lobe. In the case of 0942+355 the brightest part of the X-ray source is right in the center of the source, between the two jets. 1558+536 is a galaxy classified as LEG with diffuse, double-like morphology (Kunert-Bajraszewska et al., 2010). Only nine counts were detected in Chandra observations of this source and we did not produced an image of it. The radio and optical properties of the whole sample of LLC sources have been discussed and analyzed by us (Kunert-Bajraszewska et al., 2010; Kunert-Bajraszewska & Labiano, 2010). We suggested that they can represent a population of short-lived objects and undergo the CSS phase of activity many times before they become large scale FR I or FR II (Kunert-Bajraszewska et al., 2010; Kunert-Bajraszewska & Labiano, 2010). What is more, the evolution of the radio source seems to be independent from its radio morphology but rather determined by the properties of the central engine: strength, accretion mode, excitation level of the ionized gas. In some objects the surrounding environment could be also an important factor influencing the evolution (Cegłowski et al., 2013). Optically many of the LLC sources belong to the class of low excitation galaxies (LEGs), which are thought to be powered by the accretion of hot gas (Hardcastle et al., 2007; Buttiglione et al., 2010) and can be progenitors of large scale LEGs (Kunert-Bajraszewska & Labiano, 2010). 4.2 The optical-line emission Labiano (2008a) found that compact AGNs show a strong correlation between [O III]$\lambda$5007 line luminosity and size of the radio source suggesting a possible deceleration in the jet as it crosses the host ISM. However this correlation breaks up when including LLC sources, more specifically the compact LEGs (Kunert-Bajraszewska & Labiano, 2010). As we have already shown in the optical analysis of the whole sample of LLC sources the LLC HEGs show a $\sim 10$ times higher [O III] $\lambda$5007 luminosities than LLC LEGs. This could be caused by a stronger jet contribution to the ionization of the ISM in HEGs and/or indicates differences in the environment of HEG and LEG objects. In the pilot sample of LLC sources observed with Chandra we have found that two sources, 0810+077 and 0942+355, have the same radio and X-ray luminosities (Table 2). However, 0942+355 (HEG) has higher [OIII] emission than 0810+077 (LEG). If we compare the OII/OIII y OIII/Hb of the detections with photoionization and shock models (MAPPINGS, Allen et al., 2008), 0810+077 is consistent with 100% photoionization and 0942+355 with 80% photoionization and 20% shocks (Kunert-Bajraszewska & Labiano, 2010). 4.3 Can the central AGN power the emission line luminosity in the extended nebulae? We compared the number of ionizing photons produced by the nucleus of the source, with the number of photons needed to produce the observed emission line luminosity (see e.g. Wilson et al., 1988; Baum & Heckman, 1989; Axon et al., 2000; O’Dea et al., 2000). Assuming radiative recombination under case B conditions, the number of ionizing photons/s, $N_{\mathrm{H\beta}}$, needed to produce the observed H$\beta$ luminosity $L_{\mathrm{H\beta}}$ is: $$\hfill N_{\mathrm{H\beta}}=2.1\times 10^{52}(L_{\mathrm{H\beta}}/10^{40}% \mathrm{erg\,s}^{-1})\hfill$$ (2) We use the integrated [OIII]$\lambda 5007$ fluxes from Kunert-Bajraszewska & Labiano (2010) and scale using the typical ratio for the narrow line components in CSS sources (e.g. Gelderman & Whittle, 1994): $H\beta/[{O}{III}]~{}\lambda 5007=0.18\pm 0.02$. The number of photons/s in the continuum, between frequencies $\nu_{1}$ and $\nu_{2}$ is given by: $$\hfill N_{\mathrm{Nuc}}=4\pi D^{2}S_{\mathrm{0}}(\alpha h)^{-1}\ (\nu_{1}^{-% \alpha}-\nu_{2}^{-\alpha})\hfill$$ (3) where D is the luminosity distance, the flux density spectrum is given by $F_{\nu}=S_{0}\nu^{-\alpha}$ (we adopt $\alpha$=1, e.g. O’Dea et al., 2000) and h is Planck’s constant. We are only interested in the photons with enough energy to ionize Hydrogen, i.e. those between $\nu_{1}$ = 3.3x10${}^{15}$Hz (912Å or 13.6eV) and $\nu_{2}$ = 4.8x10${}^{17}$Hz (2 keV). For our spectral index, $\alpha$=1, higher frequencies do not add a significant number of photons. Note that this analysis is subject to the caveat that the continuum emission may not be emitted isotropically, and the extended nebulae may see a different luminosity than we do (e.g. Penston et al., 1990) The results are shown in Table 3. We find that the nucleus apparently produces enough ionizing photons to power the emission line luminosity in 0810+077 and 0942+355. This is consistent with the results from the comparison with the BPT diagrams and MAPPINGS models (Kunert-Bajraszewska & Labiano, 2010; Allen et al., 2008). 4.4 Radio/X-ray correlations The Chandra and XMM-Newton studies of GPS/CSS sources performed so far show they are strong X-ray emitters (Guainazzi et al., 2004, 2006; Vink et al., 2006; Siemiginowska et al., 2008; Tengstrand et al., 2009). The X-ray emission of CSS and GPS objects is probably a result of a recent triggering of the nuclear activity and can be characterized by an absorbed power law model with high ($>10^{22}~{}{\rm cm^{-2}}$) column densities (Guainazzi et al., 2006; Vink et al., 2006; Siemiginowska et al., 2008; Tengstrand et al., 2009). But there are also several detections of X-ray morphology in these compact objects. Extended hot 0.5-1 keV interstellar medium (ISM) has been detected in the case of two CSS sources, 3C303.1 (O’Dea et al., 2006) and 3C305 (Massaro et al., 2009), and it is interpreted as shock-heated environment gas. The X-ray jets in GPS sources and large scale X-ray emission associated with some of them have been reported by (Siemiginowska et al., 2008). These objects are classified as ’GPS sources with extended emission’ and discussed in the frame of the theory of intermittent radio activity (Stanghellini et al., 2005). However, the radio structures of GPS and CSS sources are much smaller than the spatial resolution of the current X-ray instruments in most cases, what prevents us from identification of the origin of their X-ray emission. There are several theoretical predictions of X-ray emission from evolving radio sources: i) as thermal emission emitted by the ISM of the host galaxy shock heated by the expanding radio structure (Heinz et al., 1998; O’Dea et al., 2006), or ii) that produced in the accretion disk’s hot corona (Guainazzi et al., 2004, 2006; Vink et al., 2006; Siemiginowska et al., 2008), and finally iii) as non-thermal radiation produced through IC scattering of the local thermal radiation fields off the lobe electron population (Stawarz et al., 2008; Ostorero et al., 2010) or by mini shells (Kino et al., 2013). Tengstrand et al. (2009) show that the radio versus X-ray luminosity plane can be a useful tool to derive constraints on the evolution of compact radio sources. Studies of compact radio AGN so far have been biased towards high-luminosity objects ($L_{5GHz}>10^{42}$ erg s${}^{-1}$). In this Section we extend these studies to the low-luminosity regime that our pilot Chandra study probes for the first time. Our goal is to compare the X-ray properties of different groups of radio objects, GPS, CSS and large-scale FR I and FR II sources as well as the X-ray properties of low and high power compact AGNs. For this purpose we have built the control sample of GPS/CSS sources (Siemiginowska et al., 2008; Tengstrand et al., 2009; Massaro et al., 2010, 2012) and FR I and FR II objects (Sambruna et al., 1999; Donato et al., 2004; Grandi et al., 2006; Evans et al., 2006; Balmaverde et al., 2006; Belsole et al., 2006; Hardcastle et al., 2006; Massaro et al., 2010, 2012) from results recently published in the literature. Our pilot sample of low luminosity CSS sources consist of only seven objects. That is why we have also included in it the low luminosity 3C305 described by Massaro et al. (2009). The total number of GPS/CSS sources is 40 objects, and FR I and FR IIs - 34 and 85 sources, respectively. The samples are, however, biased in terms of their redshift distribution. The GPS/CSS and FR II samples are well matched in the redshift, but the FR Is are generally at lower redshift. There are 6 GPS/CSS objects with redshift in the range $1>{\it z}<2$. All other sources from all groups have redshit ${\it z}<1$. For all plots presented in this paper we have used the total radio and X-ray luminosity in case of all group of sources. The reason for that is a lack of information of radio core fluxes of most of our compact GPS and CSS sources. Among the seven low power CSS objects only two have 5 GHz observations but without core detection (Kunert-Bajraszewska et al., 2010). Significant part of our sample of GPS/CSS sources is also unresolved in X-rays. The exceptions from the above-stated rule are a few GPS sources with extended structures (Stanghellini et al., 2005). In the case of them the radio and X-rays values used in this paper refers to their milliarcseconds VLBA structures as reported in Tengstrand et al. (2009) and Siemiginowska et al. (2008). 4.4.1 Radio/X-ray luminosity plane We have compared the X-ray luminosity of the sources from the pilot sample with their radio properties at 5 GHz and 365 MHz (Fig. 2). We have included also the control sample of GPS/CSS and FR I and FR II objects as described above. The low power objects fit well to the already established X-ray - radio luminosity correlation for AGNs and occupy the space among, weaker in the X-rays, FR I objects. This trend is visible on both plots, X-ray vs. 356 MHz and 5 GHz (Fig.2), and is independent of radio frequency. However, the 356 MHz radio luminosity versus X-ray luminosity plot shows larger scatter among observable data than in the case of 5 GHz luminosity. This is caused by the fact that the X-ray emission is mostly associated with the compact central regions of AGNs while the low frequency flux density is dominated by the extended radio structures. As has been also shown by Hardcastle et al. (1999) much of the dispersion in 5 GHz luminosity originate in beaming. Future X-ray observations of the whole sample of LLC sources would give us a definitive information about their place on the radio/X-ray luminosity plane. We have then plotted all groups of AGNs on the 5 GHz/X-ray luminosity plane (Fig. 3) with a division for high excitation galaxies (HEG) and low excitation galaxies (LEG). We took the optical identification from Buttiglione et al. (2010) in the case of FR I and FR II and indicated them as LEG and HEG/BLO. According to Buttiglione et al. (2010) the broad line objects (BLO) can be considered as members of the HEG class. Identifications of GPS/CSS objects were taken from (Kunert-Bajraszewska & Labiano, 2010) (see also Table 2 in this paper) and Table A1. We have only four LEGs among the GPS/CSS class and actually all of them have been classified as CSS sources. HEGs are found among strong GPS and CSS objects. The HEG/LEG plot confirms what we have previously found in Kunert-Bajraszewska & Labiano (2010). The HEG and LEG AGNs group in two different parts of the plot. A Pearson correlation analysis applied to both sub-samples revealed a significant X-ray/radio correlation (Table 4). In the radio versus X-ray luminosity plane (Fig. 2), objects with a different morphology are aligned along the same correlation, with an increasing fraction of large-scale FR II morphologies at higher luminosities. Compact sources are well aligned along this correlation, with weak CSS (strong CSS/GPS) closer to the parameter space occupied by FR I (FR II). However, the ionization mechanism seems to discriminate more neatly radio sources in this plane. Low versus high ionization sources occupy distinct locus in this plane, notwithstanding their evolutionary stage. This evidence agrees with a scenario whereby the X-ray emission in large-scale HEG sources is dominated by spectral components due to (obscured) accretion as opposed to LEG objects where the X-ray emission should be dominated by non-thermal synchrotron jets (Hardcastle et al., 2009; Antonucci, 2012; Son at al., 2012). As seen on the radio/X-ray luminosity diagram (Fig. 3), there are two branches, each being driven by different excitation mode and each containing compact sources. The FR morphology, as well as the GPS and CSS division, seems to be independent on the excitation modes (Buttiglione et al., 2010; Kunert-Bajraszewska & Labiano, 2010; Gendre et al., 2013). 4.4.2 Radio/X-ray luminosity ratio Another test for the AGN evolution models is a comparison of radio to the X-ray luminosity ratio with the size of the sources, and indirectly, with their age (Fig. 4). The long-term evolution of extragalactic radio sources have been investigated by a number of authors in different ways: i) as a variation of the radio power versus the total linear size (O’Dea & Baum, 1997; Kunert-Bajraszewska et al., 2010; An & Baan, 2012), or ii) dynamic evolution of FR II-like double radio sources characterized by the advance speed of the hot spots, total extent of the source and depending on the density distribution in the host galaxy along the path of the jets and lobes (Begelman & Cioffi, 1989; Begelman, 1996; Fanti et al., 1995; Kaiser & Alexander, 1997b; Kino & Kawakatu, 2005; Kawakatu & Kino, 2006; Kaiser & Best, 2007; Kawakatu et al., 2009a). The radio luminosity of the sources evolves through phases governed by the dominant energy-loss mechanism of the radiating, relativistic electrons. From the onset of the radio activity, the GPS/Compact Symmetric Object (CSO) stage, the radio power of the sources increases with time and the source size. The increase rate of radio power diminishes in the transition region (1-3 kpc from the center of the host galaxy) where the balance between adiabatic losses and synchrotron losses have been achieved. After this short period the radio power of CSS sources starts to slowly decreases with the source size. The sharp decrease in the radio power versus the total extent of the source occurs only in the large FR I and FR II objects with the FR Is being below the luminosity threshold, $L_{178MHz}\sim$ $10^{25.5}$ W H$z^{-1}$ $sr^{-1}$, on the radio power/linear size plane. If the X-ray emission in radio-loud AGNs is due to accretion only, the evolution of the radio and X-ray wavebands could be totally decoupled and the radio/X-ray luminosity ratio should reproduced the radio power evolution with the linear size of the source. The Fig. 4 shows that the above assumption is not true not only in the case of large FR Is and FR IIs but probably also in the case of young GPS and CCS sources. The less radio powerful FR Is have the radio/X-ray luminosity ratio higher than many FR IIs what may imply higher X-ray luminosity decrease with radio power in FR Is than FR IIs. This can be explained by the idea that the X-ray emission in FR Is originates from the base of a relativistic jet (Evans et al., 2006) and are thought to be synchrotron emission (Sambruna et al., 2004; Worrall et al., 2009). However, the X-ray emission of the FR IIs comes mostly from the obscured X-ray component probably associated with the accretion and in less part from the relativistic jet produced in an inverse Compton process (Balmaverde et al., 2006; Belsole et al., 2006; Evans et al., 2006). When incorporating a different division among large scale objects we notice that, on average, the low excitation radio galaxies (LERG) and narrow line radio galaxies (NLRG) have higher radio to X-ray luminosity ratio than quasars (Q) and broad line radio galaxies (BLRG). According to Hardcastle et al. (2009) the common correlation of FR II NLRGs, LERGs and FR Is indicates that the X-ray and radio emission comes from the same jet-related component. The interpretation of the place of GPS and CSS sources on the radio/X-ray luminosity ratio versus linear size plane is even more difficult because of the large scatter of the observable values. As we have already mentioned the beaming can disrupt the radio/X-ray correlations. However, at least in the case of compact steep spectrum objects this effect should be small (Wu et al., 2013). We then suggest that what we observe in the group of young GPS and CSS sources is a mix of two different types of X-ray/radio relation. We conclude that at some radio power level the compact AGNs starts to resemble the FR Is, where the X-ray emission is a synchrotron type associated with the jet. It has been already proposed by the SED modelling of two strong CSS objects that their X-ray emission can be a sum of X-ray emission from the accretion disk and non-thermal X-ray emission from the parsec scale radio jet (Kunert-Bajraszewska et al., 2009; Migliori et al., 2012). Recently, Stawarz et al. (2008) and Ostorero et al. (2010) have discussed an alternative evolutionary model for GPS sources, which predicts the dependency of the broadband Spectral Energy Distribution on the source linear size. In their model high-energy emission is produced by upscattering of various photon fields by the lobes’ electrons of the $<$1 kpc size radio source. This process can cause a decrease of the X-ray to radio luminosity ratio by 1-2 orders of magnitude when the GPS source size increases. However, with the data set gathered in this paper we cannot conclude any correlation between the radio/X-ray luminosity ratio and linear size of the sources. 4.4.3 $N_{H}$ - linear size relation Finally, we have drawn a relation between the measured column density and the total extent of the radio source for GPS and CSS sources (Fig. 5). It has been already reported (Pihlström et al., 2003; Vermeulen et al., 2003) that the small sources ($<$0.5 kpc) tend to have larger H I column density than larger sources ($>$0.5 kpc) which indicates that GPS/CSO objects evolve in a disk distribution of gas with a power-law radial density dependence. The same explanation could lay behind the (tentative) anti-correlation between X-ray column density and radio size in GPS galaxies (Tengstrand et al., 2009). Ostorero et al. (2010) reported a positive correlation between the radio and X-ray hydrogen column densities what can points toward the cospatiality of the radio and X-ray emission regions. We extended the discussion about the relation of the ${\rm N_{H}}$ versus the linear size to the CSS sources. We have noticed that the ${\rm N_{H}}$ value of the CSS sources is on average lower than that of GPS objects. However, the correlation between the X-ray column density and radio size does not hold when including larger CSS objects in the sample of small and young AGNs. 5 Summary In this paper we presented the X-ray Chandra observations of a pilot sample of low luminosity compact steep spectrum (CSS) sources. Four of them have been detected, the other three have upper limits estimations for the X-ray flux. Only for two CSS objects we were able to estimate the X-ray column density which is of the order of ${\rm 10^{21}\,cm^{-2}}$. We then expanded the sample of compact AGNs with other GPS/CSS sources with X-ray detections found in the literature and used it, together with a sample of large FR I and FR II sources, to determine the nature of the relation between morphology, X-ray properties and excitation modes in radio-loud AGNs. We found the following results: $\bullet$ We have compared the X-ray luminosity of the radio sources from all above mentioned groups with their radio properties. The large-scale FR II sources and strong GPS and CSS objects settle at higher X-ray and radio luminosities. While the low power CSSs occupy the space among, weaker in the X-rays, FR I objects. This trend is visible independently of radio frequency. $\bullet$ The HEG and LEG sources occupy distinct locus in the radio/X-ray luminosity plane, notwithstanding their evolutionary stage. This is in agreement with the postulated different origin of the X-ray emission in low and high ionization objects. Compact sources can be found in both excitation modes driven branches. $\bullet$ The less radio powerful FR Is have higher radio/X-ray luminosity ratio than many FR IIs what may imply higher X-ray luminosity decrease with radio power in FR Is than FR IIs. This is in agreement with the previous findings saying that the X-ray emission in FR Is originates from the base of a relativistic jet while the X-ray emission of FR IIs has accretion origin. The same can be true in the case of smaller radio AGNs, namely the GPS and CSS sources. The result of this study hints toward the fact that below some radio power level the compact GPS and CSS sources start to resemble the FR Is, or to be more specific, the LERG and NLRG objects. $\bullet$ The X-ray hydrogen column density of the CSS sources is on average lower than that of GPS objects. But the correlation between the X-ray column density and radio size does not hold for the whole sample of GPS and CSS objects. Acknowledgements AL wishes to thank CfA for their hospitality and support during the visit. This research has made use of data obtained by the Chandra X-ray Observatory, and Chandra X-ray Center (CXC) in the application packages CIAO, ChIPS, and Sherpa. This research is funded in part by (NASA) contract NAS8-03060. Partial support for this work was provided by the Chandra grants GO1-12124X and GO1-12145X. This research has made use of NASA’s Astrophysics Data System Bibliographic Services and of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This publication makes use of data products from the SDSS. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. 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Dimension Splitting and a Long Time-Step Multi-Dimensional Scheme for Atmospheric Transport Yumeng Chen    Hilary Weller a    \corrauth Stephen Pring and James Shaw b c b \affilnumaUniversität Hamburg \affilnumbMeteorology, University of Reading \affilnumcMet Office, UK Abstract Dimensionally split advection schemes are attractive for atmospheric modelling due to their efficiency and accuracy in each spatial dimension. Accurate long time steps can be achieved without significant cost using the flux-form semi-Lagrangian technique. The dimensionally split scheme used in this paper is constructed from the one-dimensional Piecewise Parabolic Method and extended to two dimensions using COSMIC splitting. The dimensionally split scheme is compared with a genuinely multi-dimensional, method of lines scheme with implicit time-stepping which is stable for very large Courant numbers. Two-dimensional advection test cases on Cartesian planes are proposed that avoid the complexities of a spherical domain or multi-panel meshes. These are solid body rotation, horizontal advection over orography and deformational flow. The test cases use distorted meshes either to represent sloping terrain or to mimic the distortions of a cubed sphere. Such mesh distortions are expected to accentuate the errors associated with dimension splitting, however, the dimensionally split scheme is very accurate on orthogonal meshes and accuracy decreases only a little in the presence of large mesh distortions. The dimensionally split scheme also loses some accuracy when long time-steps are used. The multi-dimensional scheme is almost entirely insensitive to mesh distortions and asymptotes to second-order accuracy at high resolution. As is expected for implicit time-stepping, phase errors occur when using long time-steps but the spatially well resolved features are advected at the correct speed and the multi-dimensional scheme is always stable. A naive estimate of computational cost (number of multiplies) reveals that the implicit scheme is the most expensive, particularly for large Courant numbers. If the multi-dimensional scheme is used instead with explicit time-stepping, the Courant number is restricted to less than one but the cost becomes similar to the dimensionally split scheme. keywords: Multi-dimensional; advection; stability; accuracy; long time-steps \runningheads Y. Chen, H. Weller, S. Pring and J. ShawDimension splitting for Advection \corraddr E-mail: <h.weller@reading.ac.uk> 1 Introduction Weather and climate models are being developed on quasi-uniform meshes in order to better exploit modern computers (eg WTC12; Lauritzen et al., 2014; ST12; Skamarock and Gassmann, 2011; K.K. et al., 2015) and so accurate and efficient transport (or advection) schemes on non-orthogonal meshes are required. There is an abundance of desirable properties of advection schemes, including: 1. Inherent local conservation of the advected quantity. 2. Stability in the presence of large Courant numbers. 3. Accuracy in the presence of large Courant numbers. 4. High order-accuracy. 5. Low computational cost, good parallel scaling and multi-tracer efficiency. 6. Low phase and dispersion errors (advection of all wavenumbers of the advected quantity at close to the correct speed). 7. Low diffusion errors (maintaining amplitude of all wavenumbers of the advected quantity). 8. Boundedness, monotonicity, positivity and maintaining correlations between multiple advected tracers. Four (important) properties are listed together in the final item because they will not be addressed here. In this paper, we address the issue of conservative, accurate, efficient advection schemes for logically rectangular, non-orthogonal meshes which are stable in the presence of large Courant numbers. These schemes would be particularly relevant for cubed-sphere meshes and for terrain following meshes. Another novel aspect of this paper is that, for simplicity, we test advection schemes entirely on planar meshes rather than on the sphere, proposing test cases to challenge advection schemes on non-orthogonal meshes without the need to implement meshes in spherical geometry. Dimensionally split schemes (operating separately in each spatial dimension) are attractive for atmospheric modelling due to their efficiency and high accuracy in each spatial dimension (eg. Lin and Rood, 1996a; Leonard et al., 1996a; Brassington and Sanderson, 1999; Putman and Lin, 2007; K.K. et al., 2015). Inherent conservation is guaranteed by using the flux-form semi-Lagrangian (FFSL or forward in time) technique (eg. Colella and Woodward, 1984a) which integrates the dependent variable over a swept distance upstream of every face in order to calculate fluxes in and out of cells. Accurate long time steps can be achieved without significant cost by calculating cumulative mass fluxes along the domain (LLM95). This is done without remapping which can be an expensive procedure, finding a conservative map between fields on overlapping meshes. One-dimensional schemes can be used with operator splitting to create dimensionally split, second-order accurate schemes (eg Leonard et al., 1996a) on logically rectangular, multidimensional meshes. Dimensionally split schemes have been found to give good accuracy on non-orthogonal meshes such as the cubed-sphere (Putman and Lin, 2007; K.K. et al., 2015) with special treatment over cube edges. Putman and Lin (2007) use the average of two one-sided schemes at cube edges whereas K.K. et al. (2015) create ghost cells outside each cube panel boundary. Without this special treatment, dimensionally split schemes do not account properly for mesh distortion and errors may occur. Accuracy between second and fourth order was found in practice on a range of test cases on the cubed-sphere by K.K. et al. (2015). The same problems occur when using dimensionally split schemes over orography since terrain following layers become non-orthogonal over orography and special treatment cannot be used everywhere where there is orography. A common solution has been to make the terrain following layers as smooth as possible, reducing non-orthogonality (eg SLF+02) or to use floating Lagrangian vertical co-ordinates (Lin04). Dimension splitting may account for some of the errors over orography reported by KUJ14 although they do not cite this as a reason for errors. Dimension splitting errors on distorted meshes can be eliminated by using genuinely multi-dimensional advection schemes. These can be either FFSL (ie swept area, eg. Lashley, 2002; Lipscomb and Ringler, 2005; Miura, 2007; Thuburn et al., 2014), method of lines (MOL, discretising space and time separetely, eg. Weller et al., 2009; Skamarock and Gassmann, 2011; K.K. et al., 2015; SWMD1x) or conservative semi-Lagrangian (with conservative re-mapping, eg Iske and Kaser, 2004; Lauritzen et al., 2010; Zerroukat et al., 2004). The FFSL and MOL multi-dimensional schemes have not previously been extended to work with Courant numbers significantly larger than one. FFSL multi-dimensional schemes could be extended to handle large time-steps by integrating the upstream swept volume over a large upstream volume, interacting with a large number of upstream cells. However the cost would be proportional to the time-step since the larger the Courant number, the more cells the upstream swept volume would need to overlap with. This technique therefore offers no advantage over using a smaller time-step. MOL multi-dimensional schemes can be extended to work with Courant numbers larger than one by using implicit time-stepping. This will increase the computational cost per tracer advected since the solution of a matrix equation would be needed for every advected tracer. Two other disadvantages of implicit time-stepping are the large phase errors when long time-steps are used (eg DB12; LWW14) and the difficulty of achieving monotonicity. This is in contrast to semi-Lagrangian or FFSL schemes which maintain accuracy with long time-steps (Pur76; PS84; LLM95) although monotonocity with long time-steps is still challenging (Bott10). Conservative semi-Lagrangian naturally extends to long time-steps but the conservative remapping is complicated and expensive, particularly on non-rectangular meshes and will not be investigated here. Lauritzen et al. (2014) described how the FFSL technique with a long time-step can be made equivalent to the conservative semi-Lagrangian. It is therefore not clear what approach should be taken for achieving long time-steps when advecting multiple tracers on distorted meshes. In this paper, we show the effect of dimension splitting errors using a FFSL dimensionally split scheme on a number of test-cases which use distorted meshes and compare with a genuinely multi-dimensional implicit MOL scheme using large and small Courant numbers. The theoretical properties of dimensionally split advection schemes are often tested on uniform, orthogonal meshes (eg Leonard et al., 1996a). On the cubed-sphere, special treatment is needed for the cube edges (eg Lin and Rood, 1996a; K.K. et al., 2015). Developing a transport scheme to the extent that it can be used on a multi-panel cubed-sphere with special treatment of cube edges is a considerable undertaking. Hence there is a need for more challenging advection test cases which are simpler to implement, without the need for spherical meshes. We therefore propose some modifications of existing test cases to use distorted meshes, or distorted co-ordinate systems, on a logically rectangular, two-dimensional plane. The long time-step permitting, dimensionally split scheme and the long time-step permitting multi-dimensional scheme are defined in section 2. In section 3 we present results of three advection test cases on distorted meshes in two-dimensional planes using Courant numbers above and below one. These are the solid body rotation test case of Leonard et al. (1996a) modified to use a mesh (or co-ordinate system) with distortions similar to a cubed-sphere (section 3.1), the horizontal advection test case over orography (SLF+02), examining sensitivity to time-step, resolution and mountain height, all on the maximally distorted basic terrain following mesh (section 3.2) and a modification of the deformational flow test case of Lauritzen et al. (2012b) for a periodic rectangular plane (section 3.3). Some estimates are made of computational cost in section 3.4 and final conclusions are drawn in section 4. 2 Transport Schemes We present two conservative advection schemes suitable for long time-steps (stable for Courant numbers significantly larger than one) for solving the linear advection equation: $$\frac{\partial\phi}{\partial t}+\nabla\cdot\mathbf{u}\phi=0$$ (1) where the dependent variable $\phi$ is advected by velocity field $\mathbf{u}(\mathbf{x},t)$. The dimensionally split scheme is the piecewise parabolic method (PPM Colella and Woodward, 1984a) which uses the flux-form semi-Lagrangian approach extended to long time-steps following LLM95 with COSMIC splitting (Leonard et al., 1996a) to extend PPM to two dimensions. The multi-dimensional scheme uses the method of lines approach (treating space and time independently). The second-order accurate spatial discretistaion of Weller and Shahrokhi (2014) is combined with implicit, Crank-Nicholson time-stepping to allow long time-steps. Neither scheme has monotonicity or positivity preservation. The code for dimensionally splitting scheme is available at https://github.com/yumengch/COSMIC-splitting. The multi-dimensional scheme is implemented using OpenFOAM16 and is available at https://github.com/AtmosFOAM/. 2.1 One-dimensional PPM with Long Time-steps We describe a long time-step version of PPM for solving the one-dimensional advection equation: $$\frac{\partial\phi}{\partial t}+\frac{\partial u\phi}{\partial x}=0.$$ (2) Colella and Woodward (1984a) defined PPM with monotonicity constraints and for variable resolution but for simplicity (and for comparison with the multi-dimensional scheme) we will define PPM without monotonicity constraints and for a fixed resolution, $\Delta x$, and time-step, $\Delta t$. With these restrictions, PPM should be fourth-order accurate in one dimension for vanishing time-step. We define the dependent variable, $\phi_{i}^{(n)}$, to be the mean value of $\phi$ in cell $i$ at time-level $n$ where $x_{i}=i\Delta x$ and $t=n\Delta t$. Since PPM is a flux-form finite volume method, $\phi_{i}^{(n)}$ is updated using: $$\phi_{i}^{(n+1)}=\phi_{i}^{(n)}+X_{C}\left(\phi\right)=\phi_{i}^{(n)}-\frac{u_% {i+1/2}\phi_{i+1/2}-u_{i-1/2}\phi_{i-1/2}}{\Delta x}$$ (3) where $X_{C}\left(\phi\right)$ is the conservative advection operator for $\phi$ in the $x$ direction. The fluxes, $u_{i\pm 1/2}\phi_{i\pm 1/2}$, are found by integrating a piecewise polynomial, $p$, along the distance travelled in each time-step upwind of cell boundary $x_{i\pm 1/2}$. The polynomial is defined in each cell, $i$, such that: $$\phi_{i}=\frac{1}{\Delta x}\int_{x_{i-1/2}}^{x_{i+1/2}}p_{i}(x)\ dx$$ (4) by $$p_{i}(x)=p_{i-1/2}+\xi\left(p_{i+1/2}-p_{i-1/2}+\left(1-\xi\right)6\left(\phi_% {i}-\frac{1}{2}\left(p_{i-1/2}+p_{i+1/2}\right)\right)\right)$$ (5) where $\xi=(x-x_{i-1/2})/\Delta x$ and $$p_{i+1/2}=\frac{7}{12}(\phi_{i}+\phi_{i+1})-\frac{1}{12}(\phi_{i+2}+\phi_{i-1})$$ (6) In order to cope with long time-steps, we follow LLM95 and divide the Courant number into a signed integer part, $c_{N}$, and a remainder, $c_{r}$. The departure point of location $x_{i-\frac{1}{2}}$ is thus computed as $x_{d}=x_{i-\frac{1}{2}}-u_{i-\frac{1}{2}}\Delta t$ and the departure cell, for cell edge $i-\frac{1}{2}$, is $i_{d}=i-c_{N}-1$ for $u_{i-\frac{1}{2}}>0$ and $i_{d}=i-c_{N}$ for $u_{i-\frac{1}{2}}<0$. Then for $u_{i-\frac{1}{2}}>0$ the flux through $x_{i-1/2}$ between times $n\Delta t$ and $(n+1)\Delta t$ is: $$u_{i-\frac{1}{2}}\phi_{i-\frac{1}{2}}=\frac{1}{\Delta t}\int_{x_{d}}^{x_{i-1/2% }}p(x)\ dx=\frac{1}{\Delta t}\left(M_{i-1/2}-M_{i-c_{N}-1/2}+\int_{x_{i-1/2-c_% {N}}-c_{r}\Delta x}^{x_{i-1/2-c_{N}}}p(x)\ dx\right)$$ (7) where $M_{i-1/2}$ is the cumulative mass from the start point to position $x_{i-1/2}$: $$M_{i-1/2}=\sum_{k<i}\Delta x\phi_{k}.$$ (8) This departure point calculation assumes that the velocity is uniform on the computational mesh which has a first-order error which could be particularly damaging for long Courant numbers, when the wrong departure cell could be found. The velocity is derived from a stream function and the Jacobian of the co-ordinate transform: $$\left(\begin{array}[]{c}u\\ v\end{array}\right)=J\left(\begin{array}[]{c}\;\Psi_{y}\\ -\Psi_{x}\end{array}\right)$$ (9) For stability, the time-step is restricted by the deformational Courant number: $$c_{d}=\Delta t\max\left(\biggl{|}\frac{\partial u}{\partial x}\biggr{|},\biggl% {|}\frac{\partial u}{\partial y}\biggr{|},\biggl{|}\frac{\partial u}{\partial z% }\biggr{|},\biggl{|}\frac{\partial v}{\partial x}\biggr{|},\biggl{|}\frac{% \partial v}{\partial y}\biggr{|},\biggl{|}\frac{\partial v}{\partial z}\biggr{% |},\biggl{|}\frac{\partial w}{\partial x}\biggr{|},\biggl{|}\frac{\partial w}{% \partial y}\biggr{|},\biggl{|}\frac{\partial w}{\partial z}\biggr{|}\right)$$ (10) (PS84) such that $c_{d}\leq 1$. 2.2 COSMIC Splitting COSMIC operator splitting (Leonard et al., 1996a) allows single stage, one-dimensional schemes such as PPM to be used stably in two or more dimensions whilst retaining conservation, constancy preservation and second-order accuracy (on orthogonal meshes). As we are now considering two spatial dimensions, we define $\phi_{ij}$, $u_{ij}$ and $v_{ij}$, the values of $\phi$ and the velocity components, $u$ and $v$ in cell $(i,j)$ where $x=i\Delta x$ and $y=j\Delta y$. COSMIC splitting uses both advective and conservative advection operators in the $x$ and $y$ directions: $$\displaystyle X_{C}(\phi)=-\frac{1}{\Delta x}\left(u_{e}\phi_{e}-u_{w}\phi_{w}\right)$$ $$\displaystyle Y_{C}(\phi)=-\frac{1}{\Delta y}\left(v_{n}\phi_{n}-v_{s}\phi_{s}\right)$$ (11) $$\displaystyle X_{A}(\phi)=X_{c}(\phi)+\frac{\phi_{ij}}{\Delta x}\left(u_{e}-u_% {w}\right)$$ $$\displaystyle Y_{A}(\phi)=Y_{c}(\phi)+\frac{\phi_{ij}}{\Delta y}\left(v_{n}-v_% {s}\right)$$ (12) where $\phi_{n}=\phi_{i,j+1/2}$, $\phi_{s}=\phi_{i,j-1/2}$, $\phi_{e}=\phi_{i+1/2,j}$ , $\phi_{w}=\phi_{i-1/2,j}$, $v_{n}=v_{i,j+1/2}$, $v_{s}=v_{i,j-1/2}$, $u_{e}=u_{i+1/2,j}$ and $u_{w}=u_{i-1/2,j}$ are the values of $\phi$, $u$ and $v$ at the cell boundaries. If COSMIC is being used to extend PPM to two spatial dimensions then $\phi_{n,s,e,w}$ are calculated from equation 7. Assuming C-grid staggering, $v_{n}$, $v_{s}$, $u_{e}$ and $u_{w}$ are dependent variables. Instead of using cell centered velocity (Lin and Rood, 1996a) or upwind velocity (Leonard et al., 1996a), the advective operators are calculated in a similar manner to Lin04. Mesh distortions can be included in the advection equation with a co-ordinate transform with Jacobian $J$: $$\frac{\partial|J|^{-1}\phi}{\partial t}+\frac{\partial|J|^{-1}u\phi}{\partial x% }+\frac{\partial|J|^{-1}v\phi}{\partial y}=0.$$ (13) The operators $X_{c}$, $Y_{c}$ $X_{A}$ and $Y_{A}$ are then: $$\displaystyle X_{C}(\phi)=-\frac{1}{\Delta x}\left(|J|_{e}^{-1}u_{e}\phi_{e}-|% J|_{w}^{-1}u_{w}\phi_{w}\right)$$ $$\displaystyle Y_{C}(\phi)=-\frac{1}{\Delta y}\left(|J|_{n}^{-1}v_{n}\phi_{n}-|% J|_{s}^{-1}v_{s}\phi_{s}\right)$$ (14) $$\displaystyle X_{A}(\phi)=X_{c}(\phi)+\frac{\phi_{ij}}{\Delta x}\left(|J|_{e}^% {-1}u_{e}-|J|_{w}^{-1}u_{w}\right)$$ $$\displaystyle Y_{A}(\phi)=Y_{c}(\phi)+\frac{\phi_{ij}}{\Delta y}\left(|J|_{n}^% {-1}v_{n}-|J|_{s}^{-1}v_{s}\right)$$ (15) which are combined to update $\phi_{ij}$ in each cell by: $$\phi_{ij}^{(n+1)}=\phi_{ij}^{n}+|J|_{ij}X_{C}\left(\phi_{ij}^{(n)}+\frac{|J|_{% ij}}{2}Y_{A}\left(\phi_{ij}^{(n)}\right)\right)+|J|_{ij}Y_{C}\left(\phi_{ij}^{% (n)}+\frac{|J|_{ij}}{2}X_{A}\left(\phi_{ij}^{(n)}\right)\right).$$ (16) where $|J|_{ij}$ is the determinant of Jacobian at the center of cell $(i,j)$, and $\Delta x$ and $\Delta y$ give the cell size in uniform computational domain. 2.3 Multi-dimensional Method of Lines (MOL) Scheme The MOL scheme uses the finite volume method for arbitrary meshes and is implemented in (OpenFOAM16). This uses a cubic upwind spatial discretisation (Weller and Shahrokhi, 2014; SWMD1x) combined with Crank-Nicholson in time. Although the interpolation uses a cubic polynomial, cell centre values are approximated as cell average values and face centre values are approximated as face average values so the method is limited to 2nd order accuracy in space. The advection scheme uses Gauss’s divergence theorem to approximate the divergence term of the advection equation: $$\nabla\cdot\mathbf{u}\phi\approx\frac{1}{V}\sum_{f\in c}\phi_{f}\mathbf{u}_{f}% \cdot\mathbf{S}_{f}$$ (17) where $V$ is the cell volume, the summation is over all faces, $f$, of cell $c$, $\phi_{f}$ is the value of the dependent variable, $\phi$ interpolated onto face $f$, $\mathbf{u}_{f}$ is the velocity at face $f$ and $\mathbf{S}_{f}$ is the vector normal to face $f$ with magnitude equal to the area of face $f$ (ie the face area vector). The advection scheme uses a fit to a 2d (or 3d) polynomial using an upwind-biased stencil of cells (fig 1) in order to interpolate from known cell values onto a face. In 2d, the cubic polynomial is: $$\phi=a+bx+cy+dx^{2}+exy+fy^{2}+gx^{3}+hx^{2}y+ixy^{2}$$ (18) omitting the $y^{3}$ term, where $x$ is the direction normal to a cell face and $y$ is perpendicular to $x$. (The $y^{3}$ term is omitted because it cannot be set with a stencil that is narrow in the direction of the flow.) Coefficients $a$ to $i$ are set from a least squares fit to the cell data in the stencil. The least-squares problem involves a $9\times m$ matrix singular value decomposition (where $m$ is the size of the stencil) for every face and for both orientations of each face . However this is purely a geometric calculation and is therefore a pre-processing activity since the mesh is fixed. This generates a set of weights for calculating $\phi_{f}$ from the cell values in the stencil, leaving $m$ multiplies for each face for each call of the advection operator. The stencils are found for three-dimensional, arbitrarily structured meshes by finding the face(s) closest to upwind of the face we are interpolating onto, taking the two cells either side of the upwind face(s) and then taking the vertex neighbours of those central cells (fig 1). For each face there are two possible stencils depending on the upwind direction. Both of the stencils are stored and the interpolation weights for both stencils are calculated. In order to ensure that the fit is accurate in the cells either side of face $f$ and to ensure that the values in these adjacent cells have the strongest control over $\phi_{f}$, rows associated with these values in the least squares fit matrix are weighted a factor of 1000 relative to the other rows (following Lashley, 2002). This does not affect the order of accuracy. Mathematically, an arbitrarily large value of the weight can be used to ensure that the fit goes exactly through the upwind and downwind cell. However if a value too large is used, the singular value problem becomes ill conditioned. We do not use any of the other stabilisation procedures as described by SWMD1x. The value $\phi_{f}$ is then calculated as a higher-order correction to first-order upwind: $$\phi_{f}=\phi_{\text{up}}+\sum_{c}w_{c}\phi_{c}$$ (19) where $w_{c}$ are the weights for each cell of the stencil calculated from the least squares fit, with $w_{\text{up}}$ reduced by 1 to make the fit a correction on upwind. 2.3.1 Implicit Solution, Matrix Solvers and Tolerances The trapezoidal implicit or Crank-Nicholson time-stepping leads to a matrix equation which needs to be solved to find all the $\phi$s at the next time-step. In order to ensure that the matrix is diagonally dominant for arbitrary time-steps, the cubic interpolation applied is a deferred correction on first-order upwind so that only the coefficient corresponding to the upwind cells are included in the matrix. This means that more than one implicit solves are needed per time-step so that the higher-order terms are solved to be second-order accurate in time. If the Courant number is less than or close to one, we use two implicit solves per time-step. Consequently, assuming that the velocity field and mesh are constant in time, the time-stepping scheme is defined as: $$\displaystyle\frac{\phi^{\prime}-\phi^{n}}{\Delta t}$$ $$\displaystyle=$$ $$\displaystyle-\frac{1}{2V}\sum_{f\in c}\left(\phi_{\text{up}}^{n}+\phi_{\text{% up}}^{\prime}+2\sum_{c}w_{c}\phi_{c}^{n}\right)\mathbf{u}_{f}\cdot\mathbf{S}_{f}$$ (20) $$\displaystyle\frac{\phi^{n+1}-\phi^{n}}{\Delta t}$$ $$\displaystyle=$$ $$\displaystyle-\frac{1}{2V}\sum_{f\in c}\left(\phi_{\text{up}}^{n}+\phi_{\text{% up}}^{n+1}+\sum_{c}w_{c}\phi_{c}^{n}+\sum_{c}w_{c}\phi_{c}^{\prime}\right)% \mathbf{u}_{f}\cdot\mathbf{S}_{f}.$$ (21) For larger Courant numbers we use four implicit solves per time-step although sensitivity to this choice has not been investigated. If this implicit scheme is applied on a logically rectangular, two-dimensional mesh with horizontal and vertical Courant numbers $c_{x}$ and $c_{z}$, then the diagonal coefficients of the matrix would be $1+c_{x}/2+c_{z}/2$ and, assuming two upwind directions, there would be exactly two off-diagonal elements, $-c_{x}/2$ and $-c_{z}/2$. Consequently the matrix is very sparse, asymmetric and diagonally dominant for all time-steps. It is solved using the OpenFOAM bi-conjugate gradient solver using DILU pre-conditioning to a tolerance of $10^{-8}$ every iteration. Sensitivity to the solver or solver tolerance have not been investigated. Information about the number of solver iterations for different test cases in given in section 3.4. 3 Results of Test Cases in Planar Geometry In order to make test cases as simple as possible without the need for incorporating spherical geometry, multi-panel meshes or non-rectangular cells, our computational domain consists of a periodic two-dimensional plane with deformations in the co-ordinate system (or mesh) mimicking the kind of distortions which are produced by a cubed-sphere mesh. We also use a two-dimensional vertical slice test case over orography using terrain following co-ordinates (or terrain following meshes). A range of test cases are undertaken on uniform and distorted meshes using the dimensionally split scheme and the multi-dimensional scheme in order to evaluate the influences of mesh distortions, the validity of using a dimensionally split scheme on a distorted mesh and the schemes’ accuracy and stability for long time-steps. 3.1 Solid Body Rotation The solid body rotation test case of Leonard et al. (1996a) is used to compare the accuracy of the dimensionally split and multi-dimensional schemes on orthogonal and non-orthogonal meshes. We define this test case on a domain that is $10^{4}\times 10^{4}\text{ m}^{2}$. The velocity is defined by numerically differentiating the streamfunction which can be defined at mesh vertices, $\mathbf{x}$, by: $$\psi(\mathbf{x},t)=A|\mathbf{x}-\mathbf{x}_{c}|^{2}$$ (22) where $\mathbf{x}_{c}$ is the centre of the domain, and $A=5\pi/3000\text{ s}^{-1}$ so that the angular velocity is $2A$. The initial tracer takes a Gaussian distribution in order to ensure that all advection schemes achieve their theoretical order of accuracy: $$\phi(\mathbf{x})=\exp\left(-\frac{1}{2}\frac{|\mathbf{x}-\mathbf{x}_{\phi}|^{2% }}{r_{\phi}^{2}}\right)$$ (23) where $\mathbf{x}_{\phi}=\mathbf{x}_{c}+r_{c\phi}\mathbf{j}$ is the initial centre of the tracer distribution, $r_{c\phi}=2500\text{ m}$, $r_{\phi}=500\text{ m}$ and $\mathbf{j}$ is the unit vector in the $y$ direction. The analytic solution has the same tracer distribution but with the centre of the tracer at: $$\mathbf{x}_{\phi}=\mathbf{x}_{c}+r_{c\phi}\left(\begin{array}[]{c}\cos(\pi/2+2% At)\\ \sin(\pi/2+2At)\end{array}\right)$$ (24) and the Gaussian rotates anti-clockwise exactly one revolution in $600\text{ s}$. The solid body rotation test case is performed on a uniform, orthogonal mesh and on a non-orthogonal mesh on a plane with non-orthogonality similar to that of a cubed-sphere mesh. For the dimensionally split scheme, non-orthogonality is achieved using the co-ordinate transform: $$\begin{array}[]{ccc}X=x&&Y=\begin{cases}y_{m}\left(1+\frac{y-f}{2y_{m}-f}% \right)&\text{ for }y\geq f\\ y_{m}\left(1+\frac{y-f}{f}\right)&\text{ for }y<f\end{cases}\end{array}$$ (25) where $y_{m}=5000\text{ m}$ and $f$ is the equation for a $y$ position of uniform $Y$ half way up the domain. In order to create angles of $120^{o}$ in the mesh, similar to a cubed-sphere, $f$ is given by: $$f=\begin{cases}y_{m}\left(1+\frac{1}{2\sqrt{3}}\right)-\frac{x}{\sqrt{3}}&% \text{ for }x\leq x_{m}\\ y_{m}\left(1-\frac{1}{2\sqrt{3}}\right)+\frac{x-x_{m}}{\sqrt{3}}&\text{ for }x% >x_{m}\end{cases}.$$ (26) where $x_{m}=5000$ m. For a $50\times 50$ mesh this gives the $x$ and $y$ co-ordinate locations as shown in figure 2. The multi-dimensional scheme model uses Cartesian co-ordinates and a distorted mesh rather than a non-orthogonal co-ordinate system on a Cartesian mesh. However this does not affect the numerical results assuming that the co-ordinate transforms are implemented in a consistent way to the distorted mesh in Cartesian co-ordinates. For the dimensionally split scheme, bi-periodic boundary conditions are applied. For the multi-dimensional scheme, it was more straightforward to impose fixed value boundary conditions of $\phi=0$ where the velocity is into the domain and zero normal gradient where the velocity is out of the domain. However $\phi$ remains almost zero near the boundaries so these boundary conditions do not affect the accuracy. Results of this test case on the orthogonal and non-orthogonal meshes of $100\times 100$ cells with $\Delta t=1\text{ s}$ are shown in figure 3 for both advection schemes (which gives a maximum Courant number close to one). The contours show the tracer value every 100 seconds and the colours show the errors summed every 100 seconds. The dimensionally split scheme outperforms the multi-dimensional scheme on both meshes due to the higher-order accuracy of the split scheme. The dimensionally split scheme introduces a small error at 300 seconds where the tracer goes through the change in direction of the mesh which would be ameliorated if we were using monotonicity constraints. The second-order, multi-dimensional scheme shows phase lag but errors are almost entirely insensitive the mesh distortions. The multi-dimensional and dimensionally split schemes take very different approaches to handling large Courant numbers. The multi-dimensional scheme uses implicit time-stepping whereas the dimensionally split scheme uses a flux-form semi-Lagrangian approach, integrating over a line of cells in order to calculate the flux across a face. Implicit schemes are known to suffer from phase errors (eg DB12; LWW14) for long time-steps whereas the accuracy of semi-Lagrangian is less sensitive to time-step (PS84). Therefore we present results of both schemes on orthogonal and non-orthogonal meshes for time-steps 10 times those used in figure 3 ($\Delta t=10\text{ s}$) giving maximum Courant numbers of around 10 using $100\times 100$ cells in figure 4. The error of the multi-dimensional scheme is again much larger than the dimensionally split scheme on both meshes. The dimensionally split scheme is accurate at large Courant numbers despite the first-order calculation of departure points. However, the dimensionally split scheme introduces oscillations on the non-orthogonal mesh, particularly where the mesh changes direction whereas for the multi-dimensional scheme, errors are not strongly affected by the non-orthogonality. Figure 4 clearly shows phase errors of the implicit time-stepping but, despite the large Courant number, the well resolved part of the profile is propagating at close to the correct speed. Dispersion analysis (LWW14) shows that high frequency oscillations (which are poorly resolved in time and space) will be slowed dramatically but fast moving features which are well resolved in space will propagate at a much more realistic speed, supporting the results shown in figure 4. In order to compare convergence with resolution of the different numerical methods, we use the $\ell_{2}$ and $\ell_{\infty}$ error norms defined in the usual way: $$\displaystyle\ell_{2}$$ $$\displaystyle=$$ $$\displaystyle\sqrt{\int_{V}\left(\phi-\phi_{T}\right)^{2}dV\bigg{/}\int_{V}% \phi_{T}^{2}\ dV}$$ (27) $$\displaystyle\ell_{\infty}$$ $$\displaystyle=$$ $$\displaystyle\max|\phi-\phi_{T}|\big{/}\max|\phi_{T}|$$ (28) where $\phi_{T}$ is the analytic solution and the integrations and maxima are over the whole domain, with volume $V$. Figure 5 shows convergence with resolution of the $\ell_{2}$ and $\ell_{\infty}$ error measures for meshes of $50\times 50$, $100\times 100$, $200\times 200$ and $400\times 400$ cells with time-steps scaled in order to maintain a maximum Courant number of 1 ($\Delta t=2,\ 1,\ 0.5,\ 0.25\text{ s}$) or scaled to achieve a maximum Courant number of 10 ($\Delta t=20,\ 10,\ 5,\ 2.5\text{ s}$). The error norms are calculated at $t=500\ \text{seconds}$, when the tracer has made $5/6$ of one revolution in order to avoid error cancellation. On both orthogonal and non-orthogonal meshes, the multi-dimensional scheme has second order convergence once errors are low enough to avoid error saturation (stable errors are bounded at around one). With a large Courant number, both schemes are less accurate. For the multi-dimensional scheme, this is due to phase errors of the implicit time-stepping whereas for the multi-dimensional scheme the first-order errors in calculating the departure point and trajectory will be emerging and there could also be significant errors from the second-order COSMIC splitting. On the orthogonal mesh, the dimensionally split scheme has third order convergence for the Courant number close to one and second order for the larger Courant number (consistent with the results of Colella and Woodward, 1984a; Leonard et al., 1996a). On the non-orthogonal mesh, the dimensionally split scheme has second order converges for both Courant numbers. The different time-stepping schemes of the two models also affect accuracy. The $\ell_{2}$ and $\ell_{\infty}$ error measures as a function of time-step for meshes of $100\times 100$ cells are shown in figure 6. The dimensionally split scheme, which uses flux-form semi-Lagrangian time-stepping, has errors reducing as time-step increases, up to a Courant number of 2 ($\Delta t=2\ \text{s}$) whereas the multi-dimensional scheme, which uses the method of lines to treat space and time separately, always has error reducing as time-steps reduces. The flux-form semi-Lagrangian technique discretises space and time together and the error is not very sensitive to time-step. However, the shorter the time-step, the more time-steps need to be taken so errors can actually accumulate more by taking more time-steps. (This is consistent with the order of accuracy of semi-Lagrangian being $\Delta x^{p}/\Delta t$ for interpolation using polynomials of degree $p$, as described by Dur10.) In summary, the dimensionally split scheme has excellent behaviour at large and small Courant numbers on the orthogonal meshes with up to third order convergence for small Courant numbers and the errors increase and order of convergence decreases on non-orthogonal meshes. In contrast, the multi-dimensional scheme is insensitive to the orthogonality, converges with second-order and suffers from phase errors at large Courant numbers. 3.2 Horizontal Advection over Orography Non-orthogonal meshes (or co-ordinate systems) are usually necessary for representing orography which could be a challenge for dimensionally split schemes. Horizontal-vertical split schemes are commonly used in this context (eg. DEE+12; WGZ+13). We present results of the SLF+02 horizontal advection over orography test case for a range of resolutions and Courant numbers for the dimensionally split and multi-dimensional schemes. All simulations use basic terrain following co-ordinates (BTF, Gal-Chen and Somerville, 1975) in order to present a challenging test case that maximises the non-orthogonality. The transformation is given by: $$\begin{array}[]{ccc}X=x&&Z=H\frac{z-h(x)}{H-h(x)}\end{array}$$ (29) where $H$ is the domain height and $h$ is the terrain height. The test case uses a domain of width 300 km, height, $H=25\text{ km}$ and a mountain range defined by $$h=\begin{cases}h_{0}\cos^{2}\frac{\pi x}{\lambda}\cos^{2}\frac{\pi x}{2a}&% \text{ for }|x|\leq a\\ 0&\text{ otherwise}\end{cases}$$ (30) with the maximum mountain height, $h_{0}=3\text{ km}$, half-width $a=25\text{ km}$ and wavelength $\lambda=8\text{ km}$. These values give a maximum terrain gradient of close to $45^{o}$. The wind is given by a streamfunction which is defined at vertices so that the wind field is discretely divergence free. The streamfunction at vertices is calculated analytically from the wind profile: $$u(z)=u_{0}\begin{cases}1&\text{ for }z_{2}\leq z\\ \sin^{2}\left(\frac{\pi}{2}\frac{z-z_{1}}{z_{2}-z_{1}}\right)&\text{ for }z_{1% }<z\leq z_{2}\\ 0&\text{ for }z<z_{1}\end{cases}$$ (31) with $u_{0}=10\text{ ms}^{-1}$, $z_{1}=4\text{ km}$ and $z_{2}=5\text{ km}$. The initial tracer position is given by: $$\phi=\begin{cases}\cos^{2}\frac{\pi r}{2}&\text{ for }r\leq 1\\ 0&\text{ otherwise}\end{cases}\text{ with }r=\sqrt{\left(\frac{x-x_{0}}{A_{x}}% \right)^{2}+\left(\frac{z-z_{0}}{A_{z}}\right)^{2}}$$ (32) with initial tracer centre, $(x_{0},z_{0})=(-50\text{ km},\ 9\text{ km})$ and halfwidths $A_{x}=25\text{ km}$, $A_{z}=9\text{ km}$. At time $t=5000\text{ s}$ the tracer is above the mountain and the simulation finishes at $t=10,000\text{ s}$ by which time the analytic solution is centred at $(50\text{km },8\text{km})$. The tracer advection over orography is shown in figure 7 for the split and multi-dimensional schemes at a resolution of $\Delta x=1\text{ km}$, $\Delta z=500\text{ m}$ and for a range of Courant numbers. The horizontal Courant number is defined as $u_{0}\Delta t/\Delta x$ and ranges from 0.25 to 10. The maximum Courant number is the maximum of the multi-dimensional Courant number which is defined for cell with faces $f$ as: $$c=\frac{1}{2V}\sum_{f}|\mathbf{u}_{f}\cdot\mathbf{S}_{f}|\Delta t$$ (33) (see section 2.3 for definitions of variables) and ranges from 0.74 to 29.6. The time-step restriction for the split scheme is based on the deformational Courant number, $c_{d}\leq 1$, (eqn 10). The maximum deformational Courant number is also given in figure 7. The contours in figure 7 show the tracer values at 0, $5000\text{ s}$ and $10,000\text{ s}$ after initialisation and the colours show the errors from the analytic solution. For horizontal Courant numbers less than one (maximum Courant number up to 3), both schemes give accurate results with the dimensionally split scheme tending to give oscillations and the multi-dimensional scheme producing more diffusion. For larger Courant numbers, when the deformational Courant number is greater than one, the split scheme is unstable while the multi-dimensional scheme produces large phase errors due to the errors associated with implicit time-stepping. The term responsible for the large deformational Courant number is $\partial u/\partial z$ where the velocity shears from $u_{0}=10$m/s at $z=z_{2}$ to zero at $z=z_{1}$. We examine the convergence with resolution in figure 8 which shows the $\ell_{2}$ and $\ell_{\infty}$ error norms as a function of $\Delta x$. These simulations all use a maximum Courant number less than one, a horizontal Courant number of 0.25, a maximum deformational Courant number of about 0.2 and fixed ratios of $\Delta x$, $\Delta z$ and $\Delta t$. Both schemes give similar accuracy with the dimensionally split scheme having faster convergence with resolution. In summary, the dimensionally split scheme has good accuracy over orography for modest Courant numbers but larger errors for larger Courant numbers and is unstable when the deformational Courant number is greater than one whereas the multi-dimensional scheme with implicit time-stepping is stable for all Courant numbers. The multi-dimensional scheme second-order convergent and the dimensionally split scheme converges faster. 3.3 Deformational Flow In deformational flow, there is no analytical solution and therefore we follow the approach of NL10; Lauritzen et al. (2012b) and define an evolving velocity field that reverses direction half way through the simulation, taking the tracer back to the initial conditions. Error norms can then be calculated by comparing the final and initial tracer fields. The deformational velocity field is added to a fixed solid body rotation that does not reverse so as to avoid error cancellation between the forwards and backwards periods (solid body here meaning horizontal wind on a periodic domain). We define a Cartesian version of the deformational, non-divergent test case of Lauritzen et al. (2012b) with a domain between $-\pi$ and $\pi$ in the $x$ direction ($L_{x}=2\pi$) and between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ in the $y$ direction ($L_{y}=\pi$) with periodic boundary conditions in the $x$ direction and zero gradient, zero flow boundary conditions in the $y$ direction. The stream function adapted to Cartesian co-ordinate is: $$\displaystyle\psi\left(x,y,t\right)$$ $$\displaystyle=$$ $$\displaystyle\frac{\hat{\psi}}{T}\left(\frac{L_{x}}{2\pi}\right)^{2}\sin^{2}% \left(2\pi\left(\frac{x}{L_{x}}-\frac{t}{T}\right)\right)\cos^{2}\left(\pi% \frac{y}{L_{y}}\right)\cos\left(\pi\frac{t}{T}\right)-L_{x}\frac{y}{T}$$ (34) where $\hat{\psi}=10$ and $T=5$ is the time for one complete revolution of the periodic domain for the solid body rotation part of the flow. In order to test the order of convergence of the advection schemes, we use the infinitely smooth Gaussian distribution for the initial tracer concentration: $$\phi=0.95\exp-\frac{|\mathbf{x}-\mathbf{x}_{0}|^{2}}{A}+0.95\exp-\frac{|% \mathbf{x}-\mathbf{x}_{1}|^{2}}{A}$$ (35) where $\mathbf{x}^{T}=(x,y)$, $\mathbf{x}_{0}^{T}=(\frac{5}{12}L_{x},0)$, $\mathbf{x}_{1}^{T}=(\frac{7}{12}L_{x},0)$ and $A=\frac{1}{5}$. A mesh with distortion similar to a cubed-sphere is defined by the co-ordinate transform: $$\begin{array}[]{ccc}X=x&&Y=\begin{cases}L_{y}\frac{y-f}{L_{y}-2f}&\text{ for }% y\geq f\\ L_{y}\frac{y-f}{L_{y}+2f}&\text{ for }y<f\end{cases}\end{array}$$ (36) where $f$ is: $$f=\begin{cases}\frac{1}{\sqrt{3}}\left(\frac{\pi}{4}-|x|\right)&\text{ for }|x% |\leq\frac{\pi}{2}\\ \frac{1}{\sqrt{3}}\left(|x|-\frac{3\pi}{4}\right)&\text{ for }|x|>\frac{\pi}{2% }\end{cases}.$$ The initial conditions and a $120\times 60$ mesh with distortion defined by this co-ordinate transform is shown in figure 9. The tracer concentrations after 1, 2, 3, 4 and 5 time units are shown in figure 10 using the dimensionally split and multi-dimensional schemes on a non-orthogonal mesh of $480\times 240$ cells, a time-step of 0.0025 units (ie 2000 time-steps to reach 5 time units) which gives a maximum Courant number of 1.03. The tracer is stretched out, wound up, advected around and then wound back into its original position with some numerical errors. Both advection schemes preserve fine filaments but suffer from some dispersion errors which generate small oscillations around zero behind sharp gradients in the direction of the flow since neither scheme is monotonic or positive preserving. The dimensionally split scheme returns the tracer to a more accurate final solution. Sensitivity to orthogonality, resolution and time-step are explored using a range of simulations with maximum Courant numbers shown in table 1. The Courant numbers on the uniform, orthogonal meshes (with no mesh distortions) are 70% of those on the non-orthogonal mesh since the non-orthogonal mesh has clustering of mesh points. The deformational Courant number is less than one for all simulations. The convergence with resolution of the $\ell_{2}$ and $\ell_{\infty}$ error norms at the final time are shown in figure 11 for both advection schemes using a maximum Courant number close to one and for a maximum Courant number close to ten. We will first consider the behaviour of the schemes at modest Courant number (maximum close to one). All of the schemes give first-order convergence with resolution at coarse resolutions due to error saturation (if the scheme is stable, errors are bounded above by about one). At higher resolution, the split scheme converges with nearly third-order for both meshes whereas the multi-dimensional scheme approaches second-order. For large Courant numbers (maximum close to 10), the split scheme converges with first-order for both mesh types due to the crude estimation of the departure point (section 2.1) whereas the multi-dimensional scheme is much less accurate but approaches second-order at high resolution. The dimensionally split scheme is sensitive to mesh distortions only for large Courant numbers. We inspect sensitivity to time-step of both schemes on orthogonal and non-orthogonal meshes of $120\times 60$ cells in figure 12 using the time-steps shown in table 1 giving maximum Courant numbers ranging from 0.25 to 10 (deformational Courant numbers always less than 0.32). This demonstrates potentially useful properties of the flux-form semi-Lagrangian time-stepping used by the split scheme. Ignoring errors in calculating the departure point, semi-Lagrangian schemes have errors proportional to $\Delta t^{-1}$ (Dur10) which explains the reduction in error as time-step increases for the split scheme. Once the maximum Courant number reaches 1 or 2 ($\Delta t=0.01$ or 0.02), errors of the split scheme do grow with the time-step, due to the deformational nature of the flow and errors in calculating the departure points. In contrast, using the multi-dimensional scheme which uses method of lines time-stepping, errors always increase as the time-step is increased. Comparisons between schemes in figures 10 and 11 used maximum Courant numbers of 1 and 10 which showed that the split scheme on the orthogonal mesh gave better accuracy than the multi-dimensional scheme. However, figure 12 shows that this advantage disappears at lower Courant numbers since the multi-dimensional scheme (method of lines) gets more accurate with lower Courant numbers whereas the split scheme (semi-Lagrangian) gets less accurate since more time-steps have to be taken. Both schemes are stable for all time-steps considered. In summary, the dimensionally split scheme has good accuracy for deformational flow independent of mesh orthogonality. The multi-dimensional scheme is competitive at small Courant numbers but the semi-Lagrangian nature of the split scheme means that errors are very low for Courant numbers close to one. 3.4 Computational Cost We cannot compare CPU time, wall clock time or parallel efficiency of the two advection schemes because the multi-dimensional scheme is written in C++ using OpenFOAM and the split scheme is written in Python, both codes have been run on different hardware and the split advection scheme code has not been parallelised. Instead we consider the number of multiply operations performed per cell per time-step for calculating the fluxes of each tracer. We appreciate that this is not a good predictor of efficiency as it does not consider memory read and write requirements or cache coherency. However all data that is multiplied has to be fetched from memory and so, assuming that all data can be arranged optimally in memory to enable fewest cache misses, the number of multiplies should be related to the wall clock time. Neither advection scheme does significantly more work per memory fetch than the other. For both schemes we consider the number of multiplies needed to calculate the flux of $\phi$ at each face. That is equation 7 for the dimensionally scheme and equation (19) for the multi-dimensional scheme. We do not consider the computational cost of updating the cell averages from the fluxes using Gauss’s theorem (eqns (3) and (17)) as these are the same for each scheme. For the multi-dimensional scheme, we do include an estimate of the amount of work done by the linear equation solver but we do not consider the scalability of this solver. 3.4.1 Dimensionally Split Scheme The computational cost of PPM with COSMIC splitting is not strongly time-step dependent due to the swept area approach of flux-form semi-Lagrangian. One-dimensional PPM uses 4 cells for interpolation to find face values used by the reconstruction. Assuming as much as possible is pre-computed, this interpolation uses 3 multiplies on a non-uniform mesh. The reconstruction then uses 6 multiplies to calculate the flux on each face. Only one additional memory access and one additional multiply are needed per cell for Courant numbers greater than one (eqn 8) making 10 multiplies per cell for applying PPM in one direction for a Cournat number greater than one. The COSMIC splitting in two dimensions involves four applications of the one-dimensional PPM. This makes 40 multiplies per cell in total for applying PPM with COSMIC splitting in two dimensions. In three dimensions, COSMIC splitting requires 12 applications of PPM (Leonard et al., 1996a) leading to 120 multiplies. 3.4.2 Multi-dimensional Scheme The number of multiplies involved in the multi-dimensional scheme includes the number of multiplies to update the higher order advection and the number of multiplications for each iteration of the linear equation solver. We will also consider the cost of an explicit version of the multi-dimensional scheme using an RK2 time-stepping scheme (eg as used by SW16) which is stable and accurate up to a Courant number of one for this spatial discretistion and gives very similar results to the implicit scheme (not shown). The explicit version of the dimensionally split scheme uses RK2 or Heun time-stepping, in which $\phi_{\text{up}}^{\prime}$ on the RHS of eqn (20) is replaced by $\phi_{\text{up}}^{n}$ and $\phi_{\text{up}}^{n+1}$ on the RHS of eqn (21) is replaced by $\phi_{\text{up}}^{\prime}$. On a logically rectangular two-dimensional mesh there are 12 cells in each stencil for each face (fig 1). Each cell has four faces and the interpolation onto each face is used to calculate the flux between two cells. This leads to 24 multiplies per cell for each RK2 stage and hence 48 multiplies per cell per time-step. When using implicit time-stepping, there will be 24 multiplies per cell for every evaluation of the right hand side of the matrix for the higher-order correction on first-order upwind. We have not explored the sensitivity of the accuracy and stability to the stencil size and shape in three dimensions. The three dimensional equivalent of the stencil of quadrilaterals in figure 1 is likely to contain 36 (rather than 12) cells although it may be possible to omit some corner cells and use a stencil of 20 cells. For a stencil of 36 cells, the number of multiplies per cell per time-step would be $36\times 3\times 2=216$. The implicit version of the multi-dimensional scheme (section 2.3.1) requires the solution of an asymmetric, diagonally dominant matrix with three non-zero elements per row for a logically rectangular two-dimensional mesh. The pre-conditioner is implemented in file DILUPreconditioner.C in OpenFOAM 3.0.1 and the solver in file PBiCG.C. From these files, we estimate that, for a mesh of quadrilaterals, the solver will use 24 multiplies (or divides) per cell, per solver iteration, including pre-conditioning. The average number of iterations of the preconditioned bi-CG solver per time-step for each of the simulations is shown in table 2 (including the number of solver iterations in each outer iteration). These simulations use two outer iterations per time-step when the maximum Courant number is $\leq 1.1$ (as in equations (20) and (21)) but, for stability, use four outer iterations per time-step for the larger Courant numbers. This partly explains the greater number of iterations for larger Courant numbers. A solver tolerance of $10^{-8}$ is used for each of the outer solves. Table 2 shows that the total number of iterations per time-step reduces slightly as resolution increases. The total number of iterations for a complete simulation is reduced by using larger Courant numbers because the number of iterations per time-step increases less than linearly with increasing Courant numbers. In fact, simulations with larger Courant numbers are considerably cheaper because there are fewer evaluations of the right hand side of the matrix equation. Combining the number of solver iterations, the number of multiply operations per solve and the number of multiply operations in calculating the explicit higher order part of the advection scheme, the total number of multiply operations per cell per time-step for the multi-dimensional scheme is shown in table 3 for the non-orthogonal mesh of $120\times 60$ cells. Table 3 also shows the number of multiplies for using the explicit, RK2 version of the multi-dimensional scheme and the dimensionally split scheme. Table 3 shows that the implicit scheme always uses more multiply operations but particularly uses more multiplies for large Courant numbers. The explicit version of the multi-dimensional scheme always uses 48 multiply operations but is not stable for all time-steps whereas the dimensionally split scheme (using flux-form semi-Lagrangian time-stepping) is stable for all Courant numbers (at this spatial resolution) and always uses the fewest number of multiply operations per cell per time-step. There is considerable flexibility in the solver configuration: the number of outer iterations per time-step determines how frequently the high-order correction on the right hand side of the matrix equation is updated, and the solver tolerance per outer iteration could be modified by using a weaker tolerance on all but the final matrix solve per time-step. These options have not been explored. It may also be beneficial to create more non-zero matrix entries rather than having the higher-order correction entirely a deferred correction on first-order upwind, but such a change would need to ensure that the matrix remains diagonally dominant. 4 Summary and Conclusions We examine the errors associated with using a dimensionally split advection scheme and a multi-dimensional advection scheme on distorted meshes. The dimensionally split scheme is very accurate on orthogonal meshes and only loses a little accuracy on highly distorted meshes, despite a first-order departure point calculation. The multi-dimensional scheme with implicit time-stepping is less accurate on orthogonal meshes than the dimensionally split scheme but the accuracy is not sensitive to mesh distortions and the stability is less sensitive to Courant number. The dimensionally split scheme is the piecewise polynomial method (PPM, Colella and Woodward, 1984a) with COSMIC splitting (Leonard et al., 1996a) that extends it to two spatial dimensions. PPM is converges with third-order in one dimension and COSMIC splitting enables second-order convergence in two orthogonal directions. PPM is a flux-form semi-Lagrangian scheme and so can handle large Courant numbers accurately without significant additional computational cost, with a time-step restriction based on the deformational Courant number. The second-order accurate multi-dimensional scheme is split in space and time (method of lines) and uses a cubic polynomial fit over a stencil of cells for spatial discretisation and trapezoidal implicit in time to retain stability for large Courant numbers. We use versions of both schemes without any monotonicity constraints in order to compare the handling of multi-dimensionality of the two schemes and order of convergence rather than comparing the limiters of the two schemes. Three two-dimensional advection test cases on Cartesian planes are proposed without the complexities of a spherical domain or multi-panel meshes but with distorted meshes to mimic the distortions of a cubed-sphere or terrain following co-ordinates. We therefore propose that these test cases could be used in the initial testing of advection schemes, before the generation of meshes on the sphere. The first test case is an extension of the Leonard et al. (1996a) solid body rotation using a distorted mesh. The second test case is the established horizontal advection over orography (SLF+02) using a basic terrain following mesh in order to maximise the distortions and a version using higher orography. The third test case is the deformational flow test case of Lauritzen et al. (2012b) adapted to a planar, Cartesian domain and a distorted mesh. We use the version of this test case with smooth initial conditions (the sum of two Gaussians) in order to examine order of convergence. The dimensionally split scheme is extremely accurate on orthogonal meshes and retains accuracy when long time-steps are used. However on distorted meshes, particularly at changes in direction such as those that appear at cube sphere edges or over orography, the split scheme loses some accuracy. In contrast, the multi-dimensional scheme is almost entirely insensitive to mesh distortion and asymptotes to second-order convergence at high resolution. As is expected for implicit time-stepping, phase errors occur when using long time-steps but the spatially well resolved features are advected at the correct speed and the multi-dimensional scheme is always stable. The matrix solver associated with the implicit time-stepping of the multi-dimensional scheme means that it is always considerably more expensive than the split scheme with cost increasing with Courant number. We haven’t investigated sensitivity to solver and pre-conditioner choices and it may be possible to do better. But the flux-form semi-Lagrangian method enables long time-steps without any matrix solutions and this will always be difficult to beat. It is possible to use the multi-dimensional scheme with explicit time-stepping such as Runge-Kutta. In that case, the Courant number is restricted to be less than one but the cost is similar to the dimensionally split scheme. The conclusions of this paper are consistent with those of K.K. et al. (2015) who found a dimensionally split scheme to be as accurate as a multi-dimensional scheme on a cubed-sphere mesh with special interpolations so that the dimensionally split scheme could cope with the cube edges. 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Neighborhoods are good communities \alignauthorDavid F. Gleich \affaddrPurdue University \affaddrComputer Science Department \alignauthorC. Seshadhri \affaddrSandia National Laboratories \affaddrLivermore, CA The author is supported by the Sandia LDRD program (under project 158477) and the applied mathematics program at the United States Department of Energy.Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. dgleich@purdue.edu scomand@sandia.gov Abstract The communities of a social network are sets of vertices with more connections inside the set than outside. We theoretically demonstrate that two commonly observed properties of social networks, heavy-tailed degree distributions and large clustering coefficients, imply the existence of vertex neighborhoods (also known as egonets) that are themselves good communities. We evaluate these neighborhood communities on a range of graphs. What we find is that the neighborhood communities often exhibit conductance scores that are as good as the Fiedler cut. Also, the conductance of neighborhood communities shows similar behavior as the network community profile computed with a personalized PageRank community detection method. The latter requires sweeping over a great many starting vertices, which can be expensive. By using a small and easy-to-compute set of neighborhood communities as seeds for these PageRank communities, however, we find communities that precisely capture the behavior of the network community profile when seeded everywhere in the graph, and at a significant reduction in total work. keywords: clustering coefficients, triangles, egonets, conductance \toappear\DeclareCaptionType copyrightbox \toappear \numberofauthors 2 \category I.5.3Pattern RecognitionClustering[Algorithms] \terms Algorithms, Theory 1 Introduction Community detection, loosely speaking, is any process that takes a graph or network and picks out sets of related nodes. An incredibly variety of techniques exist for this single task, which has a variety of names as well: community detection, graph clustering, and graph partitioning. Throughout this manuscript, we shall use the term community and cluster interchangeably. For more information about approaches for this problem, see the recent survey by Schaffer [34]. In many techniques, a community is defined as a set with a good score under a quality measure that reflects the connectivity between the set and the rest of the network. Common measures are based on density of local edges, deviance from a random null model, the behavior of random walks, or graph cuts. Mostly, these measures are NP-hard to optimize. To keep this manuscript simple, we shall evaluate communities using their conductance store. Schaeffer identified this measure as one of the most important cut-based measures and it has been studied extensively in a variety of disciplines [11, 36, 17]. Work by Leskovec et al. has recently demonstrated that, although different quality measures produce differences in terms of specific communities, strong communities persist under a variety of measures [26]. A vertex neighborhood of a vertex $v$ is the set of vertices directly connected to $v$ via an edge and $v$ itself. For example, see the green and black vertices at right. What we show here is that the presence of two commonly observed properties of modern information networks – a large global clustering coefficient [39] and a power-law degree distribution [5] – implies the existence of vertex neighborhoods with good conductance scores. We make this statement precise in Theorem 4.11. These results can be seen as an extension of the simple observation that, in the extreme case when the global clustering coefficient of a network is 1, then the network must be a union of cliques. Neighborhoods define ideal communities in this case. We mathematically show that this argument can be extended to the case when the graph has a power-law degree distribution and a large clustering coefficient. The significance of this finding is that robust community detection need not employ complicated algorithms. Instead, a straightforward approach that just involves counting triangles – a function that is easy to implement in MapReduce [12] and easy to approximate [21], suffices to identify communities. It is intriguing that arguably the two most important measurable quantities of social networks imply that communities are very easy to find. This may lead to more mathematical work explaining the success of community detection algorithms, given that the problem are in general NP-hard. We note that unfortunately, our theoretical bounds reflect a worst case behavior and are weaker than required for practical use. Consequently, in the remainder of the paper we explore the utility of neighborhood communities empirically. Section 2. The technical discussion of the manuscript begins by introducing our notation and precisely defines the quantities we examine, such as clustering coefficients, due to variability in the definitions of these measures. We also discuss the Andersen-Chung-Lang personalized PageRank clustering scheme [2] and the network whiskers from Leskovec et al. [24, 25]. We utilize the latter two algorithms as reference points for the success of our community detection. Section 3. We discuss some of the other observed properties of egonets, or vertex neighborhoods, along with other related work including overlapping communities. Section 4. We state and prove the theoretical results that graphs with a power-law degree distribution and large clustering coefficients have neighborhood communities with good conductance scores. Section 5. We review the data that will serve as the testbed for our empirical evaluation of neighborhood cuts. This comes from a variety of public sources and spans collaboration networks, social networks, technological networks, web networks, and random graph models. Section 6. Our empirical investigation of neighborhood clusters takes the following form. We first exhibit the conductance scores for the set of neighborhood communities for a few graphs (e.g. Figure 2). We find that neighborhood communities reflect the shape of the network community plot observed by Leskovec et al. [24, 25] at small size scales. We next compare the best neighborhood communities to those discovered by four other procedures: the Fiedler community, the best personalized PageRank community (§2.3), the best network whisker (§2.3), and the best clusters from metis [18]. In one third of the cases, the neighborhood community is as good as the best of any of the other algorithms. Another outcome of the theory from §4 is that large cores must exist in these graphs. (Here, a graph $k$-core is a subset of vertices where all nodes have degree at least $k$ [35].) We conclude this section by exploring the community properties of the graph $k$-cores. Section 7. Motivated by the success of the neighborhood communities at small size scales, we explore using the best vertex neighborhoods as seeds for a local greedy community expansion procedure and for the Andersen-Chung-Lang algorithm. Here, we find that these procedures, when seeded with an easy-to-identify set of neighborhood communities, produce larger clusters that decay as expected by the results in Leskovec et al. [24, 25] We make all of our algorithm and experimental code, the majority of the data for the experiments, and some extra figures that did not fit into the paper available: www.cs.purdue.edu/homes/dgleich/codes/neighborhoods These codes are easy to use. Given the adjacency matrix of a network A, the single command >> ncpneighs(A) will produce a figure analyzing the neighborhood communities in comparison to the Fiedler community (formal definition in Section 2.3). Summary of contributions • We theoretically motivate the study of neighborhood communities by showing they often have a low conductance in graphs with a power-law degree distribution and large clustering coefficients. • We empirically evaluate these neighborhood communities and find them comparable to those communities found by other algorithms at small size scales. • We find a small set of neighborhood communities that can be grown into larger communities using a PageRank based community detection algorithm. The results match those communities found with a more expensive sweep over all communities. 2 Formal setting and notation We first list out the various notations and formalisms used. All of the key notation is summarized in Table 1, and we briefly review it here. Let $G=(V,E)$ be a loop-less, undirected, unweighted graph. We denote the number of vertices by $n=|V|$ and the number of edges by $m=|E|$. In terms of the adjacency matrix, $m$ is half the number of non-zeros entries. For a vertex $v$, let $d_{v}$ be the degree of $v$. For any positive integer $d$, let $f_{d}$ be the number of vertices of degree $d$, that is, the frequency of $d$ in the degree distribution. The maximum degree is denoted by $d_{max}$. Let $D_{r}(v)$ to be the distance $r$-neighborhood of $v$. This is the set of vertices whose shortest path distance from $v$ is exactly $r$. Then, we define the ball of distance $r$ around $v$, denoted by $N_{r}(v)$, as the set $\bigcup_{i\leq r}D_{r}(v)$. 2.1 Clustering coefficients A wedge is an unordered pair of edges that share an endpoint. The center of the wedge is the common vertex between the edges. A wedge $\{(s,t),(s,u)\}$ is closed if the edge $(t,u)$ exists, and is open otherwise. We use $W$ to denote the set of wedges in $G$, and $W_{v}$ for the set of wedges centered at $V$. Note that $|W_{v}|=\binom{d_{v}}{2}$. We set $p_{v}=|W_{v}|/|W|$. Social networks often have large clustering coefficients [39]. Because of the varying definitions of this term that are used, we will denote by $\kappa$ the global clustering coefficient. This quantity is basically a normalized count of triangles. In the following, we think of $w$ drawn uniformly at random from $W$. $$\kappa=\Pr_{w\in W}[\text{$w$ is closed}]=\frac{\text{number of closed wedges}% }{|W|}$$ In terms of triangles, $\kappa={3\cdot\text{number of triangles}}/{|W|}.$ For any vertex $v$, $C_{v}$ is the local clustering coefficient of $v$. We draw $w$ uniformly at random from $W_{v}$. $$C_{v}=\Pr_{w\in W_{v}}[\text{$w$ is closed}]=\frac{\text{number of closed % wedges in $W_{v}$}}{|W_{v}|}$$ 2.2 Cuts and Conductance Given a set of vertices $S$, the set $\bar{S}$ is the complement set, $\bar{S}=V\backslash S$. For disjoint sets of vertices $S,T$, $E(S,T)$ denotes the edges between $S$ and $T$. For convenience, we denote the size of the cut induced by a set $|E(S,\overline{S})|$ by $\operatorname{\texttt{cut}}(S)$. The conductance of a cluster (a set of vertices) measures the probability that a one-step random walk starting in that cluster leaves that cluster. Let $\operatorname{\texttt{vol}}(S)$ denotes the sum of degrees of vertices in $S$ and $\operatorname{\texttt{edges}}(S)$ denotes twice the number of edges among vertices in $S$ so that $$\operatorname{\texttt{edges}}(S)=\operatorname{\texttt{vol}}(S)-\operatorname{% \texttt{cut}}(S).$$ Then the conductance of set $S$, denoted $\phi(S)$, is $$\phi(S)=\frac{\operatorname{\texttt{cut}}(S)}{\min\bigl{(}\operatorname{% \texttt{vol}}(S),\operatorname{\texttt{vol}}(\bar{S})\bigr{)}}.$$ Conductance is measured with respect to the set $S$ or $\bar{S}$ with smaller volume, and is the probability of picking an edge from the smaller set that crosses the cut. Because of this property, conductance is preserved on taking complements: $\phi(S)=\phi(\bar{S})$. For this reason, when we refer to the number of vertices in a set of conductance $\phi$, we always use the smaller set $\min(|S|,|\bar{S}|)$. Figure 1 shows a few communities and their associated cuts and conductance scores from our methods and two points of comparison. 2.3 Finding good conductance communities We briefly review three ways of identifying a community with a good conductance score. Fiedler set The well-known Cheeger inequality defines a bound between the second smallest eigenvalue of the normalized Laplacian matrix and the set of smallest conductance in a graph [11]. Formally, $$(1/2)\lambda_{2}\leq\min_{S\subset V}\phi(S)\leq\sqrt{2\lambda_{2}}$$ where $\lambda_{2}$ is the second smallest eigenvalue of the normalized Laplacian. The proof is constructive. It identifies a set of vertices that obeys the upper-bound using a sweep cut. This is the smallest conductance cut among all cuts induced by ordering vertices by increasing values of $\sqrt{d_{v}}x_{v}$, where $x_{v}$ is the component of the eigenvector associated with $\lambda_{2}$. This is the same idea used in normalized cut procedures [36]. We refer to the set identified by this procedure as the Cheeger community or Fiedler community. The latter term is based on Fiedler’s work in using the second smallest eigenvalue of the combinatorial Laplacian matrix [14]. Figure 1b shows the Fiedler community for the Les Misérables network. Personalized PageRank communities Another highly successful scheme for community detection based on conductance uses personalized PageRank vectors. A personalized PageRank vector is the stationary distribution of a random walk that follows an edge of the graph with probability $\alpha$ and “teleports” back to a fixed seed vertex with probability $1-\alpha$. We use $\alpha=0.99$ in all experiments. The essence of the induced community is that an inexact personalized PageRank vector, computed via an algorithm that “pushes” rank round the graph, will identify good bottlenecks nearby a seed vertex. These bottlenecks can be formalized in a Cheeger-like bound [2]. The procedure to find a personalized PageRank community is: i) specify a value of $\alpha$, a seed vertex $v$, and a desired clusted size $\sigma$; ii) solve the personalized PageRank problem using the algorithm from [2] until a degree-weighted tolerance of $\tau=1/(10\sigma)$; and iii) sweep over all cuts induced by the ordering of the personalized PageRank vector (normalized by degrees) and choose the best. Personalized PageRank communities (ppr communities, for short) were used to identify an interesting empirical property of communities in large networks [24, 25]. To generate these plots, those authors examined a range of values of $\sigma$ for a large number of vertices of the graph and summarized the best communities found at any size scale in a network community plot. Figure 1d shows the best personalized PageRank community for the network of character interactions in Les Misérables. Whisker communities Perhaps the best point of comparison with our approach are the whisker communities defined by Leskovec et al. [24, 25]. These communities are small dense subgraphs connected by a single edge. They can be found by looking at any subgraph connected to the largest biconnected component by a single edge. A biconnected component remains connected after the removal of any vertex. Note that the largest biconnected component is not necessarily a 2-core of the graph. Leskovec et al. observed that many of these subgraphs are rather dense. Each subgraph has a cut of exactly one, and consequently, a productive means of finding sets with low conductance is to sort these subgraphs by their volume. The best whisker cut is the single subgraph with largest volume. 3 Related work We are hardly the first to notice that vertex neighborhoods have special properties. Egonets, homophily, and structural holes In the context of social networks, vertex neighborhoods are often called egonets because they reflect the the state of the network as perceived by a single vertex. Their analysis is a key component in the study of social networks [38], especially in terms of data collection. Studies of these networks often focus on the theory of structural holes, which is the notion that an individual can derive an advantage from serving as a bridge between disparate groups [10]. These bridge roles are interesting because they contradict homophily in social ties. Homophily, or the principle that similar individuals form ties, is the mechanism that is expected to produce networks with large local clustering coefficients [28]. These social theories have prompted the development of new methods to tease apart some of these effects in real-world networks [22], and to develop network models that capture structural holes [19]. Clustering and communities Vertex neighborhoods often play a role in other techniques to find community or clustering structure in a network. Overlap in the neighborhood sets of vertices is a common vertex similarity metric used to guide graph clustering algorithms [34]. Other schemes utilize vertex neighborhoods as good seed sets for local techniques to grow communities [33, 16]. We explore using a carefully chosen set of neighborhoods for this purpose in our final empirical discussion (§7). Perhaps the most closely related work is a recent idea to utilize the connected components of ego-nets, after their ego vertex is removed, to produce a good set of overlapping communities [32]. Our theoretical results establish that these ideas are highly likely to succeed in networks with local clustering and power-law degree distributions. Graph properties Much of the modern work on networks rests on surprising empirical observations about the structure of real world connections. For instance, information networks were found to have a power-law in the degree distribution [5, 13]. These same networks were also found to have considerable local structure in the form of large clustering coefficients [39], but retained a small global diameter. Our theory shows that a third potential observation – the existence of vertex neighborhood with low conductance – is in fact implied by these other two properties. We formally show that heavy tailed degree distributions and high clustering coefficients imply the existence of large dense cores. Anomoly detection Predictable behavior in the structure of ego-nets makes them a useful tool for detecting anomalous patterns in the structure of the network. For instance, Akoglu et al. [1] compute a small collection of measures on each egonet, such as the average degree and largest eigenvalues. Outliers in this space of vertices are often rather anomalous vertices. Our work is, in contrast, a precise statement about the regularity of the ego-nets, and says that we always expect a large ego-net to be a good community. Summary Although we are not the first to study neighborhood based communities, the relationship between the local clustering, power-law degree distributions, and large neighborhoods with small conductance does not appear to have been noticed before. 4 Theoretical justification for neighborhood communities The aim of this section is to provide some mathematical justification for the success of neighborhood cuts. Our aim is to show that heavy tailed degree distributions and large clustering coefficients imply the existence of neighborhood cuts with low conductance and large dense cores. As mentioned earlier, the exact bounds we get are somewhat weak and only hold when the clustering coefficient is extremely large. Nonetheless, the proofs give significant intuition into why neighborhoods are good communities. We begin with the extreme case when the value of $\kappa$ is $1$ (so every wedge is closed). Then we have the following simple claim. Claim 4.1 Suppose the global clustering coefficient of $G$ is $1$. Then $G$ is the union of disjoint cliques. Proof 4.2. Consider two vertices $u$ and $v$ that are connected. Suppose the shortest path distance between them is $\ell>1$. Then the shortest path has at least $3$ distinct vertices (including $u$ and $v$). Take the last three vertices on this path, $v_{1},v_{2},v$. This forms a wedge at $v_{2}$, and must be closed (since the clustering coefficient is $1$). Hence, the edge $(v_{1},v)$ exists and there exists a path between $u$ and $v$ of length less than $\ell$. This is a contradiction. Hence, any two connected vertices have a shortest path distance of $1$, i.e., are connected by an edge. The graph is a disjoint union of cliques. Note that the neighborhood of any vertex in the above claim forms a clique disconnected from the rest of $G$. Therefore, all neighborhoods form perfect communities, in this extremely degenerate case. We prove this for more general settings. The quantities $p_{v}=|W_{v}|/|W|$, form a distribution over the set of vertices. Since we are performing an asymptotic analysis, we will use $o(1)$ to denote any quantity that becomes negligible as the graph size increases. We will choose $\beta$ to be a constant less than $1$. It is quite unimportant for the asymptotic analysis what this constant is. From a pratical standpoint, think of $\beta$ as a constant such that most edges are incident to a vertex of degree at least $d_{max}^{\beta}$ ($2/3$ is usually a reasonable value). Also, we will assume that the power law exponent is at most $3$, a fairly acceptable condition. Claim 4.3 Let $S$ be the set of vertices with degrees more than $d_{max}^{\beta}$. Then, $\sum_{v\in S}p_{v}=1-o(1)$. Proof 4.4. We can set $p_{v}=(2|W_{v}|)/(2|W|)$. For convenience, set $d_{1}=d_{max}^{\beta}$ and $d_{2}=d_{max}$. We have $f_{d}\approx\alpha n/d^{\gamma}$, for some constant $\alpha$ and $\gamma<3$. $$\sum_{v\in S}2|W_{v}|\approx\sum_{d=d_{1}}^{d_{2}}d^{2}f_{d}\approx\alpha n% \sum_{d=d_{1}}^{d_{2}}d^{2-\gamma}\approx\alpha^{\prime}n(d^{3-\gamma}_{1}-d^{% 3-\gamma}_{2})$$ The total number of wedges behaves like $\alpha^{\prime}nd^{3-\gamma}_{1}$ and hence, $2\sum_{v\in S}|W_{v}|=2|W|-o(|W|)$. ∎ Claim 4.5 $\sum_{v}p_{v}C_{v}=\kappa$ Proof 4.6. $$\displaystyle\sum_{v}p_{v}C_{v}$$ $$\displaystyle=$$ $$\displaystyle\sum_{v}\frac{|W_{v}|}{|W|}\cdot\frac{\textrm{number of closed % wedges in $W_{v}$}}{|W_{v}|}$$ $$\displaystyle=$$ $$\displaystyle\frac{\sum_{v}\textrm{(\# closed wedges in $W_{v}$)}}{|W|}=\kappa.\qed$$ We come to our important lemma. This argues that on the average, neighborhood cuts must have a low conductance. Lemma 4.7. $$\sum_{v}\left(p_{v}\frac{\operatorname{\texttt{cut}}(N_{1}(v))}{|W_{v}|}\right% )=2(1-\kappa)$$ Proof 4.8. We express the sum of $\operatorname{\texttt{cut}}(N_{1}(v))$ as a double summation, and perform some algebraic manipulations. $$\displaystyle\sum_{v}\operatorname{\texttt{cut}}(N_{1}(v))$$ $$\displaystyle=$$ $$\displaystyle\sum_{v}\sum_{u\in N_{1}(v)}|N_{1}(u)\setminus(N_{1}(v)\cup\{v\})|$$ $$\displaystyle=$$ $$\displaystyle\sum_{u}\sum_{v\in N_{1}(u)}|N_{1}(u)\setminus(N_{1}(v)\cup\{v\})|$$ $$\displaystyle=$$ $$\displaystyle\sum_{u}\sum_{v\in N_{1}(u)}\textrm{(\# open wedges centered}$$ $$\displaystyle\ \ \ \ \textrm{at $u$ involving edge $(u,v)$)}$$ $$\displaystyle=$$ $$\displaystyle 2\sum_{u}\textrm{(\# open wedges centered at $u$)}$$ $$\displaystyle=$$ $$\displaystyle 2(1-\kappa)|W|$$ We complete the proof with the following simple observation: $$\sum_{v}\left(p_{v}\frac{\operatorname{\texttt{cut}}(N_{1}(v))}{|W_{v}|}\right% )=\frac{\sum_{v}\operatorname{\texttt{cut}}(N_{1}(v))}{|W|}.\qed$$ Theorem 4.9. There exists a $k$-core in $G$ for $k\geq\kappa d_{max}^{\beta}/2$. Proof 4.10. By Claims 4.3 and 4.5, $$\kappa=\sum_{v}p_{v}C_{v}=\sum_{v\in S}p_{v}C_{v}+\sum_{v\in\overline{S}}p_{v}% C_{v}\leq\sum_{v\in S}p_{v}C_{v}+o(1)$$ This implies that there exists some vertex $v$ such that $d_{v}>d_{max}^{\beta}$ and $C_{v}\geq\kappa-o(1)$ (for convenience, we are going to drop the $o(1)$ lower order term). Consider $G^{\prime}$, the induced subgraph of $G$ on $N_{1}(v)$. The total number of vertices is exactly $d_{v}+1$. Because a $\kappa$-fraction of the wedges centered at $v$ are closed, the number of edges in $G^{\prime}$ is at least $\kappa\binom{d_{v}}{2}$. So $G^{\prime}$ is a dense graph, and we will show that it contains a large core. Perform a core decomposition on $G^{\prime}$. We iteratively remove the vertex of min-degree until the graph has no edges left. The total number of iterations is atmost $d_{v}$. Let the degree of the removed vertex at iteration $i$ be $e_{i}$. We have $\sum_{1\leq i\leq d_{v}}e_{i}=\kappa{d_{v}\choose 2}$. By an averaging argument, there exists some $i$ such that $e_{i}\geq\kappa(d_{v}-1)/2$. At this point, all (unremoved) vertices of $G^{\prime}$ must have a degree of at least $(d_{v}-1)/2$, forming a $k$-core with $k\geq\kappa d_{max}^{\beta}/2$. ∎ We come to our main theorem that proves the existence of a neighborhood cut with low conductance. When $\kappa=1$, we get back the statement of Claim 4.1, since we have a set of conductance $0$. But this theorem also gives non-trivial bounds for large values of $\kappa$. As we mentioned earlier, when $\kappa$ becomes small, this bound is not useful any longer. Theorem 4.11. There exists a neighborhood cut with conductance at least $4(1-\kappa)/(3-2\kappa)$. Proof 4.12. The proof uses the probabilistic method, given the bounds of Lemma 4.7 and Claim 4.5. Suppose we choose a vertex $v$ according to the probability distribution given by $p_{v}$. Let $X$ denote the random variable ${\operatorname{\texttt{cut}}(N_{1}(v))}/{|W_{v}|}$, so $\hbox{\bf E}[X]=2(1-\kappa)$ (Lemma 4.7). By Markov’s inequality, $\Pr[X>4(1-\kappa)]\leq 1/2$. Set $\alpha=2\kappa-1$, and set $\Pr[C_{v}<\alpha]=p$. $$\kappa<p\alpha+(1-p)\Longrightarrow p<(1-\kappa)/(1-\alpha)=1/2$$ By the union bound, the probability that ${\operatorname{\texttt{cut}}(N_{1}(v))}/{|W_{v}|}>4(1-\kappa)$ or $C_{v}<\alpha$ is less than $1$. Hence, there exists some vertex $v$ such that $\operatorname{\texttt{cut}}(N_{1}(v))\leq 4(1-\kappa)|W_{v}|$ and $C_{v}\geq\alpha$ (we can also show that $d_{v}\geq n^{\beta}$). Let $E$ be the set of edges in the subgraph induced on $N_{1}(v)$. Since $C_{v}\geq\alpha$, $|E|\geq\alpha|W_{v}|$. We can bound the conductance of $N_{1}(v)$, $$\displaystyle\frac{\operatorname{\texttt{cut}}(N_{1}(v))}{|E|+\operatorname{% \texttt{cut}}(N_{1}(v))}$$ $$\displaystyle\leq$$ $$\displaystyle\frac{4(1-\kappa)|W_{v}|}{\alpha|W_{v}|+4(1-\kappa)|W_{v}|}=\frac% {4-4\kappa}{3-2\kappa}.\qed$$ 5 Data Before we begin our empirical comparison, we first discuss the data we use to compare and evaluate algorithms. These come from a variety of sources. See Table 2 for a summary of the networks and their basic statistics. All networks are undirected and were symmetrized if the original data were directed. Also, any self-loops in the networks were discarded. We only look at the largest connected component of the network. There are five types of networks: Collaboration networks In these networks, the nodes represent people. The edges represent collaborations, either via a scientific publication (ca-AstroPh [23], cond-mat-2005 [31], arxiv [9], dblp [7, 8]), an email (email-Enron [25]), or a movie (hollywood-2009 [7, 8]). These networks have large mean clustering coefficients and large global clustering coefficients. Social networks The nodes are people again, and the edges are either explicit “friend” relationships (fb-Penn94 [29], fb-A [40], soc-LiveJournal [4]) or observed network activity over edges in a one-year span (fb-A-oneyear [40]). Technological networks The nodes act in a distributed communication network either as agents (p2p-Gnutella25 [27]) or as routers (oregon2 [23], as-22july06 [30], itdk0304 [37]). The edges are observed communications between the nodes. Web graphs The nodes are web-pages, and the edges are symmetrized links between the pages [25]. Forest fire models We also explore the forest fire graph model [23]. This model has large clustering coefficients and a highly skewed degree distribution. The model grows a network by adding a node at each step. On arrival, a new node picks a template uniformly at random from the existing nodes, and then the process “burns” around that node with a specified probability. Burned nodes are then connected to the new node. It has three parameters: the size of the initial clique $k$, the probability of following an edge in the burning process $p$, and the total number of nodes $n$. We specify $k=2$ and $n=25000$, and explore two choices for $p$: short-burning $p=0.4$ and long-burning $p=0.49$. 6 Empirical Neighborhood Communities To compute the conductance scores for each neighborhood in the graph, we adapt any procedure to compute all local clustering coefficients. Most of the work to compute a local clustering coefficient is performed when finding the number of triangles at the vertex. We can express the number of triangles as $\operatorname{\texttt{edges}}(D_{1}(v))/2=(\operatorname{\texttt{edges}}(N_{1}% (v))/2-2d_{v}$, that is, half the number of edges between immediate neighbors of $v$ (recall that we double-count edges). Then $\operatorname{\texttt{cut}}(N_{1}(v))=\operatorname{\texttt{vol}}(N_{1}(v))-% \operatorname{\texttt{edges}}(N_{1}(v))$. And so, given the number of triangles, we can compute the cut assuming we can compute the volume of the neighborhood. This is easy to do with any graph structure that explicitly stores the degrees. We also note that it’s easy to modify Cohen’s procedure for computing triangles with MapReduce [12] to compute neighborhood conductance scores. Two extra steps are required: i) map each triangle back to its constituent nodes, then reduce to find the number of triangles at each node; and ii) map the joined edge and degree graph to both vertices in the edge, then sum the degrees of the neighborhood in the reduce. We use the network community plot from Leskovec et al. [24] to show the information on all of the neighborhood communities. Given the conductance scores from all the neighborhood communities and their size in terms of number of vertices, we first identify the best community at each size. The network community plot shows the relationship between best community conductance and community size on a log-log scale. In Leskovec et al., they found that these plots had a characteristic shape for modern information networks: an initial sharp decrease until the community size reaches between 100 and 1000, then a considerable rise in the conductance scores for larger communities. In our case, neighborhood communities cannot be any larger than the maximum degree plus one, and so we mark this point on the graphs. We always look at the smaller side of the cut, so no community can be larger than half the vertices of the graph. We also mark this location on the plots. Each subsequent figure utilizes this size-vs-conductance plot. Note that we deliberately attempt to preserve the axes limits across figures to promote comparisons. However, some of the figures do have different axis limits to emphasize the range of data. First, we show these network community plots, or perhaps better termed neighborhood community plots for our purposes, for six of the networks in Figure 2. These figures are representative of the best and worst of our results. As a reminder, we make all summary data and codes available online. Plots for other graphs are available on the website given in the introduction. The three graphs on the left show cases where a neighborhood community is or is nearby the best Fiedler community (the red circle). The three graphs on the right highlight instances where the Fiedler community is much better than any neighborhood community. We find it mildly surprising that these neighborhood communities can be as good as the Fiedler community. The structure of the plot for both fb-A-oneyear and soc-LiveJournal1 is instructive. Neighborhoods of the highest degree vertices are not community-like – suggesting that these nodes are somehow exceptional. In fact, by inspection of these communities, many of them are nearly a star graph. However, a few of the large degree nodes define strikingly good communities (these are sets with a few hundred vertices with conductance scores of around $10^{-2}$). This evidence concurs with the intuition from Theorem 4.11. Note that all of these plots show the same shape Leskovec et al. [24] observed. Consequently, in the next set of figures, and in the remainder of the empirical investigation, we compare our neighborhood communities against those computed via the personalized PageRank community scheme employed in that work and described in Section 2.3. Second, Figure 3 compares the neighborhood communities to those computed by sweeping the local personalized PageRank algorithm over all of the vertices as described by Leskovec et al. [24]. We also show the behavior of the whisker communities in this plot as well. The plot adopts the same style of figure. The PageRank communities are in a deep blue color, and the whisker communities are show in a shade of green. Here, we see that the neighborhood communities show similar behavior at small size scales (less than 20 vertices), but the personalized PageRank algorithm is able to find larger communities of smaller, or similar conductance. In these four cases (which are representative of all of the remaining figures), one of the personalized PageRank communities was the Fiedler community. Based on this observation, we wanted to understand how the best community identified by a range of algorithms compares to the neighborhood communities. This is what our third exploration does. The results are shown in Table 3. We computed a set of communities with metis by repeatedly calling the algorithm, asking it to use more partitions each time. See our online codes for the precise details of which partitions were used. By-and-large, the Fiedler cut, personalized PageRank, whiskers, and metis all tend to identify similar communities as the best. There are sometimes small differences. An example of a large difference is in the Penn94 graph, where the Fiedler community is much larger than the best PageRank community and it has better conductance. In this comparison, the neighborhood communities fare poorly. When they identify a set of conductance that’s as good as the rest, then it is always a whisker as well. In the following full section, we explore using these neighborhood communities as seeds for the PageRank algorithms. This will let us take advantage of the observation that the neighborhood communities reflect the shape of the network community plot with PageRank communities 6.1 Empirical Core Communities In our theoretical work, we found that large $k$-cores should always exist in these networks. These should also look like good communities and we briefly investigate this idea in Figure 4. The standard procedure for computing $k$-cores is to iteratively remove in degree-sorted order using a bucket sort [6]. We additionally store the step when each vertex was removed from the graph. We sweep over all cuts induced by this ordering, and for each $k$-core, store the best conductance community. These are plotted in a line that runs from core $1$ to the largest core in the graph. The $1$ core is usually large and a bad-community. Thus, the line usually starts towards the upper-right of each network community plot. Large cores are actually rather good communities. Their conductance scores are noticeably higher than the PageRank communities, but the network plots seem to have similar shapes. We’ll exploit this property in the next section. 7 Seeded communities Many of the theorems about extracting local communities from seed sets [3, 2] require that the seed set itself be a good community. This is precisely what our theoretical results justify for neighborhood communities. Consequently, in this section, we look at growing the neighborhood communities using the local personalized PageRank community algorithm from a set of carefully chosen seeds. One of the key problems with using the personalized PageRank community algorithms is that finding a good set of seeds is not easy. For example, [15] describes a way to do this using the most popular videos on YouTube. Such a meaningful heuristic is not always available. We begin this section by empirically showing that there is an easy-to-identify set of neighborhood communities that are local extrema in the network community plot of the neighborhood communities. First, some quick terminology: we say a neighborhood community is a local minima, or locally minimal, if the conductance of the neighborhood of a vertex is smaller than the conductance of any of the adjacency neighborhood communities. Formally, $$\displaystyle\phi(N_{1}(v))\leq\phi(N_{1}(w))$$ $$\displaystyle\quad\text{ for all $w$ adjacent to $v$ }$$ is true for any locally minimal communities. We find there are only a small set of locally minimal communities with more than 6 vertices. Shown in Figure 5 are the conductance and sizes of the roughly $7000$ communities identified by this measure for the itdk0304 graph. Indeed, among all of the graphs with at least $85,000$ vertices, this heuristic picks out about 3% of the vertices as local minima. In the worst case, it picked out $100,000$ seeds for soc-LiveJournal1. Increasing the minimum size to $10$ vertices reduces this down to $50,000$ seeds. We then use these locally minimal neighborhoods as seed sets for the personalized PageRank community detection procedure. Each locally minimal neighborhoods is grown by up to 50-times its volume by solving for communities using various values of $\sigma$ up to 50. We also explore growing the $k$-cores by up to $5$ times their volume. See Figure 6 for the locally minimal communities and the best grown community from the Les Misérables graph. Figure 7 shows the results. In these figures, we leave the baseline neighborhood communities in for comparison. The key insight is that the dark black line closely tracks the the outline of the pure-PageRank based community profile. That profile was computed by using every vertex in the graph as a seed (although, some vertices were skipped after 10 other clusters had already visited that vertex). This effect is most clearly illustrated by the email-Enron dataset. The dark black line identifies almost all of the local minima from the full PageRank sweep (there are a few it misses). A weakness of these minimal seeds for PageRank is that they may not capture the largest communities. However, the $k$-core grown communities do seem to capture this region of the profile (e.g. arxiv), although ca-AstroPh is an exception. 8 Concluding discussions We recap. Community detection is the problem of finding cohesive collections of nodes in a network. We formalize this as finding vertex sets with small conductance. Modern information networks have many distinctive properties, including a large clustering coefficient and a heavy-tailed degree distribution. We derive a set of theoretical results that show these properties imply that such networks will have vertex neighborhoods that are themselves sets of small conductance. Although our theoretical bounds are weak, they suggest the following experiment: measure the conductance of vertex neighborhoods. Algorithms to compute all such conductance scores are easy to implement by modifying a routine for computing local clustering coefficients. We evaluate these communities on a set of real-world networks. In summary, our results support the idea that there are many neighborhood communities which are good communities in a conductance sense. They may be smaller than desired, however. We next investigate finding a set of locally minimal communities. These communities represent the best of the neighborhood. We find that these locally minimal communities, of which there are many fewer than vertices in the graph (usually around 3%), capture the local minimal in the network community profile plot. More importantly, they can be enlarged using a local personalized PageRank community detection procedure. Afterwards, the profile of these “grown” neighborhoods is strikingly close to the profile of the PageRank communities when seeded with all vertices individually. While we do not discuss timing due to the variability in the quality of implementations, this later procedure is much faster in our experiments. These findings have implications for future studies in community detection. One explanation for the results with the PageRank seeds is that vertex neighborhoods form the core of any good community in the network. We highlight this as a direction for future research into neighborhood communities. References [1] L. Akoglu, M. McGlohon, and C. Faloutsos. oddball: Spotting anomalies in weighted graphs. In M. Zaki, J. Yu, B. Ravindran, and V. Pudi, editors, Advances in Knowledge Discovery and Data Mining, volume 6119 of Lecture Notes in Computer Science, pages 410–421. Springer Berlin / Heidelberg, 2010. [2] R. Andersen, F. Chung, and K. Lang. 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Hand Pose Estimation via Latent 2.5D Heatmap Regression Umar Iqbal${}^{1,2}$ ${}^{1}$NVIDIA Research, ${}^{2}$University of Bonn, Germany    Pavlo Molchanov${}^{1}$ ${}^{1}$NVIDIA Research, ${}^{2}$University of Bonn, Germany    Thomas Breuel${}^{1}$ Juergen Gall${}^{2}$ ${}^{1}$NVIDIA Research, ${}^{2}$University of Bonn, Germany    Jan Kautz${}^{1}$ ${}^{1}$NVIDIA Research, ${}^{2}$University of Bonn, Germany Abstract Estimating the 3D pose of a hand is an essential part of human-computer interaction. Estimating 3D pose using depth or multi-view sensors has become easier with recent advances in computer vision, however, regressing pose from a single RGB image is much less straightforward. The main difficulty arises from the fact that 3D pose requires some form of depth estimates, which are ambiguous given only an RGB image. In this paper we propose a new method for 3D hand pose estimation from a monocular image through a novel 2.5D pose representation. Our new representation estimates pose up to a scaling factor, which can be estimated additionally if a prior of the hand size is given. We implicitly learn depth maps and heatmap distributions with a novel CNN architecture. Our system achieves the state-of-the-art estimation of 2D and 3D hand pose on several challenging datasets in presence of severe occlusions. Keywords:hand pose, 2D to 3D, 3D reconstruction, 2.5D heatmaps 1 Introduction Hand pose estimation from touch-less sensors enables advanced human machine interaction to increase comfort and safety. Estimating the pose accurately is a difficult task due to the large amounts of appearance variation, self occlusions and complexity of the articulated hand poses. 3D hand pose estimation escalates the difficulties even further since the depth of the hand keypoints also has to be estimated. To alleviate these challenges, many proposed solutions simplify the problem by using calibrated multi-view camera systems [1, 2, 3, 4, 5, 6, 7, 8, 9], depth sensors [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], or color markers/gloves [23]. These approaches are, however, not very desirable due to their inapplicability in unconstrained environments. Therefore, in this work, we address the problem of 3D hand pose estimation from RGB images taken from the wild. Given an RGB image of the hand, our goal is to estimate the 3D coordinates of hand keypoints relative to the camera. Estimating the 3D pose from a monocular hand image is an ill-posed problem due to scale and depth ambiguities. Attempting to do so will either not work at all, or results in over-fitting to a very specific environment and subjects. We address these challenges by decomposing the problem into two subproblems both of which can be solved without ambiguities. To this end, we propose a novel 2.5D pose representation and then provide a solution to reconstruct the 3D pose from 2.5D. The proposed 2.5D representation is scale and translation invariant and can be easily estimated from RGB images. It consists of 2D coordinates of the hand keypoints in the input image, and scale normalized depth for each keypoint relative to the root (palm). We perform scale normalization of the depth values such that one of the bones always have a fixed length in 3D space. Such a constrained normalization allows us to directly reconstruct the scale normalized absolute 3D pose. As a second contribution, we propose a novel CNN architecture to estimate the 2.5D pose from images. In the literature, there exists two main learning paradigms, namely heatmap regression [24, 25] and holistic pose regression [26, 27]. Heatmap regression is now a standard approach for 2D pose estimation since it allows to accurately localize the keypoints in the image via per-pixel predictions. Creating volumetric heatmaps for 3D pose estimation [28], however, results in very high computational overhead. Therefore, holistic regression is a standard approach for 3D pose estimation, but it suffers from accurate 2D keypoint localization. Since the 2.5D pose representation requires the prediction of both the 2D pose and depth values, we propose a new heatmap representation that we refer to as 2.5D heatmaps. It consists of 2D heatmaps for 2D keypoint localization and a depth map for each keypoint for depth prediction. We design the proposed CNN architecture such that the 2.5D heatmaps do not have to be designed by hand, but are learned in a latent way. We do this by a softargmax operation which converts the 2.5D heatmaps to 2.5D coordinates in a differentiable manner. The obtained 2.5D heatmaps are compact, invariant to scale and translation, and have the potential to localize keypoints with sub-pixel accuracy. We evaluate our approach on five challenging datasets with severe occlusions, hand object interactions and in-the-wild images. We demonstrate its effectiveness for both 2D and 3D hand pose estimation. The proposed approach outperforms state-of-the-art approaches by a large margin. 2 Related Work Very few works in the literature have addressed the problem of 3D hand pose estimation from a single 2D image. The problem, however, shares several properties with human body pose estimation and many approaches proposed for human body can be easily adapted for hand pose estimation. Hence, in the following, we discuss the related works for 3D articulated pose estimation in general. Model-based methods. These methods represent the articulated 3D pose using a deformable 3D shape model. This is often formulated as an optimization problem, whose objective is to find the model’s deformation parameters such that its projection is in correspondence with the observed image data [29, 30, 31, 32, 33, 34, 35]. Search-based methods. These methods follow a non-parametric approach and formulate 3D pose estimation as a nearest neighbor search problem from the large databases of 3D poses, where the matching is performed based on some low [36, 37] or high [38, 39] level features extracted from the image. From 2D pose to 3D. Earlier methods in this direction learn probabilistic 3D pose models from MoCap data and recover 3D pose by lifting the 2D keypoints  [40, 41, 42, 43]. More recent approaches, on the other hand, use deep neural networks to learn a mapping from 2D pose to 3D [44, 45, 46]. Instead of 2D keypoint locations, [46, 47] use 2D heatmaps [24, 25] as input and learn convolutional neural networks for 3D pose regression. The approach in [46] is one of the first learning based methods to estimate 3D hand pose from a single RGB image. They use an existing 2D pose estimation model [24] to first obtain the heatmaps of hand keypoints and feed them to another CNN that regresses a canonical pose representation and the camera view point. The aforementioned methods have the advantage that they do not necessarily require images with ground-truth 3D pose annotations for training, their major drawback is that they cannot handle re-projection ambiguities (a joint with positive or negative depth will have the same 2D projections). Moreover, they are sensitive to errors in 2D image measurements and the required optimization methods are often prone to local minima due to incorrect initializations. 3D pose from images. These approaches aim to learn a direct mapping from RGB images to 3D pose [48, 49, 50]. While these methods can better handle 2D projection ambiguities, their main downside is that they are prone to over-fitting to the views only present in training data. Thus, they require a large amount of training data with accurate 3D pose annotations. Collecting large amounts of training data in unconstrained environments is, however, infeasible. To this end,  [50] proposes to use Generative Adversarial Networks [51] to convert synthetically generated hand images to look realistic. Other approaches formulate the problem in a multi-task setup to jointly estimate both 2D keypoint locations and 3D pose [52, 28, 27, 53, 54]. Our method also follows this paradigm. The closest work to ours are the approaches of [27, 53, 28] in that they also perform 2.5D coordinate regression. While the approach in [27] performs holistic pose regression with a fully connected output layer, [53] follows a hybrid approach and combines heatmap regression with holistic regression. Holistic regressions is shown to perform well for human body but fails in cases where very precise localization is required, e.g., finger-tips in case of hands. In order to deal with this, the approach in [28] performs dense volumetric regression. This, however, substantially increases the model size, which in turn forces to work at a lower spatial resolution. Our approach, on the other hand, retains the input spatial resolution and allows one to localize hand keypoints with sub-pixel accuracy. It enjoys the differentiability and compactness of holistic regression-based methods, translation invariance of volumetric representations, while also providing high spatial output resolution. Moreover, in contrast to existing methods, it does not require hand-designed target heatmaps, which can arguably be sub-optimal for a particular problem, but rather implicitly learns a latent 2.5D heatmap representation and converts them to 2.5D coordinates in a differentiable way. Finally, note that given the 2.5D coordinates, the 3D pose has to be recovered. The existing approaches either make very strong assumptions such as the ground-truth location of the root [27] and the global scale of the hand in 3D is known [53], or resort to an approximate solution [28]. The approach [54] tries to directly regress the absolute depth from the cropped and scaled image regions which is a very ambiguous task. In contrast, our approach does not make any assumptions, nor does it try to solve any ambiguous task. Instead, we propose a scale and translation invariant 2.5D pose representation, which can be easily obtained using CNNs, and then provide an exact solution to obtain the absolute 3D pose up to a scaling factor and only approximate the global scale of the hand. 3 Hand Pose Estimation An overview of the proposed approach can be seen in Fig. 1. Given an RGB image $\bf{I}$ of a hand, our goal is to estimate the 2D and 3D positions of all the $K=21$ keypoints of the hand. We define the 2D hand pose as $\mathbf{p}=\{p_{k}\}_{k\in K}$ and 3D pose as $\mathbf{P}=\{P_{k}\}_{k\in K}$, where $p_{k}=(x_{k},y_{k})\in\mathbb{R}^{2}$ represents the 2D pixel coordinates of the keypoint $k$ in image $\bf{I}$ and $P_{k}=(X_{k},Y_{k},Z_{k})\in\mathbb{R}^{3}$ denotes the location of the keypoint in the 3D camera coordinate frame measured in millimeters. The Z-axis corresponds to the optical axis. Given the intrinsic camera parameters $\mathcal{K}$, the relationship between the 3D location $P_{k}$ and corresponding 2D projection $p_{k}$ can be written as follows under a perspective projection: $$\displaystyle Z_{k}\begin{pmatrix}x_{k}\\ y_{k}\\ 1\end{pmatrix}=\mathcal{K}\begin{pmatrix}X_{k}\\ Y_{k}\\ Z_{k}\\ 1\end{pmatrix}=\mathcal{K}\begin{pmatrix}X_{k}\\ Y_{k}\\ Z_{root}+Z_{k}^{r}\\ 1\end{pmatrix}\quad k\in 1,\dots K$$ (1) where $k\in 1,\dots K$, $Z_{root}$ is the depth of the root keypoint, and $Z_{k}^{r}=Z_{k}-Z_{root}$ corresponds to the depth of the $k^{th}$ keypoint relative to the root. In this work we use palm of the hand as the root keypoint. 3.1 2.5D Pose Representation Given an image $\bf{I}$, we need to have a function $\mathcal{F}$, such that $\mathcal{F}:\bf{I}\to\mathbf{P}$, and the estimated 3D hand pose $\mathbf{P}$ can be projected to 2D with the camera parameters $\mathcal{K}$. However, predicting the absolute 3D hand pose in camera coordinates is infeasible due to irreversible geometry and scale ambiguities. We, therefore, choose a 2.5D pose representation which can be recovered from a 2D image without ambiguity, and provide a solution to recover the 3D pose from the 2.5D representation. We define the 2.5D pose as $\mathbf{P}^{2.5D}_{k}=\{P^{2.5D}_{k}\}_{k\in K}$, where $P^{2.5D}_{k}=(x_{k},y_{k},Z^{r}_{k})$. The coordinates $x_{k}$ and $y_{k}$ are the image pixel coordinates of the $k^{\mathrm{th}}$ keypoint and $Z^{r}_{k}$ is its metric depth relative to the root keypoint. Moreover, in order to remove the scale ambiguities, we scale-normalize the 3D pose as follows: $$\mathbf{\hat{P}}=\dfrac{C}{s}\cdot\mathbf{P},$$ (2) where $s=\|P_{n}-P_{parent(n)}\|_{2}$ is computed for each 3D pose independently. This results in a normalized 3D pose $\hat{\mathbf{P}}$ with a constant distance $C$ between a specific pair of keypoints $(n,parent(n))$. Subsequently, our normalized 2.5D representation for keypoint $k$ becomes $\hat{P}^{2.5D}_{k}=(x_{k},y_{k},\hat{Z}^{r}_{k})$. Note that the 2D pose does not change due to the normalization, since the projection of the 3D pose remains the same. Such a normalized 2.5D representation has several advantages: it allows to effectively exploit image information; it enables dense pixel-wise prediction (Sec. 4); it allows us to perform multi-task learning so that multiple sources of training data can be used; and finally it allows us to devise an approach to exactly recover the absolute 3D pose up to a scaling factor. We describe the proposed solution to obtain the function $\mathcal{F}$ in Sec. 4, while the 3D pose reconstruction from 2.5D pose is explained in the next section. 3.2 3D Pose Reconstruction from 2.5D Given a 2.5D pose $\hat{\mathbf{P}}^{2.5D}=\mathcal{F}(\bf{I})$, we need to find the depth $\hat{Z}_{root}$ of the root keypoint to reconstruct the scale normalized 3D pose $\hat{\mathbf{P}}$ using Equation (1). While there exists many 3D poses that can have the same 2D projection, given the 2.5D pose and intrinsic camera parameters, there exists a unique 3D pose that satisfies $$(\hat{X}_{n}-\hat{X}_{m})^{2}+(\hat{Y}_{n}-\hat{Y}_{m})^{2}+(\hat{Z}_{n}-\hat{% Z}_{m})^{2}=C^{2},$$ (3) where $(n,\ m\!=\!parent(n))$ is the pair of keypoints used for normalization in Equation (2). The equation above can be rewritten in terms of the 2D projections $(x_{n},y_{n})$ and $(x_{m},y_{m})$ as follows: $$(x_{n}\hat{Z}_{n}-x_{m}\hat{Z}_{m})^{2}+(y_{n}\hat{Z}_{n}-y_{m}\hat{Z}_{m})^{2% }+(\hat{Z}_{n}-\hat{Z}_{m})^{2}=C^{2}.$$ (4) Subsequently, replacing $\hat{Z}_{n}$ and $\hat{Z}_{m}$ with $(\hat{Z}_{root}+\hat{Z}_{n}^{r})$ and $(\hat{Z}_{root}+\hat{Z}_{m}^{r})$, respectively, yields: $$\displaystyle(x_{n}(\hat{Z}_{root}+\hat{Z}_{n}^{r})-x_{m}(\hat{Z}_{root}+\hat{% Z}_{m}^{r}))^{2}+(y_{n}(\hat{Z}_{root}+\hat{Z}_{n}^{r})-y_{m}(\hat{Z}_{root}+% \hat{Z}_{m}^{r}))^{2}\\ \displaystyle+((\hat{Z}_{root}+\hat{Z}_{n}^{r})-(\hat{Z}_{root}+\hat{Z}_{m}^{r% }))^{2}=C^{2}.$$ (5) Given the 2.5D coordinates of both keypoints $n$ and $m$, $Z_{root}$ is the only unknown in the equation above. Simplifying the equation further leads to a quadratic equation with the following coefficients $$\displaystyle a$$ $$\displaystyle=(x_{n}-x_{m})^{2}+(y_{n}-y_{m})^{2}\textsc{}$$ $$\displaystyle b$$ $$\displaystyle=\hat{Z}_{n}^{r}(x_{n}^{2}+y_{n}^{2}-x_{n}x_{m}-y_{n}y_{m})+\hat{% Z}_{m}^{r}(x_{m}^{2}+y_{m}^{2}-x_{n}x_{m}-y_{n}y_{m})$$ (6) $$\displaystyle c$$ $$\displaystyle=(x_{n}\hat{Z}_{n}^{r}-x_{m}\hat{Z}_{m}^{r})^{2}+(y_{n}\hat{Z}_{n% }^{r}-y_{m}\hat{Z}_{m}^{r})^{2}+(\hat{Z}_{n}^{r}-\hat{Z}_{m}^{r})^{2}-C^{2}.$$ This results in two values for the unknown variable $Z_{root}$, one in front of the camera and one behind the camera. We choose the solution in front of the camera $$\hat{Z}_{root}=0.5(-b+\sqrt{b^{2}-4ac})/a.$$ (7) Given the value of $Z_{root}$, $\hat{\mathbf{P}}^{2.5D}$, and the intrinsic camera parameters $\mathcal{K}$, the scale normalized 3D pose can be reconstructed by back-projecting the 2D pose $\mathbf{p}$ using Eq. (1). In this paper, we use $C=1$, and use the distance between the first joint (metacarpophalangeal - MCP) of the index finger and palm (root) to calculate the scaling factor $s$. We choose these keypoints since they are the most stable in terms of 2D pose estimation. 3.3 Scale Recovery Up to this point, we have obtained the 2D and scale normalized 3D pose $\hat{\mathbf{P}}$ of the hand. In order to recover the absolute 3D pose $\mathbf{P}$, we need to know the global scale of the hand. In many scenarios this can be known a priori, however, in case it is not available, we estimate the scale $\hat{s}$ by $$\hat{s}=\underset{s}{\mathrm{argmin}}\sum_{k,l\in\mathcal{E}}(s\cdot\|\hat{P}_% {k}-\hat{P}_{l}\|-\mu_{kl})^{2},$$ (8) where $\mu_{kl}$ is the mean length of the bone between keypoints $k$ and $l$ in the training data, and $\mathcal{E}$ defines the kinematic structure of the hand. 4 2.5D Pose Regression In order to regress the 2.5D pose $\mathbf{\hat{P}}^{2.5D}$ from an RGB image of the hand, we learn the function $\mathcal{F}$ using a CNN. In this section, we first describe an alternative formulation of the CNN (Sec. 4.1) and then describe our proposed solution for regressing latent 2.5D heatmaps in Sec. 4.2. In all formulations, we train the CNNs using a loss function $\mathcal{L}$ which consists of two parts $\mathcal{L}_{xy}$ and $\mathcal{L}_{\hat{Z}^{r}}$, each responsible for the regression of 2D pose and root-relative depths for the hand keypoints, respectively. Formally, the loss can be written as follows: $$\mathcal{L}(\hat{\mathbf{P}}^{2.5D})=\mathcal{L}_{xy}(\mathbf{p},\mathbf{p}_{% gt})+\alpha\mathcal{L}_{\hat{Z}^{r}}(\hat{\bf{Z}}^{r},\hat{\bf{Z}}^{r,gt}),$$ (9) where $\hat{\bf{Z}}^{r}=\{\hat{Z}_{k}^{r}\}_{r\in K}$ and $\hat{\bf{Z}}^{r,gt}=\{\hat{Z}_{k}^{r,gt}\}_{r\in K}$ and $gt$ refers to ground-truth annotations. This loss function has the advantage that it allows to utilize multiple sources of training, i.e., in-the-wild images with only 2D pose annotations and constrained or synthetic images with accurate 3D pose annotations. While $\mathcal{L}_{xy}$ is valid for all training samples, $\mathcal{L}_{\hat{Z}^{r}}$ is enforced only when the 3D pose annotations are available, otherwise it is not considered. 4.1 Direct 2.5D Heatmap Regression Heatmap regression is the de-facto approach for 2D pose estimation [55, 24, 25, 56]. In contrast to holistic regression, heatmaps have the advantage of providing higher output resolution, which helps in accurately localizing the keypoints. However, they are scarcely used for 3D pose estimation since a 3D volumetric heatmap representation [28] results in a high computational and storage cost. We, thus, propose a novel and compact heatmap representation, which we refer to as 2.5D heatmaps. It consists of 2D heatmaps $H^{2D}$ for keypoint localization and depth maps $H^{\hat{z}^{r}}$ for depth predictions. While the 2D heatmap $H^{2D}_{k}$ represents the likelihood of the $k^{th}$ keypoint at each pixel location, the depth map $H^{\hat{z}^{r}}_{k}$ provides the scale normalized and root-relative depth prediction for the corresponding pixels. This representation of depth maps is scale and translation invariant and remains consistent across similar hand poses, therefore, it is significantly easier to be learned using CNNs. The CNN provides a $2K$ channel output with $K$ channels for 2D localization heatmaps $H^{2D}$ and $K$ channels for depth maps $H^{\hat{z}^{r}}$. The target heatmap $H_{k}^{2D,gt}$ for the $k^{\mathrm{th}}$ keypoint is defined as $$H^{2D,gt}_{k}(p)=\exp\left(-\dfrac{\|p-p^{gt}_{k}\|}{\sigma^{2}}\right),\quad p\in\Omega$$ (10) where $p^{gt}_{k}$ is the ground-truth location of the $k^{\mathrm{th}}$ keypoint, $\sigma$ controls the standard deviation of the heatmaps and $\Omega$ is the set of all pixel locations in image $\bf{I}$. Since the ground-truth depth maps are not available, we define them by $$H^{\hat{z}^{r}}_{k}=\hat{Z}_{k}^{r,gt}\cdot H^{2D,gt}_{k}$$ (11) where $\hat{Z}_{k}^{r,gt}$ is the ground-truth normalized root-relative depth value of the $k^{\mathrm{th}}$ keypoint. During inference, the 2D keypoint position is obtained as the pixel with the maximum likelihood $$p_{k}={\underset{p}{\mathrm{argmax}}~{}H_{k}^{2D}(p)},$$ (12) and the corresponding depth value is obtained as, $$\hat{Z}^{r}_{k}=H^{\hat{z}^{r}}_{k}(p_{k}).$$ (13) 4.2 Latent 2.5D Heatmap Regression The 2.5D heatmap representation as described in the previous section is, arguably, not the most optimal representation. First, the ground-truth heatmaps are hand designed and are not ideal, i.e., $\sigma$ remains fixed for all keypoints and cannot be learned due to indifferentiability of Eq. (12). Ideally, it should be adapted for each keypoint, e.g., heatmaps should be very peaky for finger-tips while relatively wide for the palm. Secondly, the Gaussian distribution is a natural choice for 2D keypoint localization, but is not very intuitive for depth prediction, i.e., the depth stays roughly the same throughout the palm but is modeled as Gaussians. Therefore, we alleviate these problems by proposing a latent representation of 2.5D heatmaps, i.e., the CNN learns the optimal representation by minimizing a loss function in a differentiable way. To this end, we consider the $2K$ channel output of the CNN as latent variables $H_{k}^{*2D}$ and $H_{k}^{*\hat{z}^{r}}$ for 2D heatmaps and depth maps, respectively. We then apply spatial softmax normalization to 2D heatmap $H_{k}^{*2D}$ of each keypoint $k$ to convert it to a probability map $$H^{2D}_{k}(p)=\dfrac{\exp(\beta_{k}H^{*2D}_{k}(p))}{\sum_{p^{\prime}\in\Omega}% \exp(\beta_{k}H^{*2D}_{k}(p^{\prime}))},$$ (14) where $\Omega$ is the set of all pixel locations in the input map $H_{k}^{*2D}$, and $\beta_{k}$ is the learnable parameter that controls the spread of the output heatmaps $H^{2D}$. Finally, the 2D keypoint position of the $k^{\mathrm{th}}$ keypoint is obtained as the weighted average of the 2D pixel coordinates as, $$p_{k}=\sum_{p\in\Omega}H^{2D}_{k}(p)\cdot p,$$ (15) while the corresponding depth value is obtained as the summation of the Hadamard product of $H^{2D}_{k}(p)$ and $H^{*\hat{z}^{r}}_{k}(p)$ as follows $$\hat{Z}^{r}_{k}=\sum_{p\in\Omega}H^{2D}_{k}(p)\circ H^{*\hat{z}^{r}}_{k}(p).$$ (16) A pictorial representation of this process can be seen in Fig. 1. The operation in Eq. (15) is known as soft-argmax in the literature [57]. Note that the computation of both the 2D keypoint location and the corresponding depth value is fully differentiable. Hence the network can be trained end-to-end, while generating latent 2.5D heatmap representation. In contrast to the heatmaps with fixed standard deviation in Sec. 4.1, the spread of the latent heatmaps can be adapted for each keypoint by learning the parameter $\beta_{k}$, while the depth maps are also learned implicitly without any ad-hoc design choices. A comparison between heatmaps obtained by direct heatmap regression and the ones implicitly learned by latent heatmap regression can be seen in Fig. 2. 5 Experiments In this section, we evaluate the performance of the proposed approach in detail and also compare it with the state-of-the-art. For this, we use five challenging datasets – namely, the Dexter+Object dataset  [22], the Ego-Dexter dataset [58], the Stereo Hand Pose dataset [59] dataset, the Rendered Hand Pose dataset [46], and the MPII+NZSL dataset [56]. The details of each dataset are as follows. Dexter+Object (D+O). The D+O dataset [22] provides 6 test video sequences with $3145$ frames in total. All sequences are recorded using a static camera with a single person interacting with an object. The dataset provides both 2D and 3D pose annotations for the finger-tips of the left hand. EgoDexter (ED). The ED dataset [58] provides both 2D and 3D pose annotations for 4 testing video sequences with $3190$ frames. The videos are recorded with body-mounted camera from egocentric viewpoints and contain cluttered backgrounds, fast camera motion, and complex interactions with various objects. Similar to D+O dataset, it only provides annotations for the finger-tips. In addition, [58] also provides the so called SynthHands dataset containing synthetic images of hands from ego-centric views with accurate 3D pose annotations. The images are provided with chroma-keyed background, that we replace with random backgrounds from NYU depth dataset [60] and use them as additional training data for testing on the ED dataset. Stereo Hand Pose (SHP). The SHP dataset [59] provides 2D and 3D pose annotations of $21$ keypoints for 6 pairs of stereo sequences with a total of 18000 stereo pairs of frames. The sequences record a single person performing a variety of gestures with different backgrounds and lighting conditions. Rendered Hand Pose (RHP). The RHP dataset [46] provides $41258$ and $2728$ images for training and testing, respectively. All images are generated synthetically using a blending software and come with accurate 2D and 3D annotations of $21$ keypoints. The dataset contains 20 different characters performing 39 actions with different lighting conditions, backgrounds, and camera viewpoints. MPII+NZSL. The MPII+NZSL dataset [56] provides $2800$ 2D hand pose annotations for in-the-wild images. The images are taken from YouTube videos of people performing daily life activities and New Zealand Sign Language exercises. The dataset is split into $2000$ and $800$ images for training and testing, respectively. In addition,  [56] also provides additional training data that contains $14261$ synthetic images and $14817$ real images. The annotations for real images are generated automatically using multi-view bootstrapping. We refer to these images as MVBS in the rest of this paper. 5.1 Evaluation Metrics For our evaluation on the D+O, ED, SHP, and RHP datesets, we use average End-Point-Error (EPE) and the Area Under the Curve (AUC) on the Percentage of Correct Keypoints (PCK). We report the performance for both 2D and 3D hand pose where the performance metrics are computed in pixels and millimeters (mm), respectively. We use the publicly available implementation of evaluation metrics from [46]. For the D+O and ED datasets, we follow the evaluation protocol proposed by [50], which requires estimating the absolute 3D pose with global scale. For SHP and RHP, we follow the protocol proposed by [46], where the root keypoints of the ground-truth and estimated poses are aligned before calculating the metrics. For the MPII+NZSL dataset, we follow [56] and report head-normalized PCK (PCKh) in our evaluation. 5.2 Implementation Details For 2.5D heatmap regression we use an Encoder-Decoder network architecture with skip connections [61, 25] and fixed number of channels (256) in each convolutional layer. The input to our model is a $128\times 128$ image, which produces 2.5D heatmaps as output with the same resolution as the input image. Further details about the network architecture and training can be found in the appendix. For all the video datasets, i.e., D+O, ED, SHP we use the YOLO detector [62] to detect the hand in the first frame of the video, and generate the bounding box in the subsequent frames using the estimated pose of the previous frame. We trained the hand detector using the training sets of all aforementioned datasets. 5.3 Ablation Studies We evaluate the proposed method under different settings to better understand the impact of different design choices. We chose the D+O dataset for all ablation studies, mainly because it does not have any training data. Thus, it allows us to evaluate the generalizability of the proposed method. Finally, since the palm (root) joint is not annotated, it makes it compulsory to estimate the absolute 3D pose in contrast to the commonly used root-relative 3D pose. We use Eq. (8) to estimate the global scale of each 3D pose using the mean bone lengths from the SHP dataset. The ablative studies are summarized in Tab. 1. We first examine the impact of different choices of CNN architectures for 2.5D pose regression. For holistic 2.5D pose regression, we use the commonly adopted [27] ResNet-50 [63] model. The details can be found in the appendix. We use the SHP and RHP datasets to train the models. Using a holistic regression approach results in an AUC of $0.41$ and $0.54$ for 2D and 3D pose, respectively. Directly regressing the 2.5D heatmaps significantly improves the performance of 2D pose estimation ($0.41$ vs. $0.57$), while also raising the 3D pose estimation accuracy from $0.54$ AUC to $0.55$. Using latent heatmap regression improves the performance even further to $0.59$ AUC and $0.57$ AUC for 2D and 3D pose estimation, respectively. While the holistic regression approach achieves a competitive accuracy for 3D pose estimation, the accuracy for 2D pose estimation is inferior to the heatmap regression due to its limited spatial output resolution. We also evaluate the impact of training the network in a multi-task setup. For this, we train the model with additional training data from [56] which provides annotations for 2D keypoints only. First, we only use the $2000$ manually annotated real images from the training set of MPII+NZSL dataset. Using additional 2D pose annotations significantly improves the performance. Adding additional $15,000$ annotations of real images, automatically generated by multi-view bootstrapping [56], improves the performance only slightly. Hence, only $2000$ real images are sufficient to generalize the model trained on synthetic data to a realistic scenario. The annotations of the finger tips in the D+O dataset are slightly different than the other datasets. In the D+O dataset, the finger tips are annotated at the middle of the tips whereas other datasets annotate it at the edge of the nails. To remove this discrepancy, we shorten the last bone of the finger tip by $0.9$. Fixing the annotation differences results in further improvements, revealing the true performance of the proposed approach. Finally, we also evaluate the impact of using multiple stages in the network, where each stage produces latent 2.5D heatmaps as output. While the first stage only uses the features extracted from the input image using the initial block of convolutional layers, each subsequent stage also utilizes the output of the preceding stage as input. This provides additional contextual information to the subsequent stages and helps in incrementally refining the predictions. Similar to [25, 24] we provide local supervision to the network by enforcing the loss at the output of each stage (see appendix for more details). Adding one extra stage to the network increases the 3D pose estimation accuracy from AUC $0.69$ to $0.71$, but decreases the 2D pose estimation accuracy from AUC $0.76$ to $0.74$. The decrease in 2D pose estimation accuracy is most likely due to over-fitting to the training datasets. Remember that we do not use any training data from the D+O dataset. In the rest of this paper, we always use networks with two stages unless stated otherwise. 5.4 Comparison to State-of-the-Art We provide a comparison of the proposed approach with state-of-the-art methods on all aforementioned datasets. Note that different approaches use different training data. We thus replicate the training setup of the corresponding approaches for a fair comparison. Fig. 2(a) and 2(b) compare the proposed approach with other methods on the D+O dataset for 2D and 3D pose estimation, respectively. In particular, we compare with the state-of-the-art approach by Zimmerman and Brox (Z&B) [46] and the contemporary work by Mueller et al. [50]. We use the same training data (SHP+RHP) for comparison with [46] (AUC $0.64$ vs $0.49$), and only use additional data for comparison with [50](AUC $0.74$ vs $0.64$). For the 3D pose estimation accuracy (Fig. 2(b)), the approach [46] is not included since it only estimates scale normalized root-relative 3D pose. Our approach clearly outperforms current RGB state-of-the-art method by Mueller et al. [50] by a large margin. The approach [50] utilizes the video information to perform temporal smoothening and also performs subject specific adaptation under the assumption that the users hold their hand parallel to the camera image plane. In contrast, we only perform frame-wise predictions without temporal filtering or user assumptions. Additionally, we report the results of the depth based approach by Sridhar et al. [22], which are obtained from [50]. While the RGB-D sensor based approach [22] still works better, our approach takes a giant leap forward as compared to the existing RGB based approaches. Fig. 2(c) compares the proposed method with existing approaches on the SHP dataset. We use the same training data (SHP+RHP) as in [46] and outperform all existing methods despite the already saturated accuracy on the dataset and the additional training data and temporal information used in [50]. Fig. 2(d) compares the 2D pose estimation accuracy on the EgoDexter dataset. While we outperform all existing methods for 2D pose estimation, none of the existing approaches report their performance for 3D pose estimation on this dataset. We, however, also report our performance in Fig. 2(e). The results on the RHP dataset are reported in Tab. 2. Our approach significantly outperforms [46] even though they use ground-truth 2D poses to estimate the 3D poses. Since the dataset provides 3D pose annotations for complete hand skeleton, we also report the performance of the proposed approach when the ground-truth depth of the root joint and the global scale of the hand is known (w. GT $\hat{Z}_{root}$ and $\hat{s}$). We can see that our approach of 3D pose reconstruction and scale recovery is very close to the ground-truth. Finally, for completeness, in Fig. 2(f) we compare our approach with [56] which is a state-of-the-art approach for 2D pose estimation. The evaluation is performed on the test set of the MPII+NZSL dataset. We follow [56] and use the provided center location of the hand and the size of the head of the person to obtain the hand bounding box. We define a square bounding box with height and width equals to $0.7\times head\textnormal{-}length$. We report two variants of our method; 1) the model trained for both 2D and 3D pose estimation using the datasets for both tasks, and 2) a model trained for only 2D pose estimation using the same training data as in [56]. In both cases we use the models trained with 2-stages. Our approach performs similar or better than [56], even though we use a smaller backbone network as compared to the 6-stage Convolutional Pose Machines (CPM) network [24] used in [56]. The CPM model with 6-stages has $51M$ parameters, while our $1$ and $2$-stage models have only $17M$ and $35M$ parameters, respectively. Additionally, our approach also infers the 3D hand pose. Some qualitative results for 3D hand pose estimation for in-the-wild images can be seen in Fig. 4. 6 Conclusion We have presented a method for 3D hand pose estimation from a single RGB image. We demonstrated that the absolute 3D hand pose can be reconstructed efficiently from a single image up to a scaling factor. We presented a novel 2.5D pose representation which can be recovered easily from RGB images since it is invariant to absolute depth and scale ambiguities. It can be represented as 2.5D heatmaps, therefore, allows keypoint localization with sub-pixel accuracy. We also proposed a CNN architecture to learn 2.5D heatmaps in a latent way using a differentiable loss function. Finally, we proposed an approach to reconstruct the 3D hand pose from 2.5D pose representation. The proposed approach demonstrated state-of-the-art results on five challenging datasets with severe occlusions, object interactions and images taken from the wild. Appendix In this appendix we provide implementation details to reproduce results in the paper (Sec. 0.A) and also provide additional ablative studies in Sec. 0.B. Appendix 0.A Implementation Details 0.A.1 Holistic 2.5D Regression We follow [27] and use a ResNet-50 [63] model for holistic regression. As in [27], we mean normalize the poses before training and use $L_{1}$ norm as the loss function. The input to the network is a $224\times 224$ image. We use $\alpha=1$ since the poses are normalized and the range of $\mathcal{L}_{xy}$ and $\mathcal{L}_{\hat{z}}$ is similar. The initial learning rate is set to $0.03$. 0.A.2 2.5D Heatmap Regression For 2.5D heatmap regression, we use an Encoder-Decoder architecture with skip connections [25] and fixed number of channels (256) in each convolutional layer. The detailed network architecture can be seen in Fig. 5. The input to our model is a $128\times 128$ image, which produces full resolution latent/direct 2.5D heatmaps as output. 0.A.2.1 Direct 2.5D Heatmap Regression: We use $\sigma=5$ to create the target heatmaps for training. We follow [24, 25] and use $L_{2}$ norm as the loss function. The initial learning rate is set to $0.0001$. 0.A.2.2 Latent 2.5D Heatmap Regression: For latent 2.5D regression, the $\mathcal{L}_{xy}$ is calculated on 2D pixel coordinates and $\mathcal{L}_{\hat{z}^{r}}$ is computed on the scale normalized root-relative depths. Therefore, a balancing factor is required. We empirically chose $\alpha=20$ such that both losses have a similar magnitude. In our experiments we also tried with $\alpha=1$ and the performance dropped insignificantly by less than $1\%$. We use $L_{1}$ norm as the loss function with a learning rate of $0.001$. The overview of the two-stage model for 2.5D heatmap regression can be seen in Fig. 6 0.A.3 Common details We train the models only for the right hand, and during inference, flip the left hand images before passing them to the network. All models are trained from scratch with a batch size of $32$ for $70$ epochs. During training, we crop the bounding box such that the hand is  $70\%$ of the image. We perform data augmentation by rotation $(0,$$)$, translation ($\pm 20$ pixel), scale (0.7,1.1), and color transformations. In addition, in order to make the models robust against object occlusions, we follow [58] and randomly add textured objects (ovals and cubes) to the training samples. We decay the learning rates for all models by a factor of $10$ after every $30$ epochs, and use SGD with $\mathrm{momentum}=0.9$. During mixed training, the training images with 2D-only or 3D annotations are sampled with equal probability. Appendix 0.B Additional ablative studies The skeleton of the hand used in this work can be seen in Fig. 6(a). We evaluate the impact of pair of keypoints (bones) selected for 3D pose normalization (eqt. 2 in Fig. 6(b). For this, we trained a separate CNN model while using a specific pair of keypoints for normalization. We can see that the performance remains consistent ($\approx 0.69$) for most of the bones. 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Social-Group-Agnostic Word Embedding Debiasing via the Stereotype Content Model Ali Omrani    Brendan Kennedy    Mohammad Atari    Morteza Dehghani University of Southern California {aomrani,btkenned,atari,mdehghan}@usc.edu Abstract Existing word embedding debiasing methods require social-group-specific word pairs (e.g., “man”–“woman”) for each social attribute (e.g., gender), which cannot be used to mitigate bias for other social groups, making these methods impractical or costly to incorporate understudied social groups in debiasing. We propose that the Stereotype Content Model (SCM), a theoretical framework developed in social psychology for understanding the content of stereotypes, which structures stereotype content along two psychological dimensions — “warmth” and “competence” — can help debiasing efforts to become social-group-agnostic by capturing the underlying connection between bias and stereotypes. Using only pairs of terms for warmth (e.g., “genuine”–“fake”) and competence (e.g., “smart”–“stupid”), we perform debiasing with established methods and find that, across gender, race, and age, SCM-based debiasing performs comparably to group-specific debiasing. 1 Introduction The societal impacts of Natural Language Processing (NLP) have stimulated research on measuring and mitigating the unintended social-group biases encoded in language models Hovy and Spruit (2016). However, the majority of this important line of work is atheoretical in nature and “fails to engage critically with what constitutes ‘bias’ in the first place” Blodgett et al. (2020). Although there is a multitude of approaches to word embedding debiasing (Bolukbasi et al., 2016; Zhao et al., 2018; Dev and Phillips, 2019), most of these approaches rely on group-specific bias subspaces. Recently, Gonen and Goldberg (2019) found that, for example, gender bias in word embeddings is more systematic than simply debiasing along the “gender” subspace. Combining well-established models of bias and stereotyping from social psychology with word embedding debiasing efforts, we follow Blodgett et al. (2020) in proposing a theory-driven debiasing approach that does not rely on a particular social group such as gender or race. We show that, by relying on a theoretical understanding of social stereotypes to define a group-agnostic bias subspace, word embeddings can be adequately debiased across multiple social attributes. Group-specific debiasing, which debiases along subspaces defined by social groups (e.g., gender or race), is not only atheoretical but also unscalable. For example, previous works’ excessive focus on gender bias in word embeddings has driven the development of resources for gender debiasing (e.g., equality sets for gender), but biases associated with other social groups remain understudied. This is due to the fact that resources developed for one social group (e.g., gender) do not translate easily to other groups. Furthermore, group-specific debiasing is limited in terms of generalizability: as shown by Agarwal et al. (2019), stereotype content in word embeddings is deep-rooted, and thus is not easily removed using explicit sets of group-specific words. In contrast, a social-group-agnostic approach would not have such restrictions. Social-group bias mitigation, as a societal problem, can benefit from social psychological theories to understand the underlying structure of language-embedded biases rather than attending to ad hoc surface patterns. The Stereotype Content Model (SCM) Fiske et al. (2002) is a theoretical framework developed in social psychology to understand the content and function of stereotypes in interpersonal and intergroup interactions. The SCM proposes that human stereotypes are captured by two primary dimensions of warmth (e.g., trustworthiness, friendliness) and competence (e.g., capability, assertiveness). From a socio-functional, pragmatic perspective, people’s perception of others’ intent (i.e., warmth) and capability to act upon their intentions (i.e., competence) affect their subsequent emotion and behavior Cuddy et al. (2009). Depending on historical processes, various social groups may be located in different stereotypic quadrants (high vs. low on warmth and competence) based on this two-dimensional model. Here, we propose that SCM-based debiasing can provide a theory-driven and scalable solution for mitigating social-group biases in word embeddings. In our experiments, we find that by debiasing with respect to the subspace defined by warmth and competence, our SCM-based approach performs comparably with group-specific debiasing. Our approach fares well both in terms of bias reduction and the preservation of embedding utility (i.e., the preservation of semantic and syntactic information) (Bolukbasi et al., 2016), while having the advantage of being social-group-agnostic. 2 Background 2.1 Post hoc Word Embedding Debiasing Our work builds on post hoc debiasing, removing biases by modifying pre-trained word embeddings. Most work we review focuses on gender-related debiasing (e.g., Bolukbasi et al., 2016; Zhao et al., 2018; Dev and Phillips, 2019); importantly, we focus our work on other social categories as well, bringing attention to these understudied groups. Originally, Bolukbasi et al. (2016) proposed Hard Debiasing (HD) for gender bias. HD removes the gender component from inherently non-gendered words and enforces an equidistance property for inherently gendered word pairs (equality sets). Two follow-ups to this work include Manzini et al. (2019), which formulated a multiclass version of HD for attributes such as race, and Dev and Phillips (2019), which introduced Partial Projection, a method that does not require equality sets and is more effective than HD in reducing bias. Extending these approaches to other social attributes is not trivial because a set of definitional word pairs has to be curated for each social group, which is a dynamic and context-dependent task because these pairs are dependent on historical moment. Gonen and Goldberg (2019) provided evidence that gender bias in word embeddings is deeper than previously thought, and methods based on projecting words onto a “gender dimension” only hide bias superficially. They showed that after debiasing, most words maintain their relative position in the debiased subspace. In this work, we address the shortcomings highlighted by Gonen and Goldberg and Agarwal et al. with a theory-driven bias subspace, rather than algorithmic improvement. 2.2 Bias and the Stereotype Content Model The bias found in language models is rooted in human biases Caliskan and Lewis (2022); thus, to alleviate such biases, we should ground our debiasing approaches in social psychological theories of stereotyping Blodgett et al. (2020). The Stereotype Content Model (SCM) Fiske et al. (2002); Cuddy et al. (2009) is a social psychological theory positing that stereotyping of different social groups can be captured along two orthogonal dimensions, “warmth” and “competence.” The warmth dimension of stereotypes has to do with people’s intentions in interpersonal interactions, while the competence dimension has to do with assessing others’ ability to act on those intentions. While there are a number of other social psychological theories capturing outgroup biases (e.g., Zou and Cheryan, 2017; Koch et al., 2016), SCM has been shown to predict emotional and behavioral reactions to societal outgroups. 2.3 The SCM and Language SCM is a well-established theoretical frameworks of stereotyping, and has begun to be applied in NLP. Recently Nicolas et al. (2021) developed dictionaries to measure warmth and competence in textual data. Each dictionary was initialized with a set of seed words from the literature which was further expanded using WordNet Miller (1995) to increase the coverage of stereotypes collected from a sample of Americans. Fraser et al. (2021) showed that, in word embeddings, SCM dictionaries capture the group stereotypes documented in social psychological research. Recently, Mostafazadeh Davani et al. (2021) applied SCM dictionaries to quantify social group stereotypes embedded in language, demonstrating that patterns of prediction biases can be explained using social groups’ warmth and competence embedded in language. 3 Methods & Evaluation There are two components to each post hoc debiasing approach: the Bias Subspace, which determines the subspace over which the algorithms operate, and the Algorithm, which is how the word embeddings are modified with respect to the bias subspace. In this section, we review the concept of bias subspaces, established algorithms for debiasing, and how bias is quantified in word embeddings. Finally, we introduce our social-group-agnostic framework; SCM-based debiasing. 3.1 Identifying a Bias Subspace Post hoc word embedding bias mitigation algorithms operate over a subspace of bias in the embedding space. Given a set $D=\{(d^{+}_{1},d^{-}_{1}),...,(d^{+}_{n},d^{-}_{n})\}$ of word pairs that define the bias concept (e.g. “father”–“mother” for gender) the bias subspace $v_{B}$ is the first $k$ principal components of matrix $C$, constructed from stacking the difference in embeddings of $d^{+}_{i}$ and $d^{-}_{i}$. 3.2 Debiasing Algorithms Method definitions below use the following notation: $W$ denotes vocabulary, $\vec{w}$ and $\vec{w^{\prime}}$ denote the embedding of word $w$ before and after debiasing. Hard Debiasing (HD)   An established approach for mitigating bias in word embeddings is Hard Debiasing (HD; Bolukbasi et al., 2016). For gender, HD removes the gender subspace from words that are not inherently gendered by projecting them orthogonal to gender subspace. For word pairs that are inherently gendered, HD equalizes them, modifying the embeddings such that they are equidistant from the inherently non-gendered words. Subtraction (Sub)    Subtraction (Sub) was introduced as a baseline by Dev and Phillips (2019) wherein the bias subpspace $v_{B}$ is subtracted from all word vectors. Formally, for all $w\in W$, $\vec{w^{\prime}}:=\vec{w}-\vec{v_{B}}$. Linear Projection (LP)   To mitigate the bias with respect to bias dimension $v_{B}$, Linear Projection (LP) projects every word $w\in W$ to be orthogonal to $v_{B}$. Formally, $\vec{w^{\prime}}:=\vec{w}-<\vec{w},\vec{v_{B}}>\vec{v_{B}}$. Partial Projection (PP)   To improve on LP, Partial Projection (PP) was developed to allow the extent of projection to vary based on the component of the given word vector which is orthogonal to the bias subspace. Intuitively, only words with unintended bias (e.g., “nurse” or “doctor”), and not words which are definitional to the bias concept (e.g., “man” or “woman”) will have a large orthogonal component to the bias subspace $v_{B}$. For all words $w\in W$, $$\displaystyle w^{\prime}$$ $$\displaystyle=\mu+r(w)+\beta\cdot f(\|r(w)\|)\cdot v_{B}$$ $$\displaystyle\beta$$ $$\displaystyle=\langle w,v_{B}\rangle-\langle\mu,v_{B}\rangle$$ where $\mu$ is the mean embedding of words used to define $v_{B}$, $r(w)=w-\langle w,v_{B}\rangle v_{B}$ is the bias-orthogonal component, and $f$ is a smoothing function, for example $f(\eta)=\frac{\sigma^{2}}{(\eta+1)^{2}}$, which helps to remove unintended bias and keep definitional bias (see Dev and Phillips, 2019). 3.3 Measures of Bias in Word Embeddings Embedding Coherence Test   Given a list of word pairs $A=\{(a^{+}_{1},a^{-}_{1}),...,(a^{+}_{k},a^{-}_{k})\}$, indicating two “poles” of a social attribute, and a set of professions $P=\{p_{1},...,p_{m}\}$, the Embedding Coherence Test (ECT; Dev and Phillips, 2019) is the Spearman rank correlation between the rank order of cosine similarities of professions with each pole’s average embedding. Ideally, if bias is completely removed, poles should achieve identical ordering of associations with professions (ECT $=1$). Embedding Quality Test   The EQT (Dev and Phillips, 2019) quantifies the improvement in unbiased analogy generation after debiasing. Similar to ECT, EQT requires a set of word pairs $A$ and a set of professions $P$. For each word pair $(a^{+}_{i},a^{-}_{i})$ the analogy $a^{+}_{i}:a^{-}_{i}::p_{j}$ is completed, if the answer is $p_{j}$ or plurals or synonyms of $p_{j}$ (via NLTK; Bird et al. 2009), it is counted as unbiased. EQT is the ratio of unbiased analogies to all analogies. An ideal unbiased model would achieve EQT$=1$ while lower values indicate a more biased model. 3.4 SCM-Based Debiasing To identify a group-agnostic bias subspace, we use the warmth and competence dictionaries from (Nicolas et al., 2021). To construct the poles of the dimensions, “high” and “low” word pairs (e.g., “able”–“unable” for competence and “sociable”–“unsociable” for warmth) were selected by down-sampling to 15 word pairs, per dimension. We use word pairs for each SCM dimension to identify an SCM subspace (see Section 3.1), and subsequently apply the methods from Sec. 3.2. 4 Experiments We test whether SCM-based debiasing can substitute for group-specific debiasing simultaneously for gender, race, and age. This is broken down into two related research questions. First, does SCM-based debiasing remove a comparable amount of bias relative to group-specific debiasing? And second, does SCM-based debiasing have more or less of a negative effect on embedding utility (Bolukbasi et al., 2016)? We compare SCM-based debiasing to group-specific debiasing using previous debiasing methods, specifically HD, Sub, LP, and PP (Section 3.2), and evaluate bias as measured by ECT and EQT following Dev and Phillips (2019). In addition, we evaluate the performance of each set of debiased embeddings on established word embedding benchmarks Jastrzebski et al. (2017). 4.1 Bias Reduction We investigate whether SCM-based debiasing can simultaneously debias word embeddings with respect to gender, race, and age. For a given bias dimension, we established baselines by applying HD, Sub, LP, and PP using the respective word pair list (e.g., for gender bias we used gender word pairs), denoted with the subscript “same.” To place an upper bound on removed bias, we perform PP using gender, race, and age word lists (PP${}_{\text{G+R+A}}$). For race and age we used the lists from Caliskan et al. (2017), while gender lists were taken from Bolukbasi et al. (2016). All methods were repeatedly applied using 30 different word pair samples, and we report each measure’s average and compare values using 95% confidence intervals. Implementation details are provided in the Appendix. Table 1 shows the results of our experiments. Overall, SCM-based debiasing performs comparably to social-group-specific debiasing across methods. Specifically for ECT, SCM-based debiasing was either better than, or not statistically different from, LP${}_{\text{same}}$ and Sub${}_{\text{same}}$, while SCM-based debiasing was only slightly out-performed by PP${}_{\text{same}}$ ($0.01$–$0.03$). In other words, these results demonstrate that warmth and competence dimensions can simultaneously capture gender, race, and age bias in word embeddings. For the EQT, results are somewhat similar to those of ECT; however, we caution against interpreting small differences in EQT due to its definition of biased analogies relying on NLTK to compile comprehensive sets of synonyms and plural forms of words (Dev and Phillips, 2019). 4.2 Word Embedding Utility Table 1 shows that PP${}_{\text{G+R+A}}$ outperformed all other methods. However, one trade-off is the reduction in word embedding utility. Table 2 shows that PP${}_{\text{SCM}}$ (PP applied to Warmth and Competence) preserves more embedding utility than PP${}_{\text{G+R+A}}$, using established benchmarks for analogy and similarity Jastrzebski et al. (2017). Due to the information removed in the debiasing process, as the number of social attributes increases, the quality of embeddings for group-specific debiasing deteriorates; however, this is not the case for PP${}_{\text{SCM}}$. 5 Conclusion We demonstrated that social-group biases in word embeddings can be adequately mitigated in a social-group-agnostic way by operating along the SCM dimensions of warmth and competence (Fiske et al., 2002), introducing a new theory-driven approach for mitigating such biases. Our work shows that the SCM subspace can be used to mitigate bias for additional social groups without lowering word embedding quality. Future directions of our work include: (1) using other social psychological frameworks for quantifying language-embedded social stereotypes (e.g., Koch et al., 2016); (2) extending these findings from English to other languages Kučera and Mehl (2022); and (3) the extension of our findings to other social group biases (e.g., nationality) Herold et al. (2022) and intersectional biases (e.g., intersection of race and gender). 6 Limitations Some limitations of the present work are worth noting. First, we note the contextual limitations of our current analysis using the proposed theory-driven framework. Specifically, the word embeddings used in this work are trained on contemporary English language and our social context overly contains explicit stereotypes encoded in English word embedding model. Stereotypes for the same group can be quite different, however, depending on the language and culture. While out of scope of the present work, cross-societal differences in human stereotyping have been shown to be explainable using the SCM framework (Cuddy et al., 2009). Thus, it is possible that our SCM-based framework generalizes to social group biases beyond those in English. Future research is encouraged to replicate our study in non-English languages. Another limitation of our work is that we only test our proposed framework on word embeddings. We acknowledge that to maximize the impact of bias mitigation efforts, these methods need to be extended to the state-of-the-art language models. Future studies are encouraged to address this limitation. Be that as it may, we emphasize that our proposed approach is the first theory-driven and generalizable approach in mitigating such social biases based on social psychological theories of stereotyping and bias. Furthermore, we would like to point out that there exists a catalogue of bias measurements for word embeddings in the field. 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Experimental Parametric Subharmonic Instability in Stratified Fluids Sylvain Joubaud sylvain.joubaud@ens-lyon.fr Laboratoire de Physique de l’École Normale Supérieure de Lyon, CNRS and Université de Lyon, 46 Allée d’Italie, 69007 Lyon, France    James Munroe jmunroe@mun.ca Department of Physics and Physical Oceanography, Memorial University of Newfoundland, St. John’s, NL A1B 3X7, Canada    Philippe Odier philippe.odier@ens-lyon.fr Laboratoire de Physique de l’École Normale Supérieure de Lyon, CNRS and Université de Lyon, 46 Allée d’Italie, 69007 Lyon, France    Thierry Dauxois thierry.dauxois@ens-lyon.fr Laboratoire de Physique de l’École Normale Supérieure de Lyon, CNRS and Université de Lyon, 46 Allée d’Italie, 69007 Lyon, France (December 2, 2020) Abstract Internal gravity waves contribute to fluid mixing and energy transport, not only in oceans but also in the atmosphere and in astrophysical bodies. An efficient way to transfer energy from large scale to smaller scale is the parametric subharmonic instability. We provide here the first experimental measurement of the growth rate of this instability. We make careful and quantitative comparisons with theoretical predictions for propagating vertical modes in laboratory experiments. pacs: 92.05.Bc, 47.35.Bb, 47.55.Hd, 47.20.-k Internal gravity waves (IGW) result from the balance of inertia and buoyancy forces in a density stratified fluid. Such waves have received a great deal of attention recently because of their relevance and ubiquity in different physical situations: they are believed to be of primary importance as they affect ocean mixing and energy transport Kunze . Although internal gravity waves do not play the dominant role in the evolution of weather and climate, their influence is non-negligible in the dynamics of the atmosphere BruceBook . From a fundamental point of view, these waves are also particularly intriguing. A striking consequence of stratification is an anisotropic dispersion relation relating the frequency to the direction of propagation of the wave and not to the wavelength. This property is also encountered for inertial waves (in presence of rotation) or plasma waves (in presence of a magnetic field). This has unexpected and interesting consequences in the propagation, reflection PRLSwinney or transmission properties of these waves PRLManiTom . Internal waves are known to be inherently unstable due to parametric subharmonic instability (PSI) Staquet02 . PSI is a type of resonant triad interaction where nonlinear terms in the equations of motion allow for efficient transfer of energy from large to small length scales where it can be dissipated. The terminology "parametric sub-harmonic" is used because, for inviscid fluids, PSI transfers energy from a primary wave to two recipient waves of half the frequency. As viscosity effects set in, the frequencies of the recipient waves diverge from half the frequency of the primary wave. In previous laboratory experiments, PSI has been qualitatively observed by driving low-order standing modes with plungers on the sides of the container Thorpe69 , with an oscillating paddle McEwan71 ; McEwan1972 or relying on the parametric forcing of the tank Benielli98 . For large amplitude forcing, “irregularities” or “traumata” were observed, which led to mixing and overturning. In ref McEwan1972 , the critical amplitude of the instability has been measured. Quantitative measurements of the growth rate of the instability have never been reported. We present here experiments performed with a wave generator, which produces sinusoidal vertical modes propagating along a rectangular tank. We measured the growth rate of the instability. This quantity is of paramount importance to single out the major mechanism in dissipation processes, a recently highly debated issue Kunze ; debate ; debatebis . We first briefly outline theoretical aspects of this instability, after which the experimental configuration is described. Then we present our experimental results and compare some of them with theoretical predictions. Theory Internal waves are characterized by the buoyancy frequency, $N=\sqrt{(-g/\bar{\rho_{0}})({\rm d}\rho_{0}/{\rm d}z)}$, in which $g$ is the acceleration of gravity, $\bar{\rho_{0}}$ the characteristic fluid density and $({\rm d}\rho_{0}/{\rm d}z)$ the density gradient in the vertical direction $z$. At large Prandtl number, the 2-D Boussinesq equations of motion can be written as $$\frac{\partial^{2}\nabla^{2}\psi}{\partial t^{2}}+N^{2}\frac{\partial^{2}\psi}% {\partial x^{2}}=\frac{\partial}{\partial t}J(\psi,\nabla^{2}\psi)-\frac{g}{% \rho_{0}}\frac{\partial}{\partial x}J(\tilde{\rho},\psi)+\nu\nabla^{4}\psi_{t}\,,$$ (1) where $\tilde{\rho}=\rho-\rho_{0}$ is the perturbation density field, $\psi$ the stream function, $J$ the Jacobian operator and $\nu$ the viscosity. Seeking wave solutions with wave number $\overrightarrow{k}=(k,m)$, Eq. (1) leads to the inviscid linear dispersion relation for frequency $\omega$, $$\omega^{2}=N^{2}\frac{k^{2}}{k^{2}+m^{2}}.$$ (2) For small amplitudes, it can be assumed that that several waves concurrently exist simply as a linear superposition. In the case of a resonant triad interaction, where three waves satisfy the spatial resonance condition $$\overrightarrow{k_{0}}=\overrightarrow{k_{1}}+\overrightarrow{k_{2}}\,,$$ (3) and the temporal resonance condition $${\omega}_{0}={\omega}_{1}+{\omega}_{2}\,,$$ (4) the nonlinear terms of Eq. (1) act as forcing terms transferring energy between the three waves. Each wave must satisfy the dispersion relation (2). A finite amplitude, large length scale, high frequency wave ($\overrightarrow{k_{0}},\omega_{0}$) can transfer energy to produce two secondary waves of smaller length scales and lower frequencies, ($\overrightarrow{k_{1}},\omega_{1}$) and ($\overrightarrow{k_{2}},\omega_{2}$). The instability results from a competition between nonlinear effects and viscous dissipation. The growth is exponential if the amplitude of the secondary waves is initially small compared to the amplitude of the primary wave Koudella2006 ; McEwan1977 . In this case, the growth rate is equal to $$\lambda=-\frac{1}{2}(T_{1}+T_{2})+\left[\frac{1}{4}(T_{1}-T_{2})^{2}+I_{1}I_{2% }\psi_{0}^{2}\right]^{1/2}\,,$$ (5) where $\psi_{0}$ is the amplitude of the stream function of the primary wave, $I_{1}$ and $I_{2}$ are the interaction coefficients $$I_{i}=\frac{k_{p}m_{q}-k_{q}m_{p}}{2\omega_{i}\kappa_{i}^{2}}\left[\omega_{i}(% \kappa_{p}^{2}-\kappa_{q}^{2})+N^{2}k_{i}\left(\frac{k_{p}}{\omega_{p}}-\frac{% k_{q}}{\omega_{q}}\right)\right]$$ (6) and $i,p,q=0,1$ or $2$ while $T_{i}=\nu\kappa_{i}^{2}/2$ is the viscous damping factor of the wave $i$ and $\kappa^{2}=k^{2}+m^{2}$. Experimental Configuration A tank, $160$ cm long and $17$ cm wide, is filled with linearly stratified salt water with constant buoyancy frequency $N$ using the standard double bucket method. A monochromatic vertical mode-1 wave is generated using the wave generator employed in previous experiments Gostiaux2007 ; Mercier2010 . The generator is composed of $50$ plates moving horizontally to impose the horizontal velocity component of a mode-1, i.e, $u(x=0,z,t)=-a\omega_{0}\cos(\pi z/H)\cos(\omega_{0}t)$, $H$ being the water depth, $\omega_{0}$ the excitation frequency and $a$ the amplitude of the oscillation of the plates. The motion of the fluid is captured by the synthetic schlieren technique using a dotted image behind the tank Dalziel00 . A camera is used to acquire images of this background at $1.875$ frames per second. The CIVx algorithm Fincham2000 computes the cross-correlation between the real-time and the $t=0$ background images, giving the variation of the horizontal, $\tilde{\rho}_{x}(x,z,t)=\partial_{x}(\rho(x,z,t)-\rho_{0}(z))$, and vertical, $\tilde{\rho}_{z}(x,z,t)=\partial_{z}(\rho(x,z,t)-\rho_{0}(z))$, density gradients. Results Snapshots of an experimental horizontal density gradient field at different times for a particular experiment are presented in Fig. 1. At early times, a pure vertical mode-1 wave can be seen propagating to the right away from the wave generator located at $x=0$: this is the primary wave. After several frequency periods $T$ (typically 30), this wave is destabilized and two secondary waves appear, with different frequencies and wave numbers from the primary wave. To see these waves more clearly, the horizontal density gradient at later times is filtered at the frequency of the primary wave, $\omega_{0}$ and at the frequencies of the two secondary waves $\omega_{1}$ and $\omega_{2}$. As described below, the frequencies $\omega_{1}$ and $\omega_{2}$ were determined from a power spectrum. The result is shown in Fig. 2. Some of the energy of the primary wave has been transferred to both secondary waves, leading to a decrease in the amplitude of the primary wave (compare the left part of Fig. 1(left) and Fig. 2(left)). These two waves have smaller frequency and also smaller wavelength. In agreement with the dispersion relation, which links the frequency to the angle of propagation of the wave, the angle of constant phase is different for the two wavelengths. For the experiment presented in Fig. 1, the three measured frequencies $(\omega_{0},\omega_{1},\omega_{2})$ are equal to $(0.95,0.38,0.57)N$, attesting that the temporal resonance condition (4) is satisfied. To justify that the spatial resonance condition (3) is also satisfied, the components of the three wavevectors have to be measured. This is done by extracting the phase of the signal at a given frequency, $\omega_{0,1,2}t\pm k_{0,1,2}x\pm m_{0,1,2}z$, using a Hilbert transform HilbertTransform . At a fixed time and $x$ (respectively $z$), the phase is linear with the position $z$ (resp. $x$). The component $m_{0,1,2}$ (resp. $k_{0,1,2}$) is given by the slope of the linear fit. Within experimental errors, the wave vectors, represented in Fig. 3, satisfy the theoretical spatial resonance condition (3). The measured density gradient fields are then analyzed using a time-frequency representation calculated at each spatial point $$S_{x}(t,\omega)=\left|\int_{-\infty}^{+\infty}{\rm d}u\,\tilde{\rho}_{x}(u)\,e% ^{i\omega u}\,h(t-u)\right|^{2}\,,$$ (7) where $h$ is a smoothing Hamming window of energy unity Flandrin99 . Good frequency resolution is provided by a large time window $h$ while good time resolution is provided by small time window $h$. To increase the signal to noise ratio, the data is averaged along a vertical line over the water depth. For large $\omega_{0}/N$ values, the dissipation length is small, so the analysis line is chosen to be close to the generator so that the amplitude is large. Fig. 4(left) presents the spectra of the density field for four different excitation amplitudes with $\omega_{0}=0.94N$. The spectra are obtained using a time window width equal to $100$ T to have good frequency resolution. $S_{x}(t,\omega)$ is then averaged over the $10$ last periods. Analyzing first the result for the amplitude $0.5$ cm, the picture emphasizes a large peak close to $N$, corresponding to the frequency of the mode-1 wave. A pair of twin peaks are observed, corresponding to secondary waves of frequencies, $\omega_{1}$ and $\omega_{2}$, smaller than $\omega_{0}$. The amplitude of each wave is then computed using a time-frequency analysis with a time window width equal to $20$ T to increase time resolution. The amplitude of the secondary wave of frequency $\omega_{1}$ is presented in Fig. 4(right). After several forcing periods, a steady state for the primary wave is reached. After a time interval, the secondary wave starts to grow and a linear increase of the amplitude on a semilogarithmic plot is observed, confirming exponential growth. The value of the growth rate $\lambda$ is measured using a linear fit, shown with the dashed lines in Fig. 4(right). The amplitude of the secondary waves eventually saturates. Comparing the different curves in Fig. 4, one observes that the amplitude has an influence not only on the location but also on the height of the peaks of the secondary waves in the spectrum. If the amplitude of the primary wave is too small, no peaks are visible and therefore no instability is observed during the experiment run time, $T_{\rm run}$. This result shows that the growth rate in this particular case has to be smaller than $1/{\rm run}$. It may also give an indication of the existence of a threshold in amplitude. As the amplitude increases, the distance between the two peaks increases and the instability occurs earlier (after fewer forcing periods) and is stronger, i.e. with a larger growth rate which is in agreement with the theoretical growth rate (5). Experiments were performed using the same stratification and an amplitude of $0.5$ cm for frequencies in the range of $0.9<\omega_{0}/N<1$. For each experiment, the value of the frequencies of the two secondary waves, $\omega_{1}$ and $\omega_{2}$, and the growth rate $\lambda$ were measured. Experimental results are presented as a function of the frequency of the primary wave, $\omega_{0}/N$, in Fig. 5. The sum of the frequencies of the two secondary waves, $\omega_{1}+\omega_{2}$, is equal to the frequency of the primary wave, $\omega_{0}$, within experimental errors, in agreement with Eq. (4). As $\omega_{0}/N$ increases, the distance between the two secondary frequencies is larger. The measured value of the growth rate is presented in Fig. 5(right). The growth rate increases to reach a maximum around $\omega_{0}=0.95N$ and then decreases as $\omega_{0}$ gets closer to $N$. To compare quantitatively the experimental results with the theoretical prediction of the growth rate, the value of the amplitude of the mode-1 wave has to be precisely known. The theoretical value of the amplitude of the streamfunction is equal to $a\omega_{0}/m_{0}$. However, the conversion efficiency from the energy of the wavemaker to the energy of the mode-1 is less than unity and depends on experimental conditions Mercier2010 . Moreover, as $\omega_{0}$ gets closer to the cut-off frequency, $N$, the value of the viscous damping increases Echeverri09 . Consequently, the efficiency is not the same for all primary frequencies $\omega_{0}$, and the amplitude of the primary wave has to be measured experimentally to compute the theoretical value of the growth rate. It is important to check that the steady-state of the mode-1 wave has been reached. However, the tank being finite in length, the measurement has to be performed before the mode-1 wave reflects back into the measurement area. Then, using a linear polarization relation, the amplitude $\psi_{0}$ of the stream function at this particular frequency and wave number is $\psi_{0}={g\omega_{0}}\partial_{x}\tilde{\rho}_{0}/(4{k_{0}^{2}\bar{\rho}N^{2}})$. The theoretical frequency pair ($\omega_{1}$,$\omega_{2}$) of the instability is defined as the one that maximizes the growth rate. Without adjustable parameters, the comparison between experimental and theoretical results, presented in Fig. 5, emphasizes a good quantitative agreement. Conclusions We have reported the first experimental measurement of the growth rate of parametric subharmonic instability in stratified fluids and we have demonstrated this effect in a systematic set of laboratory experiments allowing careful comparisons with theoretical predictions. In practice, this heavily debated mechanism debate has implications for many geophysical scenarios. Interestingly, although the generation mechanisms of oceanic IGW are quite well understood, the comprehension of the processes by which they dissipate is much more open. Consequently, determining the relative importance of parametric subharmonic instability, among the four recognized dissipation processes Kunze , is the next step in furthering our understanding of how internal waves impact ocean mixing. Quantitative measurements of the subsequent mixing together with a fundamental study of wave turbulence would be of high interest. Acknowledgements.The authors thank G. Bordes, P. Borgnat, B. Bourget, C. Staquet, for helpful discussions. This work has been partially supported by the PIWO grant (ANR-08-BLAN-0113-01) and the ONLITUR grant (ANR-2011-BS04-006-01). This work has been partially achieved thanks to the ressources of PSMN (Pôle Scientifique de Modélisation Numérique) de l’ENS de Lyon. References (1) E. Kunze, S.G. Llewellyn Smith, “The role of small scale topography in turbulent mixing of the global ocean”, Oceanography 17, 55 (2004). (2) B. Sutherland, “Internal Gravity Waves”, (Cambridge University Press, London, 2011). (3) H. P. Zhang, B. King and H. L. Swinney, “Resonant generation of internal waves on a model continental slope”, Physical Review Letters 100, 244504 (2008). (4) M. Mathur, T. Peacock, “Internal Wave Interferometry”, Physical Review Letters 104, 118501 (2010). (5) C. Staquet, J. Sommeria, “Internal gravity waves: From instabilities to turbulence”, Annual Review Fluid Mechanics 34, 559 (2002). (6) S. Thorpe, “On standing internal gravity waves of finite amplitude”, Journal of Fluid Mechanics 32, 489 (1968). (7) A. McEwan, “Degeneration of resonantly-excited standing internal gravity waves”, Journal of Fluid Mechanics 50, 431 (1971). (8) A.D. McEwan, D.W. Mander and R.K. Smith, “Forced resonant second-order interaction between damped internal waves”, Journal of Fluid Mechanics 55, 589 (1972). (9) D. Benielli, J. Sommeria, “Excitation and breaking of internal gravity waves by parametric instability”, Journal Fluid Mechanics 374, 117 (1998). (10) D. Olbers, N. Pomphrey, “Disqualifying 2 candidates for the energy-balance of oceanic internal waves”, Journal of Physical Oceanography 11, 1423 (1981). (11) J. MacKinnon, K. Winters, “Subtropical catastrophe: Significant loss of low-mode tidal energy at 28.9 degrees”, Geophysical Research Letters 32, L15605 (2005). (12) C. Koudella, C. Staquet, “Instability mechanisms of a two-dimensional progressive internal gravity wave”, Journal of Fluid Mechanics 548, 165 (2006). (13) A.D. McEwan, R.A. Plumb, “Off-resonant amplification of finite internal wave packets”, Dynamics of Atmospheres and Oceans, 2, 83 (1977). (14) L. Gostiaux, H. Didelle, S. Mercier and T. Dauxois, “A novel internal wave generator”, Experiments in Fluids 42, 123 (2007). (15) M. Mercier, D. Martinand, M. Mathur, L. Gostiaux, T. Peacock and T. Dauxois, “New wave generation”, Journal of Fluid Mechanics 657, 308 (2010). (16) S. Dalziel, G.O. Hughes, B. Sutherland, “Whole-field density measurements by ’synthetic schlieren’ ”, Experiments in Fluids 28, 322 (2000). (17) A. Fincham, G. Delerce, “Advanced optimization of correlation imaging velocimetry algorithms”, Experiments in Fluids 29:S13 (2000). (18) M. Mercier, N. Garnier, T. Dauxois, “Reflection and diffraction of internal waves analyzed with the Hilbert transform”, Physics of Fluids 20, 086601 (2008). (19) P. Flandrin, “Time-Frequency/Time-Scale Analysis”, (Academic Press, San Diego, 1999). Time-Frequency Toolbox for Matlab©, http://tftb.nongnu.org/. (20) P. Echeverri, M.R. Flynn, T. Peacock and K.B. Winters, “Low-mode internal tide generation by topography: an experimental and numerical investigation”, Journal of Fluid Mechanics 636, 91 (2009).

††thanks: Both authors contributed equally to this work. Corresponding author: manuel.algaba@meetiqm.com Corresponding author: mario.ponce@meetiqm.com††thanks: Both authors contributed equally to this work. Corresponding author: manuel.algaba@meetiqm.com Corresponding author: mario.ponce@meetiqm.com Co-Design quantum simulation of nanoscale NMR Manuel G. Algaba IQM, Nymphenburgerstr. 86, 80636 Munich, Germany    Mario Ponce-Martinez IQM, Nymphenburgerstr. 86, 80636 Munich, Germany Department of Physics and Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333 Munich, Germany    Carlos Munuera-Javaloy Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain    Vicente Pina-Canelles IQM, Nymphenburgerstr. 86, 80636 Munich, Germany    Manish Thapa IQM, Nymphenburgerstr. 86, 80636 Munich, Germany    Bruno G. Taketani IQM, Nymphenburgerstr. 86, 80636 Munich, Germany    Martin Leib IQM, Nymphenburgerstr. 86, 80636 Munich, Germany    Inés de Vega IQM, Nymphenburgerstr. 86, 80636 Munich, Germany Department of Physics and Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333 Munich, Germany    Jorge Casanova Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain IKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5, 48009 Bilbao, Spain    Hermanni Heimonen IQM, Keilaranta 19, FI-02150 Espoo, Finland Abstract Quantum computers have the potential to efficiently simulate the dynamics of nanoscale NMR systems. In this work we demonstrate that a noisy intermediate-scale quantum computer can be used to simulate and predict nanoscale NMR resonances. In order to minimize the required gate fidelities, we propose a superconducting application-specific Co-Design quantum processor that reduces the number of SWAP gates by over 90% for chips with more than 20 qubits. The processor consists of transmon qubits capacitively coupled via tunable couplers to a central co-planar waveguide resonator with a quantum circuit refrigerator (QCR) for fast resonator reset. The QCR implements the non-unitary quantum operations required to simulate nuclear hyperpolarization scenarios. I Introduction Computer simulations are the backbone of scientific research and technological development. Quantum computers promise in the long term to enable simulations of systems that are intractable to even the largest supercomputers [1, 2]. Currently, scientists have access to so-called noisy intermediate-scale quantum (NISQ) computers [3], that present limited qubit counts without error correction. While applications of error-corrected quantum computers are well established, use cases where NISQ devices might achieve quantum advantage are still elusive [4]. In the search for these early applications, the problem must fit the hardware, and the hardware must enable implementation with minimal overheads. Application-Specific Integrated Chips (ASICs) are highly specialized processors optimized for specific problems when execution speed, power efficiency, or miniaturization is of utmost importance [5]. A prominent example where computational speed and energy efficiency are optimised through the use of ASICs is training of artifical neural networks using tensor processing units [6, 7]. Building a general-purpose quantum computer capable of rivaling the most powerful classical computers has proven to be a difficult task, so it is likely that the first devices reaching useful quantum advantage will use quantum ASICs, also called Co-Design quantum computers. A good example of a problem with suitable structure for simulation by quantum computers is nanoscale nuclear magnetic resonance (NMR) [8]. The problem can be described by a number of mutually interacting spins, which natively map to the qubits of a quantum computer, thereby circumventing the overheads in mapping the problem to qubits, such as in the case of fermions [9]. In general, fast and reliable quantum simulations of interacting spin systems would improve the interpretability of solid-state NMR and electron spin resonance (ESR) spectra, where advanced numerical techniques present limited performance [10]. This shows the potential of quantum computers with a moderate number of qubits to shed light on the dynamics of these important systems. A quantum ASIC that minimizes algorithm implementation overheads could be the first method to access these simulations. Note that, other NMR problems, such as zero-field NMR [11] and Hamiltonian learning [12], have already attracted research on how quantum computers can be used to tackle them. NMR techniques have a profound impact in research areas such as material science, chemistry, biology, and medicine [13]. Recently they have approached the nanoscale through solid-state quantum sensors such as the nitrogen vacancy (NV) center in diamond [14]. This is a particularly powerful quantum device, as it enables detection and control of nearby nuclear spins with nanoscale resolution [15]. Applications of the device are, e.g., the precise determination of the structure and dynamics of nuclear ensembles such as proteins [16], finding inter-label distances (via, e.g., Bayesian analysis of the NV response) in electronically labelled biomolecules [17], and the exploration of bespoke microwave (MW) sequences that efficiently transfer NV center polarization to the nuclear environment. Hyperpolarization (i.e. polarization beyond that of a thermal state in a magnetic field) of nuclear spins in diamond presents the potential to develop new and safer contrast agents for magnetic resonance imaging. This problem, which we aim to address through simulation by a quantum computer, could lead to improved detection of different malformations in tissues –such as heart or brain– without the need to deliver ionizing radiation, in contrast to other techniques [18]. This manuscript describes a Co-Design process for a quantum ASIC to simulate nanoscale NMR scenarios. It is structured in three main parts, each of which is a crucial step in the Co-Design process: 1. Identifying the problem (Sec. II), which here is simulating a nanoscale NMR system for hyperpolarizing nuclear spins. 2. Choosing an algorithm for the nanoscale NMR problem and showing that a star-topology chip implements it with minimal overhead (Sec. III), and 3. Designing the corresponding quantum ASIC using a central resonator bus (Sec. IV). The sections are followed by results and discussions (Sec. V) and an outlook (Sec. VI). II Nanoscale NMR: Hyperpolarization Let us consider a system consisting of $M$ nitrogen-vacancy centers and $N$ carbon-13 isotopes in the presence of a driving field and an external magnetic field $\vec{B}_{Z}$. For simplicity, we consider the NV centers aligned with the external magnetic field, leading to the following Hamiltonian: $${}\begin{split}H=&\sum_{j=1}^{M}\delta_{j}\sigma^{z}_{j}-\sum_{k=1}^{N}\vec{\omega}^{c}_{k}\cdot\vec{I}_{k}+\sum_{j=1}^{M}\sum_{k=1}^{N}\frac{\sigma^{z}_{j}}{2}\vec{A}_{jk}\cdot\vec{I}_{k}+\\ &+\sum_{k>k^{\prime}}^{N}g_{k^{\prime}k}\left[I_{k^{\prime}}^{z}I_{k}^{z}-\frac{1}{4}(I_{k^{\prime}}^{+}I_{k}^{-}+I_{k^{\prime}}^{-}I_{k}^{+})\right]+\\ &+\sum_{j>j^{\prime}}^{M}h_{j^{\prime}j}\left[\sigma_{j^{\prime}}^{z}\sigma_{j}^{z}-2(\sigma_{j^{\prime}}^{+}\sigma_{j}^{-}+\sigma_{j^{\prime}}^{-}\sigma_{j}^{+})\right]+H_{\textrm{dr}}.\end{split}$$ (1) Note that, Eq. (1) is expressed in a rotating frame with respect to the free NV Hamiltonian, while $H_{\rm dr}$ represents an external driving tuned on resonance with a certain NV energy transition. A detailed derivation of Eq. (1) can be found in Appendix A. The representation of such a system for $M=1$, $N=2$ can be found in Fig. 1. In Eq. (1) $\sigma^{z}_{j}$ is the Pauli-$z$ matrix representing the $j^{{}_{\textrm{th}}}$ NV center, $I^{z}_{k}$ is the spin-$z$ operator ($I^{z}_{k}=\frac{1}{2}\sigma^{z}_{k}$) acting on the $k^{{}_{\textrm{th}}}$ nucleus, and $\sigma^{\pm}_{j}=\frac{\sigma^{x}_{j}\pm i\sigma^{y}_{j}}{2}\left(I^{\pm}_{k}=I^{x}_{k}\pm iI^{y}_{k}\right)$ are the $j^{{}_{\textrm{th}}}$ NV center ($k^{{}_{\textrm{th}}}$ nucleus) ladder operators. The term $\delta_{j}$ is the detuning of the $j^{{}_{\textrm{th}}}$ NV center with respect to the microwave drive $H_{\textrm{dr}}$. The hyperfine coupling vector $\vec{A}_{jk}$ represents the coupling between the $j^{{}_{\textrm{th}}}$ NV center and the $k^{{}_{\textrm{th}}}$ nucleus, while $\vec{\omega}_{k}^{c}=\gamma_{c}\vec{B}_{Z}-\frac{1}{2}\sum_{j=1}^{M}\vec{A}_{jk}$ is the modified Larmor frequency of the $k^{{}_{\textrm{th}}}$ nucleus with the ${}^{13}C$ gyromagnetic ratio $\gamma_{c}\approx(2\pi)\times 10.7$ MHz/T, $g_{k^{\prime}k}$ is the coupling between the $k^{{}_{\textrm{th}}}$ and $k^{\prime_{\textrm{th}}}$ nuclei, and $h_{j^{\prime}j}$ is the coupling between the $j^{{}_{\textrm{th}}}$ and $j^{\prime_{\textrm{th}}}$ NV centers. In order to hyperpolarize a diamond sample at room temperature, the NV centers are first optically polarized employing laser light, and then their state is transferred to the surrounding nuclei with the aid of a tailored microwave radiation scheme. The initial state of the nuclei in a room-temperature sample, on the other hand, is well described by a fully mixed state due to the small energy splitting of the nuclear spins. By re-initializing the NV centers and repeating this procedure, the polarization transferred into the sample can be amplified. In this paper we will consider the quantum simulation of the polarization transfer mechanism and study two different driving schemes acting on the NV centers in a room-temperature diamond. The first driving scheme is a continuous driving whose Hamiltonian in the rotating frame mentioned earlier is $H_{\textrm{dr}}=\frac{\Omega}{2}\sigma^{\phi}$, where $\sigma^{\phi}=e^{-i\phi}|1\rangle\langle 0|+e^{i\phi}|0\rangle\langle 1|=e^{-i\phi}\sigma^{-}+e^{i\phi}\sigma^{+}$, $\phi$ a phase, and $\Omega$ the Rabi frequency. NV-nucleus polarization transfer is achieved when the Rabi frequency matches the modified nuclear Larmor frequency (i.e. when $\Omega=|\vec{\omega}_{c}|$), leading to the Hartmann-Hahn double resonance condition [19]. For a single NV center and nucleus, the Hamiltonian in Eq. (1) reduces, in an interaction picture, to $H_{I}=\frac{A^{\perp}}{4}\left(|+\rangle\langle-|I^{+}+|-\rangle\langle+|I^{-}\right)$, where $\ket{\pm}=\ket{0}\pm\ket{1}$, which shows a polarization transfer mechanism with the effective transfer rate $\frac{A^{\perp}}{4}$ (a detailed derivation can be found in Appendix B). The second type of driving we consider is a pulsed-driving scheme, $H_{\textrm{dr}}=\frac{\Omega(t)}{2}\sigma^{\phi}$, where $\Omega(t)$ is a train of $\pi$-pulses, such as the Carr-Purcell-Meiboom-Gill sequence [20, 21] or the XY8 sequence [22, 23]. In this scenario, the time spacing $\tau$ between the $\pi$-pulses is the control parameter. If $\tau$ is selected such that $\tau=\frac{n\pi}{|\vec{\omega}^{c}|}$ ($n$ being an arbitrary integer number) and the pulses are evenly spaced one finds that, in an interaction picture, for a single nucleus and NV center, the Hamiltonian reduces to $H_{I}=\alpha A^{\perp}\sigma^{z}I^{x}$, where $\alpha$ is a factor that depends on the integer $n$ (See Appendix B). A phase imprinted on the pulse sequence through a time delay turns the interaction into $H_{I}=\alpha A^{\perp}\sigma^{z}I^{y}$. By combining both sequences with the appropriate rotations over the NV center, the polarization transfer interaction $H_{I}=-\frac{\alpha A^{\perp}}{4}\left(\sigma^{+}I^{-}+\sigma^{-}I^{+}\right)$ is achieved (see Appendix B and Ref. [24] for more details). Regarding common error sources, NV centers located at different positions in the diamond lattice experience stress conditions that lead to local energy deviations from the zero-field splitting. The corresponding term in Eq. (1) is the detuning $\delta_{j}$. Another common type of imperfection appears due to unavoidable fluctuations of the Rabi frequency in the driving. This fluctuation can be modelled as an Ornstein-Uhlenbeck (OU) process [25], which has been shown to be an accurate description for NV centers  [26]. Neither of the system error types lead to considerable overheads in a simulation on a quantum computer. Finally, ${}^{13}C$ nuclear spin decay is not a relevant error source on the time scale of the protocol. III Co-Design algorithm III.1 Simulation technique The best established digital quantum simulation technique is based on decomposing the time-evolution operator into single-qubit and two-qubit gates through the Lie-Trotter-Suzuki formula [27]. To simulate our problem on a quantum computer, we base our strategy on Trotterization [2] but we also explore the randomized Trotterization method qDRIFT [28] in Appendix E. Other, more NISQ-specific, simulation techniques such as the variational quantum simulator [29], the quantum assisted simulator [30], numerical quantum circuit synthesis [31], and a plethora of other quantum algorithms [4] can also be used as simulation methods. One advantage of Trotterization over some of these NISQ methods is that it closely follows the real time evolution for each time step. This is particularly important for pulsed-driving schemes, where the free evolution in between different pulses always starts with a different initial state. Variational and quantum assisted methods would then require that each interpulse evolution is solved independently, making them impractical for the problem. A second advantage of Trotterization is that its complexity and precision are straightforward to analyze. The Trotterization procedure can also be expanded to higher orders, and symmetrized expansions converge more rapidly and reduce the error with respect to the continuum time limit [32]. III.2 Hardware assumptions III.2.1 Native gates The hardware for the quantum simulation plays a major role in choosing the optimal quantum algorithm and its specific implementation. In our case, we consider a quantum computer based on superconducting qubits with the following native single-qubit gate set: $$\displaystyle R_{xy}(\phi,\theta)$$ $$\displaystyle=$$ $$\displaystyle e^{-i(\cos{\phi}X+\sin{\phi}Y)\frac{\theta}{2}};\,\,\textmd{and}$$ (2) $$\displaystyle\ R_{z}(\theta)$$ $$\displaystyle=$$ $$\displaystyle e^{-iZ\frac{\theta}{2}},$$ (3) where $X$, $Y$, and $Z$ are qubit Pauli operators. The gate $R_{z}(\theta)$ does not need to be implemented directly, but can be performed virtually by tuning the phase of the subsequent gates applied on the qubit [33]. This reduces the number of single-qubit gates (SQGs) that need to be implemented. The two-qubit gate (TQG) available for the system is a continuously-parameterized controlled-$Z$ interaction [34], which can be transformed through local virtual $R_{z}$-rotations into the form of a $ZZ$-interaction: $$\displaystyle U_{ZZ}(\phi)=\left(\begin{array}[]{cccc}e^{-i\phi}&0&0&0\\ 0&e^{i\phi}&0&0\\ 0&0&e^{i\phi}&0\\ 0&0&0&e^{-i\phi}\end{array}\right).$$ (8) Sec. IV goes into more depth on the two-qubit-gate implementation on our Co-Design quantum ASIC. III.2.2 Qubit reset A qubit reset operation can be defined by two Kraus operators: $$\displaystyle K^{\textrm{reset}}_{1}=\begin{pmatrix}{}1&0\\ 0&0\\ \end{pmatrix},\,K^{\textrm{reset}}_{2}=\begin{pmatrix}{}0&1\\ 0&0\\ \end{pmatrix}.$$ (13) Resets are necessary for implementing the re-initialization of the state of the NV centers in hyperpolarization protocols. On superconducting hardware this can be realized through connecting a quantum circuit refrigerator (QCR) to each circuit element that needs to be reset [35, 36, 37, 38]. Different reset schemes are discussed in Sec. IV.2. III.2.3 Noise and errors In this paper we show that the simulation can tolerate the noise of the quantum processing unit (QPU), and that the simulation does not require large overheads to implement imperfections in the nanoscale-NMR system, as discussed in Sec. II. We will refer by system imperfections to effects in the nanoscale NMR system only, while the noise affecting the QPU will be called noise and errors. In our simulation of the algorithm, we use the most common noise models for superconducting transmon qubits [39]. This includes an amplitude damping channel, with $T_{1}=60\,\mu s$, and a pure dephasing channel with a $1/f$ spectral function [39] corresponding to a dephasing time $T_{2}=60\,\mu s$. Additionally each gate operation is assumed to be calibrated up to a two-qubit-gate (TQG) error $\varepsilon_{\textrm{TQG}}\in[10^{-4},10^{-2}]$, with the error modelled by a depolarizing channel. Single-qubit-gate (SQG) errors $\varepsilon_{\textrm{SQG}}$ are assumed to be one order of magnitude lower than TQG errors. III.3 Algorithm components Our proposed simulation of the nanoscale NMR problem follows the general structure shown in Fig. 2a. It starts by initializing the states of the qubits, then evolving them using Trotter steps, followed by reset and re-initialization of the qubits representing NV centers. The cycle of time evolution and re-initialization is then repeated as many times as the protocol calls for. Finally the qubits are measured, and the polarization of the NV centers and nuclei are extracted as the expectation values of the qubit representing each element. In the following, we go through these steps in more detail for the case of a single NV center. III.3.1 Initial state preparation To enable the polarization transfer, it is necessary to prepare the NV center in a specific initial state that depends on the driving scheme. For the continuous-driving scheme it is the $|+\rangle$ or $|-\rangle$ state, and for the pulsed-driving scheme it is one of the two computational basis states, $|1\rangle$ or $|0\rangle$. For a diamond at room temperature, the initial state of the nuclear spins is well described by a fully mixed state $\rho_{\textrm{mixed}}=\frac{\mathbb{1}}{2^{N}}$, where $\mathbb{1}$ is the identity matrix. The state can be approximated by running the algorithm several times, each time with a different initial state obtained by applying $X$ gates randomly on the qubits representing nuclei. A faster alternative to this sampling is the random-phase-approximation-inspired method, described in [40], and introduced into quantum computing in [41]. In this method, the qubits are all prepared in an equal superposition by applying Hadamard gates, and then the phases are randomized through the application of random phase gates. The method effectively reduces the prefactor in the scaling of the sampling error [41]. III.3.2 Time evolution We choose to implement the time evolution generated by the Hamiltonian in Eq. (1) through Trotterization. For that, the Hamiltonian is rewritten in terms of qubit Pauli operators and arranged into non-commuting terms for an optimal Trotter splitting. The resulting circuit, which performs one Trotter step of the evolution, is depicted in Fig. 2b. It consists of a set of initial single-qubit gates, including the ones corresponding to the driving and the detuning of the NV center, followed by three two-qubit gates per nucleus. There are three types of interaction terms, of the form $XZ$, $YZ$ and $ZZ$, when no internuclear interactions are considered. With interactions there are a total of five TQGs. Our native gateset only includes one type of two-qubit interactions, namely of the form $ZZ$ (see Eq. (8)), so SQGs need to be introduced in order to obtain the rest of the interaction terms, as explained in Appendix C. III.3.3 Cycles and reset The dynamics of the system is known to produce an exchange of polarization between the NV center and the nuclei. This exchange is oscillatory, therefore choosing a proper stopping time is important in order to achieve an effective polarization transfer from the NV center to the nuclei. In practice, a sub-optimal transfer time can suffice, and the protocol is then repeated several times by resetting the NV center to its initial state and letting the system evolve under the drive again. Due to the re-initializations the full evolution of the system is non-unitary and a net gain of polarization of the system is enabled. This structure is represented in the quantum circuit in Fig. 2a by the repeated Trotter evolution, followed by reset operations on the qubit representing the NV center, and a single-qubit gate to prepare the initial state of the driving protocol. III.4 Layout optimization When implementing a quantum algorithm on a superconducting QPU, the planar qubit connectivity forces us to solve the qubit-routing problem by introducing additional SWAP gates to connect distant qubits. In this subsection, we study the advantages of an optimized chip topology, a star topology, over a square-grid array of qubits in terms of reducing the number of SWAP gates that must be inserted to run the algorithm in Fig. 2 on the device. Different topologies will imply different counts of SWAPs added on top of the gates arising from the algorithm itself, as shown in Fig. 3. On a NISQ device, this implies different computational precision for the same gate error magnitudes. We choose the SWAP count as our metric to compare different topologies, as commonly gates have fidelities limited by calibration. The errors could be due to crosstalk, leakage, or filtering causing disturbances to the control signals. Under this scenario we want to minimize the gate count. On the other hand, for a highly tuned up device whose gates are limited by qubit coherence times, it would be optimal to minimize the circuit depth instead of the TQG count. Assuming the gate errors are independent, the total error will be bounded by: $$\varepsilon_{\textrm{gates}}=1-{(1-\varepsilon_{\textrm{TQG}})}^{N_{\textrm{TQG}}}(1-\varepsilon_{\textrm{SQG}})^{N_{\textrm{SQG}}},$$ (14) where $N_{\textrm{TQG}}$ is the number of two-qubit gates, $N_{\textrm{SQG}}$ the number of single-qubit gates, and $\varepsilon_{\textrm{SQG}}$ is the SQG error. Consequently, reducing the gate count, especially $N_{\textrm{TQG}}$, has an exponential effect on the precision of the computations, highlighting the effect of minimizing the SWAP gate overhead. III.4.1 Square grid A common choice in superconducting quantum chips is the square grid. It has high connectivity and is suitable for performing the surface code error correction when scaled to large enough qubit counts with fast measurement and feedback [42]. The qubit routing problem on a square grid can be tackled using various numerical approaches [43, 44, 45, 46]. However, these methods are inefficient. In our case, a tailored SWAP routing method, shown in Fig. 3a, has been chosen and developed in Appendix F that can be shown to be well suited from two perspectives. First, a comparison against the cited numerical approaches (shown in Appendix F) reveals that our routing method is better in terms of number of gates. Second, it is completely deterministic and does not rely on expensive numerical optimization methods. It can also be shown not to be far from optimal: on a square grid each qubit has at most 4 nearest neighbors, implying that any SWAP operation provides at most 3 new neighbors. For an all-to-all interacting Hamiltonian there are $\frac{n^{2}}{2}$ interactions to leading order for a simulation performed on $n$ qubits (corresponding to $N$ nuclei and one NV center), implying a lower bound of at least $\frac{n^{2}}{6}$ SWAPs for any SWAP pattern on the square grid topology. This shows that our SWAP pattern with $\frac{n^{2}}{2}$ SWAPs, discussed in Appendix F, is not far from optimal. III.4.2 Star architecture A star topology allows to implement the simulation of the simplified case without internuclear interactions directly, without any SWAP gates. With internuclear interactions, we still find a reduction in SWAP gates as compared to the square grid topology, as shown in Fig. 3b. This reduction comes from the SWAP routing we implement, that consists of making the qubit $0$ in Fig. 3b interact with all the external qubits and then swap its state with that of qubit $1$ and repeat this process until all interactions have been performed. This allows us to use only $n$ SWAP gates. The percentage of SWAP gates that can be saved can be observed in Fig. 4. However, this improvement in the number of gates comes with a price to pay in the depth of the algorithm, which is $\frac{3}{2}n^{2}+\frac{15}{2}n-9$, while for a square grid it is $6n$. Such depth increase comes from the reduction in parallelization, since all gates now act via the central qubit. On the other hand, less parallelization reduces the types of possible crosstalk errors. Adding connections between external qubits reduces the depth of the circuit, since the main cause of circuit depth is the fact that the interaction of two external qubits needs to be done exclusively by the central qubit. Further studies are required to see if the addition of more external layers to this topology (such as in a spiderweb) can lead to better compromises between depth and gate count, especially for simulating systems with clusters of strongly interacting nuclei. III.5 Gate-level optimization The two-qubit interactions that appear in the algorithm are the $XZ$, $YZ$ and $ZZ$ interactions, as highlighted in section III.3.2 and Fig. 2b. When compiling the algorithm into the native gates considered here, all these interactions must be implemented in terms of some available gate set. We study in Table 1 the overhead introduced by decomposing these interactions into different examples of native TQGs of superconducting devices; namely, the parametrizable and fixed-phase $U_{ZZ}(\phi)$ gate, the parametrizable and fixed-phase cross-resonance gate $\mathrm{CR}(\phi)$, and the $\mathrm{CNOT}$ gate. We assume that the SQGs that can be implemented are the $R_{xy}$ and the $R_{z}$ gates. These numbers can be further reduced if the first and last SQGs introduced by this compilation are combined with the adjacent SQGs in the algorithm. The conclusion is that fixed-angle gates will double the number of gates that need to be physically performed. In Ref. [47], the improvements coming from the reduction of the gate count are compared to the new errors introduced by the interpolation of the calibrated phases. For two instances of a QAOA problem, it is shown that the performance is better when using parametrized TQGs. The most efficient gates are therefore the parametrizable cross-resonance gate [39, 48], and the parametrizable $U_{ZZ}(\phi)$, which is equivalent to the native controlled-$Z$ (CZ) up to two virtual $R_{z}$ gates. IV Co-Design hardware A star-architecture chip has fundamental scaling issues using a transmon as the central qubit as the number of neighbors grows. Every neighbor added to the center qubit would decrease its charging energy $E_{c}$. To keep the qubit frequency constant and anharmonicity in the transmon regime, the ratio of the qubit’s Josephson energy to its charging energy, $E_{j}/E_{c}$, must remain unaffected. Therefore we cannot afford to change its charging energy. This leads to a trade-off between the number of coupled qubits and their coupling strength to the central element. The spirit of Co-Design calls for replacing the central transmon with another object that enables this scaling in size. A way to avoid this issue is to replace the central qubit by a resonator. A resonator has no Josephson energy $E_{j}$, so the $E_{j}/E_{c}$ ratio is not changed by adding more capacitive couplings to the resonator. Only small corrections to its frequency are introduced by adding coupled qubits. As a distributed element, a co-planar waveguide resonator also has physically more space for couplings than a central transmon qubit. By elongating the resonator and choosing the mode with the target frequency, the number of qubits coupled to it can further be increased. In the device in Fig. 5 the qubits are capacitively coupled to the resonator via tunable couplers [49, 34, 50] in the proximity of a voltage maximum of a standing wave in the resonator. As the resonator is elongated, we must use higher harmonic excitations of the resonator to keep the frequency around the operational frequency of the qubits. Tunable couplers avoid the frequency crowding issues related to direct coupling [51, 52], and the linear resonator has higher connectivity in the center than ring resonator structures with quasi-all-to-all connectivities [53]. A linear resonator cannot in general be used as a qubit, since a microwave drive on it will not only populate the $\{\ket{0},\ket{1}\}$ subspace, but also higher excited states. However, two-qubit gates via tunable couplers do not cause leakage to higher excited states since no driving of the resonator is involved. The theory for performing iSWAP and CZ gates between the resonator and a qubit is developed in Sec. IV.1. Then, a resonator can be used as an effective qubit in the following way: 1. Prepare all qubits and the resonator in their ground states 2. Designate one qubit as the central qubit 3. Prepare an arbitrary state in the central qubit 4. Perform an iSWAP from the qubit to the resonator in the ground state 5. Perform CZ gates between the resonator and any other qubits 6. Perform an iSWAP of the state back to the central qubit for measurement The most theoretically straightforward protocol would be to perform a SWAP gate from the qubit to the resonator. The iSWAP, on the other hand, is a native gate that can directly be implemented on the hardware in Fig. 5b . Since the CZ gates performing the computation following the iSWAP are diagonal in the computational basis, the phase introduced by the iSWAP is uninvolved in the gate. This enables substituting the SWAP gate by an iSWAP gate in the protocol to further minimize the gate count. IV.1 Gate theory and simulations Here we demonstrate that in our star architecture CZ and iSWAP-type gates between any of the qubits and the $\{\ket{0},\ket{1}\}$ subspace of a chosen resonator mode can be implemented. The operational principles of these gates are very similar to those between two qubits coupled with a tunable coupler [49, 34, 50, 54]. The main limitation of our architecture (where one transmon is replaced by a resonator) is that iSWAP operations can only be performed on a subspace of the two-qubit computational basis (i.e. the state $|\rm 1\rangle_{r}\otimes|\rm 1\rangle$ must be excluded, where $|\rm 1\rangle_{r}$ denotes the first excited state of the resonator). IV.1.1 Conditional-Z gate The CZ operation between the resonator and the qubit is described by the unitary operator: $$\displaystyle\textrm{CZ}(\phi)=\left(\begin{array}[]{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&e^{-i\phi}\end{array}\right).$$ (19) This gate is equivalent to the $U_{ZZ}(\phi)$ gate in Eq. 8 up to two $R_{z}$-rotations. To operate a CZ gate, we initialize the resonator-coupler-qubit set up shown in Fig. 5b at the idling configuration with zero effecting coupling between the qubit and resonator. Note that coupler is also a transmon that shows a higher sensitivity to the magnetic flux noise than regular qubits. We next apply a time-dependent flux pulse that lowers the coupler frequency, turning on the effective coupling between the resonator and the qubit. Depending on the flux pulse shape, the setup collects conditional phase $\phi$ and possibly experiences population oscillations between computational and non-computational states, as a function of the time spent at the gate-operation frequency. We optimize the pulse amplitude and duration such that after the flux pulse the CZ gate fidelity is maximized. Details on the gate theory can be found in Appendix G and the considered device parameters in Table 3. In Fig. 6a we sweep the flux-pulse amplitude (which results in a coupler frequency shift by $\omega_{c}^{\mathrm{shift}}$ from the idling configuration) and the gate time $\tau$ to locate the optimal pulse configuration that minimizes the CZ($\pi$) gate error $\varepsilon_{\textrm{CZ}}=1-\big{(}\mathrm{tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\big{)}^{2}$, where $\sigma$ is the target density matrix obtained after propagating some initial state $|\Psi\rangle\langle\Psi|$ with the ideal unitary of Eq. 19 and $\rho$ the final density matrix obtained after propagating $|\Psi\rangle\langle\Psi|$ with the Lindbladian corresponding to our system defined in Eq. (57). For our device parameters, the maximal decoherence limited CZ gate error averaged over a number of random initial states is $1.6\times 10^{-3}$. Note that the system parameters in Table 3 were chosen such that they allow for the possibility to find a good idling configuration, where the residual CZ interaction vanishes before the gate operation. In our simulations, we have included environmental noise, such as amplitude damping and pure dephasing and treated them using a Lindblad master equation solver in $Qutip$. IV.1.2 iSWAP gate Just as the CZ gate, the iSWAP gate can be natively realized in superconducting quantum computing architecture [39]. The iSWAP gate operation between the resonator and the qubit is represented by the unitary operator: $$\displaystyle U_{\mathrm{iSWAP}}=\left(\begin{array}[]{cccc}1&0&0&0\\ 0&0&-i&0\\ 0&-i&0&0\\ 0&0&0&1\end{array}\right).$$ (24) With our device, we can perform high-fidelity iSWAP gates between single-excitation computational states. The capacitive coupling between the elements of the electrical circuit shown in Fig. 5b gives rise to an effective $XY$-interaction between the qubit and resonator under the rotating wave approximation. Such an interaction conserves excitation number. With only the qubit or resonator (or neither) initially populated, we stay within the single excitation subspace of the joint system, thereby minimizing leakage of quantum population into the higher excited states of the resonator. The $XY$-interaction can be turned on by first tuning the qubit in resonance with the resonator, and then applying a flux-pulse to the coupler to turn on the coupling, similar to the CZ gate operation. Fig. 6b shows iSWAP gate error landscape for the same device parameters (given in Table 3). The optimal average iSWAP gate error $\varepsilon_{\mathrm{iSWAP}}$ obtained for our device is $1.7\times 10^{-3}$. This result is obtained by averaging over a number of random initial states within the zero- and one-excitation manifolds. The results of our two-qubit-gate simulations demonstrate that our star architecture supports operating gates with similar fidelities as regular transmon qubits coupled together. The increased local connectivity of the device reduces the need for SWAP gates to simulate the nanoscale NMR problem (and others with a similar structure) and consequently in the end improves simulation fidelities. IV.2 Reset The hyperpolarization protocol described in Sec. II needs regular re-initializations of the state of the NV center. The Co-Design hardware for simulating the protocol must therefore support this operation within qubit lifetimes. This is a hardware challenge, but one with solutions in sight. In particular, the quantum circuit refrigerator (QCR) has been used to perform the reset in tens of nanoseconds [35, 36, 37, 38], which is a similar timescale to gate operations. The advantage of using a QCR for the reset is the possibility to reset the central resonator directly, without the need transfer the resonator population back to the central qubit using an iSWAP gate. Alternatively, a fast reset is possible through applying a flux drive to a qubit to SWAP its state with its measurement line [55]. This scheme has the advantage of not requiring any additional hardware not already present on the chip, but comes with a small cost in the circuit depth, as the state of the resonator must be transported using an iSWAP gate into the designated central qubit and be re-initialized there. V Results and discussion In this section we discuss the two main results of the paper: namely the predicted performance of our proposed quantum algorithm on a regular noisy QPU, as well as the performance increase obtained with our proposed Co-Design QPU. To this aim, we will focus on the polarizations of the NV center and nuclear spins, that are relevant quantities of the problem and straightforward to measure in a quantum computer. In Fig. 7 we compare the frequency response of the polarization transfer process on two different simulated devices: a QPU with realistic noise parameters, and an ideal noiseless QPU. We consider one NV center, two interacting nuclei and different driving frequencies for both continuous and pulsed driving schemes. In the simulation we ignore errors in the preparation of the fully mixed state of the qubits representing the nuclei. The blue curves show the remaining polarization in the NV center after one cycle of initialization and time evolution, while the red and the green curves correspond to the nuclear polarizations at the end of the cycle. For each nucleus there appears a resonance frequency in the system, for which the polarization transfer is optimal for said nucleus, depicted in the figure by the peaks of the curves. Both simulations include the effects of the most common imperfections in nanoscale NMR systems, i.e. energy detunings and Rabi frequency fluctuations discussed in Sec. II. The simulation of the quantum algorithm additionally includes noise and gate errors present in the QPU. It is notable that the noise affects the height and shape of the peaks more than their location. The system imperfections include a detuning of $120$ kHz of the NV center from the zero-field splitting that shifts the peaks in Fig. 7 (left) to frequencies lower than their predicted Larmor frequencies (dotted vertical black lines). Fig. 7 (right) shows how the pulsed-driving scheme XY8 [22, 23] acts as a robust dynamical decoupling sequence, eliminating such frequency shifts both in the ideal and noisy simulations. Regarding the QPU noise and errors, the amplitude damping channel causes an overall shift down of all polarizations at all driving frequencies. Dephasing noise and gate errors (as modelled by depolarizing noise) cause the curves in Fig. 7 to flatten and lose contrast. While we have discussed how the product of gate errors is minimized by reducing the SWAP overhead through Co-Design hardware, the loss of contrast can also be addressed through error mitigation techniques such as zero-noise extrapolation [56, 57, 58]. Dephasing can also be reduced through dynamical decoupling techniques [39], thus extending the system coherence and increasing the $T_{2}$ time. The simulations presented in Fig. 7 include the decoherence times and gate fidelities that can be achieved with the hardware in Sec. IV. This implies an overestimation of actual errors in the simulation, since the gate fidelities include some decoherence in reality. To quantify the advantage of our Co-Design processor, Fig. 8 shows how the reduction in TQGs improves our ability to extract relevant information from the simulation. The figure compares the star-architecture chip to qubits connected on a square grid simulating a six qubit system with one NV center and five non-interacting nuclei. On the two chips we use SWAP patterns according to the schemes discussed in Sec. III.4. First, Fig. 8a shows the average height-to-width ratio $\bar{\xi}$ of the nuclear polarization peaks obtained with star and square grid topologies with respect to an ideal error-free simulation. It serves as an indicator of how much the QPU noise degrades the simulation for each case. The ratio $\bar{\xi}$ is computed by fitting a Gaussian function on each peak, and computing: $$\bar{\xi}=\Big{\langle}\frac{h}{\sigma}\Big{\rangle},$$ (25) where $h$ is the height and $\sigma$ the variance of the fitted Gaussian function, averaged over the five nuclei. The curves for both topologies must coincide at $\bar{\xi}=0$ for a maximal-error device, and at $\bar{\xi}=\bar{\xi}_{\textrm{ideal}}$ for an error-free quantum computer, since for an a maximal-error device the output is pure noise and for an error-free quantum computer the number of SWAPs is irrelevant to the precision. For NISQ devices in between these limits, a performance difference between the architectures is observed. For systems with more nuclei and NV centers, the differences between topologies start to appear at lower errors, since the number of total operations grows. This shows how the QPU topology is of great importance for the computational precision of NISQ devices, while for fault-tolerant quantum computers the precision is unaffected by the topology. Second, Fig. 8b shows the average relative error in the central frequency of the NMR peaks: $$\bar{\Delta}_{\textrm{peak}}=\Big{\langle}\Big{|}\frac{\omega_{\textrm{noisy}}-\omega_{\textrm{ideal}}}{\omega_{\textrm{ideal}}}\Big{|}\Big{\rangle},$$ (26) where $\omega_{\textrm{noisy}}$ and $\omega_{\textrm{ideal}}$ are the peak-center frequencies extracted from the Gaussian fittings for the noisy and ideal cases, respectively. The peak centers correspond to driving frequencies that efficiently transfer polarization to different parts of the diamond lattice. With the quantum simulation we can individually identify the nuclear resonance peaks by directly measuring the polarization of each qubit. This could enable exploration of how the polarization diffuses in the lattice with single-nucleus precision. In contrast, in a standard nanoscale NMR experiment, one typically only has only access to the excitation loss of the NV ( and thus only to the average transmitted polarization). This demonstrates the advantage of simulating the system on a quantum computer, as a it provides access to the relevant microscopic details of the dynamics that are otherwise inaccessible. The figures demonstrate that the Co-Design chip is able to detect the resonance frequencies and predict the peak heights better at all considered noise levels. The power of Co-Design is particularly evident in Fig. 8b, where the square grid is shown to require two orders of magnitude lower noise levels to reach the same accuracy as the Co-Design chip. VI Conclusions and outlook We have presented a quantum algorithm to simulate a nanoscale NMR problem, namely a hyperpolarization protocol. We have simulated the proposed quantum algorithm with typical noise processes of a NISQ superconducting quantum computer with state-of-the-art parameters. We find that, despite considering a noisy QPU, our protocol still allows to identify the positions of the nuclear resonances (corresponding to the maximal polarizations) in the frequency domain, as well as the behavior in the vicinity of such resonant frequencies, thus enabling the exploration of optimized protocols and driving parameters to hyperpolarize the nuclear ensemble. Moreover, we have shown that a specific Co-Design architecture adapted to the problem provides an advantage over general-purpose designs in the NISQ era, thanks to the reduction in two-qubit-gate count. Consequently, the adapted design reduces the necessary gate fidelities to solve practical problems in nanoscale NMR. This application-specific QPU consists of a central resonator, representing an NV center, coupled to a number of qubits representing the nuclei. The design can be scaled to more NV centers and a potentially large number of qubits around them. This is an example of a shortcut to quantum advantage. Adapting more NISQ-friendly algorithm alternatives, such as those listed in [4], to the problem and to the Co-Design hardware can provide further shortcuts. Our work opens interesting directions for further investigation, since a quantum processor able to efficiently simulate nanoscale-NMR scenarios with a large number of nuclear spins would have a great impact on NMR-based applications. Fast and reliable quantum simulations of interacting spin systems would improve the interpretability of zero- and low-field NMR where spin-spin interactions become dominant [11], and nanoscale-NMR systems where a quantum sensor is strongly coupled via dipole-dipole interactions to nuclear or electron spin clusters. A possible application of the latter is the estimation of inter-label distances (via, e.g., Bayesian analysis of the NV center response) in electronically labelled biomolecules [17]. In this case, the numerical analysis of systems beyond two-electron spin labels in realistic conditions, including protein motion and decoherence channels, is already numerically challenging. Acknowledgments The authors would like to thank Caspar Ockeloen-Korppi, Alessandro Landra and Johannes Heinsoo for their help in developing the idea of the star-architecture chip, Jani Tuorila for his support in developing the gate theory, Amin Hosseinkhani and Tianhan Liu for reviewing the manuscript, and Henrikki Mäkynen and Hoang-Mai Nguyen for graphic design. J.C. additionally acknowledges the Ramón y Cajal program (RYC2018-025197-I). We further acknowledge support from Atos for support with the Quantum Learning Machine (QLM). Finally, the authors acknowledge financial support to BMBF through the Q-Exa project FZK: 13N16062. Appendix A Derivation of the system Hamiltonian The Hamiltonian in Eq. (1) can be derived from first principles. Let us first assume for simplicity a model including only two ${}^{13}C$ nuclei and one NV center (Fig. 1) with dipole-dipole interactions. For simplicity we also consider the NVs to be aligned with the external magnetic field. In that case, the Hamiltonian of the system reads: $$H=DS_{z}^{2}-\gamma_{e}B_{z}S_{z}-\gamma_{c}B_{z}\left(I_{1}^{z}+I_{2}^{z}\right)+\sum_{k=1}^{2}\frac{\hbar\mu_{0}\gamma_{e}\gamma_{c}}{2\left|\vec{r}_{k}\right|^{3}}\left[\vec{S}\cdot\vec{I}_{k}-\frac{3\left(\vec{S}\cdot\vec{r}_{k}\right)\left(\vec{I}_{k}\cdot\vec{r}_{k}\right)}{\left|\vec{r}_{k}\right|^{2}}\right]+\frac{\hbar\mu_{0}\gamma_{c}^{2}}{2\left|\vec{r}_{1,2}\right|^{3}}\left[\vec{I}_{1}\cdot\vec{I}_{2}-\frac{3\left(\vec{I}_{1}\cdot\vec{r}_{1,2}\right)\left(\vec{I}_{2}\cdot\vec{r}_{1,2}\right)}{\left|\vec{r}_{1,2}\right|^{2}}\right],$$ (27) where $S_{j}$ is the $j$-th spin component of the NV center, $I^{j}_{k}$ the $j$-th spin component of nucleus $k$, $D$ is the zero-field splitting of the NV center, $\gamma_{e}$ and $\gamma_{c}$ are the gyromagnetic factors of the NV center and the nuclei respectively, $B_{z}$ is the external magnetic field, which is aligned with the symmetry axis of the NV center $\vec{r}_{k}$ is the relative position vector between the NV center and nucleus $k$ and $\vec{r}_{1,2}$ is the relative position vector between both nuclei. Due to the large energy splitting introduced by $D$, the previous Hamiltonian reduces to: $$H=DS_{z}^{2}-\gamma_{e}B_{z}S_{z}-\gamma_{c}B_{z}\left(I_{1}^{z}+I_{2}^{z}\right)+S_{z}\left(\vec{A}_{1}\cdot\vec{I}_{1}+\vec{A}_{2}\cdot\vec{I}_{2}\right)+g_{1,2}\left[I_{1}^{z}I_{2}^{z}-\frac{1}{4}\left(I_{1}^{+}I_{2}^{-}+I_{1}^{-}I_{2}^{+}\right)\right],$$ (28) where $I_{k}^{\pm}=I_{k}^{x}\pm iI_{k}^{y}$, the hyperfine vectors are $\vec{A}_{k}=\frac{\hbar\mu_{0}\gamma_{e}\gamma_{c}}{2\left|\vec{r}_{k}\right|^{3}}\left[\hat{z}-\frac{3\left(\hat{z}\cdot\vec{r}_{k}\right)\vec{r}_{k}}{\left|\vec{r}_{k}\right|^{2}}\right]$, while the internuclear coupling constant is $g_{1,2}=\frac{\hbar\mu_{0}\gamma_{c}^{2}}{2\left|\vec{r}_{1,2}\right|^{3}}\left[1-3\left(\frac{r_{1,2}^{z}}{\left|\vec{r}_{1,2}\right|}\right)^{2}\right]$. Now, in a rotating frame with respect to $DS_{z}^{2}-\gamma_{e}B_{z}S_{z}$ we obtain: $$H_{I}=-\gamma_{c}B_{z}\left(I_{1}^{z}+I_{2}^{z}\right)+S_{z}\left(\vec{A}_{1}\cdot\vec{I}_{1}+\vec{A}_{2}\cdot\vec{I}_{2}\right)+g_{1,2}\left[I_{1}^{z}I_{2}^{z}-\frac{1}{4}\left(I_{1}^{+}I_{2}^{-}+I_{1}^{-}I_{2}^{+}\right)\right],$$ (29) We can use that $|1\rangle\langle 1|=\frac{\mathbb{1}+\sigma^{z}}{2}$ to rewrite $S_{z}$ only in terms of the $\{|0\rangle,|1\rangle\}$ subspace by dropping out the $|-1\rangle$ energy state as it will not participate in the dynamics. With this we get: $$H_{I}=-\vec{\omega}^{c}_{1}\cdot\vec{I}_{1}-\vec{\omega}^{c}_{2}\cdot\vec{I}_{2}+\frac{\sigma_{z}}{2}\left(\vec{A}_{1}\cdot\vec{I}_{1}+\vec{A}_{2}\cdot\vec{I}_{2}\right)+g_{1,2}\left[I_{1}^{z}I_{2}^{z}-\frac{1}{4}\left(I_{1}^{+}I_{2}^{-}+I_{1}^{-}I_{2}^{+}\right)\right],$$ (30) where $\vec{\omega}^{c}_{k}=-\left(\frac{A_{k}^{x}}{2},\frac{A_{j}^{y}}{2},\frac{A_{j}^{z}}{2}-\gamma_{c}B_{z}\right)$ is the modified nuclear Larmor term due to the presence of the NV center. Generalizing equation (30) to $M$ NV centers and $N$ nuclei, including the detuning of the NV centers and adding the microwave driving term we obtain precisely the Hamiltonian in Eq. (1). Appendix B Hyperpolarization sequences B.1 Hartmann-Hahn sequence Here we explain the dynamics produced by the continuous driving on the hyperpolarization protocol. To illustrate the mechanism, we consider a system including a single NV center and a single nucleus. The corresponding Hamiltonian, now including the driving term, reads: $$H=DS_{z}^{2}-\gamma_{e}B_{z}S_{z}-\gamma_{c}B_{z}I^{z}+S_{z}\vec{A}\cdot\vec{I}+S_{x}\sqrt{2}\,\Omega\cos(\omega t+\phi),$$ (31) When the microwave drive is on resonance with the NV center, $\omega=D+|\gamma_{e}|B_{z}$, and we go into a rotating frame with respect to the terms $DS_{z}^{2}-\gamma_{e}B_{z}S_{z}$ and $|1\rangle\langle 1|=\frac{\mathbb{1}+\sigma^{z}}{2}$ to rewrite $S_{z}$ only in terms of the $\{|0\rangle,|1\rangle\}$ subspace by dropping out the $|-1\rangle$ energy state, as we did in equations (29) and (30), the interaction Hamiltonian is then: $$H_{I}=-\vec{\omega}^{c}\cdot\vec{I}+\frac{\sigma^{z}}{2}\vec{A}\cdot\vec{I}+\frac{\Omega}{2}\sigma^{\phi}.$$ (32) where $\sigma^{\phi}=e^{-i\phi}|1\rangle\langle 0|+e^{i\phi}|0\rangle\langle 1|=e^{-i\phi}\sigma^{-}+e^{i\phi}\sigma^{+}$. More details about the different terms were discussed in the main text in section II. Choosing $\phi=0$ and further moving to an interaction picture with respect to the terms $-\vec{\omega}^{c}\cdot\vec{I}+\frac{\Omega}{2}\sigma^{x}$ we obtain: $$H_{I}=\frac{e^{i\frac{\Omega}{2}\sigma^{x}t}\sigma^{z}e^{-i\frac{\Omega}{2}\sigma^{x}t}}{2}e^{-i\vec{\omega}^{c}\cdot\vec{I}t}\vec{A}\cdot\vec{I}e^{i\vec{\omega}^{c}\cdot\vec{I}t}.$$ (33) We choose now $\Omega=|\vec{\omega}^{c}|$, leading to the so called Hartmann-Hahn double-resonance condition. Applying the identity $e^{i\vec{I}\cdot\hat{l}\phi}\vec{I}\cdot\vec{b}e^{-i\vec{I}\cdot\hat{l}\phi}=\vec{I}\left[(\vec{b}-(\vec{b}\cdot\hat{l})\hat{l})\cos{\phi}-\hat{l}\times\vec{b}\sin{\phi}+(\vec{b}\cdot\hat{l})\hat{l}\right]$ and the rotating-wave approximation to remove time-dependent terms, we get the flip-flop Hamiltonian: $$H_{I}=\frac{A^{\perp}}{4}\left(|+\rangle\langle-|I^{+}+|-\rangle\langle+|I^{-}\right),$$ (34) with $A^{\perp}=|\vec{A}_{x}^{\perp}|=|\vec{A}-\left(\vec{A}\cdot\hat{\omega}^{c}\right)\hat{\omega}^{c}|$ and the nuclear coordinates changed so that $\hat{x}=\hat{A}_{x}^{\perp}$ and $\hat{z}=\hat{A}_{z}^{\parallel}$ with $\vec{A}_{z}^{\parallel}=(\vec{A}\cdot\hat{\omega}^{c})\hat{\omega}^{c}$. B.2 Pulsed sequence Now we consider the pulsed case, represented by the driving term $H_{\textrm{dr}}=\frac{\Omega(t)}{2}\sigma^{\phi}$ where $\Omega(t)$ is a train of $\pi$-pulses. The Hamiltonian is already expressed in the interaction picture from Eq. (32). From there, we further move into a rotating frame with respect to the driving term. The corresponding unitary transformation is $U_{0}=(-i\sigma^{\phi})^{k}$ for the time interval between pulses $k$ and $k+1$. This leads to: $$H_{I}=-\vec{\omega}^{c}\cdot\vec{I}+F(t)\frac{\sigma^{z}}{2}\vec{A}\cdot\vec{I},$$ (35) where $F(t)$ is the so-called filter function, with value $+1$ when $k$ is even, and $-1$ when $k$ is odd. The application of regularly-spaced pulses raises to a square-wave filter function, which can be expanded in Fourier series as: $$F(t)=\sum_{n=1}^{\infty}f_{n}\cos\left(\frac{2\pi n}{T}t\right),$$ (36) for the symmetric version, with $f_{n}=0$ when $n$ is even and $f_{n}=\frac{4}{\pi n}(-1)^{\frac{n-1}{2}}$ when $n$ is odd. We choose now the resonance condition $T=\frac{2\pi n}{|\vec{\omega}^{c}|}$. Going to an interaction picture with respect to $-\vec{\omega}^{c}\cdot\vec{I}$ and repeating the procedure we used in the Hartmann-Hahn case, we get: $$H_{I}=\alpha A^{\perp}\sigma^{z}I^{x},$$ (37) where $\alpha=\frac{f_{n}}{4}$. The same analysis gives $H=\frac{g_{n}}{4}A^{\perp}\sigma^{z}I^{y}$ with $g_{n}=\frac{4}{\pi n}$ in the asymmetric case. Appendix C Hamiltonian decomposition for Trotterized time evolution In order to simulate the dynamics generated by the Hamiltonian in Eq. (1) on a quantum computer using Trotterization, we first need to express it in a suitable way. To begin with, we split the Hamiltonian into two parts: $$H=H_{\textrm{SQG}}+H_{\textrm{TQG}},$$ (38) which can be expressed in terms of qubit Pauli operators: $$H_{\textrm{SQG}}=\sum_{k=1}^{N}\Big{[}\frac{A^{x}_{k}}{2}\frac{X_{k}}{2}+\frac{A^{y}_{k}}{2}\frac{Y_{k}}{2}+\Big{(}\frac{A^{z}_{k}}{2}-\gamma_{c}B_{z}\Big{)}\frac{Z_{k}}{2}\Big{]}+\sum_{j=1}^{M}\delta_{j}Z_{j},$$ (39) $$\displaystyle\begin{split}H_{\textrm{TQG}}&=\sum_{j=1}^{M}\sum_{k=1}^{N}\Big{[}\frac{A^{x}_{k}}{2}\frac{X_{k}}{2}Z_{j}+\frac{A^{y}_{k}}{2}\frac{Y_{k}}{2}Z_{j}+\frac{A^{z}_{k}}{2}\frac{Z_{k}}{2}Z_{j}\Big{]}+\\ &+\sum_{k^{\prime}>k=1}^{N}\frac{g_{k^{\prime}k}}{4}\Big{[}Z_{k^{\prime}}Z_{k}-\frac{1}{2}X_{k^{\prime}}X_{k}-\frac{1}{2}Y_{k^{\prime}}Y_{k}\Big{]}+\\ &+\sum_{j>j^{\prime}}^{M}h_{j^{\prime}j}\Big{[}Z_{j^{\prime}}Z_{j}-X_{j^{\prime}}X_{j}-Y_{j^{\prime}}Y_{j}\Big{]}.\end{split}$$ (40) Since in the rotating frame with the drive the Hamiltonian is time independent, the time-evolution operator is simply given by: $$U=e^{-it_{f}H},$$ (41) where $t_{f}$ is the time for which the simulation runs. The time-evolution operator is split into $s$ discrete steps through Trotter decomposition: $$U=e^{-it_{f}H}=e^{-it_{f}(H_{\textrm{SQG}}+H_{\textrm{TQG}})}\approx\left[e^{-i\frac{t_{f}}{s}H_{\textrm{SQG}}}e^{-i\frac{t_{f}}{s}H_{\textrm{TQG}}}\right]^{s}+\mathcal{O}\left(\left(\frac{t_{f}}{s}\right)^{2}\right).$$ (42) The evolution operator associated with single-qubit gates in each Trotter step of equation (42) needs to be rewritten in terms of our native gate set. It is always possible to decompose any single-qubit unitary exactly, up to a global phase, into a sequence of three single-qubit rotations such as, for example, a rotation about the $y$-axis in between two rotations about the $z$-axis: $$U_{1}=R_{z}(\beta)R_{xy}(\pi/2,\gamma)R_{z}(\delta),$$ (43) where the angles $\beta,\gamma,$ and $\delta$ need to be determined from the specific entries of the unitary in question to simulate the evolution of the $p^{{}_{\textrm{th}}}$ qubit: $$U_{1}^{p}=e^{-i\frac{t_{f}}{s}\left(\frac{A^{x}_{p}}{2}\frac{X_{p}}{2}+\frac{A^{y}_{p}}{2}\frac{Y_{p}}{2}+\left(\frac{A^{z}_{p}}{2}-\gamma_{c}B_{z}\right)\frac{Z_{p}}{2}\right)}.$$ (44) From now on, we will concentrate on the case of a single NV center, which will be encoded in qubit $0$. Then, the evolution operator associated to single-qubit gates for the NV center will be: $$U^{0}_{1}=e^{-i\frac{t_{f}}{s}\delta_{0}Z_{0}}.$$ (45) Matching the entries of the matrices corresponding to the unitaries on equations (44) and (45) we get a system of equations for the angles $\beta,\gamma,$ and $\delta$ for each Trotter step $s$. There are 3 (5) types of interaction terms of the form ${XZ,YZ,ZZ,\cdots}$ in $H_{\textrm{TQG}}$ without (with) internuclear interactions. Due to the native TQG being of only $ZZ$ interaction type (see Eq. (8)), local rotations need to be introduced for simulating the rest of the TQG terms. These are $R^{\sigma_{i}\rightarrow\sigma_{j}}_{k}$, which have the effect of converting the Pauli operator $\sigma_{i}$ into the Pauli operator $\sigma_{j}$ for qubit $k$. The time-evolution operator for each Trotter step needs to be further split into terms consisting of only one operator each: $$\displaystyle\begin{split}e^{-i\frac{t_{f}}{s}H_{\textrm{TQG}}}\approx\,&e^{-i\frac{t_{f}}{s}\left(\sum_{k}\frac{A^{x}_{k}}{2}\frac{X_{k}}{2}Z_{0}\right)}e^{-i\frac{t_{f}}{s}\left(\sum_{k}\frac{A^{y}_{k}}{2}\frac{Y_{k}}{2}Z_{0}\right)}\\ &e^{-i\frac{t_{f}}{s}\left(\sum_{k}\frac{A^{z}_{k}}{2}\frac{Z_{k}}{2}Z_{0}\right)}e^{-i\frac{t_{f}}{s}\left(\sum_{k^{\prime}>k}\frac{g_{k^{\prime}k}}{4}Z_{k^{\prime}}Z_{k}\right)}\\ &e^{i\frac{t_{f}}{s}\left(\sum_{k^{\prime}>k}\frac{g_{k^{\prime}k}}{8}X_{k^{\prime}}X_{k}\right)}e^{i\frac{t_{f}}{s}\left(\sum_{k^{\prime}>k}\frac{g_{k^{\prime}k}}{8}Y_{k^{\prime}}Y_{k}\right)}\\ &+\mathcal{O}\left(\left(\frac{t_{f}}{s}\right)^{2}\right).\end{split}$$ (46) Finally, we observe that the operators $Z_{k}Z_{0}$ (and the rest of the TQG terms) commute with each other, so the exponentials can be further split without Trotterizing: $$e^{-i\frac{t_{f}}{s}(\sum_{k}\frac{A^{z}_{k}}{2}\frac{Z_{k}}{2}Z_{0})}=\Pi_{k}e^{-i\frac{t_{f}}{s}(\frac{A^{z}_{k}}{2}\frac{Z_{k}}{2}Z_{0})}.$$ (47) The time-evolution operator implementing the continuous sinusoidal driving $\sigma^{\phi}$ is: $$e^{-i\frac{t_{f}}{s}\frac{\Omega}{2}\sigma^{\phi}}=R_{xy}(-\phi,\theta=\Omega\frac{t_{f}}{s}).$$ (48) The quantum algorithm for simulating the system under a pulsed-driving scheme is somewhat more involved than the continuous-driving case, due to the two different time-dependent processes involved in the Trotter decomposition: the free dynamics of the spins and the sequence of pulses. The most crucial point to be aware of is the interplay between Trotter steps and interpulse spacing. The time interval between pulses bounds from below the minimum number of Trotter steps for the simulation. At least one Trotter step is needed for each interpulse evolution, that is, the free evolution that occurs in between two consecutive pulses, so at least as many Trotter steps are needed as pulses. Taking this interplay into account, the most straightforward setup is to choose a frequency which will determine the spacing of the pulse sequence, and to identify each interpulse evolution with a single Trotter step. If the achieved precision is not high enough, more Trotter steps can be added for each interpulse evolution. Each $\pi$-pulse itself is simply implemented as an $X$- or $Y$-gate on the qubit representing the NV center. The OU-distributed Rabi frequency fluctuations present in nanoscale NMR systems are then simulated by over- and under-rotations of the $X$- and $Y$-gates. Appendix D Rotational optimization In principle, we had a Hamiltonian with terms of the type $ZX$, $ZX$ and $ZZ$ for the case of no internuclear interactions. However, we can rotate the basis so the Hamiltonian loses the $ZX$ and $ZY$ terms, allowing to reduce the number of TQGs. To make up for this rotation, we need to introduce different constants $\vec{A}^{\textrm{rot}}_{i}$ for the problem and rotate the vector state we obtain at the end before measuring it. Now, if we want to obtain the mean value of $\sigma_{z}$: $$\begin{split}\langle\sigma_{z}\rangle=Tr(\rho(T)\sigma_{z})=Tr(U(0,T)\rho(0)U^{\dagger}(0,T)\sigma_{z}),\end{split}$$ (49) where $U(0,T)$ represents the evolution operator from $t=0$ to $t=T$. Our intention is to obtain an expression of this mean value in terms of the rotated evolution operators. Then, taking into account that the trace is invariant under a rotation $R$ we get: $$\langle\sigma_{z}\rangle=Tr(RU(0,T)\rho(0)U^{\dagger}(0,T)\sigma_{z}R^{\dagger})=Tr(RU(0,T)R^{\dagger}R\rho(0)R^{\dagger}RU^{\dagger}(0,T)R^{\dagger}R\sigma_{z}R^{\dagger}).$$ (50) This can be expressed as: $$\begin{split}\langle\sigma_{z}\rangle=Tr(U_{\textrm{rot}}(0,T)\rho_{\textrm{rot}}(0)U_{\textrm{rot}}^{\dagger}(0,T)R\sigma_{z}R^{\dagger}).\end{split}$$ (51) In our case, the initial state of each qubit is a thermal state at room temperature well approximated by a fully mixed state $\rho_{\textrm{mixed}}=\frac{\mathbb{1}}{2^{N}}$ for the nuclei. Thus, any rotation on nuclei qubits leaves the density matrix unaffected, leading to: $$\begin{split}\langle\sigma_{z}\rangle=Tr(U_{\textrm{rot}}(0,T)\rho(0)U_{\textrm{rot}}^{\dagger}(0,T)R\sigma_{z}R^{\dagger}).\end{split}$$ (52) Then we need to rotate the system previous to the measurement. By using the invariance of the trace under rotations we get: $$\begin{split}\langle\sigma_{z}\rangle=Tr(R^{\dagger}U_{\textrm{rot}}(0,T)\rho(0)U_{\textrm{rot}}^{\dagger}(0,T)R\sigma_{z}),\end{split}$$ (53) which is equivalent to introducing a counter-rotation in the circuit before measurement. Now let us focus on the specific rotation we have to implement. Since the constants multiplying the Pauli matrices in the Hamiltonian are $\frac{\vec{A_{1}}}{2}$ and $\vec{\omega^{c}_{1}}=\frac{\vec{A_{1}}}{2}-\gamma_{c}B_{z}\vec{e_{z}}$ (for nucleus 1), we can rotate the basis to obtain a representation in which the vectors have only z-component for $\vec{A_{1}}$. The vectors before and after the needed rotation can be seen in Fig. 9. To compute the new vectors (and thus the new coefficients for the gates of our algorithm), we can use Rodrigues’ rotation formula to rotate a vector $\vec{v}$ an angle $\theta$ around a unitary axis $\hat{k}$: $$\vec{v}_{rot}=\vec{v}\cos\theta+(\hat{k}\times\vec{v})\sin\theta+\hat{k}(\hat{k}\cdot\vec{v})(1-\cos\theta),$$ (54) being in our case, $\theta=\arccos{(A^{z}_{1}/|\vec{A}_{1}|)}$ and $\hat{k}=(\cos(\phi),\sin(\phi),0)$, with $\phi=-\frac{\pi}{2}+\phi_{xy}=-\frac{\pi}{2}+\arctan{(A^{y}_{1}/A^{x}_{1})}$. For implementing the counter-rotation of this in the quantum circuit, we use: $$R^{\dagger}=e^{i\frac{\theta}{2}(\cos(\phi)X-\sin(\phi)Y)}.$$ (55) Appendix E Randomized Trotter techniques As explained in section III, we chose Trotter expansion. Besides this, we can consider other simulation approaches such as the variational quantum simulator [29], the quantum assisted simulator [30], numerical quantum circuit synthesis [31], or a plethora of other quantum simulation algorithms aimed at NISQ devices [4]. In addition, other approaches like randomized Trotter have been recently shown to provide some advantage compared to standard Trotter expansions [59]. We choose one randomized approach, qDRIFT [28], that consists of the following: instead of splitting the whole evolution operator $e^{-it_{f}\sum_{j}h_{j}H_{j}}$ into simpler terms as done in full Trotterization, the method applies a random selection of such terms to the quantum circuit. This random selection is based on the probability distribution given by the weight of each term $h_{j}H_{j}$. For a certain evolution time, this set of gates can approximate the whole evolution operator by statistically drifting the state of the circuit towards the deterministic final state. The error bound for this method is given as [28]: $$\varepsilon^{\textrm{qDRIFT}}_{\textrm{sim}}\leq\frac{2\lambda^{2}t_{f}^{2}}{N_{\textrm{terms}}},$$ (56) where $\lambda=\sum_{j}h_{j}$ and $N_{\textrm{terms}}$ is the number of terms we need to implement. The advantage of qDRIFT compared to Trotterization is particularly apparent when dealing with Hamiltonians with a large number of terms with small coefficients, simulated for short times. While in the standard Trotter case, every term has to be simulated for each step no matter how small its effect is, in qDRIFT this is not required. A more thorough analysis of errors in qDRIFT and gate counts can be found in [60]. This method is particularly suitable to our problem, since the range of coefficients in the Hamiltonian of a real diamond is large due to the length scales involved. In this case, with qDRIFT the terms with smaller coefficients do not add a significant amount of gates as they would in conventional Trotterization approaches. We note that other adapted protocols such as SparSto [61] can further enhance the simulation of this type of systems. SparSto represents a compromise between Trotterization and qDRIFT, generally guaranteeing an equal or better performance than both of them. We will not go into detail on this method since Trotterization and qDRIFT are enough to illustrate the main ideas behind this work. Appendix F SWAP routing Our qubit routing method consists of mapping the square grid to a linear chain with qubits labeled from 0 to $n$. Then, in the simplified case of no internuclear interactions, the optimal SWAP method for the one-to-all interaction case on a linear chain can be used. For a single NV center the protocol goes as follows: 1. Initialize the state of the NV center in the second qubit; 2. Perform interactions with the first and third qubits; 3. SWAP the NV center qubit to the right; 4. Perform interaction with right qubit; 5. Repeat steps 3-4 until all interactions have been achieved. The pattern is seen in Fig. 3a denoted by the intense blue arrows. With internuclear interactions we need to perform a swap pattern that enables all-to-all interactions. The so-called odd-even mapping in Fig 3a is an efficient one [62] represented by green arrows in Fig. 3a. This consists of swapping first all the even qubits with their right neighbors and then swapping all the odd qubits with their right neighbors. This way, we will obtain ATA interactions with $\frac{1}{2}(n-1)(n-2)$ SWAP gates and a total TQG depth of $6n$. A summary of the TQG counts is shown in Table 2. To motivate the creation of a chip with a star topology and the use of an alternative linearized SWAP routing for a square grid instead of standard numerical approaches, a comparison between all the cases is provided in Fig. 10. A reduction in the number of SWAPs can be noticed for both the linear chain approach and the star-topology chip against standard numerical approaches for a square grid. Appendix G Qubit-resonator gate theory In the following discussion, we consider gate operation between the resonator and one of the qubits, and neglect any effects that arise from the interactions with spectator qubits and other resonator modes. The time dynamics in such a system are determined by the Hamiltonian: $$\displaystyle\begin{split}H=H_{0}+H_{rc}+H_{qc}+H_{rq},\end{split}$$ (57) where the uncoupled part of the total Hamiltonian $H_{0}=H_{r}+H_{c}+H_{q}$ is: $$\displaystyle\begin{split}H_{r}&=\hbar\omega_{r}b_{r}^{\dagger}b_{r}\\ H_{c}&=\hbar\omega_{c}b_{c}^{\dagger}b_{c}+\frac{\hbar}{2}\alpha_{c}b_{c}^{\dagger}b_{c}^{\dagger}b_{c}b_{c},\\ H_{q}&=\hbar\omega_{q}b_{q}^{\dagger}b_{q}+\frac{\hbar}{2}\alpha_{q}b_{q}^{\dagger}b_{q}^{\dagger}b_{q}b_{q},\end{split}$$ (58) where $b_{\lambda}$ and $\omega_{\lambda}$ are the annihilation operator and fundamental frequency for the mode $\lambda=\{r,c,q\}$, respectively, and $\alpha_{\gamma}$ is the anharmonicity of the mode $\gamma=\{q,c\}$. 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The near-infrared counterpart of 4U 1636–53††thanks: Based on observations collected at the European Southern Observatory, Chile, under ESO Programme ID 085.D-0456(D). D. M. Russell 1Astronomical Institute ‘Anton Pannekoek’, University of Amsterdam, P.O. Box 94249, 1090 GE Amsterdam, the Netherlands 12Instituto de Astrofísica de Canarias (IAC), Vía Láctea s/n, La Laguna E-38205, S/C de Tenerife, Spain 2russell@iac.es    K. O’Brien 3Department of Physics, University of California, Santa Barbara, California 93106, USA 3kobrien@physics.ucsb.edu    T. Muñoz-Darias${}^{,}$ 2Instituto de Astrofísica de Canarias (IAC), Vía Láctea s/n, La Laguna E-38205, S/C de Tenerife, Spain 2russell@iac.es 4School of Physics and Astronomy, University of Southampton, Southampton, Hampshire SO17 1BJ, UK 4t.munoz-darias@soton.ac.uk    P. Casella 4School of Physics and Astronomy, University of Southampton, Southampton, Hampshire SO17 1BJ, UK 4t.munoz-darias@soton.ac.uk 5INAF - Osservatorio Astronomico di Roma, Via Frascati 33, I-00040 Monteporzio Catone (Roma), Italy 5piergiorgio.casella@oa-roma.inaf.it    P. Gandhi 6ISAS, Japan Aerospace Exploration Agency, 3-1-1 Yoshinodai, chuo-ku, Sagamihara, Kanagawa 229-8510, Japan 6pgandhi@astro.isas.jaxa.jp    M. G. Revnivtsev 7Space Research Institute, Russian Academy of Sciences, Profsoyuznaya 84/32, 117997 Moscow, Russia 7revnivtsev@iki.rssi.ru (Received ?? ??, 2011; accepted ?? ??, 2011) Key Words.: Stars: neutron – X-rays: binaries – stars: infrared Abstract Context:The optical counterpart of the neutron star X-ray binary and well known X-ray burster, 4U 1636–53 (= 4U 1636–536 = V801 Ara) has been well studied for three decades. However to date, no infrared studies have been reported. Aims:Our aims are to identify and investigate the near-infrared (NIR) counterpart of 4U 1636–53. Methods:We present deep, $K_{\rm S}$-band ($2.2\mu\mathrm{m}$) imaging of the region of 4U 1636–53 taken with the Infrared Spectrometer And Array Camera (ISAAC) on the Very Large Telescope. Archival optical and UV data are used to infer the $0.2-2.2\mu\mathrm{m}$ spectral energy distribution (SED). Results:One star is located at coordinates $\alpha=16$:40:55.57, $\delta=-53$:45:05.2 (J2000; $1\sigma$ positional uncertainty of $\sim 0.3$ arcsec) which is consistent with the known optical position of 4U 1636–53; its magnitude is $K_{\rm S}=16.14\pm 0.12$. This star is also detected in the 2MASS survey in $J$-band and has a magnitude of $J=16.65\pm 0.22$. Under the assumption that the persistent emission is largely unvarying, the $0.4-2.2\mu\mathrm{m}$ de-reddened SED can be described by a power law; $F_{\nu}\propto\nu^{1.5\pm 0.3}$, with some possible curvature ($F_{\nu}\propto\nu^{\lesssim 1.5}$) at $0.2-0.4\mu\mathrm{m}$. The SED can be approximated by a blackbody of temperature $\sim 27000$ K. This is typical for an active low-mass X-ray binary, and the emission can be explained by the outer regions of a (likely irradiated) accretion disc. We therefore interpret this $K_{\rm S}$-band star as the NIR counterpart. Conclusions: 1 Introduction Low mass X-ray binaries (LMXBs) are interacting binaries where a low mass donor is transferring material onto a neutron star or a black hole. In order to transport the excess of angular momentum outwards, an accretion disc is formed. In the disc, the gravitational potential energy is transformed into mainly X-ray radiation and kinetic energy, and temperatures approach $\sim 10^{8}$ K. The mass transfer rate supplied by the donor star, $\dot{M}_{2}$, is driven by the binary/donor evolution and mass transfer rates $\dot{M}_{2}>\dot{M}_{\rm crit}\sim 10^{-9}M_{\odot}~{}\mathrm{kg}~{}\mathrm{yr% }^{-1}$ (where $M_{\odot}$ is the mass of the Sun in kg) result in persistently bright X-ray sources (King et al. 1996). There are $\sim 200$ such bright ($L_{\rm X}\simeq 10^{36}-10^{38}\mathrm{erg}~{}\mathrm{s}^{-1}$) LMXBs in the Galaxy and most of them harbour neutron stars as implied by the detection of pulsations and X-ray bursts resulting from nuclear burning caused by the accumulation of Hydrogen and Helium on their surfaces. They show energy spectra dominated by emission from their irradiated accretion discs which also dominate at optical wavelengths (e.g., van Paradijs & McClintock 1995). Even in the near-infrared (NIR), where the companion star could have an important contribution, the disc emission seems dominant in bright systems. For instance, in the prototypical persistent neutron star, Sco X–1, which given its relatively long orbital period ($\sim 19$ h) should have a large, evolved companion star, no spectral feature from the donor has been detected in the NIR (Bandyopadhyay et al. 1997). In most persistent neutron star X-ray binaries, spectral and temporal studies have favoured an X-ray heated accretion disc as the origin of the NIR emission (e.g., 4U 1705–440; Homan et al. 2009). Compact jets producing synchrotron emission typically dominate the radio emission (Migliari & Fender 2006) and their spectra can extend to higher frequencies. In the NIR, high amplitude flares from GX 17+2 (Callanan et al. 2002), a synchrotron spectrum in 4U 0614+09 (Migliari et al. 2010) and variable linear polarization in Sco X–1 (Russell & Fender 2008) have suggested a strong infrared jet contribution in these persistent neutron star X-ray binaries. One of the classical neutron star systems is 4U 1636–53 (= 4U 1636–536 = V801 Ara). It is an X-ray burster, which has been extensively studied in X-ray and optical regimes for more than three decades (e.g., Pedersen et al. 1982). It has a 3.8 h orbital period (van Paradijs et al. 1990; Giles et al. 2002) pointing to a relatively faint, late type companion star. The spectrum of the system from X-ray to optical wavelengths seems to be dominated by the emission of its bright accretion disc, the companion star only being detected by using emission lines arising from reprocessing of the strong X-ray emission ($L_{X}\sim 10^{37-38}\mathrm{erg}~{}\mathrm{s}^{-1}$) in its inner hemisphere (Casares et al. 2006). To date, no detections of 4U 1636–53 have been reported at wavelengths longer than the optical regime. At radio frequencies, upper limits of both the persistent and burst fluxes were presented in Thomas et al. (1979). Here we present the first detection of the near-infrared counterpart of 4U 1636–53. Together with available optical and UV data, we construct the intrinsic 0.2–2.2 $\mu\mathrm{m}$ spectral energy distribution (SED). 2 Very Large Telescope observations The data were acquired with the Infrared Spectrometer And Array Camera (ISAAC) on the European Southern Observatory (ESO) 8-m class Very Large Telescope UT3 (Melipal) on 2010-05-21 02:23 – 03:17 UT (MJD $55337.118\pm 0.019$) under ESO Programme ID 085.D-0456(D). Twenty-four data cubes, each of exposure time 132.5 s (comprising $530\times 0.25$ s individual exposures) were obtained of the region of 4U 1636–53 in high time resolution (FastJitter) mode using the $K_{\rm S}$-band filter centred at $2.16\mu\mathrm{m}$. The brightest object in the field has $<1000$ counts in each 0.25 s exposure, so non-linearities are not a concern. A small (up to 6″) telescope offset was applied between cubes. Average images were made of each cube, removing the sky generated from a median of the surrounding averaged cubes. The field of view of each ISAAC image is $40\arcsec\times 40\arcsec$ and the pixel scale is $0.1478\arcsec\ \mathrm{pixel}^{-1}$. The 24 average images were then aligned and stacked using IRAF to produce a deep image. The total on-source exposure time of the combined image is 3180 s. The $K_{\rm S}$-band seeing as measured from the combined image was 0.6 arcsec and conditions were clear. The camera was rotated at an angle of $127^{\circ}$ to maximise the number of reference stars on the array. The combined image is presented in Fig. 1 and can be used as a deep, high resolution $K_{\rm S}$-band finding chart. 3 Source identification Within the combined image, six stars listed in the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006) are detected, which were used to calculate the world coordinate system and achieve $K_{\rm S}$-band flux calibration. DAOPHOT in IRAF is used to perform point-spread-function (PSF) photometry on the combined image. All comparison stars were successfully fitted by a single PSF consistent with the image PSF; no blends were identified. There is one star consistent with the best optical position of 4U 1636–53, as listed in the USNO-B1 digital sky survey ($\alpha=16$:40:55.61, $\delta=-53$:45:05.1; J2000, with an error circle of 0.339″), shown as a white circle in Fig. 1. We derive a position of $\alpha=16$:40:55.57, $\delta=-53$:45:05.2 (J2000) for the infrared counterpart of 4U 1636–53, with a $1\sigma$ positional uncertainty of $\sim 0.3$ arcsec. Other optical coordinates of 4U 1636–53 in the literature (Jernigan et al. 1977; Liu et al. 2001; Samus et al. 2003) are also consistent with the USNO-B1 and ISAAC positions. No precise X-ray coordinates of 4U 1636–53 from Chandra or Swift observations exist in the literature. Some faint, blended stars lie just 1.5″ from 4U 1636–53 in the ISAAC image, but none are consistent with the optical position. The above source is the only candidate counterpart in our $K_{\rm S}$-band image. All stars within a $3\arcsec$ radius of the LMXB are $\geq 1.2$ magnitudes fainter in $K_{\rm S}$-band than this proposed counterpart. No faint blended stars are visible after subtracting the PSF of the counterpart. From the known magnitudes of the 2MASS stars within the field, we measure the magnitude of 4U 1636–53 during the time of our observations to be $K_{\rm S}=16.14\pm 0.12$. The errors are dominated by systematic errors derived from the six 2MASS star magnitudes. 4U 1636–53 is detected with a signal-to-noise ratio (S/N) of 145. The limiting magnitude in the image is $K_{\rm S}\sim 19.3$ (i.e. stars brighter than this are detected at the $3\sigma$ level). The counterpart we have found is not listed in the 2MASS point source catalogue. On inspection of the 2MASS images, the counterpart is not visible in $K$ and $H$-bands, but appears as a faint source in $J$-band. Performing PSF photometry on the $J$-band 2MASS field, we obtain a magnitude of $J=16.65\pm 0.22$ for 4U 1636–53; the S/N is 7. The aforementioned faint stars within $\sim 2\arcsec$ of the source may contribute up to $\sim 10-30$% of the flux measured, although the PSF fitting method should have removed the majority of this excess flux (which is offset from the central PSF position). This 2MASS observation was made on 1999-06-18 04:27 UT. In the 2MASS $H$-band image, the upper limit to the magnitude of 4U 1636–53 is $H>15.76$. The new VLT and 2MASS detections of 4U 1636–53 are listed in Table 1. 4 Swift UVOT observations Optical–ultraviolet observations of 4U 1636–53 were made with the UltraViolet/Optical Telescope (UVOT; Roming et al. 2005) on board the Swift satellite and the data are publicly available. UVOT observed 4U 1636–53 on 11 dates between 2005-02-13 and 2010-06-10. On most dates, one or two of the six UVOT filters were used. The image data of each filter on each date were summed using uvotimsum. Photometry of the source in individual sequences was derived with uvotsource. 4U 1636–53 is clearly detected at the position derived above, in all six filters ($5500-1900\AA$). An extraction region of radius 3″ centred at the ISAAC-derived position was adopted, which excludes a neighbouring star 6″ away from 4U 1636–53; no stars appear to contaminate the flux of 4U 1636–53 within the extraction radius. The mean magnitudes and range in each filter are given in Table 1. 5 Intrinsic spectral energy distribution We de-redden the $K_{\rm S}$, $J$-band and UVOT magnitudes using the known interstellar extinction towards 4U 1636–53; $A_{\rm V}=2.5\pm 0.3$ mag (derived from optical data; Lawrence et al. 1983) applying the extinction law of Cardelli et al. (1989). We also take optical magnitudes from Lawrence et al. (1983) and Pedersen et al. (1982); $U=17.92\pm 0.06$, $B=18.47\pm 0.03$, $V=17.89\pm 0.03$, $R=17.46\pm 0.03$, $I=16.77\pm 0.03$, and de-redden them in the same manner. Using the de-reddened fluxes (adopting the standard conversion zeropoints for optical Johnson filters, and 2MASS IR filters from the Explanationary Supplement of Skrutskie et al. 2006) we construct the intrinsic NIR–optical–UV SED of the persistent (non-burst) emission from 4U 1636–53. The SED, which spans one order of magnitude in frequency, is presented in Fig. 2. The observed, reddened fluxes are plotted in addition to the de-reddened data. The optical ($I$ to $U$-band) SED from Lawrence et al. (1983) and Pedersen et al. (1982), which has small errors, has a spectral index of $\alpha=1.5\pm 0.4$, where $F_{\nu}\propto\nu^{\alpha}$. The error on $\alpha$ is dominated by the uncertainty in the extinction. This spectral index is typical for active low-mass X-ray binaries (both neutron star and black hole systems; e.g., Hynes 2005; Russell et al. 2007) and is consistent with the outer regions of a blue accretion disc. Shih et al. (2011) showed that the optical emission is correlated with the soft X-ray flux; the optical light is dominated by the irradiated disc (X-ray reprocessing on the disc surface). The $K_{\rm S}$, $H$-band and $J$-band magnitudes correspond to de-reddened flux densities of $F_{\nu,K_{\rm S}}=0.30\pm 0.03$ mJy, $F_{\nu,H}<0.83$ mJy and $F_{\nu,J}=0.67\pm 0.14$ mJy, respectively. The optical–infrared ($K_{\rm S}$ to $U$-band) SED (neglecting the UVOT data) can be fitted with a power law with spectral index $\alpha=1.5\pm 0.3$ (solid line in Fig. 2). This is similar to the optical spectral index, and the $K_{\rm S}$-band and $J$-band fluxes are consistent with an extrapolation of the accretion disc spectrum. It is unlikely that IR emission lines (e.g. Bandyopadhyay et al. 1997) could contribute a significant fraction of the observed IR fluxes. There is some possible curvature in the spectrum at the higher frequencies, apparent in the UVOT data, and the whole SED can be approximated by a blackbody at a temperature of $\sim 27~{}000$ K (dotted line in Fig. 2). The curvature implies the blackbody of the irradiated disc spectrum may peak at around 1900 $\AA$, similar to some other LMXBs (Hynes 2005). However, the curvature (and even more so the spectral index of the power law) is sensitive to the uncertainty in the extinction, and the optical counterpart varies by $\sim 0.6$ mag amplitude (a factor of $\sim 1.7$ in flux) over weeks to months (e.g., Shih et al. 2011). We detect variability in our UVOT data over several years (Table 1). Quasi-simultaneous optical–infrared data are required to accurately measure the spectral index, and uncertainties will be decreased once the extinction towards the source is measured more accurately. There is no evidence for the companion star, or synchrotron emission from the jet (if the system has a jet) to dominate the infrared flux. Both of these components are expected to be redder than the irradiated disc component, and would produce an infrared excess above the disc spectrum. Since the spectrum between $J$ and $K_{\rm S}$-bands is blue ($\alpha\sim 1.5$), these components make a negligible contribution to $J$-band. In the most extreme scenario, the disc could have a steep spectrum, $\alpha=2.0$ between $J$ and $K_{\rm S}$-band, and therefore the disc could be as faint as 0.18 mJy in $K_{\rm S}$-band ($0.12\pm 0.03$ mJy fainter than the observed flux). If this were the case, the jet or companion could contribute up to 46% of the $K_{\rm S}$-band flux. These calculations again neglect possible variability, since the NIR observations were made on different dates. Simultaneous optical–infrared observations could constrain the level of star or jet emission more precisely. 6 Conclusions We have identified the near-infrared counterpart of the neutron star burster LMXB 4U 1636–53. Its magnitudes on the dates observed are $K_{\rm S}=16.14\pm 0.12$ (VLT / ISAAC), $J=16.65\pm 0.22$ (2MASS). The intrinsic infrared–optical–UV spectrum of the persistent (non-burst) emission is consistent with a blackbody, likely from the irradiated surface of the accretion disc. We find no evidence for emission from other components that may be expected to contribute, such as the donor star or synchrotron emission from a jet, although simultaneous optical and infrared data are needed to constrain these contributions further. Acknowledgements. This research was partly supported by a Netherlands Organisation for Scientific Research (NWO) Veni Fellowship and partly by a Marie Curie Intra European Fellowship within the 7th European Community Framework Programme under contract no. IEF 274805. The research leading to these results has received funding from the European Communitys Seventh Framework Programme (FP7/2007-2013) under grant agreement number ITN 215212 -Black Hole Universe-. Partially funded by the Spanish MEC under the Consolider-Ingenio 2010 Program grant CSD2006-00070: First Science with the GTC (http://www.iac.es/consolider-ingenio-gtc/). TMD acknowledges support by ERC advanced investigator grant 267697-4PI-SKY. PC acknowledges funding via a EU Marie Curie Intra-European Fellowship under contract no. 2009-237722. MR is supported by grants MD-1832.2011.2, RFBR 10-02-00492 and by Dynasty foundation. 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Recent high-$p_{T}$ results from STAR M. van Leeuwen for the STAR collaboration 1Universiteit Utrecht, PO Box 80000, Utrecht, Netherlands 1m.vanleeuwen1@uu.nl Abstract We present selected recent results of multi-hadron correlation measurements in azimuth and pseudorapidity at intermediate and high $p_{T}$ in Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV, from the STAR experiment at RHIC. At intermediate $p_{T}$, measurements are presented that attempt to determine the origin of the associated near-side (small $\Delta\phi$) yield at large pseudo-rapidity difference $\Delta\eta$ that is found to be present in heavy ion collisions. In addition, results are reported on new multi-hadron correlation measures at high-$p_{T}$ that use di-hadron triggers and multi-hadron cluster triggers with the goal to constrain the underlying jet kinematics better than in the existing measurements of inclusive spectra and di-hadron correlations. Keywords:heavy-ion collisions – jets – ridge pacs: 25.75.Bh The goal of research with high-energy nuclear collisions at the Relativistic Heavy Ion Collider (RHIC) is to study bulk matter systems where the strong nuclear force as described by Quantum Chromo Dynamics (QCD) is the dominant interaction. In particular, it is expected that there is a phase transition of bulk QCD matter to a deconfined state at high temperature. In heavy ion collisions, products of initial state hard scatterings, such as high-$p_{T}$ hadrons, photons and heavy mesons, can be used to probe the soft matter generated in the collision. Initial state production of high-$p_{T}$ partons is relatively unaffected by the presence of the soft medium, but the partons lose energy when traversing the medium, dominantly due to gluon radiation. The goal of high-$p_{T}$ measurements is to study these interactions and to use them to measure the density and temperature of the soft matter. It has also been found at RHIC that at intermediate $p_{T}\approx 2-6$ GeV/$c$, the baryon/meson ratio is much larger in heavy ion collisions than in proton-proton collisions Abelev:2006jr ; Abelev:2007ra . This could imply that soft production mechanisms, e.g. hadron formation by coalescence of quarks from thermal matter Lin:2002rw ; Hwa:2002tu ; Fries:2003vb , have a significant contribution to baryon (and meson) production up to $p_{T}\sim 6$ GeV/$c$. The underlying (di-)jet structure for particle production at high-$p_{T}$ can be probed using azimuthal di-hadron correlations, which measure the associated particle distribution in azimuth with respect to some higher $p_{T}$ ‘trigger’ particle Adams:2006yt . This technique has been applied with trigger hadrons from low to high $p_{T}$. In the following, we will first discuss di-hadron measurements at intermediate $p_{T}$, where the soft matter from the medium seems to contribute to correlation structures. The last part of these proceedings deals with high-$p_{T}$ measurements, where jet-quenching followed by jet-fragmentation in the vacuum seems to dominate. 1 Intermediate $p_{T}$: near-side ridge One of the striking results in di-hadron correlations at intermediate $p_{T}$ at RHIC is that near-side (small $\Delta\phi$) associated yield is observed at large $\Delta\eta$ with respect to the trigger in heavy ion collisions. This is illustrated in Fig. 1 which shows associated particle distributions in pseudo-rapidity difference $\Delta\eta$ and azimuthal angle difference $\Delta\phi$, with respect to the trigger hadron direction for associated particles with $2~{}\mathrm{GeV}/c<p_{T}^{assoc}<p_{T}^{trig}$. The upper panel shows the results for trigger hadrons $3<p_{T}^{trig}<4$ GeV/$c$ and the lower panel for $4<p_{T}^{trig}<6$ GeV/$c$. The associated particle distribution shows a clear peak at $(\Delta\eta,\Delta\phi)\approx(0,0)$ as expected from jet fragmentation. Additional associated yield is seen at large $\Delta\eta\gtrsim 1$, which is unique to heavy ion collisions. Within the experimental acceptance, the additional yield is approximately independent of $\Delta\eta$, and therefore referred to as the ridge. For higher $p_{T}^{trig}$ (lower panel of Fig. 1), the ridge yield is less prominent, because the yield in the jet-like peak increases with $p_{T}^{trig}$. Quantitative analysis of the yields shows that ridge yield is independent of $p_{T}^{trig}$ within the statistical and systematic uncertainties Putschke:2007mi . Various production mechanisms for the ridge have been proposed in the literature. Here we would like to classify these mechanisms into two broad categories, namely ’jet-like’ and ’bulk-like’. Jet-like production mechanisms are based on the idea that the ridge is formed mainly from jet fragments, which couple to the longitudinal flow Armesto:2004pt ; Majumder:2006wi ; Romatschke:2006bb . Bulk-like production mechanisms, on the other hand, start from the assumption that the passage of the jet through the longitudinally extended bulk matter locally increases the yield, for example by increasing the mean-$p_{T}$ through collisions Wong:2007pz , or heating of the medium Chiu:2005ad . A variant on these bulk-like mechanisms is that the enhancement does not need to be due to interactions between the jet and the medium, but could be the result of radial flow combined with trigger bias Voloshin:2004th ; Pruneau:2007ua . Here, we would like to highlight two recent results from STAR that may shed further light on the origin of the ridge. Figure 2 shows the p/$\pi$ ratio in the ridge and jet-like peak as a function of $p_{T}^{assoc}$ for $4<p_{T}^{trig}<6$ GeV/$c$. The inclusive p/$\pi$ ratios are also shown for comparison. Clearly, the p/$\pi$ ratio in the ridge is similar to the inclusive p/$\pi$ ratio in Au+Au events, which is much larger than in p+p events. The p/$\pi$ ratio in the jet-like peak is similar to the inclusive ratio in p+p events. These results clearly suggest that the ridge is formed from bulk matter and not from jet fragments. The second result that addresses the origin of the ridge yield is the analysis of three-particle $\Delta\eta-\Delta\eta$ correlations. Figure 3 shows the distribution of associated hadron pairs as a function of $\Delta\eta_{1}$ and $\Delta\eta_{2}$, the pseudo-rapidity difference between the trigger hadron and the first and second associated hadron, for particles with $1<p_{T}^{assoc}<3$ GeV/$c$ and $3<p_{T}^{trig}<10$ GeV/$c$. Combinatorial backgrounds have been subtracted (for details on the procedure, see Netrakanti:2008jw ). This measurement is sensitive to the event-by-event substructure of the ridge: an excess along the diagonal would indicate that ridge particles tend to be close together in $\eta$, which would be expected if the ridge hadrons are fragmentation products of radiated gluons that are carried along with the longitudinal flow. No such excess is observed, indicating that (within the statistical reach of the analysis) the particles in the ridge are distributed evenly in $\eta$ in every event. Another remarkable feature in the figure is the absence of a horizontal and vertical band (i.e. $\Delta\eta_{1}\approx 0$ or $\Delta\eta_{2}\approx 0$), which should arise from events where one associated particle is inside the jet-like peak and the other is part of the ridge. Further work is ongoing to determine how significant the absence of the various structures is. Future RHIC runs will provide increased statistics which will reduce the uncertainties and will make it possible to perform this analysis with larger $p_{T}^{trig}$ and $p_{T}^{assoc}$, where backgrounds are smaller. 2 Path length dependence To vary the path length of high-$p_{T}$ partons through the medium, measurements are performed as a function of the angle with respect to the event plane. Figure 4 shows the background subtracted distributions of associated hadrons with $1.0<p_{T}^{assoc}<1.5$ GeV/$c$ relative to a trigger hadron $3.0<p_{T}^{trig}<4.0$ GeV/$c$ in four different ranges for the angle $\phi_{s}$ between the trigger hadron and the event plane. On the near side, a decrease of the associated yield with increasing $\phi_{s}$ is visible. The recoil peak shape changes from a broad single peak at $\phi_{s}=0$ to a doubly-peaked structure for larger angles. The systematic uncertainty on the background subtraction due to the uncertainty in elliptic flow $v_{2}$ is indicated by the dashed and solid histograms. For details of the analysis, see Feng:2008an . Figure 5 shows the dependence of the near-side yield on $\phi_{s}$, for the ridge and the jet-like peak separately. It can be seen that the jet-like peak has almost no dependence on the angle with respect to the event plane, while the ridge component is significantly larger for small $\phi_{s}$, i.e. in-plane. This implies that the ridge yield is larger for smaller path lengths, while naively one would expect larger jet modifications for longer path lengths. The observed trend might be due to a trigger bias effect, for example because jets with large energy loss do not give rise to a trigger particle. Another possible interpretation is that the ridge effect may depend on the local flow velocity in the medium. Figure 6 shows the dependence of the recoil yield in two angular regions around $\Delta\phi=\pi$ and $\Delta\phi\approx 2/3\pi$ (‘double peak region’) on $\phi_{s}$. The associated yield at $\Delta\phi\approx 2/3\pi$ shows no significant dependence on $\phi_{s}$, while the yield at $\Delta\phi=\pi$ is largest in-plane ($\phi_{s}=0$) and then decreases. This indicates that the change of the shape of the recoil distribution is mostly due to changing yield at $\Delta\phi=\pi$. The doubly-peaked away-side occurs when the yield at $\Delta\phi=\pi$ is smaller than in the ‘double peak region’. It is also interesting to note that while the double peak structure is associated with large path length, the strongest increase of the yield at $\Delta\phi=\pi$ with respect to d+Au collisions is seen for the shortest path lengths, $\phi_{s}=0$. An important systematic effect in the measurement of azimuthal correlations as a function of the angle with respect to the event plane, is that the jet-like azimuthal correlations may bias the event plane. In order to reduce this effect, the event plane is reconstructed using tracks with $\Delta\eta>0.5$ with respect to the trigger hadron. Some of the observed effects, however, could be due to such an event plane bias, so further tests of this systematic effect are still ongoing. 3 High $p_{T}$ multi-hadron correlations At high $p_{T}\gtrsim 6$ GeV/$c$, both the hadrochemistry, as measured by the baryon/meson ratio Abelev:2006jr and the jet-like peak shapes in the di-hadron analysis Adams:2006yt are very similar in heavy ion collisions and p+p collisions, which suggests that at high $p_{T}$, the dominant particle production mechanism in heavy ion collisions is parton fragmentation. In-medium parton energy loss leads to a suppression of high-$p_{T}$ hadron production, which can be modeled using perturbative QCD techniques. Recently, progress has being made towards determining medium properties by systematically confronting the data with several models for energy loss Dainese:2004te ; Zhang:2007ja ; Adare:2008cg . However, inclusive spectra and di-hadron correlations are rather insensitive to details of the energy loss process, such as the probability distribution for energy loss, because the initial parton energy is not measured Renk:2007mv . As a result, a number of energy loss models are able to describe the data, despite significant differences in the underlying energy loss distributions. To further address this, more differential measurements are needed, preferable including a direct measure of the initial parton energy. Direct-photon jet measurements are especially promising in this regard Wang:1996yh ; Renk:2006qg , since the photon energy is a direct measure of the initial parton energy. Direct photon measurements, however, have the difficulty that the event-samples are smaller than for hadronic signatures and that the backgrounds are large. Another approach is to perform jet reconstruction in heavy ion collisions, where the jet energy is a measure of the initial parton energy. Results from both types of measurement will be discussed in other contributions to this volume hamed ; heinz ; putschke ; salur . Here, we would like to present two analyses which provide more differential information that the inclusive spectra and di-hadrons, but do not provide a direct measurement of the initial parton energies. 3.1 Di-hadron triggered correlations In the standard di-hadron measurement Adams:2006yt , the near-side trigger hadron $p_{T}$ selects jets within a relatively broad energy range. As a result, the $p_{T}$-cut on associated hadrons on the away-side can impose an additional bias on the underlying jet distribution. The analysis presented here aims to reduce this effect by using a di-jet trigger, i.e. selecting events with a back-to-back pair of high-$p_{T}$ hadrons. Associated hadrons can then be studied with respect to this pair. Figure 7 presents a first result from STAR for this type of analysis, using trigger hadron pairs with $5<p_{T,1}^{trig}<10$ GeV/$c$ and $4<p_{T,2}^{trig}<5$ GeV/$c$ and associated hadrons $1.5<p_{T}^{assoc}<4$ GeV/$c$. The combinatorial backgrounds have been subtracted, including jet-background cross-terms. For details, see barannikova . For comparison, the line shows the associated hadron distribution for a single trigger particle with $4<p_{T}^{trig}<5$ GeV/$c$ and the same $p_{T}^{assoc}$ selection as the di-hadron trigger. The figure shows two striking features. Firstly, the associated yields with di-hadron triggers are larger than with the single-hadron trigger. This is because the di-hadron trigger selects more energetic jets than a single hadron trigger. The second observation from the figure is that the distributions are very similar in d+Au and Au+Au events, indicating that the selected di-jets fragment like in the vacuum. This also implies that there are no events with a larger energy-difference between the two jets, as one might expect for Au+Au events with energy loss. It should be realised, however, that the fact that the $p_{T}$-selection for the two trigger hadrons are similar in the current measurement is likely to select events with small energy-asymmetry between the jets. The first model calculations for this kind of measurement are already available Renk:2008km . They indicate that by by changing the $p_{T}$-cuts for the trigger hadrons independently, one can select events with larger difference between the energies of the jets. The larger triggered data sample from RHIC run-7 may allow to perform this measurement with more asymmetric cuts. 3.2 Multi-hadron cluster triggered correlations A different approach to reduce the jet-energy bias in di-hadron correlation measurements is to cluster hadrons into a ’proto-jet’ and use this as the trigger object. A first analysis of this type is being performed in STAR, using particles with $p_{T}^{seed}>5$ GeV/$c$ as ’seed’ particles. Secondary particles with $p_{T}>3$ GeV/$c$ are added to the cluster if they are within $R=\sqrt{\Delta\eta^{2}+\Delta\phi^{2}}<0.3$ from the seed particle. The sum of the transverse momenta of the particles in the cluster is used as the multi-hadron trigger $p_{T}$. For details of the analysis, see Haag:2008gi . In particular, it should be noted that at the moment, no correction is made for ’background triggers’, i.e. random combinations of particles that form a trigger cluster. The signal-to-background ratio for trigger clusters is estimated to be 0.7 Haag:2008gi . Figure 8 shows the recoil yield for such multi-hadron cluster triggers in three different $p_{T}$ ranges. The left panel shows the measurement in 0-12% central Au+Au collisions, while the right panel shows the result for the same analysis on PYTHIA events Sjostrand:2006za . For comparison, results of the di-hadron analysis (single particle trigger) are also shown (solid symbols). It can be seen in the figure that the analysis with multi-hadron cluster triggers gives similar results to the single-hadron triggers, indicating that the underlying jet energy selection is similar in both cases. A similar trend is seen in the PYTHIA simulation. This implies that multi-hadron clusters with the current cuts do not provide a significantly better measure of the jet energy than the leading particle. Further studies with PYTHIA are ongoing to determine whether for example including electromagnetic energy in the cluster changes the result. 4 Conclusion and outlook We have presented recent results on intermediate and high-$p_{T}$ multi-hadron correlation measurements. At intermediate $p_{T}$, two results were highlighted that provide some insight in the origin of the near-side associated yield at large $\Delta\eta$, the ridge. Both the large baryon/meson ratios and the uniform event-by-event $\Delta\eta$-structure indicate that the the ridge is likely formed from bulk matter and not from jet fragments. Measurements of the associated hadron distribution as a function of the angle with the event plane show a clear evolution of the correlation structure with path length, on both the near and the away side. Comparisons to theoretical calculations are needed to interpret the interplay between trigger bias and path length dependent energy loss. At high $p_{T}$, STAR is exploring multi-hadron analyses to provide more differential measures of energy loss and better constraints on parton energy loss models. Back-to-back di-hadron triggered correlations may provide a way to select events with large energy loss for further analysis. The current data sample is not large enough to provide the $p_{T}$-reach that is needed to fully exploit this technique. Another analysis uses multi-hadron clusters as triggers to reduce the trigger bias in correlation measurements. So far, the differences between the multi-hadron triggered analysis and the analysis with single-hadron triggers are found to be small. Simulated PYTHIA events are being used to further explore the potential of this technique. 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On-sky performance during verification and commissioning of the Gemini Planet Imager’s adaptive optics system Lisa A. Poyneer\supita    Robert J. De Rosa\supitb,c    Bruce Macintosh\supitd,a    David W. Palmer\supita    Marshall D. Perrin\supite    Naru Sadakuni\supitf    Dmitry Savransky\supitg Brian Bauman\supita    Andrew Cardwell\supitf    Jeffrey K. Chilcote\supith    Daren Dillon\supiti    Donald Gavel\supiti    Stephen J. Goodsell\supitf    Markus Hartung\supitf    Pascale Hibon\supitf    Fredrik T. Rantakyrö\supitf    Sandrine Thomas\supitj,k    Jean-Pierre Véran\supitl\skiplinehalf\supitaLawrence Livermore National Lab    7000 East Ave.    Livermore    CA 94551    USA\skiplinehalf\supitbSchool of Earth and Space Exploration    Arizona State University    PO Box 871404    Tempe    AZ 85287    USA\skiplinehalf\supitcSchool of Physics    College of Engineering    Mathematics and Physical Sciences    University of Exeter    Stocker Road    Exeter    EX4 4QL    UK\skiplinehalf\supitdKavli Institute for Particle Astrophysics and Cosmology    Stanford University    Stanford    CA 94305    USA\skiplinehalf\supiteSpace Telescope Science Institute    3700 San Martin Drive    Baltimore MD 21218 USA\skiplinehalf\supitfGemini Observatory    Casilla 603    La Serena    Chile\skiplinehalf\supitgSibley School of Mechanical and Aerospace Engineering    Cornell University    Ithaca    NY 14853    USA \skiplinehalf\supithDepartment of Physics and Astronomy    UCLA    Los Angeles    CA 90095    USA\skiplinehalf\supitiUniversity of California Observatories    UC Santa Cruz    1156 High Street    Santa Cruz    CA 95064    USA\skiplinehalf\supitjNASA Ames Research Center    Moffett Field    CA 94035    USA\skiplinehalf\supitkUARC    UC Santa Cruz    1156 High Street    Santa Cruz    CA 95064 USA\skiplinehalf\supitlNational Research Council of Canada Herzberg    5071 West Saanich Road    Victoria    BC V9E 2E7    Canada Abstract The Gemini Planet Imager instrument’s adaptive optics (AO) subsystem was designed specifically to facilitate high-contrast imaging. It features several new technologies, including computationally efficient wavefront reconstruction with the Fourier transform, modal gain optimization every 8 seconds, and the spatially filtered wavefront sensor. It also uses a Linear-Quadratic-Gaussian (LQG) controller (aka Kalman filter) for both pointing and focus. We present on-sky performance results from verification and commissioning runs from December 2013 through May 2014. The efficient reconstruction and modal gain optimization are working as designed. The LQG controllers effectively notch out vibrations. The spatial filter can remove aliases, but we typically use it oversized by about 60% due to stability problems. keywords: Adaptive Optics, Gemini Planet Imager, LQG control, modal gain optimization, Spatially-filtered wavefront sensor, wavefront reconstruction \newrefformat chapChapter LABEL:#1 \newrefformatsecSection LABEL:#1 \newrefformatfigFig. LABEL:#1 \newrefformateqnEq. LABEL:#1 \newrefformattabTable LABEL:#1 \authorinfoSend correspondence to Lisa Poyneer: poyneer1@llnl.gov, 1 925 423 3360 1 INTRODUCTION The Gemini Planet Imager (GPI) [1] is a hyper-spectral coronagraphic instrument specifically designed to directly image exoplanets. GPI and the SPHERE instrument [2] are part of a new generation of instruments that feature specialized adaptive optics (AO) systems that use thousands of actuators and specialized algorithms. In this paper we discuss the algorithms and technologies that were developed for GPI, and how well they are actually working during the instrument’s verification and commissioning period at Gemini South (late 2013-summer 2014). 2 Experimental procedures and data analysis The performance of the AO system is analyzed primarily through system diagnostic data. This telemetry is dumped at the full frame rate of the AO system and can include anything from raw CCD frames, to calculated centroids, to reconstructed phases to deformable mirror commands. By synchronizing multiple data dumps, long streams of telemetry can be saved during on-sky experiments. During a typical on-sky experiment, 22-second streams of telemetry are saved. The specific control parameters in use (e.g. filter coefficients) are also saved. These data are analyzed offline using a standard telemetry analysis code. Our primary analysis technique is to examine the temporal power spectral densities (PSDs) of specific modes that the system controls. For tip, tilt and focus we analyze the coefficients directly. For the higher order phase, we begin with the residual phase as seen by the AO system directly after phase reconstruction from the centroids. For a given 22-second interval, the $x$-$y$-$t$ data cube is converted at each time step into the frequency domain to produce a $f_{x}$-$f_{y}$-t data cube. Given the time series of a modal coefficient (e.g. tip, Fourier mode $k=12$, $l=12$, where $k$ and $l$ indicate the index into a $48\times 48$ FFT of the phase), the temporal PSD is estimated by the averaged modified periodogram technique [3]. This temporal PSD represents what is measured by the AO system in closed loop. For bright targets wavefront sensor (WFS) noise is small and this is very close to the closed-loop error that should be presented at the output of the AO system to the science leg. On moderate and dim targets, however, the measurement and the error are not the same. To analyze the error, as opposed to the measurement, we use the exact same technique that the gain optimizer (see below) uses [4]. To estimate the error, we take the temporal PSD of the measurement and invert by the known system response. This is possible because the temporal response of the system is very well characterized. As shown in Figure 1, the temporal behavior for focus and a Fourier mode both closely follow our models. Once the PSD of the closed-loop measurement is converted into the PSD of the estimated open-loop measurements, we then split it into signal and noise components. The noise component should be temporally white, and it is estimated by taking a median of the PSD at the highest frequencies. This estimate is the PSD of the input noise seen by the AO system. The PSD of the signal (primarily atmosphere) seen by the AO system is obtained by subtracting the noise PSD from the estimated open-loop measurement PSD. We now have estimated temporal PSDs for the signal that the AO system wants to correct and the WFS noise that it measures. The error produced by the AO system is estimated by apply the error transfer function (ETF) to the signal PSD and the noise transfer function (NTF) to the noise PSD. This frequency domain method ignores any static level of error, which is removed before the periodogram calculation. As such (see Section 8) this term is calculated separately. Note that throughout this paper we rely on the self-reporting of the AO system for performance analysis. A provisional error budget is given in Section 8, but a rigorous comparison of science measurements to AO telemetry is beyond the scope of this present work, but will be included in a future publication. 3 Fourier transform reconstruction 3.1 Technology description and methods The concept of reconstructing the wavefront phase from slopes with a filtering approach was originally proposed in the mid 1980s [5, 6]. However, the boundary problem imposed by a circular (or annular) aperture in the square computational grid prevented the method from practical use. By solving this boundary problem, we were able to make the filtering approach feasible, calling the method Fourier Transform Reconstruction (FTR) [7]. We further defined new filters that better captured Shack-Hartmann behavior [8, 4] and showed that the Fourier basis provided a tight frame for modal control [4]. When formulated to consider the signal and noise power levels, the FTR method is a Wiener filter [9], and we showed that FTR is equivalent [9] to the Minimum Variance Unbiased approach [10] if the Fourier basis set is used. To our knowledge, the use of FTR in GPI represents the first on-sky use of a computationally efficient wavefront reconstruction method (as opposed to a parallelized vector-matrix-multiply) in an astronomical AO system. After phase reconstruction the residual is split [11] between two deformable mirrors (DMs). Low-spatial-frequency Fourier modes, which require the most stroke, are sent to the “Woofer” DM, a conventional piezo DM manufactured by Cilas. The high-order phase is sent to the “Tweeter” DM, a microelectromechanical systems (MEMS) mirror developed by Boston Micromachines. 3.2 On-sky results Use of FTR was motivated by computational complexity; a matrix multiplication of a system of the same size would require 45 times more computation. This enabled the implementation of the real-time controller (RTC) on a commercial, off-the-shelf (COTS) server running real-time linux, as opposed to customized or special-purpose hardware such as digital signal processors (DSPs) or graphics processing units (GPUs). As we see below, the use of FTR makes the residual Fourier coefficients of the wavefront available, which facilitates computationally efficient controller optimization. As of May 2014, the total delay from the end of a WFS frame integration to the application of the phase commands on the DMs is 1.7 msec. This is true for either 1 kHz or 500 Hz operation. The tip, tilt and focus commands are computed more quickly since they do not go through the FTR process. Those commands are placed on the Woofer DM and the tip-tilt stage after 1.2 msec. For both of these with first 890 microsec are taken up by the detector read. As has been noted by us elsewhere[12], use of FTR requires precise alignment of the AO system optics. This alignment is achieved through automatic processes before each acquisition [13]. The rotation of the spots grid produced by the lenslets relative to the WFS CCD pixel grid has proved problematic. As was originally noted in Section 5.2 of our prior work [12], if rotation exists in the centroids, it will be reconstructed by FTR into a shape that has a scalloping pattern around the edges. During closed-loop operation this behaves similarly to an alias - a non-physical signal is present in the slopes and makes it through to the DM, where a shape builds up to attempt to correct it. As built, there is a small but not insignificant amount of rotation in the WFS, which is nominally included in the reference slopes. The removal of rotation via reference subtraction will not work completely if some effect changes the gain of the quad cells. This could be a drift of the bias level on the CCD [14], spot size variations due to atmospheric turbulence, or potentially other subtle factors. During integration and testing we identified that bias drift was significantly reducing performance quality; the short-term solution was taking regular dark frames. Despite regular darks, during the December run we identified that rotation was not being fully removed on-sky, most likely due to spot size variations that are dependent on seeing. These spot size variations result in a large status excursion of the Tweeter actuators from flat around the edges of the pupil. To solve this problem the RTC now includes a step where rotation is explicitly removed from the centroids before FTR. This approach has been verified to work correctly, and has reduced the edge excursions. Pending results from other analysis by Greenbaum [15] and Sadakuni [14] the specific vectors used in the removal process may be modified. This is our best solution to date for this problem with FTR. On a symmetric pupil this effect is entirely due to the edge correction process where the slopes on the square grid are managed. An equivalent matrix reconstructor on a symmetric pupil would not let any rotation through. As a further complication, GPI does not have a symmetric pupil; small subregions are masked off to prevent the AO system from seeing the dead actuators, which are mostly positioned at the exterior edge of the pupil. This asymmetry allows for rotation to leak through both FTR and a matrix reconstructor. If we were using a matrix method, we would not need to do the removal step from the centroids but could instead account for the rotation internally in the matrix. 4 Modal gain optimization 4.1 Technology description and methods Originally proposed by Gendron and Lena [16], modal gain optimization is a technique where the wavefront phase is decomposed into orthogonal modes (as opposed to actuator positions in the pupil). Using either open- or closed-loop telemetry on-sky, the control loop gain of each mode is set such that it minimizes the residual error variance. Modal gain optimization has been implemented in several AO systems, including the initial demonstration in Come-On-Plus [17], which used open-loop telemetry prior to an observation. The method is used in the Altair AO system at Gemini North [18] and ESO’s NAOS system [19]. More recently, modal gain optimization for up to 400 modes has been implemented in the LBT’s FLAO system [20]. As described, that optimization is done once at the beginning of an observation, and proceeds iteratively. In GPI, the modal gain optimization procedure operates continuously as a supervisory process. As described fully elsewhere [4], the Fourier coefficients of the residual phase are buffered during the reconstruction process. Each mode is analyzed temporally, using the averaged-modified periodogram technique to estimate the closed-loop power spectral density (PSD) of the measurements. Using a fast root-finding technique, the best modal gain for each Fourier mode is calculated independently. All 2304 modal gains (which have hermitian symmetry on the spatial frequency grid, so only half need to be calculated) are used in a gain filter which is applied during the reconstruction process. Because the Fourier modal coefficients are directly available in closed loop, and because changing the modal gains involves simply changing a 2304-element vector in memory, the Fourier framework of GPI allows modal gain optimization to be conducted every eight seconds for all 2304 Fourier modes during closed loop. As such, we consider GPI to be unique in terms of the large number of modes that it optimizes, and the fact that it does so continuously during science observations. 4.2 On-sky results During the integration and testing phase, the modal gain optimization was verified through off-line re-analysis of telemetry. That is, our analysis codes estimated the modal gains from telemetry, and the results were compared to what the system itself had calculated. The primary concern of our on-sky testing is whether the modal gain optimizer improves overall AO performance. To test this, we conducted a series of experiments. For a given star magnitude and frame rate, we closed the AO loops with a fixed value of uniform modal gain. After taking telemetry we turn on the gain optimizer and let the gains settle, a process that takes less than 30 seconds. We then take another telemetry set. We repeat this uniform gain-optimized gain pairing several times, slowly stepping up the initial uniform gain from 0.05 to 0.3 with increments of 0.05. This gives us an interleaved set of measurements over a period of about 20 minutes, with six examples of uniform gain filters spanning the range of stable gains and six examples of optimized gains. During our March run we conducted four of these tests. Representative gain filters for the four cases are shown in Figure 2. As predicted by simulations [21], the gain filters span a wide range of possible gains, and show evidence of wind direction. Using the methods described in Section 2, we estimate the error for each Fourier mode and for all modes together. Figure 3 shows the estimates of total error. For each of the six uniform gain cases, the total error is marked by the black points. This total error is the sum of the error due to signal (which decreases as gain is increased) and the error due to noise (which increases as gain is increased). Analysis of the two components (not shown) confirms that the system follows this expected behavior. In the figure the blue bar indicates the full range of the total error, as estimated from the six different optimized cases. In all these cases the performance with the optimizer on (in blue) is at or very close to the minimum when the gain is adjusted by hand. This total error metric shown in Figure 3 is dominated by the optimizer’s performance at low spatial frequencies. This is because both atmospheric turbulence and the noise propagation from the Shack-Hartmann wavefront sensor produce more power at lower frequencies. To further verify that the modal gain optimization is working for all modes, we analyzed each mode individually. Figure 4 shows the test results for four different Fourier modes that are representative of the 2304 controllable modes. As shown in this figure, for a 9th magnitude star at 500 Hz, the lowest spatial frequency modes (e.g. $k=1$, $l=0$ shown at upper left) have the most total error. In this case the atmosphere dominates and the optimizer correctly drives the modal gain to maximum. For mid-frequency modes, such as $k=6$, $l=0$ at upper right and $k=10$, $l=10$ at lower left, the atmosphere and noise are more equal. For these modes the optimizer uses a high, but not maximum gain. For the highest spatial frequencies, such as $k=0$, $l=18$ shown at bottom right, noise dominates and the optimizer drives the gain lower. These modes also have significant high-temporal frequency content due to aliasing (see below), which the optimizer does not explicitly know about. In order to suppress waffle and reduce edge effects, during Integration and Test a modified Tweeter influence function filter was put in place. As discussed elsewhere [22], the estimated phase is pre-compensated by a influence function filter to correctly shape the phase on the Tweeter. Due to the broadness of the Tweeter influence function, the effective Tweeter gain for high spatial frequencies [23] is very small. Division by this small number inflates noise. As such, we artificially limit the division. As a result, the optimizer tends to drive up the gain on these under-compensated modes, as is obvious by the red regions in the corners of the gain filters shown in Figure 2. This behavior of the optimizer to correct for a miscalibration was discussed in Section 4.C of our original proposal [4]. It remains to be determined if artificially suppressing the influence function gains and then having the optimizer try to crank them back up is adversely affecting performance on-sky. 5 Spatially-filtered wavefront sensor 5.1 Technology description and methods Aliasing occurs when a signal has frequency content above half of its sampling frequency. In an adaptive optics system, the atmosphere is not-band-limited; this will cause any sensor which samples the phase, either through lenslets or pixels, to measure spurious signals. These aliases will lead to extra residual error. Rigaut first calculated this as one-third of the classical fitting error [24]. PSD treatments such as those by Jolissaint [25] show that the exact amount of aliasing error depends on the filter type (derivative vs. direct phase) and the control parameters. To prevent the aliasing error, we has proposed what we termed the spatially-filtered wavefront sensor (SFWFS) [26]. For a Shack-Hartmann sensor, this is implemented as a hard-edged field stop in a focal plane of the WFS before the lenslet array. In the case of a high-strehl system, the filter will reject phase errors that scatter light beyond the size of the field stop. In the case of a system with subaperture size $d$, the field stop with a diameter of $\lambda/d$ will reject content beyond the sampling limit, in theory producing a band-limited phase for the WFS to measure and producing a dark hole in the PSF. A first experimental demonstration of the SFWFS was done by Fusco [27], where as over-sized stop was shown to improve performance in a testbed with dynamic turbulence. In our own work we were able to demonstrate cleaning out a dark hole on static phase plate with a 32x32 MEMS mirror and sensor [28]. 5.2 Modifications from original design During the integration and testing phase of GPI, we had to make some accommodations in using the SFWFS. In particular we discovered that when sized to nominal (i.e. $\lambda/d$), we observed non-linear effects in edge sub-apertures and in the subapertures near the Tweeter’s dead actuators. Fourier optics simulations revealed that for large phase errors such as an actuator stuck more than a micron from its neighbors, the spatial filter is non-linear and will produce biased measurements. To mitigate this, we implemented a stronger leak (for an integral controller of the form $g/(1-c\mathrm{z}^{-1})$, $c$ would be set to 0.9) on the actuators immediately next to the dead actuators to bleed off this error. Another problem we encountered is poor stability at the edge of the pupil, which leads to loss of light. During integration and testing we examined this, and found it was exacerbated by a still poorly-understood effect on the partially illuminated subapertures along the outer edge of the pupil. During this phase we had also used a stronger leak ($c=0.9$) on the actuators bordering these partially-illuminated subapertures. During our May 2014 testing run we switched to simply not using measurements from these partially illuminated subapertures around the outside edge. Operation remains good in this slightly modified configuration, though it remains to be determined if this has improved spatial filter stability at all. 5.3 On-sky results We can assess the ability of the SFWFS to reject specific spatial frequencies by examining the temporal PSDs of the Fourier modes during closed loop. When frozen flow is present (which it typically is), the wind motion creates characteristic peaks in the temporal PSDs of the Fourier modes. For any Fourier mode of spatial frequency (inverse-meters) $<f_{x},f_{y}>$ a wind layer with velocity vector (m/s) $<v_{x},v_{y}>$, a peak appears in the PSD at temporal frequency (Hz) $f_{x}v_{x}+f_{y}v_{y}$. For the controllable modes, this creates a characteristic pattern (see Figure 4 of Poyneer et al. [29]). When a spatial frequency above the sampling limit of the AO system is measured, it aliases down to a lower spatial frequency, but the temporal frequency of the wind component stays the same. This results in peaks in the controllable modes from aliasing at temporal frequencies $(f_{x}\pm d^{-1})v_{x}+(f_{y}\pm d^{-1})v_{y}$, where $d$ is the subaperture size in the pupil, e.g. 18 cm for GPI. An example of this is shown in Figure 5, left side. At left the plot shows that temporal PSD of the residual phase for Fourier mode $k=6$, $l=0$. We know, from examining the other Fourier modes, that the peak at -20 Hz is from frozen flow atmosphere. The alias is visible at right from 100 to 130 Hz. In black is the residual as seen with the SF = 5.0 mm. When the filter is closed to 4.0 mm, this alias is suppressed. (The leakage of the non-focus 60 Hz vibrations are clearly seen at 60 Hz for this Fourier mode). In the right panel we see Fourier mode $k=12$, $l=0$. In this case the wind peak has moved out to temporal frequency -30 Hz. The alias has moved down to about 90 to 110 Hz. For this higher frequency the blue curve shows the residual as seen with the SF at 3.2 mm. As the filter gets smaller, it cleans up aliases for controllable Fourier modes of higher frequency. The figure shows data taken on May 14, 2014 on target HD101615. We also have data from December 11, 2013 on $\beta$ Pictoris down to SF size 2.8 mm. A qualitative analysis of the rejection of aliases of wind-blown turbulence produces estimates for the range of modes that the SFWFS “cleans up” by removing aliases. As given in Table 1, this follows the expected linear trend as a function of SF size. At size 3.2 mm, the filter cleans up modes to $k=12$, which is halfway out to the edge of the dark hole. Though as the filter narrows beyond this and more modes are cleared, overall performance is reduced due to edge effects that compromise stability. The $\beta$ Pictoris testing occurred during very good seeing (see first $\beta$ Pictoris column of  2). In this case we were able to stably close the filter to 2.8 mm, which we estimate clears out 66% of the dark hole. This performance is atypical. 6 Tip-tilt vibration correction with LQG 6.1 Technology description and methods Mathematical treatments of optimal control methods for AO date back more than a decade, such as Gavel & Wiberg’s [30] Kalman filter. Le Roux developed a Linear-Quadratic-Gaussian (LQG) framework for AO control[31], which was experimentally demonstrated in a testbed for vibration filtering by Petit [32]. Further progress has recently been made with an on-sky demonstration by Sivo of LQG for all modes in CANARY [33]. Another vibration scheme has been tested in GEMS by Guesalaga [34]. For use in GPI we have adapted the framework of Le Roux and added in the ability to work in a system with an arbitrary (i.e. non-integer frame) control delay [35], and also to correct both common-path and non-common-path vibrations [36], both of which were deemed necessary to use an LQG controller in GPI. As noted above, the tip-tilt and focus controllers have a 1.2 frame delay, which we account for explicitly in the LQG model. 6.2 Modifications from original design There are two significant differences between the tip-tilt LQG filter as initially proposed and as used in GPI. First, in our proposal we presented a high-order model for the atmospheric tip-tilt; in practice we always use a first-order auto-regressive (AR(1)) model. Second, in GPI the pointing is actually controlled with two surfaces. By using a low-order low-pass filter, the low temporal frequency part of the pointing signal is sent to the high-stroke Stage, while the higher-frequency placed on the Woofer DM in combination with the phase correction. Since each mirror has its own integrator, the result of the LQG (which is integrated internally) must be converted to a pseudo-residual that is then split and integrated on each surface. Specifically, if the Woofer integrator has control law $C(\mathrm{z})=1/(1-0.999\mathrm{z}^{-1})$, the pseudo-residual is produced with a filter that is a“leaky” differentiator: $D(z)=1-0.999\mathrm{z}^{-1}$. We have had to be very conservative on the temporal split between the two surfaces to avoid instabilities; the current cutoff for the split is at 25 Hz. A more robust solution would have been to have the LQG model know explicitly about the temporal split [37]. 6.3 On-sky results During integration and testing we identified the pointing vibrations at 60, 120 and 180 Hz as the most significant. We were unable to determine if these tip-tilt vibrations are in the common path or the non-common path of the system, so we have assumed that they are common path and generated appropriate LQG filters. After analyzing on-sky telemetry with the default integral controllers for pointing, we generated LQG filters designed to recede the total vibration at 60, 120 and 180 Hz to 1 mas. LQG filters were tested in comparison to the default integral controller in an interleaved fashion. Pointing vibrations are always larger on the y-axis; in Figure 6 we show the on-sky residual tip (top left) and tilt (top right) as seen by the WFS for the integral controller. At bottom is the results for our preferred LQG. The figure shows the AO measurements in closed loop. Annotated in red are the error levels (see Section 2 for a discussion of measurements versus error) passed on to science. The total vibration at 60, 120 and 180 Hz has been reduced to less than 1 mas for each axis with use of the LQG. The remaining pointing error is primarily residual atmosphere. 7 Focus vibration correction with LQG 7.1 Technology description and methods When GPI is on the telescope, its cryocoolers cause vibration and result in significant phase errors. his manifests as a low-order wavefront - primarily focus, spherical and trefoil - with a temporal RMS of up to 150 nm occuring at 60 Hz. It remains unclear what is vibrating; either the secondary mirror moving in both piston and deforming, or a deformation of the primary mirror; both are very lightweight. For a thorough discussion of errors and mitigations, see Hartung [38]. This phase vibration was a highly suitable candidate for LQG correction - the error was highly concentrated at 60 Hz, and analysis of the temporal variation of the phase shape indicated that it was primarily focus that oscillated in total amplitude. To correct this we simply applied the LQG filtering framework that had already been tested for tip-tilt. In the tip-tilt case, the signal is calculated and removed from the centroids and then sent to the LQG and on to the Stage and Woofer. In the case of focus we calculate focus directly from the slopes. To calculate and remove focus from the slopes via projection we use a model to make x- and y-slope vectors for focus. To apply the focus correction to the surface of the Woofer, we use the measured influence functions to generate a 9 by 9 signal which makes focus. The slope and command signals are calibrated against each other to ensure that the overall loop gain is 1. 7.2 On-sky results The focus LQG was tested on-sky during the May run. In this case a pure focus shape was pulled out; the remaining components of the 60 Hz phase (mainly trefoil and spherical) were sent through the high-order loop. Figure 7 shows three different temporal PSDs of the focus measurements. Annotated in red is the estimate error that is sent on to the science leg from the 60 Hz component. At upper left is on-sky operation with an integral controller for the FOcus control. In this case the 60 Hz error leads to a time-average of 90 nm rms wavefront error due to focus. When the cryocoolers are turned down, but not off, the amount of focus at 60 Hz is reduced to 51 nm. (When the cryocoolers are turned off completely this amount is 0 nm; due to system constraints we were not able to test this option as well on this target.) With the cryocoolers at normal and the LQG, the deep notch (see Figure 1, left panel) strongly rejects the 60 Hz signal. In this case the result (bottom panel of figure) is just 3 nm. This level is negligible in terms of the overall error budget. Two more items remain in the validation of the focus control. First, we still do not have definitive science images that show the difference between LQG on and off. Second, the exact spatial shape of the error to be corrected, that is not just focus but some combination of focus, trefoil and sphere, could be refined. 8 Error budget We have developed a provisional error budget. These numbers are generated from analysis of closed-loop AO measurements following the methods outlined in Section 2 to convert from measurements to estimated error passed on to the science leg. Error estimates are given in Table 2 for five different observations. For each observation the star name and magnitude is given. Modal gain optimization (“OFC”) was used in four of the five cases, though on targets brighter than 7th this essentially is the same as uniform gains of value 0.3. The first observation of $\beta$ Pictoris from March represents performance in very good seeing - we were able to use the spatial filter at 2.8 mm stably for several minutes, and servo lag error was estimated to be just 25 nm. During more typical seeing on a bright to moderate target with loop gains at maximum, the servo lag error is around 50 nm (second $\beta$ Pictoris, HD101615, HD141569). The HIP59563 observation (the same one used in Section 4 for testing the gain optimizer) is representative of our performance on the dimmest target required of GPI. At 500 Hz with optimized gains we have 99 nm of servo lag error and 52 nm of WFS noise. With cryocoolers at normal power with no LQG, science leg error from the 60 Hz phase is generally between 80 and 120 nm (depends on conditions and orientation). With cryocoolers off this is reduced to 0; with the LQG as tested in May on the focus shape, there are 31 nm of 60 Hz phase leftover - this is dominated by trefoil and spherical. As noted above, this may be reduced further, but it is no longer a dominant term in the AO error budget. The telemetry for HD141569 was taken co-incident with an non-coronagraphic image, which is shown in Figure 8. The Strehl of this image has not been rigorously estimated at this time. However, up to 15 Airy rings can be detected in the narrow-band image, indicating that the AO system is performing well. We estimate the fitting [41] and aliasing error[24] terms from theory using the coefficient of 0.3, such that $\sigma^{2}_{\mathrm{fitting}}=0.3(d/r_{0})^{5/3}$ and $\sigma^{2}_{\mathrm{aliasing}}=0.33\sigma^{2}_{\mathrm{fitting}}$. We assume that $r_{0}$ was 14 cm for the typical seeing conditions. For the HD141569 observation this gives us a fitting error of 54 nm and an aliasing error of 31 nm. Adding in the assumed 30 nm non-common-path errors given in Macintosh [1], we add in quadrature to the terms in the HD141569 column of the table: $(54^{2}+31^{2}+61^{2}+24^{2}+16^{2}+0^{2}+30^{2})^{1/2}=97$ nm. At 2.1 microns, this is a Strehl of 92%. We reiterate that we do not have a direct measurement of fitting error, aliasing error of non-common-path errors, but are using our best estimate at this time. In future work we plan to improve it through better estimates of seeing (and hence fitting), better examination of the aliasing error through analysis of telemetry, and improved information on the non-common-path error. These improved estimates will be accompanied by analysis of the co-incident science images, which was beyond the scope of this work. 9 Conclusions GPI’s AO system features several new technologies. During its verification and commissioning period we have systematically tested these technologies and analyzed performance. The computationally efficient FTR method of wavefront reconstruction is successful, though we have to remove rotation from the centroids to prevent driving a scalloped shape on the Tweeter when the spot size changes (due to either bias level or spot size due to seeing). The modal gain optimization method updates 2304 modal gains for the Fourier modes every 8 seconds in closed loop. Tests have shown that the optimizer is correctly finding gains for each individual mode that minimize the error, and also minimizing overall error. Modal gain filters show clear evidence of wind. The AO system uses an LQG controller for tip, tilt and focus to reject vibrations. For tip and tilt we correct common-path vibrations at 60, 120 and 180 Hz to under 1 mas per axis. By pulling focus out of the centroids and sending it through the LQG, we can very strongly reject the focus at 60 Hz , reducing it to just 3 nm rms wavefront error. There is some 60 Hz phase left over (trefoil and spherical) that could be further reduced if it helps overall performance. The spatially-filtered wavefront sensor has under-performed relative to our expectations. We have clear evidence, thanks to layers of frozen flow turbulence, that the spatial filter does reject aliases. However, we are not able to regularly use the spatial filter stably at it nominal $\lambda/d$ diameter. In very good seeing we can get it down to 2.8 mm, but it is more typical that we use it at 3.5 mm. At this size it clears out aliasing from the inner 42% of the dark hole (i.e. to radius $0.42\lambda/d$ instead of $\lambda/d$.) We have presented an provisional error budget of AO terms that we can directly measure. On bright to moderate targets (down to around 7th magnitude), performance is dominated by servo lag error. In typical seeing this term is 50 nm. On dimmer targets the gain optimizer will trade off servo lag and WFS noise. On our dimmest required target of 9th magnitude we achieved 99 nm servo lag error and 52 nm WFS noise. We will continue to work during the end of the verification and commissioning period to analyze existing science images and obtain new ones as necessary to confirm our analysis, which is here based on AO telemetry. Of particular interest are verifying the impact of the focus LQG, which on most targets removes the dominant 60 Hz term in the error budget. We also want to quantify contrast in longer exposures as a function of spatial filter size and determine if we can do better than 3.5 mm, which is oversized by about 60%. Acknowledgements.This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. The document number is LLNL-CONF-655984. The Gemini Observatory is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciência, Tecnologia e Inovação (Brazil) and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina). 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A halo bias function measured deeply into voids without stochasticity Mark C. Neyrinck${}^{1}$, Miguel A. Aragón-Calvo${}^{1,2}$, Donghui Jeong${}^{1}$, Xin Wang${}^{1}$ ${}^{1}$Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA ${}^{2}$Department of Physics and Astronomy, University of California, Riverside, CA 92521, USA E-mail:neyrinck@pha.jhu.edu Abstract We study the relationship between dark-matter haloes and matter in the MIP $N$-body simulation ensemble, which allows precision measurements of this relationship, even deeply into voids. What enables this is a lack of discreteness, stochasticity, and exclusion, achieved by averaging over hundreds of possible sets of initial small-scale modes, while holding fixed large-scale modes that give the cosmic web. We find (i) that dark-matter-halo formation is greatly suppressed in voids; there is an exponential downturn at low densities in the otherwise power-law matter-to-halo density bias function. Thus, the rarity of haloes in voids is akin to the rarity of the largest clusters, and their abundance is quite sensitive to cosmological parameters. The exponential downturn appears both in an excursion-set model, and in a new model in which fluctuations evolve in voids as in an open universe with an effective $\Omega_{m}$ proportional to a large-scale density. We also find that (ii) haloes typically populate the average halo-density field in a super-Poisson way, i.e. with a variance exceeding the mean; and (iii) the rank-order-Gaussianized halo and dark-matter fields are impressively similar in Fourier space. We compare both their power spectra and cross-correlation, supporting the conclusion that one is roughly a strictly-increasing mapping of the other. keywords: large-scale structure of Universe – cosmology: theory 1 Introduction The spatial arrangement of matter and galaxies contains a large fraction of the information available about cosmology and galaxy formation, especially at late times. The dynamics of collisionless dark matter is straightforward to model in simulations, but the distribution of the easiest-to-observe tracers of the dark matter, the galaxies, may have a complicated relationship to the distribution of dark matter. The mapping from dark matter to galaxies is often treated in two parts: the mapping from the dark-matter field to haloes, and from haloes to galaxies. The dark-matter-to-halo mapping is rather straightforward, since it only involves dark-matter physics. The halo-to-galaxy mapping involves baryonic physics, so may be quite complicated. Thankfully, the relationship seems to be simpler than it could be in principle. Relatively large galaxies seem to relate quite well to dark-matter subhaloes, through subhalo abundance matching (e.g. Brainerd & Villumsen, 1994; Kravtsov & Klypin, 1999; Neyrinck et al., 2004; Kravtsov et al., 2004; Conroy et al., 2006). Another common strategy is to populate galaxies in haloes according to a halo occupation distribution (HOD, e.g.  Berlind & Weinberg, 2002). Inferring the presence of haloes and subhaloes from dark matter can be done by (sub)halo finding in $N$-body simulations (Knebe et al., 2013, e.g.[). While this relationship is conceptually straightforward, it is useful to model the process using analytical approximations as well, for example to simply interpret observed galaxy clustering. A nontrivial relationship between matter and galaxies is called bias (e.g. Kaiser, 1984; Mo & White, 1996; Manera & Gaztañaga, 2011; Paranjape et al., 2013). In this paper, we investigate the approximation that matter and halo density fields are related by a local monotonic bias function on few-Mpc scales, that may be non-linear. The approximation that halo formation is entirely local is sure to fail at some level. This formulation in terms of a general biasing function is an idea that has been studied (e.g. Szalay, 1988; Matsubara, 1995; Sigad et al., 2000; Matsubara, 2011; Frusciante & Sheth, 2012) but current work in this area tends to formulate the bias in terms of the first few Taylor-series coefficients. Indeed, on large, linear scales, the bias takes an approximately linear form when measured through the power spectrum, i.e. $P_{g}(k)=b^{2}P_{m}(k)$, for a scale-independent bias $b$. Here, $P_{g}$ denotes the power spectrum of the galaxy overdensity field $\delta_{h}=(\rho_{h}/\bar{\rho}_{h})-1$, and $P_{m}$ is the power spectrum of the matter overdensity field $\delta_{m}$. The assumption underlying this is that the halo overdensity is a constant $b$ times the matter overdensity, i.e. $\delta_{h}=b\delta_{m}$, which is a good approximation on large scales. But this relationship is unphysical if it is assumed that it holds literally for $b\neq 1$, down to the lowest densities, since where there is no matter ($\delta_{m}=-1$), a linear bias with $b\neq 1$ predicts that $\delta_{h}$ will have positive or even negative density. Parameterizing the bias in terms of log-densities, for example letting $\ln(1+\delta_{h})=b\ln(1+\delta_{m})$, avoids these issues, giving something that in principle could hold literally. Various authors have used a power law in $(1+\delta)$ (Cen & Ostriker, 1993; de la Torre & Peacock, 2012; Kitaura et al., 2013), which is equivalent to a first-order bias in the log-density variable. Indeed, Jee et al. (2012) found that formulating bias explicitly in terms of log-densities, and weighting haloes by their masses, results in a two-second-order bias function with low scatter. Log-densities are natural density variables (Coles & Jones, 1991), and their statistics are generally better-behaved than those of the overdensity $\delta$ (Neyrinck et al., 2009; Carron, 2011). Using the local bias-function framework, a mock halo catalog can be produced from a low-resolution (with e.g. few-Mpc cells) matter density field $\delta_{m}$ by mapping $\delta_{m}$ to a ‘continuous’ halo-density field $\delta_{h}$. This continuous field can then be sampled into discrete haloes, for example in a Poisson fashion. Recently, Kitaura et al. (2013) showed that this approach is useful for producing fast mock halo catalogs that have halo power spectra consistent with those from full $N$-body simulations (see also Chan & Scoccimarro, 2012). They use Augmented Lagrangian Perturbation Theory (ALPT; Kitaura & Heß, 2012) to produce $N$-body realizations that are accurate on few-Megaparsec scales. ALPT interpolates in Fourier space between second-order Lagrangian perturbation theory on large scales, and a spherical-collapse approach (Neyrinck, 2012) on small scales, to fix 2LPT’s problems at high and low densities. This approach will be useful, for instance, to produce the vast number of simulations necessary for current and upcoming surveys to get sufficiently accurate covariance matrices for precision cosmological constraints (e.g. from baryon acoustic oscillations). To investigate the bias function, we use the MIP (Multum In Parvo, ‘many things in the same place’) ensemble of simulations (Aragón-Calvo, 2012). We use 225 simulations from this suite, all of which have the same large-scale cosmic web (built from modes with initial modes of wavelength $2\pi/k\geq 4$ $h^{-1}$ Mpc). Each simulation has a different set of initial small-scale modes (with wavelength $<4$ $h^{-1}$ Mpc). This 4 $h^{-1}$ Mpc scale is a bit arbitrary, but roughly divides scales giving dominant components of the cosmic web with scales giving finer details. Suhhonenko et al. (2011) studied the roles of different scales in producing the cosmic web at $z=0$ using simulations. Below the fiducial scale where the linear-theory density dispersion $\sigma_{R}\approx 1$, $R=8$ $h^{-1}$ Mpc, one starts to get stream-crossing and cosmic-web formation. The choice of 4 $h^{-1}$ Mpc is a factor of two smaller, enough to give the top-level hierarchy of the cosmic web (Aragon-Calvo & Szalay, 2013). This ensemble enables a sort of ‘cosmic-web occupation distribution,’ giving a set of possible ways each location in the cosmic web might be occupied by different types of haloes. The MIP-ensemble mean gives an estimate of the continuous halo-density field $\delta_{h}$ with negligible discreteness, and also largely free of halo-halo exclusion effects. Halo exclusion may suppress the halo density in clusters, but in the stacked ensemble, haloes may be arbitrarily close to each other, unlike in a single simulation. The MIP ensemble offers an ideal laboratory to explore halo-bias stochasticity (e.g.  Pen, 1998; Tegmark & Bromley, 1999; Dekel & Lahav, 1999; Matsubara, 1999; Sheth & Lemson, 1999; Taruya & Soda, 1999; Seljak & Warren, 2004; Neyrinck et al., 2005; Hamaus et al., 2010; Baldauf et al., 2013), at least with stochasticity defined in a particular way. For us, there are two distinct types of stochasticity. First, there can be fluctuations in the ensemble-mean ‘continuous’ $\delta_{h}$ away from that predicted from the matter field $\delta_{m}$ using the biasing function, $\delta_{h}(\delta_{m})$. Second, on top of that, there is necessarily a sampling stochasticity, in the way the continuous $\delta_{h}$ gets point-sampled in each realization to get the halo sample. In our case, this sampling stochasticity is imparted by the differences in modes smaller than 4 $h^{-1}$ Mpc. Each MIP simulation has 256${}^{3}$ particles in a 32 $h^{-1}$ Mpc box, and was run with vanilla $\Lambda$CDM cosmological parameters: $\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$, $h=0.73$, $\sigma_{8}=0.84$, and $n_{s}=0.93$. We analyze only the results at redshift $z=0$. The particle mass is 1.6$\times 10^{8}M_{\odot}/h$; the haloes we analyze were found using a friends-of-friends algorithm with linking length 0.2 times the Lagrangian particle spacing. The smallest haloes we use consist of 20 particles, and thus the lowest mass below is in fact 3.2$\times 10^{9}M_{\odot}/h$, even though we write the mass bin as $10^{9-10}$ for brevity. It should be noted that the small, 32-$h^{-1}$ Mpc box size implies that large-scale modes, and sheer volume, are missing that would increase the range of environments encountered in the simulation. For example, the largest clusters and voids in the ensemble are likely not particularly extreme. Voids are emerging as potentially powerful cosmological tools (Ryden, 1995; Granett et al., 2008; Biswas et al., 2010; Lavaux & Wandelt, 2012; Bos et al., 2012; Li et al., 2012; Spolyar et al., 2013; Hamaus et al., 2013). For many of these techniques, precision constraints will require precision knowledge of the relationship between haloes and matter within voids. This relationship has already seen much study (Peebles, 2001; Gottlöber et al., 2003; Furlanetto & Piran, 2006; Tinker & Conroy, 2009; Jennings et al., 2013; Sutter et al., 2013), but the MIP ensemble allows us to measure it with precision to unprecedentedly low densities. This paper is laid out as follows. In $\lx@sectionsign$2 we show measurements from the MIP ensemble of the relationship between matter and halo densities, and compare them to models described in $\lx@sectionsign$3. These include an ‘additive’-excursion-set model in $\lx@sectionsign$3.1, and a local-growth-factor model, in which the effective growth factor of small fluctuations is given by the local large-scale density. In $\lx@sectionsign$4, we test the assumption that the halo population in a given realization is a simple Poisson sampling of the continuous halo-density field. In $\lx@sectionsign$5, we visually assess factors besides the density that influence the halo-matter bias relationship. Finally, in $\lx@sectionsign$6, we compare the rank-order-Gaussianized $\delta_{m}$ and $\delta_{h}$ fields in Fourier space. If one is a strictly increasing function of the other, they should be equal after Gaussianizing each of them to the same probability density function (PDF). 2 The dark-matter-to-halo density relation Fig. 1 shows scatter plots of the halo and dark-matter densities in 2 and 4 $h^{-1}$ Mpc grid cells in the ensemble-mean MIP fields at $z=0$. Density fields were measured using nearest-grid-point (NGP) assignment in each simulation, and then these density fields were averaged across the simulation ensemble. NGP, although it is naive, makes the discreteness level in the stacked MIP ensemble obvious. The lowest-possible nonzero $(1+\delta_{h})$ corresponds to only one halo in the grid cell, across all 225 MIP realizations. We show results for haloes up to $10^{12}M_{\odot}/h$, because we want the haloes analyzed should fit in a Lagrangian box of size 4 $h^{-1}$ Mpc (and mass $\sim 7\times 10^{12}M_{\odot}/h$) in the initial conditions. This is important because we want to test for halo collapse from small-scale modes (below 4 $h^{-1}$ Mpc), unmixed with larger-scale modes that are held fixed. The additive-excursion-set (AES) and local-growth-factor (LGF) models, described in the next section, fit the curves quite well, especially at low density. In the AES model ($\lx@sectionsign$3.1), small-scale modes contribute additively to the large-scale densities, as in the usual sharp-$k$-filtered excursion-set model. In the local-growth-factor (LGF) model ($\lx@sectionsign$3.2), each patch in the simulation evolves as though it were in a homogeneous Friedman-Robertson-Walker (FRW) universe with an effective matter density parameter $\Omega_{m}^{\rm eff}=(1+\delta_{m})\Omega_{m}$. The curves were calculated from the linear power spectrum using the CosmoPy111http://www.ifa.hawaii.edu/cosmopy/ package. To estimate $\sigma(M)$ in each case, we use halo masses of $5\times 10^{9}$, $2\times 10^{10}$, and $2\times 10^{11}M_{\odot}/h$, approximately the median halo mass in each bin. Note that in translating the equations of $\lx@sectionsign$3 to this section, we use $\delta_{m}$ instead of $\delta_{\ell}$; here they both mean the matter density smoothed with 4-$h^{-1}$ Mpc Lagrangian sharp-$k$ filter. The density pixelization adds an additional Eulerian 2- or 4-$h^{-1}$ Mpc pixel window function, but this does little to the field that already has a Lagrangian filter applied. Thus, it is not surprising that the 2- and 4-$h^{-1}$ Mpc results line up with each other. For simplicity and to highlight the similarity in shape between the two models, we have set the somewhat uncertain normalization of the LGF model similarly as in the AES model, adding a factor $[\sigma(M)/\sigma_{s}(M)]^{3}$ – see Eq. (4) below – compared to what appears in Eq. (9). Note that this normalization does not go through $(0,0)$, the position of which on the plots is indicated by hollow black circles. The measured locus of points typically goes about halfway between $(0,0)$ and the predictions, so empirically, the best-fitting amplitude seems to be about the square root of the factor prescribed by the AES. The offset is the result of the zero-lag term in the local bias expansion (Schmidt et al., 2013), which comes about because the mean number density of haloes in a finite region is different from the cosmic mean. The AES and LGF curves have interestingly similar shapes at low density, especially given their quite different assumptions. We comment on this further after their descriptions. The LGF model seems to be a bit more accurate where the two curves diverge most in the low-density regime, at low halo mass (the top row). Changing the LGF normalization can change the quality of the fit, but perhaps not substantially on the low-density exponential tail. The success of the models at low densities has an interesting implication: the mass function of haloes within voids is highly sensitive to cosmology (Song & Lee, 2009; Lee, 2012). Of course, here a ‘void’ means a (matter) density depression (e.g. Platen et al., 2007; Neyrinck, 2008), not a structure entirely devoid of galaxies, in which, by definition, there would be no galaxies. In voids’ effective low-$\Omega_{m}$ universe, the high-mass cutoff in the global mass function moves to lower masses (Gottlöber et al., 2003). This makes the presence of even a globally modest-mass halo deep in a void possibly as rare as the highest-mass clusters in the Universe. Void-galaxy abundances could be used together with cluster abundances to test for primordial non-Gaussianity, as well, since both of them probe fluctuations in different large-scale density regimes. Cosmological inference from void galaxies presents a few difficulties: baryonic processes are unlikely to be negligible in determining whether void haloes become populated with galaxies; and void-halo masses are likely even harder to measure accurately than cluster masses. Still, the high abundance of small voids compared to the most extreme clusters, as well as the high volume filling fraction of voids offers hope that void-galaxy abundances could be used fruitfully. The dotted curves in the right-hand column show by-eye fits of the form $$\displaystyle\rho_{h}$$ $$\displaystyle=f\rho_{m}^{\alpha}\exp\left[\left(\rho_{m}/\rho_{\rm exp}\right)% ^{-\varepsilon}\right];$$ $$\displaystyle A_{h}$$ $$\displaystyle=\ln f+\alpha A_{m}+\exp\left[-\varepsilon\left(A_{m}-A_{\rm exp}% \right)\right].$$ (1) where $\rho\equiv 1+\delta$, $A\equiv\ln\rho$, and $f$ is a constant. Without the exponential factor, this form is $\rho_{h}=f\rho_{m}^{\alpha}$ (e.g. Kitaura et al., 2013). In the higher halo-mass bins, high-density pixels do scatter substantially down, which was neglected in making the fits. We also show vertical one-$\sigma$ error band away from the fits, assuming Poisson statistics at each $\delta_{h}(\delta_{m})$. To demonstrate the statistical power and low discreteness level of the MIP ensemble average, we show in Fig. 2 a scatter-plot measured from a single realization. It uses the most-abundant $10^{9-10}M_{\odot}/h$ halo bin, to be compared with the upper-right panel of Fig. 1. In a single realization, the minimum nonzero halo number density in a cell of volume $V$ is $1/V$, whereas in the MIP ensemble-mean, the minimum nonzero halo number density is $1/(225V)$. This high discreteness noise shows up as much-inflated vertical one-$\sigma$ error bands compared to Fig. 1. With the poor sampling in a single realization, we get barely a hint of the low-density downturn. 3 Models of the bias function In this section we discuss the two models for the dark-matter-to-halo relationship shown in the previous section. We explore two approaches: (1) an ‘additive’ excursion-set (AES) approach in which small-scale modes simply contribute additively to a large-scale density as they attempt to draw mass over the spherical-collapse barrier for halo formation; and (2) a new local-growth-factor (LGF) model, in which halo formation on small scales proceeds as though it were in a homogeneous FRW universe with an effective $\Omega_{M}^{\rm eff}$ depending on the large-scale density. In the Press-Schechter model (Press & Schechter, 1974), the average number density of haloes of mass $M$ is $$n(M)=\sqrt{\frac{2}{\pi}}\frac{\bar{\rho}}{M^{2}}\left|\frac{d\ln\sigma(M)}{d% \ln M}\right|\frac{\delta_{c}}{\sigma(M)}\exp\left(-\frac{\delta_{c}^{2}}{2% \sigma^{2}(M)}\right).$$ (2) Here, $\sigma^{2}(M)$, an integral over the power spectrum, is the variance in the linear-theory density field in a top-hat sphere encompassing mass $M$ in Lagrangian space (of radius $R(M)$), and $\delta_{c}=1.686$ is the threshold in the linearly-extrapolated density for spherical collapse. The Press-Schechter model is known to have shortcomings in halo mass-function predictions; the sharp-$k$ filter used for the random walk (Bond et al., 1991) is unnatural, although it simplifies calculations. Also, the collapse or expansion is not generally spherical. Departures from these assumptions can be modelled (e.g. Sheth et al., 2001; Corasaniti & Achitouv, 2011; Achitouv et al., 2013), although unmodelled complexities in halo formation may persist in any simple model (Ludlow & Porciani, 2011). Given these caveats, it is intriguing that the models work well in this case. 3.1 Additive excursion-set model The MIP ensemble is well-suited to a sharp-$k$ random-walk excursion-set model (Bond et al., 1991; Mo & White, 1996), since there is a decoupling between long and short modes, with a sharp-$k$ boundary at $k_{\rm cut}=2\pi/R_{\rm cut}$, where $R_{\rm cut}=4$ $h^{-1}$ Mpc. Denote the linearly-extrapolated Lagrangian initial density at a given location as $\delta^{\rm lin}$. We split this into long and short (or large-scale and small-scale) components, $\delta_{\ell}^{\rm lin}$ (the same in all MIP realizations) and $\delta_{s}^{\rm lin}$ (different in each realization). Our measurements of $\delta_{h}$ are as a function of the non-linear density $\delta_{\ell}$ as measured in the simulation. But the excursion-set model requires linear-theory densities, so we use a one-to-one mapping giving $\delta_{\ell}^{\rm lin}(\delta_{\ell})$ in the spherical-shell evolution model, given parametrically e.g. by Sheth & van de Weygaert (2004). Conceptually, holding $\delta_{\ell}$ fixed changes Eq. (2) in a few ways. It changes the barrier for collapse, $\delta_{c}\to(\delta_{c}-\delta_{\ell}^{\rm lin})$. It also changes the variance that the density field has to play with to form haloes; $\sigma^{2}(M)\to\sigma_{s}^{2}(M)$, where $\sigma_{s}^{2}(M)$ is the variance of a top-hat sphere applied to $\delta_{s}^{\rm lin}$, i.e. zeroing out modes with $k<k_{\rm cut}$. Also defining $\sigma_{\ell}^{2}$ to be the variance of $\delta_{\ell}^{\rm lin}$, if $R(M)\ll R_{\rm cut}$, then $\sigma^{2}(M)=\sigma_{\ell}^{2}+\sigma_{s}^{2}(M)$. [$R(M)\ll R_{\rm cut}$ ensures that the difference between the sharp-$k$ filter used for $\sigma_{\ell}$ and the sharp-$x$ filter used for the other two terms is negligible for $k<k_{\rm cut}$.] There is another change: since the Press-Schechter model is really a Lagrangian model, there is an added factor in the conversion to Eulerian space, $(1+\delta_{\ell})$ (implicitly there before, with $\delta_{\ell}=0$). The result can be explicitly calculated from the cumulative distribution, $$\displaystyle n(M|\delta_{\ell})=(1+\delta_{\ell})n_{\rm Lagrangian}(M|\delta_% {\ell})$$ $$\displaystyle=$$ $$\displaystyle-(1+\delta_{\ell})\frac{\bar{\rho}}{M}\frac{d}{dM}\left[\mathrm{% erfc}\left(\frac{\delta_{c}-\delta_{\ell}^{\rm lin}}{\sqrt{2(\sigma^{2}(M)-% \sigma_{\ell}^{2})}}\right)\right]$$ $$\displaystyle=$$ $$\displaystyle-(1+\delta_{\ell})\frac{\bar{\rho}}{M}\frac{d\sigma(M)}{dM}\frac{% d}{d\sigma(M)}\left[\mathrm{erfc}\left(\frac{\delta_{c}-\delta_{\ell}^{\rm lin% }}{\sqrt{2\sigma_{s}^{2}(M)}}\right)\right]$$ $$\displaystyle=$$ $$\displaystyle(1+\delta_{\ell})\sqrt{\frac{2}{\pi}}\frac{\bar{\rho}}{M^{2}}% \frac{\sigma^{2}(M)}{\sigma_{s}^{3}(M)}\left|\frac{d\ln\sigma_{M}}{d\ln M}% \right|\frac{\delta_{c}-\delta_{\ell}^{\rm lin}}{\delta_{c}}\times$$ $$\displaystyle\exp\left[-\frac{(\delta_{c}-\delta_{\ell}^{\rm lin})^{2}}{2% \sigma_{s}^{2}}\right].$$ (3) (Furlanetto & Piran, 2006) give the same expression for $n_{\rm Lagrangian}$. Note that there is also extra factor of $\sigma^{2}(M)/\sigma_{s}^{2}(M)$ compared to Eq. (2), beyond the conceptual substitutions discussed above. In Lagrangian space, this ensures that $\int n_{\rm Lag}(M|\delta_{\ell}^{\rm lin})\mathcal{P}(\delta_{\ell}^{\rm lin}% )d\delta_{\ell}^{\rm lin}=n(M)$, using the Gaussian distribution of $\delta_{\ell}^{\rm lin}$, $\mathcal{P}(\delta_{\ell}^{\rm lin})=\exp[-(\delta_{\ell}^{\rm lin}/\sigma_{% \ell})^{2}/2]/\sqrt{2\pi\sigma_{\ell}^{2}}$. Note that we integrate over $\delta_{\ell}^{\rm lin}>\delta_{c}$, which arguably should not be done since patches of this initial density have collapsed (Sheth & Lemson, 1999; Musso et al., 2012); accounting for this would change the normalization a bit. Dividing Eq. (3) by Eq. (2) gives $$\displaystyle 1+\delta_{h}(M|\delta_{\ell})=$$ $$\displaystyle(1+\delta_{\ell})\frac{\sigma^{3}(M)}{\sigma_{s}^{3}(M)}\frac{% \delta_{c}-\delta_{\ell}^{\rm lin}}{\delta_{c}}\times$$ $$\displaystyle\exp\left\{-\frac{1}{2}\left[\left(\frac{\delta_{c}-\delta_{\ell}% ^{\rm lin}}{\sigma_{s}(M)}\right)^{2}-\frac{\delta_{c}^{2}}{\sigma^{2}(M)}% \right]\right\},$$ (4) or, slightly more simply, in log-density variables $A\equiv\ln(1+\delta)$, $$\displaystyle A_{h}(M|A_{\ell})=$$ $$\displaystyle A_{\ell}+\ln\left[\frac{\sigma^{3}(M)}{\sigma_{s}^{3}(M)}\frac{% \delta_{c}-\delta_{\ell}^{\rm lin}}{\delta_{c}}\right]-$$ (5) $$\displaystyle\frac{1}{2}\left[\left(\frac{\delta_{c}-\delta_{\ell}^{\rm lin}}{% \sigma_{s}(M)}\right)^{2}-\frac{\delta_{c}^{2}}{\sigma^{2}(M)}\right].$$ (6) Note that in the MIP ensemble, there is a difference between $\delta_{\ell}$ and the matter density measured in an Eulerian cell, the ‘void density’ $\delta_{v}$, in the notation of (Furlanetto & Piran, 2006). This is because in the MIP, $\delta_{\ell}$ truly comes from a sharp-$k$ cut in Lagrangian space. In a usual simulation, or in reality, there would be no such Lagrangian cut, and $\delta_{\ell}$ as estimated on a grid would be filtered through a Eulerian pixel window function. In Fig. 1, there are two windows being applied to obtain $\delta_{m}$: a Lagrangian sharp-$k$ filter, and the Eulerian pixel window function. In comparing the model to the measurements, we are neglecting the Eulerian pixel window function, which is likely negligible compared to the Lagrangian filter. In applying Eq. (4 outside the MIP, we suspect that in calculating $\sigma_{s}^{2}(M)$, it would work adequately to filter the linear power spectrum with the Eulerian pixel window function. 3.2 Local-growth-factor model In this section, we describe a model in which fluctuations in halo number density involve a local growth factor (LGF) $D(\delta_{\ell})$ that depends on the large-scale density $\delta_{\ell}$. For instance, by Birkhoff’s theorem, inside a spherically symmetric void, the dynamics inside can be treated like an FRW universe with a modified $\Omega_{m}$ (e.g. Sheth & van de Weygaert, 2004); this concept has rather wide applicability (e.g. Baldauf et al., 2011; Sherwin & Zaldarriaga, 2012). As before, we work with the small-scale linear density field $\delta_{s}^{\rm lin}$, but we assume that it gets amplified by a factor $D(\delta_{\ell})/D_{0}$. Here, $D_{0}$ is the global growth factor, and $D(\delta_{\ell})$ is a local growth factor, estimated using $\Omega_{m}^{\rm eff}=\Omega_{m}(1+\delta_{\ell})$, and with unchanged $\Omega_{\Lambda}$. Note that one must use the non-linear $\delta_{\ell}$ here; using $\delta_{\ell}^{\rm lin}$ can give nonsense in voids, since $1+\delta_{\ell}^{\rm lin}$ is not constrained to be positive. For calculations, we used the grow$\lambda$ package (Hamilton, 2001). We also tried using the expression (Lahav et al., 1991) $$D(\delta_{\ell})=\frac{5}{2}\frac{a\Omega_{m}^{\rm eff}}{(\Omega_{m}^{\rm eff}% )^{4/7}-\Omega_{\Lambda}+(1+\Omega_{m}^{\rm eff}/2)(1+\Omega_{\Lambda}/70)};$$ (7) it gives very similar results, but there are slight visible differences. As Hamilton (2001) states, all of these growth-factor formulae are invalid for $\Omega_{m}$ sufficiently large to give a collapsing universe, so the results in the LGF model for large $\delta_{\ell}$ should be used with caution. Compared to Eq. (2), we make the following changes: $\sigma(M)\to\sigma_{s}(M)D(\delta_{\ell})/D_{0}$; and we additionally multiply the whole expression by the Lagrangian-to-Eulerian factor $(1+\delta_{\ell})$. This gives $$\displaystyle n(M|\delta_{\ell})=$$ $$\displaystyle(\delta_{\ell}+1)\sqrt{\frac{2}{\pi}}\frac{\bar{\rho}}{M^{2}}% \left|\frac{d\ln\sigma(M)}{d\ln M}\right|\frac{\delta_{c}}{\sigma_{s}(M)}\frac% {D_{0}}{D(\delta_{\ell})}$$ $$\displaystyle\times\exp\left[-\frac{1}{2}\left(\frac{\delta_{c}}{\sigma_{s}(M)% }\frac{D_{0}}{D(\delta_{\ell})}\right)^{2}\right].$$ (8) Arguably, the logarithmic derivative here should be changed to use $\sigma_{s}(M)$, but as we are about to discuss, the normalization is a bit uncertain anyway. [In the AES model, the logarithmic derivative remained intact, but contributed a factor of $\sigma(M)/\sigma_{s}(M)$.] Eq. (8) is normalized to simply give $n(M|\delta_{\ell}=0)=n(M)$, i.e. giving curves that go through $(0,0)$ in Fig. 1. This does not happen in the AES model, and the measurements too seem to depart from $(0,0)$, most notably at high masses. Ideally, there would be a true normalization done, ensuring that $n(M)=\int n(M|\delta_{\ell})\mathcal{P}(\delta_{\ell})d\delta_{\ell}$. There are two reasons we do not work through this. First, this integral is over the non-linear $\mathcal{P}(\delta_{\ell})$, a simple, accurate analytic form for which we are not aware. Second, as noted above, we do not expect the model to be accurate at high $\delta_{\ell}$, so it is likely unwise to take any integral over the full range of $\delta_{\ell}$ seriously. One strategy is to adopt the normalization from the AES model, as done in Fig. 1. Also, the Eulerian excursion-set model introduced by (Sheth, 1998) may be of use in determining the proper normalization. But we leave a thorough investigation of the normalization of this expression to future work. Dividing Eq. (8) by Eq. (2) gives $$\displaystyle 1+\delta_{h}(M,\delta_{\ell})$$ $$\displaystyle=(1+\delta_{\ell})\frac{D(\delta_{\ell})}{D_{0}}\times$$ $$\displaystyle\exp\left[-\frac{1}{2}\left\{\left(\frac{\delta_{c}}{\sigma_{s}(M% )}\frac{D_{0}}{D(\delta_{\ell})}\right)^{2}-\frac{\delta_{c}^{2}}{\sigma^{2}(M% )}\right\}\right].$$ (9) This simplifies a bit in log-density variables: $$A_{h}=A_{m}-\ln\frac{D(\delta_{\ell})}{D_{0}}-\frac{1}{2}\left[\left(\frac{% \delta_{c}}{\sigma_{s}(M)}\frac{D_{0}}{D(\delta_{\ell})}\right)^{2}-\frac{% \delta_{c}^{2}}{\sigma^{2}(M)}\right].$$ (10) The similarity in the shapes of the two predictions in Fig. 1 at low density is worth a bit of investigation. Physically, this lends credence to both models. Mathematically, their agreement implies that subtracting a linear-theory large-scale density ($\delta_{\ell}^{\rm lin}$) from $\delta_{c}$ is roughly equivalent to multiplicatively scaling $\delta_{c}$ by $D_{0}/D(\delta_{\ell})$. Setting the arguments in the exponentials of Eqs. (4) and (9) equal, $$\delta_{\ell}^{\rm lin}=\delta_{c}[1-D_{0}/D(\delta_{\ell})].$$ (11) Fig. 3 shows how this formula compares to the (presumably most accurate) parametric solution used above (Sheth & van de Weygaert, 2004). It also compares two ‘local-Lagrangian’ models (Protogeros & Scherrer, 1997), that contain a parameter $\gamma$ that equals $3/2$ in a low-$\Omega_{m}$, $\Omega_{\Lambda}=0$ fit to the spherical-collapse model (Bernardeau, 1992), that works quite well for voids (e.g. Neyrinck, 2013). The local-Lagrangian parameterization is $$\delta_{\ell}^{\rm lin}=\gamma[1-(1+\delta_{\ell})^{-1/\gamma}].$$ (12) The new model involving the growth factor does not perform badly for $1+\delta>0.1$, but the local-Lagrangian models definitely work better. Curiously, $\gamma=1.58$, between 3/2 and $\delta_{c}=1.686$, works best, giving a curve nearly indistinguishable from the parametric solution. 4 The Poisson approximation To model observable, discrete halo samples in the Universe, it is important to know statistics of the point process that produces haloes from the ‘continuous,’ ensemble-mean $\delta_{h}$. A common assumption is that the continuous halo-density field is Poisson-sampled in each pixel. This is an inhomogeneous Poisson process, in which the mean intensity varies with position. At least at the highest densities, various authors have inferred ‘super-Poisson scatter’ in this sampling (e.g. Kitaura et al., 2013). This means that the variance exceeds the mean, unlike in a Poisson process, in which the variance equals the mean. They found that super-Poisson scatter in high-density pixels was necessary to include to model the halo power spectrum accurately at small scales. The MIP ensemble allows the Poissonity assumption to be tested at low densities. Fig. 4 is a scatter plot, one dot per 2-$h^{-1}$ Mpc pixel, of $(\sigma^{2}/\mu)$ against $\mu$, where $\sigma^{2}$ and $\mu$ are the variance and mean halo densities in each pixel, across MIP simulations. Super-Poissonity (variance exceeding the mean) is prevalent throughout the density range in the two lower-mass halo bins, except at the very lowest densities. However, the ratio here loses meaning when there are only a couple of haloes in the pixel across all simulations. For example, if there is only one halo, this ratio is exactly Poisson because the halo can only belong to one simulation. The discrete allotment of haloes to simulations is the reason for the low-density patterns in each plot. But there is typically an upturn at high mean number densities, where the variance can exceed the mean by a factor of a few. However, curiously, in the highest-mass bin, Poissonity seems to be a rather good assumption. Fig. 5 goes beyond the mean and variance, and shows the full PDFs across the MIP ensemble in single 2-$h^{-1}$ Mpc cells of different mean halo number densities. As the plots show, the PDFs in each cell are rather well-fitted with both Saslaw-Hamilton (SH, Saslaw & Hamilton, 1984; Hamilton et al., 1985) and similarly-shaped negative-binomial (NB) distributions (Sheth, 1995). Kitaura et al. (2013) successfully use a pixel-by-pixel NB distribution to model the super-Poissonity in high-density pixels; based on our results, this seems to be a good strategy. The SH and NB distributions are as follows: $$\displaystyle f_{\rm SH}(N)$$ $$\displaystyle=\frac{\lambda}{N!}e^{-\lambda(1-b)-Nb}(1-b)[\lambda(1-b)+Nb)]^{N% -1};$$ (13) $$\displaystyle f_{\rm NB}(N)$$ $$\displaystyle=\frac{\lambda}{N!}\frac{\Gamma(\beta+N)}{\Gamma(\beta)(\beta+% \lambda)^{N}(1+\lambda/\beta)^{\beta}}.$$ (14) Here $\lambda$ is the mean, and the variance $\sigma^{2}$ determines the parameters $b=1-\sqrt{\lambda/\sigma^{2}}$, and $\beta=\lambda^{2}/(\sigma^{2}-\lambda)$. An NB distribution can arise in a process in which the Poisson mean parameter $\lambda$ is itself a gamma-distributed (similar to lognormal-distributed) random variable. That is, our results are consistent with a gamma (or, perhaps, lognormal) distribution of $1+\delta_{m}$ in each cell, which is then Poisson-sampled. 5 Stochasticity in the continuous halo-density field While the scatter plots in Fig. 1 are rather tight, there is substantial scatter at high $\delta_{m}$. Especially for high-mass halos, this scatter seems to be beyond the scatter from the point process in each individual pixel, i.e. the error bands in Fig. 1. The scatter in $\delta_{h}$ for different pixels of the same $\delta_{m}$ relates to elements of the pixel ‘environment’ beyond the density measured in 2 or 4 $h^{-1}$ Mpc cells. As a preliminary test of whether this stochasticity is caused by visually evident factors such as the cosmic-web environment, we show in Fig. 6 a two-dimensional slice, containing a large cluster, of MIP ensemble-mean densities, with 1 $h^{-1}$ Mpc pixels. For haloes of mass $10^{9-10}M_{\odot}/h$, Fig. 6 shows log-densities of three fields: the actual ensemble-mean halo-density field; the halo-density field as predicted from the matter-density field using the empirical fit (Eq. 1) in the upper-right panel of Fig. 1; and in the bottom panel, we plot the ‘Anscombe difference’ between the two, an estimate of the number of standard deviations away from the expected mean halo density in each cell, assuming Poisson statistics. In the bottom panel, we do not show the raw difference in log-densities, since this would mostly show the differences in voids, where both the increased discreteness and the density downturn make the fractional noise large. Instead, we use an Anscombe (1948) transform separately in each cell, which is designed to transform Poisson-distributed data into Gaussian-distributed data. The Anscombe transform is ${\rm Ansc}(x)=2\sqrt{x+3/8}$. A Poisson-distributed variable of mean $\lambda$ Anscombe-transforms into a Gaussian of mean $\mu_{\rm Ansc}(\lambda)=2\sqrt{\lambda+3/8}-1/(4\sqrt{\lambda})$, and variance 1. Explicitly, what we show is ${\rm Ansc}(N_{h})-\mu_{\rm Ansc}[\lambda=\bar{N}_{h}(1+\delta_{h}(\delta_{m}))]$, where $N_{h}$ is the number of haloes in the cell in the MIP ensemble stack, and $\bar{N}_{h}$ is the mean number of haloes. In the bottom panel of Fig. 1, except for the large cluster at the bottom, the fluctuations typically do not correlate with obvious cosmic-web elements. However, they often deviate more from zero than expected in a Gaussian (Anscombe-transformed Poisson) distribution. Recall, however, that the small pixel size (1 $h^{-1}$ Mpc) is half that of the smaller pixels used for Fig. 1, resulting in inflated dispersion. Still, this is a sign that the point process is super-Poisson, i.e. each cell is sampled in a process with variance exceeding the mean, which we explore further in $\lx@sectionsign$4. The cluster at the bottom, however, shows huge deviations in halo density away from that predicted by the matter, consistent with the substantial overdispersion of the halo density compared to Poisson in high-density areas. Morphologically, the cluster center is underpopulated with haloes compared to what $\delta_{m}$ would predict, and its outskirts are overpopulated. We suspect that this is because this plot is made from low-mass haloes, while the mass typically goes into higher-mass haloes (or perhaps a single large halo) in the cluster center. If we were to include subhaloes of the same mass, they might populate this central hole. Also, it is likely that including all haloes, and weighting them by their mass as Park et al. (2010) do, would reduce this effect. Additionally, although there is no explicit halo exclusion in the MIP ensemble-stack (i.e. haloes can get arbitrarily close, in principle), the halo exclusion in each simulation could still be showing up here. Effects like these are likely responsible for the large scatter at high densities in each halo-mass bin. The visual test shown here is inadequate to make definitive conclusions about the source of the scatter in Fig. 1, but it does seem that except for large clusters, the departure from the mean relationship does not strongly correlate with cosmic-web elements. Still, the cosmic-web morphology, and higher-order density quantities such as the tidal tensor, must drive some of this scatter, and we plan to further use the MIP ensemble to tease out these effects. 6 Gaussianization: a test of stochasticity Another way to test for systematic fluctuations in the ensemble-mean relation between $\delta_{h}$ and $\delta_{m}$ is to compare their power spectra and Fourier-space cross correlations, which we do in this section. Suppose that $\delta_{h}$ is a strictly increasing local function of $\delta_{m}$. Then, if both are mapped to give the same PDF, they will be the same fields. A natural choice of PDF to map each field onto is a Gaussian (Weinberg, 1992), since, for instance, the power spectrum of a Gaussian field has low covariance (Neyrinck et al., 2009). This benefit of a rank-order-Gaussianized field, that the result is insensitive to any monotonic transformation made on the field before it is Gaussianized, has long been exploited for topological statistics such as the genus (e.g. Weinberg et al., 1987). Denote as ${\rm Gauss}(\delta)$ the field $\delta$ after rank-order Gaussianization. Explicitly, if $\delta$ is a field defined on a finite number of pixels $N$, ${\rm Gauss}(\delta)=\sqrt{2}\sigma{\rm erf}^{-1}(2f_{<\delta}-1+1/N)$, where $f_{<\delta}$ is the fraction of pixels less-dense than $\delta$, and $\sigma$ is the standard deviation of the Gaussian that $\delta$’s PDF is mapped onto. If the function $\delta_{h}(\delta_{m})$ is strictly increasing, then $f_{<\delta_{h}(\delta_{m})}=f_{<\delta_{m}}$, so ${\rm Gauss}(\delta_{h})={\rm Gauss}(\delta_{m})$. Fig. 7 shows power spectra of both $\delta$ and ${\rm Gauss}(\delta)$, with Gaussianization done on a 2-$h^{-1}$ Mpc grid, for both the ensemble-mean matter and halo-density fields. If the power spectrum were shown from a single simulation, it would have a noticeable shot noise, but this discreteness is negligible in the ensemble mean. While each $P_{\delta}$ varies substantially from each other, the Gaussianized power spectra are nearly indistinguishable, the halo curves only beginning to depart from the matter curve for haloes in the largest-mass bin. The amplitude of each curve is the same because the standard deviation $\sigma$ used for Gaussianization is set to 1 in each case. We used the same cell size for all fields for easy comparison, but note that in the sparsest case of $10^{11-12}M_{\odot}/h$ haloes, the sparsity might prescribe a somewhat larger cell size that would likely agree better with the matter field, after Gaussianizing both in cells of that scale (Neyrinck et al., 2011). It is a bit surprising that $P_{\delta}$ for the highest-mass bin is not generally larger than the other two, but note that because of the small box, even the smallest $k$ is smaller than the comfortably linear regime. Fig. 8 shows a common measure of stochasticity, the cross-correlation coefficient $R(k)=P_{h\times m}(k)/\sqrt{P_{h}(k)P_{m}(k)}$, where $P_{h\times m}$ is the halo-matter cross spectrum. To ensure that $|R(k)|\leq 1$ (Neyrinck et al., 2005), as one might hope from the Schwarz inequality, we do not subtract the (small, in our case) shot noise from $P_{h}$ when forming this ratio. $R(k)$ is sensitive to each individual Fourier amplitude and phase, while $P(k)$ is sensitive only to Fourier amplitudes, as averaged within bins. Indeed, $R(k)$ is closer to unity (that is, the halo and matter fields are more similar) for each halo field if both the halo and matter fields are Gaussianized. This is an example of the superiority of the statistics of Gaussianized fields. $R(k)$ measured from the raw $\delta$ fields is a measure of stochasticity, invariant to a linear bias between haloes and matter. But the Fourier cross-correlation of the Gaussianized fields has even greater power as a measure of stochasticity, because it is invariant not only to a linear bias, but to any strictly-increasing biasing function. While these results show that to an excellent approximation, the ensemble-mean halo-density fields are local, strictly-increasing transformations of the matter-density field, it is important to remember that an observed galaxy-density field would typically have substantial discreteness and halo-exclusion effects, that are negligible here. The influence of these effects on Gaussianized fields can be straightforwardly measured using the MIP ensemble, but we leave their full investigation to future work. 7 Conclusions In this paper, we have measured the bias relation giving the mean halo density in terms of the dark-matter density down to unprecedentedly low densities. The new tool that made this possible was the MIP ensemble, in which a large-scale cosmic web is sampled in hundreds of ways by changing initial small-scale modes underneath fixed large-scale modes. The ensemble-mean we measure from both fields has negligible stochasticity and exclusion; it is instructive to separate out these from effects of the large-scale cosmic-web environment. The form we find for the bias relation is that of a power law and an exponential at low densities, which are fit well with two models; one is a standard excursion-set model, and another is a new ‘local growth factor’ model, in which small-scale modes grow as though they were in a homogeneous universe with $\Omega_{m}$ proportional to the local large-scale density. These models imply that the abundance of modest-size haloes in voids is quite sensitive to cosmological parameters. The strictly-increasing bias function giving $\delta_{h}$ from $\delta_{m}$ is promising for cosmological tests involving voids, which often assume that voids found in a galaxy sample correspond to dark-matter voids. In addition, the tight relationship we found suggests that the bias function can be used to directly convert a void profile found in haloes to a matter void profile. This remains to be explicitly shown, since exclusion and discreteness could corrupt the relationship, but this test would be straightforward using the MIP ensemble. A tight $\delta_{h}$-$\delta_{m}$ relationship also implies a similarity in their power spectra if a local transform is done on one or both to give the same PDF. A statistically convenient PDF is a Gaussian; we check that indeed, after rank-order Gaussianizing the ensemble-mean fields, they have high cross-correlation and impressively similar power spectra across a few halo-mass ranges. There are still many aspects of the dark-matter-to-halo mapping to explore, all of which the MIP ensemble can help to study. The current investigation is entirely in real space, but it is essential to investigate redshift space, as well; for example, it is possible that in fingers of God, haloes sample the matter in a different way than in real space. The assumption that the halo density field is a (super-)Poisson sampling of a continuous field also should be investigated in more detail; for example, in the sampling process, substantial covariance among nearby pixels could occur due to halo exclusion. The effects of discreteness on statistics like the power spectrum, and especially the Gaussianized- or log-density power spectrum, would also benefit from study with the MIP ensemble. Acknowledgments We thank Francisco-Shu Kitaura, Ixandra Achitouv and Aseem Paranjape for useful comments. MCN and MAAC are grateful for support from a New Frontiers in Astronomy and Cosmology grant from the Sir John Templeton Foundation. DJ is supported by DoE SC-0008108, NASA NNX12AE86G, and NSF 0244990. 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11institutetext: INGI, ICTEAM, Université Catholique de Louvain, Place Sainte Barbe 2, LG05.02,01, 1348 Louvain-La-Neuve, Belgium 11email: {charles-henry.bertrand,axel.legay}@uclouvain.be Malware Analysis with Symbolic Execution and Graph Kernel Charles-Henry Bertrand Van Ouytsel 11    Axel Legay 11 Abstract Malware analysis techniques are divided into static and dynamic analysis. Both techniques can be bypassed by circumvention techniques such as obfuscation. In a series of works, the authors have promoted the use of symbolic executions combined with machine learning to avoid such traps. Most of those works rely on natural graph-based representations that can then be plugged into graph-based learning algorithms such as Gspan. There are two main problems with this approach. The first one is in the cost of computing the graph. Indeed, working with graphs requires one to compute and representing the entire state-space of the file under analysis. As such computation is too cumbersome, the techniques often rely on developing strategies to compute a representative subgraph of the behaviors. Unfortunately, efficient graph-building strategies remain weakly explored. The second problem is in the classification itself. Graph-based machine learning algorithms rely on comparing the biggest common structures. This sidelines small but specific parts of the malware signature. In addition, it does not allow us to work with efficient algorithms such as support vector machine. We propose a new efficient open source toolchain for machine learning-based classification. We also explore how graph-kernel techniques can be used in the process. We focus on the 1-dimensional Weisfeiler-Lehman kernel, which can capture local similarities between graphs. Our experimental results show that our approach outperforms existing ones by an impressive factor. Keywords: Malware Analysis Symbolic Execution Malware Classification 1 Introduction According to the independent IT security institute AV-Test [5], the number of malware infections has increased significantly over the last ten years, reaching a total of 1287.32 million in 2021. With approximately 450 000 new malware every day, companies spend on average $2.4$ millions dollars [1] on defenses against such malicious software. For this reason, effective and automated malware detection and classification is an important requirement to guarantee system safety and user protection. Most malware classification approaches are based on the concept of signature and signature detection. A malware signature, which is often built manually, represents the DNA of the malware[12, 17, 6]. Consequently, deciding whether a binary file contains a specific malware boils down to checking whether the signature of such malware is present in the binary. The simplest type of signature is the syntactic signature, i.e., signatures based on syntactic properties of the malware binaries (length, entropy, number of sections, or presence of certain strings). Alternatively, behavioral signatures are based on the malware’s behavioral properties (interaction with the system and its network communications). Different types of signature give rise to different malware classification approaches. In static malware analysis approaches, the classification boils down to detecting the presence of a given static signature directly in the binary that has been disassembled. This signature often boils down to a sequence of characters[14]. The two main advantages of this approach is that it is fast and does not require executing the malware. On the other hand, static signatures are very sensitive to obfuscation techniques that modify the binary code to change its syntactic properties [21]. An illustration of those limitations is given in [6, 24] where the authors show the approach is not robust to variants of the MIRAI malware. Another classification approach is that of dynamic analysis, which executes the malware and observes if its effect on the system corresponds to some behavioral signature [20, 35]. This approach is based on the fact that a static obfuscation does not modify the behavior of the malware and therefore has no influence on the classification of a behavioral signature. To avoid infecting the analyst’s system and to prevent the malware from spreading, the malware is commonly executed in a sandbox. Unfortunately, malware can implement sandbox detection techniques to determine whether they are being executed in a sandbox. As dynamic analysis is limited to one execution, a malware can pass detection by avoiding exhibiting malicious behavior [3]. More information on static and dynamic malware analysis can be found in the following tutorial [6]. Aware of those limitations, several authors have proposed using some exploration techniques coming from the formal verification areas. This includes symbolic execution [11, 13, 9], a technique that explores possible execution paths of the binary without either concretizing the values of the variables or dynamically executing the code. As the code exploration progresses, constraints on symbolic variables are built and system calls tracked. A satisfiability-modulo-theory (SMT) checker is in charge of verifying the satisfiability of the collected symbolic constraints and thus the validity of an execution path. The advent of symbolic execution has led to the development of a new set of machine learning-based fully automatised malware classification methods. Those continue and extend the trend of applying machine learning to malware classification [19, 32]. In particular, in [24] the authors have proposed combining symbolic execution with Gspan [34], a machine learning algorithm that allows us to detect the biggest common subgraphs between two graphs. In its training phase, the algorithm collects binary calls via symbolic analysis. Such calls are then connected in a System Call Dependency Graph (SCDG), that is a graph that abstracts the flow of information between those calls. Gspan can compute the biggest common subgraphs between malware of a given family. Those then represent the signature for the family. In its classification phase, the approach extracts the SCDG from the binary and compares it with each family’s signature. In addition to being fully automatised, the approach has been shown to be more efficient than classical static and dynamic analysis approaches on a wide range of case studies. Unfortunately, the above-mentioned approach has several limitations. The first one is that it depends on the efficiency of the symbolic analysis engine. The second one is that SCDGs are built as an abstraction of the real behavior of the binary. In particular, the approach will connect two calls that have the same argument even though those calls may be from different function. Such a choice, which is motivated by efficiency reasons, may lead to a crude over-approximation of the file’s behavior and hence to misclassification. Relying on the biggest common subgraphs may exclude important but isolated calls that are specific to the malware. In addition, using graphs poses a particular challenge in the application of traditional data mining and machine learning approaches that rely on vectors. To surmount those limitations, the authors in [23] proposed using a graph kernel [22], which can be intuitively understood as a function measuring the similarity of pairs of graphs. In their work, the authors used the approach in a non-supervised process. However, such kernel can be plugged into a kernel machine, such as a support vector machine. Results in [23] show that the graph kernel outperforms Gspan in terms of accuracy. Unfortunately, the kernel used in [23] still implicitly relies on detecting the biggest commonalities between graphs. Consequently, individual important calls are still out of its scope. Our paper makes several contributions to improving symbolic analysis-based malware classification. The first contribution consists in a flexible and open source implementation of a malware analysis toolchain based on [30] (available here [4]). In addition to obtaining better performances, the flexibility of the new implementation allows us to plug in and compare various classification algorithms and symbolic execution strategies. In particular we develop and compare several efficient resource-based strategies that enable us to build compact but more informative SCDGs than those in [24, 23]. The approach is able to distinguish more SCDGs and hence obtain a finer grain in both training and classification processes. Another important contribution of this paper is the comparison of the Weisfeiler-Lehman Kernel [29] with other classifiers. Such a graph kernel is capable of comparing the graph’s local small structures by a ingenious relabelling of its vertices. Finally, a major contribution of the paper is a series of experimental results showing that our approach outperforms those in [24] and in [23] when being used in a supervised context. 2 On Graph Comparison for Malware Analysis This section briefly introduces several notions related to graphs. It also outlines the limits of graph-based representation in malware analysis and advantages of graph kernels. A graph G is defined as a pair $(V,E)$, where $V$ is a set of Vertices and $E$ a set of edges such that $\{\{u,v\}\subseteq V|u\neq v\}$. The set of edges and vertices of $G$ are given by $E(G)$ and $V(G)$, respectively. We also consider labelled graphs where a label function $l:V(G)\rightarrow\Sigma$ assigns a label from $\Sigma$ to each vertex of $G$. We use $l(v)$ to denote the label of vertices $v$. A graph G’ = (V’,E’) is a subgraph of G=(V,E) if $V^{\prime}\subseteq V$ and $E^{\prime}\subseteq E$. In this work, we are particularly interested in applying graph comparison to extract and compare malware signatures represented by SCDGs. In particular, graph isomorphism is considered to be a powerful tool that allows us to detect structural similarities between graphs that may not be identical. Two unlabelled graphs G and H are said to be isomorphic ($G\simeq H$) if there exists a bijection $\phi:V(G)\rightarrow V(H)$ such that $(u,v)\in E(G)$ if and only if (1) $(\phi(u),\phi(v))\in E(H)$ (for all $u,v\in V(G)$), and (2) $l(v)=l(v^{\prime})$ for each $(v,v^{\prime})\in\phi$. There exists a wide range of graph similarity measures. This includes, e.g., subgraph isomorphism used to compute the largest common subgraph. Checking graph isomorphism is known to be NP. Moreover, reducing the comparison of two graphs to checking their isomorphism is known to be restrictive as it requires both graphs to have same structure. This situation is rarely encountered when comparing (classes of) malware. The situation is illustrated in Figure 1, where two malware from the same family are considered to be different since Vertex $SetFilePointer$ cannot be covered by an isomorophic relationship. In order to leverage this problem, the authors in [24] proposed an approach based on common subgraph comparison. More precisely, the authors use Gspan, which is a popular algorithm for frequent graph-based pattern mining. Given a set of graphs $\mathbb{G}$ and a desired support $min\_supp$, Gspan (whose pseudo-code is given in Figure1) tries to extract all subgraphs present at least in $min\_supp$ graphs of $\mathbb{G}$. If $\mathbb{G}$ represents a set of malware from the same family, the set of common subgraphs represents their signatures. Unfortunately, relying on computing the biggest subgraphs may dismiss small but important connected components that do not belong to the biggest subgraphs. The situation is illustrated in Graph 3 of Figure 2, where important calls such as $IsDebuggerPresent$ may be ignored. An inefficient solution could be to extend the number of subgraphs. Unfortunately, when bigger graphs than in our example are involved, this approach will mostly favor a variant of the biggest connected component, as we will see in Section 4. In order to leverage this problem, we resort to the concept of Graph Kernels. 2.1 Graph kernels In machine learning, kernel methods are algorithms that allow us to compare different data points with a particular similarity measure. Consider a set of data points $X$ such as $\mathbb{R}^{m}$ and let $k:X\times X\rightarrow\mathbb{R}$ be a function. Function $k$ is a valid kernel on $X$ if there exists a Hilbert space $\mathcal{H}_{k}$ and a feature map $\phi:X\rightarrow\mathcal{H}_{k}$ such that $k(x,y)=\langle\phi(x),\phi(y)\rangle$ for $x,y\in\mathcal{X}$, where $\langle\cdot,\cdot\rangle$ denotes the inner product of $\mathcal{H}_{k}$. It is known that $\phi$ exists only if $k$ is a positive semidefinite function. A well-known kernel is the Gaussian radial basis function (RBF) kernel on $\mathbb{R}^{m}$, $m\in\mathbb{N}$, defined as: $$k_{RBF}(x,y)=exp(-\frac{\left\lVert x-y\right\rVert^{2}}{2\sigma^{2}})$$ (1) with $\sigma$, the bandwidth parameter. Observe that RBF kernel gives an explicit definition of $\phi$. In practice, this is not always required. Indeed, algorithms such as Support Vectors Machine (SVM) use the data $X$ only through inner products between data points. Having the kernel value $k(x,y)$ between each data point is thus sufficient to build an SVM-based classifier. This approach is known as the kernel trick [15]. A Gram matrix $K$, is defined with respect to a finite set of point $x_{1},..,x_{n}\in X$. Each element $K_{i,j}$ with $i,j\in\{0,..,n\}$ represents the kernel value between pairs of points $k(x_{i},x_{j})$. If the Gram Matrix $K$ of Kernel $k$ is positive semi-definite for every possible set of data points, then k is a valid kernel. It is common for kernels to compare data points using differences between data vectors. However, the structures of graphs are invariant to permutations of their representations (i.e., ordering of edges/vertices does not influence structure and distance between graphs). This motivates the need to compare graphs in ways that are permutation invariant. Moreover, to avoid strict comparison (which would be equivalent to isomorphism), it is common to use smoother metrics of comparison, such as convolutionnal kernels, for better generalization capabilities. Convolutionnal kernels divide structures (i.e., graphs in our case) into substructures (e.g., edges, subgraphs, paths, etc) and then evaluate a kernel between each pair of such substructures. In [23], the authors propose a similarity metric for malware behavior graphs based on common vertices and edges. Concretely, they define a similarity $\sigma$ between two graphs G and H as: $$\sigma(G,H)=\alpha\sigma_{vertices}(G,H)+(1-\alpha)\sigma_{edges}(G,H)$$ (2) where $\alpha$ is the vertice-edge factor allowing to adjust weights of vertices and edges in the similarity function (set to 0.25 in the conclusion of their work). The vertice similarity is defined as: $$\sigma_{vertices}(G,H)=\frac{|\mathcal{V}(G)\cap\mathcal{V}(H)|}{min(\mathcal{V}(G),\mathcal{V}(H))}$$ (3) and the edge similarity as: $$\sigma_{edges}(G,H)=\frac{|\mathcal{CC}_{max}(G\cap H)|}{min(|\mathcal{CC}_{max}(G)|,|\mathcal{CC}_{max}(H)|)}$$ (4) where $\mathcal{V}(G)$ are the set of vertices of G and $\mathcal{CC}_{max}(G)$ is the biggest connected component of G. While this approach adds information related to all nodes labels compared to Gspan, it suffers from similar drawbacks than Gspan. Indeed, it focus on the biggest connected component, neglecting edges in other connected components. This problem is illustrated on Graph 4 of Figure 2. One can see that the kernel identifies similarities between nodes of Graph $1$ and Graph $2$. However, it ignores important edge dependencies such as GetModuleHandle, CopyFileA, and GetSystemDirectoryA. To tackle this problem, a popular approach in graph kernels is the comparison of local structure. In this approach, two vertices of different graphs are considered to be similar if they share the same labels. The two vertices are considered to be more similar if, in addition, they share similar neighborhoods (i.e., vertices with the same labels). Using this approach, Shervashidze et al. [29] introduced graph kernels based on the 1-dimensional Weisfeiler-Lehman (WL). Let $G$ and $H$ be graphs, and $l:V(G)\cup V(H)\rightarrow\Sigma$ be a function giving their vertices labels. By several iterations $i=0,1,...$, the 1-WL algorithm computes a new label function $l_{i}:V(G)\cup V(H)\rightarrow\Sigma$, with each iteration allowing comparison of G and H. In the first iteration, $l_{0}=l$, and in subsequent iterations, $$l_{i}(v)=\textit{relabel}(l_{i-1}(v),\textit{sort}(l_{i-1}(u)|u\in N(v)))$$ (5) with $v\in V(G)\cup V(H)$, sort(S) returning a sorted tuple of S and function relabel(p) maps the pair p to a unique value in $\Sigma$ which is not already used in previous iterations. When the cardinality of $l_{i}$ equals the cardinality of $l_{i-1}$, the algorithm stops. The idea of the WL sub-tree kernel is to compute the previous algorithm for $h\geq 0$ and after each iteration $i$ to compute a feature vector $\phi_{i}(G)\in\mathcal{R}^{|\Sigma_{i}|}$ for each graph G, where $\Sigma_{i}\subseteq\Sigma$ denotes the image of $l_{i}$. Each component $\phi_{i}(G)_{\sigma^{i}_{j}}$ counts the number of appearances of vertices labelled with $\sigma^{i}_{j}\in\Sigma_{i}$. The overall feature vector $\phi^{WL}(G)$ is defined as the concatenation of the feature vectors of all h iterations, i.e., $$\phi^{WL}(G)=(\phi^{0}(G)_{\sigma^{0}_{1}},...,\phi^{0}(G)_{\sigma^{0}_{|\Sigma_{0}|}},\phi^{h}(G)_{\sigma^{h}_{1}},...,\phi^{h}(G)_{\sigma^{h}_{\Sigma_{h}}})$$ (6) Finally, to compute similarity between two different feature vectors, we apply the following formula: $$k_{WL}(G,G^{\prime})=\sum_{\phi\in\phi^{WL}(G)}\sum_{\phi^{\prime}\in\phi^{WL}(G^{\prime})}\delta(\phi,\phi^{\prime})$$ (7) where $\delta$ is the Dirac kernel, that is, it is $1$ when its arguments are equals and $0$ otherwise. The more labels the two graphs have in common, the higher this kernel value will be. Compared with Gspan and the kernel from [23], this kernel also targets similarities related to all nodes and edges of the biggest subgraph but also local similarities. This is illustrated in Figure 3, where dependencies between GetModuleHandle, CopyFileA, and GetSystemDirectoryA are kept in the learning process. 3 An Open Source Toolchain for Malware analysis We propose an open source toolchain for malware analysis that is based on machine learning and SCDGs (available here [4]). The toolchain, which is represented in Figure 4, relies on the following important components: the first component consists in collecting and labelling a series of binaries from different malware families. Then, Angr [30], a python framework for symbolic execution, is used to execute those files. The result is used to extract a SCDG for each such binary. One of the contributions of this paper will be to improve and adapt the symbolic engine to malware analysis as well as the construction of SCDGs. Those SCDGs are then used to train machine learning algorithms. If Gspan is used, the training will result in common subgraphs to represent signatures for each family. If SVM is used, a Gram matrix between all the malware programs is created. Finally, the toolchain also contains supervised classifiers. If Gspan is used, the SCDG of the new malware is compared with those of the signature of each family and the classifier retains the one with the closest distance. If SVM is used, a Gram matrix is created between all trained malware and the new malware. This matrix is then used in the SVM classification process. A main contribution of this paper is to compare those two types of classification. 3.1 Extraction of calls The construction of the SCDG is based on Symbolic Execution. This approach envisages the exploration of all the possible execution paths of the binary without either concretizing the values of the variables or dynamically executing the code (i.e., the binary is analyzed statically). Instead, all the values are represented symbolically. As the code exploration progresses, constraints on symbolic variables are built and system calls tracked. A satisfiability-modulo-theory (SMT) checker is in charge of verifying the satisfiability of the collected symbolic constraints and thus the validity of an execution path. A wide range of tools and techniques have been developed for efficient symbolic execution analysis. Most of those techniques agree on the fact that symbolic execution still suffers from state-space-explosion and, consequently, only a finite set of symbolic paths can be explored in a reasonable amount of time. This is particularly the case with malware analysis where the classification process must be done with very limited resources. As the calls that form the SCDG are collected directly from those symbolic paths, the choice of which paths to follow will have an impact on the machine learning process.In a recent work, authors showed how SMT solving could impact performances [27, 7, 10]. In this paper, we focus on path selection strategies. The work in [24] implements a Breadth-First Search (BFS) approach, that is, at each execution step all ongoing paths are explored simultaneously. This approach leads to an important growth of states and memory usage. As we have limited resources, we propose to explore one subset of paths at a time. We prioritize states from which one can explore new assembly instruction addresses of the program. Our Custom Breadth-First Search Strategy (CBFS-Strategy) is presented in 3. The algorithm begins by taking $L$ states for exploration from the set of available states and putting them in the list $R$ of states to explore next (line $4$). It then iterates among all other available states. If it finds a state leading to an unexplored part of the code or with a shorter path of execution (line $6$), it puts it in $R$ and takes out a state with a lower priority. After going through each state, it returns $R$ to allow ANGR to perform a new execution step on $R$’ states. In addition to BFS-Strategy, we also implemented a Custom Depth-First Search Strategy (CDFS-Stategy), which is presented in Algorithm 4 (the main difference with CBFS-Strategy being the condition to select successor state at Line $6$). Observe that symbolic execution with depth and breadth first search is not new. However, the implementation and evaluation of restricted versions within a tool for malware classification are. Another important challenge in symbolic execution is that of handling loops. Indeed, the condition of such loops may be symbolic. In addition, the loop may create an infinite repetitive behavior. In those situations, deciding between staying in the loop or exiting the loop remains a tricky choice that has been the subject of several works focusing on the possibilities, which include Loop-extended Symbolic Execution [25], Read-Write set [8], and bit-precise symbolic mapping [33]. As those approaches may be too time-consuming, we propose to reuse two intermediary heuristics from [24]. The first one applies to loops whose condition contains a symbolic value. Such loops may give rise to two states at each iteration: one that exits the loop for those symbolic values that exceed the condition and one that remains within the loop for other values, with this last state being used again to iterate on the loop. We chose to stop such iteration after four steps for loops that do not contain symbolic values, since such loop may still lead to an unbounded number of behaviors, our approach consists in limiting the execution to a finite precomputed number of steps and then forcing the execution to exit the loop. 3.2 Creating SCDGs Symbolic execution allows us to obtain several paths representing executions of a given binary. Our next step is to collect the sets of calls present on each such path as well as their addresses and arguments. Those are used to build the SCDG corresponding to this binary. ). Following [24], SCDGs are graphs where each vertex is labelled with the name of a system call; and the edges correspond to (an abstraction of) information flow between these calls. Concretely, each SCDG is built from the symbolic representation by merging and linking calls from one or more symbolic paths. Consider first the creation of a graph execution from one symbolic path. We consider three types of edge. In the first one, two calls are linked if they both share an argument with identical value. This is, for example, the case of two calls with the same file handler. The second link is established between two calls that both have an argument with the same symbolic value. An example is a symbolic file size returned by a call and passed to a second call added to another value. In addition, we consider that two calls can be linked if an argument of the first call is the calling address of the second one. This situation typically arises in dynamic loading of a library. We also label each edge with the index of the argument in both calls (return value of a call is given index $0$). The three-edges strategy is called SCDG-Strategy 1 and the one-edge strategy is called SCDG-Strategy 2. Our experiments shows that SCDG-Strategy 2 loses important dependency between calls and leads to more isolated nodes in SCDGs. Indeed, this strategy suffers from two types of problem. First, symbolic values may be modified before being passed to another call. Second, some calls used by obfuscation techniques exhibit address-arguments links. A typical example is given by GetProcAddress, used to hide real content of the import table of PE files. An example of an SCDG is given in Figure 5 with SCDG-Strategy 1 and SCDG-Strategy 2. The program first calls CreateFile, which returns a handle to the file with the specified filename. A vertex is thus constructed for CreateFile. Then, a call to SetFilePointer on the preceding file handle occurs. This leads to the creation of a new vertex (SetFilePointer). Since the returned argument of CreateFile (index 0) is the same as the first argument of SetFilePointer (index 1), an edge is added between them. Vertices ReadFile and WriteFile are created and linked following similar principles. There are situations where different calls in the same execution share the same API name and occur at the same instruction address but with distinct arguments. In such situations, one may decide to merge the two calls into one single vertex. In this case, we conserve the set of arguments of the first call observed in the execution. This merge incurs a loss of precision but leads to a more compact SCDG representation. This may be of importance when one has to train the system with a large number of different types of malware. In the rest of the paper, this merging strategy is called SCDG-Strategy 3. Merging calls gives different advantages. First, it decreases the size of the SCDG, which may lead to better classification/detection performances. In addition, it may reduce the impact of some calls in the learning phase. An example is given with the wabot malware, which uses a hundred of calls to WriteFile during its execution. Those calls are not part of the main actions of the malware. If they are not merged, they will constitute an important part of the signature and may have a negative impact on the training phase. On the other hand, there are situations where SCDG-Strategy 3 will merge several calls with different goals. This situation may result in losing part of the malware behavior. Observe that the above strategies apply to single symbolic paths only. When several symbolic paths are considered, one can decide to produce an SCDG that is composed of the disjoint union of such executions. Such a strategy is referred to as SCDG-Strategy 5 in the rest of the paper. On the other hand, SCDG-Strategy 4 consists in merging successive executions from different symbolic paths. SCDG-Strategy 5 is simplier to compute, but SCDG-Strategy 5 gives smaller graphs. According to our experimental results, SCDG-Strategy 4 may speed up the computation time by an exponential factor for families with high symbolic execution numbers. 3.3 Creating a classification model and evaluate new samples The toolchain uses SCDGs to train a classifier which is used to detect and classify malware. We have implemented two classifiers. One implementation is based on Gspan and follows the idea from [24]. Another one implements the graph kernel from [23] and the Weisfeiler-Lehman extension we outlined in Section 2. The classifier that uses Gspan implementation works by extracting signatures from malware families. We obtain the signature of each family by computing the biggest subgraphs between the SCDG of each malware. In the classification phase, we compare the SCDG of new binary with those of each signature. The file belongs to the malware family whose graph is the closest to the binary’s. For the case of graph kernel, the training phase consists in computing the feature vector that corresponds to applying the algorithm in [23] or the Weisfeiler-Lehman extension to each malware of the family. As explained in the background section, the algorithm produces a Gram matrix between all those vectors. This matrix represents an implicit version of the kernel. A support vector machine can then exploit this implicit representation. In the classification phase, we compute a Gram matrix between the feature vector of the binary under classification and the vectors of all malware used in the training set. Observe that, contrary to the Gspan approach, graph kernel does not require us to produce an explicit and hence all-encompassing representation of the signature of each family. 4 Experimental Results This section describes the methodology used to assess our toolchain’s performance in both extracting SCDGs and classifying new binaries. Our evaluation set was composed of 1874 malware divided into 15 families plus 150 cleanware samples. The data set’s exact composition is given in Table 1. In terms of origins, $64$ percent of the samples used in our data set were obtained thanks to a direct pipeline between UCLOUVAIN and Cisco between April and June 2020. The remaining $36$ percent were extracted from MalwareBazaar [2] between April 2020 and December 2020. Samples were labelled using AVClass [26], a python tool to label malware samples. This tool is fed with VirusTotal reports and outputs the most likely family of each sample. To evaluate detection performance, we used 150 open source programs found online [31]. In the rest of the section, all experiments were performed on a desktop PC with an Intel Core i7-8665U CPU (1.90GHz x 8) and 16GB RAM running Ubuntu 18.04.5. Our experimental results relied on our ability to extract SCDGs efficiently. In all experiments, we used a timeout of ten minutes for each SCDG. Note that 20 percent of the SCDGs were computed in time while 80 percent never computed entirely. For the case of BFS-Strategy, we used the same parameters as in [27] (loop threshold of 4, unlimited number of states to explore, z3 optimization enabled). However, for CDFS-Strategy and CBFS-Strategy we imposed a limit of 10 states that could be explored simultaneously. Environment modeling Proper environment modelling is a major challenge in developing efficient symbolic execution techniques. Indeed, when we apply symbolic execution we avoid exploring/executing API call code. Indeed, performing such an operation would drastically increase the computation time [18]. In ANGR, when a call to an external library occurs, the call is hooked to a simulated procedure called simprocedures that will produce the symbolic outputs for the function. A simple but crude implementation of such procedure is to assume that the external function returns a symbolic value without any constraint. In such a case, simprocedures simply returns symbolic values covering the full range of outputs given in the specification. In practice, such a solution gives good results in 26 percent of the cases. However, this solution may generate outputs that are not defined in the specification. In addition, it ignores many potential effects of the call, which include the modifications of input parameters or the number of its arguments. This may lead to incoherent executions if those parameters impact the rest of the execution (e.g., in branch choices). We propose several improvements to fix those issues. The first one consists in restricting the ranges of outputs to those given in the specification. As an example, if the output is an integer variable that can take only four values, simprocedures would generate those values instead of the full range of integers. Another one concerns the case where an execution is blocked or cheated because modifications of some arguments by the external call are not performed. This happens in situations where the external call may modify some of its inputs or even some environment variables. In such case, we emulate all potential modifications with concrete values. Finally, we also take variadic functions into account. The specification of such a function is given with a fixed number of arguments. As the number of arguments may change at execution, considering this fixed number only would have an impact on the stack. To solve this problem, we explicitly count the number of arguments passed onto the function. Finally, we also use the ANGR abstraction for file systems that allows us to reuse a file in multiple systems. Observe that this improvement work must be performed for each call that causes problems. This constitutes a tremendous amount of work. To ease the life of future developers, we have constituted a simprocedures library that is constantly enriched with new experiments and calls. We first apply Gspan to SCDGs obtained with combinations of different strategies. Signatures are obtained by sampling randomly 30% of the SCDGs of each family; those SCDGs constitute the training set. Other SCDGs are then classified to assess the quality of those signatures; those SCDGs constitute the test set. This process is repeated three times and performance is averaged. The results are reported in Table 2, from which we observe that CBFS-Strategy and BFS-Strategy generally outperform CDFS-Strategy.By inspecting the results, we observed that BFS-Strategy ran out of memory for 7 percent of the binaries, thus reducing its performance compared to CBFS-Strategy. While SCDG-strategy 4 showed improvements with SCDG-strategy 5, it should be noted that SCDG-strategy 5 entails significant overhead in SCDG building (up to 100 times slower) and signature size (5 times bigger on average). In general, the best performances were obtained by combining SCDG-strategy 2, SCDG-strategy 3, and SCDG-strategy 4. Upon inspecting the confusion matrix obtained for the best set of parameters in Figure 8, we see a lot of confusion between different classes. This can be explain by plotting similarities between signatures built with Gspan, as illustrated in Figure 6. One can see that different signatures share important similarities, leading to confusion between different malware families, as illustrated in Figure 8. This problem is directly linked to a problem exposed in Section 2, that is, Gspan focus on the computation of the biggest subgraph while neglecting other components. We now turn to applying kernel from [23]. Table 3 shows that overall performance increased compared with Gspan. Moreover, CBFS-Strategy and CDFS-Strategy outperformed both BFS-Strategy and SCDG-strategy 2, SCDG-strategy 3, and SCDG-strategy 4 strategies appeared to be more efficient. However, Figure 8 shows that several families were still indistinguishable. Weisfeiler-Lehman kernel . Finally, we investigated the SVM classifier with the Weisfeiler-Lehman kernel. The results in Table 4 clearly outperformed the others, reaching an $F_{1}$-score of 0.929 with CDFS-Strategy and SCDG-strategy 1-3-4. Malware families were better distinguished, as illustrated in the confusion matrix given in Figure 9. Those observations confirm our supposition exposed in Section 2: taking advantage of an SCDG’s local structure increases the efficiency of machine learning in malware classification. Table 5 shows that those observations also extend to the simpler operation of detecting malware. Training time We also recorded average training time for the three approaches implemented in our toolchain. In general, we observed that Weisfeiler-Lehman kernel outperforms Gspan by a factor of 15 and Kernel in  [23] by a factor of 10 000. We suspect that the overhead can be explained by the extensive use of pairwise graph mining in the similarity metric presented in Section 2. Compare to the Kernel in  [23], Gspan reduces these number of computation since it first create a signature for each family before comparing those signature with the binary to classify. These experiments also enable us to draw conclusions with respect to symbolic execution strategies. First, SCDG-strategy 1 gives overall better results than SCDG-strategy 2 with Weisfeiler-Lehman kernel. That means that the edges added by this strategy provide useful SCDG abstraction information for learning. That is not the case for the other classifier where these information seems to lead to overfitting and SCDG-strategy 2 should be preferred. Moreover, the impact of SCDG-strategy 3 varies. While it improves classification for kernels that are based on the biggest common subgraph, its impact when combined with other strategies varies. Finally, observe that while SCDG-strategy 5 leads to a considerable overhead, it does not improve performance of any classifier. On the other hand, SCDG-strategy 4 leads to more compact signatures, better computation times and good classification performances. Thus, SCDG-strategy 4 should be preferred. Regarding exploration strategies, BFS-strategy is generally outperformed by CBFS-strategy while CDFS-strategy outperforms all other exploration strategies when used with the Weisfeiler-Lehman graph kernel. Generally speaking, the experiments confirm our intuition given in Section 2 that ”better classification is possible with Weisfeiler-Lehman graph kernel.” 5 Conclusions and Future Work We propose an open-source toolchain for malware analysis based on symbolic execution and machine learning [4]. This toolchain exploits SCDGs extracted from malware of the same family to learn the common behavior shared among this family. We also developed and compare several heuristics related to binary exploration and SCDG building in order to improve the use of symbolic execution in the malware analysis domain. Finally, we demonstrate how using the Weisfeiler-Lehman kernel could improve learning from SCDGs compared with other techniques such as Gspan by exploiting information contained in those graphs better. This leads to significant improvements in the malware sample classification and detection. Directions for future work includes new exploration heuristics, such as concolic executions [28] or smart sampling [16]. Another objective is to apply our kernel in a non-supervised approach like in  [23]. We are also interested in implementing a distributed version of the toolchain. In this context, the federated learning paradigm should allow us to combine information from different contributors. In addition, we will continue to improve our toolchain with new simprocedure and plugin interfaces. Acknowledgments. Charles-Henry Bertrand Van Ouytsel is an FRIA grantee of the Belgian Fund for Scientific Research (FNRS-F.R.S.). 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Fast neutrino flavor conversion, ejecta properties, and nucleosynthesis in newly-formed hypermassive remnants of neutron-star mergers Manu George kuttan.mgc@gmail.com Institute of Physics, Academia Sinica, Taipei, 11529, Taiwan    Meng-Ru Wu mwu@gate.sinica.edu.tw Institute of Physics, Academia Sinica, Taipei, 11529, Taiwan Institute of Astronomy and Astrophysics, Academia Sinica, Taipei, 10617, Taiwan Physics Division, National Center for Theoretical Sciences, 30013 Hsinchu, Taiwan    Irene Tamborra tamborra@nbu.ku.dk Niels Bohr International Academy and DARK, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100, Copenhagen, Denmark    Ricard Ardevol-Pulpillo ricardar@mpa-garching.mpg.de Max-Planck-Institut für Astrophysik, Postfach 1317, 85741 Garching, Germany    Hans-Thomas Janka thj@mpa-garching.mpg.de Max-Planck-Institut für Astrophysik, Postfach 1317, 85741 Garching, Germany (November 18, 2020) Abstract Neutrinos emitted in the coalescence of two neutron stars affect the dynamics of the outflow ejecta and the nucleosynthesis of heavy elements. In this work, we analyze the neutrino emission properties and the conditions leading to the growth of flavor instabilities in merger remnants consisting of a hypermassive neutron star and an accretion disk during the first 10 ms after the merger. The analyses are based on hydrodynamical simulations that include a modeling of neutrino emission and absorption effects via the “Improved Leakage-Equilibration-Absorption Scheme” (ILEAS). We also examine the nucleosynthesis of the heavy elements via the rapid neutron-capture process ($r$-process) inside the material ejected during this phase. The dominant emission of $\bar{\nu}_{e}$ over $\nu_{e}$ from the merger remnant leads to favorable conditions for the occurrence of fast pairwise flavor conversions of neutrinos, independent of the chosen equation of state or the mass ratio of the binary. The nucleosynthesis outcome is very robust, ranging from the first to the third $r$-process peaks. In particular, more than $10^{-5}$ $M_{\odot}$ of strontium are produced in these early ejecta that may account for the GW170817 kilonova observation. We find that the amount of ejecta containing free neutrons after the $r$-process freeze-out, which may power early-time UV emission, is reduced by roughly a factor of 10 when compared to simulations that do not include weak interactions. Finally, the potential flavor equipartition between all neutrino flavors is mainly found to affect the nucleosynthesis outcome in the polar ejecta within $\lesssim 30^{\circ}$, by changing the amount of the produced iron-peak and first-peak nuclei, but it does not alter the lanthanide mass fraction therein. I Introduction Compact binary systems consisting of two neutron stars (NS) or a NS and a black hole (BH) can lose their angular momentum through continuous emission of gravitational waves (GW), eventually leading to the merging of the compact objects. Such merger events have long been considered to be the sites producing short gamma-ray bursts (sGRB) and synthesizing heavy elements via the rapid neutron-capture process ($r$-process) [1, 2, 3], which powers electromagnetic transients in optical and infrared wavelengths, the so-called kilonovae [4, 5, 6, 7]. The first detected GW emission from a binary neutron star merger event by the LIGO and Virgo Collaborations (GW170817) together with multi-wavelength electromagnetic observations have confirmed theoretical predictions [8, 9, 10]. Future observations, like the one of GW170817, will be able to offer further opportunities to precisely determine the population of binary NS systems and the yet-uncertain rich physics involved in binary NS mergers, including the nuclear equation of state (EoS) and the properties of neutron-rich nuclei. In order to achieve these goals, solid theoretical modeling of mergers is needed without any doubt. The interaction of neutrinos with matter and their flavor conversions in the binary NS merger environment are among the most uncertain theoretical aspects that can affect the observables. A copious amount of neutrinos and antineutrinos can be produced by the merger as matter is heated up to several tens of MeV due to the collision of two NSs. Neutrinos play an important role in determining the cooling of the merger remnant, changing the composition of the ejecta, and altering the $r$-process outcome and the kilonova emission properties. The early-time blue color of the GW170817 kilonova and the recently inferred amount of strontium production [11] both suggest that the merger ejecta contain some less neutron-rich material with electron fraction per nucleon $Y_{e}\gtrsim 0.3$. On the other hand, numerical simulations which include weak interactions of neutrinos with nucleons all suggest that neutrino emission and absorption have the effect to reduce the neutron-richness of the outflow launched at different post-merger phases, preferably in the direction perpendicular to the merger plane [12, 13, 14, 15, 16, 17, 18, 19, 20]. Meanwhile, the $\nu_{e}\bar{\nu}_{e}$ pair annihilation to $e^{+}e^{-}$ pairs above the accretion disk, although it might not be the dominant driver, can also contribute to launch the sGRB jet [2, 21, 22, 23, 24] Neutrinos additionally undergo flavor conversions above the merger remnant, altering the neutrino absorption rates on nucleons as well as their pair-annhilation rates [25, 26, 27, 28, 29, 30, 31], thus possibly affecting the interpretation of the observed signals. In particular, Ref. [30] showed that favorable conditions for the so-called “fast flavor conversion” [32, 33, 34, 35, 36, 37, 38, 39, 40] exist nearly everywhere above the merger remnant because of the disk geometry and the protonization of the merger remnant in its effort to reach a new beta-equilibrium state for the high temperatures produced in the merger process. Fast pairwise conversions can give rise to rapid flavor oscillations of neutrinos within a length scale of $\sim(G_{\text{F}}|n_{\nu_{e}}-n_{\bar{\nu}_{e}}|)^{-1}$ $\approx\mathcal{O}(1)$ cm. Subsequently, Ref. [31] adopted time-dependent neutrino emission characteristics from simulations of merger remnants consisting of a central BH with an accretion disk and also found favorable conditions for fast flavor conversion. By assuming full flavor equilibration between all neutrino flavors, it was shown that the nucleosynthesis outcome in the neutrino-driven outflow from the BH–disk remnant can be largely altered [31] In this work, we focus on the first $\simeq 10$ ms after the merger of two NSs, during which a central hypermassive NS surrounded by an accretion disk forms. By relying on recent hydrodynamical simulations with two different EoS and two different mass ratios for the binary, developed in Refs. [41, 42] and available at [43], we explore the neutrino emission properties, the conditions for the occurrence of fast flavor instabilities, and the effects of neutrino absorption and flavor conversions on the neutron-richness of the ejecta as well as the nucleosynthesis outcome. The paper is organized as follows. In Sec. II, we introduce the merger remnant models and the neutrino emission properties. In Sec. III, we outline the framework used for the linear stability analysis and present our numerical results on the occurrence of fast flavor conversions. We analyze the ejecta properties, the neutrino absorption effects on the evolution of $Y_{e}$, and the impact of flavor equipartition in Sec. IV. We summarize our findings and discuss their implications in Sec. V. We adopt natural units with $\hbar=c=1$ for all equations throughout the paper. II Neutrino emission from binary neutron star mergers II.1 Models of binary neutron star mergers We consider models of binary NS mergers simulated by a three-dimensional relativistic smoothed particle hydrodynamics code that adopts conformal flatness conditions [44]. The code has recently been coupled to an approximate neutrino transport scheme called “Improved-leakage-equilibration-absorption scheme (ILEAS),” which is implemented on a three-dimensional (3D) Cartesian coordinate grid ($x,y,z$) with the axis perpendicular to the merging plane chosen as $z-$axis. ILEAS serves as an efficient transport method for multidimensional simulations; when compared to two-moment neutrino transport results for protoneutron stars and post-merger tori, it captures well neutrino energy losses from the densest regions of the system as well as neutrino absorption in the free-streaming regime [42, 41]. Our fiducial models discussed in this section are from Ref. [42] and simulate the mergers of two non-rotating NSs with mass 1.35 $M_{\odot}$ each, with the EoS of DD2 [45, 46] and SFHo [47]. In addition, we consider in later sections cases of unequal mass binaries consisting of two NSs with 1.25 $M_{\odot}$ and 1.45 $M_{\odot}$ each, based on the same numerical scheme of Ref. [42, 41]. We note here that we have taken all simulation data averaged over the azimuthal angle. This is a fairly good assumption as the merger remnant quickly reaches an approximately axi-symmetric state after $\sim 2-3$ ms. Figure 1 shows the evolution of the energy luminosity of different neutrino species and their average energy during the first 10 ms after the merger of two NSs for our symmetric mergers 111More precisely, the time is measured with respect to the first minimum of the lapse function (see Ref. [42]).. During this time, the central object is a hypermassive NS supported by differential rotation. The energy luminosity of $\bar{\nu}_{e}$ ($L_{\bar{\nu}_{e}}\simeq 10^{53}$ erg s${}^{-1}$) is a factor of 2 or 3 larger than the one of $\nu_{e}$ throughout the whole 10 ms, leading to continuous protonization of the remnant. The energy luminosities of the heavy-lepton neutrino flavors, denoted by $\nu_{x}$ (with $\nu_{x}$ being representative of one species of heavy-lepton neutrinos), are $\sim 5\times 10^{52}$ erg s${}^{-1}$ per species. The averaged energies estimated by the leakage approximation show a clear hierarchy of $\langle E_{\nu_{e}}\rangle<\langle E_{\bar{\nu}_{e}}\rangle<\langle E_{\nu_{x}}\rangle$, reflecting the ordering of the temperature at decoupling; neutrinos that decouple in the innermost region of the remnant have higher average energy (see below and Fig. 2). When comparing the energy luminosity evolution of the models with DD2 and SFHo EoS, one sees that the latter produces higher luminosities in all flavors. This is related to the fact that the SFHo EoS is softer than the DD2 EoS, which results in a NS with smaller radius. Consequently, a more violent collision during the merging leads to higher temperatures, and correspondingly higher neutrino emission rates, when the SFHo EoS was adopted. This gives rise to a faster protonization of the remnant. Comparing Fig. 1 to Fig. 2 of Ref. [31], where the latter displays the neutrino emission properties for a BH remnant, one can see that the neutrino emission properties of the electron-flavor neutrinos are comparable in the two scenarios. However, the non-electron-flavor neutrinos are more abundant in the models investigated in this work and have average energies higher than the electron-flavor neutrinos. In addition, the neutrino energy luminosities in the case with the BH accretion disk quickly decrease after $\simeq 20$ ms (see Fig. 2 of Ref. [31]). Although we only focus on the first 10 ms after the coalescence, the neutrino energy luminosity reaches a plateau in the models where a hypermassive NS forms. The first three rows in Fig. 2 and Fig. 3 show the baryon density, temperature, and electron fraction profiles in the half $x-z$ plane for $x>0$ at 2.5, 5, 7.5, and 10 ms post merger for the models with DD2 and SFHO EoS, respectively. Also shown are the locations of the $\nu_{e}$, $\bar{\nu}_{e}$, and $\nu_{x}$ emission surfaces. The emission surface for a given neutrino species $\nu_{\alpha}$ ($\nu_{e}$, $\bar{\nu}_{e}$, or $\nu_{x}$) is defined by a surface where the energy-averaged optical depth is $\tau_{\nu_{\alpha}}=2/3$ (see Sec. II.2 for details). One can see from the density profiles that the remnant consists of a rotating deformed hypermassive NS surrounded by an accretion disk. The central object retains its initial low $Y_{e}\lesssim 0.1$ (see the third rows) and it is surrounded by a dense and neutron-rich disk, which is opaque to neutrinos for an extended radius out to $\sim 40-60$ km. Both $\nu_{e}$ and $\bar{\nu}_{e}$ decouple at locations where the matter density is approximately between $10^{11}-10^{12}$ g cm${}^{-3}$ and the temperature is $\sim 5$ MeV. The $\bar{\nu}_{e}$ emission surfaces generally sit inside the ones of $\nu_{e}$ during this period, independently of the adopted EoS. The size of the neutrino emission surfaces slightly expands as the remnant evolves, due to the settling of the post-merger object and the redistribution of matter with high angular momentum towards the equatorial plane, where the disk formation process proceeds. Moreover, as the remnant keeps protonizing, $Y_{e}$ inside the neutrino surfaces gradually increases with time. The thick disk of the remnant in the model with the softer SFHo EoS protonizes faster as discussed above, and thus has higher $Y_{e}$ inside the disk when compared to the profiles with DD2 EoS. Note that in the polar region close to the $z$-axis, a high $Y_{e}\gtrsim 0.4$ funnel forms at later times, as both the differences between the $\nu_{e}$ and $\bar{\nu}_{e}$ luminosities and average energies become smaller. In the bottom panels of both Fig. 2 and Fig. 3, we show the ratio of the difference of the number densities of $\bar{\nu}_{e}$ and $\nu_{e}$ to the sum of the $\bar{\nu}_{e}$ and $\nu_{e}$ densities, i.e., $(n_{\bar{\nu}_{e}}-n_{\nu_{e}})/(n_{\bar{\nu}_{e}}+n_{\bar{\nu}_{e}})$. Once again, as the remnant is protonizing and emitting more $\bar{\nu}_{e}$ than $\nu_{e}$, nearly any location above the $\nu_{e}$ surface in both models has this ratio larger than zero during the entire first 10 ms. The only exception is represented by the small patches in the polar region at 7.5 and 10 ms for the model with DD2 EoS. These patches are a consequence of the neutronization that takes place locally around the poles of the high-density core of the merger remnant in the DD2 case. Because the stiff EoS prevents the merger core from further contraction, the polar regions cool quickly, and the density just inside the neutrinospheres increases by gravitational settling, forcing the neutrinospheres to move inward to smaller radii. This explains the more pronounced polar trough of the neutrinospheres in the DD2 model compared to the SFHo merger. Striving for a new beta-equilibrium state, now at lower temperature, the plasma begins to neutronize again, radiating more electron neutrinos than antineutrinos in both polar directions. This leads to the excess of $\nu_{e}$ relative to $\bar{\nu}_{e}$ outside the neutrinospheres, visible as the two blue patches in the two bottom right panels of Fig. 2. Correspondingly, $\nu_{e}$ captures dominate $\bar{\nu}_{e}$ captures in this region and the polar outflow becomes more and more proton-rich (red regions of $Y_{e}>0.5$ around the $z$-axis in the right panels of the third row of Fig. 2). The same trends are visible in the SFHo simulation (Fig. 3), though less extreme and less rapidly evolving than in the DD2 run. II.2 Neutrino number densities on their emission surfaces The inner regions of the merger remnant are dense enough to trap neutrinos. This allows to define a neutrino emission surface for each species $\nu_{\alpha}$ above which $\nu_{\alpha}$ can approximately free-stream. As we will use the properties of $\nu_{e}$ and $\bar{\nu}_{e}$ on their respective surfaces to construct their angular distributions outside the $\nu_{e}$ surface in Sec. III, we discuss below the time evolution and the dependence on the adopted EoS of the neutrino densities on their emission surfaces. For any point $\bm{x}$ inside the simulation domain, the optical depth along a specific path $\gamma$ to another point $\bm{y}$ that a neutrino with an energy $E$ traverses is given by $$\tau_{\nu_{\alpha}}(E,\bm{x},\gamma)=\int_{\bm{x}}^{\bm{y}}\lambda^{-1}_{\nu_{% \alpha}}(E,\bm{x}^{\prime})ds,$$ (1) where $\bm{x}^{\prime}$ is a point, $ds$ is the differential segment along $\gamma$, and $\lambda_{\nu_{\alpha}}(E,\bm{x}^{\prime}(s))$ is the corresponding mean-free-path at $\bm{x}^{\prime}$. For each species $\alpha$, we determine the position of the neutrino decoupling surface in the same way as in Ref. [42], i.e. for every point $\bm{x}$, the minimum of the spectral-averaged $\langle\tau_{\nu_{\alpha}}(\bm{x},i)\rangle$ is computed along the six different directions ($i\in(\pm x,\pm y,\pm z)$) through the edge of the simulation domain. The neutrino decoupling surface is then defined by the location corresponding to $\langle\tau_{\nu_{\alpha}}(\bm{x},i)\rangle=2/3$ in the six directions. The emission surfaces computed in this way for $\nu_{e}$, $\bar{\nu}_{e}$, and $\nu_{x}$ are the ones shown in Figs. 2 and 3. Figure 4 shows the number densities of $\nu_{e}$, $\bar{\nu}_{e}$, and $\nu_{x}$ at their respective emission surfaces as functions of $x$ for different snapshots. Note that the $\nu_{e}$ and $\bar{\nu}_{e}$ number densities are computed by using Fermi-Dirac distribution functions with temperature and chemical potential extracted at each location on the neutrino surfaces, consistently with Ref. [42]. For $\nu_{x}$, we rescale the number density according to the procedure described in Appendix A to account for the trapping effect between the energy-surfaces and the emission surfaces, which is known to considerably reduce the $\nu_{x}$ luminosity when compared to the analogous values directly estimated through the local emission (see, e.g., [48]). The top panels show the neutrino density corresponding to the DD2 EoS, while the bottom panels display the same quantities for the SFHo EoS. As discussed in Sec. II.1, the softer SFHo EoS leads to a generally higher temperature in the merger remnant; as a consequence, the $\bar{\nu}_{e}$ luminosity (Fig. 1) and number density on the emission surface are higher than the one obtained in the model with DD2 EoS. Both $n_{\nu_{e}}$ and $n_{\bar{\nu}_{e}}$ are larger in a region close to the pole where the temperature is higher (see Figs. 2 and 3). Of relevance to the occurrence of fast flavor conversions is the fact that the $\bar{\nu}_{e}$ emission is a factor of 2–3 larger than the one of $\nu_{e}$ across the whole emission surface; it remains quite stable throughout the simulated evolution until 10 ms after the plunge. The only exception is the snapshot at 7.5 ms for the model with DD2 EoS, which shows a dip in the $\bar{\nu}_{e}$ number density around $x\simeq 10$ km. This is because the enhanced deleptonization above the poles of the high-density core (described in Sec. II.1) leads to a short, transient excess of $\nu_{e}$ over $\bar{\nu}_{e}$ near the neutrinospheres in the northern hemisphere (see Fig. 2). The effect is somewhat pathological and unusual, also because it is considerably less strongly developed in the southern hemisphere. Notably, comparing Fig. 4 with the bottom panel of Fig. 1 of Ref. [31], one can see that the neutrino-antineutrino asymmetry is more pronounced in the present models than in the BH remnant case. III flavor Instability In Sec. III.1 we first briefly introduce the theoretical formalism, viz., the dispersion relation (DR) approach [35], widely used in the literature to investigate the occurrence of neutrino flavor conversions. In Sec. III.2 we look for the conditions leading to flavor instabilities using the simulation data introduced in Sec. II. We then apply the DR formalism to investigate the occurrence of flavor conversions above the neutrino emission surfaces in our merger remnant models in Sec. III.3. III.1 Dispersion relation formalism We adopt the density matrix formalism to describe the statistical properties of the neutrino dense gas incorporating flavor mixing [49]. For a given density matrix $\varrho(\bm{p},\bm{x},t)$, its diagonal elements in the flavor basis, $\varrho_{\alpha\alpha}$, record the phase-space distributions $f_{\nu_{\alpha}}$ of a given neutrino flavor $\nu_{\alpha}$ at the space-time location $(t,\bm{x})$ and with momentum $\bm{p}$. The off-diagonal terms $\varrho_{\alpha\beta}$ carry the information about the neutrino mixing (neutrino flavor correlations). In the absence of any mixing, i.e., all neutrinos are in their flavor eigenstates, the off-diagonal elements vanish. Neglecting general-relativistic effects and collisions of neutrinos with matter, the space-time evolution of the density matrix $\varrho(\bm{p},\bm{x},t)$ is governed by a Liouville equation $$\partial_{t}\varrho(\bm{p},\bm{x},t)+\bm{v_{p}}\cdot\nabla\varrho(\bm{p},\bm{x% },t)=-i[\Omega(\bm{p},\bm{x},t),\varrho(\bm{p},\bm{x},t)],$$ (2) where $\Omega$ is the Hamiltonian that accounts for the flavor oscillations of neutrinos. On the left hand side of Eq. (2), the first term takes care of the explicit time dependence of $\varrho$ and the second term takes into account the neutrino propagation with velocity $\bm{v_{p}}\simeq{\bm{p}}/|\bm{p}|$ for ultrarelavistic neutrinos. The Hamiltonian matrix $\Omega$ on the right hand side can be decomposed as $$\Omega(\bm{p},\bm{x},t)=\Omega_{\text{vac}}+\Omega_{\text{MSW}}+\Omega_{\nu\nu},$$ (3) where the first term $\Omega_{\text{vac}}$ takes into account flavor conversions in vacuum. In a simplified two-flavor scenario, $\Omega_{\text{vac}}=\text{diag}(\omega_{\rm v}/2,-\omega_{\rm v}/2)$ in the mass basis with $\omega_{\rm v}=(m_{2}^{2}-m_{1}^{2})/2E$ being the vacuum oscillation frequency of the neutrinos with energy $E$. The second term on the right hand side of Eq. (3) embodies the effects of neutrino coherent forward scattering with electrons and nucleons. In the flavor basis, this term can be expressed as $$\Omega_{\text{MSW}}=(\sqrt{2}G_{\text{F}}n_{e})\text{diag}(1,0),$$ (4) where $n_{e}$ is the net electron number density. The last term in Eq. (3) is the effective Hamiltonian due to the $\nu$–$\nu$ interaction. For a neutrino traveling with momentum $\bm{p}$, $\Omega_{\nu\nu}$ is given by $$\Omega_{\nu\nu}=\sqrt{2}G_{\text{F}}\int\frac{d^{3}\bm{q}}{(2\pi^{3})}(1-\bm{v% _{p}\cdot v_{q}})(\varrho(\bm{q},\bm{x},t)-\bar{\varrho}(\bm{q},\bm{x},t)),$$ (5) where $\bar{\varrho}$ is the corresponding density matrix for antineutrinos. The presence of $(1-\bm{v_{p}\cdot v_{q}})$ in Eq. (5) leads to multi-angle effects, i.e., neutrinos propagating in different directions experience different $\Omega_{\nu\nu}$. The equation of motion for antineutrinos can be obtained in a similar fashion, by replacing $\omega_{\rm v}$ by $-\omega_{\rm v}$ in $\Omega_{\rm vac}$. We focus on a simplified system that deals with two neutrino flavors. Under this assumption, both the density matrix $\varrho$ and the Hamiltonian $\Omega$ are $2\times 2$ Hermitian matrices and hence can be expanded in terms of the identity matrix and three Pauli matrices. Thus, we write $\varrho=[(f_{\nu_{e}}+f_{\nu_{x}})+(f_{\nu_{e}}-f_{\nu_{x}})\xi]/2$ for neutrinos and $\bar{\varrho}=-[(f_{\bar{\nu}_{e}}+f_{\bar{\nu}_{x}})+(f_{\bar{\nu}_{e}}-f_{% \bar{\nu}_{x}})\xi^{*}]/2$ for antineutrinos. The entity $\xi$ is a matrix defined as $$\xi=\begin{pmatrix}s&&S\\ S^{*}&&-s\end{pmatrix},$$ (6) where $-1\leq s\leq 1$ and $|s|^{2}+|S|^{2}$ =1. In the absence of any flavor correlation $S=0$. Furthermore, as in previous work that studied the fast neutrino flavor conversion, we omit the vacuum oscillation term in the following discussion as $\omega_{\rm v}$ marginally affects the linear regime [50, 51, 52]. With these assumptions and introducing the metric tensor $\eta^{\mu\nu}=\text{diag}(1,~{}-1,~{}-1,~{}-1)$ and for any contra-variant vector $A^{\mu}$, $A_{\mu}=\eta_{\mu\nu}A^{\nu}$, we can recast the Hamiltonian defined in Eq. (3) into the following form $$\Omega=v^{\mu}\lambda_{\mu}\frac{\sigma_{3}}{2}+\int d\bm{\Gamma}^{\prime}v^{% \mu}v_{\mu}^{\prime}\xi(\bm{v}^{\prime})g(\bm{v}^{\prime}),$$ (7) where $\bm{v}=(\sin\theta\cos\phi,~{}\sin\theta\sin\phi,~{}\cos\theta)$ with velocity $v^{\mu}=(1,\bm{v})$, $d\bm{\Gamma}=\sin\theta d\theta d\phi$ and $\lambda^{\mu}=\sqrt{2}G_{\text{F}}n_{e}(1,\bm{v}_{\text{m}})$ with $\bm{v}_{\text{m}}$ being the vector of the fluid velocity of the background matter. Since the rate of pairwise conversion is much faster than any other inverse time scale involved in the problem, we treat the background matter as stationary and homogeneous: $\lambda_{\mu}v^{\mu}=\lambda_{0}$. The quantity $g(\bm{v})$ is related to the angular distribution of the neutrino ELN angular distribution $$g(\bm{v})=\sqrt{2}G_{\text{F}}(\bm{\Phi}_{\nu_{e}}-\bm{\Phi}_{\bar{\nu}_{e}}),$$ (8) where $\bm{\Phi}_{\nu_{\alpha}}=dn_{\nu_{\alpha}}/d\bm{\Gamma}$. To study the growth of $S$ in the linear regime, we treat the flavor correlation $S$ as a perturbation and neglect all terms of $O(S^{2})$ or higher. Taking $S(t,\bm{x})=Q(\omega,\bm{k})e^{-i(\omega t-\bm{k}\cdot\bm{x})}$, the EoM becomes $$v_{\mu}\bar{\lambda}^{\mu}Q(\omega,\bm{k})=-\int d\bm{\Gamma}^{\prime}v^{\mu}v% _{\mu}^{\prime}g(\bm{v}^{\prime})Q(\omega,\bm{k}).$$ (9) In the above Eq. (9) we have introduced the four vector $\bar{\lambda}^{\mu}=(\omega-\lambda_{0}-\epsilon_{0},\bm{k}-\bm{\epsilon})$, $\epsilon_{0}\equiv\int d\bm{\Gamma}g(\bm{v})$ and $\bm{\epsilon}\equiv\int d\bm{\Gamma}\bm{v}g(\bm{v})$. Inspecting Eq. (9), one can make the ansatz $Q(\omega,\bm{k})=v_{\mu}a^{\mu}/v_{\mu}\bar{\lambda}^{\mu}$, with $a^{\mu}$ being the coefficients of eigenfunction solutions. Thus, Eq. (9) becomes $$v_{\mu}\Pi^{\mu\nu}a_{\nu}=0,$$ (10) where we have used the definition $$\Pi^{\mu\nu}\equiv\eta^{\mu\nu}+\int d\bm{\Gamma}g(\bm{v})\frac{v^{\mu}v^{\nu}% }{v_{\sigma}\bar{\lambda}^{\sigma}}.$$ (11) The EoM defined in Eq. (10) holds for any $v^{\mu}$. Thus, we have the condition $\Pi^{\mu\nu}a_{\nu}=0$. Eigenfunctions of the latter have non-trivial solution only if $\Pi^{\mu\nu}$ satisfies the condition $$\text{det}[\Pi^{\mu\nu}(\omega,\bm{k})]=0.$$ (12) Equation (12) is the DR in flavor space. The solutions of the DR have been classified into several types [36, 38]. If $\omega$ is real for real values of $\bm{k}$, a perturbation in $S$ only propagates without growing or damping, i.e., it stays in the linear regime, meaning that no significant flavor conversion occurs. On the other hand, an imaginary solution of $\omega$ with $\text{Im}(\omega)>0$ corresponds to exponentially growing modes. In other words, the flavor correlation $|S|$ grows exponentially with time, leading to significant flavor conversion. Rigorous studies have been carried out to understand the characteristics of the above DR with respect to the ELN angular distribution of neutrinos [35, 36, 38]. It was shown that in the presence of a crossing in the ELN distribution, the DR will yield complex $\omega$ solutions for real $\bm{k}$, leading to temporal instabilities. In the following, we examine the ELN distributions above the merger remnants and the corresponding flavor instabilities. III.2 Neutrino electron lepton number angular distribution To construct the ELN distribution above the merger remnant, we follow a method similar to the one adopted in Ref. [31]. First, we assume that both $\nu_{e}$ and $\bar{\nu}_{e}$ freely propagate outside their respective emission surfaces defined in Sec. II.2. Second, we approximate their forward-peaked angular distributions at each point on the emission surfaces as $${\bm{\Phi}}_{\nu_{e},\bar{\nu}_{e}}(\theta_{n})=\frac{n_{\nu_{e},\bar{\nu}_{e}% }}{4\pi}(1+\cos\theta_{n}),$$ (13) where $\theta_{n}$ is the angle with respect to the normal direction of the location on the emission surface. Ignoring the minor effect of GR bending, we can then ray-trace the neutrino intensities from the emission surfaces to obtain their angular distributions at any location above the surfaces. Figures 5 and 6 show the obtained ELN distributions as a function of the local angular variables $\theta$ (angle with respect to the $z$-axis) and $\phi$ (angle with respect to the $x$-axis on the $x$-$y$ plane) at selected locations above the $\nu_{e}$ surface at 2.5, 5, 7.5 and 10 ms after the coalescence, for the $1.35+1.35$ $M_{\odot}$ merger models with DD2 and SFHo EoS, respectively. Here we only show the sign of the ELN distribution to highlight the crossing. The blue shade represents the region where the net ELN is positive while the red region corresponds to negative ELN. The top panel shows the ELN distribution at a representative point on the $z$ axis and the middle and bottom panels show the ELN distributions at near-center and outer regions above the $\nu_{e}$ surface. Note here that Figs. 5 and 6 only show the ELN distribution for $\phi/\pi\geq 0$ as the distributions for axi-symmetric emission surfaces possess reflection symmetry, $\Phi_{\nu_{e},\bar{\nu}_{e}}(\cos\theta,\phi)=\Phi_{\nu_{e},\bar{\nu}_{e}}(% \cos\theta,-\phi)$. The angular coverage of the $\nu_{e}$ and $\bar{\nu}_{e}$ fluxes at a point ($x,z$) is determined by the geometry of the $\nu_{e}$ and $\bar{\nu}_{e}$ emission surfaces. On the other hand, the ELN angular distribution is determined by the combination of the emission surface geometries and their respective emission properties, including the relative strength of $\nu_{e}$ and $\bar{\nu}_{e}$ and the angular dependence of the emission. For example, the ELN angular distribution at any point on the $z$ axis above the emission surface is independent of $\phi$, reflecting the (assumed) rotational symmetry of the merger remnants about the $z$ axis. As we move away from the $z$ axis, the ELN distribution becomes dependent on $\phi$ (second and third panels). For both EoS, the angular coverage in $\theta$ slightly increases with time for a given $(x,z)$, caused by the expansion of the emission surfaces. This is different from what was found in Ref. [31] where the central remnant is a BH for which the size of the $\nu_{e}/\bar{\nu}_{e}$ emitting torus surfaces shrinks with time. As $\bar{\nu}_{e}$ are more abundant than $\nu_{e}$ in most parts of the emission region during the first 10 ms (see Fig. 4), the overall shapes of the ELN crossings are qualitatively similar. The only exception is represented by the snapshot at 7.5 ms for the case with DD2 EoS, for which the inner region above the merger remnant shows a double ELN crossing structure, see the left and middle panels in the third row. This is related to the dip of the $\bar{\nu}_{e}$ density on the $\bar{\nu}_{e}$ surface at $\sim 10$ km discussed in Sec. II.2. Also, the ELN angular distribution in Fig. 6 is more extended towards $\theta=0$ compared to Fig. 5, resulting from slightly larger radii of the neutrino emission surfaces in the SFHo model. For the simulations adopting $1.25+1.45$ $M_{\odot}$ binaries, we have similarly checked the ELN distributions during the same time snapshots for both EoS. Unsurprisingly, ELN crossings appear at all times as those shown here. III.3 Flavor instabilities for fast pairwise conversion After obtaining the ELN distributions above the neutrino emitting surface, we numerically solve the DR [Eq. (12)] starting from the outer neutrino surface to inspect whether solutions containing non-zero $\rm{Im}(\omega)$ for a given $\bm{k}$ can be found222Note that the positive and negative $\rm{Im}(\omega)$ solutions always appear together.. As the ELN distributions above the merger remnants with axial-symmetry have a reflection symmetry with respect to $\phi\rightarrow-\phi$, one can obtain two different solutions that correspond to the reflection symmetry-preserving and symmetry-breaking cases [30]. We show the obtained $|\text{Im}(\omega)|$ as a function of $k_{z}$, taking $k_{x}=k_{y}=0$, at different times for a location at $(x,z)=(10,25)$ km above the emitting surfaces in Fig. 7 and the solutions for different locations corresponding to the ELN crossing shown in Fig. 8 at 5 ms. The top (bottom) panels are for cases with DD2 (SFHo) EoS while the left (right) panels show the symmetry-preserving (symmetry-breaking) solutions. Flavor instabilities with growth rate of $\mathcal{O}(1)$ cm${}^{-1}$ exist at all locations at all times for a large range of $k_{z}$. The symmetry preserving solution of the DR has two branches while the symmetry breaking solution has only one branch, similar to what was found in Ref. [30]. Comparing the results for the models with DD2 and SHFo EoS, the growth rate of the flavor instability is generally larger in the latter, as the neutrino emission is stronger with the SHFo EoS. The shape of the solutions is rather stable over time for the case with SFHo EoS. On the other hand, the model with DD2 EoS shows a somewhat stronger time-dependence. In particular, the range of $k_{z}$ that leads to non-zero $|{\rm Im}(\omega)|$ as well as the value of $|{\rm Im}(\omega)|$ both decrease at the snapshot of 7.5 ms, when the double crossing shape of the ELN appears (see Fig. 5). By comparing the solutions at different locations for the $t=5$ ms snapshot, Fig. 8 shows that $|{\rm Im}(\omega)|$ is larger closer to the $z$-axis for both the symmetry-preserving and symmetry-breaking solutions. This, again, is caused by the fact that $\Phi_{\nu_{e}}$ and $\Phi_{\bar{\nu}_{e}}$ are largest in this region (see Fig. 4). As shown in Figs. 7 and 8, the symmetry-breaking solutions at all times and all locations contain non-zero $|\text{Im}(\omega)|$ for the mode $\bm{k}=0$. We further show in Fig. 9 the contour plot of $|\text{Im}(\omega)|$ above the neutrino surface for this mode. The plots in this figure clearly show that growth rates of the instability of $\sim{\cal O}(1)$ cm${}^{-1}$ exist everywhere above the remnant at all times. The region closer to the $z$-axis just above the emission surface has the maximal growth rates. Moving away from the surface, the neutrino flux is suppressed by the geometric effects and the magnitude of the instability decreases in all cases. For the sake of comparison with the existing literature, we find $|\rm{Im}(\omega)/\mu|\sim\mathcal{O}(10^{-2})$ for all cases examined here; this value of the growth rate is similar albeit a bit smaller than the ones reported in Ref. [31, 53]. The growth rates $|\rm{Im}(\omega)/\mu|$ are generally larger towards the middle region above the $\nu_{e}$ surface, in agreement with the findings of Ref. [30, 31]. This is due to the relative strength of the positive vs. negative ELN strength, in the proximity of crossings between the $\nu_{e}$ and $\bar{\nu}_{e}$ angular distributions [54]. For a system that is more dominated by $\nu_{e}$ or $\bar{\nu}_{e}$, i.e., where the positive ELN distribution dominates the negative parts or the other way around, one expects that the value of $|\rm{Im}(\omega)/\mu|$ is smaller than in a more balanced system with similar positive and negative parts of the ELN distribution (see e.g., Ref. [38]). Looking at the ELN crossing pattern shown in Figs. 5 and 6, the locations above the middle part of the remnant have large angular area of $g(\bm{v})>0$ due to the larger separation of the $\nu_{e}$ and $\bar{\nu}_{e}$ surfaces (see Fig. 2 and 3). This, in turn, enhances the corresponding values of $|\rm{Im}(\omega)/\mu|$ relative to the ones in the inner region closer to the $z$-axis. Likewise, when comparing the values of $|\rm{Im}(\omega)/\mu|$ to the ones shown in Ref. [31], the more dominant $\bar{\nu}_{e}$ emission relative to $\nu_{e}$ here, together with the smaller separation of their emission surfaces, results in smaller values of $|\rm{Im}(\omega)/\mu|$. On the other hand, while Ref. [31] showed that the region where the flavor instability exists shrinks on a time scale of $\sim\mathcal{O}(10)$ ms as the BH-disk remnant evolves, the instability region found here remains stable within the examined 10 ms of post-merger evolution. This can have important consequences for the growth of the instabilities and seems to favor the eventual development of flavor conversions in the non-linear regime. The effect of advection hindering the growth of the flavor instabilities for systems with non-sustained and fluctuating unstable conditions shown by Ref. [54] may therefore not happen above the merger remnant, as shown in Ref. [53]. In the case of the models with unequal mass binaries, due to the similar ELN crossing features, we find unstable regions above the merger remnant disk and growth rates very similar to the symmetric models (results not shown here). We here focus on the diagnostics of flavor instabilities, but do not distinguish whether they belong to the convective or absolute type (see Refs. [55, 38]). This would require solving the full dispersion relation for the complete $(\omega,\mathbf{k})$ space. IV Merger ejecta and nucleosynthesis of the heavy elements In this section, we first analyze the properties of the material ejected during the first $\simeq 10$ ms post merger, examine how neutrino absorption affects the evolution of $Y_{e}$ of the outflow material, and discuss the nucleosynthesis outcome in the absence of neutrino flavor conversion in Sec. IV.1. In Sec. IV.2, we further explore the potential effect of fast flavor conversion on the neutrino absorption rates and the outcome of nucleosynthesis of in these ejecta. IV.1 Ejecta properties and nucleosynthesis The dynamical ejecta masses extracted at 10 ms post-merger for the $1.35+1.35$ $M_{\odot}$ merger simulations are $\sim 2.0\times 10^{-3}$ $M_{\odot}$ and $3.3\times 10^{-3}$ $M_{\odot}$ with the DD2 and SFHo EoS, represented by 783 and 1263 tracer particles, respectively. For the $1.25+1.45$ $M_{\odot}$ merger cases, the dynamical ejecta masses are $\sim 3.2\times 10^{-3}$ $M_{\odot}$ (DD2) and $8.7\times 10^{-3}$ $M_{\odot}$ (SFHo), represented by 1290 and 4398 tracer particles. In computing the evolution of $Y_{e}$ for a given tracer particle, we first post-process the neutrino emission data from Ref. [42] to calculate the absorption rates of $\nu_{e}$ and $\bar{\nu}_{e}$ on neutrons and protons, $\lambda^{0}_{\nu_{e}}$ and $\lambda^{0}_{\bar{\nu}_{e}}$ (the superscript $0$ here denotes the case where any neutrino flavor conversion is omitted), as detailed in Appendices A and B. These rates are then combined with the nuclear reaction network used in Refs. [56, 57] to compute the nucleosynthesis yields in the merger ejecta. For each tracer particle, we begin the network calculation either at a location where $T=50$ GK or just outside the disk with height of 25 km and radius of 55 km to make sure that it is outside the $\nu_{e}$ emission surface. The initial nuclear abundances are calculated by using the nuclear statistical equilibrium (NSE) condition. For $T>10$ GK, we only compute the weak reaction rates to track the $Y_{e}$ evolution while assuming NSE at each moment to obtain the abundances. When $T\leq 10$ GK, we instead follow the full evolution of all nuclear species and include the feedback due to the nuclear energy release on the ejecta temperature following Ref. [58]. In Fig. 10, we show the distributions of the ejecta mass for the $1.35+1.35$ $M_{\odot}$ models as a function of the time when the ejecta reach the radius $r=100$ km at $t(r=100~{}\rm{km})$ in panel (a), the angle $\theta_{\rm ej}$ (relative to the $z$-axis) when they leave the simulation domain in panel (b), and the radial velocity $v_{r}$ at $r=100$ km in panel (c). We also show, in panel (d), the distributions of $Y_{e}$ for the ejecta at $r=100$ km, in the absence of flavor mixing. Additionally, panels (e) and (g) [(f) and (h)] show the distributions of each tracer particle in the plane spanned by $t(r=100~{}\rm{km})$ [$v_{r}(r=100)~{}\rm{km}$] and $\theta_{\rm ej}$ with the corresponding $Y_{e}$ at 100 km labeled in color for the DD2 and SHFo EoS, respectively. These plots show that, independently of the adopted EoS, most of the material is ejected within the first $\sim 2$ ms with lower $Y_{e}\lesssim 0.3$, within $\sim 45^{\circ}$ away from the mid-plane. The ejecta launched in the second episode [$t(r=100~{}\rm{km})\simeq 3-4$ ms] are similarly distributed closer to the equatorial plane and have $Y_{e}\simeq 0.4$. After that, neutrino irradiation continues to power the mass ejection along the polar direction with $Y_{e}$ reaching $\sim 0.5$. In terms of the ejecta kinematics, because of the weak interactions, it is clear that $Y_{e}$ can be raised up to $0.3-0.4$ mostly for the material expanding more slowly with $v_{r}/c\sim 0.1-0.2$ at $r=100$ km, even around the equatorial plane, and reach $\sim 0.5$ closer to the polar direction. The main differences between the DD2 and the SFHo EoS results are the following. First, for the simulation with SHFo EoS, the amount of ejecta is larger, particularly during the first ejection episode [see panel (a)], than that with the DD2 EoS; this is a direct consequence of the more violent collision of the two NSs in the case with softer EoS (SHFo), as discussed in Ref. [44]. Second, the amount of high-velocity ejecta with $v_{r}/c\gtrsim 0.5$ for the SFHo case is also higher for the same reason [see panels (c) and (f)]. Third, the $Y_{e}$ distribution [panel (d)] for both models shows that $Y_{e}(r=100~{}\mathrm{km})$ is slightly larger for the DD2 EoS model than for the SFHo EoS one. This reflects the higher $\bar{\nu}_{e}$ luminosity obtained in the SHFo EoS model (see Fig. 1) leading to larger $\bar{\nu}_{e}$ absorption rates on protons; hence, $Y_{e}$ is raised less for the bulk of the ejecta. The high $Y_{e}$ tail extends to values $\gtrsim 0.5$ in the DD2 EoS model, while it remains $\lesssim 0.5$ in the SFHo EoS model [see panels (c) and (e)]. Interestingly, it is noticeable that the high velocity components in the SFHo model have more high-$Y_{e}$ ejecta than in the DD2 model [see panel (f)], different from the main bulk of the ejecta. This is because these ejecta are mainly driven during the early post-merger phase. During this early phase, positron capture is responsible for raising $Y_{e}$ in the ejecta, rather than neutrino absorption. Thus, in the model with SFHo EoS, the higher post-merger temperature of the remnant due to the more violent merger leads to relatively higher $Y_{e}$ material in the early fast ejecta. We show in Fig. 11 the same quantities as in Fig. 10, but for the models with $1.25+1.45$ $M_{\odot}$ mass binaries. Qualitatively, the features are similar to the ones described above, but they are quantitatively different. For instance, the division between different episodes of mass ejection is less clear, in particular for the SHFo EoS model [see panel (a)]. On average, most of the ejecta have higher velocities, peaked at $\sim 0.25c$ [see panel (c), (f) and (h)]. Moreover, a larger fraction of the ejecta has lower $Y_{e}\lesssim 0.2$, despite the fact that similarly wide distributions of $0.01\lesssim Y_{e}\lesssim 0.5$ are obtained. Figure 12 shows the resulting abundance distributions for the ejecta of the symmetric and asymmetric merger models with the DD2 and SFHo EoS, in comparison to the re-scaled solar $r$ abundance pattern [59]. Due to the wide-spread $Y_{e}$ distribution ranging from $0.02-0.5$ in all models, all three $r$-process peaks at $A\simeq 90$, $130$, and $195$ are in relatively good agreement with the solar abundance pattern. The difference of the high $Y_{e}$ component discussed above only results in abundance variations at $A\simeq 50-80$. In particular, we compute the amount of strontium present at the time of a day, which is of relevance to the potential identification in the spectral analysis of the GW170817 kilonova [11]. The total amount of strontium is $\simeq 8.9\times 10^{-5}$ $M_{\odot}$ and $\simeq 3.9\times 10^{-5}$ $M_{\odot}$ for the DD2 and SFHo EoS models with equal mass binaries. For the asymmetric merger models, the corresponding amount of strontium is $\simeq 2.66\times 10^{-4}$ $M_{\odot}$ and $3.25\times 10^{-4}$ $M_{\odot}$, respectively. These results are consistent with the findings of Ref. [11]. References [60, 58, 61, 62] found that some merger ejecta have low $Y_{e}$ and fast expansion time scale to allow for a neutron-rich freeze-out during the $r$-process nucleosynthesis. A potentially thin layer of “neutron-skin” at the outskirt of the ejecta may possibly power an early-time UV emission at $\sim$ hours post-merger due to the radioactive heating of neutron decay [60, 7]. We find that the amount of free neutrons at the end of the $r$-process, for equal-mass binaries, is $\simeq 6.2\times 10^{-6}$ $M_{\odot}$ and $6.6\times 10^{-6}$ $M_{\odot}$ with the DD2 and SFHo EoS, respectively. These numbers are roughly a factor of 10 smaller than what was found in Ref. [58], which analyzed simulation trajectories without including the weak interactions. The reduction of the amount of free neutrons at the end of the $r$-process is related to the high post-merger temperature effect raising $Y_{e}$ even for the early fast ejecta. For the unequal mass binaries, the corresponding amount of free neutrons is $\simeq 9.2\times 10^{-6}$ $M_{\odot}$ and $3.1\times 10^{-6}$ $M_{\odot}$ for the DD2 EoS model and for the SFHo model. A future (non)identification of this component may shed light on the role of weak interactions in the post-merger environments. IV.2 Impact of flavor equipartition on $Y_{e}$ and nucleosynthesis Following previous work [31, 63], we assume that fast flavor conversions lead to conditions close to flavor equipartition for neutrinos and antineutrinos. The assumption of flavor equilibration is an extreme ansatz, especially in the light of the findings of Ref. [53], which, however, relied on a simplified model of a relic merger disk. Nevertheless, our extreme assumption for the flavor ratio is useful to explore the largest possible impact that flavor conversions might have on the nucleosynthesis of the heavy elements. The corresponding neutrino absorption rates can be approximated by $$\displaystyle\lambda^{\rm osc}_{\nu_{e}}$$ $$\displaystyle=\frac{1}{3}~{}\lambda^{0}_{\nu_{e}}+\frac{2}{3}~{}\lambda_{\nu_{% x}},$$ (14) $$\displaystyle\lambda^{\rm osc}_{\bar{\nu}_{e}}$$ $$\displaystyle=\frac{1}{3}~{}\lambda^{0}_{\bar{\nu}_{e}}+\frac{2}{3}~{}\lambda_% {\bar{\nu}_{x}},$$ (15) where $\lambda_{\nu_{x}}$ and $\lambda_{\bar{\nu}_{x}}$ are the neutrino absorption rates on free nucleons assuming that all $\nu_{x}$ and $\bar{\nu}_{x}$ are converted to $\nu_{e}$ and $\bar{\nu}_{e}$, as detailed in Appendices A and B. We then perform the same nucleosynthesis calculations as in Sec. IV.1 for all tracer particles in all the merger models by replacing $\lambda^{0}_{\nu_{e}}$ and $\lambda^{0}_{\bar{\nu}_{e}}$ by $\lambda^{\rm osc}_{\nu_{e}}$ and $\lambda^{\rm osc}_{\bar{\nu}_{e}}$. Below, we only focus on the findings for the models with equal mass and different EoS because the results obtained in the unequal mass binaries are qualitatively the same, independent of the EoS. We show in Fig. 13 the comparison of the ratio of $\lambda_{\nu_{e}}/\lambda_{\bar{\nu}_{e}}$ evaluated at $r=100$ km for all tracer particles for the cases with and without flavor equipartition. The corresponding values of $\lambda_{\bar{\nu}_{e}}$ are also shown. These figures highlight that the neutrino absorption rates are orders of magnitudes larger in the polar region than close to the equator. For the case without neutrino flavor equipartition, nearly all tracer particles have $\lambda^{0}_{\nu_{e}}/\lambda^{0}_{\bar{\nu}_{e}}\lesssim 1$, reflecting the stronger $\bar{\nu}_{e}$ flux emitted from its surface. With flavor equipartition, the nearly equal contribution of the converted $\nu_{x}$ and $\bar{\nu}_{x}$ significantly changes the ratio $\lambda^{\rm osc}_{\nu_{e}}/\lambda^{\rm osc}_{\bar{\nu}_{e}}$ for both EoS. For the case with DD2 EoS, most of the trajectories with $\theta_{\rm ej}\lesssim 60^{\circ}$ or $\theta_{\rm ej}\gtrsim 120^{\circ}$ have $\lambda^{\rm osc}_{\nu_{e}}/\lambda^{\rm osc}_{\bar{\nu}_{e}}\gtrsim 1$. On the other hand, the values of $\lambda^{\rm osc}_{\nu_{e}}/\lambda^{\rm osc}_{\bar{\nu}_{e}}$ scatter around 1 for the model with SFHo EoS. The large change in $\lambda_{\nu_{e}}/\lambda_{\bar{\nu}_{e}}$ strongly affects the $Y_{e}$ distribution of the ejecta. Figures 14(a) and 15(a) show $\Delta Y_{e}\equiv Y_{e}^{({\rm osc})}-Y_{e}^{({\rm no~{}osc})}$ calculated at $r=100$ km for each tracer particles as a function of the corresponding $\theta_{\rm ej}$, where $Y_{e}^{({\rm osc})}$ and $Y_{e}^{({\rm no~{}osc})}$ are the $Y_{e}$ values with and without flavor equipartition. These results show that the flavor equipartition can significantly increase $Y_{e}$ of the ejecta up to $\sim 0.15$ for $\theta_{\rm ej}<60^{\circ}$ or $\theta_{\rm ej}>120^{\circ}$ closer to the polar directions. In particular, the increase of $Y_{e}$ due to flavor equipartition for these ejecta is more pronounced with $Y_{e}^{\rm(no~{}osc)}\simeq 0.3-0.4$. For the ejecta with $Y_{e}^{\rm(no~{}osc)}\lesssim 0.2$ or $Y_{e}^{\rm(no~{}osc)}\simeq 0.5$, $Y_{e}$ is less affected. This is because the ejecta with $Y_{e}^{\rm(no~{}osc)}\lesssim 0.2$ expand too fast for neutrino absorption to raise $Y_{e}$ either with or without flavor conversion. On the other hand, for the ejecta with $Y_{e}^{\rm(no~{}osc)}\simeq 0.5$, $\lambda_{\nu_{e}}/\lambda_{\bar{\nu}_{e}}\simeq 1$ even without flavor conversions. As for the tracer particles with $60^{\circ}\leq\theta_{\rm ej}\leq 120^{\circ}$ closer to the disk mid-plane, $Y_{e}$ is barely influenced by flavor equipartition because of the low neutrino absorption rates (see Fig. 13). Figures 14(b)-(d) and 15(b)-(d) further show the comparison of the $Y_{e}$ distribution for the cases with and without flavor conversion, for the ejecta classified into three groups according to $\theta_{\rm ej}$: the polar ejecta with $60^{\circ}<|\theta_{\rm ej}-90^{\circ}|\leq 90^{\circ}$, the middle ejecta with $30^{\circ}<|\theta_{\rm ej}-90^{\circ}|\leq 60^{\circ}$, and the equatorial ejecta with $0^{\circ}\leq|\theta_{\rm ej}-90^{\circ}|\leq 30^{\circ}$. Flavor equipartition influences the $Y_{e}$ distribution of the polar ejecta by shifting the peak from $\sim 0.4$ to $\sim 0.55$ (0.5) for the DD2 (SFHo) model. The larger (smaller) shift of the $Y_{e}$ peak in the DD2 (SFHo) model is related to the larger (smaller) values of $\lambda^{\rm osc}_{\nu_{e}}/\lambda^{\rm osc}_{\bar{\nu}_{e}}$ shown in Fig. 13. For the middle ejecta, a fraction of ejecta originally with $0.3\lesssim Y^{(\rm no~{}osc)}_{e}\lesssim 0.4$ has $Y_{e}^{\rm(osc)}\simeq 0.4-0.5$ when flavor equipartition is reached, while the distribution with $Y_{e}\lesssim 0.3$ is barely altered. As for the equatorial ejecta, the corresponding $Y_{e}$ distribution is only affected negligibly, as expected. The impact of flavor equipartition in the neutrino-driven ejecta studied in Ref. [31] is to lower $Y_{e}$ (see Fig. 11 therein), while $Y_{e}$ increases in the models investigated in this work, as discussed above. The main difference is that flavor equipartition leads to a larger reduction of $\lambda_{\nu_{e}}$ and $\lambda_{\bar{\nu}_{e}}$ in the BH–torus case, due to the vanishingly small $\nu_{x}$ fluxes. Moreover, a significant part of the ejecta in the BH–torus case is ejected on timescales of several tens of milliseconds during which the neutrino luminosities decrease substantially (see Figs. 2 and 8 in Ref. [31]); this leads to much smaller neutrino absorption rates even without assuming neutrino flavor equipartition. As a consequence, since the ejecta start out as neutron-rich material, a largely reduced $Y_{e}$ for the neutrino-driven outflow was found in the BH–torus model of Ref. [31]. We show in Fig. 16 the impact of flavor equipartition on the abundance distribution for the polar ejecta ($60^{\circ}\leq|\theta_{\rm ej}-90^{\circ}|\leq 90^{\circ}$) for the $1.35+1.35$ $M_{\odot}$ model with both EoSs. Since flavor equipartition mainly shifts the distribution of high $Y_{e}\gtrsim 0.3$ material, noticeable changes appear regarding the iron peak and the first peak elements. In particular, the amount of produced $A=56$ nuclei is enhanced by a factor of $\sim 6(27)$ for the DD2(SFHo) EoS model. The amount of lanthanides in the polar ejecta, relevant to the kilonova color, is affected negligibly for the DD2 model and reduced by $\sim$ a factor of 2 for the SHFo model. Since the polar ejecta contribute up to $\sim 10\%$ to the total ejecta mass considered here, the overall modifications induced by flavor equipartition on the total abundance yields are relatively small. For instance, the change of the produced amount of strontium at the time of a day and the amount of free neutrons at the end of the $r$-process is at the level of $\sim 10\%$. V Summary and discussion In this work, we have examined the neutrino emission properties and the conditions for the occurrence of fast neutrino flavor conversions during the first 10 ms after the coalescence of symmetric ($1.35+1.35$ $M_{\odot}$) and asymmetric ($1.25+1.45$ $M_{\odot}$) NS binaries, during which the remnant consists of a hypermassive NS surrounded by an accretion disk. We have also performed detailed analyses regarding the properties of the material ejected during this phase and nucleosynthesis calculations for cases with and without neutrino flavor mixing. Our study is based on the outputs from the general-relativistic simulations with approximate neutrino transport, performed with two different EoS (DD2 and SHFo) based on Refs. [41, 42] and available at [43]. Flavor instabilities that may lead to fast pairwise flavor conversions occur throughout the whole investigated post-merger evolution, independent of the adopted EoS or the mass ratio of the binary. This is a direct consequence of the $\bar{\nu}_{e}$ emission dominating over the $\nu_{e}$ one due to the protonization of the merger remnant, which leads to crossings in the neutrino ELN angular distributions everywhere above the neutrino emitting surfaces. Our results thus confirm the earlier conclusions of Ref. [30], which adopted a simple toy-model for the neutrino emission characteristics and emission surface geometry. However, in contrast to the results obtained in Ref. [31], which showed that the region where flavor instabilities exist shrinks on a time scale of $\sim\mathcal{O}(10)$ ms as the BH–disk remnant evolves, the flavor unstable regions reported here remain quite stable within the examined $10$ ms of post-merger evolution. Since Refs. [13, 18] reported dominating emission of $\bar{\nu}_{e}$ over $\nu_{e}$ on time scales longer than $\mathcal{O}(100)$ ms for a hypermassive NS accretion disk system, we expect that the flavor instabilities found in this paper may be sustained for even longer duration and affect the nucleosynthesis in the disk winds. As for the ejecta properties and the nucleosynthesis outcome, the ejecta contain a wide $Y_{e}$ distribution up to 0.5 due to the effect of weak interactions including neutrino absorption, allowing for the formation of heavy elements in all three $r$-process peaks. In particular, a few times $10^{-5}$ $M_{\odot}$ of strontium are synthesized in the ejecta in all models, consistent with the amount inferred from the GW170817 kilonova observation [11]. We also find that the amount of free neutrons left after the $r$-process freeze-out is roughly a factor of $10$ smaller than the one obtained in simulations without taking into account the effect of weak interactions. This has implications for the prediction of the early-time UV emission that may be powered by the decay of free neutrons [60]. By relying on the extreme ansatz that fast pairwise conversions lead to flavor equilibration, we find that flavor mixing of neutrinos mostly affects the polar ejecta within $\sim 30^{\circ}$ by changing the peak $Y_{e}$ from $\sim 0.4$ to $\sim 0.5$. The dominant effect is thus to reduce the first peak abundances while enhancing the iron peak abundances. Note that we have only examined the flavor instability above the neutrino emitting surfaces. Beyond that, future work investigating the occurrence of unstable conditions inside the neutrino-trapping regime, along the lines of recent work done in the context of core-collapse supernovae [64, 65, 66, 67, 68, 69, 70], should be carried out. Multidimensional numerical simulations tracking the flavor evolution in the presence of fast pairwise conversions (see, e.g., Refs. [71, 72, 54, 73, 74, 52, 53]), including the collisional term in the equations of motion, are essential to draw robust conclusions on the role of neutrino flavor conversion for the outcome of nucleosynthesis in the ejecta and the corresponding kilonova observables. Acknowledgements. MG and MRW acknowledge support from the Ministry of Science and Technology, Taiwan under Grant No. 107-2119-M-001-038, No. 108-2112-M-001-010, No. 109-2112-M-001-004, the Academia Sinica under project number AS-CDA-109-M11, and the Physics Division of National Center for Theoretical Sciences. IT acknowledges support from the Villum Foundation (Project No. 13164), the Danmarks Frie Forskningsfonds (Project No. 8049-00038B), the Knud Højgaard Foundation. At Garching, funding by the European Research Council through grant ERC-AdG No. 341157-COCO2CASA and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through grants SFB-1258 “Neutrinos and Dark Matter in Astro- and Particle Physics (NDM)” and under Germany’s Excellence Strategy through Excellence Cluster ORIGINS (EXC 2094)—390783311 is acknowledged. Appendix A Computing the neutrino number densities from the simulation data As the simulation outputs from Ref. [42, 41] did not directly store the $\nu_{e}$ and $\bar{\nu}_{e}$ absorption rates on free nucleons along the ejecta trajectories, we compute the rates in a post-processing fashion as follows. First, the simulations provide the local $\nu_{e}$ and $\bar{\nu}_{e}$ energy luminosities and average energies at radii of 50 and 100 km as functions of the polar angle $\theta$ (with respect to the $z$-axis) for different post-merger time snapshots. This allows to compute the $\nu_{e}$ and $\bar{\nu}_{e}$ number densities at these two radii. Then, at a given time $t$, for any spatial coordinate $\bm{x}$ on a trajectory with radius 50 km$\leq r\leq 100$ km (with an angle $\theta$), we compute the corresponding number densities of $\nu_{e}$ and $\bar{\nu}_{e}$ by linearly interpolating the logarithmic values of the densities obtained above at $r=50$ and $100$ km. Once we have the number densities of the $\nu_{e}$ and $\bar{\nu}_{e}$, the absorption rates for $50$ km$\leq r\leq 100$ km can be computed using Eqs. (22) and (23) in Appendix. B. For $r>100$ km, we extrapolate the rates assuming that they scale as $r^{-2}$: $$\lambda_{\nu_{\alpha}}(r>100{\rm~{}km})=\lambda_{\nu_{\alpha}}(r=100{\rm~{}km}% )\times\left(\frac{\rm 100~{}km}{r}\right)^{2},$$ (16) For $r<50$ km, we take a different form of extrapolation to partly account for the finite-size emission geometry: $$\displaystyle\lambda_{\nu_{\alpha}}(r<50{\rm km})$$ $$\displaystyle=$$ $$\displaystyle\lambda_{\nu_{\alpha}}(r=50{\rm km})\times$$ $$\displaystyle\left(\frac{1-\sqrt{1-(r_{0}/r)^{2}}}{1-\sqrt{1-(r_{0}/(50{\rm km% }))^{2}}}\right)^{2},$$ with $r_{0}=25$ km. We have checked that even simply taking the $1/r^{2}$ extrapolation for regions with $r<50$ km leads to nearly identical results to those obtained by using Eq. (A). Figure 17 compares the $Y_{e}$ distribution at $r=100$ km, obtained by the simulation of Ref. [42] to the one obtained by using the above neutrino absorption rates in the nuclear reaction network described in Sec. IV.1. It shows that the $Y_{e}$ distributions agree with each other reasonably well. For $\nu_{x}$, since the simulations do not store the energy luminosity and average energy in the bins at 50 and 100 km, we estimate the $\nu_{x}$ number density and average energy along each ejecta trajectory as follows. First, the locations of the $\nu_{x}$ surface and the associated temperatures for times between $2.5\leq t\leq 10$ ms are interpolated by using the data provided at $2.5,5,7.5$ and $10$ ms. Then, for a given time $t$, a $\nu_{x}$ number density on each point $\vec{x}$, ${\tilde{n}}_{\nu_{x}}(\vec{x})$, at the emission surface following the Fermi-Dirac distribution with temperature $T(\vec{x}_{i})$ and zero chemical potential can be easily calculated. We then re-normalize the total neutrino number luminosity emitted from the $\nu_{x}$ surface to the value given by simulation data, $$L_{N,\nu_{x}}=\frac{L_{\nu_{x}}}{\langle E_{\nu_{x}}\rangle}=\frac{5\cdot\xi}{% 12}\int dS{\tilde{n}}_{\nu_{x}},$$ (18) where $L_{\nu_{x}}$ and $\langle E_{\nu_{x}}\rangle$ are the energy luminosity and mean energy of $\nu_{x}$ shown in Fig. 1, respectively. The quantity $dS$ is the differential surface area on the $\nu_{x}$ surface. The factor $5/12$ accounts for the forward peaked angular profile of $\nu_{x}$ emission consistent with Eq. (13), and $\xi$ is the normalization constant. Correspondingly, the rescaled $\nu_{x}$ number density on their emission surface is given by $n_{\nu_{x}}(\vec{x})=\xi{\tilde{n}}_{\nu_{x}}(\vec{x})$. We assume the $\nu_{x}$ average energy at each location $\vec{x}$ on the emission surface to be $$\langle E_{\nu_{x}}\rangle(\vec{x})=\frac{1}{2}\left(\langle E_{\nu_{x}}% \rangle+3.15~{}T(\vec{x})\right),$$ (19) to partly account for the fact that the $\nu_{x}$-$e^{\pm}$ scatterings, which can down-scatter $\nu_{x}$, was not included in the numerical simulations of Ref. [42]. Since the local temperature on the $\nu_{x}$ emission surface within $x\lesssim 20$ km is found to be $\gtrsim 6$ MeV, the main effect of the above choice is to reduce the $\langle E_{\nu_{x}}\rangle(\vec{x})$ at the outer edge of the emission surface. We have additionally confirmed that adopting a location independent average energy of $\nu_{x}$ given by $\langle E_{\nu_{x}}\rangle$ does not qualitatively change our results shown in the main text. For $t<2.5$ ms, we simply assume that the neutrino emission surface is the same as the one at $t=2.5$ ms, and scale the number density $n_{\nu_{x}}(\vec{x},t)$ and the average energy $\langle E_{\nu_{x}}\rangle(\vec{x},t)$ in the following way $$\displaystyle n_{\nu_{x}}(\vec{x},t)$$ $$\displaystyle=n_{\nu_{x}}(\vec{x},t=2.5~{}{\rm ms})\left(\frac{L_{N,\nu_{x}}(t% )}{L_{N,\nu_{x}}(t=2.5~{}{\rm ms})}\right),$$ (20) $$\displaystyle\langle{E_{\nu_{x}}}\rangle(\vec{x},t)$$ $$\displaystyle=\langle{E_{\nu_{x}}}\rangle(\vec{x},t=2.5~{}{\rm ms})\left(\frac% {\langle E_{\nu_{x}}\rangle(t)}{\langle E_{\nu_{x}}\rangle(t=2.5~{}{\rm ms})}% \right).$$ (21) Once we have the desired quantities on the emission surface for all times, we use the same ray-tracing technique as in the main text to compute the $\nu_{x}$ number densities for the locations crossed by the trajectories. The absorption rates of the converted $\nu_{x}$ and $\bar{\nu}_{x}$ on nucleons, $\lambda_{\nu_{x}}$ and $\lambda_{\bar{\nu}_{x}}$, along all trajectories, are similarly computed as those of $\nu_{e}$ and $\bar{\nu}_{e}$ given in Appendix B by replacing the corresponding number densities, the average energies, and other higher energy moments. Appendix B Computing the neutrino absorption rates In order to compute the evolution of $Y_{e}$ for the outflows for the cases with and without flavor conversions, we first compute the number densities and average energies of $\nu_{e}$, $\bar{\nu}_{e}$ and $\nu_{x}$ (without flavor conversions) along the trajectories of all tracer particles by post-processing the simulation data as detailed in Appendix A. For the case without flavor conversions, we follow Ref. [75] to calculate the $\nu_{e}$ and $\bar{\nu}_{e}$ absorption on free nucleons: $$\displaystyle\lambda_{\nu_{e}}^{0}$$ $$\displaystyle=n_{\nu_{e}}\langle\sigma_{\nu_{e}}\rangle,$$ (22) $$\displaystyle\lambda_{\bar{\nu}_{e}}^{0}$$ $$\displaystyle=n_{\bar{\nu}_{e}}\langle\sigma_{\bar{\nu}_{e}}\rangle,$$ (23) where $\langle\sigma_{\nu_{e}}\rangle$ and $\langle\sigma_{\nu_{e}}\rangle$ are the spectrally averaged absorption cross-sections of $\nu_{e}$ and $\bar{\nu}_{e}$. By taking into account the recoil corrections and weak magnetism [76], the average neutrino capture cross sections are approximated by $$\displaystyle\langle\sigma_{\nu_{e}}\rangle$$ $$\displaystyle\simeq k\langle E_{\nu_{e}}\rangle\varepsilon_{\nu_{e}}\left[1+2% \left(\frac{\Delta}{\varepsilon_{\nu_{e}}}\right)+a_{\nu_{e}}\left(\frac{% \Delta}{\varepsilon_{\nu_{e}}}\right)^{2}\right]W_{\nu_{e}},$$ (24) $$\displaystyle\langle\sigma_{\bar{\nu}_{e}}\rangle$$ $$\displaystyle\simeq k\langle E_{\bar{\nu}_{e}}\rangle\varepsilon_{\bar{\nu}_{e% }}\left[1+2\left(\frac{\Delta}{\varepsilon_{\bar{\nu}_{e}}}\right)+a_{\bar{\nu% }_{e}}\left(\frac{\Delta}{\varepsilon_{\bar{\nu}_{e}}}\right)^{2}\right]W_{% \bar{\nu}_{e}},$$ (25) where $k=9.3\times 10^{-44}$ cm${}^{2}/$MeV${}^{2}$, $\varepsilon_{\nu_{e},\bar{\nu}_{e}}=\langle E^{2}_{\nu_{e},\bar{\nu}_{e}}% \rangle/\langle E_{\nu_{e},\bar{\nu}_{e}}\rangle$, $a_{\nu_{e},\bar{\nu}_{e}}=\langle E^{2}_{\nu_{e},\bar{\nu}_{e}}\rangle/\langle E% _{\nu_{e},\bar{\nu}_{e}}\rangle^{2}$, and $\Delta=(m_{n}-m_{p})=1.293~{}\text{MeV}$ is the neutron-proton mass difference. The weak-magnetism and recoil correction factors $W_{\nu_{e},\bar{\nu}_{e}}$ are given by $$\displaystyle W_{\nu_{e}}$$ $$\displaystyle=$$ $$\displaystyle\left[1+1.02\frac{b_{\nu_{e}}\varepsilon_{\nu_{e}}}{M}\right],$$ (26) $$\displaystyle W_{\bar{\nu}_{e}}$$ $$\displaystyle=$$ $$\displaystyle\left[1-7.22\frac{b_{\bar{\bar{\nu}}_{e}}\varepsilon_{\bar{\nu}_{% e}}}{M}\right],$$ (27) where $b_{\nu_{e},\bar{\nu}_{e}}=\langle E^{3}_{\nu_{e},\bar{\nu}_{e}}\rangle\langle E% _{\nu_{e},\bar{\nu}_{e}}\rangle/\langle E^{2}_{\nu_{e},\bar{\nu}_{e}}\rangle^{2}$ is the spectral shape factor for $\nu_{e}(\bar{\nu}_{e})$ and $M=940$ is roughly the mass of a nucleon in MeV. Note that in deriving the rates through the above equations, we have assumed zero chemical potentials for all neutrino species to compute the $i$-th neutrino energy moments $\langle E_{\nu_{\alpha}}^{i}\rangle$. 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R4 and its circumstellar nebula: evidence for a binary merger?${}^{1}$ A. Pasquali ESO/ST-ECF, Karl-Schwarzschild-Strasse 2, D-85748 Garching bei München, Germany apasqual@eso.org    A. Nota ${}^{2}$ STScI, 3700 San Martin Drive, Baltimore, MD 21218, USA nota@stsci.edu    N. Langer Institut für Physik, Universität Potsdam, Postfach 601553, D-14415 Potsdam, Germany ntl@leo.astro.physik.uni-potsdam.de    R.E. Schulte-Ladbeck University of Pittsburgh, Pittsburgh, PA 15260 rsl@phyast.pitt.edu    M. Clampin STScI, 3700 San Martin Drive, Baltimore, MD 21218, USA clampin@stsci.edu Abstract We present new, NTT longslit spectroscopy of the B[e] supergiant in the binary system R4 in the Small Magellanic Cloud. The data show extended, forbidden N and S emissions which are typical signatures of circumstellar matter. Their extension along the space axis of the slit defines an angular size of 8.6${}^{\prime\prime}$ which translates into a linear size of 2.4 pc. The N emission lines also show the velocity structure of a bipolar outflow expanding at 100 km s${}^{-1}$ on average. This implies that, for a measured radius of 1.2 pc, the outflow originated about 1.2 $\times$ 10${}^{4}$ yr ago. The line flux ratio [NII]6584/[SII]6717 indicates that the nebula is nitrogen enriched and therefore it has been ejected from the central star. This is the first bipolar, ejection nebula detected around a well-established B[e] supergiant. The bipolar morphology and the chemical enrichment shown by the nebula associated with R4 are consistent with the picture of a binary merger (Langer & Heger 1998), in which R4 was originally a system composed by a close pair and a third star (the observed A companion). The close pair merged into a single star and the merging process produced a circumstellar nebula that was later shaped by the ensueing B star wind. Since the bipolar morphology, the kinematics and the enriched chemical composition make the nebula surrounding R4 very similar to the observed LBV nebulae, our findings imply that at least a few LBV outbursts and nebulae might well be the result of the merging process of two massive stars. keywords: B[e] supergiants — Luminous Blue Variables — Circumstellar medium 11footnotetext: Based on observations obtained at the European Southern Observatory, La Silla22footnotetext: Affiliated with the Astrophysics Division, Space Science Department of the European Space Agency 1 Introduction The class of B[e] supergiants consists of about 20 luminous evolved B stars with a rich emission line spectrum and a strong infrared excess (Zickgraf et al. 1986, Lamers et al. 1998). Most of the confirmed members of this class are located in the Magellanic Clouds, mainly for two reasons: the luminosities of the Galactic objects cannot be precisely determined due to the uncertain distances, and the difficulty to resolve the objects of this class from other B-type emission line stars (Be stars, Herbig Be stars, and other types of B[e] stars). Gummersbach et al. (1995) were able to place 14 Magellanic Cloud B[e] supergiants in the HR diagram. There, they appear to define two distinct groups, one at relatively low luminosity ($L\mathrel{\hbox{\hbox to 0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}}\hbox{$<$}}}10% ^{5}\,{\rm L}_{\odot}$) and low effective temperature ($\log\,T_{\rm eff}\,\mathrel{\hbox{\hbox to 0.0pt{\hbox{\lower 4.0pt\hbox{$\sim% $}}}\hbox{$<$}}}15\,000\,$K), and the other at higher luminosities ($L\mathrel{\hbox{\hbox to 0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}}\hbox{$>$}}}10% ^{5}\,{\rm L}_{\odot}$) and temperatures ($\log\,T_{\rm eff}\,\simeq 12\,000\,$K – $30\,000\,$K). The spectral properties of the B[e] supergiants are best explained within the model by Zickgraf et al. (1985), who propose that these stars have a two component wind: a fast polar wind responsible for the high excitation UV resonance lines, and an equatorial slow and cool wind producing the narrow permitted and forbidden lines. The equatorial wind is associated with the highest mass-loss rate and usually identified with an outflowing disk where dust can condense and emit at infrared wavelengths. Such disk might be produced by rotational wind compression (Bjorkman & Cassinelli 1993, Bjorkman 1999). Nevertheless, it remains to be shown that disk inhibition due to non-radial components of the wind driving line force and gravity darkening (Owocki & Gayley 1998) can be overcome, perhaps by a combination of rotational compression and wind bi-stability proposed by Lamers & Pauldrach (1991) which predicts a sudden increase in the wind mass flux and decrease in the wind velocity at a critical temperature ($\sim 20\,000\,$K) when the stellar surface temperature decreases gradually from the pole towards the equator. Langer & Heger (1998) have connected the B[e] supergiant stage with phases in the evolution of rotating massive stars during which the star can possibly reach the $\Omega$-limit, i.e. its surface rotation rate (which also takes into account the radiation force of the star) is able to destabilize the stellar surface at the equator (Langer 1997). They found that the most luminous and hot B[e] stars might be related to core hydrogen burning models which arrive at the $\Omega$-limit due to increasing surface opacities during their main sequence evolution, which is possible for stars relatively close to the Eddington-limit even if they are slow rotators (Langer 1998). They proposed further that stars below $\sim 10^{5}\,{\rm L}_{\odot}$ could reach the $\Omega$-limit during core helium burning (on the so called blue loops) due to efficient angular momentum transport from the stellar interior to the stellar surface during this phase (Heger & Langer 1998). Finally, the outbursts of Luminous Blue Variables have been associated with these stars hitting the $\Omega$-limit (Langer 1997, Langer et al. 1999), a conjecture which is strongly supported by the bi-polarity of virtually all circumstellar nebulae of LBVs (Nota et al. 1995). Whether all massive stars go through a B[e] supergiant stage, and whether they are connected to Luminous Blue Variables is unclear. Empirically, the distribution of the group of luminous B[e] supergiants in the HR diagram overlaps with that of the LBVs (Bohannan 1997). A connection between B[e] supergiants and LBV stars has been early suggested by Shore (1990) and Schulte-Ladbeck & Clayton (1993) from their analysis of S22, in the Large Magellanic Cloud. Classified as a B[e] supergiant by Zickgraf et al. (1986), S22 shows an intrinsic polarization of 0.52 $\%$ due to electron scattering in an aspherical wind. The polarization degree is variable and this is probably linked to variations in the mass-loss rate of the star (Schulte-Ladbeck & Clayton 1993). A similar result has been found for the galactic LBV HR Carinae, which is characterized by an intrinsic continuum polarization of about 0.4$\%$, possibly variable (Clampin et al. 1995). This can again be explained as due to a non-spherical wind geometry (the presence of a circumstellar disk has been also discussed by Nota et al. 1997) and a time dependent mass loss rate. In addition, Shore (1990) has detected almost a factor of two variation in the UV flux of S22 longward of 1600 Å and a factor between 2 and 3 variation shortward of 1600 Å. The amplitude of the UV variability is quite similar to that observed in LBVs during their shell ejection phase (Pasquali & Nota 1999). As an alternative approach, to study the occurrence of the LBV phase in the evolution of massive stars, we have undertaken a longslit spectroscopy campaign of galactic and MC evolved supergiants whose stellar properties (M${}_{Bol}$ and Log T${}_{eff}$) are in the range set by confirmed LBVs. The aim of the observations is to detect the presence of circumstellar nebulae and to determine whether these are ejected by the star and possibly establish an evolutionary connection with LBVs. Here, we present the first results obtained for the R4, in the Small Magellanic Cloud. With $L\simeq 10^{5}\,{\rm L}_{\odot}$ and T${}_{eff}\mathrel{\hbox{\hbox to 0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}}\hbox{$% <$}}}27\,000\,$K (Zickgraf et al. 1996), R4 is the hottest and least luminous star within the high luminosity group of B[e] supergiants. Zickgraf et al. showed that R4 is a spectroscopic binary (a = 23 A.U.) comprising a B[e] supergiant with spectrophotometric variability characteristic of a LBV, and an evolved A type companion star which is about 10 times less luminous ($10^{4}\,{\rm L}_{\odot}$). In Sect. 2 we present the spectroscopic data taken for R4, while in Sect. 3 we describe the results obtained from our observations. A discussion of our results, of the implications for the evolutionary history of R4, and of the connection of B[e] supergiants with LBVs follows in Sect. 4. 2 Observations and data reduction We observed R4 at the 3.5m ESO/NTT telescope, on the nights of July 27 - 30, 1998. We used the EMMI spectrograph in the REMD (Red Medium Dispersion) configuration and acquired longslit spectra through gratings 6 and 7 in the following wavelength ranges: 4470 - 5820 Å, 5080 - 6420 Å and 6240 - 6870 Å. We observed in three different positions: on the star, at 3${}^{\prime\prime}$ North and 3${}^{\prime\prime}$ South, respectively. These two offsets were computed by translating the typical linear sizes of known LBV nebulae into angular sizes at the SMC distance (57.5 Kpc, van den Bergh 1989), in order to be able to detect a circumstellar nebula, if present, when no high-resolution imaging could be performed. We employed a 1${}^{\prime\prime}$ x 180${}^{\prime\prime}$ longslit oriented East - West in all the cases but in the 4470 - 5820 Å spectrum, for which the slit had been oriented North - South. The complete journal of the observations is reported in Table 1, where the wavelength range, the dispersion and effective resolution, the exposure time and the offset are summarized for each spectrum. The red arm of EMMI was equipped with a TK2048EB4 CCD, whose spatial scale is 0.27${}^{\prime\prime}$ per pixel. We obtained the usual set of bias and flat-field images, together with comparison spectra of HeAr for the wavelength calibration. The spectra were first cleaned to remove cosmic rays and bad columns, and then corrected for bias and flat-fielded. Their subsequent reduction was performed following the procedure outlined in the IRAF LONGSLIT package. We modeled the sky background in each frame by fitting a surface described by Chebyshev polinomials of low order in X and Y. This surface was then subtracted from the frame itself. The wavelength calibration was achieved in three steps: first, we identified the emission lines at the central row of the comparison spectrum and derived the dispersion correction by fitting a Chebyshev function of order 3. The same lines were then reidentified across the entire frame adopting a step of 2 rows and readjusting the dispersion correction if necessary. Finally, the dispersion corrections obtained through the frame were fitted into a surface using Chebyshev polynomials of order 5. This surface was then applied to each spectrum for the final wavelength calibration. The same procedure was of course repeated for each grating. The effective resolution is reported in Column 5 of Table 1 for each grating: a FWHM = 2.72 and 2.36 Å corresponds to a velocity resolution of 84 and 61 km s${}^{-1}$ at H$\beta$ and [NII]$\lambda$5755 respectively, while a FWHM of 1.12 Å defines a velocity resolution of 25 km s${}^{-1}$ at H$\alpha$. Measurements of the sky lines indicate that the wavelength calibration is certain at $\pm$ 10 km s${}^{-1}$ at any position within the longslit spectra. 3 Results 3.1 Nebular kinematics The long-slit spectra of the R4 region show the presence of extended nebular lines, such as [NII] $\lambda\lambda$6548, 6584 and [SII] $\lambda\lambda$6717, 6731. These lines show a well defined spatial extension and velocity structure, clearly distinguishable from the underlying emission due to the local interstellar medium. We show in Figure 1 the [NII] $\lambda$6584 line as detected at the position 3${}^{\prime\prime}$ North with respect to the star (top panel), and 3${}^{\prime\prime}$ South (bottom panel). The spatial extent of the nebular emission is approximately 32 pixels, which corresponds to an angular size of 8.6${}^{\prime\prime}$ (given the EMMI/CCD spatial scale of 1 pixel = 0.27${}^{\prime\prime}$) and to a linear size of 2.4 pc, in the assumption of a distance for the SMC of 57.5 Kpc (van den Bergh 1989). The peculiar velocity structure of the nebular lines can be immediately noted in a quick look inspection of the two-dimensional spectra: while at 3${}^{\prime\prime}$ North the bulk of the emission is redshifted, at 3${}^{\prime\prime}$ South the line is blueshifted. A remarkable symmetry is present, both along the spatial direction (North compared with South), and in the velocity structure (redshifted versus blueshifted). We used the [NII]$\lambda$6584 line to derive the nebular radial velocity map at the two offset pointings. We binned the spectra by a factor of 2 along the spatial axis (2 pixels corresponding to 0.54${}^{\prime\prime}$) and extracted an individual spectrum from each bin. We measured the peak wavelengths in the [NII] profile by multi-gaussian fitting and computed the corresponding radial expansion velocities. Our fits are characterized by a typical error of $\pm$ 4 km s${}^{-1}$. In this wavelength region, the spectra have a velocity resolution of $\simeq$ 25 km s${}^{-1}$ and the absolute wavelength calibration is within $\pm$ 10 km s${}^{-1}$ across the entire slit. The derived radial velocities, corrected for the heliocentric motion, have been plotted in Figure 2 as a function of distance from the star, in arcseconds, for both slit positions (3${}^{\prime\prime}$ North - top panel; 3${}^{\prime\prime}$ South - bottom panel). On the abscissae, East is to the left and West is to the right. The star is at position 0. From Figure 2 it is clear that the local interstellar medium dominates the velocity distribution at distances larger than $|$5$|^{\prime\prime}$ from the star, since we measure the mean radial velocity of this overall motion to be 110 km s${}^{-1}$, in agreement with the Fabry-Perot H$\alpha$ observations of le Coarer et al. (1993). However, in the distance range -5${}^{\prime\prime}<$ d $<5^{\prime\prime}$ from the star, two additional components are clearly resolved: at the position 3${}^{\prime\prime}$ North we find a component which is redshifted, and spans between 200 and 280 km s${}^{-1}$, covering the entire spatial range. At the position 3${}^{\prime\prime}$ South, a second component is also present, which varies between 20 and 90 km s${}^{-1}$. This component extends between 4${}^{\prime\prime}$ E and 4${}^{\prime\prime}$ W from the star. The remarkable symmetry of these radial velocity structures can be better appreciated in Figure 3 (top panel) where we have plotted on the same spatial scale the two velocity profiles obtained at the two positions. First, we notice that the northern velocity distribution mirrors the southern. The N component displays two radial velocity maxima at $\simeq$ 2.5${}^{\prime\prime}$ E and $\simeq$ 1.5${}^{\prime\prime}$ W. In correspondence to the same two positions, the S component shows two radial velocity minima. The N component reaches radial velocity minima in correspondence to the star and at the outer boundaries of its spatial extension. The S component follows a symmetrical trend. The peak-to-peak radial velocity amplitude of both components is very similar ($\simeq$ 80 km s${}^{-1}$). In addition, there is symmetry between the E and W regions of the nebula with respect to the central star. Indeed, the eastern portion of both velocity curves, when folded, significantly overlaps the western. The observed velocity structure is likely to be indicative of a nebula surrounding R4. The nebula is dynamically associated with the central star, for which Zickgraf et al. (1996) determined a radial velocity of 147 km s${}^{-1}$, and most likely has been ejected by R4 in a previous phase. Therefore, with respect to the central star, the northern component of the nebula turns out to be red-shifted by 84 km s${}^{-1}$ while the southern is blue-shifted by 118 km s${}^{-1}$ on average. Assuming a mean expansion velocity of 100 km s${}^{-1}$ and a full linear size of 2.4 pc, we derive a dynamical age of the nebula of $\sim$ 1.2 $\times$ 10${}^{4}$ yr. Without a direct image, and on the basis of kinematics considerations alone, it is difficult to make definite conclusions on its structure. Compared with the information available on nebulae around LBVs, the nebula around R4 appears more complicated in nature. From the kinematics, it is fair to conclude that: • the nebula is not a simple expanding shell. An expanding shell would result in a radial velocity map which has maximum dispersion in correspondence to the position of the star, and minimum velocity at the shell boundaries (eg. AG Car: Smith 1991, Nota et al. 1992). • the nebula is not strictly bipolar. Compared with the radial velocity maps derived for HR Car, a prototypical bipolar outflow (Nota et al. 1997), the situation is very different: in the case of HR Car, there is a clear demarcation between the two sides of the bipolar outflow, with the redshifted region limited to the NW quadrant, and the blueshifted to the SE quadrant (see Figure 9 in Nota et al. 1997). In order to provide a consistent explanation for the radial velocity maxima and minima observed in the R4, a more complicated structure needs to be invoked, which is symmetrical around two axes. In the bottom panel of Figure 3, we provide a cartoon of one possible structure, in which a cloverleaf morphology is aligned with the peculiar radial velocity features. In this proposed structure, the outflow occurs in four directions along two axes, perpendicular to each other, and oriented at a PA $\simeq$ 45${}^{o}$. This complicated structure, although speculative, has been observed in planetary nebulae. Only a direct image will confirm whether such speculation is correct. 3.2 Nebular composition In addition to the kinematical properties, the nebular lines provide also some information on the chemical composition of the nebula. We have computed the line ratio [NII]6584/[SII]6717 for the R4 nebula and the local interstellar medium from the long-exposure spectra which we corrected for atmospheric extinction. We derived a ratio [NII]6584/[SII]6717 of about 3 (in agreement also with the data of Zickgraf et al. 1996) and 0.3 for the R4 nebula and the local interstellar medium, respectively. Such result is inconsistent with the same line ratio measured in HII regions and SN remnants in the SMC by Russell & Dopita (1990). Typically, the [NII]6584/[SII]6717 ratio is 0.6 for the HII regions (with one exception: N84C is characterized by a value of 2.4) and varies between 0.2 and 0.4 in the case of SNR, independently of the position in the galaxy. A [NII]6584/[SII]6717 value of 0.3 is considered to reflect the intrinsic lower N content of the SMC with respect to the Galaxy and the LMC. We may then conclude that the R4 nebula is N-enriched by a factor 10 with respect to both the local interstellar medium and HII regions/SN remnants. 4 Discussion ¿From Section 3, we conclude that R4 is surrounded by a bipolar circumstellar nebula, nitrogen enriched, with a dynamical age of $\sim$ 1.2 $\times$ 10${}^{4}$yr. This nebula appears to have been ejected from the central star, and its morphological, kinematic and chemical properties are comparable to the average properties of LBV nebulae (cf., Nota et al. 1995). Although Esteban & Fernandez (1998) have detected a circumstellar nebula around the galactic B[e] star MCW$\,$137 which appears not to be chemically enriched, our findings for R4 provide the first evidence for an ejected nebula around a B[e] supergiant. 4.1 The progenitor evolution of R4 In order to investigate the implications of our finding for the connection of B[e] supergiants and LBVs, let us recall the proposed evolutionary scenarios for the B[e] supergiants. Table 3 summarises the expected stellar and circumstellar nebula properties of B[e] supergiants according to the very massive main sequence star scenario, the blue loop scenario and the binary merger scenario (Langer & Heger 1998). In the first scenario, the star reaches the $\Omega$-limit (cf. Section 1) during its main sequence evolution due to its high luminosity, i.e. its proximity to the Eddington limit (Langer 1998). This appears possible only for the most massive stars. Since at low metallicities the Eddington-limit is even higher at higher luminosity than for young stars in the solar neighborhood (Ulmer & Fitzpatrik 1998), this scenario would require an extraordinarily fast rotation to apply to the case of R4. In the second scenario, the star reaches the $\Omega$-limit on a blue loop evolving off the Hayashi line during core helium burning (Heger & Langer 1998). This scenario can not apply to R4 since stars on blue loops do not exceed effective temperatures of $\sim 20\,000\,$K (cf. Langer & Maeder 1995), i.e., the blue loops never extend to the main sequence band. In the third scenario, an equatorial disk or ring is created by mass overflow through the second Lagrangian point in a close binary system in the course of a close binary merger. This scenario was supposed to be most appropriate for the case of R4 by Langer & Heger (1998) for the following reasons. The B[e] supergiant R4 has an evolved A type companion star with a mass of about 12.9$\,\mathrm{M}_{\odot}$ (Zickgraf et al. 1996). While the B[e] star mass has been determined to a similar value ($\sim 13.2\,\mathrm{M}_{\odot}$) its bolometric luminosity outshines that of the A star by a factor of $\sim 10$. Zickgraf et al. conclude from the high luminosity of R4 ($\sim 10^{5}\,{\rm L}_{\odot}$), and from its strong surface enrichment in CNO processed material, that it must have lost large amounts of mass in a previous red supergiant phase. However, as noted by Langer & Heger (1998), even if a $\sim 20\,\mathrm{M}_{\odot}$ red supergiant at the very low metallicity of the SMC had lost about $10\,\mathrm{M}_{\odot}$ via its stellar wind (which appears unlikely for several reasons; e.g., the large envelope mass of the progenitor of supernova 1987A), there would remain a basic puzzle in the R4 binary system. The A star is clearly beyond core hydrogen exhaustion: how then can the B star have an evolved companion which is ten times less luminous? The B star should have long exhausted its fuel and exploded as a supernova. Clearly, binary mass transfer must have occured in this system. Now, the orbital separation in the system is presently about 23 A.U. or 5000$\,{\rm R}_{\odot}$ (Zickgraf et al. 1996), which may be too large to allow any mass transfer from the A star to the B star. Also, mass transfer in very wide systems is supposed to be unstable and would not leave the two stars at a large separation (e.g., Podsiadlowski et al. 1992), and can therefore be excluded. A viable binary scenario for R4 may be that the B[e] star has been formed by a recent binary merger, and the A star is not involved in any mass transfer but only serves as a suitable clock (Langer & Heger 1998). Wellstein et al. (2000) find in a parameter study of binary evolution models that a system which starts out with a 12$\,\mathrm{M}_{\odot}$ and a 11$\,\mathrm{M}_{\odot}$ star on a 40 day orbit would evolve into contact after core hydrogen exhaustion in the 12$\,\mathrm{M}_{\odot}$ star, which leads to mass overflow through the outer Lagrangian point $L_{2}$ and then most likely to a merger. The $L_{2}$ overflow, which is likely to comprise several solar masses of material (Wellstein et al. 2000), gives rise to a circumstellar nebula, which is then shaped by the ensueing B star wind, like the Homunculus nebula around $\eta$ Carinae in the wind interaction scenario of Langer et al. (1999). The large amount of angular momentum in the stellar merger remnant may lead to a disk wind according to the Bjorkman-Cassinelli mechanism, which then is responsible for the B[e] morphology of the stellar spectrum. Furthermore, the merger star would be expected to contain an overly large helium fraction in its interior — which would give it an unusually large L/M-ratio — and a surface strongly enriched in CNO products. That all these details are observational facts makes R4 the strongest massive star candidate for a binary merger. Within this picture, the system started out as a triple system with three very similar stars, a close pair of, say, 12$\,\mathrm{M}_{\odot}$ and 11$\,\mathrm{M}_{\odot}$, in a wide orbit with a $\sim 13\,\mathrm{M}_{\odot}$ star. Soon after the latter has evolved off the main sequence into an A type supergiant, the close pair would merge to form the B[e] supergiant, about 1.2 $\times$ 10${}^{4}$ yr ago. 4.2 Implications for other B[e] supergiants and LBVs In order to understand the relevance of R4 for massive stars in general, we should establish whether R4 is a peculiar or a typical B[e] supergiant. The latter case would have the dramatic implication that most B[e] supergiants might be the result of a binary merger. However, there are at least two arguments against this proposition. First, R4 has an extreme location in the HR diagram compared to all other B[e] supergiants in that its location is by far the closest to the main sequence band. Zickgraf et al. (1996) compared it with evolutionary tracks of Charbonnel et al. (1993), where R4 falls exactly on the terminal age main sequence of a 20$\,\mathrm{M}_{\odot}$ track. Second, R4 is today the only B[e] supergiant with an ejected circumstellar nebula. Before generalizing the evolutionary scenario of its progenitor, one would certainly want to detect more examples with ejected nebulae. On the other hand, the similarity of the properties of the R4 nebula and that of LBV nebulae in general — which provides a remarkably homogeneous class — is striking. These nebulae are all very similar in terms of morphological and physical properties. They are all typically 1 parsec in size, with morphologies which are mildly to extremely bipolar. They expand in the surrounding medium with velocities of the order of 50 – 100 km s${}^{-1}$. Their size and expansion velocities identify dynamical timescales which are of the order of several thousand of years. Densities are generally found to be low (500 - 1000 cm${}^{-3}$) and so are temperatures, found in the range 5000 - 10000 K. In terms of overall physical and chemical properties, the nebula surrounding R4 would fit this category well. However, two facts preclude the possibility that the R4 nebula and all LBV nebulae have been formed the same way. The first is R4’s location in the HR diagram, which of all the stars in the group of luminous B[e] supergiants is farthest away from the LBV regime. The second, even stronger argument is the fact that some LBV nebulae appear to have multiple shells (Nota et al. 2000) and thus the central stars most likely experienced multiple outbursts, which appears not possible within the binary merger scenario. Although we argue that not all B[e] supergiants are formed by stellar mergers, we conclude that at least two intrinsically very different nebula formation mechanisms, the binary merger and the single star LBV outburst mechanism, can produce nebulae with very similar properties. Seemingly, the prerequisites for the nebular structure are the same in both cases, i.e. a massive disk ejected by the central star which is then shaped by its strong wind. While the disk forms through the $L_{2}$-Roche lobe overflow in the case of the merger (hardcore hydrodynamic calculations for such events exist so far only for low mass stars, which, however, support the general idea; cf. Yorke et al. 1995), it may be formed through rotational wind compression (Bjorkman & Cassinelli 1993) in the single star case (cf. Langer et al. 1999). 5 Conclusions Our longslit spectroscopic observations reveal that R4 is embedded in a circumstellar nebula whose full spatial extension is 8.6${}^{\prime\prime}$, i.e. 2.4 pc at the SMC distance of 57.5 Kpc (van den Bergh 1989). The emission lines show a significant structure in velocity indicating that the nebula is expanding at 100 km s${}^{-1}$ on average. This implies a dynamical age of 1.2 $\times$ 10${}^{4}$ yr. The radial velocity maps obtained for the nebula at 3${}^{\prime\prime}$ North and 3${}^{\prime\prime}$ South from the central star are characterized by two velocity maxima and two velocity minima, respectively, located at the same positions with respect to the star. Hence, the two velocity distribution appear to be symmetric not only in the velocity field but also along the E-W direction (cf. Figure 3). Such a symmetry excludes that the R4 nebula is either a simple expanding shell (i.e. AG Car) or a simple bipolar outflow (i.e. HR Car). It rather suggests a cloverleaf morphology, where the outflow occurs in four directions along two axes, perpendicular to each other and oriented at PA $\simeq$ 45${}^{o}$. Since the line flux ratio [NII]6584/[SII]6717 is almost independent of electron temperature and density, it can be used to estimate any overabundance of nitrogen with respect to the ”unprocessed” sulphur. The nebular [NII]6584/[SII]6717 ratio turns out to be 3 against a value of 0.3 as measured for the local interstellar medium in the same observed spectra, and a value between 0.6 and 0.2 as derived for HII regions and SN remnants in the SMC by Russell & Dopita (1990). This factor of 10 discrepancy clearly indicates that the R4 nebula is nitrogen enriched and therefore, since it is also kinematically associated with the central B[e] supergiant, it is an ejected nebula. R4 is surrounded by a bipolar and N-enriched nebula, ejected from the central star, whose morphological, kinematical and chemical properties well compare with the mean properties of LBV nebulae. We have shown that the central star most likely formed through a binary merger, as proposed by Langer & Heger (1998). This makes R4 the strongest observational counterpart of such event among massive stars, which has long been sought for. E.g., in a comprehensive binary and single star evolution and population sysnthesis study, Podsiadlowski et al. (1992) conclude that $\sim 25$% of all massive binaries undergo a merging process, most likely so just after the initially more massive star has terminated core hydrogen burning. In Section 4.2, we concluded from the properties of R4 that two distinct mechanisms can form circumstellar nebulae of exactly that type found around LBVs. This has two consequences. First, it implies that some LBV outbursts and nebulae may in fact be due to the merging process of two massive stars. The predicted amount of expected binary mergers (see above) implies that this is in fact likely. Second, it may deepen the understanding of why bipolar circumstellar nebulae are such a frequent phenomenon. E.g., what holds for massive stars may be true for the progenitors of planetary nebulae, and in the end the dispute of whether bipolar planetaries are formed through binaries (e.g., Soker 1998) or single stars (e.g., García-Segura et al. 1999) may end in a draw. We conclude by urging for imaging observations of the R4 nebula. They will not only provide for the first time the morphological details of a nebula ejected by a massive stellar merger and will thereby allow to constrain the hydrodynamical processes at work in such phenomenon, but it will perhaps reveal clues of how to empirically discriminate binary and single star ejection mechanisms and thus allow for a better understanding of bipolar circumstellar nebulae in general. Acknowledgements.The authors wish to thank Guillermo García-Segura and the staff of the Tonantzintla Institute for their hospitality during their visit where this project was perceived. AP, AN and MC would like to thank STScI and ST-ECF for their hospitality during their visits where the paper was finalized. NL acknowledges supporting from the Deutsche Forschungsgemeinschaft through grants La 587/15-1 and 16-1. RSL acknowledges funding from the HST grant to GO proposal 6540. References (1) Bjorkman, J.E. 1999, in Variable and Non-spherical Stellar Winds in Luminous Hot Stars, eds. 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Asymptotic scaling behavior of self-avoiding walks on critical percolation clusters Niklas Fricke niklas.fricke@itp.uni-leipzig.de    Wolfhard Janke wolfhard.janke@itp.uni-leipzig.de Institut für Theoretische Physik and Centre for Theoretical Sciences (NTZ), Universität Leipzig, Postfach 100920, D–04009 Leipzig, Germany Abstract We study self-avoiding walks on three-dimensional critical percolation clusters using a new exact enumeration method. It overcomes the exponential increase in computation time by exploiting the clusters’ fractal nature. We enumerate walks of over $10^{4}$ steps, far more than has ever been possible. The scaling exponent $\nu$ for the end-to-end distance turns out to be smaller than previously thought and appears to be the same on the backbones as on full clusters. We find strong evidence against the widely assumed scaling law for the number of conformations and propose an alternative, which perfectly fits our data. pacs: 05.10.-a, 36.20.-r, 64.60.al, 64.60.De The self-avoiding walk (SAW) Madras1993 is a fundamental model in statistical mechanics and crucial for our understanding of the scaling behavior of polymers DeGennes1976 . Asymptotically, it is characterized by universal exponents, which are related to the critical exponents of spin systems and assumed to describe long, flexible polymers in good solvent condition. While much is known about SAWs on regular lattices, their behavior in disordered environments, such as porous rocks or biological cells, is less understood. The paradigmatic model for such systems are SAWs on critical percolation clusters Chakrabarti2005 ; Ben-Avraham2000 . Here the walks can only visit a random fraction of sites, whose concentration is equal to the percolation threshold of the lattice. This critical concentration may not be realistic, but it represents an important limiting case, and the effect of the critical clusters’ fractal structure is particularly intriguing Stauffer1992 . One usually considers quenched disorder averages, here denoted by square brackets: On each disorder realization (“cluster”), one takes the average over all walk conformations of length $N$. Each such conformational average contributes equally to the disorder average. It is assumed that the average number of conformations, $\left[Z\right]$, and their mean squared end-to-end distance, $\left[\left\langle R^{2}\right\rangle\right]$, follow asymptotic scaling laws similar to those for normal SAWs: $$\displaystyle\left[Z\right]$$ $$\displaystyle\sim\mu^{N}N^{\gamma-1},$$ (1) $$\displaystyle\left[\left\langle R^{2}\right\rangle\right]$$ $$\displaystyle\sim N^{2\nu}{=N^{2/d_{f}}}.$$ (2) $\gamma$ and $\nu$ are universal scaling exponents, $d_{f}$ is the SAW’s fractal (Hausdorff) dimension, and $\mu$ is a lattice dependent effective connectivity constant. While the effect of the fractal disorder on $\gamma$ and $\mu$ is still very controversial, there is convincing evidence that $\nu$ is different than on regular lattices Meir1989 . However, there is uncertainty concerning the actual value despite a considerable amount of work dedicated to the system. Analytical works have yielded conflicting results Ferber2004 ; Janssen2007 ; Janssen2012a , while accuracy and reliability of numerical investigations have been poor due to modest system and sample sizes. In most numerical studies (see for instance  Nakanishi1991 ; Vanderzande1992 ; Rintoul1994 ; Ordemann2000 ; Singh2009 ), exact enumeration was used to determine the conformational averages. Owing to exponentially increasing computation time [see Eq. (1)], the length of the walks was restricted to $30-50$. Chain-growth Monte Carlo methods may allow for more than a hundred steps Lee1988 ; Woo1991 ; Grassberger1993 ; Blavatska2008a ; Blavatska2008 , but they add statistical uncertainty and the danger of biased results Fricke2013 . We recently developed a new algorithm for exact enumeration of SAWs on two-dimensional critical percolation clusters Fricke2012a , which we have now generalized to higher dimensions. By making use of the clusters’ fractal properties, it overcomes the exponential increase in computation time that usually affects exact enumeration methods. Walks of over $10^{4}$ steps are now accessible, permitting a much more refined investigation of the system. Indeed, we think that the true asymptotic behavior may now be revealed for the first time. Our method exploits the self-similar nature of the clusters to factorize the problem hierarchically, drawing on the ideas of renormalization group theory. The key lies in the observation that the connectivity of a critical percolation cluster is extremely low on any length scale. This is best appreciated by looking at the backbone of a cluster (Fig. 1), the part that remains when all singly-connected “dangling ends” are removed. We define it as the largest bi-connected component, i.e., the largest piece from which nothing can be disconnected by removing a single site. The fragile structure of the backbone, which is the most connected part, suggests that the whole cluster can be decomposed into separate regions by removing only a small number of sites Coniglio1987 . This applies on all length scales, so that we can organize the cluster into a hierarchy of nested “blobs” as sketched in Fig. 2. Each blob should have very few ($\approx O(1)$) external connections and should not contain too much mass (colored areas in Fig. 2) that is not encapsulated in smaller blobs. We create the blob hierarchy by repeatedly fusing regions with many interconnections, aiming for an optimal balance between the two requirements. The principle idea now is to factorize the enumeration procedure by treating walk segments through different blobs separately. The self-similarity of the system suggests a recursive approach: We start by enumerating all possible walk conformations within smallest blobs (“children”) that are contained within larger ones (“parents”). We use the standard backtracking routine Nakanishi1991 ; Vanderzande1992 ; Rintoul1994 ; Ordemann2000 ; Singh2009 for this but distinguish different classes of paths depending on the the external connections they are linked to. When we later generate the paths through the parents, the children are effectively treated as single sites, but we note by which links they are accessed and left. This information is needed to properly match the paths to the segments through the children once the counting within the parents is done. Now we can delete all information concerning the children and proceed on to the “grand parents”. This procedure is repeated up to the largest blob, which is the whole cluster. These main ideas are simple, but defining the blob hierarchy and matching the path segments involves some technical challenges. The gain, however, is significant: On a present-day 3GHz processor, the number of conformations for a $10^{4}$-step SAW (typically $10^{1200}$) and their average end-to-end distance are determined in about 10 minutes on average using our current implementation. The exponential complexity has vanished, and we empirically find a polynomial time increase with an exponent around $2.4$. While the method works in any dimension, we here focus on the physically most relevant case of $D=3$. To generate the clusters we used a depth-first growth algorithm known as the Leath method Leath1976 . We only consider clusters that percolate according to the “wrapping” criterion used in Ref. Newman2001 . The backbones were identified using Tarjan’s algorithm Tarjan1974 . To avoid correlations, independent sets of clusters were used for walks of different length, which we increased by factors of $\sqrt{2}$ from $N=25$ to $N=12800$. Thanks to the method’s efficiency, we could afford samples of at least $5\times 10^{4}$ clusters for each length. The average squared end-to-end distance as a function of $N$ is shown in Fig. 3 on a double-logarithmic scale. To enhance visibility, the values are divided by $N^{1.33}$, which is close to $N^{2\nu}$ according to previous studies. The curves appear straight initially, but around $N\approx 150$ they notably start to slump, crossing over to a slightly different slope. We hence have to use a lower cutoff, $N_{\rm{min}}$, when estimating $\nu$ via a least-squares fit of Eq. (2). On the whole (“incipient”) clusters, the $\chi^{2}$-value of the fit becomes close to one ($\chi^{2}=1.3$) if we choose $N_{\rm{min}}=800$. This yields a value of $\nu_{\rm{ic}}=0.6433(4)$. The fit is shown as dotted red line $f_{1}$ in Fig. 3. For the backbones, we get a decent fit ($\chi^{2}=3.4$) with $N_{\rm{min}}=1131$ (dotted green line), which yields a similar result: $\nu_{bb}=0.643(1)$. Note, that the values of $\chi^{2}$ are meaningful since the data points are uncorrelated and the errors are purely statistical. It has often been claimed Rammal1984 ; Aharony1989 ; Ordemann2000 ; Blavatska2009a that the asymptotic statistics are determined by the backbone alone as a dangling end can only support finite SAWs. As noted in Ref. Woo1991 , this argument is questionable since dangling ends come in all sizes and have a larger fractal dimension than the backbone. Still, our results provide strong evidence that $\nu_{ic}=\nu_{bb}$ is indeed correct. As demonstrated in Fig. 3, this becomes manifest only for sufficiently long walks. The initial slope on the backbone is slightly larger, and the asymptotic behavior is approached more slowly. This is somewhat surprising: if the effects of the dangling ends vanish with $N$, it should be the other way around, unless they happen to cancel out other finite-size effects. The fact that $\nu_{bb}$ appears larger initially explains why results for $\nu$ from the most recent numerical studies, which only considered the backbones, are significantly larger than ours ($0.662(6)$ Ordemann2000 , $0.667(3)$ Blavatska2008 ). The discrepancy is clearly due to the limited system sizes that had been accessible ($30$ and $80$ steps, respectively). To verify this claim, we used an upper cutoff of $N_{\rm{max}}=84$. Since calculations in this regime are swift, we afforded a few more data points and increased the sample sizes to $5\times 10^{5}$. As can be seen in the inset of Fig. 3, a simple power-law nicely fits this data ($\chi_{bb}^{2}=2.0$, $\chi_{ic}^{2}=2.6$); one could hardly suspect a different asymptotic behavior from this perspective. The resulting backbone exponent $\nu_{bb}=0.6646(2)$ is consistent with previous findings, though $\nu_{ic}=0.6547(2)$ is slightly (but significantly) smaller. Some of the finite-size effects may be explained by higher-order corrections to Eq. (2). Better fit results over a larger range can indeed be obtained by including the next-to-leading confluent correction term, $N^{2\nu-\Delta}$. In practice, we fit $$\left[\left\langle R^{2}\right\rangle\right]=a(N+\delta N)^{2\nu}\left(1+b/(N+% \delta N)^{\Delta}\right)$$ (3) as was done for the full-lattice SAW in Ref. Clisby2010 . The small shift, which we set to $\delta N=1/2$, provides for a smoother convergence of the fit but has little effect on the actual results. From the range $N=$ 25–12800 we thus obtain $a_{ic}=1.13(2)$, $b_{ic}=-0.44(3)$; $a_{bb}=1.25(5)$, $b_{bb}=-0.60(1)$ and $$\displaystyle\nu_{ic}=0.644(2),\quad\Delta_{ic}=0.51(5);$$ (4) $$\displaystyle\nu_{bb}=0.640(3),\quad\Delta_{bb}=0.34(4)$$ (5) as our final estimates (for comparison, $\nu_{\rm{full}}=0.587\;597(7)$ and $\Delta_{\rm{full}}=0.528(12)$ were found for the regular simple-cubic lattice Clisby2010 ). The fits are shown as continuous curves $g$ in Fig. 3. The $\chi^{2}$-values are $1.2$ (incipient clusters) and $1.6$ (backbones). These estimates for $\nu$ are consistent with those from the simple fits and again support $\nu_{ic}=\nu_{bb}$. We also obtained similar (but less precise) estimates by extrapolating the successive slopes, as was done in Lee1988 ; Rintoul1994 ; Ordemann2000 . In terms of the fractal dimensions, $d_{f}=1/\nu$, the results are $d_{f,ic}=1.553(5)$, $d_{f,bb}=1.563(7)$. We now turn to the number of conformations, $Z$. Here we only discuss the results for the incipient clusters; those for the backbones are qualitatively the same. The distribution of $Z$ resembles a log-normal as can be seen in Fig. 4 where we have plotted the measured frequencies of $\ln{Z}$ for various $N$ alongside normal distributions with the same mean and variance. As noted before Grassberger1993 , such a “multifractal” distribution can be explained by the fact that $Z$ is roughly a product of random variables, namely the average coordination numbers at each step. Indeed, we find for the variance of $\ln{Z}$: $$\sigma^{2}_{\rm{\\ ln{Z}}}\sim AN^{2\chi};\>A=0.1667(3),\>\chi=0.500(1),$$ (6) which supports this picture. A similar result ($\chi=0.49(1)$) had been reported previously Rintoul1994 , but there appears to be no theoretical explanation why $\chi=1/2$ should hold exactly. These large deviations make it hard to obtain unbiased estimates for $[Z]$: the value is easily underestimated if the sample size is too small. We managed to obtain reliable data for $N\leq 200$ by pushing the number of analyzed clusters to $10^{7}$. According to Eq. (1), $\mu$ and $\gamma$ can be estimated by fitting $$\ln{[Z]}/N=\ln{a}/N+\ln{\mu}+(\gamma-1)\ln{N}/N$$ (7) as was done in Ordemann2000 ; Blavatska2009a . Trying this for different ranges within $N\leq 200$, we found that the estimates for $\mu$ ($\gamma$) systematically decrease (increase) with the upper cutoff $N_{\rm{max}}$. This suggests that the asymptotic behavior is not reached (which would not be surprising given our experience with $\nu$). From our observations one might hence only infer the following bounds: $$\mu<1.440(4),\quad\gamma>1.9(1),$$ (8) obtained from a fit over $N=25$–$200$. However, such a large value for $\gamma$ is very different from previous results (see, e.g., Table 4 in Ref. Blavatska2009a ) and would be highly unusual. For longer chains, we can only approximate $[Z]$ by assuming the distribution of $Z$ to be log-normal: $$\left[Z\right]\approx e^{[\ln{Z}]+\sigma^{2}_{\ln{Z}}/2}=\mathrel{\mathop{:}}% \left[Z_{\rm{logn}}\right].$$ (9) $\left[Z_{\rm{logn}}\right]$ can be estimated more easily since the “entropies”, $\ln{Z}$, are better behaved. In Fig. 5 we have plotted $\left[\ln{Z}\right]/N$, $\ln{\left[Z\right]}/N$, and $\ln{\left[Z_{\rm{logn}}\right]}/N$ vs $N$. As can be seen, $[Z]\approx\left[Z_{\rm{logn}}\right]$ is fulfilled well for small $N$. For larger $N$, $\ln{\left[Z\right]}$ appears to approach $[\ln{Z}]$, which is a consequence of the aforementioned bias. The exponential of the mean entropy, $[Z_{0}]\mathrel{\mathop{:}}=e^{[\ln{Z}]}$, is supposed to follow a scaling law similar to Eq. (1) Ordemann2000 . Here we have reliable data up to $N=12800$, which should in principle allow for accurate estimates of the “zeroth moments”, $\mu_{0}$ and $\gamma_{0}$. However, a fit analogous to Eq. (7) for $[\ln{Z}]/N$ yields poor results, which remain strongly dependent on the fit range. (We made a similar observation for the two-dimensional case Fricke2012a .) The data are much better described by a function of the form $$\left[\ln{Z}\right]/N=\ln{a}/N+{\left(\ln{\mu_{0}}\right)\left(1+bN^{-\zeta}% \right)}$$ (10) as can be seen in the inset of Fig. 5. Using Eq. (10), the fit results stabilize around $N_{\rm{min}}\approx 800$, which is consistent with the findings for $\nu$. From the range $N=800$–$12800$ we obtain: $a=0.7(4)$, $\ln{\mu_{0}}=0.2715(3)$, $\zeta=0.48(3)$, and $b=1.3(3)$ with $\chi^{2}=0.52$. This would imply: $$[Z_{0}]\sim\mu_{0}^{N(1+b/N^{\zeta})}\quad\rm{with}\quad\mu_{0}=1.3119(3),$$ (11) rather than a scaling law of the form of Eq. (1). There might still be a factor $N^{\gamma_{0}-1}$, but we found no numerical evidence for it. Unfortunately, we cannot do a similar fit for $\ln{[Z]}/N$ for lack of reliable data points. However, if Eq. (11) is correct for $[Z_{0}]$ and assuming Eq. (6), we can infer a similar law for $\left[Z_{\rm{logn}}\right]$ with $\mu_{\rm{logn}}=e^{\ln{\mu_{0}}+A/2}=1.4260(6)$. $[Z]\approx\left[Z_{\rm{logn}}\right]$ would then suggest a scaling law like Eq. (10) for $[Z]$ as well. That approximation is not well-founded, so Eq. (1) might still be correct, but we think that the empirical evidence against it is significant. In fact, there is also little theoretical foundation for Eq. (1) other than the analogy to the regular-lattice case where numerical and analytical support for such a scaling law is strong. The unusual correction term in Eq. (11) may arise from the non-self-averaging properties of the critical clusters. A similar law, but with $b<0$, has been found for conformations of random walks on percolation clusters Giacometti1994a . In any case, our results clearly disprove that $\mu$ results as the undiluted value times the critical concentration, $\mu\approx p_{c}\mu_{\rm{full}}=1.45958(2)$ Xu2014 ; Schram2011 , claimed in Woo1991 ; Ordemann2000 ; Blavatska2009a . By restricting the range of $N$ we again get a similar value, which is probably due to finite-size effects: Initially, the lattice defects and the self-avoidance act independently; only with increasing $N$ does their interplay and the topology of the clusters become relevant. The asymptotic behavior might, for instance, be affected by the distribution of loop sizes on the cluster (backbone), or by the spatial distribution of regions that contribute disproportionately to the entropy, which cannot be gauged by short walks. In summary, we have presented a method to exactly enumerate SAWs of over $10^{4}$ steps on three-dimensional critical percolation clusters. This enabled a firm analysis of the asymptotic scaling behavior of the end-to-end distance with unprecedented accuracy. We revised the established estimate for the leading scaling exponent, verified the hypothesis $\nu_{ic}=\nu_{bb}$, and gave a first estimate for the confluent correction exponent $\Delta$. Direct investigation of the average number of conformations, $[Z]$, was hampered by large deviations, rendering our results less conclusive here. The nature of the distribution of $\ln{Z}$, which resembles a Gaussian whose variance we found to increase linearly with $N$, suggests that information can be gleaned from the mean entropy, $[\ln{Z}]$. Surprisingly, $[\ln{Z}]$ does not behave as expected, which puts the commonly assumed scaling law for $Z$ [Eq. (1)] into question as well. Our findings show that the true asymptotic scaling behavior cannot be observed from system sizes accessible with other numerical tools. This observation may serve as a general lesson of caution regarding numerical studies of systems with strong (fractal) disorder, and calls for further applications of our new method. These may include other types of walks or media (or both). The SAW can be furnished with short-range interactions to model $\Theta$-polymers Roy1990 ; Barat1995 , possibly under stretching force Blavatska2009 ; Singh2009 . One can also add bending stiffness to study semi-flexible polymers Giacometti1992 ; Lekic2011 . The underlying idea of a scale-free partitioning is not even restricted to walk models but could be transferred to spin systems or transport processes. Of course, the necessary condition is that the medium has a weakly connected, self-similar geometry. One can obviously study percolation clusters of different dimensionality, even beyond the upper critical dimension of $D=6$, to gain deeper understanding of the role of the medium’s fractal dimensions. For $p>p_{c}$, the efficiency of our method eventually deteriorates, but it can still beat other methods near the critical concentration Fricke2013 . Further applications could include Ising and Potts clusters, DLA clusters, and possibly certain types of quantum gravity graphs Janke2000 or real-world fractal networks Song2004 . Acknowledgements. This work was funded by the Deutsche Forschungsgemeinschaft (DFG) via FOR 877, Grant No. JA 483/29-1, and SFB/TRR 102 (project B04). We are grateful for further support from Graduate School GSC 185 “BuildMoNa”, Deutsch-Französische Hochschule (DFH) under Grant No. CDFA-02-07, and an AvH Institute Partnership Grant with Lviv, Ukraine. 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Magnetic component of gluon plasma and its viscosity M.N. Chernodub LMPT, CNRS UMR 6083, Fédération Denis Poisson, Université de Tours, F-37200, Tours, France DMPA, University of Gent, Krijgslaan 281, S9, B-9000 Gent, Belgium ITEP, B. Cheremushkinskaya 25, Moscow, 117218, Russia    H. Verschelde DMPA, University of Gent, Krijgslaan 281, S9, B-9000 Gent, Belgium    V.I. Zakharov ITEP, B. Cheremushkinskaya 25, Moscow, 117218, Russia Max-Planck-Institut für Physik, Föhringer Ring 6, 80805 M̈unich, Germany Abstract We discuss the role of the magnetic degrees of freedom of the gluon plasma in its viscosity. The main assumption is that motions of the magnetic component and of the rest of the plasma can be considered as independent. The magnetic component in the deconfined phase is described by a three-dimensional (Euclidean) field theory. The parameters of the theory can be estimated phenomenologically, from the lattice data. It is not ruled out that the magnetic component is superfluid. ††preprint: ITEP-LAT/2009-04 I Introduction The interpretation of heavy ion experiments at RHIC and first-principle lattice simulations suggest that the quark-gluon plasma has quite unusual properties review ; teaney . Contrary to general expectations, at temperatures just above the critical temperature, $T_{c}$, the plasma has properties rather of an ideal fluid than of a weakly interacting gas of quarks and gluons foot1 . The unexpectedly low viscosity is in contrast with thermodynamical properties of the plasma which do not betray much unexpected. Indeed, at high enough temperatures the difference between the observed pressure (energy) density and its perturbative value can be fitted by a $g^{6}(T)$ contribution, $${p(T)_{\mathrm{full}}-p(T)_{\mathrm{pert}}\over T^{4}}~{}\approx~{}{\mathrm{% const}}\cdot g^{6}(T)~{}~{},$$ (1) where $g^{2}(T)$ s the running constant. The fit (1) is expected on theoretical grounds since in the order $g^{6}(T)$ one runs into infrared divergencies which can in fact be treated only non-perturbatively. Thus the constant in the r.h.s. of Eq. (1) is not calculable analytically at this time. Let us consider an idealized picture and assume for the moment that the deviations of the thermodynamical properties from (weak-coupling) perturbative values are small while the viscosity is much lower than its perturbative value. This would suggest that we deal with two (quantum mechanically) independent motions. Indeed if there are two independent fluid components with viscosities $\eta_{1,2}$ then we have additivity of fluidity, i.e., inverse viscosity fluidity : $${1\over\eta_{\mathrm{tot}}}~{}=~{}{(phase~{}space)_{1}\over\eta_{1}}~{}+~{}{(% phase~{}space)_{2}\over\eta_{2}}~{}~{}.$$ (2) If, say, $\eta_{2}\approx 0$ the total value $\eta_{\mathrm{tot}}$ can still be small even if the corresponding phase space factor $(phase~{}space)_{2}$ is small. In this note we will explore the possibility that the magnetic component of the Yang-Mills plasma magncom ; shuryak ; alessio ; atsushi ; sasha ; ratti . provides us with an independent motion in the sense of viscosity, see Eq. (2). Examples of “independent motions” in condensed-matter systems are well known. In the case of ordinary superconductivity, the contribution of Cooper pairs is independent of electrons in the normal state. Closer to our problem, superfluid and ordinary (or dissipative) components of liquids at low temperatures can be treated as independent lifshitz . In a crude approximation, one can understand by magnetic component the 3d field theory which determines the r.h.s. of Eq. (1) at high temperatures. Indeed it is known since long pisarski that at high temperatures it is the 3d field theory corresponding to the zero Matsubara frequency $\omega_{M}=0$ which is to be treated non-perturbatively. In this limit, the temperature dependencies of all the non-perturbative observables can be reconstructed from their dimensions. For example, the string tension of the spatial Wilson line is to be proportional $$\sigma_{3}~{}\sim~{}g_{3}^{4}\,,$$ (3) where $$g^{2}_{3}=g^{2}(T)T\,,$$ (4) is the dimensionally reduced gauge coupling, $g^{2}(T)$ is the running coupling of the 4d Yang-Mills theory. Here $g^{2}(T)$ is assumed to be small enough to serve as a small expansion parameter. Numerically, the scaling laws like (3) set in at temperatures not too much higher than the critical temperature of the deconfining phase transition $T_{c}$ although $g^{2}(T)$ does not seem yet to be small ($T_{c}/\Lambda_{QCD}\approx 1.2$ fingberg ). More precisely, the magnetic component is defined in terms of the magnetic monopoles and center vortices identified on the lattice. These degrees of freedom are commonly believed to be responsible for confinement at low temperatures, for a review see, e.g., greensite . As is argued in Refs. magncom ; shuryak ; alessio ; atsushi ; sasha at $T>T_{c}$ the magnetic degrees of freedom become a part of the Yang-Mills plasma. A subtle point is that magnetic degrees of freedom are studied on the lattices, or in Euclidean space while viscosity is defined most straightforwardly in the Minkowski space. In particular, we are going to treat the magnetic component as an “independent motion” in the Minkowski space. To substantiate (or reject) this hypothesis one needs a continuum-theory interpretation of the lattice defects. Dual models of Yang-Mills theories, see in particular sasha and references therein, seem to provide such an interpretation. Namely, the dual models are formulated in terms of strings living in extra dimensions, for a review see, e.g., aharony . Then there exist various topologically stable solutions in the dual formulation of the Yang-Mills theories. In particular, the observed properties of the lattice vortices and monopoles fit remarkably well the pattern expected within the dual models for the magnetic strings zakharov ; sasha . Moreover, the monopole and vortex pictures get unified since monopoles are to be thought about as 1d defects (trajectories) living on the 2d defects (vortices, or strings) zakharov ; adriano . The monopoles and vortices constitute the magnetic component of the gluon plasma. What is most relevant to our purposes, it is expected theoretically that the magnetic strings become time oriented at $T>T_{c}$ sasha since only time oriented magnetic strings are (nearly) tensionless at $T>T_{c}$. Then the magnetic strings reduce to their projections to a time-slice since the time dependence is trivial. Thus, the solutions of the full 4d theory are mapped into 3d solutions. Consider now the 3d medium of these topological excitations. Since we deal with solutions of the full theory we do not need to consider further, for example, their interaction with gluons. The properties of the 3d medium depend, however, on the interaction of the topological excitations between themselves which is not taken into the account yet. Similarly, in the theory of superconductivity one starts first with an (approximate) solution for the Fermi-liquid at T=0. The Cooper pairs emerge after accounting for (relatively weak) interactions near the edge of the Fermi-sphere. At present, there are no means to clarify interaction among the solutions theoretically and we will rely on the lattice phenomenology at this point. The lattice data are in the Euclidean space, however. In the static approximation for the magnetic defects the continuation from the Euclidean to Minkowski space is trivial and this is the approximation we will use. In the static approximation, the measurements reduce to the measurements on the ground state of the 3d system (which is the same in the Euclidean and Minkowski spaces). There is a spectrum of excitations which determine, in particular the time development of the system. The spectrum is obtained by quantization on the background of the classical solutions. If there is time dependence, the continuation from the Euclidean to Minkowski space is highly non-trivial and very difficult in reality. However, understanding the ground state alone allows one to decide, for example, whether we deal with a superfluid. This is our strategy here. II Magnetic component of the plasma At high temperatures and for static quantities, all the non-perturbative physics is expected to be described in terms of a three-dimensional theory pisarski : $$L~{}=~{}\frac{1}{g_{3}^{2}}\Big{(}\frac{1}{2}{\mathrm{Tr}}\,F_{ij}^{2}+|D_{i}% \Pi^{a}|^{2}+V(\Pi^{2})\Big{)}~{}~{},$$ (5) where $\Pi^{a}$ is a scalar color field ($\omega_{M}=0$ component of the potential $A_{0}^{a}$). As is mentioned above, the dimensional reduction implies simple scaling laws for various quantities. In particular, if one defines magnetic monopoles within the 3d Yang-Mills theory (which is a part of (5)) then the monopole density should scale as $g_{3}^{6}$ in terms of the dimensionally reduced coupling (4). And, indeed, in the 3d Yang-Mills theory grigorev : $$\rho_{3,{\mathrm{mon}}}~{}\approx~{}10^{-7}g_{3}^{6}~{}~{}.$$ (6) According to (4) the density is proportional to $T^{3}$ as would be also the case for massless particles at temperature $T$. However, the density (6) is not given by the Boltzmann distribution in terms of the original temperature. The temperature dependence arises because of the rescaling the fields of the original 4d theory. This trivial observation becomes crucial later. As is mentioned in the Introduction, within the dual model of Yang-Mills theory there exists an absolutely different mechanism of reducing the non-perturbative physics from four to three dimensions foot2 . It is related to the dynamics of strings (which are the basic objects of the dual formulation, or Yang-Mills theories in the infrared). To put it shortly, instead of 2d magnetic surfaces or strings percolating in 4d at $T=0$ one has at $T\geq T_{c}$ particles percolating in 3d. Such a percolation can be adequately described by 3d field theories. This conclusion, as is argued in atsushi , arises within various approaches, such as models of gauge/string dualities sasha , effective field theories, see in particular deforcrand , or approaches based on the lattice data as referred to in atsushi . For the sake of our presentation we will reiterate the main points in the language of the lattice defects, i.e. 2d surfaces (strings) or 1d trajectories (monopoles) mentioned above. It is useful to start from the $T=0$ theory of confinement. Confinement is commonly believed to be due to condensation of the magnetic degrees of freedom. Usually one understands by the magnetic degrees of freedom either Abelian monopoles or $Z_{2}$ vortices, for a review see, e.g., greensite . In fact both projections are manifestations of one and the same non-Abelian object. Phenomenologically, the monopole trajectories cover densely the vortices (2d surfaces) or, vice versa, the vortices can be defined as minimal-area surfaces spanned on the monopole trajectories, for a review see zakharov Both the vortices and monopoles percolate through the vacuum state, i.e. form infinite clusters of the 2d surfaces or 1d trajectories. Important properties of these clusters is that their total length, respectively area, scale in physical units: $$\displaystyle L_{\mathrm{tot}}^{\mathrm{mon}}$$ $$\displaystyle\approx$$ $$\displaystyle~{}{\mathrm{const}}\cdot\Lambda_{QCD}^{3}V^{(4)}_{\mathrm{tot}}\,,$$ (7) $$\displaystyle A_{\mathrm{tot}}^{\mathrm{vort}}$$ $$\displaystyle\approx$$ $$\displaystyle~{}{\mathrm{const}}\cdot\Lambda_{QCD}^{2}V^{(4)}_{\mathrm{tot}}\,,$$ (8) where $V_{\mathrm{tot}}^{(4)}$ is the total 4d volume of the lattice. The detailed picture for confinement does depend on whether one uses monopoles or vortices. The existence of two alternative languages, as we will see, is important within the context of this note. What happens at $T\geq T_{C}$ is that the 4d percolation of the defects is becoming a 3d percolation. In more detail and in the monopole language magncom ; alessio the 4d percolating cluster disappears. Which corresponds to destroying the condensate of the magnetically charged field by temperature. Instead there appear monopole trajectories which are wrapped around the periodic time direction. One can argue that the wrapped trajectories in the Euclidean space correspond to real particles in the Minkowski space magncom . The 3d density of the wrapped trajectories scales indeed in physical units alessio : $$\rho_{\mathrm{wr}}~{}\approx~{}T^{3}f(T,\Lambda_{QCD})~{},$$ (9) where the function $f(T,\Lambda_{QCD})$ is slowly varying at high temperatures. It is crucial however that this function does not depend on the lattice spacing, as it should be for physical objects. Phenomenologically the geometrical picture simplifies actually further. First, already at $T$ close (and larger than) $T_{c}$ the wrapping number is equal to $n_{\mathrm{wr}}=1$ for practically all the wrapped monopole trajectories (while generically $n_{\mathrm{wr}}$ could be equal to any integer). Moreover, the trajectories are rapidly becoming more and more static. Roughly speaking, the approximation of static trajectories is not so bad beginning with $T=T_{c}$ foot3 . As is mentioned above, in the static limit one can consider a 3d picture. In a 3d time slice the monopole trajectories become point-like excitations which can be called instantons (in resemblance to but not in an exact correspondence with the Polyakov model polyakov ). The density (9) becomes the density of the instantons Within the vortices, or string foot4 approach the geometrical picture is similar, with the corresponding change of dimensions. At $T>T_{c}$ the percolating vortices become preferably time oriented and, moreover, simply static to a reasonable approximation. In the static approximation, the 2d surfaces can be replaced by their 1d intersections with a given time-slice. The 1d defects or trajectories correspond to particles, or fields in any number of dimensions. Thus, the vortices reduce to a 3d field. While the time-direction dependence becomes trivial, the percolation in the three spatial dimensions persists. In field theoretic language this means that the corresponding 3d scalar field, $\Sigma_{M}$ has a non-vanishing vacuum expectation value: $$\langle\Sigma_{M}\rangle~{}\neq~{}0~{},$$ (10) for further details see atsushi . To summarize, the 3d physics sets in for non-perturbative effects quite early, at temperatures, just above $T_{c}$ The reason seems to be understood rather within dual models than within a field theoretic formulation. In the region, say, $$T_{c}~{}<T~{}<~{}2T_{c}$$ the parameters of the 3d field theory are to be treated phenomenologically. At larger temperatures simple scaling laws like (6) become valid in many cases. III Temperature dependence Imagine for a moment that the 3d magnetic component of plasma is indeed a liquid, as is argued on the basis of the lattice data magncom ; shuryak . Could it be a superconducting liquid? At first sight, the answer is obviously “no”. Indeed, ordinary superfluidity is destroyed at finite, but low temperature. But now we are discussing temperatures of order $T_{c}~{}\sim~{}200$ MeV. However, why does superfluidity, present at $T=0$, disappear at some $T_{0}$ despite of the fact that the two motions (superfluid and dissipative) are independent? The reason lifshitz could be called a kind of a “nonrelativistic unitarity condition”. The density of the bosons in the condensed mode $n_{0}$ diminishes with temperature, $$[n_{0}(0)-n_{0}(T)]/n_{0}(0)~{}\sim~{}T^{2}\qquad(T\ll T_{0})\,,$$ (11) and at $T=T_{0}$ the boson condensate gets evaporated, $n_{0}(T_{0})=0$. In the case of Yang-Mills theory and in the weak-coupling limit, $g^{2}(T)\to 0$ the total energy/pressure density starts with the Boltzmann’s factors for non-interacting gluons. One calculates then corrections in $g^{2}(T)$ and as less and less uncertainty is left in the perturbative sum the phase space available for the non-perturbative component (let it be superfluid) diminishes. However, as is mentioned above, the uncertainty of the perturbative series does not go down as an inverse power of $T$ foot5 but is proportional only to $g^{6}_{3}(T)$, or to $T^{3}\ln(\Lambda_{QCD}/T)^{-3}$. Thus, the r.h.s. of Eq. (1) of characterizes the phase space of the component which is actually not controlled by temperature and is determined by the physics in the infrared even at $T\to\infty$. Thus, for the phase space factor associated with the magnetic component in Eq. (2) we have $$(phase~{}factor)_{\mathrm{magnetic}}~{}\sim~{}\Big{(}{1\over\ln T}\Big{)}^{3}~% {}~{},$$ (12) which implies an amusing possibility of having superfluidity even at $T\to\infty$ provided that the 3d field theory behind the magnetic component corresponds to a superfluid foot6 . IV Dynamics of the magnetic component The dynamics of the instantons (monopoles) in 3d Yang-Mills theories has been investigated numerically in ishiguro . In the high-temperature limit this is our magnetic component as well. One assumes that monopoles can be treated as Abelian objects with a partition function of a Coulomb gas: $$Z~{}=~{}\sum_{N=0}^{\infty}{\zeta^{N}\over N!}\big{[}\prod_{a}\int d^{3}x^{(a)% }\big{]}\exp\Big{(}-{g_{m}^{2}\over 2}\sum_{a\neq b}q_{a}q_{b}D_{ab}\Big{)}\,,$$ (13) where $D_{ab}\equiv D(x^{(a)}-x^{(b)})$ is the scalar particle propagator, $q_{a,b}$ are the monopole charges in units of elementary magnetic charge $g_{m}$, $|q_{a}|=1$, and $\zeta$ is the fugacity. The model (13) is known to induce confinement of external electric charges polyakov . It was found ishiguro that the lattice data can be fit by the model (13). In particular, one can define the Debye screening mass $m_{D}$ of the magnetic plasma described by (13). However, it turns out that $\rho_{3,{\mathrm{mon}}}/m^{3}_{D}\approx 0.03$ where $\rho_{3,{\mathrm{mon}}}$ is the 3d monopole density. The latter observation is in contradiction with the mechanism of the Debye screening. Another weak point of the model is that it replaces the original non-Abelian action by its Abelian projection. In view of the observation $\rho_{\mathrm{mon}}/m_{D}^{3}\ll 1$ one might be tempted to try an opposite limit and consider the system of the 3d monopoles not as a plasma but rather as a Bose-particles system with low density. Then one of the possibility is the Bose condensation and, as a result, superfluidity lifshitz . The problem is tractable provided that the interaction region is small compared to the volume occupied by a particle on average: $$\rho_{\mathrm{mon}}a^{3}_{\mathrm{sc}}~{}\ll~{}1~{}~{},$$ (14) where $a_{\mathrm{sc}}$ is the scattering length. Also, the interaction is to be repulsive, $a_{\mathrm{sc}}>0$. Otherwise, the slow particles would attract each other and the condensation of the original particles is impossible. In the Abelian projection, monopoles and anti-monopoles attract each other at short distances, and the Bose condensation seems to be ruled out. However, it is more consistent to view the “monopoles” and “antimonopoles” as non-Abelian objects detected through the Abelian projection, see, e.g., adriano and references therein. Then their interaction at short distances should be treated phenomenologically, through the lattice studies. It was observed alessio that both monopole and monopole and monopole and anti-monopole repel each other at short distances. In the language of the scattering lengths: $$a_{\mathrm{mon-mon}}~{}>~{}0,~{}~{}a_{\mathrm{mon-antimon}}~{}>~{}0~{},$$ (15) and there is no contradiction, at this level, with the idea of the Bose condensation. A reservation is that monopole and antimonopole still attract each other at “intermediate distances”, while the monopole-monopole interaction is repulsive at all distances. The attraction, however, is not strong enough to bind the particles and in this sense can be neglected. Moreover, it turned also possible alessio to extract the interaction potentials. In case of the monopole-monopole interaction the estimate is: $$V_{\mathrm{mon-mon}}(r)~{}\sim~{}{1\over r}\exp(-r/\lambda)~{}~{},~{}~{}% \lambda~{}\approx~{}0.1~{}{\mathrm{fm}}~{}.$$ (16) Thus, one can estimate the interaction region as $$V_{\mathrm{int}}\sim(0.2~{}{\mathrm{fm}})^{3}~{}~{}.$$ (17) Whether the interaction region (17) is large or small depends on the density of the monopoles which is also provided by the lattice measurements alessio . As is mentioned above at high temperatures we expect that all the temperature dependencies are trivial. Namely, after rescaling fields and distances the 3d theory does not depend on the temperature at all. In other words, all the observables should be proportional to the corresponding powers of $g^{2}(T)\cdot T$. This should be true also for the interaction region (17) However, numerically there is no much evidence for that. It is more appropriate to say that we are dealing with estimates rather than with exact numbers. The density of the monopoles, on the other hand, is measured with good accuracy. In particular, $$\rho_{\mathrm{mon}}~{}<~{}5~{}{\mathrm{fm}}^{3},~{}~{}~{}~{}T_{c}~{}<T~{}<~{}2% T_{c}~{}.$$ Thus, in this temperature range the approximation $V_{\mathrm{int}}\cdot\rho_{\mathrm{mon}}~{}\ll~{}1$ seems granted. For higher temperatures the issue is more subtle and we postpone a detailed discussion of the numerics. The general impression is that the density is still low enough in the appropriate units. Now we come to the following question. Phenomenologically, we have two descriptions of the medium of 3d excitations. If we start from the 4d monopoles at $T=0$ then they become a 3d gas of instantons at $T>T_{c}$. On other hand, if we start from magnetic strings then they turn into a 3d magnetically charged scalar field with non-trivial vacuum expectations value (10). Both description are obtained in Abelian projections and in this sense both oversimplify the actual non-Abelian picture. However, if we are looking for a possible match of the lattice phenomenology to the theory of superfluidity, the first impression is that the pictures are mutually excluding each other and only one of them has chances to be correct, if any. The good news is that both descriptions can correspond to superfluidity and the two pictures are just dual to each other, see, e.g., schakel . This duality is well known in the theory of superfluidity. One starts with the Hamiltonian for heavy particles: $$\hat{H}~{}=~{}{p^{2}\over 2m}\hat{a}^{+}_{\bf p}\hat{a}_{\bf p}+{U_{0}\over 2V% }\Sigma\hat{a}^{+}_{\bf p_{1}^{{}^{\prime}}}\hat{a}^{+}_{\bf p_{2}^{{}^{\prime% }}}\hat{a}_{\bf p_{1}}\hat{a}_{\bf p_{2}}~{},$$ (18) where $\hat{a},\hat{a}^{+}$ are annihilation and production operators. One performs then the Bogolyubov transformation, $$\hat{a}_{\bf p}~{}=~{}u_{\bf p}\hat{b}_{\bf p}+v_{\bf p}\hat{b}^{+}_{-\bf p},~% {}~{}\hat{a}_{\bf p}^{+}~{}=~{}u_{\bf p}\hat{b}^{+}_{\bf p}+v_{\bf p}\hat{b}_{% -\bf p}~{}~{},$$ (19) where $u_{\bf p},v_{\bf p}$ are coefficients, to diagonalize the Hamiltonian. In terms of the new field (associated with the operators $\hat{b},\hat{b}^{+}$) the spectrum starts with linear, or phonon term: $$\epsilon({p})~{}\approx~{}up~{}~{},$$ where $u$ is the speed of sound. The new field, associated with the operators $\hat{b},\hat{b}^{+}$ has vacuum expectation value and we can identify this field phenomenologically with the field $\Sigma_{M}$, see Eq. (10). Thus, the existence of the two descriptions of the ground state, in terms of the gas of monopoles/instantons and in terms of an infinite cluster of trajectories, or vacuum expectation value (10) is in fact a strong argument in favor of the superfluidity of the magnetic component. Note that the linear spectrum $\epsilon(p)=u\cdot p$ corresponds to a massless field in “relativistic 3d language”. This masslessness can be traced back to the magnetic $U(1)$ symmetry of Hamiltonian (18) in terms of heavy particles. As is mentioned above the actual ‘monopole” and “antimonopole” do not interact at short distances as a particle and antiparticle (the terminology used is rooted in the Abelian projection but the actual non-Abelian dynamics is different). As a result, the would-be massless Goldstone boson does not show up in the spectrum of the 4d theory. Because confinement in the spatial directions is due to breaking of magnetic $U(1)$ to magnetic $\mathbb{Z}_{2}$ a phenomenological 3d model which seems to be more appropriate to describe the condensation (10) is the ’t Hooft model thooft : $$\displaystyle\mathcal{L}$$ $$\displaystyle=$$ $$\displaystyle\partial_{\mu}\varphi\partial_{\mu}\varphi^{*}-M^{2}\varphi% \varphi^{*}-\lambda(\varphi\varphi^{*})^{2}+\frac{\zeta}{2}\left((\varphi)^{2}% +(\varphi^{*})^{2}\right)$$ This model has the magnetic U(1) broken to $\mathbb{Z}_{2}$ by the $\zeta$-term and incorporates 3d confinement (area law for the spatial Wilson loop). There is no Goldstone particle. The model (IV) might describe condensation of the field $\Sigma_{M}$. V Conclusions It appears that the magnetic component of the Yang-Mills plasma could provide an independent component to the fluidity of the plasma. It is most remarkable that the properties of this component, especially its viscosity, are in a way independent of the temperature. The temperature does determine the phase factor which controls the contribution of this component of plasma to the total viscosity (2) but not the partial viscosity itself. The reason is that the magnetic component is directly related to the infrared divergences known since long time in high-temperature field theory. As a result the density of, say, monopoles is not given by a Bose distribution corresponding to the overall temperature and certain mass of the monopole. Instead, it is proportional to $(g^{2}(T)\cdot T)^{3}$ at high temperatures. To adjust phenomenology to this prediction of the theory one can introduce a corresponding chemical potential magncom but this is just another demonstration that the density of the magnetic degrees of freedom is not determined by the standard high-temperature dynamics. The properties of the magnetic component are determined by a 3d field theory. At very high temperature it should be the standard dimensionally reduced Yang-Mills theory. At intermediate temperatures, parameters of the 3d theory can be fitted phenomenologically. In particular, the magnetic component could be superfluid. To clarify whether such a possibility realizes we invoke lattice data. Because the data are obtained in Euclidean space they refer in fact to the ground state. Phenomenologically superfluidity of the magnetic component seems plausible although further data are required to make the evaluation more reliable. The same lattice data seem to fit known Abelian models of three-dimensional confinement as far as long-distance interaction of the constituents is concerned. At short distances, the actual non-Abelian nature of the magnetic degrees of freedom is manifested and turns crucial for the self-consistency of the models. Thus, there is a perspective that the same magnetic component of the Yang-Mills plasma could explain both the 3d confinement and low viscosity of the plasma. Acknowledgements.We are grateful to M. D’Elia, A.S. Gorsky, F.V. Gubarev, A. Nakamura and A. Niemi for useful discussions. The paper was worked out during the visits of V.I.Z. to the Universities of Gent, Belgium and Tours, France. V.I.Z is thankful for the hospitality extended to him during these visits. This work was supported by the STINT Institutional grant IG2004-2 025, by the RFBR 08-02-00661-a, DFG-RFBR 436 RUS, by a grant for scientific schools NSh-679.2008.2, by the Federal Program of the Russian Ministry of Industry, Science and Technology No. 40.052.1.1.1112 and by the Russian Federal Agency for Nuclear Power. 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Feasibility Study of Internal Conversion Electron Spectroscopy of ${}^{229m}$Th Benedict Seiferle Present address: benedict.seiferle@physik.uni-muenchen.de1Ludwig-Maximilians-Universität München, Am Coulombwall 1, Garching, Germany 1    Lars von der Wense 1Ludwig-Maximilians-Universität München, Am Coulombwall 1, Garching, Germany 1    Peter G. Thirolf 1Ludwig-Maximilians-Universität München, Am Coulombwall 1, Garching, Germany 1 (February 1${}^{\mbox{\scriptsize st}}$, 2017) Abstract With an expected energy of 7.8(5) eV, the isomeric first excited state in ${}^{229}$Th exhibits the lowest excitation energy of all known nuclei. Until today, a value for the excitation energy has been inferred only by indirect measurements. In this paper, we propose to use the internal conversion decay channel as a probe for the ground-state transition energy. MatLab-based Monte Carlo simulations have been performed to obtain an estimate of the expected statistics and to test the feasibility of the experiment. From the simulations we conclude, that with the presented methods an energy determination with a precision of better than 0.1 eV is possible. pacs: PACS-key 23.35.+g Isomer decay 1 Introduction The isomeric first excited state of ${}^{229}$Th, called ${}^{229m}$Th, is subject to current vivid research. Among all known nuclear excited states it is the only one that could allow for a direct optical laser excitation, due to its extraordinary low excitation energy of only 7.8(5) eV, corresponding to about 160 nm Beck1 ; Beck2 . This has led to a multitude of proposals for possible applications, including a nuclear optical clock PeikClock ; CampbellClock , that could provide a complementary technology to today’s existing optical atomic clocks, potentially even outperforming the present frequency standards due to the superior resilience of a nuclear clock against external perturbations. It took 40 years until the first direct identification of the ground-state decay of ${}^{229m}$Th via the observation of its internal conversion decay branch LarsNature . However, despite large experimental efforts conducted world-wide Jeet ; Stellmer ; Porsev ; PTB ; Kazakov ; Campbell ; Lars-Jinst ; Ian , the uncertainty in the excitation energy value is still too large to allow for a direct laser excitation. By now, energy values have only been acquired with indirect measurements, investigating nuclear excited states at higher energies and $\gamma$ rays emitted in their decays to the ground- and isomeric state Beck1 ; Beck2 ; 1976 ; 1990 ; 1994 . There are three decay channels of ${}^{229m}$Th to its ground state discussed in literature ElectronicTkalya ; ElectronicEnvironment2 : (i) internal conversion (IC), which proceeds via the emission of an electron with an energy of $E_{e}=E_{I}-E_{B}$, where $E_{I}$ is the isomeric energy and $E_{B}$ is the binding energy of the electron. (ii) $\gamma$ decay, where the emitted photon carries the energy of the isomer, and (iii) bound internal conversion, which proceeds via the excitation of a bound electronic shell state. While a variety of proposals and experimental attempts can be found in literature, aiming at the direct measurement of a VUV photon emitted during the ground-state decay of ${}^{229m}$Th Jeet ; Stellmer ; EPJD , this paper investigates the possibilities of an energy determination via the electron that is emitted during the already experimentally observed internal conversion decay LarsNature ; PRL . Internal conversion electron spectroscopy of ${}^{229m}$Th has several advantages compared to the photonic approach: Due to the large conversion coefficient, the decay via internal conversion is about $10^{9}$ times faster than the photonic decay. Therefore it is possible to trigger the IC decay by neutralizing a ${}^{229m}$Th ion: IC decay is only possible, if the binding energy of an electron in the surrounding of the nucleus is below the isomeric energy. Therefore IC is suppressed in ${}^{229m}$Th ions, but not in the neutral thorium atom Ionization . 2 Simulated Setup In our experimental setup, ${}^{229(m)}$Th ions are produced as $\alpha$ recoil ions from a thin extended ${}^{233}$U source. A decay branch of 2% ends up in the isomeric first excited state of ${}^{229}$Th. ${}^{229(m)}$Th ions are stopped in a buffer-gas stopping cell filled with ultra-pure helium, so that after thermalization of the $\alpha$-recoil ions in the buffer gas and their transport via RF- and DC fields to an extraction nozzle ${}^{229(m)}$Th ions can be extracted into a segmented radio frequency quadrupole (RFQ) ion guide and phase-space cooler structure. The segmented structure of the RFQ allows to form ion bunches. A subsequent quadrupole mass separator removes accompanying $\alpha$-decay daughter products. A detailed study of the experimental setup can be found in Lars_EPJA . A width of the ion bunches of 10 $\mu$s (FWHM${}_{\mbox{\scriptsize TOF}}$), with $\approx$200 ${}^{229(m)}$Th${}^{3+/2+}$ ions per bunch was achieved at a rate of 10 Hz PRL . The ion bunches contain ${}^{229m}$Th, which has already been detected with this setupLarsNature . The general idea of performing internal conversion electron spectroscopy of ${}^{229m}$Th is to guide the extracted, mass-separated and bunched ions towards an electron spectrometer, where they will be neutralized, thereby triggering the IC decay and to measure the subsequently emitted electrons. In this work, an approach is studied, where ${}^{229m}$Th ions are collected directly on a metallic catcher for neutralization. As there are only about 4 ions in the isomeric state extracted per bunch, it is advantageous to use a spectrometer with a high efficiency. Therefore, a magnetic bottle-type spectrometer MagneticBottle , providing an acceptance angle of nearly 4$\pi$, is envisaged. In such a spectrometer, electrons are collected and collimated by a magnetic gradient field. In general, the electron energy is then either inferred by a time-of-flight method or by retarding fields. In case of ${}^{229m}$Th, the time-of-flight method cannot be used, since the lifetime of the internal conversion decay is roughly 10 $\mu$s and thus long compared to the short flight time of electrons ($v\approx 6\times 10^{5}$ m/s for 1 eV electrons). By applying retarding fields, an integrated spectrum is generated, where all electrons are counted, whose energy is sufficient to pass the retarding fields that are applied with high-transmission grids. In the following, not the specific features of the magnetic bottle spectrometer are investigated, but rather measurement principles are discussed to check the feasibility of high-precision internal conversion electron spectroscopy of the ${}^{229}$Th isomer. 2.1 Solid Sample & Surface Effects A simple way to neutralize the ${}^{229m}$Th ions is to collect them on a metallic catcher. If the implantation depth is not too deep111The stopping range for Thorium ions with $E_{\mbox{\scriptsize kin}}=100$ eV ($500$ eV) in gold is $5$ Å ($8$ Å) (values are taken from SRIM simulations SRIM )., electrons emitted during the decay should be able to leave the sample and be measured by the spectrometer. A possible source of background for experiments with slow ions is electron emission via Auger processes (see Hagstrum ). When ion bunches are used, these electrons can be distinguished from electrons emitted by the isomeric decay: Due to its lifetime of $\approx$ 10 $\mu$s, the isomeric decay can be temporally separated from signals potentially generated by ionic impact PRL . It can be expected that the energy distribution of the electrons reflects the electronic structure of the catcher’s surface. The processes are similar to metastable atom electron spectroscopy (MAES) Hagstrum1 ; MAES , that is used to study the electronic structure of surfaces. 2.1.1 Surface Influence In this section, the influence of the catcher material (or sample) is investigated and a possible measurement scheme is shown. The problem of measuring the isomer’s energy is similar, but not identical to ultraviolet photo electron spectroscopy, where a surface is irradiated with UV photons of known energy Hufner . The electronic structure is then inferred from the energy of electrons emitted during the photoelectric effect. Opposed to that, when measuring the isomeric energy, it is the objective to infer the energy of the ”light source” (i.e. isomer) from the energy distribution of the electrons. In the following, a short review of the terms and measurement schemes deployed in photo electron spectroscopy is given (see also Helander ): The work function $W$ of a metallic material is defined as the potential energy difference between the local vacuum level ($E_{\mbox{\scriptsize vac}}$) and the Fermi level ($E_{F}$): $$W=E_{\mbox{\scriptsize vac}}-E_{F}.$$ (1) When two materials (for example the spectrometer and the catcher surface with work functions $W_{S}$ and $W_{C}$) are in electrical contact, their Fermi levels align. If their work functions differ, a potential difference between the local vacuum levels is generated. The contact potential difference amounts to $$\Delta E=W_{C}-W_{S}.$$ (2) In our situation, a contact potential difference may be generated between the catcher and the spectrometer, which is visualized in Fig. 1a. Therefore, if the work function of the spectrometer exceeds the catcher’s work function, an offset voltage $\Delta U$ needs to be applied to the sample in order to give the electrons enough energy to overcome the contact potential difference. Consequently, the contact potential difference is shifted by $\Delta U$ (see Fig. 1b): $$\Delta E^{\prime}=W_{C}-W_{S}+\Delta Ue.$$ (3) Photons of energy $h\nu$ may eject electrons from a metallic surface with a work function $W_{C}$, as long as $h\nu\geq W_{C}$. The energy of such a photo electron is described by a Fermi distribution with a maximum energy of $$E_{C}^{\mbox{\scriptsize max}}=h\nu-W_{C}.$$ (4) Note, that in this definition $E_{C}^{\mbox{\scriptsize max}}$ is given with respect to the local vacuum energy level of the catcher. Given the shifted contact potential difference with the spectrometer, $\Delta E^{\prime}$, the maximum kinetic energy of the electrons measured with the spectrometer amounts to $$\displaystyle E_{S}^{\mbox{\scriptsize max}}$$ $$\displaystyle=$$ $$\displaystyle h\nu-W_{C}+\Delta E^{\prime}$$ (5) $$\displaystyle=$$ $$\displaystyle h\nu-W_{C}+W_{C}-W_{S}+\Delta Ue$$ (6) $$\displaystyle=$$ $$\displaystyle h\nu-W_{S}+\Delta Ue.$$ (7) From the above equation it is obvious, that the energy of the electrons in the end does not depend on the value of the work function of the catcher, but only on the spectrometer work function and the applied offset voltage $\Delta U$. Treating the isomeric decay of ${}^{229m}$Th as a photon with energy $E_{I}=h\nu$ that is coupling to the electrons in the catcher surface, the energy of the isomer $E_{I}$ can be inferred by the following equation: $$E_{I}=E_{S}^{\mbox{\scriptsize max}}+(W_{S}-\Delta Ue),$$ (8) where the expression $(W_{S}-\Delta Ue)$ can be measured with a light source of known energy and using Eq. (7). Therefore, the only remaining surface influence of the sample on the maximum kinetic energy of an electron is the temperature dependent Fermi distribution of $E_{e}^{\mbox{\scriptsize max}}$, but not the value of the sample work function. 3 Simulations In order to get an estimate for the count rates and resulting integrated spectra that can be measured for ${}^{229m}$Th IC electrons emitted from a solid sample, Monte Carlo (MC) simulations were performed with a custom MC code. 3.1 Generation Of Density Distribution The code allows to simulate a predefined kinetic energy distribution $D(E)$ of the electrons. To perform a Monte Carlo simulation, an arbitrary uniform random number $p_{i}\in[0,\ 1]$ is mapped to a kinetic energy $E_{i}$. The abundance of particles with energies $E$ should then finally reflect the kinetic energy distribution $D(E)$. We achieve this by performing the following operations: At first, $E\in\big{[}0,\ E_{\mbox{\scriptsize max}}\big{]}$ is discretized into $N+1$ parts, equally spaced with $\Delta E=E_{\mbox{\scriptsize max}}/N$. For simplicity we define $$\displaystyle E_{i}$$ $$\displaystyle=$$ $$\displaystyle(0.5+i)\cdot\Delta E,\mbox{ and}$$ (9) $$\displaystyle D_{i}$$ $$\displaystyle=$$ $$\displaystyle D(E_{i}).$$ (10) We take $S_{k}=\sum\limits_{i=0}^{k}D_{i}$, with $0<k\leq N$, calculate the sum $I=\sum\limits_{k=0}^{N}S_{k}$ and define the normalized $\hat{S}_{i}=(1/I)\cdot S_{i}$. In this way, $\hat{S}_{i}$ is a number between 0 and 1. If now a random number $p\in[0,\ 1]$ lies between $\hat{S}_{i}$ and $\hat{S}_{i+1}$, then the output energy $E$ should be an arbitrary value between $(E_{i}-0.5\Delta E)$ and $(E_{i}+0.5\Delta E)$: $E=E_{i}+(0.5-r)\cdot\Delta E$, where $r$ is a uniform random number $\in[0,\ 1]$222Note that the values for $S_{i}$ can also be generated analytically, as long as $D(E)$ stays doubly integrable (which is not the case for a Gaussian distribution). Then $S(E)=\int\limits_{0}^{E}dE^{\prime}D(E^{\prime})$ and $I=\int\limits_{0}^{\infty}dE^{\prime}S(E^{\prime})$.. In this way it is possible to map a random number between 0 and 1 to an energy value. The procedure is visualized with an example in Fig. 2, where we used a Gaussian energy distribution ($D(E)=a\cdot\exp(-(E-\mu)^{2}/(2\sigma^{2}))$) with $\mu=2$ and $\sigma=0.5$: $D(E)=\frac{2}{\sqrt{2\pi}}\exp\big{[}-2(E-2)^{2}\big{]}$. To define the values of $\hat{S}_{i}$, the following parameters were used: $N=20$ and $E_{\mbox{\scriptsize max}}=4$. In the following simulations, the values $N=300$ and $E_{\mbox{\scriptsize max}}=4$ eV were used and a superposition of a Gaussian distribution with a Fermi distribution was used as input functions. 3.2 Description of the simulation process In the following section, the simulation process is discussed. The number of isomers per bunch is calculated by taking the 2% branching ratio to the isomeric state from the ${}^{233}$U $\alpha$ decay for the 200 ions contained in one bunch. In this way, we are left with $4$ ions in the isomeric state per bunch. We further assume that only 20% of the ions are collected in the center of the spectrometer and contribute to a spectrum. A collimation efficiency of the magnetic field of 80%, a combined grid transmission of 50% (3 grids with 80% geometrical transmission each) and a detection efficiency of 30% was used. Since a catcher surface is used, only 50% of the IC electrons that are potentially emitted in one hemisphere can be collected. In this way we are left with a total combined detection efficiency of the spectrometer of $\epsilon=6$%. General input values that were used for the simulation, such as the detection properties, count rates, resolution and temperature are listed in Table 1. Background was simulated by calculating the signal-to-background ratio and simulating the dark counts accordingly. As already mentioned in Sect. 2.1.1, the maximum energy of the electrons (with respect to the spectrometer) does not depend on the work function of the sample, but rather on the work function of the spectrometer and the sample offset voltage $(W_{S}-\Delta Ue)$, that needs to obtained from a calibration measurement with a light source of known energy. Nevertheless, the work function of the sample does play a role, since $h\nu=E_{I}\geq W_{C}$ must always be satisfied and the electron energies (with respect to the vacuum level of the sample) are distributed between 0 and $(E_{I}-W_{C})$ eV. As a typical work function of metals, $W_{C}$ was set to 5 eV. In the simulations $(W_{S}-\Delta Ue)$ was set to be equal to 5 eV (this can result no contact potential difference and $0$ V offset voltage). In this way, the electron energies are distributed over a range between 0 and $E_{I}-5$ eV. For the isomer energy $E_{I}$ two values were simulated: 7.8 eV and 7.9 eV. The two values with a difference of 0.1 eV were chosen in order to check the resolving power of this approach. Since the photoelectrons reflect the surface’s electronic structure, one cannot assume a ”bare” Fermi distribution in the simulation. This is taken into account by adding a Gaussian energy distribution to the low-energy part of the electron spectrum, so that only $\approx$ 10% of the electrons have a higher energy than 2 eV. The simulated energy distribution and resulting spectra are shown in Fig. 3. 4 Analysis Of Simulated Spectra Spectra measured with retarding field analyzers are typically differentiated to gain information on the electronic structure of the surfaces. Since we are only interested in the maximum energy of the electron (i.e. the Fermi edge) and a differentiation may lead to large relative errors, we directly fit the indefinite integral of the Fermi function to the high energy part of the integrated spectrum. When a solid sample is used for the neutralization of the ${}^{229m}$Th ions, the subsequently emitted electrons reflect the electronic structure of the surface (see sect. 2.1.1). Especially the maximum energy edge must reflect the Fermi distribution: $$f(E)=\frac{a}{e^{((E-E_{0})/b)}+1},$$ (11) with $b=k_{B}\cdot T$, $E_{0}$ as the maximum kinetic energy of the electrons and $a$ as a constant. Its antiderivative reads $$F(E)=a\cdot\Big{(}b\cdot\ln{\Big{[}\frac{e^{((E-E_{0})/b)}+1}{e^{-E_{0}/b}+1}% \Big{]}}-E\Big{)}+C,$$ (12) where $C$ is a constant. Fig. 4 shows the simulation with voltage increments of 0.04 V over a range of 1.5 V and a measurement time of 180 h (leading to 38 data points, with 4.8 h measurement time per data point) and the corresponding fit plots. For 7.8 eV (7.9 eV) isomeric energy, a maximum kinetic energy of $E=2.80\pm 0.05$ eV ($E=2.91\pm 0.05$ eV) was obtained from the fit. 5 Energy resolution The precision and accuracy of the fit method was probed, by performing 1000 simulations (each with a specific measurement time (90 h, 180 h, 270 h), 1.5 V blocking voltage range and blocking voltage increment of 0.04 V). The fit results of the maximum kinetic energy value were then subtracted from the simulated energy and the difference was filled in a histogram. The histograms and corresponding Gaussian fits333$f(x)=a\cdot\exp(-(x-\mu)^{2}/(2\sigma^{2}))$ was used for the fit function. are plotted in Fig. 5. The fit results are shown in Table 2. Taking these results, it is obvious that the precision and accuracy both improve with longer measurement times. The width $\sigma$ follows a $\sqrt{N}$ law (starting below 50 meV for a measurement time of 90 h). Although it is much smaller than the width of the distribution, there is a shift towards lower energies, which is decreasing with better statistics. 6 Conclusion and Outlook We presented a way to measure the excitation energy of the isomeric first excited state in ${}^{229}$Th via internal conversion electrons. The approach uses a metallic catcher to neutralize ${}^{229m}$Th ions to open the IC decay channel. The analysis of simulated data results in uncertainties of below 0.1 eV in a reasonable measurement time of 180 h ($\hat{=}$ 7.5 d). One needs to mention, that there is no influence of the sample material on the absolute achieved energy value, since only the maximum kinetic energies of the electrons are measured and no specific binding energies of electrons in the sample. Therefore, the cleanliness of the sample surface does not affect the energy measurements. Still different metallic materials can be probed to enhance confidence in the obtained energy value and investigate systematic shifts. Taking all together, we conclude that it is possible to measure the isomeric energy to better than 0.1 eV with the proposed method. We acknowledge fruitful discussions with S. Stellmer, M. Laatiaoui, P. Feulner, G. Dedes and J. Crespo López-Urrutia. 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Entanglement criterion via general symmetric informationally complete measurements Le-Min Lai School of Mathematical Sciences, Capital Normal University, Beijing 100048, China    Tao Li School of Science, Beijing Technology and Business University, Beijing 100048, China    Shao-Ming Fei School of Mathematical Sciences, Capital Normal University, Beijing 100048, China Max Planck Institute for Mathematics in the Sciences, 04103, Leipzig, Germany    Zhi-Xi Wang School of Mathematical Sciences, Capital Normal University, Beijing 100048, China Abstract We study the quantum separability problem by using general symmetric informationally complete measurements and present a separability criterion for arbitrary dimensional bipartite systems. We show by detailed examples that our criterion is more powerful than the existing ones in entanglement detection. Entanglement detection and Separability criterion and General symmetric informationally complete measurements I Introduction Quantum entanglement is one of the most fundamental resources in quantum information processing 1 ; 2 ; 3 . Operational and efficient criteria for the detection of entanglement are of great significance. It has been discussed that the problem of determining whether or not a given state is entangled is NP-hard 4 ; 5 ; 6 . There have been numerous criteria to distinguish quantum entangled states from the separable ones, such as positive partial transposition(PPT) criterion 7 ; 8 ; 9 , realignment criterion 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16 , covariance matrix criterion 17 , correlation matrix criterion 18 ; 19 and generalized form of the correlation matrix criterion 20 . While numerous mathematical tools have been employed in entanglement detection of quantum states, experimental implementation of entanglement detection for unknown quantum states has fewer results 21 ; 22 ; 23 ; 24 . In 25 , the authors connected the separability problem with the concept of mutually unbiased bases (MUBs) 26 for two-qubit, multipartite and continuous-variable quantum systems. These entanglement criteria are shown to be powerful and can be implemented experimentally. After that, the authors in 27 ; 28 generalized such idea and provided an entanglement criterion based on mutually unbiased measurements (MUMs) 29 . Moreover, it has been shown that the criterion based on MUMs is more effective than the criterion based on MUBs. Besides mutually unbiased bases, another intriguing topic in quantum information theory is the symmetric informationally complete positive operator-valued measures (SIC-POVMs). Most of the literature on SIC-POVMs focus on rank-$1$ SIC-POVMs that all the positive operator-valued measure (POVM) elements are proportional to rank-$1$ projectors. Nevertheless, the existence of SIC-POVMs in arbitrary dimension is still an open problem 30 . In Appleby 31 , the author introduced the general symmetric informationally complete measurements (general SIC-POVMs) in which the elements need not to be rank one, and showed that general SIC-POVMs exist in all finite dimensions. Furthermore, Gour and Kalev 32 constructed all general SIC measurements from the generalized Gell-Mann matrices. In Chen et al. and Xi et al. 33 ; 34 , the authors presented separability criteria for both $d$-dimensional bipartite and multipartite systems based on general SIC-POVMs. Very recently, Bae et al. 35 studied entanglement detection via quantum $2$-designs, which includes SIC-POVMs as a special example. In Czartowki et al.36 , the authors investigated the entanglement properties of multipartite systems with tight informationally complete measurements including SIC-POVMs. In addition, the authors in 37 considered a nonlinear entanglement criterion based on SIC-POVMs. In Shang et al.38 , the authors used the SIC-POVMs to derive the entanglement criterion and demonstrated the superiority of the criterion by various examples. In this paper, we study the quantum separability problem by using general SIC-POVMs and present a separability criterion for arbitrary high dimensional bipartite systems of a $d_{A}$-dimensional subsystem and a $d_{B}$-dimensional subsystem. The paper is organized as follows: In Sect. II, we recall some basic notions of SIC-POVMs and general SIC-POVMs. Section III provides an entanglement criterion based on the general SIC-POVMs and some remarks. In Sect. IV, we compare the criterion with the ones in 33 and 38 via detailed examples, and show that our criterion is more efficient than the existing ones. We conclude the paper in Sect. V. II SIC-POVMs and general SIC-POVMs We first review some basic knowledge of symmetric informationally complete measurements and general symmetric informationally complete measurements. A POVM $\{P_{j}\}$ with $d^{2}$ rank-$1$ operators acting on $\mathbb{C}^{d}$ is symmetric informationally complete, if $$\displaystyle P_{j}=\frac{1}{d}|\phi_{j}\rangle\langle\phi_{j}|,$$ (1) $$\displaystyle\sum_{j=1}^{d^{2}}P_{j}=\mathbb{I},$$ (2) where $j=1,2,\dots,d^{2}$, $\mathbb{I}$ is the identity operator, and the vectors $|\phi_{j}\rangle$ satisfy $|\langle\phi_{j}|\phi_{k}\rangle|^{2}={1}/({d+1})$, $j\neq k$. The existence of SIC-POVMs in arbitrary dimension $d$ is still an open problem. Only analytical solutions have been found in dimensions $d=2-24,28,30,31,35,37,39,43,48,124$, and numerical solutions have been found up to dimension $151$ 30 . The concept and constructions of general SIC measurements are introduced in Refs. 31 ; 32 . A set of $d^{2}$ positive semidefinite operators $\{P_{\alpha}\}_{\alpha=1}^{d^{2}}$ on $\mathbb{C}^{d}$ is said to be a general SIC measurements if $$\displaystyle\sum_{\alpha=1}^{d^{2}}P_{\alpha}=\mathbb{I},$$ (3) $$\displaystyle\mathrm{Tr}(P_{\alpha}^{2})=a,$$ (4) $$\displaystyle\mathrm{Tr}(P_{\alpha}P_{\beta})=\frac{1-da}{d(d^{2}-1)},$$ (5) where $\alpha,\beta\in\{1,2,\dots,d^{2}\},\alpha\neq\beta$, the parameter $a$ satisfies $\frac{1}{d^{3}}<a\leqslant\frac{1}{d^{2}}$, and $a=\frac{1}{d^{2}}$ if and only if all $P_{\alpha}$ are rank one, which gives rise to a SIC-POVM. It can be shown that $\mathrm{Tr}(P_{\alpha})=\frac{1}{d}$ for all $\alpha$. Contrasting to SIC-POVM, the general SIC-POVM can be explicitly constructed 32 . Let $\{F_{\alpha}\}_{\alpha=1}^{d^{2}-1}$ be a set of $d^{2}-1$ Hermitian, traceless operators acting on $\mathbb{C}^{d}$, satisfying $\mathrm{Tr}(F_{\alpha}F_{\beta})=\delta_{\alpha,\beta}$. Define $F=\sum_{\alpha=1}^{d^{2}-1}F_{\alpha}$. The $d^{2}$ operators $$\displaystyle P_{\alpha}=\frac{1}{d^{2}}\mathbb{I}+t[F-d(d+1)F_{\alpha}],% \alpha=1,2,\dots,d^{2}-1,$$ (6) $$\displaystyle P_{d^{2}}=\frac{1}{d^{2}}\mathbb{I}+t(d+1)F$$ (7) form a general SIC measurement. Here $t$ should be chosen such that $P_{\alpha}\geqslant 0$ and the parameter $a$ is given by $$\displaystyle a=\frac{1}{d^{3}}+t^{2}(d-1)(d+1)^{3}.$$ (8) III Entanglement detection via general SIC-POVMs Entanglement detection via SIC-POVMs had been discussed in 38 . However, the method subjects to the existence of SIC-POVMs, which is an open problem. Unlike the SIC-POVMs, the general symmetric informationally complete measurements do exist for arbitrary dimension $d$. Consider a quantum state $\rho$ and a general SIC-POVM $\mathcal{M}_{s}=\{P_{\alpha}\}_{\alpha=1}^{d^{2}}$. We have the probability $p_{\alpha}=\langle P_{\alpha}\rangle=\mathrm{Tr}(P_{\alpha}\rho)$ of outcome $\alpha$. Conversely, the quantum state $\rho$ can be reconstructed from these probabilities: $$\displaystyle\rho=\frac{d(d^{2}-1)}{ad^{3}-1}\sum_{\alpha=1}^{d^{2}}p_{\alpha}% P_{\alpha}-\frac{d(1-ad)}{ad^{3}-1}\mathbb{I}.$$ (9) Denote $(e|=(p_{1}\ p_{2}\ \cdots\ p_{d^{2}})$ and $|e)=(p_{1}\ p_{2}\ \cdots\ p_{d^{2}})^{\mathrm{T}}$. Calculation shows that $$\displaystyle\sum_{\alpha=1}^{d^{2}}p_{\alpha}^{2}$$ $$\displaystyle=$$ $$\displaystyle\frac{(ad^{3}-1)\mathrm{Tr}(\rho^{2})+d(1-ad)}{d(d^{2}-1)}$$ (10) $$\displaystyle\leqslant$$ $$\displaystyle\frac{ad^{2}+1}{d(d+1)},$$ where the upper bound is saturated iff $\rho$ is pure. Now consider a $d_{A}\times d_{B}$ bipartite state $\rho$, and two general SIC-POVMs: $\{P_{\alpha}^{A}\}_{\alpha=1}^{d_{A}^{2}}$ with parameter $a_{A}$ and $\{P_{\alpha}^{B}\}_{\alpha=1}^{d_{B}^{2}}$ with parameter $a_{B}$ for the two subsystems, respectively. The linear correlations between $P^{A}$ and $P^{B}$ read $$\displaystyle[\mathcal{P}]_{ij}=\langle P_{i}^{A}\otimes P_{j}^{B}\rangle.$$ (11) Denote $\mathcal{P}$ the matrix with entries given by $[\mathcal{P}]_{ij}$. Theorem 1. If a bipartite state $\rho$ is separable, then $$\displaystyle\|\mathcal{P}\|_{\mathbf{tr}}\leqslant\sqrt{\frac{a_{A}d_{A}^{2}+% 1}{d_{A}(d_{A}+1)}}\sqrt{\frac{a_{B}d_{B}^{2}+1}{d_{B}(d_{B}+1)}},$$ (12) where $\|\mathcal{P}\|_{\mathbf{tr}}=\mathrm{Tr}(\sqrt{\mathcal{P}\mathcal{P}^{% \dagger}})$. Proof. We consider a pure separable state $\rho=\rho_{A}\otimes\rho_{B}$ at first. We have $$\displaystyle\mathcal{P}$$ $$\displaystyle=$$ $$\displaystyle\left(\begin{array}[]{c}p_{A,1}\\ \vdots\\ p_{A,d_{A}^{2}}\\ \end{array}\right)\left(\begin{array}[]{ccc}p_{B,1}&\cdots&p_{B,d_{B}^{2}}\\ \end{array}\right)\equiv|e_{A})(e_{B}|,$$ (13) where $p_{A,i}=\mathrm{Tr}(P_{i}^{A}\rho)$ for $i=1,2,\dots,d_{A}^{2}$ and $p_{B,j}=\mathrm{Tr}(P_{j}^{B}\rho)$ for $j=1,2,\dots,d_{B}^{2}$. Then $$\displaystyle\|\mathcal{P}\|_{\mathbf{tr}}$$ $$\displaystyle=$$ $$\displaystyle(e_{A}\mid e_{A})^{\frac{1}{2}}(e_{B}\mid e_{B})^{\frac{1}{2}}$$ (14) $$\displaystyle\leqslant$$ $$\displaystyle\sqrt{\frac{a_{A}d_{A}^{2}+1}{d_{A}(d_{A}+1)}}\sqrt{\frac{a_{B}d_% {B}^{2}+1}{d_{B}(d_{B}+1)}}.$$ By employing the convexity property of the trace norm, we have $$\displaystyle\|\mathcal{P}\|_{\mathbf{tr}}\leqslant\sqrt{\frac{a_{A}d_{A}^{2}+% 1}{d_{A}(d_{A}+1)}}\sqrt{\frac{a_{B}d_{B}^{2}+1}{d_{B}(d_{B}+1)}}$$ (15) for separable states. ∎ Remark 1. If one takes $a=\frac{1}{d^{2}}$, the criterion of Theorem 1 reduces to the criterion based on SIC-POVM 38 , i.e., if a bipartite state $\rho$ is separable, then $\|\mathcal{P}\|_{\mathbf{tr}}\leqslant\sqrt{\frac{2}{d_{A}(d_{A}+1)}}\sqrt{% \frac{2}{d_{B}(d_{B}+1)}}$. Remark 2. If $d_{A}=d_{B}$ and $a_{A}=a_{B}$, we have $\|\mathcal{P}\|_{\mathbf{tr}}\leqslant\frac{ad^{2}+1}{d(d+1)}$. Furthermore, for a product state, one gets $$\displaystyle J_{a}(\rho)\leqslant\|\mathcal{P}_{s}\|_{\mathbf{tr}},$$ (16) where $J_{a}(\rho)=\sum\limits_{j=1}\limits^{d^{2}}\mathrm{Tr}(P_{j}\otimes Q_{j}\rho)$ 33 . IV Examples Let us consider some examples to illustrate the effectiveness and superiority of our criterion compared with the previously known criterion in 33 and the recently criterion in 38 . Let $\{P_{\alpha}\}_{\alpha=1}^{d^{2}}$ be a set of general SIC-POVM on $\mathbb{C}^{d}$ with the parameter $a$. Let $\bar{P}_{\alpha}$ denote the conjugation of $P_{\alpha}$. Then $\{\bar{P}_{\alpha}\}_{\alpha=1}^{d^{2}}$ is another set of general SIC-POVM with the same parameter $a$. We consider the case of $d=3$. It can be shown that for any nonzero $t\in[-0.012,0.012]$, the following nine matrices $$\displaystyle P_{\alpha}=\frac{1}{9}\mathbb{I}+t(G_{9}-12G_{\alpha}),\ \mathrm% {for}\ \alpha=1,2,\dots,8,$$ (17) $$\displaystyle P_{9}=\frac{1}{9}\mathbb{I}+4tG_{9}$$ (18) form a general SIC-POVM, where $$\displaystyle G_{1}=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}&0&0\\ 0&-\frac{1}{\sqrt{2}}&0\\ 0&0&0\\ \end{array}\right),~{}~{}G_{2}=\left(\begin{array}[]{ccc}0&\frac{1}{\sqrt{2}}&% 0\\ \frac{1}{\sqrt{2}}&0&0\\ 0&0&0\\ \end{array}\right),~{}~{}G_{3}=\left(\begin{array}[]{ccc}0&0&\frac{1}{\sqrt{2}% }\\ 0&0&0\\ \frac{1}{\sqrt{2}}&0&0\\ \end{array}\right),$$ $$\displaystyle G_{4}=\left(\begin{array}[]{ccc}0&-\frac{i}{\sqrt{2}}&0\\ \frac{i}{\sqrt{2}}&0&0\\ 0&0&0\\ \end{array}\right),~{}~{}G_{5}=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{6}}&0&% 0\\ 0&\frac{1}{\sqrt{6}}&0\\ 0&0&-\sqrt{\frac{2}{3}}\\ \end{array}\right),~{}~{}G_{6}=\left(\begin{array}[]{ccc}0&0&0\\ 0&0&\frac{1}{\sqrt{2}}\\ 0&\frac{1}{\sqrt{2}}&0\\ \end{array}\right),$$ $$\displaystyle G_{7}=\left(\begin{array}[]{ccc}0&0&-\frac{i}{\sqrt{2}}\\ 0&0&0\\ \frac{i}{\sqrt{2}}&0&0\\ \end{array}\right),~{}~{}G_{8}=\left(\begin{array}[]{ccc}0&0&0\\ 0&0&-\frac{i}{\sqrt{2}}\\ 0&\frac{i}{\sqrt{2}}&0\\ \end{array}\right),~{}~{}G_{9}=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}+% \frac{1}{\sqrt{6}}&\frac{1-i}{\sqrt{2}}&\frac{1-i}{\sqrt{2}}\\ \frac{1+i}{\sqrt{2}}&-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{6}}&\frac{1-i}{\sqrt{2% }}\\ \frac{1+i}{\sqrt{2}}&\frac{1+i}{\sqrt{2}}&-\sqrt{\frac{2}{3}}\\ \end{array}\right).$$ We can use the two general SIC-POVMs $\{P_{\alpha}\}_{\alpha=1}^{9}$ and $\{\bar{P}_{\alpha}\}_{\alpha=1}^{9}$ to recognize entanglement. Example 1. Consider the isotropic states that are locally unitarily equivalent to a maximally entangled state mixed with white noise: $$\displaystyle\rho_{\mathrm{iso}}=q\mid\Phi^{+}\rangle\langle\Phi^{+}|+(1-q)% \frac{\mathbb{I}}{d^{2}},\ \ \ 0\leqslant q\leqslant 1,$$ (19) where $\mid\Phi^{+}\rangle=\displaystyle\frac{1}{\sqrt{d}}\sum\limits_{i=0}\limits^{d% -1}\mid ii\rangle$. For $d=3$, by directly calculating the correlation entries $[\mathcal{P}]_{ij}=\langle P_{i}\otimes\bar{P}_{j}\rangle$, $i,j=1,\dots,9$, we have $\|\mathcal{P}\|_{\mathbf{tr}}-\frac{9a+1}{12}=96t^{2}(4q-1)>0$ for $\frac{1}{4}<q\leqslant 1$. Thus, our criterion can detect the entanglement of the state $\rho_{\mathrm{iso}}$ for $\frac{1}{4}<q\leqslant 1$. Example 2. Consider the Werner states 39 $$\displaystyle W_{d}\equiv\frac{1}{d^{3}-d}((d-f)\mathbb{I}_{d^{2}}+(df-1)V),$$ (20) where $-1\leqslant f\leqslant 1$, $V=\sum_{i,j=0}^{d-1}|ij\rangle\langle ji|$. $W_{d}$ is entangled if and only if $-1\leqslant f<0$. For $d=3$, by direct calculation we have $\|\mathcal{P}\|_{\mathbf{tr}}-\frac{9a+1}{12}=48t^{2}(\sqrt{(3f-1)^{2}}-2)>0$ for $-1\leq f<-\frac{1}{3}$. Thus our criterion recognizes the entanglement for $-1\leqslant f<-\frac{1}{3}$. From the criterion in 33 , one has $J_{a}(W_{3})-\frac{9a+1}{12}=\sum\limits_{j=1}\limits^{d^{2}}\mathrm{Tr}(P_{j}% \otimes\bar{P}_{j}W_{3})-\frac{9a+1}{12}=36(f-3)t^{2}<0$, since $-1\leqslant f\leqslant 1$. Hence, our criterion is more efficient than the criterion in 33 . Example 3. Consider the $3\times 3$ bound entangled state $\rho^{x}$ 9 , $$\displaystyle\rho^{x}=\left(\begin{array}[]{ccccccccc}\frac{x}{8x+1}&0&0&0&% \frac{x}{8x+1}&0&0&0&\frac{x}{8x+1}\\ 0&\frac{x}{8x+1}&0&0&0&0&0&0&0\\ 0&0&\frac{x}{8x+1}&0&0&0&0&0&0\\ 0&0&0&\frac{x}{8x+1}&0&0&0&0&0\\ \frac{x}{8x+1}&0&0&0&\frac{x}{8x+1}&0&0&0&\frac{x}{8x+1}\\ 0&0&0&0&0&\frac{x}{8x+1}&0&0&0\\ 0&0&0&0&0&0&\frac{x+1}{2(8x+1)}&0&\frac{\sqrt{1-x^{2}}}{2(8x+1)}\\ 0&0&0&0&0&0&0&\frac{x}{8x+1}&0\\ \frac{x}{8x+1}&0&0&0&\frac{x}{8x+1}&0&\frac{\sqrt{1-x^{2}}}{2(8x+1)}&0&\frac{x% +1}{2(8x+1)}\\ \end{array}\right),$$ (21) where $0<x<1$. By straightforward computation, we have that $\|\mathcal{P}\|_{\mathbf{tr}}>\frac{9a+1}{12}$ for $0<x<1$. Thus, our criterion can detect the entanglement for the whole family of $3\times 3$ bound entangled states. In Fig. 1, we plot the value of $|\mathcal{P}\|_{\mathbf{tr}}-\frac{9a+1}{12}$ as a function of $x$ and $t$. Now we add white noise to $\rho^{x}$, and consider $$\displaystyle\rho(x,q)=q\rho^{x}+\frac{(1-q)}{9}\mathbb{I},~{}~{}0\leqslant q% \leqslant 1.$$ (22) Using the same general SIC-POVMs, we have $$\displaystyle J_{a}(\rho(x,q))-\frac{9a+1}{12}$$ $$\displaystyle=\sum\limits_{j=1}\limits^{d^{2}}\mathrm{Tr}(P_{j}\otimes\bar{P}_% {j}\rho(x,q))-\frac{9a+1}{12}$$ (23) $$\displaystyle=24t^{2}(-4+\frac{q+35qx}{1+8x}).$$ From Fig. 2, one can easily find that $J_{a}(\rho(x,q))-\frac{9a+1}{12}<0$ for all permissible $x,\ q$. Thus, our criterion is shown to be more efficient in detecting entanglement of $\rho(x,q)$ than the criterion of Ref. 33 . Moreover, our criterion can successfully detect entanglement for some states that cannot be detected by the criterion in 38 . Let us consider the following three states, $\rho(0.25,0.994)$, $\rho(0.45,0.995)$ and $\rho(0.57,0.996)$, whose entanglement cannot be identified by the criterion in 38 . Denote the correlation entries by $[\mathcal{P}^{\alpha}]_{ij}=\langle P_{i}\otimes\bar{P}_{j}\rangle$, $i,j=1,\dots,9$, $\alpha=1,2,3$, for three states $\rho(0.25,0.994)$, $\rho(0.45,0.995)$ and $\rho(0.57,0.996)$, respectively. We have $\|\mathcal{P}^{1}\|_{\mathbf{tr}}>\frac{9a+1}{12}$, $\|\mathcal{P}^{2}\|_{\mathbf{tr}}>\frac{9a+1}{12}$ and $\|\mathcal{P}^{3}\|_{\mathbf{tr}}>\frac{9a+1}{12}$ in the respective fixed parameter interval, see Fig. 3. Thus, our criterion can successfully detect the entanglement of these states by suitably choosing the general SIC measurements, namely, the parameter $t$. Therefore, in this case our criterion is more efficient than the criterion in 38 . V Conclusion We have presented an entanglement detection criterion constructed from the general SIC measurements. Interestingly, this construction includes the criterion constructed by the SIC measurements as a special case that the parameter $a$ of general SIC measurements is equal to ${1}/{d^{2}}$. 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Minimum Decision Cost for Quantum Ensembles Dorje Brody 111Electronic address: d.brody@tp.ph.ic.ac.uk and Bernhard Meister 222Electronic address: b.mister@ic.ac.uk * Blackett Laboratory, Imperial College, South Kensington, London SW7 2BZ, U.K. $\dagger$ Isac Newton Institute For Mathematical Science, 20 Clarkson Road, Cambridge, CB3 0EH, U.K. (December 6, 2020) Abstract For a given ensemble of $N$ independent and identically prepared particles, we calculate the binary decision costs of different strategies for measurement of polarised spin 1/2 particles. The result proves that, for any given values of the prior probabilities and any number of constituent particles, the cost for a combined measurement is always less than or equal to that for any combination of separate measurements upon sub-ensembles. The Bayes cost, which is that associated with the optimal strategy (i.e., a combined measurement) is obtained in a simple closed form. pacs: PACS Numbers : 02.50.-r, 02.50.Le, 03.65.Bz ††preprint: Imperial/TP/94-95/46 In a problem of experimental design, the task of the experimentalist is to find an optimal observational strategy. Ordinarily, one must choose among different strategies before the data can be obtained, and hence one must perform a preposterior analysis. When the experiment involves a decision among different quantum mechanical states, such an analysis is indeed important, since, unlike the classical case, virtual sampling, i.e., repeated samplings of the same system, are not generally permitted. There are a number of different approaches for finding an optimal strategy. In the information-theoretic approach, one typically determines the strategy that maximises the mutual information (see, e.g., [1]), but this is generally difficult, owing to the nonlinear nature of the Shannon information. In the minimax approach [2], one finds the strategy that minimises the maximum cost (or loss) incurred by the decision among different strategies. When certain a priori knowledge concerning the nature of the state is available, then one may seek a strategy that minimises the expected cost, using a Bayes procedure [2], [3]. In the present Letter, we study the Bayesian approach to a binary decision problem for an ensemble of polarised spin 1/2 particles. First, we briefly introduce the Bayesian approach to quantum hypothesis testing. These notions, developed by Helstrom and others ([4], [5], and [6]), are then applied to obtain the optimal strategy for a Bayes decision between two quantum mechanical pure states, for an ensemble of polarised spin 1/2 particles. In this example, we first study the application of quantum Bayes sequential analysis to the ensemble. The result is then compared with a combined measurement of the entire ensemble, treated as a single composite system. Other strategies consisting of combined measurements of sub-ensembles are also considered. The Bayes solution to the problem demonstrates that the Bayes cost for separate sequential measurements of the individual particles is the same as that of a combined measurement. This result differs from that predicted by Peres and Wootters [7]. Any other strategy turns out to entail a higher expected cost. Nevertheless, we conclude, for the reasons given below, that a combined measurement of the entire ensemble is, in general, an optimal one. First, consider a decision problem requiring a choice among $M$ hypotheses $H_{1},\cdots,H_{M}$ concerning a quantum system. Hypothesis $H_{k}$ asserts that the density operator of the system is $\hat{\rho}_{k}$, $(k=1,\cdots,M)$, and the prior probability of the $j$-th state is $\xi_{j}$, with $$\sum_{k=1}^{M}\xi_{k}\ =\ 1\ .$$ (1) From past experience, one knows that the system is in the $j$-th state with a relative frequency $\xi_{j}$. The self-adjoint operators $\hat{\rho}_{k}$ act on the vectors of a Hilbert space $\cal{H}$, are non-negative definite, and have unit trace. A quantum decision strategy is characterised by a probability operator measure (pom) on ${\cal H}$, i.e., a set of $M$ non-negative definite self-adjoint operators $\Pi_{j}$ satisfying $$\sum_{j=1}^{M}\Pi_{j}\ =\ {\bf 1}\ .$$ (2) If this pom is applied to the system when hypothesis $H_{k}$ is true, then the conditional probability of choosing hypothesis $H_{j}$ is given by $${\rm Pr}(X=j|W=k)\ =\ {\rm Tr}(\rho_{k}\Pi_{j})\ .$$ (3) Here, $X$ denotes the random variable that is to be observed, and $W$, typically being the parameter, is the unknown state of nature. Now, let $C_{ij}$ be the cost of choosing hypothesis $H_{i}$ when $H_{j}$ is true. Then the expected cost of the observational strategy specified by the pom $\{\Pi_{j}\}$ is [4] $${\bar{C}}\ =\ \sum_{i=1}^{M}\sum_{j=1}^{M}\xi_{j}C_{ij}{\rm Tr}(\hat{\rho}_{j}% \Pi_{i})\ \equiv\ {\rm Tr}\sum_{i}^{M}R_{i}\Pi_{i}\ ,$$ (4) where the Hermitian $risk$ $operators$ $R_{i}$ are defined by $$R_{i}\ =\ \sum_{j=1}^{M}\xi_{j}C_{ij}\hat{\rho}_{j}\ .$$ (5) A set $\{\Pi_{j}\}$ of pom that minimises the cost (4), under the constraints (2), is defined as optimal and the cost is Bayes, i.e., ${\bar{C}}={\bar{C}}^{*}$ (the suprescript * here corresponds to the optimal strategy). Necessary and sufficient conditions for the optimality of a pom are known to be [5], [6] the self-adjointness of the operator $$\Upsilon\ =\ \sum_{j=1}^{M}R_{j}\Pi_{j}\ =\ \sum_{j=1}^{M}\Pi_{j}R_{j}$$ (6) and the non-negative definiteness of the operator $R_{j}-\Upsilon$ for all $j=1,\cdots,M$. The minimum expected Bayes cost is thus $${\bar{C}}^{*}(\xi,\{\Pi_{j}^{*}\})\ =\ {\rm Tr}\ \Upsilon\ .$$ (7) In a simple case where $M=2$, i.e., for binary decisions, one can easily verify [4] that the optimal pom is projection valued, and the Bayes cost becomes $$\displaystyle{\bar{C}}^{*}(\xi,\{\Pi_{j}^{*}\})$$ $$\displaystyle=$$ $$\displaystyle\ \xi_{1}C_{11}+\xi_{2}C_{12}$$ (8) $$\displaystyle\ -\xi_{2}(C_{12}-C_{22})\sum_{\eta_{i}>0}\eta_{i}\ ,$$ where $\eta_{i}$ are the eigenvalues of the operator $\hat{\rho}_{2}-\gamma\hat{\rho}_{1}$, with $$\gamma\ =\ \frac{\xi_{1}(C_{21}-C_{11})}{\xi_{2}(C_{12}-C_{22})}\ =\ \frac{\xi% }{1-\xi}\ .$$ (9) Here and in the sequel, we choose a 0-1 cost structure; $C_{ij}=1-\delta_{ij}$, i.e., assign cost 1 to an incorrect decision and 0 to a correct decision. Also, the prior probability for state 1 is given by $\xi_{1}=\xi$, and hence $\xi_{2}=1-\xi$. Now, we consider an experiment where a physicist must estimate (decide) the direction of polarisation of a given ensemble of $N$ spin 1/2 particles, using a Stern-Gerlach (s-g) device. The physicist knows that the particles have been filtered through another s-g device with a magnetic field in the $x-y$ plane at a constant angle $\theta_{1}$ or $\theta_{2}$ from the $x$-axis, and in either case the spin up state has been selected. The physicist can select the orientation angle $\phi$ of the detector relative to the $x$-axis. When the particle passes through the field of the detector magnet, the physicist observes either the spin up (head) or spin down (tail) state, whereupon he must decide between the alternatives $\theta_{1}$ (i.e., the polarisation direction $\theta=\theta_{1}$) and $\theta_{2}$. We do not specify the values of the angles $\{\theta_{k}\}$, but the difference between the two angles is given by $|\theta_{2}-\theta_{1}|=2\delta$. First, consider the case where the physicist performs sequential observations of each individual spin 1/2 particle. Suppose, for simplicity, that $N=1$. The physicist has to decide, either before or after the observation, whether the particle is polarised in the $\theta_{1}$ or $\theta_{2}$ direction. If a decision were to be chosen without any observation, then a Bayes decision against the prior distribution $\xi(W)$ of $W$ (in this case, $W=1$ or $2$) would be optimal. Suppose that $X$ (spin ‘up’ or ‘down’) is observed before a decision is chosen. Then, the decision process for the physicist follows the same procedure as the previous case. However, the difference here is that the distribution of $W$ has changed from the prior to the posterior distribution. Hence, a Bayes decision against the posterior distribution of $W$ is now optimal. The conditional probability for observing the spin up $(+1)$ state, when the state of the system is $\hat{\rho}_{k}$, is given by $$b_{k}(\phi)\ \equiv\ {\rm Pr}(X=+1|W=\theta_{k})\ =\ \cos^{2}(\frac{\theta_{k}% -\phi}{2})\ .$$ (10) If one fixes the angle $\phi$, then the experiment is entirely analogous to a classical coin tossing problem [8], with coins whose bias is given by the above $b_{k}$. However, having the freedom to choose the angle $\phi$ for each value of the prior $\xi$, the physicist must choose an optimal direction given by [9] $$\phi_{opt}(\xi)\ =\ \tan^{-1}\left(\frac{\xi\sin\theta_{1}-(1-\xi)\sin\theta_{% 2}}{\xi\cos\theta_{1}-(1-\xi)\cos\theta_{2}}\right)\ .$$ (11) Hence, we have a problem of tossing quantum coins whose bias is a function of the prior probability $\xi$. Having chosen the optimal angle $\phi_{opt}$, the Bayes decision rule specifies that $\theta_{1}$ is to be chosen if the spin up state is observed, and $\theta_{2}$ otherwise. The Bayes cost against the prior $\xi$, when $N=1$, can easily be obtained by calculating the eigenvalues of $\hat{\rho}_{2}-\gamma\hat{\rho}_{1}$, with the result [9] $${\bar{C}}^{*}(\xi,1)\ =\ \frac{1}{2}\left(1-\sqrt{2\xi^{2}-(2+\cos 2\delta)\xi% +1}\ \right)\ .$$ (12) Now, suppose that $N=2$, and the result of measurement of the first particle has been obtained. As mentioned above, the physicist must follow the same procedures as in the case $N=1$, with the posterior distribution $\xi(\pm)$ instead of the prior $\xi$. From Bayes’ theorem, the posterior probability that $\theta=\theta_{1}$ is given by $$\xi(+)\ =\ \frac{b_{1}(\phi)\cdot\xi}{b_{1}(\phi)\cdot\xi+b_{2}(\phi)\cdot(1-% \xi)}\ $$ (13) or $$\xi(-)\ =\ \frac{(1-b_{1}(\phi))\cdot\xi}{(1-b_{1}(\phi))\cdot\xi+(1-b_{2}(% \phi))\cdot(1-\xi)}\ ,$$ (14) according to the outcome ($+$ or $-$) of the first measurement. The optimal orientation angle, before performing the second measurement, is now given by $\phi_{opt}(\xi(+))$ or $\phi_{opt}(\xi(-))$, accordingly. The Bayes cost for this case ($N=2$) is given by the weighted average, i.e., $${\bar{C}}^{*}\ =\ b_{1}\xi{\bar{C}}^{*}(\xi(+),1)+b_{2}(1-\xi){\bar{C}}^{*}(% \xi(-),1)\ .$$ Next, we consider an arbitrary number $N$ of particles. Again, the procedures are the same as above, except that the prior is now replaced by one of the $2^{N-1}$ posteriors [$\xi(++\cdots+)$, $\cdots$], after observations of $N-1$ particles. In a classical Bayes decision procedure [2], [3], it is difficult (or impossible) to obtain the Bayes cost as a closed function of $N$. The reason is that, first, one must study the tree [10] of the posterior distributions, with branches proliferating as $\sim 2^{N}$. To each branch (i.e., posterior) of the tree, one associates the cost ${\bar{C}}^{*}(\cdot,1)$, and then calculates the weight (probability) for the sequence of outcomes associated with that branch. After these considerations, one can, in principle, obtain the weighted average of the cost, which involves $2^{N-1}$ terms. (Note that, for classical coins, the branches of the posterior tree do recombine and hence proliferate as $\sim N$. However, the weights associated with the branches do not recombine, and therefore one cannot avoid the consideration of $2^{N-1}$ terms.) In the case of our “quantum coins”, the situation appears even worse, since, after each observation, the physicist must turn the device in accordance with formula (11). This results in changing the bias $b_{k}(\phi)$ of the “coins” at each stage, and hence one must also incorporate the bias tree (which proliferates $\sim 2^{N}$). However, it turns out that this optimal orientation forces the posterior tree to recombine into two branches, i.e., $$\xi(n,\pm)\ =\ \frac{1}{2}\left(1\pm\sqrt{1-4\xi(1-\xi)\cos^{2(n-1)}\delta}\ % \right)\ ,$$ (15) where $\pm$ corresponds to the outcome of the last ($n-1$-th) trial being spin up $(+)$ or down $(-)$. This result can be proven by induction as follows. First, for $n=1$, it is easily verified that $\xi(1,\pm)=\xi(\pm)$ as given in (13) and (14). Next, assume that the last ($n-1$-th) outcome of the trial is $(-)$, and that the posterior is given by the above $\xi(n,-)$. Then, if the next trial outcome is $(+)$, follows from Bayes’ theorem, that the posterior distribution, after $n+1$ observations, is given by $$\xi(\cdots-+)\ =\ \frac{b_{1}(\phi)\cdot\xi(n,-)}{b_{1}(\phi)\cdot\xi(n,-)+b_{% 2}(\phi)\cdot(1-\xi(n,-))}\ ,$$ with $\phi=\phi_{opt}(\xi(n,-))$. After some algebra, one can show that the above $\xi(\cdots-+)=\xi(n+1,+)$. The other three cases [$\xi(\cdots--)$, etc.] can also be treated in the same manner. Although the weights for different branches neither recombine in the quantum case, since ${\bar{C}}^{*}(\xi(n,+),1)={\bar{C}}^{*}(\xi(n,-),1)$, the final average cost is just ${\bar{C}}^{*}(\xi(n,\pm),1)$ times the sum of all the different weights (which is just 1), and hence we finally deduce that the Bayes cost for sequential observations is $$\displaystyle{\bar{C}}^{*}(\xi,N)$$ $$\displaystyle=$$ $$\displaystyle\ {\bar{C}}^{*}(\xi(N-1,\pm),1)$$ (16) $$\displaystyle=$$ $$\displaystyle\ \frac{1}{2}\left(1-\sqrt{1-4\xi(1-\xi)\cos^{2N}\delta}\ \right)\ ,$$ for either value of the $N-1$-th outcome ($+$ or $-$). Next, consider the case where the physicist treats the entire ensemble as a single composite system. The total spin of a system with $N$ particles is just $N/2$, and the density operator for a spin $N/2$ particle polarised in the direction ${\bf n}=(\cos\theta,\sin\theta,0)$ is given by $$\left(\hat{\rho}(\theta)\right)_{mn}\ =\ 2^{-N}\sqrt{{}_{N}C_{m}\ {}_{N}C_{n}}% e^{-i(m-n)\theta}\ ,$$ (17) where $(n,m)=0,\cdots,N$. According to the result in (8), one must find the eigenvalues of the matrix $\hat{\rho}_{2}-\gamma\hat{\rho}_{1}$ in order to obtain the Bayes cost. We first show that the matrix $\hat{\rho}_{2}-\gamma\hat{\rho}_{1}$ is of rank two, and thus has only two non-zero eigenvalues. Define two vectors ${\bf u}=\{u_{n}\}$ and ${\bf v}=\{v_{n}\}$ by $$u_{n}\ \equiv\ 2^{-N/2}\sqrt{{}_{N}C_{n}}e^{in\theta_{1}}\ ,$$ (18) and $$v_{n}\ \equiv\ 2^{-N/2}\sqrt{{}_{N}C_{n}}e^{in\theta_{2}}\ .$$ (19) Then, $(\hat{\rho}_{1})_{mn}=u^{*}_{m}u_{n}$ and $(\hat{\rho}_{2})_{mn}=v^{*}_{m}v_{n}$. Since the inner product ${\bf u}\cdot{\bf u}^{*}={\bf v}\cdot{\bf v}^{*}=1$, one obtains $${\hat{\rho}}_{1}{\bf u}^{*}\ =\ \sum_{n}(\hat{\rho}_{1})_{mn}u^{*}_{n}\ =\ {% \bf u}^{*}$$ and similarly, $\hat{\rho}_{2}{\bf v}^{*}={\bf v}^{*}$. Now, let ${\bf w}$ and $\lambda$ be an eigenvector and the corresponding eigenvalue of the matrix $\hat{\rho}_{2}-\gamma\hat{\rho}_{1}$, i.e., $$(\hat{\rho}_{1}-\gamma\hat{\rho}_{2}){\bf w}\ =\ \lambda{\bf w}\ .$$ (20) We may expand the eigenvector ${\bf w}$ in terms of a basis that contains either ${\bf u}^{*}$ or ${\bf v}^{*}$, i.e., ${\bf w}=c_{1}{\bf u}^{*}+{\bf u}^{*}_{\perp}$ or ${\bf w}=c_{2}{\bf v}^{*}+{\bf v}^{*}_{\perp}$. Here, ${\bf u}^{*}_{\perp}$ denotes some vector orthogonal to ${\bf u}^{*}$, and similarly for ${\bf v}^{*}_{\perp}$. However, since $\hat{\rho}_{1}{\bf u}^{*}_{\perp}=\hat{\rho}_{2}{\bf v}^{*}_{\perp}=0$, we have $$\lambda{\bf w}\ =\ c_{1}{\bf u}^{*}-\gamma c_{2}{\bf v}^{*}\ .$$ (21) Therefore, the matrix $\hat{\rho}_{2}-\gamma\hat{\rho}_{1}$ is of rank two, as claimed. On the other hand, if we form the inner product of the two vectors ${\bf w}=c_{1}{\bf u}^{*}+{\bf u}^{*}_{\perp}$ and ${\bf u}$, we obtain $${\bf w}\cdot{\bf u}\ =\ c_{1}\ =\ \frac{c_{1}}{\lambda}-\frac{\gamma}{\lambda}% c_{2}({\bf v}^{*}\cdot{\bf u})\ ,$$ (22) and similarly, $${\bf w}\cdot{\bf v}\ =\ c_{2}\ =\ \frac{c_{1}}{\lambda}({\bf u}^{*}\cdot{\bf v% })-\frac{\gamma}{\lambda}c_{2}\ .$$ (23) Without any loss of generality, we may now set $c_{1}=1$, and then by eliminating $c_{2}$ from the above equations, we obtain the eigenvalues of the matrix $\hat{\rho}_{2}-\gamma\hat{\rho}_{1}$, i. e., $$\lambda_{\pm}\ =\ \frac{1}{2}\left\{(1-\gamma)\pm\sqrt{(1-\gamma)^{2}-4\gamma(% \Delta^{2}-1)}\right\}\ ,$$ (24) where $$\displaystyle\Delta^{2}$$ $$\displaystyle=$$ $$\displaystyle\ ({\bf v}^{*}\cdot{\bf u})({\bf u}^{*}\cdot{\bf v})$$ (25) $$\displaystyle=$$ $$\displaystyle\ \left|2^{-N}\sum_{m=0}^{N}\ {}_{N}C_{m}e^{2im\delta}\right|^{2}% \ =\ \cos^{2N}(\delta)\ .$$ Therefore, the binary Bayes decision cost for a spin $N/2$ particle is $${\bar{C}}^{*}(\xi,N)\ =\ \frac{1}{2}\left(1-\sqrt{1-4\xi(1-\xi)\cos^{2N}\delta% }\ \right)\ .$$ (26) One immediately observes that the above cost (26) is the same as that obtained from sequential analysis, given by (16). Hence, the Bayes solution to our optimisation problem states that a combined measurement is as advantageous as sequential measurements. These two strategies, however, are not the only ones, and many other partially combined measurement procedures are possible. However, in the present formalism of sequential analysis, the only effect of any intermediate measurements, either partially combined or not, consists in updating the posterior distributions. Since the Bayes cost is a monotonically decreasing function of the number of updating steps, this implies that any partially combined measurements will increase the cost. Therefore, we may now conclude that the optimal measurement strategy consists in either performing a combined measurement of the entire ensemble or performing sequential measurements of the individual particles. Any other strategies will result in higher costs. This result is quite different from that expected by Peres and Wootters, who conjectured that sequential measurements can never be as efficient as a combined measurement [7]. However, it is important to note that their conjecture is based upon an information-theoretic approach, and the solution of an optimisation problem using a Bayesian approach can yield a different result. Massar and Popescu [11], on the other hand, have proved the above mentioned conjecture explicitly for the case $N=2$. The method used therein is effectively similar to a Bayesian approach, without the use of the prior distributions. However, when a prior distribution is available, the Bayes solution is known to be optimal in general [2]. If prior knowledge is not available, one can still apply the Bayesian approach, using a non-informative prior. The analysis of such cases is, however, beyond the scope of the present Letter. Throughout the present Letter, we have only considered the cost associated with making decisions. In any practical situation, on the other hand, one must take into consideration other costs (e.g., the observational cost, the cost of analysing the results, etc.). In our example of sequential analysis, for example, at each stage before performing an observation, the physicist must analyse the previous results in order to determine the optimal turning angle. One might argue that [2] the analysing cost can be ignored, since, after all, scientists are so underpaid that the cost of their labors is usually negligible! Nonetheless, the observational costs cannot be ignored in general. Assuming the linearity of the utility function (e.g., that the total cost is just the sum of the decision cost and the observational costs), it is clear that any separate measurements will result in a higher total cost, since the decision cost for optimal sequential measurements (i.e., sequential measurements with optimal angular orientations) can never be lower than that for a combined measurement. Therefore, we conclude, after these considerations, that a combined measurement is optimal in general. In connection with the decision problem for classical coins which was briefly mentioned above, it is interesting to note that all the quantum results obtained by calculating the eigenvalues of the density operators can, in principle, be recovered from purely classical calculations, even for sequential measurements, if and only if the spins of the particles concerned are 1/2. That is, provided one does not perform any combined measurements, the results can be obtained from classical calculations. More details of this, as well as a treatment including the observational costs, may be found in [8]. (See, also [9] for a comparison between classical and quantum coin tossings.) The authors acknowledge their gratitude to J. T. Key, and J. D. Malley for useful discussions of the foregoing topics. [1] Davies, E. B., IEEE Trans. Inform. Theory. IT-24, 596 (1978). [2] Berger, J. O., Statistical Decision Theory and Bayesian Analysis (Springer-Verlag, New York, 1985). [3] DeGroot, M. H., Optimal Statistical Decisions (McGraw-Hill, New York, 1970). [4] Helstrom, C. W., Quantum Detection and Estimation Theory (Academic Press, New York, 1976). [5] Holevo, A. S., J. Multivar. Anal. 3, 337 (1973). [6] Yuen, H. P., Kennedy, R. S., and Lax, M., IEEE Trans. Inform. Theory. IT-21, 125 (1975). [7] Peres, A. and Wootters, W. K., Phys. Rev. Lett. 66, 1119 (1991). [8] Brody, D. and Meister, B., “Bayesian Inference in Quantum Systems”, Imperial College Preprint (1995), submitted to Physica A. [9] Malley, J. D. and Hornstein, J., Statist. Sci. 8, 433 (1993). [10] Lindley, D. V., Making Decisions (Wiley, London, 1971). [11] Massar, S. and Popescu, S., Phys. Rev. Lett. 74, 1259 (1995).

Representation formulae and Monotonicity of the generalized $\mathtt{k}$-Bessel functions Saiful R. Mondal Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Hasa 31982, Saudi Arabia smondal@kfu.edu.sa Abstract. This paper introduces and studies a generalization of the $\mathtt{k}$-Bessel function of order $\nu$ given by $$\mathtt{W}^{\mathtt{k}}_{\nu,c}(x):=\sum_{r=0}^{\infty}\frac{(-c)^{r}}{\Gamma_% {\mathtt{k}}\left(r\mathtt{k}+\nu+\mathtt{k}\right)r!}\left(\frac{x}{2}\right)% ^{2r+\frac{\nu}{\mathtt{k}}}.$$ Representation formulae are derived for $\mathtt{W}^{\mathtt{k}}_{\nu,c}.$ Further the function $\mathtt{W}^{\mathtt{k}}_{\nu,c}$ is shown to be a solution of a second order differential equation. Monotonicity and log-convexity properties for the generalized $\mathtt{k}$-Bessel function $\mathtt{W}^{\mathtt{k}}_{\nu,c}$ are investigated, particularly in the case $c=-1$. Several inequalities, including the Turán-type inequality are established. Key words and phrases:Generalized $\mathtt{k}$ Bessel functions, monotonicity, log-convexity, Turán-type inequality 2010 Mathematics Subject Classification: 33C10, 33B15, 34B30 1. Introductions Motivated with the repeated appearance of the expression $$x(x+\mathtt{k})(x+2\mathtt{k})\ldots(x+(n-1)\mathtt{k})$$ in the combinatorics of creation and annihilation operators [7, 8] and the perturbative computation of Feynman integrals, see [5], a generalization of the well-known Pochhammer symbols is given in [6] as $$(x)_{n,\mathtt{k}}:=x(x+\mathtt{k})(x+2\mathtt{k})\ldots(x+(n-1)\mathtt{k}),$$ for all $\mathtt{k}>0$ and called it as Pochhammer k-symbol. The closely associated functions relate with the Pochhammer symbols are the gamma and beta functions. Thus it is natural to introduce about $\mathtt{k}$-gamma and $\mathtt{k}$-beta function. The $\mathtt{k}$-gamma functions, denoted as $\Gamma_{\mathtt{k}}$, is studied in [6], and defined by $$\displaystyle\Gamma_{\mathtt{k}}(x):=\int_{0}^{\infty}t^{x-1}e^{-\frac{t^{% \mathtt{k}}}{\mathtt{k}}}dt,$$ (1.1) for $\operatorname{Re}(x)>0$. Several properties of the $\mathtt{k}$-gamma functions and it’s applications to generalize other related functions like as $\mathtt{k}$-beta functions, $\mathtt{k}$-digamma functions, can be seen in the articles [6, 17, 16] and references therein. The $k$-digamma functions defined by $\Psi_{\mathtt{k}}:=\Gamma_{\mathtt{k}}^{\prime}/\Gamma_{\mathtt{k}}$ is studied in [17]. This functions have the series representation as $$\displaystyle\Psi_{\mathtt{k}}(t):=\frac{\log(\mathtt{k})-\gamma_{1}}{\mathtt{% k}}-\frac{1}{t}+\sum_{n=1}^{\infty}\frac{t}{n\mathtt{k}(n\mathtt{k}+t)}$$ (1.2) where $\gamma_{1}$ is the Euler-Mascheroni’s constant. A calculation yields $$\displaystyle\Psi_{\mathtt{k}}^{\prime}(t)=\sum_{n=0}^{\infty}\frac{1}{(n% \mathtt{k}+t)^{2}},\quad\mathtt{k}>0\quad\text{and}\quad t>0.$$ (1.3) Clearly, $\Psi_{\mathtt{k}}$ is increasing on $(0,\infty)$. The Bessel function of order $p$ is given by $$\mathtt{J}_{p}(x):=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{\mathrm{\Gamma}{\left(k+% p+1\right)}\mathrm{\Gamma}{\left(k+1\right)}}\left(\frac{x}{2}\right)^{2k+p}$$ (1.4) is a particular solution of the Bessel differential equation $$\displaystyle x^{2}y^{\prime\prime}(x)+xy^{\prime}(x)+(x^{2}-p^{2})y(x)=0.$$ (1.5) Here $\mathrm{\Gamma}$ denote the gamma function. A solution of the modified Bessel equation $$\displaystyle x^{2}y^{\prime\prime}(x)+xy^{\prime}(x)-(x^{2}+{\nu}^{2})y(x)=0.$$ (1.6) yields the modified Bessel function $$\mathtt{I}_{\nu}(x):=\sum_{k=0}^{\infty}\frac{1}{\mathrm{\Gamma}{\left(k+\nu+1% \right)}\mathrm{\Gamma}{\left(k+1\right)}}\left(\frac{x}{2}\right)^{2k+\nu}.$$ (1.7) The Bessel function has gone through several generalizations and investigations, notably in [3, 9]. In [3], generalized Bessel function defined on the complex plane, and obtained sufficient conditions for it to be univalent, starlike, close-to-convex, and convex. This generalization is given by the power series $$\displaystyle\mathcal{W}_{p,b,c}(z)=\sum_{k=0}^{\infty}\frac{(-c)^{k}\left(% \frac{z}{2}\right)^{2k+p+1}}{\mathrm{\Gamma}\left(k+\frac{3}{2}\right)\mathrm{% \Gamma}\left(k+p+\frac{b+2}{2}\right)},\quad\quad p,b,c\in\mathbb{C}.$$ (1.8) In this article we will consider the function defined by the series $$\displaystyle\mathtt{W}_{\nu,c}^{\mathtt{k}}(x):=\sum_{r=0}^{\infty}\frac{(-c)% ^{r}}{\Gamma_{\mathtt{k}}(r\mathtt{k}+\nu+\mathtt{k})r!}\left(\frac{x}{2}% \right)^{2r+\frac{\nu}{\mathtt{k}}}.$$ (1.9) where $\mathtt{k}>0$, $\nu>-1$, and $c\in\mathbb{R}$. Since for $\mathtt{k}\to 1$, the $\mathtt{k}$-Bessel functions $\mathtt{W}_{\nu,1}^{1}$ reduce to the classical Bessel function $J_{\nu}$, while $\mathtt{W}_{\nu,-1}^{1}$ is equivalent to the modified Bessel function $I_{\nu}$. Thus, we call the function $\mathtt{W}_{\nu,c}^{\mathtt{k}}$ as the generalized $\mathtt{k}$-Bessel functions. The basic properties about the $\mathtt{k}$-Bessel functions can be seen in the work by Ghelot et. al. [10, 11, 12]. Turán [18] proved that the Legendre polynomials $P_{n}(x)$ satisfy the determinantal inequality $$\displaystyle\left|\begin{array}[]{cc}P_{n}(x)&P_{n+1}(x)\\ P_{n+1}(x)&P_{n+2}(x)\\ \end{array}\right|\leq 0,\quad-1\leq x\leq 1$$ (1.10) where $n=0,1,2,\ldots$ and equality occurs only if $x=\pm 1$. The above classical result has been extended in several directions, for example, ultraspherical polynomials, Laguerre and Hermite polynomials, Bessel functions of the first kind, modified Bessel functions, Polygamma etc. Karlin and Szegö [13] named determinants such in (1.10) as Turánians. In Section 2, representation formulae and few recurrence relation for $\mathtt{W}_{\nu,c}^{\mathtt{k}}$ will be derived. More importantly, the function $\mathtt{W}_{\nu,c}^{\mathtt{k}}$ is shown to be a solution of a certain differential equation of second order, which reduces to (1.5) and (1.6) in the case $\mathtt{k}=1$ and for particular values of $c.$ At the end of the Section 2, two type integral representations of $\mathtt{W}_{\nu,c}^{\mathtt{k}}$ are also given. Section 3 is devoted to the investigation of monotonicity and log-convexity properties involving the function $\mathtt{W}_{\nu,c}^{\mathtt{k}},$ as well as the ratio between two $\mathtt{k}$-Bessel functions of different order. As a consequence, Turán-type inequalities are deduced. 2. Representations for the $\mathtt{k}$-Bessel function 2.1. Differential equation In this section we will find the differential equations corresponding to the functions $\mathtt{W}_{\nu,c}^{\mathtt{k}}$. Differentiating both side of (1.9) with respect to $x$, it follows that $$\displaystyle x\frac{d}{dx}\mathtt{W}_{\nu,c}^{\mathtt{k}}(x)=\sum_{r=0}^{% \infty}\frac{(-c)^{r}\left(2r+\frac{\nu}{\mathtt{k}}\right)}{\Gamma_{\mathtt{k% }}(r\mathtt{k}+\nu+\mathtt{k})r!}\left(\frac{x}{2}\right)^{2r+\frac{\nu}{% \mathtt{k}}}.$$ (2.1) Recall that the $\mathtt{k}$-gamma function satisfy the relation $\Gamma_{\mathtt{k}}(z+\mathtt{k})=z\Gamma_{\mathtt{k}}(z)$. Now differentiate (2.1) with respect to $x$ and then using this property yields $$\displaystyle x^{2}\frac{d^{2}}{dx^{2}}\mathtt{W}_{\nu,c}^{\mathtt{k}}(x)+x% \frac{d}{dx}\mathtt{W}_{\nu,c}^{\mathtt{k}}(x)$$ $$\displaystyle=\sum_{r=0}^{\infty}\frac{(-c)^{r}\left(2r+\frac{\nu}{\mathtt{k}}% \right)^{2}}{\Gamma_{\mathtt{k}}(r\mathtt{k}+\nu+\mathtt{k})r!}\left(\frac{x}{% 2}\right)^{2r+\frac{\nu}{\mathtt{k}}}$$ $$\displaystyle=\sum_{r=1}^{\infty}\frac{(-c)^{r}4r\left(r+\frac{\nu}{\mathtt{k}% }\right)}{\Gamma_{\mathtt{k}}(r\mathtt{k}+\nu+\mathtt{k})r!}\left(\frac{x}{2}% \right)^{2r+\frac{\nu}{\mathtt{k}}}+\frac{\nu^{2}}{\mathtt{k}^{2}}\sum_{r=0}^{% \infty}\frac{(-c)^{r}}{\Gamma_{\mathtt{k}}(r\mathtt{k}+\nu+\mathtt{k})r!}\left% (\frac{x}{2}\right)^{2r+\frac{\nu}{\mathtt{k}}}$$ $$\displaystyle=\frac{4}{\mathtt{k}}\sum_{r=1}^{\infty}\frac{(-c)^{r}}{\Gamma_{% \mathtt{k}}(r\mathtt{k}+\nu)(r-1)!}\left(\frac{x}{2}\right)^{2r+\frac{\nu}{% \mathtt{k}}}+\frac{\nu^{2}}{\mathtt{k}^{2}}\mathtt{W}_{\nu,c}^{\mathtt{k}}(x)$$ $$\displaystyle=-\frac{cx^{2}}{\mathtt{k}}\mathtt{W}_{\nu,c}^{\mathtt{k}}(x)+% \frac{\nu^{2}}{\mathtt{k}^{2}}\mathtt{W}_{\nu,c}^{\mathtt{k}}(x).$$ Thus we have the following result. Proposition 2.1. Let $\mathtt{k}>0$ and $\nu>-k$. Then the function $\mathtt{W}_{\nu,c}^{\mathtt{k}}$ is the solution of the homogeneous differential equation $$\displaystyle\frac{d^{2}y}{dx^{2}}+x^{-1}\frac{dy}{dx}+\frac{1}{\mathtt{k}^{2}% }\left(c\;\mathtt{k}-\frac{\nu^{2}}{x^{2}}\right)y=0.$$ (2.2) 2.2. Recurrence relations From (2.1), we have $$\displaystyle x\frac{d}{dx}\mathtt{W}_{\nu,c}^{\mathtt{k}}(x)$$ $$\displaystyle=\frac{1}{\mathtt{k}}\sum_{r=0}^{\infty}\frac{(-c)^{r}\left(2r% \mathtt{k}+{\nu}\right)}{\Gamma_{\mathtt{k}}(r\mathtt{k}+\nu+\mathtt{k})r!}% \left(\frac{x}{2}\right)^{2r+\frac{\nu}{\mathtt{k}}}$$ $$\displaystyle=\frac{\nu}{\mathtt{k}}\sum_{r=0}^{\infty}\frac{(-c)^{r}\left(2r% \mathtt{k}+{\nu}\right)}{\Gamma_{\mathtt{k}}(r\mathtt{k}+\nu+\mathtt{k})r!}% \left(\frac{x}{2}\right)^{2r+\frac{\nu}{\mathtt{k}}}+2\sum_{r=1}^{\infty}\frac% {(-c)^{r}}{\Gamma_{\mathtt{k}}(r\mathtt{k}+\nu+\mathtt{k})(r-1)!}\left(\frac{x% }{2}\right)^{2r+\frac{\nu}{\mathtt{k}}}$$ $$\displaystyle=\frac{\nu}{\mathtt{k}}\mathtt{W}_{\nu,c}^{\mathtt{k}}(x)+2\sum_{% r=0}^{\infty}\frac{(-c)^{r+1}}{\Gamma_{\mathtt{k}}(r\mathtt{k}+\nu+2\mathtt{k}% )r!}\left(\frac{x}{2}\right)^{2r+2+\frac{\nu}{\mathtt{k}}}$$ $$\displaystyle=\frac{\nu}{\mathtt{k}}\mathtt{W}_{\nu,c}^{\mathtt{k}}(x)-xc% \mathtt{W}_{\nu+\mathtt{k},c}^{\mathtt{k}}(x).$$ Thus, we have the difference equation $$\displaystyle x\frac{d}{dx}\mathtt{W}_{\nu,c}^{\mathtt{k}}(x)=\frac{\nu}{% \mathtt{k}}\mathtt{W}_{\nu,c}^{\mathtt{k}}(x)-xc\mathtt{W}_{\nu+\mathtt{k},c}^% {\mathtt{k}}(x).$$ (2.3) Again rewrite (2.1) as $$\displaystyle x\frac{d}{dx}\mathtt{W}_{\nu,c}^{\mathtt{k}}(x)$$ $$\displaystyle=\frac{1}{\mathtt{k}}\sum_{r=0}^{\infty}\frac{(-c)^{r}\left(2r% \mathtt{k}+2{\nu}\right)-\nu}{\Gamma_{\mathtt{k}}(r\mathtt{k}+\nu+\mathtt{k})r% !}\left(\frac{x}{2}\right)^{2r+\frac{\nu}{\mathtt{k}}}$$ $$\displaystyle=-\frac{\nu}{\mathtt{k}}\sum_{r=0}^{\infty}\frac{(-c)^{r}}{\Gamma% _{\mathtt{k}}(r\mathtt{k}+\nu+\mathtt{k})r!}\left(\frac{x}{2}\right)^{2r+\frac% {\nu}{\mathtt{k}}}+2\sum_{r=0}^{\infty}\frac{(-c)^{r}\left(r\mathtt{k}+{\nu}% \right)}{\Gamma_{\mathtt{k}}(r\mathtt{k}+\nu+\mathtt{k})r!}\left(\frac{x}{2}% \right)^{2r+\frac{\nu}{\mathtt{k}}}$$ $$\displaystyle=-\frac{\nu}{\mathtt{k}}\mathtt{W}_{\nu,c}^{\mathtt{k}}(x)+\frac{% x}{\mathtt{k}}\sum_{r=0}^{\infty}\frac{(-c)^{r}}{\Gamma_{\mathtt{k}}(r\mathtt{% k}+\nu-\mathtt{k}+\mathtt{k})r!}\left(\frac{x}{2}\right)^{2r+\frac{\nu-\mathtt% {k}}{\mathtt{k}}}$$ $$\displaystyle=-\frac{\nu}{\mathtt{k}}\mathtt{W}_{\nu,c}^{\mathtt{k}}(x)+\frac{% x}{\mathtt{k}}\mathtt{W}_{\nu-\mathtt{k},c}^{\mathtt{k}}(x).$$ This give us the second difference equation as $$\displaystyle x\frac{d}{dx}\mathtt{W}_{\nu,c}^{\mathtt{k}}(x)=\frac{x}{\mathtt% {k}}\mathtt{W}_{\nu-\mathtt{k},c}^{\mathtt{k}}(x)-\frac{\nu}{\mathtt{k}}% \mathtt{W}_{\nu,c}^{\mathtt{k}}(x).$$ (2.4) Thus (2.3) and (2.4) leads to the following recurrence relations. Proposition 2.2. Let $\mathtt{k}>0$ and $\nu>-\mathtt{k}$. Then $$\displaystyle 2\nu\mathtt{W}_{\nu,c}^{\mathtt{k}}(x)$$ $$\displaystyle=x\mathtt{W}_{\nu-\mathtt{k},c}^{\mathtt{k}}(x)+xc\mathtt{k}% \mathtt{W}_{\nu+\mathtt{k},c}^{\mathtt{k}}(x),$$ (2.5) $$\displaystyle\mathtt{W}^{\mathtt{k}}_{\nu-\mathtt{k},c}(x)$$ $$\displaystyle=\frac{2}{x}\sum_{r=0}^{\infty}(-1)^{r}(\nu+2r\mathtt{k})\mathtt{% W}^{\mathtt{k}}_{\nu+2r\mathtt{k},c}(x)$$ (2.6) $$\displaystyle\frac{d}{dx}\left(x^{\frac{\nu}{\mathtt{k}}}\mathtt{W}^{\mathtt{k% }}_{\nu,c}(x)\right)$$ $$\displaystyle=\frac{x^{\frac{\nu}{\mathtt{k}}}}{\mathtt{k}}\mathtt{W}^{\mathtt% {k}}_{\nu-\mathtt{k},c}(x)$$ (2.7) $$\displaystyle\frac{d}{dx}\left(x^{-\frac{\nu}{\mathtt{k}}}\mathtt{W}^{\mathtt{% k}}_{\nu,c}(x)\right)$$ $$\displaystyle=-cx^{-\frac{\nu}{\mathtt{k}}}\mathtt{W}^{\mathtt{k}}_{\nu+% \mathtt{k},c}(x)$$ (2.8) $$\displaystyle\frac{d^{m}}{dx^{m}}\left(\mathtt{W}^{\mathtt{k}}_{\nu,c}(x)\right)$$ $$\displaystyle=\frac{1}{2^{m}\mathtt{k}^{m}}\sum_{n=0}^{m}(-1)^{n}\left(\begin{% array}[]{c}m\\ n\\ \end{array}\right)c^{n}\mathtt{k}^{n}\mathtt{W}^{\mathtt{k}}_{\nu-m\mathtt{k}+% 2n\mathtt{k},c}(x)\quad\text{for all}\quad m\in\mathbb{N}.$$ (2.9) Proof. The relation (2.5) follows by subtracting (2.4) from (2.3). Next to establish (2.6), lets rewrite (2.5) as $$\displaystyle\mathtt{W}_{\nu-\mathtt{k},c}^{\mathtt{k}}(x)+c\mathtt{k}\mathtt{% W}_{\nu+\mathtt{k},c}^{\mathtt{k}}(x)=2\frac{\nu}{x}\mathtt{W}_{\nu,c}^{% \mathtt{k}}(x)$$ (2.10) Now multiply both side of (2.10) by $-c\mathtt{k}$ and replace $\nu$ by $\nu+2\mathtt{k}$. Then we have $$\displaystyle-c\mathtt{k}\mathtt{W}_{\nu+\mathtt{k},c}^{\mathtt{k}}(x)-c^{2}% \mathtt{k}^{2}\mathtt{W}_{\nu+3\mathtt{k},c}^{\mathtt{k}}(x)=-2c\mathtt{k}% \frac{\nu+2\mathtt{k}}{x}\mathtt{W}_{\nu+2\mathtt{k},c}^{\mathtt{k}}(x).$$ (2.11) Similarly, multiplying both side of (2.10) by $c^{2}\mathtt{k}^{2}$ and replacing $\nu$ by $\nu+4\mathtt{k}$, we get $$\displaystyle c^{2}\mathtt{k}^{2}\mathtt{W}_{\nu+3\mathtt{k},c}^{\mathtt{k}}(x% )+c^{3}\mathtt{k}^{3}\mathtt{W}_{\nu+5\mathtt{k},c}^{\mathtt{k}}(x)=2c^{2}% \mathtt{k}^{2}\frac{\nu+4\mathtt{k}}{x}\mathtt{W}_{\nu+4\mathtt{k},c}^{\mathtt% {k}}(x).$$ (2.12) If we continue like the above and addition them leads to (2.6). From the definition (1.9) it is clear that $$\displaystyle x^{\frac{\nu}{\mathtt{k}}}\mathtt{W}_{\nu,c}^{\mathtt{k}}(x)=% \sum_{r=0}^{\infty}\frac{(-c)^{r}}{\Gamma_{\mathtt{k}}(r\mathtt{k}+\nu+\mathtt% {k})2^{2r+\frac{\nu}{\mathtt{k}}}r!}\left(x\right)^{2r+\frac{2\nu}{\mathtt{k}}}.$$ (2.13) The derivative of (2.13) with respect to $x$ yields $$\displaystyle\frac{d}{dx}\left(x^{\frac{\nu}{\mathtt{k}}}\mathtt{W}_{\nu,c}^{% \mathtt{k}}(x)\right)$$ $$\displaystyle=\sum_{r=0}^{\infty}\frac{(-c)^{r}(2r+\frac{2\nu}{\mathtt{k}})}{% \Gamma_{\mathtt{k}}(r\mathtt{k}+\nu+\mathtt{k})2^{2r+\frac{\nu}{\mathtt{k}}}r!% }\left(x\right)^{2r+\frac{2\nu}{\mathtt{k}}-1}$$ $$\displaystyle=\frac{x^{\frac{\nu}{\mathtt{k}}}}{\mathtt{k}}\sum_{r=0}^{\infty}% \frac{(-c)^{r}}{\Gamma_{\mathtt{k}}(r\mathtt{k}+\nu)r!}\left(\frac{x}{2}\right% )^{2r+\frac{\nu}{\mathtt{k}}-1}=\frac{x^{\frac{\nu}{\mathtt{k}}}}{\mathtt{k}}% \mathtt{W}_{\nu-\mathtt{k},c}^{\mathtt{k}}(x).$$ Similarly, $$\displaystyle\frac{d}{dx}\left(x^{-\frac{\nu}{\mathtt{k}}}\mathtt{W}_{\nu,c}^{% \mathtt{k}}(x)\right)$$ $$\displaystyle=\sum_{r=1}^{\infty}\frac{(-c)^{r}2r}{\Gamma_{\mathtt{k}}(r% \mathtt{k}+\nu+\mathtt{k})2^{2r+\frac{\nu}{\mathtt{k}}}r!}\left(x\right)^{2r-1}$$ $$\displaystyle=x^{-\frac{\nu}{\mathtt{k}}}\sum_{r=1}^{\infty}\frac{(-c)^{r}}{% \Gamma_{\mathtt{k}}(r\mathtt{k}+\nu+\mathtt{k})(r-1)!}\left(\frac{x}{2}\right)% ^{2r+\frac{\nu}{\mathtt{k}}-1}$$ $$\displaystyle=x^{-\frac{\nu}{\mathtt{k}}}\sum_{r=0}^{\infty}\frac{(-c)^{r+1}}{% \Gamma_{\mathtt{k}}(r\mathtt{k}+\nu+2\mathtt{k})r!}\left(\frac{x}{2}\right)^{2% r+\frac{\nu}{\mathtt{k}}+1}=-cx^{-\frac{\nu}{\mathtt{k}}}\mathtt{W}_{\nu+% \mathtt{k},c}^{\mathtt{k}}(x).$$ The identity (2.9) can be proved by using mathematical induction on $m$. Recall that $$\left(\begin{array}[]{c}r\\ r\\ \end{array}\right)=\left(\begin{array}[]{c}r\\ 0\\ \end{array}\right)=1\quad\text{and}\quad\left(\begin{array}[]{c}r\\ n\\ \end{array}\right)+\left(\begin{array}[]{c}r\\ n-1\\ \end{array}\right)=\left(\begin{array}[]{c}r+1\\ n\\ \end{array}\right)$$ For $m=1$, the proof of the identity (2.9) is equivalent to show that $$\displaystyle 2\mathtt{k}\frac{d}{dx}\mathtt{W}_{\nu,c}^{\mathtt{k}}(x)$$ $$\displaystyle=\mathtt{W}_{\nu-\mathtt{k},c}^{\mathtt{k}}(x)-c\mathtt{k}\mathtt% {W}_{\nu+\mathtt{k},c}^{\mathtt{k}}(x).$$ (2.14) This above relation can be obtained by simple adding (2.3) and (2.4). Thus, identity (2.9) hold for $m=1$. Assume that the identity (2.9) also holds for any $m=r\geq 2$, i.e. $$\displaystyle\frac{d^{r}}{dx^{r}}\left(\mathtt{W}^{\mathtt{k}}_{\nu,c}(x)\right)$$ $$\displaystyle=\frac{1}{2^{m}\mathtt{k}^{r}}\sum_{n=0}^{r}(-1)^{n}\left(\begin{% array}[]{c}r\\ n\\ \end{array}\right)c^{n}\mathtt{k}^{n}\mathtt{W}^{\mathtt{k}}_{\nu-r\mathtt{k}+% 2n\mathtt{k},c}(x).$$ This implies for $m=r+1$, $$\displaystyle\frac{d^{r+1}}{dx^{r+1}}\left(\mathtt{W}^{\mathtt{k}}_{\nu,c}(x)\right)$$ $$\displaystyle=\frac{1}{2^{r}\mathtt{k}^{r}}\sum_{n=0}^{r}(-1)^{n}\left(\begin{% array}[]{c}r\\ n\\ \end{array}\right)c^{n}\mathtt{k}^{n}\frac{d}{dr}\mathtt{W}^{\mathtt{k}}_{\nu-% r\mathtt{k}+2n\mathtt{k},c}(x).$$ $$\displaystyle=\frac{1}{2^{r+1}\mathtt{k}^{r+1}}\sum_{n=0}^{r}(-1)^{n}\left(% \begin{array}[]{c}r\\ n\\ \end{array}\right)c^{n}\mathtt{k}^{n}\bigg{(}\mathtt{W}^{\mathtt{k}}_{\nu-(r+1% )\mathtt{k}+2n\mathtt{k},c}(x)-c\mathtt{k}\mathtt{W}^{\mathtt{k}}_{\nu-(r-1)% \mathtt{k}+2n\mathtt{k},c}(x)\bigg{)}.$$ $$\displaystyle=\frac{1}{2^{r+1}\mathtt{k}^{r+1}}\sum_{n=0}^{r}(-1)^{n}\left(% \begin{array}[]{c}r\\ n\\ \end{array}\right)c^{n}\mathtt{k}^{n}\mathtt{W}^{\mathtt{k}}_{\nu-(r+1)\mathtt% {k}+2n\mathtt{k},c}(x)$$ $$\displaystyle\quad\quad\quad-\frac{1}{2^{r+1}\mathtt{k}^{r+1}}\sum_{n=0}^{r}(-% 1)^{n}\left(\begin{array}[]{c}r\\ n\\ \end{array}\right)c^{n+1}\mathtt{k}^{n+1}\mathtt{W}^{\mathtt{k}}_{\nu-(r-1)% \mathtt{k}+2n\mathtt{k},c}(x).$$ $$\displaystyle=\frac{1}{2^{r+1}\mathtt{k}^{r+1}}\bigg{[}\mathtt{W}^{\mathtt{k}}% _{\nu-(r+1)\mathtt{k},c}(x)+\sum_{n=1}^{r}(-1)^{r}\left(\left(\begin{array}[]{% c}r\\ n\\ \end{array}\right)+\left(\begin{array}[]{c}r\\ n-1\\ \end{array}\right)\right)\mathtt{W}^{\mathtt{k}}_{\nu-(r+1)\mathtt{k}+2n% \mathtt{k},c}(x)\bigg{.}$$ $$\displaystyle\quad\quad\quad\quad\quad\quad\quad\bigg{.}-(-1)^{r}c^{r+1}% \mathtt{k}^{r+1}\mathtt{W}^{\mathtt{k}}_{\nu+(r+1)\mathtt{k},c}(x)\bigg{]}$$ $$\displaystyle=\frac{1}{2^{r+1}\mathtt{k}^{r+1}}\bigg{[}\left(\begin{array}[]{c% }r+1\\ 0\\ \end{array}\right)\mathtt{W}^{\mathtt{k}}_{\nu-(r+1)\mathtt{k},c}(x)+\sum_{n=1% }^{r}(-1)^{r}\left(\begin{array}[]{c}r+1\\ n\\ \end{array}\right)\mathtt{W}^{\mathtt{k}}_{\nu-(r+1)\mathtt{k}+2n\mathtt{k},c}% (x)\bigg{.}$$ $$\displaystyle\quad\quad\quad\quad\quad\quad\bigg{.}+(-1)^{r+1}\left(\begin{% array}[]{c}r+1\\ r+1\\ \end{array}\right)c^{r+1}\mathtt{k}^{r+1}\mathtt{W}^{\mathtt{k}}_{\nu-(r+1)% \mathtt{k}+2(r+1)\mathtt{k},c}(x)\bigg{]}$$ $$\displaystyle=\frac{1}{2^{r+1}\mathtt{k}^{r+1}}\sum_{n=0}^{r+1}(-1)^{r}\left(% \begin{array}[]{c}r+1\\ n\\ \end{array}\right)\mathtt{W}^{\mathtt{k}}_{\nu-(r+1)\mathtt{k}+2n\mathtt{k},c}% (x).$$ Hence the conclusion by mathematical induction on $m$. ∎ 2.3. Integral representation Now we will derive two type integral representation of the functions $\mathtt{W}_{\nu,c}^{\mathtt{k}}$. For this purpose we need to recall the $\mathtt{k}$-Beta functions from [6]. The $\mathtt{k}$ version of the beta functions is defined by $$\displaystyle\mathtt{B}_{\mathtt{k}}(x,y)=\frac{\Gamma_{\mathtt{k}}(x)\Gamma_{% \mathtt{k}}(y)}{\Gamma_{\mathtt{k}}(x+y)}=\frac{1}{\mathtt{k}}\int_{0}^{1}t^{% \frac{x}{\mathtt{k}}-1}(1-t)^{\frac{y}{\mathtt{k}}-1}dt.$$ (2.15) Substituting $t$ by $t^{2}$ on the right hand integration in (2.15), it follows that $$\displaystyle\mathtt{B}_{\mathtt{k}}(x,y)=\frac{2}{\mathtt{k}}\int_{0}^{1}t^{% \frac{2x}{\mathtt{k}}-1}(1-t^{2})^{\frac{y}{\mathtt{k}}-1}dt.$$ (2.16) Let, $x=(r+1)\mathtt{k}$ and $y=\nu$. Then from (2.15) and (2.16), we have $$\displaystyle\frac{1}{\Gamma_{\mathtt{k}}(r\mathtt{k}+\nu+\mathtt{k})}=\frac{2% }{\Gamma_{\mathtt{k}}((r+1)\mathtt{k})\Gamma_{\mathtt{k}}(\nu)}\int_{0}^{1}t^{% 2r+1}(1-t^{2})^{\frac{\nu}{\mathtt{k}}-1}dt.$$ (2.17) Owing to [6], we have the identity $\Gamma_{\mathtt{k}}(\mathtt{k}x)=\mathtt{k}^{x-1}\Gamma(x)$. This gives $$\displaystyle\frac{1}{\Gamma_{\mathtt{k}}(r\mathtt{k}+\nu+\mathtt{k})}=\frac{2% }{\mathtt{k}^{r}\Gamma(r+1)\Gamma_{\mathtt{k}}(\nu)}\int_{0}^{1}t^{2r+1}(1-t^{% 2})^{\frac{\nu}{\mathtt{k}}-1}dt.$$ (2.18) Now (1.9) and (2.18) together yield $$\displaystyle\mathtt{W}_{\nu,c}^{\mathtt{k}}(x)$$ $$\displaystyle=\frac{2}{\Gamma_{\mathtt{k}}(\nu)}\left(\frac{x}{2}\right)^{% \frac{\nu}{\mathtt{k}}}\int_{0}^{1}t(1-t^{2})^{\frac{\nu}{\mathtt{k}}-1}\sum_{% r=0}^{\infty}\frac{(-c)^{r}}{\Gamma(r+1)r!}\left(\frac{xt}{2\sqrt{\mathtt{k}}}% \right)^{2r}dt.$$ $$\displaystyle=\frac{2}{\Gamma_{\mathtt{k}}(\nu)}\left(\frac{x}{2}\right)^{% \frac{\nu}{\mathtt{k}}}\int_{0}^{1}t(1-t^{2})^{\frac{\nu}{\mathtt{k}}-1}% \mathcal{W}_{0,1,c}\left(\frac{xt}{\sqrt{\mathtt{k}}}\right)dt,$$ (2.19) where $\mathcal{W}_{p,b,c}$ is defined in (1.8). Now for the second integral representation, substitute $x=r+\mathtt{k}/2$ and $y=\nu+\mathtt{k}/2$ in (2.16). Then, (2.17) can be rewrite as $$\displaystyle\frac{1}{\Gamma_{\mathtt{k}}(r\mathtt{k}+\nu+\mathtt{k})}=\frac{2% }{\Gamma_{\mathtt{k}}\left(\left(r+\frac{1}{2}\right)\mathtt{k}\right)\Gamma_{% \mathtt{k}}\left(\nu+\frac{\mathtt{k}}{2}\right)}\int_{0}^{1}t^{2r}(1-t^{2})^{% \frac{\nu}{\mathtt{k}}-\frac{1}{2}}dt.$$ (2.20) Again the identity $\Gamma_{\mathtt{k}}(\mathtt{k}x)=\mathtt{k}^{x-1}\Gamma(x)$ yields $$\displaystyle\Gamma_{\mathtt{k}}\left(\left(r+\frac{1}{2}\right)\mathtt{k}% \right)=\mathtt{k}^{r-\frac{1}{2}}\Gamma\left(r+\frac{1}{2}\right).$$ (2.21) Further, the Legendre duplication formula (see [1, 2]) $$\mathrm{\Gamma}{(z)}\mathrm{\Gamma}{\left(z+\tfrac{1}{2}\right)}=2^{1-2z}\;% \sqrt{\pi}\;\mathrm{\Gamma}{(2z)}$$ (2.22) shows that $$\Gamma\left(r+\frac{1}{2}\right)r!=r\Gamma\left(r+\frac{1}{2}\right)\Gamma(r)=% \frac{\sqrt{\pi}(2r)!}{2^{2r}}.$$ This along with (2.20) and (2.21) reduce the series of $\mathtt{W}^{\mathtt{k}}_{\nu,c}$ as $$\displaystyle\mathtt{W}_{\nu,c}^{\mathtt{k}}(x)$$ $$\displaystyle=\frac{2\sqrt{\mathtt{k}}}{\Gamma_{\mathtt{k}}\left(\nu+\frac{% \mathtt{k}}{2}\right)}\left(\frac{x}{2}\right)^{\frac{\nu}{\mathtt{k}}}\int_{0% }^{1}(1-t^{2})^{\frac{\nu}{\mathtt{k}}-\frac{1}{2}}\sum_{r=0}^{\infty}\frac{(-% c)^{r}}{\Gamma(r+1)r!}\left(\frac{xt}{2\sqrt{\mathtt{k}}}\right)^{2r}dt.$$ $$\displaystyle=\frac{2\sqrt{\mathtt{k}}}{\sqrt{\pi}\Gamma_{\mathtt{k}}\left(\nu% +\frac{\mathtt{k}}{2}\right)}\left(\frac{x}{2}\right)^{\frac{\nu}{\mathtt{k}}}% \int_{0}^{1}(1-t^{2})^{\frac{\nu}{\mathtt{k}}-\frac{1}{2}}\sum_{r=0}^{\infty}% \frac{(-c)^{r}}{(2r)!}\left(\frac{xt}{\sqrt{\mathtt{k}}}\right)^{2r}dt.$$ (2.23) Finally for $c=\pm\alpha^{2}$, $\alpha\in\mathbb{R}$, the representation (2.3) respectively leads to $$\displaystyle\mathtt{W}_{\nu,\alpha^{2}}^{\mathtt{k}}(x)=\frac{2\sqrt{\mathtt{% k}}}{\sqrt{\pi}\Gamma_{\mathtt{k}}\left(\nu+\frac{\mathtt{k}}{2}\right)}\left(% \frac{x}{2}\right)^{\frac{\nu}{\mathtt{k}}}\int_{0}^{1}(1-t^{2})^{\frac{\nu}{% \mathtt{k}}-\frac{1}{2}}\cos\left(\frac{\alpha xt}{\sqrt{\mathtt{k}}}\right)dt.$$ (2.24) and $$\displaystyle\mathtt{W}_{\nu,-\alpha^{2}}^{\mathtt{k}}(x)=\frac{2\sqrt{\mathtt% {k}}}{\sqrt{\pi}\Gamma_{\mathtt{k}}\left(\nu+\frac{\mathtt{k}}{2}\right)}\left% (\frac{x}{2}\right)^{\frac{\nu}{\mathtt{k}}}\int_{0}^{1}(1-t^{2})^{\frac{\nu}{% \mathtt{k}}-\frac{1}{2}}\cosh\left(\frac{\alpha xt}{\sqrt{\mathtt{k}}}\right)dt.$$ (2.25) If $\nu=\mathtt{k}/2$, then from (2.24) a computation give the relation between sine functions and generalized $\mathtt{k}$-Bessel functions by $$\sin\left(\frac{\alpha x}{\sqrt{\mathtt{k}}}\right)=\frac{\alpha}{\mathtt{k}}% \sqrt{\frac{\pi x}{2}}\mathtt{W}_{\frac{\nu}{\mathtt{k}},\alpha^{2}}^{\mathtt{% k}}(x).$$ Similarly, the relation $$\sinh\left(\frac{\alpha x}{\sqrt{\mathtt{k}}}\right)=\frac{\alpha}{\mathtt{k}}% \sqrt{\frac{\pi x}{2}}\mathtt{W}_{\frac{\nu}{\mathtt{k}},-\alpha^{2}}^{\mathtt% {k}}(x)$$ can be derive from (2.25). 3. Monotonicity and log-convexity properties This section is devoted for the discussion of the monotonicity and the log-convexity properties of the modified $\mathtt{k}$-Bessel function $\mathtt{W}_{\nu,-1}^{\mathtt{k}}=\mathtt{I}_{\nu}^{\mathtt{k}}$. As consequences of those results, we derive several functional inequalities for $\mathtt{I}_{\nu}^{\mathtt{k}}$. The following result of Biernacki and Krzyż [4] will be required. Lemma 3.1. [4] Consider the power series $f(x)=\sum_{k=0}^{\infty}a_{k}x^{k}$ and $g(x)=\sum_{k=0}^{\infty}b_{k}x^{k}$, where $a_{k}\in\mathbb{R}$ and $b_{k}>0$ for all $k$. Further suppose that both series converge on $|x|<r$. If the sequence $\{a_{k}/b_{k}\}_{k\geq 0}$ is increasing (or decreasing), then the function $x\mapsto f(x)/g(x)$ is also increasing (or decreasing) on $(0,r)$. The above lemma still holds when both $f$ and $g$ are even, or both are odd functions. We now state and prove our main results in this section. Consider the functions $$\displaystyle\mathcal{I}_{\nu}^{\mathtt{k}}(x):=\left(\frac{2}{x}\right)^{% \frac{\nu}{\mathtt{k}}}\Gamma_{\mathtt{k}}(\nu+\mathtt{k})\mathtt{I}_{\nu}^{% \mathtt{k}}(x)=\sum_{r=0}^{\infty}f_{r}(\nu)x^{2r},$$ (3.1) where, $$\displaystyle f_{r}(\nu)=\frac{\Gamma_{\mathtt{k}}{(\nu+\mathtt{k})}}{\Gamma_{% \mathtt{k}}{(r{\mathtt{k}}+\nu+\mathtt{k})}4^{r}r!}.$$ (3.2) Then we have the following properties. Theorem 3.1. Let $\mathtt{k}>0$. Following results are true for the modified $\mathtt{k}$-Bessel functions. (a) If $\nu\geq\mu>-\mathtt{k}$, then the function $x\mapsto{\mathcal{I}_{\mu}^{\mathtt{k}}(x)}/{\mathcal{I}_{\nu}^{\mathtt{k}}(x)}$ is increasing on $\mathbb{R}$. (b) The function $\nu\mapsto\mathcal{I}^{\mathtt{k}}_{\nu+\mathtt{k}}(x)/\mathcal{I}^{\mathtt{k}% }_{\nu}(x)$ is increasing on $(-\mathtt{k},\infty)$ that is, for $\nu\geq\mu>-\mathtt{k}$, the inequality $$\mathcal{I}_{\nu+\mathtt{k}}^{\mathtt{k}}(x)\mathcal{I}_{\mu}^{\mathtt{k}}(x)% \geq\mathcal{I}_{\nu}^{\mathtt{k}}(x)\mathcal{I}_{\mu+\mathtt{k}}^{\mathtt{k}}% (x)$$ (3.3) holds for each fixed $x>0$ and $\mathtt{k}>0$. (c) The function $\nu\mapsto\mathcal{I}_{\nu}^{\mathtt{k}}(x)$ is decreasing and log-convex on $(-\mathtt{k},\infty)$ for each fixed $x>0$. Proof. (a). From (3.1) it follows that $$\displaystyle\frac{\mathcal{I}^{\mathtt{k}}_{\nu}(x)}{\mathcal{I}^{\mathtt{k}}% _{\mu}(x)}=\frac{\sum_{r=0}^{\infty}f_{r}(\nu)x^{2r}}{\sum_{r=0}^{\infty}f_{r}% (\mu)x^{2r}}.$$ Denote $w_{r}:=f_{r}(\nu)/f_{r}(\mu)$. Then $$w_{r}=\frac{\Gamma_{\mathtt{k}}{(\nu+\mathtt{k})}\Gamma_{\mathtt{k}}{(r{% \mathtt{k}}+\mu+\mathtt{k})}}{\Gamma_{\mathtt{k}}{(\mu+\mathtt{k})}\Gamma_{% \mathtt{k}}{(r{\mathtt{k}}+\nu+\mathtt{k})}}.$$ Now using the properties $\Gamma_{\mathtt{k}}{(y+\mathtt{k})}=y\Gamma_{\mathtt{k}}{(y)}$ it can be shown that $$\displaystyle\frac{w_{r+1}}{w_{r}}$$ $$\displaystyle=\frac{\Gamma_{\mathtt{k}}{(r{\mathtt{k}}+\nu+\mathtt{k})}\Gamma_% {\mathtt{k}}{(r{\mathtt{k}}+\mu+2\mathtt{k})}}{\Gamma_{\mathtt{k}}{(r{\mathtt{% k}}+\mu+\mathtt{k})}\Gamma_{\mathtt{k}}{(r{\mathtt{k}}+\nu+2\mathtt{k})}}=% \frac{r{\mathtt{k}}+\mu+\mathtt{k}}{r{\mathtt{k}}+\nu+\mathtt{k}}\leq 1,$$ in view of $\nu\geq\mu>-\mathtt{k}.$ Now the result follows from the Lemma 3.1. (b). Let $\nu\geq\mu>-\mathtt{k}.$ It follows from part $(a)$ that $$\displaystyle\frac{d}{dx}\left(\frac{\mathcal{I}^{\mathtt{k}}_{\nu}(x)}{% \mathcal{I}^{\mathtt{k}}_{\mu}(x)}\right)\geq 0$$ on $(0,\infty).$ Thus $$\displaystyle\left(\mathcal{I}^{\mathtt{k}}_{\nu}(x)\right)^{\prime}\left(% \mathcal{I}^{\mathtt{k}}_{\mu}(x)\right)-\left(\mathcal{I}^{\mathtt{k}}_{\nu}(% x)\right)\left(\mathcal{I}^{\mathtt{k}}_{\mu}(x)\right)^{\prime}\geq 0.$$ (3.4) It now follows from $(\ref{rr-6})$ that $$\displaystyle\frac{x}{2}\bigg{(}\mathcal{I}^{\mathtt{k}}_{\nu+k}(x)\;\mathcal{% I}^{\mathtt{k}}_{\mu}(x)-\mathcal{I}^{\mathtt{k}}_{\mu+k}(x)\;\mathcal{I}^{% \mathtt{k}}_{\nu}(x)\bigg{)}\geq 0,$$ whence $\mathcal{I}^{\mathtt{k}}_{\nu+k}/\mathcal{I}^{\mathtt{k}}_{\nu}$ is increasing for $\nu>-\mathtt{k}$ and for some fixed $x>0$. (c). It is clear that for all $\nu>-\mathtt{k}$, $$f_{r}(\nu)=\frac{\Gamma_{\mathtt{k}}{(\nu+\mathtt{k})}}{\Gamma_{\mathtt{k}}{(r% {\mathtt{k}}+\nu+\mathtt{k})}4^{r}r!}>0.$$ A logarithmic differentiation of $f_{r}(\nu)$ with respect to $\nu$ yields $$\displaystyle\frac{f_{r}^{\prime}(\nu)}{f_{r}(\nu)}=\Psi_{\mathtt{k}}(\nu+% \mathtt{k})-\Psi_{\mathtt{k}}(r\mathtt{k}+\nu+\mathtt{k})\leq 0,$$ in view of the fact that the $\Psi_{\mathtt{k}}$ is increasing functions on $(-\mathtt{k},\infty).$ This implies that $f_{r}(\nu)$ is decreasing. Thus for $\mu\geq\nu>-\mathtt{k}$, it follows that $$\displaystyle\sum_{r=0}^{\infty}f_{r}(\nu)x^{2r}\geq\sum_{r=0}^{\infty}f_{r}(% \mu)x^{2r},$$ which is equivalent to say that the function $\nu\mapsto\mathcal{I}^{\mathtt{k}}_{\nu}$ is decreasing on $(-\mathtt{k},\infty)$ for some fixed $x>0$. The twice logarithmic differentiation of $f_{r}(\nu)$ yields $$\displaystyle\frac{\partial^{2}}{\partial\nu^{2}}\bigg{(}\log(f_{r}(\nu)\bigg{)}$$ $$\displaystyle=\Psi_{k}^{\prime}(\nu+\mathtt{k})-\Psi_{k}^{\prime}(r\mathtt{k}+% \nu+\mathtt{k})$$ $$\displaystyle=\sum_{n=0}^{\infty}\left(\frac{1}{(n\mathtt{k}+\nu+\mathtt{k})^{% 2}}-\frac{1}{(n\mathtt{k}+r\mathtt{k}+\nu+\mathtt{k})^{2}}\right)$$ $$\displaystyle=\sum_{n=0}^{\infty}\frac{r\mathtt{k}(2n\mathtt{k}+r\mathtt{k}+2% \nu+2\mathtt{k})}{(n\mathtt{k}+\nu+\mathtt{k})^{2}(n\mathtt{k}+r\mathtt{k}+\nu% +\mathtt{k})^{2}}\geq 0.$$ for all $\mathtt{k}>0$ and $\nu>-\mathtt{k}$. Thus $\nu\mapsto f_{r}(\nu)$ is log-convex on $(-\mathtt{k},\infty)$. In view of the fact that sums of log-convex functions are also log-convex, it follows that $\mathcal{I}_{\nu}^{\mathtt{k}}$ is log-convex on $(-\mathtt{k},\infty)$ for each fixed $x>0$. ∎ Remark 3.1. One of the most significance consequences of the Theorem 3.1 is the Tur$\acute{\text{a}}$n-type inequality for the function $\mathcal{I}_{\nu}^{\mathtt{k}}$. From the definition of log-convexity, it follows from Theorem 3.1 (c) that $$\displaystyle\mathcal{I}_{\alpha\nu_{1}+(1-\alpha)\nu_{2}}^{\mathtt{k}}(x)\leq% \left(\mathcal{I}_{\nu_{1}}^{\mathtt{k}}\right)^{\alpha}(x)\left(\mathcal{I}_{% \nu_{2}}^{\mathtt{k}}\right)^{1-\alpha}(x),$$ where $\alpha\in[0,1]$, $\nu_{1},\nu_{2}>-\mathtt{k}$, and $x>0$. Choose $\alpha=1/2$, and for any $\mathtt{a}\in\mathbb{R}$, let $\nu_{1}=\nu-\mathtt{a}$ and $\nu_{2}=\nu+\mathtt{a},$. Then the above inequality yields reverse Turán type inequality $$\displaystyle\left(\mathcal{I}_{\nu}^{\mathtt{k}}(x)\right)^{2}-\mathcal{I}_{% \nu-\mathtt{a}}^{\mathtt{k}}(x)\mathcal{I}_{\nu+\mathtt{a}}^{\mathtt{k}}(x)\leq 0$$ (3.5) for any $\nu\geq|a|-\mathtt{k}$. Our final result is based on the Chebyshev integral inequality [15, p. 40], which states the following: suppose $f$ and $g$ are two integrable functions and monotonic in the same sense (either both decreasing or both increasing). Let $q:(a,b)\to\mathbb{R}$ be a positive integrable function. Then $$\displaystyle\left(\int_{a}^{b}q(t)f(t)dt\right)\left(\int_{a}^{b}q(t)g(t)dt% \right)\leq\left(\int_{a}^{b}q(t)dt\right)\left(\int_{a}^{b}q(t)f(t)g(t)dt% \right).$$ (3.6) The inequality in (3.6) is reversed if $f$ and $g$ are monotonic but in the opposite sense. Follwoing function is required $$\displaystyle\mathcal{J}_{\nu}^{\mathtt{k}}(x):=\left(\frac{2}{x}\right)^{% \frac{\nu}{\mathtt{k}}}\Gamma_{\mathtt{k}}(\nu+\mathtt{k})\mathtt{J}_{\nu}^{% \mathtt{k}}(x)=\sum_{r=0}^{\infty}g_{r}(\nu)x^{2r},$$ (3.7) where, $$\displaystyle g_{r}(\nu)=\frac{(-1)^{r}\Gamma_{\mathtt{k}}{(\nu+\mathtt{k})}}{% \Gamma_{\mathtt{k}}{(r{\mathtt{k}}+\nu+\mathtt{k})}4^{r}r!}.$$ (3.8) Theorem 3.2. Let $\mathtt{k}>0$. Then for $\nu\in(-3\mathtt{k}/4,-\mathtt{k}/2]\cup[\mathtt{k}/2,\infty)$ $$\displaystyle\mathcal{I}_{\nu}^{\mathtt{k}}(x)\mathcal{I}_{\nu+\tfrac{\mathtt{% k}}{2}}^{\mathtt{k}}(x)\leq\frac{\sqrt{\mathtt{k}}}{x}\sin\left(\frac{x}{% \mathtt{k}}\right)\mathcal{I}_{2\nu+\tfrac{\mathtt{k}}{2}}^{\mathtt{k}}(x).$$ (3.9) and $$\displaystyle\mathcal{J}_{\nu}^{\mathtt{k}}(x)\mathcal{J}_{\nu+\tfrac{\mathtt{% k}}{2}}^{\mathtt{k}}(x)\leq\frac{\sqrt{\mathtt{k}}}{x}\sinh\left(\frac{x}{% \mathtt{k}}\right)\mathcal{J}_{2\nu+\tfrac{\mathtt{k}}{2}}^{\mathtt{k}}(x).$$ (3.10) The inequalities in (3.9) and (3.10) are reversed if $\nu\in(-\mathtt{k}/2,\mathtt{k}/2)$. Proof. Define the functions $p$, $f$ and $g$ on $[0,1]$ as $$q(t)=\cos\left(\tfrac{xt}{\sqrt{\mathtt{k}}}\right),\quad f(t)=(1-t^{2})^{% \tfrac{v}{\mathtt{k}}-\tfrac{1}{2}},\quad g(t)=(1-t^{2})^{\tfrac{v}{\mathtt{k}% }+\tfrac{1}{2}}.$$ Then $$\displaystyle\int_{0}^{1}q(t)dt$$ $$\displaystyle=\int_{0}^{1}\cos\left(\tfrac{xt}{\sqrt{\mathtt{k}}}\right)dt=% \frac{\sqrt{\mathtt{k}}}{x}\sin\left(\tfrac{x}{\sqrt{\mathtt{k}}}\right);$$ $$\displaystyle\int_{0}^{1}q(t)f(t)dt$$ $$\displaystyle=\int_{0}^{1}\cos\left(\tfrac{xt}{\sqrt{\mathtt{k}}}\right)(1-t^{% 2})^{\tfrac{v}{\mathtt{k}}-\tfrac{1}{2}}dt=\mathcal{I}_{\nu}^{\mathtt{k}}(x),% \quad\text{if}\quad\nu\geq-\mathtt{k};$$ $$\displaystyle\int_{0}^{1}q(t)g(t)dt$$ $$\displaystyle=\int_{0}^{1}\cos\left(\tfrac{xt}{\sqrt{\mathtt{k}}}\right)(1-t^{% 2})^{\tfrac{v}{\mathtt{k}}+\tfrac{1}{2}}dt=\mathcal{I}_{\nu+\mathtt{k}}^{% \mathtt{k}}(x),\quad\text{if}\quad\nu\geq-2\mathtt{k};$$ $$\displaystyle\int_{0}^{1}q(t)f(t)g(t)dt$$ $$\displaystyle=\int_{0}^{1}\cos\left(\tfrac{xt}{\sqrt{\mathtt{k}}}\right)(1-t^{% 2})^{\tfrac{2v}{\mathtt{k}}}dt=\mathcal{I}_{2\nu+\tfrac{\mathtt{k}}{2}}^{% \mathtt{k}}(x),\quad\text{if}\quad\nu\geq-\tfrac{3\mathtt{k}}{4};$$ Since the functions $f$ and $g$ both are decreasing for $\nu\geq\mathtt{k}/2$ and both are increasing for $\nu\in(-3\mathtt{k}/4,-\mathtt{k}/2]$, the inequality (3.6) yields (3.9). On the other hand if $\nu\in(-\mathtt{k}/2,\mathtt{k}/2)$, the function $f$ is increasing but $g$ is decreasing, and hence the inequality in (3.9) is reversed. Similarly, the inequality in (3.10) can be derived by using (3.6) by choosing $$q(t)=\cosh\left(\tfrac{xt}{\sqrt{\mathtt{k}}}\right),\quad f(t)=(1-t^{2})^{% \tfrac{v}{\mathtt{k}}-\tfrac{1}{2}},\quad g(t)=(1-t^{2})^{\tfrac{v}{\mathtt{k}% }+\tfrac{1}{2}}.$$ ∎ References [1] M. Abramowitz and I. A. Stegun, A Handbook of Mathematical Functions, New York, (1965). [2] G. E. Andrews, R. Askey and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge Univ. Press, Cambridge, 1999. [3] Á. Baricz, Generalized Bessel functions of the first kind, Lecture Notes in Mathematics, vol. 1994. Springer-Verlag, Berlin, 2010. [4] M. Biernacki and J. Krzyż, On the monotonity of certain functionals in the theory of analytic functions, Ann. Univ. Mariae Curie-Skłodowska. Sect. A. 9 (1955), 135–147 (1957). [5] P. 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Continuity of measure-dimension mappings Liangang Ma Dept. of Mathematical Sciences, Binzhou University, Huanghe 5th Road No. 391, Binzhou 256600, Shandong, P. R. China maliangang000@163.com Abstract. We study continuity and discontinuity of the upper and lower (modified) box-counting, Hausdorff, packing, (modified) correlation measure-dimension mappings under the weak, setwise and TV topology on the space of Borel measures respectively in this work. We give various examples to show that no continuity can be guaranteed under the weak, setwise or TV topology on the space of Borel measures for any of these measure-dimension mappings. However, in some particular circumstances or by assuming some restrictions on the measures, we do have some (semi-)continuity results for some of these measure-dimension mappings under the setwise topology. In the end we point out some applications of our continuity results on deciding the dimensions of measures on ambient space with some dynamical structures. The work is supported by ZR2019QA003 from SPNSF and 12001056 from NSFC 1. Introduction The dimension of a measure on a metrizable space indicates how well the mass is distributed on the space. Its definition usually depends on the dimensions of sets in the space. P. Mattila, M. Morán and J. M. Rey gave some criterions on evaluating the well-posedness of different concepts of dimensions of measures, see [MMR, p220]. In their work they treat the dimensions of measures as as a function (functional) from the space of Borel measures to $\mathbb{R}^{+}=[0,\infty)$: the measure-dimension mappings. From this point of view it is a natural question to ask the continuity of various measure-dimension mappings, with suitable topology endowed on the space of Borel measures. It would be interesting if the measure-dimension mappings could admit some continuity under some topology in case one is trying to deal with the corresponding dimensions of Borel measures. Considering the topological description of the space of Borel measures, the vague, weak, setwise and TV topology are well-known among probabilists, with their fineness in increasing order. One can refer to [Kal2] and [Kle] for the vague and weak topology, while to [FKZ1] and [Las] for the setwise and TV topology on the space of Borel measures. These topologies have lots of applications in various circumstances, for example, see the applications of the vague or weak topology in particle systems by Cox-Klenke-Perkins and Kallenberg [CKP, Kal1], the applications of the weak, setwise or TV topology in Markov decision processes by Feinberg-Kasyanov-Zgurovsky, O. Hernández-Lerma and J. Lasserre in [FKZ2, HL]. The choice of topology is subtle in different circumstances, with better properties possible under finer topology at the cost of more difficulty in guaranteeing convergence of concerning sequences of measures under the finer topology. Luckily, we find that lots of measure-dimension mappings admit some semi-continuity properties under the setwise topology on the space of Borel measures. Especially, the upper measure-dimension mapping induced from some dimensional mapping satisfying the countable stability admits lower semi-continuity under the setwise topology, in words of Falconer [Fal1, p40]. Upper semi-continuity always holds for any induced measure-dimension mappings under the setwise topology. However, some measure-dimension mappings do not admit any semi-continuity under any of these topologies, for example, the (modified) correlation measure-dimension mapping. We will also give examples to show that none of our concerning measure-dimension mappings admits any continuity if the topology is relaxed from the setwise level to weak level. There are some applications of our continuity results to Hausdorff dimensions of ergodic measures on some Riemannian manifold following Young [You] at the end of the work. 2. Notations, definitions and the main results In this section we first introduce some popular measure-dimension mappings, then we give definitions of the vague, weak, setwise and TV topology on the space of Borel measures. At the end of the section we present our results on semi-continuity of some measure-dimension mappings under the setwise topology. Let $(X,\rho)$ be a metric space with its Borel $\sigma$-algebra $\mathcal{B}$. Denote by $\mathcal{M}(X)$ to be the collection of all the Borel measures (note that it is possible that some measures are infinite in $\mathcal{M}(X)$). The collection of all the finite Borel measures on $(X,\mathcal{B})$ is denoted by $\hat{\mathcal{M}}(X)$. A dimensional mapping is a real non-negative function $dim:\mathcal{B}\rightarrow\mathbb{R}^{+}$. Of course the concept is interesting if and only if the function satisfies some reasonable properties, for example, the ones in [Fal1, p40]. We are particularly interested in those dimensional mappings satisfying the countable stability, that is, for any collection of countable sets $\{A_{i}\}_{i=1}^{\infty}$, $dim(\cup_{i=1}^{\infty}A_{i})=\sup\{dim(A_{i})\}_{i=1}^{\infty}$. Notable dimensional mappings in this work are the upper and lower (modified) box-counting, Hausdorff and packing dimensional mappings, which are denoted by $\overline{dim}_{B},\underline{dim}_{B}(\overline{dim}_{MB},\underline{dim}_{MB}),dim_{H},dim_{P}$ respectively, see [Fal1]. Denote by $dim_{B}$ ($dim_{MB}$) in case $\overline{dim}_{B}=\underline{dim}_{B}$ ($\overline{dim}_{MB}=\underline{dim}_{MB}$). A measure-dimension mapping is a real non-negative function $dim:\mathcal{M}(X)\rightarrow\mathbb{R}^{+}$. We abuse notations as one can easily distinguish its meaning of dimensional mapping or measure-dimension mapping from the contexts. Refer to [MMR, p220] on natural properties of reasonable measure-dimension mappings. The following two dual measure-dimension mappings are highlighted in various circumstances. 2.1 Definition. Let $dim$ be a dimensional mapping on $(X,\rho)$. The induced lower and upper measure-dimension mappings from $\mathcal{M}(X)$ to $[0,\infty)$ are defined respectively to be: $dim^{L}(\nu)=\inf\{dim(A):\nu(A)>0,A\in\mathcal{B}\},$ and $dim^{U}(\nu)=\inf\{dim(A):\nu(X\setminus A)=0,A\in\mathcal{B}\}$ for a measure $\nu\in\mathcal{M}(X)$. It is easy to see that $dim^{L}(\nu)\leq dim^{U}(\nu)$ for any $\nu\in\mathcal{M}(X)$. When taking the dimensional mappings to be $dim_{B},\overline{dim}_{B},\underline{dim}_{B}(\overline{dim}_{MB},\underline{dim}_{MB}),dim_{H},dim_{P}$, the corresponding lower and upper measure-dimension mappings are denoted respectively by $dim_{B}^{L},\overline{dim}_{B}^{L},\underline{dim}_{B}^{L}(\overline{dim}_{MB}^{L},\underline{dim}_{MB}^{L}),dim_{H}^{L},dim_{P}^{L}$ and $dim_{B}^{U},\overline{dim}_{B}^{U},\underline{dim}_{B}^{U}(\overline{dim}_{MB}^{U},\underline{dim}_{MB}^{U}),dim_{H}^{U},dim_{P}^{U}$. In case the lower and upper measure-dimension mappings coincide with each other at some measure $\nu\in\mathcal{M}(X)$, denote the values simply by $dim_{B}(\nu),\overline{dim}_{B}(\nu),\underline{dim}_{B}(\nu)\big{(}\overline{dim}_{MB}(\nu),\underline{dim}_{MB}(\nu)\big{)},dim_{H}(\nu),dim_{P}(\nu)$. Following Mattila-Morán-Rey, we also pay attention to the correlation measure-dimension mapping $\dim_{C}$ and its modified version $\dim_{MC}$. The correlation measure-dimension mapping is introduced by P. Grassberger and I. Procaccia [GP1, GP2]. 2.2 Definition. The correlation dimension of $\nu$ (see [Cut]) is defined to be $dim_{C}(\nu)=\lim_{r\rightarrow 0}\cfrac{\log\int\nu(B(x,r))d\nu}{\log r}$ for a measure $\nu\in\mathcal{M}(X)$, in case the above limit exists. The modified one $\dim_{MC}$ is given by Pesin [Pes]. 2.3 Definition. The modified correlation dimension of $\nu$ is defined to be $dim_{MC}(\nu)=\lim_{\delta\rightarrow 0}\sup_{\{A\in\mathcal{B}:\nu(A)\geq 1-\delta\}}\lim_{r\rightarrow 0}\cfrac{\log\int_{A}\nu(B(x,r))d\nu}{\log r}$ for a measure $\nu\in\mathcal{M}(X)$, in case the above limit exists. 2.4 Remark. Regardless of the existence of the above two limits, one can define the upper and lower correlation dimension as well as the upper and lower modified correlation dimension as [MMR, (2),(4)]. We adopt these definitions directly because all the measures in our concerns in this work have the same upper and lower (modified) correlation dimensions. By [Fal2, Proposition 10.2] and [MMR, Lemma 2.8], we have $dim_{H}^{L}(\nu)\leq dim_{C}(\nu)\leq dim_{MC}(\nu)$ for any $\nu\in\mathcal{M}(X)$. In order to discuss the continuity of the measure-dimension mappings, some topology on $\mathcal{M}(X)$ is necessary. The following concepts apply to $\mathcal{M}(X)$ with topological ambient space $X$. 2.5 Definition. The vague topology is the topology with basis $\{\varrho\in\mathcal{M}(X):|\int_{X}f(x)d\varrho-\int_{X}f(x)d\nu|<\epsilon\}$ on $\mathcal{M}(X)$ for any continuous function $f:X\rightarrow\mathbb{R}$ with compact support and any real $\epsilon>0$. Refer to [Kal2, Kle]. As the coarsest topology, no continuity can be guaranteed for any measure-dimension mappings in our concerns under it. However, we still would like to pose it here in case of particular interests for some readers. A finer topology is the weak topology. 2.6 Definition. The weak topology on $\mathcal{M}(X)$ is the topology with basis $\{\varrho\in\mathcal{M}(X):|\int_{X}f(x)d\varrho-\int_{X}f(x)d\nu|<\epsilon\}$ for any bounded continuous function $f:X\rightarrow\mathbb{R}$ and any real $\epsilon>0$. Refer to [Bil1, Bil2, Kle, Mat]. Unfortunately, still no (semi-)continuity can be guaranteed for any measure-dimension mappings in our concerns under the weak topology in general, see Theorem 3.1. 2.7 Definition. The setwise topology on $\mathcal{M}(X)$ is the topology with basis $\{\varrho\in\mathcal{M}(X):|\int_{X}f(x)d\varrho-\int_{X}f(x)d\nu|<\epsilon\}$ for any bounded measurable function $f:X\rightarrow\mathbb{R}$ and any real $\epsilon>0$. Refer to [Doo, FKZ1, GR, HL, Las, LY]. After escalating to the setwise topology, we are excited to find that some measure-dimension mappings admit some semi-continuity under it. Denote by $\nu_{n}\stackrel{{\scriptstyle v}}{{\rightarrow}}\nu,\nu_{n}\stackrel{{\scriptstyle w}}{{\rightarrow}}\nu,\nu_{n}\stackrel{{\scriptstyle s}}{{\rightarrow}}\nu,\nu_{n}\stackrel{{\scriptstyle TV}}{{\rightarrow}}\nu$ as $n\rightarrow\infty$ for sequences of measures in $\mathcal{M}(X)$ converging under the vague, weak, setwise or TV topology (to be defined later) respectively. 2.8 Theorem. Let $dim$ be a dimensional mapping satisfying the countable stability property. The induced upper measure-dimension mapping $dim^{U}$ is lower semi-continuous under the setwise topology on $\mathcal{M}(X)$, that is, if $\nu_{n}\stackrel{{\scriptstyle s}}{{\rightarrow}}\nu$ in $\mathcal{M}(X)$ as $n\rightarrow\infty$, then (2.1) $$\liminf_{n\rightarrow\infty}dim^{U}(\nu_{n})\geq dim^{U}(\nu).$$ Alternatively, we have upper semi-continuity for the lower measure-dimension mapping $dim^{L}$, which shows the two measure-dimension mappings $dim^{U}$ and $dim^{L}$ are dual to each other in some sense. 2.9 Theorem. Let $dim$ be a dimensional mapping. The induced lower measure-dimension mapping $dim^{L}$ is upper semi-continuous under the setwise topology on $\mathcal{M}(X)$, that is, if $\nu_{n}\stackrel{{\scriptstyle s}}{{\rightarrow}}\nu$ in $\mathcal{M}(X)$ as $n\rightarrow\infty$, then (2.2) $$\limsup_{n\rightarrow\infty}dim^{L}(\nu_{n})\leq dim^{L}(\nu).$$ As the finest topology in this work, the TV topology is induced from an explicit metric on $\mathcal{M}(X)$. 2.10 Definition. The TV (total-variation) metric is defined to be $\|\nu-\varrho\|_{TV}=\sup_{A\in\mathcal{B}}\{|\nu(A)-\varrho(A)|\}$ for two measures $\nu,\varrho\in\mathcal{M}(X)$. Refer to [Doo, FKZ1, HL, Las, PS]). Considering the two measure-dimension mappings $dim_{C}$ and $dim_{MC}$, they do not admit any semi-continuity under the finest topology on $\mathcal{M}(X)$ in general, see Theorem 3.13. 3. Proofs of the semi-continuity results and examples on discontinuity of the measure-dimension mappings under various topology First, to provide a general intuition on the continuity of the measure-dimension mappings in our consideration, we present the readers with the following result. 3.1 Theorem. The measure-dimension mappings $dim_{B}^{L},\overline{dim}_{B}^{L},\underline{dim}_{B}^{L}(\overline{dim}_{MB}^{L},\underline{dim}_{MB}^{L}),dim_{H}^{L},dim_{P}^{L},$ $dim_{B}^{U},\overline{dim}_{B}^{U},\underline{dim}_{B}^{U}(\overline{dim}_{MB}^{U},\underline{dim}_{MB}^{U}),dim_{H}^{U},dim_{P}^{U}$ and $dim_{C},dim_{MC}$ from $\mathcal{M}(X)$ to $[0,+\infty)$ are not continuous under the weak, setwise or TV topology in general. Proof. See for instance our Example 3.6 and 3.11. ∎ 3.2 Remark. It would be interesting to ask whether some measure-dimension mappings can be defined to be equipped with continuity under some topology, possibly at the cost of losing some known properties of some popular measure-dimension mappings. We first give an example to show that for a weakly convergent sequence $\nu_{n}\stackrel{{\scriptstyle w}}{{\rightarrow}}\nu$ as $n\rightarrow\infty$, any of the sequences (3.1) $$\begin{array}[]{cc}\{dim_{B}^{L}(\nu_{n})\}_{n=1}^{\infty},\{\overline{dim}_{B}^{L}(\nu_{n})\}_{n=1}^{\infty},\{\underline{dim}_{B}^{L}(\nu_{n})\}_{n=1}^{\infty},\{\overline{dim}_{MB}^{L}(\nu_{n})\}_{n=1}^{\infty},\\ \{\underline{dim}_{MB}^{L}(\nu_{n})\}_{n=1}^{\infty},\{dim_{H}^{L}(\nu_{n})\}_{n=1}^{\infty},\{dim_{P}^{L}(\nu_{n})\}_{n=1}^{\infty}\end{array}$$ (3.2) $$\begin{array}[]{cc}\{dim_{B}^{U}(\nu_{n})\}_{n=1}^{\infty},\{\overline{dim}_{B}^{U}(\nu_{n})\}_{n=1}^{\infty},\{\underline{dim}_{B}^{U}(\nu_{n})\}_{n=1}^{\infty},\{\overline{dim}_{MB}^{U}(\nu_{n})\}_{n=1}^{\infty},\\ \{\underline{dim}_{MB}^{U}(\nu_{n})\}_{n=1}^{\infty},\{dim_{H}^{U}(\nu_{n})\}_{n=1}^{\infty},\{dim_{P}^{U}(\nu_{n})\}_{n=1}^{\infty}\end{array}$$ and (3.3) $$\{dim_{C}(\nu_{n})\}_{n=1}^{\infty},\{dim_{MC}(\nu_{n})\}_{n=1}^{\infty}$$ may not converge. In this and all the examples below, we set $X\subset\mathbb{R}^{d}$ endowed with the Euclidean metric $\rho_{u}$ for some positive integer $d$. Let $\mathfrak{L}^{d}$ be the $d$-dimensional Lebesgue measure on $\mathbb{R}^{d}$. 3.3 Example. Define a sequence of probability measures $\{\nu_{n}\}_{n\in\mathbb{N}}$ on $[0,1]$ to be $\nu_{n}=\left\{\begin{array}[]{ll}n\mathfrak{L}^{1}|_{[0,\frac{1}{n}]}&\mbox{ if }n\mbox{ is }odd,\\ \delta_{\frac{1}{n}}&\mbox{ if }n\mbox{ is }even.\\ \end{array}\right.$ in which $\delta_{\frac{1}{n}}$ is the Dirac measure at the point $\frac{1}{n}$. Let $\nu=\delta_{0}$ be the Dirac measure at $0$. Obviously we have $\nu_{n}\stackrel{{\scriptstyle w}}{{\rightarrow}}\nu$ as $n\rightarrow\infty$ (but not setwisely). By [You, Proposition 2.1] and [MMR, Lemma 2.8], direct computations give that $\begin{array}[]{ll}&dim_{B}(\nu_{n})=dim_{MB}(\nu_{n})=dim_{H}(\nu_{n})=dim_{P}(\nu_{n})=dim_{C}(\nu_{n})=dim_{MC}(\nu_{n})\\ =&\left\{\begin{array}[]{ll}1&\mbox{ if }n\mbox{ is }odd,\\ 0&\mbox{ if }n\mbox{ is }even.\\ \end{array}\right.\end{array}$ So any of the sequences in (3.1)(3.2)(3.3) does not converge in Example 3.3. We then give an example to show that in case of $\nu_{n}\stackrel{{\scriptstyle w}}{{\rightarrow}}\nu$, even if all the sequences $\{dim_{B}(\nu_{n})\}_{n=1}^{\infty},\{dim_{MB}(\nu_{n})\}_{n=1}^{\infty},\{dim_{H}(\nu_{n})\}_{n=1}^{\infty}$, $\{dim_{P}(\nu_{n})\}_{n=1}^{\infty},\{dim_{C}(\nu_{n})\}_{n=1}^{\infty},\{dim_{MC}(\nu_{n})\}_{n=1}^{\infty}$ converge, their limits may not equal $dim_{B}(\nu),dim_{MB}(\nu),dim_{H}(\nu),dim_{P}(\nu),dim_{C}(\nu)$ and $dim_{MC}(\nu)$ respectively. The example is borrowed from [Bil1, Example 2.2] essentially. 3.4 Example (Billingsley). Define a sequence of probability measures $\nu_{n}$ on $[0,1]$ to be $\nu_{n}=\frac{1}{n}\Sigma_{i=1}^{n}\delta_{\frac{i}{n}}$ for $n\in\mathbb{N}$. Obviously $\nu_{n}\stackrel{{\scriptstyle w}}{{\rightarrow}}\mathfrak{L}^{1}|_{[0,1]}$ as $n\rightarrow\infty$ (but not setwisely). However, we have $\lim_{n\rightarrow\infty}dim_{B}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{MB}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{H}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{P}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{C}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{MC}(\nu_{n})=0$, while $dim_{B}(\mathfrak{L}^{1}|_{[0,1]})=dim_{MB}(\mathfrak{L}^{1}|_{[0,1]})=dim_{H}(\mathfrak{L}^{1}|_{[0,1]})=dim_{P}(\mathfrak{L}^{1}|_{[0,1]})=dim_{C}(\mathfrak{L}^{1}|_{[0,1]})=dim_{MC}(\mathfrak{L}^{1}|_{[0,1]})=1$. One might think that things will be better if the atomic measures are excluded, however, the following example shows that the discontinuity still exists among sequences of non-atomic measures. 3.5 Example. Suppose $k\geq 2$ is an integer. Let $S=\{s_{i}:[0,1]\rightarrow[0,1]\}_{i=1}^{k}$ be an affine IFS with the contraction ratio $0<|s_{i}|:=|s_{i}^{\prime}(x)|<1$ being a fixed number for any $1\leq i\leq k$ on $X=[0,1]$. We require it satisfies the strong separation condition: $s_{i}([0,1])\cap s_{j}([0,1])=\emptyset$ for any $1\leq i\neq j\leq k$. We do not specify the terminals of the the intervals $\{s_{i}([0,1])\}_{i=1}^{k}$ as they have no effect on the properties we would like to demonstrate through the example. Let $0<h<1$ be the unique solution of the Bowen equation $\sum_{i=1}^{k}|s_{i}|^{h}=1$. There is an unique ergodic measure (with respect to the push-forward of the shift map under the projection) $\nu$ on the attractor $J$ in this case. Let $\mathbb{N}_{k}:=\{1,2,\cdots,k\}$ be the $k$-truncation of $\mathbb{N}$. For every fixed $n\in\mathbb{N}$, let $\mathbb{N}_{k}^{n}=\{\omega:\omega=\omega_{1}\omega_{2}\cdots\omega_{n},\omega_{i}\in\mathbb{N}_{k}\mbox{\ for any }1\leq i\leq n\}$ be the collection of length-$n$ concatenation-words of $\mathbb{N}_{k}$. Let $X_{\omega}=s_{\omega}([0,1])=s_{\omega_{1}}\circ s_{\omega_{2}}\cdots\circ s_{\omega_{n}}([0,1])$ for any $\omega\in\mathbb{N}_{k}^{n}$ and any $n\in\mathbb{N}$. Define a sequence of probability measures $\{\nu_{n}\}_{n\in\mathbb{N}}$ to be $\nu_{n}=\sum_{\omega\in\mathbb{N}_{k}^{n}}|s_{\omega}|^{h}\mathfrak{L}^{1}|_{X_{\omega}}=\sum_{\omega\in\mathbb{N}_{k}^{n}}(|s_{\omega_{1}}||s_{\omega_{2}}|\cdots|s_{\omega_{n}}|)^{h}\mathfrak{L}^{1}|_{X_{\omega}}$ on $[0,1]$ for any $n\in\mathbb{N}$. It is supported on $\cup_{\omega\in\mathbb{N}_{k}^{n}}X_{\omega}$ for each $n\in\mathbb{N}$ obviously. In fact $\nu_{n}$ is the $n$-th canonical mass distribution of the linear cookie-cutter map in each cutting step (see for example, the self-similar measures [MR]). Note that none of the measures above is atomic. It is easy to show that for any open set $L\subset[0,1]$, we have $\nu_{n}(L)\geq\nu(L)$. So by [Bil1, Theorem 2.1], we have $\nu_{n}\stackrel{{\scriptstyle w}}{{\rightarrow}}\nu$ as $n\rightarrow\infty$ (but not setwisely). One can show that (left to the readers) $\lim_{n\rightarrow\infty}dim_{B}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{MB}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{H}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{P}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{C}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{MC}(\nu_{n})=1$, while $dim_{B}(\nu)=dim_{MB}(\nu)=dim_{H}(\nu)=dim_{P}(\nu)=dim_{C}(\nu)=dim_{MC}(\nu)=h$. One might also hope that escalating the strength of convergence of measures may save the continuity, but we will give an example to show that convergence of dimensions is not true even for TV convergent sequences of measures (see also Example 3.11). 3.6 Example. Define a sequence of probability measures $\nu_{n}$ on $[0,2]$ to be $\nu_{n}=\frac{n-1}{n}\delta_{0}+\frac{1}{n}\mathfrak{L}^{1}|_{[1,2]}.$ It is easy to see that $\|\nu_{n}-\delta_{0}\|_{TV}=\frac{1}{n}\rightarrow 0$ as $n\rightarrow\infty$, so $\nu_{n}\stackrel{{\scriptstyle TV}}{{\rightarrow}}\delta_{0}$. However, $\begin{array}[]{ll}&\lim_{n\rightarrow\infty}dim_{B}^{U}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{MB}^{U}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{H}^{U}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{P}^{U}(\nu_{n})=1\\ >&0=dim_{B}(\delta_{0})=dim_{MB}(\delta_{0})=dim_{H}(\delta_{0})=dim_{P}(\delta_{0}).\end{array}$ In fact, continuity of these measure-dimension mappings has no chance to be true if there is no dimensional restrictions on the sequences in (3.1)(3.2)(3.3). Luckily we do have some semi-continuity property for some measure-dimension mappings under the setwise topology. Now we prove Theorem 2.8. Proof of Theorem 2.8: Proof. Assume that (2.1) does not hold, that is, $\liminf_{n\rightarrow\infty}dim^{U}(\nu_{n})<dim^{U}(\nu)$. Then we can find a sequence of positive integers $\{n_{k}\}_{k=1}^{\infty}$ and a real number $a<dim^{U}(\nu)$, such that $\lim_{k\rightarrow\infty}dim^{U}(\nu_{n_{k}})=a$. So there exists a sequence of measurable sets $\{A_{k}\}_{k=1}^{\infty}$, such that $\nu_{n_{k}}(X\setminus A_{k})=0$ and $dim(A_{k})<a+\varepsilon<dim^{U}(\nu)$ for some small $\varepsilon>0$. Now let $A=\cup_{k=1}^{\infty}A_{k}$. Since $\nu_{n}\stackrel{{\scriptstyle s}}{{\rightarrow}}\nu$, we have $\nu(X\setminus A)=\lim_{k\rightarrow\infty}\nu_{n_{k}}(X\setminus A)=0.$ However, as $dim$ satisfies the countable stability property, then $dim(A)=\sup_{k}\{dim(A_{k})\}<a+\varepsilon<dim^{U}(\nu)$. This contradicts the definition of $dim^{U}(\nu)$, which justifies our theorem. ∎ 3.7 Remark. Theorem 2.8 does not hold for weakly convergent sequences $\nu_{n}\stackrel{{\scriptstyle w}}{{\rightarrow}}\nu$ as $n\rightarrow\infty$ in $\mathcal{M}(X)$, as one can see from our Example 3.4. Together with Example 3.5, one can see that both lower semi-continuity and upper semi-continuity are not true for general measure-dimension mappings under the weak topology. Theorem 2.8 has some interesting extensions in some special cases. 3.8 Corollary. Let $dim$ be a dimensional mapping satisfying the countable stability property. Let $dim^{U}$ be the induced upper measure-dimension mapping. For $\nu_{n}\stackrel{{\scriptstyle s}}{{\rightarrow}}\nu$ as $n\rightarrow\infty$, if $\operatorname*{ess\,sup}\{dim^{U}(\nu_{n})\}_{n=1}^{\infty}\leq dim^{U}(\nu)$, then $\lim_{n\rightarrow\infty}dim^{U}(\nu_{n})=dim^{U}(\nu)$. Proof. This follows instantly from Theorem 2.8. ∎ 3.9 Corollary. The measure-dimension mappings $dim_{MB}^{U},dim_{H}^{U},dim_{P}^{U}$ are all lower semi-continuous under the setwise topology on $\mathcal{M}(X)$. Proof. This is because they are respectively induced from dimensional mappings $dim_{MB},dim_{H},dim_{P}$ with countable stability property, refer to [Fal1]. ∎ As to $dim_{B}^{U}$ ($\overline{dim}_{B}^{U},\underline{dim}_{B}^{U}$), since the dimensional mapping $dim_{B}$ ($\overline{dim}_{B},\underline{dim}_{B}$) inducing it does not satisfy countable stability, Theorem 2.8 does not apply to it. The following example shows that it is not lower semi-continuous under the setwise topology. 3.10 Example. Let $X=\{\frac{1}{i}\}_{i=1}^{\infty}$. Consider the sequence of measures $\nu_{n}=\sum_{i=1}^{n}\frac{1}{i^{2}}\delta_{\frac{1}{i}}$ for $n\in\mathbb{N}$ on $X$. Let $\nu=\sum_{i=1}^{\infty}\frac{1}{i^{2}}\delta_{\frac{1}{i}}$. Direct computations (left to the readers) show that $\nu_{n}\stackrel{{\scriptstyle TV}}{{\rightarrow}}\nu$ while $\lim_{n\rightarrow\infty}dim_{B}^{U}(\nu_{n})=0<\frac{1}{2}=dim_{B}^{U}(\nu)$ in Example 3.10. These measures can be normalized to be probabilities. Note that the proof of Theorem 2.8 can not be applied to the measure-dimension mapping $dim^{L}$, because if we choose a sequence of sets $\{A_{k}\}_{k=1}^{\infty}$, such that $\nu_{n_{k}}(A_{k})>0$ and $dim^{L}(A_{k})<a+\varepsilon<dim^{L}(\nu)$ we can not guarantee $\nu(A)=\lim_{k\rightarrow\infty}\nu_{n_{k}}(A)>0.$ In fact one can see from the following example that lower semi-continuity of typical measure-dimension mapping $dim^{L}$ is usually not true under setwise topology on $\mathcal{M}(X)$. 3.11 Example. Define a sequence of probability measures $\{\nu_{n}\}_{n\in\mathbb{N}}$ on $[0,1]$ to be $\nu_{n}=\frac{1}{n}\delta_{0}+\mathfrak{L}^{1}|_{[\frac{1}{n},1]}$ for $n\in\mathbb{N}$. It is easy to see that $\|\nu_{n}-\mathfrak{L}^{1}|_{[0,1]}\|_{TV}=\frac{1}{n}\rightarrow 0$ as $n\rightarrow\infty$, so $\nu_{n}\stackrel{{\scriptstyle TV}}{{\rightarrow}}\delta_{0}$. However, $\lim_{n\rightarrow\infty}dim_{B}^{L}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{MB}^{L}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{H}^{L}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{P}^{L}(\nu_{n})=0<1=dim_{B}(\mathfrak{L}^{1}|_{[0,1]})=dim_{MB}(\mathfrak{L}^{1}|_{[0,1]})=dim_{H}(\mathfrak{L}^{1}|_{[0,1]})=dim_{P}(\mathfrak{L}^{1}|_{[0,1]})$. Now we prove the semi-continuity result for the measure-dimension mapping $dim^{L}$. Proof of Theorem 2.9: Proof. Accoriding to the definition of $dim^{L}(\nu)$, for any small $\varepsilon>0$, we can find a measurable set $A$ with $\nu(A)>0$ and $dim(A)\leq dim^{L}(\nu)+\varepsilon$. Since $\nu_{n}\stackrel{{\scriptstyle s}}{{\rightarrow}}\nu$ as $n\rightarrow\infty$, we can guarantee that $\nu_{n}(A)>0$ for any $n$ large enough. This implies that $\limsup_{n\rightarrow\infty}dim^{L}(\nu_{n})\leq dim^{L}(\nu)+\varepsilon$. The proof is finished by letting $\epsilon\rightarrow 0$. ∎ The following result is instant in virtue of Theorem 2.9. 3.12 Corollary. The measure-dimension mappings $dim_{B}^{L}(\overline{dim}_{B}^{L},\underline{dim}_{B}^{L}),dim_{MB}^{L},dim_{H}^{L},dim_{P}^{L}$ are all upper semi-continuous under the setwise topology on $\mathcal{M}(X)$. Considering Corollary 3.9 and 3.12, it would be interesting to ask whether the two measure-dimension mappings $dim_{C}$ or $dim_{MC}$ admits any semi-continuity. It turns out that they do not have any semi-continuity under the TV topology in general. 3.13 Theorem. The two measure-dimension mappings $dim_{C}$ and $dim_{MC}$ from $\mathcal{M}(X)$ to $[0,+\infty)$ are neither upper semi-continuous nor lower semi-continuous under the TV topology. Proof. Simple calculations show that, in Example 3.11, $dim_{C}(\nu_{n})=dim_{MC}(\nu_{n})=\lim_{r\rightarrow 0}\cfrac{\log\int\nu_{n}(B(x,r))d\nu_{n}}{\log r}=\cfrac{\log\big{(}(\frac{1}{n})^{2}+2r\frac{n-1}{n}\big{)}}{\log r}=0$, while $dim_{C}(\mathfrak{L}^{1}|_{[0,1]})=dim_{MC}(\mathfrak{L}^{1}|_{[0,1]})=1$. Thus Example 3.11 justifies the fact that the two measure-dimension mappings $dim_{C}$ and $dim_{MC}$ can not be lower semi-continuous under the TV topology. For an example violating the upper semi-continuity of $dim_{C}$ and $dim_{MC}$ under the TV topology, see Example 3.16. ∎ However, in virtue of [MMR, Theorem 2.6], if some restrictions are set on the sequence of the measures $\{\nu_{n}\}_{n=1}^{\infty}$, we have some partial semi-continuity results. 3.14 Corollary. For a sequence of measures $\{\nu_{n}\in\mathcal{M}(X)\}_{n=1}^{\infty}$, if $\nu_{n}$ has an essentially bounded density with respect to $\nu$ (which implies that $\nu_{n}$ is absolutely continuous with respect to $\nu$) for any $n\in\mathbb{N}$, then $\liminf_{n\rightarrow\infty}dim_{C}(\nu_{n})\geq dim_{C}(\nu)$. The restriction on the sequence of the measures $\{\nu_{n}\}_{n=1}^{\infty}$ can be relaxed, by [MMR, Theorem 2.10], to give the following lower semi-continuity result for $dim_{MC}$ in due course. 3.15 Corollary. For any sequence of measures $\{\nu_{n}\in\mathcal{M}(X)\}_{n=1}^{\infty}$, if $\nu_{n}$ is absolutely continuous with respect to $\nu$ for any $n\in\mathbb{N}$, then $\liminf_{n\rightarrow\infty}dim_{MC}(\nu_{n})\geq dim_{MC}(\nu)$. Considering Theorem 3.13, we still owe the readers an example of a convergent sequence of measures violating the upper semi-continuity of $dim_{C}$ and $dim_{MC}$ under the TV topology. The following example is an evolution of [MMR, Example 2.5]. 3.16 Example (Mattila-Morán-Rey). Let $a\in(0,1)$ be a real number. Consider the sequence $\{a^{n^{2}}\}_{n=0}^{\infty}\subset[0,1]$. Define a sequence of probability measures $\nu_{n}$ on $[0,1]$ to be $\nu_{n}=\cfrac{1-a}{1-a^{n+1}}\sum_{i=0}^{n}\cfrac{a^{i}}{a^{i^{2}}-a^{{(i+1)}^{2}}}\mathfrak{L}^{1}|_{[a^{i^{2}},a^{{(i+1)}^{2}}]}$ for any $n\in\mathbb{N}$. Let $\nu$ be the probability measure on $[0,1]$ with $\nu=(1-a)\sum_{i=0}^{\infty}\cfrac{a^{i}}{a^{i^{2}}-a^{{(i+1)}^{2}}}\mathfrak{L}^{1}|_{[a^{i^{2}},a^{{(i+1)}^{2}}]}$. First we show the sequence of measures converges under the TV topology. 3.17 Lemma. For the sequence of measures $\{\nu_{n}\}_{n\in\mathbb{N}}$ in Example 3.16, we have $\nu_{n}\stackrel{{\scriptstyle TV}}{{\rightarrow}}\nu$ as $n\rightarrow\infty$. Proof. In this case we have the following estimation on the total variation of the difference between the two measures $\nu_{n}$ and $\nu$, $\|\nu_{n}-\nu\|_{TV}=\max\{a^{n+1},\cfrac{a^{n+1}}{1-a^{n+1}}\}=\cfrac{a^{n+1}}{1-a^{n+1}}$ for any $n\in\mathbb{N}$. So $\lim_{n\rightarrow\infty}\|\nu_{n}-\nu\|_{TV}=0$, which justifies its convergence under the TV topology. ∎ Although the correlation dimension and modified correlation dimension appear as global concepts, they are sometimes dominated by some properties on a subset of $X$ of relatively small measure, as indicated by the following result. 3.18 Lemma. Let $\nu\in\mathcal{M}(X)$ be a probability measure. For any $r$ small enough, if there exists a subset $Y_{r}\subset X$ of positive measure, such that for a.e. $x\in Y_{r}$, the following estimation holds, (3.4) $$\nu(B(x,r))\nu(Y_{r})\geq r^{o(1)},$$ in which $o(1)$ is a positive term satisfying $\lim_{r\rightarrow 0}o(1)=0$. Then we have $dim_{C}(\nu)=dim_{MC}(\nu)=0$. Proof. Under the above assumptions, we have $\int\nu(B(x,r))d\nu(x)\geq\nu(B(x,r))\nu(Y_{r})$. So $\log\int\nu(B(x,r))d\nu(x)\geq\log\big{(}\nu(B(x,r))\nu(Y_{r})\big{)}\geq o(1)\log r$, which gives that $\cfrac{\log\int\nu(B(x,r))d\nu(x)}{\log r}\leq o(1)$. The proof ends by letting $r\rightarrow 0$ in the above estimation. ∎ Now we pay attention to the correlation and modified correlation dimensions of the limit measure of the sequence $\{\nu_{n}\}_{n\in\mathbb{N}}$ in Example 3.16. 3.19 Proposition (Mattila-Morán-Rey). For the limit measure $\nu$ in Example 3.16, we have $dim_{C}(\nu)=dim_{MC}(\nu)=0$. Proof. We achieve the conclusion by Lemma 3.18. Suppose $r$ is a number small enough such that (3.5) $$a^{n_{r}^{2}}\leq r<a^{(n_{r}+1)^{2}}$$ for some $n_{r}\in\mathbb{N}$. Let $Y_{r}=Y_{n_{r}}=[0,a^{n_{r}^{2}}]$, then we have $\nu(Y_{n_{r}})=(1-a)(a^{n_{r}}+a^{n_{r}+1}+\cdots)=(1-a)\cfrac{a^{n_{r}}}{(1-a)}=a^{n_{r}}$. For any $x\in Y_{n_{r}}$, since $Y_{n_{r}}\subset B(x,r)$, we have $\nu(B(x,r))\geq\nu(Y_{n_{r}})=a^{n_{r}}$. So $\nu(B(x,r))\nu(Y_{r})\geq a^{2n_{r}}$ for any $x\in Y_{r}$. Considering (3.5), the condition (3.4) is satisfied in this case. So by Lemma 3.18, we have $dim_{C}(\nu)=dim_{MC}(\nu)=0$. ∎ Obviously for the sequence of measures $\{\nu_{n}\}_{n\in\mathbb{N}}$ in Example 3.16, we have $dim_{C}(\nu_{n})=dim_{MC}(\nu_{n})=1$. Combining this with Lemma 3.17 and Proposition 3.19, we conclude that the measure dimension mappings $dim_{C}$ and $dim_{MC}$ can not be upper semi-continuous under the TV topology in general. We end the section by some results on comparing the dimensions between two comparable measures. 3.20 Lemma. For any two measures $\nu_{1},\nu_{2}\in\mathcal{M}(X)$, if $\nu_{1}$ is absolutely continuous with respect to $\nu_{2}$, then the induced upper measure-dimension mapping satisfies $\dim^{U}(\nu_{1})\leq\dim^{U}(\nu_{2})$. Proof. For $A\in\mathcal{B}$, if $\nu_{2}(X\setminus A)=0$, since $\nu_{1}$ is absolutely continuous with respect to $\nu_{2}$, we have $\nu_{1}(X\setminus A)=0$, so $dim^{U}(\nu_{2})=\inf\{dim(A):\nu_{2}(X\setminus A)=0\}\geq\inf\{dim(A):\nu_{1}(X\setminus A)=0\}=dim^{U}(\nu_{1})$. ∎ One is recommended to compare the lemma with [MMR, p220(a)]. These conclusions again show duality of the two measure-dimension mappings. 3.21 Corollary. For any two measures $\nu_{1},\nu_{2}\in\mathcal{M}(X)$, if $\nu_{1}$ is equivalent to $\nu_{2}$, then $\dim^{U}(\nu_{1})=\dim^{U}(\nu_{2})$ for any induced upper measure-dimension mapping. See also [HS, Lemma 3.1(3)]. 4. Applications of the semi-continuity results to dimensions of measures originated from dynamical systems Most of our results until now are set on measures on ambient space $X$ without dynamics on it. In this section we focus on probability measures carrying some dynamical structures on $X$. Let $T:X\rightarrow X$ be a transformation. Denote by $\mathcal{M}_{\sigma}(X)$ and $\mathcal{M}_{e}(X)$ respectively to be the collections of all the invariant and ergodic probability measures on $(X,\mathcal{B})$, with respect to $T$. The invariant and ergodic probability measures are crucial in many dynamical systems, while dimensions of these measures usually reflect important properties on these dynamical systems. A transformation $T:X\rightarrow X$ is said to be inverse-dimension-expanding with respect to some dimensional mapping $dim$ if there exists some $A\in\mathcal{B}$, such that $dim(T^{-1}(A))>dim(A)$. 4.1 Young’s Lemma. For a transformation $T$ on $(X,\mathcal{B})$ such that $T$ is not inverse dimension-expanding with respect to $dim$, then we have $dim^{L}(\nu)=dim^{U}(\nu)$ for any ergodic measure $\nu\in\mathcal{M}_{e}(X)$ with respect to $T$. Proof. See [You, P115]. ∎ Combining Young’s Lemma and Corollary 3.21, we have the following result. 4.2 Corollary. For two equivalent measures $\nu_{1},\nu_{2}\in\mathcal{M}(X)$ with one of them being ergodic with respect to some non-inverse-dimension-expanding transformation $T$ on $X$, we have $\dim^{L}(\nu_{1})=\dim^{U}(\nu_{1})=\dim^{L}(\nu_{2})=\dim^{U}(\nu_{2})$. This corollary suggests a possible way to deal with the dimensions $\dim^{L}(\nu)$ and $\dim^{U}(\nu)$ of some (non-invariant) measure $\nu\in\mathcal{M}(X)$ simultaneously. That is, for some concerning measure $\nu\in\mathcal{M}(X)$, if one can find an ergodic measure $\nu_{e}\in\mathcal{M}_{e}(X)$ equivalent with $\nu$, then the dimension of $\nu_{e}$ gives the dimension of $\nu$. As there are dynamical structure with respect to $\nu_{e}$, the dimension of $\nu_{e}$ may be easier to be calculated. In case of non-existence of such an ergodic measure equivalent with $\nu$, one may resort to the ergodic decomposition of $\nu$. The method still has some meaning for an invariant measure $\nu$, though an invariant measure $\nu$ absolutely continuous with respect to an ergodic one $\nu_{e}$ necessarily satisfies $\nu=\nu_{e}$ according to [Wal, P153, Remarks(1)]. In some cases it is difficult to decide the dimensions of ergodic (invariant) measures admitting some singularity (for example, when the measure has infinite entropy with respect to the transformation $T$). In these cases our semi-continuity of measures-dimension mappings under the setwise topology together with Young’s Lemma provide a systematic way to approximate the dimensions of these singular ergodic (invariant) measures by sequences of dimensions of non-singular ergodic measures converging setwisely to the singular ones. The method usually involves the following steps. 1. For ones’ target (singular or non-singular) measure $\nu$ on $X$ with a transformation $T$ being not inverse-dimension-expanding, find a sequence of non-singular measures $\{\nu_{n}\}_{n\in\mathbb{N}}$ converging to the target measure $\nu$ under the setwise topology. 2. Show the existence of ergodic measures $\{\nu_{n}^{*}\}_{n\in\mathbb{N}}$ (with respect to some appropriate dynamical structures on $X$) equivalent with the ones $\{\nu_{n}\}_{n\in\mathbb{N}}$ respectively for any $n\in\mathbb{N}$ (sometimes they coincide with each other). 3. Apply Young’s Lemma and our semi-continuity results-Theorem 2.8 and 2.9 to show the convergence of the sequence of dimensions of the measures $\{\nu_{n}\}_{n\in\mathbb{N}}$ and $\{\nu_{n}^{*}\}_{n\in\mathbb{N}}$ to the dimension of the concerning measure $\nu$. For example, as a tool to tackle the obstacle of exploding entropy of some measure with respect to some partitions on the $X$ space, we have the following result, as an extension of [You, Main Theorem]. 4.3 Corollary. Let $X$ be a (non-compact) two Riemannian manifold, $T:X\rightarrow X$ is a $C^{2}$ endomorphism. For an ergodic measure $\nu\in\mathcal{M}(X)$ with respect to $T$, if one can find a sequence of ergodic measures $\{\nu_{n}\in\mathcal{M}(X)\}_{n\in\mathbb{N}}$ with respect to a corresponding sequence of $C^{2}$ endomorphisms $\{T_{n}:X\rightarrow X\}_{n\in\mathbb{N}}$ such that $\nu_{n}\stackrel{{\scriptstyle s}}{{\rightarrow}}\nu$ as $n\rightarrow\infty$ and $0\leq h_{\nu_{n}}(T_{n}),\lambda_{1}(T_{n}),\lambda_{2}(T_{n})<\infty$ for any $n\in\mathbb{N}$, then $\begin{array}[]{ll}&dim_{H}^{L}(\nu)=dim_{H}^{U}(\nu)=\lim_{n\rightarrow\infty}dim_{H}^{L}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{H}^{U}(\nu_{n})\\ =&\lim_{n\rightarrow\infty}h_{\nu_{n}}(T_{n})\Big{(}\cfrac{1}{\lambda_{1}(T_{n})}-\cfrac{1}{\lambda_{2}(T_{n})}\Big{)},\end{array}$ in which $\lambda_{1}(T_{n})\geq\lambda_{2}(T_{n})$ are the Lyapunov exponents of the maps $T_{n}$ and $T_{n}^{-1}$ respectively (they are constants for a.e. $x\in X$ due to ergodicity of $T_{n}$). Proof. First, considering Young’s Lemma, apply [You, Main Theorem] to the measure $\nu_{n}$ with finite entropy and Lyapunov exponents, we have (4.1) $$dim_{H}^{L}(\nu_{n})=dim_{H}^{U}(\nu_{n})=h_{\nu_{n}}(T_{n})\Big{(}\cfrac{1}{\lambda_{1}(T_{n})}-\cfrac{1}{\lambda_{2}(T_{n})}\Big{)}.$$ Since $\nu_{n}\stackrel{{\scriptstyle s}}{{\rightarrow}}\nu$ as $n\rightarrow\infty$, apply Theorem 2.8 to the sequence of measures, we get (4.2) $$\liminf_{n\rightarrow\infty}dim^{U}(\nu_{n})\geq dim^{U}(\nu),$$ while applying 2.9 to the sequence of measures, we get (4.3) $$\limsup_{n\rightarrow\infty}dim^{L}(\nu_{n})\leq dim^{L}(\nu).$$ Due to the ergodicity and Young’s Lemma we have $dim^{U}(\nu)=dim^{L}(\nu)$, combing with (4.2) and (4.3) together we get (4.4) $$\limsup_{n\rightarrow\infty}dim^{L}(\nu_{n})\leq dim^{L}(\nu)=dim^{U}(\nu)\leq\liminf_{n\rightarrow\infty}dim^{U}(\nu_{n}).$$ (4.1) and (4.4) together justify the corollary. ∎ References [Bil1] P. 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Non-Markovianity over ensemble averages in quantum complex networks Johannes Nokkala jsinok@utu.fi Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014, Turun Yliopisto, Finland Sabrina Maniscalco Jyrki Piilo Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014, Turun Yliopisto, Finland Abstract We consider bosonic quantum complex networks as structured finite environments for a quantum harmonic oscillator and investigate the interplay between the network structure and its spectral density, excitation transport properties and non-Markovianity. After a review of the formalism used, we demonstrate how even small changes to the network structure can have a large impact on the transport of excitations. We then consider the non-Markovianity over ensemble averages of several different types of random networks of identical oscillators and uniform coupling strength. Our results show that increasing the number of interactions in the network tends to suppress the average non-Markovianity. This suggests that tree networks are the random networks optimizing this quantity. 1 Introduction Understanding the dynamics of open quantum systems is important in several fields of physics and chemistry including problematics dealing, e.g., with quantum to classical transition and decoherence with its harmful effects for quantum information processing and communication. In general, formulating or deriving a suitable equation of motion for the density matrix plöt for the open system is often a daunting task. Perhaps the most celebrated and most used theoretical result in this context is the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation [1, 2] $$\dfrac{d\rho_{s}(t)}{dt}=-i[H_{s},\rho_{s}(t)]+\sum_{k}\gamma_{k}\left(C_{k}% \rho_{s}(t)C_{k}^{\dagger}-\dfrac{1}{2}\left\{C_{k}^{\dagger}C_{k},\rho_{s}(t)% \right\}\right),$$ (1.1) with the associated completely positive and trace preserving dynamical map with semigroup property. Above, $H_{s}$ is the open system Hamiltonian, $\gamma_{k}$ are positive constant rates, and $C_{k}$ are the jump operators with $k$ indexing the different decoherence channels. Indeed, this master equation and the corresponding publications had recently 40th anniversary celebrations in the Symposium on Mathematical Physics in Toruń in June 2016. GKSL master equation $(1.1)$ describes Markovian memoryless open system dynamics and during the last 10-15 years there has been an increasing amount of research activities in understanding memory effects and quantifying non-Markovianity for open systems beyond the semigroup property [3, 4, 5]. A pair of complementary approaches here include a description based on quantifying the information flow between the open system and its environment [6] or the characterization of dynamical maps in terms of their divisibility properties [3, 8] while a large number of other ways to characterize non-Markovianity also exist, see e.g. [9, 10, 11, 12, 13]. Most of the research so far has focused on non-Markovianity using discrete variable open systems as examples while in the current work we are interested in the memory effects in a continuous variable (CV) open system with controlled environmental structure. Indeed, here we consider structured finite environments modeled by bosonic quantum complex networks. While this and other kinds of quantum complex networks have recieved increasing attention in recent years in the context of perfect state transfer [14, 15], quantum random walks [16, 17], efficient entanglement distribution [18, 19, 20] and the unification of classical and quantum network theory [21, 22], here the focus is on the interplay between the network structure and the reduced dynamics of an open quantum system attached to it. To this end, we investigate the impact of the structure on the network spectral density, excitation transport properties and non-Markovianity of the reduced dynamics. The paper is organized as follows. Section 2 concerns the network itself. Here we present the microscopic model and briefly discuss the connection between the network Hamiltonian and certain matrix representations of abstract graphs in classical graph theory. The dynamics of the network is given in terms of a symplectic matrix acting on the vector of operators at initial time. In Section 3, we describe how complex quantum networks can be treated in the framework of the theory of open quantum systems as tunable structured environments. We demonstrate how small changes in the network structure can have a large impact on its excitation transport properties. In Section 4, we consider the non-Markovianity of the reduced dynamics using a recently introduced witness based on non-monotonicity of the evolution of Gaussian interferometric power. Finally, conclusions are drawn in Section 5. 2 Bosonic quantum complex networks 2.1 The Hamiltonian We set $\hbar=1$ and work with position and momentum operators defined as $q=(a^{\dagger}+a)/\sqrt{2\omega}$ and $p=(a^{\dagger}-a)i\sqrt{\omega/2}$, satisfying the commutation relation $[q,p]=i$. We consider networks of $N$ unit mass quantum harmonic oscillators coupled by springlike couplings. The general form of a Hamiltonian for such networks is $$H_{E}=\dfrac{\mathbf{p}^{T}\mathbf{p}}{2}+\mathbf{q}^{T}\mathbf{Aq},$$ (2.1) where we have introduced the vectors of position and momentum operators $\mathbf{q}^{T}=\{q_{1},...,q_{N}\}$ and $\mathbf{p}^{T}=\{p_{1},...,p_{N}\}$, and where $\mathbf{A}$ is the matrix containing the coupling terms and frequencies. It has elements $\mathbf{A}_{ij}=\delta_{ij}\tilde{\omega}_{i}^{2}/2-(1-\delta_{ij})g_{ij}/2$, where $g_{ij}$ is the strength of the springlike coupling $g_{ij}(q_{i}-q_{j})^{2}/2$ between the position operators of oscillators $i$ and $j$, and $\tilde{\omega}_{i}^{2}=\omega_{i}^{2}+\sum_{j}g_{ij}$ is the effective frequency of oscillator $i$ resulting from absorbing the quadratic parts of the coupling terms into the free Hamiltonians of the oscillators. The matrix $\mathbf{A}$, which completely determines the network Hamiltonian, can be related to some of the typical matrix representations of weighted graphs, i.e. abstract networks of nodes connected by weighted edges. By weighted, we mean that a magnitude is assigned to each connection. This can be used to establish a link between the properties of the network and results from graph theory. A paradigmatic example is the adjacency matrix $\mathbf{V}$ having elements $\mathbf{V}_{ij}=w_{ij}$, where $w_{ij}$ is the weigth of the connection between nodes $i$ and $j$; a weigth of 0 corresponds to the nodes being disconnected. Another matrix that arises very naturally is the Laplace matrix $\mathbf{L}$, related to the adjacency matrix as $\mathbf{L}=\mathbf{D}-\mathbf{V}$, where $\mathbf{D}$ is diagonal with elements $\mathbf{D}_{ii}=\sum_{j}w_{ij}$. In terms of them, matrix $\mathbf{A}$ can be written as $\mathbf{A}=\mathbf{\Delta}_{\tilde{\omega}}^{2}/2-\mathbf{V}/2$ or as $\mathbf{A}=\mathbf{\Delta}_{\omega}^{2}/2+\mathbf{L}/2$, where $\mathbf{\Delta}_{\tilde{\omega}}$ and $\mathbf{\Delta}_{\omega}$ are diagonal matrices of the effective and bare frequencies of the network oscillators, respectively, and weights are given by the coupling strengths. The graph aspect of this and other kinds of quantum networks have been very recently used to, e.g., develop a local probe for the connectivity and coupling strength of a quantum complex network by using results of spectral graph theory [24], and constructing Bell-type inequalities for quantum communication networks by mapping the task to a matching problem of an equivalent unweighted bipartite graph [25]. The Hamiltonian $(2.1)$ is a special case of the quadratic Hamiltonian $H=\mathbf{x}^{T}\mathbf{Mx}$, where the vector $\mathbf{x}$ contains both the position and momentum operators and $\mathbf{M}$ is a $2N\times 2N$ matrix such that $H$ is Hermitian. It can be shown [23] that quadratic Hamiltonians can be diagonalized to arrive at an equivalent eigenmode picture of uncoupled oscillators provided that $\mathbf{M}$ is positive definite. Since $H$ is Hermitian, this is equivalent with the positivity of the eigenvalues of $\mathbf{M}$. In the case at hand, $H_{E}$ may be diagonalized with an orthogonal matrix $\mathbf{K}$ such that $\mathbf{K}^{T}\mathbf{AK}=\mathbf{\Delta}$, where the diagonal matrix $\mathbf{\Delta}$ holds the eigenvalues of $\mathbf{A}$. By defining new operators $$\begin{cases}\mathbf{Q}=\mathbf{K}^{T}\mathbf{q}\\ \mathbf{P}=\mathbf{K}^{T}\mathbf{p},\\ \end{cases}$$ (2.2) the diagonal form of $H_{E}$ reads $$H_{E}=\dfrac{\mathbf{P}^{T}\mathbf{P}}{2}+\mathbf{Q}^{T}\mathbf{\Delta}\mathbf% {Q},$$ (2.3) which is the Hamiltonian of $N$ decoupled oscillators with frequencies $\Omega_{i}=\sqrt{2\mathbf{\Delta}_{ii}}$. 2.2 The dynamics of the network A bosonic quantum complex network is also an interesting system to study in its own right. Below, we review the mathematical tools useful for the task, adopting the definitions for a commutator and anti-commutator between two operator-valued vectors used in [26]. While we will be later concerned with networks initially in the thermal state, we will also briefly discuss the case of an initial Gaussian state without displacement. For a more detailed review of Gaussian formalism in phase space, see [27]. What is presented here is straightforward to apply to the case where interactions with external oscillators is considered, and we will do so in Section $3$. Let $\mathbf{x}$ be a vector containing the position and momentum operators of the network oscillators, and define the commutator between two operator valued vectors as $[\mathbf{x}_{1},\mathbf{x}_{2}^{T}]=\mathbf{x}_{1}\mathbf{x}_{2}^{T}-(\mathbf{% x}_{2}\mathbf{x}_{1}^{T})^{T}$. Now canonical commutation relations give rise to a symplectic form $\mathbf{J}$, determined by $[\mathbf{x},\mathbf{x}^{T}]=i\mathbf{J}$. Let $\mathbf{x}^{\prime}=\mathbf{S}\mathbf{x}$, where $\mathbf{S}$ is a $2N\times 2N$ matrix of real numbers. In order for $\mathbf{S}$ to be a canonical transformation of $\mathbf{x}$, the commutation relations must be preserved. This requirement gives $i\mathbf{J}=[\mathbf{x}^{\prime},\mathbf{x}^{\prime T}]=[\mathbf{S}\mathbf{x},% (\mathbf{S}\mathbf{x})^{T}]=\mathbf{S}[\mathbf{x},\mathbf{x}^{T}]\mathbf{S}^{T% }=i\mathbf{S}\mathbf{J}\mathbf{S}^{T}$, implying that $\mathbf{SJS}^{T}=\mathbf{J}$. Such a matrix is called symplectic with respect to symplectic form $\mathbf{J}$. Symplectic matrices form the symplectic group $Sp(2N,\mathbb{R})$ with respect to matrix multiplication, which can be used to define a symplectic representation of the Gaussian unitary group, meaning that (up to an overall phase factor) the two groups are bijective. We fix $\mathbf{x}^{T}=\{\mathbf{q}^{T},\mathbf{p}^{T}\}=\{q_{1},...,q_{N},p_{1},...,p% _{N}\}$ throughout the rest of the present work. Then the symplectic form becomes $\mathbf{J}=\left(\begin{smallmatrix}0&\mathbf{I}_{N}\\ -\mathbf{I}_{N}&0\end{smallmatrix}\right)$, where $\mathbf{I}_{N}$ is the $N\times N$ identity matrix. By defining the vector of eigenmode operators to be $\mathbf{X}^{T}=\{\mathbf{Q}^{T},\mathbf{P}^{T}\}=\{Q_{1},...,Q_{N},P_{1},...,P% _{N}\}$, we can express the transformation that diagonalizes the network Hamiltonian as $\mathbf{X}=\left(\begin{smallmatrix}\mathbf{K}^{T}&0\\ 0&\mathbf{K}^{T}\end{smallmatrix}\right)\mathbf{x}$; a direct calculation shows that the matrix diagonalizing the Hamiltonian is both symplectic and orthogonal. In the eigenmode picture, the equations of motion are those of noninteracting oscillators. By defining the auxiliary diagonal matrices with elements $\mathbf{D}_{\cos ii}^{\Omega}=\cos(\Omega_{i}t)$, $\mathbf{D}_{\sin ii}^{\Omega}=\sin(\Omega_{i}t)$ and $\mathbf{\Delta}_{\Omega ii}=\Omega_{i}$, we can express them as $$\begin{pmatrix}\mathbf{Q}(t)\\ \mathbf{P}(t)\end{pmatrix}=\begin{pmatrix}\mathbf{D}_{\cos}^{\Omega}&\mathbf{% \Delta}_{\Omega}^{-1}\mathbf{D}_{\sin}^{\Omega}\\ -\mathbf{\Delta}_{\Omega}\mathbf{D}_{\sin}^{\Omega}&\mathbf{D}_{\cos}^{\Omega}% \end{pmatrix}\begin{pmatrix}\mathbf{Q}(0)\\ \mathbf{P}(0)\end{pmatrix},$$ (2.4) where the block matrix acting on the vectors is again symplectic. To recover the dynamics of the network oscillators, we may use Eq. $(2.2)$ to express $\mathbf{x}(t)$ in terms of either $\mathbf{X}(0)$ as $$\begin{pmatrix}\mathbf{q}(t)\\ \mathbf{p}(t)\end{pmatrix}=\begin{pmatrix}\mathbf{K}\mathbf{D}_{\cos}^{\Omega}% &\mathbf{K}\mathbf{\Delta}_{\Omega}^{-1}\mathbf{D}_{\sin}^{\Omega}\\ -\mathbf{K}\mathbf{\Delta}_{\Omega}\mathbf{D}_{\sin}^{\Omega}&\mathbf{K}% \mathbf{D}_{\cos}^{\Omega}\end{pmatrix}\begin{pmatrix}\mathbf{Q}(0)\\ \mathbf{P}(0)\end{pmatrix},$$ (2.5) or in terms of $\mathbf{x}(0)$ as $$\begin{pmatrix}\mathbf{q}(t)\\ \mathbf{p}(t)\end{pmatrix}=\begin{pmatrix}\mathbf{K}\mathbf{D}_{\cos}^{\Omega}% \mathbf{K}^{T}&\mathbf{K}\mathbf{\Delta}_{\Omega}^{-1}\mathbf{D}_{\sin}^{% \Omega}\mathbf{K}^{T}\\ -\mathbf{K}\mathbf{\Delta}_{\Omega}\mathbf{D}_{\sin}^{\Omega}\mathbf{K}^{T}&% \mathbf{K}\mathbf{D}_{\cos}^{\Omega}\mathbf{K}^{T}\end{pmatrix}\begin{pmatrix}% \mathbf{q}(0)\\ \mathbf{p}(0)\end{pmatrix}.$$ (2.6) Notice that the group properties of symplectic matrices quarantees that in both cases the block matrix remains symplectic. If we now restrict our attention to Gaussian states with zero mean, we may define the covariance matrix of the initial state as $$\mathrm{cov}(\mathbf{x}(0))=\tfrac{1}{2}\langle[\mathbf{x}(0),\mathbf{x}^{T}(0% )]_{+}\rangle,$$ (2.7) where the anti-commutator is defined as $[\mathbf{x}_{1},\mathbf{x}_{2}]_{+}=\mathbf{x}_{1}\mathbf{x}_{2}^{T}+(\mathbf{% x}_{2}\mathbf{x}_{1}^{T})^{T}$. If $\mathbf{x}(t)=\mathbf{S}\mathbf{x}(0)$, then the covariance matrix at time $t$ becomes $$\displaystyle\mathrm{cov}(\mathbf{x}(t))$$ $$\displaystyle=\mathrm{cov}(\mathbf{S}\mathbf{x}(0))=\tfrac{1}{2}\langle[(% \mathbf{S}\mathbf{x}(0),(\mathbf{S}\mathbf{x}(0))^{T}]_{+}\rangle$$ (2.8) $$\displaystyle=\tfrac{1}{2}\mathbf{S}\langle[\mathbf{x}(0),\mathbf{x}(0))^{T}]_% {+}\rangle\mathbf{S}^{T}=\mathbf{S}\mathrm{cov}(\mathbf{x}(0))\mathbf{S}^{T}.$$ In the present case of symplectic matrices appearing in Eqs. $(2.5)$ and $(2.6)$, the choice depends on the basis where the initial covariance matrix is defined. A particular subtlety concerns an initial thermal state for the network, where either choice might seem natural. Here, assuming the usual thermal expectation values for non-interacting oscillators in the real oscillator basis, i.e. a diagonal $\mathrm{cov}(\mathbf{x}(0))$, corresponds to the case where the interactions are suddenly switched on at $t=0+$. As here the state is not the stationary state with respect to the Hamiltonian $(2.1)$, one will see the excitations of each network oscillator evolve with time. On the other hand, if one assumes the covariance matrix to be diagonal in the eigenmode basis instead, the excitations will be frozen. In this work we are using the latter approach as it is quite natural to assume an initial stationary state for the environment of an open quantum system. While here the correlation structure in the state of the network is not studied, it is of great interest in the emerging field of continuous-varibale quantum information processing and in particular in the study of so-called cluster states [28, 29], which are multi-mode correlated states used as a resource in measurement-based quantum computing. In this context, it is typically the state, rather than the Hamiltonian, that is represented with a graph. It has been shown that specific quadratic Hamiltonians have cluster states as their ground state, which can then be adiabatically prepared by cooling a set of non-interacting modes to zero temperature and then switching on the interactions [30]. Finally, we mention the complementary viewpoint of open quantum networks, where the network is considered as the open system interacting with an environment of infinite size. The dynamics can then be described with a master equation for the network density matrix. Collective phenomena, such as synchronization, can occur in a network relaxing towards a steady state [31]. 2.3 Experimental aspect To implement an oscillator network, the basic requirements to meet are a static topology, harmonic potential and quantum regime for the oscillators. To match the form of the Hamiltonian $(2.1)$, the couplings between the oscillator position operators should be springlike, and any other interactions between them should either be eliminated or minimized. More challenging requirements include the scalability to many nodes and the ability to implement also long-range couplings in order to have a nontrivial topology. The biggest difficulties are related to the implementation of generic networks: essentially a platform reconfigurable to a desired static topology would be needed, i.e. independent control and tunability over all couplings would be necessary. A possible way to implement a simple oscillator network is to use vibrational modes of trapped ions. In this way, it is possible to implement simple oscillator chains that interact in a harmonic way via Coulomb force in single or segmented traps [32, 33]. The main limitations are related to scalability and independent control of couplings. In particular, if the couplings are mediated by Coulomb force, then they cannot be controlled in an independent way, which limits the networks that can be realized in this way. Proposals for scalable arrays of trapped ions have been made [34]. One can also consider cold atoms trapped in optical lattices. They offer a scalable platform to simulate different many-body systems, in particular the Bose-Hubbard Hamiltonian, which describes interacting bosons in a lattice. While the Hamiltonian is different, it still shares some similarities with that of an oscillator network. The parameters of the Hamiltonian can be tuned, but it cannot be used to implement an arbitrary topology. An array of coupled micro- or nanomechanical resonators acting as phonon traps is a natural candidate for an experimental realization. The setup has good scalability, as experimental implementations of arrays of up to 400 resonators have been reported [35]. In the case of mechanically coupled devices, independent control of the coupling strengths might not be possible, however a proposal of a fully reconfigurable resonator array based on optical couplings has been made [36]. Other challenges include the suppression of intrinsic nonlinearities of the devices, as well as cooling them to reduce thermal noise. First steps in this direction have been taken, as coherent phonon manipulation has been reported in a system of two resonators with a tunable mechanical coupling [37]. Perhaps the most promising alternative is the very recently proposed optical implementation of the dynamics given by the Hamiltonian $(2.1)$, based on a simultaneous downconversion of the components of an optical frequency comb from a femtosecond laser followed by pulse shaping and mode-selective measurements [38]. By mapping the Hamiltonian to quadrature operators of the optical field modes and determining the so called Bloch-Messiah decomposition of either the symplectic matrix $(2.5)$ or $(2.6)$, one will find the pulse shape and measurement basis necessary to implement it. In particular, since the network structure is mapped into the parameters of the platform, changing the network does not require a change in the optical setup. The result is a deterministic and highly reconfigurable implementation of quantum complex networks with in principle arbitrary structure. In practice, producing the required pump shape to a sufficiently good accuracy will require further theoretical and experimental work before the proposal can be tested. 3 Quantum networks as structured environments 3.1 Attaching external oscillators We consider as the open quantum system a single additional quantum harmonic oscillator interacting with one of the network oscillators. While this is sufficient to our present purposes, what follows is straightforward to extend to the case of multiple external oscillators or interactions with multiple network nodes. Moreover, we will fix the states of the open system and the network to be a Gaussian state and a thermal state of temperature $T$, respectively, assume factorizing initial conditions and work in such units that the Boltzmann constant $k_{B}=1$. The open system Hamiltonian is $H_{S}=(p_{S}^{2}+\omega_{S}^{2}q_{S}^{2})/2$, and the form of the interaction Hamiltonian reads $H_{I}=-kq_{S}q_{i}$, or equivalently, $H_{I}=-kq_{S}\sum_{j}^{N}\mathbf{K}_{ij}Q_{j}$ in the basis of eigenmodes, where $k$ is the coupling strength between the open system and the network. The total Hamiltonian is now $H=H_{S}+H_{E}+H_{I}$. By including the operators of the open system as the final elements of the vectors of operators, we may express it analogously to Hamiltonian $(2.1)$ as $$H=\dfrac{\{\mathbf{P},p_{S}\}^{T}\{\mathbf{P},p_{S}\}}{2}+\{\mathbf{Q},q_{S}\}% ^{T}\mathbf{B}\{\mathbf{Q},q_{S}\},$$ (3.1) where the matrix $\mathbf{B}$ has diagonal elements $\mathbf{B}_{ii}=\Omega_{i}^{2}/2$ for $i<N+1$ and $\mathbf{B}_{N+1,N+1}=\omega_{S}^{2}/2$, while $\mathbf{B}_{N+1,i}=\mathbf{B}_{i,N+1}=-k\mathbf{K}_{li}/2$ for $i<N+1$; here the index $l$ is the index of the network oscillator directly interacting with the open system. We may diagonalize the matrix $\mathbf{B}$ as $\mathbf{O}^{T}\mathbf{B}\mathbf{O}=\mathbf{F}$ where $\mathbf{O}$ is orthogonal and $\mathbf{F}$ diagonal with elements $F_{ii}=f_{i}^{2}/2$, where $f_{i}$ will be the frequencies of the modes in the fully diagonal picture. If we define the new operators as $$\begin{cases}\bm{\mathcal{Q}}=\mathbf{O}^{T}\{\mathbf{Q},q_{S}\}\\ \bm{\mathcal{P}}=\mathbf{O}^{T}\{\mathbf{P},p_{S}\},\\ \end{cases}$$ (3.2) the total Hamiltonian reads $$H_{E}=\dfrac{\bm{\mathcal{P}}^{T}\bm{\mathcal{P}}}{2}+\bm{\mathcal{Q}}^{T}% \mathbf{F}\bm{\mathcal{Q}}.$$ (3.3) We are now in position to write down the symplectic matrix giving the dynamics of the total Hamiltonian. By following the steps leading from Hamiltonian $(2.3)$ to Eq. $(2.6)$, we arrive at $$\begin{pmatrix}\mathbf{Q}(t)\\ q(t)\\ \mathbf{P}(t)\\ p(t)\\ \end{pmatrix}=\begin{pmatrix}\mathbf{O}\mathbf{D}_{\cos}\mathbf{O}^{T}&\mathbf% {O}\mathbf{\Delta}_{f}^{-1}\mathbf{D}_{\sin}\mathbf{O}^{T}\\ -\mathbf{O}\mathbf{\Delta}_{f}\mathbf{D}_{\sin}\mathbf{O}^{T}&\mathbf{O}% \mathbf{D}_{\cos}\mathbf{O}^{T}\end{pmatrix}\begin{pmatrix}\mathbf{Q}(0)\\ q(0)\\ \mathbf{P}(0)\\ p(0)\\ \end{pmatrix},$$ (3.4) where we have introduced the diagonal matrices $\mathbf{D}_{\cos ii}=\cos(f_{i}t)$, $\mathbf{D}_{\sin ii}=\sin(f_{i}t)$ and $\mathbf{\Delta}_{fii}=f_{i}$. As we will consider an initial thermal state for the network, throughout the rest of the present work we will consider as the initial basis the one on the R.H.S. of the equation above, where the initial covariance matrix of the network is diagonal with elements $\langle Q_{i}(0)^{2}\rangle=(n_{i}+1/2)/\Omega_{i}$ and $\langle P_{i}(0)^{2}\rangle=(n_{i}+1/2)\Omega_{i}$, where $n_{i}=(\exp(\Omega_{i}/T)-1)^{-1}$. If we are interested in the dynamics of the operators in the network basis, we may use Eq. $(2.2)$ and define the symplectic and orthogonal $N+1\times N+1$ matrix $\tilde{\mathbf{K}}$ with elements $\tilde{\mathbf{K}}_{N+1,i}=\mathbf{K}_{i,N+1}=0$ for $i<N+1$, $\tilde{\mathbf{K}}_{N+1,N+1}=1$, and $\tilde{\mathbf{K}}_{ij}=\mathbf{K}_{ij}$ otherwise. Now $$\begin{pmatrix}\mathbf{q}(t)\\ q(t)\\ \mathbf{p}(t)\\ p(t)\\ \end{pmatrix}=\begin{pmatrix}\tilde{\mathbf{K}}\mathbf{O}\mathbf{D}_{\cos}% \mathbf{O}^{T}&\tilde{\mathbf{K}}\mathbf{O}\mathbf{\Delta}_{f}^{-1}\mathbf{D}_% {\sin}\mathbf{O}^{T}\\ -\tilde{\mathbf{K}}\mathbf{O}\mathbf{\Delta}_{f}\mathbf{D}_{\sin}\mathbf{O}^{T% }&\tilde{\mathbf{K}}\mathbf{O}\mathbf{D}_{\cos}\mathbf{O}^{T}\end{pmatrix}% \begin{pmatrix}\mathbf{Q}(0)\\ q(0)\\ \mathbf{P}(0)\\ p(0)\\ \end{pmatrix}.$$ (3.5) The dynamics can be readily determined from the initial covariance matrix of the total system, as outlined in Eq. $(2.8)$, where the result will be the covariance matrix at time $t$ in the basis of either the eigenmodes or the network oscillators, depending on which symplectic matrix is used. If the open system has displacement, also the evolution of its first moments needs to be considered to determine the evolution of its state. While we are concerned with the dynamics of the open system as well as the network oscillators, we mention here the possibility to treat the network in the framework of Gaussian channels. For a general Gaussian state and for $\mathbf{x}(0)=\left(\begin{smallmatrix}q(0)\\ p(0)\end{smallmatrix}\right)$, the elements of the covariance matrix of a single mode system are $\mathrm{cov}(\mathbf{x}(0))_{ij}=\langle\mathbf{x}(0)_{i}\mathbf{x}(0)_{j}+% \mathbf{x}(0)_{j}\mathbf{x}(0)_{i}\rangle/2-\langle\mathbf{x}(0)_{i}\rangle% \langle\mathbf{x}(0)_{j}\rangle$. For any Gaussian channel taking the covariance matrix to time $t$, the transformation can be written as $$\mathrm{cov}(\mathbf{x}(t))=\mathbf{C}(t)\mathrm{cov}(\mathbf{x}(0))\mathbf{C}% (t)^{T}+\mathbf{L}(t),$$ (3.6) where $\mathbf{C}(t)$ and $\mathbf{L}(t)$ are real matrices and $\mathbf{L}(t)$ is symmetric. In terms of the elements of the symplectic matrix $\mathbf{S}$ of Eq. $(3.4)$, we may find the elements using Eq. $(2.8)$ to be $$\mathbf{C}(t)=\begin{pmatrix}\mathbf{S}_{N+1,N+1}&\mathbf{S}_{N+1,2N+2}\\ \mathbf{S}_{2N+2,N+1}&\mathbf{S}_{2N+2,2N+2}\end{pmatrix},$$ (3.7) and $$\mathbf{L}(t)=\sum_{i}\langle\mathbf{X}_{i}(0)^{2}\rangle\begin{pmatrix}% \mathbf{S}_{N+1,i}^{2}&\mathbf{S}_{N+1,i}\mathbf{S}_{2N+2,i}\\ \mathbf{S}_{N+1,i}\mathbf{S}_{2N+2,i}&\mathbf{S}_{2N+2,i}^{2}\end{pmatrix},$$ (3.8) where the sum is taken to $2N+1$ excluding $N+1$, such that $\mathbf{L}(t)$ is independent of the initial expectation values of the open system. The matrices $\mathbf{C}(t)$ and $\mathbf{L}(t)$ now completely characterize the channel, allowing, e.g. to make comparisons with channels defined by a master equation or to construct intermediate channels taking the system from time $t>0$ to $s>t$ and checking if the resulting channel is completely positive or not, as is done in a recently introduced measure of non-Markovianity for Gaussian channels [39]. The difficulty in implementing this measure in the present case is neither in the construction of the intermediate map nor checking its complete positivity, but rather in the fact that it considers the limit $s\rightarrow t$, and it is not clear how to take such a limit in the case of numerical, rather than analytical, matrices $\mathbf{C}(t)$ and $\mathbf{L}(t)$. 3.2 The spectral density One of the central concepts in the theory of open quantum systems is the spectral density of environmental couplings $J(\omega)$, which encodes the relevant information in the environment and interaction Hamiltonians into a single function of frequency. The reduced dynamics of the open system can then be determined once the initial state of the total system as well as the system Hamiltonian are fixed [40]. In particular, a heat bath is completely characterized by its spectral density and temperature. The definition of the spectral density, in terms of the environment eigenfrequencies $\Omega_{i}$ and coupling strengths to eigenmodes $g_{i}$, reads $$J(\omega)=\dfrac{\pi}{2}\sum_{i}\dfrac{g_{i}^{2}}{\Omega_{i}}\delta(\omega-% \Omega_{i}),$$ (3.9) where $\delta$ is the Dirac’s delta function. The definition is rarely used in practice, since in the case of an infinite heat bath with a continuum of frequencies the spectral density becomes a continuous function, and phenomenological spectral densities are defined instead. In the case of finite environments it is convenient to use the relation between $J(\omega)$ and the damping kernel $\gamma(t)$, the latter appearing in the generalized quantum Langevin equations giving the dynamics for the open system operators [40]. It is defined as $$\gamma(t)=\sum_{i}\dfrac{g_{i}^{2}}{\Omega_{i}^{2}}\cos(\Omega_{i}t),$$ (3.10) and the relation is given by $$J(\omega)=\omega\int_{0}^{\infty}\gamma(t)\cos(\omega t)dt,$$ (3.11) If the environment is finite, both Eq. $(3.9)$ and Eq. $(3.11)$ will result in delta spikes. However, by replacing the upper limit of integration by a finite time $t_{max}$, the intermediate form of the spectral density can be considered instead. If a quantum network defined by a Hamiltonian of the form $(2.1)$ is sufficiently symmetric, the reduced dynamics will have a regime where the system interacts with a continuum of frequencies as if the environment was infinite. This is evident from the damping kernel having a very small value during this transient, until finite size effects cause a revival of oscillations. The duration of this continuous regime of reduced dynamics depends on the structure and size of the finite environment. In the present case of quantum complex networks, the coupling strengths to eigenmodes $g_{i}$ are determined by the interaction Hamiltonian $H_{I}$ and the matrix $\mathbf{K}$ diagonalizing the Hamiltonian $(2.1)$ as $g_{i}=-k\mathbf{K}_{li}$, where $l$ is the index of the network oscillator directly interacting with the system and $k$ the interaction strength in the network basis. In Fig. 1, we show two examples of damping kernels and spectral densities for quantum networks. The symmetric network is a chain with nearest and next nearest couplings. Additionally, the chain is made homogeneous by setting the effective frequencies of the ends of the chain equal with the rest. The spectral density is continuous for the used value of $t_{max}$. If the interaction time is sufficiently short, it would not be possible to tell from the reduced dynamics of an open quantum system coupled to the network alone that the environment is in fact finite. In contrast, the disorder in the other network results in a highly structured spectral density that does not have a continuous regime. In general, it may be asked whether $J(\omega)$ of a quantum complex network can be deduced from the reduced dynamics of the system. It can be shown [41] that, provided the coupling to the network $k$ is weak and the network is in a thermal state, the system excitation number is well approximated by the expression $\langle n(t)\rangle=\exp(-\Gamma t)\langle n(0)\rangle+n(\omega_{S})(1-\exp(-% \Gamma t)$, where $\Gamma=J(\omega_{S})/\omega_{S}$ and $n(\omega_{S})=(\exp(\omega_{S}/T)-1)^{-1}$, or the thermal average boson number at system frequency $\omega_{S}$. The value of the spectral density at system frequency is then approximated by $$J(\omega_{S})=\dfrac{\omega_{S}}{t}\ln\left(\dfrac{\Delta n(0)}{\Delta n(t)}% \right),$$ (3.12) where $\Delta n(t)=n(\omega_{S})-\langle n(t)\rangle$. If $T$ is known, the local value of the spectral density can be determined by performing measurements on the system only. This is demonstrated in Fig. 1, where the dots are probed values of the spectral density with each circle corresponding to one value of the system frequency. By keeping the interaction time fixed to the used value of $t_{max}$, it can be seen that even for networks with disorder, the probed values follow the shape of $J(\omega)$. It is also worth mentioning that the machinery introduced so far can be used to approximate an infinite heat bath, determined by its spectral density, with a finite one. Together with its temperature, the finite bath is completely characterized by the coupling strengths $g_{i}$ and frequencies $\Omega_{i}$. While there is considrebale freedom in choosing $\Omega_{i}$, they should cover the non-vanishing parts of $J(\omega)$ and there should be enough of them to push the finite size effects to interaction times longer than what is being considered. Next, the couplings are determined from the spectral density as follows. From Eq. $(3.9)$, it can be seen that $\int_{0}^{\infty}\frac{2}{\pi}J(\omega)\omega d\omega=\sum_{i}g_{i}^{2}$. Approximating the integral on the left hand side with, e.g., a Riemann sum, and identifying the terms on both sides then gives $g_{i}^{2}=\frac{2}{\pi}J(\Omega_{i})\Omega_{i}\Delta\Omega_{i}$, where $\Delta\Omega_{i}=|\Omega_{i}-\Omega_{i+1}|$ is the sampling interval. The range of interaction times where the approximation is valid can be checked by comparing the damping kernels calculated for the finite bath from Eq. $(3.10)$ and for the infinite bath from the inversion of Eq. $(3.11)$, namely, $\gamma(t)=\frac{2}{\pi}\int_{0}^{\infty}\frac{J(\omega)}{\omega}\cos(\omega t)d\omega$. The two will be similar up to the point where finite size effects manifest. This can be of advantage when considering early or intermediate dynamics in the case of a strong coupling, since the dynamics given by Eqs. $(3.4)$ or $(3.5)$ is exact. 3.3 Engineering aspect and excitation transport Reservoir engineering aims to modify the properties of the environment of an open quantum system, typically to protect non-classicality of the system or to increase the efficiency of some task. In the present case, the environment is a quantum network determined by the matrix $\mathbf{A}$. To assess its properties as an environment, it is convenient to consider the effect of the structure on the spectral density $J(\omega)$, which can be returned to the effect of the structure on the eigenfrequencies $\Omega_{i}$ and coupling strengths to eigenmodes $g_{i}$. By changing the structure by, e.g., adding or removing links, one can try to effectively decouple the system from the network by finding a configuration where $J(\omega_{S})$ has a small value, or alternatively to look for structures with increases transport efficiency. In fact, assuming that the system can be freely coupled to any single node in the network, a single network can produce as many spectral densities as it has nodes. This is because a coupling to a single node corresponds to a set of coupling strengths $g_{i}$, which are in turn directly proportional to a row of the matrix $\mathbf{K}$ diagonalizing the network. On the other hand, the set of eigenfrequencies $\Omega_{i}$ are completely determined by the eigenvalues of the matrix $\mathbf{A}$ and as such are independent of where in the network the system is coupled. Even small changes to the network structure can have a large impact on both the network spectral density and excitation transport properties. Generally speaking, when the reduced dynamics has a continuous regime, the flow of energy is steady provided that the system is resonant with the network. Furthermore, excitations can freely be exchanged between different nodes in the network. On the other hand, when the degree of disorder in the network is high, the excitations typically become locked to a subset of the network nodes and cannot spread effectively. We present examples of this in Fig. 2, where the same symmetric network is considered as in Fig. 1. Rewiring randomly only a single coupling changes the path taken by the majority of excitations. Also shown is the excitation dynamics in a random network. While transport is inefficient in most random networks, a search can be carried out for exceptions, and indeed it can be shown that when sampling the distribution of random networks, some rare cases have vastly superior transport properties robust against ambient dephasing [42]. One may also ask whether there is any connection between the excitation transport properties and non-Markovianity. While in the spin-boson model non-Markovianity and the back-flow of excitations can behave similarly with respect to the environment parameters [43], there does not seem to be such a connection in the case of a continuous variable system [44]. Furthermore, even in the spin-boson model, information and excitation backflows can occur without the other [45]. 4 Non-Markovianity in complex quantum networks 4.1 Generalities The dynamics of an open quantum system can significantly deviate from the memoryless Markovian case when the interaction between the open system and the environment is strong, or if the environment is structured. Previous investigations [46] of harmonic chains with nearest neighbor couplings, having a Hamiltonian of the form $(2.1)$, show that the strongest memory effects occur when the system frequency is located near the edges of the spectral density. A $J(\omega)$ with a single band will then have two regimes of system frequency where memory effects are strong while one with band-gaps will have more. In this work, the Breuer-Laine-Piilo [47] and Rivas-Huelga-Plenio [48] measures were used. To the best of our knowledge, however, there have been no studies of non-Markovianity attempting to connect it to the structure of a complex network. While it is the case that any spectral density of an oscillator network with non-regular structure can be replicated with an oscillator chain with nearest neighbor couplings [49, 50, 51], it is nevertheless of interest to ask whether the amount of non-Markovianity could be tied to the statistical properties of complex networks by comparing the average non-Markovianity over many realizations, and whether adding more structure typically increases the non-Markovianity or not. To this end, we considered three types of random networks presented in Figure 3. For all three cases, we fixed the size of the network to be $N=30$ and assumed that the network is connected, i.e. any node can be reached from any other by following the links. The Erdős-Rényi network $G(N,p)$ [52] is constructed from the completely connected network of $N$ nodes by independently selecting each link to be part of the final network with a probability $p$. The Barabási-Albert network $G(N,l)$ [53] is constructed from a connected network of $3$ nodes and repeatedly adding a new node with $l$ links, connecting it randomly to existing nodes but favoring nodes which already have a high number of links, until the size $N$ is reached. Setting $l=1$ is an important special case, as the resulting network is a tree, i.e. it has the smallest possible number of links that a connected network of size $N$ can have. Finally, a Watts-Strogatz network $G(N,p,n)$ [54] is constructed starting from a circular network where all nodes are connected to $n$-th nearest neighbors, and then rewiring each link with the probability $p$. In this work, we fixed $n=2$. 4.2 Non-Markovianity quantified by the non-monotonicity of Gaussian interferometric power The key concept used in several witnesses and measures of non-Markovianity is to track the dynamics of a quantity that can be shown to behave differently under Markovian and non-Markovian evolutions. In this work, we consider a recently introduced measure and a witness based on the non-monotonicity of Gaussian interferometric power under non-divisible dynamical maps [55]. Gaussian inteferometric power $\mathcal{Q}$ quantifies the worst-case precision achievable in black-box phase estimation using a bipartite Gaussian probe composed of modes $A$ and $B$. It is also a measure of discord-type correlations between the two modes, as it vanishes for product states. For quantifying non-Markovianity, it is enough to consider the case where mode $A$ is subjected to a local Gaussian channel while mode $B$ remains unchanged. Then the expression for the Gaussian interferometric power $\mathcal{Q}$ has a closed form in terms of the symplectic invariants of the two-mode covariance matrix $\sigma_{AB}$ [56]. For Markovian channels, $\mathcal{Q}$ is a monotonically non-increasing function of time, implying that $\dfrac{d}{dt}\mathcal{Q}(\sigma_{AB})\leq 0$. Any period of time where this does not hold is then a sign of non-Markovianity. Once the initial covariance matrix $\sigma_{AB}$ has been fixed, the degree of non-Markovianity of the reduced dynamics can then be quantified as $$\mathcal{N}_{GIP}=\frac{1}{2}\int_{0}^{\infty}(|\mathcal{D}(t)|+\mathcal{D}(t)% )dt,$$ (4.1) where $\mathcal{D}(t)=\dfrac{d}{dt}\mathcal{Q}(\sigma_{AB})$. While the related measure is defined with a maximization over all initial states for the bi-partite system, Eq. $(4.1)$ provides a lower bound for this measure. Since there is strong numerical evidence that squeezed thermal states are particularly suited for witnessing non-Markovianity of this type [55], we fix the initial state of the two-mode system to be a squeezed thermal state with two-mode squeezing parameter $r=\frac{1}{2}\cosh^{-1}(5/2)$ and initial thermal excitations $n_{A}=n_{B}=1/2$ for both modes. The network initial state is taken to be the vacuum. Besides disorder in the network structure, additional sources of non-Markovianity include finite size effects that become stronger as the interaction time is increased, and memory effects at the boundaries of the spectral density. To better assess the non-Markovianity arising from the structure, we will restrict the interaction time to an intermediate value of $t=50$ and fix the frequency of the system to be the $15th$ eigenfrequency of the networks to ensure that it is resonant. The results are shown in Fig. 4. For all considered cases, changing the interaction strength affects the magnitude but not the behaviour of non-Markovianity against the network parameter. For Erdős-Rényi and Barabási-Albert networks, the number of couplings between network oscillators grows with the parameter, reducing the amount of non-Markovianity. On the other hand, the number of couplings in the network is constant for the Watts-Strogatz network. The results suggest that when the system is resonant with the network, non-Markovianity is highest for networks with a small amount of random couplings. For all considered coupling strengths, the highest non-Markovianity is achieved when the network is a tree. If the network is highly symmetric, as is the case with Watts-Strogatz networks with a low rewiring probability, the amount of non-Markovianity in the resonant case is very small. Non-Markovianity is increased by introducing disorder into the network through rewiring of the couplings. Besides the results we present here, we also checked that increasing the network temperature decreases the non-Markovianity. Furthermore, for comparison we determined the non-Markovianity in the simple case of a homogeneous chain with nearest-neighbor couplings only and found that even at the edges of the spectral density, where memory effects are strongest, $\mathcal{N}_{GIP}$ has a similar value than Erdős-Rényi and Barabási-Albert networks have in the resonant case. 5 Conlusions and outlook In this work, we have studied bosonic quantum complex networks in the framework of open quantum systems. After briefly investigating the effect of the network stucture on the spectral density and transport of excitations, we focused on the non-Markovianity in the reduced dynamics of an open quantum system interacting with the network. We considered non-Markovianity over ensemble averages of different types of random networks of identical oscillators and constant coupling strength between the network oscillators. Previous work shows that strong memory effects can occur in symmetric networks at the edges of the spectral density and near band gaps. Here we have shown that increasing the disorder of the network can lead to a high degree of non-Markovianity also when the system is resonant with the network, however increasing the number of interactions between network oscillators appears to suppress it, suggesting that trees optimize the ensemble averaged non-Markovianity. While here we considered only the lower bound of a single non-Markovianity measure, it would be interesting to extend the investigations to other measures such as the measure introduced by Torre, Roga and Illuminati [39]. 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Wide or Narrow? The Phenomenology of 750 GeV Diphotons Matthew R. Buckley Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854, USA Abstract I perform a combined analysis of the ATLAS and CMS diphoton data, using both Run-I and Run-II results, including those released at the 2016 Moriond conference. I find combining the ATLAS and CMS results from Run-II increases the statistical significance of the reported 750 GeV anomaly, assuming a spin-0 mediator coupling to gluons or heavy quarks with a width much smaller than the detector resolution. This significance does not decrease when the 8 TeV data is included. A spin-2 mediator is disfavored compared to the spin-0 case. The cross section required to fit the ATLAS anomaly is in tension with the aggregate data, all of which prefers a smaller value. The best fit for all models I consider is a $4.0\sigma$ local significance for a 750 GeV spin-0 mediator coupling to gluons with a cross section of 4 fb at 13 TeV (assuming narrow width) or 10 fb (assuming $\Gamma=45$ GeV). I Introduction The ATLAS announcement atlas13 of a $3.6\sigma$ local excess in diphotons with invariant masses near $m_{\gamma\gamma}\sim 750$ GeV in the first batch of LHC Run-II data, combined with the CMS Collaboration announcing cms13 a $2.6\sigma$ local excess in the same channel and invariant mass, has sent the theoretical physics community into a frenzy of model-building, with 150 papers on the topic appearing on arXiv within a month. The “$\gamma\gamma$ Resonance that Stole Christmas” Craig:2015lra is of course tremendously exciting. It is the LHC’s most statistically significant deviation from the Standard Model of particle physics made public since the discovery of the 125 GeV Higgs boson. The addition of 0.6 fb${}^{-1}$ of CMS data gathered without the presence of the solenoid field CMS_0 , as well a reanalysis of the 13 and 8 TeV data by the ATLAS Collaboration ATLAS_redo has increased the statistical preference for new physics, with CMS now reporting a $3.4\sigma$ local significance in their combined data. At this stage, the information on which theorists can build models is somewhat minimal, assuming that these results are not just a somewhat unlikely statistical fluctuation and there is actually new physics to be begin with. A $3.6\sigma$ signal is not very large, and additional slicing of the data in order to ask more detailed questions will typically yield only small statistical preference for any possible result. The goal of this paper is to investigate what information can be extracted from the combination of ATLAS and CMS data in the diphoton channel, including the most recent analyses released at the 2016 Moriond conference CMS_0 ; ATLAS_redo . In particular, I am interested in the statistical significance of the combined 13 TeV Run-II data, and in the significance of combinations with the Run-I 8 TeV data. In addition, the ATLAS Collaboration claims a better fit ($3.9\sigma$ local) to a resonance at 750 GeV with a width of 45 GeV, as opposed to a narrow resonance with width much less than the experimental resolution (as might be naively expected). Fitting this wide resonance into theoretical models has been the focus of many of the 150 papers on the topic. In this paper, I consider the combined statistical preference for this width. This paper should be viewed as a follow-up to the work of Refs. Knapen:2015dap ; Gupta:2015zzs ; Falkowski:2015swt , which are some of the earliest theoretical papers on the 750 GeV anomaly, and have largely set the parameters which later papers have adopted. Where our results overlap, my results broadly agree with those found in Refs. Knapen:2015dap ; Gupta:2015zzs ; Falkowski:2015swt ; in particular in terms of the best-fit cross sections for the anomaly. This paper goes into somewhat more detail in fitting the signal in the ATLAS and CMS data, by floating the fit to the background functional forms as signal is added. I also perform fits to spin-2 resonances, and investigate the preference of the combined data for a wide or narrow resonance. This paper does not attempt to address the implications of this anomaly for other search channels at the LHC, new physics models beyond the resonance itself, or derive any limits on fundamental couplings. Such questions have been well addressed in the literature. I restrict this paper solely to the input parameters for theoretical studies: the cross sections, widths, and statistical significance of the anomaly itself. The next section describes my combination fits to the ATLAS and CMS diphoton data, using both 13 and 8 TeV data. For simplicity (and due to the computational limitations of the event generation), I consider only local significance in my combination fits, rather than including the look-elsewhere-effect. The statistical question at hand here is the significance of the anomaly at 750 GeV specifically, thus somewhat justifying this choice. I parameterize the possible couplings of the mediator to gluons and light quarks separately, and consider both spin-0 and spin-2 resonances. I present additional results for second-generation and $b$-quark couplings in Appendix A. In Section III, I then turn to the question of the statistical preference for a wide or narrow resonance; using the best-fit parameters from Section II. I will discuss my conclusions in Section IV, but I summarize some of the more salient points here: 1. The required cross section to fit the anomaly reported by ATLAS is in tension with the 8 TeV results, as well as the required cross section to fit the CMS anomaly. 2. Combining all data sets yields a local significance of $\sim 4.0\sigma$ for a 750 GeV spin-0 resonance produced through couplings to gluons or heavy quarks. While quoted statistical significances must be taken with a grain of salt, as they are obtained using binned data without inclusion of systematic errors, I find the combination yields a net increase in the statistical significance as compared to the ATLAS data alone. 3. The spin-2 interpretation is mildly disfavored compared to a spin-0 mediator. This is due to correlations in the photon momenta which results in a relative decrease in the ATLAS acceptance compared to CMS. 4. The combination of ATLAS and CMS 13 TeV data has a slight statistical preference for a spin-0 mediator with a natural width much smaller than the experimental resolution, as compared to the $\Gamma=45$ GeV preferred by ATLAS alone. When the 8 TeV data is added, there is a slight statistical preference for a wide resonance over the narrow option, as it is easier to hide a wide resonance in the 8 TeV background. II Combination Fits In this section, I consider the ATLAS and CMS searches for diphoton resonances. There are six data sets to consider: the ATLAS atlas13 and CMS cms13 Run-II searches at 13 TeV – which contain the anomalous excess that have caused so much excitement of late – and the previous 8 TeV Run-I diphoton resonance searches from the two experiments Aad:2014ioa ; Khachatryan:2015qba . To this, CMS has a smaller data set, gathered when their solenoid magnetic field was off during the 13 TeV Run-II CMS_0 , released at the Moriond 2016 conference. Note that ATLAS had two possible Run-I diphoton searches at the time of the announcement of the Run-II anomaly: one from the Higgs working group Aad:2014ioa , and one from the exotica working group Aad:2015mna . These have slightly different event selection criteria; the exotica search was developed to target spin-2 graviton searches, while the former is designed to search for Higgs-like scalar resonances. After the reanalysis of both Run-I and Run-II data released by ATLAS at the Moriond conference, a choice of two event selection criteria was applied to both the ATLAS 13 and 8 TeV data sets, one targeted for spin-0 resonances, and the other for spin-2 ATLAS_redo . In this paper, I use the appropriate ATLAS event selection as I consider the two spin options. For the remainder of this paper, I will refer to these six data sets by the names Atlas13 atlas13 ; ATLAS_redo , Cms13 cms13 , Cms13/0T CMS_0 (for the data collected when the magnetic field was 0 T), Atlas8 ATLAS_redo , and Cms8 Khachatryan:2015qba . The selection criteria for each of the data sets varies slightly. In Table 1, I list the requirements for diphoton events to end up in the analysis region for each experimental search. In addition, each search implements isolation criteria on the photon candidates. Cms13 has two search regions, one where both photons are central (denoted “BB” here, for “barrel-barrel” events), and one where one photon is forward (“BE,” for barrel-endcap). The Cms13/0T data uses identical selection criteria. The CMS Collaboration analysis of their Cms8 results uses four signal regions, depending on whether the leading- and subleading-$p_{T}$ photons have converted in the detector. For the range of photon $p_{T}$ of interest here, this level of distinction is less important, and for simplicity I used the combined Cms8 data set. Both the Atlas13 and Atlas8 data uses separate selection criteria for their spin-0 and spin-2 searches. As a resonance search, we are interested in peak-like structures in the differential distribution of the diphoton invariant mass $m_{\gamma\gamma}$. As a first step, I verified my ability to reproduce the reported fits to the background functional forms, assuming no signal injected. In each analysis, the backgrounds are fit to a data-driven two-parameter function, assuming Poisson statistics. These functions are $$\displaystyle\textsc{Atlas8}(\textsc{Atlas13}):f(m_{\gamma\gamma};a_{0},b)$$ $$\displaystyle=$$ $$\displaystyle\left(1-\left(\frac{m_{\gamma\gamma}}{\sqrt{s}}\right)^{1/3}\right)^{b}\left(\frac{m_{\gamma\gamma}}{\sqrt{s}}\right)^{a_{0}},$$ (1) $$\displaystyle\textsc{Cms13}(\textsc{Cms13/0T}):f(m_{\gamma\gamma};a,b)$$ $$\displaystyle=$$ $$\displaystyle m_{\gamma\gamma}^{a+b\log m_{\gamma\gamma}},$$ (2) $$\displaystyle\textsc{Cms8}:f(m_{\gamma\gamma};p_{1},p_{2})$$ $$\displaystyle=$$ $$\displaystyle e^{-p_{1}m_{\gamma\gamma}}m_{\gamma\gamma}^{-p_{2}}.$$ (3) After digitizing the binned $m_{\gamma\gamma}$ data from each experiment, I fit to these functional forms, marginalizing over the two free parameters assuming Poisson statistics. While the experiments themselves obviously have access to much more information of the unbinned diphoton events, I am restricted to the public data, which is of course binned. This loss of information will result in some degradation of statistical power, as will be seen, but the difference is not large. My resulting best-fit backgrounds are shown in Figure 1 (using the spin-0 selection for the ATLAS data) overlaid with the experimental data for the six experimental searches. In all cases, I can successfully reproduce the best-fit backgrounds found by the experimental collaborations It should be noted that these functional forms are data-driven, and out of six diphoton analyses, three different functional forms were chosen. It has been noted that changing the functional forms to increase support at high invariant mass could possibly reduce the significance of the observed excess Davis:2016hlw . This is made possible by the low statistics of diphoton counts at large $m_{\gamma\gamma}$. Further, the 750 GeV diphoton excess sits near the tail of the 8 TeV ATLAS and CMS analyses. Thus, it is possible to “hide” the 13 TeV excess in the 8 TeV by lowering the background function in this region and absorbing the excess into the signal. This is especially notable when the signal is assumed to be a wide resonance, covering much of the high $m_{\gamma\gamma}$ range. After fitting the background functions to the digitized data, I then use these background-only fits to validate my simulation pipeline. I simulate the primary irreducible background of $pp\to\gamma\gamma+X$ using MadGraph5 Alwall:2014hca , matched up to two jets at $p_{T}=10$ GeV using Pythia6 Sjostrand:2006za . Detector simulation is performed using Delphes3 deFavereau:2013fsa , with the default ATLAS and CMS detector cards. A $K$-factor of between $1.4-1.8$ was needed to match the experimental yields. The resulting distributions, normalized using the $K$-factors, are also shown in Figure 1. While the simulated $m_{\gamma\gamma}$ distribution is largely in good agreement, some deviation is observed at low invariant masses. This deviation is due to the lack of box diagrams in the MadGraph5 simulation. Fortunately this occurs far from the signal region. Therefore, this simulation technique should be acceptable for the generation of signal events. I now turn to the excess at 750 GeV in the 13 TeV data. I fit the data to two possibilities: either a spin-0 or spin-2 particle decaying to two photons with a mass near 750 GeV.111Spin-1 mediators decaying to diphotons are ruled out by the Landau-Yang theorem, though it may be possible to find gauge bosons mediator solutions through sufficient theoretical model-building efforts Chala:2015cev . In particular, I will discuss the agreement of the four data sets, and the preference in the data (if any) for a wide or narrow resonance. II.1 Spin-0 Resonance Of the avalanche of theory papers discussing the Atlas13 and Cms13 diphoton anomaly, the majority have considered the spin-0 scenario. Here, I take a model-independent approach, though I do specialize to the CP-even scalar option. The CP-odd pseudoscalar is also possible (see e.g. Ref. Low:2015qep ), but will result in a very similar analysis. Using FeynRules2.3 Alloul:2013bka , I constructed MadGraph5 model files for a new scalar spin-0 particle with couplings to one of the following sets of partons: 1. Gluons (presumably mediated through a loop) via the interaction $${\cal L}_{\Phi-g}\supseteq c_{g}\Phi_{g}G^{\mu\nu,a}G_{\mu\nu}^{a}.$$ (4) 2. Valance quark-antiquark ($u/d$) pairs, through the interaction $${\cal L}_{\Phi-q}\supseteq c_{q}\Phi_{q}\left(\frac{m_{u}}{v}\bar{u}u+\frac{m_{d}}{v}\bar{d}d\right).$$ (5) 3. Charm and strange quark-antiquark pairs, through the interaction $${\cal L}_{\Phi-Q}\supseteq c_{Q}\Phi_{Q}\left(\frac{m_{c}}{v}\bar{c}c+\frac{m_{c}}{v}\bar{s}s\right).$$ (6) 4. Bottom quark-antiquark pairs, through $${\cal L}_{\Phi-b}\supseteq c_{b}\Phi_{b}\frac{m_{b}}{v}\bar{b}b.$$ (7) Here $v$ is the Standard Model Higgs vacuum expectation value $v=246$ GeV, and the $c_{i}$ are couplings which can be floated to fit to the observed cross section. Assuming proportionality of interactions to the quark masses is motivated from Minimal Flavor Violation D'Ambrosio:2002ex , but what follows is not too sensitive to this assumption. In all cases, decays to photons are the result of the interaction $${\cal L}_{\Phi-\gamma}\supseteq c_{\gamma}\Phi F^{\mu\nu}F_{\mu\nu}.$$ (8) The interactions are separated in this way to allow for more fine-grained investigation of the agreement of the 13 and 8 TeV data. If the anomaly is the result of a new particle at $\sim 750$ GeV, the expected signal strength in each experiment should be related by the ratios of the relevant parton distribution functions (p.d.f.s). Any “realistic” theory for the anomaly could have couplings to more than one of these sets of partons; in such cases one can reweight the results of this paper. I also note that full, realistic simulation of the gluon-coupling may require resolving the heavy colored particles running in the loop; this is relevant at values of the mediator $p_{T}$ comparable to the mass of the particles in the loop Buckley:2014fba . Here I assume infinite masses, which presumably is a reasonable assumption for most models, as typically the mediator will be produced nearly at rest and the new colored mediators running the loop must be heavy. I also stress that my analysis assumes that the mediator producing the diphoton excess at $m_{\gamma\gamma}\sim 750$ GeV is indeed a particle with mass near 750 GeV. It is possible that some heavier particle is produced, followed by a cascade decay resulting in the observed excess (see Ref. Knapen:2015dap ). By increasing the mass of the mediator, the constraints from the 8 TeV data can be weakened, and only the direct comparison of Atlas13 and Cms13 would be relevant. For each model, I generated $pp\to(\Phi\to\gamma\gamma)+X$ events, matched to two jets with a matching scale of 10 GeV, using the MadGraph5/Pythia6/Delphes3 simulation chain described previously. No cuts were placed on the $\Phi$ particles at the generator level. I scanned over mediator masses from 700 to 800 GeV, under two assumptions of the width: narrow and wide. The narrow width mediator has a width set by the Lagrangian terms above (typically $\Gamma\lesssim 50~{}$MeV), which much less than the diphoton invariant mass resolution. The “wide” resonance has a width of $\Gamma=45$ GeV, as this is reported best-fit width in the Atlas13 results. Scanning over the widths would be preferable to considering just these two assumptions. However, as I am considering only the binned data, the effective resolution of this scan is poor and extremely computationally intensive. I find nearly flat signal acceptance for the range of mediator masses considered, for both the narrow and wide hypotheses. The Atlas13 analysis has an efficiency of $\sim 50\%$ for spin-0 mediators – this is somewhat lower than the value quoted in Ref. Falkowski:2015swt . The combined barrel-barrel and barrel-endcap Cms13 analysis has an efficiency of $\sim 60\%$, with 40% of events ending up in the barrel-barrel category, and 20% in the barrel-endcap. The Cms8 search also has a 60% efficiency, while the Atlas8 Higgs search is slightly less than this. Using the $m_{\gamma\gamma}$ distributions constructed from the simulated $\Phi$ production and decay, I then fit the signal plus background to the data provided from each experiment, floating the normalization of the signal in terms of the production cross section times branching ratio into photons $\sigma\times\mbox{BR}_{\gamma\gamma}$. In each case, I refit the background distributions using the appropriate functional forms Eqs. (1)–(3), marginalizing over the background function parameters, and maximizing the log likelihood assuming Poissonian statistics. Production cross sections for the 8 TeV data are then reweighted to the 13 TeV results using MadGraph5 simulation to obtain the necessary p.d.f. factors. All cross sections quoted in this paper are in terms of the 13 TeV data, and are thus directly comparable. The ratio of 13 TeV cross sections to 8 TeV cross sections (for both wide and narrow resonances), are nearly independent of resonance mass in the 700-800 GeV range considered here. For scalar mediators coupling to gluons ($\Phi_{g}$), this ratio is $\sim 4.5$, for couplings to valance quarks ($\Phi_{q}$) it is $\sim 3.1$, for couplings to $s/c$ quarks ($\Phi_{Q}$) it is $\sim 4.2$, and $\sim 4.0$ for bottom quarks ($\Phi_{b}$). These cross section ratios come from two-jet matching, and so include initial states other than those that couple directly to the mediator in question. Due to the similarity of the $\Phi_{g}$, $\Phi_{b}$, and $\Phi_{Q}$ p.d.f. ratios, I will show only $\Phi_{g}$ in this section, and relegate the $\Phi_{b}$ and $\Phi_{Q}$ results to Appendix A. Using the fitting procedure described, in Figure 2, I show the best-fit values for the cross-sections time branching ratios into photons, as a function of resonance mass (again, for both choices of overall width). The statistical significance of these best-fit excesses are shown in Figure 3 (for $\Phi_{Q}$ and $\Phi_{b}$ interpretations, see Figures 8 and 9 in Appendix A). The statistical significance is obtained from the $\Delta\log$ likelihood assuming a single degree of freedom. Best-fit $\sigma\times$BR and statistical significances are shown individually for the Atlas13 and Cms13 data-sets; as is by now well-understood, these analyses show an excess near 750 GeV. Adding in the Cms13/0T data also shows some preference for a signal slightly about 750 GeV. My statistical fits must be compared with the quoted values from the ATLAS and CMS Collaborations themselves. For Atlas13, I find a local statistics-only significance for a narrow signal of $\sim 3.6\sigma$ for a particle with a mass of 750 GeV. I further find a marginal improvement to the local significance (up to $\sim 3.9\sigma$) for the $\Gamma=45$ GeV hypothesis. The full experimental analysis finds $3.6\sigma$ for the narrow width and $3.9\sigma$ for the wider resonance, using the unbinned data and including systematic errors which are not replicable in a theory analysis – though the exact agreement of my numbers with the experimental results must be seen as coincidental. For the Cms13 data, I find a local statistical significance of $2.0\sigma$ for the narrow width hypothesis, while the CMS Collaboration found a local significance of $2.6\sigma$. Thus, my results combining the data sets are also likely to be underestimations of the true significance of various combinations of data sets (despite the fact that I have neglected systematic errors), though of course one cannot be sure barring a full experimental analysis. Going further, I next consider the evidence for an excess in combinations of the data sets. Looking first at the 13 TeV data, I test the statistical significance of a single resonance with a common $\sigma\times$BR in both the Atlas13, Cms13, and Cms13/0T data. This “Combo13” result prefers a signal at 750 GeV with a cross section between the Atlas13 and Cms13 value, as expected. More interesting perhaps is the change in the statistical significance of this excess: for narrow widths, the Combo13 best-fit cross section of $\sim 5$ fb is preferred at $\sim 3.9\sigma$. This is an increase from the Atlas13 individual fit, but below the naive expectation one might have from combining the Atlas13, Cms13, and Cms13/0T significances ($\sim\sqrt{(3.6\sigma)^{2}+(2.0\sigma)^{2}+(2.5\sigma)^{2}}=4.8\sigma$), as the best-fit cross sections and masses for the three experimental analyses disagree. For wide resonances, combining the Atlas13 and Cms13 data results in a net decrease in the statistical significance relative to the Atlas13 data, to $\sim 3.4\sigma$. This is the first indication that the wide resonance seen by ATLAS is disfavored by the CMS results. The preceding set of statements is largely independent of the type of the coupling between the mediator and the proton’s partons. However, when adding in the 8 TeV data, I must specify the coupling in order to determine the 8 TeV cross section which is equivalent to the 13 TeV value. For my purposes, it suffices to discuss the coupling to gluons and compare to the coupling to the valence $u/d$ quarks, as couplings to other quark flavors have similar pd.f.s to gluons and thus have very similar conclusions. As can be seen in Figure 2, the combined 8 TeV data disfavors the Atlas13 best-fit cross section for a narrow width at $1\sigma$. However, if I instead ask for the best fit region for a single cross section fitting both the 13 and 8 TeV data, I find that there is a good fit for a narrow resonance at a 4 fb cross section, close to the Cms13 value, assuming a gluon coupling. This is within the $1\sigma$ region for Cms13, within $2\sigma$ of Atlas13, and is less than $1\sigma$ tension with the 8 TeV data. The statistical preference for this signal is identical to the combined fit to Atlas13 and Cms13, at about $4.0\sigma$. In the wide resonance interpretation, the 8 TeV data is much less constraining. While the combination of Atlas13 and Cms13 data reduces the preference for a wide signal, adding the 8 TeV data returns the total statistical significance to $\sim 4.0\sigma$ assuming couplings to gluons – about the same significance as Atlas13 alone. This is driven by the ability for the background models of the 8 TeV data to absorb the signal without resulting in any large excess above the observed smooth distribution. The insensitivity of the 8 TeV data to the broad excess is enough to hide even that larger cross sections required by the light quark coupling. II.2 Spin-2 Resonance I now turn to the spin-2 possibility. My general approach is the same as in Section II.1: I investigate individual couplings to the partons one-by-one, in order to make comparisons between the 8 and 13 TeV data. The mediator here is based on a spin-2 Kaluza-Klein graviton $K$, as implemented in MadGraph5 by Ref. deAquino:2011ix (see also Refs. Giudice:1998ck ; Han:1998sg ; Hagiwara:2008jb . I modified the relevant couplings to limit the interactions to gluons, light quarks, second generation quarks, or bottom quarks. In all cases, the coupling to photons is generated through the simplified interaction $$\displaystyle{\cal L}_{K-\gamma}$$ $$\displaystyle\supseteq$$ $$\displaystyle c_{\gamma}K^{\mu\nu}T_{\gamma,\mu\nu},$$ (9) $$\displaystyle T^{\mu\nu}_{\gamma}$$ $$\displaystyle=$$ $$\displaystyle\eta^{\mu\nu}\left(-\frac{1}{4}F^{\rho\sigma}F_{\rho\sigma}+(\partial_{\rho}\partial_{\sigma}A^{\sigma})A^{\rho}+\frac{1}{2}\partial_{\rho}A^{\rho}\partial_{\sigma}A^{\sigma}\right)$$ (10) $$\displaystyle-F^{\mu\rho}F^{\nu}_{\rho}+(\partial^{\mu}\partial_{\rho}A^{\rho})A^{\nu}+(\partial^{\nu}\partial_{\rho}A^{\rho})A^{\mu}.$$ The terms involving $A$ fields are gauge-fixing terms in the Feynman gauge. The stress-energy tensor for a quark $q$, relevant for the $K$-quarks interaction, is given by $$\displaystyle T^{\mu\nu}$$ $$\displaystyle=$$ $$\displaystyle-i\eta^{\mu\nu}\left(\bar{q}\gamma^{\rho}\partial_{\rho}q-\frac{1}{2}\partial^{\rho}(\bar{q}\gamma_{\rho}q)\right)+\frac{i}{2}\left(\bar{q}\gamma^{\mu}\partial^{\nu}q+\bar{q}\gamma^{\nu}\partial^{\mu}q-\frac{1}{2}\partial^{\mu}(\bar{q}\gamma^{\nu}q)-\frac{1}{2}\partial^{\nu}(\bar{q}\gamma^{\mu}q)\right).$$ (11) The production is through one of the following interactions: 1. Gluons, through the operator $$\displaystyle{\cal L}_{K-g}$$ $$\displaystyle\supseteq$$ $$\displaystyle c_{g}K_{g}^{\mu\nu}T_{g,\mu\nu},$$ (12) $$\displaystyle T^{\mu\nu}_{g}$$ $$\displaystyle=$$ $$\displaystyle\eta^{\mu\nu}\left(-\frac{1}{4}G^{a,\rho\sigma}G^{a}_{\rho\sigma}+(\partial_{\rho}\partial_{\sigma}G^{a,\sigma})G^{a,\rho}+\frac{1}{2}\partial_{\rho}G^{a,\rho}\partial_{\sigma}G^{a,\sigma}\right)$$ (13) $$\displaystyle-G^{a,\mu\rho}G^{a,\nu}_{\rho}+(\partial^{\mu}\partial_{\rho}G^{a,\rho})G^{a,\nu}+(\partial^{\nu}\partial_{\rho}G^{a,\rho})G^{a,\mu}.$$ 2. Light valence quarks, $u/d$, through the interaction $$\displaystyle{\cal L}_{K-q}$$ $$\displaystyle\supseteq$$ $$\displaystyle c_{q}K_{q}^{\mu\nu}T_{q,\mu\nu},$$ (14) where $T_{q}^{\mu\nu}$ is the stress-energy tensor Eq. (11) with $q=u/d$. 3. Second generation quarks, $s/c$, through $$\displaystyle{\cal L}_{K-Q}$$ $$\displaystyle\supseteq$$ $$\displaystyle c_{Q}K_{Q}^{\mu\nu}T_{Q,\mu\nu},$$ (15) where $T_{Q}^{\mu\nu}$ is the stress-energy tensor Eq. (11) with $q=s/c$. 4. The bottom quark, through $$\displaystyle{\cal L}_{K-b}$$ $$\displaystyle\supseteq$$ $$\displaystyle c_{b}K_{b}^{\mu\nu}T_{b,\mu\nu},$$ (16) where $T_{b}^{\mu\nu}$ is the stress-energy tensor Eq. (11) with $q=b$. The major difference in the analysis when compared to the spin-0 case is change in acceptance of the experiments to diphoton events. These changes are different for each of the six experimental searches I consider. After the ATLAS reanalysis from the 2016 Moriond conference, which uses a separate set of selection criteria for the spin-0 and spin-2 searches, I find that spin-2 signal events have an have an acceptance of $\sim 55\%$ in Atlas13 and Atlas8 for 750 GeV mediators. This is essentially the same as the acceptance for spin-0 mediators in these experiments. However, there are 75% more events in the spin-2 analysis. I find the barrel-barrel Cms13(Cms13/0T) signal acceptance is $\sim 35\%$, while the barrel-endcap signal acceptance is 25%, for a combined efficiency of nearly 60%, essentially the same as for the spin-0 case. The acceptance of spin-2 signals for Cms8 is 45%, a significant drop from the 60% acceptance for spin-0 signals. In Figure 4, I show the best-fit regions for signal cross section as a function of mediator mass for the $K$ particle coupling to gluons or light quarks (couplings to $c/s$ or $b$ quarks are shown in Figure 10, and are very similar to those for gluons). The statistical significance of the best-fit cross sections are shown in Figure 5. I find that the combined data sets disfavors the spin-2 mediator when compared to the spin-0, with a combined statistical significance of $\sim 3.5\sigma$ for a narrow gluon-initiated resonance. This preference appears to be largely driven by the lower statistical preference for a spin-2 mediator in the Atlas13 data. However, it must be stressed that the changes in statistical significance discussed here are less than $1\sigma$, and so are at best examples of mild preferences in the data. III Narrow or Wide? Now I attempt to address the question of whether there is any preference in the data for a wide resonance over a narrow one. For computational simplicity, I consider the best-fit scenario from Section II: a scalar mediator coupling to gluons, with a cross section of 4 fb for the narrow resonance, and 10 fb for the broad resonance. As before, I do not scan over the width, but simply compare the narrow resonance (where the LHC width is controlled entirely by the detector resolution) with a mediator with a 45 GeV width, as suggested by the Atlas13 results. In Figure 6, I show the experimental data around the 750 GeV $m_{\gamma\gamma}$ bins, compared to the predicted differential distributions of these best-fit points. It is clear from these examples why the wide resonance is in more tension with the Cms13 result, as well as why the 8 TeV data can more easily absorb this type of signal. I have already discussed some of the statistical evidence for the question of narrow or wide resonances in the previous section. The combination of Atlas13, Cms13, Cms13/0T increases the overall significance for a narrow resonance, and this significance does not decrease when the 8 TeV data is added, while the combination of 13 TeV data decreases the significance for a wide resonance (though this then increases once the 8 TeV data is included). Thus, one can say that both interpretations have equal statistical significance when combining all the data, though it is true that the combination of the 13 TeV alone mildly prefers the narrow resonance. I perform one additional statistical test, calculating the likelihood ratio for the preference for the narrow resonance over a wide resonance. This is the ratio of the probability of observing some set of data given a narrow resonance over the probability of observing that data given the wide resonance. To calculate this, I define the log-likelihood ratio $$\lambda=\log(L_{\rm wide}/L_{\rm narrow}),$$ (17) where the $L$ values are the maximum likelihoods (given an assumption of the width) marginalized over the background fit parameters. After calculating the observed $\lambda_{o}$ from the data, I estimate the probability distribution by generating pseudoexperiments: injecting either a narrow or wide signal (with the best fit cross section) over the best-fit background, and then calculating the $\lambda$ value of that particular pseudoexperiment. The likelihood ratio $R$ is then the ratio at $\lambda_{0}$ of the normalized probability distribution for narrow signal over the distribution for a wide signal. In Figure 7, I show these probability distributions along with the $\lambda_{0}$ fit to the combination of Atlas13 and Cms13 (left panel), and fit to all data (right panel). As can be seen, when considering only 13 TeV data, there is a very slight preference for a narrow resonance, with a corresponding ratio of $R\sim 1$, which indicates no significance preference for either model using the 13 TeV data only. When combining all data, the likelihood ratio for the wide resonance over the narrow width is $R\sim 20$. Thus, if one is considering only the Run-II data, there is no particular preference for a wide or narrow width using this test, while if all data is considered, the 45 GeV width is somewhat more probable. However, it certainly should not be stated that we know with any degree of confidence that the proposed 750 GeV resonance has a width much larger than one might expect from a perturbatively coupled spin-0 mediator. IV Conclusions When considering the 750 GeV diphoton excess, the theoretical community must balance its natural exuberance with the recognition that the statistical size of the anomalies are very small. As a result, any further slicing of data will yield at best modest statistical preferences for the phenomenological questions that we in the community want answers to. That said, given that this excess is the most significant seen at the LHC since the discovery of the 125 GeV Higgs, and the resulting avalanche of theoretical papers which shows no sign of slowing, it is still a useful exercise to carefully analyze the available data and determine what we do – and do not – know at this stage. While there is some useful information to be gleaned from this exercise, we are fortunate that the continuation of Run-II will be upon us shortly. From the existing data, we can conclude the following: 1. Explaining the anomaly through a spin-0 resonance is preferred over a spin-2 mediator, though this preference is less than $1\sigma$ in most cases. 2. Combining the 8 and 13 TeV data from ATLAS and CMS sets yields a $\sim 4.0\sigma$ statistical preference for a signal of $\sim 4(10)$ fb, assuming a narrow (wide) spin-0 resonance. This ignores the look-elsewhere effect, as discussed. Given that the significance of my fits to individual ATLAS and CMS data-sets are underestimates when compared to the full experimental results, it is possible that the actual statistical preferences are larger than these quoted values. However this would require a combined analysis performed by the ATLAS and CMS Collaborations. 3. The cross sections needed for the Atlas13, Cms13, and Cms13/0T data sets are incompatible at the two sigma level, though they agree in mass. The most straightforward reading of this (while maintaining a new physics explanation for the anomalies) is that the larger Atlas13 cross section constitutes a modest upward fluctuation from the “true” cross section, which is more in line with the Cms13 value.222Of course, in this interpretation, had the Atlas13 data not had an upward fluctuation the combined statistical significance would likely not have been large enough to generate the massive response from the theoretical community. Thus, one may be tempted to take an Anthropic Principle view: an upward fluctuation in at least one of the data sets was a necessary condition for this paper to exist. The reverse is also possible of course, but would bring the diphoton excess in the 13 TeV data in greater tension with the 8 TeV null results. 4. When considering only the 13 TeV data, the Cms13 data does not share the Atlas13 preference for a 45 GeV width. I find that the “wide” interpretation of the resonance has a statistical significance in the combo13 data set which is approximately $0.5\sigma$ less likely than the “narrow” interpretation. The corresponding likelihood ratio shows no preference for either width. Thus, while the theoretical challenge of a wide resonance may be appealing, the data in no way requires any new physics explanation to have the unusually large width of $\Gamma\sim 45$ GeV. 5. Combining the 13 TeV data with the 8 TeV, I find that gluon-initiated mediators are preferred, due to having the largest ratio of relevant p.d.f.s. In particular, the combination of all six data sets for a gluon-initiated narrow resonance has the same statistical preference for a signal as the Combo13 data alone does, though the best-fit cross section decreases slightly when the 8 TeV data is added ($\sim 4.0\sigma$ for a $\sim 4$ fb signal). In the narrow width assumption, heavy quark-initiated mediators have slightly smaller statistical preference, and a light-quark coupling has a fairly significant decrease in statistical preference, indicating a more serious conflict between the 13 and 8 TeV data. 6. Combining the 13 and 8 TeV data sets under the $\Gamma=45$ GeV spin-0 model increases the statistical preference for signal as compared to the Combo13 result, as the excess can be more easily absorbed by the background model here. Combining all the data sets in this way results in a $\sim 3.5\sigma$ preference for a $\sim 10$ fb signal (a $0.5\sigma$ increase over the Combo13 wide-resonance fit), with a likelihood ratio of $\sim 20$ rejecting the narrow interpretation. Again, these statistical preferences are relatively small, thus theorists are free to explore the options, but should keep in mind that the experimental results are inconclusive. The conclusions of this paper are perhaps not a surprise. There is clear tension between the Atlas13 and Cms13 results, as well as with the non-observation in 8 TeV data. The question of the width is especially puzzling; but further slicing of the data, as I have demonstrated, leads to somewhat conflicting results which do not have a clear statistical preference towards any one solution. I note that if the ATLAS excess is indeed an upward fluctuation from a signal which is more in line with the Cms13 value, then perhaps this could also give a spurious signal of large width. However, the true answers will only come with more data, though I note that, if the signal is indeed real, but on the order of 4 fb, then we may need 10-20 fb${}^{-1}$ for a single experiment to have $5\sigma$ discovery. Acknowledgements I thank Matthew Baumgart, JP Chou, Yuri Gershtein, Marco Farina, Angelo Monteux, David Shih, and Scott Thomas for useful feedback and discussions while writing this paper. Appendix A Additional Analysis In the main body of the paper, I presented the best-fit cross sections and statistical significances of spin-0 and spin-2 mediators coupling to gluons and light $u/d$ quarks. Couplings to $c/s$ and $b$ quarks in the proton are also possible, and here I present the equivalent results. As the ratio of p.d.f.s for these partons between 8 and 13 TeV is very close to that found for gluons, these results match the gluon plots in most respects. Figures 8 and 9 show the spin-0 best fit cross sections and statistical significances for $c/s$ and $b$ couplings, while the spin-2 cases are shown in Figures 10 and 11. References (1) ATLAS Collaboration, ATLAS-CONF-2015-081 (2015). (2) CMS Collaboration, CMS-PAS-EXO-15-004 (2015). (3) N. Craig, P. Draper, C. Kilic and S. Thomas, arXiv:1512.07733 [hep-ph]. (4) CMS Collaboration, CMS-PAS-EXO-16-018 (2016). (5) ATLAS Collaboration, ATLAS-CONF-2016-018 (2016). 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Linear programming on non-compact polytopes and the Kuratowski convergence with application in economics Anna Denkowska, Maciej Denkowski and Marta Kornafel (Date:: December 20th 2015, Revised: January 2nd 2017) Abstract. The aims of this article are two-fold. First, we give a geometric characterization of the optimal basic solutions of the general linear programming problem (no compactness assumptions) and provide a simple, self-contained proof of it together with an economical interpretation. Then, we turn to considering a dynamic version of the linear programming problem in that we consider the Kuratowski convergence of polyhedra and study the behaviour of optimal solutions. Our methods are purely geometric. Key words and phrases:Linear programming, tangent cone, normal cone, Kuratowski convergence 1991 Mathematics Subject Classification: 1. Introduction A classical problem in optimization theory and one that has a wide range of applications economics, is the linear programming problem (LP for short). In the canonical form it is written as: $$\begin{cases}c^{T}x\to\min\\ Ax=b\\ x\geq 0,\end{cases}$$ where $c\in\mathbb{R}^{n}$ is the cost vector, $c^{T}$ is its transposed (thus $c^{T}x=\langle c,x\rangle$ denotes the usual inner product), $A$ is the matrix of a linear function $A\colon\mathbb{R}^{n}\to\mathbb{R}^{m}$, $b\in\mathbb{R}^{m}$, $x\geq 0$ means $x_{i}\geq 0$ for $i=1,\dots,n$, and it is usually assumed that the set of feasible solutions $F_{A,b}:=\{x\in{\mathbb{R}}^{n}\mid Ax=b,x\geq 0\}$ is compact, so that a solution necessarily exists. The classical solution to this PL problem is given by the so called simplex method. Observe that even a discrete LP problem, that is one in which we consider $F^{d}_{A,b}:=F_{A,b}\cap\mathbb{Z}^{n}$ can be reduced to the above one by considering the LP problem on the convex hull $\mathrm{conv}(F^{d}_{A,b})$. It is a classical and easy to show fact that the solutions to the LP problem lie all on the boundary $\partial F_{A,b}$ (more accurately: on the relative boundary computed in the unique affine space of the lowest possible dimension containing $F_{A,b}$) and it is sufficient to look for them among the extremal points of $F_{A,b}$. Recall that given a closed, convex set $F\subset{\mathbb{R}}^{n}$, a point $x_{0}\in F$ is called extremal — we write then $x_{0}\in F^{*}$ — if $$\exists x_{1},x_{2}\in F,\exists t\in(0,1)\colon x_{0}=(1-t)x_{1}+tx_{2}% \Rightarrow x_{1}=x_{2}.$$ The following fact is well-known: Proposition 1.1. Assuming that the rank $\mathrm{rk}A=m<n$ (which is not really restrictive), a point $x\in F_{A,b}$ is extremal if and only if it is a basic feasible solution. Of course, a basic feasible solution is a point $x\in F_{A,b}$ such that either $x=0$, or the columns of $A$ corresponding to the non-zero coordinates of $x$ are linearly independent. Notation. Given an $m\times n$ matrix $A$ we denote by $A^{(i_{1},\dots,i_{k})}$ the matrix $A$ without the rows with indices $\neq i_{j}$. On the other hand $A^{j}$ will denote the $j$-th column of $A$. Finally, we write $A=(A_{1},\dots,A_{m})$ with $A_{i}\colon{\mathbb{R}}^{n}\to{\mathbb{R}}$ that are linear forms. Let us stress that we will use interchangeably the words linear polytope and polyhedron meaning actually convex polyhedron in the following sense: Definition 1.2. A nonempty set $E\subset{\mathbb{R}}^{n}$ is called a convex polyhedron or just polyhedron, if there is a non-zero linear mapping $A\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{m}$ and a vector $b\in{\mathbb{R}}^{m}$ such that $E=\{x\in{\mathbb{R}}^{n}\mid Ax\leq b\}$. Observe that this definition excludes ${\mathbb{R}}^{n}$ and that a polyhedron need not be compact (111A compact convex polyhedron is usually called a polytope.) For a point $x\in E$,we denote by $J(x)=\{i\in\{1,\dots,m\}\mid A_{i}x=b_{i}\}$ the set of active constraints at $x$. Of course, the describing linear mapping $A$ is not uniquely determined, unless we require it to be minimal in the following sense. Let $d$ be the dimension of the convex polyhedron $E$. Then there is an affine $d$-dimensional subspace $V\subset{\mathbb{R}}^{n}$ containing $E$ (the affine hull or envelope of $E$, denoted also by $\operatorname{Aff}(E)$) and such that $E=\overline{\operatorname{int}_{V}E}$. This affine hull is described by $n-d$ equations $\langle w_{j},x\rangle=u_{j}$. Now, let $f_{k}(E)$ denote the number of $k$-dimensional faces of $E$. In particular, $f_{0}(E)=\#E^{*}$ is the number of vertices or extremal points, whereas $f_{d-1}(E)$ is the number of facets (faces of maximal possible dimension) (222Note that $f_{0}(E)$ may be zero, unlike $f_{d-1}(E)$.). Then in $V\equiv{\mathbb{R}}^{d}$ we need exactly $f_{d-1}(E)$ linear inequalities $A_{i}x\leq b_{i}$ to describe $E$, as this set is the intersection of as much half-spaces as it has facets. Therefore, a minimal description of $E$ is given by $n-d$ linear equations together with $f_{d-1}(E)$ linear inequalities. Hereafter we will deal with the general linear programming problem: (G⁢L⁢P)𝐺𝐿𝑃( italic_G italic_L italic_P ) $$\begin{cases}c^{T}x\to\min\\ Ax\leq b\\ \end{cases}$$ with $A\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{m}$ linear with $m\geq n$. This is somehow motivated by the following proposition, that we prove for the convenience of the reader. Proposition 1.3. Let $E_{A,b}=\{x\in\mathbb{R}^{n}\mid Ax\leq b\}$ with $A$ as above. Then $\bar{x}\in E_{A,b}^{*}$ implies that $m\geq n$ and there are indices $i_{1}<\ldots<i_{n}$ such that $A^{(i_{1},\dots,i_{n})}\bar{x}=(b_{i_{1}},\dots,b_{i_{n}})$ and $\det A^{(i_{1},\dots,i_{n})}\neq 0$. In particular, $$\bigcap_{i\in J(\bar{x})}\{x\in{\mathbb{R}}^{n}\mid A_{i}x=b_{i}\}=\{\bar{x}\}$$ where $J(\bar{x})=\{i\in\{1,\dots,q\}\mid A_{i}x=b_{i}\}$ are the indices of the active constraints at $\bar{x}$. The point $\bar{x}\in E_{A,b}^{*}$ is called a vertex of the polytope $E_{A,b}$. In the usual terminology $\bar{x}$ is called a basic optimal solution. Proof of Proposition 1.3. The point $\bar{x}$ being extremal, it cannot lie in the interior of $E_{A,b}$. Thus, there is an index $i\in J(\bar{x})$. We may assume that $i=1$. Now, we use the fact that for linear subspaces $V,W\subset{\mathbb{R}}^{n}$ we have $\dim V\cap W\geq\dim V+\dim W-n$. The case $n=1$ being obvious, we may assume that $n\geq 2$. Moreover, no harm will be inflicted on generality, if we assume that $A_{j}\not\equiv 0$. Thus $\dim\mathrm{Ker}A_{j}=n-1$ for all $j$. Had we $A_{i}\bar{x}<b_{i}$ for all $i>2$, we would find a ball $B$ centred at $\bar{x}$ and such that $B\cap A_{1}^{-1}(b_{1})\subset E_{A,b}$. But this set has dimension $n-1>0$ and so $\bar{x}$ is not extremal. Therefore there is $i>1$ in $J(\bar{x})$. We may assume that $i=2$. Since $\dim A_{1}^{-1}(b_{1})\cap A_{2}^{-1}(b_{2})\geq n-2$, we conclude that this has to be an equality for some index $i>1$ (otherwise $\bar{x}$ would not be extremal). Then we may repeat the preceding argument in order to conclude that either there must be an index $i>2$ in $J(\bar{x})$, or $n=2$ and we have the equality sought for. It is then clear that the procedure must end and that $\bar{x}$ would not be extremal if we needed less than $n$ steps. Hence we have $A_{1},\dots,A_{n}$ such that $\bigcap_{i=1}^{n}A_{i}^{-1}(b_{i})=\{\bar{x}\}$. This in turn implies that $\bigcap_{i=1}^{n}\mathrm{Ker}A_{i}=\{0\}$ which means that $A_{1},\dots,A_{n}$ are linearly independent which ends the proof. ∎ Remark 1.4. In this article we do not assume that $E_{A,b}$ is compact. Note that in real life we often do not know exactly all the constraints (we lack data) of a given engineering or economics problem and actually we are dealing with a non-compact $E_{A,b}$. Note that the interest in matters conerning linear programming is still quite important (see e.g. [6]). Our approach is very basic, nevertheless it gives some applicable results. We have two aims: to explain under which condition the GLP problem is solvable and give a geometric solution to it, and to study what happens when we approximate the polyhedron $E_{A,b}$ by similar polyhedra, in particular — how do the solutions behave. 2. Solving the GLP problem using normal cones For a given set $E\subset{\mathbb{R}}^{n}$ and a point $a\in\overline{E\setminus\{a\}}$ we define the usual Peano tangent cone of $E$ at $a$ as the cone $$C_{a}(E)=\{v\in{\mathbb{R}}^{n}\mid\exists E\ni x_{\nu}\to a,\lambda_{\nu}>0% \colon\lambda_{\nu}(x_{\nu}-a)\to v\},$$ and the normal cone of $E$ at $a$ as the cone $$N_{a}(E)=\{w\in{\mathbb{R}}^{n}\mid\forall v\in C_{a}(E),\langle v,w\rangle% \leq 0\},$$ which means that any vector $w\in N_{a}(E)$ forms with any vector $v\in C_{a}(E)$ an angle greater than or equal to $\pi/2$. Keeping the notations introduced so far we obtain first: Lemma 2.1. Let $w\in E_{A,b}$. Then $$C_{w}(E_{A,b})=\bigcap_{i\in J(w)}\{x\in{\mathbb{R}}^{n}\mid A_{i}x\leq 0\}.$$ Proof. Both sets contain the origin. Take a non-zero vector $v$ from the tangent cone. Let $E_{A,b}\ni x_{\nu}\to w$ and $\lambda_{\nu}>0$ be the sequences yielding $\lambda_{\nu}(x_{\nu}-w)\to v$. For $i\in J(w)$ we have $$A_{i}(\lambda_{\nu}(x_{\nu}-w))=\lambda_{\nu}(A_{i}x_{\nu}-b_{i})\leq 0,$$ for $\lambda_{\nu}$ are positive. Therefore, $A_{i}$ being continuous, we obtain $A_{i}v\leq 0$, as required. Take now $v\neq 0$ belonging to the set on the right-hand side. Then for $i\in J(w)$ we have $A_{i}w=b_{i}$ and so for any $\varepsilon>0$, we obtain $$A_{i}\left({\varepsilon}v+w\right)=\varepsilon A_{i}v+b_{i}\leq b_{i}.$$ If in turn $i\notin I(w)$, then $A_{i}w<b_{i}$, and so suitably small $\varepsilon$ ensure that $$A_{i}\left({\varepsilon}v+w\right)=\varepsilon A_{i}v+A_{i}w<b_{i}$$ still holds. Now, taking $\varepsilon_{\nu}$ decreasing to zero and $\lambda_{\nu}:=\frac{1}{\varepsilon_{\nu}}$ we conclude that $x_{\nu}:=\varepsilon_{\nu}v+w\in E_{A,b}$ and $\lambda_{\nu}(x_{\nu}-w)=v$. ∎ In particular, we can reconstruct $E_{A,b}$ from its vertices: Proposition 2.2. If $E_{A,b}^{*}\neq\varnothing$, then $$E_{A,b}=\bigcap_{w\in E_{A,b}^{*}}(C_{w}(E_{A,b})+w).$$ Proof. The inclusion ‘$\subset$’ is obvious (compare with the previous proof). Take now a point $x$ from the set on the right-hand side. Then for any $i\in\bigcup_{w\in E_{A,b}^{*}}J(w)=:J$ we obtain $A_{i}(x-w)\leq 0$, i.e. $A_{i}x\leq b_{i}$. It remains to observe that if there is an index $j\in\{1,2,\dots,m\}\setminus J$, then the correspponding inequality $A_{j}x\leq b_{j}$ is superfluous in the description of $E_{A,b}$. We may thus conclude that $x\in E_{A,b}$. ∎ It follows also from the lemma above that for $w\in E_{A,b}$, $N_{w}(E_{A,b})=\{\sum_{i\in J(w)}\lambda_{i}a_{i}\mid\lambda_{i}\geq 0,i\in J(% w)\}$ where $A_{i}(x)=\langle a_{i},x\rangle$. Therefore, we easily obtain the following remark. Corollary 2.3. The polyhedron $E_{A,b}$ is unbounded iff either $E_{A,b}^{*}=\varnothing$, or $E_{A,b}^{*}\neq\varnothing$ and $$\bigcup_{w\in E_{A,b}^{*}}N_{w}(E_{A,b})\neq{\mathbb{R}}^{n}.$$ Now we are ready to prove in an elementary fashion the following basic theorem: Theorem 2.4. Let a linear mapping $A\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{m}$ of rank $n$ define a (possibly unbounded) polyhedron $E_{A,b}$. Then the functional $f(x)=c^{T}x$ attains its minimum on $E_{A,b}$, if and only if $$-c\in\bigcup_{w\in E_{A.b}^{*}}N_{w}(E_{A,b}).$$ In particular, the minimum is attained at those vertices $w\in E_{A,b}$ for which $-c\in N_{w}(E_{A,b})$. Note that such a result can of course be deduced from some much more general results in convex analysis involving subgradients and so on (compare e.g. [1]). In our opinion, however, it is rather useful – in view of the importance of linear programming – to have a straightforward and self-contained proof, based on simple geometric notions. Before proving the theorem, we note the following lemma: Lemma 2.5. Let $f(x)=c^{T}x$ and consider a nonempty closed set $C\subset{\mathbb{R}}^{n}$. Let $V$ be the affine envelope of $C$. Then $f$ attains $\inf_{x\in C}f(x)$ iff there is a point in the relative boundary $x_{0}\in\partial_{V}C$ for which $f(x_{0})=\inf_{x\in C}f(x)$. Proof of Theorem 2.4. Using the previous lemma it is easy to check the well-known fact that $f$ attains its minimum on $E_{A,b}$ iff there is a vertex $w\in E_{A,b}^{*}$ such that $f(w)=\inf_{E_{A,b}}f$. Therefore, it suffices to prove that for a given vertex $w$, $\langle c,x\rangle\geq\langle c,w\rangle$ for all $x\in E_{A,b}$ iff $-c\in N_{w}(E_{A,b})$, i.e. $\langle-c,v\rangle\leq 0$ for all $v\in C_{w}(E_{A,b})$. We begin with the ‘if’ part. By Proposition 2.2, for any $x\in E_{A,b}$ we have $x-w\in C_{w}(E_{A,b})$. Now, $\langle c,w\rangle\leq\langle c,x\rangle$ is equivalent to $\langle c,w-x\rangle\leq 0$, or in other words $\langle-c,x-w\rangle\leq 0$. The latter we know to be true. Now, for the ‘only if’ part, to prove that $-c\in N_{w}(E_{A,b})$ we take any point $v\in C_{w}(E_{A,b})$. Then we consider the approximating sequence $\lambda_{\nu}(x_{\nu}-w)\to v$ with $E_{A,b}\ni x_{\nu}\to w$ and $\lambda_{\nu}>0$. We have $\langle c,x_{\nu}\rangle\geq\langle c,w\rangle$ and this remains true when we multiply both sides by $\lambda_{\nu}$, whence $\langle c,\lambda_{\nu}(x_{\nu}-w)\rangle\geq 0$. After multiplying both sides by $-1$ and passing to the limit we obtain $\langle-c,v\rangle\leq 0$ as required. ∎ Remark 2.6. The theorem above has a straightforward real-life application. It says that any functional $c_{p}^{T}x$, where $p$ is a parameter, attains its minimum (or maximum — by duality) at a fixed vertex $w\in E_{A,b}^{*}$ as long as the cost vectors $-c_{p}$ remain in the normal cone $N_{w}(E_{A,b})$. Moreover, we do not need the compactness of $E_{A,b}$ to obtain this, which means that some of the constraints are negligible. For instance, suppose that a factory produces $n$ products selling them at prices $c_{j}$ that could vary, as dictated by the market, in the intervals $[a_{j},b_{j}]$ ($j=1,\dots,n$) and the constraints $Ax\leq b$ correspond to how the machines can be set up (and the data may be incomplete, as they are in real life, i.e. $E_{A,b}$ can be non-compact). Assuming that the set up $\bar{x}\in E_{A,b}$ is optimal for the profit $c^{T}x$ to be maximal, the theorem says precisely how may the prices evolve without raising the need of changing the set up $\bar{x}$ in order to keep the profit maximal: to do this we only need to compute the normal cone at $\bar{x}$ (or more accurately, at the vertex corresponding to this optimal point). 3. Kuratowski convergence and LP problem First, let us recall the notion of convergence of sets we will be using. We will state the definition for a natural type of nets (generalized sequences). Consider a set $E\subset{\mathbb{R}}^{k}_{t}\times{\mathbb{R}}^{n}_{x}$ and denote by $E_{t}:=\{x\in{\mathbb{R}}^{n}\mid(t,x)\in E\}$ its section at $t\in{\mathbb{R}}^{k}$. Also, let $\pi(t,x)=t$ be the natural projection and fix $t_{0}\in\overline{\pi(E)}$. Definition 3.1. We write $x\in\limsup_{t\to t_{0}}E_{t}$ iff for any neighbourhood $U\ni x$ and for any neighbourhood $V\ni t_{0}$ there exists a point $t\in V\cap\pi(E)$ different from $t_{0}$ and such that $E_{t}\cap U\neq\varnothing$. We call the resulting set the Kuratowski upper limit of $E_{t}$ at $t_{0}$. We write $x\in\liminf E_{t}$ iff for any neighbourhood $U\ni x$ there is a neighbourhood $V\ni t_{0}$ such that for all $t\in V\cap\pi(E)\setminus\{t_{0}\}$, we have $E_{t}\cap V\neq\varnothing$. We call the resulting set the Kuratowski lower limit of $E_{t}$ at $t_{0}$. We say that $E_{t}$ converges to the set $F\subset{\mathbb{R}}^{n}$ iff $$\limsup_{t\to t_{0}}E_{t}=\liminf_{t\to t_{0}}E_{t}=F.$$ We write then $F=\lim_{t\to t_{0}}E_{t}$ or $E_{t}\stackrel{{\scriptstyle K}}{{\longrightarrow}}F$ ($t\to t_{0}$). Remark 3.2. Of course, $\liminf_{t\to t_{0}}E_{t}\subset\limsup_{t\to t_{0}}E_{t}$ and both sets are closed. Moreover, they do not change, if we take $\overline{E_{t}}$ instead of $E_{t}$. Therefore, it is natural to restrict ourselves only to closed sets. Observe also that $$F=\lim_{t\to t_{0}}E_{t}\ \Longleftrightarrow\ \limsup_{t\to t_{0}}E_{t}% \subset F\subset\liminf_{t\to t_{0}}E_{t}.$$ Note that a sequence of sets $(E_{\nu})$ can be identified with the $t$-sections of the set $E=\bigcup_{\nu}\{1/\nu\}\times E_{\nu}\subset{\mathbb{R}}\times{\mathbb{R}}^{n}$ and thus the upper and lower limits of $(E_{\nu})$ for $\nu\to+\infty$ may be understood as $\limsup_{t\to 0}E_{t}$ and $\liminf_{t\to 0}E_{t}$, respectively. In this case it is easy to see that $\liminf E_{\nu}$ consists of all the possible limits of converging sequences $x_{\nu}\in E_{\nu}$, while $\limsup E_{\nu}$ consists of all the possible limits of converging subsequences $x_{\nu_{s}}\in E_{\nu_{s}}$. Remark 3.3. For compact sets, the Kuratowski convergence is exactly the convergence in the usual Hausdorff measure. Note also that for a given set $E\subset{\mathbb{R}}^{n}$ and $a\in\overline{E}$ we have $$C_{a}(E)=\limsup_{\varepsilon\to 0}\frac{E-a}{\varepsilon}.$$ We will denote by $H(a;b)$ the affine hypersurface $\langle a,x\rangle=b$, where $||a||=1$ and by $\hat{H}(a;b)$ the half-space defined by $\langle a,x\rangle\leq b$. Observe that $\hat{H}(a_{\nu};b_{\nu})\stackrel{{\scriptstyle K}}{{\longrightarrow}}\hat{H}(% a;b)$, whenever $a_{\nu}\to a$, $b_{\nu}\to b$ and the same is true for the corresponding hypersurfaces. Recall that we say that two sets $E_{1},E_{2}\subset{\mathbb{R}}^{n}$ can be separated, if there are $a,b$ such that $E_{i}\subset\hat{H}((-1)^{i}a;(-1)^{i}b)$, $i=1,2$ which for convex sets is equivalent to $0$ not being an interior point of $E_{1}-E_{2}$ (cf. [8] Theorem 2.39). Let us also note the following easy Proposition: Proposition 3.4. The Kuratowski limit of a converging sequence of convex set is a convex set. Proof. Let $C_{\nu}$ be convex sets converging to a set $C_{0}$. Take $x,y\in C_{0}$. Then, due to the convergence, these points are limits of some sequences $x_{\nu},y_{\nu}\in C_{\nu}$, respectively. But $[x_{\nu},y_{\nu}]\subset C_{\nu}$ and clearly, the limit of a sequence of segments is a segment (maybe reduced to a point). It follows easily that $[x,y]\subset C_{0}$. ∎ We start this section with a short discussion of the following question: Assume that $\varnothing\neq C\subset{\mathbb{R}}^{n}$ is a closed, convex set and $f\colon C\to{\mathbb{R}}$ a continuous function with $M:=\sup_{x\in C}f(x)<+\infty$. When does there exist a point $x_{0}\in C$ such that $f(x_{0})=M$? Of course, the question makes sense in particular for an unbounded set $C$. In general there is not much hope to obtain a positive answer: for $n=1$ and $C=[0,+\infty)$ take $f(x)=\arctan x$. If $f$ were linear, we would have a realizing point in this case. Even though $f$ is linear, such a realizing point $x_{0}$ may not exist in general, unless $C$ is a polyhedron. Take $n=2$, $C=\{(x,y)\mid x>0,y\geq 1/x\}$ and $f(x,y)=-y$. Nevertheless, the following is true: Proposition 3.5. Let $C=\{x\in{\mathbb{R}}^{n}\mid\langle a_{i},x\rangle\leq b_{i},i=1,\dots,k\}$ be a nonempty polyhedron and $f(x)=\langle c,x\rangle$ with $M:=\sup_{x\in C}f(x)<+\infty$. Then, independently of the fact whether $C$ is bounded or not, there is a point $x_{0}\in C$ such that $f(x_{0})=M$. Proof. If $f\not\equiv 0$, we may assume that $||c||=1$ and $C$ is unbounded. Then we have $C\subset\hat{H}(c;M)$ and clearly $\mathrm{dist}(f^{-1}(M),C)=0$. Take a sequence $(x_{\nu})\subset C$ for which $f(x_{\nu})\to M$. Each point $x_{\nu}$ can be written as $\frac{f(x_{\nu})}{||c||^{2}}c+z_{\nu}=f(x_{\nu})c+z_{\nu}$ where $z_{\nu}\in\mathrm{Ker}f$. This gives us points $Mc+z_{\nu}\in f^{-1}(M)$ and $y_{\nu}\in C$ realizing their distance to $C$. Then it is easily seen that $f(y_{\nu})\to M$. We may assume now that $||y_{\nu}||\to+\infty$ (otherweise the limit of a convergent subsequence yields a point in $C$ realizing $M$ for $f$). Since $y_{\nu}\in\partial C$, then passing to a subsequence we may assume furhter that $\langle a_{i},y_{\nu}\rangle=b_{i}$ for $i=1,\dots,N$ with $N\geq 1$, while $\langle a_{i},y_{\nu}\rangle<b_{i}$ for $i=N+1,\dots,k$. Then choosing a subsequence we will get $\frac{y_{\nu}-y_{1}}{||y_{\nu}-y_{1}||}\to v$ and of course $[y_{1},y_{\nu}]\subset C$ for each $\nu$. Then $\ell:=y_{1}+\mathbb{R}_{+}v\subset C$ and we obtain $\mathrm{dist}(\ell,f^{-1}(M))=0$, i.e. $\ell\subset f^{-1}(M)$. ∎ Suppose that $S_{\nu}$ is the set of solutions of $c^{T}x\to\min$ on $E_{A,b_{\nu}}$. When do these sets converge to the set of solutions of $c^{T}x\to\min$ on $E_{A,b}$ and what can guarantee that the latter is nonempty? The main theorem of the preceding section gives a possible answer to this problem. Namely, if we know how do behave the normal cones and if we know that the cost vectors are ‘nicely’ related to them, then we can say that the limit problem has a solution and even give the vertex realizing it. Theorem 3.6. Let $E_{\nu}\subset{\mathbb{R}}^{n}$ be a sequence of convex polyhedra such that $E_{\nu}\stackrel{{\scriptstyle K}}{{\longrightarrow}}E\neq\varnothing$ where $\varnothing\subsetneq E\subsetneq{\mathbb{R}}^{n}$, and one of the following conditions is satisfied: either $E$ is compact and there is a uniform bound $\#E_{\nu}^{*}\leq M$, or there is a uniform bound $f_{\dim E_{\nu}-1}(E_{\nu})\leq M$. Then (1) $E$ is a convex polyhedron, too, and $\#E^{*}\leq\#E_{\nu}^{*}$, for almost all indices; (2) For any vertex $v\in E^{*}$ there is a sequence of vertices $E_{\nu}^{*}\ni v_{\nu}\to v$ and $C_{v_{\nu}}(E_{\nu})\stackrel{{\scriptstyle K}}{{\longrightarrow}}C_{v}(E)$, as well as $N_{v_{\nu}}(E_{\nu})\stackrel{{\scriptstyle K}}{{\longrightarrow}}N_{v}(E)$; (3) If $f_{\nu}\colon{\mathbb{R}}^{n}\to{\mathbb{R}}$ is a sequence of linear forms converging to $f\colon{\mathbb{R}}^{n}\to{\mathbb{R}}$ and such that each $f_{\nu}$ attains its maximum on $E_{\nu}$, then $f$ attains its maximum on $E$; moreover, $\operatorname{argmax}f_{\nu}\stackrel{{\scriptstyle K}}{{\longrightarrow}}% \operatorname{argmax}f$, provided one of the following conditions holds: either $E$ is compact, or $\#E^{*}=\#E_{\nu}^{*}$ for indices large enough, or $\max_{E}f$ exists and is the limit of $\max_{E_{\nu}}f_{\nu}$. Remark 3.7. In the noncompact case a uniform bound on the number of vertices is in general not enough to obtain a polyhedron as the limit. Consider an approximation of the unit circle in ${\mathbb{R}}^{2}$ by $\nu$-gones inscribed in it. Embed the plane ${\mathbb{R}}^{2}\times\{0\}\to{\mathbb{R}}^{3}$ and consider the infinite cones spanned over the $\nu$-gones from the vertex at $(0,0,1)$ — these are the sets $E_{\nu}$. Of course, they are convex, non-compact polyhedra converging to the regular cone spanned over the circle from the point $(0,0,1)$. Not only the limit is no longer a polyhedron, but it has infinitely many extremal points, while $E_{\nu}^{*}=\{(0,0,1)\}$. Of course, the assumption that $\varnothing\subsetneq E\subsetneq{\mathbb{R}}^{n}$ is unavoidable, too, cf. $(-\infty,\nu]\stackrel{{\scriptstyle K}}{{\longrightarrow}}{\mathbb{R}}$, while $(-\infty,-\nu]\stackrel{{\scriptstyle K}}{{\longrightarrow}}\varnothing$ — in both cases the limit is not a polytope according to our definition. Finally, the last point can be illustrated by the following example in ${\mathbb{R}}^{2}$: let $f_{\nu}(x,y)=f(x,y)=-y$ and let $E_{\nu}=\{(x,y)\in{\mathbb{R}}^{2}\mid x,y\geq 0,\nu y\geq\nu-x\}$. Then $E_{\nu}$ converges to $E=\{(x,y)\mid x\geq 0,y\geq 1\}$, but the maximizers do not converge. In the course of the proof we shall be using the following notions. Definition 3.8. Two linear inequalities $\langle a_{i},x\rangle\leq b_{i}$ with $||a_{i}||=1$, $i=1,2$ are called inverse equivalent (i-e for short), if $a_{1}=-a_{2}$ and $b_{1}=-b_{2}$. Put together, two i-e inequalities describe the affine hypersurface $H(a_{1};b_{1})=H_{2}(a_{2},b_{2})$. Let $a_{1},a_{2}\in{\mathbb{R}}^{n}$ be non-colinear unit vectors. We put $\displaystyle v(a_{1},a_{2}):=\frac{a_{1}+a_{2}}{|||a_{1}+a_{2}||}$. Lemma 3.9. Let $V_{\nu}$ and $V$ be real cones (333I.e. $tV\subset V$ for any $t\geq 0$.) in ${\mathbb{R}}^{n}$ with $V_{\nu}\stackrel{{\scriptstyle K}}{{\longrightarrow}}V$. Then the normal cones $N(V_{\nu})$ converge to $N(V)$. Proof. Take $w\in\limsup N(V_{\nu})$ and $v\in V$. Then there is a sequence $V_{\nu}\ni v_{\nu}\to v$ and a subsequence $N(V_{\nu_{k}})\ni w_{\nu_{k}}\to w$. Since $\langle w_{\nu_{k}},v_{\nu_{k}}\rangle\leq 0$, we get $\langle w,v\rangle\leq 0$, i.e. $w\in N(V)$. Fix now $w\in N(V)$. Without loss of generality we may assume that $||w||=1$. Then $V\subset\hat{H}(w;0)$ and the type of convergence implies that for large indices, $V_{\nu}\subset\hat{H}(w,0)$. Indeed, $V_{\nu}\cap{\mathbb{R}}^{n}\setminus\{0\}=V_{\nu}\setminus\{0\}$ converge to $V\setminus\{0\}$, whence $V_{\nu}\setminus\{0\}\cap\mathrm{int}\hat{H}(w,0)$ converge to $V\setminus\{0\}\cap\mathrm{int}\hat{H}(w;0)=V\setminus\{0\}$. It follows that $w\in N(V_{\nu})$, for almost all indices, i.e. $N(V)\subset\liminf N(V_{\nu})$. ∎ Proof of Theorem 3.6. If $E$ is compact, then so are the sets $E_{\nu}$, from some index onward (this follows directly from the definition of the convergence, compare e.g. [5]). Then it is easy to see that $f_{k}(E_{\nu})\leq\binom{f_{0}(E_{\nu})}{k+1}$, since a $k$-dimensional face must contain $k+1$ affinely independent points that define it. Therefore, we will be working under the assumption that the number of facets is uniformly bounded. By passing to a subsequence, we may assume that all the polyhedra $E_{\nu}$ have the same dimension $d$ and then that the numbers $f_{k}(E_{\nu})$, $k=0,\dots,d-1$ are independent of the index, both in the compact and non-compact case. What is more, we may assume that $d=n$ due to the following argument. Let $V_{\nu}=\operatorname{Aff}(E_{\nu})$ and let $\vec{V}_{\nu}$ be the underlying vector space. Then by the Zarankiewicz Theorem (i.e. sequential compacity), after passing to a subsequence we can find a limit $\vec{V}_{0}=\lim\vec{V}_{\nu}$ which is, obviously, also a $d$-dimensional vector space. But if we fix a point $x_{0}\in E$ and take any sequence $E_{\nu}\ni x_{\nu}\to x_{0}$, then we see that $V_{\nu}$ converge to $V_{0}:=\vec{V}_{0}+x_{0}$ and of course, $V_{0}\supset E$. It follows now easily from the definition of the Kuratowski convergence that we may assume that all the sets $E_{\nu}$ lie in the same $d$-dimensional space $V_{0}$, or rather that, actually, we are dealing with $n$-dimensional polyhedra. This implies that we can describe the sets $E_{\nu}$ in the following manner: $$E_{\nu}\colon\langle a_{i,\nu},x\rangle\leq b_{i,\nu},\>i=1,\dots,N=f_{n-1}(E_% {\nu}),$$ with $||a_{i,\nu}||=1$ for all $i,\nu$. Again, passing to a subsequence, we may assume that $a_{i,\nu}\to a_{i}$ for each $i$ when $\nu\to+\infty$. Now, each sequence $(b_{i,\nu})_{\nu}$ may be bounded or unbounded. Note that since $E\neq\varnothing$, we cannot have $b_{i,\nu}\to-\infty$. On the other hand, if $b_{i,\nu}\to+\infty$, then from the set-theoretical point of view, the corresponding $i$-th constraint stops playing any role in the description, i.e. we may forget it in the limit. The only interesting case is when (for a subsequence) $b_{i,\nu}\to b_{i}\in\mathbb{R}$. Assume that, passing to a subsequence, $(b_{i,\nu})_{\nu}$ have limits $b_{i}$ for $i=1,\dots,N^{\prime}$ and diverge to $+\infty$ for $i=N^{\prime}+1,\dots,N$. Observe that there must be $N^{\prime}\geq 1$, because $E\neq{\mathbb{R}}^{n}$. Consider first the set $$E^{\prime}\colon\langle a_{i},x\rangle\leq b_{i,\nu},\>i=1,\dots,N^{\prime}.$$ It may happen that some pairs of the constraints above are i-e. Suppose that this is the case for the indices $i,j$. It may happen that $a_{i,\nu}=-a_{j,\nu}$ for all indices (but, of course, $b_{i,\nu}\neq b_{j,\nu}$ due to the assumption that $\dim E_{\nu}=n$) — we will say then that the pair of constraints $(i,j)$ is parallel i-e. Suppose, however, that it is not the case, i.e. we can assume that $a_{i,\nu},a_{j,\nu}$ are not colinear, for all indices (as usual, by extracting a subsequence). Then $v_{ij,\nu}=v(a_{i,\nu},a_{j,\nu})$ make sense and due to the type of convergence, the positive cones ${\mathbb{R}}_{+}a_{i,\nu}+{\mathbb{R}}_{+}a_{j,\nu}$ must converge to an affine half-plane. Therefore, the vectors $v_{ij,\nu}$ have a well-defined limit $v_{ij}$ (for once there is no need to extract a subsequence). In this situation, adding to the description of $E_{\nu}$ the inequality $\langle v_{ij,\nu},x\rangle\leq u_{ij,\nu}$ where $u_{ij,\nu}$ is the value at a point $x_{0}$ satisfying $\langle a_{i,\nu},x_{0}\rangle=b_{i,\nu}$ and $\langle a_{j,\nu},x_{0}\rangle=b_{j,\nu}$ (444There must necessarily exist such a point for large indices $\nu$, for by assumptions $H(a_{i,\nu};b_{i,\nu})$ and $H(a_{j,\nu};b_{j,\nu})$ are not parallel.), does not change $E_{\nu}$. We may assume that $v_{ij,\nu}\to v_{ij}$. We face again two possibilities. Namely $W_{ij,\nu}:=H(a_{i,\nu},b_{i,\nu})\cap H(a_{j,\nu},b_{j,\nu})$ may converge (after passing to a subsequence) to an affine $n-2$-dimensional subspace $W_{ij}$, or to the empty set: this depends on whether the translating vectors $w_{ij,\nu}$ in $W_{ij,\nu}=w_{ij,\nu}+\vec{W}_{ij,\nu}$ with $||w_{ij,\nu}||=\mathrm{dist}(0,W_{ij,\nu})$ have a bounded subsequence or not. Clearly, this corresponds to the behaviour of $u_{ij,\nu}$, i.e. we will obtain $W_{ij}$, provided the $u_{ij,\nu}$ converge to some $u_{ij}\in{\mathbb{R}}$. Otherwise, if $W_{ij,\nu}\stackrel{{\scriptstyle K}}{{\longrightarrow}}\varnothing$, then $\hat{H}(a_{i,\nu},b_{i,\nu})\cap\hat{H}(a_{j,\nu},b_{j,\nu})$ converge to an affine hyperplane and we do not need to bother adding the additional constraint $\langle v_{ij,\nu},x\rangle\leq u_{ij,\nu}$ to the description of $E_{\nu}$, as it does not play any role in the limit. We introduce now the set $$E^{\prime\prime}:=E^{\prime}\cap\{x\in{\mathbb{R}}^{n}\mid\langle v_{ij},x% \rangle\leq u_{ij},(i,j)\in\mathcal{I}\}$$ where $\mathcal{I}$ is the set of all pairs of indices from $\{1,\dots,N^{\prime}\}$ that are i-e but not parallel i-e and for which $u_{ij}$ is well-defined. We claim that $E=E^{\prime\prime}$. It is obvious that $E\subset E^{\prime\prime}$: for $E=\liminf E_{\nu}$, whence any $x_{0}\in E$ is the limit of some sequence of points $x_{\nu}\in E_{\nu}$ and we just pass to the limit in the description (555Remember that we are working on a subsequence of $E_{\nu}$ chosen by taking into account $\mathcal{I}$, among other conditions.). To prove the converse, take a point $x_{0}\in E^{\prime\prime}$. There is $x_{0}\in E^{\prime}$ and if we had only strict inequalities in the description, we would be able to move $a_{i}$ and $b_{i}$ to $a_{i,\nu}$ and $b_{i,\nu}$, for sufficiently large indices $\nu$, without changing the inequalities; i.e. $x_{0}\in E$ in such a case. Assume, however, that there is $$\displaystyle\langle a_{i},x_{0}\rangle=b_{i},\>i=1,\dots,N^{\prime\prime}$$ $$\displaystyle\langle a_{i},x_{0}\rangle<b_{i},\>i=N^{\prime\prime}+1,\dots,N^{% \prime},$$ where $1\leq N^{\prime\prime}\leq N^{\prime}$. We may also assume that for $\nu\gg 1$, $$\displaystyle\langle a_{i,\nu},x_{0}\rangle>b_{i,\nu}\>i=1,\dots,N^{\prime% \prime\prime}$$ $$\displaystyle\langle a_{i,\nu},x_{0}\rangle\leq b_{i,\nu},\>i=N^{\prime\prime% \prime}+1,\dots,N^{\prime},$$ for some $1\leq N^{\prime\prime\prime}\leq N^{\prime\prime}$. Observe that it implies that for the distance $d_{\nu}:=\mathrm{dist}(x_{0},E_{\nu})$ which is realized by exactly one point $x_{\nu}\in E_{\nu}$ (due to the convexity of the sets $E_{\nu}$), we necessarily have $\langle a_{i_{\nu},\nu},x_{\nu}\rangle=b_{i_{\nu},\nu}$, for some $i_{\nu}$thatnecessarily belongs to $\{1,\dots,N^{\prime\prime\prime}\}$ (the point realizing the distance has to lie on the boundary). Then we may assume that $i_{\nu}=:i_{0}$ does not depend on $\nu$, i.e., to be more specific, that we have (possibly after a permutation of $\{1,\dots,N^{\prime\prime\prime}\}$) $$\displaystyle\langle a_{i,\nu},x_{\nu}\rangle=b_{i,\nu},\>i=1,\dots,i_{0}$$ $$\displaystyle\langle a_{i,\nu},x_{\nu}\rangle<b_{i,\nu},\>i=i_{0}+1,\dots,N^{% \prime}.$$ Now, $d_{\nu}\to d:=\mathrm{dist}(x_{0},E)$, because, if $\varepsilon>0$, then $\mathbb{B}(x_{0},d-\varepsilon)\cap E=\varnothing$, while $\mathbb{B}(x_{0},d+\varepsilon)\cap E\neq\varnothing$ and these conditions hold also for $\nu\gg 1$, due to the convergence (cf. [5] Lemma 2.1). This implies $d-\varepsilon<d_{\nu}<d+\varepsilon$, $\nu\gg 1$, as required. Moreover, $(x_{\nu})$ has to be a bounded sequence, since $d_{\nu}=||x_{0}-x_{\nu}||$, so that we may assume that $x_{\nu}\to\bar{x}_{0}$. Of course, $\bar{x}_{0}\in E$ and it realizes $d$. This realizing point is unique, because $E$ is a convex set, too. Suppose that all the points $x_{\nu}$ lie on a facet of the corresponding set $E_{\nu}$, i.e. $i_{0}=1$. Then, there must be $x_{0}=x_{\nu}+d_{\nu}a_{i_{0},\nu}$, which means that $\langle a_{i_{0},\nu},x_{\nu}\rangle=b_{i_{0},\nu}$ yields $$\langle a_{i_{0},\nu},x_{0}\rangle-d_{\nu}=b_{i_{0},\nu}.$$ By passing to the limit, we get $$\langle a_{i_{0}},x_{0}\rangle-d=b_{i_{0}}.$$ But $\langle a_{i_{0}},x_{0}\rangle=b_{i_{0}}$, whence $d=0$, i.e. $x_{0}=\bar{x}_{0}\in E$. Suppose that $i_{0}>1$ and let $x_{\nu,i}$ denote the orthogonal projections of $x_{0}$ onto $H(a_{i,\nu};b_{i,\nu})$ for $i=1,\dots,i_{0}$ and $d_{\nu,i}=\mathrm{dist}(x_{0},H(a_{i,\nu};b_{i,\nu}))$. By the argument above, $d_{\nu,i}\to 0$. Now, if $\{1,\dots,i_{0}\}\times\{1,\dots,i_{0}\}\cap\mathcal{I}=\varnothing$, then this implies that $x_{\nu}\to x_{0}$, i.e. $x_{0}=\bar{x}_{0}$. Otherwise, let us consider the corresponding additional constraints $\langle v_{ij,\nu},x\rangle\leq u_{ij,\nu}$ together with the orthogonal projections $x_{\nu,ij}$ to $H(v_{ij,\nu},w_{ij,\nu})$ and the corresponding distances $d_{\nu,ij}$. Note that we necessarily have $\langle v_{ij,\nu},x_{\nu}\rangle=u_{ij,\nu}$, whence, as earlier, we obtain $d_{\nu,ij}\to 0$. Nowarguing similarly asin the proof of [2] Theorem 1.1 based on [7] Formula (13) (compare [2] Theorem 1.3;in particular the constant in this theorem is bounded), this is sufficient to conclude that $x_{\nu}\to x_{0}$ (666Essentially, what is taken care of here may be illustrated by the following simple example in the plane: let $E_{\nu}$ be given by $-y\leq 0$ and $y-(1/\nu)x\leq 0$ which are i-e constraints; these sets converge to $E=[0,+\infty)\times\{0\}$ but $E^{\prime}$ is the whole $x$-axis.). This ends the proof of (1). Once we have obtained (1) with the convergence of the facets, we directly get (2) from simple linear algebra (compare Proposition 1.3): if $E$ is $n$-dimensional, then a vertex is described by $n$ linearly independent inequalities. Then thenearby inequalities are linearly independent and it follows that they define a vertex approaching the one in question. If, however, we had some i-e inequalities so that $\dim E=k<n$, then the same kind of argument works for $k$ describing functions restricted to $\mathrm{Aff}(E)$. If we take into account also the i-e inequalities, then we see that the vertex must be a limit of vertices. Proposition 2.1 implies now the convergence of the tangent cones, while Lemma 3.9 yields the assertion concerning the normal cones. Finally, the first part of (3) holds, because $f(x)=\langle c,x\rangle$ attains a maximum on $E$ iff $\hat{H}(c/||c||,b)\supset E$ for some $b$; since this holds foreach index $\nu$, it will hold also in the limit. For the second part, the compactness of $E$ implies the compactness of $E_{\nu}$ and Theorem 2.4 gives the result. The same argument is valid, if the number of vertices is constant, since the maximum is realized in a vertex. If we know that the maxima $M_{\nu}$ converge to the maximum $M$ of $f$ on $E$, then we easily get the convergence of the maximizers $H(c_{\nu}/||c_{\nu}||;M_{\nu}/||c_{\nu}||)\cap E_{\nu}$ to $H(c/||c||,M/||c||)\cap E$ using the half-spaces (compare [8] Theorem 4.32). ∎ Remark 3.10. Let us observe that a particular case of this theorem can be directly derived from [8] Theorem 4.32 (a). Essentially, this theorem states that if $A_{\nu}\to A$ for linear maps $A_{\nu},A\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{m}$ and the sets $A({\mathbb{R}}^{n})$ and $\prod_{i=1}^{m}(-\infty,b_{i}]$ cannot be separated, then $E_{A_{\nu},b_{\nu}}\stackrel{{\scriptstyle K}}{{\longrightarrow}}E_{A,b}$ when $b_{\nu}\to b$. This, however, does not cover entirely our result. The last point of the Theorem is a particular instance of the De Giorgi-Franzoni Theorem, namely: Theorem 3.11. Assume that the vectors $c_{\nu}\in{\mathbb{R}}^{n}$ converge to $c$ and let $M_{\nu}$ denote the set of minimizers of $f_{\nu}(x)=c_{\nu}^{T}x$ in $E=E_{A,b}$. Then $M_{\nu}$ converge in the sense of Kuratowski to the set $M\subset E$ being the set of minimizers for the limiting functional $f(x)=c^{T}x$. It follows from the proof of Theorem 3.6 that for linear polytopes we have also the following strong result. Theorem 3.12. A sequence of linear polytopes $E_{\nu}$ converges iff their boundaries $\partial E_{\nu}$ converge and then $$\partial\lim_{\nu\to+\infty}E_{\nu}=\lim_{\nu\to+\infty}\partial E_{\nu}.$$ Moreover, if the polytopes $E_{\nu}$ have nonempty interiors, then $\mathbb{R}^{n}\setminus E_{\nu}$ converges to the complement of the limit of the sets $E_{\nu}$. Remark 3.13. Of course, this type of result necessarily requires at least a convexity assumption. Indeed, if $K$ is the unit disc in the plane, then $\displaystyle\overline{K\setminus\frac{1}{\nu}K}$ converges to $\overline{K}$ but the boundaries do not converge to the boundary of the limit. 4. Examples of application We end our paper with some simple examples of application. Let us start with an economical one that illustrates Theorem 2.4. Example 4.1. A factory produces $n$ articles that are sold at prices $c_{1},\dots,c_{n}$ per unit. Of course, the prices are subject to some variations. We denote by $a_{i}j$ the coefficient encoding how much of the $j$-th raw material is used to produce the $i$-th article. Let $\beta_{j}$ be an upper bound for the stock of the $j$-th raw material. As is well-known, in order to maximize the profit we have to solve a linear programming problem given by $$\begin{cases}c^{T}x\to\max\\ x\leq\beta\\ x\geq 0,\end{cases}$$ where $c=(c_{1},\dots,c_{n})$ is the cost vector and $x=(x_{1},\dots,x_{n})$ gives the number of articles produced.By passing to the dual problem, we may rewrite this as $-c^{T}x\to\min$. Here $$A=\left[\begin{array}[]{ccc}a_{11}&\dots&a_{1n}\\ \dots&\ddots&\dots\\ a_{m1}&\dots&a_{mn}\\ -1&\dots&0\\ \dots&\ddots&\dots\\ 0&\dots&-1\end{array}\right]$$ and $b=(\beta,0,\dots,0)^{T}$. Ifi $\bar{x}$ denotes an optimal point, it implies a certain regulation of the machines in the factory. Now, Theorem 2.4 tells us that this regulation is optimal (gives a maximal profit) as long as the prices represented by the cost vector $c$ do not leave the cone $N_{a}(E)$ for an appropriate choice of the vertex $a$ in the feasible set $E$. Note that computing the cones from initial data is an easy task. Example 4.2. Consider the producer’s system [4], where the production set $Y$ is given by constraints $Ay\leq b$ with the properties $Y^{*}\neq\emptyset$ and $Y\cap(-Y)\subset\{0\}$. The goal is to maximize the producer’s profit $p^{T}y$. Therefore taking $f(y)=-p^{T}y$ we look for minimum over the set $Y$. By theorem 2.4 the optimal production plan is at some $y^{*}\in Y^{*}$ and the optimal price is the one satisfying $p\in N_{y^{*}}(Y)$. In the next picture we present this example in two-dimensional space of goods. Pay attention that we do not have any returns to scale, i.e. in contrast to [4] we waive the assumption about the convexity of production set $Y$. Additionally, the constants below satisfy $b>a>0$, $b>1$(777This assumption is only technical. The fact that $a,b>0$ implies that in the considered example the production set meets the standard economic expectations. Thanks to the fact that $b>a$ and $b>1$ it is possible to determine the optimal production plan). $$\begin{array}[]{rcl}Y=Y_{1}\cup Y_{2}&=&\{(y_{1},y_{2})\in\mathbb{R}^{2}:y_{2}% \leq a\wedge y_{2}\leq-\frac{1}{2}y_{1}\wedge y_{2}\leq-2y_{1}\}\cup\\ &&\{(y_{1},y_{2})\in\mathbb{R}^{2}:y_{2}\leq-b\wedge y_{2}\leq-2y_{1}+2\}\end{array}$$ Then $Y^{*}=\{(-2a,a),(0,0),(b,-2b),(b+1,-2b)\}$ and: $$\begin{array}[]{l}N_{(-2a,a)}(Y)=\{(p_{1},p_{2})\in\mathbb{R}^{2}:p_{2}\geq 0% \wedge p_{2}\geq 2p_{1}\}\\ \par N_{(0,0)}(Y)=\{(p_{1},p_{2})\in\mathbb{R}^{2}:2p_{1}\geq p_{2}\geq\frac{1% }{2}p_{1}\}\\ \par N_{(b,-2b)}(Y)=\emptyset\\ \par N_{(b+1,-2b)}(Y)=\{(p_{1},p_{2})\in\mathbb{R}^{2}:p_{2}\geq 0\wedge p_{2}% \geq\frac{1}{2}p_{1}\}\end{array}$$ For the prices from the corresponding cones the profit from production is: $$\begin{array}[]{l}\pi_{(-2a,a)}(p_{1},p_{2})=-2ap_{1}+ap_{2}=:\pi_{1}\\ \pi_{(0,0)}(p_{1},p_{2})=0\\ \pi_{(b+1,-2b)}(p_{1},p_{2})=(b+1)p_{1}-2bp_{2}=:\pi_{2}\\ \end{array}$$ Moreover, the constraint $b>1$ implies $\pi_{1}>\pi_{2}$. Therefore the optimal producion plan is $y^{*}=\{(-2a,a)\}$ giving the maximal profit $\pi^{*}=\pi_{1}$. 4.1. Kuratowski convergence and LP problem Example 4.3. Continuing the 4.2, consider the producer’s system, in which the producer is introducing some innovations. The innovations may be understood as the employment of some new technologies into the production process, rearrengement of the existing production process in the way that increases production abilities, etc. All of them result in extension of the set of possible production plans, denoted as $Y_{\nu}$. We naturally ask about the influence of those changes on the optimal plans. When can we assure that realisation of a current producer’s optima leads to the optimal production in the final set $Y$? The positive answer is given by Theorem 3.6, provided the sets $Y_{\nu}$ converge to the set $Y$ in Kuratowski sense. To illustrate the example let’s consider again the following numerical example in two-dimensional space of goods. As before, the constants below satisfy $b>a>0$, $b>1$. $$\begin{array}[]{rcl}Y_{\nu}=Y_{1,\nu}\cup Y_{2,\nu}&=&\{(y_{1},y_{2})\in% \mathbb{R}^{2}:y_{2}\leq a\wedge y_{2}\leq-\frac{\nu}{2}y_{1}\wedge y_{2}\leq-% \frac{2}{\nu}y_{1}\}\cup\\ &&\{(y_{1},y_{2})\in\mathbb{R}^{2}:y_{2}\leq-b\wedge y_{2}\leq-\frac{2}{\nu}y_% {1}+2\}\end{array}$$ Then the Kuratowski limit of the sequence $(Y_{\nu})$ when $\nu\nearrow 1$ is the set $Y$ defined in the example 2.2. Moreover, the sequence $(Y_{\nu})$ is ascending, i.e. for $\mu>\nu$ it holds $Y_{\nu}\subset Y_{\mu}$. This represents the described expansion of production set. The candidates for $\nu$-optimal production plans are $Y^{*}_{\nu}=\{(-\frac{2a}{\nu},a),\ (0,0),\ (\nu b,-2b),\linebreak(\nu b+1,-2b)\}$, while the corresponding normal cones are: $$\begin{array}[]{l}N_{(-\frac{2a}{\nu},a)}(Y)=\{(p_{1},p_{2})\in\mathbb{R}^{2}:% p_{2}\geq 0\wedge p_{2}\geq\frac{2}{\nu}p_{1}\}\\ \par N_{(0,0)}(Y)=\{(p_{1},p_{2})\in\mathbb{R}^{2}:\frac{2}{\nu}p_{1}\geq p_{2% }\geq\frac{\nu}{2}p_{1}\}\\ \par N_{(\nu b,-2b)}(Y)=\emptyset\\ \par N_{(\nu b+1,-2b)}(Y)=\{(p_{1},p_{2})\in\mathbb{R}^{2}:p_{2}\geq 0\wedge p% _{2}\geq\frac{\nu}{2}p_{1}\}\end{array}$$ The profits generated by the production plans and price vectors from corresponding normal cones are: $$\begin{array}[]{l}\pi_{(-\frac{2a}{\nu},a)}(p_{1},p_{2})=-\frac{2a}{\nu}p_{1}+% ap_{2}=:\pi_{1,\nu}\\ \pi_{(0,0)}(p_{1},p_{2})=0\\ \pi_{(\nu b+1,-2b)}(p_{1},p_{2})=(\nu b+1)p_{1}-2bp_{2}=:\pi_{2,\nu}\\ \end{array}$$ By similar arguments as before $y^{*}_{\nu}=(-\frac{2a}{\nu},a)$ and $\pi^{*}_{\nu}=\pi_{1,\nu}$. Clearly, $\lim\limits_{\nu\nearrow 1}y^{*}_{\nu}=y^{*}$ and $\lim\limits_{\nu\nearrow 1}\pi^{*}_{\nu}=\pi^{*}$. Now we present an example in which the number of vertices is reduced in the limit passing. Example 4.4. Consider the descending sequence of production sets: $$Y_{\nu}=\{(y_{1},y_{2})\in\mathbb{R}^{2}:y_{2}\leq a\ \wedge\ y_{2}\leq-\nu y_% {1}-2a\ \wedge\ y_{2}\leq-\frac{1}{\nu}y_{1}-\frac{(\nu+1)^{2}}{\nu}\cdot a\ % \wedge\ p_{1}\leq 0\}$$ with $\nu\nearrow 1$. Then for any $1>\nu>0$ the set of veritices is $$Y^{*}_{\nu}=\left\{(-\frac{3a}{\nu},a),(-a,-(2+\nu)a),\left(0,-\frac{(\nu+1)^{% 2}}{\nu}\cdot a\right)\right\}.$$ The Kuratowski limit of the sequence $(Y_{\nu})$ when $\nu\nearrow 1$ is the set $$Y=\{(y_{1},y_{2})\in\mathbb{R}^{2}:y_{2}\leq a\ \wedge\ y_{2}\leq-y_{1}-2a\ % \wedge\ p_{1}\leq 0\},$$ for which $Y^{*}=\{(-3a,a),(0,-4a)\}.$ Clearly, for any $\nu\in(0,1)$ the optimal plans are $y^{*}_{\nu}=(-\frac{3a}{\nu},a)$, which converges to the optimal production plan in the limiting set $y^{*}=(-3a,a)\in Y$. References [1] E. Chong, S. Żak, An Instroduction to Optimization, Wiley Eds 2004; [2] C. Bergthaller, I. Singer, The distance to a polyhedron, Linear Alg. Appl. 169 (1992), 111-129; [3] G. Dal Maso, Introduction to $\Gamma$-convergence, Birkhäuser 1991; [4] G. Debreu, Theory of value, Yale University Press 1959; [5] Z. Denkowska, M. Denkowski, The Kuratowski convergence and connected components, J. Math. Anal. Appl. 387 (2012), 48-65; [6] A. Daniilidis, M. Goberna, M. Lopeza, R. Luchetti, Lower semicontinuity of the feasible set mapping of linear systems relative to their domains, preprint 2014 [7] P.-J. Laurent, B. Martinet, Méthodes duales pour le calcul du minimum d’une fonction convexe sur une intersection de convexes in Symposium on Optimization, Nice 1969, Lect. Notes in Math 132, Springer-Verlag, New York 1970, 159-180; [8] R. T. Rockafellar, R.Wets, Variational Analysis, Springer Verlag 1998; [9] J. Franklin, Methods of Mathematical Economics, Springer Verlag 1980. Addresses: (A.D. M. K.) (M.D.) Cracow University of Economics Jagiellonian University Department of Mathematics Faculty of Mathematics and Computer Science Rakowicka 27 Institute of Mathematics 31-510 Cracow, Poland Łojasiewicza 6 anna.denkowska@uek.krakow.pl 30-348 Kraków, Poland marta.kornafel@uek.krakow.pl maciej.denkowski@uj.edu.pl

Learning Dense Visual Descriptors using Image Augmentations for Robot Manipulation Tasks Christian Graf${}^{\dagger,1}$, David B. Adrian${}^{\dagger,1,2}$, Joshua Weil${}^{1,3}$, Miroslav Gabriel${}^{1}$, Philipp Schillinger${}^{1}$, Markus Spies${}^{1}$, Heiko Neumann${}^{2}$, Andras Kupcsik${}^{1}$ ${}^{\dagger}$Equal contribution, ${}^{1}$Bosch Center for Artifical Intelligence, ${}^{2}$Ulm University, ${}^{3}$KTH Royal Institute of Technology Abstract We propose a self-supervised training approach for learning view-invariant dense visual descriptors using image augmentations. Unlike existing works, which often require complex datasets, such as registered RGBD sequences, we train on an unordered set of RGB images. This allows for learning from a single camera view, e.g., in an existing robotic cell with a fix-mounted camera. We create synthetic views and dense pixel correspondences using data augmentations. We find our descriptors are competitive to the existing methods, despite the simpler data recording and setup requirements. We show that training on synthetic correspondences provides descriptor consistency across a broad range of camera views. We compare against training with geometric correspondence from multiple views and provide ablation studies. We also show a robotic bin-picking experiment using descriptors learned from a fix-mounted camera for defining grasp preferences. Keywords: self-supervised learning, computer vision, representation learning, bin-picking 1 Introduction Scene and object understanding is essential for robot manipulation tasks, including assembly or bin picking. Often, the representation of choice is task-specific segmentation or pose estimation, trained in a supervised manner with labeled data. Labeling, however, is expensive and time-consuming, which is why self-supervised learning of dense visual descriptors has recently gained substantial attention in the robotics community, inspired by the works of Schmidt et al. [1] and Florence et al. [2]. Dense Object Nets (DONs) proposed by Florence et al. [2] learn dense visual descriptors of objects fully self-supervised in a robotic environment. The learned descriptors are view-invariant, show potential for within-class generalization and they naturally apply to non-rigid objects. The dense descriptor representation can be flexibly used for various downstream robotic tasks, such as, grasping (Florence et al. [2], Kupcsik et al. [3], Adrian et al. [4]), rope manipulation (Sundaresan et al. [5]) and learning control (Manuelli et al. [6], Florence et al. [7]). Self-supervised training of DONs, however, relies on pixel correspondences across multiple camera views provided by a registered RGBD image sequence, which requires accurate camera calibration and pose recording. Furthermore, pixel correspondence tends to be inaccurate with inexpensive depth cameras, even with data preprocessing. Finally, data collection is constrained by robot kinematics and the need for an expert setting up and supervising the procedure. In this paper we relax these assumptions fundamentally and instead of a complex setup with a single, robot-mounted moving camera, or multiple static ones, we solely rely on an unordered set of RGB images to learn object descriptors, for example, recorded by a single fixed camera. In our work, instead of relying on multi-view, geometric correspondence, we use augmentations of single images to obtain alternative views and synthetic correspondence. This idea was already explored in computer vision by Thewlis et al. [8] and Novotny et al. [9], in the context of learning geometrically consistent pixel-level descriptors across multiple object classes. In this paper we show that relying on synthetic image augmentations achieves competitive performance in terms of keypoint tracking accuracy compared to a network trained with geometric correspondence. Importantly, this approach can easily be adopted to existing industrial setups with fix-mounted cameras, or with cameras too heavy to be mounted on a robotic arm, without additional engineering effort. We show such a robotic bin-picking setup in Fig. 1, with an overhead, fix-mounted camera. Our contributions are as follows: (i) we adapt existing work on training self-supervised pixel embeddings (Novotny et al. [9], Chen et al. [10]) to robotic grasping downstream tasks. (ii) we show that for robotic grasping tasks our approach is en par with state-of-the-art (Adrian et al. [4]) in terms of keypoint tracking accuracy while drastically simplifying the data collection, and finally (iii) we show a real-world robotic bin-picking experiment where human preference on grasp configuration is encoded with dense visual descriptors, with the constraint of using a single, fixed-mounted camera. 2 Related work In the following, we review recent work on self-supervised dense visual descriptor learning for robotic manipulation in more detail. We also discuss related work on self-supervised representation learning and learning from a set of single images, which are core concepts in our work. Dense visual descriptors in robotic manipulation. Inspired by the work of Schmidt et al. [1], Florence et al. [2] proposed self-supervised training of dense visual descriptors by and for robotic manipulation. Their approach was later adopted by Florence et al. [7] to learn from multi-view correspondence in dynamic scenes and showed an application for policy learning. Sundaresan et al. [5] applied the descriptor space representation to learn challenging rope manipulation in simulations. Applying the dense descriptor representation to learn model predictive controllers was shown by Manuelli et al. [6]. Vecerik et al. [11] use multi-view consistency for keypoint detection and show an application for reinforcement learning. Another line of work investigated improved training strategies of dense visual descriptors. There are multiple contributions focusing on learning multi-object and multi-class descriptors by Yang et al. [12], Hadjivelichkov and Kanoulas [13] and Adrian et al. [4]. The work by Kupcsik et al. [3] exploits known object geometry to compute optimal descriptor embeddings. Finally, Yen-Chen et al. [14] utilize NeRF to generate dense correspondence datasets from RGB images. This alleviates problems with noisy depth data and proves especially helpful for thin and reflective objects. Several papers proposed to learn directly from synthetic images composed of random backgrounds and randomly sampled, masked objects distributed over the image, see Florence et al. [2], Chai et al. [15], Yang et al. [12]. Learning from such synthetic images can be more efficient due to higher object density and ground truth correspondence, however, they rely on labeled datasets with object masks. Masking is either achieved by 3D reconstruction with a robot wrist mounted camera (Florence et al. [2], Chai et al. [15]), or a labelled RGBD dataset (Yang et al. [12]). As opposed to image composition via mask-labeled datasets, the image augmentation technique, as in this paper, only requires an unordered, unlabelled RGB dataset. This significantly simplifies data collection and opens up the possibilities to learn dense visual descriptors where no 3D reconstruction is possible, or where object masks are not available. Self-supervised descriptor learning from RGB images. An intuitive way to generate geometric correspondence is to estimate the optical flow of subsequent frames from a video. Deekshith et al. [16] adopted this technique to train DONs using contrastive learning. Thewlis et al. [8] proposed to use optical flow from videos, or image augmentations to embed pixels of objects in view-invariant coordinate frames. Novotny et al. [9] adopted this method for pretraining of geometry-oriented tasks, such as object specific part detection in images. Zhang and Maire [17] propose to learn pixel-wise descriptors from single images with augmentations by using hierarchical visual grouping of image patches based on contour. Our work follows the image augmentation technique of single RGB images to generate alternative views and synthetic correspondence. Equivariant network architectures such as proposed by Cohen and Welling [18], Wang et al. [19] could replace certain augmentations (e.g. rotation) during training and improve sample efficiency. In our setup we can easily apply affine transformation on training data and we rely on a vanilla ResNet architecture for our experiments, which achieves good $SE(2)$ equivariance, as shown in our experiments. Self-supervised visual representations learning. Instead of training on large supervised datasets, self-supervised methods have become a popular way to obtain visual representations, which can be fine-tuned to specific downstream tasks. A recent and very successful approach using contrastive learning is SimCLR by Chen et al. [10]. It aims to maximize agreement between two augmented versions of the same image, while considering all other images in the batch as negative samples. Grill et al. [20] proposed BYOL (Bootstrap your own latent) that, in comparison to contrastive methods, does not rely on the sampling of negatives. With Barlow Twins Zbontar et al. [21] also forgo negative samples by optimizing the cross-correlation matrix between embeddings from two augmented versions of the same image to be close to identity. Our approach is most similar to SimCLR as we employ the same loss formulation, but with the important difference that our batch is constituted by individual pixel descriptors instead of full image embeddings. 3 Method In this section we discuss our proposed training approach using image augmentations. We first give an overview of the whole training pipeline, then discuss image augmentation techniques and finally present the loss formulation and dataset requirements. For an illustration of the training pipeline we refer to Fig. 2. 3.1 Dense Descriptor Training with Synthetic Correspondence Inspired by the work of Novotny et al. [9] we rely on training on an unordered set of images and use image augmentations to arrive at alternative views of each image. First, we sample a minibatch of $N$ RGB images from the training data set consisting of independent RGB images. For every image $I$ in the minibatch we sample two augmented views $I^{\prime}\sim g(I)$ and $I^{\prime\prime}\sim g(I)$ by applying randomized augmentations $g:\mathbb{R}^{H\times W\times 3}\mapsto\mathbb{R}^{H\times W\times 3}$ (described in more detail below). A learned fully-convolutional network [22] model $f(\cdot;\theta),~{}f:\mathbb{R}^{H\times W\times 3}\mapsto\mathbb{R}^{H\times W\times D}$ maps the augmented images $I^{\prime}$ and $I^{\prime\prime}$ to their descriptor space embeddings $I^{\prime}_{d}$ and $I^{\prime\prime}_{d}$. The user defined parameter $D\in\mathbb{N}^{+}$ controls the resolution of the descriptor space. By keeping track of the position of each pixel in the original image $I$ during the augmentations, we sample pairs of pixel locations between $I^{\prime}$ and $I^{\prime\prime}$ that share the same position in $I$. We refer to these as synthetic correspondences, emphasizing the use of synthetic image augmentations as opposed to geometric correspondences coming from the 3D geometry of multiple camera views, as in [1, 2]. The descriptor values at the sampled pixel correspondence locations serve as positive pairs for contrastive learning. 3.2 Image Augmentations For robotic applications we require descriptors that are invariant to translations, rotations and perspective changes of the objects, as well as changes of lighting conditions. In vanilla DONs training, cf. Florence et al. [2], this is achieved by recording a diverse training set of registered RGBD image sequences, which contain sufficient variance in camera and object poses, and to some extent in lighting conditions. In our work, we achieve a similar effect purely by imposing data augmentations on single RGB images from an unordered set. We carefully select augmentations, which reflect the desired invariance properties stated above, as follows: affine transformation induce rotations and scale changes (zoom out), perspective distortions mimic view changes, resize&crop implies scale changes (zoom in), and lastly, color jitter affects brightness, contrast, hue, and saturation of the image, hence the lighting conditions of the scene. We show an illustration of these augmentations in Fig. 3. We utilize torchvision library [23] of Pytorch as reference implementations. Adrian et al. [4] already demonstrated the improved performance based on image augmentations for the training of dense visual descriptors on datasets utilizing geometric correspondences. In our work, each augmentation is not only helpful, but relevant to the ability of the model to successfully learn an invariant descriptor space. See Sec. 4.3 for an ablation on the respective impact of each augmentation on the overall performance. 3.3 Loss Function Following Chen et al. [10] we adopt the NT-Xent loss, for learning the dense descriptor representations. For a pair of corresponding descriptors $\{d_{i},d_{j}\}$, obtained from images $I^{\prime}$ and $I^{\prime\prime}$ in a minibatch of size $M$, we compare their distance to the distance of $d_{i}$ to all other sampled descriptors in the given minibatch arriving at the following individual loss term: $$l_{i,j}=-\log\frac{\exp(\mbox{dist}\left(d_{i},d_{j}\right)/\tau)}{\sum_{k=1;k\neq i}^{2M}\exp(\mbox{dist}\left(d_{i},d_{k}\right)/\tau)}.$$ (1) with temperature parameter $\tau$ which we fix to $\tau=0.07$ throughout this paper following Adrian et al. [4]. We choose the metric $\mbox{dist}\left(\cdot\right)$ to be the cosine similarity between descriptors $d_{i}$ and $d_{j}$. As the cosine similarity expects normalized vectors we normalize the descriptors $d$. The total loss is given as the mean over all individual loss terms, cf. Chen et al. [10]. Note that this loss, together with our correspondence sampling, does not distinguish background from objects, nor does it explicitly address multiple object classes and instances, as opposed to the approach by Florence et al. [2]. Instead, we follow the method of Adrian et al. [4] and sample correspondences uniformly in image plane and assume that every pixel is unique. This method is tailored to datasets depicting densely packed scenes with single object instances, for example, a heap of objects in a bin-picking scenario. The learned descriptor space does not imply semantic information on object classes or background, but still provides consistent keypoint detection and robust tracking performance, which is essential for downstream tasks. 4 Comparison of Training with Synthetic and Geometric Correspondence In this section we show an in-depth comparison between training with geometric and synthetic correspondence. We also investigate the invariance of descriptors obtained from a network trained with synthetic correspondence with respect to object-camera relative transformations. In all our evaluations we utilize a pretrained ResNet-34 with 8-stride output as used by Florence et al. [2], which yields an upsampled output matching the resolution of the input. In the supplementary material we give a brief introduction into the baseline training method we use in our comparisons relying on geometric correspondence by Adrian et al. [4]. We recorded a dataset consisting of a set of registered RGBD sequences, to enable comparison between both approaches. Despite the availability for registered image pairs, the synthetic view training only uses single RGB images for both training and validation. However, the camera poses help with the evaluation as they allow us to generate ground truth pixel matches across any two images of the same static scene without the need for manual labeling. The dataset consists of eight scenes with various object configurations, with every scene containing only one instance per object, and every object is visible to some extent in every frame. The scenes are recorded with a robot wrist mounted camera while the robot arm follows a predefined trajectory keeping the objects in view. Both approaches are evaluated on the same ground-truth image pairs and correspondences. For robustness, we perform a k-fold cross-validation, that is, each scene from our total dataset was once used as test set. One scene is chosen as validation set, with the remaining 6 scenes used for training. The averaged results are reported. We use the same loss function, training parameters and augmentations for both approaches, with the exception that augmentations are chosen with $50\%$ probability for the geometric training, as it yields better results. For synthetic training, each augmentation is always used. Training details and an ablation study of using different augmentation probabilities is given in the supplementary materials. 4.1 Keypoint Tracking Performance Given a common dataset, measuring keypoint tracking accuracy provides a task-agnostic comparison between both training methods. We define a keypoint as a location in image plane $k_{i}=(u,v)$. Each keypoint is associated with a unique descriptor $d\in\mathbb{R}^{D}$ and the keypoint tracking problem is defined as finding the pixel location $k_{i}^{*}$ closest to $d$ in the descriptor image $I_{d}$ such that $k_{i}^{*}=\arg\min_{k_{i}}\mbox{dist}(I_{d}(k_{i}),d)$. Evaluation. From the test set we sample $1000$ image pairs $\{A,B\}$ representing alternative views of the same scene. For each image pair we sample $200$ keypoints $\{k_{i}^{A},d_{A}\}$, where $~{}d_{A}=f(A;\theta)(k_{i}^{A})$ is located on one of the objects in image $A$. We also recorded keypoint locations in image $B$, $\{k_{i}^{B}\}$, such that every pair $\{k_{i}^{A},k_{i}^{B}\}$ projects from pixel to world coordinates. Then, we solve the keypoint tracking problem for every descriptor $d_{A}$ in image $B$ and record the pixel error $e=\lVert k^{B}_{i}-k_{i}^{*}\rVert_{2}$. The network parameters $\theta$ are either obtained by geometric [4] or synthetic correspondence training proposed in Sec. 3.1. Results. Fig. 3(a) shows the distribution of pixel errors for the two training approaches. The median of the corresponding distribution is highlighted as a dashed line. With a difference of only $1.9$px in median pixel error, the synthetic correspondence performs competitively to the geometric training. The percentage of pixel errors that are larger than 50 pixels are $10.1\%$ and $5.4\%$ respectively, as indicated in the bottom right part of Fig. 3(a). In Fig. 3(b) we compare the median and the 75% quantile of the two training methods with respect to the descriptor dimensions $D$. The results were obtained without k-fold cross-validation. For both training approaches the median pixel error decreases for larger descriptor dimensions. For dimensions larger than $9$ the median pixel error decreases only marginally. In contrast, the 75% quantile error still increases further until $D=64$, before improvements saturate. For additional insights into these results we refer to the supplementary material. 4.2 Invariance Tests In the following, we wish to answer the question: how well does training with image augmentations proposed in Sec. 3.2 generalize to physical camera transformations? For this purpose, we recorded different test scenes with a wrist-mounted camera and the following specific camera movements: (i) changing camera perspective (camera tilting), (ii) translation along the camera z-axis (zooming in and out), and (iii) camera rotation along the camera z-axis (see Sec. D.2 for details). We compare the performance of a network trained with synthetic and one with geometric correspondence both trained on the same data as described in Sec. 4.1 with 64 descriptor dimensions and affine, perspective and resize&crop augmentations. As in the previous section we compute the 75% quantile pixel error and use $1000$ keypoints per image pair, fix the base image $A$ and only vary image $B$, which shows the changing camera views. The results for three different types of transformations are compiled in Fig. 5. It can be seen that training with synthetic correspondence generalizes well to a large range of camera transformations, especially to those parallel to the object plane (Fig. 4(a), Fig. 4(b)). These physical camera transformations are very similar to the affine and resize&crop augmentations used during training. For generalizing to the perspective transformations with angles above $45^{\circ}$ degrees, as shown in Fig. 4(c), the synthetic correspondence training shows a clear deterioration in performance. As the perspective changes, occluded parts become visible and vice versa. This physical transformation is not well captured by the synthetic augmentations for larger angles. 4.3 Augmentations Complementing the findings in section 4.2 we study the influence of different augmentations for the synthetic correspondence training on the ability of the network to generalize to different camera transformations. Fig. 6 shows the 75% quantile of the pixel error distribution for the synthetic correspondence training with different augmentations. We find that affine transformations are most critical, as only a model trained with it shows invariance to rotations, see Fig. 5(a). This matches the expectations as standard CNN are by default not invariant to rotations. Nevertheless, we find that both resize&crop and perspective distortion both further improve the performance of just affine transformations. In particular, for camera movements that induce perspective distortions and scale changes, see Fig. 5(c) and Fig. 5(b), the error decreases considerably. Lastly, we find that color jitter further reduces the overall mean pixel error from $19.4$ to $17.1$ pixel. The improvement appears modest, but we note that our test dataset was recorded at the same time as train and validation, and the lighting conditions of the scene were not explicitly altered. For a complete table of all the combinations of augmentations, see the supplementary material. 5 Grasping Experiment with Fix-mounted Camera We demonstrate a robotic bin-picking experiment that relies on dense visual descriptors for defining grasp preferences. We use a 7-DoF Franka Emika Panda arm with a suction gripper mounted on the end-effector, see Fig. 1. Our setup uses a fix-mounted Zivid One+ camera above the bin in a robotic cell. Training a descriptor network using this setup prevents the use of geometric correspondences. Instead, we show that our proposed method can be trained on a setup as it is often present in real world applications and prove that the keypoint tracking accuracy is good enough for guiding a generic grasping method by human annotated grasp preferences. In the experiment, we consider picking ten different types of objects from one bin. However, using a suction gripper the objects are difficult to grasp at certain locations: some have cutouts on the packaging, transparent or foldable parts, and uneven surfaces (see supplementary material). In our experiments, a purely model-free grasp pose generator often predicts poses at these challenging parts of objects, ultimately reducing picking performance. We hypothesize that introducing human domain knowledge to aid grasp pose selection will lead to higher grasp success rates. Human domain knowledge is considered by highlighting parts of the RGB image where grasp poses are preferred. To do so, we show a small number of RGB images depicting the objects in different configurations in the bin and ask the human to click on pixel locations corresponding to a preferred grasp location. We track these descriptors to generate a preference heatmap, as shown in Fig. 6(c). The heatmap, in contrast to discrete single-pixel correspondences, enables intuitive consideration of matching uncertainty in the form of distance in descriptor space. Furthermore, it provides a more flexible and quantifiable basis for combination with grasp detectors. To generate the final set of grasp pose candidates we intersect the grasp preference heatmap with the detected graspable areas identified by the model-free grasp detector (Fig. 6(b)). While we make no specific assumption regarding which grasp detection method to use, we employ in the shown experiment a dense pixel-wise graspability estimation based on a fully convolutional neural network with RGB-D input, specifically UNet [24] trained on annotated pixel-wise labels of expected graspability for a wide range of different bin picking scenes. For the example image, the resulting poses are shown in Fig. 6(a) and Fig. 6(d). Note that the descriptor network is trained purely from the set of RGB images of the bin, including the objects in random configurations, recorded with the overhead camera. We require a single instance of the objects to be present in the bin. For more details on the experiment setup and heatmap generation we refer to the supplementary material. 5.1 Quantitative Evaluation We evaluate the benefit of the proposed method for robotic bin picking based on a set of sixteen manually annotated scenes. The scene images include the ten known objects, but in different configurations compared to the training set. For each evaluation image, we manually annotated object instances, pixel-wise graspable areas, and those areas that correspond to selected descriptors. In total, the evaluation dataset contains 131 graspable objects of which 103 have visible descriptor spots. We compare our descriptor-based grasping approach with a purely model-free approach that directly uses the graspable areas without accounting for descriptors. With this study, we investigate two questions regarding the descriptor-based approach. First, how effective is the approach to re-identify descriptor spots compared to grasping at these spots by chance, considering that the descriptors indicate the best way of grasping the respective object? And second, is there a considerable negative effect on the amount of objects that can be grasped, e.g., due to missing to identify graspable areas? Table 1 shows a summary of the results. Success Rate denotes the number of successful grasps compared to all grasps attempted. Descriptor Success denotes the number of grasps at those spots marked as desired grasping points compared to all grasps attempted. Object Hits denotes the percentage of all graspable objects for which at least one feasible grasp pose has been found irrespective of descriptors, including objects without visible descriptor spots. Descriptor Hits denotes the percentage of objects for which a grasping pose has been found at the respective descriptor spot, only including objects with visible descriptors. We consider descriptor spots with a tolerance of around 1cm, which corresponds to the radius of the suction gripper. It can be concluded from Table 1 that using the proposed method to encode grasp preferences is effective as it significantly increases the amount of grasps at desired spots from $50.4\%$ to $91.1\%$. Due to the challenging object geometries, this helps to raise the overall grasp success rate from $79.9\%$ to $98.9\%$. The expected downside, however, is that the proposed method finds grasp poses only for a smaller amount of objects, $63.4\%$ instead of $78.6\%$. Still, this includes objects that have no desired grasping spot visible. In some applications it can be the desired behavior to not propose a grasp pose for objects if they cannot be grasped at the preferred location. When only considering which preferred grasping spots have been covered by grasp poses, our method manages to find grasps for $78.6\%$ instead of $68.0\%$ of the visible spots. In consequence, we conclude that there is only a moderate negative effect due to limiting grasping to descriptor spots which can indeed be beneficial for some applications. 6 Limitations In the following we share more insights on the limitations of our method and discuss future research. Variance in camera poses. As seen in Fig. 5, the synthetic training ensures that descriptors are stable within a limited margin of camera transformations relative to the object. For example, in Fig. 4(c), for viewing angles steeper than $45^{\circ}$, accuracy deteriorates. This imposes a limit on descriptor consistency across images with large differences in object poses. Changing Environment. A change in environment (background, lighting) between training and inference time may have a negative influence on keypoint tracking performance. It is expected that augmentations such as color jitter and, if masks are available, background randomization can mitigate these effects. We will investigate these aspects in future research. Generalization to unseen objects. Although Florence et al. [2] demonstrated capabilities to generalize intra-class instances, our work focuses on instance specific descriptors. Given our outlined training setup, it works best on known objects from the training set. Object Edges. We observe worse descriptor consistency across images for keypoints located at the edges of objects or close to parts that are occluded by other objects. In setups where object masks are present background randomization could reduce this effect. 7 Conclusion In this paper we proposed a novel training method for learning dense visual descriptors based on image augmentations for robotic manipulation. The evaluation shows that overall our proposed method is competitive to the existing geometric training approach. For physical transformations like changing the camera perspective on the scene, which are harder to mimic by augmentations, the training method using geometric correspondences shows superior performance. Being aware of these limitations our proposed method is expected to perform well for setups where objects are mostly altered by translations, rotations parallel to the camera plane, or are slightly tilted. 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Cats: Cost aggregation transformers for visual correspondence. In M. Ranzato, A. Beygelzimer, Y. N. Dauphin, P. Liang, and J. W. Vaughan, editors, Advances in Neural Information Processing Systems 34: Annual Conference on Neural Information Processing Systems 2021, NeurIPS 2021, December 6-14, 2021, virtual, pages 9011–9023, 2021. Appendix A Overview In the following, we provide additional technical details and supporting results. The sections are organized as follows: • Sec. B reviews how training with geometric correspondences is conducted. • Sec. C provides further details for training models. • Sec. D offers an overview and example images of our datasets. • Sec. F.1 discusses further results from the augmentation ablation study, • Sec. G details grasp preference heatmap generation and the objects used in the grasping experiment. Appendix B Training with Geometric Correspondences While originally introduced by Florence et al. [2], we utilize the adapted method by [4] for training without masks in multi-object settings. The training relies on sampling a set of corresponding pixels in image $A$ and $B$, where both images observe the same static scene and objects, but from different view points. By employing a contrastive loss, the descriptors for each pixel pair are trained to have the same embedding, while separating all other pixels in latent space. The view-invariance of the descriptors is the consequence of utilizing images with different view points. Geometric correspondence training exploits the geometric prior provided by a registered RGBD sequence. As the relative pose between any two images in the sequence is known, and given the depth and camera information, one can establish the per-pixel correspondence between each image pair, allowing for straight-forward sampling correspondences. In practice, depth data can be noisy or incomplete. For example, structured light cameras struggle with transparent or black surfaces, and with higher measurement uncertainty around edges of objects. Thus, Florence et al. [2] perform a 3D-reconstruction of the scene, to render synthetic depth images which are complete and denoised, albeit not perfect ground-truth. In the original approach, which focuses on training with singulated objects, an automatic or manual mask generation of the object is performed. As we deal with multi-object scenes, we follow [4] and instead record scenes with multiple objects present and do not compute any masks. Hence, we sample correspondences anywhere in the image and do not differentiate between object or background. Instead, to sample correspondences we first prune the correspondence map from image $A$ to $B$ by occlusion and field of view masking, then we sample a set of $N$ pixel correspondences. As we apply augmentations, the process can shift or remove pixels from either image. We need to account for this in the sampling process. Given a set of sampled correspondences, we employ the same loss as described for our synthetic view training. Appendix C Training Details The training settings, see Table 2, are shared for both the geometric and synthetic training, with exceptions specified below. They follow the findings made by [4] for geometric training. They are used for all experiments shown, unless specified otherwise. We perform validation after each epoch and retain the checkpoint with the best score. The score is evaluated with respect to the area under the curve (AUC) of the PCK@K (percentage of correct keypoints). PCK@K is determined by taking a set of predictions and calculating the pixel error $e$ with respect to the corresponding ground-truth pixels. The percentage is given by the number of predictions with a pixel error $e<K$. We evaluate the AUC for the range $K\in[1\cdots 100]$. Both approaches, geometric and synthetic, are trained with the same augmentation parameters as listed in Table 3. The major difference is that for synthetic training, we sample each augmentation with probability ${p=1.0}$ , whereas for geometric training each augmentation is sampled with ${p=0.5}$. We found that the latter performed better. Appendix D Training Datasets This section gives more details on the data used during training of the described methods. Overall, two different training dataset were used. One taken with a robot-mounted camera used for comparing training with geometric and synthetic correspondence and for all results presented in section 4. And another one with a fix-mounted camera used for the grasping experiment described in section 5. D.1 Dataset with robot-mounted camera Fig. 8 shows example images from this dataset. We prepared seven objects in a static scene and recorded a stream of images with 30 frames per second, while the robot-mounted Realsense D435 camera moved in different perspectives around the scene. In total we took nine recordings like this with different configurations of the same objects. Six of those recordings were used for training, one for validation and one for testing. Each recording contains about 4400 images. D.2 Dataset for invariance tests For the invariance tests reported in Sec. 4.2 we took data with the same setup of the robot mounted camera and the same objects as reported above in Sec. D.1, but with dedicated camera movements. These camera transformations are visualized in Fig. 9. The left shows the camera rotation ($z$-axis), where the location of the camera is fixed with the camera plane parallel to the table. The camera is then rotated around the $z$-axis. In the middle we show the camera moving in/out along the $z$-axis, closer to and further away from the scene. The $x$ and $y$-positions as well as the camera orientation are kept stable. The last movement is the camera perspective movement. Here, the camera is moved in $x$ direction on a half-sphere around the scene changing the orientation to keep focus on the center of the scene. During this movement the distance of the camera to the center of the scene is fixed. D.3 Dataset with fixed-mounted camera Example images of this dataset are shown in Fig. 10. It was recorded with a fixed-mounted Zivid One+ camera. We recorded 529 images of randomly shuffled heaps of the objects presented in section G. Additional data was held back for validation and testing. Appendix E Additional Results In the main section we provide results mostly in terms of the median pixel error and the 75% quantile of pixel errors. In the following section, we provide the main results with additional metrics. Furthermore, we present results on the generalization capabilities on a small test set featuring unknown objects. E.1 Extended Main Results The results of Sec. 4.1 are summarized in the Table 4 with additional metrics. E.2 Results on Unknown Objects We further investigate the performance of our model on objects not seen during train time. Both the geometric and synthetic correspondence models were trained on the same kfold dataset splits of the main section. We test on five new objects, not previously seen during training and validation. The objects and their arrangement in the two test scenes are shown in Figure 11. We kept the training setup as in Sec. 4.1, although we note, that better generalization might be achieved with different configuration of hyper-parameters, the choice and amount of augmentation applied. However, the overall trend is evident in the results compiled in Table 5. Both approaches, SV and GC, exhibit a loss in performance, especially the geometric correspondence training. While the median changes only slightly, we find a large increase with respect to the 90% and 95% quantile for both methods. Up to 25% of the sampled keypoints are now mispredicted with an error nearly three times as high as before. Using features from a purely pre-trained backbone, without further training, fails completely. Training on a generic dataset, such as COCO, yields surprisingly good results, but still fails to work accurately. For more details on the pre-trained and SV-COCO setup, see the ablation study in Section F.3. We note that, as we train only on a set of unordered RGB images, fine-tuning the model for additional new objects is as easy as adding a few new image taken of the novel objects. Hence, despite limited generalization to completely new objects, the simple and efficient training of our proposal may effectively compensates for it. Appendix F Ablation F.1 Augmentation To investigate the impact of each augmentation, we trained the synthetic approach with each combination and tested it on the same dataset as in Section 4.1. The full results are listed in Table 6. We see that affine transformations have a strong impact on the overall performance. All other augmentations, even the combination of color jitter, perspective and resize+crops, performs considerably worse. This result confirms that for our approach affine transformations are indeed essential to obtaining invariance to rotations with a CNN-based backbone network. Generally, all combinations with affine augmentation further improve the models accuracy and robustness. An exception is the combination of color jitter and affine, which seems to find a worse solution when combined. We find that when using perspective distortion, the median typically seems to slightly decrease, while the mean, as well as the 75% and 90% quantiles improve considerably. Hence, perspective distortions seem to play an important role in improving model robustness, but requires further investigation as to why the accuracy is negatively affected. Color jitter seems to be the augmentation with the smallest impact. However, we note that while we tested on different scenes, the lighting conditions are generally the same. Hence, on datasets, and more importantly during model deployment, the impact of color jitter with respect to model robustness and reliability could be larger. Lastly, we see that the combination of all augmentations ensure the learned descriptor is not overly focused on single type of invariance, but different kinds yielding the overall best result. F.2 Probability of Augmenting and Number of Augmented Frames In this experiment we vary two hyperparameters of our training: i) the chance that any given augmentation might be applied (independently drawn), ii) number of views that will be augmented. We evaluated each configuration on the invariance test dataset, with the results shown in Figure 12. We find that augmenting just one or both images, has limited impact for both geometric and synthetic correspondence training. For geometric training we find that our setting, which is using 50% probability per augmentation, yields similar results compared to augmenting just one view. This was already observed in [4]. We reconfirm, as reported by [4], that augmenting more heavily, e.g., both frames with each augmentation at 100%, has adverse effects on the performance of geometric correspondences trained networks. In contrast, the synthetic correspondence training is most strongly impacted by the probability, less by the number of augmented images. This is not surprising, as unlike the geometric training, augmentations are essential for the synthetic training, cf. Section F.1. Without any augmentations, both views are identical and the network will only learn a trivial solution. Consequently, it is important to increase the chance, or guarantee in some ways, that at least one frame is augmented. The difference between augmenting one or both views with high probability yields nearly the same results. We note, that a more refined selection of differing probabilities per augmentation type would most likely yield even better results, rather than just one global parameter choice. F.3 Comparison to Baseline Methods and State-Of-The-Art Approaches With this additional set of experiments we validate our assumption that domain specific data and explicit augmentations for perspective changes are crucial for a good performance on the target domain. For this, we compare our method to four different baseline methods: 1. GC Specific: Using data from different viewpoints ([1, 2, 4]), as already reported in the main paper. 2. Pretrain only: Using the features from a pretrained ResNet backbone (on ImageNet), without further fine-tuning. This serves as a naïve baseline. 3. SC COCO: Using the method presented in this paper, but fine-tuned on COCO data instead of domain-specific data. 4. CATs: A state-of-the-art keypoint matching algorithm among the top ranking methods on various keypoint matching datasets [26]. We compare to the pretrained method on PF-Pascal as provided by the authors. We evaluated all methods on our view-invariance test dataset, as described in 4.2. Figure 13 shows the results. Not surprisingly, the raw pretrain-only features, exhibit little to no rotational invariance, and generally lack view-invariance on other tests. SC COCO does perform better than the pretrained-only method, especially for smaller transformation angles. However, it seems to not generalize well to larger transformation angles in our invariance tests. This indicates that in-distribution training data is important for the accuracy we need for the robotics use-case. The keypoint matching method CATs achieves very impressive results for semantic keypoint matching, where e.g., the tip of a dog’s nose will be matched to a completely different dog’s nose in a second image. Surprisingly, the method does not outperform the Pretrain only baseline on our test dataset. We account this result to the following: (a) PF-Pascal has a limited number of classes and the pretrained network overfits to those (b) The goal of CATs (and similar approaches) is semantic keypoint matching. In this goal it achieves very impressive results. However, the goal of these methods are not the very accurate matching of geometric points on target objects, which we need for robotic grasping (see the reported mean pixel error in the original CATs paper). The above experiments support our claim that that our proposed training schema with the choice of augmentations and loss together with in-distribution training data containing scenes of the target objects, plays an important role to achieve sufficiently high accuracy for robotic grasping applications. Appendix G Grasping Experiment G.1 Objects In Fig. 14 we show the objects we used in the grasping experiment. Each object is either challenging to grasp with a suction gripper while relying only on depth images and geometrical features, or successful grasps may damage the objects. Therefore, we wish to rely on human annotated grasp preferences to avoid damage and improve success chances. Fig. 13(a) and Fig. 13(b) show gloves which are only graspable on the paper label, which is also the preferred grasp location. Additionally, a cutout at the top and plastic strips in the middle of the paper label make these objects challenging to grasp. The depth camera may not recognize the small bumps in the depth image for the hangers in Fig. 13(c), therefore we would like to enforce grasping on the paper label. The non-rigid object in Fig. 13(d) is not particularly hard to grasp, but we wish to improve success chance by grasping in the middle of the object. Similarly, the white box in Fig. 13(e) is not challenging to grasp, but grasping in the middle improves the chance of success. The dense descriptor representation allows to accurately locate the center for such textureless objects. The sponge in Fig. 13(f) has a cutout in the middle of the packaging where suction grasps will fail, therefore we prefer to grasp towards the top, or bottom. As the towel in Fig. 13(g) is clearly visible in the depth image, suction grasps will often fail on the towel, but not on the label. The plastic cover in Fig. 13(h) does not show up on depth images, therefore we prefer to focus the grasp towards the top of the packaging. Finally, the wet wipe in Fig. 13(i) is not difficult to grasp, but we wish to avoid grasping by the package opening, which might be damaged when using a suction gripper. G.2 Generating Grasp Preference Heatmaps In this section we detail the computation steps of the grasp preference heatmap. The experiment consists of an offline grasp preference annotation phase, and an online autonomous operation phase performing the grasps in the bin. Annotation phase. First an RGB image $I$ showing the objects in the bin is presented. Then the human clicks at pixel locations $\{k_{i}^{j}\}$ corresponding to preferred grasp locations. The descriptor values $d_{j}=f(I;\theta)(k_{i}^{j})$ at these pixel locations are then stored into a keypoint database $\mathcal{D}=\{d_{j}\}$. See Fig. 6(a) for an example image of the objects in the bin in a random configuration. Autonomous operation phase. During autonomous operation, the latest RGB image $I$ taken of the bin is evaluated with the trained network resulting in the descriptor image $I_{d}=f(I;\theta)$. Then, keypoint heatmaps are generated from the database with $h_{j}(u,v)=\exp(-\mbox{dist}\left(I_{d},d_{j}\right)/\eta),~{}\forall d_{j}\in\mathcal{D}$, with $\eta$ as a temperature parameter that controls the width of the heatmap. Finally, the individual keypoint heatmaps are fused into a single heatmap function $h(u,v)=\sum_{j}h_{j}(u,v)/H$, with $H$ as a normalization constant (see Fig. 6(c) for an illustration).

Abelian gauge theories on compact manifolds and the Gribov ambiguity Gerald Kelnhofer Faculty of Physics University of Vienna Boltzmanngasse 5, A-1090 Vienna Austria We study the quantization of abelian gauge theories of principal torus bundles over compact manifolds with and without boundary. It is shown that these gauge theories suffer from a Gribov ambiguity originating in the non-triviality of the bundle of connections whose geometrical structure will be analyzed in detail. Motivated by the stochastic quantization approach we propose a modified functional integral measure on the space of connections that takes the Gribov problem into account. This functional integral measure is used to calculate the partition function, the Green’s functions and the field strength correlating functions in any dimension using the fact that the space of inequivalent connections itself admits the structure of a bundle over a finite dimensional torus. The Green’s functions are shown to be affected by the non-trivial topology, giving rise to non-vanishing vacuum expectation values for the gauge fields. 1. Introduction Functional integral techniques together with the Faddeev-Popov method [1] play a central role in the quantization of Yang-Mills theories. Impressive successes of this method were obtained within perturbation theory. The fundamental object in the quantized pure (non-abelian) Yang-Mills theory is the non-normalized density $$\Xi^{(P)}(A)=vol_{\Cal{A}^{(P)}}\ e^{-S_{inv}(A)},$$ 1.11.11.1 where $vol_{\Cal{A}^{(P)}}$ denotes the (formal) volume form on the space of all connections $\Cal{A}^{(P)}$ of a certain principal $G$-bundle $P$ over $M$. $S_{inv}(A)=\frac{1}{2}\int_{M}tr(F_{A}\wedge\star F_{A})$ is the gauge invariant classical Yang-Mills action, defined by the field strength $F_{A}$ and the Hodge star operator $\star$ with respect to a fixed metric on $M$. The trace $tr$ is taken along the Lie-algebra of the corresponding compact symmetry group $G$. The vacuum expectation value (VEV) of a gauge invariant observable $f\in C^{\infty}(\Cal{A}^{(P)})$ may be defined by $$<f>_{P}=\frac{\int_{\Cal{A}^{(P)}}\ \Xi^{(P)}(A)\cdot f(A)}{\int_{\Cal{A}^{(P)% }}\Xi^{(P)}(A)}.$$ 1.21.21.2 A difficulty arises because the integrands appearing in (1.2) are constant along the orbits of the corresponding gauge group $\Cal{G}^{(P)}$, which have infinite measure. This implies that the non-physical degrees of freedom must be eliminated before the theory can be quantized. According to the Faddeev-Popov approach a unique representative is selected from each $\Cal{G}^{(P)}$-orbit in a smooth way giving rise to a gauge fixing submanifold. The functional integral over the total space $\Cal{A}^{(P)}$ is then restricted to this submanifold by extracting the infinite volume of the gauge group, which is absorbed into an overall normalization constant in the end. As a result the quadratic part of the classical action $S_{inv}$ becomes invertible and the resulting functional integral leads to a consistent perturbative expansion with corresponding Feynman diagrams. However, the treatment of the infinite gauge group volume is not satisfying from a mathematical point of view. Moreover, in the non-perturbative regime it was envisaged very soon that the Faddeev-Popov formulation suffers from so called Gribov ambiguities [2]. Topologically this is related to the fact that $\Cal{A}^{(P)}$ is a non-trivial $\Cal{G}^{(P)}$-bundle over the gauge orbit space $\Cal{M}^{(P)}=\Cal{A}^{(P)}/\Cal{G}^{(P)}$ preventing the definition of a smooth global gauge fixing submanifold [3-6]. Gauge fixing is thus only locally possible and the aim is to find a constructive way to take all relevant gauge fields into account. Not only the non-abelian Yang-Mills theory suffers from this obstruction but even pure Maxwell theory on the four-dimensional torus $\mathbb{T}^{4}$ was shown to be affected by the Gribov problem using general topological arguments [7]. The question whether or how the original Faddeev-Popov approach can be modified appropriately has generated a controversial discussion during the last years. Several proposals to overcome the Gribov problem have been published (see [8] for a recent overview). In the following outline we will focus on formulations for an appropriate functional integral in the continuum theory. One direction follows the original suggestion of Gribov by restricting the functional integral to a submanifold of $\Cal{A}^{(P)}$, where the Gribov problem is absent and all gauge fields are uniquely determined [9,10]. Hence the challenge is to find a systematic way to restrict the Yang-Mills density to this domain of definition and to perform the integration explicitly, see also [11]. A different formulation avoiding the Gribov problem in the Yang-Mills theory has been proposed by some authors in [12,13], where the original functional integral is modified by the introduction of a non-local gauge fixing term by hand. The modified functional integral restricts the domain of integration appropriately, yet in this approach the infinite volume of the gauge group has to be omitted. An alternative way towards the quantization of Yang-Mills theory is to construct a functional integral directly on the gauge orbit space $\Cal{M}^{(P)}$ instead of $\Cal{A}^{(P)}$. The so called ”invariant integration” [14,15] relies methodically on the reduction of an integral of invariant functions over the total space of a finite dimensional principal fibre bundle with compact structure group to an integral over the base manifold of this bundle multiplied with the finite volume of the symmetry group. If applied literally to the Yang-Mills theory, one would encounter once again the problem with the ill-defined volume of the gauge group. Hence the idea is to define the partition function of the gauge theory completely in terms of the resulting functional integral over $\Cal{M}^{(P)}$. However, compared to the affine space $\Cal{A}^{(P)}$ the structure of $\Cal{M}^{(P)}$ is much more complicated so that an explicit computation of this integral over the gauge orbit space is often not possible. In [16] the functional integral has been constructed directly on the gauge orbit space $\Cal{M}^{(P)}$. The basic ingredient is a regularized Brownian motion governed by the Riemannian structure of $\Cal{M}^{(P)}$, which is induced by the kinetic term of the (regularized) Yang-Mills action. A further attempt to be mentioned is [17], where a patching construction for the locally restricted Faddeev-Popov densities - disregarding the infinite volume of the gauge group - has been investigated. In this paper we want to present a functional integral measure on the space of all connections that resolves the Gribov problem and provides for a mathematically reasonable treatment of the gauge degrees of freedom. A novel way to treat these two problems has been proposed some time ago within the stochastic quantization scheme [18,19]. Generally, the stochastic quantization method of Parisi and Wu [20] was introduced as a new method for quantizing field theories. It is based on concepts of non-equilibrium statistical mechanics and provides novel and alternative insights into quantum field theory (see ref. [21], for a comprehensive review and referencing). Let us comment on our proposal in brief: The gauge fields are regarded as stochastic processes with respect to a fictive so-called ”stochastic time”, which are governed by an equivalence class of stochastic differential equation. The notion of equivalence refers to the fact that stochastic correlation functions of gauge invariant observables are well-defined and unique. This equivalence allows to select a distinguished representative [18,19]. The stochastic scheme can thus be recast into a formulation in terms of a normalizable probability density as functional of the gauge fields, which has to satisfy the Fokker-Planck equation [21]. In this respect, the introduction of a damping force along the gauge degrees of freedom regularizing the volume of the gauge group is one of the main aspects of this approach. The strategy in taking care of the Gribov copies is to restrict the domain of the stochastic processes to local coordinate patches in the configuration space $\Cal{A}^{(P)}$, furthermore to construct locally defined equilibrium distributions and finally to paste them together in such a way that the physical relevant objects become independent of the particular way this pasting is provided. Distinguished by its concept the whole field content, even the gauge degrees of freedom, has to be taken into account within the stochastic quantization scheme to permit a probability interpretation. Our aim in this paper is to apply the modified functional integral [19] to abelian gauge theories of connections of principal torus bundles over $n$-dimensional compact manifolds. These theories are proved to suffer from Gribov ambiguities. So we generalize the results of [7] to a wider class of manifolds. The motivation to study this theory is twofold: First of all we want to get a more profound understanding of our new concept for the functional integral by analyzing a simple but non-trivial field theoretical model. As a consequence the interrelation between the occurrence of the Gribov problem and the necessity for a regularizing measure for the gauge group can be elucidated. However, besides serving as a laboratory for the new concept the second reason is that abelian gauge theories gained a strong interest during the last years by its own. Examples are the analysis of two dimensional gauge theories, the description of the fractional quantum Hall effect (see [22] for a comprehensive review) and questions related to the duality in field theory on three and four dimensional manifolds with and without boundary [23-26]. The paper is structured as follows: In section 2 the concept of the modified functional integral measure will be briefly reviewed. The abelian field theoretical model which we are going to consider is introduced in section 3. Sections 4 and 5 are devoted to the analysis of the abelian gauge theory on closed compact manifolds respectively on compact manifolds with a boundary. Since the calculation of the modified functional integral relies on the knowledge of the bundle geometry of the space of gauge fields, we will analyze its structure in detail in subsection 4.1 for closed manifolds and in subsection 5.1 for manifolds with a non-empty boundary. In both cases, the bundle of connections exhibits a non-trivial structure which implies that it is impossible to fix the gauge globally. It should be remarked that on closed manifolds the topology of the gauge orbit space of $\mathbb{T}^{1}$-connections has been studied for several years, often in low dimensions. In this respect some of our results regarding the structure of the gauge group and the topology of the gauge orbit space have already been displayed using different methods (see e.g. [27]). However, to our knowledge an explicit construction of the various bundle structures in terms of local sections has not appeared in the literature so far. Our geometrical results will then be used to compute the partition function, the vacuum expectation value of gauge invariant functions and the Green’s functions of the gauge fields for closed manifolds in the subsections 4.2 and 4.3. Analogous results will be displayed in subsection 5.2 for manifolds with a non-empty boundary. Furthermore our results are compared with those obtained by the conventional covariant quantization schemes. In section 6 the concept of the modified functional integral and its consequences are illustrated with two examples, namely the abelian gauge theory on the circle and the abelian gauge theory on two-dimensional closed manifolds. The paper concludes with a summary of the main results in section 7. 2. A modified functional integral measure for gauge theories In the present publication we want to shed some new light onto the question of how a reasonable partition function can be formulated for a gauge theory suffering from Gribov ambiguities. In order to ascribe a probabilistic interpretation to the Yang-Mills density according to the stochastic quantization scheme, the formal measure in (1.1) has to be modified appropriately so that it becomes integrable. The strategy is to introduce a damping force which regularizes the gauge degrees of freedom. This will be provided by a so called gauge fixing function $S_{gf}$ on the gauge group $\Cal{G}^{(P)}$, which is assumed to render the volume of the gauge group $$Vol(\Cal{G}^{(P)};S_{gf}):=\int_{\Cal{G}^{(P)}}\ vol_{\Cal{G}^{(P)}}\ e^{-S_{% gf}}$$ 2.12.12.1 finite. Here $vol_{\Cal{G}^{(P)}}$ is the (formal) left-invariant volume form on $\Cal{G}^{(P)}$. In the following we shall consider only gauge fixing functions $S_{gf}$, which satisfy $Vol(\Cal{G}^{(P)};S_{gf})=1$. The (non-abelian) gauge group $\Cal{G}^{(P)}$ acts freely on $\Cal{A}^{(P)}$, denoted by $(A,g)\mapsto A^{g}$, so that $\Cal{A}^{(P)}$ admits the structure of a principal $\Cal{G}^{(P)}$-bundle over the gauge orbit space $\Cal{M}^{(P)}:=\Cal{A}^{(P)}/\Cal{G}^{(P)}$ with projection $\pi_{\Cal{A}^{(P)}}$. (In fact, the gauge group has to be restricted appropriately to give a free action [4]). The theory is said to possess a Gribov problem if this bundle is non-trivial. There exists a family of local trivializations $(U_{a},\varphi_{a})$ given by $U_{a}\times\Cal{G}^{(P)}@>\varphi_{a}>>\pi_{\Cal{A}^{(P)}}^{-1}(U_{a})$ where $\{U_{a}\}$ is a locally finite open cover of $\Cal{M}^{(P)}$ and the local diffeomorphisms $\varphi_{a}(\pi_{\Cal{A}}(A),g)=\sigma_{a}(\pi_{\Cal{A}^{(P)}}(A))^{g}$ are generated by a family of local sections $U_{a}@>\sigma_{a}>>\pi_{\Cal{A}}^{-1}(U_{a})$. For the inverse we write $\varphi_{a}^{-1}(A)=(\pi_{\Cal{A}^{(P)}}(A),\omega_{a}(A))$. We propose that the quantization of the Yang-Mills theory is described by the following local densities on $\Cal{A}^{(P)}$ $$\Xi_{a}^{(P)}=vol_{\Cal{A}^{(P)}}\mid_{\pi_{\Cal{A}^{(P)}}^{-1}(U_{a})}\ e^{-S% _{inv}-\omega_{a}^{\ast}S_{gf}}.$$ 2.22.22.2 which - if normalized - appear as equilibrium solutions of the Fokker-Planck operator on each open set $\pi_{\Cal{A}^{(P)}}^{-1}(U_{a})\subseteq\Cal{A}^{(P)}$. Due to the Gribov ambiguity these local partition functions must be pasted together using a partition of unity on the gauge orbit space. Definition 2.1 Let $\{p_{a}\}$ denote a partition of unity on $\Cal{M}^{(P)}$ subordinate to the open cover $\{U_{a}\}$. We define a global (non-perturbative) Yang-Mills density $\Xi^{(P)}$ by $$\Xi^{(P)}:=\sum_{a}(\pi_{\Cal{A}^{(P)}}^{\ast}p_{a})\cdot\Xi_{a}^{(P)}.$$ 2.32.32.3 Accordingly, the vacuum expectation value (VEV) of a gauge invariant function $f$ is given by $$<f>_{P}=\frac{I^{(P)}(f)}{I^{(P)}(1)},\qquad I^{(P)}(f)=\int_{\Cal{A}^{(P)}}% \Xi^{(P)}\cdot f.$$ 2.42.42.4 For the partition function we take $Z^{(P)}:=I^{(P)}(1)$. It has been shown in [19] that based on this constructive procedure the VEV of gauge invariant observables This idea to patch the local Yang-Mills densities together to obtain a global functional integral in the field space takes up a suggestion raised by Singer [3] in his seminal paper. For some applications it is necessary to consider the total configuration space, which consists of disconnected components $\Cal{A}^{(P)}$ labelled by the equivalence class of bundles $P$. The set of all $\mathbb{T}^{N}$ connections over $M$, denoted by $\Cal{A}^{(M)}$, is given as disjoint union $$\Cal{A}^{(M)}=\bigsqcup_{(P)}\Cal{A}^{(P)}.$$ 2.52.52.5 Correspondingly, the partition function and the VEV of gauge invariant observables are represented by a sum over equivalence classes of principal bundles $P$, namely $$Z=\sum\limits_{(P)}Z^{(P)},\qquad<f>=\frac{\sum\limits_{P}I^{(P)}(f)}{\sum% \limits_{P}I^{(P)}(1)}.$$ 2.62.62.6 3. The geometrical setting for the abelian gauge theory In this section the abelian field theoretical model which we are going to consider in this paper is introduced. As we focus on compact abelian structure groups only, we can restrict ourselves to the $N$-dimensional torus $\mathbb{T}^{N}$ as relevant symmetry group. We shall begin with a brief review of torus bundles: Let $M$ be a $n$-dimensional connected, oriented and compact manifold with a fixed Riemannian metric. Let us now consider an arbitrary principal $\mathbb{T}^{N}$-bundle $P(M,\pi_{P},\mathbb{T}^{N})$ over $M$ with projection $\pi_{P}$. The group structure on $\mathbb{T}^{N}$ is provided by point-wise multiplication and its Lie algebra $\mathfrak{t}^{N}$ is given by $\mathfrak{t}^{N}=\sqrt{-1}\ \mathbb{R}^{N}$. A $L^{2}$ inner product can be defined on the complex $\Omega^{k}(M;\mathfrak{t}^{N})$ of $k$-forms on $M$ by $$<\upsilon_{1},\upsilon_{2}>=\sum_{\alpha=1}^{N}\int_{M}\ \upsilon_{1}^{\alpha}% \wedge\star\bar{\upsilon}_{2}^{\alpha},$$ 3.13.13.1 where $\star$ is the Hodge star operator with respect to the given metric on $M$, satisfying $\star^{2}=(-1)^{k(n-k)}$ and $\bar{\upsilon}^{\alpha}$ denotes the complex conjugate of $\upsilon=(\upsilon^{\alpha})_{\alpha=1}^{N}\in\Omega^{k}(M;\mathfrak{t}^{N})$. The $C^{\infty}$-Hilbert manifold of all connections on $P$ of a certain Sobolev class will be denoted by $\Cal{A}^{(P)}$. The gauge group $\Cal{G}^{(M)}$ is defined as the group of vertical bundle automorphisms on $P$ and can be identified with the Hilbert Lie-Group $C^{\infty}(M,\mathbb{T}^{N})$ of differentiable maps between $M$ and $\mathbb{T}^{N}$. Finally, its Lie-algebra $\mathfrak{G}^{(M)}$ is given by $\mathfrak{G}^{(M)}=C^{\infty}(M;\mathfrak{t}^{N})$. Under an arbitrary gauge transformation $g\in\Cal{G}^{(M)}$, the gauge fields transform according to $$\Cal{A}\mapsto A^{g}=A+(\pi_{P}^{\ast}g)^{\ast}\vartheta\qquad g\in\Cal{G}^{(M% )},$$ 3.23.23.2 where $\vartheta\in\Omega^{1}(\mathbb{T}^{N};\mathfrak{t}^{N})$ is the Maurer Cartan form on $\mathbb{T}^{N}$. (For notational convenience we shall not distinguish between $\pi_{P}^{\ast}g$ and $g$.) How can torus bundles be classified? The topological type of $\mathbb{T}^{N}$ torus bundles is expressed by the first Cech-cohomology $H^{1}(M;sh_{M}(\mathbb{T}^{N}))$, where $sh_{M}(\mathbb{T}^{N})$ denotes the sheaf of all $\mathbb{T}^{N}$ valued differentiable functions on $M$. The sheaves of $\mathbb{Z}^{N}$ and $\mathbb{R}^{N}$ valued differentiable functions on $M$, which are denoted by $sh_{M}(\mathbb{Z}^{N})$, $sh_{M}(\mathbb{R}^{N})$, respectively, fit into the following exact sequence of sheaves $$0\rightarrow sh_{M}(\mathbb{Z}^{N})\rightarrow sh_{M}(\mathbb{R}^{N})% \rightarrow sh_{M}(\mathbb{T}^{N})\rightarrow 1,$$ 3.33.33.3 which induces a corresponding long exact sequence in cohomology $$\dots\rightarrow\hat{H}^{1}(M,sh_{M}(\mathbb{R}^{N}))\rightarrow\hat{H}^{1}(M,% sh_{M}(\mathbb{T}^{N}))\rightarrow\hat{H}^{2}(M,sh_{M}(\mathbb{Z}^{N}))% \rightarrow\hat{H}^{2}(M,sh_{M}(\mathbb{R}^{N}))\rightarrow\ldots\ $$ 3.43.43.4 Since the sheaf $sh_{M}(\mathbb{R}^{N})$ is fine, the set $\mathfrak{P}[M,\mathbb{T}^{N}]$ of equivalence classes of principal $\mathbb{T}^{N}$ bundles over $M$ is given by $$\mathfrak{P}[M;\mathbb{T}^{N}]=\hat{H}^{1}(M,sh_{M}(\mathbb{T}^{N}))=H^{2}(M,% \mathbb{Z}^{N})=\bigoplus\limits_{i=1}^{N}H^{2}(M,\mathbb{Z}),$$ 3.53.53.5 so that any principal $\mathbb{T}^{N}$-bundle is classified by an integer cohomology class $c\in H^{2}(M,\mathbb{Z}^{N})$. Accordingly, $c=c_{1}\oplus\cdots\oplus c_{N}$, where each component $c_{\alpha}\in H^{2}(M,\mathbb{Z})$ determines a principal circle bundle $P^{\alpha}(M,\mathbb{T}^{1})$ over $M$ having $c_{\alpha}$ as its first Chern class. Thus $P$ can be equivalently viewed as $N$-fold fiber product $P^{1}\times_{M}\times\cdots\times_{M}P^{N}$ over $M$. Let $F_{A}=(F_{A}^{\alpha})_{\alpha=1}^{N}\in\Omega^{2}(M;\mathfrak{t}^{N})$ denote the field strength of the $\mathbb{T}^{N}$-connection $A$ on $P$. Each component $F_{A}^{\alpha}$ can be regarded as field strength of the $\alpha$-th principal $\mathbb{T}^{1}$-bundle $P^{\alpha}$ in the fiber product $P$. The classical gauge invariant action is defined by $$S_{inv}(A)=\frac{1}{2}\sum_{\alpha,\beta=1}^{N}\int_{M}\lambda_{\alpha\beta}F_% {A}^{\alpha}\wedge\star\bar{F}_{A}^{\beta},$$ 3.63.63.6 where $(\lambda_{\alpha\beta})_{\alpha,\beta=1}^{N}$ is a symmetric positive definite matrix with $\det\lambda=1$. This matrix determines the relative couplings between the components $A^{\alpha}$ of the $\mathbb{T}^{N}$-gauge fields $A$ on $P$. From a physical point of view some extensions of (3.6) are of particular interest: If the conventional Maxwell action is extended by a theta term the resulting partition function was shown to exhibit a non-trivial transformation behavior under electric-magnetic duality [23-26]. On the other hand, if the action (3.6) is extended by an additional Chern-Simons term in a three dimensional space-time, this model allows for a mathematical description of the fractional quantum Hall effect. The integer resulting from the evaluation of the corresponding Chern classes $c^{\alpha}$ ($\alpha=1,\ldots N$) along the 2-dimensional space admits the interpretation of the total number of electrons in the $\alpha$-th Landau level [22]. Provided by the matrix of couplings there is a second $L^{2}$ inner product on the complex $\Omega(M;\mathfrak{t}^{N})$ given by $$<\upsilon_{1},\upsilon_{2}>_{\lambda}=\sum_{\alpha,\beta=1}^{N}\int_{M}\lambda% _{\alpha\beta}\upsilon_{1}^{\alpha}\wedge\star\bar{\upsilon}_{2}^{\beta},$$ 3.73.73.7 where $\upsilon=(\upsilon^{1},\ldots,\upsilon^{N})\in\Omega^{k}(M;\mathfrak{t}^{N})$. 4. Abelian gauge theories on closed manifolds In this chapter we want to construct the modified functional integral for the abelian gauge theory on closed manifolds. We begin with an analysis of the geometrical properties of the gauge group. Based on these considerations we will then derive two results regarding the bundle structure of the space of connections. 4.1. The geometry of the abelian gauge fields The action (3.2) of the gauge group $\Cal{G}^{(M)}$ is not free possessing the non-trivial isotropy group $\mathbb{T}^{N}$, namely the subgroup of constant gauge transformations. In order to get a free action let us now choose an arbitrary but fixed reference point $x_{0}\in M$. By restricting the gauge group to the subgroup $\Cal{G}_{\ast}^{(M)}=\{g\in\Cal{G}|g(x_{0})=1\}$ which itself is diffeomorphic to $\Cal{G}^{(M)}/\mathbb{T}^{N}$ by $g\rightarrow g\cdot g(x_{0})$, we finally obtain a free action of $\Cal{G}_{\ast}^{(M)}$ on $\Cal{A}^{(P)}$. This gives rise to a smooth gauge orbit space $\Cal{M}_{\ast}^{(P)}=\Cal{A}^{(P)}/\Cal{G}_{\ast}^{(M)}$, which has to be regarded as the true configuration space of the theory. For the Maxwell theory ($N=1$) some of the results regarding the gauge group topology have been considered in [28]. Let us denote by $Z_{k}(M;\mathbb{Z})$ the subcomplex of all closed smooth singular $k$-cycles on $M$. We define the abelian group $$\Omega_{\mathbb{Z}}^{k}(M,\mathbb{R}^{N})=\{\alpha\in\Omega^{k}(M;\mathbb{R}^{% N})|\quad d\alpha=0,\quad\int_{\gamma}\alpha\in\mathbb{Z}^{N}\quad\forall% \gamma\in Z_{k}(M;\mathbb{Z})\}$$ 4.1.14.1.14.1.1 of all closed $\mathbb{R}^{N}$-valued differential $k$-forms with integer periods and denote by $H_{\mathbb{Z}}^{k}(M;\mathbb{R}^{N})$ the corresponding cohomology group. The question of how the subgroup of constant gauge transformations is related to the gauge group is answered by the following statement: Proposition 4.1 The following sequence of abelian groups is split exact $$0\rightarrow\mathbb{T}^{N}\rightarrow\Cal{G}^{(M)}@>\kappa_{(M)}>>\Omega_{% \mathbb{Z}}^{1}(M,\mathbb{R}^{N})\rightarrow 0\qquad\kappa_{(M)}(g)=\frac{1}{2% \pi\sqrt{-1}}g^{\ast}\vartheta.$$ 4.1.24.1.24.1.2 Demonstration Proof The split is given by the isomorphism of abelian groups $$\begin{split}&\displaystyle\tilde{\kappa}_{(M)}\colon\Omega_{\mathbb{Z}}^{1}(M% ,\mathbb{R}^{N})\times\mathbb{T}^{N}\rightarrow\Cal{G}^{(M)}\\ &\displaystyle\tilde{\kappa}_{(M)}(\alpha,t)(x)=t\cdot\exp{2\pi\sqrt{-1}\int_{% c_{x}}\alpha}=t\cdot\exp{2\pi\sqrt{-1}\int_{0}^{1}\ c_{x}^{\ast}\alpha}\\ &\displaystyle\tilde{\kappa}_{(M)}^{-1}(g)=(\kappa_{(M)}(g),g(x_{0})),\end{split}$$ 4.1.34.1.34.1.3 where $c_{x}\colon[0,1]\rightarrow M$ is a path in $M$ connecting $x_{0}$ with $x$. That this integral is already well-defined can be seen by choosing a different path $c_{x}^{\prime}$ connecting $x_{0}$ and $x$. Since the combined path $c_{x}^{\prime}\diamond c_{x}$ can be regarded as element in $Z_{1}(M;\mathbb{Z})$. The integration of any element in $\Omega_{\mathbb{Z}}^{1}(M,\mathbb{R}^{N})$ along this cycle gives an integer. ∎ The co-differential $d_{k}^{\ast}=(-1)^{n(k+1)+1}\star d_{n-k}\star\colon\Omega^{k}(M;\mathbb{R})% \rightarrow\Omega^{k-1}(M;\mathbb{R})$ gives rise to the Laplacian operator $\Delta_{k}=d_{k+1}^{\ast}d_{k}+d_{k-1}d_{k}^{\ast}$. Let $Harm^{k}(M)^{\bot}$ denote the orthogonal complement of the space of harmonic $k$-forms $Harm^{k}(M)$ with values in $\mathbb{R}$, then we can define the Green´s operator [29] $$G_{k}\colon\Omega^{k}(M;\mathbb{R})\rightarrow Harm^{k}(M)^{\bot},\quad G_{k}=% (\Delta_{k}|_{Harm^{k}(M)^{\bot}})^{-1}\circ\Pi^{Harm^{k}(M)^{\bot}},$$ 4.1.44.1.44.1.4 where $\Pi^{Harm^{k}(M)^{\bot}}$ is the projection of $\Omega^{k}(M;\mathbb{R})$ onto $Harm^{k}(M)^{\bot}$. By construction $\Delta_{k}\circ G_{k}=G_{k}\circ\Delta_{k}=\Pi^{Harm^{k}(M)^{\bot}}$. It is evident that the Lie algebra $\mathfrak{G}_{\ast}^{(M)}$ of the restricted gauge group $\Cal{G}_{\ast}^{(M)}$ consists of those $C^{\infty}$ maps from $M$ to $\mathfrak{t}^{N}$, which vanishes in $x_{0}$. The next result shows that the pointed gauge group $\Cal{G}_{\ast}^{(M)}$ is not connected. Proposition 4.2 The following sequence of abelian groups is split exact $$0\rightarrow\mathfrak{G}_{\ast}^{(M)}@>\exp>>\Cal{G}_{\ast}^{(M)}@>\kappa_{(M)% }^{\prime}>>H_{\mathbb{Z}}^{1}(M;\mathbb{R}^{N})\rightarrow 0,$$ 4.1.54.1.54.1.5 where $\kappa_{(M)}^{\prime}(g)=[\kappa_{(M)}(g)]$. Demonstration Proof It is easy to see that the exponential function $\exp$ is indeed a monomorphism. A split of (4.1.5) is given by the following isomorphism of abelian groups $$\begin{split}&\displaystyle\hat{\kappa}_{(M)}\colon H_{\mathbb{Z}}^{1}(M;% \mathbb{R}^{N})\times\mathfrak{G}_{\ast}^{(M)}\rightarrow\Cal{G}_{\ast}^{(M)}% \\ &\displaystyle\hat{\kappa}_{(M)}([\alpha],\xi)(x)=\exp{(2\pi\sqrt{-1}\int_{c_{% x}}\Pi^{Harm^{1}(M)}(\alpha))}\cdot\exp\xi(x)\\ &\displaystyle\hat{\kappa}_{(M)}^{-1}(g)=(\kappa_{(M)}^{\prime}(g),G_{0}d_{1}^% {\ast}g^{\ast}\vartheta-(G_{0}d_{1}^{\ast}g^{\ast}\vartheta)(x_{0})).\end{split}$$ 4.1.64.1.64.1.6 ∎ Now we will prove that even an abelian gauge theory would admit a Gribov ambiguity if the space time manifold $M$ is topologically non-trivial. This generalizes the previous result [7], where the existence of Gribov ambiguities has been shown for Maxwell theory on the four-torus. Theorem 4.3 $\Cal{A}^{(P)}$ is a flat principal bundle over $\Cal{M}_{\ast}^{(P)}$ with structure group $\Cal{G}_{\ast}^{(M)}$ and projection $\pi_{\Cal{A}^{(P)}}$. This bundle is trivializable if $H^{1}(M;\mathbb{Z})=0$. Demonstration Proof We are going to construct a bundle atlas explicitly. For this we have to define an open cover of the gauge orbit space and a family of local sections. For any fixed $l,N\in\mathbb{N}$ we consider the exact sequence of abelian groups $$0\rightarrow\mathbb{Z}^{lN}\rightarrow\mathbb{R}^{lN}@>\exp{2\pi\sqrt{-1}(.)}>% >\mathbb{T}^{lN}\rightarrow 0,$$ 4.1.74.1.74.1.7 which gives the universal covering of the $lN$-dimensional torus $\mathbb{T}^{lN}$. Let us view $\mathbb{T}^{lN}$ as the product $$\mathbb{T}^{lN}=\underbrace{\mathbb{T}^{N}\times\cdots\times\mathbb{T}^{N}}^{l% }=\underbrace{(\underbrace{\mathbb{T}^{1}\times\cdots\times\mathbb{T}^{1}}^{N}% )\times\cdots\times(\underbrace{\mathbb{T}^{1}\times\cdots\times\mathbb{T}^{1}% }^{N})}^{l}.$$ 4.1.84.1.84.1.8 We introduce an open cover $\Cal{V}$ of $\mathbb{T}^{lN}$ by the following family of open sets $$\Cal{V}=\{V_{a}|\quad a:=(a_{1},\ldots,a_{j},\ldots,a_{l}),\quad a_{j}:=(a_{j1% },\ldots,a_{j\alpha},\ldots,a_{jN}),a_{j\alpha}\in\mathbb{Z}_{2}=\{1,2\}\},$$ 4.1.94.1.94.1.9 where $V_{a}=V_{a_{1}}\times\cdots\times V_{a_{j}}\times\cdots\times V_{a_{l}}$ is a open set in $\mathbb{T}^{lN}$. Each $V_{a_{j}}$ is itself the product of open sets $V_{a_{j}}=V_{a_{j1}}\times\cdots\times V_{a_{j\alpha}}\times\cdots\times V_{a_% {jN}}$ in the $k$-th $N$-dimensional torus $\mathbb{T}^{N}$ within (4.1.8). Here $V_{1}=\mathbb{T}^{1}\backslash\{northernpole\}$ for $a_{j\alpha}=1$ and $V_{2}=\mathbb{T}^{1}\backslash\{southernpole\}$ for $a_{j\alpha}=2$ provide an open cover of each 1-torus $\mathbb{T}^{1}$. Let us choose the following two local sections of the universal covering $\mathbb{R}^{1}\rightarrow\mathbb{T}^{1}$ $$\displaystyle s_{a_{j\alpha}}(z)=\{{\frac{1}{2\pi}\arccos|_{(0,\pi]}\Re z\atop% \frac{1}{2\pi}\arccos|_{[\pi,2\pi)}\Re z}\qquad{\Im z\geq 0\atop\Im z<0},\quad a% _{j\alpha}=1$$ 4.1.104.1.104.1.10 $$\displaystyle s_{a_{j\alpha}}(z)=\{{\frac{1}{2\pi}\arccos|_{(\pi,2\pi]}\Re z% \atop\frac{1}{2\pi}\arccos|_{[2\pi,3\pi)}\Re z}\qquad{\Im z\leq 0\atop\Im z>0}% \quad a_{j\alpha}=2,$$ where $z=\Re z+\sqrt{-1}\Im z\in\mathbb{T}^{1}$. The corresponding transition functions $g_{a_{j\alpha}a_{j\alpha}^{\prime}}^{\mathbb{T}^{1}}\colon V_{a_{j\alpha}}\cap V% _{a_{j\alpha}^{\prime}}\rightarrow\mathbb{Z}$ are given by $$s_{a_{j\alpha}^{\prime}}(z_{j\alpha})=s_{a_{j\alpha}}(z_{j\alpha})+g_{a_{j% \alpha}a_{j\alpha}^{\prime}}^{\mathbb{T}^{1}}(z_{j\alpha}).$$ 4.1.114.1.114.1.11 Evidently a family of $2^{lN}$ local sections $s_{a}\colon V_{a}\subset\mathbb{T}^{lN}\rightarrow\mathbb{R}^{lN}$ can be induced by $$s_{a}=(s_{a_{1}},\cdots,s_{a_{l}})=\left((s_{a_{11}},\cdots,s_{a_{1N}}),\cdots% ,(s_{a_{l1}},\cdots,s_{a_{lN}})\right),$$ 4.1.124.1.124.1.12 where on $V_{a}\cap V_{a^{\prime}}$ the corresponding sections $s_{a}$ and $s_{a^{\prime}}$ are related by the locally constant transition functions $g_{aa^{\prime}}^{\mathbb{T}^{Nl}}\colon V_{a}\cap V_{a^{\prime}}\rightarrow% \mathbb{Z}^{Nl}$ $$\multline\displaystyle g_{aa^{\prime}}^{\mathbb{T}^{Nl}}(\vec{z}_{1},\ldots,% \vec{z}_{l})=(g_{a_{1}a_{1}^{\prime}}^{\mathbb{T}^{N}}(\vec{z}_{1}),\ldots,g_{% a_{j}a_{j}^{\prime}}^{\mathbb{T}^{N}}(\vec{z}_{j}),\ldots,g_{a_{l}a_{l}^{% \prime}}^{\mathbb{T}^{N}}(\vec{z}_{l}))=\\ \displaystyle=((g_{a_{11}a_{11}^{\prime}}^{\mathbb{T}^{1}}(z_{11}),\ldots,g_{a% _{1N}a_{1N}^{\prime}}^{\mathbb{T}^{1}}(z_{1N})),\ldots,(g_{a_{l1}a_{l1}^{% \prime}}^{\mathbb{T}^{1}}(z_{l1}),\ldots,g_{a_{lN}a_{lN}^{\prime}}^{\mathbb{T}% ^{1}}(z_{lN}))),\endmultline\displaystyle g_{aa^{\prime}}^{\mathbb{T}^{Nl}}(% \vec{z}_{1},\ldots,\vec{z}_{l})=(g_{a_{1}a_{1}^{\prime}}^{\mathbb{T}^{N}}(\vec% {z}_{1}),\ldots,g_{a_{j}a_{j}^{\prime}}^{\mathbb{T}^{N}}(\vec{z}_{j}),\ldots,g% _{a_{l}a_{l}^{\prime}}^{\mathbb{T}^{N}}(\vec{z}_{l}))=\\ \displaystyle=((g_{a_{11}a_{11}^{\prime}}^{\mathbb{T}^{1}}(z_{11}),\ldots,g_{a% _{1N}a_{1N}^{\prime}}^{\mathbb{T}^{1}}(z_{1N})),\ldots,(g_{a_{l1}a_{l1}^{% \prime}}^{\mathbb{T}^{1}}(z_{l1}),\ldots,g_{a_{lN}a_{lN}^{\prime}}^{\mathbb{T}% ^{1}}(z_{lN}))),$$ 4.1.134.1.134.1.13 for $\vec{z}_{j}=(z_{j1},\ldots,z_{jN})\in\mathbb{T}^{N}$ with $j=1,\ldots,l$. These local sections will be the building blocks for the construction of a bundle atlas. Let $Harm_{\mathbb{Z}}^{k}(M;\mathbb{R})$ denote the abelian group of harmonic $k$-forms with integer periods and let $D_{n-1}\colon H^{n-1}(M;\mathbb{Z})\rightarrow H_{1}(M;\mathbb{Z})$, $D_{n-1}(\nu)=\nu\cap[M]$ be the Poincaré duality isomorphism [30]. Here $\cap$ is the cap product and $[M]$ denotes the fundamental cycle. Since the homology of $M$ is finitely generated with rank $b_{1}$ (the first Betti number of $M$) we shall choose a set of 1-cycles $\gamma_{i}\in Z_{1}(M,\mathbb{Z})$, $i=1,\ldots,b_{1}$, whose homology classes $[\gamma_{i}]$ provides a Betti basis thus generating the free part $H_{1}(M;\mathbb{Z})/TorH_{1}(M;\mathbb{Z})$ in $H_{1}(M;\mathbb{Z})$. Here $TorH_{1}(M;\mathbb{Z})$ denotes the torsion part of the first homology group. Then $D_{n-1}^{-1}([\gamma_{i}])$ provides a basis for cohomology, from which a basis of harmonic forms $(\rho_{i}^{(n-1)})_{i=1}^{b_{n-1}}\in Harm_{\mathbb{Z}}^{n-1}(M;\mathbb{R})$ can be selected according to the following isomorphisms $$H^{n-1}(M;\mathbb{Z})/TorH^{n-1}(M;\mathbb{Z})\cong H_{\mathbb{Z}}^{n-1}(M;% \mathbb{R})\cong Harm_{\mathbb{Z}}^{n-1}(M;\mathbb{R}).$$ 4.1.144.1.144.1.14 Using the Poincaré duality and the Universal Coefficient Theorem it follows that the product $$\begin{split}\displaystyle H^{1}(M;\mathbb{Z})/TorH^{1}(M;\mathbb{Z})&% \displaystyle\times H^{n-1}(M;\mathbb{Z})/TorH^{n-1}(M;\mathbb{Z})\rightarrow% \mathbb{Z}\\ \displaystyle(\mu,\nu)&\displaystyle\mapsto<\mu,D_{n-1}(\nu)>=<\mu\cup\nu,[M]>% ,\end{split}$$ 4.1.154.1.154.1.15 gives a perfect pairing [30], where $<,>$ denotes the evaluation in cohomology. We remark that $TorH^{1}(M;\mathbb{Z})=0$. A basis $(\rho_{i}^{(1)})_{i=1}^{b_{1}}\in Harm_{\mathbb{Z}}^{1}(M;\mathbb{R})$ can be adjusted in such a way so that $$\int_{\gamma_{j}}\rho_{i}^{(1)}=\int_{M}\ \rho_{i}^{(1)}\wedge\rho_{j}^{(n-1)}% =\delta_{ij}.$$ 4.1.164.1.164.1.16 Hence $\int_{\gamma_{j}}\alpha=\int_{M}\alpha\wedge\rho_{j}^{(n-1)}$ holds for any $[\alpha]\in H^{1}(M;\mathbb{R})$. On $Harm^{1}(M;\mathbb{R})$ there exists an induced metric $$h_{jk}=<\rho_{j}^{(1)},\rho_{k}^{(1)}>.$$ 4.1.174.1.174.1.17 For any choice of an arbitrary but fixed background gauge field $A_{0}\in\Cal{A}^{(P)}$ there exists a smooth surjective map $\pi_{\Cal{M}_{\ast}^{(P)}}^{A_{0}}\colon\Cal{M}_{\ast}^{(P)}\rightarrow\mathbb% {T}^{b_{1}N}$ defined by $$\pi_{\Cal{M}_{\ast}^{(P)}}^{A_{0}}([A])=(e^{\int_{M}(A-A_{0})\wedge\rho_{1}^{(% n-1)}},\ldots,e^{\int_{M}(A-A_{0})\wedge\rho_{b_{1}}^{(n-1)}}),$$ 4.1.184.1.184.1.18 where its components can be rewritten in terms of the inner product (3.1), namely $$\int_{M}(A-A_{0})\wedge\rho_{j}^{(n-1)}=(-1)^{n}<A-A_{0},\star\rho_{j}^{(n-1)}>.$$ 4.1.194.1.194.1.19 The family of open sets $U_{a}^{A_{0}}=(\pi_{\Cal{M}_{\ast}^{(P)}}^{A_{0}})^{-1}(V_{a})$ provides a finite open cover $\Cal{U}^{A_{0}}=\{U_{a}^{A_{0}}\}$ of the infinite dimensional manifold $\Cal{M}_{\ast}^{(P)}$. Now we can construct a bundle atlas from the family of local trivializations $\varphi_{a}^{A_{0}}:U_{a}^{A_{0}}\times\Cal{G}_{\ast}^{(M)}\rightarrow(\pi_{% \Cal{A}^{(P)}})^{-1}(U_{a}^{A_{0}})$, $\varphi_{a}^{A_{0}}([A],g)=A^{(\omega_{a}^{A_{0}}(A))^{-1}g}$, and $(\varphi_{a}^{A_{0}})^{-1}(A)=(\pi_{\Cal{A}^{(P)}}(A),\omega_{a}^{A_{0}}(A))$, where $$\begin{split}&\displaystyle\omega_{a}^{A_{0}}\colon\pi_{\Cal{A}^{(P)}}^{-1}(U_% {a}^{A_{0}})\rightarrow\Cal{G}_{\ast}^{(M)}\\ &\displaystyle\omega_{a}^{A_{0}}(A)=\hat{\kappa}_{M}([\sum_{j=1}^{b_{1}}% \epsilon_{a_{j}}(A)\rho_{k}^{(1)}],\exp{G_{0}d_{1}^{\ast}(A-A_{0})}\cdot\exp{G% _{0}d_{1}^{\ast}(A-A_{0})(x_{0})}),\\ &\displaystyle\epsilon_{a_{j}}^{A_{0}}=(\epsilon_{a_{j1}}^{A_{0}},\ldots,% \epsilon_{a_{j\alpha}}^{A_{0}},\ldots,\epsilon_{a_{jN}}^{A_{0}})\colon\pi_{% \Cal{A}^{(P)}}^{-1}(U_{a})\rightarrow\mathbb{Z}^{N}\\ &\displaystyle\epsilon_{a_{j\alpha}}^{A_{0}}(A)=\frac{1}{2\pi\sqrt{-1}}\int_{M% }(A^{\alpha}-A_{0}^{\alpha})\wedge\rho_{j}^{(n-1)}-s_{a_{j\alpha}}(\exp{\int_{% M}(A^{\alpha}-A_{0}^{\alpha})\wedge\rho_{j}^{(n-1)}}),\end{split}$$ 4.1.204.1.204.1.20 for $\alpha=1,\ldots,N$. To verify that (4.1.20) indeed gives a local trivialization of the bundle, we recognize that $\frac{1}{2\pi\sqrt{-1}}\int_{\gamma_{j}}\ g^{\ast}\vartheta=:m_{j}\in\mathbb{Z% }^{N}$. With respect to the basis $(\rho_{j}^{(1)})_{j=1}^{b_{1}}$, the orthogonal projector onto $Harm^{1}(M)$ reads $$\Pi^{Harm^{1}(M)}(\alpha)=\sum_{j,k=1}^{b_{1}}h_{jk}^{-1}<\alpha,\rho_{j}^{(1)% }>\rho_{k}^{(1)},\ \forall\alpha\in\Omega^{1}(M;\mathfrak{t}^{N}).$$ 4.1.214.1.214.1.21 From $\epsilon_{a_{j}}^{A_{0}}(A^{g})=\epsilon_{a_{j}}^{A_{0}}(A)+m_{j}$ and $\Pi^{Harm^{1}(M)}(g^{\ast}\vartheta)=2\pi\sqrt{-1}\sum_{j=1}^{b_{1}}m_{j}\rho_% {j}^{(1)}$ one gets $\omega_{a}^{A_{0}}(A^{g})=\omega_{a}^{A_{0}}(A)g$. According to the transition functions $\varphi_{aa^{\prime}}^{A_{0}}\colon U_{a}^{A_{0}}\cap U_{a^{\prime}}^{A_{0}}% \rightarrow\Cal{G}_{\ast}^{(M)}$, $$\varphi_{aa^{\prime}}^{A_{0}}([A])=\hat{\kappa}_{(M)}([\sum_{j=1}^{b_{1}}g_{a_% {j}a_{j}^{\prime}}^{\mathbb{T}^{N}}(e^{\int_{M}(A-A_{0})\wedge\rho_{j}^{(n-1)}% })\rho_{j}^{(1)}],0)$$ 4.1.224.1.224.1.22 one concludes that the bundle is trivializable if $H^{1}(M;\mathbb{Z})=0$. Since the transition functions (4.1.22) are locally constant in the field space, $\Cal{A}^{(P)}$ is a flat principal bundle over $\Cal{M}_{\ast}^{(P)}$. In the next step of the proof we want to discuss the dependence on the background connection $A_{0}$. However, let $A_{0}^{\prime}$ denote another background connection which generates the open cover $\Cal{U}^{A_{0}^{\prime}}=\{U_{a}^{A_{0}^{\prime}}\}$ of the gauge orbit space. By passing to the common refinement (if necessary) $\epsilon_{a}^{A_{0}^{\prime}}$ is related to $\epsilon_{a}^{A_{0}}$ by $\epsilon_{a}^{A_{0}^{\prime}}(A)=\epsilon_{a}^{A_{0}}(A)+\hat{h}_{a}^{A_{0},A_% {0}^{\prime}}(A)$, where $$\multline\displaystyle\hat{h}_{a_{j}}^{A_{0},A_{0}^{\prime}}(A)=\frac{1}{2\pi% \sqrt{-1}}\int_{M}(A_{0}-A_{0}^{\prime})\wedge\rho_{j}^{(n-1)}+s_{a_{j}}(exp{% \int_{M}(A-A_{0})\wedge\rho_{j}^{(n-1)}})\\ \displaystyle-s_{a_{j}}(exp{(\int_{M}(A-A_{0})\wedge\rho{}_{j}^{(n-1)})}\cdot% \exp{(\int_{M}(A-A_{0}^{\prime})\wedge\rho_{k}^{(n-1)})})\endmultline% \displaystyle\hat{h}_{a_{j}}^{A_{0},A_{0}^{\prime}}(A)=\frac{1}{2\pi\sqrt{-1}}% \int_{M}(A_{0}-A_{0}^{\prime})\wedge\rho_{j}^{(n-1)}+s_{a_{j}}(exp{\int_{M}(A-% A_{0})\wedge\rho_{j}^{(n-1)}})\\ \displaystyle-s_{a_{j}}(exp{(\int_{M}(A-A_{0})\wedge\rho{}_{j}^{(n-1)})}\cdot% \exp{(\int_{M}(A-A_{0}^{\prime})\wedge\rho_{k}^{(n-1)})})$$ 4.1.234.1.234.1.23 is a locally constant gauge invariant function on $\pi_{\Cal{A}^{(P)}}^{-1}(U_{a}^{A_{0}}\cap U_{a}^{A_{0}^{\prime}})$. Since this function $\hat{h}_{a_{j}}^{A_{0},A_{0}^{\prime}}(A)\in\mathbb{Z}^{N}$, there exists a map $h_{a}^{A_{0},A_{0}^{\prime}}\colon U_{a}^{A_{0}}\cap U_{a}^{A_{0}^{\prime}}% \rightarrow\Cal{G}_{\ast}^{(M)}$ which is given by $$h_{a}^{A_{0},A_{0}^{\prime}}([A])=\hat{\kappa}([\sum_{j=1}^{b_{1}}\hat{h}_{a_{% j}}^{A_{0},A_{0}^{\prime}}(A)\rho_{j}^{(1)}],G_{0}d_{1}^{\ast}(A_{0}-A_{0}^{% \prime})-G_{0}d_{1}^{\ast}(A_{0}-A_{0}^{\prime})(x_{0}))$$ 4.1.244.1.244.1.24 resulting in $\sigma_{a}^{A_{0}^{\prime}}([A])=\sigma_{a}^{A_{0}}([A])+(h_{a}([A])^{A_{0},A_% {0}^{\prime}})^{\ast}\vartheta$. This finally proves that any different choice of the background connection gives rise to an equivalent bundle atlas of $\Cal{A}^{(P)}(\Cal{M}_{\ast}^{(P)},\pi_{\Cal{A}^{(P)}},\Cal{G}_{\ast}^{(M)})$. This concludes the proof of theorem 4.3. ∎ missing

On the computational complexity and generalization properties of multi-stage and recursive scenario programs Nikolaos Kariotoglou, Kostas Margellos and John Lygeros N. Kariotoglou and J. Lygeros are with the Automatic Control Laboratory, Department of Information Technology and Electrical Engineering, ETH Zürich, Zürich 8092, Switzerland (e-mail: karioto@control.ee.ethz.ch; lygeros@control.ee.ethz.ch)K. Margellos is with the Department of Industrial Engineering and Operations Research, UC Berkeley, Sutardja Dai Hall 330, Berkeley CA 94720, United States (e-mail: kostas.margellos@berkeley.edu)The work of N. Kariotoglou was supported by the Swiss National Science Foundation under grant number $200021\_137876$. (January 18, 2021) Abstract We discuss the computational complexity and feasibility properties of scenario based techniques for uncertain optimization programs. We consider different solution alternatives ranging from the standard scenario approach to recursive variants, and compare feasibility as a function of the total computation burden. We identify trade-offs between the different methods depending on the problem structure and the desired probability of constraint satisfaction. Our motivation for this work stems from the applicability and complexity reduction when making decisions by means of recursive algorithms. We illustrate our results on an example from the area of approximate dynamic programming. Scenario approach, randomized optimization, uncertain systems, approximate dynamic programming. I Introduction Robust optimization comes up naturally in a range of problems from finance to robotics ([1, 2, 3]). Uncertain data is often present in the formulation of a decision making problem and the optimal solution is required to be robust against any possible uncertainty realization [4]. However, uncertainty may take values from an infinite and possibly unbounded set, which we might not know analytically, giving rise to a robust optimization problem that is in general not tractable [5, 6, 7]. A significant amount of research has concentrated on robust problems with structural characteristics and uncertainty sets of specific geometry for which robust decisions can be made by means of a tractable optimization program [8]. An alternative way to deal with data uncertainty is to formulate a chance constrained variant of the initial problem where the optimal decision is allowed to violate the robust constraint on a set of pre-specified measure. The authors in [4, 9] provide explicit solutions to such problems under assumptions on the probability distribution of the uncertainty. To avoid such assumptions one can make use of uncertainty samples (either based on historical data or via a scenario generation model) and construct decisions that satisfy the system constraints only for the sampled uncertainty scenarios. The feasibility and performance properties of the solution can be generalized to quantify the confidence with which the optimizer of the scenario program satisfies the constraints for uncertainty realizations different than those used in the optimization process, providing a probabilistic link between scenario based and chance constrained optimization. The scenario approach introduced in [10, 11] can be used to provide such feasibility generalization statements for convex optimization problems. Beyond feasibility guarantees, [12, 13, 14] provide bounds on the amount of constraint violation and probabilistic performance. Generalization properties of similar nature can be obtained for non-convex optimization programs as well, using VC theoretic results [15, 16, 17]; the complexity, however, of the resulting solution depends on the so-called VC dimension (see [15] for a precise definition), which is in general difficult to compute. Here we focus on the generalization properties of scenario based convex optimization problems using the scenario approach [10, 11, 18, 19]. The scenario approach deals with robust and chance constrained convex optimization problems by solving sampled programs constructed using a finite number of samples. The method provides bounds on the number of samples needed to provide guarantees about the feasibility of the optimal solution of the sampled program with respect to the original one. The number of required samples determines the total number of constraints that, together with the number of decision variables and the type of problem (linear program, quadratic program, second-order cone program, semi-definite program, etc.), determine the overall computation effort. The given bounds scale well with certain structural quantities of the underlying problem and with the design parameters, apply to any problem under relatively mild assumptions and can be shown to be tight in a specific class of problems [11, 20]. However, when one considers optimization problems with additional structure on the constraints, the generic bounds on the number of samples are not a sufficient performance measure. The same guarantees on the feasibility of a scenario based solution may be obtained by formulating several alternative scenario programs, each with a potentially different number of decision variables and constraints and hence different computational complexity. Here we investigate these trade-offs for a class of recursive optimization problems that naturally arises in applications such as stochastic model predictive control (SMPC) [21] and approximate dynamic programming (ADP) [22]. We consider two alternative structures, one with a single convex optimization problem with multiple constraint functions and one where the constraint functions are coupled. We show how, besides the standard scenario program, other types of scenario programs can be formulated for generic problems that respect these structures. These alternatives provide the same feasibility guarantees at potentially lower computation cost. We demonstrate this trade-off by benchmarking on a particular class of algorithms (primal-dual) and a particular class of problems (robust second-order cone problems). We also show how the stage-wise confidence and the violation level, typically treated as parameters in scenario programs, can be chosen by means of a convex optimization program to reduce the overall computation time. We demonstrate our results by applying them to a particular ADP algorithm developed for reachability problems. Section II provides a statement of the problem under consideration. In Section III we present the different scenario based alternatives along with a pair of convex optimization problems that choose the stage-wise confidence and violation probability levels that result in the most favorable computational complexity. In Section IV we discuss the trade-off between the feasibility properties and the computational complexity of each alternative. Section V illustrates some features of the different algorithmic alternatives by means of a numerical example arising in ADP, while Section VI concludes the paper with some ideas on future research directions. Notation: Let $\mathbb{R}$ denote the real numbers, $\mathbb{N}$ the natural numbers and $\mathbb{N}_{+}$ the positive natural numbers. In the derivations below all uncertainty samples are extracted from a (possibly unknown) uncertainty set $\Delta$ according to a fixed, possibly unknown probability measure $\mathbb{P}$. $\mathbb{P}^{S}$ denotes the corresponding product measure for some $S\in\mathbb{N}_{+}$. We use i.i.d for identically and independently distributed uncertainty samples. Operator $|\cdot|$ denotes the cardinality of its argument, $\dim(A)$ denotes the dimension of a linear space $A$ and $x\models y$ implies that $x$ satisfies the statement in $y$. II Convex optimization programs with multiple robust constraints Consider a compact convex set $\mathcal{X}\subseteq\operatorname{\mathbb{R}}^{d}$, a possibly unbounded uncertainty set $\Delta\subseteq\operatorname{\mathbb{R}}^{w}$, a convex cost function $f:\operatorname{\mathcal{X}}\rightarrow\operatorname{\mathbb{R}}$ and a set of $M\in\operatorname{\mathbb{N}}_{+}$ convex constraint functions $g_{i}:\operatorname{\mathcal{X}}\times\Delta\rightarrow\operatorname{\mathbb{R}}$, $i=1,\dots,M$; our results also extend to non-real valued (e.g. binary valued) uncertainties as long as they take values in a probability space. We are concerned with robust convex optimization problems (RCP) of the form: $$\displaystyle\operatorname{\text{RCP}}:\begin{cases}\begin{split}\displaystyle% \min_{x\in\mathcal{X}}&\displaystyle f(x)\\ \displaystyle\text{s.t}&\displaystyle g_{i}\left(x,\delta\right)\leq 0,\ % \forall\delta\in\Delta,\ \forall i\in\{1,\dots,M\}.\end{split}\end{cases}$$ (1) The set $\Delta$ may be infinite and possibly unbounded, rendering (1) a convex, semi-infinite optimization program. For such problems there is no general algorithm to obtain a solution, unless particular assumptions on the structure of $\Delta$ and the functions $g_{i}$ are made (see for example [8, 23]). A common approach to approximate the solution is to impose the constraints on a finite number of uncertainty instances. To this end, consider $S\in\mathbb{N}_{+}$ i.i.d samples $\{\delta^{j}\}_{j=1}^{S}$ extracted from $\Delta$ according to some, possibly unknown, underlying probability distribution, and a collection $\left\{\Delta_{i}\right\}_{i=1}^{M}$ of $M$ subsets of $\{\delta^{j}\}_{j=1}^{S}$ such that for each $\delta\in\{\delta^{j}\}_{j=1}^{S}$ there exists $i$ so that $\delta\in\Delta_{i}$, i.e. the sets may be overlapping but each $\delta$ belongs in at least one of them. The interpretation is that for each $i=1,\ldots,M$, the corresponding constraint $g_{i}(x,\delta)$ should be satisfied for all $\delta\in\Delta_{i}$, but not necessarily for all $\delta\in\Delta$. Problem (1) is then approximated by a scenario convex optimization program (SCP) of the form: $$\displaystyle\text{SCP}\left[\Delta_{1},\dots,\Delta_{M}\right]:\begin{cases}% \begin{split}\displaystyle\min_{x\in\mathcal{X}}&\displaystyle f(x)\\ \displaystyle\text{s.t}&\displaystyle g_{i}\left(x,\delta\right)\leq 0,\forall% \delta\in\Delta_{i},\ \forall i\in\{1,\dots,M\}.\end{split}\end{cases}$$ (2) This is a convex optimization program with a finite number of decision variables and constraints, that can be solved to optimality by various numerical solvers (e.g. CPLEX, Gurobi, MOSEK). We impose the following assumption on $\text{SCP}[\Delta_{1},\dots,\Delta_{M}]$: Assumption 1. For any set $\{\delta^{j}\}_{j=1}^{S}$ and collection of subsets $\left\{\Delta_{i}\right\}_{i=1}^{M}$ with $S,M\in\mathbb{N}_{+}$, $\text{SCP}[\Delta_{1},\dots,\Delta_{M}]$ is feasible, its feasibility region has a non-empty interior and its minimizer $x^{*}[\Delta_{1},\ldots,\Delta_{M}]:~{}\Delta^{S}\rightarrow\mathcal{X}$ is unique. We refer to [10], [20] for details on how the feasibility and uniqueness assumption can be relaxed; however, we keep these assumptions here to streamline the presentation of our results. Measurability of the minimizer $x^{*}\left[\Delta_{1},\dots,\Delta_{M}\right]$ is assumed as needed, see [14, 24] for details. Note that problem (2) and its minimizer are parametrized by the sets $\Delta_{1},\dots,\Delta_{M}$. However, once the sets $\Delta_{i}$ are fixed, the unique (under Assumption 1) minimizer $x^{*}[\Delta_{1},\dots,\Delta_{M}]$ of $\text{SCP}[\Delta_{1},\dots,\Delta_{M}]$ is a mapping from $\Delta^{S}$ to $\mathcal{X}$ and satisfies $g_{i}(x^{*},\delta)\leq 0$, for all $\delta\in\Delta_{i}$ and $i\in\left\{1,\dots,M\right\}$. One of the challenges concerning this type of problem is to analyze the properties of $x^{*}$ in terms of feasibility (satisfiability of $g_{i}(x^{*},\delta)\leq 0$ for all $\delta\in\Delta$) and performance (optimality of $f(x^{*})$). Following the standard literature on the scenario approach [10, 11] we concentrate here on the feasibility properties of $x^{*}$ as a function of the algorithm used to construct the solution; for a discussion on performance issues see [13, 14]. We establish that, depending on problem structure, there may be different ways of formulating the scenario program as a function of the choice of the number of samples $S$ and the partition sets $\Delta_{i}$. The computation effort necessary to solve the corresponding scenario programs differs, despite the fact that solutions have comparable feasibility properties. We will investigate the trade-offs between different design choices in the context of second order cone problems (SOCP) solved via primal-dual algorithms which are in general known to be of $\mathcal{O}\left\lparen\left\lparen n+m\right\rparen^{3}\right\rparen$ complexity, where $n$ denotes the dimension of the decision space and $m$ the total number of constraints. Our motivation stems from the fact that a wide range of control-inspired optimization programs are SOCP, while primal-dual algorithms provide reliable termination and optimality conditions by iteratively reducing the duality gap. With straightforward modifications, related statements can be made for other classes of algorithms (e.g. gradient methods) and other classes of problems (linear programs, quadratic programs, semi-definite programs etc.). III Feasibility properties of scenario convex programs We introduce four different approaches to formulate the scenario program: the standard scenario approach, the multi-stage scenario approach, the recursive scenario approach using the same samples at every recursive step and the recursive scenario approach using different samples at every recursive step. The standard scenario approach is the most general and applies to all problems in the form of $\operatorname{\text{RCP}}$. The multi-stage and recursive counterparts assume particular structure on the constraint functions and exploit it to reduce computational complexity while maintaining similar feasibility properties. To streamline the comparison between different methods, we present here their main characteristics and devote the next section to discussing relative advantages. III-A The standard scenario approach Let $\bar{\Delta}=\{\delta^{j}\}_{j=1}^{S}$ and assume that $\Delta_{1}=\ldots=\Delta_{M}=\bar{\Delta}$, in other words enforce each constraint for all elements in $\bar{\Delta}$. Denote by $\text{SCP}[\bar{\Delta}]$, $x^{*}[\bar{\Delta}]$ the resulting instance of $\text{SCP}[\Delta_{1},\dots,\Delta_{M}]$ and its minimizer, respectively. For each $(x,\delta)$, let $g(x,\delta):=\max_{i=1,\ldots,M}g_{i}(x,\delta)$. The constraints $g_{i}\left(x,\delta\right)\leq 0,\ \forall\delta\in\bar{\Delta},\ \forall i\in% \{1,\dots,M\}$ are equivalent to $g(x,\delta)\leq 0,\ \forall\delta\in\bar{\Delta}$. Problem $\text{SCP}[\bar{\Delta}]$ was first studied in terms of feasibility in [10]. The following Theorem was then shown in [11]. Theorem 1 ([11], Theorem 2.4). Choose $\varepsilon,\beta\in(0,1)$ and fix $S\geq S(\operatorname{\varepsilon},\beta,d)$ where $$\displaystyle\begin{split}\displaystyle S(\varepsilon,\beta,d):=\min\left\{N% \in\operatorname{\mathbb{N}}\ \bigg{|}\ \sum_{i=0}^{d-1}\binom{N}{i}% \varepsilon^{i}(1-\varepsilon)^{N-i}\leq\beta\right\}.\end{split}$$ (3) Extract $S$ samples i.i.d from $\Delta$ according to a probability measure $\operatorname{\mathbb{P}}$, construct $\Delta_{1}=\ldots=\Delta_{M}=\bar{\Delta}$ and formulate $\operatorname{\text{SCP}}[\bar{\Delta}]$. Under Assumption 1, the minimizer $x^{*}[\bar{\Delta}]$ of $\operatorname{\text{SCP}}[\bar{\Delta}]$ satisfies the chance constraint, $$\displaystyle\operatorname{\text{CCP}}_{\operatorname{\varepsilon}}:% \operatorname{\mathbb{P}}\left[g\left\lparen x^{*}[\bar{\Delta}],\delta\right% \rparen>0\right]\leq\operatorname{\varepsilon}$$ (4) with confidence (measured with respect to $\mathbb{P}^{S}$) at least $1-\beta$. Using the satisfiability notation “$\models$” (along the lines of [14]), the statement in Theorem 1 can be compactly written as $\operatorname{\mathbb{P}}^{S}\left[x^{*}[\bar{\Delta}]\models\operatorname{% \text{CCP}}_{\operatorname{\varepsilon}}\right]\geq 1-\beta$. The interpretation of Theorem 1 is that, with certain confidence, the solution of the scenario convex program satisfies the robust constraint apart from a subset of the uncertainty space with measure at most $\operatorname{\varepsilon}$. The computational complexity associated with constructing $x^{*}[\bar{\Delta}]$, along with the feasibility properties of Theorem 1, depend on the choice of $\operatorname{\varepsilon},\beta$ and the number of decision variables $d$ that implicitly affect the number of constraints (inspect (3)). The exact effect of each parameter is discussed in Section IV. Note that Theorem 1 remains unaffected if $d$ is replaced by any upper bound on the number of the so-called support constraints (see [10] for a precise definition) other than the dimension of the decision space. Refinements along this direction are discussed in [18, 25, 26]. III-B The multi-stage scenario approach We impose here additional structure on the $\operatorname{\text{RCP}}$ by assuming that for any $\delta\in\Delta$ and for each $i=1,\ldots,M$, the constraint function $g_{i}(\cdot,\delta)$ depends on some (i.e. not necessarily all) of the decision variables. The set-up is then similar to the structure considered in [18], where the authors studied optimization programs with multiple chance constraints. For each $i=1,\ldots,M$, let $\mathcal{X}_{i}\subseteq\operatorname{\mathcal{X}}$ denote the domain of each $g_{i}(\cdot,\delta)$ and $d_{i}=\dim(\mathcal{X}_{i})$, where $\dim(\mathcal{X}_{i})$ denotes the dimension of the smallest subspace of $\mathbb{R}^{d}$ containing $\operatorname{\mathcal{X}}_{i}$. We further assume that $d_{i}<d$ for at least one $i=1,\ldots,M$ to exclude the case where all constraint functions depend on all the decision variables; if this is not the case the subsequent analysis reduces to the standard scenario approach of Section III-A. We then have the following theorem due to [18], which serves as the multi-stage counterpart of Theorem 1. Theorem 2 ([18], Theorem 4.1). For each $i=1,\ldots,M$, choose $\varepsilon_{i},\beta_{i}\in(0,1)$, and fix ${S}_{i}\geq S(\operatorname{\varepsilon}_{i},\beta_{i},d_{i})$ where $$\displaystyle\begin{split}\displaystyle S(\varepsilon_{i},\beta_{i},d_{i}):=% \min\left\{N\in\operatorname{\mathbb{N}}\ \bigg{|}\ \sum_{j=0}^{d_{i}-1}\binom% {N}{j}\varepsilon_{i}^{j}(1-\varepsilon_{i})^{N-j}\leq\beta_{i}\right\}.\end{split}$$ (5) Extract $S=\sum_{i=1}^{M}S_{i}$ samples i.i.d from $\Delta$ according to a probability measure $\operatorname{\mathbb{P}}$, construct $\{\Delta_{i}\}_{i=1}^{M}$ as in Section II with $|\Delta_{i}|=S_{i}$ and formulate $\operatorname{\text{SCP}}[\Delta_{1},\dots,\Delta_{M}]$. Under Assumption 1, for each $i=1,\ldots,M$, the minimizer $x^{*}[\Delta_{1},\dots,\Delta_{M}]$ of $\operatorname{\text{SCP}}[\Delta_{1},\dots,\Delta_{M}]$ satisfies the chance constraint, $$\displaystyle\operatorname{\text{CCP}}_{\operatorname{\varepsilon}_{i}}:% \operatorname{\mathbb{P}}\left[g_{i}\left(x^{*}[\Delta_{1},\dots,\Delta_{M}],% \delta\right)>0\right]\leq\operatorname{\varepsilon}_{i},$$ (6) with confidence (measured with respect to $\mathbb{P}^{S_{i}}$) at least $1-\beta_{i}$. As with Theorem 1, each $d_{i}$ can be replaced by a tighter upper bound on the support constraints of $g_{i}$. Equation (6) in Theorem 2 establishes the feasibility properties of $x^{*}[\Delta_{1},\dots,\Delta_{M}]$ for each separate constraint. However, no guarantees are provided on the probability that $x^{*}[\Delta_{1},\dots,\Delta_{M}]$ satisfies all constraints simultaneously, i.e. $\operatorname{\text{CCP}}_{\operatorname{\varepsilon}}$ in (4). This issue is addressed by the following corollary, that is a direct implication of Theorem 2. Corollary 1. Fix $\operatorname{\varepsilon},\beta\in(0,1)$ and select $\operatorname{\varepsilon}_{i},\beta_{i}\in(0,1)$, for all $i=1,\dots,M$, such that $\sum_{i=1}^{M}\operatorname{\varepsilon}_{i}=\operatorname{\varepsilon}$ and $\sum_{i=1}^{M}\beta_{i}=\beta$. Under the set-up of Theorem 2 and Assumption 1 we have that $\operatorname{\mathbb{P}}^{S}\left[x^{*}[\Delta_{1},\dots,\Delta_{M}]\models% \operatorname{\text{CCP}}_{\operatorname{\varepsilon}}\right]\allowbreak\geq 1-\beta$, where $\operatorname{\text{CCP}}_{\operatorname{\varepsilon}}$ is given in (4). Proof. The proof of Corollary 1 is essentially an application of the Boole-Bonferroni inequalities [27]. By Theorem 2 we have that $\operatorname{\mathbb{P}}^{S}\left[x^{*}[\Delta_{1},\dots,\Delta_{M}]\models% \operatorname{\text{CCP}}_{\operatorname{\varepsilon}_{i}}\right]\geq 1-\beta_% {i},\ \text{for all $i=1,\dots,M$}$. By the subadditivity of $\operatorname{\mathbb{P}}_{\Delta^{S}}$ we have that $\operatorname{\mathbb{P}}^{S}\left[x^{*}[\Delta_{1},\dots,\Delta_{M}]\models% \operatorname{\text{CCP}}_{\operatorname{\varepsilon}_{i}},\text{ for all }i=1% ,\ldots,M\right]\geq 1-\sum_{i=1}^{M}\beta_{i}=1-\beta.$ To complete the proof it suffices to show that $x^{*}[\Delta_{1},\dots,\Delta_{M}]\models\operatorname{\text{CCP}}_{% \operatorname{\varepsilon}_{i}}$ for all $i=1,\ldots,M$, implies that $x^{*}[\Delta_{1},\dots,\Delta_{M}]\models\operatorname{\text{CCP}}_{% \operatorname{\varepsilon}}$, where $\operatorname{\text{CCP}}_{\operatorname{\varepsilon}}$ is given in (4). By the subadditivity of $\operatorname{\mathbb{P}}$, and since $x^{*}[\Delta_{1},\dots,\Delta_{M}]\models\operatorname{\text{CCP}}_{% \operatorname{\varepsilon}_{i}}$ is equivalent to $\operatorname{\mathbb{P}}\left[g_{i}(x^{*}[\Delta_{1},\dots,\Delta_{M}],\delta% )>0\right]\allowbreak\leq\operatorname{\varepsilon}_{i}$, we have that $\operatorname{\mathbb{P}}\left[\exists i\in\{1,\ldots,M\}\text{ such that }g_{% i}(x^{*}[\Delta_{1},\dots,\Delta_{M}],\delta)>0\right]\leq\sum_{i=1}^{M}% \operatorname{\varepsilon}_{i}=\operatorname{\varepsilon}.$ Since by definition $g(x,\delta):=\max_{i=1,\ldots,M}g_{i}(x,\delta)$, the last statement implies that $\operatorname{\mathbb{P}}\left[g(x^{*}[\Delta_{1},\dots,\Delta_{M}],\delta)>0% \right]\leq\operatorname{\varepsilon}$, which is equivalent to $x^{*}[\Delta_{1},\dots,\Delta_{M}]\models\operatorname{\text{CCP}}_{% \operatorname{\varepsilon}}$ and concludes the proof. ∎ The computational complexity associated with obtaining $x^{*}[\Delta_{1},\dots,\Delta_{M}]$ with the feasibility properties of Corollary 1, depends on $\{d_{i}\}_{i=1}^{M}$ and the choices for $\{\varepsilon_{i}\}_{i=1}^{M}$,$\{\beta_{i}\}_{i=1}^{M}$. The obvious choice of $\varepsilon_{i}=\varepsilon/M$ and $\beta_{i}=\beta/M$ for $i=1,\dots,M$ will in general be suboptimal; in Section III-E we formulate convex optimization problems to compute better choices. III-C Recursive scenario approach without re-sampling In the sequel we consider $\operatorname{\text{RCP}}$ problems with specific structure on the constraint functions that enables us to tackle $\text{SCP}[\Delta_{1},\dots,\Delta_{M}]$ in a sequential manner. We assume that the constraint functions $g_{i}(\cdot,\cdot,\cdot):~{}\mathcal{X}_{i}\times\mathcal{X}_{i+1}\times\Delta% \rightarrow\mathbb{R}$ are pairwise coupled and convex with respect to their first argument, and $g_{M}(\cdot,\cdot):~{}\mathcal{X}_{M}\times\Delta\rightarrow\mathbb{R}$. As a consequence of this assumption we have by construction of $\operatorname{\text{RCP}}$ that $\mathcal{X}=\mathcal{X}_{1}\times\ldots\times\mathcal{X}_{M}$, which is a special case of the structure assumed in Section III-B. Let $x=(x_{1},\ldots,x_{M})$ where $x_{i}\in\operatorname{\mathcal{X}}_{i}$, for each $i=1,\dots,M$. We further assume that the objective function is separable, i.e. $f(x)=\sum_{i=1}^{M}f_{i}(x_{i})$. Such problem structures appear naturally in SMPC and ADP, as we demonstrate in Section V. The pairwise coupling structure can be relaxed to any form of stage-wise coupling as long as the constraint function at every stage is convex with respect to the decision variables. The separable structure assumed, motivates the decomposition of $\text{SCP}[\Delta_{1},\dots,\Delta_{M}]$ into a sequence of coupled scenario programs. For each $i=1,\ldots,M-1$ we define the following parametric scenario program $$\displaystyle\operatorname{\text{SCP}}_{i}[x_{i+1},\Delta_{i}]:\begin{cases}% \begin{split}\displaystyle\min_{x_{i}\in\mathcal{X}_{i}}&\displaystyle f_{i}(x% _{i})\\ \displaystyle\text{s.t}&\displaystyle g_{i}\left(x_{i},x_{i+1},\delta\right)% \leq 0,\ \forall\delta\in\Delta_{i}\end{split}\end{cases}$$ (7) and $\operatorname{\text{SCP}}_{M}[\Delta_{M}]$ analogously, with $g_{M}(x_{M},\delta)\leq 0$ for all $\delta\in\Delta_{M}$, replacing the corresponding constraint in (7). We assume that all stage problems in (7) satisfy Assumption 1 for any fixed $x_{i+1}\in\operatorname{\mathcal{X}}_{i+1}$; weaker assumptions are discussed in [28, Section 4]. Consider now the sequence of $M$ pairwise coupled programs $\operatorname{\text{SCP}}_{i}\left[x_{i+1},\Delta_{i}\right]$ and let all sets $\Delta_{i}$ be identical, i.e. $\Delta_{1}=\cdots=\Delta_{M}=\bar{\Delta}$. Such optimization problems were referred to as cascading programs in [28], where the authors study the feasibility properties of a solution generated by sequentially solving a pair of coupled problems using the same set of uncertain scenarios. In particular, the following is a direct consequence of [28, Theorem 7]. Theorem 3 ([28], Theorem 7). Let $d_{i}=\dim(\operatorname{\mathcal{X}}_{i})$ be the dimension of the smallest subspace of $\mathbb{R}^{d}$ containing $\operatorname{\mathcal{X}}_{i}$ and $\bar{d}=\sum_{i=1}^{M}d_{i}$. Fix $\varepsilon,\beta\in(0,1)$ and $S\geq S(\varepsilon,\beta,\bar{d})$, where $S(\varepsilon,\beta,\bar{d})$ is given by (3). Construct $x^{*}:=\left\lparen x_{1}^{*},\dots,x^{*}_{M}\right\rparen$, where each $x^{*}_{i}[\bar{\Delta}]$ is recursively computed from (7) with $\Delta_{i}=\bar{\Delta}$ for $i=1,\dots,M$. We then have that $\operatorname{\mathbb{P}}^{S}\left[x^{*}[\bar{\Delta}]\models\operatorname{% \text{CCP}}_{\operatorname{\varepsilon}}\right]\geq 1-\beta$. This recursive scenario based solution can be used to obtain feasibility properties for the solution of each step of the recursion. If $x^{*}:=\left\lparen x_{1}^{*},\dots,x^{*}_{M}\right\rparen$ is constructed according to Theorem 3, for any fixed $x_{i+1}\in\operatorname{\mathcal{X}}_{i+1}$, with probability at least $1-\beta_{i}$, $x^{*}_{i}[x_{i+1},\bar{\Delta}]$ satisfies $\operatorname{\mathbb{P}}\left[g_{i}\left\lparen x^{*}_{i}[x_{i+1},\bar{\Delta% }],x_{i+1},\delta\right\rparen>0\right]\leq\operatorname{\varepsilon}_{i},$ for any $\varepsilon_{i},\beta_{i}\in(0,1)$ satisfying the equation $S(\operatorname{\varepsilon}_{i},\beta_{i},d_{i})\leq S$ where $S$ is chosen such that $S\geq S(\operatorname{\varepsilon},\beta,\bar{d})$. In this case, however, the values of $\varepsilon_{i}$ and $\beta_{i}$ are not set a-priori and are not design choices; they are implicitly determined by the dimension of each subproblem. Consequently, the computational complexity of the recursive scenario approach only depends on $\bar{d}$ and the choice of $\operatorname{\varepsilon},\beta$. III-D Recursive scenario approach with re-sampling Consider the separable structure assumed in Section III-C and note that for a fixed $x_{i+1}\in\operatorname{\mathcal{X}}_{i+1}$, $\operatorname{\text{SCP}}_{i}[x_{i+1},\Delta_{i}]$ is in the form of $\operatorname{\text{SCP}}[\bar{\Delta}]$ considered in Section III-A. Fix $\varepsilon_{i}$ and $\beta_{i}$ and let the number of samples $S_{i}$, $i=1,\ldots,M$ be chosen according to (5). For any $x_{i+1}\in\operatorname{\mathcal{X}}_{i+1}$, $\Delta_{i}\in\Delta^{S_{i}}$, let $x^{*}_{i}[x_{i+1},\Delta_{i}]:~{}\operatorname{\mathcal{X}}_{i+1}\times\Delta^% {{S}_{i}}\rightarrow\operatorname{\mathcal{X}}_{i}$ be the minimizer of $\operatorname{\text{SCP}}_{i}[x_{i+1},\Delta_{i}]$. Theorem 1 implies that for all $i=1,\ldots,M-1$, $x^{*}_{i}[x_{i+1},\Delta_{i}]$ satisfies the chance constraint $$\displaystyle\operatorname{\text{CCP}}_{\operatorname{\varepsilon}_{i}}[x_{i+1% }]:\operatorname{\mathbb{P}}\left[g_{i}\left\lparen x^{*}_{i}[x_{i+1},\Delta_{% i}],x_{i+1},\delta\right\rparen>0\right]\leq\operatorname{\varepsilon}_{i},$$ (8) with probability at least $1-\beta_{i}$, while for $i=M$, $x_{M}^{*}[\Delta_{M}]$ satisfies $\operatorname{\text{CCP}}_{\varepsilon_{M}}$ with probability at least $1-\beta_{M}$. Using the parametrized scenario optimization problems in the form of (7) we can recursively construct a decision vector $x^{*}:=\left\lparen x_{1}^{*},\dots,x^{*}_{M}\right\rparen$, where for each $i=1,\ldots,M-1$ the optimizer $x^{*}_{i}[x_{i+1},\Delta_{i}]$ can be written as $x^{*}_{i}[\Delta_{i},\dots,\Delta_{M}]:~{}\Delta^{S_{i}}\times\cdots\times% \Delta^{S_{M}}\rightarrow\operatorname{\mathcal{X}}_{i}$, satisfying $$\displaystyle\operatorname{\mathbb{P}}^{S_{i}}\left[x^{*}_{i}[\Delta_{i},\dots% ,\Delta_{M}]\models\operatorname{\text{CCP}}_{\operatorname{\varepsilon}_{i}}% \Big{[}x^{*}_{i+1}[\Delta_{i+1},\dots,\Delta_{M}]\Big{]}\right]\geq 1-\beta_{i},$$ (9) and for $i=M$, $\operatorname{\mathbb{P}}^{S_{M}}\left[x^{*}_{M}\models\operatorname{\text{CCP% }}_{\varepsilon_{M}}\right]\geq 1-\beta_{M}$. Note that due to the recursive process, $x_{i}^{*}$ depends implicitly on all sets $\Delta_{i},\ldots,\Delta_{M}$. The following theorem can be used to compare the feasibility properties of a solution constructed in this way with a solution obtained using Theorems 1,2 and 3. Theorem 4. Fix $\operatorname{\varepsilon},\beta\in(0,1)$ and select $\operatorname{\varepsilon}_{i},\beta_{i}\in(0,1)$, for $i=1,\dots,M$, such that $\sum_{i=1}^{M}\operatorname{\varepsilon}_{i}=\operatorname{\varepsilon}$ and $\sum_{i=1}^{M}\beta_{i}=\beta$. Construct $x^{*}:=\left\lparen x_{1}^{*},\dots,x^{*}_{M}\right\rparen$, where each $x^{*}_{i}[\Delta_{i},\dots,\Delta_{M}]$ is recursively computed from (7), satisfying (9). Let $S=\sum_{i=1}^{M}{S_{i}}$ with $\{S_{i}\}_{i=1}^{M}$ chosen according to (5). We then have that $\operatorname{\mathbb{P}}^{S}\left[x^{*}[\Delta_{1},\dots,\Delta_{M}]\models% \operatorname{\text{CCP}}_{\operatorname{\varepsilon}}\right]\geq 1-\beta$. Proof. Let $\bar{S}_{i}=\sum_{k=i}^{M}S_{k}$, $\bar{\operatorname{\varepsilon}}_{i}=\sum_{k=i}^{M}\operatorname{\varepsilon}_% {k}$, $\bar{\beta}_{i}=\sum_{k=i}^{M}\beta_{k}$, $\bar{\Delta}_{i}=(\Delta_{i},\dots,\Delta_{M})$ and $\bar{x}^{*}_{i}=\left\lparen x_{i}^{*},\dots,x^{*}_{M}\right\rparen$. We claim that for all $i=1,\ldots,M$, the following statement holds $$\displaystyle\begin{split}&\displaystyle\operatorname{\mathbb{P}}^{\bar{S}_{i}% }\Big{[}\operatorname{\mathbb{P}}\big{[}g_{M}\left\lparen x^{*}_{M}[\bar{% \Delta}_{M}],\delta\right\rparen>0\text{ or }\\ &\displaystyle\exists k\in\mathbb{N}_{+},i\leq k<M:~{}g_{k}\left\lparen x^{*}_% {k}[\bar{\Delta}_{k}],x^{*}_{k+1}[\bar{\Delta}_{k+1}],\delta\right\rparen>0% \big{]}\leq\bar{\operatorname{\varepsilon}}_{i}\Big{]}\geq 1-\bar{\beta}_{i}.% \end{split}$$ (10) The statement of the claim implies that, with confidence at least $1-\bar{\beta}_{i}$, $\bar{x}^{*}_{i}$ satisfies all constraints with indices greater than or equal to $i$, with probability at least $1-\bar{\operatorname{\varepsilon}}_{i}$. If the claim holds, then for $i=1$ we get the result. We show that the claim holds using induction. For $i=M$, (10) is trivially satisfied since $\bar{\Delta}_{M}=\Delta_{M}$ and $\operatorname{\text{SCP}}_{M}[\Delta_{M}]$ is in the form of $\operatorname{\text{SCP}}[\bar{\Delta}]$ considered in Section III-A with $\bar{\Delta},S,\operatorname{\varepsilon},\beta$ replaced by $\Delta_{M},S_{M},\operatorname{\varepsilon}_{M}$ and $\beta_{M}$, respectively. Assume that (10) holds for some $1<i<M$. By (9) we have that $$\displaystyle\begin{split}&\displaystyle\operatorname{\mathbb{P}}^{S_{i-1}}% \Big{[}\operatorname{\mathbb{P}}\big{[}g_{i-1}\left\lparen x^{*}_{i-1}[\bar{% \Delta}_{i-1}],x^{*}_{i}[\bar{\Delta}_{i}],\delta\right\rparen>0\big{]}\leq% \operatorname{\varepsilon}_{i-1}\Big{]}\geq 1-\beta_{i-1}.\end{split}$$ (11) Using the fact that all samples are extracted independently, and (10), (11), hold for any uncertainty realization not in $\bar{\Delta}_{i}$ and $\bar{\Delta}_{i-1}$, respectively, (10), (11) would also hold with $\operatorname{\mathbb{P}}^{\bar{S}_{i-1}}$ in place of $\operatorname{\mathbb{P}}^{\bar{S}_{i}}$ and $\operatorname{\mathbb{P}}^{S_{i-1}}$. From the resulting statements and the subadditivity of $\operatorname{\mathbb{P}}^{\bar{S}_{i-1}}$ and $\operatorname{\mathbb{P}}$, we can then show analogously to the proof of Corollary 1 that (10) holds with $i-1$ in place of $i$. The latter implies that $\operatorname{\mathbb{P}}^{\bar{S}_{i-1}}\left[\bar{x}^{*}_{i-1}\models% \operatorname{\text{CCP}}_{\bar{\operatorname{\varepsilon}}_{i-1}}\right]\geq 1% -\bar{\beta}_{i-1}$ and proves the claim. ∎ The computational complexity associated with obtaining $x^{*}[\Delta_{1},\dots,\Delta_{M}]$ with the feasibility properties of Theorem 4 depends on $\{d_{i}\}_{i=1}^{M}$ and the choices for $\{\varepsilon_{i}\}_{i=1}^{M}$,$\{\beta_{i}\}_{i=1}^{M}$. Notice that unlike the recursive scenario approach without re-sampling, $\{\operatorname{\varepsilon}_{i}\}_{i=1}^{M}$ and $\{\beta_{i}\}_{i=1}^{M}$ are again design parameters for $i=1,\dots,M$ and can be chosen in a way that reduces the computational complexity of the algorithm used to solve the corresponding optimization problems. We deal with this issue in the next section. III-E Complexity optimization For the problems in Sections III-A and III-C, the number of decision variables $d$ and $\sum_{i=1}^{M}d_{i}$ and the overall violation and confidence levels $\varepsilon$ and $\beta$ determine the total complexity. For Sections III-B and III-D on the other hand, although the overall violation and confidence are chosen a priori, the stage-wise levels $\{\operatorname{\varepsilon}_{i}\}_{i=1}^{M}$ and $\{\beta_{i}\}_{i=1}^{M}$ are typically not fixed by the problem data and constitute a design choice that can affect the computational complexity due to the cubic dependence of SOCP solvers on the total number of samples and decision variables. Since the values of $\{d_{i}\}_{i=1}^{M}$ are fixed by problem data and generating samples from $\Delta$ can be hard, we focus on minimizing the total number of samples as an approximation to minimizing the total complexity. Throughout this section we replace the implicit sample size bound $S(\varepsilon_{i},\beta_{i},d_{i}):=\min\left\{N\in\operatorname{\mathbb{N}}\ % \bigg{|}\ \sum_{j=0}^{d_{i}-1}\binom{N}{j}\varepsilon_{i}^{j}(1-\varepsilon_{i% })^{N-j}\leq\beta_{i}\right\}$ that upper bounds the required sample size by the explicit bound $S(\varepsilon_{i},\beta_{i},d_{i})\geq\tfrac{e}{e-1}\tfrac{1}{\operatorname{% \varepsilon}_{i}}\left\lparen d_{i}-1+\ln\left\lparen\tfrac{1}{\beta_{i}}% \right\rparen\right\rparen$ due to [17]. For simplicity we treat the right-hand-side as an integer. Proposition 1. Consider the setup of Sections III-B and III-D where for each $i=1,\ldots,M$ the values of $d_{i}=\dim\left\lparen\operatorname{\mathcal{X}}_{i}\right\rparen$ and $d=\dim\left\lparen\operatorname{\mathcal{X}}\right\rparen$ are fixed by the problem data. Fix $\varepsilon,\beta\in(0,1)$. The problem of selecting $\{\varepsilon_{i},\beta_{i}\in(0,1)\}_{i=1}^{M}$ with $\sum_{i=1}^{M}\operatorname{\varepsilon}_{i}\leq\varepsilon,\ \sum_{i=1}^{M}% \beta_{i}\leq\beta$ that minimize the total number of samples $\sum_{i=1}^{M}{S(\varepsilon_{i},\beta_{i},d_{i})}$, is a convex optimization program of the form: $$\displaystyle\begin{split}\displaystyle\min_{\{\operatorname{\varepsilon}_{i},% \beta_{i}\}_{i=1}^{M}}&\displaystyle\sum_{i=1}^{M}S(\varepsilon_{i},\beta_{i},% d_{i})\\ \displaystyle\text{subject to:}&\displaystyle\sum_{i=1}^{M}\operatorname{% \varepsilon}_{i}\leq\varepsilon,\sum_{i=1}^{M}\beta_{i}\leq\beta,\operatorname% {\varepsilon}_{i},\beta_{i}>0,\forall i\in\{1,\dots,M\}.\\ \end{split}$$ (12) Proof. The function $S(\operatorname{\varepsilon}_{i},\beta_{i},d_{i})$ is convex with respect to $\operatorname{\varepsilon}_{i},\beta_{i}$ since the Hessian matrix is positive definite for any $\operatorname{\varepsilon}_{i},\beta_{i}\in(0,1)$. As a result, $\sum_{i=1}^{M}{S(\operatorname{\varepsilon}_{i},\beta_{i},d_{i})}$ is the sum of convex functions. ∎ The objective function of problem (12) is not in a standard form compatible with commercially available optimization software. As a result, one needs to implement a first or second order method to solve (12) (see for example [29]) taking advantage of the fact that both the gradient and Hessian matrix of the objective function are bounded with respect to $\varepsilon_{i},~{}\beta_{i}$ in $[\mu,1)$ for any $\mu>0$. Fixing the confidence levels $\beta_{i}$ a priori (e.g. $\beta_{i}=\beta/M$) simplifies the structure of (12) significantly and transforms the problem into a standard semi-definite program (SDP). Proposition 2. Choose $\beta\in(0,1)$ and fix the stage-wise confidence levels $\{\beta_{i}\in(0,1)\}_{i=1}^{M}$ such that $\sum_{i=1}^{M}\beta_{i}\leq\beta$. Fix $\varepsilon\in(0,1)$. For $c_{i}=\tfrac{e}{e-1}\left\lparen d_{i}-1+\ln\left\lparen\tfrac{1}{\beta_{i}}% \right\rparen\right\rparen$, $i=1,\ldots,M$, the following SDP is equivalent to (12). $$\displaystyle\begin{split}\displaystyle\min_{\{t_{i},\operatorname{\varepsilon% }_{i}\}_{i=1}^{M}}&\displaystyle\sum_{i=1}^{M}t_{i}\\ \displaystyle\text{subject to:}&\displaystyle\begin{bmatrix}t_{i}&\sqrt{c_{i}}% \\ \sqrt{c_{i}}&\operatorname{\varepsilon}_{i}\end{bmatrix}\succcurlyeq 0,\sum_{i% =1}^{M}\operatorname{\varepsilon}_{i}\leq\operatorname{\varepsilon},% \operatorname{\varepsilon}_{i}>0,\forall i\in\{1,\dots,M\}\\ \end{split}$$ (13) Proof. The objective function in (12) can be written as $\sum_{i=1}^{M}c_{i}/\operatorname{\varepsilon}_{i}$. Writing the problem in standard epigraph form and using Schur’s complement we end up with the constraints in (13). ∎ IV Discussion and trade-offs Each scenario based algorithm presented in Section III assumes a specific structure on the original $\operatorname{\text{RCP}}$ to construct a probabilistically feasible solution. Here we discuss differences between the feasibility properties of each solution and analyze the computational complexity of the associated algorithms as a function of design parameters. IV-A Structure and feasibility properties In contrast to the standard scenario approach of Section III-A, the multi-stage variant of Section III-B assumes that the domain of each constraint function in $\operatorname{\text{RCP}}$ is restricted to a subset of $\operatorname{\mathcal{X}}$. By investigating each constraint separately, Theorem 2 provides guarantees on the probability that $x^{*}[\Delta_{1},\ldots,\Delta_{M}]$ satisfies every individual constraint, something that cannot be achieved with the standard scenario approach. In the recursive methodologies of Sections III-C and III-D we further restrict the structure of $\operatorname{\text{RCP}}$ by requiring the constraint functions to be pairwise coupled. In this way we relax the assumption regarding the convexity of the constraint functions. In particular, we require $g_{i}(x_{i},x_{i+1},\delta)$ to be convex with respect to $x_{i}$, but do not require any convexity assumptions for the dependance on $x_{i+1}$. One situation where this can be of advantage is optimization programs with constraint functions that are bi-convex with respect to two decision vectors. Practically, such problems are often solved through a descent algorithm, alternating between optimizing with respect to one of the decision vectors while fixing the other decision vector to the value obtained at the preceding iteration. Theorem 3 allows us to provide probabilistic guarantees for the feasibility of the solution generated through such a descent algorithm, provided we fix a priori the number of iterations considered. Moreover, using the methodology of Section III-C which employs the same samples at every step of the recursive methodology ensures monotonicity of the objective function between consecutive steps of the recursion, that is crucial to ensure that the objective function decreases; see [28, Section 4]. Theorem 1, Corollary 1 and Theorems 3 and 4 all lead to a feasibility statement in the form of $\operatorname{\mathbb{P}}^{S}\left[x^{*}[\Delta_{1},\dots,\Delta_{M}]\models% \operatorname{\text{CCP}}_{\operatorname{\varepsilon}}\right]\geq 1-\beta$. Each method however requires a different number of samples to construct a solution and in turn the space on which the confidence related to the probability of constraint satisfaction is measured differs. In the standard scenario approach the total number of samples $S$ is determined by the value of $d$ and the choice of violation and confidence levels $\varepsilon,\beta$ (inspect (3)). Assuming the same choice of $\varepsilon$ and $\beta$, the total number of samples in the multi-stage scenario approach $\sum_{i=1}^{M}S_{i}$ can be greater or less than $S$ depending on the values of $\{d_{i}\}_{i=1}^{M}$ (inspect (5) and the first two columns in Table I). In general, if each $d_{i}$ is significantly smaller than $d$, then the total number of samples is smaller in the multi-stage scenario approach. The situation is analogous between the recursive scenario approach without and with re-sampling, where the total number of samples will be generally higher in the latter depending on the values of $\{d_{i}\}_{i=1}^{M}$ and the choices of $\{\operatorname{\varepsilon}_{i}\}_{i=1}^{M}$,$\{\beta_{i}\}_{i=1}^{M}$ (see the last two columns in Table I). Note that for the multi-stage scenario approach and the recursive scenario approach with re-sampling we can use the methods of Section III-E to optimize over $\{\operatorname{\varepsilon}_{i}\}_{i=1}^{M}$ and $\{\beta_{i}\}_{i=1}^{M}$ but there is no guarantee that this will lead to a smaller number of total samples since $\{d_{i}\}_{i=1}^{M}$ is fixed by problem data. IV-B Complexity Both the standard and multi-stage scenario approach of Sections III-A and III-B require solving a single problem of the same structure with the same number of decision variables but a potentially different number of constraints. The number of decision variables $d$ is given by problem data, while the number of constraints depends on $d$, $\{d_{i}\}_{i=1}^{M}$ and the chosen $\varepsilon,\beta$ and $\{\operatorname{\varepsilon}_{i}\}_{i=1}^{M}$, $\{\beta_{i}\}_{i=1}^{M}$. In the standard scenario approach, we use the same samples $S$ (see (3)) for each constraint function leading to a total of $MS$ constraints. In the multi-stage scenario approach, we use different samples $S_{i}$ (see (5)) for each constraint function leading to a total of $\sum_{i=1}^{M}{S_{i}}$ constraints, a number that can be minimized over $\{\operatorname{\varepsilon}_{i}\}_{i=1}^{M}$, $\{\beta_{i}\}_{i=1}^{M}$ using the methods of Section III-E. The computational complexity of each method is reported in the first two columns of Table I. Depending on the ratio between the minimum value of $\sum_{i=1}^{M}{S_{i}}$ and $MS$, either of the two methods might be preferable. The computational complexity of the recursive methodologies of Sections III-C and III-D depends on the number of decision variables and constraints per subproblem. Each subproblem involves a single constraint function and as a result the number of samples required by Theorems 3 and 4 coincides with the number of constraints. In the recursive scenario approach without re-sampling, we use the same number of samples $S$ in every subproblem which depends on $\bar{d}=\sum_{i=1}^{M}{d_{i}}$ and the choice of $\operatorname{\varepsilon},\beta$ (see Theorem 3). If $\bar{d}=d$ (as is the case for example in some ADP problems, see Section V), the number of samples coincides with that of the standard scenario approach. In general however, it might very well be that $\bar{d}>d$ (as is the case, for example, in some SMPC problems). In the recursive scenario approach with re-sampling, the number of samples $S_{i}$ in each subproblem coincides with the number of samples used in the multi-stage scenario approach and depends on $d_{i}$ and the choice of $\operatorname{\varepsilon}_{i}$, $\beta_{i}$ (see Theorem 4). As in the multi-stage scenario approach, $\sum_{i=1}^{M}{S_{i}}$ can be minimized over $\{\operatorname{\varepsilon}_{i}\}_{i=1}^{M}$, $\{\beta_{i}\}_{i=1}^{M}$ using the methods in Section III-E. The computational complexity of both recursive methods is reported in the last two columns of Table I. Whenever applicable, the recursive methods of Sections III-C and III-D can provide significant computational advantages, as illustrated in the next section. V Numerical example: Approximate Dynamic Programming Dynamic programming (DP) recursions are widely used to characterize the value function of optimal control problems [30]. For systems with continuous states, explicitly computing the value function by space discretization methods suffers from the curse of dimensionality, making the process intractable for state spaces of even moderate dimensions. This has motivated the development of sophisticated ADP methods [31]. A recently established methodology is the linear programming approach to ADP [32] which projects the optimal value function on the span of a pre-selected set of basis functions, intersected with the feasibility region determined by a set of inequality constraints. The authors in [22, 33] developed an algorithm based on the linear programming approach to ADP, specifically to approximate the value function of stochastic reachability problems. In this section we use this algorithm to investigate the relative performance of the alternative scenario program formulations of Section III. We consider a simplified planar unicycle model with additive noise $$\displaystyle\begin{bmatrix}\delta_{1}(i+1)\\ \delta_{2}(i+1)\end{bmatrix}=\begin{bmatrix}\delta_{4}(i)\cos(\delta_{3}(i))+% \delta_{1}(i)\\ \delta_{4}(i)\sin(\delta_{3}(i))+\delta_{2}(i)\end{bmatrix}+w_{i}$$ (14) where $\delta_{1},\delta_{2}$ denote linear position, $\delta_{3}$ yaw angle and $\delta_{4}$ linear velocity. We assume that $\delta_{3}$ and $\delta_{4}$ are control inputs to the system and treat $\delta_{1}$ and $\delta_{2}$ as states. The noise terms $w_{i}\in\mathbb{R}^{2}$ are assumed to be independent for different $i$ and identically distributed according to a multivariate normal distribution $\mathcal{N}(0,\Sigma)$ with diagonal covariance matrix. The combined state-action space is denoted by $\Delta=\Delta_{x}\times\Delta_{u}=\mathbb{R}^{2}\times\left([-0.5,0.5]\times[-% 2\pi,2\pi]\right)$ where for $\delta=(\delta_{1},\delta_{2},\delta_{3},\delta_{4})=(\delta_{x},\delta_{u})\in\Delta$, $\delta_{x}$ corresponds to spatial coordinates while $\delta_{u}$ to control inputs. The symbol $\delta$ is used for the state and input variables since in the sequel we will be sampling from $\Delta$. Given a target set $T=[0.8,1]^{2}$, an avoid set $A=[-0.45,0.25]\times[-0.2,0.15]$ and a collection of time indexed safe sets $\{S_{i}\}_{i=1}^{3}=\left\{[-1,1]^{2},[-0.3,1]^{2},[0.4,1]^{2}\right\}$, the three step reach-avoid problem considered here is to maximize the probability that (14) reaches $T$ while staying in the corresponding safe region $S_{i}\setminus A$ for time steps $i=1,2,3$ (see Figure 1). The authors in [34] show that this reach-avoid problem can be solved via a DP recursion: $$\displaystyle\begin{split}&\displaystyle V_{i}^{*}(\delta_{x})=\sup_{\delta_{u% }\in\Delta_{u}}\{\underbrace{\mathds{1}_{T}(\delta_{x})+\mathds{1}_{(S_{i}% \setminus A)\setminus T}(\delta_{x})\int_{\Delta_{x}}{V^{*}_{i+1}(y)Q(dy|% \delta)}}_{h(\delta_{x},\delta_{u})}\}\\ &\displaystyle V_{4}^{*}(\delta_{x})=\mathds{1}_{T}(\delta_{x}).\end{split}$$ (15) where $V_{i}^{*}$ denotes the value function at stage $i$, $Q$ denotes the transition kernel of the stochastic process in (14) and $\mathds{1}_{T},\mathds{1}_{(S_{i}\setminus A)\setminus T}$ denote the indicator functions of the sets $T$ and $(S_{i}\setminus A)\setminus T$ respectively. We follow the ADP formulation for reach-avoid problems suggested in [32] and applied in [22, 33] to approximate (15). To this end we express the value function of each step in the DP recursion as a solution to an infinite dimensional linear program: $$\displaystyle\begin{split}\displaystyle V_{i}^{*}\in\arg&\displaystyle\inf_{V(% \cdot)\in\mathcal{F}}\quad\int_{\Delta_{x}}V(\delta_{x})\nu(\text{d$\delta_{x}% $)}\\ &\displaystyle\text{subject to}\quad V(\delta_{x})\geq h(\delta_{x},\delta_{u}% ),\ \forall\delta\in\Delta\end{split}$$ (16) where $\nu$ is a (positive) measure supported on $\Delta_{x}$ and $\mathcal{F}$ denotes the space of Borel-measurable functions in which, under mild assumptions, $V^{*}_{i}$ resides [22]. Problems in the form of (16) are generally intractable and it is common in the literature to restrict the decision space to a finite dimensional subspace of $\mathcal{F}$ to approximate each $V^{*}_{i}$. As suggested in [22], we restrict the decision space to a set of Gaussian radial basis functions (RBFs) with fixed centers and variances and use their span to approximate each $V_{i}^{*}$. Let $\{d_{i}\}_{i=1}^{3}=\{200,150,100\}$ denote the cardinality of each basis set over the time horizon and $x=\{x_{i}\}_{i=1}^{3}$ with $x_{i}\in\mathbb{R}^{d_{i}}$, a collection of vectors corresponding to the weights of each RBF in the set. The reduction in the number of basis elements over the horizon is motivated by the reduction in the size of each safe set $S_{i}$. We denote by $L_{i}:\mathbb{R}^{d_{i}}\times\mathbb{R}^{d_{i+1}}\times\Delta\rightarrow% \mathbb{R}$ the functions (linear in the first and second arguments) that for $i=1,2$ and each $\delta\in\Delta$ return the difference between the approximate value function at time $i$ and the one-step-ahead reward at time $i+1$ (observe the constraints in (16)). Each $L_{i}$ implicitly depends on the safe, avoid and target regions at time $i$ and the weights $x_{i},x_{i+1}$ completely determine its value over $\Delta$. For $i=3$, the function is defined as $L_{3}:\mathbb{R}^{d_{3}}\times\Delta\rightarrow\mathbb{R}$ since the reach-avoid value function at $i=4$ is known (15). Using this notation, the approximate reach-avoid value functions can be computed via a sequence of coupled robust linear programs: $$\displaystyle\begin{split}\displaystyle\min_{x_{i}\in\mathbb{R}^{d_{i}}}&% \displaystyle x_{i}^{\top}I_{i}\\ \displaystyle\text{subject to:}&\displaystyle L_{i}(x_{i},x_{i+1},\delta)\geq 0% ,\ \forall\delta\in\Delta\end{split}$$ (17) where $I_{i}$ denotes the element-wise integral over $\Delta$ of each RBF in the basis set with respect to the measure $\nu$. The sequence of problems in (17) can be combined to a single problem as: $$\displaystyle\begin{split}\displaystyle\min_{x\in\mathbb{R}^{d_{1}+d_{2}+d_{3}% }}&\displaystyle\sum_{i=1}^{3}x_{i}^{\top}I_{i}\\ \displaystyle\text{subject to:}&\displaystyle L_{i}(x_{i},x_{i+1},\delta)\geq 0% ,\ \forall\delta\in\Delta,\ i=1,2\\ &\displaystyle L_{3}(x_{3},\delta)\geq 0,\ \forall\delta\in\Delta.\\ \end{split}$$ (18) Using the methods presented in Section III to solve (17) and (18) we can obtain an optimal solution for the weight vector $x$, possibly different for each method. Using the optimal weights, we can then directly construct the approximate value function of the stochastic reach-avoid problem for each $i=1,2,3$. We solved the problem with all methods and Table II compares the theoretical feasibility guarantees (column $\varepsilon$) with the empirical ones (column $\hat{\varepsilon}$) along with the associated complexities (columns “Sampling” and “Solver”). The empirical violation values were calculated by uniformly sampling 1000 realizations from $\Delta$, other than those used in the optimization process, and computing the ratio between the number of realizations that resulted in constraint violation and 1000. We highlight with bold the parameters that can be chosen by the user and are not fixed by the problem data; for the multi-stage scenario approach and the recursive scenario approach with re-sampling, we have chosen the violation levels $\varepsilon_{i}$ at each stage by solving the complexity optimization program in (13). The associated confidence levels $1-\beta_{i},i=1,2,3$ were all fixed to $0.99$ to achieve an overall confidence $1-\beta$ of at least $0.97$. The basis centers and variances were sampled uniformly at random from each safe set and $(0,0.01]$ respectively. All computations were done on an Intel Core i7 Q820 CPU clocked @1.73 GHz with 16GB of RAM memory, using the Gurobi optimization suite. Figure 1 shows the level sets of the approximation at time $i=1$ restricted on $[-1,1]^{2}$, constructed using the recursive scenario approach with re-sampling. Even though the optimal value function corresponds to a reach-avoid probability, the values of the approximation go above 1 since it is only an upper bound [22]. The accuracy of the approximation can be increased by increasing the number of basis elements or reducing the values of $\varepsilon,\beta$. The results in Table II indicate that in this instance it is favorable to solve problems in a recursive manner since the same overall violation levels are respected while the computation times are smaller. Notice that in the standard scenario approach, the total number of samples is three times smaller than the total number of constraints since every sample is enforced on every constraint function in the horizon separately. Moreover, in the multi-stage scenario approach and the recursive scenario approach with re-sampling we have to generate different samples for each constraint function in the horizon and thus sampling consumes more time. The reported solver times differ since differences in the sampled data affect solution time. In particular, the samples used for each of the constraints of the standard scenario approach are identical, giving structure to the problem which appears to be exploited by the solver. For the multi-stage scenario approach different samples are used for each constraint and the resulting optimization program has less structure. The differences in the reported sampling times (even when sample numbers are the same) are a consequence of the hit and run algorithm used to generate them [35]. The numbers reported are averaged over 10 runs of each method. VI Conclusion We investigated the feasibility properties of different scenario based optimization programs, involving the standard scenario approach, its multi-stage counterpart as well as recursive variants that can be employed in case the problem exhibits a separable structure. We showed how confidence and violation levels can be treated as optimization assets and can be selected by means of convex optimization problems to reduce the computation time of the associated algorithm. We verified with a numerical example that the recursive structure often encountered in sequential decision making can be exploited, leading to much shorter computation times. Our future work focuses on utilizing the insights gained in this paper in different problems where the assumed recursive structure is present. 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Some Problems Concerning Interchange of Order of Integration in Functional Integral Formalism of U(1) Gauge Field Theories Wei-Min Sun    Xiang-Song Chen and Fan Wang Department of Physics and Center for Theoretical Physics, Nanjing University, Nanjing 210093, China Abstract We show that in the functional integral formalism of U(1) gauge field theory some formal manipulation such as interchange of order of integration can yield erroneous results. The example studied is analysed by Fubini theorem. pacs: PACS: 11.15.-q,12.20-m,02.30.Cj Functional integral has now become an indispensible tool in the study of field theories, especially gauge field theories. When using the formalism of functional integral we usually assume some formal manipulations to be justified, for example, interchange of order of integration. From the general theory of integration [1] we know that this is not always correct. Usually we do not have to worry about these matters. In this paper we will show that in some circumstances naive interchange of order of integration in the manipulation of functional integral does lead to erroneous results. In the literature [2] there exists a kind of averaging procedure for constructing gauge-invariant quantities out of gauge-variant ones. This averaging procedure can be described as follows(to be specific we restrict ourself to pure U(1) gauge field theories). For an arbitrary operator $O[A]$, gauge invariant or not, we can always define an operator $G_{O}[A]=\int D\omega O[A^{\omega}]$ where $A$ denotes gauge fields, $A^{\omega}$ denotes the result of $A$ after a gauge transformation $\omega$ and $\int D\omega$ stands for functional integration over the gauge group(here we choose the normalization to be $\int D\omega=1$). Using the property of the Haar measure it can be easily proved that $G_{O}[A]$ is gauge-invariant. In [3] we proved that such a method is inapplicable for $O[A]$ which is a gauge-variant polynomial in the gauge field $A_{\mu}(x)$ because the relevant functional integral in $\omega$ is divergent. When this averaging procedure is applicable it is obvoius that the correlation function $$\langle G_{O}[A]\rangle=\int D\omega\langle O[A^{\omega}]\rangle$$ (1) is gauge independent because the correlation function of any gauge-invariant operator is always gauge independent [4]. Sometimes the operator $G_{O}[A]=\int D\omega O[A^{\omega}]$ itself does not exist but the functional integral in the right hand side of eq.(1) is convergent. Then is the quantity $\int D\omega\langle O[A^{\omega}]\rangle$ gauge independent? Naively one may expect this quantity to be gauge independent because we have averaged over the whole gauge group. In the following we will use the formalism of functional integral to give a formal proof to ”show” that the quantity studied is gauge independent, irrespective of whether the operator $G_{O}[A]$ itself exists or not. We will only investigate those $O[A]$ which can be expressed as polynomials in the gauge field $A_{\mu}(x)$. In our case of pure U(1) gauge field theory there are no problems such as Gribov ambiguity and the difficulty of rigorously defining functional measure for interacting quantum field theories. We will work in Euclidean space throughout. Recalling the Faddeev-Popov trick in quantizing a general gauge field theory we introduce a functional $\Delta_{F}[A]$ by the relation $\Delta_{F}[A]\int D\omega\delta(F[A^{\omega}])=1$, where $F[A]$ is a suitable(linear) gauge fixing functional(here we choose the normalization to be $\int D\omega=1$; this is different from the usual one but only affect an irrelevant normalization constant.), and get $$\displaystyle\int D\omega\int DAO[A^{\omega}]\Delta_{F}[A]\delta(F[A])e^{-S[A]}$$ (2) $$\displaystyle=$$ $$\displaystyle\int D\omega\int DAO[A]\Delta_{F}[A]\delta(F[A^{\omega^{-1}}])e^{% -S[A]}$$ $$\displaystyle=$$ $$\displaystyle\int DA\int D\omega O[A]\Delta_{F}[A]\delta(F[A^{\omega^{-1}}])e^% {-S[A]}$$ $$\displaystyle=$$ $$\displaystyle\int DA\int D\omega O[A]\Delta_{F}[A]\delta(F[A^{\omega}])e^{-S[A]}$$ $$\displaystyle=$$ $$\displaystyle\int DAO[A]e^{-S[A]}$$ where in the second line we have changed the variable from $A$ to $A^{\omega^{-1}}$ and used the gauge invariance of $DA$, $\Delta_{F}[A]$ and the action $S[A]$, in the third line we have changed the order of integration and in the fourth line we have used the property $\int D\omega f[\omega]=\int D\omega f[\omega^{-1}]$. Now in the above equation we change $F[A]$ into $F[A]-f$, where $f$ is an arbitary function, and obtain $$\int D\omega\int DAO[A^{\omega}]\Delta_{F}[A,f]\delta(F[A]-f)e^{-S[A]}=\int DAO% [A]e^{-S[A]}$$ (3) Noting that $\Delta_{F}[A,f]\delta(F[A]-f)=\tilde{\Delta}_{F}[A]\delta(F[A]-f)$, where $\tilde{\Delta}_{F}[A]=\det\frac{\delta F[A^{\omega}]}{\delta\omega}|_{\omega=1}$ is the familiar Faddeev-Popov determinant, multiplying both sides with a suitable weight functional $G[f]\sim e^{-\frac{1}{2\lambda}\int d^{4}x[f(x)]^{2}}$(normalized to $\int DfG[f]=1$) and integrating with respect to $f$, we obtain $$\int D\omega\int DAO[A^{\omega}]\tilde{\Delta}_{F}[A]G[F[A]]e^{-S[A]}=\int DAO% [A]e^{-S[A]}$$ (4) Setting $O[A]=1$ in the above equation we get $$\int D\omega\int DA\tilde{\Delta}_{F}[A]G[F[A]]e^{-S[A]}=\int DAe^{-S[A]}$$ (5) Recalling that $$\langle O[A^{\omega}]\rangle=\frac{\int DAO[A^{\omega}]\tilde{\Delta}_{F}[A]G[% F[A]]e^{-S[A]}}{\int DA\tilde{\Delta}_{F}[A]G[F[A]]e^{-S[A]}}$$ (6) we have $$\int D\omega\langle O[A^{\omega}]\rangle=\frac{\int DAO[A]e^{-S[A]}}{\int DAe^% {-S[A]}}$$ (7) That is to say, the quantity $\int D\omega\langle O[A^{\omega}]\rangle$ is gauge independent. Is this conclusion right? Let us see an explicit example. We take $O[A]=A_{\mu}(x)F_{\nu\rho}(y)$. Note that according to the general result in [3] $G_{O}[A]$ does not exist. It can be easily seen that $\langle O[A^{\omega}]\rangle=\langle O[A]\rangle$ because $\langle F_{\nu\rho}(y)\rangle$ vanishes. But the value of $\langle A_{\mu}(x)F_{\nu\rho}(y)\rangle$ differs in covariant and axial gauges and so the above conclusion is wrong! Then where does the above formal proof go wrong? The point is that in the third line of eq.(2) we have illegitimately interchanged the order of integration. When the operator $G_{O}[A]$ exists, that is, the functional integral $\int D\omega O[A^{\omega}]$ converges, there is no problem. But when $\int D\omega O[A^{\omega}]$ diverges problems do arise. If the functional integral $\int D\omega O[A^{\omega}]$ diverges, the iterated integral $\int DA\int D\omega O[A^{\omega}]\Delta_{F}[A]\delta(F[A])e^{-S[A]}$ also does not exist. Then from Fubini theorem [1] we know that the double integral $\int\int D\omega DAO[A^{\omega}]\Delta_{F}[A]\delta(F[A])e^{-S[A]}$ does not exist. Now we do a change of variable: $A^{\omega}=\tilde{A},\omega=\tilde{\omega}$. The Jacobian matrix $J$ of this change of variable can be written as $$\left(\begin{array}[]{cc}\frac{\partial\tilde{A}}{\partial A}&\frac{\partial% \tilde{A}}{\partial\omega}\\ 0&1\end{array}\right)$$ and $\det J=\det\frac{\partial\tilde{A}}{\partial A}=1$. Under this change of variable the double integral changes into $\int\int D\omega DAO[A]\Delta_{F}[A]\delta(F[A^{\omega^{-1}}])e^{-S[A]}$(we have changed the dummy variables $(\tilde{A},\tilde{\omega})$ back into $(A,\omega)$). Obviously this double integral does not exist because the former one does not. In this case the two iterated integrals $\int D\omega\int DAO[A]\Delta_{F}[A]\delta(F[A^{\omega^{-1}}])e^{-S[A]}$ and $\int DA\int D\omega O[A]\Delta_{F}[A]\delta(F[A^{\omega^{-1}}])e^{-S[A]}$ are not necessarily equal, even both of them exist. The above analysis tells us that when the functional integral $\int D\omega O[A^{\omega}]$ does not exist the formal proof of the gauge independence of the quantity $\int D\omega\langle O[A^{\omega}]\rangle$ does not apply. It was proved in [3] that when $O[A]$ is a gauge-variant polynomial in $A_{\mu}(x)$ the operator $G_{O}[A]$ does not exist. Now let us assume $O[A]$ is gauge-variant and directly analyse whether $\int D\omega\langle O[A^{\omega}]\rangle$ exists, and when exists, whether it is gauge independent. To investigate this problem recall that in [3] we have proved that the functional integral $\int D\omega\partial_{\mu}\theta(x)$ does not exist. Using similar reasoning it can be shown that the functional integral $\int D\omega\int dxK^{\mu}(x)\partial_{\mu}\theta(x)$ does not exist for a general $K^{\mu}(x)$. Now we will show that the functional integral $\int D\omega P[\partial_{\mu}\theta(x)]$ also does not exist for any polynomial $P[\partial_{\mu}\theta(x)]$. The proof is by mathematical induction. We already know that this statement is true for any first order polynomial in $\partial_{\mu}\theta(x)$. Assume this statement holds for any $n$-th order polynomial $P_{n}[\partial_{\mu}\theta(x)]$, let us see an $(n+1)$-th order polynomial $P_{n+1}[\partial_{\mu}\theta(x)]$. If the functional integral $\int D\omega P_{n+1}[\partial_{\mu}\theta(x)]$ exist, it should be equal to the functional integral $\int D\omega P_{n+1}[\partial_{\mu}\theta(x)+\partial_{\mu}\theta_{0}(!x)]$,where $\theta_{0}(x)$ is an arbitary function. From these we find the functional integral $\int D\omega\{P_{n+1}[\partial_{\mu}\theta(x)+\partial_{\mu}\theta_{0}(x)]-P_{% n+1}[\partial_{\mu}\theta(x)]\}$ exists and equals zero. But note that the integrand of this functional integral is an $n$-th order polynomial in $\partial_{\mu}\theta(x)$, hence a contradiction with our assumption. That is to say, the functional integral $\int D\omega P_{n+1}[\partial_{\mu}\theta(x)]$ also does not exist. From the principle of mathematical induction our statement is proved. Now let us turn to the operator $\int D\omega O[A^{\omega}]$ and the quantity $\int D\omega\langle O[A^{\omega}]\rangle$. When the operator $O[A]$ is a gauge-variant polynomial in $A_{\mu}(x)$ the operator $O[A^{\omega}]$ and the correlation function $\langle O[A^{\omega}]\rangle$ should be a polynomial in $\partial_{\mu}\theta(x)$. According to the above proved result the functional integral $\int D\omega O[A^{\omega}]$ does not exist(note that this provides another proof of the final conclusion in [3]). The functional integral $\int D\omega\langle O[A^{\omega}]\rangle$ also does not exist in general. But there is a special case: $\langle O[A^{\omega}]\rangle$ is a constant independent of $\partial_{\mu}\theta(x)$, i.e.,$\langle O[A^{\omega}]\rangle=\langle O[A]\rangle$(our explicit example $A_{\mu}(x)F_{\nu\rho}(y)$ is such a case). In other words the gauge variation of $O[A]$ vanishes after taking the vacuum expectation value. In this case the quantity $\int D\omega\langle O[A^{\omega}]\rangle$ does exist; but unfortunately it is generally gauge dependent because $O[A]$ is a gauge-variant operator. Returning back to the problem of whether we can interchange the order of integration in our functional proof we conclude that this formal manipulation is erroneous in this case. In summary we showed that when $O[A]$ is a gauge-variant polynomial in the gauge field $A_{\mu}(x)$ the quantity $\int D\omega\langle O[A^{\omega}]\rangle$ is gauge dependent in general(if it exists) even though we have averaged over the gauge group. The formal functional proof for the gauge independence of this quantity does not apply because one have illegitimately changed the order of integration when doing the functional integral. This work is supported in part by the NSF(19675018), SED and SSTD of China. References [1] See, for inatance, W.Rudin Real and Complex Analysis, 3rd edition, Ch8 (McGraw-Hill, 1987) [2] I.I.Kogan and A.Kovner, Phys. Rev. D51 (1995) 1948. [3] W.M.Sun, X.S.Chen and F.Wang, Phys. Lett. B483 (2000) 299, hep-th/0005060 [4] See, for instance, S. Weinberg The Quantum Theory of Fields, vol 2, Ch 15 (Cambridge University Press, 1996)

Correlation search for coherent pion emission in heavy ion collisions S.V. Akkelin${}^{2}$    R. Lednicky${}^{1,3}$ and Yu.M. Sinyukov${}^{1,2}$ Abstract The methods allowing to extract the coherent component of pion emission conditioned by the formation of a quasi-classical pion source in heavy ion collisions are suggested. They exploit a nontrivial modification of the quantum statistical and final state interaction effects on the correlation functions of like and unlike pions in the presence of the coherent radiation. The extraction of the coherent pion spectrum from $\pi^{+}\pi{}^{-}$ and $\pi^{\pm}\pi^{\pm}$ correlation functions and single–pion spectra is discussed in detail for large expanding systems produced in ultra-relativistic heavy ion collisions. ${}^{1}$ SUBATECH, (UMR, Universite, Ecole des Mines, IN2P3/CNRS), [0pt] 4, rue Alfred Castler, F-44070 Nantes Cedex 03, France. [0pt] ${}^{2}$ Bogolyubov Institute for Theoretical Physics, Kiev 03143, Metrologicheskaya 14b,Ukraine. [0pt] ${}^{3}$ Institute of Physics ASCR, Na Slovance 2, 18221 Prague 8, Czech Republic. [0pt] I Introduction The hadronic observables, such as single- or multi-particle hadron spectra, play an important role in the studies of ultra-relativistic heavy ion collisions. However, these observables contain rather indirect information on the initial stage of the collision process since the particle interactions result in substantial stochastization and thermalization of a system during its evolution. Nevertheless, the final hadronic state can carry some residual signals of the earlier stages of the particle production process. A partial coherence of the produced pions is supposed to be one of the important examples. The first systematic study of coherent processes in high energy hadron-nucleus ($h+A$) collisions was based on Glauber theory [1]. In this theory, the $h+A$ collision is considered as a process of subsequent scatterings of the projectile on separate nucleons of the nucleus; the projectile energies are supposed much higher than the inverse nucleus radius ($E_{h}\gg 1/R$), thus allowing to consider a linear projectile trajectory inside the nucleus (eikonal approximation). If the scattering process occurred with almost no recoil of the nucleus nucleons, i.e. with no witnesses of the individual scatterings, then the $h+A$ collision should be described by a coherent superposition of the elementary hadron-nucleon scattering amplitudes. Such a type of the collision is called coherent scattering. Since the nucleus in coherent scattering does not change its state, it manifests itself just as a particle with some form-factor. In the oscillator approximation, the nucleus form-factor can be represented by a Gaussian: $\exp(-{\bf q}^{2}R^{2}/4)$. The coherent processes are essential only for small momenta transferred from the projectile hadron to the nucleus: $\left|{\bf q}\right|<1/R.$ Then, one can neglect the recoil energy and consider the nucleus as a whole during the scattering process. There is a kinematic limitation of the minimal longitudinal momentum transfer, $\left|q_{z}\right|_{\min}\approx(M^{2}-m_{h}^{2})/(2\left|{\bf p}_{h}\right|)$, required to produce a particle or a group of particles of the invariant mass $M$. The vanishing of $\left|q_{z}\right|_{\min}$ with the increasing energy explains why the coherent processes can take place only at high enough energies. It is worth noting that the total coherent cross-section does not die out with the increasing energy (see., e.g., [2]).111We are grateful to V. L. Lyuboshitz for drawing our attention to this important point and for an interesting discussion. Typically, however, the transferred momenta are sufficient for substantial recoil effects and the excitation of the nucleus or its breakup. Then, due to a small coherence length $\sim 1/\left|{\bf q}\right|$, the nucleus does not participate in the collision as a whole and one can consider the $h+A$ collision as an incoherent superposition of elementary hadron-nucleon scatterings corresponding to random phases of the amplitudes of the latter. The resulting cross-section is then given by the sum of the moduli squared of the amplitudes (probabilities) at each of the possible scattering points (unlike to coherent scattering, when the individual amplitudes are summed up first). As a result, the particles are produced in chaotic (incoherent) states. Let us come back to the production of particles (e.g., pions) in the processes of non-elastic coherent scattering at small transferred momenta. Since the nucleus is not excited in these processes and manifests itself as a quasi-classical object, one can describe particle production using the quantum field model of interaction with a classical source [3]. It is well known that the interaction with a classical source results in the production of bosons in coherent states [4]. These states minimize the uncertainty relation and, so, are the closest to classical ones.222The coherent states have been introduced and studied in detail by Glauber [5]. The concept of coherent states was then applied to pion production in high energy processes in Refs. [6, 7, 8]. This is the main physical link between the processes of coherent scattering and particle production in coherent states. In heavy ion collisions at high energies, the average multiplicities are quite high, e.g., several thousands of pions can be produced at maximal RHIC energies. The inclusive particle spectra thus represent natural characteristics of these processes. A convenient way to account for the coherent properties of these processes consists in a model description of particle emission, rather than in detailed evaluation of the contributing amplitudes. The Gyulassy-Kauffmann-Wilson (GKW) model [8] is an example of such an approach. The model assumes that all pions are radiated by classical currents (sources) which are produced in some space–time region during the collision process. The corresponding density matrix is constructed by averaging over the unobservable positions of the centers of individual sources. The pion spectra then effectively contain both chaotic and coherent components. In fact, the chaotic component dominates in case of a large emission region, while, in the opposite limit of very small space–time extent of this region, almost all pions are produced in the coherent state. This seems to be rather general result: if the distances between the centers of pion sources are smaller than the typical wave length of the quanta (the source size), the substantial overlap of the wave packets leads to the strong correlations (indistinguishability) between the phases in pion wave functions and, thus, to the coherence[9, 10]. Recently, the coherence of multipion radiation in high energy heavy ion collisions was studied within GKW model in Ref. [11]. In the model, due to the longitudinal Lorentz contraction of the colliding nuclei, almost all pions produced with small transverse momenta $p_{t}<1/R$ in central nucleus-nucleus collisions are emitted coherently, and their momentum spectra are determined by the system’s space–time extent. Clearly, the coherence of pions can be destroyed by pion rescatterings. Nevertheless, the duration of hadron formation may happen to be long enough to allow a considerable part of the coherent pions escape from the interaction zone without rescatterings [11]. However, as noted in [11], one can expect a strong suppression of the GKW mechanism of coherent pion production if quark-gluon plasma were created: the hadronization then occurs in a thermal quark-gluon system and hadrons are produced in the chaotic state only. Note that clear signals of the thermalization and collective flows, observed at CERN SPS and RHIC energies (see, e.g., [12, 13] and references therein), point to strong rescattering effects and may reflect also the importance of the quark-gluon degrees of freedom. The new physical phenomena, expected in RHIC and LHC experiments with heavy ions, are associated with the creation of quasi-macroscopic, very dense and hot systems. In such systems, the deconfinement phase transition and the restoration of the chiral symmetry are likely to happen, possibly leading to creation of the new states of matter: quark-gluon plasma (QGP) and disoriented chiral condensate (DCC). In the latter case, another possibility for the coherent pion radiation (above the thermal background) appears. If the DCC were created at the chiral phase transition, a quasi–classical pion field $\stackrel{{\scriptstyle\rightarrow}}{{\pi}}_{cl}$ forms the ground state of the system. The subsequent system decay is accompanied by a relaxation of the ground state to normal vacuum. Such a process can be described by the quantum field model of interaction with a classical source (see, e.g. [14]), and results in the coherent pion radiation. One of the general conditions of the ground state rearrangement and formation of the quasi–classical field is a large enough system volume [15]. Therefore, such a field could be generated in heavy ion collisions at sufficiently high energies provided the spontaneous chiral symmetry breaking via DCC formation takes place. The overpopulation of the (quasi) pion medium, making it close to the Bose-Einstein condensation point, can lead to the strengthening of the coherent component conditioned by the ground state (quasi-particle vacuum) decay [16]. Since the DCC appears relatively late (at the end of the hadronization stage), the coherent radiation could partially survive and be observed. The coherent emission manifests itself in a most direct way in the inclusive correlation function $C(p,q)$ of two identical bosons in the region of very small $|{\bf q}|$; $p=(p_{1}+p_{2})/2$, $q=p_{1}-p_{2}$. In case of only chaotic contribution, the intercept of the quantum statistical (QS) Bose-Einstein part of the correlation function $C_{QS}(p,0)=2$ [17] while, in the presence of the coherent radiation, $C_{QS}(p,0)<2$. Generally, the coherence means strong phase correlations of different radiation components. In Ref. [9], a simple quantum–mechanical model of the phase–correlated one-particle wave packets with different radiation centers has been considered. In such a case (corresponding to indistinguishable correlated emitting centers), the emission amplitude $A(p)$ averaged over the event ensemble is not equal to zero, $\langle A(p)\rangle\neq 0$, and the QS correlation function intercept $C_{QS}(p,0)<2.$ In the second quantization representation (more adequate for processes of multi–boson production), the analogous results take place for inclusive averages of the quantum field operators: $\langle a(p)\rangle\neq 0$, $C_{QS}(p,0)<2$, provided the radiation has a non-zero coherent state component. The latter represents a superposition of the states of all possible boson numbers at fixed phase relations. In practice, most of the correlation measurements is done with charged particles. However, charged bosons cannot form the usual coherent state since it obviously violates the super-selection rule. To overcome this difficulty, the generalized concept of charge-constrained coherent states should be used [7, 8, 18]. Nevertheless, the correlations of charged bosons are usually described with the help of ordinary (not charge-constrained) coherent states [19, 20] (see, however, Refs. [21, 22]). Our treatment of two-pion correlations takes into account the restrictions imposed by the super-selection rule and is based on the density matrix formalism. The density matrix approach gives the possibility to describe, in a natural way, the chaotic radiation (the initial state then corresponding to a local-equilibrium statistical operator of quasi–particle excitations) and coherent emission (arising due to the interaction with a classical source). This approach can easily incorporate also the squeeze-state component of pion radiation [23], appearing due to the modification of the energy spectrum of quasi-pions as compared with that of free pions [24]. The density matrix formalism is also simply related with the Wigner function description of the multiparticle phase-space and its evolution governed by the relativistic transport equation [25], representing very useful tools with a clear classical limit. Recent development of the classical current approach to multiparticle production [23, 19] has made it closer to the density matrix formalism; particularly, the clasical current in momentum space has been shown mathematically identical with the coherent-state representation of the density matrix, the latter called ”$P$” or Glauber-Sudarshan representation [5], see also [26]. In our approach, the super-selection rule requires an averaging, in the density matrix, over all orientations of the quasi-classical pion source in the isospin space. As a consequence, the averaged pion field vanishes: $\langle a(p)\rangle=0$ whereas, for identical pions, the intercept $C_{QS}(p,0)$ is still less than 2. The correlations of non-identical pions also appear to be sensitive to the presence of the quasi–classical source. This sensitivity arises due to properties of the generalized coherent states satisfying, after the averaging over all orientations of the quasi-classical source in isospin space, the super-selection rule for charged particles. Due to isospin symmetry of the strong-interaction Hamiltonian, there are unique relations for the intercepts $C_{QS}^{ij}(p,0)$ of the pure QS correlation functions of two pions in various charge states $i,j={\pm},{0}$. For example, the coherence suppression of $C^{\pm\pm}$ determines the coherence enhancement of $C^{+-}$. The coherence phenomena can be, however, masked by a number of effects suppressing the measured correlation functions. The most important among them are the decays of long-lived particles and resonances (e.g., $\Lambda$, $K_{s}^{0}$, $\eta$, $\eta^{\prime}$, $\dots$), the single- and two–track resolution and particle contamination. In Ref.[27], the method to discriminate between the effects of coherent radiation and decays of long-lived resonances has been proposed. The method assumes the simultaneous analysis of two- and three--particle correlation functions of identical pions. The practical utilization of the method is however difficult due to a low statistics of near--threshold three-pion combinations and the problem of the three--particle Coulomb interaction; also, one has to account for the super--selection rule.333The latter problems are absent for neutral pions. However, sufficiently accurate measurements of neutral pion correlations are practically out of the present experimental possibilities. Therefore, in the present work we will restrict ourselves to the consideration of two-particle correlation functions. In addition to QS, the correlations of particles with small relative velocities are also influenced by their final state interaction (FSI). The effect of the latter on two–particle correlations is well understood and introduces no principle problems. It is important that the correlations in different two–pion systems are influenced by the QS, FSI and coherence effects in a different way. This offers a possibility to discriminate different effects suppressing the measured correlation functions and so to extract the coherent contribution using correlation functions of like and unlike pions measured at small relative momenta. In the paper we study the influence of the coherent pion radiation on the behavior of pion inclusive spectra and two–pion correlation functions and, based on it, develop the methods for the extraction of the coherent component above the chaotic background. Despite we associate the coherent radiation with the formation of the DCC (as the most probable mechanism of the coherence in ultra–relativistic A+A collisions), our results are rather general. Actually, they are based on the general properties of the coherent pion radiation: the quasi-classical nature of the coherent pion source and the constrains imposed by the charge super–selection rule. In Sec. II, we consider a general form of the density matrix of partially coherent pions, and calculate quantum statistical correlations of identical and nonidentical pions. In Sec. III, we set forth the density matrix formalism taking into account the decays of short-lived resonances and FSI of produced pions, and calculate the corresponding correlation functions. In Sec. IV, we discuss how to extract the coherent component of particle radiation from the two–pion correlation functions, particularly, in the case of large expanding systems produced in ultra-relativistic A+A collisions. A short summary and conclusion are given in Sec. V. II Quantum statistical correlations of partially coherent pions It is well known that the description of the inclusive pion spectra and two-pion correlations is based on a computation of the following averages [8]: $$\begin{array}[]{c}\omega_{{\bf p}}\frac{d^{3}N_{i}}{d^{3}{\bf p}}\equiv n_{i}(% p)=\sum\limits_{\alpha}|{\cal T}(in;p,\alpha)|^{2}=\bigl{\langle}a_{i}^{% \dagger}({p})a_{i}({p)}\bigr{\rangle},\\ \text{ }\\ \;\omega_{{\bf p}_{1}}\omega_{{\bf p}_{2}}\frac{d^{6}N_{ij}}{d^{3}{\bf p}_{1}d% ^{3}{\bf p}_{2}}\equiv n_{ij}(p_{1},p_{2})=\sum\limits_{\alpha}|{\cal T}(in;p_% {1},p_{2},\alpha)|^{2}=\bigl{\langle}a_{i}^{\dagger}({p}_{1})a_{j}^{\dagger}({% p}_{2})a_{i}({p}_{1})a_{j}({p}_{2})\bigr{\rangle},\text{ }\\ \\ C^{ij}(p,q)=n_{ij}(p_{1},p_{2})/n_{i}(p_{1})n_{j}(p_{2}),\text{ }\omega_{{\bf p% }_{i}}=\sqrt{m^{2}+{\bf p}_{i}^{2}},\end{array}$$ (1) where ${\cal T}(in;p,\alpha)$ is the normalized invariant production amplitude. The summation is done over all quantum numbers $\alpha$ of other produced particles, including integration over their momenta; $a_{i}^{\dagger}({p})$ and $a_{i}({p)}$ are respectively the creation and annihilation operators of asymptotically free pions $i={\pm},{0}$; the bracket $\langle\dots\rangle$ formally corresponds to the averaging over some density matrix $|f\rangle\langle f|.$ A special attention requires the production of particles with near-by velocities which can be strongly influenced particle interaction in the final state. In this Section, we concentrate mainly on quantum statistical correlations ignoring, for a while, the effects of resonance decays and FSI. Let us suppose that the density matrix ${\bf\rho}$ is a statistical operator describing the thermal hadronic system in a pre-decaying state on a hyper-surface of the thermal freeze-out $\sigma_{f}:t=t_{f}({\bf x})$. After the thermal freeze-out the system is out of local thermal equilibrium but still can be in a pre-decaying (interacting) state. In fact, the complete decay (neglecting the long-time scale forces) happens at some finite asymptotic times $t_{out}<\infty$. Then the formal solution of the Heisenberg equation for the pionic annihilation (creation) operators at this post thermal freeze-out stage has the form444For a space–like hypersurface $\sigma_{f}$ (an example is $\sigma_{f}=t_{f}({\bf x})=(\tau^{2}+x_{long}^{2})^{1/2}$ in the Bjorken hydrodynamic model with the proper expansion time $\tau$), the use of the covariant Tomonaga–Schwinger formalism gives the same result with the substitution $t\rightarrow t({\bf x})$.: $${\rm a}_{i,qm}({\bf p},t_{out})=[{\rm a}_{i,qm}({\bf p}{,}t_{f})+{\rm d}_{i}({% \bf p,}t_{f},t_{out})]e^{-i\omega_{{\bf p}}(t_{out}-t_{f})}.$$ (2) It formally corresponds to the sum of the general solution of the free (homogeneous) Heisenberg equation of motion for pionic field (first term), and a particular solution of the complete (inhomogeneous) Heisenberg equation with a source (second term). The value ${\rm d}_{i}({\bf p,}t_{f},t_{out})$ depends on the actual form of the source term in the Heisenberg equation. The decay of the system at this stage, $t_{f}<t<t_{out}$, can be accompanied by the coherent pion radiation due to the modification of hadron properties in hot and dense hadronic environment or - due to some peculiarities of the phase transition from QGP to hadron gas, e.g., the formation of DCC. In both cases, almost non–interacting quasiparticle excitations could be formed above a rearranged ground state (”condensate”). In the systems containing the DCC, the appearance of the quasi-classical pion field $\stackrel{{\scriptstyle\rightarrow}}{{\pi}}_{cl}$ (corresponding to the density of virtual pionic excitations of the quasi-pionic vacuum) at the thermal stage is usually described in the mean field approximation as $\pi_{i,cl}(x)=\pi_{i}(x)-\pi_{i,qm}(x),$ where the field $\pi_{i,qm}(x)$ corresponds to the quasi-pion quantum excitations above the temporary vacuum background $\pi_{i,cl}(x)$ (the order parameter). Assuming the isotopic symmetry of the Lagrangian like in the sigma model (see, e.g., [28]), we have $\pi_{i,cl}(x)=e_{i}\pi_{cl}(x)$, where ${\bf e}$ is randomly oriented unit vector, ${\bf e}^{2}=1$, in the three-dimensional isospin space. Then, for each ${\bf e-}$orientation of the quasi-pionic vacuum at the thermal freeze–out, the free quasi–pions $\stackrel{{\scriptstyle\rightarrow}}{{\pi}}_{qm}$are distributed according to the Gibbs local-equilibrium density matrix $\rho_{{\bf e}}$ above the quasi-pionic vacuum. After the thermal freeze-out, when the decay of such a thermal system happens, the quasi-pion masses approach the usual free particle values and the condensate (the temporary disoriented vacuum) tends to relax back to the normal vacuum by emitting physical pions in coherent states - the vacuum for quasi-particles becomes a coherent state for free particles. The latter process is similar to particle radiation by a classical source. Then the ”source” term in Eq. (2) takes on the form $${\rm d}_{i}({\bf p,}t_{f},t_{out})={\rm d}_{i,qm}({\bf p,}t_{f},t_{out})+e_{i}% {\rm d}_{coh}({\bf p,}t_{f},t_{out}),\quad e_{0}=\cos\theta\text{ , }e_{\pm}=% \frac{\sin\theta}{\sqrt{2}}e^{\pm i\phi},$$ (3) where ${\rm d}_{i,qm}({\bf p,}t_{f},t_{out})$ and $e_{i}{\rm d}_{coh}({\bf p,}t_{f},t_{out})$ are q- and c-value quantities respectively. While the total number of pions of momentum ${\bf p}$ radiated by a classical source is fixed by $\left|{\rm d}_{coh}({\bf p,}t_{f},t_{out})\right|^{2}$, the distribution of radiating pions in isospace is determined by the orientation of the vector ${\bf e}$; we suppose ${\bf e}$ independent of $x$. We further assume that the quasi-pion masses at the thermal freeze-out are near the physical mass, $m_{i}(t_{f})\simeq m_{out}\equiv m$, neglecting a possible mass shift which can generate squeeze-state components in particle radiation.555Squeeze-state component can arise also in a strongly inhomogeneous thermal boson system for particles with wavelengths larger than the system’s homogeneity lengths [29]. Below we will assume the pion Compton wave–length much smaller than the typical system lengths of homogeneity (e.g., hydrodynamical lengths) at the thermal freeze-out hypersurface $\sigma_{f}$. We will neglect the rescatterings at the post thermal freeze-out stage, i.e. put ${\rm d}_{i,qm}({\bf p,}t_{f},t_{out})\approx 0$, and approximately describe the production of coherent pions at this stage by the quantum field model of the interaction with a classical source [3]. Then, there is well known linear relationship between the annihilation (creation) operators diagonalizing the pion field Hamiltonian at the times $t_{f\text{ }}$ and $t_{out}$ ($i={\pm},{0}$): $${\rm a}_{i,qm}({\bf p},t_{out})=[{\rm a}_{i,qm}({\bf p}{,}t_{f})+e_{i}{\rm d}_% {coh}({\bf p,}t_{f},t_{out})]e^{-i\omega_{{\bf p}}(t_{out}-t_{f})},\quad$$ (4) where the c-value quantity ${\rm d}_{coh}({\bf p,}t_{f},t_{out})$ depends on a mechanism and the rate of the classical field decay.666It follows, from the continuity of the complete field $\pi_{i}(x)$ and its derivative at $t=t_{f}$ that, for a fast freeze-out ($t_{out}-t_{f}\rightarrow 0$), the quantity ${\rm d}_{coh}({\bf p,}t_{f},t_{out})$ is directly associated with the strength of the pion condensate. On the other hand, an adiabatically slow switch-off of the classical source yields ${\rm d}_{coh}({\bf p,}t_{f},t_{out})\approx 0$ [3]. The operators $a_{i}(p)$ of the asymptotic free pion field (with the origin of the time coordinate shifted to the point $t_{f}$) are connected with the operators ${\rm a}_{i,qm}({\bf p},t)$ taken at the asymptotic times $t_{out}$ by the relation [30] $$a_{i}(p)=\sqrt{p_{0}}e^{ip_{0}(t_{out}-t_{f})}{\rm a}_{i,qm}({\bf p},t_{out}),% ~{}~{}~{}p_{0}=\omega_{{\bf p}}.$$ (5) Eqs. (4)and (5) allow to calculate the mean values of the asymptotic operators $a_{i}(p)$ and $a_{i}^{\dagger}(p)$ for each ${\bf e}$-orientation of the quasi-pion vacuum applying the thermal Wick theorem to the operators ${\rm a}_{i,qm}({\bf p},t_{f})$ and ${\rm a}_{i,qm}^{\dagger}({\bf p},t_{f})$. The Gaussian form of the statistical operator ${\bf\rho}_{{\bf e}}$ guarantees that $\langle{\rm a}_{i,qm}({\bf p},t_{f})\rangle_{{\bf e}}=0$ for any fixed isospin orientation ${\bf e}$ of the quasi-particle vacuum. Then, $$\begin{array}[]{c}\bigl{\langle}a_{i}^{\dagger}({p}_{1})a_{j}^{\dagger}({p}_{2% })a_{i}({p}_{1})a_{j}({p}_{2})\bigr{\rangle}_{{\bf e}}=\bigl{\langle}a_{i}^{% \dagger}({p}_{1})a_{i}({p}_{1})\bigr{\rangle}_{{\bf e}}\bigl{\langle}a_{j}^{% \dagger}({p}_{2})a_{j}({p}_{2})\bigr{\rangle}_{{\bf e}}+\\ \\ \delta_{ij}\left[\bigl{\langle}a_{i}^{\dagger}({p}_{2})a_{i}({p}_{1})\bigr{% \rangle}_{{\bf e}}\bigl{\langle}a_{i}^{\dagger}({p}_{1})a_{i}({p}_{2})\bigr{% \rangle}_{{\bf e}}-\bigl{\langle}a_{i}^{\dagger}({p}_{1})\bigr{\rangle}_{{\bf e% }}\bigl{\langle}a_{i}^{\dagger}({p}_{2})\bigr{\rangle}_{{\bf e}}\bigl{\langle}% a_{i}({p}_{1})\bigr{\rangle}_{{\bf e}}\bigl{\langle}a_{i}({p}_{2})\bigr{% \rangle}_{{\bf e}}\right].\end{array}$$ (6) Here $$\bigl{\langle}a_{i}^{\dagger}({p}_{1})a_{i}({p}_{2})\bigr{\rangle}_{{\bf e}}=% \bigl{\langle}a_{i}^{\dagger}({p}_{1})a_{i}({p}_{2})\bigr{\rangle}_{ch}+\bigl{% \langle}a_{i}^{\dagger}({p}_{1})\bigr{\rangle}_{{\bf e}}\bigl{\langle}a_{i}({p% }_{2})\bigr{\rangle}_{{\bf e}},$$ (7) where the irreducible (thermal) part of the two-operator average $$\bigl{\langle}a_{i}^{\dagger}({p}_{1})a_{i}({p}_{2})\bigr{\rangle}_{ch}=\sqrt{% p_{10}p_{20}}\bigl{\langle}{\rm a}_{i,qm}^{\dagger}({\bf p}_{1},t_{f}){\rm a}_% {i,qm}({\bf p}_{2},t_{f})\bigr{\rangle}_{{\bf e}}$$ (8) does not depend on ${\bf e}$ 777Such a dependence could take place if the mass shift were non-zero and dependent on the ${\bf e}$-orientation of the quasi-pion vacuum. and $$\bigl{\langle}a_{i}({p})\bigr{\rangle}_{{\bf e}}=e_{i}d({p})\equiv e_{i}\sqrt{% p_{0}}{\rm d}_{coh}({\bf p,}t_{f},t_{out})\text{ .}$$ (9) One can introduce the one-particle Wigner function [25] $$f_{{\bf e},i}(x,p)=(2\pi)^{-3}\int d^{4}q^{\prime}\delta(q^{\prime}\cdot p)e^{% iq^{\prime}x}\bigl{\langle}a_{i}^{\dagger}(p+q^{\prime}/2)a_{i}(p-q^{\prime}/2% )\bigr{\rangle}_{{\bf e}},$$ (10) satisfying the relation $$p_{\mu}\partial^{\mu}f_{{\bf e},i}(x,p)=0$$ (11) and describing the phase-space density of the non-interacting pions at $t\geqslant$ $t_{out}$ or, in covariant formalism, at $t\geqslant\sigma_{out}=t_{out}({\bf x})$; here $\sigma_{out}$ is a space-time hypersurface where the interactions are ”switched off” and particles can be considered as free. From Eq. (10), we get $$\bigl{\langle}a_{i}^{\dagger}({p}_{1})a_{i}({p}_{2})\bigr{\rangle}_{{\bf e}}=% \int\limits_{\sigma_{out}}d\sigma_{\mu}p^{\mu}f_{{\bf e},i}(x,p)e^{-iqx},~{}~{% }q=p_{1}-p_{2},~{}~{}p=(p_{1}+p_{2})/2.$$ (12) Using Eqs. (7), (9) and (12), one can split the Wigner function into the chaotic ($ch$) and coherent ($coh$) components: $$f_{{\bf e},i}(x,p)=f_{ch}(x,p)+|e_{i}|^{2}f_{coh}(x,p).$$ (13) Integrated over $\sigma_{out}$, these components determine the operator averages $\bigl{\langle}a_{i}^{\dagger}({p}_{1})a_{i}({p}_{2})\bigr{\rangle}_{ch}$ and $\bigl{\langle}a_{i}^{\dagger}({p}_{1})\bigr{\rangle}_{{\bf e}}\bigl{\langle}a_% {i}({p}_{2})\bigr{\rangle}_{{\bf e}}$ respectively: $$\begin{array}[]{l}\bigl{\langle}a_{i}^{\dagger}({p}_{1})a_{i}({p}_{2})\bigr{% \rangle}_{ch}=\int\limits_{\sigma_{out}}d\sigma_{\mu}p^{\mu}e^{-iq\cdot x}f_{% ch}(x,p),\\ \\ \bigl{\langle}a_{i}^{\dagger}({p}_{1})\bigr{\rangle}_{{\bf e}}\bigl{\langle}a_% {i}({p}_{2})\bigr{\rangle}_{{\bf e}}=|e_{i}|^{2}d^{\ast}({p}_{1})d({p}_{2})=|e% _{i}|^{2}\int\limits_{\sigma_{out}}d\sigma_{\mu}p^{\mu}e^{-iq\cdot x}f_{coh}(x% ,p).\end{array}$$ (14) We suppose that the system has zero average charge and calculate the observables averaging over the random orientation of the quasi-pion vacuum in the isospin space ($d\Omega({\bf e})=d\cos\theta d\phi$): $$\bigl{\langle}...\bigr{\rangle}\equiv Sp(...{\bf\rho)}=(4\pi)^{-1}{\bf\int}d% \Omega({\bf e})\langle\dots\rangle_{{\bf e}}\equiv(4\pi)^{-1}{\bf\int}d\Omega(% {\bf e})Sp(\dots{\bf\rho}_{{\bf e}}).$$ (15) The observable pion field is related to the ensemble of events only, so the corresponding complete averages of the asymptotically free operators vanish, for example, $\bigl{\langle}a_{\pi^{+}}({p})\bigr{\rangle}=(4\pi)^{-1}{\bf\int}d\Omega({\bf e% )}\bigl{\langle}a_{\pi^{+}}({p})\bigr{\rangle}_{{\bf e}}=0$. The averages of these operators also vanish for charge-constrained coherent pion states $|c\rangle$, the states of a fixed electric charge and isospin - so called generalized coherent states [7, 8, 18]. This means that the density matrix $\rho$ can be represented as a weighted sum of the projection operators $|c\rangle\langle c|$ of these states. To illustrate this statement, let us consider a simple artificial case of only two sorts of oppositely charged bosons in one mode. Then the usual coherent states $\bigl{|}\alpha_{\lambda}\bigr{\rangle}$, $\lambda=\pm$, are $$\begin{array}[]{l}\bigl{|}\alpha_{\lambda}\bigr{\rangle}=\exp(-\frac{1}{2}% \left|\alpha_{\lambda}\right|^{2})\stackrel{{\scriptstyle\infty}}{{\mathrel{% \mathop{\sum}\limits_{n=0}}}}\frac{\alpha_{\lambda}^{n}}{(n!)^{1/2}}\bigl{|}n_% {\lambda}\bigr{\rangle},\quad a_{\lambda}\bigl{|}\alpha_{\lambda}\bigr{\rangle% }=\alpha_{\lambda}\bigl{|}\alpha_{\lambda}\bigr{\rangle},\\ \\ \bigl{|}n_{\lambda}\bigr{\rangle}=(n!)^{-1/2}(a_{\lambda}^{\dagger})^{n}\bigl{% |}0_{\lambda}\bigr{\rangle},\quad\left[a_{\lambda},a_{\lambda^{\prime}}^{% \dagger}\right]=\delta_{\lambda\lambda^{\prime}},\quad\alpha_{\pm}=\left|% \alpha\right|e^{\pm i\phi}.\end{array}$$ (16) These states represent superpositions of the states with different charges and so violate the super-selection rule. The charge-constrained coherent state $|c_{0}\rangle$ of charged quanta with a zero total charge may be obtained by projecting this state out from the charge-unconstrained two-component coherent state $\bigl{|}\alpha_{+}\bigr{\rangle}\bigl{|}\alpha_{-}\bigr{\rangle}$ [18]: $$|c_{0}\rangle=\frac{1}{2\pi}\stackrel{{\scriptstyle 2\pi}}{{\mathrel{\mathop{% \int}\limits_{0}}}}d\phi\bigl{|}\alpha_{+}\bigr{\rangle}\bigl{|}\alpha_{-}% \bigr{\rangle}=\exp(-\left|\alpha\right|^{2})\stackrel{{\scriptstyle\infty}}{{% \mathrel{\mathop{\sum}\limits_{n=0}}}}\frac{\left|\alpha\right|^{2n}}{n!}\bigl% {|}n_{+}\bigr{\rangle}\bigl{|}n_{-}\bigr{\rangle}.$$ (17) One may see that the zero charge state $|c_{0}\rangle$ represents a superposition of the states with the same charges (with equal numbers of particles and antiparticles) and thus satisfies the super-selection rule. Similarly, the density matrix $$\begin{array}[]{c}\widehat{\rho}=\frac{1}{2\pi}\stackrel{{\scriptstyle 2\pi}}{% {\mathrel{\mathop{\int}\limits_{0}}}}d\phi\left|\alpha_{+}\bigr{\rangle}\left|% \alpha_{-}\bigr{\rangle}\bigl{\langle}\alpha_{+}\right|\bigl{\langle}\alpha_{-% }\right|=\\ \\ \exp(-2\left|\alpha\right|^{2})\stackrel{{\scriptstyle\infty}}{{\mathrel{% \mathop{\sum}\limits_{n_{1}=0}}}}\stackrel{{\scriptstyle\infty}}{{\mathrel{% \mathop{\sum}\limits_{n_{2}=0}}}}\stackrel{{\scriptstyle\infty}}{{\mathrel{% \mathop{\sum}\limits_{n_{3}=0}}}}\stackrel{{\scriptstyle\infty}}{{\mathrel{% \mathop{\sum}\limits_{n_{4}=0}}}}\frac{\left|\alpha\right|^{n_{1}+n_{2}+n_{3}+% n_{4}}}{(n_{1}!)^{1/2}(n_{2}!)^{1/2}(n_{3}!)^{1/2}(n_{4}!)^{1/2}}\delta_{n_{1}% -n_{2},n_{3}-n_{4}}\left|n_{1,+}\bigr{\rangle}\left|n_{2,-}\bigr{\rangle}\bigl% {\langle}n_{3,+}\right|\bigl{\langle}n_{4,-}\right|\end{array}$$ (18) describes the mixture of the charge-constrained coherent states $|c_{n}\rangle$: $$\widehat{\rho}=\stackrel{{\scriptstyle\infty}}{{\mathrel{\mathop{\sum}\limits_% {n=-\infty}}}}|c_{n}\rangle\langle c_{n}|,$$ (19) where $|c_{n}\rangle$ is the coherent state of charge ”$n$”: $$|c_{n}\rangle=\exp(-\left|\alpha\right|^{2})\stackrel{{\scriptstyle\infty}}{{% \mathrel{\mathop{\sum}\limits_{n_{1}=0}}}}\stackrel{{\scriptstyle\infty}}{{% \mathrel{\mathop{\sum}\limits_{n_{2}=0}}}}\delta_{n_{1}-n_{2},n}\frac{\left|% \alpha\right|^{n_{1}+n_{2}}e^{i\phi(n_{1}-n_{2})}}{(n_{1}!)^{1/2}(n_{2}!)^{1/2% }}\bigl{|}n_{1,+}\bigr{\rangle}\bigl{|}n_{2,-}\bigr{\rangle}.$$ (20) While, in our example, the system described by the density matrix $\widehat{\rho}$ has not a definite charge, the average charge is equal to zero: $$Sp(\widehat{\rho}(a_{+}^{\dagger}a_{+}-a_{-}^{\dagger}a_{-}))=0.$$ (21) Note, that the expectation values of the annihilation operators in the corresponding coherent states are non-zero, $\bigl{\langle}\alpha_{\lambda}\bigr{|}a_{\lambda}\bigl{|}\alpha_{\lambda}\bigr% {\rangle}=\alpha_{\lambda}$, while $Sp(\widehat{\rho}a_{\lambda})=0$. Continuing the discussion of coherent pion production, we will assume the density matrix ${\bf\rho}_{{\bf e}}$ of a Gaussian-type in terms of the quasi-particle annihilation (creation) operators ${\rm a}_{i,qm}({\bf p}{,}t_{f})$, related to the free particle operators according to Eqs. (4) and (5). Then, similar to the above example, this density matrix can be expressed through the projection operators on the usual charge-unconstrained coherent states of free pion field. Averaging ${\bf\rho}_{{\bf e}}$ over all directions of the isovector ${\bf e}$ according to Eq. (15), we finally get the density matrix $\rho$ in a form of a weighted sum of the projection operators on the charge-constrained coherent states describing, in agreement with the super-selection rule, the system of a fixed average charge.888 We do not consider here the squeeze-states of the density matrix conditioned by possible mass shift of quasi-particles. Note, however, that charged pions have anyway no squeeze-state components [23]. The expressions for pion spectra in Eq. (1) thus contain the averaging over the direction of the isovector ${\bf e}$. As a result, the single-pion spectra are independent of pion charges $i={\pm},{0}$: $$\begin{array}[]{l}\omega_{{\bf p}}\frac{d^{3}N_{i}}{d^{3}{\bf p}}=(4\pi)^{-1}{% \bf\int}d\Omega({\bf e)}\int d\sigma_{\mu}p^{\mu}f_{{\bf e},i}(x,p)=\int d% \sigma_{\mu}p^{\mu}f(x,p),\\ \\ f(x,p)=f_{ch}(x,p)+\frac{1}{3}f_{coh}(x,p),\end{array}$$ (22) where we have used the equality $(4\pi)^{-1}{\bf\int}d\Omega({\bf e)}|e_{i}|^{2}=1/3.$ Note that the coherent part of the single–pion spectrum is $$\omega_{{\bf p}}\frac{d^{3}N_{coh}}{d^{3}{\bf p}}\equiv\omega_{{\bf p}}\frac{d% ^{3}N}{d^{3}{\bf p}}G(p)\equiv\omega_{{\bf p}}\frac{d^{3}N_{ch}}{d^{3}{\bf p}}% D(p)=\frac{1}{3}\int d\sigma_{\mu}p^{\mu}f_{coh}(x,p)=\frac{1}{3}|d({p})|^{2},$$ (23) where the functions $G(p)$ and $D(p)$ measure the coherent fraction: $$G(p)=\frac{D(p)}{1+D(p)}\equiv\frac{d^{3}N_{coh}/d^{3}{\bf p}}{d^{3}N/d^{3}{% \bf p}}=\frac{\frac{1}{3}\int d\sigma_{\mu}p^{\mu}f_{coh}(x,p)}{\int d\sigma_{% \mu}p^{\mu}f(x,p)},~{}~{}D(p)\equiv\frac{d^{3}N_{coh}/d^{3}{\bf p}}{d^{3}N_{ch% }/d^{3}{\bf p}}=\frac{\frac{1}{3}\int d\sigma_{\mu}p^{\mu}f_{coh}(x,p)}{\int d% \sigma_{\mu}p^{\mu}f_{ch}(x,p)}.$$ (24) The coherence influences also the quantum statistical (without FSI) correlation functions: $$C_{QS}^{ij}(p,q)=\frac{(4\pi)^{-1}{\bf\int}d\Omega({\bf e)}\bigl{\langle}a_{i}% ^{\dagger}({p}_{1})a_{j}^{\dagger}({p}_{2})a_{i}({p}_{1})a_{j}({p}_{2})\bigr{% \rangle}_{{\bf e}}}{\left((4\pi)^{-1}{\bf\int}d\Omega({\bf e)}\bigl{\langle}a_% {i}^{\dagger}({p}_{1})a_{i}({p}_{1})\bigr{\rangle}_{{\bf e}}\right)\left((4\pi% )^{-1}{\bf\int}d\Omega({\bf e)}\bigl{\langle}a_{j}^{\dagger}({p}_{2})a_{j}({p}% _{2})\bigr{\rangle}_{{\bf e}}\right)}.$$ (25) Taking into account Eqs. (6)-(9), (24) and the equalities $p_{1,2}=p\pm q/2$, we get $$\begin{array}[]{c}C_{QS}^{ij}(p,q)=1+\bigl{(}9\bigl{\langle}|e_{i}e_{j}|^{2}% \bigr{\rangle}-1-\delta_{ij}\bigr{)}G(p_{1})G(p_{2})+\delta_{ij}\bigl{\langle}% \cos(qx_{12})\bigr{\rangle}^{\prime}\\ \\ =\bigl{[}1+D(p_{1})\bigr{]}^{-1}\bigl{[}1+D(p_{2})\bigr{]}^{-1}\Bigl{\{}1+D(p_% {1})+D(p_{2})+9\bigl{\langle}|e_{i}e_{j}|^{2}\bigr{\rangle}D(p_{1})D(p_{2})\\ \\ +\delta_{ij}\bigl{\langle}\cos(qx_{12})\bigr{\rangle}^{\prime}_{ch}\bigl{[}1+{% \cal D}(p_{1},p_{2})+{\cal D}(p_{2},p_{1})\bigr{]}\Bigr{\}},\end{array}$$ (26) where the quasi–average $\langle\cos(qx_{12})\rangle^{\prime}\equiv\langle\cos(q(x_{1}-x_{2}))\rangle^{\prime}$ is defined as: $$\langle\cos(qx_{12})\rangle^{\prime}=\frac{\int d^{3}\sigma_{\mu}(x_{1})d^{3}% \sigma_{\nu}(x_{2})p^{\mu}p^{\nu}f(x_{1},p)f(x_{2},p)\cos(qx_{12})}{\int d^{3}% \sigma_{\mu}(x_{1})d^{3}\sigma_{\nu}(x_{2})p_{1}^{\mu}p_{2}^{\nu}f(x_{1},p_{1}% )f(x_{2},p_{2})}$$ (27) and similarly, with the substitution $f\rightarrow f_{ch}$, the quasi–average $\langle\cos(qx_{12})\rangle^{\prime}_{ch}$; the function $${\cal D}(p_{1},p_{2})=\frac{\frac{1}{3}\int d\sigma_{\mu}p^{\mu}f_{coh}(x,p)e^% {-iq\cdot x}}{\int d\sigma_{\mu}p^{\mu}f_{ch}(x,p)e^{-iq\cdot x}}=\frac{\frac{% 1}{3}d^{*}(p_{1})d(p_{2})}{\bigl{\langle}a_{i}^{\dagger}({p}_{1})a_{i}({p}_{2}% )\bigr{\rangle}_{ch}},~{}~{}{\cal D}(p,p)=D(p).$$ (28) Note that $$\displaystyle\langle\cos(qx_{12})\rangle^{\prime}$$ $$\displaystyle=$$ $$\displaystyle G(p_{1})G(p_{2})+\frac{1+{\cal D}(p_{1},p_{2})+{\cal D}(p_{2},p_% {1})}{[1+D(p_{1})][1+D(p_{2})]}\langle\cos(qx_{12})\rangle^{\prime}_{ch}$$ $$\displaystyle=$$ $$\displaystyle\frac{1+{\cal D}(p_{1},p_{2})+{\cal D}(p_{2},p_{1})+{\cal D}(p_{1% },p_{2}){\cal D}(p_{2},p_{1})}{[1+D(p_{1})][1+D(p_{2})]}\langle\cos(qx_{12})% \rangle^{\prime}_{ch}.$$ Calculating the averages $$\bigl{\langle}|e_{i}e_{j}|^{2}\bigr{\rangle}=(4\pi)^{-1}{\bf\int}d\Omega({\bf e% )}|e_{i}e_{j}|^{2},$$ (30) $$\bigl{\langle}|e_{0}|^{4}\bigr{\rangle}=\frac{1}{5},~{}~{}\bigl{\langle}|e_{% \pm}|^{4}\bigr{\rangle}=\bigl{\langle}|e_{+}e_{-}|^{2}\bigr{\rangle}=\frac{2}{% 15},~{}~{}\bigl{\langle}|e_{0}e_{\pm}|^{2}\bigr{\rangle}=\frac{1}{15},$$ (31) we get for the intercepts of the QS correlation functions: $$\begin{array}[]{l}C_{QS}^{++}(p,0)=2-\frac{4}{5}G^{2}(p),~{}~{}C_{QS}^{00}(p,0% )=2-\frac{1}{5}G^{2}(p),\\ \\ C_{QS}^{+-}(p,0)=1+\frac{1}{5}G^{2}(p),~{}~{}C_{QS}^{+0}(p,0)=1-\frac{2}{5}G^{% 2}(p).\end{array}$$ (32) Particularly, it follows from Eqs. (32) that the decay of the quasi-pion vacuum suppresses the correlation functions of identical charged pions and enhances the one of non-identical charged pions, the latter effect being by a factor of 4 smaller. For $G^{2}(p)=1$, the intercepts in Eqs. (32) coincide with those found in Ref. [31] in the case of a strong pion condensation. Our results however differ from the intercepts found in the model [21, 22] of pion emission in a pure quantum state, - the charge-constrained coherent state. They are recovered only for large average numbers of coherent pions. One can then replace the canonical ensemble corresponding to the pure quantum state with a fixed charge, by the grand canonical one, described by the density matrix of the ensemble with a fixed average charge. For ultra-relativistic A+A collisions, the inclusive description based on the grand canonical ensemble is a fairly adequate approach, allowing to built explicitly the density matrix for a mixture of thermal and charge-constrained coherent radiations and make some calculations analytically. One can check that the intercepts, as well as the QS correlation functions at any $q$, satisfy the relation [32] $$C_{QS}^{++}+C_{QS}^{+-}=C_{QS}^{00}+C_{QS}^{+0}.$$ (33) This relation follows from the assumed isotopically unpolarized pion emission. It is valid also for the complete correlation functions (with FSI), except for the region of very small $|{\bf q}|$ where the correlation functions of charged pions are strongly affected by the isospin non-conserving Coulomb interaction. Note that the correlation functions, as well as their QS parts, satisfy the usual normalization condition $C(p,q)\rightarrow 1$ at large $|{\bf q}|$ provided that the coherent part of the Wigner density vanishes with the increasing $|p\pm q/2|$ faster than the chaotic one, i.e. $G(p\pm q/2)\rightarrow 0$ at large $|{\bf q}|$. To get some insight in a possible behavior of the relative coherent contribution $G(p)$, consider the situation when the system decays during rather short time, $t_{out}-t_{f}\rightarrow 0$, and the partial (at a fixed ${\bf e}$) average of the pion annihilation operator has a simple Gaussian form: $$\bigl{\langle}a_{i}(p)\bigr{\rangle}_{{\bf e}}\sim\exp(-R_{coh}^{2}{\bf p}^{2}).$$ (34) According to Eq. (14), the corresponding Wigner density $$f_{coh}(x,p)\sim\exp(-2R_{coh}^{2}{\bf p}^{2}-{\bf x}^{2}/2R_{coh}^{2}),$$ (35) so the parameter $R_{coh}$ determines not only the spectrum, but also the characteristic radius of the region of the instantaneous coherent pion emission in accordance with the minimized uncertainty relation $\Delta x\Delta p=\hbar/2$. Let us assume a similar Gaussian parametrization of the chaotic component of the Wigner density in the non-relativistic momentum region: $$f_{ch}(x,p)\sim\exp(-2R_{T}^{2}{\bf p}^{2}-{\bf x}^{2}/2R_{ch}^{2}),$$ (36) where $R_{T}\equiv(4mT)^{-1/2}$ measures the characteristic size of the single–pion emitter (heat de Broglie length) and $R_{ch}\geq R_{T}$ is the characteristic radius of the region of the chaotic pion emission. In the considered rare gas limit, we then get the correlator $$\langle\cos(qx_{12})\rangle^{\prime}_{ch}=\exp(-R^{2}{\bf q}^{2}),$$ (37) where $R=(R_{ch}^{2}-R_{T}^{2})^{1/2}\approx R_{ch}$ represents (in the absence of the coherent contribution) the usual interferometry radius. The coherent fraction $G(p)=D(p)/[1+D(p)]$ and $$D(p)=\frac{d^{3}N_{coh}/d^{3}{\bf p}}{d^{3}N_{ch}/d^{3}{\bf p}}\equiv\frac{% \frac{1}{3}\int d\sigma_{\mu}p^{\mu}f_{coh}(x,p)}{\int d\sigma_{\mu}p^{\mu}f_{% ch}(x,p)}\sim\exp\left[-2\left(R_{coh}^{2}-R_{T}^{2})\right){\bf p}^{2}\right].$$ (38) We see that $G(p)\rightarrow 0$ at large $|{\bf p}|$ on a reasonable condition $R_{coh}>R_{T}$. In fact, since the quasi–classical (coherent) pion emission is conditioned by the decay of a thermal system, one may expect the effective radius for the coherent radiation, $R_{coh}$, close to that for the thermal emission, $R_{ch}$. Generally, in dynamical models, the effective radius varies with the momentum ${\bf p}$ and characterizes the size of the homogeneity region - the region of a substantial density of the pions emitted at the freeze–out time with three–momenta in the vicinity of ${\bf p}$. In this case, both the coherent and chaotic radii practically coincide with the homogeneity length of the system. Assuming $R_{coh}\approx R_{ch}$, we have ${\cal D}(p_{1},p_{2})\approx{\cal D}(p,p)=D(p)$ and, according to Eq. (II), $$~{}~{}~{}\langle\cos(qx_{12})\rangle^{\prime}\approx\frac{[1+D(p)]^{2}}{[1+D(p% +q/2)][1+D(p-q/2)]}\langle\cos(qx_{12})\rangle_{ch}^{\prime}.$$ (39) One can see that $\langle\cos(qx_{12})\rangle^{\prime}\approx\langle\cos(qx_{12})\rangle^{\prime% }_{ch}$ at small $|{\bf q}|$ or, in the case of a small coherent contribution $D(p)\ll 1$. Note that in the opposite case, $D(p)\gg 1$, a decrease of the correlation function towards unity with the increasing ${\bf q}^{2}$ is conditioned by the chaotic component $\langle\cos(qx_{12})\rangle^{\prime}_{ch}$ starting at ${\bf q}^{2}\sim R^{-2}\ln D^{2}({\bf 0})-4{\bf p}^{2}$. At smaller ${\bf q}^{2}$-values, the behavior of the correlation function is essentially flatter due to the $q$-dependence of the denominator in Eq. (39). For the extreme case of a pure coherent radiation, $D(p)\rightarrow\infty$ ($G(p)\rightarrow 1$), the function $\langle\cos(qx_{12})\rangle^{\prime}$ tends to unity at all $q$ irrespective of the assumption $R_{coh}\approx R_{ch}$: $$\langle\cos(qx_{12})\rangle^{\prime}\rightarrow\frac{\int d^{3}\sigma_{\mu}(x_% {1})d^{3}\sigma_{\nu}(x_{2})p^{\mu}p^{\nu}f_{coh}(x_{1},p)f_{coh}(x_{2},p)\cos% (qx_{12})}{\int d^{3}\sigma_{\mu}(x_{1})d^{3}\sigma_{\nu}(x_{2})p_{1}^{\mu}p_{% 2}^{\nu}f_{coh}(x_{1},p_{1})f_{coh}(x_{2},p_{2})}=1.$$ (40) The last equality in Eq. (40) follows from the definition (14) of the coherent Wigner function, both the nominator and denominator in Eq. (40) being equal to $|d(p_{1})d(p_{2})|^{2}$. Experimentally, the approach to such an extreme regime can display itself as a tendency of the intercepts of the QS correlation functions to the values defined by Eqs. (32) at $G(p)\rightarrow 1,$ and - as a flattering of the QS correlation functions within a growing ${\bf q}$-interval. The latter mimics a decrease of interferometry radii; of course, it does not mean that the real size of the system tends to zero. The effect of coherent radiation on pion spectra and $\pi^{+}\pi^{+}$ and $\pi^{+}\pi^{-}$ correlation functions is demonstrated in Figs. 1-3 for different ratios $D_{tot}=D({\bf 0})(R_{T}/R_{coh})^{3}$ of the total numbers of coherent and chaotic pions. The plots correspond to simple Gaussian Wigner functions (35), (36) with $R_{T}\equiv(4mT)^{-1/2}\approx 0.72$ fm ($T=0.135$ GeV) and $R_{coh}=R_{ch}=5$ fm. Under the assumption of a common source of coherent and chaotic pions in ultra–relativistic heavy ion collisions, characterized by a typical radius $R\sim 5-10$ fm, the coherent component in the spectra is concentrated in rather small momentum region of a characteristic width $(2R)^{-1}\sim 20-10$ MeV/c (see Fig. 1). III Correlation functions affected by final state interaction and coherence In ultrarelativistic A+A collisions, free hadrons appear mainly at the late stage of the evolution after the system expands and reaches the thermal freeze-out. After the hydrodynamic tube decays and produces final particles and resonances, particles still appear from resonance decays. Thus, more than half of pions produced in high energy heavy ion collisions is of the resonance origin. As a consequence, the pion spectra and correlations are influenced by resonance production and decay spectra, as well as - by resonance lifetimes. Particularly, the pions from the decays of long-lived resonances do not contribute to QS and FSI correlations and thus suppress the correlation function $C^{ij}(p,q)$; we will consider this suppression in next Section. However, even after the thermal (hydrodynamic) system and short–lived resonances decay, the particles in near–by phase space points continue to interact. Due to a large effective emission volume in heavy ion collisions, the particle interaction in the final state is usually dominated by the long-range Coulomb forces. To calculate the FSI effect on two-particle spectra, we will assume sufficiently small phase space density of the produced particles and use the FSI theory in the two–body approximation [8, 33, 34] for pions, neglecting the FSI of resonances. The single–pion spectrum in Eq. (1) then remains unchanged while the two–pion one (for pairs containing no pions from long–lived sources) takes the form $$\omega_{{\bf p}_{1}}\omega_{{\bf p}_{2}}\frac{d^{6}N_{ij}}{d^{3}{\bf p}_{1}d^{% 3}{\bf p}_{2}}\doteq\int d^{4}k_{1}d^{4}k_{2}d^{4}k_{1}^{\prime}d^{4}k_{2}^{% \prime}\bigl{\langle}a_{i}^{\dagger}(k_{1})a_{j}^{\dagger}(k_{2})a_{i}(k_{1}^{% \prime})a_{j}(k_{2}^{\prime})\bigr{\rangle}\Phi_{p_{1}p_{2}}^{(-)ij}(k_{1},k_{% 2})\Phi_{p_{1}p_{2}}^{(-)ij*}(k_{1}^{\prime},k_{2}^{\prime}),$$ (41) where the non–symmetrized Bethe–Salpeter amplitude $\Phi_{p_{1}p_{2}}^{(-)ij}(k_{1},k_{2})\equiv\Phi_{p_{1}p_{2}}^{(+)ij*}(k_{1},k% _{2})$ in four–momentum representation is expressed through the propagators of particles $i$ and $j$ and their scattering amplitude ${\cal F}_{ij}$ analytically continued to the unphysical region [33, 34]:999 It is important that the relation between the production amplitude and the operator product average, as given in Eq. (1), is valid also off mass shell. $$\Phi_{p_{1}p_{2}}^{(-)ij}(k_{1},k_{2})=\delta^{4}(k_{1}-p_{1})\delta^{4}(k_{2}% -p_{2})+\delta^{4}(k_{1}+k_{2}-p_{1}-p_{2})\frac{i\sqrt{p^{2}}}{\pi^{3}}\frac{% {\cal F}_{ij}^{*}(k_{1},k_{2};p_{1},p_{2})}{(k_{1}^{2}-m^{2}-i0)(k_{2}^{2}-m^{% 2}-i0)}.$$ (42) The averaging in Eq. (41) is performed with the help of the statistical operator ${\bf\rho}$ without FSI: $\langle\dots\rangle=Sp(\dots{\bf\rho})$. Introducing the Bethe-Salpeter amplitudes $\Psi_{p_{1}p_{2}}^{(-)ij}(x_{1},x_{2})$ in space-time representation: $$\Phi_{p_{1}p_{2}}^{(-)ij}(k_{1},k_{2})=(2\pi)^{-8}\int d^{4}x_{1}d^{4}x_{2}e^{% ik_{1}x_{1}+ik_{2}x_{2}}\Psi_{p_{1}p_{2}}^{(-)ij}(x_{1},x_{2}),$$ (43) one can rewrite Eq. (41) as $$\omega_{{\bf p}_{1}}\omega_{{\bf p}_{2}}\frac{d^{6}N_{ij}}{d^{3}{\bf p}_{1}d^{% 3}{\bf p}_{2}}\doteq\int d^{4}x_{1}d^{4}x_{2}d^{4}x_{1}^{\prime}d^{4}x_{2}^{% \prime}\rho^{ij}(x_{1},x_{2};x_{1}^{\prime},x_{2}^{\prime})\Psi_{p_{1}p_{2}}^{% (-)ij}(x_{1},x_{2})\Psi_{p_{1}p_{2}}^{(-)ij*}(x_{1}^{\prime},x_{2}^{\prime}),$$ (44) where the space–time density matrix $\rho^{ij}$ is just the Fourier transform of the four–operator average in Eq. (41):101010For identical particles, it differs from the space–time density matrix of ref. [33], where the effect of QS enters through the symmetrization of the Bethe–Salpeter amplitudes while, here - through the Wigner decomposition of the four–operator average in Eq. (52) below. $$\rho^{ij}(x_{1},x_{2};x_{1}^{\prime},x_{2}^{\prime})=(2\pi)^{-16}\int d^{4}k_{% 1}d^{4}k_{2}d^{4}k_{1}^{\prime}d^{4}k_{2}^{\prime}e^{ik_{1}x_{1}+ik_{2}x_{2}}e% ^{-ik_{1}^{\prime}x_{1}^{\prime}-ik_{2}^{\prime}x_{2}^{\prime}}\bigl{\langle}a% _{i}^{\dagger}(k_{1})a_{j}^{\dagger}(k_{2})a_{i}(k_{1}^{\prime})a_{j}(k_{2}^{% \prime})\bigr{\rangle}.$$ (45) Separating the phase factor due to free motion of the two–particle c.m.s.: $$\begin{array}[]{c}\Psi_{p_{1}p_{2}}^{(-)ij}(x_{1},x_{2})=e^{-iPX_{12}}\psi_{q}% ^{(-)ij}(x_{12}),\\ \\ X_{12}=\frac{1}{2}(x_{1}+x_{2}),~{}~{}x_{12}=x_{1}-x_{2},~{}~{}P\equiv 2p=p_{1% }+p_{2}\end{array}$$ (46) and integrating over the pair c.m.s. four–coordinates $X_{12}$ and $X_{12}^{\prime}$ in Eq. (44), one can express the two–particle spectrum through the reduced space–time density matrix $\rho_{P}^{ij}(x_{12};x_{12}^{\prime})$, the latter depending on the pair total four–momentum $P$ and the relative four–coordinates of the emission points only: $$\omega_{{\bf p}_{1}}\omega_{{\bf p}_{2}}\frac{d^{6}N_{ij}}{d^{3}{\bf p}_{1}d^{% 3}{\bf p}_{2}}\doteq\int d^{4}x_{12}d^{4}x_{12}^{\prime}\rho_{P}^{ij}(x_{12};x% _{12}^{\prime})\psi_{q}^{(-)ij}(x_{12})\psi_{q}^{(-)ij*}(x_{12}^{\prime}),$$ (47) $$\rho_{P}^{ij}(x_{12};x_{12}^{\prime})=(2\pi)^{-8}\int d^{4}k_{1}d^{4}k_{1}^{% \prime}e^{i(k_{1}-p)x_{12}}e^{-i(k_{1}^{\prime}-p)x_{12}^{\prime}}\bigl{% \langle}a_{i}^{\dagger}(k_{1})a_{j}^{\dagger}(P-k_{1})a_{i}(k_{1}^{\prime})a_{% j}(P-k_{1}^{\prime})\bigr{\rangle}.$$ (48) Note that in the two–particle c.m.s., where $P=\{m_{12},0,0,0\}$, $q=\{0,2{\rm k}^{*}\}$, $x_{12}=\{t^{*},{\rm r}^{*}\}$, the reduced Bethe–Salpeter amplitude $\psi_{q}^{(-)ij*}(x_{12})=\psi_{q}^{(+)ij}(x_{12})$ at $t^{*}=t_{1}^{*}-t_{2}^{*}=0$ coincides with a stationary solution $\psi_{-{\rm k}^{*}}({\rm r}^{*})$ of the scattering problem having at large distances $r^{*}$ the asymptotic form of a superposition of of the plane and outgoing spherical waves (the minus sign of the vector ${\rm k}^{*}$ corresponds to the reverse in time direction of the emission process). This amplitude can be substituted by this solution (equal time approximation) on condition [34] $|t^{*}|\ll mr^{*2}$ which is usually satisfied for particle production in heavy ion collisions. Since the resonances have finite lifetimes, their decay products are created in an essentially four–dimensional space-time region. At the post thermal freeze-out stage, the resonances are usually described by semiclassical techniques; they are considered as unstable particles moving along classical trajectories and decaying according to the exponential law [35] (see, however, [33, 36, 37]). This approximation neglects a small correlation effect in pairs of unlike pions appearing due to QS correlations of identical resonances. The resonances are supposed to be described according to the Gibbs density matrix prior to the thermal freeze-out; this guarantees the chaoticity of the decay pions.111111 Note that the chaotisation of decay pions partially happens irrespective of the form of the density matrix if pions were emitted by a large number of many different sorts of resonances. Therefore, the pions from resonance decays do not destroy the structure of the decomposition of the operator averages in Eqs. (6) and (7) into irreducible parts based on the thermal Wick theorem. After the production, the pions in near-by phase space points, chaotic as well as coherent ones, undergo a long-time scale interaction in the final state. According to Eqs. (44) or (47), the intensity of FSI interaction is conditioned by the two–particle Bethe–Salpeter amplitudes $\Psi_{p_{1}p_{2}}(x_{1},x_{2})$ or $\psi_{q}(x_{12})$ and the corresponding two–particle space–time density matrices $\rho(x_{1},x_{2};x_{1}^{\prime},x_{2}^{\prime})$ or $\rho_{P}(x_{12};x_{12}^{\prime})$. Clearly, in the case of absent FSI, the two–pion spectrum merely reduces to the Fourier transform of the space–time density matrix. It can be represented as an integral over the mean four–coordinates $\bar{x}=(x+x^{\prime})/2$ of a combination of bilinear products of single–particle chaotic and coherent emission functions $g_{ch}(\bar{x},p)$ and $g_{coh}(\bar{x},p)$, respectively defined in Eqs. (50) and (51) below. The emission function $g(\bar{x},p)$ is closely related with the Wigner phase space density $f(x,p)$ at asymptotic times $t\geqslant t_{out}$. Let us denote by $\bar{x}\equiv\{\bar{t},{\bf x}-({\bf p}/p_{0})(t-\bar{t})\}$ the space-time point, starting from which a free particle moving with velocity $p/p_{0}$ reaches a point $x$; the portion of such particles is $g(\bar{x},p)$. Collecting all the contributions (starting in our case from the thermal freeze-out time $t_{f}$), we have $$p_{0}f(x,p)=\int d^{4}\bar{x}\delta^{3}\bigl{(}\bar{{\bf x}}-{\bf x}+({\bf p}/% p_{0})(t-\bar{t})\bigr{)}g(\bar{x},p),$$ (49) where $g(\bar{x},p)=p_{0}\delta(\bar{t}-t_{f})f(\bar{x},p)+s(\bar{x},p)$ and $s(\bar{x},p)\sim\theta(t-\bar{t})\theta(\bar{t}-t_{f})$ is the density of pion emission at the post–thermal stage, $t>t_{f}$. Therefore we can rewrite the irreducible (thermal) part of the two-operator average through the chaotic emission function as: $$\begin{array}[]{c}\bigl{\langle}a_{i}^{\dagger}({p}_{1})a_{i}({p}_{2})\bigr{% \rangle}_{ch}=\int\limits_{\sigma_{out}}d\sigma_{\mu}p^{\mu}e^{-iqx}f_{ch}(x,p% )=\int d^{4}\bar{x}e^{-iq\bar{x}}g_{ch}\left(\bar{x},p\right),\\ \\ p\equiv P/2=(p_{1}+p_{2})/2,~{}~{}q=p_{1}-p_{2},\end{array}$$ (50) where we have used the equality $qx=q\bar{x}$ following from the relation $qp\equiv q_{0}p_{0}-{\bf qp}=0$. Similarly, for the coherent component of the two-operator average at fixed ${\bf e}$, we get $$\begin{array}[]{c}\bigl{\langle}a_{i}^{\dagger}({p}_{1})\bigr{\rangle}_{{\bf e% }}\bigl{\langle}a_{i}({p}_{2})\bigr{\rangle}_{{\bf e}}=|e_{i}|^{2}d^{*}(p_{1})% )d(p_{2})=\\ \\ |e_{i}|^{2}\int\limits_{\sigma_{out}}d\sigma_{\mu}p^{\mu}e^{-iqx}f_{coh}(x,p)=% |e_{i}|^{2}\int d^{4}\bar{x}e^{-iq\bar{x}}g_{coh}\left(\bar{x},p\right).\end{array}$$ (51) The results of Section II can thus be rewritten in terms of the emission functions in accordance with a formal substitution $\int\limits_{\sigma_{out}}d\sigma_{\mu}p^{\mu}f(x,p)\rightarrow\int d^{4}xg% \left(x,p\right)$. To express the four-operator average in Eq. (48) through the emission functions, we can exploit the decomposition similar to that in Eq. (6): $$\begin{array}[]{c}\bigl{\langle}a_{i}^{\dagger}({k}_{1})a_{j}^{\dagger}(P-k_{1% })a_{i}(k_{1}^{\prime})a_{j}(P-k_{1}^{\prime})\bigr{\rangle}_{{\bf e}}=\bigl{% \langle}a_{i}^{\dagger}(k_{1})a_{i}(k_{1}^{\prime})\bigr{\rangle}_{{\bf e}}% \bigl{\langle}a_{j}^{\dagger}(P-k_{1})a_{j}(P-k_{1}^{\prime})\bigr{\rangle}_{{% \bf e}}+\\ \\ \delta_{ij}\left[\bigl{\langle}a_{i}^{\dagger}(k_{1})a_{i}(P-k_{1}^{\prime})% \bigr{\rangle}_{{\bf e}}\bigl{\langle}a_{i}^{\dagger}(P-k_{1})a_{i}(k_{1}^{% \prime})\bigr{\rangle}_{{\bf e}}-\bigl{\langle}a_{i}^{\dagger}(k_{1})\bigr{% \rangle}_{{\bf e}}\bigl{\langle}a_{i}^{\dagger}(P-k_{1})\bigr{\rangle}_{{\bf e% }}\bigl{\langle}a_{i}(k_{1}^{\prime})\bigr{\rangle}_{{\bf e}}\bigl{\langle}a_{% i}(P-k_{1}^{\prime})\bigr{\rangle}_{{\bf e}}\right].\end{array}$$ (52) Using Eqs. (50) and (51) for the two-operator averages in Eq. (52), we get: $$\begin{array}[]{l}\bigl{\langle}a_{i}^{\dagger}(k_{1})a_{j}^{\dagger}(P-k_{1})% a_{i}(k_{1}^{\prime})a_{j}(P-k_{1}^{\prime})\bigr{\rangle}_{{\bf e}}=\int d^{4% }\bar{x}_{1}d^{4}\bar{x}_{2}\times\\ \\ \Bigl{\{}e^{-i(k_{1}-k_{1}^{\prime})\cdot\bar{x}_{12}}g_{{\bf e},i}\left(\bar{% x}_{1},\frac{1}{2}(k_{1}+k_{1}^{\prime})\right)g_{{\bf e},j}\left(\bar{x}_{2},% P-\frac{1}{2}(k_{1}+k_{1}^{\prime})\right)+\Bigr{.}\\ \\ \delta_{ij}e^{-i(k_{1}+k_{1}^{\prime}-P)\cdot\bar{x}_{12}}\bigl{[}g_{{\bf e},i% }\left(\bar{x}_{1},p+\frac{1}{2}(k_{1}-k_{1}^{\prime})\right)g_{{\bf e},i}% \left(\bar{x}_{2},p-\frac{1}{2}(k_{1}-k_{1}^{\prime})\right)\bigr{.}\\ \\ \Bigl{.}\bigl{.}-|e_{i}|^{4}g_{coh}\left(\bar{x}_{1},p+\frac{1}{2}(k_{1}-k_{1}% ^{\prime})\right)g_{coh}\left(\bar{x}_{2},p-\frac{1}{2}(k_{1}-k_{1}^{\prime})% \right)\bigr{]}\Bigr{\}},\end{array}$$ (53) where $\bar{x}_{12}=\bar{x}_{1}-\bar{x}_{2}$ and $$g_{{\bf e},i}(\bar{x},k)=g_{ch}(\bar{x},k)+|e_{i}|^{2}g_{coh}(\bar{x},k).$$ (54) After the averaging over the orientation of the isospin vector ${\bf e}$, we get $$\begin{array}[]{l}\bigl{\langle}a_{i}^{\dagger}({k}_{1})a_{j}^{\dagger}(P-k_{1% })a_{i}({k}_{1}^{\prime})a_{j}(P-k_{1}^{\prime})\bigr{\rangle}=\int d^{4}\bar{% x}_{1}d^{4}\bar{x}_{2}\cdot\\ \\ \Bigl{\{}e^{-i(k_{1}-k_{1}^{\prime})\cdot\bar{x}_{12}}\bigl{[}g\left(\bar{x}_{% 1},\frac{1}{2}(k_{1}+k_{1}^{\prime})\right)g\left(\bar{x}_{2},P-\frac{1}{2}(k_% {1}+k_{1}^{\prime})\right)\bigr{.}\Bigr{.}\\ \\ \bigr{.}+\bigl{(}\bigl{\langle}|e_{i}e_{j}|^{2}\bigr{\rangle}-\frac{1}{9}\bigr% {)}g_{coh}\left(\bar{x}_{1},\frac{1}{2}(k_{1}+k_{1}^{\prime})\right)g_{coh}% \left(\bar{x}_{2},P-\frac{1}{2}(k_{1}+k_{1}^{\prime})\right)\bigr{]}+\\ \\ \delta_{ij}e^{-i(k_{1}+k_{1}^{\prime}-P)\cdot\bar{x}_{12}}\bigl{[}g\left(\bar{% x}_{1},p+\frac{1}{2}(k_{1}-k_{1}^{\prime})\right)g\left(\bar{x}_{2},p-\frac{1}% {2}(k_{1}-k_{1}^{\prime})\right)\bigr{.}\\ \\ \Bigl{.}\bigl{.}-\frac{1}{9}g_{coh}\left(\bar{x}_{1},p+\frac{1}{2}(k_{1}-k_{1}% ^{\prime})\right)g_{coh}\left(\bar{x}_{2},p-\frac{1}{2}(k_{1}-k_{1}^{\prime})% \right)\bigr{]}\Bigr{\}},\end{array}$$ (55) where $$g(\bar{x},k)=g_{ch}(\bar{x},k)+\frac{1}{3}g_{coh}(\bar{x},k).$$ (56) Inserting expression (55) for the four–operator average into Eq. (48) and, integrating in the first and second term over $(k_{1}-k_{1}^{\prime})$ and $(k_{1}+k_{1}^{\prime}-P)$ respectively, one can rewrite the reduced space–time density matrix as: $$\begin{array}[]{l}\rho_{P}^{ij}(x_{12};x_{12}^{\prime})=(2\pi)^{-4}\int d^{4}% \bar{x}_{1}d^{4}\bar{x}_{2}d^{4}\kappa\cdot\\ \\ \Bigl{\{}e^{i\kappa\cdot(x_{12}-x_{12}^{\prime})}\delta^{4}\bigl{(}\frac{1}{2}% (x_{12}+x_{12}^{\prime})-\bar{x}_{12}\bigr{)}\bigl{[}g\left(\bar{x}_{1},p+% \kappa\right)g\left(\bar{x}_{2},p-\kappa\right)\bigr{.}\Bigr{.}\\ \\ \bigr{.}+\bigl{(}\bigl{\langle}|e_{i}e_{j}|^{2}\bigr{\rangle}-\frac{1}{9}\bigr% {)}g_{coh}\left(\bar{x}_{1},p+\kappa\right)g_{coh}\left(\bar{x}_{2},p-\kappa% \right)\bigr{]}+\\ \\ \delta_{ij}e^{i\kappa\cdot(x_{12}+x_{12}^{\prime})}\delta^{4}\bigl{(}\frac{1}{% 2}(x_{12}-x_{12}^{\prime})-\bar{x}_{12}\bigr{)}\bigl{[}g\left(\bar{x}_{1},p+% \kappa\right)g\left(\bar{x}_{2},p-\kappa\right)\bigr{.}\\ \\ \Bigl{.}\bigl{.}-\frac{1}{9}g_{coh}\left(\bar{x}_{1},p+\kappa\right)g_{coh}% \left(\bar{x}_{2},p-\kappa\right)\bigr{]}\Bigr{\}}.\end{array}$$ (57) According to Eq. (47) and using the equality $\psi_{q}(-\bar{x}_{12})=\psi_{-q}(\bar{x}_{12})$, the two–pion spectrum then becomes: $$\begin{array}[]{c}\omega_{{\bf p}_{1}}\omega_{{\bf p}_{2}}\frac{d^{6}N_{ij}}{d% ^{3}{\bf p}_{1}d^{3}{\bf p}_{2}}\doteq(2\pi)^{-4}\int d^{4}\bar{x}_{1}d^{4}% \bar{x}_{2}d^{4}\kappa d^{4}\epsilon\,e^{i\kappa\cdot\epsilon}\cdot\\ \\ \Bigl{\{}\bigl{[}g\left(\bar{x}_{1},p+\kappa\right)g\left(\bar{x}_{2},p-\kappa% \right)+\bigl{(}\bigl{\langle}|e_{i}e_{j}|^{2}\bigr{\rangle}-\frac{1}{9}\bigr{% )}g_{coh}\left(\bar{x}_{1},p+\kappa\right)g_{coh}\left(\bar{x}_{2},p-\kappa% \right)\bigr{]}\psi_{q}^{(-)ij}(\bar{x}_{12}+\frac{1}{2}\epsilon)\psi_{q}^{(-)% ij*}(\bar{x}_{12}-\frac{1}{2}\epsilon)\Bigr{.}\\ \\ \Bigl{.}\bigl{.}+\delta_{ij}\bigl{[}g\left(\bar{x}_{1},p+\kappa\right)g\left(% \bar{x}_{2},p-\kappa\right)-\frac{1}{9}g_{coh}\left(\bar{x}_{1},p+\kappa\right% )g_{coh}\left(\bar{x}_{2},p-\kappa\right)\bigr{]}\psi_{q}^{(-)ij}(\bar{x}_{12}% +\frac{1}{2}\epsilon)\psi_{-q}^{(-)ij*}(\bar{x}_{12}-\frac{1}{2}\epsilon)\Bigr% {\}}\\ \\ =(2\pi)^{-4}\int d^{4}\bar{x}_{1}d^{4}\bar{x}_{2}d^{4}\kappa d^{4}\epsilon\,e^% {i\kappa\cdot\epsilon}\cdot\\ \\ \Bigl{\{}\bigl{[}g_{ch}\left(\bar{x}_{1},p+\kappa\right)g_{ch}\left(\bar{x}_{2% },p-\kappa\right)+\frac{1}{3}\bigl{(}g_{ch}\left(\bar{x}_{1},p+\kappa\right)g_% {coh}\left(\bar{x}_{2},p-\kappa\right)+g_{coh}\left(\bar{x}_{1},p+\kappa\right% )g_{ch}\left(\bar{x}_{2},p-\kappa\right)\bigr{)}\bigr{]}\cdot\Bigr{.}\\ \\ \Bigl{.}\bigl{[}\psi_{q}^{(-)ij}(\bar{x}_{12}+\frac{1}{2}\epsilon)\psi_{q}^{(-% )ij*}(\bar{x}_{12}-\frac{1}{2}\epsilon)+\delta_{ij}\psi_{q}^{(-)ij}(\bar{x}_{1% 2}+\frac{1}{2}\epsilon)\psi_{-q}^{(-)ij*}(\bar{x}_{12}-\frac{1}{2}\epsilon)% \bigr{]}\Bigr{.}\\ \\ \Bigl{.}+\langle|e_{i}e_{j}|^{2}\bigr{\rangle}g_{coh}\left(\bar{x}_{1},p+% \kappa\right)g_{coh}\left(\bar{x}_{2},p-\kappa\right)\psi_{q}^{(-)ij}(\bar{x}_% {12}+\frac{1}{2}\epsilon)\psi_{q}^{(-)ij*}(\bar{x}_{12}-\frac{1}{2}\epsilon)% \Bigr{\}}.\end{array}$$ (58) If the FSI were absent, i.e. $\psi_{q}^{(-)ij}(\bar{x}_{12})=\exp(-iq\cdot\bar{x}_{12}/2)$, one would get $$\begin{array}[]{c}\omega_{{\bf p}_{1}}\omega_{{\bf p}_{2}}\frac{d^{6}N_{ij}}{d% ^{3}{\bf p}_{1}d^{3}{\bf p}_{2}}\doteq\int d^{4}\bar{x}_{1}d^{4}\bar{x}_{2}\,% \Bigl{\{}g\left(\bar{x}_{1},p_{1}\right)g\left(\bar{x}_{2},p_{2}\right)+\bigl{% (}\bigl{\langle}|e_{i}e_{j}|^{2}\bigr{\rangle}-\frac{1}{9}\bigr{)}g_{coh}\left% (\bar{x}_{1},p_{1}\right)g_{coh}\left(\bar{x}_{2},p_{2}\right)\Bigr{.}\\ \\ \Bigl{.}\bigl{.}+\delta_{ij}\bigl{[}g\left(\bar{x}_{1},p\right)g\left(\bar{x}_% {2},p\right)-\frac{1}{9}g_{coh}\left(\bar{x}_{1},p\right)g_{coh}\left(\bar{x}_% {2},p\right)\bigr{]}\cos(q\bar{x}_{12})\Bigr{\}}\\ \\ =\int d^{4}\bar{x}_{1}d^{4}\bar{x}_{2}\,\Bigl{\{}g\left(\bar{x}_{1},p_{1}% \right)g\left(\bar{x}_{2},p_{2}\right)+\bigl{(}\bigl{\langle}|e_{i}e_{j}|^{2}% \bigr{\rangle}-\frac{1}{9}(1+\delta_{ij})\bigr{)}g_{coh}\left(\bar{x}_{1},p_{1% }\right)g_{coh}\left(\bar{x}_{2},p_{2}\right)\Bigr{.}\\ \\ \Bigl{.}\bigl{.}+\delta_{ij}g\left(\bar{x}_{1},p\right)g\left(\bar{x}_{2},p% \right)\cos(q\bar{x}_{12})\Bigr{\}}\\ \\ =\int d^{4}\bar{x}_{1}d^{4}\bar{x}_{2}\,\Bigl{\{}g_{ch}\left(\bar{x}_{1},p_{1}% \right)g_{ch}\left(\bar{x}_{2},p_{2}\right)+\frac{1}{3}\bigl{(}g_{ch}\left(% \bar{x}_{1},p_{1}\right)g_{coh}\left(\bar{x}_{2},p_{2}\right)+g_{coh}\left(% \bar{x}_{1},p_{1}\right)g_{ch}\left(\bar{x}_{2},p_{2}\right)\bigr{)}\Bigr{.}\\ \\ \Bigl{.}+\bigl{\langle}|e_{i}e_{j}|^{2}\bigr{\rangle}g_{coh}\left(\bar{x}_{1},% p_{1}\right)g_{coh}\left(\bar{x}_{2},p_{2}\right)+\delta_{ij}\bigl{[}g_{ch}% \left(\bar{x}_{1},p\right)g_{ch}\left(\bar{x}_{2},p\right)+\frac{2}{3}g_{ch}% \left(\bar{x}_{1},p\right)g_{coh}\left(\bar{x}_{2},p\right)\bigr{]}\cos(q\bar{% x}_{12})\Bigr{\}}\end{array}$$ (59) and recover Eqs. (26) for the pure QS correlation functions. In the case of absent coherent emission, i.e. $d=g_{coh}=0$, and on the usual assumption ($R_{T}^{2}\ll R_{ch}^{2}$) of sufficiently smooth four–momentum dependence of the chaotic emission function $g_{ch}(\bar{x},p)$ as compared with a sharp $q$–dependence of the QS and FSI correlations (determined by the inverse characteristic distance between the emission points), the chaotic emission functions in Eq. (58) can be taken out of the integral over $\kappa$ at small values of $\kappa$, this integral thus being close to $\delta^{4}(\epsilon)$. Choosing the momentum arguments in $g_{ch}$-functions in accordance with Eq. (59) for the case of absent FSI, we get for the two–pion spectrum and the correlation function: $$\begin{array}[]{c}\omega_{{\bf p}_{1}}\omega_{{\bf p}_{2}}\frac{d^{6}N_{ij}}{d% ^{3}{\bf p}_{1}d^{3}{\bf p}_{2}}\approx\int d^{4}\bar{x}_{1}d^{4}\bar{x}_{2}% \cdot\\ \\ \left\{g_{ch}\left(\bar{x}_{1},p_{1}\right)g_{ch}\left(\bar{x}_{2},p_{2}\right% )|\psi_{q}^{(-)ij}(\bar{x}_{12})|^{2}+\delta_{ij}g_{ch}\left(\bar{x}_{1},p% \right)g_{ch}\left(\bar{x}_{2},p\right)\psi_{q}^{(-)ij}(\bar{x}_{12})\psi_{-q}% ^{(-)ij*}(\bar{x}_{12})\right\},\end{array}$$ (60) $$C_{ch}^{ij}\approx\bigl{\langle}|\psi_{q}^{(-)ij}(\bar{x}_{12})|^{2}\bigr{% \rangle}_{ch}+\delta_{ij}\bigl{\langle}\psi_{q}^{(-)ij}(\bar{x}_{12})\psi_{-q}% ^{(-)ij*}(\bar{x}_{12})\bigr{\rangle}_{ch}^{\prime},$$ (61) where the average $\langle{\cal A}\rangle_{ch}$ and quasi-average $\langle{\cal A}\rangle^{\prime}_{ch}$ are defined as: $$\bigl{\langle}{\cal A}\bigr{\rangle}_{ch}=\frac{\int d^{4}\bar{x}_{1}d^{4}\bar% {x}_{2}\,{\cal A}\,g_{ch}\left(\bar{x}_{1},p_{1}\right)g_{ch}\left(\bar{x}_{2}% ,p_{2}\right)}{\int d^{4}\bar{x}_{1}\,g_{ch}\left(\bar{x}_{1},p_{1}\right)\int d% ^{4}\bar{x}_{2}\,g_{ch}\left(\bar{x}_{2},p_{2}\right)},$$ (62) $$\bigl{\langle}{\cal A}\bigr{\rangle}_{ch}^{\prime}=\frac{\int d^{4}\bar{x}_{1}% d^{4}\bar{x}_{2}\,{\cal A}\,g_{ch}\left(\bar{x}_{1},p\right)g_{ch}\left(\bar{x% }_{2},p\right)}{\int d^{4}\bar{x}_{1}\,g_{ch}\left(\bar{x}_{1},p_{1}\right)% \int d^{4}\bar{x}_{2}\,g_{ch}\left(\bar{x}_{2},p_{2}\right)}.$$ (63) In the case of a nonzero coherent contribution, the $\epsilon/2$- and $\bar{x}_{12}$–dispersions in the pure coherent term in Eq. (58) are the same ($2R_{coh}^{2}$), contrary to usually negligible $\epsilon/2$-dispersion in the pure chaotic term: $2R_{T}^{2}\ll 2R_{ch}^{2}$. As for the mixed term, the $\epsilon/2$-dispersion would be negligible if only the characteristic size $R_{coh}$ of the coherent source were sufficiently small; with the increasing $R_{coh}$, this dispersion may become important - for $R_{coh}\approx R_{ch}$ it amounts to about half of the $\bar{x}_{12}$–dispersion. Therefore, the $\epsilon$–dependence of the Bethe–Salpeter amplitudes should be generally retained in these terms. The important exception is the case of practical interest in heavy ion collisions, when the two charged pions are created in their c.m.s. at a distance much larger than the corresponding s–wave scattering length (of a fraction of fm) and much smaller than their Bohr radius (of 387.5 fm). The two–pion FSI interaction at small $q$ is then dominated by the Coulomb FSI and depends only weakly on the space–time separation of the emission points. In this case, $$C_{ch}^{ij}\approx\bigl{\langle}|\psi_{q}^{(-)ij}(\bar{x}_{12})|^{2}\bigr{% \rangle}+\delta_{ij}\bigl{\langle}\psi_{q}^{(-)ij}(\bar{x}_{12})\psi_{-q}^{(-)% ij*}(\bar{x}_{12})\bigr{\rangle}^{\prime}+\bigl{(}9\bigl{\langle}|e_{i}e_{j}|^% {2}\bigr{\rangle}-1-\delta_{ij}\bigr{)}G(p_{1})G(p_{2})\bigl{\langle}|\psi_{q}% ^{(-)ij}(\bar{x}_{12})|^{2}\bigr{\rangle}_{coh},$$ (64) where the averages are defined as in Eqs. (62) and (63) with the substitutions $g_{ch}\rightarrow g$ or $g_{ch}\rightarrow g_{coh}$ and, the relative coherent contribution $G(p)$ - in Eq. (24) with a formal substitution $\int\limits_{\sigma_{out}}d\sigma_{\mu}p^{\mu}f(x,p)\rightarrow\int d^{4}xg% \left(x,p\right)$. IV Extracting coherent component of particle radiation Up to now, we have ignored the contributions $d^{3}N_{i}^{(l)}/d^{3}{\bf p}$ arising in the pion spectra from the decays of long–lived ($l$) sources such as $\eta$-, $\eta^{\prime}$–mesons, and also the unregistered kaons and hyperons. The pions from these sources possess no observable FSI (due to very large relative distance of the emission points) as well as no noticeable interference effect, because the corresponding correlation width is much smaller than the relative momentum resolution $q_{\min}$ of a detector. Therefore the measured correlation functions, defined in Eq. (1), can be expressed through the correlation functions $\widetilde{C}^{ij}(p,q)$ (discussed in previous Section) of all pion pairs $\pi^{i}\pi^{j}$ except for those containing pions from long–lived sources as follows [38]: $$C^{ij}(p,q)=n_{ij}(p_{1},p_{2})/n_{i}(p_{1})n_{j}(p_{2})=\Lambda^{ij}(p)% \widetilde{C}^{ij}(p,q)+1-\Lambda^{ij}(p),$$ (65) where the suppression parameter $\Lambda^{ij}(p)$ measures the fraction of pion pairs containing no pions from long--lived sources:121212 One can include in $N_{i}^{(l)}$ and the corresponding suppression parameters $\Lambda^{ij}$ the contribution of misidentified particles which also introduce practically no correlation. $$\Lambda^{ij}(p)=\left(1-\frac{d^{3}N_{i}^{(l)}/d^{3}{\bf p}}{d^{3}N_{i}/d^{3}{% \bf p}}\right)\left(1-\frac{d^{3}N_{j}^{(l)}/d^{3}{\bf p}}{d^{3}N_{j}/d^{3}{% \bf p}}\right)<1.$$ (66) In the (artificial) case of absent FSI effect, the correlation function $\widetilde{C}^{ij}(p,q)=C_{QS}^{ij}(p,q)$, and the averaging in $\langle\cos(qx_{12})\rangle^{\prime}$ in the QS correlation functions in Eqs. (26) should be applied only to the pion pairs containing no pions from long–lived sources. Then, assuming sufficiently good detector resolution, $q_{\min}\ll R^{-1}$, we can determine the intercepts $C^{ij}(p,0)$ calculating the correlation functions at $|q|\sim q_{\min}$: $$C^{ij}(p,q_{\min})=1+\Lambda^{ij}(p)\Bigl{[}\delta_{ij}+\bigl{(}9\bigl{\langle% }|e_{i}e_{j}|^{2}\bigr{\rangle}-1-\delta_{ij}\bigr{)}G^{2}(p)\Bigr{]}.$$ (67) The intercepts are lower than 2 for any system of identical pions and they are higher (lower) than 1 for $\pi^{+}\pi^{-}$ ($\pi^{\pm}\pi^{0}$) systems. Since the suppression parameters $\Lambda(p)$ are generally different for different pion pairs, e.g., due to different contributions of hyperon decays, it is impossible, using only apparent intercepts in Eq. (67), to separate the contributions of the coherent and long-lived sources, unless there is known a ratio of the suppression parameters $\Lambda(p)$ for identical and non-identical pions: $\Lambda^{ii}(p)/\Lambda^{ij}(p)$. Then, for example, from the intercepts of the $\pi^{+}\pi^{+}$ and $\pi^{+}\pi^{-}$ correlation functions, one obtains the coherent fraction squared: $$G^{2}(p)=\frac{\Lambda^{++}(p)}{\Lambda^{+-}(p)}\left[\frac{4}{5}\frac{\Lambda% ^{++}(p)}{\Lambda^{+-}(p)}+\frac{1}{5}\frac{C^{++}(p,q_{\min})-1}{C^{+-}(p,q_{% \min})-1}\right]^{-1}.$$ (68) In fact, the knowledge of the ratio $\Lambda^{ii}(p)/\Lambda^{ij}(p)$ is not of principle importance for the extraction of the coherent fraction $G(p)$. Besides the intercepts, one can exploit also the $q$ dependence of $C_{QS}(p,q)$ in sufficiently wide interval to follow Eqs. (26), and perform simultaneous or separate fits of the correlation functions $C^{ij}$, suitably parameterizing the correlator $\langle\cos(qx_{12})\rangle$ and the function $G(p\pm{q}/{2})$. For example, one can use the usual Gaussian correlator parameterization $$\langle\cos(qx_{12})\rangle_{ch}^{\prime}\simeq\exp(-q_{x}^{2}R_{x}^{2}-q_{y}^% {2}R_{y}^{2}-q_{z}^{2}R_{z}^{2})$$ (69) in the longitudinally comoving system (LCMS) in which the pion pair is emitted transverse to the collision axis ($p_{L}=0$). The components of the vector ${\bf q}$ are chosen parallel to the collision axis (z=Longitudinal), parallel to the vector ${\bf p}_{t}$ (x=Outward) and perpendicular to the production plane (z,x) of the pair (y=Sideward). Assuming the same radii also for the coherent emission region, and a transverse thermal law $\exp(-m_{t}/T)$ for the chaotic radiation with the temperature $T$ ($m_{t}$ is the pion transverse mass), we can parameterize the coherent fraction $G(p)$ similar to Eq. (38) for the non-relativistic case with [16] $$D(p)\simeq D(0)\exp\left[-2(p_{x}^{2}R_{x}^{2}+p_{y}^{2}R_{y}^{2}+p_{z}^{2}R_{% z}^{2})+\frac{m_{t}}{T})\right],$$ (70) and use Eq. (39) to calculate $\langle\cos(qx_{12})\rangle^{\prime}$. The presence of the FSI effect introduces the additional $q$–dependence of the correlation functions and thus improves, in principle, the accuracy of the extraction of the coherent contribution $G(p)$. Consider, for example, only effect of the Coulomb FSI and assume that the emission functions, $g_{ch}$ and $g_{coh}$, are localized in the regions of characteristic sizes much smaller than the two–pion Bohr radius $|a|=387.5$ fm so that the modulus of the non–symmetrized Coulomb wave function can be substituted by its value at zero separation. As a result the Coulomb effect factorizes in a form of so called Gamow or Coulomb factor $A_{c}(ak^{\ast})=\left|\psi_{q}^{coul}(0)\right|^{2}$ (see, e.g., [8]): $$\widetilde{C}(p,q)=A_{c}(ak^{\ast})C_{QS}(p,q),~{}~{}~{}A_{c}(x)=(2\pi/x)/[% \exp(2\pi/x)-1],$$ (71) where $k^{\ast}=|{\bf q}^{\ast}|/2$ is momentum of one of the two pions in their c.m.s. For the correlation functions of like ($a=|a|$) and unlike ($a=-|a|$) charged pions, we get $$\displaystyle C^{\pm\pm}(p,q)$$ $$\displaystyle=$$ $$\displaystyle\Lambda^{\pm\pm}(p)A_{c}(|a|k^{*})\left[1+\langle\cos(qx_{12})% \rangle^{\prime}-\frac{4}{5}G(p+q/2)G(p-q/2)\right]+[1-\Lambda^{\pm\pm}(p)],$$ $$\displaystyle C^{+-}(p,q)$$ $$\displaystyle=$$ $$\displaystyle\Lambda^{+-}(p)A_{c}(-|a|k^{*})\left[1+\frac{1}{5}G(p+q/2)G(p-q/2% )\right]+[1-\Lambda^{+-}(p)].$$ (72) Similar to the case of absent FSI, we can again use the parameterizations (69), (70) and the relation (39), and fit, simultaneously or separately, the correlation functions of like and unlike charged pions according to Eqs. (72). Moreover, the known $q$–dependence of the Gamow factors allows to separate the coherent fraction $G(p)$ from the suppression parameter $\Lambda(p)$ in a model independent way, without exploiting the q–dependence of $\langle\cos(qx_{12})\rangle_{ch}$ and $G(p\pm q/2)$. Indeed, one can perform the fits according to Eqs. (72) in an interval of $q_{\min}<|q|\ll R^{-1}$ guaranteeing $\langle\cos(qx)\rangle^{\prime}\approx 1$ and $G(p_{1,2})\approx G(p)$. The q–dependence of the correlation functions is then uniquely determined by the known functions $A_{c}(|a|k^{\ast})$ and $A_{c}(-|a|k^{\ast})$, and the three fitted parameters: $G({p})$, $\Lambda^{\pm\pm}(p)$ and $\Lambda^{+-}(p)$. Of course, such an analysis requires very good detector resolution and its good understanding. Note that Eqs. (72) are not applicable for very small ($\sim 1$ fm) as well as for large sources. In the former case one has to account for the strong FSI, in the latter - for the finite–size Coulomb effects. For ultra-relativistic heavy ion collisions, the strong FSI effect on two–pion correlation functions is negligible for like charge pions and small (a few percent) for unlike pions. The Coulomb finite–size effects can be approximately taken into account, substituting the Gamow factor $A_{c}(ak^{\ast})$ in Eqs. (17) by the finite size Coulomb factor $\widetilde{A}_{c}(ak^{\ast},\langle r^{\ast}\rangle/a)$ [39]. The latter represents a simple function of the arguments $ak^{\ast}$ and $\langle r^{\ast}\rangle/a$, where $\langle r^{\ast}\rangle$ is the mean distance of the pion emission points in the pair c.m.s., corresponding to a given momentum ${\bf p}$. Particularly, $\widetilde{A}_{c}\doteq A_{c}(ak^{\ast})[1+2\langle r^{\ast}\rangle/a]$ at $k^{\ast}<\sim 1/\langle r^{\ast}\rangle$. The dependence of the Coulomb factor on the unknown parameter $\bigl{\langle}r^{\ast}\bigr{\rangle}$ somewhat complicates the model-independent method for the extraction of coherent component $G(p)$ exploiting only the correlation functions in the region of very small relative momenta. Now, the simultaneous analysis of the correlation functions of like and unlike charged pions is required because their separate analysis yields the coherent contribution $G(p)$ up to a correction $\langle r^{\ast}\rangle/a$ only. As for the method based on a fit in a wide $|{\bf q}|$–interval, the quantity $\langle r^{\ast}\rangle$ being a unique function of the parameters characterizing the emission density, actually represents no new free parameter. Particularly, for a universal anisotropic Gaussian ${\bf r}^{\ast}$–distribution of the chaotic and coherent emission functions, the quantity $\langle r^{\ast}\rangle$ can be expressed analytically through the Gaussian interferometry radii $R_{y}$, $R_{z}$ and $R_{x}^{\ast}=\frac{M_{t}}{M}R_{x}$ ($M$ and $M_{t}$ are the two–pion effective and transverse masses respectively) in the case of practical interest, when $R_{x}^{\ast}\geq R_{y}\approx R_{z}$ [39]. In practice, however, the Gaussian parametrization of the relative distances between the emission points may happen to be insufficient. Particularly, it can lead to apparent inconsistencies in the treatment of QS and FSI effects because the latter is more sensitive to the tail of the distribution of the relative distances. If, for example, the $r^{*}$–distribution were represented by a sum of two Gaussians with essentially different mean squared radii, the $r^{*}$–”tail”, determined by the larger Gaussian radius, would influence the observed correlation functions in different ways. For identical pions, the ”tail” results in an additional rather narrow peak in the QS correlation function; however, this ”tail” would show up only as a suppression of the correlation function if the peak were concentrated at $q\lesssim q_{\min}$ or if one measured a given projection of the correlation function (e.g., in $q_{side}$ direction) fixing others ($q_{long}$ and $q_{out}$) in the interval exceeding the width of the narrow peak. At the same time, the $r^{*}$–”tail” would influence Coulomb correlations at small $q\gtrsim$ $q_{\min}$ since the long-distance nature of Coulomb forces leads to the observable effect conditioned by the ”tail” up to $r^{*}\sim|a|$. In such a situation, one can no more rely on the equality between $\langle r^{\ast}\rangle_{QS}$, determined by the interferometry radii, and the characteristic size $\langle r^{\ast}\rangle_{C}$ determining the Coulomb FSI effect. Generally, one has to introduce also different suppression parameters $\Lambda_{QS}<\Lambda_{C}$ corresponding to $\langle r^{\ast}\rangle_{QS}<\langle r^{\ast}\rangle_{C}$. Eqs. (72) for the correlation functions of like and unlike charged pions, with the substitution of the Gamow factor $A_{c}(ak^{\ast})$ by the finite–size Coulomb factor $\widetilde{A}_{c}(ak^{\ast},\langle r^{\ast}\rangle/a)$ [39], are then modified to the form: $$\displaystyle C^{\pm\pm}(p,q)$$ $$\displaystyle=$$ $$\displaystyle\Lambda_{QS}^{\pm\pm}(p)\widetilde{A}_{c}(|a|k^{\ast},\langle r^{% \ast}\rangle_{QS}^{\pm\pm}/|a|)\left[\langle\cos(qx)\rangle^{\prime}-\frac{4}{% 5}G(p+q/2)G(p-q/2)\right]+$$ (73) $$\displaystyle\Lambda_{C}^{\pm\pm}(p)\widetilde{A}_{c}(|a|k^{\ast},\langle r^{% \ast}\rangle_{C}^{\pm\pm}/|a|)+[1-\Lambda_{C}^{\pm\pm}(p)],$$ $$\displaystyle C^{+-}(p,q)$$ $$\displaystyle=$$ $$\displaystyle\Lambda_{QS}^{+-}(p)\widetilde{A}_{c}(-|a|k^{\ast},-\langle r^{% \ast}\rangle_{QS}^{+-}/|a|)\frac{1}{5}G(p+q/2)G(p-q/2)+$$ $$\displaystyle\Lambda_{C}^{+-}(p)\widetilde{A}_{c}(-|a|k^{\ast},-\langle r^{% \ast}\rangle_{C}^{+-}/|a|)+[1-\Lambda_{C}^{+-}(p)].$$ To simplify the analysis, one can neglect a small difference between the suppression parameters $\Lambda_{QS}$ and $\Lambda_{C}$ due to the tail of the $r^{\ast}$–distribution and also neglect a presumably small difference between $\langle r^{\ast}\rangle^{\pm\pm}$ and $\langle r^{\ast}\rangle^{+-}$. Note, that at SPS and RHIC energies the effect of strong FSI on $\pi^{+}\pi^{-}$ correlations is still quite noticeable and, when neglected, it can lead to a suppression of a fitted $\langle r^{\ast}\rangle^{+-}$ by $\sim 50\%$. Also, due to a substantial inaccuracy of the Coulomb factor $\widetilde{A}_{c}(ak^{\ast},\langle r^{\ast}\rangle/a)$ near the tailing point $k^{\ast}\sim 1/\langle r^{\ast}\rangle,$ the parameters $\langle r^{\ast}\rangle^{++}$ and $\langle r^{\ast}\rangle^{+-}$ can be respectively overestimated and underestimated if the fitted region were not sufficiently wide. Further, in the case of different chaotic and coherent emission volumes, one has to use finite–size Coulomb factors with different $\langle r^{\ast}\rangle$ in the chaotic, coherent and mixed terms. All these problems can be overcome exploiting the exact formulae for the two–pion wave functions (in the equal time approximation) and calculating the correlation functions according to the approximate Eq. (64). To control the systematic errors due to the smoothness assumption in Eq. (64), one can give up this assumption (at least in the pure coherent term) and check the results using instead the general expression for the two–pion spectrum in Eq. (58). After the extraction of the fractions $G(p)$ and $\Lambda^{++}(p)$ or $\Lambda^{--}(p)$, one can obtain the coherent part of the measured single–pion spectra $\omega_{{\bf p}}d^{3}N_{\pm}/d^{3}{\bf p}$. Using Eq. (66), and substituting $d^{3}N/d^{3}{\bf p\rightarrow(}d^{3}N_{\pm}/d^{3}{\bf p-}d^{3}N_{\pm}^{(l)}/d^% {3}{\bf p)}$ in Eq. (24), one gets: $$\omega_{{\bf p}}\frac{d^{3}N_{coh}}{d^{3}{\bf p}}\equiv\frac{1}{3}\left|d({p})% \right|^{2}=\omega_{{\bf p}}\frac{d^{3}N_{\pm}}{d^{3}{\bf p}}G(p)\sqrt{\Lambda% ^{\pm\pm}(p)}.$$ (74) The coherent part of the observed spectra is thus directly connected with the intensity $\left|d({p})\right|^{2}$ of the quasi-classical source of coherent pions. V Conclusions Using the density matrix formalism, satisfying the requirements of the isospin symmetry and the super-selection rule for generalized coherent states, and accounting for the final state interaction in the two–body approximation, we have developed methods allowing one to study the coherent component of pion radiation which, in heavy ion collisions, is likely conditioned by formation of a quasi-classical pion source. These methods are based on a nontrivial modification of the effects of quantum statistics and final state interaction on two–pion correlation functions (including those of non-identical pions) in the presence of a coherent pion radiation generated by the decay of the quasipionic ground state (”condensate”). It has been shown that the combined analysis of the correlation functions of like and unlike pions gives the possibility to discriminate between the suppression of the like–pion correlation functions conditioned by the coherent pion component and that due to the decays of long–lived sources. The methods allowing to extract the coherent pion component from $\pi^{+}\pi^{-}$ and $\pi^{\pm}\pi^{\pm}$ correlation functions and single–pion spectra have been discussed in detail for large expanding systems produced in ultra–relativistic heavy ion collisions. For such systems, the two–pion final state interaction is dominated by the Coulomb one and plays an important role in this analysis, allowing one to determine the coherent fraction using a suitable model for the coherent and chaotic emission functions and fitting the corresponding correlation functions. For rough estimations the procedure can be substantially simplified accounting for the finite–size Coulomb effects in an approximate analytic form [39]. Finally, the coherent fractions extracted from the correlation analysis, combined with the single–pion spectra, give us the possibility to determine the spectrum of the coherent pion radiation above the thermal background and, therefore, to estimate the quasipionic condensate at the pre-decaying stage of the matter evolution and discriminate between possible mechanisms of coherent production in ultra–relativistic A+A collisions. 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Neutrino and antineutrino charge-exchange reactions on ${}^{12}$C A. R. Samana${}^{1,2}$, F. Krmpotić${}^{3}$, N. Paar${}^{4}$, and C. A. Bertulani${}^{1}$ ${}^{1}$ Department of Physics, Texas A&M University Commerce, P.O.3011 Commerce, 75429 TX, USA ${}^{2}$ Departamento de Ciências Exactas e Tecnológicas, Universidade Estadual de Santa Cruz, CEP 45662-000 Ilheús, Bahia-BA, Brazil ${}^{3}$Instituto de Física La Plata, CONICET, Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, 1900 La Plata, Argentina, ${}^{4}$Physics department, Faculty of Science, University of Zagreb, Croatia (November 23, 2020) Abstract We extend the formalism of weak interaction processes, obtaining new expressions for the transition rates, which greatly facilitate numerical calculations, both for neutrino-nucleus reactions and muon capture. Explicit violation of CVC hypothesis by the Coulomb field, as well as development of a sum rule approach for the inclusive cross sections have been worked out. We have done a thorough study of exclusive (ground state) properties of ${}^{12}$B and ${}^{12}$N within the projected quasiparticle random phase approximation (PQRPA). Good agreement with experimental data achieved in this way put in evidence the limitations of standard RPA and the QRPA models, which come from the inability of the RPA in opening the $p_{3/2}$ shell, and from the non-conservation of the number of particles in the QRPA. The inclusive neutrino/antineutrino ($\nu/\tilde{\nu}$) reactions ${}^{12}$C($\nu,e^{-})^{12}$N and ${}^{12}$C($\tilde{\nu},e^{+})^{12}$B are calculated within both the PQRPA, and the relativistic QRPA (RQRPA). It is found that the magnitudes of the resulting cross-sections: i) are close to the sum-rule limit at low energy, but significantly smaller than this limit at high energies both for $\nu$ and $\tilde{\nu}$, ii) they steadily increase when the size of the configuration space is augmented, and particulary for $\nu/\tilde{\nu}$ energies $>200$ MeV, and iii) converge for sufficiently large configuration space and final state spin. The quasi-elastic ${}^{12}$C($\nu,\mu^{-})^{12}$N cross section recently measured in the MiniBooNE experiment is briefly discussed. We study the decomposition of the inclusive cross-section based on the degree of forbiddenness of different multipoles. A few words are dedicated to the $\nu/\tilde{\nu}$-${}^{12}$C charge-exchange reactions related with astrophysical applications. pacs: 23.40.-s, 25.30.Pt, 26.50.+x ††preprint: I Introduction The massiveness of neutrinos and the related oscillations are strongly sustained by many experimental works involving atmospheric, solar, reactor and accelerator neutrinos  Ath96 ; Ath98 ; Agu01 ; Fuk98 ; Aha05 ; Ara04 ; Ahn03 . The subsequent experimental goal is to determine precisely the various parameters of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mass matrix, absolute masses of different flavors of neutrinos, CP violation in neutrino sector, etc. To address these problems several analyses of neutrino oscillation data are presently going on. At the same time, several experiments presently collect data, and others are planned. Accelerator experiments, experiments with neutrinos from $\nu$-factories, $\beta$-beams, etc., are also planned and designed, as well as some experiments with natural $\nu$-sources like solar neutrinos, atmospheric neutrinos, or antineutrinos from nuclear reactors. The neutrino-nucleus scattering on ${}^{12}$C is important because this nucleus is a component of many liquid scintillator detectors. Experiments such as LSND Ath96 ; Ath98 , KARMEN Mas98 ; Arm02 , and LAMPF All90 ; Kra92 have used ${}^{12}$C to search for neutrino oscillations, and for measuring neutrino-nucleus cross sections. Present atmospheric and accelerator based neutrino oscillation experiments also involve ${}^{12}$C, and operate at neutrino energies $E_{\nu}\sim 1$ GeV in order to access the relevant regions of oscillation parameter space. This is the case of the SciBar detector SciBar , where the molecule $C_{8}H_{8}$ is involved, and the MiniBooNE detector  MiniBooNE , which uses the light mineral oil containing the molecule $CH_{2}$. The ${}^{12}$C target will be used in several planned experiments, such as the spallation neutron source (SNS) at Oak Ridge National Laboratory (ORNL) Efr05 , and the LVD (Large Volume Detector) experiment Aga07 , developed by the INFN in Gran Sasso. For the planned experimental searches of supernovae neutrino signals, which involve ${}^{12}$C as scintillator liquid detector, the precise knowledge of neutrino cross sections of ${}^{12}$N and ${}^{12}$B ground-states, i.e., of $\sigma_{e^{-}}(E_{\nu},1^{+}_{1})$, and $\sigma_{e^{+}}(E_{\tilde{\nu}},1^{+}_{1})$ is very important. In fact, in the LVD experiment Aga07 the number of events detected during the supernova explosion are estimated by convoluting the neutrino supernova flux with: i) the interaction cross sections, ii) the efficiency of the detector, and iii) the number of target nuclei. For the carbon content of the LVD detector have been used so far $\sigma_{e^{-}}(E_{\nu},1^{+}_{1})$, and $\sigma_{e^{+}}(E_{\tilde{\nu}},1^{+}_{1})$, as obtained from the Elementary Particle Treatment (EPT) Fuk88 . Moreover, as an update of the LVD experiment related to supernovae neutrinos detection (where ${}^{12}$C will also be employed), there is ongoing design study concerning large size scintillator detectors, called LAGUNA, where a 50 kt scintillator LENA is being considered Aut07 . On the other hand, as the ${}^{12}$C nucleus forms one of the onion-like shells of a large star before collapse, it is also important for astrophysics studies. Concomitantly, several authors Lun03 ; Dig03 ; Aga07 ; Dua07 ; Das08 ; Das10 ; Mez10 have recently stressed the importance of measuring supernova neutrino oscillations. They claim that a supernova explosion represents a unique scenario for further study of the PMNS matrix. The corresponding neutrinos, which carry all flavors were observed in only one occasion (SN1987A), have an energy $E_{\nu}\lesssim 100$ MeV Str06 , and are also studied through the interactions with carbon nuclei in the liquid scintillator. Thus, the main interest in the neutrino/antineutrino-${}^{12}$C charge-exchange cross sections comes from the neutrino oscillations, and precise knowledge of the cross sections in the neutrino energies going from a few MeV´s up to a few GeV´s is required. Up to quite recently the only available experimental information on reactions was that for the flux-averaged cross-sections: i) ${}^{12}$C$(\nu_{e},e^{-})^{12}$N in the DAR region: $E_{\nu_{e}}<60$ MeV Ath97 ; Aue01 ; Zei98 , and ii) ${}^{12}$C$(\nu_{\mu},\mu^{-})^{12}$N in the DIF region: $127$ MeV $\leq E_{\nu_{\mu}}\leq 300$ MeV Ath97a ; Aue02a ; LSND . In last few years, however, several experimental programs at MiniBooNE MiniBooNE Collaboration , K2K K2K Collaboration , and SciBooNE SciBooNE Collaboration yield results on the ($\nu_{\mu},^{12}$C) cross section for $0.4$ GeV $\leq E_{\nu_{\mu}}\leq 1.7$ GeV. It is well known that for $E_{\nu}$ larger than a few hundreds MeV’s, besides the quasi-elastic (QE) channel, many inelastic channels are open and pion production becomes important. In fact, there have been quite active experimental efforts to investigate neutrino-induced coherent single-pion production in the $\Delta$-excitation region of ${}^{12}$C. Starting approximately at the threshold coming from the pion and charged lepton masses ($\mathrm{m}_{\pi}$ and $\mathrm{m}_{\ell}$), the $\pi+\ell$ production cross section steadily increases with the neutrino energy becoming larger than the quasi-elastic one for $E_{\nu}\>\raisebox{-2.15pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>1.5$ GeV MiniBooNE Collaboration ; K2K Collaboration ; SciBooNE Collaboration . From the theoretical side there have been great efforts to understand the nuclear structure within the triad $\{^{12}$B,${}^{12}$C,${}^{12}$N$\}$. In the seminal work of O’Connell, Donelly, and Walecka  Con72 a unified analysis of electromagnetic, and semileptonic weak interactions was presented. To describe the nuclear dynamics they have used the particle-hole Tamm-Dancof Approximation (TDA) within a very small single-particle space 111From now on a single-particle (s.p.) space that includes all orbitals within $N$ harmonic oscillator (HO) shells will be labeled as space $S_{N}$. ($S_{2}$ $\equiv\{1s_{1/2}$, $1p_{3/2}$, $1p_{1/2}$, $1d_{5/2}$, $2s_{1/2}\}$) Don70 . To achieve agreement with experiments for the $\beta^{\pm}$-decays, and $\mu$-capture they were forced to use an overall reduction factor $\xi^{2}$ of the order of $4$ ($2$) for even (odd) parity states. They have also pointed out that this factor would become totally unnecessary with use of a better nuclear model able to open the $1p_{3/2}$ shell. Rather thorough comparisons of $2s1d$ and $2p1f$ shell-model predictions with measured allowed $\beta$-decay rates have yielded a simple, phenomenological effective axial coupling $g_{\scriptscriptstyle A}=1$ that should be used rather than the bare value Bro85 ; Cas87 ; Ost92 ; Mar96 . This observation is the basis for many nuclear model estimates of the Gamow-Teller (GT) response that governs allowed neutrino cross sections. In Ref. Con72 $g_{\scriptscriptstyle A}=1.23$ was used based on a study of neutron $\beta$-decay, and, as the analyzed processes were dominantly of the axial-vector type, the use of $g_{\scriptscriptstyle A}=1$ would have diminished the reduction factors $\xi^{2}$ in an appreciable way. In the Random Phase Approximation (RPA), besides the TDA forward-going amplitudes, the backward-going amplitudes are present as well. However, these additional RPA amplitudes did not help to open the $1p_{3/2}$ shell in the continuum RPA (CRPA) calculations of Kolbe, Langanke, and Krewald Kol94 . Thus, as in the case of the TDA used in Ref. Con72 , to get agreement with data for the ground state triplet $T=1$ ($\beta^{\pm}$-decays, $\mu$-capture, and the exclusive ${}^{12}$C$(\nu_{e},e^{-})^{12}$N reaction) their calculations were rescaled by a factor $\cong 4$. The main aim of the CRPA is to describe appropriately not only the bound states but also the virtual (quasi-bound), resonant, and continuum states, which are treated as bound states in the RPA. However, this superiority has not been evidenced so far in numerical calculations. For instance, in the case of $\mu$-capture rates in ${}^{16}$N the two methods agree with each other quite well for the $0^{-}$ and $1^{-}$ states, while the RPA result is preferred for the $2^{-}$ state  Kol94a . To open the $1p_{3/2}$ shell one has to introduce pairing correlations. This is done within the Shell Model (SM) Hay00 ; Vol00 ; Suz06 , which reproduces quite well both i) the experimental flux-averaged exclusive, and inclusive cross sections for the ${}^{12}\rm C(\nu_{e},e^{-})^{12}\rm N$ DAR Ath97 ; Aue01 ; Zei98 , and ${}^{12}\rm C(\nu_{\mu},\mu^{-})^{12}\rm N$ DIF Ath97a reactions, and ii) the $\mu^{-}+{{}^{12}\rm C}\rightarrow\nu_{\mu}+{{}^{12}\rm B}$ muon-capture modes Mil72 ; Mea01 ; Sto02 . The quasiparticle RPA (QRPA) also opens the $1p_{3/2}$ shell by means of the pairing interaction. However, it fails as well in accounting for the exclusive processes to the isospin triplet $T=1$ in ${}^{12}\rm C$, because a new problem emerges, as first observed by Volpe et al. Vol00 . They noted that within the QRPA the lowest state in ${{}^{12}\rm N}$ irremediably turned out not to be the most collective one. Later it was shown Krm02 ; Krm05 ; Sam06 that: 1) the origin of this difficulty arises from the degeneracy among the four lowest proton-neutron two-quasiparticle ($2qp$) states $|1p_{1/2}1p_{3/2}\rangle$, $|1p_{3/2}1p_{3/2}\rangle$, $|1p_{1/2}1p_{1/2}\rangle$ and $|1p_{3/2}1p_{1/2}\rangle$, which, in turn, comes from the fact that for $N=Z=6$ the quasiparticle energies $E_{1p_{1/2}}$ and $E_{1p_{3/2}}$ are very close to each other, and 2) it is imperative to use the projected QRPA (PQRPA) for a physically sound description of the weak processes among the ground states of the triad $\{{{{}^{12}\rm B},{{}^{12}\rm C},{{}^{12}\rm N}}\}$ Krm02 ; Krm05 ; Sam06 ; see Figs. 2 and 3 in Ref. Krm05 . In summary, neither the CRPA nor the QRPA are the appropriate nuclear models to describe the “fine structure” of exclusive charge-exchange processes around ${}^{12}$C, and they only can be used for global inclusive descriptions. Of course, the same is valid for the relativistic RPA (RQRPA) that has recently been applied with success in calculations of inclusive charged-current neutrino-nucleus reactions in ${}^{12}$C, ${}^{16}$O, ${}^{56}$Fe, and ${}^{208}$Pb Paa07 , and total muon capture rates on a large set of nuclei from ${}^{12}$C to ${}^{244}$Pu Mar09 . The continuum QRPA (CQRPA) would have to be superior to the QRPA for the same reasons that the CRPA would have to be better than the RPA. Nevertheless, neither this superiority has been put in evidence by numerical calculations Hag01 ; Rod08 . Finally, it is clear that the nuclear structure descriptions inspired on the Relativistic Fermi Gas Model (RFGM) Smi72 ; Nie04 ; Val06 , which do not involve multipole expansions, should only be used for inclusive quantities. When the effects due to resonant and continuum states are considered, as it is done within the CRPA and CQRPA, the spreading in strength of the hole states in the inner shells should also be taken into account for the sake of consistency. In fact, a single-particle state $j$ that is deeply bound in the parent nucleus, after a weak interacting process can become a highly excited hole-state $j^{-1}$ in the continuum of the residual nucleus. There it is suddenly mixed with more complicated configurations (2h1p, 3h2p, …excitations, collective states, and so on) spreading its strength in a relatively wide energy interval Ma85 222One should keep in mind that the mean life of ${}^{12}$N and ${}^{12}$B are, respectively, $11.0$ and $20.2$ ms, while strong interaction times are of the order of $10^{-21}$ s.. This happens, for instance, with the $1s_{1/2}$ orbital in ${}^{12}$C, that is separated from the $1p_{3/2}$ state by approximately $23$ MeV, which is enough to break the $12$ particle system, where the energy of the last excited state amounts to $11.5$ MeV in ${}^{12}$N, and $16.5$ MeV in ${}^{12}$B channels. Although the detailed structure and fragmentation of hole states are still not well known, the exclusive knockout reactions provide a wealth of information on the structure of single-nucleon states of nuclei. Excitation energies and widths of proton-hole states were systematically measured with quasifree (p, 2p) and $(e,e^{\prime}$p) reactions, which revealed the existence of inner orbital shells in nuclei Ja73 ; Fr84 ; Be85 ; Le94 ; Ya96 ; Ya01 ; Yo03 ; Ya04 ; Ko06 . In the TDA calculation of Ref. Con72 the $S_{2}$ space has been used, which extends only from $13.77$ MeV up to $30.05$ MeV, embracing, respectively, $1,2,2,1$, and $1$ negative parity states $J^{\pi}=0^{-},1^{-},2^{-},3^{-}$, and $4^{-}$, and $1,2,2$, and $1$ positive parity states $J^{\pi}=0^{+},1^{+},2^{+}$, and $3^{+}$. With such small configuration space, the neutrino cross sections $\sigma_{e}(E_{\nu})$, and $\sigma_{\mu}(E_{\nu})$ have been evaluated up to a neutrino energy $E_{\nu}$ of $0.6$ GeV, and extrapolated up to $20$ GeV. In recent years, however, large configuration spaces have been used in the evaluation of QE cross sections for $E_{\nu}\sim 1$ GeV. For instance, Amaro et al. Ama05 have employed the single-particle SM (TDA without the residual interaction) in a semirelativistic description of quasielastic neutrino reactions $(\nu_{\mu},\mu^{-})$ on ${}^{12}$C going up to $E_{\nu}=1.5$ GeV, and including multipoles $J^{\pi}\leq 47^{\pm}$. Good agreement with the RFGM was obtained for several choices of kinematics of interest for the ongoing neutrino oscillation experiments. Kolbe et al. Kol03 have also achieved an excellent agreement between the RFGM and the CRPA calculations of the total cross section and the angular distribution of the outgoing electrons in ${}^{16}{\rm O}(\nu_{e},e)X$ for $E_{\nu}\leq 0.5$ GeV. They have considered states up to $J^{\pi}=9^{\pm}$ only, and didn’t specify the configuration space used. Moreover, Valverde et al. Val06 have analyzed the theoretical uncertainties of the RFGM developed in Nie04 for the $(\nu_{e},e^{-})$, and $(\nu_{\mu},\mu^{-})$ cross sections in ${}^{12}$C, ${}^{16}$O, and ${}^{40}$Ca for $E_{\nu}\leq 0.5$ GeV. The work of Kim et al. Kim08 should also be mentioned where were studied the effects of strangeness on the $(\nu_{\mu},\mu^{-})$ and $({\tilde{\nu}}_{\mu},\mu^{+})$ cross sections in ${}^{12}$C for incident energies between $0.5$ MeV and $1.0$ GeV, within the framework of a relativistic single-particle model. Quite recently, Butkevich But10 has also studied the scattering of muon neutrino on carbon targets for neutrino energies up to $2.8$ GeV within a relativistic shell-model approach without specifying the model space. For relatively large neutrino energies ($E_{\nu_{e}}\>\raisebox{-2.15pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$% }\>\mathrm{m}_{\pi}$, and $E_{\nu_{\mu}}\>\raisebox{-2.15pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}% }$}\>\mathrm{m}_{\pi}+\mathrm{m}_{\mu}$) to the above-mentioned QE cross sections should be added the pion production cross section, as done, for instance, in Refs. Mart09 ; Lei09 . One should also note that $\sigma_{e}(E_{\nu_{e}})$ and $\sigma_{\mu}(E_{\nu_{\mu}})$ coincide with each other asymptotically. This is clearly put in evidence in the Extreme Relativistic Limit (ERL) where $|{\bf p}_{\ell}|/E_{\ell}\rightarrow 1$, and the neutrino-nucleus cross sections depend on $\mathrm{m}_{\ell}$ only trough the threshold energy, as can be seen from the Appendix C of the present work. The Figure 4 from Ref. Kur90 is also illustrative in this respect. Therefore, we focus our attention only on the quasi-elastic cross section $\sigma_{e}(E_{\nu_{e}})$ since, at muon-neutrino energies involved in the MiniBooNE experiment MiniBooNE , it is equal to $\sigma_{\mu}(E_{\nu_{\mu}})$ for all practical purposes. One of main objectives in the present study is to analyze the effect of the size of the configuration space up to neutrino energies of several hundred MeV. As in several previous works Con72 ; Mar96 ; Kol94 ; Kol94a ; Hay00 ; Vol00 ; Suz06 ; Krm02 ; Krm05 ; Sam06 ; Paa07 ; Mar09 ; Ama05 ; Kol03 ; Val06 ; Kim08 ; Kur90 this will be done in first order perturbation theory. The consequences of the particle-particle force in the S = 1, T = 0 channel, within the PQRPA will also be examined. The importance of this piece of the residual interaction was discovered more than 20 years ago by Vogel and Zirnbauer Vog86 and Cha Cha87 , and since then the QRPA became the most frequently used nuclear structure method for evaluating double $\beta$-decay rates. A few words will be devoted as well as to the non-relativistic formalisms for neutrino-nucleus scattering. The most popular one was developed by the Walecka group Con72 ; Don79 ; Hax79 ; Wal95 , where the nuclear transition matrix elements are classified as Coulomb, longitudinal, transverse electric, and transverse magnetic multipole moments. We feel that these denominations might be convenient when discussing simultaneously charge-conserving, and charge-exchange processes, but seems unnatural when one considers only the last ones. As a matter of fact, this terminology is not often used in nuclear $\beta$-decay, $\mu$-capture, and charge-exchange reactions where one only speaks of vector and axial matrix elements with different degrees of forbiddenness: allowed (GT and Fermi), first forbidden, second forbidden, etc., types Beh82 ; Krm80 . There are exceptions, of course, as for instance, is the recent work of Marketin et al. Mar09 on muon capture, where the Walecka’s classification was used. The formalism worked out by Kuramoto Kur90 is also frequently used for the evaluation of neutrino-nucleus cross-sections. It is simpler than that of Walecka, but it does not contain relativistic matrix elements, nor is applicable for muon capture rates. More recently, we have introduced another formalism Krm02 ; Krm05 ; Sam06 . Besides of being almost as simple as that of Kuramoto, it retains relativistic terms and can be used for $\mu$-capture. This formalism is briefly sketched here, including the consequences of the violation of the Conserved Vector Current (CVC) by the Coulomb field. It is furthermore simplified by classifying the nuclear matrix elements in natural and unnatural parities. We also show how within the present formalism both the sum rule approach, and the formula for ERL look like. In Section II we briefly describe the formalism used to evaluate different weak interacting processes. Some details are delegated to the Appendices: A) Contributions of natural and unnatural parity states to the transition rates, B) Sum rule approach for the inclusive neutrino-nucleus cross section, C) Formula for the inclusive neutrino-nucleus cross section at the extreme relativistic limit, and D) Formula for the muon capture rate. In Section III we present, and discuss the numerical results. Comparisons with experimental data, as well as with previous theoretical studies, are done whenever possible. Here we firstly sketch the two theoretical frameworks, namely the PQRPA and RQRPA, used in the numerical calculations. In subsections II, and III.2 we present the results for the exclusive and inclusive processes, respectively. Finally, in Section IV we give a brief summary, and final conclusions. II Formalism for the Weak Interacting Processes The weak Hamiltonian is expressed in the form Don79 ; Wal95 ; Bli66 $$\displaystyle H_{{\scriptscriptstyle{W}}}({\bf r})$$ $$\displaystyle=$$ $$\displaystyle\frac{G}{{\sqrt{2}}}J_{\alpha}l_{\alpha}e^{-i{\bf r}\cdot{\bf k}},$$ (1) where $G=(3.04545\pm 0.00006){\times}10^{-12}$ is the Fermi coupling constant (in natural units), the leptonic current $l_{\alpha}\equiv\{{\bf l},il_{\emptyset}\}$ is given by the Eq. (2.3) in Ref. Krm05 and the hadronic current operator $J_{\alpha}\equiv\{{\bf J},iJ_{\emptyset}\}$ in its nonrelativistic form reads 333 As in Ref. Krm05 we use the Walecka’s notation Wal95 with the Euclidean metric for the quadrivectors, and $\alpha=1,2,3,4$. The only difference is that we substitute his indices $(0,3)$ by our indices $(\emptyset,0)$, where we use the index $\emptyset$ for the temporal component and the index 0 for the third spherical component. $$\displaystyle J_{\emptyset}$$ $$\displaystyle=$$ $$\displaystyle g_{\mbox{\tiny V}}+(\overline{g}_{\mbox{\tiny A}}+\overline{g}_{% \mbox{\tiny P1}}){\mbox{\boldmath$\sigma$}}\cdot\hat{{\bf k}}+g_{% \scriptscriptstyle{A}}\frac{i\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$% \nabla$}}{\rm M},$$ (2) $$\displaystyle{{\bf J}}$$ $$\displaystyle=$$ $$\displaystyle-g_{\mbox{\tiny A}}{\mbox{\boldmath$\sigma$}}-i\overline{g}_{% \mbox{\tiny W}}{\mbox{\boldmath$\sigma$}}\times\hat{{\bf k}}-\overline{g}_{% \mbox{\tiny V}}\hat{{\bf k}}+\overline{g}_{\mbox{\tiny P2}}({\mbox{\boldmath$% \sigma$}}\cdot\hat{{\bf k}})\hat{{\bf k}}-g_{\scriptscriptstyle{V}}\frac{i% \mbox{\boldmath$\nabla$}}{\rm M},$$ where $\hat{{\bf k}}\equiv{{\bf k}}/{|{\bf k}|}$. The quantity $$\displaystyle k=P_{i}-P_{f}\equiv\{{\bf k},ik_{\emptyset}\}$$ (3) is the momentum transfer, ${\rm M}$ is the nucleon mass, and $P_{i}$ and $P_{f}$ are momenta of the initial and final nucleon (nucleus). The effective vector, axial-vector, weak-magnetism and pseudoscalar dimensionless coupling constants are, respectively $$\displaystyle g_{\scriptscriptstyle V}$$ $$\displaystyle=$$ $$\displaystyle 1,~{}g_{\scriptscriptstyle A}=1,~{}g_{\scriptscriptstyle M}=% \kappa_{p}-\kappa_{n}=3.70,$$ $$\displaystyle~{}g_{\scriptscriptstyle P}$$ $$\displaystyle=$$ $$\displaystyle g_{\scriptscriptstyle A}\frac{2\mathrm{M}\mathrm{m}_{\ell}}{k^{2% }+\mathrm{m}_{\pi}^{2}},$$ (4) where the following short notation has been introduced: $$\displaystyle\overline{g}_{\mbox{\tiny V}}$$ $$\displaystyle=$$ $$\displaystyle g_{\mbox{\tiny V}}\frac{{\kappa}}{2\mathrm{M}};~{}\overline{g}_{% \mbox{\tiny A}}=g_{\mbox{\tiny A}}\frac{{\kappa}}{2\mathrm{M}};~{}\overline{g}% _{\mbox{\tiny W}}=(g_{\mbox{\tiny V}}+g_{\mbox{\tiny M}})\frac{{\kappa}}{2% \mathrm{M}},$$ $$\displaystyle\overline{g}_{\mbox{\tiny P1}}$$ $$\displaystyle=$$ $$\displaystyle g_{\mbox{\tiny P}}\frac{{\kappa}}{2\mathrm{M}}\frac{k_{\emptyset% }}{\mathrm{m}_{\ell}};~{}\overline{g}_{\mbox{\tiny P2}}=g_{\mbox{\tiny P}}% \frac{{\kappa}}{2\mathrm{M}}\frac{{\kappa}}{\mathrm{m}_{\ell}},$$ (5) where ${\kappa}\equiv|{\bf k}|$. The above estimates for $g_{\scriptscriptstyle M}$ and $g_{\scriptscriptstyle P}$ come from the conserved vector current (CVC) hypothesis, and from the partially conserved axial vector current (PCAC) hypothesis, respectively. The finite nuclear size (FNS) effect is incorporated via the dipole form factor with a cutoff $\Lambda=850$ MeV, i.e., $g\rightarrow g\left[{\Lambda^{2}}/(\Lambda^{2}+k^{2})\right]^{2}$. In performing the multipole expansion of the nuclear operators $$O_{\alpha}\equiv({\bf O},iO_{\emptyset})=J_{\alpha}e^{-i{\bf k}\cdot{\bf r}},$$ (6) it is convenient 1) to take the momentum ${\bf k}$ along the $z$ axis, i.e., $$\displaystyle e^{-i{\bf k}\cdot{\bf r}}$$ $$\displaystyle=$$ $$\displaystyle\sum_{\sf L}i^{-\sf L}\sqrt{4\pi(2{\sf L}+1)}j_{\sf L}(\rho)Y_{{% \sf L}0}(\hat{{\bf r}}),$$ (7) $$\displaystyle=$$ $$\displaystyle\sum_{\sf J}i^{-\sf J}\sqrt{4\pi(2{\sf J}+1)}j_{\sf J}(\rho)Y_{{% \sf J}0}(\hat{{\bf r}}),$$ where $\rho={\kappa}r$, and 2) to define the operators ${\sf O}_{{\alpha}}$ as $$O_{\alpha}=\sqrt{4\pi}\sum_{\sf J}i^{-{\sf J}}\sqrt{2{\sf J}+1}{\sf O}_{{% \alpha}{\sf J}}.$$ (8) In this way we avoid the troublesome factor $i^{-{\sf J}}$. In spherical coordinates (${m}=-1,0,+1)$ we have $$\displaystyle J_{\emptyset}$$ $$\displaystyle=$$ $$\displaystyle g_{\mbox{\tiny V}}+(\overline{g}_{\mbox{\tiny A}}+\overline{g}_{% \mbox{\tiny P1}})\sigma_{0}+ig_{\scriptscriptstyle{A}}{\rm M}^{-1}\mbox{% \boldmath$\sigma$}\cdot\mbox{\boldmath$\nabla$}$$ $$\displaystyle J_{m}$$ $$\displaystyle=$$ $$\displaystyle-g_{\mbox{\tiny A}}\sigma_{m}+{m}\overline{g}_{\mbox{\tiny W}}% \sigma_{m}+\delta_{{m}0}[-\overline{g}_{\mbox{\tiny V}}+\overline{g}_{\mbox{% \tiny P2}}\sigma_{0}]$$ (9) $$\displaystyle-$$ $$\displaystyle ig_{\scriptscriptstyle{V}}{\rm M}^{-1}\nabla_{m},$$ and $$\displaystyle{\sf O}_{\emptyset{\sf J}}$$ $$\displaystyle=$$ $$\displaystyle j_{\sf J}(\rho)Y_{{\sf J}0}(\hat{{\bf r}})J_{\emptyset},$$ $$\displaystyle{\sf O}_{{m}{\sf J}}$$ $$\displaystyle=$$ $$\displaystyle\sum_{{\sf L}}i^{\sf J-L}F_{{\sf LJ}m}j_{\sf L}(\rho)\left[Y_{{% \sf L}}(\hat{{\bf r}})\otimes{\bf J}\right]_{\sf J},$$ (10) where $$\displaystyle F_{{\sf LJ}m}$$ $$\displaystyle\equiv$$ $$\displaystyle(-)^{\sf J+m}\sqrt{2{\sf L}+1}\left(\negthinspace\begin{array}[]{% ccc}{\sf L}&1&{\sf J}\\ 0&-m&m\end{array}\right)$$ (11) $$\displaystyle=$$ $$\displaystyle(-)^{1+m}(1,-m{\sf J}m|{\sf L}0),$$ is a Clebsch-Gordan coefficient. 444Their values are: $\displaystyle F_{{\sf J}+1,{\sf J},0}$ $\displaystyle=$ $\displaystyle-\sqrt{\frac{{\sf J}+1}{2{\sf J}+1}},~{}~{}~{}~{}F_{{\sf J}-1,{% \sf J},0}=\sqrt{\frac{{\sf J}}{2{\sf J}+1}},$ $\displaystyle F_{{\sf J}+1,{\sf J},\pm 1}$ $\displaystyle=$ $\displaystyle\sqrt{\frac{{\sf J}}{2(2{\sf J}+1)}},~{}~{}~{}~{}F_{{\sf J},{\sf J% }-1,\pm 1}=\sqrt{\frac{{\sf J}+1}{2(2{\sf J}+1)}},$ $\displaystyle F_{{\sf J},{\sf J},0}$ $\displaystyle=$ $\displaystyle 0,~{}~{}~{}~{}F_{{\sf J},{\sf J},\pm 1}=\mp{1\over\sqrt{2}}.$ Explicitly, from (9) $$\displaystyle{\sf O}_{\emptyset{\sf J}}$$ $$\displaystyle=$$ $$\displaystyle g_{\scriptscriptstyle{V}}{{\cal M}}_{\sf J}^{\scriptscriptstyle V% }+ig_{\scriptscriptstyle{A}}{{\cal M}}^{\scriptscriptstyle A}_{\sf J}+i(% \overline{g}_{\mbox{\tiny A}}+\overline{g}_{\mbox{\tiny P1}}){{\cal M}}^{% \scriptscriptstyle A}_{0{\sf J}},$$ (12) $$\displaystyle{\sf O}_{{m}{\sf J}}$$ $$\displaystyle=$$ $$\displaystyle i(\delta_{{m}0}\overline{g}_{\mbox{\tiny P2}}-g_{\mbox{\tiny A}}% +m\overline{g}_{\mbox{\tiny W}}){{\cal M}}^{\scriptscriptstyle A}_{{m}{\sf J}}$$ (13) $$\displaystyle+$$ $$\displaystyle g_{\scriptscriptstyle{V}}{{\cal M}}^{\scriptscriptstyle V}_{{m}{% \sf J}}-\delta_{{m}0}\overline{g}_{\mbox{\tiny V}}{{\cal M}}_{\sf J}^{% \scriptscriptstyle V}.$$ The elementary operators are given by $$\displaystyle{{\cal M}}^{\scriptscriptstyle V}_{\sf J}$$ $$\displaystyle=$$ $$\displaystyle j_{\sf J}(\rho)Y_{{\sf J}}(\hat{{\bf r}}),$$ $$\displaystyle{{\cal M}}^{\scriptscriptstyle A}_{\sf J}$$ $$\displaystyle=$$ $$\displaystyle{\rm M}^{-1}j_{\sf J}(\rho)Y_{\sf J}(\hat{{\bf r}})(\mbox{% \boldmath$\sigma$}\cdot\mbox{\boldmath$\nabla$}),$$ (14) and $$\displaystyle{{\cal M}}^{\scriptscriptstyle A}_{{m\sf J}}$$ $$\displaystyle=$$ $$\displaystyle\sum_{{\sf L}\geq 0}i^{{\sf J-L}-1}F_{{{\sf LJ}m}}j_{\sf L}(\rho)% \left[Y_{{\sf L}}(\hat{{\bf r}})\otimes{\mbox{\boldmath$\sigma$}}\right]_{{\sf J% }},$$ $$\displaystyle{{\cal M}}^{\scriptscriptstyle V}_{{m\sf J}}$$ $$\displaystyle=$$ $$\displaystyle{\rm M}^{-1}\sum_{{\sf L}\geq 0}i^{{\sf J-L}-1}F_{{{\sf LJ}m}}j_{% \sf L}(\rho)[Y_{\sf L}(\hat{{\bf r}})\otimes\mbox{\boldmath$\nabla$}]_{{\sf J}}.$$ The CVC relates the vector-current pieces of the operator (6) as (see Eqs. (10.45) and (9.7) from Ref. Beh82 ) $$\displaystyle{\bf k}\cdot{\bf O}^{\scriptscriptstyle V}$$ $$\displaystyle\equiv$$ $$\displaystyle{\kappa}O_{0}^{\scriptscriptstyle V}=\tilde{k}_{\emptyset}O_{% \emptyset}^{\scriptscriptstyle V},$$ (16) with $$\displaystyle\tilde{k}_{\emptyset}$$ $$\displaystyle\equiv$$ $$\displaystyle k_{\emptyset}-S(\Delta E_{\rm Coul}-\Delta\mathrm{M}),$$ (17) where $$\displaystyle\Delta E_{\rm Coul}\cong\frac{6e^{2}Z}{5R}\cong 1.45ZA^{-1/3}~{}~% {}\mbox{MeV},$$ (18) is the Coulomb energy difference between the initial and final nuclei, $\Delta\mathrm{M}=\mathrm{M}_{n}-\mathrm{M}_{p}=1.29$ MeV is the neutron-proton mass difference, and $S=\pm 1$ for neutrino and antineutrino scattering, respectively. The consequence of the CVC relation (16) is the substitution $$\displaystyle g_{\scriptscriptstyle{V}}{{\cal M}}^{\scriptscriptstyle V}_{{\sf 0% }{\sf J}}-\overline{g}_{\mbox{\tiny V}}{{\cal M}}_{\sf J}^{\scriptscriptstyle V% }\rightarrow\frac{\tilde{k}_{\emptyset}}{{\kappa}}g_{\scriptscriptstyle{V}}{{% \cal M}}^{\scriptscriptstyle V}_{\sf J},$$ (19) in (13), and ${\sf O}_{{m}{\sf J}}$ now reads $$\displaystyle{\sf O}_{{m}{\sf J}}$$ $$\displaystyle=$$ $$\displaystyle i(\delta_{{m}0}\overline{g}_{\mbox{\tiny P2}}-g_{\mbox{\tiny A}}% +m\overline{g}_{\mbox{\tiny W}}){{\cal M}}^{\scriptscriptstyle A}_{{m}{\sf J}}$$ (20) $$\displaystyle+$$ $$\displaystyle|m|g_{\scriptscriptstyle{V}}{{\cal M}}^{\scriptscriptstyle V}_{{m% }{\sf J}}+\delta_{{m}0}\frac{\tilde{k}_{\emptyset}}{{\kappa}}g_{\mbox{\tiny V}% }{{\cal M}}_{\sf J}^{\scriptscriptstyle V}.$$ The second term in (17) comes from the violation of the CVC by the electromagnetic interaction. Although it is frequently employed in the nuclear ${\beta}$-decay, as far as we know, it has never been considered before in the neutrino-nucleus scattering. $\Delta E_{\rm Coul}$ is equal to $3.8,9.8$, and $20.0$ MeV for ${}^{12}$C, ${}^{56}$Fe, and ${}^{208}$Pb, respectively, and therefore the just mentioned term could be quite significant, specially for heavy nuclei. The transition amplitude for the neutrino-nucleus reaction at a fixed value of ${\kappa}$, from the initial state $|0^{+}\rangle$ in the $(Z,N)$ nucleus to the n-th final state $|{\sf J}^{\pi}_{n}\rangle$ in the nucleus $(Z\pm 1,N\mp 1)$, reads $$\displaystyle{{\cal T}}_{{\sf J}^{\pi}_{n}}({\kappa})\equiv\sum_{s_{\ell},s_{% \nu}}\left|\langle{\sf J}_{n}^{\pi}|H_{{\scriptscriptstyle{W}}}({\kappa})|0^{+% }\rangle\right|^{2}.$$ (21) The momentum transfer here is $k=p_{\ell}-q_{\nu}$, with $p_{\ell}\equiv\{{\bf p}_{\ell},iE_{\ell}\}$ and $q_{\nu}\equiv\{{\bf q}_{\nu},iE_{\nu}\}$, and after some algebra Krm05 one gets $$\displaystyle{{\cal T}}_{{\sf J}^{\pi}_{n}}({\kappa})$$ $$\displaystyle=$$ $$\displaystyle{4\pi G^{2}}[\sum_{{\alpha}=\emptyset,0,\pm 1}|\langle{\sf J}^{% \pi}_{n}||{\sf O}_{{\alpha}{{\sf J}}}({\kappa})||0^{+}\rangle|^{2}{{\cal L}}_{% {\alpha}}$$ $$\displaystyle-$$ $$\displaystyle 2\Re\left(\langle{\sf J}^{\pi}_{n}||{\sf O}_{\emptyset{{\sf J}}}% ({\kappa})||0^{+}\rangle\langle{\sf J}^{\pi}_{n}||{\sf O}_{0{{\sf J}}}({\kappa% })||0^{+}\rangle^{*}\right){{\cal L}}_{\emptyset 0}],$$ where $$\displaystyle{{\cal L}}_{\emptyset}$$ $$\displaystyle=$$ $$\displaystyle 1+\frac{|{\bf p}_{\ell}|\cos\theta}{E_{\ell}},$$ $$\displaystyle{{\cal L}}_{0}$$ $$\displaystyle=$$ $$\displaystyle 1+\frac{2q_{0}p_{0}}{E_{\ell}E_{\nu}}-\frac{|{\bf p}_{\ell}|\cos% \theta}{E_{\ell}},$$ $$\displaystyle{{\cal L}}_{\pm 1}$$ $$\displaystyle=$$ $$\displaystyle 1-\frac{q_{0}p_{0}}{E_{\ell}E_{\nu}}\pm\left(\frac{q_{0}}{E_{\nu% }}-\frac{p_{0}}{E_{\ell}}\right)S,$$ $$\displaystyle{{\cal L}}_{\emptyset 0}$$ $$\displaystyle=$$ $$\displaystyle\frac{q_{0}}{E_{\nu}}+\frac{p_{0}}{E_{\ell}},$$ (23) are the lepton traces, with $\theta\equiv\hat{{\bf q}}_{\nu}\cdot\hat{{\bf p}}_{\ell}$ being the angle between the incident neutrino and ejected lepton momenta, and $$\displaystyle q_{0}$$ $$\displaystyle=$$ $$\displaystyle{\hat{k}}\cdot{\bf q}_{\nu}=\frac{E_{\nu}(|{\bf p}_{\ell}|\cos% \theta-E_{\nu})}{{\kappa}},$$ $$\displaystyle p_{0}$$ $$\displaystyle=$$ $$\displaystyle{\hat{k}}\cdot{\bf p}_{\ell}=\frac{|{\bf p}_{\ell}|(|{\bf p}_{% \ell}|-E_{\nu}\cos\theta)}{{\kappa}},$$ (24) the $z$-components of the neutrino and lepton momenta. The exclusive cross-section (ECS) for the state $|{\sf J}^{\pi}_{n}\rangle$, as a function of the incident neutrino energy $E_{\nu}$, is $$\displaystyle{\sigma}_{\ell}({\sf J}^{\pi}_{n},E_{\nu})$$ $$\displaystyle=$$ $$\displaystyle\frac{|{\bf p}_{\ell}|E_{\ell}}{2\pi}F(Z+S,E_{\ell})\int_{-1}^{1}% d(\cos\theta){{\cal T}}_{{\sf J}^{\pi}_{n}}(\kappa),$$ where $$\displaystyle E_{\ell}$$ $$\displaystyle=$$ $$\displaystyle E_{\nu}-{\omega}_{{\sf J}^{\pi}_{n}},~{}|{\bf p}_{\ell}|=\sqrt{(% E_{\nu}-{\omega}_{{\sf J}^{\pi}_{n}})^{2}-m_{\ell}^{2}},$$ $$\displaystyle\kappa$$ $$\displaystyle=$$ $$\displaystyle|{\bf p}_{\ell}-{\bf q}_{\nu}|$$ (26) $$\displaystyle=$$ $$\displaystyle\sqrt{2E_{\nu}(E_{\ell}-|{\bf p}_{\ell}|\cos\theta)-m_{\ell}^{2}+% {\omega}_{{\sf J}^{\pi}_{n}}^{2}},$$ and ${\omega}_{{\sf J}^{\pi}_{n}}=-k_{\emptyset}=E_{\nu}-E_{\ell}$ is the excitation energy of the state $|{\sf J}^{\pi}_{n}\rangle$ relative to the state $|0^{+}\rangle$. Moreover, $F(Z+S,E_{\ell})$ is the Fermi function for neutrino $(S=1)$, and antineutrino $(S=-1)$, respectively. As it well known the charged-current cross sections must be corrected for the distortion of the outgoing lepton wave function by the Coulomb field of the daughter nucleus. For relatively low neutrino energies ($\sim 40-50$ MeV) this correction was implemented by numerical solution of the Dirac equation for an extended nuclear charge Beh82 . At higher energies, the effect of the Coulomb field was described by the effective momentum approximation (EMA) Eng98 , in which the lepton momentum $p_{\ell}$ and energy $E_{\ell}$ are modified by the corresponding effective quantities (see (Paa07, , Eq. (34) and (45))). Here, we will also deal with inclusive cross-section (ICS), $$\displaystyle{\sigma}_{\ell}(E_{\nu})=\sum_{{\sf J}^{\pi}_{n}}{\sigma}_{\ell}(% {\sf J}^{\pi}_{n},E_{\nu}),$$ (27) as well as with folded cross-sections, both exclusive, $$\displaystyle\overline{{\sigma}}_{\ell}({{\sf J}^{\pi}_{n}})=\int dE_{\nu}{% \sigma}_{\ell}({\sf J}^{\pi}_{n},E_{\nu})n_{\ell}(E_{\nu}),$$ (28) and inclusive $$\overline{{\sigma}}_{\ell}=\int dE_{\nu}{\sigma}_{\ell}(E_{\nu})n_{\ell}(E_{% \nu}),$$ (29) where $n_{\ell}(E_{\nu})$ is the neutrino (antineutrino) normalized flux. In the evaluation of both neutrino, and antineutrino ICS the summation in (27) goes over all $n$ states with spin and parity ${\sf J}^{\pi}\leq 7^{\pm}$ in the PQRPA, and over ${\sf J}^{\pi}\leq 14^{\pm}$ in the RQRPA. In the Appendix A we show that the real and imaginary parts of the operators ${\sf O}_{{{\alpha}}{\sf J}}$, given by (12) and (20), contribute to natural and unnatural parity states, respectively. This greatly simplifies the numerical calculations, because one always deals with real operators only. In Appendix D are also shown the formula for the muon capture process within the present formalism. III Numerical results and discussion The major part of the numerical calculations have been done within the PQRPA by employing the $\delta$-interaction (in MeV fm${}^{3}$) $$V=-4\pi\left(v_{s}P_{s}+v_{t}P_{t}\right)\delta(r),$$ with singlet ($v_{s}$), and triplet ($v_{t}$) coupling constants different for the particle-hole ($ph$), particle-particle ($pp$), and pairing ($pair$) channels Sam08 . This interaction leads to a good description of single and double $\beta$-decays and it has been used extensively in the literature Hir90a ; Krm92 ; Krm93 ; Krm94 . The single-particle wave functions were approximated with those of the HO with the length parameter $b=1.67$ fm, which corresponds to the oscillator energy $\hbar\omega=45A^{-1/3}-25A^{-2/3}$  MeV. The s.p. spaces $S_{2},S_{3},S_{4}$, and $S_{6}$ will be explored. In Refs. Krm02 ; Krm05 , where the $S_{3}$ space was used, we have pointed out that the values of the coupling strengths $v^{pp}_{s}$, $v^{pair}_{s}$, and $v^{p}_{t}$ used in $N>Z$ nuclei ($v^{pp}_{s}=v^{pair}_{s}$, $v^{pp}_{t}\gtrsim v^{pp}_{s}$), might not be suitable for $N=Z$ nuclei. In fact, the best agreement with data in ${}^{12}$C is obtained for: i) the energy of the ground state in ${}^{12}$N, $E(^{12}$N), ii) the GT $B$-values in ${}^{12}$C, $B(^{12}$N) and $B(^{12}$B), and iii) the exclusive muon capture in ${}^{12}$B, $\Lambda^{\rm exc}\equiv\Lambda^{(}1^{+}_{1})$, is obtained when the $pp$ channel is totally switched off, i.e., $v^{pp}_{s}\equiv v^{pp}_{t}=0$. The adopted $ph$ coupling strengths are $v^{ph}_{s}=27$ MeV  fm${}^{3}$ and $v^{ph}_{t}=64$ MeV  fm${}^{3}$ Krm02 . For the $pp$ channel it is convenient to define the parameters $$s=\frac{v_{s}^{pp}}{v_{s}^{pair}},\hskip 17.071654ptt=\frac{v_{t}^{pp}}{v_{s}^% {pair}},$$ where $v_{s}^{pair}=(v_{s}^{pair}(p)+v_{s}^{pair}(n))/2$ Krm94 . As in our previous work on ${}^{12}$C, we will use here the same singlet and triplet $pp$ couplings, i.e., $s\equiv t$ Krm02 ; Krm05 . The states with ${\sf J}^{\pi}=0^{+}$, and ${\sf J}^{\pi}=1^{+}$ only depend on $s$, and $t$, respectively, while all remaining depend on both coupling strengths. The s.p. energies and pairing strengths for $S_{2}$, $S_{3}$, and $S_{4}$ spaces, were varied in a $\chi^{2}$ search to account for the experimental spectra of odd-mass nuclei ${}^{11}$C, ${}^{11}$B, ${}^{13}$C, and ${}^{13}$N, as explained in Ref. Krm05 . This method, however, is not practical for the space $S_{6}$ which comprises $21$ s.p. levels. Therefore in this case the energies were derived in the way done in Ref. Paa07 , while the pairing strengths were adjusted to reproduce the experimental gaps in ${}^{12}$C Sam09 , considering all the quasiparticle energies up to $100$ MeV. For the purpose of the present study, we also employ the RQRPA theoretical framework  PNVR.04 . In this case the ground state is calculated in the Relativistic Hartree-Bogoliubov model (RHB) using effective Lagrangians with density dependent meson-nucleon couplings and DD-ME2 parameterization LNVR.05 , and pairing correlations are described by the finite range Gogny force BGG.91 . Details of the formalism can be found in Refs. Paa.03 ; PVKC.07 . The RHB equations, and respective equations for mesons are usually solved by expanding the Dirac spinors and the meson fields in a spherical harmonic oscillator basis with $S_{20}$ s.p. space. In the present study of neutrino-nucleus cross sections, with energies of incoming neutrinos up to $600$ MeV, we extend the number of oscillator shells up to $N=30$ in order to accommodate s.p. states at higher energies necessary for description of cross sections involving higher energies of incoming (anti)neutrinos. The number of $2qp$ configurations in the RQRPA is constrained by the maximal excitation energy $E_{2qp}$. Within the RHB+RQRPA framework the oscillator basis is used only in RHB to determine the ground state and single-particle spectra. The resulting wave functions are converted to coordinate space for evaluation of the RQRPA matrix elements. However, it is the original HO basis employed in RHB that determines the maximal $E_{2qp}$ and the size of RQRPA configuration space. III.1 Weak interaction properties of ${}^{12}$N and ${}^{12}$B ground states Let us first compare the QRPA and PQRPA within the smallest configuration space $S_{2}$, which contains $16$ ${\sf J}^{\pi}=1^{+}$ states, and with null $pp$ coupling: $t=0.$ The PQRPA ground state energies in ${}^{12}$N, and ${}^{12}$B, are, respectively: $\omega_{+1}(1^{+})=18.319$ MeV, and $\omega_{-1}(1^{+})=12.528$ MeV, while the corresponding wave functions read $$\displaystyle|^{12}{\rm N}\rangle$$ $$\displaystyle=$$ $$\displaystyle 0.963|1p^{\pi}_{3/2}1p^{\nu}_{1/2}\rangle+0.232|1p^{\pi}_{3/2}1p% ^{\nu}_{3/2}\rangle$$ (30) $$\displaystyle+$$ $$\displaystyle 0.122|1p^{\pi}_{1/2}1p^{\nu}_{3/2}\rangle+0.105|1p^{\pi}_{1/2}1p% ^{\nu}_{1/2}\rangle$$ $$\displaystyle+$$ $$\displaystyle\cdots,$$ and $$\displaystyle|^{12}{\rm B}\rangle$$ $$\displaystyle=$$ $$\displaystyle-0.971|1p^{\pi}_{1/2}1p^{\nu}_{3/2}\rangle+0.204|1p^{\pi}_{3/2}1p% ^{\nu}_{3/2}\rangle$$ (31) $$\displaystyle-$$ $$\displaystyle 0.125|1p^{\pi}_{3/2}1p^{\nu}_{1/2}\rangle+0.090|1p^{\pi}_{1/2}1p% ^{\nu}_{1/2}\rangle$$ $$\displaystyle+$$ $$\displaystyle\cdots$$ The analogous QRPA energies are quite similar: $\omega_{+1}(1^{+})=17.992$ MeV, $\omega_{-1}(1^{+})=12.437$ MeV. However, the wave functions are quite different. The main difference is in the fact that QRPA furnishes the same wave functions for all four nuclei ${}^{12}$N, ${}^{10}$B, ${}^{14}$N, and ${}^{12}$B, being that of the ground state: $$\displaystyle|1^{+}_{GS}\rangle$$ $$\displaystyle=$$ $$\displaystyle-0.272|1p^{\pi}_{3/2}1p^{\nu}_{1/2}\rangle-0.759|1p^{\pi}_{3/2}1p% ^{\nu}_{3/2}\rangle$$ (32) $$\displaystyle+$$ $$\displaystyle 0.356|1p^{\pi}_{1/2}1p^{\nu}_{3/2}\rangle-0.472|1p^{\pi}_{1/2}1p% ^{\nu}_{1/2}\rangle$$ $$\displaystyle+$$ $$\displaystyle\cdots.$$ The difference in the wave functions is an important issue that clearly signalizes towards the need for the number projection. In fact, the PQRPA yields the correct limits ($1p^{\pi}_{3/2}\rightarrow 1p^{\nu}_{1/2}$ and $1p^{\nu}_{3/2}\rightarrow 1p^{\pi}_{1/2}$) for one-particle-one-hole (1p1h) excitations on the ${}^{12}$C ground state to reach the ${}^{12}$N, and ${}^{12}$B nuclei. All remaining configurations in (30), and (31) come from the higher order 2p2h, and 3p3h excitations. Contrary, the QRPA state (32) is dominantly the two-hole excitation $[(1p^{\pi}_{3/2})^{-1},(1p^{\nu}_{3/2})^{-1}]$, which corresponds to the ground state of ${}^{10}$B. More details on this question can be found in Figure 3 of Ref. Krm05 . The 1p1h amplitudes$[(1p^{\pi}_{3/2})^{-1},1p^{\nu}_{1/2}]$, and $[(1p^{\nu}_{3/2})^{-1},(1p^{\pi}_{1/2})]$ are dominantly present in the following QRPA states $$\displaystyle|1^{+}_{2}\rangle$$ $$\displaystyle=$$ $$\displaystyle 0.708|1p^{\pi}_{1/2}1p^{\nu}_{3/2}\rangle+0.703|1p^{\pi}_{3/2}1p% ^{\nu}_{1/2}\rangle$$ $$\displaystyle+$$ $$\displaystyle\cdots$$ $$\displaystyle|1^{+}_{4}\rangle$$ $$\displaystyle=$$ $$\displaystyle-0.476|1p^{\pi}_{3/2}1p^{\nu}_{1/2}\rangle+0.437|1p^{\pi}_{3/2}1p% ^{\nu}_{3/2}\rangle$$ (33) $$\displaystyle+$$ $$\displaystyle 0.441|1p^{\pi}_{1/2}1p^{\nu}_{3/2}\rangle-0.096|1p^{\pi}_{1/2}1p% ^{\nu}_{1/2}\rangle$$ $$\displaystyle+$$ $$\displaystyle\cdots$$ The wave functions displayed above clearly evidence the superiority of the PQRPA on the QRPA. Therefore from now on only the PQRPA results will be discussed for the exclusive observables. In Figure  1 we show the ${}^{12}$B and ${}^{12}$N ground state energies, and the corresponding GT $B$-values within the PQRPA for different s.p. spaces, as function of the $pp$-coupling $t$. One sees that the energies depend rather weakly on both, and agree fairly well with the measured energies: $E(^{12}$B$)=13.37$ MeV, and $E(^{12}$N$)=17.33$ MeV  Ajz85 , although the first one is somewhat underestimated, while the second one is somewhat overestimated. Both GT $B$-values significantly increase with $t$ and diminish when size of the s.p. space is increased. For spaces $S_{2}$ and $S_{3}$ the best overall agreement with data ($B(^{12}$B$)=0.466$, and $B(^{12}$N$)=0.526$ Al78 ) is achieved with $t=0$, while for spaces $S_{4}$ and $S_{6}$ this happens when the couplings are, respectively, $t=0.2$, and $t=0.3$. After establishing the PQRPA parametrization, we analyze the behavior of the ECSs of the ground states in ${}^{12}$N and ${}^{12}$B, as a function of the size of the configuration space. Figure 2 shows the ECSs for the reaction ${}^{12}$C($\nu,e^{-})^{12}$N (in units of $10^{-42}$ cm${}^{2}$) for several configuration spaces, and for $t=0$, within three different energy intervals. The top panel represents the DAR region, where experimental data are available Ath97 , and search for neutrino oscillations was done Ath97 ; Zei98 . The middle panel represents the region of interest for supernovae neutrinos, as pointed out in Refs. Aga07 ; Str09 , while the bottom panel shows the asymptotic behavior of the cross-section, which becomes almost constant for $E_{\nu}\simeq 200$ MeV. Within the spaces $S_{2}$ and $S_{3}$ the calculations reproduce quite well the experimental cross sections in the DAR region, as seen from the first panel. In Figure 3 we show the calculated ECSs for the reaction ${}^{12}$C($\nu,e^{-})^{12}$N within several configuration spaces, but now with different values of the $pp$-coupling. From comparison with the experimental data in the DAR region Ath97 one observes that the appropriate values for the coupling $t$ for s.p. spaces $S_{4}$, and $S_{6}$, are, respectively, $t=0.2$, and $t=0.3$, i.e., the same as those required to reproduce the experimental energies and the GR $B$-values in ${}^{12}$B, and ${}^{12}$N. This change of parametrization hint at the self-consistency of the PQRPA, and comes from the fact that in this model: i) the GT strength allocated in the ground state is moved to another $1^{+}$ states when the size of the space is increased, and ii) the effect of the $pp$ residual interaction goes in the oppositive direction, returning the GT strength to the $1^{+}_{1}$ state. Only for the space $S_{2}$ the cross-section $\sigma_{e^{-}}(E_{\nu},1^{+}_{1})$ is appreciable larger (at $E_{\nu}\gtrsim 60$ MeV) than for other spaces, which is just because of the small number of configurations in this case. In the same figure are exhibited as well the results for the ECSs evaluated within the SM Eng96 , and the EPT Fuk88 . Both of them agree well with the data and with the present calculation. The results for the reaction $({\tilde{\nu}},e^{+})$ to the ground state in ${}^{12}$B are shown in Figure 4. The cross-section $\sigma_{e^{+}}(E_{\tilde{\nu}},1^{+}_{1})$ is similar to that produced by neutrinos but significantly smaller in magnitude. When compared with the EPT result  Fuk88 , which are also shown in the same figure, one notices that they are considerable different. To some extent this is surprising as in the case of neutrinos the two models yield very similar results. One should remember that in the EPT model the axial form factor, used for both neutrinos and antineutrinos, is gauged to the average of the GR $B$-values in ${}^{12}$B, and ${}^{12}$N, which, in turn, are well reproduced by the PQRPA. Therefore it is difficult to understand why the EPT results agree with the present calculations for neutrinos and disagree for antineutrinos. In Figure 5 we show the dependence on the configuration space of the exclusive muon capture transition rate $\Lambda(1^{+}_{1})$ to the ${}^{12}$B ground state, and the electron and muon flux-averaged ECSs, given by (28), to the ${}^{12}$N ground state, i.e., $\overline{{\sigma}}_{\epsilon}(1^{+}_{1})$, and $\overline{{\sigma}}_{\mu}(1^{+}_{1})$. As in Refs. Krm02 ; Krm05 the electron neutrino distribution $n_{e}(E_{\nu})$ was approximated with the Michel energy spectrum Kol99a ; Arm02 , and for the muon neutrinos we used $n_{\mu}(E_{\nu})$ from Ref. LSND . The energy integration is carried out in the DAR interval $m_{e}+{\omega}_{J_{f}}\leq\Delta_{J_{f}}^{\rm DAR}\leq 52.8$ MeV for electrons and in the DIF interval $m_{\mu}+{\omega}_{J_{f}}\leq\Delta_{J_{f}}^{\rm DIF}\leq 300$ MeV for muons. From Figure 5, and comparison with experimental data: $\Lambda(^{12}$B$)=6.2\pm 0.3$ Mil72 , $\overline{\sigma}_{e}(^{12}$N$)=9.1\pm 0.4\pm 0.9$ Ath97 , $8.9\pm 0.3\pm 0.9$ Aue01 , $\overline{\sigma}_{\mu}(^{12}$N$)=6.6\pm 1.0\pm 1.0$ Ath97a , $5.6\pm 0.8\pm 1.0$ Aue02a , one finds out, as for results shown in Figures 1, and 3, the model self-consistency between s.p. spaces and the $pp$-couplings. That is, for larger s.p. spaces larger values of $t$ are required. In brief, the experimental data of $\overline{\sigma}_{e}(^{12}$N), and $\overline{\sigma}_{\mu}(^{12}$N) are well reproduced by the PQRPA. The same is true for the SM calculations Hay00 ; Vol00 , while in RPA, and QRPA models they are strongly overestimated, as can be seen from Table II in Ref. Vol00 , and Table 1 in Ref. Sam06 . III.2 Inclusive cross-sections ${}^{12}$C($\nu,e^{-})^{12}$N and ${}^{12}$C($\tilde{\nu},e^{+})^{12}$B, and Sum Rule In Figure 6 we confront the PQRPA results for the ICS $\sigma_{e^{-}}(E_{\nu})$ within spaces $S_{2}$, $S_{3}$, and $S_{6}$ with the corresponding sum-rules ${\sigma}_{e^{-}}^{SR}(E_{\nu})$ evaluated from (47). One immediately sees that the PQRPA results depend very strongly on the size of the employed s.p. space. On the other hand, as already mentioned in the Appendix B, the sum rule ${\sigma}_{e^{-}}^{SR}(E_{\nu})$ depends on the average energy $\overline{{\omega}_{{\sf J}^{\pi}_{n}}}$. Here we use two values $\overline{{\omega}_{{\sf J}^{\pi}_{n}}}=17.34$ MeV, which is the ground state energy ${}^{12}$N, (GT-resonance), and $\overline{{\omega}_{{\sf J}^{\pi}_{n}}}=42$ MeV, which is roughly the energy of the first forbidden resonance Kr83 . The corresponding curves in Figure 6 are labeled, respectively, as $SR_{GT}$, and $SR_{1F}$. They should be the upper limits for allowed and first forbidden transitions, respectively. The validity of these sum rules is questionable for neutrinos energies of several hundred MeV, as already pointed out by Kuramoto et al. Kur90 . In fact, we note that the cross section $SR_{GT}$ ($SR_{1F}$) exceeds the free particle cross section ${\sigma}_{6}\equiv 6{\sigma}(\nu_{e}+n\rightarrow e^{-}+p)$ for $E_{\nu}>200$ MeV ($E_{\nu}>300$ MeV) Bud03 . Several previous SM and RPA-like calculations of ${\sigma}_{e^{-}}(E_{\nu})$, employing different effective axial-vector coupling constants, and different s.p. spaces, are exhibited in Figure 6 as well, namely: a) TDA Con72 , with $g_{\scriptscriptstyle A}=1.23$, and $S_{2}$, b) SM and RPA Vol00 , with $g_{\scriptscriptstyle A}=0.88$, and $S_{3}$, c) CRPA Kol99b , with $g_{\scriptscriptstyle A}=1.26$, and $S_{4}$, d) RQRPA Paa07 , with $g_{\scriptscriptstyle A}=1.23$, $S_{20}$, and $E_{2qp}$=100 MeV. It is important to specify the values $g_{\scriptscriptstyle A}$ because the partial cross sections are predominantly of the axial-vector type (specially the allowed ones), which are proportional to $g_{\scriptscriptstyle A}^{2}$. In spite of very significant differences in $g_{\scriptscriptstyle A}$, and the s.p. spaces, the different calculations of $\sigma_{e^{-}}(E_{\nu})$ yield quite similar results for energies $E_{\nu}\lesssim 130$ MeV, lying basically in vicinity of the the sum-rule result $SR_{1F}$. But for higher energies they could become quite different, and are clearly separated in two groups at $E_{\nu}=300$ MeV. In the first group with ${\sigma}_{e^{-}}(E_{\nu})\lesssim 5\times 10^{-39}$ cm${}^{2}$ are: the SM, TDA, and PQRPA within spaces $S_{2}$, $S_{3}$, while in the second one with ${\sigma}_{e^{-}}(E_{\nu})\gtrsim 10\times 10^{-39}$ cm${}^{2}$ are: the RPA, RQRPA, CRPA and PQRPA within space $S_{6}$. Volpe et al. Vol00 have found that the difference between their SM and RPA calculations is due to differences in the correlations taken into account, and to a too small SM space. We also note that only the CRPA result approaches the sum rule limits for $E_{\nu}>200$ MeV. Similar results for the inclusive ${}^{12}$C($\tilde{\nu},e^{+})^{12}$B cross-section $\sigma_{e^{+}}(E_{\tilde{\nu}})$ are displayed in Figure 7, and analogous comments can be done here. For the comparison, we show in the figure the antineutrino-${}^{12}$C cross-sections evaluated with the CRPA Kol99b . III.3 Large configuration spaces As there are no experimental data on flux unfolded ICSs for $E_{\nu}\leq 400$ MeV we cannot conclude which of the results displayed in Figures 6, and 7 are good and which are not. We can only conclude that the ICSs strongly depend on the size of the s.p. space. In the PQRPA calculations we are not able to use spaces lager than $S_{6}$ because of numerical difficulties. Thus instead of using the PQRPA, from now on we employ the RQRPA where such calculations are feasible. It is important to note that within the RHB+RQRPA model the oscillator basis is used only in the RHB calculation in order to determine the ground state and the single-particle spectra. The wave functions employed in RPA equations are obtained by converting the original HO basis to the coordinate representation. Therefore, the size of the RQRPA configuration space and $2qp$  energy cut-offs are determined by the number of oscillator shells in the RHB model. First, we analyze the effect of the cut-off energy within the $S_{20}$ space on $\sigma_{e^{-}}(E_{\nu})$ for $E_{\nu}$ up to $600$ MeV. From the left panel in Figure 8 one sees that at high energies this cross-section increases roughly by a factor of two when $E_{2qp}$ is augmented from $100$ to $200$ MeV. The increase of the cross-section is also quite important when $E_{2qp}$ is moved from $200$ to $300$ MeV. For the limiting value of $E_{2qp}$=300 MeV, all possible configurations are included in RQRPA calculations. Next, we do the same within the $S_{30}$ space, and the resulting $\sigma_{e^{-}}(E_{\nu})$ are displayed on the right panel of Figure 8. From the comparison of both panels it is easy to figure out that up to $E_{2qp}=300$ MeV the cross sections obtained with the $S_{30}$ space are basically the same to those calculated with the $S_{20}$ space. Small differences between the cross sections using $S_{20}$ and $S_{30}$ spaces for $E_{2qp}$ up to 300 MeV are caused by modifications of positive-energy single-particle states contributing to the QRPA configuration space within the restricted $2qp$ energy window. But, for $E_{\nu}\gtrsim 400$ MeV additional transition strength appears in the $S_{30}$ space when $E_{2qp}$ is moved up to $400$ MeV, from where further increase in $E_{2qp}$ has a very small effect. We conclude therefore that the configuration space engendered by $N=20$ HO shells with $E_{2qp}=300$ MeV, is large enough to describe $\sigma_{e^{-}}(E_{\nu})$ with $E_{\nu}$ up to $400$ MeV. Similarly, the space brought about by $N=30$ HO shells with $E_{2qp}=400$ MeV is appropriate to account for $\sigma_{e^{-}}(E_{\nu})$ up to $E_{\nu}=600$ MeV. For larger neutrino energies very likely we would have to continue increasing the number of shells. Analogous results for antineutrino ICSs $\sigma_{e^{+}}(E_{\tilde{\nu}})$ are displayed in Figure 9. One notes important differences in comparison with $\sigma_{e^{-}}(E_{\nu})$ shown in Figure 8. First, here the spaces $S_{20}$ and $S_{30}$ yield almost identical results in the entire interval of antineutrino energies up to $E_{\tilde{\nu}}=600$ MeV. Second, the successive increase in the cross-sections when the cut-off $E_{2qp}$ is augmented in steps of $100$ MeV are smaller, and decrease more rapidly than in the neutrino case. This suggests that the configuration space is now sufficiently large to appropriately account for $\sigma_{e^{+}}(E_{\tilde{\nu}})$ even at antineutrino energies larger than $600$ MeV. At present, due to numerical difficulties, we cannot perform the RQRPA calculations for the full range of neutrino energies where the QE cross section was measured at MiniBooNE MiniBooNE , but only up to 0.6 GeV. However, we feel that it could still be illustrative for comparison with data. This is done in Figure 10, which is basically a piece of Figure 21 Ref. Mart09 for the QE $\sigma_{\mu^{-}}(E_{\nu})$ (see also Ref. Mart10 ), where is incorporated our result for $\sigma_{e^{-}}(E_{\nu})$ from Figure 8 for $S_{30}$ and $E_{2qp}=500$ MeV. As already pointed out in the Introduction, at relatively high energies ($E_{\nu}>300$ MeV) the electron and muon neutrino cross sections converge to each other, and therefore, in the present analysis, the electron neutrino cross section provides a reasonable upper limit estimate. One sees that we underestimate the data by almost a factor of two. But one should keep in mind that, while we use $g_{\scriptscriptstyle A}=1$ (see (4)) in the RFGM calculation done by Martini et al. Mart09 ; Mart10 $g_{\scriptscriptstyle A}=1.255$ was used. Being the axial-vector contribution dominant for the latter value of the coupling constant, one would have to re-normalize our $\sigma_{e^{-}}(E_{\nu})$ by a factor $\sim 1.5$. Such a result is also shown in Figure  10, and although the resulting cross section still underestimates somewhat the data for $\sigma_{\mu^{-}}(E_{\nu})$, it is notably superior to the pure 1p-1h result from Ref. Mart09 , where good agreement with the data is achieved only after considering additional two-body (2p-2h) and three-body (3p3h) decay channels. One should keep in mind, however, that as the weak decay Hamiltonian is one-body operator, these excitations are only feasible via the ground-state correlations (GSC), which basically redistribute the 1p-1h transition strength without increasing its total magnitude when the initial wave function is properly normalized. In the present work, as well as in all SM-like calculations, the GSC, and a normalized initial state wave function are certainly considered to all orders in perturbation theory through the full diagonalization of the hamiltonian matrix. On the other hand, in Refs. Mart09 ; Mart10 the GSC are taken into account in second order perturbation theory, but there are no references to the normalization of the ${}^{12}$C ground-state wave function. How to carry out the normalization in the framework of the infinite nuclear matter model is discussed in a recent paper related to the nonmesonic weak decay of the hypernucleus ${}^{12}_{\Lambda}$C Bau10 ; see also Refs. Ma91 ; Va92 ; Ma95 . III.4 Multipole decomposition of cross-sections We did not mention yet the contributions of different multipoles to the ICSs. Normally, the RHB model within $S_{20}$, and with ${\sf J}^{\pi}\leq 7^{\pm}$, provides converged results for RQRPA excitation spectra at incident neutrino energies $E_{\nu}\leq 300$ MeV as seen from Figure 2 in Ref. Paa07 . But this is not the case for neutrino-nucleus cross sections at energies $E_{\nu}\gtrsim 300$ MeV where one has to consider large cutoff energies $E_{2qp}$. In fact, it is necessary to consider more and more multipoles according as the configuration space is enlarged by increasing $E_{2qp}$. This is illustrated in Figure 11 for the case of $E_{2qp}=500$ MeV. One sees that are significant all multipoles up to ${\sf J}^{\pi}=14^{\pm}$ for neutrinos, and up to ${\sf J}^{\pi}=11^{\pm}$ for antineutrinos. Next we discuss the partial multipole contributions to the ICS, having in view the degree of forbiddenness of the transition matrix elements, namely, $\bullet$ Allowed: $\sigma^{A}_{e^{+}}(E_{\tilde{\nu}})$, with ${\sf J}^{\pi}=0^{+},1^{+}$, $\bullet$ First-forbidden $\sigma^{1F}_{e^{+}}(E_{\tilde{\nu}})$, with ${\sf J}^{\pi}=0^{-},1^{-},2^{-}$, $\bullet$ Second-forbidden $\sigma^{2F}_{e^{+}}(E_{\tilde{\nu}})$, with ${\sf J}^{\pi}=2^{+},3^{+}$, and $\bullet$ Third-forbidden $\sigma^{3F}_{e^{+}}(E_{\tilde{\nu}})$ with ${\sf J}^{\pi}=3^{-},4^{-}$, cross-sections. Thus, in the left panel of Figure 12 we show these individual contributions for the inclusive ${}^{12}$C($\tilde{\nu},e^{+})^{12}$B cross-section $\sigma_{e^{+}}(E_{\tilde{\nu}})$, evaluated within both the PQRPA (spaces $S_{2}$, and $S_{6}$) the RQRPA (space $S_{30}$ with $E_{2qp}=500$ MeV). The same is done for the corresponding derivatives, i.e., the spectral functions $d\sigma_{e^{+}}(E_{\tilde{\nu}})/dE_{\tilde{\nu}}$, on the right panel of the same figure. Several conclusions can be drawn. First, as in the case of total $\sigma_{e^{+}}(E_{\tilde{\nu}})$, they depend very strongly on the size of the configuration space. This dependence, in turn, increases with the degree of forbiddenness; that is, it is more pronounced for first-forbidden than for allowed transitions, and so on. Second, within the PQRPA the allowed cross-section $\sigma^{A}_{e^{+}}(E_{\tilde{\nu}})$ exhibits a resonant pattern at low energy, and is dominant for $E_{\tilde{\nu}}\lesssim 50$ MeV. For large s.p. spaces its contribution is quite significant even at $E_{\tilde{\nu}}=500$ MeV. 555The denominations here don’t have exactly the same meaning as in the low-energy $\beta$-decay, where allowed transitions are those within the same HO shell ($\Delta N=0$), while here all values of $\Delta N$ are permitted. Similarly happens with the forbidden transitions. The degrees of hindrance basically come from value of the orbital angular momenta. In the case of RQRPA, the spectral function $d\sigma^{A}_{e^{+}}(E_{\tilde{\nu}})/dE_{\tilde{\nu}}$ also displays low-energy resonant structure, and $\sigma^{A}_{e^{+}}(E_{\tilde{\nu}})$ is always smaller in magnitude than in the PQRPA case. Third, $\sigma^{1F}_{e^{+}}(E_{\tilde{\nu}})$ is peaked at $E_{\tilde{\nu}}\sim 75$ MeV, and its contribution is always larger than that of $\sigma^{A}_{e^{+}}(E_{\tilde{\nu}})$ for $E_{\tilde{\nu}}\gtrsim 150$ MeV. Fourth, $\sigma^{2F}_{e^{+}}(E_{\tilde{\nu}})$, and $\sigma^{3F}_{e^{+}}(E_{\tilde{\nu}})$ mainly contribute in the interval $150\lesssim E_{\tilde{\nu}}\lesssim 400$ MeV, and their overall contributions are of the same order of magnitude, and comparable to that of the $\sigma^{1F}_{e^{+}}(E_{\tilde{\nu}})$. Fifth, the contributions of the remaining multipoles with ${\sf J}^{\pi}=4^{+},5^{\pm},6^{\pm},7^{\pm}$ are always very small for the space $S_{2}$, but are quite sizeable for $S_{6}$ at high energies. For instance, at $E_{\tilde{\nu}}=100,300,600$ MeV they contribute, respectively with $0.02\%,0.86\%,1.18\%$ within the space $S_{2}$, and $0.04\%,14\%,20\%$ for $S_{6}$. With further increase of the single-particle basis, configurations from higher multipoles become more pronounced at higher neutrino energies. In particular, the sum of contributions coming from ${\sf J}^{\pi}=4^{+},\cdots,11^{\pm}$ multipoles when evaluated within the RQRPA using the space $S_{30}$ and maximal value of $E_{2qp}$=500 MeV, are 1.1%, 14.4%, and 33.2% at $E_{\nu}$=100, 300, and 600 MeV, respectively. Recently Lazauskas and Volpe Laz07 ; Vol04 have suggested the convenience of performing nuclear structure studies using low energy neutrino and antineutrino beams. Because of feasibility reasons the flux covers $80$ MeV only. Nevertheless, from the analysis of ${}^{16}$O, ${}^{56}$Fe, ${}^{100}$Mo and ${}^{208}$Pb nuclei within the QRPA using the Skyrme force they were able to disentangle the multipole distributions of forbidden cross-sections, showing that the forbidden multipole contribution is different for various nuclei. In this work we extend this kind of study to ${}^{12}$C. In Table 1 we show the results for the flux-averaged cross sections $\overline{{\sigma}}_{e^{+}}$ for the reaction ${}^{12}{\rm C}({\tilde{\nu}},e^{+})^{12}{\rm B}$. In (29) we have used the same antineutrino fluxes $n_{e^{+}}(E_{\tilde{\nu}})$ as in Ref. Laz07 , i.e., the DAR flux, and those produced by boosted ${}^{6}$He ions with different values of time dilation factor $\gamma=1/\sqrt{1-v^{2}/c^{2}}$. Results of two calculations are presented: i) PQRPA within $S_{6}$, and ii) RQRPA within $N=20$, and cutoff $E_{2qp}=300$ MeV. One sees that in both models, and principally in the PQRPA, the allowed transitions dominate the forbidden one, and specially for the low-energy beam with $\gamma=6$. The contributions of the second-forbidden processes are very small in all the cases, while those coming from third-forbidden ones are always negligible. All this is totally consistent with the results shown in Figure 12, from where it is clear that to study second and third forbidden reactions in ${}^{12}$C, one would need fluxes $n_{e^{+}}(E_{\tilde{\nu}})$ with $E_{\tilde{\nu}}$ at least up to $\gtrsim 150$ MeV. It should also be stressed that our results both for allowed and forbidden transitions fully agree with those obtained in Ref. Laz07 ; the difference in ${}^{16}$O comes from the double-shell closure in this nucleus. III.5 Supernova neutrinos We also address briefly the $\nu/\tilde{\nu}$-${}^{12}$C nucleus cross-sections related with astrophysical applications, the knowledge of which could have important implications. For this purpose, are evaluated the $\overline{{\sigma}}_{e^{\pm}}$ folded with supernovae $\nu/\tilde{\nu}$ spectra represented by a normalized Fermi-Dirac distribution with temperatures in the interval $T_{\nu_{e}}=2-12$ MeV, which includes the most commonly used values are $T_{\nu_{e}}=3.2$ MeV and $T_{\bar{\nu}_{e}}=5.0$ MeV. For mean energies $\langle E_{\nu}\rangle\approx 3.15\times T_{\nu}$, and zero chemical potential Woo90 ; Kei03 the neutrino flux is $$n_{e}(E_{\nu})=\frac{0.5546}{T_{\nu}^{3}}\frac{E_{\nu}^{2}}{e^{E_{\nu}/T_{\nu}% }+1},$$ (34) and similarly for antineutrinos. For the sake of simplicity we do not analyze same relevant aspects of $n_{e}(E_{\nu})$ in supernova simulation, such as the MSW effect (see, for example, Ref. Akh00 ), and the spectral swapping of the neutrino flux (Ref. Dun07 ). In Figure 13 we confront the $\nu-^{12}$C cross sections averaged over supernova $\nu$-fluxes for the range of $T_{\nu}=2-12$ MeV, obtained within following calculations: i) PQRPA within $S_{6}$, ii) RQRPA within $S_{30}$ and $E_{2qp}=500$ MeV, and iii) SM done by Suzuki et al. Suz06 with the SFO Hamiltonian (the PSDMK2 interaction yields a quite similar result). As seen from Figure 13, in the vicinity of the temperatures mentioned at the beginning ($T_{\nu}=3-5$ MeV), these three calculations yield, respectively, that: i) $\overline{{\sigma}}_{e^{+}}$ is significantly larger than $\overline{{\sigma}}_{e^{-}}$, ii) $\overline{{\sigma}}_{e^{-}}$ is only slightly larger than $\overline{{\sigma}}_{e^{+}}$, and iii) $\overline{{\sigma}}_{e^{+}}\cong\overline{{\sigma}}_{e^{-}}$. Both SM cross sections are always smaller than those obtained in the the other two calculations, and specially in comparison with the RQRPA one. IV Summary and concluding remarks The present work is a continuation of our previous works Krm02 ; Krm05 . In fact, the formalism for weak interaction processes introduced there is now elaborated more thoroughly yielding very simple expressions for the transition rates, which greatly facilitate the numerical calculation. This is done through the separation of the nuclear matrix elements into their real and imaginary parts, which, in turn, permits to split the transition rates, for neutrino-nucleus reactions (II) into natural (40) and unnatural parity (41) operators. Similar separation is done for the muon capture transition rate (58) in Eqs. (59) and (60). Moreover, consequences of explicit violation of CVC hypothesis by the Coulomb field (18) are addressed for the first time, and the sum rule approach for the inclusive cross section, proper to the present formalism, has been worked out in the Appendix B. For the sake of completeness, the extreme relativistic limit of neutrino-nucleus cross section is also presented in the Appendix C, where in the formula for transition rates turn out to be still simpler. We note that, except at very low neutrino energies, they can be used without any restriction in practical applications. We have discussed in details the inclusive properties that comprise: i) Ground state energies in ${}^{12}$B and ${}^{12}$N, and the corresponding GT $B$-values (Figure  1), ii) Exclusive ${}^{12}$C($\nu,e^{-})^{12}$N cross-section $\sigma_{e}(E_{\nu},1^{+}_{1})$, as a function of the incident neutrino energy $E_{\nu}$ (Figures 2, and  3), iii) Exclusive ${}^{12}$C(${\tilde{\nu}},e^{+})^{12}$B cross-section $\sigma_{e^{+}}(E_{\tilde{\nu}},1^{+}_{1})$, as a function of the incident antineutrino energy $E_{\tilde{\nu}}$ (Figure  4), and iv) Muon capture transition rate to the ${}^{12}$B ground state, and electron and muon folded cross-sections to the ${}^{12}$N ground state $\overline{{\sigma}}_{\epsilon}(1^{+}_{1})$, and $\overline{{\sigma}}_{\mu}(1^{+}_{1})$ (Figure  5). Special attention was paid to the interplay between the size of the configuration space, and the magnitude of the residual interaction within the $pp$-channel. It was found that as the first becomes larger, the second has to increase to obtain the agreement with the experimental data for the exclusive observables. The main purpose of our discussion of exclusive properties was to put in evidence the limitations of the RPA and the QRPA models. The basic problem in the implementation of the RPA is the lack of pairing correlations, i.e. the inability for opening the $1p_{3/2}$ shell, while deficiency of the standard QRPA is in the non-conservation of the number of particles, as evidenced by the wave functions (30), (31), and (32) presented in Sec. III. In this way we have definitively established that the SM and the PQRPA are proper theoretical frameworks to describe the ground state properties of ${}^{12}$B and ${}^{12}$N. 666 After our work has been finished, Cheoun et al. Ch10 have presented a new evaluation of the ECS in ${}^{12}$C within the QRPA. They get good agreement with data for $\overline{\sigma}_{e}(^{12}$N), which is at variance with the previous QRPA calculation Vol00 . The inclusive cross-sections ${}^{12}$C($\nu,e^{-})^{12}$N and ${}^{12}$C($\tilde{\nu},e^{+})^{12}$B have been studied within the PQRPA in the same manner as the exclusive ones for $E_{\nu_{e}}$ up to $300$ MeV. As there are no experimental available in this case the comparison is done with the previous calculations only 777 As already mentioned in the introduction the only available experimental data on ${}^{12}$C ICS is the low neutrino energy ($E_{\nu_{e}}<60$ MeV) folded one, which has already been discussed in our previous works Krm05 ; Sam06 ; Paa07 ., and displayed in Figures 6, and 7. Here, unlike within Figures 2, 3, and 4, we also show the results obtained with the other RPA-like models  Con72 ; Vol00 ; Kol99b ; Paa07 , which could be a suitable framework for describing global nuclear properties such as it is the inclusive cross-sections. When the size of the configuration space is enlarged the calculated PQRPA cross-sections, at difference with the exclusive ones, steadily increase, and particulary for neutrino energies larger than $200$ MeV, in spite of including the particle-particle interaction. At low energy they approach to the cross section of the first-forbidden sum-rule limit, but are significantly smaller at high energies both for neutrino and antineutrino. The largest space that we can deal within the number projection procedure is the one that includes all the orbitals until the $N=6$ HO shell. This is the reason why we have recurred to the RQRPA where it is possible to employ larger configuration spaces. It seems that when the number of shells is increased to $N={30}$, and the cut-off energy $E_{2qp}$ is large enough, the cross sections very likely converge as shown in Figures 8, 9, 10, and 11. The Figure 10 also indicates that the RQRPA is a promising nuclear model to reproduce the quasi elastic ($\nu_{\mu},^{12}$C) cross section in the region of $E_{\nu_{\mu}}\sim 1$ GeV which has been measured recently at MiniBooNE MiniBooNE . We do not know whether the discrepancy between the experiment and the theory comes from the non completeness of the configuration space or from the smallness of the effective axial-vector coupling constant that we are using $g_{\mbox{\tiny A}}=1$. It could also happens that we need $g_{\mbox{\tiny A}}=1$ for the low energy exclusive cross section and $g_{\mbox{\tiny A}}=1.255$ for the high energy inclusive cross section. We do not understand the reason for such a energy dependence of $g_{\mbox{\tiny A}}$, but it is consistent with the Eq. (23) in Ref. Na82 where it is shown that for the low energy $\beta$-decay $g_{\mbox{\tiny A}}$ could be much more quenched that the total GT strength. We hope to be able to say more on this matter in the next future. We have also addressed the issue of multipole composition of the inclusive cross sections, by separating them into allowed ($J^{\pi}=0^{+},1^{+}$), first-forbidden ($J^{\pi}=0^{-},1^{-},2^{-}$), second-forbidden ($J^{\pi}=2^{+},3^{+}$), and third-forbidden ($J^{\pi}=3^{-},4^{-}$) processes. The results for the antineutrino reaction ${}^{12}$C($\tilde{\nu},e^{+})^{12}$B are displayed in Figure 12 both for the PQRPA and the RQRPA. Of course, similar results are obtained also for neutrinos. We remark that the spectral functions $d\sigma^{A}_{e^{+}}(E_{\tilde{\nu}})/dE_{\tilde{\nu}}$, when evaluated within the PQRPA, clearly put into evidence the resonant structure of the allowed cross-section, which is mainly of the GT type. The study of the partial ICSs has been related with the proposal done in Refs. Laz07 ; Vol04 of performing nuclear structure studies of forbidden processes by using low energy neutrino and antineutrino beams. From the results shown in Table 1 for the flux-averaged cross sections $\overline{{\sigma}}_{e^{+}}$ in the reaction ${}^{12}{\rm C}({\tilde{\nu}},e^{+})^{12}{\rm B}$ we show that the contribution of allowed transitions decreases gradually in favor of the first forbidden transitions according with the increase of $\gamma$-boost. We conclude that to study high forbidden reactions one would need ${\tilde{\nu}}$-fluxes with $E_{\tilde{\nu}}$ up to $\gtrsim 150$ MeV in ${}^{12}C$. At the end we considered possible astrophysical applications of the $\nu/\tilde{\nu}$-${}^{12}$C nucleus folded cross sections $\overline{{\sigma}}_{e^{\pm}}$, using supernovae $\nu/\tilde{\nu}$ spectra represented by a normalized Fermi-Dirac distribution with mean energy $\langle E_{\nu}\rangle\approx 3.15\times T_{\nu}$, and zero chemical potential. It is found that for temperature $T_{\nu}=3-5$ MeV both the PQRPA and RQRPA models yield significantly larger cross sections that the previously used shell model. Acknowledgements This work was partially supported by the Argentinean agency CONICET under contract PIP 0377, and by the U.S. DOE grants DE-FG02-08ER41533, DE-FC02-07ER41457 (UNEDF, SciDAC-2) and the Research Corporation. A.R.S. thanks to W.C. Haxton and G.M. Fuller for stimulating discussion and to the Institute of Nuclear Theory of University of Washington, where part of this work was performed. N. P. acknowledges support by the Unity through Knowledge Fund (UKF Grant No. 17/08), MZOS - project 1191005-1010 and Croatian National Foundation for Science. Appendix A Contributions to ${{\cal T}}_{{\sf J}^{\pi}_{n}}({\kappa})$ of natural and unnatural parity states The real and imaginary parts of the operators ${\sf O}_{{{\alpha}}{\sf J}}$ given by (12) and (20) do not contribute simultaneously. In fact, the $\Re{\sf O}_{{{\alpha}}{\sf J}}$ ($\Im{\sf O}_{{{\alpha}}{\sf J}}$) contributes to natural (unnatural) parity states, which means that we always can work only with real operators, which greatly simplifies the calculations. To see this we note that, while the operators ${{\cal M}}^{\scriptscriptstyle V}_{\sf J}$, ${{\cal M}}^{\scriptscriptstyle A}_{\sf J}$, and $$\displaystyle{{{\cal M}}}^{\scriptscriptstyle A}_{0{\sf J}}$$ $$\displaystyle=$$ $$\displaystyle\sum_{{\sf L}={\sf J}\pm 1}(-)^{({\sf J-L}-1)/2}\ F_{{{\sf LJ}0}}% j_{\sf L}(\rho)\left[Y_{{\sf L}}(\hat{{\bf r}})\otimes{\mbox{\boldmath$\sigma$% }}\right]_{{\sf J}},$$ are real, ${{\cal M}}^{\scriptscriptstyle A}_{\pm 1{\sf J}}$ and ${{\cal M}}^{\scriptscriptstyle V}_{\pm 1{\sf J}}$ are not. Explicitly, $$\displaystyle{{\cal M}}_{\pm 1{\sf J}}^{\scriptscriptstyle A}$$ $$\displaystyle=$$ $$\displaystyle{{\cal M}}_{\pm 1\sf J}^{\scriptscriptstyle A,R}+i{{\cal M}}_{\pm 1% \sf J}^{\scriptscriptstyle A,I}$$ $$\displaystyle{{\cal M}}_{\pm 1{\sf J}}^{\scriptscriptstyle V}$$ $$\displaystyle=$$ $$\displaystyle{{\cal M}}_{\pm 1\sf J}^{\scriptscriptstyle V,R}+i{{\cal M}}_{\pm 1% \sf J}^{\scriptscriptstyle V,I}$$ (36) where $$\displaystyle{{\cal M}}^{\scriptscriptstyle A,R}_{1\sf J}$$ $$\displaystyle\equiv$$ $$\displaystyle{{\cal M}}^{\scriptscriptstyle A,R}_{-1\sf J}$$ $$\displaystyle=$$ $$\displaystyle\sum_{{\sf L}={\sf J}\pm 1}(-)^{({\sf J-L}-1)/2}\ F_{{{\sf LJ}1}}% j_{\sf L}(\rho)\left[Y_{{\sf L}}(\hat{{\bf r}})\otimes{\mbox{\boldmath$\sigma$% }}\right]_{{\sf J}},$$ $$\displaystyle{{\cal M}}^{\scriptscriptstyle A,I}_{1{\sf J}}$$ $$\displaystyle\equiv$$ $$\displaystyle-{{\cal M}}^{\scriptscriptstyle A,I}_{-1{\sf J}}=-F_{1{\sf JJ}}j_% {\sf J}(\rho)\left[Y_{{\sf J}}(\hat{{\bf r}})\otimes{\mbox{\boldmath$\sigma$}}% \right]_{{\sf J}},$$ $$\displaystyle{{\cal M}}^{\scriptscriptstyle V,R}_{1\sf J}$$ $$\displaystyle\equiv$$ $$\displaystyle{{\cal M}}^{\scriptscriptstyle V,R}_{-1\sf J}$$ $$\displaystyle=$$ $$\displaystyle\sum_{{\sf L}={\sf J}\pm 1}(-)^{({\sf J-L}-1)/2}\ F_{{{\sf LJ}1}}% j_{\sf L}(\rho)\left[Y_{{\sf L}}(\hat{{\bf r}})\otimes{\mbox{\boldmath$\nabla$% }}\right]_{{\sf J}},$$ $$\displaystyle{{\cal M}}^{\scriptscriptstyle V,I}_{1{\sf J}}$$ $$\displaystyle\equiv$$ $$\displaystyle-{{\cal M}}^{\scriptscriptstyle V,I}_{-1{\sf J}}=-F_{1{\sf JJ}}j_% {\sf J}(\rho)\left[Y_{{\sf J}}(\hat{{\bf r}})\otimes{\mbox{\boldmath$\nabla$}}% \right]_{{\sf J}},$$ (37) with ${\sf L}\geq 0$, and ${\sf J}\neq 0$. Thus $$\displaystyle{\sf O}_{\pm 1{\sf J}}$$ $$\displaystyle=$$ $$\displaystyle i(-g_{\mbox{\tiny A}}\pm\overline{g}_{\mbox{\tiny W}})({{\cal M}% }^{\scriptscriptstyle A,R}_{1\sf J}\pm i{{{\cal M}}}^{\scriptscriptstyle A,I}_% {1{\sf J}})$$ (38) $$\displaystyle+$$ $$\displaystyle g_{\scriptscriptstyle{V}}({{\cal M}}^{\scriptscriptstyle V,R}_{1% \sf J}\pm i{{{\cal M}}}^{\scriptscriptstyle V,I}_{1{\sf J}}),$$ and writing $$\displaystyle{\sf O}_{\emptyset{\sf J}}$$ $$\displaystyle=$$ $$\displaystyle{\sf O}_{\emptyset{\sf J}}^{\scriptscriptstyle R}+i{\sf O}_{% \emptyset{\sf J}}^{\scriptscriptstyle I},$$ $$\displaystyle{\sf O}_{m{\sf J}}$$ $$\displaystyle=$$ $$\displaystyle{\sf O}_{m{\sf J}}^{\scriptscriptstyle R}+i{\sf O}_{m{\sf J}}^{% \scriptscriptstyle I},$$ (39) it is not difficult to discover that: • For natural parity states , with $\pi=(-)^{J}$, i.e., $J^{\pi}=0^{+},1^{-},2^{+},3^{-},\cdots$: $$\displaystyle{\sf O}_{\emptyset{\sf J}}^{\scriptscriptstyle R}$$ $$\displaystyle=$$ $$\displaystyle g_{{\scriptscriptstyle{V}}}{{\cal M}}_{\sf J}^{% \scriptscriptstyle V},$$ $$\displaystyle{\sf O}_{{0}{\sf J}}^{\scriptscriptstyle R}$$ $$\displaystyle=$$ $$\displaystyle\frac{{\tilde{k}}_{\emptyset}}{{\kappa}}g_{\mbox{\tiny V}}{{\cal M% }}_{\sf J}^{\scriptscriptstyle V},$$ $$\displaystyle{\sf O}_{\pm 1{\sf J}}^{\scriptscriptstyle R}$$ $$\displaystyle=$$ $$\displaystyle(\pm g_{\mbox{\tiny A}}-\overline{g}_{\mbox{\tiny W}}){{{\cal M}}% }^{\scriptscriptstyle A,I}_{1{\sf J}}+g_{\scriptscriptstyle{V}}{{\cal M}}^{% \scriptscriptstyle V,R}_{1\sf J},$$ (40) • For unnatural parity states, with $\pi=(-)^{J+1}$, i.e., $J^{\pi}=0^{-},1^{+},2^{-},3^{+},\cdots$: $$\displaystyle{\sf O}_{\emptyset{\sf J}}^{\scriptscriptstyle I}$$ $$\displaystyle=$$ $$\displaystyle-g_{\scriptscriptstyle{A}}{{\cal M}}^{\scriptscriptstyle A}_{\sf J% }-(\overline{g}_{\mbox{\tiny A}}+\overline{g}_{\mbox{\tiny P1}}){{\cal M}}^{% \scriptscriptstyle A}_{0{\sf J}},$$ $$\displaystyle{\sf O}_{{0}{\sf J}}^{\scriptscriptstyle I}$$ $$\displaystyle=$$ $$\displaystyle(g_{\mbox{\tiny A}}-\overline{g}_{\mbox{\tiny P2}}){{\cal M}}^{% \scriptscriptstyle A}_{{0\sf J}},$$ $$\displaystyle{\sf O}_{\pm 1{\sf J}}^{\scriptscriptstyle I}$$ $$\displaystyle=$$ $$\displaystyle(g_{\mbox{\tiny A}}\mp\overline{g}_{\mbox{\tiny W}}){{\cal M}}^{% \scriptscriptstyle A,R}_{1\sf J}\mp g_{\mbox{\tiny V}}{{{\cal M}}}^{% \scriptscriptstyle V,I}_{1{\sf J}}.$$ (41) These operators have to be used in (II), instead of those defined in (12), and (20). The correspondence between the individual matrix elements, defined by Donnelly, and Peccei in Eq (3.31)of Ref. Don79 , and the ones used here, is: $$\displaystyle M_{\sf J}$$ $$\displaystyle\rightarrow$$ $$\displaystyle{{\cal M}}^{\scriptscriptstyle V}_{\sf J},$$ $$\displaystyle\Delta_{\sf J}$$ $$\displaystyle\rightarrow$$ $$\displaystyle\sqrt{2}{{{\cal M}}}^{\scriptscriptstyle V,I}_{1{\sf J}},$$ $$\displaystyle\Delta^{\prime}_{\sf J}$$ $$\displaystyle\rightarrow$$ $$\displaystyle-\sqrt{2}{{{\cal M}}}^{\scriptscriptstyle V,R}_{1{\sf J}},$$ $$\displaystyle\Sigma_{\sf J}$$ $$\displaystyle\rightarrow$$ $$\displaystyle\sqrt{2}{{{\cal M}}}^{\scriptscriptstyle A,I}_{1{\sf J}},$$ (42) $$\displaystyle\Sigma^{\prime}_{\sf J}$$ $$\displaystyle\rightarrow$$ $$\displaystyle-\sqrt{2}{{{\cal M}}}^{\scriptscriptstyle A,R}_{1{\sf J}},$$ $$\displaystyle\Sigma_{\sf J}"$$ $$\displaystyle\rightarrow$$ $$\displaystyle{{{\cal M}}}^{\scriptscriptstyle A}_{0{\sf J}},$$ $$\displaystyle\Omega_{\sf J}$$ $$\displaystyle\rightarrow$$ $$\displaystyle{{{\cal M}}}^{\scriptscriptstyle A}_{{\sf J}}.$$ Moreover, the correspondence between the linear combinations of these matrix elements defined in (Don79, , Eqs. (3.32)) (for ${\hat{L}}_{\sf J}$ see  (Hax79, , Eq. (14))), and the ones introduced here is: • For natural parity states : $$\displaystyle{\hat{M}}_{\sf J}$$ $$\displaystyle=$$ $$\displaystyle{\sf O}_{\emptyset{\sf J}},$$ $$\displaystyle{\hat{L}}_{\sf J}$$ $$\displaystyle=$$ $$\displaystyle{\sf O}_{0{\sf J}},$$ $$\displaystyle{\hat{T}}_{\sf J}^{\rm el}\pm{\hat{T}}_{\sf J}^{\rm mag5}$$ $$\displaystyle=$$ $$\displaystyle-\sqrt{2}{\sf O}_{\pm 1{\sf J}},$$ (43) • For unnatural parity states: $$\displaystyle{\hat{M}}_{\sf J}^{5}$$ $$\displaystyle=$$ $$\displaystyle{\sf O}_{\emptyset{\sf J}},$$ $$\displaystyle-i{\hat{L}}_{\sf J}^{5}$$ $$\displaystyle=$$ $$\displaystyle{\sf O}_{0{\sf J}},$$ $$\displaystyle i({\hat{T}}_{\sf J}^{\rm el5}\pm{\hat{T}}_{\sf J}^{\rm mag})$$ $$\displaystyle=$$ $$\displaystyle\sqrt{2}{\sf O}_{\pm 1{\sf J}}.$$ (44) The following relation can also be useful: $$\displaystyle{\sf O}_{\emptyset{\sf J}}$$ $$\displaystyle=$$ $$\displaystyle{\hat{{\cal M}}}_{\sf J},$$ (45) $$\displaystyle{\sf O}_{m\sf J}$$ $$\displaystyle=$$ $$\displaystyle\left\{\begin{array}[]{ccc}{\hat{{\cal L}}}_{\sf J},&\mbox{for}&m% =0\\ -{1\over\sqrt{2}}\left[m{\hat{{{\cal T}}}^{\rm mag}}_{\sf J}+{\hat{{\cal T}}^{% \rm el}}_{\sf J}\right],&\mbox{for}&m=\pm 1\end{array}\right.,$$ where ${\hat{{\cal M}}}_{\sf J}={\hat{M}}_{\sf J}+{\hat{M}}_{\sf J}^{5}$, ${\hat{{\cal L}}}_{\sf J}={\hat{L}}_{\sf J}+{\hat{L}}_{\sf J}^{5}$, ${\hat{{\cal T}}^{\rm el}}_{\sf J}={\hat{T}}_{\sf J}^{\rm el}+{\hat{T}}_{\sf J}% ^{\rm el5}$, and ${\hat{{\cal T}}^{\rm mag}}_{\sf J}={\hat{T}}_{\sf J}^{\rm mag}+{\hat{T}}_{\sf J% }^{\rm mag5}$. The matrix elements of Kuramoto et al. Kur90 are related with our non-relativistic operators (14) as: $$\displaystyle|\langle f|{\hat{1}}|i\rangle|^{2}$$ $$\displaystyle=$$ $$\displaystyle 4\pi\sum_{{\sf J}^{\pi}_{n}}|\langle{\sf J}^{\pi}_{n}||{{\cal M}% }^{\scriptscriptstyle V}_{\sf J}||0^{+}\rangle|^{2},$$ $$\displaystyle|\langle f|{\hat{\sigma}}|i\rangle|^{2}$$ $$\displaystyle=$$ $$\displaystyle 4\pi\sum_{{\sf J}^{\pi}_{n}}\sum_{m=0,\pm 1}|\langle{\sf J}^{\pi% }_{n}||{{\cal M}}^{\scriptscriptstyle A}_{{m\sf J}}||0^{+}\rangle|^{2},$$ $$\displaystyle\Lambda$$ $$\displaystyle=$$ $$\displaystyle\frac{4\pi}{3}\sum_{{\sf J}^{\pi}_{n}}\left[|\langle{\sf J}^{\pi}% _{n}||{{\cal M}}^{\scriptscriptstyle A}_{{0\sf J}}||0^{+}\rangle|^{2}\right.$$ (46) $$\displaystyle-$$ $$\displaystyle\left.|\langle{\sf J}^{\pi}_{n}||{{\cal M}}^{\scriptscriptstyle A% }_{{1\sf J}}||0^{+}\rangle|^{2}\right].$$ In Ref. Kur90 are neglected the relativistic operators ${{\cal M}}^{\scriptscriptstyle A}_{\sf J}$, and ${{\cal M}}^{\scriptscriptstyle V}_{{m\sf J}}$ defined in (LABEL:2.15). Appendix B Sum Rule Approach We follow here the sum-rule approach developed by Kuramoto et al. Kur90 , and adapt it to our formalism. We start from Eqs. (LABEL:2.25), and (27), and as in this work we assume that the ${\omega}_{{\sf J}^{\pi}_{n}}$ dependence of the integrand can be ignored, fixing it at a representative value $\overline{{\omega}_{{\sf J}^{\pi}_{n}}}$ . The summation over final nuclear states ${\sf J}^{\pi}_{n}$ then can be carried out by closure, and the ICS is $$\displaystyle{\sigma}_{\ell}^{SR}(E_{\nu})$$ $$\displaystyle=$$ $$\displaystyle G^{2}\frac{|{\bf p}_{\ell}|E_{\ell}}{2\pi}F(Z+S,E_{\ell})$$ (47) $$\displaystyle\times$$ $$\displaystyle\int_{-1}^{1}d(\cos\theta){{\cal T}}^{SR},$$ where the lepton energy is $E_{\ell}=E_{\nu}-\overline{{\omega}_{{\sf J}^{\pi}_{n}}}$, while the sum-rule matrix element reads: $$\displaystyle{{\cal T}}^{SR}$$ $$\displaystyle=$$ $$\displaystyle\sum_{{\alpha}=\emptyset,0,\pm 1}\langle 0^{+}|O_{\alpha}^{% \dagger}O_{\alpha}|0^{+}\rangle{{\cal L}}_{{\alpha}}$$ (48) $$\displaystyle-$$ $$\displaystyle 2\Re\left(\langle 0^{+}|O_{\emptyset}^{\dagger}O_{0}|0^{+}% \rangle{{\cal L}}_{\emptyset 0}\right).$$ The operators $O_{\alpha}$ are given by (6), and the lepton traces by (Krm05, , Eq. (2.24)). The matrix elements in (48) are proportional to $N(1-D)$, where $N_{N}$ is the number of neutrons (protons), contained in the target nucleus for the neutrino (anti-neutrino) reaction. The correlation functions $D$ come from the Pauli-exclusion-effect, and depend on the type of the operator. One gets: $$\displaystyle{{\cal T}}^{SR}$$ $$\displaystyle=$$ $$\displaystyle N_{N}\left(T_{\emptyset}{{\cal L}}_{\emptyset}+\sum_{M}T_{M}{{% \cal L}}_{M}-2T_{\emptyset 0}{{\cal L}}_{\emptyset 0}\right),$$ (49) with $$\displaystyle T_{\emptyset}$$ $$\displaystyle\equiv$$ $$\displaystyle g_{\mbox{\tiny V}}^{2}(1-D_{S})+(\overline{g}_{\mbox{\tiny A}}+% \overline{g}_{\mbox{\tiny P1}})^{2}(1-D_{L}),$$ $$\displaystyle T_{0}$$ $$\displaystyle\equiv$$ $$\displaystyle\overline{g}_{\mbox{\tiny V}}^{2}(1-D_{S})+(g_{\mbox{\tiny A}}-% \overline{g}_{\mbox{\tiny P2}})^{2}(1-D_{L}),$$ $$\displaystyle T_{1}$$ $$\displaystyle\equiv$$ $$\displaystyle(g_{\mbox{\tiny A}}-\overline{g}_{\mbox{\tiny W}})^{2}(1-D_{T}),$$ (50) $$\displaystyle T_{-1}$$ $$\displaystyle\equiv$$ $$\displaystyle(g_{\mbox{\tiny A}}+\overline{g}_{\mbox{\tiny W}})^{2}(1-D_{T}),$$ $$\displaystyle T_{\emptyset 0}$$ $$\displaystyle\equiv$$ $$\displaystyle-g_{\mbox{\tiny V}}\overline{g}_{\mbox{\tiny V}}(1-D_{S})+(% \overline{g}_{\mbox{\tiny A}}+\overline{g}_{\mbox{\tiny P1}})(g_{\mbox{\tiny A% }}-\overline{g}_{\mbox{\tiny P2}})(1-D_{L}).$$ The correlation functions $D_{S},D_{L}$ and $D_{S}$ were taken from the SM calculation done by Bell, and Llewellyn Smith Bel71 with HO wave functions, and representing the nuclear ground state by a single determinant wave function. The results for ${}^{12}$C are  (Bel71, , Table 1)): $$\displaystyle D_{S}$$ $$\displaystyle=$$ $$\displaystyle e^{-\eta}\left[1+0.148\eta^{2}\right],$$ $$\displaystyle D_{T}$$ $$\displaystyle=$$ $$\displaystyle e^{-\eta}\left[0.704+0.148\eta+0.148\eta^{2}\right],$$ $$\displaystyle D_{L}$$ $$\displaystyle=$$ $$\displaystyle e^{-\eta}\left[0.704+0.296\eta+0.148\eta^{2}\right],$$ (51) where $\eta=\frac{1}{2}b^{2}{\kappa}^{2}\cong 0.0558$. As seen from (26), the factor $|{\bf p}_{\ell}|E_{\ell}$ in (47) behaves as $(E_{\nu}-\overline{{\omega}_{{\sf J}^{\pi}_{n}}})^{2}$, and therefore ${\sigma}_{\ell}^{SR}(E_{\nu})$ depends very critically on the average value for the excitation energy $\overline{{\omega}_{{\sf J}^{\pi}_{n}}}$. $$\displaystyle E_{\ell}$$ $$\displaystyle=$$ $$\displaystyle E_{\nu}-{\omega}_{{\sf J}^{\pi}_{n}},~{}|{\bf p}_{\ell}|=\sqrt{(% E_{\nu}-{\omega}_{{\sf J}^{\pi}_{n}})^{2}-m_{\ell}^{2}},$$ $$\displaystyle\kappa$$ $$\displaystyle=$$ $$\displaystyle|{\bf p}_{\ell}-{\bf q}_{\nu}|$$ (52) $$\displaystyle=$$ $$\displaystyle\sqrt{2E_{\nu}(E_{\ell}-|{\bf p}_{\ell}|\cos\theta)-m_{\ell}^{2}+% {\omega}_{{\sf J}^{\pi}_{n}}^{2}},$$ Appendix C Extreme Relativistic Limit Using the present formalism the ERL, defined by the limit of the lepton velocity $|{\bf p}_{\ell}|/E_{\ell}\rightarrow 1$, yields $$\displaystyle{\sigma}^{ERL}_{\ell}(E_{\nu})$$ $$\displaystyle=$$ $$\displaystyle\sum_{J^{\pi}_{n}}\frac{E^{2}_{\ell}}{2\pi}F(Z+S,E_{\ell})\int_{-% 1}^{1}d(\cos\theta){{\cal T}}^{ERL}_{{\sf J}^{\pi}_{n}}(\kappa),$$ with $$\displaystyle\kappa$$ $$\displaystyle=$$ $$\displaystyle\sqrt{2E_{\nu}E_{\ell}(1-\cos\theta)+{\omega}_{{\sf J}^{\pi}_{n}}% ^{2}},$$ (54) and $$\displaystyle{{\cal T}}^{ERL}_{{\sf J}^{\pi}_{n}}({\kappa})$$ $$\displaystyle=$$ $$\displaystyle{4\pi G^{2}}\left[2\cos^{2}\frac{\theta}{2}\left|\langle{\sf J}^{% \pi}_{n}||{\sf O}_{\emptyset{{\sf J}}}({\kappa})-\frac{k_{\emptyset}}{{\kappa}% }{\sf O}_{0{\sf J}}({\kappa})||0^{+}\rangle\right|^{2}\right.$$ (55) $$\displaystyle+$$ $$\displaystyle\sum_{m=\pm 1}|\langle{\sf J}^{\pi}_{n}||{\sf O}_{m{\sf J}}({% \kappa})||0^{+}\rangle|^{2}$$ $$\displaystyle\times$$ $$\displaystyle\left(\frac{k^{2}}{{\kappa}^{2}}\cos^{2}\frac{\theta}{2}+2\sin^{2% }\frac{\theta}{2}\right.$$ $$\displaystyle+$$ $$\displaystyle 2mS\left.\left.\sin\frac{\theta}{2}\sqrt{\frac{k^{2}}{{\kappa}^{% 2}}\cos^{2}\frac{\theta}{2}+\sin^{2}\frac{\theta}{2}}\right)\right].$$ Appendix D Muon Capture rate For the sake of completeness we also show the formula for the muon capture process within the present formalism. Here ${\kappa}=E_{\nu}=\mathrm{m}_{\mu}-{\omega}_{{\sf J}^{\pi}_{n}}-\Delta\mathrm{M% }-E_{B}$, where $E_{B}^{\mu}$ is the binding energy of the muon in the $1S$ orbit, and instead of (5) one has: $$\displaystyle\overline{g}_{\mbox{\tiny V}}$$ $$\displaystyle=$$ $$\displaystyle g_{\mbox{\tiny V}}\frac{E_{\nu}}{2\mathrm{M}};~{}\overline{g}_{% \mbox{\tiny A}}=g_{\mbox{\tiny A}}\frac{E_{\nu}}{2\mathrm{M}},$$ $$\displaystyle\overline{g}_{\mbox{\tiny W}}$$ $$\displaystyle=$$ $$\displaystyle(g_{\mbox{\tiny V}}+g_{\mbox{\tiny M}})\frac{E_{\nu}}{2\mathrm{M}% };~{}\overline{g}_{\mbox{\tiny P}}=g_{\mbox{\tiny P}}\frac{E_{\nu}}{2\mathrm{M% }},$$ (56) where $\overline{g}_{\mbox{\tiny P}}=\overline{g}_{\mbox{\tiny P2}}-\overline{g}_{% \mbox{\tiny P1}}$. The muon capture transition rate reads $$\displaystyle\Lambda({\omega}_{{\sf J}^{\pi}_{n}})$$ $$\displaystyle=$$ $$\displaystyle\frac{E_{\nu}^{2}}{2\pi}|\phi_{1S}|^{2}{{\cal T}}_{\Lambda}({% \omega}_{{\sf J}^{\pi}_{n}}),$$ (57) where $\phi_{1S}$ is the muonic bound state wave function evaluated at the origin, and $$\displaystyle{{\cal T}}_{\Lambda}({\omega}_{{\sf J}^{\pi}_{n}})$$ $$\displaystyle=$$ $$\displaystyle 4\pi G^{2}\left[|\langle{\sf J}^{\pi}_{n}||{\sf O}_{\emptyset{{% \sf J}}}(E_{\nu})-{\sf O}_{0{{\sf J}}}(E_{\nu})||0^{+}\rangle|^{2}\right.$$ (58) $$\displaystyle+$$ $$\displaystyle\left.2|\langle{\sf J}^{\pi}_{n}||{\sf O}_{-1{{\sf J}}}(E_{\nu})|% |0^{+}\rangle|^{2}\right],$$ with • For natural parity states , with $\pi=(-)^{J}$, i.e., $J^{\pi}=0^{+},1^{-},2^{+},3^{-},\cdots$: $$\displaystyle{\sf O}_{\emptyset{\sf J}}-{\sf O}_{0,\sf J}$$ $$\displaystyle=$$ $$\displaystyle g_{\mbox{\tiny V}}\frac{\mathrm{m}_{\mu}-\Delta E_{\rm Coul}-E_{% B}}{E_{\nu}}{{\cal M}}^{\scriptscriptstyle V}_{\sf J},$$ $$\displaystyle{\sf O}_{-1{\sf J}}$$ $$\displaystyle=$$ $$\displaystyle-(g_{\mbox{\tiny A}}+\overline{g}_{\mbox{\tiny W}}){{{\cal M}}}^{% \scriptscriptstyle A,I}_{-1{\sf J}}+g_{\scriptscriptstyle{V}}{{\cal M}}^{% \scriptscriptstyle V,R}_{-1\sf J},$$ (59) and • For unnatural parity states, with $\pi=(-)^{J+1}$, i.e., $J^{\pi}=0^{-},1^{+},2^{-},3^{+},\cdots$: $$\displaystyle{\sf O}_{\emptyset{\sf J}}-{\sf O}_{0,\sf J}$$ $$\displaystyle=$$ $$\displaystyle g_{\scriptscriptstyle{A}}{{\cal M}}^{\scriptscriptstyle A}_{\sf J% }+\left(g_{\mbox{\tiny A}}+\overline{g}_{\mbox{\tiny A}}-\overline{g}_{\mbox{% \tiny P}}\right){{\cal M}}^{\scriptscriptstyle A}_{0{\sf J}},$$ $$\displaystyle{\sf O}_{-1{\sf J}}$$ $$\displaystyle=$$ $$\displaystyle-(g_{\mbox{\tiny A}}+\overline{g}_{\mbox{\tiny W}}){{\cal M}}^{% \scriptscriptstyle A,R}_{-1\sf J}-g_{\scriptscriptstyle{V}}{{{\cal M}}}^{% \scriptscriptstyle V,I}_{-1{\sf J}}.$$ (60) References (1) C. Athanassopoulos et al.  [LSND Collaboration], Phys. 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Technical Considerations for Semantic Segmentation in MRI using Convolutional Neural Networks Arjun D. Desai Department of Radiology Stanford University arjundd@stanford.edu &Garry E. Gold Departments of Radiology, Bioengineering, and Orthopedic Surgery Stanford University gold@stanford.edu &Brian A. Hargreaves Departments of Radiology, Electrical Engineering, and Bioengineering Stanford University bah@stanford.edu &Akshay S. Chaudhari Department of Radiology Stanford University akshaysc@stanford.edu Abstract High-fidelity semantic segmentation of magnetic resonance volumes is critical for estimating tissue morphometry and relaxation parameters in both clinical and research applications. While manual segmentation is accepted as the gold-standard, recent advances in deep learning and convolutional neural networks (CNNs) have shown promise for efficient automatic segmentation of soft tissues. However, due to the stochastic nature of deep learning and the multitude of hyperparameters in training networks, predicting network behavior is challenging. In this paper, we quantify the impact of three factors associated with CNN segmentation performance: network architecture, training loss functions, and training data characteristics. We evaluate the impact of these variations on the segmentation of femoral cartilage and propose potential modifications to CNN architectures and training protocols to train these models with confidence.   Technical Considerations for Semantic Segmentation in MRI using Convolutional Neural Networks  Submitted to Magnetic Resonance in Medicine Arjun D. Desai Department of Radiology Stanford University arjundd@stanford.edu Garry E. Gold Departments of Radiology, Bioengineering, and Orthopedic Surgery Stanford University gold@stanford.edu Brian A. Hargreaves Departments of Radiology, Electrical Engineering, and Bioengineering Stanford University bah@stanford.edu Akshay S. Chaudhari Department of Radiology Stanford University akshaysc@stanford.edu 1 Introduction Magnetic resonance imaging (MRI) provides high spatial resolution and exquisite soft tissue contrast, leading to its pervasive use for visualization of tissue anatomy. Using MR images for quantitative analysis of tissue-specific information is critical for numerous diagnostic and prognostic protocols. The gold-standard for high fidelity region annotation is manual tissue segmentation, which can be time-consuming and prone to inter-reader variations [1, 2]. Thus, there has always been great interest in developing fully-automated tissue segmentation techniques that are robust to small image variations [3, 4]. A common application of segmentation is the segmentation of articular cartilage for studying changes associated with onset and development of osteoarthritis (OA) [5, 6]. Recent advances in MRI have focused on developing non-invasive morphological and compositional biomarkers for tracking the onset and progression of OA. There is promising evidence suggesting that changes in cartilage morphology and composition may serve as early-OA biomarkers [7, 8]. Despite its potential, accurate measurement of cartilage morphology entails tedious manual segmentation of the fine structure of cartilage in hundreds of MRI slices and patients [9]. Though automatic segmentation of femoral cartilage is of great interest, the tissue’s thin morphology and low contrast with surrounding tissue structures makes automatic segmentation challenging. Traditional automatic segmentation approaches have utilized 3D statistical shape modeling or multi-atlas segmentations modulated by anisotropic regularizers [10, 11]. However, such techniques are highly sensitive to deformations in knee shape, which can be caused by variations in patient knee size and in incidence and progression of pathology [12]. Advances in deep learning and convolution neural networks (CNNs) have shown great potential for enhancing the accuracy of cartilage segmentation [13, 14]. However, due to the stochastic nature of deep learning and the multitude of training parameters (hyperparameters) that can be fine-tuned for any given problem, developing analytic estimations of network behavior is challenging [15, 16]. As a result, practical design choices for optimizing CNN performance for segmentation in MRI, especially for femoral cartilage segmentation, have been under-explored. Often, CNN architectures are modified in the hope of increasing overall accuracy and generalizability while minimizing inference time. In the case of the popular ImageNet challenge [17] for natural image classification, classification accuracy and generalizability have varied considerably with changes in network architecture [18, 19]. Additionally, while 2D architectures have been effective at slice-by-slice segmentation of medical images, recent works have also utilized volumetric architectures, which take 3D volumes as inputs to potentially add through-plane (depth) contextual cues to improve segmentation [20, 21]. However, the extent to which network structure and input depth impact semantic segmentation in medical imaging remains unclear. Variations in CNN training protocol may also affect network performance. As network weights are optimized with respect to the gradient of the training loss function, the selection of loss function may dictate network accuracy. In particular for segmentation, where foreground-background class imbalance is common, loss functions, such as weighted cross-entropy and soft Dice loss, are often chosen to minimize the impact of class imbalance [22, 23]. In addition, supervised CNN training requires both large training sets and corresponding high-fidelity segmentation masks, which are difficult to produce. In cases of limited training data, data augmentation is a common practice for artificially increasing variability of training data to reduce overfitting and promote generalizability [24, 25]. Moreover, MRI volumes can be acquired with varying fields of view (FOVs), resulting in different matrix sizes. A commonly reported advantage of fully convolutional network (FCN) CNN architectures is their ability to infer on images or volumes of arbitrary matrix sizes not specifically utilized during the training process [26]. In this study, we investigate three factors associated with the performance and generalizability of segmentation CNNs: network architecture, training loss functions, and data extent. While performance is quantified by traditional segmentation accuracy metrics, we also quantify the generalizability of a network using the sensitivity of applying trained networks to segment MR images at varying FOVs. All experiments were conducted on segmentation of femoral cartilage, a challenging localization problem and a relevant target for studying OA. We seek to quantify performance variations induced by these three factors to motivate how CNN segmentation models can be built, trained, and deployed with confidence. 2 Methods 2.1 Dataset Data for this study were acquired from the Osteoarthritis Initiative (OAI) (http://oai.epi-ucsf.org), a longitudinal study for studying osteoarthritis progression [27]. 3D sagittal double echo steady state (DESS) datasets along with their corresponding femoral cartilage segmented masks were utilized in this study (relevant scan parameters: FOV=14cm, Matrix=384$\times$307 (zero-filled to 384$\times$384), TE/TR=5/16ms, 160 slices with a thickness of 0.7mm) [27]. This dataset consisted of 88 patients with Kellgren-Lawrence (KL) OA grades between 1 and 4 [28], measured at two time points (baseline and 1 year) for a total of 176 segmented volumes. These patients were randomly split into cohorts of 60 patients for training, 14 for validation, and 14 for testing, resulting in 120, 28, and 28 volumes used during training, validation, and testing, respectively. An approximately equal distribution of KL grades was maintained among all three groups (Supporting Table S1). 2.2 Data Pre-processing All DESS volumes in §2.1 were downsampled by a factor of 2 in the slice dimension (to dimensions of 384$\times$384$\times$80) prior to network training and inference to increase SNR and reduce computational complexity, justified by previous studies reporting that approximately 1.5mm slices are adequate for cartilage morphometry [9]. Images were downsampled using sum-of-squares combinations, and the corresponding masks were downsampled by taking the union of the masks to compensate for partial volume artifacts. The volume was then cropped in-plane to 288$\times$288 by calculating the center of mass (COM) and centering the cropped region 50 pixels in the superior (up) direction and 20 pixels in the anterior (left) direction to bias the COM to femoral cartilage and away from the tibia and posterior muscles. The volume was then cropped to remove 4 slices from both the medial and lateral ends, resulting in volumes of dimensions 288$\times$288$\times$72. All scans were subsequently volumetrically zero-mean whitened ($\mu$=0, $\sigma$=1) by subtracting the image volume mean and scaling by the image volume standard deviation. All data pre-processing was conducted using MATLAB (MathWorks, Natick, MA). These training, validation, and testing sets were used for all experiments unless otherwise indicated. 2.3 Network Architecture In this experiment, we wanted to evaluate the sensitivity of the semantic segmentation task to different, popular CNN architectures. We selected three general 2D FCN architectures for analysis: U-Net, SegNet, and DeeplabV3+ [29, 30, 31]. These FCN architectures utilize variations of the encoder-decoder model for semantic segmentation for extracting features at different spatial fields of view. The U-Net architecture implements an encoder-decoder model using max-pooling and transpose convolutions to downsample and upsample feature maps (Figure  1a). In this structure, the number of network filters increases exponentially as a function of network depth. The U-Net also relies on deep skip connections by concatenating encoder outputs to the decoding layers in order to share spatial cues between the two and to propagate the loss efficiently at different network depths [32, 33]. SegNet uses a similar encoder-decoder architecture but passes pooling indices to upsampling layers to avoid the overhead of copying encoder weights (Figure  1b). In contrast to using max-pooling to promote spatial invariance and to downsample feature maps, DeeplabV3+ implements ‘Xception‘ blocks [34] and spatial pyramid pooling with dilated convolutions to capture a larger receptive field without increasing the parameter size (Figure  1c). Instead of transposed convolutions, the DeeplabV3+ decoder uses bilinear upsampling to upsample the features to the input image size. While the U-Net and SegNet have shown promise for musculoskeletal MRI semantic segmentation [13, 14], DeeplabV3+ has been primarily utilized for natural image segmentation [35] and has seen limited use in segmentation of medical images. As a baseline comparison, all architectures were trained for 100 epochs and subsequently fine-tuned for 100 epochs following training hyperparameters detailed in Table  1. 2.4 Volumetric Architectures In this experiment, we trained a 2.5D [36] and 3D U-Net architecture for femoral cartilage segmentation. The 2.5D network uses a stack of $t$ continuous slices in a scan to generate a segmentation mask for the central slice (additional details are described in the supplemental information). Three 2.5D networks with inputs of thickness $t$=3,5,7 were trained. In contrast, a 3D network outputs a segmentation on all $t$ slices. As all operations are applied in 3D, the 2$\times$ max-pooling applied in the through-plane direction constrains the input to have $t=2^{N_{p}}$ slices, where $N_{p}$ refers to the number of pooling steps. To maintain an identical number of pooling steps as the 2D and 2.5D networks ($N_{p}=5$), the 3D U-Net was trained using 32 slices of the volume as an input ($t=32$). As a result, the scans in the training dataset described in §2.2 were also cropped by an additional 4 slices from the medial and lateral ends, resulting in volumes with 64 slices. Memory constraints of the hardware necessitated that this volume be further divided into two 3D subvolumes of size 288$\times$288$\times$32 and that the batch size be reduced to 1. The 2D and 2.5D networks had an exponentially increasing number of filters ranging from 32$\xrightarrow{}$1024, while the 3D network had filters ranging from 16$\xrightarrow{}$512 to accommodate for the same network depth. All networks maintained a comparable number of weights (Supporting Figure S2). 2.5 Loss Function As trainable parameters in a network update with respect to the loss function, we hypothesize that a relevant loss function is critical for any learning task. Traditionally, binary cross-entropy has been used for binary classification tasks. However, class imbalance has shown to limit the optimal performance in cases of general cross-entropy losses [20]. We selected three additional loss functions commonly used for segmentation in cases of class imbalance for comparison: soft Dice loss [37], weighted cross-entropy (WCE), and focal loss ($\gamma$=3) [38], as described additionally in the supplementary. In this experiment, four models using the U-Net architecture were trained using the four loss functions described above with the training, validation, and testing sets described in §2.1. 2.6 Data Augmentation To qualify the effect of data augmentation on model generalizability, we trained the standard U-Net architecture with and without augmented training data. Each 2D slice and corresponding mask in the training volume were randomly augmented to add heterogeneity to the training dataset. The augmentation procedure consisted of sequential transformations of the original image and masks with: 1. zooming (between 0-10%), 2. shearing (between -15${}^{\circ}$ to 15${}^{\circ}$ in both the horizontal and vertical directions), 3. gamma variations (between 0.8-1.1 for simulating varying contrasts), and 4. motion blur (between 0-5 pixels in magnitude and 0${}^{\circ}$ to 360${}^{\circ}$ in direction). Parameters for each transformation were chosen uniformly at random within the specified ranges, with an example slice shown in Figure  2. These specific augmentation methods and magnitudes were chosen to mimic typically encountered physiological and imaging variations. No augmentations were applied to the scans in the validation and test sets. All 2D slices were augmented fourfold, resulting in the augmented dataset consisting 5x the data in the non-augmented training set. To overcome this discrepancy while training separate networks with and without augmented data, the networks trained using augmented data were trained for 5x shorter than those trained using non-augmented data (20 epochs total). 2.7 Generalizability to Multiple Fields of View In this experiment, we compare the differences in network performance on scans at different FOVs with the same underlying image resolution. In addition to the inference dataset (V0) cropped to a volume of ($288\times 288\times 72$) described in §2.1, three new test sets were created with different degrees of cropping: V1 ($320\times 320\times 80$), V2 ($352\times 352\times 80$), and V3 ($384\times 384\times 80$). As data augmentation is hypothesized to increase network generalizability, we compared the performances of the U-Net models trained using non-augmented and augmented data as specified in §2.6 among the four test sets (V0-V3). 2.8 Training Data Extent Performance of CNNs has also been shown to be limited by the extent (amount) of data available for training [39]. To explore the relationship between the extent of training data and network accuracy, we trained each of the three base network architectures in §2.3 on varying sized subsets of the training data. The original training set consisting of 60 patients was randomly sampled (with replacement) to create 3 additional sub-training sets of 5, 15, and 30 patients with similar distributions of KL grades (Supporting Table S3). The same validation and testing sets described in §2.1 (with 14 patients, each at two time points) were used to assess the generalizability of the networks. The network trained on the complete sample of training data (60 patients) was trained for 100 epochs. To ensure that all sub-sampled networks maintained an equal number of backpropagation steps to update filter weights, we scaled the number of epochs by the ratio of the fully sampled patient count (60) to the number of patients in the sub-training set. As a result, networks trained on 5, 15, and 30 patients were trained for 1200, 400, and 200 epochs respectively. Experiments were repeated 3 times each (with Python seeds 1, 2, and 3) to enhance reproducibility and to minimize the stochasticity of random network weight initializations. 2.9 Network Training Hyperparameters For all experiments, convolutional layers with rectified linear unit (ReLU) activations were initialized using the "He" initialization [40, 41]. Training was performed using the Adam optimizer with default parameters ($\beta_{1}=0.9$, $\beta_{2}=0.999$, $\epsilon$=1e-8) with random shuffling of mini-batches using a Tensorflow backend [42, 43]. All neural network computations were performed on 1 Titan Xp graphical processing unit (GPU, NVIDIA, Santa Clara, CA) consisting of 3,840 CUDA cores and 12GB of GDDR5X RAM. Due to the randomness of the training processes, we empirically determined a pseudo-optimal set of hyperparameters for training each network. To reduce large variances in training batch normalization layers caused by small mini-batch sizes, the largest mini-batch size that could be loaded on the Titan Xp GPU was used for each network. The initial learning rate and use of step decay was also empirically determined based on the network architecture. Table  1 details the hyperparameters used with each network architecture. Networks were trained using the soft Dice loss, unless otherwise specified. 2.10 Quantitative Comparisons For each experiment, the model that resulted in the best loss on the validation dataset was used for analysis on the testing dataset. During testing, output probabilities of femoral cartilage ($p_{FC}$) were thresholded at 0.5 to create binary femoral cartilage segmentations ($p_{FC}\leq 0.5\xrightarrow{}0$, $p_{FC}>0.5\xrightarrow{}1$). No additional post-processing was performed on the thresholded masks. Segmentation accuracy was measured on the testing dataset using three metrics - Dice similarity coefficient (DSC), volumetric overlap error (VOE), and coefficient of variation (CV) [44]. High accuracy segmentation methods maximize DSC (a maximum of 100%) while minimizing VOE and CV (a minimum of 0%). The segmentation masks obtained from the OAI dataset served as ground truth. Statistical comparisons between the inference accuracy of different models were assessed using Kruskal-Wallis tests, and corresponding Dunn post-hoc tests, ($\alpha=0.05$). All statistical analyses were performed using the SciPy (v1.1.0) library [45]. Additionally, changes in network performance in the slice (depth-wise) direction were visualized using graphs termed depth-wise region of interest distribution (dROId) plots. The normalized depth-wise field of view (dFOV) spanning the region of interest is defined as the ordered set of continuous slices containing femoral cartilage according to ground truth manual segmentation, where, in the set, the first slice corresponds to medial side (dFOV=0%) and the last slice corresponds to the lateral side (dFOV=100%). All volumes were mirrored to follow this convention. 3 Results All performance results (except data limitation) are summarized in Table  2. 3.1 Network Architecture Comparison A comparison of the performance of the U-Net, SegNet, and DeeplabV3+ architectures on sample slices is shown in Figure  3. All three base architectures maintained high fidelity in segmenting slices containing thick cartilage structures (Figure  3A). However, all networks had worse performance in slices containing regions of full-thickness cartilage loss and denuded subchondral bone, edge slices, and medial-lateral transition regions (Figure  3B,C). Despite lower accuracy in these regions, these networks accurately segmented slices with heterogeneous signal caused by pathology and proximity to anatomy with similar signal (Figure  3C). Performance decreased at edge regions (dFOV~[0, 10]%, [90, 100]%) and at the medial-lateral transition region (dFOV~[55, 65]%) as seen in the dROId plot in Figure  4A. There was no significant difference in the performance of U-Net, SegNet, and DeeplabV3+ models as measured by DSC (p=0.08), VOE (p=0.08), and CV (p=0.81). 3.2 Volumetric Architectures Comparison Results of the 2D, 2.5D, and 3D U-Net architectures showed no significant difference between the performance of 2D U-Net and that of the three versions ($t$=3,5,7) of the 2.5D U-Net. The 2D U-Net, however, did perform significantly better than the 3D U-Net (DSC,VOE-p<0.05). There were also no significant differences (p=1.0) in the performance between the 2.5D architectures using inputs of different depths ($t$=3,5,7). Decreased DSC at edge and medial-lateral transition regions was indicative for all models as seen on the dROId plot (Figure  4B). In the 3D U-Net model, DSC was greater in the lateral compartment of the knee (dFOV~[60,90]%) compared to that of the medial compartment (dFOV~[15, 45]%). Among 2D and 2.5D networks, performance in the lateral and medial regions was comparable. 3.3 Loss Function Comparison Performance differences between BCE, soft Dice, and focal losses were negligible, but all three losses significantly outperformed WCE (p<5e-10) across all slices (Figure  4C). Using the WCE loss model for inference, the incidence rate of false-positives (misclassifying a background pixel as femoral cartilage) was significantly greater (p<2e-10) than the incidence rate of false-negatives (misclassifying a femoral cartilage pixel as background). Over 99% of the WCE model errors were false-positives (Figure  5C). The pixel-wise error distribution, as measured on the test set (V0) appeared correlated to the output probability of femoral cartilage ($0\leq p_{FC}\leq 1$), which may be an indicator of network confidence in classifying a pixel as femoral cartilage. For BCE, soft Dice, and focal losses, the difference between the false-positive and false-negative rates were not significant (p>0.4). The incidence of errors is also symmetrically distributed around the threshold probability with medians $p^{*}_{FC}$ of 0.48, 0.84, and 0.51, respectively (Figure  5 A,B,D). The error rate in BCE was relatively uniform across all probabilities while the distribution of error rates in soft Dice loss is primarily bi-modal with peaks at $p_{FC}=0$ and $p_{FC}=1$. The focal loss error distribution was more densely centered around $p_{FC}=0.5$. 3.4 Data Augmentation Comparison The use of augmented training data significantly decreased network performance (p<0.001) compared to the augmented training data set (Figure  4B). The performance was also consistently lower at other regions of the knee. 3.5 FOV Generalizability Comparison Baseline U-Net network performance was variable across test sets consisting of scans at different fields of view (Figure  6). Inference on semi-cropped test sets (V1, V2) had significantly lower performance (p<0.01) than that on the original test set (V0). There was no significant difference (p=1.0) between performance on test set V0 and the non-cropped test set (V3). In contrast, there was no significant difference in performance of the augmented U-Net model across all four test sets (p>0.99). 3.6 Data Extent Trend Network performance for all three networks increased with increasing training data (Figure  7). The trend between the number of patients in the training dataset and network performance followed a power-law ($y=\alpha x^{\beta}$) scaling, as hypothesized previously [46], for all performance metrics (p<1e-4). Pixel-wise performance metrics, DSC and VOE, had a strong fit to the hypothesized power-law curve for all architectures ($r^{2}>0.91$ and $r^{2}>0.95$, respectively). CV had a relatively weaker, but still strong, fit ($r^{2}>0.63$). Among the different architectures, there was no significant difference in the intercept ($\alpha$) or exponent ($\beta$) of the curve fit measured at different seeds, and all exponents were less than 1 ($\beta$<1). 4 Discussion In this study, we examined how variations in FCN architecture, loss functions, and training data impacted network performance for femoral cartilage segmentation. We found no significant pixel-wise difference in the performance of U-Net, SegNet, and DeeplabV3+, three commonly used FCN frameworks for natural image segmentation. There was also no significant performance difference between the segmentations produced by 2D and 2.5D networks. We demonstrated that BCE, soft Dice, and focal losses had similar false-positive and false-negative incidence rates, while WCE biased the network toward false-positive errors. Moreover, while data augmentation reduced U-Net performance, it increased generalizability in performance among scan volumes at different fields of view. Additionally, this study verified that segmentation performance scales directly, following a power-law relationship, with increasing data size. Traditionally, training methods and architectures have been a design choice when applying CNNs for semantic segmentation. In these cases, our findings provide insight into which design choices may be most effective for knee MR image segmentation using CNNs. 4.1 Base Architecture Variations Based on network performance metrics, newer network architectures like DeeplabV3+ have slightly improved, though not significant, segmentation accuracy compared to traditionally used U-Net and SegNet models. The larger receptive field induced by using dilated convolutions in DeeplabV3+ may increase spatial awareness to foreground-background boundary regions. The expressivity of a network, often used to characterize network generalizability, is defined as the degree to which the network structure facilitates learning features that are representative for the task. As expressivity increases, performance also increases. Raghu, et al. and Bengio, et al. suggest that expressivity is highly impacted by network structures such as depth, which enables hierarchical feature representations, and regularizations, which prime the network to learn representative features that are stable across different inputs [47, 48]. While DeeplabV3+ does not follow the same sequential autoencoder structure as U-Net and SegNet, it leverages dilated convolutions to extract features at various fields of view and decodes these features to create a hierarchical feature representation as expressive as those generated by the other two architectures. Though network architecture has been closely linked with expressivity, there was no significant difference in the performance of the three network architectures, and all networks failed in similar regions of minimal, disjoint cartilage (Figure  3). The non-uniqueness in failure cases indicates that all three network models may optimize for similar deep features and consequently, segment images in a visually comparable manner. This minimal difference in performance suggests that beyond some threshold expressivity, differences in CNN architectures may have a negligible impact on the overall segmentation performance. Similar work for fully-connected neural networks (i.e. no convolutions) demonstrated network generalization is not limited by the architecture for a wide array of tasks, given that the network is expressive enough to achieve a small training error [49]. While CNNs and fully-connected neural networks are not an exhaustive representation of all forms of neural networks, the trend of the limited effect of network structure on overall expressivity indicates that improving architectures may not be as effective in training better-performing networks. 4.2 Practical Design for Volumetric Architectures In this study, the volumetric (2.5D/3D) networks had a negligible impact on segmentation accuracy and even performed worse than traditional 2D slice-wise segmentation in the case of the 3D network. The limited difference between 2.5D and 2D networks may be explained by the negligible difference in expressivity of these networks. These networks only differ at the first convolutional layer, which takes the image/volume as the input. While 2.5D networks accept an input volume ($M\times N\times t$), and 2D networks accept an input slice ($M\times N\times 1$), the output of the initial convolution layer is the same size in both networks. As a result, 2.5D networks only have more parameters in the first convolution layer (Supporting Figure S2), which is negligible when compared to the general size of the network and may not expressively represent the through-plane information 2.5D networks hope to capture. Unlike 2.5D networks, which collapse the 3D input into multiple 2D features after the first convolutional layer, 3D networks maintain the depth-wise dimension throughout the network. While this allows depth-wise features to be extracted throughout the entire network, the number of network parameters also increases, which can limit the batch size of the network. In the 3D network trained in this study, a batch size of 1 was required to fit the scan volume as an input, which may have lead to less stable feature regularization. Additionally, to allow fitting the scan volume as an input, the 3D network had approximately the same number of parameters as the 2.5D and 3D networks. However, as the number of parameters per kernel increases to maintain the extra dimension, the number of filters at the initial convolutional layers had to be curbed twofold. The fewer filters at earlier stages in the network likely contributed to lower expressivity, and consequently poorer performance, of the network. With increased computational and parallelization power, designing 3D networks with similar filter counts as 2D networks may increase network expressivity. 4.3 Selecting Loss Functions While network architectures did not significantly impact performance, U-Net models trained using BCE, soft Dice, and focal losses performed significantly better than the model trained using WCE loss. While WCE is intended to normalize loss magnitude between imbalanced classes, the artificial weighting biases the network to over-classify the rarer class. The degree of false-positive bias introduced into the network using WCE is likely modulated by the respective class weights. As the median frequency re-weighting method over-biases the network, traditional weighting protocols based on class incidence may not be the optimal weighting scheme. While optimal performance is traditionally measured by reducing the overall error, WCE loss weightings may be used to intentionally steer a network either towards additional false-positives or false-negatives, depending on the specific use case. Additionally, the different error distributions around the threshold probability ($p_{FC}$=0.5) indicate the potential success of each loss function (Figure  5). In a binary problem, the probability output of pixel $i$ ($p_{i}$) is binarized at some threshold probability $p_{T}$, typically chosen to be the midpoint ($p_{T}$=0.5). Let $\hat{y}_{i}\in\{0,1\}$ define the output of the binarization operation $\beta$ on for pixel $i$, such that $\hat{y}_{i}=\beta(p_{i})=\begin{cases}1,\;\;p_{i}>p_{T}\\ 0,\;\;p_{i}\leq p_{T}\\ \end{cases}$. Let $y_{i}\in\{0,1\}$ correspond to the ground-truth class for pixel $i$. Therefore, pixel $i$ is misclassified if $y_{i}\neq\beta(p_{i})$. If pixel $i$ is misclassified, let $dp_{i}$ be the minimum amount of shift required to $p_{i}$ to correctly classify pixel $i$ (i.e. $\beta(p_{i}+dp_{i})=y_{i}$). For the loss functions used above, the energy required to shift $p_{i}$ is directly proportional to $dp_{i}$. If $p_{i}$ is close to the limit bounds ($p_{i}\approx 0,1$), $dp_{i}$ is very large; but if $p_{i}$ is close to the threshold probability $p_{T}$, $dp_{i}$ is much smaller. Therefore, a distribution that is densely centered around $p_{T}$ minimizes $dp_{i}$ and has the most potential for reducing error rate with limited energy. Of the four error distributions induced by different loss functions, focal loss produces errors that are most densely centered around $p_{T}$, which may make it most amenable for future optimization. Focal loss likely achieves this distribution by weighting the BCE loss to be inversely proportional to the correct classification. For example, a pixel with a probability for its correct class close to 1 will be weighted less than a pixel with a probability for its correct class close to 0. As a result, well-classified pixels will not contribute to the loss, and consequently, will not be further optimized. This preserves high error rate close to $p_{FC}$=0.5, as a network trained with focal loss is most uncertain about these examples. This symmetric distribution also suggests that correcting false-positive and false-negative errors would require an equal amount of energy. 4.4 Achieving Multi-FOV Robustness through Data Augmentation As MR scan protocols can often adjust the image FOV for different sized patients, training an FCN that is generalizable to multiple FOVs may be desired. The U-Net trained on non-augmented images did not exhibit the same performance across different FOVs. Recall that test sets V1, V2, and V3 covered a larger through-plane field of view (80 slices) than V0, whose dimensions were identical to the training volumes (72 slices). Failure cases in V1 and V2 were predominately in the 8 slices not included in the volumetrically cropped testing/training sets. It is likely the network failed in these regions because these additional slices include anatomy that may not have been seen during training. In contrast, the U-Net trained using augmented training data exhibited the same performance across all FOVs. The augmentations introduced realistic variations that could occur during imaging (motion and gamma variations) and artificial variations that change the distribution of anatomy across pixels (zooming and shearing). The later set of artificial augmentations manipulate the FOV that the tissue of interest covers in the training image. As a result, the optimized network likely consists of a family of features that is robust to spatial FOV variations within the degrees of the zooming and shearing distributions used. Thus, instead of measuring the expressivity of a FCN network on a single test set, we suggest that the expressivity for multi-FOV applications should be quantified by its performance on test sets at varying FOVs for evaluating robustness to multi-FOV scans. While augmentations have been readily accepted as a method to increase network accuracy, the 2D U-Net trained with augmented data in this study had sub-optimal performance. This phenomenon likely occurred because the network trained with non-augmented data optimized features for images containing the same FOV of anatomy as the training images. In contrast, the augmented dataset may challenge the network by varying the FOV and contrast of information seen. The optimal minimum may not minimize loss as efficiently on the non-augmented datasets, and as a result, the features are not optimized to achieve a high testing loss on test set V0. However, these features likely increased the stability of the network for inference on scans of multiple FOVs. We suggest that augmentations should be meticulously curated to increase network expressivity to expected image variations, especially in regards to tissues of interest having variable sizes in potential test images. 4.5 Navigating Training with Limited Data The performance of all three networks changed at a considerably slow rate as data size increased. The rate is primarily governed by the exponent value ($\beta$) in the power-law equation. The mean exponent across three seeds $\bar{\beta}$<0.05 for all architectures indicated a slow growth in performance as a function of data size. In a recent work, Hestness, et al. empirically verified that the error rate in image classification decreases following a power-law scaling with $\beta<1$ regardless of architecture [50]. Like image classification, semantic segmentation also appeared to follow this trend, with minimal variation in $\beta$ among architectures. Moreover, this slow-growth power-law performance scaling can allow us to empirically estimate the performance of these networks as the data size increases. Based on these parameter estimates, achieving a 95% Dice accuracy for the U-Net, SegNet, and DeeplabV3+ models would require approximately 350, 440, and 300 patients, respectively. Therefore, while increasing training data does increase performance over time, the addition of each subsequent dataset diminishes marginal utility. These results suggest that even with small amounts of data, high percentage of performance can generally be obtained. 4.6 Limitations Despite the promising empirical relationships elucidated in this work, there were limitations to this study that should be addressed in future studies. Training hyperparameters for each network were empirically determined by investigating training loss curves for the initial epochs. While a robust hyperparameter search may yield a more optimal set for training, this was beyond the primary premise of this work, which aimed to explore larger tradeoffs between network architectures, loss functions, and training data. Additionally, the 3D U-Net architecture trained in the volumetric architecture experiment fixed the input depth at 32 slices, resulting in a low batch size and fewer number of filters at each network level. Future studies could modulate the number of input slices to increase batch size and number of filters to optimize network performance. Moreover, all networks performed binary segmentation, but as most loss functions allow for multi-class segmentation, it would be useful to understand the impact of this problem on performance for each tissue. 5 Conclusion In this study, we quantified the impact of variations in network architecture, loss functions, and training data for segmenting femoral cartilage from 3D MRI in order to investigate the tradeoffs involved in segmentation with CNNs. Variations in network architectures yielded minimal differences in overall segmentation accuracy. Additionally, loss functions dictate how the network weights are optimized and, as a result, influence how errors are distributed across probabilities. Moreover, realistic data augmentation methods can increase network generalizability at the cost of absolute network performance on any given test set. Limited amounts of training data may also not be the bottleneck in network performance. Acknowledgments Contract grant sponsor: National Institutes of Health (NIH); contract grant numbers NIH R01 AR063643, R01 EB002524, K24 AR062068, and P41 EB015891. Contract grant sponsor: Philips (research support). Image data was acquired from the Osteoarthritis Initiative (OAI). The OAI is a public-private partnership comprised of five contracts (N01-AR-2-2258; N01-AR-2-2259; N01-AR-2-2260; N01-AR-2-2261; N01-AR-2-2262) funded by the National Institutes of Health, a branch of the Department of Health and Human Services, and conducted by the OAI Study Investigators. Private funding partners include Merck Research Laboratories; Novartis Pharmaceuticals Corporation, GlaxoSmithKline; and Pfizer, Inc. Private sector funding for the OAI is managed by the Foundation for the National Institutes of Health. This manuscript was prepared using an OAI public use data set and does not necessarily reflect the opinions or views of the OAI investigators, the NIH, or the private funding partners. 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THE CONCEPTUAL DESIGN OF THE MAGDALENA RIDGE OBSERVATORY INTERFEROMETER David F. Buscher${}^{1}$    Michelle Creech-Eakman${}^{2}$    Allen Farris${}^{2}$    Christopher A. Haniff${}^{3}$    and John S. Young${}^{3}$ ${}^{1}$Cavendish Laboratory, University of Cambridge, J J Thompson Avenue, Cambridge, UK, dfb@mrao.cam.ac.uk ${}^{2}$MRO, New Mexico Institute of Mining and Technology, 801 Leroy Place, Socorro, NM 87801, USA, ${}^{3}$Cavendish Laboratory, University of Cambridge, J J Thompson Avenue, Cambridge, UK. ((to be inserted by publisher); (to be inserted by publisher); (to be inserted by publisher)) Abstract We describe the scientific motivation for and conceptual design of the Magdalena Ridge Observatory Interferometer, an imaging interferometer designed to operate at visible and near-infrared wavelengths. The rationale for the major technical decisions in the interferometer design is discussed, the success of the concept is appraised, and the implications of this analysis for the design of future arrays are drawn out. keywords: instrumentation: interferometers \catchline11footnotetext: Corresponding author.{history} ; ; ; 1 Introduction The last 50 years have seen a drive to enhance the angular resolution delivered by ground-based telescopes in the presence of atmospheric turbulence. Most of this effort has focused on the use of adaptive optics (AO): modern implementations can routinely deliver 100 milliarcsecond diameter point spread functions. Nevertheless, the desire for angular resolutions better than this remains a powerful driving force in astrophysics. For example, over half of the key science cases comprising the E-ELT design reference mission Liske et al. [2011] require the diffraction-limited performance at the 10 mas level to be delivered. Ground-based interferometric arrays have contributed to studies on much smaller angular scales. Typical recent studies have focused on Cepheid multiplicity [Gallenne et al., 2013; Li Causi et al., 2013], circumstellar disk physics [Kreplin et al., 2013], the orbital dynamics of close binaries [Hummel et al., 2013; Sana et al., 2013] and the fundamental parameters of single stars [Klotz et al., 2013; Arroyo-Torres et al., 2013]. Interferometric arrays currently offer the only direct means to study astrophysical phenomena on such small angular scales. However, the scientific output of the current generation of ground-based arrays appears to have converged to about 50 peer reviewed papers per year, focused almost exclusively on galactic astrophysics. This level and focus of activity has been maintained for at least five years now, and this might suggest that these major facility arrays are being constrained by the capabilities of their initial suite of instruments. The principal shortcomings of current interferometers are easily identifiable and are broadly speaking poor sensitivity and the inability, in most implementations, to routinely image targets with high fidelity. These issues formed the basis of the initial planning of the technical architecture of the Magdalena Ridge Observatory Interferometer (MROI). The MROI was conceived as part of an ambitious plan at the New Mexico Institute of Mining and Technology (NMT) to develop a new astronomical observatory at a high altitude site in central New Mexico to support a broad portfolio of programmes in research, education, and the commercial and defence arenas. In this paper we review the conceptual design of the MROI, currently under construction on the Magdalena Ridge at an elevation of 10,500 ft. From conception, the MROI was targeted towards expanding the capabilities of ground-based optical/near-infrared interferometry in three main directions: sensitivity, imaging capability, and speed of operation. However, it aimed to do this not by solving a set of extreme technical challenges, but rather by leveraging the many technical lessons that had been learnt developing the first generation of prototype interferometers and by capitalising on advances in technology that had taken place since those first arrays had been commissioned. In this sense it has always been a low-risk endeavour. More importantly, though, its architecture was guided through an iterative process whereby the scientific ambitions of its proposed user community were moderated by transparent presentations of the technical constraints implicit in interferometric imaging and the current state-of-the-art in the management of the turbulent atmosphere. The layout of our manuscript is as follows. We begin with a review of the limitations of existing arrays in section 2 and follow this with a brief resumé of the broad areas of science the MROI has been designed to tackle in section 3. We then walk through the major technical requirements in section 4 and the key design choices and trades that were made as the overall architecture of the MROI was frozen (section 5). We present the high-level error budget for the array in section 6 and evidence that we can meet the key sensitivity requirement in section 7. The main technical lessons learned during the design, prototyping and on-going deployment of the MROI are summarised in section 8 and our conclusions appear in section 9. 2 Limitations of existing interferometers The scientific capabilities of conventional optical/IR telescopes are frequently characterised by their “performance” measured against half a dozen or so key metrics. These usually include such items as spatial resolution, spectral resolving power, wavelength range, sensitivity, field-of-view and so on. Almost all of these are appropriate figures of merit for interferometric arrays, but as written they “hide” perhaps the most important shortcoming of contemporary optical/IR arrays, that is, the fact that few are able to provide images of the targets under study. In the following sub-sections we discuss the background to this problem as well as the capabilities of existing interferometric arrays with reference to two of the more conventional performance metrics listed above: sensitivity and spatial resolution. 2.1 Imaging capability The ability of an interferometric array to recover an image is limited fundamentally by how many independent measurements of the visibility function of the source, i.e. the Fourier transform of the sky brightness distribution, can be secured. If a total of $N_{uv}$ independent visibility data are collected, this implies that an image with of order $N_{uv}$ degrees-of-freedom can be reconstructed. Thus, for example, if a $5\times 5$ pixel image of an extended source is required, of order 25 independent visibility data need to be collected. In addition, the projected baselines used to secure these data should range in length by a factor of roughly 5:1. Meeting these requirements is a significant challenge for all existing optical/IR arrays. For example, at the CHARA array on Mt Wilson, the typical range in baseline lengths realised in imaging observations to date has been between 3 and 4:1 [see e.g. Baron et al., 2012] At the VLTI, baseline length ratios of up to 6:1 have been achieved but only by combining measurements made over several nights Millour et al. [2011] during which time some of the interferometer elements have been relocated. An additional issue is the long time needed to secure individual visibility “snapshots”. At the VLTI, the allocations of time recommended for securing calibrated snapshot visibility data with the AMBER (3 visibilities) and MIDI (2 visibilities) instruments are between 50 and 90 minutes. As a result of these issues, in excess of 90% of the published scientific results from optical/IR interferometers rely upon fitting either geometric or physical models to the measured visibility data. In cases where the systems under study are well known, in the sense that they can be reliably characterised by a model with a small number of degrees-of-freedom, interferometric data can provide very powerful diagnostics for the precise values of the model parameters. However, there are many cases where the models themselves are in question, e.g. “Is the dust distributed in a continuous disk or might an Archimedean spiral be preferred?”, or where the physical model used contains many parameters whose interaction is complex — an example of such a physical model might be that of a dusty disk with an inner and outer radius, an inclination, parameters describing the radial and vertical scale-lengths, a dust mass, a dust composition, a grain size distribution and a stellar temperature and luminosity. In these cases the inability to recover an image assuming no preferred structure (i. e. “model-independent imaging”) is a major shortcoming of contemporary interferometers. 2.2 Limiting sensitivity An almost equally pressing issue for the current generation of optical/IR interferometric arrays is their relatively poor sensitivity. Because ground-based arrays are constrained to operate in the presence of atmospheric fluctuations, they are fundamentally limited by the small values of Fried’s parameter, $r_{0}$, and the atmospheric coherence time. $t_{0}$. In view of this, one might expect the limiting sensitivity of an optical/IR array to be similar to that of a natural guide star (NGS) AO system, i.e. to be based on the brightness of the star being used to sense the atmospheric wavefront perturbations. At typical NGS AO systems reference stars as faint as $m_{V}\sim 12$ can used very effectively, and partial AO correction can still be realised with guide stars as faint as $m_{V}\sim 16$. In practice, all existing optical/IR arrays struggle to come even close to these sensitivity limits. The faintest optical measurements secured at the NPOI, SUSI and CHARA arrays have been from targets with visual magnitudes in the range 5–7, while in the near-infrared $K$ band typical limiting magnitudes have been in the range 6–8. In a few cases fringe data have been secured on targets as faint as $m_{K}\sim 10$ but these studies have relied upon arrays that have exploited AO-corrected 8 m-class telescopes as the basis array elements. It is this relatively poor sensitivity that means that contemporary optical/IR interferometry has has such a small impact on extra-galactic astrophysics, and that most interferometric studies have been directed towards a small number of the brightest exemplars of different classes of targets. The ability to reach a limiting sensitivity comparable to that of current NGS AO installations would be a major advance. 2.3 Angular resolution A dimension along which existing interferometric implementations have made significant inroads is that of angular resolution. As mentioned in section 1 above, all existing separated element optical/IR arrays have baselines sufficiently long — typically at least 100 m — that sensitivity to sub-10 mas angular scales is guaranteed. There seems little doubt that arrays with baselines at least five times longer than this, i.e. of order 500 m are desirable. Baldwin and Haniff (2002) provide a useful table summarising the characteristic angular scales expected for a range of targets at distances which would ensure that more than only the brightest targets would be observable. Many of these classes of object, e.g. main sequence stars, stellar gas shells, spectroscopic and interacting binaries, and AGN broad line regions, would only be expected to be resolved on baselines in excess of 300 m. What is less clear is whether arrays with baselines in excess of a kilometer are of high priority. Targets requiring such long baselines to resolve them must necessarily be small, and if so, they must be distant. In that case, it is likely they would be faint unless they had particularly high brightness temperatures. Baldwin & Haniff [2002] discuss this argument further, but notwithstanding the details, it seems probable that enhancing the imaging capability and sensitivity of existing arrays ought to take priority over extending their maximum baselines. 3 Science drivers for the MROI The science case for the MROI is predicated on three key areas, familiar to many existing interferometric facilities today. This is really a reflection of the brightness temperatures of the targets and the operational wavelengths and baselines accessible at most modern facilities. The key differences associated with the interferometric observations at MROI, however, are in more accurate imaging and in much greater sensitivities, both of which will dramatically change the fundamental scientific questions that can be answered. Our three key areas, outlined briefly below, encapsulate the scientific drivers which drive the design choices for the facility. 3.1 AGN Astrophysics Though active galactic nuclei (AGN) have been studied for many years, interferometric imaging offers the prospect of gaining important new insights into their structure. The physical scales of most interest include the broad line region (BLR) on sub 0.1 pc scales; the narrow line region (NLR) extending from 1–1000 pc and the dust torus which ranges in size from $\sim 0.5-10$ pc. In nearby AGN ($z<0.01$) these correspond to angular sizes from 0.5 mas to several seconds of arc, but it is on scales smaller than 0.1 arcseconds where observations with the MROI will have most leverage. The limiting sensitivity of modern interferometers today means that only about 1 dozen AGN have been studied at near or mid-infrared wavelengths [c.f. Tristram et al., 2009; Honig et al., 2012; Kishimoto et al., 2011]. However, even this small quantity of data has been intriguing. For instance, in the mid-infrared, Centaurus A appears to have a thin, dusty disk, the axis of which may align with its radio jet [Meisenheimer et al., 2007] while NGC 3783 and the Circinus galaxy appear clumpy [Beckert et al., 2008; Tristram et al., 2007] and NGC 1068 may have a torus and funnel structure [Raban et al., 2009]. Studying a significantly larger sample of these galaxies with an interferometer with complex imaging capabilities is required to understand the origin and ubiquity of these morphologies. A key goal for MROI will be to investigate the details and reliability of dust torus models. It is being designed to address questions such as: a) the frequency of occurrence of tori, b) the geometric and physical properties of the obscuring material, and c) whether these are consistent with “unified” schemes. In particular, model-independent images are crucial in interpreting observations of clumpy (but small) tori, which are not well-described by a small number of parameters but which are increasingly favoured. Simulations of the archetype NGC 1068 [Honig et al., 2006] predict K-band visibilities greater than 30% on baselines of up to 50 m, a regime where the most compact configurations of the MROI (with baselines from 8–45 m) will be ideally matched to the relevant scales. Furthermore it should be possible to correlate the torus axis with the larger scale radio emission. A more ambitious study to be undertaken with MROI will be to investigate the BLR/NLR transition region. Imaging the transition region between the BLR and the NLR offers the prospect of isolating the outflowing line-emitting gas and in turn determining the origin, geometric, physical and temporal characteristics of the outflow. This would result in a breakthrough in the understanding of the dynamics of AGN cores, of which little is currently known. 3.2 YSO Astrophysics A second major theme for MROI will be to advance detailed studies of star and planet formation through studies of young stellar objects (YSOs). As with the AGN, results from first-generation near-IR interferometers have led to significant new insights into the complex inner structure of protostellar/planetary disks. For example, the sizes of Herbig Ae/Be and T Tauri objects have been measured to be 3-7 times larger than predicted by geometrically-thin disk models previously used to explain the SED measurements [see e.g. Millan-Gabet et al., 2007, and references therein], and have led to the development of a new class of “puffed up” models for the flared disk emission [e.g. Dullemond et al., 2001; Isella & Natta, 2005]. Other measurements have produced tantalising evidence for structures in the inner disk relevant to the planet formation process. For example, hotspots have been detected in the disks of AB Aur [Millan-Gabet et al., 2006] and FU Orionis [Malbet et al., 2005], and evidence for gap clearing has been found in LkCa 15 [Espaillat et al., 2008]. What has been missing in all of these studies has been sufficient visibility and closure phase data to properly constrain the complex geometrical and physical structure of the inner, i. e. sub-10-AU-scale, regions of the disk. Measurements of order one hundred Fourier amplitudes and closure phases, at sensitivity levels a several magnitudes fainter than what is currently possible, will be critical to understanding the physical processes taking place there. A key goal for MROI’s early science will be to provide a census of disk properties for well-defined samples of low, intermediate and high-mass stars. By allowing hundreds of targets to be observed with excellent Fourier plane coverage, it will be possible to critically assess theoretical models which predict not only the structure expected but also the temporal evolution of the dust. This evolution is expected to be strongly impacted by the presence of any planetary or brown-dwarf companions in the inner disk, which should lead to either disk breakup or disk-clearing [Creech-Eakman et al., 2010]; such structures will certainly be detectable in active YSO disks with MROI. The higher resolution spectral modes of MROI ($R\sim 300$) will allow it to discriminate between dust, gas and molecular emission using spectral diagnostics such as the Brackett gamma line and CO bandheads [c.f. Eisner et al., 2007]. MROI will thus enable routine monitoring of the dust and gas on scales that have Keplerian rotational timescale of weeks to years. A parallel theme will be to search for low-mass companions in star forming regions, not only from the perspective of precise dynamical mass and age estimation [e.g. Konopacky et al., 2007a] but also to validate models of star formation. To date this has been undertaken using high resolution imaging methods on single telescopes [e.g. shift-and-add imaging, speckle interferometry or AO — see Konopacky et al., 2007b], but these are limited in both angular resolution and the ability to detect the lowest-mass companions. MROI, when employed in the compact configuration, will be ideally suited to make major inroads in this area: companions with a K-band flux 1/100 that of the primary star will be detectable within a night’s observations, allowing numerous candidates in, e.g., Taurus and Orion to be surveyed to well below the hydrogen burning limit. 3.3 Mass-Loss and Dynamical Systems The third key area for optical/infrared interferometry is the study of fundamental physical processes. Unique contributions can be made by imaging interferometers with sub-milliarcsecond resolution whenever complex systems/processes are being studied which are difficult to understand without direct imaging. In this case, MROI will focus on the physics of mass loss, mass transfer in binaries, and time-domain/dynamical interactions. While this type of work is in principle possible for any interferometer, it becomes much more powerful with an increase in the number of apertures (and consequent numbers of visibilities and closure-phases) that are used and with shorter timescales for producing complete images. Excellent recent examples of images in these broad scientific areas include: CHARA imaging results on the long-period eclipsing system, Epsilon Aurigae [Kloppenborg et al., 2010], and on the interacting binary system Beta Lyrae [Zhao et al., 2008] and older aperture masking results from the Keck Observatory on interacting binary/mass-losing system WR104 [Tuthill et al., 2008]. Because few actual images exist today, except for the brightest “archetypes” of each class of object, it is expected that tremendous insight into all the evolutionary stages of low to high-mass stars will be accomplished when statistical samples of objects in each class are accumulated. Some of the questions MROI will address include: a) how does the pulsational behaviour in the late stages of life of an AGB star affect the mass-loss and subsequent shaping processes of planetary nebulae; b) what are the characteristics of mass-loss processes at different stages in stellar evolution, i.e. continuous, episodic, clumpy, smooth; c) what is the interaction between the stellar surface (i. e. magnetic fields, star spots) and the mass-loss processes as traced over a variety of time-scales; d) how does mass-transfer in interacting systems trigger subsequent explosive events; e) what is the connection between stellar “shape” (i. e. non-spherical, rapidly rotating systems) and their wind structures; and f) do optical or infrared counterparts exist to trace known phenomena at X-ray/UV wavelengths, for example in the shocks or hot winds, for high-mass systems? Papers which shed some light on these questions have been published in the last few years and include: on the nova RS Ophiuchi Barry et al. [2008], on star-spots on Betelgeuse [Chiavassa et al., 2010], and on imaging of rapidly rotating stars [c.f. van Belle, 2012, and references therein]. In all cases, high-resolution images on statistical numbers of objects, and especially over a variety of timescales, will only increase our understanding of the physical processes involved. MROI’s design is uniquely developed to address many of these questions. 4 Technical requirements In order to clarify the goals for the conceptual design of the MROI, the functional and performance requirements for the array need to be defined. The major requirements are briefly summarised below, but this is only part of a much larger list. The requirements generally result from a desire to address the three primary science missions outlined above, i.e. studies of active galactic nuclei, star and planet formation, and stellar accretion and mass loss, but is important to recognise that the requirements do not solely flow top-down from the science case but result from a balancing scientific desirability and technical feasibility: it will be seen below that many scientifically desirable specifications conflict with one another when it comes to implementing them together in a single design. 4.1 Imaging The attainment of a unique imaging capability was one of the chief scientific drivers for the MROI. The imaging performance can be characterised crudely by the number of resolution elements (often called “resels” in analogy to pixels) in an image but for interferometric imaging the number of resels which can be derived is dependent not only on the angular size of the object but also on the object morphology. Objects such as stellar disks have visibility functions which fall rapidly from unity with increasing spatial frequency. On the baselines which have sufficient angular resolution to see interesting features such as surface activity, the majority of the flux in the object is resolved and the visibility can be so low that the signal-to-noise ratio of the fringes is below the level required to find the interferometric fringes even on bright targets. Our experience with the COAST interferometer was that many such objects could not be observed with more than a few resels across the image, not for the lack of long enough baselines but for lack of ability to acquire fringes on baselines beyond the first lobe of the object visibility function. A way to get around this limitation is known as “baseline bootstrapping” and is discussed in the next section, but use of this technique imposes limitations on the efficient sampling of the $(u,v)$ plane. As a result the requirements for imaging of objects which have a bright and compact core which remains unresolved when the features of interest have been resolved are different from those for imaging of so-called “resolved-core” objects like stellar disks. The requirement for the MROI was that with resolved-core objects it should be able to make images with 5$\times$5 resolution elements across the object while with compact-core objects under favourable conditions it should be able to make 10$\times$10 resel images. This exceeds the image quality of all existing optical arrays and is comparable to the images from aperture-masking arrays and many existing radio interferometers. Another important aspect of the imaging is the dynamic range of the images, i.e. the ratio of the brightest and weakest believable features in any recovered maps. This is dependent on a combination of the Fourier coverage of the observation and the calibration errors on the visibility amplitudes and phases. Aperture-masking results suggest that a dynamic range of 100:1 on bright sources is both feasible and scientifically productive, and so this specification adopted for the MROI. 4.2 Sensitivity The concept of sensitivity is usually defined for an interferometer in terms of a limiting magnitude. An object at the limiting magnitude is just bright enough that it is possible to acquire interferometric data on the object. This implies that some form of fringe acquisition and tracking is performed, as without this it is impossible to guarantee that the interferometer is observing fringes. Thus the limiting magnitude refers to the overall brightness of the object for fringe-tracking purposes: if an image with a large dynamic range can be made, it may be possible to observe objects within the field of view which are several magnitudes fainter than this. The extra-galactic component of the top-level science mission for the MROI sets the basic requirement for its desired sensitivity. At H-band magnitudes fainter than about 11 the very closest and brightest active galactic nuclei just start to become visible. However, it is not until a H-band sensitivity of 14 is reached that of order 100 targets become visible in the Northern celestial sky. In practice, such a sensitivity requirement for the array is particularly challenging but later sections will show that this can be achieved in an optimised design. 4.3 Wavelength coverage The wavelength range from 0.6–2.4 $\mu$m is key to the science goals of MROI because it includes the key H$\alpha$ line at 656 nm and also the near-infrared range where many of the top-level science targets are bright. The deterioration in wavefront quality due to atmospheric “seeing” means that it becomes rapidly more difficult to compensate for the atmosphere at shorter wavelengths, while at longer wavelengths the rising thermal background causes the sensitivity to fall rapidly. Thus the near-infrared is the regime where the best interferometric sensitivity can be realised. 4.4 Spectral resolution Spectral resolving powers of $R>5,000$ would be valuable for studying atomic line emission and absorption in stellar sources but such a capability would only be useful on relatively bright objects because of signal-to-noise-ratio issues. A higher priority is a resolving power $R\stackrel{{\scriptstyle>}}{{{}_{\sim}}}300$ that allows useful isolation of molecular features from their nearby continuum in stellar sources, and velocity-resolved imaging in the very nearest AGN, together with a lower-resolution mode $R\sim 30$ which allows crude spectral diagnostics on fainter targets. 4.5 Automation and reliability It is well established that the most productive optical/infrared interferometers have been those with highly automated sequencing and operation. Our design philosophy for the MROI implicitly assumes this model, so that tasks such as pre-observing alignment of the optical trains, self-testing of the detector and beam combination subsystems, and failure recovery, will all be carried out transparently with a minimum of operator intervention. 5 Conceptual design The requirements for the MROI emphasise its ability to image faint and complex sources in a model-independent manner, and so designing array which optimised both of these aspects was critical. Both of these goals, and especially the sensitivity goal, affect all aspects of the interferometer design and so there is a many-to-many relationship between the elements of the implementation and the design goals. In this section the major elements of the conceptual design are described, the relationship of these elements to the overall goals is described and the trade-offs made between conflicting requirements are addressed. The elements of the design are addressed roughly in order of their importance, but are also ordered by the sequence in which the stellar light impinges on each of these in its journey from the star to being detected as fringes, as shown in Figure 2. 5.1 Array configuration A critical choice for any interferometer is the number of telescopes. The cost of an interferometer increases approximately linearly with the number of telescopes, but the number of $(u,v)$ points sampled by $N$ telescopes increases faster than $N^{2}$ . Adequate $(u,v)$ coverage is critical to imaging performance, both in terms of image complexity and dynamic range, but can also be achieved (at the expense of imaging speed) through repositioning of telescopes. When imaging resolved-core sources (as defined in section 4.1), an interferometer with a larger number telescopes is superior not only in imaging speed but also in the achievable image complexity. This is because the only reliable way to access baselines which give many resels across such sources is to utilise the so-called “baseline bootstrapping” technique Pauls et al. [1998]. In baseline bootstrapping, multiple telescopes are arranged in a “chain” such that the nearest-neighbour telescopes are spaced by a distance which is less than the size of the main lobe of the visibility function. On these shortest baselines the fringe visibility is high enough to allow fringe tracking and hence the measurement of atmospheric phase differences. These differences can be integrated along the chain to track the phase perturbations on the longest baselines where the fringe visibility is too low to allow direct fringe tracking. The longest baseline which can be accessed is limited to $N-1$ times the shortest baseline. Hence the maximum number of resels across such an object rises approximately linearly with the number of telescopes. This limit cannot be improved upon by moving the telescopes. The MROI infrastructure is designed for 10 telescopes (more telescopes could be added but could not be operated simultaneously). When the telescopes are arranged in an equispaced $Y$-shaped configuration as shown in Figure 1 then the array can be viewed as a set of 3 “bootstrapping chains” each consisting of 7 telescopes, oriented at 120${}^{\circ}$ angles to one another. The ratio of the longest baseline in this array to the nearest-neighbour spacing is 5:1 and so this will allow approximately 5$\times$5 resel imaging on resolved-core objects, a capability which is unmatched by any existing optical interferometer. At the same time the “snapshot” $(u,v)$ coverage of the array is superior to any existing optical interferometer, as shown in Figure 1, giving access to 36 well-separated baselines and 36 closure phases. For compact-core objects, then a non-redundant configuration can be used, giving access to 45 baselines simultaneously. Using Earth rotation to increase the density of the $(u,v)$ coverage will then allow images with approximately 10$\times$10 resels to be reconstructed on these objects. A “Y” shaped array configuration has been used in a number of interferometers including COAST, NPOI and CHARA and offers many advantages including the ability to easily route light from the telescopes to the centre of the array while using a train of mirrors whose angles of incidence are symmetric between telescopes. However the most compelling advantage of this layout for the MROI is that the array site is on a saddle in the mountain which allows the arms of a horizontal “Y” shaped array to be built with a minimum of earth movement. The longest practicable baseline using this layout is approximately 350 meters and this gives a maximum angular resolution of 0.3 milliarcseconds at a wavelength of 600 nm. A single configuration of the array gives access to minimum and maximum angular scales with a ratio of only 5:1, whereas the range of sizes of possible science targets ranges over more than two orders of magnitude. The telescopes are therefore designed to be relocatable between a number of different arrays. The primary array configurations are scaled versions of the “bootstrapping” array, with maximum baselines ranging from 40 m to 346 m and minimum baselines ranging from 7.8 m to 67 m. The array therefore allows access to angular scales with a range of 44:1. Four “bootstrapping” configurations span this range with a scaling of approximately two between successive configurations. Different configurations have been arranged to re-use some of the telescope stations (by distorting the regular spacing of the telescopes along the arms by a few percent) so that a total of 28 stations is required. 5.2 Telescopes and adaptive optics One critical factor in the sensitivity of an interferometer is the telescopes, called “unit telescopes” here to distinguish them from the aperture synthesis telescope comprising the entire array. Large unit telescopes equipped with high-order adaptive optics (AO) would appear at first sight to be the most promising route to high sensitivity, but not only is this an expensive option for an imaging array with many telescopes, but it also conflicts with the need to pack the telescopes close together in order to sample larger-scale angular structure and thereby access targets such as nearby evolved stars and geosynchronous satellites. In addition, high-order adaptive optics are only effective if there is a bright enough reference available to sense the wavefront perturbations. The angular density of bright natural references (stars) is sufficiently low that the science target itself is most often the AO reference. Typical AGN-type targets with an near-infrared magnitudes of 14 have visible-wavelength magnitudes of around 16 and these are too faint to drive current AO systems at radial orders higher than tip-tilt Wilson & Jenkins [1996]. Tip/tilt correction gives the best limiting magnitude on telescopes which are of order 2–3$r_{0}$ in diameter Buscher [1988b] and so, in the absence of laser guide star systems at each telescope, telescopes in the 1–2 metre size range provide close to the best possible interferometric sensitivity at optical and near-infrared wavelengths. 5.3 Beam relay The parallel beams exiting from the unit telescopes are sent to the beam combining optics as a set of parallel beams, where most of the path is in vacuum, as shown in Figure 2. Vacuum beam propagation was chosen over monomode fiber optics for beam transport as it allows the propagation of the entire bandpass from the optical to the infrared to be handled in a single system rather than requiring splitting the bandpass into multiple bands for propagation, which inevitably incurs additional losses. The diameters of the beams need to be significantly larger than the Fresnel zone size $\sqrt{\lambda z}$ in order to minimise the effects of diffraction for propagation over a distance $z$. For propagation distances of order 1 km and a wavelength of 2.2 $\mu$m, this zone size is of order 5 cm and so relatively large optics are required for propagation over these distances. The propagation distances inside the beam combining area are of the order 20 m so smaller beams can be used in order to keep the size and cost of the more complex beam combination optics within reasonable bounds. A set of beam compressors between the delay lines and the beam combination optics serves to transform between these two beam sizes. 5.4 Delay lines Dynamical optical path compensation is required in order to equalise the path-lengths travelled by the starlight from the target to the detector. The path-length stroke required from these compensators scales with the size of the array, and arrays of comparable size such as CHARA and NPOI utilise a two-stage path compensation system, consisting of a switchable delay to give large stroke and a continuously-variable delay to give fine control of the path-length. The approach adopted at MROI was to have a single-stage system which introduces all the delay in a continuously-variable manner. This minimises the number of reflections needed and at the same time avoids the switching overheads associated with the two-stage systems. 5.5 Science instruments The 10 beams exiting from the delay lines are compressed and then spectrally split using optimised dichroics Hobson & Baldwin [2004] between a number of interferometric beam combiners. A visible-light combiner and a near-infrared (JHK) combiner can be operated simultaneously, and space has been left for a “guest” instrument which might substitute for one or other of these combiners. These beam combiners will be optimised for operation on faint sources, but at low light levels the signal-to-noise ratio of the fringes decreases as the number of beams which are combined simultaneously is increased, because the photons from all the telescopes contribute to the noise on all the baselines. Therefore the MROI science instrument concepts are based on a number of parallel combiners, each of which combine a different subset of the beams. A beam “switchyard” then allows the beams to be “shuffled” between combiners to allow all pairs of telescopes to be interfered with one another. One possibility would be to have two combiners accepting 4 beams each. This would produce fringes on 12 baselines and 6 independent closure phases. A series of 4 reconfigurations of the switchyard would allow access to all 45 baselines and 24 of the 36 independent closure phases possible from the array. 5.6 Fringe tracking Fringe tracking is required to overcome effects of the random path-length perturbations introduced by the atmosphere and the instrument. Fringe “cophasing” attempts to compensate for the motion of the fringes at the sub-wavelength level while “coherencing” is coarser and attempts to reduce the fringe motion to less than the coherence length of the light, which can be many microns. Fringe tracking in the MROI will mostly utilise the group delay method which is based on observing the phase differences between the fringes in multiple spectral bands. This allows fringe coherencing on sources about 10 times fainter than is possible with cophasing methods Buscher [1988a] and so is the best way of achieving the MROI faint-science goals. The MROI has been designed with a separate fringe-tracking combiner rather than using the science instrument to derive a fringe-tracking signal, because it allows each combiner to be optimised for a different role. 5.7 Alignment Misalignments of components of the interferometric beam train are a source of wavefront error in an interferometer which can lead to significant losses in light throughput and fringe visibility. Many of the components need to be realigned on at least a nightly basis to account for drifts which have occurred during the day. A significant fraction of the realignment in many interferometers requires manual intervention and this can limit the amount of realignment which can be done at the start of the night. The MROI was designed from the start to allow automated alignment of the majority of the optical train in order to minimise the time overhead and to increase the overall accuracy of the alignment. 5.8 Control software Automated operation of an interferometer is critical to its scientific productivity, as large numbers of subsystems must work together to produce the interference fringes, and the cadence between observations is necessarily short in order to allow for observations of calibrator stars as close in time to the target observations as possible. A second critical feature for interferometers is continuous recording and storing of engineering data, as the calibration of the fringe visibility can depend on many variables within the system, and some of these can only be determined after the fact. For commissioning and operations, near-real-time display of data from multiple subsystems on the same console is essential to debugging operation of the system. The control software concept is based around independently-developed sub-systems forming a distributed system communicating over Ethernet. The system as a whole is only soft-real-time as the hard-real-time elements are confined mostly within subsystems such as the fast tip/tilt system. Real-time communication between subsystems is needed only between the fringe-tracker and the delay lines and this can be provided by a dedicated communications link. 6 Error budget Given that increasing telescope size does not provide substantial gains in sensitivity, the overall efficiency of the system becomes paramount. In order to achieve the faint-object science goals for the MROI a two-fold strategy to achieve the maximum optical efficiency was adopted. The first component of the strategy was to focus on an optical design which was as simple as possible, so as to minimise the number of optical elements in the system. In some cases this meant sacrificing capabilities, for example a full polarimetric capability, which could have offered increased science performance on brighter targets. It should be noted that the polarisation fidelity of the interferometer (defined as the ability to faithfully measure the object morphology in the Stokes I component) was not sacrificed as this did not involve compromising the optical efficiency of the system Buscher et al. [2009]. This resulted in a beam train which has a low number of optical elements compared with many existing arrays: in the MROI design starlight experiences 13 reflections from the entrance of the telescope to the entrance of any of the beam combining instruments. This can be compared with the VLTI, where the light experiences 32 reflections between the equivalent locations in the beam train Puech & Gitton [2005]. The second component of the strategy was to make sure that each surface introduced the minimum optical loss consistent with an affordable budget and minimal technical risk. This can be achieved in principle by only using components with the best possible coatings and which are manufactured to the tightest possible wavefront tolerances. However, the components tend to cost exponentially more the tighter the tolerances are set, and achieving these tolerances is easier for some components than for others. From this emerges the concept of an optical “error budget”, in which global values for the allowable losses are defined and these budgets are then shared between subsystems in a way that excessive requirements are not placed on any single subsystem. In interferometric instruments, both the losses to the number of photons and losses that reduce the fringe visibility are important, and so both must be considered as part of the optical error budget. The majority of the visibility loss is caused by wavefront errors, including both temporal “piston jitter” errors caused by vibrations in the system and spatial wavefront errors such as tilt and focus errors. As a simplifying assumption, the fringe visibility loss $\gamma$ is assumed to vary as $$\gamma=\exp(-\sigma_{\rm diff}^{2}/2)$$ (1) where $\sigma_{\rm diff}$ is the spatial or temporal RMS difference of phase between the interfering wavefronts. This assumption is related to the “extended Maréchal approximation” used in aberrated optical systems and empirically found to be a reasonable approximation for $\sigma\stackrel{{\scriptstyle<}}{{{}_{\sim}}}2$ Mahajan [1983]; Hardy [1998]. If each beam train in the interferometer introduces uncorrelated phase errors with RMS value $\sigma$ then $$\gamma=\exp(-\sigma^{2}).$$ (2) If in addition each component labelled $i$ in the beam train introduces a spatial or temporal wavefront error $\sigma_{i}$ which is uncorrelated between components then the total error is the root summed squared (RSS) of the individual errors $$\sigma^{2}=\Sigma_{i}\sigma_{i}^{2}.$$ (3) As a result, the visibility loss contributed by different components can be combined multiplicatively in the same way that losses in photon throughput can be combined. Figure 3 shows an outline of the visibility loss budget for the MRO interferometer. This is shown in a “tree” form showing how the budget is split amongst various components. The loss is shown for a wavelength of 1.6 $\mu$m which is the wavelength of the highest-sensitivity fringe-tracking for MROI. The visibility losses in the error budget can be broken down into factors due to atmospheric seeing and those due to the instrument. At the reference wavelength, the MROI telescopes have a diameter $D/r_{0}\approx 2.4$ in the reference seeing conditions ($r_{0}=14$cm at a wavelength of 500 nm) and so with perfect tip/tilt correction the RMS visibility loss factor due to the uncorrected higher-order atmospheric aberrations is about 0.6 Buscher [1988b]. Residual tilts caused by the imperfect performance of the tip-tilt correction system are budgeted to allow an additional visibility loss factor of 0.9. Assuming an exposure time of 2$t_{0}$ then the visibility loss due to atmospheric temporal fluctuations is about 0.8 Buscher [1988b]. Spatial and temporal wavefront errors introduced by the interferometer optics are budgeted to each introduce a reduction in the fringe visibility by a factor 0.8, values comparable to the atmospheric losses and corresponding to $\lambda/14$ RMS wavefront errors in each case. The telescope optics are the largest elements in the optical train and the most prone to vibration, so half of the RSS wavefront error budget (corresponding to a 10% visibility loss) is allocated to the telescopes and the remaining half to the rest of the interferometer subsystems. The remaining optical train has smaller optics, but a larger number of components. Each optical element will introduce a wavefront error due to imperfections in its manufacture, but in addition the curved optical components can introduce wavefront errors due to alignment errors such as defocus. Assigning the wavefront errors equally to manufacturing errors in each element and the alignment error leads to the spatial wavefront error budget being shared between more than 20 contributors. As a result, each contributes on average only 0.5% or so to the final visibility loss. This level of loss corresponds to an RMS wavefront error of $\lambda/90$ at the fringe-tracker wavelength of 1600 nm, corresponding to $\lambda/35$ RMS at the 633 nm HeNe laser wavelength at which optics are normally tested. Small optics can be routinely polished to give a wavefront quality of $\lambda/10$ peak-to-valley and the RMS wavefront error will typically be a factor of 5 or so smaller than the peak-to-valley [Porro et al., 1999], so a $\lambda_{\rm HeNe}/10$ mirror will contribute a visibility loss well within the typical tolerances assigned. Thus it can be seen that by using an error budget, the wavefront error requirements can be shared out such that no single component has unfeasible requirements placed on it. An error budget for both photon loss and visibility loss was developed for the entire system with a starting goal of an overall visibility loss factor of $\gamma=0.276$ and a photon throughput from the top of the atmosphere to detected photoelectrons of 20%. Adjustments to the error budget based on more detailed designs of the subsystems systems has resulted in a predicted system visibility loss factor of $\gamma=0.30$ and a photon throughput of 13%. 7 Limiting magnitude The science requirements lead to a specification that the interferometer should allow fringe tracking on an object with an H magnitude of 14 which is unresolved on the nearest-neighbour “bootstrapping” baselines. With a 13% throughput and assuming a pairwise, nearest-neighbour fringe-tracking beam combiner, 1540 photons/second will be detected over the H-band from such an object in each fringe pattern, and so in a 35 ms exposure (corresponding to 2$t_{0}$ in the reference seeing conditions of $t_{0}=4.4$ ms at a wavelength of 500 nm), approximately $N=54$ photons from the target will be detected. With an RMS system visibility of $\gamma=0.30$, the fringe signal level (sometimes called the coherent flux) will therefore be $$S=\tfrac{1}{2}\gamma N=8.1\,{\rm photons}.$$ (4) There will be an additional 8.2 photons detected per frame of sky and thermal background photons giving a photon noise level of $$\sigma_{\rm phot}=\sqrt{62.2}=7.9\,{\rm photons}$$ (5) Assuming that the fringe tracker uses 4-bin “ABCD” sampling and 5 spectral channels across the H-band, then the noise due to detector read noise is $$\sigma_{\rm read}=\sigma_{\rm pix}\sqrt{20}=8.4\,{\rm photons}$$ (6) where $\sigma_{\rm pix}$ is the noise on the readout of a single pixel, assumed to be 2 electrons: near-infrared detectors with single-read noise values of this order are now available. Thus the signal-to-noise ratio for a fringe measurement in a single exposure will be $${\rm SNR}=\frac{S}{\sqrt{\sigma_{\rm phot}^{2}+\sigma_{\rm read}^{2}}}=0.70$$ (7) Simulations of fringe tracking with this number of spectral channels Buscher [1988a] show that group-delay tracking is possible when the signal-to-noise ratio per exposure is as low as 0.56, so the MROI science goal of fringe-tracking on a H=14 AGN appears to be achievable with this design. Improved margins could be obtained if detector read noise performance better than 2 electrons can be achieved using multiple non-destructive reads, as has already been demonstrated in the laboratory [Finger et al., 2012]. 8 Experience from implementing the design Because the MROI is not currently operational it is not yet possible to provide definitive evidence that by implementing the conceptual design all the scientific performance goals will definitely be met. Instead we can look at what the experience of designing (and in many cases building and testing) the subsystems on the basis of the conceptual design tells us about how well the conceptual design stands up to the reality of implementation. Here we discuss this experience for selected subsystems. The aim is not to describe the designs of these subsystems in detail, as this has been (and will be) discussed elsewhere, but rather to comment on where the design was unusual and where there were technical roadblocks due to unrealistic constraints imposed by the conceptual design. 8.1 Unit Telescopes The unit telescopes for the MROI are 1.4 m in diameter and use a relatively unusual mount design in order to improve light throughput to the interferometer. This “alt-alt” design as shown in Figure 4 allows the starlight to be captured and converted into a narrow collimated beam travelling in a fixed horizontal direction using only 3 reflections. This therefore has lower reflection losses than conventional “alt-az” mount which typically requires 7 reflections to achieve the same result. In addition, the telescope has been specified with a high wavefront quality in order to reduce fringe visibility losses due to the optics. Wavefront quality for an interferometric telescope includes also the “piston term” errors which can be induced by telescope vibrations and also pupil wander which can cause the beams from different telescopes not to overlap. The telescope mounts have been built and tested in the factory by AMOS (Advanced Mechanical and Optical Systems) and have passed all major performance tests. The OPD vibrations are less than 40 nm RMS over a 35 ms exposure and the pupil motion is less than 350 $\mu$m in radius. The telescopes are designed to be relocatable between different stations along the array. The concept adopted is to relocate the telescope and its enclosure together, using a transporter based on those used for stacking shipping containers. Despite the relatively small size of the telescopes, attaining the shortest telescope spacings in a close-packed configuration proved difficult. The enclosures were designed to allow only the minimum space for support electronics, servicing and relocation, and the minimum spacing achieved was 7.8 m, only just short enough to overlap with the baselines offered by 8-meter-class telescopes. 8.2 Fast Tip-Tilt Systems The location of the fast tip-tilt (FTT) systems in the MROI beam-train is perhaps unusual; we have opted to place the sensors on the Unit Telescope Nasmyth platforms (with corrections applied by the UT secondary mirrors). Compared with locating the sensors in the beam combining building, this increases the photon flux available for sensing atmospheric tip-tilt perturbations and thus enhances the limiting sensitivity of the fast tip-tilt systems. This sensitivity limit would otherwise be the limiting factor for observations of red objects such as AGN and YSOs. With the sensor at the telescope, we expect to achieve a residual two-axis tip-tilt error of 60.8 mas RMS on the sky at $m_{V}=16$ in the reference $0.7^{\prime\prime}$ seeing, by taking advantage of the high quantum efficiency and sub-electron read-noise offered by electron-multiplying CCD detectors. However the disadvantage to this location is the sensitivity to non-common-path errors, whether movements of the FTT optics which shift the image on the tip-tilt sensor or changes in the alignment of the beam relay mirrors which are not be seen by the tip-tilt sensor. The system has therefore been designed for high optomechanical stability ($0.015^{\prime\prime}$ on the sky for a $5\,^{\circ}$C change in ambient temperature, corresponding to a 0.5 $\mu$m image shift on the tip-tilt sensing CCD). The design of the FTT systems, including optomechanical considerations, is discussed in more detail by Young et al. [2012]. The level of intra-night stability we expect to achieve with the final design, based on tests of prototype components, is a factor 2–4 greater than originally budgeted. To accommodate this, as well as any instability in the tilts of the beam relay mirrors, we have designed a continuous alignment system to be installed in the beam combining building — this is described in section 8.7. 8.3 Beam Relay System The diameters of the beams for propagation from the telescopes to the BCA was chosen to be 95 mm based on beam propagation studies which included the effects of atmospheric turbulence Horton et al. [2001]. Inside the BCA the beam is compressed to approximately 13 mm. Due to the Y-shaped array design, only two reflections are needed between the telescopes and the delay lines, but the mirrors involved need to be outdoors with their centres 1.6 m above ground level. The concrete piers to hold these mirrors needed to be engineered to achieve the appropriate stability in the face of wind loads, and this proved technically challenging, resulting in bulky and costly piers. Although the mirror mounts will be inside vacuum enclosures, they and their piers will be subject to substantial diurnal temperature fluctuations and so the stability of the beam relay is being tested to see if they will be sufficiently stable during the night. 8.4 Delay Lines The desire to deliver high throughput and efficient operations were major drivers for the implementation of the MROI delay lines. The detailed design of these has been presented elsewhere [Fisher et al., 2010] and so here we summarise only a few of their novel features. The MROI design supports a delay range from 0 to 380 m, in vacuum, and is realised in a single stroke with only three reflections. As a result there is no need for any separate switchable “long” delays nor any longitudinal dispersion correctors. Implementing this functionality required a design which concentrated in making the delay scalable to hundreds of metres at an affordable cost. This involved innovations such as using the vacuum vessel as the surface on which the delay carriage runs, wireless control and inductive supply of power so that there are no cables to be dragged as the optical delay is adjusted. Figure 5 shows a schematic of the delay line, indicating that in addition to the normal control loop for the position of the catseye (controlling the “piston” component of the wavefront), the delay lines incorporate active control of the shear of the return beams, to allow the use of less straight and hence more affordable pipes for the vacuum vessel. Tests in Cambridge in a 25 m long evacuated test rig have demonstrated that the system can meet all the system’s top level performance requirements, including a delay jitter $<15$ nm RMS over 10 ms; $<41$ nm RMS over 35 ms, and $<55$ nm RMS over 50 ms ($\lambda/40$ RMS over $2t_{0}$). Full repositioning of the carriage over the whole OPD range can be executed in 5 min. The first delay line carriage has been delivered to the MROI site and site acceptance tests in a 100 m long delay line will take place later this year. 8.5 Fringe Tracking Beam Combiner The fringe tracking beam combiner that has been adopted for the MROI (Infrared COherencing Nearest Neighbour tracker: ICONN) is a so-called nearest-neighbour design in which the beam combiner optics only mix beams from pairs of telescopes that are closest to each other. This arrangement maximises the per-baseline signal-to-noise ratio. The design of ICONN [Jurgenson et al., 2008] has allowed for up to 10 input beams, from the three arms of the MROI, and straightforwardly manages the unusual situation of the telescope at the vertex of the array which, unlike all the other array elements, has three nearest-neighbours. Currently the performance of ICONN is being tested in a single-baseline demonstration in the laboratory using a PICNIC detector to sense the dispersed fringes. Initial results are promising, and show very high fringe visibilities and an excellent level of stability in a relatively uncontrolled environment. 8.6 Science Beam Combiner The eventual implementation will include two science beam combiners, using “visible” (0.6–1.0 $\mu$m) and near-infrared ($J$, $H$, $K$) light respectively. A decision was made to prioritise development of the near-IR science beam combiner over the visible-wavelength one, as this enables a larger fraction of the science mission and places less stringent demands on the performance of the rest of the interferometer. A range of design concepts for the near-IR science beam combiner were considered. These were described in Baron et al. [2006]. All the designs include a fast optical “switchyard” would comprise plane mirrors mounted on precision slides to enable the rapid selection of beams, such that different subsets from of the available beams are selected for entry into the combiner. This approach maximizes the instantaneous fringe signal-to-noise but incurs a small efficiency overhead associated with reconfiguring the switchyard every few minutes to measure all of the baselines. Initial tests of candidate slides suggest that it will be possible to use a look-up table to correct for errors in the mirror orientation following each reconfiguration. 8.7 Alignment The biggest sources of alignment error in the interferometer are due to drifts in the tilts of mirrors in the beam train, which leads to tilt errors in the propagating beam. Over long distances these tilt errors lead to errors in the transverse position of the beam (“beam shear”) and so control of these errors requires measuring and correcting both the tilt and the shear simultaneously. For the purposes of nightly alignment, the optical train from the telescope to the beam combiner can be conceptually split into three independent optical “sub-trains” whose optical axes must be aligned with one another, as shown in Figure 6. The optical axes of these sub-trains are defined by the inner rotation axis of the unit telescopes, by the line of motion of the delay line and by the optical axes of the relevant beam combiner respectively. These axes are connected by pairs of mirrors which can be adjusted to align the tilt and shear of the axes between sub-trains. The alignment of these sub-trains will be accomplished by sending out a reference “pilot beam” which is aligned with the delay line axis and which shines both out towards the telescopes and back towards the beam combiners. Tilt and shear sensors at the telescopes and the beam combiners will allow the pilot beam position and direction to be compared with the corresponding optical axes. Automated procedures will be used to adjust the appropriate relay mirror pairs for any misalignment [see Shtromberg et al., 2010]. Most of this activity will be automated, and so is expected to run in parallel on the beam trains for different telescopes, minimising the overhead for system alignment. For this strategy to be effective, most of the alignment system components will be located in thermally-stable environments so that drifts during and after the alignment process are kept to a minimum. At the MROI the beam combining area (BCA) is stabilised so that diurnal temperature fluctuations are at the 0.1${}^{\circ}$ C level and this zone will contain the optics for the pilot beam injection as well as all the beam compressors and the beam combination optics. The delay lines are less susceptible to thermal drifts and so are housed in a passively stabilised enclosure, the delay line area or DLA. Over the seasons the DLA will be allowed to drift in temperature by more than 10${}^{\circ}$ C but on a diurnal basis the temperature change is of the order 1${}^{\circ}$ C, and so intra-night drifts will be kept within an acceptable range. An issue which was recognised at a late stage in the design was that during the night (i. e. after the automated alignment) it will be likely that temperature-induced perturbations of the components of the beam trains mounted either on the telescope’s Nasmyth tables or the “exposed” beam relay mirrors will lead to slow ($>100$ s) drifts in alignment. To address this, there will be a secondary low-bandwidth alignment system located within the temperature-controlled BCA. This will correct slow changes in alignment by picking off light in the 940–1000 nm bandpass and monitoring the tilt and shear drifts in real-time. 8.8 Controls In implementing the MROI control system, we have had to solve one overriding issue – the need to integrate many diverse systems into an integrated whole that can be operated efficiently. The diversity arises for a variety of reasons, including intrinsic differences in the kinds of functionality provided by the sub-systems (such as capturing sensor readings, closing fast servo loops, and complex algorithms for e.g. alignment or fringe tracking), and implementation constraints (such as the need to interface specific hardware devices or run under certain operating systems). The problem of merging these systems is solved by using standardised interface software that is automatically generated from a simple high-level description of these systems. The generated interface code provides functionality for object construction and destruction, system configuration using data obtained from the central database, receiving commands, publishing monitor data, faults and alerts, and subscribing to monitor data published by other interferometer systems. The control system is capable of managing independent subsets of the interferometer array. A typical use of this feature would be to operate most of the unit telescopes as a single system, while simultaneously doing calibrations on two recently moved telescopes. For additional information see Farris et al. [2010]. 9 Conclusions Although the MROI is not yet operational, a number of conclusions can be drawn from the design of the interferometer. The first of these is the importance of imaging to the scientific productivity of interferometers and the technical implications that arise from this. It is clear from the science case for the MROI that there are a large number of science targets for which imaging would provide critical new insights. All these targets are complex, and so the number of degrees of freedom in any model used to describe the target is large. Model-independent imaging is the only way to constrain these degrees of freedom in a way which can constrain which models are appropriate and is robust to any degeneracies in the models. Given the scientific importance of imaging, we have argued that having a large number of telescopes is critical in meeting the imaging goals, not only because of the substantial increase in the speed of imaging, but because for many objects the problem of tracking atmospheric phase perturbations can only be tackled with a “bootstrapping” array, which necessarily requires many telescopes. The MROI with 10 telescopes will be able to image “resolved-core” objects with of order $5\times 5$ resels across the image; to make more detailed images of these objects will require correspondingly more telescopes — moving the telescopes around will not be able to achieve this effect. A second scientific focus of the MROI is that of sensitivity. Our analysis suggests that increasing the aperture size of the unit telescopes in the interferometer will have limited effect in increasing the faintness of the targets which can be observed, and that concentrating on designing an efficient beam train which is able to provide the required functionality may have greater benefits. Our analysis has assumed the use of natural-guide-star adaptive optics, and the availability of cheap and reliable laser-guide-star technology could change this picture. Even in this latter scenario the increase in sensitivity can only be realised with larger and so more costly telescopes and this needs to be tensioned against the smaller number of telescopes which can be deployed within a given budget. A final remark can be made about the process of the development of the conceptual design which is reflected in the structure of this paper. It is often argued that to design a new interferometer requires first the development a “killer” science case, and this then drives the technical development of the interferometer. The process described here is a more nuanced one which starts from an analysis of the scientific successes and shortcomings of existing arrays, followed by a judgement of which of the shortcomings can be overcome most fruitfully without requiring inordinate amounts of technical development. 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Improving the Transmittance of an Epsilon-Near-Zero based Wavefront Shaper G. Briere, B. Cluzel and O. Demichel olivier.demichel@u-bourgogne.fr Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303 CNRS-Université Bourgogne Franche-Comté, 21078 Dijon, France Abstract Although Epsilon-Near-Zero metamaterials (ENZ) offer many unconventional ways to play with light, the optical impedance mismatch with surroundings can limit the efficiency of future devices. We report here on the improvement of the transmittance of an Epsilon-Near-Zero (ENZ) wavefront shaper. We first address in this paper the way to enhance the transmittance of a plane wave through a layer of ENZ material thanks to a numerical optimization approach based on the Transfer Matrix Method. We then transpose the one dimensional approach to a two dimensional case where the emission of a dipole is shaped into a plane wave by an ENZ device with a design that optimizes the transmittance. As a result, we demonstrate a transmittance efficiency of 15 % that is 4 orders of magnitude higher than previous devices proposed in the literature for wavefront shaping applications. This work aims at paving the way for future efficient ENZ devices by offering new strategies to optimize the transmittance through ENZ materials. Metamaterials engheta2006metamaterials have attracted a growing interest over the last decade due to their unique ability to manage the behavior of electromagnetic waves by the mean of a careful control of their dielectric permittivity and magnetic permeability. Invisibility cloaks Pendry2006Science , perfect lenses Pendry2000PRL and perfect absorbers Cao2014ScRep are the most emblematic metamaterials applications. Among the huge diversity of metamaterials, Epsilon-Near-Zero (ENZ) materials consider the particular case where the dielectric constant reaches zero. These metamaterials have already demonstrated to behave unconventionally, with their ability to bend the light Luo2014ScRep ; Luo2012APL , to tunnel light through subwavelength channels Silveirinha2006PhysRevLett or to exalt optical nonlinearities Caspani2016PRL , promising a new paradigm for nonlinear photonics and nonlinear plasmonics. In ENZ metamaterials, the vanishing permittivity induces an effective wavelength of electromagnetic waves that becomes infinite. As a consequence, the phase velocity becomes infinite and the phase distribution becomes almost uniform within the material. The phase distribution of the wave at the end facet of an ENZ is thus constant and the wavefront of outcoming waves is entirely driven by the shape of this output facet. ENZ metamaterials can therefore act as perfect wavefront shapers Alu2007PhysRevB ; fang2013OpSp . Until now, practical implementations of ENZ-based wavefront shapers are still elusive for two main reasons. First, the elaboration of ENZ materials is still at its infancy and second, it is quite challenging to optimize the amount of transmitted light through an ENZ materials because of the large Fresnel coefficient of reflection at the ENZ/dielectric interface (r=$\frac{n_{1}-n_{2}}{n_{1}+n_{2}}\approx 1$). This latter issue has been poorly discussed in the literature despite its fundamental importance for future ENZ-based applications. Concerning the elaboration of ENZ metamaterials, two main approaches have been proposed that consist either working at the cut-off wavelength of a waveguide Edwards2008PhysRevLett or mixing materials with negative and positive permittivities to reach a vanishing mean permittivity Maas2013NarPhot . Since these methods provide an anisotropic permittivity they prevent the phase to be uniform and are not suitable for wavefront shaping applications. A recent alternative relies on the management of the plasma frequency Traviss2013APL at which the dielectric permittivity reaches zero. In solid-state photonics, recent contributions have shown the feasibility of ENZ materials at telecommunication frequencies by adjusting the free carrier concentration of transparent conducting oxides such as Indium Tin Oxide (ITO) or Aluminium Zinc Oxide (AZO) Naik201OptMatExpr ; Kinsey2015Optica ; Yoon2015ScRep , paving the way to practical implementations and to the validation of proposed ENZ concepts. In this work, we numerically address the second issue that concerns the transmission efficiency through an ENZ region in the context of ENZ-based wavefront shaping. We propose a practical implementation for shaping a dipolar emission into a plane wave with a transmission efficiency as high as 15 % that is four orders of magnitude higher than devices proposed in ref Alu2007PhysRevB . We anticipate that such improved performances could be generalized to other ENZ-based concepts and allow for the development of efficient ENZ devices. This work is divided in three parts. We first demonstrate the poor transmission of a dipolar emission through a single ENZ layer. Next, we address the transmittance of a plane wave through a layer of ENZ material with the goal to find a strategy that improves the transmittance in such a simple situation. Finally, we transfer this strategy to the dipolar case for which we also consider the shape of the input facet of the ENZ device. In the following, we employ a Finite Element Method (FEM) based on the COMSOL software for the numerical investigation of a dipolar emission in the neighbourhood of an ENZ. We apply perfectly matched layers at the model boundaries to mimic an open model and the dipole orientation is parallel to the ENZ facet as depicted in Fig 1.a. To stay in line with wavefront shaping applications, the minimal thickness required for an ENZ material has to be comparable to the free space wavelength ($\lambda$) to allow for a $2\pi$ control of the phase shift on the end facet of the ENZ materials. The ENZ layer thickness is thus set to $\lambda$ which is chosen to be equal to $1\mu m$, even if the present work is scalable to any wavelength. The Fig.1.b shows the phase distribution of the E${}_{y}$ component of the field (component parallel to the dipole) for a permittivity of 10${}^{-3}$ (with no imaginary part). It can be seen that the output wavefront at the right of the ENZ is planar and no longer presents the characteristic cylindrical shape of a dipolar emission. From this phase distribution, it is obvious that the light radiated by the dipole is efficiently transformed into a plane wave after passing through the ENZ layer. We found that the value of $10^{-3}$ for the permittivity is a threshold below which materials behave as ENZ in terms of wavefront shaping. This gives an upper bound for the permittivity required for future experimental implementations. In the following, the dielectric constant is set to 10${}^{-4}$. Note that recent contributions on conductive oxide materials for ENZ applications show a real part of the dielectric constant that goes continuously from positive to negative values Naik201OptMatExpr ; Kinsey2015Optica ; Yoon2015ScRep . Then, it is realistic to expect values as small as 10${}^{-4}$ for an adequate wavelength. We now turn to the issue challenged by this manuscript that concerns the field transmission through ENZ metamaterials. For consistency with earlier reports on similar systems Alu2007PhysRevB ; fang2013OpSp , we consider here the magnetic field component Hz of the radiated field. Figure1.c shows its amplitude distribution in a logarithmic scale, emphasizing the poor transmittance through the ENZ layer. Indeed, the amplitude of the transmitted field is decreased by at least two orders of magnitude and the energy flow transmission through the ENZ layer has an efficiency as low as 2.10${}^{-5}$. Such a low efficiency prevents any further applications, and the goal of this paper is to improve this transmittance. For this purpose, we start with a one dimensional approach where a plane wave impinges a planar thin film at normal incidence. Using a Transfer Matrix Method (TMM), we compute the transmittance of a plane wave onto a layer of ENZ material immersed in the vacuum. Before going further, we want to precise that in order to validate the accuracy of both FEM and TMM computation methods, all results showed in this part have been double checked with the FEM and no significant discrepancy was found between them. We found that the transmittance decreases as soon as the ENZ thickness increases (not shown here) and for a thickness of $\lambda$ only $9\%$ of the field intensity is transmitted. This poor transmittance results from a strong optical impedance mismatch between the ENZ and the vacuum. In order to reduce this impedance mismatch, we introduce an adaptation layer in front of the ENZ layer as depicted in Fig.2.a. We then resort to an optimization algorithm coupled to the TMM calculations in order to search for the permittivity profile of the adaptation layer that maximizes the transmittance of the system {Adaptation Layer+ENZ Layer}. For this purpose, the adaptation layer is subdivided in $100$ sub-layers being 10 nm thick and with a random dielectric constant varying from $1$ to $10^{-4}$. The optimization algorithm computes the transmittance and its local gradients in the parameter space (the permittivity of all sub-layers) for a given permittivity profile. As a result and surprisingly, a transmittance of $100\%$ is obtained after few hundreds of iterations. The robustness of the result has been successfully tested with the FEM method and the optimal permittivity profile remains always the same whatever the initial conditions. This profile is reported in Fig.2.b. and appears surprisingly simple. It consists of two ENZ layers ($\epsilon_{r}=10^{-4}$) of the same thickness $\lambda$ separated by an air gap whose length is about $e_{gap}\approx\frac{\lambda}{20}$. Spectral and angular transmittance of this ENZ device have been computed by the TMM and are plotted in red in Fig.2.c and d respectively. They both exhibit a peaked shape centered on the initial numerical conditions ($\lambda$=1 $\mu$m and normal incidence). The full width at half maximum (FWHM) are respectively of 168 nm and 22 °. In order to investigate the behaviour of such devices with real and lossy ENZ materials, the transmission of such a structure has been also computed for complex permittivities with Im($\epsilon_{r}$)=10${}^{-3}$, 5.10${}^{-3}$ and 10${}^{-2}$. Fig.2.c shows the transmission spectra with these values. The maximal transmission stays at 1$\mu$m but falls down when materials are more lossy. However the transmission stays as high as 27% for an imaginary part of 10${}^{-2}$ (3 times higher than for a single slab of perfect ENZ). Nevertheless, values reported for conducting oxides Naik201OptMatExpr ; Kinsey2015Optica ; Yoon2015ScRep are one order of magnitude higher (in the 10${}^{-1}$ – 1 range) which correspond to transmissions below 1%, indicating that efforts have to be pursued to make such materials available for wavefront shaping applications. We now focus on the case of perfect ENZ with a real permittivity. Although a 100% transmission is surprising, the whole system mimics a Fabry Perot interferometer made of two mirrors of ENZ layers with a complex reflectivity $r_{ENZ}$, separated by a gap of air with length $e_{gap}$. The condition of resonance of this interferometer is reached when the phase of the wave after one round trip in the cavity ($\Delta\phi$) is a multiple of $2\pi$. Since $r_{ENZ}$ is complex, a phase ($\phi_{r}$) is introduced at the reflection on each ENZ mirror. $\Delta\phi$ is then related to both the phase due to wave propagation inside the air gap and the phase due to reflections. Resonances are defined by the relation : $$\Delta\phi=\frac{4\pi}{\lambda}e_{gap}-2\phi_{r}=2m\pi$$ (1) To assess this interpretation, we computed the complex value of $r_{ENZ}$ as a function of the ENZ layer thickness. The figure 3.a shows the evolution of the related phase $\phi_{r}$ of $r_{ENZ}$ as a function of the ENZ layer thickness ($d$, in unit of $\lambda$). From this figure, we found $\phi_{r}\approx 0.3$ rad when $d=\lambda$. The resonance condition defined by eq.1 can be written as : $$e_{gap}=\frac{\phi_{r}}{2\pi}\lambda+m\lambda/2$$ (2) For m=0, we find $e_{gap}\approx\lambda/20$ in perfect agreement with the value obtained earlier (fig.2.b), validating the Fabry Perot description of the structure that optimizes the transmission through an ENZ layer. In figure 3.b, we plotted the transmittance of a Fabry Perot interferometer given by the standard Airy function of eq.3 as a function of the gap thickness for two different sets of mirrors: $$T=\frac{1}{1+\frac{4|r_{ENZ}|^{2}}{(1-|r_{ENZ}|^{2})^{2}}\sin^{2}\left(\frac{% \Delta\phi}{2}\right)}$$ (3) The dark curve corresponds to a mirror with a reflectivity —$r_{ENZ}$— with no phase shift at the reflection while the red one describes the ENZ situation taking into account $\phi_{r}$. As shown in Fig.3.b, Fabry Perot resonances are shifted as soon as the reflection phase is taken into account. The $m=0$ mode now corresponds to a cavity with a finite size of $e_{gap}=\lambda/20$ as discussed above. The FWHM of resonances is of 10 nm indicating the precision required for the realisation of future devices. The blue curve corresponds to the more realistic case where the ENZ has a complex dielectric constant with an imaginary part of 10${}^{-2}$. Here again, losses decrease the transmitted efficiency but not the Fabry Perot behaviour of the ENZ cavity. Interestingly, we notice that the index profile proposed here can also be seen as a strongly subwavelength cavity made of ENZ mirrors. The length of the cavity is entirely dependant on $\phi_{r}$ and therefore on the ENZ thickness. This feature could be used for building innovative subwavelength resonators. We now move forward with the optimization of the transmission of the dipolar emission through an ENZ waveshaper. We start with a direct transfer of the optimal permittivity profile for a plane wave to the dipolar case as depicted in figure 4.a. The phase distribution of the $E_{y}$ component is showed in figure 4.b. Here again, wavefronts at the right part of the ENZ device is efficiently shaped into a planar profile, confirming the device still enables wavefront shaping. Figure 4.c gives the distribution of the magnetic field amplitude with the same logarithmic colorscale as figure.1.c. It is noteworthy that the multilayer approach greatly improves the overall transmittance. To get a more quantitative insight into this improvement, we define the transmission efficiency as the ratio between the Poynting vector flow through the frontier near the end facet of the ENZ (on the right) and the total flow through all the frontiers of the model. According to this definition, we find that the optimized ENZ multilayer efficiency is close to $10^{-2}$ in comparison to the efficiency of the single ENZ layer shown in Fig.1 evaluated at 2$\times 10^{-5}$. As a result the sole optimization of the transmittance of a plane wave enables us a three order of magnitude enhancement of the ENZ transmittance. Although the overall efficiency have been strongly improved, it still remains as low as 1%. For further improvement, it is necessary to understand the origin of such a low efficiency. The strategy of introducing a multilayer structure allowed to transmit 100% of the waves that reach the structure with a normal incidence. Furthermore, figure 2.d shows a decreasing transmittance as soon as the incidence angle moves away from normal with an overall acceptance angle close to 22 degrees. It implies that most of the field radiated by the dipole is reflected by the ENZ device. To overcome this issue, we propose here to optimize the shape of the ENZ input facet in order to match the emission shape of the dipole. As shown in figure.5.a, we set the dipole near a multilayer ENZ with an air gap. The shape of the input facet is modified to form an ellipse. The ellipse semi-axis lengths, the dipole position and the air gap length have been optimized by coupling our optimization algorithm to the FEM method. Starting from $20$ sets of free parameters randomly chosen, the algorithm converges to a single solution after about $200$ iterations. The optimal parameters are $1.67\times\lambda$ (resp. 2.4 $\times\lambda$) for the horizontal (resp. vertical) ellipse semi-axis, $x_{dipole}=1.6\times\lambda$ for the dipole position, and $d\approx\lambda/25$ for the gap thickness. For this ENZ device, figure5.b shows the phase profile of the E${}_{y}$ component of the field that is adequately shaped into a plane wave. This confirms here again that the shape of output waves are governed by the output facet of the ENZ devices whatever the input profile. Figure 5.c shows the distribution of the magnetic field amplitude which is now of the same order of magnitude at both right and left frontiers and indicates an efficient transmission through the ENZ device. This has been quantitatively confirmed by computing the transmittance efficiency which is now of $1.5\times 10^{-1}$. This design provides a four orders of magnitude improvement of the intensity of the dipolar emission that is shaped through this optimized bidimensionnal ENZ multilayer compared to the single ENZ layer shown in Fig.1. It is noteworthy that the way this transmission efficiency has been defined introduces an upper limit of 50% because only half part of the dipole emission is directed towards the ENZ device and can be shaped. Although hard to quantify, we found that slight changes in the design geometry do not affect significantly the transmission, and deviations of 10 nm in the elliptical input facet only decrease the transmission of few percent while the precision of the gap thickness should be controlled with an precision close to 5 nm. Finally, when introducing a complex permittivity (Im($\epsilon_{r}$)=10${}^{-2}$), with such a design (data not shown here), the transmission efficiency is of 1-2% that is still 3 orders of magnitude higher than the initial situation. In conclusion, based on FEM numerical investigations, we determined an upper limit of $10^{-3}$ for the permittivity for wavefront shaping applications. Then, we have showed that the transmittance efficiency through an ENZ device that transforms the field radiated by a dipole into a plane wave can be strongly enhanced. We have addressed this issue in two steps: a first one by considering the optimization of the transmittance of a plane wave through an ENZ multilayer, and a second one by shaping the input facet of the ENZ device. A numerical efficiency up to 15% has been obtained which is four orders of magnitude higher than the non-optimized case. The results reported here may allow for practical implementations at optical frequencies with realistic ENZ materials such as ITO or AZO as soon as the losses of such materials can be decreased below 10${}^{-2}$, which however still requires some efforts. It also provides some general rules to improve the transmittance efficiency of electromagnetic waves through ENZ multilayer and as such could enable the future development of ENZ devices with unexpected electromagnetic properties. I Funding Information This work has been performed in the framework of the Labex Action ANR-11-LABEX-0001-01 References (1) N. Engheta and R. W. 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Differential invariants for a class of diffusion equations Elsa Dos Santos Cardoso-Bihlo${}^{\dagger}$, Alexander Bihlo${}^{\dagger}$ and Roman O. Popovych${}^{\ddagger}$ ${}^{{\dagger}}$ Department of Mathematics and Statistics, Memorial University of Newfoundland, $\phantom{{}^{\ddagger}}$St. John’s (NL) A1C 5S7, Canada ${}^{{\ddagger}}$ Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria $\phantom{{}^{\ddagger}}$ Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivska Str., 01601 Kyiv, Ukraine E-mail: ecardosobihlo@mun.ca, abihlo@mun.ca, rop@imath.kiev.ua We find the complete equivalence group of a class of (1+1)-dimensional second-order evolution equations, which is infinite-dimensional. The equivariant moving frame methodology is invoked to construct, in the regular case of the normalization procedure, a moving frame for a group related to the equivalence group in the context of equivalence transformations among equations of the class under consideration. Using the moving frame constructed, we describe the algebra of differential invariants of the former group by obtaining a minimum generating set of differential invariants and a complete set of independent operators of invariant differentiation. 1 Introduction Invariants and differential invariants of transformation groups, in particular, point symmetry groups admitted by systems of differential equations have a wide range of applications and are therefore an intensively investigated subject. Differential invariants play a central role in the invariant parameterization problem [1, 2, 30] and in the problem of invariant discretization [3, 5, 7]. They are also used to construct invariant differential equations and invariant variational problems [22, 23], as well as in computer vision, integrable systems, classical invariant theory and the calculus of variations [6, 22, 24]. Rather recently, finding differential invariants in problems related to group classification became a research topic of interest. The idea is to compute the differential invariants not for the point symmetry group of a single system of differential equations but for the equivalence group admitted by a class of such systems. The primary motivation for such a survey is to study the equivalence of systems of differential equations. Exploring equivalence, it is possible to explicitly determine point transformations among systems from a class [28]. Such a mapping between two systems of differential equations is especially helpful if for one of the systems involved wide sets of exact solutions are known. These solutions then can be mapped to solutions of the equivalent system. Another case of particular interest is the mapping between nonlinear and linear elements of a class of systems of differential equations [19]. For the solution of the equivalence problem, finding differential invariants for the equivalence group is a main ingredient. There are a number of papers where some low-order differential invariants of the equivalence groups of various physically relevant classes of systems of differential equations were computed using the Lie infinitesimal method; see, e.g., [11, 12, 13, 14, 15, 17, 32, 33, 34, 35] and references therein. In the present paper we will be concerned with differential invariants for a group111In fact, this object and the “equivalence group” of the class (1) are Lie pseudogroups of locally defined point transformations. We use the term “group” for brevity since this does not lead to any confusion. related to the equivalence group of the class of diffusion equations $$u_{t}=u_{xx}+f(u,u_{x}).$$ (1) in the context of equivalence transformations among equations of this class. This subject was originally considered in [32], using the infinitesimal method and restricting the order of differential invariants up to two. We revisit the construction of differential invariants for the class (1) from the very beginning, analyzing differential invariants of which group should be found. Then, we apply the method of equivariant moving frames in the formulation originally proposed and formulated by Fels and Olver [9, 10], which was later generalized to infinite-dimensional Lie (pseudo)groups in [6, 25, 26], and this is the setting that is needed to study differential invariants for the class (1). The advantage of moving frames is that they allow for a canonical process of invariantization, which associates to each object, such as functions, differential functions, differential forms and total differentiation operators, its invariant counterpart. For the problem of finding differential invariants of a Lie transformation (pseudo)group, this property is especially convenient. The invariantization of the jet-space coordinate functions yields the so-called normalized differential invariants. The invariantized coordinate functions whose transformed counterparts were involved in the construction of the corresponding moving frame via the normalization procedure are equal to the respective constants chosen in the course of normalization. This is why these objects are called phantom normalized differential invariants. The non-phantom normalized differential invariants constitute a complete set of functionally independent differential invariants. As a further asset, the method of moving frames also permits to study the algebra of differential invariants by deriving relations, called syzygies, between invariant derivatives of non-phantom normalized differential invariants. Finding syzygies can aid in the establishment of a minimum generating set of differential invariants. See e.g. [6, 8, 22, 25, 26] for more details and an extensive discussion on the computation of differential invariants for both finite-dimensional Lie symmetry groups and for infinite-dimensional Lie (pseudo)groups using moving frames. The further organization of this paper is as follows. In Section 2 we compute the equivalence group and the equivalence algebra of the class (1). Section 3 is devoted to the selection of a group to be considered and a preliminary analysis of equivariant moving frames associated with this group. The structure of the algebra of differential invariants is determined in the main Section 4. This includes a description of a minimum generating set of differential invariants and a complete set of independent operators of invariant differentiation, which serve to exhaustively describe the set of differential invariants. Moreover, for each $k\in\mathbb{N}_{0}$ we explicitly present a functional basis of differential invariants of order not greater than $k$. 2 The equivalence group The auxiliary system for the class (1), which is satisfied by the arbitrary element $f$, is $f_{t}=f_{x}=f_{u_{t}}=f_{u_{tt}}=f_{u_{tx}}=f_{u_{xx}}=0$. By definition [27, 28, 29, 31], the (usual) equivalence group $G^{\sim}$ of the class (1) consists of the point transformations in the space with coordinates $(t,x,u,u_{t},u_{x},u_{tt},u_{tx},u_{xx},f)$ that have the following properties: • they are projectable to the space with the coordinates $(t,x,u)$, • their components for derivatives of $u$ are found by prolongation using the chain rule, and • they map every equation from the class (1) to an equation from the same class. To begin finding the group $G^{\sim}$, we fix an arbitrary equation of the class (1), $u_{t}=u_{xx}+f(u,u_{x})$, and aim to find point transformations in the space with coordinates $(t,x,u)$, $$\tilde{t}=T(t,x,u),\quad\tilde{x}=X(t,x,u),\quad\tilde{u}=U(t,x,u),$$ (2) that transform the fixed equation to an equation of the same class, $$\tilde{u}_{\tilde{t}}=\tilde{u}_{\tilde{x}\tilde{x}}+\tilde{f}(\tilde{u},% \tilde{u}_{\tilde{x}}).$$ (3) A preliminary simplification is obtained from noting that the class (1) is a subclass of the class of second-order (1+1)-dimensional semi-linear evolution equations. Any point transformation between two equations from the latter class satisfies the constraints $T_{x}=T_{u}=X_{u}=0$, i.e., $\tilde{t}=T(t)$, $\tilde{x}=X(t,x)$, and $T_{t}X_{x}U_{u}\neq 0$. See [16, 18, 21] for further details. After taking into account the above constraints, the required transformed derivatives read $$\tilde{u}_{\tilde{t}}=\frac{1}{T_{t}}\left(\mathrm{D}_{t}U-\frac{X_{t}}{X_{x}}% \mathrm{D}_{x}U\right),\quad\tilde{u}_{\tilde{x}}=\frac{1}{X_{x}}\mathrm{D}_{x% }U,\quad\tilde{u}_{\tilde{x}\tilde{x}}=\left(\frac{1}{X_{x}}\mathrm{D}_{x}% \right)^{2}U,$$ where $\mathrm{D}_{t}$ and $\mathrm{D}_{x}$ are the usual total derivative operators with respect to $t$ and $x$, respectively. Substituting these expressions and $u_{t}=u_{xx}+f$ into Eq. (3), we split the resulting equation with respect to $u_{xx}$ yielding $T_{t}=X_{x}^{\,2}$. The remaining equation is $$\displaystyle f=\frac{1}{U_{u}}\left(T_{t}\tilde{f}-U_{t}+\frac{X_{t}}{X_{x}}(% U_{x}+U_{u}u_{x})+U_{xx}+2U_{xu}u_{x}+U_{uu}u_{x}^{2}\right).$$ (4) The differential consequences of Eq. (4) that are obtained by separate differentiations with respect to $t$ and $x$ can be split with respect to derivatives of $\tilde{f}$ since they are regarded as independent for equivalence transformations. This yields the equations $T_{tt}=X_{xt}=X_{tt}=U_{t}=U_{x}=0$. The equation (4) itself gives the $f$-component of equivalence transformations. The arbitrary element $f$ in fact depends only on $u$ and $u_{x}$. The space with coordinates $(t,x,u,u_{x},f)$ is preserved by all elements of $G^{\sim}$. This is why we can assume this space as the underlying space for $G^{\sim}$ and present merely the transformation components for its coordinates. As a result, we have proved the following theorem. Theorem 1. The equivalence group $G^{\sim}$ of the class (1) is constituted by the transformations $$\displaystyle\begin{split}&\displaystyle\tilde{t}=C_{1}^{2}t+C_{0},\quad\tilde% {x}=C_{1}x+C_{1}C_{2}t+C_{3},\quad\tilde{u}=\varphi(u),\quad\tilde{u}_{\tilde{% x}}=\frac{\varphi^{\prime}}{C_{1}}u_{x},\\ &\displaystyle\tilde{f}=\frac{1}{C_{1}^{2}}\left(\varphi^{\prime}f-C_{2}% \varphi^{\prime}u_{x}-\varphi^{\prime\prime}u_{x}^{2}\right),\end{split}$$ (5) where $C_{0},C_{1},C_{2},C_{3}\in\mathbb{R}$, $\varphi$ is an arbitrary smooth function of $u$ and $C_{1}\varphi^{\prime}\neq 0$. The infinitesimal generators of one-parameter subgroups of $G^{\sim}$, which constitute the equivalence algebra $\mathfrak{g}^{\sim}$ of the class (1), can be derived from (5) by differentiation, cf. the proof of Corollary 11 in [20] or the proof of Corollary 6 in [4]. These generators coincide with those determined in [32]. As we will later need them for the description of the algebra of differential invariants of a group related to $G^{\sim}$ in the context of the $G^{\sim}$-equivalence among equations of the class (1), we present them here. The general element of $\mathfrak{g}^{\sim}$ is $$Q=\tau\partial_{t}+\xi\partial_{x}+\phi\partial_{u}+\eta\partial_{u_{x}}+% \theta\partial_{f},$$ where the components are of the form $$\displaystyle\tau=2c_{1}t+c_{0},\quad\xi=c_{1}x+c_{2}t+c_{3},\quad\phi=\phi(u),$$ $$\displaystyle\eta=(\phi^{\prime}-c_{1})u_{x},\quad\theta=(\phi^{\prime}-2c_{1}% )f-c_{2}u_{x}-\phi^{\prime\prime}u_{x}^{2},$$ in which $c_{0}$, $c_{1}$, $c_{2}$ and $c_{3}$ are arbitrary real constants, and $\phi$ is an arbitrary smooth function of $u$. In other words, the equivalence algebra $\mathfrak{g}^{\sim}$ of the class (1) is spanned by the vector fields $$\displaystyle\partial_{t},\quad 2t\partial_{t}+x\partial_{x}-u_{x}\partial_{u_% {x}}-2f\partial_{f},\quad t\partial_{x}-u_{x}\partial_{f},\quad\phi\partial_{u% }+\phi^{\prime}u_{x}\partial_{u_{x}}+(\phi^{\prime}f-\phi^{\prime\prime}u_{x}^% {2})\partial_{f},$$ where $\phi$ runs through the set of smooth functions of $u$. 3 Preliminary analysis of moving frames Let us first clarify the space of independent and dependent variables to be used and the group to be considered. While formally the arbitrary element $f$ is a smooth function on the second-order jet space with coordinates $(t,x,u,u_{t},u_{x},u_{tt},u_{tx},u_{xx})$, practically it explicitly depends only on $u$ and $u_{x}$. This is why subsequently we will only consider the projection of the equivalence transformations to the space with coordinates $(u,u_{x},f)$. As a shorthand, we denote $v:=u_{x}$ and $\tilde{v}:=\tilde{u}_{\tilde{x}}=V(u,v):=C_{1}^{-1}\varphi^{\prime}(u)v$. In other words, we will in fact study differential invariants of the projection $G_{1}$ of $G^{\sim}$ to the space with coordinates $(u,v,f)$, where $u$ and $v$ are the independent variables and $f$ is the dependent variable. The infinitesimal counterpart of $G_{1}$ is the projection $\mathfrak{g}_{1}$ of $\mathfrak{g}^{\sim}$ to the space with coordinates $(u,v,f)$. In order to describe the algebra of differential invariants of the group $G_{1}$, we now construct a moving frame for this group. Since it is infinite-dimensional, we have to use the machinery developed for Lie (pseudo)groups, see [6, 25] for an extensive description of this subject. The first step in the construction of the moving frame is the computation of the lifted horizontal coframe, the dual of which yields the implicit total differentiation operators $\mathrm{D}_{\tilde{u}}$ and $\mathrm{D}_{\tilde{v}}$. For the equivalence transformations (5), the lifted horizontal coframe is $$\displaystyle\mathrm{d}_{\rm h}\tilde{u}=(\mathrm{D}_{u}U)\,\mathrm{d}u+(% \mathrm{D}_{v}U)\,\mathrm{d}v=\varphi^{\prime}\,\mathrm{d}u,$$ $$\displaystyle\mathrm{d}_{\rm h}\tilde{v}=(\mathrm{D}_{u}V)\,\mathrm{d}u+(% \mathrm{D}_{v}V)\,\mathrm{d}v=\frac{\varphi^{\prime\prime}}{C_{1}}v\,\mathrm{d% }u+\frac{\varphi^{\prime}}{C_{1}}\,\mathrm{d}v.$$ Computing the dual, we derive that $$\mathrm{D}_{\tilde{u}}=\frac{1}{\varphi^{\prime}}\mathrm{D}_{u}-\frac{\varphi^% {\prime\prime}}{(\varphi^{\prime})^{2}}v\mathrm{D}_{v},\quad\mathrm{D}_{\tilde% {v}}=\frac{C_{1}}{\varphi^{\prime}}\mathrm{D}_{v}$$ (6) are the required implicit differentiation operators. Acting with them on the transformation component for $f$, we find that $$\tilde{f}_{ij}=\frac{\partial^{i+j}\tilde{f}}{\partial\tilde{u}^{i}\partial% \tilde{v}^{j}}=\mathrm{D}_{\tilde{u}}^{\,\,i}\mathrm{D}_{\tilde{v}}^{\,\,j}F,$$ where $i,j\in\mathbb{N}_{0}:=\mathbb{N}\cup\{0\}$ and $\tilde{f}_{00}=\tilde{f}=F:=C_{1}^{-2}(\varphi^{\prime}f-C_{2}\varphi^{\prime}% v-\varphi^{\prime\prime}v^{2})$ is the $f$-component of equivalence transformations. In particular, the derivatives up to order 2 are exhausted by $$\displaystyle\tilde{f}_{10}=$$ $$\displaystyle\frac{1}{C_{1}^{2}\varphi^{\prime}}\left(\varphi^{\prime}f_{u}+% \varphi^{\prime\prime}(f-vf_{v})-\varphi^{\prime\prime\prime}v^{2}+2\frac{(% \varphi^{\prime\prime})^{2}}{\varphi^{\prime}}v^{2}\right),$$ $$\displaystyle\tilde{f}_{01}=$$ $$\displaystyle\frac{1}{C_{1}\varphi^{\prime}}\left(\varphi^{\prime}f_{v}-C_{2}% \varphi^{\prime}-2\varphi^{\prime\prime}v\right),$$ $$\displaystyle\tilde{f}_{20}=$$ $$\displaystyle\frac{1}{C_{1}^{2}\varphi^{\prime}}\left(f_{uu}\!-\!\frac{\varphi% ^{\prime\prime}}{\varphi^{\prime}}(f_{u}\!-\!2vf_{uv})\!+\!\left(\frac{\varphi% ^{\prime\prime}}{\varphi^{\prime}}\right)^{2}\!\!v^{2}\!f_{vv}\!+\!\bigg{(}% \frac{\varphi^{\prime\prime}}{\varphi^{\prime}}\bigg{)}^{\prime}\!(f\!-\!vf_{v% })\!-\!(\varphi^{\prime})^{2}\!\left(\frac{1}{\varphi^{\prime}}\left(\frac{1}{% \varphi^{\prime}}\right)^{\prime\prime}\right)^{\prime}\!\!v^{2}\!\right)\!,$$ $$\displaystyle\tilde{f}_{11}=$$ $$\displaystyle\frac{1}{C_{1}\varphi^{\prime 2}}\left(\varphi^{\prime}f_{uv}-% \varphi^{\prime\prime}vf_{vv}-2\varphi^{\prime\prime\prime}v+4\frac{\varphi^{% \prime\prime 2}}{\varphi^{\prime}}v\right),$$ $$\displaystyle\tilde{f}_{02}=$$ $$\displaystyle\frac{1}{\varphi^{\prime 2}}(\varphi^{\prime}f_{vv}-2\varphi^{% \prime\prime}).$$ There are a relative invariant and a relative conditional invariant which play a significant role in the following consideration. By taking the difference $\tilde{f}_{00}-\tilde{v}\tilde{f}_{01}$ we exclude the inessential constant $C_{2}$, which only arises in $\tilde{f}_{00}$ and $\tilde{f}_{01}$, $$\tilde{f}_{00}-\tilde{v}\tilde{f}_{01}=\frac{1}{C_{1}^{2}}\left(\varphi^{% \prime}(f-vf_{v})+\varphi^{\prime\prime}v^{2}\right).$$ Combining further $2(\tilde{f}_{00}-\tilde{v}\tilde{f}_{01})+\tilde{v}^{2}\tilde{f}_{02}$ to exclude $\varphi^{\prime\prime}$, we obtain $$\tilde{W}=\frac{1}{C_{1}^{2}}W,\quad\mbox{where}\quad W=2f-2vf_{v}+v^{2}f_{vv}% ,\quad\tilde{W}=2\tilde{f}-2\tilde{v}\tilde{f}_{\tilde{v}}+\tilde{v}^{2}\tilde% {f}_{\tilde{v}\tilde{v}},$$ i.e., $W$ is a relative invariant of $G_{1}$. In other words, the condition $W=0$ is preserved by any equivalence transformation in the class (1). Analogously, the combination $2\tilde{f}_{10}-v\tilde{f}_{11}$ gives $$\displaystyle\tilde{S}=\frac{1}{C_{1}^{2}}S+\frac{1}{C_{1}^{2}}\frac{\varphi^{% \prime\prime}}{\varphi^{\prime}}W,\quad\mbox{where}\quad S=2f_{u}-vf_{uv},% \quad\tilde{S}=2\tilde{f}_{\tilde{u}}-\tilde{v}\tilde{f}_{\tilde{u}\tilde{v}}.$$ (7) This means that $S$ is a relative invariant of $G_{1}$ if the condition $W=0$ is satisfied. Values of the differential functions $W$ and $S$ determine which normalization conditions should be chosen. We next find appropriate normalization conditions, which form the basis for the construction of an equivariant moving frame. As $\varphi$ arises only in $U$, we can set $U$ to any value including zero. The value of $V$ can be set to any constant excluding zero, and all these possibilities are equivalent. We find it convenient to put $V=1$ and express $\varphi^{\prime}=C_{1}/v$. The constraint $W=0$ singles out the singular case for the moving frame construction, which has to be investigated separately. Within this singular case, there is the ultra-singular subcase associated with the constraint $S=0$. Indeed, under the constraint $W=0$ the equation (7) can be solved for $C_{1}$ if and only if $S\neq 0$. 4 Differential invariants for the regular case In this paper, we only consider the regular case for moving frames of $G_{1}$, where $W\neq 0$. In this case, the following normalization conditions can be used to determine a complete moving frame $$\begin{split}&\displaystyle\tilde{u}=0,\quad\tilde{v}=1,\quad\tilde{f}=1,\quad% \tilde{f}_{01}=0,\quad\tilde{f}_{02}=0,\\ &\displaystyle\tilde{f}_{i0}=-\frac{v^{2}\varphi^{(i+2)}}{C_{1}^{\,\,2}(% \varphi^{\prime})^{i}}+\frac{1}{C_{1}^{\,\,2}}\sum_{i^{\prime}=0}^{i}\binom{i}% {i^{\prime}}\frac{1}{(\varphi^{\prime})^{i^{\prime}}}\left(\frac{\varphi^{% \prime\prime}}{(\varphi^{\prime})^{2}}\right)^{i-i^{\prime}}f_{i^{\prime},i-i^% {\prime}}+\dots=0,\ i\in\mathbb{N}.\end{split}$$ (8) In the expression for $\tilde{f}_{i0}$, we presented only the summands with the highest-order derivatives of $\varphi$ and $f$, which are $\varphi^{(i+2)}$ and $f_{i^{\prime},i-i^{\prime}}$, $i^{\prime}=0,\dots,i$, respectively. We solve the first five equations with respect to $C_{1}$, $C_{2}$, $\varphi$, $\varphi^{\prime}$ and $\varphi^{\prime\prime}$ and substitute the obtained expressions into the other equations. For each fixed $i\in\mathbb{N}$, we solve the modified equation $\tilde{f}_{i0}=0$ in view of the similar equations with lower values of $i$ and thus find an expression for $\varphi^{(i+2)}$, the explicit form of which is essential for further consideration only for $i=3$. This yields the following complete moving frame: $$\displaystyle\begin{split}&\displaystyle C_{1}=\frac{W}{2v},\quad C_{2}=f_{v}-% vf_{vv},\quad\varphi=0,\quad\varphi^{\prime}=\frac{W}{2v^{2}},\quad\varphi^{% \prime\prime}=\frac{W}{4v^{2}}f_{vv},\\ &\displaystyle\varphi^{\prime\prime\prime}=\frac{W}{4v^{4}}\Big{(}2f_{u}+(f-vf% _{v}+v^{2}f_{vv})f_{vv}\Big{)},\\ &\displaystyle\varphi^{(i+2)}=\frac{W}{2v^{1}}\sum_{i^{\prime}=0}^{i}\binom{i}% {i^{\prime}}\left(\frac{v^{2}}{W}\right)^{i-i^{\prime}}f_{i^{\prime},i-i^{% \prime}}+\cdots,\quad i=2,3,\dots\,.\end{split}$$ (9) In the expression for $\varphi^{(i+2)}$, we presented only the summands with the highest-order derivatives $f_{i^{\prime},i-i^{\prime}}$, $i^{\prime}=0,\dots,i$. The invariantization $I^{ij}=\iota(f_{ij})$ of the derivatives $f_{ij}$ of $\tilde{f}$ that are not involved in the normalization conditions (8) gives rise to a complete set of functionally independent differential invariants of $G_{1}$. The lowest-order non-phantom normalized differential invariant is $I^{11}$, and it reads $$I^{11}=-2v^{2}\frac{4f_{u}-2vf_{uv}+(2f-2vf_{v}+v^{2}f_{vv})f_{vv}}{(2f-2vf_{v% }+v^{2}f_{vv})^{2}}.$$ This differential invariant is of second order. For each tuple $(i,j)$ with $i+j\geqslant 3$ and $j\neq 0$, the maximal orders of derivatives of $f$ and $\varphi$ appearing in the expression for $\tilde{f}_{ij}$ are $i+j$ and $i+2$, respectively. This is why the maximal order of derivatives of $f$ in the expression for $\tilde{f}_{ij}$ cannot be lowered in the course of the invariantization, i.e., the order of the normalized differential invariant $I^{ij}$ is $i+j$. Therefore, there are precisely $\frac{1}{2}k(k+1)-2$ functionally independent differential $G_{1}$-invariants of order not greater than $k\geqslant 2$. They are given by the functions $I^{11}$ and $I^{ij}$ with $3\leqslant i+j\leqslant k$ and $j\neq 0$. Apart from finding the complete set of functionally independent differential invariants of $G_{1}$ for each fixed order by successively invariantizing all the derivatives $f_{ij}$, the moving frame (9) can be used to determine the operators of invariant differentiation. They are found upon invariantizing the operators of total differentiation (6) and read $$\mathrm{D}_{u}^{\mathrm{i}}=\frac{2v^{2}}{2f-2vf_{v}+v^{2}f_{vv}}\left(\mathrm% {D}_{u}-\frac{1}{2}vf_{vv}\mathrm{D}_{v}\right),\quad\mathrm{D}_{v}^{\mathrm{i% }}=v\mathrm{D}_{v}.$$ (10) We now aim to investigate the structure of the algebra of differential invariants of $G_{1}$. The starting point for this investigation is the universal recurrence relation, which relates the differentiated invariantized differential functions or differential forms with the invariantization of the respective differentiated objects. This universal recurrence relation reads [25] $$\mathrm{d}\iota(\Omega)=\iota\big{(}\mathrm{d}\Omega+Q^{(\infty)}(\Omega)\big{% )}.$$ (11) The first step in our study is the evaluation of (11) for the independent variables $u$ and $v$ and the derivatives $f_{ij}$, $i,j\in\mathbb{N}_{0}$, $$\displaystyle\mathrm{d}_{\rm h}\iota(u)=\omega^{1}+\iota(\phi),\quad\mathrm{d}% _{\rm h}\iota(v)=\omega^{2}+\iota(\eta),$$ $$\displaystyle\mathrm{d}_{\rm h}I^{ij}=\mathrm{d}_{\rm h}\iota(f_{ij})=\iota(f_% {i+1,j}\mathrm{d}u+f_{i,j+1}\mathrm{d}v+\theta^{ij})=I^{i+1,j}\omega^{1}+I^{i,% j+1}\omega^{2}+\iota(\theta^{ij}),$$ where $\omega^{1}=\iota(\mathrm{d}u)$, $\omega^{2}=\iota(\mathrm{d}v)$, and $$\begin{split}\displaystyle\theta^{ij}=&\displaystyle\mathrm{D}_{u}^{\,\,i}% \mathrm{D}_{v}^{\,\,j}(\theta-\phi f_{10}-\eta f_{01})+\phi f_{i+1,j}+\eta f_{% i,j+1}\\ \displaystyle=&\displaystyle-(j-1)\sum_{i^{\prime}=0}^{i}\binom{i}{i^{\prime}}% \phi^{(i^{\prime}+1)}f_{i-i^{\prime},j}-\sum_{i^{\prime}=1}^{i}\binom{i}{i^{% \prime}}\Big{(}\phi^{(i^{\prime})}f_{i-i^{\prime}+1,j}+v\phi^{(i^{\prime}+1)}f% _{i-i^{\prime},j+1}\Big{)}\\ &\displaystyle{}+(j-2)c_{1}f_{ij}-c_{2}\delta_{0i}(\delta_{0j}v+\delta_{1j})-% \phi^{(i+2)}(\delta_{0j}v^{2}+2\delta_{1j}v+2\delta_{2j})\end{split}$$ is the $f_{ij}$-component of the infinite prolongation of the vector field $\phi\partial_{u}+\eta\partial_{v}+\theta\partial_{f}$. Here $\delta_{ij}$ is the Kronecker delta. The respective recurrence relations then split into two kinds, the first being the so-called phantom recurrence relations. For a well-defined moving frame cross-section, they can be uniquely solved for the invariantized Maurer–Cartan forms, which arise due to the presence of the correction term $\iota\big{(}Q^{(\infty)}(\Omega)\big{)}$ in (11). Then, plugging these invariantized Maurer–Cartan forms into the second kind of recurrence relations, the non-phantom ones, gives a complete description of the relation between the normalized and differentiated differential invariants, see [6, 25] for more details. For the chosen cross-section (8), the phantom recurrence relations read $$\displaystyle 0$$ $$\displaystyle=\mathrm{d}_{\rm h}\iota(u)=\omega^{1}+\iota(\phi)=\omega^{1}+% \hat{\phi},$$ $$\displaystyle 0$$ $$\displaystyle=\mathrm{d}_{\rm h}\iota(v)=\omega^{2}+\iota(\eta)=\omega^{2}+% \hat{\phi}^{\prime}-\hat{c}_{1},$$ $$\displaystyle 0$$ $$\displaystyle=\mathrm{d}_{\rm h}I^{00}=\iota(\theta)=\hat{\phi}^{\prime}-2\hat% {c}_{1}-\hat{c}_{2}-\hat{\phi}^{\prime\prime},$$ $$\displaystyle 0$$ $$\displaystyle=\mathrm{d}_{\rm h}I^{01}=I^{11}\omega^{1}+\iota(\theta^{01})=I^{% 11}\omega^{1}-\hat{c}_{2}-2\hat{\phi}^{\prime\prime},$$ $$\displaystyle 0$$ $$\displaystyle=\mathrm{d}_{\rm h}I^{02}=I^{12}\omega^{1}+I^{03}\omega^{2}+\iota% (\theta^{02})=I^{12}\omega^{1}+I^{03}\omega^{2}-2\hat{\phi}^{\prime\prime},$$ $$\displaystyle 0$$ $$\displaystyle=\mathrm{d}_{\rm h}I^{i0}=I^{i1}\omega^{2}+\iota(\theta^{i0})$$ $$\displaystyle=I^{i1}\omega^{2}+\hat{\phi}^{(i+1)}-\hat{\phi}^{(i+2)}-\sum_{i^{% \prime}=1}^{i-1}\binom{i}{i^{\prime}}I^{i-i^{\prime},1}\hat{\phi}^{(i^{\prime}% +1)},\quad i\in\mathbb{N},$$ where the forms $\hat{c}_{1}$, $\hat{c}_{2}$ and $\hat{\phi}^{(i)}$, $i\in\mathbb{N}_{0}$, are the invariantizations of the parameters $c_{1}$, $c_{2}$ and $\phi^{(i)}$ of the infinitely prolonged general element of the projected algebra $\mathfrak{g}_{1}$, respectively, $\hat{c}_{1}=\iota(c_{1})$, $\hat{c}_{2}=\iota(c_{2})$ and $\hat{\phi}^{(i)}=\iota(\phi^{(i)})$. More rigorously, here the parameters $c_{1}$, $c_{2}$ and $\phi^{(i)}$, $i\in\mathbb{N}_{0}$, are interpreted as the coordinate functions on the infinite prolongation of $\mathfrak{g}_{1}$. Recall that under the prolongation we consider $u$ and $v$ to be the independent variables and $f$ to be the dependent variable. In other words, these coefficients are first-order differential forms in the jet space $\mathrm{J}^{\infty}(u,v|\,f)$. Hence their invariantizations are also forms, which are called invariantized Maurer–Cartan forms. The above system can be solved to yield the following invariantized Maurer–Cartan forms $$\displaystyle\begin{split}&\displaystyle\hat{c}_{1}=\left(\frac{1}{2}I^{12}-I^% {11}\right)\omega^{1}+\left(\frac{1}{2}I^{03}-1\right)\omega^{2},\quad\hat{c}_% {2}=(I^{11}-I^{12})\omega^{1}-I^{03}\omega^{2},\\ &\displaystyle\hat{\phi}=-\omega^{1},\quad\hat{\phi}^{\prime}=\left(\frac{1}{2% }I^{12}-I^{11}\right)\omega^{1}+\left(\frac{1}{2}I^{03}-2\right)\omega^{2},% \quad\hat{\phi}^{\prime\prime}=\frac{1}{2}I^{12}\omega^{1}+\frac{1}{2}I^{03}% \omega^{2},\\ &\displaystyle\hat{\phi}^{(i+2)}=\hat{\phi}^{(i+1)}-\sum_{i^{\prime}=1}^{i-1}% \binom{i}{i^{\prime}}I^{i-i^{\prime},1}\hat{\phi}^{(i^{\prime}+1)}+I^{i1}% \omega^{2},\quad i\in\mathbb{N}.\end{split}$$ (12) The explicit expression for the invariantized form $\hat{\phi}^{(i+2)}$, $i\in\mathbb{N}$, as a combination of $\omega^{1}$ and $\omega^{2}$ with coefficients being polynomials of normalized differential invariants is obtained by expanding the above expression when successively going over the values of $i$. In particular, $$\displaystyle\hat{\phi}^{\prime\prime\prime}=\frac{1}{2}I^{12}\omega^{1}+\left% (I^{11}+\frac{1}{2}I^{03}\right)\omega^{2},$$ $$\displaystyle\hat{\phi}^{(4)}=\left(\frac{1}{2}I^{12}-I^{11}I^{12}\right)% \omega^{1}+\left(I^{11}+\frac{1}{2}I^{03}+I^{21}-I^{11}I^{03}\right)\omega^{2}.$$ For $i\geqslant 3$, the greatest value of $i^{\prime}+j^{\prime}$ for the normalized differential invariants $I^{i^{\prime}j^{\prime}}$ that are involved in $\hat{\phi}^{(i+2)}$ is $i+1$, and $I^{i1}\omega^{2}$ is the only summand with this value. The non-phantom recurrence relations are $$\displaystyle\mathrm{d}_{\rm h}I^{11}$$ $$\displaystyle=I^{21}\omega^{1}+I^{12}\omega^{2}+\iota(\theta^{11})$$ $$\displaystyle{}=(I^{21}+2(I^{11})^{2}-I^{11}I^{12}-I^{12})\omega^{1}+(I^{12}-I% ^{11}I^{03}+I^{11}-I^{03})\omega^{2},$$ $$\displaystyle\mathrm{d}_{\rm h}I^{ij}$$ $$\displaystyle=I^{i+1,j}\omega^{1}+I^{i,j+1}\omega^{2}+\iota(\theta^{ij}),\quad i% +j\geqslant 3,\ j\neq 0,$$ with $$\begin{split}\displaystyle\iota(\theta^{ij})=&\displaystyle-(j-1)\sum_{i^{% \prime}=0}^{i}\binom{i}{i^{\prime}}I^{i-i^{\prime},j}\hat{\phi}^{(i^{\prime}+1% )}-\sum_{i^{\prime}=1}^{i}\binom{i}{i^{\prime}}\Big{(}I^{i-i^{\prime}+1,j}\hat% {\phi}^{(i^{\prime})}+I^{i-i^{\prime},j+1}\hat{\phi}^{(i^{\prime}+1)}\Big{)}\\ &\displaystyle{}+(j-2)I^{ij}\hat{c}_{1}-\delta_{0i}(\delta_{0j}+\delta_{1j})% \hat{c}_{2}-(\delta_{0j}+2\delta_{1j}+2\delta_{2j})\hat{\phi}^{(i+2)}.\end{split}$$ The first non-phantom recurrence relation splits into $$\mathrm{D}_{u}^{\mathrm{i}}I^{11}=I^{21}+2(I^{11})^{2}-I^{11}I^{12}-I^{12},% \quad\mathrm{D}_{v}^{\mathrm{i}}I^{11}=I^{12}-I^{11}I^{03}+I^{11}-I^{03}.$$ Therefore, the normalized differential invariants $I^{12}$ and $I^{21}$ are expressed in terms of invariant derivatives of $I^{11}$ and $I^{03}$, $$\displaystyle\begin{split}&\displaystyle I^{12}=\mathrm{D}_{v}^{\mathrm{i}}I^{% 11}+I^{11}I^{03}-I^{11}+I^{03},\\ &\displaystyle I^{21}=\mathrm{D}_{u}^{\mathrm{i}}I^{11}-2(I^{11})^{2}+(I^{11}+% 1)(\mathrm{D}_{v}^{\mathrm{i}}I^{11}+I^{11}I^{03}-I^{11}+I^{03}).\end{split}$$ (13) In view of the above discussion on the invariantize forms $\hat{\phi}^{(i^{\prime})}$, $i\in\mathbb{N}$, the expression for $\iota(\theta^{ij})$ with $i+j\geqslant 3$ and $j\neq 0$ implies that the greatest value of $i^{\prime}+j^{\prime}$ for $I^{i^{\prime}j^{\prime}}$ involved in $\iota(\theta^{ij})$ is $i+j$. Hence splitting the recurrence relation with $\mathrm{d}_{\rm h}I^{ij}$ leads to expressions for $I^{i+1,j}$ and $I^{i,j+1}$ in terms of invariant derivatives of $I^{i^{\prime}j^{\prime}}$ with $i^{\prime}+j^{\prime}\leqslant i+j$. For example, from the non-phantom recurrence relation $$\displaystyle\mathrm{d}_{\rm h}I^{03}$$ $$\displaystyle=I^{13}\omega^{1}+I^{04}\omega^{2}+\iota(\theta^{03})$$ $$\displaystyle{}=\left(I^{13}+I^{11}I^{03}-\frac{1}{2}I^{12}I^{03}\right)\omega% ^{1}+\left(I^{04}+I^{03}-\frac{1}{2}(I^{03})^{2}\right)\omega^{2}$$ we derive $$\displaystyle I^{13}=\mathrm{D}_{u}^{\mathrm{i}}I^{03}-I^{11}I^{03}+\frac{1}{2% }I^{12}I^{03},\quad I^{04}=\mathrm{D}_{v}^{\mathrm{i}}I^{03}+\frac{1}{2}(I^{03% })^{2}-I^{03}.$$ This implies by induction, where the expressions (13) for $I^{12}$ and $I^{21}$ give the base case, that any non-phantom normalized differential invariant can be expressed in terms of invariant derivatives of $I^{11}$ and $I^{03}$. To find a minimum generating set of differential invariants for the projected group $G_{1}$, we should additionally check whether $I^{03}$ can be expressed in terms of invariant derivatives of $I^{11}$. We use (11) to compute the commutator between the operators of invariant differentiation. This is done upon evaluating (11) for the basis horizontal forms $\mathrm{d}u$ and $\mathrm{d}v$, $$\displaystyle\mathrm{d}_{\rm h}\iota(\mathrm{d}u)=\iota(\phi^{\prime}\mathrm{d% }u)=\iota(\phi^{\prime})\wedge\iota(\mathrm{d}u)=\left(2-\frac{1}{2}I^{03}% \right)\omega^{1}\wedge\omega^{2}=-Y^{1}_{12}\,\omega^{1}\wedge\omega^{2},$$ $$\displaystyle\mathrm{d}_{\rm h}\iota(\mathrm{d}v)=\iota\big{(}\phi^{\prime% \prime}v\mathrm{d}u+(\phi^{\prime}-c_{1})\mathrm{d}v\big{)}=\iota(\phi^{\prime% \prime}v)\wedge\iota(\mathrm{d}u)=-\frac{1}{2}I^{03}\omega^{1}\wedge\omega^{2}% =-Y^{2}_{12}\,\omega^{1}\wedge\omega^{2}.$$ The commutation relation then evaluates as $$[\mathrm{D}_{u}^{\mathrm{i}},\mathrm{D}_{v}^{\mathrm{i}}]=Y^{1}_{12}\mathrm{D}% _{u}^{\mathrm{i}}+Y^{2}_{12}\mathrm{D}_{v}^{\mathrm{i}}=\left(\frac{1}{2}I^{03% }-2\right)\mathrm{D}_{u}^{\mathrm{i}}+\frac{1}{2}I^{03}\mathrm{D}_{v}^{\mathrm% {i}},$$ see [25] for details of the technique applied. Evaluating $[\mathrm{D}_{u}^{\mathrm{i}},\mathrm{D}_{v}^{\mathrm{i}}]I^{11}$, we can derive the following expression for $I^{03}$: $$I^{03}:=\frac{2v^{3}f_{vvv}}{2f-2vf_{v}+v^{2}f_{vv}}=2\frac{2\mathrm{D}_{u}^{% \mathrm{i}}I^{11}+[\mathrm{D}_{u}^{\mathrm{i}},\mathrm{D}_{v}^{\mathrm{i}}]I^{% 11}}{\mathrm{D}_{u}^{\mathrm{i}}I^{11}+\mathrm{D}_{v}^{\mathrm{i}}I^{11}}.$$ As a result, we have proved the following theorem. Theorem 2. The algebra of differential invariants of the group $G_{1}$, which is the projection of the equivalence group $G^{\sim}$ of the class of diffusion equations (1) to the space with coordinates $(u,v,f)$, is generated by the single differential invariant $$I^{11}=-2v^{2}\frac{4f_{u}-2vf_{uv}+(2f-2vf_{v}+v^{2}f_{vv})f_{vv}}{(2f-2vf_{v% }+v^{2}f_{vv})^{2}}$$ along with the two operators of invariant differentiation $$\mathrm{D}_{u}^{\mathrm{i}}=\frac{2v^{2}}{2f-2vf_{v}+v^{2}f_{vv}}\left(\mathrm% {D}_{u}-\frac{1}{2}vf_{vv}\mathrm{D}_{v}\right),\quad\mathrm{D}_{v}^{\mathrm{i% }}=v\mathrm{D}_{v}.$$ All other differential invariants are functions of $I^{11}$ and invariant derivatives thereof. Corollary 1. A functional basis of differential invariants of order not greater than $k\in\mathbb{N}_{0}$ in terms of invariant derivatives of non-phantom normalized differential invariants is exhausted by $$\big{(}\mathrm{D}_{u}^{\mathrm{i}}\big{)}^{i}\big{(}\mathrm{D}_{v}^{\mathrm{i}% }\big{)}^{j}I^{11},\ i+j\leqslant k-2,\quad\big{(}\mathrm{D}_{v}^{\mathrm{i}}% \big{)}^{j^{\prime}}I^{03},\ j^{\prime}\leqslant k-3.$$ Acknowledgements This research was undertaken, in part, thanks to funding from the Canada Research Chairs program, the NSERC Discovery Grant program and the InnovateNL LeverageR&D program. AB is a recipient of an APART Fellowship of the Austrian Academy of Sciences. The research of ROP and EMDSCB was supported by the Austrian Science Fund (FWF), projects P25064 and P29177. 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Out-of-phase mixed holographic gratings : a quantative analysis Martin Fally Faculty of Physics, Nonlinear Physics, University of Vienna, Boltzmanngasse 5, A-1090 Wien, Austria martin.fally@univie.ac.at http://nlp.univie.ac.at    Mostafa A. Ellabban Faculty of Physics, Nonlinear Physics, University of Vienna, Boltzmanngasse 5, A-1090 Wien, Austria Physics Department, Faculty of Science, Tanta University, Tanta 31527, Egypt    Irena Drevenšek-Olenik Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1001 Ljubljana, Slovenia J. Stefan Institute, Jamova 39, SI-1001 Ljubljana, Slovenia (November 20, 2020, CORRECTION) Abstract We show, that by performing a simultaneous analysis of the angular dependencies of the $\pm$ first and the zeroth diffraction orders of mixed holographic gratings, each of the relevant parameters can be obtained: the strength of the phase grating and the amplitude grating, respectively, as well as a potential phase between them. Experiments on a pure lithium niobate crystal are used to demonstrate the applicability of the analysis. I Introduction Recently, volume holographic gratings with a modulation of both, the absorption coefficient and the refractive index, have attracted attention in various materials such as silver halide emulsions Carretero-ol01 ; Neipp-jpd02 ; Neipp-oex02 , doped garnet crystals Ellabban-oex06 or materials with colloidal color centersShcheulin-os07 . The simplest theoretical description (two-wave-coupling theory) of light propagation in an isotropic medium with a periodic modulation of the (complex) dielectric constant was already given long ago by Kogelnik Kogelnik-bell69 . Considering periodic phase and amplitude modulations, the grating types are treated to be in phase. Later Guibelalde generalized the equations to be valid for out-of-phase gratings Guibelalde-oqe84 . The quantity of major interest usually is the (first order) diffraction efficiency $\eta_{1}$, defined as the ratio of powers between the diffracted beam and the incoming beam. For the case of high diffraction efficiencies (above 50%) or even for overmodulated gratings Neipp-joa01 ; Neipp-oex02 ; Gallego-oc03 , the so called ’transmission efficiency’ $\eta_{0}$, i.e., more correctly termed as zero order diffraction efficiency, was also employed for characterization of the grating parameters. It was suggested, that by measuring the diffraction and transmission efficiency it is possible to evaluate the refractive-index modulation $n_{1}$ and the absorption constant modulation $\alpha_{1}$ if one assumes in-phase gratings Carretero-ol01 . In this article we show how the shape of the angular dependencies for the $\pm$ first and zero order diffraction efficiencies depend characteristically on the parameters $n_{1},\alpha_{1}$ and the phase $\varphi$ between them. We generalize the formulae given in Ref. Carretero-ol01 to the case of out-of-phase gratings and demonstrate at two experimental examples that the analysis is applicable. This is important, as up to now the evaluation of mixed gratings including a phase was only conducted by beam-coupling experiments, an interferometric technique which is more demanding from an experimental point of view. II Diffraction efficiencies of zero and $\pm$ first order According to Refs. Kogelnik-bell69 ; Guibelalde-oqe84 a plane wave propagating in a (thick) medium with a one dimensional periodic complex dielectric constant, composed of its real part $n(x)=n_{0}+n_{1}\cos{(Kx)}$ and imaginary part $\alpha(x)=\alpha_{0}+\alpha_{1}\cos{(Kx+\varphi)}$, yields outgoing complex electric field amplitudes for the (zero order) forward diffracted $\hat{R}_{0}$ and (first order) diffracted $\hat{R}_{\pm 1}$ waves. These depend characteristically on the following parameters: the mean absorption constant $\alpha_{0}$, the thickness $d$ of the grating, the dephasing $\vartheta$ due to the deviation from Bragg’s law and the complex coupling constant $\kappa^{\pm}=n_{1}\pi/\lambda-i\alpha_{1}/2e^{\pm i\varphi}=\kappa_{1}-i\kappa% _{2}e^{\pm i\varphi}$ . Further, $K$ denotes the spatial frequency of the grating, $n_{0}$ the mean refractive index of the medium, and $\varphi$ a possible phase shift between the refractive-index and absorption grating. The goal of an experiment is to extract the grating parameters $n_{1},\alpha_{1},\varphi$ by varying the dephasing, e.g., through measuring the angular response of $\eta_{0}=\hat{R}_{0}\hat{R}_{0}^{*}/I$ and $\eta_{\pm 1}=\hat{R}_{\pm 1}\hat{R}_{\pm 1}^{*}/I$ where ${}^{*}$ denotes the complex conjugate and $I$ the incident intensity. For simplicity in calculations and as the most often used experimental setup we assume a symmetrical geometry, i.e., that the grating vector and the surface normal are mutually perpendicular. A schematic of the setup is shown in Figure 1. Slightly adapting the convenient notation from Ref. Carretero-ol01 the efficiencies for transmission gratings can easily be calculated to yield $$\displaystyle\eta_{\pm 1}(\theta)$$ $$\displaystyle=$$ $$\displaystyle 2A(\theta)\frac{\kappa_{1}^{2}+\kappa_{2}^{2}\pm 2\kappa_{1}% \kappa_{2}\sin{\varphi}}{z}\left(\cosh{\left[z^{1/2}D\cos{\psi}\right]}-\cos{% \left[z^{1/2}D\sin{\psi}\right]}\right)$$ (1) $$\displaystyle\eta_{0}(\theta)$$ $$\displaystyle=$$ $$\displaystyle\frac{A(\theta)}{2z}\left((z+\vartheta^{2})\cosh{\left[z^{1/2}D% \cos{\psi}\right]}+(z-\vartheta^{2})\cos{\left[z^{1/2}D\sin{\psi}\right]}\right.$$ (2) $$\displaystyle+$$ $$\displaystyle\left.2\frac{\cos{\varphi}}{|\cos{\varphi}|}\vartheta z^{1/2}% \left\{\sin{\psi}\sinh{\left[z^{1/2}D\cos{\psi}\right]}-\cos{\psi}\sin{\left[z% ^{1/2}D\sin{\psi}\right]}\right\}\right)$$ with the abbreviations $A(\theta)=\exp{\{-2\alpha_{0}D\}}$, $D=d/\cos{\theta}$ and $$\displaystyle\vartheta$$ $$\displaystyle=$$ $$\displaystyle K(\sin{\theta}-\sin{\theta_{B}})$$ (3) $$\displaystyle z$$ $$\displaystyle=$$ $$\displaystyle\left\{[\vartheta^{2}+4(\kappa_{1}^{2}-\kappa_{2}^{2})]^{2}+[8% \kappa_{1}\kappa_{2}\cos{\varphi}]^{2}\right\}^{1/2}$$ (4) $$\displaystyle 2\psi$$ $$\displaystyle=$$ $$\displaystyle\arccos{\left(-\frac{\vartheta^{2}+4(\kappa_{1}^{2}-\kappa_{2}^{2% })}{z}\right)}.$$ (5) Here, $\theta_{B}$ denotes the Bragg angle (inside the medium). Equations (1) and (2) are valid for $\theta\geq 0$; for $\theta\leq 0$ the angles and phase-shifts are replaced by their negative values, i.e., $\eta_{\pm 1}(-\theta)=\eta_{\mp 1}(\theta)$ and $\eta_{\pm 1}(-\varphi)=\eta_{\mp 1}(\varphi)$. Note, that Equation (1) is identical to Equation (11) from Ref. Guibelalde-oqe84 . Employing Equations (1) and (2) we now study the particular case of $\alpha_{0}d=1$ and $\kappa_{2}=\alpha_{0}/2$, i.e., maximal grating strength for the amplitude contribution Kogelnik-bell69 . We vary the strength of the phase grating between $\kappa_{1}=\kappa_{2}/4,\kappa_{2},4\kappa_{2}$ with different phase angles $\varphi=0,\pi/4,\pi/2,3\pi/4$ between the grating types. The angular dependencies of the zero order and $\pm$ first order diffraction efficiencies are depicted in Figure 1(a)-(d). At this point let us summarize the main characteristic features occurring in the diffraction efficiencies at the example for $\varphi=\pi/4$ to obtain a qualitative understanding of the curve shapes and their dependency on the ratio of $\kappa_{1}/\kappa_{2}$: • Zero order diffraction efficiency $\eta_{0}(\theta)$ – The curves are symmetric with respect to normal incidence, i.e., $\theta=0$. – Neither the minima nor the maxima of the curve are located at the Bragg angle, except for $\kappa_{1}=0$ or $\kappa_{2}=0$ or $\varphi=\pi/2$. In general the position and the height of the minima or maxima depend in a complex way on $\kappa_{1},\kappa_{2},\varphi$ and even the mean absorption constant $\alpha_{0}$ (see discussion for $\alpha_{0}d\gg 1$). – For $\kappa_{1}<\kappa_{2}$ the curve at the Bragg angle extends more to the region above the mean absorption curve (dash-dot line, first term in Equations (1) and (2) ) than below, i.e., as a simple approximation $\eta_{0}^{max}+\eta_{0}^{min}>2A(\theta_{B})$. The same is true vice versa for $\kappa_{1}>\kappa_{2}$ – Note, that for $|\theta|\ll\theta_{B}$ the curve resides below the mean absorption curve, for $|\theta|\gg\theta_{B}$ above • Diffraction efficiency – The maximum value of the diffraction efficiency differs for $\eta_{-1}(\theta_{B})$ and $\eta_{+1}(\theta_{B})$; in our case $\eta_{-1}(\theta_{B})<\eta_{+1}(\theta_{B})$. – The curves are symmetric with respect to $\theta_{B}$, i.e., $\eta_{1}(\theta_{B}+x)=\eta_{1}(\theta_{B}-x)$ except for their different mean absorption $A(\theta_{B}\pm x)$. – Note, that despite $\kappa_{1}<\kappa_{2}$ the diffraction efficiency $\eta_{-1}(\theta_{B},\kappa_{2}/4)>\eta_{-1}(\theta_{B},\kappa_{2})$ for the minus first diffraction order, whereas it is vice versa for the plus first diffraction order, i.e., $\eta_{+1}(\theta_{B},\kappa_{2}/4)<\eta_{+1}(\theta_{B},\kappa_{2})$ Next we would like to point out the difference between the curves for various $\varphi$ values. Figure 2(c) shows a unique case which is most instructive. For $\varphi=\pi/2$ the coupling constant $\kappa=\kappa_{1}\pm\kappa_{2},\in\mathbb{R}$ . Thus a maximum difference between $\eta_{-1}$ and $\eta_{+1}$ is obtained, culminating in the full depletion of $\eta_{-1}$ if $\kappa_{1}=\kappa_{2}$ (see appendix). Finally, we want to draw the attention to the case of $\varphi=3\pi/4>\pi/2$. Then $\eta_{\pm 1}$ gives identical results as for $\varphi-\pi/2$. The zero order diffraction efficiency $\eta_{0}$, however, approaches the mean absorption curve for $|\theta|\gg\theta_{B}$ from above in the case of $\varphi<\pi/2$ and contrary from below for $\varphi>\pi/2$ . Considering these arguments it is obvious, that only a simultaneous fit of all diffraction data, i.e., zero and $\pm$ first order diffracted intensities, allows to extract the decisive parameters $\kappa_{1},\kappa_{2},\varphi$. On the other hand these curves are therefore fingerprints of the relation between the parameters. The following recipe can help in judging about the general situation (for $\alpha_{0}d\approx 1$): • Check $\eta_{\pm 1}$: if their magnitudes differ, this is a fingerprint that mixed gratings exist that are out of phase ($\varphi\not=0$). The ratio $\eta_{+1}/\eta_{-1}$ at the Bragg position obtains a maximum value for $\varphi=\pi/2$ and for $\kappa_{1}=\kappa_{2}$ Ellabban-oex06 . • Check $\eta_{0}$: if $\eta_{0}(\theta=0)<A(0)$ then $|\varphi|<\pi/2$ and else vice versa • If $\eta_{0}^{max}+\eta_{0}^{min}>2A(\theta_{B})$, the absorptive component is dominating and else vice versa. • For overmodulated phase gratings another feature of the diffraction efficiencies becomes prominent: the side minima near the Bragg peak are lifted to nonzero values (for $\varphi\not=\pi/2$). This striking feature can already be understood in the case $\varphi=0$ where we simply add up the pure absorptive and the pure phase grating. The positions of the $s^{\rm th}$ side minima are then given by $\vartheta_{s}^{(1)}=2[(s\pi/D)^{2}-\kappa_{1}^{2}]^{1/2}$ (phase grating) and $\vartheta_{s}^{(2)}=2[(s\pi/D)^{2}+\kappa_{2}^{2}]^{1/2}$ (absorption grating). Thus, their minima considerably deviate from each other for $\kappa_{1,2}\approx s\pi/D$. Recalling, that $\kappa_{2}=\alpha_{1}/2\leq\alpha_{0}/2$ such a situation will practically occur if $\kappa_{1}\gg\pi/D$, i.e., for $s>1$ (overmodulated phase gratings exist). This is realized in various systems (see e.g., Neipp-jpd02 ; Neipp-joa01 ; Drevensek-Olenik-pre06 but did not deserve proper attention. Finally, we would like to recall that for $\varphi\to\varphi+\pi$ the complex coupling parameters $\kappa^{\pm}\to\kappa^{\mp}$ are interchanged and thus the $\eta_{\pm 1}\to\eta_{\mp 1}$. For $\eta_{0}$ the term in the second line of Equation 2 changes sign because of $\cos{\varphi}\to\cos{(\varphi+\pi)}=-\cos{\varphi}$. III Experimental and Discussion The investigations were performed on a pure congruently melted lithium niobate crystal (thickness: $5{\rm mm}$). Holographic transmission gratings were prepared by a standard two-wave mixing setup using an argon-ion laser at a recording wavelength of $\lambda_{p}=351$ nm. Two plane waves with equal intensities and parallel polarization states (s-polarization) were employed as recording beams under a crossing angle of $2\Theta_{B}=20.21^{\circ}$ (outside the medium) corresponding to a grating period of 1000 nm where the polar $c$-axis is lying in the plane of incidence. The total intensity of the writing beams was 9 mW/cm${}^{2}$. HPDLC samples were fabricated from a UV curable mixture prepared from commercially available constituents as previously reported in literature Drevensek-Olenik-pre06 . The grating period was 1216 nm, the grating thickness about $30\mu$m Fally-prl06 . After holographic recording we postcured the sample by illuminating it homogeneously with one of the UV writing beams. The grating characteristics of the samples was analyzed by monitoring the angular dependencies of the $\pm$ first and zero order diffraction efficiencies. For this purpose the samples were fixed on an accurately controlled rotation stage with an accuracy of $\pm 0.001^{\circ}$) and facultatively (HPDLC) in a heating chamber. In the case of $\rm LiNbO_{3}~{}$we used a single considerably reduced readout beam at $\lambda_{r}=\lambda_{p}=351$ nm and s-polarization, whereas for the HPDLC a He-Ne laser beam at a readout wavelength of $\lambda_{r}=543$ nm and p-polarization state was employed. Figure 3 shows the experimental curves for the $0.,\pm 1.$ diffraction orders from a grating recorded in nominally pure congruently melted $\rm LiNbO_{3}~{}$. According to the recipe given above we immediately can diagnose mixed out-of-phase refractive-index and amplitude gratings, because the $\eta_{+1}>\eta_{-1}$. Further by inspecting the zero order diffraction we come to know that the phase $0<\varphi<\pi/2$. The effects in the zero order are not so prominent for two reasons: the overall diffraction efficiency is very small and the phase grating is dominant because the zero order diffraction curve extends mostly to values below the mean absorption curve (dash-dot line in Figure 3). A simultaneous fit of Equations 1 and 2 to the measured data yielded the following parameters: $n_{1}=(3.01\pm 0.04)\times 10^{-6},\alpha_{1}=8.18\pm 0.48{\rm m}^{-1},\varphi% =1.027\pm 0.059,\alpha_{0}=118\pm 1.7{\rm m}^{-1}$ with a reduced chi-square value of $1.89\times 10^{-7}$ . From this value and Figure 3 it is obvious that the equations excellently fit the data. Finally, we intend to demonstrate the usability of the (qualitative) analysis employing an example with strong overmodulation and high extinction: holographic polymer-dispersed liquid crystals (H-PDLCs). Only recently was a preliminary beam-coupling analysis of such a system conducted, a task which is not simple from an experimental point of view Ellabban-spie07 , in particular if the experiments should be carried out under high temperatures or application of external electric fields. Figure 4 shows the diffraction curves from a grating in a HPDLC at an elevated temperature. We can understand the major characteristic features as follows: The liquid crystal (LC) component in an HPDLC is highly birefringent. Statistical alignment of the LC-droplets of about the light wavelength’s size leads to strong scattering, i.e., extinction which can be treated similar to absorption provided that multiple scattering does not play an essential role. HPDLCs basically consist of alternating regions with high and low concentration of LCs embedded in a polymer matrix. Thus, these periodically varying scatterers act as extinction gratings. In addition, of course, also the refractive index is strongly modulated (at least via the density changes). Therefore, HPDLCs are typical examples of mixed gratings. Furthermore, it is well known in literature that the light-induced refractive-index changes are extremely high and strong overmodulation occurs (see e.g. Drevensek-Olenik-pre06 ). Such an example is shown in Figure 4. From the experimental data we conclude, that combined refractive-index and extinction gratings are produced. This is consistent with our previous beam-coupling measurements Ellabban-spie07 . However, we do not dare to decide about a possible phase between them. A quantitative evaluation is not possible for this case as we are aware of the fact, that in HPDLCs the gratings are anisotropic and thus the basic equations of Ref. Kogelnik-bell69 should be replaced by the full equations given by Montemezzani and Zgonik Montemezzani-pre97 . In addition, the gratings are usually rather inhomogeneous across the sample but might be considerably improved upon further efforts during recording De-Sio-ao06 . The non-zero minima in the diffracted beams partially might originate from overmodulation as discussed above but mainly from the inhomogeneity of the gratings and a profile perpendicular to the grating vector Uchida-josa73 . However, a qualitative understanding of the changes occuring during heating or applying an electric field can still be read off from the diffraction curves like those shown in Ref. Drevensek-Olenik-pre06 . IV Remarks and Conclusion The above discussed analysis is easily applicable for $\alpha_{0}d\approx 1,\kappa_{1}\approx\kappa_{2}$ and $\eta_{1}(\theta_{B})/\eta_{0}(\theta_{B})\gtrsim 0.01$, so that with the chosen example of $\rm LiNbO_{3}~{}$above we are already at the limit. If one grating type is dominant the analysis still remains valid, however, the resulting values for $\varphi$ and the smaller component result in quite large errors. We would like to draw the attention to the fact, that for $\alpha_{0}\ll 1$ the absorptive grating strength is considerably limited, so that in general the zero order diffraction will not feel the Bragg diffraction. On the other hand, for $\alpha_{0}d\gg 1$, the forward diffracted beam will exhibit a maximum near the Bragg position, a fact which is well known in x-ray optics (anomalous transmission), see e.g. Batterman-rmp64 . We would like to point out, that the analysis of only the first diffraction orders cannot give the full information on all relevant parameters Ellabban-oex06 . However, it is sufficient to use the $\pm$ first together with the zero order diffraction and to avoid more demanding beam-coupling (interferometric) experiments. A prospective phase between the grating and the interference patternSutter-josab90 ; Kahmann-josaa93 ; Fally-josaa06 , however, cannot be determined by simple diffraction experiments. We further would like to emphasize, that the limitations of the coupled wave equations according to Ref. Kogelnik-bell69 should be kept in mind when employing Equations 1 and 2, e.g., it is assumed that the gratings are planar, purely sinusoidal and isotropic (for anisotropic gratings the theory given in Ref. Montemezzani-pre97 should be employed), $\alpha_{1}\leq\alpha_{0}$ (for violation of this condition see Shcheulin-os07 ) and only two beams are kept in the coupling scheme. If the latter is not applicable the theory of rigorous coupled waves has to be applied Moharam-josa81 , naturally with an increase of the number of coupling constants between the beams and thus with loss of simplicity. Acknowledgment We are grateful to Profs. Th. Woike and M. Imlau for providing the $\rm LiNbO_{3}~{}$sample. Financially supported by the Austrian Science Fund (P-18988) and the ÖAD in the frame of the STC program Slovenia-Austria (SI-A4/0708). We acknowledge continuous support by E. Tillmanns by making one of his labs available to us. Appendix For the particular case of $\varphi=\pi/2$ the diffraction efficiencies read: $$\displaystyle\eta_{\pm 1}(\theta)$$ $$\displaystyle=$$ $$\displaystyle A(\theta)z^{-1}4(\kappa_{1}\pm\kappa_{2})^{2}\sin^{2}{\left(z^{1% /2}D/2\right)}=$$ (6) $$\displaystyle=$$ $$\displaystyle A(\theta)z^{-1}r^{\pm 1}4(\kappa_{1}^{2}-\kappa_{2}^{2})\sin^{2}% {\left(z^{1/2}D/2\right)}$$ $$\displaystyle\eta_{0}$$ $$\displaystyle=$$ $$\displaystyle A(\theta)z^{-1}\left[\vartheta^{2}+4(\kappa_{1}^{2}-\kappa_{2}^{% 2})\cos^{2}{\left(z^{1/2}D/2\right)}\right]$$ (7) $$\displaystyle z$$ $$\displaystyle=$$ $$\displaystyle\vartheta^{2}+4(\kappa_{1}^{2}-\kappa_{2}^{2}).$$ It’s interesting to note, that for this case the diffracted and forward diffracted beams have the functional dependence of pure phase gratings with an effective coupling constant of $2[\kappa_{1}^{2}-\kappa_{2}^{2}]^{1/2}$. 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Metric-affine gravity and the Nester-Witten 2-form Marco Godina, Paolo Matteucci   & James A. Vickers${}^{\dagger}$ Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy.Faculty of Mathematical Studies, University of Southampton, Highfield, Southampton SO17 1BJ, England (UK).Corresponding author. E-mail address: p.matteucci@maths.soton.ac.uk. Abstract In this paper we redefine the well-known metric-affine Hilbert Lagrangian in terms of a spin-connection and a spin-tetrad. On applying the Poincaré-Cartan method and using the geometry of gauge-natural bundles, a global gravitational superpotential is derived. On specializing to the case of the Kosmann lift, we recover the result originally found by ?) for the metric (natural) Hilbert Lagrangian. On choosing a different, suitable lift, we can also recover the Nester-Witten 2-form, which plays an important role in the energy positivity proof and in many quasi-local definitions of mass. Introduction Conserved quantities have always represented an intriguing issue in general relativity, as was pointed out by ?) in a very famous paper. The jet bundle formalism provides an adequate framework for Lagrangian field theories and the Poincaré-Cartan method enables one to associate with each of them globally conserved charges (cf., e.g., ???). In particular, for first order theories these charges are uniquely defined, and in the second order case, although uniqueness is lost, still there is a unique canonical choice. Natural Lagrangian field theories have been known for a long time, Einstein’s general relativity being one of them. Many physical theories, though, such as Yang-Mills and Dirac theories, are non-natural, i.e. the “configuration bundle”, which is nothing but the space of the dependent variables or “fields”, is not a natural bundle. Roughly speaking, natural bundles (such as the tangent or the cotangent bundle) form a particular class of fibre bundles, where, once a coordinate change on the base manifold is given, the corresponding fibred coordinate change is known. More technically, natural bundles can be regarded as fibre bundles associated with higher order frame bundles on manifolds (cf. ?). If we aim at considering the coupling of a natural theory, such as general relativity, with a non-natural one, we are sometimes forced to “redefine” our field variables in order to make the coupling physically meaningful. In particular, if we want to describe the interaction and feedback between gravity and spinor fields, spin-tetrads, and not tetrads, are the appropriate objects to be considered (cf. ??). Gauge-natural bundles provide a suitable geometrical framework for such objects. These bundles are fibre bundles associated with “abstract” principal bundles with arbitrary structure group (cf. ?). The superpotential associated with the standard Hilbert Lagrangian for general relativity, or the “Hilbert superpotential”, was first given by ?) using ideas developed by ? himself (?) and ??). It was also derived explicitly using the Poincaré-Cartan method by ?), in a Hamiltonian (multisymplectic) framework, and ?), in the Lagrangian context111There exists an extensive literature on both the multisymplectic formulation of field theories and its Lagrangian counterpart. The interested reader is referred to ?), on page 15 and 25–6, respectively. In ?) the authors were able to reformulate the previous result of two of them in terms of tetrads. But, again, their theory was still natural, and this meant there was no real advantage of such a reformulation. Recently, two parallel papers (??) addressed the problem of re-expressing the above results in terms of spin-tetrads and coupling true general relativity with Fermionic matter, but their findings implicitly relied on a Poincaré-Cartan form associated with a particular (“quasi-natural”) lift of vector fields onto the bundle of orthonormal frames, the “Kosmann lift” (cf. ?). In this paper we redefine the metric-affine Hilbert Lagrangian in terms of a spin-connection and a spin-tetrad. The ensuing superpotential is genuinely “general”, in the sense that it is derived in a completely gauge-natural context, and also allows for the presence of torsion. Such a reformulation enables us not only to single out the aforementioned link between the Hilbert superpotential and the Kosmann lift, but also to associate the well-known Nester-Witten 2-form with another particular lift, thereby providing us with a clear-cut geometric interpretation of a rather famous but somewhat obscure integrand in general relativity. This lift turns out to be essentially the dual of the Kosmann lift. For a different characterization of the Nester-Witten 2-form see, e.g., the detailed analysis by ?). The structure of the paper is as follows: in §1 we recall the main ingredients of the Poincaré-Cartan method, in §2 we set up the geometric framework of our theory, and in §3 we derive our main results. Finally, in §4 we present a first order covariant Lagrangian for general relativity and derive the relevant superpotential. 1 Poincaré-Cartan method It is well-know that to each first order Lagrangian there corresponds a unique global Poincaré-Cartan form. Let $M$ be an (orientable, Hausdorff, paracompact, smooth) $m$-dimensional manifold and $$\left\{\begin{aligned} &\displaystyle\mathcal{L}\colon J^{1}\!B\to{\textstyle% \bigwedge}^{m}T^{*}\!M\\ &\displaystyle\mathcal{L}\colon j^{1}y\mapsto\mathcal{L}(j^{1}y)\equiv L(x^{% \lambda},y^{\mathfrak{a}},y^{\mathfrak{a}}{}_{\!\mu})\,\mathrm{d}s\end{aligned% }\right.$$ a first order Lagrangian defined on the first order jet prolongation $J^{1}\!B$ of a gauge-natural bundle $B$ over $M$ (cf. ?, §51), $(x^{\lambda},y^{\mathfrak{a}})$ being coordinates on $B$ and $\mathrm{d}s\equiv\mathrm{d}x^{0}\wedge\mathrm{d}x^{1}\wedge\dots\wedge\mathrm{% d}x^{m-1}$ the natural volume element on $M$. Define its momenta as $$f_{\mathfrak{a}}{}^{\mu}:=\frac{\partial L}{\partial y^{\mathfrak{a}}{}_{\!\mu% }}.$$ The Poincaré-Cartan form associated with $\mathcal{L}$ is then given by $$\Theta(\mathcal{L}):=\mathcal{L}+f_{\mathfrak{a}}{}^{\mu}\,\mathrm{d}_{\mathrm% {V}}y^{\mathfrak{a}}\wedge\mathrm{d}s_{\mu},$$ where $\mathrm{d}_{\mathrm{V}}$ is the vertical differential (notably, $\mathrm{d}_{\mathrm{V}}y^{\mathfrak{a}}=\mathrm{d}y^{\mathfrak{a}}-y^{% \mathfrak{a}}{}_{\!\mu}\,\mathrm{d}x^{\mu}$: cf. ? ?) and we set $\mathrm{d}s_{\mu}:=\partial_{\mu}\operatorname{\rfloor}\mathrm{d}s$, ‘$\operatorname{\rfloor}$’ denoting the inner product. The knowledge of the Poincaré-Cartan  form enables us to calculate the so-called Noether current of the Lagrangian in question. Indeed, if one has a one-parameter subgroup of automorphisms of $B$ generated by a projectable vector field $\Xi$ (with projection $\xi$ onto $M$), the Noether current associated with $\mathcal{L}$ along the vector field $\Xi$ is given by $$\displaystyle E(\mathcal{L},\Xi)$$ $$\displaystyle:=-\operatorname{Hor}[J^{1}\Xi\operatorname{\rfloor}\Theta(% \mathcal{L})]$$ $$\displaystyle\hphantom{:}=-\xi\operatorname{\rfloor}\mathcal{L}+f_{\mathfrak{a% }}{}^{\mu}\pounds{}_{\!\Xi}y^{\mathfrak{a}}\,\mathrm{d}s_{\mu},$$ where $\operatorname{Hor}$ denotes the horizontal projection (cf. ?, §3.1), $J^{1}\Xi$ is the first order jet prolongation of $\Xi$, and the well-known relation $$J^{1}\Xi\operatorname{\rfloor}\mathrm{d}_{\mathrm{V}}y^{\mathfrak{a}}=-\pounds% {}_{\!\Xi}y^{\mathfrak{a}}$$ between vertical differential and (generalized) Lie derivative is used in obtaining the second equation (cf. ?, §47). 2 Geometric framework Let $M$ be an orientable, Hausdorff, paracompact, smooth, 4-dimensional manifold. Suppose $M$ admits Lorentzian metrics of signature $-2$, i.e. assume that $M$ satisfies the topological requirements which ensure the existence on it of Lorentzian structures [$\mathrm{SO}(1,3)^{e}$-reductions]. Let $\mathbb{L}(M)$ be the (principal) bundle of linear frames over $M$ with structure group $\mathrm{GL}(4,\mathbb{R})$. Assume now that $M$ admits a free spin structure $(\Sigma,\tilde{\Lambda})$, i.e. the existence of at least one principal (fibre) bundle $\Sigma$ over $M$ with structure group $\mathrm{Spin}(1,3)^{e}\cong\mathrm{SL}(2,\mathbb{C})$, called the spin structure bundle, and at least one strong (i.e. covering the identity map) equivariant morphism $\tilde{\Lambda}\colon\Sigma\to\mathbb{L}(M)$ (?). We call the bundle map $\tilde{\Lambda}$ a spin-frame on $\Sigma$. This definition of a spin structure induces metrics on $M$. Indeed, given a spin-frame $\tilde{\Lambda}\colon\Sigma\to\mathbb{L}(M)$, we can define a metric via the reduced subbundle $\mathrm{SO}(M,g)\equiv\tilde{\Lambda}(\Sigma)$ of $\mathbb{L}(M)$. In other words, the dynamic metric $g\equiv g_{\tilde{\Lambda}}$ is defined to be the metric such that frames in $\tilde{\Lambda}(\Sigma)\subset\mathbb{L}(M)$ are $g$-orthonormal frames. It is important to stress that in our picture the metric $g$ is built up a posteriori, after a spin-frame has been determined by the field equations in a way which is compatible with the (free) spin structure one has used to define spinors. Now let $\Lambda$ be the epimorphism which exhibits $\mathrm{Spin}(1,3)^{e}$ as a twofold covering of $\mathrm{SO}(1,3)^{e}$ and consider the following left action of the group $\mathrm{GL}(4,\mathbb{R})\times\mathrm{Spin}(1,3)^{e}$ on the manifold $\mathrm{GL}(4,\mathbb{R})$ $$\left\{\begin{aligned} \displaystyle\rho&\displaystyle\colon(\mathrm{GL}(4,% \mathbb{R})\times\mathrm{Spin}(1,3)^{e})\times\mathrm{GL}(4,\mathbb{R})\to% \mathrm{GL}(4,\mathbb{R})\\ \displaystyle\rho&\displaystyle\colon((A^{\mu}{}_{\!\nu},S^{a}{}_{\!b}),u^{a}{% }_{\!\mu})\mapsto u^{\prime}{}^{a}{}_{\!\mu}:=(\Lambda(S))^{a}{}_{\!b}u^{b}{}_% {\!\nu}(A^{-1})^{\nu}{}_{\!\mu}\end{aligned}\right.$$ together with the associated bundle $\Sigma_{\rho}:=W^{1,0}(\Sigma)\times_{\rho}\mathrm{GL}(4,\mathbb{R})$, where $W^{1,0}(\Sigma):=\mathbb{L}(M)\underset{M}{\times}\Sigma$ denotes the principal prolongation of order $(1,0)$ of the principal fibre bundle $\Sigma$ (cf. ?, §52.4). The bundle $W^{1,0}(\Sigma)$ is a principal fibre bundle with structure group $\mathrm{GL}(4,\mathbb{R})\times\mathrm{Spin}(1,3)^{e}$. It turns out that $\Sigma_{\rho}$ is a fibre bundle associated with $W^{1,0}(\Sigma)$, i.e. a gauge-natural bundle of order $(1,0)$. A section of $\Sigma_{\rho}$ will be called a spin-tetrad. Recall now that a (principal) connection on a principal (fibre) bundle $P(M,G)$ may be regarded as a $G$-equivariant global section of the affine jet bundle $J^{1}\!P\to P$, where the $G$-action on $J^{1}\!P$ is induced by the first jet prolongation of the canonical (right) action of $G$ on $P$ (cf. ?, §2.7). Owing to $G$-equivariance there is a 1-1 correspondence between principal connections and global sections of the quotient bundle $J^{1}\!P/G\to M$. More specifically, let $P=\Sigma$ and let $\mathfrak{spin}(1,3)\cong\mathfrak{so}(1,3)\cong\mathfrak{sl}(2,\mathbb{C})$ denote the Lie algebra of $\mathrm{Spin}(1,3)^{e}$. Consider then the following left action on the vector space $V_{C}:=(\mathbb{R}^{4})^{*}\otimes\mathfrak{so}(1,3)$ $$\left\{\begin{aligned} \displaystyle\lambda&\displaystyle\colon(\mathrm{GL}(4,% \mathbb{R})\times T^{1}_{4}\mathrm{Spin}(1,3)^{e})\times V_{C}\to V_{C}\\ \displaystyle\lambda&\displaystyle\colon((A^{\mu}{}_{\!\nu},S^{a}{}_{\!b},S^{a% }{}_{\!b\mu}),u^{a}{}_{\!b\mu})\mapsto u^{\prime}{}^{a}{}_{\!b\mu}:=(A^{-1})^{% \nu}{}_{\!\mu}[(\Lambda(S))^{a}{}_{\!c}u^{c}{}_{\!d\nu}(\Lambda(S^{-1}))^{d}{}% _{\!b}\\ &\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad{}-(% \Lambda(S))^{a}{}_{\!c\nu}(\Lambda(S^{-1}))^{c}{}_{\!b}]\end{aligned}\right.,$$ where $(\Lambda(S))^{a}{}_{\!c\nu}$ are the components of $j^{1}_{0}(\Lambda\circ S)$, an element of $T^{1}_{4}\mathrm{SO}(1,3)^{e}$, and $S\colon\mathbb{R}^{4}\to\mathrm{Spin}(1,3)^{e}$ is a local map defined around the origin $0\in\mathbb{R}^{4}$. Hence define the associated bundle $C:=W^{1,1}(\Sigma)\times_{\lambda}V_{C}$, where $W^{1,1}(\Sigma):=\mathbb{L}(M)\underset{M}{\times}J^{1}\Sigma$ denotes the principal prolongation of order $(1,1)$ of $\Sigma$ (cf. ?, §52.4). It turns out that $C$ is a gauge-natural bundle of order $(1,1)$ isomorphic to $J^{1}(\Sigma/\mathbb{Z}_{2})/\mathrm{SO}(1,3)^{e}$. A section of $C$ will be called a spin-connection. 3 Metric-affine gravity Let $\theta^{a}{}_{\!\mu}$ be a spin-tetrad and $\omega^{a}{}_{\!b\mu}$ a spin-connection, as defined in the previous section. Set locally $$\displaystyle\theta^{a}$$ $$\displaystyle:=\theta^{a}{}_{\!\mu}\,\mathrm{d}x^{\mu},$$ $$\displaystyle e_{a}$$ $$\displaystyle:=e_{a}{}^{\mu}\partial_{\mu},$$ where $e_{a}{}^{\mu}$ is implicitly defined via the relation $\theta^{a}{}_{\!\mu}e_{b}{}^{\mu}=\delta^{a}{}_{\!b}$, and $$\displaystyle\omega^{a}{}_{\!b}$$ $$\displaystyle:=\omega^{a}{}_{\!b\mu}\,\mathrm{d}x^{\mu},$$ $$\displaystyle\Omega^{a}{}_{\!b}$$ $$\displaystyle:=\mathrm{d}_{\mathrm{H}}\omega^{a}{}_{\!b}+\omega^{a}{}_{\!c}% \wedge\omega^{c}{}_{\!b},$$ $\mathrm{d}_{\mathrm{H}}$ being the horizontal differential (cf. ?, §3.1); $\omega^{a}{}_{\!b}$ and $\Omega^{a}{}_{\!b}$ are recognized to be the (horizontal) connection 1-form and curvature 2-form, respectively. We can now “redefine” the (metric-affine) Hilbert Lagrangian as $$\left\{\begin{aligned} &\displaystyle\mathcal{L}\colon\Sigma_{\rho}\underset{M% }{\times}J^{1}C\to{\textstyle\bigwedge}^{4}T^{*}\!M\\ &\displaystyle\mathcal{L}\colon(\theta^{a}{}_{\!\mu},j^{1}\omega^{a}{}_{\!b\mu% })\mapsto\mathcal{L}(\theta^{a}{}_{\!\mu},j^{1}\omega^{a}{}_{\!b\mu}):=-\tfrac% {1}{2\kappa}\Omega_{ab}\wedge\Sigma^{ab}\end{aligned}\right.,$$ (3.1) where $\kappa:=8\pi G/c^{4}$ and $\Sigma^{ab}:=\@mathmeasure\displaystyle{\!}\@mathmeasure 4\displaystyle{\vbox t% o 0.0pt{}^{*}}\@mathmeasure 6\displaystyle{\!}\hbox to 0.0pt{}\mathord{\kern 0% .0pt}{}(\theta^{a}\wedge\theta^{b})$. The equations of motion are obtained by varying $\mathcal{L}$ with respect to $\theta^{c}$ and $\omega_{ab}$: $$\displaystyle\tfrac{\delta\mathcal{L}}{\delta\theta^{c}}$$ $$\displaystyle\equiv\tfrac{1}{2\kappa}\Omega_{ab}\wedge\Sigma^{ab}{}_{\!c}% \equiv-\tfrac{1}{\kappa}G^{a}{}_{\!c}\Sigma_{a}=0,$$ (3.2a) $$\displaystyle\tfrac{\delta\mathcal{L}}{\delta\omega_{ab}}$$ $$\displaystyle\equiv\tfrac{1}{2\kappa}\nabla\Sigma^{ab}=0,$$ (3.2b) where $\Sigma^{ab}{}_{\!c}:=e_{c}\operatorname{\rfloor}\Sigma^{ab}$, $\Sigma_{a}:=1/6\,e_{abcd}\,\theta^{b}\wedge\theta^{c}\wedge\theta^{d}$ and $\nabla$ denotes the (gauge-)covariant exterior derivative. We stress that the condition $\nabla\Sigma^{ab}=0$ is equivalent to $T^{a}\equiv\nabla\theta^{a}=0$, $T^{a}$ being the torsion 2-form. According to the definition given in §1, the appropriate Poincaré-Cartan form for Lagrangian (3.1) is $$\displaystyle\Theta(\mathcal{L})$$ $$\displaystyle\equiv\mathcal{L}+\mathrm{d}_{\mathrm{V}}\omega_{ab}\wedge\tfrac{% \partial\mathcal{L}}{\partial\mathrm{d}_{\mathrm{H}}\omega_{ab}}$$ $$\displaystyle=\mathcal{L}-\tfrac{1}{2\kappa}\mathrm{d}_{\mathrm{V}}\omega_{ab}% \wedge\Sigma^{ab},$$ (3.1) where $\partial\mathcal{L}/\partial\mathrm{d}_{\mathrm{H}}\omega_{ab}$ stands for $\partial L/\partial\omega_{ab\nu,\mu}\,\mathrm{d}s_{\mu\nu}$ and $\mathrm{d}s_{\mu\nu}:=\partial_{\nu}\operatorname{\rfloor}\mathrm{d}s_{\mu}$. Hence, the Noether current associated with a projectable vector field $\Xi$ is $$\displaystyle E(\mathcal{L},\Xi)$$ $$\displaystyle=-\xi\operatorname{\rfloor}\mathcal{L}-\tfrac{1}{2\kappa}\pounds{% }_{\!\Xi}\omega_{ab}\wedge\Sigma^{ab}$$ $$\displaystyle\equiv\tfrac{1}{2\kappa}[(\xi\operatorname{\rfloor}\Omega_{ab})% \wedge\Sigma^{ab}+\Omega_{ab}\wedge(\xi\operatorname{\rfloor}\Sigma^{ab})-% \pounds{}_{\!\Xi}\omega_{ab}\wedge\Sigma^{ab}]$$ $$\displaystyle\equiv\tfrac{1}{2\kappa}[(\xi\operatorname{\rfloor}\Omega_{ab})% \wedge\Sigma^{ab}+\xi^{c}\Omega_{ab}\wedge\Sigma^{ab}{}_{\!c}-\pounds{}_{\!\Xi% }\omega_{ab}\wedge\Sigma^{ab}].$$ (3.2) Now, our configuration bundle $B$ is $\Sigma_{\rho}\underset{M}{\times}C$, which is a gauge-natural bundle. Therefore, every (principal) automorphism $\Phi\in\mathrm{Aut}(\Sigma)$ induces an automorphism $\Phi_{B}$ on $B$. This holds also infinitesimally, i.e. for invariant (projectable) vector fields defined on $\Sigma$. Strictly speaking, an invariant vector field $\Xi\in\mathfrak{X}(\Sigma)$ defines functorially a projectable vector field $\Xi_{B}\in\mathfrak{X}(\Sigma_{\rho}\underset{M}{\times}C)$. Moreover, every $\mathrm{Spin}(1,3)^{e}$-invariant vector field $\Xi\in\mathfrak{X}(\Sigma)$ projects onto an $\mathrm{SO}(1,3)^{e}$-invariant vector field, which we denote by the same symbol $\Xi\in\mathfrak{X}(\Sigma/\mathbb{Z}_{2})$. Since the natural projection $\mathrm{pr}\colon\Sigma\to\Sigma/\mathbb{Z}_{2}$ is a covering map (locally, a diffeomorphism) of principal fibre bundles, it follows that there is a bijection between projectable $\mathrm{SO}(1,3)^{e}$-invariant vector fields on $\Sigma/\mathbb{Z}_{2}$ and projectable $\mathrm{Spin}(1,3)^{e}$-invariant vector fields on $\Sigma$ (cf. ?). If a spin-frame is given, such a bijection extends to an invariant vector field bijection between $\Sigma/\mathbb{Z}_{2}$ and $\mathrm{SO}(M,g)\equiv\tilde{\Lambda}(\Sigma)$, and, hence, between $\mathrm{SO}(M,g)$ and $\Sigma$. Yet, only the Lie derivative of the connection 1-form is needed here, so we can simply regard $\Xi_{B}$ as belonging to $\mathfrak{X}(C)$. Then, a projectable vector field $\Xi_{C}\in\mathfrak{X}(C)$ onto a vector field $\xi\equiv\xi^{\mu}\partial_{\mu}\in\mathfrak{X}(M)$ reads as $$\Xi_{C}=\xi^{\mu}\partial_{\mu}+\Xi^{a}{}_{\!b\mu}\frac{\partial}{\partial u^{% a}{}_{\!b\mu}},$$ where $$\Xi^{a}{}_{\!b\mu}:=-(\partial_{\mu}\xi^{\nu}u^{a}{}_{\!b\nu}+u^{a}{}_{\!c\mu}% \Xi^{c}{}_{\!b}-u^{c}{}_{\!b\mu}\Xi^{a}{}_{\!c}+\partial_{\mu}\Xi^{a}{}_{\!b}),$$ $\Xi\equiv\xi^{\mu}(x)\partial_{\mu}+\Xi^{a}{}_{\!b}(x)\alpha_{a}{}^{b}$ being the corresponding projectable vector field on $\Sigma/\mathbb{Z}_{2}$ and $(u^{a}{}_{\!b\mu})$ local fibre coordinates on $C$. The vector fields $\alpha_{a}{}^{b}$ are local right $\mathrm{SO}(1,3)^{e}$-invariant vector fields on $\Sigma/\mathbb{Z}_{2}$, which in a suitable chart $(x^{\mu},u_{a}{}^{b})$ read as $$\alpha_{a}{}^{b}\equiv\frac{1}{2}(\rho_{a}{}^{b}-\eta^{bc}\eta_{ad}\rho_{c}{}^% {d}),$$ $\eta$ denoting the Minkowski metric and $\rho_{a}{}^{b}:=u_{c}{}^{b}\partial/\partial u_{c}{}^{a}$. Therefore, the Lie derivative of $u^{a}{}_{\!b\mu}=\omega^{a}{}_{\!b\mu}(x)$ is just $$\pounds{}_{\!\Xi}\omega^{a}{}_{\!b\mu}=\xi^{\nu}\partial_{\nu}\omega^{a}{}_{\!% b\mu}+\partial_{\mu}\xi^{\nu}\omega^{a}{}_{\!b\nu}+\omega^{a}{}_{\!c\mu}\Xi^{c% }{}_{\!b}-\omega^{c}{}_{\!b\mu}\Xi^{a}{}_{\!c}+\partial_{\mu}\Xi^{a}{}_{\!b},$$ which can be readily recast in Cartan formalism as $$\pounds{}_{\!\Xi}\omega^{a}{}_{\!b}=\xi\operatorname{\rfloor}\Omega^{a}{}_{\!b% }+\nabla\check{\Xi}^{a}{}_{\!b},$$ (3.3) $\check{\Xi}^{a}{}_{\!b}:=\Xi^{a}{}_{\!b}+\omega^{a}{}_{\!b\mu}\xi^{\mu}$ being the vertical part of $\Xi$. On substituting (3.3) into (3.2), we finally get $$\displaystyle E(\mathcal{L},\Xi)$$ $$\displaystyle=\tfrac{1}{2\kappa}(\xi^{c}\Omega_{ab}\wedge\Sigma^{ab}{}_{\!c}-% \nabla\check{\Xi}_{ab}\wedge\Sigma^{ab})$$ $$\displaystyle=\tfrac{1}{2\kappa}[\xi^{c}\Omega_{ab}\wedge\Sigma^{ab}{}_{\!c}+% \check{\Xi}_{ab}\nabla\Sigma^{ab}-\mathrm{d}_{\mathrm{H}}(\check{\Xi}_{ab}% \Sigma^{ab})].$$ (3.4) Now, by virtue of equations of motion (3.2a) and (3.2b), $$U(\mathcal{L},\Xi):=-\frac{1}{2\kappa}\check{\Xi}_{ab}\Sigma^{ab}$$ (3.5) is recognized to be the superpotential associated with Lagrangian (3.1). This superpotential, which was derived in a completely gauge-natural context and—to the best of our knowledge—appears here for the first time, represents the most general superpotential possible in this metric-affine formulation of gravity (modulo, of course, closed 2-forms). Note that in the case of the Kosmann lift (?) we have $$\check{\Xi}_{ab}=(\check{\xi}_{\mathrm{K}}{})_{ab}\equiv-\nabla{}_{\![a}\xi_{b% ]},$$ (3.6) which, substituted in (3.5), gives $$U(\mathcal{L},\xi_{\mathrm{K}}{})=\frac{1}{2\kappa}\nabla{}_{\!a}\xi_{b}\Sigma% ^{ab},$$ (3.7) i.e. half of the well-known ?) potential, in accordance with the result found by ?) in a purely natural context. This is also the lift implicitly used by ?) in the 2-spinor formalism. Let now $\sigma_{a}{}^{AA^{\prime}}$ denote the Infeld-van der Waerden symbols, which express the isomorphism between $\operatorname{Re}[S(M)\otimes\bar{S}(M)]$ and $TM$ in the orthonormal basis induced by the spin-frame chosen (cf. ??), and consider the following lift: $$\xi^{\mu}=e_{a}{}^{\mu}\sigma^{a}{}_{\!AA^{\prime}}\lambda^{A}\bar{\lambda}^{A% ^{\prime}},\qquad\check{\Xi}_{ab}=(\check{\xi}_{\mathrm{W}}{})_{ab}:=-4\sigma_% {[a}{}^{AA^{\prime}}\sigma_{b]}{}^{BB^{\prime}}\operatorname{Re}(\bar{\lambda}% _{B^{\prime}}\nabla{}_{\!BA^{\prime}}\lambda_{A}),$$ (3.8) which will be referred to as the Witten lift. Then $$U(\mathcal{L},\xi_{\mathrm{W}}{})=\operatorname{Re}W\equiv-\frac{2}{\kappa}% \operatorname{Re}(\mathrm{i}\bar{\lambda}_{A^{\prime}}\nabla\lambda_{A}\wedge% \theta^{AA^{\prime}}),$$ (3.9) which is the (real) Nester-Witten 2-form (??). Indeed, we have222With the exception of formula (3.11) below, we shall suppress hereafter the Infeld-van der Waerden symbols and adopt the standard identification $a=AA^{\prime}$, $b=BB^{\prime}$, &c., as is customary in the current literature (cf. ?).: $$\displaystyle\check{\Xi}_{ab}\Sigma^{ab}$$ $$\displaystyle=-2\bar{\lambda}_{B^{\prime}}\nabla{}_{\!BA^{\prime}}\lambda_{A}% \Sigma^{ab}+\textsc{cc}$$ $$\displaystyle=2\mathrm{i}\@mathmeasure\displaystyle{\!}\@mathmeasure 4% \displaystyle{\vbox to 0.0pt{}^{*}}\@mathmeasure 6\displaystyle{\!}\hbox to 0.% 0pt{}\mathord{\kern 0.0pt}{}(\bar{\lambda}_{A^{\prime}}\nabla{}_{\!BB^{\prime}% }\lambda_{A})\Sigma^{ab}+\textsc{cc}$$ $$\displaystyle=2\mathrm{i}\bar{\lambda}_{A^{\prime}}\nabla{}_{\!b}\lambda_{A}% \@mathmeasure\displaystyle{\Sigma}\@mathmeasure 4\displaystyle{\vbox to 0.0pt{% }^{*}}\@mathmeasure 6\displaystyle{\Sigma}\hbox to 0.0pt{}\mathord{\kern 0.0pt% }{}^{ab}+\textsc{cc}$$ $$\displaystyle=-2\mathrm{i}\bar{\lambda}_{A^{\prime}}\nabla{}_{\!b}\lambda_{A}% \theta^{a}\wedge\theta^{b}+\textsc{cc}$$ $$\displaystyle=2\mathrm{i}\bar{\lambda}_{A^{\prime}}\nabla\lambda_{A}\wedge% \theta^{AA^{\prime}}+\textsc{cc},$$ (3.10) where we used the identities (cf. ?) $$\@mathmeasure\displaystyle{\!A}\@mathmeasure 4\displaystyle{\vbox to 0.0pt{}^{% *}}\@mathmeasure 6\displaystyle{\!A}\hbox to 0.0pt{}\mathord{\kern 0.0pt}{}_{% ab}B^{ab}=A_{ab}\@mathmeasure\displaystyle{\!B}\@mathmeasure 4\displaystyle{% \vbox to 0.0pt{}^{*}}\@mathmeasure 6\displaystyle{\!B}\hbox to 0.0pt{}\mathord% {\kern 0.0pt}{}^{ab},\qquad\@mathmeasure\displaystyle{\!A}\@mathmeasure 4% \displaystyle{\vbox to 0.0pt{}^{**}}\@mathmeasure 6\displaystyle{\!A}\hbox to % 0.0pt{}\mathord{\kern 0.0pt}{}^{ab}=-A^{ab},\qquad\@mathmeasure\displaystyle{% \!A}\@mathmeasure 4\displaystyle{\vbox to 0.0pt{}^{*}}\@mathmeasure 6% \displaystyle{\!A}\hbox to 0.0pt{}\mathord{\kern 0.0pt}{}^{ABA^{\prime}B^{% \prime}}=\mathrm{i}A^{ABB^{\prime}A^{\prime}}$$ for any two bivectors $A^{ab}$ and $B^{ab}$. Inserting (3.10) into (3.5) gives (3.9), as claimed. If we wish, it is also possible to define a complexified Witten lift as $$\xi^{\mu}=e_{a}{}^{\mu}\sigma^{a}{}_{\!AA^{\prime}}\lambda^{A}\bar{\lambda}^{A% ^{\prime}},\qquad\check{\Xi}_{ab}=(\check{\xi}_{\mathrm{W}}^{\mathbb{C}}{})_{% ab}:=-4\sigma_{[a}{}^{AA^{\prime}}\sigma_{b]}{}^{BB^{\prime}}\bar{\lambda}_{B^% {\prime}}\nabla{}_{\!BA^{\prime}}\lambda_{A}.$$ (3.11) Then, the relevant superpotential is $$U(\mathcal{L},\xi_{\mathrm{W}}^{\mathbb{C}}{})=W:=-\frac{2\mathrm{i}}{\kappa}% \bar{\lambda}_{A^{\prime}}\nabla\lambda_{A}\wedge\theta^{AA^{\prime}},$$ (3.12) which is the (complex) Nester-Witten 2-form (??). From the viewpoint of physical applications (proof of positivity of the Bondi or ADM mass, quasi-local definitions of momentum and angular momentum in general relativity, &c.), it is immaterial whether one uses (3.12) or its real part (3.9), as its imaginary part turns out to be $-1/\kappa\,\mathrm{d}_{\mathrm{H}}(\lambda_{A}\bar{\lambda}_{A^{\prime}}\theta% ^{a})$, which vanishes upon integration over a closed 2-surface. Note, though, that (3.12) appears to relate more directly to Penrose quasi-local 4-momentum, when suitable identifications are made (cf. ?, p. 432). Remark 3.1. Note also that—modulo an inessential numerical factor—the Kosmann lift is (the real part of) the dual of the (complex) Witten lift, in the sense that $$(\check{\xi}_{\mathrm{K}}{})_{ab}=-\frac{1}{2}\operatorname{Re}[\@mathmeasure% \displaystyle{\!}\@mathmeasure 4\displaystyle{\vbox to 0.0pt{}^{*}}% \@mathmeasure 6\displaystyle{\!}\hbox to 0.0pt{}\mathord{\kern 0.0pt}{}(\check% {\xi}_{\mathrm{W}}^{\mathbb{C}}{})_{ab}],$$ as can be easily checked on starting from equations (3.6) and (3.11)(2), whenever, of course, $\xi^{a}=\lambda^{A}\bar{\lambda}^{A^{\prime}}$. Remark 3.2. The theory developed herein is obviously tailored to the coupling with spinor fields described by the Dirac Lagrangian, $$\mathcal{L}_{\mathrm{D}}:=\left[\frac{\mathrm{i}}{2}(\tilde{\Psi}\gamma^{a}% \nabla{}_{\!a}\Psi-\widetilde{\nabla{}_{\!a}\Psi}\gamma^{a}\Psi)-m\tilde{\Psi}% \Psi\right]\sqrt{g}\,\mathrm{d}s.$$ In the purely metric case, the total superpotential turns out to be $$U(\mathcal{L}+\mathcal{L}_{\mathrm{D}},\Xi)=U(\mathcal{L},\Xi)+U(\mathcal{L}_{% \mathrm{D}},\Xi),$$ where $$\displaystyle U(\mathcal{L}_{\mathrm{D}},\Xi)$$ $$\displaystyle:=\tfrac{\mathrm{i}}{8}\tilde{\Psi}[(\gamma_{a}\gamma_{b}\gamma_{% c}+2\eta_{ac}\gamma_{b})\xi^{c}]\Psi\,\Sigma^{ab},$$ $$\displaystyle\hphantom{:}\equiv\tfrac{\mathrm{i}\sqrt{2}}{4}\xi_{A}^{A^{\prime% }}(\bar{\varphi}_{A^{\prime}}\varphi_{B}-\bar{\psi}_{B}\psi_{A^{\prime}})% \Sigma^{AB}+\textsc{cc}.$$ The reader is referred to ?) and ?) for further details and notation. Conversely, in the present metric-affine context, it can be readily shown that, although the Dirac Lagrangian does enter the equations of motion (notably, the “second” Einstein-Cartan equation), it does not contribute to the total superpotential. From this fact one might mistakenly conclude that the Dirac fields do not contribute to the total conserved quantities. This conclusion is wrong because, although the Dirac Lagrangian does not contribute directly to the superpotential, in order to obtain the corresponding conserved quantities, one needs integrate the superpotential on a solution, which in turn depends on the Dirac Lagrangian via its energy-momentum tensor and the second Einstein-Cartan equation. 4 First order gravity In the case of vanishing torsion ($T^{a}\equiv 0\iff\nabla\Sigma^{ab}\equiv 0$), it is easy to see that Lagrangian (3.1) can be split into a total divergence plus a first order covariant Lagrangian. In many contexts, the superpotential associated with this Lagrangian proved to give more physically reasonable answers than the Hilbert superpotential (cf. ?). For this reason and the sake of completeness, we now give the derivation of the aforementioned superpotential in the new geometrical framework outlined in §2. The first order covariant Lagrangian in question is (cf. ??) $$\displaystyle\hat{\mathcal{L}}$$ $$\displaystyle:=-\tfrac{1}{2\kappa}(\hat{\Omega}_{ab}-Q_{ac}\wedge Q^{c}{}_{\!b% })\wedge\Sigma^{ab}$$ (4.1) $$\displaystyle\hphantom{:}\equiv\mathcal{L}+\tfrac{1}{2\kappa}\mathrm{d}_{% \mathrm{H}}(Q_{ab}\wedge\Sigma^{ab}),$$ where $\mathcal{L}$ is given by (3.1), $\hat{\Omega}_{ab}:=\mathrm{d}_{\mathrm{H}}\hat{\omega}_{ab}+\hat{\omega}_{ac}% \wedge\hat{\omega}^{c}{}_{\!b}$ and $Q_{ab}:=\omega_{ab}-\hat{\omega}_{ab}$, $\hat{\omega}_{ab}$ being a “background” (non-dynamical) spin-connection. The corresponding Poincaré-Cartan form is $$\displaystyle\Theta(\hat{\mathcal{L}})$$ $$\displaystyle=\hat{\mathcal{L}}-\tfrac{1}{2\kappa}(\mathrm{d}_{\mathrm{V}}\hat% {\omega}_{ab}\wedge\Sigma^{ab}-\mathrm{d}_{\mathrm{V}}\Sigma^{ab}\wedge Q_{ab})$$ $$\displaystyle\equiv\Theta(\mathcal{L})+\tfrac{1}{2\kappa}[\mathrm{d}_{\mathrm{% H}}(Q_{ab}\wedge\Sigma^{ab})+\mathrm{d}_{\mathrm{V}}Q_{ab}\wedge\Sigma^{ab}+% \mathrm{d}_{\mathrm{V}}\Sigma^{ab}\wedge Q_{ab}],$$ Hence, the Noether current associated with a projectable vector field $\Xi$ is $$\displaystyle E(\hat{\mathcal{L}},\Xi)$$ $$\displaystyle=E(\mathcal{L},\Xi)+\tfrac{1}{2\kappa}[\pounds{}_{\!\Xi}Q_{ab}% \wedge\Sigma^{ab}+\pounds{}_{\!\Xi}\Sigma^{ab}\wedge Q_{ab}-\xi\operatorname{% \rfloor}\mathrm{d}_{\mathrm{H}}(Q_{ab}\wedge\Sigma^{ab})]$$ $$\displaystyle=E(\mathcal{L},\Xi)+\tfrac{1}{2\kappa}[\pounds{}_{\!\Xi}Q_{ab}% \wedge\Sigma^{ab}+\pounds{}_{\!\Xi}(Q_{ab}\wedge\Sigma^{ab})-\pounds{}_{\!\Xi}% Q_{ab}\wedge\Sigma^{ab}$$ $$\displaystyle\qquad\qquad\qquad\quad{}-\xi\operatorname{\rfloor}\mathrm{d}_{% \mathrm{H}}(Q_{ab}\wedge\Sigma^{ab})]$$ $$\displaystyle=E(\mathcal{L},\Xi)+\tfrac{1}{2\kappa}[\pounds{}_{\!\Xi}(Q_{ab}% \wedge\Sigma^{ab})-\xi\operatorname{\rfloor}\mathrm{d}_{\mathrm{H}}(Q_{ab}% \wedge\Sigma^{ab})],$$ (4.2) $\xi$ denoting, as usual, the projection of $\Xi$ onto $M$. Now, $$\displaystyle\pounds{}_{\!\Xi}(Q_{ab}\wedge\Sigma^{ab})$$ $$\displaystyle\equiv\pounds{}_{\!\xi}(Q_{ab}\wedge\Sigma^{ab})$$ $$\displaystyle=\xi\operatorname{\rfloor}\mathrm{d}_{\mathrm{H}}(Q_{ab}\wedge% \Sigma^{ab})+\mathrm{d}_{\mathrm{H}}[\xi\operatorname{\rfloor}(Q_{ab}\wedge% \Sigma^{ab})].$$ (4.3) On substituting (4.3) into (4.2), we get $$E(\hat{\mathcal{L}},\Xi)=E(\mathcal{L},\Xi)+\frac{1}{2\kappa}\mathrm{d}_{% \mathrm{H}}[\xi\operatorname{\rfloor}(Q_{ab}\wedge\Sigma^{ab})],$$ whence $$U(\hat{\mathcal{L}},\Xi):=U(\mathcal{L},\Xi)+\frac{1}{2\kappa}\xi\operatorname% {\rfloor}(Q_{ab}\wedge\Sigma^{ab})$$ (4.4) is recognized to be the superpotential associated with Lagrangian (4.1). Remark 4.1. Note that, contrary to what happens in the purely natural context, no additional conditions need be imposed on the vector field $\Xi$ here. Discussion This paper stresses the important role the theory of gauge-natural bundles plays in a significant issue of mathematical physics such as the definition of the gravitational energy and, more generally, of conserved quantities associated with the gravitational field, especially when coupled to spinor fields. Besides providing a new gravitational superpotential in a gauge-natural context, the paper sheds some new light on the definition of the Nester-Witten 2-form and gives it an interpretation as a further, genuine gravitational superpotential. Moreover, this paper shows that it is crucial in this context not to regard the metric as the fundamental gravitational field. Indeed, when considering the interaction between gravity and spinors one is forced to give up a purely natural formalism and instead consider a gauge-natural formalism in which one chooses a spin-tetrad (together with a spin-connection, in a metric-affine formulation) as one’s fundamental variable. A parallel and analogous method of investigation is possible when dealing, in a gauge-natural context, with Legendre and dual Legendre transforms, for which the reader is referred to the recent and fundamental papers by ?) and ?). Acknowledgements P. M. acknowledges an EPSRC research studentship and a Faculty Research Studentship from the University of Southampton. \bibname\@mkboth\bibname\bibname Dubois-Violette, M. & Madore, J. (1987). ‘Conservation laws and integrability conditions for gravitational and Yang-Mills field equations’. Comm. Math. Phys. 108, 213–23. Fatibene, L., Ferraris, M., Francaviglia, M. & Godina, M. (1996). ‘A geometric definition of Lie derivative for spinor fields’. In J. Janyška, I. Kolář & J. Slovák (Eds.), Proc. 6th International Conference on Differential Geometry and its Applications (Brno, 1995), pp. 549–58. Brno: Masaryk University. Fatibene, L., Ferraris, M., Francaviglia, M. & Godina, M. 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Linear scaling calculation of band edge states and doped semiconductors H. J. Xiang Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China    Jinlong Yang Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China    J. G. Hou Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China    Qingshi Zhu Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China (November 25, 2020) Abstract Linear scaling methods provide total energy, but no energy levels and canonical wavefuctions. From the density matrix computed through the density matrix purification methods, we propose an order-N (O(N)) method for calculating both the energies and wavefuctions of band edge states, which are important for optical properties and chemical reactions. In addition, we also develop an O(N) algorithm to deal with doped semiconductors based on the O(N) method for band edge states calculation. We illustrate the O(N) behavior of the new method by applying it to boron nitride (BN) nanotubes and BN nanotubes with an adsorbed hydrogen atom. The band gap of various BN nanotubes are investigated systematicly and the acceptor levels of BN nanotubes with an isolated adsorbed H atom are computed. Our methods are simple, robust, and especially suited for the application in self-consistent field electronic structure theory. ††thanks: Corresponding author. E-mail: jlyang@ustc.edu.cn I Introduction Traditional electronic structure algorithms calculate all eigenstates associated with discrete energy levels. The disadvantage of this approach is that it leads to a diagonalization problem that has an unfavorable cubic scaling in the computational effort. Linear scaling density functional or Hartree-Fock methods are an essential tool for the calculation of the electronic structure of large systems containing many atoms.DM_rev The key point of the success of most linear scaling methods is that only the density matrix or localized Wannier functions which span the occupied manifold is calculated. In these O(N) methods, no canonical wavefunctions or eigenvalues are available. However, in many cases, one may be interested in some eigenstates, especially the states near the Fermi level, i.e., band edge states. For instance, from the theory of frontier orbitals, many molecular properties are determined by the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO), and frontier orbitals play an important role in chemical reactions. On the other hand, there are some linear scaling algorithms such as the Kim-Mauri-Galli (KMG)KMG method need the Fermi level which can be estimated from the HOMO and LUMO energy. There are some methods which can be used to obtain band edge states. The most popular method for calculating states near a reference energy $\epsilon_{ref}$ is the folded spectrum method.Wang1994 However, in this method, by squaring the Hamiltonian, the condition number is also squared and thus the difficulty of solving the equation is also increased. To solve this problem, Tackett et al.Tackett2002 presented the Jacobi-Davidson method in which the condition number and difficulty in solving for the selected eigensolutions is the same as the original eigenvalue equation. Unfortunately, the implementation of the Jacobi-Davidson method is rather involved and its application is not widespread. In the field of computational mathematics, the shift-and-invert Lanczos algorithm is a well-known method for calculating a pair of eigenvalue and eigenvector near a reference energy. This method was used by Liang et al.Liang2003 to obtain the Fermi level in the context of linear scaling Fermi operator expansion method. In this method, the Lanczos method is applied to the so-called shift-and-invert matrix, $(H-\epsilon_{ref}I)^{-1}$, where $H$ and $I$ are the Hamiltonian and identity matrices, respectively, and $\epsilon_{ref}$ is the reference energy. These matrices are not, of course, formed explicitly. Instead, each time the Lanczos method requires a multiplication of a vector $v$ by matrix $(H-\epsilon_{ref}I)^{-1}$, a linear solver subroutine is called to solve the corresponding linear systems. If these linear systems are solved sufficiently accurately, the convergence of the Lanczos method is typically much faster compared to that when the matrix $H$ is used in the Lanczos method. The difficulty now is that accurate numerical solution of linear systems, needed on each iteration of the Lanczos method, can be costly. Besides the difficulties of these methods mentioned above, when they are applied to get the frontier orbitals, another inconvenience is that a reference energy $\epsilon_{ref}$ must be selected. Here in this work, we present an alternative simple method to get states near gap based on linear scaling density matrix methods. In our method, we do not need the reference energy. The new O(N) method is particularly useful for calculating frontier orbitals in the framework of self-consistent field (SCF) electronic structure theory. Using this method, we also propose a promising linear scaling method which can be utilized to explore the energetics, defective levels, and gemoetry of doped semiconductors. This paper is organized as follows: In Sec. II, we present our new O(N) methods for calculating band edge states and dealing with doped semiconductors. In Sec. III, we describe the details of the implementation and perform some test calculations to illustrate the rightness, robustness, and linear-scaling behavior of our methods. In Sec. IV, we use our new methods to calculate the band gap of boron nitride (BN) nanotubes and the acceptor level of a single H adsorbed BN nanotubes. Finally, our concluding remarks are given in Sec. V. II Theory II.1 Calculation of band edge states Within our method, we must first obtain density matrix $\rho$ corresponding a given Hamiltonian $H$ before we proceed to calculate band edge states. However, it is not an inconvenience in the framework of linear scaling SCF electronic structure theory. In principles, any linear scaling density matrix methods can be used to obtain the density matrix.DM_rev ; DM11 ; fd Moreover, O(N) localized orbital based methods can also be used to construct the density matrix.DM_rev ; OM1 ; OM2 In the representation of molecular canonical orbitals, density matrix $\rho$ and Hamiltonian $H$ are diagonal matrices of the following forms: $$\begin{array}[]{lll}\rho&=&diag(1,1,\ldots,1,0,0,\ldots,0),\\ H&=&diag(\epsilon_{1},\epsilon_{2},\ldots,\epsilon_{N_{e}/2},\epsilon_{N_{e}/2% +1},\ldots,\epsilon_{N_{b}}),\\ \end{array}$$ (1) where $N_{e}$ is the number of electrons of a closed-shell system, and $N_{b}$ is the number of basis functions. Without loss of generality, we assume that $$\epsilon_{1}\leq\epsilon_{2}\leq\ldots\leq\epsilon_{N_{e}/2}\leq\epsilon_{N_{e% }/2+1}\leq\ldots\leq\epsilon_{N_{b}},$$ (2) then $\epsilon_{N_{e}/2}$ and $\epsilon_{N_{e}/2+1}$ will be the HOMO and LUMO energies respectively. It can be easily seen that: $$\rho H=H\rho=diag(\epsilon_{1},\epsilon_{2},\ldots,\epsilon_{N_{e}/2},0,\ldots% ,0).$$ (3) If $\epsilon_{N_{e}/2}>0$, then $\epsilon_{N_{e}/2}$ will be the largest eigenvalue of $\rho H$. Otherwise, we can shift the Hamiltonian H to $H+\lambda I$ ($\lambda>0$) so that $\lambda+\epsilon_{N_{e}/2}>0$. Clearly, $\lambda+\epsilon_{N_{e}/2}$ is the largest eigenvalue of $\rho(H+\lambda I)$. Using the similar argument, we can prove that if $-\lambda+\epsilon_{N_{e}/2+1}<0$, $-\lambda+\epsilon_{N_{e}/2+1}$ will be the smallest eigenvalue of $(I-\rho)(H-\lambda I)$. We should note that the parameter $\lambda$ can be set to be a large positive value without degrading the efficiency of the method. In practice, we find that it is usually reliable by setting $\lambda$ to be $1$ Ry. The largest (smallest) eigenvalue and its corresponding eigenvector of $\rho(H+\lambda I)$ ($(I-\rho)(H-\lambda I)$) can be computed easily using the well-known O(N) Lanczos method. Up to now, we discuss the problem in the representation of molecular canonical orbitals $\psi$. In the representation of orthogonal basis orbitals $\phi$, molecular canonical orbitals $\psi$ can be expressed as $$\psi_{i}=\sum_{\mu}\phi_{\mu}C_{\mu i},$$ (4) where the coefficient matrix $C$ is a unitary matrix. Thus in the representation of orthogonal basis orbitals, density matrix $\rho_{or}$ and Hamiltonian $H_{or}$ can be calculated as: $$\begin{array}[]{lll}\rho_{or}&=&C\rho C^{+},\\ H_{or}&=&CHC^{+}.\\ \end{array}$$ (5) Moreover, $\rho_{or}H_{or}$ can also be obtained through a unitary transformation of $\rho H$. Since the unitary transformation of a matrix does not change its eigenvalues, we can see that the above results deduced using the representation of molecular canonical orbitals also hold in the representation of orthogonal basis orbitals. The procedure for obtaining HOMO and LUMO states are illustrated in Fig. 1(a). Since many first principles codes use non-orthogonal atomic orbitals, here we discuss the case of non-orthogonal basis. A general method is transforming the non-orthogonal basis to orthogonal basis. We achieve this by transforming the atomic orbital (AO) Hamiltonian matrix $H_{ao}$ to an orthonormal basis using $H_{or}=ZH_{ao}Z^{T}$ and obtaining the AO density matrix $\rho_{ao}$ using $\rho_{ao}=Z^{T}\rho_{or}Z$, where the inverse factor $Z=L^{-1}$, and $L$ is the Cholesky factor for which $S=LL^{T}$. The Cholesky transformation has been used in severval linear scaling densit matrix programs.We next show how to get wavefunction in the non-orthogonal basis. In non-orthogonal basis, a generalized eigenvalue problem should be solved: $$H_{ao}\psi_{ao}=\epsilon S\psi_{ao},$$ (6) where $\psi_{ao}$ is the wavefunction in the non-orthogonal basis. Given the wavefunction in the orthogonal basis $\psi_{or}$, which satisfies $$H_{or}\psi_{or}=ZH_{ao}Z^{T}\psi_{or}=\epsilon\psi_{or},$$ (7) we have $$\begin{array}[]{rll}H_{ao}Z^{T}\psi_{or}&=&\epsilon Z^{-1}Z^{-T}Z^{T}\psi_{or}% \\ &=&\epsilon SZ^{T}\psi_{or},\\ \psi_{ao}&=&Z^{T}\psi_{or}.\end{array}$$ (8) We also present another method to calculate band edge states in non-orthogonal basis without transforming to orthogonal basis. This method is particularly useful when localized orbitals based O(N) algorithms are employed. From $\rho_{ao}H_{ao}=Z^{T}\rho_{or}H_{or}Z^{-T}$, one can easily see that $\rho_{ao}H_{ao}\psi_{ao}=\epsilon\psi_{ao}$ is equivalent to $\rho_{or}H_{or}Z^{-T}\psi_{ao}=\epsilon Z^{-T}\psi_{ao}$. Thus $\rho_{ao}H_{ao}$ has the same eigenvalues as $\rho_{or}H_{or}$. We can also prove that $\rho_{ao}(H_{ao}+\lambda S)$ has the same eigenvalues as $\rho_{or}(H_{or}+\lambda)$. Thus the largest eigenvalue of $\rho_{ao}(H_{ao}+\lambda S)$ will be $\epsilon(\mathrm{HOMO})+\lambda$. We should point out that $\rho_{or}(H_{or}+\lambda)$ is hermitian, but $\rho_{ao}(H_{ao}+\lambda S)$ is not. However, this doesn’t pose any problem since the Lanczos algorithm can also be used to get the extreme eigenvalues of a non-hermitian matrix. We can see that the calculation of HOMO state is simple since only $\rho_{ao}$, $H_{ao}$, and $S$ are needed. However, the calculation of the LUMO state is a different story. We can easily prove that $(I-\rho_{ao}S)(S^{-1}H_{ao}-\lambda I)$ has the same eigenvalues as $(I-\rho_{or})(H_{or}-\lambda I)$ and $-\lambda+\epsilon(\mathrm{LUMO})$ is the smallest eigenvalue of $(I-\rho_{ao}S)(S^{-1}H_{ao}-\lambda I)$. As can be seen, to calculate the LUMO state, besides $\rho_{ao}$, $H_{ao}$, and $S$, we must also have $S^{-1}$ or $S^{-1}H_{ao}$. The inverse of $S$ is usually a formidable task. Fortunately, Gibson et al. introduced an O(N) method to calculate $S^{-1}H_{ao}$.Gibson1993 II.2 Treatment of doped semiconductors To our best knowledge, most linear scaling methods are mainly applied to semiconductors or insulators with an energy gap. When the system is metallic or gapless, these O(N) methods fail or lose of effectiveness since these methods rely on the sparsity of the density matrix and the convergence of many of these methods is determined by the magnitude of band gap. Partial occupation is another obstacle for many popular linear scaling methods due to the non-idempotence of the density matrix. Here we propose an O(N) method to deal with doped semiconductors where dopants or defects exist. Our method has the similar spirit as that proposed by Raczkowski and Fong in that a subspace larger than the occupied space is used.Raczkowski2003 In their seminal work, the subspace optimization method formulated in terms of localized nonorthogonal orbitals was employed. However, besides two O(N${}^{3}$) steps in the Grassmann conjugate gradient (GCG) algorithm, an additional O(N${}^{3}$) step of diagonalization is needed. Another problem is that when the orbital localization is used to acheive linear scaling, local minima might occur in the subspace optimization method, resulting in a stalling of GCG algorithm during the last several SCF steps.Raczkowski2003 In our method, we treat the valence bands using the density matrix method, and other defective bands are calculated using our O(N) method for band edge states. For simplicity, we consider the cases where only an electron or hole is present in a semiconductor, as shown in Fig. 2. In case of n-type doping (Fig. 2(a)), the total density matrix $\rho$ can be calculated as $$\rho=\rho_{val}+0.5|\psi_{N+1}\rangle\langle\psi_{N+1}|,$$ (9) where $\rho_{val}$ is density matrix corresponding to the valence band. In case of p-type doping (Fig. 2(b)), the total density matrix $\rho$ can be calculated as $$\rho=\rho_{val}-0.5|\psi_{N}\rangle\langle\psi_{N}|.$$ (10) Both $\psi_{N+1}$ and $\psi_{N}$ are computed through the newly developed O(N) method for band edge states. Using the block Lanczos algorithm, our method can also be used when several doped levels are present. In this case, the Fermi distribution can be used to get the occupation of doped levels. Since the valence band are well separated from the conduction band, $\rho_{val}$ is sparse, and the calculation of $\rho_{val}$ can be carried out using traditional O(N) methods, such as the trace-correcting density matrix purification (TC2) method.DM11 Since canonical orbitals $\psi_{N+1}$ and $\psi_{N}$ are usually delocalized, the total density matrix $\rho$ is much denser than $\rho_{val}$. It is difficult to deal with the full density matrix. However, we notice that in fact only a small part of the full density matrix is used in the construction of the new Hamiltonian. Thus in practice, we only construct a small part of the full density matrix. To make our O(N) method for the treatment of doped semiconductors more clear, we show the flow-chart of a typical calculation in Fig. 1(b). Our method is very simple and applicable to many doped systems. We should mention that our method is not a black-box method since some knowledge of the studied system must be known prior. For instance, we should know the doping type and number of doping levels. Typically, we can get this information from intuition or deduction from other smaller systems with similar characters. III Implementation and Results III.1 Implementation Our newly developed method has been implemented in SIESTA,siesta a standard Kohn-Sham density-functional program using norm-conserving pseudopotentials and numerical atomic orbitals as basis sets. In SIESTA, periodic boundary conditions are employed to simulate both isolated and periodic systems. Here we use the O(N) TC2 methodDM11 to get the density matrix since it is very simple, robust, and efficient. The details about the implementation of the TC2 method can be found in Ref. 13. In our O(N) method for doped semiconductors, to obtain atomic forces, it is necessary to get the energy weighted density matrix $E$ when using atomic basis sets. Take the case as shown in Fig. 2(a) as an example, $$E=E_{val}+0.5\epsilon_{N+1}|\psi_{N+1}\rangle\langle\psi_{N+1}|,$$ (11) where $E_{val}$ is calculated from $\rho_{val}$. For energy weighted density matrix $E$, we also compute and save only a part of the full matrix. To speed up the calculation, we adopt the block Lanczos method to calculate the defect levels, since the vectors produced by the previous SCF step can be reused in the subsequent step. Usually, in the last several SCF steps, we don’t need any matrix-vector multiplications in the calculation of band edge states. Thus, when a geometry optimization is performed, the extra amount for computing defect levels using our O(N) method is almost negligible. This contrasts to the method proposed by Raczkowski and Fong.OM1 ; OM2 ; Raczkowski2003 III.2 Validity and performance of the O(N) method for band edge states calculation All our calculations reported in this work are done in the local density approximation (LDA).LDA Unless otherwise stated, the double-$\zeta$ plus polarization functions (DZP) basis set is used in the calculations. We first validate our method by computing the HOMO and LUMO of H${}_{2}$O molecule. The energies of HOMO and LUMO are $-7.532$ ($-7.53257$) and $-1.375$ ($-1.37292$) eV, respectively (values in parenthesis are results from the diagonalization method). We also compare the HOMO and LUMO wavefunctions with those from the diagonalization method, and find that the agreement is remarkable. To check the performance of our method, we calculate the HOMO and LUMO of BN(5,5) nanotubes with different number of atoms in the supercells. The CPU time used is shown in Fig. 3. We can clearly see the linear scaling behavior of our new method for both single-$\zeta$ (SZ) and DZP basis sets. For the purpose of comparison, we also calculate the LUMO of BN(5,5) nanotube with 400 atoms using the folded spectrum method. The SZ basis is adopted. Since the performance of the folded spectrum method is very sensitive to the choice of the reference energy, several different reference energies varying from the midgap position to the LUMO energy are chosen. The precision of the calculation is within 3 meV with respect to the value from the diagonalization. As shown in Fig. 4, the CPU time used is very large, especially when the reference energy is close to the LUMO energy (the HOMO and LUMO energies are -7.075 and -2.577 eV respectively in the current computing parameters setting). Even when the the reference energy is chosen to be optimal, the folded spectrum method is still slower by seventeen times than our new O(N) method (387 s v.s. 22 s). III.3 Validity and performance of the O(N) method for doped semiconductors We will take BN(8,0) zig-zag nanotubes with an adsorbed H atom as an example to illustrate the correctness and efficiency of our new method. As shown by Wu et al., a H atom prefers to adsorb on a B atom, and the system is a $p$-type semiconductor.Wu2004 For a BN(8,0) nanotube (128 atoms in the supercell) with an adsorbed H atom, the energy difference between our result and that from the diagonalization method is only 6 meV. And the force differences between our result and that from the diagonalization method do not exceed 0.6 meV/Å. Both the energy and forces agreement validates our new O(N) method for doped semiconductors. We also deal successfully with a BN(8,0) nanotube with two adsorbed H atoms on two B sites, indicating that our method also works in case of systems with multi defect levels. Here we show in Fig. 5 the CPU time used in an ion step for supercells with different number of atoms. Clearly, our new method for doped semiconductors displays a linear scaling behavior. IV applications IV.1 Band gap of BN chiral nanotubes Previous study showed that for small zigzag (chiral angle $\alpha=0^{\circ}$) nanotubes the energy gap decreases rapidly with the decrease of radius, while armchair nanotubes (chiral angle $\alpha=30^{\circ}$) almost have a constant energy gap. Xiang2003 Although previous experimentszig-zag indicated a preference for zig-zag and near zig-zag BN tubes and a plausible explanation Xiang2003 was proposed, a very recent high-resolution electron diffraction study on BN nanotubes grown in a carbon-free chemical vapor deposition process revealed a dispersion of the chiral angles.BN_chiral Thus a thorough knowledge of the dependence of the band gap upon the chirality of BN nanotubes is desirable. Chiral BN nanotubes usually contain large number of atoms in a unit cell, e.g., a BN(14,1) nanotube has 844 atoms in the unit cell. These nanotubes are difficult to be treated using traditional methods. Here we calculate systematicly the band gap of BN nanotubes including chiral BN nanotubes. Whenever the system is large enough to be sampled using $\Gamma$-point, we use the new O(N) method for calculating band edge states. The results are shown in Fig. 6. Two general trends are observed: first for BN nanotubes with similar radius, BN nanotubes with larger chiral angles have larger band gaps, secondly, for BN nanotubes with chiral angles close to zero, BN nanotubes with larger radius have larger band gaps. In addition, we can see that for BN($n$,$m$) nanotubes with $n+m=k$, the band gap of BN($n$,$k-n$) does not depend monotonously on the $n$ value due to the competition of the two trends mentioned above, however, the band gap of BN($k-[\frac{k}{2}]$,$[\frac{k}{2}]$) (Here $[\frac{k}{2}]$ denotes the maximal integer no larger than $\frac{k}{2}$) nanotube is the largest, and BN($k$,0) nanotube usually has the smallest band gap except that the band gap of BN(8,2) nanotube is small than that of BN(10,0) nanotube. The band gaps of some BN nanotubes were reported previously and the results are in accord with ours,Xiang2003 ; Guo2005 and a more complete picture for the trend of the band gap of BN nanotubes is presented here. IV.2 Acceptor level of H adsorbed BN nanotubes Wu et al.Wu2004 investigated the adsorption of a hydrogen atom on zigzag BN(8,0) nanotube using a supercell containing 32 boron and 32 nitrogen atoms and found H prefers to adsorb on the boron atom which introduces an acceptor state in the gap. They also showed that the dispersion of the defect band is as large as 0.2 eV. Our test calculations in the $\Gamma$-only approximation also show that the acceptor levels of a single H adsorbed BN(8,0) nanotube using a 64-atoms or 128-atoms supercell are 1.064 eV and 1.180 eV, respectively (Here, the acceptor level is defined as the energy difference between the acceptor state and the top of the valence band). In addition, the adsorption energy of the H atom also depends on the chosen supercell: For instance, the adsorption energy is $-0.353$ ($-0.246$) eV when using a 64-atoms (320-atoms) BN(8,0) supercell and the diagonalization (our linear scaling) method. All these facts suggest that larger supercells should be used to predict the properties of BN nanotubes with an isolated adsorbed H atom. Here with the O(N) method for doped semiconductors developed in this paper, we can treat much larger radius BN nanotubes with truely isolated adsorbed H atom through using huge supercells. Three BN nanotubes are considered: BN(8,0) nanotube simulated using a supercell with 320 atoms, BN(15,0) nanotube simulated using a supercell with 720 atoms, and BN(13,2) nanotube with 796 atoms in the unit cell. Here we show the distribution of the acceptor state and the highest orbital of the valence band in Fig. 7. Clearly, the acceptor state is a relatively localized state around the adsorbed H atom, which agrees with the result reported by Wu et al.Wu2004 . However, the highest orbital of the valence band is delocalized and mainly contributed by N 2p${}_{\mathrm{z}}$ orbitals. As can be seen from Fig. 6, BN(15,0) nanotube and BN(13,2) nanotube have similar radius but different chirality, and the radius of BN(8,0) nanotube is smaller. The calculated acceptor levels introduced by an isolated H atom are 1.184 eV, 1.557 eV and 1.563 eV for BN(8,0), BN(15,0) and BN(13,2) nanotubes, respectively. Thus the position of the defect level is closer to the top of valence bands for smaller radius BN nanotubes, but does not depend significantly on the chirality. V Conclusions We present a simple O(N) method for calculating band edge states using the density matrix obtained from O(N) electronic structure methods. Based on the O(N) method for calculating band edge states, we further develop an O(N) algorithm to deal with doped semiconductors. In our methods, no reference energy is needed to obtain the band edge states, and they are especially suited for the application in SCF electronic structure theory. The O(N) behavior of the new methods is demonstrated by applying it to bare and H adsorbed BN nanotubes. The band gap of various BN nanotubes are investigated systematicly and the acceptor levels of BN nanotubes with an isolated adsorbed H atom are calculated. Our algorithms could be generalized straightforwardly to spin-unrestricted systems,Xiang2005 such as magnetic semiconductors and diluted magnetic semiconductors.spin This work is partially supported by the National Natural Science Foundation of China (50121202, 20533030, 10474087), by the USTC-HP HPC project, and by the SCCAS and Shanghai Supercomputer Center. References (1) S. Goedecker, Rev. Mod. Phys. 71, 1085 (1999); S. Y. Wu and C. S. Jayanthi, Phys. Rep. 358, 1 (2002). (2) J. Kim, F. Mauri, and G. Galli, Phys. Rev. B 52, 1640 (1995). (3) L. W. Wang and A. Zunger, J. Chem. Phys. 100, 2394 (1994). (4) A. R. Tackett and M. D. Ventra, Phys. Rev. B 66, 245104 (2002). (5) W. Z. Liang, C. Saravanan, Y. H. Shao, R. Baer, A. T. 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Gravitational Lensing from a Spacetime Perspective Volker Perlick Physics Department Lancaster University Lancaster LA1 4YB United Kingdom email: vperlick@lancaster.ac.uk () Abstract The theory of gravitational lensing is reviewed from a spacetime perspective, without quasi-Newtonian approximations. More precisely, the review covers all aspects of gravitational lensing where light propagation is described in terms of lightlike geodesics of a metric of Lorentzian signature. It includes the basic equations and the relevant techniques for calculating the position, the shape, and the brightness of images in an arbitrary general-relativistic spacetime. It also includes general theorems on the classification of caustics, on criteria for multiple imaging, and on the possible number of images. The general results are illustrated with examples of spacetimes where the lensing features can be explicitly calculated, including the Schwarzschild spacetime, the Kerr spacetime, the spacetime of a straight string, plane gravitational waves, and others. 1 Introduction In its most general sense, gravitational lensing is a collective term for all effects of a gravitational field on the propagation of electromagnetic radiation, with the latter usually described in terms of rays. According to general relativity, the gravitational field is coded in a metric of Lorentzian signature on the 4-dimensional spacetime manifold, and the light rays are the lightlike geodesics of this spacetime metric. From a mathematical point of view, the theory of gravitational lensing is thus the theory of lightlike geodesics in a 4-dimensional manifold with a Lorentzian metric. The first observation of a ‘gravitational lensing’ effect was made when the deflection of star light by our Sun was verified during a Solar eclipse in 1919. Today, the list of observed phenomena includes the following: Multiple quasars. The gravitational field of a galaxy (or a cluster of galaxies) bends the light from a distant quasar in such a way that the observer on Earth sees two or more images of the quasar. Rings. An extended light source, like a galaxy or a lobe of a galaxy, is distorted into a closed or almost closed ring by the gravitational field of an intervening galaxy. This phenomenon occurs in situations where the gravitational field is almost rotationally symmetric, with observer and light source close to the axis of symmetry. It is observed primarily, but not exclusively, in the radio range. Arcs. Distant galaxies are distorted into arcs by the gravitational field of an intervening cluster of galaxies. Here the situation is less symmetric than in the case of rings. The effect is observed in the optical range and may produce “giant luminous arcs”, typically of a characteristic blue color. Microlensing. When a light source passes behind a compact mass, the focusing effect on the light leads to a temporal change in brightness (energy flux). This microlensing effect is routinely observed since the early 1990s by monitoring a large number of stars in the bulge of our Galaxy, in the Magellanic Clouds and in the Andromeda galaxy. Microlensing has also been observed on quasars. Image distortion by weak lensing. In cases where the distortion effect on galaxies is too weak for producing rings or arcs, it can be verified with statistical methods. By evaluating the shape of a large number of background galaxies in the field of a galaxy cluster, one can determine the surface mass density of the cluster. By evaluating fields without a foreground cluster one gets information about the large-scale mass distribution. Observational aspects of gravitational lensing and methods of how to use lensing as a tool in astrophysics are the subject of the Living Review by Wambsganss [427]. There the reader may also find some notes on the history of lensing. The present review is meant as complementary to the review by Wambsganss. While all the theoretical methods reviewed in [427] rely on quasi-Newtonian approximations, the present review is devoted to the theory of gravitational lensing from a spaectime perspective, without such approximations. Here the terminology is as follows: “Lensing from a spacetime perspective” means that light propagation is described in terms of lightlike geodesics of a general-relativistic spacetime metric, without further approximations. (The term “non-perturbative lensing” is sometimes used in the same sense.) “Quasi-Newtonian approximation” means that the general-relativistic spacetime formalism is reduced by approximative assumptions to essentially Newtonian terms (Newtonian space, Newtonian time, Newtonian gravitational field). The quasi-Newtonian approximation formalism of lensing comes in several variants, and the relation to the exact formalism is not always evident because sometimes plausibility and ad-hoc assumptions are implicitly made. A common feature of all variants is that they are “weak-field approximations” in the sense that the spacetime metric is decomposed into a background (“spacetime without the lens”) and a small perturbation of this background (“gravitational field of the lens”). For the background one usually chooses either Minkowski spacetime (isolated lens) or a spatially flat Robertson–Walker spacetime (lens embedded in a cosmological model). The background then defines a Euclidean 3-space, similar to Newtonian space, and the gravitational field of the lens is similar to a Newtonian gravitational field on this Euclidean 3-space. Treating the lens as a small perturbation of the background means that the gravitational field of the lens is weak and causes only a small deviation of the light rays from the straight lines in Euclidean 3-space. In its most traditional version, the formalism assumes in addition that the lens is “thin”, and that the lens and the light sources are at rest in Euclidean 3-space, but there are also variants for “thick” and moving lenses. Also, modifications for a spatially curved Robertson–Walker background exist, but in all variants a non-trivial topological or causal structure of spacetime is (explicitly or implicitly) excluded. At the center of the quasi-Newtonian formalism is a “lens equation” or “lens map”, which relates the position of a “lensed image” to the position of the corresponding “unlensed image”. In the most traditional version one considers a thin lens at rest, modeled by a Newtonian gravitational potential given on a plane in Euclidean 3-space (“lens plane”). The light rays are taken to be straight lines in Euclidean 3-space except for a sharp bend at the lens plane. For a fixed observer and light sources distributed on a plane parallel to the lens plane (“source plane”), the lens map is then a map from the lens plane to the source plane. In this way, the geometric spacetime setting of general relativity is completely covered behind a curtain of approximations, and one is left simply with a map from a plane to a plane. Details of the quasi-Newtonian approximation formalism can be found not only in the above-mentioned Living Review [427], but also in the monographs of Schneider, Ehlers, and Falco [367] and Petters, Levine, and Wambsganss [343]. The quasi-Newtonian approximation formalism has proven very successful for using gravitational lensing as a tool in astrophysics. This is impressively demonstrated by the work reviewed in [427]. On the other hand, studying lensing from a spacetime perspective is of relevance under three aspects: Didactical. The theoretical foundations of lensing can be properly formulated only in terms of the full formalism of general relativity. Working out examples with strong curvature and with non-trivial causal or topological structure demonstrates that, in principle, lensing situations can be much more complicated than suggested by the quasi-Newtonian formalism. Methodological. General theorems on lensing (e.g., criteria for multiple imaging, characterizations of caustics, etc.) should be formulated within the exact spacetime setting of general relativity, if possible, to make sure that they are not just an artifact of approximative assumptions. For those results which do not hold in arbitrary spacetimes, one should try to find the precise conditions on the spacetime under which they are true. Practical. There are some situations of astrophysical interest to which the quasi-Newtonian formalism does not apply. For instance, near a black hole light rays are so strongly bent that, in principle, they can make arbitrarily many turns around the hole. Clearly, in this situation it is impossible to use the quasi-Newtonian formalism which would treat these light rays as small perturbations of straight lines. The present review tries to elucidate all three aspects. More precisely, the following subjects will be covered: • The basic equations and all relevant techniques that are needed for calculating the position, the shape, and the brightness of images in an arbitrary general-relativistic spacetime are reviewed. Part of this material is well-established since decades, like the Sachs equations for the optical scalars (Section 2.3), which are of crucial relevance for calculating distance measures (Section 2.4), image distortion (Section 2.5), and the brightness of images (Section 2.6). It is included here to keep the review self-contained. Other parts refer to more recent developments which are far from being fully explored, like the exact lens map (Section 2.1) and variational techniques (Section 2.9). Specifications and simplifications are possible for spacetimes with symmetries. The case of spherically symmetric and static spacetimes is treated in greater detail (Section 4.3). • General theorems on lensing in arbitrary spacetimes, or in certain classes of spacetimes, are reviewed. Some of these results are of a local character, like the classification of locally stable caustics (Section 2.2). Others are related to global aspects, like the criteria for multiple imaging in terms of conjugate points and cut points (Sections 2.7 and 2.8). The global theorems can be considerably strengthened if one restricts to globally hyperbolic spacetimes (Section 3.1) or, more specifically, to asymptotically simple and empty spacetimes (Section 3.4). The latter may be viewed as spacetime models for isolated transparent lenses. Also, in globally hyperbolic spacetimes Morse theory can be used for investigating whether the total number of images is finite or infinite, even or odd (Section 3.3). In a spherically symmetric and static spacetime, the occurrence of an infinite sequence of images is related to the occurrence of a “light sphere” (circular lightlike geodesics), like in the Schwarzschild spacetime at $r=3m$ (Section 4.3). • Several examples of spacetimes are considered, where the lightlike geodesics and, thus, the lensing features can be calculated explicitly. The examples are chosen such that they illustrate the general results. Therefore, in many parts of the review the reader will find suggestions to look at pictures in the example section. The best known and astrophysically most relevant examples are the Schwarzschild spacetime (Section 5.1), the Kerr spacetime (Section 5.8) and the spacetime of a straight string (Section 5.10). Schwarzschild black hole lensing and Kerr black hole lensing was intensively investigated already in the 1960s, 1970s, and 1980s, with astrophysical applications concentrating on observable features of accretion disks. More recently, the increasing evidence that there is a black hole at the center of our Galaxy (and probably at the center of most galaxies) has led to renewed and intensified interest in black hole lensing (see Sections 5.1 and 5.8). This is a major reason for the increasing number of articles on lensing beyond the quasi-Newtonian approximation. This introduction ends with some notes on subjects not covered in this review: Wave optics. In the electromagnetic theory, light is described by wavelike solutions to Maxwell’s equations. The ray-optical treatment used throughout this review is the standard high-frequency approximation (geometric optics approximation) of the electromagnetic theory for light propagation in vacuum on a general-relativistic spacetime (see, e.g., [279], § 22.5 or [367], Section 3.2). (Other notions of vacuum light rays, based on a different approximation procedure, have been occasionally suggested [271], but will not be considered here. Also, results specific to spacetime dimensions other than four or to gravitational theories other than Einstein’s are not covered.) For most applications to lensing the ray-optical treatment is valid and appropriate. An exception, where wave-optical corrections are necessary, is the calculation of the brightness of images if a light source comes very close to the caustic of the observer’s light cone (see Section 2.6). Light propagation in matter. If light is directly influenced by a medium, the light rays are no longer the lightlike geodesics of the spacetime metric. For an isotropic non-dispersive medium, they are the lightlike geodesics of another metric which is again of Lorentzian signature. (This “optical metric” was introduced by Gordon [179]. For a rigourous derivation, starting from Maxwell’s equation in an isotropic non-dispersive medium, see Ehlers [114].) Hence, the formalism used throughout this review still applies to this situation after an appropriate re-interpretation of the metric. In anisotropic or dispersive media, however, the light rays are not the lightlike geodesics of a Lorentzian metric. There are some lensing situations where the influence of matter has to be taken into account. For instance., for the deflection of radio signals by our Sun the influence of the plasma in the Solar corona (to be treated as a dispersive medium) is very well measurable. However, such situations will not be considered in this review. For light propagation in media on a general-relativistic spacetime, see [337] and references cited therein. Kinetic theory. As an alternative to the (geometric optics approximation of) electromagnetic theory, light can be treated as a photon gas, using the formalism of kinetic theory. This has relevance, e.g., for the cosmic background radiation. For basic notions of general-relativistic kinetic theory see, e.g., [115]. Apart from some occasional remarks, kinetic theory will not be considered in this review. Derivation of the quasi-Newtonian formalism. It is not satisfacory if the quasi-Newtonian formalism of lensing is set up with the help of ad-hoc assumptions, even if the latter look plausible. From a methodological point of view, it is more desirable to start from the exact spacetime setting of general relativity and to derive the quasi-Newtonian lens equation by a well-defined approximation procedure. In comparison to earlier such derivations [367, 362, 373] later effort has led to considerable improvements. For lenses embedded in a cosmological model, see Pyne and Birkinshaw [352] who consider lenses that need not be thin and may be moving on a Robertson–Walker background (with positive, negative, or zero spatial curvature). For the non-cosmological situation, a Lorentz covariant approximation formalism was derived by Kopeikin and Schäfer [238]. Here Minkowski spacetime is taken as the background, and again the lenses need not be thin and may be moving. 2 Lensing in Arbitrary Spacetimes By a spacetime we mean a 4-dimensional manifold $\mathcal{M}$ with a ($C^{\infty}$, if not otherwise stated) metric tensor field $g$ of signature $(+,+,+,-)$ that is time-oriented. The latter means that the non-spacelike vectors make up two connected components in the entire tangent bundle, one of which is called “future-pointing” and the other one “past-pointing”. Throughout this review we restrict to the case that the light rays are freely propagating in vacuum, i.e., are not influenced by mirrors, refractive media, or any other impediments. The light rays are then the lightlike geodesics of the spacetime metric. We first summarize results on the lightlike geodesics that hold in arbitrary spacetimes. In Section 3 these results will be specified for spacetimes with conditions on the causal structure and in Section 4 for spacetimes with symmetries. 2.1 Light cone and exact lens map In an arbitrary spacetime $(\mathcal{M},g)$, what an observer at an event $p_{\mathrm{O}}$ can see is determined by the lightlike geodesics that issue from $p_{\mathrm{O}}$ into the past. Their union gives the past light cone of $p_{\mathrm{O}}$. This is the central geometric object for lensing from the spacetime perspective. For a point source with worldline $\gamma_{\mathrm{S}}$, each past-oriented lightlike geodesic $\lambda$ from $p_{\mathrm{O}}$ to $\gamma_{\mathrm{S}}$ gives rise to an image of $\gamma_{\mathrm{S}}$ on the observer’s sky. One should view any such $\lambda$ as the central ray of a thin bundle that is focused by the observer’s eye lens onto the observer’s retina (or by a telescope onto a photographic plate). Hence, the intersection of the past light cone with the world-line of a point source (or with the world-tube of an extended source) determines the visual appearance of the latter on the observer’s sky. In mathematical terms, the observer’s sky or celestial sphere $\mathcal{S}_{\mathrm{O}}$ can be viewed as the set of all lightlike directions at $p_{\mathrm{O}}$. Every such direction defines a unique (up to parametrization) lightlike geodesic through $p_{\mathrm{O}}$, so $\mathcal{S}_{\mathrm{O}}$ may also be viewed as a subset of the space of all lightlike geodesics in $(\mathcal{M},g)$ (cf. [263]). One may choose at $p_{\mathrm{O}}$ a future-pointing vector $U_{\mathrm{O}}$ with $g(U_{\mathrm{O}},U_{\mathrm{O}})=-1$, to be interpreted as the 4-velocity of the observer. This allows identifying the observer’s sky $\mathcal{S}_{\mathrm{O}}$ with a subset of the tangent space $T_{p_{\mathrm{O}}}\mathcal{M}$, $$\mathcal{S}_{\mathrm{O}}\simeq\left\{w\in T_{p_{\mathrm{O}}}\mathcal{M}\,\big{% |}\,g(w,w)=0\mbox{~{}~{}and~{}~{}}g(w,U_{\mathrm{O}})=1\right\}.$$ (1) If $U_{\mathrm{O}}$ is changed, this representation changes according to the standard aberration formula of special relativity. By definition of the exponential map $\exp$, every affinely parametrized geodesic $s\mapsto\lambda(s)$ satisfies $\lambda(s)=\exp\bigl{(}s\,\dot{\lambda}(0)\bigr{)}$. Thus, the past light cone of $p_{\mathrm{O}}$ is the image of the map $$(s,w)\longmapsto\exp(sw),$$ (2) which is defined on a subset of $]0,\infty[\times\mathcal{S}_{\mathrm{O}}$. If we restrict to values of $s$ sufficiently close to 0, the map (2) is an embedding, i.e., this truncated light cone is an embedded submanifold; this follows from the well-known fact that $\exp$ maps a neighborhood of the origin, in each tangent space, diffeomorphically into the manifold. However, if we extend the map (2) to larger values of $s$, it is in general neither injective nor an immersion; it may form folds, cusps, and other forms of caustics, or transverse self-intersections. This observation is of crucial importance in view of lensing. There are some lensing phenomena, such as multiple imaging and image distortion of (point) sources into (1-dimensional) rings, which can occur only if the light cone fails to be an embedded submanifold (see Section 2.8). Such lensing phenomena are summarized under the name strong lensing effects. As long as the light cone is an embedded submanifold, the effects exerted by the gravitational field on the apparent shape and on the apparent brightness of light sources are called weak lensing effects. For examples of light cones with caustics and/or transverse self-intersections, see Figures 13, 25, and 26. These pictures show light cones in spacetimes with symmetries, so their structure is rather regular. A realistic model of our own light cone, in the real world, would have to take into account numerous irregularly distributed inhomogeneities (“clumps”) that bend light rays in their neighborhood. Ellis, Bassett, and Dunsby [129] estimate that such a light cone would have at least $10^{22}$ caustics which are hierarchically structured in a way that reminds of fractals. For calculations it is recommendable to introduce coordinates on the observer’s past light cone. This can be done by choosing an orthonormal tetrad $(e_{0},e_{1},e_{2},e_{3})$ with $e_{0}=-U_{\mathrm{O}}$ at the observation event $p_{\mathrm{O}}$. This parametrizes the points of the observer’s celestial sphere by spherical coordinates $(\Psi,\Theta)$, $$w=\sin\Theta\,\cos\Psi\,e_{1}+\sin\Theta\,\sin\Psi\,e_{2}+\cos\Theta\,e_{3}+e_% {0}.$$ (3) In this representation, map (2) maps each $(s,\Psi,\Theta)$ to a spacetime point. Letting the observation event float along the observer’s worldline, parametrized by proper time $\tau$, gives a map that assigns to each $(s,\Psi,\Theta,\tau)$ a spacetime point. In terms of coordinates $x=(x^{0},x^{1},x^{2},x^{3})$ on the spacetime manifold, this map is of the form $$x^{i}=F^{i}(s,\Psi,\Theta,\tau),\qquad i=0,1,2,3.$$ (4) It can be viewed as a map from the world as it appears to the observer (via optical observations) to the world as it is. The coordinates $(s,\Psi,\Theta,\tau)$ were called optical coordinates by Temple [401] and observational coordinates by Ellis [128]. A detailed discussion of their properties can be found in [130]. They are particularly useful in cosmology but can be introduced for any observer in any spacetime. It is useful to consider observables, such as distance measures (see Section 2.4) or the ellipticity that describes image distortion (see Section 2.5) as functions of the observational coordinates. Some observables, e.g., the redshift and the luminosity distance, are not determined by the spacetime geometry and the observer alone, but also depend on the 4-velocities of the light sources. If a vector field $U$ with $g(U,U)=-1$ has been fixed, one may restrict to an observer and to light sources which are integral curves of $U$. The above-mentioned observables, like redshift and luminosity distance, are then uniquely determined as functions of the observational coordinates. In applications to cosmology one chooses $U$ as tracing the mean flow of luminous matter (“Hubble flow”) or as the rest system of the cosmic background radiation; present observations are compatible with the assumption that these two distinguished observer fields coincide [43, 87, 213]. Writing map (4) explicitly requires solving the lightlike geodesic equation. This is usually done, using standard index notation, in the Lagrangian formalism, with the Lagrangian $\mathcal{L}=\frac{1}{2}g_{ij}(x)\dot{x}^{i}\dot{x}^{j}$, or in the Hamiltonian formalism, with the Hamiltonian $\mathcal{H}=\frac{1}{2}g^{ij}(x)p_{i}p_{j}$. A non-trivial example where the solutions can be explicitly written in terms of elementary functions is the string spacetime of Section 5.10. Somewhat more general, although still very special, is the situation that the lightlike geodesic equation admits three independent constants of motion in addition to the obvious one $g^{ij}(x)p_{i}p_{j}=0$. If, for any pair of the four constants of motion, the Poisson bracket vanishes (“complete integrability”), the lightlike geodesic equation can be reduced to first-order form, i.e., the light cone can be written in terms of integrals over the metric coefficients. This is true, e.g., in spherically symmetric and static spacetimes (see Section 4.3). Having parametrized the past light cone of the observation event $p_{\mathrm{O}}$ in terms of $(s,w)$, or more specifically in terms of $(s,\Psi,\Theta)$, one may set up an exact lens map. This exact lens map is analogous to the lens map of the quasi-Newtonian approximation formalism, as far as possible, but it is valid in an arbitrary spacetime without approximation. In the quasi-Newtonian formalism for thin lenses at rest, the lens map assigns to each point in the lens plane a point in the source plane (see, e.g., [367, 343, 427]). When working in an arbitrary spacetime without approximations, the observer’s sky $\mathcal{S}_{\mathrm{O}}$ is an obvious substitute for the lens plane. As a substitute for the source plane we choose a 3-dimensional submanifold $\mathcal{T}$ with a prescribed ruling by timelike curves. We assume that $\mathcal{T}$ is globally of the form $\mathcal{Q}\times\mathbb{R}$, where the points of the 2-manifold $\mathcal{Q}$ label the timelike curves by which $\mathcal{T}$ is ruled. These timelike curves are to be interpreted as the worldlines of light sources. We call any such $\mathcal{T}$ a source surface. In a nutshell, choosing a source surface means choosing a two-parameter family of light sources. The exact lens map is a map from $\mathcal{S}_{\mathrm{O}}$ to $\mathcal{Q}$. It is defined by following, for each $w\in\mathcal{S}_{\mathrm{O}}$, the past-pointing geodesic with initial vector $w$ until it meets $\mathcal{T}$ and then projecting to $\mathcal{Q}$ (see Figure 1). In other words, the exact lens map says, for each point on the observer’s celestial sphere, which of the chosen light sources is seen at this point. Clearly, non-invertibility of the lens map indicates multiple imaging. What one chooses for $\mathcal{T}$ depends on the situation. In applications to cosmology, one may choose galaxies at a fixed redshift $z=z_{\mathrm{S}}$ around the observer. In a spherically-symmetric and static spacetime one may choose static light sources at a fixed radius value $r=r_{\mathrm{S}}$. Also, the surface of an extended light source is a possible choice for $\mathcal{T}$. The exact lens map was introduced by Frittelli and Newman [154] and further discussed in [117, 116]. The following global aspects of the exact lens map were investigated in [338]. First, in general the lens map is not defined on all of $\mathcal{S}_{\mathrm{O}}$ because not all past-oriented lightlike geodesics that start at $p_{\mathrm{O}}$ necessarily meet $\mathcal{T}$. Second, in general the lens map is multi-valued because a lightlike geodesic might meet $\mathcal{T}$ several times. Third, the lens map need not be differentiable and not even continuous because a lightlike geodesic might meet $\mathcal{T}$ tangentially. In [338], the notion of a simple lensing neighborhood is introduced which translates the statement that a deflector is transparent into precise mathematical language. It is shown that the lens map is globally well-defined and differentiable if the source surface is the boundary of such a simple lensing neighborhood, and that for each light source that does not meet the caustic of the observer’s past light cone the number of images is finite and odd. This result applies, as a special case, to asymptotically simple and empty spacetimes (see Section 3.4). For expressing the exact lens map in coordinate language, it is recommendable to choose coordinates $(x^{0},x^{1},x^{2},x^{3})$ such that the source surface $\mathcal{T}$ is given by the equation $x^{3}=x^{3}_{\mathrm{S}}$, with a constant $x^{3}_{\mathrm{S}}$, and that the worldlines of the light sources are $x^{0}$-lines. In this situation the remaining coordinates $x^{1}$ and $x^{2}$ label the light sources and the exact lens map takes the form $$(\Psi,\Theta)\longmapsto(x^{1},x^{2}).$$ (5) It is given by eliminating the two variables $s$ and $x^{0}$ from the four equations (4) with $F^{3}(s,\Psi,\Theta,\tau)=x^{3}_{\mathrm{S}}$ and fixed $\tau$. This is the way in which the lens map was written in the original paper by Frittelli and Newman; see Equation (6) in [154]. (They used complex coordinates $(\eta,\overline{\eta})$ for the observer’s celestial sphere that are related to our spherical coordinates $(\Psi,\Theta)$ by stereographic projection.) In this explicit coordinate version, the exact lens map can be succesfully applied, in particular, to spherically symmetric and static spacetimes, with $x^{0}=t$, $x^{1}=\varphi$, $x^{2}=\vartheta$, and $x^{3}=r$ (see Section 4.3 and the Schwarzschild example in Section 5.1). The exact lens map can also be used for testing the reliability of approximation techniques. In [237] the authors find that the standard quasi-Newtonian approximation formalism may lead to significant errors for lensing configurations with two lenses. 2.2 Wave fronts Wave fronts are related to light rays as solutions of the Hamilton–Jacobi equation are related to solutions of Hamilton’s equations in classical mechanics. For the case at hand (i.e., vacuum light propagation in an arbitrary spacetime, corresponding to the Hamiltonian $\mathcal{H}=\frac{1}{2}g^{ij}(x)p_{i}p_{j}$), a wave front is a subset of the spacetime that can be constructed in the following way: 1. Choose a spacelike 2-surface $\mathcal{S}$ that is orientable. 2. At each point of $\mathcal{S}$, choose a lightlike direction orthogonal to $\mathcal{S}$ that depends smoothly on the foot-point. (You have to choose between two possibilities.) 3. Take all lightlike geodesics that are tangent to the chosen directions. These lightlike geodesics are called the generators of the wave front, and the wave front is the union of all generators. Clearly, a light cone is a special case of a wave front. One gets this special case by choosing for $\mathcal{S}$ an appropriate (small) sphere. Any wave front is the envelope of all light cones with vertices on the wave front. In this sense, general-relativistic wave fronts can be constructed according to the Huygens principle. In the context of general relativity the notion of wave fronts was introduced by Kermack, McCrea, and Whittaker [233]. For a modern review article see, e.g., Ehlers and Newman [119]. A coordinate representation for a wave front can be given with the help of (local) coordinates $(u^{1},u^{2})$ on $\mathcal{S}$. One chooses a parameter value $s_{0}$ and parametrizes each generator $\lambda$ affinely such that $\lambda(s_{0})\in\mathcal{S}$ and $\dot{\lambda}(s_{0})$ depends smoothly on the foot-point in $\mathcal{S}$. This gives the wave front as the image of a map $$(s,u^{1},u^{2})\longmapsto F^{i}(s,u^{1},u^{2}),\qquad i=0,1,2,3.$$ (6) For light cones we may choose spherical coordinates, $(u^{1}=\Psi,u^{2}=\Theta)$, (cf. Equation (4) with fixed $\tau$). Near $s=s_{0}$, map (6) is an embedding, i.e., the wave front is a submanifold. Orthogonality to $\mathcal{S}$ of the initial vectors $\dot{\lambda}(s_{0})$ assures that this submanifold is lightlike. Farther away from $\mathcal{S}$, however, the wave front need not be a submanifold. The caustic of the wave front is the set of all points where the map (6) is not an immersion, i.e., where its differential has rank $<3$. As the derivative with respect to $s$ is always non-zero, the rank can be $3-1$ (caustic point of multiplicity one, astigmatic focusing) or $3-2$ (caustic point of multiplicity two, anastigmatic focusing). In the first case, the cross-section of an “infinitesimally thin” bundle of generators collapses to a line, in the second case to a point (see Section 2.3). For the case that the wave front is a light cone with vertex $p_{\mathrm{O}}$, caustic points are said to be conjugate to $p_{\mathrm{O}}$ along the respective generator. For an arbitrary wave front, one says that a caustic point is conjugate to any spacelike 2-surface in the wave front. In this sense, the terms “conjugate point” and “caustic point” are synonymous. Along each generator, caustic points are isolated (see Section 2.3) and thus denumerable. Hence, one may speak of the first caustic, the second caustic, and so on. At all points where the caustic is a manifold, it is either spacelike or lightlike. For instance, the caustic of the Schwarzschild light cone in Figure 13 is a spacelike curve; in the spacetime of a transparent string, the caustic of the light cone consists of two lightlike 2-manifolds that meet in a spacelike curve (see Figure 26). Near a non-caustic point, a wave front is a hypersurface $S=\mbox{constant}$ where $S$ satisfies the Hamilton–Jacobi equation $$g^{ij}(x)\,\partial_{i}S(x)\,\partial_{j}S(x)=0.$$ (7) In the terminology of optics, Equation (7) is called the eikonal equation. At caustic points, a wave front typically forms cuspidal edges or vertices whose geometry might be arbitrarily complicated, even locally. If one restricts to caustics which are stable against perturbations in a certain sense, then a local classification of caustics is possible with the help of Arnold’s singularity theory of Lagrangian or Legendrian maps. Full details of this theory can be found in [14]. For a readable review of Arnold’s results and its applications to wave fronts in general relativity, we refer again to [119]. In order to apply Arnold’s theory to wave fronts, one associates each wave front with a Legendrian submanifold in the projective cotangent bundle over $\mathcal{M}$ (or with a Lagrangian submanifold in an appropriately reduced bundle). A caustic point of the wave front corresponds to a point where the differential of the projection from the Legendrian submanifold to $\mathcal{M}$ has non-maximal rank. For the case $\dim(\mathcal{M})=4$, which is of interest here, Arnold has shown that there are only five types of caustic points that are stable with respect to perturbations within the class of all Legendrian submanifolds. They are known as fold, cusp, swallow-tail, pyramid, and purse (see Figure 2). Any other type of caustic is unstable in the sense that it changes non-diffeomorphically if it is perturbed within the class of Legendrian submanifolds. Fold singularities of a wave front form a lightlike 2-manifold in spacetime, on a sufficiently small neighborhood of any fold caustic point. The second picture in Figure 2 shows such a “fold surface”, projected to 3-space along the integral curves of a timelike vector field. This projected fold surface separates a region covered twice by the wave front from a region not covered at all. If the wave front is the past light cone of an observation event, and if one restricts to light sources with worldlines in a sufficiently small neighborhood of a fold caustic point, there are two images for light sources on one side and no images for light sources on the other side of the fold surface. Cusp singularities of a wave front form a spacelike curve in spacetime, again locally near any cusp caustic point. Such a curve is often called a “cusp ridge”. Along a cusp ridge, two fold surfaces meet tangentially. The third picture in Figure 2 shows the situation projected to 3-space. Near a cusp singularity of a past light cone, there is local triple-imaging for light sources in the wedge between the two fold surfaces and local single-imaging for light sources outside this wedge. Swallow-tail, pyramid, and purse singularities are points where two or more cusp ridges meet with a common tangent, as illustrated by the last three pictures in Figure 2. Friedrich and Stewart [149] have demonstrated that all caustic types that are stable in the sense of Arnold can be realized by wave fronts in Minkowski spacetime. Moreover, they stated without proof that, quite generally, one gets the same stable caustic types if one allows for perturbations only within the class of wave fronts (rather than within the larger class of Legendrian submanifolds). A proof of this statement was claimed to be given in [187] where the Lagrangian rather than the Legendrian formalism was used. However, the main result of this paper (Theorem 4.4 of [187]) is actually too weak to justify this claim. A different version of the desired stability result was indeed proven by another approach. In this approach one concentrates on an instantaneous wave front, i.e., on the intersection of a wave front with a spacelike hypersurface $\mathcal{C}$. As an alternative terminology, one calls the intersection of a (“big”) wave front with a hypersurface $\mathcal{C}$ that is transverse to all generators a “small wave front”. Instantaneous wave fronts are special cases of small wave fronts. The caustic of a small wave front is the set of all points where the small wave front fails to be an immersed 2-dimensional submanifold of $\mathcal{C}$. If the spacetime is foliated by spacelike hypersurfaces, the caustic of a wave front is the union of the caustics of its small (= instantaneous) wave fronts. Such a foliation can always be achieved locally, and in several spacetimes of interest even globally. If one identifies different slices with the help of a timelike vector field, one can visualize a wave front, and in particular a light cone, as a motion of small (= instantaneous) wave fronts in 3-space. Examples are shown in Figures 14, 19, 20, 28, and 29. Mathematically, the same can be done for non-spacelike slices as long as they are transverse to the generators of the considered wave front (see Figure 31 for an example). Turning from (big) wave fronts to small wave fronts reduces the dimension by one. The only caustic points of a small wave front that are stable in the sense of Arnold are cusps and swallow-tails. What one wants to prove is that all other caustic points are unstable with respect to perturbations of the wave front within the class of wave fronts, keeping the metric and the slicing fixed. For spacelike slicings (i.e., for instantaneous wave fronts), this was indeed demonstrated by Low [264]. In this article, the author views wave fronts as subsets of the space $\mathcal{N}$ of all lightlike geodesics in $(\mathcal{M},g)$. General properties of this space $\mathcal{N}$ are derived in earlier articles by Low [262, 263] (also see Penrose and Rindler [330], volume II, where the space $\mathcal{N}$ is treated in twistor language). Low considers, in particular, the case of a globally hyperbolic spacetime [264]; he demonstrates the desired stability result for the intersections of a (big) wave front with Cauchy hypersurfaces (see Section 3.2). As every point in an arbitrary spacetime admits a globally hyperbolic neighborhood, this local stability result is universal. Figure 29 shows an instantaneous wave front with cusps and a swallow-tail point. Figure 14 shows instantaneous wave fronts with caustic points that are neither cusps nor swallow-tails; hence, they must be unstable with respect to perturbations of the wave front within the class of wave fronts. It is to be emphasized that Low’s work allows to classify the stable caustics of small wave fronts, but not directly of (big) wave fronts. Clearly, a (big) wave front is a one-parameter family of small wave fronts. A qualitative change of a small wave front, in dependence of a parameter, is called a “metamorphosis” in the English literature and a “perestroika” in the Russian literature. Combining Low’s results with the theory of metamorphoses, or perestroikas, could lead to a classsification of the stable caustics of (big) wave fronts. However, this has not been worked out until now. Wave fronts in general relativity have been studied in a long series of articles by Newman, Frittelli, and collaborators. For some aspects of their work see Sections 2.9 and 3.4. In the quasi-Newtonian approximation formalism of lensing, the classification of caustics is treated in great detail in the book by Petters, Levine, and Wambsganss [343]. Interesting related mateial can also be found in Blandford and Narayan [45]. For a nice exposition of caustics in ordinary optics see Berry and Upstill [37]. A light source that comes close to the caustic of the observer’s past light cone is seen strongly magnified. For a point source whose worldline passes exactly through the caustic, the ray-optical treatment even gives an infinite brightness (see Section 2.6). If a light source passes behind a compact deflecting mass, its brightness increases and decreases in the course of time, with a maximum at the moment of closest approach to the caustic. Such microlensing events are routinely observed by monitoring a large number of stars in the bulge of our Galaxy, in the Magellanic Clouds, and in the Andromeda Galaxy (see, e.g., [280] for an overview). In his millennium essay on future perspectives of gravitational lensing, Blandford [44] mentioned the possibility of observing a chosen light source strongly magnified over a period of time with the help of a space-born telescope. The idea is to guide the spacecraft such that the worldline of the light source remains in (or close to) the one-parameter family of caustics of past light cones of the spacecraft over a period of time. This futuristic idea of “caustic surfing” was mathematically further discussed by Frittelli and Petters [159]. 2.3 Optical scalars and Sachs equations For the calculation of distance measures, of image distortion, and of the brightness of images one has to study the Jacobi equation (= equation of geodesic deviation) along lightlike geodesics. This is usually done in terms of the optical scalars which were introduced by Sachs et al. [221, 360]. Related background material on lightlike geodesic congruences can be found in many text-books (see, e.g., Wald [425], Section 9.2). In view of applications to lensing, a particularly useful exposition was given by Seitz, Schneider and Ehlers [373]. In the following the basic notions and results will be summarized. Infinitesimally thin bundles. Let $s\longmapsto\lambda(s)$ be an affinely parametrized lightlike geodesic with tangent vector field $K=\dot{\lambda}$. We assume that $\lambda$ is past-oriented, because in applications to lensing one usually considers rays from the observer to the source. We use the summation convention for capital indices $A,B,\dots$ taking the values 1 and 2. An infinitesimally thin bundle (with elliptical cross-section) along $\lambda$ is a set $$\mathcal{B}=\left\{c^{A}Y_{A}\,\big{|}\,c^{1},c^{2}\in\mathbb{R},~{}~{}\delta_% {AB}\,c^{A}c^{B}\leq 1\right\}.$$ (8) Here $\delta_{AB}$ denotes the Kronecker delta, and $Y_{1}$ and $Y_{2}$ are two vector fields along $\lambda$ with $$\displaystyle\nabla_{K}\nabla_{K}Y_{A}$$ $$\displaystyle=$$ $$\displaystyle R(K,Y_{A},K),$$ (9) $$\displaystyle g(K,Y_{A})$$ $$\displaystyle=$$ $$\displaystyle 0,$$ (10) such that $Y_{1}(s)$, $Y_{2}(s)$, and $K(s)$ are linearly independent for almost all $s$. As usual, $R$ denotes the curvature tensor, defined by $$R(X,Y,Z)=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z.$$ (11) Equation (9) is the Jacobi equation. It is a precise mathematical formulation of the statement that “the arrow-head of $Y_{A}$ traces an infinitesimally neighboring geodesic”. Equation (10) guarantees that this neighboring geodesic is, again, lightlike and spatially related to $\lambda$. Vector fields $Y_{A}$ that satisfy Equation (9) are known as Jacobi vector fields. Sachs basis. For discussing the geometry of infinitesimally thin bundles it is usual to introduce a Sachs basis, i.e., two vector fields $E_{1}$ and $E_{2}$ along $\lambda$ that are orthonormal, orthogonal to $K=\dot{\lambda}$, and parallelly transported, $$g(E_{A},E_{B})=\delta_{AB},\qquad g(K,E_{A})=0,\qquad\nabla_{K}E_{A}=0.$$ (12) Apart from the possibility to interchange them, $E_{1}$ and $E_{2}$ are unique up to transformations $$\displaystyle\tilde{E}_{1}$$ $$\displaystyle=$$ $$\displaystyle\cos\alpha\,E_{1}+\sin\alpha\,E_{2}+a_{1}K,$$ (13) $$\displaystyle\tilde{E}_{2}$$ $$\displaystyle=$$ $$\displaystyle-\sin\alpha\,E_{1}+\cos\alpha\,E_{2}+a_{2}K,$$ (14) where $\alpha$, $a_{1}$, and $a_{2}$ are constant along $\lambda$. A Sachs basis determines a unique vector field $U$ with $g(U,U)=-1$ and $g(U,K)=1$ along $\lambda$ that is perpendicular to $E_{1}$, and $E_{2}$. As $K$ is assumed past-oriented, $U$ is future-oriented. In the rest system of the observer field $U$, the Sachs basis spans the 2-space perpendicular to the ray. It is helpful to interpret this 2-space as a “screen”; correspondingly, linear combinations of $E_{1}$ and $E_{2}$ are often refered to as “screen vectors”. Jacobi matrix. With respect to a Sachs basis, the basis vector fields $Y_{1}$ and $Y_{2}$ of an infinitesimally thin bundle can be represented as $$Y_{A}=D_{A}^{B}E_{B}+y_{A}K.$$ (15) The Jacobi matrix $\boldsymbol{D}=(D_{A}^{B})$ relates the shape of the cross-section of the infinitesimally thin bundle to the Sachs basis (see Figure 3). Equation (9) implies that $\boldsymbol{D}$ satisfies the matrix Jacobi equation $$\ddot{\boldsymbol{D}}=\boldsymbol{D}\boldsymbol{R},$$ (16) where an overdot means derivative with respect to the affine parameter $s$, and $$\boldsymbol{R}=\begin{pmatrix}\Phi_{00}&0\\ 0&\Phi_{00}\end{pmatrix}+\begin{pmatrix}-\operatorname{Re}(\psi_{0})&% \operatorname{Im}(\psi_{0})\\ \operatorname{Im}(\psi_{0})&\operatorname{Re}(\psi_{0})\end{pmatrix}$$ (17) is the optical tidal matrix, with $$\Phi_{00}=-\frac{1}{2}\operatorname{Ric}(K,K),\qquad\psi_{0}=-\frac{1}{2}C% \left(E_{1}-iE_{2},K,E_{1}-iE_{2},K\right).$$ (18) Here $\operatorname{Ric}$ denotes the Ricci tensor, defined by $\operatorname{Ric}(X,Y)=\operatorname{tr}\left(R(\cdot,X,Y)\right)$, and $C$ denotes the conformal curvature tensor (= Weyl tensor). The notation in Equation (18) is chosen in agreement with the Newman–Penrose formalism (cf., e.g., [75]). As $Y_{1}$, $Y_{2}$, and $K$ are not everywhere linearly dependent, $\det(\boldsymbol{D})$ does not vanish identically. Linearity of the matrix Jacobi equation implies that $\det(\boldsymbol{D})$ has only isolated zeros. These are the “caustic points” of the bundle (see below). Shape parameters. The Jacobi matrix $\boldsymbol{D}$ can be parametrized according to $$\boldsymbol{D}=\begin{pmatrix}\cos\psi&-\sin\psi\\ \sin\psi&\cos\psi\end{pmatrix}\begin{pmatrix}D_{+}&0\\ 0&D_{-}\end{pmatrix}\begin{pmatrix}\cos\chi&\sin\chi\\ -\sin\chi&\cos\chi\end{pmatrix}.$$ (19) Here we made use of the well-known facts that any matrix can be written as the product of an orthogonal and a symmetric matrix and that any symmetric matrix can be diagonalized by an orthogonal transformation. Our definition of infinitesimally thin bundles implies that $D_{+}$ and $D_{-}$ are non-zero almost everywhere. In the representation of Equation (19), the extremal points of the bundle’s elliptical cross-section are given by the position vectors $$\displaystyle Y_{+}$$ $$\displaystyle=$$ $$\displaystyle\cos\psi\,Y_{1}+\sin\psi\,Y_{2}\simeq D_{+}\left(\cos\chi\,E_{1}+% \sin\chi\,E_{2}\right),$$ (20) $$\displaystyle Y_{-}$$ $$\displaystyle=$$ $$\displaystyle-\sin\psi\,Y_{1}+\cos\psi\,Y_{2}\simeq D_{-}\left(-\sin\chi\,E_{1% }+\cos\chi\,E_{2}\right),$$ (21) where $\simeq$ means equality up to multiples of $K$. Hence, $|D_{+}|$ and $|D_{-}|$ give the semi-axes of the elliptical cross-section and $\chi$ gives the angle by which the ellipse is rotated with respect to the Sachs basis (see Figure 3). We call $D_{+}$, $D_{-}$, and $\chi$ the shape parameters of the bundle. This name is taken from Frittelli, Kling, and Newman [152, 151] who actually use, instead of $D_{+}$ and $D_{-}$, the equivalent quantities $D_{+}D_{-}$ and $D_{+}/D_{-}$. For the case that the infinitesimally thin bundle can be embedded in a wave front, the shape parameters $D_{+}$ and $D_{-}$ have the following interesting property (see Kantowski et al. [222, 110]). $\dot{D}_{+}/D_{+}$ and $\dot{D}_{-}/D_{-}$ give the principal curvatures of the wave front in the rest system of the observer field $U$ which is perpendicular to the Sachs basis. The notation $D_{+}$ and $D_{-}$, which is taken from [110], is convenient because it often allows to write two equations in the form of one equation with a $\pm$ sign (see, e.g., Equation (27) or Equation (98) below). The angle $\chi$ can be directly linked with observations if a light source emits linearly polarized light (see Section 2.5). For any infinitesimally thin bundle, given in terms of $Y_{1}$ and $Y_{2}$, we can choose the Sachs basis as we like. This freedom leads to two ambiguities in the definition of $D_{+}$ and $D_{-}$. Firstly, the transformation $(E_{1},E_{2})\mapsto(-E_{1},E_{2})$ results in $(D_{+},D_{-},\chi,\psi)\mapsto(-D_{+},D_{-},-\chi,\psi)$, and the analogous transformation $(E_{1},E_{2})\mapsto(E_{1},-E_{2})$ results in $(D_{+},D_{-},\chi,\psi)\mapsto(D_{+},-D_{-},-\chi,\psi)$; this shows that the signs of $D_{+}$ and $D_{-}$ are ambiguous. Secondly, the transformation $(E_{1},E_{2})\mapsto(E_{2},-E_{1})$ results in $(D_{+},D_{-},\chi,\psi)\mapsto(D_{-},D_{+},\chi,\psi+\pi/2)$; this shows that $D_{+}$ and $D_{-}$ can be interchanged. The most interesting case for us is that of an infinitesimally thin bundle that issues from a vertex at an observation event $p_{\mathrm{O}}=\lambda(0)$ into the past. For such bundles we can remove the sign ambiguity in the definition of $D_{+}(s)$ and $D_{-}(s)$ by requiring that they are positive for small positive values of $s$. The freedom of interchanging them can be removed, e.g., by requiring that $D_{+}(s)\geq D_{-}(s)$ for small positive values of $s$; for spherically symmetric and static spacetimes, however, another convention is more convenient, see Section 4.3 below. If we have chosen a convention that makes $D_{+}$ and $D_{-}$ unique along the bundle, the Sachs basis can still be changed by a transformation (13, 14). Under such a transformation the shape parameters change according to $\tilde{D}_{\pm}=D_{\pm}$, $\tilde{\chi}=\chi-\alpha$, $\tilde{\psi}=\psi$. This demonstrates the important fact that the shape and the size of the cross-section of an infinitesimally thin bundle have an invariant (observer-independent) meaning [360]. Optical scalars. Along each infinitesimally thin bundle one defines the deformation matrix $\boldsymbol{S}$ by $$\dot{\boldsymbol{D}}=\boldsymbol{D}\boldsymbol{S}.$$ (22) This reduces the second-order linear differential equation (16) for $\boldsymbol{D}$ to a first-order non-linear differential equation for $\boldsymbol{S}$, $$\dot{\boldsymbol{S}}+\boldsymbol{S}\boldsymbol{S}=\boldsymbol{R}.$$ (23) It is usual to decompose $\boldsymbol{S}$ into antisymmetric, symmetric-tracefree, and trace parts, $$\boldsymbol{S}=\begin{pmatrix}0&\omega\\ -\omega&0\end{pmatrix}+\begin{pmatrix}\sigma_{1}&\sigma_{2}\\ \sigma_{2}&-\sigma_{1}\end{pmatrix}+\begin{pmatrix}\theta&0\\ 0&\theta\end{pmatrix}.$$ (24) This defines the optical scalars $\omega$ (twist), $\theta$ (expansion), and $(\sigma_{1},\sigma_{2})$ (shear). One usually combines them into two complex scalars $\varrho=\theta+i\omega$ and $\sigma=\sigma_{1}+i\sigma_{2}$. A change (13, 14) of the Sachs basis affects the optical scalars according to $\tilde{\varrho}=\varrho$ and $\tilde{\sigma}=e^{-2i\alpha}\sigma$. Thus, $\varrho$ and $|\sigma|$ are invariant. If rewritten in terms of the optical scalars, Equation (23) gives the Sachs equations $$\displaystyle\dot{\varrho}$$ $$\displaystyle=$$ $$\displaystyle-\varrho^{2}-|\sigma|^{2}+\Phi_{00},$$ (25) $$\displaystyle\dot{\sigma}$$ $$\displaystyle=$$ $$\displaystyle-\sigma\left(\varrho+\overline{\varrho}\right)+\psi_{0}.$$ (26) One sees that the Ricci curvature term $\Phi_{00}$ directly produces expansion (focusing) and that the conformal curvature term $\psi_{0}$ directly produces shear. However, as the shear appears in Equation (25), conformal curvature indirectly influences focusing (cf. Penrose [328]). With $\boldsymbol{D}$ written in terms of the shape parameters and $\boldsymbol{S}$ written in terms of the optical scalars, Equation (22) results in $$\dot{D}_{\pm}+i\dot{\chi}D_{\pm}-i\dot{\psi}D_{\mp}=\left(\rho\pm e^{-2i\chi}% \sigma\right)D_{\pm}.$$ (27) Along $\lambda$, Equations (25, 26) give a system of 4 real first-order differential equations for the 4 real variables $\varrho$ and $\sigma$; if $\varrho$ and $\sigma$ are known, Equation (27) gives a system of 4 real first-order differential equations for the 4 real variables $D_{\pm}$, $\chi$, and $\psi$. The twist-free solutions ($\varrho$ real) to Equations (25, 26) constitute a 3-dimensional linear subspace of the 4-dimensional space of all solutions. This subspace carries a natural metric of Lorentzian signature, unique up to a conformal factor, and was nicknamed Minikowski space in [26]. Conservation law. As the optical tidal matrix $\boldsymbol{R}$ is symmetric, for any two solutions $\boldsymbol{D}_{1}$ and $\boldsymbol{D}_{2}$ of the matrix Jacobi equation (16) we have $$\dot{\boldsymbol{D}}_{1}\boldsymbol{D}_{2}^{T}-\boldsymbol{D}_{1}\dot{% \boldsymbol{D}}_{2}^{T}=\mbox{constant},$$ (28) where $(~{})^{T}$ means transposition. Evaluating the case $\boldsymbol{D}_{1}=\boldsymbol{D}_{2}$ shows that for every infinitesimally thin bundle $$\omega D_{+}D_{-}=\mbox{constant}.$$ (29) Thus, there are two types of infinitesimally thin bundles: those for which this constant is non-zero and those for which it is zero. In the first case the bundle is twisting ($\omega\neq 0$ everywhere) and its cross-section nowhere collapses to a line or to a point ($D_{+}\neq 0$ and $D_{-}\neq 0$ everywhere). In the second case the bundle must be non-twisting ($\omega=0$ everywhere), because our definition of infinitesimally thin bundles implies that $D_{+}\neq 0$ and $D_{-}\neq 0$ almost everywhere. A quick calculation shows that $\omega=0$ is exactly the integrability condition that makes sure that the infinitesimally thin bundle can be embedded in a wave front. (For the definition of wave fronts see Section 2.2.) In other words, an infinitesimally thin bundle is twist-free if and only if we can find a wave front such that $\lambda$ is one of the generators and the vector fields $Y_{1}$ and $Y_{2}$ connect $\lambda$ with infinitesimally neighboring generators. For a (necessarily twist-free) infinitesimally thin bundle, points where one of the two shape parameters $D_{+}$ and $D_{-}$ vanishes are called caustic points of multiplicity one, and points where both shape parameters $D_{+}$ and $D_{-}$ vanish are called caustic points of multiplicity two. This notion coincides exactly with the notion of caustic points, or conjugate points, of wave fronts as introduced in Section 2.2. The behavior of the optical scalars near caustic points can be deduced from Equation (27) with Equations (25, 26). For a caustic point of multiplicty one at $s=s_{0}$ one finds $$\displaystyle\theta(s)$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{2(s-s_{0})}\left(1+{\cal O}(s-s_{0})\right),$$ (30) $$\displaystyle\left|\sigma(s)\right|$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{2(s-s_{0})}\left(1+{\cal O}(s-s_{0})\right).$$ (31) By contrast, for a caustic point of multiplicity two at $s=s_{0}$ the equations read (cf. [373]) $$\displaystyle\theta(s)$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{s-s_{0}}+{\cal O}(s-s_{0}),$$ (32) $$\displaystyle\sigma(s)$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{3}\psi_{0}(s_{0})(s-s_{0})+{\cal O}\left((s-s_{0})^{2}% \right).$$ (33) Infinitesimally thin bundles with vertex. We say that an infinitesimally thin bundle has a vertex at $s=s_{0}$ if the Jacobi matrix satisfies $$\boldsymbol{D}(s_{0})=\boldsymbol{0},\qquad\dot{\boldsymbol{D}}(s_{0})=% \boldsymbol{1}.$$ (34) A vertex is, in particular, a caustic point of multiplicity two. An infinitesimally thin bundle with a vertex must be non-twisting. While any non-twisting infinitesimally thin bundle can be embedded in a wave front, an infinitesimally thin bundle with a vertex can be embedded in a light cone. Near the vertex, to within a first-order approximation with respect to $s-s_{0}$, it has a circular cross-section. If $\boldsymbol{D}_{1}$ has a vertex at $s_{1}$ and $\boldsymbol{D}_{2}$ has a vertex at $s_{2}$, the conservation law (28) implies $$\boldsymbol{D}_{2}^{T}(s_{1})=-\boldsymbol{D}_{1}(s_{2}).$$ (35) This is Etherington’s [133] reciprocity law. The method by which this law was proven here follows Ellis [127] (cf. Schneider, Ehlers, and Falco [367]). Etherington’s reciprocity law is of relevance, in particular in view of cosmology, because it relates the luminosity distance to the area distance (see Equation (47)). It was independently rediscovered in the 1960s by Sachs and Penrose (see [328, 243]). The results of this section are the basis for Sections 2.4, 2.5, and 2.6. 2.4 Distance measures In this section we summarize various distance measures that are defined in an arbitrary spacetime. Some of them are directly related to observable quantities with relevance for lensing. The material of this section makes use of the results on infinitesimally thin bundles which are summarized in Section 2.3. All of the distance measures to be discussed refer to a past-oriented lightlike geodesic $\lambda$ from an observation event $p_{\mathrm{O}}$ to an emission event $p_{\mathrm{S}}$ (see Figure 4). Some of them depend on the 4-velocity $U_{\mathrm{O}}$ of the observer at $p_{\mathrm{O}}$ and/or on the 4-velocity $U_{\mathrm{S}}$ of the light source at $p_{\mathrm{S}}$. If a vector field $U$ with $g(U,U)=-1$ is distinguished on $\mathcal{M}$, we can choose for the observer an integral curve of $U$ and for the light sources all other integral curves of $U$. Then each of the distance measures becomes a function of the observational coordinates $(s,\Psi,\Theta,\tau)$ (recall Section 2.1). Affine distance. There is a unique affine parametrization $s\longmapsto\lambda(s)$ for each lightlike geodesic through the observation event $p_{\mathrm{O}}$ such that $\lambda(0)=p_{\mathrm{O}}$ and $g\left(\dot{\lambda}(0),U_{\mathrm{O}}\right)=1$. Then the affine parameter $s$ itself can be viewed as a distance measure. This affine distance has the desirable features that it increases monotonously along each ray and that it coincides in an infinitesimal neighborhood of $p_{\mathrm{O}}$ with Euclidean distance in the rest system of $U_{\mathrm{O}}$. The affine distance depends on the 4-velocity $U_{\mathrm{O}}$ of the observer but not on the 4-velocity $U_{\mathrm{S}}$ of the light source. It is a mathematically very convenient notion, but it is not an observable. (It can be operationally realized in terms of an observer field whose 4-velocities are parallel along the ray. Then the affine distance results by integration if each observer measures the length of an infinitesimally short part of the ray in his rest system. However, in view of astronomical situations this is a purely theoretical construction.) The notion of affine distance was introduced by Kermack, McCrea, and Whittaker [233]. Travel time. As an alternative distance measure one can use the travel time. This requires the choice of a time function, i.e., of a function $t$ that slices the spacetime into spacelike hypersurfaces $t=\mbox{constant}$. (Such a time function globally exists if and only if the spacetime is stably causal; see, e.g., [193], p. 198.) The travel time is equal to $t(p_{\mathrm{O}})-t(p_{\mathrm{S}})$, for each $p_{\mathrm{S}}$ on the past light cone of $p_{\mathrm{O}}$. In other words, the intersection of the light cone with a hypersurface $t=\mbox{constant}$ determines events of equal travel time; we call these intersections “instantaneous wave fronts” (recall Section 2.2). Examples of instantaneous wave fronts are shown in Figures 14, 19, 20, 28, and 29. The travel time increases monotonously along each ray. Clearly, it depends neither on the 4-velocity $U_{\mathrm{O}}$ of the observer nor on the 4-velocity $U_{\mathrm{S}}$ of the light source. Note that the travel time has a unique value at each point of $p_{\mathrm{O}}$’s past light cone, even at events that can be reached by two different rays from $p_{\mathrm{O}}$. Near $p_{\mathrm{O}}$ the travel time coincides with Euclidean distance in the observer’s rest system only if $U_{\mathrm{O}}$ is perpendicular to the hypersurface $t=\mbox{constant}$ with $dt(U_{\mathrm{O}})=1$. (The latter equation is true if along the observer’s world line the time function $t$ coincides with proper time.) The travel time is not directly observable. However, travel time differences are observable in multiple-imaging situations if the intrinsic luminosity of the light source is time-dependent. To illustrate this, think of a light source that flashes at a particular instant. If the flash reaches the observer’s wordline along two different rays, the proper time difference $\Delta\tau_{\mathrm{O}}$ of the two arrival events is directly measurable. For a time function $t$ that along the observer’s worldline coincides with proper time, this observed time delay $\Delta\tau_{\mathrm{O}}$ gives the difference in travel time for the two rays. In view of applications, the measurement of time delays is of great relevance for quasar lensing. For the double quasar 0957+561 the observed time delay $\Delta\tau_{\mathrm{O}}$ is about 417 days (see, e.g., [343], p. 149). Redshift. In cosmology it is common to use the redshift as a distance measure. For assigning a redshift to a lightlike geodesic $\lambda$ that connects the observation event $p_{\mathrm{O}}$ on the worldline $\gamma_{\mathrm{O}}$ of the observer with the emission event $p_{\mathrm{S}}$ on the worldline $\gamma_{\mathrm{S}}$ of the light source, one considers a neighboring lightlike geodesic that meets $\gamma_{\mathrm{O}}$ at a proper time interval $\Delta\tau_{\mathrm{O}}$ from $p_{\mathrm{O}}$ and $\gamma_{\mathrm{S}}$ at a proper time interval $\Delta\tau_{\mathrm{S}}$ from $p_{\mathrm{S}}$. The redshift $z$ is defined as $$z=\!\!\underset{\Delta\tau_{\mathrm{S}}\to 0}{\lim}\frac{\Delta\tau_{\mathrm{O% }}-\Delta\tau_{\mathrm{S}}}{\Delta\tau_{\mathrm{S}}}.$$ (36) If $\lambda$ is affinely parametrized with $\lambda(0)=p_{\mathrm{O}}$ and $\lambda(s)=p_{\mathrm{S}}$, one finds that $z$ is given by $$1+z=\frac{g\left(\dot{\lambda}(0),U_{\mathrm{O}}\right)}{g\left(\dot{\lambda}(% s),U_{\mathrm{S}}\right)}.$$ (37) This general redshift formula is due to Kermack, McCrea, and Whittaker [233]. Their proof is based on the fact that $g(\dot{\lambda},Y)$ is a constant for all Jacobi fields $Y$ that connect $\lambda$ with an infinitesimally neighboring lightlike geodesic. The same proof can be found, in a more elegant form, in [59] and in [389], p. 109. An alternative proof, based on variational methods, was given by Schrödinger [368]. Equation (37) is in agreement with the Hamilton formalism for photons. Clearly, the redshift depends on the 4-velocity $U_{\mathrm{O}}$ of the observer and on the 4-velocity $U_{\mathrm{S}}$ of the light source. If a vector field $U$ with $g(U,U)=-1$ has been distinguished on $\mathcal{M}$, we may choose one integral curve of $U$ as the observer and all other integral curves of $U$ as the light sources. Then the redshift becomes a function of the observational coordinates $(s,\Psi,\Theta,\tau)$. For $s\to 0$, the redshift goes to 0, $$z(s,\Psi,\Theta,\tau)=h(\Psi,\Theta,\tau)s+{\cal O}(s^{2}),$$ (38) with a (generalized) Hubble parameter $h(\Psi,\Theta,\tau)$ that depends on spatial direction and on time. For criteria that $h$ and the higher-order coefficients are independent of $\Psi$ and $\Theta$ see [190]. If the redshift is known for one observer field $U$, it can be calculated for any other $U$, according to Equation (37), just by adding the usual special-relativistic Doppler factors. Note that if $U_{\mathrm{O}}$ is given, the redshift can be made to zero along any one ray $\lambda$ from $p_{\mathrm{O}}$ by choosing the 4-velocities $U_{\lambda(s)}$ appropriately. This shows that $z$ is a reasonable distance measure only for special situations, e.g., in cosmological models with $U$ denoting the mean flow of luminous matter (“Hubble flow”). In any case, the redshift is directly observable if the light source emits identifiable spectral lines. For the calculation of Sagnac-like effects, the redshift formula (37) can be evaluated piecewise along broken lightlike geodesics [29]. Angular diameter distances. The notion of angular diameter distance is based on the intuitive idea that the farther an object is away the smaller it looks, according to the rule $$\mbox{object diameter}=\mbox{angle}\times\mbox{distance}.$$ (39) The formal definition needs the results of Section 2.3 on infinitesimally thin bundles. One considers a past-oriented lightlike geodesic $s\longrightarrow\lambda(s)$ parametrized by affine distance, i.e., $\lambda(0)=p_{\mathrm{O}}$ and $g\bigl{(}\dot{\lambda}(0),U_{\mathrm{O}}\bigr{)}=1$, and along $\lambda$ an infinitesimally thin bundle with vertex at the observer, i.e., at $s=0$. Then the shape parameters $D_{+}(s)$ and $D_{-}(s)$ (recall Figure 3) satisfy the initial conditions $D_{\pm}(0)=0$ and $\dot{D}_{\pm}(0)=1$. They have the following physical meaning. If the observer sees a circular image of (small) angular diameter $\alpha$ on his or her sky, the (small but extended) light source at affine distance $s$ actually has an elliptical cross-section with extremal diameters $\alpha|D_{\pm}(s)|$. It is therefore reasonable to call $D_{+}$ and $D_{-}$ the extremal angular diameter distances. Near the vertex, $D_{+}$ and $D_{-}$ are monotonously increasing functions of the affine distance, $D_{\pm}(s)=s+{\cal O}(s^{2})$. Farther away from the vertex, however, they may become decreasing, so the functions $s\mapsto D_{+}(s)$ and $s\mapsto D_{-}(s)$ need not be invertible. At a caustic point of multiplicity one, one of the two functions $D_{+}$ and $D_{-}$ changes sign; at a caustic point of multiplicity two, both change sign (recall Section 2.3). The image of a light source at affine distance $s$ is said to have even parity if $D_{+}(s)D_{-}(s)>0$ and odd parity if $D_{+}(s)D_{-}(s)<0$. Images with odd parity show the neighborhood of the light source side-inverted in comparison to images with even parity. Clearly, $D_{+}$ and $D_{-}$ are reasonable distance measures only in a neighborhood of the vertex where they are monotonously increasing. However, the physical relevance of $D_{+}$ and $D_{-}$ lies in the fact that they relate cross-sectional diameters at the source to angular diameters at the observer, and this is always true, even beyond caustic points. $D_{+}$ and $D_{-}$ depend on the 4-velocity $U_{\mathrm{O}}$ of the observer but not on the 4-velocity $U_{\mathrm{S}}$ of the source. This reflects the fact that the angular diameter of an image on the observer’s sky is subject to aberration whereas the cross-sectional diameter of an infinitesimally thin bundle has an invariant meaning (recall Section 2.3). Hence, if the observer’s worldline $\gamma_{\mathrm{O}}$ has been specified, $D_{+}$ and $D_{-}$ are well-defined functions of the observational coordinates $(s,\Psi,\Theta,\tau)$. Area distance. The area distance $D_{\mathrm{area}}$ is defined according to the idea $$\mbox{object area}=\mbox{solid angle}\times\mbox{distance}^{2}.$$ (40) As a formal definition for $D_{\mathrm{area}}$, in terms of the extremal angular diameter distances $D_{+}$ and $D_{-}$ as functions of affine distance $s$, we use the equation $$D_{\mathrm{area}}(s)=\sqrt{\left|D_{+}(s)\,D_{-}(s)\right|}.$$ (41) $D_{\mathrm{area}}(s)^{2}$ indeed relates, for a bundle with vertex at the observer, the cross-sectional area at the source to the opening solid angle at the observer. Such a bundle has a caustic point exactly at those points where $D_{\mathrm{area}}(s)=0$. The area distance is often called “angular diameter distance” although, as indicated by Equation (41), the name “averaged angular diameter distance” would be more appropriate. Just as $D_{+}$ and $D_{-}$, the area distance depends on the 4-velocity $U_{\mathrm{O}}$ of the observer but not on the 4-velocity $U_{\mathrm{S}}$ of the light source. The area distance is observable for a light source whose true size is known (or can be reasonably estimated). It is sometimes convenient to introduce the magnification or amplification factor $$\mu(s)=\frac{s^{2}}{D_{+}(s)\,D_{-}(s)}.$$ (42) The absolute value of $\mu$ determines the area distance, and the sign of $\mu$ determines the parity. In Minkowski spacetime, $D_{\pm}(s)=s$ and, thus, $\mu(s)=1$. Hence, $|\mu(s)|>1$ means that a (small but extended) light source at affine distance $s$ subtends a larger solid angle on the observer’s sky than a light source of the same size at the same affine distance in Minkowski spacetime. Note that in a multiple-imaging situation the individual images may have different affine distances. Thus, the relative magnification factor of two images is not directly observable. This is an important difference to the magnification factor that is used in the quasi-Newtonian approximation formalism of lensing. The latter is defined by comparison with an “unlensed image” (see, e.g., [367]), a notion that makes sense only if the metric is viewed as a perturbation of some “background” metric. One can derive a differential equation for the area distance (or, equivalently, for the magnification factor) as a function of affine distance in the following way. On every parameter interval where $D_{+}D_{-}$ has no zeros, the real part of Equation (27) shows that the area distance is related to the expansion by $$\dot{D}_{\mathrm{area}}=\theta D_{\mathrm{area}}.$$ (43) Insertion into the Sachs equation (25) for $\theta=\varrho$ gives the focusing equation $$\ddot{D}_{\mathrm{area}}=-\left(|\sigma|^{2}+\frac{1}{2}\operatorname{Ric}(% \dot{\lambda},\dot{\lambda})\right)D_{\mathrm{area}}.$$ (44) Between the vertex at $s=0$ and the first conjugate point (caustic point), $D_{\mathrm{area}}$ is determined by Equation (44) and the initial conditions $$D_{\mathrm{area}}(0)=0,\qquad\dot{D}_{\mathrm{area}}(0)=1.$$ (45) The Ricci term in Equation (44) is non-negative if Einstein’s field equation holds and if the energy density is non-negative for all observers (“weak energy condition”). Then Equations (44, 45) imply that $$D_{\mathrm{area}}(s)\leq s,$$ (46) i.e., $1\leq\mu(s)$, for all $s$ between the vertex at $s=0$ and the first conjugate point. In Minkowski spacetime, the equation $D_{\mathrm{area}}(s)=s$ holds. Hence, the inequality (46) says that a gravitational field has a focusing, as opposed to a defocusing, effect. This is sometimes called the focusing theorem. Corrected luminosity distance. The idea of defining distance measures in terms of bundle cross-sections dates back to Tolman [404] and Whittaker [435]. Originally, this idea was applied not to bundles with vertex at the observer but rather to bundles with vertex at the light source. The resulting analogue of the area distance is the so-called corrected luminosity distance $D^{\prime}_{\mathrm{lum}}$. It relates, for a bundle with vertex at the light source, the cross-sectional area at the observer to the opening solid angle at the light source. Owing to Etherington’s reciprocity law (35), area distance and corrected luminosity distance are related by $$D^{\prime}_{\mathrm{lum}}=(1+z)D_{\mathrm{area}}.$$ (47) The redshift factor has its origin in the fact that the definition of $D^{\prime}_{\mathrm{lum}}$ refers to an affine parametrization adapted to $U_{\mathrm{S}}$, and the definition of $D_{\mathrm{area}}$ refers to an affine parametrization adapted to $U_{\mathrm{O}}$. While $D_{\mathrm{area}}$ depends on $U_{\mathrm{O}}$ but not on $U_{\mathrm{S}}$, $D^{\prime}_{\mathrm{lum}}$ depends on $U_{\mathrm{S}}$ but not on $U_{\mathrm{O}}$. Luminosity distance. The physical meaning of the corrected luminosity distance is most easily understood in the photon picture. For photons isotropically emitted from a light source, the percentage that hit a prescribed area at the observer is proportional to $1/(D^{\prime}_{\mathrm{lum}})^{2}$. As the energy of each photon undergoes a redshift, the energy flux at the observer is proportional to $1/(D_{\mathrm{lum}})^{2}$, where $$D_{\mathrm{lum}}=(1+z)D^{\prime}_{\mathrm{lum}}=(1+z)^{2}D_{\mathrm{area}}.$$ (48) Thus, $D_{\mathrm{lum}}$ is the relevant quantity for calculating the luminosity (apparent brightness) of pointlike light sources (see Equation (52)). For this reason $D_{\mathrm{lum}}$ is called the (uncorrected) luminosity distance. The observation that the purely geometric quantity $D^{\prime}_{\mathrm{lum}}$ must be modified by an additional redshift factor to give the energy flux is due to Walker [426]. $D_{\mathrm{lum}}$ depends on the 4-velocity $U_{\mathrm{O}}$ of the observer and of the 4-velocity $U_{\mathrm{S}}$ of the light source. $D_{\mathrm{lum}}$ and $D^{\prime}_{\mathrm{lum}}$ can be viewed as functions of the observational coordinates $(s,\Psi,\Theta,\tau)$ if a vector field $U$ with $g(U,U)=-1$ has been distinguished, one integral curve of $U$ is chosen as the observer, and the other integral curves of $U$ are chosen as the light sources. In that case Equation (38) implies that not only $D_{\mathrm{area}}(s)$ but also $D_{\mathrm{lum}}(s)$ and $D^{\prime}_{\mathrm{lum}}(s)$ are of the form $s+{\cal O}(s^{2})$. Thus, near the observer all three distance measures coincide with Euclidean distance in the observer’s rest space. Parallax distance. In an arbitrary spacetime, we fix an observation event $p_{\mathrm{O}}$ and the observer’s 4-velocity $U_{\mathrm{O}}$. We consider a past-oriented lightlike geodesic $\lambda$ parametrized by affine distance, $\lambda(0)=p_{\mathrm{O}}$ and $g\left(\dot{\lambda}(0),U_{\mathrm{O}}\right)=1$. To a light source passing through the event $\lambda(s)$ we assign the (averaged) parallax distance $D_{\mathrm{par}}(s)=-\theta(0)^{-1}$, where $\theta$ is the expansion of an infinitesimally thin bundle with vertex at $\lambda(s)$. This definition follows [221]. Its relevance in view of cosmology was discussed in detail by Rosquist [357]. $D_{\mathrm{par}}$ can be measured by performing the standard trigonometric parallax method of elementary Euclidean geometry, with the observer at $p_{\mathrm{O}}$ and an assistant observer at the perimeter of the bundle, and then averaging over all possible positions of the assistant. Note that the method refers to a bundle with vertex at the light source, i.e., to light rays that leave the light source simultaneously. (Averaging is not necessary if this bundle is circular.) $D_{\mathrm{par}}$ depends on the 4-velocity of the observer but not on the 4-velocity of the light source. To within first-order approximation near the observer it coincides with affine distance (recall Equation (32)). For the potential obervational relevance of $D_{\mathrm{par}}$ see [357], and [367], p. 509. In view of lensing, $D_{+}$, $D_{-}$, and $D_{\mathrm{lum}}$ are the most important distance measures because they are related to image distortion (see Section 2.5) and to the brightness of images (see Section 2.6). In spacetimes with many symmetries, these quantities can be explicitly calculated (see Section 4.1 for conformally flat spacetimes, and Section 4.3 for spherically symmetric static spacetimes). This is impossible in a spacetime without symmetries, in particular in a realistic cosmological model with inhomogeneities (“clumpy universe”). Following Kristian and Sachs [243], one often uses series expansions with respect to $s$. For statistical considerations one may work with the focusing equation in a Friedmann–Robertson–Walker spacetime with average density (see Section 4.1), or with a heuristically modified focusing equation taking clumps into account. The latter leads to the so-called Dyer–Roeder distance [112, 113] which is discussed in several text-books (see, e.g., [367]). (For pre-Dyer–Roeder papers on optics in cosmological models with inhomogeneities, see the historical notes in [223].) As overdensities have a focusing and underdensities have a defocusing effect, it is widely believed (following [428]) that after averaging over sufficiently large angular scales the Friedmann–Robertson–Walker calculation gives the correct distance-redshift relation. However, it was argued by Ellis, Bassett, and Dunsby [129] that caustics produced by the lensing effect of overdensities lead to a systematic bias towards smaller angular sizes (“shrinking”). For a spherically symmetric inhomogeneity, the effect on the distance-redshift relation can be calculated analytically [291]. For thorough discussions of light propagation in a clumpy universe also see Pyne and Birkinshaw [352], and Holz and Wald [203]. 2.5 Image distortion In special relativity, a spherical object always shows a circular outline on the observer’s sky, independent of its state of motion [325, 402]. In general relativity, this is no longer true; a small sphere usually shows an elliptic outline on the observer’s sky. This distortion is caused by the shearing effect of the spacetime geometry on light bundles. For the calculation of image distortion we need the material of Sections 2.3 and 2.4. For an observer with 4-velocity $U_{\mathrm{O}}$ at an event $p_{\mathrm{O}}$, there is a unique affine parametrization $s\longmapsto\lambda(s)$ for each lightlike geodesic through $p_{\mathrm{O}}$ such that $\lambda(0)=p_{\mathrm{O}}$ and $g\bigl{(}\dot{\lambda}(0),U_{\mathrm{O}}\bigr{)}=1$. Around each of these $\lambda$ we can consider an infinitesimally thin bundle with vertex at $s=0$. The elliptical cross-section of this bundle can be characterized by the shape parameters $D_{+}(s)$, $D_{-}(s)$ and $\chi(s)$ (recall Figure 3). As outlined in Section 2.3, we choose the convention of having $D_{+}(s)$ and $D_{-}(s)$ positive for small positive $s$. In the terminology of Section 2.4, $s$ is the affine distance, and $D_{+}(s)$ and $D_{-}(s)$ are the extremal angular diameter distances. The complex quantity $$\epsilon(s)=\left(\frac{D_{+}(s)}{D_{-}(s)}-\frac{D_{-}(s)}{D_{+}(s)}\right)e^% {2i\chi(s)}$$ (49) is called the ellipticity of the bundle. The phase of $\epsilon$ determines the position angle of the elliptical cross-section of the bundle with respect to the Sachs basis. The absolute value of $\epsilon(s)$ determines the eccentricity of this cross-section; $\epsilon(s)=0$ indicates a circular cross-section and $|\epsilon(s)|=\infty$ indicates a caustic point of multiplicity one. (It is also common to use other measures for the eccentricity, e.g., $|D_{+}-D_{-}|/|D_{+}+D_{-}|$.) From Equation (27) with $\varrho=\theta$ we get the derivative of $\epsilon$ with respect to the affine distance $s$, $$\dot{\epsilon}=2\sigma\sqrt{|\epsilon|^{2}+4}.$$ (50) The initial conditions $D_{\pm}(0)=0$, $\dot{D}_{\pm}(0)=1$ imply $$\epsilon(0)=0.$$ (51) Equation (50) and Equation (51) determine $\epsilon$ if the shear $\sigma$ is known. The shear, in turn, is determined by the Sachs equations (25, 26) and the initial conditions (32, 33) with $s_{0}=0$ for $\theta(=\varrho)$ and $\sigma$. It is recommendable to change from the $\epsilon$ determined this way to $\varepsilon=-\overline{\epsilon}$. This transformation corresponds to replacing the Jacobi matrix $\boldsymbol{D}$ by its inverse. The original quantity $\epsilon(s)$ gives the true shape of objects at affine distance $s$ that show a circular image on the observer’s sky. The new quantity $\varepsilon(s)$ gives the observed shape for objects at affine distance $s$ that actually have a circular cross-section. In other words, if a (small) spherical body at affine distance $s$ is observed, the ellipticity of its image on the observer’s sky is given by $\varepsilon(s)$. By Equations (50, 51), $\epsilon$ vanishes along the entire ray if and only if the shear $\sigma$ vanishes along the entire ray. By Equations (26, 33), the shear vanishes along the entire ray if and only if the conformal curvature term $\psi_{0}$ vanishes along the entire ray. The latter condition means that $K=\dot{\lambda}$ is tangent to a principal null direction of the conformal curvature tensor (see, e.g., Chandrasekhar [75]). At a point where the conformal curvature tensor is not zero, there are at most four different principal null directions. Hence, the distortion effect vanishes along all light rays if and only if the conformal curvature vanishes everywhere, i.e., if and only if the spacetime is conformally flat. This result is due to Sachs [360]. An alternative proof, based on expressions for image distortions in terms of the exponential map, was given by Hasse [186]. For any observer, the distortion measure $\varepsilon=-\overline{\epsilon}$ is defined along every light ray from every point of the observer’s worldline. This gives $\varepsilon$ as a function of the observational coordinates $(s,\Psi,\Theta,\tau)$ (recall Section 2.1, in particular Equation (4)). If we fix $\tau$ and $s$, $\varepsilon$ is a function on the observer’s sky. (Instead of $s$, one may choose any of the distance measures discussed in Section 2.4, provided it is a unique function of $s$.) In spacetimes with sufficiently many symmetries, this function can be explicitly determined in terms of integrals over the metric function. This will be worked out for spherically symmetric static spacetimes in Section 4.3. A general consideration of image distortion and example calculations can also be found in papers by Frittelli, Kling and Newman [152, 151]. Frittelli and Oberst [158] calculate image distortion by a “thick gravitational lens” model within a spacetime setting. In cases where it is not possible to determine $\varepsilon$ by explicitly integrating the relevant differential equations, one may consider series expansions with respect to the affine parameter $s$. This technique, which is of particular relevance in view of cosmology, dates back to Kristian and Sachs [243] who introduced image distortion as an observable in cosmology. In lowest non-vanishing order, $\varepsilon(s,\Psi,\Theta,\tau_{\mathrm{O}})$ is quadratic with respect to $s$ and completely determined by the conformal curvature tensor at the observation event $p_{\mathrm{O}}=\gamma(\tau_{\mathrm{O}})$, as can be read from Equations (50, 51, 33). One can classify all possible distortion patterns on the observer’s sky in terms of the Petrov type of the Weyl tensor [78]. As outlined in [78], these patterns are closely related to what Penrose and Rindler [330] call the fingerprint of the Weyl tensor. At all observation events where the Weyl tensor is non-zero, the following is true. There are at most four points on the observer’s sky where the distortion vanishes, corresponding to the four (not necessarily distinct) principal null directions of the Weyl tensor. For type $N$, where all four principal null directions coincide, the distortion pattern is shown in Figure 5. The distortion effect is routinely observed since the mid-1980s in the form of arcs and (radio) rings (see [367, 343, 427] for an overview). In these cases a distant galaxy appears strongly elongated in one direction. Such strong elongations occur near a caustic point of multiplicity one where $|\varepsilon|\to\infty$. In the case of rings and (long) arcs, the entire bundle cannot be treated as infinitesimally thin, i.e., a theoretical description of the effect requires an integration. For the idealized case of a point source, images in the form of (1-dimensional) rings on the observer’s sky occur in cases of rotational symmetry and are usually called “Einstein rings” (see Section 4.3). The rings that are actually observed show extended sources in situations close to rotational symmetry. For the majority of galaxies that are not distorted into arcs or rings, there is a “weak lensing” effect on the apparent shape that can be investigated statistically. The method is based on the assumption that there is no prefered direction in the universe, i.e., that the axes of (approximately spheroidal) galaxies are randomly distributed. So, without a distortion effect, the axes of galaxy images should make a randomly distributed angle with the $(\Psi,\Theta)$ grid on the observer’s sky. Any deviation from a random distribution is to be attributed to a distortion effect, produced by the gravitational field of intervening masses. With the help of the quasi-Newtonian approximation, this method has been elaborated into a sophisticated formalism for determining mass distributions, projected onto the plane perpendicular to the line of sight, from observed image distortions. This is one of the most important astrophysical tools for detecting (dark) matter. It has been used to determine the mass distribution in galaxies and galaxy clusters, and to probe the large-scale structure of the universe (see [28, 206] for reviews). From a methodological point of view, it would be desirable to analyse this important line of astronomical research within a spacetime setting. This should give prominence to the role of the conformal curvature tensor. Another interesting way of observing weak image distortions is possible for sources that emit linearly polarized radiation. This is true for many radio galaxies. (Polarization measurements are also relevant for strong-lensing situations; see Schneider, Ehlers, and Falco [367], p. 82 for an example.) The method is based on the geometric optics approximation of Maxwell’s theory. In this approximation, the polarization direction is parallel along each ray between source and observer [114] (cf., e.g., [279], p. 577). We may, thus, choose the Sachs basis $(E_{1},E_{2})$ such that the plane spanned by $K$ and $E_{1}$ gives the polarization direction. This fixes the Sachs basis up to transformations (14) with $\alpha=0$, i.e., it gives an unambiguous (observer-independent) meaning to the angle $\chi$ in Figure 3. If a light source (e.g., a galaxy) shows an approximately elliptic shape on the observer’s sky, it is reasonable to assume that at the light source the polarization direction is aligned with one of the axes, i.e., $2\chi(s)/\pi\in\mathbb{Z}$. A distortion effect is verified if the observed polarization direction is not aligned with an axis of the image, $2\chi(0)/\pi\notin\mathbb{Z}$. It is to be emphasized that such a change of the angle $\chi$ along the ray cannot be the result of a rotation; the bundles under consideration have a vertex and are, thus, twist-free. It can only be the result of successive shearing processes, governed by the behaviour of the conformal curvature tensor along the ray. Also, the effect has nothing to do with the rotation of an observer field; we have already stressed that the angle $\chi$ is observer-independent. Related misunderstandings have been clarified by Panov and Sbytov [322, 323]. So far, this distortion effect has not been observed. (Panov and Sbytov [322] have clearly shown that an anisotropy observed by Birch [42], even if real, cannot be interpreted in this way.) Its future detectability is estimated, for distant radio sources, in [396]. The effect of a gravitatational field on the polarization direction of light was first discussed by Skrotskii [383] in 1957 and is therefore sometimes called the “Skrotskii effect”. If the spacetime is conformally stationary, and if the worldlines of observers and light sources are integral curves of the conformal Killing vector field, the effect can be expressed in terms of the “Fermat geometry” of 3-space [189], see Section 4.2 below for the definition of the Fermat geometry. (Note that Figure 1 in [189] is erroneous because it ignores the fact that, in general, the principal shear directions of a bundle are not parallel along the central ray.) Relative to a frame that is parallel with respect to the Fermat metric, one finds a rotation of the polarization direction that is analogous to the well-known Faraday rotation in a magnetic field. In this analogy, the magnetic field corresponds to the rotation (twist) of the conformal Killing vector field. Because of this analogy, the Skrotskii effect is also known as the “gravitational Faraday effect”. It has been quite extensively discussed for stationary spacetimes and, in particular, for the Kerr metric (see, e.g., [178, 395, 142, 211, 310, 374]). All these articles give formulas for the rotation of the polarization direction relative to a frame distinguished by the symmetry assumptions. This rotation should not be confused with the above-mentioned motion of the polarization direction relative to the orientation of the image. The latter is a distortion effect, governed by the conformal curvature tensor; the former is a gravitomagnetic effect, governed by the rotation of a distinguished observer field. 2.6 Brightness of images For calculating the brightness of images we need the definitions and results of Section 2.4. In particular we need the luminosity distance $D_{\mathrm{lum}}$ and its relation to other distance measures. We begin by considering a point source (worldline) that emits isotropically with (bolometric, i.e., integrated over all frequencies) luminosity $L$. By definition of $D_{\mathrm{lum}}$, in this case the energy flux at the observer is $$F=\frac{L}{4\pi D_{\mathrm{lum}}{}^{2}}.$$ (52) $F$ is a measure for the brightness of the image on the observer’s sky. The magnitude $m$ used by astronomers is essentially the negative logarithm of $F$, $$m=2.5\log_{10}\left(D_{\mathrm{lum}}{}^{2}\right)-2.5\log_{10}(L)+m_{0},$$ (53) with $m_{0}$ being a universal constant. In Equation (52), $D_{\mathrm{lum}}$ can be expresed in terms of the area distance $D_{\mathrm{area}}$ and the redshift $z$ with the help of the general relation (48). This demonstrates that the magnification factor $\mu$, which is defined by Equation (42), admits the following reinterpretation. $|\mu(s)|$ relates the flux from a point source at affine distance $s$ to the flux from a point source with the same luminosity at the same affine distance and at the same redshift in Minkowski spacetime. $D_{\mathrm{lum}}$ can be explicitly calculated in spacetimes where the Jacobi fields along lightlike geodesics can be explicitly determined. This is true, e.g., in spherically symmetric and static spacetimes where the extremal angular diameter distances $D_{+}$ and $D_{-}$ can be calculated in terms of integrals over the metric coefficients. The resulting formulas are given in Section 4.3 below. Knowledge of $D_{+}$ and $D_{-}$ immediately gives the area distance $D_{\mathrm{area}}$ via Equation (41). $D_{\mathrm{area}}$ together with the redshift determines $D_{\mathrm{lum}}$ via Equation (48). Such an explicit calculation is, of course, possible only for spacetimes with many symmetries. By Equation (48), the zeros of $D_{\mathrm{lum}}$ coincide with the zeros of $D_{\mathrm{area}}$, i.e., with the caustic points. Hence, in the ray-optical treatment a point source is infinitely bright (magnitude $m=-\infty$) if it passes through the caustic of the observer’s past light cone. A wave-optical treatment shows that the energy flux at the observer is actually bounded by diffraction. In the quasi-Newtonian approximation formalism, this was demonstrated by an explicit calculation for light rays deflected by a spheroidal mass by Ohanian [313] (cf. [367], p. 220). Quite generally, the ray-optical calculation of the energy flux gives incorrect results if, for two different light paths from the source worldline to the observation event, the time delay is smaller than or approximately equal to the coherence time. Then interference effects give rise to frequency-dependent corrections to the energy flux that have to be calculated with the help of wave optics. In multiple-imaging situations, the time delay decreases with decreasing mass of the deflector. If the deflector is a cluster of galaxies, a galaxy, or a star, interference effects can be ignored. Gould [181] suggested that they could be observable if a deflector of about $10^{-15}$ Solar masses happens to be close to the line of sight to a gamma-ray burster. In this case, the angle-separation between the (unresolvable) images would be of the order $10^{-15}$ arcseconds (“femtolensing”). Interference effects could make a frequency-dependent imprint on the total intensity. Ulmer and Goodman [409] discussed related effects for deflectors of up to $10^{-11}$ Solar masses. Femtolensing has not been observed so far. However, it is an interesting future perspective for lensing effects where wave optics has to be taken into account. This would give practical relevance to the theoretical work of Herlt and Stephani [196, 197] who calculated gravitational lensing on the basis of wave optics in the Schwarzschild spacetime. Wave-optical aspects of gravitational lensing are also discussed in [294]. We now turn to the case of an extended source, whose surface makes up a 3-dimensional timelike submanifold $\mathcal{T}$ of the spacetime. In this case the radiation is characterized by the surface brightness $B$ (= luminosity $L$ per area) at the source and by the intensity $I$ (= energy flux $F$ per solid angle) at the observer. For each past-oriented light ray from an observation event $p_{\mathrm{O}}$ and to an event $p_{\mathrm{S}}$ on $\mathcal{T}$, we can relate $B$ and $I$ in the following way. By definition, the area distance $D_{\mathrm{area}}$ relates the area at the source to the solid angle at the observer, so we get from Equation (52) $I=BD_{\mathrm{area}}{}^{2}/(4\pi D_{\mathrm{lum}}{}^{2})$. As area distance and luminosity distance are related by a redshift factor, according to the general law (48), this gives the relation $$I=\frac{B}{4\pi(1+z)^{4}}.$$ (54) This result is, of course, valid only if the radiation from different parts of the emitting surface is incoherent; otherwise interference effects have to be taken into account. The most remarkable feature of Equation (54) is that all distance measures have dropped out. Save for a redshift factor, the (observed) intensity of a radiating surface is the same for all observers. The law for point sources (52) and the law for extended sources (54) refer to bolometric quantities, i.e., to integration over all frequencies. As every astronomical observation is restricted to a certain frequency range, it is actually necessary to consider frequency-specific quantities. For a point source, one writes $L=\int_{0}^{\infty}\ell(\omega_{\mathrm{S}})d\omega_{\mathrm{S}}$ and $F=\int_{0}^{\infty}f(\omega_{\mathrm{O}})d\omega_{\mathrm{O}}$, where the specific luminosity $\ell$ is a function of the emitted frequency $\omega_{\mathrm{S}}$ and the specific flux $f$ is a function of the received frequency $\omega_{\mathrm{O}}$. As $\omega_{\mathrm{S}}$ and $\omega_{\mathrm{O}}$ are related by a redshift factor, the frequency-specific version of Equation (52) reads $$f(\omega_{\mathrm{O}})=\frac{\ell\left(\omega_{\mathrm{O}}(1+z)\right)(1+z)}{4% \pi D_{\mathrm{lum}}{}^{2}}.$$ (55) Similarly, for an extended source one introduces a specific surface brightness $b$ and a specific intensity $i$ such that $B=\int_{0}^{\infty}b(\omega_{\mathrm{S}})d\omega_{\mathrm{S}}$ and $I=\int_{0}^{\infty}i(\omega_{\mathrm{O}})d\omega_{\mathrm{O}}$. Then one gets the following frequency-specific version of Equation (54). $$i(\omega_{\mathrm{O}})=\frac{b\left(\omega_{\mathrm{O}}(1+z)\right)}{4\pi(1+z)% ^{3}}.$$ (56) The results summarized in this section can also be derived from the kinetic theory of photons (see, e.g., [115]). In the photon picture, the three redshift factors in Equation (56) are easily understood: The first reflects the fact that each photon undergoes a redshift; the second relates the rate of emission (with respect to proper time at the source) to the rate of reception (with respect to proper time at the obsever); the third reflects the aberration effect on the angular size of the source in dependence of the motion of the observer. As an example for the calculation of the brightness of images we consider the Schwarzschild spactime (see Figure 18). 2.7 Conjugate points and cut points In general, the past light cone of an event forms caustics and transverse self-intersections, i.e., it is neither an embedded nor an immersed submanifold. The relevance of this fact in view of lensing was emphasized already in Section 2.1. In the following we demonstrate that caustics and transverse self-intersections of the light cone are related to extremizing properties of lightlike geodesics. A light cone with a caustic and a transverse self-intersection is shown in Figure 26. In this section and in Section 2.8 we use mathematical techniques which are related to the Penrose–Hawking singularity theorems. For background material, see Penrose [329], Hawking and Ellis [193], O’Neill [315], and Wald [425]. Recall from Section 2.2 that the caustic of the past light cone of $p_{\mathrm{O}}$ is the set of all points where this light cone is not an immersed submanifold. A point $p_{\mathrm{S}}$ is in the caustic if a generator $\lambda$ of the light cone intersects at $p_{\mathrm{S}}$ an infinitesimally neighboring generator. In this situation $p_{\mathrm{S}}$ is said to be conjugate to $p_{\mathrm{O}}$ along $\lambda$. The caustic of the past light cone of $p_{\mathrm{O}}$ is also called the “past lightlike conjugate locus” of $p_{\mathrm{O}}$. The notion of conjugate points is related to the extremizing properties of lightlike geodesics in the following way. Let $\lambda$ be a past-oriented lightlike geodesic with $\lambda(0)=p_{\mathrm{O}}$. Assume that $p_{\mathrm{S}}=\lambda(s_{0})$ is the first conjugate point along this geodesic. This means that $p_{\mathrm{S}}$ is in the caustic of the past light cone of $p_{\mathrm{O}}$ and that $\lambda$ does not meet the caustic at parameter values between 0 and $s_{0}$. Then a well-known theorem says that all points $\lambda(s)$ with $0<s<s_{0}$ cannot be reached from $p_{\mathrm{O}}$ along a timelike curve arbitrarily close to $\lambda$, and all points $\lambda(s)$ with $s>s_{0}$ can. For a proof we refer to Hawking and Ellis [193], Proposition 4.5.11 and Proposition 4.5.12. It might be helpful to consult O’Neill [315], Chapter 10, Proposition 48, in addition. Here we have considered a past-oriented lightlike geodesic because this is the situation with relevance to lensing. Actually, Hawking and Ellis consider the time-reversed situation, i.e., with $\lambda$ future-oriented. Then the result can be phrased in the following way. A material particle may catch up with a light ray $\lambda$ after the latter has passed through a conjugate point and, for particles staying close to $\lambda$, this is impossible otherwise. The restriction to particles staying close to $\lambda$ is essential. Particles “taking a short cut” may very well catch up with a lightlike geodesic even if the latter is free of conjugate points. For a discussion of the extremizing property in the global sense, not restricted to timelike curves close to $\lambda$, we need the notion of cut points. The precise definition of cut points reads as follows. As ususal, let $I^{-}(p_{\mathrm{O}})$ denote the chronological past of $p_{\mathrm{O}}$, i.e., the set of all $q\in\mathcal{M}$ that can be reached from $p_{\mathrm{O}}$ along a past-pointing timelike curve. In Minkowski spacetime, the boundary $\partial I^{-}(p_{\mathrm{O}})$ of $I^{-}(p_{\mathrm{O}})$ is just the past light cone of $p_{\mathrm{O}}$ united with $\{p_{\mathrm{O}}\}$. In an arbitrary spacetime, this is not true. A lightlike geodesic $\lambda$ that issues from $p_{\mathrm{O}}$ into the past is always confined to the closure of $I^{-}(p_{\mathrm{O}})$, but it need not stay on the boundary. The last point on $\lambda$ that is on the boundary is by definition [66] the cut point of $\lambda$. In other words, it is exactly the part of $\lambda$ beyond the cut point that can be reached from $p_{\mathrm{O}}$ along a timelike curve. The union of all cut points, along any past-pointing lightlike geodesic $\lambda$ from $p_{\mathrm{O}}$, is called the cut locus of the past light cone (or the past lightlike cut locus of $p_{\mathrm{O}}$). For the light cone in Figure 25 this is the curve (actually 2-dimensional) where the two sheets of the light cone intersect. For the light cone in Figure 26 the cut locus is the same set plus the swallow-tail point (actually 1-dimensional). For a detailed discussion of cut points in manifolds with metrics of Lorentzian signature, see [32]. For positive definite metrics, the notion of cut points dates back to Poincaré [349] and Whitehead [434]. For a generator $\lambda$ of the past light cone of $p_{\mathrm{O}}$, the cut point of $\lambda$ does not exist in either of the two following cases: 1. $\lambda$ always stays on the boundary $\partial I^{-}(p_{\mathrm{O}})$, i.e., it never loses its extremizing property. 2. $\lambda$ is always in $I^{-}(p_{\mathrm{O}})$, i.e., it fails to be extremizing from the very beginning. Case 2 occurs, e.g., if there is a closed timelike curve through $p_{\mathrm{O}}$. More precisely, Case 2 is excluded if the past distinguishing condition is satisfied at $p_{\mathrm{O}}$, i.e., if for $q\in\mathcal{M}$ the implication $$I^{-}(q)=I^{-}(p_{\mathrm{O}})\quad\Longrightarrow\quad q=p_{\mathrm{O}}$$ (57) holds. If the implication (57) is true, the following can be shown: \hangafter (P1) If, along $\lambda$, the point $\lambda(s)$ is conjugate to $\lambda(0)$, the cut point of $\lambda$ exists and it comes on or before $\lambda(s)$. \hangafter (P2) Assume that a point $q$ can be reached from $p_{\mathrm{O}}$ along two different lightlike geodesics $\lambda_{1}$ and $\lambda_{2}$ from $p_{\mathrm{O}}$. Then the cut point of $\lambda_{1}$ and of $\lambda_{2}$ exists and it comes on or before $q$. \hangafter (P3) If the cut locus of a past light cone is empty, this past light cone is an embedded submanifold of $\mathcal{M}$. For proofs see [336]; The proofs can also be found in or easily deduced from [32]. Statement 2.7 says that conjugate points and cut points are related by the easily remembered rule “the cut point comes first”. Statement 2.7 says that a “cut” between two geodesics is indicated by the occurrence of a cut point. However, it does not say that exactly at the cut point a second geodesic is met. Such a stronger statement, which truly justifies the name “cut point”, holds in globally hyperbolic spacetimes (see Section 3.1). Statement 2.7 implies that the occurrence of transverse self-intersections of a light cone are always indicated by cut points. Note, however, that transverse self-intersections of the past light cone of $p_{\mathrm{O}}$ may occur inside $I^{-}(p_{\mathrm{O}})$ and, thus, far away from the cut locus. Statement 2.7 implies that $\partial I^{-}(p_{\mathrm{O}})$ is an immersed submanifold everywhere except at the cut locus and, of course, at the vertex $p_{\mathrm{O}}$. It is known (see [193], Proposition 6.3.1) that $\partial I^{-}(p_{\mathrm{O}})$ is achronal (i.e., it is impossible to connect any two of its points by a timelike curve) and thus a 3-dimensional Lipschitz topological submanifold. By a general theorem of Rademacher (see [143], Theorem 3.6.1), this implies that $\partial I^{-}(p_{\mathrm{O}})$ is differentiable almost everywhere, i.e., that the cut locus has measure zero in $\partial I^{-}(p_{\mathrm{O}})$. Note that this argument does not necessarily imply that the cut locus is a “small” subset of $\partial I^{-}(p_{\mathrm{O}})$. Chruściel and Galloway [79] have demonstrated, by way of example, that an achronal subset $\mathcal{A}$ of a spacetime may fail to be differentiable on a set that is dense in $\mathcal{A}$. So our reasoning so far does not even exclude the possibility that the cut locus is dense in an open subset of $\partial I^{-}(p_{\mathrm{O}})$. This possibility can be excluded in globally hyperbolic spacetimes where the cut locus is always a closed subset of $\mathcal{M}$ (see Section 3.1). In general, the cut locus need not be closed as is exemplified by Figure 25. In Section 2.8 we investigate the relevance of cut points (and conjugate points) for multiple imaging. 2.8 Criteria for multiple imaging To investigate whether multiple imaging occurs in a spacetime $(\mathcal{M},g)$, we choose any point $p_{\mathrm{O}}$ (observation event) and any timelike curve $\gamma_{\mathrm{S}}$ (wordline of light source) in $\mathcal{M}$. The following cases are possible: 1. There is no past-pointing lightlike geodesic from $p_{\mathrm{O}}$ to $\gamma_{\mathrm{S}}$. Then the observer at $p_{\mathrm{O}}$ does not see any image of the light source ${\gamma_{\mathrm{S}}}$. For instance, this occurs in Minkowski spacetime for an inextendible worldline $\gamma_{\mathrm{S}}$ that asymptotically approaches the past light cone of $p_{\mathrm{O}}$. 2. There is exactly one past-pointing lightlike geodesic from $p_{\mathrm{O}}$ to ${\gamma}_{\mathrm{S}}$. Then the observer at $p_{\mathrm{S}}$ sees exactly one image of the light source ${\gamma}_{\mathrm{S}}$. This is the situation naively taken for granted in pre-relativistic astronomy. 3. There are at least two but not more than denumerably many past-pointing lightlike geodesics from $p_{\mathrm{O}}$ to ${\gamma}_{\mathrm{S}}$. Then the observer at $p_{\mathrm{O}}$ sees finitely or infinitely many distinct images of ${\gamma}_{\mathrm{O}}$ at his or her celestial sphere. 4. There are more than denumerably many past-pointing lightlike geodesics from $p$ to ${\gamma}$. This happens, e.g., in rotationally symmetric situations where it gives rise to the so-called “Einstein rings” (see Section 4.3). It also happens, e.g., in plane-wave spacetimes (see Section 5.11). If Case 3 or 4 occurs, astronomers speak of multiple imaging. We first demonstrate that Case 4 is exceptional. It is easy to prove (see, e.g., [336], Proposition 12) that no finite segment of the timelike curve $\gamma_{\mathrm{S}}$ can be contained in the past light cone of $p_{\mathrm{O}}$. Thus, if there is a continuous one-parameter family of lightlike geodesics that connect $p_{\mathrm{O}}$ and $\gamma_{\mathrm{O}}$, then all family members meet $\gamma_{\mathrm{S}}$ at the same point, say $p_{\mathrm{S}}$. This point must be in the caustic of the light cone because through all non-caustic points there is only a discrete number of generators. One can always find a point $p_{\mathrm{O}}^{\prime}$ arbitrarily close to $p_{\mathrm{O}}$ such that $\gamma_{\mathrm{S}}$ does not meet the caustic of the past light cone of $p_{\mathrm{O}}^{\prime}$ (see, e.g., [336], Proposition 10). Hence, by an arbitrarily small perturbation of $p_{\mathrm{O}}$ one can always destroy a Case 4 situation. One may interpret this result as saying that Case 4 situations have zero probability. This is, indeed, true as long as we consider point sources (worldlines). The observed rings and arcs refer to extended sources (worldtubes) which are close to the caustic (recall Section 2.5). Such situations occur with non-zero probability. We will now show how multiple imaging is related to the notion of cut points (recall Section 2.7). For any point $p_{\mathrm{O}}$ in an arbitrary spacetime, the following criteria for multiple imaging hold: \hangafter (C1) Let $\lambda$ be a past-pointing lightlike geodesic from $p_{\mathrm{O}}$ and let $p_{\mathrm{S}}$ be a point on $\lambda$ beyond the cut point or beyond the first conjugate point. Then there is a timelike curve $\gamma_{\mathrm{S}}$ through $p_{\mathrm{S}}$ that can be reached from $p_{\mathrm{O}}$ along a second past-pointing lightlike geodesic. \hangafter (C2) Assume that at $p_{\mathrm{O}}$ the past-distinguishing condition (57) is satisfied. If a timelike curve $\gamma_{\mathrm{S}}$ can be reached from $p_{\mathrm{O}}$ along two different past-pointing lightlike geodesics, at least one of them passes through the cut locus of the past light cone of $p_{\mathrm{O}}$ on or before arriving at $\gamma_{\mathrm{S}}$. For proofs see [335] or [336]. (In [335] Criterion 2.8 is formulated with the strong causality condition, although the past-distinguishing condition is sufficient.) Criteria 2.8 and 2.8 say that the occurrence of cut points is sufficient and, in past-distinguishing spacetimes, also necessary for multiple imaging. The occurrence of conjugate points is sufficient but, in general, not necessary for multiple imaging (see Figure 25 for an example without conjugate points where multiple imaging occurs). So we have the following diagram: Occurrence of: Sufficient for multiple imaging in: Necessary for multiple imaging in: cut point arbitrary spacetime past-distinguishing spacetime conjugate point arbitrary spacetime – It is well known (see [193], in particular Proposition 4.4.5) that, under conditions which are to be considered as fairly general from a physical point of view, a lightlike geodesic must either be incomplete or contain a pair of conjugate points. These “fairly general conditions” are, e.g., the weak energy condition and the so-called generic condition (see [193] for details). This result implies the occurrence of conjugate points and, thus, of multiple imaging, for a large class of spacetimes. The occurrence of conjugate points has an important consequence in view of the focusing equation for the area distance $D_{\mathrm{area}}$ (recall Section 2.4 and, in particular, Equation (44)). As $D_{\mathrm{area}}$ vanishes at the vertex $s=0$ and at each conjugate point, there must be a parameter value $s_{m}$ with $\dot{D}_{\mathrm{area}}(s_{m})=0$ between the vertex and the first conjugate point. An elementary evaluation of the focusing equation (44) then implies $$1\leq\int_{0}^{s_{m}}\!\!\!s\left(|\sigma(s)|^{2}+\left|\frac{1}{2}% \operatorname{Ric}\left(\dot{\lambda}(s),\dot{\lambda}(s)\right)\right|\right)ds.$$ (58) As the Ricci term is related to the energy density via Einstein’s field equation, (58) gives an estimate of energy-density-plus-shear along the ray. If we observe a multiple imaging situation, and if we know (or assume) that we are in a situation where conjugate points are necessary for multiple imaging, we have thus an estimate on energy-density-plus-shear along the ray. This line of thought was worked out, under additional assumptions on the spacetime, in [318]. 2.9 Fermat’s principle for light rays It is often advantageous to characterize light rays by a variational principle, rather than by a differential equation. This is particularly true in view of applications to lensing. If we have chosen a point $p_{\mathrm{O}}$ (observation event) and a timelike curve $\gamma_{\mathrm{S}}$ (worldline of light source) in spacetime $\mathcal{M}$, we want to determine all past-pointing lightlike geodesics from $p_{\mathrm{O}}$ to $\gamma_{\mathrm{S}}$. When working with a differential equation for light rays, we have to calculate all light rays issuing from $p_{\mathrm{O}}$ into the past, and to see which of them meet $\gamma_{\mathrm{S}}$. If we work with a variational principle, we can restrict to curves from $p_{\mathrm{O}}$ to $\gamma_{\mathrm{S}}$ at the outset. To set up a variational principle, we have to choose the trial curves among which the solution curves are to be determined and the functional that has to be extremized. Let $\mathcal{L}_{p_{\mathrm{O}},\gamma_{\mathrm{S}}}$ denote the set of all past-pointing lightlike curves from $p_{\mathrm{O}}$ to $\gamma_{\mathrm{S}}$. This is the set of trial curves from which the lightlike geodesics are to be singled out by the variational principle. Choose a past-oriented but otherwise arbitrary parametrization for the timelike curve $\gamma_{\mathrm{S}}$ and assign to each trial curve the parameter at which it arrives. This gives the arrival time functional $T:\mathcal{L}_{p_{\mathrm{O}},\gamma_{\mathrm{S}}}\longrightarrow\mathbb{R}$ that is to be extremized. With respect to an appropriate differentiability notion for $T$, it turns out that the critical points (i.e., the points where the differential of $T$ vanishes) are exactly the geodesics in $\mathcal{L}_{p_{\mathrm{O}},\gamma_{\mathrm{S}}}$. This result (or its time-reversed version) can be viewed as a general-relativistic Fermat principle: Among all ways to move from $p_{\mathrm{O}}$ to $\gamma_{\mathrm{S}}$ in the past-pointing (or future-pointing) direction at the speed of light, the actual light rays choose those paths that make the arrival time stationary. This formulation of Fermat’s principle was suggested in 1990 by Kovner [240], a local version (restricted to a convex normal neighborhood) can be found already in a 1938 paper by Temple [401]. The crucial idea is to refer to the arrival time which is given only along the curve $\gamma_{\mathrm{S}}$, and not to some kind of global time which in an arbitrary spacetime does not even exist. The proof that the solution curves of Kovner’s variational principle are, indeed, exactly the lightlike geodesics was given in [332]. The proof can also be found, with a slight restriction on the spacetime that simplifies matters considerably, in [367]. An alternative version, based on making $\mathcal{L}_{p_{\mathrm{O}},\gamma_{\mathrm{S}}}$ into a Hilbert manifold, is given in [334]. As in ordinary optics, the light rays make the arrival time stationary but not necessarily minimal. A more detailed investigation shows that for a geodesic $\lambda\in\mathcal{L}_{p_{\mathrm{O}}\gamma_{\mathrm{S}}}$ the following holds. (For the notion of conjugate points see Sections 2.2 and 2.7.) \hangafter (A1) If along $\lambda$ there is no point conjugate to $p_{\mathrm{O}}$, $\lambda$ is a strict local minimum of $T$. \hangafter (A2) If $\lambda$ passes through a point conjugate to $p_{\mathrm{O}}$ before arriving at $\gamma$, it is a saddle of $T$. \hangafter (A3) If $\lambda$ reaches the first point conjugate to $p_{\mathrm{O}}$ exactly on its arrival at $\gamma_{\mathrm{S}}$, it may be a local minimum or a saddle but not a local maximum. For a proof see [332] or [334]. The fact that local maxima cannot occur is easily understood from the geometry of the situation: For every trial curve we can find a neighboring trial curve with a larger $T$ by putting “wiggles” into it, preserving the lightlike character of the curve. Also for Fermat’s principle in ordinary optics, where light propagation is characterized by a positive index of refraction on Euclidean 3-space, the extremum is never a local maximum, as is mentioned, e.g., in Born and Wolf [46], p. 137. Note, however, that in the quasi-Newtonian lensing approximation with one or more deflector planes, where only broken straight lines are allowed as trial paths, local maxima do occur, see, e.g., [343]. Also, in the general formalism of ray optics, where the rays are the solutions of Hamilton’s equations with an unspecified Hamiltonian, local maxima do occur, unless a certain regularity condition is imposed on the Hamiltonian [337]. The advantage of Kovner’s version of Fermat’s principle is that it works in an arbitrary spacetime. In particular, the spacetime need not be stationary and the light source may arbitrarily move around (at subluminal velocity, of course). This allows applications to dynamical situations, e.g., to lensing by gravitational waves (see Section 5.11). If the spacetime is stationary or conformally stationary, and if the light source is at rest, a purely spatial reformulation of Fermat’s principle is possible. This more specific version of Femat’s principle is known since decades and has found various applications to lensing (see Section 4.2). A more sophisticated application of Fermat’s principle to lensing theory is to put up a Morse theory in order to prove theorems on the possible number of images. In its strongest version, this approach has to presuppose a globally hyperbolic spacetime and will be reviewed in Section 3.3. For a generalization of Kovner’s version of Fermat’s principle to the case that observer and light source have a spatial extension see [340]. An alternative variational principle was introduced by Frittelli and Newman [154] and evaluated in [155, 153]. While Kovner’s principle, like the classical Fermat principle, is a varional principle for rays, the Frittelli–Newman principle is a variational principle for wave fronts. (For the definition of wave fronts see Section 2.2.) Although Frittelli and Newman call their variational principle a version of Fermat’s principle, it is actually closer to the classical Huygens principle than to the classical Fermat principle. Again, one fixes $p_{\mathrm{O}}$ and $\gamma_{\mathrm{S}}$ as above. To define the trial maps, one chooses a set $\mathcal{W}(p_{\mathrm{O}})$ of wave fronts, such that for each lightlike geodesic through $p_{\mathrm{O}}$ there is exactly one wave front in $\mathcal{W}(p_{\mathrm{O}})$ that contains this geodesic. Hence, $\mathcal{W}(p_{\mathrm{O}})$ is in one-to-one correspondence to the lightlike directions at $p_{\mathrm{O}}$ and thus to the 2-sphere. Now let $\mathcal{W}(p_{\mathrm{O}},\gamma_{\mathrm{S}})$ denote the set of all wave fronts in $\mathcal{W}(p_{\mathrm{O}})$ that meet $\gamma_{\mathrm{S}}$. We can then define the arrival time functional $T:\mathcal{W}(p_{\mathrm{O}},\gamma_{\mathrm{S}})\longrightarrow\mathbb{R}$ by assigning to each wave front the parameter value at which it intersects $\gamma_{\mathrm{S}}$. There are some cases to be excluded to make sure that $T$ is defined on an open subset of $\mathcal{W}(p_{\mathrm{O}})\simeq S^{2}$, single-valued and differentiable. If this is the case, one finds that $T$ is stationary at $W\in\mathcal{W}(p_{\mathrm{O}})$ if and only if $W$ contains a lightlike geodesic from $p_{\mathrm{O}}$ to $\gamma_{\mathrm{S}}$. Thus, to each image of $\gamma_{\mathrm{S}}$ on the sky of $p_{\mathrm{O}}$ there corresponds a critical point of $T$. The great technical advantage of the Frittelli–Newman principle over the Kovner principle is that $T$ is defined on a finite dimensional manifold, directly to be identified with (part of) the observer’s celestial sphere. The arrival time $T$ in the Frittelli–Newman approach is directly analogous to the “Fermat potential” in the quasi-Newtonian formalism which is discussed, e.g., in [367]. In view of applications, a crucial point is that the space $\mathcal{W}(p_{\mathrm{O}})$ is a matter of choice; there are many wave fronts which have one light ray in common. There is a natural choice, e.g., in asymptotically simple spacetimes (see Section 3.4). Frittelli, Newman, and collaborators have used their variational principle in combination with the exact lens map (recall Section 2.1) to discuss thick and thin lens models from a spacetime perspective [155, 153]. Methods from differential topology or global analysis, e.g., Morse theory, have not yet been applied to the Frittelli–Newman principle. 3 Lensing in Globally Hyperbolic Spacetimes In a globally hyperbolic spacetime, considerably stronger statements on qualitative lensing features can be made than in an arbitrary spacetime. This includes, e.g., multiple imaging criteria in terms of cut points or conjugate points, and also applications of Morse theory. The value of these results lies in the fact that they hold in globally hyperbolic spacetimes without symmetries, where lensing cannot be studied by explicitly integrating the lightlike geodesic equation. The most convenient formal definition of global hyperbolicity is the following. In a spacetime $(\mathcal{M},g)$, a subset $\mathcal{C}$ of $\mathcal{M}$ is called a Cauchy surface if every inextendible causal (i.e., timelike or lightlike) curve intersects $\mathcal{C}$ exactly once. A spacetime is globally hyperbolic if and only if it admits a Cauchy surface. The name globally hyperbolic refers to the fact that for hyperbolic differential equations, like the wave equation, existence and uniqueness of a global solution is guaranteed for initial data given on a Cauchy surface. For details on globally hyperbolic spacetimes see, e.g., [193, 32]. It was demonstrated by Geroch [165] that every gobally hyperbolic spacetime admits a continuous function $t:\mathcal{M}\longrightarrow\mathbb{R}$ such that $t^{-1}(t_{0})$ is a Cauchy surface for every $t_{0}\in\mathbb{R}$. A complete proof of the fact that such a Cauchy time function can be chosen differentiable was given much later by Bernal and Sánchez [34, 35, 36]. The topology of a globally hyperbolic spacetime is determined by the topology of any of its Cauchy surfaces, $\mathcal{M}\simeq\mathcal{C}\times\mathbb{R}$. Note, however, that the converse is not true because $\mathcal{C}_{1}\times\mathbb{R}$ may be homeomorphic (and even diffeomorphic) to $\mathcal{C}_{2}\times\mathbb{R}$ without $\mathcal{C}_{1}$ being homeomorphic to $\mathcal{C}_{2}$. For instance, one can construct a globally hyperbolic spacetime with topology $\mathbb{R}^{4}$ that admits a Cauchy surface which is not homeomorphic to $\mathbb{R}^{3}$ [305]. In view of applications to lensing the following observation is crucial. If one removes a point, a worldline (timelike curve), or a world tube (region with timelike boundary) from an arbitrary spacetime, the resulting spacetime cannot be globally hyperbolic. Thus, restricting to globally hyperbolic spacetimes excludes all cases where a deflector is treated as non-transparent by cutting its world tube from spacetime (see Figure 25 for an example). Note, however, that this does not mean that globally hyperbolic spacetimes can serve as models only for transparent deflectors. First, a globally hyperbolic spacetime may contain “non-transparent” regions in the sense that a light ray may be trapped in a spatially compact set. Second, the region outside the horizon of a (Schwarzschild, Kerr, …) black hole is globally hyperbolic. 3.1 Criteria for multiple imaging in globally hyperbolic spacetimes In Section 2.7 we have considered the past light cone of an event $p_{\mathrm{O}}$ in an arbitrary spacetime. We have seen that conjugate points (= caustic points) indicate that the past light cone fails to be an immersed submanifold and that cut points indicate that it fails to be an embedded submanifold. In a globally hyperbolic spacetime $(\mathcal{M},g)$, the following additional statements are true. \hangafter (H1) The past light cone of any event $p_{\mathrm{O}}$, together with the vertex $\{p_{\mathrm{O}}\}$, is closed in $\mathcal{M}$. \hangafter (H2) The cut locus of the past light cone of $p_{\mathrm{O}}$ is closed in $\mathcal{M}$. \hangafter (H3) Let $p_{\mathrm{S}}$ be in the cut locus of the past light cone of $p_{\mathrm{O}}$ but not in the conjugate locus (= caustic). Then $p_{\mathrm{S}}$ can be reached from $p_{\mathrm{O}}$ along two different lightlike geodesics. The past light cone of $p_{\mathrm{O}}$ has a transverse self-intersection at $p_{\mathrm{S}}$. \hangafter (H4) The past light cone of $p_{\mathrm{O}}$ is an embedded submanifold if and only if its cut locus is empty. Analogous results hold, of course, for the future light cone, but the past version is the one that has relevance for lensing. For proofs of these statements see [32], Propositions 9.35 and 9.29 and Theorem 9.15, and [336], Propositions 13, 14, and 15. According to Statement 3.1, a “cut point” indicates a “cut” of two lightlike geodesics. For geodesics in Riemannian manifolds (i.e., in the positive definite case), an analogous statement holds if the Riemannian metric is complete and is known as Poincaré theorem [349, 434]. It was this theorem that motivated the name “cut point”. Note that Statement 3.1 is not true without the assumption that $p_{\mathrm{S}}$ is not in the caustic. This is exemplified by the swallow-tail point in Figure 26. However, as points in the caustic of the past light cone of $p_{\mathrm{O}}$ can be reached from $p_{\mathrm{O}}$ along two “infinitesimally close” lightlike geodesics, the name “cut point” may be considered as justified also in this case. In addition to Statements 3.1 and 3.1 one would like to know whether in globally hyperbolic spactimes the caustic of the past light cone of $p_{\mathrm{O}}$ (also known as the past lightlike conjugate locus of $p_{\mathrm{O}}$) is closed. This question is closely related to the question of whether in a complete Riemannian manifold the conjugate locus of a point is closed. For both questions, the answer was widely believed to be ‘yes’ although actually it is ‘no’. To the surprise of many, Margerin [269] constructed Riemannian metrics on the 2-sphere such that the conjugate locus of a point is not closed. Taking the product of such a Riemannian manifold with 2-dimensional Minkowski space gives a globally hyperbolic spacetime in which the caustic of the past light cone of an event is not closed. In Section 2.8 we gave criteria for the number of past-oriented lightlike geodesics from a point $p_{\mathrm{O}}$ (observation event) to a timelike curve $\gamma_{\mathrm{S}}$ (worldline of a light source) in an arbitrary spacetime. With Statements 3.1, 3.1, 3.1, and 3.1 at hand, the following stronger criteria can be given. Let $(\mathcal{M},g)$ be globally hyperbolic, fix a point $p_{\mathrm{O}}$ and an inextendible timelike curve $\gamma_{\mathrm{S}}$ in $\mathcal{M}$. Then the following is true: \hangafter (H5) Assume that $\gamma_{\mathrm{S}}$ enters into the chronological past $I^{-}(p_{\mathrm{O}})$ of $p_{\mathrm{O}}$. Then there is a past-oriented lightlike geodesic $\lambda$ from $p_{\mathrm{O}}$ to $\gamma_{\mathrm{S}}$ that is completely contained in the boundary of $I^{-}(p_{\mathrm{O}})$. This geodesic does not pass through a cut point or through a conjugate point before arriving at $\gamma_{\mathrm{S}}$. \hangafter (H6) Assume that $\gamma_{\mathrm{S}}$ can be reached from $p_{\mathrm{O}}$ along a past-oriented lightlike geodesic that passes through a conjugate point or through a cut point before arriving at $\gamma_{\mathrm{S}}$. Then $\gamma_{\mathrm{S}}$ can be reached from $p_{\mathrm{O}}$ along a second past-oriented lightlike geodesic. Statement 3.1 was proven in [408] with the help of Morse theory. For a more elementary proof see [336], Proposition 16. Statement 3.1 gives a characterization of the primary image in globally hyperbolic spacetimes. (By definition, an image is “primary” if no other image shows the light source at an older age.) The condition of $\gamma_{\mathrm{S}}$ entering into the chronological past of $p_{\mathrm{O}}$ is necessary to exclude the case that $p_{\mathrm{O}}$ sees no image of $\gamma_{\mathrm{S}}$. Statement 3.1 implies that there is a unique primary image unless $\gamma_{\mathrm{S}}$ passes through the cut locus of the past light cone of $p_{\mathrm{O}}$. The primary image has even parity. If the weak energy condition is satisfied, the focusing theorem implies that the primary image has magnification factor $\geq 1$, i.e., that it appears brighter than a source of the same luminosity at the same affine distance and at the same redshift in Minkowski spacetime (recall Sections 2.4 and 2.6, in particular the inequality (46)). For a proof of Statement 3.1 see [336], Proposition 17. 3.2 Wave fronts in globally hyperbolic spacetimes In Section 2.2 the notion of wave fronts was discussed in an arbitrary spacetime $(\mathcal{M},g)$. It was mentioned that a wave front can be viewed as a subset of the space $\mathcal{N}$ of all lightlike geodesics in $(\mathcal{M},g)$. This approach is particularly useful in globally hyperbolic spacetimes, as was demonstrated by Low [263, 264]. The construction is based on the observations that, if $(\mathcal{M},g)$ is globally hyperbolic and $\mathcal{C}$ is a smooth Cauchy surface, the following is true: \hangafter (N1) $\mathcal{N}$ can be identified with a sphere bundle over $\mathcal{C}$. The identification is made by assigning to each lightlike geodesic its tangent line at the point where it intersects $\mathcal{C}$. As every sphere bundle over an orientable 3-manifold is trivializable, $\mathcal{N}$ is diffeomorphic to $\mathcal{C}\times S^{2}$. \hangafter (N2) $\mathcal{N}$ carries a natural contact structure. (This contact structure is also discussed, in twistor language, in [330], volume II.) \hangafter (N3) The wave fronts are exactly the Legendre submanifolds of $\mathcal{N}$. Using Statement 3.2, the projection from $\mathcal{N}$ to $\mathcal{C}$ assigns to each wave front its intersection with $\mathcal{C}$, i.e., an “instantaneous wave front” or “small wave front” (cf. Section 2.2 for terminology). The points where this projection has non-maximal rank give the caustic of the small wave front. According to the general stability results of Arnold (see [14]), the only caustic points that are stable with respect to local perturbations within the class of Legendre submanifolds are cusps and swallow-tails. By Statement 3.2, perturbing within the class of Legendre submanifolds is the same as perturbing within the class of wave fronts. For this local stability result the assumption of global hyperbolicity is irrelevant because every spacelike hypersurface is a Cauchy surface for an appropriately chosen neighborhood of any of its points. So we get the result that was already mentioned in Section 2.2: In an arbitrary spacetime, a caustic point of an instantaneous wave front is stable if and only if it is a cusp or a swallow-tail. Here stability refers to perturbations that keep the metric and the hypersurface fixed and perturb the wave front within the class of wave fronts. For a picture of an instantaneous wave front with cusps and a swallow-tail point, see Figure 29. In Figure 14, the caustic points are neither cusps nor swallow-tails, so the caustic is unstable. 3.3 Fermat’s principle and Morse theory in globally hyperbolic spacetimes In an arbitrary spacetime, the past-oriented lightlike geodesics from a point $p_{\mathrm{O}}$ (observation event) to a timelike curve $\gamma_{\mathrm{S}}$ (worldline of light source) are the solutions of a variational principle (Kovner’s version of Fermat’s principle; see Section 2.9). Every solution of this variational principle corresponds to an image on $p_{\mathrm{O}}$’s sky of $\gamma_{\mathrm{S}}$. Determining the number of images is the same as determining the number of solutions to the variational problem. If the variational functional satisfies some technical conditions, the number of solutions to the variational principle can be related to the topology of the space of trial paths. This is the content of Morse theory. In the case at hand, the “technical conditions” turn out to be satisfied in globally hyperbolic spacetimes. To briefly review Morse theory, we consider a differentiable function $F:\mathcal{X}\longrightarrow\mathbb{R}$ on a real manifold $\mathcal{X}$. Points where the differential of $F$ vanishes are called critical points of $F$. A critical point is called non-degenerate if the Hessian of $F$ is non-degenerate at this point. $F$ is called a Morse function if all its critical points are non-degenerate. In applications to variational problems, $\mathcal{X}$ is the space of trial maps, $F$ is the functional to be varied, and the critical points of $F$ are the solutions to the variational problem. The non-degeneracy condition guarantees that the character of each critical point – local minimum, local maximum, or saddle – is determined by the Hessian of $F$ at this point. The index of the Hessian is called the Morse index of the critical point. It is defined as the maximal dimension of a subspace on which the Hessian is negative definite. At a local minimum the Morse index is zero, at a local maximum it is equal to the dimension of $\mathcal{X}$. Morse theory was first worked out by Morse [285] for the case that $\mathcal{X}$ is finite-dimensional and compact (see Milnor [278] for a detailed exposition). The main result is the following. On a compact manifold $\mathcal{X}$, for every Morse function the Morse inequalities $$N_{k}\geq B_{k},\qquad k=0,1,2,\dots,$$ (59) and the Morse relation $$\sum_{k=0}^{\infty}(-1)^{k}N_{k}=\sum_{k=0}^{\infty}(-1)^{k}B_{k}$$ (60) hold true. Here $N_{k}$ denotes the number of critical points with Morse index $k$ and $B_{k}$ denotes the $k$th Betti number of $\mathcal{X}$. Formally, $B_{k}$ is defined for each topological space $\mathcal{X}$ in terms of the $k$th singular homology space $H_{k}(\mathcal{X})$ with coefficients in a field $\mathbb{F}$ (see, e.g., [104], p. 32). (The results of Morse theory hold for any choice of $\mathbb{F}$.) Geometrically, $B_{0}$ counts the connected components of $\mathcal{X}$ and, for $k\geq 1$, $B_{k}$ counts the “holes” in $\mathcal{X}$ that prevent a $k$-cycle with coefficients in $\mathbb{F}$ from being a boundary. In particular, if $\mathcal{X}$ is contractible to a point, then $B_{k}=0$ for $k\geq 1$. The right-hand side of Equation (60) is, by definition, the Euler characteristic of $\mathcal{X}$. By compactness of $\mathcal{X}$, all $N_{k}$ and $B_{k}$ are finite and in both sums of Equation (60) only finitely many summands are different from zero. Palais and Smale [319, 320] realized that the Morse inequalities and the Morse relations are also true for a Morse function $F$ on a non-compact and possibly infinite-dimensional Hilbert manifold, provided that $F$ is bounded below and satisfies a technical condition known as Condition C or Palais–Smale condition. In that case, the $N_{k}$ and $B_{k}$ need not be finite. The standard application of Morse theory is the geodesic problem for Riemannian (i.e., positive definite) metrics: given two points in a Riemannian manifold, to find the geodesics that join them. In this case $F$ is the “energy functional” (squared-length functional). Varying the energy functional is related to varying the length functional like Hamilton’s principle is related to Maupertuis’ principle in classical mechanics. For the space $\mathcal{X}$ one chooses, in the Palais–Smale approach [319], the $H^{1}$-curves between the given two points. (An $H^{n}$-curve is a curve with locally square-integrable $n$th derivative). This is an infinite-dimensional Hilbert manifold. It has the same homotopy type (and thus the same Betti numbers) as the loop space of the Riemannian manifold. (The loop space of a connected topological space is the space of all continuous curves joining any two fixed points.) On this Hilbert manifold, the energy functional is always bounded from below, and its critical points are exactly the geodesics between the given end-points. A critical point (geodesic) is non-degenerate if the two end-points are not conjugate to each other, and its Morse index is the number of conjugate points in the interior, counted with multiplicity (“Morse index theorem”). The Palais–Smale condition is satisfied if the Riemannian manifold is complete. So one has the following result: Fix any two points in a complete Riemannian manifold that are not conjugate to each other along any geodesic. Then the Morse inequalities (59) and the Morse relation (60) are true, with $N_{k}$ denoting the number of geodesics with Morse index $k$ between the two points and $B_{k}$ denoting the $k$th Betti number of the loop space of the Riemannian manifold. The same result is achieved in the original version of Morse theory [285] (cf. [278]) by choosing for $\mathcal{X}$ the space of broken geodesics between the two given points, with $N$ break points, and sending $N\to\infty$ at the end. Using this standard example of Morse theory as a pattern, one can prove an analogous result for Kovner’s version of Fermat’s principle. The following hypotheses have to be satisfied: \hangafter (M1) $p_{\mathrm{O}}$ is a point and $\gamma_{\mathrm{S}}$ is a timelike curve in a globally hyperbolic spacetime $(\mathcal{M},g)$. \hangafter (M2) $\gamma_{\mathrm{S}}$ does not meet the caustic of the past light cone of $p_{\mathrm{O}}$. \hangafter (M3) Every continuous curve from $p_{\mathrm{O}}$ to $\gamma_{\mathrm{S}}$ can be continuously deformed into a past-oriented lightlike curve, with all intermediary curves starting at $p_{\mathrm{O}}$ and terminating on $\gamma_{\mathrm{S}}$. The global hyperbolicity assumption in Statement 3.3 is analogous to the completeness assumption in the Riemannian case. Statement 3.3 is the direct analogue of the non-conjugacy condition in the Riemannian case. Statement 3.3 is necessary for relating the space of trial paths (i.e., of past-oriented lightlike curves from $p_{\mathrm{O}}$ to $\gamma_{\mathrm{S}}$) to the loop space of the spacetime manifold or, equivalently, to the loop space of a Cauchy surface. If Statements 3.3, 3.3, and 3.3 are valid, the Morse inequalities (59) and the Morse relation (60) are true, with $N_{k}$ denoting the number of past-oriented lightlike geodesics from $p_{\mathrm{O}}$ to $\gamma_{\mathrm{S}}$ that have $k$ conjugate points in its interior, counted with muliplicity, and $B_{k}$ denoting the $k$th Betti number of the loop space of $\mathcal{M}$ or, equivalently, of a Cauchy surface. This result was proven by Uhlenbeck [408] à la Morse and Milnor, and by Giannoni and Masiello [170] in an infinite-dimensional Hilbert manifold setting à la Palais and Smale. A more general version, applying to spacetime regions with boundaries, was worked out by Giannoni, Masiello, and Piccione [171, 172]. In the work of Giannoni et al., the proofs are given in greater detail than in the work of Uhlenbeck. If Statements 3.3, 3.3, and 3.3 are satisfied, Morse theory gives us the following results about the number of images of $\gamma_{\mathrm{S}}$ on the sky of $p_{\mathrm{O}}$ (cf. [274]): \hangafter (R1) If $\mathcal{M}$ is not contractible to a point, there are infinitely many images. This follows from Equation (59) because for the loop space of a non-contractible space either $B_{0}$ is infinite or almost all $B_{k}$ are different from zero [378]. \hangafter (R2) If $\mathcal{M}$ is contractible to a point, the total number of images is infinite or odd. This follows from Equation (60) because in this case the loop space of $\mathcal{M}$ is contractible to a point, so all Betti numbers $B_{k}$ vanish with the exception of $B_{0}=1$. As a consequence, Equation (60) can be written as $N_{+}-N_{-}=1$, where $N_{+}$ is the number of images with even parity (geodesics with even Morse index) and $N_{-}$ is the number of images with odd parity (geodesics with odd Morse index), hence $N_{+}+N_{-}=2N_{-}+1$. These results apply, in particular, to the following situations of physical interest: Black hole spacetimes. Let $(\mathcal{M},g)$ be the domain of outer communication of the Kerr spacetime, i.e., the region between the (outer) horizon and infinity (see Section 5.8). Then the assumption of global hyperbolicity is satisfied and $\mathcal{M}$ is not contractible to a point. Statement 3.3 is satisfied if $\gamma_{\mathrm{S}}$ is inextendible and approaches neither the horizon nor (past lightlike) infinity for $t\to-\infty$. (This can be checked with the help of an analytical criterion that is called the “metric growth condition” in [408].) If, in addition Statement 3.3 is satisfied, the reasoning of Statement 3.3 applies. Hence, a Kerr black hole produces infinitely many images, under fairly generic conditions on the motion of the light source. The details of this argument are worked out, for the more general case of a Kerr-Newman black hole, in [192]. Asymptotically simple and empty spacetimes. As discussed in Section 3.4, asymptotically simple and empty spacetimes are globally hyperbolic and contractible to a point. They can be viewed as models of isolated transparent gravitational lenses. Statement 3.3 is satisfied if $\gamma_{\mathrm{S}}$ is inextendible and bounded away from past lightlike infinity $\mbox{ I}^{\,-}$. If, in addition, Statement 3.3 is satisfied, Statement 3.3 guarantees that the number of images is infinite or odd. If it were infinite, we had as the limit curve a past-inextendible lightlike geodesic that would not go out to $\mbox{ I}^{\,-}$, in contradiction to the definition of asymptotic simplicity. So the number of images must be finite and odd. The same odd-number theorem can also be proven with other methods (see Section 3.4). In this way Morse theory provides us with precise mathematical versions of the statements “A black hole produces infinitely many images” and “An isolated transparent gravitational lens produces an odd number of images”. When comparing this theoretical result with observations one has to be aware of the fact that some images might be hidden behind the deflecting mass, some might be too faint for being detected, and some might be too close together for being resolved. In conformally stationary spacetimes, with $\gamma_{\mathrm{S}}$ being an integral curve of the conformal Killing vector field, a simpler version of Fermat’s principle and Morse theory can be used (see Section 4.2). 3.4 Lensing in asymptotically simple and empty spacetimes In elementary optics one often considers “light sources at infinity” which are characterized by the fact that all light rays emitted from such a source are parallel to each other. In general relativity, “light sources at infinity” can be defined if one restricts to a special class of spacetimes. These spacetimes, known as “asymptotically simple and empty” are, in particular, globally hyperbolic. Their formal definition, which is due to Penrose [326], reads as follows (cf. [193], p. 222., and [148], Section 2.3). (Recall that a spacetime is called “strongly causal” if each neighborhood of an event $p$ admits a smaller neighborhood that is intersected by any non-spacelike curve at most once.) A spacetime $(\mathcal{M},g,)$ is called asymptotically simple and empty if there is a strongly causal spacetime $({\tilde{\mathcal{M}}},{\tilde{g}})$ with the following properties: \hangafter (S1) $\mathcal{M}$ is an open submanifold of ${\tilde{\mathcal{M}}}$ with a non-empty boundary $\partial\mathcal{M}$. \hangafter (S2) There is a smooth function $\Omega:{\tilde{\mathcal{M}}}\longrightarrow{\mathbb{R}}$ such that $\mathcal{M}=\{p\in{\tilde{\mathcal{M}}}|\Omega(p)>0\}$, $\partial\mathcal{M}=\{p\in{\tilde{\mathcal{M}}}|\Omega(p)=0\}$, $d\Omega\neq 0$ everywhere on $\partial\mathcal{M}$ and ${\tilde{g}}=\Omega^{2}g$ on $\mathcal{M}$. \hangafter (S3) Every inextendible lightlike geodesic in ${\mathcal{M}}$ has past and future end-point on $\partial\mathcal{M}$. \hangafter (S4) There is a neighborhood $\mathcal{V}$ of $\partial\mathcal{M}$ such that the Ricci tensor of $g$ vanishes on $\mathcal{V}\cap\mathcal{M}$. Asymptotically simple and empty spacetimes are mathematical models of transparent uncharged gravitating bodies that are isolated from all other gravitational sources. In view of lensing, the transparency condition 3.4 is particularly important. We now summarize some well-known facts about asymptotically simple and empty spacetimes (cf. again [193], p. 222, and [148], Section 2.3). Every asymptotically simple and empty spacetime is globally hyperbolic. $\partial\mathcal{M}$ is a ${\tilde{g}}$-lightlike hypersurface of ${\tilde{\mathcal{M}}}$. It has two connected components, denoted $\mbox{ I}^{\,+}$ and $\mbox{ I}^{\,-}$. Each lightlike geodesic in $(\mathcal{M},g)$ has past end-point on $\mbox{ I}^{\,-}$ and future end-point on $\mbox{ I}^{\,+}$. Geroch [166] gave a proof that every Cauchy surface $\mathcal{C}$ of an asymptotically simple and empty spacetime has topology $\mathbb{R}^{3}$ and that $\mbox{ I}^{\,\pm}$ has topology $S^{2}\times\mathbb{R}$. The original proof, which is repeated in [193], is incomplete. A complete proof that $\mathcal{C}$ must be contractible and that $\mbox{ I}^{\,\pm}$ has topology $S^{2}\times\mathbb{R}$ was given by Newman and Clarke [305] (cf. [304]); the stronger statement that $\mathcal{C}$ must have topology $\mathbb{R}^{3}$ needs the assumption that the Poincaré conjecture is true (i.e., that every compact and simply connected 3-manifold is a 3-sphere). In [305] the authors believed that the Poincaré conjecture was proven, but the proof they are refering to was actually based on an error. As the more recent proof of the Poincaré conjecture by Perelman [331] (cf. [281]) has been generally accepted as being correct, the matter is now settled. As $\mbox{ I}^{\,\pm}$ is a lightlike hypersurface in $\tilde{\mathcal{M}}$, it is in particular a wave front in the sense of Section 2.2. The generators of $\mbox{ I}^{\,\pm}$ are the integral curves of the gradient of $\Omega$. The generators of $\mbox{ I}^{\,-}$ can be interpreted as the “worldlines” of light sources at infinity that send light into $\mathcal{M}$. The generators of $\mbox{ I}^{\,+}$ can be interpreted as the “worldlines” of observers at infinity that receive light from $\mathcal{M}$. This interpretation is justified by the observation that each generator of $\mbox{ I}^{\,\pm}$ is the limit curve for a sequence of timelike curves in $\mathcal{M}$. For an observation event $p_{\mathrm{O}}$ inside $\mathcal{M}$ and light sources at infinity, lensing can be investigated in terms of the exact lens map (recall Section 2.1), with the role of the source surface $\mathcal{T}$ played by $\mbox{ I}^{\,-}$. (For the mathematical properties of the lens map it is rather irrelevant whether the source surface is timelike, lightlike or even spacelike. What matters is that the arriving light rays meet the source surface transversely.) In this case the lens map is a map $S^{2}\rightarrow S^{2}$, namely from the celestial sphere of the observer to the set of all generators of $\mbox{ I}^{\,-}$. One can construct it in two steps: First determine the intersection of the past light cone of $p_{\mathrm{O}}$ with $\mbox{ I}^{\,-}$, then project along the generators. The intersections of light cones with $\mbox{ I}^{\,\pm}$ (“light cone cuts of null infinity”) have been studied in [242, 241]. One can assign a mapping degree (= Brouwer degree = winding number) to the lens map $S^{2}\rightarrow S^{2}$ and prove that it must be $\pm 1$ [338]. (The proof is based on ideas of [305, 304]. Earlier proofs of similar statements – [241], Lemma 1, and [336], Theorem 6 – are incorrect, as outlined in [338].) Based on this result, the following odd-number theorem can be proven for observer and light source inside $\mathcal{M}$ [338]: Fix a point $p_{\mathrm{O}}$ and a timelike curve $\gamma_{\mathrm{S}}$ in an asymptotically simple and empty spacetime $(\mathcal{M},g)$. Assume that the image of $\gamma_{\mathrm{S}}$ is a closed subset of ${\tilde{\mathcal{M}}}\setminus\mbox{ I}^{\,+}$ and that $\gamma_{\mathrm{S}}$ meets neither the point $p_{\mathrm{O}}$ nor the caustic of the past light cone of $p_{\mathrm{O}}$. Then the number of past-pointing lightlike geodesics from $p_{\mathrm{O}}$ to $\gamma_{\mathrm{S}}$ in $\mathcal{M}$ is finite and odd. The same result can be proven with the help of Morse theory (see Section 3.3). We will now give an argument to the effect that in an asymptotically simple and empty spacetime the non-occurrence of multiple imaging is rather exceptional. The argument starts from a standard result that is used in the Penrose–Hawking singularity theorems. This standard result, given as Proposition 4.4.5 in [193], says that along a lightlike geodesic that starts at a point $p_{\mathrm{O}}$ there must be a point conjugate to $p_{\mathrm{O}}$, provided that 1. the so-called generic condition is satisfied at $p_{\mathrm{O}}$, 2. the weak energy condition is satisfied along the geodesic, and 3. the geodesic can be extended sufficiently far. The last assumption is certainly true in an asymptotically simple and empty spacetime because there all lightlike geodesics are complete. Hence, the generic condition and the weak energy condition guarantee that every past light cone must have a caustic point. We know from Section 3.1 that this implies multiple imaging for every observer. In other words, the only asymptotically simple and empty spacetimes in which multiple imaging does not occur are non-generic cases (like Minkowski spacetime) and cases where the gravitating bodies have negative energy. The result that, under the aforementioned conditions, light cones in an asymptotically simple and empty spacetime must have caustic points is due to [208]. This paper investigates the past light cones of points on $\mbox{ I}^{\,+}$ and their caustics. These light cones are the generalizations, to an arbitrary asymptotically simple and empty spacetime, of the lightlike hyperplanes in Minkowski spacetime. With their help, the eikonal equation (Hamilton–Jacobi equation) $g^{ij}\partial_{i}S\partial_{j}S=0$ in an asymptotically simple and empty spacetime can be studied in analogy to Minkowski spacetime [157, 156]. In Minkowski spacetime the lightlike hyperplanes are associated with a two-parameter family of solutions to the eikonal equation. In the terminology of classical mechanics such a family is called a complete integral. Knowing a complete integral allows constructing all solutions to the Hamilton–Jacobi equation. In an asymptotically simple and empty spacetime the past light cones of points on $\mbox{ I}^{\,+}$ give us, again, a complete integral for the eikonal equation, but now in a generalized sense, allowing for caustics. These past light cones are wave fronts, in the sense of Section 2.2, and cannot be represented as surfaces $S=\mbox{constant}$ near caustic points. The way in which all other wave fronts can be determined from knowledge of this distinguished family of wave fronts is detailed in [156]. The distinguished family of wave fronts gives a natural choice for the space of trial maps in the Frittelli–Newman variational principle which was discussed in Section 2.9. 4 Lensing in Spacetimes with Symmetry 4.1 Lensing in conformally flat spacetimes By definition, a spacetime is conformally flat if the conformal curvature tensor (= Weyl tensor) vanishes. An equivalent condition is that every point admits a neighborhood that is conformal to an open subset of Minkowski spacetime. As a consequence, conformally flat spacetimes have the same local conformal symmetry as Minkowski spacetime, that is they admit 15 independent conformal Killing vector fields. The global topology, however, may be different from the topology of Minkowski spacetime. The class of conformally flat spacetimes includes all (kinematic) Robertson–Walker spacetimes. Other physically interesting examples are some (generalized) interior Schwarzschild solutions and some pure radiation spacetimes. All conformally flat solutions to Einstein’s field equation with a perfect fluid or an electromagnetic field are known (see [388], Section 37.5.3). If a spacetime is globally conformal to an open subset of Minkowski spacetime, the past light cone of every event is an embedded submanifold. Hence, multiple imaging cannot occur (recall Section 2.8). For instance, multiple imaging occurs in spatially closed but not in spatially open Robertson–Walker spacetimes. In any conformally flat spacetime, there is no image distortion, i.e., a sufficiently small sphere always shows a circular outline on the observer’s sky (recall Section 2.5). Correspondingly, every infinitesimally thin bundle of light rays with a vertex is circular, i.e., the extremal angular diameter distances $D_{+}$ and $D_{-}$ coincide (recall Section 2.4). In addition, $D_{+}=D_{-}$ also coincides with the area distance $D_{\mathrm{area}}$, at least up to sign. $D_{+}=D_{-}$ changes sign at every caustic point. As $D_{+}$ has a zero if and only if $D_{-}$ has a zero, all caustic points of an infinitesimally thin bundle with vertex are of multiplicity two (anastigmatic focusing), so all images have even parity. The geometry of light bundles can be studied directly in terms of the Jacobi equation (= equation of geodesic deviation) along lightlike geodesics. For a detailed investigation of the latter in conformally flat spacetimes, see [341]. The more special case of Friedmann–Lemaître-Robertson–Walker spacetimes (with dust, radiation, and cosmological constant) is treated in [131]. For bundles with vertex, one is left with one scalar equation for $D_{+}=D_{-}=\pm D_{\mathrm{area}}$, that is the focusing equation (44) with $\sigma=0$. This equation can be explicitly integrated for Friedmann–Robertson–Walker spacetimes (dust without cosmological constant). In this way one gets, for the standard observer field in such a spacetime, relations between redshift and (area or luminosity) distance in closed form [273]. There are generalizations for a Robertson–Walker universe with dust plus cosmological constant [228] and dust plus radiation plus cosmological constant [94]. Similar formulas can be written for the relation between age and redshift [403]. 4.2 Lensing in conformally stationary spacetimes Conformally stationary spacetimes are models for gravitational fields that are time-independent up to an overall conformal factor. (The time-dependence of the conformal factor is important, e.g., if cosmic expansion is to be taken into account.) This is a reasonable model assumption for many, though not all, lensing situations of interest. It allows describing light rays in a 3-dimensional (spatial) formalism that will be outlined in this section. The class of conformally stationary spacetimes includes spherically symmetric and static spacetimes (see Sections 4.3) and axisymmetric stationary spacetimes (see Section 4.4). Also, conformally flat spacetimes (see Section 4.1) are conformally stationary, at least locally. A physically relevant example where the conformal-stationarity assumption is not satisfied is lensing by a gravitational wave (see Section 5.11). By definition, a spacetime is conformally stationary if it admits a timelike conformal Killing vector field $W$. If $W$ is complete and if there are no closed timelike curves, the spacetime must be a product, $\mathcal{M}\simeq\mathbb{R}\times\widehat{\mathcal{M}}$ with a (Hausdorff and paracompact) 3-manifold $\widehat{\mathcal{M}}$ and $W$ parallel to the $\mathbb{R}$-lines [185]. If we denote the projection from $\mathcal{M}$ to $\mathbb{R}$ by $t$ and choose local coordinates $x=(x^{1},x^{2},x^{3})$ on $\widehat{\mathcal{M}}$, the metric takes the form $$g=e^{2f(t,x)}\left(-(dt+\hat{\phi}_{\mu}(x)\,dx^{\mu})^{2}+\hat{g}_{\mu\nu}(x)% \,dx^{\mu}\,dx^{\nu}\right)$$ (61) with $\mu,\nu,\ldots=1,2,3$. The conformal factor $e^{2f}$ does not affect the lightlike geodesics apart from their parametrization. So the paths of light rays are completely determined by the metric $\hat{g}=\hat{g}_{\mu\nu}(x)dx^{\mu}dx^{\nu}$ and the one-form $\hat{\phi}=\hat{\phi}_{\mu}(x)dx^{\mu}$ which live on $\widehat{\mathcal{M}}$. The metric $\hat{g}$ must be positive definite to give a spacetime metric of Lorentzian signature. We call $f$ the redshift potential, $\hat{g}$ the Fermat metric and $\hat{\phi}$ the Fermat one-form. The motivation for these names will become clear from the discussion below. If $\hat{\phi}_{\mu}=\partial_{\mu}h$, where $h$ is a function of $x=(x^{1},x^{2},x^{3})$, we can change the time coordinate according to $t\longmapsto t+h(x)$, thereby transforming $\hat{\phi}_{\mu}dx^{\mu}$ to zero, i.e., making the surfaces $t=\mbox{constant}$ orthogonal to the $t$-lines. This is the conformally static case. Also, Equation (61) includes the stationary case ($f$ independent of $t$) and the static case ($\hat{\phi}_{\mu}=\partial_{\mu}h$ and $f$ independent of $t$). In Section 2.9 we have discussed Kovner’s version of Fermat’s principle which characterizes the lightlike geodesics between a point (observation event) $p_{\mathrm{O}}$ and a timelike curve (worldline of light source) $\gamma_{\mathrm{S}}$. In a conformally stationary spacetime we may specialize to the case that $\gamma_{\mathrm{S}}$ is an integral curve of the conformal Killing vector field, parametrized by the “conformal time” coordinate $t$ (in the past-pointing sense, to be in agreement with Section 2.9). Without loss of generality, we may assume that the observation event $p_{\mathrm{O}}$ takes place at $t=0$. Then for each trial path (past-oriented lightlike curve) $\lambda$ from $p_{\mathrm{O}}$ to $\gamma_{\mathrm{S}}$ the arrival time is equal to the travel time in terms of the time function $t$. By Equation (61) this puts the arrival time functional into the following coordinate form $$T(\lambda)=\int_{\ell_{1}}^{\ell_{2}}\left(\sqrt{\hat{g}_{\mu\nu}(x)\frac{dx^{% \mu}}{d\ell}\frac{dx^{\nu}}{d\ell}}-\hat{\phi}_{\mu}(x)\frac{dx^{\mu}}{d\ell}% \right)d\ell,$$ (62) where $\ell$ is any parameter along the trial path, ranging over an interval $[\ell_{1},\ell_{2}]$ that depends on the individual curve. The right-hand side of Equation (62) is a functional for curves in $\widehat{\mathcal{M}}$ with fixed end-points. The projections to $\widehat{\mathcal{M}}$ of light rays are the stationary points of this functional. In general, the right-hand side of Equation (62) is the length functional of a Finsler metric. In the conformally static case $\hat{\phi}_{\mu}=\partial_{\mu}h$, the integral over $\hat{\phi}_{\mu}(x)dx^{\mu}/d\ell$ is the same for all trial paths, so we are left with the length functional of the Fermat metric $\hat{g}$. In this case the light rays, if projected to $\widehat{\mathcal{M}}$, are the geodesics of $\hat{g}$. Note that the travel time functional (62) is invariant under reparametrization; in the terminology of classical mechanics, it is a special case of Maupertuis’ principle. It is often convenient to switch to a parametrization-dependent variational principle which, in the terminology of classical mechanics, is called Hamilton’s principle. The Maupertuis principle with action functional (62) corresponds to Hamilton’s principle with a Lagrangian $$\mathcal{L}=\frac{1}{2}\hat{g}_{\mu\nu}(x)\frac{dx^{\mu}}{d\ell}\frac{dx^{\nu}% }{d\ell}-\hat{\phi}_{\mu}\frac{dx^{\mu}}{d\ell},$$ (63) (see, e.g., Carathéodory [73], Sections 304 – 307). The pertaining Euler–Lagrange equations read $$\hat{g}_{\mu\nu}\left(\frac{d^{2}x^{\nu}}{d\ell^{2}}+\hat{\Gamma}^{\nu}_{% \sigma\tau}\frac{dx^{\sigma}}{d\ell}\frac{dx^{\tau}}{d\ell}\right)=\left(% \partial_{\nu}\hat{\phi}_{\mu}-\partial_{\mu}\hat{\phi}_{\nu}\right)\frac{dx^{% \mu}}{d\ell}$$ (64) where $\hat{\Gamma}^{\nu}_{\sigma\tau}$ are the Christoffel symbols of the Fermat metric $\hat{g}$. The solutions admit the constant of motion $$\hat{g}_{\mu\nu}(x)\frac{dx^{\mu}}{d\ell}\frac{dx^{\nu}}{d\ell}=\mbox{constant},$$ (65) which can be chosen equal to 1 for each ray, such that $\ell$ gives the $\hat{g}$-arclength. By Equation (62), the latter gives the travel time if $\hat{\phi}=0$. According to Equation (64), the Fermat two-form $$\hat{\omega}=d\hat{\phi}$$ (66) exerts a kind of Coriolis force on the light rays. This force has the same mathematical structure as the Lorentz force in a magnetostatic field. In this analogy, $\hat{\phi}$ corresponds to the magnetic (vector) potential. In other words, light rays in a conformally stationary spacetime behave like charged particles, with fixed charge-to-mass ratio, in a magnetostatic field $\hat{\omega}$ on a Riemannian manifold $(\hat{M},\hat{g})$. For linearly polarized light, the Fermat geometry can also be used for describing the propagation of the polarization plane [189]. One finds that the polarization plane undergoes a rotation similar to the Faraday rotation in a magnetic field. This observation corroborates the formal analogy between $\hat{\omega}$ and a magnetic field. The gravitational analogue of the Faraday rotation was already discussed briefly in Section 2.5 above. Fermat’s principle in static spacetimes dates back to Weyl [430] (cf. [260, 397]). The stationary case was treated by Pham Mau Quan [344], who even took an isotropic medium into account, and later, in a more elegant presentation, by Brill [60]. These versions of Fermat’s principle are discussed in several text-books on general relativity (see, e.g., [279, 147, 389] for the static and [253] for the stationary case). A detailed discussion of the conformally stationary case can be found in [333]. Fermat’s principle in conformally stationary spacetimes was used as the starting point for deriving the lens equation of the quasi-Newtonian apporoximation formalism by Schneider [366] (cf. [367]). As an alternative to the name “Fermat metric” (used, e.g., in [147, 389, 333]), the names “optical metric” (see, e.g., [176, 105, 175, 177]) and “optical (reference) geometry” (see, e.g., [4, 244, 390, 392, 201, 3]) are also used. In the conformally static case, one can apply the standard Morse theory for Riemannian geodesics to the Fermat metric $\hat{g}$ to get results on the number of $\hat{g}$-geodesics joining two points in space. This immediately gives results on the number of lightlike geodesics joining a point in spacetime to an integral curve of $W=\partial_{t}$. Completeness of the Fermat metric corresponds to global hyperbolicity of the spacetime metric. The relevant techniques, and their generalization to (conformally) stationary spacetimes, are detailed in a book by Masiello [272]. (Note that, in contrast to standard terminology, Masiello’s definition of a stationary spacetime includes the assumption that the hypersurfaces $t=\mbox{constant}$ are spacelike.) The resulting Morse theory is a special case of the Morse theory for Fermat’s principle in globally hyperbolic spacetimes (see Section 3.3). In addition to Morse theory, other standard methods from Riemannian geometry have been applied to the Fermat metric, e.g., convexity techniques [173, 174]. If the metric (61) is conformally static, $\hat{\phi}_{\mu}(x)=\partial_{\mu}h(x)$, and if the Fermat metric is conformal to the Euclidean metric, $\hat{g}_{\mu\nu}(x)=n(x)^{2}\delta_{\mu\nu}$, the arrival time functional (62) can be written as $$T(\lambda)=\int_{\ell_{1}=0}^{\ell_{2}}n(x)\,d\ell+\mbox{constant},$$ (67) where $\ell$ is Euclidean arclength. Hence, Fermat’s principle reduces to its standard optics form for an isotropic medium with index of refraction $n$ on Euclidean space. As a consequence, light propagation in a spacetime with the assumed properties can be mimicked by a medium with an appropriately chosen index of refraction. This remark applies, e.g., to spherically symmetric and static spacetimes (see Section 4.3) and, in particular, to the Schwarzschild spacetime (see Section 5.1). The analogy with ordinary optics in media has been used for constructing, in the laboratory, analogue models for light propagation in general-relativistic spacetimes (see [311]). Extremizing the functional (67) is formally analogous to Maupertuis’ principle for a particle in a scalar potential on flat space, which is discussed in any book on classical mechanics. Dropping the assumption that the Fermat one-form is a differential, but still requiring the Fermat metric to be conformal to the Euclidean metric, corresponds to introducing an additional vector potential. This form of the optical-mechanical analogy, for light rays in stationary spacetimes whose Fermat metric is conformal to the Euclidean metric, is discussed, e.g., in [7]. The conformal factor $e^{2f}$ in Equation (61) does not affect the paths of light rays. However, it does affect redshifts and distance measures (recall Section 2.4). If $g$ is of the form (61), for every lightlike geodesic $\lambda$ the quantity $g(\dot{\lambda},\partial_{t})$ is a constant of motion. This leads to a particularly simple form of the general redshift formula (36). We consider an arbitrary lightlike geodesic $s\mapsto\lambda(s)$ in terms of its coordinate representation $s\mapsto\left(t(s),x^{1}(s),x^{2}(s),x^{3}(s)\right)$. If both observer and emitter are at rest in the sense that their 4-velocities $U_{\mathrm{O}}$ and $U_{\mathrm{S}}$ are parallel to $W=\partial_{t}$, Equation (36) can be rewritten as $$\log\left(1+z(s)\right)=f\left(t(s),x(s)\right)-f\left(t(0),x(0)\right).$$ (68) This justifies calling $f$ the redshift potential. It is shown in [188] that there is a redshift potential for a congruence of timelike curves in a spacetime if and only if the timelike curves are the integral curves of a conformal Killing vector field. The notion of a redshift potential or redshift function is also discussed in [97]. Note that Equation (68) immediately determines the redshift in conformally stationary spacetimes for any pair of observer and emitter. If the 4-velocity of the observer or of the emitter is not parallel to $W=\partial_{t}$, one just has to add the usual special-relativistic Doppler factor. Conformally stationary spacetimes can be characterized by another interesting property. Let $W$ be a timelike vector field in a spacetime and fix three observers whose worldlines are integral curves of $W$. Then the angle under which two of them are seen by the third one remains constant in the course of time, for any choice of the observers, if and only if $W$ is proportional to a conformal Killing vector field. For a proof see [188]. 4.3 Lensing in spherically symmetric and static spacetimes The class of spherically symmetric and static spacetimes is of particular relevance in view of lensing, because it includes models for non-rotating stars and black holes (see Sections 5.1, 5.2, 5.3), but also for more exotic objects such as wormholes (see Section 5.4), monopoles (see Section 5.5), naked singularities (see Section 5.6), and Boson or Fermion stars (see Section 5.7). A spherically symmetric and static spacetime can also be used, as a rough approximation, to model a star cluster, a galaxy or a cluster of galaxies. Here we collect the relevant formulas for an unspecified spherically symmetric and static metric. We find it convenient to write the metric in the form $$g=e^{2f(r)}\left(-dt^{2}+S(r)^{2}\,dr^{2}+R(r)^{2}\left(d\vartheta^{2}+\sin^{2% }\vartheta\,d\varphi^{2}\right)\right).$$ (69) As Equation (69) is a special case of Equation (61), all results of Section 4.2 for conformally stationary metrics apply. However, much stronger results are possible because for metrics of the form (69) the geodesic equation is completely integrable. Hence, all relevant quantities can be determined explicitly in terms of integrals over the metric coefficients. Redshift and Fermat geometry. Comparison of Equation (69) with the general form (61) of a conformally stationary spacetime shows that here the redshift potential $f$ is a function of $r$ only, the Fermat one-form $\hat{\phi}$ vanishes, and the Fermat metric $\hat{g}$ is of the special form $$\hat{g}=S(r)^{2}\,dr^{2}+R(r)^{2}\left(d\vartheta^{2}+\sin^{2}\vartheta\,d% \varphi^{2}\right).$$ (70) This Fermat metric has several interesting applications. E.g., Gibbons and Werner [175] have derived some lensing features of a spherically symmetric static fluid ball by applying the Gauss-Bonnet theorem to the corresponding Fermat metric (or optical metric). By Fermat’s principle, the geodesics of $\hat{g}$ coincide with the projection to 3-space of light rays. The travel time (in terms of the time coordinate $t$) of a lightlike curve coincides with the $\hat{g}$-arclength of its projection. By symmetry, every $\hat{g}$-geodesic stays in a plane through the origin. From Equation (70) we read that the sphere of radius $r$ has area $4\pi R(r)^{2}$ with respect to the Fermat metric. Also, Equation (70) implies that the second fundamental form of this sphere is a multiple of its first fundamental form, with a factor $-R^{\prime}(r)\left(R(r)\,S(r)\right)^{-1}$. If $$R^{\prime}(r_{\mathrm{p}})=0,$$ (71) the sphere $r=r_{\mathrm{p}}$ is totally geodesic, i.e., a $\hat{g}$-geodesic that starts tangent to this sphere remains in it. The best known example for such a light sphere or photon sphere is the sphere $r=3m$ in the Schwarzschild spacetime (see Section 5.1). Light spheres also occur in the spacetimes of wormholes (see Section 5.4). If $R^{\prime\prime}(r_{\mathrm{p}})<0$, the circular light rays in a light sphere are stable with respect to radial perturbations, and if $R^{\prime\prime}(r_{\mathrm{p}})>0$, they are unstable like in the Schwarschild case. The condition under which a spherically symmetric static spacetime admits a light sphere was first given by Atkinson [16]. Abramowicz [1] has shown that for an observer traveling along a circular light orbit (with subluminal velocity) there is no centrifugal force and no gyroscopic precession. Claudel, Virbhadra, and Ellis [81] investigated, with the help of Einstein’s field equation and energy conditions, the amount of matter surrounded by a light sphere. Among other things, they found an energy condition under which a spherically symmetric static black hole must be surrounded by a light sphere. A purely kinematical argument shows that any spherically symmetric and static spacetime that has a horizon at $r=r_{\mathrm{H}}$ and is asymptotically flat for $r\to\infty$ must contain a light sphere at some radius between $r_{\mathrm{H}}$ and $\infty$ (see Hasse and Perlick [191]). In the same article, it is shown that in any spherically symmetric static spacetime with a light sphere there is gravitational lensing with infinitely many images. Bozza [48] investigated a strong-field limit of lensing in spherically symmetric static spacetimes, as opposed to the well-known weak-field limit, which applies to light rays that come close to an unstable light sphere. (In later papers, the term “strong-field limit” was replaced with “strong-deflection limit”. This is, indeed, more appropriate because the gravitational field, measured in terms of tidal forces, need not be particularly strong near an unstable light sphere. The characteristic feature is that the bending angle goes to infinity, i.e., that light rays make arbitrarily many turns around the center if they approach an unstable light sphere.) This limit applies, in particular, to light rays that approach the sphere $r=3m$ in the Schwarzschild spacetime (see [53] and, for illustrations, Figures 16, 17, and 18). The strong-deflection limit has also been applied to many other spherically symmetric and static metrics; several examples are discussed in Section 5 below. As demonstrated in the original article by Bozza [48], the parameters that characterize the strong-deflection limit can be used to distinguish between different black-hole metrics. These parameters were related to quasi-normal modes in [386]. Index of refraction and embedding diagrams. Transformation to an isotropic radius coordinate ${\tilde{r}}$ via $$\frac{S(r)\,dr}{R(r)}=\frac{d\tilde{r}}{\tilde{r}}$$ (72) takes the Fermat metric (70) to the form $$\hat{g}=n(\tilde{r})^{2}\left(d\tilde{r}^{2}+\tilde{r}^{2}(d\vartheta^{2}+\sin% ^{2}\vartheta\,d\varphi^{2})\right)$$ (73) where $$n(\tilde{r})=\frac{R(r)}{\tilde{r}}.$$ (74) On the right-hand side $r$ has to be expressed by $\tilde{r}$ with the help of Equation (72). The results of Section 4.2 imply that the lightlike geodesics in a spherically symmetric static spacetime are equivalent to the light rays in a medium with index of refraction (74) on Euclidean 3-space. For arbitrary metrics of the form (69), this result is due to Atkinson [16]. It reduces the lightlike geodesic problem in a spherically symmetric static spacetime to a standard problem in ordinary optics, as treated, e.g., in [266], §27, and [252], Section 4. One can combine this result with our earlier observation that the integral in Equation (67) has the same form as the functional in Maupertuis’ principle in classical mechanics. This demonstrates that light rays in spherically symmetric and static spacetimes behave like particles in a spherically symmetric potential on Euclidean 3-space (cf., e.g., [135]). If the embeddability condition $$S(r)^{2}\geq R^{\prime}(r)^{2}$$ (75) is satisfied, we define a function $Z(r)$ by $$Z^{\prime}(r)=\sqrt{S(r)^{2}-R^{\prime}(r)^{2}}.$$ (76) Then the Fermat metric (70) reads $$\hat{g}=(dR(r))^{2}+R(r)^{2}\left(d\vartheta^{2}+\sin^{2}\vartheta\,d\varphi^{% 2}\right)+(dZ(r))^{2}.$$ (77) If restricted to the equatorial plane $\vartheta=\pi/2$, the metric (77) describes a surface of revolution, embedded into Euclidean 3-space as $$(r,\varphi)\mapsto\left(R(r)\cos\varphi,R(r)\,\sin\varphi,Z(r)\right).$$ (78) Such embeddings of the Fermat geometry have been visualized for several spacetimes of interest (see Figure 12 for the Schwarzschild case and [201, 202] for other examples). This is quite instructive because from a picture of a surface of revolution one can read the qualitative features of its geodesics without calculating them. Note that Equation (72) defines the isotropic radius coordinate uniquely up to a multiplicative constant. Hence, the straight lines in this coordinate representation give us an unambiguously defined reference grid for every spherically symmetric and static spacetime. These straight lines have been called triangulation lines in [84, 85], where their use for calculating bending angles, exactly or approximately, is outlined. Light cone. In a spherically symmetric static spacetime, the (past) light cone of an event $p_{\mathrm{O}}$ can be written in terms of integrals over the metric coefficients. We first restrict to the equatorial plane $\vartheta=\pi/2$. The $\hat{g}$-geodesics are then determined by the Lagrangian $$\mathcal{L}=\frac{1}{2}\left(S(r)^{2}\left(\frac{dr}{d\ell}\right)^{2}+R(r)^{2% }\left(\frac{d\varphi}{d\ell}\right)^{2}\right).$$ (79) The Euler-Lagrange equations read $$\frac{d}{d\ell}\left(S(r)^{2}\frac{dr}{d\ell}\right)=S(r)S^{\prime}(r)\left(% \frac{dr}{d\ell}\right)^{2}+R(r)R^{\prime}(r)\left(\frac{d\varphi}{d\ell}% \right)^{2}.$$ (80) $$\frac{d}{d\ell}\left(R(r)^{2}\frac{d\varphi}{d\ell}\right)=0.$$ (81) After dividing the first equation by $R(r)^{2}(d\varphi/d\ell)^{2}$, and using the second equation, we find $$\frac{d}{d\varphi}\left(\frac{S(r)^{2}}{R(r)^{2}}\frac{dr}{d\varphi}\right)=% \frac{S(r)S^{\prime}(r)}{R(r)^{2}}\left(\frac{dr}{d\varphi}\right)^{2}+\frac{R% ^{\prime}(r)}{R(r)}.$$ (82) Equations (80) and (81) give the light rays parametrized by $\hat{g}$-arclength (which equals travel time) $\ell$, Equation (82) can be used for determining the orbits of light rays if the parametrization plays no role. For fixed radius value $r_{\mathrm{O}}$, initial conditions $$\begin{array}[]{rclrcl}r(0)&=&r_{\mathrm{O}},&\qquad\displaystyle\frac{dr}{d% \ell}(0)&=&\displaystyle\frac{\cos\Theta}{S(r_{\mathrm{O}})},\\ \varphi(0)&=&0,&\qquad\displaystyle\frac{d\varphi}{d\ell}(0)&=&\displaystyle% \frac{\sin\Theta}{R(r_{\mathrm{O}})}\end{array}$$ (83) determine a unique solution $r=\mathsf{r}(\ell,\Theta)$, $\varphi=\phi(\ell,\Theta)$ of the Euler–Lagrange equations (80) and (81). $\Theta$ measures the initial direction with respect to the symmetry axis (see Figure 6). We get all light rays issuing from the event $r=r_{\mathrm{O}}$, $\varphi=0$, $\vartheta=\pi/2$, $t=t_{\mathrm{O}}$ into the past by letting $\Theta$ range from 0 to $\pi$ and applying rotations around the symmetry axis. This gives us the past light cone of this event in the form $$(\ell,\Psi,\Theta)\longmapsto\begin{pmatrix}t_{\mathrm{O}}-\ell\\ \mathsf{r}(\ell,\Theta)\,\sin\phi(\ell,\Theta)\,\cos\Psi\\ \mathsf{r}(\ell,\Theta)\,\sin\phi(\ell,\Theta)\,\sin\Psi\\ \mathsf{r}(\ell,\Theta)\,\cos\phi(\ell,\Theta)\end{pmatrix}.$$ (84) $\Psi$ and $\Theta$ are spherical coordinates on the observer’s sky. If we let $t_{\mathrm{O}}$ float over $\mathbb{R}$, we get the observational coordinates (4) for an observer on a $t$-line, up to two modifications. First, $t_{\mathrm{O}}$ is not the same as proper time $\tau$; however, along each $t$-line they are related just by a constant, $$\frac{d\tau}{dt_{\mathrm{O}}}=e^{-f(r_{\mathrm{O}})}.$$ (85) Second, $\ell$ is not the same as the affine parameter $s$; along a ray with initial direction $\Theta$, they are related by $$\frac{ds}{d\ell}=e^{f(\mathrm{r}(\ell,\Theta))}.$$ (86) The constants of motion $$R(r)^{2}\frac{d\varphi}{d\ell}=R(r_{\mathrm{O}})\sin\Theta,\qquad S(r)^{2}% \left(\frac{dr}{d\ell}\right)^{2}+R(r)^{2}\left(\frac{d\varphi}{d\ell}\right)^% {2}=1$$ (87) give us the functions $\mathsf{r}(\ell,\Theta)$, $\phi(\ell,\Theta)$ in terms of integrals, $$\displaystyle\ell$$ $$\displaystyle=$$ $$\displaystyle\int_{r_{\mathrm{O}}\dots}^{\dots\mathsf{r}(\ell,\Theta)}\frac{R(% r)\,S(r)\,dr}{\sqrt{R(r)^{2}-R(r_{\mathrm{O}})^{2}\,\sin^{2}\Theta}},$$ (88) $$\displaystyle\phi(\ell,\Theta)$$ $$\displaystyle=$$ $$\displaystyle R(r_{\mathrm{O}})\,\sin\Theta\int_{r_{\mathrm{O}}\dots}^{\dots% \mathsf{r}(\ell,\Theta)}\frac{S(r)\,dr}{R(r)\sqrt{R(r)^{2}-R(r_{\mathrm{O}})^{% 2}\,\sin^{2}\Theta}}.$$ (89) Here the notation with the dots is a short-hand; it means that the integral is to be decomposed into sections where $\mathsf{r}(\ell,\Theta)$ is a monotonous function of $\ell$, and that the absolute value of the integrals over all sections have to be added up. Turning points occur at radius values where $R(r)=R(r_{\mathrm{O}})\sin\Theta$ and $R^{\prime}(r)\neq 0$ (see Figure 10). If the metric coefficients $S$ and $R$ have been specified, these integrals can be calculated and give us the light cone (see Figure 13 for an example). Having parametrized the rays with $\hat{g}$-arclength (= travel time), we immediately get the intersections of the light cone with hypersurfaces $t=\mbox{constant}$ (“instantaneous wave fronts”); see Figures 14, 19, and 20. Exact lens map and various approximation methods. Recall from Section 2.1 that the exact lens map [154] refers to a chosen observation event $p_{\mathrm{O}}$ and a chosen “source surface” $\mathcal{T}$. In general, for $\mathcal{T}$ we may choose any 3-dimensional submanifold that is ruled by timelike curves. The latter are to be interpreted as wordlines of light sources. In a spherically symmetric and static spacetime, we may take advantage of the symmetry by choosing for $\mathcal{T}$ a sphere $r=r_{\mathrm{S}}$ with its ruling by the $t$-lines. This restricts the consideration to lensing for static light sources. Note that, for an observer at $r_{\mathrm{O}}$, all static light sources at radius $r_{\mathrm{S}}$ undergo the same redshift, $\log(1+z)=f(r_{\mathrm{S}})-f(r_{\mathrm{O}})$. Without loss of generality, we place the observation event $p_{\mathrm{O}}$ on the 3-axis. This gives us the past light cone in the representation (84). To each ray from the observer, with initial direction characterized by $\Theta$, we can assign the total angle $\Phi(\Theta)$ the ray sweeps out on its way from $r_{\mathrm{O}}$ to $r_{\mathrm{S}}$ (see Figure 6). $\Phi(\Theta)$ is given by Equation (89), $$\Phi(\Theta)=R(r_{\mathrm{O}})\,\sin\Theta\int_{r_{\mathrm{O}}\dots}^{\dots r_% {\mathrm{S}}}\frac{S(r)\,dr}{R(r)\sqrt{R(r)^{2}-R(r_{\mathrm{O}})^{2}\,\sin^{2% }\Theta}},$$ (90) where the same short-hand notation is used as in Equation (89). $\Phi(\Theta)$ is not necessarily defined for all $\Theta$ because some light rays that start at $r_{\mathrm{O}}$ may not reach $r_{\mathrm{S}}$. Also, $\Phi(\Theta)$ may be multi-valued because a light ray may intersect the sphere $r=r_{\mathrm{S}}$ several times. Equation (84) gives us the (possibly multi-valued) lens map $$(\Psi,\Theta)\longmapsto\begin{pmatrix}r_{\mathrm{S}}\,\sin\Phi(\Theta)\,\cos% \Psi\\ r_{\mathrm{S}}\,\sin\Phi(\Theta)\,\sin\Psi\\ r_{\mathrm{S}}\,\cos\Phi(\Theta)\end{pmatrix}.$$ (91) This version of the exact lens map in spherically symmetric and static spacetimes was first considered in [339]. It is interesting to compare it with the standard lens map (or lens equation) in the quasi-Newtonian approximation formalism, see e.g. Wambsganss [427], Section 3.1. In both cases, rotational symmetry about the axis through the observer has the effect that in essence the lens map reduces to a map from an angle to another angle; the first angle, here $\Theta$, determines the position of the image on the observer’s sky, the second angle, here $\Phi(\Theta)$, gives the actual position of the light source. If the metric coefficients $R(r)$ and $S(r)$ are given, the integrals in Equation (90) can be numerically calculated and from the result all lensing features can be determined with arbitrary accuracy. As an example, the exact lens map will be evaluated for the Schwarzschild metric in Section 5.1 below. In [339]), the examples of an Ellis wormhole (cf. Section 5.4) and of a Barriola-Vilenkin monopole (cf. Section 5.5) were treated. – Note that $\Phi(\Theta)$ may take any value between 0 and infinity. A value $\Phi(\Theta)>2\pi$ occurs whenever a light ray makes more than one full turn around the center. For each image we can define the order $$i(\Theta)=\min\left\{m\in\mathbb{N}\,\big{|}\,\Phi(\Theta)<m\pi\right\},$$ (92) which counts how often the ray has crossed the axis. (If the lens map is multi-valued, one should introduce an index to label different images that correspond to the same angle $\Theta$). In accordance with the terminology introduced in Section 3.1, an image of order 0 is called primary, an image of order 1 is called secondary, and so on. The standard example where images of arbitrarily high order occur is the Schwarzschild spacetime (see Section 5.1). For a light source which is not perfectly aligned with the observer and the center, images of even order have even parity and line up on one side of the direction towards the center; images of odd order have odd parity and line up on the other side of the direction towards the center. In the case of perfect alignment, a sequence of Einstein rings is seen. An Einstein ring of order 0 is called primary, an Einstein ring of order 1 is called secondary, and so on. – We can rewrite the exact lens map in a spherically symmetric and static spacetime in a form more similar to the standard quasi-Newtonian lens map if we make two additional assumptions which are satisfied in many, though not all, situations of interest: • The spacetime is asymptotically flat and both $r_{\mathrm{O}}$ and $r_{\mathrm{S}}$ are very large. • The source is almost exactly opposite to the observer, i.e., $\Phi(\Theta)$ is close to an odd multiple of $\pi$. The first assumption makes sure that the lens map is single-valued, and both assumptions together imply that along each (past-oriented) light ray from the observer to the source the radius coordinate has precisely one turning-point. For a light ray with turning point at $r_{m}(\Theta)$, the asymptotic assumption allows to approximate Equation (90) by $$\Phi(\Theta)=2\int_{r_{m}(\Theta)}^{\infty}\frac{S(r)\,dr}{R(r)\sqrt{R(r)^{2}-% R(r_{\mathrm{O}})^{2}\,\sin^{2}\Theta}}.$$ (93) To link up with the notation of the standard lens map, we introduce distances $D_{d}$ and $D_{ds}$ and angles $\theta=\pi-\Theta$, $\beta(\theta)$ and $\hat{\alpha}(\theta)$ according to Figure 7. The alignment assumption implies that $\beta(\theta)$ is small, and the asymptotic condition implies that the bending angle $\hat{\alpha}$ can be approximated as $$\hat{\alpha}(\theta)=\Phi(\Theta)-\pi$$ (94) After some elementary geometry, one finds that $$\mathrm{tan}\,\beta(\theta)=\mathrm{tan}\,\theta-\frac{D_{ds}}{D_{d}+D_{ds}}% \Big{(}\mathrm{tan}\,\theta+\mathrm{tan}\big{(}\hat{\alpha}(\theta)-\theta\big% {)}\Big{)}.$$ (95) This is the lens equation of Virbhadra and Ellis [420] (cf. [422] for an earlier version). Equation (95) gives a well-defined (single-valued) lens map $\theta\mapsto\beta(\theta)$ if we insert Equations (94) and (93). The Virbhadra-Ellis lens map may be called “almost exact”. It is based on approximations as to the positions of source and observer, but it is not restricted to the case that the bending angle is small. As a matter of fact, the bending angle may be arbitrarily large; $\hat{\alpha}(\theta)$ diverges to infinity if the turning point $r_{m}(\Theta)$ approaches an unstable light sphere. (It was already mentioned that an unstable light sphere occurs at a radius value $r_{\mathrm{p}}$ if and only if $R^{\prime}(r_{\mathrm{p}})=0$ and $R^{\prime\prime}(r_{\mathrm{p}})>0$; the standard example is the sphere at $r_{p}=3m$ in the Schwarzschild spacetime, see Section 5.1). It was shown by Bozza [48] that, whenever an unstable light sphere is approached, the divergence of the bending angle is logarithmic. The Virbhadra-Ellis lens equation was originally introduced for the Schwarzschild metric [420] where it approximates the exact treatment remarkably well within a wide range of validity [150]. In comparison to the exact lens map, the Virbhadra-Ellis lens map has the appealing property of resembling the standard quasi-Newtonian lens map as much as possible. On the other hand, neither analytical nor numerical evaluation of the “almost exact lens map” is significantly easier than that of the exact lens map. The Virbhadra-Ellis lens map was successfully applied to many spherically symmetric and static spacetimes, several examples are considered in Section 5 below. Bozza [50] compared the Virbhadra-Ellis lens equation with other approximate lens equations that had been proposed for spherically symmetric and static spacetimes and, in particular, for the Schwarzschild metric: • the Ohanian lens equation, which was implicitly contained in Ohanian’s pioneering work [314] on Schwarzschild lensing, • a modification of the Virbhadra-Ellis lens equation, introduced by Dab̧rowski and Schunck [93] in their treatment of lensing by a boson star, • a lens equation introduced by Bozza and Sereno [58] that is essentially equivalent to the Ohanian lens equation but replaces an angle centered at the lens by an angle centered at the observer, • a new lens equation that is, again, a slight modification of the Ohanian lens equation. All these lens equations relax the alignment condition but retain some kind of asymptotic assumption. After discussing the accuracy of these various lens equations in realistic situations, Bozza argues in favour of the Ohanian lens equation and its modifications. – In addition to approximative lens equations, several other approximation techniques have been developed for lensing in spherically symmetric and static spacetimes. Amore and Arceo [9] expressed the bending angle analytically as a rapidly convergent series; this approach was further developed in [10, 11]. Keeton and Petters [230] expanded corrections to the weak-deflection limit as a Taylor series in the gravitational radius of the lens. In two follow-up papers, they applied this formalism to post-Newtonian metrics [231] and to braneworld black holes [232]. A major purpose of all approximation methods mentioned is to test general relativity by comparing Schwarzschild lensing to lensing in alternative theories of gravity, see Section 5.1. Distance measures, image distortion and brightness of images. For calculating image distortion (see Section 2.5) and the brightness of images (see Section 2.6) we have to consider infinitesimally thin bundles with vertex at the observer. In a spherically symmetric and static spacetime, we can apply the orthonormal derivative operators $\partial_{\Theta}$ and $\sin\Theta\,\partial_{\Psi}$ to the representation (84) of the past light cone. Along each ray, this gives us two Jacobi fields $Y_{1}$ and $Y_{2}$ which span an infinitesimally thin bundle with vertex at the observer. $Y_{1}$ points in the radial direction and $Y_{2}$ points in the tangential direction (see Figure 8). The radial and the tangential direction are orthogonal to each other and, by symmetry, parallel-transported along each ray. Thus, we can choose the Sachs basis $(E_{1},E_{2})$ such that $Y_{1}=D_{+}E_{1}$ and $Y_{2}=D_{-}E_{2}$. The coefficients $D_{+}$ and $D_{-}$ are unique if we require them to be positive near the vertex. $D_{+}$ and $D_{-}$ are the extremal angular diameter distances of Section 2.4 with respect to a static observer (because the $(\Psi,\Theta)$-grid refers to a static observer). In the case at hand, they are called the radial and tangential angular diameter distances. They can be calculated by normalizing $Y_{1}$ and $Y_{2}$, $$\displaystyle D_{+}(\ell,\Theta)$$ $$\displaystyle=$$ $$\displaystyle e^{f(\mathsf{r}(\ell,\Theta))}\,R(r_{\mathrm{O}})\,\cos\Theta\,% \sqrt{R\left(\mathsf{r}(\ell,\Theta)\right)^{2}-R(r_{\mathrm{O}}^{2})\,\sin^{2% }\Theta}$$ (96) $$\displaystyle\times\int_{r_{\mathrm{O}}\dots}^{\dots\mathsf{r}(\Theta,\ell)}\!% \!\!\!\!\frac{S(r)\,R(r)dr}{\sqrt{R(r)^{2}-R(r_{\mathrm{O}})^{2}\,\sin^{2}% \Theta}^{\,3}},$$ $$\displaystyle D_{-}(\ell,\Theta)$$ $$\displaystyle=$$ $$\displaystyle e^{f(\mathsf{r}(\ell,\Theta))}\,R\left(\mathsf{r}(\ell,\Theta)% \right)\frac{\sin\phi(\ell,\Theta)}{\sin\Theta}.$$ (97) These formulas have been derived first for the special case of the Schwarzschild metric by Dwivedi and Kantowski [110] and then for arbitrary spherically symmetric static spacetimes by Dyer [111]. (In [111], Equation (97) is erroneously given only for the case that, in our notation, $e^{f(r)}R(r)=r$.) From these formulas we immediately get the area distance $D_{\mathrm{area}}=\sqrt{|D_{+}D_{-}|}$ for a static observer and, with the help of the redshift $z$, the luminosity distance $D_{\mathrm{lum}}=(1+z)^{2}D_{\mathrm{area}}$ (recall Section 2.4). In this way, Equation (96) and Equation (97) allow to calculate the brightness of images according to the formulas of Section 2.6. Similarly, Equation (96) and Equation (97) allow to calculate image distortion in terms of the ellipticity $\varepsilon$ (recall Section 2.5). In general, $\varepsilon$ is a complex quantity, defined by Equation (49). In the case at hand, it reduces to the real quantity $\varepsilon=D_{-}/D_{+}-D_{+}/D_{-}$. The expansion $\theta$ and the shear $\sigma$ of the bundles under consideration can be calculated from Kantowski’s formula [222, 110], $$\dot{D}_{\pm}=\left(\theta\pm\sigma\right)D_{\pm},$$ (98) to which Equation (27) reduces in the case at hand. The dot (= derivative with respect to the affine parameter $s$) is related to the derivative with respect to $\ell$ by Equation (86). Evaluating Equations (96, 97) in connection with the exact lens map leads to quite convenient formulas, for static light sources at $r=r_{\mathrm{S}}$. Setting $\mathsf{r}(\ell,\Theta)=r_{\mathrm{S}}$ and $\phi(\ell,\Theta)=\Phi(\Theta)$ and comparing with Equation (90) yields (cf. [339]) $$\displaystyle D_{+}(\Theta)$$ $$\displaystyle=$$ $$\displaystyle e^{f(r_{\mathrm{S}})}\,\sqrt{R(r_{\mathrm{S}})^{2}-R(r_{\mathrm{% O}})^{2}\,\sin^{2}\Theta}\;\Phi^{\prime}(\Theta),$$ (99) $$\displaystyle D_{-}(\Theta)$$ $$\displaystyle=$$ $$\displaystyle e^{f(r_{\mathrm{S}})}\,R(r_{\mathrm{S}})\,\sin\Phi(\Theta).$$ (100) These formulas immediately give image distortion and the brightness of images if the map $\Theta\mapsto\Phi(\Theta)$ is known. Caustics of light cones. Quite generally, the past light cone has a caustic point exactly where at least one of the extremal angular diameter distances $D_{+}$, $D_{-}$ vanishes (see Sections 2.2, 2.3, and 2.4). In the case at hand, zeros of $D_{+}$ are called radial caustic points and zeros of $D_{-}$ are called tangential caustic points (see Figure 9). By Equation (97), tangential caustic points occur if $\phi(\ell,\Theta)$ is a multiple of $\pi$, i.e., whenever a light ray crosses the axis of symmetry through the observer (see Figure 9). Symmetry implies that a point source is seen as a ring (“Einstein ring”) if its worldline crosses a tangential caustic point. By contrast, a point source whose wordline crosses a radial caustic point is seen infinitesimally extended in the radial direction. The set of all tangential caustic points of the past light cone is called the tangential caustic for short. In general, it has several connected components. In accordance with the order of images, as defined in Equation (92), these connected components can be labeled as primary, secondary, etc. tangential caustics. Each connected component is a spacelike curve in spacetime which projects to (part of) the axis of symmetry through the observer. The radial caustic is a lightlike surface in spacetime unless at points where it meets the axis; its projection to space is rotationally symmetric around the axis. The best known example for a tangential caustic, with infinitely many connected components, occurs in the Schwarzschild spacetime (see Figure 13). It is also instructive to visualize radial and tangential caustics in terms of instantaneous wave fronts, i.e., intersections of the light cone with hypersurfaces $t=\mbox{constant}$. Examples are shown in Figures 14, 19, and 20. By symmetry, a tangential caustic point of an instantaneous wave front can be neither a cusp nor a swallow-tail. Hence, the general result of Section 2.2 implies that the tangential caustic is always unstable. The radial caustic in Figure 20 consists of cusps and is, thus, stable. 4.4 Lensing in axisymmetric stationary spacetimes Axisymmetric stationary spacetimes are of interest in view of lensing as general-relativistic models for rotating deflectors. The best known and most important example is the Kerr metric which describes a rotating black hole (see Section 5.8). For non-collapsed rotating objects, exact solutions of Einstein’s field equation are known only for the idealized cases of infinitely long cylinders (including string models; see Section 5.10) and disks (see Section 5.9). Here we collect, as a preparation for these examples, some formulas for an unspecified axisymmetric stationary metric. The latter can be written in coordinates $(y^{1},y^{2},\varphi,t)$, with capital indices $A,B,\dots$ taking the values 1 and 2, as $$g=g_{tt}(y)\,dt^{2}+2g_{t\varphi}(y)\,dt\,d\varphi+g_{\varphi\varphi}(y)\,d% \varphi^{2}+g_{AB}(y)\,dy^{A}\,dy^{B},$$ (101) where all metric coefficients depend on $y=(y^{1},y^{2})$ only. We assume that the integral curves of $\partial_{\varphi}$ are closed, with the usual $(2\pi)$-periodicity, and that the 2-dimensional orbits spanned by $\partial_{\varphi}$ and $\partial_{t}$ are timelike. Then the Lorentzian signature of $g$ implies that $g_{AB}(y)$ is positive definite. In general, the vector field $\partial_{t}$ need not be timelike and the hypersurfaces $t=\mbox{constant}$ need not be spacelike. Our assumptions allow for transformations $(\varphi,t)\mapsto(\varphi+\Omega t,t)$ with a constant $\Omega$. If, by such a transformation, we can achieve that $g_{tt}<0$ everywhere, we can use the purely spatial formalism for light rays in terms of the Fermat geometry (recall Section 4.2). Comparison of Equation (101) with Equation (61) shows that the redshift potential $f$, the Fermat metric $\hat{g}$, and the Fermat one-form $\hat{\phi}$ are $$\displaystyle e^{2f}$$ $$\displaystyle=$$ $$\displaystyle-g_{tt},$$ (102) $$\displaystyle\hat{g}$$ $$\displaystyle=$$ $$\displaystyle-\frac{g_{AB}}{g_{tt}}\,dx^{A}\,dx^{B}+\frac{g_{t\varphi}^{2}-g_{% tt}\,g_{\varphi\varphi}}{g_{tt}^{2}}\,d\varphi^{2},$$ (103) $$\displaystyle\hat{\phi}$$ $$\displaystyle=$$ $$\displaystyle-\frac{g_{t\varphi}}{g_{tt}}\,d\varphi,$$ (104) respectively. If it is not possible to make $g_{tt}$ negative on the entire spacetime domain under consideration, the Fermat geometry is defined only locally and, therefore, of limited usefulness. This is the case, e.g., for the Kerr metric where, in Boyer–Lindquist coordinates, $g_{tt}$ is positive in the ergosphere (see Section 5.8). Variational techniques related to Fermat’s principal in stationary spacetimes are detailed in a book by Masiello [272]. Note that, in contrast to standard terminology, Masiello’s definition of stationarity includes the assumption that the surfaces $t=\mbox{constant}$ are spacelike. For a rotating body with an equatorial plane (i.e., with reflectional symmetry), the Fermat metric of the equatorial plane can be represented by an embedding diagram, in analogy to the spherically symmetric static case (recall Figure 12). However, one should keep in mind that in the non-static case the lightlike geodesics do not correspond to the geodesics of $\hat{g}$ but are affected, in addition, by a sort of Coriolis force produced by $\hat{\phi}$. For a review on embedding diagrams, including several examples (see [201]). 5 Examples 5.1 Schwarzschild spacetime The (exterior) Schwarzschild metric $$g=-\left(1-\frac{2m}{r}\right)dt^{2}+\left(1-\frac{2m}{r}\right)^{-1}dr^{2}+r^% {2}\left(d\vartheta^{2}+\sin^{2}\vartheta\,d\varphi^{2}\right)$$ (105) has the form (69) with $$e^{2f(r)}=S(r)^{-1}=1-\frac{2m}{r},\qquad R(r)=\frac{r}{\sqrt{1-\frac{2m}{r}}}.$$ (106) It is the unique spherically symmetric vacuum solution of Einstein’s field equation. At the same time, it is the most important and best understood spacetime in which lensing can be explicitly studied without approximations. Schwarzschild lensing beyond the weak-field approximation has astrophysical relevance in view of black holes and neutron stars. The increasing evidence that there is a supermassive black hole at the center of our Galaxy (see [136] for background material) is a major motivation for a detailed study of Schwarzschild lensing (and of Kerr lensing; see Section 5.8). In the following we consider the Schwarzschild metric with a constant $m>0$ and we ignore the region $r<0$ for which the singularity at $r=0$ is naked. The Schwarzschild metric is static on the region $2m<r<\infty$. (The region $r<0$ for $m>0$ is equivalent to the region $r>0$ for $m<0$. It is usually considered as unphysical but has found some recent interest in connection with lensing by wormholes; see Section 5.4.) Historical notes. Shortly after the discovery of the Schwarzschild metric by Schwarzschild [372] and independently by Droste [106], basic features of its lightlike geodesics were found by Flamm [144], Hilbert [198], and Weyl [430]. Detailed studies of its timelike and lightlike geodesics were made by Hagihara [183] and Darwin [95, 96]. For a fairly complete list of the pre-1979 literature on Schwarzschild geodesics see Sharp [380]. All modern text-books on general relativity include a section on Schwarzschild geodesics, but not all of them go beyond the weak-field approximation. For a particularly detailed exposition see Chandrasekhar [75]. Redshift and Fermat geometry. The redshift potential $f$ for the Schwarzschild metric is given in Equation (106). With the help of $f$ we can directly calculate the redshift via Equation (68) if observer and light source are static (i.e., $t$-lines). If the light source or the observer does not follow a $t$-line, a Doppler factor has to be added. Independent of the velocity of observer and light source, the redshift becomes arbitrarily large if the light source is sufficiently close to the horizon. For light source and observer freely falling, the redshift formula was discussed by Bażański and Jaranowski [30]. If projected to 3-space, the light rays in the Schwarzschild spacetime are the geodesics of the Fermat metric which can be read from Equation (70) (cf. Frankel [147]), $$\hat{g}=\frac{dr^{2}}{(1-\frac{2m}{r})^{2}}+\frac{r^{2}(d\vartheta^{2}+\sin% \vartheta\,d\varphi^{2})}{1-\frac{2m}{r}}.$$ (107) The metric coefficient $R(r)$, as given by Equation (106), has a strict minimum at $r=3m$ and no other extrema (see Figure 10). Hence, there is an unstable light sphere at this radius (recall Equation (71)). The existence of circular light rays at $r=3m$ was noted already by Hilbert [198]. The relevance of these circular light rays in view of lensing was clearly seen by Darwin [95, 96] and Atkinson [16]. They realized, in particular, that a Schwarzschild black hole produces infinitely many images of each light source, corresponding to an infinite sequence of light rays whose limit curve asymptotically spirals towards a circular light ray. The circular light rays at $r=3m$ are also associated with other physical effects such as centrifugal force reversal and “locking” of gyroscopes. These effects have been discussed with the help of the Fermat geometry (= optical reference geometry) in various articles by Abramowicz and collaborators (see, e.g., [5, 4, 6, 2]). Index of refraction and embedding diagrams. We know from Section 4.3 that light rays in any spherically symmetric and static spacetime can be characterized by an index of refraction. This requires introducing an isotropic radius coordinate $\tilde{r}$ via Equation (72). In the Schwarzschild case, $\tilde{r}$ is related to the Schwarzschild radius coordinate $r$ by $${\tilde{r}}=\frac{1}{2}\left(\sqrt{r^{2}-2mr}+r-m\right),\qquad r=\frac{\left(% 2\tilde{r}+m\right)^{2}}{4\tilde{r}}.$$ (108) $\tilde{r}$ ranges from $m/2$ to infinity if $r$ ranges from $2m$ to infinity. In terms of the isotropic coordinate, the Fermat metric (107) takes the form $$\hat{g}=n({\tilde{r}})^{2}\left(d{\tilde{r}}^{2}+{\tilde{r}}^{2}\left(d% \vartheta^{2}+\sin^{2}\vartheta\,d\varphi^{2}\right)\right)$$ (109) with $$n(\tilde{r})=\left(1+\frac{m}{2\tilde{r}}\right)^{3}\left(1-\frac{m}{2\tilde{r% }}\right)^{-1}.$$ (110) Hence, light propagation in the Schwarzschild metric can be mimicked by the index of refraction (110); see Figure 11. The index of refraction (110) is known since Weyl [432]. It was employed for calculating lightlike Schwarzschild geodesics, exactly or approximately, e.g., in [16, 296, 134, 258]. This index of refraction can be modeled by a fluid flow [358]. The embeddability condition (75) is satisfied for $r>2.25m$ (which coincides with the so-called Buchdahl limit). On this domain the Fermat geometry, if restricted to the equatorial plane $\vartheta=\pi/2$, can be represented as a surface of revolution in Euclidean 3-space (see Figure 12). The entire region $r>2m$ can be isometrically embedded into a space of constant negative curvature [3]. Lensing by a Schwarzschild black hole. To get a Schwarzschild black hole, one joins at $r=2m$ the static Schwarzschild region $2m<r<\infty$ to the non-static Schwarzschild region $0<r<2m$ in such a way that ingoing light rays can cross this surface but outgoing cannot. If the observation event $p_{\mathrm{O}}$ is at $r_{\mathrm{O}}>2m$, only the region $r>2m$ is of relevance for lensing, because the past light cone of such an event does not intersect the black-hole horizon at $r=2m$. (For a Schwarzschild white hole see below.) Such a light cone is depicted in Figure 13 (cf. [236]). The picture was produced with the help of the representation (84) which requires integrating Equation (88) and Equation (89). For the Schwarzschild case, these are elliptical integrals. Their numerical evaluation is an exercise for students (see [65] for a MATHEMATICA program). Note that the evaluation of Equation (88) and Equation (89) requires knowledge of the turning points. In the Schwarzschild case, there is at most one turning point $r_{m}(\Theta)$ along each ray (see Figure 10), and it is given by the cubic equation $$r_{m}(\Theta)^{3}\,(r_{\mathrm{O}}-2m)-r_{m}(\Theta)\,r_{\mathrm{O}}^{3}\,\sin% ^{2}\Theta+2mr_{\mathrm{O}}^{3}\,\sin^{2}\Theta=0.$$ (111) The representation (84) in terms of Fermat arclength $\ell$ (= travel time) gives us the intersections of the light cone with hypersurfaces $t=\mbox{constant}$. These “instantaneous wave fronts” are depicted in Figure 14 (cf. [184]). With the light cone explicitly known, one can analytically verify that every inextendible timelike curve in the region $r>2m$ intersects the light cone infinitely many times, provided it is bounded away from the horizon and from (past lightlike) infinity. This shows that the observer sees infinitely many images of a light source with this worldline. The same result can be proven with the help of Morse theory (see Section 3.3), where one has to exclude the case that the worldline meets the caustic of the light cone. In the latter case the light source is seen as an Einstein ring. Note that a moving source might appear simultaneously as a point image and as an Einstein ring on the observer’s sky. For static light sources (i.e., $t$-lines), however, either all images are Einstein rings or none. For such light sources we can study lensing in the exact-lens-map formulation of Section 4.3 (see in particular Figure 6). Also, Section 4.3 provides us with formulas for distance measures, brightness, and image distortion which we just have to specialize to the Schwarzschild case. For another treatment of Schwarzschild lensing with the help of the exact lens map, see [150]. We place our static light sources at radius $r_{\mathrm{S}}$. If $r_{\mathrm{O}}<r_{\mathrm{S}}$ and $3m<r_{\mathrm{S}}$, only light rays with $\Theta<\delta$, $$\sin\delta:=\frac{R(3m)}{R(r_{\mathrm{O}})}=\sqrt{\frac{27m^{2}(r_{\mathrm{O}}% -2m)}{r_{\mathrm{O}}^{3}}},$$ (112) can reach the radius value $r_{\mathrm{S}}$ (see Figure 10). Rays with $\Theta=\delta$ asymptotically spiral towards the light sphere at $r=3m$. $\delta$ lies between 0 and $\pi/2$ for $r_{\mathrm{O}}<3m$ and between $\pi/2$ and $\pi$ for $r_{\mathrm{O}}>3m$. The escape cone defined by Equation (112) is depicted, for different values of $r_{\mathrm{O}}$, in Figure 15. It gives the domain of definition for the lens map. The lens map is graphically discussed in Figure 16. The pictures are valid for $r_{\mathrm{O}}=5m$ and $r_{\mathrm{S}}=10m$. Qualitatively, however, they look the same for all cases with $r_{\mathrm{S}}>r_{\mathrm{O}}$ and $r_{\mathrm{S}}>3m$. From the diagram one can read the position of the infinitely many images for each light source which, for the two light sources on the axis, degenerate into infinitely many Einstein rings. For each fixed source, the images are ordered by the number $i$ ($=0,1,2,3,\dots$) which counts how often the ray has crossed the axis. This coincides with ordering according to travel time. With increasing order $i$, the images come closer and closer to the rim at $\Theta=\delta$ (see Figure 16) and their brightness decreases rapidly (see Figure 18). For a light source not on the axis, images of even order are upright and line up on one side of the direction towards the center, images of odd order are side-inverted (see Figure 17) and line up on the other side of the direction towards the center (see Figure 16). These basic features of Schwarzschild lensing are known since pioneering papers by Darwin [95] and Atkinson [16] (cf. [265, 314, 254]). Various methods of how multiple imaging by a black hole could be discovered, directly or indirectly, have been discussed [265, 254, 21, 20, 342, 99]. Related work has also been done for Kerr black holes (see Section 5.8). An interesting suggestion was made in [204]. A Schwarzschild black hole, somewhere in the universe, would send photons originating from our Sun back to the vicinity of our Sun (“boomerang photons” [394]). If the black hole is sufficiently close to our Solar system, this would produce images of our own Sun on the sky that could be detectable. Quite generally, one speaks of retrolensing when a gravitating mass sends light back into approximately the same direction from which it has come in. Retrolensed images have not been observed so far, the perspectives are discussed, e.g., in [99, 125]. – The lensing effect of a Schwarzschild black hole has been visualized in two ways: 1. by showing the visual appearance of some background pattern as distorted by the black hole [89, 365, 300, 18, 290], also cf. [288, 289]. 2. by showing the visual appearence of an accretion disk around the black hole [265, 161, 21, 20], also cf. [62, 63, 67]. In the course of time the ray tracing programs on which these visualizations are based have become more and more advanced, taking not only redshift and magnification (including higher-order images) but also Fraunhofer diffraction (due to the finite aperture of the observer’s eye) or scattering into account. Ray tracing programs have also been developed for the more general case of the Kerr metric, see Section 5.8. – Interest in Schwarzschild lensing (and Kerr lensing) beyond the weak-field approximation has greatly increased with the growing evidence that there is a supermassive black hole at the center of our galaxy, and probably at the center of most galaxies. Higher-order images, where a light ray makes at least one full turn around the center, have not been observed so far, but they are thought to be relevant for future observations. It was already emphasized that, even if the bending angles are arbitrarily large, all lensing properties of a Schwarzschild black hole can be calculated exactly, in terms of elliptic integrals; then these integrals can be evaluated numerically with arbitrary accuracy and the results can be discussed graphically, as exemplified in Figures 16, 17 and 18. However, for practical purposes many authors found it convenient to develop approximation methods that go beyond, or are complementary to, the weak-field approximation, rather than to work with the exact formulas. Two approximation methods have proven particularly useful: Virbhadra and Ellis [420] developed a lens equation that applies to the case that source and observer are in the asymptotic region and approximately aligned with the center, but is not restricted to light rays that remain in the asymptotic region. Bozza et al. [53, 48] introduced a strong-field limit (or strong-deflection limit) that describes light rays the better the more turns they make around the center. Both methods, along with other approximation techniques [9, 10, 11, 230, 231, 232] for light bending in spherically-symmetric and static spacetimes, have already been discussed in Section 4.3 above. Especially for the Schwarzschild metric, Iyer and Petters [215] have demonstrated that, by combining a strong-deflection series expansion and a weak-deflection series expansion, one gets an approximation that is within $1\,$% of the exact bending angle value for light rays traversing anywhere between the photon sphere and infinity. A main goal of all these endeavours is to provide a new test of general relativity with the help of higher-order images, once they have been observed. As shown by Bozza [48], the separation of higher-order images and their decrease in magnitude can be used for discriminating between different black holes. Hence, the observation of higher-order images would reveal if the bending object can be modeled as a Schwarzschild black hole, or if an alternative model has to be used. An example for such an alternative model is the Reissner-Nordström black hole (see Section 5.3). Other spherically-symmetric and static black hole models have found some interest because their existence is predicted by alternative theories of gravity. E.g., the bending properties have been worked out for black holes from string theory [39], from braneworld gravity [224, 433, 122, 123, 41], from Einstein-Born-Infeld theory [124], from dilaton theory [286, 168] and from Hořava-Lifshitz gravity [76]. Up to now there is no observational indication that any of these black holes exist in nature. The future observation of higher-order images could help to find out if they exist. For the time being, all observations are in agreement with the assumption that the existing black holes are Schwarzschild or Kerr black holes, as predicted by standard general relativity. Schwarzschild lensing as a tool for probing the supermassive objects at the center of galaxies is discussed in detail by Virbhadra [419]. For a recent review on black hole lensing in general see Bozza [52]. Lensing by a non-transparent Schwarzschild star. To model a non-transparent star of radius $r_{*}$ one has to restrict the exterior Schwarzschild metric to the region $r>r_{*}$. Lightlike geodesics terminate when they arrive at $r=r_{*}$. The star’s radius cannot be smaller than $2m$ unless it is allowed to be time-dependent. The qualitative features of lensing depend on whether $r_{*}$ is bigger than $3m$. Stars with $2m<r_{*}\leq 3m$ are called ultracompact [214]. Their existence is speculative. The lensing properties of an ultracompact star are the same as that of a Schwarzschild black hole of the same mass, for observer and light source in the region $r>r_{*}$. In particular, the apparent angular radius $\delta$ on the observer’s sky of an ultracompact star is given by the escape cone of Figure 15. Also, an ultracompact star produces the same infinite sequence of images of each light source as a black hole. For $r_{*}>3m$, only finitely many of the images survive because the other lightlike geodesics are blocked. A non-transparent star has a finite focal length $r_{\mathrm{f}}>2m$ in the sense that parallel light from infinity is focused along a line that extends from radius value $r_{\mathrm{f}}$ to infinity. $r_{\mathrm{f}}$ depends on $m$ and on $r_{*}$. For the values of our Sun one finds $r_{\mathrm{f}}=550$ au (1 au = 1 astronomical unit = average distance from the Earth to the Sun). An observer at $r\geq r_{\mathrm{f}}$ can observe strong lensing effects of the Sun on distant light sources. The idea of sending a spacecraft to $r\geq r_{\mathrm{f}}$ was occasionally discussed in the literature [424, 301, 407]. The lensing properties of a non-transparent Schwarzschild star have been illustrated by showing the appearance of the star’s surface to a distant observer. For $r_{*}$ bigger than but of the same order of magnitude as $3m$, this has relevance for neutron stars (see [436, 324, 160, 355, 275, 307]). $r_{*}$ may be chosen time-dependent, e.g., to model a non-transparent collapsing star. A star starting with $r_{*}>2m$ cannot reach $r=2m$ in finite Schwarzschild coordinate time $t$ (though in finite proper time of an observer at the star’s surface), i.e., for a collapsing star one has $r_{*}(t)\to 2m$ for $t\to\infty$. To a distant observer, the total luminosity of a freely (geodesically) collapsing star is attenuated exponentially, $L(t)\propto\exp\left(-t(3\sqrt{3}m)^{-1}\right)$. This formula was first derived by Podurets [348] with an incorrect factor 2 under the exponent and corrected by Ames and Thorne [8]. Both papers are based on kinetic photon theory (Liouville’s equation). An alternative derivation of the luminosity formula, based on the optical scalars, was given by Dwivedi and Kantowski [110]. Ames and Thorne also calculated the spectral distribution of the radiation as a function of time and position on the apparent disk of the star. All these analyses considered radiation emitted at an angle $\leq\pi/2$ against the normal of the star as measured by a static (Killing) observer. Actually, one has to refer not to a static observer but to an observer comoving with the star’s surface. This modification was worked out by Lake and Roeder [251]. An interesting approximation formula was derived by Beloborodov [33]. He showed that a light ray that is emitted at radius $r_{\mathrm{S}}$ at an angle $\alpha$ with respect to the radial direction escapes to infinity at an angle $\psi$, approximately given by $1-\mathrm{cos}\,\alpha=\big{(}1-\mathrm{cos}\,\psi\big{)}\big{(}1-2m/r_{% \mathrm{S}}\big{)}$. As an application, he discusses the light bending of pulsars. Another approximation formula for the bending angle was found by Mutka and Mähöhnen [292, 293]. Lensing by a transparent Schwarzschild star. To model a transparent star of radius $r_{*}$ one has to join the exterior Schwarzschild metric at $r=r_{*}$ to an interior (e.g., perfect fluid) metric. Lightlike geodesics of the exterior Schwarzschild metric are to be joined to lightlike geodesics of the interior metric when they arrive at $r=r_{*}$. The radius $r_{*}$ of the star can be time-independent only if $r_{*}>2m$. For $2m<r_{*}\leq 3m$ (ultracompact star), the lensing properties for observer and light source in the region $r>r_{*}$ differ from the black hole case only by the possible occurrence of additional images, corresponding to light rays that pass through the star. Inside such a transparent ultracompact star, there is at least one stable photon sphere, in addition to the unstable one at $r=3m$ outside the star (cf. [191]). In principle, there may be arbitrarily many photon spheres [227]. For $r_{*}>3m$, the lensing properties depend on whether there are light rays trapped inside the star. For a perfect fluid with constant density, this is not the case; the resulting spacetime is then asymptotically simple, i.e., all inextendible light rays come from infinity and go to infinity. General results (see Section 3.4) imply that then the number of images must be finite and odd. The light cone in this exterior-plus-interior Schwarzschild spacetime is discussed in detail by Kling and Newman [236]. (In this paper the authors constantly refer to their interior metric as to a “dust” where obviously a perfect fluid with constant density is meant.) Effects on light rays issuing from the star’s interior have been discussed already earlier by Lawrence [257]. The “escape cones”, which are shown in Figure 15 for the exterior Schwarzschild metric have been calculated by Jaffe [216] for points inside the star. The focal length of a transparent star with constant density is smaller than that of a non-transparent star of the same mass and radius. For the mass and the radius of our Sun, one finds 30 au for the transparent case, in contrast to the above-mentioned 550 au for the non-transparent case [301]. Radiation from a spherically symmetric homogeneous dust star that collapses to a black hole is calculated in [379], using kinetic theory. A collapsing inhomogeneous spherically symmetric dust configuration may form a naked singularity. Its visual appearance, and other observable features, are discussed in [109, 103, 306, 295]. This analysis was generalized from the dust case to more general matter models in [169, 102]. Lensing by a Schwarzschild white hole. To get a Schwarzschild white hole one joins at $r=2m$ the static Schwarzschild region $2m<r<\infty$ to the non-static Schwarzschild region $0<r<2m$ at $r=2m$ in such a way that outgoing light rays can cross this surface but ingoing cannot. In analogy to the gravitational collapse of a spherically symmetric star into a black hole, one can consider the outburst of a white hole into a spherically symmetric star. The observable effects for an observer in the region $r>2m$ are discussed in [141, 298, 299, 107, 249, 250]. 5.2 Kottler spacetime The Kottler metric $$g=-\left(1-\frac{2m}{r}-\frac{\Lambda r^{2}}{3}\right)dt^{2}+\frac{dr^{2}}{1-% \frac{2m}{r}-\frac{\Lambda r^{2}}{3}}+r^{2}\left(d\vartheta^{2}+\sin^{2}% \vartheta\,d\varphi^{2}\right)$$ (113) is the unique spherically symmetric solution of Einstein’s vacuum field equation with a cosmological constant $\Lambda$. It has the form (69) with $$e^{2f(r)}=S(r)^{-1}=1-\frac{2m}{r}-\frac{\Lambda r^{2}}{3},\qquad R(r)=\frac{r% }{\sqrt{1-\frac{2m}{r}-\frac{\Lambda r^{2}}{3}}}.$$ (114) It is also known as the Schwarzschild–deSitter metric for $\Lambda>0$ and as the Schwarzschild–anti-deSitter metric for $\Lambda<0$. The Kottler metric was found independently by Kottler [239] and by Weyl [431]. For $\Lambda=0$, it reduces to the Schwarzschild metric (105). In the following we consider the Kottler metric with a constant $m>0$ and we ignore the region $r<0$ for which the singularity at $r=0$ is naked, for any value of $\Lambda$. For $\Lambda<0$, there is one horizon at a radius $r_{\mathrm{H}}$ with $0<r_{\mathrm{H}}<2m$; the staticity condition $e^{f(r)}>0$ is satisfied on the region $r_{\mathrm{H}}<r<\infty$. For $0<\Lambda<(3m)^{-2}$, there are two horizons at radii $r_{\mathrm{H1}}$ and $r_{\mathrm{H2}}$ with $2m<r_{\mathrm{H1}}<3m<r_{\mathrm{H2}}$; the staticity condition $e^{f(r)}>0$ is satisfied on the region $r_{\mathrm{H1}}<r<r_{\mathrm{H2}}$. For $\Lambda>(3m)^{-2}$ there is no horizon and no static region. At the horizon(s), the Kottler metric can be analytically extended into non-static regions. For $\Lambda<0$, the resulting global structure is similar to the Schwarzschild case. For $0<\Lambda<(3m)^{-2}$, the resulting global structure is more complex (see [248]). The extreme case $\Lambda=(3m)^{-2}$ is discussed in [347]. For any value of $\Lambda$, the Kottler metric has a light sphere at $r=3m$. Escape cones and embedding diagrams for the Fermat geometry (optical geometry) can be found in [391, 201] (cf. Figures 15 and 12 for the Schwarzschild case). Similarly to the Schwarzschild spacetime, the Kottler spacetime can be joined to an interior perfect-fluid metric with constant density. Embedding diagrams for the Fermat geometry (optical geometry) of the exterior-plus-interior spacetime can be found in [393]. For the optical appearance of a Kottler black hole see [18], and for the optical appearance of a Kottler white hole see [249]. The shape of infinitesimally thin light bundles in the Kottler spacetime is determined in [111]. In view of gravitational lensing, the Kottler metric is of particular interest because it can be used to answer the question of how the bending angle of light is affected by a cosmological constant. To that end one has to consider the orbits of light rays in the Kottler spacetime and to investigate how they differ from the orbits of light rays in the Schwarzschild spacetime. The first person who looked into this question was, surprisingly late, Islam in 1983 [212]. He found that the bending angle of light is not affected at all by a cosmological constant. His conclusion, which eventually turned out to be erroneous, was based on the (correct) observation that the differential equation for the orbits of light rays in the Kottler spacetime is exactly the same as in the Schwarzschild spacetime. To verify this, it suffices to insert the metric coefficients $S(r)$ and $R(r)$ from Equation (114) into Eq. (82). This results in the differential equation $$\frac{d^{2}r}{d\varphi}-\frac{2}{r}\left(\frac{dr}{d\varphi}\right)^{2}-r+3m=0$$ (115) which is, indeed, independent of $\Lambda$. Hence, the orbits of light rays in the Kottler spacetime are given by exactly the same coordinate equations as in the Schwarzschild spacetime. On the basis of this result, it was generally accepted for more than two decades that the gravitational bending of light is unaffected by a cosmological constant. (See, however, Lake [247] for an interesting caveat, as to the question of whether the constant $m$ has the same physical meaning in the Kottler case as in the Schwarzschild case.) Only in 2007 was it shown, in a paper by Rindler and Ishak [356], that this conclusion was incorrect. The fact that the coordinate expressions for the orbits of light rays in the Kottler spacetime are the same as in the Schwarzschild spacetime does not imply that the bending angles are the same. The reason is that physically measured angles differ from coordinate angles; the physically measured angles involve the metric, and the metric does depend on $\Lambda$. The analysis of Rindler and Ishak showed that, in contrast to earlier belief, a positive cosmological constant would have a diminishing effect on the bending angle. This is in perfect agreement with the intuitive idea that a positive cosmological constant has a repelling effect (i.e., that it tends to weaken the gravitational attraction). In terms of the Fermat metric (or optical metric), recall Equation (70), the Rindler-Ishak result can be rephrased in the following way [177]: The Fermat metric of the Kottler spacetime is projectively equivalent to the Fermat metric of the Schwarzschild spacetime (i.e., the unparametrized geodesics are the same), but not conformally equivalent (i.e., angles are different). Sereno supported (and slightly modified) the results of Rindler and Ishak in two papers. In the first one [375] he analyzed the influence of a cosmological constant on the bending of light in the weak deflection limit. He found that, in the case of a positive $\Lambda$, the radius of an Einstein ring decreases and, in a multiple imaging situation, the images are demagnified and the time delay increases. In the second paper [376] he demonstrated that the influence of a cosmological constant on the lens equation can be partly (but not completely) absorbed by an appropriate redefinition of the angular diameter distance. He argued that, for physical reasons, one should express all results in terms of angular diameter distances, rather than in terms of the radial Schwarzschild-like coordinate, as in the Rindler-Ishak paper. In the same paper, Sereno also calculated the influence of a cosmological constant on the redshifts in a multiple imaging situation. Further contributions to the subject were made by Schücker [369]. In contrast to Sereno, Schücker deliberately avoided any reference to a lens equation; instead, he concentrated on the difference between coordinate angles and physical angles. The Rindler-Ishak paper has caused a fairly large number of follow-up papers. Although some of them were critical, it sems fair to say that, by now, it is generally accepted that a positive cosmological constant has a diminishing effect on the bending angle of light. However, there is still a controversy about the question of whether this effect is actually observable, in realistic astrophysical situations. In order to answer this question, it is not sufficient to analyze the light bending in the Kottler metric, which describes the gravitational field around an isolated mass in a world with a cosmologal constant. It is rather necessary to take the influence of a cosmological background spacetime into account. This has been done by applying the Einstein-Straus method with a cosmological constant, i.e., by matching a Kottler vacuole at an outer boundary to a Robertson-Walker spacetime. Calculations of light bending in such a composed spacetime were undertaken by Ishak et al. [210], and then, e.g., by Schücker [370]. Whereas these papers come to the conclusion that the effect of $\Lambda$ on the light bending by some galaxy clusters could be observable, some other authors feel that this effect is negligibly small, see e.g. [382]. For a recent review article on the topic the reader may consult Ishak and Rindler [209]. 5.3 Reissner–Nordström spacetime The Reissner–Nordström metric $$g=-\left(1-\frac{2m}{r}+\frac{e^{2}}{r^{2}}\right)dt^{2}+\frac{dr^{2}}{1-\frac% {2m}{r}+\frac{e^{2}}{r^{2}}}+r^{2}\left(d\vartheta^{2}+\sin^{2}\vartheta\,d% \varphi^{2}\right)$$ (116) is the unique spherically symmetric and asymptotically flat solution of the Einstein–Maxwell equations. It has the form (69) with $$e^{2f(r)}=S(r)^{-1}=1-\frac{2m}{r}+\frac{e^{2}}{r^{2}},\qquad R(r)=\frac{r}{% \sqrt{1-\frac{2m}{r}+\frac{e^{2}}{r^{2}}}}.$$ (117) It describes the field around an isolated spherical object with mass $m$ and charge $e$. The Reissner–Nordström metric was found independently by Reissner [354], Weyl [430], and Nordström [309]. A fairly complete list of the pre-1979 literature on Reissner–Nordström geodesics can be found in [380]. A detailed account of Reissner–Nordström geodesics is given in [75]. (The Reissner–Nordström spacetime can be modified by introducing a cosmological constant. This generalized Reissner–Nordström spacetime, whose global structure is investigated in [256], will not be considered here.) We assume $m>0$ and ignore the region $r<0$ for which the singularity at $r=0$ is naked, for any value of $e$. Two cases are to be distinguished: 1. $0\leq e^{2}\leq m^{2}$; in this case the staticity condition $e^{f(r)}>0$ is satisfied on the regions $0<r<m-\sqrt{m^{2}-e^{2}}$ and $m+\sqrt{m^{2}-e^{2}}<r<\infty$, i.e., there are two horizons. 2. $m^{2}<e^{2}$; then the staticity condition $e^{f(r)}>0$ is satisfied on the entire region $0<r<\infty$, i.e., there is no horizon and the singularity at $r=0$ is naked. By switching to isotropic coordinates, one can describe light propagation in the Reissner–Nordström metric by an index of refraction (see, e.g., [135]). The resulting Fermat geometry (optical geometry) is discussed, in terms of embedding diagrams for the black-hole case and for the naked-singularity case, in [244, 3] (cf. [201]). The visual appearance of a background, as distorted by a Reissner–Nordström black hole, is calculated in [276]. Lensing by a charged neutron star, whose exterior is modeled by the Reissner–Nordström metric, is the subject of [91, 92]. The lensing properties of a Reissner–Nordström black hole are qualitatively (though not quantitatively) the same as that of a Schwarzschild black hole. The reason is the following. For a Reissner–Nordström black hole, the metric coefficient $R(r)$ has one local minimum and no other extremum between horizon and infinity, just as in the Schwarzschild case (recall Figure 10). The minimum of $R(r)$ indicates an unstable light sphere towards which light rays can spiral asymptotically, thereby defining the “shadow” of a Reissner-Nordström black hole. The existence of this minimum, and of no other extremum, was responsible for all qualitative features of Schwarzschild lensing. Correspondingly, Figures 16, 17, and 18 also qualitatively illustrate lensing by a Reissner–Nordström black hole. In particular, there is an infinite sequence of images for each light source, corresponding to an infinite sequence of light rays whose limit curve asymptotically spirals towards the light sphere. One can consider the “strong-field limit” [53, 48] of lensing for a Reissner–Nordström black hole, in analogy to the Schwarzschild case which is indicated by the asymptotic straight line in the middle graph of Figure 16. Bozza [48] investigates whether quantitative features of the “strong-field limit”, e.g., the slope of the asymptotic straight line, can be used to distinguish between different black holes. For the Reissner–Nordström black hole, image positions and magnifications have been calculated in [126], and travel times have been calculated in [255]. In both cases, the authors use the “almost exact lens map” of Virbhadra and Ellis [420] (recall Section 4.3) and analytical methods of Bozza et al. [53, 48, 56]. The question of whether the “shadow” of a Reissner-Nordström black hole can be observationally distinguished from that of a Schwarzschild black hole is discussed in [437]. 5.4 Morris–Thorne wormholes We consider a spacetime whose metric is of the form (69) with $e^{f(r)}S(r)=1$, i.e., $$g=-e^{2f(r)}dt^{2}+dr^{2}+e^{2f(r)}R(r)^{2}\left(d\vartheta^{2}+\sin^{2}% \vartheta\,d\varphi^{2}\right),$$ (118) where $r$ ranges from $-\infty$ to $\infty$. We call such a spacetime a Morris–Thorne wormhole (see [283]) if $$f(r)\underset{r\to\pm\infty}{\longrightarrow}0,\qquad r^{-2}R(r)^{2}\underset{% r\to\pm\infty}{\longrightarrow}1,$$ (119) such that the metric (118) is asymptotically flat for $r\to-\infty$ and for $r\to\infty$. A particular example of a Morris–Thorne wormhole is the Ellis wormhole [132] where $$f(r)=0,\qquad R(r)=\sqrt{r^{2}+a^{2}}$$ (120) with a constant $a$. The Ellis wormhole has an unstable light sphere at $r=0$, i.e., at the “neck” of the wormhole. It is easy to see that every Morris–Thorne wormhole must have an unstable light sphere at some radius between $r=-\infty$ and $r=\infty$. This has the consequence [191] that every Morris–Thorne wormhole produces an infinite sequence of images of an appropriately placed light source. This infinite sequence corresponds to infinitely many light rays whose limit curve asymptotically spirals towards the unstable light sphere. Lensing by the Ellis wormhole was discussed in [77]; in this paper the authors identified the region $r>0$ with the region $r<0$ and they developed a scattering formalism, assuming that observer and light source are in the asymptotic region. Lensing by the Ellis wormhole was also discussed in [339] in terms of the exact lens map. The resulting features are qualitatively very similar to the Schwarzschild case, with the radius values $r=-\infty$, $r=0$, $r=\infty$ in the wormhole case corresponding to the radius values $r=2m$, $r=3m$, $r=\infty$ in the Schwarzschild case. With this correspondence, Figures 16, 17, and 18 qualitatively illustrate lensing by the Ellis wormhole. More generally, the same qualitative features occur whenever the metric function $R(r)$ has one minimum and no other extrema, as in Figure 10. Lensing by the Ellis wormhole (and other types of wormholes) is also discussed in [297]. For a detailed discussion of lensing by Morris-Thorne wormholes, including visualizations, see [287, 288]. If observer and light source are on the same side of the wormhole’s neck, and if only light rays in the asymptotic region are considered, lensing by a wormhole can be studied in terms of the quasi-Newtonian approximation formalism [235]. However, as wormholes are typically associated with negative energy densities [283, 284], the usual assumption of the quasi-Newtonian approximation formalism that the mass density is positive cannot be maintained. This observation has raised some interest in lensing by negative masses, in particular in the question of whether negative masses can be detected by their (“microlensing”) effect on the energy flux from sources passing behind them. So far, related calculations [86, 361] have been done only in the quasi-Newtonian approximation formalism. 5.5 Barriola–Vilenkin monopole The Barriola–Vilenkin monopole [27] is given by the metric $$g=-dt^{2}+dr^{2}+k^{2}r^{2}(d\vartheta^{2}+\sin^{2}\vartheta\,d\varphi^{2}),$$ (121) with a constant $k<1$. There is a deficit solid angle and a singularity at $r=0$; the plane $t={\mbox{constant}}$, $\vartheta=\pi/2$ has the geometry of a cone. (Similarly, for $k>1$ one gets a surplus solid angle.) The Einstein tensor has non-vanishing components $G_{tt}=-G_{rr}=(1-k^{2})/r^{2}$. The metric (121) was briefly mentioned as an example for a conical singularity by Sokolov and Starobinsky [384]. Barriola and Vilenkin [27] realized that this metric can be used as a model for monopoles that might exist in the universe, resulting from breaking a global ${\cal O}(3)$ symmetry. They also discussed the question of whether such monopoles could be detected by their lensing properties which were characterized on the basis of some approximative assumptions (cf. [108]). However, such approximative assumptions are actually not necessary. The metric (121) has the nice property that the geodesics can be written explicitly in terms of elementary functions. This allows to write down explicit expressions for image positions and observables such as angular diameter distances, luminosity distances, image distortion, etc. (see [339]). Note that because of the deficit angle the metric (121) is not asymptotically flat in the usual sense. (It is “quasi-asymptotically flat” in the sense of [312].) For this reason, the “almost exact lens map” of Virbhadra and Ellis [420] (see Section 4.3), is not applicable to this case, at least not without modification. The metric (121) is closely related to the metric of a static string (see metric (139) with $a=0$). Restricting metric (121) to the hyperplane $\vartheta=\pi/2$ and restricting metric (139) with $a=0$ to the hyperplane $z=\mbox{constant}$ gives the same (2 + 1)-dimensional metric. Thus, studying light rays in the equatorial plane of a Barriola–Vilenkin monopole is the same as studying light rays in a plane perpendicular to a static string. Hence, the multiple imaging properties of a Barriola–Vilenkin monopole can be deduced from the detailed discussion of the string example in Section 5.10. In particular Figures 25 and 26 show the light cone of a non-transparent and of a transparent monopole if we interpret the missing spatial dimension as circular rather than linear. This makes an important difference. While in the string case the cone of Figures 25 has a 2-dimensional set of transverse self-intersection points, the corresponding cone for the monopole has a 1-dimensional radial caustic. The difference is difficult to visualize in spacetime pictures; it is therefore recommendable to use a purely spatial visualization in terms of instantaneous wave fronts (intersections of the light cone with hypersurfaces $t=\mbox{constant}$) (compare Figures 19 and 20 with Figures 28 and 29). 5.6 Janis–Newman–Winicour spacetime The Janis–Newman–Winicour metric [217] can be brought into the form [418] $$g=-\left(1-\frac{2m}{\gamma r}\right)^{\gamma}dt^{2}+\frac{dr^{2}}{\left(1-% \frac{2m}{\gamma r}\right)^{\gamma}}+\frac{r^{2}\left(d\vartheta^{2}+\sin^{2}% \vartheta\,d\varphi^{2}\right)}{\left(1-\frac{2m}{\gamma r}\right)^{\gamma-1}},$$ (122) where $m$ and $\gamma$ are constants. It is the most general spherically symmetric static and asymptotically flat solution of Einstein’s field equation coupled to a massless scalar field. For $\gamma=1$ it reduces to the Schwarzschild solution; in this case the scalar field vanishes. For $m>0$ and $\gamma\neq 1$, there is a naked curvature singularity at $r=2m/\gamma$. Lensing in this spacetime was studied in [422, 421, 417]. The main motivation was to find out whether the lensing characteristics of such a naked singularity can be distinguished from lensing by a Schwarschild black hole. The result is that the qualitative features of lensing remain similar to the Schwarzschild case as long as $1/2<\gamma<1$. However, if $\gamma$ drops below $1/2$, they become quite different. The reason is easily understood if we write Equation (122) in the form (69). The metric coefficient $$R(r)=r\left(1-\frac{2m}{\gamma r}\right)^{\frac{1}{2}-\gamma}$$ (123) has a minimum between the singularity and infinity as long as $\frac{1}{2}<\gamma<1$ (see Figure 21). This minimum indicates an unstable light sphere (recall Equation (71)), as in the Schwarzschild case at $r=3m$. All qualitative features of lensing carry over from the Schwarzschild case, i.e., Figures 16, 17, and 18 remain qualitatively unchanged. Clearly, the precise shape of the graph of $\Phi$ in Figure 16 changes if $\gamma$ is changed. The question of how the logarithmic asymptote (“strong-field limit”) depends on $\gamma$ is dicussed in [48]. If $\gamma$ drops below $1/2$, $R(r)$ has no longer an extremum, i.e., there is no light sphere. Owing to a general result proven in [191], this implies that only finitely many images are possible. In [421] naked singularities of spherically symmetric spacetimes are called weakly naked if they are surrounded by a light sphere (cf. [81]). In a nutshell, weakly naked singularities show the same qualitative lensing features as black holes. A generalization of this result to spacetimes without spherical symmetry has not been worked out so far. 5.7 Boson and fermion stars Spherically symmetric static solutions of Einstein’s field equation coupled to a scalar field may be interpreted as (uncharged, non-rotating) boson stars if they are free of singularities. Because of the latter condition, the Janis–Newman–Winicour metric (see Section 5.6) does not describe a boson star. The theoretical concept of boson stars goes back to [229, 359]. The analogous idea of a fermion star, with the scalar field replaced by a spin 1/2 (neutrino) field, is even older [270]. Until today there is no observational evidence for the existence of either a boson or a fermion star. However, they are considered, e.g., as hypothetical candidates for supermassive objects at the center of galaxies (see [371, 405] for the boson and [416, 406] for the fermion case). For the supermassive object at the center of our own galaxy, evidence points towards a black hole, but the possibility that it is a boson or fermion star cannot be completely excluded so far. Exact solutions that describe boson or fermion stars have been found only numerically (in 3 + 1 dimensions). For this reason there is no boson star model for which the lightlike geodesics could be studied analytically. Numerical studies of lensing have been carried out by Da̧browski and Schunck [93] for a transparent spherically symmetric static maximal boson star, and by Bilić, Nikolić, and Viollier [40] for a transparent spherically symmetric static maximal fermion star. For the case of a fermion-fermion star (two components) see [220]. In all three articles the authors use the “almost exact lens map” of Virbhadra and Ellis (see Section 4.3) which is valid for observer and light source in the asymptotic region and almost aligned. Da̧browski and Schunck [93] also discuss how the alignment assumption can be dropped. The lensing features found in [93] for the boson star and in [40] for the fermion star have several similarities. In both cases, there is a tangential caustic and a radial caustic (recall Figure 9 for terminology). A (point) source on the tangential caustic (i.e., on the axis of symmetry through the observer) is seen as a (1-dimenional) Einstein ring plus a (point) image in the center. If the (point) source is moved away from the axis the Einstein ring breaks into two (point) images, so there are three images altogether. Two of them merge and vanish if the radial caustic is crossed. So the qualitative lensing features are quite different from a Schwarzschild black hole with (theoretically) infinitely many images (see Section 5.1). The essential difference is that in the case of a boson or fermion star there are no circular lightlike geodesics towards which light rays could asymptotically spiral. 5.8 Kerr spacetime Next to the Schwarzschild spacetime, the Kerr spacetime is the physically most relevant example of a spacetime in which lensing can be studied explicitly in terms of the lightlike geodesics. The Kerr metric is given in Boyer–Lindquist coordinates $(r,\vartheta,\varphi,t)$ as $$g=\varrho(r,\vartheta)^{2}\left(\frac{dr^{2}}{\Delta(r)}+d\vartheta^{2}\right)% +(r^{2}+a^{2})\sin^{2}\vartheta\,d\varphi^{2}-dt^{2}+\frac{2mr}{\varrho(r,% \vartheta)^{2}}\left(a\sin^{2}\vartheta\,d\varphi-dt\right)^{2},$$ (124) where $\varrho$ and $\Delta$ are defined by $$\varrho(r,\vartheta)^{2}=r^{2}+a^{2}\cos^{2}\vartheta,\qquad\Delta(r)=r^{2}-2% mr+a^{2},$$ (125) and $m$ and $a$ are two real constants. We assume $0<a<m$, with the Schwarzschild case $a=0$ and the extreme Kerr case $a=m$ as limits. Then the Kerr metric describes a rotating uncharged black hole of mass $m$ and specific angular momentum $a$. (The case $a>m$, which describes a naked singularity, will be briefly considered at the end of this section.) The domain of outer communication is the region between the (outer) horizon at $$r_{+}=m+\sqrt{m^{2}-a^{2}}$$ (126) and $r=\infty$. It is joined to the region $r<r_{+}$ in such a way that past-oriented ingoing lightlike geodesics cannot cross the horizon. Thus, for lensing by a Kerr black hole only the domain of outer communication is of interest unless one wants to study the case of an observer who has fallen into the black hole. The lensing properties of a Kerr black hole will be reviewed below. For the effect of a Kerr black hole on the propagation of the polarization plane of light (cf. Section 2.5) see, e.g., [178, 395, 142, 211, 310, 374]. Historical notes. The Kerr metric was found by Kerr [234]. The coordinate representation (124) is due to Boyer and Lindquist [47]. The literature on lightlike (and timelike) geodesics of the Kerr metric is abundant (for an overview of the pre-1979 literature, see Sharp [380]). Detailed accounts on Kerr geodesics can be found in the books by Chandrasekhar [75] and O’Neill [316]. Fermat geometry. The Killing vector field $\partial_{t}$ is not timelike on that part of the domain of outer communication where $\varrho(r,\vartheta)^{2}\leq 2mr$. This region is known as the ergosphere. Thus, the general results of Section 4.2 on conformally stationary spacetimes apply only to the region outside the ergosphere. On this region, the Kerr metric is of the form (61), with redshift potential $$e^{2f(r,\vartheta)}=1-\frac{2mr}{\varrho(r,\vartheta)^{2}},$$ (127) Fermat metric $$\hat{g}=\frac{\varrho(r,\vartheta)^{4}}{\varrho(r,\vartheta)^{2}-2mr}\left(% \frac{dr^{2}}{\Delta(r)}+d\vartheta^{2}\right)+\frac{\varrho(r,\vartheta)^{4}% \,\Delta(r)\,\sin^{2}\vartheta\,d\varphi^{2}}{\left(\varrho(r,\vartheta)^{2}-2% mr\right)^{2}},$$ (128) and Fermat one-form $$\hat{\phi}=\frac{2mra\,\sin^{2}\vartheta\,d\varphi}{\varrho(r,\vartheta)^{2}-2% mr}.$$ (129) (Equation (128) corrects a misprint in [333], Equation (66), where a square is missing.) With the Fermat geometry at hand, the optical-mechanical analogy (Fermat’s principle versus Maupertuis’ principle) allows to write the equation for light rays in the form of Newtonian mechanics (cf. [7]). Certain embedding diagrams for the Fermat geometry (optical reference geometry) of the equatorial plane have been constructed [390, 201]. However, they are less instructive than in the static case (recall Figure 12) because they do not represent the light rays as geodesics of a Riemannian manifold. First integrals for lightlike geodesics. Carter [74] discovered that the geodesic equation in the Kerr metric admits another independent constant of motion $K$, in addition to the constants of motion $L$ and $E$ associated with the Killing vector fields $\partial_{\varphi}$ and $\partial_{t}$. This reduces the lightlike geodesic equation to the following first-order form: $$\displaystyle\varrho(r,\vartheta)^{2}\dot{t}$$ $$\displaystyle=$$ $$\displaystyle a\left(L-Ea\,\sin^{2}\vartheta\right)+\frac{(r^{2}+a^{2})\left((% r^{2}+a^{2})E-aL\right)}{\Delta(r)},$$ (130) $$\displaystyle\varrho(r,\vartheta)^{2}\dot{\varphi}$$ $$\displaystyle=$$ $$\displaystyle\frac{L-Ea\,\sin^{2}\vartheta}{\sin^{2}\vartheta}+\frac{(r^{2}+a^% {2})aE-a^{2}L}{\Delta(r)},$$ (131) $$\displaystyle\varrho(r,\vartheta)^{4}\dot{\vartheta}^{2}$$ $$\displaystyle=$$ $$\displaystyle K-\frac{(L-Ea\,\sin^{2}\vartheta)^{2}}{\sin^{2}\vartheta},$$ (132) $$\displaystyle\varrho(r,\vartheta)^{4}\dot{r}^{2}$$ $$\displaystyle=$$ $$\displaystyle-K\Delta(r)+\left((r^{2}+a^{2})E-aL\right)^{2}.$$ (133) Here an overdot denotes differentiation with respect to an affine parameter $s$. This set of equations allows writing the lightlike geodesics in terms of elliptic integrals [22]. Clearly, $\dot{\vartheta}$ and $\dot{r}$ may change sign along a ray; thus, the integration of Equation (132) and Equation (133) must be done piecewise. The determination of the turning points where $\dot{\vartheta}$ and $\dot{r}$ change sign is crucial for numerical evaluation of these integrals and, thus, for ray tracing in the Kerr spacetime (see, e.g., [411, 353, 138]). With the help of Equations (132, 133) one easily verifies the following important fact (cf. [192]). Through each point of the region $$\mathcal{K}\,:\,\left(2r\Delta(r)-(r-m)\,\varrho(r,\vartheta)^{2}\right)^{2}% \leq 4a^{2}r^{2}\Delta(r)\,\sin^{2}\vartheta$$ (134) there is spherical light ray, i.e., a light ray along which $r$ is constant (see Figure 22). These spherical light rays are unstable with respect to radial perturbations. For the spherical light ray at radius $r_{\mathrm{p}}$ the constants of motion $E$, $L$, and $K$ satisfy $$\displaystyle a\frac{L}{E}$$ $$\displaystyle=$$ $$\displaystyle r_{\mathrm{p}}^{2}+a^{2}-\frac{2r_{\mathrm{p}}\Delta(r_{\mathrm{% p}})}{r_{\mathrm{p}}-m},$$ (135) $$\displaystyle\frac{K}{E^{2}}$$ $$\displaystyle=$$ $$\displaystyle\frac{4r_{\mathrm{p}}^{2}\Delta(r_{\mathrm{p}})}{(r_{\mathrm{p}}-% m)^{2}}.$$ (136) The region $\mathcal{K}$ is the Kerr analogue of the “light sphere” $r=3m$ in the Schwarzschild spacetime. Light cone. With the help of Equations (130, 131, 132, 133), the past light cone of any observation event $p_{\mathrm{O}}$ can be explicitly written in terms of elliptic integrals. In this representation the light rays are labeled by the constants of motion $L/E$ and $K/E^{2}$. In accordance with the general idea of observational coordinates (4), it is desirable to relabel them by spherical coordinates $(\Psi,\Theta)$ on the observer’s celestial sphere. This requires choosing an orthonormal tetrad $(e_{0},e_{1},e_{2},e_{3})$ at $p_{\mathrm{O}}$. It is convenient to choose $e_{1}\propto\partial_{\vartheta}$, $e_{2}\propto\partial_{\varphi}$, $e_{3}\propto\partial_{r}$ and, thus, $e_{0}$ perpendicular to the hypersurface $t=\mbox{constant}$ (“zero-angular-momentum observer”). For an observation event in the equatorial plane, $\vartheta_{\mathrm{O}}=\pi/2$, at radius $r_{\mathrm{O}}$, one finds $$\displaystyle\frac{L}{E}$$ $$\displaystyle=$$ $$\displaystyle a+\frac{\left((r_{\mathrm{O}}(r_{\mathrm{O}}^{2}+a^{2})\,\sin% \Theta\,\sin\Psi-ar_{\mathrm{O}}\sqrt{\Delta(r_{\mathrm{O}})}\right)}{r_{% \mathrm{O}}\sqrt{\Delta(r_{\mathrm{O}})}+2ma\,\sin\Theta\,\sin\Psi},$$ (137) $$\displaystyle\frac{K}{E^{2}}$$ $$\displaystyle=$$ $$\displaystyle\frac{r_{\mathrm{O}}^{2}\left(r_{\mathrm{O}}^{2}+a^{2}-a\sqrt{% \Delta(r_{\mathrm{O}})}\,\sin\Theta\,\sin\Psi\right)^{2}-r_{\mathrm{O}}^{3}% \left(r_{\mathrm{O}}(r_{\mathrm{O}}^{2}+a^{2})+2ma^{2}\right)\cos^{2}\Theta}{% \left(r_{\mathrm{O}}\sqrt{\Delta(r_{\mathrm{O}})}+2ma\,\sin\Theta\,\sin\Psi% \right)^{2}}.$$ (138) As in the Schwarzschild case, some light rays from $p_{\mathrm{O}}$ go out to infinity and some go to the horizon. In the Schwarzschild case, the borderline between the two classes corresponds to light rays that asymptotically approach the light sphere at $r=3m$. In the Kerr case, it corresponds to light rays that asymptotically approach a spherical light ray in the region $\mathcal{K}$ of Figure 22. The constants of motion for such light rays are given by Equation (135, 136), with $r_{\mathrm{p}}$ varying between its extremal values $r_{+}^{\mathrm{ph}}$ and $r_{-}^{\mathrm{ph}}$ (see again Figure 22). Thereupon, Equation (137) and Equation (138) determine the celestial coordinates $\Psi$ and $\Theta$ of those light rays that approach a spherical light ray in $\mathcal{K}$. The resulting curve on the observer’s celestial sphere gives the apparent shape of the Kerr black hole (see Figure 23). For an observation event on the axis of rotation, $\sin\vartheta_{\mathrm{O}}=0$, the Kerr light cone is rotationally symmetric. The caustic consists of infinitely many spacelike curves, as in the Schwarzschild case. A light source passing through a point of the caustic is seen as an Einstein ring. For observation events not on the axis, the light cone has no rotational symmetry and the caustic structure is quite different from the Schwarzschild case. The caustic still consists of infinitely many connected subsets (a primary caustic and infinitely many higher-order caustics), but these are no longer spacelike curves. This fact is somewhat disguised if one restricts to light rays in the equatorial plane $\vartheta=\pi/2$ (which is possible, of course, only if the observation event is in the equatorial plane). Then the resulting 2-dimensional light cone looks indeed qualitatively similar to the Schwarzschild cone of Figure 13 (cf. [184]), where intersections of the light cone with hypersurfaces $t=\mbox{constant}$ are depicted. However, in the Kerr case the transverse self-intersection of this 2-dimensional light cone does not occur on an axis of symmetry. Therefore, the caustic of the full (3-dimensional) light cone is more involved than in the Schwarzschild case. The primary caustic turns out to be not a spatially straight line, as in the Schwarzschild case, but rather a tube, with astroid cross-section, that winds a certain number of times around the black hole; for $a\to m$ it approaches the horizon in an infinite spiral motion. The primary caustic of a Kerr light cone, with vertex in the equatorial plane far away from the black hole, was numerically calculated and depicted, for $a=m$, by Rauch and Blandford [353]. A detailed study of primary and higher-order caustics, for a Kerr light cone with vertex far away from the black hole but not necessarily in the equatorial plane, was presented by Bozza [51]. This work, which contains several pictures of Kerr caustics in 3+0 dimensions, is based on numerical calculations. The results are in good agreement with analytical approximation methods for studying the caustics. Two such methods exist which are complementary to each other in the sense that the first is valid for light rays that come close to a spherical light ray in the region $\mathcal{K}$ and the second is valid for light rays that stay far away from the black hole: The first method is due to Bozza, de Luca, Scarpetta and Sereno [55, 57] who analytically studied higher-order caustics in the strong deflection limit; this approach is not applicable to the primary caustic. The second method is due to Sereno and de Luca [377] who developed an analytic formula for the primary caustic that is valid up to fourth order in $m/b$ and $a/b$, where $b$ is the impact parameter. Taking all this together, a fairly clear picture of the caustics of Kerr light cones has now emerged. Also, attempts have been made to visualize the Kerr light cones in terms of their intersections with hypersurfaces $t=\mbox{constant}$, see Figure 1 in [182]. From the study of light cones one may switch to the study of arbitrary wave fronts. (For the definition of wave fronts see Section 2.2.) Pretorius and Israel [351] determined all axisymmetric wave fronts in the Kerr geometry. In this class, they investigated in particular those members that are asymptotic to Minkowski light cones at infinity (“quasi-spherical light cones”) and they found, rather surprisingly, that they are free of caustics. Special families of wave fronts in the Kerr spacetime are also considered, e.g., in [145, 194, 17]. Lensing by a Kerr black hole. For an observation event $p_{\mathrm{O}}$ and a light source with worldline $\gamma_{\mathrm{S}}$, both in the domain of outer communication of a Kerr black hole, several qualitative features of lensing are unchanged in comparison to the Schwarzschild case. If $\gamma_{\mathrm{S}}$ is past-inextendible, bounded away from the horizon and from (past lightlike) infinity, and does not meet the caustic of the past light cone of $p_{\mathrm{O}}$, the observer sees an infinite sequence of images; for this sequence, the travel time (e.g., in terms of the time coordinate $t$) goes to infinity. These statements have been proven in [192] with the help of Morse theory (cf. Section 3.3). On the observer’s sky the sequence of images approaches the apparent boundary of the black hole which is shown in Figure 23. This follows from the fact that • the infinite sequence of images must have an accumulation point on the observer’s sky, by compactness, and • the lightlike geodesic with this initial direction cannot go to infinity or to the horizon, by assumption on $\gamma_{\mathrm{S}}$. If $\gamma_{\mathrm{S}}$ meets the caustic of $p_{\mathrm{O}}$’s past light cone, the image is not an Einstein ring, unless $p_{\mathrm{O}}$ is on the axis of rotation. It has only an “infinitesimal” angular extension on the observer’s sky. As always when a point source meets the caustic, the ray-optical calculation gives an infinitely bright image. Numerical studies show that in the Kerr spacetime, where the caustic is a tube with astroid cross-section, the image is very bright whenever the light source is inside the tube [353]. In principle, with the lightlike geodesics given in terms of elliptic integrals, image positions on the observer’s sky can be calculated explicitly. This has been worked out for several special wordlines $\gamma_{\mathrm{S}}$. The case that $\gamma_{\mathrm{S}}$ is a circular timelike geodesic in the equatorial plane of the extreme Kerr metric, $a=m$, was treated by Cunningham and Bardeen [90, 23]. This example is of relevance in view of accretion disks. Viergutz [411] developed a formalism for the case that $\gamma_{\mathrm{S}}$ has constant $r$ and $\vartheta$ coordinates, i.e., for a light source that stays on a ring around the axis. One aim of this approach, which could easily be rewritten in terms of the exact lens map (recall Section 2.1), was to provide a basis for numerical studies. The case of a stationary light source (i.e., the case that $\gamma_{\mathrm{S}}$ is an integral curve of $\partial_{t}$) was investigated in great detail in a series of papers by Bozza, de Luca, Scarpetta and Sereno [49, 55, 54, 57]. In all these papers the authors derive analytic approximation formulas using the strong-deflection limit, i.e., the approximation is good for light rays that undergo a deflection of $\pi$ or more. Such light rays come close to one of the spherical light rays in the region $\mathcal{K}$, recall Figure 22. The first two papers in the series make the additional assumption that the light source and the observer are in the equatorial plane and that not only the observer but also the light source is far away from the black hole; in the last two papers these assumptions are relaxed. This series of papers gives a fairly complete analysis of Kerr lensing for stationary light sources under the strong deflection hypothesis. An alternative approach to Kerr lensing with stationary light sources, partly based on numerical results, was brought forward by Vazquez and Esteban [410]. All these articles also calculate the brightness of images. This requires determining the cross-section of infinitesimally thin bundles with a vertex, e.g., in terms of the shape parameters $D_{+}$ and $D_{-}$ (recall Figure 3). For a bundle around an arbitrary light ray in the Kerr metric, all relevant equations were worked out analytically by Pineault and Roeder [345]. However, the equations are much more involved than for the Schwarzschild case and will not be given here. Lensing by a Kerr black hole has been visualized (i) by showing the apparent distortion of a background pattern [346, 381] and (ii) by showing the visual appearence of an accretion disk [346, 350, 381, 31]. The main difference, in comparison to the Schwarzschild case, is in the loss of the left-right symmetry. In view of observations, Kerr black holes are considered as candidates for active galactic nuclei (AGN) since many years. In particular, the X-ray variability of AGN is interpreted as coming from a “hot spot” in an accretion disk that circles around a Kerr black hole. Starting with the pioneering work in [90, 23], many articles have been written on calculating the light curves and the spectrum of such “hot spots”, as seen by a distant observer (see, e.g., [98, 15, 225, 218, 138]). The spectrum can be calculated in terms of a transfer function that was tabulated, for some values of $a$, in [88] (cf. [411, 412]). A Kerr black hole is also considered as the most likely candidate for the supermassive object at the center of our own galaxy. (For background material see [136].) In this case, the predicted angular diameter of the black hole on our sky, in the sense of Figure 23, is about 30 microarcseconds; this is not too far from the reach of current VLBI technologies [137]. Also, the fact that the radio emission from our galactic center is linearly polarized gives a good motivation for calculating polarimetric images as produced by a Kerr black hole [64]. The calculation is based on the geometric-optics approximation according to which the polarization vector is parallel along the light ray. In the Kerr spacetime, this parallel-transport law can be explicitly written with the help of constants of motion [83, 345, 395] (cf. [75], p. 358). As to the large number of numerical codes that have been written for calculating imaging properties of a Kerr black hole the reader may consult [226, 411, 353, 138]. Notes on Kerr naked singularities. The Kerr metric with $a>m$ describes a naked singularity. Until now there is no observational indication that such objects exist in nature. The lightlike geodesics in a Kerr spacetime with $a>m$ have been studied in [68, 70] (cf. [75], p. 375). Observable effects of accretion disks around a Kerr naked singularity, in comparison to a Kerr black hole, were discussed in [400]. The “shadow” of a Kerr naked singularity was calculated in [100, 199] and, under different assumptions, in [19]. Notes on the Kerr–Newman spacetime. The Kerr–Newman spacetime (charged Kerr spacetime) is usually thought to be of little astrophysical relevance because the net charge of celestial bodies is small. For the lightlike geodesics in this spacetime the reader may consult [69, 71]. Embeddability diagrams of the equatorial plane of a Kerr-Newman spacetime can be found in [392]. The shadow of a Kerr-Newman black hole, and of a Kerr-Newman naked singularity, was discussed in [100, 399]. A Morse-theoretical analysis of lensing in the Kerr-Newman spacetime can be found in [192]. 5.9 Rotating disk of dust The stationary axisymmetric spacetime around a rigidly rotating disk of dust was first studied in terms of a numerical solution to Einstein’s field equation by Bardeen and Wagoner [24, 25]. The exact solution was found much later by Neugebauer and Meinel [303]. It is discussed, e.g., in [302]. The metric cannot be written in terms of elementary functions because it involves the solution to an ultraelliptic integral equation. It depends on a parameter $\mu$ which varies between zero and $\mu_{\mathrm{c}}=4.62966\dots$. For small $\mu$ one gets the Newtonian approximation, for $\mu\to\mu_{\mathrm{c}}$ the extreme Kerr metric ($a=m$) is approached. The lightlike geodesics in this spacetime have been studied numerically and the appearance of the disk to a distant observer has been visualized [429]. It would be desirable to support these numerical results with exact statements. From the known properties of the metric, only a few qualitative lensing features of the disk can be deduced. As Minkowski spacetime is approached for $\mu\to 0$, the spacetime must be asymptotically simple and empty as long as $\mu$ is sufficiently small. (This is true, of course, only if the disk is treated as transparent.) The general results of Section 3.4 imply that in this case the gravitational field of the disk produces finitely many images of each light source, and that the number of images is odd, provided that the worldline of the light source is past-inextendible and does not go out to past lightlike infinity. For larger values of $\mu$, this is no longer true. For $\mu>0.5$ there are two counter-rotating circular lightlike geodesics in the equatorial plane, a stable one at a radius $\tilde{\rho}_{1}$ inside the disk and an unstable one at a radius $\tilde{\rho}_{2}$ outside the disk. (This follows from [13] where it is shown that for $\mu>0.5$ timelike counter-rotating circular geodesics do not exist in a radius interval $[\tilde{\rho}_{1},\tilde{\rho}_{2}]$. The boundary values of this interval give the radii of lightlike circular geodesics.) The existence of circular light rays has the consequence that the number of images must be infinite; this is obviously true if light source and observer are exactly on the spatial track of such a circular light ray and, by continuity, also in a neighborhood. For a better understanding of lensing by the disk of dust it is desirable to investigate, for each value of $\mu$ and each event $p_{\mathrm{O}}$: Which past-oriented lightlike geodesics that issue from $p_{\mathrm{O}}$ go out to infinity and which are trapped? Also, it is desirable to study the light cones and their caustics. 5.10 Straight spinning string Cosmic strings (and other topological defects) are expected to exist in the universe, resulting from a phase transition in the early universe (see, e.g., [415] for a detailed account). So far, there is no direct observational evidence for the existence of strings. In principle, they could be detected by their lensing effect. The general perspective is discussed in [207, 164, 282]. The object CSL-1, which consists of a pair of galaxies, was discussed as a candidate for lensing by a string for some time [364]. However, more recent observations by the Hubble Space Telescope led to the conclusion that it is not a lensed image [363]. Basic lensing features for various string configurations are briefly summarized in [12]. Here we consider the simple case of a straight string that is isolated from all other masses. This is one of the most attractive examples for investigating lensing from the spacetime perspective without approximations. In particular, studying the light cones in this metric is an instructive exercise. The geodesic equation is completely integrable, and the geodesics can even be written explicitly in terms of elementary functions. We consider the spacetime metric $$g=-(dt-a\,d\varphi)^{2}+dz^{2}+d\rho^{2}+k^{2}\rho^{2}\,d{\varphi}^{2},$$ (139) with constants $a$ and $k>0$. As usual, the azimuthal coordinate $\varphi$ is defined modulo $2\pi$. For $a=0$ and $k=1$, metric (139) is the Minkowski metric in cylindrical coordinates. For any other values of $a$ and $k$, the metric is still (locally) flat but not globally isometric to Minkowski spacetime; there is a singularity along the $z$-axis. For $a=0$ and $0<k<1$, the plane $t=\mbox{constant}$, $z=\mbox{constant}$ has the geometry of a cone with a deficit angle $$\delta=(1-k)2\pi.$$ (140) (see Figure 24); for $k>1$ there is a surplus angle. Note that restricting the metric (139) with $a=0$ to the hyperplane $z=\mbox{constant}$ gives the same result as restricting the metric (121) of the Barriola–Vilenkin monopole to the hyperplane $\vartheta=\pi/2$. The metric (139) describes the spacetime around a straight spinning string. The constant $k$ is related to the string’s mass-per-length $\mu$, in Planck units, via $$k=1-4\mu,$$ (141) whereas the constant $a$ is a measure for the string’s spin. Equation (141) shows that we have to restrict to the deficit-angle case $k<1$ to have $\mu$ positive. One may treat the string as a line singularity, i.e., consider the metric (139) for all $\rho>0$. (This “wire approximation”, where the energy-momentum tensor of the string is concentrated on a 2-dimensional submanifold, is mathematically delicate; see [167].) For a string of finite radius $\rho_{*}$ one has to match the metric (139) at $\rho=\rho_{*}$ to an interior solution, thereby getting a metric that is regular on all of $\mathbb{R}^{4}$. In view of lensing it is important to distinguish between a transparent string, where light rays are allowed to pass through the interior solution, and a non-transparent string, where light rays are blocked at the boundary of the string. Historical notes. With $a=0$, the metric (139) and its geodesics were first studied by Marder [267, 268]. He also discussed the matching to an interior solution, without, however, associating it with strings (which were no issue at that time). The same metric was investigated by Sokolov and Starobinsky [384] as an example for a conic singularity. Later Vilenkin [413, 414] showed that within the linearized Einstein theory the metric (139) with $a=0$ describes the spacetime outside a straight non-spinning string. Hiscock [200], Gott [180], and Linet [261] realized that the same is true in the full (non-linear) Einstein theory. Basic features of lensing by a non-spinning string were found by Vilenkin [414] and Gott [180]. The matching to an interior solution for a spinning string, $a\neq 0$, was worked out by Jensen and Soleng [219]. Already earlier, the restriction of the metric (139) with $a\neq 0$ to the hyperplane $z=0$ was studied as the spacetime of a spinning particle in 2 + 1 dimensions by Deser, Jackiw, and ’t Hooft [101]. The geodesics in this (2 + 1)-dimensional metric were first investigated by Clément [82] (cf. Krori, Goswami, and Das [245] for the (3 + 1)-dimensional case). For geodesics in string metrics one may also consult Galtsov and Masar [162]. The metric (139) can be generalized to the case of several parallel strings (see Letelier [259] for the non-spinning case, and Krori, Goswami, and Das [245] for the spinning case). Clarke, Ellis and Vickers [80] found obstructions against embedding a string model close to metric (139) into an almost-Robertson–Walker spacetime. This is a caveat, indicating that the lensing properties of “real” cosmic strings might be significantly different from the lensing properties of the metric (139). Redshift and Fermat geometry. The string metric (139) is stationary, so the results of Section 4.2 apply. Comparison of metric (139) with metric (61) shows that the redshift potential vanishes, $f=0$. Hence, observers on $t$-lines see each other without redshift. The Fermat metric $\hat{g}$ and Fermat one-form $\hat{\phi}$ read $$\displaystyle\hat{g}$$ $$\displaystyle=$$ $$\displaystyle dz^{2}+d\rho^{2}+k^{2}\rho^{2}\,d\varphi^{2},$$ (142) $$\displaystyle\hat{\phi}$$ $$\displaystyle=$$ $$\displaystyle-a\,d\varphi.$$ (143) As the Fermat one-form is closed, $d\hat{\phi}=0$, the spatial paths of light rays are the geodesics of the Fermat metric $\hat{g}$ (cf. Equation (64)), i.e., they are not affected by the spin of the string. $\hat{\phi}$ can be transformed to zero by changing from $t$ to the new time function $t-a\varphi$. Then the influence of the string’s spin on the travel time (62) vanishes as well. However, the new time function is not globally well-behaved (if $a\neq 0$), because $\varphi$ is either discontinuous or multi-valued on any region that contains a full circle around the $z$-axis. As a consequence, $\hat{\phi}$ can be transformed to zero on every region that does not contain a full circle around the $z$-axis, but not globally. This may be viewed as a gravitational analogue of the Aharonov–Bohm effect (cf. [385]). The Fermat metric (142) describes the product of a cone with the $z$-line. Its geodesics (spatial paths of light rays) are straight lines if we cut the cone open and flatten it out into a plane (see Figure 24). The metric of a cone is (locally) flat but not (globally) Euclidean. This gives rise to another analogue of the Aharonov–Bohm effect, to be distinguished from the one mentioned above, which was discussed, e.g., in [146, 38, 195]. Light cone. For the metric (139), the lightlike geodesics can be explicitly written in terms of elementary functions. One just has to apply the coordinate transformation $(t,\varphi)\longmapsto(t-a\varphi,k\varphi)$ to the lightlike geodesics in Minkowski spacetime. As indicated above, the new coordinates are not globally well-behaved on the entire spacetime. However, they can be chosen as continuous and single-valued functions of the affine parameter $s$ along all lightlike geodesics through some chosen event, with $\varphi$ taking values in $\mathbb{R}$. In this way we get the following representation of the lightlike geodesics that issue from the observation event $(\rho=\rho_{0},\varphi=0,z=0,t=0)$ into the past: $$\displaystyle\rho(s)$$ $$\displaystyle=$$ $$\displaystyle\sqrt{s^{2}\,\sin^{2}\Theta+2s\rho_{0}\,\sin\Theta\,\cos\Psi+\rho% _{0}^{2}},$$ (144) $$\displaystyle\mathrm{tan}\left(k\varphi(s)\right)$$ $$\displaystyle=$$ $$\displaystyle\frac{s\,\sin\Theta\,\sin\Psi}{\rho_{0}+s\,\sin\Theta\,\cos\Psi},$$ (145) $$\displaystyle z(s)$$ $$\displaystyle=$$ $$\displaystyle s\cos\Theta,$$ (146) $$\displaystyle t(s)$$ $$\displaystyle=$$ $$\displaystyle-s+a\varphi(s).$$ (147) The affine parameter $s$ coincides with $\hat{g}$-arclength $\ell$, and $(\Psi,\Theta)$ parametrize the observer’s celestial sphere, $$\frac{d}{ds}\left.\begin{pmatrix}\rho(s)\,\cos\varphi(s)\\ \rho(s)\,\sin\varphi(s)\\ z(s)\end{pmatrix}\right|_{s=0}\!\!\!\!\!=\begin{pmatrix}\cos\Psi\,\sin\Theta\\ \sin\Psi\,\sin\Theta\\ \cos\Theta\end{pmatrix}.$$ (148) Equations (144, 145, 146, 147) give us the light cone parametrized by $(s,\Theta,\Psi)$. The same equations determine the intersection of the light cone with any timelike hypersurface (source surface) and thereby the exact lens map in the sense of Frittelli and Newman [154] (recall Section 2.1). For $k=0.8$ and $a=0$, the light cone is depicted in Figure 25; intersections of the light cone with hypersurfaces $t=\mbox{constant}$ (“instantaneous wave fronts”) are shown in Figure 28. In both pictures we consider a non-transparent string of finite radius $\rho_{*}$, i.e., the light rays terminate if they meet the boundary of the string. Figures 26 and 29 show how the light cone is modified if the string is transparent. This requires matching the metric (139) to an interior solution which is everywhere regular and letting light rays pass through the interior. For the non-transparent string, the light cone cannot form a caustic, because the metric is flat. For the transparent string, light rays that pass through the interior of the string do form a caustic. The special form of the interior metric is not relevant. The caustic has the same features for all interior metrics that monotonously interpolate between a regular axis and the boundary of the string. Also, there is no qualitative change of the light cone for a spinning string as long as the spin $a$ is small. Large values of $a$, however, change the picture drastically. For $a^{2}>k^{2}\rho_{*}^{2}$, where $\rho_{*}$ is the radius of the string, the $\varphi$-lines become timelike on a neighborhood of the string. As the $\varphi$-lines are closed, this indicates causality violation. In this causality-violating region the hypersurfaces $t=\mbox{constant}$ are not everywhere spacelike and, in particular, not transverse to all lightlike geodesics. Thus, our notion of instantaneous wave fronts becomes pathological. Lensing by a non-transparent string. With the lightlike geodesics known in terms of elementary functions, positions and properties of images can be explicitly determined without approximation. We place the observation event at $\rho=\rho_{0}$, $\varphi=0$, $z=0$, $t=0$, and we consider a light source whose worldline is a $t$-line at $\rho=\rho_{\mathrm{S}}$, $\varphi=\varphi_{\mathrm{S}}$, $z=z_{\mathrm{S}}$ with $0\leq\varphi_{\mathrm{S}}\leq\pi$. From Equations (144, 145, 146) we find that the images are in one-to-one correspondence with integers $n$ such that $$\left|\varphi_{\mathrm{S}}+2n\pi\right|<\pi/k.$$ (149) They can be numbered by the winding number $n$ in the order $n=0,-1,1,-2,2,\dots$ The total number of images depends on $k$. Let $N_{1}(k)$ be the largest integer and $N_{2}(k)$ be the smallest integer such that $N_{1}(k)\leq 1/k<N_{2}(k)$. Of the two integers $N_{1}(k)$ and $N_{2}(k)$, denote the odd one by $N_{\mathrm{odd}}(k)$ and the even one by $N_{\mathrm{even}}(k)$. Then we find from Equation (149) $$\displaystyle 0\leq\varphi_{\mathrm{S}}<\left|N_{\mathrm{even}}-1/k\right|\pi:$$ $$\displaystyle~{}N_{\mathrm{odd}}(k)\mbox{ images},$$ (150) $$\displaystyle\left|N_{\mathrm{even}}-1/k\right|\pi<\varphi_{\mathrm{S}}\leq\pi:$$ $$\displaystyle~{}N_{\mathrm{even}}(k)\mbox{ images}.$$ (151) Thus, the number of images is even in a wedge-shaped region behind the string and odd everywhere else. If the light source approaches the boundary between the two regions, one image vanishes behind the string (see Figure 24 for the case $1\leq 1/k<2$). (If the non-transparent string has finite thickness, there is also a region with no image at all, in the “shadow” of the string.) The coordinates $(\Psi_{n},\Theta_{n})$ on the observer’s sky of an image with winding number $n$ and the affine parameter $s_{n}$ at which the light source is met can be determined from Equations (144, 145, 146). We just have to insert $\rho(s)=\rho_{\mathrm{S}}$, $\varphi(s)=\varphi_{\mathrm{S}}+2n\pi$, $z(s)=z_{\mathrm{S}}$ and to solve for $\mathrm{tan}\Psi=\mathrm{tan}\Psi_{n}$, $\mathrm{tan}\Theta=\mathrm{tan}\Theta_{n}$, $s=s_{n}$: $$\displaystyle\mathrm{tan}\Psi_{n}$$ $$\displaystyle=$$ $$\displaystyle\frac{\rho_{\mathrm{S}}\,\sin\left(k(\varphi_{\mathrm{S}}+2n\pi)% \right)}{\rho_{\mathrm{S}}\,\cos\left(k(\varphi_{\mathrm{S}}+2n\pi)\right)-% \rho_{0}},$$ (152) $$\displaystyle\mathrm{tan}\Theta_{n}$$ $$\displaystyle=$$ $$\displaystyle\frac{\sqrt{\rho_{\mathrm{S}}^{2}+\rho_{0}^{2}-2\rho_{\mathrm{S}}% \rho_{0}\,\cos\left(k(\varphi_{\mathrm{S}}+2n\pi)\right)}}{z_{\mathrm{S}}},$$ (153) $$\displaystyle s_{n}$$ $$\displaystyle=$$ $$\displaystyle\sqrt{z_{\mathrm{S}}^{2}+\rho_{\mathrm{S}}^{2}+\rho_{0}^{2}-2\rho% _{\mathrm{S}}\rho_{0}\,\cos\left(k(\varphi_{\mathrm{S}}+2n\pi)\right)}.$$ (154) The travel time follows from Equation (147): $$T_{n}=s_{n}-a(\varphi_{\mathrm{S}}+2n\pi).$$ (155) It is the only relevant quantity that depends on the string’s spin $a$. With the observer on a $t$-line, the affine parameter $s$ coincides with the area distance, $D_{\mathrm{area}}(s)=s$, because in the (locally) flat string spacetime the focusing equation (44) reduces to $\ddot{D}_{\mathrm{area}}=0$. For observer and light source on $t$-lines, the redshift vanishes, so $s$ also coincides with the luminosity distance, $D_{\mathrm{lum}}(s)=s$, owing to the general law (48). Hence, Equation (154) gives us the brightness of images (see Section 2.6 for the relevant formulas). The string metric produces no image distortion because the curvature tensor (and thus, the Weyl tensor) vanishes (recall Section 2.5). Realistic string models yield a mass density $\mu$ that is smaller than $10^{-4}$. So, by Equation (141), only the case $N_{\mathrm{odd}}(k)=1$ and $N_{\mathrm{even}}(k)=2$ is thought to be of astrophysical relevance. In that case we have a single-imaging region, $0\leq\varphi_{\mathrm{S}}<2\pi-\pi/k$, and a double-imaging region, $2\pi-\pi/k<\varphi_{\mathrm{S}}\leq\pi$ (see Figure 24). The occurrence of double-imaging and of single imaging can also be read from Figure 25. In the double-imaging region we have a (“primary”) image with $n=0$ and a (“secondary”) image with $n=-1$. From Equations (153, 154) we read that the two images have different latitudes and different brightnesses. However, for $k$ close to 1 the difference is small. If we express $k$ by Equation (140) and linearize Equations (152, 153, 154, 155) with respect to the deficit angle (140), we find $$\displaystyle\Psi_{0}$$ $$\displaystyle=$$ $$\displaystyle\frac{\rho_{0}\pi-\rho_{\mathrm{S}}\varphi_{\mathrm{S}}}{\rho_{% \mathrm{S}}+\rho_{0}}-\frac{\varphi_{\mathrm{S}}\rho_{\mathrm{S}}\delta}{(\rho% _{\mathrm{S}}+\rho_{0})2\pi}$$ (156) $$\displaystyle\Psi_{-1}$$ $$\displaystyle=$$ $$\displaystyle\Psi_{0}+\frac{\rho_{\mathrm{S}}\delta}{\rho_{\mathrm{S}}+\rho_{0% }},$$ (157) $$\displaystyle\Theta_{-1}-\Theta_{0}$$ $$\displaystyle=$$ $$\displaystyle 0,$$ (158) $$\displaystyle s_{-1}-s_{0}$$ $$\displaystyle=$$ $$\displaystyle 0,$$ (159) $$\displaystyle T_{-1}-T_{0}$$ $$\displaystyle=$$ $$\displaystyle 2a\pi.$$ (160) Hence, in this approximation the two images have the same $\Theta-$coordinate; their angular distance $\Delta$ on the sky is given by Vilenkin’s [414] formula $$\Delta=\frac{\rho_{\mathrm{S}}\delta\,\sin\Theta_{0}}{\rho_{\mathrm{S}}+\rho_{% 0}},$$ (161) and is thus independent of $\varphi_{\mathrm{S}}$; they have equal brightness and their time delay is given by the string’s spin $a$ via Equation (160). All these results apply to the case that the worldlines of the observer and of the light source are $t$-lines. Otherwise redshift factors must be added. Lensing by a transparent string. In comparison to a non-transparent string, a transparent string produces additional images. These additional images correspond to light rays that pass through the string. We consider the case $a=0$ and $1<1/k<2$, which is illustrated by Figures 25 and 26. The general features do not depend on the form of the interior metric, as long as it monotonously interpolates between a regular axis and the boundary of the string. In the non-transparent case, there is a single-imaging region and a double-imaging region. In the transparent case, the double-imaging region becomes a triple-imaging region. The additional image corresponds to a light ray that passes through the interior of the string and then smoothly slips over one of the cusp ridges. The point where this light ray meets the worldline of the light source is on the sheet of the light cone between the two cusp ridges in Figure 26, i.e., on the sheet that does not exist in the non-transparent case of Figure 25. From the picture it is obvious that the additional image shows the light source at a younger age than the other two images (so it is a “tertiary image”). A light source whose worldline meets the caustic of the observer’s past light cone is on the borderline between single-imaging and triple-imaging. In this case the tertiary image coincides with the secondary image and it is particularly bright (even infinitely bright according to the ray-optical treatment; recall Section 2.6). Under a small perturbation of the worldline the bright image either splits into two or vanishes, so one is left either with three images or with one image. 5.11 Plane gravitational waves A plane gravitational wave is a spacetime with metric $$g=-2\,du\,dv-\left(f(u)(x^{2}-y^{2})+2g(u)xy\right)du^{2}+dx^{2}+dy^{2},$$ (162) where $f(u)^{2}+g(u)^{2}$ is not identically zero. For any choice of $f(u)$ and $g(u)$, the metric (162) has vanishing Ricci tensor, i.e., Einstein’s vacuum field equation is satisfied. The lightlike vector field $\partial_{v}$ is covariantly constant. Non-flat spacetimes with a covariantly constant lightlike vector field are called plane-fronted waves with parallel rays or pp-waves for short. They made their first appearance in a purely mathematical study by Brinkmann [61]. In spite of their high idealization, plane gravitational waves are interesting mathematical models for studying the lensing effect of gravitational waves. In particular, the focusing effect of plane gravitational waves on light rays can be studied quite explicitly, without any weak-field or small-angle approximations. This focusing effect is reflected by an interesting light cone structure. The basic features with relevance to lensing can be summarized in the following way. If the profile functions $f$ and $g$ are differentiable, and the coordinates $(x,y,u,v)$ range over $\mathbb{R}^{4}$, the spacetime with the metric (162) is geodesically complete [118]. With the exception of the integral curves of $\partial_{v}$, all inextendible lightlike geodesics contain a pair of conjugate points. Let $q$ be the first conjugate point along a past-oriented lightlike geodesic from an observation event $p_{\mathrm{O}}$. Then the first caustic of the past light cone of $p_{\mathrm{O}}$ is a parabola through $q$. (It depends on the profile functions $f$ and $g$ whether or not there are more caustics, i.e., second, third, etc. conjugate points.) This parabola is completely contained in a hyperplane $u=\mbox{constant}$. All light rays through $p_{\mathrm{O}}$, with the exception of the integral curve of $\partial_{v}$, pass through this parabola. In other words, the entire sky of $p_{\mathrm{O}}$, with the exception of one point, is focused into a curve (see Figure 30). This astigmatic focusing effect of plane gravitational waves was discovered by Penrose [327] who worked out the details for “sufficiently weak sandwich waves”. (The name “sandwich wave” refers to the case that $f(u)$ and $g(u)$ are different from zero only in a finite interval $u_{1}<u<u_{2}$.) Full proofs of the above statements, for arbitrary profile functions $f$ and $g$, were given by Ehrlich and Emch [120, 121] (cf. [32], Chapter 13). The latter authors also demonstrate that plane gravitational wave spacetimes are causally continuous but not causally simple. This strengthens Penrose’s observation [327] that they are not globally hyperbolic. (For the hierarchy of causality notions see [32].) The generators of the light cone leave the boundary of the chronological past $I^{-}(p_{\mathrm{O}})$ when they reach the caustic. Thus, the above-mentioned parabola is also the cut locus of the past light cone. By the general results of Section 2.8, the occurrence of a cut locus implies that there is multiple imaging in the plane-wave spacetime. The number of images depends on the profile functions. We may choose the profile functions such that there is no second caustic. (The “sufficiently weak sandwich waves” considered by Penrose [327] are of this kind.) Then Figure 30 demonstrates that an appropriately placed worldline (close to the caustic) intersects the past light cone exactly twice, so there is double-imaging. Thus, the plane waves demonstrate that the number of images need not be odd, even in the case of a geodesically complete spacetime with trivial topology. The geodesic and causal structure of plane gravitational waves and, more generally, of pp-waves is also studied in [205, 72]. One often considers profile functions $f$ and $g$ with Dirac-delta-like singularities (“impulsive gravitational waves”). Then a mathematically rigorous treatment of the geodesic equation, and of the geodesic deviation equation, is delicate because it involves operations on distributions which are not obviously well-defined. For a detailed mathematical study of this situation see [387, 246]. Garfinkle [163] discovered an interesting example for a pp-wave which is singular on a 2-dimensional worldsheet. This exact solution of Einstein’s vacuum field equation can be interpreted as a wave that travels along a cosmic string. Lensing in this spacetime was numerically discussed by Vollick and Unruh [423]. The vast majority of work on lensing by gravitational waves is done in the weak-field approximation. 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Viscous lock-exchange in rectangular channels [ Univ. Pierre et Marie Curie-Paris6, Univ. Paris-Sud, CNRS. Lab FAST, Bat. 502, Rue du Belvedere, Campus Univ., Orsay, F-91405, France. (November 21, 2020) Abstract In a viscous lock-exchange gravity current, which describes the reciprocal exchange of two fluids of different densities in a horizontal channel, the front between two Newtonian fluids spreads as the square root of time. The resulting diffusion coefficient reflects the competition between the buoyancy driving effect and the viscous damping, and depends on the geometry of the channel. This lock-exchange diffusion coefficient has already been computed for a porous medium, a $2D$ Stokes flow between two parallel horizontal boundaries separated by a vertical height, $H$, and, recently, for a cylindrical tube. In the present paper, we calculate it, analytically, for a rectangular channel (horizontal thickness $b$, vertical height $H$) of any aspect ratio ($H/b$) and compare our results with experiments in horizontal rectangular channels for a wide range of aspect ratios ($1/10-10$). We also discuss the $2D$ Stokes-Darcy model for flows in Hele-Shaw cells and show that it leads to a rather good approximation, when an appropriate Brinkman correction is used. J. Martin et al. ] J\ls.\nsM\lsA\lsR\lsT\lsI\lsN, N\ls.\nsR\lsA\lsK\lsO\lsT\lsO\lsM\lsA\lsL\lsA\lsL\lsA\ls, L\ls.\nsT\lsA\lsL\lsO\lsN and D\ls.\nsS\lsA\lsL\lsI\lsN 1 Introduction The lock-exchange configuration refers to the release, under gravity, of the interface between two fluids of different densities, confined in the section of a horizontal channel. This physical process has prompted renewed interest, as a part of the carbon dioxide sequestration issues ([Neufeld & Huppert(2009)]). The top of Fig. 1 shows the initial lock-exchange situation of a so-called full-depth release. The two fluids, initially separated by a vertical barrier (the lock gate), fill the whole section of the tank. When the gate is withdrawn (bottom of Fig. 1), buoyancy drives the denser fluid along the bottom wall, while the lighter one flows in the opposite direction at the top of the channel. The so-called lock-exchange results in the elongation of the interface between the two fluids along the horizontal direction. Different regimes have been reported for the velocity and shape of the elongating interface. The slumping phase refers to the initial regime where inertia dominates over viscous forces, which typically applies for the case of salted and fresh water in a tank. In this regime, [Benjamin(1968)], and more recently [Shin et al.(2004)Shin, Dalziel & Linden] showed that in the presence of a small density contrast (i.e. in the Boussinesq approximation $\Delta\rho<<\rho$), the two opposite currents traveled at the same constant velocity. When the Boussinesq approximation does not apply, [Lowe et al.(2005)Lowe, Rottman & Linden], [Birman et al.(2005)Birman, Martin & Meiburg], [Cantero et al.(2007)Cantero, Lee, Balachandar & Garcia] and [Bonometti et al.(2008)Bonometti, Balachandar & Magnaudet] showed that the two opposite fronts did travel at constant, but with different velocities. However this interface elongation, proportional to the time, is slowed down at later stages, in the viscous phase, where dissipation prevails over inertia. In the latter regime, the interface elongates as $t^{\alpha}$, where the exponent $\alpha$, smaller than unity, may take different values depending on the geometry and the confinement of the flow ([Didden & Maxworthy(1982), Huppert(1982), Gratton & Minotti(1990), Cantero et al.(2007)Cantero, Lee, Balachandar & Garcia, Takagi & Huppert(2007), Hallez & Magnaudet(2009)]). In porous media, [Bear(1988)] and [Huppert & Woods(1995)] predicted an interface spreading proportionally to the square root of time that [Séon et al.(2007)Séon, Znaien, Salin, Hulin, Hinch & Perrin] observed in a horizontal cylindrical tube. Such a spreading can be quantified with a diffusion coefficient, which reflects the balance between the buoyancy driving and the viscous damping. This coefficient, which depends on the nature and the geometry of the flow, has been computed for a porous medium by [Huppert & Woods(1995)], for a $2D$ Stokes flow between two parallel horizontal boundaries separated by a vertical height, $H$, by [Hinch(2007)] and [Taghavi et al.(2009)Taghavi, Seon, Martinez & Frigaard], and for a cylindrical tube by [Séon et al.(2007)Séon, Znaien, Salin, Hulin, Hinch & Perrin]. However, to our knowledge, such a diffusion coefficient has not been derived for a rectangular channel (horizontal thickness $b$, vertical height $H$, Fig. 1), for which one expects to recover the porous medium regime for $b\ll H$, and, possibly, the Stokes flow regime for $b\gg H$. In order to gather the limiting cases in the same paper, we first recall the results for porous media and $2D$ Stokes flows, together with the tube case, for the sake of comparison. Then we compute, for a rectangular channel of aspect ratio, $H/b$, the dependence of the interface $h(x,t)$ and the corresponding viscous lock-exchange diffusion coefficient. We also test the so-called Stokes-Darcy $2D$ model to this lock-exchange configuration. Finally, we test and validate our theoretical results with experiments in horizontal rectangular channels for a wide range of aspect ratios ($1/10-10$). 2 Lock-exchange in different geometries Let us first recall the basic hypotheses on the viscous gravity currents, common to the different geometries, used for instance by [Huppert & Woods(1995)] or [Hinch(2007)]). As sketched in Fig. 1, the interface between the two fluids is assumed independent on the $y$ direction. Its distance from the bottom boundary of the vessel is denoted $h(x,t)$. This interface can be a pseudo-interface between two miscible fluids for which molecular diffusion can be neglected or between two immiscible fluids, provided that the interfacial tension can be neglected. The flow is assumed to be quasi-parallel to the horizontal $x$ axis. This is a key hypothesis. Neglecting accordingly the vertical component of the fluid velocity implies that the vertical pressure gradient follows the hydrostatic variations: $\partial p/\partial z=-\rho g$. This hypothesis is violated at short times, immediately after the opening of the gate, but should become valid at later stages, as soon as the interface has slumped over a distance larger than $H$, thus ensuring a small enough local slope $\partial h/\partial x$. Then, the pressures, $P_{+}$ and $P_{-}$, in the lower layer, $0<z<h(x,t)$, and in the upper one, $h(x,t)<z<H$, respectively, write $$P_{+}=p(x,t)-\rho_{+}gz\;\;;\;P_{-}=P_{+}+\Delta\rho g(z-h(x,t))$$ (1) where $p(x,t)$ denotes the pressure at the lower wall, $z=0$. The difference between the horizontal pressure gradients in the two fluids is therefore linked to the interface slope by: $$\frac{\partial P_{+}}{\partial x}-\frac{\partial P_{-}}{\partial x}=\Delta\rho g% \frac{\partial h}{\partial x}$$ (2) The time evolution of the interface, $h(x,t)$, is governed by the mass conservation of each fluid (see Fig. 1 for notations). For instance, for the heavier bottom layer we have: $$\frac{\partial h}{\partial t}+\frac{\partial q}{\partial x}=0$$ (3) where $q(x,t)=q_{+}(x,t)$ is the horizontal flux ($m^{2}/s$) of the denser fluid at the location $x$: $$q(x,t)=q_{+}(x,t)=\int_{0}^{h}\frac{1}{b}\int u_{x}(x,y,z,t)dydz$$ (4) with $u_{x}(x,y,z,t)$ the $x-$velocity component and $b$ the spanwise length (the $y$ integration is along this spanwise length). Moreover, in our configuration of uniform section along the horizontal axis $x$, $u_{x}(x,y,z,t)$ must also satisfy the no net flux condition: $$q_{+}(x,t)+q_{-}(x,t)=\int_{0}^{H}\frac{1}{b}\int u_{x}(x,y,z,t)dydz=0$$ (5) We will see in the following that, in the viscous regime of interest, the horizontal velocity component $u_{x}$, solution of either a Darcy or a Stokes equation, is proportional to the pressure gradient in each fluid layer. Such solutions, combined with eq. (2), eq. (4) and eq. (5) allow then to eliminate the pressure gradients and to derive an expression of the flux $q$, of the form: $$q=-Df(\frac{h}{H})\frac{\partial h}{\partial x}$$ (6) where $D$ writes $$D=\tau\frac{\Delta\rho g}{\eta}$$ (7) and where the constant $\tau$ (scaling with a volume) and the function $f$ depend on the geometry and the flow equation and $\eta$ is the dynamic viscosity. Using the expression (6) for the flux, eq. (3) admits a self-similar solution $h(\zeta)=H\,\psi(\zeta)$ with the similarity variable $\zeta=x/\sqrt{Dt}$, which obeys: $$-\zeta\frac{d\psi}{d\zeta}=2\,\frac{d}{d\zeta}(f(\psi)\frac{d\psi}{d\zeta})$$ (8) This equation may alternatively be rewritten, in terms of $\zeta(\psi)$: $$\zeta(\frac{d\zeta}{d\psi})^{2}-2\,f\frac{d^{2}\zeta}{d\psi^{2}}+2\frac{df}{d% \psi}(\frac{d\zeta}{d\psi})=0$$ (9) The solution of the above equations can be found analytically or numerically, depending on the complexity of the normalized flux function $f(\psi)$. In the following, we will first recall the case of porous media, treated by [Huppert & Woods(1995)] and the $2D$ Stokes flow, addressed by [Hinch(2007)] (unpublished) and [Taghavi et al.(2009)Taghavi, Seon, Martinez & Frigaard]. We note that the latter paper included the effects of the rheological properties of the fluids. However, in order to focus on the geometrical aspects, we will assume in the following that both fluids are Newtonian and have the same viscosity. 2.1 Lock-exchange in porous media For a homogeneous layer of porous medium of permeability $\kappa$ (see for instance [Huppert & Woods(1995)]), the flow in each fluid is given by Darcy’s law which relates the velocity in each phase to the local pressure gradient : $$u_{x\pm}=-\frac{\kappa}{\eta}\frac{\partial P_{\pm}}{\partial x}$$ (10) At a given location $x$, the velocity is then uniform in each layer, and the no net flux condition (eq. (5)) simply writes: $hu_{x+}+(H-h)u_{x-}=0$. The latter equation, combined with eq. (10) and eq. (2), leads to eq. (6), and thus (combined with eq. (3)) to eq. (8) with a diffusion coefficient and a flux function: $$\displaystyle D_{PM}=\kappa\,H\frac{\Delta\rho g}{\eta}$$ (11) $$\displaystyle f_{PM}(\psi)=\psi(1-\psi)$$ (12) The solution of eq. (8), in the similarity variable $\zeta=x/\sqrt{D_{PM}t}$, is then a linear profile ([Huppert & Woods(1995)]): $$\psi=h(x,t)/H=(1+\zeta)/2$$ (13) The so-obtained front profile in homogeneous porous media is displayed in Fig. 2 (straight line) together with the ones for rectangular cells (referred to in subsection 2.3). The leading ($\psi=0$, $\zeta=-1$) and trailing ($\psi=1$, $\zeta=1$) edges spread as $\sqrt{D_{PM}t}$. Therefore, the lock-exchange diffusion coefficient for porous media is $D_{PM}$. It should be noticed that [Bear(1988)] reported a numerical integration of eq. (8) indicating that the gravity current spreads as the square root of time. Note that the corresponding result for a Hele-Shaw cell, that is two parallel plates of height, $H$, separated by a tiny gap $b$ ($b\ll H$), is obtained using the permeability $\kappa=b^{2}/12$: $$D_{HS}=\frac{b^{2}H\Delta\rho g}{12\eta}$$ (14) 2.2 Lock-exchange for a $2D$ Stokes flow between two horizontal boundaries For a $2D$ Stokes flow between two horizontal parallel boundaries, separated by a height $H$ in the plane ($z-x$) (assuming invariance along the $y-$direction), the flow in each fluid is given by the Stokes equation: $$\eta\;\nabla^{2}u_{x\pm}(x,z)=\frac{\partial P_{\pm}}{\partial x}$$ (15) At a given location $x$, the velocity profile consists of two parabola profiles matching the no slip boundary conditions at the bottom and the top boundaries ($u_{x+}(x,0)=u_{x-}(x,H)=0$) and the continuity of the velocity ($u_{x-}(x,h)=u_{x+}(x,h)$) and of the shear stress ($\eta\;\partial u_{x-}(x,h)/\partial z=\eta\;\partial u_{x+}(x,h)/\partial z$) at the interface, $z=h$. Using the no net flux condition (eq. (5)) and eq. (2), we obtain eq. (6), which enables to rewrite eq. (3), in the form of eq. (8) or eq. (9), with: $$\displaystyle D=\frac{H^{3}\Delta\rho g}{3\eta}$$ (16) $$\displaystyle f(\psi)=\psi^{3}(1-\psi)^{3}$$ (17) Note that the polynomial development of the solution of eq. (9) around $\psi=0$ gives: $\zeta=-\zeta_{0}+2\psi^{3}/(3\,\zeta_{0})$. Thus, the location, $\zeta(\psi=0)=-\zeta_{0}$, of the leading edge of the interface is indeed constant in the similarity variable. Moreover, the development shows that the slope of the interface is vertical at the bottom wall ($\psi=0$). This is also the case at the upper wall ($\psi=1$), as the problem is symmetric with respect to the centre of the cell. We note that in the presence of such a vertical slope, our (horizontal) quasi-parallel flow assumption falls locally, but it is still valid upstream and downstream, where the slope of the interface remains small. The solution of eq. (9) can be found numerically using a shooting method similar to the one used by [Hinch(2007)] and [Taghavi et al.(2009)Taghavi, Seon, Martinez & Frigaard]. It was computed starting the integration of eq. (9) from $(\psi=0.5,\zeta=0)$ and matching the asymptotic development in the vicinity of $\psi=0$. From the so-obtained solution, one can deduce the spreading diffusion coefficient between the leading edge ($h=0$, $-\zeta_{0}=-0.1607$) and the trailing edge ($h=H$, $\zeta_{0}$) of the front, from $[0.5\,(x(h=0)-x(h=H))]^{2}=D\,t$, which gives $D_{2D}=D\,\zeta_{0}^{2}$, so that: $$D_{2D}=0.0086\,H^{3}\,\frac{\Delta\rho g}{\eta}$$ (18) This result is in agreement with the one found by [Hinch(2007)]. [Taghavi et al.(2009)Taghavi, Seon, Martinez & Frigaard] provide five $\psi(\zeta)$ plots in their Fig. 9, corresponding to different viscosity ratios and including our case. From that figure we may obtain a value of their similarity variable, $\eta_{0}\sim 0.09$, which is consistent with our finding $\zeta_{0}=0.1607$, when taking into account their definition of the similarity variable, $\zeta=\eta\,\sqrt{3}$. For completeness, $D_{2D}$ may be compared to the result for a cylindrical tube of diameter $d$ ([Séon et al.(2007)Séon, Znaien, Salin, Hulin, Hinch & Perrin]): $$D_{T}=0.0054\,d^{3}\,\frac{\Delta\rho g}{\eta}$$ (19) The above expression is indeed very close to the $2D$ result, with $d$ playing the role of $H$. 2.3 Lock-exchange in a rectangular cross-section channel This article aims to extend the computation of the lock-exchange diffusion coefficient to rectangular cells of arbitrary cross-sections $H\times b$ (see Fig. 1). In the following, the cross-section aspect ratio will be denoted $\Gamma=H/b$. As in the previous section, a quasi-parallel flow approximation is assumed (i.e. small interface slope) which leads to eq. (2) for the pressure gradient. We will also assume the invariance of the interface location along the gap direction $y$. This requires that the deformation of the interface, induced by the flow profile along the direction $y$, relaxes much more quickly than the gradient along $x$. The flow in each fluid obeys a $3D$ Stokes equation: $$\eta\;\nabla^{2}u_{x\pm}(x,y,z)=\frac{\partial P_{\pm}}{\partial x}$$ (20) In order to solve this equation, we follow the series decomposition in Fourier modes of the velocity field used by [Gondret et al.(1997)Gondret, Rakotomalala, Rabaud, Salin & Watzky]. This paper addressed the issue of the parallel flow of two fluids of different viscosities in a rectangular cell. This issue is very closed to ours, as it requires to solve the Poisson equation (eq. (20)), but with different viscosities and the same pressure gradient for both fluids in [Gondret et al.(1997)Gondret, Rakotomalala, Rabaud, Salin & Watzky]. The method used was to split the velocity into two terms, $u_{x\pm}(x,y,z)=u_{x\pm}^{*}(x,y)+u_{x\pm}^{**}(x,y,z)$. Here, the first term, $u_{x\pm}^{*}(x,y)=\frac{b^{2}}{8\eta}\frac{\partial P_{\pm}}{\partial x}[1-(% \frac{2y}{b})^{2}]$ is the Poiseuille-like unperturbed velocity far away from the interface. The second term satisfies the Laplace equation, $\nabla^{2}u_{x\pm}^{**}(x,y,z)=0$ and vanishes far away from the interface. Its expression in terms of a sum of Fourier modes leads to a velocity profile of the form: $$\displaystyle u_{x\pm}(x,y,z)=\frac{b^{2}}{8\eta}\frac{\partial P_{\pm}}{% \partial x}{\left\{1-(\frac{2y}{b})^{2}\right.}\\ \displaystyle\left.{+\sum_{n=1}^{\infty}\frac{32(-1)^{n}\,(a_{\pm n}\,e^{(2n-1% )\frac{\pi(z-H/2)}{b}}+b_{\pm n}\,e^{-(2n-1)\frac{\pi(z-H/2)}{b}})}{{\left(2\,% n-1\right)}^{3}\,{\pi}^{3}}\cos[(2n-1)\frac{\pi y}{b}]}\right\}$$ (21) in which the no slip boundary conditions at the two vertical walls ($u_{x\pm}(x,y=\pm b/2,z)=0$) have been taken into account. Each Fourier mode, ($(2n-1)\pi/b)$, involves two constants for each fluid, $a_{\pm n}$ and $b_{\pm n}$. These four constants are determined by using the no slip boundary conditions at the bottom and top of the cell ($u_{x+}(x,y,z=0)=u_{x-}(x,y,z=H)=0$) and the continuity of the velocity ($u_{x+}(x,y,z=h)=u_{x-}(x,y,z=h)$) and of the shear stress ($\eta\,\partial u_{x+}(x,y,z=h)/\partial z=\eta\,\partial u_{x-}(x,y,z=h)/\partial z$) at the interface. Combining the so-obtained expressions for the velocity with eq. (2) and the no net flux condition (eq. (5)), one obtains the horizontal flux of the heavy fluid (eq. (6)): $$q=-D_{HS}f_{\Gamma}(h/H)\frac{\partial\,h}{\partial\,x}$$ (22) with $$\displaystyle D_{HS}=\frac{b^{2}H\Delta\rho g}{12\eta}$$ (23) $$\displaystyle f_{\Gamma}(\psi)=\frac{\psi+\alpha_{\Gamma}(\psi)}{1-\gamma_{% \Gamma}}(1-\psi-\alpha_{\Gamma}(\psi)-\gamma_{\Gamma})-\delta_{\Gamma}(\psi)$$ (24) where $$\displaystyle\alpha_{\Gamma}(\psi)$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{\Gamma}\sum_{n=1}^{\infty}\frac{96\,\left(1+e^{\Gamma% \left(1-\psi\right)\,\left(2\,n-1\right)\,\pi}\right)\,\left(1-e^{\Gamma\;\psi% \,\left(2\,n-1\right)\,\pi}\right)}{\left(1+e^{\Gamma\,\left(2\,n-1\right)\,% \pi}\right)\,{\left(2\,n-1\right)}^{5}\,{\pi}^{5}}$$ (25) $$\displaystyle\delta_{\Gamma}(\psi)$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{\Gamma}\sum_{n=1}^{\infty}\frac{48\,{\left(1-e^{\Gamma% \left(1-\psi\right)\,\left(2\,n-1\right)\,\pi}\right)}^{2}\,{\left(1-e^{\Gamma% \;\psi\,\left(2\,n-1\right)\,\pi}\right)}^{2}}{\left(-1+e^{2\,\Gamma\,\left(2% \,n-1\right)\,\pi}\right)\,{\left(2\,n-1\right)}^{5}\,{\pi}^{5}}$$ (26) $$\displaystyle\gamma_{\Gamma}$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{\Gamma}\sum_{n=1}^{\infty}\frac{192\,\tanh\,(\frac{% \Gamma\,\left(2\,n-1\right)\,\pi}{2})}{{\left(2\,n-1\right)}^{5}\,{\pi}^{5}}$$ (27) Eq. (22) admits a self-similar solution, $h(\zeta)=H\,\psi(\zeta)$, with the similarity variable $\zeta=x/\sqrt{D_{HS}\,t}$, which obeys eq. (8) or eq. (9). As previously, it is easier to compute the solution $\zeta(\psi)$ of eq. (9) subject to the corresponding asymptotics, $\zeta=-\zeta_{0}+8\,\Gamma^{2}\,\psi^{3}/(3\,\zeta_{0})$ in the vicinity of the boundary, $\psi=0$. We solve this equation using the shooting method previously described and using Mathematica Software. The solutions $h(\zeta)$ are plotted in Fig. 2 for different values of the cell aspect ratio $\Gamma$. We notice that, in contrast with Darcy predictions (straight line in Fig. 2), but similarly to the case of the $2D$ Stokes flow, the profiles, $h(\zeta)$, exhibit vertical slopes at the edges of the cell. We note also that such vertical slopes were observed in the experiments by [Séon et al.(2007)Séon, Znaien, Salin, Hulin, Hinch & Perrin] and [Huppert & Woods(1995)]. When comparing their experiments in a Hele-Shaw cell with Darcy predictions, the latter authors reported that ”Some discrepancies develop near the leading edge of the current as a result of the increasing importance of the bottom friction at the nose” (Fig. 2 of [Huppert & Woods(1995)]). This mismatch will be addressed in the next section. According to the so-obtained profiles, stationary in the similarity variable $\zeta$, the leading and trailing edges of the front spread as the square root of time, and a lock-exchange diffusion coefficient, dependent on the cell aspect ratio, can be defined: $$D_{R}=\frac{b^{2}H\Delta\rho g}{12\eta}\,F(\frac{H}{b})=D_{HS}\,F(\frac{H}{b})$$ (28) Fig. 3 displays a log-log plot of the normalized rectangular cell lock-exchange diffusion coefficient, $D_{R}/D_{HS}=F(H/b)$, versus the aspect ratio $\Gamma=H/b$. At small aspect ratios, $\Gamma<1$, the diffusion coefficient falls on top of the full line of slope $2$, which corresponds to the $2D$ Stokes flow between boundaries distant of $H$ ($b\rightarrow\infty$, eq. (18)). At large aspect ratios, $\Gamma\rightarrow 100$, the diffusion coefficient approaches the dashed line, $D_{R}/D_{HS}=1$, obtained for the $2D$ homogeneous porous medium case (eq. (14)). We note that the latter case, which corresponds to a Hele-Shaw cell of infinite aspect ratio, overestimates the lock-exchange diffusion coefficient, by a relative amount of about $30\%$, for aspect ratios as large as $\Gamma=10-20$. 3 $2D$ Stokes-Darcy model for lock-exchange in a rectangular cross-section channel The above-mentioned failures of the $2D$ Darcy model at finite aspect ratios may come from the velocity slip condition at the bottom and top edges of the cell ($z=0$ and $z=H$, respectively). This non physical condition is indeed required by the use of Darcy equation for the flow, which neglects the momentum diffusion in the presence of velocity gradients, in the plane of the cell ($z-x$ plane). The momentum diffusion may however be taken into account in $2D$, through the so-called Stokes-Darcy equation (see [Bizon et al.(1997)Bizon, Werne, Predtechensky, Julien, McCormick, Swift & Swinney, Ruyer-Quil(2001), Martin et al.(2002a)Martin, Rakotomalala & Salin, Zeng et al.(2003)Zeng, Yortsos & Salin]), which is similar to the Darcy-Brinkman equation used in porous media (see [Brinkman(1947)]). This $2D$ model enables to handle discontinuities such as cell edges, gap heterogeneities and fluid interfaces ([Ruyer-Quil(2001), Martin et al.(2002a)Martin, Rakotomalala & Salin, Zeng et al.(2003)Zeng, Yortsos & Salin, Talon et al.(2003)Talon, Martin, Rakotomalala, Salin & Yortsos]) and was successfully applied in the study of Rayleigh-Taylor instability ([Martin et al.(2002a)Martin, Rakotomalala & Salin, Fernandez et al.(2002)Fernandez, Kurowski, Petitjeans & Meiburg, Graf et al.(2002)Graf, Meiburg & Härtel]), of dispersion in heterogeneous fractures ([Talon et al.(2003)Talon, Martin, Rakotomalala, Salin & Yortsos]) and of chemical reaction fronts ([Martin et al.(2002b)Martin, Rakotomalala, Salin & Böckmann]). Although our present case of interest can be handled with $3D$ Stokes calculations, it is of interest to test the applicability of the $2D$ Stokes-Darcy model to the case of deep and narrow cells. Indeed, such a $2D$ model, once validated, could be a useful tool to address the issue of more complicated cases, such as gravity currents in the presence of viscosity contrasts, or in fractures with aperture heterogeneities. In this model, the flow in the rectangular cell (Fig. 1) is assumed to be parallel to the plates ($\vec{u}(x,y,z)=(u_{x}(x,y,z),0,u_{z}(x,y,z)$) with a Poiseuille parabolic profile across the gap (the key assumption). Using the Stokes equation with this $y$ dependency, the gap-averaged fluid velocity $\vec{U}(x,z)=\frac{1}{b}\int_{-b/2}^{b/2}\vec{u}(x,y,z)dy$, follows a Stokes-Darcy (SD) equation which reads here for the horizontal component of the velocity: $$-\frac{12\eta}{b^{2}}U_{x\pm}(x,z)+\beta\;\eta\nabla^{2}U_{x\pm}(x,z)=\frac{% \partial P_{\pm}}{\partial x}$$ (29) The first term on the left hand side of eq. (29) and the pressure gradient correspond to the Darcy’s law (eq. (10)) with a permeability $\kappa=\frac{\displaystyle{b^{2}}}{\displaystyle{12}}$ for the Hele-Shaw cell as mentioned above (eq. (14)). The second term on the left hand side of eq. (29) is the Brinkman correction to the Darcy equation (see [Brinkman(1947)]), which involves an effective viscosity, $\beta\eta$. This effective viscosity may be taken equal to the one of the fluid ($\beta=1$) for the sake of simplicity (or to enable the matching with a $2D$ Stokes regime at $\Gamma\rightarrow 0$). However, [Zeng et al.(2003)Zeng, Yortsos & Salin] showed that in the Hele-Shaw cell regime (at large $\Gamma$), the effective viscosity was slightly higher, with $\beta=12/\pi^{2}\simeq 1.215$. At a given location $x$, integrating eq. (29) leads to the two velocity profiles matching the no slip boundary conditions at the bottom and the top boundaries ($u_{x+}(x,0)=u_{x-}(x,H)=0$) and the continuity of the velocity ($u_{x-}(x,h)=u_{x+}(x,h$)) and of the shear stress ($\beta\,\eta\,\partial u_{x-}(x,h)/\partial z=\beta\,\eta\,\partial u_{x+}(x,h)% /\partial z$) at the interface. Using the no net flux condition, $\int_{0}^{h}u_{x+}dz+\int_{h}^{H}u_{x-}dz=0$, and eq. (2), we obtain the horizontal flux (eq. (4)): $$q(x)=-D_{HS}\,f_{SD}(h/H)\,\frac{\partial\,h}{\partial\,x}$$ (30) where $D_{HS}$ was already given in eq. (14) and the reduced flux function is equal to: $$\displaystyle f_{SD}(\psi)=\frac{1}{4d(d-\tanh d)}\left\{2+4\,d^{2}\,(1-\psi)% \,\psi-d\,\frac{3\,\cosh(2\,d)+\cosh(2\,d\,(1-2\,\psi))}{\sinh(2\,d)}\right.\\ \displaystyle\left.+4\,d\,\frac{(1-\psi)\,\cosh(2\,d\,(1-\psi))+\psi\,\cosh(2% \,d\,\psi)}{\sinh(2\,d)}\,-2\,\frac{\cosh(d\,(1-2\,\psi))}{\cosh(d)}\right\}$$ (31) where $$d=\sqrt{\frac{H^{2}}{4\kappa\beta}}=\sqrt{\frac{3}{\beta}}\,\Gamma$$ (32) and $\kappa=b^{2}/12$. A comparison of the full $3D$ calculations for a rectangular channel of aspect ratio $\Gamma=H/b$ with this $2D$ approximation can be performed on the flux functions, $f_{\Gamma}(\psi)$ (eq. (24)) and $f_{SD}(\psi)$ (eq. (31)). These two flux functions are close to each other, within a few per cents. In order to address the comparison in the range of interest for the Hele-Shaw assumption, i.e. $\Gamma\gg 1$, let us analyze the limit $\Gamma\rightarrow\infty$ ($d\rightarrow\infty$), which gives $$f_{SD,\Gamma\rightarrow\infty}\simeq\psi(1-\psi)-(\frac{3}{4}-\psi(1-\psi))% \sqrt{\frac{\beta}{3}}\;\frac{b}{H}+O\left((\frac{b}{H})^{2}\right)$$ (33) for the Stokes-Darcy flux and $$f_{\Gamma,\Gamma\rightarrow\infty}\simeq\psi(1-\psi)-(\frac{3}{4}-\psi(1-\psi)% )\frac{186\,Zeta(5)}{\pi^{5}}\;\frac{b}{H}+O\left((\frac{b}{H})^{2}\right)$$ (34) for the full $3D$ rectangular cell flux (with $Zeta(5)=\sum_{1}^{\infty}n^{-5}=1.03693$, the value of the Riemann-Zeta function). We note that the leading term of both series corresponds to the expected porous media Darcy limit (eq. (12)) with a permeability $\kappa=b^{2}/12$. However, the next order term ($O(b/H)$) is not the same, unless one chooses for the factor $\beta$, $$\beta=3\;(\frac{186\,Zeta(5)}{\pi^{5}})^{2}\simeq 1.192$$ (35) which is very close to the value $12/\pi^{2}\simeq 1.215$ found by [Zeng et al.(2003)Zeng, Yortsos & Salin] and to the value $6/5$ proposed by [Ruyer-Quil(2001)]. The lock-exchange diffusion coefficient has been computed, with the same procedure as above, by integrating eq. (9), using $f_{SD}(\psi)$, from $\psi=0.5$, and matching the asymptotics, $\zeta=-\zeta_{0}+8\,d^{2}\,\psi^{3}/(9\,\zeta_{0})$ in the vicinity of the boundary, $\psi=0$. The so-obtained lock-exchange diffusion coefficients, calculated for two different values of $\beta$ ($\beta=1$ and $\beta=12/\pi^{2}$) are compared to the $3D$ calculations in Fig. 4. The data for both values of $\beta$ are indeed very close to the $3D$ data over the whole range of aspect ratios, $\Gamma=1-100$. The inset of Fig. 4 gives the percentage of error for the two values of $\beta$. We note that these data were obtained by the difference between values of accuracy of the order of a few $10^{-3}$, which results in the small dispersion observed in the inset of Fig. 4. As expected, the results at large $\Gamma$ (in the Hele-Shaw regime) are closer to the $3D$ full problem for $\beta=12/\pi^{2}$ than for $\beta=1$. We point out that, whereas the Brinkman term does bring a significant correction, the exact value of the Brinkman viscosity factor $\beta$ is however not crucial: For instance, for $\Gamma=10$, we obtained a diffusion coefficient $3.5\%$ smaller than the $3D$ value for $\beta=1$ and $0.1\%$ larger for $\beta=12/\pi^{2}$, to be compared to the $30\%$ of error if the cell was assumed to be of infinite aspect ratio (Hele-shaw limit) as in [Huppert & Woods(1995)]. In conclusion of this comparison, we have shown that the $2D$ Stokes-Darcy model for lock-exchange in a rectangular cell captures quite accurately the effect of the finiteness of the cross-section aspect ratio. By using the correct $\beta$ value, the error in the model is smaller than $5\%$ for aspect ratios larger than $\Gamma>1$. 4 Experiments In this section, we will present experimental measurements of the diffusion coefficient in Hele-Shaw cells of different aspect ratios and we will compare them with our computed values. We used borosilicate rectangular cells of height $H$ and thickness $b$ and typical length $30\,cm$ (Fig. 1). The rectangular cross-sections of the cells were (in $mm^{2}$): $2\times 6$, $2\times 12$, $2\times 20$, $3\times 3$, $3\times 9$, $3\times 30$, $4\times 6$, $4\times 10$, $6\times 6$. Each cell was used with one side or the other held vertically, leading to two aspect ratios per cell. With such values, we covered a wide range of aspect ratios, from $\Gamma=H/b=1/10$ to $10$. We used, as Newtonian miscible fluids, aqueous solutions of natrosol and calcium chlorite. The fluids had equal viscosities, which were fixed by the polymer concentration and measured with an accuracy of $1\%$. The fluid densities were adjusted by addition of salt and measured with an accuracy of $0.01\%$. The overall accuracy in $D_{HS}$ was typically $5\%$, when taking into account the above accuracies in viscosities and densities and the inherent temperature variations during the experiments. The viscosities and the densities of the fluids were chosen to satisfy two experimental requirements. The experiments must be fast enough in order to prevent any significant molecular mixing of the fluids and one should be able to put the two fluids in contact without mixing. The latter condition requires a rather large density contrast and large viscosities. With our cell sizes, a good compromise was obtained with a density contrast of about $1\%$ and typical viscosities in the range, $10-50\,mPa.s$, leading to a lock-exchange diffusion coefficient ranging from $10^{-3}cm^{2}/s$ to $1cm^{2}/s$. The typical Reynolds number, built with the gap of the cell of these experiments is smaller than $0.1$. For each experiment, the cell was first held with its axis $Ox$ vertical. The fluids were successively slowly injected, with the lighter fluid on top of the heavier. Then the cell was closed and put in the desired position, with its axis $Oz$ vertical in a few seconds. The development of the lock-exchange pseudo-interface was then recorded thanks to a video camera. Typical pictures (side view in the plane $z-x$) are given in Fig. 5 for cells of different aspect ratios. The horizontal axis is scaled with $\sqrt{D_{HS}\,t}$, so that one can see the decrease of $\zeta_{0}$ as $\Gamma$ decreases. With this representation using the self-similar variable $\zeta=x/\sqrt{D_{HS}\,t}$, the profiles are stationary. One may notice that the trailing edge is fuzzy. This can be attributed to the stick condition at the upper wall: The dark dense fluid does stay at the walls for a long time, in particular in the corners of the cross-section. The same phenomenon takes place at the bottom of the cell, but the presence of transparent light fluid has little effect on the turbidity of the heavy dark fluid, and is therefore not noticeable on the pictures. It is worth noting that the shape of the leading edge evolves from an edge at large aspect ratios $\Gamma$ to a more and more step-like shape as $\Gamma$ decreases. It should be noticed that for small aspect ratios, although it is rather difficult to take pictures, a top view of the cell reveals a mild spanwise dependency of the interface, but we do not observe the spanwise lobe-and-cleft instability reported by [Simpson(1972)]. For each experiment, the locations of the leading and trailing edges of the front were measured in time. Fig. 6 gives the variations of the square of the spreading distance versus time for five cells. It is worth noting that the dependency is almost linear: Therefore a linear fit provided the lock-exchange diffusion coefficient, with a typical accuracy of $20\%$. Fig. 7 (top) displays the so-obtained normalized lock-exchange diffusion coefficient as a function of $\Gamma$. One can see that the agreement with the $3D$ calculations over the two decades of our measurements is rather good. We note that for the large aspect ratio limit of the experiments (up to $\Gamma=10$), the Hele-Shaw cell limit is not reached, and would underestimate, by $30\%$, the lock-exchange diffusion coefficient. This result thus confirms that for such aspect ratios, one should either compute the full $3D$ Stokes equation or use the Stokes-Darcy model to obtain the correct behaviour. We also note that our calculation still holds for aspect ratios as small as $\Gamma=0.1$. This result is quite unexpected since for such aspect ratios, some spanwise dependency of the profile was observed, and the hypothesis of the interface surface, $h(x,y,z)$, invariant in the $y$ direction is certainly broken. The bottom of Fig. 7 displays the superimposition of the theoretical and the experimental interfaces between the fluids, for an aspect ratio $\Gamma=4$. The agreement between the two is rather good, thus validating our model. Such an agreement is rather surprising as our small slope assumption is violated at the edges of the gravity current. This agreement, already emphasized by [Huppert(1982)] and [Séon et al.(2007)Séon, Znaien, Salin, Hulin, Hinch & Perrin], is likely to be common to viscosity dominated gravity current without surface tension. 5 Conclusion The viscous lock-exchange diffusion coefficient reflects the competition between the buoyancy driving effect and the viscous damping, and depends on the geometry of the channel. We give the backbone to calculate this coefficient in different configurations: We recall its computation for a porous medium already found by [Huppert & Woods(1995)], and compute it for a $2D$ Stokes flow between two parallel horizontal boundaries separated by a vertical height, $H$. This result is in agreement with [Hinch(2007)] (unpublished) and in reasonable agreement with recent computations by [Taghavi et al.(2009)Taghavi, Seon, Martinez & Frigaard]. Using a quasi-parallel flow assumption, we have calculated the pseudo-interface profile between the two fluids and the diffusion coefficient of viscous lock-exchange gravity currents for a rectangular channel (horizontal thickness $b$, vertical height $H$) of any aspect ratio ($H/b$). This analysis provides a cross-over between the $2D$ Stokes flow between two parallel horizontal boundaries separated by a vertical height, $H$, and the Hele-Shaw cell limit (applying for $H/b>100$). Moreover, the shape of our profiles allows to account for the discrepancy observed at the nose of the gravity current in the experiments by [Huppert & Woods(1995)]. The agreement, obtained despite the failure of the lubrication assumption at the edges of the current, should deserve however further theoretical investigation. Our calculations of the diffusion coefficient and of the shape of the profile have also been convincingly compared to new experiments carried out in cells of various aspect ratios ($1/10-10$). We have also calculated the lock-exchange diffusion coefficient for the same rectangular cells, using the $2D$ Stokes-Darcy model. This model is shown to apply to aspect ratios $H/b>1$, provided that the appropriate Brinkman correction is used. Such a $2D$ model may be useful to describe gravity currents with a finite volume of release, with fluids of different viscosities, or in heterogeneous vertical fractures. 6 Acknowledgement This work was partly supported by CNES (No 793/CNES/00/8368), ESA (No AO-99-083), by Réseaux de Thématiques de Recherches Avancées ”Triangle de la physique”, by the Initial Training Network (ITN) ”Multiflow” and by French Research National Agency (ANR) through the ”Captage et Stockage du CO${}_{\mbox{2}}$” program (projet CO-LINER No ANR-08-PCO2-XXX). 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Momentum-Resolved Exciton Coupling and Valley Polarization Dynamics in Monolayer WS${}_{2}$ Alice Kunin    Sergey Chernov    Jin Bakalis Department of Chemistry, Stony Brook University, Stony Brook, New York 11794, USA.    Ziling Li    Shuyu Cheng Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA.    Zachary H. Withers Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794, USA.    Michael G. White Department of Chemistry, Stony Brook University, Stony Brook, New York 11794, USA. Chemistry Division, Brookhaven National Laboratory, Upton 11973 New York, USA.    Gerd Schönhense Johannes Gutenberg-Universität, Institut für Physik, D-55099 Mainz, Germany.    Xu Du Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794, USA.    Roland K. Kawakami Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA.    Thomas K. Allison thomas.allison@stonybrook.edu Department of Chemistry, Stony Brook University, Stony Brook, New York 11794, USA. Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794, USA. Abstract Coupling between exciton states across the Brillouin zone in monolayer transition metal dichalcogenides can lead to ultrafast valley depolarization. Using time- and angle-resolved photoemission, we present momentum- and energy-resolved measurements of exciton coupling in monolayer WS${}_{2}$. By comparing full 4D ($k_{x},k_{y},E,t$) data sets after both linearly and circularly polarized excitation, we are able to disentangle intervalley and intravalley exciton coupling dynamics. Recording in the exciton binding energy basis instead of excitation energy, we observe strong mixing between the B${}_{1s}$ exciton and A${}_{n>1}$ states. The photoelectron energy and momentum distributions observed from excitons populated via intervalley coupling (e.g. K${}^{-}$ $\rightarrow$ K${}^{+}$) indicate that the dominant valley depolarization mechanism conserves the exciton binding energy and center-of-mass momentum, consistent with intervalley Coulomb exchange. On longer timescales, exciton relaxation is accompanied by contraction of the momentum space distribution. Monolayer transition metal dichalcogenides (TMDs) have garnered significant interest in the last 10 years following the discovery of valley-selective circular dichroism in these novel, atomically thin, direct band gap semiconductors [1, 2, 3, 4, 5, 6, 7, 8]. Right ($\sigma^{+}$) and left ($\sigma^{-}$) circularly polarized light selectively excites interband transitions in the inequivalent K${}^{+}$ and K${}^{-}$ valleys, respectively, where the band extrema are located [9, 10]. Strong Coulomb forces and spin-orbit coupling in these materials yield two series of tightly bound exciton states of opposite spin character in each K valley, the A and B excitons, giving rise to the potential for long-lived, spin-valley locked excitons [11, 12, 13, 14, 15, 16]. These unique excitons provide a promising platform for novel optoelectronic and valleytronic device applications [17, 18, 19, 20, 21, 22]. In TMD monolayers, the same strong Coulomb forces that give exciton binding energies on the order of $\sim$0.5 eV [23, 24] can also give rise to substantial interactions between exciton states, both within the same valley (intra-valley coupling) and between different valleys (inter-valley coupling). In particular, the Coulomb exchange interaction couples bright excitons of opposite spin character, coupling A and B excitons within the same valley (A${}^{+}$ $\xleftrightarrow{}$ B${}^{+}$) or degenerate excitons in opposite valleys (A${}^{+}$ $\xleftrightarrow{}$ A${}^{-}$, B${}^{+}$ $\xleftrightarrow{}$ B${}^{-}$), as illustrated in Fig. 1a) [25, 26, 27, 28]. Due to this strong coupling, the exciton eigenstates are, in general, a combination of exciton states with mixed spin and valley characters [27, 26, 29, 30, 31, 32]. Optical excitation addresses only the bright states, in which the electron and hole occupy the same valley with small total momentum $\mathbf{Q}=\mathbf{k}_{e}-\mathbf{k}_{h}$ and have net spin zero. Photoexcitation thus creates a superposition of eigenstates which then rapidly evolves in time, leading effectively to relaxation of the initial excitation and valley depolarization. The strength of the eigenstate splitting due to Coulomb exchange, and thus its contribution to valley depolarization, is disputed among theoretical models [26, 33, 27, 34, 30, 35]. The additional role of exciton-phonon interactions in both intervalley and intravalley exciton dynamics is also non-negligible [36, 37, 38, 39]. Many optical spectroscopy techniques have been employed to investigate depolarization lifetimes in monolayer TMDs, including photoluminescence [5, 40, 41, 42, 43], differential transmission [44, 45], time-resolved Kerr and Faraday rotation [29, 46, 47, 48, 49, 50, 51, 52], and multidimensional spectroscopies [30, 53, 32, 54, 55], among others [56]. Valley polarization lifetimes ranging from a few picoseconds [41, 42, 45] to hundreds [44, 46, 48] or tens [54] of femtoseconds have been reported, depending on the system under study and the spectroscopy method. Interpreting this body of work has been the subject of considerable debate [57, 45, 58, 59, 60, 54]. Optical measurements record the excitation energy of the bright states (Fig. 1b)), rendering discernment of the role of dark states difficult. Critically, optical measurements are also momentum integrated, and can only distinguish between different excitons via the excitation energy and polarization selection rules. Recently, technological advancements in time- and angle-resolved photoemission spectroscopy (tr-ARPES) have enabled the technique to be applied to small monolayer TMD samples [62, 63, 64, 65, 66], providing direct momentum-space visualization of exciton wavefunctions as well as previously inaccessible dark states. In this article, we present comprehensive tr-ARPES measurements of the exciton dynamics in monolayer WS${}_{2}$ following excitation at 2.4 eV, the nominal B exciton resonance [61]. We measure full 4D ($k_{x},k_{y},E,t$) photoelectron distributions after both linearly polarized and circularly polarized photoexcitation. Resolving exciton binding energy (Fig. 1c)) instead of excitation energy, we observe previously unseen strong mixing between A${}_{n>1}$ and B${}_{\scriptsize\mbox{1s}}$ excitons in the initial photoexcited spectrum. With parallel momentum detection across the full Brillouin zone, we provide the first reported momentum-space visualizations of circular dichroism and ultrafast valley depolarization in the monolayer TMDs. We also observe that the exciton relaxation is accompanied by significant contraction of the initial exciton distribution in momentum space. These measurements report on the time-, energy-, and momentum-dependence of intervalley and intravalley exciton coupling, providing new insights on exciton formation in TMDs and the many-exciton coupled wave function. Our measurement scheme is shown in Fig. 1d). Linearly and circularly polarized pump pulses ($h\nu_{\scriptsize\mbox{pump}}=2.4$ eV) and $p$-polarized extreme ultraviolet (XUV) probe pulses ($h\nu_{\scriptsize\mbox{probe}}=$ 20$-$30 eV) with variable delay illuminate the sample and photoelectrons are collected by a custom time-of-flight momentum microscope [67, 68]. High data rates are enabled by conducting the experiment at 61 MHz repetition rate with XUV probe pulses produced via cavity-enhanced high-harmonic generation (CE-HHG). The laser system and HHG beamline have been previously described in detail in [69, 70, 71]. The sample is an exfoliated monolayer of WS${}_{2}$ stacked on an exfoliated buffer layer of hexagonal boron nitride on a silicon substrate. We use the spatial imaging capabilities of the momentum microscope [72] to isolate the photoelectron signal from the $\sim$10$\times$10 $\upmu$m${}^{2}$ monolayer region of interest of the sample. The valence band structure of the sample for a cut along the K-$\Gamma$-K axis of WS${}_{2}$ is shown in Fig. 1e)). The measured band structure shows that the valence band maximum (VBM) is located at the K${}^{+}$ and K${}^{-}$ valleys at the edges of the WS${}_{2}$ Brillouin zone, as expected for a monolayer sample. The energy resolution is broadened to approximately 160 meV due to sample inhomogeneity [62], but the spin-orbit splitting of the valence bands at the K valley is still clearly resolved. Additional sample characterization and experimental details can be found in the Supplemental Material [73]. All measurements are done at room temperature unless stated otherwise. The 2.4 eV pump pulses produce photoexcited signals at the K${}^{+}$ and K${}^{-}$ valleys (Fig. 1f)). In contrast to previous studies on monolayer WSe${}_{2}$/hBN and strongly pumped WS${}_{2}$ on bare silicon [62, 64], the signals we observe at the $\Sigma$ valleys are centered $\sim$100 meV higher than the K valley signal, and are much weaker in intensity than previously reported in WSe${}_{2}$ [62]. We find that the $\Sigma$/K intensity ratio depends strongly on the probe photon energy, but is always more than 2.5$\times$ smaller than that found in similar measurements of bulk WS${}_{2}$ [73, 74], where the $\Sigma$ valleys are lower in energy than the K valleys but the photoemission matrix elements are similar. Thus, we believe there is only minor involvement of excitons with electrons at $\Sigma$ and focus here on the K valley excitons. By varying the excitation fluence between 1.3 $\upmu$J/cm${}^{2}$ and 29 $\upmu$J/cm${}^{2}$, we find the tr-ARPES signals to be fluence independent below 5 $\upmu$J/cm${}^{2}$ [73]. Thus, all measurements reported here are conducted at 5 $\upmu$J/cm${}^{2}$ excitation fluence, corresponding to an excitation density of approximately 7 x 10${}^{11}$ carriers/cm${}^{2}$ at our pump energy [61]. The cross-correlation of the pump and probe pulses yields a Gaussian instrument response function with 200 $\pm$ 20 fs FWHM. Photoexcitation with linearly polarized light populates the K${}^{+}$ and K${}^{-}$ valleys equally and both valleys show the same dynamics. The time-resolved photoelectron spectrum recorded with s-polarized excitation is shown in Fig. 2a). No intensity is ever observed in the conduction band at $E_{\scriptsize\mbox{VBM}}+h\nu_{\scriptsize\mbox{pump}}=2.4$ eV, indicating the direct formation of bound excitons. Exciton signals appear below the conduction band due to the exciton binding energy [75, 76, 62], as illustrated in Fig. 1c) and the leftmost scales in Fig. 2. The most prominent feature at early pump-probe delays is the large intensity at energies between 2.05$-$2.3 eV above the VBM in the K valley. This corresponds to exciton binding energies compatible with excited A excitons (A${}_{n>1}$) [23, 24]. At longer delays, a lower energy feature centered at approximately 1.93 eV grows in and persists beyond the longest pump-probe delays recorded (25 ps). This lower energy feature appears at binding energies compatible with those expected for both the A${}_{1s}$ and B${}_{1s}$ excitons, which are expected to have similar binding energy [24, 77, 78]. Similar results are obtained with $p$-polarized excitation, indicating that excitation of spin-forbidden intravalley excitons by the out-of-plane component of the electric field has a negligible effect on the observed signals, as expected due to the much smaller transition dipole for these excitations [79, 80, 73]. The spectrum of Fig. 2a) consists of multiple overlapping components. To deconvolve the overlapping spectral and temporal components of the experimental data, we have applied global analysis (GA) [81, 82, 83, 84, 85], which reduces the signal to a few principal spectral components $S_{i}(E)$, each with simple exponential time dynamics $f_{i}(t)$ convolved with the instrument response, viz. $I(E,t)=\sum_{i}^{N}S_{i}(E)f_{i}(t)$. We find an excellent fit with only $N=2$ components as shown in Fig. 2b). Component 1 corresponds to the initially excited population and is peaked at $E-E_{\scriptsize\mbox{VBM}}=$ 2.15 eV but also shows a long tail to lower photoelectron energies (larger binding energies) covering the region of the B${}_{1s}$ exciton. We assign this to an initially excited mixture of A${}_{n>1}$ and B${}_{1s}$ excitons. Despite initial photoexcitation of the B exciton resonance, we clearly observe strong weighting towards lower binding energies consistent with population of the A${}_{n>1}$ excited states. This is seen both in the GA results and in the raw data, with both much more weighted towards the A${}_{n>1}$ states than the B exciton than what would be expected from the optical absorption spectrum [61]. This indicates very strong mixing of the B${}_{1s}$ states with A${}_{n>1}$ states, such that photoexcitation of what is nominally the B exciton resonance promptly populates A${}_{n>1}$ exciton states as well. Such A/B mixing due to intravalley Coulomb exchange has been discussed before [32], although the degree of mixing we observe here is much larger than suggested by this previous work. Component 1 decays with a time constant of 378 $\pm$ 40 fs, giving rise to component 2, shown in Fig. 2b). Component 2 is centered at the energy of the long-delay photoelectron spectrum and has a GA lifetime longer than 50 ps. We assign component 2 to a mixture of relaxed bright and dark 1s excitons with binding energies of approximately 0.35 eV. We find adding additional components beyond $N=2$ does not improve the quality of the global fit or offer additional physical insight. More details of the GA can be found in the Supplemental Material [73]. The dynamics observed under linearly polarized excitation can be due to a mixture of both intervalley and intravalley relaxation mechanisms. To disentangle their relative contributions, we use circularly polarized pump pulses to prepare valley-polarized excitons. We excite the sample with both $\sigma^{+}$ and $\sigma^{-}$ polarizations, which preferentially excite K${}^{+}$ and K${}^{-}$ valleys, respectively. Figs. 3a) and 3b) show the integrated K${}^{+}$ and K${}^{-}$ valley signals under $\sigma^{+}$ and $\sigma^{-}$ polarizations, respectively. Fig. 3c) shows the valley asymmetry, $\rho$(t), defined by: $$\rho(t)=\frac{I_{K^{+}}(t)-I_{K^{-}}(t)}{I_{K^{+}}(t)+I_{K^{-}}(t)},$$ where I${}_{K^{+}}$ and I${}_{K^{-}}$ denote the integrated intensity in the K${}^{+}$ and K${}^{-}$ valleys, respectively. The valley asymmetry decays in approximately 250 fs, limited by the instrument response. We observe similar time scales for the decay of $\rho(t)$ for low-temperature data recorded at 126 K [73], suggesting exciton-phonon coupling is not a main driver of the dynamics. For comparison, we also show the s-polarized data in Fig. 3c), which shows no valley asymmetry. The K${}^{+}$ and K${}^{-}$ valley signals following s-polarized photoexcitation can be found in the Supplemental Material [73]. Figs. 4a) and 4b) show the time-resolved photoelectron spectra and the $S_{1}(E)$ GA spectral components for the K${}^{-}$ and K${}^{+}$ valleys after $\sigma^{-}$ excitation. Strikingly, the spectrum in the unpumped K${}^{+}$ valley does not show any appreciable difference to that of the initially pumped K${}^{-}$ valley, except an approximately 50 fs delay between the population of the two valleys. We quantify this by applying the same GA described above to the K${}^{+}$ and K${}^{-}$ valleys independently in the circularly polarized data. For the unpumped valley, we allow for a shift, $\Delta t$, in the onset of the time dynamics $f_{i}(t)\rightarrow f_{i}(t-\Delta t)$. We find the spectral components and exponential rates in the K${}^{+}$ and K${}^{-}$ valleys to be similar to one another and also to those found under $s$-polarized excitation. The delayed onset captured by $\Delta t$ was found to be the singular notable difference between the dynamics in the two valleys. From the GA fitting, we find $\Delta t=43\pm 4$ fs for $\sigma^{-}$ excitation and $\Delta t=53\pm 6$ fs for $\sigma^{+}$. These 50 fs shifts are also apparent in the integrated signals of Figs. 3a) and 3b). As a control, we analyzed the $s$-polarized data in the same way and find $\Delta t=6\pm 5$ fs [73]. The small 50 fs shift, indicating very rapid valley depolarization, is consistent with the $\sim$250 fs time scale on which $\rho(t)$ becomes zero when the instrument response is considered. The integrated GA model results are also shown as the lines in Fig. 3. Importantly, the prompt valley depolarization we observe in the tr-ARPES signal is not accompanied by energy relaxation. This is evident from both the data of Fig. 4a) as well as the GA analysis in Fig. 4b), with $S_{1,\scriptsize\mbox{K}^{+}}(E)$ closely resembling $S_{1,\scriptsize\mbox{K}^{-}}(E)$. This is consistent with valley depolarization driven by intervalley Coulomb exchange, which couples energetically degenerate bright exciton states, A${}^{\pm}$ $\xleftrightarrow{}$ A${}^{\mp}$, B${}^{\pm}$ $\xleftrightarrow{}$ B${}^{\mp}$ [33, 26, 28, 30, 8], but is in contrast to other recently proposed non-degenerate intervalley depolarization mechanisms that couple A${}^{\pm}$ $\xleftrightarrow{}$ B${}^{\mp}$, B${}^{\pm}$ $\xleftrightarrow{}$ A${}^{\mp}$ [45, 86, 60, 87, 88]. The observed timescale is also consistent with calculations of intervalley exchange matrix elements. For large $\sim$0.1 Å${}^{-1}$ center-of-mass momentum, valley depolarization via the exchange interaction is expected to be extremely efficient, with eigenstate energy splittings of 10s of meV [28] and corresponding valley depolarization predicted in several 10s of fs [33]. In Fig. 5, we additionally examine the momentum distributions of the photoelectrons. The data shown are recorded after $\sigma^{+}$ excitation with 30 eV probe energy. A representative image of the initial momentum distribution of the K${}^{+}$ valley signal is shown in Fig. 5a). At 5 ps, the distribution has relaxed to the narrower one in Fig. 5b). We quantify the extent of the photoelectron momentum distributions in the K${}^{+}$ and K${}^{-}$ valleys as a function of time by fitting the energy-integrated K valley signal with a Gaussian, $\exp[-(1/2)|\mathbf{k}-K|^{2}/(\Delta k)^{2}]$, and report the standard deviation, $\Delta k$, in Fig. 5. We observe that the initial photoelectron momentum distribution encompasses nearly twice the extent of the relaxed photoelectron population at approximately 5 ps delay time. The final distribution width of $\Delta k\sim$ 0.07 Å${}^{-1}$ is commensurate with the recent experimental measurement of relaxed exciton states of WSe${}_{2}$ at 90 K [63]. Remarkably, no large differences are observed in the momentum distributions in the K${}^{+}$ and K${}^{-}$ valleys. For example, the initial K${}^{+}$ valley distribution with $\Delta k=0.12$  Å${}^{-1}$  arrives at the K${}^{-}$ valley 50 fs later with the same width. The Coulomb exchange interaction conserves the total exciton momentum $\mathbf{Q}=\mathbf{k}_{e}-\mathbf{k}_{h}$. While we do not measure $\mathbf{Q}$ directly in this experiment, we conjecture that the width of the distribution in $\mathbf{Q}$ is correlated with the width of our photoelectron distributions. Thus, the conservation of the photoelectron momentum distribution after intervalley coupling suggests conservation of the exciton momentum, consistent with the intervalley exchange coupling mechanism of valley depolarization. While energy conservation and momentum conservation during valley depolarization are both consistent with intervalley Coulomb exchange coupling, the similarity of the energy and momentum distributions between the pumped and unpumped valleys suggests that rate of transfer does not appear to depend strongly on the exciton binding energy or exciton momentum. The strength of the exchange interaction is expected to scale as $|\mathbf{Q}|$ and the square of the electron-hole wavefunction overlap [33, 28, 89]. This would indicate faster transfer for excitons with larger momentum or tighter electron-hole binding. However, within our experimental resolution, we do not observe such $\mathbf{Q}$- or E-dependence in the population transfer. In this work, we have used time-of-flight momentum microscopy combined with ultrashort XUV pulses at 61 MHz repetition rate to image the exciton dynamics in monolayer WS${}_{2}$. Our measurements record the dynamics in the natural momentum-space basis in which theory and calculations are formulated, and shed new light on the ultrafast intervalley and intravalley coupling dynamics in monolayer TMDs. While these dynamics have been the subject of extensive optical spectroscopy, to our knowledge these are the first reported momentum-space measurements of valley depolarization in the monolayer TMDs. Future work with higher resolution can address the energy- and momentum-dependence of exciton coupling in further detail and also study these phenomena in 2D heterostructures. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under award number DE-SC0022004 and the Air Force Office of Scientific Research under FA9550-20-1-0259. R.K.K. acknowledges support from the U.S. National Science Foundation under Grant No. CHE-1935885. 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\newdateformat monthyeardate\monthname[\THEMONTH] \THEYEAR Universal Gibbons-Hawking-York term for theories with curvature, torsion and non-metricity Johanna Erdmenger Institute for Theoretical Physics and Astrophysics, Julius-Maximilians-Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter ct.qmat Bastian Heß Institute for Theoretical Physics and Astrophysics, Julius-Maximilians-Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter ct.qmat Ioannis Matthaiakakis Corresponding author: ioannis.matthaiakakis@edu.unige.it Dipartimento di Fisica, Università di Genova, via Dodecaneso 33, I-16146, Genova, Italy I.N.F.N. - Sezione di Genova, via Dodecaneso 33, I-16146, Genova, Italy Institute for Theoretical Physics and Astrophysics, Julius-Maximilians-Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter ct.qmat René Meyer Institute for Theoretical Physics and Astrophysics, Julius-Maximilians-Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter ct.qmat Zusammenfassung Motivated by establishing holographic renormalization for gravitational theories with non-metricity and torsion, we present a new and efficient general method for calculating Gibbons-Hawking-York (GHY) terms. Our method consists of linearizing any nonlinearity in curvature, torsion or non-metricity by introducing suitable Lagrange multipliers. Moreover, we use a split formalism for differential forms, writing them in $(n-1)+1$ dimensions. The boundary terms of the action are manifest in this formalism by means of Stokes’ theorem, such that the compensating GHY term for the Dirichlet problem may be read off directly. We observe that only those terms in the Lagrangian that contain curvature contribute to the GHY term. Terms polynomial solely in torsion and non-metricity do not require any GHY term compensation for the variational problem to be well-defined. We test our method by confirming existing results for Einstein-Hilbert and four-dimensional Chern-Simons modified gravity. Moreover, we obtain new results for Lovelock-Chern-Simons and metric-affine gravity. For all four examples, our new method and results contribute to a new approach towards a systematic hydrodynamic expansion for spin and hypermomentum currents within AdS/CFT.   \EdefEscapeHextoc.sectiontoc.section\EdefEscapeHexInhaltsverzeichnisInhaltsverzeichnis\hyper@anchorstarttoc.section\hyper@anchorend Inhaltsverzeichnis 1 Introduction 2 Geometric setup and summary of the main results 3 Examples for Gibbons-Hawking-York terms 3.1 Einstein-Hilbert gravity $\mathcal{L}\propto R$ 3.2 4d Chern-Simons modified gravity $\mathcal{L}\propto\Omega^{2}$ 3.3 Lovelock-Chern-Simons gravity 4 Derivation of the Gibbons-Hawking-York term 4.1 Frame decomposition 4.2 Adapted frame decomposition 4.3 Decomposition of metric and connection 4.4 Foundations of the 3+1 formalism in general frames 4.5 Decomposition of the field strengths 4.6 Post-Riemannian Gibbons-Hawking-York term 4.6.1 Gibbons-Hawking-York term for metric compatible theories 5 Summary and discussion 6 Acknowledgments A GHY term for metric-affine gravity (MAG) A.1 Variation of Hodge stars and interior products A.2 Variation of the MAG Lagrangian A.3 GHY term for metric compatible MAG B GHY term for metric compatible theories   1 Introduction Curved Riemannian and pseudo-Riemannian spacetimes are of immediate relevance for astrophysics, cosmology and high energy physics. In most cases, the spacetime metric is the only dynamical field associated to the geometry, while further geometric structures are argued to be irrelevant by appealing to current experimental evidence. Recently, however, general spacetimes with non-vanishing curvature111$\Omega\indices{{}^{\mu}_{\nu}}$ is the curvature two-form, whose components are the Riemann tensor $R\indices{{}^{\mu}_{\nu}{}_{\rho}{}_{\sigma}}$. $\Omega\indices{{}^{\mu}_{\nu}}$ and torsion $T^{\mu}$ found applications in condensed matter systems—where lattice deformations generate non-trivial $\Omega\indices{{}^{\mu}_{\nu}}$ and $T^{\mu}$ directly coupled to the electronic degrees of freedom—as a means to simulate high-energy phenomena in tabletop experiments [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. In particular, torsion has been shown to alter the the hydrodynamic expansion by introducing terms relevant for both heavy-ion physics and electron flows in condensed matter [15, 17, 18, 19, 16, 20, 21]. We may, however, go even further beyond spacetimes with curvature and torsion to the more general metric-affine spacetimes hosting a non-trivial non-metricity one-form $Q_{\mu\nu}$. In condensed matter systems, this may be achieved by the introduction of dislocations and cuts in the lattice of a given material [22]. For our purposes, the importance of considering metric-affine spacetimes arises from the coupling of torsion and non-metricity to matter fields. Both torsion and non-metricity may be seen as the sources of conserved matter currents. In particular, $T^{\mu}$ is the source of the spin tensor $S_{\mu}^{\leavevmode\nobreak\ \nu\rho}$, while $Q_{\mu\nu}$ is the source of the hypermomentum tensor $\Delta_{\rho\mu\nu}$ [23]. The spin tensor is well-known and has been studied in depth: In particular, spin transport was recently suggested to play an important role in hadron-hadron collisions at LHC [24, 25, 18], as well as in tabletop experiments on electronic systems [26, 27]. In contrast, the hypermomentum tensor is not widely studied in the physics literature. The trace of this tensor, $J^{D}_{\mu}=\Delta\indices{{}_{\mu}{}_{\nu}^{\nu}}$, however, is referred to as dilation current and has found applications in the context of trace anomalies in condensed matter systems [3]. The source interpretation of torsion and non-metricity is the most relevant in the present context, due to its role in the AdS/CFT correspondence [28, 29, 30]. The AdS/CFT correspondence or holography is a duality between a weakly curved theory of gravity and a strongly coupled quantum field theory (QFT) living in one dimension lower, at the boundary of the curved spacetime. The sources coupled to the QFT determine the boundary conditions of the dynamical fields of the gravitational theory, while the on-shell gravitational action acts as the QFT’s generating functional. Holography has widely been used to shed light on the transport properties of matter, for instance of the quark-gluon plasma or of graphene [31, 32]. These transport properties, however, concern only non-trivial energy-momentum and electric charge transfer in the QFT, i.e. only a dynamical metric and gauge field in the gravitational theory. It is our goal to extend the correspondence to include spin and hypermomentum transport in strongly coupled systems. The former arises from the intrinsic spin of particles, while the latter arises from the modification of the causal structure of spacetime due to matter [33]. Causality may exclude hypermomentum transport in relativistic systems of particle and astrophysics, since it may modify the light-cone structure and turn spacelike separated events into timelike separated ones under parallel transport. Condensed matter systems are not bound to such constraints, since their causal structure arises only as an emergent description of the system’s electronic bands. We therefore expect hypermomentum to lead to new transport phenomena and perhaps novel phase transitions in this context. In order to extend holography to the realm of spin and hypermomentum transport, the dual gravitational theory must contain dynamical torsion and non-metricity tensors, i.e. we must consider dynamical theories of metric-affine gravity (MAG) on spacetimes with boundary. The introduction of boundaries into the spacetime considered raises issues of fundamental importance that need to be addressed before applications can be considered. In particular, we have to ensure that the boundary conditions of the MAG fields are compatible with a well-defined variational problem. This is achieved in general relativity by the inclusion of the Gibbons-Hawking-York (GHY) boundary term. The importance of the GHY boundary term in theories of gravity cannot be overstated. In the simplest of applications of gravitational theories, in terms of a Lagrangian, the GHY term makes the variational formulation well-posed [34, 35, 36]. In addition, within the Hamiltonian formalism, the GHY term allows us to correctly define the Hamiltonian as well as the asymptotic conserved charges of the theory, such as energy [36]. The GHY terms also play a crucial role within holography: First, the asymptotic charges evaluated at the boundary are precisely the charges of the dual QFT. Second, the on-shell gravitational action (and QFT generating functional) suffers from divergences. The GHY terms act as counterterms and provide (part of) the regularization necessary to define a finite gravitational action [37]. Thus, in order to use holography to derive QFT observables and understand spin and hypermomentum transport, we have to derive the GHY term for every MAG Lagrangian. We carry out this derivation in the present paper. The GHY term we find provides the starting point for holographic renormalization [38] that we plan to address in the future in the present context. Holographic renormalization in turn will be the starting point for a fluid-gravity hydrodynamic expansion [39] including torsion and non-metricity. Also beyond holography, our results are of interest to both the general relativity and condensed matter communities. The first may use our results as a stepping stone towards analyzing extensions of general relativity to the MAG framework and their compatibility with cosmological data. For the second, in addition to forthcoming results on spin hydrodynamics that we expect to be obtainable based on the analysis presented here, our terms become important when considering the dynamics of defects that give rise to torsion and non-metricity on systems with boundaries. In this context, our GHY terms describe the dynamics of the defects on the boundary which are consistent with the dynamics in the bulk. Such terms may also become important when considering topological field theories on spaces with boundaries, perhaps with quantum anomalies, since they describe the emergent degrees of freedom at the boundary. The main result of this work is a new and efficient method for deriving GHY terms for actions formulated in the language of differential forms. In particular, our generalized GHY term applies for any theory which is allowed to have curved, torsional and non-metric degrees of freedom in arbitrary polynomial combinations. We achieve this generality by formulating our result in terms of auxiliary fields which are calculated for generic theories by taking variations of their Lagrangian $n$-form with respect to suitable Lagrange multipliers. One of our main findings is that our generalized GHY term receives contributions from the variation of only those terms in the Lagrangian that contain curvature. Terms that are solely built from torsion and non-metricity do not contribute a GHY term, assuming that any derivatives of torsion and non-metricity have been converted into curvature polynomials by means of the Bianchi identities. Using our method we confirm the GHY term for the Einstein-Hilbert action in arbitrary dimensions. We furthermore verify a result for 4d Chern-Simons modified gravity up to a factor of 2. Moreover, we present new results for Lovelock-Chern-Simons and metric-affine gravity. These new results have the form which we expect from comparison of the Lagrangians with those of Einstein-Hilbert and 4d Chern-Simons modified gravity. We begin the main part of this paper with section 2, where we briefly set up the geometric framework used. This allows us to present the main results of our work in a concise manner in the same section. Subsequently, we apply these general results to the special cases of Einstein-Hilbert and four-dimensional Chern-Simons modified gravity in section 3 to find familiar results as a check of our method. In the same section, we apply our method to the case of Lovelock-Chern-Simons gravity and derive new results for its GHY term. An explicit derivation of our results is given in section 4. The more involved case of MAG is considered in appendix A. 2 Geometric setup and summary of the main results In the current section, we present our main result for the GHY term for any MAG theory in (12). To fully grasp the meaning of each term in (12), however, we must first give a lightning review of our geometric setup. For the interested reader, more details are given in section 4. As mentioned in the introduction, the geometry that we consider in a dynamical setting is that of a metric-affine spacetime. This spacetime is an $n$-dimensional manifold $\mathcal{M}$ equipped with a coframe basis $\theta^{\mu}$, a metric $\mathrm{d}s^{2}=g_{\mu\nu}\theta^{\mu}\otimes\theta^{\nu}$ and a connection one-form $\omega^{\mu}_{\nu}=\Gamma^{\mu}_{\rho\nu}\theta^{\rho}$, where Greek indices take values in $\left\{0,\dots,n-1\right\}$. These fields are independent of each other and may be thought of as the kinematic variables of the spacetime in the same sense that a U(1) gauge field provides the kinematic variables for a U(1) gauge theory. Following this analogy further, we may define field strengths for $\theta^{\mu}$, $g_{\mu\nu}$ and $\omega^{\mu}_{\nu}$. To preserve diffeomorphism invariance, we construct these field strengths in terms of the exterior derivative $\mathrm{d}$ and the exterior covariant derivative $D=\mathrm{d}+\omega^{\mu}_{\nu}\rho(L)^{\nu}_{\mu}\wedge$, where $\rho(L)^{\nu}_{\mu}$ is the appropriate representation of the GL(n,$\mathbb{R}$) generators $L$. In particular, the field strengths for the fields $\omega^{\mu}_{\nu}$, $\theta^{\mu}$, $g_{\mu\nu}$ are $$\begin{aligned} &\text{the curvature two-form}\vphantom{\frac{1}{2}}\\ &\text{the torsion two-form}\vphantom{\frac{1}{2}}\\ &\text{the non-metricity one-form}\vphantom{\frac{1}{2}}\end{aligned}\quad\begin{aligned} &\Omega\indices{{}^{\mu}_{\nu}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{d}\omega^{\mu}_{\nu}+\omega^{\mu}_{\rho}\wedge\omega^{\rho}_{\nu}=\frac{1}{2}R\indices{{}^{\mu}_{\nu}{}_{\rho}{}_{\sigma}}\theta^{\rho}\wedge\theta^{\sigma}\,,\\ &T^{\mu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=D\theta^{\mu}=\mathrm{d}\theta^{\mu}+\omega^{\mu}_{\nu}\wedge\theta^{\nu}=\frac{1}{2}T\indices{{}^{\mu}_{\rho}{}_{\sigma}}\theta^{\rho}\wedge\theta^{\sigma}\,,\\ &Q_{\mu\nu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-Dg_{\mu\nu}=-(\mathrm{d}g_{\mu\nu}-\omega^{\rho}_{\mu}g_{\rho\nu}-\omega^{\rho}_{\nu}g_{\mu\rho})=Q_{\mu\nu\rho}\theta^{\rho}\,,\vphantom{\frac{1}{2}}\end{aligned}$$ (1) respectively. The curvature $R\indices{{}^{\mu}_{\nu}{}_{\rho}{}_{\sigma}}$ is the Riemann tensor, while $T\indices{{}^{\mu}_{\rho}{}_{\sigma}}$ is the torsion and $Q_{\mu\nu\rho}$ the non-metricity tensor. All three of the fields strengths satisfy a corresponding Bianchi identity $$\displaystyle\begin{split}D\Omega\indices{{}^{\mu}_{\nu}}&=0\,,\\ DT\indices{{}^{\mu}}&=\Omega\indices{{}^{\mu}_{\nu}}\wedge\theta^{\nu}\,,\\ DQ_{\mu\nu}&=\Omega_{\mu\nu}+\Omega_{\nu\mu}\,.\end{split}$$ (2) We introduce a boundary $\partial\mathcal{M}$ in $\mathcal{M}$ and a geometry on $\partial\mathcal{M}$ via two sets of vector fields. First, we define $\partial\mathcal{M}$ through the normal vector $n^{\mu}$, normalized as $n_{\mu}n^{\mu}=\varepsilon=-1$ for spacelike and $n_{\mu}n^{\mu}=\varepsilon=+1$ for timelike boundaries. Second, we introduce the vielbein $e^{\mu}_{a}$ as a basis of tangent vectors on $\partial\mathcal{M}$, $n_{\mu}e^{\mu}_{a}=0$, where Latin indices take values in $\left\{0,\dots,n-2\right\}$. The vielbein and its dual $e^{a}_{\mu}$ induce a geometry on $\partial\mathcal{M}$ via pullback from $\mathcal{M}$. For example, the boundary coframe basis $\phi^{a}$ is obtained from the manifold coframe basis $\theta^{\mu}$ as $e^{a}_{\mu}\theta^{\mu}=\phi^{a}$, while the induced metric on $\partial\mathcal{M}$ is $\gamma_{ab}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{\mu}_{a\vphantom{b}}e^{\nu\vphantom{\mu}}_{b}g_{\mu\nu}$. While the metric and coframe project to the boundary via pullback, the connection is related to its boundary value $\omega^{a}_{b}$ in a more involved way. In particular, we assume that $\omega^{\mu}_{\nu}$ projects to $\omega^{a}_{b}$ via the vielbein postulate $$\displaystyle\omega^{b}_{a}=e^{b}_{\mu}\left(\mathrm{d}e^{\mu}_{a}+\omega^{\mu}_{\nu}e^{\nu}_{a}\right)\,.$$ (3) We prove the transformation property (3) within the MAG framework [23, 40] in section 4.3. To our knowledge this is the most general theory that includes curvature, torsion and non-metricity. It is thus natural to assume that (3) holds for a large range of theories featuring fewer fields or more symmetries. Apart from the curvature, torsion and non-metricity associated to the projected connection, coframe and metric on the boundary, the embedding of $\partial\mathcal{M}$ in $\mathcal{M}$ allows for additional quantities that describe the geometry of $\partial\mathcal{M}$ relative to that of $\mathcal{M}$. For example, in the following we use extensively the extrinsic curvatures $$\displaystyle K^{a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{a}_{\mu}Dn^{\mu}\qquad\text{and}\qquad\tilde{K}_{a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{\mu}_{a}Dn_{\mu}=K_{a}-e^{\mu}_{a}n^{\nu}Q_{\mu\nu}\,.$$ (4) The two definitions of the extrinsic curvature in (4) differ only by a deformation in the $n-e_{a}$ plane due to non-metricity. Thus, for metric compatible connections satisfying $Dg_{\mu\nu}=0$ it suffices to consider only one of the extrinsic curvature definitions. The tensor components of $K^{a}$ tangent to the boundary give the extrinsic curvature familiar from the literature, $K\indices{{}^{a}_{b}}=e^{a}_{\mu}e_{b}^{\nu}\nabla_{\nu}n^{\mu}$, where $\nabla_{\nu}$ is the covariant derivative on $\mathcal{M}$  [41, 36, 42]. This concludes our brief review of the geometry on $\mathcal{M}$ and the induced geometry on $\partial\mathcal{M}$. For investigating the dynamics of this geometry, we consider the most general action possible, $$\displaystyle S_{\mathrm{orig}}[g_{\mu\nu},\omega^{\mu}_{\nu},\theta^{\mu}]=\int_{\mathcal{M}}\mathcal{L}(\Omega\indices{{}^{\mu}_{\nu}},T^{\mu},Q_{\mu\nu})\,,$$ (5) where the Lagrangian $n$-form $\mathcal{L}(\Omega\indices{{}^{\mu}_{\nu}},T^{\mu},Q_{\mu\nu})$ is an arbitrary function of the curvature, torsion and non-metricity defined in (1). Not fixing a particular form for $\mathcal{L}$ ensures that our results hold for any action of interest and may be applied to a specific system by choosing $\mathcal{L}$ appropriately. Note that we have restricted ourselves to actions which are polynomial in the field strengths but do not involve their derivatives. This is not a severe restriction since the Bianchi identities (2) tell us that derivatives of curvature vanish, while those of torsion and non-metricity can be reduced to polynomial terms in curvature. Our goal is to make the variation of the action (5) well-defined by adding the appropriate boundary terms to it. To find these terms we vary $S_{\rm orig}$ and isolate the terms on the boundary that do not vanish after enforcing the boundary conditions on the fields. To be precise, the boundary conditions we consider are $\left.\delta g_{\mu\nu}\right|_{\mathrm{\partial\mathcal{M}}}=0$, $\left.\delta\theta^{\mu}\right|_{\mathrm{\partial\mathcal{M}}}=0$ and $\left.\delta\omega^{\mu}_{\nu}\right|_{\mathrm{\partial\mathcal{M}}}=0.$ However, according to [36] it suffices to demand that the Dirichlet boundary conditions $$\displaystyle\delta\gamma_{ab}=0\,,\leavevmode\nobreak\ \leavevmode\nobreak\ \delta\phi^{a}=0\,,\leavevmode\nobreak\ \leavevmode\nobreak\ \delta\omega^{a}_{b}=0$$ (6) hold since the original conditions may be reinstated by gauge transformations on $\partial\mathcal{M}$. In principle, we may now obtain the boundary terms from a direct variation of the action $S_{\rm orig}$. However, general covariance requires that we write the boundary terms in terms of geometric quantities on $\partial\mathcal{M}$. To achieve this we need to perform a $3+1$ decomposition222While we work in general $n$ dimensions and perform a full $(n-1)+1$ decomposition, we use the term $3+1$ decomposition for compactness throughout. of $S_{\rm orig}$ which is impossible to write down if $\mathcal{L}$ is not specified. We circumvent this issue by the method of Lagrange multipliers expounded in [43] for the case of $\mathcal{L}(\Omega\indices{{}^{\mu}_{\nu}})$ gravity. This method essentially makes the action linear in the field strengths without losing any information of the dynamics induced by $\mathcal{L}$. In order to isolate the boundary terms, we introduce the Lagrange multipliers $\varphi\indices{{}_{\mu}^{\nu}}$, $t_{\mu}$ and $q^{\mu\nu}$ and turn the Langrangian into a linear function of $\Omega\indices{{}^{\mu}_{\nu}}$, $T^{\mu}$ and $Q_{\mu\nu}$. In particular, we consider the gravitational action $$\displaystyle\begin{split}S[g_{\mu\nu},\omega^{\mu}_{\nu},\theta^{\mu},\varphi\indices{{}^{\mu}_{\nu}},\varrho\indices{{}^{\mu}_{\nu}},t_{\mu},\tau^{\mu},q^{\mu\nu},&\sigma_{\mu\nu}]\\ =\int_{\mathcal{M}}&\left[\mathcal{L}(\varrho\indices{{}^{\mu}_{\nu}},\tau^{\mu},\sigma_{\mu\nu})+*\varphi\indices{{}_{\mu}^{\nu}}\wedge(\Omega\indices{{}^{\mu}_{\nu}}-\varrho\indices{{}^{\mu}_{\nu}})\right.\\ &\left.\vphantom{*\varphi\indices{{}_{\mu}^{\nu}}}+*t_{\mu}\wedge\left(T^{\mu}-\tau^{\mu}\right)+*q^{\mu\nu}\wedge\left(Q_{\mu\nu}-\sigma_{\mu\nu}\right)\right]\,,\end{split}$$ (7) where $\varrho\indices{{}^{\mu}_{\nu}}$, $\tau^{\mu}$ and $\sigma_{\mu\nu}$ are auxiliary fields and $*$ is the Hodge duality. Expressing $S$ as in (7) allows us to directly access $\Omega\indices{{}^{\mu}_{\nu}}$, $T^{\mu}$ and $Q_{\mu\nu}$ regardless of the explicit form of $\mathcal{L}$. We choose the Lagrange multipliers to be independent of each other and of the fields $\omega^{\mu}_{\nu}$, $\theta^{\mu}$ and $g_{\mu\nu}$. In this way we ensure that the equations of motion for $\Omega\indices{{}^{\mu}_{\nu}}$, $T^{\mu}$ and $Q_{\mu\nu}$ are kept unchanged if we first impose the equations of motion $$\displaystyle\Omega\indices{{}^{\mu}_{\nu}}=\varrho\indices{{}^{\mu}_{\nu}}\,,\qquad T^{\mu}=\tau^{\mu}\,,\qquad Q_{\mu\nu}=\sigma_{\mu\nu}$$ (8) for the Lagrange multipliers $\varphi\indices{{}_{\mu}^{\nu}}$, $t_{\mu}$ and $q^{\mu\nu}$, respectively. To that end, we additionally demand these Lagrange multipliers to have the exact same symmetries as the corresponding field strengths. In order to express the boundary terms in terms of a diffeomorphic invariant Lagrangian on the boundary, we consider projections of the Lagrange multipliers on $\partial\mathcal{M}$. In particular, we project each index of the Lagrange multipliers either to the boundary by contraction with the vielbein $e^{a}_{\mu}$ or normal to the boundary by contraction with $n^{\mu}$. We abbreviate these contractions as $\varphi_{a\mathbf{n}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e_{a}^{\mu}n^{\nu}\varphi_{\mu\nu}$ for instance. These projections may be regarded as an extension of the $3+1$ decomposition in general relativity and can be derived by expanding every contraction of Greek indices in (7) using $$\displaystyle\delta^{\mu}_{\nu}=e^{\mu}_{a}e^{a}_{\nu}+\varepsilon n^{\mu}n_{\nu}\,.$$ (9) For example $A^{\mu}\wedge B_{\mu}=A^{a}\wedge B_{a}+\varepsilon A_{\mathbf{n}}\wedge B_{\mathbf{n}}$ for generic differential forms $A^{\mu},$ $B_{\mu}$. The explicit form of the action $S$ in terms of the $3+1$ decomposition is rather lengthy and not that enlightening. So we leave (7) unchanged and keep in mind that for further calculations any of the Greek indices is $3+1$ decomposed as just described. While the equations of motion (8) of the Lagrange multipliers $\varphi\indices{{}_{\mu}^{\nu}}$, $t_{\mu}$ and $q^{\mu\nu}$ ensure the equivalence of $S$ and $S_{\rm orig}$, the equations of motion of the additional fields $\varrho\indices{{}^{\mu}_{\nu}}$, $\tau^{\mu}$ and $\sigma_{\mu\nu}$ yield constraints which enable us to determine the explicit forms of $\varphi\indices{{}_{\mu}^{\nu}},t_{\mu}$ and $q^{\mu\nu}$ in terms of $\mathcal{L}$. Among these constraints the ones which yield $*\varphi^{\mathbf{n}a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=n^{\alpha}e_{\beta}^{a}*\varphi\indices{{}_{\alpha}^{\beta}}$, $*\varphi_{a\mathbf{n}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{\alpha}_{a}n_{\beta}*\varphi\indices{{}_{\alpha}^{\beta}}$ and $*\varphi_{\mathbf{n}\mathbf{n}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=n^{\alpha}n_{\beta}*\varphi\indices{{}_{\alpha}^{\beta}}$ are particularly important for us. These are $$\displaystyle\begin{split}*\varphi^{\mathbf{n}a}\wedge\delta\varrho_{\mathbf{n}a}&=\varepsilon\delta_{\varrho_{\mathbf{n}a}}\mathcal{L}(\varrho_{\mathbf{n}a},\varrho^{a\mathbf{n}},\varrho_{\mathbf{n}\mathbf{n}},\dots)\,,\\ *\varphi_{a\mathbf{n}}\wedge\delta\varrho^{a\mathbf{n}}&=\varepsilon\delta_{\varrho^{a\mathbf{n}}}\mathcal{L}(\varrho_{\mathbf{n}a},\varrho^{a\mathbf{n}},\varrho_{\mathbf{n}\mathbf{n}},\dots)\,,\\ *\varphi^{\mathbf{n}\mathbf{n}}\wedge\delta\varrho_{\mathbf{n}\mathbf{n}}&=\hphantom{\varepsilon}\delta_{\varrho_{\mathbf{n}\mathbf{n}}}\mathcal{L}(\varrho_{\mathbf{n}a},\varrho^{a\mathbf{n}},\varrho_{\mathbf{n}\mathbf{n}},\dots)\,,\end{split}$$ (10) obtained from varying (7) with respect to $\varrho_{\mathbf{n}a}$, $\varrho_{a\mathbf{n}}$ and $\varrho_{\mathbf{n}\mathbf{n}}$.333Only these variations are relevant, since the boundary conditions (6) do not fix $\omega^{a}_{\mathbf{n}}$, $\omega^{\mathbf{n}}_{a}$ and $\omega^{\mathbf{n}}_{\mathbf{n}}$. To put it simple, $\varphi\indices{{}^{\mu}_{\nu}}$ is the Hodge dual of the equations of motion for $\Omega\indices{{}^{\mu}_{\nu}}$.444Note that these are not Einstein’s equations which result from a variation with respect to the connection. In addition to the $3+1$ decomposition of the Lagrange multipliers and auxiliary fields we also need to introduce the $3+1$ decomposition of the curvature, torsion and non-metricity into the action $S$ in (7). The full details of the derivation are give in section 4. Here we mention the $3+1$ decomposition of curvature which plays an important role in the examples we consider in section 3. We have $$\begin{aligned} \vphantom{\frac{1}{2}}e_{\mu}^{a}e^{\nu}_{b}\Omega\indices{{}^{\mu}_{\nu}}&=\Omega\indices{{}^{a}_{b}}-\varepsilon K^{a}\wedge\tilde{K}_{b}\,,\\ e^{a}_{\mu}n^{\nu}\Omega\indices{{}^{\mu}_{\nu}}&=DK^{a}+\frac{\varepsilon}{2}K^{a}\wedge Q_{\mathbf{n}\mathbf{n}}\,,\end{aligned}\qquad\begin{aligned} n_{\mu}e^{\nu}_{a}\Omega\indices{{}^{\mu}_{\nu}}&=D\tilde{K}_{a}+\frac{\varepsilon}{2}\tilde{K}_{a}\wedge Q_{\mathbf{n}\mathbf{n}}\,,\\ n_{\mu}n^{\nu}\Omega\indices{{}^{\mu}_{\nu}}&=\frac{1}{2}DQ_{\mathbf{n}\mathbf{n}}+K^{a}\wedge\tilde{K}_{a}\,,\end{aligned}$$ (11) where we define $\Omega\indices{{}^{a}_{b}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{d}\omega^{a}_{b}+\omega^{a}_{c}\wedge\omega^{c}_{b}$ for the curvature on the boundary.555Note the components of $\Omega\indices{{}^{a}_{b}}$ also include normal contributions in general. The purely hypersurface curvature, $\Omega\indices{{}^{(n-1)a}_{b}}$, is found by means of projecting the components of $\Omega\indices{{}^{a}_{b}}$ to the hypersurface, that is $\Omega\indices{{}^{(n-1)a}_{b}}=\Omega\indices{{}^{a}_{b}}(\varphi_{c},\varphi_{d})\phi^{c}\wedge\phi^{d}$, with $\varphi_{a}$ the dual of $\phi^{a}$ (see (44)). The result (11) may be considered a generalized version of the Gauß-Codazzi equations. Finally, let us mention the strategy we use to obtain the GHY term for the generic action (5) after the $3+1$ decomposition has been made. Expressing the action in a $3+1$ form results in two types of terms appearing in the Lagrangian; one containing the fields $\omega^{\mu}_{\nu}$, $\theta^{\mu}$, $g_{\mu\nu}$ and their projection to the boundary and another one containing their derivatives. Due to the Dirichlet boundary conditions (6) the first type does not contribute at the boundary and, thus, only the second type of terms can lead to boundary contributions under variation. To calculate them, we employ Stokes’ theorem to rewrite these terms as a pure boundary integral. We then vary the boundary integral and isolate the terms that do not vanish after variation. These are the terms which have to be subtracted from $S$ by means of the GHY term. Carrying out this calculation in section 4 we find the explicit form of the GHY term for a MAG theory with Lagrangian $\mathcal{L}$ to be $$\displaystyle\begin{split}S_{\mathrm{GHY}}&=-\int_{\partial\mathcal{M}}\left.\left(-\varepsilon\tilde{K}_{a}\wedge*\varphi^{\mathbf{n}a}+\varepsilon K^{a}\wedge*\varphi_{a\mathbf{n}}+\frac{1}{2}Q_{\mathbf{n}\mathbf{n}}\wedge*\varphi_{\mathbf{n}\mathbf{n}}\right)\right|_{\mathrm{\partial\mathcal{M}}}\\ &=-\int_{\partial\mathcal{M}}\left.\left(\varepsilon K^{a}\wedge*\left(\varphi_{a\mathbf{n}}-\varphi_{\mathbf{n}a}\right)+\varepsilon Q_{\mathbf{n}a}\wedge*\varphi^{\mathbf{n}a}+\frac{1}{2}Q_{\mathbf{n}\mathbf{n}}\wedge*\varphi_{\mathbf{n}\mathbf{n}}\right)\right|_{\mathrm{\partial\mathcal{M}}}\,,\end{split}$$ (12) where $\varphi_{a\mathbf{n}},$ $\varphi_{\mathbf{n}a}$ and $\varphi_{\mathbf{n}\mathbf{n}}$ are implicitly expressed in terms of $\mathcal{L}$ through the constraints (10). A couple of comments about $S_{\rm GHY}$ are in order. First, we note that our result should be expected. All terms in (12) depend on the first derivatives of the metric as in typical general relativity. That torsion does not appear explicitly in the GHY terms should also be expected. Torsion may contribute boundary terms only via the derivative of the frame field it contains, see (1). This derivative measures the non-holonomicity of the frame and is not a true geometric invariant of the theory. Therefore, in a generally covariant theory, vanishing non-holonomicity is locally enforceable and as a result the torsion two-form is only a polynomial in $\omega^{\mu}_{\nu}$ and $\theta^{\mu}$. This is why the torsion two-form cannot appear explicitly in the GHY terms. Of course, this does not forbid turning the GHY term from a function of the curvature to a function of torsion if additional constraints between the fields strengths are taken into account as in teleparallel theories of gravity for example [44]. Second, in the important case of vanishing non-metricity, $Q_{\mu\nu}=0$, the GHY term (12) simplifies considerably to $$\displaystyle S_{\mathrm{GHY}}^{Q=0}=2\int_{\mathrm{\partial\mathcal{M}}}\left.\varepsilon K^{a}\wedge*\varphi_{\mathbf{n}a}\right|_{\mathrm{\partial\mathcal{M}}}\,.$$ (13) That is, the GHY term is a direct generalization of the GHY term of general relativity which is proportional to the extrinsic curvature of the boundary. Finally, some technical notes are in place. First, for the calculation we considered $\varrho_{\mathbf{n}a}$ and $\varrho_{a\mathbf{n}}$ as independent although curvature fulfills $\Omega_{\mu\nu}+\Omega_{\nu\mu}=DQ_{\mu\nu}$ according to (2) and $\varrho_{\mu\nu}$ has the same symmetry as $\Omega_{\mu\nu}$. This simplifies our calculation considerably without altering the final result. Alternatively, it is possible to invoke the symmetry relation satisfied by $\varrho_{\mu\nu}$ prior to the variation. In appendix B we explicitly show that this yields the same resulting GHY term (77). Second, we worked entirely in the first order formalism where all fields are independent. If one wants to consider a second order one, the equations of motion must be used to express all fields in terms of the independent ones. In traditional general relativity, this involves solving for $\omega^{\mu}_{\nu}$ in terms $g_{\mu\nu}$. Substituting the solution into $\mathcal{L}$ yields a new Lagrangian $\tilde{\mathcal{L}}$ for the independent fields. If $\tilde{\mathcal{L}}$ is a function of the remaining independent field strengths, say $\tilde{\mathcal{L}}=\tilde{\mathcal{L}}(\Omega\indices{{}^{\mu}_{\nu}})$, our algorithm may still be applied mutatis mutandis to obtain the GHY term for $\tilde{\mathcal{L}}$. We have implicitly used this observation when writing down (13) for the GHY terms of metric compatible theories in the following section. To recap, the calculation of the GHY term for any MAG theory follows these steps: First, choose the Lagrangian $\mathcal{L}$ of the MAG of interest. Second, express $\mathcal{L}$ in a $3+1$ decomposed form (see section 4). Third, derive the Lagrange multiplier $\varphi_{\mu\nu}$ through the constraint equations (10). Finally, substitute the result into the general form (12) of the GHY term. In the following section, we apply this algorithm to several particular theories of gravity as a consistency check of our results. 3 Examples for Gibbons-Hawking-York terms In the current section we apply the algorithm explained in the previous section to calculate the GHY terms for several theories of gravity. The examples we consider involve the Einstein-Hilbert action in general dimensions, the action of 4d Chern-Simons modified gravity and the 5d Lovelock-Chern-Simons action. We also calculate the GHY term for the most general MAG Lagrangian quadratic in the field strengths. Since both the calculation and the result for the MAG GHY terms are quite lengthy, but do not add any additional insight regarding the formalism, we relegate them to appendix A. 3.1 Einstein-Hilbert gravity $\mathcal{L}\propto R$ As a very basic consistency check of our algorithm, we first consider the Einstein-Hilbert action $$\displaystyle S^{\mathrm{EH}}=\frac{1}{2\kappa}\int\mathrm{d}^{4}x\sqrt{-g}R=\frac{1}{2\kappa}\int\eta^{\mu\nu}\wedge\Omega_{\mu\nu}\,,$$ (14) where $\eta^{\mu\nu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=*\left(\theta^{\mu}\wedge\theta^{\nu}\right)$. The Lagrangian for this action is $\mathcal{L}=\mathcal{L}(\Omega\indices{{}^{\mu}_{\nu}})=\eta^{\mu\nu}\wedge\Omega_{\mu\nu}/2\kappa$. Hence, the general form of the extended action (7) including the auxiliary fields and Lagrange multipliers reads $$\displaystyle S^{\rm EH}_{\rm aux}=\int\left(\frac{1}{2\kappa}\eta^{\mu\nu}\wedge\varrho_{\mu\nu}+*\varphi\indices{{}_{\mu}^{\nu}}\wedge(\Omega\indices{{}^{\mu}_{\nu}}-\varrho\indices{{}^{\mu}_{\nu}})\right)$$ (15) in the case of Einstein-Hilbert gravity. The relevant terms in the $3+1$ decomposition of $\mathcal{L}(\varrho_{\mu\nu})$ for our calculation of the GHY term are $$\displaystyle\mathcal{L}(\varrho_{\mu\nu})=\frac{1}{2\kappa}\eta^{\mu\nu}\wedge\varrho_{\mu\nu}\simeq\frac{1}{2\kappa}\varepsilon\eta_{\mathbf{n}a}\wedge\varrho^{\mathbf{n}a}\,,$$ (16) where $\simeq$ omits irrelevant terms of the decomposition. According to (10), the equations of motion for $\varrho^{\mathbf{n}a}$ read $$\displaystyle*\varphi_{\mathbf{n}a}\wedge\delta\varrho^{\mathbf{n}a}=\varepsilon\delta_{\varrho^{\mathbf{n}a}}\mathcal{L}(\varrho)=\frac{1}{2\kappa}\eta_{\mathbf{n}a}\wedge\delta\varrho^{\mathbf{n}a}$$ (17) which fix the form of $\varphi_{\mathbf{n}a}$ to $$\displaystyle*\varphi_{\mathbf{n}a}=\frac{1}{2\kappa}\eta_{\mathbf{n}a}\,.$$ (18) Substituting (18) into our general result (13) for the GHY term, we obtain $$\displaystyle S_{\mathrm{GHY}}^{\mathrm{EH}}=2\int_{\mathrm{\partial\mathcal{M}}}\left.\varepsilon K^{a}\wedge*\varphi_{\mathbf{n}a}\right|_{\mathrm{\partial\mathcal{M}}}=\frac{\varepsilon}{\kappa}\int_{\partial\mathcal{M}}\mathrm{d}^{3}x\sqrt{|\gamma|}K\indices{{}^{a}_{a}}$$ (19) which is the well-known result for the GHY term in general relativity [34, 35]. Note that we used the $4$d Einstein-Hilbert action, but since the differential geometric formulation of the action (14) holds for an arbitrary number of dimensions, our result is actually more general. In particular, the GHY term $$\displaystyle S_{\mathrm{GHY}}^{\mathrm{EH}}=\frac{\varepsilon}{\kappa}\int_{\partial\mathcal{M}}\mathrm{d}\text{Vol}_{\partial\mathcal{M}}\sqrt{|\gamma|}K\indices{{}^{a}_{a}}$$ (20) solves the Dirichlet problem for Einstein-Hilbert gravity on any $n$-manifold $\mathcal{M}$. The abbreviation $\mathrm{d}\text{Vol}_{\partial\mathcal{M}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\phi^{0}\wedge\dots\wedge\phi^{n-2}$ is equal to the standard integral measure $\mathrm{d}^{n-1}y$ in holonomic coordinates $\phi^{a}=\mathrm{d}y^{a}$. 3.2 4d Chern-Simons modified gravity $\mathcal{L}\propto\Omega^{2}$ As a next check of our formalism we turn to the four-dimensional Chern-Simons action as used in Chern-Simons modified gravity [45]. This provides a non-trivial check since the action is quadratic in curvature and the GHY term was already derived in [46]. The Chern-Simons part of the full action is666In the differential geometric version of (21) we could alternatively use a notation which involves the Hodge dual of one of the curvature two-forms. We refrain from that since the variation of Hodge duals is non-trivial. For details on the variation of Hodge duals as well as interior products and their inclusion into the framework of this paper, see appendix A.1. $$\displaystyle S^{\mathrm{CS}}=\frac{\kappa}{4}\int\mathrm{d}^{4}x\sqrt{-g}\,\theta\,*RR=\frac{\kappa}{2}(-1)^{\mathrm{ind}g}\int\theta\,\Omega\indices{{}^{\mu}_{\nu}}\wedge\Omega\indices{{}^{\nu}_{\mu}}\,,$$ (21) where the background scalar field $\theta$ must not be confused with the coframe basis $\theta^{\mu}$. We read off $\mathcal{L}(\Omega\indices{{}^{\mu}_{\nu}})=\frac{\kappa}{2}(-1)^{\mathrm{ind}g}\theta\,\Omega\indices{{}^{\mu}_{\nu}}\wedge\Omega\indices{{}^{\nu}_{\mu}}$ from the action (21) so that the equation of motion (10) for $\varrho^{\mathbf{n}a}$ yields $$\displaystyle*\varphi_{\mathbf{n}a}\wedge\delta\varrho^{\mathbf{n}a}=\varepsilon\delta_{\varrho^{\mathbf{n}a}}\mathcal{L}=(-1)^{\mathrm{ind}g}\kappa\theta\varrho_{a\mathbf{n}}\wedge\delta\varrho^{\mathbf{n}a}$$ (22) which leads to $*\varphi^{\mathbf{n}a}=(-1)^{\mathrm{ind}g}\kappa\theta\varrho^{a\mathbf{n}}$. We observe that $\varphi^{\mathbf{n}a}$ depends explicitly on $\varrho^{a\mathbf{n}}$, unlike the Einstein-Hilbert action. To proceed, we use the equations of motion (8) of $\varphi\indices{{}^{\mu}{}^{\nu}}$ to fix $\varrho_{\mu\nu}=\Omega_{\mu\nu}$. Subsequently, we employ the $3+1$ decomposition of curvature which is given by the Gauß-Codazzi equations (11) to express $\varphi^{\mathbf{n}a}$ in terms of geometric quantities on $\partial\mathcal{M}$. We find $$\displaystyle\left.\vphantom{(-1)^{\mathrm{ind}g}}*\varphi^{\mathbf{n}a}\right|_{\varrho_{\mu\nu}=\Omega_{\mu\nu}}=\left.(-1)^{\mathrm{ind}g}\kappa\theta\varrho^{a\mathbf{n}}\right|_{\varrho_{\mu\nu}=\Omega_{\mu\nu}}=(-1)^{\mathrm{ind}g}\kappa\theta DK^{a}$$ (23) which evaluated on the boundary reads $$\displaystyle\left.*\varphi^{\mathbf{n}a}\right|_{\partial\mathcal{M}}=(-1)^{\mathrm{ind}g}\kappa\theta\nabla_{a}K\indices{{}^{c}_{b}}\phi^{a}\wedge\phi^{b}\,.$$ (24) Here we define the boundary covariant derivative as $\nabla_{a}K\indices{{}^{c}_{b}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\partial_{a}K\indices{{}^{c}_{b}}+\Gamma^{c}_{ad}K\indices{{}^{d}_{b}}-\Gamma^{d}_{ab}K\indices{{}^{c}_{d}}$, where the connection coefficients are related to the boundary connection (3) as $\omega^{a}_{b}=\Gamma^{a}_{cb}\phi^{c}$. To match the conventions of [46] we choose $|g|=-g$, $|\gamma|=\gamma$ as well as torsion freedom. This, in conjunction with (13), leads to the GHY term $$\displaystyle S_{\mathrm{GHY}}^{\mathrm{CS}}=2\kappa\int_{\mathcal{M}_{3}}\mathrm{d}^{3}x\sqrt{\gamma}\theta\epsilon^{ijk}K\indices{{}_{i}^{l}}\nabla_{j}K_{kl}\,,$$ (25) where $\epsilon^{ijk}$ denotes the totally antisymmetric tensor which differs from the $\varepsilon^{ijk}$-symbol by a factor of $1/\sqrt{\gamma}$. The result (25) coincides with the result of [46] up to a factor of 2. However, a calculation of this GHY term via the method of [43] supports our result (25) including the factor of 2. We now turn to the derivation of the GHY term for Lovelock-Chern-Simons gravity which is not known in literature as far as we are aware. 3.3 Lovelock-Chern-Simons gravity For our final example, we consider Lovelock-Chern-Simons gravity in a $5$d spacetime with boundary. This example involves both a non-trivial polynomial Lagrangian for curvature and non-vanishing torsion. In particular, we investigate the action $$\displaystyle S^{\mathrm{LCS}}=\kappa\int_{\mathcal{M}_{5}}\epsilon_{\mu\nu\alpha\beta\sigma}\left(\Omega^{\mu\nu}\wedge\Omega^{\alpha\beta}+\frac{2}{3}\Omega^{\mu\nu}\wedge\theta^{\alpha}\wedge\theta^{\beta}+\frac{1}{5}\theta^{\mu}\wedge\theta^{\nu}\wedge\theta^{\alpha}\wedge\theta^{\beta}\right)\wedge\theta^{\sigma}$$ (26) employed in [47, 16]. As in the previous examples we identify the integrand in (26) as $\mathcal{L}$ and use the equations of motion (8), (10) for $\varphi\indices{{}^{\mu}_{\nu}}$ and $\varrho\indices{{}^{\mu}_{\nu}}$ to obtain $$\displaystyle\left.\vphantom{(-1)^{\mathrm{ind}g}}*\varphi_{\mathbf{n}a}\right|_{\varrho_{\mu\nu}=\Omega_{\mu\nu}}=2\varepsilon\kappa\epsilon_{\mu\nu\alpha\beta\sigma}n^{\mu}e^{\nu}_{a}\left(\Omega^{\alpha\beta}+\frac{1}{3}\theta^{\alpha}\wedge\theta^{\beta}\right)\wedge\theta^{\sigma}\,.$$ (27) Note that since $\epsilon_{\mu\nu\alpha\beta\sigma}$ in (27) is contracted with $n^{\mu}$, the $\alpha,\beta$ and $\sigma$ indices are all on $\partial\mathcal{M}$ which means that they need to be contracted with $e^{\mu}_{a}$ when evaluating the $3+1$ decomposition of $\varphi_{\mathbf{n}a}$. Thus, we may write (27) as $$\displaystyle\left.\vphantom{(-1)^{\mathrm{ind}g}}*\varphi_{\mathbf{n}a}\right|_{\varrho_{\mu\nu}=\Omega_{\mu\nu}}=2\varepsilon\kappa\epsilon_{abcd}\left(e^{b}_{\alpha}e^{c}_{\beta}\Omega^{\alpha\beta}+\frac{1}{3}\phi^{b}\wedge\phi^{c}\right)\wedge\phi^{d}\,.$$ (28) Employing the Gauß-Codazzi equations (11) for vanishing non-metricity, we evaluate $\varphi_{\mathbf{n}a}$ in terms of geometric quantities on the boundary as $$\displaystyle\left.\vphantom{(-1)^{\mathrm{ind}g}}*\varphi_{\mathbf{n}a}\right|_{\mathcal{M}_{4}}=2\varepsilon\kappa\epsilon_{abcd}\left.\left(\Omega^{bc}-\varepsilon K^{b}\wedge K^{c}+\frac{1}{3}\phi^{b}\wedge\phi^{c}\right)\wedge\phi^{d}\right|_{\mathcal{M}_{4}}\,,$$ (29) where $\Omega\indices{{}^{a}_{b}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{d}\omega^{a}_{b}+\omega^{a}_{c}\wedge\omega^{c}_{b}$ was defined in (11). By virtue of (13) the GHY term of Lovelock-Chern-Simons gravity is then $$\displaystyle S_{\mathrm{GHY}}^{\mathrm{LCS}}=-4\kappa\int_{\mathcal{M}_{4}}\left.\epsilon_{abcd}\left(\Omega^{ab}-\varepsilon K^{a}\wedge K^{b}+\frac{1}{3}\phi^{a}\wedge\phi^{b}\right)\wedge\phi^{c}\wedge K^{d}\right|_{\mathcal{M}_{4}}\,.$$ (30) In components (30) takes the form $$\displaystyle S_{\mathrm{GHY}}^{\mathrm{LCS}}=-4\kappa\int_{\mathcal{M}_{4}}\mathrm{d}\mathrm{Vol}_{\mathcal{M}_{4}}\sqrt{-\gamma}\left[3!\left(\frac{1}{2}R\indices{{}^{a}{}^{b}_{[ab}}-\varepsilon K\indices{{}^{a}_{[a}}K\indices{{}^{b}_{b}}\right)K\indices{{}^{c}_{c]}}+2K\indices{{}^{a}_{a}}\right]\,.$$ (31) General considerations regarding (30) show it is consistent with expectations. To see this, consider the Lovelock-Chern-Simons action (26) in more detail. This action consists of three terms. The first of them which is a quadratic curvature term in five dimensions is new to us so far. Because of its quadratic curvature form we expect it to yield a GHY term of a new form which is of higher order in the extrinsic curvature just as we observed it in the example of $4$d Chern-Simons modified gravity in section 3.2. We immediately observe this behaviour from the generalized Gauß-Codazzi equations (11) which give the curvature $3+1$ decomposition. The third term in (26) has no curvature contribution and is not expected to yield a GHY contribution since these contributions all relate to variations of curvature, see (12). The remaining second term we already know. In fact, we rewrite it as $$\displaystyle S^{\mathrm{LCS,\leavevmode\nobreak\ 2nd\leavevmode\nobreak\ term}}=\kappa\int_{\mathcal{M}_{5}}\epsilon_{\mu\nu\alpha\beta\sigma}\frac{2}{3}\Omega^{\mu\nu}\wedge\theta^{\alpha}\wedge\theta^{\beta}\wedge\theta^{\sigma}=8\kappa\int_{\mathcal{M}_{5}}\sqrt{-g}\,\eta^{\mu\nu}\wedge\Omega_{\mu\nu}\,,$$ (32) where $\eta^{\mu\nu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=*\left(\theta^{\mu}\wedge\theta^{\nu}\right)$ was defined in section 3.1. Note that this is nothing but a 5-dimensional Einstein-Hilbert action. In section 3.1 we already noticed that actions of this form always have the same GHY term regardless of their dimension. Hence, we expect a contribution to the total Lovelock-Chern-Simons GHY term of the type $$\displaystyle S_{\mathrm{GHY}}^{\mathrm{LCS,\leavevmode\nobreak\ 2nd\leavevmode\nobreak\ term}}\propto\int_{\partial\mathcal{M}}\mathrm{d}\text{Vol}_{\partial\mathcal{M}}\sqrt{|\gamma|}K\indices{{}^{a}_{a}}\,.$$ (33) We only write a proportionality instead of an equality in the latter equation since there could be additional contributions from the quadratic curvature term which we omit in this consistency consideration. We observe all of these expected contributions in our result (31). Some results regarding the GHY term for Lovelock-Chern-Simons theory already exist in the literature. Namely, the action of Lovelock-Chern-Simons (26) was renormalized in [47]. To achieve this, the authors of [47] considered a finite Fefferman-Graham expansion and compensated the divergent terms by addition of counterterms to the action. The non-divergent terms in this procedure were not given explicitly in [47] but calculated in [16], where they were interpreted as the GHY term of Lovelock-Chern-Simons gravity. This term constitutes an on-shell result whereas our result (30) is the universal GHY term for the Lovelock-Chern-Simons action (26). From this point of view, our result (30) may be considered as a generalization of the Lovelock-Chern-Simons GHY term in [16]. Our results for the GHY term of Lovelock-Chern-Simons constitute a new result in the literature. An additional novel result can be found in appendix A, where we derive the GHY term for the most general MAG Lagrangian quadratic in the fields. In the same appendix we include details on how to treat interior products and Hodge duals under the variations necessary to apply our algorithm. These allow to work directly in the differential form version of the Lagrangian, which usually provides the most economical expression for the Lagrangian. Before proceeding to appendix A, we suggest reading section 4. This section contains the explicit derivation of the general formula for the GHY term (12) along with some useful formulae regarding the $3+1$ decomposition of the field strengths. 4 Derivation of the Gibbons-Hawking-York term In this section we present the explicit calculation leading to the GHY term (12) given in section 2. The essence of the calculation of the GHY term involves the $3+1$ decomposition of the fundamental fields, $\omega^{\mu}_{\nu}$, $g_{\mu\nu}$ and $\theta^{\mu}$, and their field strengths, $\Omega\indices{{}^{\mu}_{\nu}}$, $Q_{\mu\nu}$ and $T^{\mu}$ respectively. In order to derive such a decomposition, we first consider the geometric setup for spacetime foliations introduced in section 2 in more detail. This involves an investigation of how the frame fields $\theta^{\mu}$, $n^{\mu}$ and $e^{a}_{\mu}$ decompose with respect to the spacetime foliation. 4.1 Frame decomposition In this subsection, we first introduce frame decompositions on manifolds in the vielbein formalism. These frame decompositions constitute the basis for the subsequent investigation of foliations in the $3+1$ formalism [48, 42, 41], in which the manifold is described as a stack of hypersurfaces. We assume that neither torsion nor non-metricity vanishes and use differential forms throughout to simplify the analysis (see e.g. [40, 23, 49]). We consider an $n$-dimensional manifold $\mathcal{M}$ equipped with a generic coframe $\theta^{\mu}$, $\mu\in\{0,\dots,n-1\}$. Locally, a coframe is a basis of the cotangent space at each point $p$ of $\mathcal{M}$. In order to describe how the boundary is embedded in $\mathcal{M}$, we aim at decomposing all geometric quantities on $\mathcal{M}$ into contributions tangent and normal to the boundary. For this reason, we assume that $\mathcal{M}$ is foliated by a family of hypersurfaces $\{\Sigma_{\lambda}\}_{\lambda}$, where $\lambda=$const defines each hypersurface $\Sigma_{\lambda}$. In order to be able to describe the embedding of the boundary as well as the boundary components of the manifold’s geometric quantities torsion, curvature, and non-metricity, we choose this foliation such that the asymptotic boundary $\partial\mathcal{M}$ is one of the hypersurfaces. The most basic quantity to decompose in this foliation is the coframe $\theta^{\mu}$. We associate a frame field $\vartheta_{\mu}$ to this coframe by means of $\theta^{\mu}(\vartheta_{\nu})=\delta^{\mu}_{\nu}=\vartheta_{\nu}(\theta^{\mu})$, and decompose both according to $$\displaystyle\theta^{\mu}$$ $$\displaystyle=e^{\mu}_{a}\tilde{\phi}^{a}+t^{\mu}\phi\,,$$ (34a) $$\displaystyle\vartheta_{\mu}$$ $$\displaystyle=E^{a}_{\mu}\varphi_{a}+\frac{\varepsilon}{N}n_{\mu}\tilde{\varphi}\,,$$ (34b) where $a\in\{0,\dots,n-2\}$. The new tensors introduced in this decomposition are given by $$\displaystyle e^{\mu}_{a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\theta^{\mu}(\varphi_{a})\,,\qquad E^{a}_{\mu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\vartheta_{\mu}(\tilde{\phi}^{a})\,,\qquad t^{\mu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\theta^{\mu}(\tilde{\varphi})\,.$$ (35) Using the induced metric on $\Sigma_{\lambda}$, $\gamma_{ab}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{\mu}_{a}e^{\nu}_{b}g_{\mu\nu}$, we define $$\displaystyle\varepsilon\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{sgn}\left(\frac{\det g_{\mu\nu}}{\det\gamma_{ab}}\right)\,,\qquad N\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=1/\sqrt{|g^{\mu\nu}\vartheta_{\mu}(\phi)\vartheta_{\nu}(\phi)|}\,,\qquad n_{\mu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\varepsilon N\vartheta_{\mu}(\phi)\,.$$ (36) The lapse function $N$ is introduced to normalize $n_{\mu}$, $n_{\mu}n^{\mu}=\varepsilon$. Moreover, the sign $\varepsilon$ in $n_{\mu}$ is a convenient choice for the direction of $n_{\mu}$ (see [41]), such that $\varepsilon^{2}=1$. We impose orthogonality for the frame decomposition (4.1) by virtue of $$\displaystyle\tilde{\phi}^{a}(\varphi_{b})=\delta^{a}_{b}=\varphi_{b}(\tilde{\phi}^{a})\,,\qquad\phi(\tilde{\varphi})=1=\tilde{\varphi}(\phi)\,,$$ (37) with all other pairings between the $\phi$ and $\varphi$ frames vanishing, for instance $\phi(\varphi_{a})=0$. These pairings imply that the tensors $e^{\mu}_{a}$, $E^{a}_{\mu}$, $t^{\mu}$ and $n_{\mu}$ are not independent of each other. To see this, we insert the decompositions of frame and coframe (4.1) into their defining relation $\theta^{\mu}(\vartheta_{\nu})=\delta^{\mu}_{\nu}$ to obtain $$\displaystyle\delta^{\mu}_{\nu}=e^{\mu}_{a}E^{a}_{\nu}+\frac{\varepsilon}{N}t^{\mu}n_{\nu}\,.$$ (38) Next, we contract this equation with $n^{\mu}$ as well as $e_{\nu}^{a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\gamma^{ab}g_{\mu\nu}e^{\mu}_{b}$ to find the relations $$\displaystyle t^{\mu}=Nn^{\mu}+N^{a}e^{\mu}_{a}\quad\text{ and }\quad E^{a}_{\mu}=e^{a}_{\mu}-\frac{\varepsilon}{N}n_{\mu}t^{\nu}e^{a}_{\nu}=e^{a}_{\mu}-\frac{\varepsilon}{N}n_{\mu}N^{a}-\varepsilon n_{\mu}n^{\nu}e^{a}_{\nu}\,,$$ (39) where we define the shift vector $N^{a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-Nn^{\mu}E^{a}_{\mu}$. Note that the expressions (39) relate the fields $e^{\mu}_{a}$, $E^{a}_{\mu}$, $t^{\mu}$ and $n_{\mu}$ with each other. Nevertheless, their relation may be examined further by means of777Equation (40) can be proven by expressing the completeness relation (38) in terms of $e^{\mu}_{a}$, $n^{\mu}$ via (39) to find $\delta^{\mu}_{\nu}=e^{\mu}_{a}e^{a}_{\nu}+\varepsilon n^{\mu}n_{\nu}-\varepsilon n^{\rho}e_{\rho}^{a}e_{a}^{\mu}n_{\nu}$. Using this, we can write $n_{\mu}e^{\mu}_{a}=n_{\mu}\delta^{\mu}_{\nu}e^{\nu}_{a}=n_{\mu}e^{\mu}_{a}(\varepsilon n^{\rho}e_{\rho}^{b}n_{\nu}e^{\nu}_{b})$. This implies $n_{\mu}e^{\mu}_{a}=0$, since $1=\varepsilon n^{\rho}e^{a}_{\rho}n_{\nu}e^{\nu}_{a}$ leads to $1=0$ when we take the trace of the completeness relation. $$\displaystyle n_{\mu}e^{\mu}_{a}=0$$ (40) which implies that $t^{\mu}E_{\mu}^{a}=0$ holds, as well. Using the orthogonality conditions (37) and (40), we may revisit the relations (38) and (39) and rewrite them as $$\displaystyle\delta^{\mu}_{\nu}$$ $$\displaystyle=e^{\mu}_{a}e^{a}_{\nu}+\varepsilon n^{\mu}n_{\nu}\,,$$ (41a) $$\displaystyle t^{\mu}$$ $$\displaystyle=Nn^{\mu}+N^{a}e^{\mu}_{a}\,,$$ (41b) $$\displaystyle E^{a}_{\mu}$$ $$\displaystyle=e^{a}_{\mu}-\frac{\varepsilon}{N}n_{\mu}N^{a}\,.$$ (41c) In the following section we use these equations to find a version of the frame decomposition (4.1) which is adapted to the foliation of the manifold $\mathcal{M}$ by means of hypersurfaces. 4.2 Adapted frame decomposition In the frame decomposition (4.1) we have introduced four a priori independent tensors $e^{\mu}_{a}$, $E^{a}_{\mu}$, $t^{\mu}$ and $n_{\mu}$. We imposed orthogonality conditions in (37) and saw that they allow to express $E^{a}_{\mu}$ and $t^{\mu}$ in terms of $e^{\mu}_{a}$ and $n_{\mu}$ as in (4.1). Thus, we use the relations (4.1) to eliminate $E^{a}_{\mu}$ and $t^{\mu}$ in the original frame decomposition (4.1), to obtain $$\displaystyle\theta^{\mu}$$ $$\displaystyle=e^{\mu}_{a}\phi^{a}+Nn^{\mu}\phi\,,$$ (42a) $$\displaystyle\vartheta_{\mu}$$ $$\displaystyle=e^{a}_{\mu}\varphi_{a}+\frac{\varepsilon}{N}n_{\mu}\varphi\,,$$ (42b) where we introduced the adapted frame $$\displaystyle\phi^{a}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\tilde{\phi}^{a}+N^{a}\phi\,,\qquad\varphi\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\tilde{\varphi}-N^{a}\varphi_{a}\,.$$ (43) The pairings (37) of the $(\phi^{a},\phi,\varphi_{a},\varphi)$ frames are invariant under this transformation and read $$\displaystyle\phi^{a}(\varphi_{b})=\delta^{a}_{b}=\varphi_{b}(\phi^{a})\,,\qquad\phi(\varphi)=1=\varphi(\phi)$$ (44) with all remaining pairings vanishing. Hence, the new frame fields $(\phi^{a},\phi,\varphi_{a},\varphi)$ are orthogonal as well. To gain an intuition for what these fields are, consider a generic one-form $A=A_{\mu}\theta^{\mu}$ and a vector $B=B^{\mu}\vartheta_{\mu}$. We use (4.2) to expand $A$ and $B$ as $$\displaystyle A$$ $$\displaystyle=A_{\mu}e^{\mu}_{a}\phi^{a}+NA_{\mu}n^{\mu}\phi\,,$$ (45a) $$\displaystyle B$$ $$\displaystyle=B^{\mu}e^{a}_{\mu}\varphi_{a}+\frac{\varepsilon}{N}B^{\mu}n_{\mu}\varphi\,.$$ (45b) Recall that a tensor or differential form is called tangent (normal) to a hypersurface if it vanishes when any of its indices is contracted with $n_{\mu}$ ($e^{\mu}_{a}$). We apply these definition to (4.2) and find that tangent one-forms and vectors take the form $$\displaystyle A=A_{\mu}e^{\mu}_{a}\phi^{a}\,,\qquad B=B^{\mu}e^{a}_{\mu}\varphi_{a}$$ (46) while normal ones are expanded as $$\displaystyle A=NA_{\mu}n^{\mu}\phi\,,\qquad B=\frac{\varepsilon}{N}B^{\mu}n_{\mu}\varphi\,.$$ (47) Combining these notions of being tangent and normal with the orthogonality conditions (44), we may interpret $\varphi_{a}$ as a frame basis on $\Sigma_{\lambda}$ with associated coframe basis $\phi^{a}$. Hence, the frame decomposition (4.2) is adapted to the foliation and we proceed to work with it in the following. Having set up the frame decomposition, we proceed by investigating how fields transform. 4.3 Decomposition of metric and connection In the previous subsections we have investigated the decomposition of the frame fields in the foliation of the manifold $\mathcal{M}$ by means of hypersurfaces. We use this decomposition next to investigate how the basic geometric objects on $\mathcal{M}$ decompose in that foliation. In particular, apart from the frame field, we consider the metric tensor $g_{\mu\nu}$ as well as the connection one-form $\omega^{\mu}_{\nu}$ as independent dynamical fields. We start by investigating the decomposition of $g_{\mu\nu}$ by applying the adapted frame decomposition (4.2). Contracting (4.2) with $e^{\mu}_{a}$ and $n_{\mu}$, we exploit the normality condition (40), $n_{\mu}e^{\mu}_{a}=0$, to identify $\phi=\frac{\varepsilon}{N}n_{\mu}\theta^{\mu}$ and $\phi^{a}=e^{a}_{\mu}\theta^{\mu}$. We use the manifold metric $g_{\mu\nu}$ and the hypersurface metric $\gamma_{ab}=e^{\mu}_{a}e^{\nu}_{b}g_{\mu\nu}$ to raise and lower indices in $e^{a}_{\mu}=\gamma^{ab}g_{\mu\nu}e^{\nu}_{b}$. Using these relations, we rewrite the metric tensor by means of (4.2) as $$\displaystyle\mathrm{d}s^{2}=g_{\mu\nu}\theta^{\mu}\otimes\theta^{\nu}=\left(\gamma_{ab}e^{a}_{\mu}e^{b}_{\nu}+\varepsilon n_{\mu}n_{\nu}\right)\theta^{\mu}\otimes\theta^{\nu}\,.$$ (48) From (48) we can read of the $3+1$ decomposition of the metric tensor $g_{\mu\nu}$ as $$\displaystyle g_{\mu\nu}=\gamma_{ab}e^{a}_{\mu}e^{b}_{\nu}+\varepsilon n_{\mu}n_{\nu}$$ (49) in hypersurface tangent and normal contributions. The decomposition (49) is used heavily in the following discussion. For deriving the transformation of the connection one-form $\omega^{\mu}_{\nu}$ to the hypersurfaces, we use the transformation law of connections with respect to the metric-affine group. For $\Lambda\indices{{}^{\mu}_{\nu}}\in$ GL(n,$\mathbb{R}$), this transformation is given by [23] $$\displaystyle\omega^{\mu}_{\nu}\rightarrow{\omega^{\prime}}^{\mu}_{\nu}=\Lambda\indices{{}^{-1}{}^{\mu}_{\rho}}\omega^{\rho}_{\sigma}\Lambda\indices{{}^{\sigma}_{\nu}}+\Lambda\indices{{}^{-1}{}^{\mu}_{\rho}}\mathrm{d}\Lambda\indices{{}^{\rho}_{\nu}}\,.$$ (50) Motivated by the frame decomposition (4.2) we choose the transformation $$\displaystyle\begin{split}\Lambda\indices{{}^{\mu}_{\nu}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=&e^{\mu}_{a}\delta^{a}_{\nu}+Nn^{\mu}\delta_{\nu}^{n-1}\,,\\ \Lambda\indices{{}^{-1}{}^{\mu}_{\nu}}=&e^{a}_{\nu}\delta^{\mu}_{a}+\frac{\varepsilon}{N}n_{\nu}\delta^{\mu}_{n-1}\,.\end{split}$$ (51) After inserting this transformation into (50), we only consider the components $\mu,\nu=(0,\dots,n-2)$ to find $$\displaystyle\omega^{b}_{a}=e^{b}_{\mu}\left(\mathrm{d}e^{\mu}_{a}+\omega^{\mu}_{\nu}e^{\nu}_{a}\right)\qquad\Leftrightarrow\qquad e^{b}_{\mu}De^{\mu}_{a}=0\,,$$ (52) where we suppressed the prime on the left hand side since the indices immediately clarify which connection is meant here. This is the transformation of the connection on $\mathcal{M}$ to the connection on $\Sigma_{\lambda}$. Essentially, the above argument is a proof of the vielbein postulate for the frame field on the hypersurface for MAG, along the lines of a similar proof in [23]. We interpret the hypersurface connection coefficients by means of (52) as the contribution of $De^{\mu}_{a}$ which is tangent to the hypersurfaces. It is, thus, natural to consider its normal contributions next to obtain an expression for $De^{\mu}_{a}$, which constitutes a differential form version of the Gauß-Weingarten equation. 4.4 Foundations of the 3+1 formalism in general frames In this section we derive the fundamental equations of the $3+1$ formalism in terms of differential forms. This involves the decomposition of $De^{\mu}_{a}$ into contributions normal and tangent to the hypersurfaces in the foliation of the manifold $\mathcal{M}$, which can be considered as a generalized version of the Gauß-Weingarten equations. We furthermore derive a similar equation for the normal vector $n^{\mu}$, and take exterior covariant derivatives of these relations to derive generalized versions of the Ricci identities. This set of equations is the foundation of the $3+1$ formalism in differential forms. We start our calculation by investigating $De^{\mu}_{a}$. While we found the tangent contributions of this one-form in (52), we still need to calculate its hypersurface normal parts. To that end we define a one-form corresponding to the extrinsic curvature tensor $K\indices{{}^{a}_{b}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{a}_{\mu}e^{\nu}_{b}\nabla_{\nu}n^{\mu}$. Due to non-vanishing non-metricity, we furthermore define extrinsic curvature in a slightly different manner by $\tilde{K}_{ab}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e_{a}^{\mu}e^{\nu}_{b}\nabla_{\nu}n_{\mu}$, and we work with both definitions interchangeably in the following. Thus, we also define two different extrinsic curvature one-forms by $$\displaystyle\begin{split}K^{a}&\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{a}_{\mu}Dn^{\mu}\quad\text{and}\\ \tilde{K}_{a}&\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{\mu}_{a}Dn_{\mu}\,.\end{split}$$ (53) We use the frame decomposition (4.2) to calculate the components of these quantities in our foliation, $$\displaystyle\begin{split}K^{a}&=K\indices{{}^{a}_{b}}\phi^{b}+Ne^{a}_{\mu}a^{\mu}\phi\,,\\ \tilde{K}_{a}&=\tilde{K}_{ab}\phi^{b}+Ne^{\mu}_{a}\tilde{a}_{\mu}\phi\,,\end{split}$$ (54) where we defined $a^{\mu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\nabla_{\mathbf{n}}n^{\mu}$ and $\tilde{a}_{\mu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\nabla_{\mathbf{n}}n_{\mu}$ analogous to the different notions of extrinsic curvature. With the extrinsic curvature one-form at hand, we now investigate the covariant derivative of generic forms $A^{\mu}=e_{a}^{\mu}A^{a}$ tangent to $\Sigma_{\lambda}$. We use the $3+1$ decomposition of the metric (49) and the vielbein postulate (52) to evaluate $$\displaystyle DA^{\mu}=D\left(g^{\mu}_{\nu}A^{\nu}\right)=\left(e^{\mu}_{a}e^{a}_{\nu}+\varepsilon n^{\mu}n_{\nu}\right)DA^{\nu}=e^{\mu}_{a}DA^{a}-\varepsilon n^{\mu}e^{\nu}_{a}Dn_{\nu}\wedge A^{a}\,.$$ (55) At the same time $A^{\mu}$ being tangent to the hypersurface implies $$\displaystyle DA^{\mu}=D\left(e_{a}^{\mu}A^{a}\right)=De^{\mu}_{a}\wedge A^{a}+e^{\mu}_{a}DA^{a}\,.$$ (56) We compare both equations and noting they hold for arbitrary $A^{a}$, we find $$\displaystyle De^{\mu}_{a}=-\varepsilon n^{\mu}\tilde{K}_{a}\,.$$ (57) This is the expression of the Gauß-Weingarten equation in terms of differential forms. We can derive a similar expression like the Gauß-Weingarten equation for the normal vector $n^{\mu}$. To that end, we first need to define the non-metricity one-form, $$\displaystyle Q_{\mu\nu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-Dg_{\mu\nu}=Q_{\mu\nu\rho}\theta^{\rho}\,,$$ (58) with its tensor components given by $Q_{\mu\nu\rho}=-\nabla_{\rho}g_{\mu\nu}$. Abbreviating $Q_{\mathbf{n}\mathbf{n}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=n^{\mu}n^{\nu}Q_{\mu\nu}$, we investigate $0=D\varepsilon$ to obtain $$\displaystyle n_{\mu}Dn^{\mu}=\frac{1}{2}Q_{\mathbf{n}\mathbf{n}}\,.$$ (59) We use this result for the calculation of $Dn^{\mu}$ and follow the same steps as described in the case of $De^{\mu}_{a}$ above to find $$\displaystyle Dn^{\mu}=e^{\mu}_{a}K^{a}+\frac{\varepsilon}{2}n^{\mu}Q_{\mathbf{n}\mathbf{n}}\,.$$ (60) This is the hypersurface normal Gauß-Weingarten equation. The last missing fundamental equations of the differential geometric $3+1$ decomposition are the Ricci identities for $e^{\mu}_{a}$ and $n^{\mu}$. Straightforward evaluation of the second covariant exterior derivatives of $e^{\mu}_{a}$ and $n^{\mu}$ yields $$\displaystyle D^{2}e^{\mu}_{a}$$ $$\displaystyle=e^{\nu}_{a}\Omega\indices{{}^{\mu}_{\nu}}-e^{\mu}_{b}\Omega\indices{{}^{b}_{a}}\,,$$ (61a) $$\displaystyle D^{2}n^{\mu}$$ $$\displaystyle=\Omega\indices{{}^{\mu}_{\nu}}n^{\nu}\,,$$ (61b) where the curvature two-form on $\mathcal{M}$ is given by $$\displaystyle\Omega\indices{{}^{\mu}_{\nu}}$$ $$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{d}\omega^{\mu}_{\nu}+\omega^{\mu}_{\rho}\wedge\omega^{\rho}_{\nu}=\frac{1}{2}R\indices{{}^{\mu}_{\nu}{}_{\rho}{}_{\sigma}}\theta^{\rho}\wedge\theta^{\sigma}\,,$$ (62) and we defined $\Omega\indices{{}^{a}_{b}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{d}\omega^{a}_{b}+\omega^{a}_{c}\wedge\omega^{c}_{b}$, the boundary curvature, analogously. The tensor components $R\indices{{}^{\mu}_{\nu}{}_{\rho}{}_{\sigma}}$ on the right hand side of (62) are those of the Riemann curvature tensor. The equations we have derived in this section are sufficient to derive the Gauß-Codazzi equations, which are at the heart of the $3+1$ formalism. They describe how curvature decomposes into contributions normal and tangent to the hypersurfaces of the foliation. We perform the derivation of these equations and their analogues for torsion and non-metricity in the following subsection. We begin by investigating the $3+1$ decomposition of non-metricity. 4.5 Decomposition of the field strengths In order to examine which boundary terms the non-metricity one-form $Q_{\mu\nu}$ admits, we decompose this differential form into contributions normal and tangent to the hypersurfaces given by the foliation of $\mathcal{M}$. For this derivation we make extensive use of the $3+1$ decompositions derived in the previous subsections, among which the generalized Gauß-Weingarten equations (57), (60) are particularly important. We start our investigation by using (41a) to decompose uncontracted manifold indices of a differential form $A_{\mu}$ as $$\displaystyle A_{\mu}=\delta^{\nu}_{\mu}A_{\nu}=e_{\mu}^{a}e^{\nu}_{a}A_{\nu}+\varepsilon n_{\mu}n^{\nu}A_{\nu}\,.$$ (63) Analogous decompositions hold for contravariant indices. We apply this index decomposition to the non-metricity one-form $Q_{\mu\nu}$ to obtain888We use symmetrization as $A_{(\mu_{1}\dots\mu_{p})}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{1}{p!}\sum_{\sigma\in\mathcal{S}}A_{\sigma(\mu_{1})\dots\sigma(\mu_{p})}$ over all permutations $\sigma$ in the permutation group $\mathcal{S}$. Likewise, we define $A_{[\mu_{1}\dots\mu_{p}]}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{1}{p!}\sum_{\sigma\in\mathcal{S}}\mathrm{sgn}(\sigma)A_{\sigma(\mu_{1})\dots\sigma(\mu_{p})}$ as antisymmetrization, where $\mathrm{sgn}(\sigma)$ equals $+1$ if $\sigma$ consists of an even number of transpositions and $-1$ else. $$\displaystyle Q_{\mu\nu}=e_{\mu}^{a}e_{\nu}^{b}\left(e_{a}^{\alpha}e_{b}^{\beta}Q_{\alpha\beta}\right)+2\varepsilon e_{(\mu}^{a}n_{\nu)}\left(e_{a}^{\alpha}n^{\beta}Q_{\alpha\beta}\right)+n_{\mu}n_{\nu}\left(n^{\alpha}n^{\beta}Q_{\alpha\beta}\right)\,.$$ (64) To make the decomposition (64) useful, we recall the definition of non-metricity (58) and use the $3+1$ decomposition of the metric (49) along with with repeated use of the generalized Gauß-Weingarten equation (57) and its normal counterpart (60). We obtain $$\displaystyle Q_{\mu\nu}=e^{a}_{\mu}e^{b}_{\nu}(-D\gamma_{ab})+2\varepsilon e_{(\mu}^{a}n_{\nu)}\left(K_{a}-\tilde{K}_{a}\right)+2n_{\mu}n_{\nu}n_{\alpha}Dn^{\alpha}\,.$$ (65) This result decomposes the indices of $Q_{\mu\nu}$ into contributions normal and tangent to the hypersurfaces in the foliation of $\mathcal{M}$. In contrast to (64), (65) expresses these contributions solely in terms of fundamental fields on the hypersurface. Hence, the comparison of (65) and (64) yields the $3+1$ decomposition of the non-metricity one-form that reads $$\displaystyle e^{\mu}_{a}e^{\nu}_{b}Q_{\mu\nu}=-D\gamma_{ab}\,,\qquad e^{\mu}_{a}n^{\nu}Q_{\mu\nu}=K_{a}-\tilde{K}_{a}\,,\qquad n^{\mu}n^{\nu}Q_{\mu\nu}=2n_{\mu}Dn^{\mu}\,.$$ (66) This is the decomposition of non-metricity we aim for. The projections on the left hand sides of (66) are expressed through boundary data on the right hand sides. Note that $n_{\mu}Dn^{\mu}$ is a manifold scalar and does not have to be express through boundary data any further. Next we derive the corresponding $3+1$ decompositions for torsion and curvature. Torsion is defined as the field strength of the coframe field as $$\displaystyle T^{\mu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=D\theta^{\mu}=\frac{1}{2}T\indices{{}^{\mu}_{\rho}{}_{\sigma}}\theta^{\rho}\wedge\theta^{\sigma}\,.$$ (67) As in the case of non-metricity, we need to decompose the torsion two-form into contributions which are normal and tangent to the hypersurfaces w.r.t. the foliation of the manifold. The calculation proceeds identically to that of the non-metricity decomposition above. Its result is that the torsion two-form is decomposed as $$\displaystyle e^{a}_{\mu}T^{\mu}$$ $$\displaystyle=D\phi^{a}+NK^{a}\wedge\phi=T^{a}+NK^{a}\wedge\phi\,,$$ (68a) $$\displaystyle n_{\mu}T^{\mu}$$ $$\displaystyle=-\tilde{K}_{a}\wedge\phi^{a}+\varepsilon D(N\phi)+\frac{N}{2}Q_{\mathbf{n}\mathbf{n}}\wedge\phi\,,$$ (68b) with $T^{a}$ the torsion 2-form on the hypersurfaces. The final $3+1$ decomposition of the field strengths to consider is the one of the curvature two-form defined in (62). In addition to the methods used for non-metricity and torsion we exploit the Ricci identities (4.4) for $e^{\mu}_{a}$ and $n^{\mu}$ in the derivation of the curvature decomposition. Apart from that, the analysis proceeds similar to the one of non-metricity and yields $$\begin{aligned} \vphantom{\frac{1}{2}}e_{\mu}^{a}e^{\nu}_{b}\Omega\indices{{}^{\mu}_{\nu}}&=\Omega\indices{{}^{a}_{b}}-\varepsilon K^{a}\wedge\tilde{K}_{b}\,,\\ e^{a}_{\mu}n^{\nu}\Omega\indices{{}^{\mu}_{\nu}}&=DK^{a}+\frac{\varepsilon}{2}K^{a}\wedge Q_{\mathbf{n}\mathbf{n}}\,,\end{aligned}\qquad\begin{aligned} n_{\mu}e^{\nu}_{a}\Omega\indices{{}^{\mu}_{\nu}}&=-D\tilde{K}_{a}+\frac{\varepsilon}{2}\tilde{K}_{a}\wedge Q_{\mathbf{n}\mathbf{n}}\,,\\ n_{\mu}n^{\nu}\Omega\indices{{}^{\mu}_{\nu}}&=\frac{1}{2}DQ_{\mathbf{n}\mathbf{n}}+K^{a}\wedge\tilde{K}_{a}\,.\end{aligned}$$ (69) These equations are generalizations of the Gauß-Codazzi equations. As we have seen in section 3, they are particularly relevant for the calculation of GHY terms if a specific Lagrangian $\mathcal{L}$ is considered. In particular, the calculation of GHY terms which are of second or higher order in curvature always involves terms like $e^{a}_{\mu}n^{\nu}\Omega\indices{{}^{\mu}_{\nu}}$ which are evaluated by means of (69). Moreover, as we show next, the $3+1$ decompositions of curvature, torsion and non-metricity which we have just derived are sufficient for calculating the GHY term for generic Lagrangians which are polynomials or Taylor series of these field strengths. 4.6 Post-Riemannian Gibbons-Hawking-York term This section follows [43] to derive a generalized GHY term from the foliations of curvature, torsion and non-metricity. For the sake of generality we consider an action of the form $$\displaystyle S_{\mathrm{orig}}[g_{\mu\nu},\omega^{\mu}_{\nu},\theta^{\mu}]=\int_{\mathcal{M}}\mathcal{L}(\Omega\indices{{}^{\mu}_{\nu}},T^{\mu},Q_{\mu\nu})\,,$$ (70) where the Lagrangian $\mathcal{L}$ is a polynomial in curvature $\Omega\indices{{}^{\mu}_{\nu}}$, torsion $T^{\mu}$ and non-metricity $Q_{\mu\nu}$.999Derivatives of $\Omega\indices{{}^{\mu}_{\nu}}$, $T^{\mu}$ or $Q_{\mu\nu}$ may be reduced to polynomials by means of the Bianchi identities (2). We introduce Lagrange multipliers $\varphi\indices{{}_{\mu}^{\nu}}$, $t_{\mu}$ and $q^{\mu\nu}$ in order to linearize $\cal L$ and derive the GHY term more easily. $$\displaystyle\begin{split}S[g_{\mu\nu},\omega^{\mu}_{\nu},\theta^{\mu},\varphi\indices{{}^{\mu}_{\nu}},\varrho\indices{{}^{\mu}_{\nu}},t_{\mu},\tau^{\mu},q^{\mu\nu},&\sigma_{\mu\nu}]\\ =\int_{\mathcal{M}}&\left[\mathcal{L}(\varrho\indices{{}^{\mu}_{\nu}},\tau^{\mu},\sigma_{\mu\nu})+*\varphi\indices{{}_{\mu}^{\nu}}\wedge(\Omega\indices{{}^{\mu}_{\nu}}-\varrho\indices{{}^{\mu}_{\nu}})\right.\\ &\left.\vphantom{*\varphi\indices{{}_{\mu}^{\nu}}}+*t_{\mu}\wedge\left(T^{\mu}-\tau^{\mu}\right)+*q^{\mu\nu}\wedge\left(Q_{\mu\nu}-\sigma_{\mu\nu}\right)\right]\,.\end{split}$$ (71) Integrating out the multipliers shows that the action (71) yields the same equations of motion as (70). The remaining fields $\varrho\indices{{}^{\mu}_{\nu}}$, $\tau^{\mu}$ and $\sigma_{\mu\nu}$ in (71) introduced are auxiliary fields that we demand to have the same symmetries as the corresponding field strength. This symmetry identification is also required for the Lagrange multipliers $\varphi\indices{{}_{\mu}^{\nu}}$, $t_{\mu}$ and $q^{\mu\nu}$ in order to allow equivalence of the equations of motion of (70) and (71) [43]. To proceed we substitute the $3+1$ decompositions of $\Omega\indices{{}^{\mu}_{\nu}}$, $T^{\mu}$ and $Q_{\mu\nu}$ in the action (71) in order to decompose $S$ into boundary tangent and normal contributions and, subsequently, isolate the boundary terms. For this, we write by means of (41a) for the linearized terms in $S$ $$\displaystyle\begin{split}*\varphi\indices{{}_{\mu}^{\nu}}\wedge\Omega\indices{{}^{\mu}_{\nu}}=&\left(e^{\alpha}_{a}e_{\beta}^{b}*\varphi\indices{{}_{\alpha}^{\beta}}\right)\wedge\left(e_{\mu}^{a}e^{\nu}_{b}\Omega\indices{{}^{\mu}_{\nu}}\right)+\varepsilon\left(e^{\alpha}_{a}n_{\beta}*\varphi\indices{{}_{\alpha}^{\beta}}\right)\wedge\left(e_{\mu}^{a}n^{\nu}\Omega\indices{{}^{\mu}_{\nu}}\right)\\ &+\varepsilon\left(n^{\alpha}e_{\beta}^{a}*\varphi\indices{{}_{\alpha}^{\beta}}\right)\wedge\left(n_{\mu}e^{\nu}_{a}\Omega\indices{{}^{\mu}_{\nu}}\right)+\left(n^{\alpha}n_{\beta}*\varphi\indices{{}_{\alpha}^{\beta}}\right)\wedge\left(n_{\mu}n^{\nu}\Omega\indices{{}^{\mu}_{\nu}}\right)\,,\end{split}$$ (72a) $$\displaystyle\begin{split}*t_{\mu}\wedge T^{\mu}=&\left(e^{\nu}_{a}*t_{\nu}\right)\wedge\left(e_{\mu}^{a}T^{\mu}\right)+\varepsilon\left(n^{\nu}*t_{\nu}\right)\wedge\left(n_{\mu}T^{\mu}\right)\,,\end{split}$$ (72b) $$\displaystyle\begin{split}*q^{\mu\nu}\wedge Q_{\mu\nu}=&\left(e_{\alpha}^{a}e_{\beta}^{b}*q^{\alpha\beta}\right)\wedge\left(e^{\mu}_{a}e^{\nu}_{b}Q_{\mu\nu}\right)+\varepsilon\left(e_{\alpha}^{a}n_{\beta}*q^{\alpha\beta}\right)\wedge\left(e_{a}^{\mu}n^{\nu}Q_{\mu\nu}\right)\\ &+\varepsilon\left(n_{\alpha}e^{a}_{\beta}*q^{\alpha\beta}\right)\wedge\left(n^{\mu}e^{\nu}_{a}Q_{\mu\nu}\right)+\left(n_{\alpha}n_{\beta}*q^{\alpha\beta}\right)\wedge\left(n^{\mu}n^{\nu}Q_{\mu\nu}\right)\,.\end{split}$$ (72c) The right hand side of each of these equations can be written in terms of the $3+1$ decompositions of $Q_{\mu\nu}$, $T^{\mu}$ and $\Omega\indices{{}^{\mu}_{\nu}}$ shown in (66), (4.5) and (69). For example, (72a) takes the form $$\displaystyle\begin{split}*\varphi\indices{{}_{\mu}^{\nu}}&\wedge\Omega\indices{{}^{\mu}_{\nu}}\\ &=\left(e^{\alpha}_{a}e_{\beta}^{b}*\varphi\indices{{}_{\alpha}^{\beta}}\right)\wedge\left(\Omega\indices{{}^{a}_{b}}-\varepsilon K^{a}\wedge\tilde{K}_{b}\right)+\varepsilon\left(e^{\alpha}_{a}n_{\beta}*\varphi\indices{{}_{\alpha}^{\beta}}\right)\wedge\left(DK^{a}+\frac{\varepsilon}{2}K^{a}\wedge Q_{\mathbf{n}\mathbf{n}}\right)\\ &+\varepsilon\left(n^{\alpha}e_{\beta}^{a}*\varphi\indices{{}_{\alpha}^{\beta}}\right)\wedge\left(D\tilde{K}_{a}+\frac{\varepsilon}{2}\tilde{K}_{a}\wedge Q_{\mathbf{n}\mathbf{n}}\right)+\left(n^{\alpha}n_{\beta}*\varphi\indices{{}_{\alpha}^{\beta}}\right)\wedge\left(\frac{1}{2}DQ_{\mathbf{n}\mathbf{n}}+K^{a}\wedge\tilde{K}_{a}\right)\,.\end{split}$$ (73) We insert this decomposition and the analogues for torsion and non-metricity into the action (71). The derivative terms yield by means of Stokes’ theorem [50, 51] a boundary action, which reads $$\displaystyle\begin{split}S_{\mathrm{bdy}}=&\int_{\partial\mathcal{M}}\left[\omega^{a}_{b}\wedge\left(e^{\alpha}_{a}e_{\beta}^{b}*\varphi\indices{{}_{\alpha}^{\beta}}\right)+\phi^{a}\wedge\left(e^{\nu}_{a}*t_{\nu}\right)-\gamma_{ab}\left(e_{\alpha}^{a}e_{\beta}^{b}*q^{\alpha\beta}\right)\vphantom{\frac{1}{2}}\right.\\ &\left.\left.\vphantom{\frac{1}{2}}-\varepsilon\tilde{K}_{a}\wedge\left(n^{\alpha}e_{\beta}^{a}*\varphi\indices{{}_{\alpha}^{\beta}}\right)+\varepsilon K^{a}\wedge\left(e^{\alpha}_{a}n_{\beta}*\varphi\indices{{}_{\alpha}^{\beta}}\right)+\frac{1}{2}Q_{\mathbf{n}\mathbf{n}}\wedge\left(n^{\alpha}n_{\beta}*\varphi\indices{{}_{\alpha}^{\beta}}\right)\right]\right|_{\partial\mathcal{M}}\,.\end{split}$$ (74) Note that we use the restriction of the integrand in (74) to $\partial\mathcal{M}$ as abbreviation for its pullback to $\partial\mathcal{M}$. We finally construct the GHY term which cancels the boundary contributions to the action (71). Only some of the terms in (74) actually contribute boundary terms when we consider variations of the action. In particular, the variational principle assumes that $\left.\delta g_{\mu\nu}\right|_{\mathrm{\partial\mathcal{M}}}=0$, $\left.\delta\theta^{\mu}\right|_{\mathrm{\partial\mathcal{M}}}=0$ and $\left.\delta\omega^{\mu}_{\nu}\right|_{\mathrm{\partial\mathcal{M}}}=0$. According to [36] it equivalently suffices to demand that the Dirichlet boundary conditions $\delta\gamma_{ab}=0$, $\delta\phi^{a}=0$ and $\delta\omega^{a}_{b}=0$ hold since the original conditions may be reinstated by gauge transformations on $\partial\mathcal{M}$.101010By the same argument the residual terms of the curvature, torsion and non-metricity foliations which are not exact forms do not contribute to the GHY term. We have, hence, disregarded them in the consideration above. This implies that the GHY term of (70) only has to cancel the contributions from the second line of (74) and, thus, takes the form $$\displaystyle S_{\mathrm{GHY}}=-\int_{\partial\mathcal{M}}\left.\left(-\varepsilon\tilde{K}_{a}\wedge*\varphi^{\mathbf{n}a}+\varepsilon K^{a}\wedge*\varphi_{a\mathbf{n}}+Q_{\mathbf{n}\mathbf{n}}\wedge\frac{1}{2}*\varphi_{\mathbf{n}\mathbf{n}}\right)\right|_{\mathrm{\partial\mathcal{M}}}\,,$$ (75) where we abbreviated $*\varphi^{\mathbf{n}a}\equiv n^{\alpha}e_{\beta}^{a}*\varphi\indices{{}_{\alpha}^{\beta}}$, $*\varphi_{a\mathbf{n}}\equiv e^{\alpha}_{a}n_{\beta}*\varphi\indices{{}_{\alpha}^{\beta}}$ and $*\varphi_{\mathbf{n}\mathbf{n}}\equiv n^{\alpha}n_{\beta}*\varphi\indices{{}_{\alpha}^{\beta}}$. The result (75) is the general form of the GHY term in theories with torsion and non-metricity we aim for. It is the main result of this paper. In particular, (75) is the GHY term for any theory which is constituted by a Lagrangian $\mathcal{L}$ that is, possibly an infinite, polynomial in curvature, torsion and non-metricity. To apply our result to a specific Lagrangian $\mathcal{L}$ in (70), we need to know the Lagrange multipliers $\varphi^{\mathbf{n}a}$, $\varphi_{a\mathbf{n}}$ and $\varphi_{\mathbf{n}\mathbf{n}}$. To obtain these, we write the action (71) in a $3+1$ form in terms of $(\varphi^{\mathbf{n}a},\leavevmode\nobreak\ \varphi_{a\mathbf{n}},\leavevmode\nobreak\ \varphi_{\mathbf{n}\mathbf{n}},\dots)$ instead of $\varphi\indices{{}_{\mu}^{\nu}}$, using (4.6). Then we can read the required components of $\varphi$ from the equations of motion of $\varrho_{\mu\nu}$ $$\displaystyle\begin{split}*\varphi^{\mathbf{n}a}\wedge\delta\varrho_{\mathbf{n}a}&=\varepsilon\delta_{\varrho_{\mathbf{n}a}}\mathcal{L}(\varrho_{\mathbf{n}a},\varrho^{a\mathbf{n}},\varrho_{\mathbf{n}\mathbf{n}},\dots)\,,\\ *\varphi_{a\mathbf{n}}\wedge\delta\varrho^{a\mathbf{n}}&=\varepsilon\delta_{\varrho^{a\mathbf{n}}}\mathcal{L}(\varrho_{\mathbf{n}a},\varrho^{a\mathbf{n}},\varrho_{\mathbf{n}\mathbf{n}},\dots)\,,\\ *\varphi^{\mathbf{n}\mathbf{n}}\wedge\delta\varrho_{\mathbf{n}\mathbf{n}}&=\hphantom{\varepsilon}\delta_{\varrho_{\mathbf{n}\mathbf{n}}}\mathcal{L}(\varrho_{\mathbf{n}a},\varrho^{a\mathbf{n}},\varrho_{\mathbf{n}\mathbf{n}},\dots)\end{split}$$ (76) derived from (71). Through these constraints, we may calculate $S_{\mathrm{GHY}}$ for any given action polynomial in curvature, torsion and non-metricity. We emphasize that (75) and (76) are the only equations necessary to calculate the GHY term for a specific action. The advantage of our method is, thus, not only the universality of our results but furthermore the efficiency of the calculation. We have already observed this efficiency in the examples examined in section 3. Finally we note that the GHY term (75) simplifies considerably if non-metricity is absent as shown below. 4.6.1 Gibbons-Hawking-York term for metric compatible theories Let us simplify the above result for theories which are metric compatible, that is, in which the non-metricity one-form $Q_{\mu\nu}$ vanishes. For metric compatible theories, the Bianchi identity of non-metricity (2), $\Omega_{\mu\nu}+\Omega_{\nu\mu}=DQ_{\mu\nu}$, forces the curvature two-form to be antisymmetric. Since the corresponding Lagrange multipliers $\varphi$ and $\varrho$ are required to have the same symmetry as $\Omega\indices{{}^{\mu}_{\nu}}$, it follows that $\varphi$ is antisymmetric in its two indices. This allows to simplify the Gibbons-Hawking York term as $$\displaystyle S_{\mathrm{GHY}}^{Q=0}=2\int_{\mathrm{\partial\mathcal{M}}}\left.\varepsilon K^{a}\wedge*\varphi_{\mathbf{n}a}\right|_{\mathrm{\partial\mathcal{M}}}$$ (77) in the metric-compatible case. Note that due to the antisymmetry of the curvature two-form, formally we must consider the constraints (76) as not independent. Nevertheless, we may forego this caveat if we treat $\delta\varrho_{\mathbf{n}a}$ and $\delta\varrho_{a\mathbf{n}}$ as independent variations and use the symmetry conditions only after the variational calculus. In appendix B we show explicitly that both methods yield the same result. This concludes the explicit derivation of the GHY boundary term for any action polynomial in curvature, torsion and non-metricity, as anticipated in section 2. We refer to the interpretation of our results in section 2. 5 Summary and discussion In conclusion, in this paper we have accomplished the first step towards the full holographic renormalization of metric affine gravity (MAG) theories, namely the derivation of the Gibbons-Hawking-York term (12) in closed form for any given polynomial action (5) involving curvature, torsion and non-metricity.111111This includes non-polynomial Lagrangians that can be Taylor expanded in the field strengths. In addition, our method for calculating the GHY term is very efficient in practical terms, since it amounts to evaluating a single variation, (10). Interpreting the main result (12), we notice that only explicitly curvature-related terms in the Lagrangian contribute to the GHY boundary terms. If a Lagrangian depends solely on torsion and non-metricity, we do not need to introduce GHY terms in order for the Dirichlet variational problem to be well-defined.121212Of course, boundary terms may arise when these theories must satisfy additional constraints, such as setting the curvature two-form $\Omega\indices{{}^{\mu}_{\nu}}=0$ as in teleparallel theories of gravity [44]. We tested our method explicitly for two theories with known GHY terms, Einstein-Hilbert gravity and 4d Chern-Simons modified gravity, in sections 3.1 and 3.2 respectively. Moreover, we used our method to derive the GHY term for Lovelock-Chern-Simons theory in section 3.3 and for metric-affine gravity in appendix A. Our new result for the Lovelock-Chern-Simons GHY term meets intuitive expectations: Since the Lovelock-Chern-Simons action has a form similar to the Einstein-Hilbert and 4d Chern-Simons modified ones, we expect their GHY terms to also be comparable. This expectation bears out as explained in more detail in section 3.3. We emphasize that for Lovelock-Chern-Simons theory, our result generalizes that of [16] to off-shell field configurations. It would be interesting to evaluate our GHY term on the solution of [16] as a consistency check. The next step towards understanding spin and hypermomentum transport by means of the AdS/CFT correspondence is now to complete the holographic renormalization procedure for MAG theories.131313Recall spin and hypermomentum are dual to torsion and non-metricity respectively [23]. Evaluating the MAG action and the associated GHY term given in appendix A on the torsionful and non-metric AdS Reissner-Nordström solution [52] will provide a hint on what kind of divergences may appear in this procedure. In general, it will be necessary to find the asymptotic expansion for torsion and non-metricity coupled to an asymptotically AdS metric. This will provide us with the holographic dictionary for torsion and non-metricity. We then have to follow e.g. [53] and calculate the regularized on-shell action in order to find all possible divergences, and to construct appropriate counterterms to cancel them. This will allow us to derive the thermodynamic properties of black holes with spin and hypermomentum. It may also enable us to study Hawking-Page type phase transitions [54] relying entirely on spin and hypermomentum. In fact we expect such transitions to exist, since the entropy of a black hole generically decreases when it starts rotating [55]. Holographic renormalization then sets the stage for applying the fluid/gravity correspondence [39, 56] to MAG and for deriving the complete hydrodynamic expansion of strongly-coupled holographic systems with non-trivial spin and hypermomentum transport. We note that non-trivial spin transport in hydrodynamics has already been discussed in [15, 17, 18, 19, 16, 20, 21], but hypermomentum transport is still mostly unexplored. The inclusion of hypermomentum is particularly interesting, since it contains degrees of freedom largely left unexplored in the literature.141414See [57] for a discussion on non-metricity of the Weyl type. These degrees of freedom are expected to lead to interesting changes in the transport properties of matter, since their source, non-metricity, is interpreted geometrically as the modification of the causal structure of spacetime. Since non-metricity is by definition first order in the hydrodynamic derivative expansion, we expect it to contribute to hydrodynamic transport only at second order in derivatives or higher. This makes conformal fluids, where the second order derivative expansion is well-known [58], the most suitable candidate for carrying out this procedure. Apart from the AdS/CFT correspondence, MAG theories are relevant in their own right. For example, they provide us with extensions of Einstein’s theory of relativity based on a gauge principle [23]. Thus, they are a promising standing point for exploring deviations from general relativity in search for clues and constraints helpful for the construction of consistent quantum gravity models. Moreover, torsion appears naturally in supergravity and, hence, top-down string theory constructions [59]. In addition, non-metricity may play a role for conformal gravity as follows: When the non-metricity is pure trace, i.e. $Q_{\mu\nu}=\sigma g_{\mu\nu}$, it may be absorbed into the covariant derivative. This derivative is then metric-compatible and covariant under both diffeomorphisms and Weyl transformations (see e.g. [57]). We expect such derivatives to be useful for the construction of non-local actions such as the Riegert action that gives rise to the Euler term in the four-dimensional conformal anomaly [60], and for obtaining Weyl covariant differential operators [61, 62] and curvature scalars [63]. Finally, restricting ourselves to the realm of classical gravity, our results may be applied to MAG theories to test cosmological implications of deviations from Einstein gravity along the lines suggested in [64]. To conclude, we note that our results may also be useful within the context of studying topological field theories in the presence of torsion and non-metricity. Topological field theories constitute a convenient way of deriving non-dissipative transport properties, in this case for spin and hypermomentum, without the necessity of solving the underlying electronic dynamics (see [7] for instance). For example, we may follow [65] instead of a more complicated entropy current analysis. Our results and specifically our hypersurface decomposition for torsion and non-metricity allows writing down such theories in a spacetime with a timelike boundary. We expect them to be relevant for describing systems relevant for condensed matter physics. 6 Acknowledgments We thank Daniel Grumiller, Umut Gürsoy, Stefan Theisen, and Zhou-Yu Xian for useful discussions. The work in Würzburg is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through Project-ID 258499086—SFB 1170 „ToCoTronics“ and through the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter – ct.qmat Project-ID 390858490—EXC 2147. The research of B.H. is funded by the Bundesministerium für Bildung und Forschung (BMBF, German Federal Ministry of Education and Research) through the Cusanuswerk - Bischöfliche Studienförderung. I.M.’s research is supported by the „Curiosity Driven Grant 2020“ of the University of Genova and by the INFN Scientific Initiatives SFT: „Statistical Field Theory, Low-Dimensional Systems, Integrable Models and Applications“. Anhang A GHY term for metric-affine gravity (MAG) Metric-affine gravity contains non-vanishing non-metricity in addition to curvature and torsion. According to [23, 52] the most general Lagrangian density which is at most quadratic in these fields and parity conserving is $$\displaystyle\begin{split}V_{\mathrm{MAG}}=\frac{1}{2\kappa}\left[\vphantom{\frac{1}{2\kappa}}\right.&-a_{0}R^{\alpha\beta}\wedge\eta_{\alpha\beta}-2\Lambda\eta+T^{\alpha}\wedge*\left(\sum\limits_{I=1}^{3}a_{I}\,{}^{(I)}T_{\alpha}\right)\\ &+2\left(\sum\limits_{I=2}^{4}c_{I}\,{}^{(I)}Q_{\alpha\beta}\right)\wedge\theta^{\alpha}\wedge*T^{\beta}+Q_{\alpha\beta}\wedge*\left(\sum\limits_{I=1}^{4}b_{I}\,{}^{(I)}Q^{\alpha\beta}\right)\\ &+b_{5}\left({}^{(3)}Q_{\alpha\gamma}\wedge\theta^{\alpha}\right)\wedge*\left({}^{(4)}Q^{\beta\gamma}\wedge\theta_{\beta}\right)\left.\vphantom{\frac{1}{2\kappa}}\right]\\ &-\frac{1}{2\rho}R^{\alpha\beta}\wedge*\left[\vphantom{\frac{1}{2\kappa}}\right.\sum\limits_{I=1}^{6}w_{I}\,{}^{(I)}W_{\alpha\beta}+w_{7}\theta_{\alpha}\wedge(e_{\gamma}\rfloor{}^{(5)}W\indices{{}^{\gamma}_{\beta}})\\ &+\sum\limits_{I=1}^{5}z_{I}\,{}^{(I)}Z_{\alpha\beta}+z_{6}\,\theta_{\gamma}\wedge(e_{\alpha}\rfloor{}^{(2)}Z\indices{{}^{\gamma}_{\beta}})+\sum\limits_{I=7}^{9}z_{I}\,\theta_{\alpha}\wedge(e_{\gamma}\rfloor{}^{(I-4)}Z\indices{{}^{\gamma}_{\beta}})\left.\vphantom{\frac{1}{2\kappa}}\right]\,,\end{split}$$ (78) where  $a_{0},\dots,a_{3},b_{1},\dots,b_{5},c_{2},c_{3},c_{4},w_{1},\dots,w_{7},z_{1},\dots,z_{9}$  are dimensionless constants and the curvature two-form $R\indices{{}_{\nu}^{\mu}}$ relates to the definition in (62) as $R\indices{{}_{\nu}^{\mu}}=\Omega\indices{{}^{\mu}_{\nu}}$. The various terms account for the 4+3+11 irreducible contributions to $Q_{\alpha\beta}$, $T^{\alpha}$ and $R\indices{{}_{\alpha}^{\beta}}$ under the (pseudo)orthogonal group, respectively. The explicit form of these irreducible decompositions is given in appendix B of [23]. $e_{\mu}$ denotes the basis dual to $\theta^{\mu}$ and $\rfloor$ is the interior product so that $e_{\mu}\rfloor\theta^{\nu}=\delta^{\nu}_{\mu}$. While $\kappa$ is related to Newton’s constant $G_{n}$ in $n$ dimensions as $\kappa=8\pi G_{n}$, the coefficient $\rho$ controls the curvature squared terms and is, thus, called strong gravity coupling constant. Let us consider the terms in (78) which are proportional to $(2\kappa)^{-1}$ first. The first one of them, $-a_{0}R^{\alpha\beta}\wedge\eta_{\alpha\beta}$, is exactly the Einstein-Hilbert Lagrangian which we already discussed in section 3.1. The remainder of the terms proportional to $(2\kappa)^{-1}$ does not contain curvature and therefore does not contribute to the GHY term as we already noticed in sections 2 and 4. Therefore, the interesting part of the MAG Lagrangian (78) in view of the GHY term is $$\displaystyle\begin{split}V_{\mathrm{MAG},\rho}=&-\frac{1}{2\rho}R^{\alpha\beta}\wedge*\left[\vphantom{\frac{1}{2\kappa}}\right.\sum\limits_{I=1}^{6}w_{I}\,{}^{(I)}W_{\alpha\beta}+w_{7}\theta_{\alpha}\wedge(e_{\gamma}\rfloor{}^{(5)}W\indices{{}^{\gamma}_{\beta}})\\ &+\sum\limits_{I=1}^{5}z_{I}\,{}^{(I)}Z_{\alpha\beta}+z_{6}\,\theta_{\gamma}\wedge(e_{\alpha}\rfloor{}^{(2)}Z\indices{{}^{\gamma}_{\beta}})+\sum\limits_{I=7}^{9}z_{I}\,\theta_{\alpha}\wedge(e_{\gamma}\rfloor{}^{(I-4)}Z\indices{{}^{\gamma}_{\beta}})\left.\vphantom{\frac{1}{2\kappa}}\right]\,.\end{split}$$ (79) Since the number of terms in this action is large compared to the actions considered in 2, some preliminary considerations are in place before we calculate its GHY term. A.1 Variation of Hodge stars and interior products For deriving the GHY term for the MAG Lagrangian we need to insert the explicit forms of the Lagrange multipliers $\varphi^{\mathbf{n}a}$, $*\varphi_{a\mathbf{n}}$ and $\varphi_{\mathbf{n}\mathbf{n}},\dots)$ into the generic GHY result (75). We derive these explicit expressions using the constraints (76) which relate $\varphi^{\mu\nu}$ to the variation of the Lagrangian with respect to $\varrho_{\mu\nu}$. Considering the relevant part of the MAG Lagrangian (79) it is immediately obvious that its variation involves variations of Hodge duals as well as interior products. Therefore, we investigate variations of these objects next. To that effect, we consider differential forms $A,\leavevmode\nobreak\ B$ of the same degree $p$ on an $n$-manifold. For evaluating expressions like $\delta*A\wedge B$ we use the relation [66, 67, 68] $$\displaystyle*A\wedge B=*B\wedge A\,.$$ (80) Furthermore, we employ the result $$\displaystyle\begin{split}(\delta*-*\delta)A=\delta\theta^{\alpha}\wedge(e_{\alpha}\rfloor*A)&-*\left[\delta\theta^{\alpha}\wedge(e_{\alpha}\rfloor A)\right]\\ &+\delta g_{\alpha\beta}\left[\theta^{(\alpha}\wedge(e^{\beta)}\rfloor*A)-\frac{1}{2}g^{\alpha\beta}*A\right]\end{split}$$ (81) of [67] for commuting $\delta$ and $*$. The right hand side of (81) does not contribute at the boundary in the variational principle. Hence, it comes in handy to define $\simeq$ as being equivalent at the boundary and omitting terms which are irrelevant there. (81) therefore implies $$\displaystyle\delta*A\simeq*{}\delta A$$ (82) which we combine with (80) to find $$\displaystyle\delta*A\wedge B\simeq*B\wedge\delta A\,.$$ (83) For the further evaluation of variations in combination with the Hodge duality and the interior product we use the relations [66, 67, 68] $$\displaystyle e_{\alpha}\rfloor A$$ $$\displaystyle=(-1)^{n(p-1)+\mathrm{ind}g}*(\theta_{\alpha}\wedge*A)\,,$$ (84a) $$\displaystyle*(e_{\alpha}\rfloor A)$$ $$\displaystyle=(-1)^{p-1}\theta_{\alpha}\wedge*A\,,$$ (84b) $$\displaystyle e_{\alpha}\rfloor*A$$ $$\displaystyle=*(A\wedge\theta_{\alpha})\,,$$ (84c) $$\displaystyle**A$$ $$\displaystyle=(-1)^{p(n-p)+\mathrm{ind}g}A\,,$$ (84d) where $\mathrm{ind}g$ is the number of negative signs in the signature of $g$. The combination of (82) with (84a) immediately implies $$\displaystyle\delta\left(e_{\mu}\rfloor A\right)\simeq e_{\mu}\rfloor\delta A$$ (85) for the variation of interior products. These are all the expressions for the variation of Hodge duals and interior product which we need to calculate the variations of the terms in the GHY relevant part of the MAG Lagrangian (79). Therefore, we proceed by considering variations of the involved terms separately. A.2 Variation of the MAG Lagrangian As first step of the irreducible decomposition of curvature we follow [23] and decompose the symmetric and antisymmetric parts of the curvature two-form as $$\displaystyle R_{\mu\nu}=W_{\mu\nu}+Z_{\mu\nu}\,,$$ (86a) $$\displaystyle\text{where}\quad W_{\mu\nu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=R_{[\mu\nu]},\quad Z_{\mu\nu}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=R_{(\mu\nu)}\,.$$ (86b) From this decomposition we immediately read off $$\displaystyle\delta R_{\mu\nu}=\delta W_{\mu\nu}+\delta Z_{\mu\nu}$$ (87) which we use to consider the variations of $W$ and $Z$ instead of the $R$ variation. We start our investigation by considering variations of the terms in (79) which include the antisymmetric part of curvature, $W_{\mu\nu}=R_{[\mu\nu]}$. Performing the variations of the irreducible decompositions of $W_{\mu\nu}$ using the methods from the previous subsections we obtain $$\displaystyle R^{\alpha\beta}\wedge\delta*{}^{(I)}W_{\alpha\beta}$$ $$\displaystyle\simeq\delta W^{\alpha\beta}\wedge*{}^{(I)}W_{\alpha\beta}$$ (88) for all $I\in\{1,2,\dots,6\}$, where $\simeq$ omits terms irrelevant for the GHY term calculation. (88) is a non-trivial result as we observe in the case of the term with coefficient $w_{7}$. For this term we obtain $$\displaystyle R^{\alpha\beta}\wedge\delta*$$ $$\displaystyle\left(\theta_{\alpha}\wedge\left(e_{\gamma}\rfloor{}^{(5)}W\indices{{}^{\gamma}_{\beta}}\right)\right)\simeq-\frac{(-1)^{n+\mathrm{ind}g}}{n-2}\delta W_{\mu\nu}\wedge\Theta\indices{{}^{\nu}{}^{\mu}_{[\beta\gamma]}{}^{\gamma}_{\alpha}}\left(*R^{\alpha\beta}\right)\,.$$ (89) The operator $\Theta$ is introduced such that for any differential form $A$ we have $$\displaystyle\Theta_{\mu}(A)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=*\left(\theta_{\mu}\wedge A\right)\qquad\text{and}\qquad\Theta_{\mu_{p}\dots\mu_{1}}(A)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=*\theta_{\mu_{p}}\wedge\Theta_{\mu_{p-1}\dots\mu_{1}}(A)\,.$$ (90) This definition fulfills $\Theta_{\mu_{p}\dots\mu_{1}}\left(\Theta_{\nu_{q}\dots\nu_{1}}(A)\right)=\Theta_{\mu_{p}\dots\mu_{1}\nu_{q}\dots\nu_{1}}(A)$ which we use in the variation of the symmetric curvature contributions that we consider next. We calculate the relevant terms of the $Z_{\mu\nu}$ variation like for the $W_{\mu\nu}$ one. In contrast to the latter each irreducible component of $Z_{\mu\nu}$ appears in two terms in the Lagrangian (79). Before we give the results of the variation we introduce the decomposition $Z_{\alpha\beta}=\cancelto{}{Z}_{\alpha\beta}+\frac{1}{n}g_{\alpha\beta}Z$, where $Z\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=Z\indices{{}^{\alpha}_{\alpha}}$ is the curvature trace. As in the case of $W_{\mu\nu}$ some of the variations simplify significantly as $$\displaystyle R^{\alpha\beta}\wedge\delta*{}^{(I)}Z_{\alpha\beta}$$ $$\displaystyle\simeq\delta\cancelto{}{Z}^{\alpha\beta}\wedge*{}^{(I)}Z_{\alpha\beta}$$ (91) for $I\in\{2,4,5\}$. Prominently, $I=3$ is missing in this list. For the according term we obtain $$\displaystyle R^{\alpha\beta}\wedge\delta*{}^{(3)}Z_{\alpha\beta}$$ $$\displaystyle\simeq\frac{(-1)^{n+\mathrm{ind}g}}{n^{2}-4}\delta\cancelto{}{Z}_{\mu\nu}\wedge\Theta^{\nu\mu}\left(*\left(n(n-2)\Delta-2Z\right)\right)\,.$$ (92) This result does not simplify in the same way as the other terms did since the combination of terms in ${}^{(3)}Z_{\alpha\beta}$ is not trivial. In particular, one of the terms involved in ${}^{(3)}Z_{\alpha\beta}$ is proportional to $g_{\alpha\beta}$ and in the contraction with $R^{\alpha\beta}$ thus yields the factor of $Z\equiv Z\indices{{}^{\alpha}_{\alpha}}$ which we observe in the result. But for casting the result in the way one naively expects from comparison with (91), (92) may solely depend on $\cancelto{}{Z}_{\alpha\beta}$ but not on $Z$. Hence, the structure of ${}^{(3)}Z_{\alpha\beta}$ makes it impossible to rewrite (92) in the same way as the remaining terms. Accordingly, the variation of ${}^{(1)}Z_{\alpha\beta}$ is non-trivial, too, and needs to be evaluated according to $$\displaystyle\begin{split}R^{\alpha\beta}\wedge\delta*{}^{(1)}Z_{\alpha\beta}&=R^{\alpha\beta}\wedge\left[\delta*Z_{\alpha\beta}-\sum_{I=2}^{5}\delta*{}^{(I)}Z_{\alpha\beta}\right]\,.\end{split}$$ (93) The rest of the combinations of curvature terms in the MAG action mixes not only $\cancelto{}{Z}_{\alpha\beta}$ and $Z$ but furthermore $Z_{\alpha\beta}$ and $W_{\alpha\beta}$ terms in general. Hence, there is no simplification like the ones presented above. The results for these variations are $$\displaystyle\begin{split}R^{\alpha\beta}\wedge\delta*&\left(\theta^{\gamma}\wedge\left(e_{\alpha}\rfloor{}^{(2)}Z_{\gamma\beta}\right)\right)\\ \simeq&\frac{1}{2}\delta\cancelto{}{Z}_{\mu\nu}\wedge\theta^{\nu}\wedge\left[-(-1)^{\mathrm{ind}g}*\left(\theta_{\alpha}\wedge\theta_{\gamma}\wedge*\left(\theta^{(\gamma|}\wedge*R^{\alpha|\mu)}\right)\right)\right.\\ &\hskip 62.0pt\left.+\frac{1}{n-2}\Theta\indices{{}^{\mu}_{\beta}}\left(*\left(\theta_{\alpha}\wedge\theta_{\gamma}\wedge*\left(\theta^{(\gamma|}\wedge*R^{\alpha|\beta)}\right)\right)\right)\right]\,,\end{split}$$ (94a) $$\displaystyle\begin{split}R^{\alpha\beta}\wedge\delta*&\left(\theta_{\alpha}\wedge\left(e^{\gamma}\rfloor{}^{(3)}Z_{\gamma\beta}\right)\right)\\ \simeq&\frac{1}{n^{2}-4}\delta\cancelto{}{Z}_{\mu\nu}\wedge\Theta\indices{{}^{\mu}{}^{\nu}_{\beta}}\left[n(-1)^{n+\mathrm{ind}g}\Theta\indices{{}_{\gamma}^{(\gamma|}{}_{\alpha}}\left(*R^{\alpha|\beta)}\right)-2\Theta_{\alpha}\left(*R^{\alpha\beta}\right)\right]\,,\end{split}$$ (94b) $$\displaystyle\begin{split}R^{\alpha\beta}\wedge\delta*&\left(\theta_{\alpha}\wedge\left(e^{\gamma}\rfloor{}^{(4)}Z_{\gamma\beta}\right)\right)\simeq\frac{(-1)^{n+\mathrm{ind}g}}{n}\delta Z\wedge\Theta_{\beta\alpha}\left(*R^{\alpha\beta}\right)\,,\end{split}$$ (94c) $$\displaystyle\begin{split}R^{\alpha\beta}\wedge\delta*&\left(\theta_{\alpha}\wedge\left(e^{\gamma}\rfloor{}^{(5)}Z_{\gamma\beta}\right)\right)\\ \simeq&\frac{(-1)^{n}}{n}\delta\cancelto{}{Z}_{\mu\nu}\wedge\Theta^{\nu}\left[2\Theta\indices{{}_{\gamma}^{(\gamma|}{}_{\alpha}}\left(*R^{\alpha|\mu)}\right)+\Theta\indices{{}^{\mu}_{\beta}{}_{\gamma}^{(\gamma|}{}_{\alpha}}\left(*R^{\alpha|\beta)}\right)\right]\,.\end{split}$$ (94d) For the construction of the full GHY term of MAG we need to proceed as in the examples in section 3. In particular, with the variations above it is straightforward to derive the required Lagrange multipliers from (10) and apply the Gauß-Codazzi equations (11) to simplify the remaining curvature forms in these Lagrange multipliers. Subsequently, the multipliers simplified that way need to be inserted into (12) to obtain the GHY term for MAG. We outline how this calculation is performed using the variations in (88- A.2) for vanishing non-metricity next. A.3 GHY term for metric compatible MAG For constructing the GHY term for MAG let us consider the Lagrangian (79) again. For simplicity we only consider the case of vanishing non-metricity here, $Q_{\mu\nu}\equiv-Dg_{\mu\nu}=0$. From the relation $\Omega_{\mu\nu}+\Omega_{\nu\mu}=DQ_{\mu\nu}$ for the curvature two-form $\Omega\indices{{}^{\mu}_{\nu}}$ we obtain that $\Omega_{(\mu\nu)}=0$ in this case. Comparison with the curvature decomposition in (A.2) implies that $Z_{\mu\nu}=0$. Thus, we are left with the relevant part of the MAG Lagrangian (79) for the calculation of the metric compatible GHY term to be $$\displaystyle\begin{split}V_{\mathrm{MAG},\rho}^{Q=0}=&-\frac{1}{2\rho}R^{\alpha\beta}\wedge*\left[\vphantom{\frac{1}{2\kappa}}\sum\limits_{I=1}^{6}w_{I}\,{}^{(I)}W_{\alpha\beta}+w_{7}\theta_{\alpha}\wedge(e_{\gamma}\rfloor{}^{(5)}W\indices{{}^{\gamma}_{\beta}})\right]\,.\end{split}$$ (95) This is the part of the MAG Lagrangian (78) in the presence of torsion and vanishing non-metricity which contributes new terms to the GHY term. Hence, we use (95) next to determine the GHY term for metric compatible MAG with the method presented in chapter 2. This method requires us to calculate the Lagrange multiplier $*\varphi^{\mathbf{n}a}$ first, since it is the only term which contributes to the metric compatible GHY term (13). We calculate $*\varphi^{\mathbf{n}a}$ by means of the constraints (10) to obtain $$\displaystyle\begin{split}\left.*\varphi^{\mathbf{n}a}\right|_{\varrho=R}=-\frac{1}{2\rho}&\left[\vphantom{\frac{(-1)^{n+\mathrm{ind}g}}{n-2}}2\sum\limits_{I=1}^{6}w_{I}\,*{}^{(I)}W_{\alpha\beta}+w_{7}(-1)^{n+\mathrm{ind}g}\Theta_{\mathbf{n}\gamma}\left(*{}^{(5)}W\indices{{}^{\gamma}_{|a]}}\right)\right.\\ &\hskip 4.0pt\left.-w_{7}\frac{(-1)^{n+\mathrm{ind}g}}{n-2}\Theta\indices{{}_{[a\mathbf{n}][\mu\gamma]}^{\gamma}{}_{\nu}}\left(*R^{\nu\mu}\right)\right]\,,\end{split}$$ (96) where we used the variational results (88) and (89). We already used the equations of motion for $\varphi^{\mu\nu}$ that are $\varrho_{\mu\nu}=R_{\mu\nu}$. As next step, we need to $3+1$ decompose the curvature indices by means of (41a) and impose the Gauß-Codazzi equations to replace the curvature decomposition terms. We note that $R_{\mu\nu}=W_{\mu\nu}$ is implicit in the irreducible components ${}^{(I)}W_{\mu\nu}$ of $W_{\mu\nu}$. For the considered case of vanishing non-metricity, the generalized Gauß-Codazzi equations (11) simplify to $$\displaystyle\begin{split}\vphantom{\frac{1}{2}}e_{\mu}^{a}e^{\nu}_{b}R\indices{{}^{\mu}_{\nu}}&=-\mathrm{d}\omega^{a}_{b}-\omega^{a}_{c}\wedge\omega^{c}_{b}+\varepsilon K^{a}\wedge K_{b}\,,\\ n_{\mu}e^{\nu}_{a}R\indices{{}^{\mu}_{\nu}}&=-e_{a}^{\mu}n^{\nu}R\indices{{}_{\mu}{}_{\nu}}=DK^{a}\,,\end{split}$$ (97) where the extrinsic curvature one-form $K^{a}$ was defined in (4). Using (12) we finally write the GHY term of (95) as $$\displaystyle\begin{split}S_{\mathrm{GHY\leavevmode\nobreak\ MAG},\rho}^{Q=0}=2\varepsilon\int_{\mathrm{\partial\mathcal{M}}}\left.\ K^{a}\wedge\left.*\varphi_{\mathbf{n}a}\right|_{\varrho=R,\,\text{Gauß-Codazzi}}\right|_{\mathrm{\partial\mathcal{M}}}\,,\end{split}$$ (98) where $\left.*\varphi_{\mathbf{n}a}\right|_{\varrho=R,\,\text{Gauß-Codazzi}}$ is understood as evaluation of (96) using the Gauß-Codazzi equations (97). This evaluation is straightforward but yet obviously cumbersome so we state only the implicit result here and leave the full evaluation to computer algebra systems. The same holds for the generalization to the full MAG Lagrangian in presence of non-metricity. Anhang B GHY term for metric compatible theories In section 2 we comment on the calculation of the GHY term in the case of vanishing non-metricity, $Q_{\mu\nu}\equiv-Dg_{\mu\nu}=0$. The latter condition enforces the curvature two-form $\Omega\indices{{}^{\mu}_{\nu}}$ to be antisymmetric which is obtained from the Bianchi identity (2) of non-metricity, $\Omega_{\mu\nu}+\Omega_{\nu\mu}=DQ_{\mu\nu}$. The Lagrange multipliers $\varphi_{\mu\nu}$ and $\varrho_{\mu\nu}$ are assumed to have the same symmetries as $\Omega_{\mu\nu}$, but we recommend to consider $\varrho_{\mathbf{n}a}$ and $\varrho_{a\mathbf{n}}$ as being independent for the variational calculation in section 2 nevertheless. In the current section we examine the differences if one does not follow this method for the calculation but uses $\varrho_{\mathbf{n}a}=-\varrho_{a\mathbf{n}}$ in the variational process instead. As a first step, we need to employ the antisymmetry of $\varrho_{\mu\nu}$ in the $3+1$ decomposition of $*\varphi^{\mu\nu}\wedge\delta\varrho_{\mu\nu}$. In contrast to the constraints for the calculation of $\varphi_{\mu\nu}$ we have found in (10) we now obtain $$\displaystyle*\varphi^{\mathbf{n}a}\wedge\delta\varrho_{\mathbf{n}a}=\frac{\varepsilon}{2}\delta_{\varrho_{\mathbf{n}a}}\mathcal{L}(\varrho_{\mathbf{n}a},\dots)$$ (99) as the only relevant constraint. Note the factor $\frac{1}{2}$ on the right hand side of (99) in contrast to (10). This factor results from using the $\varrho_{\mu\nu}$ antisymmetry. By the same argument we obtain a factor of $2$ from using the antisymmetry of $\varrho_{\mu\nu}$ in the calculation of $\delta_{\varrho_{\mathbf{n}a}}\mathcal{L}(\varrho_{\mathbf{n}a},\dots)$ for a specific theory. This factor of $2$ cancels the factor $\frac{1}{2}$ in (99). For this reason we recommended to do the variational calculus without assuming the antisymmetry of $\varrho_{\mu\nu}$ in section 2. Note that this assumption is made only in the variational calculus when calculating $\varphi^{\mathbf{n}a}$ by means of the constraints (10). It can be considered as a method which simplifies the calculation. Of course, for the asymmetry method to be valid, the GHY terms in both methods need to coincide. Indeed, using the antisymmetry of $\Omega\indices{{}^{\mu}_{\nu}}$ and $\varphi_{\mu\nu}$ yields $$\displaystyle S_{\mathrm{GHY}}^{Q=0}=2\int_{\mathrm{\partial\mathcal{M}}}\left.\varepsilon K^{a}\wedge*\varphi_{\mathbf{n}a}\right|_{\mathrm{\partial\mathcal{M}}}$$ (100) as GHY term in coincidence with the result (13) from the asymmetry assumption. The discussion in this appendix generalizes straightforwardly if non-metricity is not vanishing. 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Testing the Universality of Free Fall at ${10^{-10}}$ level by Comparing the Atoms in Different Hyperfine States with Bragg Diffraction Ke Zhang    Min-Kang Zhou zmk@hust.edu.cn    Yuan Cheng    Le-Le Chen    Qin Luo    Wen-Jie Xu    Lu-Shuai Cao    Xiao-Chun Duan    Zhong-Kun Hu zkhu@hust.edu.cn MOE Key Laboratory of Fundamental Physical Quantities Measurements, Hubei Key Laboratory of Gravitation and Quantum Physics, School of Physics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China (December 3, 2020) Abstract We have performed a precision atomic interferometry experiment on testing the universality of free fall (UFF) considering atoms’ spin degree of freedom. Our experiment employs the Bragg atom interferometer with ${}^{87}$Rb atoms either in hyperfine state $\left|{F=1,{m_{F}}=0}\right\rangle$ or $\left|{F=2,{m_{F}}=0}\right\rangle$, and the wave packets in these two states are diffracted by one pair of Bragg beams alternatively, which can help suppress the common-mode systematic errors. We have obtained an E$\rm{\ddot{o}}$tv$\rm{\ddot{o}}$s ratio $\eta=\left({0.9\pm 2.7}\right)\times{10^{-10}}$, and set a new record on the precision with a nearly 5 times improvement. Our experiment gives stronger restrictions on the possible UFF breaking mechanism. pacs: 37.25.+k, 03.75.Dg, 04.80.Cc The General Relativity (GR) has made imperial success in modern physics for describing gravity. The huge success of GR has also inspired extensive research on the extension of the theory, which bears the hope for, e.g., unifying the fundamental interactions Dam96 ; Cap11 . The validity of universality of free fall (UFF), being one of the fundamental postulations of GR Mis73 , has excited a huge amount of experiments Sch08 ; Wil04 under various circumstances to search for the sign of the extended GR theory. The most accurate tests for UFF to date were provided by the MICROSCOPE satellite mission Tou17 at the relative precision of $10^{-14}$ level. Other space-born experiments have also been proposed Tin13 ; Agu14 ; Bar16 . The UFF test has also been extended to the domain of quantum technology based on matter-wave interferences Bor89 ; Pet99 ; Mer10 ; Sch14 ; Bon13 ; Fra04 ; Zho15 ; Tar14 ; Dua16 ; Ros17 ; Gei18 . Testing UFF with quantum method was performed between different atomic species like Rb and K Sch14 , or different isotopes of one species Bon13 ; Fra04 ; Zho15 . For example, a precision level of $10^{-8}$ was reached in an atomic fountain containing the isotope of rubidium Zho15 , and experiments with much higher precision were proposed Har15 ; Kov15 . Quantum test of UFF is not only advancing in the potential improvements of precision, but also particularly interesting in searching possible spin-gravity coupling and torsion of space time. Atoms possessing well defined spin properties, like the fermionic and bosonic isotopes of Sr Tar14 , the ${}^{87}$Rb with opposite spin orientations Dua16 , the ${}^{85}$Rb in different hyperfine states Fra04 , were employed as test masses in the UFF experiments. An experimental implementation using entangled atoms of ${}^{85}$Rb and ${}^{87}$Rb has also been proposed Gei18 . Especially, a relative precision of low $10^{-9}$ has been achieved by ${}^{87}$Rb atoms prepared in two hyperfine states and in their superposition Ros17 . In this letter, we present an improved UFF test at precision of $2.7\times{10^{-10}}$ through the comparison of the free fall of ${}^{87}$Rb atoms in different hyperfine states. In this experiment, we perform Bragg interferometry measurements of the gravity acceleration difference between Rb atoms in states $\left|{5{S_{1/2}},F=1,{m_{F}}=0}\right\rangle$ and $\left|{5{S_{1/2}},F=2,{m_{F}}=0}\right\rangle$, termed as $\Delta g={g_{F=1}}-{g_{F=2}}$. As shown in Fig. 1(a), ${}^{87}$Rb atoms are initially prepared in the magnetic-insensitive state $\left|{{m_{F}}=0}\right\rangle$ either populated in states $\left|{5{S_{{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}},F=1}\right\rangle$ or $\left|{5{S_{{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}},F=2}\right\rangle$. Provided a proper laser frequency detuning $\Delta$, both states can be coupled to the same Bragg laser beam and manipulated between the momentum states $\left|{{p_{0}}}\right\rangle$ and $\left|{{p_{0}}+2n\hbar k}\right\rangle$, i.e. the states with momenta ${p_{0}}$ and ${p_{0}}+2n\hbar k$ respectively, via the Bragg diffraction Mul08 ; Mull08 ; Alt13 ; Est15 ; Maz15 ; Ami16 ; Har16 . Then we can construct two Bragg atom interferometers with corresponding labeled hyperfine states in Fig. 1(b), and get the interference phase ${\phi_{F=1}}$ and ${\phi_{F=2}}$, respectively. The free fall acceleration ${g_{F=1}}$ (or ${g_{F=2}}$) with atoms in different spin states is proportional to the corresponding interference phase. Then the UFF signal can be extracted from the phase difference (${\phi_{F=1}}-{\phi_{F=2}}$) of the two interferometers. The main advantages for our scheme are that, i) the Bragg atom interferometer does not change the internal hyperfine state, which makes this method intrinsically insensitive to the noise arising from the external electromagnetic field, and ii) both the two interferometers labeled by $F=1$ and $F=2$ share the same Bragg beam, therefore some systematic effects coupled with the driven laser can be significantly common rejected. We propose that our experiment can give a new constraint on the possible breaking mechanism of UFF due to the spin degree of freedomObu01 ; Pap08 ; Obu01 ; Lam06 . In our previous experiment Dua16 , we have performed tests on the UFF between ${}^{87}$Rb in states of $\left|{5{S_{1/2}},{m_{F}}=+1}\right\rangle$ and $\left|{5{S_{1/2}},{m_{F}}=-1}\right\rangle$, which corresponds to the test of the possible UFF breaking mechanism due to the spin projected to the direction of the gravity force. There still remains an open question, whether the spin projected to the perpendicular plane of the gravity force can break the UFF or not. Our current experiment addresses this open question and contributes to a more complete knowledge on the breaking of UFF due to the spin degree of freedom. To test the possible breaking of UFF against the spin projected to the plane perpendicular to the gravity force, it is natural to assume that the breaking effect only depends on the amplitude but not the polarization of the spin in the plane, $e.g.$ assuming a rotation symmetry in the perpendicular plane. A Hamiltonian term corresponding to such a breaking mechanism reads: $${V_{\bot}}(z)=\tilde{k}{\left|{{F_{\bot}}}\right|^{2}}mgz,$$ (1) where the gravity force is taken along the z direction, and ${\left|{{F_{\bot}}}\right|^{2}}$ is the amplitude of the spin projection in the horizontal plane of the gravity force which equals to 2 and 6 for $F=1$ and $F=2$. Our experiment can then give an upper bound of $\left({-0.2\pm 0.7}\right)\times{10^{-10}}$ to the strength of the UFF breaking term $\tilde{k}$. Because atoms are prepared in different hyperfine states with different internal energy, our experiment also corresponds to a test of the diagonal terms of the possible breaking operator of the UFF as ${r_{1}}-{r_{2}}{\rm{=}}(0.9\pm 2.7)\times{10^{{\rm{-10}}}}$ according to the mass-energy equivalence, which correspond to an improvement over previous results of about a factor of 5Ros17 ; Zyc17 . Firstly, we offer a detailed description of the experiment setup. The key point for realizing two Bragg atom interferometers with different hyperfine states is that, we have to ensure the effective Rabi frequency ${\Omega_{{\rm{eff}}}}$ of $F=1$ equals to that of $F=2$ when both of them couple to the same Bragg laser beams. For the Gaussian shape Bragg pulses, the nth-order effective Rabi frequency Mul08 also depends on the normal two-photon Rabi frequency $\Omega$, where $\Omega$ is inversely proportional to the single photon frequency detuning $\Delta$ Kas92 . We denote this two photon Rabi frequency as ${\Omega_{1}}$ and ${\Omega_{2}}$ for $F=1$ and $F=2$ state, respectively. For the requirements of ${\Omega_{1}}={\Omega_{2}}$, and considering the couplings of hyperfine states in $\left|{5{P_{{3\mathord{\left/{\vphantom{32}}\right.\kern-1.2pt}2}}}}\right\rangle$, the detuning $\Delta$ should be set around 3.1817 GHz. In this configuration, Bragg beams are red (blue) detuned for $F=1$ ($F=2$) state. Atom interferometer with Bragg diffraction requires high power laser beams. The frequency doubling method is employed to produce more than 1 W Bragg laser beam at 780 nm (Fig. 1(c)). A narrow-linewidth distributed feedback (DFB) seed laser at the telecom wavelength is amplified to 30 W by an erbium-doped fiber amplifier (EDFA, IPG photonics). Then the output beam from the EDFA passes through a periodically poled lithium niobate (PPLN) crystal, which can double the laser frequency from 1560 nm to 780 nm San12 . The 780 nm laser is split into two beams (beam1 and beam2) and are frequency shifted by two pieces of acousto-optical modulators (AOM1 and AOM2) respectively. The two counter-propagating Bragg beams are composed by beam 1 and 2 with perpendicular polarization. The frequency difference of the two beams is adjusted by either AOM1 or AOM2 to match the resonance condition, which is noted as $\Delta\omega={\omega_{1}}-{\omega_{2}}=2{\bf{k}}\cdot{{\bf{v}}_{a}}+4n{\omega_% {r}}$, where ${\omega_{r}}$ is the single photon recoil frequency shift. The Doppler frequency shift $2{\bf{k}}\cdot{{\bf{v}}_{a}}$ due to free fall can also be compensated by one of the AOM, where ${\bf{k}}$ and ${{\bf{v}}_{a}}$ represent the wave vector of single Bragg beam and the atom’s free fall velocity, respectively. In order to maximize the diffraction efficiency, the shape of Bragg pulses is programmed to a Gaussian form Mull08 according to AOM3. Before injecting into the vacuum chamber, about 80 mW Bragg beams are overlapped with the blow-away beams in a single mode polarization-maintaining (PM) fiber. The Raman beams which are employed on the velocity selection in the vertical direction are produced by a fiber electro-optic modulator (EOM) with a source of 6.83 GHz before the EDFA. With shutting down the AOM2, they pass through the AOM1 and AOM3. So the Bragg beams and Raman beams share a same optical path and are able to switch alternately by turning on or off the driven sources of EOM and AOM2. The ${e^{-2}}$ diameter of both the Bragg and Raman beams is about 19 mm. All of the beams are aligned and injected into the vacuum chamber through the top window, passing through a quarter wave plate, and retro-reflected by a reference mirror on a vibration isolator. The interference for matter-wave is implemented based on a cold ${}^{87}$Rb atom fountain that can be found elsewhere Hu13 . The total height of the fountain is 0.66 m. The initial state preparation that promises atoms either in $\left|{F=1,{m_{F}}=0}\right\rangle$ or in $\left|{F=2,{m_{F}}=0}\right\rangle$ is necessary before they fly into the interferometer chamber. This is realized with the microwave $\pi$-pulses between the hyperfine states, the repumping laser, the blow away beam of lower state ($\left|{5{S_{{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}},F=1}\right\rangle$ to $\left|{5{P_{{3\mathord{\left/{\vphantom{32}}\right.\kern-1.2pt}2}}},F=0}\right\rangle$), and the blow away beam of upper state ($\left|{5{S_{{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}},F=2}\right\rangle$ to $\left|{5{P_{{3\mathord{\left/{\vphantom{32}}\right.\kern-1.2pt}2}}},F=3}\right\rangle$). By compositing the above microwave pulses and the blow away laser beams, we can prepare atoms into the two target states alternately shot by shot. A Doppler-sensitive Raman $\pi$-pulse 80 $\mu$s long further prepares atoms in a narrow vertical momentum width less than 0.37 $\hbar k$. After atoms enter in to the magnetic shielding region, a series of Bragg pulse in the form of ${\pi\mathord{\left/{\vphantom{\pi 2}}\right.\kern-1.2pt}2}-\pi-{\pi\mathord{% \left/{\vphantom{\pi 2}}\right.\kern-1.2pt}2}$ manipulate the atomic wavepacket regardless of their internal states. All the Bragg pulses are programmed to a Gaussian shape. The typical width for the $\pi$ pulse is about 42 $\mu$s. A diffraction order $n=1$ is selected for both states by carefully setting $\Delta\omega=2{\bf{k}}\cdot{{\bf{v}}_{a}}+4{\omega_{r}}$. The laser frequency detuning $\Delta$ is well adjusted, thus interferences can happen between $\left|{F=1,{p_{0}}}\right\rangle$($\left|{F=2,{p_{0}}}\right\rangle$) and $\left|{F=1,{p_{0}}+2\hbar k}\right\rangle$($\left|{F=2,{p_{0}}+2\hbar k}\right\rangle$). The first order diffraction efficiency both for $\left|{F=1}\right\rangle$ and $\left|{F=2}\right\rangle$ can reach $88\%$ with a single Bragg $\pi-$pulse. The free evolution time $T$ between two Bragg pulses is 150 ms. When atoms fall back to the detection region, the two atomic clouds in the two interference paths are still partly overlapped in vertical direction, and can’t be easily distinguished with the normal time of flight method. Here we use the Doppler-sensitive Raman spectroscopy method Kas91 ; chengyuan18 with Raman $\pi$ pulses to get the population of $\left|{{p_{0}}}\right\rangle$ and $\left|{{p_{0}}+2\hbar k}\right\rangle$ in momentum space through a fluorescence measurement. The frequency difference between these two states due to the Doppler effect in the spectrum is about 30 kHz, while the resolution of our Raman spectroscopy can be better than 0.3 kHz, which is good enough to distinguish the $\left|{{p_{0}}}\right\rangle$ and $\left|{{p_{0}}+2\hbar k}\right\rangle$ states and measure their population. For the Bragg atom interferometer labeled by $\left|{F=1}\right\rangle$, the Raman spectrum is obtained by sweeping the Raman laser’s frequency, and the probability of finding atoms in state $\left|{F=1,{p_{0}}+2\hbar k}\right\rangle$ is given by the amplitude ratio of the two peaks corresponding to $\left|{F=1,{p_{0}}}\right\rangle$ and $\left|{F=1,{p_{0}}+2\hbar k}\right\rangle$. The two peaks can be found with two fixed Raman frequencies and the probability is primarily sensitive to the peak values, therefore two measurements with two shots are required to get one probability. Each measurement including the MOT loading, state preparation, interference stage and detection takes 1 s. The detection for the momentum states labeled by $\left|{F=2}\right\rangle$ is also performed with the same strategy. The probability of finding atoms in $\left|{{p_{0}}+2\hbar k}\right\rangle$ state depends on the interferometry phase, and can be written as $P={{\left({1-\cos\left({n\left({{k_{{\rm{eff}}}}g-\alpha}\right){T^{2}}}\right% )}\right)}\mathord{\left/{\vphantom{{\left({1-\cos\left({n\left({{k_{{\rm{eff}% }}}g-\alpha}\right){T^{2}}}\right)}\right)}2}}\right.\kern-1.2pt}2}$. Here $\alpha$ is Bragg beam’s frequency chirp rate for compensating the Doppler shift, the effective wave vector ${\vec{k}_{{\rm{eff}}}}={\vec{k}_{1}}-{\vec{k}_{2}}$ relies on the wave number of upper and down shooting Bragg beams. The matter-wave interference is performed either in state $F=1$ or $F=2$ by using Bragg diffraction with a time separation of 2 s alternately. As shown in Fig. 2(a), fringes for $F=1$ and $F=2$ with similar contrast are obtained by slightly modulating the driven frequency of AOM2. Each fringe contains two periods corresponding to a phase interval of $4\pi$ and taking 160 s totally. The differential acceleration ($\Delta g={g_{F=1}}-{g_{F=2}}$) in Fig. 2(a) is used to test the UFF with atoms in different hyperfine states. Fig. 2(b) shows the Allan deviation of the differential acceleration measurements by this state-labeled Bragg atom interferometer. The short-term sensitivity of $\Delta g$ is $1.2\times{10^{-7}}g/{\rm{H}}{{\rm{z}}^{{1\mathord{\left/{\vphantom{12}}\right.% \kern-1.2pt}2}}}$. The resolution of the differential measurement scales at $t^{-1/2}$, and can be better than $1\times{10^{-9}}g$ at 20000 s, which promise a UFF test with our scheme at $10^{-10}$ level . A test of the UFF is then performed by continuously measuring the gravity acceleration with this state-labeled atom interferometer. As shown in Fig. 3, about 63 hours data is recorded by the apparatus. Each point in this data is the mean result of 400 s. Both the two interferometers with $\left|{F=1}\right\rangle$ and $\left|{F=2}\right\rangle$ can precisely map the gravity tides, which are displayed in Fig. 3(a) by subtracting a constant offset ${g_{{\rm{offset}}}}$. What we care about in the UFF test is the differential acceleration $\Delta g$ shown by blue squares in Fig. 3(b). By averaging all the data, we get $\Delta g=\left({-1.2\pm 2.6}\right)\times{10^{-10}}g$ where the uncertainty is the standard deviation of the weighted mean. In our experiment, atoms in $F=1$ and $F=2$ are prepared in the same way and only one pair of Bragg beams is employed to diffract atoms both in $F=1$ and $F=2$. Therefore some systematic effects can be common rejected by differential measurement, such as the gravity gradient effect, the Coriolis effect and the wavefront aberration. The fluctuations of these effects between two states due to the alternatively measurements only contribute to the noise of the differential measurement which is included in the statistical uncertainty. The main systematic effects for $\Delta g$ are listed in Table 1. The magnetic field inhomogeneity contribute with a bias to $\Delta g$, because of the spatial separation of the two arms of the Bragg atom interferometer and the opposite sign of the Landé g-factor for $F=1$ and $F=2$ state. As atoms are prepared in ${m_{F}}=0$ state, we only have to consider the quadratic Zeeman effect. The magnetic field in the interferometry region is measured precisely by the Raman spectroscopy method Zho10 , and the Zeeman effect on the UFF test due to spatial separation of wavepacket is evaluated to be $\left({-2.1\pm 0.5}\right)\times{10^{-10}}g$ giving a dominant systematic impact. This effect is also confirmed with modulation experiments by measuring the differential acceleration $\Delta g$ in different magnetic bias field. As shown in Fig.4, when increasing the magnetic field, the value of $\Delta g$ performs as a quadratic increase, which is consistent with the evaluation values based on the magnetic field distribution. Because atoms are in a same internal state for Bragg type interferometer, the ac Stark shifts caused by the spatial intensity gradients of the Bragg lasers Ros17 and the intensity fluctuation between the first and third Bragg pulses Alt13 are both less than $1\times{10^{-11}}g$ in our experiment. Limited by the accuracy of absolute frequency of the Bragg lasers, the maximum frequency deviation of 1 MHz from the detuning $\Delta=3.1817$ GHz will contribute less than $1\times{10^{-11}}g$ due to the two-photon light shift Gau08 ; Gie16 . The local gravity variation due to the tides will induce a systematic error as the measurement in $F=1$ is always 2 s after $F=2$. This effect is evaluated at the level of $3\times{10^{-12}}g$ and can be neglected in the present test. As we select the first order Bragg diffraction to manipulate atoms without other unwanted momentum states, there should be no parasitic interference Est15 ; Par16 . After considering and correction of these systematic effects described above which are summarized in Table 1 , the E$\rm{\ddot{o}}$tv$\rm{\ddot{o}}$s ratio given by $${\eta_{1-2}}=2{{\left({{g_{F=1}}-{g_{F=2}}}\right)}\mathord{\left/{\vphantom{{% \left({{g_{F=1}}-{g_{F=2}}}\right)}{\left({{g_{F=1}}+{g_{F=2}}}\right)}}}% \right.\kern-1.2pt}{\left({{g_{F=1}}+{g_{F=2}}}\right)}}$$ (2) is finally determined to be ${\eta_{1-2}}=\left({0.9\pm 2.7}\right)\times{10^{-10}}$, which means the UFF between atoms in different hyperfine states is still valid at the precision of $10{{}^{-10}}$ level. Quantitatively, a direct upper bound of $\tilde{k}$ is given by $\tilde{k}=-({g_{F=2}}-{g_{F=1}})/4g=-\eta/4=\left({-0.2\pm{\rm{0}}{\rm{.7}}}% \right)\times{10^{{\rm{-10}}}}$. The diagonal terms of the possible breaking operator of a UFF violation is also estimated to be ${r_{1}}-{r_{2}}{\rm{=}}(0.9\pm 2.7)\times{10^{{\rm{-10}}}}$. In conclusion, we have realized the Bragg atom interferometers with different hyperfine states, and demonstrated its application in high precision measurements of gravitational acceleration. Due to the property of coupling to the same Bragg beams, various systematic effects can be common rejected in the present precision. With this state-labeled ${}^{87}$Rb atom interferometer, a precise quantum test on the UFF between different hyperfine states is performed at ${10^{-10}}$ level, gain about 5 times improvements on the accuracy and still see no violation of UFF. 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Temporal Dark Solitons in Nonuniform Bose-Einstein Condensates T. Hong, Y. Z. Wang and Y. S. Huo Joint Laboratory for Quantum Optics, Shanghai Institute of Optics and Fine Mechanics, Academia Sinica, P.O.Box 800-211, Shanghai 201800, China (Received 3 October 1997 by Phys. Rev. A Vol.58, No.4) Abstract We discuss temporal dark solitons in confined nonuniform Bose condensates. As a kind of localized high excitations, these solitons can be viewed as macroscopic quasiparticles, having relative motion to the background condensate. We get analytic expression for one dark soliton under slowly varying approximation and discuss its special propagation properties in nonuniform condensate, then we numerically prove that this approximation is reasonable and this kind of solitons exhibit their propagation properties in the nonuniform condensate. Finally, we simulate the generation of dark-soliton-like pulses in the condensate, and indicate that the excitation experiment, done by W. Ketterle’s group MITSound , can also be interpreted in terms of temporal dark soliton. pacs: 03.75.Fi, 42.65.Tg, 42.81.Dp Bose Einstein condensation(BEC) has been observed in dilute alkali atomic vapors JILABEC ; MITBEC ; RICEBEC . Recently, further experiments have also demonstrated that the condensate can be perfectly described by the Gross-Pitaevskii equation JILAExp ; MITExp ; Theories ; PRuprecht . For a ground-state condensate, the kinetic energy in this equation is usually negligible as compared with the s-wave scattering interaction energy that is the cubic nonlinear term. This indicates that the scattering term dominates main properties of the ground-state condensate, although it is a weakly interacting system. But for an excited condensate in which macroscopic number of atoms are collectively excited from ground state into high modes, the kinetic energy may become comparable to the scattering interaction energy. It could be noticed that the Gross-Pitaevskii equation, except for the confining potential, takes the form of the cubic nonlinear Schrödinger equation (CNLSE) which is well-known for the existence of soliton solutions ZakDSoliton . Naturally, one will ask whether solitons can exist in a condensate or not. In the previous works, A. Mysyrowicz et al. have already found the soliton-like propagation of the condensed excitons in $Cu_{2}O$ crystal AMysyrowicz , which obeys a similar Gross-Pitaevskii equation predicted by E. Hanamura EHanamura . Weiping Zhang et al., G. Lenz et al., and P. A. Ruprecht et al. have proved the existence of bright soliton in coherent atomic waves WZhang ; GLenz and condensate PRuprecht respectively. S. A. Morgan et al. have analyzed the solitary wave, synchronously moving with the background condensate SMorgan . The purpose of this paper is to discuss temporal dark solitons, as a kind of macroscopic quasiparticles, which have relative motion to the background condensate, and it is different from the work in reference SMorgan . This kind of solitons is actually localized high-mode excitations in the condensate consisting of atoms with positive scattering length, such as ${}^{87}Rb$ or ${}^{23}Na$. First, we will consider a single temporal dark soliton in an idealized uniform condensate. Second, we will consider it in a nonuniform condensate and discuss its specific propagation features in this case. Third, we will numerically simulate the propagation and collision of this kind of solitons in nonuniform condensate in order to prove that the dark solitons can exist in the nonuniform condensate steadily, exhibiting those propagation features, and the approximation adopted in the analytical solution is reasonable. Finally, we will also simulate the generation of dark-solition-like pulses by arbitrary perturbations, using the dark solitons’ nonlinear superposition properties, and indicate that the excitation experiment, done by W. Ketterle’s group MITSound , can also be interpreted in terms of temporal dark soliton. For a highly excited condensate in which macroscopic number of atoms are excited into high excited modes, the creation and destruction operators of these modes can be considered to be able to commute with each other GDMahan . Therefore, in an idealized uniform condensate, its atomic field operator can be treated as scalar, called macroscopic wave function. The macroscopic wave function can be described by the following Gross-Pitaevskii equation NoziPine , $$i\hbar\frac{\partial\Psi\left({\bf r},t\right)}{\partial t}=-\frac{\hbar^{2}}{% 2m}\nabla^{2}\Psi\left({\bf r},t\right)+V_{\rm exc}\Psi\left({\bf r},t\right)+% U_{0}\left|\Psi\left({\bf r},t\right)\right|^{2}\Psi\left({\bf r},t\right)$$ (1) where $\Psi\left({\bf r},t\right)$ is the macroscopic wave function of the condensate, including not only the ground sate but also the macroscopically populated high excited modes. $V_{exc}$ is an external flat potential. $U_{0}=4\pi\hbar^{2}a_{sc}/m$ is a scattering constant, where $a_{sc}$ is the s-wave scattering length and $m$ is the atomic mass. Eq. (1) is a CNLSE that possesses soliton solutions. When $U_{0}>0$ for positive scattering length, the solutions should be nonlinear superposition of dark solitons, according to the soliton theorems ZakDSoliton . With the inverse scattering transform method ZakDSoliton , we can solve this equation and get one-soliton solution, $$\Psi\left({\bf r},t\right)=\Psi_{0}\frac{1+\left(\mu_{s}-i\nu_{s}\right)^{2}e^% {-2\Gamma}}{1+e^{-2\Gamma}}e^{-i\left(V_{exc}+U_{0}\left|\Psi_{0}\right|^{2}% \right)t/\hbar}$$ (2) where $\Psi_{0}$ is the background amplitude of $\Psi\left({\bf r},t\right)$, $\mu_{s}$ and $\nu_{s}$ are real constants, satisfying $\mu_{s}^{2}+\nu_{s}^{2}=1$, and $\mu_{s}$ is called the eigenvalue of this dark soliton. In this equation, $$\Gamma=\nu_{s}\left[\sqrt{U_{0}\left|\Psi_{0}\right|^{2}m/\hbar^{2}}{\bf k}_{s% }\cdot\left({\bf r}-{\bf r}_{0}\right)+\mu_{s}U_{0}\left|\Psi_{0}\right|^{2}t/% \hbar\right]$$ (3) where ${\bf k_{s}}$ is a unit vector in the propagation direction of the soliton, ${\bf r_{0}}$ is the center coordinate of the soliton. Therefore, the atomic number density distribution is $$\left|\Psi\left({\bf r},t\right)\right|^{2}=\left|\Psi_{0}\right|^{2}\left(1-% \nu_{s}^{2}sech^{2}\Gamma\right)$$ (4) where $\nu_{s}^{2}$ is called the darkness of the dark soliton. Additionally, it is worthy to be emphasized that dark soliton possesses no threshold, so it can be stimulated easily. According to the theorems of dark soliton SAGredeskul , an arbitrary perturbation of the wave function can be described as a nonlinear superposition of solitons, which indicates that soliton has its universality in the space described by the CNLSE (1). From expression (4), we can find that the soliton is a localized function, because when $z\rightarrow\pm\infty$ , its dark density decreases to zero quickly. Thus we can derive its full width at half peak darkness, $$\triangle_{\rm s\left(FWHM\right)}=\frac{arccosh\sqrt{2}}{\nu_{s}\sqrt{\pi a_{% sc}\left|\Psi_{0}\right|^{2}}}$$ (5) This expression has a similar form with the healing length of the Bose condensate EPGross , except for the characteristic constant of the soliton. It is known that a two-dimensional vortex core in the Bose condensate has a similar size to the healing length. Consequently, their common features tell us that the dark soliton could be viewed as a kind of one-dimensional vortex core. Although there is no circular current around this kind of core due to its noncircularly-connected feature, the both sides of the soliton are connected by its bottom current density, which we will give more discussion later. Similar to a two-dimensional vortex, this one-dimensional vortex has a “ vortex line”, or more exactly, it may be called vortex plane, which is the center plane of the soliton perpendicular to the vector ${\bf k_{s}}$. Corresponding to the circulation around a vortex line in two-dimensional case, there is also a similar quantity for the one-dimensional vortex core, which is just the multiply of the width and the velocity of the soliton and is directly determined by its characteristic constant. However, for a dark soliton, its characteristic constant is usually not a quantized, but a continuously valued number, this feature indicates that the one-dimensional vortex is not quantized for an uniform Bose condensate, and therefore the corresponding one-dimensional flow should exhibit classical fluidity instead of superfluidity. But, this does not mean that it won’t become quantized for the externally confined nonuniform Bose condensate, because in considering the solitonic nonlinear dynamics at the vicinity of the confined boundaries, this problem always becomes very complicated and couldn’t be solved analytically at present. With Eq. (5), we can calculate that when the atomic number density of the condensate $\left|\Psi_{0}\right|^{2}$ is $10^{14}cm^{-3}$ , the width of a black ($\nu_{s}=1$) soliton is $0.95\mu m$ for condensate of ${}^{23}Na$. Obviously, it can be much smaller than the real size of a nonuniform condensate, for example, $17\mu m$ in radial direction and $300\mu m$ in axial direction, realized by W. Ketterle’s group MITBEC . So the previous idealized uniform condensate is reasonable for very dark solitons. Subsequently, we consider a more real and special case in which the condensate, confined in external potential, is nonuniform, and it has a shape of cigar. Similar to the above assumption, a macroscopic number of atoms are still assumed to be excited, and the commutation relations of the atomic field operators are still valid. Thus, the Gross-Pitaevskii equation can be expressed as $$i\hbar\frac{\partial\Psi\left({\bf r},t\right)}{\partial t}=-\frac{\hbar^{2}}{% 2m}\nabla^{2}\Psi\left({\bf r},t\right)+V_{\rm ext}\left({\bf r}\right)\Psi% \left({\bf r},t\right)+U_{0}\left|\Psi\left({\bf r},t\right)\right|^{2}\Psi% \left({\bf r},t\right)$$ (6) where $V_{\rm ext}\left({\bf r}\right)=V_{r}\left({\bf R}\right)+V_{a}\left(z\right)$ is the external confining potential, which can be divided into two parts, the radial $V_{r}\left({\bf R}\right)$ and the axial $V_{a}\left(z\right)$ in cylindrical coordinates. Additionally, We assume the wave function $\Psi\left({\bf r},t\right)$ can be divided into the background ground-state condensate $\Phi\left({\bf r}\right)$ and a soliton function $\beta\left(z,t\right)$ propagating along $z$ axis, $$\Psi\left({\bf r},t\right)=\beta\left(z,t\right)\Phi\left({\bf r}\right)e^{-i% \mu_{c}t/\hbar}$$ (7) where $\mu_{c}$ is the chemical potential of the stationary ground-state of the condensate. Then Eq. (6) is transformed into $$\displaystyle\left[i\hbar\frac{\partial\beta\left(z,t\right)}{\partial t}+% \frac{\hbar^{2}}{2m}\frac{\partial^{2}\beta\left(z,t\right)}{\partial z^{2}}+% \frac{\hbar^{2}}{m}\frac{\partial\beta\left(z,t\right)}{\partial z}\frac{% \partial ln\Phi\left({\bf r}\right)}{\partial z}\right]\Phi\left({\bf r}\right)$$ $$\displaystyle+\left[-U_{0}\left|\beta\left(z,t\right)\Phi\left({\bf r}\right)% \right|^{2}\beta\left(z,t\right)+U_{0}\left|\Phi\left({\bf r}\right)\right|^{2% }\beta\left(z,t\right)\right]\Phi\left({\bf r}\right)$$ $$\displaystyle=\beta\left(z,t\right)\left[-\mu_{c}\Phi\left({\bf r}\right)-% \frac{\hbar^{2}}{2m}\nabla^{2}\Phi\left({\bf r}\right)+V_{ext}\left({\bf r}% \right)\Phi\left({\bf r}\right)+U_{0}\left|\Phi\left({\bf r}\right)\right|^{2}% \Phi\left({\bf r}\right)\right]$$ (8) Because $\Phi\left({\bf r}\right)$ is the stationary ground-state wave function of the condensate, it must satisfy $$\mu_{c}\Phi\left({\bf r}\right)=-\frac{\hbar^{2}}{2m}\nabla^{2}\Phi\left({\bf r% }\right)+V_{ext}\left({\bf r}\right)\Phi\left({\bf r}\right)+U_{0}\left|\Phi% \left({\bf r}\right)\right|^{2}\Phi\left({\bf r}\right)$$ (9) Consequently, Eq. (8) is reduced to $$\displaystyle i\hbar\frac{\partial\beta\left(z,t\right)}{\partial t}+\frac{% \hbar^{2}}{2m}\frac{\partial^{2}\beta\left(z,t\right)}{\partial z^{2}}+\frac{% \hbar^{2}}{m}\frac{\partial\beta\left(z,t\right)}{\partial z}\frac{\partial ln% \Phi\left({\bf r}\right)}{\partial z}$$ $$\displaystyle-U_{0}\left|\beta\left(z,t\right)\Phi\left({\bf r}\right)\right|^% {2}\beta\left(z,t\right)+U_{0}\left|\Phi\left({\bf r}\right)\right|^{2}\beta% \left(z,t\right)=0$$ (10) Furthermore, we assume $\left|\Phi\left({\bf r}\right)\right|^{2}$ is a slowly varying function of coordinate $z$, as compared with $\beta\left(z,t\right)$ , satisfying $$\frac{1}{2}\left|\frac{\partial^{2}\beta\left(z,t\right)}{\partial z^{2}}% \right|\gg\left|\frac{\partial\beta\left(z,t\right)}{\partial z}\frac{\partial ln% \Phi\left({\bf r}\right)}{\partial z}\right|$$ (11) Therefore, we can ignore the third term in Eq. (10) , and get $$i\hbar\frac{\partial\beta\left(z,t\right)}{\partial t}+\frac{\hbar^{2}}{2m}% \frac{\partial^{2}\beta\left(z,t\right)}{\partial z^{2}}-U_{0}\left|\beta\left% (z,t\right)\Phi\left({\bf r}\right)\right|^{2}\beta\left(z,t\right)+U_{0}\left% |\Phi\left({\bf r}\right)\right|^{2}\beta\left(z,t\right)=0$$ (12) Because $\left|\Phi\left({\bf r}\right)\right|^{2}$ is a slowly varying function in contrast to a soliton, when we solve this equation with inverse scattering transform method for a single soliton solution, we can take $\left|\Phi\left({\bf r}\right)\right|^{2}$ as a constant. Subsequently, we can routinely get one soliton expression, $$\beta\left(z,t\right)=\frac{1+\left(\mu_{s}-i\nu_{s}\right)^{2}e^{-2\Gamma}}{1% +e^{-2\Gamma}}$$ (13) where $$\Gamma=\nu_{s}\left[\sqrt{U_{0}\left|\Phi\left({\bf r}\right)\right|^{2}m/% \hbar^{2}}{\bf k}_{s}\cdot\left({\bf r}-{\bf r}_{0}\right)+\mu_{s}U_{0}\left|% \Phi\left({\bf r}\right)\right|^{2}t/\hbar\right]$$ (14) $\mu_{s}$ and $\nu_{s}$ still follow the previous definition. $z_{0}$ is the center coordinate of this soliton. The full width at half peak darkness of the soliton is $$\triangle_{\rm s\left(FWHM\right)}=\frac{arccosh\sqrt{2}}{\nu_{s}\sqrt{\pi a_{% sc}\left|\Phi\left({\bf r}\right)\right|^{2}}}$$ (15) which is different from Eq. (5) for $\left|\Phi\left({\bf r}\right)\right|^{2}$ being a slowly varying function of $z$. This means $\triangle_{\rm s\left(FWHM\right)}$ is to be increased when the soliton moves from the “top” to the “downhill” of the ground condensate wave function. From Eq. (14) we can derive the velocity of the soliton, $${\bf v}\left({\bf r}\right)=-\mu_{s}\sqrt{\frac{U_{0}\left|\Phi\left({\bf r}% \right)\right|^{2}}{m}}{\bf k}$$ (16) where ${\bf k}$ is the unit vector in the direction of $z$. ${\bf v}\left({\bf r}\right)$ varies with $\Phi\left({\bf r}\right)$, which is consistent with the local speed of sound given by Bogoliubov NBogoliubov and Lee, Huang, and Yang LeeHuangYang . Additionally, Eq. (16) tells us that the velocity of this temporal dark soliton is also relevant to the eigenvalue, and it is, more explicitly, usually less than the absolute value of the corresponding sound speed in the nonuniform Bose condensate. However, this property can not be gotten in the density perturbation theory of hydrodynamics, such as ref. EZaremba . In considering the local density dependent property of the velocity, we can find that there is a slow feedback process between the velocity and the displacement of the soliton, and the temporal dark soliton behaves as an oscillator in the nonuniform Bose condensate THong . Additionally, for those radial tightly confined Bose condensates, the velocity of the soliton varies rapidly with the density in the radial direction. This leads to very serious radial dispersion of the temporal dark soliton, to which a similar phenomenon has been observed in the experiments done by W. Ketterle’s group MITSound . As it is known in nonlinear optics, the $1+1$ dimensional dark solitons are usually instable under their transverse perturbations, and always evolve into dark vortices BLDavies . The analogy between this solitonic excitation in Bose condensate and the optical dark solitons enlighten us that the temporal dark soliton must be instable under the stretch of transverse dispersion, and it may also evolve into vortices. Due to the cylindrical symmetry of the ground state condensate, it is more likely to evolve into vortex rings NoziPine , because they can still preserve the cylindrical symmetry. The circulation of those stable vortex rings must be quantized, which is a direct consequence of the coherent property of the Bose condensate, relevant to the nature of the superfluidity of the Bose condensate. Our further work on this topic is still in progress. Assuming that the radial variation could be neglected for some cases, we will find the product $$\triangle_{\rm s\left(FWHM\right)}\cdot v\left({\bf r}\right)=-2arccosh\sqrt{2% }\frac{\hbar\mu_{s}}{m\nu_{s}}$$ (17) should be a constant for a darkness-fixed soliton. As we have described above, analogous to the circulation around the vortex line of a two-dimensional vortex in the fluid, this quantity may also be called a one-dimensional circulation, which characterizes the properties of the one-dimensional dark soliton vortex core directly. But the more important thing is that it is very realistic for an experimental measurement of this quantity. Therefore, Eq. (17) may offer us a useful method to judge whether a dark pulse in the condensate can be very precisely interpreted as a temporal dark soliton. From Eq. (7) we can also get the atomic current density carried by the soliton $${\bf j}=-\frac{i\hbar}{2m}\left(\Psi^{\ast}\left({\bf r},t\right)\nabla\Psi% \left({\bf r},t\right)-\Psi\left({\bf r},t\right)\nabla\Psi^{\ast}\left({\bf r% },t\right)\right)={\bf k}\mu_{s}\nu_{s}^{2}U_{0}^{1/2}m^{-1/2}\Phi^{3}\left({% \bf r}\right)sech^{2}\Gamma$$ (18) which is pulsed. The width of this pulse is proportional to that of the soliton. It is worthy to be noticed that the direction of the current density pulse is opposite to the propagation direction of the dark soliton, however, the propagation direction and the velocity of the current density pulse are both same to those of the soliton. When $\nu_{s}=0$ or $\nu_{s}=1$, the amplitude of this current density pulse reaches its minimum $j_{\rm min}=0$. When $\nu_{s}=2\sqrt{3}/9$, it gets to its maximum $$j_{\rm max}=\frac{4\sqrt{69}}{243}U_{0}^{1/2}m^{-1/2}\Phi^{3}\left({\bf r}\right)$$ (19) The existence of this maximum shows that each distribution of the current density usually corresponds to two possible soliton eigenvalues and therefore two one-dimensional dark soliton vortices. This property is different from the usual two- dimensional vortex case. Additionally, from Eq. (18), we can find that the relationship of the current density and the soliton eigenvalue $\mu_{s}$ is in the shape of S. In considering some nonlinear dynamical process, such as the solitonic excitation or the soliton collision with a boundary. This relationship may provide us an opportunity of finding bistable phenomena of the temporal dark soliton. In the above derivation, we have adopted slowly varying approximation (11). To make sure this approximation is reasonable and temporal dark solitons can really exhibit their propagation properties in a confined nonuniform condensate, we numerically simulate the propagation and collision of this kind of solitons. We assume the ground-state condensate is composed of $N=5\times 10^{6}$ of ${}^{23}Na$ atoms with $a_{sc}=27.5\AA$ ETiesinga , and the radial trapping frequency is $\omega_{r}=2\pi\times 1800Hz$, the axial trapping frequency is $\omega_{a}=2\pi\times 18Hz$. During the simulation, we adopt one-dimensional Thomas-Fermi approximation in the radial direction of the cylindrically symmetric condensate. Consequently, we can approximate Eq. (6) as one-dimensional Gross-Pitaevskii equation by integration in the radial cross section of the condensate, $$i\hbar\frac{\partial\psi_{a}\left(z,t\right)}{\partial t}+\frac{2}{3}\mu_{c}% \psi_{a}\left(z,t\right)=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}\psi_{a}\left(% z,t\right)}{\partial z^{2}}+V_{a}\left(z\right)\psi_{a}\left(z,t\right)+\frac{% 2}{3}U_{0}\left|\psi_{a}\left(z,t\right)\right|^{2}\psi_{a}\left(z,t\right)$$ (20) where $\psi_{a}\left(z,t\right)$ is the axial wave function of the condensate, distributed along the axial direction $z$, and $V_{a}\left(z\right)=m\omega_{a}^{2}z^{2}/2$. Although the one-dimensional Thomas-Fermi approximation is still not strict, it can still keep the axial wave function $\psi_{a}\left(z,t\right)$ without slowly varying approximation (11) in the direction of $z$, and therefore Eq. (20) is sufficient to give the approximation a test. We numerically solve Eq. (20) by split-Fourier transform method MEguchi . As an example, two solitons’ collision process is illustrated in Fig.1. The two soliton, $\nu_{s1}=0.5$, $\nu_{s2}=0.5$, set initially on the “downhill” of the condensate wave function with distance $6.2\mu m$, shown in Fig.1.(a). It should be noted that both the wave function and the atomic number density, shown in all figures in this paper, have already been transformed into dimensionless forms with respect to their ground-state center values at $z=0.0\mu m$. These two solitons propagate and collide with each other, and then they separate and propagate independently, as shown in Fig.1.(b). Before and after the collision, they can conserve their shape steadily. From the contour graph of this collision, as shown in Fig.1.(c), the widths of the solitons vary with their position, which indicates the locality of their widths. The velocities of the two solitons are all about $v=15\mu m/ms$. There is no damage occurred to both of these solitons, so the collision must be elastic. More numerical experiments show that so long as the numerical precision is enough, the solitons can propagate steadily as far as one can compute. Therefore, we can draw a conclusion that the slowly varying approximation (11) is reasonable, and the solitons indeed have the propagation properties, indicated above in nonuniform condensate confined by external potential. Additionally, as we have seen, the properties of these temporal dark solitons in the condensate are very similar to those of conventional particles, so we can consider them as a kind of quasiparticles. But they are different from phonons, because every one of them is macroscopic whereas a phonon is usually microscopic, and furthermore every soliton must be a coherent structure as described in Eq. (13). According to the dark soliton theorems ZakDSoliton , dark solitons have nonlinear superposition properties. We may use these properties to generate temporal dark solitons in nonumiform condensate. To illustrate this possibility, we numerically simulate the generation of soliton-like pulses by arbitrary perturbations in the nonuniform condensate. As an example, here we give two reversed Gaussian function perturbations in the condensate, shown in Fig.2.(a). The darkness of these two perturbations are $0.44$ and $0.64$ respectively, and the full widths of them are $2.47\mu m$ and $0.99\mu m$ respectively, and their distance is $10\mu m$. As shown in Fig.2.(b), each of these two dark perturbations splits quickly into more pairs of dark pulses, propagating with different velocity depending on its own darkness. The collision of the two out-splitted dark pulses must be elastic because the two pulses can still keep even symmetry with their own twins. Obviously, these properties are very similar to those of solitons. From Fig.2.(c), which is the contour graph of Fig.2.(a), we can find that all the velocities of these out-splitted soliton-like pulses are around $V=20\mu m/ms$. Although we can’t make sure that these out-splitted pulses won’t split anymore ( in fact, the pair, splited by the left one, have already begun to split again into more pairs of pulses), with the above elastic collision evidence, we can at least draw a conclusion that the initial dark pulses can be approximated as multi-solitons, each of which is composed of a number of solitons. The recent experiment, done by W. Ketterle’s group MITSound , has shown a similar phenomenon as we have simulated here. In their experiment, one dark perturbation splits into two, and then these two out-splitted pulses continue to propagate with spreading widths. Although there is no description about observation of further splitting of the out-split dark pulse, the out-split dark pulse must be composed of more than one solitons. The spreading of the pulses can be interpreted as the radial dispersion as we have described above, but one can’t exclude some other possible reasons, such as further splitting of the multi-solitons and the widening with the decrement of the background condensate as described by Eq. (15). And it can’t be usually simply explained only as the last one, except that one can, by chance, get twin strict dark solitons, every one of which must have the form as we have described analytically in Eq. (13). However, further splitting may not be observable, because the size of the condensate is limited, the velocities differences are all to decrease to zero due to the Eq. (16), and they may therefore have no enough distance to propagate for further splitting. To test the soliton-like properties of the dark pulses in the present experiment, according to our numerical simulation, we suggest that one can generate two dark pulses propagating in opposite direction, and observe their collision. One can also test these properties by measuring the relationship between the velocity of every steadily propagating out-split dark pulse and its width, and Eq. (17) can be used as a judgment. Certainly, more precise interpretation of the experiment would need more study in details and more realistic simulation which can’t be included in this short paper. But, as we have briefly discussed above, the temporal dark soliton theorems are promising tools to interpret the experiment done by W. Ketterle’s group MITSound . In conclusion, we have derived the temporal dark soliton in nonuniform condensate under slowly varying approximation. Viewed as the one-dimensional vortex cores, these temporal solitons may provide deeper insight into the superfluidity in the cigar-shaped Bose-Einstein condensate. Then we have numerically simulated the propagation and collision of this kind of solitons, and proved that the slowly varying approximation is reasonable and the solitons can exhibit the special propagation properties in the nonuniform condensate. We have finally simulated the generation of soliton-like pulses by two arbitrary dark pulses in the nonuniform condensate, and with this simulation, we have given the experimental results of W. Ketterle’s group a qualitative interpretation in terms of temporal dark soliton. We would like to thank Dr. Jaren Liu, Dr. Xueru Zhang, Dr. Xunming Liu, Dr. Xinqi Wang, Dr. Fusheng Li, and Mr. Wenbao Wang for their fruitful discussions and helps. This work is supported by National Science Foundation of China, under Grant No.19392503. References (1) M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269, 198(1995). 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The origin of Weyl gauging in metric-affine theories Dario Sauro dario.sauro@phd.unipi.it Università di Pisa and INFN - Sezione di Pisa, Largo Bruno Pontecorvo 3, 56127 Pisa, Italy    Omar Zanusso omar.zanusso@unipi.it Università di Pisa and INFN - Sezione di Pisa, Largo Bruno Pontecorvo 3, 56127 Pisa, Italy Abstract In the first part, we discuss the interplay between local scale invariance and metric-affine degrees of freedom from few distinct points of view. We argue, rather generally, that the gauging of Weyl symmetry is a natural byproduct of requiring that scale invariance is a symmetry of a gravitational theory that is based on a metric and on an independent affine structure degrees of freedom. In the second part, we compute the Nöther identities associated with all the gauge symmetries, including Weyl, Lorentz and diffeomorphisms invariances, for general actions with matter degrees of freedom, exploiting a gauge covariant generalization of the Lie derivative. We find two equivalent ways to approach the problem, based on how we regard the spin-connection degrees of freedom, either as an independent object or as the sum of two Weyl invariant terms. The latter approach, which rests upon the use of a Weyl-covariant connection with desirable properties, denoted $\hat{\nabla}$, is particularly convenient and constitutes one of our main results. I Introduction The idea that either rigid scale or local Weyl invariances could be fundamental symmetries of nature has a very long history weyl1922space ; Coleman:1970je ; charap1974gauge ; Smolin:1979uz . In fact, the idea has been resurrected multiple times and in various forms, both for particle physics Meissner:2006zh ; tHooft:2011aa ; Ghilencea:2018thl and for the gravitational interactions Bekenstein:1980jw ; Ghilencea:2015mza ; Ghilencea:2018dqd . The most attractive feature of a scale or Weyl invariant spacetime would be that the invariance could encompass the fundamental limitation given by the Planck mass, that is, the dimensionful quantity that makes us think at general relativity as an effective theory. The assumption that spacetime is fundamentally scale invariant at high energies could provide meaningful insights to a theory of the traditional geometric degrees of freedom, metric and connection, which works at arbitrarily high energies Mannheim:2011ds . Such theory could be renormalizable, and even asymptotically free Fradkin:1985am or safe Reuter:1996cp , as long as scale invariance is broken in the low energy regimes Wetterich:2014gaa ; Salvio:2017qkx . A description of geometry like the above would also cast gravity in a form that is closer to the one of the other interactions of the standard model of particle physics, which is scale invariant for the most part of its interactions. Consequently, it would be a convenient feature for a possible complete unification of forces tHooft:2010mvw . However, we need not forget that our understanding of general relativity, and of the standard model for what matters, is, now more than ever, leaning towards the fact that they are effective theories Baldazzi:2021kaf , that work well on certain intervals of scales, while eluding us outside of them. Assuming that general relativity's metric and connection effective degrees of freedom are also fundamental ones should be regarded as an endeavor, instead of an obvious natural assumption. The elusive part, that we discuss at length in this paper, is that if we try to enforce in the same construction both the usual degrees of freedom of metric-affine gravity (MAG), that is to say the aforementioned metric and connection as well as local scale invariance, we are lead naturally to a geometric theory in which the metric-affine connection is complemented with a gauge potential for the Weyl symmetry. This could be seen as a theoretical prediction, or a price to pay, depending on the reader's point of view. In this paper, we discuss the naturalness of the inclusion of an Abelian potential for the Weyl gauge symmetry in the context of general relativity and metric-affine gravity, and we give particular emphasis on torsion degrees of freedom and their conformal properties. Some of these aspects and their cosmological implications have already been investigated in the $f(T)$ literature, see e.g. Bahamonde:2015zma ; Cai:2015emx ; Krssak:2018ywd ; Hohmann:2019nat ; Gakis:2019rdd ; Capozziello:2021pcg , and the gauging of Weyl symmetry emerges quite naturally in the context of noncommutative gravity deCesare:2018cjr . We give several arguments that motivate the form of the gauging of scale invariance Iorio:1996ad , ranging from the requirement that geodesics are not changed by a conformal transformation, to the rethinking of the original Palatini's discussion that a dynamical torsionless connection must be Levi-Civita's on-shell Palatini:1919 . We find two equivalent ways of describing our metric-affine geometric setup, one of which makes more transparent the Weyl covariance of the construction. Bottom-up and top-down views on the naturalness of Weyl gauging are discussed in Sects. II and III, respectively, while a direct connection with Palatini's approach to metric-affine gravity is drawn in Sect. IV. In the bottom-up approach, we show that the requirements of Weyl invariance and nonvanishing torsion force us to include a Weyl potential, enhancing the local gauge group from the Lorentz one to $SO(3,1)\times D(1)$, where $D(1)$ is the Abelian group of local Weyl rescalings. On the other hand, in the top-down appraoch, we start from a metric-affine viewpoint and analyze how the irreducible components of a general affine connection transform under Weyl rescalings. We eventually specialize to vanishing traceless nonmetricity, which allows us to focus on co-frame $e^{a}{}_{\mu}$, spin-connection $\omega^{a}{}_{b\mu}$ and Weyl potential $S_{\mu}$ as the natural gravitational field variables. Armed with the necessary geometric toolkit, we also discuss the implications that Weyl gauging, together with all other symmetries, have on the coupling of matter fields with gravity. We do so by obtaining the most general Nöther identities associated with the coupling of matter fields on-shell in terms of the currents that couple to the gauge potentials of our construction. These currents are energy-momentum tensor, dilation-vector and spin-current. We divide the derivation of Nöther identities in two parts, based on two equivalent but distinct approaches that come out naturally from the introductory discussion. We refer to the approaches as the Cartan-Weyl, that features a conformally covariant torsion, and the Einstein-Weyl, that features instead a conformally invariant torsion, which are given in Sects. V and VI, respectively. The main difference between the two viewpoints is that, from the in Cartan-Weyl perspective the spin-connection is completely general, while from the Einstein-Weyl one the connection is split into two contributions $\omega^{a}{}_{b}=\hat{\omega}^{a}{}_{b}+\hat{\Omega}^{a}{}_{b}$. The splitting is chosen in such a way that $\hat{\omega}^{a}{}_{b}$ is a function only of co-frame and the Weyl gauge potential, $\hat{\omega}^{a}{}_{b}=\hat{\omega}^{a}{}_{b}(e^{a},S)$, transforming affinely under the local Lorentz group, while $\hat{\Omega}^{a}{}_{b}$ is a Lorentz tensor and is regarded as an independent field variable. We recover the traditional conservation laws of the energy-momentum tensor for vanishing dilation- and spin-currents. Our analysis is complemented by an in-depth discussion of the relevant geometric quantities and the use of a particular generalization of the Lie derivative, introduced initially in Sect. V.2, which allows us to modify the generators of the diffeomorphis group in a way that makes them covariant under all gauge symmetries, including both local Weyl and Lorentz invariance. The appendices contain further discussions of some geometrical aspects that would have overburdenend the main text. Appendix A includes relevant formulas for the commutators of the covariant derivatives of the main text, the nontrivial contractions of the curvature tensors, as well as the Bianchi identities associated to the curvatures. Appendix B clarifies some aspects of the covariant integration by parts of connections in presence of Weyl gauging and including torsional degrees of freedom. Appendix C explores further the algebra associated to the covariant Lie derivative that is used extensively in the main text. II Weyl transformations vs independent connections: a bottom-up approach In this section we present a bottom-up approach to Weyl gauging. The presentation is going to be introductive and motivates the notion of Weyl gauging as natural in the context of a formalism that accounts simultaneously for an independent connection (including, e.g., torsional degrees of freedom) and for conformal Weyl rescalings of the metric. This section also introduces much of the notation that is necessary for the rest of the paper. II.1 Holonomic vs anholonomic degrees of freedom To set the stage and part of the notation for the dicussion, consider the two equivalent approaches towards general relativity and metric-affine gravity: the holonomic approach in which one works with a symmetric metric tensor $g_{\mu\nu}$ and a holonomic connection $\Gamma^{\lambda}{}_{\nu\mu}$, and the anholonomic approach (see e.g. 1980ASIB…58..489C ; Scholz:2018iuc ; Kibble:1961ba ; Gronwald:1995em ), in which one works with a co-frame $e^{a}=e^{a}{}_{\mu}dx^{\mu}$ and a spin-connection $\omega^{a}{}_{b}=\omega^{a}{}_{b\mu}dx^{\mu}$. The components of metric and co-frame are related by the requirement that $g_{\mu\nu}=\eta_{ab}e^{a}{}_{\mu}e^{b}{}_{\nu}$, where $\eta_{ab}$ is the Minkowski metric. We can switch from one approach to the other by means of the tetrad postulate Yepez:2011bw , which is the requirement that the full covariant derivative of the co-frame vanishes $$\nabla_{\mu}e^{a}{}_{\nu}=\partial_{\mu}e^{a}{}_{\nu}+\omega^{a}{}_{b\mu}e^{b}{}_{\nu}-\Gamma^{\lambda}{}_{\nu\mu}e^{a}{}_{\lambda}=0\,,$$ (1) implying that the full connection is compatible with the co-frame. We can use this equation to express either one of the two connections in terms of the vierbein, its inverse $E^{\mu}{}_{a}$, and the other connection. For example, we have the relation $$\omega^{a}{}_{b\mu}=E^{\nu}{}_{a}\Gamma^{\lambda}{}_{\nu\mu}e^{a}{}_{\lambda}-E^{\nu}{}_{a}\partial_{\mu}e^{a}{}_{\nu}\,.$$ (2) Notice that, in general, the holonomic-connection belongs to the algebra of the group of general linear transformations, $\Gamma_{\mu}\in gl(4)$, because we are implicitly assuming that it is compatible with the metric, $\nabla_{\mu}g_{\mu\nu}=0$, but it is not necessarily symmetric. However, the requirement of metricity severely restricts the generality of the affine connection in that it can contain torsion components. Likewise, $\omega^{a}{}_{b\mu}$ is also general, yet, by definition, it must belong to the adjoint representation of the Lorentz algebra, $\omega_{\mu}\in so(3,1)$. We anticipate that in the next section we are going to temporarily depart from the condition of metric compatibility to accommodate the effect of Weyl transformations, but also ultimately restore it introducing an additional gauge component. The two formulations are completely equivalent and simplify considerably in the case of pure gravity with a Lagrangian density that is proportional to the scalar curvature, i.e. with the Einstein-Hilbert Lagrangian. In this case, it can be shown that the field equations of the spin-connection yield the torsion-free condition (see, for example, Ref. gasperini2013theory ), which implies that the holonomic connection is symmetric. In a similar way, a symmetric holonomic connection is metric compatible on-shell Palatini:1919 . As a consequence, both connections can be expressed in terms of the vierbein and its derivatives in the anholonomic case, or the metric and its derivatives in the holonomic one. We denote $\mathring{\nabla}_{\mu}$ the covariant derivative with components $\mathring{\Gamma}^{\mu}{}_{\rho\nu}=\frac{1}{2}g^{\mu\lambda}(\partial_{\nu}g_{\lambda\rho}+\partial_{\rho}g_{\lambda\nu}-\partial_{\lambda}g_{\nu\rho})$, that is the unique symmetric Christoffel connection, and $\mathring{\omega}^{a}{}_{b\mu}$, that is the corresponding spin-connection obtained by the tetrad postulate as in Eq. (1), $\mathring{\omega}^{a}{}_{b\mu}=E^{\nu}{}_{a}\mathring{\Gamma}^{\lambda}{}_{\nu\mu}e^{a}{}_{\lambda}-E^{\nu}{}_{a}\partial_{\mu}e^{a}{}_{\nu}$. In presence of some type of matter degrees of freedom, especially spinors, the torsion $2$-form, denoted ${\cal T}^{a}$, which is defined as the covariant curl of the co-frame, ${\cal T}^{a}=de^{a}+\omega^{a}{}_{b}\wedge e^{b}$, might not vanish. As a consequence, the holonomic connection is not generally symmetric unless we force it to be. The connections can be written as before modulo the contortion tensor $$\omega^{a}{}_{b\mu}=\mathring{\omega}^{a}{}_{b\mu}+\Omega^{a}{}_{b\mu}\,,\qquad\Gamma^{\mu}{}_{\rho\nu}=\mathring{\Gamma}^{\mu}{}_{\rho\nu}+K^{\mu}{}_{\rho\nu}\,.$$ (3) In the above expressions, $\Omega^{a}{}_{b\mu}$ is the torsional part of the spin-connection, which is a tensor in both Lorentz and coordinate indices, and $K^{\lambda}{}_{\nu\mu}$ is the contortion tensor. The two are easily related by the tetrad postulate, $$\Omega^{a}{}_{b\mu}=E^{\nu}{}_{b}K^{\lambda}{}_{\nu\mu}e^{a}{}_{\lambda}\,.$$ (4) The contortion tensor is related to the components $T^{\mu}{}_{\nu\rho}=E^{\mu}{}_{a}{\cal T}^{a}{}(\partial_{\nu},\partial_{\rho})=2\Gamma^{\mu}{}_{[\rho\nu]}$ of the torsion through the relation $K^{\lambda}{}_{\mu\nu}=\frac{1}{2}\bigl{(}T_{\mu}{}^{\lambda}{}_{\nu}+T_{\nu}{}^{\lambda}{}_{\mu}-T^{\lambda}{}_{\mu\nu}\bigr{)}$. II.2 Weyl transformations vs independent connections Weyl tansformations, often referred to as conformal transformations in the literature of General Relativity Wald:1984rg , are defined as the Abelian group of transformations in which the metric is rescaled by a conformal factor, $g_{\mu\nu}\to g^{\prime}_{\mu\nu}={\rm e}^{2\sigma}g_{\mu\nu}$, where $\sigma=\sigma(x)$ is a local function over spacetime. The transformations are easily and unambiguously extended to the tetrads $$g_{\mu\nu}\rightarrow{\rm{e}}^{2\sigma}g_{\mu\nu}\,,\qquad e^{a}{}_{\mu}\rightarrow{\rm{e}}^{\sigma}e^{a}{}_{\mu}\,,\qquad E^{\mu}{}_{a}\rightarrow{\rm{e}}^{-\sigma}E^{\mu}{}_{a}\,.$$ (5) The Christoffel connection and the associated spin-connection transform, for they are expressed in terms of metric's and co-frame's components $$\displaystyle\mathring{\Gamma}^{\mu}{}_{\rho\nu}$$ $$\displaystyle\rightarrow$$ $$\displaystyle\mathring{\Gamma}^{\mu}{}_{\rho\nu}+\delta^{\mu}{}_{\nu}\partial_{\rho}\sigma+\delta^{\mu}{}_{\rho}\partial_{\nu}\sigma-g_{\nu\rho}g^{\mu\lambda}\partial_{\lambda}\sigma\,,$$ (6) $$\displaystyle\mathring{\omega}^{a}{}_{b\mu}$$ $$\displaystyle\rightarrow$$ $$\displaystyle\mathring{\omega}^{a}{}_{b\mu}+E^{\nu}{}_{b}e^{a}{}_{\mu}\partial_{\nu}\sigma-E^{\nu a}e_{b\mu}\partial_{\nu}\sigma\,.$$ (7) One seemingly innocuous fact is that the Weyl transformation of the Christoffel connection contains three contributions, while the one of the associated spin-connection contains two. This happens because $\mathring{\omega}^{a}{}_{b\mu}$ is antisymmetric in the Latin indices and so must be its transformation. To make it more transparent, it is sufficient to raise one index using the metric $\mathring{\omega}^{ab}{}_{\mu}=\mathring{\omega}^{a}{}_{c\mu}\eta^{cb}$, in which case $$\displaystyle\mathring{\omega}^{ab}{}_{\mu}$$ $$\displaystyle\rightarrow$$ $$\displaystyle\mathring{\omega}^{ab}{}_{\mu}+2E^{\nu[a}\delta^{b]}{}_{c}e^{c}{}_{\mu}\partial_{\nu}\sigma\,,$$ (8) where square brackets denote antisymmetrization with a factor. Of course, this is a fundamental property if we want the transformation of $\mathring{\omega}^{ab}{}_{\mu}$ to be an element of the local Lorentz algebra as it should. The same does not hold true for $\mathring{\Gamma}^{\mu}{}_{\rho\nu}$, which has a symmetric term proportional to $\delta^{\mu}{}_{\rho}$ in its transformation. Now we must face the problem of how to extend the conformal properties to the holonomic and anholonomic independent connections Iosifidis:2018zwo . There are two ``natural'' routes that we could follow. The first one is to enforce that the holonomic connection transforms like the Christoffel one, e.g. $\delta{\Gamma}^{\mu}{}_{\rho\nu}=\delta^{\mu}{}_{\nu}\partial_{\rho}\sigma+\delta^{\mu}{}_{\rho}\partial_{\nu}\sigma-g_{\nu\rho}g^{\mu\lambda}\partial_{\lambda}\sigma$, from which we deduce that the contortion tensor must not transform, $\delta K^{\mu}{}_{\rho\nu}=0$ (one index must be up). Likewise, the torsional part of the spin-connection does not transform either, $\delta\Omega^{a}{}_{b\mu}=0$. As a consequence, conformal symmetry cannot be a symmetry of the autoparallel equation, $\ddot{x}^{\mu}+\Gamma^{\mu}{}_{\rho\nu}\dot{x}^{\nu}\dot{x}^{\rho}=0$. Geodesics of the resulting geometry will consequently change according to the action of the Weyl group.111Notice that here we are using the symbol $\delta$ to denote finite transformations, not infinitesimal ones. Note that even the covariant derivative of tensor that is not charged under Weyl would not be Weyl invariant anymore. The second route, which is the one that we are concentrating on the most in this paper, is to require that the full connection does not transform, $\delta{\Gamma}^{\mu}{}_{\rho\nu}=0$, which does not change the autoparallel equation. One straightforward reason to follow this route is that, in the metric-affine formalism, $g_{\mu\nu}$ and ${\Gamma}^{\mu}{}_{\rho\nu}$ are two independent objects, so, a priori, there is no reason why the Weyl transformations of the two fields should be related with each other Iosifidis:2018zwo . In fact, this is the most natural choice if we aim at constructing a conformal (more precisely Weyl invariant) theory that generalizes General Relativity above a certain energy scale (e.g. the Planck mass). Such type of transformations are classified in Iosifidis:2018zwo , where they are called conformal transformation. However, now comes the crucial point: we are going to argue that the presence of torsion and the requirement of Weyl invariance are incompatible unless we promote the Weyl symmetry from a local symmetry to a full gauge one. II.3 Affine Weyl transformation of the torsion tensor We have already written the transformation properties of the Christoffel symbols $\mathring{\Gamma}^{\mu}{}_{\rho\nu}$; since ${\Gamma}^{\mu}{}_{\rho\nu}$ is required to be invariant, the contortion tensor must transform in the opposite way as $\mathring{\Gamma}^{\mu}{}_{\rho\nu}$ $$K^{\mu}{}_{\rho\nu}\rightarrow K^{\mu}{}_{\rho\nu}-\delta^{\mu}{}_{\nu}\partial_{\rho}\sigma-\delta^{\mu}{}_{\rho}\partial_{\nu}\sigma+g_{\nu\rho}g^{\mu\lambda}\partial_{\lambda}\sigma\,.$$ (9) Using the relation between the contortion tensor and the torsionful part of the spin-connection, $\Omega^{a}{}_{b\mu}=E^{\nu}{}_{b}K^{\lambda}{}_{\nu\mu}e^{a}{}_{\lambda}$, we deduce the Weyl transformation of $\Omega_{\mu}{}^{a}{}_{b}$ $$\Omega^{a}{}_{b\mu}\overset{?}{\to}\Omega^{a}{}_{b\mu}-e^{a}{}_{\mu}E^{\rho}{}_{b}\partial_{\rho}\sigma+e_{b\mu}E^{\rho a}\partial_{\rho}\sigma-\delta^{a}{}_{b}\partial_{\mu}\sigma\,.$$ (10) The origin of the incompatibility should now be transparent: the last term of the above equation does not belong to the adjoint representation of the algebra of the local Lorentz group. Instead, $\delta^{a}{}_{b}$ belongs to a $1$-dimensional trace-part of a symmetric rank-$2$ tensor, rather than an antisymmetric one, and the algebra is extended by one generator to an Abelian dilatation group, say $D_{1}$. This feature of Weyl transformations has already been noticed in deCesare:2016mml . If we assume that fundamental fields must transform according to irreducible representations of the Lorentz group, we should amend the gauge transformation to eliminate the trace part. In order to do so, the straighforward option is to introduce a new Abelian gauge potential, which we refer to as the Weyl connection $S_{\mu}$ (see for example charap1974gauge ), that accounts for covariance under local scale symmetry and belongs to this representation. We take $S_{\mu}$ to transform as $$S_{\mu}\rightarrow S_{\mu}-\partial_{\mu}\sigma\,,$$ (11) and we modify the relation (4) between the contortion tensor and the torsional part of the spin connection by replacing ${K}^{\mu}{}_{\rho\nu}$ with a new tensor ${\Phi}^{\mu}{}_{\rho\nu}$ as $$\Phi^{\mu}{}_{\rho\nu}=E^{\mu}_{a}\Bigl{(}\Omega^{a}{}_{b\nu}e_{\rho}^{b}+S_{\nu}e_{\rho}^{a}\Bigr{)}\,.$$ (12) Since we have changed the relation, we stress that $\Phi^{\mu}{}_{\rho\nu}$ does not have the same properties as the contortion ${K}^{\mu}{}_{\rho\nu}$ that appeared in Eq. (3), as it is not antisymmetric in its first two indices. We shall elaborate further on the actual difference in a moment. We see that, even if we assume that the tensor transforms as the old one (to balance $\mathring{\Gamma}^{\mu}{}_{\rho\nu}$), $\delta\Phi^{\mu}{}_{\rho\nu}=-\delta^{\mu}{}_{\nu}\partial_{\rho}\sigma-\delta^{\mu}{}_{\rho}\partial_{\nu}\sigma+g_{\nu\rho}g^{\mu\lambda}\partial_{\lambda}\sigma$, now the trace part of (10) is reproduced by (11), while $\Omega^{a}{}_{b\mu}$ remains an antisymmetric element of the algebra under transformations $$\Omega^{a}{}_{b\mu}\to\Omega^{a}{}_{b\mu}-e^{a}{}_{\mu}E^{\rho}{}_{b}\partial_{\rho}\sigma+e_{b\mu}E^{\rho a}\partial_{\rho}\sigma\,,$$ (13) which replaces (10). The question is: what have we given up as compared to the discussion of Sect. II.1? Notice first the holonomic relation of the new connection including $\Phi^{\mu}{}_{\rho\nu}$, which we denote $$\breve{\nabla}_{\mu}v^{\nu}=\partial_{\mu}v^{\nu}+\breve{\Gamma}^{\nu}{}_{\rho\mu}v^{\rho}=\partial_{\mu}v^{\nu}+\mathring{\Gamma}^{\nu}{}_{\rho\mu}v^{\rho}+K^{\nu}{}_{\rho\mu}v^{\rho}+S_{\mu}v^{\nu}\,,$$ (14) or, in other words, $\breve{\Gamma}^{\nu}{}_{\rho\mu}=\mathring{\Gamma}^{\nu}{}_{\rho\mu}+\Phi^{\nu}{}_{\rho\mu}=\mathring{\Gamma}^{\nu}{}_{\rho\mu}+K^{\nu}{}_{\rho\mu}+S_{\mu}\delta^{\nu}{}_{\rho}$, with $K^{\nu}{}_{\rho\mu}=E^{\nu}{}_{a}\Omega^{a}{}_{b\mu}e^{b}{}_{\rho}$ as in (4). The new contribution to the connection is a special case of distortion tensor, $\breve{L}^{\nu}{}_{\rho\mu}=S_{\mu}\delta^{\nu}{}_{\rho}$, which implies that we have added nonmetricity to the original geometrical construction. The nonmetricity tensor of $\breve{\nabla}$ is defined $\breve{Q}_{\mu\nu\rho}=-\breve{\nabla}_{\mu}g_{\nu\rho}=2S_{\mu}g_{\nu\rho}$. II.4 Take one: Weyl gauging to restore compatibility of the metric Given the special form of the nonmetricity of the connection introduced in (14) there is a natural way to extend it to a metric compatible connection. Assume that any field, say ${\cal V}^{A}$, where $A$ is an arbitrary collection of holonomic and anholonomic indices, carries an Abelian charge equal to its conformal weight $w({\cal V}^{A})=w_{\cal V}$ (in units of length), which is generally nonzero (notice that the weight depends on the covariance properties of the tensor ${\cal V}^{A}$). If we take the gauge field of the new Abelian group to be $S_{\mu}$ itself, i.e. the Weyl potential, then we can extend $\breve{\nabla}$ in (14) as $$\tilde{\nabla}_{\mu}{\cal V}^{A}=\breve{\nabla}_{\mu}{\cal V}^{A}+w_{{\cal V}}S_{\mu}{\cal V}^{A}\,.$$ (15) The resulting connection is covariant under Weyl transformations: for a transformation of the tensors as ${\cal V}^{A}\to{\rm e}^{w_{{\cal V}}\sigma}{\cal V}^{A}$, it is straightforward to see that $\tilde{\nabla}_{\mu}{\cal T}^{V}\to{\rm e}^{w_{{\cal V}}\sigma}\tilde{\nabla}_{\mu}{\cal V}^{A}$. In practice, covariant fields carry labels of both the local Lorentz group and $GL(4)$ in $A$, as well as the Weyl group (the latter being their weight), and we have gauged the latter. Naturally, the metric $g_{\mu\nu}$ carries the Weyl charge equal to its weight too, $w(g_{\mu\nu})=w_{g}=2$. Using the definitions (15) and (14) of the connections and the weight, it is straightforward to show that $\tilde{\nabla}$ is compatible with $g_{\mu\nu}$ $$\tilde{\nabla}_{\mu}g_{\mu\nu}=0\,.$$ (16) In the next section we discuss a more top-down approach to the construction of the same connection, which is both more general and unveils independently Weyl covariant contributions that are hidden in the general relation (15). The motivation for such an approach is that, even though the torsion that is used in this section is Weyl covariant, it is not isomorphic (meaning in one-to-one correspondence) to the contortion tensor, which has an affine Weyl transformation. This feature comes from the torsion-vector contribution stemming from the presence of the Weyl potential in (14). In fact, in the approach of this section the torsion transforms as a Weyl gauge potential, which has allowed for applications in which the torsion vector itself (modulo a constant) plays the role of $S_{\mu}$, see Karananas:2015eha ; Karananas:2021gco ; Karananas:2021zkl . Later on, we are also going to elaborate further on the properties of the Weyl covariant connection. III Weyl transformations vs independent connections: the Weyl covariant formulation In the following we introduce a different and more convenient way to deal with torsion in presence of Weyl gauging (as compared to the procedure given in Sect. II). The approach provided here should also be regarded as a top-down point of view on the construction of a Weyl covariant theory with MAG degrees of freedom. In order to give some further insight on the origin of the gauging of the Weyl group, we spend some time surveying some features of metric-affine theories of gravity, so we temporarily reset our notation, but, in due time, we reconnect with Sect. II. III.1 MAG interlude: general connections and the requirement of Weyl invariance In MAG theories, the affine connection is an independent field variable from the onset. For our purpose, it is a $1$-form $\Gamma_{\mu}$ with values in the algebra of the gauge group $GL(4)$. Whenever a metric tensor $g_{\mu\nu}$ is defined on the spacetime manifold, the affine connection can be split as $$\Gamma^{\lambda}{}_{\nu\mu}=\mathring{\Gamma}^{\lambda}{}_{\nu\mu}+\Phi^{\lambda}{}_{\nu\mu}\,,$$ (17) and $\mathring{\Gamma}$ is the symmetric compatible Levi-Civita connection, which is symmetric in the lower indices and metric compatible. Traditionally, the tensor $\Phi$ is further decomposed as $\Phi^{\lambda}{}_{\nu\mu}=N^{\lambda}{}_{\nu\mu}+K^{\lambda}{}_{\nu\mu}$, where $N$ is the distortion tensor and $K$ is the contortion tensor, which account for nonmetricity and torsion contributions, respectively. The tensor $K^{\lambda}{}_{\nu\mu}$ is antisymmetric in the first indices and written only in terms of the torsion $T^{\lambda}{}_{\mu\nu}=\Gamma^{\lambda}{}_{\nu\mu}-\Gamma^{\lambda}{}_{\mu\nu}$, while $N^{\lambda}{}_{\nu\mu}$, which is sometimes called con-metricity Floerchinger:2021uyo , is symmetric in the lower indices and written solely in terms of the nonmetricity $Q_{\lambda\mu\nu}=-\nabla_{\lambda}g_{\mu\nu}$. The explicit forms are $$\displaystyle K^{\lambda}{}_{\nu\mu}$$ $$\displaystyle=\,\frac{1}{2}\left(T_{\mu}{}^{\lambda}{}_{\nu}+T_{\nu}{}^{\lambda}{}_{\mu}-T^{\lambda}{}_{\nu\mu}\right)\,,$$ (18a) $$\displaystyle N^{\lambda}{}_{\nu\mu}$$ $$\displaystyle=\,\frac{1}{2}\left(Q_{\mu}{}^{\lambda}{}_{\nu}+Q_{\nu}{}^{\lambda}{}_{\mu}-Q^{\lambda}{}_{\mu\nu}\right)\,.$$ (18b) In order to understand the role of Weyl transformations on the components of $\Phi$, however, we need a slightly different decomposition to begin with. The tensor $\Phi$, which is a rank-$3$ tensor under coordinate transformations, can be split into irreducible parts with respect to the first two indices. We focus on the first pair of indices because they provide the information on the subgroups of $GL(4)$ (different choices of splitting can be found in the literature, and the relationship with one of them will be discussed below). Therefore, we perform the decomposition $$\Phi_{\lambda\nu\mu}=\Phi_{(\lambda\nu)\mu}+\Phi_{[\lambda\nu]\mu}\,,$$ (19) where we also notice that we have made use of the metric to lower the first index. The symmetric term can be expressed as the sum of trace and traceless parts, thence we have $$\Phi^{\lambda}{}_{\nu\mu}=\delta^{\lambda}{}_{\nu}V_{\mu}+D^{\lambda}{}_{\nu\mu}+M^{\lambda}{}_{\nu\mu}\,.$$ (20) We call $D_{\lambda\nu\mu}=D_{(\lambda\nu)\mu}$ the traceless distortion tensor, owing to the property $D^{\lambda}{}_{\lambda\mu}=0$; $M_{\lambda\nu\mu}=M_{[\lambda\nu]\mu}$ is the generalized contortion tensor; finally $V_{\mu}$ is the vector-distortion. We have that $M^{\lambda}{}_{\nu\mu}$ reduces to the usual contortion tensor in the limit in which both $D^{\lambda}{}_{\nu\mu}$ and the $V_{\mu}$ vanish, which should explain its name Hehl:1976kt . Let us further split the traceless distortion tensor as $$D^{\lambda}{}_{\nu\mu}=\frac{2}{9}\left(B_{\nu}\delta^{\lambda}{}_{\mu}+B^{\lambda}g_{\mu\nu}-\frac{1}{2}\delta^{\lambda}{}_{\nu}B_{\mu}\right)+C^{\lambda}{}_{\nu\mu}\,,$$ (21) where the co-vector $B_{\nu}$ is defined as $B_{\nu}\equiv D^{\lambda}{}_{\nu\lambda}$ and all the traces of $C^{\lambda}{}_{\nu\mu}$ vanish identically. We can ask ourselves what are the consequences of requiring that the whole affine connection is Weyl invariant, similarly to the discussion of Sect. II. In order to properly answer this question, we first need to introduce a conjugation in the algebra of $GL(4)$ induced by the metric $\Phi^{\lambda}{}_{\nu\mu}\leftrightarrow g_{\nu\rho}g^{\lambda\kappa}\Phi^{\rho}{}_{\kappa\mu}$. This comes in handy to perform (anti)symmetrization of the first two indices, while preserving the covariance properties of $\Phi$ as a $GL(4)$ algebra element. As in the previous section, we start from the assumption that the whole affine connection be Weyl invariant. Focusing on the Weyl variation of the linear combinations of $\Phi$ and its conjugate we get $$\displaystyle\delta^{W}_{\sigma}\left(\Phi^{\lambda}{}_{\nu\mu}\pm g_{\nu\rho}g^{\lambda\kappa}\Phi^{\rho}{}_{\kappa\mu}\right)$$ $$\displaystyle=\,\delta^{W}_{\sigma}\Phi^{\lambda}{}_{\nu\mu}\pm g_{\nu\rho}g^{\lambda\kappa}\delta^{W}_{\sigma}\Phi^{\rho}{}_{\kappa\mu}$$ (22) $$\displaystyle=\,-\delta^{\lambda}{}_{\nu}\left(1\pm 1\right)\partial_{\mu}\sigma-\delta^{\lambda}{}_{\mu}\left(1\mp 1\right)\partial_{\nu}\sigma+g_{\mu\nu}\left(1\mp 1\right)\partial^{\lambda}\sigma\,.$$ Therefore, if we consider the symmetric combination (upper signs), we see that it is natural to attribute the Weyl transformation only to the trace part of the symmetric combination, i.e. to the vector distortion $V_{\mu}$, because the transformation is proportional to $\delta^{\lambda}{}_{\nu}$. This implies naturally that $\delta^{W}_{\sigma}V_{\mu}=-\partial_{\mu}\sigma$ and $\delta^{W}_{\sigma}D^{\lambda}{}_{\nu\mu}=0$. Similarly, if we consider the antisymmetric combination, we can read the Weyl transformation of the generalized contortion tensor, $\delta^{W}_{\sigma}M^{\lambda}{}_{\nu\mu}=g_{\mu\nu}\partial^{\lambda}\sigma-\delta^{\lambda}{}_{\mu}\partial_{\nu}\sigma$. These choices ensure that the structure of the decomposition is unaltered by the action of the Weyl transformation.222Notice that the covariance of the indices is important, as raising or lowering them would change the action of $\delta^{W}_{\sigma}$. This is also the reason why in (22) we adopted that specific structure for the conjugate of $\Phi$. Consequently, we see indirectly that the traceless distortion, if present, is Weyl invariant if we demand that Weyl invariance is a symmetry of the full affine connection. We return shortly to the consequences of these equations. Now notice that the generalized contortion term automatically drops if we write the covariant derivative of the metric tensor, i.e., the nonmetricity $$Q_{\mu\nu\rho}\equiv-\nabla_{\mu}g_{\nu\rho}=2g_{\nu\rho}V_{\mu}+2D_{\nu\rho\mu}\,.$$ (23) The first term on the right hand side can be reabsorbed by a suitable redefinition of the covariant derivative, provided that we require the existence of a new $1$-dimensional, noncompact, Abelian symmetry group, in which case $V_{\mu}$ plays the role of its gauge connection as done in Sect. II.4. The Weyl transformation of $V_{\mu}$ deduced above complies with this notion. Therefore, the previous equation could be seen as merely expressing the need to write the covariant derivative of the metric tensor in a gauge-invariant way. In contrast, the same logic cannot be extended to the traceless distortion tensor, which always gives rise to ``true'' nonmetricity. Proceeding further, we can inspect the two inequivalent traces of the nonmetricity tensor $$\displaystyle Q_{\mu}{}^{\lambda}{}_{\lambda}=$$ $$\displaystyle\,8V_{\mu}\,,$$ (24a) $$\displaystyle Q_{\lambda}{}^{\lambda}{}_{\mu}=$$ $$\displaystyle\,2V_{\mu}+2B_{\mu}\,.$$ (24b) Therefore, we could restate the previous result as follows. The trace of the nonmetricity in the second and third indices can be interpreted as the Abelian gauge-potential of a noncompact scale symmetry group, and similarly brought to the left hand side of $\nabla g=-Q$, making the whole expression gauge-invariant. Now we focus our attention on the generalized contortion tensor. We denote its unique nontrivial contraction as $\chi_{\nu}\equiv M^{\lambda}{}_{\nu\lambda}$. We also separate the totally antisymmetric contribution defining $\theta_{\nu}\equiv M^{\lambda}{}_{\rho\mu}\varepsilon_{\nu\lambda}{}^{\rho\mu}$, and introduce $\kappa^{\lambda}{}_{\nu\mu}$, which is the trace-free and not totally antisymmetric part of the contortion tensor. We write the full decomposition as $$M^{\lambda}{}_{\nu\mu}=\frac{1}{3}\left(\chi_{\nu}\delta^{\lambda}{}_{\mu}-\chi^{\lambda}g_{\nu\mu}\right)+\frac{1}{6}\varepsilon_{\nu\mu}{}^{\lambda\rho}\theta_{\rho}+\kappa^{\lambda}{}_{\nu\mu}\,.$$ (25) This decomposition becomes useful if we concentrate on the torsion $2$-form, which, in holonomic indices, is defined by $$T^{\lambda}{}_{\mu\nu}\equiv\Gamma^{\lambda}{}_{\nu\mu}-\Gamma^{\lambda}{}_{\mu\nu}=\left(M^{\lambda}{}_{\nu\mu}-M^{\lambda}{}_{\mu\nu}\right)+\left(V_{\nu}\delta^{\lambda}{}_{\mu}-V_{\mu}\delta^{\lambda}{}_{\nu}\right)+\left(D^{\lambda}{}_{\nu\mu}-D^{\lambda}{}_{\mu\nu}\right)\,.$$ (26) Notice that all the tensors defined from the generalized contortion also enter in the equation of the torsion $2$-form. To have a better insight on this feature, let us write down the expressions for the three irreducible components of the torsion tensor (vector, axial-vector and tensor) $$\displaystyle\tau_{\mu}\equiv$$ $$\displaystyle\,T^{\lambda}{}_{\mu\lambda}=-\chi_{\mu}-3V_{\mu}-B_{\mu}\,,$$ (27a) $$\displaystyle\Theta_{\nu}\equiv$$ $$\displaystyle\,\varepsilon_{\nu\lambda}{}^{\rho\mu}T^{\lambda}{}_{\rho\mu}=\,2\theta_{\nu}\,,$$ (27b) $$\displaystyle t^{\lambda}{}_{\mu\nu}=$$ $$\displaystyle\,\kappa^{\lambda}{}_{\nu\mu}-\kappa^{\lambda}{}_{\mu\nu}+C^{\lambda}{}_{\nu\mu}-C^{\lambda}{}_{\mu\nu}\,.$$ (27c) Even though only the generalized contortion enters the expression of the axial torsion-vector, distortion generally appears in the other two expressions. In order to explicitly make contact with the previous section, we are going to discard the traceless distortion $D_{\mu\nu\rho}$ of the decomposition (21) henceforth. In this limit, we can easily identify $V_{\mu}$ with the connection $1$-form $S_{\mu}$ that is associated to local scale and Weyl invariances, as argued before and done in Sect. II. Given the gauge-potential for scale transformations, which transforms as $S_{\mu}\rightarrow S_{\mu}-\partial_{\mu}\sigma$, we notice that we can make the following ``redefinition'' of the contortion $$K^{\lambda}{}_{\nu\mu}=\hat{K}^{\lambda}{}_{\nu\mu}+\delta^{\lambda}{}_{\mu}S_{\nu}-g_{\mu\nu}S^{\lambda}\,.$$ (28) This definition has the advantage that a Weyl invariant term is singled out, in fact, the tensor $\hat{K}^{\lambda}{}_{\nu\mu}$ can thus be understood as a Weyl invariant generalization of the contortion tensor. Using the Weyl invariant contortion, the tensor part of the affine connection can be rewritten $$\Phi^{\lambda}{}_{\nu\mu}=\delta^{\lambda}{}_{\nu}S_{\mu}+K^{\lambda}{}_{\nu\mu}=\delta^{\lambda}{}_{\nu}S_{\mu}+\delta^{\lambda}{}_{\mu}S_{\nu}-g_{\mu\nu}S^{\lambda}+\hat{K}^{\lambda}{}_{\nu\mu}\equiv L^{\lambda}{}_{\nu\mu}+\hat{K}^{\lambda}{}_{\nu\mu}\,,$$ (29) which defines the tensor $L^{\lambda}{}_{\nu\mu}=\delta^{\lambda}{}_{\nu}S_{\mu}+\delta^{\lambda}{}_{\mu}S_{\nu}-g_{\mu\nu}S^{\lambda}$, see Smolin:1979uz ; Cheng:1988zx (recall that we have set $D_{\mu\nu\rho}=0$ of (21) after having identified $S_{\mu}$). The original distortion $N^{\lambda}{}_{\nu\mu}$ given in (18b) obviously differs from $L^{\lambda}{}_{\nu\mu}$. The tensor $L^{\lambda}{}_{\nu\mu}$ is precisely the contribution that appears in the traditional construction of the gauged Weyl-covariant derivative, which takes into account the fact that dilatations do not commute with spacetime transformations Iorio:1996ad ; Karananas:2015eha , though here it emerges from the decomposition under local scale transformations of a more general MAG connection. The first term on the right hand side of (29) is symmetric in the lower indices, as such it does not give rise to any torsion, whereas the second one is the Weyl invariant contortion tensor and can be used to find the torsion tensor itself. However, it is important to stress at this stage that $L^{\lambda}{}_{\nu\mu}$ is not a distortion tensor, since it is not symmetric in the first two indices. Nevertheless, it contains a distortion part, i.e. the one which is proportional to $\delta^{\lambda}{}_{\nu}$. III.2 Take two: Weyl invariant torsion The discussion of Sect. II on the bottom-up approach to motivate the Weyl gauging procedure culminated in Sect. II.4 where we assumed that the potential of the Abelian symmetry be the Weyl potential itself, extending the covariant derivative in (15). It is natural to make the same choice in the second approach just outlined. The added bonus coming from this section is that we have separated the Weyl invariant contortion $\hat{K}$. The new covariant derivative is defined as $$\tilde{\nabla}_{\mu}{\cal V}^{A}=\hat{\nabla}_{\mu}{\cal V}^{A}+(\hat{K}_{\mu})^{A}{}_{B}{\cal V}^{B}+w_{{\cal V}}S_{\mu}{\cal V}^{A}\,,$$ (30) where $w_{{\cal V}}$ is the Weyl weight of the tensor ${\cal V}^{A}$, and, again, $A$ is a collective multi-index standing for both Lorentz, coordinate and other internal indices. The notation $(\hat{K}_{\mu})^{A}{}_{B}{\cal V}^{B}$ stands symbolically for the presence of a contortion contribution to the full covariant derivative (the components $\hat{K}^{\nu}{}_{\rho\mu}$ act on each index, as usual, with a plus sign for every contravariant coordinate index and a minus sign for every covariant one). The connection is covariant under Weyl transformations: for a transformation of the tensors as ${\cal V}^{A}\to{\rm e}^{w_{{\cal V}}\sigma}{\cal V}^{A}$, it is straightforward to see that $\tilde{\nabla}_{\mu}{\cal V}^{A}\to{\rm e}^{w_{{\cal V}}\sigma}\tilde{\nabla}_{\mu}{\cal V}^{A}$. As a consequence, covariant fields carry labels of both the local Lorentz group in $A$ as well as the Weyl group, the latter being their weights. In fact, the covariant derivative (30) coincides with (15). The connection $\tilde{\nabla}_{\mu}$ is metric compatible and satisfies the tetrad postulate. Metric compatibility follows easily from the antisymmetry of the contortion tensor in its first two indices and the definition of $L^{\rho}{}_{\nu\mu}$ $$\tilde{\nabla}_{\mu}g_{\alpha\beta}=\mathring{\nabla}_{\mu}g_{\alpha\beta}-L^{\lambda}{}_{\alpha\mu}g_{\lambda\beta}-L^{\lambda}{}_{\beta\mu}g_{\alpha\lambda}+w_{g}S_{\mu}g_{\alpha\beta}=0\,,$$ (31) where we also used the original compatibility of the Levi-Civita connection $\mathring{\nabla}$ and the fact that the metric has weight two, $w_{g}=2$. III.2.1 Holonomic splitting and the covariant derivative $\hat{\nabla}$ To give an explicit example of the holonomic action of $\tilde{\nabla}$, we take a spacetime vector $v^{\mu}$ with weight $w_{v}$. The connection acts as $$\displaystyle\tilde{\nabla}_{\mu}v^{\nu}$$ $$\displaystyle=$$ $$\displaystyle\partial_{\mu}v^{\nu}+\tilde{\Gamma}^{\nu}{}_{\sigma\mu}v^{\sigma}+w_{v}S_{\mu}v^{\nu}=\partial_{\mu}v^{\nu}+\hat{\Gamma}^{\nu}{}_{\sigma\mu}v^{\sigma}+\hat{K}^{\nu}{}_{\sigma\mu}v^{\sigma}+w_{v}S_{\mu}v^{\nu}\,.$$ (32) The convenience of the approach discussed in this section should be apparent from the previous equation. The full holonomic connection $\tilde{\Gamma}^{\nu}{}_{\sigma\mu}$ is split in the torsion-free $\hat{\Gamma}^{\nu}{}_{\sigma\mu}$ and torsionful $\hat{K}^{\nu}{}_{\sigma\mu}$ parts, each of them being Weyl invariant. In fact, we can write $$\displaystyle\tilde{\nabla}_{\mu}v^{\nu}$$ $$\displaystyle=$$ $$\displaystyle\hat{\nabla}_{\mu}v^{\nu}+\hat{K}^{\nu}{}_{\sigma\mu}v^{\sigma}\,,$$ (33) where $\hat{\nabla}$, which does not include the torsional part $\hat{K}^{\nu}{}_{\sigma\mu}$, is the connection acting as $$\displaystyle\hat{\nabla}_{\mu}v^{\nu}=\partial_{\mu}v^{\nu}+\hat{\Gamma}^{\nu}{}_{\rho\mu}v^{\rho}+w_{v}S_{\mu}v^{\nu}\,,$$ (34) with components $$\displaystyle\hat{\Gamma}^{\nu}{}_{\sigma\mu}=$$ $$\displaystyle\,\mathring{\Gamma}^{\nu}{}_{\sigma\mu}+L^{\nu}{}_{\sigma\mu}$$ (35) $$\displaystyle=$$ $$\displaystyle\,\mathring{\Gamma}^{\nu}{}_{\sigma\mu}+\left(\delta^{\nu}{}_{\sigma}S_{\mu}+\delta^{\nu}{}_{\mu}S_{\sigma}-g_{\mu\sigma}S^{\nu}\right)$$ $$\displaystyle=$$ $$\displaystyle\,\frac{1}{2}\,g^{\nu\rho}\left(\hat{D}_{\mu}g_{\rho\sigma}+\hat{D}_{\sigma}g_{\rho\mu}-\hat{D}_{\rho}g_{\mu\sigma}\right)\,.$$ The connection $\hat{\nabla}$ becomes Weyl covariant thanks to the contribution of the Weyl potential and reproduces Iorio:1996ad ; Karananas:2015eha . Among other things, this connection is symmetric and compatible, but it can also be integrated by parts maintaining covariance (provided that the weight of the scalar that is integrated is $-4$, as it should to balance the weight of $\sqrt{-g}$, see Appendix B). It can be thought of as a torsionless version of $\tilde{\nabla}$, because they differ by the Weyl invariant contortion $\hat{K}$. We believe that the new procedure to obtain $\hat{\nabla}$, as well as the procedure to couple it to torsional degrees of freedom, are the most important results of our work, as we are trying to motivate in the following. The components of the torsion of the affine connection $\tilde{\nabla}$ are $$\tilde{T}^{\sigma}{}_{\mu\nu}=2\tilde{\Gamma}^{\sigma}{}_{[\nu\mu]}=2\hat{K}^{\sigma}{}_{[\nu\mu]}\,,$$ (36) while the contortion tensor is written in terms of the torsion as in Eq. (18a), i.e. $$\hat{K}^{\rho}{}_{\nu\mu}=\frac{1}{2}\left(\tilde{T}_{\nu}{}^{\rho}{}_{\mu}+\tilde{T}_{\mu}{}^{\rho}{}_{\nu}-\tilde{T}^{\rho}{}_{\nu\mu}\right)\,.$$ (37) We also note that, defining the torsion-vector as $$\tilde{\tau}_{\mu}\equiv\tilde{T}^{\nu}{}_{\mu\nu}\,,$$ (38) we have $\hat{K}^{\nu}{}_{\mu\nu}=-\tilde{\tau}_{\mu}$. III.2.2 Anholonomic splitting Up to now we have considered the splitting into torsion-free and torsionful pieces of the holonomic affine connection. We repeat the same reasoning for the anholonomic case as well. Given the gauge transformations of the co-frame, the torsion $2$-form can be defined as its covariant exterior-derivative (i.e. its covariant-curl) $$\tilde{T}^{a}=\tilde{D}e^{a}\,,$$ (39) where $\tilde{D}$ is the part of $\tilde{\nabla}$ in which only the gauge-potentials $\omega^{a}{}_{b\mu}$ and $S_{\mu}$ are considered. In general, we adopt the notation in which the capitalized letter $D$ includes only the gauge connections, as opposed to $\nabla$ which contains the full connection. On the anholonomic side, and in absence of Weyl gauging, the splitting of the spin-connection, induced by the presence of a metric tensor, is usually written as $\omega^{a}{}_{b}=\mathring{\omega}^{a}{}_{b}+\Omega^{a}{}_{b}$ (see Sect. II). Here $\mathring{\omega}^{a}{}_{b}$ solves the torsion-free condition $de^{a}+\mathring{\omega}^{a}{}_{b}\wedge e^{b}=0$, i.e., $\mathring{\omega}^{a}{}_{b}$ can be expressed in terms of $e^{a}$. The torsion-free condition is Weyl covariant, because the infinitesimal Weyl transformation of $\mathring{\omega}^{a}{}_{b}$ takes the form $\delta^{W}_{\sigma}\mathring{\omega}^{a}{}_{b}=(\partial^{b}\sigma)e^{a}-(\partial^{a}\sigma)e^{b}$. Therefore, if we insist with the above splitting, the Weyl covariant torsion $2$-form of the full Weyl invariant connection reads $$\tilde{T}^{a}=de^{a}+(\mathring{\omega}+\Omega)^{a}{}_{b}\wedge e^{b}+S\wedge e^{a}=\Omega^{a}{}_{b}\wedge e^{b}+S\wedge e^{a}\,.$$ (40) Since the last term is not Weyl covariant, $\Omega^{a}{}_{b}$ itself cannot be Weyl invariant to balance out. In absence of Weyl gauging, this gives rise to a noncovariant torsion $2$-form Fabbri:2011vk . The affine transformation properties of $\Omega^{a}{}_{b\mu}$ can be understood better by realizing that it is isomorphic (meaning in one-to-one correspondence) to the noncovariant contortion tensor of Sect. II, i.e., $\Omega^{a}{}_{b\mu}=e^{a}{}_{\lambda}E^{\nu}{}_{b}K^{\lambda}{}_{\nu\mu}$. Similarly to the case of the full holonomic connection, we can take into account a more elaborate splitting, which is the anholonomic counterpart of (28): $$\omega^{a}{}_{b}=\mathring{\omega}^{a}{}_{b}+e^{a}(E_{b}\cdot S)-e_{b}(E^{a}\cdot S)+\hat{\Omega}^{a}{}_{b}\equiv\hat{\omega}^{a}{}_{b}+\hat{\Omega}^{a}{}_{b}\,.$$ (41) We are going to see in a moment that $\hat{\Omega}^{a}{}_{b}$ is related to $\hat{K}$, $\hat{\Omega}^{a}{}_{b\mu}=e^{a}{}_{\lambda}E^{\nu}{}_{b}\hat{K}^{\lambda}{}_{\nu\mu}$ in anholonomic form. Thus, using $e_{b}\wedge e^{b}=0$, the covariant torsion $2$-form takes the form $$\tilde{T}^{a}=de^{a}+\mathring{\omega}^{a}{}_{b}\wedge e^{b}+e^{a}\wedge S+\hat{\Omega}^{a}{}_{b}\wedge e^{b}+S\wedge e^{a}=\hat{\Omega}^{a}{}_{b}\wedge e^{b}\,,$$ (42) where we used the defining equation of $\mathring{\omega}^{a}{}_{b}$ and the antisymmetry of the wedge. We have that $\hat{\Omega}^{a}{}_{b}$ is Weyl invariant, likewise $\hat{K}^{\mu}{}_{\nu\rho}$, as expected. We find a very similar expression, as well as the interpolating cases in Ref. Izaurieta:2020kuy , though the authors did not gauge the Weyl group, thus their results can be obtained from ours using the pure gauge limit of the Weyl potential ($S_{\mu}$ is a total derivative). III.2.3 General comments on dependences and compatibility in view of the Nöther identities Consider Eqs. (40) and (42). In both cases, the splitting can be seen as a simple redefinition of the components of the connections themselves, in which we rearranged the dependence on their parts. In the first case, the three ``natural'' field variables become $(e^{a},\Omega^{a}{},S)$, while in the second case they are $(e^{a},\hat{\Omega}^{a}{}_{b},S)$. The two choices are possible, and both natural, to an extent. However, we claim that the second one is somewhat preferable, since $\hat{\Omega}^{a}{}_{b}$ is Weyl invariant and it is in one-to-one correspondence with the covariant torsion $2$-form, thus it can be more easily physically and geometrically interpreted. Clearly, Eq. (41) is in complete analogy with Eq. (28) given above; in fact, it is straightforward to prove that there is actually an isomorphism, $$\hat{K}^{\lambda}{}_{\nu\mu}=E^{\lambda}{}_{b}e^{a}{}_{\nu}\hat{\Omega}^{a}{}_{b\mu}\,,$$ (43) relating the two tensors. After the splitting of the spin-connection, the full covariant derivative of the co-frame takes the form $$\tilde{\nabla}_{\mu}e^{a}{}_{\nu}=\partial_{\mu}e^{a}{}_{\nu}-\mathring{\Gamma}^{\lambda}{}_{\nu\mu}e^{a}{}_{\lambda}+\hat{\omega}^{a}{}_{b\mu}e^{b}{}_{\nu}-\hat{K}^{\lambda}{}_{\nu\mu}e^{a}{}_{\lambda}-L^{\lambda}{}_{\nu\mu}e^{a}{}_{\lambda}+\hat{\Omega}^{a}{}_{b\mu}e^{b}{}_{\nu}+w_{e}S_{\mu}e^{a}{}_{\nu}\,,$$ (44) which vanishes for $w_{e}=1$ by virtue of the other compatibility, $\mathring{\nabla}_{\mu}e^{a}{}_{\nu}=0$, by use of Eq. (43), and by the definitions of $\hat{\omega}^{a}{}_{b}$ and $L^{\lambda}{}_{\nu\mu}$ (see Eqs. (28), (29) and (41)). IV Weyl invariance and Palatini's approach A natural question is whether the lengthy discussions of Sects. II and III are relevant in a more traditional context closer to standard general relativity. We briefly revisit here an argument that appeared recently in Ref. Wheeler:2022ggm , adapting it to our purpose. Consider the simplest Einstein-Hilbert action with independent variables $g_{\mu\nu}$ and $\Gamma^{\mu}{}_{\nu\rho}$ $$S[g,\Gamma]=\int\sqrt{-g}g^{\mu\nu}R^{\rho}{}_{\mu\rho\nu}\,,$$ (45) where $R^{\mu}{}_{\nu\rho\theta}$ is the curvature tensor of $\Gamma^{\mu}{}_{\nu\rho}$, which we take to be symmetric as in the Palatini's approach. The variation of $S[g,\Gamma]$ with respect to the metric reproduces Einstein's equations straightforwardly, but with the independent symmetric connection. In a famous paper Palatini:1919 , Palatini has shown that the metricity condition on $\Gamma^{\mu}{}_{\nu\rho}$ has dynamical origin. The equations of motion coming from the variation of the connection are $$\nabla_{\mu}\left(\sqrt{-g}g^{\nu\rho}\right)=0\,.$$ (46) The action of $\nabla_{\mu}$ on $\sqrt{-g}$ can be deduced by the fact that any operator, say $\delta$, that satisfies the Leibniz rule acts on the determinant as $\delta\sqrt{-g}=-\frac{1}{2}\sqrt{-g}g_{\rho\nu}\delta g^{\rho\nu}$. Taking $\delta=\nabla_{\mu}$, we can rewrite the field equations (46) as $$\nabla_{\mu}g^{\nu\rho}-\frac{1}{2}g^{\nu\rho}g_{\lambda\sigma}\nabla_{\mu}g^{\lambda\sigma}=0$$ (47) Notice that we are not extending the connection to densities, but we are simply considering the determinant for what it is. Contracting the previous equation with $g_{\nu\rho}$ and specializing to a manifold of dimension $d\neq 2$, one finds $g_{\nu\rho}\nabla_{\mu}g^{\nu\rho}=0$, which can be inserted back in (47) to obtain $\nabla_{\mu}g^{\nu\rho}=0$. In short, we have that the connection has to be compatible with the metric on-shell $$\nabla_{\mu}g_{\nu\rho}=0\,.$$ (48) The traditional interpretation is that the unique solution to the above equation is $\Gamma^{\mu}{}_{\nu\rho}=\mathring{\Gamma}^{\mu}{}_{\nu\rho}$, that is, the Levi-Civita connection, which makes the Palatini action equivalent to Einstein-Hilbert one on-shell. However, this is entirely true only in absence of Abelian gauge symmetries under which the metric has a nontrivial charge. To show this, assume that the metric tensor has some charge $w_{g}$ under an Abelian gauge symmetry and that it transforms as $$g_{\mu\nu}\rightarrow{\rm e}^{w_{g}\sigma}g_{\mu\nu}\,.$$ (49) The Abelian group can be either compact or noncompact, according to the value of $w_{g}$. The relation with Weyl invariance should be obvious from the form of the transformation for $w_{g}=2$, in which case the group would be noncompact. Denoting with $S_{\mu}$, as in the previous sections, the Abelian gauge potential which transforms as $S_{\mu}\rightarrow S_{\mu}-\partial_{\mu}\sigma$, we have that the gauge covariant (under the Abelian group) derivative of the metric tensor is $$\hat{D}_{\mu}g_{\nu\rho}=\partial_{\mu}g_{\nu\rho}+w_{g}S_{\mu}g_{\nu\rho}\,.$$ (50) Recall that we reserve the capitalized letter $D$ for purely gauge covariant derivatives, which, in this case, contain the new Abelian charge $w_{g}$. In fact, the latter equation is not coordinate covariant, because it must be supplemented by the connection $\Gamma^{\mu}{}_{\nu\rho}$ as in $$\hat{\nabla}_{\mu}g_{\nu\rho}=\partial_{\mu}g_{\nu\rho}-\Gamma^{\lambda}{}_{\nu\mu}g_{\lambda\rho}-\Gamma^{\lambda}{}_{\rho\mu}g_{\nu\lambda}+w_{g}S_{\mu}g_{\nu\rho}\,,$$ (51) which is now covariant under both coordinate and Abelian transformations. It is convenient to split $\Gamma^{\mu}{}_{\nu\rho}$ in terms of the Levi-Civita connection $\mathring{\Gamma}^{\mu}{}_{\nu\rho}$ and an additional symmetric tensor, i.e. $\Gamma^{\mu}{}_{\nu\rho}=\mathring{\Gamma}^{\mu}{}_{\nu\rho}+L^{\mu}{}_{\nu\rho}$, which, to some extent, plays the role of a distortion. The tensor $L^{\mu}{}_{\nu\rho}$ inherits the symmetry property of the original Palatini connection. Looking back at the compatibility condition, we find $$\hat{\nabla}_{\mu}g_{\nu\rho}\equiv\mathring{\nabla}_{\mu}g_{\nu\rho}-L^{\lambda}_{\phantom{\lambda}\nu\mu}g_{\lambda\rho}-L^{\lambda}_{\phantom{\lambda}\rho\mu}g_{\nu\lambda}+w_{g}S_{\mu}g_{\nu\rho}=0\,.$$ (52) By construction, the Levi-Civita connection is compatible, $\mathring{\nabla}_{\mu}g_{\nu\rho}=0$, so we have $L_{\rho\mu\nu}+L_{\nu\mu\rho}-w_{g}S_{\mu}g_{\nu\rho}=0$. We can apply the standard manipulation of the fundamental theorem of Riemannian geometry (cycling the three indices, summing two expressions and subtracting the third) to find $$L_{\rho\mu\nu}=\frac{w_{g}}{2}\left(S_{\mu}g_{\nu\rho}+S_{\nu}g_{\mu\rho}-S_{\rho}g_{\mu\nu}\right)\,.$$ (53) The ``hat'' connection that we have just written down coincides with the one given in Sect. III.2 if we assign the weight $w_{g}=2$ that is required for compatibility. The full affine connection can be written in a manifestly gauge-invariant way $$\Gamma^{\rho}_{\phantom{\rho}\mu\nu}=\frac{1}{2}g^{\rho\lambda}\bigl{(}\hat{D}_{\mu}g_{\nu\lambda}+\hat{D}_{\nu}g_{\mu\lambda}-\hat{D}_{\lambda}g_{\mu\nu}\bigr{)}\,,$$ (54) which represents a departure from the traditional Palatini solution, $\Gamma^{\mu}{}_{\nu\rho}=\mathring{\Gamma}^{\mu}{}_{\nu\rho}$. We stress that the departure is possible only in presence of an additional Abelian symmetry besides that of general covariance. V Nöther identities in the Cartan-Weyl formalism Now we couple a matter field to our geometrical construction, the objective being to find the general Nöther identities. Consider an arbitrary matter field $\phi$ that transforms according to some linear representation of the Lorentz group and with a general Weyl weight. We take a general Lorentz, Weyl and diffeomorphism invariant action of the form $$S_{\rm m}=S_{\rm m}[\phi,e^{a},\omega^{a}{}_{b},S]\,.$$ (55) We refer to this form as the Cartan-Weyl form because the coframe and the gauge connections are regarded as independent, and the torsional degrees of freedom are inside $\omega^{a}{}_{b}$, which has affine transformation properties under the Lorentz group. In the next section we discuss equivalent identities that come from rearranging the dependencies of $S_{\rm m}$ in a different way. The general functional differential of $S_{\rm m}$ can be expressed as $$\delta S_{\rm m}=\int\frac{\delta S_{\rm m}}{\delta\phi}\delta\phi+\int\underline{e}\,T^{\mu}{}_{a}\delta e^{a}{}_{\mu}+\int\underline{e}\,\Sigma^{\mu}{}_{ab}\delta\omega^{ab}{}_{\mu}+\int\underline{e}\,\Delta^{\mu}\delta S_{\mu}\,,$$ (56) where we have introduced the scalar density $\underline{e}\equiv\det(e^{a}{}_{\mu})=\sqrt{-g}$. Most importantly, we define the energy-momentum tensor $T^{\mu}{}_{a}$, the spin-current $\Sigma^{\mu}{}_{ab}$, and the dilation-current $\Delta^{\mu}$ as variations of $S_{\rm m}$ with respect to the gravitational fields, $$T^{\mu}{}_{a}=\frac{1}{\underline{e}}\frac{\delta S_{\rm m}}{\delta e^{a}{}_{\mu}}\,,\qquad\Sigma^{\mu}{}_{ab}=\frac{1}{\underline{e}}\frac{\delta S_{\rm m}}{\delta\omega^{ab}{}_{\mu}}\,,\qquad\Delta^{\mu}=\frac{1}{\underline{e}}\frac{\delta S_{\rm m}}{\delta S_{\mu}}\,.$$ (57) The sign-convention of our definition of the energy-momentum tensor adheres with that of Weinberg's textbook Weinberg:1972kfs . In the MAG literature, when the connection is completely general, the functional derivative of the matter action is referred to as hypermomentum Hehl:1976kt , but, in our formalism, it splits into the sum of the spin-current and the dilation-current. On-shell, we can insert a solution to the equations of motion, $\frac{\delta S_{\rm{m}}}{\delta\phi}=0$, thus canceling the first term of (56). Then, the differentials can be evaluated along the aforementioned symmetry transformations, yielding the respective Nöther identities among the tensors. V.1 Local Lorentz and Weyl symmetries An infinitesimal local Lorentz transformation has the form $\Lambda^{a}{}_{b}=\delta^{a}{}_{b}+\alpha^{a}{}_{b}$, for a local antisymmetric matrix, $\alpha^{ab}=-\alpha^{ba}$. The transformation involves the Latin indices and is inhomogeneous for the spin-connection. The local Lorentz variations of co-frame, spin-connection and Weyl potential are $$\begin{split}&\delta_{\alpha}^{L}\,e^{a}{}_{\mu}=\alpha^{a}{}_{b}e^{b}{}_{\mu}\,,\\ &\delta_{\alpha}^{L}\,\omega^{a}{}_{b\mu}=-\partial_{\mu}\alpha^{a}{}_{b}-[\omega_{\mu},\alpha]^{a}{}_{b}=-D_{\mu}\alpha^{a}{}_{b}\,,\\ &\delta_{\alpha}^{L}\,S_{\mu}=0\,.\end{split}$$ (58) As in the previous sections, we have used the symbol $D_{\mu}$ to stress that the corresponding covariant derivative is purely gauge, because, in general, the transformation of a connection, such as $\omega^{a}{}_{b\mu}$, can be rewritten as a gauge covariant derivative acting on the transformation parameter itself (up to a convention-dependent sign). Since the matrix $\alpha^{a}{}_{b}$ carries only Lorentz indices, we have $D_{\mu}\alpha^{a}{}_{b}=\tilde{\nabla}_{\mu}\alpha^{a}{}_{b}$ by properly assigning Weyl weight zero to $\alpha^{a}{}_{b}$. Using (58) in (56), going on-shell and integrating by parts, we find $$\begin{split}\delta^{L}_{\alpha}S_{\rm m}&=\int\underline{e}\,T^{\mu}{}_{a}\alpha^{a}{}_{b}e^{b}{}_{\mu}-\int\underline{e}\,\Sigma^{\mu}{}_{ab}\tilde{\nabla}_{\mu}\alpha^{ab}=\int\underline{e}\,\alpha^{ab}\left\{T^{\mu}{}_{a}e_{b\mu}+\tilde{\nabla}_{\mu}\Sigma^{\mu}{}_{ab}+\Sigma^{\mu}{}_{ab}\tilde{\tau}_{\mu}\right\}\,.\end{split}$$ (59) The integration by parts of a nonmetric connection has to be treated with care, so we refer to the discussion of appendix B for more insights. From the above equation we see that Lorentz symmetry implies a generalized conservation law, which states that the antisymmetric part of the energy-momentum tensor is the divergence of the spin-current, modulo a torsion-vector contribution Kibble:1961ba . Denoting $T_{ab}=T^{\mu}{}_{b}\eta_{ac}e^{c}{}_{\mu}$, we have $$T_{[ab]}=(\tilde{\nabla}_{\mu}+\tilde{\tau}_{\mu})\Sigma^{\mu}{}_{ab}\,.$$ (60) (see Kibble:1961ba , as well as Eq. (44) in Penrose:1983mf , where the author deals with the Einstein-Cartan-Sciama-Kibble theory). The presence of an antisymmetric part is not unexpected, in fact, if the tensor $T_{\mu\nu}$ were computed using the connection $\mathring{\omega}^{a}{}_{b}$ the result would be symmetric on-shell, but the connection ${\omega}^{a}{}_{b}$ in this formulation is independent and does not guarantee such property. Now we turn to Weyl gauge invariance. We take the infinitesimal version of the Weyl transformations, seen also in the previous sections, that is, $g_{\mu\nu}\to g^{\prime}_{\mu\nu}=(1+2\sigma)g_{\mu\nu}$ for an infinitesimal local function $\sigma=\sigma(x)$. The transformations are $$\begin{split}&\delta_{\sigma}^{W}e^{a}{}_{\mu}=\sigma e^{a}{}_{\mu}\,,\qquad\delta_{\sigma}^{W}\omega^{a}{}_{b\mu}=0\,,\qquad\delta_{\sigma}^{W}S_{\mu}=-\partial_{\mu}\sigma\,,\end{split}$$ (61) where the fact that the spin-connection $\omega^{a}{}_{b\mu}$ is Weyl invariant reflects our assumption that the gauge-group is $SO(3,1)\times D(1)$, i.e., a direct product (the same gauge transformations are taken into account in Karananas:2015eha ). Inserting (61) in (56) with $\phi$ on-shell and integrating by parts, we find $$\delta_{\sigma}^{W}S_{\rm m}=\int\underline{e}\sigma\left\{T^{\mu}{}_{a}e^{a}{}_{\mu}+(\tilde{\nabla}_{\mu}+\tilde{\tau}_{\mu})\Delta^{\mu}\right\}\,.$$ (62) Furthermore, using the definition of $\tilde{\nabla}_{\mu}$, given in (32), in terms of $\hat{\nabla}_{\mu}$, given in (34), and $\hat{K}_{\mu}$, we see that the vector-torsion contribution cancels when switching to $\hat{\nabla}_{\mu}$. Thus, the consequence of Weyl gauge invariance is that the trace of the energy-momentum tensor is the negative of the divergence of the dilation current $$T^{\mu}{}_{\mu}=-\hat{\nabla}_{\mu}\Delta^{\mu}\,,$$ (63) if expressed in the Weyl covariant ``hatted'' connection. Some comments are in order. Whereas Eq. (60) is widely known in the literature and represents the first step of the Belinfante procedure for the improvement of the energy-momentum tensor belinfante1940current , Eq. (63) usually appears in a different form, though we found a very similar expression in Floerchinger:2021uyo . This happens because Weyl symmetry is not generally gauged, even though it is a local symmetry either way. Enforcing the invariance without gauging leads to $T^{\mu}{}_{\mu}=0$, which is a signature of Weyl symmetry and also of conformal symmetry, to some extent. Instead, $T^{\mu}{}_{\mu}$ is the divergence of a vector, modulo the torsion-vector term. In the flat space limit, this is the signature of a scale invariant (i.e. a rigid Weyl invariant) theory ORaifeartaigh:1996hvx . The vector $\Delta^{\mu}$ for the case of a scale invariant theory in flat space is generally referred to as virial current Coleman:1970je ; Nakayama:2013is . V.2 Diffeomorphism invariance and improved transformations We finally turn our attention to the consequences of the diffeomorphism invariance of the theory. To discuss the last symmetry we find convenient to introduce a different, yet equivalent, type of transformations, which we refer to as ``improved'', because they let us write down formulas that are covariant with repect to all the gauge symmetries at every step. The diffeomorphism Einstein's variations are parametrized by a contravariant vector field locally written as $\xi=\xi^{\mu}\partial_{\mu}$ and all gauge potentials transform infinitesimally according to their Lie derivatives $$\begin{split}&\delta^{E}_{\xi}e^{a}{}_{\mu}=\pounds_{\xi}e^{a}{}_{\mu}\,,\qquad\delta^{E}_{\xi}\omega^{a}{}_{b\mu}=\pounds_{\xi}\omega^{a}{}_{b\mu}\,,\qquad\delta^{E}_{\xi}S_{\mu}=\pounds_{\xi}S_{\mu}\,.\end{split}$$ (64) The algebra of infinitesimal diffeomorphisms satisfies the commutation relation $$\begin{split}\left[\delta^{E}_{\xi},\delta^{E}_{\zeta}\right]=\delta^{E}_{[\xi,\zeta]}\,,\end{split}$$ (65) where $[\xi,\zeta]$ are the Lie brackets of the two generators with components $[\xi,\zeta]^{\mu}=\xi^{\nu}\partial_{\nu}\zeta^{\mu}-\zeta^{\nu}\partial_{\nu}\xi^{\mu}=\xi^{\nu}\tilde{\nabla}_{\nu}\zeta^{\mu}-\zeta^{\nu}\tilde{\nabla}_{\nu}\xi^{\mu}+\xi^{\nu}\zeta^{\rho}\tilde{T}^{\mu}{}_{\rho\nu}$, assuming that the vectors $\xi^{\mu}$ and $\zeta^{\mu}$ have Weyl weight zero. The algebraic structure is a direct consequence of the same property holding for the Lie derivative. V.2.1 Improved transformations The Lie derivative is insensitive to the internal gauge indices of a given field or potential. As a consequence, the diffeomorphism transformation of tensors that are not gauge singlets is not covariant with respect to the Lorentz group. Take for example a spin vector $v^{a}$ with weight $w_{v}$, we obviously have that the action of diffeomorphisms does not ``see'' the Latin index and the Weyl weight, $\delta^{E}_{\xi}v^{a}=\pounds_{\xi}v^{a}=\xi^{\mu}\partial_{\mu}v^{a}$. For further insight, we can rewrite the partial derivative as the covariant one and subtract the connection terms $$\begin{split}\delta^{E}_{\xi}v^{a}=\xi^{\mu}\partial_{\mu}v^{a}&=\xi^{\mu}\tilde{\nabla}_{\mu}v^{a}-\xi^{\mu}\omega^{a}{}_{b\mu}v^{b}-w_{v}\xi^{\mu}S_{\mu}v^{a}\\ &=\xi^{\mu}\tilde{\nabla}_{\mu}v^{a}-(\xi\cdot\omega)^{a}{}_{b}v^{b}-w_{v}(\xi\cdot S)v^{a}\,,\end{split}$$ (66) where we introduced the shorthands $(\xi\cdot\omega)^{a}{}_{b}=\xi^{\mu}\omega_{\mu}{}^{a}{}_{b}$ and $\xi\cdot S=\xi^{\mu}S_{\mu}$. It is an innocuous observation that the noncovariant terms can be written as Lorentz and a Weyl transformations $$\begin{split}\delta^{E}_{\xi}v^{a}=\xi^{\mu}\partial_{\mu}v^{a}&=\xi^{\mu}\tilde{\nabla}_{\mu}v^{a}-\delta^{L}_{\xi\cdot\omega}v^{a}-\delta^{W}_{\xi\cdot S}v^{a}\,,\end{split}$$ (67) but the transformations are unusual in that their parameters contain the connections of the respective gauge groups. The combination $\delta^{E}_{\xi}v^{a}+\delta^{L}_{\xi\cdot\omega}v^{a}+\delta^{W}_{\xi\cdot S}=\xi^{\mu}\tilde{\nabla}_{\mu}v^{a}$ is covariant under all symmetry groups. We are led to the definition of the following ``improved'' Einstein variations, which we distinguish over the other by a tilde $$\tilde{\delta}^{E}_{\xi}\equiv\delta^{E}_{\xi}+\delta^{L}_{\xi\cdot\omega}+\delta^{W}_{\xi\cdot S}\,.$$ (68) which differ from the standard one by the terms that are necessary to make the result covariant under all gauge groups.333If additional internal gauge groups with gauge potentials $A^{i}$ are present ($i$ runs over all the simple factors), then Eq. (68) can be naturally generalized to include them as $\tilde{\delta}^{E}_{\xi}\equiv\delta^{E}_{\xi}+\delta^{L}_{\xi\cdot\omega}+\delta^{W}_{\xi\cdot S}+\sum_{i}\delta^{G^{i}}_{\xi\cdot A^{i}}$. Since the usual Einstein variations are the ordinary Lie derivatives, we are going to refer to the improved Einstein variations as the covariant Lie derivaties, $\widetilde{\pounds}_{\xi}\equiv\tilde{\delta}^{E}_{\xi}$. Our definition is consistent with other applications in the metric-affine literature, in which the covariant Lie derivative is used to extend the Lie derivative to fields of arbitrary spin by incorporating the local Lorentz factor Gronwald:1997jd ; Obukhov:2006ge . One important remark is in order: even though the covariant Lie derivative is defined in a such a way that it yields (gauge) covariant quantities, it is different in nature from an ordinary covariant derivative and, in fact, cannot replace it. One simple way to see this is by noticing that it is not a directional derivative, $\tilde{\pounds}_{\alpha\xi}\neq\alpha\tilde{\pounds}_{\xi}$ for $\alpha=\alpha(x)$ a scalar function over the spacetime. Directionality over the first argument is a crucial for any covariant derivative. We elaborate more on the properties of $\tilde{\pounds}$ in appendix C. To begin with, we stress again that some of the gauge parameters of the improved transformation (68) are not generic, but, instead, are the contraction of the vector field $\xi^{\mu}$ with the gauge potentials. For this reason, the transformations form an algebra, but not a Lie-algebra; we will come back to this point after having studied the algebra of commutators. With this choice, the improved Einstein variation for our protagonist fields are $$\begin{split}&\tilde{\delta}^{E}_{\xi}e^{a}{}_{\mu}=\pounds_{\xi}e^{a}{}_{\mu}+(\xi\cdot\omega)^{a}{}_{b}e^{b}{}_{\mu}+(\xi\cdot S)e^{a}{}_{\mu}\,,\\ &\tilde{\delta}^{E}_{\xi}\omega^{a}{}_{b\mu}=\pounds_{\xi}\,\omega^{a}{}_{b\mu}-D_{\mu}(\xi\cdot\omega)^{a}{}_{b}\,,\\ &\tilde{\delta}^{E}_{\xi}S_{\mu}=\pounds_{\xi}S_{\mu}-D_{\mu}(\xi\cdot S)\,,\end{split}$$ (69) which, after some manipulation of the indices and using the tetrad postulate, can be rewritten as $$\begin{split}&\tilde{\delta}^{E}_{\xi}e^{a}{}_{\mu}=\xi^{\nu}\tilde{T}^{a}{}_{\nu\mu}+e^{a}{}_{\nu}\tilde{\nabla}_{\mu}\xi^{\nu}\,,\\ &\tilde{\delta}^{E}_{\xi}\omega^{a}{}_{b\mu}=\xi^{\nu}R^{a}{}_{b\nu\mu}\,,\\ &\tilde{\delta}^{E}_{\xi}S_{\mu}=\xi^{\nu}W_{\nu\mu}\,.\end{split}$$ (70) In the above form the transformations highlight their geometrical origin in terms of the curvature tensors (the torsion tensor can be thought of as the curvature of the translations Scholz:2018iuc ). It is straightforward to see that all the previous variations are both gauge- and coordinate-covariant. Ultimately, if an action is invariant under the separate transformations in (68), then it is also invariant under (70), so, assuming local Lorentz and Weyl invariance, the invariance under the improved transformations is equivalent to the invariance under traditional diffeomorphisms. We stress that the main use of covariant Lie derivatives is that of obtaining the associated Nöther identities in a full-covariant way. Now we take a relatively long detour in discussing some geometrical properties of the improved transformations, the reader purely interested in their application can skip directly to Sect. V.2.4. V.2.2 Algebra of the improved transformations Now we turn our attention to the algebra of the new transformations; we are going to be succint with the computations and refer to Appendix C for more details. In order to understand the effect of the connection-dependent generators on the algebra of (68), we first notice that the Lorentz and Weyl subalgebras are ``twisted'' by the connections $$\left[\delta^{L}_{\xi\cdot\omega},\delta^{L}_{\zeta\cdot\omega}\right]=\delta^{L}_{[\xi,\zeta]\cdot\omega}+\delta^{L}_{{\cal R}(\xi,\zeta)}\,,\qquad\left[\delta^{W}_{\xi\cdot S},\delta^{W}_{\zeta\cdot S}\right]=\delta^{W}_{[\xi,\zeta]\cdot S}+\delta^{W}_{{\cal W}(\xi,\zeta)}\,,$$ (71) where ${\cal R}(\xi,\zeta)^{a}{}_{b}=\xi^{\mu}\zeta^{\nu}R^{a}{}_{b\mu\nu}$ and ${\cal W}(\xi,\zeta)=\xi^{\mu}\zeta^{\nu}W_{\mu\nu}$. The right hand sides of (71) ``feel'' the presence of the curvature two-forms. In particular, the twisted Weyl subalgebra is not even Abelian. The commutator of two improved transformations can be computed with some work. Using (71) and (171), one can show that $$\bigl{[}\widetilde{\pounds}_{\xi},\widetilde{\pounds}_{\zeta}\bigr{]}=\widetilde{\pounds}_{[\xi,\zeta]}+\delta^{L}_{\mathcal{R}(\xi,\zeta)}+\delta^{W}_{\mathcal{W}(\xi,\zeta)}\,,$$ (72) where the first term of the right hand side is completely analogous to the one of the traditional transformation, while the additional two terms are implied by the twists (71). Again, the commutator is sensitive to both Lorentz and Weyl curvatures. In appendix C we prove that the commutation rule (72) holds for all fields, including gauge-connections, besides the tensors transforming covariantly under all gauge groups, so it is completely general. In appendix C we also prove that the Jacobi identities are satisfied for covariant Lie derivatives as well $${\rm Cycl}_{\xi,\zeta,\chi}\big{[}\widetilde{\pounds}_{\xi},\big{[}\widetilde{\pounds}_{\zeta},\widetilde{\pounds}_{\chi}\big{]}\big{]}=0\,,$$ (73) which sums over the cycles of the three vectors. The algebra is thus closed, as expected. V.2.3 Geometric interpretation à la Cartan The construction of the improved Einstein variations can be carried over in a manifestly diffeomorphism-invariant, geometric manner, using Cartan's formalism. We start from the variation of the co-frame and make use of Cartan's magic formula for Lie derivatives acting on $p$-forms, $\pounds_{\xi}=d\circ\iota_{\xi}+\iota_{\xi}\circ d$, where $\iota_{\xi}$ is the contraction of the $p$-form with $\xi$ from the left.444The action of $\iota_{\xi}$ can be defined iteratively as the $\mathcal{C}^{\infty}(M)$-linear map $\iota_{\xi}:\Omega^{p}(M)\rightarrow\Omega^{p-1}(M)$ that satisfies $\iota_{\xi}(dx^{\mu})=\xi^{\mu}$ and $\iota_{\xi}(dx^{\mu}\wedge dx^{\nu})=(\iota_{\xi}(dx^{\mu}))dx^{\nu}-(\iota_{\xi}(dx^{\nu}))dx^{\mu}$. It is easy to see that the variation of $e^{a}{}_{\mu}$ in (69) can be written as $$\tilde{\delta}^{E}_{\xi}e^{a}=\left(\iota_{\xi}\circ d+d\circ\iota_{\xi}\right)e^{a}+(\iota_{\xi}\omega^{a}_{\phantom{a}b})e^{b}+(\iota_{\xi}S)e^{a}\,.$$ (74) The geometric interpretation can be unveiled using the following relations $$\begin{split}(\iota_{\xi}S)e^{a}&=\iota_{\xi}(S\wedge e^{a})+S\,\iota_{\xi}e^{a}\,,\\ (\iota_{\xi}\omega^{a}_{\phantom{a}b})e^{b}&=\iota_{\xi}(\omega^{a}{}_{b}\wedge e^{b})+\omega^{a}_{\phantom{a}b}\,\iota_{\xi}e^{b}\,,\end{split}$$ (75) which highlight the analogy between the linear operators $\iota_{\xi}$ and $d$ acting on p-forms. Such relations can be exploited to obtain $$\tilde{\delta}^{E}_{\xi}e^{a}=\left(\iota_{\xi}\circ\tilde{D}+\tilde{D}\circ\iota_{\xi}\right)e^{a}=\iota_{\xi}\tilde{\mathcal{T}}^{a}+\tilde{D}\xi^{a}\,,$$ (76) where $\tilde{D}$ is the covariant exterior derivative on forms that extends $d$. The above equation is very interesting because it suggests us how to define, in general, a gauge-covariant Lie derivative of a given $p$-form in the fundamental representation of the gauge group. We simply replace the exterior-derivative with the covariant exterior-derivative. We find the same relation in Obukhov:2006ge . The same ``magic formula'' cannot hold for the Lie derivative of gauge-potentials, since it does not make sense to talk about their covariant exterior derivatives. Indeed, in the case of the spin-connection we find $$\begin{split}\tilde{\delta}^{E}_{\xi}\omega^{a}{}_{b}&=\left(\iota_{\xi}\circ d+d\circ\iota_{\xi}\right)\omega^{a}{}_{b}-d\circ\iota_{\xi}(\omega^{a}{}_{b})-\omega^{a}{}_{c}\iota_{\xi}(\omega^{c}{}_{b})+\omega^{c}{}_{b}\iota_{\xi}(\omega^{a}{}_{c})=\iota_{\xi}\mathcal{R}^{a}{}_{b}\,,\end{split}$$ (77) and similarly for the Weyl potential $$\tilde{\delta}^{E}_{\xi}S=\iota_{\xi}\mathcal{W}\,.$$ (78) As a result of the change of basis (68) in the space of functional variations, the field strength $2$-forms have appeared on the right hand sides of (70). This is a general feature, and we would see the same properties at work for other internal simple gauge factors as well, had we included them. Analogously, the covariant torsion $2$-form, which is the field strength of the co-frame, stands on the right-hand side of the variation of the co-frame itself (70). As we have shown above, the same fact is easily understood in the geometric framework. V.2.4 Conservation laws for diffeomorphisms We have developed all the necessary machinery for taking into account the consequences of diffeomorphism invariance. Using (68) in (56) and integrating by parts, we find $$\delta^{\tilde{E}}_{\xi}S=\int\underline{e}\,\xi^{\nu}\left\{-(\tilde{\nabla}_{\mu}+\tilde{\tau}_{\mu})(T^{\mu}{}_{a}e^{a}{}_{\nu})+T^{\mu}{}_{a}\tilde{T}^{a}{}_{\nu\mu}+\Sigma^{\mu}{}_{ab}R^{ab}{}_{\nu\mu}+\Delta^{\mu}W_{\nu\mu}\right\}\,.$$ (79) The consequence is that the covariant divergence of the energy-momentum tensor in a theory invariant under both Lorentz and Weyl gauge symmetries reads $$\tilde{\nabla}_{\mu}T^{\mu}{}_{\nu}=-\,\tilde{\tau}_{\mu}T^{\mu}{}_{\nu}+T^{\mu}{}_{a}\tilde{T}^{a}{}_{\nu\mu}+\Sigma^{\mu}{}_{ab}R^{ab}{}_{\nu\mu}+\Delta^{\mu}W_{\nu\mu}\,.$$ (80) This is the generalization of the Nöther identity associated to diffeomorphism invariance for torsionful spacetimes Kibble:1961ba ; Obukhov:2006ge in presence of Weyl gauging. We found the analog of the previous equation in Neeman:1996zcr , where it is derived in the metric-affine context, with the so-called metric energy-momentum playing the role of the dilation current and the non-metricity $1$-form that of the Weyl $2$-form. Analogous metric-affine derivations of such Nöther identity can be found in Hehl:1994ue ; Iosifidis:2020gth . Including another simple gauge factor in the internal gauge group simply adds a term $j^{\mu}_{I}F^{I}_{\nu\mu}$ to the right hand side of (80), where $j^{\mu}_{I}\equiv\frac{1}{\underline{e}}\frac{\delta S_{\rm m}}{\delta A_{\mu}^{I}}$ is the gauge-current and $F^{I}_{\nu\mu}$ is the gauge field-strength with adjoint index $I$. To gain more insight on the conservation law, we start by analyzing it in some limiting cases. V.2.5 Case $\Delta^{\mu}=\Sigma^{\mu\nu\rho}=0$ As a first limit, let us consider the simplest situation in which $\Delta^{\mu}=0$ and $\Sigma^{\mu\nu\rho}=0$, i.e. the matter Lagrangian does not explicitly depend on the Weyl and Lorentz gauge potentials. Under these assumptions, the energy-momentum tensor is automatically symmetric and traceless thanks to the vanishing of the spin- and dilation-currents (see eqs. (60) and (63)). Since the energy-momentum tensor is symmetric, now Eq. (80) reads $$\tilde{\nabla}_{\mu}T^{\mu\nu}=-\,\tilde{\tau}_{\mu}T^{\mu\nu}+\frac{1}{2}T^{\mu\rho}\left(\tilde{T}_{\rho}{}^{\nu}{}_{\mu}+\tilde{T}_{\mu}{}^{\nu}{}_{\rho}\right)\,.$$ (81) We further split the covariant-derivative on the left hand side singling-out the contortion contributions (symbolically $\tilde{\nabla}_{\mu}=\hat{\nabla}_{\mu}+\hat{K}_{\mu}$). Using $\hat{K}^{\mu}{}_{\nu\mu}=-\tilde{\tau}_{\nu}$ and $T^{\mu\rho}=T^{\rho\mu}$, and the expression of the contortion tensor in terms of the covariant torsion, yields $$\displaystyle\tilde{\nabla}_{\mu}T^{\mu\nu}$$ $$\displaystyle=\,\hat{\nabla}_{\mu}T^{\mu\nu}+\tilde{K}^{\mu}{}_{\lambda\mu}T^{\lambda\nu}+\tilde{K}^{\nu}{}_{\lambda\mu}T^{\mu\lambda}$$ (82) $$\displaystyle=\,\hat{\nabla}_{\mu}T^{\mu\nu}-\tilde{\tau}_{\mu}T^{\mu\nu}+\frac{1}{2}T^{\mu\rho}\left(\tilde{T}_{\rho}{}^{\nu}{}_{\mu}+\tilde{T}_{\mu}{}^{\nu}{}_{\rho}\right)\,.$$ Therefore, in absence of the spin- and dilation-currents, the Nöther identity implied by the requirement of diffeomorphism-invariance is simply the conservation of the energy-momentum tensor from the viewpoint of the ``hatted'' symmetric affine connection $$\hat{\nabla}_{\mu}T^{\mu\nu}=0$$ (83) Using again the symmetry of both the energy-momentum tensor and the affine connection, as well as $T^{\mu}{}_{\mu}=0$ and $w(T)=-6$, the previous relations becomes $$\displaystyle\hat{\nabla}_{\mu}T^{\mu\nu}=$$ $$\displaystyle\,\mathring{\nabla}_{\mu}T^{\mu\nu}+L^{\mu}{}_{\lambda\mu}T^{\lambda\nu}+L^{\nu}{}_{\lambda\mu}T^{\mu\lambda}-6S_{\mu}T^{\mu\nu}$$ $$\displaystyle=$$ $$\displaystyle\,\mathring{\nabla}_{\mu}T^{\mu\nu}+4S_{\lambda}T^{\lambda\nu}+2S_{\mu}T^{\mu\nu}-6S_{\mu}T^{\mu\nu}$$ $$\displaystyle=$$ $$\displaystyle\,\mathring{\nabla}_{\mu}T^{\mu\nu}\,,$$ (84) yielding automatically the usual conservation law of the energy-momentum tensor $$\mathring{\nabla}_{\mu}T^{\mu\nu}=0\,.$$ (85) We remark the fundamental role of the trace-free nature of the energy-momentum tensor. If $T^{\mu\nu}$ is symmetric and trace-full and carries weight $w=-6$,555By construction, $T^{\mu}{}_{\nu}$ has weight $-4$, implying that the trace is naturally integrated over with a weight $4$ density. Raising the lower index as in $T^{\mu\nu}$ gives weight $-6$. it can be decomposed in the direct-sum $\underline{9}\oplus\underline{1}$, which reads $$T^{\mu\nu}=\Theta^{\mu\nu}+\frac{1}{4}g^{\mu\nu}\chi\,,$$ (86) where $w(\chi)=-4$ and $\chi=T^{\mu}{}_{\mu}$. Thus, the ``hatted'' covariant derivative of such a tensor reads $$\hat{\nabla}_{\mu}T^{\mu\nu}=\mathring{\nabla}_{\mu}\Theta^{\mu\nu}+\frac{1}{4}g^{\mu\nu}\left(\partial_{\mu}\chi-4S_{\mu}\chi\right)\,,$$ (87) In practice, the scalar component of the energy-momentum tensor feels the presence of the Weyl structure, whereas the $\underline{9}$-dimensional tensor component does not. V.2.6 Case $\Delta^{\mu}\neq 0$ and $\Sigma^{\mu\nu\rho}=0$ The second limiting case that we want to study includes a nontrivial coupling to the Weyl potential, but no interaction with the Lorentz one, that is, $\Delta^{\mu}\neq 0$, $\Sigma^{\mu\nu\rho}=0$. Now the energy-momentum tensor $T^{\mu\nu}$ is symmetric, but not necessarily traceless, though its trace is a total divergence. Following similar steps, we find $$\hat{\nabla}_{\mu}T^{\mu\nu}=\Delta^{\mu}W^{\nu}{}_{\mu}\,.$$ (88) Therefore, in general, the usual local conservation of the energy and momentum $\hat{\nabla}_{\mu}T^{\mu\nu}=0$ is obtained only when the Weyl $2$-form strictly vanishes, i.e., when the Weyl potential is a pure gauge one. Splitting the contributions of the symmetric trace-free and trace-full parts of the energy-momentum tensor $$T^{\mu\nu}=\Theta^{\mu\nu}+\frac{1}{4}g^{\mu\nu}\chi\,,$$ (89) with $\chi=-\hat{\nabla}_{\mu}\Delta^{\mu}$. Further exploiting the results of Appendix A, eqs. (135) and (A), we can rewrite the conservation law as $$\hat{\nabla}_{\mu}\Theta^{\mu\nu}=\frac{1}{4}g^{\mu\nu}\left(\hat{\nabla}_{\rho}\hat{\nabla}_{\mu}\Delta^{\rho}-\widehat{Ric}_{\rho\mu}\Delta^{\rho}\right)\,.$$ (90) Thus, the conservation of the trace-free part of the energy-momentum tensor holds if and only if the right hand side vanishes identically, that is, for $$\hat{\nabla}_{\mu}\hat{\nabla}_{\nu}\Delta^{\mu}=\widehat{Ric}_{\mu\nu}\Delta^{\mu}$$ (91) V.2.7 Case $\Delta^{\mu}=0$ and $\Sigma^{\mu\nu\rho}\neq 0$ The last limiting case that we are focusing on is that of vanishing dilation-current and unknown spin-current. With these assumptions, Eq. (80) reads $$\tilde{\nabla}_{\mu}T^{\mu}{}_{\nu}=-\,\tilde{\tau}_{\mu}T^{\mu}{}_{\nu}+T^{\mu}{}_{a}\tilde{T}^{a}{}_{\nu\mu}+\Sigma^{\mu}{}_{ab}R^{ab}{}_{\nu\mu}$$ (92) In presence of a nonvanishing spin-current, the energy-momentum tensor is asymmetric, the antisymmetric being given by Eq. (60). On the other hand, in absence of dilation-current the symmetric part of the energy-momentum tensor is automatically trace-free. Thence, we can split $T^{\mu\nu}$ as $$T^{\mu\nu}=T^{(\mu\nu)}+T^{[\mu\nu]}=\Theta^{\mu\nu}+(\tilde{\nabla}+\tilde{\tau})_{\rho}\Sigma^{\rho\mu\nu}\,,$$ (93) where $\Theta^{\mu\nu}\in\underline{9}$ and $T^{[\mu\nu]}\in\underline{6}$. Repeating the same expansion of the left hand side of the conservation equation considered in the special case of vanishing currents for $\Theta^{\mu\nu}$, we see that it still cancels out with the respective torsion-terms on the right hand side. Thus, the Nöther identity takes the form $$\hat{\nabla}_{\mu}\Theta^{\mu\nu}+(\tilde{\nabla}+\tilde{\tau})_{\mu}T^{[\mu\nu]}=\frac{1}{2}T^{[\mu\rho]}\left(\tilde{T}_{\rho}{}^{\nu}{}_{\mu}-\tilde{T}_{\mu}{}^{\nu}{}_{\rho}\right)+\Sigma^{\mu}{}_{\rho\lambda}\tilde{R}^{\rho\lambda}{}_{\nu\mu}\,.$$ (94) To gain further insight, we expand the covariant derivative of the antisymmetric part of the energy-momentum tensor $$\tilde{\nabla}_{\mu}T^{[\mu\nu]}=\hat{\nabla}_{\mu}T^{[\mu\nu]}-\tilde{\tau}_{\mu}T^{[\mu\nu]}-\frac{1}{2}\tilde{T}^{\nu}{}_{\rho\mu}T^{[\mu\rho]}\,,$$ (95) which combines with both the vector-torsion contribution and the first term on the right hand side to give $$\hat{\nabla}_{\mu}\Theta^{\mu\nu}=-\hat{\nabla}_{\mu}T^{[\mu\nu]}-\hat{K}_{\mu\rho}{}^{\nu}T^{[\mu\rho]}+\Sigma^{\mu}{}_{\rho\lambda}\tilde{R}^{\rho\lambda}{}_{\nu\mu}\,.$$ (96) Exploiting again Eq. (60), we finally obtain $$\displaystyle\hat{\nabla}_{\mu}\Theta^{\mu\nu}=$$ $$\displaystyle\,-\hat{\nabla}_{\lambda}\hat{\nabla}_{\mu}\Sigma^{\lambda\mu\nu}-2W_{\lambda\mu}\Sigma^{\lambda\mu\nu}+\Sigma^{\lambda\mu\rho}\hat{R}_{\mu\rho}{}^{\nu}{}_{\lambda}$$ (97) $$\displaystyle-$$ $$\displaystyle\,\hat{\nabla}_{\mu}\left(\hat{K}^{\mu}{}_{\rho\lambda}\Sigma^{\lambda\rho\nu}-\hat{K}^{\nu}{}_{\rho\lambda}\Sigma^{\lambda\rho\mu}+\hat{K}_{\lambda\rho}{}^{\nu}\Sigma^{\mu\lambda\rho}\right)+\Sigma^{\lambda\mu\rho}\hat{\nabla}^{\nu}\hat{K}_{\mu\rho\lambda}\,.$$ In this case the conservation equation looks more cumbersome; thus, in general, there is no straightforward limit in which the symmetric trace-free energy-momentum tensor $\Theta^{\mu\nu}$ is conserved. V.2.8 Nöther identities in geometrical language For the sake of completeness, we also give the Nöther identities stemming from Weyl, Lorentz and diffeomorphisms invariances in a geometric, coordinate-free notation. In this approach the gravitational field variables are given by the connection $1$-forms $\omega^{a}{}_{b}=\omega^{a}{}_{b\mu}dx^{\mu}$ and $S=S_{\mu}dx^{\mu}$, and by the co-frame $e^{a}=e^{a}{}_{\mu}dx^{\mu}$. The associated curvature tensors are the Riemann $2$-form $\mathcal{R}^{a}{}_{b}=\frac{1}{2}R^{a}{}_{b\mu\nu}dx^{\mu}\wedge dx^{\nu}$, the Weyl (or homothetic curvature) $2$-form $\mathcal{W}=\frac{1}{2}W_{\mu\nu}dx^{\mu}\wedge dx^{\nu}$ and the torsion $2$-form $\tilde{\mathcal{T}}^{a}=\tilde{D}e^{a}$, where $\tilde{D}$ is the covariant exterior derivative. In order to write the Nöther identities in the geometric language, we adopt the operation $\lfloor$, which contracts contravariant indices with covariant ones from the left. The general rule associated to such contraction can be obtained generalizing the following example: let $\xi=\xi^{\mu}\partial_{\mu}$ be a vector field, then we can contract it with the Riemann $2$-form as $$\xi\lfloor\mathcal{R}^{a}{}_{b}\equiv\mathcal{R}^{a}{}_{b}(\xi,\,)=\xi^{\mu}R^{a}{}_{b\mu\nu}dx^{\nu}\,,$$ (98) where in the second step only the first argument of the curvature $2$-form is evaluated on $\xi$, thus resulting in a $1$-form. We also use the dot symbol for contracting a covariant derivative with a contra-variant vector field, and denote the covariant divergence $$\hat{\nabla}\cdot\xi\equiv\hat{\nabla}_{\mu}\xi^{\mu}\,.$$ (99) Similarly we contract a $1$-form, for example $\tilde{\tau}=\tilde{\tau}_{\mu}dx^{\mu}$, with a vector field $\xi$ $$\tilde{\tau}\cdot\xi\equiv\tilde{\tau}_{\mu}\xi^{\mu}\,.$$ (100) Having fixed the notation, it is time to come back to the physics. Given the matter action $S_{m}=S_{m}[\phi,e^{a},\omega^{a}{}_{b},S]$, where $\phi$ is the matter field as before, we have that the gravitational currents are the tensor-valued vector-fields defined in coordinate free notation as $$\displaystyle T_{a}=\frac{1}{\underline{e}}\frac{\delta S_{m}}{\delta e^{a}}\,,\qquad\Sigma_{a}{}^{b}=\frac{1}{\underline{e}}\frac{\delta S_{m}}{\delta\omega^{a}{}_{b}}\,,\qquad\Delta=\frac{1}{\underline{e}}\frac{\delta S_{m}}{\delta S}\,,$$ (101) in analogy with Eq. (57). Using these definitions, the Nöther identities associated in order to Weyl (63), Lorentz (60) and diffeomorphism invariance (80) can be written in the index-free form as $$\displaystyle\hat{\nabla}\cdot\Delta+T{}_{a}\lfloor\,e^{a}=0\,,$$ (102a) $$\displaystyle(\hat{\nabla}+\tilde{\tau})\cdot\Sigma_{[ab]}+T{}_{[a}\lfloor\,e_{b]}=0\,,$$ (102b) $$\displaystyle(\tilde{\nabla}+\tilde{\tau})\cdot T{}_{a}+E_{a}\lfloor\,T{}_{b}\lfloor\,\tilde{\mathcal{T}}^{b}+E_{a}\lfloor\,\Sigma_{bc}\lfloor\,\mathcal{R}^{bc}+E_{a}\lfloor\,\Delta\lfloor\,\mathcal{W}=0\,.$$ (102c) VI Nöther identities in the Einstein-Weyl formalism One way to carry out the Belinfante improvement of the energy-momentum tensor belinfante1940current is through the splitting of the spin-connection in two pieces: the first one, $\mathring{\omega}^{a}{}_{b}$, is the torsion-free spin-connection, while the other one, $\Omega^{a}{}_{b}$, is a tensor with Lorentz indices. Such a splitting is usually worked out requiring the tetrad postulate be fulfilled by two, separate, equalities. Nevertheless, in our approach the independent object is $\hat{\Omega}^{a}{}_{b}$ – see the discussion after Eq. (42) – whereas $\hat{\omega}^{a}{}_{b}$ depends on both the co-frame and the Weyl potential, yielding the Weyl invariant splitting discussed in Sect. III $$\omega^{a}{}_{b\mu}=\hat{\omega}^{a}{}_{b\mu}+\hat{\Omega}^{a}{}_{b\mu}\,.$$ (103) The explicit expression of the ``hatted'' spin-connection $\hat{\omega}^{a}{}_{b}$ is $$\displaystyle\hat{\omega}^{a}{}_{b\mu}=$$ $$\displaystyle E^{\nu}{}_{b}\Bigl{(}\hat{\Gamma}^{\lambda}{}_{\nu\mu}e^{a}{}_{\lambda}-\partial_{\mu}e^{a}{}_{\nu}-S_{\mu}e^{a}{}_{\nu}\Bigr{)}$$ (104) $$\displaystyle=$$ $$\displaystyle E^{\nu}{}_{b}\Bigl{(}\mathring{\Gamma}^{\lambda}{}_{\nu\mu}e^{a}{}_{\lambda}-\partial_{\mu}e^{a}{}_{\nu}\Bigr{)}+\Bigl{(}e^{a}{}_{\mu}E^{\nu}{}_{b}-e_{b\mu}E^{\nu a}\Bigr{)}S_{\nu}\,.$$ The first contribution in the last line give rise to the Levi-Civita spin-connection $\mathring{\omega}^{a}{}_{b}$, whereas the remaining ones are the new contribution coming from the Weyl gauging. For the purpose of obtaining the Nöther identities, we now want to take into account the implicit dependence of $\hat{\omega}^{a}{}_{b}$ on both the co-frame and the Weyl gauge-potential. Therefore, we need to compute the variation of $\hat{\omega}^{a}{}_{b}$, and we are going to start from the Levi-Civita piece. The variation of $\mathring{\omega}^{a}{}_{b}$ is given by $$\displaystyle\delta\mathring{\omega}^{ab}{}_{\mu}$$ $$\displaystyle=\frac{1}{2}E^{\nu b}\left(\mathring{\nabla}_{\nu}\delta e^{a}{}_{\mu}-\mathring{\nabla}_{\mu}\delta e^{a}{}_{\nu}\right)+\frac{1}{2}E^{\nu a}\left(\mathring{\nabla}_{\mu}\delta e^{b}{}_{\nu}-\mathring{\nabla}_{\nu}\delta e^{b}{}_{\mu}\right)$$ (105) $$\displaystyle+\frac{1}{2}E^{\rho a}E^{\nu b}e_{c\mu}\left(\mathring{\nabla}_{\nu}\delta e^{c}{}_{\rho}-\mathring{\nabla}_{\rho}\delta e^{c}{}_{\nu}\right)\,.$$ The second term has the following variation $$\displaystyle\delta\left[\left(e^{a}{}_{\mu}E^{\nu b}-e^{b}{}_{\mu}E^{\nu a}\right)S_{\nu}\right]=$$ $$\displaystyle\,E^{\nu b}S_{\nu}\delta e^{a}{}_{\mu}-E^{\nu a}S_{\nu}\delta e^{b}{}_{\mu}+E^{\nu}{}_{c}E^{\rho a}e^{b}{}_{\mu}S_{\nu}\delta e^{c}{}_{\rho}-E^{\nu}{}_{c}E^{\rho b}e^{a}{}_{\mu}S_{\nu}\delta e^{c}{}_{\rho}$$ $$\displaystyle+$$ $$\displaystyle\,\left(e^{a}{}_{\mu}E^{\nu b}-e^{b}{}_{\mu}E^{\nu a}\right)\delta S_{\nu}\,.$$ (106) After some manipulations and combining these results, we find $$\displaystyle\delta\hat{\omega}^{a}{}_{b\mu}$$ $$\displaystyle=\,\frac{1}{2}E^{\nu b}\left(\hat{\nabla}_{\nu}\delta e^{a}{}_{\mu}-\hat{\nabla}_{\mu}\delta e^{a}{}_{\nu}\right)+\frac{1}{2}E^{\nu a}\left(\hat{\nabla}_{\mu}\delta e^{b}{}_{\nu}-\hat{\nabla}_{\nu}\delta e^{b}{}_{\mu}\right)$$ (107) $$\displaystyle+\,\frac{1}{2}E^{\rho a}E^{\nu b}e_{c\mu}\left(\hat{\nabla}_{\nu}\delta e^{c}{}_{\rho}-\hat{\nabla}_{\rho}\delta e^{c}{}_{\nu}\right)\,.$$ As a consequence, the energy-momentum tensor $T^{\mu}{}_{a}$ – see Eq. (57) – gets some correction terms stemming from the implicit dependence on $e^{a}$ in $\hat{\omega}^{a}{}_{b}$; likewise, the form of the dilation current $\Delta^{\mu}$ is modified. Using the definition of the spin-current $\Sigma^{\mu}{}_{ab}$ in (57), we get the following contribution to the total variation of the matter action $$\displaystyle\delta S_{\rm m}\supset\int\underline{e}\,\Sigma^{\mu}{}_{ab}\delta\hat{\omega}^{ab}{}_{\mu}=\int\underline{e}\,\left(\hat{\nabla}_{\nu}\Sigma^{\mu\nu\rho}+\hat{\nabla}_{\nu}\Sigma^{\rho\nu\mu}-\hat{\nabla}_{\nu}\Sigma^{\nu\mu\rho}\right)e_{a\rho}\delta e^{a}{}_{\mu}+2\int\underline{e}\,\Sigma^{\nu}{}_{\nu}{}^{\mu}\delta S_{\mu}\,.$$ (108) The first two terms in round brackets modify the symmetric part of the energy-momentum term, whereas the third term contributes to the antisymmetric part. Since we assume that our action functional depends on $e^{a}{}_{\mu}$, $\hat{\Omega}^{ab}{}_{\mu}$, $S_{\mu}$, besides some generic matter fields collectively denoted by $\phi$, a general variation with $\phi$ on-shell can be written as $$\displaystyle\delta S_{\rm m}=\int\underline{e}\,\Theta^{\mu}{}_{a}\delta e^{a}{}_{\mu}+\int\underline{e}\,\Sigma^{\mu}{}_{ab}\delta\hat{\Omega}^{ab}{}_{\mu}+\int\underline{e}\,\mathfrak{D}^{\mu}\delta S_{\mu}\,,$$ (109) where $\Theta^{\mu}{}_{a}$ is the full energy-momentum tensor, and $\mathfrak{D}^{\mu}$ is the full dilation-current. We remark that in general $\Theta^{\mu\nu}$ does not have definite symmetry properties, and it should not be confused with the symmetric trace-free tensor of the previous section. Since the splitting of the spin-connection into torsion-free and torsionful parts is linear, $\Sigma^{\mu}{}_{ab}$ remains unchanged when compared to the previous section. From the above calculations, we deduce that the improved energy-momentum tensor and dilation current are $$\displaystyle\Theta^{\mu}{}_{a}$$ $$\displaystyle=T^{\mu}{}_{a}+e_{a\rho}\left(\hat{\nabla}_{\nu}\Sigma^{\mu\nu\rho}+\hat{\nabla}_{\nu}\Sigma^{\rho\nu\mu}-\hat{\nabla}_{\nu}\Sigma^{\nu\mu\rho}\right)$$ (110a) $$\displaystyle\mathfrak{D}^{\mu}$$ $$\displaystyle=\Delta^{\mu}+2\,\Sigma^{\nu}{}_{\nu}{}^{\mu}\,.$$ (110b) We found an analogous improvement of the energy-momentum tensor in Hehl:1994ue ; Iosifidis:2020gth ; Floerchinger:2021uyo , in the more general arena of metric-affine theories. In Hehl:1994ue such improvement is derived in the geometric language of differential forms, whereas in Iosifidis:2020gth ; Floerchinger:2021uyo the derivation is coordinate-based. We must stress that the energy-momentum tensor is best written in holonomic indices, defining $\Theta^{\mu}{}_{\nu}=\Theta^{\mu}{}_{a}e^{a}{}_{\nu}$. The reason is that we know that the energy-momentum tensor must become an energy density, i.e. it must give the energy per unit volume at a given spacetime point. As such, it has mass dimension $4$, implying that its Weyl weight is $-4$, which is precisely weight with holonomic indices is $\Theta^{\mu}{}_{\nu}$ (one index up and one down). The above form of the energy-momentum tensor is also consistent with the result usually given in classical field theory, i.e. $$\Theta^{\mu}{}_{\nu}=\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi^{i})}\partial_{\nu}\phi^{i}-\delta^{\mu}{}_{\nu}\mathcal{L}\,.$$ For completeness, we notice that the same logic would allow us to consider also $\Theta^{a}{}_{b}$, for $w(\Theta^{a}{}_{b})=w(\Theta^{\mu}{}_{\nu})$. We are not going to speculate further on this choice in the present paper. VI.1 Lorentz and Weyl invariances Having derived the total variation of the matter action with matter fields on-shell, and having defined the improved currents associated with the three field variables $e^{a}{}_{\mu}$, $\hat{\Omega}^{a}{}_{b\mu}$ and $S_{\mu}$, we are ready to inspect the Nöther identities implied by Lorentz, Weyl and diffeomorphisms invariances. We start with Lorentz invariance as in the previous section, the only difference with respect to (58) is that $\hat{\Omega}^{a}{}_{b}$ is a tensor under Lorentz indices, thus $$\begin{split}&\delta_{\alpha}^{L}\,e^{a}{}_{\mu}=\alpha^{a}{}_{b}e^{b}{}_{\mu}\,,\\ &\delta_{\alpha}^{L}\,\hat{\Omega}^{a}{}_{b\mu}=\alpha^{a}{}_{c}\hat{\Omega}^{c}{}_{b\mu}-\alpha^{c}{}_{b}\hat{\Omega}^{a}{}_{c\mu}=[\alpha,\hat{\Omega}_{\mu}]^{a}{}_{b}\,,\\ &\delta_{\alpha}^{L}\,S_{\mu}=0\,.\end{split}$$ (111) Using these variations in the first equation of (VI) yields $$\delta^{L}_{\alpha}S_{\rm m}=\int\underline{e}\alpha^{ab}\left\{\Theta^{\mu}{}_{a}e_{b\mu}-\Sigma^{\mu}{}_{ac}\hat{\Omega}^{c}{}_{b\mu}+\Sigma^{\mu}{}_{bc}\hat{\Omega}^{c}{}_{a\mu}\right\}\,.$$ (112) Therefore, Lorentz invariance imposes the following constraint on the antisymmetric part of the improved energy-momentum tensor $$\Theta_{[\mu\nu]}=\hat{K}^{\lambda}{}_{\mu\rho}\Sigma^{\rho}{}_{\nu\lambda}-\hat{K}^{\lambda}{}_{\nu\rho}\Sigma^{\rho}{}_{\mu\lambda}=\tilde{\nabla}_{\rho}\Sigma^{\rho}{}_{\mu\nu}-\hat{K}^{\rho}{}_{\lambda\rho}\Sigma^{\lambda}{}_{\mu\nu}-\hat{\nabla}_{\rho}\Sigma^{\rho}{}_{\mu\nu}\,.$$ (113) On the other hand, from Eq. (VI), we observe that $$\Theta_{[\mu\nu]}=T_{[\mu\nu]}-\hat{\nabla}_{\rho}\Sigma^{\rho}{}_{\mu\nu}\,.$$ (114) Using $\hat{K}^{\rho}{}_{\lambda\rho}=-\tilde{\tau}_{\lambda}$, which relates the trace of the contortion to the vector torsion, Eq. (113) becomes $$\Theta_{[\mu\nu]}=\left(\tilde{\nabla}_{\rho}+\tilde{\tau}_{\rho}\right)\Sigma^{\rho}{}_{\mu\nu}-\hat{\nabla}_{\rho}\Sigma^{\rho}{}_{\mu\nu}\,,$$ (115) which is fully consistent with Eq. (60). The consequence is that imposing Lorentz invariance in the Einstein-Weyl formalism or in Cartan's approach is completely equivalent, as we could have naively expected. We remark that, in both cases, the energy-momentum tensor is symmetric if the hypermomentum current is zero, $\Sigma^{\rho\mu\nu}=0$. Moving on to Weyl symmetry, the variations of the field variables under Weyl gauge transformations are exactly the same as in the Cartan formalism – see Eq. (61) – thanks to the fact that we have built $\hat{\Omega}^{a}{}_{b}$ in such a way that it is Weyl invariant. Therefore, the total variation of the matter action is $$\delta^{W}_{\sigma}S_{\rm m}=\int\underline{e}\,\sigma\left\{\Theta^{\mu}{}_{a}e^{a}{}_{\mu}+\hat{\nabla}_{\mu}\mathfrak{D}^{\mu}\right\}\,,$$ (116) which gives rise to the Nöther identity for the dilation-current $$\hat{\nabla}_{\mu}\mathfrak{D}^{\mu}=-\Theta^{\mu}{}_{\mu}\,.$$ (117) From the first equation in (VI), we know that the trace of the improved energy-momentum tensor becomes $$\Theta^{\mu}{}_{\mu}=T^{\mu}{}_{\mu}+2\,\hat{\nabla}_{\mu}\Sigma^{\nu\mu}{}_{\nu}\,,$$ (118) whereas the second one gives us the explicit form of the improved dilation-current. Putting all these formulae together, we can rewrite Eq. (117) as $$\hat{\nabla}_{\mu}\Delta^{\mu}+2\hat{\nabla}_{\mu}\Sigma^{\nu}{}_{\nu}{}^{\mu}=-T^{\mu}{}_{\mu}-2\hat{\nabla}_{\mu}\Sigma^{\nu\mu}{}_{\nu}\,,$$ (119) which also implies the Nöther identity that was previously found in the Cartan formalism (63). So, the two approaches yield the same result for Weyl invariance as well. VI.2 Diffeomorphism invariance Having established the Nöther identities which follow from Lorentz and Weyl gauge invariances, we finally turn our attention to those derived from diffeomorphism invariance. As we did for the Cartan formalism in the previous section, we are going to derive the diffeomorphisms conservation laws using the improved Einstein variations, i.e. the covariant Lie derivatives defined in Sect. V.2 and further explored in Appendix C. Given that we have split the spin-connection into torsion-free and torsionful parts, now $\tilde{\delta}^{E}_{\xi}$ takes the form $$\tilde{\delta}^{E}_{\xi}=\pounds_{\xi}+\delta^{L}_{\xi\cdot\hat{\omega}}+\delta^{L}_{\xi\cdot\hat{\Omega}}+\delta^{W}_{\xi\cdot S}\,,$$ (120) which includes the ``hatted'' connection. Therefore, using the symmetry of the holonomic connection $\hat{\Gamma}_{\mu}=\mathring{\Gamma}_{\mu}+L_{\mu}$ and $\hat{\nabla}_{\nu}e^{a}{}_{\mu}=0$, we can write the variations of the gravitational field variables as $$\begin{split}&\tilde{\delta}_{\xi}^{E}\,e^{a}{}_{\mu}=e^{a}{}_{\nu}\hat{\nabla}_{\mu}\xi^{\nu}+\xi^{\nu}\hat{\Omega}^{a}{}_{b\nu}e^{b}{}_{\mu}\,,\\ &\tilde{\delta}_{\xi}^{E}\,\hat{\Omega}^{a}{}_{b\mu}=\xi^{\nu}\hat{\nabla}_{\nu}\hat{\Omega}^{a}{}_{b\mu}+\hat{\Omega}^{a}{}_{b\nu}\hat{\nabla}_{\mu}\xi^{\nu}+\xi^{\nu}[\hat{\Omega}_{\nu},\hat{\Omega}_{\mu}]^{a}{}_{b}\,,\\ &\tilde{\delta}_{\xi}^{E}\,S_{\mu}=\xi^{\nu}W_{\nu\mu}\,.\end{split}$$ (121) Exploiting the rules for the integration by parts discussed in Appendix (B), the improved Einstein variation of the matter action with on-shell matter fields takes the form $$\begin{split}\tilde{\delta}^{E}_{\xi}S_{\rm m}=\int\underline{e}\,\xi^{\nu}\Bigl{\{}-\hat{\nabla}_{\mu}\Theta^{\mu}{}_{\nu}+\Theta^{\mu}{}_{\lambda}\hat{K}^{\lambda}{}_{\mu\nu}+\Sigma^{\mu}{}_{\lambda}{}^{\rho}\hat{\nabla}_{\nu}\hat{K}^{\lambda}{}_{\rho\mu}\\ -\hat{\nabla}_{\mu}\bigl{(}\Sigma^{\mu}{}_{\lambda}{}^{\rho}\hat{K}^{\lambda}{}_{\rho\nu}\bigr{)}+\Sigma^{\mu}{}_{\lambda}{}^{\rho}[\hat{K}_{\nu},\hat{K}_{\mu}]^{\lambda}{}_{\rho}+\mathfrak{D}^{\mu}W_{\nu\mu}\Bigr{\}}\,.\end{split}$$ (122) The consequence is that the conservation law following from covariant diffeomorphism invariance is $$\hat{\nabla}_{\mu}\Theta^{\mu}{}_{\nu}=\Theta^{\mu}{}_{\lambda}\hat{K}^{\lambda}{}_{\mu\nu}+\Sigma^{\mu}{}_{\lambda}{}^{\rho}\hat{\nabla}_{\nu}\hat{K}^{\lambda}{}_{\rho\mu}-\hat{\nabla}_{\mu}\left(\Sigma^{\mu}{}_{\lambda}{}^{\rho}\hat{K}^{\lambda}{}_{\rho\nu}\right)+\Sigma^{\mu}{}_{\lambda}{}^{\rho}[\hat{K}_{\nu},\hat{K}_{\mu}]^{\lambda}{}_{\rho}+\mathfrak{D}^{\mu}W_{\nu\mu}\,.$$ (123) Let us notice that, in the first term on the right hand side of the above equation, only the antisymmetric part of $\Theta_{[\mu\nu]}$ appears. We can then exploit the Eq. (113) to write it in terms of the spin-density and the contortion tensor. A striking simplification occurs, for this term cancels exactly with the next-to-last in (123). As a consequence, we can rewrite the Nöther identity (123) $$\hat{\nabla}_{\mu}\Theta^{\mu}{}_{\nu}=\Sigma^{\mu}{}_{\lambda}{}^{\rho}\hat{\nabla}_{\nu}K^{\lambda}{}_{\rho\mu}-\hat{\nabla}_{\mu}\left(\Sigma^{\mu}{}_{\lambda}{}^{\rho}K^{\lambda}{}_{\rho\nu}\right)+\mathfrak{D}^{\mu}W_{\nu\mu}\,.$$ (124) The next step is provided by decomposing the energy-momentum tensor on the left hand side into its symmetric traceless, trace and antisymmetric irreducible components, using both Lorentz and Weyl invariance. Denoting the symmetric traceless part by $\theta^{\mu\nu}=\Theta^{\mu\nu}-\frac{1}{4}\Theta^{\rho}{}_{\rho}g^{\mu\nu}$, we have $$\Theta^{\mu\nu}=\theta^{\mu\nu}-\frac{1}{4}g^{\mu\nu}\hat{\nabla}_{\rho}\mathfrak{D}^{\rho}+K^{\lambda\mu}{}_{\rho}\Sigma^{\rho\nu}{}_{\lambda}-K^{\lambda\nu}{}_{\rho}\Sigma^{\rho\mu}{}_{\lambda}\,.$$ (125) Thus, we can express the Nöther identity in terms as an equation for the divergence of the symmetric trace-free part of the energy-momentum tensor $$\displaystyle\hat{\nabla}_{\mu}\theta^{\mu\nu}=$$ $$\displaystyle\,\hat{\nabla}_{\mu}\left(K^{\lambda\nu}{}_{\rho}\Sigma^{\rho\mu}{}_{\lambda}-K^{\lambda\mu}{}_{\rho}\Sigma^{\rho\nu}{}_{\lambda}\right)+\Sigma^{\mu\lambda\rho}\hat{\nabla}^{\nu}K_{\lambda\rho\mu}-\hat{\nabla}_{\mu}\left(\Sigma^{\mu\lambda\rho}K_{\lambda\rho}{}^{\nu}\right)$$ $$\displaystyle+$$ $$\displaystyle\,\frac{1}{4}\hat{\nabla}^{\nu}\hat{\nabla}_{\mu}\mathfrak{D}^{\mu}+\mathfrak{D}_{\mu}W^{\nu\mu}$$ (126) The special cases, studied in the previous section, of vanishing spin-current in Sect. V.2.6 or zero dilation-current in Sect. V.2.7, can similarly be read off from the second and first line of the previous equation, respectively. VI.2.1 The complete equivalence of Cartan-Weyl and Einstein-Weyl formalisms We have already observed that the Cartan-Weyl and Einstein-Weyl approaches yield equivalent Nöther identities for Lorentz and Weyl invariance. Now we aim at proving that the same goes for diffeomorphism-invariance as well. In order to do so, we need to derive the Nöther identity of Cartan's formalism (80) from Eq. (123). We start by rewriting the left hand side of Eq. (123), using the first equation of (VI) $$\displaystyle\hat{\nabla}_{\mu}\Theta^{\mu\nu}=$$ $$\displaystyle\,\hat{\nabla}_{\mu}T^{\mu\nu}+\hat{\nabla}_{\mu}\hat{\nabla}_{\rho}\left(\Sigma^{\mu\rho\nu}+\Sigma^{\nu\rho\mu}-\Sigma^{\rho\mu\nu}\right)$$ (127) $$\displaystyle=$$ $$\displaystyle\,\tilde{\nabla}_{\mu}T^{\mu\nu}+\tilde{\tau}_{\mu}T^{\mu\nu}+\hat{K}_{\lambda}{}^{\nu}{}_{\mu}T^{\mu\lambda}+\frac{1}{2}\big{[}\hat{\nabla}_{\mu},\hat{\nabla}_{\rho}\big{]}\left(\Sigma^{\mu\rho\nu}+\Sigma^{\nu\rho\mu}-\Sigma^{\rho\mu\nu}\right)\,,$$ where we have used the antisymmetry in the exchange of two indices, $\mu\leftrightarrow\rho$, in the expression in round parentheses in the first line. The first term on the right hand side of (123) can be combined with the part of the third one in which $\hat{\nabla}_{\mu}$ acts on the spin-current $$\displaystyle\Theta^{\mu}{}_{\lambda}\hat{K}^{\lambda}{}_{\mu}{}^{\nu}-\hat{K}^{\lambda}{}_{\rho}{}^{\nu}\hat{\nabla}_{\mu}\Sigma^{\mu}{}_{\lambda}{}^{\rho}=$$ $$\displaystyle\,T^{\mu}{}_{\lambda}\hat{K}^{\lambda}{}_{\mu}{}^{\nu}+\hat{K}_{\lambda\mu}{}^{\nu}\hat{\nabla}_{\rho}\left(\Sigma^{\mu\rho\lambda}+\Sigma^{\lambda\rho\mu}-\Sigma^{\rho\mu\lambda}\right)-\hat{K}^{\lambda}{}_{\rho}{}^{\nu}\hat{\nabla}_{\mu}\Sigma^{\mu}{}_{\lambda}{}^{\rho}$$ (128) $$\displaystyle=$$ $$\displaystyle\,T^{\mu}{}_{\lambda}\hat{K}^{\lambda}{}_{\mu\nu}\,.$$ In fact, the last two terms of the first line cancel each other, whereas the first two in the round parenthesis drop since the symmetrized indices are contracted with the antisymmetric ones of the contortion tensor. We note that the third term in the last line of (127) combines with (128) to give the required contraction of the energy-momentum tensor with the covariant torsion tensor. Going back to the commutator of covariant derivatives in (127), after some algebra we find $$\displaystyle\frac{1}{2}\big{[}\hat{\nabla}_{\mu},\hat{\nabla}_{\rho}\big{]}\left(\Sigma^{\mu\rho\nu}+\Sigma^{\nu\rho\mu}-\Sigma^{\rho\mu\nu}\right)=$$ $$\displaystyle\,\frac{1}{2}g^{\nu\sigma}\left\{\hat{R}_{\sigma\lambda\rho\mu}\Sigma^{\lambda\rho\mu}-4W_{\rho\mu}\Sigma^{\mu\rho}{}_{\sigma}-2W_{\rho\mu}\Sigma_{\sigma}{}^{\rho\mu}\right\}\,.$$ (129) Let us focus on the first term on the right hand side of the last equation. Using repeatedly the first Bianchi identity given in (141) for the ``hatted'' Riemann tensor, and exploit both Eq. (137) and the symmetry properties of the spin-current, yields $$\displaystyle\hat{R}_{\sigma\lambda\rho\mu}\Sigma^{\lambda\rho\mu}=$$ $$\displaystyle\,-2\hat{R}_{\mu\lambda\sigma\rho}\Sigma^{\lambda\rho\mu}+4g_{\mu\lambda}W_{\sigma\rho}\Sigma^{\lambda\rho\mu}+2W_{\rho\mu}\Sigma_{\sigma}{}^{\rho\mu}$$ (130) $$\displaystyle=$$ $$\displaystyle\,2\hat{R}_{\mu\rho\lambda\sigma}\Sigma^{\lambda\rho\mu}+2\hat{R}_{\mu\sigma\rho\lambda}\Sigma^{\lambda\rho\mu}+4W_{\sigma\rho}\Sigma^{\lambda\rho}{}_{\lambda}+2W_{\rho\mu}\Sigma_{\sigma}{}^{\rho\mu}$$ $$\displaystyle=$$ $$\displaystyle\,2\hat{R}_{\rho\mu\sigma\lambda}\Sigma^{\lambda\rho\mu}-2\hat{R}_{\sigma\mu\rho\lambda}\Sigma^{\lambda\rho\mu}+4g_{\mu\sigma}W_{\rho\lambda}\Sigma^{\lambda\rho\mu}+4W_{\sigma\rho}\Sigma^{\lambda\rho}{}_{\lambda}+2W_{\rho\mu}\Sigma_{\sigma}{}^{\rho\mu}\,.$$ The second term in the last line vanishes because of the first Bianchi identity, whereas the third and fifth are equal and opposite to those appearing in (129). Consequently, we have $$\frac{1}{2}\big{[}\hat{\nabla}_{\mu},\hat{\nabla}_{\rho}\big{]}\left(\Sigma^{\mu\rho\nu}+\Sigma^{\nu\rho\mu}-\Sigma^{\rho\mu\nu}\right)=\hat{R}^{\rho\lambda\nu}{}_{\mu}\Sigma^{\mu}{}_{\rho\lambda}+2W^{\nu}{}_{\mu}\Sigma^{\lambda\mu}{}_{\lambda}\,.$$ (131) Using the explicit form of the improved dilation-current (VI), the last term in Eq. (123) reads $$\mathfrak{D}_{\mu}W^{\nu\mu}=\Delta_{\mu}W^{\nu\mu}+2\Sigma^{\lambda}{}_{\lambda\mu}W^{\nu\mu}\,,$$ (132) and the second term on the right hand side cancels out with the last term in (131). All those terms which have not been considered up to now can be recognized straightforwardly as the torsional part of the full curvature tensor. Therefore, adding up all the contributions, we reobtain the Nöther identity that we have found in the Cartan formalism and repeate here for convenience $$\tilde{\nabla}_{\mu}T^{\mu}{}_{\nu}=-\tilde{\tau}_{\mu}T^{\mu}{}_{\nu}+T^{\mu}{}_{\rho}\tilde{T}^{\rho}{}_{\nu\mu}+\Sigma^{\mu}{}_{\rho\lambda}\tilde{R}^{\rho\lambda}{}_{\nu\mu}+\Delta^{\mu}W_{\nu\mu}\,.$$ (133) VII Conclusions We have given arguments to support the idea that Weyl symmetry should be gauged in a gravitational theory with independent metric and connection degrees of freedom. This is especially important if we assume that local scale invariance is a fundamental symmetry of nature at high energies, as it predicts the existence of a vector gauge potential for Weyl symmetry. Our exploration has been mostly based on the fact that the independent connection does not have to transform under Weyl rescalings as the Levi-Civita one, which is symmetric and metric compatible, does. In fact, it should not transform in that way, for example, if we are trying to have a Weyl invariant notion of geodesic. The path that we have followed is not dissimilar to the one of other authors, and, in fact, we have achieved similar conclusions at some stages. This can be seen in the construction that we have referred to as ``Cartan-Weyl'' formalism, which deals with a Weyl covariant torsion and a special form of nonmetricity that cancels exactly with the local gauging when the covariant derivative is gauged in the Weyl symmetry. However, an important achievement of our analysis is that we have shown how to define a generalization of torsion and contortion which is Weyl invariant, rather than covariant, referred to as the ``Einstein-Weyl'' formalism. This has resulted in the definition of a new covariant derivative, $\hat{\nabla}$, which has several desirable features, besides the compatibility with the metric. In particular, this affine-connection induces a new splitting of the spin-connection $\omega^{a}{}_{b}=\hat{\omega}^{a}{}_{b}+\hat{\Omega}^{a}{}_{b}$, where $$\hat{\omega}^{a}{}_{b}=\frac{1}{2}\left(E_{b}\lfloor De^{a}-E^{a}\lfloor De_{b}+e_{c}\,E^{a}\lfloor E_{b}\lfloor De^{c}\right)$$ (134) is the torsion-free Weyl-invariant piece with affine Lorentz transformation, whereas $\hat{\Omega}^{a}{}_{b}$ is the Weyl-invariant torsion-full Lorentz tensor (a formal definition of floor operator $\lfloor$ is given in Sect. V.2.8). One important property of $\hat{\nabla}$ is that we can integrate it by parts on scalars with Weyl weight $-4$ keeping Weyl invariance manifest. We believe that this connection could be a staple in future discussions of Weyl symmetry in the context of metric-affine theories of gravity. One natural reason to incorporate Weyl symmetry in gravity and metric-affine theories is to try to construct a theory that is ``complete'' above some ultraviolet scale, which is generally associated with the Planck mass. In fact, metric and metric-affine theories are generally interpreted as effective ones Baldazzi:2021kaf , because, among other things, it is not clear if they are predictive at all energy scales, especially if quantum mechanical effects are taken into account. Needless to say, the formalism developed in this paper can also have potentially interesting implications in the context of theories equivalent to general relativity, e.g. Jimenez:2019woj ; Jimenez:2021nup , and it would be interesting to what geometrical model a Weyl gauged theory could be equivalent to. In the second part of the paper we have discussed how matter degrees of freedom should couple to the Weyl gauged geometry. In a completely general way, we have derived the Nöther identities associated to Weyl, Lorentz and diffeomorphisms invariances. The identities constrain the currents that couple the matter fields on-shell with the curvatures and covariant derivatives. For the discussion of diffeomorphism invariance, we have shown how the use of a generalization of the Lie derivative, occasionally known as covariant Lie derivative, is particularly convenient. Mathematically, the covariant Lie derivative allows us to maintain covariance under all gauge symmetries at any moment of the computations, and results in a deformed version of the algebra of diffeomorphisms in which parts associated with local Weyl and Lorentz algebras are twisted by the presence of the connections. The covariant Lie derivative generates what we referred to as ``improved'' diffeomorphisms, which we explored at length through the paper. The prominent future perspective of our work would be to put into practice the geometrical construction that we have pushed forward here. The Weyl gauging potential $S_{\mu}$, which couples to the charges given by the weights of all fields, can offer the opportunity to construct vector-tensor theories in which the space of all parameters is heavily constrained by Weyl symmetry. Since we have introduced a notion of Weyl invariant torsion, we are then free to either include or exclude torsional degrees of freedom, depending on the prescriptions that we want to follow when model-building. In this direction, it would be interesting to find out which models with Weyl gauging are renormalizable and, eventually, asymptotically free, because they would pair well with the remaining interactions of the standard model of particle physics. More pragmatically, as we discussed at length in the main text, a theory that is invariant under gauged Weyl transformations is scale invariant because the trace of its energy momentum tensor is a total divergence in the flat-space limit (of the so-called virial current, which couples to the Weyl gauge potential). The natural applications of the framework are thus systems in which scale or almost-scale invariances are realized naturally. These includes cosmological models of inflation, in which case the formalism has the potential to give an interpretation of scale invariance through fundamental fields among which torsion can play a pivotal role Karananas:2021gco . Another important potential application is in the context of asymptotically safe gravity Reuter:1996cp , which is an attempt to complete the (quantum) ultraviolet behavior of general relativity through a nonperturbatively renormalizable scale invariant theory with gravitational degrees of freedom such as the metric and the connection. We hope to elaborate more on these applications in future work. Appendix A Commutators of covariant derivatives and Bianchi identities In this appendix we give some relevant identities for the two covariant derivatives used in the main text, $\tilde{\nabla}$ and $\hat{\nabla}$, introduced in Sects. II.4 and III.2. For both of them, we give the commutators of two covariant derivatives, we consider the nontrivial contractions of the curvature tensors, and we write down the appropriate generalizations of the two Bianchi identities. The symmetric Weyl invariant connection We begin with the torsion-free connection $\hat{\nabla}$. Given a vector field $v^{\mu}$ with Weyl weight $w_{v}$, the commutator of two covariant derivatives is $$\bigl{[}\hat{\nabla}_{\rho},\hat{\nabla}_{\mu}\bigr{]}v^{\lambda}=\hat{R}^{\lambda}{}_{\sigma\rho\mu}v^{\sigma}+w_{v}W_{\rho\mu}v^{\lambda},$$ (135) where $W_{\rho\mu}=\partial_{\rho}S_{\mu}-\partial_{\mu}S_{\rho}$ is the Weyl $2$-form, and $\hat{R}^{\lambda}{}_{\sigma\rho\mu}$ is the Weyl invariant curvature tensor constructed from the full connection $\hat{\Gamma}_{\mu}=\mathring{\Gamma}_{\mu}+L_{\mu}$. In terms of the curvature of the metric compatible symmetric connection, we have $$\displaystyle\hat{R}^{\lambda}{}_{\sigma\rho\mu}$$ $$\displaystyle=\mathring{R}^{\lambda}{}_{\sigma\rho\mu}+\mathring{\nabla}_{\rho}L^{\lambda}{}_{\sigma\mu}-\mathring{\nabla}_{\mu}L^{\lambda}{}_{\sigma\rho}+\bigl{[}L_{\rho},L_{\mu}\bigr{]}^{\lambda}{}_{\sigma}$$ (136) $$\displaystyle=\mathring{R}^{\lambda}{}_{\sigma\rho\mu}+\delta^{\lambda}{}_{\sigma}W_{\rho\mu}+\delta^{\lambda}{}_{\mu}\mathring{\nabla}_{\rho}S_{\sigma}-\delta^{\lambda}{}_{\rho}\mathring{\nabla}_{\mu}S_{\sigma}-g_{\sigma\mu}\mathring{\nabla}_{\rho}S^{\lambda}+g_{\sigma\rho}\mathring{\nabla}_{\mu}S^{\lambda}$$ $$\displaystyle+\delta^{\lambda}{}_{\rho}\left(S_{\mu}S_{\sigma}-g_{\mu\sigma}S^{2}\right)-\delta^{\lambda}{}_{\mu}\left(S_{\rho}S_{\sigma}-g_{\rho\sigma}S^{2}\right)+g_{\mu\sigma}S_{\rho}S^{\lambda}-g_{\rho\sigma}S_{\mu}S^{\lambda}\,.$$ Notice that the full curvature tensor does not have definite symmetry properties. Indeed, even though it is antisymmetric in the pair $(\rho,\mu)$, it contains both symmetric and antisymmetric terms in the pair $(\lambda,\sigma)$. However, $\delta^{\lambda}{}_{\sigma}W_{\rho\mu}$ is the only term symmetric under the exchange of $(\lambda,\sigma)$, in fact it is a pure trace and, clearly, Weyl invariant. Therefore, since the whole expression is Weyl invariant, $\hat{R}^{\lambda}{}_{\sigma\rho\mu}-\delta^{\lambda}{}_{\sigma}W_{\rho\mu}$ is also a Weyl invariant tensor, which is antisymmetric in $(\lambda,\sigma)$. We notice that under exchange of the first two indices the curvature satisfies $$\hat{R}^{\lambda\rho}{}_{\mu\nu}=2g^{\lambda\rho}W_{\mu\nu}-\hat{R}^{\rho\lambda}{}_{\mu\nu}\,.$$ (137) We have three nonvanishing contractions of the curvature tensor $$\displaystyle\hat{R}^{\lambda}{}_{\lambda\rho\mu}=4W_{\rho\mu}\,,$$ (138a) $$\displaystyle\hat{R}^{\lambda}{}_{\sigma\lambda\mu}\equiv\widehat{Ric}_{\sigma\mu}=\mathring{R}_{\sigma\mu}+W_{\sigma\mu}-g_{\sigma\mu}\mathring{\nabla}_{\rho}S^{\rho}-2\mathring{\nabla}_{\mu}S_{\sigma}+2S_{\mu}S_{\sigma}-2g_{\mu\sigma}S^{2}\,,$$ (138b) $$\displaystyle\hat{R}^{\lambda}{}_{\sigma\rho\mu}g^{\sigma\mu}g_{\lambda\nu}=\widehat{Ric}_{\nu\rho}-2W_{\nu\rho}\,.$$ (138c) The first contraction charap1974gauge , which is known in the literature with the name of homothetic curvature tensor, is nonvanishing as it often happens with theories in which the local symmetry group is enhanced (e.g., to $GL(4)$) Vazirian:2013baa . In our geometry the homothetic curvature is proportional to the Weyl curvature $2$-form, which is the field strength of local scale transformations. We also notice that the last contraction in (A) is a linear superposition of the previous two, so we concentrate on the second. Whereas the first contraction is clearly antisymmetric in the two indices, the second one is still reducible. To highlight this feature, we give the explicit form of its symmetric and antisymmetric parts $$\displaystyle\widehat{Ric}_{(\mu\sigma)}$$ $$\displaystyle=\mathring{R}_{\mu\sigma}-\left(\mathring{\nabla}_{\mu}S_{\sigma}+\mathring{\nabla}_{\sigma}S_{\mu}+g_{\mu\sigma}\mathring{\nabla}_{\rho}S^{\rho}\right)+2\left(S_{\mu}S_{\sigma}-g_{\mu\sigma}S^{2}\right)\,,$$ (139a) $$\displaystyle\widehat{Ric}_{[\mu\sigma]}$$ $$\displaystyle=2W_{\mu\sigma}\,.$$ (139b) Only the symmetric part contributes to the trace that results in the Weyl covariant scalar curvature $$\hat{R}=\mathring{R}-6\mathring{\nabla}_{\rho}S^{\rho}-6S^{2}\,.$$ (140) As a consistency check, we note that if we take the Weyl gauge potential to be a pure gauge of the form $S_{\mu}=\partial_{\mu}\omega$, the previous equation gives us the transformation rule of the Levi-Civita scalar curvature, $\mathring{R}$, when we apply a standard Weyl transformation $g_{\mu\nu}\rightarrow e^{2\omega(x)}g_{\mu\nu}$ to the metric. Since the affine connection $\hat{\Gamma}$ is symmetric, we expect that the first Bianchi identities follow straightforwardly. In fact, it becomes $$\displaystyle\hat{R}^{\lambda}{}_{[\sigma\mu\nu]}=\delta^{\lambda}{}_{[\sigma}W_{\mu\nu]}+\delta^{\lambda}{}_{[\nu}W_{\mu\sigma]}=0\,.$$ (141) Using again the symmetry and of the affine connection as well as the Weyl invariance of both curvature tensors, we obtain two more Bianchi identities for the curvature $2$-forms of the Weyl geometry $$\displaystyle\hat{\nabla}_{[\mu}\hat{R}^{\lambda}{}_{|\sigma|\nu\rho]}=0\,,$$ (142a) $$\displaystyle\hat{\nabla}_{[\mu}W_{\nu\rho]}=0\,.$$ (142b) Exploiting the second Bianchi identities for the curvature tensor, the contractions (A) and the second equation in (A), we obtain the proper generalization of the contracted Bianchi identities $$\hat{\nabla}^{\mu}\left(\widehat{Ric}_{\mu\rho}-W_{\mu\rho}-\frac{1}{2}g_{\mu\rho}\hat{R}\right)=0\,.$$ (143) Notice that, taking the symmetric and antisymmetric parts of the Ricci tensor, we can rewrite the previous equation as $$\hat{\nabla}^{\mu}\left(\widehat{Ric}_{(\mu\rho)}-\frac{1}{2}g_{\mu\rho}\hat{R}\right)=-\hat{\nabla}^{\mu}W_{\mu\rho}\,.$$ (144) The nonsymmetric torsionful connection The commutator of two covariant derivatives on a vector field $v^{\rho}$ with weight $w_{v}$ is $$\bigl{[}\tilde{\nabla}_{\mu},\tilde{\nabla}_{\nu}\bigr{]}v^{\rho}=\tilde{R}^{\rho}{}_{\sigma\mu\nu}v^{\sigma}+\tilde{T}^{\sigma}{}_{\nu\mu}\tilde{\nabla}_{\sigma}v^{\rho}+w_{v}W_{\mu\nu}v^{\rho}\,,$$ (145) where the Weyl invariant curvature tensor of $\tilde{\nabla}$ is $$\displaystyle\tilde{R}^{\rho}{}_{\lambda\mu\nu}=$$ $$\displaystyle\,\hat{R}^{\rho}{}_{\lambda\mu\nu}+\hat{\nabla}_{\mu}\hat{K}^{\rho}{}_{\lambda\nu}-\hat{\nabla}_{\nu}\hat{K}^{\rho}{}_{\lambda\mu}+\big{[}\hat{K}_{\mu},\hat{K}_{\nu}\big{]}^{\rho}{}_{\lambda}$$ (146) $$\displaystyle=$$ $$\displaystyle\,\hat{R}^{\rho}{}_{\lambda\mu\nu}+\big{[}\hat{K}_{\mu},\hat{K}_{\nu}\big{]}^{\rho}{}_{\lambda}+\mathring{\nabla}_{\mu}\hat{K}^{\rho}{}_{\lambda\nu}-\mathring{\nabla}_{\nu}\hat{K}^{\rho}{}_{\lambda\mu}+S_{\lambda}\tilde{T}^{\rho}{}_{\mu\nu}+S^{\kappa}\left(g_{\lambda\mu}\hat{K}^{\rho}{}_{\kappa\nu}-g_{\lambda\nu}\hat{K}^{\rho}{}_{\kappa\mu}\right)$$ $$\displaystyle\,+\left(\delta^{\rho}{}_{\mu}S_{\kappa}-g_{\kappa\mu}S^{\rho}\right)\hat{K}^{\kappa}{}_{\lambda\nu}-\left(\delta^{\rho}{}_{\nu}S_{\kappa}-g_{\kappa\nu}S^{\rho}\right)\hat{K}^{\kappa}{}_{\lambda\mu}$$ $$\displaystyle=$$ $$\displaystyle\,\mathring{R}^{\rho}{}_{\lambda\mu\nu}+\delta^{\rho}{}_{\lambda}W_{\mu\nu}+\delta^{\rho}{}_{\nu}\mathring{\nabla}_{\mu}S_{\lambda}-\delta^{\rho}{}_{\mu}\mathring{\nabla}_{\nu}S_{\lambda}+g_{\lambda\mu}\mathring{\nabla}_{\nu}S^{\rho}-g_{\lambda\nu}\mathring{\nabla}_{\mu}S^{\rho}$$ $$\displaystyle\,+\delta^{\rho}{}_{\mu}\left(S_{\lambda}S_{\nu}-g_{\lambda\nu}S^{2}\right)-\delta^{\rho}{}_{\nu}\left(S_{\lambda}S_{\mu}-g_{\lambda\mu}S^{2}\right)+g_{\lambda\nu}S_{\mu}S^{\rho}-g_{\lambda\mu}S_{\nu}S^{\rho}$$ $$\displaystyle\,+\big{[}\hat{K}_{\mu},\hat{K}_{\nu}\big{]}^{\rho}{}_{\lambda}+\mathring{\nabla}_{\mu}\hat{K}^{\rho}{}_{\lambda\nu}-\mathring{\nabla}_{\nu}\hat{K}^{\rho}{}_{\lambda\mu}+\delta^{\rho}{}_{\mu}S_{\kappa}\hat{K}^{\kappa}{}_{\lambda\nu}-\delta^{\rho}{}_{\nu}S_{\kappa}\hat{K}^{\kappa}{}_{\lambda\mu}$$ $$\displaystyle\,+g_{\mu\lambda}S^{\kappa}\hat{K}^{\rho}{}_{\kappa\nu}-g_{\nu\lambda}S^{\kappa}\hat{K}^{\rho}{}_{\kappa\mu}+S_{\lambda}\tilde{T}^{\rho}{}_{\mu\nu}-S^{\rho}\tilde{T}_{\lambda\mu\nu}\,,$$ where we have also used the results of the previous subsection to expand the ``hatted'' Riemann tensor, as well as the explicit form of the covariant contortion-tensor. The commutation relations have the same form when applied to a Lorentz vector. For example, if we take $v^{a}=e^{a}{}_{\rho}v^{\rho}$, then we have $$\bigl{[}\tilde{\nabla}_{\mu},\tilde{\nabla}_{\nu}\bigr{]}v^{a}=\tilde{R}^{a}{}_{b\mu\nu}v^{b}+\tilde{T}^{\lambda}{}_{\nu\mu}\tilde{\nabla}_{\lambda}v^{a}+(w_{v}-1)W_{\mu\nu}v^{a}\,,$$ (147) where $\tilde{R}^{a}{}_{b\mu\nu}=e^{a}{}_{\rho}E^{\sigma}{}_{b}\tilde{R}^{\rho}{}_{\sigma\mu\nu}$ are the components of the curvature $2$-form, $\tilde{{\cal R}}^{a}{}_{b}=\frac{1}{2}\tilde{R}^{a}{}_{b\mu\nu}dx^{\mu}\wedge dx^{\nu}$, $W_{\mu\nu}=\partial_{\mu}S_{\nu}-\partial_{\nu}S_{\mu}$ are the components of the Weyl $2$-form, ${\cal W}=dS=\frac{1}{2}W_{\mu\nu}dx^{\mu}\wedge dx^{\nu}$, and $\tilde{T}^{\sigma}{}_{\mu\nu}=E^{\sigma}{}_{a}\tilde{T}^{a}{}_{\mu\nu}$ those of the torsion $2$-form, $\tilde{\mathcal{T}}^{a}=\tilde{D}e^{a}$. Notice that the weight of the tensor depends on the position (upper or lower) of its holonomic indices, because both the tetrad and the metric have nonzero Weyl weight. For example, if we define $z_{\mu}=g_{\mu\nu}v^{\nu}$, then $w_{z}=w_{v}+2$ and $$\bigl{[}\tilde{\nabla}_{\mu},\tilde{\nabla}_{\nu}\bigr{]}z_{\rho}=-\tilde{R}^{\sigma}{}_{\rho\mu\nu}z_{\sigma}+\tilde{T}^{\sigma}{}_{\nu\mu}\tilde{\nabla}_{\sigma}z^{\rho}+(w_{v}+2)W_{\mu\nu}z^{\rho}\,.$$ (148) Now we turn our attention to the nonvanishing contractions of the full curvature tensor, exploiting the definition of the torsion-vector (38) to simplify the expression. There are three, but only two are truly independent $$\displaystyle\tilde{R}^{\rho}{}_{\rho\mu\nu}=$$ $$\displaystyle 4W_{\mu\nu}\,,$$ (149a) $$\displaystyle\tilde{R}^{\mu}{}_{\lambda\mu\nu}\equiv$$ $$\displaystyle\widetilde{Ric}_{\lambda\nu}=\mathring{R}_{\lambda\nu}-g_{\lambda\nu}\mathring{\nabla}_{\mu}S^{\mu}+W_{\lambda\nu}-2\mathring{\nabla}_{\nu}S_{\lambda}+2S_{\lambda}S_{\nu}-2g_{\lambda\nu}S^{2}$$ (149b) $$\displaystyle+\mathring{\nabla}_{\mu}\hat{K}^{\mu}{}_{\lambda\nu}+\mathring{\nabla}_{\nu}\tilde{\tau}_{\lambda}-\tilde{\tau}_{\mu}\hat{K}^{\mu}{}_{\lambda\nu}-\hat{K}^{\mu}{}_{\rho\nu}\hat{K}^{\rho}{}_{\lambda\mu}$$ $$\displaystyle+S_{\mu}\left(\tilde{T}^{\mu}{}_{\nu\lambda}+\tilde{T}_{\nu}{}^{\mu}{}_{\lambda}\right)-S_{\lambda}\tilde{\tau}_{\nu}+g_{\lambda\nu}S_{\mu}\tilde{\tau}^{\mu}\,,$$ $$\displaystyle\tilde{R}^{\rho}{}_{\lambda\mu\nu}g^{\lambda\nu}g_{\rho\sigma}=$$ $$\displaystyle\widetilde{Ric}_{\sigma\mu}-2W_{\sigma\mu}\,,$$ (149c) but, in analogy with the torsionless case, only two are truly independent. As in the previous case, we give the expressions of the symmetric and antisymmetric parts of the Ricci tensor $$\displaystyle\widetilde{Ric}_{(\lambda\nu)}=$$ $$\displaystyle\,\mathring{R}_{\lambda\nu}-\left(\mathring{\nabla}_{\lambda}S_{\nu}+\mathring{\nabla}_{\nu}S_{\lambda}+g_{\lambda\nu}\mathring{\nabla}_{\rho}S^{\rho}\right)+2\left(S_{\lambda}S_{\nu}-g_{\lambda\nu}S^{2}\right)$$ (150a) $$\displaystyle+\,\frac{1}{2}\left(\mathring{\nabla}_{\mu}+S_{\mu}-\tilde{\tau}_{\mu}\right)\left(\tilde{T}_{\nu}{}^{\mu}{}_{\lambda}+\tilde{T}_{\lambda}{}^{\mu}{}_{\nu}\right)$$ $$\displaystyle+\,\frac{1}{2}\left(\mathring{\nabla}_{\nu}\tilde{\tau}_{\lambda}+\mathring{\nabla}_{\lambda}\tilde{\tau}_{\nu}-S_{\nu}\tilde{\tau}_{\lambda}-S_{\lambda}\tilde{\tau}_{\nu}\right)+g_{\lambda\nu}S_{\mu}\tilde{\tau}^{\mu}$$ $$\displaystyle+\,\frac{1}{4}\left(\tilde{T}_{\lambda}{}^{\mu\rho}\tilde{T}_{\nu\mu\rho}-\tilde{T}_{\lambda}{}^{\rho}{}_{\mu}\tilde{T}_{\rho}{}^{\mu}{}_{\nu}-\tilde{T}_{\nu}{}^{\rho}{}_{\mu}\tilde{T}_{\rho}{}^{\mu}{}_{\lambda}\right)\,,$$ $$\displaystyle\widetilde{Ric}_{[\lambda\nu]}=$$ $$\displaystyle\,2W_{\lambda\nu}-\frac{1}{2}\left(\mathring{\nabla}_{\mu}-\tilde{\tau}_{\mu}+2S_{\mu}\right)\tilde{T}^{\mu}{}_{\lambda\nu}+\frac{1}{2}S_{\mu}\left(\tilde{T}_{\nu}{}^{\mu}{}_{\lambda}-\tilde{T}_{\lambda}{}^{\mu}{}_{\nu}\right)$$ (150b) $$\displaystyle+\,\frac{1}{2}\left(\mathring{\nabla}_{\nu}\tilde{\tau}_{\lambda}-\mathring{\nabla}_{\lambda}\tilde{\tau}_{\nu}+S_{\nu}\tilde{\tau}_{\lambda}-S_{\lambda}\tilde{\tau}_{\nu}\right)+\frac{1}{4}\left(\tilde{T}_{\lambda}{}^{\mu}{}_{\rho}\tilde{T}^{\rho}{}_{\mu\nu}-\tilde{T}_{\nu}{}^{\mu}{}_{\rho}\tilde{T}^{\rho}{}_{\mu\lambda}\right)\,.$$ The Weyl covariant scalar curvature in presence of torsion is thus $$\tilde{R}=\mathring{R}-6\mathring{\nabla}_{\rho}S^{\rho}-6S^{2}+2\mathring{\nabla}_{\mu}\tilde{T}^{\nu\mu}{}_{\nu}-\tilde{T}^{\mu}{}_{\rho\mu}\tilde{T}^{\nu\rho}{}_{\nu}+\frac{1}{4}\tilde{T}^{\mu\nu\rho}\tilde{T}_{\mu\nu\rho}+\frac{1}{2}\tilde{T}^{\mu\nu\rho}\tilde{T}_{\rho\nu\mu}+4S_{\nu}\tilde{T}^{\mu\nu}{}_{\mu}\,.$$ (151) We can use the relation $\hat{\nabla}_{\mu}\tilde{T}^{\nu\mu}{}_{\nu}=\mathring{\nabla}_{\mu}\tilde{T}^{\nu\mu}{}_{\nu}+2S_{\mu}\tilde{T}^{\nu\mu}{}_{\nu}$, which allows to write the scalar curvature in a manifestly Weyl covariant way $$\tilde{R}=\hat{R}+2\hat{\nabla}_{\mu}\tilde{T}^{\nu\mu}{}_{\nu}-\tilde{T}^{\mu}{}_{\rho\mu}\tilde{T}^{\nu\rho}{}_{\nu}+\frac{1}{4}\tilde{T}^{\mu\nu\rho}\tilde{T}_{\mu\nu\rho}+\frac{1}{2}\tilde{T}^{\mu\nu\rho}\tilde{T}_{\rho\nu\mu}\,.$$ (152) The Bianchi identities for the connection $\tilde{\nabla}$ with both torsion and Weyl gauging are slightly more complicated, mostly because of the presence of the Weyl $2$-form $W_{\mu\nu}$. We find $$\displaystyle\,\tilde{\nabla}_{[\mu}\tilde{T}^{\lambda}{}_{\nu\rho]}$$ $$\displaystyle=$$ $$\displaystyle\tilde{R}^{\lambda}{}_{[\mu\nu\rho]}+\tilde{T}^{\kappa}{}_{[\mu\nu}\tilde{T}^{\lambda}{}_{\rho]\kappa}\,,$$ (153) $$\displaystyle\,\tilde{\nabla}_{[\mu}\tilde{R}^{\lambda}{}_{|\sigma|\nu\rho]}$$ $$\displaystyle=$$ $$\displaystyle\tilde{T}^{\kappa}{}_{[\mu\nu}R^{\lambda}{}_{|\sigma|\rho]\kappa}\,,$$ (154) $$\displaystyle\,\tilde{\nabla}_{[\mu}W_{\nu\rho]}$$ $$\displaystyle=$$ $$\displaystyle\tilde{T}^{\lambda}{}_{[\mu\nu}W_{\rho]\lambda}\,.$$ (155) The first equation could be derived by expressing the torsion tensors on the left hand side as differences of contortion tensors and exploiting $\hat{R}^{\lambda}{}_{[\mu\nu\rho]}=0$. Nevertheless, the most efficient way of deriving all the above relations is using Cartan's differential-form formalism, in which they read $$\displaystyle\tilde{D}\,\tilde{\cal T}^{a}$$ $$\displaystyle=$$ $$\displaystyle\tilde{\cal R}^{a}{}_{b}\wedge e^{b}+{\cal W}\wedge e^{a}\,,$$ (156) $$\displaystyle\tilde{D}\,\tilde{\cal R}^{a}{}_{b}$$ $$\displaystyle=$$ $$\displaystyle 0\,,$$ (157) $$\displaystyle\tilde{D}\,{\cal W}$$ $$\displaystyle=$$ $$\displaystyle 0\,.$$ (158) The only caveat for passing from the latter set of equations to the former is that $\tilde{D}E^{\lambda}{}_{a}$ does not vanish, which complicates slightly the derivation of the first relation (153). Notice that, since the torsion $2$-form $\tilde{\mathcal{T}}^{a}$ has Weyl weight $w_{\tilde{\mathcal{T}}^{a}}=1$, the Weyl $2$-form appears on the right hand side of the corresponding Bianchi identity. On the other hand, the holonomic torsion tensor, as well as the Riemann and Weyl curvatures, are Weyl invariant, and this is the reason why no such contributions appears on the right hand side of (153). Let us finally remark that passing from the differential form to the holonomic formalism results in new terms on the right hand side, in which the curvature tensors are contracted with the torsion itself. Contracting two pairs of indices in (154), we obtain the contracted Bianchi identities for the curvature tensor $$\tilde{\nabla}_{\mu}\tilde{R}-2\tilde{\nabla}^{\nu}\widetilde{Ric}_{\nu\mu}+2\tilde{\nabla}^{\nu}W_{\nu\mu}=2\tilde{T}^{\rho\nu}{}_{\mu}\left(\widetilde{Ric}_{\nu\rho}-W_{\nu\rho}\right)+\tilde{T}^{\rho}{}_{\nu\sigma}\tilde{R}^{\nu\sigma}{}_{\mu\rho}\,.$$ (159) Since we now have another nontrivial differential Bianchi identity (153), we also have one more contracted identity, which can be derived from (153) and by contracting the upper index with any of the lower ones. Denoting the torsion vector by $\tilde{\tau}_{\mu}\equiv\tilde{T}^{\nu}{}_{\mu\nu}$ (as in the main text), the identity becomes $$(\tilde{\nabla}+\tilde{\tau})_{\nu}\left(\tilde{T}^{\nu}{}_{\mu\rho}+\delta^{\nu}{}_{\mu}\tilde{\tau}_{\rho}-\delta^{\nu}{}_{\rho}\tilde{\tau}_{\mu}\right)=2W_{\mu\rho}\,,$$ (160) where we recognize the modified torsion tensor on the left hand side. A brief comment on the coupling with fermionic fields before closing in with the appendix. In the Einstein-Cartan formalism, the modified torsion-tensor is ``sourced'' by the spin-density of fermionic fields (see, for example, the Sciama-Kibble field equations in Trautman:2006fp ). In the case of Dirac fields, it has only vector and axial-vector irreducible components (see, for example, Freidel:2005sn ; gasperini2013theory ; Karananas:2021zkl ). Furthermore, it is also well-known that the Weyl potential decouples from the Dirac Lagrangian Oda:2020yyv . On the other hand, in a Weyl gauged analog of Einstein-Cartan theory, the algebraic equations of motion for the torsion tensor read $$\phi^{2}\left(\tilde{T}^{\nu}{}_{\mu\rho}+\delta^{\nu}{}_{\mu}\tilde{\tau}_{\rho}-\delta^{\nu}{}_{\rho}\tilde{\tau}_{\mu}\right)=a\Sigma^{\nu}{}_{\mu\rho}\,,$$ (161) with some unspecified numerical factor $a$. Applying $(\tilde{\nabla}+\tilde{\tau})_{\nu}$ on both sides of the previous equation and exploiting Eq. (60), we obtain the antisymmetric part of the energy-momentum tensor in a Weyl gauged Einstein-Cartan theory $$aT_{[\mu\rho]}=2\phi^{2}W_{\mu\rho}+\left(\tilde{T}^{\nu}{}_{\mu\rho}+\delta^{\nu}{}_{\mu}\tilde{\tau}_{\rho}-\delta^{\nu}{}_{\rho}\tilde{\tau}_{\mu}\right)(\hat{\nabla}+\tilde{\tau})_{\nu}\phi^{2}\,.$$ (162) Appendix B Integration by parts The derivations of the Nöther identites in the main text require multiple uses of the integration by parts of independent connections, which is not as straightforward as with the unique symmetric metric compatible connection $\mathring{\nabla}$. Mathematically speaking, integration on a $d$-dimensional manifold $\mathcal{M}$ is defined by appropriately gluing together the integration of a $d$-form over local charts with the aid of the partition of the unity. In this context, integration by parts is simply Stokes' theorem. Given a $(d-1)$-form $\zeta$, we have $\int_{\mathcal{M}}d\zeta=\int_{\partial\mathcal{M}}\zeta$, up to an orientation dependent sign. Therefore, requiring that $\zeta$ vanish on the boundary of the manifold (for example at spatial infinity in general relativity), Stokes' theorem becomes $\int_{\mathcal{M}}d\zeta=0$, and we can safely integrate by parts. If the boundary does not exist, $\partial\mathcal{M}=\emptyset$, the same result holds. For simplicity, we assume that we can cover our manifold $\mathcal{M}$ with a unique coordinate chart and that the coordinates range on the entire real axis (the generalization follows straightforwardly using the standard tools to prove Stokes' theorem). Physically speaking, we need a volume form, generally chosen $\underline{e}=\sqrt{-g}$, and integration must be defined over scalar densities. When considering the connection $\mathring{\nabla}$ and the explicit form of its components $\mathring{\Gamma}^{\mu}{}_{\nu\mu}$, it is trivial to show $$\mathring{\nabla}_{\mu}v^{\mu}=\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}v^{\mu}\right)\,.$$ (163) The consequence is that the volume form simplifies when we integrate and are left with a partial derivative $$\int\sqrt{-g}\,\mathring{\nabla}_{\mu}v^{\mu}=\int\partial_{\mu}\left(\sqrt{-g}v^{\mu}\right)dx^{1}\cdots dx^{d}=\sum_{\mu}\int\left(\sqrt{-g}v^{\mu}\right)\bigg{|}^{\infty}_{-\infty}dx^{1}\cdots\widehat{dx^{\mu}}\cdots dx^{d}\,,$$ (164) where the notation indicates with a hat that $dx^{\mu}$ is excluded from the final expression. With hindsight, in order to integrate by parts, we need our covariant differential operator to display the crucial property (163). Now we take into account the ``hatted'' covariant derivative defined in Eq. (34). The trace of the distortion tensor is $L^{\mu}{}_{\nu\mu}=4S_{\nu}$ (the numerical factor comes from the dimension of spacetime $d=4$). Considering a vector field $v^{\mu}$ with Weyl weight $w_{v}$ and using (163), we expand its covariant divergence as $$\hat{\nabla}_{\mu}v^{\mu}=\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}v^{\mu}\right)+(4+w_{v})S_{\mu}v^{\mu}=\frac{1}{\sqrt{-g}}\hat{D}_{\mu}\left(\sqrt{-g}v^{\mu}\right),$$ (165) where in the last step we used the fact that $w(\sqrt{-g})=4$ (again, the dimension of spacetime $d=4$), and $\hat{D}$ is the gauge-covariant derivative (acting only as gauge derivative, but blind to coordinate indices). We clearly see that integration by parts with respect to the ``hatted'' differential operator can be safely performed only for vector fields $v^{\mu}$ with Weyl weight $w_{v}=-4$, that is, when $\hat{D}_{\mu}\left(\sqrt{-g}v^{\mu}\right)=\partial_{\mu}\left(\sqrt{-g}v^{\mu}\right)$. Fortunately, this is the case in all the relevant calculations carried out in the main text. It is important to realize that, contrary to $\hat{\nabla}$, the complete covariant derivative $\tilde{\nabla}$ does not possess the crucial property (163) which is needed for a straightforward integration by parts. Indeed, considering the covariant divergence of a vector field with Weyl weight $w_{v}$ $$\tilde{\nabla}_{\mu}v^{\mu}=\hat{\nabla}_{\mu}v^{\mu}+\hat{K}^{\mu}{}_{\nu\mu}v^{\nu}=\frac{1}{\sqrt{-g}}\hat{D}_{\mu}\left(\sqrt{-g}v^{\mu}\right)+\tilde{T}^{\mu}{}_{\mu\nu}v^{\nu}\,.$$ (166) We see that the last term, proportional to the torsion vector, cannot be written in the form (163) and, in general, does not vanish. Consequently, we cannot use the full covariant derivative $\tilde{\nabla}$ to integrate by parts. This is the reason why, throughout the whole paper, we have always used either $\mathring{\nabla}$ or $\hat{\nabla}$ to integrate by parts. It is most convenient to use $\hat{\nabla}$ instead of $\mathring{\nabla}$, for the former maintains Weyl covariance manifest. Notwithstanding the ``bad'' behavior of $\tilde{\nabla}$, we might be interested in finding the general rule for appropriately integrating it by parts. To work it out, let us consider a vector-field $v^{\nu}$ and a tensor field $z^{\mu}{}_{\nu}$, such that the sum of their Weyl weights is $w_{v}+w_{z}=-4$. Expressing schematically $\tilde{\nabla}_{\mu}=\hat{\nabla}_{\mu}+\hat{K}_{\mu}$, we find $$\displaystyle\int\sqrt{-g}v^{\nu}\tilde{\nabla}_{\mu}z^{\mu}{}_{\nu}$$ $$\displaystyle=\int\sqrt{-g}v^{\nu}\hat{\nabla}_{\mu}z^{\mu}{}_{\nu}+\int\sqrt{-g}v^{\nu}\left(\hat{K}^{\mu}{}_{\lambda\mu}z^{\lambda}{}_{\nu}-\hat{K}^{\lambda}{}_{\nu\mu}z^{\mu}{}_{\lambda}\right)$$ (167) $$\displaystyle=-\int\sqrt{-g}z^{\mu}{}_{\nu}\hat{\nabla}_{\mu}v^{\nu}+\int\sqrt{-g}z^{\mu}{}_{\nu}\left(\hat{K}^{\lambda}{}_{\mu\lambda}v^{\nu}-\hat{K}^{\nu}{}_{\lambda\mu}v^{\lambda}\right)$$ $$\displaystyle=-\int\sqrt{-g}z^{\mu}{}_{\nu}\tilde{\nabla}_{\mu}v^{\nu}+\int\sqrt{-g}z^{\mu}{}_{\nu}\hat{K}^{\lambda}{}_{\mu\lambda}v^{\nu}\,.$$ The result that we have just found is valid even if we trade the holonomic index $\nu$ for an arbitrary collection of Latin and Greek indices. To highlight this feature, we are going to label this general set of indices with a multi-index $I$. Therefore, we have the rule $$\int\sqrt{-g}v^{I}\tilde{\nabla}_{\mu}z^{\mu}{}_{I}=-\int\sqrt{-g}z^{\mu}{}_{I}\tilde{\nabla}_{\mu}v^{I}+\int\sqrt{-g}v^{I}z^{\mu}{}_{I}\hat{K}^{\lambda}{}_{\mu\lambda}\,.$$ (168) Using the fact that $\hat{K}^{\lambda}{}_{\mu\lambda}=-\tilde{\tau}_{\mu}$ (see Eq.  (38)), we can rewrite the previous equation compactly $$\int\sqrt{-g}v^{I}(\tilde{\nabla}+\tilde{\tau})_{\mu}z^{\mu}{}_{I}=-\int\sqrt{-g}z^{\mu}{}_{I}\tilde{\nabla}_{\mu}v^{I}\,,$$ (169) which thus becomes the go-to formula for integration by parts of $\tilde{\nabla}$ (recall that $w_{v}+w_{z}=-4$). Appendix C Covariant Lie derivatives and the extended algebra In the main text, specifically Sects. V.2 and VI.2, we have defined and applied a covariant generalization of the Lie derivative, $\widetilde{\pounds}_{\xi}$. Such extension goes under the name of covariant Lie derivative and can be found, mutatis mutandis, in the literature of metric-affine gravity (see, for example, Gronwald:1997jd ). In the present paper, $\widetilde{\pounds}_{\xi}$ provides an ``improvement'' to the standard infinitesimal diffeomorphisms, $\tilde{\delta}^{E}_{\xi}=\widetilde{\pounds}_{\xi}$, which is covariant under all gauge groups. We use the two symbols interchangeably. Since its application to Lorentz-Weyl gauge theories has never been carried out, at least to our knowledge, we prove some of the main properties in this appendix. Before diving into the details of the algebra generated by $\widetilde{\pounds}$, it is important to stress the main differences between the covariant Lie derivative and (any) ordinary covariant derivative. The first observation is that we can give meaning to the covariant Lie derivative of gauge potentials, whereas it is meaningless to speak about their covariant derivatives since they are connections. Secondly, we emphasize that $\tilde{\pounds}_{\alpha\xi}\neq\alpha\tilde{\pounds}_{\xi}$, where $\alpha$ is an arbitrary scalar function. Thus, $\tilde{\pounds}_{\xi}$ does not possess the directional property, whence it cannot be interpreted as a covariant derivative. The difference becomes clearer if one takes into account $\tilde{\pounds}_{\xi}T$, for some arbitrary tensor field $T$ which has, at least, one holonomic index. Indeed, such expression depends on the (covariant) derivative of $\xi$, i.e. it is only local in $\xi$, whereas $\tilde{\nabla}_{\xi}$ is always punctual in $\xi$. Algebra properties of the covariant Lie derivatives The algebraic properties of the full group of infinitesimal transformations, and especially of the covariant ones, hinges on the proof of Eq. (72). The interplay between the covariant Lie derivative and ordinary gauge variations can be summarized in the commutator $$\bigl{[}\widetilde{\pounds}_{\xi},\delta^{L}_{\alpha}\bigr{]}=\delta^{L}_{\xi\cdot D\alpha}\,,\qquad\bigl{[}\widetilde{\pounds}_{\xi},\delta^{W}_{\sigma}\bigr{]}=\delta^{W}_{\xi\cdot\partial\sigma}\,,$$ (170) An important difference between $\pounds_{\xi}={\delta}^{E}_{\xi}$ and $\widetilde{\pounds}_{\xi}$ is that $\pounds_{\xi}$ has trivial commutators $$\big{[}\delta^{E}_{\xi},\delta^{L}_{\zeta\cdot\omega}\big{]}=0\,,\qquad\big{[}\delta^{E}_{\xi},\delta^{W}_{\zeta\cdot S}\big{]}=0\,,$$ (171) instead, the improved transformation does not, as seen in Eq. (71) of the main text. Commutator acting on tensors First of all, we want to prove the following commutator of two covariant Lie derivatives acting on a Lorentz tensor with Weyl weight $w_{A}$ $$\big{[}\widetilde{\pounds}_{\xi},\widetilde{\pounds}_{\zeta}\big{]}A^{a\phantom{b}\rho}_{\phantom{a}b\phantom{\rho}\kappa}=\widetilde{\pounds}_{[\xi,\zeta]}A^{a\phantom{b}\rho}_{\phantom{a}b\phantom{\rho}\kappa}+\delta^{L}_{\mathcal{R}(\xi,\zeta)}A^{a\phantom{b}\rho}_{\phantom{a}b\phantom{\rho}\kappa}+\delta^{W}_{\mathcal{W}(\xi,\zeta)}A^{a\phantom{b}\rho}_{\phantom{a}b\phantom{\rho}\kappa}\,,$$ (172) where the tensor is chosen to have covariant and contravariant, holonomic and anholonomic indices (the extension to an arbitrary number of indices is straightforward). A further simplification comes from noticing that Lie derivatives, exterior derivatives and contractions are intrinsic operations on a generic manifold, so we can suppress coordinate indices henceforth. Thus, we need to prove that $$\big{[}\widetilde{\pounds}_{\xi},\widetilde{\pounds}_{\zeta}\big{]}A^{a}{}_{b}=\widetilde{\pounds}_{[\xi,\zeta]}A^{a}{}_{b}+\delta^{L}_{\mathcal{R}(\xi,\zeta)}A^{a}{}_{b}+\delta^{W}_{\mathcal{W}(\xi,\zeta)}A^{a}{}_{b}\,,$$ (173) where $A^{a}{}_{b}$ is a generic $(p,q)$-tensor. We need the following formula, which is easy to show $$\pounds_{\xi}(\zeta\cdot S)-\pounds_{\zeta}(\xi\cdot S)=dS(\xi,\zeta)+\big{[}\xi,\zeta\big{]}\cdot S\,,$$ (174) and is valid for any $1$-form, that is, we can use it for the spin-connection by replacing $S_{\mu}$ with $\omega^{a}{}_{b\mu}$. Using the definition of covariant Lie derivative, we write $$\displaystyle\big{[}\widetilde{\pounds}_{\xi},\widetilde{\pounds}_{\zeta}\big{]}A^{a}{}_{b}$$ $$\displaystyle=\left(\pounds_{\xi}+\delta^{L}_{\xi\cdot\omega}+\delta^{W}_{\xi\cdot S}\right)\left(\widetilde{\pounds}_{\zeta}A^{a}{}_{b}\right)-(\zeta\leftrightarrow\xi)\,,$$ (175) and exploit the gauge and coordinate covariance of $\widetilde{\pounds}_{\zeta}A^{a}{}_{b}$ to obtain $$\displaystyle\big{[}\widetilde{\pounds}_{\xi},\widetilde{\pounds}_{\zeta}\big{]}A^{a}{}_{b}$$ $$\displaystyle=\pounds_{\xi}\left(\widetilde{\pounds}_{\zeta}A^{a}{}_{b}\right)+(\xi\cdot\omega)^{a}{}_{c}\left(\widetilde{\pounds}_{\zeta}A^{c}{}_{b}\right)$$ (176) $$\displaystyle-(\xi\cdot\omega)^{c}{}_{b}\left(\widetilde{\pounds}_{\zeta}A^{a}{}_{c}\right)+w_{A}(\xi\cdot S)\left(\widetilde{\pounds}_{\zeta}A^{a}{}_{b}\right)-\left(\zeta\leftrightarrow\xi\right)$$ We focus on the first term on the right hand side and apply the Leibniz rule to the ordinary Lie derivative, $$\displaystyle\pounds_{\xi}\left(\widetilde{\pounds}_{\zeta}A^{a}{}_{b}\right)$$ $$\displaystyle=\pounds_{\xi}\left(\pounds_{\zeta}\right)A^{a}{}_{b}+\overline{\pounds_{\zeta}\left(\pounds_{\xi}A^{a}{}_{b}\right)}+\pounds_{\xi}\left(\zeta\cdot\omega\right)^{a}{}_{c}A^{c}{}_{b}+\underline{\left(\zeta\cdot\omega\right)^{a}{}_{c}\left(\pounds_{\xi}A^{c}{}_{b}\right)}$$ (177) $$\displaystyle-\pounds_{\xi}\left(\zeta\cdot\omega\right)^{c}{}_{b}A^{a}{}_{c}-\underline{\left(\zeta\cdot\omega\right)^{c}{}_{b}\left(\pounds_{\xi}A^{a}{}_{c}\right)}+w_{A}\pounds_{\xi}\left(\zeta\cdot S\right)A^{a}{}_{b}+\underline{w_{A}\left(\zeta\cdot S\right)\left(\pounds_{\xi}A^{a}{}_{b}\right)}\,.$$ The remaining three terms on the right hand side of Eq. (176) give $$\displaystyle(\xi\cdot\omega)^{a}{}_{c}\left(\widetilde{\pounds}_{\zeta}A^{c}{}_{b}\right)$$ $$\displaystyle=(\xi\cdot\omega)^{a}{}_{c}\left\{\underline{\pounds_{\zeta}A^{c}{}_{b}}+\left(\zeta\cdot\omega\right)^{c}{}_{d}A^{d}{}_{b}-\overline{\left(\zeta\cdot\omega\right)^{d}{}_{b}A^{c}{}_{d}}+\overline{\overline{w_{A}\left(\zeta\cdot S\right)A^{c}{}_{b}}}\right\}\,,$$ (178a) $$\displaystyle(\xi\cdot\omega)^{c}{}_{b}\left(\widetilde{\pounds}_{\zeta}A^{a}{}_{c}\right)$$ $$\displaystyle=(\xi\cdot\omega)^{c}{}_{b}\left\{\underline{\pounds_{\zeta}A^{a}{}_{c}}+\overline{\left(\zeta\cdot\omega\right)^{a}{}_{d}A^{d}{}_{c}}-\left(\zeta\cdot\omega\right)^{d}{}_{c}A^{a}{}_{d}+\underline{\underline{w_{A}\left(\zeta\cdot S\right)A^{a}{}_{c}}}\right\}\,,$$ (178b) $$\displaystyle(\xi\cdot S)\left(\widetilde{\pounds}_{\zeta}A^{a}{}_{b}\right)$$ $$\displaystyle=(\xi\cdot S)\left\{\underline{\pounds_{\zeta}A^{a}{}_{b}}+\overline{\overline{\left(\zeta\cdot\omega\right)^{a}{}_{c}A^{c}{}_{b}}}-\underline{\underline{\left(\zeta\cdot\omega\right)^{c}{}_{b}A^{a}{}_{c}}}+w_{A}\left(\zeta\cdot S\right)A^{a}{}_{b}\right\}\,.$$ (178c) Upon antisymmetrization in $\xi\leftrightarrow\zeta$ many terms combine. All the terms double underlining or overlining cancel out against each others upon antisymmetrization. Furthermore, the underlined terms in Eqs. (178a), (178b) and (178c) with the ordinary Lie derivatives, simplify with the underlined ones in (177). Finally, the last term in (178c) drops, since Weyl symmetry is Abelian. Thus, using $\left[\pounds_{\xi},\pounds_{\zeta}\right]A^{a}{}_{b}=\pounds_{[\xi,\zeta]}A^{a}{}_{b}$, we can write $$\displaystyle\big{[}\widetilde{\pounds}_{\xi},\widetilde{\pounds}_{\zeta}\big{]}A^{a}{}_{b}=$$ $$\displaystyle\,\pounds_{[\xi,\zeta]}A^{a}{}_{b}+\left[\pounds_{\xi}\left(\zeta\cdot\omega\right)^{a}{}_{c}-\pounds_{\zeta}\left(\xi\cdot\omega\right)^{a}{}_{c}\right]A^{c}{}_{b}$$ (179) $$\displaystyle-\left[\pounds_{\xi}\left(\zeta\cdot\omega\right)^{c}{}_{b}-\pounds_{\zeta}\left(\xi\cdot\omega\right)^{c}{}_{b}\right]A^{a}{}_{c}$$ $$\displaystyle+w_{A}\left[\pounds_{\xi}\left(\zeta\cdot S\right)-\pounds_{\zeta}\left(\xi\cdot S\right)\right]A^{a}{}_{b}$$ $$\displaystyle+\left[(\xi\cdot\omega)^{a}{}_{c}\left(\zeta\cdot\omega\right)^{c}{}_{d}-(\zeta\cdot\omega)^{a}{}_{c}\left(\xi\cdot\omega\right)^{c}{}_{d}\right]A^{d}{}_{b}$$ $$\displaystyle-\left[(\xi\cdot\omega)^{c}{}_{b}\left(\zeta\cdot\omega\right)^{d}{}_{c}-(\zeta\cdot\omega)^{c}{}_{b}\left(\xi\cdot\omega\right)^{d}{}_{c}\right]A^{a}{}_{d}\,.$$ Further, exploiting the result (174), we get $$\displaystyle\big{[}\widetilde{\pounds}_{\xi},\widetilde{\pounds}_{\zeta}\big{]}A^{a}{}_{b}=$$ $$\displaystyle\,\pounds_{[\xi,\zeta]}A^{a}{}_{b}+\left(\big{[}\xi,\zeta\big{]}\cdot\omega\right)^{a}{}_{c}A^{c}{}_{b}-\left(\big{[}\xi,\zeta\big{]}\cdot\omega\right)^{c}{}_{b}A^{a}{}_{c}+\left(\big{[}\xi,\zeta\big{]}\cdot S\right)A^{a}{}_{b}$$ (180) $$\displaystyle+\,\mathcal{R}(\xi,\zeta)^{a}{}_{c}A^{c}{}_{b}-\mathcal{R}(\xi,\zeta)^{c}{}_{b}A^{a}{}_{c}+w_{A}\mathcal{W}(\xi,\zeta)A^{a}{}_{b}$$ $$\displaystyle=$$ $$\displaystyle\,\left(\widetilde{\pounds}_{[\xi,\zeta]}+\delta^{L}_{\mathcal{R}(\xi,\zeta)}+\delta^{W}_{\mathcal{W}(\xi,\zeta)}\right)A^{a}{}_{b}\,,$$ as given in the main text. Commutator acting on connections The previous proof holds for Lorentz and Weyl tensor, however, we know that physical fields can transform in a much more general way, e.g., as connections. It might be unclear whether the same structure of the algebra holds for the connections as well, so we are going to show that the same commutation rule hold for the spin connection. The proof for any other gauge-connection relies on the same steps, even though for a given affine connection $\Gamma^{\rho}{}_{\nu\mu}$ the same result happens to be a consequence of the tetrad postulate, the Leibniz rule and the fact that $\Gamma^{\rho}{}_{\nu\mu}$ is Lorentz and Weyl invariant from the onset. Using the result given in Eq. (70) of the main text, we know that $$\widetilde{\pounds}_{\xi}\omega^{a}{}_{b\mu}=\xi^{\nu}R^{a}{}_{b\nu\mu}\,,$$ (181) so, we have to prove that $$\big{[}\widetilde{\pounds}_{\xi},\widetilde{\pounds}_{\zeta}\big{]}\omega^{a}{}_{b\mu}=\big{[}\xi,\zeta\big{]}^{\nu}R^{a}{}_{b\nu\mu}-D_{\mu}(\mathcal{R}^{a}_{\phantom{a}b}(\xi,\zeta)),$$ (182) where $D_{\mu}$ is the gauge-covariant derivative. Since the Riemann $2$-form and the spin-connection are Weyl invariant, the Weyl variation will not appear in the following equations. We have $$\displaystyle\big{[}\widetilde{\pounds}_{\xi},\widetilde{\pounds}_{\zeta}\big{]}\omega^{a}{}_{b\mu}=$$ $$\displaystyle\,\widetilde{\pounds}_{\xi}\left(\zeta^{\nu}R^{a}{}_{b\nu\mu}\right)-\widetilde{\pounds}_{\zeta}\left(\xi^{\nu}R^{a}{}_{b\nu\mu}\right)$$ (183) $$\displaystyle=$$ $$\displaystyle\,\pounds_{\xi}(\zeta^{\nu})R^{a}{}_{b\nu\mu}+\zeta^{\nu}\pounds_{\xi}R^{a}{}_{b\nu\mu}+\zeta^{\nu}\delta^{L}_{\xi\cdot\omega}R^{a}{}_{b\nu\mu}$$ $$\displaystyle-\,\pounds_{\zeta}(\xi^{\nu})R^{a}{}_{b\nu\mu}-\xi^{\nu}\pounds_{\zeta}R^{a}{}_{b\nu\mu}-\xi^{\nu}\delta^{L}_{\zeta\cdot\omega}R^{a}{}_{b\nu\mu}$$ $$\displaystyle=$$ $$\displaystyle\,2\big{[}\xi,\zeta\big{]}^{\nu}R^{a}{}_{b\nu\mu}\,+$$ $$\displaystyle+\,\zeta^{\nu}\xi^{\rho}\left[\partial_{\rho}R^{a}{}_{b\nu\mu}-\partial_{\nu}R^{a}{}_{b\rho\mu}+\omega^{a}{}_{c\rho}R^{c}{}_{b\nu\mu}-\omega^{c}{}_{b\rho}R^{a}{}_{c\nu\mu}-\omega^{a}{}_{c\nu}R^{c}{}_{b\rho\mu}+\omega^{c}{}_{b\nu}R^{a}{}_{c\rho\mu}\right]$$ $$\displaystyle+\,\zeta^{\nu}R^{a}{}_{b\rho\mu}\partial_{\nu}\xi^{\rho}+\zeta^{\nu}R^{a}{}_{b\nu\rho}\partial_{\mu}\xi^{\rho}-\xi^{\nu}R^{a}{}_{b\rho\mu}\partial_{\nu}\zeta^{\rho}-\xi^{\nu}R^{a}{}_{b\nu\rho}\partial_{\mu}\zeta^{\rho}$$ Notice that the first and third terms in the last line can be combined in $[\zeta,\xi]^{\nu}R^{a}{}_{b\nu\mu}$, thus canceling the factor $2$ which appears in the commutator of two lines above. Another manipulation comes by adding and subtracting $\zeta^{\nu}\xi^{\rho}\partial_{\mu}R^{a}{}_{b\rho\nu}$, $\zeta^{\nu}\xi^{\rho}\omega^{a}{}_{c\mu}R^{c}{}_{b\nu\rho}$ and $-\,\zeta^{\nu}\xi^{\rho}\omega^{c}{}_{b\mu}R^{a}{}_{c\nu\rho}$. The added terms combine with those in the square brackets to give the second Bianchi identities for the Riemann tensor, $D_{[\mu}R^{a}{}_{|b|\nu\rho]}=0$. Instead, the subtracted terms add up with those in the last line of (183) to give the gauge-covariant derivative, which appears in the right hand side of (182), thus completing the proof. With hindsight, we notice that the commutation rule applies for all gauge potentials, provided that the second Bianchi identities hold. Jacobi identities The improved transformations are a field-dependent generalizations of a Lie algebra, which is closed if the Jacobi identities hold. For the algebra to be closed, we thus need to prove that the identities hold. We consider three vector fields $\xi$, $\zeta$ and $\chi$. We have that $$\displaystyle{\rm Cycl}_{\xi,\zeta,\chi}\big{[}\widetilde{\pounds}_{\xi},\widetilde{\pounds}_{\zeta}\big{]}\left(\widetilde{\pounds}_{\chi}A^{a}{}_{b}\right)={\rm Cycl}_{\xi,\zeta,\chi}\widetilde{\pounds}_{\xi}\left(\widetilde{\pounds}_{[\zeta,\chi]}A^{a}{}_{b}+\delta^{L}_{\mathcal{R}(\zeta,\chi)}A^{a}{}_{b}+\delta^{W}_{\mathcal{W}(\zeta,\chi)}A^{a}{}_{b}\right)\,,$$ (184) where the notation ${\rm Cycl}_{\xi,\zeta,\chi}$ stands for the sum over a cyclic permutation of the three vectors. 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Bayesian Ensembles of Crowds and Deep Learners for Sequence Tagging Edwin Simpson and Iryna Gurevych Ubiquitious Knowledge Processing Lab Department of Computer Science Technische Universität Darmstadt https://www.ukp.tu-darmstadt.de {simpson,gurevych}@ukp.informatik.tu-darmstadt.de Abstract Current methods for sequence tagging, a core task in NLP, are data hungry. Crowdsourcing is a relatively cheap way to obtain labeled data, but the annotators are unreliable, so redundant labeling and aggregation techniques are required. We evaluate multiple models of annotator reliability and develop a Bayesian method for aggregating sequence labels from multiple annotators. Typically, the process of data collection, aggregation and training a sequence tagger is a pipeline of discrete steps. We integrate these steps by training black-box sequence taggers as components in the aggregation model and accounting for their unreliability. We evaluate our model on named entity recognition and information extraction tasks, showing that our method outperforms previous methods, particularly in small data scenarios that are encountered at the beginning of a crowdsourcing process. Our code is published to encourage adaptation and reuse. Bayesian Ensembles of Crowds and Deep Learners for Sequence Tagging Edwin Simpson and Iryna Gurevych Ubiquitious Knowledge Processing Lab Department of Computer Science Technische Universität Darmstadt https://www.ukp.tu-darmstadt.de {simpson,gurevych}@ukp.informatik.tu-darmstadt.de 1 Introduction The high demand for labeled training data in current NLP methods, particularly deep learning, is widely recognized (Zoph et al., 2016; Rastogi et al., 2016; Gormley et al., 2014). A common NLP task that has benefited from deep learning is sequence tagging, which involves classifying sequences of tokens for tasks such as named entity recognition (NER), part-of-speech tagging (POS), or information extraction (IE). Neural network sequence taggers are typically trained on tens of thousands of documents  (Ma and Hovy, 2016; Lample et al., 2016). This requirement for large labeled datasets presents a challenge when facing new domains or tasks, where obtaining labels is often time-consuming or costly. One way to obtain labeled data relatively cheaply is crowdsourcing, in which large numbers of untrained workers annotate documents instead of more expensive experts. However, this requires aggregating multiple unreliable labels for each document. We could also obtain noisy labels from models trained on different domains, multiple experts, or users of applications who click on and interact with text. Probabilistic methods for aggregating unreliable classifications have been shown to be more accurate than simple heuristics such as majority voting (Raykar et al., 2010; Sheshadri and Lease, 2013; Rodrigues et al., 2013; Hovy et al., 2013). However, work on sequence tagging is less extensive and existing methods cannot model some common annotator error patterns or the effects of the order of annotators’ labels (Rodrigues et al., 2014; Nguyen et al., 2017). The sequence labeling tasks we consider in this paper follow a beginning, inside, outside (BIO) scheme, in which the first token in a span of type ‘x’ is labeled ‘B-x’, subsequent tokens in the same span are labeled ‘I-x’, and tokens outside spans are labeled ‘O’. We propose an aggregation method that takes advantage of the sequential dependencies between BIO tags to learn the reliability of individual annotators and predict the true sequence. When learning from noisy or small datasets, commonly-used methods based on maximum likelihood estimation may produce over-confident predictions (Xiong et al., 2011; Srivastava et al., 2014). We therefore apply a Bayesian treatment to our method to account for model uncertainty in our predictions. The resulting posterior probabilities facilitate active learning (Settles, 2010), which aims to reduce the number of labels required to train a model by iteratively selecting the most informative data points to label. When aggregating crowdsourced data, we can improve performance and make predictions for unlabeled documents by modeling the text features as well as the annotators (Simpson et al., 2015; Felt et al., 2016). For complex tasks such as sequence tagging, we may wish to exploit existing state-of-the-art models, such as neural networks that do not account for model uncertainty. In this paper, we show how to integrate existing black box methods into the aggregation model to construct ensembles of deep learners and human annotators. Our method learns the reliability of each black box method, since they may not always perform well, particularly given small training datasets, and avoids the need to aggregate crowdsourced data using a separate pre-processing step before training a sequence tagger. This paper provides the following contributions: • A theoretical comparison of annotator reliability models and evaluation on sequence tagging tasks • Bayesian sequence combination (BSC), a method for aggregating sequence labels from multiple annotators that can model sequential dependencies between tags • A technique for wrapping existing black-box sequence taggers into the aggregation model to improve the quality of aggregated labels The following sections discuss related work, then detail annotator models for sequence tagging, and present our variational approach that enables us to integrate existing classifiers. We then describe the modular implementation of our proposed method, which is made public with all of our experimental code111http://github.com/ukplab/arxiv2018-bayesian-ensembles and can easily be extended to new aggregation problems. The next sections compare different aggregation methods with simulated annotators and two crowdsourced NLP datasets, showing that our Bayesian aggregation method consistently outperforms the previous state-of-the-art. Our experiments evaluate both active and passive learning scenarios with varying dataset sizes, analyze types of errors, and visualize the annotator models learned by our method. Finally, we give conclusions and ideas for future work. 1.1 Related Work A number of works have investigated methods for aggregating non-sequential classifications from crowds, including Sheshadri and Lease Sheshadri and Lease (2013), who benchmarked several aggregation methods. They found the most consistent performance from the method of Raykar et al. Raykar et al. (2010), which employs probabilistic confusion matrices to model the reliability of individual annotators, as proposed by Dawid and Skene (Dawid and Skene, 1979). In this paper, we develop and compare variations of this model for sequence tagging, including a variant based on MACE (Hovy et al., 2013). We focus on the core annotator representation, rather than extensions for clustering annotators (Venanzi et al., 2014; Moreno et al., 2015), modeling their dynamics (Simpson et al., 2013), adapting to task difficulty (Whitehill et al., 2009; Bachrach et al., 2012), or time spent (Venanzi et al., 2016). For aggregating sequence tags, Rodrigues et al. Rodrigues et al. (2014) proposed a CRF-based model, CRF-MA, that assumes only one annotator is correct for any given label. Recently, Nguyen et al. Nguyen et al. (2017) proposed an approach that outperformed CRF-MA, based on hidden Markov models (HMMs), called HMM-crowd. Both CRF-MA and HMM-crowd use simpler annotator models than Dawid and Skene Dawid and Skene (1979) that do not capture the effect of sequential dependencies on annotator reliability. Neither CRF-MA nor HMM-crowd use a fully Bayesian approach, which has been shown to be effective for handling uncertainty due to noise in crowdsourced data for non-sequential classification (Kim and Ghahramani, 2012; Simpson et al., 2013; Venanzi et al., 2014; Moreno et al., 2015). In this paper, we develop a sequential annotator model and a fully Bayesian method for aggregating sequence labels that improves performance over previous approaches. The HMM adapted by Nguyen et al Nguyen et al. (2017) uses only a simple conditional independence model of text features. The authors also show how to train neural network sequence taggers directly on crowdsourced data using an additional network layer to handle worker reliability, similar to Rodrigues and Pereira Rodrigues and Pereira (2018). However, the proposed approaches did not outperform either CRF-MA (Rodrigues and Pereira, 2018) or HMM-crowd(Nguyen et al., 2017). Albarqouni et al. Albarqouni et al. (2016) integrate a CNN classifier for image annotation into an aggregation method based on expectation maximization (EM) (Dempster et al., 1977). Yang et al. Yang et al. (2018) adapt a Bayesian neural network so that it can be trained concurrently with an annotator model, also using EM. In contrast to previous work, we do not require neural networks to be adapted, nor assume that their predictions are reliable when aggregating annotations. Instead, we propose to learn the reliability of existing sequence taggers using a variational approach, allowing untrusted, off-the-shelf sequence taggers to enhance the performance of the aggregation method. 2 Modeling Sequential Annotators When combining multiple annotators with varying skill levels, we can improve performance by modeling their individual reliability. Several models have previously been applied that do not consider dependencies between a sequence of annotations. In this section, we describe these existing models and provide an extension that captures sequential dependencies. Each of the approaches presented employs a particular function, $A$, to model the likelihood of the annotator choosing the label $c_{\tau}$ given the true label, $t_{\tau}$, for token $\tau$. Accuracy model (acc): simply models the annotator’s accuracy, $\pi$, as follows: $$\displaystyle A=p(c_{\tau}\!=\!i|t_{\tau}\!=\!j,\pi)=\left.\begin{cases}\pi&\!% \!\!\text{ where }i=j\\ \frac{1-\pi}{J-1}&\!\!\!\text{ otherwise}\end{cases}\right\}\!,$$ (1) where $c_{\tau}$ is the label given by the annotator for token $\tau$, $t_{\tau}$ is its true label and $J$ is the number of classes. This is the basis of several previous methods (Donmez et al., 2010; Rodrigues et al., 2013). The limitation of this approach is that it assumes reliability is constant, which means that in domains where one class label is far more common than others, a spammer who always selects the most common label will nonetheless have a high $\pi$. MACE spamming model (Hovy et al., 2013): This method again assumes a constant annotator accuracy, $\pi$, but also models the case where annotators are incorrect by assuming they label according to a spamming distribution, $\boldsymbol{\xi}$, that is independent of the true label, $t_{\tau}$. $$\displaystyle A$$ $$\displaystyle=p(c_{\tau}=i|t_{\tau}=j,\pi,\boldsymbol{\xi})$$ $$\displaystyle=\left.\begin{cases}\pi+(1-\pi)\xi_{j}&\text{ where }i=j\\ (1-\pi)\xi_{j}&\text{ otherwise}\end{cases}\right\}.$$ (2) While MACE can capture spamming patterns, it does not explicitly model different rates of errors per class. This could be an issue for sequence tagging using the BIO encoding, for example, if an annotator frequently labels longer spans than the true spans by starting the spans one or two tokens early. In this case, they may more frequently mis-label the ‘B’ tokens than the ‘I’ or ‘O’ tokens, but this cannot be modeled by MACE. Confusion vector (CV): this approach learns a separate accuracy for each class label (Nguyen et al., 2017) using a parameter vector, $\boldsymbol{\pi}$, of size $J$: $$\displaystyle A=p(c_{\tau}\!\!=\!i|t_{\tau}\!=\!j,\boldsymbol{\pi})=\left.% \begin{cases}\pi_{j}&\text{ where }i\!=\!j\\ \frac{1\!-\!\pi_{j}}{J-1}&\text{ otherwise}\end{cases}\!\right\}\!.$$ (3) For the incorrect label cases where $i\!\neq\!j$, $p(c_{\tau}\!\!=\!i|t_{\tau}\!=\!j,\boldsymbol{\pi})$ is constant for all values of $i$. Therefore, this model does not explicitly capture spamming patterns where one of the incorrect labels has a much higher likelihood than the others. Confusion matrix (CM) (Dawid and Skene, 1979): this model can be seen as an expansion of the confusion vector so that $\boldsymbol{\pi}$ becomes a $JxJ$ matrix with values given by: $$\displaystyle A=p(c_{\tau}\!\!=\!i|t_{\tau}\!=\!j,\boldsymbol{\pi})=\pi_{j,i}.$$ (4) This requires a larger number of parameters, $J^{2}$, compared to the $J+1$ parameters of MACE or $J$ parameters of the confusion vector. The confusion matrix therefore represents the probability of each individual mistake, so it can model spammers who frequently chose one label regardless of the ground truth. It can also model annotators in sequence tagging tasks who have different error rates for ‘B-x’, ‘I-x’ and ‘O’ labels, for example, if an annotator is better at detecting type ‘x’ spans than type ‘y’, or if they frequently mis-label the start of a span as ‘O’ when the true label is ‘B-x’, but are otherwise accurate. However, the confusion matrix ignores the dependencies between annotations in a sequence that affect these probabilities. For instance, it is usually not possible for an annotator to assign an ‘I’ label that is preceded by ‘O’. Sequential Confusion Matrix (seq): we introduce a new extension to the confusion matrix to model the dependency of each label in a sequence on its predecessor. The likelihood of a label can now be written as follows: $$\displaystyle A=p(c_{\tau}\!\!=\!i|c_{\tau-1}\!=\!\iota,t_{\tau}\!=\!j,% \boldsymbol{\pi})=\pi_{j,\iota,i},$$ (5) where $\boldsymbol{\pi}$ is now three-dimensional with size $J\times J\times J$. In the case of disallowed transitions, e.g. from $c_{\tau-1}=$‘O’ to $c_{\tau}=$‘I’, the value $\pi_{j,c_{\tau-1},c_{\tau}}=0$, $\forall j$ does not need to be learned. The sequential model can capture phenomena such as a tendency toward overly long sequences, by learning that $\pi_{O,O,O}>\pi_{O,I,O}$, or a tendency to split spans by inserting ‘B’ in place of ‘I’ by increasing the value of $\pi_{I,I,B}$ without affecting $\pi_{I,B,B}$ and $\pi_{I,O,B}$. The annotator models described above are extensions of one another that can be used as part of the model for aggregating sequential annotations described in the next section. The experiments in Section 6 test whether the more expressive seq annotator model, which has more parameters to learn, is beneficial in a realistic setting. 3 A Model for Bayesian Sequence Combination The generative story for our approach, Bayesian sequence combination (BSC), is as follows. We assume a transition matrix, $\boldsymbol{T}$, where each entry is $T_{j,\iota}=p(t_{\tau}=\iota|t_{\tau-1}=j)$. We draw each row of the transition matrix, $T_{j}\sim\mathrm{Dir}(\boldsymbol{\gamma}_{j})$, where $\mathrm{Dir}$ is the Dirichlet distribution. For each document, $n$, in a set of $N$ documents, we draw a sequence of class labels, $\boldsymbol{t}_{n}=[t_{n,1},...,t_{n,L_{n}}]$, of length $L_{n}$, from a categorical distribution: $t_{n,\tau}\sim\mathrm{Cat}(\boldsymbol{T}_{t_{n,\tau-1}})$. The set of all labels for all documents is referred to as $\boldsymbol{t}=\{\boldsymbol{t}_{1},...,\boldsymbol{t}_{N}\}$. For each of $K$ annotators, we choose one of the annotator models defined in Section 2. The number of parameters depends on the choice of model: for acc, only one parameter, $\pi^{(k)}$, is drawn for annotator $k$; for MACE, we draw a single value $\pi^{(k)}$ and a vector $\xi^{(k)}$ of length $J$, while for CV we draw $J$ independent values of $\pi_{j}^{(k)}$, and for CM we draw a vector $\boldsymbol{\pi}^{(k)}_{j}$ of size $J$ for each true label value $j\in\{1,...,J\}$; in the case of seq, we draw vectors $\boldsymbol{\pi}^{(k)}_{j,\iota}$ for each true label value and each previous label value, $\iota$. All parameters of these annotator models are probabilities, so are drawn from Dirichlet priors. We refer to the set of hyperparameters for $k$’s annotator model as $\boldsymbol{\alpha}^{(k)}$. Given its parameters, the annotator model defines a likelihood function, $A^{(k)}(t_{n,\tau},\boldsymbol{c}_{n,\tau},\boldsymbol{c}_{n,\tau-1})$. The argument $\boldsymbol{c}_{n,\tau-1}$ is only required if $A^{(k)}$ is an instance of seq and is ignored by the other annotator models. We draw annotator $k$’s label for each token $\tau$ in each document $n$ according to: $$\displaystyle c^{(k)}_{n,\tau}\sim\mathrm{Cat}([A^{(k)}(t_{n,\tau},1,% \boldsymbol{c}_{n,\tau-1}^{(k)}),...,$$ $$\displaystyle                             A^{(k)}(t_{n,\tau},J,\boldsymbol{c}_% {n,\tau-1}^{(k)})]).$$ (6) The annotators are assumed to be conditionally independent of one another given the true labels, $\boldsymbol{t}$, which means that their errors are assumed to be uncorrelated. This is a strong assumption when considering that the annotators have to make their decisions based on the same input data. However, in practice, dependencies do not usually cause the most probable label to change (Zhang, 2004), hence the performance of classifier combination methods is only slightly degraded, while avoiding the complexity of modeling dependencies between annotators (Kim and Ghahramani, 2012). Black-box Sequence Taggers: As an optional extension to our model, we can integrate $S$ automated methods as additional noisy annotators. In comparison to human annotators, sequence taggers can quickly label large numbers of documents, providing a cheap source of additional annotations across the whole dataset. We model each sequence tagger, $s$, using an annotator model, $B^{(s)}$, of one of the types described in Section 2 (analagous to $A^{(k)}$ for a human annotator), with hyperparameters $\boldsymbol{\beta}^{(s)}$. Each sequence tagger generates a sequence of labels, $\boldsymbol{d}_{n}^{(s)}$, for each document $n$ (analogous to $\boldsymbol{c}_{n}^{(k)}$ produced by annotators) according to: $$\displaystyle d_{n,\tau}^{(s)}\sim\mathrm{Cat}([B^{(s)}(\boldsymbol{t}_{n,\tau% },1,d_{n,\tau-1}^{(s)}),...,$$ $$\displaystyle                             B^{(s)}(\boldsymbol{t}_{n,\tau},J,d_% {n,\tau-1}^{(s)})]).$$ (7) In the generative model, we draw a sequence of text tokens, $\boldsymbol{x}_{n}$, from a likelihood, $p\left(\boldsymbol{x}_{n}|\boldsymbol{d}_{n}^{(s)},\boldsymbol{\theta}^{(s)}\right)$, given internal parameters, $\boldsymbol{\theta}^{(s)}$, and label sequence, $\boldsymbol{d}_{n}^{(s)}$. This likelihood is defined by the black-box sequence tagger. If the sequence tagger is Bayesian, its parameters, $\boldsymbol{\theta}^{(s)}$, may also be drawn from an unknown prior distribution. However, since we are treating the tagger as a black box, we do not need to know these internal details. In the next section, we explain how we can avoid computing this likelihood explicitly during inference, and instead use only the sequence tagger’s existing training and prediction functions to learn $\boldsymbol{\theta}^{(s)}$ in parallel with the parameters of the BSC model. Like the human annotators, each sequence tagger is assumed to produce labels that are conditionally independent of the other sequence taggers given $\boldsymbol{t}$. Due to the fact that sequence taggers will typically use the same features, i.e. the text of the documents, this independence assumption may be violated, yet as with the human annotators, comparable assumptions in other models have been shown not to hamper performance in many practical situations (Zhang, 2004). Joint distribution: the complete model can be represented by the joint distribution over its random variables, given by: $$\displaystyle p(\boldsymbol{t},\boldsymbol{A},\boldsymbol{B},\boldsymbol{T},% \boldsymbol{\theta},\boldsymbol{c},\boldsymbol{d},\boldsymbol{x}|\boldsymbol{% \alpha},\boldsymbol{\beta},\boldsymbol{\gamma})$$ (8) $$\displaystyle=\prod_{k=1}^{K}\left\{p(A^{(k)}|\boldsymbol{\alpha}^{(k)})\prod_% {n=1}^{N}p(\boldsymbol{c}_{n}^{(k)}|A^{(k)},\boldsymbol{t})\right\}$$ $$\displaystyle\prod_{j=1}^{J}p(\boldsymbol{T}_{j}|\boldsymbol{\gamma}_{j})\prod% _{n=1}^{N}\prod_{\tau=1}^{L_{n}}p(\boldsymbol{t}_{n}|\boldsymbol{T}_{t_{n,\tau% -1}})\prod_{s=1}^{S}\bigg{\{}p(\boldsymbol{\theta}^{(s)})$$ $$\displaystyle p(B^{(s)}|\boldsymbol{\beta}^{(s)})\prod_{n=1}^{N}\!\left\{p(% \boldsymbol{x}|\boldsymbol{d}^{(s)},\boldsymbol{\theta}^{(s)})p(\boldsymbol{d}% ^{(s)}|B^{(s)},\boldsymbol{t})\right\}\!\!\bigg{\}}$$ 4 Inference using Variational Bayes Given a set of annotations, $\boldsymbol{c}=\{\boldsymbol{c}^{(1)},...,\boldsymbol{c}^{(K)}\}$ from $K$ annotators, our aim is to obtain a posterior distribution over the sequence labels, $\boldsymbol{t}$. We present an approximate inference method using variational Bayes (VB) (Attias, 2000). In comparison to other Bayesian approaches such as Markov chain Monte Carlo (MCMC), VB is often faster, readily allows incremental learning, and provides easier ways to determine convergence (Bishop and Nasrabadi, 2007). Unlike maximum likelihood methods such as standard expectation maximization (EM), VB considers prior distributions and accounts for parameter uncertainty in a Bayesian manner. The trade-off is that to apply VB to our BSC model, we need to approximate the posterior distribution over $\boldsymbol{t}$ and the model parameters, $\boldsymbol{T}$, $\boldsymbol{\theta}=\{\boldsymbol{\theta}^{(1)},...,\boldsymbol{\theta}^{(S)}\}$, $\boldsymbol{A}=\{A^{(1)},...,A^{(K)}\}$ and $\boldsymbol{B}=\{B^{(1)},...,B^{(S)}\}$. The labels produced by the sequence taggers, $\boldsymbol{d}$, can be marginalized, so do not appear in the approximate posterior, which is given by: $$\displaystyle p(\boldsymbol{t},\boldsymbol{A},\boldsymbol{B},\boldsymbol{T},% \boldsymbol{\theta}|\boldsymbol{c},\boldsymbol{x},\boldsymbol{\alpha},% \boldsymbol{\beta},\boldsymbol{\gamma})\approx\prod_{k=1}^{K}q(A^{(k)})$$ $$\displaystyle\prod_{j=1}^{J}q(\boldsymbol{T}_{j})\prod_{n=1}^{N}q(\boldsymbol{% t}_{n})\prod_{s=1}^{S}\Big{\{}q(B^{(s)})q(\boldsymbol{\theta}^{(s)})\Big{\}}.$$ (9) The variational approximation factorizes between subsets of parameters or latent variables, so that each subset, $z$, has a variational distribution $q(z)$. Due to our choice of conjugate priors, the variational factors for BSC all have the same form as their prior distributions defined in Section 3, and the parameters of each variational distribution can be computed in terms of expectations over the other subsets of variables. The inference algorithm works by updating each of these variational factors, $q(z)$, in turn, taking expectations with respect to the current estimates of the other variational factors. Each iteration reduces the KL-divergence between the true and approximate posteriors of Equation 4, and hence optimizes a lower bound on the log marginal likelihood (also called evidence lower bound or ELBO), as described in  (Bishop and Nasrabadi, 2007; Attias, 2000). A summary of the VB procedure for BSC is shown in Algorithm 1. The remainder of this section provides the variational factors, which can be used to approximate the marginal posterior distributions for the parameters and sequence labels, and explains how to incorporate existing sequence taggers into the algorithm. Variational factor for $\boldsymbol{t}$, true sequence labels: $$\displaystyle\ln q(\boldsymbol{t}_{n})=\sum_{n=1}^{N}\sum_{\tau=1}^{L_{n}}% \bigg{\{}\mathbb{E}_{\boldsymbol{T}}\left[\ln T_{t_{n,\tau-1},t_{n,\tau}}\right]$$ $$\displaystyle+\sum_{k=1}^{K}\mathbb{E}_{\boldsymbol{A}}\left[\ln A^{(k)}(t_{n,% \tau},c_{n,\tau}^{(k)},c_{n,\tau-1}^{(k)})\right]$$ $$\displaystyle+\sum_{s=1}^{S}\mathbb{E}_{\boldsymbol{B},\boldsymbol{d}^{(s)}}\!% \left[\ln B^{(s)}(t_{n,\tau},d_{n,\tau}^{(s)},d_{n,\tau-1}^{(s)})\right]\bigg{% \}}\!+\!\mathrm{const}.$$ (10) To compute $q(\boldsymbol{T}_{j})$, $q(\boldsymbol{A}^{(k)})$, and $q(\boldsymbol{B}^{(s)})$, we require expectations for the individual true labels $r_{n,\tau,j}=\mathbb{E}_{\boldsymbol{T},\boldsymbol{A},\boldsymbol{B},% \boldsymbol{d}}[p(t_{n,\tau}=j|\boldsymbol{c})]$ and transitions from one each label to the next, $s_{n,\tau,j,\iota}=\mathbb{E}_{\boldsymbol{T},\boldsymbol{A},\boldsymbol{B},% \boldsymbol{d}}[p(t_{n,\tau-1}=j,t_{n,\tau}=\iota|\boldsymbol{c})]$. These terms can be computed using the forward-backward algorithm (Ghahramani, 2001), which consists of two passes. The forward pass is run for each document $n$, starting from $\tau=1$, and computes for each value of $\tau$ the posterior given crowdsourced annotations for tokens $\leq\tau$. $$\displaystyle\ln r^{-}_{n,\tau,j}=\ln\sum_{\iota=1}^{J}\left\{r^{-}_{n,\tau-1,% \iota}\exp\mathbb{E}[\ln T_{\iota,j}]\right\}$$ $$\displaystyle                          +ll(j,n,\tau),$$ (11) where the log likelihood $ll(n,\tau)$ of the annotations from the crowd and sequence taggers for token $\tau$ in document $n$ is: $$\displaystyle ll(j,n,\tau)=\sum_{k=1}^{K}\mathbb{E}_{\boldsymbol{A}}\left[\ln A% ^{(k)}\left(j,c_{n,\tau}^{(k)},c_{n,\tau-1}^{(k)}\right)\right]+$$ $$\displaystyle\sum_{s=1}^{S}\sum_{i=1}^{J}\sum_{\iota=1}^{J}\mathbb{E}_{% \boldsymbol{B}}\left[\ln B^{(s)}\left(j,i,\iota\right)\right]\hat{p}(d_{n,\tau% }^{(s)}=i)$$ $$\displaystyle\hat{p}(d_{n,\tau}^{(s)}=\iota),$$ (12) where $\hat{p}(d_{n,\tau}^{(s)}=i)$ is the probability of label $d_{n,\tau}^{(s)}$ produced by the sequence tagger $s$, which we explain in more detail below (see Equation 26). For the first token in each sequence, $r^{-}_{n,0,\iota}=1$ where $\iota$ corresponds to the ‘O’ label and is $0$ otherwise. After the forward pass is complete, the backwards pass starts from $\tau=L_{n}$ and scrolls backwards, computing the likelihoods of the annotations at positions from $\tau+1$ to $L_{n}$, as follows: $$\displaystyle\ln\lambda_{n,L_{n},j}=0$$ $$\displaystyle\ln\lambda_{n,\tau,j}=\ln\sum_{\iota=1}^{J}\exp\bigg{\{}\ln% \lambda_{i,\tau+1,\iota}+\mathbb{E}[\ln T_{j,\iota}]$$ $$\displaystyle+ll(\iota,n,\tau+1)\bigg{\}}.$$ (13) Since the terms may become small over a long sequence, we normalize $r^{-}_{n,\tau,j}$ and $\lambda_{n,\tau,j}$ after each iteration of the forward and backward pass by dividing by their sum over $j$. By taking the exponents and applying Bayes’ rule we arrive at the terms $r_{n,\tau,j}$ and $s_{n,\tau,j,\iota}$: $$\displaystyle r_{n,\tau,j}=\frac{r^{-}_{n,\tau,j}\lambda_{n,\tau,j}}{\sum_{% \iota=1}^{J}r^{-}_{n,\tau,\iota}\lambda_{n,\tau,\iota}}$$ (14) $$\displaystyle\ln\tilde{s}_{n,\tau,j,\iota}=\ln r^{-}_{n,\tau-1,j}+\ln\lambda_{% n,\tau,\iota}+\mathbb{E}[\ln T_{j,\iota}]$$ $$\displaystyle                          +ll(\iota,n,\tau)$$ (15) $$\displaystyle s_{n,\tau,j,\iota}=\frac{\tilde{s}_{n,\tau,j,\iota}}{\sum_{j=1}^% {J}\sum_{\iota=1}^{J}\tilde{s}_{n,\tau,j,\iota}}$$ (16) The $r_{i,\tau,j}$ terms provide the output predictions of the class labels. Variational factor for $\boldsymbol{T}$: each row of the transition matrix has a separate factor: $$\displaystyle\ln q(\boldsymbol{T}_{j})=\sum_{\iota=1}^{J}N_{j,\iota}+\ln% \mathrm{Dir}(\boldsymbol{T}_{j}|\boldsymbol{\gamma}_{j})+\mathrm{const}$$ $$\displaystyle=\ln\mathrm{Dir}\left(\left[N_{j,\iota}+\gamma_{j,\iota},\forall% \iota\in\{1,...,J\}\right]\right),$$ (17) where $N_{j,\iota}=\sum_{n=1}^{N}\sum_{\tau=1}^{L_{n}}s_{n,\tau,j,\iota}\ln T_{j,\iota}$ is the pseudo-count of the number of times that label $\iota$ follows label $j$. The variational factor $q(\boldsymbol{t})$ requires the following expectations for the transition matrix: $$\displaystyle\mathbb{E}[\ln T_{j,\iota}]=\Psi\left(N_{j,\iota}+\gamma_{j,\iota% }\right)$$ $$\displaystyle                 -\Psi\left(\sum_{\iota=1}^{J}(N_{j,\iota}+\gamma% _{j,\iota})\right),$$ (18) where $\Psi$ is the digamma function. Variational factors for $\boldsymbol{A}$ and $\boldsymbol{B}$: The variational factor for each annotator model is a distribution over its parameters, which differs between models. For seq, the variational factor is given by: $$\displaystyle\ln q\left(A^{(k)}\right)=\sum_{j=1}^{J}\sum_{l=1}^{J}\bigg{\{}% \sum_{m=1}^{J}N_{j,l,m}^{(k)}\ln\pi_{j,l,m}^{(k)}$$ $$\displaystyle                       +\ln p\left(\boldsymbol{\pi}_{j,l}^{(k)}|% \boldsymbol{\alpha}_{j,l}^{(k)}\right)\bigg{\}}+\mathrm{const},$$ $$\displaystyle=\sum_{j=1}^{J}\sum_{l=1}^{J}\mathrm{Dir}\left(\left[\boldsymbol{% N}_{j,l,m}^{(k)}\!+\alpha_{j,l,m}^{(k)},\!\forall m\!\in\!\{1,..,J\}\right]% \right),$$ (19) $$\displaystyle N^{(k)}_{j,l,m}=\sum_{n=1}^{N}\sum_{\tau=1}^{L_{n}}r_{n,\tau,j}% \delta_{l,c^{(k)}_{n,\tau-1}}\delta_{m,c^{(k)}_{n,\tau}},$$ (20) where $\delta$ is the Kronecker delta. For the CM model, the variational factor is simplified to: $$\displaystyle\ln q\left(A^{(k)}\right)=\sum_{j=1}^{J}\mathrm{Dir}\bigg{(}\bigg% {[}\sum_{n=1}^{N}\sum_{\tau=1}^{L_{n}}r_{n,\tau,j}\delta_{m,c^{(k)}_{n,\tau}}$$ $$\displaystyle                 +\alpha_{j,m}^{(k)},\!\forall m\!\in\!\{1,..,J\}% \bigg{]}\bigg{)}.$$ (21) For MACE, CV and acc, the factors follow a similar pattern of summing pseudo-counts of correct and incorrect answers. For reasons of space, we omit the equations for these variants. The variational factor $q(\boldsymbol{t})$ also requires the following expectation terms for seq models: $$\displaystyle\mathbb{E}\left[\ln A^{(k)}(j,l,m)\right]=\Psi\left(N^{(k)}_{j,l,% m}+\alpha^{(k)}_{j,l,m}\right)$$ $$\displaystyle                 -\Psi\left(\sum_{m^{\prime}=1}^{J}\left(N^{(k)}_% {j,l,m^{\prime}}+\alpha^{(k)}_{j,l,m^{\prime}}\right)\right).$$ (22) For CM, the equation can be adapted by omitting the $l$ subscripts on the right-hand side, which refer to the previous annotation in the sequence. The varational factor, $q(B^{(s)})$, for each sequence tagger’s annotator model follows the same form as $q(A^{(k)})$, substituting $\delta_{l,c^{(k)}_{n,\tau-1}}$ for $\hat{p}(d_{n,\tau}^{(s)}=i)$, as defined in below in Equation 26. Black-box sequence taggers: Our inference approach can incorporate either pre-trained sequence taggers, or train the sequence tagger using the crowdsourced data while performing inference over the complete BSC model. In both cases, the tagger’s reliability will be modeled by an annotator model, $B^{(s)}$, so it is possible to incorporate noisy sequence taggers into the ensemble. With pre-trained sequence taggers, we assume that the tagger’s parameters, $\boldsymbol{\theta}^{(s)}$, or their distribution are already fixed and we do not update the variational factor $q(\boldsymbol{\theta}^{(s)})$. For sequence taggers that we wish to train as part of our VB algorithm, the variational factor is: $$\displaystyle\ln q(\boldsymbol{\theta}^{(s)})=\ln p(\boldsymbol{\theta}^{(s)})% +\mathbb{E}_{\boldsymbol{d}_{n}^{(s)}}\left[\ln p(\boldsymbol{x}|\boldsymbol{% \theta}^{(s)},\boldsymbol{d}_{n}^{(s)})\right]$$ $$\displaystyle\approx\ln p(\boldsymbol{\theta}^{(s)})+\ln p\left(\boldsymbol{x}% |\boldsymbol{\theta}^{(s)},\mathbb{E}\left[\boldsymbol{d}_{n}^{(s)}|B^{(s)},% \boldsymbol{t}_{n}\right]\right)$$ (23) The approximation above enables us to train the sequence tagger using its standard training or fitting function: we compute $\ln q(\boldsymbol{\theta}^{(s)})$ by running the training function of the black-box sequence tagger, passing in a set of expectations over the labels in place of gold labels: $$\displaystyle\tilde{p}(d_{n,\tau})=\mathbb{E}\left[p(d_{n,\tau}^{(s)}=i|B^{(s)% },t_{n,\tau})\right]$$ $$\displaystyle=\sum_{j=1}^{J}\sum_{\iota=1}^{J}r_{n,\tau,j}\tilde{p}(d_{n,\tau-% 1})\mathbb{E}[B^{(s)}(j,i,\iota)]$$ (24) The term $d_{n,\tau}^{(s)}$ can be marginalized without recourse to its own variational factor. since it is independent of all other variables given $t_{n,\tau}$, $\boldsymbol{x}_{n}$, $B^{(s)}$, $d_{n,\tau-1}^{(s)}$ and $\boldsymbol{\theta}^{(s)}$. Depending on its implementation, it may be necessary to train the sequence tagger using discrete labels, in which case we take the most probable values at each token instead of Equation 24: $$\displaystyle\tilde{d}_{n,\tau}^{(s)}=\operatorname*{\arg\!\max\!}_{i}\;% \mathbb{E}\left[p(d_{n,\tau}^{(s)}=i|B^{(s)},t_{n,\tau})\right].$$ (25) If we use discrete labels to train a sequence tagger, our inference procedure becomes a hybrid between VB and a maximum a posteriori (MAP) expectation maximization (EM) solution (Bishop and Nasrabadi, 2007). Similarly, if the sequence tagger may not employ an explicit prior, $p(\boldsymbol{\theta}^{(s)})$, or may optimize point values for the parameters in $\boldsymbol{\theta}^{(s)}$, rather than marginalizing them. This is typically the case for most neural network methods, which perform maximum likelihood optimization. When integrating such sequence taggers, the complete procedure becomes a hybrid between maximum likelihood EM for $\boldsymbol{\theta}^{(s)}$ and VB for the other variables. The forward and backward passes used to update $q(\boldsymbol{t})$ require expectations over $\boldsymbol{d}_{n}^{(s)}$, defined as: $$\displaystyle\hat{p}(d_{n,\tau}^{(s)}=i)=\mathbb{E}_{\boldsymbol{\theta}^{(s)}% }\left[p(d_{n,\tau}^{(s)}=i|\boldsymbol{x}_{n},\boldsymbol{\theta}^{(s)})% \right].$$ (26) If possible, we obtain this posterior through the prediction function of the sequence tagger. However, some sequence tagger implementations may output only discrete predictions of the following form: $$\displaystyle\hat{d}_{n,\tau}^{(s)}(i)=\operatorname*{\arg\!\max\!}_{i}\;p% \left(d_{n,\tau}^{(s)}=i|\boldsymbol{x}_{n},\hat{\boldsymbol{\theta}}^{(s)}% \right),$$ (27) where $\hat{\boldsymbol{\theta}}^{(s)}$ is the value of $\boldsymbol{\theta}^{(s)}$ learned using maximum likelihood or MAP optimization. As in Equation 25, we can use these discrete predictions in place of probabilities to perform an M-step from maximum likelihood-EM in place of taking expectations over $\boldsymbol{d}^{(s)}$. Our method requires only training and prediction functions to integrate a sequence tagger. Its annotator model, $B^{(s)}$, accounts for the sequence tagger’s error rates and provides confidence estimates based on their reliability. This means we can treat sequence taggers as black boxes and ignore their internal details, even if their predictions are noisy or overly confident, as may be the case when a tagger is not optimized for the current domain. 4.1 Predicting the Sequence Labels Two types of output from the BSC inference algorithm are of particular interest: (1) posterior probabilities of the true labels, $\mathbb{E}[\boldsymbol{t}]$, which provide confidence estimates for the labels; (2) the most probable sequence of labels, $\hat{\boldsymbol{t}}$. The latter can be computed using the Viterbi algorithm using the converged variational factors to compute the transition matrix, $\mathbb{E}[\boldsymbol{T}]$, and the likelihood or emission probabilities as a function of $\mathbb{E}[\boldsymbol{A}]$, $\mathbb{E}[\boldsymbol{B}]$ and $\hat{p}(d_{n,\tau}^{(s)}=i)$, $\forall s,\forall n,\forall\tau,\forall i$. The most probable sequence is particularly useful because, unlike $\mathbb{E}[\boldsymbol{t}]$, the sequence will be consistent with any transition constraints imposed by the priors on the transition matrix $\boldsymbol{T}$, such as preventing ‘O’$\rightarrow$‘I’ transitions by assigning them zero probability. We can also make predictions for unlabelled documents in a similar manner. In this case, we omit the human annotations, $\boldsymbol{c}$, and rely only on the black-box sequence taggers. 5 Modular Implementation of Variational Inference The variational inference method described in Section 4 is naturally suited to a modular implementation. We divide the BSC model, as defined in Section 3 and Equation 8, into three modules: (a) the true label model, (b) the annotator model, and (c) black-box sequence taggers. The true label model defines the distribution over sequences of labels, $q(\boldsymbol{t}_{n})$, and implements lines 2, 3 and 8 in Algorithm 1. The annotator model may be one of those described in Section 2 and implements lines 6 and 7. The black-box sequence taggers are existing implementations that provide training and prediction functions to predict true labels given text tokens, and are used in lines 4 and 5. The true label model exposes methods to compute $r_{n,\tau,j}$ and $s_{n,\tau,j,\iota}$, $\forall n,\forall\tau,\forall j,\forall\iota$. In BSC, the true label model learns a transition matrix, $\boldsymbol{B}$, which assumes a first-order Markov chain, but true label models with longer memory could also be used, as long as they provide the terms $r_{n,\tau,j}$ and $s_{n,\tau,j,\iota}$ required by the other components. The annotator models must provide methods to initialise the variational distributions $q(\boldsymbol{A})$ and $q(\boldsymbol{B})$, update $q(\boldsymbol{A})$ and $q(\boldsymbol{B})$ during the VB algorithm, and compute expectations involving $\boldsymbol{A}$ and $\boldsymbol{B}$ required for other parts of the model. As discussed in Section 2, various annotator models are possible. By allowing individual functions to be replaced without rewriting the inference method, the modular implementation makes it easier to adapt the model to different types of annotations, and to test each component part. For example, new annotator models could, in future, be used to aggregate continuous-valued ratings or pairwise preferences. 6 Experiments We evaluate Bayesian sequence combination (BSC) against alternative methods to test (a) the different annotator models described in Section 3, (b) the performance of BSC on unreliable or small training sets, and (c) the benefits of including sequence taggers into the probabilistic model. The first experiment uses simulated annotators to investigate the effects of different annotator flaws on aggregation methods. We then introduce two NLP datasets to test aggregation performance in passive and active learning scenarios, analyze errors, visualize the learned annotator models, and test LSTM sequence taggers (Lample et al., 2016) trained using our proposed method. 6.1 Evaluated Methods As established non-sequential baselines, we include token-level majority voting (MV), MACE (Hovy et al., 2013), Dawid-Skene (DS) (Dawid and Skene, 1979) and independent Bayesian classifier combination (IBCC) (Kim and Ghahramani, 2012), a Bayesian treatment of Dawid-Skene. We also test the sequential HMM-crowd method (Nguyen et al., 2017), which uses a combination of maximum a posteriori (or smoothed maximum likelihood) estimates for a confusion vector (CV) annotator model and variational inference for an integrated hidden Markov model. We also introduce a clustering baseline, that aggregates spans from multiple annotators by grouping them together using kernel density estimation (Rosenblatt, 1956). BSC is tested with each of the different annotator models described in Section 2 and integrating different text models. As the default set-up, we integrate a simple black-box classifier that treats all text features as conditionally independent of each other and of the sequence of labels. This set-up is tested with all annotator models. The BSC-seq variant is also tested without a text model (notext), and with an integrated LSTM (Lample et al., 2016), labeled BSC-seq+LSTM. We also use HMM-crowd and BSC-seq to produce training labels for the LSTM as a separate pre-processing step, labeled in our results as $\rightarrow$LSTM. The MACE and IBCC methods are non-sequential variants of BSC-MACE and BSC-CM, and serve to show the benefits of the sequential BSC model. The HMM-Crowd and DS methods also allow us to compare non-Bayesian methods against their Bayesian variants, BSC-CV and IBCC, respectively. 6.2 Simulated Annotators Simulated data allows us to test the effect of one type of error in the crowdsourced data, while keeping other characteristics of the data constant. We generate crowds of 10 annotators for four experiments, which test the effect of varying (a) average annotator accuracy, (b) short span bias, i.e. the probability of not including the last tokens in a span, (c) missed span bias, i.e. the probability of missing a span entirely, and (d) the ratio of good to uninformative annotators in the crowd. We simulate annotators using the generative model of BSC-seq, drawing annotator labeling probabilities from Dirichlet distributions. By default, Dirichlet parameters corresponding to incorrect answers are 1, those for correct answers are 2.5, and disallowed transitions (O$\rightarrow$I) are close to 0. We then change the parameters of these Dirichlet distributions to obtain the variations described above. We repeat each experiment 25 times, in each case generating 25 documents of 100 tokens each. Figure 1 shows the F1-scores for our tested methods. Where annotator accuracy is high, majority voting and clustering are less accurate than methods that model individual annotator behavior, although the difference decreases as we introduce more errors. Clustering performs better with high short span bias, as density estimation can compensate for short spans but may over-estimate those of the correct length. Among the BSC variants, performance increases with the complexity of the annotator model, from BSC-acc to BSC-seq, suggesting that this richer model can be successfully learned on a small dataset. There are some benefits for the Bayesian approaches, IBCC and BSC-CV, over the similar models, DS and HMM-crowd, respectively, in handling all four types of annotator error. 6.3 Crowdsourced Datasets We use two datasets containing both crowdsourced and gold sequential annotations. The CoNLL 2003 named-entity recognition dataset (Tjong Kim Sang and De Meulder, 2003), NER, contains gold labels for four named entity categories (PER, LOC, ORG, MISC), with crowdsourced labels provided by (Rodrigues et al., 2014). PICO (Nguyen et al., 2017), consists of medical paper abstracts that have been annotated by a crowd to indicate text spans that identify the population enrolled in a clinical trial. Further information about the datasets is shown in Table 1. Note that NER spans are typically much shorter than those in PICO. 6.4 Evaluation Metrics For NER we use the CoNLL 2003 F1-score, which considers only exact span matches to be correct. For PICO, we use the relaxed F1-measure (Nguyen et al., 2017), which counts the matching fractions of spans when computing precision and recall. Since the spans in PICO are longer than those of NER, partial matches may still contain much of the required information. We also compute the cross entropy error (CEE) at the level of tokens to compare the probability estimates produced by aggregation methods, which are useful for decision-making tasks such as active learning. 6.5 Aggregating Crowdsourced Labels In this task, we use the aggregation methods to combine multiple crowdsourced labels and predict the true labels for the same documents. For both datasets, we provide all the crowdsourced labels as input to the aggregation method. In both cases, we split the gold-labelled documents into 50% validation and test sets. For NER, we use the split given by Nguyen et al.  Nguyen et al. (2017), while for PICO, the split was not available so our results are not directly comparable to theirs. We tune the hyperparameters using a validation set. To limit the number of hyperparameters to tune, we optimize only three values for BSC. Hyperparameters of the transition matrix, $\boldsymbol{\gamma}_{j}$, are set to the same value, $\gamma_{0}$, except for disallowed transitions, (O$\rightarrow$I, transitions between types, e.g. I-PER$\rightarrow$I-ORG), which are set to 0.1. For the annotator models (both $\boldsymbol{A}$ and $\boldsymbol{B}$), all values are set to $\alpha_{0}$, except for disallowed transitions, which are set to 0.1, then $\epsilon_{0}$ is added to hyperparameters corresponding to correct annotations (e.g. diagonal entries in a confusion matrix). We use validation set F1-scores to choose values from $[0.1,1,10,100]$, training on a small subset of 250 documents for NER and 500 documents for PICO. The results of this task are shown in Table 2. Although DS and IBCC do not consider sequence information nor the text itself, they both perform well against HMM-crowd on NER, and BSC-CM variants on PICO. The improvement of DS over the results given by Nguyen et al.  Nguyen et al. (2017) may be due to implementation differences. Neither BSC-acc nor BSC-MACE perform strongly, with F1-scores sometimes falling below MV. The annotator models of BSC-CV and BSC-CM are better, although BSC-CM performs worse on PICO. The sequential annotator model of BSC-seq performs strongly, despite having a larger number of parameters to learn. When the text model is removed, BSC-seq-notext performs worse than BSC-seq, suggesting that incorporating even a simple text model provides a valuable boost. Using the predictions from HMM-crowd or BSC-seq to train an LSTM produces a small improvement, but is outperformed by BSC-seq+LSTM. To get a deeper understanding of the key methods, we categorize the errors they make and list the counts for each category in Table 3. All machine learning methods shown reduce the number of spans that were completely missed by majority voting. BSC-seq+LSTM increases the number of exact span matches on NER, but reduces this number substantially on PICO while increasing the number of partial matches and false postives (where no true span was present). This appears to be due to a larger number of split spans, where a ’B’ token is inserted incorrectly inside a span. Therefore, while BSC-seq outperforms the alternatives in terms of F1-score and missing spans, further work may be required to improve the distinction between ’B’ and ’I’ tokens. Table 2 shows a benefit of using the sequential annotator model over CM, CV and acc. To understand how BSC uses the richer model in practice, we plot the learned annotator models for PICO as probabilistic confusion matrices in Figure 2. To enable us to visualize the large number of annotator models, we clustered annotators into five groups by applying K-means to their posterior expected values. In all clusters, BSC-CV has different heights for the diagonal entries for B, I and O, showing that it learns differences in accuracy for each of these label values. BSC-CM has more distinctive clusters and the first, fourth and fifth have off-diagonal values with different heights for the same true label value. The second cluster for BSC-CM appears to encode very weakly informative labelers who usually choose ’O’ regardless of the ground truth. Unlike BSC-CM, BSC-seq improved performance on PICO over BSC-CV. Its confusion matrices are very different depending on the worker’s previous annotation. Each column in the figure shows the confusion matrices corresponding to the same cluster of annotators. The first column, for example, shows annotators with a tendency toward I$\rightarrow$I or O$\rightarrow$O transitions, while the following clusters indicate very different labeling behavior. The model therefore appears able to learn distinct confusion matrices for different workers given previous labels, which supports the use of sequential annotator models. 6.6 Small Data and Active Learning We investigate the performance of the aggregation methods with smaller datasets, and the effectiveness of active learning at improving performance with fewer annotations. Two set-ups were evaluated on NER and PICO: the first tests our methods on random subsamples of crowdsourced data of increasing size; the second starts with a random initial subsample, then uses uncertainty sampling, a well-established active learning heuristic (Settles, 2010), to iteratively select additional crowd labels given posterior label predictions from a model trained on the previous subset. To compute the uncertainty of each document, we take the mean shannon entropy $H$ of the labels, $H(\boldsymbol{t}|\boldsymbol{c}_{\mathrm{current}})/L_{n}$, where $\boldsymbol{c}_{\mathrm{current}}$ is the current set of crowd labels. We used the same random samples for all methods and repeated the experiments ten times with different initializations. Figure 3 plots the F1 score at each iteration of the random sampling and active learning procedures. BSC performs best with smaller datasets, where it may benefit from a Bayesian approach. Uncertainty sampling appears to have a greater improvement over random sampling on NER after around $7000$ labels have been obtained, suggesting that a different strategy could be beneficial while the dataset is very small. On PICO, with its smaller sample sizes, the effect of active learning is only observed with BSC-seq+LSTM. BSC-seq$\rightarrow$LSTM and HMM-crowd$\rightarrow$LSTM are effective on NER with smaller datasets, improving over BSC-seq and HMM-crowd methods that use only a simple independent text model to make predictions for unlabeled data. However, on PICO, they underperform BSC-seq and HMM-crowd respectively. BSC-seq+LSTM accounts for uncertainty in the predictions of the integrated LSTM, enabling it to outperform BSC-seq$\rightarrow$LSTM when active learning aquires more than $10000$ labels. We observe that BSC-seq$\rightarrow$LSTM learns different values for the accuracy of the integrated LSTM depending on the true class label, even with only 1486 tokens labeled by the crowd. 6.7 Prediction with Crowd-Trained LSTMs We compare the LSTM sequence taggers (Lample et al., 2016) trained by HMM-crowd and BSC-seq on test data from NER and PICO. For NER, we use the original CoNLL English test set (Tjong Kim Sang and De Meulder, 2003), while for PICO, we train the aggregators on the $3,649$ documents without gold labels, then evaluate on the gold-labelled test data split used in Section 6.5. The results in Table 4 show that the LSTM trained with BSC-seq predictions outperforms that trained using the outputs of HMM-crowd, the previous state-of-the-art (Nguyen et al., 2017). However, while BSC-seq+LSTM also outperforms HMM-crowd$\rightarrow$LSTM and produces the lowest cross entropy error, its F1-scores are lower than those of BSC-seq$\rightarrow$LSTM. 7 Conclusions Previous work has demonstrated the benefits of modeling annotator reliability when aggregating noisy data, such as crowdsourced labels. We proposed BSC-Seq, a fully Bayesian approach to aggregating sequence labels, which models the effect of label sequences on annotator reliability, and showed how it improves the state-of-the-art, particularly with small datasets. 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Multivariate Uncertainty in Deep Learning Rebecca L. Russell and Christopher Reale This work was carried out with funding from DARPA/MTO (HR0011-16-S-0001.) Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors. The authors are with The Charles Stark Draper Laboratory, Inc., Cambridge, MA 02139 (e-mail: rrussell@draper.com; creale@draper.com). Abstract Deep learning is increasingly used for state estimation problems such as tracking, navigation, and pose estimation. The uncertainties associated with these measurements are typically assumed to be a fixed covariance matrix. For many scenarios this assumption is inaccurate, leading to worse subsequent filtered state estimates. We show how to model multivariate uncertainty for regression problems with neural networks, incorporating both aleatoric and epistemic sources of heteroscedastic uncertainty. We train a deep uncertainty covariance matrix model in two ways: directly using a multivariate Gaussian density loss function, and indirectly using end-to-end training through a Kalman filter. We experimentally show in a visual tracking problem the large impact that accurate multivariate uncertainty quantification can have on Kalman filter estimation for both in-domain and out-of-domain evaluation data. Deep learning, covariance matrices, Kalman filters, neural networks, uncertainty I Introduction Uncertainty quantification is an important challenge for applications of deep learning within systems. Uncertainty that can vary from sample to sample within a data domain is termed heteroscedastic. Though it is frequently neglected, heteroscedasticity is inherent to uncertainty in nearly all sources of natural data. Heteroscedastic uncertainty in deep learning can be modeled from two sources: epistemic uncertainty and aleatoric uncertainty [1]. Epistemic uncertainty reflects uncertainty in the model parameters and has been addressed by recent work to develop fast approximate Bayesian inference for deep learning [2, 3, 4]. Accurate estimation of epistemic uncertainty enables systems to perform more reliably in out-of-domain situations. Aleatoric uncertainty reflects the noise inherent to the data and is irreducible with additional training. Accurate estimation of aleatoric uncertainty enables systems achieve maximum performance. Finally, the uncertainties of predictions of multiple values are often correlated, so it is important to account for the full multivariate uncertainty. Heteroscedastic and correlated multivariate uncertainty is illustrated in Figure 1. Accurate estimates of the uncertainty of a neural network “measurement” (i.e. prediction) enable a down-stream system to better fuse measurements or make decisions based on them. One of the most common examples is a system that relies on a probabilistic filter, such as a Kalman filter [5], to recursively estimate a probability distribution over the system’s state from uncertain measurements and a model of the state’s transition. Kalman filters are optimal estimators when the state transition model is correct and measurement noise is Gaussian and uncorrelated between measurements. They are widely used in variety of navigation and tracking problems. These problems have been impacted by recent work in deep learning, where measurement quantities are directly regressed from raw sensor data such as visual odometry [6], object detection [7], human pose estimation [8], object pose estimation [9], or camera pose estimation [10]. While these measurement predictions are highly optimized, the uncertainty estimation and the final performance of the full state-estimating system is not. Uncertainty and noise in visual measurements is often both heteroscedastic and highly correlated between regressed outputs, both of which can have a significant impact on overall system performance. In this work, we study the quantification of heteroscedastic uncertainty including multivariate correlations for regression problems with the goal of improving overall system performance. II Related Work Heteroscedastic noise is an important topic in the filtering literature. Mehra [11] developed the adaptive Kalman filter, which is able to estimate the process and measurement noise covariance matrices online based on the measurement innovations. The adaptive Kalman filter was demonstrated for navigation by Mohamed and Schwarz [12] and can work well when noise properties vary slowly in time. In contrast, multiple model adaptive estimation [13] uses a bank of filters with different noise properties and dynamically chooses between them, which can work well when there are a small number of regimes with different noise properties [14]. Covariance estimation techniques for specific applications, such as the iterative closest point algorithm [15] and simultaneous localization and mapping [16], have been developed but do not generalize well. Recently, learning approaches, both parametric and non-parametric, have been used model heteroscedastic noise from prior data. Kersting et al. [17] used Gaussian process regression to predict noise variance. Wilson and Ghahramani [18] developed a stochastic process to model covariance matrices. Vega-Brown et al. [19] used the nearest-neighbor algorithm to predict covariances based on previous data in a given feature space. Tallavajhula et al. [20] used non-parametric distribution regression to model sensor noise variance. Hu and Kantor [21] learned parametric models of noise covariance from linear combinations of features. Liu et al. [22] used neural networks to model the measurement noise of sensors. Kendall and Gal [1] showed how to predict the variance of neural network outputs including epistemic uncertainty, but focused on applications with high-dimensional outputs like depth estimation in which it is more practical to ignore the correlations between the uncertainty of different outputs. Several works have also investigated the direct learning of neural network models in probabilistic filters. Haarnoja et al. [23] trained measurement models through a Kalman filter and showed an improvement on two 2D regression problems over using an independently-trained measurement model. Jonschkowski et al. [24] used a differentiable particle filter to learn prediction and measurement models. Coskun et al. [25] learned a motion model and noise models with neglected correlations though a Kalman filter, achieving good results on pose estimation tasks. All of these works demonstrated the practicality of filter-based training, but none attempted to account for epistemic uncertainty or studied the impact of the uncertainty estimation part itself. Additionally, none of these works included methods to learn accurate covariance prediction without doing full filter-based training, which can be slow and delicate. We build on this body of work by showing how to predict multivariate uncertainty from both epistemic and aleatoric sources without neglecting correlations, training either through a Kalman filter or independently from one. III Multivariate Uncertainty Prediction We present two methods for training a neural network to estimate the uncertainty covariance of either its own regressed outputs or those of another measurement system. The first is based on direct training using a Gaussian maximum likelihood loss function (Section III-A) and the second is indirect end-to-end training through a Kalman filter (Section III-B.) These two methods can be either used alone or in conjunction, depending on the exact application and availability of data. For training a neural network to estimate its own uncertainty, we also present a method to approximately incorporate epistemic uncertainty at test time (Section III-C.) III-A Gaussian maximum likelihood training In this first method, we directly learn to predict covariance matrix parameters that describe the distribution of training data labels with respect to a neural network’s outputs. We assume that the uncertainty on the $k$-dimensional output of a model $\bm{f}$ for a given input $\bm{x}$ can be approximated by a multivariate Gaussian distribution $$\displaystyle p\left(\bm{y}\mid\bm{x}\right)=$$ $$\displaystyle\frac{1}{\sqrt{(2\pi)^{k}\left|\bm{\Sigma}(\bm{x})\right|}}\times$$ (1) $$\displaystyle\exp\left[-\frac{1}{2}\left(\bm{y}-\bm{f}(\bm{x})\right)^{\mathsf% {T}}\bm{\Sigma}({\bm{x}})\left(\bm{y}-\bm{f}(\bm{x})\right)\right],$$ where $\bm{y}$ is the label corresponding to $\bm{x}$ and $\bm{\Sigma}({\bm{x}})$ is the covariance matrix model. The negative log likelihood, which we use as our loss function, is then $$\mathcal{L}=\frac{1}{2}\left(\bm{y}-\bm{f}(\bm{x})\right)^{T}\bm{\Sigma}(\bm{x% })^{-1}\left(\bm{y}-\bm{f}(\bm{x})\right)+\frac{1}{2}\ln\left|\bm{\Sigma}(\bm{% x})\right|.$$ (2) This loss function allows us to train both $\bm{f}(\bm{x})$ and $\bm{\Sigma}(\bm{x})$, either simultaneously or separately. Typically, we use a single base model with two heads, one for $\bm{f}$ and one for $\bm{\Sigma}$, and train them simultaneously. The $\bm{\Sigma}$ model should output $k$ values $\bm{s}$, which we use to define the variances along the diagonal $$\Sigma_{ii}=\sigma_{i}^{2}=g_{v}(s_{i})$$ (3) and $k(k-1)/2$ additional values $\bm{r}$, which, along with $\bm{s}$, define the off-diagonal covariances $$\Sigma_{ij}=\rho_{ij}\sigma_{i}\sigma_{j}=g_{\rho}(r_{ij})\sqrt{g_{v}(s_{i})g_% {v}(s_{j})},$$ (4) where $\Sigma_{ij}=\Sigma_{ji}$ for $j<i$. We use the $g_{v}=\exp$ activation for the variances, $\sigma^{2}_{i}$, and $g_{\rho}=\tanh$ activation for the Pearson correlation coefficients, $\rho_{ij}$, to stabilize training and help encourage prediction of valid positive-definite covariance matrices. Additional tricks to provide numerical stability during training and a PyTorch [26] implementation of themodel output formatting and loss function are given in Appendix A. III-B Kalman-filter training Our second method of training a neural network to estimate multivariate uncertainty uses indirect training through a Kalman filter, illustrated in Figure 2. The Kalman filter [5] is a state estimator for linear Gaussian systems that fuses information from measurements and predicted states. The relative contribution of these two sources is determined by their covariance uncertainty estimates. Similarly to Section III-A, we use a neural network to estimate the measurement covariance uncertainty for each measurement, but instead of training with the Eq. 2 loss function, we train using the error of the Kalman state estimate relative to some labels. In practice, we find that an L1 loss (mean absolute error) provides the most stable training. The measurement covariance $\bm{\Sigma}$ enters the Kalman filter in the calculation of the innovation covariance, the sum of $\bm{\Sigma}$ with the covariance of the measurement prediction. The innovation itself is the difference between the measurement and the measurement prediction and directly affects the direction of the state update. The innovation covariance determines the Kalman gain, which modifies and size and direction of the state update. Thus, $\bm{\Sigma}$ directly enters in the estimation of the state in the Kalman filter by means of straightforward linear algebra. Thus, we can backpropagate errors from any part of the state through the Kalman filter to train the model that estimates $\bm{\Sigma}$. In Section IV-B, we detail this training approach for a specific Kalman filter implementation for a visual tracking problem. In some situations, this indirect Kalman-based training has a significant advantage over the direct Gaussian MLE training described in Section III-A, as it means that hidden parts of the state can be used as labels during training, rather than the measurements themselves (for which it may be difficult to obtain true values.) Since the Kalman filter is optimal for Gaussian measurement noise, this method of training should yield equivalent results to the MLE training provided the rest of the Kalman filter assumptions are met. Even if not, in situations where the Kalman filter is the desired end-usage of the neural network, it can be preferable to optimize for the overall end-to-end performance. On the other hand, training through a Kalman filter can be slow or prone to instability for many applications, so the more direct approach is preferable when the appropriate labels are available, at the minimum as a pre-training stage. III-C Incorporation of epistemic uncertainty Epistemic uncertainty, also known as “model uncertainty”, represents uncertainty in the neural network model parameters themselves. Like aleatoric uncertainty, epistemic uncertainty can vary dramatically from measurement-to-measurement. Epistemic uncertainty is a particular concern for neural networks given their many free parameters, and can be large for data that is significantly different from the training set. Thus, for any real-world application of neural network uncertainty estimation, it is critical that it be taken into account. Numerous approaches for Bayesian inference have been developed that allow for the estimation of this uncertainty. The easiest and most practical approach is to use dropout Monte Carlo [3]. This approach trades off accuracy for speed and convenience. Recent Bayesian ensembling approaches [4] driven by the empirical success of ensembling for estimating uncertainty [27] are also promising. If epistemic uncertainty can be estimated from $N$ samples of $f(\bm{x})$, the total predictive covariance estimate should be calculated by $$\begin{split}\displaystyle\bm{\Sigma}_{\textit{pred}}=&\displaystyle\bm{\Sigma% }_{\textit{epistemic}}+\bm{\Sigma}_{\textit{aleatoric}}\\ \displaystyle\approx&\displaystyle\frac{1}{N}\sum_{n=1}^{N}f_{n}(\bm{x})f_{n}(% \bm{x})^{T}\\ &\displaystyle-\frac{1}{N^{2}}\left(\sum_{n=1}^{N}f_{n}(\bm{x})\right)\left(% \sum_{n=1}^{N}f_{n}(\bm{x})\right)^{T}\\ &\displaystyle+\frac{1}{N}\sum_{n=1}^{N}\bm{\Sigma}_{n}(\bm{x}).\end{split}$$ (5) As long as epistemic uncertainty for the training data is small relative to aleatoric uncertainty, this formulation only needs to be used at test time. However, if epistemic uncertainty is significant for data in the training set after training, the predicted $\bm{\Sigma}$ directly from the neural network will incorporate this uncertainty in its prediction, making it challenging to separate. We found it possible to calculate the epistemic uncertainty covariances for the training data set and tune the model covariance prediction to predict the residual from that with Eq. 2 while holding the main model $\bm{f}$ constant. However, this tuning is less stable and our best results were generally achieved by training until the epistemic uncertainty had become small for the training set. IV Experiments We evaluated our methods developed in Section III on a problem that is simple to simulate but also contains many complexities and is a reasonable proxy for many practical applications: 3D object tracking from video data. For this problem, we trained a neural network to regress the $x$-$y$-$z$ position of a predetermined object in a single image, and used a Kalman filter to fuse the individual measurements over the video into a full track. This application allows the neural network to handle the challenging computer vision component of the problem, while the Kalman filter builds in our knowledge of physics, geometry, and statistics. By using simulated data, we were able to carefully evaluate our methods of learning multivariate uncertainty on both in-domain and out-of-domain data. IV-A Simulated 3D visual tracking problem We generated data using Blender [28] to render frames of objects moving through 3D space. Each track was randomly and uniformly initialized on one of the four camera frustrum sides, within a range of depth values from the camera (0.25 to 5 meters.) The track velocity was then generated uniformly at random within the frustum at a range of speeds (from 0.01 m/s to 0.2 m/s.) The tracks had constant velocity, though a more complex kinematic model or process noise could easily be added without changing the experiments significantly. The object orientation was sampled uniformly independently for each frame in order to add an additional source of visual noise. Example tracks rendered with a cube object are shown in Figure 3. The images were downsampled to $64\times 64$ pixels to make the problem more challenging. IV-B Kalman filter for evaluation and training A tracking Kalman filter allows us to both experiment with Kalman filter training and evaluate the quality of the uncertainty prediction methods. An example of a Kalman filter being used on our visual tracking problem is shown in Figure 4, with both the neural network measurement uncertainty $\bm{\Sigma}$ and the Kalman state estimate uncertainty $\mathbf{P}$ at each frame shown. We represent the state for our tracking problem in a Kalman filter by $$\bm{z}=(p_{x},p_{y},p_{z},v_{x},v_{y},v_{z}),$$ (6) the position and velocity of our object in space. Then, our constant-velocity state-transition model is given by $$\mathbf{F}=\left[\begin{matrix}\mathbf{I}_{3}&\Delta t\mathbf{I}_{3}\\ \bm{0}_{3}&\mathbf{I}_{3}\end{matrix}\right]$$ (7) and the observation model by $$\mathbf{H}=\left[\begin{matrix}\bm{\mathbf{I}}_{3}&\bm{0}_{3}\\ \bm{0}_{3}&\bm{0}_{3}\end{matrix}\right],$$ (8) where $\bm{\mathbf{I}_{3}}$ and $\bm{\mathbf{0}_{3}}$ are, respectively, the identity and zero matrices of size 3. Given an observed image $\bm{x}_{t}$ at time step $t$, the updated state estimate is $$\hat{\bm{z}}_{t}=\mathbf{F}\hat{\bm{z}}_{t-1}+\mathbf{K}_{t}\left(\bm{f}(\bm{x% }_{t})-\mathbf{H}\mathbf{F}\hat{\bm{z}}_{t-1}\right)$$ (9) and its covariance is $$\mathbf{P}_{t}=\left(\mathbf{I}-\mathbf{K}_{t}\mathbf{H}\right)\mathbf{F}% \mathbf{P}_{t-1}\mathbf{F}^{\intercal},$$ (10) which depend on the Kalman gain $$\mathbf{K}_{t}=\mathbf{F}\,\mathbf{P}_{t-1}(\mathbf{H}\,\mathbf{F})^{\intercal% }\mathbf{S}_{t}^{-1}$$ (11) and in turn the innovation covariance $$\mathbf{S}_{t}=\mathbf{H}\,\mathbf{F}\,\mathbf{P}_{t-1}(\mathbf{H}\,\mathbf{F}% )^{\intercal}+\mathbf{\Sigma}(\bm{x}_{t}).$$ (12) Again, $\bm{f}(\bm{x}_{t})$ and $\bm{\Sigma}(\bm{x}_{t})$ are our neural network’s predictions of the object’s position and covariance, respectively. During evaluation, the full $\bm{\Sigma}_{\textit{pred}}$ given in Eq. 5 should be used to incorporate epistemic uncertainty in the Kalman filter’s error handling. Like our neural network, our Kalman filter was implemented in PyTorch [26], allowing for the straightforward batching of data over many tracks and automatic differentiation through the full filter. When training through the Kalman filter, we used the mean absolute error over the full Kalman state in Eq. 6 as the loss function. However, a loss based on only a subset or measurement of the state could also be used, for example, to allow for self-supervised training when position labels are unavailable. IV-C Results Our experiments used a ResNet-18 [29] with the final linear layer replaced by a $512\times 512$ linear layer with dropout and our size-3 position and size-6 covariance heads. The model was trained on random batches of image frames using the covariance-predicting loss function given by Eq. 2. To provide a fair comparison between different uncertainty prediction methods, we then froze the position prediction results from the model. Four uncertainty estimation methods were compared for representing the observation noise $\bm{\Sigma}$ in a Kalman filter: 1. Fixed covariance: As a baseline, we calculated the position error covariance over the full dataset. This is the conventional approach to estimating $\bm{\Sigma}$ for a Kalman filter. 2. MLE-learned variance: The neural network covariance estimation head was replaced by a size-3 variance-estimating output and trained to convergence using a simplified version of Eq. 2 assuming no correlation between outputs, equivalent to the loss function given in Ref. [1]. 3. MLE-learned covariance: The original covariance head, tuned with Eq. 2 to convergence. 4. Kalman-learned covariance: The covariance estimation head is replaced by a new size-6 covariance-estimating head. It is trained to convergence by backpropagation of the mean absolute error of the Kalman state through the Kalman filter and back to the neural network. These four methods were evaluated using the track velocity estimation of the Kalman filter on a test set of in-domain track data. The results, shown in Table I, indicate that moving from the fixed covariance to heteroscedastic covariance estimation yields a large improvement in the quality of the filter estimates, even though the measurements themselves are identical. Both learned covariance methods further dramatically improve the results, indicating that accounting for the correlations within the measurements can be very important. The MLE-based and Kalman-based covariance learning methods were generally consistent with each other. The improvement over the baseline fixed covariance method is plotted versus the number of tracked measurements in Figure 5. To test the quality of the uncertainty estimation methods when epistemic uncertainty is significant, we simulated out-of-domain data by randomly jittering the input image color channel, a form of data augmentation not seen during training. The results for the MLE-trained variance and covariance, as well as their break down into aleatoric and epistemic uncertainty, are shown in Table II and Figure 6. The “in-domain covariance” results when just the uncertainty estimation model is trained on the out-of-domain position predictions are added to provide a best-case-scenario point of comparison. When evaluated on out-of-domain data, the performances of the aleatoric-only uncertainty estimates are greatly diminished, and the incorporation of correlation into the estimation no longer seems to help. However, when the epistemic and aleatoric uncertainties are combined, the results are close to the in-domain best-case-scenario and accounting for the correlation in uncertainty again gives a large improvement. These results illustrate how critical the incorporation of epistemic uncertainty in real applications of neural networks is. V Conclusions We have provided two methods for training a neural network to predict its own correlated multivariate uncertainty as well as shown how to incorporate epistemic uncertainty during test time. The choice of which method is best depends on the application and available training data. Our experiments show that these methods yield accurate uncertainty estimates and can dramatically improve the performance of a probabilistic filter that uses them. Significant improvement in filter state estimation came from accounting for both the heteroscedasticity in and correlation between the model outputs uncertainty. 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[29] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 770–778, 2016. Appendix A Covariance Prediction Implementation Here we include details on the covariance matrix formatting from neural network outputs and the implementation of the Eq. 2 loss function. To ensure numerical stability during training, we found it useful to multiply all of the off-diagonal covariances by a constant $(1-\epsilon)$, to force the Pearson correlation coefficients to not get too close to 1. Likewise, to make sure that the $\tanh$ activation functions do not saturate quickly for the correlation coefficients, we multiply them by a value $\alpha<1$. Our PyTorch 1.2 implementation to format the neural network output into a $k$-dimensional prediction and $k\times k$ uncertainty covariance prediction is given by: ⬇ def process_output(output, k, alpha=0.05, eps=1e-3):     """ ␣␣␣␣Format␣neural␣network␣output␣into␣mean␣and␣covariance ␣␣␣␣Args: ␣␣␣␣␣␣␣␣output␣:␣(batch,␣k*(k␣+␣3)/2)␣tensor ␣␣␣␣␣␣␣␣k␣:␣integer␣dimension␣of␣regression␣problem ␣␣␣␣␣␣␣␣alpha␣:␣saturation␣coefficient ␣␣␣␣␣␣␣␣eps␣:␣small␣value␣for␣numerical␣stability ␣␣␣␣"""     assert output.shape[1] == k*(k + 3)//2     mean = output[:, :k]     var = output[:, k:2*k].exp()     var_mat = torch.sqrt(var.unsqueeze(2)*var.unsqueeze(1))     # Build correlation matrix     rhos = (1 - eps)*torch.tanh(alpha*output[:, 2*k:])     rho_mat = torch.ones_like(var_mat)     for i in range(k):         for j in range(i + 1, k):             # Flattened upper diagonal indexing             rho = rhos[:, k*i - i*(i + 3)//2 + j - 1]             rho_mat[:, i, j] = rho             rho_mat[:, j, i] = rho     return mean, rho_mat*var_mat For our loss function, we clamp the determinant of $\bm{\Sigma}$ to a small positive value to ensure the logarithm is always defined. Our implementation of the loss function, which uses the output processing function above, is given by: ⬇ def multivariate_loss(output, target):     """ ␣␣␣␣Calculate␣multivariate␣uncertainty␣loss ␣␣␣␣Args: ␣␣␣␣␣␣␣␣output␣:␣(batch,␣k*(k␣+␣3)/2)␣tensor ␣␣␣␣␣␣␣␣target␣:␣(batch,␣k)␣tensor ␣␣␣␣"""     k = target.shape[1]     mean, covar = process_output(output, k)     err = (mean - target).unsqueeze(-1)     term1 = err * covar.inverse().bmm(err)     term2 = torch.log(covar.det().clamp(min=1e-10))     return torch.mean(term1.sum(1) + term2) / 2

Joint Linear and Nonlinear Computation across Functions for Efficient Privacy-Preserving Neural Network Inference Qiao Zhang,Tao Xiang, Chunsheng Xin, Biwen Chen, and Hongyi Wu Qiao Zhang, Tao Xiang, and Biwen Chen are with College of Computer Science, Chongqing University, Chongqing, 400044, China. E-mail: {qiaozhang, txiang, macrochen}@cqu.edu.cn. Chunsheng Xin is with Department of Electrical and Computer Engineering, Old Dominion University, Norfolk, VA, 23529, USA. E-mail: cxin@odu.edu. Hongyi Wu is with Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ, 85721, USA. E-mail: mhwu@arizona.edu.Manuscript received April 19, 2005; revised August 26, 2015. Abstract While it is encouraging to witness the recent development in privacy-preserving Machine Learning as a Service (MLaaS), there still exists a significant performance gap for its deployment in real-world applications. We observe the state-of-the-art frameworks follow a compute-and-share principle for every function output where the summing in linear functions, which is the last of two steps for function output, involves all rotations (which is the most expensive HE operation), and the multiplexing in nonlinear functions, which is also the last of two steps for function output, introduces noticeable communication rounds. Therefore, we challenge the conventional compute-and-share logic and introduce the first joint linear and nonlinear computation across functions that features by 1) the PHE triplet for computing the nonlinear function, with which the multiplexing is eliminated; 2) the matrix encoding to calculate the linear function, with which all rotations for summing is removed; and 3) the network adaptation to reassemble the model structure, with which the joint computation module is utilized as much as possible. The boosted efficiency is verified by the numerical complexity, and the experiments demonstrate up to 13$\times$ speedup for various functions used in the state-of-the-art models and up to 5$\times$ speedup over mainstream neural networks. Index Terms: Machine Learning as a service; Privacy-preserving machine learning; Cryptographic inference; Joint computation. 1 Introduction Deep Learning (DL) has dramatically evolved in the recent decade [1, 2, 3, 4] and been successfully deployed in many applications such as image classification [5], voice recognition [6] and financial evaluation [7]. Due to the need for massive training data and adequate computation resources [8], it is often impractical for an end user (the client $\mathcal{C}$) to train her DL model. To this end, the Machine Learning as a Service (MLaaS) emerges as a feasible alternative where a server $\mathcal{S}$ in the cloud owns a neural network that is well trained on plenty of data, and the client uploads her input to the server to obtain the prediction result. However, noticeable privacy concerns raise in MLaaS since the client must send her private data, which could be sensitive, to the server. In many cases, the client prefers to obtain the prediction without letting other parties, including the cloud server, know her data. In fact, regulations have been enforced to forbid the disclosure of private data, e.g., the Health Insurance Portability and Accountability Act (HIPAA) [9] for medical data and the General Data Protection Regulation (GDPR) [10] for business data. Meanwhile, the cloud server intends to hide the proprietary parameters of its well-trained neural network, and only return the model output in response to the client’s prediction request. Privacy-preserving MLaaS takes both legal and ethical concerns for the client’s private data and the server’s model parameters into account, which aims to ensure that 1) the server learns nothing about the client’s private data and 2) the client learns nothing about the server’s model parameters beyond what can be returned from the network output, e.g., the predicted class. The key challenge in privacy-preserving MLaaS is how to efficiently embed cryptographic primitives into function computation of neural networks, which otherwise may lead to prohibitively high computation complexity and/or degraded prediction accuracy due to large-size circuits and/or function approximations. To achieve usable privacy-preserving MLaaS, a series of recent works have made inspiring progress towards the system efficiency [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. Specifically, the inference speed has gained several orders of magnitude from CryptoNets [11] to the recent frameworks. At a high level, these privacy-preserving frameworks carefully consider and adopt several cryptographic primitives (e.g., the Homomorphic Encryption (HE) [26, 27, 28] and Multi-Party Computation (MPC) techniques [29] (such as Oblivious Transfer (OT) [30], Secret Sharing (SS) [31] and Garbled Circuits (GC) [32, 33])) to compute the linear (e.g., dot product and convolution) and nonlinear (e.g., ReLU) functions, which are repeatedly stacked and act as the building blocks in a neural network. Among them, the mixed-protocol approaches that utilizes HE to compute linear functions while adapting MPC for nonlinear functions show more efficiency advantages for the privacy-preserving MLaaS with two-party computation [12, 13, 17]. For example, CrypTFlow2 [17] has shown significant speedup compared with other state-of-the-art schemes such as GAZELLE [13] and DELPHI [16]. While it is encouraging to witness the recent development in privacy-preserving MLaaS, there still exists a significant performance gap for its deployment in real-world applications. For example, our benchmark has shown that CrypTFlow2 takes 115 seconds and 147 seconds to run the well-known DL networks VGG-19 [3] and ResNet-34 [4] on the Intel(R) Xeon(R) E5-2666 v3 $@$ 2.90GHz CPU (see the detailed experimental settings and results in Sec. 4). It is worth pointing out that the constraints of response time in many practical ML-driven applications (such as speech recognition and wearable health monitoring) are within a few seconds or up to one minute [34, 35]. This performance gap motivates us to further improve the efficiency of privacy-preserving MLaaS especially for the mixed-protocol approaches. We begin with analyzing the computation logic in these state-of-the-art privacy-preserving frameworks. Concretely, the basic logic in all designs is to firstly calculate the output of a function, based on specific cryptographic primitives. That securely computed output is in an “encrypted” form and is then shared between $\mathcal{C}$ and $\mathcal{S}$. The respective share at $\mathcal{C}$ and $\mathcal{S}$ acts as the input of next function. As a neural network consists of stacked linear and nonlinear functions, this compute-and-share logic for each function output is sequentially repeated until the last one. For example, the initial input of CrypTFlow2 is $\mathcal{C}$’s private data, which is encrypted and sent to $\mathcal{S}$. $\mathcal{S}$ conducts HE-based computation for the linear function where the HE addition, multiplication, and rotation (which are three basic operators over encrypted data) are performed between $\mathcal{C}$-encrypted data and $\mathcal{S}$’s model parameters. It produces an encrypted function output which is then shared (in plaintext) between $\mathcal{C}$ and $\mathcal{S}$, and those shares serve as the input of the following OT-based computation for subsequent nonlinear function, whose corresponding output shares act as the input for the next function. This computation mode is repeated in function wise until $\mathcal{C}$ gets the network output. While this compute-and-share mode for each function output is seemly logical and all of the aforementioned works follow this principle, we observe the fact that the summing in linear functions, which is the last of two steps (where the first step is multiplication) to get the function output, involves all rotations (which is the most expensive HE operation), and the multiplexing in nonlinear functions, which is also the last of two steps (where the first step is comparison) to get the function output, introduces noticeable communication rounds (e.g., about one third in OT-based communication rounds111One round is the communication trip from source node to sink node and then from sink node back to source node, and 0.5 round is either communication trip from source node to sink node or the one from sink node to source node. [17]). Therefore we ask the following natural question: Is it possible to efficiently circumvent the compute-and-share logic for the function output such that we can evade expensive r- otations and noticeable communication rounds at the last step of function computation to achieve more efficient privacy-preservi-  ng MLaaS? Our Contributions. In this paper, we give an affirmative answer to this question. To this end, we challenge and break the conventional compute-and-share logic for function output, and propose the new share-in-the-middle logic towards function computation for efficient privacy-preserving MLaaS. In particular, we come up with the first joint linear and nonlinear computation across functions where the expensive rotations and noticeable communication rounds at the last step of function computation are efficiently removed, via a careful utilization of the intermediates during the function computation. The underlying bases of our joint computation across functions feature with the following three novel designs: 1. the PHE triplet for computing the intermediates of nonlinear function, with which the communication cost for multiplexing is totally eliminated. Meanwhile, it is not only offline-oriented (i.e., its generation is completely independent of $\mathcal{C}$’s private input) but also non-interactive (i.e., its generation is fully asynchronous between $\mathcal{C}$ and $\mathcal{S}$), which is much more flexible compared with other offline-but-synchronous generation [12, 16]. 2. the matrix encoding to calculate the intermediates of linear function, with which all rotations for summing is totally removed. For instance, 2048 rotations are needed to calculate one of the convolutions in  ResNet [4] while we save this cost via the proposed matrix encoding. Furthermore, the PHE triplet is integrated with the matrix encoding to form our joint computation block (to be discussed in Sec. 3.2), which shows better efficiency from both numerical and experimental analysis. 3. the network adaptation to reassemble the DL architecture (e.g., adjust the function orders and decompose the function computation to make it integrated with both previous and subsequent functions) such that the recombined functions are more suitable for applying the proposed PHE triplet and matrix encoding, and thus can take more advantages of our proposed joint computation block, which finally contributes to the overall efficiency improvement. The boosted efficiency of our protocol is demonstrated by both the numerical complexity analysis and the experimental results. For example, we achieve up to 13$\times$ speedup for various functions used in the state-of-the-art DL models and up to $5\times$ speedup over mainstream neural networks, compared with the state-of-the-art privacy-preserving frameworks. Furthermore, we lay our work as an initial attempt for the share-in-the-middle computation, which could inspire more works in this new lane for efficient privacy-preserving MLaaS. On the other hand, we also notice two most recent works that demonstrate comparable efficiency as ours. The first one is COINN [36] which features with the well-designed model customization and ciphertext execution, and our computation can be built on top of COINN’s quantized models to gain more speedup. The second one is Cheetah [37] which utilizes the accumulative property of polynomial multiplication and the ciphertex extraction to eliminate rotations and relies on VOLE-style OT to boost the nonlinear computation. While we can also benefit from the optimized OT to compute nonlinear functions, we differentiate our work from Cheetah’s rotation elimination by exploiting the accumulative nature in matrix-vector multiplication where the output can be viewed as the linear combination of all columns in the weight matrix. Furthermore, both COINN and Cheetah are under the compute-and-share logic while we set our work as the first one for share-in-the-middle computation. The rest of the paper is organized as follows. In Sec. 2, we introduce the system setup and the primitives that are adopted in our protocol. Sec. 3 elaborates the design of our joint linear and nonlinear computation. The experimental results are illustrated and discussed in Sec. 4. Finally, we conclude the paper in Sec. 5. 2 Preliminaries Notations. We denote $[k]$ as the set of integers $\{0,1,\dots,k-1\}$. $\lceil\;\rceil$ and $\left\lfloor\;\right\rfloor$ denote the ceiling and flooring function, respectively. Let 1$\{\mathcal{I}\}$ denote the indicator function that is 1 when $\mathcal{I}$ is true and 0 when $\mathcal{I}$ is false. $r$ $\scriptstyle\overset{\$}{\leftarrow}$ $\mathcal{D}$ denotes randomly sampling a component $r$ from a set $\mathcal{D}$. $``\cdots,(i_{1}:i_{2}),\cdots"$ includes indices from $i_{1}$ to $(i_{2}-1)$ for one dimension while $``\cdots,:,\cdots"$ contains all indices in that dimension. $``|"$ represents the concatenation of matrices or numbers. 2.1 System Model We consider the context of cryptographic inference (as shown in Figure 1) where $\mathcal{C}$ holds a private input $\bm{x}$ and $\mathcal{S}$ holds the neural network with proprietary model parameters $\bm{W}$. After the inference, $\mathcal{C}$ learns two pieces of information: the network architecture (such as the number, types and dimensions of involved functions) and the network output222Note that this learnt information is commonly assumed in the state-of-the-art frameworks such as MiniONN [12] and Cheetah [37]., while $\mathcal{S}$ learns nothing. The neural network processes $\bm{x}$ through a sequence of linear and nonlinear functions to finally classify $\bm{x}$ into one of the potential classes. Specifically, we target at the widely-applied Convolutional Neural Network (CNN) and describe its included functions as follows. Convolution (Conv). The Conv operates between a three-dimension input $\bm{a}\in\mathcal{R}^{c_{i}\times{h_{i}}\times{w_{i}}}$ (the $\mathcal{R}$ is defined in Sec. 2.3.1) and the kernel $\textbf{K}\in\mathcal{R}^{c_{o}\times{c_{i}}\times{f_{h}}\times{f_{w}}}$ with a stride $s\in\mathbb{N}^{+}$ where $c_{i}$ and $c_{o}$ are the number of input and output channels, $h_{i}$ and $w_{i}$ are the height and width of each two-dimension input channel, and $f_{h}$ and $f_{w}$ are the height and width (which are always equal) of each two-dimension filter of K. The output is a three-dimension matrix $\bm{y}=\textsf{Conv}(\bm{a},\textbf{K})\in\mathcal{R}^{c_{o}\times{h_{i}^{\prime}}\times{w_{i}^{\prime}}}$ where $h_{i}^{\prime}=\left\lceil\frac{h_{i}}{s}\right\rceil$ and $w_{i}^{\prime}=\left\lceil\frac{w_{i}}{s}\right\rceil$ are the height and width of the two-dimension output channel. Such strided convolution is mathematically expressed as $$\bm{y}_{\alpha,\beta,\gamma}=\sum_{\lambda\in{[c_{i}]}}\sum_{\ell_{1},\ell_{2}\in\bm{\delta}}\bm{a}_{\lambda,s\beta+\ell_{1},s\gamma+\ell_{2}}\textbf{K}_{\alpha,\lambda,\ell_{1}+\Delta,\ell_{2}+\Delta}$$ where $\bm{\delta}=\mathbb{Z}\cap[-\frac{f_{h}-1}{2},\frac{f_{h}-1}{2}]$, $\Delta=\frac{f_{h}-1}{2}$ and $\bm{a}_{\lambda,s\beta+\ell_{1},s\gamma+\ell_{2}}=0$ if $s\beta+\ell_{1}<0$ or $s\gamma+\ell_{2}<0$. The summing process in the cryptographic inference inevitably introduces a series of expensive rotations which limits the inference efficiency, a carefully designed matrix encoding is proposed in this work, together with the joint computation with the nonlinear function, which efficiently removes all the expensive rotations (see Sec. 3). Dot Product (Dot). The input to the dot product is a $n_{i}$-sized vector $\bm{a}\in\mathcal{R}^{n_{i}}$ and the output is the $n_{o}$-sized vector $\bm{y}=\textsf{Dot}(\textbf{W},\bm{a})\in\mathcal{R}^{n_{o}}$ where $$\bm{y}_{j}=\sum_{\lambda\in{[n_{i}]}}\textbf{W}_{j,\lambda}\bm{a}_{\lambda}$$ (1) and $\textbf{W}\in\mathcal{R}^{n_{o}\times{n_{i}}}$. As the dot product and convolution are both intrinsically weighted sums, we treat dot product similarly with convolution in the cryptographic inference where the proposed matrix encoding for Dot, along with the joint computation with the nonlinear function (see Sec. 3), efficiently eliminates all the expensive rotations. Batch Normalization (BN). In the neural network inference, the BN scales and shifts each two-dimension input channel by a constant $\bm{\mu}_{\beta}\in\mathcal{R}$ and $\bm{\theta}_{\beta}\in\mathcal{R}$, respectively, given the input $\bm{a}\in\mathcal{R}^{c_{i}\times{h_{i}}\times{w_{i}}}$. This operation is mathematically expressed as $$\bm{a}_{\beta,:,:}\leftarrow\bm{\mu}_{\beta}\bm{a}_{\beta,:,:}+\bm{\theta}_{\beta}$$ As the BN always follows behind the convolution, we integrate it with the convolution (see Sec. 3), together with our joint computation with nonlinear function, to improve the efficiency for calculating the Conv+BN. ReLU. For a value $a\in\mathcal{R}$, the ReLU is calculated as $\textsf{ReLU}(a)=a\cdot\textbf{1}\{a\}$. In the context of cryptographic inference, the multiplication between $a$ and $\textbf{1}\{a\}$ (denoted by $\textsf{DReLU}(a)$ thereafter) involves the multiplexing with noticeable communication rounds (e.g., about one third for OT-based communication rounds in [17]). In this work, we combine the ReLU computation with Conv (see Sec. 3) to totally relieve that cost. Mean Pooling (MeanPool). Given the input $\bm{a}\in\mathcal{R}^{c_{i}\times{h_{i}}\times{w_{i}}}$, the MeanPool sums and averages the components in each $s_{\textsf{mp}}\times{s_{\textsf{mp}}}$ pooling window where $s_{\textsf{mp}}\in\mathbb{N}^{+}$, and returns all mean values as output. In this way, the output size becomes $c_{i}\times{\left\lceil\frac{h_{i}}{s_{\textsf{mp}}}\right\rceil}\times{\left\lceil\frac{w_{i}}{s_{\textsf{mp}}}\right\rceil}$. Since the MeanPool always appears between Conv and ReLU, we decouple its summing and averaging to ReLU and Conv (see Sec. 3) to utilize our joint computation block for a more efficient cryptographic inference. Max Pooling (MaxPool). The MaxPool works similarly as MeanPool except that the returned value is the maximum in each $s_{\textsf{mxp}}\times{s_{\textsf{mxp}}}$ pooling window where $s_{\textsf{mxp}}\in\mathbb{N}^{+}$. The state-of-the-art framework deals with MaxPool in each $s_{\textsf{mxp}}\times{s_{\textsf{mxp}}}$ pooling window via $(s_{\textsf{mxp}}^{2}-1)$ sequential comparisons [17] and we further parallel this process by considering the independent nature of two comparisons among four values (see Sec. 3). ArgMax. The ArgMax is usually the last function in a neural network to classify $\mathcal{C}$’s input to a potential class. ArgMax operates similarly as the MaxPool except that its pooling window embraces all values of the input and it finally returns the index of the maximum. The calculation of ArgMax is traditionally in a sequential manner [17], which is optimized in a similar way as MaxPool. 2.2 Threat Model Our protocol involves two parties namely the client $\mathcal{C}$ and the server $\mathcal{S}$, and we follow in this work the static semi-honest security definition [38] for secure two-party computation. Static Semi-Honest Security. There are two parties denoted by $\mathcal{C}$ and $\mathcal{S}$. Let $f_{\mathcal{C}}(\bm{x},\bm{W})$ and $f_{\mathcal{S}}(\bm{x},\bm{W}))$ be the output for $\mathcal{C}$ and $\mathcal{S}$ in the ideal functionality $\mathcal{F}$, respectively, while $f(\bm{x},\bm{W})=(f_{\mathcal{C}}(\bm{x},\bm{W}),f_{\mathcal{S}}(\bm{x},\bm{W})))$ be the joint output. Let the view of $\mathcal{C}$ and $\mathcal{S}$ during an execution of $\Pi$ on inputs $(\bm{x},\bm{W})$ be $\textsf{view}_{\mathcal{C}}^{\Pi}(\bm{x},\bm{W})$ and $\textsf{view}_{\mathcal{S}}^{\Pi}(\bm{x},\bm{W})$ that consist of the private input $\bm{x}$ and model parameters $\bm{W}$, as well as the contents of internal random tap and the messages received at $\mathcal{C}$ and $\mathcal{S}$ during the execution, respectively. Similarly, $\textsf{output}_{\mathcal{C}}^{\Pi}(\bm{x},\bm{W})$ and $\textsf{output}_{\mathcal{S}}^{\Pi}(\bm{x},\bm{W})$ are the output of $\mathcal{C}$ and $\mathcal{S}$ during an execution of $\Pi$ on inputs $(\bm{x},\bm{W})$ and can be computed from the $\Pi$’s view. The joint output of both parties is $\textsf{output}^{\Pi}(\bm{x},\bm{W})=(\textsf{output}_{\mathcal{C}}^{\Pi}(\bm{x},\bm{W}),\textsf{output}_{\mathcal{S}}^{\Pi}(\bm{x},\bm{W}))$. Definition 1. A protocol $\Pi$ securely computes $\mathcal{F}$ against static semi-honest adversaries $\mathcal{A}$ if there exist probabilistic polynomial-time (PPT) algorithms $\textsf{Sim}_{\mathcal{C}}$ and $\textsf{Sim}_{\mathcal{S}}$ such that $$\{\textsf{Sim}_{\mathcal{C}}(\bm{x},f_{\mathcal{C}}(\bm{x},\bm{W})),f(\bm{x},\bm{W})\}\overset{\textsf{c}}{\equiv}\{\textsf{view}_{\mathcal{C}}^{\Pi}(\bm{x},\bm{W}),\textsf{output}^{\Pi}(\bm{x},\bm{W})\},$$ $$\{\textsf{Sim}_{\mathcal{S}}(\bm{W},f_{\mathcal{S}}(\bm{x},\bm{W})),f(\bm{x},\bm{W})\}\overset{\textsf{c}}{\equiv}\{\textsf{view}_{\mathcal{S}}^{\Pi}(\bm{x},\bm{W}),\textsf{output}^{\Pi}(\bm{x},\bm{W})\}.$$ The main ideal functionalities for our protocol are $\mathcal{F}_{\textsf{DReLU}}$ and $\mathcal{F}_{\textsf{ReConv}}$ as shown in Algorithm 3 and Algorithm 4, and we prove in Sec. 3.2.5 that our designed protocol $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$ presented in Algorithm 2 securely realizes $\mathcal{F}_{\textsf{ReConv}}$ in the $\{\mathcal{F}_{\textsf{DReLU}}\}$-hybrid model, in the presence of semi-honest adversaries. Furthermore, we do not defend against attacks, such as the API attacks [39, 40], which are based purely on the inference results and other orthogonal techniques, such as differential privacy [41, 42], can be utilized to provide more privacy guarantee [37]. 2.3 Cryptographic Primitives 2.3.1 Packed Homomorphic Encryption (PHE). PHE is a primitive that supports various vector operations over encrypted data without decryption, and generates an encrypted result which matches the corresponding operations on plaintext [26, 27, 28]. Specifically, given a vector $\bm{x}=(x_{0},x_{1},\dots,x_{n-1})\in\mathcal{R}^{n}$ over a ring $\mathcal{R}=\mathbb{Z}_{p}=\mathbb{Z}\cap(-p/2,p/2]$, it is encrypted into a ciphertext ct${}_{\bm{x}}^{\textsf{pk}_{\mathcal{C}}}=$ Enc${}^{\textsf{pk}_{\mathcal{C}}}(\bm{x})$ where pk${}_{\mathcal{C}}$ denotes the public key of the client $\mathcal{C}$ and we thereafter use ct${}_{\bm{x}}^{{\mathcal{C}}}=$ Enc${}^{{\mathcal{C}}}(\bm{x})$ for brevity. Similar logic is applied for the ciphertext ct${}_{\bm{x}}^{{\mathcal{S}}}=$ Enc${}^{{\mathcal{S}}}(\bm{x})$ where $\bm{x}$ is encrypted by the public key of the server $\mathcal{S}$. The correctness of PHE is firstly guaranteed by a decryption process such that $\bm{x}=$ Dec${}^{\textsf{sk}_{\star}}(\textsf{ct}_{\bm{x}}^{\star})$ where ${\star}\in\{\mathcal{C},\mathcal{S}\}$, $\textsf{sk}_{\star}$ is the secret key of either $\mathcal{C}$ or $\mathcal{S}$, and Dec${}^{\star}()$ is used henceforth to simplify the notation for decryption. Second, the PHE system can securely evaluate an arithmetic circuit consisting of addition and multiplication gates by leveraging the following operations where $\bm{u}=(u_{0},\dots,u_{n-1})$ (resp. $\bm{v}=(v_{0},\dots,v_{n-1})$) $\in\mathcal{R}^{n}$, and $``+"$ (resp. $``-"$ and $``\cdot"$) is the component-wise addition (resp. subtraction and multiplication). • Homomorphic addition $(\oplus)$ and subtraction $(\ominus)$: Dec${}^{\star}(\textsf{ct}_{\bm{u}}^{\star}$ $\oplus$ $\textsf{ct}_{\bm{v}}^{\star})=\bm{u}+\bm{v}$ (resp. Dec${}^{\star}(\textsf{ct}_{\bm{u}}^{\star}\oplus{\bm{v}})=\bm{u}+\bm{v}$) and Dec${}^{\star}(\textsf{ct}_{\bm{u}}^{\star}\ominus\textsf{ct}_{\bm{v}}^{\star})=\bm{u}-\bm{v}$ (resp. Dec${}^{\star}(\textsf{ct}_{\bm{u}}^{\star}\ominus{\bm{v}})=\bm{u}-\bm{v}$). • Homomorphic multiplication $(\otimes)$: Dec${}^{\star}(\textsf{ct}_{\bm{u}}^{\star}\otimes\textsf{ct}_{\bm{v}}^{\star})=\bm{u}\cdot\bm{v}$ (resp. Dec${}^{\star}(\textsf{ct}_{\bm{u}}^{\star}\otimes{\bm{v}})=\bm{u}\cdot\bm{v}$). • Homomorphic rotation (Rot): Dec${}^{\star}(\textsf{Rot}(\textsf{ct}_{\bm{u}}^{\star};\ell))=\bm{u}_{\ell}$ where $\bm{u}_{\ell}=\rho(\bm{u};\ell)=(u_{\ell},\dots,u_{n-1},u_{0},\dots,u_{\ell-1}$ $)$ and $\ell\in[n]$. Note that a rotation by $(-\ell)$ is the same as a rotation by $(n-\ell)$. Given the above four operations, the runtime complexity of Rot is significantly larger than that of $\oplus$, $\ominus$ and $\otimes$ [37, 24]. Meanwhile, the runtime complexity of homomorphic multiplication between two ciphertext namely $\textsf{ct}_{\bm{u}}^{\star}$ $\otimes$ $\textsf{ct}_{\bm{v}}^{\star}$ is also larger than that of homomorphic multiplication between one ciphertext and one plaintext namely $\textsf{ct}_{\bm{u}}^{\star}$ $\otimes$ ${\bm{v}}$. While the neural network is composed of a stack of linear and nonlinear functions, the PHE is preferably utilized to calculate the linear functions such as Dot and Conv in the privacy-preserving neural networks, due to its intrinsic support for linear arithmetic computation [25]. Such computation process involves calling aforementioned PHE operations to get the encrypted output of linear function, which is then shared between $\mathcal{C}$ and $\mathcal{S}$ to serve as the input for computing next function, and this compute-and-share logic for the output of linear function is undoubtedly adopted in the state-of-the-art privacy-preserving frameworks [13, 16]. Unfortunately, this compute-and-share process requires a series of expensive Rot due to the needed summing in linear functions [13], and the PHE based linear calculation dominates the overall cost of neural model computation [18]. Therefore it makes the rotation elimination a significant and challenging task to achieve the next-stage boost towards computation efficiency. In this work, we jointly consider the computation for both linear and nonlinear functions, which enables us to feed the specifically- formed PHE intermediates from the linear function to the computation for the subsequent nonlinear function. Such joint design, rather than traditionally separate computation for each function output, contributes to efficiently eliminate all Rot333Our design also involves no ciphertext-ciphertext but faster ciphertext-plaintext multiplication. in the entire process of cryptographic inference. 2.3.2 Oblivious Transfer (OT). In this work, we rely on OT to compute the comparison-based nonlinear functions such as ReLU, MaxPool and ArgMax. Specifically, in the 1-out-of-$k$ OT, $(\mathop{}_{1}^{k})$-OT${}_{\ell}$, the sender inputs $k$ $\ell$-length strings $m_{0},\dots,m_{k-1}$, and the receiver’s input is a value $i\in[k]$. The receiver gets $m_{i}$ after the OT functionality while the sender obtains nothing [43]. The state-of-the-art computation for a comparison-based nonlinear function mainly involves the OT-based DReLU functionality along with the OT-based multiplexer (Mux) to get the corresponding shares of such function output, which act as the input to compute next function [17]. While such process has relatively light computation and transmission load, the Mux accounts for nearly one third of the total communication rounds [17], which hinders the further optimization towards overall efficiency. In this paper, we totally remove the required Mux by a joint computation for both nonlinear and subsequent (linear) functions. Instead of getting the output shares with Mux, our design directly utilizes the nonlinear intermediates (e.g., DReLU) to generate the input of subsequent function. 2.3.3 Secret Sharing (SS). The additive SS over the ring $\mathcal{R}=\mathbb{Z}_{p}$ is adopted in this paper to form the output shares of linear or nonlinear functions, which serve as the input to compute next function. Here $p$ is either the plaintext modulus determined by the PHE for the linear functions or two for the boolean computation of nonlinear functions. Specifically, the SS algorithm $\textsf{Shr}(x)=(\langle x\rangle^{\mathcal{C}}_{p},\langle x\rangle^{\mathcal{S}}_{p})$ inputs an element $x$ in $\mathbb{Z}_{p}$ and generates shares of $x$, $\langle x\rangle^{\mathcal{C}}_{p}$ and $\langle x\rangle^{\mathcal{S}}_{p}$, over $\mathbb{Z}_{p}$. The shares are randomly sampled in $\mathbb{Z}_{p}$ such that $\langle x\rangle^{\mathcal{C}}_{p}\boxplus\langle x\rangle^{\mathcal{S}}_{p}=x$ where $``\boxplus"$ is the addition over $\mathbb{Z}_{p}$ and we similarly denote $``\boxminus"$ and $``\boxdot"$ as the subtraction and multiplication over $\mathbb{Z}_{p}$. Furthermore, we denote the reconstruction of $x$ as $\textsf{Rec}(\langle x\rangle^{\mathcal{C}}_{p},\langle x\rangle^{\mathcal{S}}_{p})=\langle x\rangle^{\mathcal{C}}_{p}\boxplus\langle x\rangle^{\mathcal{S}}_{p}=x$. Conventional $\textsf{Shr}(x)$ functionality in the state-of-the-art frameworks needs to exactly obtain the encrypted $x$ with massive amount of either computation-intensive Rot for a linear function or round-intensive Mux for the nonlinear function [17]. By carefully sharing specific intermediates in the process of computing adjacent functions, we efficiently eliminate all Rot and Mux in the cryptographic inference. 2.4 Linear and Nonlinear Transformations Deevashwer Rathee et al. [17] have recently proposed efficient computation for linear and nonlinear functions $\Pi_{\textsf{ReLU+Conv}}^{\textrm{ring},p}$ as shown in Algorithm 1, which contributes to high-performance neural network inference. Specifically, there are mainly two parts in $\Pi_{\textsf{ReLU+Conv}}^{\textrm{ring},p}$ where the first one is to get the shares of ReLU based on the shares of previous function (as shown in the gray block of Algorithm 1), and the second one is to subsequently get the shares of Conv based on the shares of ReLU (as shown in the blue block of Algorithm 1). Here, the ReLU and Conv are analyzed together as they are always adjacent and serve as a repeatable module to form modern neural networks [3, 4]. Among the state-of-the-art frameworks [12, 13, 14, 15, 16, 17, 37, 36], the above two parts are independently optimized such that the efficiency-unfriendly Mux and Rot are extensively involved to produce shares of one function, which serve as the input to compute subsequent function. While it is definitely logical to address the efficiency optimization in function wise, we show in the following section that our design enables efficient elimination of Mux and Rot by jointly computing the ReLU and Conv. Generally, specific intermediates of ReLU are utilized to directly compute Conv and vice versa. Such joint computation produces a new building block $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$ that combines the computation process of stacked functions and achieves more efficient privacy-preserving neural network inference. 3 System Description 3.1 Overview Figure 2 illustrates the overview of our proposed framework for cryptographic inference. Specifically, a neural network contains a stack of linear and nonlinear functions. As for a CNN, we group these functions into six blocks as: 1) the Conv block for $\mathcal{C}$’s input (i.e., the first function in the CNN); 2) the $\textsf{ReLU}+\textsf{MaxPool}+\textsf{Conv}$ block; 3) the $\textsf{ReLU}+\textsf{Conv}+\textsf{BN}$ block; 4) the $\textsf{ReLU}+\textsf{Conv}$ block; 5) the $\textsf{ReLU}+\textsf{FC}$ block; and 6) the ArgMax block for network’s output (i.e., the last function in the CNN). Such six blocks are able to construct many state-of-the-art CNN architectures such as VGG [3] and ResNet [4]. Compared with the function-wise optimization in existing schemes, which inevitably involves noticeable amount of expensive Rot and round-intensive Mux, we are motivated to jointly consider the computation process of functions in each block, aiming to efficiently eliminate the Rot and Mux, and thus to boost the performance of privacy-preserving inference. As for the $\textsf{ReLU}+\textsf{Conv}$ block, we design the joint computation module $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$ (to be discussed in Sec. 3.2), which serves as the core component in our framework. The $\textsf{ReLU}+\textsf{MaxPool}+\textsf{Conv}$ block is converted into the combination of $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$ and our optimized module for MaxPool, $\Pi_{\textsf{tMaxPool}}^{\textrm{ring},p}$ (to be explained in Sec. 3.3). For neural network inference, the BN in $\textsf{ReLU}+\textsf{Conv}+\textsf{BN}$ block is integrated into the Conv, and it enables us to compute such transformed block by $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$. The calculation towards Conv block for $\mathcal{C}$’s input and $\textsf{ReLU}+\textsf{FC}$ block works similarly with $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$. Finally, the ArgMax block for network’s output is realized by our optimized module for ArgMax, $\Pi_{\textsf{tArgMax}}^{\textrm{ring},p}$ (to be marrated in Sec. 3.3). In the following, we first detail the construction of $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$ in Sec. 3.2 and then elaborate other optimizations in Sec. 3.3. Additionally, we follow the truncation strategy over $\mathbb{Z}_{p}$ [17] to deal with the floating-point computation. 3.2 The Joint Computation Module: $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$ For a lucid illustration, we begin with revisiting traditional calculation for the $\textsf{ReLU}+\textsf{Conv}$ block, where the PHE Triplet is proposed in Sec. 3.2.1 for Mux-free ReLU and our Matrix Encoding is described in Sec. 3.2.2 for Rot-free Conv. These two components form our joint computation module $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$. Different from the compute-and-share process in state-of-the-art function-wise computation, which exhaustively obtains encrypted output of current function and then shares that result to serve as the input for next function, $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$ shares specifically formed intermediates to finish computing $\textsf{ReLU}+\textsf{Conv}$ and we refer this methodology as share-in-the-middle logic (SIM). SIM enables efficient elimination of Rot and Mux, which are expensive in the state-of-the-art solutions. The overall process of $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$ is summarized in Sec. 3.2.3, which is followed by the corresponding complexity analysis in Sec. 3.2.4. The security of $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$ is justified in Sec. 3.2.5. 3.2.1 PHE Triplet Generation in ReLU. Recall in Algorithm 1 the traditional computation for $\textsf{ReLU}+\textsf{Conv}$, $\Pi_{\textsf{Relu+Conv}}^{\textrm{ring},p}$, where the input is the shares of previous function, $\langle{\bm{a}}\rangle_{p}^{\mathcal{C}}$ and $\langle{\bm{a}}\rangle_{p}^{\mathcal{S}}$. The conventional logic to compute ReLU includes the OT-based DReLU and subsequent Mux to obtain the output shares namely $\langle{\bar{\bm{a}}}\rangle_{p}^{\mathcal{C}}$ and $\langle{\bar{\bm{a}}}\rangle_{p}^{\mathcal{S}}$, which serve as the input of Conv (see the gray block in Algorithm 1). Since almost one third of the communication rounds are consumed by Mux to obtain the output shares of ReLU [17], we utilize the intermediates from DReLU to directly enable the subsequent Conv. This design efficiently removes the Mux in the process of $\textsf{ReLU}+\textsf{Conv}$. Specifically, given the shares of $\textsf{DReLU}(\bm{a})$, $\langle{\hat{\bm{a}}}\rangle_{2}^{\mathcal{C}}$ and $\langle{\hat{\bm{a}}}\rangle_{2}^{\mathcal{S}}$, the ReLU of $\bm{a}$ is computed according to $$\displaystyle\textsf{ReLU}(\bm{a})=\textsf{DReLU}(\bm{a})\boxdot\bm{a}$$ (2) $$\displaystyle=\underbrace{\{\langle{\hat{\bm{a}}}\rangle_{2}^{\mathcal{C}}\boxplus\langle{\hat{\bm{a}}}\rangle_{2}^{\mathcal{S}}\boxminus(\bm{2}\boxdot\langle{\hat{\bm{a}}}\rangle_{2}^{\mathcal{C}}\boxdot\langle{\hat{\bm{a}}}\rangle_{2}^{\mathcal{S}})\}}_{\leavevmode\hbox to8.5pt{\vbox to8.5pt{\pgfpicture\makeatletter\raise-1.9964pt\hbox{\hskip 4.25195pt\lower-4.25195pt\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@moveto{4.25195pt}{0.0pt}\pgfsys@curveto{4.25195pt}{2.34831pt}{2.34831pt}{4.25195pt}{0.0pt}{4.25195pt}\pgfsys@curveto{-2.34831pt}{4.25195pt}{-4.25195pt}{2.34831pt}{-4.25195pt}{0.0pt}\pgfsys@curveto{-4.25195pt}{-2.34831pt}{-2.34831pt}{-4.25195pt}{0.0pt}{-4.25195pt}\pgfsys@curveto{2.34831pt}{-4.25195pt}{4.25195pt}{-2.34831pt}{4.25195pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{-2.25555pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{\scriptsize{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}1}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}:\;\textrm{each}\;\textrm{operation}\;\textrm{is}\;\textrm{over}\;\mathbb{Z}_{p}^{c_{i}\times{h_{i}}\times{w_{i}}}}\boxdot(\langle{\bm{a}}\rangle_{p}^{\mathcal{C}}\boxplus\langle{\bm{a}}\rangle_{p}^{\mathcal{S}})$$ (3) $$\displaystyle=\underbrace{(\langle{\bm{a}}\rangle_{p}^{\mathcal{C}}\boxdot\langle{\hat{\bm{a}}}\rangle_{2}^{\mathcal{C}}\boxminus{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\bm{r}^{\mathcal{C}}}})}_{\bm{h}_{1}}\boxplus\{\underbrace{\langle{\bm{a}}\rangle_{p}^{\mathcal{C}}\boxdot\{\bm{1}\boxminus(\bm{2}\boxdot\langle{\hat{\bm{a}}}\rangle_{2}^{\mathcal{C}})\}}_{\bm{h}_{2}}\boxdot\underbrace{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\langle{\hat{\bm{a}}}\rangle_{2}^{\mathcal{S}}}}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{h}_{3}}}\}\boxplus$$ $$\displaystyle\{\underbrace{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\langle{\bm{a}}\rangle_{p}^{\mathcal{S}}\boxdot\{\bm{1}\boxminus(\bm{2}\boxdot\langle{\hat{\bm{a}}}\rangle_{2}^{\mathcal{S}})\}}}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{h}_{4}}}\boxdot\underbrace{\langle{\hat{\bm{a}}}\rangle_{2}^{\mathcal{C}}}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{h}_{5}}}\}\boxplus(\underbrace{\langle{\bm{a}}\rangle_{p}^{\mathcal{S}}\boxdot\langle{\hat{\bm{a}}}\rangle_{2}^{\mathcal{S}}}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{h}_{6}}})\boxplus\underbrace{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{r}^{\mathcal{C}}}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{h}_{7}}}.$$ (4) Here $``\boxdot"$, $``\boxplus"$ and $``\boxminus"$ are element-wise plaintext multiplication, addition and subtraction, respectively. The item 1 converts the shares of $\textsf{DReLU}(\bm{a})$ from $\mathbb{Z}_{2}^{c_{i}\times{h_{i}}\times{w_{i}}}$ to $\mathbb{Z}_{p}^{c_{i}\times{h_{i}}\times{w_{i}}}$. By rearranging and modifying the terms in Eq. (3), we finally get five to-be-added parts in Eq. (4) to obtain ReLU$(\bm{a})$: 1) the $\mathcal{C}$-computed $\bm{h}_{1}$ where $\bm{r}^{\mathcal{C}}{\scriptstyle\overset{\$}{\leftarrow}}\,\mathbb{Z}_{p}^{c_{i}\times{h_{i}}\times{w_{i}}}$; 2) the multiplication between $\mathcal{C}$-computed $\bm{h}_{2}$ and $\bm{h}_{3}$ namely $\langle{\hat{\bm{a}}}\rangle_{2}^{\mathcal{S}}$; 3) the multiplication between $\mathcal{S}$-computed $\bm{h}_{4}$ and $\bm{h}_{5}$ namely $\langle{\hat{\bm{a}}}\rangle_{2}^{\mathcal{C}}$; 4) the $\mathcal{S}$-computed $\bm{h}_{6}$; and 5) the $\bm{h}_{7}$ namely $\bm{r}^{\mathcal{C}}$. Given the input-independent nature of shares $\langle{\hat{\bm{a}}}\rangle_{2}^{\mathcal{S}}$, $\langle{{\bm{a}}}\rangle_{p}^{\mathcal{S}}$ and $\bm{r}^{\mathcal{C}}$ which are pregenerated by either $\mathcal{S}$ or $\mathcal{C}$ before the inference process, the $\bm{h}_{3}$, $\bm{h}_{4}$, $\bm{h}_{6}$ and $\bm{h}_{7}$ are pre-determined. Meanwhile, $\mathcal{C}$ would encrypt its share of $\textsf{ReLU}(\bm{a})$ and send it to $\mathcal{S}$ for obtaining encrypted $\textsf{ReLU}(\bm{a})$ and thus enabling the subsequent HE-based Conv. By carefully encrypting the pre-determined terms, without introducing Mux, we are able to make $\mathcal{S}$ obtain the encrypted input for Conv. Therefore, we define the PHE triplet: $$\{\textsf{ct}^{\mathcal{S}}_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{h}_{3}},\textsf{ct}^{\mathcal{S}}_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{h}_{4}},\textsf{ct}^{\mathcal{C}}_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{h}_{7}}\}$$ where the first two terms are generated by $\mathcal{S}$ and sent to $\mathcal{C}$ while the last term is generated by $\mathcal{C}$ and sent to $\mathcal{S}$. Meanwhile, it is worth pointing out that each component in the above triplet is non-interactively formed and sent to either $\mathcal{C}$ or $\mathcal{S}$ offline. As such, $\mathcal{S}$ precomputes $$\textsf{ct}_{\bm{h}_{8}}^{\mathcal{C}}=\bm{h}_{6}\oplus{\textsf{ct}^{\mathcal{C}}_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{h}_{7}}}$$ (5) while $\mathcal{C}$ obtains $$\textsf{ct}_{\bm{h}_{9}}^{\mathcal{S}}=\bm{h}_{1}\oplus{(\bm{h}_{2}\otimes\textsf{ct}^{\mathcal{S}}_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{h}_{3}})\oplus({\textsf{ct}^{\mathcal{S}}_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{h}_{4}}\otimes{\bm{h}_{5}}})}$$ (6) and sends it to $\mathcal{S}$ right after the DReLU computation. $\mathcal{S}$ then conducts the decryption as $$\bm{h}_{9}=\textsf{Dec}^{\mathcal{S}}(\textsf{ct}_{\bm{h}_{9}}^{\mathcal{S}})=\bm{h}_{1}\boxplus{(\bm{h}_{2}\boxdot{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{h}_{3}})\boxplus({{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bm{h}_{4}}\boxdot{\bm{h}_{5}}})}.$$ (7) It is obvious that $\textsf{ReLU}(\bm{a})=\bm{h}_{8}\boxplus\bm{h}_{9}$. Thus $\mathcal{S}$ obtains the $\mathcal{C}$-encrypted $\textsf{ReLU}(\bm{a})$ as $$\textsf{ct}_{\bar{\bm{a}}}^{\mathcal{C}}=\textsf{ct}_{\bm{h}_{8}}^{\mathcal{C}}\oplus\bm{h}_{9},$$ (8) which is utilized to compute Conv to be elaborated next. Note that the proposed PHE triplet enables us to eliminate the Mux and directly get the encrypted ReLU at $\mathcal{S}$ for computing the subsequent Conv, with only half a round after the DReLU. This helps to save all computation and communication cost for calling Mux. 3.2.2 Matrix Encoding for Conv. Based on the encrypted ReLU $\textsf{ct}_{\bar{\bm{a}}}^{\mathcal{C}}$ in Eq. (8), $\mathcal{S}$ is supposed to utilize its kernel K to compute the Conv (see the blue block in Algorithm 1). In other words, $\mathcal{S}$ should obtain Enc${}^{\mathcal{C}}(\bm{y})=\textsf{Conv}(\textsf{ct}_{\bar{\bm{a}}}^{\mathcal{C}},\textrm{{K}})$ and share it with $\mathcal{C}$. The state-of-the-art frameworks extensively call the PHE operations to get the encrypted output of Conv, which is then shared between $\mathcal{C}$ and $\mathcal{S}$ to compute the subsequent function. However, it inevitably involves a series of Rot due to the needed summing process [13]. We totally relieve the need for Rot via a carefully designed matrix encoding. Our main idea lays in the observation that the dot product between a matrix and a vector in Eq. (1) can be viewed as the linear combination of all columns in that matrix. As such, if the $\textsf{ReLU}(\bm{a})$ and kernel K were respectively such matrix and vector, we could construct $\textsf{ct}_{\bar{\bm{a}}}^{\mathcal{C}}$ as a set of ciphertext each of which encrypts one column in $\textsf{ReLU}(\bm{a})$, and the Conv output, Enc${}^{\mathcal{C}}(\bm{y})$, could be obtained by $\mathcal{S}$ via only HE multiplication and addition. In the following, we describe in detail the feasibility of above conjecture for Rot-free Conv, and Sec. 3.2.4 shows the efficiency advantages of our design compared with the state-of-the-art solution. Specifically, we first transform the Conv $\bm{y}=\textsf{Conv}(\bar{\bm{a}},\textbf{{K}})$ $\in\mathcal{R}^{c_{o}\times h_{i}^{\prime}\times{w_{i}^{\prime}}}$ into corresponding dot product $\tilde{\bm{y}}=\textsf{Dot}(\tilde{\bm{a}},\tilde{\textbf{{K}}})$ $\in\mathcal{R}^{h_{i}^{\prime}w_{i}^{\prime}\times{c_{o}}}$ based on the im2col operator [44]. Recall that $\bar{\bm{a}}\in\mathcal{R}^{c_{i}\times h_{i}\times w_{i}}$ and ${\textbf{K}}\in\mathcal{R}^{c_{o}\times c_{i}\times f_{h}\times f_{w}}$, they are then respectively converted into $\tilde{\bm{a}}\in\mathcal{R}^{h_{i}^{\prime}w_{i}^{\prime}\times{c_{i}f_{h}f_{w}}}$ and $\widetilde{\textbf{K}}\in\mathcal{R}^{c_{i}f_{h}f_{w}\times{c_{o}}}$ such that $\tilde{\bm{y}}_{\alpha,\beta}=\bm{y}_{\beta,\gamma,\eta}$ where $\alpha\in[h_{i}^{\prime}w_{i}^{\prime}]$, $\beta\in[c_{o}]$, $\gamma=\left\lfloor\frac{\alpha}{w_{i}^{\prime}}\right\rfloor$, $h_{i}^{\prime}=\left\lceil\frac{h_{i}}{s}\right\rceil$, $w_{i}^{\prime}=\left\lceil\frac{w_{i}}{s}\right\rceil$, $\eta=\alpha$ mod $w_{i}^{\prime}$. In other words, each $c_{i}f_{h}f_{w}$ elements along the first dimension of K forms one column in $\widetilde{\textbf{K}}$ while each $c_{i}f_{h}f_{w}$ values in $\bar{\bm{a}}$ that are weighted and summed for one number of $\bm{y}$ forms a row in $\tilde{\bm{a}}$. Thereafter, we denote the mapping from ${\bm{y}}$ to $\tilde{\bm{y}}$ as $\phi:\bm{y}\mapsto{\tilde{\bm{y}}}$. Note that the conversion towards $\bar{\bm{a}}$ is completed during the ReLU process which is computed in element wise and is independent with element locations. Meanwhile the conversion towards K is easily performed by $\mathcal{S}$ since it’s in plaintext. We also summarize the overall procedure in Sec. 3.2.3. Recall that we are supposed to construct $\textsf{ct}_{\bar{\bm{a}}}^{\mathcal{C}}$ such that the computation for Enc${}^{\mathcal{C}}(\bm{y})$ doesn’t involve any Rot. Given $\tilde{\bm{a}}$, a matrix encoding is proposed which produces a set of ciphertext and enables Rot-free Conv. Concretely, we define the encoding mapping $\iota:\mathcal{R}^{h_{i}^{\prime}w_{i}^{\prime}\times{c_{i}f_{h}f_{w}}}\mapsto\mathcal{R}^{d\times{n}}$ by $$\iota:\tilde{\bm{a}}\mapsto{\textbf{{A}}}$$ where $\textbf{A}_{j,\zeta}=\tilde{\bm{a}}_{\tau,\lambda}$, $\lambda=j\xi+\left\lfloor\frac{\zeta}{h_{i}^{\prime}w_{i}^{\prime}}\right\rfloor$, $j\in[d]$, $\zeta\in[n]$, $\xi=\left\lfloor\frac{n}{h_{i}^{\prime}w_{i}^{\prime}}\right\rfloor$, $d=\left\lceil\frac{c_{i}f_{h}f_{w}}{\xi}\right\rceil$, and $\tau=\zeta$ mod $h_{i}^{\prime}w_{i}^{\prime}$. In this way, $\iota$ maps $\xi$ columns of $\tilde{\bm{a}}$ into one row of A. Meanwhile, $\forall\beta\in[c_{o}]$, we have $\tilde{\bm{y}}_{:,\beta}=\sum_{\lambda}\tilde{\bm{a}}_{:,\lambda}\widetilde{\textbf{{K}}}_{\lambda,\beta}$, which means that the $\beta$-th output channel after Conv is the summation of all weighted columns in $\tilde{\bm{a}}$ namely the $\lambda$-th cloumn $\tilde{\bm{a}}_{:,\lambda}$ is multiplied with $\widetilde{\textbf{{K}}}_{\lambda,\beta}$. Therefore, $\forall\beta$, $\sum_{j}(\textbf{A}_{j,:}\boxdot\overline{\textbf{K}}_{j,\beta})\in\mathcal{R}^{n}$ contains $\xi$ finally summed columns derived from $\{\tilde{\bm{a}}_{:,\lambda}\tilde{\textbf{{K}}}_{\lambda,\beta}\}_{\lambda}$ where $\overline{\textbf{K}}_{j,\beta}\in\mathcal{R}^{n}=\{\widetilde{\textbf{{K}}}_{j\xi,\beta}\}^{h_{i}^{\prime}w_{i}^{\prime}}|\cdots|\{\widetilde{\textbf{{K}}}_{(j+1)\xi-1,\beta}\}^{h_{i}^{\prime}w_{i}^{\prime}}|\{0\}^{n-h_{i}^{\prime}w_{i}^{\prime}\xi}$ and note that $\widetilde{\textbf{{K}}}_{\varphi,\beta}=0$ if $\varphi\geq{c_{i}f_{h}f_{w}}$. Based on the element-wise operation of $\sum_{j}(\textbf{A}_{j,:}\boxdot\overline{\textbf{K}}_{j,\beta})$, we are ready to obtain a Rot-free Conv. In particular, each $\textbf{A}_{j,:}$ is encrypted into the ciphertext $\textsf{ct}_{\textbf{A}_{j,:}}^{\mathcal{C}}$, then $\forall\beta$, we obtain a $\textsf{ct}_{\textbf{B}_{\beta}}^{\mathcal{C}}=\oplus_{j}(\textsf{ct}_{\textbf{A}_{j,:}}^{\mathcal{C}}\odot\overline{\textbf{K}}_{j,\beta})$ which encrypts $\xi$ finally summed columns derived from $\{\tilde{\bm{a}}_{:,\lambda}\tilde{\textbf{{K}}}_{\lambda,\beta}\}_{\lambda}$. Here $\textbf{B}_{\beta}\in\mathcal{R}^{n}=\sum_{j}({\textbf{A}_{j,:}}\boxdot\overline{\textbf{K}}_{j,\beta})$. Considering that $\textbf{B}_{\beta}$ contains all partial sums for $\tilde{\bm{y}}_{:,\beta}$ and $\{\textsf{ct}^{\mathcal{C}}_{\textbf{B}_{\beta}}\}_{\beta}$ are obtained by $\mathcal{S}$ without Rot, $\mathcal{S}$ then directly shares $\{\textsf{ct}^{\mathcal{C}}_{\textbf{B}_{\beta}}\}_{\beta}$ by $\{\textsf{ct}^{\mathcal{C}}_{\textbf{C}_{\beta}}=\textsf{ct}^{\mathcal{C}}_{\textbf{B}_{\beta}}\ominus{\bm{r}^{\mathcal{S}}_{\beta}}\}_{\beta}$ where $\textbf{C}_{\beta}\in\mathcal{R}^{n}=\textbf{B}_{\beta}\boxminus\bm{r}^{\mathcal{S}}_{\beta}$ and $\bm{r}^{\mathcal{S}}_{\beta}{\scriptstyle\overset{\$}{\leftarrow}}\,\mathbb{Z}_{p}^{n}$. Then $\mathcal{S}$ sends $\{\textsf{ct}^{\mathcal{C}}_{\textbf{C}_{\beta}}\}_{\beta}$ to $\mathcal{C}$, which decrypts $\{\textsf{ct}^{\mathcal{C}}_{\textbf{C}_{\beta}}\}_{\beta}$ into $\{\textsf{Dec}^{\mathcal{C}}(\textsf{ct}^{\mathcal{C}}_{\textbf{C}_{\beta}})=\textbf{C}_{\beta}\}_{\beta}$ and gets $$\left\langle\bm{y}\right\rangle_{p}^{\mathcal{C}}=\phi^{-1}(\left\langle\tilde{\bm{y}}_{:,0}\right\rangle_{p}^{\mathcal{C}}|\left\langle\tilde{\bm{y}}_{:,1}\right\rangle_{p}^{\mathcal{C}}|\cdots|\left\langle\tilde{\bm{y}}_{:,c_{o}-1}\right\rangle_{p}^{\mathcal{C}})$$ (9) where $\left\langle\tilde{\bm{y}}_{:,\beta}\right\rangle_{p}^{\mathcal{C}}=\{\boxplus_{\chi=0}^{\xi-1}\textbf{C}_{\beta,\{h_{i}^{\prime}w_{i}^{\prime}\chi:h_{i}^{\prime}w_{i}^{\prime}(\chi+1)\}}\}^{\top}$. Meanwhile, $\mathcal{S}$ computes $$\left\langle\bm{y}\right\rangle_{p}^{\mathcal{S}}=\phi^{-1}(\left\langle\tilde{\bm{y}}_{:,0}\right\rangle_{p}^{\mathcal{S}}|\left\langle\tilde{\bm{y}}_{:,1}\right\rangle_{p}^{\mathcal{S}}|\cdots|\left\langle\tilde{\bm{y}}_{:,c_{o}-1}\right\rangle_{p}^{\mathcal{S}})$$ (10) where $\left\langle\tilde{\bm{y}}_{:,\beta}\right\rangle_{p}^{\mathcal{S}}=\{\boxplus_{\chi=0}^{\xi-1}\bm{r}^{\mathcal{S}}_{\beta,\{h_{i}^{\prime}w_{i}^{\prime}\chi:h_{i}^{\prime}w_{i}^{\prime}(\chi+1)\}}\}^{\top}$. Here $\mathcal{S}$ is able to get $\left\langle\bm{y}\right\rangle_{p}^{\mathcal{S}}$ offline. As such, $\mathcal{C}$ and $\mathcal{S}$ respectively obtain their shares of Conv, which act as the input of subsequent function. 3.2.3 Putting Things Together. With the proposed PHE triplet in ReLU and the matrix encoding for Conv, the Mux in ReLU and the Rot in Conv are totally eliminated, which forms our building block $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$ to compute consecutive ReLU and Conv. The complexity analysis to be elaborated in Sec. 3.2.4 demonstrates the numerical advantages of $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$ compared with the state-of-the-art function-wise design [17]. Algorithm 2 summaries the joint computation process. Specifically, we have an offline phase for precomputation and an online phase for getting $\langle{\bm{y}}\rangle_{p}^{\mathcal{C}}$ based on the shares of DReLU. Since the dimension of $\bar{\bm{a}}$ is the same as the ones of seven items in Eq. (4), the structures of $\bm{h}_{1}$ to $\bm{h}_{7}$ are correspondingly tuned to transform $\textsf{ct}_{\bar{\bm{a}}}^{\mathcal{C}}$ to $\{\textsf{ct}_{\textbf{{A}}_{j,:}}^{\mathcal{C}}\}_{j}$. Concretely, the structures of $\bm{h}_{6}$ and $\bm{h}_{7}$ are mapped to the one of A. Meanwhile, as $\bm{h}_{9}$ is homomorphically obtained by $\mathcal{C}$ based on Eq. 6 and is then decrypted by $\mathcal{S}$ based on Eq. 7, the structures of $\bm{h}_{1}$ to $\bm{h}_{5}$ can be as tight as possible (see the mapping $\psi$ in Algorithm 2) to enable minimal calls of PHE operations at both $\mathcal{S}$ and $\mathcal{C}$. Furthermore, the mapped $\bm{h}_{9}$ at $\mathcal{S}$ has the same structure as A, which enables $\mathcal{S}$ to compute $\{\textsf{ct}_{\textbf{{A}}_{j,:}}^{\mathcal{C}}\}_{j}$ based on Eq. 8. After that, $\mathcal{S}$ gets $\{\textsf{ct}^{\mathcal{C}}_{\textbf{C}_{\beta}}\}_{\beta}$ which are later decrypted by $\mathcal{C}$ as $\{\textbf{C}_{\beta}\}_{\beta}$. $\mathcal{C}$ finally gets $\left\langle\bm{y}\right\rangle_{p}^{\mathcal{C}}$ based on Eq. 9. The offline computation mainly involves the generation of our PHE triplet whose non-interactive and input-independent nature relieves the necessary synchronization required in other state-of-the-art offline processes [12, 16]. After $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$, the shares of Conv $\bm{y}$ act as the input of subsequent function. Note that the bias $\bm{b}\in\mathcal{R}^{c_{o}}$ in the Conv is combined with $\{\bm{r}^{\mathcal{S}}_{\beta}\}_{\beta}$ such that $$\left\langle\tilde{\bm{y}}_{:,\beta}\right\rangle_{p}^{\mathcal{S}}=\{\boxplus_{\chi=0}^{\xi-1}\bm{r}^{\mathcal{S}}_{\beta,(h_{i}^{\prime}w_{i}^{\prime}\chi:h_{i}^{\prime}w_{i}^{\prime}(\chi+1))}\}^{\top}\boxplus\bm{b}_{\beta}.$$ Furthermore, our joint computation block is easily adapted for the $\textsf{ReLU}+\textsf{FC}$ block. Concretely, the $\bar{\bm{a}}=\textsf{ReLU}(\bm{a})$ becomes an $n_{i}$-size vector and the mapping $\iota$ becomes $$\iota:\bar{\bm{a}}\mapsto{\textbf{A}}$$ where $\textbf{A}\in\mathcal{R}^{{n}}=\{\bar{\bm{a}}\}^{\xi}|\{0\}^{n-n_{i}\xi}$. The weight matrix $\textbf{W}\in\mathcal{R}^{n_{o}\times{n_{i}}}$ is transformed to $\bar{\textbf{W}}\in\mathcal{R}^{d_{1}\times{n}}$ where $d_{1}\in{[\left\lceil\frac{n_{o}}{\xi}\right\rceil]}$, $\xi=\left\lfloor\frac{n}{n_{i}}\right\rfloor$ and $\bar{\textbf{W}}_{j,:}=\textbf{W}_{j\xi,:}|\cdots|\textbf{W}_{(j+1)\xi-1,:}|\{0\}^{n-n_{i}\xi}$, $j\in[d_{1}]$, $\textbf{W}_{\phi,:}=\{0\}^{n_{i}}$ if $\phi\geq{n_{o}}$. In this way, the $\textsf{ct}^{\mathcal{C}}_{\bar{\bm{a}}}$ obtained by $\mathcal{S}$, based on Eq. 8, turns out to be $\textsf{ct}^{\mathcal{C}}_{\textbf{A}}$ that encrypts $\xi$ copies of $\bar{\bm{a}}$ and the ciphertext $\textsf{ct}^{\mathcal{C}}_{\textbf{B}_{j}}=\textsf{ct}^{\mathcal{C}}_{\textbf{A}}\otimes{\bar{\textbf{W}}_{j,:}}$ contains all partial sums of $j\xi$-th to $(j\xi+\xi-1)$-th components in $\bm{y}=\textsf{Dot}(\textbf{W},\bar{\bm{a}})\in\mathcal{R}^{n_{o}}$. Without introducing the Rot, $\mathcal{S}$ shares $\{\textsf{ct}^{\mathcal{C}}_{\textbf{B}_{j}}\}_{j}$ by $\{\textsf{ct}^{\mathcal{C}}_{\textbf{C}_{j}}=\textsf{ct}^{\mathcal{C}}_{\textbf{B}_{j}}\ominus{\bm{r}^{\mathcal{S}}_{j}}\}_{j}$ and sends $\{\textsf{ct}^{\mathcal{C}}_{\textbf{C}_{j}}\}_{j}$ to $\mathcal{C}$, which decrypts $\{\textsf{ct}^{\mathcal{C}}_{\textbf{C}_{j}}\}_{j}$ into $\{\textsf{Dec}^{\mathcal{C}}(\textsf{ct}^{\mathcal{C}}_{\textbf{C}_{j}})=\textbf{C}_{j}\}_{j}$ and gets $$\left\langle\bm{y}\right\rangle_{p}^{\mathcal{C}}=\left\langle{\bm{y}}_{0}\right\rangle_{p}^{\mathcal{C}}|\left\langle{\bm{y}}_{1}\right\rangle_{p}^{\mathcal{C}}|\cdots|\left\langle{\bm{y}}_{n_{o}-1}\right\rangle_{p}^{\mathcal{C}}$$ (11) where $\left\langle{\bm{y}}_{\beta\in[n_{o}]}\right\rangle_{p}^{\mathcal{C}}=\boxplus_{\chi=0}^{n_{i}-1}\textbf{C}_{\tau,n_{i}\lambda+\chi}$, $\tau=\left\lfloor\frac{\beta}{\xi}\right\rfloor$, $\lambda=\beta\;\textrm{mod}\;\xi$. Meanwhile, $\mathcal{S}$ computes $$\left\langle\bm{y}\right\rangle_{p}^{\mathcal{S}}=\left\langle{\bm{y}}_{0}\right\rangle_{p}^{\mathcal{S}}|\left\langle{\bm{y}}_{1}\right\rangle_{p}^{\mathcal{S}}|\cdots|\left\langle{\bm{y}}_{n_{o}-1}\right\rangle_{p}^{\mathcal{S}}$$ (12) in offline and here $\left\langle{\bm{y}}_{\beta}\right\rangle_{p}^{\mathcal{S}}=\boxplus_{\chi=0}^{n_{i}-1}\bm{r}^{\mathcal{S}}_{\tau,n_{i}\lambda+\chi}$. Similar with Conv, the bias $\bm{b}\in\mathcal{R}^{n_{o}}$ is combined with $\{\bm{r}^{\mathcal{S}}_{j}\}_{j}$ such that $$\left\langle{\bm{y}}_{\beta}\right\rangle_{p}^{\mathcal{S}}=\{\boxplus_{\chi=0}^{n_{i}-1}\bm{r}^{\mathcal{S}}_{\tau,n_{i}\lambda+\chi}\}\boxplus\bm{b}_{\beta}.$$ 3.2.4 Complexity Analysis. We now give the complexity analysis444We mainly analyze the computation and communication complexity in the online phase when $\mathcal{C}$ feeds her private input to compute the output of neural network. Similar analysis is applied for the offline computation. for the proposed $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$, and demonstrate its numerical advantages over the function-wise computation for consecutive ReLU and Conv in the state-of-the-art framework [17]555Note that [17] gives optimal complexity under the basic PHE and OT primitives with full security and intact neural networks among the state-of-the-art frameworks [12, 13, 16, 17]. as shown in Algorithm 1. Specifically, [17] introduces the Mux to get the shares of $\textsf{ReLU}(\bm{a})$ after the DReLU, see the line 2 in Algorithm 1, which involves four communication rounds666Two rounds in parallel. with $2c_{i}h_{i}w_{i}(\kappa+2\left\lceil\log{p}\right\rceil)$ bits where the $\kappa$ is the security parameter. After that $\mathcal{C}$ encrypts her share of ReLU to more than $\sigma$ ciphertext which are sent to $\mathcal{S}$. $\mathcal{S}$ then conducts more than $\sigma$ PHE addition $(\textsf{Add})$ to get encrypted ReLU, which serves as the input of Conv. In our protocol, $\mathcal{C}$ sends $\sigma$ ciphertext to $\mathcal{C}$ after $2\sigma$ PHE multiplication $(\textsf{Mult})$ and $2\sigma$ Add, as illustrated in line 4 of Algorithm 2. $\mathcal{C}$ subsequently conducts $\sigma$ PHE decryption $(\textsf{Dec})$ and $d$ Add to get $d$ ciphertext that act as the input of Conv (see line 5 in Algorithm 2). The complexity before Conv is summarized in Table I. Since the complexity of Enc is larger than Dec [24], the cost of more than $\sigma$ Enc in [17] is more expensive than that of our $\sigma$ Dec. On the other hand, less than $(\sigma+d)$ Add are additionally required in our protocol which have negligible complexity overhead as the Add is much cheaper than other PHE operations. Meanwhile, we only need half a communication round before Conv while [17] involves extra four rounds. Furthermore, the extra $2\sigma$ Mult in our protocol are offset for Conv computation to be discussed in the following. As for complexity of Conv, given the encrypted ReLU as input, $\mathcal{S}$ in [17] involves about $dc_{o}$ Mult, $dc_{o}$ Add and more than $\{(f_{h}f_{w}-1)\sigma+c_{o}-\left\lceil\frac{c_{o}}{\left\lfloor\frac{n}{h_{i}w_{i}}\right\rfloor}\right\rceil\}$ Rot to enable $\mathcal{C}$ to finally obtain the share of Conv with $\left\lceil\frac{c_{o}}{\left\lfloor\frac{n}{h_{i}w_{i}}\right\rfloor}\right\rceil$ Dec. By comparison, our joint computation eliminates the Rot with similar amount of Mult and Add, as well as $c_{o}$ Dec (see lines 5 and 6 in Algorithm 2). The complexity is also summarized in Table I. On the one hand, if the needed $(c_{o}-\left\lceil\frac{c_{o}}{\left\lfloor\frac{n}{h_{i}w_{i}}\right\rfloor}\right\rceil)$ Rot in [17] are replaced by the same amount of Dec, it becomes the Dec complexity of our protocol. On the other hand, given the involved $2\sigma$ Mult before Conv in our protocol, the another $(f_{h}f_{w}-1)\sigma$ Rot for Conv in [17] is larger than $2\sigma$ as long as the kernel sizes $f_{h}$ and $f_{w}$ are larger than one. Since Dec and Mult is cheaper than Rot, and $f_{h}$ and $f_{w}$ are mainly three or more in many mainstream neural networks [2, 3], our $2\sigma$ Mult are cheapter than $(f_{h}f_{w}-1)\sigma$ Rot in [17]. As such, we numerically demonstrates the computation advantage of proposed $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$ over the state-of-the-art framework. Furthermore, $\mathcal{S}$ needs to send $c_{o}$ ciphertext to $\mathcal{C}$ in our protocol, which is about ${\left\lfloor\frac{n}{h_{i}w_{i}}\right\rfloor}$ times more than that in [17]. This overhead is offset by the $2c_{i}h_{i}w_{i}(\kappa+2\left\lceil\log{p}\right\rceil)$ bits and extra round cost before Conv required in [17], as well as our further optimization to be elaborated in Sec. 3.3. 3.2.5 Security Analysis. we now prove the semi-honest security of $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$ in the $\{$ $\mathcal{F}_{\textsf{DReLU}}$ (as shown in Algorithm 3) $\}$-hybrid model. Theorem 1. The protocol $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$ presented in Algorithm 2 securely realizes $\mathcal{F}_{\textsf{ReConv}}$ (as shown in Algorithm 4) in the $\{\mathcal{F}_{\textsf{DReLU}}\}$-hybrid model, given the presence of semi-honest adversaries. Proof. We construct $\textsf{Sim}_{\mathcal{C}}$ and $\textsf{Sim}_{\mathcal{S}}$ to simulate the views of corrupted client $\mathcal{C}$ and corrupted server $\mathcal{S}$ respectively. Corrupted client. Simulator $\textsf{Sim}_{\mathcal{C}}$ simulates a real execution in which the client $\mathcal{C}$ is corrupted by the semi-honest adversary $\mathcal{A}$. $\textsf{Sim}_{\mathcal{C}}$ obtains $\left\langle\bm{a}\right\rangle_{p}^{\mathcal{C}}$ from $\mathcal{A}$, externally sends it to $\mathcal{F}_{\textsf{ReConv}}$ and waits for the output from $\mathcal{F}_{\textsf{ReConv}}$. Meanwhile, $\textsf{Sim}_{\mathcal{C}}$ waits for the $\{\textsf{ct}^{\mathcal{C}}_{\textbf{H}_{j,:}}\}_{j}$ from $\mathcal{A}$. $\textsf{Sim}_{\mathcal{C}}$ constructs another $\overline{\textbf{H}}$ and $\widetilde{\textbf{H}}$ with all zeros and all ones respectively, chooses the key $\textsf{pk}_{\textsf{Sim}}$ and encrypts the new $\overline{\textbf{H}}$ and $\widetilde{\textbf{H}}$ to $\{\textsf{ct}^{\textsf{Sim}}_{\overline{\textbf{H}}_{\nu,:}}\}_{\nu}$ and $\{\textsf{ct}^{\textsf{Sim}}_{\widetilde{\textbf{H}}_{\nu,:}}\}_{\nu}$ which are sent to $\mathcal{A}$. $\textsf{Sim}_{\mathcal{C}}$ randomly chooses a new $\langle{\hat{\bm{a}}}\rangle_{2}^{\mathcal{C}}\scriptstyle\overset{\$}{\leftarrow}\,\mathbb{Z}_{2}^{c_{i}\times{h_{i}}\times{w_{i}}}$ as share, simulates $\mathcal{F}_{\textsf{DReLU}}$ sending it to $\mathcal{A}$, and waits $\{\textsf{ct}^{\textsf{Sim}}_{\acute{\textbf{H}}_{\nu,:}}\}_{\nu}$ from $\mathcal{A}$. After obtaining the output $\left\langle\bm{y}\right\rangle_{p}^{\mathcal{C}}$ from $\mathcal{F}_{\textsf{ReConv}}$, $\textsf{Sim}_{\mathcal{C}}$ maps it to $\left\langle\tilde{\bm{y}}\right\rangle_{p}^{\mathcal{C}}$ via $\phi$ (obtained from $\mathcal{A}$) and splits each column $\left\langle\tilde{\bm{y}}_{:,\beta\in[c_{o}]}\right\rangle_{p}^{\mathcal{C}}$ to the sum of $\chi$ random vectors each with size $h_{i}^{\prime}w_{i}^{\prime}$. Then $\textsf{Sim}_{\mathcal{C}}$ concatenates each group of those $\chi$ random vectors to form a new $\textbf{C}_{\beta}$ and encrypts it to $\textsf{ct}^{\mathcal{C}}_{\textbf{C}_{\beta}}$ via $\textsf{pk}_{\mathcal{C}}$ (obtained from $\mathcal{A}$). After that $\textsf{Sim}_{\mathcal{C}}$ sends $\{\textsf{ct}^{\mathcal{C}}_{\textbf{C}_{\beta}}\}_{\beta}$ to $\mathcal{A}$. We argue that the output of $\textsf{Sim}_{\mathcal{C}}$ are indistinguishable from the real view of $\mathcal{C}$ by the following hybrids: $\textsf{Hyb}_{0}$: $\mathcal{C}$’s view in the real protocol. $\textsf{Hyb}_{1}$: Same as $\textsf{Hyb}_{0}$ except that the $\{\textsf{ct}^{\mathcal{S}}_{\overline{\textbf{H}}_{\nu,:}}\}_{\nu}$ and $\{\textsf{ct}^{\mathcal{S}}_{\widetilde{\textbf{H}}_{\nu,:}}\}_{\nu}$ are replaced by $\{\textsf{ct}^{\textsf{Sim}}_{\overline{\textbf{H}}_{\nu,:}}\}_{\nu}$ and $\{\textsf{ct}^{\textsf{Sim}}_{\widetilde{\textbf{H}}_{\nu,:}}\}_{\nu}$ constructed by $\textsf{Sim}_{\mathcal{C}}$. The security of PHE guarantees the view in simulation is computationally indistinguishable from the view in the real protocol. $\textsf{Hyb}_{2}$: Same as $\textsf{Hyb}_{1}$ except that $\langle{\hat{\bm{a}}}\rangle_{2}^{\mathcal{C}}$ is replaced by the randomly-chosen one from $\textsf{Sim}_{\mathcal{C}}$. The security protocol $\mathcal{F}_{\textsf{DReLU}}$ [17] guarantees the view in simulation is computationally indistinguishable from the view in the real protocol. $\textsf{Hyb}_{3}$: Same as $\textsf{Hyb}_{2}$ except that the $\{\textsf{ct}^{\mathcal{C}}_{\textbf{C}_{\beta}}\}_{\beta}$ is replaced by the ones constructed by $\textsf{Sim}_{\mathcal{C}}$. The randomness of $\textbf{C}_{\beta}$ guarantees the view in simulation is computationally indistinguishable from the view in the real protocol and the $\left\langle\bm{y}\right\rangle_{p}^{\mathcal{C}}$ in real world is identical to that in simulation. The hybrid is the view output by $\textsf{Sim}_{\mathcal{C}}$. Corrupted server. Simulator $\textsf{Sim}_{\mathcal{S}}$ simulates a real execution in which the $\mathcal{S}$ is corrupted by the semi-honest adversary $\mathcal{A}$. $\textsf{Sim}_{\mathcal{S}}$ obtains $\langle{\bm{a}}\rangle_{p}^{\mathcal{S}}$ and K from $\mathcal{A}$ and forwards them to the ideal functionality $\mathcal{F}_{\textsf{ReConv}}$. $\textsf{Sim}_{\mathcal{S}}$ constructs a new H filling with zeros, chooses the key $\textsf{pk}_{\textsf{Sim}}$, encrypts the new H to $\{\textsf{ct}^{\textsf{Sim}}_{\textbf{H}_{j,:}}\}_{j\in[d]}$ and sends them to $\mathcal{A}$. $\textsf{Sim}_{\mathcal{S}}$ waits for the $\{\textsf{ct}^{\mathcal{S}}_{\overline{\textbf{H}}_{\nu,:}}\}_{\nu}$ and $\{\textsf{ct}^{\mathcal{S}}_{\widetilde{\textbf{H}}_{\nu,:}}\}_{\nu}$ from $\mathcal{A}$. As for the $\mathcal{F}_{\textsf{DReLU}}$, $\mathcal{A}$ does not need to obtain output from it, and thus $\textsf{Sim}_{\mathcal{S}}$ does nothing. $\textsf{Sim}_{\mathcal{S}}$ constructs a new $\acute{\textbf{H}}$ filling with random numbers, encrypts it to $\{\textsf{ct}^{\mathcal{S}}_{\acute{\textbf{H}}_{\nu,:}}\}_{\nu}$ via $\textsf{pk}_{\mathcal{S}}$ (obtained from $\mathcal{A}$) and sends them to $\mathcal{A}$. $\textsf{Sim}_{\mathcal{S}}$ waits the $\{\textsf{ct}^{\textsf{Sim}}_{{\textbf{C}}_{\beta}}\}_{\beta}$ from $\mathcal{A}$. We argue that the output of $\textsf{Sim}_{\mathcal{S}}$ are indistinguishable from the view of $\mathcal{S}$ by the following hybrids: $\textsf{Hyb}_{0}$: $\mathcal{S}$’s view in the real protocol. $\textsf{Hyb}_{1}$: Same as $\textsf{Hyb}_{0}$ except that $\{\textsf{ct}^{\mathcal{S}}_{\textbf{H}_{j,:}}\}_{j\in[d]}$ is replaced by the $\textsf{Sim}_{\mathcal{S}}$-constructed $\{\textsf{ct}^{\textsf{Sim}}_{\textbf{H}_{j,:}}\}_{j\in[d]}$. The security of PHE guarantees the view in simulation is computationally indistinguishable from the view in the real protocol. $\textsf{Hyb}_{2}$: Same as $\textsf{Hyb}_{1}$ except that the $\{\textsf{ct}^{\mathcal{S}}_{\acute{\textbf{H}}_{\nu,:}}\}_{\nu}$ is replaced by the ones constructed by $\textsf{Sim}_{\mathcal{S}}$. The randomness of $\acute{\textbf{H}}$ guarantees the view in simulation is computationally indistinguishable from the view in the real protocol. The hybrid is the view output by $\textsf{Sim}_{\mathcal{S}}$. 3.3 Further Optimizations 3.3.1 Deal with $\mathcal{C}$’s Input. Once $\mathcal{C}$ has a query to get the output of neural network, her input is firstly feed to Conv rather than ReLU. Therefore we address the Conv with $\mathcal{C}$’s input by $\Pi_{\textsf{iConv}}^{\textrm{ring},p}$ which is derived from the computation for Conv in $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$ (as shown in Algorithm 5). Specifically, $\mathcal{C}$ directly forms $\{\textsf{ct}_{\textbf{A}_{j,:}}^{\mathcal{C}}\}_{j\in[d]}$ by treating her private input as the output of ReLU namely $\bar{\bm{a}}$. Those $d$ ciphertext are sent to $\mathcal{S}$ which conducts $dc_{o}$ Mult and $dc_{o}$ Add to get $\{\textsf{ct}^{\mathcal{C}}_{\textbf{C}_{\beta}}\}_{\beta\in[c_{o}]}$ which are sent to $\mathcal{C}$. $\mathcal{C}$ decrypts $\{\textsf{ct}^{\mathcal{C}}_{\textbf{C}_{\beta}}\}_{\beta}$ to $\{\textbf{C}_{\beta}\}_{\beta}$, and gets $\left\langle\bm{y}\right\rangle_{p}^{\mathcal{C}}$ based on Eq. 9 (see lines 5 and 6 in Algorithm 2). The complexity is summarized in Table II. Similar with the complexity analysis for Conv in $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$, the $c_{o}$ Dec in our protocol is cheaper than that in [17] by combining [17]’s $c_{o}-\left\lceil\frac{c_{o}}{\left\lfloor\frac{n}{h_{i}w_{i}}\right\rfloor}\right\rceil$ Rot with its Dec. On the other hand, if the another $(f_{h}f_{w}-1)\sigma$ Rot in [17] is replaced by equivalent amount of Enc, its Enc complexity becomes $$\{(f_{h}f_{w}-1)\sigma+\left\lceil\frac{c_{i}}{\left\lfloor\frac{n}{h_{i}w_{i}}\right\rfloor}\right\rceil\}\approx{d},$$ which is the Enc complexity of ours. As the Enc is relatively cheaper than Rot, we thus have numerical advantage for computing the first Conv with $\mathcal{C}$’s input. 3.3.2 Compute the MaxPool before ReLU. The MaxPool always appears as a form of ReLU+MaxPool+Conv and thus we propose to convert the ReLU+MaxPool+Conv to MaxPool+ReLU+Conv where the proposed $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$ is applied to compute ReLU+Conv and the MaxPool is calculated by $\Pi_{\textsf{tMaxPool}}^{\textrm{ring},p}$ to be described in Sec. 3.3.5. On the one hand, the correctness is guaranteed by $$\textsf{ReLU}(\textsf{MaxPool}(\bm{a}))=\textsf{MaxPool}(\textsf{ReLU}(\bm{a}))$$ where $\bm{a}\in{\in{\mathbb{Z}_{p}^{c_{i}\times{h_{i}}\times{w_{i}}}}}$. On the other hand, we also enjoy the efficiency benefit from this conversion. Specifically, we denote the comparison process between two numbers as Comp and the pooling size of MaxPool as $s_{\textsf{mxp}}\in\mathbb{N}^{+}$. In the $\textsf{ReLU}(\bm{a})$, there are $c_{i}h_{i}w_{i}$ Comp calls for DReLU$(\bm{a})$ where the dimension of $\textsf{ReLU}(\bm{a})$ is $c_{i}\times{h_{i}}\times{w_{i}}$. Since $(s_{\textsf{mxp}}^{2}-1)$ Comp calls are needed to compute the subsequent MaxPool in each $s_{\textsf{mxp}}\times{s_{\textsf{mxp}}}$ block of $\textsf{ReLU}(\bm{a})$, the total Comp calls for $\textsf{MaxPool}(\textsf{ReLU}(\bm{a}))$ is $$c_{i}h_{i}w_{i}+\left\lceil\frac{h_{i}}{s_{\textsf{mxp}}}\right\rceil\left\lceil\frac{w_{i}}{s_{\textsf{mxp}}}\right\rceil(s_{\textsf{mxp}}^{2}-1).$$ In contrast, while $\left\lceil\frac{h_{i}}{s_{\textsf{mxp}}}\right\rceil\left\lceil\frac{w_{i}}{s_{\textsf{mxp}}}\right\rceil(s_{\textsf{mxp}}^{2}-1)$ Comp calls are needed for $\textsf{MaxPool}(\bm{a})$, the size for each output channel becomes $\left\lceil\frac{h_{i}}{s_{\textsf{mxp}}}\right\rceil\times\left\lceil\frac{w_{i}}{s_{\textsf{mxp}}}\right\rceil$. Therefore there are $c_{i}\left\lceil\frac{h_{i}}{s_{\textsf{mxp}}}\right\rceil\left\lceil\frac{w_{i}}{s_{\textsf{mxp}}}\right\rceil$ Comp calls to compute the subsequent ReLU and the total Comp calls for $\textsf{ReLU}(\textsf{MaxPool}(\bm{a}))$ becomes $$c_{i}\left\lceil\frac{h_{i}}{s_{\textsf{mxp}}}\right\rceil\left\lceil\frac{w_{i}}{s_{\textsf{mxp}}}\right\rceil+\left\lceil\frac{h_{i}}{s_{\textsf{mxp}}}\right\rceil\left\lceil\frac{w_{i}}{s_{\textsf{mxp}}}\right\rceil(s_{\textsf{mxp}}^{2}-1),$$ which reduces the Comp calls for ReLU computation about $s_{\textsf{mxp}}^{2}$ times. While this ReLU-MaxPool conversion has been applied in optimizations for GC-based and SS-based nonlinear functions [45, 46], or even in plaintext neural networks [47], they only consider the sequential computation for the stacked functions in a neural network. We further gain the efficiency boost on top of the function-wise optimization via our joint linear and nonlinear computation. 3.3.3 Decouple the Mean for MeanPool. Similar with the MaxPool, the MeanPool appears as a form of ReLU+MeanPool+Conv. Given the summing and averaging of the MeanPool, we decouple these two operations such that the summing is integrated with the ReLU while the averaging is fused with the Conv. In this way, we are able to apply our $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$ to directly compute the ReLU+MeanPool+Conv. Specifically, we proceed the summing within the $s_{\textsf{mp}}\times{s_{\textsf{mp}}}(s_{\textsf{mp}}\in\mathbb{N}^{+})$ pooling window in Eq. 4 such that the local addition in each $s_{\textsf{mp}}\times{s_{\textsf{mp}}}$ block is conducted from $\bm{h}_{6}$ to $\bm{h}_{7}$, which produces new $\bm{h}_{6}$ and $\bm{h}_{7}$. Meanwhile, $\mathcal{S}$ performs the same local addition for $\bm{h}_{9}$, and the corresponding mapping or process in Algorithm 2 is adapted to these new matrices with shrinking size. Since each component after the above summing need to multiplied with the same number namely ${1}/{s_{\textsf{mp}}^{2}}$, we premultiply ${1}/{s_{\textsf{mp}}^{2}}$ with all components in K such that $$\textbf{K}\leftarrow{\textbf{K}}/{s_{\textsf{mp}}^{2}},$$ which achieves the averaging of MeanPool by averaging the subsequent kernel K for Conv. Furthermore, the $s_{\textsf{mp}}$ is usually two which results in the finite decimal 0.25 for ${1}/{s_{\textsf{mp}}^{2}}$. 3.3.4 Integrate the BN with Conv. As the BN usually appears after the Conv and it is specified by a constant tuple $(\bm{\mu},\bm{\theta})$ in neural network inference [37] where $\bm{\mu},\bm{\theta}\in\mathcal{R}^{c_{o}}$. Therefore, it is easily fused into the Conv such that $$\textbf{K}_{\beta,:,:,:}\leftarrow\bm{\mu}_{\beta}{\textbf{K}_{\beta,:,:,:}}\;\textrm{and}\;\bm{b}_{\beta}\leftarrow\bm{\theta}_{\beta}+{\bm{b}_{\beta}}.$$ In this way, the Conv+BN becomes a new Conv, which can be combined with ReLU to perform our $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$ or be processed via $\Pi_{\textsf{iConv}}^{\textrm{ring},p}$. 3.3.5 Parallel the MaxPool and ArgMax. Built upon the optimized Comp, the state-of-the-art framework sequentially conducts $(s_{\textsf{mxp}}^{2}-1)$ Comp calls in each $s_{\textsf{mxp}}\times{s_{\textsf{mxp}}}$ pooling window to get one component of MaxPool [17]. We further improve the efficiency via $\left\lceil\log{s_{\textsf{mxp}}^{2}}\right\rceil$ paralleling Comp calls. Specifically, the $s_{\textsf{mxp}}^{2}$ values is divided into $\left\lceil s_{\textsf{mxp}}^{2}/2\right\rceil$ pairs each have two values. Each of these pairs independently calls the Comp to get the maximum. The $\left\lceil s_{\textsf{mxp}}^{2}/2\right\rceil$ maximums are again divided into $\left\lceil s_{\textsf{mxp}}^{2}/2^{2}\right\rceil$ pairs and each pair again independently calls the Comp to get its maximum. The process ends with only one pair calling one Comp to finally get the shares of maximum among the $s_{\textsf{mxp}}^{2}$ values. Meanwhile, all of the $s_{\textsf{mxp}}\times{s_{\textsf{mxp}}}$ blocks are able to simultaneously perform $\left\lceil\log{s_{\textsf{mxp}}^{2}}\right\rceil$ paralleling Comp calls as they are mutually independent and we refer this paralleled computation as $\Pi_{\textsf{tMaxPool}}^{\textrm{ring},p}$. On the other hand, $(s_{\textsf{amx}}-1)$ sequential Comp calls are involved in the ArgMax to get the output of a neural network [17] where the $s_{\textsf{amx}}$ is the number of classes in the neural network. We similarly treat the $s_{\textsf{amx}}$ values as the leaves of a binary tree such that $\left\lceil\log{s_{\textsf{amx}}}\right\rceil$ paralleling Comp calls are performed and we refer this paralleled process as $\Pi_{\textsf{tArgMax}}^{\textrm{ring},p}$. 3.3.6 Performance Bonus by Channel/Layer-Wise Pruning. It is well-known that many mainstream neural networks have noticeable redundancy and numerous pruning techniques have been proposed to slim the networks while maintaining the model accuracy [48]. Among various pruning techniques, the channel-wise pruning contributes to reduce the number of input/output channels namely $c_{i}$/$c_{o}$ in the neural networks [49] while the layer-wise pruning deletes certain layers (i.e., removes specific functions) in the neural networks [50]. Note that the above pruning does not affect the computation logic for the neural networks but with fewer functions or smaller dimensions in the functions, which serves as a performance bonus to improve the efficiency when we run the privacy-preserving protocol in the mainstream neural networks. Meanwhile, almost all the state-of-the-art frameworks rely on the intact neural networks without considering the pruned versions [12, 13, 16, 17, 36, 37], and we left it as a performance bonus for practical deployment of privacy-preserving MLaaS. 4 Evaluation In this section, we present the performance evaluation and experimental results. We first introduce the experimental setup in Sec. 4.1, and then discuss in Sec. 4.2 and 4.3 about how efficient is our protocol to speed up the function computation and what is the prediction latency and communication cost on practical DL models by using our protocol compared with the state-of-the-art framework [17]777Code available at https://github.com/mpc-msri/EzPC., respectively. 4.1 Experimental Setup We run all experiments on two Amazon AWS c4.xlarge instances possessing the Intel(R) Xeon(R) CPU E5-2666 v3 $@$ 2.90GHz, with 7.5GB of system memory. In the LAN setting, the client $\mathcal{C}$ and server $\mathcal{S}$ were executed on such two instances both located in the us-east-1d (Northern Virginia) availability zone. In the WAN setting, $\mathcal{C}$ and $\mathcal{S}$ were executed on such two instances respectively located in the us-east-1d (Northern Virginia) and us-east-2c (Ohio) availability zone. $\mathcal{C}$ and $\mathcal{S}$ each used an 4-thread execution. These experiential settings are similar with those used for the evaluation of the state-of-the-art frameworks [13, 16]. Furthermore, we evaluate on the MNIST [51] and CIFAR10[52] datasets with architectures LeNet [1], AlexNet [2], VGG-11 [3], VGG-13 [3], VGG-16 [3], VGG-19 [3], ResNet-18 [4], and ResNet-34 [4]. 4.2 Microbenchmarks We first benchmark the performance of ReLU+Conv in [17] and our $\Pi_{\textsf{ReConv}}^{\textrm{ring},p}$. In Table III, we evaluate the cost of various ReLU+Conv used in LeNet, AlexNet, VGG-11, VGG-13, VGG-16, VGG-19, ResNet- 18, and ResNet-34. The key takeaway from Table III is that our online time is over 2$\times$ to 13$\times$ smaller than CrypTFlow2’s. Note that our online communication cost is higher than CrypTFlow2’s. However, our protocol has noticeable overall performance gain (due to the reduced computation cost as analyzed in Sec. 3.2.4 and the lower communication cost before Conv) to offset the total communication. 4.3 Performance on Modern DL Models We test the performance of our protocol on various DL models used in practice. The overall evaluation is shown in Table IV. Specifically, in the LAN setting, our protocol has a speedup of $5.3\times$, $2\times$, $1.97\times$, $1.95\times$, $1.94\times$, $1.93\times$, $3.63\times$, $2.94\times$ over CrypTFlow2 on LeNet, AlexNet, VGG-11, VGG-13, VGG-16, VGG-19, ResNet-18, and ResNet-34. The speedup is respectively $3.3\times$, $2\times$, $1.9\times$, $1.89\times$, $1.88\times$, $1.86\times$, $3.18\times$, $2.6\times$ in the WAN setting. Here we can see that our protocol has a relatively larger data load to be transmitted in the online phase compared with CrypTFlow2. As the bandwidth is a relatively low cost in today’s transmission link, e.g., Amazon AWS can easily keep a bandwidth around Gigabit, the reduction of computation and communication round in our protocol brings an overall performance boost that offsets the transmission overhead. Meanwhile, the communication overhead of our protocol in offline phase is lighter and comparable to that of CrypTFlow2 (in online phase), which has proved to be communication-reduced for the involved parties [17]. Furthermore, it is worth reiterating that our protocol’s offline phase is totally non-interactive, which dose not need the involved parties to synchronously exchange any calculated data. Therefore, it eliminates the interactive offline computation that is often used in the state-of-the-art frameworks such as DELPHI [16] and MiniONN [12]. Next we test the runtime breakdown of each layer in our evaluated eight DL models as shown in Fig. 3, which allows us to have detailed observations. Specifically, the runtime for each layer includes all overhead for linear and nonlinear functions. Meanwhile, the layer index also includes the pooling, e.g., mean pooling. In LeNet, our protocol has noticeable speedup in each convolution layer (which includes Conv and ReLU) and the speedup is larger at last layer as our protocol only needs one HE multiplication while CrypTFlow2 involves a series of HE rotations. Similar observations are found in the last layer of other networks. In AlexNet, the large kernel width and height in the first layer, i.e., $f_{h}=f_{w}=11$, result in more rotations needed in CrypTFlow2 while the counterpart in our protocol is the same amount of multiplications, which is more efficient (see more details at Sec. 3.2). At the same time, the stride of 4 in the first layer involves 16 non-stride convolutions in CrypTFlow2 while our protocol benefits from the decreased data size to be transmitted and from the smaller computation overhead for strided convolution. As such our protocol gains a larger speedup. In VGG-11, VGG-13, VGG-16 and VGG-19, the speedup is larger in layer 11, 13, 15 and 17 (except the last layer) respectively. This is because large $c_{o}$ (i.e., the number of output channels) makes the gap between rotation and decryption more significant. As decryption is generally cheaper than multiplication, the speedup correspondingly increases. In ResNet-18 and ResNet-34, our protocol’s speedup is lager in layers 9, 15, 21 (except the last layer) with strided convolution, which is in line with the speedup for first layer in AlexNet. Similar observations are found in the WAN setting. Finally, we show in Figure 4 the breakdown of communication overhead in each layer in our evaluated eight DL models. As demonstrated in Figure I, the gap in communication cost (i.e., the amount of data to be transmitted) between our protocol and CrypTFlow2 is proportional to the number of output channel $c_{o}$. As such we can see an increased difference between their communication cost as $c_{o}$ increases along the layers in all networks. But as we discussed earlier, despite the larger amount of data for communication, our protocol has noticeable overall performance gain (due to the reduced computation cost as analyzed in Sec. 3.2 and the lower communication cost before Conv) to offset the increased communication cost. 5 Conclusions In this paper, we have challenged and broken the conventional compute-and-share logic for function output, and have proposed the share-in-the-middle logic, with which we have introduced the first joint linear and nonlinear computation across functions that features by 1) the PHE triplet for computing the intermediates of nonlinear function, with which the communication cost for multiplexing is eliminated; 2) the matrix encoding to calculate the intermediates of linear function, with which all rotations for summing is removed; and 3) the network adaptation to reassemble the model structure such that the recombined functions are able to take advantages of our proposed joint computation block as much as possible. 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Automatic winning shifts Jarkko Peltomäki111Corresponding author. E-mail addresses: r@turambar.org (J. Peltomäki), vosalo@utu.fi (V. Salo). The Turku Collegium for Science, Medicine and Technology TCSMT, University of Turku, Turku, Finland Turku Centre for Computer Science TUCS, Turku, Finland University of Turku, Department of Mathematics and Statistics, Turku, Finland Ville Salo University of Turku, Department of Mathematics and Statistics, Turku, Finland Abstract To each one-dimensional subshift $X$, we may associate a winning shift $W(X)$ which arises from a combinatorial game played on the language of $X$. Previously it has been studied what properties of $X$ does $W(X)$ inherit. For example, $X$ and $W(X)$ have the same factor complexity and if $X$ is a sofic subshift, then $W(X)$ is also sofic. In this paper, we develop a notion of automaticity for $W(X)$, that is, we propose what it means that a vector representation of $W(X)$ is accepted by a finite automaton. Let $S$ be an abstract numeration system such that addition with respect to $S$ is a rational relation. Let $X$ be a subshift generated by an $S$-automatic word. We prove that as long as there is a bound on the number of nonzero symbols in configurations of $W(X)$ (which follows from $X$ having sublinear factor complexity), then $W(X)$ is accepted by a finite automaton, which can be effectively constructed from the description of $X$. We provide an explicit automaton when $X$ is generated by certain automatic words such as the Thue-Morse word. Keywords: winning shift, combinatorial game, abstract numeration system, automatic sequence, regular language, sofic shift 1 Introduction Consider a target set $X$ of words of a common length $n$ written over an alphabet $A$. Let $\alpha_{1}\dotsm\alpha_{n}$ be a choice sequence of integers in $\{0,1,\ldots,|A|-1\}$. Given the choice sequence, Alice and Bob play a game as follows. On round $j$, $1\leq j\leq n$, Alice chooses a subset $A_{j}$ of $A$ of size $\alpha_{j}+1$. Then Bob picks a letter $a_{j}$ from $A_{j}$. After $n$ rounds, a word $a_{1}\dotsm a_{n}$ is built. If this word is in $X$, then Alice wins, and Bob wins otherwise. Let $W(X)$ be the set of choice sequences for which Alice has a winning strategy. This set is called the winning set of $X$. For example, if $X=\{000,110,111\}$ and the choice sequence is $101$, then Bob has a winning strategy. Indeed on the first turn Alice must allow Bob to choose from $A_{1}=\{0,1\}$ (since this is the only subset of size $2$ of the alphabet), and Bob may pick the letter $0$. Alice must then select the subset $A_{1}=\{0\}$ since no word in $X$ begins with $01$. On the final round Bob again has two choices, and he may select $1$. This results in the word $001$ which is not in $X$. Hence $101\notin W(X)$. It is straightforward to verify that $W(X)=\{000,001,100\}$. More generally, the game can be allowed to have infinitely many turns, and the winning set makes sense for sets $X$ containing words of different lengths by setting $$W(X)=\bigcup_{n\in\mathbb{N}\cup\{\mathbb{N}\}}W(X\cap A^{n}).$$ The winning set was introduced in the paper [29] by the second author and I. Törmä. The paper [29] contains rigorous definitions of the preceding concepts and proofs of the basic properties of winning sets. A key property of $W(X)$ is that it is hereditary (or downward-closed): if $u$ and $v$ are words of equal length satisfying $u\leq v$ and $v\in W(X)$, then $u\in W(X)$ (here $\leq$ is the coordinatewise order induced by the natural order on $\mathbb{N}$). Most importantly, if $X$ is a subshift, then $W(X)$, now called the winning shift of $X$, is also a subshift. As a terminological point, in [29] the term “winning shift” and the notation $W(X)$ refers to a slightly different subshift in the case of a nonbinary alphabet, and our $W(X)$ is denoted by $\tilde{W}(X)$. Several properties of $X$ are inherited by $W(X)$. For example, if $X$ is a regular language, then $W(X)$ is also regular. Surprisingly factor complexity is preserved: there are equally many words of length $n$ in $X$ and $W(X)$ for all $n$. In a sense, the mapping $X\mapsto W(X)$ is a factor complexity preserving map that reorganizes $X$ to a hereditary set. This property was used in [24] to rederive results of [9] that allow to determine the factor complexity of fixed points of marked uniform substitutions. Additionally, it was shown that $W(X)$ has a substitutive structure when $X$ is a subshift generated by a marked uniform substitution. In [18], Marcus and Törmä study the descriptional complexity of $W(X)$ when $X$ is regular. A concept equivalent to winning sets has been introduced in the context of set systems in [2]. In [28], winning shifts are utilized to show that positive entropy implies that a subshift contains a “steadily branching binary tree” (which in the case of a binary alphabet implies positive independence entropy). The aim of this paper is to study the structure of the winning shift $W(X)$ of a subshift $X$ generated by an automatic word. Automatic words are a well-studied class of words which are generated by finite state machines. In fact, the above-mentioned fixed points of marked uniform substitutions are automatic words. Our setup is as follows. We consider representations of integers in an abstract numeration system $\mathcal{S}$ and define $\mathcal{S}$-automatic words as words obtained by feeding representations of integers to finite automata with output. We then develop the notion of an $\mathcal{S}$-codable set of infinite words over $\mathbb{N}$ as a set of words whose supports, represented in $\mathcal{S}$, can be recognized by a finite automaton. Our main result states that if the addition relation of $\mathcal{S}$ is rational and $X$ is a subshift generated by an $\mathcal{S}$-automatic word with sublinear factor complexity, then its winning shift $W(X)$ is $\mathcal{S}$-codable. We obtain as a corollary that if $\mathcal{S}$ is a Pisot numeration system and $X$ is a subshift generated by an $\mathcal{S}$-automatic word, then $W(X)$ is $\mathcal{S}$-codable. Pisot-automatic words include the well-known $k$-automatic words as well as Fibonacci-automatic words (such as the Fibonacci word). We also discuss the necessity of the assumptions of our main result. For the result, it is crucial that $W(X)$ is countable. We give an example due to J. Cassaigne of a subshift $X$ generated by an $\mathcal{S}$-automatic word with superlinear factor complexity for which $W(X)$ is uncountable, and we exhibit a subshift $X$ with sublinear factor complexity such that $W(X)$ is uncountable. The proof of the main result is constructive, so in principle an automaton for the winning shift can be found algorithmically. We determine the automaton explicitly for the winning shifts of the subshifts generated by the following $2$-automatic words: the Thue-Morse word, the regular paperfolding word, the Rudin-Shapiro word, and the period-doubling word. This paper also includes some general results on the $\mathcal{S}$-codability of winning shifts of sofic shifts, as well as basic robustness properties of the class of $\mathcal{S}$-codable subshifts, in particular closure under topological conjugacy. 2 Preliminaries We assume that the reader is familiar with basic formal language theory. We point the reader to [27] for information on automata and regular languages and [17] on combinatorics on words. As a terminological point, “regular” and “rational” both refer to regular languages, but we use the word “rational” to emphasize that a language encodes a relation. In this paper, $\mathbb{N}=\{0,1,\ldots\}$, and we index words from $0$ unless stated otherwise. Let $A$ be an alphabet and $A^{\mathbb{N}}$ be the set of right-infinite words over $A$. When $\mathbf{x}=x_{0}x_{1}\dotsm$ with $x_{i}\in A$, then with $\mathbf{x}[i,j]$, $i\leq j$, we refer to the factor $x_{i}\dotsm x_{j}$ of $\mathbf{x}$. We use the shorthand $\mathbf{x}[i]$ for $\mathbf{x}[i,i]$. The set of factors, the language of $\mathbf{x}$ is denoted by $\mathcal{L}(\mathbf{x})$. The factor complexity function $\rho_{\mathbf{x}}$ counts the number of distinct factors of length $n$ in $\mathbf{x}$, that is, $\rho_{\mathbf{x}}(n)=|\{\mathbf{x}[p,p+n-1]:p\geq 0\}|$. If $\rho_{\mathbf{x}}(n)=\mathcal{O}(n)$, then we say that $\mathbf{x}$ has sublinear factor complexity. Otherwise we say that $\mathbf{x}$ has superlinear factor complexity. A subshift $X$ is a shift-invariant subset of $A^{\mathbb{N}}$ which is compact in the product topology on $A^{\mathbb{N}}$ with the discrete topology on a finite set $A$. We also consider subshifts in $\mathbb{N}^{\mathbb{N}}$, and they are defined analogously by letting $\mathbb{N}$ to have the discrete topology. Compactness implies that the alphabet must actually be finite, that is, up to renaming of letters, we have $X\subseteq A^{\mathbb{N}}$ for a finite $A$. Indeed, otherwise the open cover $\{\{x_{0}x_{1}\dotsm\in X:x_{0}=n\}:n\in\mathbb{N}\}$ of $X$ has no finite subcover. A subshift $X$ is determined by its language $\mathcal{L}(X)$ which is the set of factors of all words in $X$. By $\overline{\mathcal{O}(\mathbf{x})}$ we mean the closure of the orbit of $\mathbf{x}$ under the shift map. If $X=\overline{\mathcal{O}(\mathbf{x})}$, then we say that $X$ is generated by $\mathbf{x}$. A subshift $X$ is transitive if it is generated by $\mathbf{x}$ for some $\mathbf{x}\in X$. Suppose that $A$ has a total order $<$. A subset $X$ of $A^{n}$ with $n\in\mathbb{N}\cup\{\mathbb{N}\}$ is hereditary if $v\in X$ and $u\leq v$ coordinatewise imply $u\in X$ for all $v\in X$ and all $u\in A^{n}$. If $X$ is a subshift, then the smallest hereditary subshift containing $X$ is called the hereditary closure of $X$. In the earlier papers [29, 24, 18], the word downward-closed was used, but we opt to use hereditary which seems more common in symbolic dynamics [13, 14]. Each word in the winning set $W(X)$ of $X$ corresponds to at least one winning strategy for Alice. It is useful to think of the winning strategies as strategy trees. When $$X=\{0010,0011,0100,0101,0110,1001,1010,1011,1100,1101\},$$ then $$W(X)=\{0000,0001,0010,0100,0101,1000,1001,1010,1100,1101\}.$$ Alice’s winning strategy for the choice sequence $1101$ is depicted as the strategy tree of Figure 1. The interpretation is that on the first turn Alice has to offer two choices for Bob, so the tree branches. The same branching happens on the second round. The letters immediately following the second branching dictate which letter Alice should force Bob to choose on the third round. As the choice sequence ends with $1$, the tree then branches once more. The paths from the root to the leaves form all possible words played when Alice uses this strategy, and indeed they all belong to $X$. 3 Numeration Systems and $\mathcal{S}$-automatic Words 3.1 Numeration Systems Let $L$ be an infinite language with a fixed total order $\prec$ of order type $\omega$, and let $\mathcal{S}=(L,\prec)$. We call the pair $\mathcal{S}$ an abstract numeration system which we abbreviate as ANS. Usually it is required in an ANS $\mathcal{S}$ that $L$ is regular and $\prec$ is the radix order induced by a total order on the alphabet meaning short words come before long ones and words of the same length are ordered lexicographically. Our definition is slightly more general since our main results work for any order $\prec$ under the orthogonal (but morally stronger) assumption of “addability”; see below for the definition. See [15], [3, Ch. 3], and the recent paper [7] and its references for more on ANS. The order $\prec$ naturally yields a bijection $\theta\colon L\to\mathbb{N}$. For $w\in L$ and $n\in\mathbb{N}$, we set $\mathrm{val}_{\mathcal{S}}(w)=\theta(w)$ and $\mathrm{rep}_{\mathcal{S}}(n)=\theta^{-1}(n)$. We omit the subscripts $\mathcal{S}$ whenever they are clear from context. A common example of an ANS is a positional numeration system. Let $(U_{i})$ be a strictly increasing sequence of positive integers with $U_{0}=1$. Represent an integer $n$ greedily as $n=d_{k}U_{k}+\dotsm+d_{0}U_{0}$ and set $\mathrm{rep}(n)=d_{k}\dotsm d_{0}$. Then $(\mathrm{rep}(\mathbb{N}),\prec)$ is an ANS where $\prec$ is induced by the natural order on $\mathbb{N}$. The usual base-$k$ representation of an integer is obtained by letting $U_{i}=k^{i}$ for all $i$. More generally if $(U_{i})$ satisfies a linear recurrence such that the characteristic polynomial of the recurrence is the minimal polynomial of a Pisot number (a number whose conjugates $\alpha$ satisfy $|\alpha|<1$), then we obtain a Pisot numeration system. If $(U_{i})$ is the sequence of Fibonacci numbers, then we obtain the Fibonacci numeration system which is an example of a Pisot numeration system. Pisot numeration systems enjoy many good properties; see, e.g., [25, Ch. 2.4] and [3, Ch. 2] and their references. Next we extend the functions $\mathrm{val}_{\mathcal{S}}$ and $\mathrm{rep}_{\mathcal{S}}$ to tuples of words and integers. Let $A$ be an alphabet and $$\begin{bmatrix}w_{1}\\ \vdots\\ w_{d}\end{bmatrix}$$ be an element of $A^{*}\times\dotsm\times A^{*}$ for a positive integer $d$. The words $w_{1}$, $\ldots$, $w_{d}$ might be of distinct lengths and we cannot feed them to an automaton in parallel, so we pad them to equal length as follows. If $\#$ is a letter that does not belong to $A$, then we set $$\begin{bmatrix}w_{1}\\ \vdots\\ w_{d}\end{bmatrix}^{\#}=\begin{bmatrix}\#^{M-|w_{1}|}w_{1}\\ \vdots\\ \#^{M-|w_{d}|}w_{d}\end{bmatrix}$$ where $M=\max\{|w_{1}|,\ldots,|w_{d}|\}$. Thus by padding, we may work on tuples of words as if they are words over the alphabet $(A\cup\{\#\})^{d}$. Let $\mathcal{S}$ be an ANS, $\#$ be a letter not appearing in the language $L$ of $\mathcal{S}$, $d$ be a positive integer, and $\mathbf{L}=(L^{d})^{\#}$. We extend $\mathrm{rep}_{\mathcal{S}}$ and $\mathrm{val}_{\mathcal{S}}$ to elements of $\mathbb{N}^{d}$ and $\mathbf{L}$ as follows: $$\mathrm{rep}_{\mathcal{S}}\colon\mathbb{N}^{d}\to\mathbf{L},\begin{bmatrix}n_{1}\\ \vdots\\ n_{d}\end{bmatrix}\mapsto\begin{bmatrix}\mathrm{rep}_{\mathcal{S}}(n_{1})\\ \vdots\\ \mathrm{rep}_{\mathcal{S}}(n_{d})\end{bmatrix}^{\#}$$ and $$\mathrm{val}_{\mathcal{S}}\colon\mathbf{L}\to\mathbb{N}^{d},\begin{bmatrix}w_{1}\\ \vdots\\ w_{d}\end{bmatrix}\mapsto\begin{bmatrix}\mathrm{val}_{\mathcal{S}}(\tau_{\#}(w_{1}))\\ \vdots\\ \mathrm{val}_{\mathcal{S}}(\tau_{\#}(w_{d}))\end{bmatrix}$$ where $\tau_{\#}$ is a substitution erasing the letter $\#$. Notice that if $L$ is regular, then $(L^{d})^{\#}$ is also regular. The following definition is central to this paper. It defines what it means that a set of integers or integer tuples is accepted by a finite automaton. Definition 3.1. Let $\mathcal{S}$ be an ANS. A subset $Y$ of $\mathbb{N}^{d}$ is $\mathcal{S}$-recognizable if $\mathrm{rep}_{\mathcal{S}}(Y)$ is regular. If $Y\subseteq\mathbb{N}^{\leq d}=\mathbb{N}\cup\mathbb{N}^{2}\cup\dotsm\cup\mathbb{N}^{d}$, we say that $Y$ is $\mathcal{S}$-recognizable if each $Y\cap\mathbb{N}^{i}$, $i=1,\ldots,d$, is $\mathcal{S}$-recognizable. In the usual base-$k$ numeration system $\mathcal{B}$, the addition relation is rational meaning that the set $\{(x,y,x+y):x,y\in\mathbb{N}\}$ is $\mathcal{B}$-recognizable. This also holds for Pisot numeration systems [10], but it is not a property of a general ANS $\mathcal{S}$. Indeed, even a constant multiple $tY$ of an $\mathcal{S}$-recognizable set $Y$ is not always $\mathcal{S}$-recognizable; see [3, Section 3.3.3]. We give the following definitions. Definition 3.2. An ANS $\mathcal{S}$ is regular if the set $\mathbb{N}$ is $\mathcal{S}$-recognizable. An ANS $\mathcal{S}$ is comparable if the set $\{(x,y):x\leq y\}$ is $\mathcal{S}$-recognizable. An ANS $\mathcal{S}$ is addable if the set $\{(x,y,x+y):x,y\in\mathbb{N}\}$ is $\mathcal{S}$-recognizable. Lemma 3.3. If an ANS is addable, then it is comparable. If an ANS is comparable, then it is regular. Proof. Up to (easy) padding issues, comparability follows from addability by projecting to the first and third component, and regularity follows from comparability by projecting to the first component. ∎ The literature contains many examples of ANS that are not Pisot and not addable, but we are unaware of any ANS using the radix order which is not Pisot and is addable. 3.2 $\mathcal{S}$-automatic Words We only give the definition of $\mathcal{S}$-automatic words. For a more comprehensive introduction to the subject, we refer the reader to [1]. A deterministic finite automaton with output, or DFAO, is a finite automaton with an output function $\tau$ associated with the set of states. When a word $w$ is fed to the automaton and state $q$ is reached, the output of the automaton with input $w$ is defined as $\tau(q)$. Definition 3.4. Let $\mathcal{S}$ be an ANS. An infinite word $\mathbf{x}$ is $\mathcal{S}$-automatic if there exists a DFAO $\mathcal{A}$ such that $\mathbf{x}[n]$ equals the output of $\mathcal{A}$ with input $\mathrm{rep}_{S}(n)$ for all $n\geq 0$. 4 Weakly $\mathcal{S}$-codable and $\mathcal{S}$-codable Sets In this section, we show how to represent sequences in $\mathbb{N}^{\mathbb{N}}$ so that their supports can be recognized by a finite automaton. Consider an infinite word $\mathbf{x}$ in $\mathbb{N}^{\mathbb{N}}$. If $\mathbf{x}=x_{0}x_{1}\dotsm$, $x_{i}\in\mathbb{N}$, then we define $$\sum\mathbf{x}=\sum_{i\in\mathbb{N}}x_{i}.$$ The support $\mathrm{supp}(\mathbf{x})$ of $\mathbf{x}$ is the set $\{n\in\mathbb{N}:x_{n}\neq 0\}$. Definition 4.1. Let $\mathbf{x}$ in $\mathbb{N}^{\mathbb{N}}$ be such that $\sum\mathbf{x}=d<\infty$. Let $\nu(\mathbf{x})$ be the unique vector $(n_{1},\ldots,n_{d})$ in $\mathbb{N}^{d}$ such that $n_{i}\leq n_{i+1}$ for all $i$ and $x_{j}=|\{k:n_{k}=j\}|$ for all $j\in\mathbb{N}$. For example, if $\mathbf{x}=1002010^{\omega}$, then $\nu(\mathbf{x})=(0,3,3,5)$. In other words, we repeat the indices in the support as many times as indicated by the letters: at index $0$ we have $1$, so the index $0$ is repeated once; at index $3$ we have $2$, so the index $3$ is repeated twice; etc. Clearly $\mathbf{x}$ can be uniquely reconstructed from $\nu(\mathbf{x})$: $\mathbf{x}$ is the sum of the characteristic functions of singleton sets $\{i\}$ where $i$ takes on the values in $\nu(\mathbf{x})$ (with repetitions). Definition 4.2. Let $Y$ be a subset of $\mathbb{N}^{\mathbb{N}}$. Its coding dimension is the smallest integer $d\in\mathbb{N}$ such that $\sum\mathbf{y}\leq d$ for all $\mathbf{y}\in Y$, if such a $d$ exists. Definition 4.3. Let $\mathcal{S}$ be an ANS. A subset $Y$ of $\mathbb{N}^{\mathbb{N}}$ is weakly $\mathcal{S}$-codable if for all nonnegative integers $k$ the set $$\left\{\nu(\mathbf{y}):\mathbf{y}\in Y,\sum\mathbf{y}\leq k\right\}$$ is $\mathcal{S}$-recognizable. The subset $Y$ is $\mathcal{S}$-codable if it is weakly $\mathcal{S}$-codable and has finite coding dimension. Notice that if $Y$ is $\mathcal{S}$-codable, then $\nu(Y)$ is $\mathcal{S}$-recognizable since the union of finitely many regular languages is regular. Notice also that weak $\mathcal{S}$-codability considers only words with finite support. Thus if $Y$ consists of infinite words over $\{0,1\}$ having infinitely many occurrences of $1$, then $Y$ is trivially weakly $\mathcal{S}$-codable for any $\mathcal{S}$ since the sets in the above definition are all empty or contain an empty vector. The notion makes most sense for subshifts where configurations with finite sum are dense. For example all hereditary subshifts have this property. $\mathcal{S}$-codability and weak $\mathcal{S}$-codability are relatively robust notions, see Section 9. 5 Weakly $\mathcal{S}$-codable Winning Shifts In this section, we introduce the necessary results and prove that the winning shift of an $\mathcal{S}$-automatic word is weakly $\mathcal{S}$-codable when $\mathcal{S}$ is addable; see Theorem 5.6. First we introduce another way to represent words in $\mathbb{N}^{\mathbb{N}}$. This representation allows more convenient proofs and leads to the same $\mathcal{S}$-codability properties as in Section 4. Let $\mathbf{x}\in\mathbb{N}^{\mathbb{N}}$, and say $\mathbf{x}$ has finite support. View the support $\mathrm{supp}(\mathbf{x})$ as a vector $(n_{1},\ldots,n_{k})$ with $n_{i}<n_{i+1}$ for all $i$, and denote this vector by $s(\mathbf{x})$. Let moreover $e(\mathbf{x})$ be the word obtained from $\mathbf{x}$ by erasing all letters $0$ (this notion naturally extends to words with infinite supports). Definition 5.1. Let $v$ be a word over $\mathbb{N}_{>0}$ and $Y$ be a subset of $\mathbb{N}^{\mathbb{N}}$. Define $$Q_{v}(Y)=\{\mathbf{y}\in Y:e(\mathbf{y})=v\}\quad\text{and}\quad P_{v}(Y)=s(Q_{v}(Y)).$$ For example, the set $P_{111}(Y)$ is the set of $3$-tuples $(n_{1},n_{2},n_{3})$ such that $n_{1}<n_{2}<n_{3}$ and there exists $\mathbf{y}$ in $Y$ with letters $1$ at positions $n_{1}$, $n_{2}$, and $n_{3}$ while other positions of $\mathbf{y}$ equal $0$. Lemma 5.2. Let $\mathcal{S}$ be an ANS and $Y$ be a subset of $\mathbb{N}^{\mathbb{N}}$. Then $Y$ is weakly $\mathcal{S}$-codable if and only if $P_{v}(Y)$ is $\mathcal{S}$-recognizable for all words $v\in\mathbb{N}_{>0}^{*}$. Proof. Suppose that $Y$ is weakly $\mathcal{S}$-codable. Let $v\in\mathbb{N}_{>0}^{*}$, and suppose that the sum of letters of $v$ equals $k$. Let $Y_{k}=\left\{\nu(\mathbf{y}):\mathbf{y}\in Y,\sum\mathbf{y}=k\right\}$. By assumption, there exists an automaton $\mathcal{A}$ accepting $\mathrm{rep}(Y_{k})$. Based on the letters of $v$, we modify $\mathcal{A}$ as follows. If the first letter of $v$ is $i$, then we accept only those words whose $i$ first components are equal. If the second letter of $v$ equals $j$, then we accept only the words whose components $i+1$, $\ldots$, $i+j$ are equal and greater than $i$. We repeat this for each letter and obtain an automaton $\mathcal{A}^{\prime}$. Then we project the words accepted by $\mathcal{A}^{\prime}$ in a suitable way: we project the first $i$ components to one component, the next $j$ components to one component, and so on. The resulting language is regular and it equals $\mathrm{rep}(P_{v}(Y))$. When $\mathrm{rep}(P_{v}(Y))$ is regular for a given $v\in\mathbb{N}_{>0}^{*}$, we can reverse the projection made in the previous paragraph to obtain a regular language $\mathcal{L}_{v}$. We have $$\mathrm{rep}(Y_{k})=\bigcup_{\begin{subarray}{c}v\in\mathbb{N}_{>0}^{*}\\ \sum v=k\end{subarray}}\mathcal{L}_{v},$$ so $\mathrm{rep}(Y_{k})$ is regular as the union is finite. It follows that $Y$ is weakly $\mathcal{S}$-codable. ∎ Notice that it follows from 5.2 that $Y$ is $\mathcal{S}$-codable if and only if $P_{v}(Y)$ is $\mathcal{S}$-recognizable for all words $v\in\mathbb{N}_{>0}^{*}$ and $P_{v}(Y)$ is nonempty for only finitely many $v$. Our aim is to show that when $X$ is a subshift generated by an $\mathcal{S}$-automatic word, then $P_{v}(W(X))$ can be described by a formula expressed in first order logic. When $\mathcal{S}$ is addable, the existence of such a formula implies that $P_{v}(W(X))$ is $\mathcal{S}$-recognizable. By 5.2, this means that $W(X)$ is weakly $\mathcal{S}$-codable; see Theorem 5.6. Proving the existence of the logical formula for a general $v$ leads to a proof with complicated notation. We thus opt to prove the existence when $v=111$ and $X$ is a binary subshift. The proof has the flavor of the general proof, and it should convince the reader that the main ideas can be carried out more generally. Next we begin defining the formulas. We advise the reader to begin reading the proof of 5.3 before attempting to grasp the meaning of these formulas. We now fix $v=111$. Let $\mathbf{x}$ be an infinite binary word. Let $n_{000}$, $n_{001}$, $\ldots$, $n_{111}$ be variables so that $n_{def}$ is one of these variables when $d,e,f\in\{0,1\}$ are given. Let $a$, $b$, $c$ be free variables. Define a formula $\varphi_{def}$ as follows: $$\varphi_{def}=(\mathbf{x}[n_{def}+a]=d\land\mathbf{x}[n_{def}+b]=e\land\mathbf{x}[n_{def}+c]=f).$$ Moreover, we define formulas $\varphi_{0}(i)$, $\varphi_{1}(i)$, and $\varphi_{2}(i)$: $$\displaystyle\varphi_{0}(i)$$ $$\displaystyle=\;$$ $$\displaystyle($$ $$\displaystyle\mathbf{x}[n_{000}+i]=\mathbf{x}[n_{001}+i]=\mathbf{x}[n_{010}+i]=\mathbf{x}[n_{011}+i]=$$ $$\displaystyle\mathbf{x}[n_{100}+i]=\mathbf{x}[n_{101}+i]=\mathbf{x}[n_{110}+i]=\mathbf{x}[n_{111}+i]),$$ $$\displaystyle\varphi_{1}(i)$$ $$\displaystyle=\;$$ $$\displaystyle($$ $$\displaystyle\mathbf{x}[n_{000}+i]=\mathbf{x}[n_{001}+i]=\mathbf{x}[n_{010}+i]=\mathbf{x}[n_{011}+i]\land$$ $$\displaystyle\mathbf{x}[n_{100}+i]=\mathbf{x}[n_{101}+i]=\mathbf{x}[n_{110}+i]=\mathbf{x}[n_{111}+i]),$$ $$\displaystyle\varphi_{2}(i)$$ $$\displaystyle=\;$$ $$\displaystyle($$ $$\displaystyle\mathbf{x}[n_{000}+i]=\mathbf{x}[n_{001}+i]\land\mathbf{x}[n_{010}+i]=\mathbf{x}[n_{011}+i]\land$$ $$\displaystyle\mathbf{x}[n_{100}+i]=\mathbf{x}[n_{101}+i]\land\mathbf{x}[n_{110}+i]=\mathbf{x}[n_{111}+i]).$$ Let finally $$\displaystyle\psi(a,b,c)=((a<b<c)\land(\exists n_{000},n_{001},...,n_{111}\colon$$ $$\displaystyle(\forall d,e,f\in\{0,1\}\colon\varphi_{def})\land$$ $$\displaystyle(\forall i\in[0,a)\colon\varphi_{0}(i))\land$$ $$\displaystyle(\forall i\in(a,b)\colon\varphi_{1}(i))\land$$ $$\displaystyle(\forall i\in(b,c)\colon\varphi_{2}(i)))).$$ Proposition 5.3. Let $X$ be the orbit closure of $\mathbf{x}$ in $\{0,1\}^{\mathbb{N}}$. Then $$P_{111}(W(X))=\{(a,b,c)\in\mathbb{N}^{3}:\psi(a,b,c)\}.$$ Proof. Suppose that there is a word $\mathbf{y}$ in the winning shift of $X$ such that it has letter $1$ at positions $a$, $b$, $c$ with $a<b<c$ and letter $0$ elsewhere so that $s(\mathbf{y})=(a,b,c)\in P_{111}(W(X))$. By the definition of the winning shift, there exist words $w_{000},w_{001},\ldots,w_{111}\in\mathcal{L}(X)$, all of length $c+1$, such that the rooted partial binary tree formed by these words has outdegree $2$ at nodes on depths $a$, $b$, $c$, and outdegree $1$ elsewhere. This tree is the strategy tree for Alice’s winning strategy. Since $X$ is over the alphabet $\{0,1\}$, at every branching we have one branch starting with $0$ and one starting with $1$. By ordering the words suitably, we may assume that $$w_{def}=u\cdot d\cdot u_{d}\cdot e\cdot u_{de}\cdot f$$ for all $d,e,f\in\{0,1\}$, and for some words $u,u_{d},u_{de}$, where $|u|=a$, $|u_{d}|=b-a-1$, $|u_{de}|=c-b-1$, and, as indicated by the subscripts, $u$ is the same for all $w_{def}$, $u_{d}$ only depends on $d$ and $u_{de}$ only depends on $d$ and $e$. By the assumption that $X=\overline{\mathcal{O}(\mathbf{x})}$, there exist $n_{def}\in\mathbb{N}$ such that $\mathbf{x}[n_{def},n_{def+c}]=w_{def}$ for all $d,e,f\in\{0,1\}$. These $n_{def}$ satisfy all of the $\forall$-statements in $\psi$. The first statement $$\forall d,e,f\in\{0,1\}\colon(\mathbf{x}[n_{def}+a]=d\land\mathbf{x}[n_{def}+b]=e\land\mathbf{x}[n_{def}+c]=f)$$ holds because $w_{def}[a]=d\implies\mathbf{x}[n_{def}+a]=d$, so $\mathbf{x}[n_{def}+a]=d$ holds for all $d$, $e$, $f$. Similarly $\mathbf{x}[n_{def}+b]=e$ and $\mathbf{x}[n_{def}+c]=f$ hold by the definition of the words $w_{def}$. The second statement $$\displaystyle\forall i\in[0,a)$$ $$\displaystyle\colon$$ $$\displaystyle($$ $$\displaystyle\mathbf{x}[n_{000}+i]=\mathbf{x}[n_{001}+i]=\mathbf{x}[n_{010}+i]=\mathbf{x}[n_{011}+i]=$$ $$\displaystyle\mathbf{x}[n_{100}+i]=\mathbf{x}[n_{101}+i]=\mathbf{x}[n_{110}+i]=\mathbf{x}[n_{111}+i])$$ holds because $\mathbf{x}[n_{def}+i]=w_{def}[i]=u[i]$ for all $i\in[0,a)$ and $d,e,f\in\{0,1\}$, so all eight values $\mathbf{x}[n_{def}+i]$ are the same. The last two $\forall$-statements correspond to $u_{d}$ and $u_{de}$ and are justified similarly. Conversely, if the $\forall$-statements hold for some choices of the $n_{def}$, then the words $\mathbf{x}[n_{def},n_{def+c}]$ are of the desired form $w_{def}=u\cdot d\cdot u_{d}\cdot e\cdot u_{de}\cdot f$, and this proves that $(a,b,c)\in P_{111}(W(X))$. ∎ The following corollary can be deduced from the previous proposition using standard results concerning $\mathcal{S}$-automatic sequences, but we provide the proof in our special case. See the discussion after Theorem 5.5 for the full story. Corollary 5.4. Let $\mathcal{S}$ be an addable ANS, and let $\mathbf{x}$ in $\{0,1\}^{\mathbb{N}}$ be $\mathcal{S}$-automatic. Then $P_{111}(W(\overline{\mathcal{O}(\mathbf{x})}))$ is $\mathcal{S}$-recognizable. Proof. Let $\mathcal{S}=(L,\prec)$. We explain the steps for constructing a finite automaton that accepts $\mathrm{rep}(P_{111}(W(\overline{\mathcal{O}(\mathbf{x})})))$. Let $\mathcal{A}$ be a DFAO for $\mathbf{x}$. Let $(a,b,c)\in P_{111}(W(\overline{\mathcal{O}(\mathbf{x})}))$. By the definition of an ANS, the condition $a<b<c$ is equivalent with $$\mathrm{rep}(a)\prec\mathrm{rep}(b)\prec\mathrm{rep}(c)\land\mathrm{rep}(a)\neq\mathrm{rep}(b)\land\mathrm{rep}(b)\neq\mathrm{rep}(c).$$ and this can be checked at the end. The check is possible because addability implies that $\mathcal{S}$ is comparable (i.e., the relation $\prec$ is rational) by 3.3. Consider a tuple $$(w_{a},w_{b},w_{c},w_{000},w_{001},\ldots,w_{111})$$ and define $a=\mathrm{val}(w_{a})$, $b=\mathrm{val}(w_{b})$, $c=\mathrm{val}(w_{c})$, and $n_{def}=\mathrm{val}(w_{def})$ for $d,e,f\in\{0,1\}$. Since regular languages over a Cartesian product alphabet are closed under projection, we can eliminate existentially quantified words, and thus it is enough to show that the language of tuples such that $a$, $b$, $c$, $n_{def}$ satisfy $\psi$ is regular. Notice that $L$ is regular by addability, so we can check that all of the words are in $L$ by intersecting with the Cartesian product language $L^{11}$ (with padding). Using addability, we can construct an automaton $A_{d,e,f,0}$ such that $A_{d,e,f,0}$ accepts the word $(w_{a},w_{b},w_{c},w_{000},...,w_{111})$ over $\{0,1,\#\}^{11}$ if and only if $$\mathbf{x}[n_{def}+a]=d.$$ Namely, $A_{d,e,f,0}$ is accepted by $\mathcal{A}\circ(+)\circ\pi_{0,3+4d+2e+f}$ by taking the final states to be those where $\mathcal{A}$ outputs the symbol $d$, where $(+)\colon L^{2}\to L$ is the sum transduction corresponding to $\mathcal{S}$ and $\pi_{i_{1},\ldots,i_{k}}$ is the projection to components $i_{1}$, $\ldots$, $i_{k}$ of the alphabet. Similarly, we can construct automata $A_{d,e,f,1}$ and $A_{d,e,f,2}$ that accept when $\mathbf{x}[n_{def}+b]=e$ and when $\mathbf{x}[n_{def}+a]=d$, respectively. Now, let $L_{0}$ be the (finite) intersection of the languages of $A_{d,e,f,i}$ for $d,e,f\in\{0,1\}$, $i\in\{0,1,2\}$. The language $L_{0}$ is regular by closure properties of regular languages. Clearly $L_{0}$ contains precisely those tuples $(w_{a},w_{b},w_{c},w_{000},\ldots,w_{111})$ such that the corresponding numbers $a$, $b$, $c$, $n_{def}$, $d,e,f\in\{0,1\}$, satisfy the first $\forall$-statement of $\psi$. The other three $\forall$-statements are actual $\forall$-quantifications (rather than shorthands for a finite conjunction). We explain how to handle the middle one $$\displaystyle\forall i\in(a,b)$$ $$\displaystyle\colon$$ $$\displaystyle($$ $$\displaystyle\mathbf{x}[n_{000}+i]=\mathbf{x}[n_{001}+i]=\mathbf{x}[n_{010}+i]=\mathbf{x}[n_{011}+i]\land$$ $$\displaystyle\mathbf{x}[n_{100}+i]=\mathbf{x}[n_{101}+i]=\mathbf{x}[n_{110}+i]=\mathbf{x}[n_{111}+i]),$$ the others being similar. We first define an auxiliary automaton that accepts those $12$-tuples $(w_{a},w_{b},w_{c},w_{000},...,w_{111},w_{i})$ such that the formula holds with $i=\mathrm{val}(w_{i})$ and then $\forall$-eliminate $w_{i}$. To accept such $12$-tuples, observe again that we can compute the value $\mathbf{x}[n_{000}+i]$ by an automaton (just like in the definition of $A_{d,e,f,j}$). Thus, we can also perform a comparison of these finitely many values, and compute the conjunction $\land$ of these values at the end of the computation. Then $(w_{a},w_{b},w_{c},w_{000},...,w_{111},w_{i})$ is accepted in the correct situations whenever $\mathrm{rep}(w_{i})\in(a,b)$. Since we want to eliminate a $\forall$-quantifier, we want the automaton to accept all other values of $i$. For this, we use the rationality of $\prec$. The automaton can simply check whether $w_{i}\in L$ and $w_{a}\prec w_{i}\prec w_{b},w_{i}\neq w_{a},w_{i}\neq w_{b}$ holds, and accept if this is not the case. Elimination of $\forall$-quantifiers is dual to elimination of $\exists$-quantifiers. Simply complement, project to all but the $12$th coordinate, and complement again. Now, let $L_{0}$, $L_{1}$, $L_{2}$, and $L_{3}$ be the four regular sublanguages of $L^{11}$ corresponding to the $\forall$-quantifiers. Then $\pi_{0,1,2}(L_{0}\cap L_{1}\cap L_{2}\cap L_{3})$ is the desired regular language. ∎ As indicated above, 5.3 and the previous corollary are straightforward to generalize. The assumption of a binary alphabet has no effect on the formulas in 5.3, but there are some minor changes in the proof ($d$, $e$, $f$ should range over some cardinality two subsets of the alphabet). To cover a general word $v$, one should draw a directed tree where all nodes on depth $i$ have out-degree $v_{i}+1$. These nodes can be indexed naturally by words $u$ such that $u\leq v$ (with elementwise comparison), and using a variable $n_{u}$ for each such $u$ one can then easily program the analogue $\varphi_{v}$ of the formula $\varphi_{def}$ by writing out in first-order logic that the words starting at the positions $n_{u}$ form a strategy tree. We obtain the following result. Theorem 5.5. If $X$ is a subshift generated by $\mathbf{x}$ and $v$ is a word of length $k$ in $\mathbb{N}_{>0}^{*}$, then there exists a first-order formula $\psi(n_{1},\ldots,n_{k})$ on $\mathcal{S}$-recognizable predicates such that $$P_{v}(W(X))=\{(n_{1},\ldots,n_{k})\in\mathbb{N}^{k}:\psi(n_{1},\ldots,n_{k})\}.$$ The conclusion of $\mathcal{S}$-recognizability of 5.4 is obtained whenever a predicate is formed from $\mathcal{S}$-recognizable predicates using the logical connectives $\land$, $\lor$, $\lnot$, $\implies$, $\iff$ and the quantifiers $\forall$, $\exists$ on variables describing elements of $\mathbb{N}$. In the case of the usual base-$b$ numeration system, this is the famous Büchi-Bruyère Theorem (see [4] for a proof). The extension to general ANS is straightforward and is sketched in [7, Sect. 7.3]. The important point here is that addability and existence of an automaton for $\mathbf{x}$ ensures that the predicates $\varphi_{def}$, $\varphi_{0}$, $\varphi_{1}$, and $\varphi_{2}$ are $\mathcal{S}$-recognizable. Theorem 5.6. Let $\mathcal{S}$ be an addable ANS. Let $\mathbf{x}$ be an $\mathcal{S}$-automatic word and $X$ its orbit closure. Then the winning shift $W(X)$ of $X$ is weakly $\mathcal{S}$-codable. Proof. By 5.2, it suffices to prove that $P_{v}(W(X))$ is $\mathcal{S}$-recognizable for all $v\in\mathbb{N}_{>0}^{*}$. By Theorem 5.5, the set $P_{v}(W(X))$ is defined by a first-order formula $\psi$ on $\mathcal{S}$-recognizable predicates. Similar to 5.4, it follows from the theory of $\mathcal{S}$-automatic sequences with addable $\mathcal{S}$ that $\mathrm{rep}(P_{v}(W(X)))$ is regular. ∎ 6 $\mathcal{S}$-codable Winning Shifts In this section, we consider conditions that ensure that a winning shift of a subshift generated by an $\mathcal{S}$-automatic word is $\mathcal{S}$-codable. A factor $w$ of a word $\mathbf{x}$ is right special if $wa,wb\in\mathcal{L}(\mathbf{x})$ for distinct letters $a$ and $b$. Proposition 6.1. If $X$ is a transitive subshift with sublinear factor complexity, then $W(X)$ is countable and has finite coding dimension. Proof. Let $X$ be a transitive subshift with sublinear factor complexity. Let $s_{\mathbf{x}}(n)$ be the number of right special factors of length $n$ in $\mathcal{L}(\mathbf{x})$ for $\mathbf{x}\in X$. Assume that $W(X)$ has infinite coding dimension. Since $W(X)$ is hereditary, there exists a strictly increasing sequence $(B_{i})$ of positive integers such that there exists $\mathbf{x}_{i}\in W(X)$ with $\sum\mathbf{x}_{i}=B_{i}$ for all $i$. Moreover, we may assume that each $\mathbf{x}_{i}$ contains only the letters $0$ and $1$. Each strategy tree corresponding to $\mathbf{x}_{i}$ branches exactly $B_{i}$ times. Clearly the final branching of a strategy tree corresponds to $2^{B_{i}-1}$ right special words of a common length in the language of $X$. Since $(B_{i})$ is unbounded, it follows by transitivity that $s_{\mathbf{y}}(n)$ is unbounded for some $\mathbf{y}\in X$. A famous result of Cassaigne [6, Thm. 1], [3, Thm. 4.9.3] states that the infinite word $\mathbf{y}$ has sublinear factor complexity if and only $s_{\mathbf{y}}(n)$ is bounded. Therefore $\mathbf{y}$ has superlinear factor complexity, and we conclude that $X$ has superlinear factor complexity as well. If $W(X)$ has finite coding dimension, then $W(X)$ is clearly countable. The claim follows. ∎ Putting 6.1 and Theorem 5.6 together gives the following result which is the main result of the paper. Theorem 6.2. Let $\mathcal{S}$ be an addable ANS. Suppose that $X$ is a subshift generated by an $\mathcal{S}$-automatic word having sublinear complexity. Then $W(X)$ is $\mathcal{S}$-codable. We have the following immediate corollary for Pisot numeration systems. Notice that the usual base-$b$ numeration system is a Pisot numeration system. Corollary 6.3. Let $\mathcal{S}$ be a Pisot numeration system. If $X$ is a subshift generated by an $\mathcal{S}$-automatic word, then $W(X)$ is $\mathcal{S}$-codable. Proof. Let $\mathbf{x}$ be an $\mathcal{S}$-automatic word. The subshift generated by $\mathbf{x}$ is transitive by definition. By [19, Thm. 3.4] all Parry-automatic words have sublinear factor complexity. Since all Pisot numeration systems are Parry numeration systems [19, Remark 3], it follows that $\mathbf{x}$ has sublinear factor complexity. It is a well-know fact that Pisot numeration systems are addable [10], so the claim follows from Theorem 6.2. ∎ We note that we cannot prove a result analogous to 6.3 for Parry-automatic words since addition is not always rational in a Parry numeration system. See [11, Ex. 3]. Lemma 6.4. There exists a hereditary subshift $X$ with sublinear factor complexity such that $X$ has infinite coding dimension. Proof. Let $(n_{i})$ be a sequence of positive integers, and set $m_{j}=\sum_{i=1}^{j}(n_{i}+1)$ for $j\geq 1$. In order to make the arguments below work out, we choose the sequence $(n_{i})$ to satisfy $m_{j-1}=\mathcal{O}(\log n_{j})$. Let $\mathbf{x}=\prod_{i=1}^{\infty}0^{n_{i}}1$, and let $X$ be the hereditary closure of the subshift generated by $\mathbf{x}$. It is clear that $X$ has infinite coding dimension, so it suffices to show that $X$ has sublinear factor complexity. Suppose that $n$ is such that $n_{j}+2\leq n\leq n_{j+1}+1$ for some $j\geq 2$. When $p\geq m_{j-1}$, the factor $\mathbf{x}[p,p+n-1]$ contains at most one letter $1$, so these positions contribute a total of $n+1$ to the factor complexity of $\mathbf{x}$. Moreover, downgrading a letter (using hereditarity) does not increase the number of factors. Therefore we need to show that the contribution from the earlier positions, including downgrading, is $\mathcal{O}(n)$. Before downgrading, we have at most $m_{j-1}$ distinct factors of length $n$. Since $n\leq n_{j+1}+1$, each early factor contains at most $j-1$ letters $1$. Downgrading each letter independently thus produces at most $\smash[t]{\sum_{i=1}^{j-1}2^{i}=2^{j}-2}$ new factors. Now $\log\log n_{j}=\Omega(\log m_{j-1})$. Since $m_{j}$ clearly grows at least exponentially, we find that $\log m_{j-1}=\Omega(j)$, so $j=\mathcal{O}(\log\log n_{j})$. As $n_{j}<n$, we thus have $j=\mathcal{O}(\log\log n)$. Since $\smash[t]{n+1+(2^{j}-2)m_{j-1}=n+1+\mathcal{O}(\log^{2}n)}$, we thus have $\mathcal{O}(n)$ factors of length $n$. ∎ A simple argument using the pumping lemma shows that the word in the proof of 6.4 is not Pisot-automatic. Next we provide an example of a substitutive subshift associated with a regular ANS which does not have finite coding dimension. This example was suggested to us by J. Cassaigne. We show below that the associated ANS is not addable, so we cannot deduce that the conclusion of Theorem 6.2 fails without the assumption of sublinear factor complexity. Let $\sigma$ be the substitution defined by $a\mapsto abab$, $b\mapsto b$. Let $$\mathbf{z}=ababbababbbababbababbbbababbababbbababbababbbbbababbababbbababbababbbb\dotsm$$ be its infinite fixed point and $Z=\overline{\mathcal{O}(\mathbf{z})}$. It follows from [3, Thm. 4.7.66] that the factor complexity of $\mathbf{z}$ is in $\Theta(n^{2})$. (For a direct proof of the lower bound, consider words of the form $uab^{n}av$ where $b^{n}$ covers the central position, and observe that one may choose the positions of the two $a$’s freely to obtain $m^{2}$ words of length $2m$.) Our aim is to prove the following result. Proposition 6.5. The winning shift $W(Z)$ of $Z$ has infinite coding dimension. For this, we need information on right special factors in $\mathcal{L}(Z)$ and the following general lemma. Lemma 6.6. Let $X$ be a subshift that satisfies the following properties: (i) Whenever $w$ is a long enough right special factor in $\mathcal{L}(X)$, there exists a letter $a$ such that $wa$ is right special in $\mathcal{L}(X)$. (ii) Right special words are dense in $X$ meaning that each word in $X$ is a limit of a sequence of right special words of $\mathcal{L}(X)$. Then $W(X)$ has infinite coding dimension. Proof. Suppose that $\mathbf{x}=u0^{\omega}$ and $\mathbf{x}\in W(X)$. The condition (ii) implies that there are arbitrarily long right special words in $\mathcal{L}(X)$, so we may assume that $\sum u>0$. Consider the strategy tree corresponding to $u$. Let $w_{0}$, $\ldots$, $w_{k-1}$ be the words in $\mathcal{L}(X)$ that correspond to paths from the root to the leaves. The condition (ii) guarantees that each $w_{i}$ can be extended to a right special factor $u_{i}$. Moreover, the condition (i) implies that each $u_{i}$ can be extended to a right special factor of length $\ell$ where $\ell$ does not depend on $i$. Consider the words of length $|v_{i}|+1$ obtained from the words $v_{i}$ by extending each $v_{i}$ in at least two ways. By considering the tree corresponding to these words, we find that it is a strategy tree for $u0^{\ell-|u|}1$. In other words, the word $u0^{\ell-|u|}10^{\omega}$ belongs to $W(X)$. By repeating the argument on this word, we find words in $W(X)$ with arbitrarily large sums. The claim follows. ∎ It is easy to show that $\mathbf{z}=\lim_{k}p_{k}$ where $p_{1}=a$ and $p_{k+1}=p_{k}b^{k}p_{k}$. From this, it is straightforward to deduce the following. Lemma 6.7. The right special factors in $\mathcal{L}(Z)$ ending with $ab^{k}$ are exactly the suffixes of $b^{k}p_{k}b^{k}$ for $k\geq 1$. Proof of 6.5. It suffices to verify the conditions of 6.6. Let $w$ be a right special factor in $\mathcal{L}(Z)$. For the condition (i), it suffices to prove that $wb$ is also right special in $\mathcal{L}(Z)$. If $w=b^{i}$, then it is clear that $w$ and $wb$ are right special. Otherwise there exists $k\geq 1$ such that the word $w$ is a suffix of $b^{k}p_{k}b^{k}$ by 6.7. The word $b^{k+1}p_{k+1}b^{k+1}$ is right special according to 6.7, and it has suffix $b^{k}p_{k}b^{k+1}$ meaning that $wb$ is right special. The condition (ii) is immediate as $Z=\overline{\mathcal{O}(\mathbf{z})}$ and $\mathbf{z}$ has arbitrarily long right special prefixes by 6.7. ∎ The sums grow slowly, the shortest factor in $W(Z)$ with sum at least $4$ is $1010^{4}10^{197}1$. Let us then describe an ANS $\mathcal{Z}$ associated with $\sigma$ and $\mathbf{z}$. We set $\mathrm{rep}_{S}(0)=\varepsilon$. Let $n$ be a positive integer and $\ell$ be the smallest integer such that $$|\sigma^{\ell}(a)|\leq n<|\sigma^{\ell+1}(a)|=|\sigma^{\ell}(aba)|+1.$$ Define $$\displaystyle m$$ $$\displaystyle=\begin{cases}|\sigma^{\ell}(a)|,&\text{if $n=|\sigma^{\ell}(a)|$},\\ |\sigma^{\ell}(ab)|,&\text{if $|\sigma^{\ell}(ab)|\leq n<|\sigma^{\ell}(aba)|$},\\ |\sigma^{\ell}(aba)|,&\text{if $n=|\sigma^{\ell}(aba)|$}\end{cases}\quad\text{and}$$ $$\displaystyle\delta$$ $$\displaystyle=\begin{cases}1,&\text{if $n=|\sigma^{\ell}(a)|$},\\ 2,&\text{if $|\sigma^{\ell}(ab)|\leq n<|\sigma^{\ell}(aba)|$},\\ 3,&\text{if $n=|\sigma^{\ell}(aba)|$}.\end{cases}$$ We define recursively $\mathrm{rep}_{S}(n)=\delta\alpha$ where $\alpha$ is the word $\mathrm{rep}_{S}(n-m)$ with enough initial zeros to make $\mathrm{rep}_{S}(n)$ a word of length $\ell+1$. What happens here is that we find which of the words $\sigma^{\ell}(a)$, $\sigma^{\ell}(ab)$, $\sigma^{\ell}(aba)$ is the longest prefix of the prefix $p$ of $\mathbf{z}$ of length $n$, write a symbol in $\{1,2,3\}$ according to the word, and repeat the procedure for the remaining suffix of $p$. The remaining suffix is a prefix of $\mathbf{z}$ because the square of the word $\sigma^{\ell}(a)$ is always a prefix of $\mathbf{z}$. The representations of the integers $1$ to $22$ are given in Table 1. The numeration system $\mathcal{Z}$ is the Dumont-Thomas numeration system associated with the substitution $\sigma$. See [25, Sect. 2.1] for more details about these numeration systems. Notice that the language $\mathrm{rep}_{S}(\mathbb{N})$ is regular as it has the regular expression $$2(0+2)^{*}+2(0+2)^{*}(1+3)0^{*}+30^{*}+10^{*}.$$ Moreover, the order relation $\prec$ is given by the radix order based on the total order $0\prec 1\prec 2\prec 3$ on the alphabet. The fixed point $\mathbf{z}$ is a $\mathcal{Z}$-automatic word. If we use the convention that the first letter of $\mathbf{z}$ equals the output of an automaton fed with $\mathrm{rep}_{S}(1)$, this is easy to see by consulting Table 1. Indeed, the words $\sigma(a)$ and $\sigma(b)$ both end with $b$, so $a$ needs to be outputted exactly when the representation ends with $1$ or $3$. The word $\mathbf{z}$ is still $\mathcal{Z}$-automatic if we index from $0$. While a DFAO could be written for $\mathbf{z}$ in this case, it is easiest to make the conclusion using well-known closure properties of $\mathcal{Z}$-automatic words; see [26, Prop. 14]. The point is that the successor function maps regular sets to regular sets; see the proof of [15, Prop. 16]. Combining the above with 6.5, we obtain the following result. Proposition 6.8. There exists a comparable ANS $\mathcal{S}$ and an $\mathcal{S}$-automatic word $\mathbf{x}$ such that $W(\overline{\mathcal{O}(\mathbf{x})})$ has infinite coding dimension. Let us next prove that the ANS $\mathcal{Z}$ is not addable. Proposition 6.9. The ANS $\mathcal{Z}$ is not addable. Proof. To simplify notation, we write $u+v$ in place of $\mathrm{rep}(\mathrm{val}(u)+\mathrm{val}(v))$ when $u,v\in\mathrm{rep}(\mathbb{N})$. Notice that if $n>0$, then $$2^{n}30^{m-n}0^{\ell}+1=2^{n-1}30^{m-n+1}0^{\ell}$$ since there is no word in $\mathrm{rep}(\mathbb{N})$ strictly between the two in lexicographic order. Now consider $w$ in $\mathrm{rep}(\mathbb{N})$ such that $|w|\leq\ell$ and $\mathrm{val}(w)>n+1$. Then by induction we have $$\displaystyle 2^{n}30^{m-n}0^{\ell}+w$$ $$\displaystyle=30^{m}0^{\ell}+\mathrm{rep}(\mathrm{val}(w)-n)$$ $$\displaystyle=100^{m}0^{\ell}+\mathrm{rep}(\mathrm{val}(w)-n-1)$$ $$\displaystyle=200^{m}0^{\ell}+\mathrm{rep}(\mathrm{val}(w)-n-2).$$ By the special form of the language, the left quotient $200^{m}\setminus\mathrm{rep}(\mathbb{N})$ is equal to $\mathrm{rep}(\mathbb{N})$, from which we obtain that if $u=\mathrm{rep}(\mathrm{val}(w)-n-2)$ then $$200^{m}0^{\ell}+u=200^{m}0^{\ell-|u|}u.$$ Assume for a contradiction that $\mathcal{Z}$ is addable, that is, assume that the addition relation $R$, which is a subset of $\mathrm{rep}(\mathbb{N})^{3}$, is regular and accepted by an automaton $\mathcal{A}$ with $Q$ states. Pick $m>Q$, $\mathrm{val}(w)>m+1$, and $\ell\geq|w|$. Then by the above argument, $R$ contains the triple $$\begin{bmatrix}2^{n}30^{m-n}0^{\ell}\\ w\\ 200^{m}0^{\ell-|\mathrm{rep}(\mathrm{val}(w)-n-2)|}\mathrm{rep}(\mathrm{val}(w)-n-2)\end{bmatrix}^{\#}$$ for each $n$ such that $0\leq n\leq m$. By the pigeonhole principle, there exist $n_{1}$ and $n_{2}$ such that $0\leq n_{1}<n_{2}\leq m$ and in some accepting runs for $$t_{1}=\begin{bmatrix}2^{n_{1}}30^{m-n_{1}}0^{\ell}\\ w\\ 200^{m}0^{\ell-|\mathrm{rep}(\mathrm{val}(w)-n_{1}-3)|}\mathrm{rep}(\mathrm{val}(w)-n_{1}-2))\end{bmatrix}^{\#}$$ and $$t_{2}=\begin{bmatrix}2^{n_{2}}30^{m-n_{2}}0^{\ell}\\ w\\ 200^{m}0^{\ell-|\mathrm{rep}(\mathrm{val}(w)-n_{2}-3)|}\mathrm{rep}(\mathrm{val}(w)-n_{2}-2)\end{bmatrix}^{\#}$$ the automaton $\mathcal{A}$ is in the same state $q$ after reading the prefix of length $m+2$. This means that also $$t=\begin{bmatrix}2^{n_{1}}30^{m-n_{1}}0^{\ell}\\ w\\ 200^{m}0^{\ell-|\mathrm{rep}(\mathrm{val}(w)-n_{2}-3)|}\mathrm{rep}(\mathrm{val}(w)-n_{2}-2)\end{bmatrix}^{\#}$$ is in $R$ since we can use the accepting run for $t_{1}$ for $m+2$ steps and then use the accepting run for $t_{2}$ for the remaining steps. However, $t\notin R$, since $$\displaystyle 2^{n_{1}}30^{m-n_{1}}0^{\ell}+w$$ $$\displaystyle=200^{m}0^{\ell-|\mathrm{rep}(\mathrm{val}(w)-n_{1}-3)|}\mathrm{rep}(\mathrm{val}(w)-n_{1}-3))$$ $$\displaystyle\neq 200^{m}0^{\ell-|\mathrm{rep}(\mathrm{val}(w)-n_{2}-3)|}\mathrm{rep}(\mathrm{val}(w)-n_{2}-3)).$$ This is a contradiction. ∎ 7 Automata for the Winning Shifts of Certain Automatic Words Theorem 6.2 is constructive: it is in principle possible to find an automaton accepting a coding of $W(X)$. There are software packages like Walnut [21] that can do this automatically. In our attempt to directly input the formula of Theorem 5.5, Walnut quickly ran out of memory even in the case of one of the simplest automatic words, the Thue-Morse word. Using a logically equivalent form of the formulas, the computation becomes manageable. Let $\mathbf{x}$ be a binary $\mathcal{S}$-automatic word for an addable ANS $\mathcal{S}$ and $X$ the subshift generated by $\mathbf{x}$. Let $\operatorname{factorEq}(i,n,m)$ be a predicate that is true if and only if $\mathbf{x}[n,n+i-1]=\mathbf{x}[m,m+i-1]$. Let $\operatorname{isRS}(i,n)$ be a predicate that is true if and only if the factor $\mathbf{x}[n,n+i-1]$ is right special. Consider a predicate $\operatorname{extRS2}(i,j,n)$ defined by the formula $$\displaystyle\operatorname{extRS2}(i,j,n)$$ $$\displaystyle=i<j\land\exists m_{1},m_{2}\colon$$ $$\displaystyle(\operatorname{isRS}(j,m_{1})\land\operatorname{isRS}(j,m_{2})\land$$ $$\displaystyle\operatorname{factorEq}(i,m_{1},m_{2})\land\operatorname{factorEq}(i,n,m_{1})\land$$ $$\displaystyle\mathbf{x}[m_{1}+i]\neq\mathbf{x}[m_{2}+i]).$$ The predicate is true if and only if at position $n$ of $\mathbf{x}$ there is a right special factor $w$ of length $i$ such that both $w0$ and $w1$ can be extended to a right special factor of length $j$. In other words, the predicate $\exists n\operatorname{extRS2}(i,j,n)$ is true if and only if there is a strategy tree with branchings at positions $i$ and $j$. The latter is equivalent with $0^{i}10^{j-i-1}10^{\omega}\in W(X)$. The formula $$\displaystyle\operatorname{extRS3}(i,j,k,n_{1},n_{2})$$ $$\displaystyle=i<j\land\operatorname{extRS2}(j,k,n_{1})\land\operatorname{extRS2}(j,k,n_{2})\land$$ $$\displaystyle\hskip 11.99998pt\operatorname{factorEq}(i,n_{1},n_{2})\land\mathbf{x}[n_{1}+i]\neq\mathbf{x}[n_{2}+i]$$ encodes strategy trees with three branchings. Here $n_{1}$ and $n_{2}$ are positional variables that indicate starting positions of right special factors in $\mathbf{x}$. There are two of them because for three branchings there are two right special factors of a common length whose extensions can all be extended to right special factors of a common length. The following formula encodes strategy trees with four branchings: $$\displaystyle\operatorname{extRS4}(i,j,k,\ell,n_{1},n_{2},n_{3},n_{4})$$ $$\displaystyle=i<j\land\operatorname{extRS3}(j,k,\ell,n_{1},n_{2})\land\operatorname{extRS3}(j,k,\ell,n_{3},n_{4})\land$$ $$\displaystyle\hskip 11.99998pt\operatorname{factorEq}(i,n_{1},n_{2})\land\operatorname{factorEq}(i,n_{2},n_{3})\land\operatorname{factorEq}(i,n_{3},n_{4})\land$$ $$\displaystyle\hskip 11.99998pt\mathbf{x}[n_{1}+i]=\mathbf{x}[n_{2}+i]\land\mathbf{x}[n_{2}+i]\neq\mathbf{x}[n_{3}+i]\land$$ $$\displaystyle\hskip 11.99998pt\mathbf{x}[n_{3}+i]=\mathbf{x}[n_{4}+i].$$ The advantage of this formulation is that the number of positional free variables is cut to one quarter. It also helps that we can directly use previously generated automata for $\operatorname{extRS1}$, $\operatorname{extRS2}$, $\ldots$ instead of encoding all previous steps into a single formula. Let now $\mathbf{x}$ be the Thue-Morse word, the fixed point $\mu^{\omega}(0)$ of the substitution $\mu\colon 0\mapsto 01,1\mapsto 10$. It is straightforward to prove that the $n$th letter of $\mathbf{t}$ equals the number of $1$’s in the binary representation of $n$ modulo $2$, so $\mathbf{x}$ is a $2$-automatic word. The following expresses the above predicates in Walnut’s syntax for $\mathbf{x}$. def factorEq "Ai (0 <= i & i < k) => T[n+i] = T[m+i]": def isRS "Em1,m2 $factorEq(k,n,m1) & $factorEq(k,n,m2) & T[m1+k] != T[m2+k]": def extRS1 "En $isRS(i,n)": def extRS2 "i < j & Em1,m2 $isRS(j,m1) & $isRS(j,m2) & $factorEq(i,m1,m2) & $factorEq(i,n,m1) & T[m1+i] != T[m2+i]": def extRS3 "i < j & $extRS2(j,k,n1) & $extRS2(j,k,n2) & $factorEq(i,n1,n2) & T[n1+i] != T[n2+i]": def extRS4 "i < j & $extRS3(j,k,l,n1,n2) & $extRS3(j,k,l,n3,n4) & $factorEq(i,n1,n2) & $factorEq(i,n2,n3) & $factorEq(i,n3,n4) & T[n1+i] = T[n2+i] & T[n2+i] != T[n3+i] & T[n3+i] = T[n4+i]": When we input def L4 "En1,n2,n3,n4 $extRS4(i,j,k,l,n1,n2,n3,n4)": to Walnut, we obtain an automaton that accepts $\mathrm{rep}(i,j,k,\ell)$ for those $4$-tuples $(i,j,k,\ell)$ for which the predicate $\exists n_{1},n_{2},n_{3},n_{4}\operatorname{extRS4}(i,j,k,\ell,n_{1},n_{2},n_{3},n_{4})$ is true. This automaton in fact rejects its all inputs, so there is no strategy tree with four branchings in $X$. Analogous computation for strategy trees with three branchings produces an automaton that accepts some $3$-tuples $(i,j,k)$. This means that the coding dimension of $X$ is $3$. Both computations finished in a few seconds. Let us then design an automaton accepting the encodings of all words in $W(X)$. We deviate slightly from Section 4 and encode the occurrences of the letter $1$ in a word of $W(X)$ as a triple $(a,b,c)$ as follows. We use $1$-based indexing and reserve the value $0$ to indicate that an occurrence is missing. For example, $(0,b,c)$ with $b,c\neq 0$ means that a word has exactly two occurrences of $1$ in positions $b$ and $c$. We require that $a\leq b\leq c$ and equality occurs only if both values are $0$. For example the words $10001000000010^{\omega}$ and $001000000010^{\omega}$, both of which are in $W(X)$, are respectively encoded as $(1,5,13)$ and $(0,3,11)$. The representations of the $3$-tuples in base $2$ are obtained by representing each component in base $2$ and padding with $0$ to achieve common length. The following Walnut code builds an automaton accepting the encodings $(a,b,c)$ of words of $W(X)$ as described. def tm "(a = 0 & b = 0 & c = 0) | (a = 0 & b = 0 & c > 0 & En $isRS(c-1,n)) | (a = 0 & b > 0 & c > 0 & En $extRS2(b-1,c-1,n)) | (a > 0 & b > 0 & c > 0 & En1,n2 $extRS3(a-1,b-1,c-1,n1,n2))": The automaton produced by Walnut is depicted in Figure 2. We saw above that the coding dimension of $W(X)$ is $3$. By a careful study of the automaton, we recover the following characterization of $W(X)$ described in [24, Sect. 2]. • The coding dimension of $W(X)$ is $3$. • If $\mathbf{x}$ in $W(X)$ contains three occurrences of $1$ at positions $a$, $b$, $c$ with $a<b<c$, then $a=1$, $c-b=2^{k}$ for some $k\geq 1$, and $b-1\leq 2^{k-1}$. • If $\mathbf{x}$ in $W(X)$ contains exactly two occurrences of $1$ at positions $b$, $c$ with $b<c$, then either $b=1$ or $c-b=2^{k}$ for some $k\geq 1$ and $b-1\leq 2^{k-1}$. • If $\mathbf{x}$ in $W(x)$ contains exactly one occurrence of $1$, then this occurrence can be at any position. The above procedure can be repeated for other automatic words. We did this for the regular paperfolding word, the Rudin-Shapiro word [1, Sect. 5.1], and the period-doubling word [1, Ex. 6.3.4] . The winning shift associated with the regular paperfolding word has coding dimension $3$, and the automaton of Figure 4 accepts it. The winning shift for the Rudin-Shapiro word has coding dimension $4$ and automaton of Figure 5. Finally, the winning shift for the period-doubling word has coding dimension $2$ and automaton of Figure 3. This automaton yields the following characterization for the winning shift $W(X)$ of the period-doubling word: • The coding dimension of $W(X)$ is $2$. • If $\mathbf{x}$ in $W(X)$ contains exactly two occurrences of $1$ at positions $a$, $b$ with $a<b$, then $b-a=2^{k}$ for some $k\geq 1$ and $a-1\leq 2^{k-1}$. • If $\mathbf{x}$ in $W(x)$ contains exactly one occurrence of $1$, then this occurrence can be at any position. 8 Winning Shifts of Sofic Shifts and $\omega$-regular Languages Let us first provide a proof that $W(X)$ is regular whenever $X$ is. This follows from [29, Prop. 3.8] in the case of a binary alphabet. We provide a general proof for completeness. It is a straightforward generalization of the arguments of [29, Prop. 3.8]. See [29, Fig. 2] for example automata. For the basics on sofic shifts, we refer the reader to [16]. Observe that the sofic shifts considered here are one-sided. Proposition 8.1. If $X$ is a regular language, then $W(X)$ is a regular language. In particular if $X$ is a sofic shift, then $W(X)$ is a sofic shift. Proof. Denote by $\mathcal{A}$ a deterministic finite automaton $(Q,\Sigma,\delta,q_{0},F)$ accepting a regular language $X$. Based on $\mathcal{A}$, we build a boolean automaton $\mathcal{B}$ (sometimes called an alternating automaton) accepting $W(X)$. Since boolean automata accept exactly the regular languages, we see that $W(X)$ is regular. For the proof of this fact and rigorous definition of boolean automata, see [5] for example. Let $Q=\{q_{1},\ldots,q_{n}\}$. The set of states of $\mathcal{B}$ is $Q$, and its set of final states is $F$. Let $\tau\colon Q\times\Sigma\to\mathbb{B}^{Q}$ be the transition function mapping a state and a letter to a boolean function. We define $$\tau(q,a)=\bigvee_{\begin{subarray}{c}\{c_{1},\ldots,c_{a+1}\}\subseteq\Sigma\\ |\{c_{1},\ldots,c_{a+1}\}|=a+1\end{subarray}}\bigwedge_{i=1}^{a+1}\delta(q,c_{i}).$$ For example, if $\Sigma=\{0,1\}$ and $a=1$, then $\tau(q,a)=s_{0}\land s_{1}$ where $s_{0}=\delta(q,0)$ and $s_{1}=\delta(q,1)$. The interpretation is that the formula is true and the input $aw$ is accepted if and only if the computations on two copies of $\mathcal{A}$ with respective initial states $s_{0}$ and $s_{1}$ and input $w$ are both accepted. In other words, we substitute $s_{0}$ with $1$ if $\mathcal{A}$ accepts $w$ from $s_{0}$ and $0$ otherwise. We do the same for $s_{1}$ and then evaluate $s_{0}\land s_{1}$. The function $\tau$ extends to $Q\times A^{*}$ recursively by setting $\tau(q,aw)$ to be the boolean function $f_{q,a}(\tau(q_{1},w),\ldots,\tau(q_{n},w))$ where $f_{q,a}=\tau(q,a)$ and $\tau(q,\varepsilon)=q$. We accept a word $w$ from state $q$ if and only if $\tau(q,w)$ evaluates to $1$. Let $L_{q}^{\mathcal{X}}$ be the language an automaton $\mathcal{X}$ accepts from state $q$. First of all, we have $\varepsilon\in W(L_{q}^{\mathcal{A}})$ if and only if $\varepsilon\in L_{q}^{\mathcal{B}}$ because the final states in $\mathcal{A}$ and $\mathcal{B}$ are the same. Let $a$ be a letter and $w$ a word. Then $$\displaystyle aw\in W(L_{q}^{\mathcal{A}})$$ $$\displaystyle\quad\Longleftrightarrow\quad\exists C\subseteq\Sigma\colon|C|=a+1\land\forall c\in C\colon w\in W\left(L_{\delta(q,c)}^{\mathcal{A}}\right)$$ $$\displaystyle\quad\Longleftrightarrow\quad\exists C\subseteq\Sigma\colon|C|=a+1\land\forall c\in C\colon w\in L_{\delta(q,c)}^{\mathcal{B}}$$ $$\displaystyle\quad\Longleftrightarrow\quad\bigvee_{\begin{subarray}{c}\{c_{1},\ldots,c_{a+1}\}\subseteq\Sigma\\ |\{c_{1},\ldots,c_{a+1}\}|=a+1\end{subarray}}\bigwedge_{i=1}^{a+1}\tau(\delta(q,c_{i}),w)=1$$ $$\displaystyle\quad\Longleftrightarrow\quad\tau(q,aw)=1$$ $$\displaystyle\quad\Longleftrightarrow\quad aw\in L_{q}^{\mathcal{B}}$$ where the second equivalence follows from the induction hypothesis. The first claim follows. The latter claim follows since $W(X)$ is shift-invariant and closed whenever $X$ is. (This is easy to show; see [29, Section 3] for details.) ∎ If the automaton $\mathcal{A}$ in the preceding proof is a Büchi automaton accepting a $\omega$-language $X$, then the constructed automaton $\mathcal{B}$ is a boolean Büchi automaton accepting $W(X)$. Since there is an equivalent Büchi automaton for each boolean Büchi automaton [20], we obtain the following result. Proposition 8.2. Let $X$ be a subset of $A^{\mathbb{N}}$. If $X$ is $\omega$-regular, then $W(X)$ is $\omega$-regular. The natural definition of $W(X)$ in this case is through branching structures of trees [28], and this is exactly analogous to how acceptance is defined in boolean $\omega$-automata, so the proof is exactly analogous. There is a small technical detail: Unlike in [5], the definition of a boolean automaton in [20] does not allow an arbitrary boolean combination but only a single conjunction or disjunction at each state (and $\varepsilon$-transitions are not allowed either). However, since $\omega$-regular languages are closed under inverse substitutions, it is enough to show that the image of $W(X)$ under the substitution $s\mapsto s\#$ is $\omega$-regular, allowing the simulation of an expression of the form $\bigvee\bigwedge$. That is, we use two transitions per one letter and switch back and forth between an automaton with conjunctions and an automaton with disjunctions. Notice that when $W(X)$ is $\mathcal{S}$-codable, it is not true that $W(X)$ is necessarily $\omega$-regular. Consider for example the winning shift $W(X)$ of the Thue-Morse word studied in Section 7. Suppose for a contradiction that there exists a Büchi automaton $\mathcal{A}$ recognizing $W(X)$. We first observe that if in a successful run a final state is visited twice, then the path corresponding to this cycle has label in $0^{*}$. Otherwise by pumping we could produce infinitely many occurrences of $1$ to a word in $W(X)$ and this is impossible as the coding dimension of $W(X)$ is $3$. Since we may assume that each final state is visited at least twice in some successful run, it follows that $\mathcal{A}$, when consider as a finite automaton, accepts $L0^{*}$ where $$L=\{w\in\{0,1\}^{*}1:w0^{\omega}\in W(X)\}.$$ Hence $L$ is regular. Let $L^{\prime}$ be the regular language of words belonging to $L$ and having exactly three occurrences of $1$. By the characterization provided in Section 7, we have $$L^{\prime}=\{10^{n}10^{m}1:\text{$m-1=2^{k}$ for some $k\geq 1$ and $n\leq 2^{k-1}$}\}.$$ A standard argument using the pumping lemma shows that $L^{\prime}$ is not regular; a contradiction. Definition 8.3. Let $\mathcal{S}$ be an ANS and $Y$ be a subset of $A^{\mathbb{N}}$. We say that $Y$ is (weakly) $\mathcal{S}$-codable if there exists a bijection $\pi\colon A\to\{0,1,\ldots,|A|-1\}$ such that $0\in\pi(A)$ and $\pi(Y)$ is (weakly) $\mathcal{S}$-codable. We call $\pi^{-1}(0)$ the zero symbol of $Y$ and $\pi^{-1}(0^{\omega})$ the zero point associated with $Y$. All other symbols are nonzero symbols. Definition 8.4. We say that a set is (weakly) $1$-codable if it is (weakly) $\mathcal{S}$-codable for the ANS $\mathcal{S}$ with language $0^{*}$ and radix order. When we say that a set is $1$-recognizable, we simply mean that it is $\mathcal{S}$-recognizable. We show in Theorem 8.8 that for any sofic shift $X$, the winning shift $W(X)$ is weakly $1$-codable, and in the countable case it is $1$-codable. Lemma 8.5. If $X$ is a sofic shift with a unique periodic point, then $X$ is countable and has finite coding dimension. Proof. Say $X$ is a sofic shift. We may assume that $X\subseteq\mathbb{N}^{\mathbb{N}}$. If $X$ has a periodic point with period greater than $1$, then it does not have unique periodic point. We may thus assume that the unique periodic point is $0^{\omega}$, and we take $0$ as the zero symbol. Suppose for a contradiction that there are words in $X$ with arbitrarily large sums. Recall that by the definition of soficity, $X$ is the set of edge labels of right-infinite paths in some finite graph. Fix such a graph defining $X$, and let $M$ be the number of vertices in this graph. Let $w_{n}$ be a word in the language of $X$ with sum at least $n$. In particular, because the alphabet of $X$ is finite, the support of $w_{n}$ tends to infinity in $n$. It is easy to see that if the support of $w_{n}$ is strictly larger than $M$, then the path corresponding to $w_{n}$ contains a cycle with a nonzero label (apply the pigeonhole principle to positions with a nonzero label). This cycle can be used to construct a periodic point which is not equal to $0^{\omega}$, and thus there are at least two periodic points, a contradiction. If $X$ has finite coding dimension, then $X$ is clearly countable. ∎ In the proof, one can also use Ogden’s lemma [23] from formal language theory. We recall the connection between sofic shifts and $\omega$-regular languages as this is needed in the following sections. (This would allow one to deduce the latter claim of 8.1 from 8.2.) Lemma 8.6. A subshift is sofic if and only if it is $\omega$-regular. Proof. A subshift $X$ is sofic if and only if it is the set of labels of right-infinite paths in a finite graph. If $X$ is sofic, we can directly use this graph as a deterministic Büchi automaton for $X$ as an $\omega$-language. Conversely, suppose $X$ is $\omega$-regular, so that it is accepted by a Büchi automaton. Without loss of generality, we may assume that from every state of the automaton there is a path to a final state. Then it is clear that making all the states final does not change the language (since $X$ is topologically closed) and making all states initial does not change the language either (since $X$ is shift-invariant). What is left is nothing but a finite edge-labeled graph, and the labels of its right-infinite paths form precisely $X$. ∎ Proposition 8.7. A subshift is sofic with a unique periodic point if and only if it is $1$-codable. Proof. Let $X$ be a subshift. We may assume that $X\subseteq\mathbb{N}^{\mathbb{N}}$ and $X$ has zero symbol $0$. Say $X$ is $1$-codable. Since $X$ has finite coding dimension, it must contain a unique periodic point. By $1$-codability, the sets $P_{v}(X)$ are $1$-recognizable for all $v\in\mathbb{N}_{>0}^{*}$. Let $v$ in $\mathbb{N}_{>0}^{*}$ be fixed and write $v=v_{0}\dotsm v_{\ell-1}$ with $\ell=|v|$. We have a finite automaton accepting the tuples $$(\#^{\ell-1-n_{0}}1^{n_{0}},\#^{\ell-1-n_{1}}1^{n_{1}},\ldots,1^{\ell-1})$$ such that $0^{n_{0}}v_{0}0^{n_{1}-n_{0}-1}v_{1}\dotsm v_{\ell-1}0^{\omega}\in Q_{v}(X)$ (recall that $Q_{v}(X)=e^{-1}(v)\cap X$). Because regular languages are closed under reversal, we also have an automaton $\mathcal{A}$ accepting the reversals $$(1^{n_{0}}\#^{\ell-1-n_{0}},1^{n_{1}}\#^{\ell-1-n_{1}},\ldots,1^{\ell-1})$$ of such word tuples. Clearly there is a bijective transducer that reduces $\omega$-words of the form $0^{n_{0}}v_{0}0^{n_{1}-n_{0}-1}v_{1}\dotsm v_{\ell-1}0^{\omega}$ to words of the form $(1^{n_{0}}\#^{\omega},1^{n_{1}}\#^{\omega},\ldots,1^{\ell-1}\#^{\omega})$, and the automaton $\mathcal{A}$ yields an automaton accepting these $\omega$-words. It follows that $Q_{v}(X)$ is $\omega$-regular. Then $X$ is also $\omega$-regular as a finite union of such $Q_{v}(X)$. Since $X$ is a subshift, it is sofic by 8.6. The other direction is exactly analogous. If $X$ is sofic, it is $\omega$-regular, and from its automaton we obtain another automaton accepting finite words $w=0^{n_{0}}v_{0}0^{n_{1}-n_{0}-1}v_{1}\dotsm v_{\ell-1}$ such that $w0^{\omega}\in Q_{v}(X)$ by additionally keeping track of the word formed by the nonzero symbols. After a transduction, we can accept the codings $(1^{n_{0}}\#^{\omega},1^{n_{1}}\#^{\omega},\ldots,1^{\ell-1}\#^{\omega})$ of such words, and after a reversal we see that $X$ is weakly $1$-codable. It follows from 8.5 that $X$ has finite coding dimension. Therefore $X$ is $1$-codable. ∎ Of course, the proof really shows that a set of $1$-codable if and only if it is $\omega$-regular and has finite coding dimension, but our interest in this paper is in closed sets. Theorem 8.8. If $X$ is a countable sofic shift, then $W(X)$ is $1$-codable. More generally, $W(X)$ is weakly $1$-codable for every sofic shift $X$. Proof. We may again assume that $X\subseteq\mathbb{N}^{\mathbb{N}}$ and $X$ has zero symbol $0$. A sofic shift is countable if and only if it has zero entropy. Thus, when $X$ is a countable sofic shift, $W(X)$ is a hereditary sofic shift with zero entropy [29, Prop. 5.7]. Say $W(X)$ has a periodic point other than $0^{\omega}$. Suppose that the word corresponding to the minimum period $p$ of this periodic point contains $\ell$ occurrences of nonzero letters. By downgrading each of these nonzero letters independently, we find that there are at least $2^{k\ell}$ factors of length $kp$ in words of $X$ for all $k$. It follows that $W(X)$ has positive entropy; a contradiction. Thus the only periodic point of $W(X)$ is $0^{\omega}$. It follows from 8.5 that $X$ has finite coding dimension. For a general sofic shift $X$, the winning shift $W(X)$ is sofic by 8.1. To see that $W(X)$ is weakly $1$-codable, it is enough to show that its restriction to points with sum at most $d$ is $1$-codable for all $d$. This restriction is clearly a sofic shift with a unique periodic point, and 8.7 applies. ∎ By the results of [8], this means that the winning shift $W(X)$ of a countable sofic shift $X$ is $\mathcal{S}$-codable for every regular ANS $\mathcal{S}$ with radix order. 9 Robustness Theorem 6.2 is only interesting if one agrees that the class of $\mathcal{S}$-codable subshifts is a natural one. In this section, we prove some closure properties for the class of (weakly) $\mathcal{S}$-codable sets for addable $\mathcal{S}$, which imply that this class is relatively robust. First, while $\mathcal{S}$-codability is defined combinatorially, it is a property of the abstract dynamical system in the sense that it is preserved under conjugacy. Proposition 9.1. Let $\mathcal{S}$ be an addable ANS, and consider a (weakly) $\mathcal{S}$-codable subshift $X$ such that $X\subseteq\mathbb{N}^{\mathbb{N}}$. If $Y$ is conjugate to $X$ and the conjugating map maps the zero point associated with $Y$ to the zero point $0^{\omega}$, then $Y$ is (weakly) $\mathcal{S}$-codable. Our main examples of $\mathcal{S}$-codable subshifts are winning shifts, which are always hereditary, so the following seems natural to prove. Proposition 9.2. Let $\mathcal{S}$ be an addable ANS and $Y$ be a subset of $\mathbb{N}^{\mathbb{N}}$. If $Y$ is $\mathcal{S}$-codable, then the hereditary closure of $Y$ is $\mathcal{S}$-codable. Finally, our class contains a subclass of sofic shifts generalizing one direction of 8.7. Proposition 9.3. Let $\mathcal{S}$ be an addable ANS. Every sofic shift with a unique periodic point is $\mathcal{S}$-codable. More generally, every sofic shift is weakly $\mathcal{S}$-codable with respect to any zero symbol. Note that the latter claim is not as weak as it may seem: if $(Y,\sigma)$ is sofic, so is $(Y,\sigma^{n})$ for any $n\geq 1$ (here $\sigma$ is the shift map), and thus these are also weakly $\mathcal{S}$-codable. Since eventually periodic points are dense in a sofic shift, in a sense the weak codability of these subshifts codes all of its points. To have a more useful statement, one might want to strengthen weak codability with uniformity conditions, but this is beyond the scope of the present paper. We deduce the above three propositions from a more abstract 9.4. Let us use infix notation for relations. For sets $X\subseteq A^{\mathbb{N}},Y\subseteq B^{\mathbb{N}}$ and a relation $R\subseteq A^{\mathbb{N}}\times B^{\mathbb{N}}$, define $$xR=\{y\in B^{\mathbb{N}}:(x,y)\in R\}\subseteq B^{\mathbb{N}}\quad\text{and}\quad XR=\bigcup_{x\in X}xR.$$ Symmetrically we define $Ry\subseteq A^{\mathbb{N}}$ for $y\in Y$ and $RY$. To such $R$ we also associate a relation $R^{\Sigma}$ in $\mathbb{N}\times\mathbb{N}$ by setting $$mR^{\Sigma}n\iff\exists(x,y)\in R\colon\sum x=m\wedge\sum y=n.$$ A relation $R^{\prime}\subseteq\mathbb{N}\times\mathbb{N}$ is locally finite if $|mR^{\prime}|<\infty$ and $|R^{\prime}n|<\infty$ for all $m,n\in\mathbb{N}$. We say $R\subseteq A^{\mathbb{N}}\times B^{\mathbb{N}}$ is locally finite if $R^{\Sigma}$ is locally finite. Proposition 9.4. Let $X\subseteq A^{\mathbb{N}},Y\subseteq B^{\mathbb{N}}$ be subshifts and $R\subseteq A^{\mathbb{N}}\times B^{\mathbb{N}}$ a sofic shift such that $R\cap X\times Y$ is locally finite. If $Y=XR$ and $X$ is weakly $\mathcal{S}$-codable, then $Y$ is weakly $\mathcal{S}$-codable. Proof. We may assume that $A,B\subseteq\mathbb{N}$ and $0$ is the zero symbol in both $X$ and $Y$. It suffices to show that $P_{v}(Y)$ is $\mathcal{S}$-recognizable for each $v\in\mathbb{N}_{>0}^{*}$. Fix $v\in\mathbb{N}_{>0}^{*}$. By the assumption of local finiteness, $Q_{v}(Y)$ is a union of finitely many sets of the form $Q_{u}(X)R\cap Q_{v}(B^{\mathbb{N}})$ where $u\in A^{*}$, so it suffices to show that each $Q_{u}(X)R\cap Q_{v}(B^{\mathbb{N}})$ is $\mathcal{S}$-codable. Since $P_{v}(Q_{u}(X)R\cap Q_{v}(B^{\mathbb{N}}))=P_{v}(Q_{u}(X)R)$, it suffices to prove that $P_{v}(Q_{u}(X)R)$ is $\mathcal{S}$-recognizable for all $u\in A^{*}$. Suppose $|u|=k,|v|=\ell$, and define $$\displaystyle T=\{(m_{1},\ldots,m_{k},n_{1},\ldots,n_{\ell}):\exists\mathbf{x}\in X,\mathbf{y}\in Y\colon e(\mathbf{x})$$ $$\displaystyle=u,e(\mathbf{y})=v,$$ $$\displaystyle s(\mathbf{x})$$ $$\displaystyle=(m_{1},\ldots,m_{k}),s(\mathbf{y})=(n_{1},\ldots,n_{\ell})\}.$$ Clearly it suffices to show that $T$ is $\mathcal{S}$-recognizable since $P_{v}(Q_{u}(X)R)$ is just the projection of $T$ to the last $\ell$ coordinates. To accept $\mathrm{rep}(T)$, we fix an ordering for the components. Given an element $(b_{1},\ldots,b_{k+\ell})$ of $T$, there exists a permutation $P$ that transforms this element to $(c_{1},\ldots,c_{k+\ell})$ that $c_{i}=b_{P(i)}$ for all $i$ and $c_{1}\leq\ldots\leq c_{k+\ell}$. Moreover, if a component $b_{i}$ of $(b_{1},\ldots b_{k})$ equals a component $b_{j}$ of $(b_{k+1},\ldots,b_{k+\ell})$, we may require that $P^{-1}(i)<P^{-1}(j)$. This makes $P$ unique since by definition $m_{1}<\ldots<m_{k}$ and $n_{1}<\ldots<n_{\ell}$. Let $T_{P}$ the set of elements of $T$ with a common permutation $P$. The set $T_{P}$ is $\mathcal{S}$-recognizable if and only if $P(T_{P})$ is $\mathcal{S}$-recognizable as a simple transducer can reorder components when $P$ is fixed. As $T$ is a union of the sets $T_{P}$ and there are only finitely many permutations $P$, it suffices to prove that $P(T_{P})$ is $\mathcal{S}$-recognizable for a fixed $P$. Define $L=\mathrm{rep}(P(T_{P}))$ so that $L$ has elements $(w_{1},\ldots,w_{k+\ell})^{\#}$ with $\mathrm{val}(w_{i})=c_{i}$ for all $i$. Let us perform a transduction to transform $(w_{1},\ldots,w_{k+\ell})^{\#}$ to $(w_{1},t_{2},\ldots,t_{k+\ell})^{\#}$ where $\mathrm{val}(w_{i})+\mathrm{val}(t_{i+1})=\mathrm{val}(w_{i+1})$ for all $i$. Such a transduction is possible because $\mathcal{S}$ is addable. Since regular languages are closed under inverse transductions, the language $L$ is regular if and only if its transduction $L^{\prime}$ is regular. The tuple $(w_{1},t_{2},...,t_{k+\ell})^{\#}$ in $L^{\prime}$ represents the support of an infinite word over the alphabet $A\times B$, and the first nonzero symbol appears at position $\mathrm{val}(w_{1})$, the second at $\mathrm{val}(w_{1})+\mathrm{val}(t_{2})$, the third at $\mathrm{val}(w_{1})+\mathrm{val}(t_{2})+\mathrm{val}(t_{3})$, and so on. The permutation $P$ tells us which of these positions have a symbol on the first or second component of the alphabet $A\times B$. An important interpretation detail is that elements of $(A\setminus\{0\})\times(B\setminus\{0\})$ are represented by having $t_{i+1}=0$ and having exactly one of the values $P(i),P(i+1)$ be in $\{1,\ldots,k\}$. This follows from the definition of $P$. In a sense, the permutation $P$ represents a shuffle of the words $u$ and $v$. We now proceed to run the finite-state automaton defining the sofic shift $R$ on the encoded input. From 8.6, we see that sofic shifts are $\omega$-regular, so let $\mathcal{A}$ be a deterministic automaton for $R$ with any standard acceptance condition. Observe that a run of $\mathcal{A}$ on an infinite word with support of size at most $k+\ell$ is determined entirely by its states corresponding to at most $k+\ell$ nonzero letters. Given input $\mathrm{rep}(n)$, we can compute the state $q^{\prime}$ the automaton $\mathcal{A}$ is in after reading input $0^{n}$ from any state $q$. If $n$ is small enough that $\mathcal{A}$ does not enter a loop, then we can use a lookup table. Suppose $n$ is not small. Let $K$ be the least common multiple of the lengths of the loops $\mathcal{A}$ may enter. Simulate $\mathcal{A}$ for $k$ steps until a loop is reached, compute in parallel $n-k$ modulo $K$ (possible since $\mathcal{S}$ is addable), and use again a lookup table. We describe an NFA $\mathcal{B}$ as follows. On each component of the input $(w_{1},t_{2},\ldots,t_{k+\ell})^{\#}$, the automaton $\mathcal{B}$ simulates a computation of $\mathcal{A}$ in parallel. On the $i$th component, $i\geq 2$, we guess a state $q^{\prime}_{i}$ of $\mathcal{A}$ and find a state $q$ such that $\mathcal{A}$ is at state $q$ after reading input $0^{\mathrm{val}(t_{i})}$ like in the previous paragraph. Let $x$ be the letter over $A\times B$ such that the infinite word encoded by the input has letter $x$ at position $\mathrm{val}(w_{1})+\mathrm{val}(t_{1})+\dotsm+\mathrm{val}(t_{i})$. Based on $P$, $u$, and $v$, we can check if there is a transition to state $q_{i}$ with letter $x$ (if not, we reject the computation). On the first component, we do the same, but we read $0^{\mathrm{val}(w_{1})}$ and start from the initial state of $\mathcal{A}$. When the computations have finished, we can verify if $q^{\prime}_{i}=q_{i-1}$ for $i=2,\ldots,k+\ell$. If this is the case, there is a path from the initial state of $\mathcal{A}$ to state $q_{k+\ell}$ whose label corresponds to the prefix of length $k+\ell+\mathrm{val}(w_{1})+\mathrm{val}(t_{1})+\dotsm+\mathrm{val}(t_{k+\ell})$ of the encoded infinite word. Finally, we check if $q_{k+\ell}$ belongs to the finite set of states of $\mathcal{A}$ which accept when only zeros are read and accept if and only this is so. (Note that in all the standard acceptance conditions, acceptance is a shift-invariant tail event, even if the set of valid runs need not be shift-invariant in general.) By construction, the automaton $\mathcal{B}$ accepts, with respect to $P$, $u$, and $v$, the encodings of infinite words in $R$ such that the support of the first component spells the word $u$ and the support of the second component the word $v$. In other words, the automaton accepts the language $L^{\prime}$. The claim follows. ∎ We can now prove our claims. Proof of 9.1. Consider first weak $\mathcal{S}$-codability. Suppose that $\phi\colon Y\to X$ is a conjugacy and $\phi$ maps the zero point associated with $Y$ to $0^{\omega}$. We select $R$ to be the graph of the conjugacy. By 9.4, it suffices to verify that $R$ is locally finite. Let $r$ be a biradius of $\phi$ (a radius common to both $\phi$ and $\phi^{-1}$). Since the zero point associated with $Y$ maps to $0^{\omega}$, the mapping $\phi$ can produce at most $2r+1$ new nonzero letters for each nonzero letter in a word of $Y$. Therefore if $\mathbf{y}$ in $Y$ contains $k$ nonzero letters, then $\phi(\mathbf{y})$ contains at most $(2r+1)k$ nonzero letters. Therefore $|mR^{\Sigma}|\leq(2r+1)m$ for all $m$. An analogous argument shows that $|R^{\Sigma}m|\leq(2r+1)m$ for all $m$, so it follows that $R^{\Sigma}$ is locally finite. For the complete claim, simply observe, again, that if $\phi\colon X\to Y$ is a conjugacy, $X$ has bounded sums if and only if $Y$ does. ∎ Proof of 9.3. Let $Y$ be a sofic shift with a unique periodic point. It is clear that this periodic point must be $a^{\omega}$ for a letter $a$ for otherwise we would have more than one periodic point. Select a permutation $\pi$ such that $\pi(a)=0$. By 8.5, $\pi(Y)$ has finite coding dimension. For the first claim, it now suffices to to show that $\pi(Y)$ is weakly $\mathcal{S}$-codable. Select $R=\{0^{\omega}\}\times\pi(Y)$. Since $\pi(Y)$ has finite coding dimension, it is plain that $|0R^{\Sigma}|$ and $|R^{\Sigma}0|$ are finite, so $R$ is locally finite. The claim follows from 9.4 as $\{0^{\omega}\}$ is clearly a weakly $\mathcal{S}$-codable sofic shift. For the general case, say $Y$ is sofic and $a^{\omega}\in Y$ for a letter $a$. Select a permutation $\pi$ such that $\pi(a)=0$. The restriction $Z$ of $\pi(Y)$ to words with sum at most $d$ is also sofic for all $d$. By the above, $Z$ is $\mathcal{S}$-codable, and it follows that $Y$ is weakly $\mathcal{S}$-codable. ∎ Proof of 9.2. Suppose that $Y$ is $\mathcal{S}$-codable, and let $\tilde{Y}$ be its hereditary closure. Define $$R=\{(\mathbf{x},\mathbf{y})\in\mathbb{N}^{\mathbb{N}}\times\mathbb{N}^{\mathbb{N}}:\text{$\mathbf{x}=x_{0}x_{1}\dotsm$, $\mathbf{y}=y_{0}y_{1}\dotsm$, and $x_{i}\geq y_{i}$ for all $i$}\}.$$ The set $R$ is clearly a sofic shift. Moreover, the relation $R\cap(Y\times\tilde{Y})$ is locally finite since the coding dimension of $Y$ is finite. The claim follows from 9.4. ∎ 9.2 is not true under weak $\mathcal{S}$-codability. One can obtain examples by assuming that $Y$ has no words with finite support. It is possible to imagine slightly modified definitions that disallow this, so we give a more interesting example. Example 9.5. Let $N$ be a subset of $2\mathbb{N}$, and let $X_{N}$ be the smallest subshift containing the infinite words $$0^{m}10^{2n}\prod_{i=1}^{n}10^{2n-2i+1}\cdot 0^{\omega}$$ where $m\in\mathbb{N}$, $2n\in N$. Clearly the hereditary closure $\tilde{X}_{N}$ of $X_{N}$ contains the infinite word $10^{2n}10^{\omega}$ if and only if $2n\in N$. Let $$Z=\{\mathbf{x}\in\tilde{X}_{N}:\sum\mathbf{x}=2\}.$$ Since there are uncountably many choices for $N$ and there are only countably many $\mathcal{S}$-codable subsets of $\{0,1\}^{\mathbb{N}}$, we can choose $N$ so that $Z$ is not $\mathcal{S}$-codable. Since $Z$ is $\mathcal{S}$-codable if $\tilde{X}_{N}$ is weakly $\mathcal{S}$-codable, we find the desired counterexample if we show that $X_{N}$ is always weakly $\mathcal{S}$-codable. Let $Y_{n}$ be the set $\{\mathbf{x}\in X_{n}:\sum\mathbf{x}\leq n\}$. It suffices to show that $Y_{n}$ is $\mathcal{S}$-codable for all $n$. It is straightforward to see that there exist finitely many words $w$ such that the words of $Y_{n}$ are of the form $0^{m}w0^{\omega}$ for $m\in\mathbb{N}$. The closure of the union of words of such form is clearly a sofic shift with exactly one periodic point. 9.3 implies that $Y_{n}$ is $\mathcal{S}$-codable when $\mathcal{S}$ is addable. Remark 9.6. We have used the assumption of addability of the ANS $\mathcal{S}=(L,\prec)$ in this section, since this is the natural generality for computing winning shifts. However, the proof of 9.4 (and thus all the results in this section) really only need the property that given two words $u$ and $v$ in $L$, the difference $\mathrm{val}(u)-\mathrm{val}(v)$ can be computed modulo a constant by a finite-state machine, or equivalently the value $\mathrm{val}(u)$ can be computed modulo a constant for a single word. It is known that this is true in the classical case where $\prec$ is the radix order [15]. We believe that this generalizes to every comparable ANS, but this is beyond the scope of this paper. Applied to the ANS on $0^{*}$ with radix order, this could be used to give another proof for the implication “sofic with a unique periodic point $\implies$ $1$-codable” of 8.7. 10 Open problems Question 10.1. Is there an addable ANS $\mathcal{S}$ such that some $\mathcal{S}$-automatic word has a winning shift with unbounded sums? If the answer is negative, then one can drop the assumption of sublinear complexity in Theorem 6.2 (the only point of this property is that it is the most natural word-combinatorial assumption we know that implies bounded sums). Cassaigne’s example provides only a comparable ANS with this property. For a general substitutive subshift we do not know how to describe even the bounded sum restrictions of winning shifts in general. It seems natural to guess that the winning shift is somehow “substitutive” in this case too, and we propose the following question: Problem 10.2. Devise an effective procedure that, given a fixed point $\mathbf{x}$ of a substitution $\tau$ and a word $v$, computes a finite description of $P_{v}(W(X))$ where $X$ is the subshift generated by $\mathbf{x}$. An obvious candidate for a finite description would be to show recognizability with respect to the corresponding Dumont-Thomas numeration system for $P_{v}(W(X))$, but we do not see why this would hold. When the winning shift of a subshift has infinite coding dimension, we do not obtain finite description of the winning shift even in the addable case, only weak $\mathcal{S}$-codability. Problem 10.3. Find a finitary way to describe the winning shift in the weakly $\mathcal{S}$-recognizable case. In particular, describe $W(Z)$ for the subshift $Z$ from 6.5. Besides these theoretical problems, it would be of interest to try to extend the practical computations in Section 6 to examples where the winning shift has larger coding dimension. We expect that the methods scale very badly, but this intuition has often turned out to be wrong in the setting of automatic theorem-proving; see [12, Remark 3] and [22] and its references. We also mention two somewhat tangential problems arising from the considerations in Section 9. First, if $X$ and $Y$ are conjugate subshifts, it is not clear to us whether $W(X)$ and $W(Y)$ are somehow related to each other as well (admittedly, it is not clear what a negative answer could be). Second, we conjecture that the results of Section 9 extend to arbitrary comparable ANS $\mathcal{S}$. Acknowledgements The first author is grateful for his postdoc position in TCSMT which allowed him to freely focus on research in 2018–2021. The second author was supported by the Academy of Finland grant 2608073211. References [1] Jean-Paul Allouche and Jeffrey Shallit “Automatic Sequences” Cambridge University Press, 2003 [2] R.P. 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Hyperfine interaction mediated electric-dipole spin resonance: The role of the frequency modulation Rui Li rl.rueili@gmail.com Quantum Physics and Quantum Information Division, Beijing Computational Science Research Center, Beijing 100193, China (November 26, 2020) Abstract The electron spin in semiconductor quantum dot can be coherently controlled by an external electric field, an effect called electric-dipole spin resonance (EDSR). There are several mechanisms underlie the EDSR effect, among which there is a hyperfine mechanism, where the spin-electric coupling is mediated by the electron-nucleus hyperfine interaction. Here, we investigate the influence of the frequency modulation (FM) on the spin-flip efficiency. Our results reveal that FM plays an important role in the hyperfine mechanism. Without FM, the electric field almost cannot flip the electron spin, the spin-flip probability is only about 20%. While under the FM, the spin-flip probability can be improved approximately to 70%. Especially, we find that the modulation amplitude has a lower bound, which is related to the width of the fluctuated hyperfine field. I Introduction The electron spin confined in semiconductor quantum dot is a good qubit candidate Hanson ; Buluta . Since the pioneer work of Loss and DiVincenzo in 1998 Loss , spin quantum computing has become an important research area. There are many theoretical and experimental advances in the following decades. Qubit initialization Johnson ; Petta , coherent manipulation Koppens ; Foletti , and single shot readout Elzerman ; Barthel ; Morello can be achieved with high precision. Quantum dot spin qubit has many advantages. It has a relative long coherence time (in comparison with the qubit manipulation time) Bluhm , the two-qubit manipulation can be easily achieved by using the Heisenberg exchange interaction Burkard ; Xuedong , and it is more or less much easier to achieve the qubit scalability. The single spin manipulation can be achieved by using the traditional electron spin resonance technique Koppens . This method not only needs a static Zeeman field, but also needs an ac magnetic field which is perpendicular to the static one Awschalom . When the frequency of the ac field matches the electron Zeeman splitting, one achieves the coherent spin rotation. However, it is not easy to produce a local ac magnetic field experimentally Koppens . Recently, instead of an ac magnetic field, an ac electric field can also be used to manipulate a single electron spin, an effect called electric-dipole spin resonance (EDSR) Tokura ; Pioro ; Nowack ; Nadj1 ; Nadj2 ; Rashba ; Golovach ; RuiLi ; Laird ; Rashba3 . It is much easier to generate a local ac electric field in experiments. We can explain the quantum dot EDSR in the following simple way. In the absence of the electric driving, the spin and the orbital degrees of freedom of the electron are mixed due to some mechanisms, e.g., the slanting magnetic field Yoneda ; Palyi ; Szechenyi ; Scarlino , the spin-orbit coupling Ban ; Yang ; Nowak ; Khomitsky ; Echeverria ; Romhanyi , and the electron-nucleus hyperfine interaction Shafiei ; Osika ; Chesi . Such that an ac electric field can induce the electric-dipole transitions between the electron Zeeman levels. The hyperfine interaction mediated EDSR was first observed experimentally in 2007 Laird . The hallmarks of this mechanism are listed as follows. First, under the electric driving, no Rabi oscillation is observed. The spin-flip probability monotonically increases to a saturating value, there is no periodical revival of the spin polarization Laird . Second, the Rabi frequency is independent of the Zeeman field Laird . Third, if there is no FM to the driving electric field, the electron spin can not be flipped Shafiei . The first theoretical investigation to the hyperfine mechanism is given by Rashba Rashba3 , who provides deep insights into the physics behind the EDSR. However, the developed theory was built on the mean field approximation, and especially the effects of the FM were not explored. In this paper, we readdress the mechanism of the hyperfine interaction mediated EDSR in the presence of FM. Our main findings are summarized as follows. First, we derive an analytical expression for the spin-flip probability where we do not make any mean field approximation. The spin-flip probability is expressed as a summation over a large number of periodic functions. Such that the oscillating behavior of each periodic function is hided by the summation. Second, under the fixed-frequency electric driving, the spin cannot be flipped. Because the electron-nucleus hyperfine interaction brings an inhomogeneous broadening to the spin splitting. The fixed driving frequency cannot match the inhomogeneously broadened spin splitting. Third, the FM is used to broaden the frequency spectrum of the driving electric field. When the width of the frequency spectrum is larger than the width of the fluctuated hyperfine field, the spin-flip probability can be greatly improved. This paper is organized as follows. In Sec. II, we focus on the EDSR in the quantum dot with quasi-1D confinement. Special emphases are given to the role of the FM. In Sec. III, we discuss some general properties of EDSR in quantum dot with quasi-2D confinement. Finally, we give a brief summary in Sec. IV. II EDSR in quantum dot with quasi-1D confinement In order to explicitly show the underlying physics of the hyperfine interaction mediated EDSR, we first study this phenomena in a simple quasi-1D quantum dot. On the other hand, quantum dot with quasi-1D confinement, e.g., InAs Schroer ; Nadj1 or InSb Nadj2 nanowire quantum dot, already can be fabricated experimentally. Therefore, at the moment, we confront the following total Hamiltonian $$\displaystyle H_{\rm tot}$$ $$\displaystyle=$$ $$\displaystyle\frac{p^{2}_{x}}{2m_{e}}+\frac{1}{2}m_{e}\omega^{2}_{0}x^{2}+\sum% ^{N}_{l=1}\frac{A}{\hbar^{2}}\textbf{S}\cdot\textbf{I}_{l}\delta(x-x_{l})$$ (1) $$\displaystyle+\gamma_{e}BS_{x}+eEx\cos\left[\int^{t}_{0}dt^{\prime}\nu(t^{% \prime})\right],$$ where $p_{x}=-i\hbar\partial_{x}$, $m_{e}$ is the effective electron mass, $\hbar\omega_{0}$ is the orbital energy of the quantum dot, $A$ is the strength of the electron-nucleus hyperfine coupling, $\gamma_{e}$ is the electron gyromagnetic ratio, S ($S=1/2$) and $\textbf{I}_{l}$ ($I=1/2$) are the electron and the $l$-th nuclear spin operators respectively, $N$ is the total number of the nuclear spins in the dot, $x_{l}$ is the site of the $l$-th nuclear spin, and $\nu\,(t^{\prime})$ is the modulated frequency of the driving electric field. We have introduced the concept of FM in Eq. (1), where the frequency of the driving field is time dependent. In this paper, we will study both the fixed-frequency driving case, i.e., $\nu(t^{\prime})=\nu_{0}$, and the modulated-frequency driving case, i.e., $\nu(t^{\prime})=\nu_{0}+\delta\nu\cos(\nu_{\rm fm}t^{\prime})$, where $\delta\nu$ and $\nu_{\rm fm}$ are the modulation amplitude and the modulation frequency respectively. II.1 The mixing of the spin and orbital degrees of freedom owing to the hyperfine interaction Our first step is to calculate the spin splitting and the corresponding wavefunctions in the quantum dot in presence of both the external magnetic field and the hyperfine field. The Hamiltonian of the quantum dot in absence of the driving term can be divided into two parts $$\displaystyle H$$ $$\displaystyle=$$ $$\displaystyle H_{0}+H_{1},$$ $$\displaystyle H_{0}$$ $$\displaystyle=$$ $$\displaystyle\frac{p^{2}_{x}}{2m_{e}}+\frac{1}{2}m_{e}\omega^{2}_{0}x^{2}+% \gamma_{e}BS_{x},$$ $$\displaystyle H_{1}$$ $$\displaystyle=$$ $$\displaystyle\sum^{N}_{l=1}\frac{A}{\hbar^{2}}\left[S_{x}I^{x}_{l}+\frac{1}{2}% (S_{+}I^{-}_{l}+S_{-}I^{+}_{l})\right]\delta(x-x_{l}),$$ (2) where $S_{\pm}=S_{y}\pm\,iS_{z}$ and $I^{\pm}_{l}=I_{y}\pm\,iI_{z}$ are the electron and the $l$-th nuclear spin raising/lowering operators respectively. Note that the spin quantized direction here is along the $x$-axis. The full quantum mechanical solution to the time-independent Schrödinger equation is almost impossible because of the coordinate-dependent hyperfine interaction term $H_{1}$. However, when the Zeeman field $\gamma_{e}B$ is strong enough, i.e., the Zeeman field is much larger than the hyperfine field, we can solve the energy spectrum by treating $H_{1}$ as a perturbation. The related physical quantities should satisfy the perturbation condition: $$\sum^{N}_{l=1}(A_{l}/4)\ll\gamma_{e}B\ll\hbar\omega_{0},$$ (3) where $A_{l}$ is the $l$-th electron-nuclear hyperfine coupling coefficient. The definition of this coefficient is given in latter Eq. (7). We are easy to find the zeroth-order eigenvalues and the corresponding eigenfunctions: $$\displaystyle E^{0}_{n\sigma_{e}\chi_{m}}$$ $$\displaystyle=$$ $$\displaystyle(n+1/2)\hbar\omega_{0}+(1/2)(-1)^{\sigma_{e}}\gamma_{e}B,$$ $$\displaystyle|\Psi^{0}_{n\sigma_{e}\chi_{m}}\rangle$$ $$\displaystyle=$$ $$\displaystyle\psi_{n}(x)|\sigma_{e}\rangle\otimes|\chi_{m}\rangle,$$ (4) where $n=0,1,2\ldots$ is the main quantum number, $\sigma_{e}=0,1$ is the electron spin quantum number, with $|0\rangle\rightarrow|\!\uparrow_{x}\rangle$ and $|1\rangle\rightarrow|\!\downarrow_{x}\rangle$, $|\chi_{m}\rangle=|\sigma^{m}_{1}\sigma^{m}_{2}\cdots\sigma^{m}_{N}\rangle$ is the $N$-bit binary representation of a decimal number $m=0,1,\ldots(2^{N}-1)$, with $\sigma^{m}_{l}=0,1$ being the $l$-th nuclear spin quantum number, and $\psi_{n}(x)$ is the eigenfunction of the harmonic oscillator, e.g., $$\displaystyle\psi_{0}(x)$$ $$\displaystyle=$$ $$\displaystyle 1/(\pi^{1/4}x^{1/2}_{0}){\rm exp}\left[-x^{2}/(2x^{2}_{0})\right],$$ $$\displaystyle\psi_{1}(x)$$ $$\displaystyle=$$ $$\displaystyle 2^{1/2}/(\pi^{1/4}x^{1/2}_{0})(x/x_{0}){\rm exp}\left[-x^{2}/(2x% ^{2}_{0})\right],$$ (5) with $x_{0}=\sqrt{\hbar/(m_{e}\omega_{0})}$ being the characteristic length of the quantum dot. The electron in the quantum dot should occupy the lowest orbital level, i.e., $n=0$. Therefore, in the following we only focus on $n=0$ Zeeman levels $|\Psi_{0\sigma_{e}\chi_{m}}\rangle$. It should be noted that, up to the zeroth order, each energy level has a degeneracy of $2^{N}$ [see Eq. (4)]. However, the perturbation $H_{1}$ does not mix the states in the degenerate subspace. Such that in our following calculation we can use the non-degenerate perturbation formulas Landau $$\displaystyle E$$ $$\displaystyle=$$ $$\displaystyle E^{0}+\langle\Psi^{0}|H_{1}|\Psi^{0}\rangle,$$ $$\displaystyle|\Psi\rangle$$ $$\displaystyle=$$ $$\displaystyle|\Psi^{0}\rangle+\sum_{\Psi^{\prime}}\frac{\langle\Psi^{\prime 0}% |H_{1}|\Psi^{0}\rangle}{E^{0}-E^{\prime 0}}|\Psi^{\prime 0}\rangle.$$ (6) Up to the first-order perturbation, the Zeeman energies have the following rectifications: $$\displaystyle E_{00\chi_{m}}$$ $$\displaystyle=$$ $$\displaystyle(1/2)\hbar\omega_{0}+(1/2)\gamma_{e}B+\sum^{N}_{l=1}(A_{l}/4)(-1)% ^{\sigma^{m}_{l}},$$ $$\displaystyle E_{01\chi_{m}}$$ $$\displaystyle=$$ $$\displaystyle(1/2)\hbar\omega_{0}-(1/2)\gamma_{e}B-\sum^{N}_{l=1}(A_{l}/4)(-1)% ^{\sigma^{m}_{l}},$$ (7) where $A_{l}=A\psi^{2}_{0}(x_{l})$ is the $l$-th electron-nucleus hyperfine coupling coefficient. The hyperfine energy structure of the quantum dot is schematically shown in Fig. 1. The corresponding first-order perturbation wavefunctions read $$\displaystyle|\Psi_{00\chi_{m}}\rangle$$ $$\displaystyle=$$ $$\displaystyle|\Psi^{0}_{00\chi_{m}}\rangle+\sum^{N}_{l=1}\frac{A\psi_{1}(x_{l}% )\psi_{0}(x_{l})}{2(\gamma_{e}B-\hbar\omega_{0})}\sigma^{m}_{l}$$ $$\displaystyle\times\psi_{1}(x)|1\rangle\otimes|\sigma^{m}_{1}\ldots\sigma^{m}_% {l-1}0\sigma^{m}_{l+1}\ldots\sigma^{m}_{N}\rangle,$$ $$\displaystyle|\Psi_{01\chi_{m}}\rangle$$ $$\displaystyle=$$ $$\displaystyle|\Psi^{0}_{01\chi_{m}}\rangle+\sum^{N}_{l=1}\frac{A\psi_{1}(x_{l}% )\psi_{0}(x_{l})}{2(-\gamma_{e}B-\hbar\omega_{0})}\frac{(-1)^{\sigma^{m}_{l}}+% 1}{2}$$ (8) $$\displaystyle\times\psi_{1}(x)|0\rangle\otimes|\sigma^{m}_{1}\ldots\sigma^{m}_% {l-1}1\sigma^{m}_{l+1}\ldots\sigma^{m}_{N}\rangle.$$ Thus, it can be seen clearly from Eq. (8), owing to the hyperfine interaction, the spin and the orbital degrees of freedom of the electron are mixed. Therefore, it is expected that the electric field will induce the transitions between the up Zeeman level and the down Zeeman level. II.2 The spin-flip probability under the fixed-frequency driving In this subsection, we study the EDSR effect under the fixed-frequency driving. The driving Hamiltonian is written as $eEx\cos(\nu_{0}t)$, where $\nu_{0}$ is the driving frequency. Although the electron spin splitting is broadened by the hyperfine interaction [see Eq. (7) and Fig. 1], we still choose the driving frequency to satisfy the ‘resonant’ condition, i.e., $\hbar\nu_{0}=\gamma_{e}B$. As we have shown in Fig. 1, each Zeeman (both the up and the down) level has been broadened to a series of sublevels, where $m$ is the sublevel index. So what is the initial state for the coupled electron-nuclear system? Because the nuclear Zeeman splitting is much less than the Boltzmann energy $k_{B}T$, with $T\sim\,mK$ being the experimental temperature. It is a good approximation to assume that the nuclear spin ensemble is totally unpolarized. In other words, in the initial mixed state of the total electron-nucleus system, each pure state $|\Psi_{00\chi_{m}}\rangle$ has the equal probability $$\rho(0)=(1/2^{N})\sum^{2^{N}-1}_{m=0}|\Psi_{00\chi_{m}}\rangle\langle\Psi_{00% \chi_{m}}|.$$ (9) Here, we have let the electron initially be in the spin-up state $|0\rangle$ ($|\uparrow_{x}\rangle$). Under the electric driving, we want to calculate the electron spin-flip rate for the initial mixed state given in Eq. (9). Our first step is to calculate the spin-flip rate for the initial pure state $|\Psi_{00\chi_{m}}\rangle$. From Eq. (8), we can calculate the electric-dipole transition element between the state $|\Psi_{00\chi_{m}}\rangle$ and the state $|\Psi_{01\chi^{\prime}_{m}}\rangle$ $$\displaystyle g^{m}_{l}$$ $$\displaystyle=$$ $$\displaystyle\langle\Psi_{00\chi_{m}}|eEx|\Psi_{01\chi^{\prime}_{m}}\rangle$$ (10) $$\displaystyle=$$ $$\displaystyle-\frac{eEx_{0}\times\hbar\omega_{0}}{\sqrt{2}\left(\hbar^{2}% \omega^{2}_{0}-\gamma^{2}_{e}B^{2}\right)}\sigma^{m}_{l}A\psi_{1}(x_{l})\psi_{% 0}(x_{l})$$ $$\displaystyle\approx$$ $$\displaystyle-\frac{eEx_{0}}{\sqrt{2}\hbar\omega_{0}}\sigma^{m}_{l}A\psi_{1}(x% _{l})\psi_{0}(x_{l}),$$ where $|\chi_{m}\rangle=|\sigma^{m}_{1}\cdots\sigma^{m}_{l}\cdots\sigma^{m}_{N}\rangle$ and $|\chi^{\prime}_{m}\rangle\equiv|\sigma^{m}_{1}\cdots\sigma^{m}_{l-1}0\sigma^{m% }_{l+1}\cdots\sigma^{m}_{N}\rangle$. Similar results were also obtained in Refs. Rashba3, ; Rudner, . The above equation also tells us that, the electron-nucleus spin flip-flop process is assisted by the driving electric field (see Fig. 2). There is an exchange of the spin polarization between the electron and the $l$-th nucleus. It should be noted that the value of $\sigma^{m}_{l}$ in Eq. (10) automatically guarantees the feasibility of this flip-flop process, i.e., only when $\sigma^{m}_{l}=1$, then $g^{m}_{l}\neq 0$. The energy difference between these two states reads $$E_{00\chi_{m}}-E_{01\chi^{\prime}_{m}}\approx\gamma_{e}B+\sum^{N}_{l=1}(A_{l}/% 2)(-1)^{\sigma^{m}_{l}}.$$ (11) There arises another problem, under the electric driving, there are many quantum states with which the state $|\Psi_{00\chi_{m}}\rangle$ can flip-flop, as long as the $l$-th bit in $|\chi_{m}\rangle$ has the value $\sigma^{m}_{l}=1$ $(l\in\{1,\ldots\,N\})$. In spite of this, the spin-flip probability for the initial pure sate $|\Psi_{00\chi_{m}}\rangle$ still has a simple expression (for details see Appendix A) $$P^{m}_{\downarrow_{x}}(t)=\frac{\sum^{N}_{l=1}(g^{m}_{l})^{2}}{\hbar^{2}\Omega% ^{2}_{m}}\sin^{2}\frac{\Omega_{m}t}{2},$$ (12) where $$\displaystyle\Delta_{m}$$ $$\displaystyle=$$ $$\displaystyle-\sum^{N}_{l=1}(A_{l}/2)(-1)^{\sigma^{m}_{l}},$$ (13a) $$\displaystyle\Omega_{m}$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{\hbar}\left[\Delta^{2}_{m}+\sum^{N}_{l=1}(g^{m}_{l})^{2}% \right]^{1/2}.$$ (13b) Here, $\Delta_{m}$ is the detuning introduced by the hyperfine field and $\Omega_{m}$ is the Rabi frequency in presence of the detuning. Therefore, for the initial pure state $|\Psi_{00\chi_{m}}\rangle$, we obtain the simple Rabi formula Scully for the spin-flip probability. As we can see from Eqs. (10) and (13b), the Rabi frequency $\Omega_{m}$ is independent of the magnetic field, which is consistent with the experimental observations Laird ; Shafiei . Since we already obtain the spin-flip probability for the initial pure state, the generalization to the initial mixed state (9) is simply. The spin-flip probability for the initial mixed state reads $$P_{\downarrow_{x}}(t)=(1/2^{N})\sum^{2^{N}-1}_{m=0}\frac{\sum^{N}_{l=1}(g^{m}_% {l})^{2}}{\hbar^{2}\Omega^{2}_{m}}\sin^{2}\frac{\Omega_{m}t}{2}.$$ (14) It is just summed over all the spin-flip probability for each pure state. Because Eq. (14) is summed over a large number of periodic functions, $P_{\downarrow_{x}}(t)$ may not contain the oscillating behaviour when $N$ is very large. In Fig. (3), we show the spin-flip probability as a function of the driving time. As we can see, under the fixed frequency driving ($\hbar\nu_{0}=\gamma_{e}B$), the electron spin almost can not be flipped, the maximal spin-down probability is only $0.2$. This very low spin-flip probability can not be observed in experiment Shafiei . Therefore, under the fixed-frequency driving, the EDSR actually can not occur. We will explain why the spin-flip probability is very low in the next subsection. II.3 The spin-flip probability under the modulated-frequency driving As we have explicitly shown in the above subsection, each pure state $|\Psi_{00\chi_{m}}\rangle$ in the initial mixed state [see Eq. (9)] brings a detuning $\Delta_{m}$ to the spin-flip probability $P_{\downarrow_{x}}(t)$ [see Eqs. (13a) and (14)]. When $m$ runs one by one from $0$ to $2^{N}-1$, the value of the detuning $\Delta_{m}$ has a distribution. Actually, this distribution describes the fluctuation of the hyperfine field. The distribution function approximately reads Merkulov $$P(\Delta)=\frac{1}{(2\pi)^{1/2}\Delta_{\rm flu}}e^{-\Delta^{2}/(2\Delta^{2}_{% \rm flu})},$$ (15) where $\Delta_{\rm flu}=(1/2)\sqrt{\sum^{N}_{l=1}A^{2}_{l}}$ is the width of the hyperfine field fluctuation. Therefore, under the fixed-frequency driving (e.g., $\gamma_{e}B=\hbar\nu_{0}$), the detuning $\Delta$ is not fixed, it obeys the distribution as indicated in Eq. (15). This is the essential reason that leads to the very low spin-flip probability. Frequency modulation (FM) is a concept in telecommunication and signal processing. As we will show in the following, FM can solve the above very low spin-flip probability problem. Here, we take the simplest example to explain the role the FM plays in the hyperfine mechanism. We consider the instantaneous frequency of the driving electric field is modulated to the following form $$\nu(t^{\prime})=\nu_{0}+\delta\nu\cos(\nu_{\rm fm}t^{\prime}),$$ (16) where $\nu_{0}$ is the central frequency, $\delta\nu$ is modulation amplitude, and $\nu_{\rm fm}$ is the modulation frequency. In order to understand the FM straightforwardly, we also give the experimental parameters here, where $\nu_{0}=2.9$ GHz, $\delta\nu=36$ MHz, and $\nu_{\rm fm}=3$ kHz Laird . In presence of the FM, the driving Hamiltonian can be written as Haykin $$eEx\cos\left[\int^{t}_{0}dt^{\prime}\nu(t^{\prime})\right]=\sum^{\infty}_{l=-% \infty}eExJ_{l}(\beta)\cos\left[(\nu_{0}+l\nu_{\rm fm})t\right],$$ (17) where we have made a Fourier transformation, $\beta=\delta\nu/\nu_{\rm fm}\gg 1$ is the modulation index (we only consider the wideband modulation here), and $$J_{l}(\beta)=\frac{1}{2\pi}\int^{\pi}_{-\pi}dx\,{\rm exp}[i(\beta\sin\,x-lx)]$$ is the $n$-th order Bessel function of the first kind. In Fig. 4, we show the Bessel function as a function of the index $l$ for the large modulation index $\beta=200$. As we can see, when $|l|>\beta$, $J_{l}(\beta)\approx 0$. Therefore, actually, the summation of $l$ in Eq. (17) is constrained to $-\beta\leq\,l\leq\beta$. From Eq. (17), we are easy to find that the frequency spectrum of the driving electric-field has been broadened by the FM. The frequency of the driving now can run from $\nu_{0}-\delta\nu$ to $\nu_{0}+\delta\nu$, where $\delta\nu=\beta\nu_{\rm fm}$ can be considered as the width of the frequency spectrum. This frequency spectrum broadening is very useful. As we have shown in Eq. (15), the detuning $\Delta$, i.e., the hyperfine field, also has a distribution which approximately ranged from $-\sqrt{2}\Delta_{\rm flu}$ to $\sqrt{2}\Delta_{\rm flu}$. Therefore, the inhomogeneously broadened spin splitting is ranged from $\gamma_{e}B-\sqrt{2}\Delta_{\rm flu}$ to $\gamma_{e}B+\sqrt{2}\Delta_{\rm flu}$. When the width of the frequency spectrum $\delta\nu$ is larger than the width of the fluctuating hyperfine field $\sqrt{2}\Delta_{\rm flu}$, i.e., $$\sqrt{2}\Delta_{\rm flu}<\delta\nu,$$ (18) the detuing $\Delta$ will not appear in the spin-flip probability $P_{\downarrow_{x}}(t)$. Because for each initial pure state $|\Psi_{00\chi_{m}}\rangle$, the broadened spin splitting is $\gamma_{e}B-\Delta_{m}$ [see Eq. (11)], we always can find a special mode $l$ in the driving field matches this splitting, where $$\gamma_{e}B-\Delta_{m}=\nu_{0}+l\nu_{\rm fm}.$$ (19) Thus, the $l$-th mode $eExJ_{l}(\beta)\cos\left[(\nu_{0}+l\nu_{\rm m})t\right]$, where $l=-\Delta_{m}/\nu_{\rm fm}$, is in resonance with this broadened spin splitting $\gamma_{e}B-\Delta_{m}$. Therefore, the spin-flip probability under the FM can be written as $$P_{\downarrow_{x}}(t)=\frac{1}{2^{N}}\sum^{2^{N}-1}_{m=0}\sin^{2}\frac{\Omega^% {\rm fm}_{m}t}{2},$$ (20) where $\Omega^{\rm fm}_{m}=\frac{1}{\hbar}J_{-(\Delta_{m}/\nu_{\rm fm})}(\beta)\sqrt{% \sum^{N}_{l=1}(g^{m}_{l})^{2}}$ is the Rabi frequency under the FM. The detuning $\Delta_{m}$ is absent from the Rabi frequency, the spin-flip probability can been greatly improved. In Fig. 5, we show the spin-flip probability as a function of the driving time under the FM. As we can find, the maximal spin-flip probability has been increased from 0.2 with the fixed-frequency driving (see Fig. 3) to 0.7 with the modulated-frequency driving (see Fig. 5). This result is consistent with the experiment observation that the EDSR can only be observed under the FM Shafiei . We also mention that the experimentally observed maximal spin-flip probability is also about 0.7 Shafiei . Because the spin-flip probability $P_{\downarrow_{x}}(t)$ is still expressed as a summation over a large number of periodical functions [see Eq. (20)]. The Rabi oscillation may not appear when $N$ is larger. If $N$ is not large enough, e.g., $N=30$ or $35$, as we show in Figs. 3 and 5, there is still an oscillation for the spin-flip probability, but the amplitude of the oscillation is small. III EDSR in quantum dot with quasi-2D confinement Most gated quantum dots made from semiconductor heterostructures, e.g., GaAs/AlGaAs heterostructures, are confined in 2D. Here we move to consider the EDSR effect in quantum dot with quasi-2D confinement. As we already explored the physics of the EDSR in a quasi-1D quantum dot, the generalization from 1D to 2D is natural. As we will show in the following, the underlying physics does not change, except the calculations are a little more complicated. We focus on the following Hamiltonian $$\displaystyle H_{\rm tot}$$ $$\displaystyle=$$ $$\displaystyle\frac{p^{2}_{x}+p^{2}_{y}}{2m_{e}}+\frac{m_{e}\omega^{2}_{0}(x^{2% }+y^{2})}{2}+\sum_{l}\frac{A}{\hbar^{2}}\textbf{S}\cdot\textbf{I}_{l}\delta(% \textbf{r}-\textbf{r}_{l})$$ (21) $$\displaystyle+\gamma_{e}BS_{x}+eEx\cos\left[\int\,dt^{\prime}\nu(t^{\prime})% \right],$$ where $p_{x,y}=-i\hbar\partial_{x,y}$, $\textbf{r}=x\hat{e}_{x}+y\hat{e}_{y}$, and $\textbf{r}_{l}$ is the site of the $l$-th nuclear spin. The in-plane magnetic field is applied along the $x$ direction, the ac electric-field is also applied along this direction. This is the simplest Hamiltonian captures the main physics of EDSR in the quantum dot with quasi-2D confinement. The calculations here are completely similar to the 1D case, the dimension $y$ in Eq. (21) only introduces the calculation complexities. The transition element defined in Eq. (10) for the 2D case is modified to $$g^{m}_{l}\approx-\frac{eEx_{0}}{\sqrt{2}\hbar\omega}\sigma^{m}_{l}A\psi^{2}_{0% }(y_{l})\psi_{1}(x_{l})\psi_{0}(x_{l}).$$ (22) Also, the hyperfine coupling coefficient for the quantum dot with 2D confinement is defined as $$A_{l}=A\psi^{2}_{0}(x_{l})\psi^{2}_{0}(y_{l}).$$ (23) Under the fixed-frequency driving, the expression of the spin-flip probability is exactly the same as that given by Eqs. (12)-(14). Also, under the FM, the expression of the spin-flip probability still has the form shown in Eq. (20). For a realistic gated GaAs quantum dot, the total number of the nuclear spins in the quantum dot is about $N=10^{5}$. This number is too huge. Therefore, it is impossible to calculate the spin-flip probability $P_{\downarrow_{x}}(t)$ via Eqs. (14) and (20). In the following, we only discuss some general properties of EDSR in a quasi-2D quantum dot. These properties are closely related to the experimental observations. First, for quantum dot with quasi-2D confinement, the Rabi frequency is still independent of the external magnetic field. As we can see from Eqs. (13b) and (20), where $g^{m}_{l}$ and $A_{l}$ are defined in Eqs. (22) and (23) respectively. Both $\Omega_{m}$ and $\Omega^{\rm fm}_{m}$ do not depend on magnetic field strength $B$. This result is consistent with the experimental observations Laird ; Shafiei . Second, in a gated GaAs quantum dot, the width of the hyperfine field fluctuation is about $\Delta_{\rm flu}\approx 24$ MHz (0.1 $\mu$eV) Merkulov . This quantity sets a lower bound on the modulation amplitude where $$\sqrt{2}\Delta_{\rm flu}\approx 34~{}{\rm MHz}<\delta\nu.$$ (24) In order to observe the EDSR more efficiently, it is better to let the modulation amplitude be larger than $34$ MHz. This result is also consistent with the experimental observations. The early experiment used a 36 MHz modulation amplitude Laird and the recent experiment used both a 40 and a 75 MHz modulation amplitudes Shafiei . IV Summary The electron-nucleus hyperfine interaction, which commonly exists in III-V semiconductor quantum dot, is often considered as a nuisance because it can lead to spin decoherence Khaetskii ; Coish ; Witzel ; Yao ; Deng ; Cywinski ; RuiLi2 . While on the other hand, we also can make use of the hyperfine interaction. The hyperfine interaction mediates an interaction between the spin and an external electric field, which can facilitate the single spin manipulation. In this paper, we give a detailed theoretical investigation to the mechanism of the hyperfine interaction mediated EDSR. We emphasize the importance of the FM to the driving electric field in this hyperfine mechanism. The spin-flip probability is expressed as a summation over a large number of periodical functions, such that there is no Rabi oscillation for the spin-flip probability. Because the hyperfine field gives an inhomogeneous broadening to the spin splitting, the fixed-frequency driving almost cannot flip the electron spin. FM to the driving field can greatly improve the spin-flip efficiency, there is approximately a 50% improvement for the spin-flip probability. Also, the width of hyperfine field fluctuation sets a lower bound on the modulation amplitude. Our theory is in qualitatively good agreement with the experimental observations. Acknowledgements This work is supported by National Natural Science Foundation of China Grant No. 11404020 and Postdoctoral Science Foundation of China Grant No. 2014M560039. Appendix A The spin-flip probability for the initial pure state $|\Psi_{00\chi_{m}}\rangle$ In the following, for brevity, we make some abbreviations such as $|0\rangle\equiv|\Psi_{00\chi_{m}}\rangle$ and $|1_{l}\rangle\equiv\psi_{0}(x)|1\rangle|\sigma^{m}_{1}\cdots\sigma^{m}_{l-1}0% \sigma^{m}_{l+1}\cdots\sigma^{m}_{N}\rangle$. Under the fixed-frequency driving $eEx\cos(\nu_{0}t)$, there is a transition element $g^{m}_{l}$ between the state $|0\rangle$ and the state $|1_{l}\rangle$ (see Fig. 6). We have the following total Hamiltonian $$H=\hbar\omega\sum^{N}_{l=1}|1_{l}\rangle\langle\,1_{l}|+\sum^{N}_{l=1}g^{m}_{l% }(|0\rangle\langle\,1_{l}|+|1_{l}\rangle\langle\,0|)\cos(\nu_{0}\,t),$$ (25) where $\hbar\omega=-\gamma_{e}B-\sum^{N}_{l=1}(A_{l}/2)(-1)^{\sigma^{m}_{l}}$ and the transition amplitude $g^{m}_{l}$ is defined in Eq. (10). It should be noted that we have set the energy of state $|0\rangle$ to 0. The frequency of the driving electric field is in resonance with the Zeeman splitting, i.e., $\gamma_{e}B=\hbar\nu_{0}$. In interaction picture, the Hamiltonian reads (under the rotating-wave approximation) $$H_{\rm int}=\Delta_{m}\sum^{N}_{l=1}|1_{l}\rangle\langle\,1_{l}|+\sum^{N}_{l=1% }\frac{g^{m}_{l}}{2}(|0\rangle\langle\,1_{l}|+|1_{l}\rangle\langle\,0|),$$ (26) where $$\Delta_{m}=-\gamma_{e}B-\sum^{N}_{l=1}(A_{l}/2)(-1)^{\sigma^{m}_{l}}+\hbar\nu_% {0}$$ is the detuning. The system initially is in state $|\phi(0)\rangle=|0\rangle$, i.e., the electron spin is in spin up state. We want to find the spin-down probability $P_{\downarrow_{x}}(t)$ under the electric driving. We let $|\phi(t)\rangle=c_{0}(t)|0\rangle+\sum^{N}_{l=1}c_{l}(t)|1_{l}\rangle$, where $c_{l}(t)$ ( $l=0,1\ldots\,N$) are the coefficients to be determined. The equations of motion for the coefficients can be derived from the Schrödinger equation $i\hbar\partial_{t}|\phi(t)\rangle=H_{\rm int}|\phi(t)\rangle$. We find the coefficients satisfy $$\displaystyle\dot{c}_{0}(t)$$ $$\displaystyle=$$ $$\displaystyle-i\sum^{N}_{l=1}\frac{g^{m}_{l}}{2\hbar}c_{l}(t),$$ $$\displaystyle\dot{c}_{l}(t)$$ $$\displaystyle=$$ $$\displaystyle-i\frac{\Delta_{m}}{\hbar}\,c_{l}(t)-i\frac{g^{m}_{l}}{2\hbar}c_{% 0}(t).$$ (27) The solution to this equation array is very simple, we obtain $$\displaystyle c_{0}(t)$$ $$\displaystyle=$$ $$\displaystyle e^{-i\frac{\Delta_{m}}{2\hbar}t}\left[\cos\frac{\Omega_{m}t}{2}+% i\frac{\Delta_{m}}{\hbar\Omega_{m}}\sin\frac{\Omega_{m}t}{2}\right],$$ $$\displaystyle c_{l}(t)$$ $$\displaystyle=$$ $$\displaystyle-i\frac{g^{m}_{l}}{\hbar\Omega_{m}}e^{-i\frac{\Delta_{m}}{2\hbar}% t}\sin\frac{\Omega_{m}t}{2},$$ (28) where the Rabi frequency is defined as $$\Omega_{m}=\frac{1}{\hbar}\sqrt{\Delta^{2}_{m}+\sum^{N}_{l=1}(g^{m}_{l})^{2}}.$$ (29) Therefore, the spin down probability is given by $$P_{\downarrow_{x}}(t)=\sum^{N}_{l=1}|c_{l}(t)|^{2}=\frac{\sum_{l}(g^{m}_{l})^{% 2}}{\hbar^{2}\Omega^{2}_{m}}\sin^{2}\frac{\Omega_{m}t}{2}.$$ (30) So we derive the result given in Eq. (12). 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Spectral Theorem for definitizable normal linear operators on Krein spaces [5mm] Michael Kaltenbäck [6mm] Abstract:In the present note a spectral theorem for normal definitizable linear operators on Krein spaces is derived by developing a functional calculus $\phi\mapsto\phi(N)$ which is the proper analogue of $\phi\mapsto\int\phi\,dE$ in the Hilbert space situation. Mathematics Subject Classification (2010): 47A60, 47B50, 47B15. Keywords: Krein space, definitizable operators, normal operators, spectral theorem 1 Introduction A bounded linear operator $N$ on a Krein space $({\mathcal{K}},[.,.])$ is called normal, if $N$ commutes with its Krein space adjoint $N^{*}$, i.e. $NN^{*}=N^{*}N$. This is equivalent to the fact that its real part $A:=\frac{N+N^{*}}{2}$ and its imaginary part $B:=\frac{N-N^{*}}{2i}$ commute. We call $N$ definitizable whenever the selfadjoint operators $A$ and $B$ are both definitizable in classical sense, i.e. there exist so-called definitizing polynomials $p(z)$ and $q(z)$ such that $[p(A)x,x]\geq 0$ and $[q(B)x,x]\geq 0$ for all $x\in{\mathcal{K}}$; see [L]. In the Hilbert space setting the spectral theorem for bounded linear, normal operators is a well-known functional analysis result. In fact, it is almost as as folklore as the older spectral theorem for bounded linear, selfadjoint operators. In the Krein space world there exists no similar result for general selfadjoint operators. But assuming in addition definitizability a spectral theorem could be shown by Heinz Langer; cf. [L]. This theorem became an important starting point for various spectral results. The main difference to selfadjoint operators on Hilbert spaces is the appearance of (finitely many) critical points, where the spectral projections no longer behave like a measure. Only a rather small number of publications dealt with the situation of a normal (definitizable) operators in a Krein space. The Pontryagin space case was studied up to a certain extent for example in [XiCh] and [LS]. Special normal operators on Krein spaces were considered for example in [AS] and [PST]. But until now no adequate version of a spectral theorem on normal definitizable operators in Krein spaces has been found. In the present paper we present a spectral theorem for bounded linear, normal, definitizable operators formulated in terms of a functional calculus generalizing the functional calculus $\phi\mapsto\int\phi\,dE$ in the Hilbert space case. In order to achieve this goal, we use the methods developed in [KP] for definitizable selfadjoint operators and extend them for two commuting definitizable selfadjoint operators. Let us anticipate a little more explicitly what happens in this note. Denoting by $p(z)$ and $q(z)$ the definitizing real polynomials for $A$ and $B$, respectively, we build a Hilbert space ${\mathcal{V}}$ which is continuously and densely embedded in the given Krein space ${\mathcal{K}}$ such that $TT^{*}=p(A)+q(B)$, where $T:{\mathcal{V}}\to{\mathcal{K}}$ denotes that adjoint of the embedding mapping. Then we use the $*$-homomorphism $\Theta:(TT^{*})^{\prime}\ (\subseteq B({\mathcal{K}}))\to(T^{*}T)^{\prime}\ (% \subseteq B({\mathcal{V}}))$, $C\mapsto(T\times T)^{-1}(C)$, studied in [KP], in order to drag our normal operator $N\in(TT^{*})^{\prime}\subseteq B({\mathcal{K}})$ into $(T^{*}T)^{\prime}\ (\subseteq B({\mathcal{V}})$. The resulting normal operator $\Theta(N)$ acts in a Hilbert space, and therefore has a spectral measure $E(\Delta)$, where $\Delta$ are Borel subsets of ${\mathbb{C}}$. The proper family ${\mathcal{F}}_{N}$ of functions suitable for the aimed functional calculus are bounded and measurable functions on $$\big{(}\sigma(\Theta(N))\cup(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})\big{)% }\dot{\cup}Z^{i}\ (\subseteq{\mathbb{C}}\dot{\cup}{\mathbb{C}}^{2})\,.$$ Here $Z^{{\mathbb{R}}}_{p}=p^{-1}\{0\}\cap{\mathbb{R}}$ and $Z^{{\mathbb{R}}}_{q}=q^{-1}\{0\}\cap{\mathbb{R}}$ denote the real zeros of $p(z)$ and $q(z)$, respectively, and $Z^{i}=(p^{-1}\{0\}\times q^{-1}\{0\})\setminus({\mathbb{R}}\times{\mathbb{R}})$. Moreover, the functions $\phi\in{\mathcal{F}}_{N}$ assume values in ${\mathbb{C}}$ on $\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$, values in ${\mathbb{C}}^{{\mathfrak{d}}_{p}(\operatorname{Re}z)\cdot{\mathfrak{d}}_{q}(% \operatorname{Im}z)+2}$ at $z\in Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q}$ and values in ${\mathbb{C}}^{{\mathfrak{d}}_{p}(\xi)\cdot{\mathfrak{d}}_{q}(\eta)}$ at $z=(\xi,\eta)\in Z^{i}$. Here ${\mathfrak{d}}_{p}(w)$ (${\mathfrak{d}}_{q}(w)$) denotes $p$’s ($q$’s) degree of zero at $w$. Finally, $\phi\in{\mathcal{F}}_{N}$ satisfies a growth regularity condition at all points from $Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q}$ which are not isolated in $\sigma(\Theta(N))\cup(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$. Any polynomial $s(z,w)\in{\mathbb{C}}[z,w]$ can be seen as a function $s_{N}\in{\mathcal{F}}_{N}$. The nice thing about these, somewhat tediously defined functions $\phi\in{\mathcal{F}}_{N}$ is that $$\phi(z)=s_{N}(z)+(p_{N}+q_{N})(z)\cdot g(z),\ z\in\sigma(\Theta(N))\,,$$ (1.1) where $s\in{\mathbb{C}}[z,w]$ is a suitable polynomial in two variables and $g:\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})\to{% \mathbb{C}}$ is bounded and measurable and $g:\sigma(\Theta(N))\cap(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})\to{\mathbb% {C}}^{2}$. We then define $\phi(N):=s(A,B)+T\int_{\sigma(\Theta(N))}^{R_{1},R_{2}}g\,dET^{*}$, show that this operator does not depend on the actual decomposition (1.1) and that $\phi\mapsto\phi(N)$ is indeed a $*$-homomorphism. Here $\int_{\sigma(\Theta(N))}^{R_{1},R_{2}}g\,dE$ is the integral of $g$ with respect to the spectral measure $E$ taking into account the fact that $g$ has values in ${\mathbb{C}}^{2}$ on $\sigma(\Theta(N))\cap(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$. If $\phi$ is stems from a characteristic function corresponding to a Borel subset $\Delta$ of ${\mathbb{C}}$ such that no point of $Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q}$ belongs to the boundary of $\Delta$, then $\phi(N)$ is a selfadjoint projection on ${\mathcal{K}}$. In fact, it can be seen as the corresponding special projection for $N$. 2 Multiple embeddings For the present section we fix a Krein space $({\mathcal{K}},[.,.])$ and Hilbert spaces $({\mathcal{V}},[.,.])$, $({\mathcal{V}}_{1},[.,.])$ and $({\mathcal{V}}_{2},[.,.])$. Moreover, let $T_{1}:{\mathcal{V}}_{1}\to{\mathcal{K}}$, $T_{2}:{\mathcal{V}}_{2}\to{\mathcal{K}}$ and $T:{\mathcal{V}}\to{\mathcal{K}}$ be bounded linear, injective mappings such that $$TT^{*}=T_{1}T_{1}^{*}+T_{2}T_{2}^{*}$$ holds true. Since for $x\in{\mathcal{K}}$ we have $$\displaystyle[T^{*}x,T^{*}x]_{{\mathcal{V}}}=[TT^{*}x,x]=\\ \displaystyle[T_{1}T_{1}^{*}x,x]+[T_{2}T_{2}^{*}x,x]=[T_{1}^{*}x,T_{1}^{*}x]_{% {\mathcal{V}}_{1}}+[T_{2}^{*}x,T_{2}^{*}x]_{{\mathcal{V}}_{2}}\,,$$ one easily concludes that $T^{*}x\mapsto T_{j}^{*}x$ constitutes a well-defined, contractive linear mapping from $\operatorname{ran}T^{*}$ onto $\operatorname{ran}T_{j}^{*}$ for $j=1,2$. By $(\operatorname{ran}T^{*})^{\bot}=\ker T=\{0\}$ and $(\operatorname{ran}T_{j}^{*})^{\bot}=\ker T_{j}=\{0\}$ these ranges are dense in the Hilbert spaces ${\mathcal{V}}$ and ${\mathcal{V}}_{j}$. Hence, there is a unique bounded linear continuation of $T^{*}x\mapsto T_{j}^{*}x$ to ${\mathcal{V}}$, which has dense range in ${\mathcal{V}}_{j}$. Denoting by $R_{j}$ for $j=1,2$ the adjoint mapping of this continuation we clearly have $T_{j}=TR_{j}$ and $\ker R_{j}=(\operatorname{ran}R_{j}^{*})^{\bot}=\{0\}$. From $TT^{*}=T_{1}T_{1}^{*}+T_{2}T_{2}^{*}$ we conclude $$T(\ I\ )T^{*}=TT^{*}=TR_{1}R_{1}^{*}T^{*}+TR_{2}R_{2}^{*}T^{*}=T(\ R_{1}R_{1}^% {*}+R_{2}R_{2}^{*}\ )T^{*}\,.$$ $\ker T=\{0\}$ and the density of $\operatorname{ran}T^{*}$ yields $R_{1}R_{1}^{*}+R_{2}R_{2}^{*}=I$. If $T_{1}T_{1}^{*}$ and $T_{2}T_{2}^{*}$ commute, then by $TT^{*}=T_{1}T_{1}^{*}+T_{2}T_{2}^{*}$ also $T_{j}T_{j}^{*}$ and $TT^{*}$ commute. Moreover, in this case $$T(\ T^{*}TR_{j}R_{j}^{*}\ )T^{*}=TT^{*}T_{j}T_{j}^{*}=T_{j}T_{j}^{*}TT^{*}=T(% \ R_{j}R_{j}^{*}T^{*}T\ )T^{*}\,.$$ Employing again $T$’s injectivity and the density of $\operatorname{ran}T^{*}$ we see that $R_{j}R_{j}^{*}$ and $T^{*}T$ commute for $j=1,2$. From this we get $$T_{j}^{*}T_{j}R_{j}^{*}R_{j}=R_{j}^{*}(T^{*}TR_{j}R_{j}^{*})R_{j}=R_{j}^{*}(R_% {j}R_{j}^{*}T^{*}T)R_{j}=R_{j}^{*}R_{j}T_{j}^{*}T_{j}\,.$$ Thus, we showed 2.1 Lemma. \thlabel existtreans With the above notations and assumptions there exist injective contractions $R_{1}:{\mathcal{V}}_{1}\to{\mathcal{V}}$ and $R_{2}:{\mathcal{V}}_{2}\to{\mathcal{V}}$ such that $T_{1}=TR_{1}$, $T_{2}=TR_{2}$ and $R_{1}R_{1}^{*}+R_{2}R_{2}^{*}=I$. If $T_{1}T_{1}^{*}$ and $T_{2}T_{2}^{*}$ commute, then the operators $R_{j}R_{j}^{*}$ and $T^{*}T$ on ${\mathcal{V}}$ commute as well as the operators $R_{j}^{*}R_{j}$ and $T_{j}^{*}T_{j}$ on ${\mathcal{V}}_{j}$ for $j=1,2$. ${\mathcal{V}}$ ${\mathcal{V}}_{1}$ $R_{1}$ ${\mathcal{V}}_{2}$ $R_{2}$${\mathcal{K}}$$T$$T_{1}$$T_{2}$ By $\Theta_{j}:(T_{j}T_{j}^{*})^{\prime}\ (\subseteq B({\mathcal{K}}))\to(T_{j}^{*% }T_{j})^{\prime}\ (\subseteq B({\mathcal{V}}_{j})),\ j=1,2$, and by $\Theta:(TT^{*})^{\prime}\ (\subseteq B({\mathcal{K}}))\to(T^{*}T)^{\prime}\ (% \subseteq B({\mathcal{V}}))$ we shall denote the $*$-algebra homomorphisms mapping the identity operator to the identity operator as in \threfthetadefeig from [KP] corresponding to the mappings $T_{j},\ j=1,2$, and $T$: $$\Theta_{j}(C_{j})=(T_{j}\times T_{j})^{-1}(C_{j})=T_{j}^{-1}C_{j}T_{j},\ C_{j}% \in(T_{j}T_{j}^{*})^{\prime}\,,$$ $$\Theta(C)=(T\times T)^{-1}(C)=T^{-1}CT,\ C\in(TT^{*})^{\prime}\,.$$ (2.1) We can apply \threfthetadefeig in [KP] also to the bounded linear, injective $R_{j}:{\mathcal{V}}_{j}\to{\mathcal{V}},\ j=1,2$, and denote the corresponding $*$-algebra homomorphisms by $\Gamma_{j}:(R_{j}R_{j}^{*})^{\prime}\ (\subseteq B({\mathcal{V}}))\to(R_{j}^{*% }R_{j})^{\prime}\ (\subseteq B({\mathcal{V}}_{j}))$: $$\Gamma_{j}(D)=(R_{j}\times R_{j})^{-1}(D)=R_{j}^{-1}DR_{j},\ D\in(R_{j}R_{j}^{% *})^{\prime}\,.$$ 2.2 Proposition. \thlabel comreg With the above notations and assumptions we have $(T_{1}T_{1}^{*})^{\prime}\cap(T_{2}T_{2}^{*})^{\prime}\subseteq(TT^{*})^{\prime}$ and $\Theta((T_{1}T_{1}^{*})^{\prime}\cap(T_{2}T_{2}^{*})^{\prime})\subseteq(R_{1}R% _{1}^{*})^{\prime}\cap(R_{2}R_{2}^{*})^{\prime}\cap(T^{*}T)^{\prime}$, where in fact ($j=1,2$) $$\Theta(C)R_{j}R_{j}^{*}=R_{j}\Theta_{j}(C)R_{j}^{*}=R_{j}R_{j}^{*}\Theta(C),\ % \ C\in(T_{1}T_{1}^{*})^{\prime}\cap(T_{2}T_{2}^{*})^{\prime}\,.$$ (2.2) Moreover, $$\Theta_{j}(C)=\Gamma_{j}\circ\Theta(C),\ \ C\in(T_{1}T_{1}^{*})^{\prime}\cap(T% _{2}T_{2}^{*})^{\prime}\,.$$ (2.3) Proof. $(T_{1}T_{1}^{*})^{\prime}\cap(T_{2}T_{2}^{*})^{\prime}\subseteq(TT^{*})^{\prime}$ is clear from $TT^{*}=T_{1}T_{1}^{*}+T_{2}T_{2}^{*}$. According to \threfthetadefeig in [KP] we have $\Theta_{j}(C)T_{j}^{*}=T_{j}^{*}C$ and $T^{*}C=\Theta(C)T^{*}$ for $C\in(T_{1}T_{1}^{*})^{\prime}\cap(T_{2}T_{2}^{*})^{\prime}$. Therefore, $$\displaystyle T(\ R_{j}\Theta_{j}(C)R_{j}^{*}\ )T^{*}=T_{j}\Theta_{j}(C)T_{j}^% {*}=T_{j}T_{j}^{*}C=\\ \displaystyle TR_{j}R_{j}^{*}T^{*}C=T(\ R_{j}R_{j}^{*}\Theta(C)\ )T^{*}\,.$$ $\ker T=\{0\}$ and the density of $\operatorname{ran}T^{*}$ yield $R_{j}\Theta_{j}(C)R_{j}^{*}=R_{j}R_{j}^{*}\Theta(C)$ for $j=1,2$. Applying this equation to $C^{*}$ and taking adjoints yields $R_{j}\Theta_{j}(C)R_{j}^{*}=\Theta(C)R_{j}R_{j}^{*}$. In particular, $\Theta(C)\in(R_{j}R_{j}^{*})^{\prime}$. Therefore, we can apply $\Gamma_{j}$ to $\Theta(C)$ and get $$\Gamma_{j}\circ\Theta(C)=R_{j}^{-1}T^{-1}CTR_{j}=T_{j}^{-1}CT_{j}=\Theta_{j}(C% )\,.$$ ∎ For the following assertion note that by (2.3) and by the fact that $\Gamma_{j}$ is a $*$-algebra homomorphism mapping the identity operator to the identity operator, we have ($j=1,2$) $$\sigma(\Theta(C))\subseteq\sigma(\Theta_{j}(C)),\ C\in(T_{1}T_{1}^{*})^{\prime% }\cap(T_{2}T_{2}^{*})^{\prime}\,.$$ (2.4) 2.3 Corollary. \thlabel normtransf With the above notations and assumptions let $N\in(T_{1}T_{1}^{*})^{\prime}\cap(T_{2}T_{2}^{*})^{\prime}$ be normal. Then $\Theta(N),\Theta_{1}(N),\Theta_{2}(N)$ are all normal operators in the Hilbert spaces ${\mathcal{V}}$, ${\mathcal{V}}_{1}$, ${\mathcal{V}}_{2}$, respectively. If $E$ ($E_{1}$,$E_{2}$) denotes the spectral measure for $\Theta(N)$ ($\Theta_{1}(N)$, $\Theta_{2}(N)$), then $E(\Delta)\in(R_{1}R_{1}^{*})^{\prime}\cap(R_{2}R_{2}^{*})^{\prime}\cap(T^{*}T)% ^{\prime}$ and $$\Gamma_{j}(E(\Delta))=E_{j}(\Delta),\ \ j=1,2\,,$$ for all Borel subsets $\Delta$ of ${\mathbb{C}}$, where $E_{j}(\Delta)\in(R_{j}^{*}R_{j})^{\prime}\cap(T_{j}^{*}T_{j})^{\prime}$. Moreover, $\int h\,dE\in(R_{1}R_{1}^{*})^{\prime}\cap(R_{2}R_{2}^{*})^{\prime}\cap(T^{*}T% )^{\prime}$ and $$\Gamma_{j}\left(\int h\,dE\right)=\int h\,dE_{j}$$ for any bounded and measurable $h:\sigma(\Theta(N))\to{\mathbb{C}}$, where $\int h\,dE_{j}$ belongs to $(R_{j}^{*}R_{j})^{\prime}\cap(T_{j}^{*}T_{j})^{\prime}$. Proof. The normality of $\Theta(N),\Theta_{1}(N)$ and $\Theta_{2}(N)$ is clear, since $\Theta$, $\Theta_{1},\Theta_{2}$ are $*$-homomorphisms. From \threfcomreg we know that $\Theta(N)\in(R_{1}R_{1}^{*})^{\prime}\cap(R_{2}R_{2}^{*})^{\prime}\cap(T^{*}T)% ^{\prime}$. According to the well known properties of $\Theta(N)$’s spectral measure we obtain $E(\Delta)\in(R_{1}R_{1}^{*})^{\prime}\cap(R_{2}R_{2}^{*})^{\prime}\cap(T^{*}T)% ^{\prime}$ and, in turn, $\int h\,dE\in(R_{1}R_{1}^{*})^{\prime}\cap(R_{2}R_{2}^{*})^{\prime}\cap(T^{*}T% )^{\prime}$. In particular, $\Gamma_{j}$ can be applied to $E(\Delta)$ and $\int h\,dE$. Similarly, $\Theta_{j}(N)\in(T_{j}^{*}T_{j})^{\prime}$ implies $E_{j}(\Delta),\int h\,dE_{j}\in(T_{j}^{*}T_{j})^{\prime}$ for a bounded and measurable $h$. Recall from \threfthetadefeig in [KP] that $\Gamma_{j}(D)R_{j}^{*}x=R_{j}^{*}D$ for $D\in(T^{*}T)^{\prime}$. For $x\in{\mathcal{V}}$ and $y\in{\mathcal{V}}_{j}$ we therefore get $$[\Gamma_{j}(E(\Delta))R_{j}^{*}x,y]=[R_{j}^{*}E(\Delta)x,y]=[E(\Delta)x,R_{j}y]$$ and, in turn, $$\displaystyle\int_{{\mathbb{C}}}s(z,\bar{z})\,d[\Gamma_{j}(E)R_{j}^{*}x,y]=% \int_{{\mathbb{C}}}s(z,\bar{z})\,d[Ex,R_{j}y]=[s(\Theta(N),\Theta(N)^{*})x,R_{% j}y]=\\ \displaystyle[R_{j}^{*}s(\Theta(N),\Theta(N)^{*})x,y]=[\Gamma_{j}(s(\Theta(N),% \Theta(N)^{*}))R_{j}^{*}x,y]$$ for any trigonometric polynomial $s(z,\bar{z})\in{\mathbb{C}}[z,\bar{z}]$. By (2.3) and the fact that $\Gamma_{j}$ is a $*$-homomorphism we have $\Gamma_{j}(s(\Theta(N),\Theta(N)^{*}))=s(\Theta_{j}(N),\Theta_{j}(N)^{*})$. Consequently, $$\int_{{\mathbb{C}}}s(z,\bar{z})\,d[\Gamma_{j}(E)R_{j}^{*}x,y]=\int_{{\mathbb{C% }}}s(z,\bar{z})\,d[E_{j}R_{j}^{*}x,y]\,.$$ Since $E({\mathbb{C}}\setminus K)=0$ and $E_{j}({\mathbb{C}}\setminus K)=0$ for a certain compact $K\subseteq{\mathbb{C}}$ and since ${\mathbb{C}}[z,\bar{z}]$ is densely contained in $C(K)$, we obtain from the uniqueness assertion of the Riesz Representation Theorem $$[\Gamma_{j}(E(\Delta))R_{j}^{*}x,y]=[E_{j}(\Delta)R_{j}^{*}x,y],\ x\in{% \mathcal{V}},\,y\in{\mathcal{V}}_{j}\,,$$ for all Borel subsets $\Delta$ of ${\mathbb{C}}$. Due to the density of $\operatorname{ran}R_{j}^{*}$ in ${\mathcal{V}}_{j}$ we even have $[\Gamma_{j}(E(\Delta))z,y]=[E_{j}(\Delta)z,y],\ y,z\in{\mathcal{V}}_{j}$ and, in turn, $\Gamma_{j}(E(\Delta))=E_{j}(\Delta)$. Since $\Gamma_{j}$ maps into $(R_{j}^{*}R_{j})^{\prime}$ we have $E_{j}(\Delta)\in(R_{j}^{*}R_{j})^{\prime}$ and, in turn, $\int h\,dE_{j}\in(R_{j}^{*}R_{j})^{\prime}$ for any bounded and measurable $h$. If $h:\sigma(\Theta(N))\to{\mathbb{C}}$ is bounded and measurable, then, clearly, also its restriction to $\sigma(\Theta_{j}(N))=\sigma(\Gamma_{j}\circ\Theta(N))$ is bounded and measurable; see (2.4). Due to $E_{j}(\Delta)R_{j}^{*}=\Gamma_{j}(E(\Delta))R_{j}^{*}=R_{j}^{*}E(\Delta)$ for $x\in{\mathcal{V}}$ and $y\in{\mathcal{V}}_{j}$ we have $$\displaystyle[\Gamma_{j}\left(\int h\,dE\right)R_{j}^{*}x,y]=[R_{j}^{*}\left(% \int h\,dE\right)x,y]=[\left(\int h\,dE\right)x,R_{j}y]=\\ \displaystyle\int h\,d[Ex,R_{j}y]=\int h\,d[E_{j}R_{j}^{*}x,y]=[\left(\int h\,% dE_{j}\right)R_{j}^{*}x,y]\,.$$ Again the density of $\operatorname{ran}R_{j}^{*}$ yields $\Gamma_{j}\left(\int h\,dE\right)=\int h\,dE_{j}$. ∎ Recall from \threfXidefeig in [KP] the mappings ($j=1,2$) $$\Xi_{j}:(T_{j}^{*}T_{j})^{\prime}\ (\subseteq B({\mathcal{V}}_{j}))\to(T_{j}T_% {j}^{*})^{\prime}\ (\subseteq B({\mathcal{K}})),\ \Xi_{j}(D_{j})=T_{j}D_{j}T_{% j}^{*}\,,$$ (2.5) and $\Xi:(T^{*}T)^{\prime}\ (\subseteq B({\mathcal{V}}))\to(TT^{*})^{\prime}\ (% \subseteq B({\mathcal{K}})),\ \Xi(D)=TDT^{*}$. By ($j=1,2$) $$\Lambda_{j}:(R_{j}^{*}R_{j})^{\prime}\ (\subseteq B({\mathcal{V}}_{j}))\to(R_{% j}R_{j}^{*})^{\prime}\ (\subseteq B({\mathcal{V}})),\ \Lambda_{j}(D_{j})=R_{j}% D_{j}R_{j}^{*}\,,$$ we shall denote the corresponding mappings outgoing from the mappings $R_{j}:{\mathcal{V}}_{j}\to{\mathcal{V}}$. By \threfexisttreans we have $$\Xi_{j}(D_{j})=T_{j}R_{j}D_{j}R_{j}^{*}T_{j}^{*}=\Xi\circ\Lambda_{j}(D_{j})\ % \text{ for }\ D_{j}\in(R_{j}^{*}R_{j})^{\prime}\cap(T_{j}^{*}T_{j})^{\prime}\,.$$ According to \threfXidefeig in [KP], $\Lambda_{j}\circ\Gamma_{j}(D)=DR_{j}R_{j}^{*}$. Hence, using the notation from \threfnormtransf $$\Xi_{j}(\int h\,dE_{j})=\Xi\circ\Lambda_{j}\circ\Gamma_{j}\left(\int h\,dE% \right)=\Xi(R_{j}R_{j}^{*}\int h\,dE)\,.$$ (2.6) Finally, $T^{-1}T_{j}T_{j}^{*}T=T^{-1}TR_{j}R_{j}^{*}T^{*}T=R_{j}R_{j}^{*}T^{*}T$. In case that $T_{1}T_{1}^{*}$ and $T_{2}T_{2}^{*}$ commute we have $T_{1}T_{1}^{*},T_{2}T_{2}^{*}\in(TT^{*})^{\prime}$ and the later equality can be expressed as ($j=1,2$) $$\Theta(T_{j}T_{j}^{*})=R_{j}R_{j}^{*}T^{*}T\,.$$ (2.7) 3 Normal definitizable operators 3.1 Definition. \thlabel definitnordef We will call a bounded linear and normal operator $N$ on a Krein Space definitizable if its real part $A:=\frac{N+N^{*}}{2}$ and its imaginary part $B:=\frac{N-N^{*}}{2i}$ are both definitizable, i.e. there exist real polynomials $p,q\in{\mathbb{R}}[z]$ such that $p$ is definitizing for $A$ ($[p(A)x,x]\geq 0,\,x\in{\mathcal{K}}$) and such that $q$ is definitizing for $B$ ($[q(B)x,x]\geq 0,\,x\in{\mathcal{K}}$); see [L]. ∎ By \threfanderedefinitz in [KP] the definitizability of $A$ and $B$ is equivalent to the concept of definitizability in [KP]. Also note that in Pontryagin spaces any bounded linear and normal operator is definitizable in the above sense; see \threfpontrunit in [KP]. 3.2 Proposition. \thlabel defNspaces Let $A$ and $B$ be commuting, bounded linear, selfadjoint and definitizable operators on a Krein space $({\mathcal{K}},[.,.])$ with definitizing polynomials $p\in{\mathbb{R}}[z]$ for $A$ and $q\in{\mathbb{R}}[z]$ for $B$. Then there exist Hilbert spaces $({\mathcal{V}}_{1},[.,.])$, $({\mathcal{V}}_{2},[.,.])$, $({\mathcal{V}},[.,.])$ and bounded linear and injective operators $T_{1}:{\mathcal{V}}_{1}\to{\mathcal{K}}$, $T_{2}:{\mathcal{V}}_{2}\to{\mathcal{K}}$, $T:{\mathcal{V}}\to{\mathcal{K}}$ such that $$T_{1}T_{1}^{*}=p(A),\ T_{2}T_{2}^{*}=q(B),\ TT^{*}=p(A)+q(B)=T_{1}T_{1}^{*}+T_% {2}T_{2}^{*}$$ with commuting $T_{1}T_{1}^{*}$ and $T_{2}T_{2}^{*}$. Moreover, if $\Theta:(TT^{*})^{\prime}\ (\subseteq B({\mathcal{K}}))\to(T^{*}T)^{\prime}\ (% \subseteq B({\mathcal{V}}))$ is as in (2.1) and $R_{j}:{\mathcal{V}}_{j}\to{\mathcal{V}}$ ($j=1,2$) are as in \threfexisttreans, then $$\displaystyle p(\Theta(A))=R_{1}R_{1}^{*}\big{(}p(\Theta(A))+q(\Theta(B))\big{% )},$$ (3.1) $$\displaystyle q(\Theta(B))=R_{2}R_{2}^{*}\big{(}p(\Theta(A))+q(\Theta(B))\big{% )}\,,$$ where $R_{1}R_{1}^{*}$ and $R_{2}R_{2}^{*}$ commute with $p(\Theta(A))+q(\Theta(B))$. Proof. Let $({\mathcal{V}}_{1},[.,.])$ be the Hilbert space completion of ${\mathcal{K}}/\ker p(A)$ with respect to $[p(A).,.]$ and let $T_{1}:{\mathcal{V}}_{1}\to{\mathcal{K}}$ be the adjoint of the embedding of ${\mathcal{K}}$ into ${\mathcal{V}}_{1}$. Since $T_{1}^{*}$ has dense range, $T_{1}$ is injective. Analogously let $({\mathcal{V}}_{2},[.,.])$ be the Hilbert space completion of ${\mathcal{K}}/\ker q(B)$ with respect to $[q(B).,.]$ and denote by $T_{2}:{\mathcal{V}}_{2}\to{\mathcal{K}}$ the injective adjoint of the embedding of ${\mathcal{K}}$ into ${\mathcal{V}}_{2}$. Finally, let $({\mathcal{V}},[.,.])$ be the Hilbert space completion of ${\mathcal{K}}/(\ker p(A)+q(B))$ with respect to $[(p(A)+q(B)).,.]$ and let $T:{\mathcal{V}}\to{\mathcal{K}}$ be the injective adjoint of the embedding of ${\mathcal{K}}$ into ${\mathcal{V}}$. From $[TT^{*}x,y]=[T^{*}x,T^{*}y]_{{\mathcal{V}}}=[x,y]_{{\mathcal{V}}}=[(p(A)+q(B))% x,y]$, $[T_{1}T_{1}^{*}x,y]=[T_{1}^{*}x,T_{1}^{*}y]_{{\mathcal{V}}_{1}}=[x,y]_{{% \mathcal{V}}_{1}}=[p(A)x,y]$ and $[T_{2}T_{2}^{*}x,y]=[q(B)x,y]$ for all $x,y\in{\mathcal{K}}$ we conclude that $$T_{1}T_{1}^{*}=p(A),\ T_{2}T_{2}^{*}=q(B),\ TT^{*}=p(A)+q(B)\,,$$ where $p(A)=T_{1}T_{1}^{*}$ and $q(B)=T_{2}T_{2}^{*}$ commute, because $A$ and $B$ do. From (2.7) and \threfthetadefeig in [KP] we get $$\displaystyle p(\Theta(A))=\Theta(p(A))=\Theta(T_{1}T_{1}^{*})=R_{1}R_{1}^{*}T% ^{*}T=R_{1}R_{1}^{*}\Theta(TT^{*})=\\ \displaystyle R_{1}R_{1}^{*}\Theta(p(A)+q(B))=R_{1}R_{1}^{*}\big{(}p(\Theta(A)% )+q(\Theta(B))\big{)}\,.$$ Similarly, $q(\Theta(B))=R_{2}R_{2}^{*}(p(\Theta(A))+q(\Theta(B)))$. Finally, $R_{1}R_{1}^{*}$ and $R_{2}R_{2}^{*}$ commute with $T^{*}T=p(\Theta(A))+q(\Theta(B))$ by \threfexisttreans. ∎ The fact that a normal operator is definitizable implies certain spectral properties of $\Theta(N)$. 3.3 Lemma. \thlabel speknorm With the notion of \threfdefNspaces applied to the real part $A:=\frac{N+N^{*}}{2}$ and the imaginary part $B:=\frac{N-N^{*}}{2i}$ of a bounded linear, normal and definitizable operator $N$ we have $$\{z\in{\mathbb{C}}:|p(\operatorname{Re}z)|>\|R_{1}R_{1}^{*}\|\cdot|p(% \operatorname{Re}z)+q(\operatorname{Im}z)|\}\subseteq\rho(\Theta(N))\,,$$ and $$\{z\in{\mathbb{C}}:|q(\operatorname{Im}z)|>\|R_{2}R_{2}^{*}\|\cdot|p(% \operatorname{Re}z)+q(\operatorname{Im}z)|\}\subseteq\rho(\Theta(N))\,.$$ In particular, the zeros of $p(\operatorname{Re}z)+q(\operatorname{Im}z)$ are contained in $\rho(\Theta(N))\cup\{z\in{\mathbb{C}}:p(\operatorname{Re}z)=0=q(\operatorname{% Im}z)\}$. Proof. We are going to show the first inclusion. The second one is shown in the same manner. For this let $n\in{\mathbb{N}}$ and set $$\Delta_{n}:=\{z\in{\mathbb{C}}:|p(\operatorname{Re}z)|^{2}>\frac{1}{n}+\|R_{1}% R_{1}^{*}\|^{2}\cdot|p(\operatorname{Re}z)+q(\operatorname{Im}z)|^{2}\}\,.$$ For $x\in E(\Delta_{n})({\mathcal{V}})$ we then have $$\displaystyle\|p(\Theta(A))x\|^{2}=\int_{\Delta_{n}}|p(\operatorname{Re}\zeta)% |^{2}\,d[E(\zeta)x,x]\geq\\ \displaystyle\int_{\Delta_{n}}\frac{1}{n}\,d[E(\zeta)x,x]+\|R_{1}R_{1}^{*}\|^{% 2}\int_{\Delta_{n}}|p(\operatorname{Re}\zeta)+q(\operatorname{Im}\zeta)|^{2}\,% d[E(\zeta)x,x]\\ \displaystyle\geq\frac{1}{n}\|x\|^{2}+\|R_{1}R_{1}^{*}\big{(}p(\Theta(A))+q(% \Theta(B))x\|^{2}\,.$$ By (3.1) this inequality can only hold for $x=0$. Since $\Delta_{n}$ is open, by the Spectral Theorem for normal operators on Hilbert spaces we have $\Delta_{n}\subseteq\rho(N)$. The asserted inclusion now follows from $$\{z\in{\mathbb{C}}:|p(\operatorname{Re}z)|>\|R_{1}R_{1}^{*}\|\cdot|p(% \operatorname{Re}z)+q(\operatorname{Im}z)|\}=\bigcup_{n\in{\mathbb{N}}}\Delta_% {n}\,.$$ ∎ 3.4 Corollary. \thlabel korvda With the notation and assumptions from \threfspeknorm we have $$\displaystyle R_{1}R_{1}^{*}\,E\{z\in{\mathbb{C}}:p(\operatorname{Re}z)\not=0% \text{ or }q(\operatorname{Im}z)\not=0\}=\\ \displaystyle\int_{\{z\in{\mathbb{C}}:p(\operatorname{Re}z)\not=0\text{ or }q(% \operatorname{Im}z)\not=0\}}\frac{p(\operatorname{Re}z)}{p(\operatorname{Re}z)% +q(\operatorname{Im}z)}\,dE(z)$$ and $$\displaystyle R_{2}R_{2}^{*}\,E\{z\in{\mathbb{C}}:p(\operatorname{Re}z)\not=0% \text{ or }q(\operatorname{Im}z)\not=0\}=\\ \displaystyle\int_{\{z\in{\mathbb{C}}:p(\operatorname{Re}z)\not=0\text{ or }q(% \operatorname{Im}z)\not=0\}}\frac{q(\operatorname{Re}z)}{p(\operatorname{Re}z)% +q(\operatorname{Im}z)}\,dE(z)$$ Proof. First note that the integrals on the right hand sides exist as bounded operators, because by \threfspeknorm we have $|p(\operatorname{Re}z)|\leq\|R_{1}R_{1}^{*}\|\cdot|p(\operatorname{Re}z)+q(% \operatorname{Im}z)|$ and $|q(\operatorname{Im}z)|\leq\|R_{2}R_{2}^{*}\|\cdot|p(\operatorname{Re}z)+q(% \operatorname{Im}z)|$ on $\sigma(\Theta(N))$. Clearly, both sides vanish on the range of $E\{z\in{\mathbb{C}}:p(\operatorname{Re}z)=0=q(\operatorname{Im}z)\}$. Its orthogonal complement $$\displaystyle{\mathcal{H}}:=\operatorname{ran}E\{z\in{\mathbb{C}}:p(% \operatorname{Re}z)=0=q(\operatorname{Im}z)\}^{\bot}=\\ \displaystyle\operatorname{ran}E\{z\in{\mathbb{C}}:p(\operatorname{Re}z)\not=0% \text{ or }q(\operatorname{Im}z)\not=0\}\,,$$ is invariant under $\int\big{(}p(\operatorname{Re}z)+q(\operatorname{Im}z)\big{)}\,dE(z)=\big{(}p(% \Theta(A))+q(\Theta(B))\big{)}$. By \threfspeknorm the restriction of this operator to ${\mathcal{H}}$ is injective, and hence, has dense range in ${\mathcal{H}}$. If $x$ belongs to this dense range, i.e.  $x=\big{(}p(\Theta(A))+q(\Theta(B))\big{)}y$ with $y\in{\mathcal{H}}$, then $$\int_{\{z\in{\mathbb{C}}:p(\operatorname{Re}z)\not=0\text{ or }q(\operatorname% {Im}z)\not=0\}}\frac{p(\operatorname{Re}z)}{p(\operatorname{Re}z)+q(% \operatorname{Im}z)}\,dE(z)x=$$ $$\displaystyle\int_{\{z\in{\mathbb{C}}:p(\operatorname{Re}z)\not=0\text{ or }q(% \operatorname{Im}z)\not=0\}}p(\operatorname{Re}z)\,dE(z)y=p(\Theta(A))y=\\ \displaystyle R_{1}R_{1}^{*}\big{(}p(\Theta(A))+q(\Theta(B))\big{)}y=R_{1}R_{1% }^{*}x\,.$$ By a density argument the first asserted equality of the present corollary holds true on ${\mathcal{H}}$ and in turn on ${\mathcal{V}}$. The second equality is shown in the same manner. ∎ 4 The proper function class In order to introduce a functional calculus we have to introduce an algebra structure on ${\mathcal{A}}_{m,n}:=(\mathbb{C}^{m}\otimes\mathbb{C}^{n})\times\mathbb{C}^{2}% \simeq\mathbb{C}^{m\cdot n+2}$ and on ${\mathcal{B}}_{m,n}:=\mathbb{C}^{m}\otimes\mathbb{C}^{n}\simeq\mathbb{C}^{m% \cdot n}$ for $m,n\in{\mathbb{N}}$. For notational convenience we also set ${\mathcal{A}}_{0,0}:=\mathbb{C}$. 4.1 Definition. \thlabel muldefb1 Firstly, let ${\mathcal{A}}_{0,0}={\mathbb{C}}$ be provided with the usual addition, scalar multiplication, multiplication and conjugation. Secondly, in case that $m,n\in{\mathbb{N}}$ we provide ${\mathcal{A}}_{m,n}$ with the componentwise addition and scalar multiplication. Moreover, for $a=(a_{k,l})_{(k,l)\in I_{m,n}},b=(b_{k,l})_{(k,l)\in I_{m,n}}$ with $I_{m,n}:=(\{0,\dots,m-1\}\times\{0,\dots,n-1\})\cup\{(m,0),(0,n)\}$ we set $$a\cdot b:=\Big{(}\sum_{c=0}^{k}\sum_{d=0}^{l}a_{c,d}b_{k-c,l-d}\Big{)}_{(k,l)% \in I_{m,n}}\ \text{ and }\ \overline{a}:=\big{(}\bar{a}_{k,l}\big{)}_{(k,l)% \in I_{m,n}}\,.$$ On ${\mathcal{B}}_{m,n}$ we define addition, scalar multiplication, multiplication and conjugation in the same way only neglecting the the entries with indices $(m,0)$ and $(0,n)$. Finally, for $m,n\in{\mathbb{N}}$ we introduce the projection $\pi:{\mathcal{A}}_{m,n}\to{\mathcal{B}}_{m,n}$, $(a_{k,l})_{(k,l)\in I_{m,n}}\mapsto(a_{k,l})_{\begin{subarray}{c}0\leq k\leq m% -1\\ 0\leq l\leq n-1\end{subarray}}$. On ${\mathcal{B}}_{m,n}$ we assume $\pi$ to be the identity. ∎ 4.2 Remark. \thlabel z30f3 It is easy to check that ${\mathcal{A}}_{m,n}$ and ${\mathcal{B}}_{m,n}$ are commutative, unital $*$-algebras. Setting $e_{0,0}=1$ and $e_{k,l}=0,\ (k,l)\neq(0,0)$, it is easy to verify that $\big{(}e_{k,l}\big{)}_{(k,l)\in I_{m,n}}$ is the multiplicative unite in ${\mathcal{A}}_{m,n}$ and $\big{(}e_{k,l}\big{)}_{\begin{subarray}{c}0\leq k\leq m-1\\ 0\leq l\leq n-1\end{subarray}}$ is the multiplicative unite in ${\mathcal{B}}_{m,n}$. We shall denote these unites by $e$. Moreover, it is straight forward to check that an element $(a_{k,l})$ of ${\mathcal{A}}_{m,n}$ (of ${\mathcal{B}}_{m,n}$) has a multiplicative inverse in ${\mathcal{A}}_{m,n}$ (in ${\mathcal{B}}_{m,n}$) if and only if $a_{0,0}\neq 0$. ∎ For the rest of the paper assume that $N$ bounded linear, normal and definitizable operator in a Krein space ${\mathcal{K}}$ with real part $A$ and imaginary part $B$. Moreover, we fix definitizing polynomials $p\in{\mathbb{R}}[z]$ for $A$ and $q\in{\mathbb{R}}[z]$ for $B$. 4.3 Definition. \thlabel muldefb2 We define functions ${\mathfrak{d}}_{p},{\mathfrak{d}}_{q}:{\mathbb{C}}\to{\mathbb{N}}\cup\{0\}$ such that ${\mathfrak{d}}_{p}(z)$ is $p$’s degrees of the zero at $z$ and ${\mathfrak{d}}_{q}(z)$ is $q$’s degrees of the zero at $z$. Moreover, we shall denote the set of their real zeros by $Z^{{\mathbb{R}}}_{p}$ and $Z^{{\mathbb{R}}}_{q}$, i.e. $$Z^{{\mathbb{R}}}_{p}:=p^{-1}\{0\}\cap{\mathbb{R}},Z^{{\mathbb{R}}}_{q}:=q^{-1}% \{0\}\cap{\mathbb{R}}\,,$$ and we set $Z^{i}:=(p^{-1}\{0\}\times q^{-1}\{0\})\setminus({\mathbb{R}}\times{\mathbb{R}})$. Now we are going to introduce class of functions: $(i)$ By ${\mathcal{M}}_{N}$ we denote the set of functions $\phi$ defined on $$\big{(}\sigma(\Theta(N))\cup(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})\big{)% }\dot{\cup}Z^{i}$$ with $\phi(z)\in{\mathfrak{C}}(z)$, where ${\mathfrak{C}}(z):={\mathcal{B}}_{{\mathfrak{d}}_{p}(\xi),{\mathfrak{d}}_{q}(% \eta)}$ for $z=(\xi,\eta)\in Z^{i}$ and where ${\mathfrak{C}}(z):={\mathcal{A}}_{{\mathfrak{d}}_{p}(\operatorname{Re}z),{% \mathfrak{d}}_{q}(\operatorname{Im}z)}$ for $z\in\sigma(\Theta(N))\cup(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$. $(ii)$ We provide ${\mathcal{M}}_{N}$ pointwise with scalar multiplication, addition and multiplication, where the operations on ${\mathcal{A}}_{{\mathfrak{d}}_{p}(\operatorname{Re}z),{\mathfrak{d}}_{q}(% \operatorname{Im}z)}$ or ${\mathcal{B}}_{{\mathfrak{d}}_{p}(\xi),{\mathfrak{d}}_{q}(\eta)}$ are as in \threfmuldefb1. We also define a conjugate linear involution $.^{\#}$ on ${\mathcal{M}}_{N}$ by $$\displaystyle\phi^{\#}(z)=\overline{\phi(z)},\ z\in\sigma(\Theta(N))\cup(Z^{{% \mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q}),\\ \displaystyle\phi^{\#}(\xi,\eta)=\overline{\phi(\bar{\xi},\bar{\eta})},\ (\xi,% \eta)\in Z^{i}\,.$$ $(iii)$ By ${\mathcal{R}}$ we denote the set of all elements $\phi\in{\mathcal{M}}_{N}$ such that $\pi(\phi(z))=0$ for all $z\in(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})\dot{\cup}Z^{i}$. ∎ With the operations introduced in \threfmuldefb2 ${\mathcal{M}}_{N}$ is a commutative $*$-algebra as can be verified in a straight forward manner. Moreover, ${\mathcal{R}}$ is an ideal of ${\mathcal{M}}_{N}$. 4.4 Definition. \thlabel feinbetef Let $f:\operatorname{dom}f\to{\mathbb{C}}$ be a function with $\operatorname{dom}f\subseteq{\mathbb{C}}^{2}$ such that $\tau\big{(}\sigma(\Theta(N))\cup(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})% \big{)}\subseteq\operatorname{dom}f$, where $\tau:{\mathbb{C}}\to{\mathbb{C}}^{2},\ (x+iy)\mapsto(x,y)$, such that $f\circ\tau$ is sufficiently smooth – more exactly, at least $\max_{x,y\in{\mathbb{R}}}{\mathfrak{d}}_{p}(x)+{\mathfrak{d}}_{q}(y)-1$ times continuously differentiable – on an open neighbourhood of $Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q}$, and such that $f$ is holomorphic on an open neighbourhood of $Z^{i}$. Then $f$ can be considered as an element $f_{N}$ of ${\mathcal{M}}_{N}$ by setting $f_{N}(z):=f\circ\tau(z)$ for $z\in\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$, by $$f_{N}(z):=\big{(}\frac{1}{k!l!}\,\frac{\partial^{k+l}}{\partial x^{k}\partial y% ^{l}}f\circ\tau(z)\big{)}_{(k,l)\in I_{{\mathfrak{d}}_{p}(\operatorname{Re}z),% {\mathfrak{d}}_{q}(\operatorname{Im}z)}}$$ for $z\in Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q}$, and by $$f_{N}(\xi,\eta):=\big{(}\frac{1}{k!l!}\,\frac{\partial^{k+l}}{\partial z^{k}% \partial w^{l}}f(\xi,\eta)\big{)}_{\begin{subarray}{c}0\leq k\leq{\mathfrak{d}% }_{p}(\xi)-1\\ 0\leq l\leq{\mathfrak{d}}_{q}(\eta)-1\end{subarray}}\,,$$ for $(\xi,\eta)\in Z^{i}$. ∎ 4.5 Remark. \thlabel bweuh30 By the Leibniz rule $f\mapsto f_{N}$ is compatible with multiplication. Obviously, it is also compatible with addition and scalar multiplication. If we define for a function $f$ as in \threffeinbetef the function $f^{\#}$ by $f^{\#}(z,w)=\overline{f(\bar{z},\bar{w})},\ (z,w)\in\operatorname{dom}f$, then we also have $(f^{\#})_{p,q}=(f_{N})^{\#}$. Note that in general $(\bar{f})_{p,q}\not=(f_{N})^{\#}$. Finally, note that $\mathds{1}_{N}(z)$ is the multiplicative unite in ${\mathfrak{C}}(z)$ for all $z\in\big{(}\sigma(\Theta(N))\cup(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})% \big{)}\dot{\cup}Z^{i}$. ∎ 4.6 Example. \thlabel fuinm0pre For the constant one function $\mathds{1}$ on ${\mathbb{C}}^{2}$ we have $\mathds{1}_{N}(z)=e$ for all $z\in\big{(}\sigma(\Theta(N))\cup(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})% \big{)}\dot{\cup}Z^{i}$, where $e$ is the multiplicative unite in ${\mathfrak{C}}(z)$; see \threfz30f3. ∎ 4.7 Example. \thlabel fuinm0 $p(z)$ considered as an element of ${\mathbb{C}}[z,w]$ is clearly holomorphic on ${\mathbb{C}}^{2}$. Hence, we can consider $p_{N}$ as defined in \threffeinbetef. It satisfies $p_{N}(z)_{k,l}=0$, $(k,l)\in I_{{\mathfrak{d}}_{p}(\operatorname{Re}z),{\mathfrak{d}}_{q}(% \operatorname{Im}z)}\setminus\{({\mathfrak{d}}_{p}(\operatorname{Re}z),0)\}$, and $$p_{N}(z)_{{\mathfrak{d}}_{p}(\operatorname{Re}z),0}=\frac{1}{{\mathfrak{d}}_{p% }(\operatorname{Re}z)!}p^{({\mathfrak{d}}_{p}(\operatorname{Re}z))}(% \operatorname{Re}z)$$ for all $z\in Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q}$. Since $\operatorname{Re}z$ is a zero of $p$ of degree exactly ${\mathfrak{d}}_{p}(\operatorname{Re}z)$ the entries with index $({\mathfrak{d}}_{p}(\operatorname{Re}z),0)$ do not vanish. Moreover, $p_{N}(\xi,\eta)=0$ for all $(\xi,\eta)\in Z^{i}$. In particular, $p_{N}\in{\mathcal{R}}$. Similarly, if $q(w)$ is considered as an element of ${\mathbb{C}}[z,w]$, then $q_{N}(z)_{k,l}=0$, $(k,l)\in I_{{\mathfrak{d}}_{p}(\operatorname{Re}z),{\mathfrak{d}}_{q}(% \operatorname{Im}z)}\setminus\{(0,{\mathfrak{d}}_{q}(\operatorname{Im}z))\}$, and $$q_{N}(z)_{0,{\mathfrak{d}}_{q}(\operatorname{Im}z)}=\frac{1}{{\mathfrak{d}}_{q% }(\operatorname{Im}z)!}q^{({\mathfrak{d}}_{q}(\operatorname{Im}z))}(% \operatorname{Im}z)\neq 0$$ for all $z\in Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q}$. Also here $q_{N}(\xi,\eta)=0$ for all $(\xi,\eta)\in Z^{i}$ and, in turn, $q_{N}\in{\mathcal{R}}$. ∎ We need an easy algebraic lemma based in the Euclidean algorithm. 4.8 Lemma. \thlabel einbett2pre For $a(z),b(z)\in{\mathbb{C}}[z]$ we denote by $a^{-1}\{0\}$ and $b^{-1}\{0\}$ the set of all zeros of $a$ and $b$ in ${\mathbb{C}}$, and by ${\mathfrak{d}}_{a}(z)$ (${\mathfrak{d}}_{b}(z)$) $a$’s ($b$’s) degree of zero at $z\in{\mathbb{C}}$. Denote by $m$ ($n$) the degree of the polynomial $a$ ($b$). Then any $s\in{\mathbb{C}}[z,w]$ can be written as $$s(z,w)=a(z)u(z,w)+b(w)v(z,w)+r(z,w)$$ with $u(z,w),v(z,w),r(z,w)\in{\mathbb{C}}[z,w]$ such that $r$’s $z$-degree is less than $m$ and its $w$-degree is less than $n$. Here $u(z,w),v(z,w),r(z,w)$ can be found in ${\mathbb{R}}[z,w]$ if $a(z),b(z)\in{\mathbb{R}}[z],\ s\in{\mathbb{R}}[z,w]$. If we define $\varpi:{\mathbb{C}}[z,w]\to{\mathbb{C}}^{m\cdot n}$ by $$\varpi(s)=\left(\Big{(}\frac{1}{k!l!}\,\frac{\partial^{k+l}}{\partial z^{k}% \partial w^{l}}s(z,w)\Big{)}_{\begin{subarray}{c}0\leq k\leq{\mathfrak{d}}_{a}% (z)-1\\ 0\leq l\leq{\mathfrak{d}}_{b}(w)-1\end{subarray}}\right)_{z\in a^{-1}\{0\},w% \in b^{-1}\{0\}}\,,$$ then $s\in\ker\varpi$ if and only if $s(z,w)=a(z)u(z,w)+b(w)v(z,w)$ for some $u(z,w),v(z,w)\in{\mathbb{C}}[z,w]$. Moreover, $\varpi$ restricted to the space of all polynomials from ${\mathbb{C}}[z,w]$ with $z$-degree less than $m$ and $w$-degree less than $n$ is bijective. Proof. Applying the Euclidean algorithm to $s(z,w)\in{\mathbb{C}}[z,w]$ and $a(z)$ we get $s(z,w)=a(z)u(z,w)+t(z,w)$, where $u(z,w),t(z,w)\in{\mathbb{C}}[z,w]$ such that $t$’s $z$-degree is less than $m$. Applying the Euclidean algorithm to $t(z,w)$ and $b(w)$ we get $$s(z,w)=a(z)u(z,w)+b(w)v(z,w)+r(z,w)$$ with $v(z,w),r(z,w)\in{\mathbb{C}}[z,w]$ such that $r$’s $z$-degree is less than $m$ and its $w$-degree is less than $n$. The resulting polynomials $u(z,w),t(z,w),v(z,w),r(z,w)$ belong to ${\mathbb{R}}[z,w]$ if $a(z),b(z)\in{\mathbb{R}}[z],\ s(z,w)\in{\mathbb{R}}[z,w]$. In any case it is easy to check that then $\varpi(s)=\varpi(r)$. Hence, $r(z,w)=0$ yields $s(z,w)\in\ker\varpi$. On the other hand, if $0=\varpi(s)=\varpi(r)$, then for each fixed $\zeta\in a^{-1}\{0\}$ and $k\in\{0,\dots,{\mathfrak{d}}_{a}(\zeta)-1\}$ the function $w\mapsto\frac{\partial^{k}}{\partial z^{k}}r(\zeta,w)$ has zeros at all $w\in b^{-1}\{0\}$ with multiplicity at least ${\mathfrak{d}}_{b}(w)$. Since $w\mapsto\frac{\partial^{k}}{\partial z^{k}}r(\zeta,w)$ is of $w$-degree less than $n$, it must be identically equal to zero. This implies that for any $\eta\in{\mathbb{C}}$ the polynomial $z\mapsto r(z,\eta)$ has zeros at all $\zeta\in a^{-1}\{0\}$ with multiplicity at least ${\mathfrak{d}}_{a}(\zeta)$. Since the degree of this polynomial in $z$ is less than $m$, we obtain $r(z,\eta)=0$ for any $z\in{\mathbb{C}}$. Thus, $r\equiv 0$. Our description of $\ker\varpi$ shows in particular that $\varpi$ restricted to the space of all polynomials from ${\mathbb{C}}[z,w]$ with $z$-degree less than $m$ and $w$-degree less than $n$ is one-to-one. Comparing dimensions shows that this restriction of $\varpi$ is also onto. ∎ 4.9 Corollary. \thlabel existbe With the notation from \threfmuldefb2 for any $\phi\in{\mathcal{M}}_{N}$ we find an $s\in{\mathbb{C}}[z,w]$ such that $\phi-s_{N}\in{\mathcal{R}}$. Proof. By \threfeinbett2pre there exists an $s\in{\mathbb{C}}[z,w]$ such that $\varpi(s)_{\operatorname{Re}z,\operatorname{Im}z}=\pi(\phi(z))$ for all $z\in Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q}$, and such that $\varpi(s)_{\xi,\eta}=\phi(\xi,\eta)$ for all $(\xi,\eta)\in Z^{i}$. According to ${\mathcal{R}}$’s definition we obtain $\phi-s_{N}\in{\mathcal{R}}$. ∎ 4.10 Remark. \thlabel durchdiv Recall from \threfspeknorm that $p(\operatorname{Re}z)+q(\operatorname{Im}z)=0$ with $z\in\sigma(\Theta(N))$ implies $p(\operatorname{Re}z)=0=q(\operatorname{Im}z)$, i.e. $z\in Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q}$. If $\phi\in{\mathcal{R}}$, then we find a function $g$ on $\sigma(\Theta(N))$ with $g(z)\in{\mathbb{C}}$ for $z\in\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$ and $g(z)\in{\mathbb{C}}^{2}$ for $z\in\sigma(\Theta(N))\cap(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$, such that $\phi(z)=(p_{N}+q_{N})(z)\cdot g(z),\ z\in\sigma(\Theta(N))$; see \threffuinm0. Here $(p_{N}+q_{N})(z)\cdot g(z)$ is the usual multiplication on ${\mathbb{C}}$ for $z\in\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$, whereas $$\big{(}(p_{N}+q_{N})(z)\cdot g(z)\big{)}_{k,l}=0,\ k=0,\dots,{\mathfrak{d}}_{p% }(\operatorname{Re}z)-1;l=0,\dots,{\mathfrak{d}}_{q}(\operatorname{Im}z)-1\,,$$ and $$\big{(}(p_{N}+q_{N})(z)\cdot g(z)\big{)}_{{\mathfrak{d}}_{p}(\operatorname{Re}% z),0}=(p_{N}+q_{N})(z)_{{\mathfrak{d}}_{p}(\operatorname{Re}z),0}\,\cdot g_{1}% (z)\,,$$ $$\big{(}(p_{N}+q_{N})(z)\cdot g(z)\big{)}_{0,{\mathfrak{d}}_{q}(\operatorname{% Im}z)}=(p_{N}+q_{N})(z)(z)_{0,{\mathfrak{d}}_{q}(\operatorname{Im}z)}\,\cdot g% _{2}(z)\,.$$ for $z\in\sigma(\Theta(N))\cap(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$. In fact, we simply set $g(z):=\frac{\phi(z)}{p(\operatorname{Re}z)+q(\operatorname{Im}z)}$ for $z\in\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$ and $$g_{1}(z):=\frac{{\mathfrak{d}}_{p}(\operatorname{Re}z)!\,\phi(z)_{({\mathfrak{% d}}_{p}(\operatorname{Re}z),0)}}{p^{({\mathfrak{d}}_{p}(\operatorname{Re}z))}(% \operatorname{Re}z)},\ g_{1}(z):=\frac{{\mathfrak{d}}_{q}(\operatorname{Im}z)!% \,\phi(z)_{(0,{\mathfrak{d}}_{q}(\operatorname{Im}z))}}{q^{({\mathfrak{d}}_{q}% (\operatorname{Im}z))}(\operatorname{Im}z)}$$ for $z\in\sigma(\Theta(N))\cap(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$. ∎ We are going to introduce a subclass of ${\mathcal{M}}_{N}$, which will be the proper class, in order to build up our functional calculus. 4.11 Definition. \thlabel FdefklM With the notation from \threfmuldefb2 we denote by ${\mathcal{F}}_{N}$ the set of all elements $\phi\in{\mathcal{M}}_{N}$ such that $z\mapsto\phi(z)$ is Borel measurable and bounded on $\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$, and such that for each $w\in\sigma(\Theta(N))\cap(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$ $$\frac{\phi(z)-\sum_{k=0}^{{\mathfrak{d}}_{p}(\operatorname{Re}w)-1}\sum_{l=0}^% {{\mathfrak{d}}_{q}(\operatorname{Im}w)-1}\phi(w)_{k,l}\operatorname{Re}(z-w)^% {k}\operatorname{Im}(z-w)^{l}}{\max(|\operatorname{Re}(z-w)|^{{\mathfrak{d}}_{% p}(\operatorname{Re}w)},|\operatorname{Im}(z-w)|^{{\mathfrak{d}}_{q}(% \operatorname{Im}w)})}$$ (4.1) is bounded for $z\in\sigma(\Theta(N))\cap U(w)\setminus\{w\}$, where $U(w)$ is a sufficiently small neighbourhood of $w$. ∎ Note that (4.1) is immaterial if $w$ is an isolated point of $\sigma(\Theta(N))$. 4.12 Example. \thlabel fedela For $\zeta\in(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})\dot{\cup}Z^{i}$ and $a\in{\mathfrak{C}}(\zeta)$ consider the functions $a\delta_{\zeta}\in{\mathcal{M}}_{N}$ which assumes the value $a$ at $\zeta$ and the value zero on the rest of $\big{(}\sigma(\Theta(N))\cup(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})\big{)% }\dot{\cup}Z^{i}$ If $\zeta$ belongs to $Z^{i}$ or if $\zeta$ is an isolated point of $\sigma(\Theta(N))\cup(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$, then $a\delta_{\zeta}$ belongs to ${\mathcal{F}}_{N}$. ∎ 4.13 Remark. \thlabel taylormehrdimrem Let $h$ be defined on an open subset $D$ of ${\mathbb{R}}^{2}$ with values in ${\mathbb{C}}$. Moreover, assume that for given $m,n\in{\mathbb{N}}$ the function $h$ is $m+n$ times continuously differentiable. Finally, fix $w\in D$. The well-known Taylors Approximation Theorem from multidimensional calculus then yields $$h(z)=\sum_{j=0}^{m+n-1}\sum_{\stackrel{{\scriptstyle k,l\in{\mathbb{N}}_{0}}}{% {k+l=j}}}\frac{1}{k!l!}\frac{\partial^{j}h}{\partial x^{k}\partial y^{l}}(w)% \operatorname{Re}(z-w)^{k}\operatorname{Im}(z-w)^{l}+O(|z-w|^{m+n})$$ for $z\to w$. Since $|z-w|^{m+n}\leq 2^{m+n}\max(|\operatorname{Re}(z-w)|^{m+n},|\operatorname{Im}(% z-w)|^{m+n})=O(\max(|\operatorname{Re}(z-w)|^{m},|\operatorname{Im}(z-w)|^{n}))$ and since $\operatorname{Re}(z-w)^{k}\operatorname{Im}(z-w)^{l}=O(\max(|\operatorname{Re}% (z-w)|^{m},|\operatorname{Im}(z-w)|^{n}))$ for $k\geq m$ or $l\geq n$, we also have $$\displaystyle h(z)=\sum_{k=0}^{m-1}\sum_{l=0}^{n-1}\frac{1}{k!l!}\frac{% \partial^{k+l}h}{\partial x^{k}\partial y^{l}}(w)\operatorname{Re}(z-w)^{k}% \operatorname{Im}(z-w)^{l}+\\ \displaystyle O(\max(|\operatorname{Re}(z-w)|^{m},|\operatorname{Im}(z-w)|^{n}% ))\,.$$ ∎ 4.14 Lemma. \thlabel gehzuF Let $f:\operatorname{dom}f\ (\subseteq{\mathbb{C}}^{2})\to{\mathbb{C}}$ be a function with the properties mentioned in \threffeinbetef. Then $f_{N}$ belongs to ${\mathcal{F}}_{N}$. Proof. For fixed $w\in\sigma(\Theta(N))\cap(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$ and $z\in\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$ by \threftaylormehrdimrem the expression $$f_{N}(z)-\sum_{k=0}^{{\mathfrak{d}}_{p}(\operatorname{Re}w)-1}\sum_{l=0}^{{% \mathfrak{d}}_{q}(\operatorname{Im}w)-1}f_{N}(w)_{k,l}\operatorname{Re}(z-w)^{% k}\operatorname{Im}(z-w)^{l}=$$ $$f\circ\tau(z)-\sum_{k=0}^{{\mathfrak{d}}_{p}(\operatorname{Re}w)-1}\sum_{l=0}^% {{\mathfrak{d}}_{q}(\operatorname{Im}w)-1}\frac{1}{k!l!}\frac{\partial^{k+l}f% \circ\tau}{\partial x^{k}\partial y^{l}}(w)\operatorname{Re}(z-w)^{k}% \operatorname{Im}(z-w)^{l}$$ is a $O(\max(|\operatorname{Re}(z-w)|^{{\mathfrak{d}}_{p}(\operatorname{Re}w)},|% \operatorname{Im}(z-w)|^{{\mathfrak{d}}_{q}(\operatorname{Im}w)}))$ for $z\to w$. Therefore, $f_{N}\in{\mathcal{F}}_{N}$. ∎ In order to be able to prove spectral results for our functional calculus, we need that with $\phi$ also $z\mapsto\phi(z)^{-1}$ belongs to ${\mathcal{F}}_{N}$ if $\phi$ is bounded away from zero. 4.15 Lemma. \thlabel einduF If $\phi\in{\mathcal{F}}_{N}$ is such that $\phi(z)$ is invertible in ${\mathfrak{C}}(z)$ (see \threfz30f3) for all $z\in\big{(}\sigma(\Theta(N))\cup(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})% \big{)}\dot{\cup}Z^{i}$ and such that $0$ does not belong to the closure of $\phi\big{(}\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{% q})\big{)}$, then $\phi^{-1}:z\mapsto\phi(z)^{-1}$ also belongs to ${\mathcal{F}}_{N}$. Proof. By the first assumption $\phi^{-1}$ is a well-defined object belonging to ${\mathcal{M}}_{N}$. Clearly, with $\phi$ also $z\mapsto\phi(z)^{-1}=\frac{1}{\phi(z)}$ is measurable on $\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$. By the second assumption $z\mapsto\phi(z)^{-1}=\frac{1}{\phi(z)}$ is bounded on this set. It remains to verify the boundedness of (4.1) on a certain neighbourhood of $w$ for each $w\in\sigma(\Theta(N))\cap(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$ for $\phi^{-1}$. To do so, we calculate for $z\in\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$ $$\phi^{-1}(z)-\sum_{k=0}^{{\mathfrak{d}}_{p}(\operatorname{Re}w)-1}\sum_{l=0}^{% {\mathfrak{d}}_{q}(\operatorname{Im}w)-1}\phi^{-1}(w)_{k,l}\operatorname{Re}(z% -w)^{k}\operatorname{Im}(z-w)^{l}=$$ (4.2) $$\frac{1}{\phi(z)}-\frac{1}{\sum_{k=0}^{{\mathfrak{d}}_{p}(\operatorname{Re}w)-% 1}\sum_{l=0}^{{\mathfrak{d}}_{q}(\operatorname{Im}w)-1}\phi(w)_{k,l}% \operatorname{Re}(z-w)^{k}\operatorname{Im}(z-w)^{l}}+$$ (4.3) $$\displaystyle\frac{1}{\sum_{k=0}^{{\mathfrak{d}}_{p}(\operatorname{Re}w)-1}% \sum_{l=0}^{{\mathfrak{d}}_{q}(\operatorname{Im}w)-1}\phi(w)_{k,l}% \operatorname{Re}(z-w)^{k}\operatorname{Im}(z-w)^{l}}-\\ \displaystyle\sum_{k=0}^{{\mathfrak{d}}_{p}(\operatorname{Re}w)-1}\sum_{l=0}^{% {\mathfrak{d}}_{q}(\operatorname{Im}w)-1}\phi^{-1}(w)_{k,l}\operatorname{Re}(z% -w)^{k}\operatorname{Im}(z-w)^{l}$$ (4.4) The expression in (4.3) can be written as $$\displaystyle\frac{1}{\phi(z)}\cdot\frac{1}{\sum_{k=0}^{{\mathfrak{d}}_{p}(% \operatorname{Re}w)-1}\sum_{l=0}^{{\mathfrak{d}}_{q}(\operatorname{Im}w)-1}% \phi(w)_{k,l}\operatorname{Re}(z-w)^{k}\operatorname{Im}(z-w)^{l}}\cdot\\ \displaystyle\left(\phi(z)-\sum_{k=0}^{{\mathfrak{d}}_{p}(\operatorname{Re}w)-% 1}\sum_{l=0}^{{\mathfrak{d}}_{q}(\operatorname{Im}w)-1}\phi(w)_{k,l}% \operatorname{Re}(z-w)^{k}\operatorname{Im}(z-w)^{l}\right)$$ Here $\frac{1}{\phi(z)}$ is bounded by assumption. The assumed invertibility of $\phi(w)$ means $\phi(w)_{0,0}\neq 0$. Hence, $$\frac{1}{\sum_{k=0}^{{\mathfrak{d}}_{p}(\operatorname{Re}w)-1}\sum_{l=0}^{{% \mathfrak{d}}_{q}(\operatorname{Im}w)-1}\phi(w)_{k,l}\operatorname{Re}(z-w)^{k% }\operatorname{Im}(z-w)^{l}}=O(1)$$ for $z\to w$. From $\phi\in{\mathcal{F}}_{N}$ we then conclude that (4.3) is a $O(\max(|\operatorname{Re}(z-w)|^{{\mathfrak{d}}_{p}(\operatorname{Re}w)},|% \operatorname{Im}(z-w)|^{{\mathfrak{d}}_{q}(\operatorname{Im}w)}))$ for $z\to w$. Because of $\phi(w)\cdot\phi^{-1}=e$ (see \threfz30f3), (4.4) can be rewritten as $$\displaystyle-\frac{1}{\sum_{k=0}^{{\mathfrak{d}}_{p}(\operatorname{Re}w)-1}% \sum_{l=0}^{{\mathfrak{d}}_{q}(\operatorname{Im}w)-1}\phi(w)_{k,l}% \operatorname{Re}(z-w)^{k}\operatorname{Im}(z-w)^{l}}\cdot\\ \displaystyle\Big{(}\sum_{k=0}^{{\mathfrak{d}}_{p}(\operatorname{Re}w)-1}\sum_% {l=0}^{{\mathfrak{d}}_{q}(\operatorname{Im}w)-1}\operatorname{Re}(z-w)^{k}% \operatorname{Im}(z-w)^{l}\cdot\sum_{c=0}^{k}\sum_{d=0}^{l}\phi(w)_{c,d}\cdot% \phi^{-1}(w)_{k-c,l-d}\\ \displaystyle+O(\max(|\operatorname{Re}(z-w)|^{{\mathfrak{d}}_{p}(% \operatorname{Re}w)},|\operatorname{Im}(z-w)|^{{\mathfrak{d}}_{q}(% \operatorname{Im}w)}))-1\Big{)}=$$ $$\displaystyle O(1)\cdot O(\max(|\operatorname{Re}(z-w)|^{{\mathfrak{d}}_{p}(% \operatorname{Re}w)},|\operatorname{Im}(z-w)|^{{\mathfrak{d}}_{q}(% \operatorname{Im}w)}))=\\ \displaystyle O(\max(|\operatorname{Re}(z-w)|^{{\mathfrak{d}}_{p}(% \operatorname{Re}w)},|\operatorname{Im}(z-w)|^{{\mathfrak{d}}_{q}(% \operatorname{Im}w)}))$$ for $z\to w$. Altogether (4.2) is a $O(\max(|\operatorname{Re}(z-w)|^{{\mathfrak{d}}_{p}(\operatorname{Re}w)},|% \operatorname{Im}(z-w)|^{{\mathfrak{d}}_{q}(\operatorname{Im}w)}))$. Therefore, $\phi^{-1}\in{\mathcal{F}}_{N}$. ∎ 5 Functional Calculus In this section we employ the same assumptions and notation as in the previous one. 5.1 Lemma. \thlabel aufspaltb For any $\phi\in{\mathcal{F}}_{N}$ there exists a polynomial $s\in\mathbb{C}[z,w]$ and a function $g$ on $\sigma(\Theta(N))$ with values in ${\mathbb{C}}$ on $\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$ and values in ${\mathbb{C}}^{2}$ on $\sigma(\Theta(N))\cap(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$ such that $\phi-s_{N}\in{\mathcal{R}}$, such that $g$ is bounded and measurable on $\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$, and such that $$\phi(z)=s_{N}(z)+(p_{N}+q_{N})(z)\cdot g(z),\ z\in\sigma(\Theta(N))\,,$$ (5.1) where the multiplication here has to be understood in the sense of \threfdurchdiv. Proof. According to \threfexistbe there exists an $s\in\mathbb{C}[z,w]$ such that $\phi-s_{N}\in{\mathcal{R}}$, and by \threfdurchdiv we then find a function $g$ such that (5.1) holds true. The measurability of $$g(z)=\frac{\phi(z)-s(\operatorname{Re}z,\operatorname{Im}z)}{p(\operatorname{% Re}z)+q(\operatorname{Im}z)}\ \text{ on }\ \sigma(\Theta(N))\setminus(Z^{{% \mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$$ follows from the assumption $\phi\in{\mathcal{F}}_{N}$; see \threfFdefklM. In order to show $g$’s boundedness, first recall from \threfspeknorm that $$\max(|p(\operatorname{Re}z)|,|q(\operatorname{Im}z)|)\leq\max(\|R_{1}R_{1}^{*}% \|,\|R_{2}R_{2}^{*}\|)\,|p(\operatorname{Re}z)+q(\operatorname{Im}z)|$$ for $z\in\sigma(\Theta(N))$. Hence, $$\displaystyle\frac{\max(|p(\operatorname{Re}z)|,|q(\operatorname{Im}z)|)}{|p(% \operatorname{Re}z)+q(\operatorname{Im}z)|}\leq\max(\|R_{1}R_{1}^{*}\|,\|R_{2}% R_{2}^{*}\|),\\ \displaystyle z\in\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb% {R}}}_{q})\,.$$ As $\phi\in{\mathcal{F}}_{N}$ we find for each $w\in\sigma(\Theta(N))\cap(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$ an open neighbourhood $U(w)$ of $w$ such that (4.1) is bounded for $z\in U(w)\setminus\{w\}$. Clearly, we can make the neighbourhoods $U(w)$ smaller so that they are pairwise disjoint. Since for $w\in\sigma(\Theta(N))\cap(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$ the real number $\operatorname{Re}w$ ($\operatorname{Im}w$) is a zero of $p(\operatorname{Re}z)$ ($q(\operatorname{Im}z)$) with multiplicity ${\mathfrak{d}}_{p}(\operatorname{Re}w)$ (${\mathfrak{d}}_{q}(\operatorname{Im}w)$), we have $$c|\operatorname{Re}(z-w)|^{{\mathfrak{d}}_{p}(\operatorname{Re}w)}\leq|p(% \operatorname{Re}z)|,\ \ d|\operatorname{Im}(z-w)|^{{\mathfrak{d}}_{q}(% \operatorname{Im}w)}\leq|q(\operatorname{Im}z)|$$ for $z\in U(w)$ with constants $c,d>0$. Hence, $$\frac{\max(|\operatorname{Re}(z-w)|^{{\mathfrak{d}}_{p}(\operatorname{Re}w)},|% \operatorname{Im}(z-w)|^{{\mathfrak{d}}_{q}(\operatorname{Im}w)})}{\max(|p(% \operatorname{Re}z)|,|q(\operatorname{Im}z)|)}\leq C_{w}$$ on $\sigma(\Theta(N))\cap U(w)\setminus\{w\}$ for some $C_{w}>0$. By what was said in \threftaylormehrdimrem and we also have $$\displaystyle s(\operatorname{Re}z,\operatorname{Im}z)=\sum_{k=0}^{{\mathfrak{% d}}_{p}(\operatorname{Re}w)-1}\sum_{l=0}^{{\mathfrak{d}}_{q}(\operatorname{Im}% w)-1}\phi(w)_{k,l}\operatorname{Re}(z-w)^{k}\operatorname{Im}(z-w)^{l}+\\ \displaystyle O\big{(}\max(|\operatorname{Re}(z-w)|^{{\mathfrak{d}}_{p}(% \operatorname{Re}w)},|\operatorname{Im}(z-w)|^{{\mathfrak{d}}_{q}(% \operatorname{Im}w)})\big{)}\,,$$ because $\phi-s_{N}\in{\mathcal{R}}$ implies $\phi(w)_{k,l}=\frac{1}{k!l!}\frac{\partial^{k+l}s}{\partial x^{k}\partial y^{l% }}(\operatorname{Re}w,\operatorname{Im}w)$. Using the boundedness of (4.1) we altogether obtain the boundedness of $$g(z)=\frac{\phi(z)-s(\operatorname{Re}z,\operatorname{Im}z)}{p(\operatorname{% Re}z)+q(\operatorname{Im}z)}=$$ (5.2) $$\displaystyle\frac{\max(|p(\operatorname{Re}z)|,|q(\operatorname{Im}z)|)}{p(% \operatorname{Re}z)+q(\operatorname{Im}z)}\cdot\\ \displaystyle\frac{\max(|\operatorname{Re}(z-w)|^{{\mathfrak{d}}_{p}(% \operatorname{Re}w)},|\operatorname{Im}(z-w)|^{{\mathfrak{d}}_{q}(% \operatorname{Im}w)})}{\max(|p(\operatorname{Re}z)|,|q(\operatorname{Im}z)|)}% \cdot\\ \displaystyle\frac{\phi(z)-s(\operatorname{Re}z,\operatorname{Im}z)}{\max(|% \operatorname{Re}(z-w)|^{{\mathfrak{d}}_{p}(\operatorname{Re}w)},|% \operatorname{Im}(z-w)|^{{\mathfrak{d}}_{q}(\operatorname{Im}w)})}$$ for $z\in\sigma(\Theta(N))\cap U(w)\setminus\{w\}$. Since by \threfspeknorm the function $\frac{1}{p(\operatorname{Re}z)+q(\operatorname{Im}z)}$ is continuous, and hence bounded on $\sigma(\Theta(N))\setminus\bigcup_{w\in\sigma(\Theta(N))\cap(Z^{{\mathbb{R}}}_% {p}+iZ^{{\mathbb{R}}}_{q})}U(w)$, we see that (5.2) is even bounded for $z\in\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$. ∎ 5.2 Definition. \thlabel funcaldef For any $\phi\in{\mathcal{F}}_{N}$ we define $$\phi(N):=s(A,B)+\Xi\left(\int^{R_{1},R_{2}}_{\sigma(\Theta(N))}g\,dE\right)\,,$$ where $s\in{\mathbb{C}}[z,w]$ and $g$ is a function on $\sigma(\Theta(N))$ with the properties mentioned in \threfaufspaltb, and where $$\displaystyle\int^{R_{1},R_{2}}_{\sigma(\Theta(N))}g\,dE:=\int_{\sigma(\Theta(% N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})}g\,dE\ +\\ \displaystyle\hskip 14.226378pt\sum_{w\in\sigma(\Theta(N))\cap(Z^{{\mathbb{R}}% }_{p}+iZ^{{\mathbb{R}}}_{q})}\big{(}g(w)_{1}R_{1}R_{1}^{*}E\{w\}+g(w)_{2}R_{2}% R_{2}^{*}E\{w\}\big{)}\,.$$ ∎ First we shall show that $\phi(N)$ is well defined. 5.3 Theorem. \thlabel welldef Let $\phi\in{\mathcal{F}}_{N}$, $s,\tilde{s}\in\mathbb{C}[z,w]$ and functions $g,\tilde{g}$ on $\sigma(\Theta(N))$ be given, such that the assertion of \threfaufspaltb holds true for $s,g$ as well as for $\tilde{s},\tilde{g}$. Then $$s(A,B)+\Xi\left(\int^{R_{1},R_{2}}_{\sigma(\Theta(N))}g\,dE\right)=\tilde{s}(A% ,B)+\Xi\left(\int^{R_{1},R_{2}}_{\sigma(\Theta(N))}\tilde{g}\,dE\right)\,.$$ Proof. By assumption we have $\phi-s_{N},\phi-\tilde{s}_{N}\in{\mathcal{R}}$. Subtracting these functions yields $\tilde{s}_{N}-s_{N}\in{\mathcal{R}}$. Using the notation of \threfeinbett2pre this gives $\varpi(\tilde{s}-s)_{\xi,\eta}=0$ for $(\xi,\eta)\in p^{-1}\{0\}\times q^{-1}\{0\}$. According to \threfeinbett2pre we then get $$\tilde{s}(z,w)-s(z,w)=p(z)u(z,w)+q(w)v(z,w)$$ (5.3) for some $u(z,w),v(z,w)\in{\mathbb{C}}[z,w]$. By \threfXidefeig in [KP] we have $$\Xi_{1}\big{(}u(\Theta_{1}(A),\Theta_{1}(B))\big{)}=\Xi_{1}\big{(}\Theta_{1}(u% (A,B))\big{)}=p(A)u(A,B)\,,$$ $$\Xi_{2}\big{(}v(\Theta_{2}(A),\Theta_{2}(B))\big{)}=\Xi_{2}\big{(}\Theta_{2}(v% (A,B))\big{)}=q(B)v(A,B)\,,$$ where $\Xi_{j},\ j=1,2$, are as defined in (2.5). Since $u(\Theta_{1}(A),\Theta_{1}(B))=\int u(\operatorname{Re}z,\operatorname{Im}z)\,% dE_{1}(z)$, we get from (2.6) $$\Xi_{1}\big{(}u(\Theta_{1}(A),\Theta_{1}(B))\big{)}=\Xi\big{(}R_{1}R_{1}^{*}% \int u(\operatorname{Re}z,\operatorname{Im}z)\,dE(z)\big{)}\,.$$ Similarly, $\Xi_{2}\big{(}v(\Theta_{2}(A),\Theta_{2}(B))\big{)}=\Xi\big{(}R_{2}R_{2}^{*}% \int v(\operatorname{Re}z,\operatorname{Im}z)\,dE(z)\big{)}$. Therefore, employing \threfkorvda we get $$\displaystyle\tilde{s}(A,B)-s(A,B)=p(A)u(A,B)+q(B)v(A,B)=\\ \displaystyle\Xi\big{(}R_{1}R_{1}^{*}\int u(\operatorname{Re}z,\operatorname{% Im}z)\,dE(z)+R_{2}R_{2}^{*}\int v(\operatorname{Re}z,\operatorname{Im}z)\,dE(z% )\big{)}=$$ (5.4) $$\displaystyle\Xi\left(\int_{\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ% ^{{\mathbb{R}}}_{q})}\hskip-14.226378pt\frac{p(\operatorname{Re}z)u(% \operatorname{Re}z,\operatorname{Im}z)+q(\operatorname{Im}z)v(\operatorname{Re% }z,\operatorname{Im}z)}{p(\operatorname{Re}z)+q(\operatorname{Im}z)}\,dE(z)+% \right.\\ \displaystyle\left.\sum_{w\in\sigma(\Theta(N))\cap(Z^{{\mathbb{R}}}_{p}+iZ^{{% \mathbb{R}}}_{q})}\hskip-14.226378pt\big{(}u(\operatorname{Re}w,\operatorname{% Im}w)R_{1}R_{1}^{*}E\{w\}+v(\operatorname{Re}w,\operatorname{Im}w)R_{2}R_{2}^{% *}E\{w\}\big{)}\right)\,.$$ On the other hand, since (5.1) holds true for $s,g$ and $\tilde{s},\tilde{g}$, we have $$(\tilde{s}_{N}-s_{N})(z)=(p_{N}+q_{N})(z)\cdot(g(z)-\tilde{g}(z)),\ z\in z\in% \sigma(\Theta(N))\,.$$ (5.5) For $z\in\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$ by (5.3) this means $$\displaystyle p(\operatorname{Re}z)u(\operatorname{Re}z,\operatorname{Im}z)+q(% \operatorname{Im}z)v(\operatorname{Re}z,\operatorname{Im}z)=\\ \displaystyle\tilde{s}(\operatorname{Re}z,\operatorname{Im}z)-s(\operatorname{% Re}z,\operatorname{Im}z)=(p(\operatorname{Re}z)+q(\operatorname{Im}z))\cdot(g(% z)-\tilde{g}(z))$$ and, in turn, $$g(z)-\tilde{g}(z)=\frac{p(\operatorname{Re}z)u(\operatorname{Re}z,% \operatorname{Im}z)+q(\operatorname{Im}z)v(\operatorname{Re}z,\operatorname{Im% }z)}{p(\operatorname{Re}z)+q(\operatorname{Im}z)}\,.$$ Considering for $z\in\sigma(\Theta(N))\cap(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$ the entries of (5.5) with indices $({\mathfrak{d}}_{p}(\operatorname{Re}z),0)$ and $(0,{\mathfrak{d}}_{q}(\operatorname{Im}z))$ together with (5.3) we get $$\displaystyle\frac{1}{{\mathfrak{d}}_{p}(\operatorname{Re}z)!}\,p^{({\mathfrak% {d}}_{p}(\operatorname{Re}z))}(\operatorname{Re}z)\,u(\operatorname{Re}z,% \operatorname{Im}z)=\\ \displaystyle\frac{1}{{\mathfrak{d}}_{p}(\operatorname{Re}z)!}\,\frac{\partial% ^{{\mathfrak{d}}_{p}(\operatorname{Re}z)}}{\partial x^{{\mathfrak{d}}_{p}(% \operatorname{Re}z)}}(\tilde{s}(\operatorname{Re}z,\operatorname{Im}z)-s(% \operatorname{Re}z,\operatorname{Im}z))=\\ \displaystyle\frac{1}{{\mathfrak{d}}_{p}(\operatorname{Re}z)!}\,p^{({\mathfrak% {d}}_{p}(\operatorname{Re}z))}(\operatorname{Re}z)\,(g(z)_{1}-\tilde{g}(z)_{1})$$ and $$\displaystyle\frac{1}{{\mathfrak{d}}_{q}(\operatorname{Im}z)!}\,q^{({\mathfrak% {d}}_{q}(\operatorname{Im}z))}(\operatorname{Im}z)\,v(\operatorname{Re}z,% \operatorname{Im}z)=\\ \displaystyle\frac{1}{{\mathfrak{d}}_{q}(\operatorname{Im}z)!}\,\frac{\partial% ^{{\mathfrak{d}}_{q}(\operatorname{Im}z)}}{\partial y^{{\mathfrak{d}}_{q}(% \operatorname{Im}z)}}(\tilde{s}(\operatorname{Re}z,\operatorname{Im}z)-s(% \operatorname{Re}z,\operatorname{Im}z))=\\ \displaystyle\frac{1}{{\mathfrak{d}}_{q}(\operatorname{Im}z)!}\,q^{({\mathfrak% {d}}_{q}(\operatorname{Im}z))}(\operatorname{Im}z)\,(g(z)_{2}-\tilde{g}(z)_{2})$$ where we employed the product rule and the fact that $p^{(k)}(\operatorname{Re}z)=0=q^{(l)}(\operatorname{Im}z)$ for $0\leq k<{\mathfrak{d}}_{p}(\operatorname{Re}z),\,0\leq l<{\mathfrak{d}}_{q}(% \operatorname{Im}z)$. Since $p^{({\mathfrak{d}}_{p}(\operatorname{Re}z))}(\operatorname{Re}z)$ and $q^{({\mathfrak{d}}_{q}(\operatorname{Im}z))}(\operatorname{Im}z)$ do not vanish for $z\in\sigma(\Theta(N))\cap(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$, we get $u(\operatorname{Re}z,\operatorname{Im}z)=g(z)_{1}-\tilde{g}(z)_{1}$ and $v(\operatorname{Re}z,\operatorname{Im}z)=g(z)_{2}-\tilde{g}(z)_{2}$. Therefore, we can write (5.4) as $$\tilde{s}(A,B)-s(A,B)=\Xi\left(\int^{R_{1},R_{2}}_{\sigma(\Theta(N))}\big{(}g-% \tilde{g}\big{)}\,dE\right)\,,$$ showing the asserted equality. ∎ 5.4 Theorem. \thlabel mimalvertr The mapping $\phi\mapsto\phi(N)$ constitutes a $*$-homomorphism from ${\mathcal{F}}_{N}$ into $\{N,N^{*}\}^{\prime\prime}\ (\subseteq B({\mathcal{K}}))$ with $s_{N}(N)=s(A,B)$ for all $s\in{\mathbb{C}}[z,w]$. Proof. $s_{N}(N)=s(A,B)$ for all $s\in{\mathbb{C}}[z,w]$ follows from \threfwelldef because we have $s_{N}=s_{N}+(p_{N}+q_{N})(z)\cdot 0,\,z\in\sigma(\Theta(N))$. Assume that for $\phi,\psi\in{\mathcal{F}}_{N}$ we have $s,r\in{\mathbb{C}}[z,w]$ and functions $g,h$ on $\sigma(\Theta(N))$ such that $\phi-s_{N},\psi-r_{N}\in{\mathcal{R}}$, such that $g$ and $h$ are bounded and measurable on $\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$, and such that (5.1) as well as $$\psi(z)=r_{N}(z)+(p_{N}+q_{N})(z)\cdot h(z),\ z\in\sigma(\Theta(N))\,,$$ hold true; see \threfaufspaltb. Then for $\lambda,\mu\in{\mathbb{C}}$ we get from \threfbweuh30 $$(\lambda\phi+\mu\psi)(z)=(\lambda s+\mu r)_{N}(z)+(p_{N}+q_{N})(z)\cdot(% \lambda g(z)+\mu h(z)),\ z\in\sigma(\Theta(N))\,,$$ where $\lambda\phi+\mu\psi-(\lambda s+\mu r)_{N}=\lambda(\phi-s_{N})+\mu(\psi-r_{N})% \in{\mathcal{R}}$, and where $\lambda g+\mu h$ is bounded and measurable on $\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$. Since the definition of $\phi(N)$ in \threffuncaldef depends linearly on $s$ and $g$, we conclude from \threfwelldef that $$(\lambda\phi+\mu\psi)(N)=\lambda\phi(N)+\mu\psi(N)\,.$$ Similarly, we get $\phi^{\#}(z)=(s^{\#})_{N}(z)+(p_{N}+q_{N})(z)\cdot\bar{g}(z),\ z\in\sigma(% \Theta(N))$; see \threfbweuh30. Thereby $\phi^{\#}-(s^{\#})_{N}=(\phi-s_{N})^{\#}\in{\mathcal{R}}$ holds true due to the fact that ${\mathfrak{d}}_{p}(\xi)={\mathfrak{d}}_{p}(\bar{\xi})$ and ${\mathfrak{d}}_{q}(\eta)={\mathfrak{d}}_{q}(\bar{\eta})$ for all $(\xi,\eta)\in Z^{i}$. Since $\bar{g}$ is bounded and measurable on $\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$, and since $$\phi(N)^{*}=s^{\#}(A,B)+\Xi\left(\int^{R_{1},R_{2}}_{\sigma(\Theta(N))}\bar{g}% \,dE\right)\,,$$ we again obtain from \threfwelldef that $\phi^{\#}(N)=\phi(N)^{*}$. Concerning the compatibility with $\cdot$, first note that by \threfbweuh30 $$\phi(z)\cdot\psi(z)=(s\cdot r)_{N}(z)+(p_{N}+q_{N})(z)\cdot\omega(z),\ z\in% \sigma(\Theta(N))\,.$$ Here we have $\omega(z)=s(z)h(z)+r(z)g(z)+g(z)h(z)(p(\operatorname{Re}z)+q(\operatorname{Im}% z))$ for $z\in\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$ and $\omega(z)_{j}=s(z)g(z)_{j}+r(z)h(z)_{j},\ j=1,2$ for $z\in\sigma(\Theta(N))\cap(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$ because $a,b\in\ker\pi$ implies $a\cdot b=0$ and, in turn, $(p_{N}+q_{N})(z)\cdot(p_{N}+q_{N})(z)=0$ for $z\in\sigma(\Theta(N))\cap(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$. On the other hand, by \threfXidefeig in [KP] we have $\Xi(D)C=\Xi(D\Theta(C))$, $C\Xi(D)=\Xi(\Theta(C)D)$, and $\Xi(D_{1})\Xi(D_{2})=\Xi(D_{1}D_{2}T^{*}T)$, where $T^{*}T=p(A)+q(B)$. Hence, $$\phi(N)\ \psi(N)=$$ $$s(A,B)\,r(A,B)+\Xi\left(\int^{R_{1},R_{2}}_{\sigma(\Theta(N))}g\,dE\right)r(A,% B)+s(A,B)\,\Xi\left(\int^{R_{1},R_{2}}_{\sigma(\Theta(N))}h\,dE\right)+$$ $$\Xi\left(\int^{R_{1},R_{2}}_{\sigma(\Theta(N))}g\,dE\right)\,\Xi\left(\int^{R_% {1},R_{2}}_{\sigma(\Theta(N))}h\,dE\right)=$$ $$\displaystyle(s\cdot r)(A,B)+\Xi\left(\int^{R_{1},R_{2}}_{\sigma(\Theta(N))}(g% \cdot r+h\cdot s)\,dE+\right.\\ \displaystyle\left.\int_{\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{% \mathbb{R}}}_{q})}\big{(}p(\operatorname{Re}(.))+q(\operatorname{Im}(.))\big{)% }\cdot h\cdot g\,dE\right)=$$ $$(s\cdot r)(A,B)+\Xi\left(\int^{R_{1},R_{2}}_{\sigma(\Theta(N))}\omega\,dE% \right)\,.$$ Here $\omega$ is bounded and measurable on $\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$ and, using the fact that ${\mathcal{R}}$ is an ideal, $$\phi\cdot\psi-(s\cdot r)_{N}=(\phi-s_{N})\cdot\psi+(\psi-r_{N})\cdot s_{N}\in{% \mathcal{R}}\,.$$ Hence, we again obtain from \threfwelldef that $\phi(N)\cdot\psi(N)=\big{(}\phi\cdot\psi\big{)}(N)$. Finally, we shall show that $\phi(N)\in\{N,N^{*}\}^{\prime\prime}$. Clearly, $s(A,B)\in\{A,B\}^{\prime\prime}=\{N,N^{*}\}^{\prime\prime}$. If $C\in\{A,B\}^{\prime}\subseteq\big{(}p(A)+q(B)\big{)}^{\prime}=(TT^{*})^{\prime}$, then $\Theta(C)\in\{\Theta(A),\Theta(B)\}^{\prime}$ because $\Theta$ is a homomorphism. By the spectral theorem for normal operators $\Theta(C)$ commutes with $$D:=\int^{R_{1},R_{2}}_{\sigma(\Theta(N))}g\,dE\,.$$ According to \threfXidefeig in [KP] we then get $$\Xi(D)C=\Xi(D\Theta(C))=\Xi(\Theta(C)D)=C\Xi(D)\,.$$ Hence, $\Xi(D)\in\{A,B\}^{\prime\prime}=\{N,N^{*}\}^{\prime\prime}$, and altogether $\phi(N)\in\{A,B\}^{\prime\prime}=\{N,N^{*}\}^{\prime\prime}$. ∎ 5.5 Remark. \thlabel rieszproj For $\zeta\in Z^{i}$ or for an isolated $\zeta\in\sigma(\Theta(N))\cup(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$ we saw in \threffedela that $a\delta_{\zeta}\in{\mathcal{F}}_{N}$. If $a$ is the unite $e\in{\mathfrak{C}}(\zeta)$ (see \threfz30f3), then $(e\delta_{\zeta})\cdot(e\delta_{\zeta})=(e\delta_{\zeta})$ together with \threfmimalvertr shows that $(e\delta_{\zeta})(N)$ is a projection. It is a kind of Riesz projection corresponding to $\zeta$. We set $\xi:=\operatorname{Re}\zeta,\ \eta:=\operatorname{Im}\zeta$ if $\zeta\in\sigma(\Theta(N))\cup(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$ and $(\xi,\eta):=\zeta$ if $\zeta\in Z^{i}$. For $\lambda\in{\mathbb{C}}\setminus\{\xi+i\eta\}$ and for $s(z,w):=z+iw-\lambda$ we then have $s_{N}\cdot(e\delta_{\zeta})=\big{(}s_{N}(\zeta)\big{)}\delta_{\zeta}$, where the entry $s(\xi,\eta)$ of $s_{N}(\zeta)$ with index $(0,0)$ does not vanish. By \threfz30f3 it therefore has a multiplicative inverse $b\in{\mathfrak{C}}(\zeta)$. We then obtain $$s_{N}\cdot(e\delta_{\zeta})\cdot(b\delta_{\zeta})=e\delta_{\zeta}\,.$$ From $s_{N}(N)=N-\lambda$ we then get that $N|_{\operatorname{ran}(e\delta_{\zeta})(N)}-\lambda$ has $(b\delta_{\zeta})(N)|_{\operatorname{ran}(e\delta_{\zeta})(N)}$ as its inverse operator. Thus, $\sigma(N|_{\operatorname{ran}(e\delta_{\zeta})(N)})\subseteq\{\xi+i\eta\}$. ∎ 5.6 Lemma. \thlabel deth56 If for $\phi\in{\mathcal{F}}_{N}$ we have $\phi(z)=0$ for all $$z\in\big{(}\sigma(\Theta(N))\cup((Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})% \cap\sigma(N))\big{)}\dot{\cup}\{(\alpha,\beta)\in Z^{i}:\alpha+i\beta,\bar{% \alpha}+i\bar{\beta}\in\sigma(N)\}\,,$$ then $\phi(N)=0$. Proof. Since any $\zeta\in(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})\setminus\sigma(N)$ is isolated in $\sigma(\Theta(N))\cup(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$, we saw in \threfrieszproj that for $$\zeta\in\underbrace{\big{(}(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})% \setminus\sigma(N)\big{)}}_{=:Z_{1}}\dot{\cup}\underbrace{\{(\alpha,\beta)\in Z% ^{i}:\alpha+i\beta\in\rho(N)\}}_{=:Z_{2}}$$ the expression $(e\delta_{\zeta})(N)$ is a bounded projection commuting with $N$. Hence, $(e\delta_{\zeta})(N)$ also commutes $(N-(\xi+i\eta))^{-1}$, where $\xi:=\operatorname{Re}\zeta,\ \eta:=\operatorname{Im}\zeta$ if $\zeta\in Z_{1}$ and $(\xi,\eta):=\zeta$ if $\zeta\in Z_{2}$. Consequently, $N|_{\operatorname{ran}(e\delta_{\zeta})(N)}-(\xi+i\eta)$ is invertible on $\operatorname{ran}(e\delta_{\zeta})(N)$, i.e. $\xi+i\eta\not\in\sigma(N|_{\operatorname{ran}(e\delta_{\zeta})(N)})$. By \threfrieszproj we have $\sigma(N|_{\operatorname{ran}(e\delta_{\zeta})(N)})\subseteq\{\xi+i\eta\}$. Hence, $\sigma(N|_{\operatorname{ran}(e\delta_{\zeta})(N)})=\emptyset$, which is impossible for $\operatorname{ran}(e\delta_{\zeta})(N)\neq\{0\}$. Thus, $(e\delta_{\zeta})(N)=0$. For $(\xi,\eta)\in Z_{3}:=\{(\alpha,\beta)\in Z^{i}:\bar{\alpha}+i\bar{\beta}\in% \rho(N)\}$ we get $(\bar{\xi},\bar{\eta})\in Z_{2}$. Hence, $$0=(e\delta_{(\bar{\xi},\bar{\eta})})(N)^{*}=(e\delta_{(\xi,\eta)})(N)\,.$$ By our assumption $\phi$ is supported on $Z_{1}\cup Z_{2}\cup Z_{3}$. Hence, $$\phi(N)=(\hskip-5.690551pt\sum_{\zeta\in Z_{1}\cup Z_{2}\cup Z_{3}}\hskip-5.69% 0551pt\phi(\zeta)\delta_{\zeta}\hskip 5.690551pt)(N)=\sum_{\zeta\in Z_{1}\cup Z% _{2}\cup Z_{3}}\phi(\zeta)(e\delta_{\zeta})(N)=0\,.$$ ∎ 5.7 Remark. \thlabel fhwr34 As a consequence of \threfdeth56 for $\phi\in{\mathcal{F}}_{N}$ the operator $\phi(N)$ only depends on $\phi$’s values on $$\displaystyle\sigma_{N}:=\big{(}\sigma(\Theta(N))\cup((Z^{{\mathbb{R}}}_{p}+iZ% ^{{\mathbb{R}}}_{q})\cap\sigma(N))\big{)}\dot{\cup}\\ \displaystyle\{(\alpha,\beta)\in Z^{i}:\alpha+i\beta,\bar{\alpha}+i\bar{\beta}% \in\sigma(N)\}$$ Thus, we can re-define the function class ${\mathcal{F}}_{N}$ for our functional calculus so that the elements $\phi$ of ${\mathcal{F}}_{N}$ are functions on this set with $\phi(z)\in{\mathfrak{C}}(z)$ such that $z\mapsto\phi(z)$ is measurable and bounded on $\sigma(\Theta(N)\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$ and such that (4.1) is bounded locally at $w$ for all $w\in\sigma(\Theta(N)\cap(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})$. ∎ 5.8 Lemma. \thlabel wannboundinv If $\phi\in{\mathcal{F}}_{N}$ is such that $\phi(z)$ is invertible in ${\mathfrak{C}}(z)$ (see \threfz30f3) for all $z\in\sigma_{N}$ and such that $0$ does not belong to the closure of $\phi\big{(}\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{% q})\big{)}$, then $\phi(N)$ is a boundedly invertible operator on ${\mathcal{K}}$. Proof. We think of $\phi$ as a function on $\big{(}\sigma(\Theta(N))\cup(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})\big{)% }\dot{\cup}Z^{i}$ by setting $\phi(z)=e$ (see \threfz30f3) for all $z$ not belonging to $\sigma_{N}$. Then all assumptions of \threfeinduF are satisfied. Hence $\phi^{-1}\in{\mathcal{F}}_{N}$, and we conclude from \threfmimalvertr and \threffuinm0pre that $$\phi^{-1}(N)\phi(N)=\phi(N)\phi^{-1}(N)=(\phi\cdot\phi^{-1})(N)=\mathds{1}_{N}% (N)=I\,.$$ ∎ 5.9 Corollary. \thlabel sigmaN If $N$ is a definitizable normal operator on the Krein space ${\mathcal{K}}$, then $\sigma(N)$ equals to $$\displaystyle\sigma(\Theta(N))\cup((Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q}% )\cap\sigma(N))\cup\\ \displaystyle\{\alpha+i\beta:(\alpha,\beta)\in Z^{i},\alpha+i\beta,\bar{\alpha% }+i\bar{\beta}\in\sigma(N)\}$$ (5.6) Proof. Since $\Theta$ is a homomorphism, we have $\sigma(\Theta(N))\subseteq\sigma(N)$. Hence, (5.6) is contained in $\sigma(N)$. For the converse, consider the polynomial $s(z,w)=z+iw-\lambda$ for a $\lambda$ not belonging to (5.6). We conclude that for any $z\in\sigma_{N}$ the first entry $(s_{N}(z))_{0,0}$ of $s_{N}(z)\in{\mathfrak{C}}(z)$ does not vanish, i.e. is invertible in ${\mathfrak{C}}(z)$. $(s_{N}(\sigma(\Theta(N))))_{0,0}=\sigma(\Theta(N))-\lambda$ being compact, $0$ does not belong to the closure of $s_{N}\big{(}\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_% {q})\big{)}$. Applying \threfwannboundinv we see that $s_{N}(N)=(N-\lambda)$ is invertible. ∎ 5.10 Corollary. For $\phi\in{\mathcal{F}}_{N}$ we have $$\sigma(\phi(N))\subseteq\overline{\phi(\sigma_{N})_{0,0}}\,.$$ Proof. For $\lambda\notin\overline{\phi(\sigma_{N})_{0,0}}$ and any $z\in\sigma_{N}$ we have $(\phi(z)-\lambda\mathds{1}_{N}(z))_{0,0}=\phi(z)_{0,0}-\lambda\neq 0$. Hence $\phi(z)-\lambda\mathds{1}_{N}(z)$ is invertible in ${\mathfrak{C}}(z)$. Moreover, $0$ does not belong to the closure of $\phi\big{(}\sigma(\Theta(N))\setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{% q})\big{)}-\lambda=(\phi-\lambda\mathds{1}_{N})\big{(}\sigma(\Theta(N))% \setminus(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})\big{)}_{0,0}$. Therefore, we can apply \threfwannboundinv to $\phi-\lambda\mathds{1}_{N}$, and get $\lambda\in\rho(\phi(N))$. ∎ 5.11 Remark. \thlabel prf55o1 For any characteristic function $\mathds{1}_{\Delta}$ of a Borel subset $\Delta\subseteq{\mathbb{C}}$ such that $(Z^{{\mathbb{R}}}_{p}+iZ^{{\mathbb{R}}}_{q})\cap\sigma(N)\cap\partial_{{% \mathbb{C}}}\Delta=\emptyset$ the function $(\mathds{1}_{\tau(\Delta)})_{N}$ belongs to ${\mathcal{F}}_{N}$; see \threffeinbetef and \threfgehzuF. Since this function is idempotent and satisfies $(\mathds{1}_{\tau(\Delta)})_{N}^{\#}=(\mathds{1}_{\tau(\Delta)})_{N}$, $(\mathds{1}_{\tau(\Delta)})_{N}(N)$ is a bounded and self-adjoint projection on the Krein space ${\mathcal{K}}$. These projections constitute the family of spectral projections for $N$. ∎ References [1] [AS] T.Ya.Azizov, V.A. Strauss: Spectral decompositions for special classes of self-adjoint and normal operators on Krein spaces, Spectral analysis and its applications, 45–67, Theta Ser. Adv. Math., 2, Theta, Bucharest, 2003. [KP] M. Kaltenbäck, R. Pruckner: Functional Calculus for definitizable selfadjoint linear relations on Krein spaces, Preprint. [L] H.Langer: Spectral functions of definitizable operators in Krein spaces, Lecture Notes in Mathematics Volume 948, 1982, 1-46. [LS] H. Langer, F.H. Szafraniec: Bounded normal operators in Pontryagin spaces, Operator theory in Krein spaces and nonlinear eigenvalue problems, 231–251, Oper. Theory Adv. Appl., 162, Birkhäuser, Basel, 2006. [PST] F. Philipp, V.A. Strauss, C. Trunk: Local spectral theory for normal operators in Krein spaces, Math. Nachr. 286 (2013), no. 1, 42–58. [XiCh] C. Xiao Man, H. Chao Cheng: Normal operators on $\Pi_{\kappa}$ space,  Northeast. Math. J. 1 (1985), no. 2, 247–252.

August, 2023 New duality-invariant models for nonlinear supersymmetric electrodynamics Sergei M. Kuzenko and Jake C. Stirling Department of Physics M013, The University of Western Australia 35 Stirling Highway, Perth W.A. 6009, Australia Email: sergei.kuzenko@uwa.edu.au, jake.stirling@research.uwa.edu.au We propose a new family of $\mathsf{U}(1)$ duality-invariant models for nonlinear ${\cal N}=1$ supersymmetric electrodynamics coupled to supergravity. This family includes the Cribiori-Farakos-Tournoy-van Proeyen supergravity-matter theory for spontaneously broken local supersymmetry with a novel Fayet-Iliopoulos term without gauged $R$-symmetry. We present superconformal duality-invariant models. 1 Introduction The general theory of duality invariance for nonlinear models with ${\cal N}=1$ and ${\cal N}=2$ vector supermultiplets was developed in 2000 [1, 2] and soon after extended to the locally supersymmetric case [3, 4, 5]. This formalism is a generalisation of the classic results on the structure of self-dual models for nonlinear electrodynamics in four dimensions [6, 7, 8, 9, 10] (see [2, 11] for a review) in conjunction with the self-duality properties [12, 13] of the ${\cal N}=1$ supersymmetric Born-Infeld action [14, 15, 16] and its generalisations. Since this paper is devoted to constructing new duality-invariant ${\cal N}=1$ supersymmetric models for a vector multiplet, it is useful to give a summary of the corresponding formalism introduced in [1]. Let $S[W,\bar{W}]$ be the action describing the dynamics of a single vector supermultiplet in Minkowski superspace. It is assumed that the action is a functional of the gauge-invariant chiral spinor field strength $W_{\alpha}=-\frac{1}{4}\bar{D}^{2}D_{\alpha}V$ and its conjugate $\bar{W}_{\dot{\alpha}}=-\frac{1}{4}D^{2}\bar{D}_{\dot{\alpha}}V$, where $V=\bar{V}$ is a gauge prepotential [17]. In order for this theory to possess $\mathsf{U}(1)$ duality invariance, the action must be a solution of the so-called self-duality equation [1] $$\displaystyle{\rm Im}\int{\rm d}^{4}x{\rm d}^{2}\theta\,\Big{\{}W^{\alpha}W_{\alpha}+M^{\alpha}M_{\alpha}\Big{\}}=0~{},\qquad M_{\alpha}:=-2{\rm i}\,\frac{\delta}{\delta W^{\alpha}}\,S[W,{\bar{W}}]~{}.$$ (1.1) Since the equation (1.1) is nonlinear, its solutions are difficult to generate. Inspired by the bosonic approach due to Ivanov and Zupnik [18, 19, 20, 21], new formulations were developed for ${\cal N}=1$ and ${\cal N}=2$ supersymmetric duality-invariant theories coupled to supergravity ten years ago [22]. The method makes use of an auxiliary unconstrained chiral superfield (a spinor in the ${\cal N}=1$ case and a scalar for ${\cal N}=2$) and is characterised by the fundamental property that $\mathsf{U}(1)$ duality invariance is equivalent to the manifest $\mathsf{U}(1)$ invariance of the self-interaction. In the ${\cal N}=1$ rigid supersymmetric case, analogous results were independently obtained in [23]. It is assumed in (1.1) that $W_{\alpha}$ is an unrestricted chiral spinor superfield, and the action $S[W,\bar{W}]$ is “analytically” continued from the original functional, which depends on $W_{\alpha}$ satisfying the Bianchi identity $D^{\alpha}W_{\alpha}=\bar{D}_{\dot{\alpha}}\bar{W}^{\dot{\alpha}}$, to a functional of the unrestricted chiral spinor $W_{\alpha}$. Such a continuation is obviously not unique, and additional conditions are required to fix it. For instance, consider a supersymmetric theory of the form $$\displaystyle S[W,{\bar{W}}]$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta\,W^{2}+{\rm c.c.}+\frac{1}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta{\rm d}^{2}\bar{\theta}\,W^{2}\,\bar{W}^{2}\Omega\left(u,\bar{u},D^{\alpha}W_{\alpha}\right)~{},$$ (1.2) where $W^{2}=W^{\alpha}W_{\alpha}$ and $u=\frac{1}{8}D^{2}W^{2}$. A possible way to extend this functional to the case when the Bianchi identity $D^{\alpha}W_{\alpha}=\bar{D}_{\dot{\alpha}}\bar{W}^{\dot{\alpha}}$ is no longer required, consists in replacing $\Omega\left(u,\bar{u},D^{\alpha}W_{\alpha}\right)\to\Omega\left(u,\bar{u},\gamma D^{\alpha}W_{\alpha}+\bar{\gamma}\bar{D}_{\dot{\alpha}}\bar{W}^{\dot{\alpha}}\right)$, for a complex parameter $\gamma$ such that $\gamma+\bar{\gamma}=1$. The ambiguity with analytic continuation is naturally resolved for the family of nonlinear models studied in [1]: $$\displaystyle S[W,{\bar{W}}]$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta\,W^{2}+{\rm c.c.}+\frac{1}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta{\rm d}^{2}\bar{\theta}\,W^{2}\,\bar{W}^{2}\Lambda\left(u,\bar{u}\right)~{}.$$ (1.3) Here the self-duality equation (1.1) implies that $$\displaystyle{\rm Im}\Big{\{}\Gamma-\bar{u}\Gamma^{2}\Big{\}}=0~{},\qquad\Gamma:=\partial_{u}(u\Lambda)~{}.$$ (1.4) This equation coincides in form with that arising in $\mathsf{U}(1)$ duality-invariant nonlinear electrodynamics $L(F_{ab})$ (see [2] for the technical details), $$\displaystyle{\rm Im}\left\{\frac{\partial(\omega\,\Lambda)}{\partial\omega}-\bar{\omega}\,\left(\frac{\partial(\omega\,\Lambda)}{\partial\omega}\right)^{2}\right\}=0~{},$$ (1.5) provided the Lagrangian is expressed in terms of invariants of the electromagnetic field $$\displaystyle L(F_{ab})=-\frac{1}{2}\,\Big{(}\omega+\bar{\omega}\Big{)}+\omega\,\bar{\omega}\;\Lambda(\omega,\bar{\omega})~{},$$ (1.6a) where we have introduced $$\displaystyle\omega=\alpha+{\rm i}\,\beta~{},\qquad\alpha=\frac{1}{4}\,F^{ab}F_{ab}~{},\qquad\quad\beta=\frac{1}{4}\,F^{ab}\tilde{F}_{ab}~{}.$$ (1.6b) Therefore, every $\mathsf{U}(1)$ duality-invariant model for nonlinear electrodynamics (1.6) possesses the ${\cal N}=1$ supersymmetric extension (1.3) which is also $\mathsf{U}(1)$ duality invariant. Instead of worrying about a procedure for analytic continuation to start with, one can follow a different path proposed by Ivanov, Lechtenfeld and Zupnik [23]. Their starting point is the assumption that some procedure of analytic continuation has been chosen, and for the unconstrained chiral spinor $W_{\alpha}$ the action reads $$\displaystyle S[W,{\bar{W}}]$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta\,W^{2}+{\rm c.c.}+\frac{1}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta{\rm d}^{2}\bar{\theta}\,W^{2}\,\bar{W}^{2}\Lambda\left(u,\bar{u},p,\bar{p}\right)~{},$$ (1.7) where $p:=D^{\alpha}W_{\alpha}$. For such a model to be $\mathsf{U}(1)$ duality-invariant, the action must satisfy the equation (1.1), and the latter implies that $$\displaystyle{\rm Im}\Big{\{}\Gamma-\bar{u}\Gamma^{2}+2u\bar{u}(\partial_{p}\Lambda)^{2}\Big{\}}=0~{}.$$ (1.8) To the best of our knowledge, no solution of (1.8) has so far been found with $\partial_{p}\Lambda\neq 0$. In the present paper, we propose a family of solutions to the self-duality equation (1.8). Before turning to the technical part of this paper, it is necessary to point out several recent developments. Although the idea to combine $\mathsf{U}(1)$ duality invariance with ${\cal N}=2$ superconformal symmetry was put forward in 2000 [1], the first duality-invariant and (super)conformal theories have only recently been derived in closed form for ${\cal N}<2$. Bandos, Lechner, Sorokin and Townsend [25] constructed the so-called ModMax theory, which is a unique nonlinear duality-invariant conformal extension of Maxwell’s equations (see [26] for a related analysis). Its ${\cal N}=1$ supersymmetric extension was proposed in [27, 28]. Ref. [28] also derived the ${\cal N}=2$ superconformal $\mathsf{U}(1)$ duality-invariant model proposed to describe the low-energy effective action for ${\cal N}=4$ super-Yang-Mills theory, thus completing the program initiated in [1]. Duality-invariant (super)conformal higher-spin models were constructed for ${\cal N}\leq 2$ in [29] on arbitrary conformally flat backgrounds. A supersymmetric nonlinear $\sigma$-model analogue of the ModMax theory was constructed in [30] building on the concept of self-dual supersymmetric nonlinear $\sigma$-models [31]. Supersymmetric duality-invariant theories have found numerous applications in the framework of $T\bar{T}$ deformations, see [32, 33, 34] and references therein. A remarkable relation has been established between helicity conservation for the tree-level scattering amplitudes and the electric-magnetic duality[35]. This paper is organised as follows. In section 2 we briefly review the ${\cal N}=1$ results of [1, 3, 23] and then present our new family of $\mathsf{U}(1)$ duality-invariant models for nonlinear ${\cal N}=1$ supersymmetric electrodynamics coupled to supergravity. In section 3 we provide a brief review of the ${\cal N}=1$ auxiliary superfield formulation of [22] and then recast the novel features of [23] (as compared with [22]) in a locally supersymmetric framework. Section 4 is devoted to superconformal $\mathsf{U}(1)$ duality-invariant models. Our two-component spinor notation and conventions follow [36], and are similar to those adopted in [37]. We make use of the Grimm-Wess-Zumino superspace geometry [38] as described in [36, 39]. 2 Duality-invariant supersymmetric models We consider a dynamical system describing an Abelian ${\cal N}=1$ vector multiplet in curved superspace and denote by $S[W,{\bar{W}}]$ the corresponding action functional. The action is assumed to depend on the chiral spinor field strength $W_{\alpha}$ and its conjugate ${\bar{W}}_{\dot{\alpha}}$ which are constructed in terms of a real unconstrained gauge prepotential $V$ [24] as $$\displaystyle W_{\alpha}=-\frac{1}{4}\,(\bar{\cal D}^{2}-4R){\cal D}_{\alpha}V~{},\qquad\bar{\cal D}_{\dot{\beta}}W_{\alpha}=0~{}.$$ (2.1) Here ${\cal D}_{\alpha}$ and $\bar{\cal D}_{\dot{\alpha}}$ are the spinor covariant derivatives in curved superspace, and $R$ is the chiral scalar torsion tensor.111Our normalisation of the torsion tensors of the Grimm-Wess-Zumino geometry follows [36, 39]. The prepotential is defined modulo gauge transformations $$\displaystyle\delta_{\xi}V=\xi+\bar{\xi}~{},\qquad\bar{\cal D}_{\dot{\alpha}}\xi=0~{},$$ (2.2) such that $\delta_{\xi}W_{\alpha}=0$. The gauge-invariant field strengths $W_{\alpha}$ and ${\bar{W}}_{\dot{\alpha}}$ obey the Bianchi identity $$\displaystyle{\cal D}^{\alpha}W_{\alpha}=\bar{\cal D}_{\dot{\alpha}}{\bar{W}}^{\dot{\alpha}}~{},$$ (2.3) and thus $W_{\alpha}$ is a reduced chiral superfield. We assume that $S[W,{\bar{W}}]$ does not involve the combination ${\cal D}^{\alpha}W_{\alpha}$ as an independent variable, and therefore it can unambiguously be defined as a functional of a general chiral superfield $W_{\alpha}$ and its conjugate ${\bar{W}}_{\dot{\alpha}}$. Then, introducing a covariantly chiral spinor superfield $M_{\alpha}$, $$\displaystyle{\rm i}\,M_{\alpha}:=2\,\frac{\delta}{\delta W^{\alpha}}\,S[W,{\bar{W}}]~{},\qquad\bar{\cal D}_{\dot{\beta}}M_{\alpha}=0~{},$$ (2.4) the equation of motion for $V$ is $$\displaystyle{\cal D}^{\alpha}M_{\alpha}={\bar{\cal D}}_{\dot{\alpha}}{\bar{M}}^{\dot{\alpha}}~{}.$$ (2.5) The variational derivative $\delta S/\delta W^{\alpha}$ in (2.4) is defined by $$\displaystyle\delta S=\int{\rm d}^{4}x\,{\rm d}^{2}\theta\,{\cal E}\,\delta W^{\alpha}\frac{\delta S}{\delta W^{\alpha}}~{}+~{}{\rm c.c.}~{},$$ (2.6) where ${\cal E}$ denotes the chiral integration measure, and $W^{\alpha}$ is assumed to be an unrestricted covariantly chiral spinor. Since the Bianchi identity (2.3) and the equation of motion (2.5) have the same functional form, one may consider $\mathsf{U}(1)$ duality rotations $$\displaystyle\delta W_{\alpha}=\lambda M_{\alpha}~{},\qquad\delta M_{\alpha}=-\lambda W_{\alpha}~{},$$ (2.7) with $\lambda\in{\mathbb{R}}$ a constant parameter. The condition for duality invariance is the so-called self-duality equation $$\displaystyle{\rm Im}\int{\rm d}^{4}x{\rm d}^{2}\theta\,{\cal E}\Big{\{}W^{\alpha}W_{\alpha}+M^{\alpha}M_{\alpha}\Big{\}}=0~{},$$ (2.8) in which $W_{\alpha}$ is chosen to be a general chiral spinor. In what follows, we shall treat $W_{\alpha}$ and $\bar{W}_{\dot{\alpha}}$ as unconstrained chiral superfields which are not subjected to the Bianchi identity (2.3). Now let us introduce the following general model for nonlinear ${\cal N}=1$ supersymmetric electrodynamics $$\displaystyle S[W,{\bar{W}}]$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta\,{\cal E}\,W^{2}+{\rm c.c.}+\frac{1}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta{\rm d}^{2}\bar{\theta}\,E\,W^{2}\,\bar{W}^{2}\,\Lambda\left(u,\bar{u},p,\bar{p}\right)~{},$$ (2.9) where the complex variables $u$ and $p$ are defined by $$\displaystyle u:=\frac{1}{8}({\cal D}^{2}-4\bar{R})W^{2}~{},\qquad p:={\cal D}^{\alpha}W_{\alpha}~{}.$$ (2.10) For this model the self-duality equation (2.8) amounts to $$\displaystyle{\rm Im}\int{\rm d}^{4}x{\rm d}^{2}\theta{\rm d}^{2}\bar{\theta}\,E\,W^{2}\,\bar{W}^{2}\Big{\{}\Gamma-\bar{u}\Gamma^{2}+2u\bar{u}(\partial_{p}\Lambda)^{2}\Big{\}}=0~{},\qquad\Gamma:=\partial_{u}(u\Lambda)~{}.$$ (2.11) In this equation the covariantly chiral spinor $W_{\alpha}$ has to be completely arbitrary, and therefore we conclude that the equation (1.8) holds. A super-Weyl invariant equivalent of the model (2.9) is given by $$\displaystyle S[W,{\bar{W}};\Upsilon]$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta\,{\cal E}\,W^{2}+{\rm c.c.}$$ (2.12) $$\displaystyle+\frac{1}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta{\rm d}^{2}\bar{\theta}\,E\,\frac{W^{2}\,\bar{W}^{2}}{\Upsilon^{2}}\Lambda\left(\frac{u}{\Upsilon^{2}},\frac{\bar{u}}{\Upsilon^{2}},\frac{p}{\Upsilon},\frac{\bar{p}}{\Upsilon}\right)~{},$$ where $\Upsilon$ is a nowhere vanishing real scalar with the super-Weyl transformation $$\displaystyle\delta_{\sigma}\Upsilon=(\sigma+\bar{\sigma})\Upsilon~{},\qquad\bar{\cal D}_{\dot{\beta}}\sigma=0~{},$$ (2.13) with $\sigma$ being the super-Weyl parameter. The transformation law (2.13) implies that (2.12) is super-Weyl invariant. One may readily check that if $\Lambda\left(u,\bar{u},p,\bar{p}\right)$ is a solution of the equation (1.8), then $$\displaystyle\widetilde{\Lambda}\left(u,\bar{u},p,\bar{p};\Upsilon\right):=\frac{1}{\Upsilon^{2}}\Lambda\left(\frac{u}{\Upsilon^{2}},\frac{\bar{u}}{\Upsilon^{2}},\frac{p}{\Upsilon},\frac{\bar{p}}{\Upsilon}\right)$$ (2.14) is also a solution of the same equation, and thus the model (2.12) is $\mathsf{U}(1)$ duality-invariant. The family of $\mathsf{U}(1)$ duality-invariant theories analysed in [22] is given by $$\displaystyle S[W,{\bar{W}};\Upsilon]$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta\,{\cal E}\,W^{2}+{\rm c.c.}$$ (2.15) $$\displaystyle+\frac{1}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta{\rm d}^{2}\bar{\theta}\,E\,\frac{W^{2}\,\bar{W}^{2}}{\Upsilon^{2}}\Lambda\left(\frac{u}{\Upsilon^{2}},\frac{\bar{u}}{\Upsilon^{2}}\right)~{}.$$ In this case the self-duality equation (1.8) turns into (1.4). We now present several simple solutions to the self-duality equation (1.8). Consider a polynomial interaction homogeneous in $p$ and $\bar{p}$, $$\displaystyle\Lambda^{(n)}(u,\bar{u},p,\bar{p})=\frac{1}{u\bar{u}}\sum_{k=0}^{n}a_{k}p^{k}\bar{p}^{n-k}~{},\qquad a_{n-k}=\bar{a}_{k}=a_{k}~{}.$$ (2.16) The self-duality equation (1.8) amounts to the following conditions on the coefficients $a_{k}$: $$\displaystyle k^{2}a_{k}^{2}-(n-k+1)^{2}a_{n-k+1}^{2}$$ $$\displaystyle=$$ $$\displaystyle 0~{},\qquad k=1,\ldots,n~{},$$ (2.17a) $$\displaystyle kla_{k}a_{l}-(n-k+1)(n-l+1)a_{n-k+1}a_{n-l+1}$$ $$\displaystyle=$$ $$\displaystyle 0~{},\qquad l\neq k~{}.$$ (2.17b) We end up with $$\displaystyle a_{j}=\frac{n!}{j!(n-j)!}a_{n}~{},\qquad j=\left\{\begin{aligned} &1,\ldots,\frac{n}{2}\quad\text{for $n$ even}\\ &1,\ldots,\frac{n+1}{2}\quad\text{for $n$ odd}\\ \end{aligned}\right.~{}.$$ (2.18) As a result, the interaction (2.16) can be written in the form $$\displaystyle\Lambda^{(n)}(u,\bar{u},p,\bar{p})=\frac{\zeta}{u\bar{u}}\left(\frac{p+\bar{p}}{2}\right)^{n}~{},$$ (2.19) with $\zeta$ a coupling constant. Of special interest is the $n=1$ case in (2.19), since it corresponds to the model introduced in [40] and describes a Fayet-Iliopoulos term in supergravity without gauged $R$-symmetry.222 Models with more general Fayet-Iliopoulos terms described in [41] do not appear to be duality invariant. The duality-invariant model constructed, eq. (2.19), can be generalised as follows $$\displaystyle\Lambda(u,\bar{u},p,\bar{p})=\frac{1}{u\bar{u}}{\mathfrak{D}}\left(\frac{p+\bar{p}}{2}\right)~{},$$ (2.20) where $\mathfrak{D}(x)$ is a real function of a real argument. It is easy to see that (2.20) is a solution to the self-duality equation (1.8). Much more general solutions of the self-duality equation (1.8) may be obtained. Let $\Lambda(u,\bar{u};g)$ be a solution of (1.4), which depends on a real duality-invariant parameter $g$. Then the following self-interaction $$\displaystyle\Lambda\big{(}u,\bar{u},p,\bar{p}\big{)}:=\Lambda\left(u,\bar{u};{\mathfrak{D}}\Big{(}\frac{p+\bar{p}}{2}\Big{)}\right)$$ (2.21) is a solution of the self-duality equation (1.8), for any real function $\mathfrak{D}(x)$ of a real variable. 3 Auxiliary superfield formalism In this section we describe an alternative formulation for the self-dual models of the ${\cal N}=1$ vector multiplet described in the previous section. In particular, we extend the results obtained in [23] to the case of curved superspace. We start with a brief summary of the construction given in [22]. Consider an auxiliary action of the form $$\displaystyle S[W,\bar{W},\eta,\bar{\eta}]=\int{\rm d}^{4}x{\rm d}^{2}\theta\,{\cal E}\Big{\{}\eta W-\frac{1}{2}\eta^{2}-\frac{1}{4}W^{2}\Big{\}}+{\rm c.c.}+{\mathfrak{S}}_{\rm int}[\eta,\bar{\eta}]~{}.$$ (3.1) Here the spinor superfield $\eta_{\alpha}$ is constrained to be covariantly chiral, $\bar{\cal D}_{\dot{\beta}}\eta_{\alpha}=0$, but otherwise it is completely arbitrary. By definition, the second term on the right, ${\mathfrak{S}}_{\rm int}[\eta,\bar{\eta}]$, contains cubic, quartic and higher powers of $\eta_{\alpha}$ and its conjugate. The above model is equivalent to a theory with action $$\displaystyle S[W,\bar{W}]=\frac{1}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta\,{\cal E}\,W^{2}+{\rm c.c.}+S_{\rm int}[W,\bar{W}]~{},$$ (3.2) describing the dynamics of the vector multiplet. Indeed, under reasonable assumptions the equation of motion for $\eta^{\alpha}$ $$\displaystyle W_{\alpha}=\eta_{\alpha}-\frac{\delta}{\delta\eta^{\alpha}}{\mathfrak{S}}_{\rm int}[\eta,\bar{\eta}]$$ (3.3) allows one to express $\eta_{\alpha}$ as a functional of $W_{\alpha}$ and its conjugate, $\eta_{\alpha}=\Psi_{\alpha}[W,\bar{W}]$. Plugging this functional and its conjugate into (3.1) leads to a vector-multiplet model of the form (3.2). If $S[W,\bar{W}]$ is a solution of the self-duality equation (2.8), then the self-interaction ${\mathfrak{S}}_{\rm int}[\eta,\bar{\eta}]$ in (3.1) proves to be invariant under rigid $\mathsf{U}(1)$ phase transformations of $\eta_{\alpha}$ and its conjugate, $$\displaystyle{\mathfrak{S}}_{\rm int}[{\rm e}^{-{\rm i}\lambda}\eta,{\rm e}^{{\rm i}\lambda}\bar{\eta}]={\mathfrak{S}}_{\rm int}[\eta,\bar{\eta}]~{},\qquad\lambda\in{\mathbb{R}}~{}.$$ (3.4) The duality rotation (2.7) acts on the chiral spinor $\eta_{\alpha}$ as $$\displaystyle\delta\eta_{\alpha}=-{\rm i}\lambda\eta_{\alpha}~{},$$ (3.5) see [22] for the technical details. We now restrict our attention to a subclass of the models (3.1) of the form: $$\displaystyle S[W,\bar{W},\eta,\bar{\eta}]$$ $$\displaystyle=$$ $$\displaystyle\int{\rm d}^{4}x{\rm d}^{2}\theta\,{\cal E}\Big{\{}\eta W-\frac{1}{2}\eta^{2}-\frac{1}{4}W^{2}\Big{\}}+{\rm c.c.}$$ (3.6a) $$\displaystyle+\frac{1}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta{\rm d}^{2}\bar{\theta}\,E\,\eta^{2}\bar{\eta}^{2}{\mathfrak{F}}(v,\bar{v},q,\bar{q})~{},$$ in which $$\displaystyle v:=\frac{1}{8}({\cal D}^{2}-4\bar{R})\eta^{2}~{},\qquad q:={\cal D}^{\alpha}\eta_{\alpha}$$ (3.6b) and $\mathfrak{F}(v,\bar{v},q,\bar{q})$ is a real function. Such models in a flat background were analysed in [23]. We aim to integrate out from (3.6a) the auxiliary spinor variables $\eta_{\alpha}$ and $\bar{\eta}_{\dot{\alpha}}$ in order to bring the action to the form (2.9). The equation of motion for $\eta^{\alpha}$ is $$\displaystyle W_{\alpha}$$ $$\displaystyle=$$ $$\displaystyle\eta_{\alpha}\left\{1+\frac{1}{8}(\bar{\cal D}^{2}-4R)\Big{[}{\bar{\eta}}^{2}\Big{(}\mathfrak{F}+\frac{1}{8}({\cal D}^{2}-4\bar{R})\big{(}\eta^{2}\,\partial_{v}\mathfrak{F}\big{)}\Big{)}\Big{]}\right\}$$ (3.7) $$\displaystyle-\frac{1}{16}(\bar{\cal D}^{2}-4R)\left[\bar{\eta}^{2}{\cal D}_{\alpha}(\eta^{2}\partial_{q}\mathfrak{F})\right]~{}.$$ Its immediate implications are $$\displaystyle\eta W$$ $$\displaystyle=$$ $$\displaystyle\eta^{2}\left\{1+\frac{1}{8}(\bar{\cal D}^{2}-4R)\Big{[}\bar{\eta}^{2}\Big{(}\partial_{v}(v\mathfrak{F})+\frac{1}{2}q\partial_{q}\mathfrak{F}\Big{)}\Big{]}\right\}~{},$$ (3.8a) $$\displaystyle W^{2}$$ $$\displaystyle=$$ $$\displaystyle\eta^{2}\Bigg{\{}\Big{[}1+\frac{1}{8}(\bar{\cal D}^{2}-4R)\left\{\bar{\eta}^{2}\partial_{v}(v\mathfrak{F})\right\}\Big{]}^{2}+\frac{1}{8}(\bar{\cal D}^{2}-4R)\left(\bar{\eta}^{2}q\partial_{q}\mathfrak{F}\right)$$ (3.8b) $$\displaystyle+\frac{1}{64}(\bar{\cal D}^{2}-4R)\left(\bar{\eta}^{2}\partial_{v}(v{\mathfrak{F}})\right)(\bar{\cal D}^{2}-4R)\left(\bar{\eta}^{2}q\partial_{q}{\mathfrak{F}}\right)$$ $$\displaystyle+\frac{1}{128}(\bar{\cal D}^{2}-4R)\left(\bar{\eta}^{2}({\cal D}^{\alpha}\eta^{\beta})\partial_{q}{\mathfrak{F}}\right)(\bar{\cal D}^{2}-4R)\left(\bar{\eta}^{2}({\cal D}_{\alpha}\eta_{\beta})\partial_{q}{\mathfrak{F}}\right)\Bigg{\}}~{},$$ $$\displaystyle W^{2}\bar{W}^{2}$$ $$\displaystyle=$$ $$\displaystyle\eta^{2}\bar{\eta}^{2}{\mathfrak{H}}\bar{\mathfrak{H}}~{},$$ (3.8c) where we have introduced $$\displaystyle{\mathfrak{H}}:=\left[1+\partial_{v}(v\bar{v}{\mathfrak{F}})\right]^{2}+\bar{v}q\partial_{q}{\mathfrak{F}}\left[1+\partial_{v}(v\bar{v}{\mathfrak{F}})\right]-2v\bar{v}^{2}(\partial_{q}{\mathfrak{F}})^{2}~{}.$$ (3.9) It should be noted that in deriving (3.8b) we have made use of the identity $$\displaystyle\eta^{2}({\cal D}_{\alpha}\eta^{\beta})({\cal D}^{\alpha}\eta_{\beta})=4v\eta^{2}~{}.$$ (3.10) Eq. (3.8b) and (3.7) imply that $$\displaystyle u$$ $$\displaystyle\approx$$ $$\displaystyle v{\mathfrak{H}}~{},$$ (3.11) $$\displaystyle p$$ $$\displaystyle\approx$$ $$\displaystyle q\left[1+\partial_{v}(v\bar{v}{\mathfrak{F}})\right]-4v\bar{v}\partial_{q}{\mathfrak{F}}$$ (3.12) respectively. The symbol $\approx$ is used to indicate that the result holds modulo terms proportional to $\eta_{\alpha}$ and $\bar{\eta}_{\dot{\alpha}}$ (or, equivalently, to $W_{\alpha}$ and $\bar{W}_{\dot{\alpha}}$). The equations (3.11) and (3.12) are the “effective” relations used to relate the auxiliary variables $(v,\bar{v},q,\bar{q})$ to the original multiplet variables $(u,\bar{u},p,\bar{p})$. The identities (3.8) may be used to derive the following integral relations $$\displaystyle\int{\rm d}^{4}x{\rm d}^{2}\theta\,{\cal E}\,\eta W$$ $$\displaystyle=$$ $$\displaystyle\int{\rm d}^{4}x{\rm d}^{2}\theta\,{\cal E}\,\eta^{2}$$ (3.13a) $$\displaystyle-\frac{1}{2}\int{\rm d}^{4}x{\rm d}^{2}\theta{\rm d}^{2}\bar{\theta}\,E\,\eta^{2}\bar{\eta}^{2}\Big{(}\partial_{v}(v{\mathfrak{F}})+\frac{1}{2}q\partial_{q}{\mathfrak{F}}\Big{)}~{},$$ $$\displaystyle\int{\rm d}^{4}x{\rm d}^{2}\theta\,{\cal E}\,W^{2}$$ $$\displaystyle=$$ $$\displaystyle\int{\rm d}^{4}x{\rm d}^{2}\theta\,{\cal E}\,\eta^{2}$$ (3.13b) $$\displaystyle-\int{\rm d}^{4}x{\rm d}^{2}\theta{\rm d}^{2}\bar{\theta}\,E\,\eta^{2}\bar{\eta}^{2}\Big{\{}\partial_{v}(v{\mathfrak{F}})+\frac{1}{2}\bar{v}[\partial_{v}(v{\mathfrak{F}})]^{2}$$ $$\displaystyle+\frac{1}{2}q\partial_{q}{\mathfrak{F}}(1+\partial_{v}(v\bar{v}{\mathfrak{F}}))-v\bar{v}(\partial_{q}{\mathfrak{F}})^{2}\Big{\}}~{}.$$ These relations along with (3.8c) allow us to rewrite the action (3.6a) in terms of the vector multiplet, $$\displaystyle S[W,\bar{W}]$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta\,{\cal E}\,W^{2}+{\rm c.c.}+\frac{1}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta{\rm d}^{2}\bar{\theta}\,E\,W^{2}\bar{W}^{2}\Lambda(u,\bar{u},p,\bar{p})~{},$$ (3.14) where we have introduced $$\displaystyle\Lambda(u,\bar{u},p,\bar{p}):=\frac{{\mathfrak{F}}+{\mathfrak{G}}+\bar{\mathfrak{G}}}{{\mathfrak{H}}\bar{\mathfrak{H}}}~{},$$ (3.15) and $$\displaystyle{\mathfrak{G}}:=\bar{v}[\partial_{v}(v{\mathfrak{F}})]^{2}+q\partial_{q}{\mathfrak{F}}\partial_{v}(v\bar{v}{\mathfrak{F}})-2v\bar{v}(\partial_{q}{\mathfrak{F}})^{2}~{}.$$ (3.16) The super-Weyl invariant version of the model (3.6) is given by $$\displaystyle S[W,{\bar{W}},\eta,\bar{\eta};\Upsilon]$$ $$\displaystyle=$$ $$\displaystyle\int{\rm d}^{4}x{\rm d}^{2}\theta\,{\cal E}\Big{\{}\eta W-\frac{1}{2}\eta^{2}-\frac{1}{4}W^{2}\Big{\}}+{\rm c.c.}$$ (3.17) $$\displaystyle+\frac{1}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta{\rm d}^{2}\bar{\theta}\,E\,\frac{\eta^{2}\,\bar{\eta}^{2}}{\Upsilon^{2}}{\mathfrak{F}}\left(\frac{v}{\Upsilon^{2}},\frac{\bar{v}}{\Upsilon^{2}},\frac{q}{\Upsilon},\frac{\bar{q}}{\Upsilon}\right)~{},$$ in which the auxiliary variable $\eta_{\alpha}$ transforms as $$\displaystyle\delta_{\sigma}\eta_{\alpha}=\frac{3}{2}\sigma\eta_{\alpha}~{},$$ (3.18) in conjunction with the transformation of $\Upsilon$, eq. (2.13). The condition of $\mathsf{U}(1)$ duality invariance (2.8) in the auxiliary variable formalism is equivalent to (3.4). For the model (3.6a) this means manifest $\mathsf{U}(1)$ invariance of the auxiliary interaction function ${\mathfrak{F}}$ and thus $$\displaystyle{\mathfrak{F}}(v,\bar{v},q,\bar{q})=f\Big{(}\frac{v}{q^{2}},\frac{\bar{v}}{\bar{q}^{2}},q\bar{q}\Big{)}~{}.$$ (3.19) To demonstrate how the construction discussed above works, we provide here a simple example. Consider the following $\mathsf{U}(1)$-invariant auxiliary interaction $$\displaystyle{\mathfrak{F}}^{(0)}(v,\bar{v},q,\bar{q})=\frac{\kappa}{v\bar{v}}~{},\qquad\kappa\in{\mathbb{R}}~{}.$$ (3.20) The effective relations (3.11) and (3.12) are trivial, $$\displaystyle u\approx v~{},\qquad p\approx q~{}.$$ (3.21) The self-interaction defined by (3.15) takes the form $$\displaystyle\Lambda^{(0)}(u,\bar{u},p,\bar{p})=\frac{\kappa}{u\bar{u}}~{},$$ (3.22) which coincides with the $n=0$ case of (2.19) when $\kappa=\zeta$. The corresponding action $$\displaystyle S[W,{\bar{W}}]$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{2}\int{\rm d}^{4}x{\rm d}^{2}\theta\,{\cal E}\,W^{2}+\frac{\kappa}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta{\rm d}^{2}\bar{\theta}\,E\,\frac{W^{2}\,\bar{W}^{2}}{u\bar{u}}$$ (3.23) contains a cosmological term, $$\displaystyle\kappa\int{\rm d}^{4}x\,e~{},\qquad e=\det(e_{m}{}^{a})~{},$$ (3.24) at the component level, where $e_{m}{}^{a}(x)$ is the vielbein. 4 Superconformal duality-invariant model We now turn to presenting a new superconformal duality-invariant model derived using the formalism described in the previous section. The duality-invariant model defined by eqs. (3.17) and (3.19) is superconformal if the action is independent of $\Upsilon$. The most general duality-invariant and superconformal model is described by $$\displaystyle{\mathfrak{F}}_{\rm SC}(v,\bar{v},q,\bar{q})=\frac{1}{q\bar{q}}\varphi\Big{(}\frac{v}{q^{2}},\frac{\bar{v}}{\bar{q}^{2}}\Big{)}~{},$$ (4.1) for some function $\varphi(z,\bar{z})$. Choosing $\varphi(z,\bar{z})={\kappa}/\sqrt{z\bar{z}}$, with $\kappa\in{\mathbb{R}}$, gives the model [28] $$\displaystyle{\mathfrak{F}}_{\rm SC}(v,\bar{v})=\frac{\kappa}{\sqrt{v\bar{v}}}~{},$$ (4.2) which is the only member of the family (4.1) without dependence on $q$ and $\bar{q}$. Eliminating the auxiliary chiral $\eta_{\alpha}$ and antichiral $\bar{\eta}_{\dot{\alpha}}$ variables leads to the supersymmetric ModMax theory [27, 28]. Here we will study a different duality-invariant and superconformal model defined by $\varphi(z,\bar{z})={\kappa}/(z\bar{z})$, with a real coupling constant $\kappa$, which leads to $$\displaystyle{\mathfrak{F}}_{\text{SC}}(v,\bar{v},q,\bar{q})=\kappa\frac{q\bar{q}}{v\bar{v}}~{}.$$ (4.3) In this case the effective relations (3.11) and (3.12) become $$\displaystyle u$$ $$\displaystyle\approx$$ $$\displaystyle v+\kappa q\bar{q}-2\kappa^{2}{\bar{q}}^{2}~{},\qquad\bar{u}\approx\bar{v}+\kappa q\bar{q}-2\kappa^{2}q^{2}~{},$$ (4.4) $$\displaystyle p$$ $$\displaystyle\approx$$ $$\displaystyle q-4\kappa\bar{q}~{},\qquad\bar{p}\approx\bar{q}-4\kappa q~{}.$$ (4.5) Using the effective relations (4.5), we can express the auxiliary variables $q$ and $\bar{q}$ in terms of the multiplet variables $p$ and $\bar{p}$, $$\displaystyle q\approx\frac{p+4\kappa\bar{p}}{1-(4\kappa)^{2}}~{},\qquad\bar{q}\approx\frac{\bar{p}+4\kappa p}{1-(4\kappa)^{2}}~{}.$$ (4.6) Substituting these expressions (4.6) into (4.4) allows one to express the remaining auxiliary variables $v$ and $\bar{v}$ purely in terms of the multiplet variables as $$\displaystyle v$$ $$\displaystyle\approx$$ $$\displaystyle u+\frac{4\kappa^{2}p^{2}(8\kappa^{2}-1)-2\kappa^{2}{\bar{p}}^{2}-\kappa p\bar{p}}{(1-(4\kappa)^{2})^{2}}~{},$$ $$\displaystyle\bar{v}$$ $$\displaystyle\approx$$ $$\displaystyle\bar{u}+\frac{4\kappa^{2}{\bar{p}}^{2}(8\kappa^{2}-1)-2\kappa^{2}p^{2}-\kappa p\bar{p}}{(1-(4\kappa)^{2})^{2}}~{}.$$ (4.7) With the aid of these relations (4.6) and (4), we can read off the self-interaction (3.15) as a function of the multiplet variables, $$\displaystyle\Lambda(u,\bar{u},p)=\left(\frac{\kappa}{1-4\kappa}\right)\frac{p^{2}}{u\bar{u}}~{}.$$ (4.8) It should be noted that for the purposes of our analysis, we have treated $p$ and $\bar{p}$ as independent, and only at the end of the calculation is the Bianchi identity (2.3) imposed. The model derived above (4.8) corresponds to the $n=2$ case of the family of duality-invariant solutions in (2.19) with $p=\bar{p}$ and $\zeta=\kappa/(1-4\kappa)$. The outcome of our analysis is the superconformal duality-invariant model, $$\displaystyle S[W,\bar{W}]=\frac{1}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta\,{\cal E}\,W^{2}+{\rm c.c.}+\frac{\zeta}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta{\rm d}^{2}\bar{\theta}\,E\,W^{2}\bar{W}^{2}\frac{({\cal D}W)^{2}}{u\bar{u}}~{}.$$ (4.9) It is a member of the family of the superconformal vector multiplet models [42] $$\displaystyle S[W,\bar{W}]$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta\,{\cal E}\,W^{2}+{\rm c.c.}$$ (4.10) $$\displaystyle+\frac{1}{4}\int{\rm d}^{4}x{\rm d}^{2}\theta{\rm d}^{2}\bar{\theta}\,E\,\frac{W^{2}\,{\bar{W}}^{2}}{({\cal D}W)^{2}}\,\Lambda\left(\frac{u}{({\cal D}W)^{2}},\frac{\bar{u}}{({\cal D}W)^{2}}\right)~{},$$ where $\Lambda(\omega,\bar{\omega})$ is a real function of one complex variable. 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Spectra of formulas with bounded quantifier alternations 111M.E. Zhukovskii is supported by the Ministry of Education and Science of the Russian Federation (the Agreement number 02.a03.21.0008) and grants of the Russian Foundation for Basic Research No. 16-31-60052 and No. 15-01-03530. A.D. Matushkin, M.E. Zhukovskii 222Moscow Institute of Physics and Technology, Laboratory of Advanced Combinatorics and Network Applications; RUDN University Abstract Spectrum of a first order sentence is the set of all $\alpha$ such that $G(n,n^{-\alpha})$ does not obey zero-one law w.r.t. this sentence. We have proved that the minimal number of quantifier alternations of a first order sentence with infinite spectrum equals $3$. 1 Previous results on zero-one laws In this paper, we consider first order sentences about graphs (a signature consists of two predicates $\sim$ (adjacency) and $=$ (equality) of arity $2$) [1, 2]. Recall that a quantifier depth ${\mathrm{q}}(\phi)$ of a formula $\phi$ is the number of quantifiers in the longest past of nested quantifiers in this formula. Let $G(n,p)$ be a binomial random graph [3, 4] with $n$ vertices and the probability $p$ of appearing of an edge. We say that $G(n,p)$ obeys zero-one law w.r.t. a first order sentence $\phi$, if either a.a.s. (asymptotically almost surely) $G(n,p)\models\phi$, or a.a.s. $G(n,p)\models\neg(\phi)$. Let $S(\phi)$ be the set of all $\alpha>0$ such that $G(n,n^{-\alpha})$ does not obey zero-one law w.r.t. $\phi$. This set is called a spectrum of $\phi$. In 1988 [5], S. Shelah and J. Spencer proved that there are only rational numbers in $S(\phi)$ for any first order sentence $\phi$. In 1991 [6], J. Spencer proved that there exists first order sentence with an infinite spectrum and the quantifier depth $14$. In 2012 [7], M. Zhukovskii proved that, for any first order sentence $\phi$ with the quantifier depth $3$, $S(\phi)\cap(0,1)=\varnothing$. Moreover, for any first order sentence $\phi$ with the quantifier depth $4$, $S(\phi)\cap(0,1/2)=\varnothing$. Later [8], it was proved that, for any first order sentence $\phi$, the set $S(\phi)\cap(1,\infty)$ is finite. In [9], a first order sentence with the quantifier depth $5$ and an infinite spectrum was obtained. This formula is given in the statement below. Theorem 1 Let $m\in\mathbb{N}$, $\alpha=\frac{1}{2}+\frac{1}{2(m+1)}$ and $p=n^{-\alpha}$. Then the random graph $G(n,p)$ does not obey zero-one law w.r.t. the sentence $$\phi=\exists x_{1}\exists x_{2}\,\left[\left(\exists x_{3}\exists x_{4}\,\left% (\bigwedge_{1\leq i<j\leq 4}(x_{i}\sim x_{j})\right)\right)\wedge(\varphi(x_{1% },x_{2}))\right],$$ where $$\varphi(x_{1},x_{2})=\forall y_{1}\,([y_{1}\sim x_{1}]\vee[y_{1}\sim x_{2}]% \vee[\forall y_{2}(\neg[(y_{2}\sim x_{1})\wedge(y_{2}\sim y_{1})])]\vee$$ $$[\exists z\,(z\sim x_{1})\wedge(z\sim x_{2})\wedge(\forall u\,[(\neg[(u\sim z)% \wedge(u\sim y_{1})])\vee(u\sim x_{1})\vee(u\sim x_{2})])]).$$ So, a minimal quantifier depth of a first order sentence with an infinite spectrum equal either $4$, or $5$. Note that the maximal number of quantifier alternations over all sequences of nested quantifiers in $\phi$ equals $3$ (we call this value the number of quantifier alternations of $\phi$). It is essential that all the negations are applied to atomic formulas only. A prenex normal form of $\phi$ with the quantifier depth $8$ is given below $$\tilde{\phi}=\exists x_{1}\exists x_{2}\exists x_{3}\exists x_{4}\forall y_{1}% \forall y_{2}\exists z\forall u\,\left[\left(\bigwedge_{1\leq i<j\leq 4}(x_{i}% \sim x_{j})\right)\wedge(\tilde{\varphi}(x_{1},x_{2},y_{1},y_{2},z,u))\right],$$ (1) where $$\tilde{\varphi}(x_{1},x_{2},y_{1},y_{2},z,u)=[y_{1}\sim x_{1}]\vee[y_{1}\sim x% _{2}]\vee[\neg((y_{2}\sim x_{1})\wedge(y_{2}\sim y_{1}))]\vee$$ $$[(z\sim x_{1})\wedge(z\sim x_{2})\wedge(\neg[(u\sim z)\wedge(u\sim y_{1})])]% \vee[u\sim x_{1}]\vee[u\sim x_{2}].$$ This raises the following questions. 1. What is the minimal quantifier depth of a first order sentence with an infinite spectrum, 4 or 5? 2. What is the minimal number of quantifier alternations of a first order sentence with an infinite spectrum, 3 or less? 3. What is the minimal quantifier depth of a first order sentence in a prenex normal form with an infinite spectrum, 4, 5, 6, 7 or 8? We partially answer these questions in Sections 4, 5. 2 Existence and extension statements Let $\phi$ be a first order sentence in a prenex normal form. We call $\phi$ an existence sentence, if all quantifiers of $\phi$ equal $\exists$. We call $\phi$ an extension sentence, if the sequence of all quantifiers of $\phi$ equals $\forall\ldots\forall\exists\ldots\exists$. We say that an existence sentence expresses an existence property, and an extension sentence expresses an extension property. An asymptotical behavior of probabilities of the random graph existence and extension properties was widely studied in [10, 11, 12, 13]. We summarize this study in the result given below. Let $G,H$ be two graphs such that $H\subset G$, $V(H)=\{a_{1},\ldots,a_{s}\}$, $V(G)\setminus V(H)=\{b_{1},\ldots,b_{m}\}$, $s,m\geq 1$. Let $\rho(H)$ be a maximal fraction $e(Q)/v(Q)$ over all subgraphs $Q\subset H$ ($\rho(H)$ is called the maximal density of $H$). Here $e(Q),v(Q)$ denote the numbers of edges and vertices in $Q$ respectively. Let $\rho(G,H)$ be a maximal fraction $(e(Q)-e(H))/(v(Q)-v(H))$ over all $Q$ such that $H\subset Q\subset G$. We say that a graph has the $(G,H)$-extension property, if, for any its distinct vertices $y_{1},\ldots,y_{s}$, there exist distinct vertices $x_{1},\ldots,x_{m}$ such that, for all $i\in\{1,\ldots,s\}$, $j\in\{1,\ldots,m\}$, $y_{i}\neq x_{j}$ and the adjacency relation $a_{i}\sim b_{j}$ implies the adjacency relation $y_{i}\sim x_{j}$. Theorem 2 Let $\rho(H)\neq 0$. If $p\gg n^{-1/\rho(H)}$, then a.a.s. in $G(n,p)$ there is an induced copy of $H$. If $p\ll n^{-1/\rho(H)}$, then a.a.s. in $G(n,p)$ there is no copy of $H$. Let $\rho(G,H)\neq 0$. If $p\gg n^{-1/\rho(G,H)}$, then a.a.s. $G(n,p)$ has the $(G,H)$-extension property. If $p\ll n^{-1/\rho(G,H)}$, then a.a.s. $G(n,p)$ does not have the $(G,H)$-extension property. It is not difficult to see that Theorem 2 implies finiteness of spectra of all existence and extension sentences (see Section 4). The next step is to consider sentences in prenex normal form that have 2 alternations. We call $\phi$ a double-extension sentence, if the sequence of all quantifiers of $\phi$ equals $\forall\ldots\forall\exists\ldots\exists\forall\ldots\forall$ (the respective properties are called double-extension as well). An asymptotical behavior of probabilities of the random graph double-extension properties was studied in [14, 15]. Let $W,G,H$ be three graphs such that $H\subset G\subset W$, $V(H)=\{a_{1},\ldots,a_{s}\}$, $V(G)\setminus V(H)=\{b_{1},\ldots,b_{m}\}$, $V(W)\setminus V(G)=\{c_{1},\ldots,c_{r}\}$, $s\geq 0$, $r,m\geq 1$. Assume that in $W$ there are edges between each connected component of $W|_{\{c_{1},\ldots,c_{r}\}}$ and $W|_{\{b_{1},\ldots,b_{m}\}}$. Let $\mathcal{W}$ be a finite set of graphs such that all $W\in\mathcal{W}$ satisfy the above conditions (but $r$ depends on $W$). We say that a graph has the $(\mathcal{W},G,H)$-double-extension property, if, for any its distinct vertices $y_{1},\ldots,y_{s}$, there exist distinct vertices $x_{1},\ldots,x_{m}$ such that, for all $W\in\mathcal{W}$ and all distinct vertices $z_{1},\ldots,z_{r(W)}$, • for all $i\in\{1,\ldots,s\}$, $j\in\{1,\ldots,m\}$, $y_{i}\neq x_{j}$ and the adjacency relation $a_{i}\sim b_{j}$ implies the adjacency relation $y_{i}\sim x_{j}$, • either there exists $h\in\{1,\ldots,r(W)\}$ and $i\in\{1,\ldots,s\}$ such that $[z_{h}=y_{i}]\vee[(z_{h}\nsim y_{i})\wedge(c_{h}\sim a_{i})]$, or there exist $h\in\{1,\ldots,r(W)\}$ and $j\in\{1,\ldots,m\}$ such that $[z_{h}=x_{j}]\vee[(z_{h}\nsim x_{j})\wedge(c_{h}\sim b_{j})]$. Theorem 3 Let, for all $W\in\mathcal{W}$, $\rho(W,G)>\rho(G,H)>0$ and $n^{-1/\rho(W,G)}\gg p\gg n^{-1/\rho(G,H)}$. Then a.a.s. $G(n,p)$ has the $(\mathcal{W},G,H)$-double-extension property. We have proved that Theorems 2, 3 imply the finiteness of spectra of double-extension sentences (see Section 4 as well). So, we generalize the well-known results about existence, extension and double-extension properties and prove that spectra of all first order sentences with at most $2$ quantifier alternations are finite. 3 Logical preliminaries 3.1 Some notations Recall that a rooted tree $T_{R}$ is a tree with one distinguished vertex $R$, which is called the root. If $R,\ldots,x,y$ is a simple path in $T_{R}$, then $x$ is called a parent of $y$ and $y$ is called a child of $x$. The relation of being a descendant is the transitive and reflexive closure of the relation of being a child. If $v\in V(T_{R})$, then $T_{R}[v]$ denotes the subforest of $T_{R}$ spanned by the set of all descendants of $v$ (children of $v$ are its roots). For two first order formulas $\phi_{1}(x_{1},\ldots,x_{s}),\phi_{2}(y_{1},\ldots,y_{s})$ ($s\in\{0,1,2,\ldots\}$), we say that they are (asymptotically) equivalent (and write $\varphi_{1}\cong\varphi_{2}$), if there exists $n\in\mathbb{N}$ such that for any graph $G$ on at least $n$ vertices and any its vertices $v_{1},\ldots,v_{s}$ either $G\models(\phi_{1}(v_{1},\ldots,v_{s}))\wedge(\phi_{2}(v_{1},\ldots,v_{s}))$, or $G\models(\neg(\phi_{1}(v_{1},\ldots,v_{s}))\wedge(\neg(\phi_{2}(v_{1},\ldots,v% _{s})))$. We say that a set of graphs $C$ is a (asymptotical) first order property of a graph, if there exists a first order sentence $\phi$ and $n\in\mathbb{N}$ such that, for any $G$ on at least $n$ vertices, $G\in C$ if and only if $G\models\phi$ (in this case, we say that $\phi$ expresses $C$). 3.2 Language $\mathcal{F}$ It is easy to see that any first order formula (not necessarily sentence) is equivalent to a formula constructed of the following symbols: variables, relational symbols $\sim,=,\nsim,\neq$, conjunctions $\wedge$, disjunctions $\vee$ and quantifiers $\forall,\exists$. We denote the set of formulas in this language by $\mathcal{F}$. Let us state a simple observation of formulas in $\mathcal{F}$. Lemma 1 Let $Z\in\{\wedge,\vee\}$, $z_{1},z_{2}\in\{\forall,\exists\}$. Then, for any two formulas $\varphi_{1}(x_{1},\ldots,x_{s})$, $\varphi_{2}(x_{1},\ldots,x_{m})\in\mathcal{F}$ (not necessarily with $s$ and $m$ free variables respectively), $$[z_{1}x_{1}\ldots z_{1}x_{s}\,(\varphi_{1}(x_{1},\ldots,x_{s}))]Z[z_{2}x_{1}% \ldots z_{2}x_{m}\,(\varphi_{2}(x_{1},\ldots,x_{m}))]\cong$$ $$z_{1}x_{1}\ldots z_{1}x_{s}z_{2}x_{s+1}\ldots z_{2}x_{s+m}\,([\varphi_{1}(x_{1% },\ldots,x_{s})]Z[\varphi_{2}(x_{s+1},\ldots,x_{s+m})]).$$ For a formula $\phi\in\mathcal{F}$, define its nesting forest $F(\phi)$ in the following way. • If $\phi$ is an atomic formula, then its nesting forest is an empty graph. • Consider a formula $\varphi(x)$. If it has an empty nesting forest, then the nesting forest of the formula $\phi=\exists x\,(\varphi(x))$ (the formula $\phi=\forall x\,(\varphi(x))$) is an isolated vertex labeled by $\exists$ (by $\forall$). This vertex is a trivial tree rooted in its only vertex. Otherwise, let $F(\varphi(x))=T^{1}_{t_{1}}\sqcup\ldots\sqcup T^{m}_{t_{m}}$, where $T^{1}_{t_{1}},\ldots,T^{m}_{t_{m}}$ are trees rooted in $t_{1},\ldots,t_{m}$ respectively. Then the nesting forest of the formula $\phi=\exists x\,(\varphi(x))$ (the formula $\phi=\forall x\,(\varphi(x))$) is a tree obtained by adding a vertex $t$ (which is the root of this three) labeled by $\exists$ (by $\forall$) to $F(\varphi(x))$ and edges from $t$ to each of $t_{1},\ldots,t_{m}$. • If $\phi=(\varphi_{1})\wedge(\varphi_{2})$ (or $\phi=(\varphi_{1})\vee(\varphi_{2})$), then $F(\phi)$ is the disjoint union of $F(\varphi_{1})$, $F(\varphi_{2})$. Consider a formula $\phi(x_{1},\ldots,x_{s})\in\mathcal{F}$ and its nesting forest $F(\phi)=T^{1}_{t_{1}}\sqcup\ldots\sqcup T^{m}_{t_{m}}$ consisting of trees $T^{1}_{t_{1}},\ldots,T^{m}_{t_{m}}$ rooted in $t_{1},\ldots,t_{m}$ respectively. Let $v$ be a vertex of $T^{i}_{t_{i}}$ for some $i\in\{1,\ldots,m\}$. Consider the forest $T^{i}_{t_{i}}[v]$. Let $V$ be the set of all vertices of $T^{i}_{t_{i}}$ such that $v$ is a descendant for each of them, $[V]:=V\cup\{v\}$. Obviously, $T^{i}_{t_{i}}|_{[V]}$ is the path $t_{i}\ldots v$. Each of the vertices of this path corresponds to a bound variable of $\phi$. Let $y_{1},\ldots,y_{r}$ be these variables ($y_{i+1},y_{i}$ corresponds to a child and a parent respectively). Then $T^{i}_{t_{i}}[v]$ is the nested forest of a subformula $\varphi(x_{1},\ldots,x_{s},y_{1},\ldots,y_{r})$ of $\phi$. The formula $\varphi(x_{1},\ldots,x_{s},y_{1},\ldots,y_{r})$ is called a nested subformula of $\phi$, the forest $F(\varphi(x_{1},\ldots,x_{s},y_{1},\ldots,y_{r}))$ is called a nested subforest of $F(\phi)$. Note that the quantifier depth of $\phi$ is the length of the longest path starting in a root (we denote it by ${\mathrm{q}}(\phi)$). For a path in $F(\phi)$ starting in a root consider the number of labels alternations (the number of (unordered) pairs of neighbors $\forall\exists$ and $\exists\forall$). For example, the number of labels alternations of the path $\exists\forall\forall\exists\exists\forall$ equals $3$. The maximal number of labels alternations over all paths starting in roots of $F(\phi)$ is called the number of quantifier alternations of $\phi$ (we denote it by ${\mathrm{ch}}(\phi)$). 3.3 Normal forms A formula $\phi\in\mathcal{F}$ is in prenex normal form (PNF) (we also say that $\phi$ is a ${\rm{PNF}}$ formula or a ${\rm{PNF}}$ sentence), if $F(\phi)$ is a path (all quantifiers are in the beginning of the formula). We say that $\hat{\phi}$ is a ${\rm{PNF}}$ of $\phi$, if $\hat{\phi}\in\mathcal{F}$, $\hat{\phi}$ is in PNF and $\hat{\phi}\cong\phi$. It is known [16], that for any first order formula (which is not necessarily in $\mathcal{F}$) there exists an equivalent first order formula in PNF. This immediately implies that $\phi$ has a PNF. The formula $\phi$ is in no-equivalence prenex normal form (NEPNF) (we also say that $\phi$ is ${\mathrm{NE}}{\rm{PNF}}$ formula or a ${\mathrm{NE}}{\rm{PNF}}$ sentence), if $\phi$ is in PNF, and is constructed as follows. Consider an arbitrary sequence $z=(z_{1},\ldots,z_{m})$ of symbols from $\{\forall,\exists\}$. Let a formula $\phi_{1}(x_{1},\ldots,x_{m})\in\mathcal{F}$ has no quantifiers and no relations $=$ and $\neq$. For each $j\in\{1,\ldots,m-1\}$ a formula $\phi_{j+1}(x_{1},\ldots,x_{m})$ is obtained from $\phi_{j}(x_{1},\ldots,x_{m})$ in the following way: $$\phi_{j+1}(x_{1},\ldots,x_{m})=(x_{j+1}\neq x_{j})\wedge\ldots\wedge(x_{j+1}% \neq x_{1})\wedge(\phi_{j}(x_{1},\ldots,x_{m})),\quad\text{if }z_{j+1}=\exists,$$ $$\phi_{j+1}(x_{1},\ldots,x_{m})=(x_{j+1}=x_{j})\vee\ldots\vee(x_{j+1}=x_{1})% \vee(\phi_{j}(x_{1},\ldots,x_{m})),\quad\text{if }z_{j+1}=\forall.$$ Finally, $\phi=z_{1}x_{1}\ldots z_{m}x_{m}\,(\phi_{m}(x_{1},\ldots,x_{m}))$. We say that $(\phi_{1}(x_{1},\ldots,x_{m}),z)$ is ${\mathrm{NE}}$-basis of $\phi$. Lemma 2 For any PNFsentence $\phi\in\mathcal{F}$ there exists an ${\mathrm{NE}}{\rm{PNF}}$ sentence $\hat{\phi}\in\mathcal{F}$ with the same sequence of quantifiers such that $\phi\cong\hat{\phi}$. Proof. Let $\phi=z_{1}x_{1}\ldots z_{m}x_{m}\,(\varphi(x_{1},\ldots,x_{m}))$, where $z_{1},\ldots,z_{m}$ is a sequence of symbols from $\{\forall,\exists\}$. Set $\hat{\phi}_{1}(x_{1},\ldots,x_{m})=\varphi(x_{1},\ldots,x_{m})$. For each $j\in\{1,\ldots,m-1\}$ a formula $\hat{\phi}_{j+1}(x_{1},\ldots,x_{m})$ is obtained from $\hat{\phi}_{j}(x_{1},\ldots,x_{m})$ in the following way. First, $\hat{\phi}_{j+1}^{0}(x_{1},\ldots,x_{m})$ is obtained from $\hat{\phi}_{j}(x_{1},\ldots,x_{m})$ by assuming that all $$x_{1}=x_{j+1},\ldots,x_{j}=x_{j+1}$$ are false, and all $$x_{1}\neq x_{j+1},\ldots,x_{j}\neq x_{j+1}$$ are true. For any $i\in\{1,\ldots,j+1\}$, $\hat{\phi}_{j}^{i}(x_{1},\ldots,x_{m}))$ is obtained from $\hat{\phi}_{j}(x_{1},\ldots,x_{m})$ by assuming that all $$x_{1}=x_{j+1},\ldots,x_{i-1}=x_{j+1},x_{i}\neq x_{j+1},x_{i+1}=x_{j+1},\ldots,% x_{j}=x_{j+1}$$ are false, and all $$x_{1}\neq x_{j+1},\ldots,x_{i-1}\neq x_{j+1},x_{i}=x_{j+1},x_{i+1}\neq x_{j+1}% ,\ldots,x_{j}\neq x_{j+1}$$ are true. Second, if $z_{j+1}=\exists$, then $$\hat{\phi}_{j+1}(x_{1},\ldots,x_{m})=\hat{\phi}_{j}^{0}(x_{1},\ldots,x_{m})% \vee\left[\bigvee_{i=1}^{j}(\hat{\phi}_{j}^{i}(x_{1},\ldots,x_{j},x_{i},x_{j+2% },\ldots,x_{m}))\right].$$ Otherwise, $$\hat{\phi}_{j+1}(x_{1},\ldots,x_{m})=\hat{\phi}_{j}^{0}(x_{1},\ldots,x_{m})% \wedge\left[\bigwedge_{i=1}^{j}(\hat{\phi}_{j}^{i}(x_{1},\ldots,x_{j},x_{i},x_% {j+2},\ldots,x_{m}))\right].$$ Let $\hat{\phi}$ be the NEPNF formula with the NE-basis $(\hat{\phi}_{m}(x_{1},\ldots,x_{m}),(z_{1},\ldots,z_{m}))$. It is easy to see that $\phi\cong\hat{\phi}$. Both formulas have the same sequence of quantifiers. $\Box$ We will frequently use the following corollary. Lemma 3 Let $\phi=\exists x\,(\varphi(x))\in\mathcal{F}$. Then there exists an ${\mathrm{NE}}{\rm{PNF}}$ formula $\hat{\phi}=\exists x\,(\hat{\varphi}(x))\in\mathcal{F}$ such that $\phi\cong\hat{\phi}$ and ${\mathrm{ch}}(\phi)={\mathrm{ch}}(\hat{\phi})$. Proof. Let $F$ be a nesting forest of a formula with the quantifier depth $q$. Moreover, let $F$ be a rooted tree with a root $t(F)$. Denote by $t^{r}_{1}(F),\ldots,t^{r}_{a(r,F)}(F)$ all the vertices of $F$ which are at the distance $r-1$ from $t(F)$, where $r\in\{1,\ldots,q\}$, $a(r,F)\in\{1,2,\ldots\}$. Obviously, $a(1,F)=1,$ $a(r,F)\geq 1$ for all $r\in\{2,\ldots,q\}$. Let $r$ be the first positive integer such that $a(r,F)>1$ (if there is no such $r$, then set $r=q+1$). Let $$\mu[F]=q+1-r.$$ Note that if $F$ is a simple path with an end-point $t(F)$, then $\mu[F]=0$. Consider a sentence $\phi=\exists x\,(\varphi(x))\in\mathcal{F}$ such that ${\mathrm{ch}}(\phi)=k$. By Lemma 2, it is sufficient to prove that there exists a PNF sentence $\hat{\phi}=\exists x\,(\hat{\varphi}(x))\in\mathcal{F}$ such that $\phi\cong\hat{\phi}$ and ${\mathrm{ch}}(\hat{\phi})=k$. If $\mu[F(\phi)]=0$, then we are done ($\hat{\phi}=\phi$). Suppose that $\mu[F(\phi)]=m\in\mathbb{N}$, and that for any formula $\zeta$ (not necessarily closed and with an arbitrary first quantifier) with $\mu[F(\zeta)]<m$ the existence of an equivalent PNF sentence with the same number of quantifier alternations and the same first quantifier is already proven. Let $\phi=z_{1}x_{1}\ldots z_{s}x_{s}\,(\varphi(x_{1},\ldots,x_{s}))$, where $s=q-\mu[F(\phi)]$, $z_{1}=\exists$, $z_{2},\ldots,z_{s}\in\{\forall,\exists\}$, and the first symbol of $\varphi(x_{1},\ldots,x_{s})$ is not a quantifier. The formula $\varphi(x_{1},\ldots,x_{s})$ is a logical combination $L$ (disjunctions and conjunctions) of formulas $$\exists x\,(\hat{\varphi}_{i}(x_{1},\ldots,x_{s},x)),\quad\forall x\,(\hat{% \varphi}^{j}(x_{1},\ldots,x_{s},x)).$$ Let $I=\{1,\ldots,|I|\}$ be the set of all such $i$s and $J=\{1,\ldots,|J|\}$ be the set of all such $j$s. So, $$\varphi(x_{1},\ldots,x_{s})=L(\exists x\,(\hat{\varphi}_{i}(x_{1},\ldots,x_{s}% ,x)),i\in I;\,\,\forall x\,(\hat{\varphi}^{j}(x_{1},\ldots,x_{s},x)),j\in J).$$ Obviously, for all $i\in I,$ $j\in J$, $$\mu[F(\hat{\varphi}_{i}(x_{1},\ldots,x_{s},x))]<m,\quad\mu[F(\hat{\varphi}^{j}% (x_{1},\ldots,x_{s},x))]<m.$$ By the induction hypothesis, for all $i\in I,$ $j\in J$ there exist PNF formulas $$\exists x\,(\tilde{\varphi}_{i}(x_{1},\ldots,x_{s},x))\cong\exists x\,(\hat{% \varphi}_{i}(x_{1},\ldots,x_{s},x)),$$ $$\forall x\,(\tilde{\varphi}^{j}(x_{1},\ldots,x_{s},x))\cong\forall x\,(\hat{% \varphi}^{j}(x_{1},\ldots,x_{s},x)),$$ such that $${\mathrm{ch}}(\exists x\,(\tilde{\varphi}_{i}(x_{1},\ldots,x_{s},x)))={\mathrm% {ch}}(\exists x\,(\hat{\varphi}_{i}(x_{1},\ldots,x_{s},x))),$$ $${\mathrm{ch}}(\forall x\,(\tilde{\varphi}^{j}(x_{1},\ldots,x_{s},x)))={\mathrm% {ch}}(\forall x\,(\hat{\varphi}^{j}(x_{1},\ldots,x_{s},x))).$$ Let $$\tilde{\psi}(x_{1},\ldots,x_{s})=L(\exists x\,(\tilde{\varphi}_{i}(x_{1},% \ldots,x_{s},x)),i\in I;\,\,\forall x\,(\tilde{\varphi}^{j}(x_{1},\ldots,x_{s}% ,x)),j\in J).$$ Then the formulas $\phi$ and $$\psi=z_{1}x_{1}\ldots z_{s}x_{s}\,(\tilde{\psi}(x_{1},\ldots,x_{s}))$$ are equivalent and have the same numbers of quantifier alternations. Moreover, $F(\psi)$ is a rooted tree with exactly one vertex with a degree greater than $2$. The distance between this vertex and the root $t(F(\psi))$ is $s-1$. Let the distance between this vertex and a vertex with the biggest distance from the root equal $r$. Let us construct a formula $\psi^{0}\cong\psi$ such that ${\mathrm{ch}}(\psi^{0})={\mathrm{ch}}(\psi)$, $F(\psi^{0})$ is a rooted tree with at most one vertex with a degree greater than $2$, and the distance between this vertex (if it exists) and a vertex with the biggest distance from the root is less than $r$. Obviously, we get the target formula $\hat{\phi}$ after applying such a construction at most $r$ times. For all $i\in I,$ $j\in J$ let us find positive integers $d_{i},d^{j}$ such that $$\exists x^{1}\,(\tilde{\varphi}_{i}(x_{1},\ldots,x_{s},x^{1}))=\exists x^{1}% \ldots\exists x^{d_{i}}\,(\tilde{\psi}_{i}(x_{1},\ldots,x_{s},x^{1},\ldots,x^{% d_{i}})),$$ $$\forall x^{1}\,(\tilde{\varphi}^{j}(x_{1},\ldots,x_{s},x^{1}))=\forall x^{1}% \ldots\forall x^{d^{j}}\,(\tilde{\psi}^{j}(x_{1},\ldots,x_{s},x^{1},\ldots,x^{% d^{j}})),$$ where the formulas $\tilde{\psi}_{i}(x_{1},\ldots,x_{s},x^{1},\ldots,x^{d_{i}})$, $\tilde{\psi}^{j}(x_{1},\ldots,x_{s},x^{1},\ldots,x^{d^{j}})$ either have no quantifiers, or $\forall,\exists$ are the quantifier symbols they begin from respectively. Set $D_{I}=\sum_{i\in I}d_{i}$, $D_{J}=\sum_{j\in J}d^{j}$. Without loss of generality, assume $z_{s}=\exists$. By Lemma 1, there exists a formula (if $z_{s}=\forall$, then this formula starts with $\forall$) $$\tilde{\psi}^{0}(x_{1},\ldots,x_{s})=$$ $$\exists x_{s+1}\ldots\exists x_{s+D_{I}}\forall x_{s+D_{I}+1}\ldots\forall x_{% s+D_{I}+D_{J}}\,(\hat{\psi}(x_{1},\ldots,x_{s+D_{I}+D_{J}}))\cong\tilde{\psi}(% x_{1},\ldots,x_{s})$$ such that $${\mathrm{ch}}(z_{1}x_{1}\ldots z_{s}x_{s}\,(\tilde{\psi}^{0}(x_{1},\ldots,x_{s% })))={\mathrm{ch}}(z_{1}x_{1}\ldots z_{s}x_{s}\,(\tilde{\psi}(x_{1},\ldots,x_{% s}))).$$ Moreover, $F(z_{1}x_{1}\ldots z_{s}x_{s}\,(\tilde{\psi}^{0}(x_{1},\ldots,x_{s})))$ is a tree with exactly one vertex with a degree greater than $2$, and the distance between this vertex and a vertex with the biggest distance from the root is less than $r$. Finally, set $$\psi^{0}=z_{1}x_{1}\ldots z_{s}x_{s}\,(\tilde{\psi}^{0}(x_{1},\ldots,x_{s})).\quad\Box$$ 3.4 Ehrenfeucht games We consider three modification of Ehrenfeucht game. 1. The game ${\mathrm{EHR}}(G,H,q)$ is played on graphs $G$ and $H$. There are two players (Spoiler and Duplicator) and a fixed number of rounds $q$. At the $\nu\mbox{-}$th round ($1\leq\nu\leq q$), Spoiler chooses either a vertex $x_{\nu}$ of $G$ or a vertex $y_{\nu}$ of $H$ (which does not coincide with any of chosen vertices). Duplicator chooses a vertex of the other graph (which does not coincide with any of chosen vertices as well). At the end of the game, the distinct vertices $x_{1},...,x_{q}$ of $G$, $y_{1},...,y_{q}$ of $H$ are chosen. Duplicator wins if and only if the map $f(x_{i})=y_{i}$, $i\in\{1,\ldots,q\}$, is an isomorphism of $G|_{\{x_{1},\ldots,x_{q}\}}$ and $B|_{\{y_{1},\ldots,y_{q}\}}$. 2. In the game ${\mathrm{EHR}}(G,H,q,\leq k)$, there are $q$ rounds as well. The only difference with the game ${\mathrm{EHR}}(G,H,q)$ is that Spoiler can alternate at most $k$ times (if in the $i$-th round Spoiler chooses a vertex, say, in $G$, and in the $i+1$-th round — in $H$ (or vice versa), then we say that he alternates). 3. The most strict rules (for Spoiler) are in the game ${\mathrm{EHR}}(G,H,q,k)$. The only difference with the game ${\mathrm{EHR}}(G,H,q,\leq k)$ is that Spoiler must alternates exactly $k$ times. Our results on first order properties of random graphs are based on the following typical arguments on the connection between an elementary equivalence and Ehrenfeucht game. Lemma 4 The following two properties are equivalent: 1) Spoiler has a winning strategy in ${\mathrm{EHR}}(G,H,q)$; 2) there is $\phi\in\mathcal{F}$ with ${\mathrm{q}}(\phi)=q$ such that $G\models\phi$, $H\models\neg(\phi)$. This statement is a particular case of Ehrenfeucht theorem [17]. The next two lemmas have typical proofs. We give it here for the sake of convenience. Lemma 5 The following two properties are equivalent: 1) Spoiler has a winning strategy in ${\mathrm{EHR}}(G,H,q,\leq k)$; 2) there is $\phi\in\mathcal{F}$ with ${\mathrm{q}}(\phi)=q$ and ${\mathrm{ch}}(\phi)\leq k$ such that $G\models\phi$, $H\models\neg(\phi)$. Proof. First, let us prove that 2) implies 1). Let $\phi\in\mathcal{F}$ be a sentence such that ${\mathrm{ch}}(\phi)\leq k$, ${\mathrm{q}}(\phi)=q$, $G\models\phi$ and $H\models\neg(\phi)$. We will describe a winning strategy of Spoiler by an induction on the number of played rounds. The sentence $\phi$ is a logical combination (disjunctions and conjunctions) of sentences $\varphi_{i}=\exists x\,(\hat{\varphi}_{i}(x))$ and $\varphi^{j}=\forall x\,(\hat{\varphi}^{j}(x))$. Obviously, one of these sentences is true for $G$ and not true for $H$. Let $\beta_{1}$ be the root of the nesting forest (tree) of this sentence. If, say, $$G\models\exists x\,(\hat{\varphi}_{1}(x)),\quad H\models\neg(\exists x\,(\hat{% \varphi}_{1}(x))),$$ then set $\varphi_{1}(x):=\hat{\varphi}_{1}(x)$. Spoiler in the first round chooses a vertex $v_{1}$ such that $G\models\varphi_{1}(v_{1})$. Duplicator chooses a vertex $u_{1}$. Obviously, $H\models\neg(\varphi_{1}(u_{1}))$. Denote the root of $F(\phi)$ by $\beta_{1}$. If, say, $$G\models\forall x\,(\hat{\varphi}^{1}(x)),\quad H\models\neg(\forall x\,(\hat{% \varphi}^{1}(x))),$$ then set $\varphi_{1}(x):=\hat{\varphi}^{1}(x)$. Spoiler in the first round chooses a vertex $u_{1}$ such that $H\models\neg(\varphi_{1}(u_{1}))$. Duplicator chooses a vertex $v_{1}$. Obviously, $G\models\varphi_{1}(v_{1})$. Fix $m\in\{2,\ldots,k\},\ell=\ell(m-1)\in\{1,\ldots,m-1\}$ and vertices $v_{1},\ldots,v_{m-1}$, $u_{1},\ldots,u_{m-1}$ (not necessarily distinct) in the graphs $G,H$ respectively. Suppose that $v_{i_{1}},\ldots,v_{i_{\ell}}$, $u_{i_{1}},\ldots,u_{i_{\ell}}$ are all distinct vertices of $v_{1},\ldots,v_{m-1}$, $u_{1},\ldots,u_{m-1}$ respectively. Moreover, $v_{j}=v_{i_{r}}$ if and only if $u_{j}=u_{i_{r}}$. Suppose that $\ell$ rounds are played, and the vertices $v_{i_{1}},\ldots,v_{i_{\ell}}$, $u_{i_{1}},\ldots,u_{i_{\ell}}$ are chosen in the graphs $G,H$ respectively. Moreover, suppose that in $\phi$ there exists a nested subformula $\varphi_{m-1}(x_{1},\ldots,x_{m-1})$ such that ${\mathrm{q}}(\varphi_{m-1}(x_{1},\ldots,x_{m-1}))=q-m+1$, $$G\models\varphi_{m-1}(v_{1},\ldots,v_{m-1}),\quad H\models\neg(\varphi_{m-1}(u% _{1},\ldots,u_{m-1})).$$ The formula $\varphi_{m-1}(x_{1},\ldots,x_{m-1})$ is a logical combination (disjunctions and conjunctions) of formulas $$\exists x_{m}\,(\hat{\varphi}_{i}(x_{1},\ldots,x_{m})),\quad\forall x_{m}\,(% \hat{\varphi}^{j}(x_{1},\ldots,x_{m})).$$ Obviously, (at least) one of these formulas is true for $G$ on $v_{1},\ldots,v_{m-1}$ and not true for $H$ on $u_{1},\ldots,u_{m-1}$. Let $\beta_{m}$ be the root of the nesting forest of such a formula. If, say, $$G\models\exists x_{m}\,(\hat{\varphi}_{1}(v_{1},\ldots,v_{m-1},x_{m})),\quad H% \models\neg(\exists x_{m}\,(\hat{\varphi}_{1}(v_{1},\ldots,v_{m-1},x_{m}))),$$ then we find a vertex $v_{m}$ such that $G\models\hat{\varphi}_{1}(v_{1},\ldots,v_{m-1},v_{m})$ and set $\varphi_{m}(x_{1},\ldots,x_{m})=\hat{\varphi}_{1}(x_{1},\ldots,x_{m})$. If $v_{m}\in\{v_{1},\ldots,v_{m-1}\}$, then Spoiler “skips” this round, and we set $u_{m}=u_{j}$, where $j\in\{1,\ldots,m-1\}$ is a number such that $v_{j}=v_{m}$. Otherwise, Spoiler chooses a vertex $v_{i_{\ell+1}}=:v_{m}$ and Duplicator chooses a vertex $u_{i_{\ell+1}}=:u_{m}$. Obviously, in both cases, $H\models\neg(\varphi_{m}(u_{1},\ldots,u_{m}))$. If, say, $$G\models\forall x_{m}\,(\hat{\varphi}^{1}(v_{1},\ldots,v_{m-1},x_{m})),\quad H% \models\neg(\forall x_{m}\,(\hat{\varphi}^{1}(v_{1},\ldots,v_{m-1},x_{m}))),$$ then fix a vertex $u_{m}$ such that $H\models\neg(\hat{\varphi}^{1}(u_{1},\ldots,u_{m-1},u_{m}))$ and set $\varphi_{m}(x_{1},\ldots,x_{m})=\hat{\varphi}^{1}(x_{1},\ldots,x_{m})$. If $u_{m}\in\{u_{1},\ldots,u_{m-1}\}$, then Spoiler “skips” this round, and we set $v_{m}=v_{j}$, where $j\in\{1,\ldots,m-1\}$ is a number such that $u_{j}=u_{m}$. Otherwise, Spoiler chooses a vertex $u_{i_{\ell+1}}=:u_{m}$ and Duplicator chooses a vertex $v_{i_{\ell+1}}=:v_{m}$. Obviously, in both cases, $G\models\varphi_{m}(v_{1},\ldots,v_{m})$. This strategy is winning for Spoiler in ${\mathrm{EHR}}(G,H,q)$. Moreover, it is easy to see that Spoiler alternates $\tilde{k}$ times, where $\tilde{k}\leq k$ is the number of labels alternations in the path $\beta_{1}\beta_{\ell(2)}\ldots\beta_{\ell(q)}$. It remains to prove that 1) implies 2). Let Spoiler have a winning strategy in the game ${\mathrm{EHR}}(G,H,k,q)$ with a first move in $G$. Let us construct a sentence $\phi\in\mathcal{F}$ such that ${\mathrm{ch}}(\phi)=k$, ${\mathrm{q}}(\phi)=q$, $G\models\phi$ and $H\models\neg(\phi)$. Let, after $q$ rounds, distinct vertices $v_{1},\ldots,v_{q}$ in $G$ and $u_{1},\ldots,u_{q}$ in $H$ be chosen. As Spoiler wins in $q$ rounds, there is a formula $\varphi_{q}(x_{1},\ldots,x_{q})\in\mathcal{F}$ such that ${\mathrm{q}}(\varphi_{q}(x_{1},\ldots,x_{q}))=0$ and $G\models\varphi_{q}(v_{1},\ldots,v_{q})$, $H\models\neg(\varphi_{q}(u_{1},\ldots,u_{q}))$. Fix $m\in\{0,\ldots,q-1\}$. Let after $m$ rounds, distinct vertices $v_{1},\ldots,v_{m}$ in $G$ and $u_{1},\ldots,u_{m}$ in $H$ be chosen. In the $m+1$-th round, Spoiler chooses, say, a vertex $v_{m+1}\in V(G)$ (according to his winning strategy). Suppose that, for any choice of Duplicator (denote it by $u_{m+1}$), there is a formula $\varphi_{m+1}^{u_{m+1}}(x_{1},\ldots,x_{m+1})\in\mathcal{F}$ such that ${\mathrm{q}}(\varphi_{m+1}^{u_{m+1}}(x_{1},\ldots,x_{m+1}))=q-m-1$ and $$G\models\varphi_{m+1}^{u_{m+1}}(v_{1},\ldots,v_{m+1}),\quad H\models\neg(% \varphi_{m+1}^{u_{m+1}}(u_{1},\ldots,u_{m+1})).$$ Note that, for a fixed number of free variables, there is only a finite number of representatives of $\cong$-equivalence classes of formulas in $\mathcal{F}$ with a fixed quantifier depth (see, e.g., [16]). Therefore, there are a positive constant $C$ (which does not depend on $|V(G)|$, $|V(H)|$) and a set $\mathcal{U}\subset V(H)$ with $|\mathcal{U}|\leq C$ such that the following property holds. For any $u_{m+1}\in V(H)$, there exists $u\in\mathcal{U}$ such that $\varphi_{m+1}^{u}(x_{1},\ldots,x_{m+1})\cong\varphi^{u_{m+1}}_{m+1}(x_{1},% \ldots,x_{m+1})$. Set $$\varphi_{m}(x_{1},\ldots,x_{m})=\exists x_{m+1}\,\left(\bigwedge_{u\in\mathcal% {U}}(\varphi^{u}_{m+1}(x_{1},\ldots,x_{m+1}))\right).$$ Obviously, $G\models\varphi_{m}(v_{1},\ldots,v_{m})$ and $H\models\neg(\varphi_{m}(u_{1},\ldots,u_{m}))$. Finally, let Spoiler choose a vertex $u_{m+1}\in V(H)$ and, for any choice of Duplicator $v_{m+1}\in V(G)$, there exists a formula $\varphi_{m+1}^{v_{m+1}}(x_{1},\ldots,x_{m+1})\in\mathcal{F}$ such that ${\mathrm{q}}(\varphi_{m+1}^{v_{m+1}}(x_{1},\ldots,x_{m+1}))=q-m-1$ and $$G\models\varphi_{m+1}^{v_{m+1}}(v_{1},\ldots,v_{m+1}),\quad H\models\neg(% \varphi_{m+1}^{v_{m+1}}(u_{1},\ldots,u_{m+1})).$$ As in the previous case, there are a positive constant $C$ (which does not depend on $|V(G)|$, $|V(H)|$) and a set $\mathcal{V}\subset V(H)$ with $|\mathcal{V}|\leq C$ such that the following property holds. For any $v_{m+1}\in V(G)$, there exists $v\in\mathcal{V}$ such that $\varphi_{m+1}^{v}(x_{1},\ldots,x_{m+1})\cong\varphi^{v_{m+1}}_{m+1}(x_{1},% \ldots,x_{m+1})$. Set $$\varphi_{m}(x_{1},\ldots,x_{m})=\forall x_{m+1}\,\left(\bigvee_{v\in\mathcal{V% }}(\varphi^{v}_{m+1}(x_{1},\ldots,x_{m+1}))\right).$$ By the induction, we get that $\phi=\phi_{0}$ is the required sentence which is true for $G$ and false for $H$. Obviously, ${\mathrm{ch}}(\phi)\leq k$. $\Box$. Lemma 6 The following two properties are equivalent: 1) Spoiler has a winning strategy in ${\mathrm{EHR}}(G,H,q,k)$; 2) there is $\phi\in\mathcal{F}$ with ${\mathrm{q}}(\phi)=q$ such that a number of labels alternations in any path of $F(\phi)$ on $q$ vertices starting in a root equals $k$, and $G\models\phi$, $H\models\neg(\phi)$. Proof. First, let us prove that 2) implies 1). The winning strategy of Spoiler is absolutely the same as in the proof of Lemma 5. The only thing we should prove is that Spoiler alternates exactly $k$ times. If $\ell(q)<q$, then consider a path $\beta_{1}\ldots\beta_{\ell(q)}\beta_{\ell(q)+1}\ldots\beta_{q}$ in $F(\phi)$. The number of labels alternations in this path equals $k$. Therefore, $k-\tilde{k}\leq q-\ell(q)$. So, Spoiler can choose graphs (and an arbitrary vertex) in each of the remaining rounds in a way such that he will alternate $k$ times overall. If $\ell(q)=q$, then, obviously, $\tilde{k}=k$. It remains to prove that 1) implies 2). The formula $\phi$ is constructed in the same way as in the proof of Lemma 5. We only need to prove that ${\mathrm{ch}}(\phi)=k$. Consider an arbitrary path $\beta_{1}\ldots\beta_{q}$ in $F(\phi)$ starting in a root. Note that $\beta_{i}$ is labeled by $\exists$ if and only if there exists a Duplicator’s strategy such that in the $i$-th round Spoiler chooses $G$. Therefore, the number of labels alternations in this path equals $k$. $\Box$ 4 Spectra of formulas with small numbers of alternations Let us start this section with the following simple observation. Lemma 7 If $\phi\in\mathcal{F}$ and $\alpha\in S(\phi)$, then there exists an ${\mathrm{NE}}{\rm{PNF}}$ sentence $\hat{\phi}$ such that ${\mathrm{ch}}(\phi)={\mathrm{ch}}(\hat{\phi})$ and $\alpha\in S(\hat{\phi})$ as well. Proof. By Lemma 3, it is enough to prove that there exists a sentence $\hat{\phi}=\exists x\,(\varphi(x))\in\mathcal{F}$ such that $\alpha$ belongs to its spectrum. As $\alpha\in S(\phi)$, there exist $\varepsilon>0$ and sequences $n_{i},m_{i}$ such that, for any $i\in\mathbb{N}$, $$\min\left\{{\sf P}\left(G(n_{i},n_{i}^{-\alpha})\models\phi\right),{\sf P}% \left(G(m_{i},m_{i}^{-\alpha})\models\neg(\phi)\right)\right\}>\varepsilon.$$ Fix $i\in\mathbb{N}$. Let $G,H$ be graphs on $n_{i},m_{i}$ vertices respectively such that $G\models\phi$, $H\models\neg(\phi)$. The formula $\phi$ is a logical combination (disjunctions and conjunctions) of formulas $$\exists x\,(\varphi_{j}(x)),\quad\forall x\,(\varphi^{j}(x)).$$ Let $N$ be the number of all formulas in this combination. Obviously, there exists either $j$ such that $G\models\exists x\,(\varphi_{j}(x))$, $H\models\neg(\exists x\,(\varphi_{j}(x)))$ or $j$ such that $G\models\forall x\,(\varphi^{j}(x))$, $H\models\neg(\forall x\,(\varphi^{j}(x)))$. Therefore, there exists $\varphi(x)=\varphi(x,i)\in\mathcal{F}$ such that ${\mathrm{ch}}(\exists x\,(\varphi(x)))\leq{\mathrm{ch}}(\phi)$, ${\mathrm{q}}(\exists x\,(\varphi(x)))\leq{\mathrm{q}}(\phi)$ and $$\min\left\{{\sf P}\left(G(n_{i},n_{i}^{-\alpha})\models\exists x\,(\varphi(x))% \right),{\sf P}\left(G(m_{i},m_{i}^{-\alpha})\models\neg(\exists x\,(\varphi(x% )))\right)\right\}>\varepsilon/N.$$ Set $\hat{\phi}_{i}=\exists x\,(\varphi(x,i))$. As there is only a finite number of representatives of $\cong$-equivalence classes of sentences in $\mathcal{F}$ with a fixed quantifier depth (see, e.g., [16]), there is only a finite number of representatives of $\cong$-equivalence classes in $\{\hat{\phi}_{i},i\in\mathbb{N}\}$ as well. Therefore, there exists a sentence $\hat{\phi}=\exists x\,(\varphi(x))$ and a sequence $i_{j}$ such that, for all $j\in\mathbb{N}$, $$\min\left\{{\sf P}\left(G(n_{i_{j}},n_{i_{j}}^{-\alpha})\models\hat{\phi}% \right),{\sf P}\left(G(m_{i_{j}},m_{i_{j}}^{-\alpha})\models\neg(\hat{\phi})% \right)\right\}>\varepsilon/N.$$ So, $\alpha\in S(\hat{\phi})$. $\Box$ Below, we state the main result of this section, which implies the following answer on Q2: the minimal number of quantifier alternations of a first order sentence with an infinite spectrum equals $3$. Theorem 4 The minimal $k$ such that there exists $\phi\in\mathcal{F}$ with infinite $S(\phi)$ and ${\mathrm{ch}}(\phi)=k$ equals $3$. Proof. By Lemma 7 and Theorem 1, it is enough to prove that, for any $k\in\{0,1,2\}$ and any NEPNF sentence $\phi=\exists x\,(\varphi(x))\in\mathcal{F}$ with ${\mathrm{ch}}(\phi)=k$, the set $S(\phi)$ is finite. Note that $S(\phi)=S(\neg(\phi))$. Therefore, equivalently, we may prove that spectra of sentences $\forall x\,(\varphi(x))$ are finite. Obviously, $k\in\{0,1\}$ are subcases of $k=2$. However, below we consider $k=0$, $k=1$ alone for the sake of convenience. Let $\phi_{H}\in\mathcal{F}$ be an existence sentence which expresses the property of containing ad induced subgraph isomorphic to $H$. 4.1 No alternations Let ${\mathrm{ch}}(\phi)=0$, where $\phi=\exists x\,(\varphi(x))\in\mathcal{F}$ is an NEPNF sentence. Obviously, there exists a finite set $\mathcal{G}$ of graphs such that $G\models\phi$ if and only if in $G$ there is an induced subgraph which is isomorphic to some $H\in\mathcal{G}$. We get $$\phi\cong\bigvee_{H\in\mathcal{G}}(\phi_{H}).$$ By Theorem 2, either $\rho:=\min_{H\in\mathcal{G}}\{\rho(H)\}>0$ and $S(\phi)\subset\{1/\rho\}$, or $\rho=0$ and $S(\phi)=\varnothing$. 4.2 One alternation Let ${\mathrm{ch}}(\phi)=1$, where $$\phi=\forall y_{1}\ldots\forall y_{s}\exists x_{1}\ldots\exists x_{m}\,(% \varphi(y_{1},\ldots,y_{s},x_{1},\ldots,x_{m}))\in\mathcal{F}$$ has the quantifier depth $s+m$. Obviously, there exists a finite set $\mathcal{G}$ of graphs on a set of vertices $\{a_{1},\ldots,a_{s}\}$ and, for each $A\in\mathcal{G}$, there exists a finite set $\mathcal{H}(A)$ of graphs on a set of vertices $\{a_{1},\ldots,a_{s},b_{1},\ldots,b_{m}\}$ such that • for any $A\in\mathcal{G}$ and $B\in\mathcal{H}(A)$, $A=B|_{\{a_{1},\ldots,a_{s}\}}$, • $G\models\phi$ if and only if, for any distinct vertices $y_{1},\ldots,y_{s}\in V(G)$, there exist distinct vertices $x_{1},\ldots,x_{m}\in V(G)$ ($x_{j}\neq y_{i}$) and graphs $A\in\mathcal{G}$, $B\in\mathcal{H}(A)$ such that the map $f:B\rightarrow G|_{\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{m}\}}$, $f(a_{i})=y_{i}$, $f(b_{j})=x_{j}$, is an isomorphism. Let all graphs $A_{1},\ldots,A_{M}$ of $\mathcal{G}$ be ordered in a way such that $$\rho_{1}:=\rho(A_{1})\geq\ldots\geq\rho(A_{M})=:\rho_{M}.$$ (2) For each $i\in\{1,\ldots,M\}$, let $\rho^{i}=\min\{\rho(B,A_{i}),B\in\mathcal{H}(A_{i})\}.$ Suppose that $1/\alpha$ is not equal to any of $\rho_{i},\rho^{i}$, $i\in\{1,\ldots,M\}$. If there is a graph on the set of vertices $\{a_{1},\ldots,a_{s}\}$ which does not belong to $\mathcal{G}$ such that its maximal density is less than $1/\alpha$, then, by Theorem 2, $G(n,p)\models\neg(\phi)$ (a.a.s.). Suppose that the above property does not hold. This implies that $\rho_{M}=0$. Set $\rho_{0}=\infty$, $1/\rho_{0}=0$ and $1/\rho_{M}=\infty$. Let $i_{0}\in\{0,1,\ldots,M-1\}$ be chosen in the following way: $1/\rho_{i_{0}}<\alpha<1/\rho_{i_{0}+1}$. If for some $i\in\{i_{0}+1,\ldots,M\}$ the inequality $\rho^{i}>1/\alpha$ holds, then, by Theorem 2, $G(n,p)\models\neg(\phi)$ (a.a.s.). Otherwise, $G(n,p)\models\phi$ (a.a.s.). Thus, $S(\phi)\subseteq\{1/\rho_{1},\ldots,1/\rho_{M},1/\rho^{1},\ldots,1/\rho^{M}\}$, and so $|S(\phi)|<\infty$. 4.3 Two alternations In this case, it is not enough to define sets of graphs as above. We divide the proof into four parts. Only the first part “Transition to sets of graphs” is similar to the previous cases. 4.3.1 Transition to sets of graphs Let a sentence $$\phi=\forall y_{1}\ldots\forall y_{s}\exists x_{1}\ldots\exists x_{m}\forall w% _{1}\ldots\forall w_{r}\,(\varphi(y_{1},\ldots,y_{s},x_{1},\ldots,x_{m},w_{1},% \ldots,w_{r}))\in\mathcal{F}$$ has the quantifier depth $s+m+r$. Obviously, there exists a set of vertices $\Sigma=\Sigma_{a}\sqcup\Sigma_{b}\sqcup\Sigma_{c}$, where $\Sigma_{a}=\{a_{1},\ldots,a_{s}\}$, $\Sigma_{b}=\{b_{1},\ldots,b_{m}\}$, $\Sigma_{c}=\{c_{1},\ldots,c_{r}\}$, and — a finite set of graphs $\mathcal{G}$ on the set of vertices $\Sigma_{a}$, — for each $A\in\mathcal{G}$, a finite set $\mathcal{H}(A)$ on the set of vertices $\Sigma_{a}\sqcup\Sigma_{b}$, — for each $A\in\mathcal{G}$ and $B\in\mathcal{H}(A)$, a finite set of graphs $\mathcal{K}(B)$ on the set of vertices $\Sigma$, such that the following properties hold. • For any $A\in\mathcal{G}$, $B\in\mathcal{H}(A)$, $C\in\mathcal{K}(B)$, we have $A=B|_{\Sigma_{a}}$, $B=C|_{\Sigma_{a}\sqcup\Sigma_{b}}$. • $G\models\phi$ if and only if for any pairwise distinct $y_{1},\ldots,y_{s}$ from $V(G)$ there exist pairwise distinct $x_{1},\ldots,x_{m}$ from $V(G)\setminus\{y_{1},\ldots,y_{s}\}$ such that for any pairwise distinct $w_{1},\ldots,w_{r}$ from $V(G)\setminus\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{m}\}$ the graph $G$ has the property $P(y_{1},\ldots,y_{s},x_{1},\ldots,x_{m},w_{1},\ldots,w_{r})$ (which is defined below). Let us say that $G$ has the property $P(y_{1},\ldots,y_{s},x_{1},\ldots,x_{m},w_{1},\ldots,w_{r})$, if there exist graphs $A\in\mathcal{G}$, $B\in\mathcal{H}(A)$, $C\in\mathcal{K}(B)$ such that the map $f:C\rightarrow G|_{\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{m},w_{1},\ldots,w_{r}\}}$ which preserves the orders of the vertices ($f(a_{i})=y_{i}$, $f(b_{j})=x_{j}$, $f(c_{h})=w_{h}$) is an isomorphism. Theorem 3 from [7] implies that $\alpha\notin S(\phi)$ for any $\alpha<\frac{1}{s+m+r-2}$. Therefore, for any positive integer $N$, the set of numbers from $S(\phi)$ with a numerator at most $N$ is finite. So, we may assume that the numerator of $\alpha$ is large enough. As in the case of one alternation, we assume that any graph on the set of vertices $\Sigma_{a}$ with a maximal density less than $1/\alpha$ belongs to $\mathcal{G}$. 4.3.2 Dense neighbourhood and its structure Let $\Gamma$ be an arbitrary graph on a set of vertices $V$ with the following property. There is $A\in\mathcal{G}$ and pairwise distinct vertices $y_{1},\ldots,y_{s}\in V$ such that the map $A\rightarrow\Gamma|_{\{y_{1},\ldots,y_{s}\}}$ which preserves the orders of the vertices is an isomorphism. Let $Y_{0}=\Gamma|_{\{y_{1},\ldots,y_{s}\}}$. For each $i\geq 0$, let us construct an induced subgraph $Y_{i+1}$ of $\Gamma$ on the union of $V(Y_{i})$ with some additional vertices (for a step $\tilde{i}$, this process halts, set $Y=Y_{\tilde{i}}$). For a step $i$ the process does not halt, if there exists a subgraph $W\subset\Gamma$ such that $W\supset Y_{i}$, $v(W)-v(Y_{i})\leq r$ and $\rho(W,Y_{i})>1/\alpha$. For such a graph $W$, set $Y_{i+1}=W$. The graph $Y=Y(\Gamma;y_{1},\ldots,y_{s})$ is constructed. Before proceeding with the next part of the proof, let us study a structure of $Y$ and introduce some notations for describing this structure. • Let $\mathcal{U}=\mathcal{U}(A)=\{B_{1},\ldots,B_{\beta}\}$ be the set of all graphs $B$ on the set of vertices $\Sigma_{a}\cup\Sigma_{b}$ such that $B|_{\Sigma_{a}}=A$. Obviously, $\beta=2^{C_{m}^{2}+sm}$. • Let $x^{0}_{1},\ldots,x^{0}_{m}$ be arbitrary vertices which are not in $V(Y)$ (and even not necessarily in $V$). • Let $\ell\in\{1,\ldots,\beta\}$, $\tilde{m}\in\{0,\ldots,m\}$. Consider the set $\mathcal{X}_{\ell,\tilde{m}}$ of all collections of vertices $x_{1},\ldots,x_{\tilde{m}}\in V(Y)\setminus\{y_{1},\ldots,y_{s}\}$ such that there exists a graph $W$ on the set of vertices $V(Y)\cup\{x^{0}_{\tilde{m}+1},\ldots,x^{0}_{m}\}$ and an isomorphism $f:B_{\ell}\to W|_{\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{\tilde{m}},x^{0}_{% \tilde{m}+1},\ldots,x^{0}_{m}\}}$ which preserves the orders of the vertices ($f(a_{i})=y_{i}$, $f(b_{j})\in\{x_{j},x^{0}_{j}\}$). • For each $\ell\in\{1,\ldots,\beta\}$, $\tilde{m}\in\{0,\ldots,m\}$, $(x_{1},\ldots,x_{\tilde{m}})\in\mathcal{X}_{\ell,\tilde{m}}$, consider the set $\mathcal{S}_{\ell}(x_{1},\ldots,x_{\tilde{m}})$ of all graphs $W$ on the set of vertices $V(Y)\cup\{x^{0}_{\tilde{m}+1},\ldots,x^{0}_{m}\}$ such that $W|_{V(Y)}=Y$, $\rho(W,Y)<1/\alpha$ and the map $f:B_{\ell}\to W|_{\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{\tilde{m}},x^{0}_{% \tilde{m}+1},\ldots,x^{0}_{m}\}}$ which preserves the orders of the vertices ($f(a_{i})=y_{i}$, $f(b_{j})\in\{x_{j},x^{0}_{j}\}$) is an isomorphism. Moreover, for each $W\in\mathcal{S}_{\ell}(x_{1},\ldots,x_{\tilde{m}})$ consider the set $\mathcal{N}_{\ell}(W;x_{1},\ldots,x_{\tilde{m}})$ of all graphs $C$ on the sets of vertices $\Sigma_{a}\cup\Sigma_{b}\cup\{c_{1},\ldots,c_{\tilde{r}}\}$, where $\tilde{r}\leq r$, such that there exists a subgraph $Z\subset W$ containing the vertices $y_{1},\ldots,y_{s}$, $x_{1},\ldots,x_{\tilde{m}}$, $x^{0}_{\tilde{m}+1},\ldots,x^{0}_{m}$, and the following two properties hold. First, there exist vertices $w_{1},\ldots,w_{\tilde{r}}\in V(Y)$ and an isomorphism $f:C\rightarrow Z$ which preserves the orders of the vertices ($f(a_{i})=y_{i}$, $f(b_{j})\in\{x_{j},x^{0}_{j}\}$, $f(c_{h})=w_{h}$). Second, $$\rho\left(Z,Z|_{\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{\tilde{m}},x^{0}_{\tilde{% m}+1},\ldots,x^{0}_{m}\}}\right)>1/\alpha.$$ • For each $\ell\in\{1,\ldots,\beta\}$, denote by $(\mathcal{N})_{\ell}[Y;y_{1},\ldots,y_{s}]$ a maximal set of pairwise distinct sets among $\mathcal{N}_{\ell}(W;x_{1},\ldots,x_{\tilde{m}})$, $W\in\mathcal{S}_{\ell}$. The vector $\mathbf{N}=((\mathcal{N})_{1}[Y;y_{1},\ldots,y_{s}],\ldots,(\mathcal{N})_{% \beta}[Y;y_{1},\ldots,y_{s}])$ defines the structure of $Y$. 4.3.3 Existence of a bounded graph with the same structure Let $\{y_{1},\ldots,y_{s}\}$ be an arbitrary set of vertices, and $A\in\mathcal{G}$. Consider an arbitrary graph $\Gamma$ which contains the vertices $y_{1},\ldots,y_{s}$ such that the map $A\rightarrow\Gamma|_{\{y_{1},\ldots,y_{s}\}}$ (preserving the orders of the vertices) is an isomorphism. Let $\ell\in\{1,\ldots,\beta\}$ (where $\beta$ is the cardinality of $\mathcal{U}(A)=\{B_{1},\ldots,B_{\beta}\}$). Determine the vector $(\mathcal{N})_{\ell}:=(\mathcal{N})_{\ell}[Y(\Gamma;y_{1},\ldots,y_{s});y_{1},% \ldots,y_{s}]$. Let $\mathbf{Y}=\mathbf{Y}(\Gamma;y_{1},\ldots,y_{s})$ be the set of all graphs $Y$ such that $Y|_{\{y_{1},\ldots,y_{s}\}}=\Gamma|_{\{y_{1},\ldots,y_{s}\}}$, and $(\mathcal{N})_{\ell}=(\mathcal{N})_{\ell}[Y;y_{1},\ldots,y_{s}]$ for all $\ell\in\{1,\ldots,\beta\}$. Let the graph $Y_{\min}(\mathbf{Y};y_{1},\ldots,y_{s})$ has a minimal number of vertices among the graphs in the set $$\{Y\in\mathbf{Y}:\quad\forall\tilde{Y}\in\mathbf{Y}\,(\rho(\tilde{Y})\geq\rho(% Y))\}$$ (and, of course, belongs to this set). Note that the set $\mathbf{Y}(\Gamma;y_{1},\ldots,y_{s})$ is defined by the vector $\mathbf{N}=((\mathcal{N})_{1},\ldots,(\mathcal{N})_{\beta})$ only. Therefore, for the vertices $y_{1},\ldots,y_{s}$ there exist only finite set of pairwise distinct sets $\mathbf{Y}(\cdot;y_{1},\ldots,y_{s})$. So, the set of pairwise distinct graphs $Y_{\min}(\cdot;y_{1},\ldots,y_{s})$ is finite. Let $$Y_{\min}^{1}(y_{1},\ldots,y_{s}),\ldots,Y_{\min}^{\theta}(y_{1},\ldots,y_{s})$$ be all such graphs. 4.3.4 Finiteness of the spectrum Recall that the numerator of the irreducible fraction $\alpha=\frac{R}{P}$ is large enough (see Section 4.3.1). So, we assume that $R>\max\{s+m+r,N\}$, where $$N:=\max\left\{v(Y_{\min}^{1}(y_{1},\ldots,y_{s})),\ldots,v(Y_{\min}^{\theta}(y% _{1},\ldots,y_{s}))\right\}.$$ Note that $N$ does not depend on a choice of $y_{1},\ldots,y_{s}$. Theorems 2, 3 imply that a.a.s. the random graph $G(n,n^{-\alpha})$ has the following properties: G1 for any $H$ with $\rho(H)>1/\alpha$ and $v(H)\leq s+r(sP+1)$, there is no subgraph isomorphic to $H$; G2 for any $H\subset G$ with $v(G)\leq s+m+r$ and $\rho(G,H)<1/\alpha$, there is the $(G,H)$-extension property; G3 for any $H\subset G$ with $v(G)\leq\max\{N,m+s+rsP\}$, $\rho(G,H)<1/\alpha$ and set $\mathcal{W}$ of graphs $W$ on a fixed set of vertices such that – $G\subset W$, $1\leq v(W)-v(G)\leq r+m$, $\rho(W,G)>1/\alpha$, – $W\setminus G$ is connected, – there are edges between $W\setminus G$ and $G$ in $W$, there is the $(\mathcal{W},G,H)$-double-extension property. Let us prove that if the graphs $\Gamma,\Upsilon$ have the properties G1, G2 and G3, then either $\phi$ is true for both of them, or $\phi$ is false for both of them. This would imply that $\alpha\notin S(\phi)$. Assume that $\Gamma\models\neg(\phi)$, $\Upsilon\models\phi$. By the property G1, a maximal density of any subgraph of $\Gamma$ on $s$ vertices is less than $1/\alpha$. All graphs on the set of vertices $\Sigma_{a}$ with such a maximal density are in $\mathcal{G}$ (see Section 4.3.1). Therefore, there exist $A\in\mathcal{G}$ and pairwise distinct $y_{1},\ldots,y_{s}\in V(\Gamma)$ such that the map $A\rightarrow\Gamma|_{\{y_{1},\ldots,y_{s}\}}$ which preserves the orders of the vertices is an isomorphism, and $\Gamma$ with distinguished vertices $y_{1},\ldots,y_{s}$ does not have the property (EXT), which is defined below. (EXT): there exist pairwise distinct $x_{1},\ldots,x_{m}\in V(\Gamma)\setminus\{y_{1},\ldots,y_{s}\}$ such that for any pairwise distinct $w_{1},\ldots,w_{r}\in V(\Gamma)\setminus\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{m}\}$ there exist graphs $B\in\mathcal{H}(A)$, $C\in\mathcal{K}(B)$ and an isomorphism $f:C\rightarrow\Gamma|_{\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{m},w_{1},\ldots,w_% {r}\}}$ which preserves the orders of the vertices ($f(a_{i})=y_{i}$, $f(b_{j})=x_{j}$, $f(c_{h})=w_{h}$). Construct the graph $Y=Y(\Gamma;y_{1},\ldots,y_{s})$ as it is done in Section 4.3.2. Let us prove that $v(Y)\leq s+rsP$. Assume that the opposite inequality is true. By the definition of $Y$, there is a subgraph $X\subset Y$ on at most $s+r(sP+1)$ vertices such that, for some $v_{1},\ldots,v_{sP+1}\in\{1,\ldots,r\}$, $$\rho(X)\geq\frac{(1/\alpha)v_{1}+\ldots+(1/\alpha)v_{sP+1}+\frac{sP+1}{R}}{s+v% _{1}+\ldots+v_{sP+1}}=\frac{1}{\alpha}+\frac{1}{R(s+v_{1}+\ldots+v_{sP+1})}>% \frac{1}{\alpha}.$$ This contradicts the property G1. So, $v(Y)\leq s+rsP$, and, therefore, $\rho(Y)\leq 1/\alpha$. Consider the graph $Y_{\min}=Y_{\min}(\mathbf{Y}(\Gamma;y_{1},\ldots,y_{s});y_{1},\ldots,y_{s})$. We have $\rho(Y_{\min})\leq\rho(Y)\leq 1/\alpha$. As $v(Y_{\min})\leq N<R$, the equality $\rho(Y_{\min})=1/\alpha$ is impossible, and so $\rho(Y_{\min})<1/\alpha$. By the property G3, in $\Upsilon$ there is an induced subgraph $Y^{\Upsilon}\cong Y_{\min}$ such that in $\Upsilon$ there is no subgraph $W\supset Y^{\Upsilon}$ with $v(W)-v(Y^{\Upsilon})\leq r+m$ and $\rho(W,Y^{\Upsilon})>1/\alpha$. Let $f:Y_{\min}\rightarrow Y^{\Upsilon}$ be an isomorphism. Set $f(y_{i})=y^{\Upsilon}_{i}$, $i\in\{1,\ldots,s\}$. As $\Upsilon\models\phi$, $\Upsilon$ with distinguished vertices $y^{\Upsilon}_{1},\ldots,y^{\Upsilon}_{s}$ has the property (EXT). Let $x_{1}^{\Upsilon},\ldots,x_{m}^{\Upsilon}\in V(\Upsilon)\setminus\{y^{\Upsilon}% _{1},\ldots,y^{\Upsilon}_{s}\}$ and $B\in\mathcal{H}(A)$ be such that for any pairwise distinct $w^{\Upsilon}_{1},\ldots,w^{\Upsilon}_{r}\in V(\Upsilon)\setminus\{y^{\Upsilon}% _{1},\ldots,y^{\Upsilon}_{s},x^{\Upsilon}_{1},\ldots,x^{\Upsilon}_{m}\}$ there exist a graph $C\in\mathcal{K}(B)$ and an isomorphism $g:C\rightarrow\Upsilon|_{\{y^{\Upsilon}_{1},\ldots,y^{\Upsilon}_{s},x^{% \Upsilon}_{1},\ldots,x^{\Upsilon}_{m},w^{\Upsilon}_{1},\ldots,w^{\Upsilon}_{r}\}}$ which preserves the orders of the vertices ($g(a_{i})=y^{\Upsilon}_{i}$, $g(b_{j})=x^{\Upsilon}_{j}$, $g(c_{h})=w^{\Upsilon}_{h}$). From the property G1 it follows that $$\rho\left(\Upsilon|_{V(Y^{\Upsilon})\cup\{x_{1}^{\Upsilon},\ldots,x_{m}^{% \Upsilon}\}},Y^{\Upsilon}\right)<1/\alpha$$ if at least one of the vertices $x_{1}^{\Upsilon},\ldots,x_{m}^{\Upsilon}$ is not in $Y^{\Upsilon}$. Indeed, there is no equality, because $v\left(\Upsilon|_{V(Y^{\Upsilon})\cup\{x_{1}^{\Upsilon},\ldots,x_{m}^{\Upsilon% }\}}\right)-v\left(Y^{\Upsilon}\right)\leq m$. Let $x_{1},\ldots,x_{\tilde{m}}\in Y^{\Upsilon}$, $x_{\tilde{m}+1},\ldots,x_{m}\in V(\Upsilon)\setminus V(Y^{\Upsilon})$, where $\tilde{m}\in\{0,1,\ldots,m\}$. From the property G3, the definitions of $Y$ and $Y_{\min}$ it follows that there exist vertices $x_{1},\ldots,x_{\tilde{m}}\in V(Y)$, $x_{\tilde{m}+1},\ldots,x_{m}\in V(\Gamma)\setminus V(Y)$ such that the following properties hold. Q1 There exists an isomorphism $f:B\rightarrow\Gamma|_{\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{m}\}}$ which preserves the orders of the vertices ($f(a_{i})=y_{i}$, $f(b_{j})=x_{j}$). Q2 There is no $W\subset\Gamma$ such that $W\supset\Gamma|_{V(Y)\cup\{x_{1},\ldots,x_{m}\}}$, $$v(W)-v\left(\Gamma|_{V(Y)\cup\{x_{1},\ldots,x_{m}\}}\right)\leq r\quad\text{% and}\quad\rho\left(W,\Gamma|_{V(Y)\cup\{x_{1},\ldots,x_{m}\}}\right)>1/\alpha.$$ Q3 Let $C$ be a graph on a set of vertices $\{a_{1},\ldots,a_{s},b_{1},\ldots,b_{m},c_{1},\ldots,c_{\tilde{r}}\}$ (where $\tilde{r}\leq r$). Let $Z\subseteq\Gamma$ be a graph consisting of the vertices $y_{1},\ldots,y_{s},x_{1},\ldots,x_{m}$ and some vertices $w_{1},\ldots,w_{\tilde{r}}\in V(Y)$. Moreover, let the map $f:C\rightarrow Z$ which preserves the orders of the vertices ($f(a_{i})=y_{i}$, $f(b_{j})=x_{j}$, $f(c_{h})=w_{h}$) be an isomorphism, and $$\rho\left(Z,Z|_{\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{m}\}}\right)>1/\alpha.$$ Then, in $\Upsilon$ there is a subgraph $Z^{\Upsilon}$ consisting of the vertices $y^{\Upsilon}_{1},\ldots,y^{\Upsilon}_{s}$, $x^{\Upsilon}_{1},\ldots,x^{\Upsilon}_{m}$ and some vertices $w^{\Upsilon}_{1},\ldots,w^{\Upsilon}_{\tilde{r}}\in V(Y^{\Upsilon})$ such that the map $f:C\rightarrow Z^{\Upsilon}$ which preserves the orders of the vertices ($f(a_{i})=y^{\Upsilon}_{i}$, $f(b_{j})=x^{\Upsilon}_{j}$, $f(c_{h})=w^{\Upsilon}_{h}$) is an isomorphism. By our assumption, there exist $w_{1},\ldots,w_{r}\in V(\Gamma)\setminus\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{m}\}$ such that for any $C\in\mathcal{K}(B)$ the map $f:C\rightarrow\Gamma|_{\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{m},w_{1},\ldots,w_% {r}\}}$ which preserves the orders of the vertices ($f(a_{i})=y_{i}$, $f(b_{j})=x_{j}$, $f(c_{h})=w_{h}$) is not an isomorphism. If $$\rho\left(\Gamma|_{\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{m},w_{1},\ldots,w_{r}% \}},\Gamma|_{\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{m}\}}\right)<1/\alpha,$$ (3) then by the property G2 in $\Upsilon$ there are vertices $w^{\Upsilon}_{1},\ldots,w^{\Upsilon}_{r}$ such the the map $$f:\Gamma|_{\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{m},w_{1},\ldots,w_{r}\}}\to% \Upsilon|_{\{y_{1}^{\Upsilon},\ldots,y_{s}^{\Upsilon},x_{1}^{\Upsilon},\ldots,% x_{m}^{\Upsilon},w_{1}^{\Upsilon},\ldots,w_{r}^{\Upsilon}\}}$$ (4) which preserves the orders of the vertices ($f(y_{i})=y_{i}^{\Upsilon}$, $f(x_{j})=x_{j}^{\Upsilon}$, $f(w_{h})=w_{h}^{\Upsilon}$) is an isomorphism — a contradiction. If $w_{1},\ldots,w_{r}\in V(\Gamma)\setminus V(Y)$, then Inequality (3) holds (there is no equality, because $v\left(\Gamma|_{\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{m},w_{1},\ldots,w_{r}\}}% \right)-v\left(\Gamma|_{\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{m}\}}\right)=r<R$). If $w_{1},\ldots,w_{r}\in V(Y)$ and $$\rho\left(\Gamma|_{\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{m},w_{1},\ldots,w_{r}% \}},\Gamma|_{\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{m}\}}\right)>1/\alpha,$$ then, the definition of $Y^{\Upsilon}$ implies the existence of vertices $w^{\Upsilon}_{1},\ldots,w^{\Upsilon}_{r}$ such that the map (4) which preserves the orders of the vertices is an isomorphism — a contradiction. Finally, let some (not all) of the vertices $w_{1},\ldots,w_{r}$ be in $V(Y)$ (say, $w_{1}\ldots,w_{\tilde{r}}\in V(Y)$, $w_{\tilde{r}+1},\ldots,w_{r}\in V(\Gamma)\setminus V(Y)$). In $Y^{\Upsilon}$ there exist vertices $w^{\Upsilon}_{1},\ldots,w^{\Upsilon}_{\tilde{r}}$ such that the map $f:\Gamma|_{\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{m},w_{1},\ldots,w_{\tilde{r}}% \}}\to\Upsilon|_{\{y_{1}^{\Upsilon},\ldots,y_{s}^{\Upsilon},x_{1}^{\Upsilon},% \ldots,x_{m}^{\Upsilon},w_{1}^{\Upsilon},\ldots,w_{\tilde{r}}^{\Upsilon}\}}$ which preserves the orders of the vertices is an isomorphism. Moreover, $$\rho\left(\Gamma|_{\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{m},w_{1},\ldots,w_{r}% \}},\Gamma|_{\{y_{1},\ldots,y_{s},x_{1},\ldots,x_{m},w_{1},\ldots,w_{\tilde{r}% }\}}\right)<1/\alpha.$$ By the property G2, in $\Upsilon$ there exist vertices $w^{\Upsilon}_{\tilde{r}+1},\ldots,w^{\Upsilon}_{r}$ such that the map (4) which preserves the orders of the vertices is an isomorphism — a contradiction. $\Box$ 5 Spectra of formulas with small quantifier depths Theorem 4 answers the second question of Section 1. We do not have an answer on the third question. However, in our second main result, we get a new lower bound on the minimal quantifier depth of PNF sentence with an infinite spectrum. Theorem 5 The minimal $q$ such that there exists a ${\rm{PNF}}$ sentence $\phi\in\mathcal{F}$ with infinite $S(\phi)$ and ${\mathrm{q}}(\phi)=q$ is at least 5. The proof is based on the statement on Ehrenfeucht game which is given below. For a positive integer $k$, consider a set $\tilde{S}(k)$ of $\alpha>0$ such that there exist $\varepsilon>0$ and increasing sequences $n_{i},m_{i}$ of positive integers such that, for any $i\in\mathbb{N}$, $${\sf P}\left(\text{Spoiler has a winning strategy in EHR}\left(G(n_{i},n_{i}^{% -\alpha}),G(m_{i},m_{i}^{-\alpha}),k,k-1\right)\right)>\varepsilon^{2}.$$ Lemma 8 The set $\tilde{S}(4)\cap(1/2,1)$ is finite. Proof. Case 1. Let $p=n^{-\alpha}$, $\alpha\in(1/2,10/19)$. Let $x_{1},x_{2},x_{3}$ be vertices of an arbitrary graph $G$. For any $i,j\in\{\{0\},\mathbb{N}\}$, we say that $(x_{1},x_{2},x_{3})$ has the type $(i,j)$, if a number of common neighbors of $x_{1},x_{3}$ (which are not adjacent to $x_{2}$) is in $i$, and a number of common neighbors of $x_{2},x_{3}$ (which are not adjacent to $x_{1}$) is in $j$. Introduce a linear order $\leq$ on the set $\mathcal{I}$ of all pairs of elements from $\{\{0\},\mathbb{N}\}$ in the following way: $(\{0\},\{0\})\leq(\{0\},\mathbb{N})\leq(\mathbb{N},\{0\})\leq(\mathbb{N},% \mathbb{N})$. For any vertices $x_{1},x_{2}$, denote by $n(x_{1},x_{2})$ the number of all pairs of adjacent common neighbors of $x_{1},x_{2}$. Denote the set of all common neighbors of $x_{1},x_{2}$ by $N(x_{1},x_{2})$. Denote the set of all common neighbors $x_{3}$ of $x_{1},x_{2}$ such that $x_{1},x_{2},x_{3}$ have no common neighbors by $U(x_{1},x_{2})$. We say that a graph has the triangle property, if, for any $s\in\{0,1,2\}$, any vertex $x_{1}$, any $x,y\in\mathcal{I}$ and any $\delta\in\{\sim,\nsim\}$, there is a vertex $x_{2}$ in the graph such that • $x_{1}\delta x_{2}$, • $n(x_{1},x_{2})\leq 1$, • there is no $K_{4}$ containing $x_{1},x_{2}$, • if $n(x_{1},x_{2})=1$, then $|U(x_{1},x_{2})|=\min\{s,1\}$, • if $n(x_{1},x_{2})=0$, then $|U(x_{1},x_{2})|=s$, • for any $x_{3}\in U(x_{1},x_{2})$, $(x_{1},x_{2},x_{3})$ has the type $x$, • for any $x_{3}\in N(x_{1},x_{2})\setminus U(x_{1},x_{2})$, $(x_{1},x_{2},x_{3})$ has the type $y$. By Theorem 3, a.a.s. $G(n,p)$ has the triangle property. Moreover, by Theorem 3, a.a.s. $G(n,p)$ has the sparse extension property, which is described below. For any $m\geq 1$ and any distinct vertices $v_{1},\ldots,v_{m}$, there are vertices $z_{1},z_{2}$ such that • $z_{1}$ is adjacent to $v_{1}$ and not adjacent to any of $v_{2},\ldots,v_{m}$, $z_{2}$ is not adjacent to any of $v_{1},\ldots,v_{m}$, • for any $i\in\{1,\ldots,m\}$, $s\in\{1,2\}$, $z_{s}\neq v_{i}$ and $z_{s}$ has no common neighbors with $v_{i}$. Finally, by Theorem 3, a.a.s., in $G(n,p)$, there exists a vertex $x_{1}$ such that • there is no $K_{4}$ containing $x_{1}$, • for any vertex $x_{2}$, $n(x_{1},x_{2})\leq 1$ (in such a case, we say that a graph has the sparse subgraph property). Let $G,H$ be graphs with the triangle property, the sparse extension property and the sparse subgraph property. Let us describe a winning strategy of Duplicator in ${\mathrm{EHR}}(G,H,4,3)$. In the first round, Spoiler chooses, say, an arbitrary vertex $v_{1}\in V(G)$. Duplicator chooses an arbitrary vertex $u_{1}\in V(H)$ such that there is no $K_{4}$ in $H$ containing $u_{1}$ and, for any vertex $u_{2}$, $n(u_{1},u_{2})\leq 1$. Such a vertex exists because $H$ has the sparse subgraph property. In the second round, Spoiler chooses a vertex $u_{2}\in V(H)$. If the set $N(u_{1},u_{2})\setminus U(u_{1},u_{2})$ is non-empty, then denote by $y\in\mathcal{I}$ the least element of the set of types of $(u_{1},u_{2},u_{3})$ over all $u_{3}\in N(u_{1},u_{2})\setminus U(u_{1},u_{2})$. If the set $U(u_{1},u_{2})$ is non-empty, then denote by $x\in\mathcal{I}$ the least element of the set of types of $(u_{1},u_{2},u_{3})$ over all $u_{3}\in U(u_{1},u_{2})$. Consider two cases. 1. $u_{1}\sim u_{2}$. Duplicator chooses $v_{2}\in V(G)$ such that • $v_{1}\sim v_{2}$, • there is no $K_{4}$ containing $v_{1},v_{2}$ in $G$, • for any $s\in\{0,1\}$, $v_{1},v_{2}$ have exactly $s$ common neighbors if and only if $u_{1},u_{2}$ have exactly $s$ common neighbors, • if $u_{1},u_{2}$ have 2 common neighbors, then $v_{1},v_{2}$ have exactly 2 common neighbors, • if $N(u_{1},u_{2})\neq\varnothing$, then the types of $(v_{1},v_{2},v_{3})$ equal $x$ for all common neighbors $v_{3}$ of $v_{1},v_{2}$. 2. $u_{1}\nsim u_{2}$. Duplicator chooses $v_{2}\in V(G)$ such that • $v_{1}\nsim v_{2}$, • $n(v_{1},v_{2})=n(u_{1},u_{2})$, • if $n(u_{1},u_{2})=1$, then the types of $(v_{1},v_{2},v_{3}^{1})$, $(v_{1},v_{2},v_{3}^{2})$ equal $y$, where $v_{3}^{1}\sim v_{3}^{2}$ are common neighbors of $v_{1},v_{2}$, • if $n(u_{1},u_{2})=1$ and $U(u_{1},u_{2})\neq\varnothing$, then $U(v_{1},v_{2})=\{v_{3}\}$ and the type of $(v_{1},v_{2},v_{3})$ equals $x$, • if $n(u_{1},u_{2})=0$ and $|U(u_{1},u_{2})|\geq 2$, then $U(v_{1},v_{2})=\{v_{3}^{1},v_{3}^{2}\}$ and the types of $(v_{1},v_{2},v_{3}^{1}),$ $(v_{1},v_{2},v_{3}^{2})$ equal $x$, • if $n(u_{1},u_{2})=0$ and $|U(u_{1},u_{2})|=1$, then $U(v_{1},v_{2})=\{v_{3}\}$ and the type of $(v_{1},v_{2},v_{3})$ equals $x$, • if $N(u_{1},u_{2})=\varnothing$, then $N(v_{1},v_{2})=\varnothing$. Such a vertex exists because 1) after the first round, there is no $K_{4}$ containing $u_{1}$ and $n(u_{1},u_{2})\leq 1$ for all $u_{2}$; 2) $G$ has the triangle property. In the third round, Spoiler chooses a vertex $v_{3}\in V(G)$. If $v_{3}\sim v_{1},v_{3}\sim v_{2}$, then Duplicator chooses a vertex $u_{3}\in V(H)$ such that • if $v_{3}\in U(v_{1},v_{2})$, then $u_{3}\in U(u_{1},u_{2})$ and $(u_{1},u_{2},u_{3})$ has the type $x$, • if $v_{3}\in N(v_{1},v_{2})\setminus U(v_{1},v_{2})$, then $u_{3}\in N(u_{1},u_{2})\setminus U(u_{1},u_{2})$ and $(u_{1},u_{2},u_{3})$ has the type $y$. Otherwise, Duplicator chooses a vertex $u_{3}\in V(H)$ such that • $v_{1}\sim v_{3}$ if and only if $u_{1}\sim u_{3}$, • $v_{2}\sim v_{3}$ if and only if $u_{2}\sim u_{3}$, • for any $j\in\{1,2\}$, the vertices $u_{j},u_{3}$ have no common vertices. Such a vertex exists because $H$ has the sparse extension property. In the fourth round, Spoiler chooses a vertex $u_{4}\in V(H)$. Obviously, if $u_{4}$ is a common neighbor of $u_{1},u_{2},u_{3}$, then $u_{1}\nsim u_{2}$ and $u_{3}\in N(u_{1},u_{2})\setminus U(u_{1},u_{2})$. Therefore, $v_{3}\in N(v_{1},v_{2})\setminus U(v_{1},v_{2})$. So, there exists a common neighbor $v_{4}\in V(G)$ of $v_{1},v_{2},v_{3}$. Assume that $u_{4}$ is not a common neighbor of $u_{1},u_{2},u_{3}$. If $u_{4}\in N(u_{1},u_{2})$, then $v_{1},v_{2}$ have at least $1$ common neighbor. So, if $u_{3}\notin N(u_{1},u_{2})$, there is $v_{4}\in N(v_{1},v_{2})$ such that $v_{4}\neq v_{3}$. If $u_{3}\in N(u_{1},u_{2}$, then $v_{1},v_{2}$ have at least $2$ common neighbors, and so there is $v_{4}\in N(v_{1},v_{2})$ such that $v_{4}\neq v_{3}$ as well. If $u_{4}\in N(u_{1},u_{3})$ (or $u_{4}\in N(u_{2},u_{3})$), then $u_{3}\in N(u_{1},u_{2})$ and $(u_{1},u_{2},u_{3})$, $(v_{1},v_{2},v_{3})$ has the same type. Therefore, there exists a vertex $v_{4}$ such that $v_{4}\in N(v_{1},v_{3})$ (or $v_{4}\in N(v_{2},v_{3})$). In all the above cases, Duplicator chooses $v_{4}$. Finally, if $u_{4}$ is adjacent to at most one vertex of $u_{1},u_{2},u_{3}$, then Duplicator has a winning strategy because $G$ has the sparse extension property. Case 2 Let $p=n^{-\alpha}$, $\alpha\in(10/19,1)$ be rational and not equal to any fraction $a/b$ with $a\leq 20$. Note that there is only a finite number of forbidden fractions $a/b$. Moreover, let $q$ be the denominator of $\alpha$. Let $H_{2}\subset H_{1}$, $V(H_{2})=\{a_{1},\ldots,a_{s}\}$, $V(H_{1})\setminus V(H_{2})=\{b_{1},\ldots,b_{\ell}\}$. We say that a graph $G$ has the 1-generic $(H_{1},H_{2})$-extension property if for any its distinct vertices $x_{1},\ldots,x_{s}$ there exist distinct vertices $y_{1},\ldots,y_{\ell}$ such that • $\forall i,j\in\{1,\ldots,\ell\}$ $(y_{i}\sim y_{j})\Leftrightarrow(b_{i}\sim b_{j})$, • $\forall i\in\{1,\ldots,k\},j\in\{1,\ldots,\ell\}$ $(x_{i}\sim y_{j})\Leftrightarrow(a_{i}\sim b_{j})$, • if there exists a vertex $z$ such that $\rho(G|_{\{x_{1},\ldots,x_{s},y_{1},\ldots,y_{\ell},z\}},G|_{\{x_{1},\ldots,x_% {s},y_{1},\ldots,y_{\ell}\}})>1/\alpha$, then there are no edges between $z$ and any of $y_{1},\ldots,y_{\ell}$. Let $\mathcal{S}$ be a set of all graphs $G$, that satisfy the following properties. (1) There exists a vertex $x$ in $G$ such that there are no subgraphs $W\supset X$ with $x\in V(W)$, $\rho(W,(\{x\},\varnothing))>1/\alpha$ and $v(W)\leq 21$. (2) For any graphs $H_{2}\subset H_{1}$ with $\rho(H_{1},H_{2})<1/\alpha$, $v(H_{1})\leq 22$, $G$ has the 1-generic $(H_{1},H_{2})$-extension property. By Theorems 2 and 3, $\lim_{n\rightarrow\infty}{\sf P}(G(n,p)\in\mathcal{S})=1$, and it is sufficient to describe a Duplicator’s winning strategy in ${\mathrm{EHR}}(G,H,4,3)$ for $G,H\in\mathcal{S}$. In the first round, Spoiler chooses, say, a vertex $v_{1}\in V(G)$. By the property (1), there is a vertex $u_{1}\in V(H)$ such that in $H$ there is no subgraph $W$ with $u_{1}\in V(W)$, $v(W)\leq 21$, $\rho(W,(\{u_{1}\},\varnothing))>1/\alpha$. In the second round, Spoiler chooses a vertex $u_{2}\in V(H)$. Consider a maximal sequence of graphs $H|_{\{u_{1},u_{2}\}}=H_{0}\subset H_{1}\subset\ldots\subset H_{L}\subset H$ with each $v(H_{i})-v(H_{i-1})=1$ and $\rho(H_{i},H_{i-1})>1/\alpha$. Note that $L\leq 19$. Indeed, if $L>19$, then $$\rho(H_{20},(\{u_{1}\},\varnothing))\geq\frac{40}{21}>\frac{19}{10}>1/\alpha,$$ that is impossible by the choice of $u_{1}$. By the choice of $u_{1}$, we have $\rho(H_{L},(\{u_{1}\},\varnothing))\leq 1/\alpha$. Moreover, $\alpha$ is not equal to any fraction $a/b$ with $a\leq 20$, hence, the inequality is strict: $\rho(H_{L},(\{u_{1}\},\varnothing))<1/\alpha$. Set $Y=H_{L}$. By (2), there exists a subgraph $X$ in $G$ such that $Y\cong X$, there is an isomorphism $f:Y\rightarrow X$ such that $f(u_{1})=v_{1}$, and there is no subgraph $W\subset G$ such that $X\subset W$, $v(W)=v(X)+1$, $\rho(W,X)>1/\alpha$. Duplicator chooses $v_{2}=f(u_{2})$. In the third round, Spoiler chooses a vertex $v_{3}\in V(G)$. Consider two cases. If $v_{3}\in V(X)$, then Duplicator chooses $u_{3}=f^{-1}(v_{3})$. If after that Spoiler chooses a vertex $u_{4}\in V(Y)$, then Duplicator chooses $v_{4}=f(u_{4})$, and she wins. If Spoiler chooses $u_{4}\notin V(Y)$, then $\rho(H|_{\{u_{1},u_{2},u_{3},u_{4}\}},H|_{\{u_{1},u_{2},u_{3}\}})<1/\alpha$. So, the property (2) implies the existence of $v_{4}\in V(G)$ such that $v_{4}\sim v_{i}$ if and only if $u_{4}\sim u_{i}$ for all $i\in\{1,2,3\}$. Duplicator chooses such a $v_{4}$ and wins. If $v_{3}\notin V(X)$, then $\rho(G|_{V(X)\cup\{v_{3}\}},X)<1/\alpha$. So, by (2) there exists a vertex $u_{3}$ in $H$ such that there exists an isomorphism $\tilde{f}:G|_{V(X)\cup\{v_{3}\}}\rightarrow H|_{V(Y)\cup\{u_{3}\}}$, $\tilde{f}(v_{i})=u_{i}$ for $i\in\{1,2,3\}$. Moreover, there is no subgraph $W\subset H$ such that $H|_{V(Y)\cup\{u_{3}\}}\subset W$, $v(W)=v(Y)+1$, $\rho(W,Y)>1/\alpha$. Duplicator chooses that $u_{3}$. Denote $\tilde{X}=G|_{V(X)\cup\{v_{3}\}}$, $\tilde{Y}=H|_{V(Y)\cup\{u_{3}\}}$. If in the fourth round Spoiler chooses a vertex $u_{4}\in\tilde{Y}$, then Duplicator chooses $\tilde{f}^{-1}(u_{4})$ and wins. If Spoiler chooses $u_{4}\notin\tilde{Y}$, then $(H|_{\{u_{1},u_{2},u_{3},u_{4}\}},H|_{\{u_{1},u_{2},u_{3}\}})<1/\alpha$. In this case Duplicator’s winning strategy is the same as in the first case. $\Box$ Proof of Theorem 5. From Theorem 4, it follows that it is enough to prove that $|S(\phi)|<\infty$ if $\phi$ is in PNF, ${\mathrm{q}}(\phi)=4$, ${\mathrm{ch}}(\phi)=3$. Let $\phi\in\mathcal{F}$ be a PNF sentence such that ${\mathrm{q}}(\phi)=4$, ${\mathrm{ch}}(\phi)=3$. Let $\alpha\in S(\phi)$. Obviously, there exist $\varepsilon>0$ and sequences $n_{i},m_{i}$ such that, for any $i\in\mathbb{N}$, $$\min\left\{{\sf P}\left(G(n_{i},n_{i}^{-\alpha})\models\phi\right),{\sf P}% \left(G(m_{i},m_{i}^{-\alpha})\models\neg(\phi)\right)\right\}>\varepsilon.$$ By Lemma 6, $${\sf P}\left(\text{Spoiler has a winning strategy in EHR}\left(G(n_{i},n_{i}^{% -\alpha}),G(m_{i},m_{i}^{-\alpha}),4,3\right)\right)>\varepsilon^{2}.$$ Therefore, $\alpha\in\tilde{S}(\phi)$. By Lemma 8, $|\tilde{S}(4)\cap(1/2,1)|<\infty$. Moreover, the random graph $G(n,n^{-\alpha})$ obeys zero-one 4-law if $\alpha<1/2$, and the set $S(\phi)\cap(1,\infty)$ is finite (see Section 1). Therefore, $S(\phi)=S(\phi)\cap[1/2,\infty)$ is finite. $\Box$ Note that as the formula (1) with an infinite spectrum is in PNF, Theorem 5 implies that a minimal quantifier depth of a PNF sentence with an infinite spectrum is in $\{5,6,7,8\}$. Finally, it is easy to see that Lemma 8 and Theorem 4 have a more general corollary which is given below. Theorem 6 Let $\phi\in\mathcal{F}$, ${\mathrm{q}}(\phi)=4$. If either all paths of $F(\phi)$ starting in a root have $3$ labels alternations, or all paths of $F(\phi)$ starting in a root have at most $2$ labels alternations, then $|S(\phi)|<\infty$. From Theorem 6, we get that if there exists a sentence $\phi=\exists x\,\varphi(x)\in\mathcal{F}$ with ${\mathrm{q}}(\phi)=4$ and an infinite spectrum, then $F(\phi)$ has both types of paths starting in the root: with maximal number of labels alternations and with less number of labels alternations. So, we still do not have an answer on the first question of Section 1, but we know much more about sentences with ${\mathrm{q}}(\phi)=4$ and finite spectra. References [1] J.H. Spencer, The Strange Logic of Random Graphs, Number 22 in Algorithms and Combinatorics, Springer-Verlag, Berlin, 2001. [2] M.E. Zhukovskii, A.M. Raigorodskii, Random graphs: models and asymptotic characteristics, Russian Mathematical Surveys, 70(1): 33–81, 2015. [3] B. Bollobás, Random Graphs, 2nd Edition, Cambridge University Press, 2001. [4] S. Janson, T. Łuczak, A. Rucinski, Random Graphs, New York, Wiley, 2000. [5] S. Shelah, J.H. Spencer, Zero-one laws for sparse random graphs, J. Amer. Math. Soc., 1988, 1:97–115. [6] Spencer J.H., “Infinite spectra in the first order theory of graphs”, Combinatorica, 10:1 (1990), 95–102. [7] M.E. Zhukovskii, Zero-one $k$-law, Discrete Mathematics, 2012, 312: 1670–1688. [8] L.B. Ostrovskii, M.E. Zhukovskii, First-order and monadic Properties of highly sparse random graphs, Doklady Mathematics, 2016, 94(2): 555–557 (Russian Original: Doklady Akademii Nauk, 2016, 470(5): 499–501). [9] Zhukovskii M., “On Infinite Spectra of First Order Properties of Random Graphs”, Moscow Journal of Combinatorics and Number Theory, 6:4 (2016), 73–102. [10] P. Erdős, A. Rényi, On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl. 5: 17–61, 1960. [11] B. Bollobás, Threshold functions for small subgraphs, Math. Proc. Camb. Phil. Soc. 90: 197–206, 1981. [12] Ruciński A., Vince A., “Balanced graphs and the problem of subgraphs of a random graph”, Congressus Numerantim, 49 (1985), 181–190. [13] Spencer J.H., “Threshold functions for extension statements”, J. of Comb. Th. Ser A, 53 (1990), 286–305. [14] N. Alon, J.H. Spencer, The Probabilistic Method, John Wiley $\&$ Sons, 2000. [15] M.E. Zhukovskii, Estimation of the number of maximal extensions in the random graph, Discrete Mathematics and Applications, 2012, 24(1): 79–107. [16] H.-D. Ebbinghaus, J. Flum, Finite model theory, Perspectives in Mathematical Logic. Springer-Verlag, Berlin, second edition, 1999. [17] A. Ehrenfeucht, An application of games to the completness problem for formalized theories, Warszawa, Fund. Math. 49: 121–149, 1960.

Perfectly Secure Steganography: Capacity, Error Exponents, and Code Constructions Ying Wang,  and Pierre Moulin,  This work was supported by NSF under grants CCR 02-08809 and CCR 03-25924, and presented in part at the 40th Conference on Information Sciences and Systems (CISS), Princeton, NJ, March 2004. Ying Wang was with the Department of Electrical and Computer Engineering at the University of Illinois at Urbana-Champaign and is now with Qualcomm Flarion Technologies, Bridgewater, NJ 08807 USA (e-mail: yingw@qualcomm.com). Pierre Moulin is with the Beckman Institute, Coordinate Science Lab and Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail: moulin@ifp.uiuc.edu). (February 16, 2007. Revised September 30, 2007) Abstract An analysis of steganographic systems subject to the following perfect undetectability condition is presented in this paper. Following embedding of the message into the covertext, the resulting stegotext is required to have exactly the same probability distribution as the covertext. Then no statistical test can reliably detect the presence of the hidden message. We refer to such steganographic schemes as perfectly secure. A few such schemes have been proposed in recent literature, but they have vanishing rate. We prove that communication performance can potentially be vastly improved; specifically, our basic setup assumes independently and identically distributed (i.i.d.) covertext, and we construct perfectly secure steganographic codes from public watermarking codes using binning methods and randomized permutations of the code. The permutation is a secret key shared between encoder and decoder. We derive (positive) capacity and random-coding exponents for perfectly-secure steganographic systems. The error exponents provide estimates of the code length required to achieve a target low error probability. In some applications, steganographic communication may be disrupted by an active warden, modelled here by a compound discrete memoryless channel. The transmitter and warden are subject to distortion constraints. We address the potential loss in communication performance due to the perfect-security requirement. This loss is the same as the loss obtained under a weaker order-1 steganographic requirement that would just require matching of first-order marginals of the covertext and stegotext distributions. Furthermore, no loss occurs if the covertext distribution is uniform and the distortion metric is cyclically symmetric; steganographic capacity is then achieved by randomized linear codes. Our framework may also be useful for developing computationally secure steganographic systems that have near-optimal communication performance. Steganography, watermarking, secret communication, timing channels, capacity, reliability function, error exponents, binning codes, randomized codes, universal codes. I Introduction Information embedding refers to the embedding of data within a cover object (also referred to as covertext) such as image, video, audio, graphics, text, or packet transmission times [1, 2, 3, 4, 5]. Applications include copyright protection, database annotation, transaction tracking, traitor tracing, timing channels, and multiuser communications. These applications often impose the requirement that embedding only slightly perturbs the covertext. The name watermarking has been widely used to describe information embedding techniques that are perceptually transparent, i.e., the marked object (after embedding) is perceptually similar to the cover object. In some applications, the presence of the embedded information should be kept secret (see applications below). Then perceptual transparency is not sufficient, because statistical analysis could reveal the presence of hidden information. The problem of embedding information that is hard to detect is called steganography, and the marked object is called stegotext [3, 4, 6, 7, 8]. Steganography differs from cryptography in that the presence of the message needs to remain secret, rather than the value of the message. The dual problem to steganography is steganalysis, that is, detection of hidden information within a stegotext. A classical model for steganography is Simmons’ prisoner problem [9]. Alice and Bob are locked up in different cells but are allowed to communicate under the vigilant eye of Willie, the prison warden. If Willie detects the presence of hidden information in the transmitted data, he terminates their communication and subjects them to a punishment. Willie is a passive warden if he merely observes and analyzes the transmitted data. He is an active warden if he introduces noise to make Alice and Bob’s task more difficult. In the information age, there are several application scenarios for steganography. 1. Steganography may be used to communicate over public networks such as the Internet. One may embed bits into inconspicuous files that are routinely sent over such networks: images, video, audio files, etc. Users of such technology may include intelligence and military personnel, people that are subject to censorship, and more generally, people who have a need for privacy. 2. Steganography may also be used to communicate over private networks. For instance, confidential documents within a commercial or governmental organization could be marked with identifiers that are hard to detect. The purpose is to trace unauthorized use of a document to a particular person who received a copy of this document. The recipient of the marked documents should not be aware of the presence of these identifiers. 3. Timing channels can be used to leak out information about computers. A pirate could modify the timing of packets sent by the computer, encoding data that reside on that computer. The pirate wishes to make this information leakage undetectable to avoid arousing suspicion. To disrupt potential information leakage, the network could jam packet timings — hence the network plays the role of an active warden. The channel over which the stegotext is transmitted could be noiseless or noisy, corresponding to the case of a passive and an active warden, respectively. Moreover, the steganographer’s ability to choose the covertext is often limited if not altogether nonexistent. In the private-network application above, the covertext is generated by a content provider, not by the steganographer (i.e., the authority responsible for document security). Similarly in the timing-channel application, the covertext is generated by the computer, not by the pirate. In view of these applications, the four basic attributes of a steganographic code are: 1. detectability: quantifying Willie’s ability to detect the presence of hidden information; 2. transparency (fidelity): closeness of covertext and stegotext under an appropriate distortion (fidelity) metric; 3. payload: the number of bits embedded in the covertext; and 4. robustness: quantifying decoding reliability in presence of channel noise (i.e., when Willie is an active warden). If Alice had complete freedom for choosing the covertext, the transparency requirement would be immaterial. A covertext would not even be needed: it would suffice for Alice to generate objects that follow a prescribed covertext distribution. This model has two shortcomings: (a) as mentioned above, in some applications Alice has little or no control over the choice of the covertext; (b) even if she has, covertexts have complicated distributions, and generating a size-$M$ steganographic code by sampling the covertext distribution would be highly impractical for large $M$. Information theory is a natural framework for studying steganography and steganalysis. Assuming a statistical model is available for covertexts, the only truly secure strategy from the steganographer’s point of view is to ensure that the probability distributions of the covertext and stegotext are identical. This strong notion of security was proposed by Cachin [10] and is the steganographic counterpart of Shannon’s notion of perfect security in cryptography. We refer to steganography that satisfies this strong property as perfectly secure. If Alice is allowed to select the covertext and Willie is passive, Alice may use the following perfectly secure steganographic code [10]. Alice and Bob agree on a hash function, and the value of the hashed stegotext is the message to be transmitted. Alice searches a database of covertexts until she finds one that matches the desired hash value. This approach is perfectly secure irrespective of the distribution of the covertext. The disadvantages are that the search is computationally infeasible for large message sets (communication rate is extremely low), and the underlying communication model is limited, as discussed above. Cachin also proposed two less stringent requirements for steganographic codes  [10]. One is $\epsilon$-secure steganographic codes, where the Kullback-Leibler divergence between the covertext and stegotext probability distributions is smaller than $\epsilon$ (perfect security requires $\epsilon=0$). For random processes he redefined perfectly secure steganography by requiring that the above Kullback-Leibler divergence, normalized by the length $N$ of the covertext sequence, tends to zero as $N\to\infty$. Unfortunately this does not preclude the possibility that Kullback-Leibler divergence remains bounded away from zero, even grows to infinity (at a rate slower than $N$) as $N\to\infty$. If such is the case, Willie’s error probability tends to zero asymptotically, and therefore the perfect-security terminology is misleading. While Cachin focused on security and not on communication performance in terms of payload, robustness and fidelity, Kullback-Leibler divergence has become a popular metric for assessing the security of practical steganographic schemes subject to transparency, payload, and robustness requirements [11, 12, 13, 14, 15, 16, 17, 18]. Other metrics are studied in [19, 20, 21, 22, 23, 24, 25]. The tradeoffs between detectability, fidelity, payload, and robustness can be studied in an information-theoretic framework. The basic mathematical model for steganography is communications with side information at the encoder [26]. Moulin and O’Sullivan studied a general information-theoretic framework for information hiding and indicated its applicability to steganography [27, Section VII.C]. However, they did not study perfectly secure steganography and did not derive expressions for steganographic capacity. Galand and Kabatiansky [28] constructed steganographic binary codes, but the code rate vanishes as $\frac{\log N}{N}$. Fridrich et al. [29, 30] proposed positive-rate “wet paper” codes, which permit a change from the original cover distribution to a new stegodistribution. However they did not analyze the fundamental tradeoffs between payload, robustness, and detectability. The goal of this paper is therefore to study the information-theoretic limits of perfectly undetectable steganography. As a first step towards this problem, we assume that covertext samples are independently and identically distributed (i.i.d.) over a finite alphabet. In practice the i.i.d. model could be applied to transform coefficients or to blocks of coefficients. While this is just a simplifying approximation to actual statistics, it allows us to derive tangible mathematical results and to understand the effects of the perfect security constraint on transparency, payload, and robustness. Our first result is a connection between public watermarking codes [27, 31, 32] and perfectly secure steganographic codes. Given any public watermarking code that preserves the first-order statistics of the covertext (this property will be referred to as order-1 security), we show that a perfectly secure steganographic code with the same error probability can be constructed using randomization over the set of all permutations of $\{1,2,\cdots,N\}$. We use this construction to derive capacity and random-coding exponent formulas for perfectly secure steganography. The codes that achieve capacity and random-coding exponents are stacked-binning schemes as proposed in [33] for general problems of channel coding with side information. The random-coding exponent yields an asymptotic upper bound on achievable error probability and therefore serves as an estimate of the code length required to achieve a target low error probability. A stacked-binning code consists of a stack of variable-size codeword arrays indexed by the type of the covertext sequence, and the corresponding decoder is a maximum penalized mutual information (MPMI) decoder. The analysis is based on the method of types [34, 35]. Due to the added perfect-security constraint, capacity and random-coding exponent for steganography cannot exceed those of the corresponding public watermarking problem. Nevertheless, we have identified a class of problems where the covertext probability mass function (PMF) is uniform and the distortion function is symmetric, with the property that the perfect undetectability constraint does not cause any capacity loss. One special example in the general class is the case of Bernoulli($\frac{1}{2}$) covertexts with the Hamming distortion metric [36]. For the binary-Hamming case, the perfect security condition has no effect on both the capacity and random-coding error exponent. Steganographic capacity is achieved by randomized nested linear codes. This paper is organized as follows. Section II describes the notation, and Section III the problem statement. Section IV shows how perfectly secure steganographic codes can be constructed from codes with the much weaker order-1 security. Section V presents our main theorems on capacity and random-coding error exponent. Section VI discusses the role of secret keys in steganographic codes. Simplified results for the no-attack case are stated in Section VII. A class of steganography problems for which perfect security comes at no cost is studied in Section VIII. As an example of this class, the binary-Hamming problem is studied in Section IX. The paper concludes with a discussion in Section X. II Notation We use uppercase letters for random variables, lowercase letters for their individual values, and boldface letters for sequences. The PMF of a random variable $X\in{\mathcal{X}}$ is denoted by $p_{X}=\{p_{X}(x),\,x\in{\mathcal{X}}\}$. The entropy of a random variable $X$ is denoted by $H(X)$, and the mutual information between two random variables $X$ and $Y$ is denoted by $I(X;Y)=H(X)-H(X|Y)$. Should the dependency on the underlying PMFs be explicit, we use the PMFs as subscripts, e.g., $H_{p_{X}}(X)$ and $I_{p_{X},p_{Y|X}}(X;Y)$. The Kullback-Leibler divergence between two PMFs $p$ and $q$ is denoted by $D(p||q)$; the conditional Kullback-Leibler divergence of $p_{Y|X}$ and $q_{Y|X}$ given $p_{X}$ is denoted by $D(p_{Y|X}||q_{Y|X}|p_{X})=D(p_{Y|X}\,p_{X}||q_{Y|X}\,p_{X})$. Let $p_{\mathbf{x}}$ denote the empirical PMF on ${\mathcal{X}}$ induced by a sequence ${\mathbf{x}}\in{\mathcal{X}}^{N}$. Then $p_{\mathbf{x}}$ is called the type of ${\mathbf{x}}$. The type class $T_{\mathbf{x}}$ associated with $p_{\mathbf{x}}$ is the set of all sequences of type $p_{\mathbf{x}}$. Likewise, we define the joint type $p_{{\mathbf{x}}{\mathbf{y}}}$ of a pair of sequences $({\mathbf{x}},{\mathbf{y}})\in{\mathcal{X}}^{N}\times{\mathcal{Y}}^{N}$ and the type class $T_{{\mathbf{x}}{\mathbf{y}}}$ associated with $p_{{\mathbf{x}}{\mathbf{y}}}$. The conditional type $p_{{\mathbf{y}}|{\mathbf{x}}}$ of a pair of sequences (${\mathbf{x}},{\mathbf{y}}$) is defined as $\frac{p_{{\mathbf{x}}{\mathbf{y}}}(x,y)}{p_{{\mathbf{x}}}(x)}$ for all $x\in{\mathcal{X}}$ such that $p_{{\mathbf{x}}}(x)>0$. The conditional type class $T_{{\mathbf{y}}|{\mathbf{x}}}$ given ${\mathbf{x}}$ is the set of all sequences $\tilde{{\mathbf{y}}}$ such that $({\mathbf{x}},\tilde{{\mathbf{y}}})\in T_{{\mathbf{x}}{\mathbf{y}}}$. We denote by $H({\mathbf{x}})$ the empirical entropy for ${\mathbf{x}}$, i.e., the entropy of the empirical PMF $p_{{\mathbf{x}}}$. Similarly, we denote by $I({\mathbf{x}};{\mathbf{y}})$ the empirical mutual information for the joint PMF $p_{{\mathbf{x}}{\mathbf{y}}}$. The above notation for types is adopted from Csiszár and Körner [34]. We let $\mathbb{U}(\Omega)$ denote the uniform PMF over a finite set $\Omega$. We let ${\mathcal{P}}_{X}$ and ${\mathcal{P}}^{N}_{X}$ represent the set of all PMFs and all empirical PMFs, respectively, on the alphabet ${\mathcal{X}}$. Likewise, ${\mathcal{P}}_{Y|X}$ and ${\mathcal{P}}^{N}_{Y|X}$ denote the set of all conditional PMFs and all empirical conditional PMFs on the alphabet ${\mathcal{Y}}$. We use $\mathsf{E}$ to denote mathematical expectation. The shorthands $a_{N}\doteq b_{N}$, $a_{N}\mbox{$\>\stackrel{{\scriptstyle\centerdot}}{{\leq}}\>$}b_{N}$, and $a_{N}\mbox{$\>\stackrel{{\scriptstyle\centerdot}}{{\geq}}\>$}b_{N}$ are used to denote asymptotic equalities and inequalities in the exponential scale for $\lim_{N\to\infty}\frac{1}{N}\log\frac{a_{N}}{b_{N}}=0$, $\limsup_{N\to\infty}\frac{1}{N}\log\frac{a_{N}}{b_{N}}\leq 0$, and $\liminf_{N\to\infty}\frac{1}{N}\log\frac{a_{N}}{b_{N}}\geq 0$, respectively. We define $|t|^{+}\triangleq\max(t,0)$, $\exp_{2}(t)\triangleq 2^{t}$, and the binary entropy function $$h(t)\triangleq-t\log t-(1-t)\log(1-t),\quad t\in[0,1].$$ We use $\ln x$ to denote the natural logarithm of $x$, and the logarithm $\log x$ is in base 2 if not specified otherwise. The notation $\mathds{1}_{\{A\}}$ is the indicator function of the event $A$: $$\mathds{1}_{\{A\}}=\left\{\begin{array}[]{ll}1&A\mbox{ is true;}\\ 0&\mbox{ else.}\end{array}\right.$$ Finally, we adopt the notional convention that the minimum (resp. maximum) of a function over an empty set is $+\infty$ (resp. 0). III Problem Statement Referring to Fig. 1, the covertext is modelled as a sequence ${\mathbf{S}}=(S_{1},\cdots,S_{N})$ of i.i.d. samples drawn from a PMF $\{p_{S}(s)$, $s\in{\mathcal{S}}\}$. A message $M$ is to be embedded in ${\mathbf{S}}$ and transmitted to a decoder; $M$ is uniformly distributed over a message set ${\mathcal{M}}$. The encoder produces a stegotext ${\mathbf{X}}$ through a function $f_{N}({\mathbf{S}},M)$, in an attempt to transmit the message $M$ to the decoder reliably. The covertext and stegotext are required to be close according to some distortion metric. The distortion model is motivated by the fact that stegotext and covertext represent physical signals (such as images, text, etc.) which can be modified to some extent without affecting perceptual quality [27]. The strength of the transparency constraint is controlled by a distortion parameter. A steganalyzer observes ${\mathbf{X}}$ and tests whether ${\mathbf{X}}$ is drawn i.i.d. from $p_{S}$. If not, Willie, the steganalyzer terminates the transmission, and obviously the decoder is unable to retrieve $M$. If ${\mathbf{X}}$ is deemed innocuous, Willy may simply forward it to the decoder. In this case, Willie is a passive warden. To be on the safe side for preventing reliable transmission of hidden messages, Willie may want to pass ${\mathbf{X}}$ through some attack channel $p_{{\mathbf{Y}}|{\mathbf{X}}}({\mathbf{y}}|{\mathbf{x}})$, thereby producing a corrupted text ${\mathbf{Y}}$. In this case, Willie is an active warden, and the corrupted text and the stegotext are also required to be close according to some distortion metric. The alphabets ${\mathcal{S}}$, ${\mathcal{X}}$ and ${\mathcal{Y}}$ for $S$, $X$ and $Y$, respectively, are assumed to be identical. The decoder does not know $p_{{\mathbf{Y}}|{\mathbf{X}}}$ selected by the steganalyzer and does not have access to the original covertext ${\mathbf{S}}$. The decoder produces an estimate $\hat{M}=\phi_{N}({\mathbf{Y}})\in{\mathcal{M}}$ of the transmitted message. We assume that the encoder/decoder pair $(f_{N},\phi_{N})$ is randomized, i.e., the choice of $(f_{N},\phi_{N})$ is a function of a random variable known only to the encoder and decoder but not to the steganalyzer. We can think of this random variable as a secret key as in [27, 31, 32]. Note that in generic information-hiding games, this secret key provides some protection against adversaries with arbitrary memory and unlimited computational resources [4, Section X]. In steganography, the secret key plays a fundamental role in ensuring perfect undetectability: the covertext and the stegotext have the same PMF when the secret key is carefully designed. The randomized code will be denoted by $(F_{N},\Phi_{N})$ with a joint distribution $p(f_{N},\phi_{N})$. III-A Steganographic Codes A distortion function is any nonnegative function ${\mathsf{d}}:{\mathcal{S}}\times{\mathcal{S}}\to\mathbb{R}^{+}\cup\{0\}$. This definition is extended to length-$N$ vectors using ${\mathsf{d}}^{N}({\mathbf{s}},{\mathbf{x}})=\frac{1}{N}\sum_{i=1}^{N}{\mathsf{% d}}(s_{i},x_{i})$. Let $D_{\max}=\max_{s,x}{\mathsf{d}}(s,x)$. We assume without loss of generality that ${\mathsf{d}}(s,x)\geq 0$, with equality if $s=x$. Definition 1 A length-$N$ perfectly secure steganographic code with maximum distortion $D_{1}$ is a triple $({\mathcal{M}},F_{N},\Phi_{N})$, where • ${\mathcal{M}}$ is the message set of cardinality $|{\mathcal{M}}|$; • $(F_{N},\Phi_{N})$ has a joint distribution $p(f_{N},\phi_{N})$; • $f_{N}~{}:~{}{\mathcal{S}}^{N}\times{\mathcal{M}}\to{\mathcal{S}}^{N}$ maps covertext ${\mathbf{s}}$ and message $m$ to stegotext ${\mathbf{x}}=f_{N}({\mathbf{s}},m)$. The mapping is subject to the maximum distortion constraint $${\mathsf{d}}^{N}({\mathbf{s}},f_{N}({\mathbf{s}},m))\leq D_{1}\mbox{ almost % surely }$$ (1) and the perfect undetectability constraint $$p_{{\mathbf{X}}}=p_{{\mathbf{S}}};$$ (2) • $\phi_{N}~{}:~{}{\mathcal{S}}^{N}\to{\mathcal{M}}$ maps the received sequence ${\mathbf{y}}$ to a decoded message $\hat{m}=\phi_{N}({\mathbf{y}})$. The above definition is similar to the definitions for a length-$N$ data-embedding or watermarking code in [27, 31, 32], with the additional steganographic constraint of (2) which requires perfect matching of $N$-dimensional distributions. Also observe that the distortion constraint is inactive if $D_{1}\geq D_{\max}$, i.e., the covertext ${\mathbf{S}}$ available to Alice plays no role. Given $p_{S}$, define the set of conditional PMFs $p_{X|S}$ such that the marginals of $p_{S}p_{X|S}$ are equal ($p_{X}=p_{S}$) and the expected distortion between $S$ and $X$ does not exceed $D_{1}$: $${\mathcal{Q}}_{1}^{Steg}(p_{S},D_{1})\triangleq\left\{p_{X|S}~{}:~{}\sum_{s,x}% p_{X|S}(x|s)\,p_{S}(s)\,{\mathsf{d}}(s,x)\leq D_{1},\quad p_{X}(x)=\sum_{s}p_{% X|S}(x|s)\,p_{S}(s)=p_{S}(x),\,\forall x\in{\mathcal{S}}\right\}.$$ (3) Also recall that in Def. 1, randomization of $(F_{N},\Phi_{N})$ is realized via a cryptographic key shared by encoder and decoder. Next, we define CCC and RM codes which will be used to construct perfectly secure steganographic codes. Definition 2 (CCC Code). A length-$N$ code with conditionally constant composition, order-1 steganographic property, and maximum distortion $D_{1}$ is a quadruple $({\mathcal{M}},\Lambda,F_{N},\Phi_{N})$, where $\Lambda$ is a mapping from ${\mathcal{P}}_{S}^{[N]}$ to ${\mathcal{P}}_{X|S}^{[N]}$. The transmitted sequence ${\mathbf{x}}=f_{N}({\mathbf{s}},m)$ has conditional type $p_{{\mathbf{x}}|{\mathbf{s}}}=\Lambda(p_{{\mathbf{s}}})$. Moreover, $\Lambda(p_{{\mathbf{s}}})\in{\mathcal{Q}}_{1}^{Steg}(p_{{\mathbf{s}}},D_{1})$. Observe that such a code matches the first-order empirical marginal PMF of the covertext, but not necessarily higher-order empirical marginals. Hence such a code generally does not satisfy the perfect-undetectability property. Definition 3 (RM Code). A length-$N$ randomly modulated code is the randomized code defined via permutations of a prototype ($f_{N},\phi_{N}$): $$\displaystyle{\mathbf{x}}=f_{N}^{\pi}({\mathbf{s}},m)$$ $$\displaystyle\triangleq$$ $$\displaystyle\pi^{-1}f_{N}(\pi{\mathbf{s}},m)$$ (4) $$\displaystyle\phi_{N}^{\pi}({\mathbf{y}})$$ $$\displaystyle\triangleq$$ $$\displaystyle\phi_{N}(\pi{\mathbf{y}}),$$ (5) where $\pi$ is drawn uniformly from the set $\Pi$ of all $N!$ permutations and is not revealed to Willie. The sequence $\pi{\mathbf{x}}$ is obtained by applying $\pi$ to the elements of ${\mathbf{x}}$. Definition 4 Given alphabets ${\mathcal{S}}$ and ${\mathcal{U}}$, a steganographic channel $p_{XU|S}(x,u|s)$ subject to distortion $D_{1}$ is a conditional PMF whose conditional marginal $p_{X|S}$ belongs to ${\mathcal{Q}}_{1}^{Steg}(p_{S},D_{1})$ of (3). We denote by ${\mathcal{Q}}^{Steg}(L,p_{S},D_{1})$ the set of steganographic channels subject to distortion $D_{1}$ when the alphabet ${\mathcal{U}}$ has cardinality $L$. If the channel $p_{XU|S}$ satisfies the distortion constraint $D_{1}$ but not necessarily the steganographic constraint $p_{X}=p_{S}$, $p_{XU|S}$ is simply a covert channel in the sense of [27, 31]. We shall denote by ${\mathcal{Q}}(L,p_{S},D_{1})$ the set of all such covert channels. Clearly, ${\mathcal{Q}}^{Steg}(L,p_{S},D_{1})\subseteq{\mathcal{Q}}(L,p_{S},D_{1})$. III-B Attack Channels A passive warden simply produces ${\mathbf{Y}}={\mathbf{X}}$. An active warden passes ${\mathbf{X}}$ through a discrete memoryless channel (DMC), producing a degraded sequence ${\mathbf{Y}}$. Definition 5 A discrete memoryless attack channel $p_{Y|X}$ is feasible if the expected distortion between $X$ and $Y$ is at most $D_{2}$: $$\sum_{x,y}p_{X}(x)\,p_{Y|X}(y|x)\,{\mathsf{d}}(x,y)\leq D_{2}.$$ (6) Then the joint conditional PMF is given by $$p_{{\mathbf{Y}}|{\mathbf{X}}}({\mathbf{y}}|{\mathbf{x}})=\prod_{i=1}^{N}p_{Y|X% }(y_{i}|x_{i}).$$ We denote by $${\mathcal{A}}(p_{X},D_{2})=\left\{p_{Y|X}\in{\mathcal{P}}_{Y|X}~{}:~{}\sum_{x,% y}p_{X}(x)\,p_{Y|X}(y|x)\,{\mathsf{d}}(x,y)\leq D_{2}\right\}$$ the set of all such feasible DMCs. This set is a compound DMC family. As an alternative to Def. 5, one may consider attack channels that have arbitrary memory but are subject to an almost sure distortion constraint [31, 32, 33]. In this case, the set of feasible attack channels is given by $${\mathcal{A}}^{\prime}(p_{\mathbf{x}},D_{2})=\left\{p_{{\mathbf{Y}}|{\mathbf{X% }}}\in{\mathcal{P}}^{N}_{Y|X}:Pr\left[{\mathsf{d}}^{N}({\mathbf{y}},{\mathbf{x% }})\leq D_{2}\right]=1\right\}.$$ There are three reasons why only memoryless channels are considered in this paper. First, it is shown in [33] that for watermarking problems, both DMCs with expected distortion and arbitrary memory attack channels with almost sure distortion result in the same capacity formula, and the former allows a smaller random-coding error exponent when $D_{2}$ is the same. Thus, in terms of minimizing the random-coding exponent, selecting $p_{Y|X}$ from the compound DMC class ${\mathcal{A}}(p_{X},D_{2})$ is a better strategy for the warden than selecting $p_{{\mathbf{Y}}|{\mathbf{X}}}$ from ${\mathcal{A}}^{\prime}(p_{\mathbf{x}},D_{2})$. Second, the assumption of memorylessness simplifies the presentation of main ideas. Finally, note that the proofs for the compound DMC provide the basis for the proofs in the case of channels with arbitrary memory [32, 33]. III-C Steganographic Capacity and Reliability Function The probability of error for a randomized code ($F_{N}$, $\Phi_{N}$) under a channel $p_{Y|X}$ is given by $$\displaystyle P_{e,N}(F_{N},\Phi_{N},p_{Y|X})=Pr(\hat{M}\neq M),$$ (7) where the average is over all possible covertexts ${\mathbf{S}}$, messages $M$, and codes ($F_{N}$, $\Phi_{N}$). Definition 6 A rate $R$ is achievable if there exists a randomized code $(F_{N},\Phi_{N})$ such that $|{\mathcal{M}}|\geq 2^{NR}$ and $$\sup_{p_{Y|X}}P_{e,N}(F_{N},\Phi_{N},p_{Y|X})\to 0\quad\mbox{ as }N\to\infty.$$ (8) Definition 7 The steganographic capacity $C^{Steg}(D_{1},D_{2})$ is the supremum of all achievable rates. Definition 8 The steganographic reliability function $E^{Steg}(R)$ is defined as $$E^{Steg}(R)=\liminf_{N\to\infty}\left[-\frac{1}{N}\log\inf_{F_{N},\Phi_{N}}% \sup_{p_{Y|X}}P_{e,N}(F_{N},\Phi_{N},p_{Y|X})\right].$$ (9) IV From Order-1 to Perfectly Secure Steganographic Codes Codes with conditionally constant composition (Def. 2) and randomly modulated codes (Def. 3) play a central role in our code constructions and coding theorems. The following proposition suggests a general construction for perfectly secure steganographic codes: first select some deterministic prototype $f_{N}$ with the CCC and order-1 steganographic properties and maximum distortion $D_{1}$ (Def. 2), second construct a RM code from that prototype. In Section V we show that this strategy is an optimal one. The proof of the proposition appears in the appendix. Proposition 1 Let $({\mathcal{M}},F_{N},\Phi_{N})$ be a RM code whose prototype $(f_{N},\phi_{N})$ has conditionally constant composition, order-1 security, and maximum distortion $D_{1}$. Then $({\mathcal{M}},F_{N},\Phi_{N})$ is a perfectly secure steganographic code with maximum distortion $D_{1}$ and same error probability as the prototype ($f_{N},\phi_{N}$). V Steganographic Capacity and Random Coding Error Exponent The steganographic codes in our achievability proofs are randomly-modulated binning codes with conditionally constant composition. The existence of a good deterministic prototype is established using a random coding argument. An arbitrarily large integer $L$ is selected, defining an alphabet ${\mathcal{U}}=\{1,2,\cdots,L\}$ for the auxiliary random variable $U$ in the binning construction. Given the covertext ${\mathbf{s}}$ and the message $m$, the encoder selects an appropriate sequence ${\mathbf{u}}$ in the binning code and then generates the stegotext randomly according to the uniform distribution over an optimized type class $T_{{\mathbf{x}}|{\mathbf{u}},{\mathbf{s}}}$. Proofs of the theorem and propositions in this section appear in Appendices B-F. The following difference between two mutual informations: $$J_{L}(p_{S},p_{XU|S},p_{Y|XUS})\triangleq I(U;Y)-I(U;S)$$ (10) plays a fundamental role in the analysis. Theorem 1 Under Def. 1 for steganographic codes and Def. 5 for the compound attack channel, steganographic capacity is given by $$\displaystyle C^{Steg}(D_{1},D_{2})$$ $$\displaystyle=$$ $$\displaystyle\lim_{L\to\infty}C_{L}^{Steg}(D_{1},D_{2}),$$ (11) where $$\displaystyle C_{L}^{Steg}(D_{1},D_{2})\triangleq\max_{p_{XU|S}\in{\mathcal{Q}% }^{Steg}(L,p_{S},D_{1})}\min_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}J_{L}(p_{S}% ,p_{XU|S},p_{Y|X})$$ (12) and $(U,S)\to X\to Y$ forms a Markov chain. The proof of Theorem 1 is given in two parts. The converse part is proved in Appendix B. The direct part is a corollary of a stronger result stated in Proposition 2 below, which provides a lower bound on the achievable error exponent (hence an upper bound on the average probability of error) and is proved in Appendix C. Proposition 2 Under Def. 1 for steganographic codes and Def. 5 for the compound attack channel, the following random-coding error exponent is achievable: $$E_{r}^{Steg}(R)=\lim_{L\to\infty}E_{r,L}^{Steg}(R),$$ (13) where $$\displaystyle E_{r,L}^{Steg}(R)$$ $$\displaystyle\triangleq$$ $$\displaystyle\min_{\tilde{p}_{S}\in\mathcal{P}_{S}}\max_{p_{XU|S}\in{\mathcal{% Q}}^{Steg}(L,\tilde{p}_{S},D_{1})}\min_{\tilde{p}_{Y|XUS}\in{\mathcal{P}}_{Y|% XUS}}\min_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}$$ $$\displaystyle\left[D(\tilde{p}_{S}\,p_{XU|S}\,\tilde{p}_{Y|XUS}||p_{S}\,p_{XU|% S}\,p_{Y|X})+\left|J_{L}(\tilde{p}_{S},p_{XU|S},\tilde{p}_{Y|XUS})-R\right|^{+% }\right].$$ Moreover, $E_{r}^{Steg}(R)=0$ if and only if $R\geq C^{Steg}$. Remark 1: The capacity and error exponent formulas in (11)—(2) coincide with those for public watermarking [33, 32], the only difference being that here the maximization over $p_{XU|S}$ is subject to a steganographic constraint. Clearly $E_{r,L}^{Steg}(R)\leq E_{r,L}^{PubWM}(R)$ and $C^{Steg}\leq C^{PubWM}$. Remark 2: Proposition 2 is proved using a random binning technique. First we establish the existence of a deterministic prototype CCC code with order-1 steganographic property, maximum distortion $D_{1}$, and error exponent $E^{Steg}(R)$. The decoder is an MPMI decoder. The main steps in this part of the proof are similar to those in the proof of Theorem 3.2 in [33], with the additional order-1 steganographic constraint on the encoder. The second part of the proof is an application of Proposition 1: random modulation of the CCC prototype code yields a perfectly-secure steganographic code with maximum distortion $D_{1}$ and error exponent $E^{Steg}(R)$. Remark 3. As mentioned earlier, the covertext plays no role in the special case $D_{1}\geq D_{\max}$, and so Alice can generate ${\mathbf{X}}$ independently of ${\mathbf{S}}$. The capacity formula (11) becomes simply $$C^{Steg}=\min_{p_{Y|S}\in{\mathcal{A}}(p_{S},D_{2})}I(S;Y),$$ and the random-coding exponent is $$E_{r}^{Steg}(R)=\min_{{\tilde{p}}_{S}}\min_{{\tilde{p}}_{Y|S}\in{\mathcal{P}}_% {Y|S}}\min_{p_{Y|S}\in{\mathcal{A}}(p_{S},D_{2})}\left[D({\tilde{p}}_{Y|S}\,{% \tilde{p}}_{S}\|p_{Y|S}\,p_{S})+|I_{{\tilde{p}}_{S}{\tilde{p}}_{Y|S}}(S;Y)-R|^% {+}\right].$$ The binning codes are degenerate in this case; the expressions for capacity and random-coding exponents reduce to classical formulas for compound DMCs without side information [34] and are achieved using constant-composition codes. Further specializing this result to the case of a passive warden ($D_{2}=0$, hence $p_{Y|X}=\mathds{1}_{\{Y=X\}})$, we obtain $C^{Steg}=H(S)$ and $E_{r}^{Steg}(R)$ is given by (21), see Section VII. The operation of the deterministic prototype code is illustrated in Fig. 2. The codebook ${\mathcal{C}}$ consists of a stack of codeword arrays indexed by the possible covertext sequence types. Given an input ${\mathbf{s}}$, the encoder evaluates its type $p_{{\mathbf{s}}}$ and selects the corresponding codeword array $${\mathcal{C}}(p_{\mathbf{s}})=\{{\mathbf{u}}(l,m,p_{\mathbf{s}}),\;1\leq l\leq 2% ^{N\rho(p_{\mathbf{s}})},\;1\leq m\leq|{\mathcal{M}}|\},$$ (15) in which the codewords are drawn from an optimized type class $T_{\mathbf{u}}\triangleq T_{U}^{*}(p_{\mathbf{s}})$. Each array ${\mathcal{C}}(p_{\mathbf{s}})$ has $|{\mathcal{M}}|$ columns and $2^{N\rho(p_{\mathbf{s}})}$ rows, where $\rho(p_{\mathbf{s}})$ is a function of the corresponding covertext type $p_{\mathbf{s}}$ and is termed the depth parameter of the array. Given ${\mathbf{y}}$, the decoder seeks a codeword in ${\mathcal{C}}=\bigcup_{p_{\mathbf{s}}}{\mathcal{C}}(p_{\mathbf{s}})$ that maximizes the penalized empirical mutual information and outputs its column index as the estimated message: $$\hat{m}=\arg\max_{m}\max_{l,p_{\mathbf{s}}}\left[I({\mathbf{u}}(l,m,p_{\mathbf% {s}});{\mathbf{y}})-\rho(p_{\mathbf{s}})\right].$$ (16) By letting $\rho(p_{\mathbf{s}})=I({\mathbf{u}};{\mathbf{s}})+\epsilon$, where $T_{{\mathbf{u}}{\mathbf{s}}}\triangleq T_{US}^{*}(p_{\mathbf{s}})$ is an optimized joint type and $\epsilon$ is an arbitrarily small positive number, an optimal balance between the probability of encoding error and the probability of decoding error is achieved. The former vanishes double-exponentially while the latter vanishes at a rate given by the random coding error exponent in (2). The above MPMI decoder can be thought of as an empirical generalized maximum a posterior (MAP) decoder [33, Section 3.1]. VI Secret Key In standard information-hiding problems with a compound DMC attack channel, deterministic codes are enough to achieve capacity; random coding is used as a method of proof to establish the existence of a deterministic code without actually specifying the code [37]. In our steganography problem, a randomized code is used to satisfy the perfect-undetectability condition of (2). Without the secret key, a deterministic code generally could not satisfy the perfect-undetectability condition. Also note that a randomized code is generally needed if the attacks have arbitrary memory [31, 32, 33]. For example, in watermarking games, knowing a deterministic code the adversary would decode and remove the message; deterministic codes are vulnerable to this kind of “surgical attack” [4]. For randomized codes, the secret key shared between encoder and decoder is the source of common randomness. For RM codes, the secret key specifies the value of the permutation $\pi$. The entropy rate of the secret key is $$\displaystyle H_{K}^{RM}=\frac{1}{N}\log_{2}N!<\log_{2}N.$$ (17) VII Passive Warden A passive warden introduces no degradation to the stegotext; in this case, $D_{2}=0$ and $Y=X$, i.e., $$p_{Y|X}=\mathds{1}_{\{Y=X\}}.$$ (18) This results in simplified expressions for the perfectly secure steganographic capacity in (11) and the random-coding error exponent in (13), see Propositions 19 and 4 below. The proofs of these propositions appear in Appendices D and E, respectively. Proposition 3 For the passive-warden case ($D_{2}=0$), the maximization in (12) is achieved by $U=X$ and $$C^{Steg}(D_{1},0)=\max_{p_{X|S}\in{\mathcal{Q}}_{1}^{Steg}(p_{S},D_{1})}H(X|S).$$ (19) Remark. Since $H(X|S)=H(X)-I(S;X)=H(S)-I(S;X)$, we have $$C^{Steg}(D_{1},0)=H(S)-\min_{p_{X|S}\in{\mathcal{Q}}^{Steg}(p_{S},D_{1})}I(S;X).$$ For the problem of encoding a source $S$ subject to distortion $D_{1}$, the minimum rate for representing the source is given by the rate-distortion function $$R_{S}(D_{1})=\min_{p_{X|S}~{}:~{}{\mathsf{E}}\,{\mathsf{d}}(S,X)\leq D_{1}}I(S% ;X)\leq\min_{p_{X|S}\in{\mathcal{Q}}_{1}^{Steg}(p_{S},D_{1})}I(S;X)$$ where the inequality holds because $p_{X|S}\in{\mathcal{Q}}_{1}^{Steg}(p_{S},D_{1})$ implies ${\mathsf{E}}\,{\mathsf{d}}(S,X)\leq D_{1}$. Hence $$C^{Steg}(D_{1},0)\leq H(S)-R_{S}(D_{1})$$ (20) and the capacity-achieving codes for the passive-warden case are analogous to rate-distortion codes. Equality holds in (20) if the distribution that achieves the rate-distortion bound satisfies the steganographic property $p_{X}=p_{S}$. Proposition 4 For the passive-warden case ($D_{2}=0$), the random-coding exponent is given by $$E_{r}^{Steg}(R)=\min_{\tilde{p}_{S}\in\mathcal{P}_{S}}\max_{p_{X|S}\in{% \mathcal{Q}}_{1}^{Steg}(\tilde{p}_{S},D_{1})}\left[D(\tilde{p}_{S}||p_{S})+% \left|H_{\tilde{p}_{S},p_{X|S}}(X|S)-R\right|^{+}\right].$$ (21) VIII Penalty for Perfect Security The capacity expressions for public watermarking in [27, 32] and for steganography in (11) take the same form, except that here the maximization of $p_{XU|S}$ is subject to the steganographic constraint. Consequently, we have $$C^{Steg}\leq C^{PubWM}$$ (22) and similarly $$E_{r}^{Steg}(R)\leq E_{r}^{PubWM}(R).$$ (23) For some special cases, it is possible that the optimal covert channel for public watermarking automatically satisfies the perfect security condition, and (22) and (23) hold with equality. Proposition 5 below states sufficient conditions on the covertext PMF $p_{S}$ and the distortion function ${\mathsf{d}}(\cdot,\cdot)$ that ensure the perfect security constraint causes no penalty in communication performance. We consider ${\mathcal{S}}=\mathbb{Z}_{q}=\{0,\,1,\,2,\,\cdots,\,q-1\}$, which is a group under addition modulo $q$. We shall use the notation $\underline{k}\triangleq k\mod q$. The covertext $S$ is uniformly distributed over $\mathbb{Z}_{q}$, i.e., $$p_{S}=\mathbb{U}({\mathcal{S}}).$$ The associated distortion function ${\mathsf{d}}:{\mathcal{S}}\times{\mathcal{S}}\to\mathbb{R}^{+}\cup\{0\}$ satisfies $${\mathsf{d}}(i,\,i)=0\mbox{ and }{\mathsf{d}}(i,\,j)={\mathsf{d}}(0,\,% \underline{j-i}),$$ If we write $\{{\mathsf{d}}(i,\,j)\}_{i,\,j=0}^{q-1}$ in a matrix form, the distortion matrix is cyclic-Toeplitz. Definition 9 Let ${\mathcal{V}}\triangleq\{0,\,1,\,\cdots,\,L-1\}$, $p_{S}=\mathbb{U}({\mathcal{S}})$, and ${\mathcal{U}}\triangleq\{0,\,1,\,2,\,\cdots,\,qL-1\}$. Given any covert channel $p_{XV|S}\in{\mathcal{Q}}(L,p_{S},D_{1})$, where $v\in{\mathcal{V}}$, we define an associated covert channel $p_{XU|S}\in{\mathcal{P}}_{XU|S}$, where $U\in{\mathcal{U}}$, by $$p_{XU|S}\left(x,\,qv+i\big{|}s\right)=\frac{1}{q}\,p_{XV|S}\left({\underline{x% -i}},\,v\big{|}{\underline{s-i}}\right),\;\forall\,v\in{\mathcal{V}},\;\forall% \,i,\;s,\;x\in{\mathcal{S}}.$$ (24) For any stochastic matrix $p_{XV|S}\in{\mathcal{Q}}(L,p_{S},D_{1})$, by (24), the new channel $p_{XU|S}$ contains all of its $q$ cyclically shifted versions (with respect to $X$ and $S$) and these shifted versions are equally likely. Since the distortion function is cyclic, it is easy to verify that $$\mathsf{E}_{p_{S},p_{XU|S}}[{\mathsf{d}}(S,X)]=\mathsf{E}_{p_{S},p_{XV|S}}[{% \mathsf{d}}(S,X)]\leq D_{1}.$$ Moreover, the marginal PMF $\hat{p}_{X}$ induced by $p_{S}=\mathbb{U}({\mathcal{S}})$ and $p_{XU|S}$ is given by $$\hat{p}_{X}(x)=\frac{1}{q}\sum_{i=0}^{q-1}p_{X}(\underline{x-i})=\frac{1}{q}% \equiv p_{S}(x),\quad\forall\,x\in{\mathcal{S}},$$ (25) where $p_{X}$ is the marginal PMF induced by $p_{S}=\mathbb{U}({\mathcal{S}})$ and $p_{XV|S}\in{\mathcal{Q}}(L,p_{S},D_{1})$. That is, $$p_{XU|S}\in{\mathcal{Q}}^{Steg}(qL,p_{S},D_{1}).$$ Definition 10 The class ${\mathcal{Q}}_{cyc}^{Steg}(qL,p_{S},D_{1})$ is the set of all such $p_{XU|S}$ defined in (24). Clearly, we have $${\mathcal{Q}}_{cyc}^{Steg}(qL,p_{S},D_{1})\subset{\mathcal{Q}}^{Steg}(qL,p_{S}% ,D_{1})\subset{\mathcal{Q}}(qL,p_{S},D_{1}).$$ (26) Definition 11 The class of cyclic attack channels subject to distortion $D_{2}$ is defined as $$\displaystyle{{\mathcal{A}}}_{cyc}(D_{2})$$ $$\displaystyle\triangleq$$ $$\displaystyle\Big{\{}p_{Y|X}\in{\mathcal{P}}_{Y|X}:\;p_{Y|X}(y|x)=p_{Y|X}(% \underline{y-x}\,|\,0\,),\quad\forall\,x,\,y\in{\mathcal{S}},$$ (27) $$\displaystyle\quad\quad\quad\quad\mbox{ and }\quad\frac{1}{q}\sum_{y=0}^{q-1}p% _{Y|X}(y|0)\,{\mathsf{d}}(y,0)\leq D_{2}\Big{\}}.$$ Any stochastic matrix $p_{Y|X}\in{{\mathcal{A}}}_{cyc}(D_{2})$ is cyclic-Toeplitz. Also note that for any $p_{X}\in{\mathcal{P}}_{X}$, $${{\mathcal{A}}}_{cyc}(D_{2})\subset{\mathcal{A}}(p_{X},D_{2}).$$ (28) Proposition 5 For the above $q$-ary information-hiding problem, the capacities for both the perfectly secure steganography game and the public watermarking game are the same. That is, the perfect security constraint in (2) does not cause any capacity loss. Moreover, there is no loss of optimality in restricting the maximization in (12) to ${\mathcal{Q}}^{Steg}_{cyc}(qL,p_{S},D_{1})$ and the minimization to ${{\mathcal{A}}}_{cyc}(D_{2})$: $$\displaystyle C^{PubWM}(D_{1},D_{2})$$ $$\displaystyle=$$ $$\displaystyle C^{Steg}(D_{1},D_{2})$$ (29) $$\displaystyle=$$ $$\displaystyle\lim_{L\to\infty}\max_{p_{XU|S}\in{\mathcal{Q}}^{Steg}_{cyc}(qL,p% _{S},D_{1})}\min_{p_{Y|X}\in{{\mathcal{A}}}_{cyc}(D_{2})}J_{L}(p_{S},p_{XU|S},% p_{Y|X}).$$ The proof is given in Appendix F. IX Example: Binary-Hamming Case We illustrate the above results through the following example, where ${\mathcal{S}}=\{0,1\}$, and the covertext is Bernoulli($\frac{1}{2}$) sequence, i.e., $$Pr[S=1]=Pr[S=0]=\frac{1}{2}.$$ The Hamming distortion metric is used: ${\mathsf{d}}(x,y)=\mathds{1}_{\{x\neq y\}}$. IX-A Capacity The capacity in the public watermarking game setting is given in [33] as follows $$C=\left\{\begin{array}[]{ll}\frac{D_{1}}{d_{D_{2}}}[h(d_{D_{2}})-h(D_{2})],&% \mbox{if}\;0\leq D_{1}\leq d_{D_{2}};\\ h(D_{1})-h(D_{2}),&\mbox{if}\;d_{D_{2}}\leq D_{1}\leq 1/2;\\ 1-h(D_{2}),&\mbox{if}\;D_{1}>1/2,\end{array}\right.$$ (30) where $d_{D_{2}}=1-2^{-h(D_{2})}$. When $D_{2}=0$, $$C=\left\{\begin{array}[]{ll}h(D_{1})&\mbox{if}\;0\leq D_{1}\leq 1/2;\\ 1&\mbox{if}\;D_{1}\geq 1/2.\\ \end{array}\right.$$ (31) Fig. 3 shows the above two capacity functions. The optimal attack channel is a binary symmetric channel (BSC) with crossover probability $D_{2}$. If $d_{D_{2}}\leq D_{1}\leq 1/2$, the optimal covert channel is also a binary symmetric channel: BSC($D_{1}$) (i.e., $|{\mathcal{U}}|=2$, $U=X$, and $p_{XU|S}=p_{X|S}$); otherwise, the capacity is achieved by time sharing: no embedding on a fraction of $1-\frac{D_{1}}{d_{D_{2}}}$ samples and embedding with the optimal covert channel BSC($d_{D_{2}}$) on the rest of samples. Since the covertext ${\mathbf{S}}$ is a Bernoulli($\frac{1}{2}$) sequence, the output of the above optimal BSC($p$) covert channel is also Bernoulli($\frac{1}{2}$). That is, the optimal covert channel for the public watermarking game satisfies $p_{X}=p_{S}$, and the perfect security constraint does not cause any loss in capacity, as stated by Proposition 5. IX-B Random-Coding Exponent In [33], we numerically computed the random-coding exponent for public watermarking in the case of $D_{1}=0.4$, $D_{2}=0.2$, and $|\mathcal{U}|=2$ as shown in Fig. 4. We found that the optimal covert channel is still a BSC($D_{1}$) ($p_{XU|S}=p_{X|S}$) with the time sharing strategy. It implies that at least for the case of $|{\mathcal{U}}|=2$, $p_{X}=p_{S}$ and the perfect security constraint causes no loss in random-coding exponent either. IX-C Randomized Nested Linear Codes—A Capacity-Achieving Code Construction For information-embedding problems with a fixed attack channel BSC($D_{2}$), deterministic nested binary linear codes were proposed to achieve capacity, where ${\mathcal{C}}_{1}$, a good source code with Hamming distance $D_{1}$, is nested in ${\mathcal{C}}_{2}$, a good channel code over BSC$(D_{2})$ [38, 39]. When $|{\mathcal{C}}_{2}|\doteq 2^{N\left[1-h(D_{2})\right]}$ and $|{\mathcal{C}}_{1}|\doteq 2^{N\left[1-h(D_{1})\right]}$, the asymptotic code rate $$R=\lim_{N\to\infty}\frac{1}{N}\log_{2}\frac{|{\mathcal{C}}_{2}|}{|{\mathcal{C}% }_{1}|}=h(D_{1})-h(D_{2})$$ is equal to the capacity in the regime $D_{1}\geq d_{D_{2}}$. In the regime $D_{1}<d_{D_{2}}$, the time-sharing strategy of (30) is applied. These nested linear codes apply to both public watermarking and steganography because BSC($D_{2}$) is the optimal discrete memoryless attack channel. The stegotext codewords are elements of ${\mathcal{C}}_{2}$ [38, 39]. In the passive-warden case ($D_{2}=0$), we simply let ${\mathcal{C}}_{2}=\mathbb{F}_{2}^{N}$, and perfect security is achieved even without a secret key. In the active-warden case, ${\mathcal{C}}_{2}$ is a subgroup of $\mathbb{F}_{2}^{N}$, and randomization via the secret key plays an essential role in achieving perfect security. The strategy described below makes the transmitted stegotext uniformly distributed over $\mathbb{F}_{2}^{N}$. The resulting code is a randomized nested binary linear code. Partition the whole space $\mathbb{F}_{2}^{N}$ into a disjoint union of ${\mathcal{C}}_{2}$ and its cosets: $$\mathbb{F}_{2}^{N}=\bigcup_{{\mathbf{c}}\in\Omega_{2}}{\mathcal{C}}_{2}\oplus{% \mathbf{c}},$$ (32) where ${\mathcal{C}}_{2}\oplus{\mathbf{c}}$ is a coset of ${\mathcal{C}}_{2}$, the element ${\mathbf{c}}\in\Omega_{2}$ is a coset leader, and the set $\Omega_{2}$ contains all coset leaders. We have $$|\Omega_{2}|=\frac{2^{N}}{|{\mathcal{C}}_{2}|}\doteq 2^{Nh(D_{2})}.$$ (33) Let the secret key ${\mathbf{K}}$ be uniformly distributed over $\Omega_{2}$. For any ${\mathbf{k}}\in\Omega_{2}$, the encoder output is defined as $${\mathbf{x}}=f_{N}^{{\mathbf{k}}}(m,{\mathbf{s}})=f_{N}^{\mathbf{0}}(m,{% \mathbf{s}}\oplus{\mathbf{k}})\oplus{\mathbf{k}},$$ (34) where $f_{N}^{\mathbf{0}}(\cdot,\cdot)$ is the deterministic encoder used for the information-embedding or watermarking problem. The decoding function is $$\hat{m}=\phi_{N}^{{\mathbf{k}}}({\mathbf{y}})=\phi_{N}^{\mathbf{0}}({\mathbf{y% }}\oplus{\mathbf{k}}),$$ (35) where $\phi_{N}^{\mathbf{0}}(\cdot)$ is the decoder associated with $f_{N}^{\mathbf{0}}(\cdot)$. Since the output of the deterministic encoder is uniformly distributed over ${\mathcal{C}}_{2}$ and the secret key ${\mathbf{K}}$ is uniformly distributed over $\Omega_{2}$, the coset decomposition property (32) ensures that the randomized encoder output of (34) is uniformly distributed over $\mathbb{F}_{2}^{N}$. Hence perfect security is achieved. By (33) the entropy rate of the secret key is $h(D_{2})$, unlike the $\log_{2}N$ growth required for general RM codes in (17). X Conclusion A strict definition of perfect security has been adopted in this paper, implying that even a warden with unlimited computational resources is unable to reliably detect the presence of a hidden message. We have studied the Shannon-theoretic limits of communication performance under this perfect-security requirement and studied the structure of codes that asymptotically achieve those limits. The main results are summarized below. • Perfectly secure steganography is closely related to the public watermarking problem of [27, 33]. Positive capacity and random-coding exponents are achieved using stacked-binning codes and an MPMI decoder. • Randomized codes are generally needed to achieve perfect security. The common randomness is provided by a secret key shared between the encoder and decoder. For i.i.d. covertexts, Proposition 1 shows that perfectly secure steganographic codes can be constructed using randomized permutations of a prototype CCC watermarking code that merely has an order-1 security property, i.e., the prototype code matches the first-order marginals of the covertext and stegotext, but not the full $N$-dimensional statistics. • The cost of perfect security in terms of communication performance is the same as the cost of order-1 security. However, if the covertext distribution is uniform and the distortion metric is cyclically symmetric, the security constraint does not cause any loss of performance. Computational Security. This paper has focused on the interplay between communication performance and information-theoretic security, where security is achieved using a private key that is uniformly distributed over a group ${\mathcal{G}}^{sub}$. A more practical setup would involve a public-key system, in which a reduced set of representers of ${\mathcal{G}}^{sub}$ is selected, each corresponding to a value of the key. Assume that the uniform distribution over this reduced set is computationally indistinguishable (in a sense to be precisely defined) from the uniform distribution over ${\mathcal{G}}^{sub}$. The resulting steganographic code is no longer perfectly secure but inherits the computational security of the key generation mechanism. Thus the framework analyzed in this paper can form the basis for constructing computationally secure steganographic codes that have near-optimal communication performance. Extensions. Our basic framework can also be used to analyze complex problems involving covertexts with Markov dependencies and covertexts defined over continuous alphabets [40, Sec. X]. While such extensions are technically challenging, we hope that the mathematical structure of optimal codes identified in this paper under simplifying assumptions will shed some light on the development of practical codes with high communication performance. Appendix A Proof of Prop 1 First we verify the perfect security condition. For RM codes (Def. 3), we have $$p_{{\mathbf{X}}|\pi,{\mathbf{S}},M}({\mathbf{x}}|\pi,{\mathbf{s}},m)=\mathds{1% }_{\{\pi{\mathbf{x}}=f_{N}(\pi{\mathbf{s}},m)\}}.$$ Also note that for any ${\mathbf{x}},{\mathbf{z}}\in T_{{\mathbf{s}}}$, there exists a permutation $\pi_{0}$ such that ${\mathbf{x}}=\pi_{0}{\mathbf{z}}$. Hence the value of the sum $\sum_{\pi}\mathds{1}_{\{\pi{\mathbf{x}}={\mathbf{z}}\}}$ is independent of ${\mathbf{z}}$ (conditioned on ${\mathbf{z}}\in T_{{\mathbf{s}}}$), and so $$\sum_{\pi}\mathds{1}_{\{\pi{\mathbf{x}}={\mathbf{z}}\}}=\frac{1}{|T_{{\mathbf{% s}}}|}\sum_{{\mathbf{z}}\in T_{{\mathbf{s}}}}\sum_{\pi}\mathds{1}_{\{\pi{% \mathbf{x}}={\mathbf{z}}\}}=\frac{1}{|T_{{\mathbf{s}}}|}\sum_{\pi}1=\frac{N!}{% |T_{{\mathbf{s}}}|}.$$ (36) Hence for any type class $T_{{\mathbf{s}}}$ we have $$\displaystyle p_{{\mathbf{X}}|T_{{\mathbf{s}}}}({\mathbf{x}}|T_{{\mathbf{s}}})$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{N!}\sum_{\pi}\frac{1}{|{\mathcal{M}}|}\sum_{m\in{% \mathcal{M}}}\frac{1}{|T_{{\mathbf{s}}}|}\sum_{{\mathbf{s}}^{\prime}\in T_{{% \mathbf{s}}}}p_{{\mathbf{X}}|\pi,{\mathbf{S}},M}({\mathbf{x}}|\pi,{\mathbf{s}}% ^{\prime},m)$$ (37) $$\displaystyle=$$ $$\displaystyle\frac{1}{N!}\sum_{\pi}\frac{1}{|{\mathcal{M}}|}\sum_{m\in{% \mathcal{M}}}\frac{1}{|T_{{\mathbf{s}}}|}\sum_{{\mathbf{s}}^{\prime}\in T_{{% \mathbf{s}}}}\mathds{1}_{\{\pi{\mathbf{x}}=f_{N}(\pi{\mathbf{s}}^{\prime},m)\}}$$ $$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}$$ $$\displaystyle\frac{1}{N!}\sum_{\pi}\frac{1}{|{\mathcal{M}}|}\sum_{m\in{% \mathcal{M}}}\frac{1}{|T_{{\mathbf{s}}}|}\sum_{{\mathbf{s}}^{\prime\prime}\in T% _{{\mathbf{s}}}}\mathds{1}_{\{\pi{\mathbf{x}}=f_{N}({\mathbf{s}}^{\prime\prime% },m)\}}$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{|{\mathcal{M}}|}\sum_{m\in{\mathcal{M}}}\frac{1}{|T_{{% \mathbf{s}}}|}\sum_{{\mathbf{s}}^{\prime\prime}\in T_{{\mathbf{s}}}}\frac{1}{N% !}\sum_{\pi}\mathds{1}_{\{\pi{\mathbf{x}}=f_{N}({\mathbf{s}}^{\prime\prime},m)\}}$$ $$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}$$ $$\displaystyle\frac{1}{|{\mathcal{M}}|}\sum_{m\in{\mathcal{M}}}\frac{1}{|T_{{% \mathbf{s}}}|}\sum_{{\mathbf{s}}^{\prime\prime}\in T_{{\mathbf{s}}}}\frac{1}{|% T_{{\mathbf{s}}}|}\,\mathds{1}_{\{{\mathbf{x}}\in T_{{\mathbf{s}}}\}}$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{|T_{{\mathbf{s}}}|}\,\mathds{1}_{\{{\mathbf{x}}\in T_{{% \mathbf{s}}}\}},$$ where in (a) we have made the change of variables ${\mathbf{s}}^{\prime\prime}=\pi{\mathbf{s}}^{\prime}$, and in (b) we have used (36) with ${\mathbf{z}}=f_{N}({\mathbf{s}}^{\prime\prime},m)$. From (37) we obtain $$p_{{\mathbf{X}}}({\mathbf{x}})=\sum_{T_{{\mathbf{s}}}}p_{{\mathbf{S}}}(T_{{% \mathbf{s}}})\,p_{{\mathbf{X}}|T_{{\mathbf{s}}}}({\mathbf{x}}|T_{{\mathbf{s}}}% )=\sum_{T_{{\mathbf{s}}}}p_{{\mathbf{S}}}(T_{{\mathbf{s}}})\,\frac{1}{|T_{{% \mathbf{s}}}|}\,\mathds{1}_{\{{\mathbf{x}}\in T_{{\mathbf{s}}}\}}=p_{{\mathbf{% S}}}({\mathbf{x}}),\quad\forall{\mathbf{x}}\in{\mathcal{S}}^{N},$$ hence the perfect security condition (2) is satisfied. Now verifying the maximum-distortion constraint (1), for every $\pi$ we have $${\mathsf{d}}^{N}({\mathbf{s}},f_{N}^{\pi}({\mathbf{s}},m))\stackrel{{% \scriptstyle(a)}}{{=}}{\mathsf{d}}^{N}({\mathbf{s}},\pi^{-1}f_{N}(\pi{\mathbf{% s}},m))\stackrel{{\scriptstyle(b)}}{{=}}{\mathsf{d}}^{N}(\pi{\mathbf{s}},f_{N}% (\pi{\mathbf{s}},m))\stackrel{{\scriptstyle(c)}}{{\leq}}D_{1}$$ where (a) uses the definition of $f_{N}^{\pi}$ in (4), (b) holds because the distortion measure is additive, and (c) holds because of our initial assumption on the prototype $f_{N}$. Therefore (1) holds. Finally, let us evaluate the error probability for the RM code. Since the covertext source and the attack channel are memoryless, we have $$p_{S}^{N}({\mathbf{s}})=p_{S}^{N}(\pi{\mathbf{s}})\quad\mathrm{and}\quad p_{Y|% X}^{N}({\mathbf{y}}|{\mathbf{x}})=p_{Y|X}^{N}(\pi{\mathbf{y}}|\pi{\mathbf{x}})$$ (38) for any permutation $\pi$. The error probability for the prototype code takes the form $$\displaystyle P_{e,N}(f_{N},\phi_{N},p_{Y|X})$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{|{\mathcal{M}}|}\sum_{m\in{\mathcal{M}}}\sum_{{\mathbf{s% }}\in{\mathcal{S}}^{N}}p_{S}^{N}({\mathbf{s}})\sum_{{\mathbf{x}}\in{\mathcal{S% }}^{N}}\mathds{1}_{\{{\mathbf{x}}=f_{N}({\mathbf{s}},m)\}}\sum_{{\mathbf{y}}% \in{\mathcal{S}}^{N}}p_{Y|X}^{N}({\mathbf{y}}|{\mathbf{x}})\,\mathds{1}_{\{% \phi_{N}({\mathbf{y}})\neq m\}}.$$ For the prototype code modulated with permutation $\pi$, we have $$\displaystyle P_{e,N}(f_{N}^{\pi},\phi_{N}^{\pi},p_{Y|X})$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{|{\mathcal{M}}|}\sum_{m\in{\mathcal{M}}}\sum_{{\mathbf{s% }}\in{\mathcal{S}}^{N}}p_{S}^{N}({\mathbf{s}})\sum_{{\mathbf{x}}\in{\mathcal{S% }}^{N}}\mathds{1}_{\{\pi{\mathbf{x}}=f_{N}(\pi{\mathbf{s}},m)\}}\sum_{{\mathbf% {y}}\in{\mathcal{S}}^{N}}p_{Y|X}^{N}({\mathbf{y}}|{\mathbf{x}})\,\mathds{1}_{% \{\phi_{N}(\pi{\mathbf{y}})\neq m\}}$$ (39) $$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}$$ $$\displaystyle\frac{1}{|{\mathcal{M}}|}\sum_{m\in{\mathcal{M}}}\sum_{{\mathbf{s% }}\in{\mathcal{S}}^{N}}p_{S}^{N}(\pi{\mathbf{s}})\sum_{{\mathbf{x}}\in{% \mathcal{S}}^{N}}\mathds{1}_{\{\pi{\mathbf{x}}=f_{N}(\pi{\mathbf{s}},m)\}}\sum% _{{\mathbf{y}}\in{\mathcal{S}}^{N}}p_{Y|X}^{N}(\pi{\mathbf{y}}|\pi{\mathbf{x}}% )\,\mathds{1}_{\{\phi_{N}(\pi{\mathbf{y}})\neq m\}}$$ $$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}$$ $$\displaystyle\frac{1}{|{\mathcal{M}}|}\sum_{m\in{\mathcal{M}}}\sum_{\pi^{-1}{% \mathbf{s}}^{\prime}\in{\mathcal{S}}^{N}}p_{S}^{N}({\mathbf{s}}^{\prime})\sum_% {\pi^{-1}{\mathbf{x}}^{\prime}\in{\mathcal{S}}^{N}}\mathds{1}_{\{{\mathbf{x}}^% {\prime}=f_{N}({\mathbf{s}}^{\prime},m)\}}\sum_{\pi^{-1}{\mathbf{y}}^{\prime}% \in{\mathcal{S}}^{N}}p_{Y|X}^{N}({\mathbf{y}}^{\prime}|{\mathbf{x}}^{\prime})% \,\mathds{1}_{\{\phi_{N}({\mathbf{y}}^{\prime})\neq m\}}$$ $$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}$$ $$\displaystyle\frac{1}{|{\mathcal{M}}|}\sum_{m\in{\mathcal{M}}}\sum_{{\mathbf{s% }}^{\prime}\in{\mathcal{S}}^{N}}p_{S}^{N}({\mathbf{s}}^{\prime})\sum_{{\mathbf% {x}}^{\prime}\in{\mathcal{S}}^{N}}\mathds{1}_{\{{\mathbf{x}}^{\prime}=f_{N}({% \mathbf{s}}^{\prime},m)\}}\sum_{{\mathbf{y}}^{\prime}\in{\mathcal{S}}^{N}}p_{Y% |X}^{N}({\mathbf{y}}^{\prime}|{\mathbf{x}}^{\prime})\,\mathds{1}_{\{\phi_{N}({% \mathbf{y}}^{\prime})\neq m\}}$$ $$\displaystyle=$$ $$\displaystyle P_{e,N}(f_{N},\phi_{N},p_{Y|X}),$$ where (a) holds because of (38), (b) is obtained using the change in variables ${\mathbf{s}}^{\prime}=\pi{\mathbf{s}}$, ${\mathbf{x}}^{\prime}=\pi{\mathbf{x}}$, ${\mathbf{y}}^{\prime}=\pi{\mathbf{y}}$, and (c) holds because the three sums run over all elements (${\mathbf{s}}^{\prime},{\mathbf{x}}^{\prime},{\mathbf{y}}^{\prime}$) of ${\mathcal{S}}^{N}\times{\mathcal{S}}^{N}\times{\mathcal{S}}^{N}$, and so the order of summation is inconsequential. Since (39) holds for every permutation $\pi$, the error probability for the RM code is equal to $$P_{e,N}(F_{N},\Phi_{N},p_{Y|X})=\frac{1}{N!}\sum_{\pi}P_{e,N}(f_{N}^{\pi},\phi% _{N}^{\pi},p_{Y|X})=P_{e,N}(f_{N},\phi_{N},p_{Y|X}).$$ This completes the proof. Appendix B Converse Proof of Theorem 1 The converse is an extension of the proof in [33, Section 7]. Our upper bound on achievable rates is derived by • replacing the perfect-security constraint with a weaker order-1 security constraint on the encoder: $$p_{{\mathbf{x}}}=p_{{\mathbf{s}}}\quad\forall\,m,{\mathbf{s}},{\mathbf{x}}=f_{% N}({\mathbf{s}},m)$$ (40) (matching the types of input ${\mathbf{s}}$ and output ${\mathbf{x}}=f_{N}(s,m)$ of the encoder $f_{N}$), • replacing the almost-sure distortion constraint with an expected distortion constraint on the encoder: $$\frac{1}{|{\mathcal{M}}|}\sum_{{\mathbf{s}}\in{\mathcal{S}}^{N}}p_{S}^{N}({% \mathbf{s}})\,{\mathsf{d}}^{N}({\mathbf{s}},f_{N}({\mathbf{s}},m)),$$ (41) • and providing the decoder with knowledge of the attack channel $p_{Y|X}$. Clearly any upper bound we derive under these assumptions is an upper bound on capacity as well. For any rate-$R$ code $(f_{N},\phi_{N})$ and DMC $p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})$, we have $$\displaystyle NR=H(M)$$ $$\displaystyle=$$ $$\displaystyle H(M|{\mathbf{Y}})+I(M;{\mathbf{Y}})$$ $$\displaystyle\leq$$ $$\displaystyle 1+P_{e}(f_{N},\phi_{N},p^{N}_{Y|X})\,NR+I(M;{\mathbf{Y}}),$$ where the inequality is due to Fano’s inequality. In order for $P_{e}$ not to be bounded away from 0, rate $R$ needs to satisfy $$\displaystyle NR-1\leq\min_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}I(M;{\mathbf{% Y}}).$$ (42) The joint PMF of $(M,{\mathbf{S}},{\mathbf{X}},{\mathbf{Y}})$ is given by $$\displaystyle p_{M{\mathbf{S}}{\mathbf{X}}{\mathbf{Y}}|f_{N}}=p_{M}\,p_{S}^{N}% \,p_{Y|X}^{N}\,\mathds{1}_{\{{\mathbf{X}}=f_{N}({\mathbf{S}},M)\}}.$$ (43) Owing to (43), for any $1\leq i\leq N$, $(M,{\mathbf{S}},\{Y_{j}\}_{j\neq i})\to X_{i}\to Y_{i}$ forms a Markov chain and so does $$\displaystyle(W_{i},S_{i})\to X_{i}\to Y_{i},$$ (44) where the random variable $W_{i}$ is defined as $$\displaystyle W_{i}=(M,S_{i+1},\cdots,S_{N},Y_{1},\cdots,Y_{i-1}).$$ (45) Using the same set of inequalities as in [26, Lemma 4], we obtain $$\displaystyle I(M;{\mathbf{Y}})\leq\sum_{i=1}^{N}\,[I(W_{i};Y_{i})-I(W_{i};S_{% i})].$$ (46) We define a time sharing random variable $T$, which is uniformly distributed over $\{1,\cdots,N\}$ and independent of all other random variables, and define the quadruple of random variables $(W,S,X,Y)$ as $(W_{T},S_{T},X_{T},Y_{T})$. With this definition, the order-1 security constraint (40) becomes $p_{X}=p_{S}$, and the expected distortion constraint (41) becomes $\sum_{s,x}p_{S}(s)p_{X|S}(x|s)\,{\mathsf{d}}(s,x)\leq D_{1}$. Therefore $p_{X|S}\in{\mathcal{Q}}_{1}^{Steg}(p_{S},D_{1})$. By (45), the random variable $W$ is defined over an alphabet of cardinality $\exp_{2}\left\{N\left[R+\log|S|\,\right]\right\}$. Moreover $(W,S)\to X\to Y$ forms a Markov chain. Combining (42) and (46), we further derive $$\displaystyle R$$ $$\displaystyle\leq$$ $$\displaystyle\frac{1}{N}\min_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}I(M;{% \mathbf{Y}})$$ (47) $$\displaystyle\leq$$ $$\displaystyle\frac{1}{N}\min_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}\sum_{i=1}^% {N}[I(W_{i};Y_{i})-I(W_{i};S_{i})]$$ $$\displaystyle=$$ $$\displaystyle\min_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}\left[I(W;Y|T)-I(W;S|T% )\right]$$ $$\displaystyle=$$ $$\displaystyle\min_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}\left[I(W,T;Y)-I(W,T;S% )-I(T;Y)+I(T;S)\right]$$ $$\displaystyle\leq$$ $$\displaystyle\min_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}\left[I(U;Y)-I(U;S)% \right],$$ where $U=(W,T)$ is defined over an alphabet of cardinality $$L(N)=N\exp_{2}\{N\left[R+\log|S|\,\right]\},$$ (48) and the last inequality is due to $I(T;Y)\geq 0$ and $I(T;S)=0$ (since $T$ is independent of $S$). Since $p_{X|S}\in{\mathcal{Q}}_{1}^{Steg}(p_{S},D_{1})$, we have $p_{XU|S}\in{\mathcal{Q}}^{Steg}(L(N),p_{S},D_{1})$. Recall that $J_{L}(p_{S},p_{XU|S},p_{Y|X})\triangleq I(U;Y)-I(U;S)$ when $|{\mathcal{U}}|=L$, and that $$C_{L}^{Steg}\triangleq\max_{p_{XU|S}\in{\mathcal{Q}}^{Steg}(L,p_{S},D_{1})}% \min_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}J_{L}(p_{S},p_{XU|S},p_{Y|X}).$$ Following the same arguments as in [33], the sequence $C_{L}^{Steg}$ is nondecreasing and converges to a finite limit $$C^{Steg}\triangleq\lim_{L\to\infty}C_{L}^{Steg}=\lim_{L\to\infty}\max_{p_{XU|S% }\in{\mathcal{Q}}^{Steg}(L,p_{S},D_{1})}\min_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_% {2})}J_{L}(p_{S},p_{XU|S},p_{Y|X}).$$ Therefore, continuing with (47), $R$ is bounded by $$\displaystyle R$$ $$\displaystyle\leq$$ $$\displaystyle\min_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}\left[I(U;Y)-I(U;S)\right]$$ (49) $$\displaystyle=$$ $$\displaystyle\min_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}J_{L(N)}(p_{X},p_{XU|S% },p_{Y|X})$$ $$\displaystyle\leq$$ $$\displaystyle\sup_{L}\max_{p_{UX|S}\in{\mathcal{Q}}^{Steg}(L,p_{S},D_{1})}\min% _{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}J_{L}(p_{S},p_{XU|S},p_{Y|X})$$ $$\displaystyle=$$ $$\displaystyle\lim_{L\to\infty}\max_{p_{XU|S}\in{\mathcal{Q}}^{Steg}(L,p_{S},D_% {1})}\min_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}J_{L}(p_{S},p_{XU|S},p_{Y|X})$$ $$\displaystyle=$$ $$\displaystyle C^{Steg}.$$ This proves the converse part of Theorem 1. Appendix C Proof of Proposition 2 We have $$E_{r,L}^{Steg}(R)\leq E_{r,L}^{PubWM}(R).$$ Recall from [33, Lemma 3.1] that the sequence $E_{r,L}^{PubWM}(R)$ is nondecreasing and converges to a finite limit $E_{r}^{PubWM}(R)$ as $L\to\infty$. Using the same arguments as in [33, Lemma 3.1], it follows that the sequence $E_{r,L}^{Steg}(R)$ is nondecreasing and converges to a finite limit $E_{r}^{Steg}(R)$ as $L\to\infty$. Hence for any $\epsilon>0$ and $R$, there exists $L(\epsilon)$ such that $$E_{r,L}^{Steg}(R)\geq E_{r}^{Steg}(R)-\epsilon,\quad\forall\,L\geq L(\epsilon).$$ We next prove that for any $L$, a sequence of deterministic codes $(f_{N},\phi_{N})$ with order-1 steganographic security exist with the property that $$\lim_{N\to\infty}\left[-\frac{1}{N}\log\max_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{% 2})}P_{e}(f_{N},\phi_{N},p_{Y|X})\right]=E_{r,L}^{Steg}(R).$$ To prove the existence of such a code, we construct a random ensemble $\mathscr{C}$ of binning codes $(f_{N},\phi_{N})$ with auxiliary alphabet $\mathcal{U}\triangleq\{1,2,\cdots,L\}$ and show that the error probability averaged over $\mathscr{C}$ vanishes at rate $E_{r,L}^{Steg}(R)$ as $N$ goes to infinity. The proof is based on that of [33, Theorem 3.2] with special treatment on the encoder construction for perfect security. Assume that $R<C_{L}^{Steg}-\epsilon$. For any covertext type $p_{\mathbf{s}}$ and conditional type $p_{{\mathbf{x}}{\mathbf{u}}|{\mathbf{s}}}$, define the function $$\displaystyle E_{L,N}(R,p_{\mathbf{s}},p_{{\mathbf{x}}{\mathbf{u}}|{\mathbf{s}% }})$$ $$\displaystyle\triangleq$$ $$\displaystyle\min_{p_{{\mathbf{y}}|{\mathbf{x}}{\mathbf{u}}{\mathbf{s}}}}\min_% {p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}\Big{[}D(p_{\mathbf{s}}\,p_{{\mathbf{x}}% {\mathbf{u}}|{\mathbf{s}}}\,p_{{\mathbf{y}}|{\mathbf{x}}{\mathbf{u}}{\mathbf{s% }}}||p_{S}\,p_{{\mathbf{x}}{\mathbf{u}}|{\mathbf{s}}}\,p_{Y|X})$$ (50) $$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+|I({% \mathbf{u}};{\mathbf{y}})-I({\mathbf{u}};{\mathbf{s}})-\epsilon-R|^{+}\Big{]}.$$ Define ${\mathcal{Q}}^{Steg}(N,L,p_{{\mathbf{s}}},D_{1})$ as the set of conditional types $p_{{\mathbf{x}}|{\mathbf{u}}{\mathbf{s}}}$ that also belong to the set ${\mathcal{Q}}^{Steg}(L,p_{{\mathbf{s}}},D_{1})$ of feasible steganographic channels. If $p_{{\mathbf{x}}|{\mathbf{u}}{\mathbf{s}}}\in{\mathcal{Q}}^{Steg}(N,L,p_{{% \mathbf{s}}},D_{1})$ then (1) $p_{\mathbf{x}}=p_{\mathbf{s}}$, i.e., the stegotext sequence has the same type as the covertext sequence and the order-1 security condition is satisfied; (2) ${\mathsf{d}}^{N}({\mathbf{x}},{\mathbf{s}})\leq D_{1}$, i.e., distortion is no greater than $D_{1}$ for any choice of ${\mathbf{s}}$ and $m$. The set ${\mathcal{Q}}^{Steg}(N,L,p_{{\mathbf{s}}},D_{1})$ includes $p_{{\mathbf{x}}|{\mathbf{u}}{\mathbf{s}}}=\mathds{1}_{\{{\mathbf{x}}={\mathbf{% s}}\}}$ and is therefore nonempty. Now denote by $p_{{\mathbf{x}}|{\mathbf{u}}{\mathbf{s}}}$ the maximizer of (50) over the set ${\mathcal{Q}}^{Steg}(N,L,p_{{\mathbf{s}}},D_{1})$. As a result of this optimization, we may associate • to any covertext type $p_{\mathbf{s}}$, a type class $T^{*}_{U}(p_{\mathbf{s}})\triangleq T_{\mathbf{u}}$ and a mutual information $I^{*}_{US}(p_{\mathbf{s}})\triangleq I({\mathbf{u}};{\mathbf{s}})$; • to any covertext sequence ${\mathbf{s}}$, a conditional type class $T^{*}_{U|S}({\mathbf{s}})\triangleq T_{{\mathbf{u}}|{\mathbf{s}}}$; • to any sequences ${\mathbf{s}}$ and ${\mathbf{u}}\in T^{*}_{US}(p_{\mathbf{s}})$, a conditional type class $T^{*}_{X|US}({\mathbf{u}},{\mathbf{s}})\triangleq T_{{\mathbf{x}}|{\mathbf{u}}% {\mathbf{s}}}$. A random codebook ${\mathcal{C}}$ is the union of codeword arrays ${\mathcal{C}}(p_{\mathbf{s}})$ indexed by the covertext sequence type $p_{\mathbf{s}}$. Let $\rho(p_{\mathbf{s}})\triangleq I^{*}_{US}(p_{\mathbf{s}})+\epsilon$. The codeword array ${\mathcal{C}}(p_{\mathbf{s}})$ is obtained by drawing $2^{N(R+\rho(p_{\mathbf{s}}))}$ random vectors independently and uniformly from the corresponding type class $T^{*}_{U}(p_{\mathbf{s}})$, and arranging them in an array with $2^{N\rho(p_{\mathbf{s}})}$ rows and $2^{NR}$ columns indexed by messages. C-A Encoder $f_{N}$ Given a codebook ${\mathcal{C}}$, a covertext sequence ${\mathbf{s}}$, and a message $m$, the encoder finds in ${\mathcal{C}}(p_{\mathbf{s}})$ an $l$ such that ${\mathbf{u}}(l,m)\in T^{*}_{U|S}({\mathbf{s}})$. If more than one such $l$ exists, pick one of them randomly (with uniform distribution). Let ${\mathbf{u}}={\mathbf{u}}(l,m)$. If no such $l$ is available, the encoder declares an error and draws ${\mathbf{u}}$ from the uniform distribution over the conditional type class $T^{*}_{U|S}({\mathbf{s}})$. Then ${\mathbf{x}}$ is drawn from the uniform distribution over the conditional type class $T^{*}_{X|US}({\mathbf{u}},{\mathbf{s}})$. Recalling the discussion below (50), $f_{N}$ satisfies both the order-1 steganographic security constraint and the maximum distortion constraint. C-B Decoder $\phi_{N}$ Given ${\mathbf{y}}$ and the same codebook ${\mathcal{C}}$ used by the encoder, the decoder first seeks a covertext type $p_{{\mathbf{s}}}$ and $\hat{\mathbf{u}}\in{\mathcal{C}}(p_{\mathbf{s}})$ that maximizes the penalized mutual information criterion $$\!\!\max_{p_{\mathbf{s}}}\max_{{\mathbf{u}}\in{\mathcal{C}}(p_{\mathbf{s}})}[I% ({\mathbf{u}};{\mathbf{y}})-\rho(p_{\mathbf{s}})].$$ (51) The decoder then outputs the column index $\hat{m}$ that corresponds to $\hat{\mathbf{u}}$. If there exist maximizers with more than one column index, the decoder declares an error. C-C Error Probability Analysis The probability of error is given by $$P_{e,N}\triangleq\max_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}Pr(M\neq\hat{M})=% \max_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}P_{e}(f_{N},\phi_{N},p_{Y|X}).$$ Following the steps in [33, Section 5], the encoding error vanishes double-exponentially and only the decoding error contributes to $P_{e,N}$ on the exponential scale: $$\displaystyle P_{e,N}\mbox{$\>\stackrel{{\scriptstyle\centerdot}}{{\leq}}\>$}% \exp_{2}\left\{-N\min_{p_{\mathbf{s}}}\max_{p_{{\mathbf{x}}{\mathbf{u}}|{% \mathbf{s}}}}E_{L,N}(R,p_{\mathbf{s}},p_{{\mathbf{x}}{\mathbf{u}}|{\mathbf{s}}% })\right\}.$$ (52) As $N\to\infty$, by [33, Lemma 2.2], the above error exponent converges to $$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!E_{r,L}^{Steg}(R)$$ $$\displaystyle\!\!\!\!=$$ $$\displaystyle\min_{\tilde{p}_{S}\in{\mathcal{P}}_{S}}\max_{p_{XU|S}\in{% \mathcal{Q}}^{Steg}(L,\tilde{p}_{S},D_{1})}\min_{\tilde{p}_{Y|XUS}\in{\mathcal% {P}}_{Y|XUS}}\min_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}$$ (53) $$\displaystyle\!\!\!\!\left[D(\tilde{p}_{S}\,p_{XU|S}\,\tilde{p}_{Y|XUS}||p_{S}% \,p_{XU|S}\,p_{Y|X})+\left|J_{L}(\tilde{p}_{S},p_{XU|S},\tilde{p}_{Y|XUS})-R% \right|^{+}\right].$$ Clearly, $E_{r,L}^{Steg}(R)\geq 0$, with equality if and only if the following conditions are met: • the minimizing PMF $\tilde{p}_{S}$ is equal to $p_{S}$; • the minimizing conditional PMF $\tilde{p}_{Y|XUS}$ is equal to $p_{Y|X}$; and • $R\geq\max_{p_{XU|S}\in{\mathcal{Q}}^{Steg}(L,p_{S},D_{1})}\min_{p_{Y|X}\in{% \mathcal{A}}(p_{X},D_{2})}J_{L}(p_{S},p_{XU|S},p_{Y|X})=C_{L}^{Steg}$. Therefore, $E_{r,L}^{Steg}(R)>0$ and the error probability vanishes for any $R<C_{L}^{Steg}(D_{1},D_{2})$. This implies that the capacity is lower-bounded by $$\lim_{L\to\infty}C_{L}^{Steg}(D_{1},D_{2}).$$ C-D Perfect Security Having established the achievability of $E_{r,L}^{Steg}(R)$ and $C_{L}^{Steg}$ for a deterministic code $(f_{N},\phi_{N})$ with order-1 security and maximum distortion $D_{1}$, we invoke Proposition 1 to claim that the randomly modulated code with prototype $(f_{N},\phi_{N})$ achieves the same error probability (hence error exponent) and distortion as the prototype. Appendix D Proof of Proposition 19 By (18), $J_{L}(p_{S},p_{XU|S},p_{Y|X})$ is reduced to $$J_{L}(p_{S},p_{XU|S},p_{Y|X})=I(U;X)-I(U;S).$$ Coosing $U=X$ yields the lower bound $$\displaystyle C^{Steg}(D_{1},0)$$ $$\displaystyle\geq$$ $$\displaystyle\max_{p_{X|S}\in{\mathcal{Q}}_{1}^{Steg}(p_{S},D_{1})}I(X;X)-I(X;S)$$ (54) $$\displaystyle=$$ $$\displaystyle\max_{p_{X|S}\in{\mathcal{Q}}_{1}^{Steg}(p_{S},D_{1})}H(X|S).$$ On the other hand, $$\displaystyle J_{L}(p_{S},p_{XU|S},p_{Y|X})$$ $$\displaystyle=$$ $$\displaystyle I(U;X)-I(U;S)$$ $$\displaystyle\leq$$ $$\displaystyle I(U;X|S)$$ $$\displaystyle=$$ $$\displaystyle H(X|S)-H(X|U,S)$$ $$\displaystyle\leq$$ $$\displaystyle H(X|S).$$ (56) Note that (D) follows from the chain rule of mutual information $$I(U;XS)=I(U;X)+I(U;S|X)=I(U;S)+I(U;X|S)$$ and $I(U;S|X)\geq 0$. Choosing $U=X$ achieves equality in both (D) and (56). From (56), we obtain $$\displaystyle C^{Steg}(D_{1},0)$$ $$\displaystyle=$$ $$\displaystyle\lim_{L\to\infty}\max_{p_{XU|S}\in{\mathcal{Q}}^{Steg}(L,p_{S},D_% {1})}J_{L}(p_{S},p_{XU|S},p_{Y|X})$$ (57) $$\displaystyle\leq$$ $$\displaystyle\lim_{L\to\infty}\max_{p_{XU|S}\in{\mathcal{Q}}^{Steg}(L,p_{S},D_% {1})}H(X|S)$$ $$\displaystyle=$$ $$\displaystyle\max_{p_{X|S}\in{\mathcal{Q}}^{Steg}(p_{S},D_{1})}H(X|S).$$ Combining (54) and (57) yields (19) and proves the proposition. Appendix E Proof of Proposition 4 Since $p_{Y|X}=\mathds{1}_{\{Y=X\}}$, the term $D(\tilde{p}_{S}\,p_{XU|S}\,\tilde{p}_{Y|XUS}||p_{S}\,p_{XU|S}\,p_{Y|X})$ in (2) is infinite if $\tilde{p}_{Y|XUS}\not=p_{Y|X}$. Hence, the minimizing $\tilde{p}_{Y|XUS}$ in (2) is given by $$\tilde{p}^{*}_{Y|XUS}=p_{Y|X}=\mathds{1}_{\{Y=X\}}.$$ Consequently, the two terms of the cost function of (2) are reduced to $$D(\tilde{p}_{S}\,p_{XU|S}\,\tilde{p}^{*}_{Y|XUS}||p_{S}\,p_{XU|S}\,p_{Y|X})=D(% \tilde{p}_{S}||p_{S})$$ and $$\left|J_{L}(\tilde{p}_{S},p_{XU|S},\tilde{p}^{*}_{Y|XUS})-R\right|^{+}=\left|J% _{L}(\tilde{p}_{S},p_{XU|S},p_{Y|X})-R\right|^{+},$$ respectively. This yields $$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!E_{r}^{Steg}(R)$$ $$\displaystyle=$$ $$\displaystyle\!\!\!\!\!\min_{\tilde{p}_{S}\in\mathcal{P}_{S}}\left[D(\tilde{p}% _{S}||p_{S})+\lim_{L\to\infty}\max_{p_{XU|S}\in{\mathcal{Q}}^{Steg}(L,\tilde{p% }_{S},D_{1})}|J_{L}(\tilde{p}_{S},p_{XU|S},p_{Y|X})-R|^{+}\right].$$ (58) Similarly to the steps in the proof of Proposition 19, we derive that $$\forall\;L\geq 2:\;\max_{p_{XU|S}\in{\mathcal{Q}}^{Steg}(L,\tilde{p}_{S},D_{1}% )}|J_{L}(\tilde{p}_{S},p_{XU|S},p_{Y|X})-R|^{+}=\max_{p_{X|S}\in{\mathcal{Q}}_% {1}^{Steg}(p_{S},D_{1})}|H_{\tilde{p}_{S},p_{X|S}}(X|S)-R|^{+}.$$ (59) The maximum on the left side is achieved by $U=X$. Combining (58) and (59) proves the proposition. Appendix F Proof of Proposition 5 We prove Proposition 5 in two parts. We first establish that the right-hand side of (29) is an upper bound on the public watermarking capacity $C^{PubWM}$. Then we prove that the right-hand side of (29) is at the same time a lower bound on the perfectly secure steganographic capacity $C^{Steg}$. We start with the following lemma on the properties of $p_{XU|S}\in{\mathcal{Q}}^{Steg}_{cyc}(qL,p_{S},D_{1})$, which are used throughout this proof. Lemma 1 Any $p_{XU|S}\in{\mathcal{Q}}^{Steg}_{cyc}(qL,p_{S},D_{1})$ generated by (24) from its corresponding $p_{XV|S}\in{\mathcal{Q}}(L,p_{S},D_{1})$ has the following properties: (i) $p_{S|U}\big{(}s\big{|}qv+i\big{)}=p_{S|V}\big{(}{\underline{s-i}}\big{|}v\big{)}$, $\forall\,i,\,s\in{\mathcal{S}}$ and $\forall\,v\in{\mathcal{V}}$; (ii) $p_{X|U}\big{(}x\big{|}qv+i\big{)}=p_{X|V}\big{(}{\underline{x-i}}\big{|}v\big{)}$, $\forall\,i,\,x\in{\mathcal{S}}$ and $\forall\,v\in{\mathcal{V}}$; (iii) $p_{U}(qv+i)=\frac{1}{q}p_{V}(v)$, $\forall\,i\in{\mathcal{S}}$, $v\in{\mathcal{V}}$, where $p_{U}$ (resp. $p_{V}$) is the marginal PMF of $U$ (resp. $V$) induced from $p_{XU|S}$ (resp. $p_{XV|S}$) and $p_{S}=\mathbb{U}({\mathcal{S}})$; and (iv) $\hat{p}_{X}=\mathbb{U}({\mathcal{S}})$, where $\hat{p}_{X}$ is the marginal PMF of $X$ induced from $p_{XU|S}$ and $p_{S}=\mathbb{U}({\mathcal{S}})$. It is straightforward to verify Lemma 1(i)-(iv) from (24). F-A Upper Bound For the capacity of the public watermarking game, $$\displaystyle C^{PubWM}(D_{1},D_{2})$$ $$\displaystyle=$$ $$\displaystyle\lim_{L\to\infty}\max_{p_{XV|S}\in{\mathcal{Q}}(L,p_{S},D_{1})}% \min_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}J_{L}(p_{S},p_{XV|S},p_{Y|X})$$ (60) $$\displaystyle\leq$$ $$\displaystyle\lim_{L\to\infty}\max_{p_{XV|S}\in{\mathcal{Q}}(L,p_{S},D_{1})}% \min_{p_{Y|X}\in{\mathcal{A}}_{cyc}(D_{2})}J_{L}(p_{S},p_{XV|S},p_{Y|X}),$$ since ${\mathcal{A}}_{cyc}(D_{2})\subset{\mathcal{A}}(p_{X},D_{2})$ by (28). Given any $p_{XV|S}\in{\mathcal{Q}}(L,p_{S},D_{1})$ and its associated $p_{XU|S}\in{\mathcal{Q}}_{cyc}^{Steg}(qL,p_{S},D_{1})$, we first verify that $$I(S;U)=I(S;V).$$ (61) From $p_{S}=\mathbb{U}({\mathcal{S}})$ and $p_{XV|S}$, we obtain $$\displaystyle H(S|V)$$ $$\displaystyle=$$ $$\displaystyle-\sum_{v=0}^{L-1}p_{V}(v)\sum_{s=0}^{q-1}p_{S|V}(s|v)\log p_{S|V}% (s|v).$$ (62) From $p_{S}=\mathbb{U}({\mathcal{S}})$ and $p_{XU|S}$, we have $$\displaystyle H(S|U)$$ $$\displaystyle=$$ $$\displaystyle-\sum_{v=0}^{L-1}\sum_{i=0}^{q-1}\sum_{s=0}^{q-1}p_{U}(qv+i)\,p_{% S|U}(s|qv+i)\,\log p_{S|U}(s|qv+i)$$ (63) $$\displaystyle=$$ $$\displaystyle-\sum_{v=0}^{L-1}\sum_{i=0}^{q-1}\sum_{s=0}^{q-1}\frac{1}{q}\,p_{% V}(v)\,p_{S|V}\big{(}\underline{s-i}\big{|}v)\,\log p_{S|V}\big{(}\underline{s% -i}\big{|}v)$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{q}\sum_{i=0}^{q-1}H(S|V)=H(S|V),$$ (64) where (63) is obtained by using Lemma 1(i) and (iii). Since $I(S;U)=H(S)-H(S|U)$ and $I(S;V)=H(S)-H(S|V)$, (61) follows from (64). For the pair $\left(p_{XV|S},p_{Y|X}\right)\in{\mathcal{Q}}(L,p_{S},D_{1})\times{\mathcal{A}% }_{cyc}(D_{2})$ and its associated pair $\left(p_{XU|S},p_{Y|X}\right)\in{\mathcal{Q}}_{cyc}^{Steg}(qL,p_{S},D_{1})% \times{\mathcal{A}}_{cyc}(D_{2})$, we have the following lemma that is proved in Appendix G. Lemma 2 $$I_{p_{S},p_{XV|S},p_{Y|X}}(Y;V)\leq I_{p_{S},p_{XU|S},p_{Y|X}}(Y;U).$$ (65) From (61), Lemma 65, and the definition of $J_{L}$ in (10), we obtain $$J_{L}(p_{S},p_{XV|S},p_{Y|X})\leq J_{qL}(p_{S},p_{XU|S},p_{Y|X}),$$ (66) which yields $$\displaystyle\lim_{L\to\infty}\max_{p_{XV|S}\in{\mathcal{Q}}(L,p_{S},D_{1})}% \min_{p_{Y|X}\in{\mathcal{A}}_{cyc}(D_{2})}J_{L}(p_{S},p_{XV|S},p_{Y|X})$$ (67) $$\displaystyle\leq$$ $$\displaystyle\lim_{L\to\infty}\max_{p_{XU|S}\in{\mathcal{Q}}_{cyc}^{Steg}(qL,p% _{S},D_{1})}\min_{p_{Y|X}\in{\mathcal{A}}_{cyc}(D_{2})}J_{qL}(p_{S},p_{XU|S},p% _{Y|X}).$$ Therefore, (60) and (67) yield $$\displaystyle C^{Steg}(D_{1},D_{2})$$ $$\displaystyle\leq$$ $$\displaystyle C^{PubWM}(D_{1},D_{2})$$ (68) $$\displaystyle\leq$$ $$\displaystyle\lim_{L\to\infty}\max_{p_{XU|S}\in{\mathcal{Q}}_{cyc}^{Steg}(qL,p% _{S},D_{1})}\min_{p_{Y|X}\in{{\mathcal{A}}}_{cyc}(D_{2})}J_{L}(p_{S},p_{XU|S},% p_{Y|X}).$$ F-B Lower Bound Using the same argument at the end of Appendix B for the sequence $\{C_{L}^{Steg}(D_{1},D_{2})\}$, we can argue that the sequence $\{C_{L}^{PubWM}(D_{1},D_{2})\}$ is also nondecreasing and bounded by $\log|{\mathcal{S}}|$. Therefore, $\{C_{L}^{PubWM}(D_{1},D_{2})\}$ and any of its subsequences converge to the same limit. That is $$\displaystyle C^{PubWM}(D_{1},D_{2})$$ $$\displaystyle=$$ $$\displaystyle\lim_{L\to\infty}\max_{p_{XU|S}\in{\mathcal{Q}}(L,p_{S},D_{1})}% \min_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}J_{L}(p_{S},p_{XU|S},p_{Y|X})$$ (69) $$\displaystyle=$$ $$\displaystyle\lim_{L\to\infty}\max_{p_{XU|S}\in{\mathcal{Q}}(qL,p_{S},D_{1})}% \min_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}J_{L}(p_{S},p_{XU|S},p_{Y|X}).$$ Similarly, $$\displaystyle C^{Steg}(D_{1},D_{2})$$ $$\displaystyle=$$ $$\displaystyle\lim_{L\to\infty}\max_{p_{XU|S}\in{\mathcal{Q}}^{Steg}(L,p_{S},D_% {1})}\min_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}J_{L}(p_{S},p_{XU|S},p_{Y|X})$$ (70) $$\displaystyle=$$ $$\displaystyle\lim_{L\to\infty}\max_{p_{XU|S}\in{\mathcal{Q}}^{Steg}(qL,p_{S},D% _{1})}\min_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}J_{L}(p_{S},p_{XU|S},p_{Y|X}).$$ From (26), $${\mathcal{Q}}_{cyc}^{Steg}(qL,p_{S},D_{1})\subset{\mathcal{Q}}^{Steg}(qL,p_{S}% ,D_{1})\subset{\mathcal{Q}}(qL,p_{S},D_{1}).$$ Thus, we have $$\displaystyle C^{PubWM}(D_{1},D_{2})$$ $$\displaystyle\geq$$ $$\displaystyle C^{Steg}(D_{1},D_{2})$$ (71) $$\displaystyle\geq$$ $$\displaystyle\lim_{L\to\infty}\max_{p_{XU|S}\in{\mathcal{Q}}_{cyc}^{Steg}(qL,p% _{S},D_{1})}\min_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}J_{L}(p_{S},p_{XU|S},p_% {Y|X}).$$ Given $p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})$, we define $q$ conditional PMFs: $$\displaystyle p^{m}_{Y|X}(y|x)=p_{Y|X}(\underline{y-m}|\underline{x-m}),\quad% \forall\,x,\,y\in{\mathcal{S}},\,0\leq m<q.$$ (72) Since the distortion matrix $\{{\mathsf{d}}(i,\,j)\}_{i,\,j=0}^{q-1}$ is cyclic, it is easy to verify that all the $q$ conditional PMFs $p^{m}_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})$. The conditional PMF $p_{Y|U}^{m}$ induced by $\left(p_{XU|S},p^{m}_{Y|X}\right)\in{\mathcal{Q}}_{cyc}^{Steg}(qL,p_{S},D_{1})% \times{\mathcal{A}}(p_{X},D_{2})$ is given by $$\displaystyle p_{Y|U}^{m}(y|qv+i)$$ $$\displaystyle=$$ $$\displaystyle\sum_{x=0}^{q-1}p_{X|U}(x|qv+i)p_{Y|X}^{m}(y|x)$$ (73) $$\displaystyle=$$ $$\displaystyle\sum_{x=0}^{q-1}p_{X|U}(x|qv+i)p_{Y|X}(\underline{y-m}|\underline% {x-m})$$ $$\displaystyle=$$ $$\displaystyle\sum_{x=0}^{q-1}p_{X|V}(\underline{x-i}|v)p_{Y|X}(\underline{y-m}% |\underline{x-m})$$ (74) $$\displaystyle=$$ $$\displaystyle\sum_{x=0}^{q-1}p_{X|U}(\underline{x-m}|qv+\underline{i-m})p_{Y|X% }(\underline{y-m}|\underline{x-m})$$ (75) $$\displaystyle=$$ $$\displaystyle p_{Y|U}(\underline{y-m}|qv+\underline{i-m}),\quad\forall\,y,\,i% \in{\mathcal{S}},\;v\in{\mathcal{V}},$$ (76) where (73) follows from the definition (72), and both (74) and (75) follow by applying Lemma 1(ii). We also obtain the marginal PMF of $Y$ as $$\displaystyle p_{Y}^{m}(y)$$ $$\displaystyle=$$ $$\displaystyle\sum_{v=0}^{L-1}\sum_{i=0}^{q-1}p_{U}(qv+i)p_{Y|U}^{m}(y|qv+i)$$ (77) $$\displaystyle=$$ $$\displaystyle\sum_{v=0}^{L-1}\sum_{i=0}^{q-1}p_{U}(qv+\underline{i-m})p_{Y|U}(% \underline{y-m}|qv+\underline{i-m})$$ $$\displaystyle=$$ $$\displaystyle p_{Y}(\underline{y-m}),\quad\forall\,y\in{\mathcal{S}},$$ (78) where (77) follows from Lemma 1(iii) and (76). From (76) and (78), we obtain $$I_{p_{S},p_{XU|S},p_{Y|X}}(Y;U)=I_{p_{S},p_{XU|S},p^{m}_{Y|X}}(Y;U)$$ (79) and hence $$J_{L}(p_{S},p_{XU|S},p_{Y|X})=J_{L}(p_{S},p_{XU|S},p^{m}_{Y|X}),$$ (80) for $0\leq m<q$. Let $\bar{p}_{Y|X}\triangleq\frac{1}{q}\sum_{m=0}^{q-1}p^{m}_{Y|X}$. It is easy to check that $\bar{p}_{Y|X}\in{{\mathcal{A}}}_{cyc}(D_{2})$. Also, $$\displaystyle J_{L}(p_{S},p_{XU|S},p_{Y|X})$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{q}\sum_{m=0}^{q-1}J_{L}(p_{S},p_{XU|S},p^{m}_{Y|X})$$ (81) $$\displaystyle\geq$$ $$\displaystyle J_{L}\left(p_{S},p_{XU|S},\frac{1}{q}\sum_{m=0}^{q-1}p^{m}_{Y|X}% \right)=J_{L}(p_{S},p_{XU|S},\bar{p}_{Y|X}),$$ (82) where the inequality comes from the fact that for fixed $p_{S}$ and $p_{XU|S}$, $J_{L}(p_{S},p_{XU|S},p_{Y|X})$ is convex in $p_{Y|X}$ [27, Proposition 4.1(iii)]. Therefore, from (82) we have $$\displaystyle C^{PubWM}(D_{1},D_{2})$$ $$\displaystyle\geq$$ $$\displaystyle C^{Steg}(D_{1},D_{2})$$ (83) $$\displaystyle\geq$$ $$\displaystyle\lim_{L\to\infty}\max_{p_{XU|S}\in{\mathcal{Q}}_{cyc}^{Steg}(qL,p% _{S},D_{1})}\min_{p_{Y|X}\in{\mathcal{A}}(p_{X},D_{2})}J_{L}(p_{S},p_{XU|S},p_% {Y|X})$$ $$\displaystyle\geq$$ $$\displaystyle\lim_{L\to\infty}\max_{p_{XU|S}\in{\mathcal{Q}}_{cyc}^{Steg}(qL,p% _{S},D_{1})}\min_{p_{Y|X}\in{{\mathcal{A}}}_{cyc}(D_{2})}J_{L}(p_{S},p_{XU|S},% p_{Y|X}).$$ Combining the upper bound inequality in (68) and the lower bound inequality in (83), we prove the claim $$\displaystyle C^{PubWM}(D_{1},D_{2})$$ $$\displaystyle=$$ $$\displaystyle C^{Steg}(D_{1},D_{2})$$ (84) $$\displaystyle=$$ $$\displaystyle\lim_{L\to\infty}\max_{p_{XU|S}\in{\mathcal{Q}}_{cyc}^{Steg}(qL,p% _{S},D_{1})}\min_{p_{Y|X}\in{{\mathcal{A}}}_{cyc}(D_{2})}J_{L}(p_{S},p_{XU|S},% p_{Y|X}),$$ which means that the perfectly secure steganographic constraint does not cause any capacity loss. Appendix G Proof of Lemma 65 For the pair $\left(p_{XV|S},p_{Y|X}\right)\in{\mathcal{Q}}(L,p_{S},D_{1})\times{\mathcal{A}% }_{cyc}(D_{2})$, the conditional PMF of $Y$ given $V$ is $$\displaystyle p_{Y|V}(y|v)$$ $$\displaystyle=$$ $$\displaystyle\sum_{x=0}^{q-1}p_{X|V}(x|v)\,p_{Y|X}(y|x)$$ (85) $$\displaystyle=$$ $$\displaystyle\sum_{x=0}^{q-1}p_{X|V}(x|v)\,p_{Y|X}(\underline{y-x}|\,0),\quad% \forall\;y\in{\mathcal{S}},\;v\in{\mathcal{V}},$$ where (85) follows from (27) in Definition 11 for $p_{Y|X}\in{\mathcal{A}}_{cyc}(D_{2})$. The conditional entropy of $Y$ given $V$ is $$\displaystyle H(Y|V)$$ $$\displaystyle=$$ $$\displaystyle-\sum_{v=0}^{L-1}p_{V}(v)\sum_{y=0}^{q-1}p_{Y|V}(y|v)\log p_{Y|V}% (y|v).$$ (86) For the associated pair $\left(p_{XU|S},p_{Y|X}\right)\in{\mathcal{Q}}_{cyc}^{Steg}(qL,p_{S},D_{1})% \times{\mathcal{A}}_{cyc}(D_{2})$, the conditional PMF of $Y$ given $U$ is $$\displaystyle p_{Y|U}\left(y|qv+i\right)$$ $$\displaystyle=$$ $$\displaystyle\sum_{x=0}^{q-1}p_{X|U}(x|qv+i)p_{Y|X}(y|x)$$ (87) $$\displaystyle=$$ $$\displaystyle\sum_{x=0}^{q-1}p_{X|V}\left(\underline{x-i}\big{|}v\right)p_{Y|X% }\left(\underline{y-i-(x-i)}\,\Big{|}\,0\right)$$ $$\displaystyle=$$ $$\displaystyle p_{Y|V}(\underline{y-i}|v),\quad\forall\;y,\,i\in{\mathcal{S}},% \;v\in{\mathcal{V}},$$ (88) where to obtain (87) we have used Lemma 1(ii) and (27) in Definition 11 for $p_{Y|X}\in{\mathcal{A}}_{cyc}(D_{2})$; and (88) follows from (85). The marginal PMF of $Y$ is given by $$\displaystyle\hat{p}_{Y}(y)$$ $$\displaystyle=$$ $$\displaystyle\sum_{v=0}^{L-1}\sum_{i=0}^{q-1}p_{U}(qv+i)\,p_{Y|U}(y|qv+i)$$ (89) $$\displaystyle=$$ $$\displaystyle\sum_{v=0}^{L-1}\sum_{i=0}^{q-1}\frac{1}{q}p_{V}(v)\,p_{Y|V}(% \underline{y-i}|v)$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{q}\sum_{j=0}^{q-1}p_{Y}(\underline{j-i})=\frac{1}{q},$$ (90) where (89) follows from Lemma 1(iii) and (88). The conditional entropy of $Y$ given $U$ is $$\displaystyle H(Y|U)$$ $$\displaystyle=$$ $$\displaystyle-\sum_{v=0}^{L-1}\sum_{i=0}^{q-1}p_{U}(qv+i)\sum_{y=0}^{q-1}p_{Y|% U}(y|qv+i)\,\log p_{Y|U}(y|qv+i)$$ (91) $$\displaystyle=$$ $$\displaystyle-\sum_{v=0}^{L-1}\sum_{i=0}^{q-1}\frac{1}{q}p_{V}(v)\sum_{y=0}^{q% -1}p_{Y|V}(\underline{y-i}|v)\,\log p_{Y|V}(\underline{y-i}|v)$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{q}\sum_{j=0}^{q-1}H(Y|V)=H(Y|V),$$ (92) where (91) follows from Lemma 1(iii) and (88), and (92) follows from (86). 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Planck intermediate results. XXXIX. The Planck list of high-redshift source candidates Planck Collaboration: P. A. R. Ade 80School of Physics and Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff, CF24 3AA, U.K.80Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    N. Aghanim 55Institut d’Astrophysique Spatiale, CNRS (UMR8617) Université Paris-Sud 11, Bâtiment 121, Orsay, France55Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    M. Arnaud 69Laboratoire AIM, IRFU/Service d’Astrophysique - CEA/DSM - CNRS - Université Paris Diderot, Bât. 709, CEA-Saclay, F-91191 Gif-sur-Yvette Cedex, France69Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    J. Aumont 55Institut d’Astrophysique Spatiale, CNRS (UMR8617) Université Paris-Sud 11, Bâtiment 121, Orsay, France55Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    C. Baccigalupi 79SISSA, Astrophysics Sector, via Bonomea 265, 34136, Trieste, Italy79Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    A. J. Banday 87Université de Toulouse, UPS-OMP, IRAP, F-31028 Toulouse cedex 4, France87 8 8    R. B. Barreiro 60Instituto de Física de Cantabria (CSIC-Universidad de Cantabria), Avda. de los Castros s/n, Santander, Spain60Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    N. Bartolo 26Dipartimento di Fisica e Astronomia G. Galilei, Università degli Studi di Padova, via Marzolo 8, 35131 Padova, Italy26 61 61    E. Battaner 88University of Granada, Departamento de Física Teórica y del Cosmos, Facultad de Ciencias, Granada, Spain88 89 89    K. Benabed 56Institut d’Astrophysique de Paris, CNRS (UMR7095), 98 bis Boulevard Arago, F-75014, Paris, France56 86 86    A. Benoit-Lévy 20Department of Physics and Astronomy, University College London, London WC1E 6BT, U.K.20 56 56 86 86    J.-P. Bernard 87Université de Toulouse, UPS-OMP, IRAP, F-31028 Toulouse cedex 4, France87 8 8    M. Bersanelli 29Dipartimento di Fisica, Università degli Studi di Milano, Via Celoria, 16, Milano, Italy29 45 45    P. Bielewicz 76Nicolaus Copernicus Astronomical Center, Bartycka 18, 00-716 Warsaw, Poland76 8 8 79 79    A. Bonaldi 63Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester, M13 9PL, U.K.63Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    L. Bonavera 60Instituto de Física de Cantabria (CSIC-Universidad de Cantabria), Avda. de los Castros s/n, Santander, Spain60Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    J. R. Bond 7CITA, University of Toronto, 60 St. George St., Toronto, ON M5S 3H8, Canada7Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, SpainDepartment of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    J. Borrill 11Computational Cosmology Center, Lawrence Berkeley National Laboratory, Berkeley, California, U.S.A.11 83 83    F. R. Bouchet 56Institut d’Astrophysique de Paris, CNRS (UMR7095), 98 bis Boulevard Arago, F-75014, Paris, France56 81 81    F. Boulanger 55Institut d’Astrophysique Spatiale, CNRS (UMR8617) Université Paris-Sud 11, Bâtiment 121, Orsay, France55Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    C. Burigana 44INAF/IASF Bologna, Via Gobetti 101, Bologna, Italy44 27 27 46 46    R. C. Butler 44INAF/IASF Bologna, Via Gobetti 101, Bologna, Italy44INAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    E. Calabrese 85Sub-Department of Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, U.K.85UPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    A. Catalano 70Laboratoire de Physique Subatomique et Cosmologie, Université Grenoble-Alpes, CNRS/IN2P3, 53, rue des Martyrs, 38026 Grenoble Cedex, France70 68 68    H. C. Chiang 23Department of Physics, Princeton University, Princeton, New Jersey, U.S.A.23 6 6    P. R. Christensen 77Niels Bohr Institute, Blegdamsvej 17, Copenhagen, Denmark77 32 32    D. L. Clements 52Imperial College London, Astrophysics group, Blackett Laboratory, Prince Consort Road, London, SW7 2AZ, U.K.52Institut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    L. P. L. Colombo 19Department of Physics and Astronomy, Dana and David Dornsife College of Letter, Arts and Sciences, University of Southern California, Los Angeles, CA 90089, U.S.A.19 62 62    F. Couchot 67LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France67Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    A. Coulais 68LERMA, CNRS, Observatoire de Paris, 61 Avenue de l’Observatoire, Paris, France68Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    B. P. Crill 62Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California, U.S.A.62 9 9    A. Curto 60Instituto de Física de Cantabria (CSIC-Universidad de Cantabria), Avda. de los Castros s/n, Santander, Spain60 5 5 65 65    F. Cuttaia 44INAF/IASF Bologna, Via Gobetti 101, Bologna, Italy44INAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    L. Danese 79SISSA, Astrophysics Sector, via Bonomea 265, 34136, Trieste, Italy79Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    R. D. Davies 63Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester, M13 9PL, U.K.63Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    R. J. Davis 63Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester, M13 9PL, U.K.63Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    P. de Bernardis 28Dipartimento di Fisica, Università La Sapienza, P. le A. Moro 2, Roma, Italy28Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    A. de Rosa 44INAF/IASF Bologna, Via Gobetti 101, Bologna, Italy44INAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    G. de Zotti 41INAF - Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, Padova, Italy41 79 79    J. Delabrouille 1APC, AstroParticule et Cosmologie, Université Paris Diderot, CNRS/IN2P3, CEA/lrfu, Observatoire de Paris, Sorbonne Paris Cité, 10, rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France1African Institute for Mathematical Sciences, 6-8 Melrose Road, Muizenberg, Cape Town, South AfricaAstrophysics & Cosmology Research Unit, School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Westville Campus, Private Bag X54001, Durban 4000, South AfricaDepartamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, SpainDepartment of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    C. Dickinson 63Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester, M13 9PL, U.K.63Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    J. M. Diego 60Instituto de Física de Cantabria (CSIC-Universidad de Cantabria), Avda. de los Castros s/n, Santander, Spain60Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    H. Dole 55Institut d’Astrophysique Spatiale, CNRS (UMR8617) Université Paris-Sud 11, Bâtiment 121, Orsay, France55 54 54    O. Doré 62Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California, U.S.A.62 9 9    M. Douspis 55Institut d’Astrophysique Spatiale, CNRS (UMR8617) Université Paris-Sud 11, Bâtiment 121, Orsay, France55Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    A. Ducout 56Institut d’Astrophysique de Paris, CNRS (UMR7095), 98 bis Boulevard Arago, F-75014, Paris, France56 52 52    X. Dupac 33European Space Agency, ESAC, Planck Science Office, Camino bajo del Castillo, s/n, Urbanización Villafranca del Castillo, Villanueva de la Cañada, Madrid, Spain33Gran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    F. Elsner 20Department of Physics and Astronomy, University College London, London WC1E 6BT, U.K.20 56 56 86 86    T. A. Enßlin 74Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany74Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    H. K. Eriksen 58Institute of Theoretical Astrophysics, University of Oslo, Blindern, Oslo, Norway58Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    E. Falgarone 68LERMA, CNRS, Observatoire de Paris, 61 Avenue de l’Observatoire, Paris, France68Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    F. Finelli 44INAF/IASF Bologna, Via Gobetti 101, Bologna, Italy44 46 46    I. Flores-Cacho 8CNRS, IRAP, 9 Av. colonel Roche, BP 44346, F-31028 Toulouse cedex 4, France8 87 87    M. Frailis 43INAF - Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, Trieste, Italy43INAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    A. A. Fraisse 23Department of Physics, Princeton University, Princeton, New Jersey, U.S.A.23Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    E. Franceschi 44INAF/IASF Bologna, Via Gobetti 101, Bologna, Italy44INAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    S. Galeotta 43INAF - Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, Trieste, Italy43INAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    S. Galli 64Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA64Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    K. Ganga 1APC, AstroParticule et Cosmologie, Université Paris Diderot, CNRS/IN2P3, CEA/lrfu, Observatoire de Paris, Sorbonne Paris Cité, 10, rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France1African Institute for Mathematical Sciences, 6-8 Melrose Road, Muizenberg, Cape Town, South AfricaAstrophysics & Cosmology Research Unit, School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Westville Campus, Private Bag X54001, Durban 4000, South AfricaDepartamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, SpainDepartment of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    M. Giard 87Université de Toulouse, UPS-OMP, IRAP, F-31028 Toulouse cedex 4, France87 8 8    Y. Giraud-Héraud 1APC, AstroParticule et Cosmologie, Université Paris Diderot, CNRS/IN2P3, CEA/lrfu, Observatoire de Paris, Sorbonne Paris Cité, 10, rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France1African Institute for Mathematical Sciences, 6-8 Melrose Road, Muizenberg, Cape Town, South AfricaAstrophysics & Cosmology Research Unit, School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Westville Campus, Private Bag X54001, Durban 4000, South AfricaDepartamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, SpainDepartment of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    E. Gjerløw 58Institute of Theoretical Astrophysics, University of Oslo, Blindern, Oslo, Norway58Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    J. González-Nuevo 16Departamento de Física, Universidad de Oviedo, Avda. Calvo Sotelo s/n, Oviedo, Spain16 60 60    K. M. Górski 62Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California, U.S.A.62 90 90    A. Gregorio 30Dipartimento di Fisica, Università degli Studi di Trieste, via A. Valerio 2, Trieste, Italy30 43 43 49 49    A. Gruppuso 44INAF/IASF Bologna, Via Gobetti 101, Bologna, Italy44INAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    J. E. Gudmundsson 23Department of Physics, Princeton University, Princeton, New Jersey, U.S.A.23Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    F. K. Hansen 58Institute of Theoretical Astrophysics, University of Oslo, Blindern, Oslo, Norway58Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    D. L. Harrison 57Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, U.K.57 65 65    G. Helou 9California Institute of Technology, Pasadena, California, U.S.A.9Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, SpainDepartment of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    C. Hernández-Monteagudo 10Centro de Estudios de Física del Cosmos de Aragón (CEFCA), Plaza San Juan, 1, planta 2, E-44001, Teruel, Spain10 74 74    D. Herranz 60Instituto de Física de Cantabria (CSIC-Universidad de Cantabria), Avda. de los Castros s/n, Santander, Spain60Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    S. R. Hildebrandt 62Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California, U.S.A.62 9 9    E. Hivon 56Institut d’Astrophysique de Paris, CNRS (UMR7095), 98 bis Boulevard Arago, F-75014, Paris, France56 86 86    M. Hobson 5Astrophysics Group, Cavendish Laboratory, University of Cambridge, J J Thomson Avenue, Cambridge CB3 0HE, U.K.5Astrophysics & Cosmology Research Unit, School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Westville Campus, Private Bag X54001, Durban 4000, South AfricaDepartamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, SpainDepartment of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    A. Hornstrup 13DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 327, DK-2800 Kgs. Lyngby, Denmark13Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, SpainDepartment of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    W. Hovest 74Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany74Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    K. M. Huffenberger 21Department of Physics, Florida State University, Keen Physics Building, 77 Chieftan Way, Tallahassee, Florida, U.S.A.21Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    G. Hurier 55Institut d’Astrophysique Spatiale, CNRS (UMR8617) Université Paris-Sud 11, Bâtiment 121, Orsay, France55Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    A. H. Jaffe 52Imperial College London, Astrophysics group, Blackett Laboratory, Prince Consort Road, London, SW7 2AZ, U.K.52Institut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    T. R. Jaffe 87Université de Toulouse, UPS-OMP, IRAP, F-31028 Toulouse cedex 4, France87 8 8    E. Keihänen 22Department of Physics, Gustaf Hällströmin katu 2a, University of Helsinki, Helsinki, Finland22Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    R. Keskitalo 11Computational Cosmology Center, Lawrence Berkeley National Laboratory, Berkeley, California, U.S.A.11Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, SpainDepartment of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    T. S. Kisner 72Lawrence Berkeley National Laboratory, Berkeley, California, U.S.A.72Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    J. Knoche 74Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany74Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    M. Kunz 14Département de Physique Théorique, Université de Genève, 24, Quai E. Ansermet,1211 Genève 4, Switzerland14 55 55 2 2    H. Kurki-Suonio 22Department of Physics, Gustaf Hällströmin katu 2a, University of Helsinki, Helsinki, Finland22 39 39    G. Lagache 4Aix Marseille Université, CNRS, LAM (Laboratoire d’Astrophysique de Marseille) UMR 7326, 13388, Marseille, France4 55 55    J.-M. Lamarre 68LERMA, CNRS, Observatoire de Paris, 61 Avenue de l’Observatoire, Paris, France68Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    A. Lasenby 5Astrophysics Group, Cavendish Laboratory, University of Cambridge, J J Thomson Avenue, Cambridge CB3 0HE, U.K.5 65 65    M. Lattanzi 27Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, Via Saragat 1, 44122 Ferrara, Italy27Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    C. R. Lawrence 62Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California, U.S.A.62Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    R. Leonardi 33European Space Agency, ESAC, Planck Science Office, Camino bajo del Castillo, s/n, Urbanización Villafranca del Castillo, Villanueva de la Cañada, Madrid, Spain33Gran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    F. Levrier 68LERMA, CNRS, Observatoire de Paris, 61 Avenue de l’Observatoire, Paris, France68Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    P. B. Lilje 58Institute of Theoretical Astrophysics, University of Oslo, Blindern, Oslo, Norway58Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    M. Linden-Vørnle 13DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 327, DK-2800 Kgs. Lyngby, Denmark13Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, SpainDepartment of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    M. López-Caniego 33European Space Agency, ESAC, Planck Science Office, Camino bajo del Castillo, s/n, Urbanización Villafranca del Castillo, Villanueva de la Cañada, Madrid, Spain33 60 60    P. M. Lubin 24Department of Physics, University of California, Santa Barbara, California, U.S.A.24Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    J. F. Macías-Pérez 70Laboratoire de Physique Subatomique et Cosmologie, Université Grenoble-Alpes, CNRS/IN2P3, 53, rue des Martyrs, 38026 Grenoble Cedex, France70Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    B. Maffei 63Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester, M13 9PL, U.K.63Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    G. Maggio 43INAF - Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, Trieste, Italy43INAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    D. Maino 29Dipartimento di Fisica, Università degli Studi di Milano, Via Celoria, 16, Milano, Italy29 45 45    N. Mandolesi 44INAF/IASF Bologna, Via Gobetti 101, Bologna, Italy44 27 27    A. Mangilli 55Institut d’Astrophysique Spatiale, CNRS (UMR8617) Université Paris-Sud 11, Bâtiment 121, Orsay, France55 67 67    M. Maris 43INAF - Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, Trieste, Italy43INAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    P. G. Martin 7CITA, University of Toronto, 60 St. George St., Toronto, ON M5S 3H8, Canada7Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, SpainDepartment of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    E. Martínez-González 60Instituto de Física de Cantabria (CSIC-Universidad de Cantabria), Avda. de los Castros s/n, Santander, Spain60Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    S. Masi 28Dipartimento di Fisica, Università La Sapienza, P. le A. Moro 2, Roma, Italy28Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    S. Matarrese 26Dipartimento di Fisica e Astronomia G. Galilei, Università degli Studi di Padova, via Marzolo 8, 35131 Padova, Italy26 61 61 36 36    A. Melchiorri 28Dipartimento di Fisica, Università La Sapienza, P. le A. Moro 2, Roma, Italy28 47 47    A. Mennella 29Dipartimento di Fisica, Università degli Studi di Milano, Via Celoria, 16, Milano, Italy29 45 45    M. Migliaccio 57Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, U.K.57 65 65    S. Mitra 51IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, Pune 411 007, India51 62 62    M.-A. Miville-Deschênes 55Institut d’Astrophysique Spatiale, CNRS (UMR8617) Université Paris-Sud 11, Bâtiment 121, Orsay, France55 7 7    A. Moneti 56Institut d’Astrophysique de Paris, CNRS (UMR7095), 98 bis Boulevard Arago, F-75014, Paris, France56Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    L. Montier  Corresponding author: L. Montier, Ludovic.Montier@irap.omp.eu87Université de Toulouse, UPS-OMP, IRAP, F-31028 Toulouse cedex 4, France87 8 8    G. Morgante 44INAF/IASF Bologna, Via Gobetti 101, Bologna, Italy44INAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    D. Mortlock 52Imperial College London, Astrophysics group, Blackett Laboratory, Prince Consort Road, London, SW7 2AZ, U.K.52Institut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    D. Munshi 80School of Physics and Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff, CF24 3AA, U.K.80Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    J. A. Murphy 75National University of Ireland, Department of Experimental Physics, Maynooth, Co. Kildare, Ireland75Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    F. Nati 23Department of Physics, Princeton University, Princeton, New Jersey, U.S.A.23Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    P. Natoli 27Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, Via Saragat 1, 44122 Ferrara, Italy27 3 3 44 44    N. P. H. Nesvadba 55Institut d’Astrophysique Spatiale, CNRS (UMR8617) Université Paris-Sud 11, Bâtiment 121, Orsay, France55Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    F. Noviello 63Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester, M13 9PL, U.K.63Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    D. Novikov 73Lebedev Physical Institute of the Russian Academy of Sciences, Astro Space Centre, 84/32 Profsoyuznaya st., Moscow, GSP-7, 117997, Russia73Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    I. Novikov 77Niels Bohr Institute, Blegdamsvej 17, Copenhagen, Denmark77 73 73    C. A. Oxborrow 13DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 327, DK-2800 Kgs. Lyngby, Denmark13Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, SpainDepartment of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    L. Pagano 28Dipartimento di Fisica, Università La Sapienza, P. le A. Moro 2, Roma, Italy28 47 47    F. Pajot 55Institut d’Astrophysique Spatiale, CNRS (UMR8617) Université Paris-Sud 11, Bâtiment 121, Orsay, France55Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    D. Paoletti 44INAF/IASF Bologna, Via Gobetti 101, Bologna, Italy44 46 46    B. Partridge 38Haverford College Astronomy Department, 370 Lancaster Avenue, Haverford, Pennsylvania, U.S.A.38Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    F. Pasian 43INAF - Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, Trieste, Italy43INAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    T. J. Pearson 9California Institute of Technology, Pasadena, California, U.S.A.9 53 53    O. Perdereau 67LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France67Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    L. Perotto 70Laboratoire de Physique Subatomique et Cosmologie, Université Grenoble-Alpes, CNRS/IN2P3, 53, rue des Martyrs, 38026 Grenoble Cedex, France70Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    V. Pettorino 37HGSFP and University of Heidelberg, Theoretical Physics Department, Philosophenweg 16, 69120, Heidelberg, Germany37Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    F. Piacentini 28Dipartimento di Fisica, Università La Sapienza, P. le A. Moro 2, Roma, Italy28Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    M. Piat 1APC, AstroParticule et Cosmologie, Université Paris Diderot, CNRS/IN2P3, CEA/lrfu, Observatoire de Paris, Sorbonne Paris Cité, 10, rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France1African Institute for Mathematical Sciences, 6-8 Melrose Road, Muizenberg, Cape Town, South AfricaAstrophysics & Cosmology Research Unit, School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Westville Campus, Private Bag X54001, Durban 4000, South AfricaDepartamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, SpainDepartment of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    S. Plaszczynski 67LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France67Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    E. Pointecouteau 87Université de Toulouse, UPS-OMP, IRAP, F-31028 Toulouse cedex 4, France87 8 8    G. Polenta 3Agenzia Spaziale Italiana Science Data Center, Via del Politecnico snc, 00133, Roma, Italy3 42 42    G. W. Pratt 69Laboratoire AIM, IRFU/Service d’Astrophysique - CEA/DSM - CNRS - Université Paris Diderot, Bât. 709, CEA-Saclay, F-91191 Gif-sur-Yvette Cedex, France69Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    S. Prunet 56Institut d’Astrophysique de Paris, CNRS (UMR7095), 98 bis Boulevard Arago, F-75014, Paris, France56 86 86    J.-L. Puget 55Institut d’Astrophysique Spatiale, CNRS (UMR8617) Université Paris-Sud 11, Bâtiment 121, Orsay, France55Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    J. P. Rachen 17Department of Astrophysics/IMAPP, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands17 74 74    M. Reinecke 74Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany74Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    M. Remazeilles 63Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester, M13 9PL, U.K.63 55 55 1 1    C. Renault 70Laboratoire de Physique Subatomique et Cosmologie, Université Grenoble-Alpes, CNRS/IN2P3, 53, rue des Martyrs, 38026 Grenoble Cedex, France70Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    A. Renzi 31Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, Italy31 48 48    I. Ristorcelli 87Université de Toulouse, UPS-OMP, IRAP, F-31028 Toulouse cedex 4, France87 8 8    G. Rocha 62Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California, U.S.A.62 9 9    C. Rosset 1APC, AstroParticule et Cosmologie, Université Paris Diderot, CNRS/IN2P3, CEA/lrfu, Observatoire de Paris, Sorbonne Paris Cité, 10, rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France1African Institute for Mathematical Sciences, 6-8 Melrose Road, Muizenberg, Cape Town, South AfricaAstrophysics & Cosmology Research Unit, School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Westville Campus, Private Bag X54001, Durban 4000, South AfricaDepartamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, SpainDepartment of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    M. Rossetti 29Dipartimento di Fisica, Università degli Studi di Milano, Via Celoria, 16, Milano, Italy29 45 45    G. Roudier 1APC, AstroParticule et Cosmologie, Université Paris Diderot, CNRS/IN2P3, CEA/lrfu, Observatoire de Paris, Sorbonne Paris Cité, 10, rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France1 68 68 62 62    J. A. Rubiño-Martín 59Instituto de Astrofísica de Canarias, C/Vía Láctea s/n, La Laguna, Tenerife, Spain59 15 15    B. Rusholme 53Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, U.S.A.53Institut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    M. Sandri 44INAF/IASF Bologna, Via Gobetti 101, Bologna, Italy44INAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    D. Santos 70Laboratoire de Physique Subatomique et Cosmologie, Université Grenoble-Alpes, CNRS/IN2P3, 53, rue des Martyrs, 38026 Grenoble Cedex, France70Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    M. Savelainen 22Department of Physics, Gustaf Hällströmin katu 2a, University of Helsinki, Helsinki, Finland22 39 39    G. Savini 78Optical Science Laboratory, University College London, Gower Street, London, U.K.78Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    D. Scott 18Department of Physics & Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia, Canada18Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    L. D. Spencer 80School of Physics and Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff, CF24 3AA, U.K.80Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    V. Stolyarov 5Astrophysics Group, Cavendish Laboratory, University of Cambridge, J J Thomson Avenue, Cambridge CB3 0HE, U.K.5 84 84 66 66    R. Stompor 1APC, AstroParticule et Cosmologie, Université Paris Diderot, CNRS/IN2P3, CEA/lrfu, Observatoire de Paris, Sorbonne Paris Cité, 10, rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France1African Institute for Mathematical Sciences, 6-8 Melrose Road, Muizenberg, Cape Town, South AfricaAstrophysics & Cosmology Research Unit, School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Westville Campus, Private Bag X54001, Durban 4000, South AfricaDepartamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, SpainDepartment of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    R. Sudiwala 80School of Physics and Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff, CF24 3AA, U.K.80Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    R. Sunyaev 74Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany74 82 82    A.-S. Suur-Uski 22Department of Physics, Gustaf Hällströmin katu 2a, University of Helsinki, Helsinki, Finland22 39 39    J.-F. Sygnet 56Institut d’Astrophysique de Paris, CNRS (UMR7095), 98 bis Boulevard Arago, F-75014, Paris, France56Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    J. A. Tauber 34European Space Agency, ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands34Gran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    L. Terenzi 35Facoltà di Ingegneria, Università degli Studi e-Campus, Via Isimbardi 10, Novedrate (CO), 22060, Italy35 44 44    L. Toffolatti 16Departamento de Física, Universidad de Oviedo, Avda. Calvo Sotelo s/n, Oviedo, Spain16 60 60 44 44    M. Tomasi 29Dipartimento di Fisica, Università degli Studi di Milano, Via Celoria, 16, Milano, Italy29 45 45    M. Tristram 67LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France67Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    M. Tucci 14Département de Physique Théorique, Université de Genève, 24, Quai E. Ansermet,1211 Genève 4, Switzerland14Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, SpainDepartment of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    M. Türler 50ISDC, Department of Astronomy, University of Geneva, ch. d’Ecogia 16, 1290 Versoix, Switzerland50Institut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    G. Umana 40INAF - Osservatorio Astrofisico di Catania, Via S. Sofia 78, Catania, Italy40INAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    L. Valenziano 44INAF/IASF Bologna, Via Gobetti 101, Bologna, Italy44INAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    J. Valiviita 22Department of Physics, Gustaf Hällströmin katu 2a, University of Helsinki, Helsinki, Finland22 39 39    B. Van Tent 71Laboratoire de Physique Théorique, Université Paris-Sud 11 & CNRS, Bâtiment 210, 91405 Orsay, France71Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    P. Vielva 60Instituto de Física de Cantabria (CSIC-Universidad de Cantabria), Avda. de los Castros s/n, Santander, Spain60Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    F. Villa 44INAF/IASF Bologna, Via Gobetti 101, Bologna, Italy44INAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    L. A. Wade 62Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California, U.S.A.62Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    B. D. Wandelt 56Institut d’Astrophysique de Paris, CNRS (UMR7095), 98 bis Boulevard Arago, F-75014, Paris, France56 86 86 25 25    I. K. Wehus 62Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California, U.S.A.62Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    N. Welikala 85Sub-Department of Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, U.K.85UPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    D. Yvon 12DSM/Irfu/SPP, CEA-Saclay, F-91191 Gif-sur-Yvette Cedex, France12Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, SpainDepartment of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    A. Zacchei 43INAF - Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, Trieste, Italy43INAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland    A. Zonca 24Department of Physics, University of California, Santa Barbara, California, U.S.A.24Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, U.S.A.Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, DenmarkGran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’Aquila, ItalyHelsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, FinlandINAF - Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, ItalyINAF/IASF Milano, Via E. Bassini 15, Milano, ItalyINFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, ItalyINFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, ItalyINFN, Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, ItalyINFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, ItalyInstitut Universitaire de France, 103, bd Saint-Michel, 75005, Paris, FranceIstituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, I-35131 Padova, ItalyKavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, RussiaSorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014, Paris, FranceSpace Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, Moscow, 117997, RussiaSpace Sciences Laboratory, University of California, Berkeley, California, U.S.A.Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian Republic, 369167, RussiaUPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago, F-75014, Paris, FranceUniversity of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, SpainWarsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland Key Words.: Galaxies: high-redshift, clusters, evolution, star formation – Submillimeter: galaxies – Gravitational lensing: strong The Planck mission, thanks to its large frequency range and all-sky coverage, has a unique potential for systematically detecting the brightest, and rarest, submillimetre sources on the sky, including distant objects in the high-redshift Universe traced by their dust emission. A novel method, based on a component-separation procedure using a combination of Planck and IRAS data, has been applied to select the most luminous cold submillimetre sources with spectral energy distributions peaking between 353 and 857 GHz at 5′ resolution. A total of 2151 Planck high-$z$ source candidates (the “PHZ”) have been detected in the cleanest 26 % of the sky, with flux density at 545 GHz above 500 mJy. Embedded in the cosmic infrared background close to the confusion limit, these high-$z$ candidates exhibit colder colours than their surroundings, consistent with redshifts $z>2$, assuming a dust temperature of $T_{\rm{xgal}}=35$ K and a spectral index of $\beta_{\rm{xgal}}=1.5$. First follow-up observations obtained from optical to submillimetre wavelengths have confirmed that this list consists of two distinct populations. A small fraction (around 3 %) of the sources have been identified as strongly gravitationally lensed star-forming galaxies, which are amongst the brightest submm lensed objects (with flux density at 545 GHz ranging from 350 mJy up to 1 Jy) at redshift 2 to 4. However, the vast majority of the PHZ sources appear as overdensities of dusty star-forming galaxies, having colours consistent with $z>2$, and may be considered as proto-cluster candidates. The PHZ provides an original sample, which is complementary to the Planck Sunyaev-Zeldovich Catalogue (PSZ2); by extending the population of the virialized massive galaxy clusters detected with their SZ signal below $z<1.5$ to a population of sources at $z>1.5$, the PHZ may contain the progenitors of today’s clusters. Hence the Planck List of High-redshift Source Candidates opens a new window on the study of the early ages of structure formation, and the understanding of the intensively star-forming phase at high-$z$. 1 Introduction Developing an understanding of the birth and growth of the large-scale structures in the Universe enables us to build a bridge between cosmology and astrophysics. The formation of structures in the nonlinear regime is still poorly constrained, because of the complex interplay between dark matter halos and baryonic cooling at early times, during this transition from the epoch of first galaxy formation to the virialization of massive halos. Hence the analysis of a large sample of high-redshift ($z>2$) objects is crucial for placing new constraints on both cosmological and astrophysical models. Galaxy clusters, as the largest virialized structures in the Universe, are ideal laboratories for studying the intense star-formation occurring in dark matter halos, and providing observational constraints on galaxy assembly, quenching, and evolution, driven by the halos’ environment. The first discoveries of strongly gravitationally lensed galaxies at very high redshift (e.g., Walsh1979; Soucail1987) opened another window onto the early stages of these intensively star-forming galaxies, and provided new information on the early star-formation phase (Danielson2011; Swinbank2011; Combes2012), allowing us to probe spatial details at scales well below 1 kpc (e.g., Swinbank2010; Swinbank2011). From the cosmological point of view, galaxy clusters, considered as the direct descendants of primordial fluctuations on Mpc scales, provide a powerful tool for probing structure formation within the $\Lambda$CDM model (Brodwin2010; Hutsi2010; Williamson2011; Harrison2012; Holz2012; Waizmann2012; Trindade2013). More specifically, planck2013-p11, planck2013-p15, and planck2014-a30 recently highlighted some tension between the cosmological and astrophysical results concerning the determination of the $\Omega_{\mathrm{M}}$ and $\sigma_{8}$ parameters, which still needs to be resolved and properly understood. Galaxy clusters in the local Universe can be efficiently traced by their dominant red sequence galaxies (e.g., Gladders2005; Olsen2007), by their diffuse X-ray emission from the hot gas of the intra-cluster medium (e.g., Ebeling2001; Fassbender2011) or by the Sunyaev-Zeldovich effect (e.g., Foley2011; Menanteau2012; planck2012-I; planck2013-p05a; Brodwin2015) up to $z\thinspace{\simeq}\thinspace 1.5$. The standard methods to search for clusters have yielded only a handful of objects at $z>1.5$ (e.g., Henry2010; Tanaka2010; Santos2011), consistent with the prediction of the concordance model that cluster-size objects virialize late. Searching for high-$z$ large-scale structures means we are looking at the progenitors of local galaxy clusters, the so-called proto-clusters, at the early stages of their evolution, where not enough processed baryonic material was available to be detected by standard methods. These proto-clusters, likely lying at $z>2$, are assumed to be in an active star-forming phase, but not yet fully virialized. To investigate these earlier evolutionary stages we need different approaches, such as the one presented in this paper. During the past decade, more and more proto-cluster candidates have been detected through different techniques, using X-ray signatures, stellar mass overdensities, Ly$\alpha$ emission, and association with radio galaxies (e.g., Brodwin2005; Miley2006; Nesvadba2006; Doherty2010; Papovich2010; Hatch2011; Gobat2011; Stanford2012; Santos2011; Santos2013; Santos2014; Brodwin2010; Brodwin2011; Brodwin2013). However, only a few detections have been done in “random” fields (e.g., Steidel1998; Steidel2005; Toshikawa2012; rettura2014), and most of these detections are biased towards radio galaxies or quasars (pentericci2000; Kurk2000; Kurk2004; venemans2002; venemans2004; venemans2007; galametz2010; galametz2013; rigby2013; wylezalek2013a; trainor2012; cooke2014), or obtained over very limited fractions of the sky, e.g., in the COSMOS field, which is 1.65 $\rm{deg}^{2}$ (capak2011; Cucciati2014; Chiang2014) and in the Hubble Space Telescope Ultra-Deep Field (Beckwith2006; mei2014) with its $200{\arcsec}\times 200{\arcsec}$ area. Since the expected surface density of such strongly lensed high-$z$ galaxies or massive proto-clusters is fairly small, a few times $10^{-2}$ $\rm{deg}^{-2}$ (Negrello2007; Negrello2010; paciga2009; lima2010; Bethermin2011; Hezaveh2012), performing an unbiased analysis of this population of sources requires us to explore much larger regions of the sky. This has been initiated, for example with the Spitzer SPT Deep Field survey covering 94 $\rm{deg}^{2}$ and yielding the detection of 300 galaxy cluster candidates with redshifts $1.3<z<2$ (rettura2014). The submm and mm sky has proved to be an efficient window onto star-forming galaxies with redshifts between 1and 6, since it allows us to detect the redshifted modified blackbody emission coming from the warm dust in galaxies. Taking advantage of the so-called “negative k-correction” in the submm (Franceschini1991), which compensates for the cosmological dimming at high redshift in the submm, many samples of high-$z$ galaxies and also proto-cluster candidates have been identified or discovered in this frequency range in the last two decades (e.g., Lagache2005; Beelen2008; Smail2014). This process has been accelerated with the observations of larger fields in the submm and mm range. The South Pole Telescope experiment (carlstrom2011) which covers 1300 $\rm{deg}^{2}$ at 1.4 and 2 mm has built a unique sample of high-$z$ dusty star-forming objects(Vieira2010), which have been shown to be strongly lensed galaxies at a median redshift $z\simeq 3.5$ (Vieira2013; Weiss2013; Hezaveh2013). A population of 38 dusty galaxies at $z>4$ has been discovered by Dowell2014 in the HerMES survey (26 $\rm{deg}^{2}$) with the Herschel-SPIRE instrument (Griffin2010). Furthermore, as predicted by Negrello2005, Clements2014 showed that the proto-cluster population can be efficiently detected in the submm as overdensities of dusty star-forming galaxies. The Planck satellite111Planck (http://www.esa.int/Planck) is a project of the European Space Agency (ESA) with instruments provided by two scientific consortia funded by ESA member states and led by Principal Investigators from France and Italy, telescope reflectors provided through a collaboration between ESA and a scientific consortium led and funded by Denmark, and additional contributions from NASA (USA). combines two of the main requirements for efficiently detecting high-$z$ sources, namely the spatial and spectral coverage. Planck’s combination of the High Frequency Instrument (HFI) and Low Frequency Instrument (LFI) provides full-sky maps from 857 down to 30 GHz,222Although we use frequency units for Planck channels here, since most of the relevant literature for submillimetre spectra uses wavelengths, we will typically discuss the bands in order of decreasing frequency. which allows coverage of the redshifted spectral energy distribution (SED) of potential dusty star-forming galaxies over a large fraction of the sky. The moderate resolution (5′ to 10′ in the HFI bands) of Planck, compared to other submm experiments, such as Herschel-SPIRE (18″ to 36″) or SCUBA-2 (15″ at 353 GHz), appears as a benefit when searching for clustered structures at high redshift: a 5′ beam corresponds to a physical size of 2.5 Mpc at $z=2$, which matches the expected typical size of proto-clusters in their early stages. We present in this work the Planck List of High-redshift Source Candidates (the “PHZ”), which includes 2151 sources distributed over 26 % of the sky, with redshifts likely to be greater than 2. This list is complementary to the Planck Catalogue of Compact Sources (PCCS2; planck2014-a35), which has been built in each of the Planck-HFI and LFI bands. The PHZ takes advantage of the spectral coverage in the HFI bands, between 857 and 353 GHz, to track the redshifted emission from dusty galaxies using an appropriate colour-cleaning method (Montier2010) and colour-colour selection. It also covers a different population of sources than the galaxy clusters of the Planck Sunyaev-Zeldovich Catalogue (PSZ2; planck2013-p05a), with redshifts likely below 1.5, by tracking the dust emission from the galaxies instead of searching for a signature of the hot intracluster gas. Because of the limited sensitivity and resolution of Planck, the PHZ entries will point to the rarest and brightest submm excess spots in the extragalactic sky, which could be either statistical fluctuations of the cosmic infrared background, single strongly-lensed galaxies, or overdensities of bright star-forming galaxies in the early Universe. This list of source candidates may provide important information on the evolution of the star-formation rate in dense environments: the submm luminosity of proto-clusters will obviously be larger if the star-formation in member galaxies is synchronous and the abundance of protoclusters detected at submm wavelengths depends on the duration of the active star-formation phase. The data that we use and an overview of the processing are presented in Sect. 2. The component separation and point source detection steps are then detailed in Sects. 3 and 4, respectively. The statistical quality of the selection algorithm is characterized in Sect. 5. The final PHZ is described in Sect. 6, followed by a discussion on the nature of the PHZ sources in Sect. 7. 2 Data and processing overview 2.1 Data This paper is based on the Planck 2015 release products corresponding to the full mission of HFI, i.e., five full-sky surveys. We refer to planck2014-a08 and planck2014-a09 for the generic scheme of time-ordered information (TOI) processing and mapmaking, as well as for the technical characterization of the Planck frequency maps. The Planck channel maps are provided in HEALPix (Gorski2005) format, at $N_{\rm{side}}=2048$ resolution. Here we approximate the Planck beams by effective circular Gaussians (planck2014-a08), reported in Table 1. The noise in the channel maps is assumed to be Gaussian, with a standard deviation of 8.8, 9.1, 8.5, and 4.2 kJy sr${}^{-1}$ at 857, 545, 353, and 217 GHz, respectively (planck2014-a09). The absolute gain calibration of HFI maps is known to better than 5.4 and 5.1 % at 857 and 545 GHz, and 0.78 and 0.16 % at 353 and 217 GHz (see table 6 in planck2014-a09). The mean level of the CIB emission has already been included in the Planck frequency maps of the 2015 release, based on theoretical modelling by Bethermin2012, so that the zero-levels of these maps are compatible with extragalactic studies. For further details on the data reduction and calibration scheme, see planck2014-a08 and planck2014-a09. In this work we make use of the “half-ring maps,” which correspond to two sets of maps built with only half of the data as described in planck2014-a09. These can be used to obtain an estimate of the data noise by computing the half-ring difference maps. We combine the Planck-HFI data at 857, 545, 353, and 217 GHz with the 3 THz IRIS data (Miville2005), the new processing of the IRAS 3 THz data (Neugebauer1984). All maps are smoothed at a common FWHM of 5′. 2.2 Mask We define a mask at high Galactic latitude to minimize the contamination by Galactic dusty structures and to focus on the fraction of the sky dominated by CIB emission. As recommended in planck2013-p06b, we used the $E(B-V)_{\rm{xgal}}$ map, released in 2013 in the Planck Legacy Archive,333http://www.cosmos.esa.int/web/planck/pla as an optimal tracer of the neutral hydrogen column density in diffuse regions. After convolving with a FWHM of $5^{\circ}$, we selected regions of the sky with a column density $N_{\rm{H}}<3\times 10^{20}\mathrm{cm^{-2}}$, which translates into $E(B-V)_{\rm{xgal}}<0.0432$. We also reject the stripes over the sky that were not covered by the IRAS satellite, and which are filled-in in the IRIS version of the data using an extrapolation of the DIRBE data at lower resolution (see Miville2005). These undefined regions of the IRAS map have been masked to avoid spurious detections when combining with the Planck maps. The resulting mask leaves out the cleanest 25.8 % of the sky, approximately equally divided between the northern and southern Galactic hemispheres. As shown in Fig. 1, this fraction of the sky remains heterogeneous, due to elongated Galactic structures with low column density. 2.3 Data processing overview The purpose of this work is to find extragalactic sources traced by their dust emission in the submillimetre range (submm). The further away these sources are located, the more redshifted their dust spectral energy distribution (SED) will be, or equivalently the colder they appear. The challenge is to separate this redshifted dust emission from various foreground or background signals and to extract these sources from the fluctuations of the cosmic infrared background (CIB) itself. The data processing is divided into two main steps. The first one is a component separation on the Planck and IRAS maps (see Sect. 3), and the second deals with the compact source detection and selection (see Sect. 4). The full processing can be summarized in the following steps: (i) CMB cleaning – we clean maps to remove the CMB signal in all submm bands using a CMB template (see Sect. 3.2); (ii) Galactic cirrus cleaning – we clean maps at 857 to 217 GHz for Galactic cirrus emission using a Galactic template combined with the local colour of the maps (see Sect. 3.3); (iii) excess map at 545 GHz – looking for sources with redshifted SEDs and peaking in the submm range, we construct an excess map at 545 GHz, revealing the cold emission of high-$z$ sources, using an optimized combination of all cleaned maps (see Sect. 3.4); (iv) point source detection in the 545 GHz excess map – the point source detection is applied on the excess map at 545 GHz (see Sect. 4.1); (v) multi frequency detection in the cleaned 857, 545, and 353 GHz maps – simultaneous detections in the cleaned maps at 857, 545, and 353 GHz are also required to consolidate the detection and enable photometry estimates in these bands (see Sect. 4.1); (vi) colour-colour selection – complementary to the map-processing aimed at emphasizing the cold emission from high redshifted sources, we apply a colour-colour selection based on the photometry (see Sect. 4.3); (vii) flux density cut – a last selection criterion is applied on the flux density to deal with the flux boosting affecting our photometry estimates (see Sect. 4.3). Notice that the first two steps, i.e., CMB and Galactic cleaning, are also applied independently on the first and last half-ring maps (planck2014-a09) in all bands, to provide robust estimates of the noise in the cleaned maps, which are then used during the photometry processing. After carrying out this full processing on the Planck and IRAS maps, we end-up with a list of 2151 Planck high-$z$ source candidates, distributed over the cleanest 25.8 % of the sky. We detail in the following sections the various steps of the processing, the construction of the final list and a statistical validation of its quality. 3 Component separation 3.1 Astrophysical emissions Owing to the negative k-correction, high-z sources (typically $z=1$–$4$) have very “red” submillimetre colours. Superimposed onto the emission from these sources are other astrophysical signals, such as the CIB fluctuations, the CMB anisotropies, and the Galactic foreground dust emission, each with a different spectral energy distribution (SED). A broad frequency coverage from the submm to mm range is thus mandatory in order to separate these astrophysical components, so that we can extract faint emission from high-$z$ candidates. Combined with IRAS (Neugebauer1984) data at 3 THz, the Planck-HFI data, which spans a wide spectral range from 100 to 857 GHz, represents a unique set of data that is particularly efficient for separating Galactic from extragalactic and CMB components, as illustrated in Fig. 2. The Galactic cirrus emission at high latitude is modelled with a modified blackbody, with a dust temperature of $T_{\rm{d}}=17.9\thinspace\rm{K}$ and a spectral index $\beta=1.8$ (planck2011-7.12). The SED of Galactic cirrus is normalized at 3 THz using an averaged emissivity (estimated by planck2011-7.12, at high Galactic latitude) of $\epsilon_{100}=0.5\thinspace\rm{MJy}\thinspace\rm{sr}^{-1}/10^{20}\rm{cm}^{-2}$ and a mean column density of $N_{\rm{H}}=2\times 10^{20}\thinspace\rm{cm}^{-2}$. The grey shaded region in Fig. 2 shows the $\pm 2\thinspace\sigma$ domain of the Galactic cirrus fluctuations estimated at 3 THz by computing the integral of the power spectrum $P(k)$ over the IRAS maps between multipoles $\ell=200$ and $\ell=2000$, as done in planck2011-6.6 for the CIB. This procedure gives $\sigma_{\rm{gal}}^{2}=2\pi\int{P(k)\times kdk}$, where $P(k)$ is the 2-D power spectrum obtained in small patches of 100 deg${}^{2}$, leading to a value of $\sigma_{\rm{gal}}=0.28\thinspace\rm{MJy}\thinspace\rm{sr}^{-1}$ at 3 THz. The CIB emission is given by the model of Bethermin2011, with 2$\thinspace\sigma$ values taken from planck2011-6.6 and defined for spatial scales of $200<\ell<2000$. The anisotropies of the CMB, $\Delta$CMB, have been normalized at 143 GHz to correspond to a 2$\thinspace\sigma$ level fluctuations, with $\sigma_{\rm{CMB}}=65\ \mu\rm{K_{CMB}}$, equivalent to $0.05\thinspace\rm{MJy}\thinspace\rm{sr}^{-1}$. Typical SEDs of extragalactic sources are also indicated on Fig. 2 using a modified blackbody emission law with a temperature of 30 K and a dust spectral index of 1.5, and redshifted to $z=1$, 2, 3, and 4. All SEDs have been normalized to a common brightness at 545 GHz, equivalent to a flux density of 1 Jy for objects as large as 5′ FWHM. As shown in Fig. 2, the Galactic cirrus emission, which appears warmer than the other components and peaks at around 2 THz, is well traced by the 3 THz band of IRAS, as well as the CIB emission which peaks around 1 THz. The CMB anisotropies are well mapped by the low frequency bands of Planck-HFI, at 100 and 143 GHz. Finally the emission from high-$z$ sources is dominant in the four bands, from 857 to 217 GHz, covered by HFI. This illustrates that the IRAS plus Planck-HFI bands are well matched to separate the far-IR emission of high-$z$ ULIRGs from that of the CMB, Galactic cirrus, and CIB fluctuations. Because of the special nature of the compact high-$z$ sources, presenting SEDs peaking between the Galactic dust component, the CIB component, and the CMB signal, we had to develop a dedicated approach to component separation, which is detailed below. This algorithm enables us to clean first for the CMB component, then for the Galactic and low-$z$ CIB component, and finally to optimize the excess at 545 GHz. 3.2 CMB cleaning The CMB component is removed from the 3000, 857, 545, 353, and 217 GHz IRIS and Planck maps using a CMB template, which is extrapolated to the other bands according to a CMB spectrum. To do this we take into account the spectral bandpass of each channel, as described in planck2013-p03d. The cleaning is performed in the HEALPix pixel basis, so that the intensity of each pixel after CMB cleaning is given by $$I_{\nu}^{\rm C}=I_{\nu}-I_{\rm{CMB}}\times\Delta B_{\nu}^{\rm{CMB}}\thinspace,$$ (1) where $I_{\nu}$ is the intensity of a pixel of the input map at frequency $\nu$, $I_{\nu}^{\rm C}$ is the intensity after CMB cleaning, $I_{\rm{CMB}}$ is the intensity of the CMB template, and $\Delta B_{\nu}^{\rm{CMB}}$ is the intensity of the CMB fluctuations integrated over the spectral bandpass of the band at frequency $\nu$. The choice of the CMB template has been driven by the aim of working as close as possible to the native 5′ resolution of the Planck high frequency maps, in order to match as well as possible the expected physical size of proto-clusters at high redshift, i.e., around 1 to 2 Mpc at $z=2$. Among the four methods applied to the Planck data to produce all-sky foreground-cleaned CMB maps (planck2014-a11), only two of them provide temperature CMB maps at 5′ resolution, i.e., NILC (Basak2012; Basak2013) and SMICA (Cardoso2008). Since the latter has been shown in planck2014-a11 to be the least contaminated by foregrounds for high $\ell$ ($\ell>2000$), it has been chosen as the CMB template in this work. The overall agreement between all four methods on the temperature CMB maps is very good, with an amplitude of pairwise difference maps below 5 $\thinspace\mu$K${}_{\rm{CMB}}$ over most of the sky on large scales, and below 1$\thinspace\sigma$ at high $\ell$. However, it is clearly stated that the these maps are not fully cleaned of high-$\ell$ foregrounds, such as extragalactic point sources or Sunyaev-Zeldovich (SZ) emission. Hence this CMB template may be used to clean efficiently the Planck and IRIS maps for CMB signal at large and intermediate scales, but not at small scales. Actually, these residual emission components – including synchrotron emission from strong radio sources, thermal emission from Galactic cold dust, or SZ signal from galaxy clusters – in the CMB template are extrapolated to the IRIS and Planck bands with a CMB spectrum during the CMB-cleaning procedure, which may impact the rest of the analysis, as we investigate in Sects. 3.5 and 3.6. In order to avoid such issues, it would have been possible to use the 143 GHz Planck map as a CMB template. In this case, the presence of non-CMB signal could have been more easily quantified; however, the common resolution of all IRIS and Planck maps would then have to have been degraded to the 7.3′ resolution of the 143 GHz map, which is not convenient when looking for compact objects. As a test case, we have performed a comparison between the two CMB cleaning options at 8′ resolution to study the impact on the flux density estimates towards PHZ sources, see Appendix A. 3.3 Galactic cirrus cleaning In order to clean the Galactic cirrus emission at high latitude, we apply the colour-cleaning method introduced by Montier2010. In this method, the 3 THz IRIS map, considered as a template of the Galactic dust emission, is extrapolated to the lower frequencies using the local colour around each pixel and is removed from the current map. Hence the intensity of a pixel in the output map at frequency ${\nu}$ is given by $$I_{\nu}^{\rm D}=I_{\nu}^{\rm C}-I_{3000}^{\rm C}\times\left<\frac{I_{\nu}^{\rm C% }}{I_{3000}^{\rm C}}\right>_{R_{\rm{cirrus}}},$$ (2) where $I_{\nu}^{\rm C}$ and $I_{\nu}^{\rm D}$ are the intensities of the pixel after the CMB and Galactic cirrus cleaning, respectively, and the $\big{<}\big{>}_{R_{\rm{cirrus}}}$ operator is the median estimate over a ring between radius $R_{\rm{cirrus}}^{\rm{in}}=20$′ and $R_{\rm{cirrus}}^{\rm{out}}=30$′ around the central pixel. The extension of the ring has been chosen, following the prescriptions of Montier2010, to maximize (at a beam scale) the signal of pixels with abnormal colours compared to the background, i.e., by cleaning structures larger than 20′ using the local colour of the background estimated up to 30′. The ratio $I_{\nu}^{\rm C}/I_{3000}^{\rm C}$ is defined as the colour index. More generally, this method of cleaning the Galactic dust emission at high latitude allows us to subtract all the “warm” dust components present in the 3 THz map, compared to the “cold” dust components, which will preferentially peak at lower frequencies. A structure with the same colour index as the average background within a 30′ radius will vanish from the cleaned map. A structure appearing colder than the background will present a colour index larger than the average background, and will produce a positive residual. On the other hand, a structure warmer than the average will be characterized by a negative residual after this colour cleaning. We stress that the definition of “warm” or “cold” at any frequency is determined relative to the local background colour, which is a mixture of Galactic cirrus emission and CIB emission at this location. Where the emission is dominated by Galactic cirrus, this method will mainly clean the “warm” Galactic dust emission; where the sky is dominated by CIB emission, it will clean the low-$z$ component of the CIB and it will enhance the high-$z$ part as positive emission. Notice also that real strong “warm” sources present in the 3 THz map will produce extremely negative residuals in the cleaned maps, so that, more generally, the statistics of the negative pixels in the cleaned maps should not be correlated with these of positive pixels, both tracing different phases of the observed sky. 3.4 Excess maps The SEDs of sources located at high redshift will exhibit an excess of power at lower frequencies, located at their dust emission peak. In order to enhance this effect, we build the excess map at 545 GHz by subtracting from the cleaned map at 545 GHz a linear interpolation between the two surrounding bands, i.e., the 857 and 353 GHz maps, as written below: $$\displaystyle I_{545}^{\rm X}=I_{545}^{\rm D}-\Bigg{\{}\big{<}I_{857}^{\rm D}% \big{>}_{R_{\rm{x}}}+$$ $$\displaystyle\qquad\qquad\qquad\left(\big{<}I_{353}^{\rm D}\big{>}_{R_{\rm{x}}% }-\big{<}I_{857}^{\rm D}\big{>}_{R_{\rm{x}}}\right)\cdot\frac{(545-857)}{(353-% 857)}\Bigg{\}}\thinspace,$$ (3) where $I_{545}^{\rm X}$ is the intensity in the excess map at 545 GHz, $I_{\nu}^{\rm D}$ is the intensity after CMB and Galactic cirrus cleaning at frequency $\nu$, and the $\big{<}\big{>}_{R_{\rm{x}}}$ operator here is the median estimate over a disk of radius ${R_{\rm{x}}}=6$′. The value of the radius $R_{\rm{x}}$ has been determined on simulations to optimize the signal-to-noise ratio of the output signal in the excess map. The full process of cleaning is illustrated in Fig. 3 for the Planck high-$z$ candidate PHZ G095.50$-$61.59, which has been confirmed by spectroscopic follow-up as a proto-cluster candidate (Florescacho2015). In Fig. 3 each row corresponds to a step in the cleaning, from original maps smoothed at 5′ (first row), to CMB-cleaned maps (second), Galactic cirrus-cleaned maps (third), and finally yielding the excess map at 545 GHz (fourth). 3.5 Impact of cleaning on high-$z$ candidates The cleaning process allows us to perform an efficient component separation to isolate the extragalactic point sources, but also impacts the original SEDs of these high-$z$ sources. The fraction of emission coming from extragalactic sources present in the CMB and 3 THz templates are extrapolated and subtracted from the other bands. Concerning the CMB template, since the amount of residual emission coming from the extragalactic sources remains unknown, we bracket the impact by making two extreme assumptions. On the one hand, the CMB template is assumed to be perfect, i.e., without any foreground residual emission. In that case the CMB cleaning has no impact on the cleaned SEDs. On the other hand, since the CMB template is mainly dominated by the signal of the 143-GHz band (where the signal-to-noise ratio of the CMB is the strongest compared to the other astrophysical components), we assume in the worse case that it includes a residual emission equivalent to the expected intensity at 143 GHz of the extragalactic high-$z$ source. The impact of cleaning is illustrated in Fig. 4, where the SEDs of extragalactic sources at five redshifts (from 0.5 to 4) are modelled by modified blackbody emission with a temperature of $T_{\rm{xgal}}=30$ K, and a spectral index $\beta_{\rm{xgal}}=1.5$, normalized at 1 MJy $\rm{sr}^{-1}$ for 857 GHz. The Galactic cirrus cleaning has been performed assuming a balanced mixture of CIB and Galactic dust emission. Cleaned SEDs are shown shown for the two cases of CMB template quality, i.e., ideal or highly foreground-contaminated. The SEDs of low-$z$ ($<1$) sources are strongly affected by the cleaning from Galactic cirrus, as expected, while the SEDs at higher redshifts ($z=4$) are potentially more affected by the CMB cleaning. This has to be kept in mind when computing the photometry for any such sources detected in the cleaned Planck maps. Note that cleaning will tend to remove some of the flux of real sources. We define the relative attenuation coefficient in each Planck-HFI band due to the cleaning process, $A_{\nu}^{\rm{clean}}$, as $$A_{\nu}^{\rm{clean}}=\frac{I_{\nu}-I_{\nu}^{\rm{D}}}{I_{\nu}}\thinspace.$$ (4) Again this attenuation coefficient ranges between two extreme cases, depending on the level of contamination by extragalactic foregrounds in the CMB template. An estimate of this relative attenuation coefficient is shown in Fig. 5 as a function of redshift for the 857, 545, 353, and 217 GHz Planck bands. We observe that, in the worse case (lower panel), flux densities at 857 and 545 GHz are barely impacted by the cleaning for redshifts $z>2$, while for the 353-GHz band the attenuation reaches 5 % to 20 %. The attenuation for the 217-GHz band is much larger, ranging between 30 % and 40 %. When the CMB template is assumed to be ideal (upper panel), the attenuation remains small for $z>2$ in all bands. At low redshifts ($<1$), the attenuation coefficient reaches 100 % in both cases, which means that the cleaning process fully removes these sources from the maps. In the intermediate range of redshifts ($1<z<2$), the situation is less clear and requires more realistic simulations to provide a reliable assessment of the detection of such sources, as performed in Sect. 5. We emphasize that this attenuation coefficient strongly depends on the SED type and the redshift of each source. Simply changing the temperature of the source $T_{\rm{xgal}}$ from 30 K to 40 K shifts the transition zone from redshift 1–2 to 2–3 (see Fig. 5), making it hard to predict the actual attenuation coefficients. 3.6 Contamination by foreground astrophysical sources 3.6.1 Thermal emission from cold Galactic dust Because of the degeneracy between the temperature of a source and its redshift, cold clouds at high latitude represent an important contaminant for the detection of high-$z$ sources, Indeed, the SED of a Galactic cold source modelled by a modified blackbody with a temperature $T_{\rm{dust}}=10$ K (blue curve of Fig. 6) will mimic the same spectral trend in the submm range as the SED of a warm source ($T_{\rm{xgal}}=30$ K) redshifted to $z=2$ (green curve of Fig. 4). This can only be disentangled by taking into account other properties of such Galactic sources, such as the H i column density or the structure of its surroundings, which may be associated with Galactic components. For this reason, with each detection there will be associated an estimate of the local extinction at the source location and in the background, as a tracer of the local H i column density. This is further discussed in Sect. 6.7, in the analysis of the cross-correlation between the list of high-$z$ source candidates and the catalogue of Planck Galactic Cold Clumps (PGCC; planck2014-a37). 3.6.2 Synchrotron emission from radio sources The typical SED of the synchrotron emission from radio sources is observed in the submm range as a power law with a spectral index around –0.5 (planck2011-6.1), as shown as a red solid line in Fig. 6. While the slope of the cleaned SED is accentuated by the cleaning when the CMB template is assumed to be ideal (red dotted line), if the CMB template is highly contaminated by extragalactic foregrounds at high $\ell$, the cleaned SED (red dashed line) exhibits a positive bump in the 857 and 545-GHz bands, and a deficit in the 353- and 217-GHz bands, which mimics the excess at 545 GHz expected for the high-$z$ sources. This artefact can be identified by looking at the intensity in the 100-GHz band, which remains strongly positive in the case of synchrotron emission, compared to the expected emission of high-$z$ dusty galaxies at 100 GHz. For this reason we provide a systematic estimate of the 100 GHz flux density and compare it to the flux densities at higher frequencies in order to reject spurious detections of radio sources. 3.6.3 SZ emission from galaxy clusters The SZ effect (Sunyaev70) is a distortion of the CMB due to the inverse Compton scattering induced by hot electrons of the intra-cluster medium. It generates a loss of power at frequencies below 217 GHz, and a gain above this frequency. An SZ spectrum after removal of the CMB monopole spectrum is shown as a black solid line in Fig. 6, using a typical integrated Compton parameter $Y_{500}=10^{-3}$ (see planck2013-p05a; planck2014-a36). Along the direction towards galaxy clusters, if the CMB template is not fully cleaned for SZ emission, the CMB cleaning method will artificially enhance the signal of the resulting SED (black dashed line) by subtracting the (negative) SZ signal at 143 GHz. This produces a clear bump of the cleaned SED in the 353-GHz band, as expected for the SED of a dusty source ($T_{\rm{xgal}}=30$ K) at $z=7.5$, which is not likely to be detected at 5′ resolution. Hence the SZ SED does not properly reproduce the expected colours of the dusty galaxies at high $z$, and should not be detected by our algorithm. However, it may represent an important contaminant if a galaxy cluster and a high-$z$ dusty source lie along the same line of sight. This is addressed in Sect. 6.7, in the analysis of the cross-correlation between this list and the Planck Catalogue of SZ sources (PSZ; planck2014-a36). 4 Point source detection We describe in this section how the point source detection is performed and the photometry estimates are obtained. We also detail the final selection process, based on both a colour-colour analysis and a flux density threshold. 4.1 Detection method The point source detection algorithm requires positive detections simultaneously within a 5′ radius in the 545 GHz excess map, and the 857, 545, and 353 GHz cleaned maps. It also requires a non-detection in the 100 GHz cleaned maps, which traces emission from synchrotron sources. As already mentioned in Sect. 3.3, negative pixels in the cleaned and excess maps represent the locally warmer phase of the high-latitude sky, which may statistically strongly differ from the one of the positive pixels tracing the colder phase. For this reason negative pixels are masked afterwards, so that we characterize the significance of a detection by comparing the value of each pixel to the statistics of positive pixels only. Hence the local noise is estimated as the median absolute deviation estimate over the positive pixels of each map within a radius $R_{\rm{det}}=60$′ around each pixel. A disk of $1^{\circ}$ radius covers about 150 times the beam of 5′, providing enough statistics to obtain a reliable estimate of the standard deviation. It also covers twice the typical scale of any Galactic cirrus structures filtered by the cleaning process (using a radius $R_{\rm{cirrus}}=30$′, see Sect. 3.3). A detection is then defined as a local maximum of the signal-to-noise ratio (S/N) above a given threshold in each map, with a spatial separation of at least 5′ being required between two local maxima. A threshold of $\rm{S/N}>5$ is adopted for detections in the 545 GHz excess map, while this is slightly relaxed to $\rm{S/N}>3$ for detections in the cleaned maps, because the constraint imposed by the spatial consistency between detections in all three bands is expected to reinforce the robustness of a simultaneous detection. Concerning the 100-GHz band, we adopt a similar threshold by requiring the absence of any local maximum with $\rm{S/N}>3$ within a radius of 5′. Notice also that this criterion is applied on the 100 GHz map, which is only cleaned from CMB after convolving the CMB template and 100 GHz maps at a common 10′ resolution. A detection is finally defined by the following simultaneous criteria: $$\left\{\begin{array}[]{l c l}I_{545}^{\rm{X}}/\sigma^{\rm{X}}_{545}&>&5% \thinspace;\\ I_{\nu}^{\rm{D}}\thinspace\thinspace/\thinspace\thinspace\sigma^{\rm{D}}_{\nu}% &>&3\thinspace,\quad{\rm for}\thinspace\nu=857,545{\rm{,\thinspace and}}% \thinspace 353\thinspace{\rm{GHz}}\thinspace;\\ I_{100}^{\rm{C}}/\sigma^{\rm{C}}_{100}&<&3\thinspace.\end{array}\right.$$ (5) 4.2 Photometry The photometry is computed at the location of the detections in the cleaned 857, 545, 353, and 217 GHz maps. It is performed in two steps: (i) determination of the extension of the source in the 545 GHz cleaned map; and (ii) aperture photometry in all bands in the cleaned maps. We perform an elliptical Gaussian fit in the 545 GHz cleaned map at the location of the detection in order to find the exact centroid coordinates, the major and minor axis FWHM values, and the position angle, with associated uncertainties. Flux densities, $S_{\nu}^{\rm{D}}$, are obtained consistently in all four bands via an aperture photometry procedure using the elliptical Gaussian parameters derived above in the cleaned maps. The accuracy of the flux densities, $\sigma_{\nu}^{S}$, can be decomposed into three components: $\sigma_{\nu}^{\rm{geom}}$ comes from the uncertainty of the elliptical Gaussian fit; $\sigma_{\nu}^{\rm{sky}}$ represents the level of the local CIB fluctuations that dominate the signal at high latitude; and $\sigma_{\nu}^{\rm{data}}$ is due to the noise measurement of the Planck data and estimated using half-ring maps. An estimate of the elliptical Gaussian fit accuracy, $\sigma_{\nu}^{\rm{geom}}$, is obtained by repeating the aperture photometry in 1000 Monte Carlo simulations, where the elliptical Gaussian parameters are allowed to vary within a normal distribution centred on the best-fit parameters and a $\sigma$-dispersion provided by the fit. The uncertainty $\sigma_{\nu}^{\rm{geom}}$ is defined as the mean absolute deviation over the 1000 flux density estimates. We use the first and last half-ring maps, which have been cleaned following the same process as the full maps, to obtain an estimate of the accuracy of the photometry related to the noise in the data. This is computed as the absolute half difference of the photometry estimates, $S_{\nu}^{\rm{first}}$ and $S_{\nu}^{\rm{last}}$, obtained from the first and last half-ring cleaned maps, respectively. Since this quantity follows a half-normal distribution, the estimate of the noise measurement in the full survey is finally given by $$\sigma_{\nu}^{\rm{data}}=\sqrt{\frac{\pi}{2}}\thinspace\left|\frac{S_{\nu}^{% \rm{first}}-S_{\nu}^{\rm{last}}}{2}\right|\thinspace.$$ (6) The local level of the CIB fluctuations, $\sigma_{\nu}^{\rm{sky}}$, is obtained by computing the standard deviation over 400 flux density estimates obtained by an aperture photometry with the nominal elliptical Gaussian shape parameters in the cleaned maps at 400 random locations within a radius of $1^{\circ}$ around the centroid coordinates. Those random locations are chosen among the positive pixels of the excess maps, for the same reason as given in Sect. 4.1, i.e., to explore the same statistics as the detection pixels. Notice that this estimate of $\sigma_{\nu}^{\rm{sky}}$ also includes the noise of the data, even if the latter is shown to be low compared to the CIB fluctuation level. We stress that the flux densities are computed using the cleaned maps, since their S/N values are higher than in the original maps, where the high-$z$ source candidates are embedded in Galactic cirrus, CIB structures, and CMB fluctuations. Nevertheless they still suffer from several potential systematic effects: (1) attenuation due to the cleaning; (2) contamination by the Sunyaev-Zeldovich effect (SZ) discussed in Sect. 6.7; and (3) the flux boosting effect presented in Sect. 5.3. 4.3 Colour-colour selection and flux cut A colour-colour selection is applied to the cleaned flux densities in order to keep only reliable high-$z$ candidates. This aims to reject Galactic cold clumps and radio sources, if still present in the detected sample. We use the three highest frequency Planck bands in which detections at $\rm{S/N}>3$ are simultaneously required. The colour-colour space is thus defined by the $S_{545}/S_{857}$ and $S_{353}/S_{545}$ colours. Firstly, we require $S_{545}/S_{857}>0.5$, to reject potential Galactic cold sources, which exhibit colour ratios ranging from 0.2 to 0.5 for dust temperatures ranging between 20 K and 10 K (with a spectral index equal to 2). It is found that 98.5 % of the cold clumps in the PGCC catalogue (planck2014-a37) have a colour $S_{545}/S_{857}<0.5$. We emphasize that this criterion can be safely applied to the colour ratio $S_{545}^{\rm{D}}/S_{857}^{\rm{D}}$ obtained on cleaned maps, as quantified with Monte Carlo simulations (see Sect. 5.3). Secondly, it is common to constrain $S_{353}/S_{545}$ to be less than 1 in order to avoid contamination from radio sources, which have negative spectral indices (e.g., see planck2014-a37). However, this criterion has to be adapted when using the photometry based on the cleaned maps. As already mentioned in Sect. 3.6.2, typical SEDs of radio sources are transformed after cleaning, so that they no longer have $S_{353}/S_{545}>1$. While SEDs of extremely redshifted dusty galaxies may present colour ratios larger than 1, their cleaned SEDs will be strongly affected by the cleaning process, so that their colour ratio goes below 0.9 whatever the redshift (as discussed in Sect. 5.3). This remains the case for galaxy clusters with an SZ signature, which produces an excess of the flux density at 353 GHz after the cleaning process, so that this colour ratio would be larger than 1. Hence the criterion is finally set to $S_{353}^{\rm{D}}/S_{545}^{\rm{D}}<0.9$, so that dusty galaxies are not rejected, but SZ contamination is. In order to properly propagate the uncertainties of the flux density estimates in all three bands during the colour-colour selection process, we construct for each source the probability for the two colour ratios to lie within the high-$z$ domain, given the 1$\thinspace\sigma$ error bars associated with the flux densities: $$\mathcal{P}\left(\thinspace\thinspace\frac{S_{545}^{\rm{D}}}{S_{857}^{\rm{D}}}% \thinspace>\thinspace 0.5\quad{\rm and}\ \quad\frac{S_{353}^{\rm{D}}}{S_{545}^% {\rm{D}}}\thinspace<\thinspace 0.9\thinspace\thinspace\right)\thinspace.$$ (7) This probability is built numerically by simulating for each source 100 000 flux densities including noise in the 857-, 545-, and 353-GHz bands, ($S^{\rm I}_{857}$, $S^{\rm I}_{545}$, and $S^{\rm I}_{353}$), using the cleaned flux density estimates and their 1$\thinspace\sigma$ uncertainties. The flux density uncertainties used to build these noise realizations are defined as the quadratic sum of the data noise, $\sigma_{\nu}^{\rm{data}}$, and the elliptical Gaussian fit accuracy, $\sigma_{\nu}^{\rm{geom}}$, so that only proper noise components of the uncertainty are included, but not the confusion level from CIB fluctuations. The probability estimate $\mathcal{P}$ for each source is then defined as the ratio between the number of occurrences satisfying the two colour criteria of Eq. (7) and the total number of realizations. The colour-colour selection criterion has been finally set up as the condition $\mathcal{P}>0.9$, based on the Monte Carlo analysis described in Sect. 5. This approach is far more robust than a simple cut based on the two colour criteria. It also enables us to reject sources that might satisfy the criteria owing to poor photometry alone. 5 Monte Carlo quality assessment 5.1 Monte Carlo simulations In order to assess the impact of the cleaning method on the recovered flux densities of the Planck high-$z$ candidates and to explore the selection function of the algorithm, we have performed Monte Carlo simulations. A total of 90 sets of mock IRIS plus Planck maps have been built by injecting 10 000 simulated high-$z$ point sources into the original Planck and IRIS maps, yielding a total of 900 000 fake injected sources. The SEDs of these sources are modelled via modified blackbody emission with a spectral index $\beta_{\rm{xgal}}=1.5$, and four equally probable values of the temperature, $T_{\rm{xgal}}=20$, 30, 40, and 50 K. The redshift of these sources is uniformly sampled between $z=0$ and $z=5$. The flux density distribution follows a power law with an index equal to the Euclidean value ($-2.5$) between 200 mJy and 5 Jy at 545 GHz. Each source is modelled as an elliptical Gaussian with a FWHM varying uniformly between 5′ and 8′, and a ratio between the major and minor axes ranging uniformly between 1 and 2. The point sources are then injected into the real IRIS and Planck maps (already convolved at 5′ resolution), excluding the regions within 5′ of true detections of high-$z$ source candidates. The full cleaning, extraction, photometry, and colour-colour selection processing described in Sects. 3 and 4 is performed on this set of mock maps, yielding a sub-sample of about 70 000 detected sources from the 900 000 injected. Notice that the cut on the 545 GHz flux density has been omitted in this analysis in order to explore the completeness of the detection algorithm beyond this flux density limit. Furthermore, we have tested two options of the CMB template during the cleaning processing: an ideal template, which consists in the SMICA 5′ CMB map; and a highly contaminated template, which has been built by injecting the expected flux densities at 143 GHz into the SMICA 5′ CMB template before cleaning, assuming here that the signal from the extragalactic source is still fully included in the CMB template. This allowed us to quantify the maximum impact of the uncleaned foregrounds present in the CMB template we use for the official cleaning. Finally, we stress that the fraction of total detections over the total number of injected sources cannot be considered as an estimate of the overall recovery rate of the algorithm, because of the unrealistic statistics of the injected population in terms of temperature, redshift or flux density. However, these mock simulations allow us to build the a posteriori uncertainties on the properties of the recovered sources, and the selection function due to the detection algorithm. 5.2 Geometry accuracy We first analyse the positional accuracy of the detected sources and show the results in Fig. 7. Recall that the centroid coordinates of the elliptical Gaussian are obtained through a fit on the cleaned 545 GHz map. Hence 68 % of the sources exhibit a positional offset smaller than 1.${}^{\scriptstyle\prime}$2 and 95 % of them within 2.${}^{\scriptstyle\prime}$9, which are not negligible values compared to the 5′ resolution of the maps. This positional uncertainty is mainly due to the confusion level of the CIB in which these sources are embedded. More problematic is the efficiency of the FWHM recovery, which drives the computation of the aperture photometry (see Fig. 8). Recall that the FWHM is defined as the geometric mean of the minor and major FWHM, FWHM=$(\rm{FWHM}_{\rm{min}}\times\rm{FWHM}_{\rm{maj}})^{1/2}$. The recovered FWHM is overestimated compared with the input FWHM over the whole range of S/N of detection in the 545 GHz cleaned map (see left panels), by an average value of 3.5 % at high S/N, and up to 15 % at low S/N (below 10). When looking more carefully at the distribution of recovered versus injected FWHMs, it appears that the largest FWHM bin, the one close to 8′, is better recovered than the smallest FWHM values, which are strongly overestimated by up to 30 %. In fact, the distribution of the recovered FWHM peaks around 6.${}^{\scriptstyle\prime}$5, while the input values were uniformly distributed between 5′ and 8′. In addition to this, the dispersion of the ratio between the recovered and the injected FWHMs does not significantly decrease with the S/N of the detection, and lies around 7 %. This is larger than the level of uncertainty provided by the elliptical Gaussian fit, which is about 1.5 % at maximum. Indeed the uncertainty on the FWHM is mainly dominated by the confusion level of the CIB. The same analysis is performed on the ellipticity of the sources, defined as $$\varepsilon=\sqrt{1-\left(\frac{\theta_{\rm{min}}}{\theta_{\rm{maj}}}\right)^{% 2}}\thinspace,$$ (8) where $\theta_{\rm{min}}$ and $\theta_{\rm{maj}}$ are the minor and major axis of the ellipse, respectively. When looking at the ratio between the recovered and injected ellipticity as a function of the S/N of the 545 GHz flux density (right panels of Fig. 8), the estimates do not seem biased for S/N larger than 5. However, the recovered versus injected ellipticity comparison shows that low ellipticities are systematically overestimated. The average ellipticity estimates are greater than 0.6 over the whole range of input ellipticity. Recall that an ellipticity $\varepsilon=0.6$ corresponds to a major axis 1.25 times larger than the minor axis. Such an error of 25 % between minor and major FWHMs is fully compatible with the level of uncertainty of the recovered FWHM, pointed above. Again this effect is probably explained by the CIB confusion. We have observed that these results are totally independent of the choice of the CMB template (ideal or highly foreground-contaminated) for the cleaning processing, because the geometry parameters are obtained in the 545 GHz cleaned map, which are barely impacted by the CMB cleaning. 5.3 Photometry quality We first recall that the recovered photometry, $S_{\nu}^{\rm{D}}$, is obtained on cleaned maps and suffers from the noise and the CIB confusion, but also from the attenuation effect due to the cleaning process. For each Planck band, the ratio of the recovered to input flux density ($S_{\nu}^{\rm{D}}$/$S_{\nu}^{\rm{I}}$) is shown in the top row of Fig. 9 as a function of the S/N of the flux density, defined here as the ratio of the recovered flux density to the uncertainty due to CIB confusion, $S_{\nu}^{\rm{D}}/\sigma_{\nu}^{\rm{sky}}$. This is shown for the two options of the CMB template, i.e., ideal (squares) or highly contaminated (crosses). When assuming a very low level of foreground contamination in the CMB template, flux density estimates in all Planck bands are recovered with a very good accuracy, as expected according to theoretical predictions of the attenuation effect of Fig. 5. The fact that all flux density estimates appear statistically slightly underestimated by about 4 % for $\rm{S/N}>5$ is related to the quality of the source shape recovery. On the contrary, when the CMB template is assumed to be highly contaminated by the extragalactic foregrounds, flux density estimates are more impacted by the cleaning process, especially at 217 GHz. In this band, the attenuation factor due to cleaning reaches a level of 47 % at high S/N, which is compatible with the predictions of Sect. 3.5. The attenuation at 353 GHz is about 17 % at high S/N. Below a S/N of around 5 two other effects appear: a much larger overestimation of the FWHM, up to 30 % at very low S/N, as discussed in Sect. 5.2; and the so-called flux boosting effect, which represents the tendency to overestimate the flux densities of faint sources close to the CIB confusion because of noise upscatters being more likely than downscatters (see Hogg1998). While the latter can be addressed using a Bayesian approach (Coppin2005; Coppin2006; Scott2008) for intermediate S/N (i.e., $\rm{S/N}>8$), we used this set of Monte Carlo simulations, as done by Scott2002 and Noble2012, to assess its impact on photometry estimates. As observed in the bottom panels of Fig. 9, flux densities of faint sources are strongly overestimated, producing a plateau around 0.5 Jy at 545 GHz. This is consistent with the confusion noise levels predicted by Negrello2004 in the Planck bands. Because of the complex interplay between the attenuation due to the cleaning process, the geometry recovery, and the flux boosting effect, any simple Bayesian approach for flux de-boosting would be difficult to implement. For this reason, the flux density estimates of the Planck high-$z$ candidates presented in this work are not corrected for flux boosting or cleaning attenuation. However, in order to minimize the impact of flux boosting when building the final list, we will apply a minimal threshold on the 545 GHz flux density estimates, which has been set to 500 mJy, as determined through these simulations. 5.4 Colour selection accuracy It is important to notice that the colour ratios of the detected sources are relatively well preserved by the cleaning and photometry processing, which is crucial to ensure the quality of the colour-colour selection of these high-$z$ candidates. The dependence with the S/N of the detection in the excess map of the ratio between the recovered to input colour ratios is shown for both $S_{353}/S_{545}$ and $S_{545}/S_{857}$ in the left panels of Fig. 10. Note that for this analysis we include all the sources detected before applying any colour-colour selection, in order to assess the robustness of the latter selection. Again, in this analysis, the ideal and highly contaminated cases of the CMB template are explored. When assuming an ideal CMB template, the recovered $S_{353}^{\rm{D}}/S_{545}^{\rm{D}}$ ratio (top left panel of Fig. 22) is unbiased on average for S/N larger than 15 when compared to the injected values. More precisely, when looking at the recovered versus injected trend (bottom left panel), it appears that the higher the $S_{353}^{\rm{I}}/S_{545}^{\rm{I}}$ ratio, the more underestimated the output colour, so that the recovered $S_{353}^{\rm{D}}/S_{545}^{\rm{D}}$ ratio always remains below 1 (within 1$\thinspace\sigma$) for input ratios $S_{353}^{\rm{I}}/S_{545}^{\rm{I}}<1$. The case is even worse when assuming a highly-contaminated CMB template, yielding an underestimate of the recovered $S_{353}^{\rm{D}}/S_{545}^{\rm{D}}$ ratio by 17 % to 7 %, from low to high S/N. This is well explained by the attenuation coefficient, which may differ between the 545- and 353-GHz bands. This effect has been taken into account when setting the colour-colour criteria in Sect. 4.3 in a conservative way. The recovered $S_{545}^{\rm{D}}/S_{857}^{\rm{D}}$ ratio (right panels of Fig. 22) does not appear as strongly biased on average, but is still underestimated for high $S_{545}^{\rm{I}}/S_{857}^{\rm{I}}$ inputs ($>0.8$); this does not impact the overall colour-colour selection, since in this case the recovered ratio still satisfies the selection criterion ($>0.5$). We have also used these Monte Carlo simulations to check the accuracy of the colour-colour selection process. The probability $\mathcal{P}$, introduced in Sect. 4.3, and based on the recovered colour ratios $S_{545}^{\rm{D}}/S_{857}^{\rm{D}}$ and $S_{353}^{\rm{D}}/S_{545}^{\rm{D}}$, has been compared to the exact probability that the input colour ratios $S_{545}^{\rm{I}}/S_{857}^{\rm{I}}$ and $S_{353}^{\rm{I}}/S_{545}^{\rm{I}}$ satisfy the colour criteria. Hence requesting a probability of 0.84 to find the true colour values inside the expected colour-colour domain (which is equivalent to a 1$\thinspace\sigma$ constraint on a half-bounded domain), gives a minimal threshold of $\mathcal{P}>0.9$ based on the recovered colour values. This is what has been applied to build the official list. 5.5 Selection function We now focus on the sample of detected sources, obtained after applying the S/N criteria in all bands and the colour-colour criteria of Sect. 4.3, in agreement with the criteria used for the true extraction. This allows us to quantify the selection function of our detection algorithm by computing the completeness of the detected sources as a function of redshift, extinction, and flux density. Here we define the completeness as the ratio between the initial number of injected sources and the number of detected sources in the same bin for a given property. In Fig. 11 the completeness is presented as a function of both redshift and input flux density in all Planck bands for each category of dust temperature of the extragalactic source, $T_{\rm{xgal}}$. Of course, the completeness is highly dependent on the input temperature of the extragalactic source ($T_{\rm{xgal}}$), because of the temperature-redshift degeneracy. Sources with a high temperature (50 K) are only detected when located at high redshift ($>3$), while sources with a low temperature (20 K) can be detected up to redshift $z=1$. To solve for this well known degeneracy, Greve2012 have used a prior on the temperature built on a sample of 58 unlensed and 14 lensed high-$z$ submm sources. They state that the median temperatures of the unlensed and lensed population of sources at $z>1$ are $T_{\rm{xgal}}=34$ K and $T_{\rm{xgal}}=46$ K, respectively, and range from 15 to 80 K, and 30 to 80 K, respectively. Studies have shown similar ranges of temperature with Herschel, SCUBA-2 ad other instruments (Chapin2009; Chapin2011; Chapman2010; Magnelli2012; Symeonidis2013; Swinbank2014). As a confirmation, the mean temperature of the dusty star forming galaxies discovered by SPT and confirmed with ALMA observations as strongly lensed sources has been estimated at $T_{\rm{xgal}}=38$ K (Weiss2013). Furthermore, first confirmations of sources of this list have shown median temperature of 44 K for lensed candidates (Canameras2015), and 32 K for proto-cluster candidates (Florescacho2015). However, we have to keep in mind that this degeneracy cannot be broken for all other sources of this list without any direct measurement of the redshift. The completeness exhibits a very sharp cut-off on the lowest redshift side (e.g., at $z>1.5$ for $T_{\rm{xgal}}=30$ K), dropping suddenly to zero below this limit. On the high redshift side, after a plateau, it goes back smoothly to zero, because of the impact of the attenuation due to the cleaning, which becomes more and more important with higher redshifts. Focusing again on the $T_{\rm{xgal}}=30$ K case, the completeness reaches 100 % for strong sources ($S_{545}>3$ Jy) and $2<z<3$. However, the completeness drops quickly for fainter sources, reaching a maximum of about 50 % at a flux density of 700 mJy and redshifts between 1 and 3. Our detection method therefore operates as a filter in redshift by selecting sources peaking in the submm range. For an average dust temperature of $T_{\rm{xgal}}=30$ K, this redshift window ranges from about 1.5 to 4.5. Finally, as shown in Fig. 12, there is no dependence of the completeness on extinction, which implies that the cleaning method and the presence of Galactic structures do not affect the ability of the detection algorithm to extract high-$z$ candidates (at least over the cleanest 26 % of the sky). This does not prevent the possible presence of some spurious detections due to Galactic cirrus, which can be addressed by looking at the $\rm{H}_{\rm{I}}$ column density, as discussed in Sect. 6.7. 6 The PHZ 6.1 Building the source list The full procedure of CMB and Galactic cirrus cleaning is performed on the set of Planck and IRAS data, enabling us to build the 545 GHz excess map on which the detection criterion $S_{545}^{\rm{X}}/\sigma_{545}^{\rm{X}}>5$ is applied, combined with the requirements $S_{\nu}^{\rm{D}}/\sigma_{\nu}^{\rm{D}}>3$ in all 857-, 545-, and 353-GHz-cleaned maps simultaneously, and the requirement $S_{100}^{\rm{C}}/\sigma_{100}^{\rm{C}}<3$ to reject contamination by radio sources. This yields a first sample of 9052 source candidates for which the photometry in the 857-, 545-, and 353-GHz-cleaned maps is computed with associated uncertainties due to noise measurement and CIB confusion. Notice that 44 sources have been rejected during this first step because of their clear detection at 100 GHz, confirming the possible contamination by radio sources as discussed in Sect. 3.6.2. The colour-colour selection is performed by requiring a probability of 90 % to satisfy both colour criteria, $S_{545}^{\rm{D}}/S_{857}^{\rm{D}}>0.5$ and $S_{353}^{\rm{D}}/S_{545}^{\rm{D}}<0.9$. In addition to the colour-colour selection, we also apply a cut in flux density, $S_{545}>500~{}\mathrm{mJy}$, to ensure a minimum bias due to the flux boosting effect, following the prescriptions motivated by the numerical simulations detailed in Sect. 5.3. This leads to a final number of 2151 high-$z$ source candidates present in the Planck List of High-redshift Source Candidates (PHZ). The all-sky distribution of the PHZ sources is shown in Fig. 13, where it can be seen that they span the whole northern and southern caps. The distribution shown does not exhibit any evidence of contamination by the extended Galactic structures. A full description of the content of the PHZ is given in Table 3. We stress that the flux densities provided in this list have been obtained on the cleaned maps and may be strongly affected by attenuation due to the cleaning process, depending on their SED type and redshift, which are still unknown. For this reason these flux density estimates have to be taken with some caution. In order to help the user to assess the reliability of the PHZ sources, we also provide cutouts ($1^{\circ}\times 1^{\circ}$) of the excess map at 545 GHz and the cleaned maps at 857, 545, 353, and 217 GHz, available soon through the Planck Legacy Archive444http://www.cosmos.esa.int/web/planck/pla and the MuFFInS555http://muffins.irap.omp.eu (Multi Frequency Follow-up Inventory Service) portal. 6.2 Statistical description The statistics of the main properties of the Planck high-$z$ candidates are shown in Fig. 14: S/N of the detection on the excess map at 545 GHz; FWHM and ellipticity of the Gaussian elliptical fit; average local extinction $E(B-V)_{\rm{xgal}}$; and flux densities in all cleaned bands. The S/N of the detection in the 545 GHz excess map does not extend to values larger than 10, peaking close to 5 (i.e., the threshold imposed by the detection criteria), while the S/N of the detection in the cleaned maps at 857, 545, and 353 GHz have 80 % to 90 % of their values below 6. The PHZ sources are not extremely high S/N detections. The distribution of the FWHM peaks around 7.${}^{\scriptstyle\prime}$9. As has been shown with Monte Carlo simulations (see Sect. 5.2), the FWHM are statistically overestimated by 20 % at low S/N (below 10), which is the case for most of the detections. This means that the actual size distribution of PHZ sources is probably centred around 6.${}^{\scriptstyle\prime}$3, leading to a real average size of 3.${}^{\scriptstyle\prime}$8 after deconvolution by the 5′ Planck beam. Concerning the ellipticity distribution, Monte Carlo simulations have shown that it is artificially stretched to an average ellipticity of 0.65, because of the confusion with the CIB in which the PHZ sources are embedded. However, the actual distribution peaks at even larger ellipticities, around 0.8, suggesting that the PHZ sources are not compact or spherical but somewhat extended objects. The distribution of the Galactic extinction $E(B-V)_{\rm{xgal}}$ (bottom right panel of Fig. 14) is similar to the statistics of the whole mask. This is entirely consistent with what has been observed in Monte Carlo simulations in Sect. 5.5, i.e., our detection algorithm is not sensitive to the Galactic foreground level, thanks to the efficient Galactic cirrus cleaning. The distribution of the flux density estimates at 545 GHz is sharply cutoff at 500 mJy because of the threshold applied to avoid too strong a flux boosting effect, and extending to 2.5 Jy. In the other bands the distribution peaks around 0.8 Jy, 250 mJy, and 70 mJy at 857, 353, and 217 GHz, respectively. 6.3 Colour-colour domain The distribution of the PHZ sources in the colour-colour diagram is shown in Fig. 15, and compared to the loci of a few typical high-$z$ astrophysical sources: the Galactic cold clumps of the PGCC catalogue; a subset of nine dusty star forming galaxies (DSFG) discovered with the South Pole Telescope (SPT; Vieira2010) and followed-up with SABOCA and LABOCA (Greve2012); and the submm galaxy SMMJ2135$-$0102, the “Cosmic Eyelash,” located at $z=2.33$ (Swinbank2011; Ivison2010; Danielson2011). The contours of the pixel distribution inside the full mask and towards Galactic cirrus in the initial Planck maps are also shown, including 99.9 %, 50 %, and 10 % of the distribution. Hence the Galactic cirrus pixels (defined as those pixels with an extinction $E(B-V)_{\rm{xgal}}$ larger than 0.03 inside the mask), as well as the Galactic cold sources of the PGCC, occupy very distinct domains compared with the high-$z$ candidates, as ensured by the colour criteria on the $S_{545}/S_{857}$ colour ratio. Furthermore it can be seen that the above criteria allow us to separate the high-$z$ ($>2$) from the intermediate and low-$z$ ($<2$) component of the CIB, which dominates the distribution of the full mask. Comparing now to the loci of known high-$z$ objects, the PHZ sources span a quite different domain; this is fully explained by the impact of attenuation on the flux density estimates obtained on cleaned maps, as has been investigated using numerical simulations in Sect. 5.3. The $S_{353}^{\rm{D}}/S_{545}^{\rm{D}}$ colour ratio is especially affected by the cleaning for high redshift sources, i.e., at high intrinsic $S_{353}/S_{545}$ colour ratio, so that the measured $S_{353}^{\rm{D}}/S_{545}^{\rm{D}}$ ratio lies between 0.2 and 0.6 even for redshifts as high as 4. That is why we cannot use this colour ratio to obtain an estimate of the redshift of the PHZ sources. On the contrary the second colour ratio $S_{545}^{\rm{D}}/S_{857}^{\rm{D}}$ is not affected by the cleaning, up to a value of 0.8, and then slightly underestimated by about 10 % for an intrinsic colour ratio of 1. This can then be used as a direct tracer of the redshift combined with the dust temperature of the detected sources. The fact that 73 % of the PHZ sources exhibit a colour ratio $S_{545}^{\rm{D}}/S_{857}^{\rm{D}}$ between 0.5 and 0.8 is mainly due to the efficiency of the detection algorithm in this colour range. The 27 % of sources with $S_{545}^{\rm{D}}/S_{857}^{\rm{D}}>0.8$ represents an interesting sample of highly redshifted or extremely cold sources. 6.4 Redshift estimates We performed a photometric redshift determination for each source, assuming simple SED modelling given by a modified blackbody emission with a dust spectral index $\beta_{\rm{xgal}}=1.5$ and six different cases of the dust temperature, namely $T_{\rm{xgal}}=25$, 30, 35, 40, 45, and 50 K. In order to take into account the impact of the cleaning algorithm introduced in Sect. 3.5, we built a grid of attenuated flux densities modelled for each value of the redshift ($0<z<8$) and the dust temperature. A $\chi^{2}$ analysis based on this grid yields the best fit of the redshift together with 1$\thinspace\sigma$ lower and upper limits. The accuracy of the redshift estimate processing has been analysed on Monte Carlo simulations (see Appendix B). The average uncertainties associated with these photometric redshift estimates are about 0.5, given a specific dust temperature. Of course the degeneracy between the redshift and the dust temperature may induce much larger uncertainties on those sources without spectroscopic data. The distribution of redshift estimates can be seen in Fig. 16 for each case of the extragalactic dust temperature. For an average dust temperature of 35 K, which is consistent with the latest analyses (e.g., Greve2012; Weiss2013; Magnelli2014; Swinbank2014), the distribution exhibits a median value of $z=2.5$, with 95 % of sources lying between 1.5 and 3.7. This is in perfect agreement with the outcomes of the Monte Carlo analysis of Sect. 5. Because of the degeneracy between the redshift and the dust temperature, this redshift range shifts to $2.6<z<5.7$ for the highest temperature, $T_{\rm{xgal}}=50$ K. 6.5 FIR luminosities and SFRs Given the redshift estimates, we derive for each source the FIR bolometric luminosity associated with the six different assumptions made on the dust temperature. This is computed as the integral of the redshifted modified blackbody emission between 300 GHz and 37.5 THz. Following the prescription of kennicutt1998 and assuming that the contribution from the AGN is negligible for these objects, we finally derive an estimate of the star formation rate as SFR [$\rm{M}_{\odot}\rm{yr}^{-1}$] = $1.7\times 10^{-10}L_{\rm{FIR}}$[$\rm{L}_{\odot}$]. The distributions of bolometric luminosity and SFR are shown in Fig. 17, for three options of the dust temperature, $T_{\rm{xgal}}=30$, 35, and 40 K, lying in the most probable range of temperature expected for dusty submm galaxies. The FIR bolometric luminosity distribution peaks around $2\times 10^{14}$ $\rm{L}_{\odot}$ (assuming $T_{\rm{xgal}}=35$ K), with an associated SFR around 3200 $\rm{M}_{\odot}\rm{yr}^{-1}$, which is not really compatible with the expected luminosities of single submm galaxies at high-$z$, typically $10^{11}$–$3\times 10^{13}$ $\rm{L}_{\odot}$ (kovacs2006; Chapin2011; Geach2013; Swinbank2014; Casey2014). Only strongly lensed galaxies may reach such high apparent luminosities, because of the magnification. The brightest strongly lensed dusty galaxies detected by SPT exhibit intrinsic FIR luminosities ranging between 1.9 and $6.9\times 10^{13}$$\mu^{-1}\rm{L}_{\odot}$, where $\mu$ is the unknown magnification factor (Vieira2013; Hezaveh2013), which represents the lowest tail of our sample distribution. Canameras2015 reported intrinsic FIR luminosities of (0.5–1.7)$\times 10^{14}$ $\mu^{-1}\rm{L}_{\odot}$ towards 11 high-$z$ strongly lensed star-forming galaxies selected using Planck data and confirmed with Herschel (see Appendix C). Focusing now on the four sources of the latter sample with a counterpart in the final PHZ, we observe that these sources exhibit an apparent FIR luminosity about 3 to 5 times larger in Planck than in Herschel. Assuming now that the Planck PHZ sources are composed of multiple galaxies, the range of FIR luminosities derived above may be compared to recent estimates obtained by integrating the submm emission of galaxy members towards proto-cluster candidates at high-$z$, e.g., about $10^{13}$ $\rm{L}_{\odot}$ at redshift $1\thinspace{<}\thinspace z\thinspace{<}\thinspace 1.5$ (Brodwin2013), or (0.5–7)$\times 10^{13}$ $\rm{L}_{\odot}$ at $z\simeq 2$ (Clements2014). Using the dedicated Herschel follow-up of 228 Planck candidates (planck2014-XXVII) described in Appendix C, it also appears that the Planck FIR luminosity estimates are about 2 to 3 times larger than the integrated luminosities of the galaxy members identified with Herschel inside the elliptical Gaussian profiles of the Planck PHZ sources. Despite the precaution we made by applying a flux density threshold at 500 mJy at 545 GHz, the flux boosting effect can still reach 20 % for flux density estimates around 0.5 Jy; this may explain a fraction of the discrepancy between Planck and Herschel, but not all. This remaining discrepancy suggests that the Planck estimates integrated over a 5′ beam include a component that is barely traced by SPIRE because of confusion. As characterized by Viero2015, this effect is even stronger for sources at high redshift, and can reach 50 % of enhancement when going from Herschel-SPIRE resolution to Planck resolution. Hence Planck flux densities allow us to recover an estimate of the overall budget of the submm emission at high-$z$, by including a population of faint sources contributing to the Planck flux, but undetected in Herschel’s higher resolution data. 6.6 Number counts The reliability of the flux density estimates in the cleaned maps has already been discussed above. It is impacted by the overestimation of the extension of the sources, but also by the CIB fluctuations, and more seriously by the attenuation due to the cleaning process, which may strongly affect the flux density estimates (depending on the dust temperature, the redshift of the sources, and the level of foreground contamination of the CMB template). A theoretical approach has shown that the flux densities at 353 and 217 GHz can be underestimated on average by about 10 % and 40 %, respectively, while the 857- and 545-GHz bands are not affected. The numerical analysis of Sect. 5 pointed out an additional bias of 3.5 %. However, these biases are both compensated at low flux densities by the flux boosting effect. We stress that an exact correction for this attenuation effect for each individual source could only be carried out by knowing its SED and redshift. Despite this warning, it is interesting to perform a crude number counts analysis on the PHZ sources. The number counts are shown in Fig. 18 for all channels. The population of PHZ sources appear extremely bright compared to the predictions of Bethermin2012 for three types of individual galaxies: main sequence (MS); starburst (SB); and lensed sources. For this analysis the models of Bethermin2012 have been integrated in the range of redshift $1.5<z<4$, according to the expected detection range of our algorithm. Three versions of the PHZ number counts are shown, depending of the assumed number of individual objects composing the Planck source, namely $n=1$, 3, and 30. If we assume the Bethermin2012 model represent the PHZ contents, then the “$n=3$” counts being closest to the model suggests that the PHZ candidate sources typically include multiple galaxies. The PHZ number counts at 353 GHz may also be compared with analytical predictions by Negrello2005 that explore the impact of clustering when building number counts with large beams such as those of Planck or Herschel. Those authors considered three scenarios for the clustering, associated with the 3-point correlation function (see Fig. 19). It appears that the PHZ distribution is more or less consistent with the modelling that is the most realistic of the three, assuming no evolution with redshift for the amplitude of the 3-point correlation function. Again, two versions of the PHZ number counts are shown, depending on the assumed typical number of internal objects composing the Planck source, i.e., $n=1$ or $n=3$. This may suggest that a fraction of the PHZ sources are combinations of objects located along the line of sight either by chance or because they belong to the same cosmic filament. 6.7 Cross-check with Planck  catalogues We performed a cross-check between the 2151 sources of the PHZ and the other catalogues made available with this Planck 2015 release (see Table 2): the Planck Catalogue of Compact Sources (PCCS2; planck2014-a35); the Planck Catalogue of SZ sources (PSZ2; planck2014-a36); and Planck Catalogue of Galactic Cold Clumps (PGCC; planck2014-a37). We counted only three associations between the PHZ and the PSZ2, which confirms the different astrophysical nature of these two populations of objects. Sources from the PSZ2 catalogue are virialized galaxy clusters traced by their Sunyaev-Zeldovich signal due to the hot intergalactic gas, while sources from the PHZ are traced by their dust submm emission coming from the high-$z$ galaxies located inside the Planck beam. The probable nature of the PHZ sources will be discussed in Sect. 7. The cross-match with the PCCS2 has been performed with the catalogues extracted in all nine individual Planck-HFI and LFI bands, but also with two band-merged catalogues: the HFI band-merged catalogue is defined as the PCCS2 sources with simultaneous detections in the 857, 545, and 353 GHz HFI bands; and the LFI band-merged catalogue requires detection in all LFI bands, i.e., 70, 44, and 30 GHz (see planck2014-a35). The HFI and LFI band-merged catalogues trace two different populations, dusty submm sources and radio sources, respectively. As shown in Table 2, the overlap between the PCCS2 and the PHZ is extremely small. Taking into account the redundancy between bands, a total of 35 sources are present in both catalogues, while no radio sources (from the LFI band-merged catalogue) and only two dusty submm sources (from the HFI band-merged catalogue) are found in the PHZ sample. Notice that the sources of the PCCS2 bands are divided into two categories, depending on their reliability, namely high reliability sources (zone 0) or unvalidated sources (zones 1, 2, and 3), where the 0-3 zones correspond to quantified-reliability zone, filament zone, Galactic zone, and Galactic filaments zone, respectively. Matches between the PHZ and PCCS2 only happen in the quantified-reliability zone, suggesting that the PHZ sources are quite clean from the cirrus contamination traced by the PCCS2 masks. When looking at the individual low-frequency matches between PHZ and PCCS2 sources, the dust emission signature in the HFI bands is clear, but may be associated with radio emission observed in the LFI bands. The PHZ is thus seen to be complementary to the PCCS2, by picking out the faintest and coldest objects at high latitude. The PGCC catalogue has been built over the whole sky, but focuses on the Galactic objects by rejecting any possible associations with extragalactic sources. This purification step was performed using three independent methods (see planck2014-a37): cross-correlation with well-characterized catalogues of extragalactic sources; identification with galaxies in optical data; and colour-colour selection. Among the 87 PGCC sources lying in the high-latitude mask used in this work, 19 are found to be correlated with PHZ sources within 5′. These 19 cross-matched sources exhibit very low temperature in the PGCC catalogue (with a median around 9 K), and are associated with low ${H\textsc{i}}$ column densities (amongst the lowest 10 % of the PGCC catalogue). On the PHZ side, these 19 sources exhibit a similar distribution of flux density at 545 GHz as the whole PHZ, with extinction values spanning the full mask statistics, suggesting that the PHZ population does not consist of the faintest component of the PGCC population. Despite this, it is still hard to determine if these sources are Galactic or extragalactic, and they are flagged in both catalogues accordingly. This analysis can be used to disentangle the possible contamination of the PHZ by cirrus. Because of the degeneracy between redshift and temperature, the PHZ sources can be interpreted as “cold” or “red” sources. For the analysis here, we assume that each PHZ source is located inside the Galaxy, i.e., $z=0$. We derive its temperature from the flux density estimates at 857, 545, and 353 GHz, assuming a dust spectral index of 2, as is observed for dense regions with temperature below 10 K. We compute the column density of each source by applying the same recipe as for the PGCC sources (see planck2014-a37). Hence the PHZ source candidates, assumed to lie at $z=0$, exhibit temperatures around 8 K and mean column densities of about $5\times 10^{19}\thinspace{\rm cm}^{-2}$. The relation between the temperature (assuming a dust spectral index of 2) and the column density of the PGCC sources and the PHZ sources assumed to be Galactic objects is shown in 20. For PGCC sources, the lower the temperature, the higher the column density, as expected for the dense Galactic medium. However, the opposite trend is observed for the PHZ sources, which are located in a very distinct domain compared to the PGCC. Similarly, the $E(B-V)_{\rm{}xgal}$ distribution of the PHZ sources has been shown to perfectly follow the distribution inside the full mask (see top right panel of Fig. 14), without showing any bias towards denser regions associated with cirrus. This reinforces the fact that the PHZ source candidates are not linked to dense Galactic structures located in cirrus, but lie at high redshift instead, and represent a complementary sample of sources to the PGCC catalogue. 7 Discussion and conclusions We have applied an original multi-frequency detection algorithm on the Planck-HFI plus 3 THz IRAS data set to build the List of Planck High-redshift Source Candidates (the PHZ), comprising 2151 objects selected by their dust emission excess in the 545-GHz band, over the 25.8 % cleanest part of the sky. We have fully characterized our detection algorithm using Monte Carlo simulations. This has enabled us to assess the quality of the flux densities provided in this list, and, more specifically, the impact of the attenuation due to the cleaning process, which tends to statistically underestimate the flux densities by 4 % to 40 %, depending on the frequency. However, we have demonstrated the robustness of the colour-colour selection process, which allows us to efficiently reject Galactic cold clumps, low-$z$ dusty sources, and contaminants such as radio galaxies or low-$z$ galaxy clusters exhibiting strong SZ signatures. The algorithm has been shown to preferentially detect dusty sources located at redshifts between 1.5 and 4, depending on their intrinsic temperature (ranging from 20 to 40 K), reaching a completeness levels of about 50 %, 80 %, and 100 % for sources with $S_{\rm{545}}=1$, 2, and 3 Jy, respectively. Despite the reliability of the high-$z$ dusty signature for all the PHZ sources, the astrophysical nature of these candidates is still uncertain. They could first of all be statistical fluctuations of the CIB, i.e., chance alignments of field galaxies along the line of sight (Negrello2005; Negrello2010; Chiang2013; Chiang2014). Given the flux density threshold of 500 mJy applied at 545 GHz, all the PHZ detections have been obtained at more than 3.7 and 3.3 times the confusion noise estimated for a Poisson plus clustering contribution with two different correlation models (Negrello2004). Assuming a Gaussian distribution for the Poisson plus clustering fluctuations as a first guess, the associated probabilities to find such CIB fluctuations at a 5′ scale become $0.012$ and $0.061$ $\rm{deg}^{-2}$, respectively, to be compared with the density of the PHZ sources which is about $0.21$ $\rm{deg}^{-2}$. Hence the PHZ source density is 17.5 and 3.5 times larger than chance alignment expectations derived in the two clustering cases of Negrello2004. While it has been shown with other Herschel analysis that this chance alignment may be larger than expected, the population of the PHZ sources is still hard to explain by chance alignment alone, even if this cannot be fully rejected yet for some fraction of the candidates. First hints about the nature of the Planck high-$z$ candidates have been obtained with Herschel follow-up observations. Negrello2007 and Bethermin2012 predicted that a small fraction of the very bright sources at high redshift ($z>2$) are expected to be lensed dusty starburst galaxies. Hence the source H-ATLAS J114637.9$-$001132, simultaneously detected in the Herschel H-ATLAS survey field, in the ERCSC catalogue (planck2011-1.10), and in a previous incarnation of the Planck list of high-$z$ candidates, was confirmed to be a gravitationally lensed galaxy at $z\thinspace{=}\thinspace 3.3$ (Fu2012; Herranz2013). Similarly the source HLS J091828.6+514223, discovered in the Herschel Lensing Survey (Egami2010) and independently detected in Planck data, was confirmed to be a strongly lensed galaxy at $z=5.2$ (Combes2012). Furthermore a dedicated Herschel follow-up programme on a sub-sample of 228 Planck high-$z$ source candidates (planck2014-XXVII), described in more detail in Appendix C, provided unique information on the nature of this sample. While 3 % of the Herschel fields show clear evidence of single bright sources inside the Planck beam, further follow-up observations in optical, Far-IR and the submm of 11 candidates confirmed that these objects are Planck-discovered strongly lensed galaxies. They exhibit flux densities at 350 $\mu$m larger than 350 mJy and up to 1 Jy, with spectroscopic redshifts ranging from 2.2 to 3.6 (Canameras2015). Compared to the properties of the recent discoveries by Herschel and the South-Pole Telescope (SPT) of large sets of strongly gravitationally lensed submm galaxies with flux densities between 100 and 200 mJy (e.g., Negrello2010; Vieira2013; Wardlow2013), these Planck high-$z$ lensed sources are amongst the brightest lensed galaxies in the submm range. Complementary to this population of strongly lensed galaxies, planck2014-XXVII states that more than 93 % of the Planck high-$z$ sources followed-up with Herschel are overdensities of around 10 red sources on average, with SEDs peaking at 350 $\mu$m. This confirms, on a small sub-sample of sources, what was suggested by the number counts analysis performed on the whole list (see Sect. 6.6), i.e., PHZ sources are preferentially structures of multiple sources instead of single red objects. This statement is in agreement with the predictions by Negrello2005 on the detectability of such overdensities of high-$z$ dusty star forming galaxies in the submm, and with recent works (e.g., Gobat2011; Santos2011; Santos2013; Santos2014; Clements2014) providing the first observations. The first newly discovered PHZ proto-cluster candidate with spectroscopic confirmation is the source PHZ G095.50$-$61.59, which consists of two systems at $z=1.7$ and $z=2.0$ (Florescacho2015). Spectroscopic redshifts have been obtained towards four and eight galaxies, associated with each one of the two structures, respectively, within a comoving radius of 1 Mpc, consistent with sizes of local cluster and recently discovered proto-clusters at $z>1.5$ (Castellano2007; Andreon & Huertas-Company, 2011; Gobat2013). With an integrated SFR of 2000–3000 $\rm{M_{\odot}}\rm{yr}^{-1}$ over the Planck beam and a mass of $4.5\times 10^{14}$ $\rm{M}_{\odot}$, this object fits into the galaxy cluster category. Despite the fact that this source has turned out to be a line of sight combination of two structures, it nevertheless has acted as a pointer towards high-$z$ objects. This indicates that the PHZ will be useful for finding such structures, even if a fraction of the sources are multiple objects; the reason is that the selection process ensures that something along the line of sight has to be red, i.e., has to have the colours of star-forming galaxies. Considering the above option of a proto-cluster population, it is interesting to compare the expected surface density of massive halos at high redshift with the one of the PHZ sources, i.e., $0.21$ $\rm{deg}^{-2}$. From the Tinker2010 halo model we derive a surface density of dark matter halos with $M>10^{14}\thinspace{\rm M}_{\odot}$ at $z>2$ of about $0.5$ $\rm{deg}^{-2}$. Given the detection efficiency of our algorithm (depending on the redshift and flux density), and the fact that only a fraction of these dark matter halos may be observed during their star-forming phase, the total number of PHZ source candidates and the expected numbers of massive high-$z$ galaxy clusters are about the same order of magnitude. Moreover the submm photometric redshift distribution of the PHZ sources, likely ranging from $z=1.5$ to 4, corresponds to the expected redshifts of the star-formation peak activity of such proto-cluster objects. The fact that no associations have been found between the PHZ and the Planck Sunyaev-Zeldovich Catalogue (PSZ2) also reveals that the population traced by the PHZ does not exhibit any clear feature in the SZ effect, which means that these objects may still be in a very early stage of their evolution and not virialized yet. It is interesting to notice that the PHZ number counts are compatible with predictions of clump number counts made earlier by Negrello2005. This Planck list of high-$z$ candidates opens a new window on the brightest and rarest structures at high redshift, which remain unaccessible to other detection methods. It is the largest list of proto-cluster candidates at $z>2$, detected in a homogeneous way over more than 25 % of the sky. It is a unique and powerful sample of particular interest for structure formation studies. The full characterization of the PHZ sample is challenging and it will require a huge effort to follow-up these objects and constrain their nature. A comparison with detailed structure formation models could then be performed in order to reveal what can be learned from this population of high-$z$ objects about the early ages of our Universe. Acknowledgements. The Planck Collaboration acknowledges the support of: ESA; CNES and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MINECO, JA, and RES (Spain); Tekes, AoF, and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); ERC and PRACE (EU). A description of the Planck Collaboration and a list of its members, indicating which technical or scientific activities they have been involved in, can be found at http://www.cosmos.esa.int/web/planck/planck-collaboration. References Andreon & Huertas-Company (2011) Andreon, S. & Huertas-Company, M., Red sequence determination of the redshift of the cluster of galaxies JKCS 041: $z\sim 2.2$. 2011, A&A, 526, A11, Appendix A Cleaning with the Planck CMB 8′ map Complementary to the Monte Carlo analysis performed in Sect. 5 to study the impact of the CMB template quality on the detection and photometry processing, we used the Planck 143 GHz map as a CMB template to assess the level of extragalactic foregrounds included in the SMICA CMB component map, and its possible impact on the PHZ. All Planck, IRIS, and SMICA CMB component map have been first smoothed at a common resolution 8′ in order to be compatible with the 143 GHz map. On this alternative set of maps, we applied the full processing of cleaning, detection, photometry, and colour selection, to build two new lists of high-$z$ source candidates at 8′, using either the SMICA CMB component map or the Planck 143 GHz map as a CMB template, counting 1121 and 1038 high-$z$ source candidates, respectively. The two catalogues have about 80 % of their sources in common. The 20 % of non-matches correspond to sources with S/N close to the detection thresholds, which is explained by the fact that the SMICA CMB component map and the Planck 143 GHz map do not exhibit the same noise properties. The level of extragalactic foreground contamination in the SMICA CMB template can be seen by comparing the flux densities toward to 1121 sources of the list based on the SMICA CMB component map at 8′ and obtained on the two versions of the cleaned maps at 857, 545, 353, and 217 GHz maps, as shown in Fig. 21. The flux density estimates of both cases are fully consistent in the 857- and 545-GHz bands, as is expected for the range of redshift of the PHZ sources ($1<z<4$, see Sect. 3.5). The attenuation becomes important in the 353- and 217-GHz bands. The flux densities obtained using the SMICA CMB component map appear statistically larger than when using the Planck 143 GHz map, which confirms that they are less affected by the attenuation. However, they do not entirely follow the statistical expectation of unattenuated flux density estimates shown in blue dashed line of Fig. 21. This discrepancy may come from the diversity of the SEDs that have been assumed to follow a modified blackbody emission law with a dust spectral index of 1.5 in our modelling. It can also be due to a residual of extragalactic foregrounds in the SMICA CMB component map, yielding up to 5 % of attenuation in the 353-GHz band, instead of the 10 % expected in the worse case. Unfortunately this residual emission is hard to quantify, and has to be included in the photometric uncertainties. It should be noted that earlier versions of the PHZ, which were used to select targets for follow-up observations, such as the Herschel follow-up described in Appendix C, were all built using the 143 GHz map as a CMB template, because no CMB component maps were available at this time at 5′ resolution. However, the 8${}^{\scriptstyle\prime}$ and 5${}^{\scriptstyle\prime}$ PHZ lists do not exactly cover the same population. Only 458 objects match both lists within 5′. This is explained by the fact that compact sources detected at 5′ may be diluted in an 8′ beam, yielding no detection in the latter case. On the other hand, extending structures integrated within a 8′ beam may not exhibit any 5′ features, yielding no detection in the 5′ list. Appendix B Redshift estimate accuracy We have tested the accuracy of the photometric redshift estimate processing (see Sect. 6.4) using the Monte Carlo simulations presented in Sect. 5.1. We applied the same SED-fitting algorithm based on the recovered flux densities at 857, 545, 353, and 217 GHz for each injected and detected source of the mock catalogue. In order to check the impact of the cleaning process and the photometric accuracy on these redshift estimates, we have compared the recovered redshift estimates with the input values injected in the Monte Carlo simulations, assuming the correct injected temperature. As shown in Fig. 22, the photometric redshift estimates are not reliable over the full range of redshift. Even if the CMB template is assumed to be ideal for both the simulations and the SED modelling used to fit the redshift, the photometric redshift estimates are systematically high for the lowest detectable redshifts, and are underestimated for the largest detectable redshifts, for each range of dust temperature. However, in the intermediate range of redshift, where most of the sources are detected, the accuracy is about 10 %, which is sufficiently accurate for our purpose. When assuming an ideal CMB template in computing the theoretical attenuation coefficients for each Planck band before the SED fitting, Fig. 22 shows that when the estimate is actually highly contaminated by extragalactic foregrounds (bottom panels), the associated photometric redshift estimates are statistically underestimated by 15 to 20 %. This last number gives the maximum impact due to the contamination of the CMB template on the redshift estimates. This simple analysis, of course, does not take into account all other uncertainties impacting any photometric redshift estimate, such as the degeneracy between the redshift and the dust temperature, or the SED assumption. For all these reasons, the photometric redshift estimates delivered in this list are provided as basic estimates only, and should be used with caution. Appendix C The Herschel sub-sample A dedicated follow-up of the Planck high-$z$ candidates has been carried out with the Herschel-SPIRE instrument, culminating in three accepted programmes during the OT1 (10 sources, PI: Montier), OT2 (70, PI: Dole), and Must-Do (106, HPASSS, PI: Dole) calls. A total of 228 sources were selected from the Planck data: 204 sources were selected using an algorithm similar to the one described in this work, but applied at 8′ resolution on earlier versions of the Planck data before the completion of the full mission, and 24 others were selected from the Planck Catalogue of Compact Sources (PCCS; planck2013-p05). From this sample, 25 (16 from PHZ plus 9 from PCCS) sources are now outside the mask defined for Galactic extinction, and 83 (82 from PHZ plus 1 from PCCS) sources remain in the final PHZ. Without including the sub-sample of sources selected separately from the PCCS, 120 of the observed sources are not in the final PHZ list, which is explained by two main factors: improvement of the data quality; and evolution of the detection method. The S/N of the Planck maps has been improved thanks to the completion of the full mission and a better control of the systematics, so that previous detections may now fall at lower S/N. To characterize this effect, we produced a larger list of Planck sources by relaxing the S/N criteria of the detection to 1 in all bands (excess and cleaned maps), and we find associations in this deep list for almost 90 % (182 sources) of the 203 sources of the Herschel sample present inside the mask. The S/N distribution of the Herschel sample is shown as a blue histogram in Fig. 23, while the sub-sample of sources present in the PHZ is given in orange. It appears that most of the sources of the Herschel sample that have not been selected in the final PHZ exhibit a S/N close to the threshold criteria in at least one band, so that they are rejected when simultaneously constraining detections in all bands. Hence only 10 % of the sources fail in more than one band. Furthermore the detection algorithm has been improved compared to the first incarnations of the method, especially when applying the colour-colour criteria. We now use a probability to reject sources not satisfying the colour-colour criteria, while a simple threshold-cut on each colour was applied before. This enabled us to improve the robustness of the final product. All of these investigations show that sources of the Herschel sample are not likely to be spurious if they happen not to be included in the PHZ, but are simply at lower significance. The statistics of the FWHM, ellipticity and extinction of the sub-sample of 83 candidates followed-up with Herschel and present in the final PHZ are shown in Fig. 24. They spans the same range of properties as the full list (dashed line). However, this Herschel sub-sample is characterized by statistically higher S/N, smaller FWHM, smaller ellipticities, and lower extinctions than the full PHZ. This can be explained by the process of selection applied to obtain robust target lists for the three various Herschel calls, which tended to bias the selection towards cleaner regions of the high-latitude sky, and to preferentially pick high S/N compact sources, i.e., with small sizes and regular shapes. Another way to probe the reliability of the Planck candidates followed-up with Herschel but not present in the final PHZ is to compare, via a stacking analysis, the statistical properties of two sub-samples, namely sources included or not included in the final PHZ. Thus we have performed the stacking of the Herschel-SPIRE $20{\arcmin}\times 20{\arcmin}$ cutouts at 500 $\mu$m, over the 83 sources included in the PHZ on the one hand, and over the 120 sources no longer included in the PHZ on the other hand. The resulting stacked maps and the associated profiles are shown in the first row of Fig. 25. The overdensity of Herschel sources appears slightly more compact for the sub-sample of sources still included in the PHZ, with a radial profile presenting a plateau within about 2′. This is consistent with the fact that the PHZ has been built at 5′ resolution, while the initial selection of the Herschel sample was based on a first list built at 8′ resolution. However, the overdensity of sources is still clearly identified in both sub-samples. Furthermore, we have performed on Herschel-SPIRE maps a similar process as the one applied on Planck maps to show the red excess at 500 $\mu$m. For each source, the background colour is first estimated between the 250 and 500 $\mu$m maps on a region defined outside the Planck peak emission at 545 GHz. This background colour is used to extrapolate the 250 $\mu$m map at 500 $\mu$m, and this is then removed from the original 500 $\mu$m map, yielding the red excess map at 500 $\mu$m: $$M_{500}^{\rm{RX}}=M_{500}-\left\langle\frac{M_{500}}{M_{250}}\right\rangle_{% \rm{bkg}}M_{250}\thinspace,$$ (9) where $M_{250}$ and $M_{500}$ are the Herschel-SPIRE maps at 250 and 500 $\mu$m, and $\left\langle\right\rangle_{\rm{bkg}}$ means the average over the background region defined above. Positive pixels in this kind of red excess map are associated with colours redder than the background, and potentially associated with higher redshift structures, while negative pixels are bluer and mostly associated with lower redshift structures. By stacking the red excess maps over the two Herschel sub-samples, we get the stacked maps shown in the second row of Fig. 25, which exhibit a clear excess of red colours for both samples. The radial profile obtained with the sample of Herschel sources included in the PHZ presents a red excess larger in the central part compared to the other sample. This is linked to the left panel of Fig. 24, where it can be seen that the 83 Herschel sources included in the PHZ exhibit larger S/N in the Planck excess and cleaned maps than the other sources. This analysis demonstrates firstly that the PHZ represents a sample of sources with a larger reliability than the initial selection made for the Herschel follow-up, and secondly that the sub-sample of sources without any counter-parts in the PHZ are not spurious detections, but simply have lower significance, as already stressed above. From the 11 sources of the Herschel sub-sample confirmed as strongly lensed star-forming galaxies (Canameras2015), four sources (over the five previously selected with a similar algorithm used in this work) are present in the final PHZ. Two other sources, confirmed at redshifts 2.2 and 2.4, did not pass the colour-colour criteria, while a third one exhibits a S/N on the 545 GHz excess map just below the required threshold of 5. The last five sources, which have been selected from the PCCS catalogue, have no counter part in the PHZ. Additionally, the first spectroscopically confirmed Planck-discovered proto-cluster candidate, PHZ G095.50$-$61.59 (Florescacho2015), exhibits one of the smallest S/N values for the 545 GHz excess in the PHZ. Finally, it is worth remarking that because the Herschel-SPIRE follow-up of the Planck high-$z$ source candidates and the final PHZ are not fully consistent (in a statistical sense), it is hard to draw definitive conclusions about the nature of the PHZ sources based on the Herschel analysis. Appendix D List description In this last appendix we present a description of the PHZ. Table 3 gives the names, units and, explanation of the contents of each column.

Ages and Metallicities of Extragalactic Globular Clusters from Spectral and Photometric Fits of Stellar Population Synthesis Models Marsha J. Wolf11affiliation: Current address: University of Wisconsin - Madison, Department of Astronomy, 475 N Charter St, Madison, WI 53706, mwolf@astro.wisc.edu , Niv Drory, Karl Gebhardt, and Gary J. Hill University of Texas at Austin, Department of Astronomy, 1 University Station, Austin, TX 78712 Abstract Spectra of galaxies contain an enormous amount of information about the relative mixture of ages and metallicities of constituent stars. We present a comprehensive study designed to extract the maximum information from spectra of data quality typical in large galaxy surveys. These techniques are not intended for detailed stellar population studies that use high quality spectra. We test techniques on a sample of globular clusters, which should consist of single stellar populations and provide good test cases, using the Bruzual-Charlot 2003 high resolution simple stellar population synthesis models to simultaneously estimate the ages and metallicities of 101 globular clusters in M31 and the Magellanic Clouds by fitting their integrated spectra and photometry. The clusters cover a wide range of ages and metallicities, 4 Myr $<$ t${}_{age}$ $<$ 20 Gyr and $-$1.6 $<$ [Fe/H] $<$ +0.3, estimated by other methods in the literature. We compare results from model fits to both the spectra and photometry and find that fits to continuum-normalized spectra over the entire range available, typically 3500 Å to 1 $\mu$m for this sample, provides the best results. For clusters older than 1 Gyr we agree with literature ages to 0.16 dex (35%) and [Fe/H] to 0.12 dex. For younger clusters we agree with literature ages to 0.3 dex (63%), but cannot constrain the metallicity. It is particularly important to use the entire continuum-normalized spectrum to avoid problems with model continua for young objects and to break age-metallicity degeneracies of broadband photometry. Our required S/N is 15-30 Å${}^{-1}$ for 20% age uncertainties and 30-55 Å${}^{-1}$ for 10% uncertainties over spectral resolutions of $\Delta\lambda$ = 5-25 Å. This technique should work well for the age-metallicity parameter space expected for early-type galaxies at z$\sim$1, although individual galaxy spectral S/N may require the coaddition of a few like objects. Lack of accurate flux calibration in large surveys is not an issue for the continuum-normalized spectra. galaxies: evolution — galaxies: fundamental parameters — galaxies: individual(M31) — galaxies: star clusters — galaxies: stellar content — Magellanic Clouds ††slugcomment: to appear in ApJ, January 20, 2007, v655 1 Introduction Understanding when and how different types of galaxies formed in the universe is an age old problem. In recent years the hierarchical merging scenario (White & Rees, 1978; Peebles, 1980; White & Frenk, 1991), where large galaxies are built up over time through mergers of smaller objects, has been the leading theory for the mechanism by which structure formed in the universe. Large galaxy surveys have been undertaken to provide observational evidence for how galaxies formed. From their observations, we are compiling a snapshot of galaxies througout time. At z$\sim$3 we find both massive starbursting galaxies in highly clustered regions (Neri et al., 2003; Blain et al., 2004; Chapman et al., 2004) and actively star forming galaxies elsewhere (Papovich, Dickinson, & Ferguson, 2001). At z$\sim$2 we see galaxies that are still forming stars, but have redder colors and larger masses from continued star formation since z$\sim$3 (Steidel et al., 2004). At z$\sim$1-2 a population of extremely red objects (EROs) are seen by near infrared surveys such as FIRES (Labbé et al., 2003), K20 (Cimatti et al., 2002), and MUNICS (Drory et al., 2001), many of which appear to contain older stellar populations that have experienced passive evolution (Förster Schreiber et al., 2004; Daddi et al., 2002; Saracco et al., 2003). At z$\sim$1 we see a decrease in global star formation rates (Lilly et al., 1996; Madau et al., 1996, 1998; Steidel et al., 1999; Barger et al., 2000; Thompson et al., 2001; Wolf et al., 2005), an increase of stellar mass densities in early-types (Hogg et al., 2002; Bell et al., 2003; Dickinson et al., 2003; Kauffmann et al., 2003b; Rudnick et al., 2003; Cimatti et al., 2004; Wolf et al., 2005), and a change in galaxy morphologies to more ordered systems (Strateva et al., 2001; Dickinson et al., 2003). At z$\lesssim$1 we see early-type galaxies that appear to have formed at z$\sim$1.5-2 in the field (Im et al., 2002; Gebhardt et al., 2003) and at z$\sim$2-3 in clusters (van Dokkum et al., 1998; Thomas et al., 2005), possibly from those that were seen as star forming galaxies at z$\sim$2 and starbursts at z$\sim$2-3. This observed galaxy demography, however, is not matched by the semi-analytic hierarchical galaxy formation models. Although the models do well on predicting the global star formation rate and stellar mass densities, they do not match observations on the correct proportions of galaxy types as a function of redshift (Somerville, 2004). In particular, not enough EROs or submm galaxies are formed at z$\sim$1-3 in the models. We do not yet understand the details of how galaxies assembled their mass, although the epoch of z$\sim$1-2 continues to stand out as an important transitional period. Detailed studies of individual galaxies around this epoch will uncover essential clues about how they assembled into the distribution of galaxies that we see today. One such detail for study is determining ages or formation redshifts of galaxies in this epoch. Various techniques have been used in the past to determine an overall age from integrated galaxy light, including using colors to estimate age from the amounts of old (red) and young (blue) stars, using the equivalent widths (EWs) of specific spectral lines or breaks that are known to be sensitive to stellar age (Trager et al., 2000a, b; Kauffmann et al., 2003a, b), matching galaxy spectra to stellar spectra (Stockton et al., 1995; Dunlop et al., 1996; Spinrad et al., 1997), using fits of stellar population synthesis models to spectral energy distributions (SEDs) obtained from broadband or narrowband photometry (Papovich, Dickinson, & Ferguson, 2001; Cimatti et al., 2004; Drory et al., 2004; McCarthy et al., 2004), and fitting galaxy spectra with model spectra (Yi et al., 2000; Nolan et al., 2003; Cimatti et al., 2004; Ferreras & Yi, 2004; McCarthy et al., 2004). We will refer to photometric SEDs throughout the paper as “SEDs,” not to be confused with spectra. Photometry has been widely available and is a good way to cover a larger spectral range than typically available with spectroscopy to get information about different stellar populations, but it lacks spectral details and suffers from more degeneracy between age, metallicity, and dust content. Recent surveys with spectra of hundreds or thousands of galaxies, in addition to the photometry, provide more information for detailed analyses. Furthermore, the release of population synthesis models at high spectral resolution (Bruzual & Charlot, 2003) allows one to take advantage of the detailed information contained in individual lines over the entire observed spectral range, rather than relying on only a few lines or on more degenerate broadband colors. In this work, we evaluate age and metallicity estimation techniques on globular clusters by comparing results from Bruzual & Charlot (2003) simple stellar population (SSP) model fits to the spectra, broadband photometry, and line indices of the clusters. Globular clusters provide a simpler test case than galaxies since their stars likely provide a coeval and nearly homogeneous metallicity population that has no internal dust extinction. We test the utility of using these models to estimate ages and metallicities of high redshift galaxies by first applying them to a sample of extragalactic globular clusters from the Santos et al. (2002) public database that have integrated spectra, photometry, and age or metallicity estimates obtained by other methods. Galaxies will be discused in a future paper. This paper is arranged as follows. Globular cluster spectra and photometry are described in §2, comparison ages and metallicities from the literature are described in §3, the stellar population synthesis models are described in §4, our model fitting procedures are explained in §5, age and metallicity results are presented in §6, a discussion of issues is given in §7, and conclusions are summarized in §8. 2 The Cluster Data The Santos et al. (2002) online spectral database provides our sample of extragalactic globular clusters that cover a significant range of age and metallicity, making their results applicable to evolving galaxies. We chose those clusters that also have photometry available in the literature, which includes 79 clusters in the Large Magellanic Cloud (LMC), 4 clusters in the Small Magellanic Cloud (SMC), and 18 clusters in M31. The spectra were obtained with the 1.52 and 2.2-meter telescopes at ESO in La Silla (Bica et al., 1986a, 1987a, 1987b, 1990, 1994), the 2.15-m CASLEO telescope in Argentina (Santos et al., 1995), the 3.6-m CFHT in Hawaii (Jablonka et al., 1992), and the 4.2-m William Herschel Telescope in La Palma (Jablonka et al., 1998). The data include near ultraviolet, optical, and near infrared spectra, covering a total wavelength range of 3200-10000 Å for some clusters, and varying in spectral resolution from 6 to 23 Å. We should note that for all clusters except a few in M31, this resolution element is larger than the $\Delta\lambda$ = 10 Å required for Lick/IDS line indices. All available spectral segments for each object are used to utilize maximum wavelength coverage, but since they originate from a number of different sources, flux calibration between regions may not be consistent. Table 1 lists each object with its specific wavelength coverage, spectral resolution, and source references. We compile photometry of the clusters from various sources in the literature. Bica et al. (1996) provide UBV photometry for the LMC clusters with $\sim$20% of the objects taken from van den Bergh (1981) and the rest re-observed by the authors at CTIO or CASLEO. Persson et al. (1983) provide VJHK photometry for LMC clusters from observations on 1-2.5 meter telescopes at Las Campanas and Cerro Tololo observatories. For the SMC clusters, Persson et al. (1983) provide VJHK and UBV are from Santos et al. (1995) who use photometry from van den Bergh (1981) and a few other sources. For the M31 clusters, Battistini et al. (1993) supply BVRI from observations on the 152-cm telescope in Loiano, Bologna and on the 4-m KPNO telescope. Barmby et al. (2000) provide UBJHK from observations on the 1.2-m telescope at the Fred L. Whipple Observatory. 3 Comparison Ages and Metallicities Even for a given globular cluster, a wide range of age and metallicity estimates typically exists in the literature. This is not completely surprising given the different methods employed to derive the parameters, some based on broadband colors and others on very narrowband spectral line strengths, but the dispersion in estimates is quite large for many clusters. We will compare our results to age and metallicity values from a number of sources and different techniques to show where our values lie in the total spread. Literature ages and metallicities for specific clusters can be found in Table 4, along with our derived values. One of the oldest techniques for measuring the age of a star cluster is using its integrated UBV colors. When placed on color-color plots, globular clusters follow a sequence that can be related to age. We use MC cluster U-B color ages that were determined by Bica et al. (1990) and Santos et al. (1995). When photometry is available for individual stars in a cluster, the location of the main sequence turnoff (MSTO) on a color-magnitude diagram (CMD) can be used to determine the age of the cluster. We use a large compilation of MSTO ages for MC clusters from Hodge (1983). Refinement of the MSTO technique is achieved by fitting theoretical isochrones to the CMDs. We take CMD fit ages from Elson & Fall (1988) and Girardi et al. (1995, 1998). We also add comparison ages determined from photometric fits to GISSEL96 model SEDs (Bruzual & Charlot, 1993) in the BATC filter system, using 15 intermediate bandwidth passbands, by Jiang et al. (2003), and from the line index ratios H$\delta$ / Fe I $\lambda$4045 and Ca II H/K by Leonardi & Rose (2003), which were developed to work better on younger objects than the typical Lick line indices for use on older objects. We consider the Leonardi & Rose line index ratio ages to be some of the most secure, given that they are determined from high quality spectra and are tailored for the younger ages of many clusters in this sample. Santos & Piatti (2004) utilize diagnostic diagrams constructed from EW sums of metal (Ca II K, G band, Mg I) and Balmer (H$\delta$, H$\gamma$, H$\beta$) lines, which are calibrated against literature values and placed on homogeneous age and metallicity scales, to determine ages and metallicites of MC clusters. We also include these in our comparison values of age and metallicity. Fitting theoretical ishchrones to cluster CMDs is generally accepted as the most secure age determination possible when using photometry, however the results do vary between sets of ishochrones. The absolute derived age depends on model zeropoints and uncertainties. Schiavon et al. (2002) find that Padova isochrones with AGB stars included underpredict the luminosity function of giants by 0.2-0.3 dex when compared to observations of 47 Tuc, resulting in an overprediction in the spectroscopic age of 2-3 Gyr for old stellar populations. Young stellar populations are much less affected. Additionally, the Padova isochrone uncertainty in T${}_{eff}$ of 75 K leads to a 1 Gyr age uncertainty or 0.1 dex uncertainty in the [Fe/H] scale. Vazdekis et al. (2001) also find discrepancies between CMD and spectroscopic ages of 47 Tuc due to the exclusion of $\alpha$-enhancement and atomic diffusion in evolutionary models, causing a zeropoint offset. If 47 Tuc is representative of other old stellar populations, for clusters in the 10-14 Gyr age range these model isochrone issues could lead to spectroscopic based age discrepancies of 20-40 % when compared to CMD based values. Large homogeneous samples of globular cluster metallicities are harder to find, since metallicity cannot be as accurately determined from photometry and typical methods require high quality spectra. However, some photometric techniques have been calibrated to spectra. Of these we use [Fe/H] values from VJK colors by Cohen & Matthews (1994), fits to CMDs (Seggewiss & Richtler, 1989; Sagar & Pandey, 1989; Suntzeff et al., 1992; Girardi et al., 1995; Dirsch et al., 2000; Johnson et al., 2001; Piatti et al., 2002), and fits to GISSEL96 model SEDs by Jiang et al. (2003). Our more secure comparison metallicites come from spectra-based techniques. For these we use [Fe/H] from the Leonardi & Rose (2003) line index ratios and their comparison literature values that were derived from EWs of Ca, Fe, Mg, and Na lines (Cohen, 1982), and from EWs of the Ca II triplet at $\lambda$8500 (Olszewski et al., 1991; Da Costa & Hatzidimitriou, 1998). We incorporate the comparison literature values used by Santos & Piatti (2004), which include metal line abundances (Jasniewicz & Thévenin, 1994; Hill et al., 2000), line indices (Beasley et al., 2002), and an infrared index at 1.6 $\mu$m (Oliva & Origlia, 1998). And finally we use a combination of line indices and spectral breaks from Huchra et al. (1991): D(4000), CNB $\lambda$3883 Å, G-band $\lambda$4300 Å, MgH, Mg b, and Fe $\lambda$5270 Å. 4 Stellar Population Synthesis Models We use the high spectral resolution (R$\sim$2000) SSP “standard model” from Bruzual & Charlot (2003; hereafter BC03) that utilizes the STELIB/BaSeL 3.1 spectral library, the Padova 1994 evolutionary tracks, and the Chabrier (2003) initial mass function (IMF) with stellar mass limits of 0.1 and 100 M${}_{\sun}$. They provide model spectra at [Fe/H] = $-$2.25, $-$1.65, $-$0.64, $-$0.33, +0.093, and +0.56, which we linearly interpolate at each wavelength point to a grid of spectra that spans [Fe/H] of $-$2.2 to +0.5 in increments of 0.1 dex, and ages of 1 Myr to 20 Gyr in increments of 2 Myr for 1$<t_{age}<$10 Myr, 20 Myr for 10$<t_{age}<$100 Myr, 200 Myr for 100$<t_{age}<$1000 Myr, and 1 Gyr for 1$<t_{age}<$20 Gyr. The BC03 models include thermally-pulsing asymptotic branch stars (TP-AGB), making use of the multi-metallicity models of Vassiliadis & Wood (1993) that have been calibrated on stars in the Galaxy, LMC, and SMC. The importance of these stars, which have a strong influence on the integrated near infrared light from star clusters of certain ages, will become apparent later in the paper. This phase can dredge up carbon in the stellar atmospheres, leading to carbon-rich stars, which are also very red and can dominate the near infrared light from some clusters. Although no simple prescription can be expected to match all clusters, these stars must somehow be included. Bruzual & Charlot (2003) achieve this by defining the transition to carbon stars and the duration of this phase with the models of Groenewegen & de Jong (1993) and Groenewegen et al. (1995). This semi-empirical prescription for TP-AGB and carbon stars has been tested on and provides good agreement with observed colors of Magellanic Cloud clusters and with optical and near infrared surface brightness fluctuations of metal-poor Galactic globular clusters and more metal-rich nearby elliptical galaxies (Liu et al., 2000). 5 Data Preparation and Model-Fitting Procedures Our approach is to use the maximum amount of information possible for each object. Broadband photometry often covers a much wider spectral range than spectroscopy, providing better constraints on the possible mixes of stellar types, but suffers from age-metallicity degeneracy. However, by combining broadband information with detailed spectral analysis, we hope to better constrain model fits and thus derived ages and metallicities. We investigate the ability of both broadband and detailed spectral features of the models to estimate age and metallicity of the globular clusters by fitting models to the full spectrum, to the continuum-normalized spectrum, to the continuum shape, to photometry, and to spectral line indices of the clusters. Each of these fitting procedures is described in detail in the following sections. • Full spectrum fits utilize information from both the line strengths and the continuum shape. • Continuum-normalized (CN) spectrum fits use only the information contained in the lines, losing important continuum information, but also removing any adverse effects of inaccurate flux calibration in the data or continuum shape errors in the models. Model continua problems are particularly possible in the near infrared due to the inability to properly account for the number of TP-AGB or carbon stars that can contribute much of the light at these wavelengths. • Continuum fits use only the continuum shape that was removed from the spectrum. • Photometry fits cover a much broader wavelength range, but lose detailed spectral information and may also be affected by continuum shape problems in the data or models. • Line index fits move from broad to narrow and focus only on specific spectral lines that are known to be sensitive to metallicity or age. We evaluate these specific lines in the models by calculating the indices directly from model spectra and comparing those to line indices calculated from the cluster spectra in the same way. 5.1 Spectra The spectra for each object in the Santos et al. (2002) database consist of near ultraviolet, optical, and near infrared segments from multiple sources at spectral resolutions of 6-23 Å. We splice all spectra together to cover the widest possible range for each cluster. No noise or sky spectra are included in the database, so we must estimate the noise from the spectra. To do this we calculate the standard deviation of the spectrum in 100 Å bins and scale the resultant signal-to-noise ratio (S/N) spectrum to match the average S/N value quoted in the source publications for some objects. For those objects that do not have quoted S/N values, the standard deviation spectrum provides the noise. This likely overestimates the noise since absorption lines will increase the standard deviation of a bin, but the required scaling is typically less than 10%. We prepare the data for fitting by marking regions that contain emission lines, noise spikes, and sky background residuals, as well as two regions for which the STELIB stellar library used in the models has problems with telluric features (6850-6950 Å and 7550-7725 Å). These regions are masked and deweighted for each cluster by significantly lowering its S/N spectrum over the affected wavelengths. We smooth the grid of model spectra, which have an intrinsic resolution of 3 Å, to the wavelength dependent resolution of each cluster and fit its spectrum using two free parameters, age and metallicity, and a S/N-weighted normalization constant, $\alpha_{n}$, which is uniquely determined for each model. We select the best fitting model by calculating the summed $\chi^{2}$ for each model spectrum of a given age and metallicity, relative to the cluster spectrum, and selecting the model in the grid with the minimum value. In the $\chi^{2}$ calculation, we use the previously determined cluster noise spectrum as the $\sigma_{i}$ for the flux, f${}_{i}$, at each wavelength bin, i, for each model, n, as given in Equations 1 and 2. Once the best fitting model is determined for a cluster, we inflate its noise until the $\chi^{2}_{reduced}$=1 to include any additional noise sources and to better estimate confidence contours and error bars. If, after this process, the estimated errors are smaller than the distance to the next model grid point, an error bar of half this distance is assigned to the cluster for both age and metallicity. $$\alpha_{n}=\frac{\sum_{i}f_{data_{i}}\left(\frac{S}{N}\right)_{i}}{\sum_{i}f_{% model_{i}}\left(\frac{S}{N}\right)_{i}}$$ (1) $$\chi^{2}_{n}=\sum_{i}\left(\frac{f_{data_{i}}-\alpha_{n}f_{model_{i}}}{\sigma_% {i}}\right)^{2}$$ (2) Errors on age and metallicity estimates are determined from the $\Delta\chi^{2}$ contours. The S/N of the data controls the size of these contours, and thus the confidence of the derived parameters. Typical levels for the globular clusters are S/N$\sim$100 per resolution element. Figure 1 shows a simulation of the effect of S/N on the confidence of the derived age and metallicity for NGC 419 in the SMC, achieved by artificially adding different levels of gaussian noise to its spectrum and fitting models to each of the noisy continuum-normalized spectra. The simulated S/N per resolution element from top left to lower right in Figure 1 is 99, 22, 12, 8, 6, and 4. Solid, dashed, and dotted lines represent 1, 2, and 3$\sigma$ contours. The color bars give the $\Delta\chi^{2}$ values for confidence levels of 68.3% ($\Delta\chi^{2}$=2.3), 90%, 95.4%, 99%, 99.73%, and 99.99% ($\Delta\chi^{2}$=18.4) for two degrees of freedom. Black dots mark the locations of models in the age-metallicity grid. The best fitting model is marked by a black cross in each panel. The age and metallicity from the best fitting model in the highest S/N case is marked by a yellow circle in each subsequent panel for comparison. Two things become apparent from these plots. First, as the confidence contours grow with decreasing S/N, metallicity becomes harder to constrain than age. In some low S/N conditions (e.g. the last panel) multiple islands of age-metallicity combinations arise as equally likely answers. The best fitting model in each panel does not always overlap the 1$\sigma$ contours of all other panels, however, this mostly happens in the [Fe/H] dimension and those values are less secure than the ages. This is our first indication that metallicity is poorly constrained, compared to ages. The second point is that we find a limiting S/N below which the estimated age and metallicity become unreliable. This happens somewhere between S/N = 22 and 12 in these plots, at which point the contours in the low S/N cases do not overlap the estimates from the highest S/N case. We find that the S/N must be $\gtrsim$15 per resolution element (S/N$\sim$5 Å${}^{-1}$) for $\Delta\lambda$=13-23 Å to adequately constrain the age. Metallicity is harder to constrain under all conditions. A more detailed analysis of limiting S/N and spectral resolution can be found in §6.5. 5.2 Continuum-Normalization Errors in flux calibration of the spectra can induce errors in derived ages and metallicities. Additionally, model continua can be off in the near infrared due to stochastic contributions from TP-AGB stars in $\sim$ 0.1 to 1 Gyr old populations and from carbon stars in populations of age 0.3 to 2.5 Gyr (Frogel et al., 1990; Marigo et al., 1996; Girardi et al., 1998). These variations can result in differences of nearly 2 magnitudes in V-K (Bruzual & Charlot 2003, Figure 8). AGB stars in the thermally pulsing phase can contribute over 80% of the K-band light (Maraston, 2005), the exact amount of which is hard to predict since this phase involves ejection of the outer stellar envelopes and geometry-dependent obscuration of the central stars. The BC03 models include TP-AGB and carbon stars semi-empirically, but in any given cluster the K-band light can vary depending on the numbers of stars it actually has in these phases. To avoid these issues, we normalize out the continua with a median-binning routine that uses bin sizes tailored to work for different classes of objects. The young clusters of the Magellanic Clouds have strong Balmer discontinuities and need to be sampled by small bins to correctly trace this sharp continuum break, while older objects in M31 require larger bins to smooth over absorption troughs in the red part of the spectrum. The bins that we use are 100 Å wide blueward of 4000 Å and 300-700 Å wide redward of 4000 Å. The points in each bin are sorted by flux values, the lower 1/2 of the points are ignored to reduce the weight of absorption lines, the highest few points are rejected to ignore noise spikes, and the median of the remaining flux values is assigned to that bin. The medianed bin fluxes are linearly connected and the object spectrum is divided by the result, providing a flat continuum. The same binning parameters used for an object are also used on each model spectrum in the grid. Removal of the continuum shape should not affect the values of line indices measured on the spectrum, providing a good check of our procedure. We applied this test, as suggested by the referee, by choosing a random sample of 25 globular clusters spanning the entire age range and measured line indices before and after continuum normalization. The indices used were Mgb, Fe5270, Fe5335, H$\beta$, H$\delta_{A}$, H$\delta_{F}$, H$\gamma_{A}$, H$\gamma_{F}$, G4300, $<$Fe$>$, and [MgFe]${}^{\prime}$. Only the positive index values that could be reliably measured were used, as in §6.4. On average the values change by less than 1%, and all line index median differences for the 25 clusters are below 3% with no correlation to the derived cluster age. The continuum shape that we divide out is also fit by model continua and referred to throughout the paper as “continuum fits.” Different types of spectral fits are illustrated in Figure 2 for an old cluster, G177 in M31, and a young cluster, NGC 1711 in the LMC. 5.3 Photometry To test ages and metallicities derived from broadband features over a wide baseline, we fit models to the UBVJHK photometry from the literature, using all of the bands that are available for each cluster. Our model fitting procedure requires scalable fluxes, so we convert cluster magnitudes to fluxes using the UBV zeropoints from Bessell (1979) and the JHK zeropoints from Wamsteker (1981). We obtain model fluxes for each band by convolving filter transmission curves with the model spectra. These fluxes are scaled to those of the object with an overall S/N-weighted normalization parameter during the fit. The best fitting model is selected by $\chi^{2}$ minimization, using Equations 1 and 2 with i now representing each photometric band. Errors are not given for all literature photometry, so we initially assume a 5% photometric flux error in all bands for each cluster, which is then adjusted to make $\chi^{2}_{reduced}$=1 for the best-fitting model. Fitting parameters are age and metallicity. Photometric fits are also illustrated in Figure 2. 5.4 Line Indices Lick line indices have been used for some time (Faber et al., 1985; Burstein et al., 1986; Gorgas et al., 1993; Worthey et al., 1992, 1994; Worthey, 1994; Worthey & Ottaviani, 1997; Trager et al., 1998). They are based on a very specific set of data and their application to new objects is somewhat complicated. To properly use Lick/IDS indices, some of the same stars must be observed with the instrumental setup of the new objects and the spectra of the new objects must be convolved to exactly the same wavelength dependent resolution as the original Lick data, which varied from run to run during the development of the indices. Very small differences in the wavelengths can cause large errors in the indices. To investigate the robustness of these specific lines in the model spectra, we calculate line indices directly from the models, similar to the calibrations done in Bruzual & Charlot (2003). Direct application of the models for this purpose would avoid the step of convolving the object spectra to the estimated Lick resolutions. If the models are smoothed to the same spectral resolution as the cluster data, indices can be calculated directly from both spectra in the same manner. However, the bandpasses of the Lick indices were originally chosen for data with $\Delta\lambda$ = 8-12 Å (Worthey & Ottaviani, 1997). Specific lines in lower resolution spectra may begin to lose age and metallicity information. Nevertheless, given the long history of the use of line indices, we wanted to attempt a variation of this technique on data with resolution and S/N representative of that from typical galaxy surveys. First, we broaden the model spectra to match the resolution of each object, which may vary for wavelength regions obtained on different instruments. Then, we calculate indices for the lines in the Lick system using the passbands defined in Trager et al. (1998) and Worthey & Ottaviani (1997). To these we add $<$Fe$>$, an average of Fe $\lambda$5270 and Fe $\lambda$5335; D${}_{n}$(4000), the 4000 Å break strength using the narrow passbands defined in Balogh et al. (1999); CNB $\lambda$3883 Å, MgH, and Ca II H+K, as defined by Brodie & Huchra (1990); and [MgFe]${}^{\prime}$ = $\sqrt{{\rm Mg}b(0.72\times{\rm Fe}5270+0.28\times{\rm Fe}5335)}$, as defined by Thomas et al. (2003). The [MgFe]${}^{\prime}$ index has been found to be a good tracer of total metallicity and to be insensitive to $\alpha$/Fe enhancement (for spectral resolution near 10 Å), which may be important for some of the globular clusters in this sample. We calculate the indices as described in Trager et al. (1998), by connecting the average flux values of the pseudocontinua sidebands with a straight line and integrating the flux under that line over the index passband. Indices (or break strengths) are calculated for CNB, H$\delta_{A}$, H$\delta_{F}$, CN${}_{1}$, CN${}_{2}$, Ca4227, G4300, H$\gamma_{A}$, H$\gamma_{F}$, Fe4383, Ca4455, Fe4531, C${}_{2}$4668, H$\beta$, MgH, Fe5015, Mg${}_{1}$, Mg${}_{2}$, Mgb, Fe5270, Fe5335, Fe5406, Fe5709, Fe5782, Na D, TiO${}_{1}$, TiO${}_{2}$, [MgFe]${}^{\prime}$, $<$Fe$>$, and D${}_{n}$(4000). Indices for model spectra are calculated in the same manner. We select the model with the smallest sum of absolute value residuals between its index values and the corresponding indices derived from a cluster’s data. The age and metallicity of this closest matching model are adopted as the estimates for the cluster; no interpolation is performed between model grid points. To facilitate comparisons with the literature, we repeated the model selection using different subsets of the indices listed above: all indices simultaneously; Huchra et al.’s (1991) group of 6, CNB, D${}_{n}$(4000), G4300, MgH, Mgb, and Fe5270; and H$\beta$-[MgFe]${}^{\prime}$ as in Thomas et al. (2003). 6 Results Examples of model fits are shown in Figure 2 for old and young clusters. Panels a,c,&e are the cluster G177 in M31 and panels b,d,&f are NGC 1711 in the LMC. The full spectrum fits are shown in a&b, the continuum-normalized spectrum fits are shown in c&d, and the continuum fits are shown in e&f. Photometry fits are also shown in a&b with U, B, V, J, and K bands for G177 and U, B, V, J, H, and K bands for NGC 1711. The circles are the object photometry and triangles are model fluxes from the best photometric fit. The solid lined black spectra are the clusters, the dashed cyan lines are best fitting models from the spectral fits, and the dotted green lines are spectra corresponding to the best fitting models from the photometry fits. The model spectra corresponding to the best fitting photometry are normalized to the cluster spectra at V band in these plots. In the continuum fits, black lines are the data and green lines are the models. The resulting ages and metallicities from the different types of fits are given in the plot labels. For these two clusters the ages derived from the different types of model fits nearly agree at 20 Gyr for G177 and about 0.1 Gyr for NGC 1711. The agreement between photometric and spectroscopic fits seen here is not always the case. G177 in Figure 2a illustrates a case where the photometry through K band reveals a redder SED than the optical spectrum. In this case, the best fitting photometry model has a higher metallicity than the model that best fits the spectrum to make the object redder, an effect of age-metallicity degeneracy. A similar effect is seen in some of the young clusters, presumably due to a near infrared excess from TP-AGB or carbon stars in that case. Some examples of such clusters within the affected age range of 0.1-2 Gyr are shown in Figure 3. 6.1 Different Model Fitting Techniques To evaluate the different model fitting techniques, we first compare their results to each other. In the following sections we compare our results to the literature. Figure 4 compares the ages derived from CN spectrum, full spectrum, photometry, and continuum fits. Clusters with average literature ages $\geq$ 1 Gyr are maked with yellow circles and $<$ 1 Gyr with cyan diamonds. Although the general trends of the ages agree, there are differences in the results. When compared to CN spectrum fits, the methods that contain the continuum overestimate the ages of many clusters younger than 1 Gyr. We believe that this is due to red supergiant stars for clusters in the age range of a few to tens of Myrs and TP-AGB stars for clusters in the age range of 0.1 to 1 Gyr. Both of these types of stars can have stocastic effects in star clusters because of their extremely high luminosities. If more light from them appears in the integrated spectra of the cluster than is included in the models, the cluster SED will appear redder than the model, forcing an older redder model SED as the best fit. All fitting methods that contain model continua will have this same bias. This effect is discussed further in comparison with literature ages in §6.2. For clusters with average literature ages $<$ 1 Gyr our tightest estimated age correlation is seen between the full spectrum and continuum fits (Figure 4c), both of which contain the spectral shape. The median fractional age offset, relative to the average literature age, between these fits is 0.04 for 63 clusters. The fact that the continuum-full spectrum correlation is tighter than the CN-full spectra correlation (fractional offset of 2.40) suggests that the continuum shape has a larger effect than the spectral lines in constraining the ages of these young globular clusters for our employed fitting methods. Therefore, any problems in the continuum shape of the data or models will have grave effects on the derived ages. For the clusters older than 1 Gyr our tightest estimated age correlation is between the CN spectrum and photometry fits (Figure 4b), with a median fractional offset of 0.18 for 12 clusters. For young clusters the metallicities obtained from our different techniques show virtually no correlation, with a tendency for the continuum to give higher metallicities than the full spectrum. For clusters older than 1 Gyr there is a very weak correlation between metallicities obtained with continuum and full spectrum fits, possibly indicating that the derived metallicity has a stronger dependence on the continuum shape than the spectral lines for our fitting method. 6.2 Ages In Figure 5 we compare our globular cluster ages to those obtained by other methods in the literature. The thick cyan horizontal bars connect points that represent the same cluster, but that have ages from multiple sources in the literature. These illustrate the large spread in previously derived ages. The thin horizontal bars are the quoted errors on the literature ages and the thin vertical bars are our 1$\sigma$ errors on ages derived from model fits. Note that the errors are much larger for the photometry fits, reflecting more degeneracy when using broadband colors than when using spectra or the continua shapes. Over the entire age range spanned by the clusters and considering all literature age estimation techniques, our CN spectrum fit ages best match those from the literature. The average offsets and rms scatter our ages from groups of literature values based on different techniques are given in Table 2, where column 1 gives values for all clusters in the sample that have literature ages and column 2 excludes two outliers (for reasons explained in §6.2.1) and the clusters that only have literature values from photometric fits to BC96 models (because these used only a few fixed metallicity values in the model grid, which could skew the derived ages, and they are based on old models). Excluding these points, our CN spectrum fits compared to the averages of the literature values for each object have an average offset of 0.29 dex (0.19 Gyr) with a dispersion of 0.36 dex (1.00 Gyr). Our photometric ages have an average offset of 0.59 dex (0.48 Gyr) and dispersion of 0.76 dex (1.19 Gyr). There are some systematic differences between the two sets of results. The scatter of the CN spectrum fit points is more uniformly distributed around the literature values, unlike the photometric points that tend to overestimate the age, particularly for the younger clusters. The full spectrum fits, which contain the continuum, also overestimate the ages below 1 Gyr, with an average offset of 0.69 dex (0.54 Gyr) and a dispersion of 0.9 dex (3.2 Gyr). This is the same effect that was seen when comparing our fits with and without the continua included. We now return to stars that can strongly affect the continua shapes. Detailed studies of such stochastic effects in star clusters can be found in Bruzual & Charlot (2003), Cerviño et al. (2000, 2001, 2002, 2006), and Maraston (2005). Our photometric fits, in particular, show two different age ranges where our best-fit ages depart from literature ages on the high side (Figure 5c): 4 to 100 Myr and 0.1 to 1 Gyr. The age range of 0.1 to 1 Gyr would be affected by TP-AGB stars. If the model prescription for these stars does not exactly match those in the clusters, then the older ages from fitting methods that include the continuum could be explained by older, redder spectra giving the lowest $\chi^{2}$ for clusters that have this near infrared excess. Our derived ages also scatter to older values for clusters younger than 0.1 Gyr when using models that include the continuum shape. This effect is likely due to stocastic red supergiants, which have been observed with estimated masses up to 120 M${}_{\sun}$ in the LMC (Massey et al., 2005). These stars will affect clusters younger than a few tens of Myrs. Either the number of red supergiants is underestimated in the BC03 models, or the upper mass cutoff of 100 M${}_{\sun}$ for the Chabrier IMF is not quite high enough for the LMC. See §6.2.1 for a discussion of two LMC clusters that are dominated by supergiants, which are also our extreme outliers in the CN spectrum fits of Figure 5a. Because of the large age range covered by the clusters, it is perhaps useful to look at the behavior of the fractional age errors relative to the literature values, which are plotted in Figure 6. Overall, the CN spectrum fits provide ages with fractional offsets from the literature of 61%, while the photometry does much worse at 1048%. If we consider younger and older clusters separately, the CN spectrum fits do better than photometry on the average age offsets for both young and old clusters, but the photometry does better on fractional errors for the older objects. For clusters with ages $<$ 1 Gyr, the CN spectrum fits have an average offset of 63% (0.30 dex or 0.09 Gyr) and dispersion of 0.37 dex (0.35 Gyr), while the older clusters have an offset of 35% (0.16 dex or 1.63 Gyr) and dispersion of 0.17 dex (3.78 Gyr). The dispersion is higher below ages of 0.1 Gyr even though the CN spectra are not affected by the continua effects mentioned earlier. This increased dispersion is likely due to the degeneracy of many spectral lines in this age range (see Figure 10 and §6.4). The photometry fits do much worse fractionally on young clusters with an average offset of 1115% (0.62 dex or 0.38 Gyr) and dispersion of 0.78 dex (1.02 Gyr), and better on the fractional errors of older clusters with an offset of 23% (0.10 dex or 2.06 Gyr) and dispersion of 0.11 dex (2.67 Gyr). The lowest fractional errors for the CN spectrum fits occur for the Leonardi & Rose (2003) line index ratios and for the clusters older than 1 Gyr, both at 35%. Photometry fits do slightly better on the older objects with errors of 23%, but these errors rise substantially for the young objects to 1115%. The largest fractional age offsets occur in the clusters less than 1 Gyr old for the full spectrum (3450%), continuum (4432%), and photometry fits. This is likely due to AGB stars. Furthermore, the full spectrum and continuum fits have larger age offsets in the older clusters (425% and 480%) than do the CN spectrum or photometry fits, which may be due to flux calibration issues, since both of these methods include the observed spectral continuum shape. To take care of both data and model continua issues, it appears that continuum-normalized spectral fits are the most robust in producing accurate age estimates for the clusters of all ages. The age errors do not correlate with metallicity, so they seem to be dominated by the age regime of the cluster. Although photometry does a decent job on older objects, it should be noted that after excluding the BC96 photometry literature points, we only have 4 clusters in the $>$ 1 Gyr bin at ages of 1.2, 2.2, 12.4, and 13.1 Gyrs. It could just be that the two younger ones have no problems with boosted near infrared emission. In general, since there is not a good way to determine whether the 1-2 Gyr old objects are affected by the AGB star problem, using the CN spectrum fits would avoid this issue. Some combination of CN spectrum and photometry fits will likely provide the most robust answer. This will be explored further in future work. Other studies have compared results from their cluster age estimation techniques to the literature. Rafelski & Zaritsky (2004) use colors formed from UBVI photometry along with Starburst99 (Leitherer et al., 1999) and GALEV (Anders & Fritze-v. Alvensleben, 2003) models to derive ages and compare to those in the literature from integrated colors (van den Bergh, 1981; Hunter et al., 2003) and isochrone fitting (Pietrzyński & Udalski, 1999; de Oliveira et al., 2000; Mighell et al., 1998; Rich et al., 2000) in Figure 8 of their paper. They find an age correlation with dispersion of 0.76 Gyr, which drops to 0.49 Gyr when considering only the more secure literature ages derived from CMDs. Over a similar age range our CN spectrum fits provide ages with an overall dispersion of 1.0 Gyr about literature comparison values, with 0.12 Gyr for UBV colors and 1.06 Gyr for CMD fits. Our photometry fits have dispersions of 1.04 Gyr for UBV colors and 0.85 Gyr for CMD fits. Although Rafelski & Zaritsky (2004) use clusters in the SMC, only two overlap with our sample. Our literature ages come from different sources as well, so we cannot make a direct comparison to this work. To summarize our results for globular cluster age estimation, the CN spectrum model fits best match the entire range of age estimates from the literature (Figure 5 and Table 2), with average errors of 0.16 dex (35%) for older clusters and 0.3 dex (63%) for younger clusters. The full spectrum, continuum, and photometry fits overestimate the ages of many clusters below 1 Gyr, apparently suffering from an excess of AGB stars making the cluster spectra redder than the models and forcing older aged models to provide the best fits. Although photmetric fits seem to do better for clusters older than 1 Gyr (0.1 dex or 23% error), we will show in §6.3 that CN spectrum fits are superior for simultaneously providing both age and metallicity for these older objects. Furthermore, the uncertainty of whether the clusters will have boosted near infrared emission due to AGB stars makes using the photometry alone a less reliable technique than using spectra. Comparing the fitting methods (Figure 4), we find a tighter correlation between ages derived from the full spectrum and continuum fits than from the CN and full spectrum fits, suggesting that the ages are more strongly driven by the continuum shape than by the spectral lines. Because the derived ages are so strongly influenced by the continua and we have seen signs of problems in the models matching the cluster continua, we conclude that the best method for deriving accurate ages for globular clusters of all ages, especially when simultaneously determining metallicity, is fitting models to their continuum-normalized spectra. 6.2.1 The Outliers There are two extreme outliers in the CN spectrum age plot of Figure 5a. The spectra of these outlying objects are shown in Figure 7. The upper leftmost outlier in the CN spectrum age plot is NGC 2092 in the LMC (Figure 7a,b), with a literature age of 4-12 Myr (from MSTOs and UBV colors) and for which we derive an age of 10 Gyr from both the CN and full spectrum fits. Its spectral coverage is only over the optical range of 3500-5870 Å. The shape of its spectrum is fairly flat with emission lines and few absorption features. The emission lines include [OII]$\lambda$3727, [OIII]$\lambda\lambda$4959,5007, and [NeIII]$\lambda$3869 nebular emission. The emission lines are masked out in our fits, which leaves very few distinguishing features. The cluster’s UBV photometry is also essentially flat. Our best fit metallicity of this object is bottomed out at the minimum in the model grid, [Fe/H]=$-$2.2, for both the CN and full spectrum fits, likely because nothing would fit well and the lowest metallicity model that has very few metal lines in the red produced the lowest, although high, $\chi^{2}$. Santos et al. (1995) list this object as a cluster embedded in a star forming complex with a flat continuum probably caused by red supergiants, which are stocastic in nature even for large star clusters. This appears to be a case that cannot be fit well with either spectral or photometric techniques using the BC03 SSP models. The other outlier in the upper lefthand corner of the CN spectrum age plot in Figure 5a is NGC 2096 in the LMC. This object has a literature age of 49 Myr from its U-B color. Its spectrum (Figure 7c,d) contains a moderate sized 4000Å break of D${}_{n}$(4000)=1.28, which is consistent with an age of at least 1 Gyr from index plots in Bruzual & Charlot (2003), and shows no evidence of the characteristic spectral shape of a young object with a strong Balmer discontinuity that would be expected for an age of only 49 Myr. For this object, our age is 1.2 Gyr (full spectrum and photometry fits) to 6 Gyr (CN spectrum fit). However, Santos et al. (1995) note this object as one where a few luminous intermediate temperature supergiants dominate the cluster’s integrated spectrum and claim that its age is even younger at 6-12 Myr. This may be a stocastic case where the supergiant dominated spectrum is not properly matched by the models. Two out of 101 globular clusters in this sample could not be properly fit by the BC03 models. Both are very young clusters that show evidence of supergiant dominance in their integrated spectra. The stocastic nature of these stars make it difficult for us to derive accurate ages for clusters younger than a few tens of Myrs. 6.3 Metallicites We compare our derived cluster metallicites to the literature values in Figure 8. There are fewer points on these plots than for the ages in Figure 5 because fewer of the clusters had metallicity estimates in the literature. The magnitudes of the metallicity errors relative to literature values are a stronger function of cluster age than [Fe/H]. This is illustrated in Figure 9 where the [Fe/H] errors are plotted against average literature ages. Table 3 gives the average [Fe/H] offsets and dispersions for different groups of clusters. Metallicity is harder to constrain than age. This was first seen in Figure 1 when the $\Delta\chi^{2}$ contours grew primarily in the metallicity dimension as noise was added to the cluster spectrum. Nevertheless, our CN spectrum fits do well on metallicity estimates for the older clusters. In Figure 8 we see a tight correlation of metallicity from these fits with the literature for clusters older than 1 Gyr. The large symbols mark literature estimates that are based on spectra. There is an overall [Fe/H] scale offset between M31 (blue pluses) and the MC (red circles, black crosses, black inverted triangles). If this offset of 0.45 dex is removed from the MC cluster literature metallicities, our metallicites agree to 0.12 dex with those from the Huchra et al. (1991) line indices (blue pluses), literature values on a homogeneous scale from Santos & Piatti (2004) (black inverted triangles), and Leonardi & Rose (2003) line index ratios (red circles) and their comparison literature values (black crosses). The small symbols have literature metallicity estimates that are based on colors. There is more scatter in these values. The weaker correlation with VJK colors (green triangles) can be seen, as well as its break down at low metallicities of [Fe/H] $\lesssim-$0.7. The Jiang et al. (2003) metallicities (magenta squares) are for older clusters in M31 and were derived from photometric fits to BC96 models using only three metallicities, [Fe/H] = 0.0, $-$0.7, $-$1.7. Their discrete metallicity steps and any differences between the 1996 and 2003 models are the reasons for the large dispersion of these points. For the clusters younger than 1 Gyr, our metallicites from CN spectrum fits show no correlation with literature metallicity values. We believe this is because continuum information is also necessary for determining metallicities of young clusters, particularly since younger clusters likely have higher metallicity with more line blanketing. A hint that the continuum information might be helping to determine the metallicity of some young clusters can be seen in the plots of our full spectrum fits in Figs. 8 and 9. In Figure 8c, if we ignore the points below the 1:1 correlation line for the moment, there does seem to be a metallicity trend with the literature for many of the other points. All of the clusters that we ignored below the line show possible signs of a near infrared excess in their photometry over the model that best fits their optical spectra. Some of these spectra are shown in Figure 3. These clusters are in the correct age range, 0.1-2 Gyr, to be affected by TP-AGB stars. It is not clear why our derived metallicities of these objects appear too low, since it seems that redder spectra would be better fit by models with higher metallicities, but maybe it is more an indication that these results cannot be trusted. Fits to the continua shape alone (Figure 8e) result in much less correlation to the literature, however, 5 of the 6 outliers below the 1:1 correlation line in this case are common to those in the full spectrum plot. Therefore, although it seems that correct continuum information might allow metallicity estimates to be made for young objects, the uncertainty in proper modeling of the continua shapes makes the full spectrum fits unreliable. We expect our metallicity estimates that are based on broadband photometry to be less well constrained than those based on spectra. Three of our comparison literature sources make cluster metallicity estimates based on photometry (denoted in our plots by smaller sized symbols): the Cohen & Matthews (1994) VJK colors, the Jiang et al. (2003) BC96 colors, and two of the Santos & Piatti (2004) homogenized literature values. Our photometrically derived metallicities (Figure 8d) do not correlate with these literature values, except for possibly a very weak relation with the VJK colors (green triangles). As expected, our metallicities derived from broadband photometry and the continuum shape show no correlation to any values from the literature that are based on spectral lines. Santos & Piatti (2004) use empirical relationships between the sums of EWs of Balmer and metal lines to the literature age and metallicity values to estimate ages and metallicities of clusters in the Magellanic Clouds and in the Galaxy. They find that EW(H$\delta$+H$\gamma$+H$\beta$) and EW(CaK+Gband+Mg) are both sensitive to age for clusters younger than 10 Gyr, while EW(CaK+Gband+Mg) is sensitive to [Fe/H] only for clusters older than 10 Gyr. Neither of these EW sums correlate with the ages of old clusters (t${}_{age}>$ 10 Gyr) or with the metallicities of young (t${}_{age}<$ 10 Gyr) clusters. Our CN spectrum fits are similar to this technique in that they use line strengths, but over the entire spectral range of the data. Our fits produce metallicities that agree with literature values only for the older clusters, supporting the Santos & Piatti (2004) result, but extending the minimum age from 10 Gyr down to $\sim$ 1 Gyr for valid metallicity estimates. To summarize our results for metallicity estimates, we agree very well with the literature metallicities that are based on spectra for clusters older than 1 Gyr. Those based on colors have much more scatter, which suggests that our CN spectrum fits are more robust in estimating [Fe/H] than methods using colors. For clusters younger than 1 Gyr, metallicity is hard to constrain. Our full spectrum fits hint that correct continua shapes would aid in metallicity estimates of younger clusters, but given the uncertainties in modeling this shape, we cannot accurately derive metallicity estimates for clusters younger than $\sim$ 1 Gyr. 6.4 Using Model Line Indices We compare line indices measured on the BC03 model spectra to those measured on the globular cluster spectra in Figure 10 for H$\beta$-[MgFe]${}^{\prime}$ and $<$Fe$>$-Mgb. Panels a-d show model grids calculated at the spectral resolutions of the data, 6 Å, 12 Å, 16 Å, and 23 Å  respectively. Symbols mark the indices that could be measured for the clusters in each resolution group (indices that came out negative due to noise or emission lines were ignored). The 6 and 23 Å resolution groups (squares and crosses) are older clusters in M31, while the rest are younger clusters in the Magellanic Clouds. The H$\beta$-[MgFe]${}^{\prime}$ index grids clearly show that the model indices become degenerate at $t_{age}<$ 100 Myr for $-$1.0 $<$ [Fe/H] $<$ +0.5, and at $t_{age}\lesssim$ 1 Gyr for [Fe/H] $<$ $-$2.0. Most of the young MC clusters fall in the degenerate regions of the grids. This degeneracy also occurs for the traditional Lick indices (tabulated with the BC03 model package) in this region of parameter space, as shown by the grids in Figure 10e. This plot only includes points from clusters that have spectra with resolutions near that of Lick indices (11 and 12 Å). Bruzual & Charlot (2003) investigated how well line indices calculated directly from a library of their models containing complex stellar populations with a range of star formation histories could match galaxy spectra from the Sloan Digital Sky Survey (SDSS). They concluded that those models could simultaneously fit observed strengths of H$\beta$, H$\gamma_{A}$, H$\delta_{A}$, [MgFe]${}^{\prime}$, [Mg${}_{1}$Fe], [Mg${}_{2}$Fe], and D${}_{\textit{n}}$(4000) in high quality galaxy spectra with S/N${}_{med}>$ 30 per pixel. Their sample of moderately low redshift SDSS galaxies probes a different region of age-metallicity space than does our sample of globular clusters. Local galaxies are generally old with considerably higher metallicity than these clusters, a region of parameter space where the models do well. In contrast, the M31 globular clusters are old and metal poor and the MC globular clusters are young and metal poor. Bruzual & Charlot note that their SSP models using the STELIB/BaSeL 3.1 library do poorer on line strengths for [Fe/H] $<$ $-$0.7 and our index plots add model degeneracy for $t_{age}<$ 100 Myr, both of which are parts of parameter space occupied by globular clusters in this sample. Figure 10f shows $<$Fe$>$ vs. Mg b with model grids included for the minimum and maximum spectral resolutions. Some of the clusters depart from the model grid such that [Mg/Fe] may be enhanced (Maraston et al., 2003). Since we see signs of $\alpha$-enhancement and the BC03 models use scaled solar abundance ratios, the preferred metal index here is [MgFe]${}^{\prime}$, which was shown by Thomas et al. (2003) to be sensitive to metallicity while insensitive to $\alpha$-enhanced abundance ratios at the Lick/IDS spectral resolution. However, when [MgFe]${}^{\prime}$ is plotted against Balmer indices, which are affected by $\alpha$-enhancement because of metal lines in the index passbands, many of the clusters fall outside of the model grids with lower Balmer indices than the models (Figure 10a-d). One might suspect that the lower H$\beta$ indices are caused by the lines being partially filled in by emission, however, many of these clusters are old ones from M31 (squares and crosses) and unlikely to have emission. The younger MC clusters, on the other hand, could be affected by emission. Nevertheless, since our purpose in calculating line indices is for comparison of using the model indices to using the entire spectrum, and the affected MC clusters fall in the degenerate region of the model grid with unusable indices, we do not correct for emission. Further possible causes of the H$\beta$ index discrepancy could be $\alpha$-enhancement or the presence of blue horizontal branch (BHB) stars in the clusters. The M31 clusters do show signs of $\alpha$-enhancement in Figure 10f. Line indices are changed by $\alpha$-enhancement, but in the wrong sense to explain the high H$\beta$ in the BC03 scaled solar abundance ratio model grids. H$\beta$ increases with [$\alpha$/Fe] (Tantalo & Chiosi, 2004a, b; Thomas et al., 2003), due to extra absorption in the blue pseudo-continuum and some in the central passband. Therefore, if the clusters are $\alpha$-enhanced their H$\beta$ indices should be higher than the model grid. The second possible culprit, BHB stars in old clusters with low metallicities, also affect H$\beta$, but in the wrong sense. H$\beta$ should increase if a BHB exists in the cluster. We cannot explain the H$\beta$ behavior of the clusters relative to the models by either of these causes. This same situation is seen by Le Borgne et al. (2004) in their Figs. 12 and 13 where they calculate Lick indices directly from the PEGASE-HR model spectra, which use the same stellar library as the BC03 models, and compare to globular clusters of intermediate to old ages in Andromeda, M31, M33, M81, and M87. They claim that this behavior of H$\beta$ is due to the clusters being extremely metal-poor, a regime where the interpolated stellar libraries suffer from large uncertainties due to lack of sufficient numbers of stars. This same regime of [Fe/H]$<-$0.7 was noted as a less reliable region by Bruzual & Charlot. The metallicites of our M31 clusters have literature values of $-$1.35$<$[Fe/H]$<$0.29, with an average of $-$0.73, so many do fall in the unreliable metallicity regime of the models. That, coupled with the index degeneracy below ages of 100 Myr, make indices calculated directly from the models unusable for most of the globular clusters in our sample. Because calculating line indices from the models is troublesome for the age-metallicity parameter space occupied by many of the clusters in this sample, we take a different approach to investigating the importance of these spectral features. We fit model spectra to the clusters only over regions that include and exclude some of the typically used line indices and breaks. These features are $D_{n}$(4000), H$\delta$, G4300, H$\gamma$, H$\beta$, Mg b, Fe5270, and Fe5335. The index definitions for the extents of the pseudocontinua passbands of lines, or for the flux ratio windows for breaks, define our wavelength ranges over which to fit the models to cluster spectra. Figure 11 shows the results of these fits, where a&b use only the regions including these features and c&d exclude these regions. It is clear that these features are very important for deriving cluster parameters from the models, but also that fitting the entire spectral range (refer to Figure 5 and 8) does a better overall job in estimating age and metallicity. 6.5 S/N and Spectral Resolution Spectra of z$\sim$1 galaxies from large surveys will most certainly have lower S/N than the spectra of the globular clusters used in the present study. We determine the lowest S/N value that can reliably be used to estimate ages by adding gaussian noise of different levels to cluster spectra via the noao.artdata.mknoise task in IRAF and fitting CN model spectra to them over the wavelength range of 3325-9000 Å. We first use actual cluster spectra to make the test realistic and avoid any biases that might be introduced by model spectra. The clusters in this test were chosen for their high S/N spectra and availability of both age and metallicity estimates in the literature. Further tests including effects of spectral resolution were then conducted on smoothed model spectra with added noise. The resulting ages and metallicities as a function of S/N Å${}^{-1}$ for each of four clusters, G158, NGC 419, NGC 2134, and NGC 1818, are shown by different symbols in Figure 12. The average spectral resolutions of these data are 23Å, 13Å, 13Å, and 13Å, respectively. The derived ages of the clusters hover around the same values as S/N decreases until $\sim$5 Å${}^{-1}$, below which the derived ages become unstable. The horizontal line associated with each set of symbols is the average of the derived ages or metallicites for that cluster from its S/N$\geq$5 spectra. For comparison, the spread of literature ages and metallicities for each cluster are shown as vertical lines of the same styles on the right-hand side of the plots. Our derived parameters for these four clusters fall within the literature age ranges. If the 0.45 dex [Fe/H] scale offset is subtracted from the MC cluster literature values, then our average [Fe/H] lines fall within the literature values for NGC 419 and G158, which are the clusters older than 1 Gyr. We do not match the literature [Fe/H] of the younger clusters. We find that the spectra need to have S/N $\geq$ 5 Å${}^{-1}$ to achieve stable age estimates from CN spectral fits. Examples of fits to NGC 419 with added noise and S/N = 99, 22, and 12 RE${}^{-1}$ (27, 6, 3 Å${}^{-1}$) are shown in Figure 13. Note that the last one is below the limiting S/N value. We have also begun a study of the effects of spectral resolution on derived ages and metallicities. To accomplish this we smooth model spectra to $\Delta\lambda$= 5, 10, 15, 20, 25, and 30 Å, add noise in 10 realizations at each S/N level up to 60 Å${}^{-1}$ for a total of 300-500 simulations at each $\Delta\lambda$, and fit CN model spectra to them. The lowest resolution of $\Delta\lambda$=30 Å was chosen to match the lowest typically achieved in galaxy surveys. We select two input models that are consistent with age and metallicity combinations occupied by some of the globular clusters in this sample and that may also be useful for high redshift galaxies. These combinations are: 1 Gyr, [Fe/H]=0.0; and 10 Gyr, [Fe/H]=0.0 (similar to G158 in Figure 12 at 18 Gyr, [Fe/H]$\sim-$0.1, $\Delta\lambda$=23 Å). The results of these fits are presented in Figures 14-16. Plots in Figure 14 show the best fit ages (top rows) and metallicities (bottom rows) as a function of S/N Å${}^{-1}$ for spectral resolutions of $\Delta\lambda$ = 5, 15, 25 Å (to conserve space $\Delta\lambda$ = 10, 20, and 30 Å are not shown) for the 1and 10 Gyr, [Fe/H] = 0 inputs. Input parameters are marked by solid lines, dashed lines represent uncertainties of 10% in age and 0.1 dex in [Fe/H], and dotted lines mark age uncertainties of 20%. Small filled circles show individual simulations, open circles are the average derived parameters in S/N bins of width 2, and vertical cyan bars mark the 1$\sigma$ variation within each bin. Age-metallicity degeneracy in the derived parameters can clearly be seen in Figure 15 for the two input models at the same spectral resolutions presented in Figure 14. The increased difficulty of distinguishing older ages is also apparent from the much larger spread in derived ages for this regime where age changes result in small spectral differences. As spectral resolution degrades, derived ages and metallicities for the lower S/N spectra tend to be driven to the model grid limits more frequently. We choose limiting S/N values for each spectral resolution corresponding to the point at which the standard deviation in derived ages for a S/N bin begins to increase beyond 10 or 20% of the input model age. Figure 16 summarizes the required S/N Å${}^{-1}$ as a function of $\Delta\lambda$ for the 1 Gyr case on the left and the 10 Gyr case on the right. Solid lines denote derived ages within 10% and dashed lines within 20% of the input. Recall that the S/N is calculated for these spectra by taking the standard deviation of the flux in wavelength bins along the spectrum, which has the disadvantage that spectral features increase the apparent “noise” in the spectrum. Because of this fact, there is a limit to the maximum S/N calculated in this manner. These artificial limits are plotted as dotted lines in Figure 16, and therefore the open upward facing triangles are lower limits to the required S/N at those resolutions. The 10 Gyr object requires higher S/N than the 1 Gyr object at almost all resolutions, implying a strong dependence on the object’s age. However, because the age is unknown a priori, the 10 Gyr model must be used to determine the minimum required S/N. The expected trend is seen for ages within 10% of the input: higher S/N is required for lower spectral resolution. At the 20% level, however, somewhat puzzling behavior is seen for the 10 Gyr old object. At $\Delta\lambda>$ 15 Å the required S/N actually decreases. This does not appear to be an aritfact of the simulations, as we see no correlations between the number of points within S/N bins and the standard deviation in derived ages. Interestingly, the globular cluster G158 at $\Delta\lambda$ = 23 Å in Figure 12 may demonstrate similar behavior. Although only one noise realization was performed at each S/N level there, its correct age is obtained at much lower S/N than the younger clusters in the plot. Perhaps this trend is real. One could envision a scenario in which with decreasing resolution model mismatches of narrow lines in the spectrum become less important than broad features typical of old spectra, such as the 4000 Å break and the Mg absorption trough. In order to get the correct age, high S/N is required, but once the allowed uncertainty is increased, the broader features can provide enough information in lower S/N spectra. Clearly, this is only an initial guide for required S/N and more investigation needs to be done into the cause of this unexpected feature. At the high resolution end, our results are consistent with those of Mathis et al. (2006) who also fit continuum-removed spectra (via a data compression algorithm) and find required S/N$\sim$11-17 Å${}^{-1}$ (20-30 per 3 Å pixel) at a spectral resolution of 3 Å. Extrapolating our 5 Å point down to 3 Å would give S/N$\sim$7-22 Å${}^{-1}$. In general, at the spectral resolutions of the globular clusters in Figure 12, $\Delta\lambda\sim$ 13-23 Å, our simulations indicate that a S/N $\gtrsim$ 25 Å${}^{-1}$ is required for a 20% age uncertainty and $\gtrsim$ 45 Å${}^{-1}$ for a 10% uncertainty. This is higher than the S/N$\sim$5 Å${}^{-1}$ obtained from adding noise to actual cluster spectra. This discrepancy is not currently understood and will be investigated further in future work. 7 Discussion 7.1 Different Spectral Regions In §6.4 we looked at the results of fits to specific regions of the spectrum around typically used line indices. Here we test the broader effects of the regions blueward and redward of the 4000 Å break on derived ages and metallicities. This will be important for fitting galaxies at varying redshifts, since at z$\sim$1 optical spectra typically reach just beyond the 4000 Å break in the galaxy’s rest frame. For the clusters in our sample that include near ultraviolet spectra, we analyze the range of 3200-5650 Å, which is the maximum extent common to all clusters in this group. Within this range, we fit models to the blue ($\lambda<$ 4000 Å), the red ($\lambda>$ 4000 Å), and the region right around the break (3750-4250 Å, as defined in the empirical calibration of the break by Gorgas et al. (1999)). The clusters in this study are listed in Table 5 with their ages derived from different parts of the spectrum. Correlations of ages from the different spectral regions are plotted in Figure 17. Fits to different regions give different results. For CN spectrum fits, the red part of the spectrum drives the result. The red ages correlate with the ages from the entire spectrum and are slightly older than the values in the literature. The blue region gives younger ages than both the red region and the literature. Ages from the break region correlate to the red ages with about the same amount of scatter as we saw in the red-entire spectrum correlation. Since the continuum is normalized out for these fits, the break region is left with only the signatures of strong features like the Ca H and K lines and H$\delta$, similar to the D${}_{n}$(4000)-H$\delta$ analysis done by Kauffmann et al. (2003a) on SDSS galaxies. The rough agreement of ages from this region with those from the entire spectrum emphasizes the importance for the 4000 Å break to be in the spectrum used for estimating the age of an object. The only correlation between [Fe/H] is a weak one between the entire spectrum and the red spectrum (not shown in Figure 17), not surprising since most metallicity sensitive lines typically used are in the red part of the spectrum. For full spectrum fits, the trends change and the blue region carries more weight, demonstrating the importance of information in the continuum shape of the blue part of the spectrum. This is especially true for young objects with strong Balmer discontinuities. Ages from the blue and red regions agree, but with some scatter. However, now the entire spectrum agrees equally well with the blue and the red regions. The 4000 Å break ages agree much better with the blue than with the red ages. The red, blue, and break ages all correlate with the literature ages, but the correlation is tightest for the break region, even more so than for the full spectrum fits over the entire range. There is no correlation between our [Fe/H] derived from the different regions, but the red [Fe/H] has a weak correlation with the literature. Because of the importance of the continuum information blueward of the 4000 Å break, we also test hybrid fits that use the full spectrum over $\lambda<$ 4250 Å and the CN spectrum over $\lambda>$ 3750 Å. Wavelengths below 4000 Å will not be affected by the TP-AGB and carbon stars that dominate the near infrared, so as long as there are no continuum problems with the data this type of fit should be valid. We expected this approach to do better on very young objects. However, the hybrid fits match the full spectrum fits that used the entire wavelength range. The continuum shape dominates the fits and spreads out the age correlation with the literature for objects younger than $\sim$0.1 Gyr. This is the same problem seen earlier with the continua of very young clusters being redder than the models, causing older best-fitting ages. In summary, the continuum shape contains a lot of age information. In the blue part of the spectrum the continuum is more important than the lines for deriving age, while in the red part of the spectrum the lines alone are sufficient to derive the age. The most robust age estimates are obtained by using the entire CN spectrum of an object ($\sim$ 3200-8200 Å), but the 4000 Å break is an essential feature to be contained within that spectrum. When using CN spectra to avoid problems in model and data continua, ages from only the blue region are younger than those obtained by the red region or the entire range. Age biases introduced by using different regions of the spectrum must be considered when combining objects that have different spectral coverage or different redshifts that change the rest frame wavelength coverage significantly. Additionally, for objects that cannot be considered a single stellar population, the ages determined from different spectral ranges may not be straighforward to interpret. Nonetheless, near infrared spectra of high redshift galaxies will be important to provide rest frame wavelengths around and redward of the 4000 Å break, particularly when fitting their CN spectra. 7.2 TP-AGB and Carbon Stars The effects of TP-AGB stars on the near infrared luminosity of objects in which they reside have long been known and discussed by groups creating stellar population synthesis models (Renzini, 1992; Bruzual & Charlot, 1993; Girardi et al., 1998; Maraston, 1998). Recent models by Maraston (2005) incorporate TP-AGB stars through an empirical spectral library of C- and O-type stars by Lançon & Mouhcine (2002). The effects of these stars have been reanalyzed and calibrated on MC cluster colors in the new models. They show that over 80% of the K band light can be contributed by AGB stars between the ages of 0.1 and 2 Gyr for solar and lower metallicities. This effect is down to 20% in the V band. Maraston fits photometric SEDs of these models to MC cluster SEDs. Two objects in common with our sample, NGC 1783 and NGC 419, show discrepancies between the Maraston models that best fit the photometric SEDs and BC03 models that best fit the spectra. These inconsistencies are the very point being address by the new models, which show that younger ages may be obtained for objects that have a near infrared excess due to TP-AGB stars if these stars are properly included. However, analysis of the spectra may reveal a different story. The CN spectra of the two clusters (shown in Figure 18b) are very similar to each other. NGC 1783 has D${}_{n}$(4000)=1.295 and NGC 419 has D${}_{n}$(4000)=1.246. We derive similar ages for the two clusters by fitting their CN spectra: 1.5 Gyr and [Fe/H]=$-$0.8 for NGC 1783, and 1.5 Gyr and [Fe/H]=$-$1.2 for NGC 419. The Maraston results for NGC 1783 are 0.3 Gyr and [Fe/H]=$-$0.33, and for NGC 419 are 1 Gyr and [Fe/H]=$-$1.35. BC03 models are overplotted on the CN spectra in Figure 18c,d for all the best-fitting models. The older age of 1.5 Gyr is a better fit to the CN spectra of both clusters. Our photometric fit age for NGC 1783 is 2.0 Gyr and [Fe/H]=$-$0.6, and for NGC 419 is 20 Gyr and [Fe/H]=$-$1.4. This much older photometric age for NGC 419 could be due to the presence of TP-AGB stars making the continuum redder. The near infrared photometry of NGC 419 (see Figure 3) does show an excess over best-fitting models to the spectra. It is likely that the BC03 models do not include the enough TP-AGB stars in this case. On the other hand, the presence of AGB stars was detected in NGC 1783 by Frogel et al. (1990), but apparently their inclusion in the BC03 models is sufficient for this case. We conclude that the detailed information contained in spectra is necessary for breaking degeneracies that can occur when using only broadband photometry through the near infrared. In a more recent paper Maraston et al. (2006) apply the models with enhanced TP-AGB stars to high redshift galaxies and find that some degeneracies, particularly between TP-AGB stars and reddening due to dust, can be broken when mid-infrared photometry is added from Spitzer. These new models look very promising for young objects if infrared data are available, however their spectral resolution is not high enough to fit to galaxy spectra. 7.3 Implications for Galaxies The ages expected for early-type galaxies at z$\sim$1 fall within the range spanned by the globular clusters in this sample. The youngest early-type galaxies at these redshifts will likely be field galaxies that were the products of mergers. If the bulk of stars in these galaxies were formed in mergers at redshifts of z$\sim$1.5-2, which are the formation redshifts found for field ellipticals (Im et al., 2002; Gebhardt et al., 2003), they would be at least 1.5 Gyr old by z$\sim$1 (for $\Omega_{M}$ = 0.3, $\Omega_{\Lambda}$=0.7, $H_{0}$=70 km s${}^{-1}$Mpc${}^{-1}$). Our CN spectrum fits estimate ages to within 0.16 dex (35%) of the literature values for globular clusters older than 1 Gyr in our sample. The metallicities of z$\sim$1 galaxies will likely be higher than many of the clusters in this sample, however, a few clusters spanned most of the range expected for galaxies. At redshifts of 0.3$<$z$<$1.0, galaxies in GOODS-N have metallicities from 0.3 to 2.5 times solar, or $-$0.5 $<$ [Fe/H] $<$ +0.4 (Kobulnicky & Kewley, 2004). We were able to determine the [Fe/H] of globular clusters older than 1 Gyr to within 0.12 dex over the range of $-$1.6 $<$ [Fe/H] $<$ +0.3. The only region we have not tested is the highest metallicities of [Fe/H] $>$ +0.3. Spectra at a resolution of $\Delta\lambda$ = 15 Å (R $\sim$ 400 at 6000 Å) must have S/N $\sim$ 30-45 Å${}^{-1}$ in order to derive ages with 20 or 10% uncertainties, respectively, for old galaxies and S/N $\sim$ 10-20 Å${}^{-1}$ for young galaxies. This is higher than typically achieved for many objects in modern galaxy surveys such as MUNICS, DEEP, and TKRS. Our initial checks on TKRS spectra from DEIMOS on Keck of GOODS-N galaxies at z$\sim$0.8-1.0 with magnitudes of R=23-23.5 find S/N $\sim$ 5 Å${}^{-1}$ for individual objects. Therefore, most galaxies will require coaddition of a few similar objects to reach sufficient S/N. For galaxies at z$\sim$1 typical optical spectra covering $\sim$4000-9000 Å will provide rest frame wavelengths of $\sim$2000-4500 Å. Our tests have shown that when using CN spectra, data to longer wavelengths will provide more robust age constraints. This means that near infrared spectra of galaxies will be important data to obtain for accurate age estimates. Many early-type galaxies appear to have $\alpha$-element abundance ratios enhanced above solar (Trager et al., 2000a, b; Thomas et al., 2003, 2005). This enhancement is a sign of short duration bursts of star formation that may well be typical for early-type galaxies formed during mergers, as we expect to find at z$\sim$1. We see indications of $\alpha$-enhancement in some of the M31 globular clusters in this sample and we still derive ages within 35% of the literature values when fitting the entire CN spectra. This suggests that results from the entire continuum-normalized spectrum, effectively averaging over all the lines, may be less affected by $\alpha$-enhancement than detailed line index methods that rely on only a few affected lines in the spectrum. Cassisi et al. (2004) find that broadband colors in the blue are significantly affected by $\alpha$-enhancement, but removal of the continuum shape must mitigate these effects to some extent. Model fitting techniques will have to be modified for galaxies. Simple stellar populations will no longer be sufficient descriptions of these complex systems. Fits will require models with complex star formation histories, multiple stellar populations, and dust. However, this work has verified that the reliable regions of model parameter space cover most of that expected to be occupied by galaxies. We have also proven techniques by which model fits to galaxy spectra can be expected to work. 8 Summary and Conclusions We have investigated the utility of using the Bruzual & Charlot (2003) high resolution simple stellar population synthesis models for estimating the ages and metallicities of globular clusters. We chose clusters from M31 and the Magellanic Clouds that cover a wide range of age and metallicity values, $\sim$ 0.004$<$t${}_{age}$(Gyr)$<$20 and $-$1.6$<$[Fe/H]$<$+0.3, to explore model performance in different regions of parameter space. We tested various techniques of fitting the models to data using spectra with and without their continua and photometry. Results were compared to age and metallicity estimates in the literature from other methods to evaluate our fitting techniques. The models are reliable in regions of parameter space where t${}_{age}>$0.1 Gyr and [Fe/H] $\gtrsim-$0.7. These regions are limited by the incompleteness of the stellar libraries used in the models. For objects between the ages of 0.1 and 2.5 Gyr, uncertainties in the amount of near infrared light coming from TP-AGB and carbon stars make it hard to match the continuum shape of individual globular clusters. Red supergiant stars affect clusters younger than 0.1 Gyr. Continuum-normalized spectra can be used to get around the TP-AGB star problem, as well as any flux calibration issues in the data, with the requirement that high spectral resolution models are used. The red supergiant star problem limits accurate age estimates to objects older than $\sim$0.1 Gyr. Ages from our CN spectrum model fits best match the entire range of age estimates from the literature (Figure 5 and Table 2), with average differences of 0.16 dex (35%) for clusters older than 1 Gyr and 0.3 dex (63%) for younger clusters. The methods that depend on the continuum shape – the full spectrum, continuum, and photometry fits – overestimate the ages of many clusters below 1 Gyr. We believe this to be due to a near infrared excess from TP-AGB stars making the cluster spectra redder than the models and forcing older aged models to provide the best fits for cluster ages of 0.1-1 Gyr. Red supergiant stars similarly affect clusters younger than 0.1 Gyr. Although photometric fits seem to do well for clusters older than 1 Gyr (0.1 dex or 23% age error relative to the literature), the CN spectrum fits are better for simultaneously providing both age and metallicity for these older objects. Fits to photometry alone cannot reproduce the correct metallicities. Comparing the fitting methods (Figure 4), we find a tighter correlation between ages derived from the full spectrum and continuum fits than from the full spectrum and CN spectrum fits for clusters younger than 1 Gyr, suggesting that these derived ages are more strongly driven by the continuum shape than by the spectral lines. Clusters older than 1 Gyr show a tighter correlation between ages derived from the CN spectrum and photometry fits. Because the derived ages of young clusters are so strongly influenced by the continua and we have seen signs of problems in the models matching the cluster continua, we conclude that the best method for deriving accurate ages for globular clusters, especially when simultaneously determining metallicity, is fitting models to their continuum-normalized spectra. On metallicities of clusters older than 1 Gyr, we agree to 0.12 dex with the literature values that are based on spectra. Those based on colors have much more scatter, which suggests that CN spectrum fits are more robust in estimating [Fe/H] than methods using colors. We cannot accurately derive metallicity estimates for clusters younger than $\sim$ 1 Gyr. Blueward of the 4000 Å break the continuum shape is more important than the lines for deriving age. Redward of the break the lines alone in a continuum-normalized spectrum are sufficient to derive the age. Many of the globular clusters tested here have spectra out to rest wavelengths of 1 $\mu$m. If fitting models to continuum-normalized spectra of z$\sim$1 galaxies, this means that near infrared spectra through K band will be important to provide similar spectral coverage and constraints on age and metallicity. The 4000 Å break is a very important feature to be contained in the spectrum. It will be contained in typical z$\sim$1 spectra, but higher redshifts may require near infrared spectra to correctly constrain ages and metallicites. We should note, however, that although a strong age indicator for globular clusters and early type galaxies, the 4000 Å break can be significantly affected by dust in star forming galaxies. For example, MacArthur (2005) find dust imparted age errors as large as 9.5 Gyr for a 0.5 Gyr BC03 model SSP when ages are derived from the D${}_{n}$(4000)-Fe4668 plane. The spectral S/N required to derive ages by fitting the entire CN spectrum is similar to the 30-40 Å${}^{-1}$ typically required when using line indices on old objects. For a 10 Gyr old object at $\Delta\lambda$ = 5-25 Å our required S/N is 15-30 Å${}^{-1}$ for a 20% age uncertainty and 30-55 Å${}^{-1}$ for a 10% uncertainty. A 1 Gyr year old object requires S/N $\sim$ 3-15 Å${}^{-1}$ for a 20% age uncertainty and 10-25 Å${}^{-1}$ for a 10% uncertainty. At $\Delta\lambda$ = 30 Å both ages require S/N $\gtrsim$ 60 Å. Fairly low resolution, R$\sim$240-1200, spectra can be used, at the cost of increasing required S/N with decreasing resolution. We have successfully tested most of the age-metallicity space expected to be occupied by galaxies at z$\sim$1, except for the highest metallicities of [Fe/H]$>$+0.3. Fitting the continuum-normalized spectra of these galaxies should work well for spectra without accurate flux calibration, typical when faint galaxies are observed on multiple masks over long periods of time during surveys. The required S/N and spectral resolution for determining ages and metallicities matches that typically achieved in recent galaxy surveys, as long as a few like objects are coadded to boost the S/N. The techniques presented here will be applied to galaxies in future work. We wish to thank the anonymous referee for very useful comments and suggestions. MJW was partially funded for this work by NASA GSRP grant number NGT 5-50301. References Anders & Fritze-v. 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Liquid argon scintillation response to electronic recoils between $2.8$–$1275~{}{\rm keV}$ in a high light yield single-phase detector M.Kimura masato@kylab.sci.waseda.ac.jp K.Aoyama M.Tanaka K.Yorita kohei.yorita@waseda.jp Waseda University, 3-4-1, Okubo, Shinjuku, Tokyo, 169-8555, Japan Abstract We measure the liquid argon scintillation response to electronic recoils in the energy range of $2.82$ to $1274.6~{}{\rm keV}$. The single-phase detector with a large optical coverage used in this measurement yields $12.6\pm 0.3~{}(11.1\pm 0.3)~{}{\rm photoelectron}/{\rm keV}$ for $661.7~{}{\rm keV}$ $\gamma$-ray events based on a photomultiplier tube single photoelectron response modeling with a Gaussian plus an additional exponential terms (with only Gaussian term). It is exposed to a variety of calibration sources such as ${}^{137}{\rm Cs}$ and ${}^{241}{\rm Am}$ $\gamma$-ray emitters, and ${}^{252}{\rm Cf}$ fast neutron emitter that induces quasimonoenergetic $\gamma$-rays through a $(n,n^{\prime}\gamma)$ reaction with ${}^{19}{\rm F}$ in polytetrafluoroethylene. In addition, the high light yield enables identification of the $2.82~{}{\rm keV}$ peak of ${}^{37}{\rm Ar}$, a cosmogenic isotope in atmospheric argon. The scintillation yield and energy resolution of the detector are obtained by the full-absorption peaks. We find up to approximately $25\%$ shift in the scintillation efficiency across the energy range and less than $3\%$ of the energy resolution for the $661.7~{}{\rm keV}$ line. The energy dependent scintillation quenching can be attributed by the electron-ion recombination process, and is discussed by an analogy to the dependence of liquid xenon. The Thomas-Imel Box model with its constant parameter $\varsigma=0.07^{+0.03}_{-0.02}$ is found to explain the results below $200~{}{\rm keV}$. keywords: Liquid argon, Scintillation, Vacuum ultraviolet light ††journal: Elsevier 1 Introduction A liquid argon (LAr) scintillation detector has several features that make it attractive for use in various physics experiments to detect ionization particles: it has efficient conversion of energy deposition into a scintillation light signal, powerful discrimination between electronic recoil (ER) and nuclear recoil (NR) events based on its scintillation pulse shape, and benefits from the fact that large quantities of argon are cheaply available. One promising application of the detector is to search and identify the NR signal possibly induced by a dark matter candidate, weakly interacting massive particles (WIMPs) agnes2018darkside ; amaudruz2018first . The typical energy of the signal is in the range of a few keV to several hundreds of keV. Burdensome backgrounds in this search are ER events caused by $\beta$-rays from diffused isotopes (such as ${}^{39}{\rm Ar}$ and ${}^{85}{\rm Kr}$) in LAr and $\gamma$-rays from radio-impurities in detector components. Predicting the measured signal from these background sources is necessary to estimate its contamination in the signal region of interest. In this context, characterization of the detector response to ER events is crucial for achieving lower energy threshold, suppressing systematic uncertainty related to background contamination, and hence enhancing physics sensitivity of the search. In the LAr detector, a charged particle interaction excites and ionizes the detector medium, resulting in the formation of self-trapped exciton states, ${\rm Ar}_{2}^{*}$, through the collision and recombination processes. The excimer is formed in either a singlet or a triplet state, both of which decay radiatively with vast different lifetimes of approximately $7~{}{\rm ns}$ and $1.6~{}{\rm\mu s}$, respectively hitachi1983effect . The scintillation light spectra from both radiative decays lie in the vacuum ultraviolet (VUV), peaked at $128~{}{\rm nm}$ heindl2010the . As direct detection of the VUV photon at LAr temperature (around $87~{}{\rm K}$) is technically challenging, it is often downshifted to the visible region where most cryogenic photosensors exhibit peak sensitivity using a wavelength shifter such as 1,1,4,4-tetraphenyl-1,3-butadiene (TPB) burton1973fluorescence ; porter1970nanosecond . The recoiled particle and its energy are inferred from the observed photon signal waveform. In this work, we measure the LAr scintillation response to ER ranging from $2.82$ to $1274.6~{}{\rm keV}$ using a high light yield single-phase detector at null electric field. The measurement is performed with a variety of calibration sources including the $2.82~{}{\rm keV}$ line of cosmic-ray induced ${}^{37}{\rm Ar}$. We present the energy dependence of the scintillation yield, as well as the basic properties of this detector such as the absolute scintillation light yield and energy resolutions of the full-absorption peaks. The energy dependence of the scintillation efficiency down to a few ${\rm keV}$ is discussed by comparing a model prediction, which is allowed by the use of the ${}^{37}{\rm Ar}$ source. 2 Experimental apparatus The measurement presented here is performed at the surface laboratory at Waseda University. Figure 1 shows the argon handling system used in this work. It mainly consists of a stainless-steel cryostat of diameter $50~{}{\rm cm}$ and height $100~{}{\rm cm}$, in which a scintillation detector sits. The argon filled in the cryostat is cooled by the recirculation system, which extracts hot gas from the cryostat and passes it through the liquefier with a $200~{}{\rm W}$ GM-cryocooler (SUMITOMO CH-110). The argon is maintained at a typical pressure of $1.4~{}{\rm atm}$ and at a liquid level that varies by no more than $1~{}{\rm mm}$ throughout the data collection period. Impurities in the argon (such as water, oxygen, and nitrogen) affect the scintillation properties, resulting in a reduced signal yield acciarri2010oxygen ; acciarri2010effects ; jones2013the . In order to remove adsorbed impurities and outgassing from the detector components, the whole system is pumped to vacuum over about $10~{}{\rm days}$ before the measurement. The pressure of the cryostat reaches below $1.0\times 10^{-3}~{}{\rm Pa}$. Then commercial LAr fills the system via a single path through a liquid filter consisting of a molecular sieve and reduced cooper which removes electronegative impurities. Additional purification is continuously performed by the getters (SAES MICROTORR MC1500-902 and PURERON GP-5) in the recirculation system. Several measurements performed in this system confirm the concentrations of these impurities are negligible in this measurement; water and oxygen contaminations of sub-ppb level and nitrogen contamination of sub-ppm level. The scintillation detector, shown in Fig. 2, is designed to minimize the loss of scintillation photons in their path and maximize light-collection efficiency. The cylindrical fiducial volume of the detector has a diameter $6.4~{}{\rm cm}$ and a length $5~{}{\rm cm}$, contained within an approximately $3~{}{\rm cm}$ thick polytetrafluoroethylene (PTFE) sleeve. A multilayer plastic-foil reflector (3M ESR) lines the inner surface of the PTFE sleeve. Each end of the cylindrical volume is capped by $3$-inch HAMAMATSU R11065 photomultiplier tubes (PMTs), with around $30\%$ of quantum efficiency (QE) for blue light. Since these PMTs are operated with a negative bias voltage of $-1570~{}{\rm V}$, field-shaping rings with the same bias voltage are embed in the PTFE bulk and ensure electric field inside the fiducial volume less than $1~{}{\rm V/cm}$. The TPB wavelength shifter is deposited on both the reflector and the PMT windows using a vacuum-evaporation technique. The amounts of deposited TPB are approximately $40~{}{\rm\mu g/cm^{2}}$ and $30~{}{\rm\mu g/cm^{2}}$, respectively, corresponding to the deposited-layer thicknesses of $\mathcal{O}(1~{}{\rm\mu m})$. These are confirmed by a quartz crystal microbalance sensor and a stylus profiler, as with a procedure similar to that reported in Ref. broerman2017application . The whole of the sleeve is immersed in a LAr bath contained in the cryostat. Four $2$-inch PMTs (HAMAMATSU R6041-506) are implemented to view the LAr bath surrounding the fiducial volume, as shown in Fig. 2. These PMTs are located $20~{}{\rm cm}$ above the fiducial volume and just below the liquid surface so that additional energy deposition in the outer region is tagged by a coincident scintillation signal. The windows of the PMTs are also coated with TPB. A passive shield against ambient $\gamma$-rays surrounds the cryostat, which consists of roughly $2~{}{\rm cm}$ thick oxygen-free copper and $10~{}{\rm cm}$ thick lead. The data acquisition (DAQ) system used in this experiment consists of a $14$-bit, $250~{}{\rm MS/s}$ flash ADC (Struck SIS3316). The signals from two fiducial-viewing PMTs and four outer-bath PMTs are digitized and recorded. The length of the digitizer records is set to $25~{}{\rm\mu s}$ ($5~{}{\rm\mu s}$ before a trigger point and $20~{}{\rm\mu s}$ after), longer enough than the lifetime of the slow component of LAr scintillation light. The trigger is given by the coincidence, within $1~{}{\rm\mu s}$, of the two fiducial PMTs with pulses above a threshold, which is set just above the baseline noise and below a typical single photoelectron (${\rm p.e.}$) pulse. The coincidence decision is internally made by the flash ADC board itself. An inhibition time of $100~{}{\rm\mu s}$ is introduced after each trigger to prevent re-triggering of the afterpulse of the PMTs, which mainly occurs after events with far greater energies than the region of interest (e.g., cosmic-ray events). The trigger efficiency is evaluated by a Monte Carlo (MC) simulation, validated using waveforms from the LAr data sample. It is found to be consistent with unity for ER signals larger than $25~{}{\rm p.e.}$, as shown in Fig. 10. 3 Event analysis 3.1 PMT calibration The gain of the fiducial PMTs is calibrated using a blue LED powered by a pulse generator. Light pulses from the LED, characterized by a width of approximately $20~{}{\rm ns}$ at tenth maximum, are injected into the fiducial volume through optical fiber, while the generator simultaneously triggers the DAQ system and the corresponding waveforms from each PMT are recorded over a window of $\pm 1~{}{\rm\mu s}$. A baseline ADC count is determined by the first $0.6~{}{\rm\mu s}$ of the window, and its subtraction is applied waveform-by-waveform. The charge response of the PMT is measured by integrating the waveforms within a $48~{}{\rm ns}$ window starting $20~{}{\rm ns}$ prior to the photoelectron pulse arrival time. The gain value is determined by fitting the charge distribution to model functions. In this analysis, two models are considered to describe the PMT response. One expression of the models (Gain-Model A) as a function of the integrated charge $q$ is followed to that used in Ref. Alexander2013light : $$\displaystyle f(q)$$ $$\displaystyle=$$ $$\displaystyle\sum_{n}P(n;\lambda)\times f_{n}(q),$$ (1) $$\displaystyle f_{n}(q)$$ $$\displaystyle=$$ $$\displaystyle\rho(q)*\psi_{1}^{n*}(q),$$ $$\displaystyle\rho(q)$$ $$\displaystyle=$$ $$\displaystyle G(q;x_{0},\sigma_{\rm ped}),$$ $$\displaystyle\psi_{1}(q)$$ $$\displaystyle=$$ $$\displaystyle\frac{p_{E}}{\tau}\exp(-q/\tau)+(1-p_{E})G(q;x_{m},\sigma_{m})$$ where $P(n;\lambda)$ is a Poisson distribution with mean $\lambda$, $G(q;x,\sigma)$ is a Gaussian distribution with mean $x$ and standard division $\sigma$, $*$ denotes a convolution, $\psi_{1}(q)$ is the PMT single photoelectron response, and $\psi_{1}^{n*}(q)$ is the $n$-fold convolution of $\psi_{1}(q)$ with itself. This model consists of two components comprising the PMT response: a simple Gaussian term, which accounts for a photoelectron signal fully amplified by the dynode chain, and an exponential term characterized by a parameter $\tau$, which accounts for under-amplified photoelectrons and/or feedback from the dynode photoemission signal. The fraction of the single photoelectron response found to be the under-amplified terms is $p_{E}$. Another expression (Gain-Model B) is simpler, consisting of only the Gaussian term; i.e., the fraction $p_{E}$ in Eq. (1) is fixed to $0$. This assumes that there is no under-amplified or dynode-feedback response in a PMT and that the photoelectron response is perfectly described by Gaussians. Figure 3 shows the charge distribution and fit for an LED calibration run with the Gain-Model A (which has a non-zero fraction $p_{E}$), where $1~{}{\rm count}\cdot{\rm sample}$ corresponds to an output charge of $9.8\times 10^{-15}~{}{\rm C}$. The mean charge for a single photoelectron $g$, defined as $$g=p_{E}\tau+(1-p_{E})x_{m},$$ (2) is approximately $2.0\times 10^{6}~{}e^{-}/{\rm p.e.}$ with a bias voltage of $1570~{}{\rm V}$. The fit with the Gain-Model B (i.e., simple convolution of Gaussian functions) returns a $12\%$ higher gain value than Gain-Model A. This difference is nearly consistent with the result reported in Ref. Alexander2013light . While we do not have enough data to determine which model is more appropriate to describe the PMT response, the Gain-Model A is adopted as baseline and the result from the model is used in the later analysis. This calibration is performed every $12~{}{\rm hours}$ during a data collection period lasting $7~{}{\rm days}$. No significant time dependences of the gain are observed for the two PMTs. 3.2 Signal analysis and selection criteria The analysis of the LAr scintillation signal is performed following a photoncounting algorithm. For each waveform, this algorithm first calculates the baseline from the pre-trigger window; once that baseline is subtracted, all samples above a software threshold are grouped with three neighboring samples ($1$ bin before and $2$ bins after). The software threshold is set based on the baseline noise and is below a typical single photoelectron PMT pulse. The signal detection time is identified as the first sampling time above a threshold of $50\%$ peak amplitude. Detected scintillation light is defined as the integrated charge in the time interval between $-0.04$ and $7.0~{}{\rm\mu s}$. A pulse shape discrimination (PSD) parameter is also defined as the fraction of light detected after $0.1~{}{\rm\mu s}$ of the scintillation signal (termed “Slow/Total”). A set of data quality cuts is applied to remove instrumental effects and event pileups. The selection criteria are as follow. (1) Software imposes a $10~{}{\rm ms}$ veto after events that contain signals greater than $\approx$$2.0\times 10^{4}$ ($\approx$$5.0\times 10^{3}$ ${\rm p.e.}$ for datasets taken with a $\gamma$-ray source with $>$$100~{}{\rm keV}$ ($<$$100~{}{\rm keV}$) its energy. This aims to remove the unstable period of the PMT after outputting a large charge signal. (2) The event has a stable baseline noise and no more than $0.7~{}{\rm p.e.}$ pulses in the pre-trigger window. (3) The sum of the pulses present after the signal-integration window is consistent within about four times its expectation. (4) The event does not occur near the PMT and is more likely to be a LAr scintillation signal than Cherenkov light on the PMT window. The signal asymmetry, defined as $A=(N_{\rm p.e.}^{1}-N_{\rm p.e.}^{2})/(N_{\rm p.e.}^{1}+N_{\rm p.e.}^{2})$ in which $N_{\rm p.e.}^{1}$ and $N_{\rm p.e.}^{2}$ are the observed photoelectron signal in each PMT, is used to evaluate the interacting position. The cut value is selected to contain approximately $99\%$ of the LAr signal. (5) The PSD parameter of the event is consistent with that of the ER. The band of the parameter used in this cut is determined by ${}^{22}{\rm Na}$ data to contain $95\%$ of ER events, as shown in Fig. 4. 3.3 Energy calibration with a Cesium-137 source Energy calibration of the detector is performed using a ${}^{137}{\rm Cs}$ $\gamma$-ray source placed on the outside surface of the cryostat wall. Figure 5 shows the observed scintillation spectrum obtained with the source. The full-absorption peak of the $661.7~{}{\rm keV}$ line of the ${}^{137}{\rm Cs}$ source is fit with a Gaussian with mean $\mu$ and width $\sigma$. The continuous background components around the peak, mainly coming from the Compton edge and degraded tails, are modeled with error and linear functions and added to the fit function. The fit shown in Fig. 5 returns $\chi^{2}/ndf=62.5/56$. It is monitored every day throughout the data collection period, and no significant time dependence is observed. The observed light yield obtained in Fig. 5 contains extra charge from PMT afterpulses and systematic effect from the photoncounting algorithm. A correction for these effects is thus applied to reconstruct the observed light yield per ER energy. This correction is based on an independent study of the PMT response as well as a MC simulation of the LAr signal. It is relatively small, approximately $1\%$ for the ${}^{137}{\rm Cs}$ line and less than $3\%$ for the whole of energy region of interest of this analysis, where the amount of afterpulse is estimated $2\%$–$4\%$ of the photoelectron signal and the algorithm can systematically slightly underestimate the charge signal. The resulting light yield is $12.6\pm 0.3~{}{\rm p.e.}/{\rm keV}$ ($11.1\pm 0.3~{}{\rm p.e.}/{\rm keV}$) when using the PMT calibration obtained by the Gain-Model A (Gain-Model B). The uncertainty includes the estimation of PMT afterpulses, systematic error in the correction, and variance of the light yield between daily calibrations. 4 Measurement of scintillation response with calibration sources 4.1 Sodium-22 source The detector is exposed to $511.0$ and $1274.6~{}{\rm keV}$ $\gamma$-rays using a ${}^{22}{\rm Na}$ radioactive source of approximately $1~{}{\rm MBq}$. The source is placed near the cryostat wall with an NaI(Tl) scintillator ($2~{}{\rm inch}\times 2~{}{\rm inch}$ cylinder). This additional scintillator is located opposite to the site of the cryostat to tag the backwards-traveling $511.0~{}{\rm keV}$ $\gamma$-ray (back-to-back tagging). The distance between the cryostat wall and the source is set to $15~{}{\rm cm}$, and that between the source and the scintillator to $25~{}{\rm cm}$. Figure 6 shows the scintillation spectra obtained with the ${}^{22}{\rm Na}$ source before and after requiring the coincidence detection of the $511.0~{}{\rm keV}$ $\gamma$-ray signal in the NaI(Tl) scintillator. Since the $1274.6~{}{\rm keV}$ $\gamma$-ray is considered to have no angular correlation with back-to-back $\gamma$-rays, the corresponding peak appears only in the former spectrum. Each peak is fit with a Gaussian plus background model function consisting of error and linear functions, as overlaid in Fig. 6. 4.2 Barium-133 source The detector is exposed to $356.0~{}{\rm keV}$ $\gamma$-ray using ${}^{133}{\rm Ba}$ radioactive source with approximately $1~{}{\rm MBq}$. The spectrum obtained with a ${}^{133}{\rm Ba}$ source is shown in Fig. 7. Due to its lower energy and the relatively high intensity of the other $\gamma$-ray lines (such as those at $383.9~{}{\rm keV}$ with a branching ratio (BR) of $8.9\%$ and $302.9~{}{\rm keV}$ with a BR of $18.3\%$, as compared with that at $356.0~{}{\rm keV}$ with a BR of $62.1\%$), the Compton edge is significantly degraded and the full-absorption peak is not individually resolved. Thus, we fit the spectrum with two Gaussians corresponding to $356.0$ and $383.9~{}{\rm keV}$ and the background components as with the fit for other spectra. 4.3 Californium-252 source exploiting $\gamma$-rays through the $(n,n^{\prime}\gamma)$ reaction with fluorine-19 Measurements of the scintillation responses for the $109.8$ and $197.1~{}{\rm keV}$ quasimonoenergetic lines are performed using $\gamma$-rays emitted from the $(n,n^{\prime}\gamma)$ reaction with ${}^{19}{\rm F}$ rogers1974Inelastic . As an external fast neutron source, a ${}^{252}{\rm Cf}$ source with a spontaneous fission rate of approximately $1\times 10^{5}~{}{\rm fission/s}$ is used. The distance between the center of the fiducial volume and the source is set to $90~{}{\rm cm}$. The NaI(Tl) scintillator is placed beside the source to detect associated $\gamma$-rays from the spontaneous fission and to provide timing information. Fast neutrons from ${}^{252}{\rm Cf}$ generate $(n,n^{\prime}\gamma)$ reaction with ${}^{19}{\rm F}$ in the PTFE bulk, producing quasimonoenergetic $\gamma$-rays. Although the intensities of each quasimonoenergetic line depend upon their incident neutron energy, $109.8$ and $197.1~{}{\rm keV}$ lines are major channels for the range of neutron energy from ${}^{252}{\rm Cf}$. Time differences between the NaI(Tl) and fiducial signals (time of flight; ToF) are used to remove $\gamma$-ray events that come directly from the fission. Figure 8 shows the spectrum and fitting results for corresponding peaks. Each peak is fit by a Gaussian plus exponential function. 4.4 Americium-241 source To expose the detector to $59.5~{}{\rm keV}$ $\gamma$-rays, an ${}^{241}{\rm Am}$ source of approximately $40~{}{\rm Bq}$ is used. The radioactive source is deposited on a $100~{}{\rm\mu m}$ thick platinum foil installed at the outer surface of the PTFE bulk. It decays into an excited level of ${}^{237}{\rm Np}$ via $\alpha$-ray transition, and subsequent de-excitation of the ${}^{237}{\rm Np}$ emits $\gamma$-rays with a major line of $59.5~{}{\rm keV}$. The $\alpha$-ray from the primary disintegration is detected by the outer-bath PMTs, allowing the $\gamma$-ray interaction to be proved in the fiducial volume. Figure 9 shows the scintillation spectrum after requiring the detection of $\alpha$-ray signals in the outer region. Due to the relatively low energy of the $\gamma$-ray from ${}^{241}{\rm Am}$ and the passive components between the source and the fiducial volume, the spectrum does not exhibit a clear full-absorption peak. The tail of the peak comes from $\gamma$-rays that reach the fiducial volume via single or multiple scattering from any materials in their path. The scintillation response to a $59.5~{}{\rm keV}$ $\gamma$-ray is evaluated via MC simulation of the experimental setup based on the Geant4 toolkit agostinelli2003geant4 ; allison2006geant4 . The MC simulation takes into account the detector geometry and composition inside the LAr bath, as well as the radioisotope mounting structure. It proceeds by generating $\gamma$-rays from ${}^{241}{\rm Am}$ with a random momentum direction and calculating the energy deposition in the fiducial volume. The fit of the energy deposition spectrum to the ${}^{241}{\rm Am}$ data provides the scintillation efficiency for $59.5~{}{\rm keV}$ $\gamma$-rays. This fit is performed by converting the energy deposition to the observed scintillation yield with a constant scintillation efficiency and Gaussian resolution. The best fit spectrum is also shown in Fig. 9; although the fit is performed only around the $59.5~{}{\rm keV}$ peak ($700$–$900~{}{\rm p.e.}$), reasonable agreement between data and MC is found down to around $400~{}{\rm p.e.}$ 4.5 Argon-37 source Measurement for ERs of a few ${\rm keV}$ is performed using ${}^{37}{\rm Ar}$, which is the second most abundant radioactive isotope in atmospheric argon, comprising an abundance of $\approx$$1.3\times 10^{-20}$ saldanha2019cosmogenic . It decays via electron capture to the ground state of ${}^{37}{\rm Cl}$ with a half-life of $35~{}{\rm days}$, producing x-rays and Auger electrons with a total energy release of $2.82~{}{\rm keV}$ (for L-shell capture), $0.27~{}{\rm keV}$ (for K-shell capture), or $0.02~{}{\rm keV}$ (for M-shell capture) cleveland1998measurement ; TabRad_v7 . Since the production of ${}^{37}{\rm Ar}$ is mainly due to cosmogenic activation of atmospheric argon saldanha2019cosmogenic , it is expected to reach equilibrium and the decay rate of ${}^{37}{\rm Ar}$ in the detector is expected to be constant from the argon filling time to the end of measurement. The data used in this measurement come from approximately $27~{}{\rm hours}$ of detector operation without any external sources. Figure 10 shows the scintillation spectrum for this measurement. The spectrum consists of events that do not have associated scintillation signals in any of the four outer-bath PMTs. The peak around $25~{}{\rm p.e.}$ is attributed to the energy release of $2.82~{}{\rm keV}$ from ${}^{37}{\rm Ar}$. No structures corresponding to the K- or M-shell capture could be seen, probably due to the large amount of random coincidence background and the lack of photostatistics. The spectrum with ${}^{37}{\rm Ar}$ is fitted with the sum of the Gaussian, exponential, and constant terms that describes the signal and low energy background model. The rate of ${}^{37}{\rm Ar}$ decays returned by the fit is approximately $25~{}{\rm mBq/kg}$, which is compatible with literature values saldanha2019cosmogenic ; benetti2017Ar ; agnes2018Low . The goodness of fit for the peak is $\chi^{2}/ndf=82.21/84$. 5 Scintillation yield and energy resolution The upper panel of Fig. 11 summarizes the mean values of the number of detected photoelectron divided by corresponding incident energies, measured by the set of radioactive sources described in the previous section. Nonlinear response on the scintillation yield is seen, which peaks around $200~{}{\rm keV}$. This trend can be attributed to the energy dependence of the ionization electron-ion recombination probability. The Thomas-Imel Box (TIB) model thomas1987recombination and Doke-Birks’ law doke1988let can presumably explain the data, as is the case for the liquid xenon (LXe) scintillation detector szydagis2011nest . For higher energy range, the Doke-Birks’ law is generally applied to deal with relativistic and longer range tracks and to predict the decrease of the probability as the track energy increases (or $dE/dx$ decreases). On the other hand, for lower energy range, typically less than several hundred ${\rm keV}$, it is known that the TIB model is suitable for modeling the data because it is based on the low energy recoiled track whose range is comparable to or shorter than the mean ionization electron-ion thermalization distance. In the following section, we focus on the scintillation response in lower energy range, less than $200~{}{\rm keV}$, and thus employ the TIB model to explain our data. Further study for quantitative evaluation and its modeling of the response will be discussed in Sec. 6. The energy resolution of the detector is also characterized based on the full-absorption peaks and is shown in the lower panel of Fig. 11. The set of points is fit to the function $$\frac{\sigma}{\mu}=\sqrt{\frac{\sigma_{s}^{2}}{E_{\gamma}}+\sigma_{c}^{2}},$$ (3) where $\sigma_{s}$ accounts for stochastic fluctuation and $\sigma_{c}$ accounts for the variance of the mean value of mono-energy deposition. The values are found to be $\sigma_{s}=0.37\pm 0.03$ and $\sigma_{c}=0.021\pm 0.002$, respectively. Several sources are expected to degrade the energy resolution. The contribution of each source are examined and listed in Table 1. Upon examination, it is clear that the statistical fluctuation for the scintillation process is the dominant source of the stochastic term, where it is assumed as Poissonian. Convoluting the terms listed in Table 1 explains approximately $90\%$ of the stochastic term observed in the data; the rest of the term is currently unknown. The constant term ($\sigma_{c}$) is believed to mainly consist of the geometrical effect. We should note that the scintillation process is suggested to have a smaller fluctuation than that would be expected under the Poissonian assumption due to the Fano effect doke1976estimation ; however, this measurement has little sensitivity to this, due to the small light-collection efficiency (roughly $30\%$, which is mainly limited by the PMT QE) and the relatively large degradation of the resolution from other sources. The result is subjected to several systematic uncertainty sources which stem from both the detector response and the analysis procedure, as listed in Table 2. The former is the PMT afterpulse, explored by the PMT response study using both LAr data and a property measurement after the LAr detector operation, and the time stability of the detector complex, monitored by the regular calibrations throughout the data collection period. The later mainly comes from the photoncounting algorithm part and the related correction of the analysis. We assign the size of the correction as the uncertainty. Relatively small uncertainty is attributed to the fit of the full-absorption peak, which is estimated by refitting the peak with a simple Gaussian function. The trigger efficiency is an additional uncertainty source for the ${}^{37}{\rm Ar}$ line analysis. We refit the peak without the correction, and assign the corresponding uncertainty as the variation between these results. The uncertainty of the energy resolution is considered as typically $10\%$ in total, mainly from the fitting modeling. 6 TIB model interpretation on scintillation response As discussed in the previous sections the measurable quantity in this analysis is the scintillation yield, defined as the detected photoelectron signal of the full-absorption peak divided by the corresponding incident energy, $E_{\gamma}$. On the other hand, the scintillation efficiency is expected to depend on the recoiled electron energy, not directly on the selected full-absorbed $\gamma$-ray energy because of multiple scattering inside the fiducial volume. Thus we should note that the definition of the real interaction energy (averaged energy) depends on the geometry and size of the detector. In fact the $\gamma$-ray cross section for argon (upper panel of Fig. 12) xcom and the average number of the interaction points for each calibration point calculated by a Geant4 MC simulation (lower panel of Fig. 12) indicate that our selected full-absorbed events generally have one or more interaction points by Compton scattering before photoelectric absorption inside the fiducial volume. As the incident $\gamma$-ray energy increases, the Compton scattering cross section and thus the number of the interaction increase accordingly. Therefore, for the higher energy points, contributions from lower energy recoiled electron to the total deposited energy have to be considered to determine invariant quantity for interaction energy. In the following, since the number of the interaction point of the ${}^{37}{\rm Ar}$ events is one due to its low energy deposition and decay mode mainly consisting of Auger electrons cleveland1998measurement , all the data points are normalized by the response of the ${}^{37}{\rm Ar}$ and the effect of multiple scattering for each energy point is estimated and corrected by using the Geant4 MC simulation and TIB model: $$\displaystyle n_{ph}$$ $$\displaystyle=$$ $$\displaystyle\frac{E_{\rm er}}{W}(N_{ex}+rN_{i})=\frac{E_{\rm er}}{W}\frac{1+r% }{1+\alpha},$$ (4) $$\displaystyle r$$ $$\displaystyle=$$ $$\displaystyle 1-\frac{1}{N_{i}\varsigma}\ln(1+N_{i}\varsigma),$$ which predicts the number of produced scintillation photons, $n_{ph}$, for recoil electron energy $E_{\rm er}$. In Eq. (4), $W=19.5~{}{\rm eV}$ is the effective work function doke1988let , $N_{ex}$ and $N_{i}$ are the numbers of produced excitons or electron-ion pairs, respectively, $\alpha=0.21$ is the initial ratio of the average of $N_{ex}$ to $N_{i}$ miyajima1974average , and $\varsigma$ is a constant parameter of the model. Parameter $\varsigma$ in the TIB model is evaluated by fitting the measured points below $200~{}{\rm keV}$ to the MC simulation as shown in Fig. 13. The fit returns a value of $\varsigma=0.07^{+0.03}_{-0.02}$, where the uncertainty includes both statistical and systematical terms. The shift of the scintillation efficiency for the same energy of full-absorbed $\gamma$-ray (red line of Fig. 13) and single electron track (green line) is estimated as $2\%$–$3\%$ around $100~{}{\rm keV}$ in this detector. The TIB model reasonably explains our data up to $200~{}{\rm keV}$ by taking into multiple scattering effect account. 7 Conclusion The energy dependence of the scintillation efficiency for electronic recoils ranging from $2.82$ to $1274.6~{}{\rm keV}$ is measured using a single-phase high light yield detector exposed to a variety of calibration sources. The scintillation detector with the TPB wavelength shifter is immersed in purified LAr and yields $12.6\pm 0.3~{}{\rm p.e.}/{\rm keV}$ ($11.1\pm 0.3~{}{\rm p.e.}/{\rm keV}$) based on the PMT calibration assuming a PMT single photoelectron response model with an additional exponential term (with only Gaussian term). The LAr scintillation response is investigated by the full-absorption peaks of external $\gamma$-ray sources, as well as an ${}^{37}{\rm Ar}$ source with $2.82~{}{\rm keV}$ line. These measurements demonstrate that the efficiency peaks at around $200~{}{\rm keV}$ and energy resolution is found to be less than $3\%$ for the $661.7~{}{\rm keV}$ $\gamma$-ray. In order to provide an invariant property of the LAr scintillation response, we investigate the scintillation quenching by analogy with the LXe scintillation detector response, where the electron-ion recombination probability is attributed to the energy dependence of the response. For the energy below $200~{}{\rm keV}$, the TIB model provides a good description of the observed scintillation quenching by the parameter $\varsigma$, and we obtain it as $\varsigma=0.07^{+0.03}_{-0.02}$. This work is primary intended for use in the direct WIMP dark matter search. In this field, low energy electronic background is one of the most severe sources disturbing the lower energy threshold, hence reducing WIMP sensitivity. The result presented here makes use of the precise estimation of background contamination in the low energy region and suppression of the systematic uncertainty. In addition, to our knowledge, the energy resolution of the LAr scintillation detector for the ${\rm keV}$ to ${\rm MeV}$ range has not been explicitly discussed thus far. The measurement in this work provides useful information for applying the LAr detector to other fields, such as astrophysical MeV Gamma-ray observation aramaki2020dual . The results presented here would help with the design, operation, and analysis of a wide variety of astrophysical and particle physics experiments in the near future to enhance their physical reach. Acknowledgments This work is a part of the outcome of research performed under the Waseda University Research Institute for Science and Engineering (project number 2016A-507), supported by JSPS Grant-in-Aid for Scientific Research on Innovative Areas (15H01038/17H05204), Grant-in-Aid for Scientific Research(B) (18H01234), and Grant-in-Aid for JSPS Research Fellow (18J13018). The authors would like to thank the Material Characterization Central Laboratory at Waseda University for granting us access to their stylus profiler. The authors acknowledge the support of the Institute for Advanced Theoretical and Experimental Physics, Waseda University. References (1) P. Agnes et al., Phys. Rev. D 98 (2018) 102006. (2) P.-A. Amaudruz et al., Phys. Rev. Lett. 121 (2018) 071801. (3) A. Hitachi et al., Phys. Rev. B 27 (1983) 5279–5285. (4) T. Heindl et al., Europhysics Letters 91 (6) (2010) 62002. (5) W. M. Burton, B. A. Powell, Appl. Opt. 12 (1) (1973) 87–89. (6) G. Porter, M. R. 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Spectra of symmetric powers of graphs and the Weisfeiler-Lehman refinements Afredo Alzaga Departamento de Matemática Universidad Nacional del Sur (UNS) Bahía Blanca, Argentina    Rodrigo Iglesias Departamento de Matemática Universidad Nacional del Sur (UNS) Bahía Blanca, Argentina    Ricardo Pignol Departamento de Matemática Universidad Nacional del Sur (UNS) Bahía Blanca, Argentina Abstract The $k$-th power of a $n$-vertex graph $X$ is the iterated cartesian product of $X$ with itself. The $k$-th symmetric power of $X$ is the quotient graph of certain subgraph of its $k$-th power by the natural action of the symmetric group. It is natural to ask if the spectrum of the $k$-th power –or the spectrum of the $k$-th symmetric power– is a complete graph invariant for small values of $k$, for example, for $k=O(1)$ or $k=O(\log n)$. In this paper, we answer this question in the negative: we prove that if the well known $2k$-dimensional Weisfeiler-Lehman method fails to distinguish two given graphs, then their $k$-th powers –and their $k$-th symmetric powers– are cospectral. As it is well known, there are pairs of non-isomorphic $n$-vertex graphs which are not distinguished by the $k$-dim WL method, even for $k=\Omega(n)$. In particular, this shows that for each $k$, there are pairs of non-isomorphic $n$-vertex graphs with cospectral $k$-th (symmetric) powers. 1 Introduction Many fundamental graph invariants arise from the study of random walks of a particle on a graph. Most of these invariants can be described in terms of the spectrum of the adjacency or the Laplacian matrix. Since the graph spectrum fails to distinguish many non-isomorphic graphs, it is interesting to study the properties of walks (or quantum walks) of $k$ particles, as a means to construct more powerful invariants. This led Audenaert et al [2] to define the $k$-th symmetric power $X^{\{k\}}$ of a graph $X$: each vertex of $X^{\{k\}}$ represents a $k$-subset of vertices of $X$, and two $k$-subsets are joined if and only if their symmetric difference is an edge of $X$. They show that the spectra of these graphs is a family of invariants stronger than the ordinary graph spectra. For $k=2$, they provide examples of cospectral graphs $X$ and $Y$ such that $X^{\{2\}}$ and $Y^{\{2\}}$ are not cospectral. On the other hand, they prove that if $X$ and $Y$ are strongly-regular cospectral graphs then $X^{\{2\}}$ and $Y^{\{2\}}$ are cospectral. For $k=3$, the authors reported computational evidence suggesting that the spectra of the symmetric cube may be a strong invariant. They did not find any pair of non-isomorphic graphs with cospectral $3$-symmetric powers, upon inspection of all strongly regular graphs of up to $36$ vertices. In this paper we prove that for each $k$ there are pairs of non-isomorphic graphs such that their $k$-th symmetric powers are cospectral by showing how these invariants are related to the well known $k$-dimensional Weisfeiler-Lehman (WL) algorithm. The automorphism group of the graph acts on the set of $k$-tuples of vertices. The $k$-WL method is a combinatorial algorithm that attempts to find the associated orbit partition (see for example [3], [5]). It starts by classifying the $k$-tuples according to the isomorphism type of their induced graphs, then an iteration is performed attaching to the previous color of a $k$-tuple, the multiset of colors of the the neighboring $k$-tuples. In this way, the partition of the $k$-tuples is refined in each step until a stable partition is reached. The multiset of colors of the stable partition is a graph invariant. Our main result is the following theorem. Theorem 1. If the $2k$-dim Weisfeiler-Lehman algorithm fails to distinguish two given graphs, then their $k$-th symmetric powers are cospectral. In fact, the result remains true if we consider $k$-th powers of graphs (associated to walks of $k$ labelled particles), instead of symmetric powers. In [3], Cai, Immerman and Fürer showed how to construct pairs of non-isomorphic $n$-vertex graphs which are not distinguished by the $k$-WL method, even for $k=\Omega(n)$. Then, our result implies that Theorem 2. If we require the $k$-th symmetric power spectrum to determine all $n$-vertex graphs then, necessarily, $k=\Omega(n)$. Nevertheless, the spectrum of the $k$-th power of a graph is a strong invariant with remarkable computational features. Since it is determined by the characteristic polynomial of a matrix of $0$s and $1$s of polynomial size (for fixed $k$), it can be computed in polylogarithmic time by a randomized parallel algorithm. This contrasts with the inherently sequential nature of the $k$-dim WL algorithm; in [4], Grohe proved that finding the $k$-dim WL stable partition is a $P$-complete problem ($k\geq 2$). This suggest that the $2k$-WL method ($k\geq 1$) is strictly more powerful than the $k$-th power (or $k$-th symmetric power) spectrum, since the complexity class $RNC$ is expected to be strictly contained in $P$. Besides power graph spectra, there are other families of graph invariants in the literature for which it is not known whether they distinguish any pair of non-isomorphic graphs or not. As it turns out, the WL-refinements provide a natural benchmark to compare other graph invariants and it is reasonable to expect that arguments of the kind we use in this work would show the limitations of some of them. The paper is organized as follows. In Section 2 we define the $k$-th power $X^{k}$ and the $k$-th symmetric power $X^{\{k\}}$ of a graph $X$. In Section 3 we recall the general notion of quotient of a graph by the action of a group, and we describe the $k$-th symmetric power as a quotient of the restricted $k$-th power $X^{(k)}$. For later use, we prove some formulas concerning the walk generating function of quotient graphs. In Section 4 we define precisely the $k$-Weisfeiler-Lehman algorithm. The heart of the proof of Theorem $1$ is in Section 6. Essentially, we show that the $2k$-WL method is stronger than the spectra of the $k$-th power $X^{k}$. Since the idea of the proof is easier to exhibit in the case $k=1$, we write this special case separately in Section 5. Finally, the proof of Theorem $1$ is given in Section 7, by passing to the quotient $X^{\{k\}}$. In order to achieve this, we exploit the structure of the set of $k$-tuples and the formulas for quotient graphs presented in Section 3. 2 Powers of graphs In this section we present the notion of the $k$-th symmetric power of a graph, as introduced in [2], and some other related constructions. Throught the paper, a graph $G$ is a finite set $V$ of vertices toghether with a set $E$ of unordered pairs $(v,w)$ of vertices with $v\neq w$. We denote by $A_{G}$ the adjacency matrix of G. Since we do not assume an order on $V$, we consider $A_{G}$ as a function $A_{G}:V\times V\rightarrow\mathbb{Z}$, defined by $A_{G}(v,w)=1$ if $(v,w)\in E$, and $A_{G}(v,w)=0$ otherwise. A k-tuple $(i_{1}...i_{k})$ of vertices is a function from $\{1,...,k\}$ to $V$. Let $\mathcal{U}_{k}$ be the set of all $k$-tuples and let $\mathcal{D}_{k}\subset\mathcal{U}_{k}$ denote the set of those $k$-tuples of pairwise distinct vertices. The symmetric group $S_{k}$ acts naturally on $\mathcal{D}_{k}$ by $\sigma(i_{1}...i_{k})=(i_{\sigma^{-1}(1)}...i_{\sigma^{-1}(k)}),$ for $\sigma\in S_{k}$. The orbits are identified with the k-subsets of vertices. The $k$-th symmetric power of $G$, denoted by $G^{\{k\}}$, has the $k$-subsets of $V$ as its vertices; two $k$-subsets are adjacent if their symmetric difference –elements in their union but not in their intersection– is an edge of $G$. The picture behind this construction is borrowed from the physical realm: start with $k$ undistinguishable particles occupying $k$ different vertices of $G$ and consider the dynamics of a walk through the graph in which, for each step, any single particle is allowed to move to an unoccupied adjacent vertex. In this way, a $k$-walk on $G$ corresponds to a $1$-walk on $G^{\{k\}}$. The connection between symmetric powers and quantum mechanics exchange Hamiltonians is further explored in [2]. Likewise, one can define the cartesian product $G\times H$ of two graphs as follows: $$A_{G\times H}(i_{1}i_{2},j_{1}j_{2})=\left\{\begin{array}[]{ll}1&\mbox{if }\ A% _{G}(i_{1},j_{1})=1\ \mbox{and }\ i_{2}=j_{2}\\ &\mbox{or else }\ A_{G}(i_{2},j_{2})=1\ \mbox{and }\ i_{1}=j_{1}\\ 0&\mbox{otherwise}\end{array}\right.$$ The $k$-th power $G^{k}$ of a graph is defined as the iterated cartesian product of $G$ with itself. The set of its vertices is $\mathcal{U}_{k}$ and its adjacency matrix $A_{G^{k}}$ is given by: $$A_{G^{k}}(i_{1}i_{2}\ldots i_{k},j_{1}j_{2}\ldots j_{k})=\left\{\begin{array}[% ]{ll}1&\mbox{if there exists }\ u\in\{1,\ldots,k\}\ \mbox{such that }\\ &A_{G}(i_{u},j_{u})=1\ \mbox{and }\ i_{l}=j_{l}\ \mbox{for }\ l\neq u\\ 0&\mbox{otherwise}\end{array}\right.$$ In the physical cartoon of the particles, the $k$-th power correspond to the situation in which the $k$ particles are labeled, and more than one particle is allowed to occupy the same vertex at the same time. Given a graph $G$, the walk generating function of $G$ is the power series $$\sum_{r=0}^{\infty}{t^{r}(A_{G})^{r}}$$ The coefficient of $t^{r}$ in the $(i,j)$-entry counts the number of paths of length $r$ from the vertex $i$ to the vertex $j$. See [2] for further properties. The trace of the walk generating function is a graph invariant, and we denote it by $$F(G,t)=Tr\sum_{r=0}^{\infty}{t^{r}(A_{G})^{r}}$$ Since the spectrum of two matrices $A$ and $B$ coincides if and only if $Tr(A^{r})=Tr(B^{r})$ for all $r$, two graphs $G$ and $H$ are cospectral if and only if $F(G,t)=F(H,t)$. In particular, they cannot be distinguished by the spectrum of their $k$-th symmetric powers if and only if $F(G^{\{k\}},t)=F(H^{\{k\}},t)$. 3 Quotient graphs The $k$-th symmetric power $G^{\{k\}}$ can be constructed from $G^{k}$ in two steps. First, we cut $G^{k}$, deleting all those vertices which are not in $\mathcal{D}_{k}$. In this way we obtain the restricted $k$-th power, denoted by $G^{(k)}$, defined as the subgraph of $G^{\{k\}}$ whose vertices are the $k$-tuples in $\mathcal{D}_{k}$. Second, we take the quotient of $G^{\{k\}}$ by the natural action of $S_{k}$ on the restricted $k$-th power $G^{(k)}$. Let us give the general definition of a quotient graph and discuss some properties. Given a graph $X$ and a group $\Gamma$ acting on $X$ by automorphisms, the quotient $X/\Gamma$ is a directed graph, in general with multiple edges and loops, defined as follows. The vertices of $X/\Gamma$ are the orbits of the vertices of $X$, and given two orbits $U$ and $W$, there are as many arrows from $U$ to $W$ as edges in $X$ connecting a fixed element $u\in U$ with vertices in $W$. We are interested in the case where this quotient has no loops and no multiple edges; we say that the quotient $X/\Gamma$ is simply laced if 1. $(u,v)\in E$ implies that $u$ and $v$ are not in the same orbit. 2. $(u,v)\in E$ and $(u,w)\in E$ implies that $v$ and $w$ are not in the same orbit. If $X/\Gamma$ is simply laced, we can consider it an ordinary graph, where $(U,W)$ is an edge if and only if there is an arrow in $X/\Gamma$ connecting them. In the simply laced case, every path on $X/\Gamma$ can be lifted to an essencially unique path on $X$. This fact simplifies the task of path-counting, and allows to derive a simple formula for the walk generating function of a quotient graph. We apply it to the symmetric power $G^{\{k\}}$ to obtain a formula that will be useful later. Proposition 1. Let $X$ be a graph, $X/\Gamma$ a simply laced quotient, and let $U$ and $W$ be two orbits. Then, the $r$-th power of the adjacency matrix of $X/\Gamma$ is given by $$A_{X/\Gamma}^{r}(U,W)=\frac{1}{\left|U\right|}\sum_{u\in U}\sum_{w\in W}A_{X}^% {r}(u,w)$$ Proof: The entry $A_{X/\Gamma}^{r}(U,W)$ equals the number of paths of length $r$ on $X/\Gamma$ from $U$ to $W$. Fix an element $u_{0}\in U$ and let $V_{0},V_{1},V_{2},...,V_{r}$ be a path of length $r$ on $X/\Gamma$, with $U=V_{0}$ and $V_{r}=W$. Since there is at most one edge in $X$ connecting a vertex in $X$ to a vertex in a different orbit, there is a unique path $v_{0},v_{1},v_{2},...,v_{r}$ in $G$ such that $v_{0}=u_{0}$ and $v_{j}\in V_{j}$ for $0\leq j\leq r$. Then, $$A_{X/\Gamma}^{r}(U,W)=\sum_{w\in W}A_{X}^{r}(u_{0},w)$$ The set of paths of length $r$ from $u_{0}$ to $W$ is carried bijectively to the set of paths from any $u\in U$ to $W$ via some automorphism in $\Gamma$. Then, the sum $$\sum_{w\in W}A_{X}^{r}(u,w)$$ does not depend on $u$, and this proves the formula of the proposition. ∎ Observe that this formula implies that if $X/\Gamma$ is a connected, simply laced quotient, then all the orbits have the same size. Let $M_{X/\Gamma}$ be the matrix with rows and columns indexed by the vertices of $X$, defined by $$M_{X/\Gamma}(v,w)=\left\{\begin{array}[]{ll}\left|U\right|&\mbox{if }\ v\ % \mbox{and }\ w\ \mbox{are in the same orbit}\ U\\ \par 0&\mbox{otherwise}\end{array}\right.$$ From Prop. 1 it follows: Proposition 2. Let $X/\Gamma$ be simply laced quotient, and let $M_{X/\Gamma}$ be defined as above. Then, $$Tr(A_{X/\Gamma}^{r})=Tr(A_{X}^{r}M_{X/\Gamma}).$$ Now we set $X=G^{(k)}$ and $\Gamma=S_{k}$, acting in the natural way on $G^{(k)}$. The quotient $G^{(k)}/S_{k}$ is isomorphic to the $k$-th symmetric power $G^{\{k\}}$, and it is easily seen to be a simply laced quotient. In this case, the matrix $M_{X/\Gamma}$ is the matrix $M_{k}$, with rows and columns indexed by $k$-tuples in $\mathcal{D}_{k}$, given by $$M_{k}(i_{1}...i_{k},j_{1}...j_{k})=\left\{\begin{array}[]{ll}k!&\mbox{if }\ \{% i_{1}...i_{k}\}\ \mbox{and }\ \{j_{1}...j_{k}\}\ \mbox{are equal as sets}\\ 0&\mbox{otherwise}\end{array}\right.$$ From Prop. 2 we obtain: Proposition 3. Let $G^{(k)}$ and $G^{\{k\}}$ be the restricted $k$-th power and the $k$-th symmetric power of a graph $G$, respectively. Let $M_{k}$ be the matrix defined as above. Then, $$Tr(A_{G^{\{k\}}}^{r})=Tr(A_{G^{(k)}}^{r}M_{k})$$ 4 The Weisfeiler-Lehman algorithm A natural approach to graph isomorphism testing is to develop algorithms to compute the vertex orbits of the automorphism group of a graph. In particular, if the orbits of the union of two graphs are known, one can decide if there is an isomorphism between them. As a first approximation to the orbit partition of a given graph, one can assign different colors to the vertices according to their degrees. We can refine this partition iteratively, by attaching to the previous color of a vertex, the multiset of colors of its neighbors. After at most $n=\left|V\right|$ steps, the partition stabilizes. For most graphs, this method distinguishes all the vertices [1], but it does not work in general. For example, it clearly fails if the vertex degrees are all equal to each other. A more powerful method, generalizing the previous one, is obtained by coloring the $k$-tuples of vertices (single vertices are implicit as $k$ repetitions of the same vertex). We start classifying the $k$-tuples according to the isomorphism type of their induced labelled graphs. Next, we apply an iteration attaching to the previous color of a $k$-tuple, the multiset of colors of the the neighboring $k$-tuples. This is the so called $k$-dimensional Weisfeiler-Lehman refinement. For fixed $k\geq 1$ the partition of the $k$-tuples is no longer refined after $n^{k}$ steps, so the algorithm runs in polynomial time. This type of combinatorial methods have been investigated since the seventies, and for some time there was hope in solving the graph isomorphism problem provided that $k=O(\log n)$ or $k=O(1)$. In [3], Cai, Immermann and Fuhrer, disposed of such conjectures; they proved that, for large $n$, $k$ must be greater than $cn$ for some constant $c$, if we require the $k$-WL refinement to reach the orbit partition of any $n$-vertex graph. Despite of this limitation, the method works with $k$ constant when restricted to some important families, such as planar or bounded genus graphs [5]. Let us define the $k$-WL algorithm more precisely. Let $G$ be a graph and $V$ its set of vertices. We define an equivalence relation on the set of $k$-tuples: we say that $(i_{1}\dots i_{k})$ and $(j_{1}\dots j_{k})$ are equivalent if 1. $i_{l}=i_{l^{\prime}}$ if and only if $j_{l}=j_{l^{\prime}}$ 2. $(i_{l},i_{l^{\prime}})\in E$ if and only if $(j_{l},j_{l^{\prime}})\in E$ We define the type $\>tp\>(i_{1}\dots i_{k})$ of a $k$-tuple as its equivalence class. Let $S_{1}$ be the set of all different types of $k$-tuples. This is the initial set of colors. We define the set $S$ of colors by $$S=\bigcup_{k=1}^{\infty}S_{k}$$ where elements of $S_{r+1}$ are finite sequences or finite multisets of elements of $\bigcup_{k=0}^{r}S_{k}$. In practice, it suffices to work with as many colors as $k$-tuples: in order to preserve the length of their names, the colors can be relabelled in each round (using a rule not depending on $G$). Nevertheless, this relabelling plays no role in our arguments. We denote the color assignment of the $k$-WL iteration in its $r$-th round, applied to the graph $G$, by $W_{G,k}^{r}:\mathcal{U}_{k}\rightarrow S$. Evaluated at the $k$-tuple $(i_{1}\dots i_{k})$ it gives the color $W_{G,k}^{r}(i_{1}\dots i_{k})\in S$. Initially, for $r=1$, it is defined by $$W_{G,k}^{1}(i_{1}\dots i_{k})=\>tp\>(i_{1}\dots i_{k}).$$ The iteration is given by $$W_{G,k}^{r+1}(i_{1}\dots i_{k})=\sum_{m\in V}\left(\>tp\>(i_{1}\dots i_{k}\>m)% ,S_{G,k}^{r}(i_{1}\dots i_{k}\>m)\right)$$ (1) where $S_{G,k}^{r}(i_{1}\dots i_{k}\>m)$ is the sequence $$\left(W_{G,k}^{r}(i_{1}\dots\>m),\dots,W_{G,k}^{r}(i_{1}\dots m\dots i_{k}),% \dots,W_{G,k}^{r}(m\dots i_{k})\right).$$ The summation symbol in (1) must be interpreted as a formal sum, so that it denotes a multiset. For example, if $x_{1}=x_{3}=x_{4}=a$ and $x_{2}=x_{5}=b$, then $\sum_{i=1}^{5}x_{i}$ is the multiset $\{a,a,a,b,b\}$. For each round, a certain number of different colors is attained. We say that the coloring scheme stabilizes in the $r$-th round if the number of different colors does not increase in the $r+1$-th iteration. In order to compare the invariant $F(G^{\{k\}},t)$ with the $k$-Weisfeiler-Lehman refinement, we define a graph invariant $I_{G,k}$ which captures the result of the $k$-WL coloring and, at the same time, it is a combinatorial analogue of $F(G^{k},t)$. For each round $r$, we collect all the resulting colors in the multiset $$M_{G,\;k}^{r}=\sum_{(i_{1}...\;i_{k})\;\in\mathcal{U}_{k}}W_{G,\;k}^{r}(i_{1}.% ..i_{k})$$ Then we define the formal power series $$I_{G,\;k}(t)=\sum_{r=0}^{\infty}t^{r}M_{G,\;k}^{r}$$ The following technical proposition will be used later. Proposition 4. Let $G$ and $H$ be two graphs with $n$ vertices. Then, $I_{G,\;k}(t)=I_{H,\;k}(t)$ if and only if there is a permutation $\sigma$ of the set of $k$-tuples such that $W_{G,\;k}^{r}(i_{1}...i_{k})=W_{H,\;k}^{r}(\sigma(i_{1}...i_{k}))$ for all $r\geq 1$. In particular, $$\>tp\>(i_{1}...i_{k})=\>tp\>(\sigma(i_{1}...i_{k})).$$ Proof: The if part is immediate. Conversely, assume $I_{G,\;k}(t)=I_{H,\;k}(t)$. The coefficient of $t^{r}$, when $r=n^{k}$, implies the existence of a permutation $\sigma$ of the set of $k$-tuples such that $$W_{G,\;k}^{n^{k}}(i_{1}...i_{k})=W_{H,\;k}^{n^{k}}(\sigma(i_{1}...i_{k}))$$ (2) Whenever Eq. 2 holds for some particular round $r_{0}$, it holds for all $1\leq r\leq r_{0}$. Then, $$W_{G,\;k}^{r}(i_{1}...i_{k})=W_{H,\;k}^{r}(\sigma(i_{1}...i_{k}))$$ (3) for all $1\leq r\leq n^{k}$. In addition, since the $WL$ refinement stabilizes after the $n^{k}$ round, we see that Eq. 3 is true for $r\geq n^{k}$. The last assertion is obtained by setting $r=1$ in Eq. 3. ∎ 5 Graph spectrum is weaker than the $2$-WL refinement As a warm-up we start by showing that the spectrum of a graph is a weaker invariant than the 2-Weisfeiler-Lehman coloring algorithm. This case displays the essential ingredients of the proof for arbitrary $k$. Theorem 3. Let $G$ and $H$ be two graphs with adjacency matrices $A_{G}$ and $A_{H}$, respectively. If $W_{G,2}^{r}(i,j)=W_{H,2}^{r}(p,q)$ then $A_{G}^{r}(i,j)=A_{H}^{r}(p,q).$ Proof: We use induction on the number of rounds $r$. The base case ($r=1$) is trivial. Assume the statement is valid for $r$, and suppose that $$W_{G,2}^{r+1}(i,j)=W_{H,2}^{r+1}(p,q).$$ Then, by the definition of the WL coloring, $$\sum_{m}(tp_{G}(i,j,m),W_{G,2}^{r}(i,m),W_{G,2}^{r}(m,j))=\sum_{m}(tp_{H}(p,q,% m),W_{H,2}^{r+1}(p,m),W_{H,2}^{r+1}(m,q)).$$ This is an equality of multisets. This means that there exists a permutation $\sigma$ of $\{1,2,...,n\}$ such that $$\left\{\begin{array}[]{cccc}&tp_{G}(i,j,m)&=&tp_{H}(p,q,\sigma(m)),\\ &W_{G,2}^{r}(i,m)&=&W_{H,2}^{r}(p,\sigma(m)),\\ &W_{G,2}^{r}(m,j)&=&W_{H,2}^{r}(\sigma(m),q)).\\ \end{array}\right.$$ By the induction hypothesis, this implies $$\left\{\begin{array}[]{ll}&A_{G}(i,m)=A_{H}(p,\sigma(m)),\;\;A_{G}(m,j)=A_{H}(% \sigma(m),q),\\ &A^{r}_{G}(i,m)=A^{r}_{H}(p,\sigma(m))\\ &A^{r}_{G}(m,j)=A^{r}_{H}(\sigma(m),q)\\ \end{array}.\right.$$ Summing over $m$, we have $$\sum_{m}A_{G}(i,m)A^{r}_{G}(m,j)=\sum_{m}A_{H}(p,m)A^{r}_{H}(m,q),$$ that is, $A^{r+1}_{G}(i,j)=A^{r+1}_{H}(p,q)$ ∎ Theorem 4. Let $G$ and $H$ be two graphs. If $I_{G,\;2}(t)=I_{H,\;2}(t)$, then $G$ and $H$ are cospectral. Proof: Assume $I_{G,\;2}(t)=I_{H,\;2}(t)$. By Prop. 4, there is a permutation $\sigma$ of the set of $2$-tuples such that, for every $2$-tuple $ij$, $$W_{G,\;2}^{r}(ij)=W_{H,\;2}^{r}(\sigma(ij))$$ for $r\geq 1$. When $r=1$, this is $$\>tp\>(ij)=\>tp\>(\sigma(ij)).$$ In particular, $\sigma$ sends the diagonal of $W_{G,\;2}^{r}$ to the diagonal of $W_{H,\;2}^{r}$, that is, $$\sigma(ii)=pp$$ for some element $p$. Then, collecting all the colors in the diagonal, we have $$\sum_{i}W_{G,\;2}^{r}(ii)=\sum_{i}W_{H,\;2}^{r}(\sigma(i)\sigma(i))$$ By Theorem 3, this implies $$\sum_{i}A^{r}_{G}(i,i)=\sum_{i}A^{r}_{H}(\sigma(i),\sigma(i))$$ that is, $TrA^{r}_{G}=TrA^{r}_{H}$ for $r\geq 1$. Then, $F(G,t)=F(H,t)$ and this means that $G$ and $H$ are cospectral. ∎ 6 Spectra of $k$-th powers For each round $r$, we think of the $2k$-WL coloring as a matrix of colors: the rows and columns are indexed by $k$-tuples, with the color $W_{G,\;k}^{r}(i_{1}...i_{k}j_{1}...j_{k})$ in the entry $(i_{1}...i_{k},j_{1}...j_{k})$. Theorem 5. Let $G^{k}$ and $H^{k}$ be the $k$-th powers of two graphs $G$ and $H$ respectively. Let $A_{G^{k}}^{r}$ and $A_{H^{k}}^{r}$ be the $r$-th powers of their adjacency matrices. If $$W_{G,2k}^{r}(i_{1}\dots i_{k}\ j_{1}\dots j_{k})=W_{H,2k}^{r}(p_{1}\dots p_{k}% \ q_{1}\dots q_{k}),$$ then $$A_{G^{k}}^{r}(i_{1}\dots i_{k},j_{1}\dots j_{k})=A_{H^{k}}^{r}(p_{1}\dots p_{k% },q_{1}\dots q_{k}).$$ Proof: The proof goes along the lines of Theorem 3. Let $r=1$. Suppose that $$A_{G^{k}}(i_{1}\dots i_{k},j_{1}\dots j_{k})=1.$$ Then $i_{l}=j_{l}$ for all $l$ except for a unique value $l_{0}$, for which $A_{G}(i_{l_{0}},j_{l_{0}})=1$. By hypothesis, $$W_{G,2k}^{1}(i_{1}\dots i_{k}\ j_{1}\dots j_{k})=W_{H,2k}^{1}(p_{1}\dots p_{k}% \ q_{1}\dots q_{k}),$$ that is, $$tp(i_{1}\dots i_{k}\ j_{1}\dots j_{k})=tp(p_{1}\dots p_{k}\ q_{1}\dots q_{k}).$$ By the definition of type, this implies that $p_{l}=q_{l}$ for $l\neq l_{0}$ and $A_{H}(p_{l_{0}},q_{l_{0}})=1$. Then $A_{H^{k}}(p_{1}\dots p_{k},q_{1}\dots q_{k})=1$. The argument can be reversed, proving that $$A_{G^{k}}(i_{1}\dots i_{k},j_{1}\dots j_{k})=A_{H^{k}}(p_{1}\dots p_{k},q_{1}% \dots q_{k}).$$ This prove the case $r=1$. Now assume the statement is valid for $r$, and suppose that $$W_{G,2k}^{r+1}(i_{1}\dots i_{k}\ j_{1}\dots j_{k})=W_{H,2k}^{r+1}(p_{1}\dots p% _{k}\ q_{1}\dots q_{k}).$$ By the definition of the WL coloring, $$\sum_{m\in V}\left(tp_{G}(i_{1}\dots i_{k}\ j_{1}\dots j_{k}\ m),S_{G,2k}^{r}(% i_{1}\dots i_{k}\ j_{1}\dots j_{k}\ m)\right)=$$ $$=\sum_{m\in V}\left(tp_{H}(p_{1}\dots p_{k}\ q_{1}\dots q_{k}\ m),S_{H,2k}^{r}% (p_{1}\dots p_{k}\ q_{1}\dots q_{k}\ m)\right).$$ Therefore there exists a permutation $\sigma$ of $\{1,2,...,n\}$ such that $$\left\{\begin{array}[]{cccc}&tp_{G}(i_{1}\dots i_{k}\ j_{1}\dots j_{k}\ m)&=&% tp_{H}(p_{1}\dots p_{k}\ q_{1}\dots q_{k}\ \sigma(m)),\\ &W_{G,2k}^{r}(i_{1}\dots i_{k}\ j_{1}\dots j_{k-1}\ m)&=&W_{H,2k}^{r}(p_{1}% \dots p_{k}\ q_{1}\dots q_{k-1},\sigma(m)),\\ &&\dots&\\ &W_{G,2k}^{r}(m\ i_{2}\dots i_{k}\ j_{1}\dots j_{k})&=&W_{H,2k}^{r}(\sigma(m)% \ p_{2}\dots p_{k}\ q_{1}\dots q_{k}).\\ \end{array}\right.$$ The induction hypothesis implies $$\left\{\begin{array}[]{ll}&A_{G}(i_{t},m)=A_{G}(p_{t},\sigma(m))\quad\quad% \mbox{for }t=1,\dots,k.\\ &A^{r}_{G^{k}}(i_{1}\dots m\dots i_{k},j_{1}\dots j_{k})=A^{r}_{H^{k}}(p_{1}% \dots\sigma(m)\dots p_{k},q_{1}\dots q_{k}).\end{array}\right.$$ Our goal is to show that $$A^{r+1}_{G^{k}}(i_{1}...i_{k},j_{1}...j_{k})=A^{r+1}_{H^{k}}(p_{1}...p_{k},q_{% 1}...q_{k}).$$ We have $$A^{r+1}_{G^{k}}(i_{1}...i_{k},j_{1}...j_{k})=\sum_{s_{1}...s_{k}}A_{G^{k}}(i_{% 1}...i_{k},s_{1}...s_{k})A^{r}_{G^{k}}(s_{1}...s_{k},j_{1}...j_{k})$$ (4) Observe that $A_{G^{k}}(i_{1}...i_{k},s_{1}...s_{k})=0$ unless there exists an index $t$ such that $A_{G}(i_{t},s_{t})=1$ and $i_{l}=s_{l}$ for all $l\neq t$. Hence $$A^{r+1}_{G^{k}}(i_{1}\dots i_{k},j_{1}\dots j_{k})=\sum_{m\in V}\sum_{t=1}^{k}% A_{G}(i_{t},m)A^{r}_{G^{k}}(i_{1}\dots m\dots i_{k},j_{1}\dots j_{k})$$ $$=\sum_{m\in V}\sum_{t=1}^{k}A_{H}(p_{t},\sigma(m))A^{r}_{H^{k}}(p_{1}\dots% \sigma(m)\dots p_{k},q_{1}\dots q_{k})=A^{r+1}_{H^{k}}(p_{1}\dots p_{k},q_{1}% \dots q_{k})$$ ∎ Theorem 6. Let $G$ and $H$ be two graphs. If $I_{G,\;2k}(t)=I_{H,\;2k}(t)$, then $$F(G^{k},t)=F(H^{k},t).$$ In other words, if the $2k$-th WL refinement cannot distinguish $G$ from $H$, then their $k$-th powers are cospectral. Proof: Assume $I_{G,\;2k}(t)=I_{H,\;2k}(t)$. By Prop. 4, there is a permutation $\sigma$ of the set of $2k$-tuples such that, for every $2k$-tuple $i_{1}...i_{k}j_{1}...j_{k}$, $$W_{G,\;2k}^{r}(i_{1}...i_{k}j_{1}...j_{k})=W_{H,\;2k}^{r}(\sigma(i_{1}...i_{k}% j_{1}...j_{k}))$$ for $r\geq 1$. When $r=1$, this is $$\>tp\>(i_{1}...i_{k}j_{1}...j_{k})=\>tp\>(\sigma(i_{1}...i_{k}j_{1}...j_{k}).$$ In particular, $\sigma$ sends the diagonal of $W_{G,\;2k}^{r}$ to the diagonal of $W_{H,\;2k}^{r}$, that is, $$\sigma(i_{1}...i_{k}i_{1}...i_{k})=p_{1}...p_{k}p_{1}...p_{k}$$ for some $k$-tuple $p_{1}...p_{k}$. Then, collecting all the colors in the diagonal, we have $$\sum_{i_{1}...i_{k}}W_{G,\;2k}^{r}(i_{1}...i_{k}i_{1}...i_{k})=\sum_{i_{1}...i% _{k}}W_{H,\;2k}^{r}(\sigma(i_{1}...i_{k})\sigma(i_{1}...i_{k}))$$ By Theorem 5, this implies $$\sum_{i_{1}...i_{k}}A^{r}_{G^{k}}(i_{1}...i_{k},i_{1}...i_{k})=\sum_{i_{1}...i% _{k}}A^{r}_{H^{k}}(\sigma(i_{1}...i_{k}),\sigma(i_{1}...i_{k}))$$ that is, $TrA^{r}_{G^{k}}=TrA^{r}_{H^{k}}$ for $r\geq 1$. Then, $F(G^{k},t)=F(H^{k},t)$. ∎ Our goal is to prove the analogue of Theorem 6 for the $k$-th symmetric powers. As an intermediate step, we prove analogues of Theorem 5 and Theorem 6 for the restricted $k$-th powers. Theorem 7. Let $G^{(k)}$ and $H^{(k)}$ be the $k$-th restricted powers of two graphs $G$ and $H$. Let $A_{G^{(k)}}^{r}$ and $A_{H^{(k)}}^{r}$ be the $r$-th powers of their adjacency matrices. Assume that $i_{1}\dots i_{k}$, $j_{1}\dots j_{k}$, $p_{1}\dots p_{k}$, and $q_{1}\dots q_{k}$ are $k$-tuples in $\mathcal{D}_{k}$. If $$W_{G,2k}^{r}(i_{1}\dots i_{k}\ j_{1}\dots j_{k})=W_{H,2k}^{r}(p_{1}\dots p_{k}% \ q_{1}\dots q_{k}),$$ then $$A_{G^{(k)}}^{r}(i_{1}\dots i_{k},j_{1}\dots j_{k})=A_{H^{(k)}}^{r}(p_{1}\dots p% _{k},q_{1}\dots q_{k}).$$ Proof: The proof mimics that of Theorem 5. The case $r=1$ is unaltered, so we assume the proposition is valid for $r$ and we suppose that $$W_{G,2k}^{r+1}(i_{1}\dots i_{k}\ j_{1}\dots j_{k})=W_{H,2k}^{r+1}(p_{1}\dots p% _{k}\ q_{1}\dots q_{k}).$$ This means that there is a permutation $\sigma$ of $\{1,2,...,n\}$ such that $$\left\{\begin{array}[]{cccc}&tp_{G}(i_{1}\dots i_{k}\ j_{1}\dots j_{k}\ m)&=&% tp_{H}(p_{1}\dots p_{k}\ q_{1}\dots q_{k}\ \sigma(m)),\\ &W_{G,2k}^{r}(i_{1}\dots i_{k}\ j_{1}\dots j_{k-1}\ m)&=&W_{H,2k}^{r}(p_{1}% \dots p_{k}\ q_{1}\dots q_{k-1},\sigma(m)),\\ &&\dots&\\ &W_{G,2k}^{r}(m\ i_{2}\dots i_{k}\ j_{1}\dots j_{k})&=&W_{H,2k}^{r}(\sigma(m)% \ p_{2}\dots p_{k}\ q_{1}\dots q_{k}).\\ \end{array}\right.$$ From the first of these equations, we observe that $m=i_{t}$ implies $\sigma(m)=p_{t}$. Therefore, the $k$-tuple $(i_{1}...i_{l-1}\;m\;i_{l+1}...i_{k})$ is in $\mathcal{D}_{k}$ if and only if $$(p_{1}...p_{l-1}\;\sigma(m)\;p_{l+1}...p_{k})$$ is in $\mathcal{D}_{k}$. This observation shows that, if we assume $m\neq i_{t}$ for $t=1,...,k$, we are allowed to apply the induction hypothesis to obtain $$\left\{\begin{array}[]{ll}&A_{G}(i_{t},m)=A_{G}(p_{t},\sigma(m))\quad\quad% \mbox{for }t=1,\dots,k.\\ &A^{r}_{G^{k}}(i_{1}\dots m\dots i_{k},j_{1}\dots j_{k})=A^{r}_{H^{k}}(p_{1}% \dots\sigma(m)\dots p_{k},q_{1}\dots q_{k}).\end{array}\right.$$ Then $$A^{r+1}_{G^{(k)}}(i_{1}...i_{k},j_{1}...j_{k})=\sum_{(s_{1}...s_{k})\in% \mathcal{D}_{k}}A_{G^{(k)}}(i_{1}...i_{k},s_{1}...s_{k})A^{r}_{G^{(k)}}(s_{1}.% ..s_{k},j_{1}...j_{k})$$ (5) $$=\sum_{m\notin\{i_{1},...,i_{k}\}}\sum_{t=1}^{k}A_{G}(i_{t},m)A^{r}_{G^{(k)}}(% i_{1}\dots m\dots i_{k},j_{1}\dots j_{k})$$ $$=\sum_{\sigma(m)\notin\{p_{1},...,p_{k}\}}\sum_{t=1}^{k}A_{H}(p_{t},\sigma(m))% A^{r}_{H^{(k)}}(p_{1}\dots\sigma(m)\dots p_{k},q_{1}\dots q_{k})$$ $$=A^{r+1}_{G^{(k)}}(p_{1}\dots p_{k},q_{1}\dots q_{k})\qed$$ Theorem 8. If the $2k$-th WL refinement fails to distinguish $G$ from $H$, then their restricted $k$-th powers are cospectral. Proof: The proof is analogous to that of Theorem 6. Assume $I_{G,\;2k}(t)=I_{H,\;2k}(t)$. Let $\sigma$ be the permutation of the set of $2k$-tuples given by Proposition 4. Since $\sigma$ preserves the type of the $2k$-tuples, if $i_{1}...i_{k}$ is in $\mathcal{D}_{k}$, then $$\sigma(i_{1}...i_{k}i_{1}...i_{k})=p_{1}...p_{k}p_{1}...p_{k}$$ for some $k$-tuple $p_{1}...p_{k}$ $\in\mathcal{D}_{k}$. Then, $$\sum_{(i_{1}...i_{k})\in\mathcal{D}_{k}}W_{G,\;2k}^{r}(i_{1}...i_{k}i_{1}...i_% {k})=\sum_{(i_{1}...i_{k})\in\mathcal{D}_{k}}W_{H,\;2k}^{r}(\sigma(i_{1}...i_{% k})\sigma(i_{1}...i_{k}))$$ By Theorem 7, this implies $$\sum_{(i_{1}...i_{k})\in\mathcal{D}_{k}}A^{r}_{G^{(k)}}(i_{1}...i_{k},i_{1}...% i_{k})=\sum_{(i_{1}...i_{k})\in\mathcal{D}_{k}}A^{r}_{H^{(k)}}(\sigma(i_{1}...% i_{k}),\sigma(i_{1}...i_{k}))$$ that is, $TrA^{r}_{G^{(k)}}=TrA^{r}_{H^{(k)}}$ for $r\geq 1$. Then, $F(G^{(k)},t)=F(H^{(k)},t)$. ∎ 7 Proof of Theorem 1 We can restate Theorem 1 as follows: Theorem 9. Let $G$ and $H$ be two graphs. If $I_{G,\;2k}(t)=I_{H,\;2k}(t)$, then $$F(G^{\{k\}},t)=F(H^{\{k\}},t).$$ Proof: Assume $I_{G,\;2k}(t)=I_{H,\;2k}(t)$. Again, by Prop. 4, there is a permutation $\sigma$ of the set of $2k$-tuples such that $$W_{G,\;2k}^{r}(i_{1}...i_{2k})=W_{H,\;2k}^{r}(\sigma(i_{1}...i_{2k}))$$ (6) for all $r\geq 1$. Since $$\>tp\>(i_{1}...i_{2k})=\>tp\>(\sigma(i_{1}...i_{2k})),$$ we can restrict $\sigma$ in the following way. If $\theta$ is a permutation in $S_{k}$, we denote by $\theta(i_{1}...i_{k})$ the $k$-tuple $(i_{\theta(1)}...i_{\theta(k)})$. Let us write the $2k$-tuples as pairs of $k$-tuples: $(i_{1}...i_{k},j_{1}...j_{k})$. Observe that if a $2k$-tuple is of the form $$(i_{1}...i_{k},\theta(i_{1}...i_{k}))$$ ,where $(i_{1}...i_{k})\in\mathcal{D}_{k}$ and $\theta\in S_{k}$, then (due to the type-conservation) $\sigma$ sends it to a $2k$-tuple of the form $(j_{1}...j_{k},\theta(j_{1}...j_{k}))$, for some $(j_{1}...j_{k})\in\mathcal{D}_{k}$. Thus, there is a permutation $\omega$ of the set $\mathcal{D}_{k}$ such that for every $(i_{1}...i_{k})\in\mathcal{D}_{k}$ $$W_{G,\;2k}^{r}(i_{1}...i_{k},\theta(i_{1}...i_{k}))=W_{H,\;2k}^{r}(\omega(i_{1% }...i_{k}),\theta(\omega(i_{1}...i_{k})))$$ (7) By Theorem 7, it follows that $$A_{G^{(k)}}^{r}(i_{1}...i_{k},\theta(i_{1}...i_{k}))=A_{H^{(k)}}^{r}(\omega(i_% {1}...i_{k}),\theta(\omega(i_{1}...i_{k}))).$$ (8) In particular, $$\sum_{(i_{1}...i_{k})\in\mathcal{D}_{k}}\sum_{\theta\in S_{k}}A_{G^{(k)}}^{r}(% i_{1}...i_{k},\theta(i_{1}...i_{k}))=\sum_{(i_{1}...i_{k})\in\mathcal{D}_{k}}% \sum_{\theta\in S_{k}}A_{H^{(k)}}^{r}(\omega(i_{1}...i_{k}),\theta(\omega(i_{1% }...i_{k}))).$$ (9) Since $\omega$ is a bijection, we can drop it from this last equation, and we have $$\sum_{(i_{1}...i_{k})\in\mathcal{D}_{k}}\sum_{\theta\in S_{k}}A_{G^{(k)}}^{r}(% i_{1}...i_{k},\theta(i_{1}...i_{k}))=\sum_{(i_{1}...i_{k})\in\mathcal{D}_{k}}% \sum_{\theta\in S_{k}}A_{H^{(k)}}^{r}(i_{1}...i_{k},\theta(i_{1}...i_{k}))$$ (10) Let $M^{k}$ be the matrix of Prop. 3. This last equation can be written as $$Tr(A_{G^{(k)}}^{r}M_{k})=Tr(A_{H^{(k)}}^{r}M_{k})$$ By Prop. 3, this is equivalent to $$Tr(A_{G^{\{k\}}}^{r})=Tr(A_{H^{\{k\}}}^{r})$$ (11) Since this is true for all $r$, then $F(G^{\{k\}},t)=F(H^{\{k\}},t).$ References [1] L. Babai, P. Erdös, and S. Selkow, Random graph isomorphism, SIAM Journal on Computing, 9, 1980, 628-635. [2] K. Audenaert, C. Godsil, G. Royle, T. Rudolph, Symmetric Squares of Graphs, Journal of Combinatorial Theory, Series B, Volume 97, Issue 1, January 2007, 74-90. [3] J-Y. Cai, M. Fürer, N. Immerman. An optimal lower bound on the number of variables for graph identification, Combinatorica, Volume 12, Number 4, December 1992, 389-410. [4] M. Grohe, Equivalence in finite-variable logics is complete for polynomial time, Proceedings of the 37th Annual Symposium on Foundations of Computer Science, FOCS 1996, 264. [5] M. Grohe, Isomorphism testing for embeddable graphs through definability, Proceedings of the Thirty-Second Annual ACM Symposium on theory of Computing, STOC 2000, 63-72.

Simple Theory of Ionic Activity in Concentrated Electrolytes Sven Schlumpberger Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA    Martin Z. Bazant Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA bazant@mit.edu (December 5, 2020) Abstract The Debye-Hückel formula for ionic activity coefficients is extended for concentrated solutions by solving a simple model of many-body Coulomb correlations and adding the Born solvation energy. Given the bulk permittivity, our formula is able to fit activity data for diverse electrolytes with only one parameter to adjust the correlation length, which interpolates between the Bjerrum length and the mean ion spacing. The results show that ionic activity in most electrolytes is dominated by three types of electrostatic forces: (i) mean-field charge screening, (ii) solvation, and (iii) Coulomb correlations, both “over-screening” (charge oscillations) and “under-screening” (extending beyond the Debye screening length). The theory of ionic activity has a long history Lewis and Randall (1921); Onsager (1933); Kirkwood (1934); Scatchard (1936); Stokes and Robinson (1948); Pitzer (1977); Newman (1989) since the seminal paper of Debye and Hückel (DH) in 1923 Debye and Hückel (1923), but a simple physical model for concentrated solutions remains elusive. Thermodynamic calculations May and Rowland (2017); Zemaitis Jr et al. (2010); Tester and Modell (1997); Bromley (1973) and engineering models Newman and Thomas-Alyea (2012) are usually based on empirical formulae, such as the Pitzer equation with fitted second virial coefficients Pitzer (1973); Kim and Frederick Jr (1988); Clegg et al. (1992). On the other hand, statistical theories are too complicated to solve analytically, even for charged hard spheres Hansen and McDonald (1986); Sloth and Soerensen (1990); Molero et al. (1992); Outhwaite et al. (1993); Simonin et al. (1998); Caccamo (1996), and still contain adjustable parameters. The DH formula successfully predicts the activity coefficient $\gamma_{i}$ or excess chemical potential, $\mu_{i}^{ex}=k_{B}T\ln\gamma_{i}$, of species $i$ in a dilute solution via a mean-field approximation of the Coulomb energy between each ion and its correlated “screening cloud” of excess counter-charge Debye and Hückel (1923), $$\ln\gamma_{i}^{DH}=-\frac{\left(z_{i}e\right)^{2}}{8\pi\varepsilon k_{B}T\left% (a_{i}+\lambda_{D}\right)}$$ (1) in terms of the effective ionic radius $a_{i}$ (a fitting parameter), the ionic charge $z_{i}e$, the solution permittivity $\varepsilon$, and the Debye screening length, $\lambda_{D}=\sqrt{\varepsilon k_{B}T/2e^{2}I}$, where $I=\frac{1}{2}\sum_{i}z_{i}^{2}s_{i}c_{0}$ is the ionic strength, $k_{B}T$ the thermal energy, $c_{0}$ the bulk salt concentration, and $\bar{c}_{i}=s_{i}c_{0}$ the bulk concentration of species $i$. The DH activity coefficient decreases with increasing salt concentration, as attractive Coulomb correlations become stronger, but breaks down for $c_{0}>0.1$M and fails to predict the enhanced activity of most (but not all) electrolytes at high concentrations. In 1925, Hückel Huckel (1925) proposed adding the change in self-energy of solvation to the DH formula, $$\ln\gamma_{i}^{H}=\ln\gamma_{i}^{DH}+\frac{\Delta E_{i}^{B}}{k_{B}T}$$ (2) in the approximation just introduced by Born Born (1920) $$\Delta E_{i}^{B}=\frac{(z_{i}e)^{2}}{8\pi a_{i}}\left(\frac{1}{\varepsilon}-% \frac{1}{\varepsilon_{s}}\right)$$ (3) which is the change in electrostatic potential of an isolated charged sphere, as the permittivity varies from $\varepsilon_{s}$ for pure solvent ($=78.36\varepsilon_{0}$ for water) to $\varepsilon$ for the solution. Since the local electric field of each ion aligns nearby solvent dipoles, the solution permittivity decreases with increasing concentration, thereby hindering solvation and increasing the activity. Hückel linearized the Born energy for small dielectric decrements, $\Delta E_{i}^{B}\sim k_{i}I$, and fitted early activity data Huckel (1925). Although Hückel’s theory was later found to be consistent with measured permittivities Hasted et al. (1948), the “extended DH equation” with a linear term in ionic strength Meier (1982); Rowland and May (2014) and the “specific ion interaction theory” (SIT) with linear terms for each ion concentration May and Rowland (2017), are widely viewed as empirical  Wright (2007), having lost their original connection with solvation energy, Eq. (2). Since the 1960s, theorists shifted their attention from solvation energy to Coulomb correlations and excluded volume, using molecular simulations and more complicated statistical theories. Many calculations based on the Primitive Model Friedman (1960); Hansen and McDonald (1986) (charged hard spheres in an implicit solvent of constant permittivity) predicted activity coefficients that were larger than DH and sometimes increasing Molero et al. (1992); Outhwaite et al. (1993) but typically decreasing  Simonin et al. (1998); Caccamo (1996); Sloth and Soerensen (1990) with ionic strength. The field has recently come full circle, as Vincze et al. Vincze et al. (2010) discovered that adding the Born energy could reverse this trend and bring hard-sphere simulations in line with experimental data. More sophisticated “molecular DH theories” including dielectric response have since emerged Xiao and Song (2011); Xiao (2015), but still no formula of comparable simplicity and ease of interpretation as DH, with the notable exception of Fraenkel’s “smaller ion shell” (SIS) extension of DH Fraenkel (2010, 2011); Vincze et al. (2011); Fraenkel (2012, 2014), which approximates packing constraints in the screening cloud for size-dissimilar ions at moderate dilution. After almost a century, the time is ripe to generalize DH for concentrated electrolytes, with the advent of an analytically tractable model for many-body Coulomb correlations. Inspired by Santangelo’s analysis of the one-component plasma at intermediate coupling Santangelo (2006), Bazant, Storey and Kornyshev (BSK) proposed a Ginzburg-Landau-like model of Coulomb correlations in ionic liquids Bazant et al. (2011) and concentrated electrolytes Bazant et al. (2009); Storey and Bazant (2012), which leads to a fourth-order Poisson equation that predicts over-sceening phenomena in diverse situations Jiang et al. (2014, 2016); Stout and Khair (2014); Alijó et al. (2014); Kornyshev et al. (2014); Yochelis (2014); Elbourne et al. (2015); Kondrat et al. (2015); Moon et al. (2015); Wang et al. (2017). Liu and Eisenberg solved the same equation numerically as two second-order equations Liu and Eisenberg (2015a) for nonlocal solvent polarization Xie et al. (2016); Kornyshev (1981); Kornyshev and Sutmann (1996); Kornyshev et al. (1997) and applied it to ion channels Liu and Eisenberg (2013) and ionic activity (including the Born energy) Liu and Eisenberg (2015b), fitting activity data for NaCl and CaCl${}_{2}$ with a correlation length close to the ion size. In this Letter, we derive an activity formula based on BSK screening, augmented by the Born energy. We begin by comparing the classical theory with recent experiments. Zuber et al. Zuber et al. (2014) fitted permittivity data to the empirical formula, $$\varepsilon=\frac{\varepsilon_{s}}{1+\sum_{i}\alpha_{i}x_{i}}=\frac{% \varepsilon_{s}}{1+\alpha x}$$ (4) where $\alpha_{i}$ is an ion-dependent parameter, $x_{i}$ is the mole fraction of ion $i$, $x$ is the mole fraction of the salt, and $\alpha=\sum_{i}s_{i}\alpha_{i}$. Using measured permittivities, we show in Fig.  1 that adding the Born energy, Eq. (2), (without Hückel’s linearization) greatly improves the fitting of activity data for concentrated solutions compared to the DH formula (1). Interestingly, the largest discrepancies remain for multivalent cations (Ca${}^{2+}$, Mg${}^{2+}$, La${}^{3+}$), which suggests the need to better account for Coulomb correlations. The excess chemical potential can be approximated as a cluster expansion, $$\mu_{i}^{ex}=\mu_{i}^{0}+2\pi\sum_{j}\int_{0}^{\infty}K_{ij}(r)g_{ij}(r)\bar{c% }_{j}r^{2}dr+...$$ (5) where we keep only the ion-solvent self energy, $\mu_{i}^{0}(\{\bar{c}_{j}\})$, and the two-body energy, where $K_{ij}(r)$ is the ion-ion pair potential and $g_{ij}(r)$ is the pair correlation function. Following Hückel, we set $\mu_{i}^{0}=\Delta E^{B}_{i}$, and neglect entropic effects of ion crowding Bazant et al. (2009), which become important at high concentrations Kilic et al. (2007) and in ionic liquids Kornyshev (2007). Following DH, we consider only Coulomb forces, $K_{ij}=\frac{(z_{i}e)(z_{j}e)}{4\pi\varepsilon r}$, neglect short-range (e.g. hydration) forces, and calculate $g_{ij}\approx 1-\frac{z_{i}e\psi}{k_{B}T}$ for the screening cloud around central ion of radius $a_{i}$ in local equilibrium with a small fluctuating pair potential, $\psi(r)$. In place of the DH mean-field approximation, we capture some many-body correlations via the linearized BSK equation, which takes the dimensionless form, $$\left(1-\delta_{c}^{2}\tilde{\nabla}^{2}\right)\tilde{\nabla}^{2}\tilde{\psi}=% \tilde{\psi}$$ (6) where $\tilde{\psi}=z_{i}e\psi/k_{B}T$ , $\tilde{\nabla}=\lambda_{D}\nabla$, and $\delta_{c}=\ell_{c}/\lambda_{D}$. Motivated by BSK Bazant et al. (2011), we set the correlation length, $\ell_{c}$, proportional to the harmonic mean of the Bjerrum length, $\ell_{B}=\frac{e^{2}}{4\pi\varepsilon k_{B}T}$, in dilute solutions and the mean ion spacing, $c_{0}^{-1/3}$, in concentrated solutions, $$\ell_{c}=2\xi\left(\ell_{B}^{-1}+c_{0}^{1/3}\right)^{-1}$$ (7) with one adjustable parameter, $\xi$. The general decaying solution of (6) in spherical coordinates is $$\tilde{\psi}=c_{1}\frac{e^{-b_{1}\tilde{r}}}{\tilde{r}}+c_{2}\frac{e^{-b_{2}% \tilde{r}}}{\tilde{r}}$$ (8) where $$b_{1}=\sqrt{\frac{1-\sqrt{1-4\delta_{c}^{2}}}{2\delta_{c}^{2}}},\ b_{2}=\sqrt{% \frac{1+\sqrt{1-4\delta_{c}^{2}}}{2\delta_{c}^{2}}}.$$ (9) Gauss’ law for nonlocal BSK polarization requires Bazant et al. (2011), $$-\hat{n}\cdot\left(1-\delta_{c}^{2}\tilde{\nabla}^{2}\right)\tilde{\nabla}% \tilde{\psi}\left|{}_{\tilde{r}=\tilde{a_{i}}}\right.=\tilde{q}_{i}=\frac{z_{i% }e^{2}\lambda_{D}}{4\pi a_{i}^{2}\varepsilon k_{B}T}$$ (10) where $\tilde{q}_{i}$ is the dimensionless surface charge, $\tilde{a}_{i}=a_{i}/\lambda_{D}$, and $\hat{n}=\hat{r}$. The fourth-order Poisson equation requires another boundary condition, which controls electrostatic correlations at the surface. When this approach was first considered for electrolytes, Bazant et al. Bazant et al. (2009) proposed a mixed Stern-layer boundary condition, interpolating between $$\hat{n}\cdot\nabla\nabla^{2}\psi=0\mbox{ (fixed potential)}$$ (11) for a surface of fixed potential without adsorbed charge, as used in modeling electrode double layers Bazant et al. (2011), and $$\hat{n}\cdot\nabla\psi=0\mbox{ (fixed charge)}$$ (12) for a surface of fixed charge, as derived by Santangelo Santangelo (2006) for the one-component plasma near a charged wall. Since our spherical ion has fixed charge, we choose Eq. (12) and obtain $$c_{1}=\frac{\tilde{q}_{i}\tilde{a}_{i}\exp{(b_{1}\tilde{a}_{i})}}{\delta_{c}^{% 2}\left(b_{1}\tilde{a}_{i}+1\right)\left(b_{2}^{2}-b_{1}^{2}\right)},\ c_{2}=% \frac{\tilde{q}_{i}\tilde{a}_{i}\exp{(b_{2}\tilde{a}_{i})}}{\delta_{c}^{2}% \left(b_{2}\tilde{a}_{i}+1\right)\left(b_{1}^{2}-b_{2}^{2}\right)}.$$ (13) Below we also consider the opposite limit of Eq. (11). Using these results to perform the integral in Eq. (5) and enforcing bulk electroneutrality, $\sum_{i}z_{i}s_{i}=0$, we arrive at our main result: $$\ln\gamma_{i}=\ln\gamma_{i}^{BSK}+\frac{\Delta E^{B}_{i}}{k_{B}T},\ \ \mbox{where}$$ (14) $$\ln\gamma_{i}^{BSK}=\frac{z_{i}\tilde{q}_{i}\tilde{a}_{i}^{2}}{2\delta_{c}^{2}% \left(b_{1}^{2}-b_{2}^{2}\right)}\left[\frac{1}{b_{1}(b_{1}\tilde{a}_{i}+1)}-% \frac{1}{b_{2}(b_{2}\tilde{a}_{i}+1)}\right].$$ As shown in Fig.  1, this simple formula is able to predict activity data for diverse aqueous electrolytes with remarkable accuracy, considering it has only one adjustable parameter, $\xi$. The fitted correlation lengths $\ell_{c}\propto\xi$ scale roughly with cation valence squared, $\xi\approx\xi_{0}z_{+}^{2}$, which supports the BSK interpretation of Eq. (6) in terms of ion-ion correlations, as opposed to nonlocal solvent polarization with $\ell_{c}=$ constant Liu and Eisenberg (2015b). In hindsight, for aqueous solutions it is natural to multiply $\ell_{c}$ by $z_{+}^{2}$ in Eq. (7) since cation correlations are favored by hydration chemistry, as negative oxygen atoms can order around cations more easily than do positive di-hydrogens around anions. With this choice, we could fit all the data in Fig.  1 with a single universal parameter, $\xi_{0}=0.08$, which is comparable to the value suggested by BSK Bazant et al. (2011, 2012) for $\ell_{c}$ in the high concentration limit of ionic liquids, $\xi_{0}^{rcp}=\left(\frac{3\Phi^{rcp}}{4\pi}\right)^{1/3}=\frac{1}{2(0.83)^{1/% 3}}=0.53$ for random close packing of spheres at volume fraction, $\Phi^{rcp}=0.63$. Our formula is also able to fit data for organic Li-ion battery electrolytes, as shown in Fig.  2. Since activity coefficients were extracted from electrochemical signals by fitting an engineering model Valøen and Reimers (2005), permittivities were not directly measured, so we reverse the procedure above by setting $\xi=1$ and fitting the data to Eq. (14) using the permittivity relation of Vincze et al. Vincze et al. (2010), $$\varepsilon(c_{0},T)=\varepsilon_{s}(T)-\delta_{s}(T)c_{0}+b_{s}(T)c_{0}^{3/2},$$ (15) where we estimate solvent permittivity, $\varepsilon_{s}(T)$, from the literature values Seward and Vieira (1958); Chernyak (2006); Lee and Park (2013). With only two adjustable parameters, $\delta_{s}(T)$ and $b_{s}(T)$, our formula is able to fit the data across a wide range of temperatures and concentrations, and the inferred dielectric decrements, $\delta_{s}(T)$, are consistent with values for aqueous solutions Vincze et al. (2010). Besides providing a simple formula for fitting thermodynamic data, our theory also sheds light on the complex physics of screening. At moderate dilution, activity increases from the DH dilute limit (since $\gamma_{i}^{DH}<1$), $$\displaystyle\frac{\ln\gamma_{i}^{BSK}}{\ln\gamma_{i}^{DH}}$$ $$\displaystyle\sim$$ $$\displaystyle 1-\left(\frac{1+\tilde{a_{i}}+\tilde{a}_{i}^{2}}{\tilde{a}_{i}+% \tilde{a}_{i}^{2}}\right)\delta_{c}^{2}\ \mbox{ for }\delta_{c}\ll\frac{1}{2}$$ (16) because the screening length, $\lambda_{s}=\lambda_{D}/(\mbox{Re }b_{1})$, increases slightly $\tilde{\lambda}_{s}\sim 1+\delta_{c}^{2}+\ldots$ for $\delta_{c}\ll 1$, thus reducing attractive Coulomb correlations. In the opposite limit of high ionic strength, $I>(32\pi\ell_{B}\ell_{c}^{2})^{-1}$ or $\delta_{c}>\frac{1}{2}$, “over-screening” charge oscillations arise Bazant et al. (2011) with radial wavelength, $\lambda_{o}=\frac{2\pi\lambda_{D}}{\mbox{Im }b_{1}}$, and the screening length diverges, $$\tilde{\lambda}_{s}\sim\frac{\tilde{\lambda}_{o}}{2\pi}\sim\sqrt{2\delta_{c}}% \ \mbox{ for }\delta_{c}\gg\frac{1}{2},$$ (17) yielding a critically damped long-range oscillation, $\lambda_{s}\sim\frac{\lambda_{o}}{2\pi}\sim\sqrt{2\ell_{c}\lambda_{D}}$. A similar phenomenon of “underscreening” Lee et al. (2017); Gebbie et al. (2017) was recently discovered by surface-force measurements in ionic liquids Gebbie et al. (2013, 2015) and concentrated electrolytes Smith et al. (2016) and attributed to short-range attractive forces that trap mobile ions in Bjerrum pairs Gebbie et al. (2013); Adar et al. (2017); Goodwin and Kornyshev (2017); Gavish et al. (2017), despite the high conductivity and double-layer capacitance Andersson et al. (2017) and seemingly weak specific forcesLee et al. (2014). Interestingly, our theory predicts the observed universal scaling of the underscreening length, $\tilde{\lambda}_{s}$ vs $\tilde{a}_{i}\approx\delta_{c}$, albeit with smaller magnitude, $\lambda_{s}\approx 1-10$nm, and exponent $\frac{1}{2}$ in Eq. (17) instead of $\approx 3$ (Fig. 4 of Smith et al.  Smith et al. (2016)), based only on many-body Coulomb correlations among mobile ions. Over-screening (charge oscillations) and underscreening (extended range) both weaken attractive Coulomb correlations, thereby increasing the activity relative to DH: $$\displaystyle\frac{\ln\gamma_{i}^{BSK}}{\ln\gamma_{i}^{DH}}$$ $$\displaystyle\sim$$ $$\displaystyle\frac{1+\tilde{a}_{i}}{\sqrt{2\delta_{c}}}\left(1-\frac{1}{4% \delta_{c}^{2}}\right)\ \mbox{ for }\delta_{c}\gg\frac{1}{2}$$ (18) As shown in Fig.  1, however, the activity of concentrated electrolytes cannot be attributed solely to underscreening, as conjectured by Lee et al. Lee et al. (2017) and tested against data for NaCl without considering solvation energy or dielectric decrement. Our analysis also supports the interpretation of boundary conditions for BSK theory in terms of ion-image correlations, as in colloids Hatlo and Lue (2008). Repeating the screening calculation with vanishing third-derivative (11), we obtain a modified formula Schlumpberger (2016), $$\ln\gamma_{i}^{mBSK}=\frac{z_{i}\tilde{q}_{i}\tilde{a}_{i}^{2}}{2\delta_{c}^{2% }\left(b_{1}^{2}-b_{2}^{2}\right)}\left[\frac{b_{2}^{2}}{b_{1}(b_{1}\tilde{a}_% {i}+1)}-\frac{b_{1}^{2}}{b_{2}(b_{2}\tilde{a}_{i}+1)}\right]$$ (19) which predicts lower activity than DH theory, the opposite trend of Eq. (14) for vanishing first derivative (12). Equation (12) thus captures repulsive ion-image forces for an ideally non-polarizable surface of fixed charge, which raise the activity of an ion (14) compared to DH, while Equation (11) describes attractive ion-images forces for an ideally polarizable surface of fixed potential, which lower the activity. The modified BSK formula (19) may find applications to metal nanoparticles, e.g. describing their solubility in ionic liquids Dupont and Scholten (2010). Although our theory captures much of the physics of ionic activity, it neglects short-range specific interactions that become important at high concentration (including the “solvent-in-salt” limit Suo et al. (2013, 2015)) especially for certain ions, as shown in Fig. 3. For aqueous KNO${}_{3}$, the DH$+$Born model over-estimates the mean activity coefficient, so the positive BSK correction for fixed-charge ions (14) cannot improve the fit. Interestingly, the negative correction for an ideally polarizable“metallic ion” (19) could fit the data, which seems consistent with the fact that nitrate ions have a labile hydration shell with hydrogen bonds fluctuating at the time scale as ion polarization Thøgersen et al. (2013). In contrast, hydroxyl ions have a tightly bound hydration shell Botti et al. (2003), which could interfere with the Grotthuss mechanism of proton hopping de Grotthuss (1806); Bernal and Fowler (1933); Agmon (1995) and raise the hydroxyl activity by lowering entropy, as shown in Fig. 3 for aqueous NaOH. It is possible to extend our theory to include short-range interactions Levy et al. (2017) using the regular-solution approximation of Goodwin and Kornyshev Goodwin and Kornyshev (2017), which makes the activity formula more cumbersome, but still physics-based, and thus more predictive than SIT or other empirical relations May and Rowland (2017); Zemaitis Jr et al. (2010); Tester and Modell (1997); Bromley (1973). In conclusion, we arrive at a physical picture of ionic activity governed by three types of electrostatic forces: (i) mean-field ion-ion correlations (DH screening), (ii) ion-solvent self-energy (Born solvation), and (iii) many-body ion-ion and ion-image correlations (BSK screening). Our activity formula (14) captures this physics in a simple way, which could be extended for nonlocal solvent polarization Levy et al. (2017); Xie et al. (2016); Kornyshev (1981), SIS discrete screening Fraenkel (2010, 2014), solvent-mediated short-range forces Goodwin and Kornyshev (2017); Levy et al. (2017), and steric constraints Bazant et al. (2009), with goal of describing the transition from concentrated electrolytes to ionic liquids. This work was supported by a Professor Amar G. Bose Research Grant. The authors thank Amir Levy for references and helpful discussions. References Lewis and Randall (1921) G. N. 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Comment on ”Composite excitation of Josephson phase and spin waves in ferromagnetic Josephson junctions” (S.Hikino, M.Mori, S.Takahashi, and S.Maekawa, arXiv:cond-mat 1009.3551) A.F. Volkov${}^{1,2}$ ${}^{1}$Theoretische Physik III, Ruhr-Universität Bochum, D-44780 Bochum, Germany ${}^{2}$Institute for Radioengineering and Electronics of Russian Academy of Sciences,11-7 Mokhovaya str., Moscow 125009, Russia Abstract We clarify the applicability of the quasistatic approximation used in Ref. [VE, ], where coupled spin and Josephson plasma waves have been predicted to exist in SIFS Josephson junctions. We show, contrary to the claim of the authors of Ref. [Maekawa, ], that this approximation is very accurate in realistic systems studied experimentally. pacs: 74.50.+r, 75.70.-i, 74.20.Rp The dynamics of SIFS (or SFIFS) Josephson junctions with ferromagnetic layers has been studied theoretically in Ref. [VE, ]. Here S, I and F stand for a superconducting, thin insulating or ferromagnetic layer, respectively. In particular, it has been shown that weakly damped magneto-plasma oscillations are possible in such a system. That is, oscillations of the magnetic moment  $M$ in the F layer and Josephson ”plasma” waves turn out to be coupled. The coupled modes (spin and Josephson plasma waves) may result in the peaks on the I-V characteristics of the junction in addition to the Fiske steps. The same problem has been studied in a recent paper [Maekawa, ]. The authors claim that the electromagnetic (EM) fields in the F layer which excite spin waves in F have been neglected in Ref. [VE, ]. In this Comment, we would like to clarify that: A) contrary to the statement of Ref. [Maekawa, ], the EM fields in the F film are taken into account in Ref. [VE, ]. Indeed, there could be no coupling between magnetic and plasma modes otherwise. B) these fields are considered in the quasistatic approximation, which describes the dynamics of realistic junctions rather accurately, while the effects of ac electric fields $E$, accounted for in Ref. [Maekawa, ], is negligible. The only important ac field is the magnetic induction $B$ which is described by the London equation. The skin effect due to quasiparticle current driven by $E$ can be neglected. Estimations justifying this approximation are not presented in Ref. [VE, ] for lack of space. Here we give these simple estimations and present a physical explanation of the coupling between the spin and Josephson plasma waves. The equation for the phase difference $\varphi$ (Eq.(4) in Ref. [VE, ]) is obtained from the Maxwell equation $$(\mathbf{\nabla}\times\mathbf{B})_{z}=\frac{4\pi}{c}j_{z}$$ (1) written in the superconducting regions S (where the magnetic induction $\mathbf{B}$ coincides with the magnetic field $\mathbf{H}$) and from the usual expression for the current through the Josephson junction. The displacement current $j_{dis}=(\epsilon/c)\partial E/\partial t$ is dropped because in metals it is negligible in comparison with the quasiparticle current ($\omega\ll\sigma_{Q}$), where at $T\precsim\Delta$ the quasiparticle conductance $\sigma_{Q}\approx\sigma_{Dr}\exp(-\Delta/T)$ with $\sigma_{Dr}=(e^{2}n\tau/m)\approx 10^{17}s^{-1}$ for the mean free path $l=v\tau\approx 10^{-6}cm.$ In the quasistatic approximation the expression for $\mathbf{B(}z,t\mathbf{)}$ is given by Eq.(3) in Ref. [VE, ] $$\mathbf{B}_{\perp}(z,t)=\{\frac{\Phi_{0}}{4\pi\lambda_{L}}\mathbf{n}_{z}% \mathbf{\times\nabla}_{\perp}\varphi-\frac{2\pi\tilde{d}_{F}}{\lambda_{L}}% \mathbf{M}_{\perp}\}\exp(-\frac{(z-d_{F})}{\lambda_{L}}).$$ (2) It relates the magnetic field in the superconductors and the phase difference $\varphi(t).$ The second term in the curly brackets appears due to the magnetic moment in the F layer(s). Integrating this expression over the square of the superconductors (in the $(x,z)$-plane perpendicular to the magnetic field) and adding the magnetic moment of the F layer $4\pi\tilde{d}_{F}L_{x}M,$, we obtain the usual law of the quantization of the magnetic moment in Josephson junctions: $\Phi\equiv\Phi_{S}+\Phi_{F}=\Phi_{0}n$, where $L_{x}$ is the length of the superconductors in the $x$-direction, $\Phi_{0}$ is the magnetic flux quantum and $n$ is an integer. Due to the second term in Eq.(2) the Josephson mode is coupled to the spin waves. Eq.(2) for $\mathbf{B}_{\perp}$ is obtained by using expression for the current $\mathbf{j}_{\perp}$ (Eq.(1) in Ref. [VE, ]) which is written in the dirty limit ($\omega\tau\ll 1,kl\ll 1,$ where $\omega,k$ are characteristic frequency and wave vector, respectively, $l=v\tau$ is the mean free path). The quasiparticle current $\mathbf{j}_{Q\perp}=\sigma_{Q}(\omega)\mathbf{E}_{\perp}$ and therefore the transverse electric field $\mathbf{E}_{\perp}$ ($\mathbf{\nabla}\times\mathbf{E}_{\perp}\neq 0$) is neglected. This approximation is valid if the skin depth $\delta_{sk}$ is much larger than the London penetration length $\lambda_{L}.$ If the current $\mathbf{j}_{Q\perp}$ is taken into account, then $\lambda_{L}$ in Eq.(3) of Ref. [VE, ] should be replaced by $\lambda_{\omega}=1/\sqrt{\lambda_{L}^{-2}+4\pi i\omega\sigma_{Q}/c^{2}}$. The second term is small in comparison with the first one if the frequency $\omega=2\pi\nu$ is not very high $$\nu\ll\frac{1}{8\pi^{2}\sigma_{Q}}(\frac{c}{\lambda_{L}})^{2}\approx 5\cdot 10% ^{12}\exp(\frac{\Delta}{T})Hz$$ (3) where we take $\lambda_{L}\approx 5\cdot 10^{-6}cm$. For the realistic SIFS junctions, where the frequency $\nu$ typically is less than one hundred gigahertz [Weides, ], this condition is easily fulfilled. The currents induced in the F layer also change the magnetic induction $\mathbf{B}$. However this change, $\delta\mathbf{B}_{F}$, is even smaller than the change $\delta\mathbf{B}_{SQ}$ caused by the quasiparticle current in S. Indeed, the change $\delta\mathbf{B}_{F}$ is determined by the total Meissner current in the F layer $j_{FMeis}d_{F}$ which is much smaller than $j_{SMeis}\lambda_{L}$ because $d_{F}\ll\lambda_{L}$ (by assumption) and $j_{FMeis}\ll j_{SMeis}$ since the condensate density in F is significantly lower than the density of Cooper pairs in S. The change $\delta\mathbf{B}_{F}$ due to skin effect can be neglected if the frequency satisfies the condition $$\nu\ll\frac{1}{8\pi^{2}\sigma_{F}}(\frac{\lambda_{L}}{d_{F}})(\frac{c}{\lambda% _{L}})^{2}$$ (4) This condition is fulfilled even easier than the condition (2). Therefore, the currents and ac electric fields in F accounted for in Ref. [Maekawa, ] can be neglected. The only EM field in the F layer, which is essential, is the induction $\mathbf{B}$ determined by Eq.(3) of Ref. [VE, ]. The quasistatic approximation used in deriving this equation is fulfilled for realistic systems (see Ref. [Weides, ]) with a great accuracy. We thank SFB 491 for financial support. References (1) A.F. Volkov and K. B. Efetov, Phys. Rev. Lett. 103, 037003 (2009). (2) S. Hikino, M. Mori, S Takahashi, and S. Maekawa, arXiv: cond-mat. 1009.3551 (3) J. Pfeiffer et al, Phys. Rev. B 77, 214506 (2008)

Idempotent reduction for the finitistic dimension conjecture Diego Bravo Instituto de Matemática y Estadística ”Rafael Laguardia”, Universidad de la República, Julio Herrera y Reissig 565, Montevideo, Uruguay dbravo@fing.edu.uy  and  Charles Paquette Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, ON K7K 7B4, Canada charles.paquette.math@gmail.com Abstract. In this note, we prove that if $\Lambda$ is an Artin algebra with a simple module $S$ of finite projective dimension, then the finiteness of the finitistic dimension of $\Lambda$ implies that of $(1-e)\Lambda(1-e)$ where $e$ is the primitive idempotent supporting $S$. We derive some consequences of this. In particular, we recover a result of Green-Solberg-Psaroudakis: if $\Lambda$ is the quotient of a path algebra by an admissible ideal $I$ whose defining relations do not involve a certain arrow $\alpha$, then the finitistic dimension of $\Lambda$ is finite if and only if that of $\Lambda/\Lambda\alpha\Lambda$ is. The second author was supported by the Natural Sciences and Engineering Research Council of Canada. 1. Introduction In this paper, $\Lambda$ stands for an Artin algebra, that we will assume to be basic. Since we are mainly interested in homological properties of the finitely generated modules over $\Lambda$, this is not a restriction. We denote by ${\rm mod}\,\Lambda$ the category of finitely generated left $\Lambda$-module. All modules considered are left modules, unless otherwise stated. We let $e$ denote an idempotent of $\Lambda$ and $S_{e}$ be the semi-simple module supported at $e$. If ${\rm rad}\Lambda$ is the Jacobson radical of $\Lambda$, then $S_{e}\cong\Lambda e/{\rm rad}\Lambda e$. Note that $S_{e}$ is simple if and only if $e$ is primitive. We will consider the Artin algebra $(1-e)\Lambda(1-e)$, which will be denoted by $\Gamma$. In other words, $\Gamma$ is the endomorphism algebra over $\Lambda$ of the projective module $\Lambda(1-e)$. It is desirable to relate the homological properties of $\Lambda$ and $\Gamma$. However, one has to impose some conditions on $e$ as in general, the Artin algebras $\Lambda$ and $\Gamma$ can be very far apart, from a homological perspective. This has been achieved in [3, 7, 8], for instance. In [7], it was shown that when $e$ is primitive and $\Lambda$ is finite dimensional over an algebraically closed field, the finiteness of the global dimensions of $\Lambda$ and $\Gamma$ are equivalent, when all higher self-extension groups of $S_{e}$ vanish. Moreover, the latter condition happens to be necessary for both $\Lambda$ and $\Gamma$ having finite global dimension. We let $${\rm findim}\,\Lambda={\rm sup\,}\{{\rm pd}_{\Lambda}M\mid M\in{\rm mod}% \Lambda,\,{\rm pd}_{\Lambda}M<\infty\}$$ denote the (little) finitistic dimension of $\Lambda$, that is, the supremum of the projective dimensions of those finitely generated $\Lambda$-modules having finite projective dimension. It has been conjectured (and publicized by Bass) in the sixties that the finitistic dimension of an Artin algebra is always finite. This is a very important problem in representation theory of algebras, as a positive answer to it implies the validity of many other important homological conjectures, including the Gorenstein symmetry conjecture, the Nakayama conjecture, the generalized Nakayama conjecture, Nunke’s condition, the Auslander-Reiten conjecture and the vanishing conjecture; see [13]. In this note, we are mainly interested in comparing the finitistic dimensions of closely related algebras such as $\Lambda$ and $\Gamma$, and we will be particularly interested in the case where $e$ is primitive. Our first main result is the following. Theorem 1.1. Let $e$ be a primitive idempotent such that ${\rm pd}_{\Lambda}S_{e}$ is finite. Then ${\rm findim}\,\Gamma\leq 2\,{\rm findim}\,\Lambda-\ell+1$ where $1\leq\ell\leq{\rm pd}_{\Lambda}S_{e}$. As we will see later, the condition that $e$ is primitive can be slightly relaxed. Note also that the condition that ${\rm pd}_{\Lambda}S_{e}<\infty$ is not that restrictive, due to the following result by Auslander [2], later generalized by Iyama [10]. Proposition 1.2 (Auslander). Given an Artin algebra $\Gamma$, there exists an Artin algebra $\Lambda$ with ${\rm gl.dim}\,\Lambda<\infty$ and $\Gamma=(1-e)\Lambda(1-e)$ for some idempotent $e$ of $\Lambda$. In particular, in this proposition, we have ${\rm pd}_{\Lambda}S_{e}<\infty$ and ${\rm findim}(\Lambda)<\infty$. Therefore, if we could extend the above theorem to $e$ not necessarily primitive, that would yield a complete proof of the finitistic dimension conjecture. In the second part of the note, we provide applications of the above theorem. Our first application is the following result, that was first proven in [12] when $x$ is an arrow. Here, $k$ is a field, $Q=(Q_{0},Q_{1})$ a finite quiver and $I$ an admissible ideal of the path algebra $kQ$. Proposition 1.3. Let $\Lambda=kQ/I$ be such that $I$ is generated by relations not involving a given element $x$ of $Q_{0}\cup Q_{1}$. Let $J$ be the two sided ideal of $\Lambda$ generated by $x$. Then ${\rm findim}\,\Lambda\leq 2\,{\rm findim}\,\Lambda/J+4\leq 2\,{\rm findim}\,% \Lambda+4$. Our second application is the following, where $e$ is a primitive idempotent; compare [14, Corollary 3.2]. Proposition 1.4. Let $\Lambda=kQ/I$ as above. Let $J$ be a submodule of $\Lambda e$ with $J{\rm rad}\Lambda=0$. Thus, $J$ is a two-sided ideal. Assume further that ${\rm pd}_{\Lambda/J}J<\infty$. Then ${\rm findim}\,\Lambda\leq 2\,{\rm findim}\,\Lambda/J+4$. 2. Main result We let $F:={\rm Hom}_{\Lambda}(\Lambda(1-e),-)$ be the exact functor from ${\rm mod}\,\Lambda$ to ${\rm mod}\,\Gamma$. Observe that $F$ can be extended naturally to an exact functor from the corresponding bounded derived categories $D^{b}({\rm mod}\,\Lambda)\cong K^{-,b}({\rm proj}\Lambda)$ and $D^{b}({\rm mod}\,\Gamma)\cong K^{-,b}({\rm proj}\Gamma)$, by taking the right derived functor $R{\rm Hom}_{\Lambda}(\Lambda(1-e),-)$, that will also be denoted by $F$. Now, consider the functor $G:=\Lambda(1-e)\otimes_{\Gamma}-:{\rm mod}\,\Gamma\to{\rm mod}\,\Lambda$. Similarly, $G$ can be extended naturally to an exact functor from the corresponding derived categories $D({\rm mod}\,\Gamma)$ and $D({\rm mod}\,\Lambda)$, by taking the left derived functor $\Lambda(1-e)\otimes_{\Gamma}^{L}-$, that will also be denoted by $G$. Notice that unbounded derived categories need to be used. Note also that as a functor between module categories, $G$ is not exact. It is well known that $(G,F)$ is an adjoint pair, both at the module category level and at the derived category level. Lemma 2.1. The co-unit $\eta:1\to FG$ of the adjunction at the module category level is a natural isomorphism. Proof. The co-unit is such that for $X\in{\rm mod}\,\Gamma$ and $x\in X$, we have $\eta_{X}(x)(1-e)=(1-e)\otimes x$, where $\eta_{X}:X\to{\rm Hom}_{\Lambda}((1-e)\Lambda,\Lambda(1-e)\otimes_{\Gamma}X)$. Using this, one can show that $\eta_{X}$ is an isomorphism. ∎ This implies that $G$ is fully faithful. Let $\mathcal{B}$ be the full subcategory of ${\rm mod}\,\Lambda$ generated by the $GX$ for $X$ in ${\rm mod}\,\Gamma$. The above implies that ${\rm mod}\,\Gamma$ is equivalent to $\mathcal{B}$ through $G$ and $F$. Finally, observe that for an idempotent $e^{\prime}$ of $\Lambda$ with $e^{\prime}(1-e)=e^{\prime}$, we have $G(\Gamma e^{\prime})\cong\Lambda e^{\prime}$ and $F(\Lambda e^{\prime})\cong\Gamma e^{\prime}$. Therefore, $G$ sends projective $\Gamma$-modules to projective $\Lambda$-modules while $F$ sends projective modules in add$(\Lambda(1-e))$ to projective $\Gamma$-modules. Observe that the objects in add$(\Lambda(1-e))$ are precisely the projective objects in $\mathcal{B}$. Let $Y(e)$ denote the Ext-algebra of $S_{e}$. It is a positively graded algebra whose degree $i$ piece is ${\rm Ext}_{\Lambda}^{i}(S_{e},S_{e})$. It is an Artin algebra when, for instance, ${\rm pd}_{\Lambda}S_{e}<\infty$. We say that a graded Artin algebra $\Lambda$ has uniform graded (left) loewy length if all the indecomposable (graded) projective modules in ${\rm mod}\,\Lambda$ have the same graded loewy length. Here, the graded loewy length of a graded module $M=(M_{i})_{i\in\mathbb{Z}}$ is the difference $r-s$ where $r$ is maximal with $M_{r}\neq 0$ and $s$ is minimal with $M_{s}\neq 0$. The reason why this concept of uniform graded loewy length is important lies in the following. Lemma 2.2. Let $Y(e)$ be an Artin algebra of uniform graded loewy length $\ell$. Let $M$ be a left $\Lambda/\Lambda(1-e)\Lambda$-module. Then, as $\Lambda$-modules, we have ${\rm Ext}_{\Lambda}^{\ell-1}(M,S_{e})\neq 0$ and ${\rm Ext}_{\Lambda}^{i}(M,S_{e})=0$ for $i\geq\ell$. Proof. The condition implies that for any indecomposable direct summand $S$ of $S_{e}$, we have ${\rm Ext}_{\Lambda}^{\ell-1}(S,S_{e})\neq 0$ and ${\rm Ext}_{\Lambda}^{i}(S,S_{e})=0$ for $i\geq\ell$. The rest of the proof is an induction on the length of $M$, using that the composition factors of $M$ are all direct summands of $S_{e}$. ∎ The following lemma is crucial for obtaining our main theorem. Lemma 2.3. Assume that ${\rm findim}(\Lambda)=r<\infty$, that ${\rm pd}_{\Lambda}S_{e}<\infty$ and that $Y(e)$ has uniform graded loewy length $\ell$. Let $N$ be a $\Gamma$-module of finite projective dimension $s$. Then the minimal projective resolution of the complex $GN$ has no cohomology in degree $i$ where $i\leq{\rm min}(-1,-r+\ell-1)$. Proof. Consider a minimal projective resolution $$0\to P_{-s}\to P_{-s+1}\to\cdots\to P_{-1}\to P_{0}$$ of $N$. Then $$C:=G(P_{-s})\to G(P_{-s+1})\to\cdots\to G(P_{-1})\to G(P_{0})$$ is a minimal projective resolution of $GN$. Note that $C$ need not be a module. Note also that each cohomology of $C$ occurring in negative degree has the structure of an $\Lambda/\Lambda(1-e)\Lambda$-module. Let $-s\leq t_{1}\leq 0$ be the least degree for which the above complex has non-zero cohomology $Z_{1}$. If $t_{1}=0$, then there is nothing to prove. So assume $t_{1}<0$. Note that $Z_{1}$ has finite projective dimension as a $\Lambda$-module, since we are assuming that ${\rm pd}_{\Lambda}S_{e}<\infty$. Moreover, since $Y(e)$ has uniform graded loewy length $\ell$, it follows from the above lemma that ${\rm Ext}^{i}_{\Lambda}(Z_{1},S_{e})=0$ for all $i\geq\ell$ and ${\rm Ext}^{\ell-1}_{\Lambda}(Z_{1},S_{e})\neq 0$. By minimality of $t_{1}$, we have a canonical morphism $Z_{1}[-t_{1}]\to C$, which induces an isomorphism in degree $t_{1}$ cohomologies. There is a corresponding exact triangle $$Z_{1}[-t_{1}]\to C\to C_{1}$$ and it follows that $C_{1}$ has a finite (possibly non-minimal) projective resolution of shape $$\to G(P_{t_{1}-2})\oplus Q_{-1}\to G(P_{t_{1}-1})\oplus Q_{0}\to G(P_{t_{1}})% \to G(P_{t_{1}+1})\to\cdots\to G(P_{-1})\to G(P_{0})$$ $$\cdots\to Q_{-t_{1}-s}\to G(P_{-s})\oplus Q_{-t_{1}-s+1}\to G(P_{-s+1})\oplus Q% _{-t_{1}-s+2}\to\cdots$$ where $\cdots\to Q_{-1}\to Q_{0}$ is the minimal projective resolution of $Z_{1}$, and where we have set $Q_{i}=0$ for $-i>{\rm pd}_{\Lambda}Z_{1}$. Also, $C_{1}$ is now exact in degrees $\leq t_{1}$. We continue inductively: suppose that $C_{i}$ has been constructed and let $t_{i+1}<0$ be the least integer, if any, such that $C_{i}$ has non-zero cohomology $Z_{i+1}$ in degree $t_{i+1}$. Then ${\rm Ext}^{j}_{\Lambda}(Z_{i+1},S_{e})=0$ for all $j\geq\ell$ and ${\rm Ext}^{\ell-1}_{\Lambda}(Z_{i+1},S_{e})\neq 0$. Consider the canonical morphism $Z_{i+1}[-t_{i+1}]\to C_{i}$ with exact triangle $$Z_{i+1}[-t_{i+1}]\to C_{i}\to C_{i+1}$$ where $C_{i+1}$ is exact in degrees $\leq t_{i+1}$. Now, there is some $m\geq 1$ such that $C_{m}$ is quasi-isomorphic to a module with finite projective dimension. Assume to the contrary that $GN\cong C$ has cohomology in degree $j$ where $j\leq{\rm min}(-1,-r+\ell-1)$. That means $t_{1}\leq j\leq{\rm min}(-1,-r+\ell-1)$. So $t_{1}$ is negative and $t_{1}\leq-r+\ell-1$. By applying the functor ${\rm Hom}(-,S_{e})$ and using that for all $i\geq 1$, we have ${\rm Hom}(Z_{i},S_{e}[\ell-1])\neq 0$ and that ${\rm Hom}(Z_{i},S_{e}[q])=0$ for all $q\geq\ell$, we get ${\rm Hom}(C_{1},S_{e}[-t_{1}+\ell])\cong{\rm Hom}(Z_{1}[-t_{1}],S_{e}[-t_{1}+% \ell-1])\neq 0$ and ${\rm Hom}(C_{1},S_{e}[-t_{1}+q+1])\cong{\rm Hom}(Z_{1}[-t_{1}],S_{e}[-t_{1}+q]% )=0$ for all $q\geq\ell$. By induction, we get ${\rm Hom}(C_{i},S_{e}[-t_{1}+\ell])\cong{\rm Hom}(C_{i+1},S_{e}[-t_{1}+\ell])\neq 0$ and ${\rm Hom}(C_{i+1},S_{e}[-t_{1}+q+1])=0$ for all $q\geq\ell$. In particular, we get ${\rm Hom}(C_{m},S_{e}[-t_{1}+\ell])\neq 0$, so ${\rm pd}_{\Lambda}C_{m}\geq-t_{1}+\ell\geq r-\ell+1+\ell=r+1$, a contradiction. ∎ Corollary 2.4. Assume that ${\rm findim}\,\Lambda=r<\infty$, that ${\rm pd}_{\Lambda}S_{e}<\infty$ and that $Y(e)$ has uniform graded loewy length $\ell$. Then ${\rm findim}\,\Gamma\leq 2r-\ell+1$. Proof. Let $N$ be a $\Gamma$-module with projective dimension $s$. By lemma 2.3, the minimal projective resolution of $GN$ has no cohomology in degree $i$ where $i\leq{\rm min}(-1,-r+\ell-1)$. Therefore, we get a $\Lambda$ module with projective dimension equal to $s-(r-\ell+1)$ whenever $s-r+\ell-1$ is non-negative. Since ${\rm findim}\,\Lambda=r$, we get $s-r+\ell-1\leq r$ when $s-r+\ell-1\geq 0$, so $s\leq 2r-\ell+1$. If $s-r+\ell-1<0$, then $s\leq r-\ell$ so $s\leq 2r-\ell+1$ as well. ∎ Note that if $e$ is primitive, then $Y(e)$ always has uniform graded loewy length, when ${\rm pd}_{\Lambda}S_{e}$ is finite. Theorem 2.5. Assume that ${\rm findim}\,\Lambda=r<\infty$ and ${\rm pd}_{\Lambda}S_{e}<\infty$ where $e$ is primitive. Then ${\rm findim}\,\Gamma\leq 2r-\ell+1$ where $\ell<\infty$ is the greatest positive integer with ${\rm Ext}^{\ell-1}_{\Lambda}(S_{e},S_{e})\neq 0$. 3. Applications In this section, we assume that $\Lambda$ is finite-dimensional over an algebraically closed field $k$. For our applications, there is no loss of generality in assuming that $\Lambda$ is basic. Therefore, there exists a finite quiver $Q$ and an admissible ideal $I$ of $kQ$ such that $\Lambda$ is isomorphic to $kQ/I$. We will therefore assume that $\Lambda=kQ/I$. 3.1. Projective ideals We start with the following result, whose proof is essentially due to Ágoston, Happel, Lukács and Unger; see [1]. Proposition 3.1. Let $e$ be primitive such that $J:=\Lambda e\Lambda$ is projective as a left module. Then $(1)$ ${\rm findim}\Lambda\leq{\rm findim}\Lambda/J+2$. $(2)$ ${\rm findim}\Lambda/J\leq{\rm findim}\Lambda.$ $(3)$ ${\rm findim}\Lambda<\infty$ if and only if ${\rm findim}\Lambda/J<\infty$. Proof. The last part is a consequence of the first two parts. For the first part, see [1, Theorem 2.2]. Let us prove the second part. Assume that ${\rm findim}\Lambda$ is finite and equal to $r$. Let $M$ be a finitely generated left $\Lambda/J$-module (that is, a finitely generated left $\Lambda$-module with $JM=0$) of finite projective dimension. Observe that the indecomposable projective $\Lambda/J$-modules are $\Lambda e_{j}/Je_{j}$, for $1\leq j\leq n$. Since $Je_{j}$ is projective, we get that a projective $\Lambda/J$-module has projective dimension at most one, when seen as a $\Lambda$-module. This means that $M$ has finite projective dimension as a $\Lambda$-module, so ${\rm pd}_{\Lambda}M\leq r$, which implies ${\rm pd}_{\Lambda/J}M\leq r$ by [11, Lemma 1.2]. ∎ Remark 3.2. Note that if $e$ is a general idempotent with $\Lambda e\Lambda$ projective as a left module, then $\Lambda e\Lambda$ is a stratifying ideal. It is well known that in this case, there is a recollement $$\xymatrix{D^{-}(\Lambda/\Lambda e\Lambda)\ar[r]&D^{-}(\Lambda)\ar@<8pt>[l]\ar@% <-8pt>[l]\ar[r]&D^{-}(e\Lambda e)\ar@<8pt>[l]\ar@<-8pt>[l]}$$ of the derived categories. Note that the fact that $\Lambda e\Lambda$ is projective as a left module implies that $e\Lambda$ is a projective $e\Lambda e$-module. If we assume further that $\Lambda e$ has finite projective dimension as a right $e\Lambda e$-module, then the above recollement restricts to a recollement $$\xymatrix{D^{b}(\Lambda/\Lambda e\Lambda)\ar[r]&D^{b}(\Lambda)\ar@<8pt>[l]\ar@% <-8pt>[l]\ar[r]&D^{b}(e\Lambda e)\ar@<8pt>[l]\ar@<-8pt>[l]}$$ and in this case Happel has shown in [13] that ${\rm findim}\,\Lambda<\infty$ if and only if ${\rm findim}\,\Lambda/\Lambda e\Lambda<\infty$ and ${\rm findim}\,e\Lambda e<\infty$. Note that when $e$ is primitive, one has ${\rm findim}\,e\Lambda e=0$. However, the condition that $\Lambda e$ has finite projective dimension as a right $e\Lambda e$-module is not automatically satisfied. Therefore, when $e$ is primitive, Proposition 3.1 is stronger than the above fact. An element $r\in kQ$ is called uniform if it is a linear combination of parallel paths. Let us fix a finite set $\{r_{1},r_{2},\ldots,r_{t}\}$ of uniform generators of $I$. If $x$ is an element of $Q_{0}\cup Q_{1}$ (which can naturally be thought of as an element of $kQ$ or of $\Lambda$) and $r\in kQ$ is such that no term of $r$ can be factorized through $x$, then we say that $r$ does not involve $x$, or that $x$ does not appear in $r$. In what follows, we will need Gr$\ddot{o}$bner bases. We refer the reader to [4, 5] for the notions needed concerning Gr$\ddot{o}$bner bases. We fix an admissible order on the set of all paths of $Q$ and all Gr$\ddot{o}$bner bases will be with respect to that given order. The tip of $x\in kQ$ is the largest path occurring in $x$, with respect to that order. Lemma 3.3. Assume that $x\in Q_{0}\cup Q_{1}$ is such that the $r_{i}$ do not involve $x$. Then there is a Gr$\ddot{o}$bner basis of the ideal $I$ consisting of uniform elements of $kQ$, all of which do not involve $x$. Proof. We start with the generators $\{r_{1},\ldots,r_{t}\}$ of $I$ and we apply Corollary 2.11 of [5] to get a finite set $\{s_{1},\ldots,s_{m}\}$ of uniform tip-reduced elements of $kQ$ that generate $I$. This new set of generators will still have the same property that the $s_{i}$ do not involve $x$, since they are obtained from the $r_{i}$ by applying simple reductions. Recall that an overlap relation between $x,y\in kQ$ is an element $\lambda_{x}xm-\lambda_{x}ny$ where $m,n$ are paths of length at least one, the length of $m$ is strictly less than that of tip$(y)$, ${\rm tip}(x)m=n{\rm tip}(y)$ and $\lambda_{x},\lambda_{y}$ are the non-zero coefficients of ${\rm tip}(x),{\rm tip}(y)$ in $x,y$, respectively. We first observe that any overlap relation between $s_{i},s_{j}$ is such that the paths $m,n$ do not involve $x$. If such a relation cannot be reduced to zero, it can be reduced to a linear combination of paths still not involving $x$. Now, according to Theorem 2.13 in [5] and the remark before it (see also Section 2.4.1 in [4]), we see that after a possibly infinite number of steps, we can get a (possibly infinite) Gr$\ddot{o}$bner basis of $I$ consisting only of linear combinations of paths not involving $x$. ∎ Corollary 3.4. Assume that $x\in Q_{0}\cup Q_{1}$ is such that the $r_{i}$ do not involve $x$. Then the algebra $\Lambda=kQ/I$ has a basis $\mathcal{N}$ consisting of residue classes of paths such that if $p_{1},p_{2}$ lie in that basis, then so does $p=p_{1}xp_{2}$. Proof. By Lemma 2.6 in [5], a basis $\mathcal{N}$ of $\Lambda$ comes from the residue class of paths $t$ such that no subpath of $t$ is the tip of an element in the constructed Gr$\ddot{o}$bner basis. Assume that both $p_{1},p_{2}$ lie in that basis $\mathcal{N}$. Since the tip of any element in our Gr$\ddot{o}$bner basis is a path not involving $x$, we see that $p$ has no subpath equal to the tip of an element in our Gr$\ddot{o}$bner basis. ∎ Proposition 3.5. Let $\Lambda=kQ/I$ be such that $I$ is generated by relations not involving a given element $x$ of $Q_{0}\cup Q_{1}$. Then the two-sided ideal generated by $x+I$ in $\Lambda$ is projective as a left and as a right $\Lambda$-module. Proof. We only prove the left version where $x=\alpha$ is an arrow. Assume $\alpha:s\to t$ and let $\mathcal{N}$ be the basis from the above corollary. Let $p_{1},\ldots,p_{r}$ be the paths in $\mathcal{N}$ ending in $s$ and let $q_{1},\ldots,q_{s}$ be the paths in $\mathcal{N}$ starting in $t$. The $p_{i}$ form a basis of $e_{s}\Lambda$ and the $q_{j}$ form a basis of $\Lambda e_{t}$. By the corollary, we see that the $q_{j}\alpha p_{i}$ form a basis of $\Lambda\alpha\Lambda$. Let $\pi:(\Lambda e_{t})^{r}\to\Lambda\alpha\Lambda$ be the morphism given by $\pi(a_{1},\ldots,a_{r})=\sum a_{i}\alpha p_{i}$. It is clear that $\pi$ is surjective. Now, the remark above implies it is injective. This proves that $\Lambda\alpha\Lambda$ is a projective left $\Lambda$-module. ∎ Now, assume that $\alpha:s\to t$ is an arrow of $Q$. Let $Q^{\prime}$ be the quiver obtained from $Q$ by removing the arrow $\alpha$ and replacing it with a path of length two $\alpha_{2}\alpha_{1}$ where $\alpha_{1}:s\to u$ and $\alpha_{2}:u\to t$, and where $u$ is a new vertex of $Q^{\prime}$. We let $I$ be the same ideal, but seen as an ideal of $kQ^{\prime}$. We let $B=kQ^{\prime}/I$. Observe that $B/Be_{u}B\cong\Lambda/\Lambda\alpha\Lambda$. Moreover, it is clear that $Be_{u}B$ is a projective left $B$-module, since the $r_{i}$ do not involve $u$. It follows from Proposition 3.1 that findim$\,B<\infty$ if and only if findim$\,B/Be_{u}B<\infty$. Therefore, we get the following, which has first been proven in [12], in case $x$ is an arrow. Proposition 3.6. Let $\Lambda=kQ/I$ be such that $I$ is generated by relations not involving a given element $x$ of $Q_{0}\cup Q_{1}$. Let $J$ be the two sided ideal generated by $x$. Then $(1)$ ${\rm findim}\Lambda\leq 2\,{\rm findim}\Lambda/J+4$. $(2)$ ${\rm findim}\Lambda/J\leq{\rm findim}\Lambda.$ $(3)$ ${\rm findim}\Lambda<\infty$ if and only if ${\rm findim}\Lambda/J<\infty$. Proof. Assume that ${\rm findim}\,\Lambda/J=r<\infty$. Then ${\rm findim}\,B/Be_{u}B=r$, so ${\rm findim}\,B\leq r+2$ by Proposition 3.1. Observe that $\Lambda=(1-e_{u})B(1-e_{u})$ and that the simple $B$-module at $u$ has projective dimension $1$. Therefore, by Theorem 2.5, we have ${\rm findim}\,\Lambda\leq 2(r+2)-\ell+1\leq 2(r+2)$ since the graded loewy length $\ell$ of $Y(e)$ is at least one. This proves (1). Statement (2) follows from the fact that $J$ is projective; see Proposition 3.1, part (2). ∎ 3.2. Almost vanishing ideals In this subsection, we consider a left ideal $J$ such that $J=Je$ for some primitive idempotent $e$. We assume further that $J{\rm rad}\Lambda=0$. This implies that $J$ is actually a two-sided ideal. If $J$ is not included in the radical of $\Lambda$, then $J=\Lambda e$ with $J{\rm rad}\Lambda=0$ so $e$ has to be a source vertex. It is easy to see that ${\rm findim}\,\Lambda<\infty$ if and only if ${\rm findim}\,\Lambda/J<\infty$ in that case. Therefore, we will assume that $J\subseteq{\rm rad}\Lambda$. Hence, $J$ is an $\Lambda/J$-module. Now, there are uniform elements $\{r_{1},\ldots,r_{t}\}$ of $kQ$ with $r_{i}=r_{i}e$ for $1\leq i\leq t$ that generate $J$ and such that $r_{i}{\rm rad}\Lambda=0$ for all $i$. We may assume that $S:=\{r_{1},\ldots,r_{t}\}$ is a minimal generating set of $J$. We construct a new quiver $Q_{J}$ by adding a new vertex $x$ and the following arrows to $Q$. We add an arrow $\alpha:v\to x$. For each $r_{i}\in S$, we add an arrow $\beta_{i}:x\to t(r_{i})$, where $t(r_{i})$ is the terminal vertex of $r_{i}$ (recall $r_{i}$ is uniform). We define an ideal $I_{J}$ of $Q_{J}$ by adding relations to $I\subseteq kQ_{J}$ as follows. For each $r_{i}\in S$, we add $r_{i}-\beta_{i}\alpha$; for each $\gamma\in(Q_{J})_{1}$ with $t(\gamma)=v$, we add $\alpha\gamma$. Finally, we impose that $\sum_{i=1}^{t}a_{i}\beta_{i}\in I_{J}$ if and only if $\sum_{i=1}^{t}a_{i}r_{i}\in I$, where the $a_{i}$ are in $kQ\subset kQ_{J}$. We set $B=kQ_{J}/I_{J}$. Lemma 3.7. With the notations of the previous paragraph, $Be_{x}B$ is a projective left ideal of $B$ with $B/Be_{x}B\cong\Lambda/J$. Proof. The fact that $B/Be_{x}B\cong\Lambda/J$ follows from the definition of $B$. Observe that $Be_{x}B=Be_{x}B(e+e_{x})=Be_{x}Be\oplus Be_{x}$, where the second summand is projective. Therefore, it remains to show that $Be_{x}Be=B\alpha\cong Be_{x}$, which amounts to proving that if $b=be_{x}$ is non-zero in $B$, then $b\alpha$ is non-zero in $B$. Therefore, assume that $b=be_{x}$ is such that $b\alpha\in I_{J}$. In particular, $b$ is represented by a linear combination of paths of positive lengths in $kQ_{J}$. Therefore, $b=b_{1}\beta_{1}+\cdots+b_{t}\beta_{t}+I_{J}$ and where the $b_{i}$ are in $kQ$. Now, $b\alpha=b_{1}r_{1}+\cdots b_{t}r_{t}$ lies in $I$, which is equivalent to $b_{1}\beta_{1}+\cdots b_{t}\beta_{t}$ lying in $I_{J}$, a contradiction. ∎ Proposition 3.8. Let $J$ be a submodule of $\Lambda e$ for $e$ primitive and assume that $J{\rm rad}\Lambda=0$. Thus, $J$ is a two-sided ideal. Assume further that ${\rm pd}_{\Lambda/J}J<\infty$. Then ${\rm findim}\,\Lambda\leq 2\,{\rm findim}\,\Lambda/J+4$. Proof. Assume that ${\rm findim}\,\Lambda/J=r<\infty$. Consider the algebra $B$ as constructed above, with idempotent $e_{x}$ such that $\Lambda/J\cong B/Be_{x}B$. We first observe that ${\rm findim}\,B\leq r+2$. This follows from Proposition 3.1 and Lemma 3.7. Consider the simple module $S_{x}$ supported at $e_{x}$ in $B$. Observe that its first syzygy is $\Omega=\sum_{i}B\beta_{i}$. Note that $e_{x}\Omega=0$, so ${\rm pd}_{B}\Omega<\infty$ if and only if ${\rm pd}_{B/Be_{x}B}\Omega<\infty$. However, through the isomorphism $B/Be_{x}B\cong\Lambda/J$, $\Omega$ corresponds to $J$. Therefore, by using the hypothesis, this yields ${\rm pd}_{B}\Omega<\infty$, so $S_{x}$ has finite projective dimension. By Theorem 2.5, this yields ${\rm findim}\,\Lambda\leq 2(r+2)-\ell+1\leq 2r+4$. ∎ We end with an example to illustrate this result. Example 3.9. Let $Q$ be the quiver given by $$\xymatrixrowsep{10pt}\xymatrixcolsep{10pt}\xymatrix{&1\ar[dl]_{\alpha}\ar[dr]^% {\gamma}&\\ 2\ar[dr]_{\beta}&&3\ar[dl]^{\delta}\\ &4\ar[uu]^{\epsilon}&}$$ with admissible ideal $I=\langle\beta\alpha-\delta\gamma,\epsilon\delta,\gamma\epsilon,\alpha\epsilon\beta\rangle$. Let $\Lambda=kQ/I$. Consider the two-sided ideal $J=\langle\alpha\epsilon\rangle$. It clearly satisfies the first hypothesis of Proposition 3.8, since $J$ is a one-dimensional radical ideal with $J=Je_{4}$. Observe that as an $\Lambda/J$-module, $J$ has projective resolution $$0\to\frac{\Lambda e_{4}}{Je_{4}}\to\frac{\Lambda e_{2}}{Je_{2}}\to J\to 0$$ where $Je_{4}=J$ and $Je_{2}=0$. Therefore, Proposition 3.8 yields that ${\rm findim}\,\Lambda\leq 2\,{\rm findim}\,\Lambda/J+4$. It is easily checked that the global dimension of $\Lambda/J$ is $4$. Therefore, ${\rm findim}\,\Lambda\leq 12$. One easy consequence of Proposition 3.8 is the following. Corollary 3.10. 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Schroll, Algebras and varieties, preprint, arXiv:1707:07877. [6] E. Green, ${\O}$. Solberg and C. Psaroudakis, Reduction techniques for the finitistic dimension, preprint, arXiv:1808.03564. [7] C. Ingalls and C. Paquette, Homological dimensions for co-rank one idempotent subalgebras, Trans. AMS 369 (2017), no. 8, 5317–5340. [8] C. Ingalls and C. Paquette, Homological behavior of idempotent subalgebras and Ext algebras, to appear in Sci. China Math., 2018, arXiv:1703.08725. [9] K. Igusa, S. Liu and C. Paquette, A proof of the strong no loop conjecture, Adv. Math. 228 (2011), no. 5, 2731–2742. [10] O. Iyama Finiteness of representation dimension, Proc. of the AMS 131 (2002), no. 4, 1011–1014. [11] S. Liu and C. Paquette, Some homological conjectures for quasi-stratified algebras, Journal of Algebra 301 (2006), 240–255. [12] E. Green, Ø. Solberg and C. Psaroudakis, Reduction techniques for the finitistic dimension, preprint, arXiv:1808.03564. [13] D. 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Coexistence of Majorana and topologically nontrivial Andreev bound states in 1D superconductors Pasquale Marra pasquale.marra@keio.jp    Muneto Nitta nitta@phys-h.keio.ac.jp Department of Physics, and Research and Education Center for Natural Sciences, Keio University, 4-1-1 Hiyoshi, Yokohama, Kanagawa 223-8521, Japan Abstract The unambiguous detection of Majorana bound states in nanowires and in magnetic atom chains is hindered by the possible presence of near-zero-energy Andreev bound states which have similar experimental signatures. These near-zero energy states are expected to be topologically trivial. Here, we report the theoretical prediction of topologically nontrivial Andreev bound states in one-dimensional superconductors with spatially varying magnetic fields. These states correspond to a novel topological invariant defined in a synthetic two-dimensional space, the particle-hole Chern number, which is an analogue of the spin Chern number in quantum spin Hall systems. Topologically nontrivial Andreev bound states and Majorana bound states have distinct features and are topologically nonequivalent. Yet, they can coexist in the same system, have similar spectral signatures, and materialize with the concomitant opening of the particle-hole gap. Consequently, the simultaneous observation of a zero-bias peak and the closing and reopening of the gap cannot be considered an exclusive fingerprint of Majorana bound states. In contrast to Majorana states, which appear simultaneously at both edges and at zero energy, nontrivial Andreev states may appear with different energies at the opposite edges of the system. Majorana bound stateskitaev_unpaired_2001 are topologically protected zero-energy quasiparticle excitations which localize at the edges or at topological defects of nontrivial superconductorsqi_topological_2011 ; alicea_new_2012 ; leijnse_introduction_2012 ; beenakker_search_2013 ; stanescu_majorana_2013 ; sato_topological_2017 ; aguado_majorana_2017 . Besides their fundamental importance as a solid state counterpart of Majorana fermions in high-energy physics, their non-abelian statistics makes them ideal building blocks for decoherence-free and fault-tolerant topological quantum computationnayak_non-abelian_2008 . Their theoretically predicted features have been experimentally observed in both spin-orbit coupled Rashba nanowiresmourik_signatures_2012 ; das_zero-bias_2012 ; rokhinson_fractional_2012 ; deng_majorana_2016 ; albrecht_exponential_2016 ; zhang_quantized_2018 and in magnetic atom chainsnadj-perge_observation_2014 ; pawlak_probing_2016 ; feldman_high-resolution_2017 . Despite this experimental evidence, their signatures, in particular the zero-bias peak and the quantized conductance, are consistent with alternative theoretical explanations, such as topologically trivial Andreev bound states, which can be induced by disorder or by spatial variations of the gate potential used to confine the nontrivial phaseasano_phenomenological_2004 ; tanaka_theory_2005 ; golubov_andreev_2009 ; tanaka_anomalous_2010 ; liu_zero-bias_2012 ; kells_near-zero-energy_2012 ; roy_topologically_2013 ; stanescu_disentangling_2013 ; cayao_sns_2015 ; san-jose_majorana_2016 ; liu_andreev_2017 ; liu_distinguishing_2018 ; moore_two-terminal_2018 ; moore_quantized_2018 ; fleckenstein_decaying_2018 ; awoga_supercurrent_2019 . Here, we report the theoretical prediction of a novel time-reversal-symmetry breaking nontrivial topological phase in one-dimensional (1D) superconductors, which is topologically distinct from the well-known Majorana phase, and exhibits topologically nontrivial Andreev bound states. This new topological phase can be realized in the presence of amplitude-modulated Zeeman fieldskjaergaard_majorana_2012 ; klinovaja_transition_2012 ; li_manipulating_2016 ; marra_controlling_2017 . In magnetic atom chains, for instance, the superposition of the intrinsic magnetic helical order and an externally applied field can produce a total Zeeman field which is harmonically amplitude-modulated. In this case, the field felt by the superconducting electrons depends on the phase-offset of the harmonic modulation, which can be regarded as an additional synthetic dimensionkraus_four-dimensional_2013 ; celi_synthetic_2014 ; price_four-dimensional_2015 ; marra_fractional_2015 ; kraus_quasiperiodicity_2016 ; marra_fractional_2017 ; zilberberg_photonic_2018 ; lohse_exploring_2018 ; ozawa_topological_2019 ; park_fractional_2016 ; thakurathi_fractional_2018 . Consequently, the system sits into two different entries of the periodic table of nontrivial phasesschnyder_classification_2009 ; kitaev_periodic_2009 ; ryu_topological_2010 corresponding to 1 and 2 dimensions in the class D. Hence, it can be characterized by two topological invariants: The Majorana number $M$, defined in the 1D space, and the particle-hole (PH) Chern number $C_{ph}$, defined in the synthetic 2D space. These two topological invariants correspond to two different kind of edge states: The familiar Majorana bound states (MBS) and the topologically protected Andreev bound states (ABS). These novel kind of nontrivial ABS are dispersive, fully spin and PH polarized, protected by the PH symmetry, but distinct from and topologically nonequivalent to MBS. The resulting topological phase space is a remarkable example of coexistence of multiple nontrivial phases in the same system, which does not originate from symmetry breaking but from the presence of an additional synthetic dimension. Moreover, nontrivial ABS can be in principle more problematic than trivial ABS, for the fact that they are topologically protected, and hence they can materialize only after the closing and reopening of the PH gap. This can in principle invalidate the unambiguous detection of MBS, since the simultaneous observations of the opening of the gap and of a zero-bias or near-zero-bias peak at the edges cannot be considered as conclusive evidence of MBSgrivnin_concomitant_2019 . To illustrate these findings, we consider a chain of magnetic Yu-Shiba-Rusinov statesklinovaja_topological_2013 ; braunecker_interplay_2013 ; vazifeh_self-organized_2013 ; pientka_topological_2013 ; poyhonen_majorana_2014 ; pientka_unconventional_2014 ; choy_majorana_2011 ; nadj-perge_proposal_2013 on the surface of a conventional superconductor, in the presence of a helical spin order and an externally applied Zeeman field, as in Fig. 1. This system can be modeled by a Bogoliubov-de Gennes tight-binding Hamiltonian, which reads $$\displaystyle H=\frac{1}{2}\sum_{n}\boldsymbol{\Psi}_{n}^{\dagger}\!\cdot\!% \begin{bmatrix}-\mu\sigma_{0}+\mathbf{b}_{n}\cdot\boldsymbol{\sigma}&\Delta\ii% \sigma_{y}\\ -(\Delta\ii\sigma_{y})^{*}&\mu\sigma_{0}-(\mathbf{b}_{n}\cdot\boldsymbol{% \sigma})^{*}\end{bmatrix}\!\cdot\!\boldsymbol{\Psi}_{n}+$$ $$\displaystyle-\frac{1}{2}\sum_{n}\boldsymbol{\Psi}_{n}^{\dagger}\!\cdot\!% \begin{bmatrix}t\sigma_{0}-\lambda\ii\sigma_{y}&\!\!\!\!\!\!\!\!0\\ \!\!\!\!0&\!\!\!\!\!\!\!\!\!\!-(t\sigma_{0}-\lambda\ii\sigma_{y})\end{bmatrix}% \!\cdot\!\boldsymbol{\Psi}_{n+1}+\text{h.\leavevmode\nobreak\ c.},$$ (1) where $\boldsymbol{\Psi}_{n}^{\dagger}=[c^{\dagger}_{n{\uparrow}},c^{\dagger}_{n{% \downarrow}},c_{n{\uparrow}},c_{n{\downarrow}}]$ is the Nambu spinor, $\mu$ the chemical potential, $t$ the hopping parameter, $\lambda$ the intrinsic spin-orbit coupling due to the inversion symmetry breaking at the surface, $\Delta$ the superconducting pairing induced by the substrate, and $\mathbf{b}_{n}$ the total field Zeeman on each site. The helical spin order is induced by the Ruderman-Kittel-Kasuya-Yosida (RKKY) coupling between localized magnetic moments of the chain, and is resonantly enhanced by the perfect nesting between Fermi momenta $\pm k_{\mathrm{F}}$ in 1D systems. This nesting condition fixes the spatial frequency $\theta$ of the helix as $\theta=2k_{\mathrm{F}}$braunecker_nuclear_2009 ; braunecker_spin-selective_2010 , which mandates $\mu=-2t\cos{(\theta/2)}$, which we assume hereafter. Moreover, in the absence of externally applied fields, the direction of the spin helix is fixed by symmetry. Indeed, the spin-orbit coupling (along $y$) breaks the $SU(2)$ spin-rotation symmetry down to $U(1)$ rotations in the $zx$ plane: Hence, the helical order becomes preferentially pinned to the $zx$ plane, and remains pinned for applied fields smaller than the spin-orbit energy splittingli_manipulating_2016 . For simplicity, we assume that the helical order is independent of the externally applied field. Thus, the total field can be written as $\mathbf{b}_{n}=\mathbf{b}+\Re(e^{\ii(n\theta+\phi)}\boldsymbol{\delta}\mathbf{% b})$, where $\mathbf{b}$ is the applied field and $\boldsymbol{\delta}\mathbf{b}=(-\ii\delta b,0,\delta b)$. Here, $\theta$, $\phi$, and $\delta b$ are respectively the spatial frequency, phase-offset, and magnitude of the rotating field induced by the helical spin order. The cases with $\delta b=0$ and $b=0$ reduce respectively to the well-known regimes where only the uniformoreg_helical_2010 ; lutchyn_majorana_2010 and the helical fieldchoy_majorana_2011 ; kim_helical_2014 ; nadj-perge_proposal_2013 are present. If the applied and helical fields are not perpendicular $\boldsymbol{\delta}\mathbf{b}\cdot\mathbf{b}\neq 0$, the total field is amplitude-modulated, $|\mathbf{b}_{n}|^{2}=b^{2}+\delta b^{2}+2\Re(e^{\ii(n\theta+\phi)}\boldsymbol{% \delta}\mathbf{b}\cdot\mathbf{b})$, and depends explicitly on the phase-offset $\phi$, which cannot be absorbed by local or global unitary rotations of the spin basisbraunecker_spin-selective_2010 ; choy_majorana_2011 . Thus, the energy spectrum, and in particular the PH gap, depends on the phase-offset $\phi$. Since we are interested in this regime, we assume that the applied field is coplanar with the helical field. Besides, one can always rotate the spin basis such that the applied field is parallel to the $z$ axis, which we assume hereafter. Notice that, assuming a rigid and uniformly rotating spin helix, the magnetic order is degenerate in the phase-offset $\phi$, even in the presence of external fields $\mathbf{b}\neq 0$, since the coupling between applied field and the helical order $\propto\mathbf{b}\cdot\mathbf{m}$ vanishes, being the total magnetization $\mathbf{m}=\sum_{n}\Re(e^{\ii(n\theta+\phi)}\boldsymbol{\delta}\mathbf{b})=0$. However, the effect of the applied field on the spin helix may induce a finite magnetization and break the $U(1)$ invariance. If these effects are negligible, the phase-offset $\phi$ will become pinned by arbitrarily small local variations of the Zeeman field or by defects and impurities along the chain. In order to define the topological invariants, it is useful to Fourier-transform the Hamiltonian (1), which yield $$\displaystyle H=\frac{1}{2}\sum_{k}\boldsymbol{\Psi}_{k}^{\dagger}\!\cdot\!% \begin{bmatrix}h(k)&\Delta\ii\sigma_{y}\\ -(\Delta\ii\sigma_{y})^{*}&-h(-k)^{*}\end{bmatrix}\!\cdot\!\boldsymbol{\Psi}_{% k}+$$ $$\displaystyle+\frac{1}{4}\sum_{k}\boldsymbol{\Psi}_{k+\theta}^{\dagger}\!\cdot% \!\begin{bmatrix}e^{\ii\phi}\boldsymbol{\delta}\mathbf{b}\cdot\boldsymbol{% \sigma}&0\\ 0&-e^{\ii\phi}\boldsymbol{\delta}\mathbf{b}\cdot\boldsymbol{\sigma}^{*}\end{% bmatrix}\!\cdot\!\boldsymbol{\Psi}_{k}+\text{h.\leavevmode\nobreak\ c.},$$ (2) where $\boldsymbol{\Psi}_{k}^{\dagger}=[c^{\dagger}_{k{\uparrow}},c^{\dagger}_{k{% \downarrow}},c_{-k{\uparrow}},c_{-k{\downarrow}}]$ and $h(k)=\mathbf{b}\!\cdot\!\boldsymbol{\sigma}-(\mu+2t\cos{k})\sigma_{0}+2\lambda% \sin{k}\,\sigma_{y}$. We notice that for $\Delta=\lambda=0$, Eq. 2 reduces to the Harper-Hofstadter Hamiltonian realized in topological quantum pumpsharper_single_1955 ; hofstadter_energy_1976 ; thouless_quantization_1983 ; hatsugai_energy_1990 . Due to the coupling between different momenta, the Hamiltonian is invariant up to momenta translations $k\to k+\theta$. Assuming a spatial frequency commensurate to the lattice, i.e., $\theta=2\pi p/q$ with $p,q$ integer coprimes, this symmetry induces a periodicity in momentum space $\Delta k=2\pi/q$ and a folding of the energy levels into a reduced Brillouin zone (BZ) $[0,2\pi/q]$. The model exhibits PH symmetry $\Xi H\Xi^{\dagger}=-H$ (with $\Xi^{2}=1$), with broken time-reversal symmetry, and belongs to the Altland-Zirnbauerschnyder_classification_2009 ; kitaev_periodic_2009 ; ryu_topological_2010 symmetry class D. Hence, gapped phases are characterized by a $\mathbb{Z}_{2}$ topological invariant, the Majorana numberkitaev_unpaired_2001 . The nontrivial phase can be realized at any finite spin-orbit coupling (intrinsic $\lambda>0$, or effective, induced by the helical field $\delta b>0$), and exhibits MBS localized at the edges. In order to calculate the Majorana number, it is convenient to rewrite Hamiltonian Eq. 2 in terms of Majorana operators $\gamma^{A}_{k{\uparrow}{\downarrow}}=(c_{k{\uparrow}{\downarrow}}^{\dagger}+c_% {k{\uparrow}{\downarrow}})$ and $\gamma^{B}_{k{\uparrow}{\downarrow}}=\ii(c_{k{\uparrow}{\downarrow}}^{\dagger}% -c_{k{\uparrow}{\downarrow}})$. Hence, the Majorana number $M$ can be defined as usual as the sign of the product of the Pfaffian of the Hamiltonian $\widetilde{H}$ in the Majorana basis evaluated at the time-reversal symmetry points $k=0,\pi/q$ in the reduced BZ, i.e., $M=\operatorname{sign}(\operatorname{pf}(\ii\widetilde{H}_{0})\operatorname{pf}% (\ii\widetilde{H}_{\pi/q}))$ where $\widetilde{H}_{k}=\sum_{nm=0}^{q-1}P_{k+n\theta}\widetilde{H}P_{k+m\theta}^{\dagger}$ are the projections of $\widetilde{H}$ onto the subspace spanned by momenta $k+n\theta$, with $P_{k}$ the projector operators. Note that, in general, the Majorana number depend periodically on the phase-offset $\phi$. Phase transitions between trivial and nontrivial phases are determined by the closing of the PH gap, i.e., $E(k,\phi)=0$ for either $k=0$ or $\pi/q$. For clarity, we will focus here only on the phases which are globally gapped, i.e., where the PH gap is finite for any value of the phase-offset $\phi$. We define the global PH gap as $E_{G}=\min_{k,\phi}E(k,\phi)$. Globally gapped phases $E_{G}>0$ are either trivial or nontrivial. Conversely, phases which are not globally gapped $E_{G}=0$, i.e., where the PH gap closes for some values of the phase-offset, may be trivial $M=1$ and nontrivial $M=-1$ depending on the phase-offset $\phi$ (see Ref. marra_controlling_2017 ). In Fig. 2(a) we plot the value of the global PH gap $E_{G}$ as a function of the helical field magnitude $\delta b$ and applied field $b$, calculated by direct numerical diagonalization of the Hamiltonian Eq. 2 for $\theta=\pi/2$ (i.e., $q=4$). The globally gapped phases $E_{G}>0$ are separated by domains where the global PH gap vanishes. We then calculate the Majorana number numerically for each globally gapped phase. Due to the lack of broken time-reversal symmetry, the phase at zero field $b=\delta b=0$ (and at small fields $b\approx\delta b\approx 0$) is obviously trivial. At larger fields, there are two separated (but topologically equivalent) nontrivial phases with $M=-1$, which are realized respectively for strong applied fields and small (or zero) helical fields $b\gg\delta b$, and for strong helical fields and small (or zero) applied fields $\delta b\gg b$. The two separated phases with $M=-1$ reduce to the well-known regimes where only a uniform fieldoreg_helical_2010 ; lutchyn_majorana_2010 (with $b>0$ and $\delta b=0$), or the helical fieldchoy_majorana_2011 ; nadj-perge_proposal_2013 (with $b=0$ and $\delta{b}>0$) are present. In these regimes, topological superconductivity is realized respectively for $b^{-}<b<b^{+}$, with $b^{\pm}=[(|\mu|\pm 2t)^{2}+\Delta^{2}]^{1/2}$ [the $\delta b=0$ axis in Fig. 1(a)] and for $b_{\mathrm{eff}}^{-}<\delta b<b_{\mathrm{eff}}^{+}$ where $b_{\mathrm{eff}}^{\pm}=[(|\mu|\pm 2t_{\mathrm{eff}})^{2}+\Delta^{2}]^{1/2}$ and $t_{\mathrm{eff}}=t\cos{(\theta/2)}-\lambda\sin{(\theta/2)}$ [the $b=0$ axis in Fig. 1(a)], as one can show by unitary rotating Eq. 1 and calculating the Majorana number directly. Nontrivial phases with $M=-1$ exhibits MBS at zero energy, as in Fig. 2(b), where we show the energy spectra for $b=\delta b/3$, calculated by direct diagonalization of Eq. 1 with open nonperiodic boundary conditions. For amplitude-modulated fields $\boldsymbol{\delta}\mathbf{b}\cdot\mathbf{b}\neq 0$, the Hamiltonian in Eq. 2 depends periodically on the phase-offset $\phi$, which can be regarded as an additional synthetic (nonspatial) dimensionkraus_four-dimensional_2013 ; celi_synthetic_2014 ; price_four-dimensional_2015 ; marra_fractional_2015 ; kraus_quasiperiodicity_2016 ; marra_fractional_2017 ; zilberberg_photonic_2018 ; lohse_exploring_2018 ; ozawa_topological_2019 . The 1D chain is thus embedded in a 2D parameter space, which coincides with a synthetic BZ spanned by the momentum $k\in[0,\pi/q]$ and by the phase-offset $\phi\in[0,2\pi]$. Topological phases in 2D and symmetry class D are described by a $\mathbb{Z}$ topological invariant. We notice that the total Chern number is zero due to PH symmetry. Therefore, to describe the nontrivial globally gapped phases of the model, we shall introduce the PH Chern number, defined as the PH analogue of the spin Chern numberkane_quantum_2005 ; kane_$z_2$_2005 ; sheng_spin_2005 ; sheng_quantum_2006 ; bernevig_quantum_2006 . For any globally gapped phase $E_{G}>0$, which does not close when the superconducting paring is adiabatically turned off $\Delta\to 0$, we define $${C}^{\pm}=\frac{1}{2\pi}\int_{\mathrm{0}}^{2\pi/q}\!\!\!\dd k\int_{0}^{2\pi}\!% \!\!\dd\phi\,\left[\Omega_{p}(k,\phi)\pm\Omega_{h}(k,\phi)\right],$$ (3) where $\Omega_{p,h}(k,\phi)=\sum_{i\in p,h}2\Theta(-E_{i})\Im\braket{\partial_{k}\Psi% _{i}}{\partial_{\phi}\Psi_{i}}$ are the total Berry curvatures in the synthetic BZ, defined respectively for the two PH sectors of the Hamiltonian in Eq. 2 as a sum over all bands with $E_{i}<0$. Since particle states $\ket{\Psi_{i}}$ and their conjugate hole states $\Xi\ket{\Psi_{i}}=\ket{\Psi_{i}}^{*}$ give opposite contributions to the Berry curvature, one has that $\Omega_{h}(k,\phi)=-\Omega_{p}(k,\phi)$. Hence, the Chern number vanishes $C={C}^{+}=0$, whereas the PH Chern number $C_{ph}=C^{-}$ can be nonzero, and it is given by $$C_{ph}=2\times\frac{1}{2\pi}\int_{\mathrm{0}}^{2\pi/q}\!\!\!\dd k\int_{0}^{2% \pi}\!\!\!\dd\phi\,\Omega_{p}(k,\phi).$$ (4) The PH Chern number is thus an even integer due to PH symmetry. Notice that the PH Chern number is well-defined only if the phase with $\Delta>0$ can be adiabatically and continuously mapped into a phase with $\Delta=0$, without closing the global PH gap $E_{G}>0$. Only in this case indeed, the phase $\Delta>0$ is homeomorphic to the phase $\Delta=0$, where the Hamiltonian becomes block-diagonal in the PH sectors, and the PH Berry curvatures become well-defined. As a counterexample, notice the PH Chern number in Eq. 4 is not well defined for the nontrivial phase with $M=-1$, where the gap closes for $\Delta\to 0$. If the helical and applied fields are comparable, the model may realize a nontrivial phase characterized by a nonzero PH Chern number, as shown in Fig. 2(a). Using the Fukui-Hatsugai-Suzuki numerical methodfukui_chern_2005 applied separately in the PH sectors, we find that the PH Chern number of this globally gapped phase is ${C_{ph}}=-2$. The emergence of a nontrivial phase $C_{ph}\neq 0$ can be understood in terms of a band inversion induced by the applied field, and by considering the relation between this model and the Harper-Hofstadter model. Considering a continuous transformation $\Delta\to 0$, $\lambda\to 0$, and $\boldsymbol{\delta}\mathbf{b}=(-\ii\delta b,0,\delta b)\to(0,0,\delta b)$, each of the PH and spin up-down sectors of the Hamiltonian in Eq. 2 reduce to an Harper-Hofstadter Hamiltonian of spinless electrons on 1D lattice with harmonic potential $-\mu+\pm\delta b\cos{(\theta n+\phi)}$, with $\pm$ for spin up and down respectively. Thus, if the transformation does not close the global PH gap, the PH Chern number $C_{ph}$ can be obtained as the sum of the corresponding Chern numbers of the Hofstadter butterfly. Since opposite gaps $\pm\delta b$ of the butterfly have opposite Chern numbers, spin up and down contributions have opposite sign. Hence, if we define $j_{\uparrow}$ and $j_{\downarrow}$ as the intraband indexes of the particle spin up and down sectors of Eq. 2, using the diophantine equation characterizing the Hofstadter butterfly Chern numbers, Eq. 4 yields $$\displaystyle C_{ph}=2(C_{j_{\uparrow}}-C_{j_{\downarrow}}),$$ $$\displaystyle\text{where}\,\,pC_{j}\equiv j\!\!\!\mod q,\quad\text{with}\,\,|C% _{j}|<q/2.$$ (5) Here, $C_{j}$ are the Chern numbers labeling each of the intraband gaps $j$ of the Hofstadter butterflybellissard_noncommutative_1994 ; osadchy_hofstadter_2001 . Since the Hofstadter Chern numbers take all possible integer values $|C_{j}|<q/2$, the PH Chern number can take all possible even integer values $|C_{ph}|<q$. At zero applied field $b=0$, spin up and down bands are degenerate, and thus $j_{\uparrow}=j_{\downarrow}$, resulting in a trivial phase $C_{ph}=0$. However, spin degeneracy breaks at finite applied fields, and thus bands with $C_{j_{\uparrow}}\neq C_{j_{\downarrow}}$ can align at zero energy. Hence, the band inversion driven by the applied field $b$ can induce a nontrivial phase with $C_{ph}\neq 0$. These nontrivial phases correspond to the presence of nontrivial ABS localized at the edges. Nontrivial ABS are midgap excitations, and are completely PH and spin polarized. Due to bulk-edge correspondencehatsugai_chern_1993 ; imura_bulk-edge_2018 , each edge exhibits a number ${C_{ph}}/2$ of particle-like edge states, and the same number of hole-like edge states, which are PH conjugates one of the other. This has to be contrasted with MBS, which appear as a single zero-energy fermionic state localized at two opposite edges, and which are consequently PH symmetric, i.e., being their own PH conjugates. In Fig. 2(c) we show the energy spectra in the nontrivial phase ${C_{ph}}=-2$ (with $b=3\delta b$) calculated by direct diagonalization of Eq. 1 with open nonperiodic boundary conditions. The spectra show $2\times 2$ PH symmetric, nontrivial ABS inside the PH gap, with 2 edge states for each boundary of the chain. Despite the fundamental difference between MBS and nontrivial ABS, there are some similarities that need to be emphasized. Nontrivial ABS are midgap excitations, and can have zero energy only for fine-tuned values $\phi^{*}$ of the phase-offset. However, their energies can be lower than the experimental resolution, and thus the resulting near-zero bias peak can be erroneously attributed to the MBS. Most importantly, being topologically protected, they can materialize only concomitantly with the closing and reopening of the PH gap. Hence, the simultaneous probe of bulk and edge conductance, with the observation of the closing of the gap accompanied by the emergence of a zero-bias peak at the edgesgrivnin_concomitant_2019 , cannot be considered as conclusive evidence of MBS. However, nontrivial ABS do not necessarily appear simultaneously with the same energy at the two opposite edges of the nontrivial phase, contrarily to the case of the zero-bias peak induced by the MBS. In order to highlight the differences and similarities between nontrivial ABS and MBS, we show in Fig. 3 the spectra and the local density of states (LDOS) as a function of the applied field $b$ through the two nontrivial phases $M=-1$ and $C_{ph}=-2$, calculated as $\rho_{n}(E)=-\Im\bra{n}G(E)\ket{n}/\pi$ with $G(E)$ the unperturbed Green’s function. As shown, both the $M=-1$ and the $C_{ph}=-2$ nontrivial phases are realized, respectively at low $b\lesssim t$ and large applied fields $b\gtrsim t$, respectively with MBS and nontrivial ABS localized at the edges. The closing and reopening of the global PH gap coincides with the appearance of nontrivial ABS. Notice that, contrarily to the case of the MBS, the energies of the 2 nontrivial ABS at the opposite edges are uncorrelated. Moreover, whereas MBS have equal spectral weights in the PH sectors (they are PH symmetric), nontrivial ABS are completely PH and spin polarized. In summary, we found that in the presence of amplitude-modulated fields, a 1D superconductor may exhibit two distinct kind of nontrivial phases corresponding to two distinct topological invariants, i.e., the Majorana number $M$ and the particle-hole Chern number $C_{ph}$, defined respectively in the 1D and in a synthetic 2D BZ. 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ARMOR: A Model-based Framework for Improving Arbitrary Baseline Policies with Offline Data Tengyang Xie University of Illinois at Urbana-Champaign Mohak Bhardwaj University of Washington Nan Jiang University of Illinois at Urbana-Champaign Ching-An Cheng Microsoft Research Abstract We propose a new model-based offline RL framework, called Adversarial Models for Offline Reinforcement Learning (ARMOR), which can robustly learn policies to improve upon an arbitrary baseline policy regardless of data coverage. Based on the concept of relative pessimism, ARMOR is designed to optimize for the worst-case relative performance when facing uncertainty. In theory, we prove that the learned policy of ARMOR never degrades the performance of the baseline policy with any admissible hyperparameter, and can learn to compete with the best policy within data coverage when the hyperparameter is well tuned, and the baseline policy is supported by the data. Such a robust policy improvement property makes ARMOR especially suitable for building real-world learning systems, because in practice ensuring no performance degradation is imperative before considering any benefit learning can bring. 1 Introduction Offline reinforcement learning (RL) is a technique for learning decision-making policies from logged data (Jin et al., 2021; Xie et al., 2021a). In comparison with alternate learning techniques, such as off-policy RL and imitation learning, offline RL reduces the data assumption needed to learn good policies and does not require collecting new data. Theoretically, offline RL can learn the best policy that the given data can explain: as long as the offline data includes all scenarios that executing a near-optimal policy would encounter, an offline RL algorithm can learn a near-optimal policy, even when the data is collected by highly sub-optimal policies or is not diverse. Such robustness to data coverage quality makes offline RL a promising technique for solving real-world problems, because collecting diverse or expert-quality data in practice is often expensive or simply infeasible. The fundamental principle behind offline RL is the concept of pessimism in face of uncertainty, which considers worst-case outcomes for scenarios without data. In implementation, this is realized by (explicitly or implicitly) constructing performance lower bounds in policy learning, which penalizes the agent to take uncertain actions. Various designs have been proposed to construct such lower bounds, including behavior regularization (Fujimoto et al., 2019; Kumar et al., 2019; Wu et al., 2019; Laroche et al., 2019; Fujimoto and Gu, 2021), point-wise pessimism based on negative bonuses or truncation (Kidambi et al., 2020; Jin et al., 2021), value penalty (Kumar et al., 2020; Yu et al., 2020), or two-player games (Cheng et al., 2022; Xie et al., 2021a; Uehara and Sun, 2021). Conceptually, the more accurate the lower bound is, the better the learned policy would perform. Despite these advances, offline RL still has not been widely adopted to build learning-based decision systems in practice. One reason we posit is that achieving high performance in the worst case is not the full picture of designing real-world learning agents. Usually, we apply machine learning to applications that are not completely unknown, but have some running policies. These policies are the decision rules that are currently used in the system (e.g., an engineered autonomous driving rule, or a heuristic-based system for diagnosis), and the goal of applying a learning algorithm is often to further improve upon these baseline policies. As a result, it is imperative that the policy learned by the agent does not lead to performance degradation. This criterion is especially critical for applications where the poor decision outcomes cannot be tolerated (such as health care, autonomous driving, and commercial resource allocation). Although optimizing for absolute or relative performance is the same when full information is available, they can lead to different policies when we only have partial data coverage. In this case, the policy that has the best worst-case performance (which most offline RL algorithms aim to recover) would not necessarily perform better than the baseline policies when deployed in the real environment. Such performance degradation happens when the data does not cover all behaviors of the baseline policies, which can be due to finite samples or a coverage mismatch between the baselines and the data collection policies. As a result, running policies learned by existing offline RL algorithms could risk degrading performance. In this work, we propose a new model-based offline RL framework, called Adversarial Models for Offline Rinforcement Learning (ARMOR), which can robustly learn policies improving upon an arbitrary baseline policy. ARMOR is designed based on the concept of relative pessimism (Cheng et al., 2022), which aims to optimize for the worst-case relative performance when facing uncertainty. In theory, we prove that the learned policy from ARMOR never degrades the performance of the baseline policy for a range of hyperparameters which is given beforehand, a property known as Robust Policy Improvement (RPI) (Cheng et al., 2022). In addition, we prove that, when the right hyperparameter is chosen, and the baseline policy is covered by the data, the learned policy of ARMOR can also compete with any policy within data coverage in an absolute sense. To our knowledge, RPI property of offline RL has so far been limited to comparing against the data collection policy (i.e. the behavior policy) (Cheng et al., 2022; Fujimoto et al., 2019; Kumar et al., 2019; Wu et al., 2019; Laroche et al., 2019; Fujimoto and Gu, 2021). However, it is common that the baseline policy of interest is different from the behavior policy. For example, in robotics manipulation, we often have a dataset of activities different from the target task. In this case, comparing against the behavior policy is meaningless, as these policies do not have meaningful performance in the target task. In ARMOR, by using models, we extend the technique of relative pessimism to achieve RPI with arbitrary baseline policies, regardless of whether they collected the data or not. Finally, based on RPI, we discuss and compare different solution concepts for offline RL (such as relative pessimism here as well as other approaches like absolute pessimism and minimax regret). We show that while these concepts are the same in online RL, in general they lead to different results in offline RL because of the undiminishable uncertainty due to missing data coverage. Our discussion reveals some interesting observations and important implications to offline RL algorithm design, which we feel that many in the offline RL community are not actively aware of. 2 Preliminaries Markov Decision Process We consider an agent acting in an infinite-horizon discounted Markov Decision Process (MDP) $M$ defined by the tuple $\langle\mathcal{S},\mathcal{A},\mathcal{P},R,\gamma\rangle$ where $\mathcal{S}$ is the state space, $\mathcal{A}$ is the action space, $\mathcal{P}:\mathcal{S}\times\mathcal{A}\rightarrow\Delta\left(\mathcal{S}\right)$ is the transition dynamics, $R:\mathcal{S}\times\mathcal{A}\rightarrow\left[0,1\right]$ is a scalar reward function and $\gamma\in[0,1)$ is the discount factor. The learner selects actions using a policy $\pi:\mathcal{S}\rightarrow\Delta\left(\mathcal{A}\right)$. We denote by $\Pi$ the space of all Markovian policies. Let, $d_{M}^{\pi}(s,a)$ denote the discounted state-action distribution obtained by running policy $\pi$ on $M$, i.e $d_{M}^{\pi}(s,a)=\left(1-\gamma\right)\mathbb{E}\left[\sum_{t=0}^{\infty}\gamma^{t}\mathbbm{1}\left(s_{t}=s,a_{t}=a|a_{t}\sim\pi\left(\cdot|s_{t}\right)\right)\right]$. Let $J_{M}(\pi)=\mathbb{E}_{\pi,M}\left[\sum_{t=0}^{\infty}\gamma^{t}r_{t}|a_{t}\sim\pi\right]$ be the expected discounted return of policy $\pi$ on $M$. The goal of reinforcement learning is to find the policy that maximizes $J$. We define the value function as $V^{\pi}_{M}(s)=\mathbb{E}_{\pi,M}\left[\sum_{t=0}^{\infty}\gamma^{t}r_{t}|s_{0}=s\right]$, and the related state-action value function (i.e., Q-function) as $Q^{\pi}_{M}(s,a)=\mathbb{E}_{\pi,M}\left[\sum_{t=0}^{\infty}\gamma^{t}r_{t}|s_{0}=s,s_{0}=a\right]$. We use $[0,V_{\max}]$ as the range of value functions. Offline RL The aim of offline RL is to output strong policies from a fixed dataset collected using a behavior policy without further environmental interactions. We assume the dataset $\mathcal{D}$ consists of $\{\left(s_{i},a_{i},r_{i},s_{i+1}\right)\}_{i=1}^{N}$, where $(s_{i},a_{i})$ is sampled i.i.d. from some distribution $\mu$. We also abuse $\mu$ as discounted state-action occupancy of behavior policy, i.e., $\mu\equiv d^{\mu}_{M}$, and we use $a\sim\mu(\cdot|s)$ to denote sampling from that behavior policy. This paper is concerned with the model-based offline RL problem, and we use $\mathcal{M}$ to denote the model class. For each $M\in\mathcal{M}$, we use $P_{M}:\mathcal{S}\times\mathcal{A}\to\Delta(\mathcal{S})$ and $R_{M}:\mathcal{S}\times\mathcal{A}\to[0,1]$ to denote the corresponding transition and reward function of $M$. Assumption 1 (Realizability). We assume the ground truth model $M^{\star}$ is in the model class $\mathcal{M}$. 3 Adversarial Models for Offline Reinforcement Learning (ARMOR) In this section, we introduce our proposed approach, Adversarial Models for Offline Reinforcement Learning (ARMOR), in Algorithm 1, and present the main theoretical results. ARMOR can be viewed as a model-based extension of the ATAC algorithm by Cheng et al. (2022). In the next sections, we illustrate that ARMOR is not only able to compete with the best data-covered policy as prior works (e.g., Xie et al., 2021a; Uehara and Sun, 2021; Cheng et al., 2022), but also enjoys a stronger robust policy improvement guarantee than (Cheng et al., 2022). Below we analyze ARMOR theoretically and present guarantees on its absolute performance and the policy improvement over the reference policy $\pi_{\sf ref}$. Before presenting the detailed guarantees, we introduce generalized single-policy concentrability, which measures the distribution shift over some arbitrary policy $\pi$ and data distribution $\mu$. Definition 1 (Generalized Single-policy Concentrability). We define the generalized single-policy concentrability for policy $\pi$ for model class $\mathcal{M}$ and offline data distribution $\mu$ as $$\displaystyle\mathfrak{C}_{\mathcal{M}}(\pi)\coloneqq\sup_{M\in\mathcal{M}}\frac{{\mathbb{E}}_{d^{\pi}}\left[D_{\rm TV}\left(P_{M}(\cdot|s,a),P^{\star}(\cdot|s,a)\right)^{2}+\left(R_{M}(s,a)-R^{\star}(s,a)\right)^{2}\right]}{d_{\mu}\left[D_{\rm TV}\left(P_{M}(\cdot|s,a),P^{\star}(\cdot|s,a)\right)^{2}+\left(R_{M}(s,a)-R^{\star}(s,a)\right)^{2}\right]}.$$ Note that $\mathfrak{C}_{\mathcal{M}}(\pi)$ is always upper bounded by the standard single-policy concentrability coefficient $\|d^{\pi}/\mu\|_{\infty}$ (e.g., Jin et al., 2021; Rashidinejad et al., 2021; Xie et al., 2021b), but it can be smaller in general with model class $\mathcal{M}$. It can also be viewed as a model-based analog of the one in Xie et al. (2021a), and the detailed discussion around $\mathfrak{C}_{\mathcal{M}}(\pi)$ refers to Uehara and Sun (2021). We are now ready to present the absolute performance guarantee of ARMOR. Theorem 1 (Absolute performance). Under Assumption 1, there is an absolute constant $c$ such that for any $\delta\in(0,1]$, if we set $\alpha=c\cdot(\log(\nicefrac{{|\mathcal{M}|}}{{\delta}}))$ in Algorithm 1, then for any reference policy $\pi_{\sf ref}$ and comparator policy ${\pi^{\dagger}}\in\Pi$, with probability $1-\delta$, the policy $\widehat{\pi}$ of Algorithm 1 satisfies $$\displaystyle J({\pi^{\dagger}})-J(\widehat{\pi})\leq$$ $$\displaystyle~{}\mathcal{O}\left(\left[\sqrt{\mathfrak{C}_{\mathcal{M}}({\pi^{\dagger}})}+\sqrt{\mathfrak{C}_{\mathcal{M}}(\pi_{\sf ref})}\right]\cdot\frac{V_{\max}}{1-\gamma}\sqrt{\frac{\log(\nicefrac{{|\mathcal{M}|}}{{,}}{\delta})}{n}}\right).$$ Roughly speaking, Theorem 1 shows that $\widehat{\pi}$ learned by Algorithm 1 could compete with any policy ${\pi^{\dagger}}$ with a large enough dataset, as long as the offline data $\mu$ has good coverage on ${\pi^{\dagger}}$ (since the reference policy $\pi_{\sf ref}$ is the input of Theorem 1, one can set $\pi_{\sf ref}=\mu$ (data collection policy) as $\mathfrak{C}_{\mathcal{M}}(\mu)\leq\mathfrak{C}_{\mathcal{M}}({\pi^{\dagger}})$). Compared to the closest model-based offline RL work (Uehara and Sun, 2021), if we set $\pi_{\sf ref}=\mu$ (data collection policy), Theorem 1 leads to almost the same guarantee as Uehara and Sun (2021, Theorem 1) (up to constant factors). In addition to the guarantee on the absolute performance, below we show that, if Assumption 1 is satisfied and $\pi_{\sf ref}\in\Pi$, ARMOR always improves over $J(\pi_{\sf ref})$ for a wide range choice of pessimistic parameter $\alpha$. Compared with the model-free ATAC algorithm in (Cheng et al., 2022, Prop. 6), Theorem 2 removes the concentration errors of $O(\sqrt{1/N})$ as ARMOR is model-based. Theorem 2 (Robust strong policy improvement). Under Assumption 1, there exists an absolute constant $c$ such that for any $\delta\in(0,1]$, if: i) $\alpha\geq c\cdot(\log(\nicefrac{{|\mathcal{M}|}}{{\delta}}))$ in Algorithm 1; ii) $\pi_{\sf ref}\in\Pi$, then with probability $1-\delta$, the policy $\widehat{\pi}$ learned by Algorithm 1 satisfies $J(\pi_{\sf ref})\leq J(\widehat{\pi})$. 4 Robust Policy Improvement (RPI) 4.1 How to formally define RPI? Improving over some reference policy has been long studied in the literature. To highlight the advantage of ARMOR, we formally give the definition of different policy improvement properties. Definition 2 (Robust policy improvement). Suppose $\widehat{\pi}$ is the learned policy from an algorithm. We say the algorithm has the policy improvement (PI) guarantee if $J(\pi_{\sf ref})-J(\widehat{\pi})\leq\nicefrac{{o(N)}}{{N}}$ is guaranteed for some reference policy $\pi_{\sf ref}$ with offline data $\mathcal{D}\sim\mu$, where $N=|\mathcal{D}|$. We use the following two criteria w.r.t. $\pi_{\sf ref}$ and $\mu$ to define different kinds PI: (i) The PI is strongstrongstrong if $\pi_{\sf ref}$ can be selected arbitrarily from policy class $\Pi$ regardless of the choice data-collection policy $\mu$; otherwise, PI is weakweakweak (i.e., $\pi_{\sf ref}\equiv\mu$ is required). (ii) The PI is robustrobustrobust if it can be achieved by a range of hyperparameters with a known subset. Weak policy improvement is also known as safe policy improvement in the literature (Fujimoto et al., 2019; Laroche et al., 2019). It requires the reference policy to be also the behavior policy that collects the offline data. In comparison, strong policy improvement imposes a stricter requirement, which requires policy improvement regardless of how the data were collected. This condition is motivated by the common situation where the reference policy is not the data collection policy. Finally, since we are learning policies offline, without online interactions, it is not straightforward to tune the hyperparameter directly. Therefore, it is desirable that we can design algorithms with these properties in a robust manner in terms of hyperparameter selection. Formally, Definition 2 requires the policy improvement to be achievable by a set of hyperparameters that is known before learning. Theorem 2 indicates the robust strong policy improvement of ARMOR. On the other hand, algorithms with robust weak policy improvement are available in the literature (Cheng et al., 2022; Fujimoto et al., 2019; Kumar et al., 2019; Wu et al., 2019; Laroche et al., 2019; Fujimoto and Gu, 2021); this is usually achieved by designing the algorithm to behave like imitation learning (IL) for a known set of hyperparameter (e.g., behavior regularization algorithms have a weight that can turn off the RL behavior and regress to IL). However, deriving guarantees of achieving the best data-covered policy of the IL-like algorithm is challenging due to its imitating nature. To our best knowledge, ATAC (Cheng et al., 2022) is the only algorithm that achieves both robust (weak) policy improvement as well as guarantees absolute performance. 4.2 When RPI actually improves? Given ARMOR’s ability to improve over an arbitrary policy, the following questions naturally arise: Can ARMOR nontrivially improve the output policy of other algorithms (e.g., such as those based on absolute pessimism (Xie et al., 2021a)), including itself? Note that outputting $\pi_{\sf ref}$ itself always satisfies RPI, but such result is trivial. By “nontrivially” we mean a non-zero worst-case improvement. If the statement were true, we would be able to repeatedly run ARMOR to improve over itself and then obtain the best policy any algorithm can learn offline. Unfortunately, the answer is negative. Not only ARMOR cannot improve over itself, but it also cannot improve over a variety of algorithms. In fact, the optimal policy of an arbitrary model in the version space is unimprovable (see Corollary 4)! Our discussion reveals some interesting observations (e.g., how equivalent performance metrics for online RL can behave very differently in the offline setting) and their implications (e.g., how we should choose $\pi_{\sf ref}$ for ARMOR). Despite their simplicity, we feel that many in the offline RL community are not actively aware of these facts (and the unawareness has led to some confusion), which we hope to clarify below. Setup We consider an abstract setup where the learner is given a version space $\mathcal{M}_{\alpha}$ that contains the true model and needs to choose a policy $\pi\in\Pi$ based on $\mathcal{M}_{\alpha}$. We use the same notation $\mathcal{M}_{\alpha}$ as before, but emphasize that it does not have to be constructed as in Eq. 1 and Eq. 2. In fact, for the purpose of this discussion, the data distribution, sample size, data randomness, and estimation procedure for constructing $\mathcal{M}_{\alpha}$ are all irrelevant, as our focus here is how decisions should be made with a given $\mathcal{M}_{\alpha}$. This makes our setup very generic and the conclusions widely applicable. To facilitate discussion, we define the fixed point of ARMOR’s relative pessimism step: Definition 3. Consider Eq. 3 as an operator that maps an arbitrary policy $\pi_{\sf ref}$ to $\widehat{\pi}$. A fixed point of this relative pessimism operator is, therefore, any policy $\pi\in\Pi$ such that $\pi\in\mathop{\mathrm{argmax}}_{\pi^{\prime}\in\Pi}\min_{M\in\mathcal{M}_{\alpha}}J_{M}(\pi^{\prime})-J_{M}(\pi)$. Given the definition, relative pessimism cannot improve over a policy if it is already a fixed point. Below we show a sufficient and necessary condition for being a fixed point, and show a number of concrete examples (some of which may be surprising) that are fixed points and thus unimprovable. Lemma 3 (Fixed-point Lemma). For any $\mathcal{M}\subseteq\mathcal{M}_{\alpha}$ and any $\psi:\mathcal{M}\to\mathbb{R}$, consider the policy $$\displaystyle\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{eq:candidate fixed-point policies}}{e}q:candidatefixed-pointpolicies}\pi\in\mathop{\mathrm{argmax}}_{\pi^{\prime}\in\Pi}\min_{M\in\mathcal{M}}J_{M}(\pi^{\prime})+\psi(M)$$ (4) Then $\pi$ is a fixed point in Definition 3. Conversely, for any fixed point $\pi$ in Definition 3, there is a $\psi:\mathcal{M}\to\mathbb{R}$ such that $\pi$ is a solution to Eq. 4. Corollary 4. The following are fixed points of relative pessimism (Definition 3): 1. Absolute-pessimism policy, i.e., $\psi(M)=0$. 2. Relative-pessimism policy for any reference policy, i.e., $\psi(M)=-J_{M}(\pi_{\sf ref})$. 3. Regret-minimization policy, i.e., $\psi(M)=-J_{M}(\pi_{M}^{*})$, where $\pi_{M}^{*}\in\mathop{\mathrm{argmax}}_{\pi\in\Pi}J_{M}(\pi)$. 4. Optimal policy of an arbitrary model $M\in\mathcal{M}_{\alpha}$, $\pi_{M}^{*}$, i.e., $\mathcal{M}=\{M\}$. This would include the optimistic policy, that is, $\mathop{\mathrm{argmax}}_{\pi\in\Pi,M\in\mathcal{M}_{\alpha}}J_{M}(\pi)$ Return maximization and regret minimization are different in offine RL We first note that these four examples generally produce different policies, even though some of them optimize for objectives that are traditionally viewed as equivalent in online RL (the “worst-case over $\mathcal{M}_{\alpha}$” part of the definition does not matter in online RL), e.g., absolute pessimism optimizes for $J_{M}(\pi)$, which is the same as minimizing the regret $J_{M}(\pi_{M}^{\star})-J_{M}(\pi)$ for a fixed $M$. However, their equivalence in online RL relies on the fact that online exploration can eventually resolve any model uncertainty when needed, so we only need to consider the performance metrics w.r.t. the true model $M=M^{\star}$. In offline RL with an arbitrary data distribution (since we do not make any coverage assumptions), there will generally be model uncertainty that cannot be resolved, and worst-case reasoning over such model uncertainty (i.e., $\mathcal{M}_{\alpha}$) separates apart the definitions that are once equivalent. Moreover, it is impossible to compare return maximization and regret minimization and make a claim about which one is better. They are not simply an algorithm design choice, but are definitions of the learning goals and the guarantees themselves—thus incomparable: if we care about obtaining a guarantee for the worst-case return, the return maximization is optimal by definition; if we are more interested in obtaining a guarantee for the worst-case regret, then again, regret minimization is trivially optimal. We also note that analyzing algorithms under a metric that is different from the one they are designed for can lead to unusual conclusions. For example, Xiao et al. (2021) show that optimistic/neutral/pessimistic algorithms111Incidentally, optimistic/neutral policies correspond to #4 in Corollary 4. are equally minimax-optimal in terms of their regret guarantees in offline multi-armed bandits. However, the algorithms they consider are optimistic/pessimistic w.r.t. the return—as commonly considered in the offline RL literature—not w.r.t. the regret which is the performance metric they are interested in analyzing. $\pi_{\sf ref}$ is more than a hyperparameter—it defines the performance metric and learning goal Corollary 4 shows that ARMOR (with relative pessimism) has many different fixed points, some of which may seem quite unreasonable for offline learning, such as greedy w.r.t. an arbitrary model or even optimism (#4). From the above discussion, we can see that this is not a defect of the algorithm. Rather, in the offline setting with unresolvable model uncertainty, there are many different performance metrics/learning goals that are generally incompatible/incomparable with each other, and the agent designer must make a choice among them and convey the choice to the algorithm. 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Mopo: Model-based offline policy optimization. Advances in Neural Information Processing Systems, 33:14129–14142, 2020. Zhang [2006] Tong Zhang. From $\varepsilon$-entropy to kl-entropy: Analysis of minimum information complexity density estimation. The Annals of Statistics, 34(5):2180–2210, 2006. Appendix Appendix A Proofs for [UNDEFINED] A.1 Technical Tools Lemma 5 (Simulation lemma). Consider any two MDP model $M$ and $M^{\prime}$, and any $\pi:\mathcal{S}\to\Delta(\mathcal{A})$, we have $$\displaystyle\left|J_{M}(\pi)-J_{M^{\prime}}(\pi)\right|\leq\frac{V_{\max}}{1-\gamma}{\mathbb{E}}_{d^{\pi}}\left[D_{\rm TV}\left(P_{M}(\cdot|s,a),P_{M^{\prime}}(\cdot|s,a)\right)\right]+\frac{1}{1-\gamma}{\mathbb{E}}_{d^{\pi}}\left[\left|R_{M}(s,a)-R_{M^{\prime}}(s,a)\right|\right].$$ Lemma 5 is the standard simulation lemma in model-based reinforcement learning literature, and its proof can be found in, e.g., Uehara and Sun [2021, Lemma 7]. A.2 Guarantees about Version Space Lemma 6. Let $M^{\star}$ be the ground truth model. Then, with probability at least $1-\delta$, we have $$\displaystyle\max_{M\in\mathcal{M}}\mathcal{L}_{\mathcal{D}}(M)-\mathcal{L}_{\mathcal{D}}(M^{\star})\leq\mathcal{O}\left(\log(\nicefrac{{|\mathcal{M}|}}{{\delta}})\right),$$ where $\mathcal{L}_{\mathcal{D}}$ is defined in Eq. 2. Proof of Lemma 6. By Lemma 8, we know $$\displaystyle\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{eq:Vspace_Mstar_P}}{e}q:Vspace_{M}star_{P}}\max_{M\in\mathcal{M}}\log\ell_{\mathcal{D}}(M)-\log\ell_{\mathcal{D}}(M^{\star})\leq\log(\nicefrac{{|\mathcal{M}|}}{{\delta}}).$$ (5) In addition, by Xie et al. [2021a, Theorem A.1] (with setting $\gamma=0$), we know w.p. $1-\delta$, $$\displaystyle\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{eq:Vspace_Mstar_R}}{e}q:Vspace_{M}star_{R}}\sum_{(s,a,r,s^{\prime})\in\mathcal{D}}\left(R^{\star}(s,a)-r\right)^{2}-\min_{M\in\mathcal{M}}\sum_{(s,a,r,s^{\prime})\in\mathcal{D}}\left(R_{M}(s,a)-r\right)^{2}\lesssim\log(\nicefrac{{|\mathcal{M}|}}{{\delta}}).$$ ($$1$$) Combining the Eqs. 5 and $1$, we have w.p. $1-\delta$, $$\displaystyle~{}\max_{M\in\mathcal{M}}\mathcal{L}_{\mathcal{D}}(M)-\mathcal{L}_{\mathcal{D}}(M^{\star})$$ $$\displaystyle\leq$$ $$\displaystyle~{}\max_{M\in\mathcal{M}}\log\ell_{\mathcal{D}}(M)-\min_{M\in\mathcal{M}}\sum_{(s,a,r,s^{\prime})\in\mathcal{D}}\left(R_{M}(s,a)-r\right)^{2}-\mathcal{L}_{\mathcal{D}}(M^{\star})$$ $$\displaystyle\lesssim$$ $$\displaystyle~{}\log(\nicefrac{{|\mathcal{M}|}}{{\delta}}).$$ This completes the proof. ∎ Lemma 7. For any $M\in\mathcal{M}$, we have with probability at least $1-\delta$, $$\displaystyle{\mathbb{E}}_{\mu}\left[D_{\rm TV}\left(P_{M}(\cdot|s,a),P^{\star}(\cdot|s,a)\right)^{2}+\left(R_{M}(s,a)-R^{\star}(s,a)\right)^{2}\right]$$ $$\displaystyle\leq\mathcal{O}\left(\frac{\max_{M^{\prime}\in\mathcal{M}}\mathcal{L}_{\mathcal{D}}(M^{\prime})-\mathcal{L}_{\mathcal{D}}(M)+\log(\nicefrac{{|\mathcal{M}|}}{{\delta}})}{n}\right),$$ where $\mathcal{L}_{\mathcal{D}}$ is defined in Eq. 2. Proof of Lemma 7. By Lemma 9, we have w.p. $1-\delta$, $$\displaystyle\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{lem:Vspace_MP}}{l}em:Vspace_{M}P}n\cdot{\mathbb{E}}_{\mu}\left[D_{\rm TV}\left(P_{M}(\cdot|s,a),P^{\star}(\cdot|s,a)\right)^{2}\right]\lesssim\log\ell_{\mathcal{D}}(M^{\star})-\log\ell_{\mathcal{D}}(M)+\log(\nicefrac{{|\mathcal{M}|}}{{\delta}}).$$ (6) Also, we have $$\displaystyle~{}n\cdot{\mathbb{E}}_{\mu}\left[\left(R_{M}(s,a)-R^{\star}(s,a)\right)^{2}\right]$$ (7) $$\displaystyle=$$ $$\displaystyle~{}n\cdot{\mathbb{E}}_{\mu}\left[\left(R_{M}(s,a)-r\right)^{2}\right]-n\cdot{\mathbb{E}}_{\mu}\left[\left(R^{\star}(s,a)-r\right)^{2}\right]$$ [see, e.g., Xie et al., 2021a, Eq. (A.10) with $$\gamma=0$$] $$\displaystyle\lesssim$$ $$\displaystyle~{}\sum_{(s,a,r,s^{\prime})\in\mathcal{D}}\left(R_{M}(s,a)-r\right)^{2}-\sum_{(s,a,r,s^{\prime})\in\mathcal{D}}\left(R^{\star}(s,a)-r\right)^{2}+\log(\nicefrac{{|\mathcal{M}|}}{{\delta}}),$$ where the last inequality is a direct implication of Xie et al. [2021a, Lemma A.4] and $1=1$. Combining Eqs. 6 and 7, we obtain $$\displaystyle~{}n\cdot{\mathbb{E}}_{\mu}\left[D_{\rm TV}\left(P_{M}(\cdot|s,a),P^{\star}(\cdot|s,a)\right)^{2}+\left(R_{M}(s,a)-R^{\star}(s,a)\right)^{2}\right]$$ $$\displaystyle\lesssim$$ $$\displaystyle~{}\log\ell_{\mathcal{D}}(M^{\star})-\sum_{(s,a,r,s^{\prime})\in\mathcal{D}}\left(R^{\star}(s,a)-r\right)^{2}-\log\ell_{\mathcal{D}}(M)+\sum_{(s,a,r,s^{\prime})\in\mathcal{D}}\left(R_{M}(s,a)-r\right)^{2}+\log(\nicefrac{{|\mathcal{M}|}}{{\delta}})$$ $$\displaystyle=$$ $$\displaystyle~{}\mathcal{L}_{\mathcal{D}}(M^{\star})-\mathcal{L}_{\mathcal{D}}(M)+\log(\nicefrac{{|\mathcal{M}|}}{{\delta}})$$ $$\displaystyle\leq$$ $$\displaystyle~{}\max_{M^{\prime}\in\mathcal{M}}\mathcal{L}_{\mathcal{D}}(M^{\prime})-\mathcal{L}_{\mathcal{D}}(M)+\log(\nicefrac{{|\mathcal{M}|}}{{\delta}}).$$ This completes the proof. ∎ A.3 MLE Guarantees We use $\ell_{\mathcal{D}}(M)$ to denote the likelihood of model $M=(P,R)$ with offline data $\mathcal{D}$, where $$\displaystyle\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{eq:def_LD}}{e}q:def_{L}D}\mathfrak{\ell}_{\mathcal{D}}(M)=$$ $$\displaystyle~{}\prod_{(s,a,r,s^{\prime})\in\mathcal{D}}P_{M}(s^{\prime}|s,a).$$ (8) For the analysis around maximum likelihood estimation, we largely follow the proving idea of Agarwal et al. [2020], Liu et al. [2022], which is inspired by Zhang [2006]. The next lemma shows that the ground truth model $M^{\star}$ has a comparable log-likelihood compared with MLE solution. Lemma 8. Let $M^{\star}$ be the ground truth model. Then, with probability at least $1-\delta$, we have $$\displaystyle\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{eq:mle_lemma}}{e}q:mle_{l}emma}\max_{M\in\mathcal{M}}\log\ell_{\mathcal{D}}(M)-\log\ell_{\mathcal{D}}(M^{\star})\leq\log(\nicefrac{{|\mathcal{M}|}}{{\delta}}).$$ (9) Proof of Lemma 8. The proof of this lemma is obtained by a standard argument of MLE [see, e.g., van de Geer, 2000]. For any $M\in\mathcal{M}$, $$\displaystyle{\mathbb{E}}\left[\exp\left(\log\ell_{\mathcal{D}}(M)-\log\ell_{\mathcal{D}}(M^{\star})\right)\right]=$$ $$\displaystyle~{}{\mathbb{E}}\left[\frac{\ell_{\mathcal{D}}(M)}{\ell_{\mathcal{D}}(M^{\star})}\right]$$ $$\displaystyle=$$ $$\displaystyle~{}{\mathbb{E}}\left[\frac{\prod_{(s,a,r,s^{\prime})\in\mathcal{D}}\mathbb{P}_{M}(s^{\prime}|s,a)}{\prod_{(s,a,r,s^{\prime})\in\mathcal{D}}\mathbb{P}_{M^{\star}}(s^{\prime}|s,a)}\right]$$ $$\displaystyle=$$ $$\displaystyle~{}{\mathbb{E}}\left[\prod_{(s,a,r,s^{\prime})\in\mathcal{D}}\frac{\mathbb{P}_{M}(s^{\prime}|s,a)}{\mathbb{P}_{M^{\star}}(s^{\prime}|s,a)}\right]$$ $$\displaystyle=$$ $$\displaystyle~{}{\mathbb{E}}\left[\prod_{(s,a)\in\mathcal{D}}{\mathbb{E}}\left[\frac{\mathbb{P}_{M}(s^{\prime}|s,a)}{\mathbb{P}_{M^{\star}}(s^{\prime}|s,a)}~{}\middle|~{}s,a\right]\right]$$ $$\displaystyle=$$ $$\displaystyle~{}{\mathbb{E}}\left[\prod_{(s,a)\in\mathcal{D}}\sum_{s^{\prime},r}\mathbb{P}_{M}(s^{\prime}|s,a)\right]$$ $$\displaystyle\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{eq:markov_eq_exp}}{e}q:markov_{e}q_{e}xp}=$$ $$\displaystyle~{}1.$$ (10) Then by Markov’s inequality, we obtain $$\displaystyle~{}\mathbb{P}\left[\left(\log\ell_{\mathcal{D}}(M)-\log\ell_{\mathcal{D}}(M^{\star})\right)>\log(\nicefrac{{1}}{{\delta}})\right]$$ $$\displaystyle\leq$$ $$\displaystyle~{}\underbrace{{\mathbb{E}}\left[\exp\left(\log\ell_{\mathcal{D}}(M)-\log\ell_{\mathcal{D}}(M^{\star})\right)\right]}_{=1\text{ by \lx@cref{creftype~refnum}{eq:markov_eq_exp}}}\cdot\exp\left[-\log(\nicefrac{{1}}{{\delta}})\right]=\delta.$$ Therefore, taking a union bound over $\mathcal{M}$, we obtain $$\displaystyle\mathbb{P}\left[\left(\log\ell_{\mathcal{D}}(M)-\log\ell_{\mathcal{D}}(M^{\star})\right)>\log(\nicefrac{{|\mathcal{M}|}}{{\delta}})\right]\leq\delta.$$ This completes the proof. ∎ The following lemma shows that, the on-support error of any model $M\in\mathcal{M}$ can be captured via its log-likelihood (by comparing with the MLE solution). Lemma 9. For any $M=(P,R)$, we have with probability at least $1-\delta$, $$\displaystyle{\mathbb{E}}_{\mu}\left[D_{\rm TV}\left(P(\cdot|s,a),P^{\star}(\cdot|s,a)\right)^{2}\right]\leq\mathcal{O}\left(\frac{\log\ell_{\mathcal{D}}(M^{\star})-\log\ell_{\mathcal{D}}(M)+\log(\nicefrac{{|\mathcal{M}|}}{{\delta}})}{n}\right),$$ where $\ell_{\mathcal{D}}(\cdot)$ is defined in Eq. 8. Proof of Lemma 9. By Agarwal et al. [2020, Lemma 25], we have $$\displaystyle\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{eq:P_tvnorm}}{e}q:P_{t}vnorm}{\mathbb{E}}_{\mu}\left[D_{\rm TV}\left(P(\cdot|s,a),P^{\star}(\cdot|s,a)\right)^{2}\right]\leq$$ $$\displaystyle~{}-2\log{\mathbb{E}}_{\mu\times P^{\star}}\left[\exp\left(-\frac{1}{2}\log\left(\frac{P^{\star}(s^{\prime}|s,a)}{P(s^{\prime}|s,a)}\right)\right)\right]$$ (11) $$\displaystyle{\mathbb{E}}_{\mu}\left[D_{\rm TV}\left(R(\cdot|s,a),R^{\star}(\cdot|s,a)\right)^{2}\right]\leq$$ $$\displaystyle~{}-2\log{\mathbb{E}}_{\mu\times R^{\star}}\left[\exp\left(-\frac{1}{2}\log\left(\frac{R^{\star}(r|s,a)}{R(r|s,a)}\right)\right)\right],$$ where $\mu\times P^{\star}$ and $\mu\times R^{\star}$ denote the ground truth offline joint distribution of $(s,a,s^{\prime})$ and $(s,a,r)$. Let $\widetilde{\mathcal{D}}=\{(\widetilde{s}_{i},\widetilde{a}_{i},\widetilde{r}_{i},\widetilde{s}_{i}^{\prime})\}_{i=1}^{n}\sim\mu$ be another offline dataset that is independent to $\mathcal{D}$. Then, $$\displaystyle~{}-n\cdot\log{\mathbb{E}}_{\mu\times P^{\star}}\left[\exp\left(-\frac{1}{2}\log\left(\frac{P^{\star}(s^{\prime}|s,a)}{P(s^{\prime}|s,a)}\right)\right)\right]$$ $$\displaystyle=$$ $$\displaystyle~{}-\sum_{i=1}^{n}\log{\mathbb{E}}_{(\widetilde{s}_{i},\widetilde{a}_{i},\widetilde{s}_{i}^{\prime})\sim\mu}\left[\exp\left(-\frac{1}{2}\log\left(\frac{P^{\star}(\widetilde{s}_{i}^{\prime}|\widetilde{s}_{i},\widetilde{a}_{i})}{P(\widetilde{s}_{i}^{\prime}|\widetilde{s}_{i},\widetilde{a}_{i})}\right)\right)\right]$$ $$\displaystyle=$$ $$\displaystyle~{}-\log{\mathbb{E}}_{\widetilde{\mathcal{D}}\sim\mu}\left[\exp\left(\sum_{i=1}^{n}-\frac{1}{2}\log\left(\frac{P^{\star}(\widetilde{s}_{i}^{\prime}|\widetilde{s}_{i},\widetilde{a}_{i})}{P(\widetilde{s}_{i}^{\prime}|\widetilde{s}_{i},\widetilde{a}_{i})}\right)\right)~{}\middle|~{}\mathcal{D}\right]$$ $$\displaystyle\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{eq:P_tvnorm_ub}}{e}q:P_{t}vnorm_{u}b}=$$ $$\displaystyle~{}-\log{\mathbb{E}}_{\widetilde{\mathcal{D}}\sim\mu}\left[\exp\left(\sum_{(s,a,s^{\prime})\in\widetilde{\mathcal{D}}}-\frac{1}{2}\log\left(\frac{P^{\star}(s^{\prime}|s,a)}{P(s^{\prime}|s,a)}\right)\right)~{}\middle|~{}\mathcal{D}\right].$$ (12) We use $\ell_{P}(s,a,s^{\prime})$ as the shorthand of $-\frac{1}{2}\log\left(\frac{P^{\star}(s|s,a)}{P(s^{\prime}|s,a)}\right)$, for any $(s,a,s^{\prime})\in\mathcal{S}\times\mathcal{A}\times\mathcal{S}$. By Agarwal et al. [2020, Lemma 24] [see also Liu et al., 2022, Lemma 15], we know $$\displaystyle{\mathbb{E}}_{\mathcal{D}\sim\mu}\left[\exp\left(\sum_{(s,a,s^{\prime})\in\mathcal{D}}\ell_{P}(s,a,s^{\prime})-\log{\mathbb{E}}_{\widetilde{\mathcal{D}}\sim\mu}\left[\exp\left(\sum_{(s,a,s^{\prime})\in\widetilde{\mathcal{D}}}\ell_{P}(s,a,s^{\prime})\right)~{}\middle|~{}\mathcal{D}\right]-\log|\mathcal{M}|\right)\right]\leq 1.$$ Thus, we can use Chernoff method as well as a union bound on the equation above to obtain the following exponential tail bound: with probability at least $1-\delta$, we have for all $(P,R)=M\in\mathcal{M}$, $$\displaystyle\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{eq:P_tvnorm_tail}}{e}q:P_{t}vnorm_{t}ail}-\log{\mathbb{E}}_{\widetilde{\mathcal{D}}\sim\mu}\left[\exp\left(\sum_{(s,a,s^{\prime})\in\widetilde{\mathcal{D}}}\ell_{P}(s,a,s^{\prime})\right)~{}\middle|~{}\mathcal{D}\right]\leq-\sum_{(s,a,s^{\prime})\in\mathcal{D}}\ell_{P}(s,a,s^{\prime})+2\log(\nicefrac{{|\mathcal{M}|}}{{\delta}}).$$ (13) Plugging back the definition of $\ell_{P}$ and combining Eqs. 11, 12 and 13, we obtain $$\displaystyle\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{eq:Pmle_final}}{e}q:Pmle_{f}inal}n\cdot{\mathbb{E}}_{\mu}\left[D_{\rm TV}\left(P(\cdot|s,a),P^{\star}(\cdot|s,a)\right)^{2}\right]\leq$$ $$\displaystyle~{}\frac{1}{2}\sum_{(s,a,s^{\prime})\in\mathcal{D}}\log\left(\frac{P^{\star}(s|s,a)}{P(s^{\prime}|s,a)}\right)+2\log(\nicefrac{{|\mathcal{M}|}}{{\delta}}).$$ (14) By the same steps of obtaining to Eq. 14, we also have $$\displaystyle\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{eq:Rmle_final}}{e}q:Rmle_{f}inal}n\cdot{\mathbb{E}}_{\mu}\left[D_{\rm TV}\left(R(\cdot|s,a),R^{\star}(\cdot|s,a)\right)^{2}\right]\leq$$ $$\displaystyle~{}\frac{1}{2}\sum_{(s,a,r^{\prime})\in\mathcal{D}}\log\left(\frac{R^{\star}(s|s,a)}{R(s^{\prime}|s,a)}\right)+2\log(\nicefrac{{|\mathcal{M}|}}{{\delta}}).$$ (15) Combining Eqs. 14 and 15, we obtain $$\displaystyle~{}n\cdot{\mathbb{E}}_{\mu}\left[D_{\rm TV}\left(P(\cdot|s,a),P^{\star}(\cdot|s,a)\right)^{2}+D_{\rm TV}\left(R(\cdot|s,a),R^{\star}(\cdot|s,a)\right)^{2}\right]$$ $$\displaystyle\lesssim$$ $$\displaystyle~{}\sum_{(s,a,s^{\prime})\in\mathcal{D}}\log\left(\frac{P^{\star}(s|s,a)}{P(s^{\prime}|s,a)}\right)+\sum_{(s,a,r^{\prime})\in\mathcal{D}}\log\left(\frac{R^{\star}(s|s,a)}{R(s^{\prime}|s,a)}\right)+\log(\nicefrac{{|\mathcal{M}|}}{{\delta}})$$ $$\displaystyle=$$ $$\displaystyle~{}\log\ell_{\mathcal{D}}(M^{\star})-\log\ell_{\mathcal{D}}(M)+\log(\nicefrac{{|\mathcal{M}|}}{{\delta}}).$$ ($$\ell_{\mathcal{D}}(\cdot)$$ is defined in Eq. 8) This completes the proof. ∎ A.4 Proof of Main Theorems Proof of Theorem 1. By the optimality of $\widehat{\pi}$ (from Eq. 3), we have $$\displaystyle J({\pi^{\dagger}})-J(\widehat{\pi})=$$ $$\displaystyle~{}J({\pi^{\dagger}})-J(\pi_{\sf ref})-\left[J(\widehat{\pi})-J(\pi_{\sf ref})\right]$$ $$\displaystyle\leq$$ $$\displaystyle~{}J({\pi^{\dagger}})-J(\pi_{\sf ref})-\min_{M\in\mathcal{M}_{\alpha}}\left[J_{M}(\widehat{\pi})-J_{M}(\pi_{\sf ref})\right]$$ (by Lemma 8, we have $$M^{\star}\in\mathcal{M}_{\alpha}$$) $$\displaystyle\leq$$ $$\displaystyle~{}J({\pi^{\dagger}})-J(\pi_{\sf ref})-\min_{M\in\mathcal{M}_{\alpha}}\left[J_{M}({\pi^{\dagger}})-J_{M}(\pi_{\sf ref})\right],\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{eq:perfbdeq1}}{e}q:perfbdeq1}$$ (16) where the last step is because of ${\pi^{\dagger}}\in\Pi$ By the simulation lemma (Lemma 5), we know for any policy $\pi$ and any $M\in\mathcal{M}_{\alpha}$, $$\displaystyle\left|J(\pi)-J_{M}(\pi)\right|\leq$$ $$\displaystyle~{}\frac{V_{\max}}{1-\gamma}{\mathbb{E}}_{d^{\pi}}\left[D_{\rm TV}\left(P_{M}(\cdot|s,a),P^{\star}(\cdot|s,a)\right)\right]+\frac{1}{1-\gamma}{\mathbb{E}}_{d^{\pi}}\left[\left|R_{M}(s,a)-R^{\star}(s,a)\right|\right]$$ $$\displaystyle\leq$$ $$\displaystyle~{}\frac{V_{\max}}{1-\gamma}\sqrt{{\mathbb{E}}_{d^{\pi}}\left[D_{\rm TV}\left(P_{M}(\cdot|s,a),P^{\star}(\cdot|s,a)\right)^{2}\right]}+\frac{1}{1-\gamma}\sqrt{{\mathbb{E}}_{d^{\pi}}\left[\left(R_{M}(s,a)-R^{\star}(s,a)\right)^{2}\right]}$$ $$\displaystyle\lesssim$$ $$\displaystyle~{}\frac{V_{\max}}{1-\gamma}\sqrt{{\mathbb{E}}_{d^{\pi}}\left[D_{\rm TV}\left(P_{M}(\cdot|s,a),P^{\star}(\cdot|s,a)\right)^{2}+\left(R_{M}(s,a)-R^{\star}(s,a)\right)^{2}\right]}$$ ($$a\lesssim b$$ means $$a\leq\mathcal{O}(b)$$) $$\displaystyle\leq$$ $$\displaystyle~{}\frac{V_{\max}\sqrt{\mathfrak{C}_{\mathcal{M}}(\pi)}}{1-\gamma}\sqrt{{\mathbb{E}}_{\mu}\left[D_{\rm TV}\left(P_{M}(\cdot|s,a),P^{\star}(\cdot|s,a)\right)^{2}+\left(R_{M}(s,a)-R^{\star}(s,a)\right)^{2}\right]}$$ $$\displaystyle\lesssim$$ $$\displaystyle~{}\frac{V_{\max}\sqrt{\mathfrak{C}_{\mathcal{M}}(\pi)}}{1-\gamma}\sqrt{\frac{\max_{M^{\prime}\in\mathcal{M}}\mathcal{L}_{\mathcal{D}}(M^{\prime})-\mathcal{L}_{\mathcal{D}}(M)+\log(\nicefrac{{|\mathcal{M}|}}{{\delta}})}{n}}$$ (by Lemma 7) $$\displaystyle\lesssim$$ $$\displaystyle~{}\frac{V_{\max}\sqrt{\mathfrak{C}_{\mathcal{M}}(\pi)}}{1-\gamma}\sqrt{\frac{\log(\nicefrac{{|\mathcal{M}|}}{{,}}{\delta})}{n}}\addcontentsline{lla}{section}{\numberline{\string\crtrefnumber{eq:perfbdeq2}}{e}q:perfbdeq2}$$ (17) where the last step is because $\max_{M^{\prime}\in\mathcal{M}}\mathcal{L}_{\mathcal{D}}(M^{\prime})-\mathcal{L}_{\mathcal{D}}(M)\leq\alpha=\mathcal{O}(\log(\nicefrac{{|\mathcal{M}|}}{{\delta}})/n$ by Eq. 1. Combining Eqs. 16 and 17, we obtain $$\displaystyle J({\pi^{\dagger}})-J(\widehat{\pi})\lesssim$$ $$\displaystyle~{}\left[\sqrt{\mathfrak{C}_{\mathcal{M}}({\pi^{\dagger}})}+\sqrt{\mathfrak{C}_{\mathcal{M}}(\pi_{\sf ref})}\right]\cdot\frac{V_{\max}}{1-\gamma}\sqrt{\frac{\log(\nicefrac{{|\mathcal{M}|}}{{,}}{\delta})}{n}}.$$ This completes the proof. ∎ Proof of Theorem 2. $$\displaystyle J(\pi_{\sf ref})-J(\widehat{\pi})=$$ $$\displaystyle~{}J(\pi_{\sf ref})-J(\pi_{\sf ref})-\left[J(\widehat{\pi})-J(\pi_{\sf ref})\right]$$ $$\displaystyle\leq$$ $$\displaystyle~{}-\min_{M\in\mathcal{M}_{\alpha}}\left[J_{M}(\widehat{\pi})-J_{M}(\pi_{\sf ref})\right]$$ (by Lemma 8, we have $$M^{\star}\in\mathcal{M}_{\alpha}$$) $$\displaystyle=$$ $$\displaystyle~{}-\max_{\pi\in\Pi}\min_{M\in\mathcal{M}_{\alpha}}\left[J_{M}(\pi)-J_{M}(\pi_{\sf ref})\right]$$ (by the optimality of $$\widehat{\pi}$$ from Eq. 3) $$\displaystyle\leq$$ $$\displaystyle~{}-\min_{M\in\mathcal{M}_{\alpha}}\left[J_{M}(\pi_{\sf ref})-J_{M}(\pi_{\sf ref})\right]$$ ($$\pi_{\sf ref}\in\Pi$$) $$\displaystyle=$$ $$\displaystyle~{}0.$$ ∎ Appendix B Proofs for Section 4 Proof of Lemma 3. We prove the result by contradiction. First notice $\min_{M\in\mathcal{M}}J_{M}(\pi^{\prime})-J_{M}(\pi^{\prime})=0$. Suppose there is $\overline{\pi}\in\Pi$ such that $\min_{M\in\mathcal{M}_{\alpha}}J_{M}(\bar{\pi})-J_{M}(\pi^{\prime})>0$, which implies that $J_{M}(\bar{\pi})>J_{M}(\pi^{\prime})$, $\forall M\in\mathcal{M}_{\alpha}$. Since $\mathcal{M}\subseteq\mathcal{M}_{\alpha}$, we have $$\displaystyle\min_{M\in\mathcal{M}}J_{M}(\bar{\pi})+\psi(M)$$ $$\displaystyle>\min_{M\in\mathcal{M}}J_{M}(\pi^{\prime})+\psi(M)=\max_{\pi\in\Pi}\min_{M\in\mathcal{M}}J_{M}(\pi)+\psi(M)$$ which is a contradiction of the maximin optimality. Thus $\max_{\pi\in\Pi}\min_{M\in\mathcal{M}_{\alpha}}J_{M}(\bar{\pi})-J_{M}(\pi^{\prime})=0$, which means $\pi^{\prime}$ is a solution. For the converse statement, suppose $\pi$ is a fixed point. We can just let $\psi(M)=-J_{M}(\pi)$. Then this pair of $\pi$ and $\psi$ by definition of the fixed point satisfies Eq. 4. ∎

Abstract In this work, we give a mathematical description of a fully chiral gapless edge of a 2d topological order (without symmetry). We show that the observables on the 1+1D world sheet of such an edge consist of a family of topological edge excitations, boundary CFT’s and walls between boundary CFT’s. These observables can be described by a chiral algebra and an enriched monoidal category. This mathematical description automatically includes that of gapped edges as special cases. Therefore, it gives a unified framework to study both gapped and gapless edges. Moreover, the boundary-bulk duality also holds for gapless edges. More precisely, the unitary modular tensor category that describes the 2d bulk phase is exactly the Drinfeld center of the enriched monoidal category that describes the gapless/gapped edge. We propose a classification of all gapped and fully chiral gapless edges of a given bulk phase. In the end, we explain how modular-invariant bulk conformal field theories naturally emerge on certain gapless walls between two trivial phases. Gapless edges of 2d topological orders and enriched monoidal categories Liang Kong${}^{a}$,  Hao Zheng${}^{b}$   111Emails: kong.fan.liang@gmail.com, hzheng@math.pku.edu.cn [1em] ${}^{a}$ Department of Mathematics and Statistics University of New Hampshire, Durham, NH 03824, USA [0.5em] ${}^{b}$ Department of Mathematics, Peking University Beijing 100871, China Contents 1 Introduction 2 Basics of boundary-bulk CFT’s 3 Observables on a fully chiral gapless edge 4 A canonical fully chiral gapless edge 5 General fully chiral gapless edges 6 Classification of gapless/gapped edges 7 0d defects and boundary-bulk CFT’s 8 Summary and outlooks A Appendix A.1 Algebras in unitary modular tensor categories A.2 Enriched monoidal categories and a canonical construction 1 Introduction It is well-known that the fusing-braiding properties of topological excitations in a 2d topological order (without symmetry) can be described by a unitary modular tensor category (UMTC) (see [FrRS, FG] and a review [Ki1, Appendix E]). A 2d topological order is uniquely determined by a pair $(\mathcal{C},c)$, where $\mathcal{C}$ is the UMTC of topological excitations and $c$ is the chiral central charge. It was also known that if the topological order $(\mathcal{C},0)$ has a gapped edge, the gapped edge can be mathematically described by a unitary fusion category (UFC) $\mathcal{M}$ such that the Drinfeld center $Z(\mathcal{M})$ of $\mathcal{M}$ coincides with $\mathcal{C}$ [KK, FSV, Ko5, LW, KWZ2]. The fact that the bulk phase is uniquely determined by the gapped edge as its Drinfeld center is also called boundary-bulk duality. In general, 2d topological orders (without symmetry), such as quantum Hall systems, have topologically protected gapless edges [Ha, W1, MR, W2] (see reviews [W2, W3, NSSFS] and references therein). A gapless edge is significantly richer than a gapped edge because gapless edge modes are described by 1+1D rational conformal field theories (RCFT’s) [BPZ, MSei], the mathematical structures of which are much richer than a UMTC [MSei, Tu, KLM, Hu3]. As far as we know, the precise mathematical description of a gapless edges is still not known. In the last twenty years, the mathematical theory of boundary-bulk (or open-closed) RCFT’s has been successfully developed from at least three different perspectives (see the conformal-net approach in [LR, R1, R2, KLM], the 2+1D-TQFT approach in [FFFS, FS1, FRS1, FjFRS] and the vertex-operator-algebra approach in [Hu1, HK1, HK2, Ko3] and references therein). These mathematical developments have revealed a universal phenomenon: the mathematical structures of a boundary-bulk RCFT can be split into two parts. One part consists of a unitary rational chiral algebra $V$ (or a conformal net in the first approach), also called a unitary rational vertex operator algebra (or simply a VOA) (see for example [LL, DL]) in mathematics, such that the category $\operatorname{Mod}_{V}$ of $V$-modules is a UMTC [Hu3]. The other part is a pure categorical structure containing certain algebras in $\operatorname{Mod}_{V}$ and its Drinfeld center $Z(\operatorname{Mod}_{V})$. This suggests that it might be possible to describe a fully chiral gapless edge of a 2d topological order $(\mathcal{C},c)$ by a pair $(V,{\mathcal{M}^{\sharp}})$, where $V$ is a VOA and ${\mathcal{M}^{\sharp}}$ is a purely categorical structure that can be constructed from $\operatorname{Mod}_{V},\mathcal{C}$ and perhaps additional categorical data. The main goal of this paper is to show that this is indeed possible. In Section 2, we recall some basic facts of boundary-bulk CFT’s. We explain that if a boundary-bulk CFT preserves a chiral symmetry given by a VOA $V$ on the boundary, then there is a very simple but equivalent categorical description of such a boundary-bulk CFT as certain algebras in UMTC’s. We summarize the results in a physics/mathematics dictionary at the end of this section. In Section 3, we explain that observables on a fully chiral gapless edge of a 2d topological order consist of a family of topological edge excitations, boundary CFT’s and walls between these boundary CFT’s. These boundary CFT’s and walls are required to preserve a chiral symmetry given by a VOA $V$ such that $\operatorname{Mod}_{V}=\mathcal{B}$ as UMTC’s. This symmetry condition allows us to describe all boundary CFT’s and walls equivalently by certain algebras and objects in $\mathcal{B}$. As a consequence, all these observables organize themselves into a $\mathcal{B}$-enriched monoidal category ${\mathcal{X}^{\sharp}}$ (see Def. A.16). We denote such a gapless edge by the pair $(V,{\mathcal{X}^{\sharp}})$. In Section 4, we describe a canonical gapless edge $(V,{\mathcal{B}^{\sharp}})$ of a 2d bulk phase $(\mathcal{B},c)$, where the enriched monoidal category ${\mathcal{B}^{\sharp}}$ is canonically obtained from $\mathcal{B}$. The boundary-bulk duality holds in this case. Namely, the Drinfeld center $Z({\mathcal{B}^{\sharp}})$ of ${\mathcal{B}^{\sharp}}$ coincides with the UMTC $\mathcal{B}$ [KZ2]. In Section 5, we explain that we can obtain a new gapless edge of a bulk phase $(\mathcal{C},c)$ by fusing a gapped domain wall $\mathcal{M}$ between two bulk phases $(\mathcal{C},c)$ and $(\mathcal{B},c)$ with the canonical gapless edge $(V,{\mathcal{B}^{\sharp}})$ of $(\mathcal{B},c)$. This new edge can again be described by a pair $(V,{\mathcal{M}^{\sharp}})$, where ${\mathcal{M}^{\sharp}}$ is a $\mathcal{B}$-enriched monoidal category canonically constructed from the pair $(\mathcal{B},\mathcal{M})$. Moreover, the boundary-bulk duality holds, i.e. $Z({\mathcal{M}^{\sharp}})=\mathcal{C}$. This mathematical description of gapless edges (as pairs $(V,{\mathcal{M}^{\sharp}})$) automatically includes that of gapped edges as special cases. In this way, we have obtained a unified mathematical theory of gapless and gapped edges. This leads us to propose, in Section 6, a mathematical classification of all gapped and fully chiral gapless edges of a given bulk phase $(\mathcal{C},c)$. In Section 7, we explain how a modular-invariant bulk CFT naturally emerge as a hom space in an enriched monoidal category describing certain gapless walls between two trivial 2d topological orders. We also discuss briefly 0d defects between edges. In Section 8, we give a summary and outlooks. In Appendix, we recall the mathematical definitions of various algebras in a UMTC and the definition of an enriched monoidal category. Assumptions: We assume that all UMTC’s can be realized as the categories of modules over some VOA’s. Acknowledgement: We thank Maissam Barkeshli, Meng Cheng, Yuan-Ming Lu, Chetan Nayak, Xiao-Gang Wen and Yi-Zhuang You for very helpful discussions. We thank Huan He for many suggestions for improvement. HZ is supported by NSFC under Grant No. 11131008. 2 Basics of boundary-bulk CFT’s In this section, we briefly review the categorical description of various ingredients of boundary-bulk CFT’s [FRS1, Ko3] (see a review [Ko4] and references therein). The mathematically definitions of various algebras in UMTC’s are given in Section A.1. The most important ingredient of a 2d CFT is a chiral algebra $V$ [BPZ, MSei]. It was defined rigorously in mathematics as a vertex operator algebra (VOA) (see for example [LL]). It is a graded vector space $V=\oplus_{i=0}^{\infty}V_{(0)}$. The grading $n$ for $V_{(n)}$ is called the conformal weight, which is also an eigenvalue of the Hamiltonian operator $L(0)$, i.e. $L(0)\cdot V_{(n)}=n\cdot V_{(n)}$. The graded dimension, for $q\in\mathbb{C}^{\times}$, $$\chi_{V}(q)=\sum_{i=0}^{\infty}\dim V_{(n)}q^{n}$$ is called the partition function of $V$. By the state-field correspondence, for each $\phi\in V$, there is a unique chiral field $\phi(z)$ associated to $\phi\in V$. The chiral field $\phi(z)$ depends on $z$ holomorphically. More precisely, $\phi(z)$ can be expanded as $\phi(z)=\sum_{n\in\mathbb{Z}}\phi_{n}z^{-n-1}$, where $\phi_{n}$ is a linear operator that maps $V_{(m)}$ into $V_{(k-n-1+m)},\forall m$ if $\phi\in V_{(k)}$. There is a distinguished weight 2 element $T\in V_{(2)}$. The chiral field $T(z)$ is the called the energy-stress tensor and can be expanded as follows: $$T(z)=\sum_{n\in\mathbb{Z}}L(n)z^{-n-2},$$ where $L(n),n\in\mathbb{Z}$ generate a Lie algebra called Virasoro algebra defined by $$[L(m),L(n)]=(m-n)L(m+n)+\delta_{m+n,0}\frac{m^{3}-m}{12}c\operatorname{Id}_{V},$$ where $c$ is a complex number called the central charge of $V$. Chiral fields in $V$ have operator product expansions (OPE), i.e. $$\phi(z_{1})\psi(z_{2})\sim\frac{(\phi_{k}\psi)(z_{2})}{(z_{1}-z_{2})^{k+1}}+% \frac{(\phi_{k-1}\psi)(z_{2})}{(z_{1}-z_{2})^{k}}+\cdots.$$ The OPE is commutative, i.e. $\phi(z_{1})\psi(z_{2})\sim\psi(z_{2})\phi(z_{1})$. In mathematics, this OPE and its properties were rigorously defined as the data and axioms of a VOA. A simple $V$-module $W$ (or an irreducible representation of $V$) is again a graded vector space, i.e. $W=\oplus_{i=0}^{\infty}W_{(n)}$, where $L(0)\cdot W_{(n)}=(h_{W}+n)W_{(n)}$ for some $h_{W}\in\mathbb{R}$. Note that the partition function $\chi_{W}(q)=\sum_{i=0}^{\infty}\dim W_{(n)}q^{n+h_{W}}$ has monodromy if $h_{W}$ is not an integer. For example, the Ising VOA $V$ has three simple $V$-modules, $\mathbf{1},\psi,\sigma$, where $\mathbf{1}$ is just $V$ itself and $h_{\mathbf{1}}=0,h_{\psi}=\frac{1}{2},h_{\sigma}=\frac{1}{16}$. The set of all $V$-modules and $V$-module maps (linear maps that intertwine the $V$-actions) form the category of $V$-modules, denoted by $\operatorname{Mod}_{V}$. The states (or chiral fields) in two different $V$-modules can be fused into the third $V$-module according to the so called chiral vertex operators222The notion of a chiral vertex operator was introduced by Moore and Seiberg [MSei], and was latter defined and constructed rigorously by the name of an intertwining operator in mathematics by Huang [Hu2]., which also have OPE [MSei]. This OPE was mathematically rigorously constructed, and was shown to provide a monoidal structure on $\operatorname{Mod}_{V}$ [HL1, HL2, HL3, HL4]. Namely, two $V$-modules $x$ and $y$ can be fused to given a new $V$-module $x\otimes y=y\otimes x$, which is a direct sum of simple $V$-modules. For example, for Ising VOA $V$, we have $$\mathbf{1}\otimes\psi=\psi,\quad\mathbf{1}\otimes\sigma=\sigma,\quad\psi% \otimes\sigma=\sigma,\quad\mathbf{1}=\mathbf{1}\otimes\mathbf{1}=\psi\otimes% \psi,\quad\sigma\otimes\sigma=1\oplus\psi.$$ The fusion product $\otimes$ and the braidings $x\otimes y\xrightarrow{c_{x,y}}y\otimes x$ provide $\operatorname{Mod}_{V}$ with a braided monoidal structure. Moreover, when the VOA $V$ is unitary [DL] and rational [Hu3], the category $\operatorname{Mod}_{V}$ is a unitary modular tensor category (UMTC) [Hu3, Hu4]. The tensor unit $\mathbf{1}$ in $\operatorname{Mod}_{V}$ is nothing but $V$ itself. Each object $x$ in $\operatorname{Mod}_{V}$ has a dual object denoted by $x^{\ast}$ and the duality maps $u_{x}:\mathbf{1}\to x\otimes x^{\ast}$ and $v_{x}:x^{\ast}\otimes x\to\mathbf{1}$. For example, for Ising VOA $V$, all objects in $\operatorname{Mod}_{V}$ are self-dual, i.e. $\mathbf{1}=\mathbf{1}^{\ast},\psi=\psi^{\ast},\sigma=\sigma^{\ast}$. In a boundary CFT [C1, C2, CL], the chiral fields on the 0+1D boundary (also called boundary fields) also have OPE, which forms a mathematical structure called open-string vertex operator algebra (OSVOA) [HK1]. A VOA is automatically an OSVOA. In general, an OSVOA $A$ is $\mathbb{R}$-graded $A=\oplus_{i\in\mathbb{R}}A_{(i)}$. An OSVOA $A$ always contains a VOA generated by the energy-stress tensor $T_{A}\in A_{(2)}$ and $T_{A}(r)=\sum_{n\in\mathbb{Z}}L(n)r^{-n-2}$, where $r$ is the coordinate of the boundary and can be chosen to be the real numbers, i.e. $r\in\mathbb{R}$. Since $r\in\mathbb{R}$, a boundary field $\phi(r)=\sum_{n\in\mathbb{R}}\phi_{n}r^{-n-1}$ can have non-integer powers of $r$. In general, the OPE of an OSVOA is not commutative. Therefore, an OSVOA can be viewed as a non-commutative generalization of a VOA. An OSVOA $A$ is called an OSVOA over $V$ if it is an extension of a VOA $V$ by $V$-modules and boundary fields in it are all chiral vertex operators of $V$. This VOA $V$ should be viewed as the chiral symmetry of $A$. When the chiral symmetry $V$ is unitary and rational, $\operatorname{Mod}_{V}$ is not only a UMTC, but also a vertex tensor category [HL1, Hu4], which allows us to reduce the complicated OPE structures in $A$ to a simple categorical structure: an algebra $A$ in the UMTC $\operatorname{Mod}_{V}$ [HK1]. By definition, an algebra $A$ in $\operatorname{Mod}_{V}$ is just an object $A$ (i.e. a $V$-module), together with a multiplication morphism $m:A\otimes A\to A$ and a unit morphism $\iota:\mathbf{1}\to A$, i.e. a triple $(A,m,\iota)$, such that $$m\circ(m\otimes\operatorname{Id}_{A})=m\circ(\operatorname{Id}_{A}\otimes m),% \quad\quad m\circ(\iota\otimes\operatorname{Id}_{A})=\operatorname{Id}_{A}=m% \circ(\operatorname{Id}_{A}\otimes\iota).$$ The information of OPE is completely encoded in the morphism $m:A\otimes A\to A$; and that of the chiral symmetry is encoded in $\iota:\mathbf{1}\to A$ (i.e. an embedding $V\hookrightarrow A$). This miraculous simplification is the key to the success of the categorical classification of boundary-bulk RCFT’s in [FRS1, FjFRS, Ko3, KR2]. Using this equivalence, it is very easy to construct OSVOA’s over $V$. For example, let $x$ be a $V$-module, i.e. an object in $\operatorname{Mod}_{V}$. Then the internal hom $[x,x]=x\otimes x^{\ast}$, together with $$m:x\otimes x^{\ast}\otimes x\otimes x^{\ast}\xrightarrow{\operatorname{Id}_{x}% \otimes v_{x}\otimes\operatorname{Id}_{x^{\ast}}}x\otimes x^{\ast}\quad\quad and% \quad\quad\iota:\mathbf{1}\xrightarrow{u_{x}}x\otimes x^{\ast},$$ (2.1) gives an algebra in $\operatorname{Mod}_{V}$, i.e. an OSVOA over $V$. Remark 2.1. If $A$ happens to be a VOA extension of $V$, then it is equivalent to a commutative algebra in $\operatorname{Mod}_{V}$ [HKL], i.e. an algebra such that $m\circ c_{A,A}=m$. For example, $[\mathbf{1},\mathbf{1}]=\mathbf{1}$ is the simplest commutative algebra in $\operatorname{Mod}_{V}$. A boundary CFT $A_{\mathrm{bdy}}$ over $V$ is an OSVOA over $V$ equipped with a non-degenerate invariant bilinear form, which upgrade $A_{\mathrm{bdy}}$ to a symmetric Frobenius algebra in $\operatorname{Mod}_{V}$ [Ko3] (see Def. A.6). A bulk CFT contains bulk fields $\phi(z,\bar{z})$ that depends on both the holomorphic variable $z$ and the anti-holomorphic variable $\bar{z}$. Bulk fields also have OPE, which forms a mathematical structure called a full field algebra [HK2]. Let $U$ be a VOA, and let $\overline{V}$ be the same VOA as $V$ but consisting of only anti-chiral fields $\psi(\bar{z})$ for $\psi\in V$. The tensor product $U\otimes_{\mathbb{C}}\overline{V}$ with $\phi(z,\bar{z})=u(z)\otimes_{\mathbb{C}}v(\bar{z})$ for $\phi=u\otimes_{\mathbb{C}}v\in U\otimes_{\mathbb{C}}\overline{V}$ gives an example of full field algebra. A full field algebra is called a full field algebra over $U\otimes_{\mathbb{C}}\overline{V}$ if it is an extension of $U\otimes_{\mathbb{C}}\overline{V}$ by objects in $\operatorname{Mod}_{U}\boxtimes\overline{\operatorname{Mod}_{V}}$ and all bulk fields are chiral vertex operators of the VOA $U\otimes_{\mathbb{C}}V$. $U$ is called the chiral symmetry and $V$ is called the anti-chiral symmetry. A full field algebra over $U\otimes_{\mathbb{C}}\overline{V}$ is equivalent to a commutative algebra in $\operatorname{Mod}_{U}\boxtimes\overline{\operatorname{Mod}_{V}}$ [Ko1]. A modular-invariant bulk CFT over $U\otimes_{\mathbb{C}}\overline{V}$ is a full field algebra over $U\otimes_{\mathbb{C}}\overline{V}$ equipped with a non-degenerate invariant bilinear form and a unique vacuum such that its genus-one correlation functions are all modular-invariant [HK3]. It is equivalent to a Lagrangian algebra in $\operatorname{Mod}_{U}\boxtimes\overline{\operatorname{Mod}_{V}}$ [KR2] (see Def. A.3). For example, the charge-conjugate modular-invariant bulk CFT over $V\otimes_{\mathbb{C}}\overline{V}$ is given by the Lagrangian algebra $\oplus_{i}i\boxtimes i^{\ast}$ in $\operatorname{Mod}_{V}\boxtimes\overline{\operatorname{Mod}_{V}}$, where $i$ are simple objects in $\operatorname{Mod}_{V}$. Its partition function $\sum_{i}\chi_{i}(q)\chi_{i^{\ast}}(\bar{q})$ is modular invariant. A boundary-bulk CFT consists of a boundary CFT $A_{\mathrm{bdy}}$ and a modular-invariant bulk CFT $A_{\mathrm{bulk}}$ satisfying some compatibility conditions [Ko3], one of which requires that the chiral symmetry of $A_{\mathrm{bdy}}$ coincides with both the chiral and anti-chiral symmetries of $A_{\mathrm{bulk}}$ (see [Ko2, Def. 1.25]). This symmetry condition implies that $A_{\mathrm{bdy}}$ is a boundary CFT over $V$ and $A_{\mathrm{bulk}}$ is a bulk CFT over $V\otimes_{\mathbb{C}}\overline{V}$. If we include all compatibility conditions (such as the Cardy condition), then $A_{\mathrm{bdy}}$ must be a connected special symmetric Frobenius algebra (CSSFA) in $\operatorname{Mod}_{V}$ (see Def. A.6 & Remark A.7), and $A_{\mathrm{bulk}}$ must be the Lagrangian algebra (see Def. A.3) given by the full center $Z(A_{\mathrm{bdy}})$ of $A_{\mathrm{bdy}}$ in $\operatorname{Mod}_{V}\boxtimes\overline{\operatorname{Mod}_{V}}$ [FjFRS, KR2]. For example, the internal hom algebra $[x,x]$ for $x\in\operatorname{Mod}_{V}$ defined in Eq. (2.1) are CSSFA’s in $\operatorname{Mod}_{V}$. They are all Morita equivalent and share the same full center $Z([x,x])=Z(\mathbf{1})$, which is nothing but the charge conjugate Lagrangian algebra $\oplus_{i}i\boxtimes i^{\ast}$ in $\operatorname{Mod}_{V}\boxtimes\overline{\operatorname{Mod}_{V}}$. It turns out that mapping CSSFA’s in $\operatorname{Mod}_{V}$ to their full centers defines a one-to-one correspondence between the set of Morita classes of CSSFA’s in $\operatorname{Mod}_{V}$ and that of Lagrangian algebras in $\operatorname{Mod}_{V}\boxtimes\overline{\operatorname{Mod}_{V}}$. The categorical descriptions of ingredients in a boundary-bulk CFT are summarized by the following dictionary: Physical terminologies Mathematical terminologies a unitary rational chiral algebra a unitary VOA $$V$$ s.t. $$\operatorname{Mod}_{V}$$ is a UMTC chiral vertex operators intertwining operators boundary fields OPE an open-string VOA (OSVOA) boundary fields OPE an OSVOA over $$V$$ with the $$V$$-symmetry = an algebra in $$\operatorname{Mod}_{V}$$ a boundary CFT over $$V$$ a symmetric Frobenius algebra in $$\operatorname{Mod}_{V}$$ a modular-invariant bulk CFT a Lagrangian algebra $$A_{\mathrm{bulk}}$$ over $$U\otimes_{\mathbb{C}}\overline{V}$$ in $$\operatorname{Mod}_{U}\boxtimes\overline{\operatorname{Mod}_{V}}$$ boundary-bulk CFT over $$V$$ contains: 1. a boundary CFT over $$V$$ a CSSFA $$A_{\mathrm{bdy}}$$ in $$\mathcal{C}:=\operatorname{Mod}_{V}$$ 2. a modular invariant bulk CFT over $$V\otimes_{\mathbb{C}}\overline{V}$$ a Lagrangian algebra $$A_{\mathrm{bulk}}$$ in $$\mathcal{C}\boxtimes\overline{\mathcal{C}}$$ 3. boundary-bulk duality $$A_{\mathrm{bulk}}=Z(A_{\mathrm{bdy}})$$, where $$Z(A_{\mathrm{bdy}})$$ is the full center of $$A$$ 3 Observables on a fully chiral gapless edge A 2d topological order is described by a UMTC $\mathcal{C}$ and the chiral central charge $c$, i.e. a pair $(\mathcal{C},c)$. The chiral central charge is defined by the difference between the central charges of the right movers and the left movers, i.e. $c=c_{R}-c_{L}$. In this work, we only study fully chiral gapless edges. Objects in $\mathcal{C}$ are topological excitations. The tensor unit $\mathbf{1}_{\mathcal{C}}$ of $\mathcal{C}$ represents the trivial topological excitation. We denote the fusion product in $\mathcal{C}$ by $\otimes$. The simplest UMTC is the category $\mathbf{H}$ of finite dimensional Hilbert spaces. The pair $(\mathbf{H},0)$ describes the trivial 2d topological order. Suppose that the 2d bulk phase $(\mathcal{C},c)$ is realized on an open 2-disk, and the edge of the 2-disk is gapless. We depict the 1+1D world sheet of the edge as a cylinder in Figure 1. The chiral edge modes that can live on the entire 1+1D world sheet are states in a chiral algebra with the central charge $c$, or a VOA $U$ with the central charge $c$. This VOA describes the OPE of chiral fields associated to the chiral edge modes. The category of $U$-modules is a UMTC, and is denoted by $\operatorname{Mod}_{U}$. In general, $\operatorname{Mod}_{U}$ might not coincide with $\mathcal{C}$. This VOA $U$ is not the only observable on the gapless edge. In Figure 1, at $t=0$, a topological bulk excitation $c$ is moved to the edge, it creates a topological edge excitation (or a defect) $x$ on the edge at $t=0$. This excitation is different from those gapless excitations in the chiral edge modes, and should be viewed as certain super-selection sector, and is similar to a topological edge excitation on a gapped edge [KK]. There are chiral fields (also called defect fields) living on $x$ at $t=0$ (see for example [W1, WW, WWH]). They form a vector space $M_{x}$. In general, non-trivial topological bulk excitations might condense on the edge, and there might be more topological edge excitations than those from the bulk. Note that the topological edge excitation $x$ is also a wall between the $t>0$ part of the world line (the blue line in Figure 1) and $t<0$ part. The chiral fields on the $t>0$ world line supported on $x$ are potentially different from those in $U$. According to [AL1], they should form a boundary CFT $A_{x}$. The OPE of chiral fields in $A_{x}$ alone forms an OSVOA (see Section 2). We denote the trivial topological edge excitation by $\mathbf{1}$. It is clear that $A_{\mathbf{1}}=U$. In order to be a well-defined boundary CFT, this OSVOA $A_{x}$ is required to satisfy certain compatibility conditions with $U$. More precisely, fusing chiral fields in $U$ into those in $A_{x}$ along a path $\gamma$ defines an OSVOA map $\iota_{\gamma}:U\to A_{x}$ (see Figure 2). The minimal symmetry requirement for $A_{x}$ to be a consistent boundary CFT is that $\iota_{\gamma}$ should preserve the conformal symmetry [C1]. More precisely, let $\langle T_{U}\rangle$ and $\langle T_{A_{x}}\rangle$ be the sub-VOA’s in $U$ and $A_{x}$ generated by the energy-stress tensors $T_{U}\in U$ and $T_{A_{x}}\in A_{x}$, respectively. Preserving the conformal symmetry means that $\iota_{\gamma}|_{\langle T_{U}\rangle}:\langle T_{U}\rangle\to\langle T_{A_{x}}\rangle$ is a VOA isomorphism and independent of paths. More generally, one can require $\iota_{\gamma}$ to preserve a larger chiral symmetry given by a sub-VOA $V$ of $U$, i.e. $\iota_{\gamma}|_{V}:V\hookrightarrow A_{x}$ is an injective OSVOA homomorphism and independent of paths (see for example [HK1, Ko2]). This independence-of-path condition implies that $V$ is the chiral symmetry of $A_{x}$, or equivalently, $A_{x}$ must be a boundary CFT over $V$ (see Section 2). This chiral symmetry $V$, or the $V$-symmetry, is an important data in describing the gapless edge. We assume that $V$ is unitary rational so that $\operatorname{Mod}_{V}$ is a UMTC. Then the boundary CFT $A_{x}$ should be a symmetric Frobenius algebra in $\operatorname{Mod}_{V}$. Moreover, as we will show in Section 7, $A_{x}$ is necessary to be the boundary CFT in a boundary-bulk CFT over $V$. Therefore, we expect $A_{x}$ to a CSSFA in $\operatorname{Mod}_{V}$. It is also possible to change a topological edge excitation $x$ to another $y$ on the same world line at $t=t_{1}>0$ as depicted in both Figure 1 and Figure 2. For example, one can move a topological bulk excitation $b\in\mathcal{C}$ to the world line at $t=t_{1}$ to give $y=b\otimes x$. This process creates a wall (at $t=t_{1}$) between two boundary CFT’s $A_{x}$ (on $\{t<t_{1}\}$) and $A_{y}$ (on $\{t>t_{1}\}$). The defect fields on the wall are a special kind of chiral vertex operators called boundary condition changing operators (see for example [AL2]). They form a vector space $M_{x,y}$. It is clear that we should have $M_{\mathbf{1},x}=M_{x}$ and $M_{x,x}=A_{x}$. Similar to $A_{x}$, the wall $M_{x,y}$ should also preserve the $V$-symmetry. This condition means that $M_{y,x}$ must be an object in $\operatorname{Mod}_{V}$. Moreover, defect fields in $M_{x,y}$ can be fused with those in $M_{y,z}$ to give defect fields in $M_{x,z}$. This fusion should commute with the $V$-actions. Therefore, it can be described by a morphism $M_{y,z}\otimes M_{x,y}\to M_{x,z}$ in $\operatorname{Mod}_{V}$. When $x=y=z$, this morphism is nothing but the multiplication morphism $A_{x}\otimes A_{x}\to A_{x}$ that defines the algebra structure on $A_{x}$. Remark 3.1. In general, $V\subsetneq U$ (see Remark 5.1). Note that $U$ also acts on $M_{x,y}$. When $U$ is viewed as a commutative algebra in $\operatorname{Mod}_{V}$, $M_{x,y}$ is a left (or right) $U$-module. The braidings in $\operatorname{Mod}_{V}$ provide a canonical $U$-$U$-bimodule structure on $M_{x,y}$. When $M_{x,y}=A_{x}$, these two-side actions amount to acting on $A_{x}$ along two paths $\gamma_{1}$ and $\gamma_{2}$ in Figure 2. But $M_{x,y}$ is not necessarily a local $U$-module ([BEK, KO]) unless $V=U$. So $M_{x,y}$ is not a VOA-module over $U$ in general. Remark 3.2. We are not aware of any earlier works that have studied the boundary CFT’s on the gapless edges of quantum Hall systems. But using boundary CFT’s to study 0d defects (or impurities) in condensed matter systems has a long history [AL1, AL2]. In summary, all the observables on 1+1D world sheet of a gapless edge of a given 2d bulk phase $(\mathcal{C},c)$ can be described by a pair $(V,{\mathcal{X}^{\sharp}})$, where ${\mathcal{X}^{\sharp}}$ is a categorical structure: • objects in ${\mathcal{X}^{\sharp}}$ are topological edge excitations; • for each pair $x,y$ objects in ${\mathcal{X}^{\sharp}}$, there is a space $\hom_{{\mathcal{X}^{\sharp}}}(x,y):=M_{x,y}$ which is an object in $\operatorname{Mod}_{V}$; • an identity map $V=\mathbf{1}_{\operatorname{Mod}_{V}}\to M_{x,x}=A_{x}$ is a morphism in $\operatorname{Mod}_{V}$ defined by the canonical embedding $V\hookrightarrow A_{x}$; • a composition map $M_{y,z}\otimes M_{x,y}\to M_{x,z}$ is a morphism in $\operatorname{Mod}_{V}$, satisfying some natural physical properties, such as the unit property of the identity map and the associativity of the composition map. As a result, this categorical structure ${\mathcal{X}^{\sharp}}$ is nothing but a category enriched in $\operatorname{Mod}_{V}$, or an $\operatorname{Mod}_{V}$-enriched category [Ke] (see Def. A.10). The last piece of structure is the fusion (on the same time slide) of two topological edge excitations $x^{\prime}$ and $x$, denoted by $x^{\prime}\otimes x$ as depicted in Figure 3. It automatically provides a fusion between observables on two parallel world lines. This fusion provides • a morphism $M_{x^{\prime},y^{\prime}}\otimes M_{x,y}\to M_{x^{\prime}\otimes x,y^{\prime}% \otimes y}$ in $\operatorname{Mod}_{V}$ for objects $x,y,x^{\prime},y^{\prime}$ in ${\mathcal{X}^{\sharp}}$, satisfying some natural properties. This fusion structure upgrades ${\mathcal{X}^{\sharp}}$ to an $\operatorname{Mod}_{V}$-enriched monoidal category [MP] (see Def. A.16). Remark 3.3. The boundary-bulk duality333It is related to but different from the bulk-edge correspondence (see [MR, GWW, SV, LWWW] and references therein). for topological orders in arbitrary dimensions was proved formally in [KWZ1] under some natural assumptions. It says that the bulk topological order should be given by the center of the boundary phase regardless how we describe the bulk and the boundary phases mathematically. This formal proof also works for the cases in which the boundary phase is gapless (see [KWZ1, Remark 5.7]). Therefore, if the enriched monoidal category ${\mathcal{X}^{\sharp}}$ indeed gives a mathematical description of the gapless edge, then we expect that the Drinfeld center $Z({\mathcal{X}^{\sharp}})$ of the enriched monoidal category ${\mathcal{X}^{\sharp}}$, a notion which was recently introduced in [KZ2, Def. 2.1], gives exactly the UMTC $\mathcal{C}$, i.e. $Z({\mathcal{X}^{\sharp}})=\mathcal{C}$. 4 A canonical fully chiral gapless edge Let $\mathcal{B}$ be a UMTC with the tensor unit $\mathbf{1}_{\mathcal{B}}$. The creation and annihilation of a particle-antiparticle pair are described by the duality morphisms $$u_{x}:\mathbf{1}_{\mathcal{B}}\to x\otimes x^{*},\quad\quad\quad v_{x}:x^{*}% \otimes x\to\mathbf{1}_{\mathcal{B}}.$$ (4.1) We denote the braiding isomorphisms by $c_{x,y}:x\otimes y\to y\otimes x$ for $x,y\in\mathcal{B}$. We use $\overline{\mathcal{B}}$ to denote the UMTC that is the same monoidal category as $\mathcal{B}$ but equipped with the braidings given by the anti-braidings in $\mathcal{B}$. In this section, we focus on the simplest gapless edge of $(\mathcal{B},c)$, called the canonical gapless edge. We describe all the observables on this canonical gapless edge below. • Topological edge excitations are all obtained from moving topological bulk excitations to the edge without any condensation. They can be labeled by objects in $\mathcal{B}$. • $\mathcal{B}=\operatorname{Mod}_{V}$. • $V=U$, namely, all the boundary CFT’s $A_{x}$ and wall between them $M_{x,y}$ are required to preserve the largest chiral symmetry $V$. As a consequence, $M_{x,y}$ are objects in $\mathcal{B}$. • $M_{x}=x$ as a $V$-module. For each $x\in\mathcal{B}$, the boundary CFT $A_{x}$ is given by the internal hom $[x,x]:=x\otimes x^{\ast}$, which is a CSSFA in $\mathcal{B}$ [FFFS, FRS1, Ko2, Ko3]. Its algebraic structures are defined in Eq. (2.1). When $x=\mathbf{1}_{\mathcal{B}}$, we have $A_{\mathbf{1}_{\mathcal{B}}}=\mathbf{1}_{\mathcal{B}}=V$. • $M_{x,y}=[x,y]=y\otimes x^{\ast}$ [FFRS2]. In particular, $M_{\mathbf{1}_{\mathcal{B}},x}=M_{x}=[\mathbf{1}_{\mathcal{B}},x]=x$. • Defect fields in $[x,y]$ can be fused with those in $[y,z]$ to give defect fields in $[x,z]$. This amounts to a morphism $[y,z]\otimes[x,y]\to[x,z]$ in $\mathcal{B}$, which is defined as follows: $$[y,z]\otimes[x,y]=z\otimes y^{\ast}\otimes y\otimes x^{\ast}\xrightarrow{% \operatorname{Id}_{z}\otimes v_{y}\otimes\operatorname{Id}_{x^{\ast}}}z\otimes x% ^{\ast}=[x,z].$$ (4.2) It is clear that $[x,y]$ is automatically a $[y,y]$-$[x,x]$-bimodule. • The fusion of topological edge excitations (on the same time slide) depicted in Figure 3 demands a morphism $M_{x^{\prime},y^{\prime}}\otimes M_{x,y}\to M_{x^{\prime}\otimes x,y^{\prime}% \otimes y}$ in $\mathcal{B}$. It is defined as follows: $$[x^{\prime},y^{\prime}]\otimes[x,y]=y^{\prime}\otimes x^{\prime*}\otimes y% \otimes x^{*}\xrightarrow{\operatorname{Id}_{y^{\prime}}\otimes c_{x^{\prime*}% ,y\otimes x^{*}}}(y^{\prime}\otimes y)\otimes(x^{\prime}\otimes x)^{*}=[x^{% \prime}\otimes x,y^{\prime}\otimes y].$$ (4.3) This canonical edge of $(\mathcal{B},c)$ is the most studied edge in physics. But our description of the complete set of observables on this edge, especially the explicit construction of boundary CFT’s and walls between them, is new. Remark 4.1. Note that CSSFA’s $[x,x]$ for $x\in\mathcal{B}$ are all Morita equivalent to each other [FFRS2, KR1]. In general, there are more CSSFA’s in $\mathcal{B}$ (not Morita equivalent to $[x,x]$) that can occur on a different gapless edge (see Remark 7.2). These observables on the canonical edge of $(\mathcal{B},c)$ can be summarized by a pair $(V,{\mathcal{B}^{\sharp}})$, where ${\mathcal{B}^{\sharp}}$ is a categorical structure: • An object in ${\mathcal{B}^{\sharp}}$ is a topological edge excitation, i.e. an object $x$ in $\mathcal{B}$; • the hom space $\hom_{\mathcal{B}^{\sharp}}(x,y)=[x,y]=y\otimes x^{*}$; • a distinguished morphism $\operatorname{id}_{x}:\mathbf{1}_{\mathcal{B}}\to[x,x]=x\otimes x^{*}$ defined by $u_{x}:\mathbf{1}_{\mathcal{B}}\to x\otimes x^{*}$. • a composition map $[y,z]\otimes[x,y]\to[x,z]$ defined by Eq. (4.2). • a morphism: $[x^{\prime},y^{\prime}]\otimes[x,y]\to[x^{\prime}\otimes x,y^{\prime}\otimes y]$ defined by Eq. (4.3). It was proved in [MP] that this categorical structure ${\mathcal{B}^{\sharp}}$ is a $\mathcal{B}$-enriched monoidal category. We denote this canonical edge by the pair $(V,{\mathcal{B}^{\sharp}})$. It is explained in Example A.17 that this enriched monoidal category ${\mathcal{B}^{\sharp}}$ is exactly one obtained from the pair $(\mathcal{B},\mathcal{B})$ via the canonical construction. Therefore, we can also denote ${\mathcal{B}^{\sharp}}$ by a pair $(\mathcal{B},\mathcal{B})$, i.e. ${\mathcal{B}^{\sharp}}=(\mathcal{B},\mathcal{B})$, where the second $\mathcal{B}$ is viewed as a UFC equipped with the unitary braided monoidal functor $\phi_{\mathcal{M}}:\overline{\mathcal{B}}\to\overline{\mathcal{B}}\boxtimes% \mathcal{B}=Z(\mathcal{B})$. Remark 4.2. The notion of an enriched monoidal category is a generalization of the usual notion of a monoidal category. For example, an ordinary UFC $\mathcal{M}$ can be viewed as the $\mathbf{H}$-enriched monoidal category obtain from the pair $(\mathbf{H},\mathcal{M})$ via the same canonical construction given in Example A.17, i.e. $\mathcal{M}=(\mathbf{H},\mathcal{M})$. Our categorical description of the canonical gapless edge need pass an important consistence check: boundary-bulk duality (recall Remark 3.3). We expect the Drinfeld center $Z({\mathcal{B}^{\sharp}})$ of ${\mathcal{B}^{\sharp}}$ to coincide with $\mathcal{B}$ as UMTC’s. Indeed, we introduced the notion of Drinfeld center of an enriched monoidal category in [KZ2, Def. 2.1] and proved that $Z({\mathcal{B}^{\sharp}})=\mathcal{B}$ [KZ2, Cor. 2.5]. The conclusion of this section is that the complete mathematical description of the canonical gapless edge of $(\mathcal{B},c)$ is given by a pair $(V,{\mathcal{B}^{\sharp}})$, where $V$ is a VOA such that $\operatorname{Mod}_{V}=\mathcal{B}$, and ${\mathcal{B}^{\sharp}}$ is the $\mathcal{B}$-enriched monoidal category obtained from the pair $(\mathcal{B},\mathcal{B})$ via the canonical construction given in Remark A.17. 5 General fully chiral gapless edges Similar to gapped edges, in general, there are more than one fully chiral gapless edges for a given bulk phase $(\mathcal{C},c)$ (see for example [PMN, CCBCN, CCMNPY, BN]). Similar to the fusion of gapped domain walls [KK, FS2, LWW, HW, AKZ, Ka], we can obtain a new gapless edge of the 2d bulk phase $(\mathcal{C},c)$ by fusing the canonical edge $(V,{\mathcal{B}^{\sharp}})$ of $(\mathcal{B},c)$ with a gapped wall $\mathcal{M}$ between $(\mathcal{B},c)$ and $(\mathcal{C},c)$. We denote the new gapless edge obtained from this fusion by $(V,{\mathcal{B}^{\sharp}})\boxtimes_{(\mathcal{B},c)}\mathcal{M}$, or graphically as follows: $$(V,{\mathcal{B}^{\sharp}})\boxtimes_{(\mathcal{B},c)}\mathcal{M}\quad\quad% \xrightarrow{\mbox{represented graphically as}}\quad\quad\raisebox{-30.0pt}{ \begin{picture}(100.0,75.0)\put(0.0,10.0){\scalebox{0.5}{\includegraphics[]{% pic-edge-eps-converted-to.pdf}}} \put(0.0,10.0){ \put(0.0,0.0){ \put(95.0,40.0){\scriptsize$(\mathcal{C},c)$} \put(33.0,50.0){\scriptsize$(\mathcal{B},c)$} \put(18.0,70.0){\scriptsize$(V,{\mathcal{B}^{\sharp}})$} \put(38.0,21.0){\scriptsize$\mathcal{M}$} }} \end{picture}}\,.$$ (5.1) When $\mathcal{C}=\mathcal{B}=\mathcal{M}$, we must have $(V,{\mathcal{B}^{\sharp}})=(V,{\mathcal{B}^{\sharp}})\boxtimes_{(\mathcal{B},c% )}\mathcal{B}$. We will give a detailed analysis of this new edge $(V,{\mathcal{B}^{\sharp}})\boxtimes_{(\mathcal{B},c)}\mathcal{M}$ in [KZ3]. We simply state the results here. Recall that the gapped wall $\mathcal{M}$ between $(\mathcal{B},c)$ and $(\mathcal{C},c)$ can be described mathematically by a UFC, still denoted by $\mathcal{M}$, equipped with a unitary braided equivalence $\psi_{\mathcal{M}}:\overline{\mathcal{B}}\boxtimes\mathcal{C}\to Z(\mathcal{M})$ [KK, KZ1, AKZ] (see also [FSV, Ko5, LWW, Ka] for equivalent descriptions). Since the topological edge excitations on the edge $(V,{\mathcal{B}^{\sharp}})$ are labeled by objects in $\mathcal{B}=\operatorname{Mod}_{V}$, it is clear that those on the new edge are labeled by objects in $\mathcal{B}\boxtimes_{\mathcal{B}}\mathcal{M}=\mathcal{M}$. The tensor unit $\mathbf{1}_{\mathcal{M}}=\mathbf{1}\boxtimes_{\mathcal{B}}\mathbf{1}_{\mathcal% {M}}$ in $\mathcal{M}$ is the trivial topological edge excitation. Since there is a unitary braided monoidal functor $\phi_{\mathcal{M}}:\overline{\mathcal{B}}\hookrightarrow\overline{\mathcal{B}}% \boxtimes\mathcal{C}\xrightarrow{\psi_{\mathcal{M}}}Z(\mathcal{M})$, for $x,y\in\mathcal{M}$, the internal hom $[x,y]$ is a well-defined object in $\mathcal{B}$ (see Remark A.17). The chiral algebra $U=A_{\mathbf{1}_{\mathcal{M}}}$ on this new edge is given by the internal hom $[\mathbf{1}_{\mathcal{M}},\mathbf{1}_{\mathcal{M}}]$ in $\mathcal{B}$, i.e. $U=A_{\mathbf{1}_{\mathcal{M}}}=[\mathbf{1}_{\mathcal{M}},\mathbf{1}_{\mathcal{% M}}]$. It turns out that $[\mathbf{1}_{\mathcal{M}},\mathbf{1}_{\mathcal{M}}]$ is a connected commutative separable algebra in $\mathcal{B}$ (see Def. A.2). According to [HKL], $U$ is a VOA extension of $V$. In general, $U\neq V$ (see Remark 5.1). The boundary CFT $A_{x}$ is given by the internal hom $[x,x]$ in $\mathcal{B}$. The defect fields $M_{x,y}$ between the boundary CFT’s $[x,x]$ and $[y,y]$ are given by the internal hom $[x,y]$ in $\mathcal{B}$. As a consequence, the new edge $(V,{\mathcal{B}^{\sharp}})\boxtimes_{(\mathcal{B},c)}\mathcal{M}$ is described by $V$ and a $\mathcal{B}$-enriched monoidal category ${\mathcal{M}^{\sharp}}$, where ${\mathcal{M}^{\sharp}}$ is uniquely determined by the pair $(\mathcal{B},\mathcal{M})$ via the canonical construction given in Example A.17. Moreover, we proved in [KZ2, Cor. 3.3] that the Drinfeld center $Z({\mathcal{M}^{\sharp}})$ of ${\mathcal{M}^{\sharp}}$ is exactly $\mathcal{C}$, i.e. $Z({\mathcal{M}^{\sharp}})=\mathcal{C}$. Namely, the boundary-bulk duality still holds for this new gapless edge. Remark 5.1. In general, $U\neq V$. For example, let $A$ be a connected commutative separable algebra [BEK, KO, DMNO] in $\mathcal{B}$ and $A\neq\mathbf{1}$. Let $(\mathcal{C},c)$ be the 2d bulk phase obtained by condensing $A$ in the 2d bulk phase $(\mathcal{B},c)$ [BS], i.e. $\mathcal{C}=\mathcal{B}_{A}^{0}$ [Ko5], where $\mathcal{B}_{A}^{0}$ is the category of local $A$-modules in $\mathcal{B}$ [BEK, KO, DMNO]. This condensation process creates between two phases a gapped domain wall $\mathcal{M}$ [BS], described mathematically by the category $\mathcal{B}_{A}$ of $A$-modules in $\mathcal{B}$, i.e. $\mathcal{M}=\mathcal{B}_{A}$ [Ko5]. In this case, we have $U=[\mathbf{1}_{\mathcal{M}},\mathbf{1}_{\mathcal{M}}]=A\neq\mathbf{1}=V$. Remark 5.2. On this new edge, all boundary CFT’s $[x,x]$ and walls $[x,y]$ preserve only the $V$-symmetry instead of the $U$-symmetry. As a consequence, ${\mathcal{M}^{\sharp}}$ is enriched in $\mathcal{B}=\operatorname{Mod}_{V}$ instead of in $\operatorname{Mod}_{U}$ (recall Remark 3.1). We will give a detailed explanation of this phenomenon in [KZ3]. Since ${\mathcal{M}^{\sharp}}$ is uniquely determined by the pair $(\mathcal{B},\mathcal{M})$, we denote the new edge $(V,{\mathcal{B}^{\sharp}})\boxtimes_{(\mathcal{B},c)}\mathcal{M}$ by the triple $(V,\mathcal{B},\mathcal{M})$. This notation has a lot of advantages. First, notice that the canonical edge $(V,{\mathcal{B}^{\sharp}})$ of $(\mathcal{B},c)$, in the new notation, is just the triple $(V,\mathcal{B},\mathcal{B})$. Secondly, this notation automatically include the mathematical description of gapped edges as special cases (recall Remark 4.2). For example, if $\mathcal{N}$ is a gapped edge of a 2d bulk phase $(\mathcal{D},0)$, it can be expressed as a triple $(\mathbb{C},\mathbf{H},\mathcal{N})$, where $\mathbb{C}$ denotes the field of complex numbers, viewed as the simplest VOA with 0 central charge. Thirdly, the fusion product $(V,{\mathcal{B}^{\sharp}})\boxtimes_{(\mathcal{B},c)}\mathcal{M}$ can be easily recovered by the following fusion formula $$(V,{\mathcal{B}^{\sharp}})\boxtimes_{(\mathcal{B},c)}\mathcal{M}=(V,\mathcal{B% },\mathcal{B})\boxtimes_{(\mathcal{B},c)}(\mathbb{C},\mathbf{H},\mathcal{M}):=% (V\otimes_{\mathbb{C}}\mathbb{C},\mathcal{B}\boxtimes\mathbf{H},\mathcal{B}% \boxtimes_{\mathcal{B}}\mathcal{M})=(V,\mathcal{B},\mathcal{M})=(V,{\mathcal{M% }^{\sharp}}).$$ (5.2) Since domain walls can be viewed as special cases of edges by the well-known folding trick, above construction also works for gapless/gapped domain walls. More precisely, for two given bulk phases $(\mathcal{C},c_{1})$ and $(\mathcal{D},c_{1}+c_{2})$ (see Figure 4), a gapless domain wall between them can be obtained by fusing a gapped 1d defect $\mathcal{M}$ at the intersection of $\mathcal{C},\mathcal{A},\mathcal{D}$ with the canonical edge $(U,{\mathcal{A}^{\sharp}})$ of the bulk phase $(\mathcal{A},c_{2})$. This new wall is nothing but $(U,{\mathcal{A}^{\sharp}})\boxtimes_{(\mathcal{A},c_{2})}\mathcal{M}=(U,% \mathcal{A},\mathcal{M})$. Now we consider the fusion of two such walls $(U,\mathcal{A},\mathcal{M})$ and $(V,\mathcal{B},\mathcal{N})$ depicted in Figure 4. It is clear that we have the following fusion formula (which generalizes Eq. (5.2)): $$(U,\mathcal{A},\mathcal{M})\boxtimes_{(\mathcal{D},c_{1}+c_{2})}(V,\mathcal{B}% ,\mathcal{N})=(U\otimes_{\mathbb{C}}V,\mathcal{A}\boxtimes\mathcal{B},\mathcal% {M}\boxtimes_{\mathcal{D}}\mathcal{N}),$$ (5.3) where the relative tensor product $\mathcal{M}\boxtimes_{\mathcal{D}}\mathcal{N}$ is defined in [Ta, ENO3, BBJ1, KZ1]. We want to remark that $\mathcal{M}\boxtimes_{\mathcal{D}}\mathcal{N}$ is, in general, not a UFC but a unitary multi-fusion categories even if $\mathcal{M}$ and $\mathcal{N}$ are both UFC’s [KWZ1, AKZ]. It describes an unstable 1d gapped defect [KWZ1, AKZ]. 6 Classification of gapless/gapped edges In Section 3, we have explained that a fully chiral gapless edge of $(\mathcal{C},c)$ should be described by a pair $(V,{\mathcal{X}^{\sharp}})$, where ${\mathcal{X}^{\sharp}}$ is an $\operatorname{Mod}_{V}$-enriched monoidal category for a VOA $V$. Let $\mathcal{B}=\operatorname{Mod}_{V}$. The objects in ${\mathcal{X}^{\sharp}}$ are topological edge excitations. For $x,y\in\mathcal{M}^{\$}$, the hom space $\operatorname{Hom}_{{\mathcal{X}^{\sharp}}}(x,x)$ is a boundary CFT, and $\operatorname{Hom}_{{\mathcal{X}^{\sharp}}}(x,y)$ is a wall between two boundary CFT’s $\operatorname{Hom}_{{\mathcal{X}^{\sharp}}}(x,x)$ and $\operatorname{Hom}_{{\mathcal{X}^{\sharp}}}(y,y)$. These boundary CFT’s and walls between them all preserve the $V$-symmetry. In general, the chiral algebra $U=\hom_{{\mathcal{X}^{\sharp}}}(\mathbf{1},\mathbf{1})$, where $\mathbf{1}$ is the trivial topological edge excitation, is a non-trivial extension of $V$ (recall Remark 5.1). Mathematically, it is known, by a nice work [MP], that any $\mathcal{B}$-enriched monoidal category ${\mathcal{X}^{\sharp}}$ is equivalent to the one obtained from the canonical construction from a pair $(\mathcal{B},\mathcal{X})$, where $\mathcal{X}$ is a monoidal category of ${\mathcal{X}^{\sharp}}$ equipped with a braided oplax-monoidal functor $\phi_{\mathcal{X}}:\overline{\mathcal{B}}\to Z(\mathcal{X})$, i.e. ${\mathcal{X}^{\sharp}}=(\mathcal{B},\mathcal{X})$. We have explained this canonical construction in Example A.17. In [KZ3], we will argue that the only physical relevant cases are those $(\mathcal{B},\mathcal{X})$ such that $\mathcal{X}$ is a UFC and $\phi_{\mathcal{M}}$ is a unitary braided monoidal functor. Note that $\phi_{\mathcal{M}}$ is automatically fully faithful by [DMNO, Corollary 3.26]. Moreover, ${\mathcal{X}^{\sharp}}$ should satisfy the boundary-bulk duality [KWZ2]. Namely, the Drinfeld center $Z({\mathcal{X}^{\sharp}})$ of ${\mathcal{X}^{\sharp}}$ should coincide with the UMTC $\mathcal{C}$. By [KZ2, Cor. 3.3], we have $Z({\mathcal{X}^{\sharp}})=\mathcal{C}$ if and only if $Z(\mathcal{X})=\overline{\mathcal{B}}\boxtimes\mathcal{C}$. This implies that the UFC $\mathcal{X}$ can be physically realized by a gapped wall between $(\mathcal{B},c)$ and $(\mathcal{C},c)$. Therefore, the gapless edge $(V,{\mathcal{X}^{\sharp}})$ of $(\mathcal{C},c)$ should be nothing but the triple $(V,\mathcal{B},\mathcal{X})$ constructed in Section 5. Therefore, we propose a classification of all gapped and fully chiral gapless stable edges associated to the same 2d bulk phase $(\mathcal{C},c)$: Gapped and fully chiral gapless stable edges of $(\mathcal{C},c)$ one-to-one correspond to triples $(V,\mathcal{B},\mathcal{M})$, where $V$ is a VOA $V$ with central charge $c$ such that $\mathcal{B}=\operatorname{Mod}_{V}$, and $\mathcal{M}$ is a unitary fusion category equipped with a unitary braided equivalence $\psi_{\mathcal{M}}:\overline{\mathcal{B}}\boxtimes\mathcal{C}\xrightarrow{% \simeq}Z(\mathcal{M})$. The edge is gapped if $V=\mathbb{C}$ and gapless if otherwise. Equivalently, one can replace the data $\mathcal{M}$ by a Lagrangian algebra $A_{\mathcal{M}}$ in $Z(\mathcal{M})$. If we allow to include unstable edges, we can simply replace “fusion” to “multi-fusion” in the description of $\mathcal{M}$ [KWZ1, AKZ]. Remark 6.1. A gapless edge of a bulk phase $(\mathcal{C},0)$ is gappable if there is UFC $\mathcal{M}$ such that $Z(\mathcal{M})=\mathcal{C}$. An example of such gappable edges is given in Section 7. Remark 6.2. There is a nice classification of gapless edges for Abelian 2d topological orders given in [CCMNPY]. It will be very interesting to compare our classification with that in [CCMNPY]. Constructing new gapless edges via the anyon condensations of a non-abelian bulk phase $(\mathcal{C},c)$ was considered in some special cases in [BN]. In our constructions of gapless edges, the Witt equivalence relation between $\mathcal{C}$ and $\mathcal{B}$ not only includes the cases, in which $(\mathcal{B},c)$ is obtained from $(\mathcal{C},c)$ via an anyon condensation [Ko5], but also more general ones, in which both $(\mathcal{B},c)$ and $(\mathcal{C},c)$ are obtained from anyon condensations of another bulk phase $(\mathcal{D},c)$ [DMNO, Ko5]. Remark 6.3. By the folding trick, we automatically obtain a classification of the gapless/gapped walls between two 2d bulk phases $(\mathcal{C},c_{1})$ and $(\mathcal{D},c_{2})$. If $c_{1}=c_{2}$ and $\mathcal{C},\mathcal{D}$ are Witt equivalent [DMNO, Ko5], then there are gapped walls; if otherwise, then all the walls are necessarily gapless. Let $\overline{V}$ be the same VOA as $V$ but defined by the anti-holomorphic complex variable $\bar{z}$. The time reverse (or the mirror reflection) of an edge $(V,\mathcal{B},\mathcal{N})$ of the bulk phase $(\mathcal{C},c)$ is given by $(\overline{V},\overline{\mathcal{B}},\mathcal{N}^{\mathrm{rev}})$, where $\mathcal{N}^{\mathrm{rev}}$ is the same unitary multi-fusion category as $\mathcal{N}$ but equipped with the reversed fusion product $a\otimes^{\mathrm{rev}}b:=b\otimes a$ for $a,b\in\mathcal{N}$. The triple $(\overline{V},\overline{\mathcal{B}},\mathcal{N}^{\mathrm{rev}})$ is automatically the canonical edge of the bulk phase $(\overline{\mathcal{C}},-c)$. 7 0d defects and boundary-bulk CFT’s In this section, we briefly discuss 0d defects between different gapless edges and the appearance of modular-invariant bulk CFT’s. Let us consider the situation depicted in Figure 5 (a). Let $\mathcal{M}$ be a gapped wall between two bulk phases $(\mathcal{C},c)$ and $(\mathcal{C},c)$. We must have $Z(\mathcal{M})=\mathcal{C}\boxtimes\overline{\mathcal{C}}=Z(\mathcal{C})$ as UMTC’s. Let $(V,\mathcal{C},\mathcal{C})$ be the canonical edge of $(\mathcal{C},c)$ (i.e. $\operatorname{Mod}_{V}=\mathcal{C}$), and $(\overline{V},\overline{\mathcal{C}},\mathcal{C}^{\mathrm{rev}})$ its time reversal. In this case, $\mathcal{M}$ can be realized by the category $\mathcal{C}_{W|W}$ of $W$-$W$-bimodules, where $W$ is a CSSFA in $\mathcal{C}$ [O, ENO2, FS1, FRS1] and an OSVOA extension of $V$ [HK1]. As a consequence, $\mathcal{M}$ can be viewed as a subcategory of $\mathcal{C}$. This CSSFA $W$ is not unique, but it is unique up to Morita equivalence. Different choices of $W$ in the Morita class amount to relabeling the objects in $\mathcal{M}$. We can regard the Morita class of $W$ as a physical observable. Moreover, there is a one-to-one correspondence between the set of UFC’s $\mathcal{M}$ (up to equivalences) such that $Z(\mathcal{M})=Z(\mathcal{C})$ and that of the Morita classes of CSSFA’s in $\mathcal{C}$. Fusing these two gapless edges with the gapped wall $\mathcal{M}$, we obtain a 1d gapless wall between two trivial phases defined by $$(V,\mathcal{C},\mathcal{C})\boxtimes_{\mathcal{C}}(\mathbb{C},\mathbf{H},% \mathcal{M})\boxtimes_{\mathcal{C}}(\overline{V},\overline{\mathcal{C}},% \mathcal{C}^{\mathrm{rev}})=(V\otimes_{\mathbb{C}}\overline{V},\mathcal{C}% \boxtimes\overline{\mathcal{C}},\mathcal{M}).$$ (7.1) Its 1+1D world sheet is depicted in Figure 5 (b). The fields that live on the world line supported on the trivial topological excitation $\mathbf{1}_{\mathcal{M}}$ are given by the internal hom $[\mathbf{1}_{\mathcal{M}},\mathbf{1}_{\mathcal{M}}]_{\mathcal{C}\boxtimes% \overline{\mathcal{C}}}$ in $\mathcal{C}\boxtimes\overline{\mathcal{C}}$. This internal hom is a Lagrangian algebra and is also called the full center of $\mathbf{1}_{\mathcal{M}}$, denoted by $Z(\mathbf{1}_{\mathcal{M}})$. It is not a VOA but a modular-invariant bulk CFT over $V\otimes_{\mathbb{C}}\overline{V}$ [FjFRS, KR2]. When $\mathcal{M}=\mathcal{C}$, $Z(\mathbf{1}_{\mathcal{C}})=\oplus_{i\in I(\mathcal{C})}\,i\boxtimes i^{\ast}$, where $I(\mathcal{C})$ is the set of simple objects in $\mathcal{C}$, is nothing but the charge-conjugate modular invariant bulk CFT. The map $\mathcal{M}\mapsto Z(\mathbf{1}_{\mathcal{M}})$ defines a one-to-one correspondence between gapped walls $\mathcal{M}$ and modular-invariant bulk CFT’s over $V\otimes_{\mathbb{C}}\overline{V}$. Therefore, Figure 5 provides a physical explanation of the one-to-one correspondences among the following three sets: the set of Lagrangian algebras in $Z(\mathcal{C})$, that of gapped walls between $(\mathcal{C},c)$ and $(\mathcal{C},c)$, and that of simple modular-invariant bulk CFT’s in $\mathcal{C}\boxtimes\overline{\mathcal{C}}$. Remark 7.1. More general heterotic modular invariant bulk CFT’s will be discussed in Remark 7.4. The correspondence between Lagrangian algebras and modular-invariant bulk CFT’s was known long ago [KR2, Thm. 3.4]. The physical explanation of this correspondence was provided for abelian anyon systems by Levin in [Le], which was built on some earlier works on the modular invariant partitions for gapless edge modes in quantum Hall systems (see for example [RZ, CZ, CVZ, CV]). Using the general anyon condensation theory [Ko5] (see also [ERB, HW]), Levin’s arguments can be generalized to non-abelian anyon systems. In this work, we have generalized all these earlier works, and have made this correspondence more precise by an explicit formula Eq. (7.1) and its generalization in Remark 7.4 in terms of a magic mathematical structure: enriched monoidal categories. There is a canonical 0d edge of the gapped wall $\mathcal{M}$ (see Figure 5 (a)). It is given by a pair $(W,{\mathcal{M}^{\sharp}})$, where the category ${\mathcal{M}^{\sharp}}$ is enriched in $\mathcal{M}$. More precisely, the objects in ${\mathcal{M}^{\sharp}}$ are the same as those in the UFC $\mathcal{M}$, and the hom spaces $\operatorname{Hom}_{{\mathcal{M}^{\sharp}}}(x,z)$ are given by the internal hom $[x,z]=z\otimes x^{\ast}$ in $\mathcal{M}$. The internal homs $[x,x]$ for $x\in\mathcal{M}$ are CSSFA’s in $\mathcal{M}$, and automatically OSVOA extensions of $W$. They are automatically boundary CFT’s over $V$ [FFRS2, KR1, KR2]. The internal homs $[x,z]$ are walls between boundary CFT’s. By passing to Figure 5 (b), this 0d edge $(W,{\mathcal{M}^{\sharp}})$ becomes a 0d edge of the 1d gapless wall $(V\otimes_{\mathbb{C}}\overline{V},\mathcal{C}\boxtimes\overline{\mathcal{C}},% \mathcal{M})$ (between two trivial phases). Its 0+1 world line is depicted as the left boundary of the 1+1D world sheet in Figure 5 (b). Boundary CFT’s on this world line are CSSFA’s $\mathbf{1}_{\mathcal{M}},[x,x],[z,z]$ in $\mathcal{M}$, and they must share the same bulk CFT $Z(\mathbf{1}_{\mathcal{M}})$. Mathematically, it means that they must have the same full center, i.e. $Z([x,x])=Z([z,z])=Z(\mathbf{1}_{\mathcal{M}})$. It is guaranteed because they are all Morita equivalent [FFRS2, KR1, D1]. Moreover, these internal homs $[x,x]$ in $\mathcal{M}$ for all $x\in\mathcal{M}$ realize all boundary CFT’s over $V$ that share the same bulk CFT $Z(\mathbf{1}_{\mathcal{M}})$ [KR1, KR2]. Remark 7.2. In particular, when $\mathcal{M}=\mathcal{C}$, $[x,x]=x\otimes x^{\ast}$ for $x\in\mathcal{C}$ are all the CSSFA’s in $\mathcal{C}$ that are allowed on the canonical edge $(V,{\mathcal{C}^{\sharp}})$ (recall Remark 4.1). When $\mathcal{M}\neq\mathcal{C}$, $[x,x]\in\mathcal{M}=\mathcal{C}_{W|W}$ recovers CSSFA’s in $\mathcal{C}$ in a Morita class different from that of $\mathbf{1}_{\mathcal{C}}$. In this way, we have recovered naturally all boundary-bulk CFT’s over $V$ from 2d topological orders. Remark 7.3. Note that $(V\otimes_{\mathbb{C}}\overline{V},\mathcal{C}\boxtimes\overline{\mathcal{C}},% \mathcal{M})$ is a gapless wall between two trivial phases. The Drinfeld center of the enriched monoidal category $(\mathcal{C}\boxtimes\overline{\mathcal{C}},\mathcal{M})$ is exactly the trivial UMTC $\mathbf{H}$ [KZ2]. Such a gapless wall is called gappable (recall Remark 6.1). By [Z, Corollary 4.3], the enriched monoidal category $(Z(\mathcal{M}),\mathcal{M})$ is Morita equivalent to $(\mathbf{H},\mathbf{H})$, and the Morita equivalence is exactly defined by the invertible $(\mathbf{H},\mathbf{H})$-$(Z(\mathcal{M}),\mathcal{M})$-bimodule ${\mathcal{M}^{\sharp}}$. Remark 7.4. More generally, bulk phases on the two sides of the gapped wall $\mathcal{M}$ in Figure 5 (a) can be different, say $(\mathcal{C},c)$ and $(\mathcal{D},c)$. There is a VOA $A$ with two VOA extensions $U$ and $V$ such that $\operatorname{Mod}_{U}=\mathcal{C}$, $\operatorname{Mod}_{V}=\mathcal{D}$ [DMNO]. The internal hom $[\mathbf{1}_{\mathcal{M}},\mathbf{1}_{\mathcal{M}}]_{\mathcal{C}\boxtimes% \overline{\mathcal{D}}}=Z(\mathbf{1}_{\mathcal{M}})$ is a Lagrangian algebra in $\mathcal{C}\boxtimes\overline{\mathcal{D}}$. The map $\mathcal{M}\mapsto Z(\mathbf{1}_{\mathcal{M}})$ defines a bijection between the set of gapped walls between $(\mathcal{C},c)$ and $(\mathcal{D},c)$, that of Lagrangian algebras in $\mathcal{C}\boxtimes\overline{\mathcal{D}}$, which are precisely those simple modular-invariant bulk CFT’s in $Z(\operatorname{Mod}_{A})$ containing $U\otimes_{\mathbb{C}}\overline{V}$ as a full field subalgebra [FFRS1, D2, DMNO]. The fusion category $\mathcal{M}$ can be realized by $\mathcal{E}_{W|W}$, where $W$ is a CSSFA in $\mathcal{E}:=(\operatorname{Mod}_{A})_{U|V}$. The 0d edge of the wall $\mathcal{M}$ is still given by the pair $(W,{\mathcal{M}^{\sharp}})$, where ${\mathcal{M}^{\sharp}}$ is the $\mathcal{M}$-enriched category consisting of the same objects as those in $\mathcal{M}$ and $\operatorname{Hom}_{{\mathcal{M}^{\sharp}}}(x,z):=[x,z]=z\otimes x^{\ast}\in% \mathcal{M}$. All $[x,z]$ in $\mathcal{M}$ can be interpreted as boundary CFT’s (or walls) over $A$ and share the same bulk CFT $Z(\mathbf{1}_{\mathcal{M}})$. We will give more details in [KZ3]. Remark 7.5. We will give a detailed study and a classification of all 0d walls in [KZ3]. Some examples of 0d walls were also studied in [CCBCN]. 8 Summary and outlooks In this work, we have shown that a gapless edge of a 2d bulk phase $(\mathcal{C},c)$ can be mathematically described/classified by an enriched monoidal category ${\mathcal{M}^{\sharp}}$ together with a VOA $V$. This description includes that of gapped edges as special cases. Therefore, we have found a unified mathematical theory for both gapped and gapless edges. We have left out a few important issues that will be discussed in [KZ3]. We briefly mention them below. • Some gapless edges given in our classification are actually gappable. A gapless edge is gappable if it shares the same bulk with a gapped edge. Mathematically, a gapless edge $(V,\mathcal{B},\mathcal{M})$ is gappable if its Drinfeld centers $Z(\mathcal{B},\mathcal{M})$ coincide with that of a UFC $\mathcal{N}$, or equivalently, $(\mathcal{B},\mathcal{M})$ is Morita equivalent to $(\mathbf{H},\mathcal{N})$ [Z]. We will explain them in [KZ3]. • In [KZ1], the complete mathematical statement of the boundary-bulk duality for gapped edges can be stated as a functor mapping UFC’s to their Drinfeld centers. Moreover, the functor is fully faithful. We will prove a similar result for gapless/gapped edges. Enriched monoidal categories are also useful in the study of topological order with symmetries (SPT’s/SET’s). Our results shed light on how to describe gapless/gapped edges for SPT’s and SET’s. We will come back to this point in the future. The unified mathematical theory of gapless/gapped edges/walls also allows us to compute global observables for topological orders with gapless edges/walls via factorization homology [AKZ]. It will be interesting to explore them thoroughly in the future. Some partial results in this direction will be given in [KZ3]. It is very important to study how to obtain a gapless edge from another via pure edge phase transitions (see for example [PMN, CCBCN, CCMNPY] and references therein). Appendix A Appendix A.1 Algebras in unitary modular tensor categories In this subsection, we recall the definitions of a few types of algebras in a UFC or a UMTC. A unitary fusion category (UFC) $\mathcal{C}$ is a unitary finite abelian rigid monoidal category [ENO1, EGNO]. We will not recall its definition, but only recall a few important ingredients of it. 1. there are finitely many simple objects, and all objects are direct sums of simple objects; 2. the hom spaces $\hom_{\mathcal{C}}(x,y)$ are all finite dimensional Hilbert spaces; For every morphism $f:x\to y$, there is an adjoint $f^{\ast}:y\to x$; and $f^{\ast}\circ f=0$ implies that $f=0$; 3. there is an associative tensor product $\otimes:\mathcal{C}\times\mathcal{C}\to\mathcal{C}$, i.e. $x\otimes(y\otimes z)\simeq(x\otimes y)\otimes z$, and a distinguished object $\mathbf{1}$ called the tensor unit; $\mathbf{1}$ is simple; 4. each object $x$ has a (two-side) dual object $x^{\ast}$, together with duality maps $u_{x}:\mathbf{1}\to x\otimes x^{\ast}$ and $v_{x}:x^{\ast}\otimes x\to\mathbf{1}$. A UFC $\mathcal{C}$ is a unitary braided fusion category if it is also equipped with a braiding $c_{x,y}:x\otimes y\xrightarrow{\simeq}y\otimes x$ for $x,y\in\mathcal{C}$ satisfying the Hexagon relations. A UMTC is a unitary braided fusion category such that $\mathbf{1}$ is the only simple object that is symmetric to all objects [Tu]. In other words, if $x$ is simple and $c_{y,x}\circ c_{x,y}=\operatorname{Id}_{x\otimes y},\forall y\in\mathcal{C}$, then $x\simeq\mathbf{1}$. Definition A.1. Let $\mathcal{C}$ be a monoidal category. An algebra in $\mathcal{C}$ is a triple $(A,m,\iota)$, where $A$ is an object in $\mathcal{C}$, $m:A\otimes A\to A$ and $\iota:\mathbf{1}\to A$ are morphisms in $\mathcal{C}$, such that $$m\circ(m\otimes\operatorname{Id}_{A})=m\circ(\operatorname{Id}_{A}\otimes m),% \quad\quad m\circ(\iota\otimes\operatorname{Id}_{A})=\operatorname{Id}_{A}=m% \circ(\operatorname{Id}_{A}\otimes\iota).$$ If $\mathcal{C}$ is also braided, then $A$ is called commutative if $m\circ c_{A,A}=m$. Definition A.2. An algebra $(A,m,\iota)$ in a monoidal category $\mathcal{C}$ is called separable if there is an $A$-$A$-bimodule map $e:A\to A\otimes A$ such that $m\circ e=\operatorname{Id}_{A}$. A separable algebra $(A,m,\iota)$ is called connected if $\hom_{\mathcal{C}}(\mathbf{1},A)=\mathbb{C}$. If $\mathcal{C}$ is a UMTC, then a commutative connected separable algebra is called an étale algebra [DMNO], or a condensable algebra because it determines an anyon condensation in a 2d topological order [Ko5]. Definition A.3 ([DMNO]). A Lagrangian algebra in a UMTC $\mathcal{C}$ is a condesable algebra $A$ such that $(\dim A)^{2}=\dim(\mathcal{C})$. Definition A.4. Let $\mathcal{C}$ be a monoidal category. A co-algebra in $\mathcal{C}$ is a triple $(A,\Delta,\epsilon)$, where $A$ is an object in $\mathcal{C}$, $\Delta:A\to A\otimes A$ and $\epsilon:A\to\mathbf{1}$ are morphisms in $\mathcal{C}$, such that $$(\Delta\otimes\operatorname{Id}_{A})\circ\Delta=(\operatorname{Id}_{A}\otimes% \Delta)\circ\Delta,\quad\quad(\epsilon\otimes\operatorname{Id}_{A})\circ\Delta% =\operatorname{Id}_{A}=(\operatorname{Id}_{A}\otimes\epsilon)\circ\Delta.$$ Remark A.5. If $\mathcal{C}$ is UFC and $(A,m,\iota)$ an algebra in $\mathcal{C}$, then the triple $(A,m^{\ast},\iota^{\ast})$ is automatically a co-algebra in $\mathcal{C}$. Definition A.6 ([FRS1]). Let $\mathcal{C}$ be a monoidal category. A Frobenius algebra in $\mathcal{C}$ is a quintuple $(A,m,\iota,\Delta,\epsilon)$ such that $(A,m,\iota)$ is an algebra and $(A,\Delta,\epsilon)$ is a co-algebra and $$(m\otimes\operatorname{Id}_{A})\circ(\operatorname{Id}_{A}\otimes\Delta)=% \Delta\circ m=(\operatorname{Id}_{A}\otimes m)\circ(\Delta\otimes\operatorname% {Id}_{A}).$$ Such a Frobenius algebra is called special if $m\circ\Delta=\operatorname{Id}_{A}$. When $\mathcal{C}$ is a UFC, a Frobenius algebra in $\mathcal{C}$ is called symmetric if $$(\operatorname{Id}_{A^{\ast}}\otimes(\epsilon\circ m))\circ(u_{A^{\ast}}% \otimes\operatorname{Id}_{A})=((\epsilon\circ m)\otimes\operatorname{Id}_{A^{% \ast}})\circ(\operatorname{Id}_{A}\otimes u_{A}).$$ We abbreviate a connected special symmetric Frobenius algebra to a CSSFA. Remark A.7. A special Frobenius algebra is automatically a separable algebra. Conversely, a connected separable algebra in a UFC has a unique structure of a CSSFA. Let $\mathcal{C}$ be a UFC. Let $\mathcal{M}$ be a left $\mathcal{C}$-module, which is a unitary category $\mathcal{M}$ equipped with a unital and associative action $\odot:\mathcal{C}\times\mathcal{M}\to\mathcal{M}$. In other words, for $a,b\in\mathcal{C},x\in\mathcal{M}$, we have $b\odot x\in\mathcal{M}$, $\mathbf{1}\odot x\simeq x$ and $a\odot(b\odot x)\simeq(a\otimes b)\odot x$. For $x,y\in\mathcal{M}$, the internal hom $[x,y]$ is an object in $\mathcal{C}$ defined by the following isomorphisms for all $a\in\mathcal{C}$, $$\gamma_{a,x,y}:\hom_{\mathcal{M}}(a\odot x,y)\xrightarrow{\simeq}\hom_{% \mathcal{C}}(a,[x,y]),$$ (A.1) which are natural with respect to all three variables $a,x,y$. Equivalently, one can define the internal hom $[x,y]$ by its universal property. When $a=[x,y]$, $\rho:=\gamma_{[x,y],x,y}^{-1}(\operatorname{Id}_{[x,y]})$ is a morphism $[x,y]\odot x\to y$. The pair $([x,y],\rho)$ satisfies the following universal property. If $(u,f)$ is another such a pair, i.e. $f:u\odot x\to y$, then there is a unique morphism $f^{\prime}:u\to[x,y]$ in $\mathcal{C}$ such that the following diagram $$\xymatrix@R=1.5em{&[x,y]\odot x\ar[rd]^{\rho}&\\ u\odot x\ar[rr]^{f}\ar[ru]^{f^{\prime}\odot\operatorname{Id}_{x}}&&y}$$ (A.2) is commutative. This universal property of internal hom determines the pair $([x,y],\rho)$ up to isomorphisms. This universal property also provides a canonical morphism $\operatorname{ev}:[y,z]\otimes[x,y]\to[x,z]$ induced from the action $([y,z]\otimes[x,y])\odot x\to[y,z]\odot y\to z$, and a canonical morphism $\mathbf{1}\to[x,x]$ induced from the unital action $\mathbf{1}\odot x\simeq x$. These morphisms provide $[x,x]$ with a structure of an algebra in $\mathcal{C}$ and $[x,y]$ with a structure of a $[y,y]$-$[x,x]$-bimodule. Example A.8. When $\mathcal{M}=\mathcal{C}$ is viewed as a left $\mathcal{C}$-module, then $[x,y]=y\otimes x^{\ast}$ and $$\rho:[x,y]\odot x=y\otimes x^{\ast}\otimes x\xrightarrow{\operatorname{Id}_{y}% \otimes v_{x}}x,$$ In this case, it is very easy to show that such defined $[x,y]$ satisfies the universal property of internal hom. More precisely, by the rigidity of $\mathcal{C}$, there is a canonical isomorphism $\phi:\hom_{\mathcal{C}}(u\otimes x,y)\simeq\hom_{\mathcal{C}}(u,y\otimes x^{% \ast})$. For any $f:u\otimes x\to y$, we obtain $f^{\prime}=\phi(f):u\to y\otimes x^{\ast}$ such that diagram in Eq. (A.2). Moreover, in this case, the canonical morphism $\operatorname{ev}:[y,z]\otimes[x,y]\to[x,z]$ is explicitly defined by $$z\otimes y^{\ast}\otimes y\otimes x^{\ast}\xrightarrow{\operatorname{Id}_{z}% \otimes v_{y}\operatorname{Id}_{x^{\ast}}}z\otimes x^{\ast},$$ and the $\mathbf{1}\to[x,x]$ is defined by $u_{x}:\mathbf{1}\to x\otimes x^{\ast}$. It turns out that $[x,x]$ for $x\in\mathcal{M}$ are CSSFA’s in $\mathcal{M}$. Example A.9. Let $\mathcal{M}$ be a UFC. Its Drinfeld center $Z(\mathcal{M})$ is a UMTC. $\mathcal{M}$ is also a left $Z(\mathcal{M})$-module. The internal hom $[\mathbf{1}_{\mathcal{M}},\mathbf{1}_{\mathcal{M}}]$ in $Z(\mathcal{M})$ is a condensable algebra (see Def. A.2), and is also called the full center of $\mathbf{1}_{\mathcal{M}}$, denoted by $Z(\mathbf{1}_{\mathcal{M}})$. When $\mathcal{M}$ is a UMTC, $Z(\mathcal{M})\simeq\mathcal{M}\boxtimes\overline{\mathcal{M}}$, and $Z(\mathbf{1}_{\mathcal{M}})=\oplus_{i}i\boxtimes i^{\ast}$, where $i$ are simple objects in $\mathcal{C}$, is a Lagrangian algebra in $Z(\mathcal{M})$. More generally, the notion of the full center $Z(A)$ can be defined for any algebra $A$ in $\mathcal{M}$ [D1]. If $A$ is a CSSFA in $\mathcal{M}$, then the full center $Z(A)$ of $A$ in $Z(\mathcal{M})$ is a Lagrangian algebra in $Z(\mathcal{M})$ [FjFRS, KR1, KR2]. A.2 Enriched monoidal categories and a canonical construction In this subsection, we recall the definition of an enriched monoidal category and a canonical construction of it. Definition A.10. Let $\mathcal{B}$ be a monoidal category. A category $\mathcal{C}^{\sharp}$ enriched in $\mathcal{B}$, or a $\mathcal{B}$-enriched category, consists of a set of objects $Ob(\mathcal{C}^{\sharp})$, an object $\operatorname{Hom}_{\mathcal{C}^{\sharp}}(x,y)$ in $\mathcal{B}$ for every pair $x,y\in\mathcal{C}^{\sharp}$, and a composition morphism $\circ:\operatorname{Hom}_{\mathcal{C}^{\sharp}}(y,z)\otimes\operatorname{Hom}_% {\mathcal{C}^{\sharp}}(x,y)\to\operatorname{Hom}_{\mathcal{C}^{\sharp}}(x,z)$ for $x,y,z\in\mathcal{C}^{\sharp}$, such that there exists a morphism $\operatorname{id}_{x}:\mathbf{1}\to\operatorname{Hom}_{\mathcal{C}^{\sharp}}(x% ,x)$ for $x\in\mathcal{C}^{\sharp}$ rendering the following diagrams commutative for $x,y,z,w\in\mathcal{C}^{\sharp}$: $$\xymatrix@!C=15ex{&\operatorname{Hom}_{\mathcal{C}^{\sharp}}(x,y)\otimes% \operatorname{Hom}_{\mathcal{C}^{\sharp}}(x,x)\ar[rd]^{\circ}\\ \operatorname{Hom}_{\mathcal{C}^{\sharp}}(x,y)\ar[rr]^{\operatorname{Id}}\ar[% ru]^{\operatorname{Id}\otimes\operatorname{id}_{x}}&&\operatorname{Hom}_{% \mathcal{C}^{\sharp}}(x,y),\\ }$$ (A.3) $$\xymatrix@!C=15ex{&\operatorname{Hom}_{\mathcal{C}^{\sharp}}(y,y)\otimes% \operatorname{Hom}_{\mathcal{C}^{\sharp}}(x,y)\ar[rd]^{\circ}\\ \operatorname{Hom}_{\mathcal{C}^{\sharp}}(x,y)\ar[rr]^{\operatorname{Id}}\ar[% ru]^{\operatorname{id}_{y}\otimes\operatorname{Id}}&&\operatorname{Hom}_{% \mathcal{C}^{\sharp}}(x,y),\\ }$$ (A.4) $$\xymatrix{\operatorname{Hom}_{\mathcal{C}^{\sharp}}(z,w)\otimes\operatorname{% Hom}_{\mathcal{C}^{\sharp}}(y,z)\otimes\operatorname{Hom}_{\mathcal{C}^{\sharp% }}(x,y)\ar[r]^{-}{\operatorname{Id}\otimes\circ}\ar[d]_{\circ\otimes% \operatorname{Id}}&\operatorname{Hom}_{\mathcal{C}^{\sharp}}(z,w)\otimes% \operatorname{Hom}_{\mathcal{C}^{\sharp}}(x,z)\ar[d]^{\circ}\\ \operatorname{Hom}_{\mathcal{C}^{\sharp}}(y,w)\otimes\operatorname{Hom}_{% \mathcal{C}^{\sharp}}(x,y)\ar[r]^{-}\circ&\operatorname{Hom}_{\mathcal{C}^{% \sharp}}(x,w).\\ }$$ (A.5) Remark A.11. In this work, we distinguish two notations $\operatorname{Id}_{x}$ and $\operatorname{id}_{x}$, where $\operatorname{Id}_{x}$ is the usual identity morphism $x\to x$ in an ordinary category, but $\operatorname{id}_{x}$ is reserved for the morphism $\mathbf{1}\to\hom_{\mathcal{C}^{\sharp}}(x,x)$ in $\mathcal{B}$ for a $\mathcal{B}$-enriched category ${\mathcal{C}^{\sharp}}$. Example A.12. (Canonical construction I): Let $\mathcal{B}$ be a UFC and $\mathcal{M}$ a left $\mathcal{B}$-module. The categorical structure ${\mathcal{M}^{\sharp}}$ consisting of 1. objects in ${\mathcal{M}^{\sharp}}$ are objects in $\mathcal{M}$, i.e. $Ob({\mathcal{M}^{\sharp}})=Ob(\mathcal{M})$; 2. $\hom_{\mathcal{M}^{\sharp}}(x,y)=[x,y]\in\mathcal{B}$; 3. $\operatorname{id}_{x}:\mathbf{1}\to[x,x]$ is the morphism canonically induced from the unital action $\mathbf{1}\odot x\simeq x$; 4. $\circ:[y,z]\otimes[x,y]\to[x,z]$ is the morphism $\operatorname{ev}$ canonically induced from the action $([y,z]\otimes[x,y])\odot x\to[y,z]\odot y\to z$. It is well-known that this ${\mathcal{M}^{\sharp}}$ is a $\mathcal{B}$-enriched category [Ke]. Definition A.13. The underlying category $\mathcal{C}$ of a $\mathcal{B}$-enriched category ${\mathcal{C}^{\sharp}}$ is an ordinary category defined as follows: $Ob(\mathcal{C})=Ob({\mathcal{C}^{\sharp}})$ and $\hom_{\mathcal{C}}(x,y):=\hom_{\mathcal{B}}(\mathbf{1},\hom_{\mathcal{C}^{% \sharp}}(x,y))$ for $x,y\in Ob(\mathcal{C})$. The identity morphism in $\hom_{\mathcal{C}}(x,x)$ is $\operatorname{id}_{x}$ and the composition of morphisms is naturally induced from that of ${\mathcal{C}^{\sharp}}$. Definition A.14. An enriched functor $F:\mathcal{C}^{\sharp}\to\mathcal{D}^{\sharp}$ between $\mathcal{B}$-enriched categories consists of a map $F:Ob(\mathcal{C}^{\sharp})\to Ob(\mathcal{D}^{\sharp})$ and a morphism $F:\operatorname{Hom}_{\mathcal{C}^{\sharp}}(x,y)\to\operatorname{Hom}_{% \mathcal{D}^{\sharp}}(F(x),F(y))$ for every pair $x,y\in\mathcal{C}^{\sharp}$ such that the following diagrams commute for $x,y,z\in\mathcal{C}^{\sharp}$ $$\xymatrix{&\mathbf{1}\ar[ld]_{\operatorname{id}_{x}}\ar[rd]^{\operatorname{id}% _{F(x)}}\\ \operatorname{Hom}_{\mathcal{C}^{\sharp}}(x,x)\ar[rr]^{-}F&&\operatorname{Hom}% _{\mathcal{D}^{\sharp}}(F(x),F(x)),\\ }$$ $$\xymatrix{\operatorname{Hom}_{\mathcal{C}^{\sharp}}(y,z)\otimes\operatorname{% Hom}_{\mathcal{C}^{\sharp}}(x,y)\ar[r]^{-}\circ\ar[d]_{F\otimes F}&% \operatorname{Hom}_{\mathcal{C}^{\sharp}}(x,z)\ar[d]^{F}\\ \operatorname{Hom}_{\mathcal{D}^{\sharp}}(F(y),F(z))\otimes\operatorname{Hom}_% {\mathcal{D}^{\sharp}}(F(x),F(y))\ar[r]^{-}\circ&\operatorname{Hom}_{\mathcal{% D}^{\sharp}}(F(x),F(z)).\\ }$$ (A.6) It is clear that the composition of two enriched functors is again an enriched functor. The enriched functor $F:{\mathcal{C}^{\sharp}}\to{\mathcal{D}^{\sharp}}$ naturally induces an ordinary functor $F:\mathcal{C}\to\mathcal{D}$ between two underlying categories. Definition A.15. An enriched natural transformation $\xi:F\to G$ between two enriched functors $F,G:\mathcal{C}^{\sharp}\to\mathcal{D}^{\sharp}$ consists of a morphism $\xi_{x}:\mathbf{1}\to\hom_{\mathcal{D}^{\sharp}}(F(x),G(x))$ for $x\in\mathcal{C}$ such that the following diagram commutes for $x,y\in\mathcal{C}^{\sharp}$: $$\xymatrix@!C=30ex{\operatorname{Hom}_{\mathcal{C}^{\sharp}}(x,y)\ar[r]^{-}{G}% \ar[d]_{F}&\operatorname{Hom}_{\mathcal{D}^{\sharp}}(G(x),G(y))\ar[d]^{-\circ% \xi_{x}}\\ \operatorname{Hom}_{\mathcal{D}^{\sharp}}(F(x),F(y))\ar[r]^{-}{\xi_{y}\circ-}&% \operatorname{Hom}_{\mathcal{D}^{\sharp}}(F(x),G(y)).\\ }$$ (A.7) Note that the composition of two enriched natural transformations $\xi:F\to G$ and $\eta:G\to H$ is defined by $(\eta\circ\xi)_{x}=\eta_{x}\circ\xi_{x}:\mathbf{1}\to\operatorname{Hom}_{% \mathcal{D}^{\sharp}}(F(x),H(x))$. An enriched natural transformation $\xi$ is called an enriched natural isomorphism if each $\xi_{x}$ is an isomorphism. Now we assume that $\mathcal{B}$ is a braided monoidal category equipped with braiding $c_{x,y}:x\otimes y\to y\otimes x$ for $x,y\in\mathcal{B}$. Let $\mathcal{C}^{\sharp}$ and $\mathcal{D}^{\sharp}$ be $\mathcal{B}$-enriched categories. The Cartesian product $\mathcal{C}^{\sharp}\times\mathcal{D}^{\sharp}$ is a $\mathcal{B}$-enriched category defined as follows [MP]: • $Ob(\mathcal{C}^{\sharp}\times\mathcal{D}^{\sharp})=Ob(\mathcal{C}^{\sharp})% \times Ob(\mathcal{D}^{\sharp})$; • $\operatorname{Hom}_{\mathcal{C}^{\sharp}\times\mathcal{D}^{\sharp}}((x,y),(x^{% \prime},y^{\prime}))=\operatorname{Hom}_{\mathcal{C}^{\sharp}}(x,x^{\prime})% \otimes\operatorname{Hom}_{\mathcal{D}^{\sharp}}(y,y^{\prime})$; • the composition $$\displaystyle\circ:\operatorname{Hom}_{\mathcal{C}^{\sharp}\times\mathcal{D}^{% \sharp}}((x^{\prime},y^{\prime}),(x^{\prime\prime},y^{\prime\prime}))\otimes% \operatorname{Hom}_{\mathcal{C}^{\sharp}\times\mathcal{D}^{\sharp}}$$ $$\displaystyle((x,y),(x^{\prime},y^{\prime}))$$ $$\displaystyle\to$$ $$\displaystyle\operatorname{Hom}_{\mathcal{C}^{\sharp}\times\mathcal{D}^{\sharp% }}((x,y),(x^{\prime\prime},y^{\prime\prime}))$$ is given by $$\displaystyle\operatorname{Hom}_{\mathcal{C}^{\sharp}}(x^{\prime},x^{\prime% \prime})\otimes\operatorname{Hom}_{\mathcal{D}^{\sharp}}(y^{\prime},y^{\prime% \prime})\otimes\operatorname{Hom}_{\mathcal{C}^{\sharp}}(x,x^{\prime})\otimes% \operatorname{Hom}_{\mathcal{D}^{\sharp}}(y,y^{\prime})$$ $$\displaystyle         \xrightarrow{\operatorname{Id}\otimes c^{-1}\otimes% \operatorname{Id}}\operatorname{Hom}_{\mathcal{C}^{\sharp}}(x^{\prime},x^{% \prime\prime})\otimes\operatorname{Hom}_{\mathcal{C}^{\sharp}}(x,x^{\prime})% \otimes\operatorname{Hom}_{\mathcal{D}^{\sharp}}(y^{\prime},y^{\prime\prime})% \otimes\operatorname{Hom}_{\mathcal{D}^{\sharp}}(y,y^{\prime})$$ (A.8) $$\displaystyle         \xrightarrow{~{}~{}\circ\otimes\circ~{}~{}}\operatorname% {Hom}_{\mathcal{C}^{\sharp}}(x,x^{\prime\prime})\otimes\operatorname{Hom}_{% \mathcal{D}^{\sharp}}(y,y^{\prime\prime}).$$ Definition A.16. A $\mathcal{B}$-enriched monoidal category consists of a $\mathcal{B}$-enriched category $\mathcal{C}^{\sharp}$, a distinguished object $\mathbf{1}_{\mathcal{C}^{\sharp}}$, an enriched functor $\otimes:\mathcal{C}^{\sharp}\times\mathcal{C}^{\sharp}\to\mathcal{C}^{\sharp}$, and enriched isomorphisms $\lambda:\mathbf{1}_{\mathcal{C}^{\sharp}}\otimes-\to\operatorname{Id}_{% \mathcal{C}^{\sharp}}$, $\rho:-\otimes\mathbf{1}_{\mathcal{C}^{\sharp}}\to\operatorname{Id}_{\mathcal{C% }^{\sharp}}$, $\alpha:\otimes\circ(\otimes\times\operatorname{Id}_{\mathcal{C}^{\sharp}})\to% \otimes\circ(\operatorname{Id}_{\mathcal{C}^{\sharp}}\times\otimes)$ such that the underlying category $\mathcal{C}$, together with $\otimes$, $\lambda,\rho,\alpha$, defines a monoidal category. Example A.17 (Canonical construction II). Let $\mathcal{M}$ be a monoidal category and $\mathcal{B}$ a braided monoidal such that there is a braided oplax-monoidal functor $\phi_{\mathcal{M}}:\overline{\mathcal{B}}\to Z(\mathcal{M})$, where $Z(\mathcal{M})$ is the Drinfeld center of $\mathcal{M}$. The oplax-monoidal structure of $\phi_{\mathcal{M}}$ is a morphism $\beta_{a,b}:\phi(a\otimes b)\to\phi(a)\otimes\phi(b)$ satisfying some natural conditions. If $\beta_{a,b}$ for $a,b\in\mathcal{B}$ are isomorphisms, then an oplax-monoidal functor becomes a monoidal functor. In general, $\beta_{a,b}$ are not isomorphisms. There is a functor $\odot:\overline{\mathcal{B}}\times\mathcal{M}\to\mathcal{M}$ defined by $\overline{\mathcal{B}}\times\mathcal{M}\xrightarrow{\phi_{\mathcal{M}}\times% \operatorname{Id}_{\mathcal{M}}}Z(\mathcal{M})\times\mathcal{M}\to\mathcal{M}% \times\mathcal{M}\xrightarrow{\otimes}\mathcal{M}$. There is a canonical construction of a $\mathcal{B}$-enriched monoidal category ${\mathcal{M}^{\sharp}}$ from the pair $(\mathcal{C},\mathcal{M})$ [MP]: • objects in ${\mathcal{M}^{\sharp}}$ are objects in $\mathcal{M}$, i.e. $Ob({\mathcal{M}^{\sharp}}):=Ob(\mathcal{M})$; • For $x,y\in\mathcal{M}$, $\operatorname{Hom}_{\mathcal{M}^{\sharp}}(x,y):=[x,y]$ in $\overline{\mathcal{B}}$ (or in $\mathcal{B}$); • $\operatorname{id}_{x}:\mathbf{1}\to[x,x]$ is the morphism in $\mathcal{B}$ canonically induced from the unital action $\mathbf{1}\odot x\simeq x$; • $\circ:[y,z]\otimes[x,y]\to[x,z]$ is the morphism canonically induced from the action $([y,z]\otimes[x,y])\odot x\to[y,z]\odot y\to z$. • $\otimes:[x^{\prime},y^{\prime}]\otimes[x,y]\to[x^{\prime}\otimes x,y^{\prime}% \otimes y]$ is the morphism in $\mathcal{B}$ canonically induced from the action $$\displaystyle([x^{\prime},y^{\prime}]\otimes[x,y])\odot x^{\prime}\otimes x$$ $$\displaystyle=\phi_{\mathcal{M}}([x^{\prime},y^{\prime}]\otimes[x,y])\otimes x% ^{\prime}\otimes x\to\phi_{\mathcal{M}}([x^{\prime},y^{\prime}])\otimes\phi_{% \mathcal{M}}([x,y])\otimes x^{\prime}\otimes x$$ $$\displaystyle\xrightarrow{\operatorname{Id}\otimes b_{\phi_{\mathcal{M}}([x,y]% ),x^{\prime}}\otimes\operatorname{Id}_{x}}\phi_{\mathcal{M}}([x^{\prime},y^{% \prime}])\otimes x^{\prime}\otimes\phi_{\mathcal{M}}([x,y])\otimes x\to y^{% \prime}\otimes y.$$ (A.9) References [AL1] I. Affleck, A.W.W. Ludwig, Exact conformal-field-theory results on the multichannel Kondo effect: Single-fermion Green’s function, self-energy, and resistivity, Phys. 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The phase transition in hot $\Lambda$ hypernuclei within relativistic Thomas-Fermi approximation Jinniu Hu hujinniu@nankai.edu.cn School of Physics, Nankai University, Tianjin 300071, China    Zhaowen Zhang Department of Physics and Astronomy and Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai Jiao Tong University, Shanghai 200240, China    Shishao Bao School of Physics, Nankai University, Tianjin 300071, China    Hong Shen School of Physics, Nankai University, Tianjin 300071, China (December 8, 2020) Abstract A self-consistent description for hot $\Lambda$ hypernuclei in hypothetical big boxes is developed within the relativistic Thomas-Fermi approximation in order to investigate directly the liquid-gas phase coexistence in strangeness finite nuclear systems. We use the relativistic mean-field model for nuclear interactions. The temperature dependence of $\Lambda$ hyperon density, $\Lambda$ hyperon radius, excitation energies, specific heat, and the binding energies of $\Lambda$ hypernuclei from ${}^{16}_{\Lambda}$O to ${}^{208}_{\Lambda}$Pb in phase transition region are calculated by using the subtraction procedure in order to separate the hypernucleus from the surrounding baryon gas. The $\Lambda$ central density is very sensitive to the temperature. The radii of $\Lambda$ hyperon at high temperature become very large. In the relativistic Thomas-Fermi approximation with the subtraction procedure, the properties of hypernuclei are independent of the size of the box in which the calculation is performed. The level density parameters of hypernuclei in the present work are confirmed to be almost constant at low temperature. It is also found that the single-$\Lambda$ binding energies of $\Lambda$ hypernuclei are largely reduced with increasing temperature. Relativistic Thomas-Fermi approximation, Hypernuclei, Finite temperature pacs: 21.10.Dr, 21.60.Jz, 21.80.+a ††preprint: I Introduction Theoretical studies of hypernuclei are continuously boosted by new and upgraded experimental facilities LL1 ; LL2 ; LL3 ; LL4 ; LL5 ; Gal10 ; Bot12 ; Tam12 . It is generally believed that from them one could derive various features of the underlying hyperon interactions Bod87 ; Usm04 ; Usm06 ; Hiy10 ; Hiy10s ; Hiy12 ; Gal11 ; Li13 . They are also related to the dense stellar matter studies Li07 ; Hu13 , as an alternative way of obtaining the matter apart from astrophysical observations and/or quite developed many-body schemes for infinite strongly interacting systems, for example, the widely used microscopic Brueckner-Hartree-Fock (BHF) theory Bur11 . Lattice QCD calculation should be an ideal tool for investigating hypernucleus structure since it retains all the fundamental characters of QCD theory. Indeed, the first calculation of hypernuclei with baryon number $A>2$ has been performed recently, for ${}^{4}_{\Lambda}$He and ${}^{4}_{\Lambda\Lambda}$He lqcd . However, a detailed and precise structure description is still beyond its reach. Few-body calculations in cluster or shell-model approach are awaited for not-so-light hypernuclei ($A>10$). Significant progress in the auxiliary field diffusion Monte Carlo method Lon13 has been achieved in the calculation of closed shell $\Lambda$ hypernuclei from $A=5$ to $91$. For a more feasible way of the systematic study of both light and heavy hypernuclei, effective models are generally employed. Among them, many models are for single-$\Lambda$ hypernuclei, for example, the quark mean-field model Shen02 , the relativistic mean-field (RMF) approach shen06 ; Xu12 , the Skyrme-Hartree-Fock model Li13 ; Sch13 ; Gul12 , the quark-meson coupling model Gui08 , a relativistic point-coupling model Tan12 , the quark mass density-dependent model Wu13 , and the density-dependent RMF theory from relativistic BHF theory Hu14 . The experiments, $(\pi,K)$, $(e,e^{\prime}K)$ and $(\gamma,K)$, are the most popular reactions used to produce hypernuclei  hashimoto06 . Recently, the heavy ion collision is suggested as one way to generate hypernuclei  gaitanos09 ; botvina11 , such as the high energy Au+Au collision steinheimer12 , which can be considered as a liquid-gas phase transition in hypermatter. The lifetime of hypernuclei in such reactions are usually very short and the production of hypernuclei should be strongly dependent on the temperature. Therefore, it is very interesting to investigate the properties of hot hypernuclei in the liquid-gas coexistence region. The matter generated from the collision of relativistic heavy ions has some probabilities to break up as the nuclear-fragment and hyper-fragment production, which can be described by the statistical multifragmentation model bondorf95 . This model was also extended to the study of hypernuclei produced in heavy ion collisions botvina07 ; gupta09 . Accordingly, we want to investigate the hot hypernuclei from the aspect of the liquid-gas phase coexistence in this work. Since the hot hypernucleus formed in nucleus-nucleus collisions is thermodynamically unstable against the emission of baryons, an external pressure has to be exerted on the hypernucleus to compensate for the tendency of baryon emission. This pressure is assumed to be exerted by a surrounding gas representing evaporated baryons, which is in equilibrium with the hot hypernucleus. In order to separate the nucleus from the surrounding gas, a subtraction procedure was first proposed in Hartree-Fock framework BLV85 for normal nucleus, and then used in the Thomas-Fermi approach TF87 . The subtraction procedure is based on the existence of two solutions to the equations of motion of nucleons. One solution corresponds to the nucleon gas alone ($G$), and the other to the nuclear liquid phase in equilibrium with the surrounding gas ($NG$). The density profile of the nucleus ($L$) is then given by subtracting the gas density from that of the liquid-plus-gas phase. Finally, the physical quantities of the isolated nucleus obtained using such subtraction procedure could be independent of the size of the box in which the calculation is performed. In the past decades, this subtraction procedure has been widely applied in the non-relativistic Thomas-Fermi approximation with Skyrme force Sub01 ; Sub02 ; Sub07 ; Sub12 ; PLB12 ; Sub14 . The relativistic Thomas-Fermi approximation with RMF Lagrangian has been developed and applied to study various subjects at the subnuclear densities, such as, droplet formation RTF99a ; RTF99b and nuclear pasta phases Mene08 ; Mene10 ; Gril12 . This method is considered to be self-consistent in the treatment of surface effects and nucleon distributions. The relativistic Thomas-Fermi approximation was also adopted to describe finite nuclei RTF02 ; RTF01 and non-uniform nuclear matter for supernova simulations Zhang14 . In Refs. RTF02 ; RTF01 , the thermodynamic properties of finite nuclei were calculated within the relativistic Thomas-Fermi approximation, and the results obtained were found to depend on the input freeze-out volume, which was actually the size of the box for performing the calculation. Recently, we developed a relativistic Thomas-Fermi model for the description of hot nuclei by employing the subtraction procedure, and investigated the temperature dependence of the symmetry energy of finite nuclei zhang14b . Actually, the results obtained from subtraction procedure are independent of the size of the box. In this work, we would like to extend the relativistic Thomas-Fermi model with subtraction procedure to describe the hot $\Lambda$ hypernuclei, which are most known in experiment and theoretical calculation among various hypernuclei. For the nuclear interaction and $\Lambda N$ interaction, we adopt the RMF model, which has been successfully used to study various phenomena in nuclear physics Sero86 ; Ring90 ; Meng06 . The thermodynamic properties of hot $\Lambda$ hypernuclei, such as excitation energies, specific heat, and level density parameters of hypernuclei will be investigated. In Sec. II, we briefly derive the relativistic Thomas-Fermi approximation using the subtraction procedure for the description of hot $\Lambda$ hypernuclei. In Sec. III, the numerical results are shown for the properties of $\Lambda$ hypernuclei from ${}^{16}_{\Lambda}$O to ${}^{208}_{\Lambda}$Pb at finite temperature. A summary is given in Sec. IV. II Relativistic Thomas-Fermi approximation for hot $\Lambda$ hypernuclei In the RMF model, the baryons (nucleons and hyperons) interact through the exchange of various mesons. The mesons considered are the isoscalar scalar and vector mesons ($\sigma$ and $\omega$) and isovector vector meson ($\rho$). The baryon Lagrangian density reads, $$\displaystyle\mathcal{L}_{\rm{RMF}}$$ $$\displaystyle=$$ $$\displaystyle\sum_{i=p,n}\bar{\psi}_{i}\left[i\gamma_{\mu}\partial^{\mu}-M_{i}% -g_{\sigma N}\sigma-g_{\omega N}\gamma_{\mu}\omega^{\mu}-\frac{g_{\rho N}}{2}% \gamma_{\mu}\tau_{a}\rho^{a\mu}-e\gamma_{\mu}\frac{(1-\tau_{3})}{2}A^{\mu}% \right]\psi_{i}$$ $$\displaystyle+\bar{\psi}_{\Lambda}(i\gamma_{\mu}\partial^{\mu}-M_{\Lambda}-g_{% \sigma\Lambda}\sigma-g_{\omega\Lambda}\gamma_{\mu}\omega^{\mu}){\psi}_{\Lambda}$$ $$\displaystyle+\frac{1}{2}\partial_{\mu}\sigma\partial^{\mu}\sigma-\frac{1}{2}m% ^{2}_{\sigma}\sigma^{2}-\frac{1}{3}g_{2}\sigma^{3}-\frac{1}{4}g_{3}\sigma^{4}$$ $$\displaystyle-\frac{1}{4}W_{\mu\nu}W^{\mu\nu}+\frac{1}{2}m^{2}_{\omega}\omega_% {\mu}\omega^{\mu}+\frac{1}{4}c_{3}\left(\omega_{\mu}\omega^{\mu}\right)^{2}$$ $$\displaystyle-\frac{1}{4}R^{a}_{\mu\nu}R^{a\mu\nu}+\frac{1}{2}m^{2}_{\rho}\rho% ^{a}_{\mu}\rho^{a\mu},$$ where $W^{\mu\nu}$ and $R^{a\mu\nu}$ are the antisymmetric field tensors for $\omega^{\mu}$ and $\rho^{a\mu}$, respectively. $g_{\sigma N},~{}g_{\omega N}$ and $g_{\rho N}$ are the coupling constants between $\sigma,\omega,\rho$ and nucleon, respectively, while $g_{\sigma\Lambda}$ and $g_{\omega\Lambda}$ are the coupling constants between $\sigma,~{}\omega$ and $\Lambda$ hyperon. Here, the tensor coupling between $\omega$ and $\Lambda$ hyperon is not taken into account, which just generates the large spin-orbit splitting of $\Lambda$ hyperon. However, in the Thomas-Fermi approximation, the single particle level at different spin states cannot be obtained. Furthermore, such tensor coupling does not change the total energy of $\Lambda$ hypernuclei very much. Therefore, we ignore this tensor coupling term in present work. The electromagnetic coupling constant is $e=\sqrt{4\pi/137}$. In the RMF approach, meson fields are treated as classical fields and the field operators are replaced by their expectation values. For a static system, the nonvanishing expectation values are $\sigma=\left\langle\sigma\right\rangle$, $\omega=\left\langle\omega^{0}\right\rangle$, and $\rho=\left\langle\rho^{30}\right\rangle$, where $0$ represents the time component in Dirac space and $3$ represents the third component in isospin space for $\rho$ meson. Using the relativistic Thomas-Fermi approximation with the subtraction procedure BLV85 ; TF87 , we study a hot $\Lambda$ hypernucleus based on the thermodynamic potential of the isolated hypernucleus, which is defined by $$\displaystyle\Omega=\Omega^{NG}-\Omega^{G}+E_{C},$$ (2) where $\Omega^{NG}$ and $\Omega^{G}$ are the baryon thermodynamic potentials in the liquid phase with the surrounding gas ($NG$) and the gas phase alone ($G$), respectively. We employ the RMF Lagrangian to obtain the thermodynamic potential $\Omega^{a}$ ($a=NG$ or $G$), which can be given as $$\displaystyle\Omega^{a}=E^{a}-TS^{a}-\sum_{i=p,n,\Lambda}\mu_{i}N^{a}_{i}.$$ (3) Here, the energy $E^{a}$, entropy $S^{a}$, and particle number $N^{a}_{i}$ in the phase $a$ are obtained by $$\displaystyle E^{a}$$ $$\displaystyle=$$ $$\displaystyle\int\varepsilon^{a}(r)d^{3}r,$$ (4) $$\displaystyle S^{a}$$ $$\displaystyle=$$ $$\displaystyle\int s^{a}(r)d^{3}r,$$ $$\displaystyle N^{a}_{i}$$ $$\displaystyle=$$ $$\displaystyle\int n^{a}_{i}(r)d^{3}r,$$ where $\varepsilon^{a}(r)$, $s^{a}(r)$, and $n^{a}_{i}(r)$ are the local energy density, entropy density, and particle number density defined in the RMF model. The local energy density derived from the Lagrangian density (II) without Coulomb force is written as $$\displaystyle{\varepsilon}(r)$$ $$\displaystyle=$$ $$\displaystyle\displaystyle{\sum_{i=p,n,\Lambda}\frac{1}{\pi^{2}}\int_{0}^{% \infty}dk\,k^{2}\,\sqrt{k^{2}+{M_{i}^{\ast}}^{2}}\left(f_{i+}^{k}+f_{i-}^{k}% \right)}$$ $$\displaystyle+\frac{1}{2}(\nabla\sigma)^{2}+\frac{1}{2}m_{\sigma}^{2}\sigma^{2% }+\frac{1}{3}g_{2}\sigma^{3}+\frac{1}{4}g_{3}\sigma^{4}$$ $$\displaystyle-\frac{1}{2}(\nabla\omega)^{2}-\frac{1}{2}m_{\omega}^{2}\omega^{2% }-\frac{1}{4}c_{3}\omega^{4}+g_{\omega N}\omega\left(n_{p}+n_{n}\right)+g_{% \omega\Lambda}\omega n_{\Lambda}$$ $$\displaystyle-\frac{1}{2}(\nabla\rho)^{2}-\frac{1}{2}m_{\rho N}^{2}\rho^{2}+% \frac{g_{\rho}}{2}\rho\left(n_{p}-n_{n}\right),$$ where $M_{i}^{\ast}=M_{i}+g_{\sigma i}\sigma$ is the effective baryon mass, and $n_{i}$ is the number density of species $i$ ($i=p,n$ or $\Lambda$). The entropy density is given by $$\displaystyle s(r)=\displaystyle{\sum_{i=p,n,\Lambda}\frac{1}{\pi^{2}}\int_{0}% ^{\infty}dk\,k^{2}}$$ $$\displaystyle\left[-f_{i+}^{k}\ln f_{i+}^{k}-\left(1-f_{i+}^{k}\right)\ln\left% (1-f_{i+}^{k}\right)\right.$$ $$\displaystyle\left.-f_{i-}^{k}\ln f_{i-}^{k}-\left(1-f_{i-}^{k}\right)\ln\left% (1-f_{i-}^{k}\right)\right].$$ Here $f_{i+}^{k}$ and $f_{i-}^{k}$ are the occupation probabilities of the particle and antiparticle at momentum $k$, respectively. Their detailed form will be determined by variational principle self-consistently later. The number density of proton ($i=p$), neutron ($i=n$) or $\Lambda$ hyperon ($i=\Lambda$) at position $r$ is given by $$\displaystyle n_{i}(r)=\frac{1}{\pi^{2}}\int_{0}^{\infty}dk\,k^{2}\,\left(f_{i% +}^{k}-f_{i-}^{k}\right).$$ (7) The Coulomb energy is calculated from the subtracted proton density as $$\displaystyle E_{C}=\int\left[e\left(n_{p}^{NG}-n_{p}^{G}\right)A_{0}-\frac{1}% {2}(\nabla A_{0})^{2}\right]d^{3}r,$$ (8) where $A_{0}$ is the electrostatic potential. The equilibrium state of the isolated hypernucleus can be obtained by minimization of the thermodynamic potential $\Omega$ defined in Eq. (2). The meson mean fields in the $NG$ phase satisfy the variational equation $$\displaystyle\frac{\delta\Omega}{\delta\phi^{NG}}=0,\hskip 28.452756pt\phi^{NG% }=\sigma^{NG},\,\omega^{NG},\,\rho^{NG},$$ (9) which leads to the following equations of motion for meson mean fields in the $NG$ phase, $$\displaystyle-\nabla^{2}\sigma^{NG}+m_{\sigma}^{2}\sigma^{NG}+g_{2}\left(% \sigma^{NG}\right)^{2}+g_{3}\left(\sigma^{NG}\right)^{3}=-g_{\sigma N}\left(n_% {s,p}^{NG}+n_{s,n}^{NG}\right)-g_{\sigma\Lambda}n_{s,\Lambda}^{NG},$$ (10) $$\displaystyle-\nabla^{2}\omega^{NG}+m_{\omega}^{2}\omega^{NG}+c_{3}\left(% \omega^{NG}\right)^{3}=g_{\omega N}\left(n_{p}^{NG}+n_{n}^{NG}\right)+g_{% \omega\Lambda}n_{\Lambda}^{NG},$$ $$\displaystyle-\nabla^{2}\rho^{NG}+m_{\rho}^{2}\rho^{NG}=\frac{g_{\rho N}}{2}% \left(n_{p}^{NG}-n_{n}^{NG}\right).$$ The occupation probability $f_{i+}^{k,NG}$ ($f_{i-}^{k,NG}$) of species $i$ ($i=p,n$ or $\Lambda$) can be derived from the variational equation, $$\displaystyle\frac{\delta\Omega}{\delta f_{i\pm}^{k,NG}}=0,$$ (11) which results in the Fermi-Dirac distribution of particle and antiparticle for proton or neutron as, $$\displaystyle f_{i\pm}^{k,NG}$$ $$\displaystyle=$$ $$\displaystyle\left\{1+\exp\left[\left(\sqrt{k^{2}+\left(M_{i}^{\ast,NG}\right)% ^{2}}+g_{\omega N}\omega^{NG}+\frac{g_{\rho N}}{2}\tau_{3}\rho^{NG}+e\frac{% \tau_{3}+1}{2}A_{0}\mp\mu_{i}\right)/T\right]\right\}^{-1},$$ (12) and the one for $\Lambda$ hyperon $$\displaystyle f_{\Lambda\pm}^{k,NG}$$ $$\displaystyle=$$ $$\displaystyle\left\{1+\exp\left[\left(\sqrt{k^{2}+\left(M_{\Lambda}^{\ast,NG}% \right)^{2}}+g_{\omega\Lambda}\omega^{NG}\mp\mu_{\Lambda}\right)/T\right]% \right\}^{-1}.$$ (13) Similarly, we obtain the equations of motion for meson mean fields in the $G$ phase, $$\displaystyle-\nabla^{2}\sigma^{G}+m_{\sigma}^{2}\sigma^{G}+g_{2}\left(\sigma^% {G}\right)^{2}+g_{3}\left(\sigma^{G}\right)^{3}=-g_{\sigma N}\left(n_{s,p}^{G}% +n_{s,n}^{G}\right)-g_{\sigma\Lambda}n_{s,\Lambda}^{G},$$ (14) $$\displaystyle-\nabla^{2}\omega^{G}+m_{\omega}^{2}\omega^{G}+c_{3}\left(\omega^% {G}\right)^{3}=g_{\omega N}\left(n_{p}^{G}+n_{n}^{G}\right)+g_{\omega\Lambda}n% _{\Lambda}^{G},$$ $$\displaystyle-\nabla^{2}\rho^{G}+m_{\rho}^{2}\rho^{G}=\frac{g_{\rho N}}{2}% \left(n_{p}^{G}-n_{n}^{G}\right),$$ and the occupation probability in the $G$ phase for proton or neutron, $$\displaystyle f_{i\pm}^{k,G}$$ $$\displaystyle=$$ $$\displaystyle\left\{1+\exp\left[\left(\sqrt{k^{2}+\left(M_{i}^{\ast,G}\right)^% {2}}+g_{\omega N}\omega^{G}+\frac{g_{\rho N}}{2}\tau_{3}\rho^{G}+e\frac{\tau_{% 3}+1}{2}A_{0}\mp\mu_{i}\right)/T\right]\right\}^{-1},$$ (15) and the one for $\Lambda$ hyperon, $$\displaystyle f_{\Lambda\pm}^{k,G}$$ $$\displaystyle=$$ $$\displaystyle\left\{1+\exp\left[\left(\sqrt{k^{2}+\left(M_{\Lambda}^{\ast,G}% \right)^{2}}+g_{\omega\Lambda}\omega^{G}\mp\mu_{\Lambda}\right)/T\right]\right% \}^{-1}.$$ (16) In the equations for meson mean fields, $n_{s,i}^{a}$ and $n_{i}^{a}$ denote respectively the scalar and number densities of species $i$ ($i=p,n$ or $\Lambda$) in the $a$ ($a=NG$ or $G$) phase zhang14b . By minimizing $\Omega$ with respect to the electrostatic potential $A_{0}$, we obtain the Poisson equation for $A_{0}$ as $$\displaystyle-\nabla^{2}A_{0}=e\left(n_{p}^{NG}-n_{p}^{G}\right).$$ (17) The inclusion of the Coulomb energy in $\Omega$ leads to a coupling between the two sets of equations for the $NG$ and $G$ phases. Therefore, the coupled equations (10), (14), and (17) should be solved simultaneously at given temperature $T$ and chemical potentials $\mu_{p},~{}\mu_{n}$ and $\mu_{\Lambda}$. For a hypernucleus with $N_{p}$ protons, $N_{n}$ neutrons and $N_{\Lambda}$ hyperons at temperature $T$, the proton, neutron and $\Lambda$ hyperon chemical potentials $\mu_{p}$, $\mu_{n}$ and $\mu_{\Lambda}$ can be determined from given $N_{p}$, $N_{n}$ and $N_{\Lambda}$. Once the chemical potentials are known, the occupation probabilities and density distributions can be obtained easily. In practice, we solve self-consistently the coupled equations (10), (14), and (17) under the constraints of given $N_{p},~{}N_{n}$ and $N_{\Lambda}$. After getting the solutions for the $NG$ and $G$ phases, we can extract the properties of the hot hypernucleus based on the subtraction procedure. The proton, neutron, and $\Lambda$ hyperon numbers, $N_{p},~{}N_{n}$ and $N_{\Lambda}$, are given by $$\displaystyle N_{i}=N^{NG}_{i}-N^{G}_{i}=\int n_{i}(r)d^{3}r,\hskip 28.452756% pti=p,\,n,\,\Lambda,$$ (18) where $n_{i}(r)=n^{NG}_{i}(r)-n^{G}_{i}(r)$ is the local density of the isolated hypernucleus, which decreases to zero at large distances. Therefore, physical quantities of the isolated hypernucleus could be independent of the size of the box in which the calculation is done. The total energy including Coulomb contributions for the hot hypernucleus is given by $$\displaystyle E=E^{NG}-E^{G}+E_{C},$$ (19) where $E^{NG}$ and $E^{G}$ are the baryon energies without Coulomb interaction in the $NG$ and $G$ phases, which are calculated from Eq. (4). The Coulomb energy $E_{C}$ is given by Eq. (8). The entropy and other extensive quantities of the isolated hypernucleus can be calculated by subtracting the contribution of the $G$ phase from the one of the $NG$ phase. The excitation energy of hot hypernuclei is a very important thermodynamic quantity. For a hypernucleus at temperature $T$, its excitation energy is defined as $$\displaystyle E^{\ast}(T)=E(T)-E(T=0).$$ (20) The center-of-mass correction of $\Lambda$ hypernucleus is taken into account by a conventional phenomenological way Xu12 , $$\displaystyle E_{\textrm{c.m.}}=-\frac{3}{4}\times 41\left(N_{n}+N_{p}+N_{% \Lambda}\right)^{-1/3}\text{MeV}.$$ (21) III Results and discussions The properties of hot $\Lambda$ hypernuclei are investigated within the relativistic Thomas-Fermi approximation using the subtraction procedure in this section. For the nuclear interaction, we adopt the RMF model with TM1 parametrization TM1 , which was determined by the ground-state properties of finite nuclei and properties of nuclear matter from relativistic BHF theory. It was successfully applied to calculate the equation of state for supernova simulations and characters of neutron stars Shen11 ; Shen02 . As for the meson-$\Lambda$ hyperon couplings, it is well known that the properties of $\Lambda$ hypernuclei are very sensitive to the ratios of the meson-$\Lambda$ hyperon couplings to the meson-nucleon couplings $R_{\sigma}=g_{\sigma\Lambda}/g_{\sigma N}$ and $R_{\omega}=g_{\omega\Lambda}/g_{\omega N}$. We take the relative $\omega$ coupling as $R_{\omega}=2/3$ from the naive quark counting and the relative $\sigma$ coupling as $R_{\sigma}=0.621$ given in Ref. shen06 . With this choice, the experimental $\Lambda$ binding energies of single-$\Lambda$ hypernuclei can be reproduced very well in the RMF model shen06 . The coupled equations (10), (14) and (17) are solved self-consistently with given baryon numbers of $\Lambda$ hypernuclei, $N_{n},~{}N_{p}$ and $N_{\Lambda}$ from Eq. (18) in a spherical box with radius $R$. In this section, we take two single-$\Lambda$ hypernuclei, ${}^{40}_{\Lambda}$Ca and ${}^{208}_{\Lambda}$Pb, as numerical examples to investigate the properties of hot hypernuclei within relativistic Thomas-Fermi approximation. In the subtraction procedure, the properties of hot hypernuclei should be independent of the size of the box, when the box radius $R$ is generally taken to be sufficiently large. In Fig. 1, the density distributions of $\Lambda$ hyperon from ${}^{208}_{\Lambda}$Pb for $G$ and $NG$ phases at $T=8$ MeV with different box sizes, $R=16,~{}18$, and $20$ fm are shown in order to check if the results depend on the size of the box. At the central region of the hypernucleus, these distributions are identical, while they have different behaviors approaching the box boundary. However, the behaviors of the $G$ phase at boundary are in accordance with the one of the $NG$ phase, which will generate their subtraction, i.e. the densities of the $L$ phase, to be independent of the size of the box. In Figs. 2 and 3, the density distributions of $\Lambda$ hyperon, neutron and proton for ${}^{40}_{\Lambda}$Ca and ${}^{208}_{\Lambda}$Pb at $T=0,~{}4$, and $8$ MeV from left panels to right panels are presented, which are obtained with the box radius, $R=20$ fm. From top to bottom, the results of the liquid-plus-gas phase ($NG$), gas phase ($G$), and subtracted liquid phase ($L$) are displayed, respectively. The $\Lambda$ hyperon density distributions are multiplied by 10 and 20 in ${}^{40}_{\Lambda}$Ca and ${}^{208}_{\Lambda}$Pb, respectively to adopt the same scales of neutron and proton in these two figures. Firstly, we can see that subtracted $\Lambda$ densities in the isolated hypernucleus ($L$) vanish at large distances. Therefore, the physical quantities of the hypernucleus will be independent of the size of the box. The $\Lambda$ densities of the $G$ phase are found to be exactly zero at zero temperature, while these densities are finite but very small at low temperature ($T=4$ MeV). As temperature increases, the $\Lambda$ hyperon densities of the $G$ phase increase obviously. On the other hand, the $\Lambda$ densities at the center of the hypernucleus are reduced largely and the nuclear surface becomes more diffuse with increasing $T$ as shown in the top and bottom panels. The $\Lambda$ hyperon density in the center region at $T=8$ MeV is just about $30\%$ of the value at $T=0$ MeV. It is easier to be influenced by the temperature for a single-$\Lambda$ hyperon compared with a large nucleus composed of many protons and neutrons whose center densities are less sensitive to the temperature as shown in Figs. 1 and 2 in Ref. zhang14b . Moreover, the $\Lambda$ density at the center of ${}^{40}_{\Lambda}$Ca is about 5 times of the one of ${}^{208}_{\Lambda}$Pb. This is because the $\Lambda$ density in a single-$\Lambda$ hypernucleus is inversely proportional to the baryon number, $n_{\Lambda}\propto\frac{1}{A}$, if we consider the hypernucleus as a liquid drop. For the neutron and proton densities in ${}^{40}_{\Lambda}$Ca and ${}^{208}_{\Lambda}$Pb, they were almost not changed by $\Lambda$ hyperon compared with the ones of ${}^{39}$Ca and ${}^{207}$Pb without $\Lambda$ hyperon as shown in Ref. zhang14b . This is because that the magnitude of single $\Lambda$ hyperon density is very small, just $5\%\sim 10\%$ of nucleons. It will not change the solutions of Eq.(10) and Eq.(14) so much in the cases of nuclei with and without $\Lambda$ hyperon. In Fig. 4, we display the root-mean-square (rms) radii of neutrons, protons and $\Lambda$ hyperon, $R_{n},~{}R_{p}$ and $R_{\Lambda}$, as a function of the temperature $T$ for ${}^{40}_{\Lambda}$Ca (left panel) and ${}^{208}_{\Lambda}$Pb (right panel), which are defined as, $$\displaystyle R_{i}=\sqrt{\frac{\int d^{3}rr^{2}n_{i}(r)}{\int d^{3}rn_{i}(r)}% },~{}~{}~{}i=n,~{}p,~{}\Lambda.$$ (22) It is shown that $R_{n}$ and $R_{p}$ slowly increase with temperature due to the diffusion of nuclear densities at high temperature. However, the radii of $\Lambda$ hyperon at low temperature are much smaller than the ones of neutrons and protons, while they are very close at high temperature. This is because the $\Lambda$ density distribution becomes much diffuser at high temperature and is more easily influenced by temperature as discussed above. The scalar and vector potentials of $\Lambda$ hyperon in ${}^{40}_{\Lambda}$Ca and ${}^{208}_{\Lambda}$Pb at $T=0,~{}4,$ and $8$ MeV are shown in Fig. 5, which are defined as $U^{\Lambda}_{S}=g_{\sigma\Lambda}\sigma$ and $U^{\Lambda}_{V}=g_{\omega\Lambda}\omega$. The magnitudes of the scalar and vector potentials reduce with temperature significantly. Especially, at higher temperature, this tendency becomes more obvious. These potentials in the center regions of hypernuclei at $T=8$ MeV are reduced by $20\%$ compared to the cases at $T=0$ MeV. The attractive scalar potential is slightly larger than the repulsive vector potential, and their differences at the center of hypernuclei are about $15-25$ MeV, which result in the bound states of $\Lambda$ hypernuclei. The excitation energies of hot $\Lambda$ hypernuclei can be calculated from Eq. (20). The temperature $T$ as functions of the excitation energy per particle $E^{\ast}/A$ (caloric curve) are plotted in Fig. 6 for ${}^{40}$Ca, ${}^{40}_{\Lambda}$Ca, ${}^{208}$Pb, and ${}^{208}_{\Lambda}$Pb. We can see that $E^{\ast}/A$ increases slowly at low temperature, while it rises more rapidly as $T$ increases. The excitation energy of $\Lambda$ hypernuclei is larger than the one of normal nuclei with the same baryon number. This is mainly because the single-$\Lambda$ hyperon is more easily excited than a nucleon which has more correlation with other nucleons considering a nucleus as a collective mode. By the same reason, the excitation energy of heavy nuclei is smaller than that of light nuclei at same temperature. We also find that there exists a limiting temperature $T_{\rm{lim}}$ for a hot hypernucleus, which is strongly dependent on the interaction and the size of the box. Generally, the limiting temperature is above $8$ MeV in the Thomas-Fermi calculations TF87 ; zhang14b . Therefore, the results of hot $\Lambda$ hypernuclei in the present work are only shown up to $T\sim 8$ MeV. The specific heat $C_{v}$ per particle is a very useful thermodynamic quantity for hot nucleus, which is defined at a fixed volume as, $$\displaystyle C_{v}=\left.\frac{d(E^{\ast}(T)/A)}{dT}\right|_{V}.$$ (23) We show in Fig. 7 the specific heat as functions of temperature for ${}^{40}_{\Lambda}$Ca and ${}^{208}_{\Lambda}$Pb. In Ref. RTF01 , the specific heat was studied with relativistic Thomas-Fermi approximation for hot nuclei, by introducing a freeze-out volume to treat the density diffusing in the surface of nuclei at finite temperature. Therefore, the specific heat was strongly dependent on the freeze-out volume. When the subtraction procedure is used to isolate the $\Lambda$ hypernucleus from the surrounding baryon gas, the properties of hot hypernucleus are independent of the size of the box. In the left panel of Fig. 7, the specific heat of ${}^{208}_{\Lambda}$Pb is shown with different box sizes $R=15$ fm and $R=20$ fm. We can find that they are identical until $T=8$ MeV. The results of specific heat for ${}^{40}_{\Lambda}$Ca and ${}^{208}_{\Lambda}$Pb are compared in the right panel. It is shown that $C_{v}$ of ${}^{40}_{\Lambda}$Ca is larger than the one of ${}^{208}_{\Lambda}$Pb. This is because the caloric curve of ${}^{40}_{\Lambda}$Ca is stiffer. It is demonstrated that light hypernuclei are more easily excited than heavy one. The properties of single-$\Lambda$ hypernuclei, $\Lambda$ hyperon radii, center density of $\Lambda$ hyperon, excitation energy per particle, single-$\Lambda$ binding energy, and the level density parameter for ${}^{40}_{\Lambda}$Ca and ${}^{208}_{\Lambda}$Pb at different temperatures are listed in Table 1 and Table 2, respectively. In the low-temperature Fermi gas approximation, the level density parameters $a$, which is related to the density of state of an excited state, can be expressed as, $S/2T,~{}E^{*}/T^{2}$ , or $S^{2}/4E^{*}$ BLV85 , where $E^{*}$ is the excitation energies from Eq. (20). In our calculation, the level density parameters with different definitions in single-$\Lambda$ hypernuclei are almost temperature independent for $T\leq 4$ MeV. Their magnitudes are also consistent with each other. The level density parameter for the light nucleus is smaller than the heavy one. To distinguish the excitation energy in Eq. (20), we would like to use the $\Lambda$ binding energy instead of $\Lambda$ excitation energy in this paper, although the Lambda hyperon may also occupy an excited state. The single-$\Lambda$ binding energy is a very important property of $\Lambda$ hypernuclei, which is obtained by the subtraction of the binding energy of $\Lambda$ hypernucleus from its core energy without hyperon. In Fig. 8, we present the single-$\Lambda$ binding energies of some typical spherical single-$\Lambda$ hypernuclei at different temperatures from ${}^{16}_{\Lambda}$O to ${}^{208}_{\Lambda}$Pb and compare them with the experimental data in term of $\Lambda$ $1s$ states at zero temperature hashimoto06 . It is seen that the single-$\Lambda$ binding energies decrease with temperature. At high temperature, such reduction becomes faster. At $T=0$ MeV, the single-$\Lambda$ binding energy can be measured at different spin states. The experimental value of the $1s$ state of ${}^{208}_{\Lambda}$Pb is $26.3\pm 0.8$ MeV hashimoto06 . In the present study, we use the relativistic Thomas-Fermi approximation to describe the hypernucleus and we do not solve the Dirac equation for the nucleon and $\Lambda$ hyperon. Therefore, the single-$\Lambda$ binding energies in this approximation cannot be distinguished from different spin states. The $\Lambda$ binding energy of ${}^{208}_{\Lambda}$Pb obtained in the present calculation at $T=0$ MeV is 27.41 MeV, which is consistent with the experiment data. For the light hypernuclei, like ${}^{16}_{\Lambda}$O, our results of $\Lambda$ binding energies are slightly overestimated in comparison with experimental data. IV Conclusion The relativistic Thomas-Fermi approximation has been applied to the investigation of hot single-$\Lambda$ hypernuclei using the RMF model for the interaction of baryons. The subtraction procedure has been employed in order to separate the hypernucleus from the surrounding baryon gas. With such treatment, the properties of hot $\Lambda$ hypernucleus are independent of the size of the box in which the calculation is performed. The nucleon and $\Lambda$ hyperon interact via the exchange of the $\sigma$ and $\omega$ mesons, whose coupling constants are determined by experimental $\Lambda$ binding energies in the RMF model. We have studied two single-$\Lambda$ hypernuclei, ${}^{40}_{\Lambda}$Ca and ${}^{208}_{\Lambda}$Pb, as numerical examples in this work. At high density, the $\Lambda$ gas density becomes visible and increases with temperature. On the other hand, the $\Lambda$ density at the center of $\Lambda$ hypernuclei is reduced largely with temperature. The temperature dependence of $\Lambda$ densities is more remarkable than that of proton and neutron, since one hyperon is more easily excited than nucleus which are compounded of many nucleons. Furthermore, the magnitudes of $\Lambda$ densities at the center of hypernuclei are almost inverse to the baryon numbers. The rms radius of $\Lambda$ hyperon is clearly different from those of proton and neutron at zero temperature. However, it increases rapidly with temperature and becomes comparable with the radii of proton and neutron, which is due to the diffusion of $\Lambda$ distribution at high temperature. The scalar and vector potentials of $\Lambda$ hyperon have been found to be reduced with temperature so that the $\Lambda$ binding energies become small at high temperature. The specific heat defined as the derivation of excitation energy with respect to temperature was found to be independent of the size of the box by employing the subtraction procedure, which is different from introducing the freeze-out volume to consider the temperature effect. Finally we also gave the single-$\Lambda$ binding energies from ${}^{16}_{\Lambda}$O to ${}^{208}_{\Lambda}$Pb at different temperatures. The binding energies are consistent with the results obtained in the RMF model at zero temperature for heavy hypernuclei. As temperature increases, the $\Lambda$ binding energies decrease significantly. We have systematically studied the properties of hot single-$\Lambda$ hypernuclei above mediate mass. 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The Chemical Composition of Carbon C(N) stars C. Abia, I. Domínguez    R. Gallino, S. Masera    M. Busso    O. Straniero    P. de Laverny    B. Plez Abstract A chemical study of normal Galactic C(N) carbon stars is presented. Abundances of Li, CNO isotopes and $s$-elements are derived. The derived abundances of $s$-elements nicely agree with theoretical s-process nucleosynthesis predictions during the AGB phase. However, the figures obtained for Li and the ${}^{12}$C/${}^{13}$C ratios might imply the existence of a non-standard mixing process during the AGB phase operating preferentially in low mass stars. The intrinsic or extrinsic nature of C(N) stars is also discussed. Dpt. Física Teórica y del Cosmos. Universidad de Granada. 18071 Granada. Spain Dipartimento di Fisica Generale. Universitá di Torino. 10125 Torino. Italy Dipartimento di Fisica. Universitá di Perugia. 06123 Perugia. Italy Osservatorio di Collurania. 64100 Teramo. Italy Observatoire de la Cote d’Azur. 06528 Nice. France GRAAL. Université de Montpellier II. Montpellier. France 1. Introduction It is commonly accepted that the spectral sequence along the asymptotic giant branch (AGB) phase (M$\rightarrow$MS$\rightarrow$S$\rightarrow$SC$\rightarrow$C) is, in fact, a chemical sequence in which the carbon content in the stellar envelope continuously increases due to the operation of the so-called 3${}^{th}$ dredge-up (TDU) mechanism. The TDU transports to the envelope fresh carbon immediately after each thermal pulse (TP) of the He-shell. In such a way, an AGB star of M-type with a ratio in the envelope C/O$\sim 0.5$ becomes a carbon C(N) star when this ratio exceeds unity, C/O$>1$. Another important consequence of TDU is the enrichment of the envelope in s-elements. The necessary neutrons for the $s$-process are released by two reactions: ${}^{13}$C$(\alpha,n)^{16}$O, which provides the bulk of the neutron flux at low neutron densities ($N_{n}\mathrel{\hbox{\hbox to 0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}}\hbox{$<$}% }}10^{7}$ cm${}^{-3}$), and ${}^{22}$Ne$(\alpha,n)^{25}$Mg, which is activated at temperatures $T\mathrel{\hbox{\hbox to 0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}}\hbox{$>$}}}3.% 0\times 10^{8}$ K, providing a high peak neutron density ($N_{n}\sim 10^{10}$ cm${}^{-3}$) and is responsible of the production of $s$-nuclei controlled by reaction branchings (see e.g. Busso, Gallino, & Wasserburg 1999). The abundances of $s$-nuclei are known to increase along the above mentioned spectral sequence, as the star gradually ascends the AGB. Evidence of this was provided during the last few decades by several studies on AGB stars of different spectral types (see Busso et al. 2001 and references therein). The importance of the study of AGB stars is obvious: i) they are excellent laboratories to test the theory of stellar evolution and nucleosynthesis (note, in addition, that they are progenitors of planetary nebulae and white dwarfs) ii) in particular C(N) stars are the main producers of the heavy s-elemnts (A$\geq 90$) in the Galaxy as well as of a substantial fraction of ${}^{12}$C, ${}^{14}$N, ${}^{7}$Li and ${}^{13}$C. However, an evaluation of such chemical yields firstly needs accurate abundance determinations. Second, in order to understand the role played by these stars in the chemical evolution of the Galaxy it is mandatory to know their typical masses and the frequency of binarity among them. In this work, we will address these questions. 2. Observations and Analysis So far the only available abundances were still those by Utsumi (1985) based on photographic plates and/or low resolution spectra. C(N) stars are very difficult to analyse spectroscopically because of their very crowded spectra. The spectrum of these stars is dominated by molecular absorptions (CN, C${}_{2}$, CH…), which require the knowledge of accurate spectroscopic information of these molecules for the chemical analysis. The use of high resolution and high signal to noise ratio spectra is also mandatory and even in that case, only a few spectral windows are available for abundance analysis: $\lambda 4750-4950$ Å, for $s$-elementsand metals, $\lambda\sim 7800$ Å for the Rb analysis, $\lambda 8000-8050$ Å for the ${}^{12}$C/${}^{13}$C ratios and several Li I lines. In our analysis of a sample of $\sim 40$ galactic C(N), we used high resolution ($\lambda/\Delta\lambda=4\times 10^{4}-2\times 10^{5}$) spectra obtained with the 4.2 m WHT at El Roque de los Muchahos and with the 2.2 m telescope at Calar Alto. We use the spectrum synthesis technique to derive all the abundance figures. The determination of the stellar parameters was made as in Abia et al. (2001) (see this paper for details). 3. Results 3.1. S-elements C(N) stars show overabundances in s-process elements. For the low atomic mass number (Sr,Y,Zr) $s$-elements(ls) we found a mean enhancement with respect to the metallicity [$<$ls$>$/M]=$+0.70\pm 0.20$, while for the high atomic mass number (Ba,La,Nd,Ce,Sm) $s$-elements(hs) the mean overabundance is [$<$hs$>$/M]=$+0.52\pm 0.29$. These enhancements are significantly lower than the previous estimations by Utsumi (1985)111For an explanation of the differences with respect to Utsumi’s data see Abia et al. (2001). and set the s-element overabundances in C(N) stars at the same level or slightly higher than those found in S stars. From the theoretical point of view, this result can be easily understood considering that the overwhelming majority of the C(N) stars analysed here have a C/O ratio very close to the unity ($<$C/O$>=1.05\pm 0.02$): in a single episode of TP and TDU, a S-star (C/O$\sim 0.9$) can become a C(N) star without increasing significantly the s-element enhancement in the envelope. C(N) stars with larger C/O ratios and s-element overabundances must lokely exist, however they escape optical detection because of the formation of a thick circumstellar envelope due to high mass-loss rates. Our C(N) star sample is probably observationally biased in this sense. In Figure 1 we compare the [hs/ls] vs. [M/H] ratios obtained with theoretical s-process nucleosynthesis models in a representative case of a 1.5 M${}_{\odot}$ AGB star (see Gallino et al. 1988; Busso et al. 2001 for details). Solid line shows the prediction corresponding to a standard (ST) case which assumes the radiative burning of $4\times 10^{-6}$ M${}_{\odot}$ of ${}^{13}$C in a tiny pocket in the intershell region. The upper and lower curves (dashed lines) limit the region allowed by the models according to the different ${}^{13}$C-pocket choices scaled to the ST case. Model predictions and observations are in nice agreement. Note that the Utsumi’s data are marginally in agreement with theoretical models. 3.2. The mass Theoretical predictions for s-process as those showed in Figure 1 do not significantly differ for a 3 $M_{\odot}$. Remarkable differences exist for more massive stars, but they are not sufficient to allow a clear discrimination of the stellar mass. To infer the mass of C(N) stars we can make use of a neutron density sensitive element: Rb. Depending of the neutron density in the s-process (i.e. depending on whether the ${}^{13}$C (M$\leq 3$ $M_{\odot}$) or ${}^{22}$Ne (M$\geq 4$ $M_{\odot}$) neutron sources provides the bulk of neutrons), a very different abundance pattern between Rb and its neighbours (Sr,Y and Zr) is expected. Figure 2 shows the result of such a comparison. It is evident that the predictions for a 1.5 $M_{\odot}$ model fit much better the observed [Rb/X] ratios and we can conclude that the majority of C(N) stars studied here are low-mass stars (M$\leq 3$ $M_{\odot}$). This figure concerning the most probable mass for C(N) stars is reinforced from the ${}^{12}$C/${}^{13}$C ratios derived. A large number of the carbon stars studied here have ${}^{12}$C/${}^{13}$C$<40$. These ratios cannot be explained by the standard evolutionary models on the AGB. Such low ratios might be explained if an extra-mixing process is operating during the AGB phase preferably in low mass stars. This non-standard mixing process, perhaps induced by rotation, has been named cool bottom processing (Wasserburg, Boothroyd & Sackmann 1995; Nollet, Busso & Wasserburg 2003) and might explain the low ${}^{12}$C/${}^{13}$C ratios found in many low mass RGB stars. 3.3. Lithium Abia et al. (1993) derived Li abundances in a sample of 230 galactic carbon stars. The most important result of this work was the confirmation that a small fraction (2-3$\%$) are super Li-rich carbon stars, showing Li abundances as high as log $\epsilon$(Li)$\sim 5$. In addition, a significant number ($\sim 10\%$) can be considered as Li-rich C-stars, with Li abundances in the range 1-2. When interpreting these observations on the basis of theoretical models some problems appear. First, how reliable are the Li abundances derived? Yakovina, Pavlenko & Abia (2002) have recently derived Li abundances in super Li-rich C-stars using four different Li lines. They show that even considering corrections by N-LTE effects the Li abundances derived from different lines show a high dispersion. They interpreted this as an evidence of the difficulty and uncertainties still present in the modelling of the atmospheres of AGB stars and put a note of caution when estimating the Li yield from AGB stars. Other figures are also difficult to understand: i) At solar metallicity, the super Li-rich phenomenon is expected for O-rich AGB stars, not for C-rich objects. ii) Some super Li-rich stars are fainter (M${}_{bol}>-6$) than theoretically predicted. iii) Theoretical models predict maximum Li abundances $\sim 4$, however a few stars show peak Li abundances close to 5 (even considering the large error bars). iv) How can we explain the large fraction of Li-rich C-stars? Some of these stars are classified as J-type carbon stars which might not be AGB stars, but a significant number of them are indeed normal (N) carbon stars. v) Why many galactic O-rich and Li-rich AGB stars do not show s-process element enhancements (García-Lario et al. 1999)? Considering that most of Galactic C-stars are of low mass, for which no Li production is predicted (see Sackmann & Boothroyd 1992), it seems evident that also an additional non-standard mixing process affecting Li might play a role in low mass AGB stars. Theoretical modelling is therefore required, but models like those discussed by Nollet, Busso & Wasserburg (2003) can explain in a unique scenario high Li abundances and low ${}^{12}$C/${}^{13}$C ratios, also predicting anomalies in the oxygen isotopic ratios and consistent production of ${}^{26}$Al. 3.4. Intrinsic or Extrinsic C-stars Let us finally address the question whether the stars studied here are intrinsic or extrinsic AGB stars, ie: their abundances are locally produced by the occurrence of TP during the AGB phase, or generated by mass transfer in binary systems. As far as we know no significant radial velocity variations have been detected in any of them. Furthermore, $\sim 60\%$ of the sample stars show ${}^{99}$Tc, an incontestable signature of the TP and TDU operation, thus, of their intrinsic nature. A further test can be done looking for infrared excess as has been done by Jorissen et al. (1993) in S-stars. If the stars are TP-AGB stars they should be high mass-losing stars; the formation of dust and, in consequence, infrared excess are expected. This excess can be measured with the flux ratio R$=$F(12 $\mu$m)/F(2.2 $\mu$m). High mass-losing stars (ie. TP-AGB-stars) typically show R$>0.1$. The large majority of our sample stars have R$>0.1$, Tc-yes and Tc-no stars being indistinguishable in the R parameter. We might conclude that probably all carbon stars in our sample are in fact of intrinsic nature and when they are classified as no-Tc stars, this is just a consequence of the difficult and uncertain analysis of the Tc I $5924$ Å blend on which this study is based. 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Channel-wise Distillation for Semantic Segmentation Changyong Shu${}^{1}$    ${}^{*}$       Yifan Liu${}^{2}$          Jianfei Gao${}^{1}$ First two authors contributed equally.          Lin Xu${}^{1}$          Chunhua Shen${}^{2}$ ${}^{1}$ Shanghai Em-Data Technology Co.    China         ${}^{2}$ The University of Adelaide    Australia Abstract Knowledge distillation (KD) has been proven to be a simple and effective tool for training compact models. Almost all KD variants for semantic segmentation align the student and teacher networks’ feature maps in the spatial domain, typically by minimizing point-wise and/or pair-wise discrepancy. Observing that in semantic segmentation, some layers’ feature activations of each channel tend to encode saliency of scene categories (analogue to class activation mapping), we propose to align features channel-wise between the student and teacher networks. To this end, we first transform the feature map of each channel into a distribution using softmax normalization, and then minimize the Kullback–Leibler (KL) divergence of the corresponding channels of the two networks. By doing so, our method focuses on mimicking the soft distributions of channels between networks. In particular, the KL divergence enables learning to pay more attention to the most salient regions of the channel-wise maps, presumably corresponding to the most useful signals for semantic segmentation. Experiments demonstrate that our channel-wise distillation outperforms almost all existing spatial distillation methods for semantic segmentation considerably, and requires less computational cost during training. We consistently achieve superior performance on three benchmarks with various network structures. 1 Introduction Semantic segmentation is a fundamental task in computer vision, which requires to assign pre-defined classes to each pixels in an input image. Its applications include autonomous driving, video surveillance, virtual reality, and so on. Thus, improving the performance of lightweight networks has drawn much attention. Effectively training lightweight networks has been studied in previous works through knowledge distillation (KD), in which, a compact network is trained with the supervision of a large teacher network, and can achieve better performance. Previous works mainly focus on image classification [48, 34, 22, 3], and only a few works study KD approaches for semantic segmentation. Semantic segmentation is a per-pixel prediction problem, which is more complex and challenging than image-level classification. Directly transferring the KD methods [23, 2] in classification to semantic segmentation may not lead to satisfactory results. Strictly aligning the point-wise classification scores or the feature activations between the teacher network and the compact student network may enforce overly strict constraints and can lead to sub-optimal solutions. Recent works [33, 32, 26] pay attention to enforce the correlations among different spatial locations, namely structural knowledge distillation. As shown in Figure 1(a), let ${\it{y}^{T}}$ and ${\it{y}^{S}}$ be the feature maps (or the score maps) from the teacher network and student network, respectively. Then the original knowledge distillation function ${\varphi}$, can be formulated as: ${\varphi({{y}^{T}},{{y}^{S}})=\varphi(\phi({\it{y}^{T}_{i}}),\phi({\it{y}^{S}_% {i}}))}$, where ${\phi}(\cdot)$ is some transformations applied to the features, typically being nonlinear; and ${\it{i}}$ indicates the pixel index. ${\it{y}^{T}_{i}}\in{\cal R}^{C}$ and ${\it{y}^{S}_{i}}\in{\cal R}^{C}$ are vectors for a spatial location. $C$ is the number of channels. Such methods may work better than the point-wise alignment in capturing spatial structure information, and improve the performance of the student network. Almost all previous methods, referred to as spatial distillation, take the activation vectors along the channel axis at one spatial location as the feature vectors, either for point-wise or pair-wise correlation alignment. We argue that such a requirement may be overly harsh. Usually the teacher network’s feature maps contain redundant and possibly noisy information which may not contribute to the final prediction. It is better that KD can transfer the useful signals to the student network while discarding the noises in the feature maps. To alleviate those limitations, here we propose a novel channel-wise distillation paradigm for semantic segmentation from a new perspective. Inspired by the observation in Figure 1(c), the feature activations of each channel tend to encode saliency of scene categories of an input image. When learning using knowledge distillation, the compact network is expected to focus more on the salient activations associated to the semantic parts, rather than other regions which may be the background or noises of a specific channel. We want to transfer the informative regions at each channel of the teacher network’s feature maps to the student network. To do so, we first transfer the activations of each channel into a distribution by using the softmax normalization. Similar to the original KD, we employ the KL divergence to minimize the discrepancy between channel-wise activation distributions of the teacher network and the student network. To summarize, we propose the efficient and effective channel-wise distillation paradigm for segmentation segmentation, where the discrepancy of the channel-wise activation distributions between the teacher and student network is minimized for each channel individually. As shown in Figure 1(b), the channel-wise distillation loss can be formulated as: ${\varphi({\it{y}^{T}},{\it{y}^{S}})=\varphi(\phi({\it{y}^{T}_{c}}),\phi({\it{y% }^{S}_{c}}))}$, where ${\it{c}}$ indicates the channel index. ${\it{y}^{T}_{c}}$ and ${\it{y}^{S}_{c}}$ are the activations for each channel with the spatial size of $W\times H$. The main contributions can be summarized as follows: • Unlike those existing spatial distillation approaches, we propose a novel channel-wise distillation paradigm for knowledge distillation for semantic segmentation. • The proposed channel-wise distillation significantly outperforms seven state-of-the-art KD methods for semantic segmentation and requires less computational cost during training. • We show consistent improvements on a few benchmark datasets with various network structures, demonstrating that our method is general. Given its simplicity and effectiveness, we believe that our method can serve as a strong baseline KD method for semantic segmentation. 1.1 Related Work Early works on knowledge distillation focus on classification tasks [23, 5, 35, 43, 45, 44, 18, 49, 19], and it has been extended to other tasks such as semantic segmentation [21, 13, 32, 26], object detection [38, 8, 15, 25], re-identification [6, 41, 4], face recognition[27], style transfer [12], GAN [9, 29] and so on. Our work aims to study efficient and effective distillation methods for semantic segmentation, beyond naively applying pixel-wise distillation as done in classification. Previous methods usually align spatial structure information among pixels. The feature map of the student network is forced to be similar to that of the teacher network via $L_{2}$ loss [13]. In [42], a local similarity map is constructed to minimize the discrepancy of segmented boundary information between the teacher and student network, where the euclidean distance among the center pixel and the 8-neighborhood pixels is used as knowledge for transferring. Liu \etal [32] first propose the concept of structure knowledge distillation. They propose two approaches to capture the structure information among pixels, including pair-wise similarity between pixels and holistic correlations captured by a discriminator. As the pixels far apart usually exhibit low similarity, the connection range and the granularity of each node in pairwise loss are further studied in [33]. The work in [40] only focuses on the intra-class feature variation among the pixels with same label, where the set of cosine distance between each pixel’s feature and its corresponding class-wise prototype is constructed to transfer the structural knowledge. Besides, an auto-encoder is used to compress features [21], and the feature adaptor is employed to mitigate the feature mismatching between the teacher and student networks. All these methods consider the $N$ channel activation values of a spatial location as the feature vectors to operate on. Here, we take a different route and perform channel-wise KD. A recent independent work [53] also pays attention to the knowledge lying in each channel. Zhou \etalcalculate the mean of the activation in each channel, and align a weighted difference for each channel in classification. Therefore, a class-independent correlation are encoded. Different from [53], we perform channel-wise distillation for semantic segmentation. Our work aligns the distributions of each channel, which guides the student network to learn the informative/salient locations of the activation maps. 2 Method In this section, we first present a brief introduction to spatial knowledge distillation, and then we describe our proposed channel-wise distillation. 2.1 Spatial Distillation Existing KD methods often employ a point-wise alignment or align structured information among spatial locations, which can be formulated as: $$\ell=\ell_{ce}(y,y^{S})+{\alpha}\cdot\varphi(\phi({\it{y}^{T}_{i}}),\phi({\it{% y}^{S}_{i}})).$$ (1) Here the cross-entropy loss $\ell_{ce}$ is still applied for semantic segmentation with $y$ being the ground-truth labels. $\alpha$ is a hyper-parameter. For better illustrating the knowledge distillation functions ${\varphi(\cdot)}$ and the transformation ${\phi(\cdot)}$ used in the literature, representative spatial distillation methods are listed in Table 1. MIMIC transfers the feature of each pixel via $L_{2}$ loss [2, 30]. Attention Transfer (AT) [47] uses an attention mask to squeeze the feature maps into a single channel for distillation. The pixel-wise loss [24] directly aligns the point-wise class probabilities. The local affinity [42] is computed by the distance between the center pixel and its $8$ neighborhood pixels. The pairwise affinity [33, 21, 32] is employed to transfer the similarity between pixel pairs. The similarity between each pixel’s feature and its corresponding class-wise prototype is computed to transfer the structural knowledge [40]. The holistic loss [33, 32] use the adversarial scheme to align the high-order relations between feature maps from the two networks. Note that, the last four terms consider the correlation among pixels. Existing KD methods as shown in Table 1 are all spatial distillation methods. 2.2 Channel-wise Distillation Inspired by Figure 1(c), the activation of different channels encodes the saliency of scene categories of an input image. Besides, a well-trained teacher network tends to produce activation maps of clearer category-specific masks for each channel—which is expected—as illustrated on the right part of Figure 2. We propose a novel channel-wise distillation paradigm to guide the student to directly learn the channel-wise distribution from the well-trained teacher. As illustrated in Figure 2, our method consists of the teacher network, student network, and channel-wise distillation module. The teacher and student network are denoted as $T$ and $S$, and the activation maps from $T$ and $S$ are $y^{T}$ and $y^{S}$, respectively. The channel-wise distillation loss can be formulated as: $$\varphi(\phi({\it{y}^{T}}),\phi({\it{y}^{S}}))=\varphi(\phi({\it{y}^{T}_{c}}),% \phi({\it{y}^{S}_{c}})).$$ (2) Here, we give a simple but effective scheme for $\phi$ and $\varphi$ in Equation (2). First, we apply a softmax normalization to transfer the activation values in each channel into a distribution $\phi_{cw}$: $$\phi_{cw}{\left(y_{c}\right)}=\frac{\textup{exp}{(\frac{y_{c,i}}{\mathcal{T}})% }}{\sum_{i=1}^{W\cdot H}\textup{exp}{(\frac{y_{c,i}}{\mathcal{T}})}},$$ (3) where $c=1,...,C$. $C$ is the number of channels and $\mathcal{T}$ is the temperature. The distribution will be softer if we use a larger $\mathcal{T}$, which means we focus on more regions in each channel. By applying the softmax normalization, we remove the influences of different magnitude scales between the large networks and the compact networks, which will benefit the knowledge distillation as shown in the previous work [39]. A $1\times 1$ convolution layer will be employed to upsampling the number of channels for the student network if the number of channels is different between the teacher and the student. The spatial locations with higher probability are the most salient regions, presumably corresponding to the most useful signals for semantic segmentation. We use the KL divergence to minimize the discrepancy of the channel-wise distribution between the teacher and student network, and the knowledge distillation function $\varphi_{cw}$, can be formulated as: $$\varphi_{cw}{\left(y^{T},y^{S}\right)}=\sum_{c=1}^{C}\sum_{i=1}^{W\cdot H}\phi% (y^{T}_{c,i})\cdot\log\Bigl{[}\frac{\phi(y^{T}_{c,i})}{\phi(y^{S}_{c,i})}\Bigr% {]}.$$ (4) The KL divergence is an asymmetric metric. From Equation (4), we can see if $\phi(y^{T}_{c,i})$ is large, the $\phi(y^{S}_{c,i})$ should be as large as $\phi(y^{T}_{c,i})$ to minimize the KL divergence; otherwise, if $\phi(y^{T}_{c,i})$ is very small, the KL divergence will pay less attention to minimize the $\phi(y^{S}_{c,i})$. Thus, the student network tend to produce similar activation distribution in the foreground sciency parts and ignore the redundancy background. 2.3 Optimization To train our network, we define a loss function in Equation (5) including two terms: a multi-label cross entropy loss $\ell_{ce}$, and the channel-wise distillation loss $\varphi_{cw}$: $$\ell=\ell_{ce}(y_{,}y^{S})+\alpha\cdot\varphi_{cw}(y^{T},y^{S}),$$ (5) where $\alpha$ is a balance hyper-parameter and we set to $35$ in all of our experiments. Following other distillation frameworks, we employ a well-trained large network as our teacher net and fix the weight of the teacher during the training phase. We only optimize the parameters of the student network during the training phase. 3 Experiments In this section, we first describe the implementation details and conduct ablation studies. Then, we compare our channel-wise distillation method with other state-of-the-art distillation methods. Finally, we show consistent improvements in different typical semantic segmentation datasets and student network structures. 3.1 Experimental Settings Datasets. Three representative semantic segmentation benchmarks, i.e., Cityscapes [14], ADE20K [52] and Pascal VOC [17] are considered. The Cityscapes dataset is used for semantic urban scene understanding. It contains 5,000 finely annotated images with 2,975/500/1,525 images for training/validation/testing respectively, where 30 common classes are provided and 19 classes are used for evaluation and testing. The size of each image is $2048\times 1024$ pixels. And there are all gathered from 50 different cities. The coarsely labeled data is not used in our experiments. The Pascal VOC dataset contains 1,464/1,449/1,456 images for training/validation/testing. It contains 20 foreground objects classes and an extra background class. In addition, the dataset is augmented by extra coarse labeling, which resulting in 10,582 images for training. The training split is used for training, and the final performance is measured on the validation set across 21 classes. The ADE20K dataset covers 150 classes of diverse scenes, where the annotation is detailed for semantic parsing. It contains 20K/2K/3K images for training, validation, and testing. In our experiments, we report the segmentation accuracy on the validation set. Evaluation Metrics. To evaluate the performance and efficiency of our proposed channel-wise distillation method, we test each strategy via the mean Intersection-over-Union (mIoU) to indicate the segmentation accuracy in all experiments under a single-scale setting. Besides, the mean class Accuracy (mAcc) is listed for Pascal VOC and ADE20K. The parameter number is computed by summing the parameters in the model and the floating-point operations per second (FLOPs) are calculated with a fixed input size (512 $\times$ 1024). Implementation Details. The teacher network is PSPNet with ResNet101 (PSPNet-R101) as the backbone for all experiments. We employ several different architectures, including PSPNet, Deeplab with the backbones of ResNet18, and MobileNetV2 as student networks to verify the effectiveness of the channel-wise distillation. In the ablation study, we analyze the effectiveness of our method based on PSPNet with ResNet18 (PSPNet-R18). Unless otherwise indicated, the training image for the student is randomly cropped into $512\times 512$, the batch size is set to $8$, and the number of the training step is $40$k. 3.2 Ablation study We discuss the choice of the hyper-parameters and the effect of the asymmetric KL divergence in this section. The baseline student model is PSPNet-R18, and the teacher model is the PSPNet-R101. All the results are evaluated on the validation set of Cityscapes. Impact of the temperature parameter. We conduct experiments to change the channel-wise distribution by adjusting the temperature parameter $\mathcal{T}$. The results are shown in Figure 3. The loss weight is set to $35$, and $\mathcal{T}\in[0.01,1000]$. The distribution tends to be softer if we increase $\mathcal{T}$, which means we focus on more regions. We get the best performance when $\mathcal{T}=1$. Besides, in a certain range, the performance is stable. The performance will drop a lot if $\mathcal{T}$ is extremely small, which means we only focus on limited salient pixels. Impact of the loss weight. The experiment results of adjusting the loss weight are shown in Figure 3. We fix $\mathcal{T}=1$, and adjust the loss weight in $[0,50]$. We can see that the distillation results are not sensitive to the loss weight. We set $\alpha=35$ in all experiments. Impact of the distribution metric. The channel-wise distribution and asymmetric KL divergence play an important role in our distillation method. We conduct experiments with three different metrics to show the effectiveness of proposed methods in Table 3. The Bhattacharyya distance [7] is a symmetric distribution measurement, which aligns the discrepancy in each channel. The $L_{2}$ norm considers the difference at all locations in all channels equally, which is similar to the previous MIMIC method. From Table 3, we can see that the asymmetric KL divergence considering the channel-wise discrepancy achieves the best performance. Note that as the KL divergence is asymmetric, the input of the student and teacher can not be swapped. We conduct the experiment by changing the order of the input in the KL divergence, and the training does not converge. 3.3 Comparison with Spatial Distillation In this subsection, we first analyze the individual comparison of previous state-of-the-art spatial distillation methods, then study the benefits of combining with previous spatial distillation methods. Individual comparison. To verify the effectiveness of our proposed channel-wise distillation, we compare our method with current spatial distillation methods listed below: • MIMIC: Transfer the feature of each pixel via $L_{2}$ loss [2, 30]. • Squeeze attention transfer for distillation (AT): Sergey [47] utilizes attention transfer to squeeze an attention map to distill. • Local affinity study (LOCAL): For each pixel, a local similarity map is constructed, which consider the correlations between itself and its 8 neighborhood pixels [42]. • Pixelwise logits distillation (PI): KL divergence is used to align the predicted logits for each pixel [33, 32, 40, 11]. • Pairwise relation distillation (PA): The correlations between each pixel pair are considered [33, 21, 32]. • Intra-class feature variation distillation (IFVD): The set of similarity between the feature of each pixel and its corresponding class-wise prototype is regarded as the intra-class feature variation to transfer the structural knowledge [40]. • Holistic distillation (HO): The holistic embeddings of feature maps are computed by a discriminator, which are used to minimize the discrepancy between high-order relations [33, 32, 40]. The conventional multi-class cross-entropy loss is applied in all experiments. The computational complexity and performance of different spatial distillation methods are studied. Given the input feature map (score map) with the size of $h_{f}\times w_{f}\times c$ ($h_{s}\times w_{s}\times n$), where $h_{f}(h_{s})\times w_{f}(w_{s})$ is the shape of the feature map (score map). $c$ is the number of channels and $n$ is the number of classes. As shown in Table 2, all distillation methods can improve the performance of the student network. Our channel-wise distillation method outperforms all spatial distillation methods, it outperforms the best spatial distillation method (AT) by $1.15\%$. Moreover, channel-wise distillation is very efficient as it requires less computational cost than other methods during the training phase. The Mimic has the same complexity as our proposed method, but the performance is lower than the proposed channel-wise distillation. The channel-wise distillation on the feature map works better than on the score map, which may due to that the channel number is larger on the feature map and may contain more detailed salient objects. To better show the improvement, we illustrate the results of Table 2 in Figure 4. The attention transfer [47] squeezes all the channels into a single channel attention map (e.g., find the max activation or the summation). On the contrary, the channel-wise distillation aligns the distribution in each channel (remove the influences of magnitude scales) and then calculates the difference in all channels. To demonstrate the effectiveness, we compare our method with current state-of-the-art spatial distillation methods, including PA and IFVD. These methods propose to transfer structure information in semantic segmentation. We list the detailed class IoU of three methods in Table 4. Our methods significantly improve the class accuracy of several objects, such as traffic light, terrain, wall, truck, bus, and train, indicating that the channel-wise alignment can transfer the structural knowledge better. We also present the visualization results in Figure 5 to intuitively demonstrate that, the channel-wise distillation method (CW) outperforms the spatial distillation strategy. Moreover, to evaluate the effectiveness of the proposed channel-wise distillation, we visualize the channel-wise distribution of the student network under three paradigms, i.e., original network, distilled result by attention transfer (AT) and channel-wise distillation respectively, in Figure 6. Impact of combinations. We evaluate the performance under a few different loss combinations, as shown in Table 5. First we combine the channel-wise distillation applying to the feature map and the score map. The results can be improved to 73.71%. Current state-of-the-art methods [33, 26] combine the pixel-wise distillation (PI), the holistic distillation (HO) on the score map and PA/IFVD on the feature map for further improving the performance. To make a fair comparison, we also combine the proposed channel-wise distillation with PI and HO on the score map. As shown in Table 5, we can see that the proposed channel-wise distillation method is not contradicted to previous spatial distillation methods and outperforms the state-of-the-art distillation methods [26, 33] when combining with previous spatial distillation methods. 3.4 Results We demonstrate that our proposed method can bring consistent improvement compared to state-of-the-art semantic segmentation distillation methods, i.e., structural knowledge distillation for segmentation/dense prediction (SKDS [32] /SKDD [33]) and intra-class feature variation distillation (IFVD [40]), under various student networks. Cityscapes. We first evaluate the performance of our method on the Cityscapes dataset. As the SKDS/SKDD (IFVD) take PI+HO+PA${}_{f}$ (PI+HO+IFVD${}_{f}$) in their methods, for a fair comparison, we also employ PI and HO in our method. Various student networks with different encoders and decoders are used to verify the effectiveness of our method. Encoders include ResNet18 (initialized with or without the weights pre-trained on ImageNet, a channel-halved variant of ResNet18 [20]) and MobileNetV2 [37], and decoders include PSPhead [51] and ASPPhead [10]. Table 6 summarizes our method’s performance applied for different student models on Cityscapes Experiment results on Pascal VOC and ADE20K are shown in the supplementary materials). Our method outperforms SKD and IFVD on seven student networks, which further indicates that the channel-wise distillation is effective for semantic segmentation distillation and can complement the spatial distillation. For the student with the same architectural type as the teacher, i.e., PSPNet-R18${}^{\diamond}$(0.5), PSPNet-R18${}^{\diamond}$ and PSPNet-R18${}^{\star}$, the improvement is more significant. As for the student with different architectural types with the teacher, i.e., PSPNet-MBV2${}^{\star}$, Deeplab-R18${}^{\diamond}$(0.5), Deeplab-R18${}^{\star}$ and Deeplab-MBV2${}^{\star}$, our method achieves consistent improvement compared with SKDS and IFVD, which proves that our channel-wise distillation is more effective and can generalize well between different teacher and student networks. The student network of a compact model capacity (PSPNet-R18${}^{\diamond}$(0.5)) shows inferior distillation performance (67.26%) compared to the student with large parameters (PSPNet-R18${}^{\star}$) (74.87%). This may be attributed to the fact that the capability of small networks is limited compared with the teacher network and can not sufficiently absorb the knowledge of the current task. For PSPNet-R18, the student initialized by the weight trained on ImageNet obtains the best distillation performance (improved from 70.04% to 74.87%), further demonstrating that the well-initialized parameters help the distillation. Thus, the better student lead to better distillation performance, but the improvement is smaller as the gap between the teacher and student network is smaller. 4 Conclusion In this paper, we have summarized previous segmentation distillation methods as the spatial distillation paradigm, and a novel structural knowledge transfer strategy, i.e., channel-wise distillation, is proposed. Experimental results show that the proposed channel-wise distillation method consistently outperforms almost all existing KD methods on three public benchmark datasets with various network backbones. Additionally, our experiments demonstrate the efficiency and effectiveness of our channel-wise distillation, and it can further complement the spatial distillation methods. 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Formation of antideuterons in heavy ion collisions B.L. Ioffe, I.A. Shushpanov and K.N. Zyablyuk Institute of Theoretical and Experimental Physics, B.Cheremushkinskaya 25, 117218 Moscow, Russia, ioffe@vitep1.itep.ru, shushpan@heron.itep.ru, zyablyuk@heron.itep.ru () Abstract The antideuteron production rate at high-energy heavy ions collisions is calculated basing on the concept of $\bar{d}$ formation by antinucleons which move in the mean field of the fireball constituents (mainly pions). The explicit formula is presented for the coalescence parameter $B_{2}$ in terms of deuteron binding energy and fireball volume. PACS numbers: 25.75.-q, 25.75.Dw, 12.38.Mh 1.Introduction. Recent measurements have reported the production of antideuterons in heavy ion collisions [1], [2], [3]. The theoretical description of this interesting effect is complicated, because the antideuterons are produced at intermediate stage of fireball evolution, when the density of hadronic matter is rather low, but the particle collisions are still important. In other words they are produced at the ”dense gas” stage of the fireball evolution. Here we present the theoretical picture of this stage and calculate the ${\bar{d}}$ production. The basic ideas of our approach are the following. The dominant mechanism of ${\bar{d}}$ production is the formation of antideuterons through the fusion reaction $\bar{p}+\bar{n}\to\bar{d}$. The fusion reaction is not possible if all participating particles are on mass shell. However, in the fireball at the ”dense gas” stage of its evolution $\bar{p}$, $\bar{n}$, $\bar{d}$ are not on mass shell, since they interact with surrounding matter. The interaction with the fireball constituents leads to appearance of the mass shift and widths of all particles propagating in the medium (or width broadening for unstable ones), analogous to refraction and attenuation indices in case of photon propagation. The fusion reaction rate is strongly enhanced in comparison with the main process of ${\bar{d}}$ production in vacuum $\bar{p}+\bar{n}\to\bar{d}+\pi$. Another important ingredient of the theoretical picture is the balance of the deuteron formation and desintegration rates. This balance is achieved because of large number of produced pions and high rate of $\pi+{\bar{d}}$ collisions leading to ${\bar{d}}$ desintegration. The balance does not imply a statistical equilibrium, but rather a stationary process, like a balance in the isotope concentrations in a radiative chain. The formation rate $\bar{p}+\bar{n}\to\bar{d}$ vanishes, when ${\bar{d}}$ size increases, i.e. its binding energy $\varepsilon\to 0$. This fact explicitly manifests itself in our calculations. The previous theoretical investigations of the problem were performed in statistical models [4, 5], in the model of (anti)nucleon sources in the fireball [6] and in the Wigner function approach (see [7, 8] and references therein). In all these approaches the interaction of nucleons, forming ${\bar{d}}$ (or $d$) with the fireball constituens as well as ${\bar{d}}$ ($d$) desintegration was not accounted (in [4, 5] the results do not depend on $\varepsilon$). According to the dominant coalescence mechanism it is convenient to characterize $\bar{d}$ production in heavy ion collisions by the coalescence parameter: $$B_{2}\,=\,E_{\bar{d}}\frac{d^{3}N_{\bar{d}}}{d^{3}p_{\bar{d}}}\left(E_{\bar{p}% }\frac{d^{3}N_{\bar{p}}}{d^{3}p_{\bar{p}}}\,E_{\bar{n}}\frac{d^{3}N_{\bar{n}}}% {d^{3}p_{\bar{n}}}\right)^{-1},$$ (1) where we can put $d^{3}N_{\bar{p}}/d^{3}p_{\bar{p}}=d^{3}N_{\bar{n}}/d^{3}p_{\bar{n}},~{}p_{\bar% {p}}=p_{\bar{n}}=p_{\bar{d}}/2$. In what follows, we will consider only the central heavy ion collisions. 2. Theory. Consider the ”dense gas” stage of fireball evolution, which follows after so-called ”chemical freeze-out” stage [9, 10]. Assume, that particle propagations at this stage may be described classically using kinetic equations. We use the notation $q_{i}(x,p)$, $i=\bar{p}$, $\bar{n}$, $\bar{d}$, $\pi$ $\ldots$ for the double densities in coordinate and momentum spaces, $n_{i}(x)=\int~{}q_{i}(x,p)d^{3}p$ are the densities. ($q_{i}(x,p)$ – are Lorentz invariant.) Let us work in the c.m. system of colliding ions. The kinetic equation for $q_{\bar{d}}(p_{\bar{d}},x)$ reads: $$\frac{m_{\bar{d}}}{E_{\bar{d}}}\frac{\partial q_{\bar{d}}(p_{\bar{d}},x)}{% \partial x_{\mu}}\,u_{\mu}^{\bar{d}}\,=\,\frac{\partial q_{\bar{d}}}{\partial t% }\,+\,{\bf v}_{\bar{d}}\nabla q_{\bar{d}}\,=\,\int d^{3}p_{\bar{p}}\,d^{3}p_{% \bar{n}}\,q_{\bar{p}}(p_{\bar{p}})\,q_{\bar{n}}(p_{\bar{n}})\,\sigma_{\bar{p}% \bar{n}\to\bar{d}}\,v^{rel}_{\bar{p}\bar{n}}\,\times$$ $$\times\delta^{3}(p_{\bar{p}}+p_{\bar{n}}-p_{\bar{d}})\,-\,q_{\bar{d}}(p_{\bar{% d}})\left[\,\int d^{3}p_{\pi}\,q_{\pi}(p_{\pi})\,\sigma_{\pi\bar{d}}\,v^{rel}_% {\pi\bar{d}}\,+\,\ldots\,\right]$$ (2) where $u_{\mu}^{\bar{d}}=(1,{\bf v}_{\bar{d}})/\sqrt{1-v^{2}_{\bar{d}}}$ is the $\bar{d}$ 4-velocity, the ellipsis mean similar terms for collisions of $\bar{d}$ with other constituents of the fireball ($p,n$ etc.) and $v^{rel}_{\bar{p}\bar{n}}$, $v^{rel}_{\pi\bar{d}}$ are the differences of $\bar{p}$, $\bar{n}$ and $\pi,\bar{d}$ velocities $v^{rel}_{\bar{p}\bar{n}}=|{\bf v}_{\bar{p}}-{\bf v}_{\bar{n}}|$ etc. The terms, when $\bar{d}$ appears in the momentum interval $p_{\bar{d}}+\Delta p_{\bar{d}}$ due to elastic collisions are neglected. Necessary applicability condition of (2) is $\lambda=p^{-1}_{i}\ll d$, where $d$ is the mean distance between fireball constituents. The cross section $\sigma_{\bar{p}\bar{n}\to\bar{d}}=\sigma_{pn\to d}$ is equal to: $$\sigma_{pn\to d}=\frac{3}{4}\cdot\frac{\pi}{4}~{}\frac{g^{2}}{E_{p}E_{n}E_{d}}% ~{}\frac{1}{v^{rel}_{pn}}~{}\delta(E_{p}+E_{n}-E_{d}),$$ (3) where $E_{p},E_{n},E_{d}$ are $p,n$ and $d$ total energies, 3/4 is the spin factor and $g$ is the coupling constant of low energy effective $pnd$ intraction (in the $d$ c.m. system). The value of $g^{2}$ was found by Landau [11] from the requirement of coincidence (at the deuteron pole) of the $pn$ scattering amplitude in effective theory with the amplitude in the Bethe-Peierls theory of the low-energy $pn$-scattering [12]. In the limit of zero range of nuclear forces $g^{2}$ is $$g^{2}\,=\,128\pi\,m_{N}\sqrt{m_{N}\varepsilon},$$ (4) where $m_{N}$ is the nucleon mass, $\varepsilon=2.2\,{\rm MeV}$ is the deuteron binding energy. The account of non-zero range $r_{0}$ increases $g^{2}$ by a factor of $(1-\sqrt{m_{N}\varepsilon}r_{0})^{-1}\approx 1.6$ [12, 13]. The mass of the particle moving in medium is shifted being compared with its vacuum value. Similarly, due to interaction with medium constutuents, the width $\Gamma$ appears (or width broadening, if the particle has its proper width). The mass shift $\Delta m(E)$ and $\Gamma(E)$ are expressed through the forward scattering amplitude $f(E)$ of the particle on medium constituent (see [14, 15] and references therein). $$\Delta\,m(E)\,=\,-\,2\pi\,\frac{n}{m}\,{\rm Re}\,f(E)$$ (5) $$\Gamma(E)\,=\,4\pi\,\frac{n}{m}{\rm Im}\,f(E)\,=\,\frac{np}{m}\,\sigma(E),$$ (6) where $E$, $p$ and $m$ are particle energy, momentum and mass, $n$ is the density of the constutuent in medium. Eqs. (5),(6) take place in the system, where constutuents are at rest. In case of moving constituents the corresponding Lorentz boost must be done. (By definition $\Delta m$ and $\Gamma$ are Lorentz invariant, for details see [16]). Therefore, $\bar{p}$, $\bar{n}$ and $\bar{d}$ in the reaction $\bar{p}+\bar{n}\to\bar{d}$ can be considered as Breit-Wigner resonances with varying masses distributed according to the Breit-Wigner formula. In process of the fireball expansion these Breit-Wigner resonances smoothly evolve to their stable counterparts. So we integrate the first term in the r.h.s. of (2) after substituting (3) over the masses $m^{\prime}$ of the Breit-Wigner resonances: $$\displaystyle I$$ $$\displaystyle=$$ $$\displaystyle\int\,dm_{\bar{p}}^{\prime}\,dm_{\bar{n}}^{\prime}\,dm_{\bar{d}}^% {\prime}\,{\Gamma_{\bar{p}}/2\pi\over(m_{\bar{p}}^{\prime}-m_{\bar{p}})^{2}+% \Gamma_{\bar{p}}^{2}/4}\,{\Gamma_{\bar{n}}/2\pi\over(m_{\bar{n}}^{\prime}-m_{% \bar{n}})^{2}+\Gamma_{\bar{n}}^{2}/4}\,{{\tilde{\Gamma}}_{\bar{d}}/2\pi\over(m% _{\bar{d}}^{\prime}-m_{\bar{d}})^{2}+{\tilde{\Gamma}}_{\bar{d}}^{2}/4}$$ (7) $$\displaystyle\times\,{3\pi\over 16}\,{g^{2}\over E_{\bar{d}}^{\prime}}\,\int{d% ^{3}p_{\bar{p}}\over E_{\bar{p}}^{\prime}}\,{d^{3}p_{\bar{n}}\over E_{\bar{n}}% ^{\prime}}\,q_{\bar{p}}(p_{\bar{p}})\,q_{\bar{n}}(p_{\bar{n}})\,\delta^{3}(p_{% \bar{p}}+p_{\bar{n}}-p_{\bar{d}})\,\delta(E_{\bar{p}}^{\prime}+E_{\bar{n}}^{% \prime}-E_{\bar{d}}^{\prime})$$ where $E_{\bar{p}}^{\prime}=\sqrt{p_{\bar{p}}^{2}+m_{\bar{p}}^{\prime}{}^{2}}$ etc. We assume that the widths $\Gamma\ll m$ are much smaller than the typical momenta in ${\bar{p}}$, ${\bar{n}}$ distributions. Than the distributions $q_{\bar{p}}(p_{\bar{p}})=q_{\bar{n}}(p_{\bar{n}})$ can be taken out from the integral sign at the values $p_{\bar{p}}=p_{\bar{n}}=p_{\bar{d}}/2$. The result of calculation is given by $$I\,=\,\frac{3\pi^{2}}{16\,E_{\bar{d}}}\,g^{2}\,\sqrt{\frac{\Gamma_{\bar{p}}+% \Gamma_{\bar{n}}+{\tilde{\Gamma}}_{\bar{d}}}{m_{N}}}\,q^{2}_{\bar{p}}(p_{\bar{% p}})$$ (8) (the mass difference $\Delta m=m_{\bar{d}}-m_{\bar{p}}-m_{\bar{n}}\sim 30\,{\rm MeV}$ is small in comparison with the width $\Gamma\sim 300\,{\rm MeV}$ and neglected in (8)). Later we assume $\Gamma_{\bar{p}}=\Gamma_{\bar{n}}\equiv\Gamma$. ${\tilde{\Gamma}}_{\bar{d}}$ generally is not equal to the antideuteron width $\Gamma_{\bar{d}}\approx 2\Gamma\sim 600\,{\rm MeV}$. The ${\bar{p}}{\bar{n}}$ system with ${\bar{d}}$ quantum numbers at high excitations will not evolve to ${\bar{d}}$ in the process of fireball expansion, but may decay in other ways. One may expect ${\tilde{\Gamma}}_{\bar{d}}<\Gamma_{\bar{d}}$. We shall keep the ratio $a\equiv{\tilde{\Gamma}}_{\bar{d}}/\Gamma_{\bar{d}}$ as free parameter in the calculations. However, the results weakly depend on this ratio: the variation within the limits $0<a<1$ may change the coalescence parameter (1) by at most $\sqrt{2}$ times, but in real cases about $20\%$. This uncertainty is within accuracy of the whole method, estimated as $50\%$. The contributions of direct processes ${\bar{p}}+{\bar{p}}\to{\bar{d}}+\pi^{-}$, ${\bar{n}}+{\bar{n}}\to{\bar{d}}+\pi^{+}$ and ${\bar{p}}+{\bar{n}}\to{\bar{d}}+\pi^{0}$ are small, alltogether about $20\%$ in comparison with (8). If these processes were essential, the coalescence parameter $B_{2}$ (1) would be meaningless, since the antideuteron distribution $d^{3}N_{\bar{d}}/d^{3}p_{\bar{d}}$ is given by complicated integral over the antinucleon distributions in this case. At large $p_{{\bar{p}}\bot}$ the ${\bar{p}}$ spectrum decreases steeply, so the approximation $p_{\bar{p}}=p_{\bar{d}}/2$ becomes inaccurate which leads to an underestimation of $B_{2}$. The case of large $p_{{\bar{p}}\bot}>1\,{\rm GeV}$ is not considered here. Using (6) and performing Lorentz boost to heavy ion c.m. frame, the term in square brackets in (2) can be brought to the form $(m_{d}/E_{\bar{d}})\,\Gamma_{\bar{d}}$, where $$\Gamma_{\bar{d}}\,=\,\sum_{i}\int d^{3}p_{i}\,q_{i}(p_{i})\,v^{rel}_{i\bar{d}}% \,\sigma_{i\bar{d}}(\bar{d}~{}\mbox{at rest})$$ (9) Suppose, that the rate of antideuteron collisions with other constituents of the fireball resulting in antideuteron desintegration is much larger than the rate of fireball expansion. This happens at collisions of heavy nuclei at high energies, when the fireball size is large because of large number of produced pions per nucleon. In this case one may expect the balance: the first term in the r.h.s. of (2) is equal to the second one and $$q_{\bar{d}}(p_{\bar{d}})\,=\,\frac{I}{\Gamma_{\bar{d}}(m_{\bar{d}}/E_{\bar{d}}% )}\,=\,\frac{3\pi^{2}}{32\,m_{N}}\,\sqrt{1+a\over 2\,\Gamma\,m_{N}}\,g^{2}\,q^% {2}_{\bar{p}}(p_{\bar{p}})$$ (10) The momentum distribution $d^{3}N_{\bar{d}}/d^{3}p_{\bar{d}}$ entering (1) is obtained from (10) by integration over the fireball volume $$\frac{d^{3}N_{\bar{d}}(p_{\bar{d}})}{d^{3}p_{\bar{d}}}\,=\,\int d^{3}x\,q_{% \bar{d}}(p_{\bar{d}},x)$$ (11) Using (1), (10), (11) and (4) (with $r_{0}$ correction) we find for the coalescence parameter $$B^{th}_{2}\,=\,\frac{24\pi^{3}}{E_{\bar{p}}}\times 1.6\,\sqrt{(1+a)\,% \varepsilon\over 2\,\Gamma}\,\,{\int d^{3}x\,q^{2}_{\bar{p}}(p_{\bar{p}},x)% \over\left[\int d^{3}x\,q_{\bar{p}}(p_{\bar{p}},x)\right]^{2}}$$ (12) Since the $x$-dependence of $q_{\bar{p}}(p_{\bar{p}},x)$ is not known, we replace (12) by: $$B^{th}_{2}\,=\,\frac{24\pi^{3}}{E_{\bar{p}}}\times 1.6\,\sqrt{(1+a)\,% \varepsilon\over 2\,\Gamma}\,\frac{2}{V}\,\frac{\overline{n^{2}}_{\bar{p}}}{(% \bar{n}_{\bar{p}})^{2}}$$ (13) where $V$ is the fireball volume, $\bar{n}_{p}$ and $\overline{n^{2}}_{p}$ are the mean and mean square $\bar{p}$ densitites in the fireball. (The coordinate dependence of $\sqrt{\Gamma}$ is neglected). $B^{th}_{2}$ is Lorenz invariant, as it should be. The volume $V$ may be understood as a mean value of the fireball volume at a stage, where, on one side, hadrons are already formed, i.e., mean distances between them are larger than the confinement radius $R_{c}\sim 1/m_{\rho}\sim(1/4)\,{\rm fm}$, but on the other side, hadron interactions are still essential. The antinucleon distributions $n_{\bar{p}}({\bf r}),n_{\bar{n}}({\bf r})$ inside the fireball are nonuniform: at the dense gas stage and before it the antinucleons strongly annihilate in the internal part of the fireball and in much less extent in its external layer of the thickness of order $\bar{p}(\bar{n})$ annihilation length $l_{ann}$ (this effect was considered in [6]). For this reason $\overline{n^{2}}_{p}/\bar{n}^{2}_{p}$ may be remarkable larger than 1. For the same reason the antinucleons and antideuterons from the backside of the fireball (relative to the observer) are absorbed in the fireball and cannot reach the detector (see Fig. 1). Therefore, only one half of the fireball volume contributes to the number of registered ${\bar{p}}$, ${\bar{n}}$ and ${\bar{d}}$. The corresponding factor approximately equal to 2 is accounted in (13). In fact, the fireball evolution after the balance may reduce the antideuteron number as $N_{\bar{d}}\to N_{\bar{d}}e^{-\Gamma_{\bar{d}}\Delta t}$, where $\Delta t$ is a typical time, required for the antideuteron to leave the interaction region. (Such $\approx 50\%$ reduction of $K^{-}$ mesons was observed in [17].) However, this effect does not change the coalescence parameter $B_{2}$. Indeed, in this case $B_{2}$ should be multiplied on the factor $e^{-\Gamma_{\bar{d}}\Delta t}/(e^{-\Gamma_{\bar{p}}\Delta t})^{2}$; the time $\Delta t$ is the same both for antideuteron and antiproton in consideration, because they move with equal velocities. But since $\Gamma_{\bar{d}}=2\Gamma_{\bar{p}}$ with good accuracy (we checked it explicitly by eq (9)), this factor is close to 1 regardless of the evolution details. $\Gamma$ may be calculated if the spectrum and densities of the fireball constituents at hadronic gas stage are known. At high energy of heavy ion collisions (SPS, RHIC) the main conributions to $\Gamma_{\bar{p}}$, $\Gamma_{\bar{n}}$, $\Gamma_{\bar{d}}$ come from the collisions of ${\bar{p}}$, ${\bar{n}}$, ${\bar{d}}$ with pions. Therefore $\Gamma_{\bar{p}}$, $\Gamma_{\bar{n}}$, $\Gamma_{\bar{d}}$ essentially depend on pionic density in the fireball, the dependence of $\Gamma$ on the densities of other fireball constituents is much weaker. Also weak is the dependence of $\Gamma$ on the spectrum of the fireball constituents, since the main contribution to $\Gamma_{\bar{p}}$ ($\Gamma_{\bar{n}}$, $\Gamma_{\bar{d}}$) arises from the collisions at high energies in c.m. system of ${\bar{p}}({\bar{n}},{\bar{d}})+\pi$, where the cross sections are approximately constant. For this reason, without a serious error, for the calculation of $\Gamma$ we can take the spectra from the experimental data, i.e. corresponding to the final stage of the fireball evolution. Moreover, since the widths enters as $\sqrt{\Gamma}$ in (13), the errors are reduced twice. If $\Gamma$ is known, then by comparison with the data the parameter $V^{-1}(\overline{n^{2}}_{\bar{p}}/\bar{n}^{2}_{\bar{p}})$ can be found, what would allow to check various models of fireball evolution. 3. Comparison with the data. Consider the NA44 experiment at SPS (CERN): $Pb+Pb$ collisions at $\sqrt{s}=17A\,{\rm GeV}$ [1]. Antideuterons were observed at $0.6<p_{\bar{d}t}<1.6\,{\rm GeV}$ and in the rapidity interval 1.9 to 2.1 in lab. system, which corresponds to $\bar{p}_{\bar{p}t}=0.55\,{\rm GeV}$, $(E_{\bar{p}})_{c.m.}=1.5\,{\rm GeV}$. The spectra and particle yields at such collisions are given in [18]. The number of active nucleons, participating in collision (”wounded” nucleons) $N_{N}$ and the number of produced pions are presented in [19]: $N_{N}=362$, $N_{\pi}=1890$, $Q_{\pi}=N_{\pi}/N_{N}=5.2$ (see also [20] for the review of the data on heavy ion collisions). We accept the following model for the dense gas stage of fireball evolution [15]. (A related model had been suggested long ago [21, 22]: it may be called Fermi–Pomeranhuk model). Neglect for a moment contributions of all particles except for nucleons and pions. Assume that any participant – nucleon or pion occupies the volume $v_{N}$ or $v_{\pi}$, respectively. Then $$n_{N}\,=\,\frac{N_{N}}{V}\,=\,\frac{n^{0}_{N}}{1+Q_{\pi}\beta}\;,\qquad n_{\pi% }\,=\,\frac{N_{\pi}}{V}\,=\,\frac{n^{0}_{N}Q_{\pi}}{1+Q_{\pi}\beta}$$ (14) where $n^{0}_{N}=1/v_{N}$, $\beta=v_{\pi}/v_{N}$. For numerical estimations we take $n^{0}_{N}=0.26\,{\rm fm}^{-3}$, 1.5 times standard nucleus density and $\beta=(r_{\pi}/r_{N})^{3}\approx 0.55$, where $r_{\pi}=0.66\,{\rm fm}$ and $r_{N}=0.81\,{\rm fm}$ are pion and nucleon electric radii. It must be stressed, that $n^{0}_{N}$ is the only essential uncertain parameter in our approach. Even if $Q_{\pi}$ at dense gas stage differs from ones at the final stage, the arising error is essentially compensated by the appearance of $Q_{\pi}$ both in numerator and denominator in (14). (As was already mentioned, the nucleon contribution to $\Gamma$ is small.) Check first the applicability conditions of our approach. We have: $n=n_{N}+n_{\pi}\approx 0.42\,{\rm fm}^{-3}$ and the mean distance between the fireball constutuents is $d=1/n^{1/3}=1.3\,{\rm fm}$. Evidently, the condition $\lambda_{\bar{p}}=1/p_{\bar{p}}\ll d$ is well satisfied. The calculation of $\Gamma$ according to (9) ($\Gamma=\Gamma_{\bar{d}}/2$) gives $\Gamma\approx 300\,{\rm MeV}$. (Only inelastic cross sections were accounted, the pion contribution comprises about $75\%$, the nucleon one about $25\%$. Note, that the value of $\Gamma$ is close to the momentum integration interval in the Wigner function approach, $\Delta P\approx 200-300\,{\rm MeV}$, found in [8].) Check now the balance condition – that the probability of deuteron desintegration exceeds the fireball expansion rate. The former is given by $2\Gamma(m_{N}/E_{\bar{p}})$. The estimation for the escape rate (or fireball expansion) is $w\sim(1/4)\,{\rm fm}^{-1}$. We have: $2\Gamma(m_{N}/E_{p})\approx 2.0\,{\rm fm}^{-1}\gg 0.25\,{\rm fm}^{-1}$. So, this condition is also fulfilled. Even more, the balance condition would be fullfilled at much lower hadronic densities than chosen above, up to $n^{0}_{N}\approx 0.05\,{\rm fm}^{3}$, i.e. up to densities not much higher, than supposed for thermal freeze-out [10], [23], [24]. However, such low densities would lead to much lower values of $B_{2}$, than ones obtained in experiments. Eq.6 is legitimate, if $Imf(E)\ll d$ [14, 15]. Since $Im\,f\approx 1\,{\rm fm}$, this condition is not well satisfied. For this reason the value of $\Gamma$, presented above, has a large (may be 50%) uncertainty, amd, probably, is overestimated (the effect of screening). This fact, however, does not influence too much the value $B^{th}_{2}$, since $\sqrt{\Gamma}$ enters (12). One may expect, that because of their slightly larger velocities in comparison with nucleons, pions form a halo around the fireball. This effect also may lead to an overestimation of $\Gamma$. At the parameters used above the fireball volume comes out to be: $V=6.2\times 10^{3}\,{\rm fm}^{3}$ (15% correction for other particles, except for pions and nucleons were accounted). This value is about 2 times larger, than the ones found in [9] at chemical freeze-out and about 2 times smaller, than at thermal freeze-out [23, 24]. (Note, that the dense gas stage is an intermediate between these two.) In the case of sphere its radius is equal to $R=11.4\,{\rm fm}$. If we assume, that antiprotons are mainly concentrated in the outer shell of the fireball of the thickness of $l_{ann}\approx 3\,{\rm fm}$, then $\overline{n^{2}}/\bar{n}^{2}\approx 2$ and we get for the coalescence parameter $$B^{th}_{2}\,=\,3.0\times 10^{-4}\,{\rm GeV}^{2}$$ (15) (We put ${\tilde{\Gamma}}_{\bar{d}}=\Gamma$, or $a=1/2$.) Experimentally [1], for the average value of the most central 10% events it was found: $B^{exp}_{2}=(4.4\pm 1.3)\times 10^{-4}\,{\rm GeV}^{2}$. However, $B^{exp}_{2}$ strongly depends on centrality: the results for $0-5\%$ centrality are about 1.5 times lower. Taking in mind all uncertaintlies – theoretical and experimental, we believe, that the NA44 data for coalescence parameter are not in contradiction with theoretical expectation. Turn now to the STAR experiment at RHIC: $Au+Au$ collisions at $\sqrt{s}=130A\,{\rm GeV}$ [2]. Antideuterons were measured at $0.5<p_{t}<0.8\,{\rm GeV}$ and in the rapidity interval $|\Delta y_{c.m.}|<0.3$, $18\%$ of central collisioins were collected. We take $\bar{E}_{\bar{p},c.m.}=1.05\,{\rm GeV}$. The number of ”wounded” nucleons in the $18\%$ central $Au+Au$ collisions can be estimated as $N_{N}=320$ [25]. Multiplicity of negative hadrons $\bar{h}$ (mainly, pions) was measured in [26] at pseudorapidity $\eta=0$ only and it was found an increasing of $dh^{-}/d\eta\mid_{\eta=0}$ by $52\%$ comparing with the SPS data at $\sqrt{s}=17A\,{\rm GeV}$. But it is known that $dh/d\eta/_{\eta=0}$ increase faster with energy than the total multiplicity. We estimate $Q_{\pi}=N_{\pi}/N_{N}\approx 7\pm 1$. (A value close to the presented above, can be found from the data compilation [27]). At $N_{N}=320$ with account of $20\%$ correction for $K$-mesons and hyperons $V=7.2\times 10^{3}\,{\rm fm}^{3}$. The coalescence parameter is equal to $$B^{th}_{2}=3.8\times 10^{-4}\,{\rm GeV}^{2}$$ (16) ($\Gamma=320\,{\rm MeV}$, $\bar{n}^{2}/(\bar{n})^{2}$ was put to be 2). Experimentally, STAR found $B^{exp}_{2}=(4.5\pm 0.3\pm 1.0)\times 10^{-4}\,{\rm GeV}^{2}$. The main uncertainty of $B_{2}^{th}$ comes from the fireball volume $V$ which was calculated by (14). However, the width $\Gamma$ also depends on the fireball volume, so that $B_{2}^{th}\sim 1/\sqrt{V}$, which suppresses this uncertainty twice. We expect the accuracy of our estimations (15), (16) to be about $50\%$. In E864 experiment [3] at AGS the antideuterons were observed in $Au+Pt$ collisions at $\sqrt{s}=4.8\,A\,{\rm GeV}$. 10% of central collisions we selected. From the data we take: $p_{\bar{p}t}=0.17\,{\rm GeV}$, $\overline{E}_{\bar{p},c.m.}=0.99\,{\rm GeV}$. The number of ”wounded” nucleons and $\pi/N$ ratio are $N_{N}=350$, $Q_{\pi}=1.6$ (see [20] and references herein). In the same way as before, we find: $V=2.8\times 10^{3}\,{\rm fm}^{3}$, $\Gamma=220\,{\rm MeV}$, $l_{ann}=1.2\,{\rm fm}$. In this case the validity conditions of our approach are at the edge of their applicability. So, the theoretical expectations for $B_{2}$ are valid only by the order of magnitude: $$B^{th}_{2}\sim 1.5\times 10^{-3}\,{\rm GeV}^{2}$$ (17) in comparison with $B^{exp}_{2}=(4.1\pm 2.9\pm 2.3)\times 10^{-3}\,{\rm GeV}^{2}$. 4. Summary and Acknowledgements. The coalescence parameter $B_{2}$ for the antideuteron production in heavy ions collisions was calculated. It was supposed, that the $\bar{d}$ production proceeds at the stage, when the fireball may be treated as a dense gas of interacting hadrons. The $\bar{d}$ production is described as the formation process $\bar{p}+\bar{n}\to\bar{d}$, where $\bar{p}$, $\bar{n}$, $\bar{d}$ are moving in the mean field of the fireball constituents (mainly pions). It was shown, that in case of large $N_{\pi}/N_{N}$ ratio one may expect a balance: the number of produced antideuterons is equal to the number of desintegrated $\bar{d}$ due to collisions with pions. The balance condition determines $\bar{d}$ production rate and the value of coalescence parameter $B_{2}$. The later is expressed in terms of deuteron binding energy and mean fireball volume at this stage. The comparison with data demonstrates, that ${\bar{d}}$-production proceeds at the stage intermediate between chemical and thermal freeze-out – the dense gas stage of the fireball evolution. 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Non-trivial gravitational waves and structure formation phenomenology from dark energy Jose Beltrán Jiménez jose.beltran@uam.es Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de Madrid, Cantoblanco, Madrid, 28049, Spain. Departamento de Física Fundamental, Universidad de Salamanca, E-37008 Salamanca, Spain.    Lavinia Heisenberg lavinia.heisenberg@eth-its.ethz.ch Institute for Theoretical Studies, ETH Zurich, Clausiusstrasse 47, 8092 Zurich, Switzerland (December 3, 2020,  ; December 3, 2020) Abstract The detection of the GW170817/GRB170817A event improved the constraints on the propagation speed of gravitational waves, thus placing possible variations caused by dark energy under restraint. For models based on scalar fields belonging to the family of Horndeski Lagrangians, non-minimal derivative couplings are now severely constrained, entailing a substantially limited phenomenology. In this work we want to stress that there is still a plethora of dark energy models that get around this obstacle while still providing interesting phenomenologies able to distinguish them from the standard cosmology. We focus on a class involving vector fields as a proxy, but our discussion is extensible to a broader class of models. In particular, we show the possibility of having a non-minimal derivative coupling giving a non-trivial effect on scalar modes without affecting gravitational waves and the possibility of having a second tensor mode that can oscillate into gravitational waves. We also present a novel class of configurations breaking rotational invariance but with an energy-momentum tensor that is isotropic on-shell. This peculiar feature makes the scalar and vector sectors of the perturbations mix so that, even in a perfectly isotropic background cosmology, preferred direction effects can appear in the perturbations. We also comment on models that give rise to isotropic solutions when averaging over rapid oscillations of the vector fields. The explored models are classified according to distinctive field configurations that provide inequivalent realisations of the Cosmological Principle. I Introduction General Relativity (GR) stands out as the most compelling contender to describe the gravitational interaction in a wide range of scales, from sub-milimiter to Solar System scales Will:2014kxa . Most experiments designed to test GR have focused on its non-radiative sector, aiming to finding deviations from Newton’s law, the equivalence principle or, in general, in any of the Parameterised Post-Newtonian (PPN) parameters. Until recently, the radiative sector, namely Gravitational Waves (GWs), has remained largely less constrained from direct means, partly due to its elusive character. However, this difficulty has not prevented to have constraints from indirect probes, being the variation in the period of binary pulsar systems the most clear indirect evidence for the existence and properties of the GWs (see e.g. Wex:2014nva ). In fact, these measurements already allowed to infer that the gravitational radiation is predominantly quadrupolar with an amplitude in perfect agreement with the GR predictions, implying that GWs have spin 2, and its propagation speed $c_{\rm gw}$ could only differ from the speed of light $c$ at the $10^{-2}-10^{-3}$ level, with the corresponding bounds for modified gravity theories Yagi:2013qpa ; Jimenez:2015bwa . The speed of (subluminal) GWs was also constrained from the absence of Cherenkov radiation for cosmic rays to the $10^{-15}$ level ($10^{-19}$ for cosmic rays of extra-galactic origin) Moore:2001bv . These Cherenkov radiation constraints were already used in Kimura:2011qn to set bounds within the Horndeski scalar-tensor theories relevant for dark energy models. The situation improved dramatically after the first direct detections of GWs by the LIGO team, clearly confirming their physical reality and proving the feasibility of GWs astronomy. In August 2017, the VIRGO team joined the LIGO network, increasing the sensitivity to the polarisation of the GWs that provided a direct confirmation of its spin-2 nature111The analysis was performed by assuming pure tensor, pure vector or pure scalar GWs with the case of pure tensor being strongly favoured. This however does not exclude the existence of additional polarisations that could be subdominant (binary pulsars also indicate that a possible radiation in scalar or vector modes must be strongly suppressed with respect to the quadrupolar emission in GWs). It has also been argued in Allen:2018khw that a pure vector could, in principle, fit the LIGO/VIRGO signal, although a specific polarisation evolution needs to be assumed and it is unclear if it can be put into effect in a realistic theoretical framework. of GWs after only one detection Abbott:2017oio . Another major (and perhaps the most outstanding so far) discovery in GWs astronomy was the GW170817 event corresponding to a merger of two neutron stars TheLIGOScientific:2017qsa . What made this detection even more exciting was the possibility of identifying and observing the same event in the electromagnetic channel as the GRB170817A signal. Among many other outstanding consequences, this multi-messenger detection provided a direct constraint on the propagation speed of GWs to be $|c_{\rm gw}/c-1|\lesssim 10^{-15}$ in agreement with the previous indirect bounds and confirming once again the predictions of GR. This constraint is many orders of magnitude tighter than the ones obtained from binary pulsars and it is at the same level as the ones drawn from the absence of Cherenkov radiation, although it directly applies to much lower frequencies. The constraints from GWs astronomy (even if only a few events are available so far) together with the indirect observations of binary pulsars and the classical tests of GR on Solar System scales do not show any deviations from the GR predictions and, thus, theories of modified gravity are tightly constrained in the infrared nowadays Berti:2015itd . This is specially important for models of dark energy based on modified gravity theories because it seems harder and harder to have models that can provide accelerated expansion on cosmological scales while being compatible with all local gravity tests. A way out to this dichotomy was the existence of screening mechanisms that could allow to hide the dark energy effects on small scales (see e.g. Joyce:2014kja for a comprehensive review on this subject). As a paradigmatic class of modified gravity theories for dark energy we can consider scalar-tensor theories and, in particular, the extensively studied family of Horndeski Lagrangians Horndeski:1974wa and some of its extensions beyondH . These theories are characterised by the presence of second order derivative self-interactions of the scalar field that, in turn, require derivative non-minimal couplings to the gravity sector in order to ensure the absence of additional ghost-like propagating modes. These non-minimal couplings give rise to a very rich phenomenology for dark energy models (which is nicely captured in the effective field theory of dark energy EFTDE ), but the very same operators that drive the interesting phenomenology for structure formation are responsible for a variation of the propagation speed of GWs. The main aim of the present work is to put forward a class of dark energy models realising one of the following features • Non-minimal derivative couplings with effects on the scalar perturbations without affecting the GWs sector. • Non-trivial predictions for GWs astronomy without immediately conflicting with $c_{\rm gw}=1$. The first condition contrasts with the findings for dark energy models based on the Horndeski Lagrangians discussed above, where non-minimal derivative couplings of the scalar field gives rise to a modification of $c_{\rm gw}$ in the presence of a time-dependent background for the scalar field and, therefore, are subject to the constraint imposed by the GW170817/GRB170817A observation Ezquiaga:2017ekz . On the other hand, the second condition shows that there is still room to probe dark energy models with GWs astronomy in a non-trivial way without being in tension with $c_{\rm gw}=1$. In particular, the models that we will discuss give rise to a possible oscillation of GWs into additional tensor modes. Throughout this work we will use theories with vector fields as proxies to show the different possibilities, but the same can be applied to other models, as we will discuss in the last section. These different possibilities will be fundamentally characterised by the crucial property of providing inequivalent realisations of the Cosmological Principle, i.e., the homogeneity and isotropy of the universe will be achieved from different symmetries owed to different field configurations. Before proceeding to the main discussion, let us fix some notation. Given a set of vector fields $A^{a}{}_{\mu}$ with $a$ denoting some internal index and $\mu$ a Lorentz/spacetime index, we will define the field strengths as $F^{a}_{\mu\nu}=\partial_{\mu}A^{a}_{\nu}-\partial_{\nu}A^{a}_{\mu}$ and $\tilde{F}^{a\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F^{a}_{\rho\sigma}$ the corresponding dual tensor. From these objects we can define the electric and magnetic components as $E_{i}=F_{0i}$ and $B_{i}=\tilde{F}_{0i}=\frac{1}{2}\epsilon_{ijk}F_{jk}$. In the case of gauge fields, the field strength will be denoted by $\mathcal{F}^{a}_{\mu\nu}=F^{a}_{\mu\nu}+gf^{abc}A^{b}{}_{\mu}A^{c}{}_{\nu}$, with $g$ the coupling constant and $f^{abc}$ the structure constants of the corresponding gauge group. We will also use the symmetrised covariant derivative given by $S^{a}_{\mu\nu}=\nabla_{\mu}A^{a}_{\nu}+\nabla_{\nu}A^{a}_{\mu}$. The norm of the vector fields will be denoted by $Y=A^{a}{}_{\mu}A^{a\mu}$. Furthermore, we will consider cosmological models described by the FLRW metric $\mathrm{d}s^{2}=\mathrm{d}t^{2}-a^{2}(t)\mathrm{d}\vec{x}^{2}$ with $H=\dot{a}/a$ the corresponding Hubble expansion rate. We will use both a bar and a subscript $0$ to denote the background value of some quantity. II Dark Energy in multi-Proca and Yang-Mills theories As explained in the introduction, we will use theories featuring vector fields to illustrate the dark energy models complying with our requirements. More specifically, we will consider theories characterised by the presence of a set of $N$ vector fields $A^{a}{}_{\mu}$, with $a=1,\cdots N$. For simplicity we will only consider $N=3$ and, in order to ensure the existence of homogeneous and isotropic solutions, we will assume an internal global $SO(3)$ symmetry. Thus, our Lagrangians will be built out of $SO(3)$ and Lorentz scalars involving the vector fields and up to their first derivatives, i.e., $A^{a}_{\mu}$, $F^{a}_{\mu\nu}$ and $S^{a}_{\mu\nu}$. For a detailed spelled out of the possible terms in the Lagrangian we refer to Allys:2016kbq ; Jimenez:2016upj . In most of this work we will however focus on the simplest term given by an arbitrary function $\mathcal{K}$ involving only the vector fields $A^{a}_{\mu}$ and their field strengths $F^{a}_{\mu\nu}$, unless otherwise stated. The global symmetry can also be promoted to a gauge symmetry (that can then be identified with an $SU(2)$ gauge symmetry) in which case we will be dealing with Yang-Mills theories. In that case, the theories can be built in terms of the $SU(2)$ scalars involving ${\mathcal{F}}^{a}_{\mu\nu}$. Although we will restrict ourselves to these groups, it is worth mentioning that any larger group within which these can be embedded will also work. For instance, we could consider GUT groups such as $SU(5)$ or $SO(10)$ that contain our desired groups as subgroups. Of course, the precise low energy phenomenology will depend on the specific symmetry breaking pattern, but what concerns us here will be to have an internal symmetry that can be traded by the breaking of spacetime symmetries so that we can eventually have a residual $ISO(3)$ symmetry complying with the Cosmological Principle. Within this set-up, we can now consider different field configurations to achieve a homogeneous and isotropic cosmological background as we discuss in detail in the following. II.1 Pure temporal configuration We will start with the simplest homogeneous and isotropic configuration for the fields given by $$A^{a}{}_{\mu}=\phi^{a}(t)\delta^{0}{}_{\mu},$$ (1) with $\phi^{a}(t)$ arbitrary functions of time. This field configuration actually corresponds to a version of several single Proca fields and, in fact, no internal symmetry is required to have a cosmological background. Thus, it suffices to analyse the single field case to pinpoint the relevant phenomenology so that we will drop the internal index in the following. The interest of this configuration is the existence of a non-minimal derivative interaction for the vector field of the form222The multi-vector case simply amounts to adding internal indices. $$\mathcal{L}\supset G_{6}(Y)L^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}-G^{% \prime}_{6}(Y)\tilde{F}^{\alpha\beta}\tilde{F}^{\mu\nu}S_{\alpha\mu}S_{\beta% \nu},$$ (2) with $L^{\mu\nu\alpha\beta}=\frac{1}{4}\epsilon^{\mu\nu\rho\sigma}\epsilon^{\alpha% \beta\gamma\delta}R_{\rho\sigma\gamma\delta}$ the double dual Riemann tensor, $G_{6}$ an arbitrary scalar function depending on $Y$ and a prime standing for derivative w.r.t. its argument. The case $G^{\prime}_{6}=0$ corresponds to the vector-tensor interaction already found by Horndeski333Some cosmological consequences and stability properties were explored in Barrow:2012ay ; Jimenez:2013qsa . Horndeski:1976gi . This non-minimal coupling has no analogous in the Horndeski scalar-tensor Lagrangians and, thus, it can give rise to novel phenomenological effects. Since the background configuration has an identically vanishing field strength $\bar{F}_{\mu\nu}=0$, the contribution to the quadratic Lagrangian for the perturbations is simply $$\displaystyle\mathcal{L}^{(2)}\supset$$ $$\displaystyle\Big{[}G_{6}(Y)R^{\mu\nu\alpha\beta}+\frac{G^{\prime}_{6}(Y)}{2}S% ^{\mu\alpha}S^{\nu\beta}\Big{]}_{0}\delta\tilde{F}_{\mu\nu}\delta\tilde{F}_{% \alpha\beta}$$ $$\displaystyle=$$ $$\displaystyle-\frac{4H^{2}}{a^{2}}\left(\bar{G}_{6}+2\phi^{2}\bar{G}^{\prime}_% {6}\right)\delta\vec{E}^{2}$$ $$\displaystyle+\frac{4}{a^{4}}\left[(H^{2}+\dot{H})\bar{G}_{6}+2H\phi\dot{\phi}% \bar{G}^{\prime}_{6}\right]\delta\vec{B}^{2}.$$ (3) This expression clearly shows that there are no effects on the GWs sector at this order. In fact, all metric perturbations are trivially absent from this term and only the vector field perturbations contribute, which however have a non-trivial impact on the perturbations. Obviously, it gives rise to propagating vector perturbations (possibly together with other terms in the full Lagrangian as e.g. a standard Maxwell kinetic term). Concerning the scalar sector, it gives a non-trivial contribution to the effective gravitational Newton’s constant. The crucial point to understand how these effects are generated is to remember that the temporal component of the vector is an auxiliary field in these theories. Thus, although metric perturbations do not contribute in (3), the perturbation of $A_{0}$ does and, after integrating it out, it will give rise to a modification of the scalar sector. We should say however that a genuine contribution to the effective Newton’s constant from the Horndeski interaction requires the presence of a term like $G_{3}(A^{2})\nabla_{\mu}A^{\mu}$ that will give rise to a braiding similar to the KGB models for scalar fields Deffayet:2010qz , but with additional operators contributing to it. Let us also mention that this term is also necessary for having a dynamical background evolution for the vector field. In order to illustrate and clarify these points, let us consider the Lagrangian444We could also add a term $G(Y)\tilde{F}^{\mu\alpha}\tilde{F}^{\nu}{}_{\alpha}S_{\mu\nu}$ which does not modify the background evolution but can affect the perturbations. This term does not add anything crucially new to our discussion and, nevertheless, we want to focus on the non-minimal derivative coupling. Let us also mention that going to the multi-vector extension allows to introduce additional terms linear in $S^{a}_{\mu\nu}$ that are not present in the single field case, as e.g. $\epsilon^{\alpha\beta\gamma\delta}\tilde{F}^{a}_{\alpha\lambda}S^{b\lambda}{}_% {\beta}A^{a}_{\gamma}A^{b}_{\delta}$ or $S^{a\mu\nu}A_{\mu}^{b}A_{\nu}^{d}A_{\alpha}^{c}A^{e\alpha}\delta_{de}\epsilon_% {abc}$. All these terms could give rise to additional interesting features, but they are not crucial for our purpose here. $$\displaystyle\mathcal{L}=$$ $$\displaystyle\mathcal{K}(Y,\cdots)+G_{3}(Y)\nabla_{\mu}A^{\mu}$$ $$\displaystyle+G_{6}(Y)L^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}-G^{\prime% }_{6}(Y)\tilde{F}^{\alpha\beta}\tilde{F}^{\mu\nu}S_{\alpha\mu}S_{\beta\nu}\,,$$ (4) where the dots stand for terms involving $F_{\mu\nu}$ that will not contribute to the background equations of motion, although they will affect the perturbations. For the case of one single vector field (as we are considering here), there are only two possibilities, namely $F_{\mu\nu}F^{\mu\nu}$ and $A^{\mu}A^{\nu}F_{\mu\alpha}F^{\alpha}{}_{\nu}$, but the multi-vector case allows for more terms some of which can also contribute to the background equations very much like $Y$. One might worry that the non-minimal coupling to the double dual Riemann tensor will introduce modifications to $c_{\rm gw}$ whenever the field strength has a non-vanishing profile, as it would be expected on sub-Hubble scales, and this could conflict with the tight constraint on $c_{\rm gw}$. We however see this as good news since it gives the possibility to test this coupling with GWs. With only one measurement it is difficult to say anything definitive, since the signal might have travelled without encountering any relevant vector field profile. However, more multimessenger observations will put interesting constraints on backgrounds with a non-vanishing profile of the vector field strength. Notice however that it is crucial to have a background vector field generating a non-trivial electric and/or magnetic component. The background vector field equations of motion for a purely temporal background derived from (4) reduce to $$\phi\Big{(}3\phi HG^{\prime}_{3}+\mathcal{K}_{Y}\Big{)}=0\,,$$ (5) where we see that we need $G^{\prime}_{3}\neq 0$ in order to have an evolving $\phi(t)$. By solving the non-trivial branch ($\phi\neq 0$) of the above equation, we obtain $\phi=\phi(H)$, which can then be inserted in the Friedmann equation, modifying that way the way in which matter fields affect the expansion of the universe555Interestingly, these theories can be obtained in quadratic gravity theories formulated in Weyl Jimenez:2014rna and general vector distorted geometries Jimenez:2015fva . For some cosmological consequences and applications to inflation and dark energy models see also Jimenez:2016opp . DeFelice:2016yws . This is a general result whenever there are auxiliary fields in the gravitational sector (see e.g. auxiliary for some other realisations). Let us begin by arguing why it is not possible to obtain any anomalous slip parameter666The slip parameter can be defined as the difference between the Newtonian and the lensing potentials or, equivalently, as the ratio of the two gravitational potentials in the Newtonian gauge. The precise definition is not important for us.. A departure from the GR value of the slip parameter can be traced to the presence of anisotropic stresses, since the off-diagonal spatial gravitational equations lead to $\Phi-\Psi\propto\sigma$, being $\Phi$ and $\Psi$ the gravitational potentials and $\sigma$ the scalar potential of the anisotropic stress. Thus, it is easy to convince oneself that we need to have an energy-momentum tensor such that $T_{ij}$ is not proportional to $\delta_{ij}$. Let us then see what quantities could contribute to $T_{ij}$ in the considered background. Since $F_{ij}$ is antisymmetric and its background value vanishes, it is obvious that it can not contribute to the scalar perturbation of $T_{ij}$ at first order. Quantities involving only the vector without derivatives will contribute as $A_{i}A_{j}$ so, again, no first order contributions are possible. Thus, we are left with contributions involving $S_{\mu\nu}$. If we want to keep $c_{\rm gw}=1$, then only the aforementioned term $G_{3}\nabla_{\mu}A^{\mu}$ can be included. However, the variation of this term w.r.t. the metric gives $$\delta\big{(}\sqrt{-g}G_{3}\nabla_{\mu}A^{\mu}\big{)}=\delta G_{3}\partial_{% \mu}(\sqrt{-g}A^{\mu})-\partial_{\mu}G_{3}\delta(\sqrt{-g}A^{\mu})$$ (6) up to integrations by parts. We thus see that it is not possible to obtain a term in the energy-momentum tensor of the form $\partial_{i}\delta A_{j}$. Having explored the different possible contributions to the anisotropic stress and obtaining that none of them can actually contribute in the temporal background, we then conclude that no anomalous slip parameter can arise by maintaining $c_{\rm gw}=1$. This is along the lines of the conclusions reached in Saltas:2014dha . Let us now turn to non-trivial effects. Even though the non-minimal coupling does not modify $c_{\rm gw}$ with the pure temporal configuration of the vector field, it can affect the formation of large scale structures through a modification of the effective Newton’s constant $G_{{\rm eff}}$, which, in the deep quasi-static approximation, takes the form DeFelice:2016yws $$\frac{G_{\rm eff}}{G}=\frac{(\alpha_{1}G_{3}^{\prime}+\alpha_{2}G_{3}^{\prime% \prime})(G_{6}+2\phi^{2}G_{6}^{\prime})+\mathcal{F}_{1}}{(\beta_{1}G_{3}^{% \prime}+\beta_{2}G_{3}^{\prime\prime})(G_{6}+2\phi^{2}G_{6}^{\prime})+\mathcal% {F}_{2}}\,,$$ (7) where $G$ is the usual Newton’s constant, $\alpha_{1,2}$ and $\beta_{1,2}$ are functions of $\{\phi,\dot{\phi},H,\dot{H}\}$ and $\mathcal{F}_{1,2}$ are functions of those same variables and also depend on $\mathcal{K}$ and its derivatives. The specific form of these functions is not relevant for us here, but we have explicitly spelled out the dependence on $G_{3}$ and $G_{6}$ that confirms our statement above, namely, that it is crucial to have $G_{3}$ for the non-minimal coupling controlled by $G_{6}$ to affect the scalar perturbations. Since this is also the condition to have an evolving field background $\phi(t)$, it is in fact the expected case in the most interesting cosmologies. In Amendola:2017orw it is shown that the effective Newton’s constant within these theories is generically larger than $G$, implying an enhancement in the clustering of dark matter777The general result reported in Amendola:2017orw that $G_{\rm eff}\geqslant G$ applies to general background solutions that are either asymptotically attracted to a de Sitter solution or have $w_{\rm DE}\leqslant-1$. The latter condition implies that is necessary to have non-phantom behaviour in order to have $G_{\rm eff}<G$. In the former case, having a background solution connecting an early phase with $G_{\rm eff}<G$ and the de Sitter attractor necessary implies crossing a point where $G_{\rm eff}$ diverges (although this divergence could occur in the future). However, more general accelerating solutions not necessarily ending in a de Sitter attractor exist for which we could have $G_{\rm eff}<G$ without ever encountering said divergence.. It is important to emphasise however that the contribution from $G_{6}$ allows to introduce an additional scale to which the background evolution is oblivious, but the perturbations are sensitive to. This feature is characteristic of these vector field theories and has no analogue in the scalar field theories belonging to the Horndeski family. Although the effective Newton’s constant is generically larger than $G$, let us comment on a possible extension where $G_{\rm eff}$ can be lowered based on a coupling of dark matter to an effective metric that depends on the vector field888A similar construction was employed in BeltranJimenez:2013fca to develop a symmetron screening mechanism for a vector field. See also BeltranJimenez:2018tfy for theories with a vector coupled to the energy-momentum tensor.. The simplest of such couplings is to $\tilde{g}_{\mu\nu}=g_{\mu\nu}+A_{\mu}A_{\nu}$ so that, at linear order and around a purely temporal background, a coupling of the form $\phi(t)\delta A_{0}\delta\rho_{\rm DM}$ is generated, which is analogous to the usual coupling of a vector field to charged particles. Since the charges in this case are all alike and given by the mass, this extra coupling will give rise to a repulsion in the dark matter that will result in a weakening of the structure formation. This mechanism is somewhat similar to some models of self-interacting dark matter (see e.g. DMselfinteractions ). We will finalise our discussion on the temporal configuration by noticing that, besides the effects on the scalar perturbations discussed thus far, it is important to keep in mind that distinctive effects will also arise in the vector perturbations and that the considered non-minimal coupling will again have an impact in them DeFelice:2016yws . II.2 Triad configuration After discussing the purely temporal configuration, we will consider the so-called triad configuration given by $$A^{a}{}_{\mu}{}=A(t)\delta^{a}{}_{\mu}$$ (8) that is still compatible with a FLRW background and represents a genuine cosmological solution for the multi-vector models that is not possible in the single field case. This configuration amounts to having three vector fields pointing along orthogonal directions that can then be identified with the three spatial axes. Thus, it breaks both the internal $SO(3)$ symmetry and the external spatial rotations, but it leaves a linear combination of them unbroken, which is then responsible for the rotational symmetry of the background. This configuration is the natural one to have cosmological solutions for non-abelian Yang-Mills theories Cervero:1978db ; Galtsov:1991un ; Darian:1996mb . The simplest case of $SU(2)$ has been used for inflationary models supported by interactions of dimension higher than 4 in gauge-flation Maleknejad:2011jw (see Maleknejad:2012fw for a nice review) or by Horndeski-like non-minimal couplings Davydov:2015epx (although these come at the price of tensor instabilities BeltranJimenez:2017cbn ). Needless to say that these configurations used to develop inflationary solutions can also be employed to construct dark energy models. Although there have been some proposals to describe dark energy purely in terms of Yang-Mills fields999There are also models with accelerated expansion where the gauge fields are assisted either by other fields as in e.g. Rinaldi:2015iza or non-minimal couplings as in e.g. Bamba:2008xa . Notice however that the latter case suffers from ghost instabilities, unless the non-minimal coupling is of the Horndeski form to the double dual Riemann tensor as in Davydov:2015epx ; BeltranJimenez:2017cbn . with the triad configuration Zhao:2005bu ; Galtsov:2008wkj ; Mehrabi:2015lfa , this has remained largely less explored than models based on scalar fields. The triad configuration (8) can also be used in theories without the non-abelian gauge symmetries discussed above, provided there is still a global internal $SO(3)$ symmetry allowing for a residual rotational invariance. This was already explored as a dark energy model based on a set of vector fields with a certain potential in ArmendarizPicon:2004pm (see also Wei:2006tn ). The triad configuration was also used for models of inflation supported by massive vector fields Golovnev:2008cf . Another class of models where this configuration is relevant is provided by the theories that extend the generalised Proca interactions Heisenberg:2014rta ; Allys:2015sht ; Jimenez:2016isa (or its extensions Heisenberg:2016eld ) to the case of multiple vector fields Allys:2016kbq ; Jimenez:2016upj . The possibility of having new interesting cosmological scenarios with the triad configuration in these theories was discussed in Jimenez:2016upj . In Rodriguez:2017wkg some of these novel interactions have been used to show the possibility of having dark energy solutions. However, these models use non-minimal couplings that modify the propagation speed of GWs in this field configuration and, thus, their observational viability is jeopardised. For the triad configuration (8), the non-minimal coupling to the double dual Riemann tensor discussed in the previous section gives rise to a modification of the GWs propagation speed already at the linear order and, as a consequence, it will be tightly constrained by the GW170817/GRB170817A event. Thus, we will disregard it for the triad configuration. This however does not mean that there will be a trivial effect on the propagation of GWs and, in fact, these models can actually be probed by GWs astronomy. The reason precisely roots in the symmetry breaking pattern of these models where the unbroken diagonal $SO(3)$ symmetry consists of a linear combination of the internal and the external rotations, what allows for a second tensor mode associated to the vector fields. In this case, performing the usual helicity decomposition of the perturbations (now in terms of the irreducible representations of the unbroken diagonal $SO(3)$ symmetry of the background) leads to two tensor modes, namely: the usual metric perturbation $h_{ij}=\delta_{T}g_{ij}/a^{2}$ and, in addition, the tensor mode built as $t_{ij}=\delta^{a}_{i}\delta_{T}A^{a}{}_{j}$, where $\delta_{T}$ stands for the transverse traceless perturbation. Notice that the perturbations of the vector fields arrange into the tensor mode $t_{ij}$ thanks to the background field $\bar{A}^{a}{}_{i}=A(t)\delta^{a}_{i}$ that allows to identify the spacetime and internal indices. The interesting feature of these models is that these two tensor modes mix in a non-trivial way and can give rise to an oscillation of GWs into the second tensor mode, among other interesting effects. Thus, this phenomenon will give a signature of these models of dark energy that can be probed with GWs without being immediately ruled out by the tight constraint on the speed of GWs. This type of oscillations for the case of Yang-Mills fields has been explored in GWoscillationsGF . The phenomenon of GWs oscillations also occurs in theories of massive bi-gravity GWoscillationsbiG . However, the small value of the required mass makes such oscillations very small and the forecasted effect very difficult to detect. The important point we want to make here is that all the dark energy models based on the triad configuration will affect in a safe way the propagation of GWs. Since we do not invoke non-minimal couplings at all, the propagation speed of GWs will be determined by the usual Einstein-Hilbert term and, thus, all these models will give rise to $c_{\rm gw}=1$. However, despite being minimally coupled to gravity, the presence of the non-trivial triad configuration that breaks the spacetime and internal rotations to the diagonal component leads to a mixing of both tensor modes. For instance, a dependence on $Y$ of the Lagrangian will lead to contributions to the quadratic action for the perturbations as $$\mathcal{L}^{(2)}\supset\bar{A}^{a}_{\mu}\delta A^{a}{}_{\nu}\delta g^{\mu\nu}% \supset A(t)\delta A_{ij}\delta g^{ij}$$ (9) where $\delta A_{ij}=\delta^{a}_{i}\delta A^{a}{}_{j}$. We thus clearly see how the tensor mode associated to the vector fields mixes with the GWs through the simple $Y$-dependence of the Lagrangian. If we look at contributions arising from a kinetic dependence on $F^{a}{}_{\mu\nu}F^{a\mu\nu}$ we obtain terms like $$\displaystyle\mathcal{L}^{(2)}\supset$$ $$\displaystyle\bar{F}^{a}{}_{\mu\nu}\delta F^{a}{}_{\alpha\beta}\bar{g}^{\mu% \alpha}\delta g^{\nu\beta}\supset\dot{A}\partial_{0}\delta A_{ij}\delta g^{ij}$$ (10) where we see a coupling between both tensor modes that involves time derivatives. The large freedom in the choice of the building blocks of the Lagrangian leads to a very rich phenomenology for GWs, as the aforementioned oscillation explored in GWoscillationsGF or the GWs opacity studied in Caldwell:2018feo . In any case, it becomes clear that all these models can still be probed by using GWs astronomy without modifying $c_{\rm gw}$. II.3 Temporally extended triad configuration The configurations considered in the two previous sections give rise to a background field configuration that respects some rotational invariance, either purely spatial or a combination of internal and spatial rotations. In this section we will consider a combination of these two configurations given by $$A^{a}_{\mu}=\phi^{a}\delta^{0}_{\mu}+A(t)\delta^{a}_{\mu}.$$ (11) This configuration does not respect any rotational symmetry, even if the theory is provided with an internal $SO(3)$ symmetry. However, as shown in Jimenez:2016upj , it is possible to restrict the interactions so that the energy momentum tensor becomes isotropic on-shell and, therefore, exact FLRW background solutions are still possible. This is a dramatically different mechanism to achieve homogeneous and isotropic solutions from those considered so far and, thus, it provides a new realisation of the Cosmological Principle. Let us show the working mechanism with a very simple model that will serve as a proof-of-concept. The crucial property to guarantee that homogeneous and isotropic solutions exist for the configuration (11) is that the Lagrangian only depends on the vector field without derivatives through $Y$ so that we will consider the Lagrangian $\mathcal{L}=\mathcal{K}(Y,Z_{i})$, where $Z_{i}$ stands for the 11 possible Lorentz and $SO(3)$-scalars built out of the field strengths $F^{a}{}_{\mu\nu}$ (see for instance Piazza:2017bsd for their explicit form, which is not relevant for our purposes here). Since $\phi^{a}(t)$ does not contribute to the field strengths, the sector containing the $Z_{i}$’s will automatically be isotropic and the only concern comes from the $Y$ sector. Variations of the corresponding action $\,S=\int\mathrm{d}^{4}x\sqrt{-g}\mathcal{K}$ will thus take the form $$\delta\mathcal{S}=\int\mathrm{d}^{4}x\sqrt{-g}\Big{[}\mathcal{K}_{Y}\delta Y+% \mathcal{K}_{Z_{i}}\delta Z_{i}-\frac{1}{2}\mathcal{K}g_{\mu\nu}\delta g^{\mu% \nu}\Big{]}\,.$$ (12) The last two terms in this variation are identically isotropic and, therefore, will not be relevant for the following discussion. The only anisotropic contribution to the energy-momentum tensor in the FLRW metric with the configuration (11) is then $$T_{0i}\propto\mathcal{K}_{Y}A(t)\phi^{a}(t)\delta^{a}{}_{i}.$$ (13) On the other hand, the equations for the $\phi^{a}$’s come from the same term and are given by $$\mathcal{K}_{Y}\phi^{a}=0,$$ (14) which clearly shows that the energy-momentum tensor is isotropic when the equations of the temporal components are satisfied. In fact, we can see the general property that $T_{0i}$ is proportional to the equation of motion of the temporal component in a homogeneous background (see also Jimenez:2009ai ). The branch with $\phi^{a}=0$ corresponds to the triad configuration of the previous section, while the branch with $\mathcal{K}_{Y}=0$ is the genuine extended triad configuration, that we are considering in this subsection. This configuration does not present new phenomenological consequences in the tensor sector with respect to the pure triad. The reason for this is that the $Z_{i}$-sector is exactly (off-shell) isotropic also in the extended triad configuration while the $Y$-sector does not contribute to the tensor modes. However, the isotropy violation of the extended triad introduces a preferred direction in the background that will reflect in the vector and scalar perturbations, giving a very distinctive signature of these models. A full account of this phenomenology will be presented elsewhere, but the main feature can be easily understood by considering our proxy Lagrangian, whose quadratic form will read $$\displaystyle\mathcal{L}^{(2)}\supset$$ $$\displaystyle\mathcal{K}_{Y}\delta^{(2)}Y+\mathcal{K}_{Z_{i}}\delta^{(2)}Z_{i}$$ $$\displaystyle+\frac{1}{2}\mathcal{K}_{Z_{i}Z_{j}}\delta Z_{i}\delta Z_{j}+% \frac{1}{2}\mathcal{K}_{Z_{i}Y}\delta Z_{i}\delta Y+\frac{1}{2}\mathcal{K}_{YY% }(\delta Y)^{2}$$ (15) where $\delta$ and $\delta^{(2)}$ stand for the first and second order perturbations respectively. Again, the $Z_{i}$ sector exactly respects the background rotational invariance of the energy-momentum tensor and, consequently, there will not be any mixing from that sector. The above expression also shows explicitly our previous statement that the $Y$ sector does not contribute to the tensor perturbations because $\delta^{(2)}Y$ enters multiplied by $\mathcal{K}_{Y}$, that vanishes on shell, and $\delta Y$ does not receive contributions from the tensor modes. In fact, the first order perturbation of $Y$ is $$\displaystyle\delta Y=$$ $$\displaystyle\phi^{2}\delta g^{00}+A^{2}\delta^{ij}\delta g_{ij}+2A\phi^{a}% \delta g^{0a}$$ $$\displaystyle+2\bar{g}^{00}\phi^{a}\delta A^{a}{}_{0}+2A\bar{g}^{ij}\delta A_{% ij}\,,$$ (16) where $\phi^{2}=|\phi^{a}|^{2}$ and $\delta A_{ij}=\delta^{a}_{i}\delta A^{a}{}_{j}$. We then confirm that tensor modes do not contribute. Furthermore, we explicitly see the characteristic feature of this configuration that scalar and vector modes can mix via the background preferred direction provided by $\phi^{a}$ in the quadratic Lagrangian (15) through the term $\mathcal{K}_{YY}(\delta Y)^{2}$. II.4 Gaugid configuration A particular class of models that allows for a different type of solutions is when the Lagrangian is provided with a set of abelian gauge symmetries, which was dubbed gaugid in Piazza:2017bsd . For the multi-Proca interactions this can be easily achieved by setting all mass terms to zero, i.e., the action is only built out of the field strengths101010There can be another alternative where the set of abelian gauge symmetries are non-linearly realised. If the gauge symmetry only involves up to first derivatives of the gauge parameter, the symmetry is the usual $U(1)$ up to field redefinitions Wald:1986bj . $F^{a}_{\mu\nu}$. For the non-abelian Yang-Mills theories with an $SU(2)$ symmetry, the same can be achieved by sending the gauge coupling constant $g\to 0$ so that the full $SU(2)$ factorises into three $U(1)$’s, i.e., we have ${\mathcal{F}}^{a}_{\mu\nu}\rightarrow F^{a}_{\mu\nu}$. In both cases, the Lagrangian will be a function of the Lorentz- and $SO(3)$-scalars built out of the field strengths $F^{a}_{\mu\nu}$ mentioned above and whose explicit form can be found in Piazza:2017bsd . These models allow for solutions with the triad configuration that will lead to an electric gaugid since, in that configuration, we have $F^{a}_{0i}=\dot{A}\delta^{a}_{i}$ and $F^{a}_{ij}=0$, so only the electric part of the fields contribute. The phenomenology for these models will differ from the one considered above in that the temporal components do not play any role because they become Lagrange multipliers instead of auxiliary fields. This will in turn result in fewer scalar modes for the perturbations. So far, we have only considered homogeneous background configurations for the fields so that the required translational symmetry of the FLRW solutions is trivially realised. However, the gaugid models make it possible to achieve good cosmological solutions with inhomogeneous fields, provided the three $U(1)$ gauge symmetries compensate for the inhomogeneities in the field configuration. This is analogous to what happens in solid inflation111111Needless to say that the set-up of solid inflation could also be used to construct solid dark energy models. Endlich:2012pz (see also Gruzinov:2004ty ), which is based on three scalar fields $\phi^{a}$ with the configuration $\phi^{a}\propto x^{a}$ and where an internal shift symmetry for each scalar field makes up for the inhomogeneous field configuration. A similar construction can be used to have cosmological solutions with inhomogeneous vector fields. This was recently pursued in the model of gaugid inflation considered in Piazza:2017bsd . The considered inhomogeneous configuration for the gauge fields is of the form $$A^{a}{}_{\mu}=\frac{1}{2}B\epsilon^{a}{}_{i\mu}x^{i}$$ (17) with $B$ a constant. In this configuration we have $F^{a}_{0i}=0$ and $F^{a}_{ij}=B\epsilon^{a}{}_{ij}$ so that $B$ represents a constant magnetic field and the configuration is therefore called magnetic gaugid. The Lagrangian is required to enjoy an internal $SO(3)$ symmetry together with three $U(1)$ gauge symmetries. The symmetry breaking pattern then goes as121212Technically, the magnetic gaugid configuration does not break time translations, so that its corresponding generator would not be broken. However, in the late time universe, the presence of other matter fields (radiation, baryons, dark matter,…) will break time translations and that is why we do not include it in the final symmetry group. $ISO(3,1)\times SO(3)\times[U(1)]^{3}\to ISO(3)$ so that each broken spatial translation can be compensated by a $U(1)$ transformation (analogously to the shift symmetry employed in solid inflation) and the broken spatial rotations are compensated by an internal one. Although this model was considered as an inflationary model, it can of course be used as a dark energy model as well so that it is also possible to have magnetic gaugid dark energy. An interesting property of this configuration is that, as shown in Piazza:2017bsd , it naturally gives rise to a Chern-Simons type of interaction of the form $\epsilon^{ijk}\partial_{i}t_{jm}h_{mk}$ between both tensor modes. A very remarkable phenomenological feature of dark energy models based on scalar fields is that non-linear interactions can lead to the appearance of screening mechanisms that allow to decouple the cosmological evolution of the scalar from its behaviour on small scales. Of course, the same will apply for the models considered in this work. We will highlight a mechanism that can be well-motivated within models of interacting dark matter and could naturally inscribe within models with the gaugid configuration. Some of those models can help alleviating some of the claimed small problems of $\Lambda$CDM and they rely either on self-interactions of the DM particles or interactions mediated by some gauge boson DMselfinteractions . This gauge boson can be associated to gaugid dark energy, thus giving rise to interactions in the dark sector with specific signatures. In that case, we can imagine the gaugid to couple to some conserved current $J_{a}^{\mu}$ carried by the dark matter particles, which are assumed to share the same charge. It is then natural to expect that the charge density of dark matter will be proportional to its energy density. If we take a proxy model for the gaugid sector with Lagrangian $\mathcal{L}=\mathcal{K}(Z)$ and including an interaction through a coupling to the conserved current as $\mathcal{L}_{\rm int}\propto A^{a}_{\mu}J^{\mu}_{a}$, the field equations around a static and spherically symmetric source with $J^{a}_{\mu}=\rho^{a}(r)\delta^{0}_{\mu}$ will be $$\vec{\nabla}\cdot(\mathcal{K}_{Z}\vec{E}^{a})=\alpha\rho^{a}\,$$ (18) where $\alpha$ measures the strength of the coupling. As usual, we can integrate the above equation around a spherical shell comprising the source to obtain $$|\mathcal{K}_{Z}\vec{E}^{a}|=\frac{\alpha q^{a}}{4\pi r^{2}}$$ (19) with $q^{a}=\int\rho^{a}\mathrm{d}^{3}x$ the total charge. Given our assumptions above, the total charge is expected to be proportional to the total mass of dark matter inside the spherical shell. We find then the expected screened solution when higher order interactions are included in the gaugid sector. Let us illustrate it with the simple Lagrangian $\mathcal{K}=-1/4Z(1+\frac{1}{\Lambda^{4}}Z)$ where $\Lambda$ is some scale controlling the non-linearities. At large distances we have $Z/\Lambda^{4}\ll 1$ and we recover the usual Coulombian potential behaviour $|\vec{E}^{a}|\propto r^{-2}$. However, below the non-linear scale determined by $r_{\rm NL}=\Lambda\sqrt{|\frac{\alpha q}{4\pi}|}$, the higher order term will take over and we have the screened solution $|\vec{E}^{a}|\propto r^{-2/3}$. This opens up the possibility for having different phenomenological signatures on small scales (inside dark matter haloes) and large scales and establishes a natural framework for new dark matter-dark energy interacting models. II.5 Approximate isotropic solutions: Oscillating fields In the precedent sections, we have focused on cosmological solutions where the required homogeneity and isotropy of the background solutions are exactly realised, i.e., the background field configurations exactly realise a residual $ISO(3)$ symmetry, which could happen to be realised only on-shell as in the extended triad configuration. However, other possibilities also exist where the background field configurations do not exactly respect those symmetries but deviations are sufficiently small as to be compatible with observations. We will be concerned here with the isotropy of the background configuration131313Inhomogeneities are anyways considered in the perturbations and they must account for structure formation. for which the CMB sets a constraint $\sim 10^{-3}$ for a dipole-like deviation and $\sim 10^{-5}$ for a quadrupolar deviation. This applies to models leading to Bianchi I universes as, e.g., anisotropic dark energy AnDE or models with some background preferred direction as it could be magnetic fields Campanelli:2006vb , moving dark energy Moving or vector fields Koivisto:2008xf . We want however to highlight another less explored possibility that relies on oscillating fields. It is clear that a homogeneous spacelike vector field introduces some preferred direction (that could even vary in time depending on the polarisation of the vector field). However, it was shown in Cembranos:2012kk that, as long as the oscillations of the vector field are fast enough as compared to the expansion rate of the universe, the corresponding energy-momentum tensor averaged over several oscillations of the field becomes isotropic. The same result also applies to non-Abelian Yang-Mills theories Cembranos:2012ng (it was even shown for arbitrary spin in Cembranos:2013cba ). These results were used in Cembranos:2016ugq to show that oscillating coherent light vector fields can be good candidates for dark matter, similarly to axion-like dark matter models. We will not elaborate much further here on this possibility but we simply want to point out that these oscillating configurations can also give rise to accelerating cosmologies. In particular, for a power law potential of the form $V\propto A^{n}$, having an equation of state close to $-1$ requires a very small value of the exponent $n$, so that we essentially have a cosmological constant. However, the richer interactions structure of generalised Proca Lagrangians calls for a dedicated analysis as to explore which (if any) terms can easily give accelerating cosmologies with oscillating fields. This would open new interesting signatures since the background oscillating fields will affect non-trivially the perturbations, for instance sourcing the gravitational waves, generating a non-trivial slip or mixing different helicity modes Cembranos:2016ugq . It is worth saying that the very oscillations of the background fields make this scenario quite cumbersome already at the background level, and the study of the perturbations quickly becomes a very challenging task, both analytically and numerically. III Discussion In this work we have argued that the very restrictive bound on the GWs propagation speed inferred from GW170817/GRB170817A still leaves room for a wide class of dark energy models with interesting phenomenological consequences for structure formation and GWs probes. Throughout this work we have considered theories with vector fields and surveyed different configurations. We have shown that a purely temporal background field permits a non-minimal derivative coupling (with no analogous in the dark energy models based on scalar fields) which does not affect the GWs propagation but gives a non-trivial contribution to the scalar and vector perturbations. We have also considered the case of a triad configuration featuring a second tensor mode that can oscillate into GWs, giving distinctive signatures. We have extended the triad configuration to include temporal components and shown that this configuration can still support exact FLRW solutions while the breaking of isotropy in the background fields configuration induces a coupling between scalar and vector perturbations. We have discussed the case of gaugid configurations that can provide dark energy models supported by inhomogeneous fields configurations. Finally, we have also commented upon the possibility of having background field configurations that only induce small deviations from isotropy and, in particular, the interesting case of oscillating fields, whose averaged energy-momentum tensor is in fact isotropic. The results presented in this work make it apparent that dark energy models can still give a rich and interesting phenomenology without being in conflict with $c_{\rm gw}=1$. We have considered theories with vector fields as a proof-of concept, but our results are not limited to that case. As an example, given the duality between vector fields and 3-forms in 4 dimensions, analogous configurations to the ones considered here are possible for 3-form dark energy models, with potentially similar phenomenologies. In particular, one can consider models with non-abelian $p-$forms with analogous symmetry breaking patterns as the ones discussed here. Dark energy models with additional types of fields like scalars Heisenberg:2018acv or massive gravity Heisenberg:2017qka will of course share some of these properties. Another main message of this work is that interesting dark energy models are still possible without having to necessarily resort to contrived higher derivative and non-minimal couplings as those of the Horndeski Lagrangians and its extensions. It is perhaps more fructuous to turn to allegedly simpler models, as the ones discussed here, that explore fundamentally different dark energy models based on different symmetry breaking patterns or, equivalently, inequivalent realisations of the Cosmological Principle. In this respect, the classification presented in Nicolis:2015sra or the approach discussed in e.g. EFTfluids can provide a useful guidance. 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On reverberation and cross-correlation estimates of the size of the broad-line region in active galactic nuclei A. V. Melnikov  and I. I. Shevchenko Pulkovo Observatory of the Russian Academy of Sciences, Pulkovskoje ave. 65/1, St.Petersburg 196140, Russia E-mail: melnikov@gao.spb.ru Abstract It is known that the dependence of the emission-line luminosity of a typical cloud in the active galactic nuclei (AGN) broad-line regions (BLRs)upon the incident flux of ionizing continuum can be nonlinear. We study how this nonlinearity can be taken into account in estimating the size of the BLR by means of the “reverberation” methods. We show that the BLR size estimates obtained by cross-correlation of emission-line and continuum light curves can be much (up to an order of magnitude) less than the values obtained by reverberation modelling. This is demonstrated by means of numerical cross-correlation and reverberation experiments with model continuum flares and emission-line transfer functions and by means of practical reverberation modelling of the observed optical spectral variability of NGC 4151. The time behaviour of NGC 4151 in the H$\alpha$ and H$\beta$ lines is modelled on the basis of the observational data by Kaspi et al. (1996) and the theoretical BLR model by Shevchenko (1984, 1985). The values of the BLR parameters are estimated that allow to judge on the size and physical characteristics of the BLR. The small size of the BLR, as determined by the cross-correlation method from the data of Kaspi et al. (1996), is shown to be an artifact of this method. So, the hypothesis that the BLR size varies in time is not necessitated by the observational data. Key words: galaxies: active — galaxies: nuclei — galaxies: Seyfert – galaxies: individual: NGC 4151 1 Introduction In the early 1970’s, in the course of observations of rapid variability of the optical spectrum of the Seyfert galaxy NGC 4151, the time lag of variations in the H$\alpha$ line with respect to variations in the optical continuum was discovered (Lyutyi & Cherepashchuk, 1971; Cherepashchuk & Lyutyi, 1973). The time lag was interpreted by Lyutyi (1977, 1982) as a consequence of the fact that the emission-line clouds are at some distance from the ionizing radiation source. Later on, Antonucci & Cohen (1983) observed much smaller time lags in variations of the H$\beta$ and H$\gamma$ lines. Such a difference in the time lag, in H$\alpha$ greater than in H$\beta$, is observed in other active galactic nuclei (AGN) as well (see Table 6 in Peterson et al. 2004). According to Shevchenko (1984, 1985a), this difference in time lag is due to an essential nonlinearity in the dependence of the H$\alpha$ luminosity of an individual cloud upon the ionizing continuum flux incident on the cloud, the dependence in the higher order Balmer lines being close to linear. This explanation was made with the assumption that the duration of the emission-line flare is much greater than the duration of the flare in the ionizing continuum, and the duration of the latter one allows its description by $\delta$-function. However, continuum variations are not so rapid usually; therefore, in order to extract physical information from the observed emission-line variations, it is necessary, in addition to theoretical estimates, to use numerical modelling taking into account the timescale of continuum variations. In the present paper, we study theoretically how the nonlinearity in the emission-line luminosity, $L_{\mathrm{l}}$, of the broad-line region (BLR) cloud, in its dependence on the ionizing continuum flux, $F_{\mathrm{i}}$, incident on the cloud, can be taken into account in estimating the BLR sizes by means of the “reverberation” methods. We show that the BLR size estimates obtained by straightforward cross-correlation of emission-line and continuum light curves can be much (up to an order of magnitude) less than those obtained by reverberation modelling. First of all, we demonstrate this by means of abstract representative numerical cross-correlation and reverberation experiments with model continuum flares and emission-line transfer functions. Then we accomplish practical numerical modelling of the light curves of NGC 4151 in the H$\alpha$ and H$\beta$ lines on the basis the observational data by Kaspi et al. (1996) and the theoretical model of the BLR by Shevchenko (1984, 1985a). This model is characterized by allowing for thick geometries of the BLR, taking into account the anisotropy of line emission of individual clouds and, most important, taking into account the nonlinearity of the “$L_{\mathrm{l}}$–$F_{\mathrm{i}}$” relation. This nonlinearity allows one to explain the differences in the time lags for different lines. Cross-correlation estimates of the BLR size are also made. They turn out to be small in comparison to the estimates obtained by the direct reverberation modelling. The values of the parameters of the BLR model are derived directly from the reverberation modelling, and that is why we do not use any specific numeric results of modern photoionization models of the emission-line spectra of AGN. We use only the fact that, according to these models, the emission-line response of an individual cloud (particularly, in H$\alpha$ and H$\beta$) can be nonlinear. We also allow for a constant emission-line component. Though the presented theoretical inferences on the time lags can be of general interest, our primary goal is to apply them to explaining the time behaviour of the H$\alpha$ and H$\beta$ lines in the emission-line spectrum of NGC 4151, in the framework of a simple uniform one-component model. Of course, explaining the time behaviour of the whole emission-line spectrum would require much more complicated multi-component models. 2 Effective Stratification of a Homogeneous BLR The broad-line region (BLR) of an active galactic nucleus, according to the “standard model”, see, e.g. Peterson 1988, represents an aggregate of line-emitting clouds under the effect of ionizing radiation of the central source. The dependence of the emission-line luminosity $L_{\mathrm{l}}$ of an individual cloud upon the value of the incident ionizing flux $F_{\mathrm{i}}$, in accordance with the photoionization models of spectra of active galactic nuclei (see, e.g., Kwan 1984; Mushotzky & Ferland 1984), is described by a power law: $L_{\mathrm{l}}\propto F_{\mathrm{i}}^{s}$, where $s\geq 0$. The rate of heat input in a gas cloud optically thick in the ionizing continuum is directly proportional to the value of the ionizing flux incident on this cloud. Kwan (1984) noted that therefore the cloud’s emission-line luminosity should be, in a first approximation, directly proportional to the ionizing flux; different lines, however, behave differently. For example, in the case of L$\alpha$ the dependence is somewhat weaker than linear. The L$\alpha$ quanta leaving the cloud are produced in the traditional H ii zone. At the high ionization parameters typical of AGNs the collisional ionization from the excited levels of hydrogen (in particular, from the second level) are effective even in the H ii zone. With increasing ionization parameter their efficiency grows, and this leads to weakening of the specified dependence (Kwan & Krolik, 1981; Kwan, 1984). So, the emission-line response of an individual cloud can be nonlinear. This fact was recognized already in the first successful photoionization models. According to them, quanta in many lines are produced mainly not in the traditional H ii zone, but deeper, in the so-called “deep partly ionized zone”. Successful modelling of stationary optical emission-line spectra of AGN requires the following two circumstances to be taken into account (Kwan & Krolik, 1979): the power-law shape of the spectrum of ionizing continuum (i.e., the fact that the major fraction of ionizing quanta is in the X-ray part of spectrum) and the big column densities of the clouds emitting in lines. If the X-ray luminosity of the ionizing source is great enough in comparison with the UV one, the “deep partly ionized zone” is formed in the cloud. Taking into account the contribution of this zone increases the luminosity of the cloud in Balmer lines, whereas the luminosity in L$\alpha$ is stabilized at the level of the luminosity of the H ii zone. So, the collisional amplification of Balmer and Paschen lines takes place in the “deep partly ionized zone”. Inside this zone the excitation temperatures of these lines increase with optical depth, but ultimately attain some limiting values. The limiting values are insensitive to variation of the ionizing flux, because the Balmer and Paschen continua dominate in cooling at such depths (Kwan, 1984). According to Kwan & Krolik (1981), when collisional ionization becomes the main source of ionizations and cooling, the rate of cooling increases with increasing electron temperature approximately as $\exp(-32\times 10^{4}\,{\rm K}/T_{\mathrm{e}})$. In the standard model by Kwan & Krolik (1981), $T_{\mathrm{e}}\approx 8000$ K in “the deep zone”; the steep dependence of the rate of cooling on temperature, as Kwan and Krolik noted, provides only weak variation of $T_{\mathrm{e}}$ with depth and insensitivity of $T_{\mathrm{e}}$ to variation of the model parameters, in particular, the ionization parameter. Increasing the ionizing flux makes higher levels of hydrogen attain the limiting excitation temperatures; the luminosity of the “deep zone” in the relevant lines then ceases to react to changes of the ionizing flux, i.e., in this limit they are constant. In the Balmer series, the approach to the limiting temperatures affects first of all the H$\alpha$ line, then H$\beta$, and so on. Thus, according to the photoionization models (Kwan & Krolik, 1981; Kwan, 1984), the dependence of the cloud’s emission-line luminosity on the incident ionizing flux for the H$\alpha$ line is weaker than for H$\beta$, for H$\beta$ is weaker than for H$\gamma$, and so on. Due to the difference between H$\alpha$ and H$\beta$ in the value of the $s$ parameter, the Balmer decrement increases with increasing distance of clouds away from the central source, therefore a photographic BLR image (if such an image could be obtained) would be larger in H$\alpha$ than in H$\beta$. A formula for an effective BLR radius in a line with an arbitrary $s$ value in the homogeneous model of the cloud aggregate was deduced in (Shevchenko, 1985a). This effective stratification is explained by differences between emission lines in the degree of nonlinearity of the $L_{\mathrm{l}}(F_{\mathrm{i}})$ function. Shevchenko (1988) showed that within the framework of the homogeneous model of the cloud aggregate, if one takes into account the results of the photoionization calculations of the emission-line spectra of active galactic nuclei (Kwan, 1984; Mushotzky & Ferland, 1984), it is possible to explain the observed time lags and amplitudes of variations in major optical and ultraviolet (UV) emission lines in the spectrum of NGC 4151. After the first successes of the photoionization computations of the AGN emission-line spectra, significant progress was made in this field; see, e.g., reviews by Ferland (2003) and Leighly & Casebeer (2007). Multi-component models were proposed and studied (Collin-Souffrin et al., 1988; Collin-Souffrin & Lasota, 1988; Korista et al., 1997), which allowed to reproduce the relative fluxes in high-ionization and low-ionization lines simultaneously. Evidence was found for the presence of optically thin line-emitting gas (Ferland, Korista & Peterson, 1990; Shields, Ferland & Peterson, 1995). This progress promoted much deeper understanding of the AGN emission-line spectra — it turned out that the uniform models are too simple to reproduce the whole spectra. However, in what follows, our study concerns only Balmer lines. We aim to explain the time behaviour of the H$\alpha$ and H$\beta$ lines in the emission-line spectrum of NGC 4151, in the framework of a simple uniform one-component model. Of course, explaining the behaviour of the emission-line spectrum in total may require much more complicated multi-component models. Effective, not physical, stratification is present in our one-component model, due to the nonlinearity in each cloud’s line emission. The alternative to a homogeneous BLR with effective stratification is a physically stratified BLR. Investigating variability of the UV lines of NGC 4151, Ulrich et al. (1984) offered the BLR model consisting of three zones with different physical characteristics (see Table 2 in their article). Gaskell & Sparke (1986) proposed a model consisting of two zones (see Table 1 in their article). These models are not considered henceforth; we adopt the effective stratification picture as implied by the nonlinearity in cloud’s line emission. 3 The Reverberation Model Blandford & McKee (1982) offered a procedure to recover the BLR structure by analysis of line and continuum lightcurves. This is the so-called method of “reverberation mapping”. Its essence consists in the following: the observed light curve in a line is supposed to represent a convolution of two curves: the transfer function describing physical characteristics and the geometry of the BLR and the light curve in ionizing continuum. The emission-line luminosity of an individual cloud was supposed to depend linearly on the incident ionizing flux. Shevchenko (1984, 1985a) found necessary and sufficient conditions for the existence of a time lag of a maximum of an emission-line flare in relation to a (short duration) continuum flare when the BLR structure is isotropic with respect to the central source; these conditions are: the typical cloud should emit in the line mainly from the side facing the central source, and, either a central cavity should be effectively present in BLR, or the $s$ parameter in the formula $L_{\mathrm{l}}\propto F_{\mathrm{i}}^{s}$ should be less than one. These conditions set useful reference points for our modelling. In the 1990’s the effect of nonlinear response as well as of anisotropy of the individual cloud emission in specific BLR models, as applied to the cross-correlation analysis, was studied in detail by Sparke (1993) and O’Brien, Goad & Gondhalekar (1994, 1995). We adopt the homogeneous model of the cloud aggregate (Shevchenko, 1984, 1985a). The effective BLR radius $R$ is defined by screening of the peripheral part of the aggregate by the clouds situated closer to its centre: $R=(\sigma n)^{-1}$, where $\sigma$ [cm${}^{2}$] is the mean geometrical cloud section orthogonal to direction to the central source, $n$ [cm${}^{-3}$] is the cloud concentration (number of clouds in a unit volume). Generally, the BLR can contain a central cloud-free cavity of radius $R_{0}$. Let us remark that the Balmer quanta, unlike the ionizing quanta, can leave the BLR freely even at large cloud-covering factors of the “sky” of the central source, because the dispersion of the cloud velocities in the BLR is assumed to be great; the latter fact is testified by the large width of the observed emission lines. The model of a homogeneous (outside the central cavity) distribution of the clouds is equivalent, in what concerns the transfer function form, to a model with zero covering factor but with $n\propto e^{-r/R}$, an exponential decrease in the cloud concentration with increasing distance from the centre. In both interpretations, $R$ characterizes the BLR radius for all lines. In the first case, it is the radius of the “lit” zone in the homogeneous aggregate, and in the second case it is the $e$-folding scale of the cloud concentration. We assume that a typical BLR cloud represents a flat “pancake” emitting lines solely from the side facing the ionizing radiation source, and, what is more, emitting orthotropically. See discussion in (Shevchenko, 1985b) on the physical basis for this assumption. The planes of the clouds are either orthogonal to the direction to the central source, or are oriented randomly. The phase function, describing the phase angle dependence of the cloud’s line emission, is different in these two cases. For random orientation, the effective phase function (the phase function of a volume unit containing many clouds) coincides with the phase function of a spherical cloud, provided (Shevchenko, 1985b): the cloud is completely opaque in the line, the line quanta are produced at small optical depths, and the cloud surface emits in the line orthotropically. One should make a reservation that the pancake shaped cloud, as well as a uniform cloud aggregate model itself, is a physical idealization that can be used only as an approximation for the real arrangement of line-emitting material in the BLR. The real structure might be closer to a combination of a disk and an outflowing wind (Emmering, Blandford & Shlosman, 1992; Murray et al., 1995; Chiang & Murray, 1996; Bottorff et al., 1997; Elvis, 2000). We adopt the pancake shape for the BLR cloud exclusively for convenience in mathematical modelling: indeed, according to Shevchenko (1985b), the phase function of the pancake cloud with the plane orthogonal to the ionizing source direction provides the maximum anisotropy of line emission, i.e., this is a physical limit worth theoretical examination, while the phase function of a spherical cloud (or, equivalently, randomly oriented pancakes) gives an approximation for the phase function of randomly oriented optically thick line-emitting material, i.e., it describes a situation that is expected to be closer to reality. If the “pancakes” are orthogonal to the central source direction, the transfer function representing the dependence of the observed integrated emission-line flux $f(t)$ on time $t$ counted from the moment of the $\delta(t)$–flare of the central source in continuum, is as follows (Shevchenko, 1984, 1985a): $$f(t)\propto\left\{\begin{array}[]{cl}0,&0\leq t\leq{\displaystyle R_{0}},\\ &\\ {\displaystyle R^{-1}\int\limits_{R_{0}}^{t}}g(r,t)\,dr,&{\displaystyle R_{0}% \leq t\leq 2R_{0}},\\ &\\ {\displaystyle R^{-1}\int\limits_{t/2}^{t}}g(r,t)\,dr,&t\geq{\displaystyle 2R_% {0}},\end{array}\right.$$ (1) where $$g(r,t)=\left(\frac{t}{r}-1\right)r^{1-2s}e^{-r/R},$$ and $r$, $R$, $R_{0}$ are measured in the light-travel time units. In the case when the planes of clouds are oriented randomly, their mean phase function coincides with the phase function of a spherical cloud. This function is as follows (Shevchenko, 1985b): $$j(\theta)\propto(1+\cos\theta)\left(1+\frac{s}{2}\cos\theta\right),$$ (2) where $\theta$ is the “ionizing source – cloud – observer” angle, $0\leq\theta\leq\pi$, $0\leq s\leq 2$. General formula (3) in (Shevchenko, 1984) for the transfer function, after substitution of phase function (2), becomes: $$f(t)\propto\left\{\begin{array}[]{cl}{\displaystyle\frac{t}{R}\int\limits_{R_{% 0}}^{\infty}}g(r,t)\,dr,&0\leq t\leq{\displaystyle 2R_{0}},\\ &\\ {\displaystyle\frac{t}{R}\int\limits_{t/2}^{\infty}}g(r,t)\,dr,&t\geq{% \displaystyle 2R_{0}},\end{array}\right.$$ (3) where $$g(r,t)=\left(1+{\displaystyle\frac{s}{2}\left(\frac{t}{r}-1\right)}\right)r^{-% 2s}e^{-r/R}\>.$$ Transfer functions (1) and (3) can be expressed through incomplete $\gamma$ functions. The behaviour of the transfer functions with different values of the $s$ parameter (while $R_{0}=0$, $R=15$ lt-days) is demonstrated in Fig. 1. The qualitative difference in the behaviour of the functions with $s$ less and greater than unity is clearly seen. In particular, $f(t)$ peaks at $t=0$ for $s\geq 1$ and at $t>0$ for $s<1$. The model emission-line light curve is determined by the convolution formula: $$F_{\mathrm{l}}(t)=a\int_{0}^{\infty}f(\tau)F_{\mathrm{c}}^{s}(t-\tau)d\tau\>,$$ (4) where $F_{\mathrm{l}}$ is the integrated flux in the line, $F_{\mathrm{c}}$ is the observed flux in continuum, $a$ is the normalizing dimensional factor. Since it is the optical continuum, not the ionizing one, that is observed, we use an assumption, formulated below, on a relation between the continua. A difference of expression (4) from those usually used (valid in the case of the linear “$L_{\mathrm{l}}$–$F_{\mathrm{i}}$” relation; see, e.g., Blandford & McKee 1982 and Horne et al. 2004) consists in raising of $F_{\mathrm{c}}$ to the power $s$. Let us remark that, according to (4), on taking $f(\tau)$ in the form of $\delta$-function, one gets $F_{\mathrm{l}}(t)\propto F_{\mathrm{c}}^{s}(t)$, i.e., the dependence for the case of quasi-stationary spectrum; see (Shevchenko, 1988). 4 The time lag and the cross-correlation method Techniques for cross-correlation analysis of AGN emission-line variability have demonstrated remarkable progress during the last decade. The methods of calculation of the basic properties of the cross-correlation function (CCF), namely, the lags of the CCF peak and CCF centroid and their uncertainties, were greatly improved (e.g., White & Peterson 1994; Peterson et al. 1995, 1998; Welsh 1999). In particular, it was realized that the CCF peaks and centroids underestimate the BLR size (Pérez et al., 1992; Welsh, 1999), and that taking into account the continuum variability time scale is important for correct estimation of the BLR size (e.g., Edelson & Krolik 1988). Let us consider the time lag as determined by means of cross-correlation analysis in the case of nonlinear emission-line response of an individual cloud. In this Section, we measure the time lag in model numerical experiments and study the dependence of the time lag on the parameters of a model transfer function and duration of the continuum flares. We consider the case of a single flare of various durations. As the model transfer function we take Eq. (1) corresponding to the case of the “pancake” clouds orthogonal to the central source direction. The central cavity in the BLR is set to be absent: $R_{0}=0$. The model light curve in the continuum is assumed to have the form of the bell-like function $F_{\mathrm{c}}(t)=\mbox{sech}((t-t_{0})/T)$, where $t_{0}=50$ d and $T$ is effective duration of the flare, $t$ is time in days. The model emission-line light curves are computed on the time interval of 500 d with the step of $0.05$ d. In the upper part of Fig. 2, the model light curve in the continuum ($T=1$ d) and the computed emission-line light curve are presented. The latter curve has been obtained by means of convolution of the light curve in the continuum and the transfer function (1) with $R=15$ lt-days, $R_{0}=0$, $s=1$. In the lower part of Fig. 2, the normalized cross-correlation function of these curves is plotted. The shift of the peak of the cross-correlation function is clearly visible; as determined numerically, $\Delta t_{\mathrm{peak}}\simeq 3.8$ d. Note that the exponential-like decay of the resulting curves in both plots reflects the radial structure of the line-emitting region, and the rise in the CCF reflects the shape of the continuum flare. One may argue that the linear response model is just a linearization of the nonlinear response model, and that any BLR radius estimate made in the linear response model is therefore an approximation that might be not far from reality. However, one should take into account, firstly, that the “equal time-travel” paraboloidal surface inside the BLR covers a whole range of distances from the ionizing source just after the ionizing flare, secondly, that with increasing time after the flare this surface retreats from the source to larger distances. The slopes of the linearized dependences for individual clouds on the surface vary significantly in both space and time. The response slopes might be averaged on the surface, but then the change of the averaged slope with time should be taken into account; the latter is never done in practice. So, it is not surprising that the nonlinear response model, as compared to the linear one, can give very different quantitative results on the BLR radius. This directly follows from the qualitative differences in the response function for different values of the $s$ parameter, as seen in Fig. 1. A vivid manifestation of the insufficiency of the linearized response model is that increasing the time lag value in the nonlinear response model can be achieved either by increasing the BLR radius or by specific increasing the response nonlinearity, namely, by decreasing the $s$ parameter value in relation to unity (see relation (5)), while in the linear response model only the BLR radius can be varied. Shevchenko (1985a, 1994) obtained an approximate theoretical relation of the time lag of the maximum of the emission-line light curve to the $s$ parameter in the homogeneous model of the cloud aggregate with or without a central cavity ($R_{0}\geq 0$). The continuum flare was described by a $\delta$ function. This relation is as follows: $$\Delta t=\left\{\begin{array}[]{ll}W(1-s)R,&{0\leq s\leq{1-2R_{0}/(WR)},}\\ 2R_{0},&{s\geq{1-2R_{0}/(WR)},}\end{array}\right.$$ (5) where the constant $W$ depends on the choice of phase function; $W=3.19$ in the considered case of clouds with regular orientation. In the case of phase function (2) one has $W=2$. By examining the shifts of peaks of cross-correlation functions at various values of parameters we can assess how well relation (5) works when the ionizing flare has a finite duration. Consider first how the time lag varies with $s$ at fixed $R$ (Fig. 3). The time lag, as defined here, is the value of the distance along the time axis from $t=0$ up to the first maximum of the cross-correlation function; i.e., it is $\Delta t_{\mathrm{peak}}$. We do not examine the shift of the centroid of the cross-correlation function here. The parameter $s$ is varied from $0.1$ to $2.0$ with the step of $0.01$. The ionizing flare duration is fixed at $T=1$ d. The curves for $R=5$, 15 and 25 lt-days are plotted. Theoretical dependences (5) for $R=5$, 15 and 25 lt-days and $R_{0}=0$ are plotted as straight line segments. It is clear that the theoretical and numerical estimates of the time lag at such a relatively small duration of the continuum flare are in a good agreement. Now we consider the dependence of the time lag on the BLR radius $R$ (Fig. 4). The ionizing flare duration is the same, $T=1$ d. The dependences for $s=0.5$ and $s=1$ are plotted. We see that the linear character of theoretical relation (5), valid for the $\delta(t)$ ionizing flare, is preserved in the case of $s=0.5$. In what concerns the $s=1$ case, it is completely different. From the viewpoint of relation (5), this case is degenerate, and the predicted value of $\Delta t_{\mathrm{peak}}$ is constant (zero). In reality we observe the “$\Delta t_{\mathrm{peak}}$–$R$” relation similar to a logarithmic one. So, as follows from Fig. 3 and Fig. 4, for $s\geq 1$ the CCF $\Delta t_{\mathrm{peak}}$ value depends only weakly on $R$. The cross-correlation peak time lag for $s\geq 1$ is small in comparison with $R$ expressed in the light-travel time units. The difference can reach an order of magnitude. For $s=1$, a similar phenomenon was observed by Pérez et al. (1992) for the CCF centroid estimates of the BLR size in an isotropic (with respect to the ionizing source) model of the line-emitting cloud distribution. They found that such estimates can be less than the real size of the BLR, the difference reaching two times. This is in accord with our findings. What is the role of the ionizing flare duration in the degenerate case $s=1$? The dependences of the time lag on $R$ in this case for various fixed values of $T$ are plotted in Fig. 5. We see that this role is far greater than that of the BLR radius. The observed “$\Delta t_{\mathrm{peak}}$–$R$” dependences seem to be logarithmic indeed; one can verify that, unlike the rational and power-law functions, the functions of the form $a+b\ln(R+c)$, where $a$, $b$, $c$ are fitting parameters, provide ideal visual description of the observed curves. Note that further work is required to explore how the logarithmic dependence found here is vulnerable to the choice of the ionizing flare shape. In summary, according to our numerical-experimental findings, the BLR radius is only weakly related to the measured $\Delta t_{\mathrm{peak}}$ value in the mathematically degenerate but observationally most common case of $s=1$. The role of the timescale of variability is far greater. Therefore, the lines with $s\approx 1$ are of little help in determining the BLR size by means of cross-correlation techniques; instead, the lines with $s$ essentially less than 1, such as H$\alpha$, should be used for this purpose. This conclusion has been obtained in a model framework and thus may be model-dependent in some way. However, it makes clear that there are no general theoretical grounds to believe that $\Delta t_{\mathrm{peak}}$ is mostly determined by the BLR size. Note that solely the CCF peak offset has been examined. The CCF centroid offset should be examined as well in a future study to check whether it exhibits the same behaviour. 5 Reverberation modelling of the emission-line light curves of NGC 4151 In this Section we examine the effect of taking into account the nonlinearity in a cloud’s line emission in practical modelling of emission-line variability of an AGN. We model the emission-line variability of the Seyfert galaxy NGC 4151. The observational data of Kaspi et al. (1996) on variability of the nucleus of this galaxy in H$\alpha$, H$\beta$ and optical continuum is used. The observations of Kaspi et al. (1996), performed in the framework the AGN Watch Programme, cover the time interval of approximately three months, from 1993 November until 1994 February. While we use the light curve continuum data for the whole time span of the observations, the data on the fluxes in the lines during the first five days and during the last five days of the observational time span are excluded, following the usual practice of eliminating the border effects (see Maoz et al. (1991)). The error bars of the individual observations, defining the weights of the observations, are taken into account in the modelling. We optimize the model parameters by using a non-linear least-squares method (Levenberg, 1944; Marquardt, 1963) to minimize $\chi^{2}$, thereby finding best-fit parameter values and their standard errors (see Press et al. (1997)). We find that the iterations converge to the same solution for random starting values in ranges specified below, with the deepest minimum of $\chi^{2}$ in all cases, though several other local minima exist, as demonstrated below. We expand the standard errors by the square root of the reduced $\chi^{2}$, because the best-fit reduced $\chi^{2}$ is greater than unity in all best-fit models found. As the transfer functions, we use Eqs. (1) and (3). The best approximation of the computed model light curve to the observed one is found by means of the modelling. We vary six parameters: the BLR radius $R$, the nonlinearity parameters $s_{\mathrm{opt}}$ for H$\alpha$ and H$\beta$, the normalizing coefficients $a$ for H$\alpha$ and H$\beta$, the flux $F_{\alpha\mathrm{n}}$ in the narrow component of H$\alpha$ (on the $F_{\beta\mathrm{n}}$ value for H$\beta$ see below). The radius $R_{0}$ of the central cavity (a zone free from line-emitting clouds) has not been varied because it is already known to be most probably small. A comparison of the known values of time lags in different Balmer lines of NGC 4151 allowed Shevchenko (1985a) to conclude that the upper bound of the cavity radius $R_{0}$ is 4–5 times less than the effective BLR radius. The deduction that the central cavity is small is in agreement with conclusions by Maoz et al. (1991) and Xue & Cheng (1998). Note that the radius of the accretion disc in the centre of the nuclear region is estimated to be equal to $0.6$–2 lt-days (Lyutyi, 2005; Sergeev et al., 2005, 2006). Taking into account the uncertainty of this estimate and the uncertainty of the $R_{0}$ estimates, the probable existence of this component of the nuclear region does not at all contradict the conclusions on the small relative size of the central cavity in the BLR. For the continuum flux we take the flux at the wavelengths of 4560–4640Å (the “4600Å” region), because this region corresponds to the shortest wavelengths at which Kaspi et al. (1996) measured the continuum flux. The series of the observed values of the continuum flux are recalculated for presentation on the uniform time grid by means of cubic spline interpolation. The constant contribution to the integrated flux in H$\alpha$ and H$\beta$ due to the narrow components of the lines has been taken into account in the following way. We set $F_{\alpha}(t)=F_{\mathrm{c}}^{s_{\alpha}}(t)+F_{\alpha\mathrm{n}}$ and $F_{\beta}(t)=F_{\mathrm{c}}^{s_{\beta}}(t)+F_{\beta\mathrm{n}}$. The contributions of the narrow components $F_{\alpha\mathrm{n}}$ and $F_{\beta\mathrm{n}}$ are connected to each other via the constant Balmer decrement $D_{\mathrm{n}}=F_{\alpha\mathrm{n}}/F_{\beta\mathrm{n}}$. Therefore in the course of searching for the best model it is enough to vary the value of $F_{\alpha\mathrm{n}}$; the value of $F_{\beta\mathrm{n}}$ is determined via the Balmer decrement. For each of the considered cases of orientation of the planes of clouds we have accomplished the modelling twice, namely, for the two reported values of the Balmer decrement in the narrow components: $D_{\mathrm{n}}=4.47$ as given in Table 1 in (Ferland & Mushotzky, 1982) and $D_{\mathrm{n}}=7.55$ as given in Table 1 in (Sergeev et al., 2001). The difference in the observed values of the decrement $D_{\mathrm{n}}$ may reflect either the difficulty in its evaluation or its probable long-term variability. The contribution of the stellar component to the observed continuum flux has been taken into account by its subtraction from the observed flux prior to modelling. According to Peterson et al. (1995) and Kaspi et al. (1996), the stellar component contribution at the wavelength of 4600Å and the aperture used at their observations is approximately equal to $2.2\times 10^{-14}\mbox{ erg cm${}^{-2}$ s${}^{-1}$ \AA${}^{-1}$}$. As it is known from observations (see, e.g., Crenshaw et al. 1996; Peterson et al. 2002), the light curves of an active galactic nucleus in the optical and UV continua can be rather different, and the relation between the continua is nonlinear. For the Seyfert galaxy NGC 5548, most studied in this respect, the slow components of variability in the optical and UV continua are connected by the power law $F_{\mathrm{opt}}\propto F_{\mathrm{UV}}^{\gamma}$, where $\gamma\approx{0.56}$ (Peterson et al., 2002). Basing on this relation, we find the real values of the $s$ parameter from the values obtained in our modelling of the optical light curves (we designate these values by $s_{\mathrm{opt}}$) by means of the formula $s=\gamma s_{\mathrm{opt}}$, where we set $\gamma=0.6$. The value of $R$ does not depend on the line choice. The values of $s_{\mathrm{opt}}$ for different lines are generally different, and the same is true for $a$. The initial data for the iterations of the Levenberg–Marquardt algorithm have been taken randomly in the following limits: $R$ — from 1 to 30 lt-days; $s_{\mathrm{opt}}$ in the both lines — from $0.2$ to $2.0$; $a$ — from $0.01$ to $2.0$; $F_{\alpha\mathrm{n}}$ — from 0 to $30\times 10^{-12}\mbox{ erg cm${}^{-2}$ s${}^{-1}$}$. In the case when the planes of clouds are orthogonal to the direction to the central source, the best-fit model light curves are presented in Fig. 6 for two different values of $D_{\mathrm{n}}$. The circles designate the observed values of the flux in optical continuum and the observed integrated emission-line fluxes according to the data in (Kaspi et al. 1996, Table 2). In both parts of the Figure, the continuous curve in the upper plot is the spline interpolation of the continuum flux. The flux is given in the units of $10^{-14}\mbox{ erg cm${}^{-2}$ s${}^{-1}$ \AA${}^{-1}$}$. In the lower two plots, the best model light curves in H$\alpha$ and H$\beta$ are presented as continuous curves; here the integrated emission-line flux is given in the units of $10^{-12}\mbox{ erg cm${}^{-2}$ s${}^{-1}$}$. In Fig. 7, the best model emission-line light curves are presented for the case when the planes of clouds are oriented randomly. To demonstrate the effect of variation of different parameters in determining the best-fit models, $\chi^{2}/N$ (where $N$ is the number of degrees of freedom) is shown in Figs. 8, 9 and 10 in dependence on the main parameters. Fig. 8 gives the dependence of the reduced $\chi^{2}$ on the BLR radius $R$ with all other parameters assigned to their best-fit values. Figs. 9 and 10 show the reduced $\chi^{2}$ in dependence on the $s$ parameters for H$\alpha$ and H$\beta$ with all other parameters assigned to their best-fit values. (Note that these graphs are presented here for illustrative purposes solely; for quantitative analysis, when offsetting one parameter, one should re-optimize all the other parameters while holding the one parameter fixed at the offset value.) In the presented graphs one can see that the $\chi^{2}$ dependence on $R$ is characterized by a single well-defined minimum. The dependences of $\chi^{2}$ on the $s$ parameters have four minima each. The Levenberg–Marquardt algorithm finds the deepest one. In Tables 1 and 2, the obtained values of the model parameters, corresponding to the model emission-line light curves in Fig. 6 and Fig. 7, are presented together with the reduced $\chi^{2}$ values of the models. In total, the results of our modelling give the following values of the radius of the BLR of NGC 4151: $R=11$–14 lt-days, with the uncertainty of 5–8 lt-days. At $D_{\mathrm{n}}=4.47$, the best-fit values of the $s$ parameter are: $s\approx 0.6$ for H$\alpha$ and $s\approx 0.85$ for H$\beta$. From comparison of Tables 1 and 2 one can see that at increasing decrement $D_{\mathrm{n}}$ from $4.47$ up to $7.55$ the difference in the computed values of the $s$ parameter for H$\alpha$ and H$\beta$ becomes less, though does not seem to disappear completely. Mushotzky & Ferland (1984) elaborated photoionization models of stationary AGN optical spectra. In the framework of the photoionization modelling they carried out calculations, in our equivalent terms representing the calculations of the functions $L_{\mathrm{l}}(F_{\mathrm{i}})$ for the BLR clouds. According to the results of their calculations, $s\approx 0.6$ for H$\alpha$, and $s\approx 0.8$ for H$\beta$. Thus, there exists a satisfactory agreement of the results of our modelling with the data of Mushotzky & Ferland (1984), especially in the case of $D_{\mathrm{n}}=4.47$. Let us remark that the values of $s$ can be nonconstant inside the BLR, varying from cloud to cloud, because they depend on physical characteristics of the clouds. As a result of our modelling we obtain some “effective” values of $s$. The model of a homogeneous distribution of clouds implies that the covering factor is close to one, but, as it has been noted above, an interpretation of the adopted model is possible as a model with an exponential decrease of the cloud concentration with distance away from the centre; then the covering factor can be small. 6 Discussion There are several ways of estimating the BLR size. Besides the reverberation and cross-correlation methods, discussed above, there exists a technique based on estimating the ionization parameter from modelling the stationary emission-line spectra of AGN. Using this technique, Mushotzky & Ferland (1984) obtained an estimate of the radius of the BLR of NGC 4151, equal to approximately 16 lt-days. By a similar argument, Cassidy & Raine (1997) found the inner and outer radii equal to 6 and 40 lt-days in their theoretical model of the BLR of this galaxy. At the same time when the photoionization estimate was made by Mushotzky & Ferland (1984), the reverberation estimate $R\simeq 15$ lt-days was obtained independently by Shevchenko (1984), in agreement with the photoionization estimate by Mushotzky & Ferland (1984). This reverberation estimation was performed within the framework of the model of a homogeneous isotropic distribution of line-emitting matter around the central ionizing source, on the basis of the observational data of Lyutyi & Cherepashchuk (1971) and Cherepashchuk & Lyutyi (1973) on the time lags in the H$\alpha$ line variations. Cross-correlation estimates are usually less than the “photoionization” values. Cross-correlation analysis by Peterson & Cota (1988) (see also discussion by Peterson 1988), accomplished on the basis of their own observational data and the data of Antonucci & Cohen (1983) on variability in the lines H$\beta$ and He ii $\lambda 4686$, gave $\sim 6$ lt-days as the estimate for the radius of the BLR of NGC 4151. Similar cross-correlation estimates of the BLR size were recovered by Clavel et al. (1990) on the basis of the IUE (International Ultraviolet Explorer) data on variability of the major UV lines: $R=4\pm 3$ lt-days. These values correspond to the peak CCF time lags; the centroid ones are greater by about two days. Wandel et al. (1999) find similar centroid CCF time lags, $4\pm 3$ d, for the H$\beta$ line. Clavel et al. (1990) note that their cross-correlation estimates of the BLR size for NGC 4151 are an order of magnitude less than the typical “photoionization” estimates for Seyfert galaxies. By means of cross-correlation analysis of their own data, Kaspi et al. (1996) found that the time lag of variations in the H$\alpha$ and H$\beta$ lines in relation to continuum is 0–3 d; thus the cross-correlation estimate of the BLR radius is 0–3 lt-days. According to the modern analysis of these data accomplished by Metzroth et al. (2006) and Bentz et al. (2006), the value of the cross-correlation time lag for the data of Kaspi et al. (1996) has no clear-cut statistical bounds. In total, the cross-correlation estimates of the BLR radius of NGC 4151 are all in the range of 0–6 lt-days. Direct reverberation modelling, in comparison with the cross-correlation analysis, give very different values of $R$ similar to the given above “photoionization” estimates. According to the results of Maoz et al. (1991), who carried out reverberation modelling of light curves of NGC 4151 in H$\alpha$ and H$\beta$, the weighted-mean (by the local emission-line luminosity of clouds) BLR radius $\approx 16$–18 lt-days for the best found model, and the central cavity radius $R_{0}\approx 2$ lt-days. The linear character of the $L_{\mathrm{l}}(F_{\mathrm{i}})$ dependence was assumed, as in practically all modern research on this subject. Xue & Cheng (1998) numerically recovered the BLR transfer functions on the basis of the data of Maoz et al. (1991) and Kaspi et al. (1996). They obtained the following estimates: $R\approx 10$ lt-days, $R_{0}\leq 1$ lt-day. As mentioned above, the reverberation estimate $R\simeq 15$ lt-days was obtained in (Shevchenko, 1984). All these reverberation estimates are in agreement with our reverberation modelling results presented in Tables 1 and 2. So, the known reverberation estimates of the BLR size of NGC 4151 are in agreement with “photoionization” estimates, and they all are much greater than the cross-correlation estimates. The strong difference between the BLR radii found by reverberation modelling, on one side, and its estimates following from cross-correlation analysis, on the other side, (10–18 lt-days versus 0–6 lt-days) underlines the conditional character of the cross-correlation estimates. Such a difference is no surprise: the size identified as the value of the observed time lag can be much (an order of magnitude) less than the true size of the BLR in lt-days (Section 4). For example, if the cloud aggregate is uniform, the time lag of variation of a line with $s\approx 1$ with respect to an ionizing flare is small compared to the BLR radius $R$ in light travel time units, and depends on $R$ only weakly. The ultimate cause of this phenomenon is the degeneracy of relation (5) at $s\geq 1$. This degeneracy means that in practice there are no rigourous theoretical grounds to believe that the $\Delta t_{\mathrm{peak}}$ value is mostly determined by the BLR size, if $\Delta t_{\mathrm{peak}}$ is calculated for a typical line (i.e., a line with $s\approx 1$). However, cross-correlation analysis by Kaspi et al. (1996) of their observational data indicated that the cross-correlation time lag was small not only for H$\beta$ (the line with $s\approx 1$ presumably), but for H$\alpha$ as well (the line with $s$ definitely less than one). To clarify this point, we have examined cross-correlations between the splined curve in continuum and our theoretical model light curves in H$\alpha$ (presented in Figs. 6 and 7; linear trends have been subtracted prior to the analysis). The analysis has shown that the cross-correlation time lag is 0–3 d, practically equal to the cross-correlation time lag found by Kaspi et al. (1996) for the observed emission-line light curves; the computed cross-correlation functions are shown in Fig. 11. So, substitution of the observed emission-line light curve by a theoretical one calculated for a definite value of $R$ (equal to 11–14 lt-days) in the transfer function does not change the cross-correlation estimate of $R$ (0–3 lt-days). It is much less than the value adopted as the parameter of the transfer function. This means that the cross-correlation method may provide inadequate results not only at $s\approx 1$, but at $s<1$ as well. The cause is not all-together clear, but one may speculate that it is related to the strong dependence of $\Delta t_{\mathrm{peak}}$ on the variability time scale (Fig. 5) and/or to the small amplitudes of variability in this particular set of observational data. Let us underline that, contrary to the cross-correlation analysis, the reverberation modelling turns out to be immune, as we have seen, to these unfavourable conditions, and provides adequate values of $R$. A hypothesis on possible variability of the BLR size of NGC 4151 was put forward by Kaspi et al. (1996), Peterson et al. (2002) and other researchers on the basis of cross-correlation analysis of optical spectral variability data at different time intervals of observations. Our reverberation modelling of the H$\alpha$ light curve data of Kaspi et al. (1996) gives the value of the BLR radius matching the majority of the BLR size estimates of other authors. This removes necessity in any special physical interpretation of the small value of the cross-correlation time lag in H$\alpha$ for these light curve data. In particular, the hypothesis by Kaspi et al. (1996) that the physical size of the BLR at the moment of their observations was an order of magnitude less than usually is not necessitated. 7 Conclusions We have studied how the nonlinearity in the “$L_{\mathrm{l}}$–$F_{\mathrm{i}}$” relation (the emission-line luminosity, $L_{\mathrm{l}}$, of the BLR cloud in dependence on the ionizing continuum flux, $F_{\mathrm{i}}$, incident on the cloud) can be taken into account in estimating the size of the BLR in active galactic nuclei by means of “reverberation” methods. We have shown that the BLR size estimates obtained by cross-correlation peaks of emission-line and continuum light curves can be much (up to an order of magnitude) less than the values obtained by reverberation modelling. This has been demonstrated by means of abstract representative numerical cross-correlation and reverberation experiments with model continuum flares and emission-line transfer functions and by means of practical reverberation modelling of the observed emission-line variability of NGC 4151. The modelling of the observed light curves of NGC 4151 in H$\alpha$ and H$\beta$ has been accomplished on the basis of the observational data by Kaspi et al. (1996) and the theoretical broad-line region model by Shevchenko (1984, 1985a). In the abstract representative numerical cross-correlation and reverberation experiments with model continuum flares and emission-line transfer functions, we have found that the value of the cross-correlation peak time lag $\Delta t_{\mathrm{peak}}$ for $s\geq 1$ is small in comparison with the BLR size $R$ expressed in the light-travel time units and depends on $R$ only weakly. We have shown that in the case of $s\geq 1$ the effect of the ionizing flare duration on the $\Delta t_{\mathrm{peak}}$ value is far greater than that of the BLR radius. In other words, the BLR radius has little effect on the measured value of the $\Delta t_{\mathrm{peak}}$ value in the mathematically degenerate but observationally most common case of $s=1$; the role of the timescale of variability is far greater. Therefore, the lines with $s\approx 1$ seem to be of little help in determining the size of the BLR by means of estimating the cross-correlation peak time lag. The presence of a noticeable time lag of variations of NGC 4151 in H$\alpha$ (Lyutyi & Cherepashchuk, 1971; Cherepashchuk & Lyutyi, 1973) and significantly shorter time lags in other Balmer lines (Antonucci & Cohen, 1983) with respect to variations in the optical continuum has been attributed, in agreement with conclusions by Shevchenko (1984, 1985a), to the effect of essential nonlinearity in the “$L_{\mathrm{l}}$–$F_{\mathrm{i}}$” relation for H$\alpha$. The low value of the power-law index, $s\approx 0.6$, distinguishes this line from the other Balmer lines. The values of the model parameters of the BLR of NGC 4151 have been estimated. In particular, estimates of the BLR radius have been made. Our reverberation modelling of the emission-line variability based on the observational data by Kaspi et al. (1996) gives values of the BLR radius agreeing with the majority of its known “reverberation” and “photoionization” estimates. Much smaller $R$ values obtained by means of the cross-correlation method have been shown to be an artifact of this method. The hypothesis by Kaspi et al. (1996) that the size of the BLR of NGC 4151 at the time interval of their observations were an order of magnitude less than usually is not necessitated. Concluding, a power-law emission-line response model and simple spherically symmetric thick geometries of the BLR cloud distribution, taken here as a basis for modelling the emission-line AGN variability, gives the size of the BLR of NGC 4151 equal to 11–14 lt-days, with the uncertainty of 5–8 lt-days. This agrees satisfactorily with BLR size estimates in photoionization models fitting emission line strengths in the mean spectrum of NGC 4151. Much shorter time lags found in the cross-correlation analysis of the emission-line and continuum light curves of NGC 4151 correspond to the size of a smaller emission-line region that is reverberating. 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The use of Generalised Functions and Distributions in General Relativity R Steinbauer†and J A Vickers‡ †Department of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Wien, Austria. Email: roland.steinbauer@univie.ac.at   ‡School of Mathematics, University of Southampton, Southampton, SO17 1BJ, UK. Email: J.A.Vickers@maths.soton.ac.uk Abstract In this paper we review the extent to which one can use classical distribution theory in describing solutions of Einstein’s equations. We show that there are a number of physically interesting cases which cannot be treated using distribution theory but require a more general concept. We describe a mathematical theory of nonlinear generalised functions based on Colombeau algebras and show how this may be applied in general relativity. We end by discussing the concept of singularity in general relativity and show that certain solutions with weak singularities may be regarded as distributional solutions of Einstein’s equations. type: Topical Review pacs: 0420 -q ††: Class. Quant. Grav. 1 Introduction Idealisations play an important role in modelling a wide range of physical situations. In many field theories a particularly useful idealisation is to replace an extended source which is concentrated in a small region of space by a point charge. Such an approximation is physically reasonable provided the internal structure of the source can be neglected. In a similar way sources concentrated near a line or a surface can be described in terms of strings or shells of matter. On trying to describe this idealisation mathematically it is natural to use a delta function to describe the source, and hence in this context distributions arise in a natural way. Distributions are also used to describe a number of other important physical scenarios such as the description of shock waves and the junction conditions between matter and vacuum regions. In the case of linear field theories such as electrodynamics, distribution theory in fact furnishes a consistent framework which has the following two important features. Firstly, since the Maxwell equations are linear with respect to both sources and fields, one can allow distributional solutions of the field equations as well as classical smooth solutions. Secondly it is guaranteed that smooth charge densities that are close to those of a point charge (in the sense of distributional convergence) produce fields that are close to the Coulomb field. While the first property permits one to have a mathematically sound formulation of concentrated sources it is precisely the latter notion of “limit consistency” which renders the idealisation physically reasonable. One would like to have a similar mathematical description of concentrated sources in the theory of general relativity. However, general relativity is different from other field theories in two important respects. Firstly the Einstein field equations are nonlinear, so that one cannot simply pass from smooth solutions of the field equations to weak solutions as one can in a linear theory. More precisely the curvature tensor is a nonlinear function of the metric and its first two derivatives. Thus the metric needs to be at least $C^{2}$ to guarantee that the curvature is continuous. Mathematically one can weaken this condition to allow a $C^{2-}$ (i.e. first derivative Lipschitz continuous) metric and this is sufficient for most results in differential geometry to remain valid. In certain special situations one can lower the differentiability requirements still further and formulate the field equations in a way which avoids ill-defined products of distributions. For example it is possible to describe shells of matter [36], [15] and gravitational radiation [39] within the context of classical distribution theory. Using the standard definition of the curvature $$R^{a}_{\phantom{a}bcd}=\Gamma^{a}_{db,c}-\Gamma^{a}_{cb,d}+\Gamma^{a}_{cf}% \Gamma^{f}_{db}-\Gamma^{a}_{df}\Gamma^{f}_{cb}$$ (1) one sees that one wants the connection to have a (weak) derivative and be locally square integrable in order for the left hand side to make sense as a distribution. This led Geroch and Traschen [26] to introduce a class of “regular metrics” for which the components of the curvature are well defined as a distribution. Geroch and Traschen went on to show that such regular metrics can only have curvature with singular support on a submanifold of co-dimension at most one. Thus for a 4-dimensional spacetime a metric representing a shell of matter could belong to this class but a string or particle could not. We will consider such distributional solutions of Einstein’s equations in more detail in §2. The second important respect in which general relativity is different from other field theories is that one does not have a fixed background metric, but instead the geometry is determined by the field equations. Spacetime is described by a manifold together with a Lorentz metric which is assumed to be sufficiently differentiable for Einstein’s equations to be defined. One then detects the presence of singularities by showing that the spacetime is incomplete in some sense. The problem with this approach is that many physically reasonable spacetimes contain singularities according to this definition. For example solutions to Einstein’s equations representing cosmic strings are singular. What one would like to do is to lower the differentiability required of the metric to permit a wide class of “distributional geometries” which represent physically reasonable solutions and are also mathematically tractable. Owing to the nonlinear nature of Einstein’s equations such distributional geometries will in general require some nonlinear theory of generalised functions. Moreover, as pointed out by Isham [35] distributional solutions are not only important in allowing one to deal with concentrated sources or describing weak singularities but (based on our experience with linear field theories) should also be included in any path integral formulation of quantum gravity. Although there has long been a desire to allow distributional geometries in general relativity it is only comparatively recently that any real progress has been made in realising this aim. Many of the early attempts (e.g. [68], [75], [4], [5]) used methods which were specifically adapted to the particular problem being considered and whose general applicability was uncertain. More recently a number of different authors (for an overview see [92], [28, Ch. 5.2]) have investigated distributional geometries using an approach based on Colombeau algebras. These were developed by J.F. Colombeau in the 1980’s and contain the smooth functions as a subalgebra and the distributions as a linear subspace. The key idea of this approach is regularisation through smoothing and the use of asymptotic estimates with respect to a regularisation parameter. In particular, it provides a mathematically consistent way of multiplying distributions and a unified view on calculations involving various regularisation procedures. An important feature of these algebras is the notion of association which gives a correspondence between elements of the algebra and distributions. This allows one to use the power of the algebras to do mathematical calculations but then use the concept of association to interpret the final result in terms of classical distributions and give the solutions a physical interpretation. In §2 we review the extent to which it is possible to incorporate nonlinear operations into classical distribution theory. We will show that some very limited operations are permitted and that some apparently reasonable “multiplication rules” lead to inconsistencies. We then consider the implications of this for distributional solutions of Einstein’s equations and end the section by looking in detail at the properties of the Geroch-Traschen regular metrics. In §3 we give a brief review of Colombeau theory and explain how it is able to circumvent the result of Laurent Schwartz on the impossibility of multiplying distributions. Historically one of the first singular solutions of Einstein’s equations to be studied from a distributional viewpoint was the Schwarzschild solution which was considered by Parker in 1979 [68]. The key observation is that when written in Kerr-Schild coordinates the coefficients of the metric are locally integrable. The Schwarzschild and Kerr solutions have been studied in these coordinates by Balasin and Nachbagauer in a number of papers and in §4 we describe their work using the language of the (special) Colombeau algebra. If one boosts the Schwarzschild solution with velocity $v$ then one obtains, in the limit that the velocity is that of light, the “ultrarelativistic Schwarzschild solution”. This metric was investigated by Aichelburg and Sexl [1] who showed that it could be thought of as a gravitational shock wave. When one does a similar calculation with the ultrarelativistic limit of the Reissner-Nordstrøm solution one obtains a solution with vanishing electromagnetic field but $\delta$-function energy density. In §5 we show how Steinbauer was able to explain this physically surprising result using Colombeau algebras. In §6 we consider geodesic equations for impulsive gravitational wave spacetimes and show how Colombeau algebras provide an appropriate way of obtaining the results expected on physical ground without the need to make use of ad hoc “rules” for the multiplication of distributions. The study of conical singularities is another area where various authors had used a variety of regularisation procedures to obtain the physically plausible result that a cone has $\delta$-function curvature at the vertex. However this result appeared to be at odds with the results of Geroch and Traschen. In §7 we review the analysis and resolution of the problem by Clarke, Vickers and Wilson using the (full) Colombeau algebra. In §8 we discuss questions of coordinate invariance of Colombeau algebras and review work on global and diffeomorphism invariant versions of the construction leading to a “nonlinear distributional geometry”. In §9 we look at “generalised hyperbolicity” and how weak singularities may be regarded as distributional solutions of Einstein’s equations. Finally we give some conclusions and an outlook to future work in §10. 2 Classical Distribution Theory and General Relativity In this section we briefly review the fundamental problems encountered when one tries to incorporate nonlinear operations into classical distribution theory. We will then examine the inherent limitations this imposes on distributional products and review the consequences for distributional solutions of Einstein’s equations. There has been a long history of using generalised function ideas in physics to model point sources and discontinuous phenomena. Such generalised functions were put on a sound mathematical footing by the development of the theory of distributions through the work of S. L. Sobolev [79] and L. Schwartz [77]. The basic idea is to make distributions dual to a space of smooth “test functions”. To introduce some notation we let ${\mathcal{D}}({\mathbb{R}}^{n})$ denote the space of smooth functions of compact support on ${\mathbb{R}}^{n}$. If $S$ is a linear form $S:{\mathcal{D}}({\mathbb{R}}^{n})\to{\mathbb{C}}$ then we will denote the action of $S$ on $\phi\in{\mathcal{D}}$ by $\langle S,\phi\rangle$. The vector space of distributions ${\mathcal{D}}^{\prime}({\mathbb{R}}^{n})$ is then defined to be the space of continuous linear forms on ${\mathcal{D}}({\mathbb{R}}^{n})$. In a similar way distributions on an orientable manifold $M$ are defined as continuous linear functionals on the space of compactly supported $n$-forms i.e. ${\mathcal{D}}^{\prime}(M):=[\Omega^{n}_{c}(M)]^{\prime}$. A rich theory of distributional tensor fields (and sections of more general vector bundles) has been developed by Marsden in [62]; for a pedagogical introduction see [28, Ch. 3.1]. The theory of distributions quickly proved to be extremely successful both in applications to the study of linear partial differential equations and in justifying the use of generalised functions in physics. For example the Malgrange-Ehrenpreis theorem shows that any linear PDE with constant coefficients has a fundamental solution in the space of distributions. However the theory soon displayed its natural limitations when in 1957 H. Lewy [52] gave his famous example of a linear partial differential equation with smooth (non-constant) coefficients which does not have a distributional solution. A second difficulty with the theory is that the definition of a distribution as a linear functional does not make it easy to define the product of distributions. This prevents one from using distributions to investigate nonlinear PDEs with singular data or coefficients. Although the Lewy example of a linear PDE without a distributional solution came as a great surprise, the difficulties that one encounters with the multiplication of distributions are much easier to understand and can be seen by looking at some simple cases. In the following we shall briefly discuss such examples concentrating on the powers of the Heaviside function $H$ and its product with Dirac’s delta function $\delta$. If we regard $H$ as a discontinuous function then $H^{m}=H$ ($m\in{\mathbb{N}}$). However if we differentiate this formula and use the Leibniz rule for the derivative the following one-line calculation implies the vanishing of the delta function: $$2H\delta=(H^{2})^{\prime}=(H^{3})^{\prime}=3H^{2}\delta=3H\delta,\ \mbox{so}\ % H\delta=0\ \mbox{hence}\ \delta=0.$$ (2) Another popular “multiplication rule” is $$H\delta={1\over 2}\ \delta.$$ (3) We demonstrate the problematic nature of this rule (and in fact any rule of the form $H\delta=c\delta$, $c\in{\mathbb{R}}$) by considering the simple ODE $$y^{\prime}(t)=\delta(t)y(t),\qquad y(-\infty)=1,$$ (4) (for an amusing discussion of this equation see [31]). Using the ansatz $y(t)=1+\alpha H(t)$ and (3) we find $\alpha\delta=(1+\alpha/2)\delta$, hence $\alpha=2$ and the solution takes the form $$y(t)=1+2H(t).$$ (5) On the other hand a different approach motivated by the requirement of stability under perturbation is to regularise the singular coefficient by a sequence $\delta_{n}$ weakly converging to $\delta$. Then the solution to the regularised equation is given by $y_{n}(t)=\exp(\int_{-\infty}^{t}\delta_{n}(s)ds)+1$ which converges to $$\tilde{y}(t)=1+(e-1)H(t),$$ (6) which obviously does not coincide with the previous solution. The deeper reason behind these and all other inconsistencies in this realm is the incompatibility of the laws of a (commutative) differential algebra with the formulae $H^{m}=H$ and $H^{\prime}\not=0$. This insight was put into stringent form by L. Schwartz himself in his incompatibility result [76], which says that if the vector space ${\mathcal{D}}^{\prime}$ of distributions is embedded into a differential algebra $({\mathcal{A}},+,\circ)$ then the following properties are mutually contradictory: (i) ${\mathcal{D}}^{\prime}$ is linearly embedded into ${\mathcal{A}}$ and $f(x)\equiv 1$ is the unity in ${\mathcal{A}}$. (ii) There exist linear derivation operators $\partial_{i}:{\mathcal{A}}\to{\mathcal{A}}$ satisfying the Leibniz rule. (iii) $\partial_{i}|_{{\mathcal{D}}^{\prime}}$ is the usual partial derivative. (iv) $\circ|_{C^{k}\times C^{k}}$ (for $k$ finite) is the usual pointwise product of functions. It was this result that led to the idea that it was impossible to multiply distributions. However given that repeatedly differentiating a $C^{k}$ function eventually produces a distribution it is perhaps unreasonable to insist on (iv) but instead we should only require that the product of smooth functions is the usual pointwise product. We will see in §3 that this is precisely the condition satisfied by Colombeau algebras. In the rest of this section we examine the extent to which one can apply linear distribution theory in the context of general relativity. After summarising work done in a number of special cases we review a classical paper by Geroch and Traschen [26] in which they set up a “maximal” distributional framework by finding the “largest possible” class of spacetime metrics which allow for a distributional formulation of the field equations, and we discuss its limitations. Spacetimes involving an energy-momentum tensor supported on a hypersurface of spacetime have long been used in general relativity (see [50, 51, 20] and [36, 66], as well as the references therein). Consider a submanifold $S$ of codimension one dividing spacetime into a “lower” and “upper” part and let the metric be smooth up to and including $S$ from each of its sides but allow for a jump of its first derivatives across $S$. Writing out the Einstein equations in terms of the extrinsic curvature of $S$ one finds junction conditions very similar to the ones in electrodynamics (see e.g. [74], Ch. 3.7). More precisely, the jump of the extrinsic curvature is interpreted as the surface stress-energy of a surface layer located at $S$. In the case of $S$ being timelike this arrangement represents a thin shell of matter, while if $S$ is null it may be interpreted as a thin shell of radiation (see e.g. [39]). In [36] W. Israel has given a general formulation of this widely applied approach with the practical advantage that no reference to any special coordinate system is required; the four-dimensional coordinates may be chosen freely and hence may be adapted to possibly different symmetries in the upper and lower part of spacetime. On the other hand Lichnerowicz [53], has given an alternative description using tensor distributions assuming the existence of an admissible continuous coordinate system across $S$. This formalism was used by Lichnerowicz [54, 55] and Choquet-Bruhat [13] to study gravitational shock waves. They derived algebraic conditions on the metric across the shock (the “gravitational Rankine-Hugoniot conditions”) as well as equations governing the propagation of the discontinuities. The respective formalisms of Israel and Lichnerowicz were shown to be equivalent in [60]. The description of gravitational sources supported on two-dimensional submanifolds of spacetime, however, is more delicate. Israel [37] has given conditions under which a sensible treatment of the field of a “thin massive wire” is possible. He isolated a class of “simple line sources” which possess a linear energy-momentum tensor and hence allow a well-defined limit as the wire’s radius shrinks to zero. We will look at line sources in more detail below. On the other hand Taub [84] has claimed to have generalised Lichnerowicz’s formalism to include gravitational sources supported on submanifolds of arbitrary codimension in spacetime. However, he had to fix the ill-defined products by “multiplication rules”, in particular, by using equation (3). We now begin to review the systematic approach by Geroch and Traschen in analysing the structure of the nonlinearities of the field equations to see how far one can get avoiding ill-defined distributional products. More precisely, the quest is for a class of metrics allowing for a distributional formulation of the Einstein tensor in order to assign to the spacetime—via the field equations—a distributional energy-momentum tensor representing the “concentrated” source. Note that there are two contradictory demands on this class of metrics: on the one hand these metrics should be “nice enough” to permit the distributional calculation of the curvature entities, while on the other hand they should be “bad enough” to have the Einstein tensor and hence the energy-momentum tensor concentrated on a submanifold of a high codimension in spacetime. We write out the coordinate formula of the Riemann curvature tensor in terms of the Levi-Civita connection and the connection in terms of the metric $$R_{abc}\,^{d}\,=\,2\Gamma^{d}_{e[b}\Gamma^{e}_{a]c}+2\partial_{[b}\Gamma^{d}_{% a]c}\,,$$ (7) $$\Gamma^{a}_{bc}\,=\,g^{ae}(\partial_{(b}g_{c)e}-\frac{1}{2}\partial_{c}g_{bc})% \,\,.$$ (8) and try to “save” these equations by putting just as much restrictions on the metric tensor as needed to allow for a distributional interpretation of the respective right hand sides. For the first term in (7) it is obviously sufficient to assume $\Gamma^{a}_{bc}$ to be locally square integrable. Since $L_{\mbox{\rm\small loc}}^{2}\subseteq L_{\mbox{\rm\small loc}}^{1}$ this requirement actually also suffices to interpret the second term in (7) as the weak derivative of the regular distribution $\Gamma^{a}_{bc}$. Furthermore from equation (8) we see that it is sufficient to demand $g^{ab}$ to be bounded locally almost everywhere in order to produce locally square integrable $\Gamma^{a}_{bc}$ from locally square integrable first weak derivatives of $g_{ab}$. This motivates the following definition. 2.1 Definition. A symmetric tensor field $g_{ab}$ on a four-dimensional manifold $M$ is called a gt-regular metric if $g_{ab}$ and $g^{ab}\in L_{\mbox{\rm\small loc}}^{\infty}\cap H_{\mbox{\rm\small loc}}^{1}$. In the above definition $L_{\mbox{\rm\small loc}}^{\infty}$ denotes the space of locally bounded functions and $H_{\mbox{\rm\small loc}}^{1}$ denotes the Sobolev space of functions which are locally square integrable and also have locally square integrable first (weak) derivative. Note that although the above definition appears to be stronger than that in [26] it is actually equivalent to the original one where it was merely demanded that $g_{ab}$ as well as $g^{ab}$ are locally bounded almost everywhere, and the first weak derivatives of $g_{ab}$ are locally square integrable. In fact, these assertions imply $\partial_{c}g^{ab}\in L_{\mbox{\rm\small loc}}^{2}$ since $g^{ab}=\mbox{cof}(g_{ab})/\det g_{ab}$, where $\mbox{cof}(g_{ab})$ denotes the cofactor of $g_{ab}$. We also note that the above conditions will always hold for a $C^{1-}$ metric, for such a metric will always admit a locally bounded weak derivative. For a detailed discussion of the relationship between gt-regularity and some other regularity conditions in the context of axial and cylindrical symmetry we refer to [94], chaps. 2.3–2.5. A further important remark on the notion of gt-regular metrics is in order. While the definition is coordinate invariant with the manifold fixed beforehand, in the case we are most interested in i.e. when we are dealing with a singular spacetime in general relativity, the situation is different. We are not given in advance a coordinate system that includes the singularity. So the question of whether a singular metric is gt-regular or not depends crucially on the choice of the differentiable structure which is imposed on the manifold to include the singular region. To see that gt-regular metrics actually allow for the distributional formulation of Einstein’s equations we show that one can build a distributional Einstein tensor. To do this we need to show that the tensor product of the contravariant metric with the Riemann tensor makes sense as a distribution. By writing the terms involving the second derivative as a total derivative we may write this in the form $$g^{ef}R_{abc}\,^{d}\,=\,2g^{ef}\Gamma^{d}_{m[b}\Gamma^{m}_{a]c}+2\partial_{[b}% (g^{ef}\Gamma^{d}_{a]c})-2(\partial_{[b}g^{ef})\Gamma^{d}_{a]c}\,\,.$$ (9) Now the first term involves a product $L_{\mbox{\rm\small loc}}^{\infty}\times L_{\mbox{\rm\small loc}}^{1}$ hence stays locally integrable. The second term involves a weak derivative of an $L_{\mbox{\rm\small loc}}^{1}$-tensor field hence may be interpreted as a distribution and the third term is a product of two locally square integrable fields so is also locally integrable. Before discussing convergence for gt-regular metrics we briefly introduce tensor distributions. Distributional sections of vector bundles and, in particular, distributional tensor fields can be defined as continuous linear forms on suitable test section spaces but are most easily viewed just as sections with distributional coefficients, that is $${{\mathcal{D}}^{\prime}}^{r}_{s}(M)={\mathcal{D}}^{\prime}(M)\otimes{\mathcal{% T}}^{r}_{s}(M),$$ (10) where ${{\mathcal{D}}^{\prime}}^{r}_{s}$ and ${\mathcal{T}}^{r}_{s}$ denote the spaces of distributional and smooth $(r,s)$-tensor fields respectively. We can now discuss an appropriate notion of convergence for gt-regular metrics. As already indicated above we would like the Einstein tensor of a sequence of metrics approximating a gt-regular one to approximate the Einstein tensor of the gt-regular metric. The natural notion of weak convergence for a sequence of locally square integrable tensor fields $((\mu^{i_{1}\dots i_{r}}_{j_{1}\dots j_{s}})_{n})_{n}$ is convergence locally in square integral, i.e. $$(\mu^{i_{1}\dots i_{r}}_{j_{1}\dots j_{s}})_{n}\to 0\quad(n\to\infty)\mbox{ % iff }\int(\mu^{i_{1}\dots i_{r}}_{j_{1}\dots j_{s}})_{n}(\mu^{k_{1}\dots k_{r}% }_{l_{1}\dots l_{s}})_{n}\,t^{j_{1}\dots j_{s}l_{1}\dots l_{s}}_{i_{1}\dots i_% {r}k_{1}\dots k_{r}}\to 0\ \in{\mathbb{C}}$$ (11) for all smooth compactly supported $(2s,2r)$-tensor densities $t^{j_{1}\dots j_{s}l_{1}\dots l_{s}}_{i_{1}\dots i_{r}k_{1}\dots k_{r}}$. Using this notion of convergence one may prove the following theorem. 2.2 Theorem. (Convergence of gt-regular metrics) Let $g_{ab}$ and $((g_{ab})_{n})_{n}$ be a gt-regular metric and a sequence of gt-regular metrics respectively and let (i) $((g_{ab})_{n})_{n}$ and $((g^{ab})_{n})_{n}$ be locally uniformly bounded, and (ii) $(g_{ab})_{n}\to g_{ab}$, $(g^{ab})_{n}\to g^{ab}$, and $(\partial_{a}g_{bc})_{n}\to\partial_{a}g_{bc}$ locally in square integral. Then $(R_{abc}\,^{d})_{n}\to R_{abc}\,^{d}$ in ${{\mathcal{D}}^{\prime}}^{1}_{3}(M)$ and hence $(G_{ab})_{n}\to G_{ab}$ in ${{\mathcal{D}}^{\prime}}^{0}_{2}(M)$. We note that the space of gt-regular metrics is complete with respect to the notion of convergence defined by hypotheses (i) and (ii). Moreover, let $g_{ab}$ be a continuous gt-regular metric then there exists a sequence of smooth metrics $((g_{ab})_{n})_{n}$ converging to $g_{ab}$ in the sense of (i) and (ii). Before actually checking which class of gravitational sources may be described by gt-regular metrics we start with the following heuristic consideration of the behaviour of gt-regular metrics. Suppose $S$ is a $d$ dimensional submanifold of a 4-dimensional spacetime $M$ and the metric $g_{ab}$ is smooth on $M\setminus S$ but some of its components diverge as one approaches $S$. What order of divergence is allowed if $g_{ab}$ is to be gt-regular? Let $r$ be a typical distance from $S$ measured by some background Riemannian metric $h_{ab}$ and suppose the components of $g_{ab}$ diverge at the rate of $r^{-s}$ for some positive number $s$. Then the weak derivatives of $g_{ab}$ diverge like $r^{-1-s}$ while the volume element is proportional to $r^{3-d}$. In order for the derivatives of the metric to be locally square integrable we therefore require that $2(-s-1)+3-d>-1$, and hence that $$s<1-\frac{d}{2}\,\,.$$ (12) Hence we see that the components of gt-regular metrics must grow more slowly than a rate of $r^{-1+d/2}$ as one approaches a $d$-dimensional submanifold in 4-dimensional spacetime. In particular, the larger the codimension of the submanifold the more strongly the components of the metric may diverge. However, as shown by the following theorem, there are severe constraints on the dimension of $S$. 2.3 Theorem. (Concentrated sources from gt-regular metrics) Let $S$ be a submanifold of dimension $d=(0,1,2,3)$ of a four-dimensional manifold $M$ and let $T^{i_{1}\dots i_{r}}_{j_{1}\dots j_{s}}\not=0$ a tensor distribution satisfying (i) $\mathrm{supp}(T^{i_{1}\dots i_{r}}_{j_{1}\dots j_{s}})\subseteq S$, and (ii) $T^{i_{1}\dots i_{r}}_{j_{1}\dots j_{s}}$ is the sum of a locally integrable tensor field and the weak derivative of a locally square integrable tensor field (hence is of the form of the Riemann tensor of a gt-regular metric). Then $d=3$. This theorem fits the picture described earlier in this section, i.e. that gravitating sources with their support concentrated on a $3$-dimensional submanifold have been treated successfully in the literature while sources concentrated on submanifolds of higher codimension have turned out to be more subtle to deal with. However, it should also be emphasised that although the gt-regularity conditions are sufficient for the curvature to make sense as a distribution they are certainly not necessary as shown by the examples below, so that the above result should not be interpreted as implying that a spacetime with distributional curvature can only have the source confined to a 3-dimensional submanifold. Indeed in certain algebraically special situations—and using a preferred coordinate system—some of the curvature quantities may be defined for non gt-regular metrics. For example impulsive $pp$-waves [69] have been treated extensively using distributions. Summing up, gt-regularity provides us with a large class of “badly behaved” metrics which nevertheless allows one to formulate the field equations in a “stable” way. We are, however, sailing close to the wind as may be seen from the fact that energy conservation may not be formulated in general for gt-regular metrics. Indeed, the left hand side of the Bianchi identities $\nabla_{[a}R_{bc]de}$ involves a product of the distributional coefficients of the Riemann tensor with the non-smooth Christoffel symbols and this is only well defined if one imposes additional conditions on the metric. Similarly, gt-regular metrics may not be used to raise or lower the indices of a general tensor distribution since the tensor product would again involve a multiplication of distributions. By looking in more detail at the combination of terms that one has in the expression for the curvature, Garfinkle in [25] has generalised the formalism of Geroch and Traschen to include a slightly more general class of metrics. However, in extending the class of metrics in this way one can no longer establish the convergence theorem which one has for gt-metrics. One is therefore forced to give up the requirement of “limit consistency”. This is another indication that with the Geroch Traschen definition of regularity one has gone about as far as possible using conventional distribution theory. Staying strictly within the mathematically and physically consistent setting given by this theory, one has to restrict oneself to a class of metrics that excludes physically interesting cases such as strings and point particles. If one wants to describe more general gravitational sources the nonlinearity of the field equations forces one to go beyond the limits of classical distribution theory and face true conceptional problems. A consistent framework allowing for nonlinear operations on singular (e.g. distributional) objects is provided by Colombeau’s algebras of generalised functions, which we introduce in the next section. 3 A brief review of Colombeau theory In this section we give a brief introduction to Colombeau algebras. For more details see [17], [18], [12], [28]. As we said in the previous section the definition of distributions as linear functionals is not well suited to formulate a definition of multiplication. However it is common to visualise the Dirac delta function as the limit of a sequence of smooth functions, all with integral one, whose support gets concentrated at the origin. In fact it is possible to give these ideas a precise mathematical formulation and an alternative sequential approach to distribution theory was developed by Mikusiński [64] as early as 1948 (see also Temple [85]). In this approach a distribution is an equivalence class of weakly converging sequences of smooth functions modulo weak zero-sequences. Working with a more subtle quotient construction Colombeau was able to construct a differential algebra ${\mathcal{A}}$ satisfying conditions (i)–(iii) of §2 but with (iv) replaced by (iv’) $\circ|_{C^{\infty}\times C^{\infty}}$ is the usual pointwise product of smooth functions. In order to introduce the basic concepts we will start by describing the special (or simplified) Colombeau algebra on ${\mathbb{R}}^{n}$. The basic idea is to consider generalised functions as 1-parameter families of smooth functions $\{f_{\epsilon}\}$. Our basic space will thus be $${\mathcal{E}}({\mathbb{R}}^{n})=\{\{f_{\epsilon}\}:\ 0<\epsilon\leq 1,\ \ f_{% \epsilon}\in C^{\infty}({\mathbb{R}}^{n})\}.$$ (13) We now want to consider how we can represent a function $f$ of finite differentiability as an element of this space. If we start with some $\Phi\in{\mathcal{D}}({\mathbb{R}}^{n})$ with integral one then we can rescale this to obtain a family of functions $$\Phi_{\epsilon}(x)={1\over{\epsilon^{n}}}\Phi\left({x\over\epsilon}\right)$$ (14) with the property that $\Phi_{\epsilon}\to\delta$ in ${\mathcal{D}}^{\prime}$ as $\epsilon\to 0$. Hence if we take the convolution of $f$ with $\Phi_{\epsilon}$ we obtain a family $$f_{\epsilon}(x)={1\over{\epsilon^{n}}}\int f(y)\Phi\left({{y-x}\over{\epsilon}% }\right){\rm d}^{n}y.$$ (15) of smooth functions that converge to $f$ in ${\mathcal{D}}^{\prime}$ as $\epsilon$ tends to zero. We will refer to $\Phi$ in the above expression as a molifier. Of course we can also apply the above formula to a smooth function $f$. However for a smooth function we can also represent $f$ as an element of ${\mathcal{E}}$ by considering the constant family $f_{\epsilon}(x)=f(x)$. For the case of a smooth function we would like both these possible representations to be equivalent. Using a Taylor series expansion to compare the difference between these expressions we are lead to define two representations to be equivalent if they differ by a “negligible function” which is defined as a 1-parameter family of functions which on any compact set vanishes faster than any given positive power of $\epsilon$. Since we are trying to construct a differential algebra we also require the derivatives of $f$ to satisfy this property and the resulting set ${\mathcal{N}}$ to be an ideal. Clearly ${\mathcal{N}}$ is not an ideal in ${\mathcal{E}}({\mathbb{R}}^{n})$, but by restricting this space to “moderate functions” ${\mathcal{E}}_{M}({\mathbb{R}}^{n})$ which grow no faster than some inverse power of $\epsilon$, one does have an ideal and we may define the differential algebra ${\mathcal{G}}$ as the quotient. 3.1 Definition. (i) (Moderate functions) $$\displaystyle{\mathcal{E}}_{M}({\mathbb{R}}^{n})$$ $$\displaystyle:=$$ $$\displaystyle\big{\{}\{f_{\epsilon}\}:\ \forall K\subset\!\subset{\mathbb{R}}^% {n},\forall\alpha\in{\mathbb{N}}^{n}_{0},\exists p\in{\mathbb{N}}\hbox{ such that}$$ $$\displaystyle\hphantom{\{\{f_{\epsilon}\}:\ }\sup\limits_{x\in K}|D^{\alpha}f_% {\epsilon}(x)|\leqslant O(\epsilon^{-p})\hbox{ as }\epsilon\to 0\big{\}}.$$ (ii) (Negligible functions) $$\displaystyle{\mathcal{N}}({\mathbb{R}}^{n})$$ $$\displaystyle:=$$ $$\displaystyle\big{\{}\{f_{\epsilon}\}:\ \forall K\subset\!\subset{\mathbb{R}}^% {n},\forall\alpha\in{\mathbb{N}}^{n}_{0},\forall q\in{\mathbb{N}}\hbox{ such that}$$ $$\displaystyle\hphantom{\{\{f_{\epsilon}\}:\ }\sup\limits_{x\in K}|D^{\alpha}f_% {\epsilon}(x)|\leqslant O(\epsilon^{q})\hbox{ as }\epsilon\to 0\big{\}}.$$ (iii) (Special algebra) $${\mathcal{G}}({\mathbb{R}}^{n}):={\mathcal{E}}_{M}({\mathbb{R}}^{n})/{\mathcal% {N}}({\mathbb{R}}^{n})$$ Note $K\subset\!\subset{\mathbb{R}}^{n}$ indicates that $K$ is compact and we have also employed the standard multi-index notation for $D^{\alpha}f$. Thus a nonlinear generalised function $f$ denoted by $f=[\{f_{\epsilon}\}]$ is an equivalence class of moderate sequences of smooth functions modulo negligible ones; it is represented by a moderate sequence of smooth functions $\{f_{\epsilon}\}$. The space ${\mathcal{E}}_{M}({\mathbb{R}}^{n})$ is a differential algebra with pointwise operations and since the space of negligible functions is a differential ideal, ${\mathcal{G}}$ is also a commutative differential algebra. The vector space of distributions is now embedded into the algebra ${\mathcal{G}}$ by convolution with a molifier: More precisely we choose a molifier $\Phi$ which for technical reasons (not to be discussed here) is a Schwartz function and has all moments vanishing i.e. $\int\Phi(x)x^{\alpha}\,dx=0\ \forall|\alpha|\geq 1$. Then we embed $T\in{\mathcal{D}}^{\prime}$ with compact support as $$\iota(T)=[\{T*\Phi_{\epsilon}\}].$$ (16) Distributions which are not compactly supported are embedded via a localised version of (16) using a standard sheaf theoretic construction. As remarked earlier one of the advantages of the Colombeau approach is that one may frequently interpret the results in terms of distributions using the concept of association or weak equivalence. A generalised function $f$ is said to be associated to a distribution $T\in{\mathcal{D}}^{\prime}$ if for one (hence any) representative $\{f_{\epsilon}\}$ we have $$\forall\phi\in{\mathcal{D}},\quad\lim_{\epsilon\to 0}\int f_{\epsilon}(x)\phi(% x){\rm d}^{n}x=\langle T,\phi\rangle$$ (17) and we then write $f\approx T$. Note that not all elements of ${\mathcal{G}}$ are associated to distributions. More generally we say for two generalised functions $f\approx g$ if $$\forall\phi\in{\mathcal{D}},\quad\lim_{\epsilon\to 0}\int(f_{\epsilon}(x)-g_{% \epsilon}(x))\phi(x){\rm d}^{n}x=0$$ (18) for one (hence any) pair of representatives $\{f_{\epsilon}\}$, $\{g_{\epsilon}\}$. Association is an equivalence relation which respects addition and differentiation. It also respects multiplication by smooth functions but by the Schwartz impossibility results cannot respect multiplication in general. The algebra presented above is the simplest of the Colombeau algebras and can be readily generalised to arbitrary manifolds (see §8). However it does suffer from the disadvantage that the embedding $\iota$ of distributions and of functions of finite differentiability is not canonical but depends on the choice of molifier $\Phi$ (see above). Thus one has to appeal to mathematical or physical arguments outside the theory to justify a particular representation of a non-smooth function. We discuss these matters in §8 where we also present the construction of so-called full Colombeau algebras which do posses a canonical embedding of distributions. In the following sections however we will discuss applications of Colombeau algebras to general relativity using the language of the special version. 4 The Schwarzschild and Kerr Spacetimes In this section we review (linear and nonlinear) distributional treatments of the Schwarzschild and Kerr spacetimes. We use the language of the special Colombeau algebra although strictly speaking we should be using the special version of the theory of generalised tensor fields on manifolds, which we will introduce in §8. However the precise details will not be needed as we aim at presenting the main ideas and concepts in the most elementary way. Balasin and Nachbagauer considered rotating, charged, Kerr-Newman black-hole solutions in a number of papers ([4, 5, 6, 7, 9, 10]). The solutions considered have the feature that they are all examples of Kerr-Schild geometries. Such metrics may be written in the form $$g_{ab}=\eta_{ab}+fk_{a}k_{b}$$ (19) where $\eta_{ab}$ is the flat Minkowski metric, $f$ is an arbitrary function, $k_{a}=\eta_{ab}k^{b}$ and $k^{a}$ is null and geodetic with respect to $\eta_{ab}$. The simplest example of such a solution is the Schwarzschild solution which in the standard Minkowski coordinates has Kerr-Schild form given by $$f={2m\over r},\quad k^{a}=(1,x^{i}).$$ (20) The expression for the Ricci tensor of a Kerr-Schild metric takes the surprisingly simple form $$R^{a}_{b}={\scriptstyle{1\over 2}}\eta^{cd}\eta^{ea}[\partial_{e}\partial_{c}(% fk_{d}k_{b})+\partial_{b}\partial_{c}(fk_{d}k_{e})+\partial_{c}\partial_{d}(fk% _{e}k_{b})].$$ (21) The energy momentum tensor is then given by Einstein’s equations as $$T^{a}_{b}=R^{a}_{b}-{\scriptstyle{1\over 2}}\delta^{a}_{b}R.$$ (22) One may then calculate the energy momentum tensor as follows. The Kerr-Schild form is regarded as being valid on the whole of Minkowski space with ${2m\over r}$ replaced by some suitable regularised function. One now considers this as an element of ${\mathcal{G}}$ and computes the components of $T^{a}_{b}$ (in Minkowski coordinates) as elements of ${\mathcal{G}}$. Finally one can show that $$\displaystyle T^{0}_{0}$$ $$\displaystyle\!\approx$$ $$\displaystyle\!-m\delta^{(3)}$$ (23) $$\displaystyle T^{a}_{b}$$ $$\displaystyle\!\approx$$ $$\displaystyle\!0\quad\hbox{otherwise.}$$ (24) Different authors ([68, 40, 67]) using various regularization procedures have also assigned a distributional energy-momentum tensor to the Schwarzschild geometry. These approaches have been compared using the language of the special algebra in [34]. The calculation of the energy-momentum tensor for the Kerr solution is significantly harder. Unlike the Schwarzschild case we cannot easily write the Kerr solution as the limit of a one parameter family of regular Kerr-Schild metrics. The problem arises because of the topology of the maximal analytic extension of the Kerr solution which leads to a branch singularity when the metric is written using the standard “flat” Kerr-Schild decomposition. Balasin [9] avoided this problem by considering metrics of generalised Kerr-Schild form. These are metrics which can be written $$g_{ab}={\hat{g}}_{ab}+fk_{a}k_{b}$$ (25) where ${\hat{g}}_{ab}$ is now a background metric, $k_{a}={\hat{g}}_{ab}k^{b}$ and $k^{a}$ is null and geodetic with respect to ${\hat{g}}_{ab}$ (and also $g_{ab}$ because of the form of the metric). One now has $$R^{a}_{b}={\hat{R}}^{a}_{b}+{\hat{g}}^{cd}{\hat{g}}^{ef}{\hat{R}}^{a}_{% \phantom{a}ceb}fk_{d}k_{f}+{\scriptstyle{1\over 2}}{\hat{g}}^{cd}{\hat{g}}^{ea% }[\partial_{e}\partial_{c}(fk_{d}k_{b})+\partial_{b}\partial_{c}(fk_{d}k_{e})+% \partial_{c}\partial_{d}(fk_{e}k_{b})]$$ (26) where ${\hat{R}}^{a}_{\phantom{a}bcd}$ is the curvature of ${\hat{g}}_{ab}$. We may write the Kerr metric in generalised Kerr-Schild form by taking the background metric to have the form $${\hat{g}}_{ab}dx^{a}dx^{b}=-dt^{2}+{\Sigma\over{r^{2}+a^{2}}}dr^{2}+\Sigma d% \theta^{2}+(r^{2}+a^{2})d\phi^{2}$$ (27) where $\Sigma=r^{2}+a^{2}\cos^{2}\theta$ and $(t,r,\theta,\phi)\in{\mathbb{R}}^{2}\times S^{2}$. Performing all the calculations in the special algebra one may compute $\sqrt{\hat{g}}R^{a}_{b}$ which is found to have the following associated distribution $$\sqrt{\hat{g}}R^{a}_{b}\approx 2\pi\delta(\cos\theta)(-a\delta(u)\partial^{a}_% {u}du_{b}+\partial^{a}_{\theta}d\theta_{b}+m\delta^{\prime}(u)(\partial^{a}_{t% }-(1/a)\partial^{a}_{\phi})(dt_{b}+ad\phi_{b}).$$ (28) The above calculation was carried out by Balasin using the special algebra. This is probably the only practical way of doing the calculation given the topology of the manifold and the complexity of the metric. It suffers from the usual problem when using the special algebra of a non-canonical embedding. Rather than embed using a convolution, the embedding has been chosen to preserve the generalised Kerr-Schild form. However the embedding used is not unique within this class and it would be desirable to show that any reasonable embedding which preserved the form of the decomposition gave the same result. It is even less clear that a “natural regularisation” in some other coordinate system would give the same result. Nevertheless the calculation is an impressive example of the complicated calculations that can be performed in general relativity using the special algebra. 5 Ultrarelativistic Black Holes In 1971 Aichelburg and Sexl [1] derived the ultrarelativistic limit of the Schwarzschild geometry. Below we give a description of the limit using the language of the special algebra. We start by considering the Kerr-Schild form of the Schwarzschild metric written in double null coordinates $u=t-x$ and $w=t=x$, $$ds^{2}=dudw-dy^{2}-dz^{2}+{2m\over r}k_{a}k_{b}dx^{a}dx^{b}$$ (29) where in these coordinates $k^{a}=((r-x)/r,(r+x)/r,y/r,z/r)$. The Minkowski background enables us to have a well defined concept of boost and we may therefore boost the solution by velocity $v$ along the $x$-axis. We therefore write $$u=\sqrt{{{1+v}\over{1-v}}}{\bar{u}}\qquad w=\sqrt{{{1-v}\over{1+v}}}{\bar{w}}$$ (30) and to keep the energy of the “particle” finite we rescale the mass according to the special relativistic formula $$m=(1-v^{2})^{{\scriptstyle{1\over 2}}}p.$$ (31) Substituting this into (29) gives a 1-parameter family of metrics depending on the boost velocity $v$. We are interested in the ultrarelativistic limit in which $v$ reaches the speed of light (i.e. $v\to 1$), so we replace $v$ by $1-\epsilon$ and regard $\tilde{g}_{ab}:=[g_{(\epsilon)ab}]$ as an element of the special algebra. It is readily shown that most of the terms in the perturbation ${2m\over r}k_{a}k_{b}$ are associated to zero. The only surviving term is ${2m\over r}k_{0}k_{0}$. Using calculations very similar to Steinbauer [81] we find this is associated to $8p\ln\rho\delta(u)$ and hence $\tilde{g}_{ab}\approx g_{(0)ab}$ where $$g_{(0)ab}dx^{a}dx^{b}=8p\ln\rho\delta(u)du^{2}+dudw-dy^{2}-dz^{2}.$$ (32) This metric describes a pp-wave and is flat everywhere except on the null plane $u=0$ which contains the “particle”. This line element was first derived by Aichelburg and Sexl [1], who started with Schwarzschild written in isotropic coordinates and simultaneously boosted the solution and made a $v$-dependent coordinate transformation to compute the limiting metric. It should be pointed out that this result is entirely consistent with the calculation of the energy-momentum tensor of the Schwarzschild solution given in the previous section. The ultrarelativistic limit of the latter in the $(u,w,x,y)$ coordinate system is associated to $$\delta(u)\delta^{(2)}(y,z)p_{a}p^{b}\quad\hbox{where }p_{a}=(1,0,0,0),$$ (33) which is just the energy-momentum tensor of $\tilde{g}_{ab}$. Indeed this observation was used by Balasin and Nachbagauer [6] to derive the ultrarelativistic limit of the Schwarzschild and Kerr geometries (see also [11]). It is also possible to calculate the ultrarelativistic limit of the Reissner-Nordstrøm solution. However in this case it is also necessary to rescale the charge according to the formula $$e^{2}=(1-v^{2})^{\scriptstyle{1\over 2}}f^{2},\quad(f\hbox{ a constant})$$ (34) in order to obtain a distributional limit at all as $v$ tends to the speed of light. The limiting metric was found by Loustó and Sánchez [58] using the methods of [1] to be $$ds^{2}=\left\{8p\ln\rho+{{3\pi f^{2}}\over{2\rho}}\right\}\delta(u)du^{2}+dudw% -dy^{2}-dz^{2}.$$ (35) This was confirmed using a calculation in ${\mathcal{G}}$ by Steinbauer [80]. The solution obtained again represents a pp-wave and is flat everywhere except on the null plane $u=0$. The ultrarelativistic limit of the electromagnetic energy-momentum tensor of Reissner-Nordstrøm is also found to be compatible with the energy-momentum tensor of (35). However the rescaling of the charge has the rather unexpected effect that while the electromagnetic field vanishes in the ${\mathcal{D}}^{\prime}$-limit the ultrarelativistic energy-momentum tensor does not (cf [58]). Steinbauer in [81] rephrased this fact using association in ${\mathcal{G}}$ i.e. showed that all the components of the electromagnetic field were associated to zero while the $00$-component of the energy-momentum tensor was associated with a multiple of the delta function. However it must be stressed that regarded as elements of ${\mathcal{G}}$ the electromagnetic field components are non-zero (even though they are associated to zero). Calculating the energy-momentum tensor of this field within ${\mathcal{G}}$ gives the correct result. There is nothing unusual within Colombeau theory in having objects in ${\mathcal{G}}$ which are associated to zero having products which are not associated to zero. This is simply a reflection of the fact that association does not respect multiplication in general. The procedure of Aichelburg and Sexl has also been used to derive the ultrarelativistic limit of the Kerr metric by several authors see [57, 59, 24, 33]. In fact a number of different sources such as cosmic strings, domain walls and monopoles have been boosted to obtain ultrarelativistic spacetimes of impulsive pp-waves which in turn have been used to describe (quantum) scattering processes of highly energetic particles (see [87, 88] for an overview). On the other hand Dray and ’t Hooft [22] have generalised Penrose’s “scissors and paste” method [69] (see also §6) to non-flat backgrounds and used it as an alternative way to derive the Aichelburg-Sexl geometry as well as more general gravitational shock waves. Using this method Dray and ’t Hooft derived the spherical shock wave due to a massless particle moving at the speed of light along the horizon of a Schwarzschild black hole which was used to study the influence of matter, falling into the black hole on its Hawking-radiation. These ideas lie at the heart of ’t Hooft’s S-matrix approach to quantum gravity [86]. 6 Geodesics for impulsive gravitational wave spacetimes In the previous section we showed how the ultrarelativistic limit of the Schwarzschild solution lead to an impulsive gravitational wave spacetime. We now consider geodesics in such spacetimes with the metric taking the form $$ds^{2}=f(x^{A})\delta(u)du^{2}+dudw-\delta_{AB}dx^{A}dx^{B}$$ (36) where $A,\ B=2,\ 3$ denote the transverse coordinates. These spacetimes have been constructed by Penrose using his vivid ”scissors and paste” approach (see [70]). Geodesics for these impulsive gravitational wave spacetimes have been considered by Ferrari, Pendenza and Veneziano [23], Balasin [8] and Steinbauer [82]. As one might expect, one can regularise the geodesic equations, and show that the geodesics consist of broken and refracted straight lines. A more rigorous derivation of these results using existence and uniqueness theorems within the Colombeau algebra has been given by Kunzinger and Steinbauer in [42]. More precisely they replaced the delta function in (36) by a generalised function $D$ possessing a so called strict delta net $\{\rho_{\epsilon}\}$ as a representative i.e. they considered the generalised line element $$\hat{d}s^{2}=f(x^{A})D(u)du^{2}+dudw-\delta_{AB}dx^{A}dx^{B}$$ (37) with $D=[\{\rho_{\epsilon}\}]$ and $\mbox{\rm supp}(\rho_{\epsilon})\to\{0\}$ and $\int\rho_{\epsilon}(x)\,dx\to 1$ as $\epsilon\to 0$ and $||\rho_{\epsilon}||_{L^{1}}$ uniformly bounded in $\epsilon$. They were able to show that the geodesic as well as the geodesic deviation equation for the metric (37) may be solved uniquely in ${\mathcal{G}}$. Moreover these unique generalised solutions possess the physically expected associated distributions. Note that diffeomorphism invariance of these results is assured by diffeomorphism invariance of the class of strict delta nets. Note further that strictly speaking these calculations have been performed using the concept of generalised functions taking values in a manifold, cf §8, since geodesics are curves from an interval into spacetime. In the literature impulsive pp-waves have frequently been described in different coordinates where the metric tensor is actually continuous (cf [70]). In the special case of a plane wave, $f(x,y)=x^{2}-y^{2}$ and $u_{+}$ denoting the kink function, $$ds^{2}\,=\,(1+u_{+})^{2}dX^{2}+(1-u_{+})^{2}dY^{2}-dudV\,.$$ (38) Clearly a transformation relating (38) and (36) cannot even be continuous, hence in addition to involving ill-defined products of distributions it changes the topological structure of the manifold. However, the two mathematically distinct spacetimes are equivalent from a physical point of view, i.e. the geodesics and the particle motion agree on a heuristic level (see [83]). Using their results on the geodesic equation in ${\mathcal{G}}$, Kunzinger and Steinbauer in [43] succeeded in showing that the discontinuous change of coordinates is just the associated distributional map of a generalised coordinate transform. More precisely modelling the (distributional form of the) impulsive pp-wave metric in a diffeomorphism invariant way by the generalised metric (37) the latter may be subject to a generalised change of coordinates $T$. In either coordinates the associated distributional metric may be computed giving the distributional (respectively the continuous) form of the pp-wave metric. Physically speaking the two forms of the impulsive metric arise as the (distributional) limits of a sandwich wave in different coordinate systems. Hence impulsive pp-waves are indeed sensibly modelled by the generalised spacetime metric (37). However in the different coordinate systems a different distributional picture is obtained. A similar situation arises in the case of impulsive spherical waves which have also been introduced in [70]. In this case the distributional form of the metric arises as an impulsive limit of type-N Robinson-Trautman solutions, which however due to the fact that the metric is quadratic in the profile function formally involves the square of the delta function. An explicit discontinuous coordinate transformation relating this form of the metric with the continuous form was given in [72]. A study of this situation using Colombeau methods relies on a better understanding of the geodesics of these spacetimes; a study of these has been initiated in [73]. 7 Conical Singularities and Cosmic Strings An important example of a two dimensional singularity is provided by the conical spacetime $$ds^{2}=dt^{2}-dr^{2}-A^{2}r^{2}d\phi^{2}-dz^{2}.$$ (39) where $A\neq 1$ and $\phi$ is a standard $2\pi$-periodic angular coordinate. This spacetime is locally flat for $r\neq 0$ and heuristic arguments suggest that it has delta-function like curvature on the $r=0$ axis. The corresponding energy-momentum tensor then describes a string with stress equal to the mass per unit length $\mu$, where $\mu=2\pi(1-A)$. This is precisely the form of the energy-momentum tensor of a cosmic string in the weak field thin string limit. Such spacetimes are also important mathematically as they provide simple examples of quasi-regular singularities (i.e. singularities for which the components of the Riemann tensor measured in a parallely propagated frame tend to a well defined limit, see e.g. Vickers [89] for further details). Unfortunately such spacetimes are not gt-regular, so one cannot expect the curvature to be well-defined using conventional distribution theory. Furthermore Geroch and Traschen showed how it was possible to obtain different values of the mass per unit length by taking different regularisations. Because of this Clarke, Vickers and Wilson [16] chose to investigate the curvature of the cone using the “full” Colombeau algebra where one has a canonical embedding of distributions. See §8 below for further details. The singular part of the curvature arises from the conical singularity in the 2-cone $$ds^{2}=dr^{2}+A^{2}r^{2}d\phi^{2}$$ (40) and for simplicity we will describe the calculation of the curvature (density) of this metric. Because polar coordinates are not well defined at the origin one must first write the metric in a regular coordinate system which includes the origin. To make things simple we will choose to work in Cartesian coordinates but as will be shown below the final result is independent of the coordinates used. In Cartesian coordinates one has $$g_{ab}={\scriptstyle{1\over 2}}(1+A^{2})\delta_{ab}+{\scriptstyle{1\over 2}}(1% -A^{2})h_{ab},$$ (41) where $$h_{ab}=\left(\begin{array}[]{cc}{{x^{2}-y^{2}}\over{x^{2}+y^{2}}}&{{2xy}\over{% x^{2}+y^{2}}}\\ {{2xy}\over{x^{2}+y^{2}}}&{{y^{2}-x^{2}}\over{x^{2}+y^{2}}}\end{array}\right).$$ (42) We now regard the metric as an element of ${\mathcal{E}}_{M}({\mathbb{R}}^{2})$ by taking the convolution of the components $g_{ab}$ with an arbitrary molifier. To find these we need to calculate $$\tilde{h}_{\epsilon}(x,y)={1\over{\epsilon^{2}}}\int_{{\mathbb{R}}^{2}}h(u+x,v% +y)\Phi((u/\epsilon),(v/\epsilon)){\rm d}u{\rm d}v$$ (43) where $h(x,y):=e^{2i\phi}$. By expanding in circular harmonics, using the compactness of the support of the molifier and using the residue theorem one can obtain a Fourier series for $\tilde{h}$ and hence for ${\tilde{g}}_{(\epsilon)ab}$. One may now estimate the curvature $\tilde{R}_{\epsilon}$ of ${\tilde{g}}_{(\epsilon)ab}$, and use the Gauss-Bonnet theorem to show that $$[{\tilde{R}}\sqrt{\tilde{g}}]\approx 4\pi(1-A)\delta^{(2)}.$$ (44) It is important to stress that the methods described here may be used to calculate the curvature of the full four dimensional cone, although one may no longer use the Gauss-Bonnet theorem to calculate the curvature and therefore requires more delicate estimates (see Wilson [94] for details). One then obtains the heuristically expected energy-momentum tensor. Furthermore it is not hard to modify the results to deal with a metric which is not exactly conical, but approaches one quadratically as $r\to 0$. A second point to note is that the result does not depend upon the coordinates in which the calculation is carried out so long as they are smoothly related to Cartesians. This may be shown by explicitly transforming to new coordinates $X=X(x,y)$ and $Y=Y(x,y)$ and doing the whole calculation in the new coordinates. Again more delicate estimates are required but one can show that the resulting curvature density transforms in exactly the same way as a delta-function (see Vickers and Wilson [91] for details). The above calculation shows that the curvature of a 4-dimensional cone, when calculated in the full Colombeau algebra, has a curvature which is associated to a delta-function which gives a mass per unit length equal to the deficit angle as one would expect from the heuristic calculation. Note that this result does not say that the curvature is equal to a delta function, but simply that it is associated to a delta function. Thus if one works at the level of association this result shows that regularisations based upon smoothing convolutions give an unambiguous answer for the curvature of the cone. Indeed a number of authors have applied different regularisations and obtained the same result. Balasin and Nachbagauer [4] used a regularisation based on a family of smooth hyperboloidal surfaces converging to a cone. A number of authors including Marder [61] and Geroch and Traschen [26] have looked at “rounding off” the point of a cone with a spherical cap, and Louko and Sorkin [56] have looked at a regularisation based on a different coordinate system. Unfortunately things are not quite as straightforward as one would hope. It is also possible to choose regularisations which do not yield a mass per unit length equal to the deficit angle. An important example of this was given by Geroch and Traschen [26]. They first formed a regularisation sequence ${\tilde{g}}_{(\epsilon)ab}$ which gave the standard answer for the mass per unit length. They then introduced a second regularisation sequence ${\hat{g}}_{(\epsilon)ab}$ which was related to the first sequence by a conformal factor $${\hat{g}}_{(\epsilon)ab}=\Omega^{2}{\tilde{g}}_{(\epsilon)ab},$$ (45) where $$\Omega=\exp(\lambda f(r/\epsilon))$$ (46) and $f$ is a smooth function whose support is $[1/2,1]$. They then found that the mass per unit length of the limiting spacetime was dependent on both $\lambda$ and $f$: $$\mu=2\pi(1-A)-2\pi\int_{1/2}^{1}\lambda^{2}\sin(\gamma r)f^{\prime}(r)^{2}dr.$$ (47) From the point of view of Colombeau theory this “bad” behaviour of the above regularisation is not unexpected since although $\Omega^{2}\approx 1$, it is not equal to the unity function in ${\mathcal{G}}({\mathbb{R}}^{4})$ and hence ${\hat{g}}_{\epsilon}$ does not represent the conical spacetime in the full Colombeau algebra. Wilson [94] looked at general regularisations of conical spacetimes and gave a condition that ensured that a given regularisation sequence gave the standard result for the mass per unit length. He showed that provided that the difference between the connection one-forms of the regularised spacetime and the original spacetime diverged no faster than $1/r$ then the answer would be the standard result. Thus although it is possible to obtain regularisations which give a different mass per unit length, the geometry of these bad regularisations diverges strongly from that of a cone as one approaches the axis. A significant generalisation of the curvature calculations for a conical spacetime was obtained by Wilson [95] who extended the results to a four dimensional time dependent cosmic string. He considered cylindrically symmetric pure radiation solutions of Einstein’s equations. These metrics may be written in the form $$ds^{2}=e^{2\gamma(t-r)}(dt^{2}-dr^{2})+r^{2}d\phi^{2}+dz^{2}$$ (48) and naively one would expect a delta-function contribution to the curvature due to the angular deficit of $2\pi(1-e^{-\gamma(t)})$. However, because of the time dependence of the angular deficit, the singularity cannot be quasi-regular but must be stronger. In fact it is an example of an “intermediate singularity” and the components of the Riemann tensor have a limit in a special choice of frame. Wilson calculated the energy-momentum tensor of this spacetime by first writing the metric in null Cartesian coordinates and then proceeded to obtain a smooth metric by convolution, estimate the curvature of the smooth metric and show that it is associated to a distribution. This result shows that even in the radiating case the mass per unit length is given by the expected formula $$\mu(u,z)\approx 2\pi(1-e^{-\gamma(u)}).$$ (49) Furthermore by applying the methods of Ashtekar et al. [3] one may show that this result agrees with the asymptotically measured mass per unit length. However unlike the case of the static string it is unclear whether the calculation is coordinate invariant. The $(u,x,y,z)$ coordinates used are natural for investigating radiating systems but have the disadvantage that Minkowski space appears to be singular on the axis when written in these coordinates. However the correct mass per unit length for a static cosmic string is obtained, and this vanishes for Minkowski space. 8 Nonlinear distributional geometry An underlying principle of general relativity is that the measurement of physical quantities should be independent of the coordinate system used. Mathematically this is reflected in the fact that the theory is formulated in terms of tensor fields on manifolds. The minimum differentiability of the coordinate transformations and the dependence on the differential structure of the manifold is quite subtle, but at the very least the theory should be invariant under smooth diffeomorphisms. The definition of the special algebra introduced in §2 may be generalised in a more or less straight forward manner to yield a special Colombeau algebra ${\mathcal{G}}(M)$ on a differentiable manifold $M$ [21]. This setup was extended by Kunzinger and Steinbauer in [44] to a theory of generalised sections of vector bundles. Furthermore in [45] the study of (pseudo-)Riemannian geometry in the generalised setting was initiated and in [46] generalised connections and curvature in general principal and vector bundles were studied. However both geodesics and diffeomorphisms involve considering functions with values in a manifold and if one wishes to consider these objects in the generalised setting one is forced to consider manifold valued generalised functions (a concept which it is not possible to deal with using distributions). The study of such functions was first looked at in [41], where the space ${\mathcal{G}}[X,Y]$ of generalised functions defined on the manifold $X$ and taking values in the manifold $Y$ was defined. This work was extended to a functorial theory in [47] where several global characterisations of the notions of moderateness and negligibility for generalised functions from $X$ to $Y$ are given. These characterisations provide the key to proving that composition of generalised functions between manifolds can be carried out unrestrictedly. Using these ideas it is possible to consider generalised geodesics and flows as well as singular ODEs on manifolds (see [48] for details). An overview over this “nonlinear distributional geometry” can be found in [28, Ch. 3.2]. However in all these constructions the embedding of distributions and functions of finite differentiability is non-canonical; in addition to the dependence on the choice of a molifier (see §2), any embedding into ${\mathcal{G}}(M)$ will not be diffeomorphism invariant. So how can this setting be of use in a diffeomorphism invariant theory like general relativity? Firstly in some applications there will be a natural physical parameter, such as a coupling constant, that may be used to directly provide a representation of the singular objects involved in ${\mathcal{G}}$. Hence there will be no need for using any embedding at all. Another possibility is to model a singular spacetime metric by a whole class of generalised metrics that is by definition diffeomorphism invariant. This has been done in case of the description of impulsive pp waves in ${\mathcal{G}}$ (cf §6). Finally one may regard the Colombeau algebra as an intermediate calculational tool for obtaining distributional answers. One picks some coordinate system, embeds the components of the metric into the Colombeau algebra (in the given coordinates) and calculates the components of the curvature and energy-momentum tensor etc. One then tries to show that these objects are associated to distributions. Finally one repeats the entire calculation in some other (smoothly related) coordinate system and tries to show that the answer transforms in the expected way for a distribution. Although such calculations can be useful in some situations it would be preferable to make the embedding into the algebra coordinate invariant. To explain how this can be done we first introduce the full Colombeau algebra ${\mathcal{G}}^{e}({\mathbb{R}}^{n})$ [17] which, unlike the special version, has a canonical embedding of distributions. This is achieved by substituting the index set $(0,1]$ by a suitable class of molifiers. More precisely we introduce the following grading on the space of all molifiers $$\displaystyle{\cal A}_{0}$$ $$\displaystyle:=$$ $$\displaystyle\{\Phi\in{\cal D}({\mathbb{R}}^{n})\,:\,\int\Phi(x)\,dx=1\}$$ $$\displaystyle{\cal A}_{q}$$ $$\displaystyle:=$$ $$\displaystyle\{\Phi\in{\cal A}_{0}\,:\,\int\Phi(x)x^{\alpha}\,dx=0\,,\,1\leq|% \alpha|\leq q\}\,\,\,(q\in{\mathbb{N}})$$ and take the basic space to be $${\mathcal{E}}^{e}:=\{f:\ {\cal A}_{0}\times{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}% |\ f\ \mbox{smooth in}\ x\}.$$ Now the respective spaces of moderate and negligible functions may be defined as follows (again $\Phi_{\epsilon}(x):=(1/\epsilon^{n})\Phi(x/\epsilon)$). 8.1 Definition. (i) (Moderate functions) $$\displaystyle{\cal E}^{e}_{M}({\mathbb{R}}^{n})$$ $$\displaystyle:=$$ $$\displaystyle\{f\in{\cal E}^{e}:\forall K\subset\subset{\mathbb{R}}^{n}\ % \forall\alpha\in{\mathbb{N}}_{0}^{n}\ \exists p\in{\mathbb{N}}_{0}\ \forall% \Phi\in{\cal A}_{p}:$$ $$\displaystyle\qquad\sup\limits_{x\in K}|D^{\alpha}f(\Phi_{\epsilon},x)|=O(% \epsilon^{-p})\mbox{ as }\epsilon\to 0\}$$ (ii) (Negligible functions) $$\displaystyle{\cal N}^{e}({\mathbb{R}}^{n})$$ $$\displaystyle:=$$ $$\displaystyle\{f\in{\cal E}(\Omega):\forall K\subset\subset{\mathbb{R}}^{n}\ % \forall\alpha\in{\mathbb{N}}_{0}^{n}\ \forall p\in{\mathbb{N}}_{0}\ \exists q% \ \forall\Phi\in{\cal A}_{q}:$$ $$\displaystyle\qquad\sup\limits_{x\in K}|D^{\alpha}R(\Phi_{\epsilon},x)|=O(% \epsilon^{p})\mbox{ as }\epsilon\to 0\}$$ (iii) (Full algebra) $${\cal G}^{e}({\mathbb{R}}^{n}):={\cal E}^{e}_{M}({\mathbb{R}}^{n})\,/\,{\cal N% }^{e}({\mathbb{R}}^{n})\,.$$ Distributions are now simply embedded into ${\cal G}^{e}$ by convolution with the molifiers i.e. $$\iota(T)=[T*\Phi]$$ (50) and one obtains a differential algebra of generalised functions on ${\mathbb{R}}^{n}$ (or open subsets thereof) just as in the special version with the additional benefit of a canonical embedding of the space of distributions. Unfortunately this construction cannot be generalised to the manifold setting in a simple way as the definition of the spaces ${\cal A}_{q}$ is not invariant and moreover the embedding is not diffeomorphism invariant since convolution again depends on the linear structure of ${\mathbb{R}}^{n}$. However an invariant embedding can be achieved by demanding that the molifiers $\Phi$ transform in an appropriate way. Colombeau and Meril in [19] made the first decisive steps towards a diffeomorphism invariant full Colombeau algebra by weakening the moment conditions to only hold asymptotically (which makes them invariant) and enlarging ${\mathcal{A}}_{q}$ to a space of bounded paths $\epsilon\mapsto\Phi^{\epsilon}\in{\mathcal{D}}({\mathbb{R}}^{n})$. A flaw in this construction was found and removed by Jelínek [38] who developed an improved version of the theory which involved some subtle changes of definitions and established a number of important technical results. These ideas were then fully developed and given a firm mathematical basis in two papers dealing with the foundations of nonlinear generalised functions [27] where the first diffeomorphism invariant full Colombeau algebra on (open sets of) ${\mathbb{R}}^{n}$ was constructed. For applications in general relativity it is moreover desirable to have a geometric and global version of the theory rather than simply giving transformation rules for the local theory. Such a construction was given in [29] (for an overview see [30]). Here we only mention that the key idea is to replace the scaled and unbounded paths $(1/\epsilon)\Phi^{\epsilon}$ which are employed in the definition of moderateness and negligibility in the local theory by smoothing kernels $\Phi$ which are $C^{\infty}$ maps from $(0,1]\times M$ to compactly supported $n$-forms on $M$. In this way one obtains a geometrically constructed full Colombeau algebra on a differential manifold $M$ where the canonical embedding of distributions commutes with Lie derivatives. However this theory still lacks a canonical embedding of distributional tensors. Although the work of [29] described above provides one with an invariant embedding of scalars, one cannot simply apply this to the components of a tensor and obtain an invariant embedding of the tensor. Embedding the components of a tensor and then transforming would in general give a different answer from transforming and then embedding since multiplication by a smooth function does not in general commute with the embedding. The problem really stems from the way the convolution integrates the components of the tensor at different points of the manifold. The solution to this is to introduce some additional structure which enables one to first transport the tensor fields to the same point $p$ in $M$ so that one can then do the integration in a meaningful way. Such a transport is naturally provided by specifying a background connection, and in keeping with the spirit of the full algebra this is made an argument of the generalised tensor field. Thus a generalised tensor field depends upon a background connection $\gamma$, a smoothing kernel $\Phi$ and the point $p$. This is currently work in progress but see [90] and [49] for a more detailed description of this approach. This approach now provides a ”nonlinear distributional geometry” plus a canonical and invariant embedding of distributional objects. For example one can canonically embed the conical metric of the previous section to obtain a generalised metric. One then finds that this metric has a curvature tensor which is associated to the Dirac distribution and this shows that in any coordinate system a conical spacetime has a generalised Einstein tensor which satisfies $$\tilde{G}_{ab}\approx\tilde{T}_{ab}$$ (51) where $\tilde{T}_{ab}$ is the embedding into the Colombeau algebra of the energy-momentum tensor of a (thin) cosmic string. 9 Weak singularities and Generalised hyperbolicity As discussed earlier the definition of a singularity in general relativity is different from other field theories where one has a background metric. In general relativity one detects the presence of singularities by showing that the spacetime is incomplete in some sense. In the standard approach to singularities (see e.g. Hawking and Ellis, [32]), a singularity is regarded as an obstruction to extending a geodesic. However this definition does not correspond very closely to ones physical intuition of a singularity. This led to a consideration of whether physical objects would be subjected to unbounded deformations as one approached the singularity and was formulated mathematically in terms of strong curvature conditions. Unfortunately it is hard to model the behaviour of real physical objects in a strong gravitational field and because of this Clarke [14] suggested that one should consider instead the behaviour of physical fields (for which one has a precise mathematical description) near the singularity. According to the philosophy of “generalised hyperbolicity” one should regard singularities as obstructions to the Cauchy development of these fields rather than an obstruction to the extension of geodesics. However even a mild singularity is an obstruction if one uses the standard theory of distributions. By considering solutions to the wave equation in the Colombeau algebra one finds a certain class of weak singularities which do not prevent the evolution of test fields (Vickers and Wilson, [93]). In this section we will look at solving the wave equation on a spacetime with a locally bounded singular metric (such as the conical spacetime considered in §7). The standard Cauchy problem takes the form $$\displaystyle\square u(t,x^{\alpha})$$ $$\displaystyle=$$ $$\displaystyle 0$$ (52) $$\displaystyle u(0,x^{\alpha})$$ $$\displaystyle=$$ $$\displaystyle v(x^{\alpha})$$ $$\displaystyle{\partial_{t}u}(0,x^{\alpha})$$ $$\displaystyle=$$ $$\displaystyle w(x^{\alpha})$$ with initial data $(v,w)$ lying in the Sobolev spaces $H^{1}(S)\times H^{0}(S)$ prescribed on the initial surface $S$, given by $t=0$. Because of the form of the metric one would expect the solution (if it exists) to be defined as a distribution. This however will cause difficulty in interpreting $$\square u=(-g)^{-1/2}\partial_{a}\bigl{(}(-g)^{1/2}g^{ab}\partial_{b}u\bigr{)}$$ (53) as a distribution in the framework of classical distribution theory because the above equation has (non-constant) singular coefficients and involves ill-defined products. Again we overcome these difficulties by using the nonlinear generalised function theory of Colombeau. For an overview of the treatment of PDEs with singular coefficients, data and solutions in this setting see [65]. We first canonically embed the metric $g_{ab}$ into the full Colombeau algebra ${\mathcal{G}}^{e}(M)$ by using a convolution integral (15) as in §7 to obtain a representative $(\tilde{g}_{(\epsilon)ab})\in{\mathcal{E}}_{M}^{e}(M)$. Since the initial data $(v,w)$ does not have to be smooth, we must also embed it into the algebra as $(V,W)$ represented again by convolution integrals denoted ${v}_{\epsilon}{}$ and ${w}_{\epsilon}{}$ respectively. The generalised function wave operator acting on a generalised function $U$ represented by $u_{\epsilon}$ may then be written as $$\square^{\epsilon}{u}_{\epsilon}{}=(-{g}_{\epsilon}{})^{-1/2}\partial_{a}\bigl% {(}(-{g}_{\epsilon}{})^{1/2}{g}^{ab}_{\epsilon}{}\partial_{b}{u}_{\epsilon}{}% \bigr{)}.$$ (54) We would like to then be able to solve the Cauchy problem in the space ${\mathcal{G}}^{e}(M)$ $$\displaystyle\square U(t,x^{\alpha})$$ $$\displaystyle=$$ $$\displaystyle 0$$ (55) $$\displaystyle U(0,x^{\alpha})$$ $$\displaystyle=$$ $$\displaystyle V(x^{\alpha})$$ $$\displaystyle{\partial_{t}U}(0,x^{\alpha})$$ $$\displaystyle=$$ $$\displaystyle W(x^{\alpha})$$ and obtain a solution $U\in{\mathcal{G}}^{e}(M)$ which is associated to a distribution. In practice one works with the equivalent problem in ${\mathcal{E}}_{M}^{e}(M)$ $$\displaystyle\square^{\epsilon}{u}_{\epsilon}{}(t,x^{\alpha})$$ $$\displaystyle=$$ $$\displaystyle{f}_{\epsilon}{}(t,x^{\alpha})$$ (56) $$\displaystyle{u}_{\epsilon}{}(0,x^{\alpha})$$ $$\displaystyle=$$ $$\displaystyle{v}_{\epsilon}{}(x^{\alpha})$$ $$\displaystyle{\partial_{t}{u}_{\epsilon}{}}(0,x^{\alpha})$$ $$\displaystyle=$$ $$\displaystyle{w}_{\epsilon}{}(x^{\alpha})$$ where $({f}_{\epsilon}{})$ is negligible. In [93] it is shown how to estimate solutions ${u}_{\epsilon}{}$ of (56) and its derivatives in terms of powers of $\epsilon$, given the moderate and null bounds of ${f}_{\epsilon}{}$, ${v}_{\epsilon}{}$ and ${w}_{\epsilon}{}$, using a method of energy estimates following Hawking and Ellis [32] and Clarke [14]. The additional complication in this situation is that one needs more explicit bounds because one needs to know the precise way in which the constants depend upon $\epsilon$. This is accomplished by working with function spaces defined using higher order energy estimates (related to the super-energy tensors of Senovilla [78]) for which one has bounds in terms of the covariant derivatives of the curvature. Using this method one can show that the Cauchy problem (55) has a unique solution $U\in{\mathcal{G}}^{e}(M)$ which is independent of the representation chosen for $g_{ab}$, $V$ or $W$. In the case of a conical spacetime one can go further and show that this solution is actually associated to a distributional solution. Thus a conical spacetime satisfies the condition of “generalised hyperbolicity” as claimed. It seems likely that these techniques may then be used to show that a much wider class of spacetimes with weak singularities satisfy the generalised hyperbolicity condition. See ([63]) for some recent results in this direction. 10 Conclusion In this review we have looked at the extent to which it is possible to use conventional distribution theory to look at solutions of Einstein’s equations. Although there is an important class of new solutions that can be obtained by going beyond the confines of $C^{2-}$ metrics the largest class that one can work with that is “stable” is given by the gt-regular metrics. Such metrics can be used to describe solutions with singular support of the curvature on a hypersurface but are unable to deal with singularities of higher codimension. To deal with these it is necessary to go beyond distributions and work with a theory of nonlinear generalised functions. We have shown that such an appropriate description of nonlinear generalised functions is given by the theory of Colombeau algebras. The special theory provides a straightforward computational tool for calculating the distributional curvature of a number of singular metrics and throwing some light on the physical nature of the singularity. Furthermore it has also been possible to define generalised functions taking values in a manifold and this allows one to talk about generalised geodesics and generalised symmetries (see e.g. [2]) of a spacetime. Unfortunately due to lack of space we have had to omit from this review both this latter topic and the topic of impulsive pp-waves and ultrarelativistic black holes with non-vanishing cosmological constant (see e.g. [71]). However the special algebra does not provide one with a canonical embedding of distributions so there is always a question about the extent to which the answer depends upon the particular embedding that is used. The full Colombeau algebra rectifies this problem and a global formulation that is independent of the coordinate system has been given. 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MUFASA: Multimodal Fusion Architecture Search for Electronic Health Records Zhen Xu$\ast$, David R. So, Andrew M. Dai Equal contribution Abstract One important challenge of applying deep learning to electronic health records (EHR) is the complexity of their multimodal structure. EHR usually contains a mixture of structured (codes) and unstructured (free-text) data with sparse and irregular longitudinal features – all of which doctors utilize when making decisions. In the deep learning regime, determining how different modality representations should be fused together is a difficult problem, which is often addressed by handcrafted modeling and intuition. In this work, we extend state-of-the-art neural architecture search (NAS) methods and propose MUltimodal Fusion Architecture SeArch (MUFASA) to simultaneously search across multimodal fusion strategies and modality-specific architectures for the first time. We demonstrate empirically that our MUFASA method outperforms established unimodal NAS on public EHR data with comparable computation costs. In addition, MUFASA produces architectures that outperform Transformer and Evolved Transformer. Compared with these baselines on CCS diagnosis code prediction, our discovered models improve top-5 recall from 0.88 to 0.91 and demonstrate the ability to generalize to other EHR tasks. Studying our top architecture in depth, we provide empirical evidence that MUFASA’s improvements are derived from its ability to both customize modeling for each data modality and find effective fusion strategies. 1 Introduction In recent years, hospitals have begun adopting electronic health record (EHR) systems (Adler-Milstein et al. 2015). This digitization of large amounts of medical data offers an unprecedented opportunity for deep learning to improve healthcare, such as by predicting diagnoses (Lipton et al. 2015; Miotto et al. 2016), reducing healthcare costs (Bates et al. 2014; Krumholz 2014), and modeling the temporal correlation among medical events (Che et al. 2018; Xue et al. 2020). However, EHR data’s intrinsic longitudinal and multimodal nature adds distinct complexity that is absent from common academic datasets, such as ImageNet and WMT, that are often used to develop machine learning models. Reflecting the complexity of real-world medical information, EHR data contains multiple modalities, both structured (codes and labs) and unstructured (free-text) (Figure 1). For instance, EHR usually contains: (1) contextual features, such as patient age and sex; (2) longitudinal categorical features, such as procedure codes, medication codes, and condition codes; (3) longitudinal continuous features, such as blood pressure, body temperature, and heart rate; and (4) longitudinal free-text clinical notes, which are often lengthy and contain a lot of medical terminology. These data types differ not only in feature spaces and dimensionalities, but also in data generation processes and measurement frequencies. For example, lab tests and procedures are ordered at the physician’s discretion, while blood pressure and body temperature can be monitored on an hourly basis. Synthesizing complimentary information across these multiple modalities allows doctors to make higher quality decisions. For instance, lab tests (continuous features) provide detailed information about a patient’s physiological condition, while diagnosis codes (categorical features) capture a system-level view of the patient’s state. These modalities have complex interactions; for example, a patient’s lab results and diagnoses should be considered when trying to predict what effect a blood pressure medication would have on them. Doctors consider this diverse data, as well as previous doctors’ notes, when making decisions. Thus, modeling these modalities jointly has strong machine learning potential, but needs to be done with care, as adding modalities naively risks making overall model performance worse (Ramachandram and Taylor 2017; Baltrušaitis, Ahuja, and Morency 2018). Three questions that guide multimodal modeling are: (1) What model architecture best suits a given modality? For example, convolutional architectures are commonly applied to images, while recurrent neural networks are typically used for temporal data. These decisions are usually based on researchers’ intuitions. (2) Which modalities should be fused together? “Fusion” refers to the joint modeling of multiple modalities at once by combining their feature embeddings; popular deep learning fusion operations include addition and concatenation. An example policy is given by Neverova et al. (2015), who argue that highly correlated modalities should be fused together. (3) When we do perform modality fusion, at what point in modeling should it occur? Available fusion strategies are data-level or early fusion (Valada et al. 2016; Rajkomar et al. 2018); intermediate-level or hybrid fusion (Liang et al. 2014; Liu et al. 2014); and classifier-level or late fusion (Kahou et al. 2016; Simonyan and Zisserman 2014) (See Figure 2 for an example of each strategy). Thus far, these questions have been addressed by expert hand-designed architectures, which vary from task to task. In this work, we aim to create a generalizable framework to automatically perform this multimodal modeling without meticulous human design. To do this, we look towards neural architecture search (NAS) (Yao 1999), which has recently produced state-of-the-art results on academic datasets (Real et al. 2019). However, existing NAS works have primarily focused on unimodal data, and so our focus on applying these techniques to real-world EHR data requires us to offer solutions to tackle the complexities of multimodality. We propose MUltimodal Fusion Architecture SeArch (MUFASA), which expands the contemporary NAS paradigm to simultaneously optimize multimodal fusion strategies; that is, we jointly search for multiple independent modality-specific architectures, as well as the fusion strategy to combine those architectures at the right representation level. We base our searched models on the Transformer (Vaswani et al. 2017) because recent works have shown it can implicitly leverage EHR’s internal structure (Choi et al. 2018, 2019). Our experimental results show that our discovered MUFASA models outperform Transformer, Evolved Transformer, RNN variants, and models discovered using traditional NAS, on public EHR data. Specifically, compared with Transformer on Clinical Classifications Software (CCS) diagnosis code prediction, MUFASA architectures improve test set top-5 recall from 0.8756 to 0.9075. In addition, we empirically demonstrate that MUFASA outperforms unimodal NAS by customizing each modality specifically – an ability not available to traditional NAS. Comparing search performance with unimodal NAS on the CCS task, MUFASA improves validation top-5 recall from 0.9025 to 0.9134 with comparable search costs. What’s more, MUFASA architectures demonstrate more effective transfer to ICD-9, a different EHR task that we do not search on directly. Our contributions are summarized as follows: • MUFASA, the first multimodal NAS that jointly optimizes fusion strategy and modality-specific architectures. • A novel search space that jointly searches unique architectures for distinct modalities and the best strategies to fuse those architectures at the right representation level. • Empirical evidence demonstrating that MUFASA is superior to traditional NAS for EHR data with comparable computation costs. This includes showing that MUFASA architectures indeed achieve improvements by customizing modeling for each modality. 2 Related Works Recently, machine learning researchers have begun to leverage the multimodal nature of EHR to improve prediction performance (Shin et al. 2019). Xu et al. (2018) use both continuous patient monitoring data, such as electrocardiograms, and discrete clinical events to better forecast the length of ICU stays. Qiao et al. (2019) propose multimodal attentional neural networks to combine information from medical codes and clinical notes, which improves diagnosis prediction. As highlighted by these authors, there are few works integrating streaming and discrete EHR data (Xu et al. 2018), or clinical text and discrete EHR data (Qiao et al. 2019). Our work focuses on integrating clinical notes, continuous data, and discrete data all together. Additionally, in contrast to these manually designed multimodal architectures, we explore NAS to automatically learn architectures that leverage the multimodal nature of EHR. Our work also builds upon neural architecture search (NAS). Recent results shows that automatically designed deep learning models can achieve state-of-the-art performance on academic benchmarks (Zoph and Le 2016; Real et al. 2019), as well as offer practical usage (Tan and Le 2019). One-shot NAS methods (Bender et al. 2018) attempt to radically reduce the amount of compute needed to run searches by not training each candidate individually; here, we use a relatively low compute task and so do not need to employ these methods. Within this NAS field, our work is most closely related to two others. The first is So, Liang, and Le (2019), who apply NAS to search for a Transformer architecture on NLP data. Our work differs in that we expand our search to include multimodal fusion strategies and modality-specific architectures; in Section 5 we compare our search methodology to their unimodal setup and our resulting architectures to the product of their search, the Evolved Transformer. The second comparable work is Pérez-Rúa et al. (2019), who use architecture search to optimize multimodal feature fusion in image classification models. However, they use off-the-shelf pretrained models as building blocks and only search over their fusion points. In contrast, we are the first work to jointly optimize multimodal fusion strategies and modality-specific architectures together; this allows us to not only optimize how modalities are fused, but also the type of deep learning computation applied to each modality. Additionally, our focus is on sequence models for EHR data, not convolution-based models for images. 3 Methods In this section, we briefly describe evolutionary NAS (Real et al. 2019) and the building blocks of our search space. We also describe MUFASA, our main methodological contribution. 3.1 Evolutionary Neural Architecture Search We use the tournament selection evolutionary architecture search algorithm proposed by Real et al. (2019). In this framework, candidate architectures are represented as the gene encodings of individuals; see Section 3.3 for a description of these encodings. An initial population is created of random or pseudo-random individuals; in our case we use the warm-start NAS method (So, Liang, and Le 2019) by seeding the initial population with a known strong architecture, the Transformer. From there, evolution begins by assigning every individual in the population a fitness. These fitnesses are determined by building the architectures described by each individual’s gene encoding and training the resulting models on training data. The models are then evaluated on validation data to determine the individuals’ fitnesses. Once fitnesses are assigned, a tournament is conducted by sampling $T$ random individuals from the population and selecting the one with the highest fitness to be a parent. This parent is mutated, with its gene encoding fields randomly changed according to a mutation rate, to produce a child. The child is assigned a fitness in the same fashion as the parent. Then another tournament is conducted by sampling $T$ random individuals from the population and having the one with the lowest fitness killed, meaning removed from the population. The newly created and evaluated child is then added to the population in the killed individual’s place. This cycle of child creation and weak individual removal is repeated, creating a population of high fitness individuals, which for NAS means strongly performing architectures (Algorithm 1 in Appendix). 3.2 MUFASA To adapt to multimodal data, we reformulate the NAS search space to also include fusion strategy search. To do this, instead of searching for a single architecture, we search for several architectures simultaneously: one for each individual data modality and a special fusion architecture that is responsible for fusing data modalities together and performing further processing. Put formally, the standard NAS objective is to find an optimal neural network function (architecture) $f_{A}(x;\theta)$, parameterized by weights $\theta$, that transforms the input $x$ to a representation that is more amenable for a target task. A majority of NAS work, which has focused on unimodal datasets, searches for a single monolithic $f^{\prime}$. Likewise, several EHR works treat modalities identically, combining all $M$ data modalities together via a simplistic combiner function, such as vector concatenation (Lipton et al. 2015; Rajkomar et al. 2018; Choi et al. 2016; Li et al. 2019), before passing them to one cohesive model (early fusion): $$\displaystyle f_{A}(x;\theta)=f^{\prime}(\operatorname{concat}(x_{0},x_{1},\dots x_{M-1});\theta)$$ Here, we decompose our target architecture into a series of modality architectures, $g_{i}$, that are applied independently to each corresponding $i$th data modality. We additionally define $h$ as the special fusion architecture that takes the outputs of each $g_{i}$ and jointly transforms them into the final output: $$f_{A}(x;\theta)=h(g_{0}(x_{0};\theta_{0}),\dots g_{M-1}(x_{M-1};\theta_{M-1});\theta_{h})$$ During search, MUFASA searches for the fusion architecture, $h$, and every modality architecture, $g_{i}$ (Figure 3). This reformulates the search space, distinguishing MUFASA from previous NAS works. The basis for this reformulation is the notion that for complex data such as EHR, deep learning transformations should be specific to their input modalities; this is represented by the independent modality architectures. As previously mentioned, EHR categorical features, continuous features and clinical notes have different data representations and generative processes; that they would each benefit from distinct types of modeling is intuitive. Joint modeling across modalities is also beneficial, but needs to be applied at the right depth; the fusion architecture embodies this mentality. In Section 5, we share empirical evidence that supports these ideas that (i) distinct modeling for each modality is beneficial, (ii) proper fusion strategy is critical to model performance, and (iii) the MUFASA search space is superior to the unimodal NAS space for EHR data, when controlling for search costs. The utility of MUFASA’s multi-architecture search is its ability to jointly represent and search over several fusion strategies while performing regular architecture search. In the next subsection, we describe how we construct architectures using typical NAS blocks. Note here that every architecture can be reduced to an identity transformation or, in the case of the fusion architecture, a simple concatenation to perform early fusion. There is a shared parameter budget for the entire model, but there is no explicit constraint on how those parameters can be allocated. 3.3 Architecture Blocks Similar to previous works, each of the architectures in our search space is composed of blocks (Figure 5). Each block receives two hidden state inputs and generates a new hidden state output. The block is a computation unit that transforms each input separately and then combines two transformed outputs together to generate the final block output. The computation applied to each input is called a branch. The outputs of both branches are combined via the ‘combiner function’. The search space for a single block contains $1$ block-level search field (combiner function) and $5$ branch-level search fields (input, normalization, layers, output dimension, and activation) for each of the two branches (10 branch-level fields total). ‘Input’ specifies which previously generated hidden state will be fed into the branch. Different from previous architecture search work, MUFASA defines two types of blocks, as depicted in Figure 5. For modality-specific architecture blocks, only hidden states from the same modality can be inputs. For fusion architecture blocks, both fusion architecture hidden states and modality architecture states can be inputs. Fusion architecture blocks are constructed after the modality-specific blocks have been constructed. These input constraints ensure 1) an independent set of blocks for each modality and 2) that the fusion architecture can access the modality architectures at any representation level, as described by the multi-architecture search space in Section 3.2. Any orphaned hidden outputs are then fused with the model output. A gene encoding for a single block is represented as {left input, left normalization, left layer, left relative output dimension, left activation, right input, right normalization, right layer, right relative output dimension, right activation, combiner function}. In total, MUFASA has a search space of $1.76\times 10^{23}$ models (See Appendix for more vocabulary and search space information). The fusion architecture can incorporate the modality architectures’ outputs at any point in its own architecture; for instance, the fusion architecture can pass the Mode 1 output through its very first neural network layer, and still delay inclusion of the Mode 3 output until the final model layer. It is through being able to freely adjust these two aspects of architecture - parameter allocation and modality inclusion points - that all multimodal fusion strategies can be expressed and searched for via evolution. See Figure 4 for an example of how early, hybrid, and late fusion are all achieved. Note, not only can all fusion strategies be represented, but different fusion strategies can be assigned to different modalities. In Section 5 we detail the particularly interesting case in which the strongest model we found using MUFASA applies two different fusion strategies to the same modality (Figure 6). 4 Experiment Setup 4.1 Dataset and Prediction Tasks Dataset We use the Medical Information Mart for Intensive Care (MIMIC-III) (Johnson et al. 2016) dataset. It contains single-center real-world EHR data for 53,423 hospital admissions, admitted to critical care units from 2001 to 2012 (See Table 8 in Appendix). We represent patient’s medical histories in Fast Healthcare Interoperability Resources (FHIR) format (Mandel et al. 2016), as described by Rajkomar et al. (2018). After data pre-processing, we have 40,511 patients and 51,081 admissions. For all tasks, data is randomly split into train, validation and test sets in an $8:1:1$ ratio. Feature Modalities We use three feature modalities for the sequence data: (1) Categorical sequence features, including diagnosis and procedure codes; medication request and administration codes; and admission sources. (2) Continuous sequence features, including lab test results and vital signs such as heart rate, respiratory rate, blood pressure, body temperature, and sodium levels, when they are available. (3) Free-text clinical notes. Before feeding this data to our models, we embed the categorical sequence features and clinical notes (trained from scratch). We normalize continuous feature values to Z-scores using training set statistics and clamp outliers 10 standard deviations away from the mean. For values that are missing at particular time steps, we use the last observed value for that signal. The outputs of the searched architectures constitute the sequence representations. After concatenating these representations with additional context features (such as age), we feed the output into dense layers to generate the final task predictions. More details can be found in Appendix Section 8.5. Prediction Tasks Our experiments focus on two diagnosis code prediction tasks at discharge time for each encounter: • CCS:   Predicting the primary Clinical Classifications Software (CCS) diagnosis code (Elixhauser 1998). This is a multiclass problem and each hospital encounter has only one primary CCS code. Because there are over 250 possible diagnosis codes, we use top-5 recall (recall@5) as the main evaluation metric. • ICD-9:   Predicting the International Classification of Diseases, 9th Revision (ICD-9) diagnosis code (Slee 1978). This is a multilabel problem, as one hospital encounter could have several of the 14,000 available ICD-9 diagnosis codes. We use AUCPR as the main evaluation metric. 4.2 Baseline Search Algorithms and Model Architectures The baseline models that we compare against are the original Transformer (Vaswani et al. 2017), LSTMs (Rajkomar et al. 2018), attentional bidirectional LSTMs (Qiao et al. 2019) and the Transformer NAS variant, the Evolved Transformer, which was searched for on translation data. To demonstrate the effectiveness of our MUFASA search method, we compare it against the same unimodal NAS setup that was used by So, Liang, and Le (2019), but using our search space vocabulary (Appx. Table 5). This baseline NAS method is not amenable to multimodal inputs and so we concatenate the inputs together before feeding them into each candidate model, as is standard practice in EHR literature (Lipton et al. 2015; Rajkomar et al. 2018; Choi et al. 2016; Li et al. 2019). 4.3 Architecture Search Configuration We conduct architecture searches on MIMIC CCS. The search configurations are almost identical for MUFASA and our unimodal search baseline. Both employ the same search space and tournament selection NAS algorithm as described in Section 3. Each search uses 200 CPU workers for evaluating candidate models asynchronously. The population size is 100 and the tournament size is 30. We independently mutate each encoding field with a probability of $1.875\%$ and uniform randomly select its replacement from the possible vocabulary. For each individual architecture search, we train 5000 child models, which in every case appeared to reach convergence. In both searches the parameters for candidate models were not allowed to exceed 76 million. The total search times for both unimodal NAS and MUFASA are approximately the same: roughly 3 days. Unimodal NAS uses the early fusion Transformer to warm-start the search. To control for maximum model depth, MUFASA uses the Transformer hybrid fusion seed (Figure 2). See the Appendix for more details. 111Code is available at https://github.com/Google-Health/records-research/tree/master/multimodal-architecture-search. 5 Results 5.1 MUFASA vs. Unimodal NAS We first compare MUFASA to the equivalent unimodal NAS setup on MIMIC CCS. All relevant configurations, such as compute and hyperparameters, are identical for both searches. We run each search three times and calculate the fitness mean and standard deviation of the best models for each search method. Compared with unimodal search, MUFASA improves CCS validation top-5 recall from $0.9025~{}(0.0041)$ to $0.9134~{}(0.0034)$; this improvement is statistically significant under independent two-sample t-test with p-value threshold $0.05$. Figures 6 depicts the best architectures from each of the searches. Both architectures are substantially different from the original Transformer seeds, but in very different ways. The unimodal NAS output is “wider” than Transformer; it utilizes several wide convolutions and at many points processes the same hidden state through parallel neural network layers. Observing this pattern across multiple unimodal NAS searches, we interpret this as the architecture creating multiple “perspectives” for the same concatenated input, as it is unable to model the modalities individually. On the other hand, the MUFASA architecture scarcely performs parallel computation on the same state, but substantially silos the different modalities and assigns them each unique fusion strategies. For example, the continuous features are processed independently and are only joined with the other modalities at the very end via late fusion. Clinical notes, on the other hand, are processed independently at first, but then are joined with categorical data and processed jointly via hybrid fusion. The most interesting case is the categorical data architecture, which utilizes both hybrid and late fusion. 5.2 Architecture Study Having demonstrated that MUFASA generates better search results than unimodal NAS, we now take a closer look at the architecture it produced and explore what makes that architecture effective. We begin with a comparison to other architectures that have been applied to EHR data, as well as other NAS baselines including the Evolved Transformer and our best unimodal NAS model (Table 1). Note that for the Transformer baselines, which we run at both the default Tensor2Tensor 2 layer size and a size comparable to those found by the searches, fusion strategy has a big impact on performance. Hybrid and late fusion are comparable, but significantly outperform early fusion; fusion strategy being key to model performance is one of the chief motivations for this work. In fact, fusion strategy is so crucial that in this particular case, early fusion performs worse than training on just categorical features or just clinical notes alone (Table 3); this may be due to the lack of neural network layers that are capable of doing efficient computation across disparate feature representations with different underlying distributions. The model learned by unimodal NAS outperforms all other baselines that use early fusion; this illustrates the power of EHR-specific modeling. By combining the benefits of both searching for a critically important fusion strategy and modality-specific modeling, MUFASA produces an architecture that substantially outperforms all other baselines. The question still stands, however, as to how tailored the MUFASA architecture is to the individual EHR modalities; its performance on the target task is clearly strong, but how much of that comes from custom modeling for each data modality? To test this, we train three models with the same architecture but with alternative inputs, thereby highlighting the importance of each data modality being fed into its optimized path (Table 2). First, we perform a forced early fusion, whereby all input modalities are concatenated and fed into all input paths; this eliminates the isolated modeling of each modality. This experiment shows that the MUFASA architecture’s separate input processing is customized for each individual modality; passing all modalities into every input path together hurts performance. Second, we randomly shuffle the input modalities, passing each modality to a different modality’s input path; specifically, categorical features are fed into the continuous features path, continuous features are fed into the clinical notes path, and clinical notes are fed into the categorical features path. This maintains individual modeling for each modality, but changes that modeling from what MUFASA designates, providing evidence that customized modeling matters. Lastly, we strike a midway point between the first two experiments by performing a partial early fusion, adding just one modality to another modality’s input path; we randomly try (I) adding categorical features to the clinical notes input path and, in a separate training, (II) adding clinical notes to the categorical features path (for experimental symmetry). Table 2 shows these improper input routings can cause a statistically significant drop in quality. The only exception is Partial (Categorical$\rightarrow$Notes); however, this is likely because categorical features are stronger features than clinical notes (Table 3), so having clinical notes “share” parameters with categorical features is not as harmful. Note, this still supports the idea that modality-specific processing is sensitive, as the mirrored “sharing,” Partial (Notes$\rightarrow$Categorical), is significantly worse. Jointly, these results confirm our hypothesis that MUFASA’s improvements come not only from its independently modeling modalities, but also its customized modeling specifically for those modalities. To understand if the discovered MUFASA modeling is effective beyond our target task, we test the generalizability of our discovered architectures on the yet unseen ICD-9 task. As shown in Table 4, MUFASA demonstrates a statistically significant improvement over the strongest baseline model. The unimodal NAS architecture also seems to generalize, outperforming the strongest early fusion baseline, but is unable to improve over the hybrid fusion Transformer, highlighting its limitations when fusion strategy plays an important role. 6 Conclusion Effective modelling of EHR data has great potential to advance healthcare, from improving diagnoses to suggesting treatments. However, its complex multimodal nature has required human experts to hand-design unorthodox models or use one-size-fits-all models – an approach that does not scale as medical data becomes richer. To address this, we proposed MUFASA to automatically design deep learning architectures that directly account for the uniqueness of different modalities. Our empirical results have shown (1) MUFASA is superior to unimodal NAS on MIMIC-III CCS; (2) the discovered MUFASA architectures can outperform commonly used baseline architectures and transfer improvements to other EHR tasks; and (3) the effectiveness of MUFASA is derived from its ability to specifically model various modalities and find effective fusion strategies. Future work can investigate applying MUFASA to other types of medical data modalities, including medical images, waveforms, and genomics. 7 Ethics We see this work as being a positive step towards the democratization of AI, as it will allow non-experts to develop machine learning models for complex multimodal datasets. 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The encoding for a single block contains $1$ block-level search field (combiner function) and $5$ branch-level search fields (input, normalization, layers, output dimension, and activation) for each of the two branches (10 branch-level fields total). ‘Combiner function’ is the function applied to combine the outputs of both branches. ‘Input’ specifies which previously generated hidden state will be fed into the branch. ‘Normalization’ is the normalization immediately applied to the input, before any other transformation. ‘Layer’ is the neural network layer applied after normalization. ‘Output dimension’ defines the relative output dimension of the layer transformation (So, Liang, and Le 2019). ‘Activation’ is the non-linearity applied to the output of each layer. Together these components define the entire searchable architecture space. The vocabulary for our gene encodings is shown in Table 5. Because MUFASA searches for multiple architectures simultaneously, we maintain separate sets of blocks for each individual architecture. Specifically, for our EHR tasks there are 3 distinct modes (Subsection 4.1) and so we maintain 4 distinct block sets: one for each of the 3 independent modality architectures and one for the fusion architecture. Unlike previous works that leverage similar spaces (Zoph et al. 2018; Real et al. 2019), we do not have the concept of a larger ‘cell’ that is stacked; our dataset is relatively small and so we find having additional stacked cells does not help. 8.2 Relationship Between Figures 3-5 Figures 3-5 all describe the same MUFASA search space, but at varying levels of abstraction. Figure 3 is the highest in abstraction, giving an overview of how the different types of MUFASA architectures (modality and fusion) interact; the modality architectures are independent, and the fusion architecture combines them together. Figure 4 gives a more concrete example with some “layers” exposed, showing how different fusion strategies can be achieved at a macro level. Figure 5 is the most granular, “zooming in” to what a search space block looks like and what inputs different MUFASA blocks can take while preserving the independence assumptions of the modality blocks. 8.3 Training Details All training parameters are optimized on top of the default Tensor2Tensor hyperparameters (Vaswani et al. 2018) for a Transformer trained with an Adam optimizer (Kingma and Ba 2015). We additionally tune the learning rate schedule and batch size for all model trainings outside of searches. The possible learning rate schedules are constant, linear decay, exponential decay, single cycle cosine decay, and inverse-square-root decay. The possible batch sizes range from 16 to 512. We use a Gaussian process bandit optimization algorithm (Desautels, Krause, and Burdick 2014) to tune hyperparameters to maximize the model performance using evaluation results on the validation set. For search, we use the best baseline Transformer hyperparameters for all models; that is a single-cycle cosine decay learning rate schedule without warm-up, batch size 32, and an initial learning rate of $4.23\times 10^{-4}$. All models are trained until convergence. 8.4 Unimodal v.s. MUFASA Search Configurations The two ways in which the unimodal NAS and MUFASA configurations differ are the number of searchable blocks and starting seeds. Unimodal NAS uses 8 blocks, straightforwardly reimplementing the early fusion Transformer (Figure 2) to warm-start the search. MUFASA uses a Transformer hybrid fusion seed as shown in the middle of Figure 2. It has 3 blocks for each of the 3 modality architectures and 5 blocks for the fusion architecture, for a total of 14 blocks. This number of MUFASA blocks was not tuned, but rather was the minimum number of blocks needed to reimplement a Transformer with hybrid fusion while maintaining the same maximum achievable layer depth as the Unimodal NAS baseline. In preliminary experiments, we found that model depth has a big impact on model performance. So, in the interest of fairness, we enforce all Transformer seeds to have the same depth and a similar number of parameters. Examples of early, hybrid and late fusion seeds are shown in Figure 2. 8.5 Dataset and Data Preprocessing We use critical care data from the Medical Information Mart for Intensive Care (MIMIC-III) (Johnson et al. 2016) in our empirical studies. It contains single-center real-world EHR data for 53,423 hospital admissions, admitted to critical care units from 2001 to 2012. The mean length of stay for all encounters is around 10 days. We represent patient’s medical histories in Fast Healthcare Interoperability Resources (FHIR) format (Mandel et al. 2016), as described by Rajkomar et al. (2018). Our study cohort is comprised of adult patients hospitalized for at least 24 hours. Detailed statistics are shown in Table 8. The extracted data includes encounter information such as admission types, status and sources, diagnosis and procedure codes, observation features such as vital signs and laboratory measurements (See detailed list in Table 9), medication orders, and free-text clinical notes. Each patient record is represented as a time series. We normalize continuous feature values to Z-scores using training set statistics and clamp outliers 10 standard deviations away from the mean. Additionally, we embed categorical features. To reduce sequence length and normalize different feature frequencies, we group each feature’s values into fixed-length time periods, called bags. We aggregate all embeddings or continuous values within the same bag for each feature. In our experiments, we use daily bagging. Lastly, we concatenate the bagged embeddings and continuous values of all features into a single representation for each time period in the patient record. We ignore empty bags, which do not contain any values for the given time period. In this way, the whole patient history is represented as a sequence example. We feed the sequence examples into the searched architectures and their outputs constitute the sequence representations. After concatenating these representations with additional context features such as age, we pass the outputs into dense layers to generate the final task predictions. 8.6 Computation Cost for Multiple Modalities The number of MUFASA search fields grows linearly with the number of modalities. To justify this increase of search space size, we perform an empirical comparison between MUFASA and unimodal NAS, which has no additional search fields for multiple modalities. MUFASA strongly outperforms unimodal NAS when controlling for the number of searched individuals and both searches cost comparable amounts of time ( 3 days). The compute cost for individual architectures grows linearly with respect to the number of modalities, because additional modalities add additional feature embedding matrices. This is the same for both MUFASA and unimodal NAS. Because we control for the number of candidate model parameters, we did not find a large difference in the training time of different architecture candidates (across both MUFASA and unimodal NAS). 8.7 Discussion on Architecture Search Experimental results in Table 3 shows that the Evolved Transformer, which was searched for using NAS on translation data, does not outperform the baseline Transformer; our interpretation of this is that the inductive biases learned by NAS do not transfer across domains that are significantly disparate, a phenomenon that has been observed in other domains such as computer vision as well (Bruno et al. 2018). This is alarming because, although academic datasets such as ImageNet and WMT have nice properties for conducting machine learning research, they differ significantly from many real world datasets, such as MIMIC-III. However, while our compute cost for a search is not trivial ($2\sim 3$ CPU years in our setup but potentially much faster on GPU or TPU), this is magnitudes smaller than previous NAS works, and yields reasonable results for our particular domain. We believe this is indication that small scale NAS may be well equipped to help model smaller, real world datasets that have not received as much attention from the academic community. 8.8 Observation Features We list the observation features and their units, which are used in the experiments, in Table 9.

Limits of surface analysis of thin film compounds using LEIS Andrey A. Zameshin Industrial Focus XUV Optics Group, MESA+ Institute for Nanotechnology, University of Twente, Drienerlolaan 5, 7522 NB Enschede, The Netherlands Andrey E. Yakshin Industrial Focus XUV Optics Group, MESA+ Institute for Nanotechnology, University of Twente, Drienerlolaan 5, 7522 NB Enschede, The Netherlands Jacobus M. Sturm Industrial Focus XUV Optics Group, MESA+ Institute for Nanotechnology, University of Twente, Drienerlolaan 5, 7522 NB Enschede, The Netherlands Cristiane Stilhano Vilas Boas Industrial Focus XUV Optics Group, MESA+ Institute for Nanotechnology, University of Twente, Drienerlolaan 5, 7522 NB Enschede, The Netherlands Fred Bijkerk Industrial Focus XUV Optics Group, MESA+ Institute for Nanotechnology, University of Twente, Drienerlolaan 5, 7522 NB Enschede, The Netherlands (06 September 2018) Abstract Low Energy Ion Scattering (LEIS) was employed to study the surface composition of thin films of Ru on B, C and B4C films at different stages of growth. Effects of surface segregation of C were observed. Previously unknown matrix effects were observed in these samples, expressed in the decrease of LEIS signals of Ru, B and C at low Ru concentrations. The effect disappears for Ru-rich surfaces. Measurements with different He+ ion energies prove that the characteristic velocities of the elements involved change with surface composition. We suggest that these matrix effects appear due to the changes in neutralization efficiency in quasiresonant neutralization from the valence band (VB-qRN). This neutralization channel is present in elemental C and B due to a wide valence band with energy states as low as -20 eV, which are in a (quasi-)resonance with the He 1s level. This mechanism was earlier reported for graphitic carbon. We suggest that it can be applied to a much wider range of materials, leading to potential matrix effects in LEIS from a variety of surfaces, containing B, C and potentially O and N atoms, e.g. borides, carbides, oxides and nitrides, as well as alloys with B and C. This hypothesis is supported by additional LEIS measurements on oxidized Ru which show matrix effect in Ru-O LEIS signals as well. We argue that it is possible to avoid the matrix effects from compounded surfaces within certain ranges of composition by a proper choice of reference samples, while for other compositions knowledge of characteristic velocities is required for reliable quantification. 1 Introduction Low Energy Ion Scattering (LEIS) is a surface analysis technique with extremely low information depth, normally only the topmost atomic layer [1, 2, 3]. Extreme surface sensitivity is given by very efficient neutralization processes of noble gas ions with energies below 8 keV. The downside of this is the dependence of the LEIS signal on neutralization efficiency. Usually the neutralization efficiency of He+ from a given surface atom does not depend on the surrounding atoms, in which case quantification of surface composition with LEIS is possible by comparing LEIS signals between given a sample and a reference sample with known surface composition and density [1]. However, occasionally neutralization processes and therefore LEIS signals can depend on the surrounding atoms. This phenomenon is called a matrix effect, and it can severely complicate the surface quantification. It is often considered that cases of matrix effects are few and limited to very specific situations. In an earlier paper LEIS was used for studying the growth process of magnetron sputtered thin films of Ru on amorphous Si [4]. In this paper we extend this research to growth of thin films of Ru on several other amorphous substrates, namely B, C and B4C. The application of Ru thin films include catalysis [5], seeding and substrate layers for Cu [6, 7] or graphene [8, 9], bottom and top electrodes for DRAM capacitors [10]. Growth of Ru thin films on B, C and B4C substrates is important for a variety of multilayer structures for reflection of soft and hard X-Rays [11, 12, 13]. Interface width and composition of such systems is important to their performance, therefore they can benefit from the development of new techniques of thin film interface analysis. Analogous to the LEIS studies of the Ru/Si combination [4], the thin film deposition and LEIS measurements were performed without breaking vacuum. This allows to perform almost in-situ quantification of the surface composition for different thicknesses of Ru, and therefore compose a deposition depth profile (growth profile) [4], as opposed to the more conventional sputter depth profile, which is produced by removing the top layer by sputtering and is consequently affected by sputter-induced intermixing and preferential sputtering. An important aspect of this paper is related to the discovery of new matrix effects in Ru-B and Ru-C material combinations, and the limitations that these matrix effects impose on the quantification of Ru-B and Ru-C based thin films. We also study the origin of these matrix effects on the example of the Ru-B combination, and further extrapolate the findings to a much wider range of compounds. The results of this work are separated into two parts. In the first part (Section 3), LEIS measurements with 3 keV He+ for different Ru thicknesses are used to obtain deposition depth profiles, but it becomes evident that in a thickness range below 1 nm the quantification procedure cannot be trusted. In the second part of the paper (Section 4), the suspected matrix effect is studied for the Ru-B system by measuring LEIS signals at different energies. 2 Experimental All samples studied in this work have been produced in a home-designed UHV magnetron sputtering chamber with a base pressure up to $2\times 10^{-10}$ mbar and target-to-substrate distance of 8.6 cm. Depositions were performed with $1\times 10^{-3}$ mbar of Kr as a sputter gas. DC magnetron sputtering was used for Ru, B4C and C, while RF magnetron sputtering was chosen for B and Si. The thicknesses of the deposited thin films were controlled by quartz crystal microbalances (QCM). QCM readings were calibrated in advance, using a set of calibration thin films of which the thicknesses were extracted from fitting of CuK$\alpha$ Grazing Incidence X-Ray Reflectivity curves. The deposition chamber was connected with a UHV transfer system to the LEIS chamber, sample transfer taking about 10 minutes between the deposition and the measurement. The LEIS setup was a Qtac100 instrument by ION-TOF, with a base pressure down to $1\times 10^{-10}$ mbar, equipped with a double toroidal electrostatic analyzer and an electron impact ion source, with ion incidence angle normal to the surface of a sample and a scattering angle of 145${}^{\circ}$. All LEIS spectra are obtained with He+ primary ions. The default He+ energy was 3 keV, and in a separate set of experiments with varied ion energies He+ ions from 1 to 5 keV were used. Typical ion current during analysis was 1 - 4 nA. Whenever ion sputtering was performed, a separate ion gun with 0.5 keV Ar+ ions at angle of incidence of 59${}^{\circ}$ and ion current of 100 nA was used. To obtain the deposition depth profiles for Ru on B, B4C and C a series of samples with a varied thickness of Ru layer was produced for each material combination. The general composition of samples was the following: the initial substrate was a superpolished Si wafer covered with its native oxide, then it was covered with 4 nm of amorphous Si, then 5 nm of the chosen substrate layer (B, C or B4C), then 0 – 35 nm of Ru. Immediately after deposition each sample was transferred to the LEIS chamber and analyzed, the delay between deposition and analysis was within 10 – 15 minutes. The results of these measurements are presented below. It is worth noting that we distinguish individual samples of each series by the as-deposited thickness of Ru film, which is obtained from calibrated signal of a quartz crystal microbalance. Due to interaction of Ru with its substrate the density of the grown film is different from a bulk Ru film used for calibration. Therefore, the as-deposited Ru thickness is not equal to the actual Ru film thickness, but instead serves as a measure of the deposited amount of Ru. 3 LEIS measurements with 3 keV He+ 3.1 LEIS spectra Raw LEIS spectra of each material combination are shown in Fig. 1. Each spectrum has elemental surface peaks corresponding to 3 keV He+ backscattering from atoms of each element in the first atomic layer. Samples with Ru have a Ru “tail” at energies below the Ru peak, which represents scattering from Ru atoms located in deeper layers, followed by reionisation when the scattered particle leaves the surface. Except for three elements deposited (Ru, B, C), there are other elements observed on the surface. A small oxygen peak is always present in the spectra in roughly the same amount, suggesting a slight surface contamination during sample transfer time. B4C films have a negligible amount of Ca contamination, which may be present due to machining of the B4C target and is only visible on the surface of B4C due to the low surface energy of Ca [14]. Boron substrate films have some Fe contamination, which can be expected to happen in a stainless steel chamber with a stainless steel magnetron chimney due to the non-localized nature of the plasma during RF magnetron sputtering. Finally, most spectra feature a background exponentially rising at lower energies, which consists of ions that were sputtered from the sample surface by He+ ions. 3.2 Surface peaks in LEIS spectra The integral area of a surface peak of an element i, expressed in counts per primary ion dose (here counts/nC), is given by the following equation [1]: $$S_{i}=\frac{1}{e}\xi RP_{i}^{+}\frac{d\sigma_{i}}{d\Omega}N_{i},$$ (1) where $e$ is the electron charge, $\xi$ is the analyzer and detector instrumental factor, $R$ is a roughness factor, $P_{i}^{+}$ is the ion fraction, $\frac{d\sigma_{i}}{d\Omega}$ is the differential cross-section in area per solid angle and $N_{i}$ is the surface atomic density in (atoms/area). The quantification of surface composition is simple when most of these parameters are constant or known. In the ideal case, we expect $S_{i}\propto N_{i}$. Such proportionality is obtained under the following assumptions. First, $\xi=\mathrm{const}$, because the same setup is used for the measurements, and analyzer and detector configurations are fixed. Second, $\frac{d\sigma_{i}}{d\Omega}=\mathrm{const}$ and only depends on the ion-target atom combination, scattering geometry and ion energy. Third, as long as our thin films do not exhibit island growth, and we mostly deal with amorphous surfaces, we can assume $R=\mathrm{const}$. This factor is most probably (slightly) varying during growth, but without additional in-situ information of high frequency roughness development we have assumed a constant roughness. Finally, in the absence of matrix effects $P_{i}^{+}=\mathrm{const}$ if measurements are done at the same energy. The involvement of matrix effects will be a subject of the Section 4. When matrix effects are absent, the incident ion energies are fixed and roughness is constant, Eq. 1 becomes a simple proportionality $S_{i}\propto N_{i}$. Following Brongersma et. al. [1], we convert $N_{i}$ into surface coverage ${\vartheta}_{i}=\frac{N_{i}}{N_{i}^{ref}}=\frac{S_{i}}{S_{i}^{ref}}$. Another surface quantity is the surface atomic fraction $x_{i}=\frac{N_{i}}{\sum_{j}N_{j}}$. The surface atomic fraction $x_{i}$ is a more representative quantity than surface coverage ${\vartheta}_{i}$, because $\sum_{i}x_{i}=1$ by definition, while $\sum_{i}{\vartheta}_{i}=1$ only in absence of compaction/expansion during compound formation. To calculate $N_{i}$, $\vartheta_{i}$ and $x_{i}$ we need $S_{i}^{\mathrm{ref}}$ and $N_{i}^{\mathrm{ref}}$ of every element present in the spectra. Reference samples with sputter cleaned surfaces of Ru, B and C were prepared to obtain $S_{\mathrm{Ru}}^{\mathrm{ref}}$, $S_{\mathrm{B}}^{\mathrm{ref}}$ and $S_{\mathrm{C}}^{\mathrm{ref}}$. Using this method, we obtain $S_{\mathrm{Ru}}^{\mathrm{ref}}=23600$ counts/nC, $S_{\mathrm{B}}^{\mathrm{ref}}=356$ counts/nC and $S_{\mathrm{C}}^{\mathrm{ref}}=76$ counts/nC. However, we will see in the next paragraph that there are problems with the $S_{i}^{\mathrm{ref}}$ obtained this way. The values of $S_{\mathrm{O}}^{\mathrm{ref}}$ and $N_{\mathrm{O}}^{\mathrm{ref}}$ were taken from reference [4]. Surface atomic densities $N_{i}^{\mathrm{ref}}$ for other elements were taken as 95% of their tabulated bulk values to account for the less dense films formed by magnetron sputtering. Before calculation of $N_{i}$, $\vartheta_{i}$ and $x_{i}$ we can perform an internal calibration of $S_{\mathrm{Ru}}^{\mathrm{ref}}$, $S_{\mathrm{C}}^{\mathrm{ref}}$ and $S_{\mathrm{B}}^{\mathrm{ref}}$ with the help of a so-called “matrix effect check”, in which two LEIS signals are plotted against each other. If we ignore a minor contribution of O contamination, Ru-C and Ru-B become binary systems, and in the absence of matrix effects and compaction/expansion we expect $\vartheta_{\mathrm{Ru}}+\vartheta_{\mathrm{C}}=1$ and $\vartheta_{\mathrm{Ru}}+\vartheta_{\mathrm{B}}=1$. “Matrix effect checks” for the Ru-C and Ru-B pairs are shown in Fig. 2, where they are compared with the “matrix effect check” for the Ru-Si pair [4]. Ru-Si pair exhibits a behavior very close to linear: $\vartheta_{\mathrm{Ru}}+\vartheta_{\mathrm{Si}}=1$. The Ru-C pair exhibits more complicated behavior: at low Ru coverages both Ru and C signals increase together, but after approximately 50% Ru coverage it switches to the same linear dependence as the Ru-Si pair. The behavior of the Ru-B pair has similarity with the Ru-C pair. A strong non-linear behavior suggests the presence of a matrix effect. The proper way to study matrix effects involves measuring LEIS signals with different ion energies, which will be done in Section 4. Before the detailed study we can already assume that there are two different regimes of behavior of Ru-C and Ru-B curves in Fig. 2, separated by certain critical amount of Ru $\vartheta_{\mathrm{Ru}}\approx 0.5$, achieved at $\thicksim 1$ nm of Ru. Any measurement that falls on the left side of the curve is suspected to have a matrix effect, and any measurement on the right side follows a straight line, which is expected in absence of matrix effects. If this assumption is correct, the right side of the curve can be used for internal calibration of $S_{\mathrm{Ru}}^{\mathrm{ref}}$ and $S_{\mathrm{C}}^{\mathrm{ref}}$ when not affected by the matrix effect. For the Ru-C pair, the extrapolation gives $S_{\mathrm{Ru}}^{\mathrm{ref}}=21600$ counts/nC and $S_{\mathrm{C}}^{\mathrm{ref}}=592$ counts/nC. The value for Ru is not very different from a clean sputtered reference, however for C the difference is almost 8 times. These references values can be used to quantify the linear part of the Ru-C binary system. For the Ru-B pair we extract $S_{\mathrm{Ru}}^{\mathrm{ref}}=21970$ counts/nC and $S_{\mathrm{B}}^{\mathrm{ref}}=738$ counts/nC. The new reference value for B is two times higher than the original. For Ru/B4C samples we use the reference signals of Ru, B and C as for Ru/B and Ru/C separately. 3.3 Deposition depth profiles After the discussion of the aspects of quantification of the surface composition, we demonstrate the deposition depth profiles of Ru on different substrates (Fig. 3). Both surface coverages $\vartheta_{i}$ and surface atomic fractions $x_{i}$ for each element are presented. All data points that are subject to matrix effect are marked differently, and not used in the discussion. There are several observations to be made from these depth profiles. The fraction of Ru on the surface is monotonously increasing with the amount of Ru, the fraction of the substrate atoms is monotonously dropping – B in Ru/B, C in Ru/C, fractions of B and C added together in Ru/B4C. It is unexpected that these changes are occurring relatively slow – in the worst case, i.e. in Ru/C, some carbon atoms are still visible after 30 nm of Ru is deposited on top of the C film. There are three possible explanations: it can be caused an extremely deep intermixing, growth of Ru in large islands, or surface segregation of C. Island growth is disproved by AFM measurements of the samples at different stages of growth. The root mean squared roughness was measured from 1.5 and 15 nm Ru films on C, B and B4C. All of these samples exhibited stochastic roughness with an RMS value from 0.14 to 0.21 nm, similar to the initial substrate roughness, or to Ru grown on Si [4]. Therefore, island growth modes can be excluded. To prove that a very slow disappearance of C from the surface cannot be explained by simple intermixing of C with Ru, we created a sample with a drastically reduced amount of C. A thickness of 3.2 nm of Ru was deposited on top of 0.4 nm of B4C on a Si substrate. The LEIS spectrum of the surface of this sample is shown in Fig. 4. The surface coverage of C in this spectrum is about 10%. A simple estimation shows that if all of the C atoms from the B4C layer were spread homogenously in the Ru layer, it would result in 3% volume concentration of C. This means that the C content within the Ru film cannot be constant or decaying toward the surface, but must increase at some point. Therefore, a surface segregation phenomenon is required to explain the observed effect. The surface segregation of C on Ru is a well-known phenomenon in graphene growth [9], and we can apply it to Ru on C growth as well. Even after the interface transition from C to Ru is finished and the Ru layer becomes bulk, some C still continues to stay on the surface, only very slowly dissolving in the Ru matrix under the incoming flux of Ru atoms. Previous experiments [4] showed that Si is also segregating on top of Ru. We cannot exclude the possibility of surface segregation of B as well. Another notable feature, which we will call “uplifting”, is related to the behavior of C is found in Ru/B4C deposition depth profiles (Fig. 3). While the fraction of Ru increases and the fraction of B decreases, the fraction of C has a more complex behavior – initially it increases, and only starts to drop when B disappears. This effect goes well beyond the thickness range involved in the matrix effect. We can assume that this feature is associated to surface segregation of C as well. If we assume that surface segregation only occurs on certain sites on a Ru surface, then the initial increase of the C signal is associated with the increase of the amount of the available sites. Upward diffusion of C atoms slows down with increasing thickness of the Ru layer, and after $\thicksim 2.5$ nm of Ru it is overcome by the process of removal of already segregated C atoms by arriving flux of fresh Ru atoms. Due to the fact that B and C fractions behave independently in Fig. 3, we conclude that B4C is decomposing during the deposition of Ru. Decomposition of B4C was already shown by XPS studies of this layer system [15]. The last observation from Fig. 3 is about the presence of C in the Ru/B deposition depth profile. In a pure B sample there is no detectable amount of C, but a small and nevertheless certain amount of C appears after 2 nm of Ru are deposited (also beyond the thickness range involved in the matrix effect). For comparison, no C signal was detected in Ru/Si growth [4]. The B target that was used for deposition always has 1-3% C doping. This amount of C is undetectable in a pure B sample. Its appearance in later samples is a sign of surface segregation of C on Ru, which brings C atoms to the surface. 4 LEIS measurements with different He+ energies The matrix effects in Ru-B and Ru-C in Fig. 2 require additional attention. In this section we study the origin of matrix effects with focus on Ru-B material combination. We follow the established technique to study matrix effects in LEIS [16, 17], which we earlier used to study matrix effects in La-containing surfaces. The measure of neutralization efficiency is a so-called characteristic velocity $v_{c}$, which is in an integrated electron transfer rate from the surface to an ion. To obtain characteristic velocity for a given ion-target combination, LEIS signal needs to be measured at different incident ion energies and then plotted as a function of ion velocity in the special coordinate set given by Eq. (5) in [17]. To measure characteristic velocity of pure Ru and pure B as reference samples, 20 nm Ru and 15 nm B thin films were deposited on Si wafers. LEIS signals of Ru or B were measured with several incident ion energies ranging from 1 to 5 keV. Before each measurement the surface was sputtered by a 0.5 keV Ar${}^{+}$ beam until signal saturation. The data for Ru is shown in Fig. 5 and the data for B is shown in Fig. 5. The next step was to perform similar measurements for the mixtures of Ru and B of different compositions, for which samples of 0.9 and 0.5 nm Ru on 5 nm B on Si wafer were chosen. To extend the range of measured surface compositions without depositing more samples, after the initial LEIS measurement each of these samples was sputtered with 0.5 keV Ar${}^{+}$ ion beam of $3\times 10^{14}$ atoms/cm2 fluence and measured again. The next step of data analysis is to extract characteristic velocities $v_{c,\mathrm{Ru}}$ and $v_{c,\mathrm{B}}$ from the slopes of linear fits of the inversed velocity plots and the atomic surface densities $N_{\mathrm{Ru}}$ and $N_{B}$ from the intersection points of the linear fits, following the same procedure as in [17]. The results are shown in Fig. 6 separately for Ru and for B. Additional measurements were also performed on a reverse layer system as well: a sample of 0.5 nm B on 10 nm Ru was deposited and measured in the same fashion. To achieve different surface compositions for the reverse system, instead of a single sputter step, 3 sputter steps were done: with 1.5, 3 and 4.5 $\times 10^{14}$ atoms/cm2 fluences. The data for B/Ru films is shown in Fig. 6 as well. A simple conclusion from Fig. 6 is that the characteristic velocities of both Ru and B strongly change with the surface composition regardless of the layer ordering. This proves the existence of matrix effect in the Ru-B material combination and associates it with the composition of the surface and not a specific atomic arrangement. To understand the mechanism of this matrix effect we need to study its behavior. $v_{c,\mathrm{B}}$ changes almost linearly with composition, while $v_{c,\mathrm{Ru}}$ shows noticeable deviation from a linear behavior. However, $v_{c,\mathrm{Ru}}$ can be plotted as a function of $N_{B}$, as it is done in Fig. 7. In these coordinates both $v_{c,\mathrm{B}}$ and $v_{c,\mathrm{Ru}}$ change linearly with the amount of B on the surface. The slopes of both lines are also rather close, which suggests that the relative changes of characteristic velocities of Ru and B happen together, i.e. these changes depend on the surface composition but are not element-specific. It means that neutralization mechanism responsible for the matrix effect is of non-local nature, and also justifies the use of the Hagstrum model for the calculation of the inversed velocity from the incident ion energy [17] in Fig. 5. Due to the abovementioned issues the work function cannot explain the observed matrix effect. It means that among two matrix effects observed for La in Chapter [17] the low work function effect (matrix effect in C-RN) is not involved; and oscillatory matrix effect matrix effect (matrix effect in qRCT) is not applicable in the current state as is expressed in changes of structuring of the ion yield curve, while our present matrix effect changes characteristic velocity. While this disqualifies matrix effect in qRCT, we should note that we cannot exclude presence of weak oscillations in Ru and B ion yields simply because of the lack of density of experimental points. However, none of these elements have atomic levels in proximity of He 1s level [18], which are required for presence of qRCT in the first place. 5 (Quasi-)resonant neutralization from the valence band Matrix effects in C-RN and qRCT are not the only two matrix effects recognized in LEIS field. There is also a separately classified matrix effect in graphitic carbon, which appears due to a neutralization mechanism solely mentioned in connection to this matrix effect: (quasi-)resonant neutralization from the valence band of the target to the ground state of the projectile [19, 20] (Fig. 8). Due to the lack of standardized abbreviation we will call it VB-qRN in this thesis. VB-qRN is a non-local mechanism, because much like C-RN or CIN it originates from a wide band which spans multiple atoms in the solid. VB-qRN can only occur when the valence band is wide enough to become resonant with the ground state of the projectile; as demonstrated in [20] this condition is satisfied for graphitic carbon. For most other materials, resonance between the bottom of the valence band and the projectile ground state is only possible at very small distances due to promotion of projectile levels in a close collision (CIN, collision-induced neutralization, or resonant neutralization in close collision). VB-qRN can be viewed as an intermediate case between CIN and qRCT. It is an extreme case of CIN with a threshold energy close to zero, when no promotion is required. It is also an extreme case of qRCT with a very wide band [21], which severely reduces the probability of reionization due to small hole lifetime within the band, and therefore makes for more efficient neutralization. It also causes severe dampening of ion yield oscillations as a function of inversed ion velocity [20]. VB-qRN relies on the unusually wide valence band of the target. We performed a literature search for calculations of the density of states (DOS) of valence electrons of different states of C, B and Ru, and the results are shown in Table 1. Instead of comparing the full DOS, we only focus on the difference between the energy of the lowest state of the valence band and the Fermi level. We will call this energy $E_{\mathrm{lowestVB}}$. The uncertainty of such calculations is significant, therefore comparisons of $E_{\mathrm{lowestVB}}$ obtained with different techniques can be misleading. For an example of graphite, we were able to find $E_{\mathrm{lowestVB}}$ from 26 eV [22], obtained from AES measurements, to 20 eV [23], calculated by the tight binding model. Such uncertainties cannot be avoided, but in several cases a relative comparison of $E_{\mathrm{lowestVB}}$ can be done for the calculations performed in the same source. Assuming typical value of work function of 4 eV, the valence band will be resonant with He 1s level as long as $E_{\mathrm{lowestVB}}>24.6-4=20.6$ eV. By this measure graphitic C will be subject to VB-qRN mechanism. Metal carbides, e.g. MoC [24], have much narrower valence band and will not have this neutralization channel. Therefore C in graphite will have much higher characteristic velocity than C in MoC, leading to a matrix effect. Comparison of graphitic C with diamond and amorphous C [23] shows that all of these states of C have $E_{\mathrm{lowestVB}}>20$ eV, which means that VB-qRN should be present in all elemental C and not only in graphite. We believe that all of the conclusions from the work of Průsa et al. [20] can be applied to diamond and amorphous C as well. This allows to explain the behavior of Ru-C LEIS signal pair in Fig. 2. The range of compositions starts with a pure amorphous C film, which has strong valence band neutralization, which reduces all LEIS signal (similar to situation with graphitic C). This effect is non-local, since it originates from the valence band, therefore Ru signal is affected as well. Ru carbides do not form at normal conditions [31, 32], therefore with increasing amount of Ru on the surface of the sample amount of electrons originating from valence band of C film proportionally reduces, and additional neutralization channel weakens. The decrease in the strength of neutralization causes an increase of the signals of both Ru and C, which is reason for a positive slope of the left side of Ru-C curve in Fig. 2. The right part of the curve behaves differently. Our assumption is that after a certain critical amount of Ru ($\vartheta_{\mathrm{Ru}}\gtrapprox 0.5$, achieved at $\thicksim 1$ nm of Ru) is deposited on the surface of the sample, no carbon atoms are bonded to each other in sufficient quantity to form a wide valence band, and additional neutralization channel disappears; matrix effect no longer present. This is represented by linear Ru-C dependence on the right side of Ru-C curve in Fig. 2. Table 1 shows that while the $E_{\mathrm{lowestVB}}$ of elemental B also varies between allotropes and different calculations, it can be expected in the range of 16 to 22 eV. Most of the calculations feature one or several sub-bands below the valence band, which may or may not be separated in the actual sample due to additional broadening of the levels. It means that bottom of the valence band of elemental B can potentially be in the strict resonance with He 1s level, and otherwise is in the quasiresonance with an energy mismatch of up to 4 eV. For qRN such energy mismatch still yields strong resonances [1, 17], therefore we can expect VB-qRN in elemental B as well. B-rich ruthenium borides can have high $E_{\mathrm{lowestVB}}$ [29], which is noticeably decreasing with decreasing amount of B (from 20 for RuB4 to 14 for RuB2). The strength of quasiresonance decreases with increasing energy mismatch, therefore VB-qRN will gets weaker with decreasing amount of B. Ru has a much smaller $E_{\mathrm{lowestVB}}$ [30], for which VB-qRN is impossible. It means that in the transition B - RuBx - Ru strength of VB-qRN is constantly decreasing, leading to a constant decrease of characteristic velocity, leading to a matrix effect in Ru-B material combination. With the help of VB-qRN we can explain additional details of the matrix effect in Ru-B that were observed in the section 4. First of all, matrix effect is observed in both Ru and B LEIS signals, and relative change of characteristic velocity with composition is the same for B and Ru (Fig. 7). VB-qRN is a non-local neutralization mechanism, which explains why it has the same effect on He ions scattered from both Ru and B atoms. Second, we can explain why $v_{c,\mathrm{B}}$ in Fig. 6 changes linearly with $N_{\mathrm{B}}$, while $v_{c,\mathrm{Ru}}$ deviates from a straight line in Fig. 6 but behaves more linearly in Fig. 7. The reason for this is that VB-qRN originates from the bottom part of the valence band, which is formed by B-B and Ru-B bonds [29] and therefore depends on the amount of B more than Ru. In this section we established that the matrix effect in Ru-B mixtures occurs in a wide range of compositions (Fig. 7), from approximately 1:1 mixture to pure B, but from this figure we are unable to say at which critical concentration of B in the mixture the matrix effect disappears. It appears that there are $v_{c,\mathrm{Ru}}$ and $v_{c,\mathrm{B}}$ change linearly with B concentration, but if the linear trend is extrapolated to $N_{\mathrm{B}}=0$, it would lead to conclusion that a small amount of B added to Ru leads to change in $v_{c}$. However, at $E_{\mathrm{lowestVB}}$ or pure Ru a small change $E_{\mathrm{lowestVB}}$ should not lead to appearance of VB-qRN. Therefore, there must be a critical concentration at which the linear trend is no longer sustained. The only estimation of critical concentration remains from Fig. 2, therefore a question of quantification of Ru-rich mixtures with LEIS can be considered open until more measurements of characteristic velocities are done. 6 Discussion Ru-B and Ru-C material combinations are not the only cases where matrix effects related to a band structure can be observed. Technically, all borides and carbides should exhibit such matrix effects if corresponding pure elements are chosen as reference samples. This is simply because VB-qRN is present in elemental B and C but absent in other elements, therefore $P^{+}$ of at least one of the elements will be different in the mixture. The matrix effects should also propagate to situations where both elements have VB-qRN, for example a boron-carbon mixture (B4C), because valence band shape and therefore neutralization efficiency in B4C is different from any elemental B or C. We expect that in a lot of cases matrix effects can be avoided by a proper choice of reference samples. For example, when measuring surface composition of certain metal carbide, one can use a pure surface of that metal as a reference for its LEIS signal, but reliable reference of C should only be obtained from extrapolation of a linear part of the pairwise signal comparison (Fig 2), and not from any form of elemental carbon. This approach will work for any metal-rich surface composition that does not yet exhibit VB-qRN. For very carbon-rich surfaces in the presence of VB-qRN, the neutralization efficiency will be changing as a function of surface composition, and therefore the only reliable quantification method will be to measure the LEIS signal at different incident ion energies and extrapolate to infinite energy. With regard to transition metal borides with very delocalized B 2s-2p states, similar to RuBx (OsBx [29], CoBx, RhBx and IrBx [33], ZrB2 and HfB2 [34, 35]), the compositional range of a constant neutralization efficiency might be narrow (see Section 4), therefore the quantification of surface composition of transition metal borides will most probably require measurements at different energies. A slightly different situation arises with regard to hexaborides, as well as metal oxides and nitrides. Alkaline earth and rare earth hexaborides have much more localized B 2s-2p states [36, 37], which do not belong to a wide continuous valence band, yet are located at energies of 16 eV below the Fermi level. Metal oxides have O 2s states with similar energies: 19 eV for RuO2 in different phases [38], 19 eV for TiO2 in different phases [39], 17 eV for ZrO2 [40], 19 eV for MgO and 20 eV for Al2O3 [41]. Metal nitrides have N 2s states at 17 eV for Cr-Ti nitrides [42], 16-18 eV for different Ru nitrides [30], 16-17 eV for different phases of MoN [43] or 14 eV for MoN [24], 16 eV for CrN and TiN [24], 18 eV for VN [24]. These isolated bands from B 2s-2p, O 2s or N 2s states can also be in a (quasi-)resonance with He 1s. This is the charge exchange process commonly known as qRN or qRCT. In the paper [17] we treated the target levels associated with qRCT as atomic levels, without taking their broadening into account. However, being a part of a solid, these levels can be treated as electron bands of a certain width. It is known that width of a band involved in a resonance affects the effectiveness of neutralization channel [21], which is expressed in dampening the oscillations of ion yield and increasing the characteristic velocity. It is also known that the radial distribution function of electron probability distribution is more narrow for d-electrons than for p- and s-electrons, which also means that energy splitting of d-electrons is lower and d-bands are more localized in space and in energy compared to p-bands. All strong oscillations in LEIS ion yield were produced by resonances with d-bands only [1]. In our earlier work [17] we observed much weaker oscillation of ion yield in He+ scattering from La in different chemical states. We attributed the oscillations to quasiresonance between He 1s and La 5p levels, but the reason for severe dampening of oscillations was unclear. Now we can say that the broadening of p-orbitals compared to d-orbitals is the most likely explanation, that does not involve the use of an orbital symmetry rule, contrary to, for example, Souda et. al. [44]. There is a lot of similarity between qRN from deep lying valence band states and qRN from relatively wide (i.e. s- or p-) non-valence bands. Changes in width and position of those bands can also affect the neutralization efficiency. Damplening of oscillations and increase of neutralization efficiency are caused by the same effect [21]. In the earlier paper [17] we explained the changes in $v_{\mathrm{c}}$ of La by low work function matrix effect, however part of these changes might originate from qRN from B 2s-2p in LaB6 and O 2s in La2O3. For the surfaces that are not subject to low work function matrix effect, for example for transition metal oxides, the change of characteristic velocity due to qRN can become dominant. Therefore, we can potentially expect qRN-related matrix effects in a variety of metal borides, carbides, oxides and nitrides. To experimentally verify this prediction at least for Ru oxide, a simple proof-of-principle experiment was performed. A pure thin film of Ru was deposited by magnetron sputtering, and without breaking the vacuum exposed to atomic oxygen from a plasma source at a pressure of $1\times 10^{-4}$ mbar for ¿1 hr at room temperature. The procedure should result in the formation of stoichiometric RuO2 thin film of $\approx$1 nm thickness. LEIS measurements of sample surfaces after the treatment did not show any quantifiable amount of Ru on the surface, which can be attributed to additional adsorbed O atoms on the surface. After mild 0.5 keV Ar+ sputtering LEIS signals of both Ru and O atoms can be reliably measured. The results of the measurements are shown in Fig. 9 for both Ru and O peaks. There is a noticeable difference in characteristic velocities of pure Ru and oxidized Ru. There is also a difference between characteristic velocities of O from untreated surface (RuO2 + adsorbed O) and sputter-cleaned surface (RuO2 only). Understanding of the actual behavior of neutralization mechanisms in Ru oxides goes beyond the scope of this work, but this proof-of-principle experiment supports our prediction about existence of matrix effects related to neutralization O 2s levels. 7 Conclusions In this work 3 keV He+ scattering was used to measure the surface composition of thin films of Ru on B, C and B4C films at different stages of growth. Surface segregation of C on Ru and strong intermixing were observed in all cases. Matrix effects in He+ scattering from surfaces with a low Ru concentration ($\vartheta_{\mathrm{Ru}}\lessapprox 0.5$, achieved at $\thicksim 1$ nm of Ru) were observed. Dedicated studies of matrix effects were performed on the Ru-B material combination. After measuring the Ru and B signals with several He+ energies we observed a roughly linear increase of the characteristic velocities of Ru and B with the increase of B content. This phenomenon cannot be explained by a low work function matrix effect, which originates in from resonant neutralization from the conduction band of the target to excited levels of He+. Yet, we explain this effect by the presence of quasiresonant neutralization from the valence band of B and RuBx targets. Elemental B and Ru borides have wide valence bands with low lying states ($E_{\mathrm{lowestVB}}=20$ to 14 eV) that can be in quasiresonance with the He 1s level, while the valence band of elemental Ru cannot participate in such quasiresonance ($E_{\mathrm{lowestVB}}=7.5$ eV). This difference results in the changes of neutralization efficiency and therefore in LEIS signals, which gives the matrix effect in Ru-B surfaces. A similar matrix effect is observed for Ru-C surfaces. We further predict that qRN-related matrix effects of a similar mechanism can be applied to a much larger variety of compounds, i.e. metal borides, carbides, oxides and nitridies. A proof-of-principle experiment on oxidized Ru film shows a matrix effect in the Ru-O material combination as well, which can be attributed to qRN from O 2s levels. Further validation of this prediction necessitates systematic comparisons of characteristic velocities between pure elements and their compounds. We suggest that such measurements are necessary for reliable surface quantification in LEIS. We hope that this work will encourage more research on matrix effects originating from VB-qRN and qRN, and henceworth the limits of quantification of compounds by LEIS imposed by the presence of matrix effects. 8 Acknowledgements This work was carried out in the Industrial Focus Group XUV Optics at the MESA+ Institute for Nanotechnology at the University of Twente, and we acknowledge the support of the industrial partners ASML, Carl Zeiss SMT, Malvern Panalytical, and TNO as well as the Province of Overijssel and the Dutch Organization for Scientific Research NWO. 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Experimental quantum repeater without quantum memory Zheng-Da Li Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China    Rui Zhang Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China    Xu-Fei Yin Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China    Yi Hu Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China    Yu-Qiang Fang Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China    Yue-Yang Fei Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China    Xiao Jiang Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China    Jun Zhang Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China    Li Li Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China    Nai-Le Liu Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China    Feihu Xu feihuxu@ustc.edu.cn Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China    Yu-Ao Chen yuaochen@ustc.edu.cn Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China    Jian-Wei Pan pan@ustc.edu.cn Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China Abstract Quantum repeaters – important components of a scalable quantum internet – enable the entanglement to be distributed over long distances. The standard paradigm for a quantum repeater relies on a necessary demanding requirement of quantum memory. Despite significant progress, the limited performance of quantum memory makes practical quantum repeaters still a great challenge. Remarkably, a proposed all-photonic quantum repeater avoids the need for quantum memory by harnessing the graph states in the repeater nodes. Here we perform an experimental demonstration of an all-photonic quantum repeater using linear optics. By manipulating a 12-photon interferometer, we implement a $2\times 2$ parallel all-photonic quantum repeater, and observe an 89% enhancement of entanglement-generation rate over the standard parallel entanglement swapping. These results open a new way towards designing repeaters with efficient single-photon sources and photonic graph states, and suggest that the all-photonic scheme represents an alternative path – parallel to that of matter-memory-based schemes – towards realizing practical quantum repeaters. Recent years have seen enormous interest in quantum communication driven by its remarkable features of secure communication Xu et al. (2019), quantum teleportation Bouwmeester et al. (1997) and distributed quantum computing Ladd et al. (2010). Photons are considered to be the optimal medium for quantum communication because of their flying nature and compatibility with current telecommunications networks. However the maximum communication distance is currently severely limited by photon loss in quantum channels, such as optical fibres. One viable solution is to use satellites as relays to transmit photons over a free-space channel Yin et al. (2017); Liao et al. (2018). In fibre-based telecommunications networks, quantum repeaters are believed to be the most promising way to overcome the distance limit Sangouard et al. (2011). The standard paradigm for a quantum repeater Briegel et al. (1998); Duan et al. (2001) consists of three basic technologies namely entanglement swapping Żukowski et al. (1993); Pan et al. (1998), entanglement purification Pan et al. (2001, 2003) and quantum memory Chou et al. (2007); Moehring et al. (2007); Yuan et al. (2008). Recently, significant progress has been made both theoretically Zwerger et al. (2012); Munro et al. (2012); Muralidharan et al. (2014) and experimentally Chen et al. (2017); Xu et al. (2017); Kalb et al. (2017). However, the limited performance of current quantum memories Yang et al. (2016) remains a major obstacle in realizing practical quantum repeaters unless there is a future experimental breakthrough. An all-photonic quantum repeater Azuma et al. (2015) eliminates the need of matter quantum memories. The main concept is analogous to the idea behind measurement-based quantum computation Raussendorf and Briegel (2001). Unlike conventional repeaters (e.g., Fig. 1a), the all-photonic scheme introduces an explicit construction of a repeater graph state (RGS) consisting of a complete subgraph of $K$ core photons each connected to an additional photon to form $K$ external arms. Fig. 1b shows an example with $K=4$. This approach presents the resilience against photon loss, and also avoids the coherence time limitations of quantum memories and the long-distance heralding requirement. These features have led to all-photonic quantum repeaters attracting much attention recently Bruschi et al. (2014); Pant et al. (2017); Buterakos et al. (2017); Ewert et al. (2016); Ewert and van Loock (2017); Hasegawa et al. (2019). Here, we demonstrate an all-photonic quantum repeater experimentally by manipulating 12 photons generated by 6 independent spontaneous parametric down-conversion (SPDC) crystals. We construct a 12-photon interferometer and verify the ability of manipulating 12 photons by measuring the photon distribution in the $Z$ basis, where $Z$ is the Pauli matrix $\sigma_{Z}$. We successfully demonstrate the concept of all-photonic quantum repeater by realizing an enhancement of entanglement generation rate compared with the conventional parallel entanglement swapping. First, we explain the theory behind the all-photonic scheme by considering an entanglement swapping protocol, which is an important way of sharing Einstein-Podolsky-Rosen (EPR) pairs over long distances. To obtain higher success rates and longer distribution distances, we can employ $M$ parallel channels and $N$ repeater nodes. Fig. 1a shows a simple example with $M=2$ channels and $N=1$ repeater node (Charlie). In such parallel schemes, the entanglement generation rate still decays exponentially with respect to the number of nodes $N$, because a successful event is produced only when all EPR pairs in the same row survive. If each EPR pair has a survival probability $\eta$, the overall entanglement generation rate is $M\eta^{N+1}$. Remarkably, if the repeater nodes can generate an RGS, as shown in Fig. 1b, the RGS can serve as a switch that connects EPR pairs in different parallel channels. This enables the entanglement generation rate to reach $M^{N+1}\eta^{N+1}$, which represents an $\mathcal{O}(M^{N})$ increase over conventional repeaters without memory. In theory, a large RGS can replace matter quantum memories in a repeater node, thus allowing us to realise an all-photonic quantum repeater Azuma et al. (2015); Pant et al. (2017); Buterakos et al. (2017). Demonstrating all-photonic quantum repeater experimentally is challenging, due to the difficulty of preparing a large RGS. To simplify the implementation, we create an experimentally feasible scheme, as shown in Figs. 1c and d. In the original proposal (see Fig. 1b), the essence of RGS at the repeater nodes is to switch between two functions, (i) establishing entanglement if the entanglement generation succeeds; and (ii) disentangling the qubit if the entanglement generation fails. We utilize the GHZ state and the passive choice measurement (PCM) to realize the switching between those two functions. On the one hand, GHZ state is local-unitary equivalent to a complete graph state. If we perform the Bell state measurement (BSM) between a qubit composing an $m$-partite GHZ state and a qubit composing another $n$-partite GHZ state, we obtain an ($m+n-2$)-partite GHZ state, which establishes the entanglement. If we perform the $X$-basis measurement on the GHZ state, we can disentangle the unwanted qubit. One the other hand, we design the PCM to perform the switching passively, in order to make it possible with fewer single photons. Specifically, the PCM performs a BSM when a coincident detection occurs in the two outputs of the circular polarisation beam splitter (CPBS), whereas it automatically performs a standard projection measurement in the $X$ basis when the photon is detected only in one of the two outputs of the CPBS (see Supplementary Information). By doing so, we do not need the active feed-forward operation to decide which photon would be connected or disconnected. Note also that the repeater node, Charlie, uses a delayed preparation of the GHZ-state which means that Charlie prepares the GHZ state just before the arrival of photons from Alice and Bob. This could enable us to assume that the GHZ state is lossless compared with the photons sent from distant nodes of Alice and Bob. Fig. 2 shows an overview of the experimental setup. A pulsed ultraviolet laser with a central wavelength of $390~{}\text{nm}$, a pulse duration of $150~{}\text{fs}$ and a repetition rate of $80~{}\text{MHz}$ is subsequently focused on six sandwich-like combinations of $\beta$-barium borate crystals (C-BBO) to generate six EPR pairs $\ket{\Phi^{+}_{i,j}}$ via the SPDC processing. Each C-BBO consists of a half wave plate (HWP) sandwiched between two 2-mm-thick, identically cut $\beta$-barium borate crystals. Here, $\ket{\Phi^{+}_{i,j}}=(\ket{HH}+\ket{VV})/\sqrt{2}$, where $\ket{H}$ and $\ket{V}$ denote the horizontal and vertical polarisation states of a single photon. To remove frequency distinguishability among the independent photons, we apply narrow bandpass filters with full width at half maximum (FWHM) wavelengths of $\lambda_{\text{FWHM}}=3$ nm and $\lambda_{\text{FWHM}}=8$ nm to the e-ray and o-ray photons respectively. With filtering, the overall system efficiency is quantified to be $38\%$ on average. We typically operate each C-BBO at a down-conversion probability $p=0.0344\pm 0.0001$, obtaining a twofold coincidence counting rate of $3.97\times 10^{5}~{}\text{s}^{-1}$ for each of the 6 EPR pairs with a pump power of $500~{}\text{mW}$. The fidelity of each EPR pair is above $96\%$. Of the six EPR pairs, $\ket{\Phi^{+}_{1,2}}$ and $\ket{\Phi^{+}_{3,4}}$ ($\ket{\Phi^{+}_{9,10}}$ and $\ket{\Phi^{+}_{11,12}}$) belong to Alice (Bob), while $\ket{\Phi^{+}_{5,6}}$ and $\ket{\Phi^{+}_{7,8}}$ belong to Charlie. And the EPR pairs of Charlie are used to prepare the four-photon GHZ state. By overlapping the photons 5&7 on a PBS, we obtain the four-photon GHZ state $\ket{\text{GHZ}_{4}}=(\ket{HHHH}+\ket{VVVV})/\sqrt{2}$. After preparation, photons 2, 3, 6 and 7 (5, 8, 9 and 12) are send to the node C1 (C2). Then, four PCMs are performed on photons 2&6, 3&7, 5&9, 8&12. Here, we use movable prisms to adjust the time delays of independent photons. The CPBS in each PCM device is realised by a normal PBS with four HWPs at $22.5{}^{\circ}$. A quarter-wave plate, an HWP and a PBS are placed at each CPBS output to perform $Z$ measurements on the photons. Depending on the PCM results, we can obtain final entangled states involving different photon pairs, namely 1&11, 4&11, 1&10 or 4&10 (see Supplementary Information). So far, the whole setup can be thought of as a 12-photon interferometer. To verify its ability to manipulate 12 photons, we measure the photon distribution in the $Z$ basis and show these results in Supplementary Information. First, we carry out full tomographic measurements on the four-photon GHZ state to reconstruct the density matrix $\rho_{\text{GHZ}_{4}}^{\text{re}}$. Here, we choose the measurement bases $\ket{k}\ket{l}\ket{m}\ket{n}$, where $\ket{k},\ket{l},\ket{m},\ket{n}\in[\ket{H},\ket{V},\ket{D},\ket{A},\ket{L},% \ket{R}]$, with $\ket{D/A}=1/\sqrt{2}(\ket{H}\pm\ket{V})$ and $\ket{R/L}=1/\sqrt{2}(\ket{H}\pm{i}\ket{V})$. In principle, a total of $1296$ measurement settings are needed. In practice, however, the orthogonal states can be measured simultaneously, meaning that only $81$ measurement settings are required in our experiment. For each setting, we record fourfold coincidences for $60~{}\text{s}$, yielding a coincidence rate of $370~{}\text{s}^{-1}$. This enables us to reconstruct the density matrix $\rho_{\text{GHZ}_{4}}^{\text{re}}$ from the measured data using the maximum likelihood method James et al. (2001). The results are shown in Fig. 3a. Ideally, the density matrix of a four-photon GHZ state would consist of only four real nonzero terms $\ket{0000}\bra{0000}$, $\ket{0000}\bra{1111}$, $\ket{1111}\bra{0000}$ and $\ket{1111}\bra{1111}$, and we can clearly see from Fig. 3a that the structure of the experimental density matrix is close to the ideal. We also use reconstructed density matrix to calculate the fidelity $F=\bra{GHZ_{4}}\rho_{\text{GHZ}_{4}}^{\text{re}}\ket{GHZ_{4}}=0.896$, which indicates that the prepared four-photon GHZ state is genuinely four-partite entangled. Next, we characterise the four PCMs experimentally. As is well known that PCM devices can be completely characterised by a measurement operator. If the PCM conduct single-qubit projection measurements, its performance is determined by the extinction ratio of PBS and HWP. In our case, the high extinction ratio of (nearly) $1000:1$ guarantees that the PCM device’s projection measurements are correct. When two photons arrive simultaneously at a PCM device, the frequency distinguishability (when not eliminated by filters), is the dominant factor affecting the fidelity. To reconstruct the measurement operators $M_{\Phi^{+}}^{\text{re}}$ and $M_{\Psi^{+}}^{\text{re}}$, we perform quantum detector tomography on the PCM Luis and Sánchez-Soto (1999) device, by preparing 16 quantum states $\ket{lm}$, where $\ket{l},\ket{m}\in\{\ket{H},\ket{V},\ket{D},\ket{R}\}$, sending them into the device, and recording the coincidence counts for each state being in $\ket{\Phi^{+}}$ or $\ket{\Psi^{+}}$ for $60~{}\text{s}$. Again, we reconstruct the measurement operators via maximum-likelihood estimation method, and the results are shown in Fig. 3b-f. For the PCM device with 3 nm filters, the fidelities of $M_{\Phi^{+}}^{\text{re}}$ and $M_{\Psi^{+}}^{\text{re}}$ are $F_{\Phi^{+}}=0.815\pm 0.011$ and $F_{\Psi^{+}}=0.834\pm 0.004$, and with 8 nm filters, they are $F_{\Phi^{+}}=0.819\pm 0.015$ and $F_{\Psi^{+}}=0.813\pm 0.015$. The high fidelities of the PCM devices are crucial in implementing all-photonic quantum repeater. In our experiments, we define the ratio $r$ as entanglement generation rate between the all-photonic scheme and the conventional parallel entanglement swapping. In order to exclude higher-order noise, we just register eight-photon coincidence events. Then, the relationship between $r$ and the down-conversion probability $p$ can be written as $r=2-4p+2p^{2}$; in the limit as $p$ tends to $0$, $r$ tends to 2 (see Supplementary Information). We record the eightfold coincidence events for the $2\times 2$ parallel quantum repeater with $p=0.0344\pm 0.0001$ over $39$ hours. For comparison, we also record these events for the upper (lower) channel of a conventional parallel entanglement swapping by removing CPBS${}_{2}$ and CPBS${}_{5}$ (CPBS${}_{3}$ and CPBS${}_{4}$) with the same $p$ and duration. We evaluate the counting ratio to be $r=1.89\pm 0.10$. Then, we increase the power of the pump laser to $720~{}\text{mW}$ and repeat this experiment with $p=0.0483$ for a duration of $22$ hours, finding a ratio of $r=1.74\pm 0.07$. These results are shown in Fig. 4a. Further, to determine the fidelity of entanglement states shared between Alice and Bob, we decompose the density matrix in terms of Pauli matrices: $$\displaystyle\ket{\Phi^{+}}\bra{\Phi^{+}}=\frac{1}{4}(I+XX-YY+ZZ),$$ (1) where $Z=\ket{H}\bra{H}-\ket{V}\bra{V}$, $X=\ket{D}\bra{D}-\ket{A}\bra{A}$, and $Y=\ket{R}\bra{R}-\ket{L}\bra{L}$. This means that we only need to measure the state in three bases, i.e., $XX$, $YY$ and $ZZ$. For each bases we record eight-fold coincidence over $39$ hours. The measured fractions for the final entangled photon pairs 1&11, 4&11, 1&10 and 4&10 are shown in Fig. 4b-e, respectively. Here, it is important to note that the measured fractions of photon pairs 4&11, 1&10 are far away from the uniform distribution, whereas those fractions for the conventional parallel entanglement swapping are uniform without any entanglement. The overall fidelity is $0.606\pm 0.010$, which clearly indicates that the final shared state is genuinely entangled. We also measured the fidelity for $p=0.0483\pm 0.0001$, finding a value of $0.546\pm 0.006$ (see Supplementary Information). Thus, we fully demonstrate a $2\times 2$ parallel all-photonic quantum repeater. To sum up, we have successfully manipulated 12 photons experimentally and accomplished a proof-of-principle demonstration of the all-photonic quantum repeater. Our experiment adopted a GHZ state and a passive scheme to realize the adaptive Bell measurement in the repeater nodes, and achieved an enhancement in entanglement generation rate over the conventional parallel entanglement swapping. Although the all-photonic scheme can remove quantum memories at the intermediate repeater nodes, quantum memories at the end nodes are still needed if Alice and Bob demand a quantum output state. Even so, the memory time at the end nodes required in the all-photonic scheme scales only linearly with communication distance Azuma et al. (2015), while the memory time of conventional quantum repeaters scales polynomial or sub-exponential with the communication distance. Note however that, several protocols, such as quantum key distribution Lo et al. (2014) and non-local measurements, do not demand strictly a quantum output state, but a shared information. Then, the memories at the end nodes can be removed by using delay-choice entanglement swapping. That is, Alice and Bob measure the local qubits first and wait for the classical signals from intermediate nodes later. In future, the all-photonic scheme is possible to be combined with the matter-memory-based scheme: a RGS can relax the requirement of long coherent time of quantum memory, while a quantum memory can reduce the requirement of large size of RGS. Overall, we believe that all-photonic and matter-memory-based schemes are two important parallel research directions towards a practical quantum repeater. This work was supported by the National Key Research and Development (R&D) Plan of China (under Grants No. 2018YFB0504303 and No. 2018YFA0306501), the National Natural Science Foundation of China (under Grants No. 11425417, No. 61771443 and U1738140), Fundamental Research Funds for the Central Universities (WK2340000083), the Anhui Initiative in Quantum Information Technologies and the Chinese Academy of Sciences. The authors particularly thank Hoi-Kwong Lo for insightful discussions and comments. Supplementary Information for “Experimental quantum repeater without quantum memory" I Memory-based and all-photonic quantum repeaters A schematic diagram of the standard memory-based quantum repeaters Briegel et al. (1998); Duan et al. (2001); Sangouard et al. (2011) is shown in Fig. 5a. The essence is the quantum memory at the intermediate nodes $C_{i}$, which allows the realization of the selective Bell state measurement (BSM) only on qubits that have successfully shared entanglement with distant nodes. Specifically, the quantum memory at $C_{i}$ allows the following two functions. First, the memory enables $C_{i}$ to keep entanglement until it is informed/heralded of the success/failure of the entanglement generation processes between neighbour nodes. Second, the independent accessibility to them enables the node $C_{i}$ to selectively apply the Bell measurement only to successfully entangled pairs. In the original protocol of all-photonic quantum repeaters Azuma et al. (2015), illustrated in Fig. 5b, one could implement the two functions by using the repeater graph states (RGS) and local feed-forward from the 2nd-leaf to the 1st-leaf qubits. Particularly, if the BSM between the signal photon and the 2nd-leaf photon succeeds, then we perform $X$ measurement on the corresponding 1st-leaf photon in order to connect the distant photons into the RGS. While the BSM fails, we perform $Z$ measurement on the corresponding 1st-leaf photon to break the 2nd-leaf photon off. It is important to note that the local heralding signals are sent and received within the same nodes, which reduces the waiting time to zero, in principle. Therefore, the RGS can replace the quantum memory to realize the selective BSM in a time-reversed manner. For the end nodes of Alice and Bob, if they demand strictly a shared quantum output state, quantum memories at the end nodes are required in both the standard memory-based scheme and the all-photonic scheme to determine which two photons are finally entangled Azuma et al. (2015). Note however that in the all-photonic scheme, the heralding signals for connecting different channels are sent and received within the same nodes, so the transmission time of the heralding signals could be nearly zero. Therefore, the memory time at the end nodes in the all-photonic scheme scales only linearly with communication distance Azuma et al. (2015). However, the memory time in the standard memory-based scheme scales polynomial or subexponential with the communication distance Sangouard et al. (2011); Duan et al. (2001). In our implementation, we use post-selection to determine the final entanglement at the end nodes. Tab. 1 shows the post-selection results. We remark that several quantum information science protocols, such as quantum key distribution Lo et al. (2014) and non-local measurements Vaidman (2003), do not demand strictly a quantum output state, but a shared correlated information. Then, the memories at the end nodes can be removed in the all-photonic scheme by introducing the method of delay-choice entanglement swapping. That is, Alice and Bob could measure the local qubits first and wait for the classical signals from intermediate nodes later. This would not lead to redundant errors. II Passive implementation of the selective BSM In the all-photonic scheme, the RGS at the intermediate nodes enables the selective BSM in a time-reversed manner, i.e., switching between two functions: (i) establishing entanglement if the local BSM succeeds; and (ii) disentangling the qubit if the local BSM fails. To simplify the implementation, we create an experimentally feasible scheme. We replace the RGS with the GHZ state, and design the passive-choice-measurement (PCM) to realize the switching passively. On the one hand, the GHZ state is local-unitary equivalent to a RGS. If we perform the Bell measurement between a qubit composing an $m$-partite GHZ state and a qubit composing another $n$-partite GHZ state, we obtain an ($m+n-2$)-partite GHZ state, which establishes the entanglement, i.e., function (i). If we perform the $X$-basis measurement on the GHZ state, we can disentangle the unwanted qubit, i.e., function (ii). One the other hand, we design the PCM to perform the switching between function (i) and function (ii) passively, rather than actively, in order to make it possible with fewer single photons. Specifically, the PCM performs a BSM of function (i), when a coincident detection occurs in the two outputs of the circular polarisation beam splitter (CPBS), whereas it performs a standard projection measurement in the $X$ basis of function (ii), when the photon is detected only in one of the two outputs of the CPBS. We emphasize that when the BSM fails, the PCM device will automatically perform single-bit measurement in $X$ basis. It means that in our passive scheme, we do not need the active feed-forward gate operation to decide which photon would be connected or disconnected. Therefore, we successfully realized the selective BSM in the intermediate nodes, thus demonstrating one of the essences of the all-photonic scheme. Next, we discuss the implementation of PCM in more detail. We use a circular PBS (CPBS), two normal PBS, and four single photon detectors (SPD) to construct the PCM, as shown in Fig. 1d of the main text. When only one photon from an $N$-qubit GHZ state, and no photon from the EPR pair, is subjected into the CPBS, the GHZ state evolves as $$\displaystyle\ket{\text{GHZ}^{+}_{N}}\bra{\text{GHZ}^{+}_{N}}\xrightarrow{% \text{CPBS}}\ket{D^{L}}\bra{D^{L}}\otimes\ket{\text{GHZ}^{+}_{N-1}}\bra{\text{% GHZ}^{+}_{N-1}}+\ket{A^{R}}\bra{A^{R}}\otimes\ket{\text{GHZ}^{-}_{N-1}}\bra{% \text{GHZ}^{-}_{N-1}},$$ (2) where $\ket{\text{GHZ}^{+}_{N}}={1}/{\sqrt{2}}(\ket{H^{\otimes N}}+\ket{V^{\otimes N}})$ and the superscript $L$ ($R$) denotes the photon come out from the left (right) output port of CPBS. In this case, as shown in the Fig. 6a-b, the CPBS serves as a single qubit projector in $X$ basis. It is obviously that when the photon is detected only in the right (left) output port, the other photons collapse to a smaller GHZ state $\ket{\text{GHZ}^{+}_{N-1}}$ ($\ket{\text{GHZ}^{-}_{N-1}}$) and the entanglement is not destroyed. When two photon (one comes from the multipartite GHZ state and another one comes from an EPR pair) are subjected into the CPBS, then they evolve as $$\displaystyle\ket{\text{EPR}}\bra{\text{EPR}}\otimes\ket{\text{GHZ}^{+}_{N}}% \bra{\text{GHZ}^{+}_{N}}$$ (3) $$\displaystyle\xrightarrow{\text{CPBS}}$$ $$\displaystyle\frac{1}{2}\ket{\text{GHZ}^{+}_{N}}\bra{\text{GHZ}^{+}_{N}}% \otimes\ket{\Phi^{+}_{2,3}}\bra{\Phi^{+}_{2,3}}+\frac{1}{2}\ket{\text{GHZ}^{+}% _{N}}\bra{\text{GHZ}^{+}_{N}}\otimes\ket{\Psi^{+}_{2,3}}\bra{\Psi^{+}_{2,3}}$$ $$\displaystyle+$$ $$\displaystyle\frac{1}{2}\ket{A_{1}}\bra{A_{1}}\otimes\ket{\text{GHZ}^{+}_{N-1}% }\bra{\text{GHZ}^{+}_{N-1}}\otimes\ket{A^{L}_{2}D^{L}_{3}}\bra{A^{L}_{2}D^{L}_% {3}}$$ $$\displaystyle+$$ $$\displaystyle\frac{1}{2}\ket{D_{1}}\bra{D_{1}}\otimes\ket{\text{GHZ}^{-}_{N-1}% }\bra{\text{GHZ}^{-}_{N-1}}\otimes\ket{D^{R}_{2}A^{R}_{3}}\bra{D^{R}_{2}A^{R}_% {3}},$$ where $\ket{\text{EPR}}=1/\sqrt{2}(\ket{H_{1}H_{2}}+\ket{V_{1}V_{2}})$. The four terms of Eq. (3) can be distinguished by the CPBS as shown in Fig. 6c-f. In conclusion, we can divide all results above into two cases: (i) When the photons are detected in two output ports, the PCM serves as a bell state analyzer; (ii) When the photons are detected only in left or right output port, the PCM serves as a projector in X basis. It indicates that if the BSM fails, the entanglement of other photons may be not destroyed. The smaller GHZ state $\ket{\text{GHZ}^{+}_{N-1}}$ or $\ket{\text{GHZ}^{-}_{N-1}}$ can also be used to connect other EPR pairs. The analysis above is based on that the detector is perfect (the efficiency is 100%). However, in a practical apparatus, the efficiency should not be $100\%$. With the imperfect device, we introduce an extra error into the result. In our experiment, the overall system efficiency is 38% in average and the down-conversion probability is 0.0344 and 0.0483. Then the error rate can be estimated as 1.04% and 1.45%. It can be ignored in our experiment. III The corresponding final states with different results of four PCMs In our scheme, with different results of PCM, we may obtain final entangled states with different photon pairs, namely 1&11, 4&11, 1&10 or 4&10. The table of respective final state according to different results of four PCMs is shown in Table 1. IV The signal-to-noise ratio of 12 photons In our experiment, the whole setup can be considered as a twelve-photon interferometer. To obtain a higher coincidence rate, just bandpass filters with $\lambda_{\text{FWHM}}=30~{}\text{nm}$ are set before the single photon detector to remove the small sidebands. With a pump power of 700 mW we obtain a 12-photon coincidence rate of 1.2 $\text{h}^{-1}$. To verify the ability of experimentally manipulating twelve photons, we turn the circular PBS to a normal PBS by setting the surrounding HWPs at $0{}^{\circ}$. Then the whole setup serves for generating twelve-photon mix states $$\displaystyle\rho_{12}=v(\ket{H^{\otimes 12}}+\ket{V^{\otimes 12}})(\bra{H^{% \otimes 12}}+\bra{V^{\otimes 12}})+\frac{(1-v)}{2}(\ket{H^{\otimes 12}}\bra{H^% {\otimes 12}}+\ket{V^{\otimes 12}}\bra{V^{\otimes 12}}).$$ (4) To characterize this state, we measure it in $Z^{\otimes 12}$ basis and show that under the condition of registering a 12-fold coincidence only the $\ket{H^{\otimes 12}}$ and $\ket{V^{\otimes 12}}$ components can be observed, but no others. The experimental data are shown in Fig. 7. The signal-to-noise ratio defined as the ratio of the average of the desired components to that of the non-desired ones is $1420:1$. It indicates that the twelve photons can be well manipulated in our set-up. V Detail results with down-conversion probability $p=0.0483$ Due to the higher-order noise of SPDC process, the ratio $r$ of entanglement generation rate between all-photonic quantum repeater and conventional scheme is also influenced by the down-conversion probability, $p$. To verify the relationship of $r$ and $p$, we increase the power of pump laser to $720~{}mW$ and acquire another data point with $p=0.049$ (shown in Fig. 4a in main text). For a detailed characterization of the $2\times 2$ all-photonic repeater with $p=0.049$, we also do tomographic measurements James et al. (2001) on the four-photon GHZ state. The reconstructed matrix is shown in Fig. 8 and the fidelity is calculated as $F=0.877$, which indicates the 4-photon GHZ states are genuine entangled. At last, we also examine the entanglement of final photon pairs. In experiment We register $22~{}\text{h}$ eight-fold coincidence in XX, YY and ZZ basis respectively. 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Evolutionary Hamiltonian Graph Theory Zh.G. Nikoghosyan111G.G. Nicoghossian (up to 1997) Abstract We present an alternative domain concerning mathematics to investigate universal evolution mechanisms by focusing on large cycles theory (LCT) - a simplified version of well-known hamiltonian graph theory. LCT joins together a number of $NP$-complete cycle problems in graph theory. $NP$-completeness is the kay factor insuring (by conjecture of Cook) the generation of endless developments and great diversity around large cycles problems. Originated about 60 years ago, the individuals (claims, propositions, lemmas, conjectures, theorems, and so on) in LCT continually evolve and adapt to their environment by an iterative process from primitive beginnings to best possible theorems based on inductive reasoning. LCT evolves much more rapidly than biosphere and has a few thousand pronounced species (theorems). Recall that life on earth with more than 2 million species was originated about 3.7 billion years ago and evolves extremely slowly. We show that all theorems in LCT have descended from some common primitive propositions such as ”every complete graph is hamiltonian” or ”every graph contains a cycle of length at least one” via improvements, modifications and three kinds of generalizations - closing, associating and extending. It is reasonable to review Darwinian mechanisms in light of LCT evolution mechanisms (especially inductive reasoning) including the origin and macroevolution disputable phenomena in the biosphere. Key words. Evolutionary theory; microevolution; macroevolution; evolution vs. creationism; fundamental theorems; hamiltonian graph theory; large cycles theory. 1 Introduction Evolutionary concepts appeared in some early Greek writings, e.g., in the works of Thales, Empedocles, Anaximander, and Aristotle. Under the influence of the Church, no evolutionary theories developed during some 15 centuries. However, much data was accumulated that was to be utilized by later theorists. With the growth of scientific observation and experimentation, there began to appear from about the middle of the 16th century glimpses of the theory of evolution that emerged in the mid 19th century. Charles Robert Darwin set forth the scientific concepts of the evolutionary theory concerning the developments of plants and animals in biosphere that came to be known as Darwinism. In 1859 appeared the first edition of Darwin’s ”Origin of Species” [21]. Projections for the total number of species on Earth range from 2 million to 50 million. Scientists have observed that over time, the descendants of living things may change slightly, called ”microevolution”. Evolutionists teach that small changes accumulated slowly over billions of years and produced the big changes, called ”macroevolution”. Darwinism postulates that all organisms on the Earth have descended from a common ancestor over vast periods of time by means of ”extremely slight modifications”. To many, this claim sounds reasonable: if small changes can occur within a species, why should not evolution produce big changes over long periods of time. Charles Darwin was the first to formulate the theory of evolution by means of natural selection. But how these variations initially arise or are transmitted to offspring, and hence to subsequent generations, was not understood by Darwin. Natural selection is the only known cause of adaptation, but not the only known cause of evolution. Other, nonadaptable causes of evolution include mutation and genetic drift. In the early 20th century, genetics was integrated with Darwin’s theory of evolution by natural selection through the discipline of population genetics. The science of genetics provided a satisfactory explanation for the transmission of variation concerning microevolution. The teaching of macroevolution is built on the claim that mutations - random changes in the genetic code of plants and animals, can produce new species of plants and animals. The gene is the carrier of heredity and determines the attributes of the individual; thus changes in the genes can be transmitted to the offspring and produce new or altered attributes in the new individual. One of the hottest controversy in the science is the creation vs. evolution controversy - the Intelligent Design challenge to the theory of evolution. Creationist arguments against evolution theory can sound as follows: • no one has observed that the small changes we can observe (microevolution) implied that much bigger changes (macroevolution) are also possible, • scientists worldwide have unearthed and cataloged some 200 million large fossils and billions of small fossils and many researchers agree that this vast and detailed record shows that all the major groups of animals appeared suddenly and remained virtually unchanged, with many species disappearing as suddenly as they arrived. At the same time, evolutionists argue that creationism is not a scientific theory because it cannot be tested by the scientific method - direct observations and experiments. This controversy continues to this day and scientists continue to study various aspects of evolution by forming and testing hypotheses, constructing scientific theories, using observational data, and performing experiments in both the field and the laboratory. Donald Campbell [16] was one of the first authors to formulate a generalized Darwinian concept to explain evolution in a wide variety of other domains, including psychology, economics, culture, language, medicine, computer science and physics. In result, a number of new areas have been appeared such as evolutionary psychology, culture, language, economics, computation, algorithm, genetic algorithms, programming and so on, inspired mainly by the human intellect. Some of these subjects (psychology, culture, language, economics and so on) trace their origins to the beginning of mankind history and are not well defined. Some others (computation, algorithm, genetic algorithms, programming and so on) are created entirely for practical use to solve optimization problems by direct application of Darwinian mechanisms. In this paper we present an alternative domain concerning mathematics to investigate universal evolution mechanisms. We will focus on one of the most heavily studied areas in graph theory that joins together a number of $NP$-complete cycle problems in discrete mathematics, called large cycles theory - a simplified version of well-known hamiltonian graph theory. $NP$-completeness is the kay factor insuring (by conjecture of Cook) the generation of endless developments and great diversity around large cycles problems providing an alternative domain comparable with biosphere. We show that the individuals (claims, propositions, lemmas, conjectures, theorems, and so on) in large cycles theory continually evolve and adapt to their environment by an iterative process from simplicity to complexity, from primitive beginnings such as ”every complete graph is hamiltonian” or ”every graph has a cycle of length at least one” to best possible theorems. We distinguish some evolutionary mechanisms that control this process: improvements, modifications and three kinds of generalizations - closing, associating and extending. Macroevolution (big changes) can be considered as incorporation of a new parameter in a proposition in result of closing or extending generalization in gradual improvement process based on inductive reasoning. Large cycles theory evolves much more rapidly than biosphere and has pronounced species (theorems) with well defined beginning and hereditary microevolution-macroevolution mechanisms. As an application, we show that all sharp theorems in large cycles theory have descended from some common ancestors (called fundamental theorems) through extending generalizations. So, large cycles theory provides an exclusive environment where evolution is evident literally, that allows to escape the traditional creation-evolution controversy debates concerning the origin and macroevolution phenomena. Large cycles theory can be considered as a simplified evolutionary model concerning human intellect with a number of certain advantages with respect to biology, and its specific evolutionary mechanisms can be useful towards better understanding the evolution mechanisms in biology, as well as the universal mechanisms to explain evolution in a wide variety of domains outside of biology. • Large cycles theory, originated about 60 years ago, has a few thousand pronounced species (theorems) and evolves much more rapidly than living forms on Earth with more than 2 million species, originated about 3.7 billion years ago. • The origins of theorems in large cycles theory can be strongly determined by exact branchings of the tree of improvements and generalizations. • Genetic units and hereditary mechanisms in large cycles theory are much more simpler than gene structures of living forms. • It is quite reasonable to review Darwinian evolutionary mechanisms in light of improvements, modifications and three kinds generalizations - closing, associating and extending, especially inductive reasoning. In the next section, we give necessary terminology and notations. Section 3 is devoted to complexity classes of computational problems, including $NP$-complete problems. The general environment (large cycles theory) and the patterns evolved in this environment (theorems in large cycles theory) are introduced in Section 4. The structure of theorems and their sharpness are described in Sections 5 and 6. Evolution mechanisms are classified in Sections 7 and 8. Finally, the definition and the list of ”fundamental theorems” are presented in Sections 9 and 10. 2 Terminology Throughout this article we consider only finite undirected graphs without loops or multiple edges. A good reference for any undefined terms is [14]. Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The degree and the neighborhood of a vertex $x\in V(G)$ are denoted by $d(x)$ and $N(x)$, respectively. A simple cycle (or just a cycle) $C$ of length $t$ is a sequence $v_{1}v_{2}...v_{t}v_{1}$ of distinct vertices $v_{1},...,v_{t}$ with $v_{i}v_{i+1}\in E(G)$ for each $i\in\{1,...,t\}$, where $v_{t+1}=v_{1}$. When $t=2$, the cycle $C=v_{1}v_{2}v_{1}$ on two vertices $v_{1},v_{2}$ coincides with the edge $v_{1}v_{2}$, and when $t=1$, the cycle $C=v_{1}$ coincides with the vertex $v_{1}$. So, by this standard definition, all vertices and edges in a graph can be considered as cycles of lengths 1 and 2, respectively. If $Q$ is a cycle then we use $|Q|$ to denote the length of $Q$, that is $|Q|=|V(Q)|$. A path (cycle) on $n$ vertices is denoted by $P_{n}$ ($C_{n}$, respectively). A Hamilton cycle of a graph is a cycle which passes through every vertex of the graph exactly once, and a graph is hamiltonian if it contains a Hamilton cycle. Large cycle structures are centered around well-known Hamilton (spanning) cycles. Other types of large cycles were introduced for different situations when the graph contains no Hamilton cycles or it is difficult to find it. Generally, a cycle $C$ in a graph $G$ is a large cycle if it dominates some certain subgraph structures in $G$ in a sense that every such structure has a vertex in common with $C$. When $C$ dominates all vertices in $G$ then $C$ is a Hamilton cycle. When $C$ dominates all edges in $G$ then $C$ is called a dominating cycle introduced by Nash-Williams [52]. Further, if $C$ dominates all paths in $G$ of length at least some fixed integer $\lambda$ then $C$ is a $PD_{\lambda}$(path dominating)-cycle introduced by Bondy [13]. Finally, if $C$ dominates all cycles in $G$ of length at least $\lambda$ then $C$ is a $CD_{\lambda}$(cycle dominating)-cycle, introduced in [57]. We reserve $n$, $q$, $\delta$, $\kappa$ and $\alpha$ to denote the number of vertices (order), the number of edges (size), minimum degree, connectivity and the independence number of a graph, respectively. The length $c$ of a longest cycle in a graph is called the circumference. For $C$ a longest cycle in $G$, let $\overline{p}$ and $\overline{c}$ denote the lengths of a longest path and a longest cycle in $G\backslash C$, respectively. Put $$\delta_{t}=\min_{v,u}\{\max\{d(v),d(u)\}:v,u\in E(G),d(v,u)=t\}.$$ Finally, we define $\sigma_{t}=+\infty$ if $\alpha<\kappa$. Otherwise $$\sigma_{t}=\min\left\{\sum^{t}_{i=1}d(x_{i}):\{x_{1},x_{2},...,x_{t}\}\ \mbox{% is an independent set of vertices in}\ G\right\}.$$ Let $s(G)$ denote the number of components of a graph $G$. A graph $G$ is $t$-tough if $|S|\geq ts(G\backslash S)$ for every subset $S$ of the vertex set $V(G)$ with $s(G\backslash S)>1$. The toughness of $G$, denoted $\tau(G)$, is the maximum value of $t$ for which $G$ is $t$-tough (taking $\tau(K_{n})=\infty$ for all $n\geq 1$). Woodall [66] defined the binding number $b(G)$ of a graph $G$ as follows: $$b(G)=\min_{X\in F}\frac{|N(x)|}{|X|},$$ where $F=\{X:\emptyset\not=X\subseteq V(G)\}$ and $N(X)=\cup_{x\in X}N(x)$. Let $N_{i,j,k}$ be the graph which is obtained by identifying each vertex of a triangle with an endvertex of one of three vertex-disjoint paths of lengths $i,j,k$. If $H_{1},...,H_{t}$ are graphs, then a graph $G$ is said to be $H_{1},...,H_{t}$-free if $G$ contains no copy of any of the graph $H_{1},...,H_{t}$ as induced subgraphs. The graphs $H_{1},...,H_{t}$ will be also referred to in this context as forbidden subgraphs. Denote by $P_{t}$ the path on $t$ vertices. A graph $G$ is said to be planar if $G$ is embeddable into the plane without crossing edges. A projective plane, sometimes called a twisted sphere, is a surface without boundary derived from a usual plane by addition of a line at infinity. Just as a straight line in projective geometry contains a single point at infinity at which the endpoints meet, a plane in projective geometry contains a single line at infinity at which the edges of the plane meet. A projective plane can be constructed by gluing both pairs of opposite edges of a rectangle together giving both pairs a half-twist. It is a one-sided surface, but cannot be realized in three-dimensional space without crossing itself. A graph $G$ is the intersection graph of subgraphs $H_{1},...,H_{m}$ of a graph $H$ if the vertices of $G$ one-to-one correspond to the subgraphs $H_{1},...,H_{m}$ and two vertices of $G$ are adjacent if and only if the corresponding subgraphs intersect. A graph is an interval graph if and only if it is an intersection graph of subpaths of a path. Next, a graph is a split graph if and only if it is an intersection graph of subtrees of a star, i.e., a graph $K_{1,m}$. Further, a graph is chordal if and only if it is an intersection graph of subtrees of a tree. Finally, a comparability graph is a graph whose edges can be transitively oriented (i.e. if $x>y$ and $y>z$, then $x>z)$; a cocomparability graph $G$ is a graph whose complement $G$ is a comparability graph. Spider graphs are the intersection graphs of subtrees of subdivisions of stars. Thus, spider graphs are chordal graphs that form a common superclass of interval and split graphs. Let $a,b,t,k$ be integers with $k\leq t$. We use $H(a,b,t,k)$ to denote the graph obtained from $tK_{a}+\overline{K}_{t}$ by taking any $k$ vertices in subgraph $\overline{K}_{t}$ and joining each of them to all vertices of $K_{b}$. Let $L_{\delta}$ be the graph obtained from $3K_{\delta}+K_{1}$ by taking one vertex in each of three copies of $K_{\delta}$ and joining them each to other. For odd $n\geq 15$, construct the graph $G_{n}$ from $\overline{K}_{\frac{n-1}{2}}+K_{\delta}+K_{\frac{n+1}{2}-\delta}$, where $n/3\leq\delta\leq(n-5)/2$, by joining every vertex in $K_{\delta}$ to all other vertices and by adding a matching between all vertices in $K_{\frac{n+1}{2}-\delta}$ and $(n+1)/2-\delta$ vertices in $\overline{K}_{\frac{n-1}{2}}$. It is easily seen that $G_{n}$ is 1-tough but not hamiltonian. A variation of the graph $G_{n}$, with $K_{\delta}$ replaced by $\overline{K}_{\delta}$ and $\delta=(n-5)/2$, will be denoted by $G^{*}_{n}$. 3 Complexity classes of computational problems Computational complexity theory focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other. Graph theory and combinatorics focus on particular problems and their real difficulties. Significant progress has been made in combinatorics and graph theory toward improving our understanding of the inherent difficulty in computational problems and what can be computed efficiently. Today, most problems of known interest have been classified as to whether they are polynomial-time solvable or $NP$-complete. An algorithm is said to be polynomial time if its running time is upper bounded by a polynomial expression in the size of the input for the algorithm. Problems for which a polynomial time algorithm exists belong to the complexity class $P$, which is central in the field of computational complexity theory. Polynomial time is a synonym for ”tractable”, ”feasible”, ”efficient”, or ”fast”. The following problems are polynomial-time solvable: shortest path problem, minimum spanning three problem, linear programming, matching, Eulerian cycle problem, network flow problem and so on. An algorithm is deterministic if at each step there is only one choice for the next step given the values of the variables at that step. An algorithm is non-deterministic if there is a step that involves parallel processing. A problem is said to be in the class $NP$ of problems if it can be solved by an algorithm which is non-deterministic and has a time complexity function which is polynomial. $NP$ problems are recognized by the fact that their solutions can be checked for correctness by a deterministic polynomial time algorithm. Every problem in $P$ is also in $NP$. The non-deterministic algorithm that can be used is ”guess the answer”. The guess can be checked in polynomial time by the algorithm which solves the problem. A famous and long standing open problem is whether or not $P=NP$. There is a collection of problems with the property that any polynomial time deterministic algorithm which solves one of them can be converted to a polynomial time algorithm which solves any other one of them (they are said to be polynomially equivalent problems) and if such an algorithm existed for any one of them, then $P=NP$. These problems are called $NP$- hard problems. $NP$-hard problems may or may not be $NP$ problems. Those that are $NP$ are called $NP$-complete problems. An example of an $NP$-complete problem is the Traveling Salesman Problem. Today, most of important developments in discrete mathematics are centered on various $NP$-complete problems in trying to find different ”effective layers” or ”effective subspaces” in structures of $NP$-complete problems. In fact, by Cook’s conjecture [19], $NP$-complete problems cannot be covered by such layers. Today, after intensive investigations, many $NP$-complete problems are like unbreakable rock fragments with numerous cuttings and bore-holes - tracks of investigations. 4 General environment and individuals Human intellect plays the role of a general environment including large cycles theory as a subarea. Various statements, including claims, propositions, lemmas conjectures and theorems in large cycles theory, play the role of individuals forming a population. Large cycles theory traces its origins to 1855. Irish physicist, astronomer and mathematician Sir William Rowan Hamilton (1805-1865) invented the ”Icosian Calculus”, a noncommutative algebra so called because it involved a planar embedding of the graph of a dodecahedron, which has 20 vertices. The system has two operations: $L$ and $R$, standing for ”left” and ”right” respectively, the idea being that if one has just arrived at a vertex, one can choose to go left or right, with the value 1 being reserved for an expression which returns to one’s point of origin. For example, a path that turns right twice and then left once can be expressed as the term $R2L$. Similarly, since each face of a dodecahedron is pentagonal, we know that $R5=L5=1$. Hamilton showed that symmetry notwithstanding, the equation $$LLLRRRLRLRLLLRRRLRLR=1$$ defines the only Hamiltonian Cycle on a dodecahedron. Since $LR\not=RL$, the Icosian Calculus is clearly noncommutative. However, it is associative. For example, $(LR)L=L(RL)$. Hamilton’s first communication about his Icosian Calculus was to his friend Robert Graves in a letter dated Oct. 7th, 1856. However, Hamilton Cycles should not have been named after Hamilton at all. In fairness, they should be called ”Kirkman Cycles” after Thomas Penyngton Kirkman, the man who actually first discovered them. His interest in polyhedra led him to discover Hamilton cycles in a paper received by the Royal Society on Aug. 6th, 1855, predates Hamilton’s earliest communication, let alone his first publication on the subject, by more than a year. However, precedence is not the only argument on Kirkman’s side. Whereas Hamilton considered only the one special case of cycles in the dodecahedron, Kirkman’s result was much more general, because he pondered the existence of Hamiltonian Cycles in all graphs corresponding to planar embeddings of solid shapes. In addition, Kirkman was the first to discover an infinite class of non-hamiltonian polyhedra. He showed that any bipartite graph with an odd number of vertices must be non-hamiltonian. He gave an example of a planar, 3-connected, bipartite, non-hamiltonian graph. Classic hamiltonian problem; determining when a graph contains a Hamilton cycle, is one of the most central notions in graph theory and is one of the most attractive and most investigated problems among $NP$-complete problems that Karp listed in his seminal paper [45]. Cook [19] conjectured that one cannot hope for a simple classification of hamiltonian graphs. In other words, it seems to be impossible to obtain a criterion for a graph to be hamiltonian which implies a polynomial-time algorithm. This fact gave rise to a growing number of conditions that are either necessary or sufficient. Today, this conjecture seems much more reasonable motivated by the fact that the developments arising around various $NP$-complete problems in discrete mathematics have undergone a natural gradual growth and evolution, generating a great diversity. This exclusive property of $NP$-complete problems force to think that the diversity arising around such problems potentially should not concede the diversity of living forms in biosphere. If a graph $G$ does not satisfy a sufficient condition for hamiltonicity, we cannot guarantee the existence of a Hamilton cycle. But if $G$ is close to satisfy the condition, we may hope find some ”hamiltonian-like” structures such as long cycles and hamiltonian paths. Further extensions of these notions lead to cycle and path covers, maximum matching, spanning trees with smallest number of leaves and many others that are rather far from their origins. Actually, each of these questions is really a part of the general area called ”hamiltonian graph theory”. Large cycles theory can be considered as a simplified alternative to hamiltonian graph theory concerning the main ”hamiltonian-like” cycle structures in graphs. In fact, large cycles theory is a natural extension of classic hamiltonian problem including Hamilton cycles, longest cycles, dominating cycles, as well as some generalized cycles including Hamilton and dominating cycles as special cases. Systematic investigations of Hamilton cycles began only in 1952 when Swiss mathematician Gabriel Andrew Dirac (1925-1984) [23] discovered the first sufficient condition for the existence of a Hamilton cycle and the first lower bound for the length of a longest cycle in graphs, based on two simplest graph invariants - order $n$ and minimum degree $\delta$. In the last 60 years, the developments in large cycles theory gave rise to a wide variety of theorems (species, kinds) [8], [36], [37]. The following 19 theorems include some generalizations and modifications of Dirac’s initial theorems with some progressive tendency, where $\kappa\leq\delta\leq\frac{1}{2}\sigma_{2}\leq\frac{1}{3}\sigma_{3}$ and $\delta\leq\delta_{2}$. (T1)  $c\geq\delta+1$       ([23], 1952) (T2)  $\kappa\geq 2$     $\Rightarrow$     $c\geq\min\{n,2\delta\}$     ([23], 1952) (T3)  $\kappa\geq 2$     $\Rightarrow$     $c\geq\min\{n,\sigma_{2}\}$     ([12], 1971) (T4)  $\kappa\geq 3$, $\delta\geq\alpha$     $\Rightarrow$     $c\geq\min\{n,3\delta-3\}$     ([39], 1978) (T5)  $\kappa\geq 3$     $\Rightarrow$     $c\geq\min\{n,3\delta-\kappa\}$     ([54], 1981) (T6)  $\kappa\geq 2$     $\Rightarrow$     $c\geq\min\{n,2\delta_{2}\}$     ([28], 1984) (T7)  $\kappa\geq 3$, $G$ is $\delta$-regular     $\Rightarrow$     $c\geq\min\{n,3\delta\}$     ([29], 1985) (T8)  $\kappa\geq 4$, $\delta\geq\alpha$    $\Rightarrow$     $c\geq\min\{n,4\delta-2\kappa\}$    ([55], 1985) (T9)  $\tau\geq 1$     $\Rightarrow$     $c\geq\min\{n,2\delta+2\}$     ([4], 1986) (T10)  $\tau\geq 1$     $\Rightarrow$     $c\geq\min\{n,\sigma_{2}+2\}$     ([4], 1986) (T11)  $c\geq(\overline{p}+2)(\delta-\overline{p})$    ([61], 1998) (T12)  $c\geq(\overline{c}+1)(\delta-\overline{c}+1)$    ([61], 1998) (T13)   $\kappa\geq 2$     $\Rightarrow$     $c\geq\min\{n,(\tau+1)(\delta+1)-1\}$     ([42], 1999) (T14)   $\overline{c}\geq\kappa$     $\Rightarrow$     $c\geq\frac{(\overline{c}+1)\kappa}{\overline{c}+\kappa+1}(\delta+2)$    ([56], 2000) (T15)  $\kappa\geq 3$     $\Rightarrow$     $c\geq\min\{n,\sigma_{3}-\kappa\}$     ([68], 2007) (T16)   $\kappa\geq 3$, $G$ is claw-free     $\Rightarrow$     $c\geq\min\{6\delta-15\}$    ([50], 2009) (T17)  $\kappa\geq\lambda+2$, $\delta\geq\alpha+\lambda-1$   $\Rightarrow$   $c\geq\min\{n,(\lambda+2)(\delta-\lambda)\}$   ([57], 2009) (T18)  $\kappa\geq 4$, $\delta\geq\alpha$    $\Rightarrow$     $c\geq\min\{n,4\delta-\kappa-4\}$    ([53], 2011) (T19)  $\tau>1$     $\Rightarrow$     $c\geq\min\{n,2\delta+5\}$ or $G=$Petersen graph     ([60], 2012) As a result of the massive amount of evidence for evolution accumulated in large cycles theory over the last 60 years, we can safely conclude that evolution has occurred and continues to occur in this area. Moreover, evolving around an $NP$-complete longest cycle problem, the list of Theorems (T1)-(T19), by conjecture of Cook [19], is not unchanging end-product and will grow generating continually growing diversity. 5 The structure of theorems Informally, theorem is of the form of an indicative conditional: $$\mbox{If}\ A\ \mbox{then}\ B.$$ (1)1( 1 ) In this case, $A$ is called the hypothesis (conditions) of Theorem (1) and $B$ the conclusion. Conclusion $B$ indicates the existence of possible types of large cycle structures in a graph $G$. In large cycles theory, conclusion $B$ usually appears in any of the following forms: $(a1)$ $G$ has a Hamilton cycle, $(a2)$ $G$ has a dominating cycle, $(a3)$ every longest cycle in $G$ is a dominating cycle, $(a4)$ $G$ has a $CD_{\lambda}$-cycle, $(a5)$ every longest cycle in $G$ is $CD_{\lambda}$-cycle, $(a6)$ a lower bound for the circumference. Sometimes, $B\equiv B_{1}\vee B_{2}$ where $$B_{1}\in\{(a1),(a2),(a3),(a4),(a5)\},\ \ B_{2}\equiv(a6).$$ As for hypothesis $A$, generally it can be presented as $A_{1}\wedge A_{2}\wedge...\wedge A_{m}$ where for each $i\in\{1,...,m\}$, $A_{i}$ appears in the following forms: $(b1)$ $A_{i}$ is an algebraic (numerical) relation $f_{1}\geq f_{2}$ between two algebraic expressions $f_{1},f_{2}$, $(b2)$ $A_{i}$ is a structural limitation defined by forbidden subgraphs (examples: forbidden triangle, claw, $P_{6}$, and so on), $(b3)$ $A_{i}$ is a structural limitation defined by direct description (examples: conditions for a graph to be regular, bipartite, interval, chordal, and so on). If $A=A_{1}\vee A_{2}$ then Theorem (1) can be partitioned into two independent theorems ”if $A_{1}$ then B” and ”if $A_{2}$ then $B$”. The hypotheses and conclusions defined by $(a_{1})-(a6)$ and $(b_{1})-(b_{3})$, carry the genetic information (genome) of a theorem in forms of initial graph invariants, generalized invariants, forbidden subgraphs and special graph classes. There are a number of well-known basic (initial) invariants of a graph $G$ occurring in various hamiltonian results and having significant impact on large cycle structures, namely order $n$, size $q$, minimum degree $\delta$, connectivity $\kappa$, binding number $b(G)$, independence number $\alpha$, toughness $\tau$ and the lengths of a longest path and a longest cycle in $G\backslash C$ for a given longest cycle $C$, denoted by $\overline{p}$ and $\overline{c}$, respectively. Some of these basic gene elements, especially minimum degree $\delta$, have been generalized (evolved) in terms of degree sequences, degree sums, generalized degree, neighborhood unions and so on, giving rise many generalized theorems. 6 Relaxation and strengthening Evolutions mechanisms in large cycles theory are based on relaxation and strengthening. Definition 1. Let $f_{1}\geq f_{2}$ be a condition in (1) defined by $(b_{1})$. We say that the condition $f_{1}\geq f_{2}$ can be relaxed in (1) if it can be replaced by $f_{1}\geq f_{2}-\epsilon$ for some positive $\epsilon$. Definition 2. Let ”$G$ is $H_{1}$-free” be a condition in (1) defined by $(b_{2})$. We say that ”$G$ is $H_{1}$-free” is stronger than ”$G$ is $H_{2}$-free” if $H_{1}$ is an induced subgraph of $H_{2}$. For example, ”$G$ is $P_{4}$-free” is stronger than ”$G$ is $P_{5}$-free” or ”$G$ is $P_{6}$-free”. Further, ”$G$ is $N_{0,0,0}$-free” is stronger than ”$G$ is $H$-free” for each $$H\in\{N_{0,0,1},N_{0,0,2},N_{0,1,1},N_{0,0,3},N_{0,1,2},N_{1,1,1}\}.$$ If a theorem is not sharp (best possible, tight) then clearly it is incomplete and need further improvement through relaxation and strengthening. Definition 3. A theorem is said to be sharp in all respects (partly, respectively) if its conclusion cannot be strengthened and each condition (some condition, respectively) in it cannot be relaxed under the same conclusion. According to Definition 3, algebraic relations (see $(b1)$ in previous section) can be gradually (smoothly) relaxed or strengthened forming the best type of hypotheses for relaxing or strengthening. Structural limitations defined by forbidden subgraphs (see $(b2)$ in previous section), form the next type of well defined hypotheses in view of relaxing or strengthening. Consider the following theorem based on structural limitations of this type. Theorem A (Broersma and Veldman [15], 1997). Every 2-connected $\{K_{1,3},P_{6}\}$-free graph is hamiltonian. Generally, it is difficult to check the sharpness related to forbidden subgraphs. However, the following result essentially simplifies this procedure in Theorem A. Theorem B (Faudree and Gould [31], 1997). Let $R$ and $S$ be connected graphs $(R,S\not=P_{3})$ and $G$ be a 2-connected graph of order $n\geq 10$. Then $G$ is $(R,S)$-free implies $G$ is hamiltonian if and only if $R=K_{1,3}$ and $S$ is one of the graphs: $P_{4},P_{5},P_{6},N_{0,0,0},N_{0,0,1},N_{0,0,2},N_{0,1,1},N_{0,0,3},N_{0,1,2}$ or $N_{1,1,1}$. By Theorem B, the condition ”$G$ is $P_{6}$-free” in Theorem A cannot be relaxed by replacing it with ”$G$ is $H$-free” for each $$H\in\{P_{4},P_{5},N_{0,0,0},N_{0,0,1},N_{0,0,2},N_{0,1,1},N_{0,0,3},N_{0,1,2},% N_{1,1,1}\}.$$ Further, the condition ”$G$ is $\{K_{1,3},P_{6}\}$-free” in Theorem A cannot be relaxed by replacing it with ”$G$ is $K_{1,3}$-free” or ”$G$ is $P_{6}$-free” by the following theorem. Theorem C (Faudree and Gould [31], 1997). Let $R$ be a connected graph and $G$ be a 2-connected graph. Then $G$ is $R$-free implies $G$ is hamiltonian if and only if $R=P_{3}$. Finally, the graph $2K_{\delta}+K_{1}$ shows that the condition $\kappa\geq 2$ in Theorem A cannot be replaced by $\kappa\geq 1$. So, Theorem A, as well as Theorems 21-25 in Section 9, are best possible. Now consider the third type of conditions providing special graph environments (see $(b3)$ in previous section) such as regular, bipartite, interval, chordal, line, spider, split and transitive graphs, powers of graphs and so on. If the condition cannot be gradually relaxed, it must be removed from the list of conditions as an extraordinary sort of relaxation. Clearly, $r$-regularity and $(r+1)$-regularity are noncomparable and when we want to relax a condition such as ”$G$ is $r$-regular”, we have to remove this condition. By relaxing the condition ”$G$ is bipartite” we get a trivial case when ”$G$ is one-partite” or empty graph. Planarity can be interpreted both in view of forbidden subgraphs and embedding in a plane without crossings. The following well-known theorem is similar to Theorem G and shows that in both cases we get a non planar graph when we try to relax the planarity condition . Theorem D (Kuratowski [49], 1930). A graph is planar if and only if it does not contain a subgraph that is homeomorphic to $K_{5}$ or $K_{3,3}$. 7 Evolution mechanisms in large cycles theory All theorems in large cycles theory have descended from trivial (primitive) propositions such as: $(c1)$ every complete graph is hamiltonian, $(c2)$ every graph contains a cycle of length at least one, $(c3)$ every graph with $n=1,2,...,10$ and $\delta\geq n-1$ is hamiltonian, $(c4)$ every graph with $\delta\geq n-1\geq 1$ is hamiltonian, $(c5)$ every graph with $\alpha\leq 1$ is hamiltonian, $(c6)$ every graph with $q\geq n(n-1)/2$ is hamiltonian, via the following evolutionary mechanisms: • improvements (vertical evolution), • modifications (horizontal evolution), • closing generalizations, • associating generalizations, • extending generalizations. 7.1 Improvements Improvement is a progressive (vertical) iterative process in evolution toward finding better results. Definition 4. Improvement is one of the following procedures: $(d1)$ relaxing one of the conditions and preserving the conclusion, $(d2)$ strengthening the conclusion and preserving the conditions. Improvements are applicable only to the trivial or incomplete (not sharp) results such as $(c3)-(c6)$. For example, $(c3)$ can be iteratively improved to ”every graph with $n=10$ and $\delta\geq i$ is hamiltonian” for $i=8,7,6,5$. The best result in this process can be formulated as follows $(c7)$ every graph with $n=10$ and $\delta\geq 5$ is hamiltonian. Furthermore, $(c7)$ can be iteratively improved to ”every graph with $n=i$ and $\delta\geq i/2$ is hamiltonian” for $i=11,12,...$. 7.2 Modifications Modification is a horizontal developmental process in evolution generating noncomparable results. Definition 5. Modification is one of the following procedures: $(e1)$ relaxing of some conditions, at the same time strengthening some others, under the same conclusion, $(e2)$ relaxing of some conditions, at the same time relaxing the conclusion, $(e3)$ strengthening of some conditions, at the same time strengthening the conclusion. Observing that $\tau\geq 1$ is stronger than $\kappa\geq 2$, and ”$G$ is hamiltonian” is stronger than ”$G$ contains a dominating cycle”, we can state that the following theorems, by Definition 5, are modifications. $(f1)$ If $\kappa\geq 2$ and $\delta\geq(n+2)/3$ then $G$ contains a dominating cycle. [52] $(f2)$ If $\tau\geq 1$ and $\delta\geq n/3$ then $G$ contains a dominating cycle. [9] $(f3)$ If $\kappa\geq 2$ and $\delta\geq(n+\kappa)/3$ then $G$ is hamiltonian. [54] 7.3 Closing generalizations Definition 6. Closing generalization is an improvement process which yields a best possible result. Often, it is based on inductive reasoning which generates new parameters. As noted above, $(c7)$ can be iteratively improved to ”every graph with $n=i$ and $\delta\geq i/2$ is hamiltonian” for $i=11,12,...$. Inductive reasoning allows to obtain a best possible theorem involving the order $n$ as a new parameter. $(c8)$ If $\delta\geq n/2$ then $G$ is hamiltonian. [23] Inductive reasoning is also known as induction: a kind of reasoning that constructs or evaluates propositions that are abstractions of observations of individual instances. 7.4 Associating generalizations Definition 7. Associating generalization joins together closely related noncomparable results for special values of some parameter $\lambda=1,2,...$, based on inductive reasoning. For example, the following theorem $(g1)$ if $\kappa\geq\lambda\geq 1$ and $\delta\geq(n+2)/(\lambda+1)+\lambda-2$ then $G$ contains a $CD_{\min\{\lambda,\delta-\lambda+1\}}$-cycle [57], associates the following noncomparable results for $\lambda=1,2,3$. $(g2)$ if $\kappa\geq 1$ and $\delta\geq n/2$ then $G$ is hamiltonian [23], $(g3)$ if $\kappa\geq 2$ and $\delta\geq(n+2)/3$ then $G$ contains a dominating cycle [52], $(g4)$ if $\kappa\geq 3$ and $\delta\geq(n+6)/4$ then $G$ contains a $CD_{3}$-cycle [41]. 7.5 Extending generalizations Since the best possible theorems cannot be improved, they can be involved only by extensions of some notions. Definition 8. Extensions of some concepts in best possible theorems generate a new kind of so called extending generalizations. The concepts $\sigma_{t}$ and $\delta_{t}$ $(t\geq 1)$ are two extensions of the minimum degree $\delta$ with $\sigma_{1}=\delta_{1}=\delta$ and therefore, the following two theorems $(h1)$ if $\sigma_{2}\geq n$ then $G$ is hamiltonian [63], $(h2)$ if $\delta_{2}\geq n/2$ then $G$ is hamiltonian [28], are two extending generalizations of $(c8)$. 8 Microevolution and macroevolution Microevolution in large cycles theory is a gradual improvement process based on numerical expressions. Macroevolution occurs in result of a closing generalization which determines a new result with a new involved parameter based on inductive reasoning. In other words, macroevolution is a transition from numerical subexpressions to parametrical subexpressions. All intermediate microevolution changes in improvement process are immediately forgotten and eliminated preserving only the final result as a macroevolution big change. In result, the evolution process in large cycles theory seems discrete with large breaking-offs. Conversely, each new parameter can be incorporated into the theorem a result of some closing generalization. Observe that along with new involved parameters, improvement process preserves the hereditary information in forms of earlier involved parameters. Graph invariants and their various extensions, combined in convenient relations as parameters, contain global and general information about a graph and its cycle structures like gene structures. They form the hereditary information of theorems in large cycles theory. The order $n$ and size $q$ as gene elements one by one are neutral graph invariants with respect to cycle structures. Meanwhile, they become more active combined together (as in Theorem 1). The minimum degree $\delta$ plays a central role in majority of hamiltonian results. It is not too primitive and not too complicated, becoming the most flexible invariant for various possible generalizations. Minimum degree is a more essential invariant than the order and size, providing some dispersion of the edges in a graph. The combinations between order n and minimum degree become much more fruitful especially under some additional connectivity conditions. The impact of some relations on cycle structures can be strengthened under additional conditions of the type $\delta\geq\alpha\pm i$ if for appropriate integer $i$. Determining the independence number $\alpha$ is shown in [35] to be $NP$-hard problem. Connectivity is the most valuable research tool toward cognation of large cycle structures. In [25], it was proved that connectivity $\kappa$ can be determined in polynomial time. Many graph theorists think that the connectivity is at the heart of all path and cycle questions providing comparatively more uniform dispersion of the edges. The binding number $b(G)$ is a measure of how well-knot a graph is. Like the connectivity, the binding number also can be computed in polynomial time, using network techniques [20]. An alternate connectedness measure is toughness $\tau$ - the most powerful and less investigated graph invariant introduced by Chvátal [17] as a means of studying the cycle structure of graphs. Moreover, it was proved [2] that for any positive rational number $t$, recognizing $t$-tough graphs (in particular 1-tough graphs) is an $NP$-hard problem. Chvátal [17] conjectured that there exists a finite constant $i_{0}$ such that every $i_{0}$-tough graph is hamiltonian. This conjecture is still open. For a given cycle $C$, the idea of using $G\backslash C$ appropriate structures lies in the base of almost all existing proof techniques in trying to construct longer cycles in graphs by the following standard procedure: choose an initial cycle $C_{0}$ in $G$ and try to enlarge it by replacing a segment $P^{\prime}$ of $C_{0}$ with a suitable path $P^{\prime\prime}$ longer than $P^{\prime}$, having the same end vertices and passing through $G\backslash C_{0}$. To find suitable $P^{\prime}$ and $P^{\prime\prime}$, one can use the paths or cycles (preferably large) in $G\backslash C_{0}$ and connections (preferably high) between these paths (cycles) and $C_{0}$. The latter are closely related to $\overline{p},\overline{c}$, as well as minimum degree $\delta$ (local connections) and connectivity $\kappa$ (global connections). Forbidden small subgraphs provide the next powerful gene element of structural nature that directly force the graph to have large cycles. For example, 2-connected $P_{3}$-free graphs are hamiltonian since they are complete graphs. The most common of forbidden subgraphs is the claw $K_{1,3}$. Finally, some special graph classes, that can be defined by direct description, provide convenient environments to construct large cycles in graphs. They are regular graphs, planar graphs, bipartite graphs, chordal graphs, interval graphs and so on. 9 On fundamental results in large cycles theory What makes a theorem (problem, conjecture) beautiful? By G.H. Hardy, ”The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics”. In [11], Bondy introduced some criteria to classify conjectures, which can be applicable for theorems as well: • Simplicity: short, easily understandable statement relating basic concepts. • Element of Surprise: links together seemingly disparate concepts. • Generality: valid for a wide variety of objects. • Centrality: close ties with a number of existing theorems and/or conjectures. • Longevity: at least twenty years old. • Fecundity: attempts to prove the conjecture have led to new concepts or new proof techniques. However, the first formal criterion toward classifying the theorems and conjectures is the property to be best possible (sharp, tight), widely applicable in combinatorics and graph theory. This criterion after some improvement can be applicable in other areas of science. The next formal criterion to distinguish some special kind of theorems is presented in [62] by focusing on pure relations between simplest graph invariants and large cycles structures. These simplest kind of relations having no forerunners in the area, actually form a source from which nearly all possible hamiltonian results can be developed further by various additional new ideas, generalizations, extensions, restrictions and structural limitations. In this paper we introduce the third formal criterion to distinguish some top theorems in large cycles theory called ”fundamental” based on all exact branchings of the tree of generalizations. By this approach, all results in large cycles theory have descended from a number of common ancestors (fundamental result) through extending generalizations. Fundamental results cannot be directly improved and can be evolved only by modifications and generalizations (associating and extending). The term ”fundamental result” is used in various fields of science to characterize mainly the central and most important results in the area, based on subjective perception. In this paper, this term is used according to the second much more important mean: ”forming the source or base from which everything else is made; not able to be divided any further”. Observe also that in general, there are no physical and abstract units in the nature, lying in the base of all material or abstract notions. However, every notion in large cycles theory has certain origins due to certain frames of this theory. 10 The list of fundamental results 10.1 Hamilton cycles Theorem 1 (Erdös and Gallai, 1959) [27] Every graph is hamiltonian if $$q\geq\frac{n^{2}-3n+5}{2}.$$ Example for sharpness. To see that the size bound $(n^{2}-3n+5)/2$ in Theorem 1 is best possible, note that the graph formed by joining one vertex of $K_{n-1}$ to $K_{1}$, contains $(n^{2}-3n+4)/2$ edges and is not hamiltonian. Theorem 2 (Erdös, 1962) [26] Every graph is hamiltonian if $1\leq\delta\leq n/2$ and $$q>\max\left\{\frac{(n-\delta)(n-\delta-1)}{2}+\delta^{2},\frac{\left(n-\lfloor% \frac{n-1}{2}\rfloor\right)\left(n-\lfloor\frac{n-1}{2}\rfloor-1\right)}{2}+% \left\lfloor\frac{n-1}{2}\right\rfloor^{2}\right\}.$$ Example for sharpness. The graph consisting of a complete graph on $n-\delta$ vertices, $\delta$ of which are joined to each of $\delta$ independent vertices, shows that the condition in Theorem 2 cannot be weakened. Theorem 3 (Moon and Moser, 1963) [51] Every balanced bipartite graph is hamiltonian if $$q\geq\frac{n^{2}-2n+5}{4}.$$ Examples for sharpness. Clearly, the condition ”$G$ is balanced” in Theorem 3 cannot be removed. The graph obtained from $K_{t,t}$ by deleting $t-1$ edges with a common vertex, shows that the condition $q\geq(n^{2}-2n+5)/4$ in Theorem 3 cannot be replaced by $q\geq(n^{2}-2n+4)/4$. Theorem 4 (Moon and Moser, 1963) [51] Every balanced bipartite graph is hamiltonian if $$q>\frac{n(n-2\delta)}{4}+\delta^{2}.$$ Examples for sharpness. Clearly, the condition ”$G$ is balanced” in Theorem 4 cannot be removed. Consider the balanced bipartite graph $G=(X,Y;E)$ with vertex classes of the form $X=P\cup Q$, $Y=R\cup S$, where $|P|=|R|=\delta$, $|Q|=|S|=n/2-\delta$, $N_{G}(x)=R$ for all $x\in P$, and $N_{G}(x)=Y$ for all $x\in Q$. This example shows that Theorem 4 is best possible. Theorem 5 (Nikoghosyan, 2011) [58] Every graph is hamiltonian if $$q\leq\delta^{2}+\delta-1.$$ Example for sharpness. $K_{1}+2K_{\delta}$. Theorem 6 (Dirac, 1952) [23] Every graph is hamiltonian if $$\delta\geq\frac{n}{2}.$$ Example for sharpness. $2K_{\delta}+K_{1}$. Theorem 7 (Moon and Moser, 1963) [51] Every balanced bipartite graph is hamiltonian if $$\delta\geq\frac{n+1}{4}.$$ Examples for sharpness. Clearly, the condition ”$G$ is balanced” in Theorem 7 cannot be removed. Since $n$ is even, the condition $\delta\geq(n+1)/4$ in Theorem 7 yields a stronger condition $\delta\geq(n+2)/4$. Let $P_{i}=x_{i}y_{i}z_{i}w_{i}$ $(i=1,2,3)$ be three disjoint paths. Form a graph from $P_{1},P_{2},P_{3}$ by identifying $x_{1},x_{2},x_{3}$ in one vertex and $w_{1},w_{2},w_{3}$ in another vertex. The resulting graph shows that the condition $\delta\geq(n+1)/4$ in Theorem 7 cannot be replaced by $\delta\geq n/4$. Theorem 8 (Jung, 1978) [39] Every graph is hamiltonian if $n\geq 11$, $\tau\geq 1$ and $$\delta\geq\frac{n-4}{2}.$$ Examples for sharpness. Petersen graph; $K_{\delta,\delta+1}$; $G^{*}_{n}$. Theorem 9 (Nikoghosyan, 2012) [60] Every graph is hamiltonian if $\tau>4/3$ and $$\delta\geq\frac{n-5}{2}.$$ Examples for sharpness. The Petersen graph shows that the condition $\tau>4/3$ in Theorem 9 cannot be replaced by $\tau=4/3$. Let $H_{1}$ be a complete graph with vertex set $V(H_{1})=\{x_{1},x_{2},x_{3},x_{4},x_{5}\}$ and $H_{2}$ a complete bipartite graph with bipartition $(V_{1},V_{2})$, where $V_{1}=\{y_{1},y_{2},y_{3},y_{4},y_{5}\}$ and $|V_{2}|=2$. The graph obtained from disjoint graphs $H_{1}$ and $H_{2}$ by adding the edges $x_{i}y_{i}$ $(i=1,2,3,4,5)$, shows that the condition $\delta\geq(n-5)/2$ in Theorem 9 cannot be replaced by $\delta\geq(n-6)/2$. Theorem 10 (Nikoghosyan, 1981) [54] Every graph is hamiltonian if $\kappa\geq 2$ and $$\delta\geq\frac{n+\kappa}{3}.$$ Examples for sharpness. $2K_{\delta}+K_{1}$; $H(1,\delta-\kappa+1,\delta,\kappa)$ $(2\leq\kappa<n/2)$. Theorem 11 (Bauer and Schmeichel, 1991) [5] Every graph is hamiltonian if $\tau\geq 1$ and $$\delta\geq\frac{n+\kappa-2}{3}.$$ Examples for sharpness. $K_{\delta,\delta+1}$; $L_{\delta}$. Theorem 12 (Nash-Williams, 1971) [52] Every graph is hamiltonian if $\kappa\geq 2$ and $$\delta\geq\max\left\{\frac{n+2}{3},\alpha\right\}.$$ Examples for sharpness. $(\lambda+1)K_{\delta-\lambda+1}+K_{\lambda}$ $(\delta\geq 2\lambda)$; $(\lambda+2)K_{\delta-\lambda}+K_{\lambda+1}$ $(\delta\geq 2\lambda+1)$; $H(\lambda,\lambda+1,\lambda+3,\lambda+2)$. Theorem 13 (Bigalke and Jung, 1979) [9] Every graph is hamiltonian if $\tau\geq 1$ and $$\delta\geq\max\left\{\frac{n}{3},\alpha-1\right\}.$$ Examples for sharpness. $K_{\delta,\delta+1}$ $(n\geq 3)$; $L_{\delta}$ $(n\geq 7)$; $K_{\delta,\delta+1}$ $(n\geq 3)$. Theorem 14 (Fraisse, 1986) [34] Let $G$ be a graph and $\lambda$ a positive integer. Then $G$ is hamiltonian if $\kappa\geq\lambda+1$ and $$\delta\geq\max\left\{\frac{n+2}{\lambda+2}+\lambda-1,\alpha+\lambda-1\right\}.$$ Examples for sharpness. $(\lambda+1)K_{\delta-\lambda+1}+K_{\lambda}$ $(\delta\geq 2\lambda)$; $(\lambda+2)K_{\delta-\lambda}+K_{\lambda+1}$ $(\delta\geq 2\lambda+1)$; $H(\lambda,\lambda+1,\lambda+3,\lambda+2)$. Theorem 14 can be considered as a union (not a generalization) of fundamental results for all possible values of $\lambda$. Theorem 15 (Yamashita, 2008) [69] Every graph is hamiltonian if $\kappa\geq 3$ and $$\delta\geq\max\left\{\frac{n+\kappa+3}{4},\alpha\right\}.$$ Examples for sharpness. $3K_{\delta-1}+K_{2}$; $H(2,n-3\delta+3,\delta-1,\kappa)$; $H(1,2,\kappa+1,\kappa)$. Theorem 16 (Chvátal and Erdös, 1972) [18] Every graph is hamiltonian if $$\kappa\geq\alpha.$$ Example for sharpness. $K_{\delta,\delta+1}$. Theorem 17 (Woodall, 1973) [66] Every graph $G$ is hamiltonian if $$b(G)\geq\frac{3}{2}.$$ Example for sharpness. $aK_{2}+\overline{K}_{a-1}$. Theorem 18 (Fleischner, 1974) [33] The square of every 2-connected graph is hamiltonian. Examples for sharpness. Clearly, the power of a graph cannot be reduced to one in Theorem 18, since there are 2-connected nonhamiltonian graphs. Next, 2-connectivity condition in Theorem 18 cannot be relaxed since the square of a graph $G$ is not hamiltonian if $G-x$ has at least three nontrivial components in which $x$ has exactly one neighbor. Theorem 19 (Tutte, 1956) [65] Every 4-connected planar graph is hamiltonian. Examples for sharpness. Tutte’s graph shows that 4-connectivity condition in Theorem 19 cannot be relaxed. Complete bipartite graph $K_{4,5}$ shows that planarity is a necessary condition in Theorem 19. Theorem 20 (R. Thomas and X. Yu, 1994) [64] Every 4-connected projective-plane graph is hamiltonian. Examples for sharpness. The simplest non-orientable surface on which the Petersen graph can be embedded without crossings is the projective plane. The Petersen graph shows that 4-connectivity condition in Theorem 20 cannot be relaxed. On the other hand, there are 4-connected non hamiltonian graphs that cannot be embedded on projective plane (otherwise, all 4-connected graphs are hamiltonian), implying that the condition ”$G$ is projective plane graph” cannot be removed in Theorem 20. Theorem 21 (Faudree and Gould, 1997) [31] Every 2-connected $P_{3}-$free graph is hamiltonian. Examples for sharpness. See Subsection 5.3. Theorem 22 (Broersma, Veldman, 1997) [15] Every 2-connected $\{K_{1,3},P_{6}\}-$free graph is hamiltonian. Examples for sharpness. See Subsection 5.3. Theorem 23 (Faudree, Gould, Ryjáček and Schiermeyer, 1997) [32] Every 2-connected $\{K_{1,3},N_{0,0,3}\}-$free graph with $n\geq 10$ is hamiltonian. Examples for sharpness. See Subsection 5.3. Theorem 24 (Bedrossian, 1997) [7] Every 2-connected $\{K_{1,3},N_{0,1,2}\}-$free graph is hamiltonian. Examples for sharpness. See Subsection 5.3. Theorem 25 (Duffus, Jakobson and Gould, 1997) [24] Every 2-connected $\{K_{1,3},N_{1,1,1}\}-$free graph is hamiltonian. Examples for sharpness. See Subsection 5.3. Theorem 26 (Keil, 1985) [46] Every 1-tough interval graph is hamiltonian. Examples for sharpness. Star graphs are interval nonhamiltonian graphs with $\tau<1$, implying that 1-toughness condition in Theorem 26 cannot be relaxed. The Petersen graph shows that the condition ”$G$ is interval graph” in Theorem 26 cannot be removed. Theorem 27 (Kratsch, Lehel and Müller, 1996) [48] Every $3/2-$tough split graph is hamiltonian. Examples for sharpness. In [48], $(3/2-\epsilon)$-tough split graphs are constructed that are not hamiltonian. There are non hamiltonian graphs with $\tau=9/4-\epsilon>3/2$, implying that the condition ”$G$ is split graph” in Theorem 27 cannot be removed. Theorem 28 (Deogun, Kratsch and Steiner, 1997) [22] Every 1-tough cocomparability graph is hamiltonian. Examples for sharpness. Clearly, any complete graph is a comparability graph and hence, any empty graph is a cocomparability graph with $\tau<1$, implying that the condition ”$G$ is 1-tough” in Theorem 28 cannot be relaxed. On the other hand, there are 1-tough non hamiltonian non cocomparability graphs (otherwise, all 1-tough graphs are hamiltonian), implying that the condition ”$G$ is cocomparability graph” in Theorem 28 cannot be removed. Theorem 29 (Böhme, Harant and Tkáč, 1999) [10] Every chordal, planar graph with $\tau>1$ is hamiltonian. Examples for sharpness. In [10], it is proved that for any $\epsilon>0$, there is a 1-tough chordal planar graph $G_{\epsilon}$ such that the length of a longest cycle of $G_{\epsilon}$ is less than $\epsilon|V(G_{\epsilon})|$, implying that the condition $\tau>1$ in Theorem 29 cannot be relaxed. Chvátal [17] obtained $(3/2-\epsilon)$-tough graphs without a 2-factor, implying that the planarity condition in Theorem 29 cannot be removed. Finally, Harant [38] found 3/2-tough planar nonhamiltonian graphs, implying that the condition ”$G$ is chordal” in Theorem 29 cannot be removed. Theorem 30 (Kaiser, Král and Stacho, 2007) [43] Every $3/2$-tough spider (intersection) graph is hamiltonian. Examples for sharpness. In [43], Kaiser, Král and Stacho constructed $(3/2-\epsilon)$-tough spider graphs that do not contain a Hamilton cycle, implying that the condition ”$G$ is 3/2-tough” in Theorem 30 cannot be relaxed. On the other hand, the condition ”$G$ is spider graph” in Theorem 30 cannot be removed since there are 3/2-tough nonhamiltonian graphs. 10.2 Dominating cycles Theorem 31 (Nikoghosyan, 2011) [59] Let $G$ be a graph. Then each longest cycle in $G$ is a dominating cycle if $\kappa\geq 2$ and $$q\leq\left\{\begin{array}[]{lll}8&\mbox{if}&\mbox{ }\delta=2,\\ \frac{3(\delta-1)(\delta+2)-1}{2}&\mbox{if}&\mbox{ }\delta\geq 3.\end{array}\right.$$ Examples for sharpness. To show that Theorem 31 is sharp, suppose first that $\delta=2$. The graph $K_{1}+2K_{2}$ shows that the connectivity condition $\kappa\geq 2$ in Theorem 31 cannot be relaxed by replacing it with $\kappa\geq 1$. The graph with vertex set $\{v_{1},v_{2},...,v_{8}\}$ and edge set $$\{v_{1}v_{2},v_{2}v_{3},v_{3}v_{4},v_{4}v_{5},v_{5}v_{6},v_{6}v_{1},v_{1}v_{7}% ,v_{7}v_{8},v_{8}v_{4}\},$$ shows that the size bound $q\leq 8$ cannot be relaxed by replacing it with $q\leq 9$. Finally, the graph $K_{2}+3K_{1}$ shows that the conclusion ”each longest cycle in $G$ is a dominating cycle” cannot be strengthened by replacing it with ”$G$ is hamiltonian”. Analogously, we can use $K_{1}+2K_{\delta}$, $K_{2}+3K_{\delta-1}$ and $K_{\delta}+(\delta+1)K_{1}$, respectively, to show that Theorem 31 is sharp when $\delta\geq 3$. So, Theorem 31 is best possible in all respects. Theorem 32 (Nash-Williams, 1971) [52] Let $G$ be a graph. Then each longest cycle in $G$ is a dominating cycle if $\kappa\geq 2$ and $$\delta\geq\frac{n+2}{3}.$$ Examples for sharpness. $2K_{3}+K_{1}$; $3K_{\delta-1}+K_{2}$; $H(1,2,4,3)$. The graph $2K_{3}+K_{1}$ shows that the connectivity condition $\kappa\geq 2$ in Theorem 32 cannot be replaced by $\kappa\geq 1$. The second graph shows that the minimum degree condition $\delta\geq(n+2)/3$ cannot be replaced by $\delta\geq(n+1)/2$. Finally, the third graph shows that the conclusion ”is a dominating cycle” cannot be strengthened by replacing it with ”is a Hamilton cycle”. Theorem 33 (Bigalke and Jung, 1979) [9] Let $G$ be a graph. Then each longest cycle in $G$ is a dominating cycle if $\tau\geq 1$ and $$\delta\geq\frac{n}{3}.$$ Examples for sharpness. $2(\kappa+1)K_{2}+\kappa K_{1}$; $L_{3}$; $G^{*}_{n}$. Theorem 34 (Yamashita, 2008) [69] Let $G$ be graph. Then each longest cycle in $G$ is a dominating cycle if $\kappa\geq 3$ and $$\delta\geq\frac{n+\kappa+3}{4}.$$ Examples for sharpness. $3K_{\delta-1}+K_{2}$; $H(2,n-3\delta+3,\delta-1,\kappa)$; $H(1,2,\kappa+1,\kappa)$. 10.3 $CD_{\lambda}$-cycles Theorem 35 (Jung, 1990) [41] Let $G$ be a graph. Then each longest cycle in $G$ is a $CD_{3}$-cycle if $\kappa\geq 3$ and $$\delta\geq\frac{n+6}{4}.$$ Examples for sharpness. $\lambda K_{\lambda+1}+K_{\lambda-1}$ $(\lambda\geq 2)$ ; $(\lambda+1)K_{\delta-\lambda+1}+K_{\lambda}$ $(\lambda\geq 1)$ ; $H(\lambda-1,\lambda,\lambda+2,\lambda+1)$ $(\lambda\geq 2)$. Theorem 36 (Nikoghosyan, 2009) [57] Let $G$ be a graph and $\lambda$ a positive integer. Then each longest cycle in $G$ is a $CD_{\min\{\lambda,\delta-\lambda+1\}}$-cycle if $\kappa\geq\lambda$ and $$\delta\geq\frac{n+2}{\lambda+1}+\lambda-2.$$ Examples for sharpness. $\lambda K_{\lambda+1}+K_{\lambda-1}$ $(\lambda\geq 2)$ ; $(\lambda+1)K_{\delta-\lambda+1}+K_{\lambda}$ $(\lambda\geq 1)$ ; $H(\lambda-1,\lambda,\lambda+2,\lambda+1)$ $(\lambda\geq 2)$. 10.4 Long cycles Theorem 37 (Dirac, 1952) [23] In every graph, $$c\geq\delta+1.$$ Example for sharpness. Join two copies of $K_{\delta+1}$ by an edge. Theorem 38 (Kouider, 1994) [47] In every graph, $$c\geq\frac{n}{\left\lceil\alpha/\kappa\right\rceil}.$$ Example for sharpness. Complete bipartite graph with $\kappa=\alpha$ shows that the bound in Theorem 38 is sharp. The original result is formulated for 2-connected graphs. However, Theorem 38 is true under assumption that each vertex (edge) is a cycle of length one (two, respectively). Theorem 39 (Nikoghosyan, 1998) [61] Let $G$ be a graph and $C$ a longest cycle in $G$. Then $$|C|\geq(\overline{p}+2)(\delta-\overline{p}).$$ Example for sharpness. $(\kappa+1)K_{\delta-\kappa+1}+K_{\kappa}$. Theorem 40 (Nikoghosyan, 2000) [61] Let $G$ be a graph and $C$ a longest cycle in $G$. Then $$|C|\geq(\overline{c}+1)(\delta-\overline{c}+1).$$ Example for sharpness. $(\kappa+1)K_{\delta-\kappa+1}+K_{\kappa}$. Theorem 41 (Nikoghosyan, 2000) [56] Let $G$ be a graph with $\kappa\geq 2$ and $C$ a longest cycle in $G$. If $\overline{c}\geq\kappa$ then $$|C|\geq\frac{(\overline{c}+1)\kappa}{\overline{c}+\kappa+1}(\delta+2).$$ Otherwise, $$|C|\geq\frac{(\overline{c}+1)\overline{c}}{2\overline{c}+1}(\delta+2).$$ Example for sharpness. $(\kappa+1)K_{\delta-\kappa+1}+K_{\kappa}$. 10.5 Hamilton cycles and long cycles Theorem 42 (Woodall, 1976) [67] Let $G$ be a graph and $\lambda,t,r$ be integers with $n=t(\lambda-1)+r+1$, where $\lambda\geq 2$, $t\geq 0$ and $0\leq r<\lambda-1$. If $$q>t\left({}^{\lambda}_{2}\right)+\left({}^{r+1}_{2}\right)$$ then $$c>\lambda.$$ Example for sharpness. The result is best possible, in view of the graph consisting of $t$ copies of $K_{\lambda}$ and one copy of $K_{r+1}$, all having exactly one vertex in common. Theorem 43 (Fan, Lv and Wang, 2004) [30] Let $G$ be a 2-connected graph and let $2\leq\lambda\leq n-1$. If $$q>\max\left\{f(n,2,\lambda),f(n,\left\lfloor\frac{\lambda}{2}\right\rfloor,% \lambda)\right\}$$ then $$c>\lambda,$$ where $f(n,t,\lambda)=(\lambda+1-t)(\lambda-t)/2+t(n-\lambda-1+t)$ and $2\leq t\leq\lambda/2$. Examples for sharpness. The result is best possible, in view of the graph obtained from $K_{\lambda+1-t}$ by adding $n-(\lambda+1-t)$ isolated vertices, each joined to the same $t$ vertices of $K_{\lambda+1-t}$. Theorem 44 (Alon, 1986) [1] Let $G$ be a graph and $\lambda$ a positive integer. If $\delta\geq\frac{n}{\lambda+1}$ then $$c\geq\frac{n}{\lambda}.$$ Examples for sharpness. $(\lambda+1)K_{\lambda}+K_{1}$; $\lambda K_{\lambda+1}$. Theorem 45 (Dirac, 1952) [23] Let $G$ be a graph. If $\kappa\geq 2$ then $$c\geq\min\{n,2\delta\}.$$ Examples for sharpness. $(\lambda+1)K_{\lambda+1}+K_{\lambda}$ $(\lambda\geq 1)$; $(\lambda+3)K_{\lambda-1}+K_{\lambda+2}$ $(\lambda\geq 2)$; $(\lambda+2)K_{\lambda}+K_{\lambda+1}$ $(\lambda\geq 1)$. Theorem 46 (Kaneko and Yoshimoto, 1952) [44] Let $G$ be a 2-connected balanced bipartite graph. Then $$c\geq\min\{n,4\delta-2\}.$$ Examples for sharpness. Clearly, the condition ”$G$ is balanced” in Theorem 46 cannot be removed. Consider the balanced bipartite graph $G=(X,Y;E)$ with vertex classes of the form $X=P\cup Q$, $Y=R\cup S$ with $z\in Q$, where $|P|=|R|=|Q|=|S|=n/4$, $N_{G}(x)=R$ for all $x\in P$, $N_{G}(x)=S$ for all $x\in Q-z$ and $N_{G}(z)=Y$. This example shows that 2-connectivity condition in Theorem 46 cannot be weakened. Next, consider the balanced bipartite graph $G=(X,Y;E)$ with vertex classes of the form $X=P\cup Q$, $Y=R\cup S$, where $|P|=|R|=|Q|=|S|=n/4$, $N_{G}(x)=R$ for all $x\in P$, and $N_{G}(x)=Y$ for all $x\in Q$. This example shows that the bound $4\delta-2$ in Theorem 46 cannot be improved. Theorem 47 (Bauer and Schmeichel, 1987) [4] Let $G$ be a graph. If $\tau\geq 1$ then $$c\geq\min\{n,2\delta+2\}.$$ Examples for sharpness. $K_{\delta,\delta+1}$; $L_{2}$. Theorem 48 (Nikoghosyan, 2012) [60] Let $G$ be a graph. If $\tau>4/3$ then $$c\geq\min\{n,2\delta+5\}.$$ Examples for sharpness. The Petersen graph shows that the condition $\tau>4/3$ in Theorem 48 cannot be replaced by $\tau=4/3$. Let $H_{1}$ be a complete bipartite graph with bipartition $V_{1}=\{x_{1},x_{2},x_{3},x_{4},x_{5}\}$ and $V_{2}=\{y_{1},y_{2}\}$, and let $H_{2}$ be a complete graph with vertex set $V=\{z_{1},z_{2},z_{3},z_{4},z_{5}\}$. The graph obtained from disjoint graphs $H_{1}$ and $H_{2}$ by adding the edges $x_{i}z_{i}$ $(i=1,...,5)$, shows that the bound $c\geq 2\delta+5$ in Theorem 48 cannot be replaced by $c\geq 2\delta+6$. Theorem 49 (Nikoghosyan, 1981) [54] Let $G$ be a graph. If $\kappa\geq 3$ then $$c\geq\min\{n,3\delta-\kappa\}.$$ Examples for sharpness. $3K_{\delta-1}+K_{2}$; $H(1,\delta-\kappa+1,\delta,\kappa)$. Theorem 50 (Jung, 1978) [39] Let $G$ be a graph. If $\kappa\geq 3$ and $\delta\geq\alpha$ then $$c\geq\min\{n,3\delta-3\}.$$ Examples for sharpness. $(\lambda+2)K_{\lambda+2}+K_{\lambda+1}$; $(\lambda+4)K_{\lambda}+K_{\lambda+3}$; $(\lambda+3)K_{\lambda+1}+K_{\lambda+2}$. Theorem 51 (Nikoghosyan, 2009) [57] Let $G$ be a graph and $\lambda$ a positive integer. If $\kappa\geq\lambda+2$ and $\delta\geq\alpha+\lambda-1$ then $$c\geq\min\{n,(\lambda+2)(\delta-\lambda)\}.$$ Examples for sharpness. $(\lambda+2)K_{\lambda+2}+K_{\lambda+1}$; $(\lambda+4)K_{\lambda}+K_{\lambda+3}$; $(\lambda+3)K_{\lambda+1}+K_{\lambda+2}$. Theorem 52 (M.Zh. Nikoghosyan and Zh.G. Nikoghosyan, 2011) [53] Let $G$ be a graph. If $\kappa\geq 4$ and $\delta\geq\alpha$ then $$c\geq\min\{n,4\delta-\kappa-4\}.$$ Examples for sharpness. $4K_{\delta-2}+K_{3}$; $H(1,2,\kappa+1,\kappa)$; $H(2,n-3\delta+3,\delta-1,\kappa)$. Theorem 53 (Bauer, Morgana, Schmeichel and Veldman, 1989) [3] Let $G$ be a graph. If $\kappa\geq 2$ and $\delta\geq\frac{n+2}{3}$ then $$c\geq\min\{n,n+\delta-\alpha\}.$$ Examples for sharpness. $2K_{\delta}+K_{1}$; $3K_{\delta-1}+K_{2}$; $K_{2\delta-2,\delta}$. Theorem 54 (Bauer, Schmeichel and Veldman, 1988) [6] Let $G$ be a graph. If $\tau\geq 1$ and $\delta\geq\frac{n}{3}$ then $$c\geq\min\{n,n+\delta-\alpha+1\}.$$ Examples for sharpness. $K_{\delta,\delta+1}$; $L_{\delta}$; $G^{*}_{n}$. 10.6 Dominating cycles and long cycles Theorem 55 (Jung, 1981) [40] Let $G$ be a graph. If $\kappa\geq 3$ then either each longest cycle in $G$ is a dominating cycle or $$c\geq 3\delta-3.$$ Examples for sharpness. $(\lambda+1)K_{\lambda+1}+K_{\lambda}$ $(\lambda\geq 1)$; $(\lambda+3)K_{\lambda-1}+K_{\lambda+2}$ $(\lambda\geq 2)$; $(\lambda+2)K_{\lambda}+K_{\lambda+1}$ $(\lambda\geq 1)$. Theorem 56 (M.Zh. Nikoghosyan and Zh.G. Nikoghosyan, 2011) [53] Let $G$ be a graph. If $\kappa\geq 4$ then either each longest cycle in $G$ is a dominating cycle or $$c\geq 4\delta-\kappa-4.$$ Examples for sharpness. $4K_{\delta-2}+K_{3}$; $H(2,\delta-\kappa+1,\delta-1,\kappa)$; $H(1,2,\kappa+1,\kappa)$. 10.7 $CD_{\lambda}$-cycles and long cycles Theorem 57 (Nikoghosyan, 2009) [57] Let $G$ be a graph and $\lambda$ a positive integer. 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Institute for Informatics and Automation Problems National Academy of Sciences P. Sevak 1, Yerevan 0014, Armenia E-mail: zhora@ipia.sci.am

Energy transfer model and large periodic boundary value problem for the quintic nonlinear Schrödinger equations Hideo Takaoka Department of Mathematics, Kobe University Kobe, 657-8501, Japan takaoka@math.kobe-u.ac.jp This work was supported by JSPS KAKENHI Grant Number 18H01129. () Abstract We study a dynamics and energy exchanges between a linear oscillator and a nonlinear interaction state for the one dimensional, quintic nonlinear Schrödinger equation. Grébert and Thomann [9] proved that there exist solutions with initial data built on four Fourier modes, that confirms the conservative exchange of wave energy. Captured multi resonance in multiple Fourier modes, we simulate a similar energy exchange in long-period waves. $2010$ Mathematics Subject Classification Numbers. 35Q55, 42B37. Key Words and Phrases. Nonlinear Schrödinger Equation, Energy Transfer, Long-Period Waves. 1 Introduction In this paper, we consider the defocusing quintic nonlinear Schrödinger equation $$\displaystyle i\partial_{t}u+\partial_{x}^{2}u=|u|^{4}u,\quad t\in\mathbb{R},~% {}x\in\mathbb{T}_{L}=[-L/2,L/2],$$ (1.1) where $L>0,~{}u=u(t,x)\,:\,\mathbb{R}\times\mathbb{T}_{L}\to\mathbb{C}$ is a complex-valued function and the spatial domain $\mathbb{T}_{L}$ is taken to be a torus of length $L$, i.e., we assume the periodic boundary condition. In the case when $L=2\pi$, we denote $\mathbb{T}=\mathbb{T}_{2\pi}$ as usual. Our aim of this paper is to consider the long periodic solutions ($L\gg 1$) to (1.1), while there are exchanges of resonant energy at particular frequencies. The sign of nonlinearity ($+1$ for the defocusing case and $-1$ for the focusing case) will not play the central role in the present discussion. For simplicity, we focus on the situation of the defocusing case. The equation (1.1) satisfies the mass $M[u]$ and energy $E[u]$ conservations laws; $$\displaystyle M[u](t)=\int_{\mathbb{T}_{L}}|u(t,x)|^{2}\,dx,$$ (1.2) $$\displaystyle E[u](t)=\int_{\mathbb{T}_{L}}\left(\frac{1}{2}|\partial_{x}u(t,x% )|^{2}+\frac{1}{6}|u(t,x)|^{6}\right)\,dx,$$ (1.3) which impose the constraints on a dynamics of mass density of solutions. We briefly recall known results concerning the Cauchy problem for the quintic NLS. In the non-periodic scenario, i.e., $x\in\mathbb{R}$, the equation is called mass-critical or $L^{2}$-critical from the viewpoint of scaling. Indeed, the one-dimensional quintic nonlinear Schrödinger equations with non-periodic boundary condition is left invariant by the scaling $$u\mapsto u_{\lambda};\quad u(t,x)\mapsto u_{\lambda}(t,x)=\lambda^{1/2}u(% \lambda^{2}t,\lambda x),\qquad\lambda>0,$$ which preserves the homogeneous Sobolev norm $\dot{H}^{s}(\mathbb{R})$ with $s=0$. On $\mathbb{R}$-case, the local well-posedness was proved by Cazenave and Weissler [2] for data in $L^{2}$ (see also [8] and [13]). Notice that in [2], the existence time of solution depends on the position of data and not only on its size. One can also prove the global well-posedness in $L^{2}$ provided that the initial data in $L^{2}$ is sufficiently small. Concerning the local well-posedness theory in fractional Sobolev spaces, we refer to the paper [8]. The global well-posedness and scattering in the threshold space $L^{2}$ was obtained by Dodson [6]. More precisely, it is shown that for all $u_{0}\in L^{2}(\mathbb{R})$, there exist a unique time global solution to (1.1) and $u_{\pm}\in L^{2}(\mathbb{R})$ satisfying that $$\|e^{-it\partial_{x}^{2}}u(t)-u_{\pm}\|_{L^{2}}\to 0$$ as $t\to\pm\infty$. It is possible to consider the global well-posedness and scattering for the nonlinear Schrödinger equations with energy sub-critical nonlinearities. This is established in [5] for initial data below the energy norm. We now turn to the case of periodic boundary conditions. The local well-posedness was proved by Bourgain [1] for data in $H^{s}(\mathbb{T})$, with $s>0$. This combined with the $H^{1}$-energy conservation law (an a priori estimate for solutions) leads to global well-posedness in $H^{1}(\mathbb{T})$. Similar results hold for the equation in the $L$-periodic boundary condition case $\mathbb{T}_{L}$ for any $L>0$, without having to change the proof. Definition 1.1. For the function $\phi:\mathbb{T}_{L}\to\mathbb{C}$, we define the Fourier transform $$\widehat{\phi}(\xi)=\int_{-L/2}^{L/2}e^{-ix\xi}\phi(x)\,dx,\quad\xi\in 2\pi% \mathbb{Z}/L.$$ Then we have the representation $\phi(x)=\int e^{ix\xi}\widehat{\phi}(\xi)\,(d\xi)_{L}$ by $$\int e^{ix\xi}\widehat{\phi}(\xi)\,(d\xi)_{L}=\frac{1}{L}\sum_{\xi\in 2\pi% \mathbb{Z}/L}e^{ix\xi}\widehat{\phi}(\xi).$$ The Sobolev $H^{s}(\mathbb{T}_{L})$ norm is given by $$\|\phi\|_{H^{s}(\mathbb{T}_{L})}=\left(\frac{1}{L}\sum_{\xi\in 2\pi\mathbb{Z}/% L}\langle\xi\rangle^{2s}|\widehat{\phi}(\xi)|^{2}\right)^{1/2}.$$ In this paper, we want to understand the interaction of mass for each frequency $\xi\in 2\pi\mathbb{Z}/L$. In fact, the $L^{2}$-norm conservation law (1.2) constrains this object. For the periodic boundary value problem, i.e., $x\in\mathbb{T}$ (the case of $L=2\pi$) with replacing the nonlinearity $|u|^{4}u$ by $\nu|u|^{4}u$ with $\nu>0$ $$\displaystyle i\partial_{t}u+\partial_{x}^{2}u=\nu|u|^{4}u,\quad(t,x)\in% \mathbb{R}\times\mathbb{T},$$ (1.4) Grébert and Thomann [9] examined the dynamics exhibited by the solution of (1.4). More precisely, they proved the following theorem. Theorem 1.1 (Grébert and Thomann [9]). Let $k\in\mathbb{Z}\backslash\{0\}$ and $n\in\mathbb{Z}$. ${\cal A}$ is a set of the form ${\cal A}=\{a_{2},~{}a_{1},~{}b_{2},~{}b_{1}\}$ where $a_{2}=n,~{}a_{1}=n+3k,~{}b_{2}=n+4k,~{}b_{1}=n+k$. There exist $T>0,~{}\lambda_{0}>0$ and a $2T$-periodic function $K_{*}:\mathbb{R}\mapsto(0,1)$ which satisfies $K_{*}(0)\leq 1/4$ and $K_{*}(T)\geq 3/4$ so that if $0<\nu<\nu_{0}$, there exists a solution to (1.4) satisfying for all $0\leq t\leq\nu^{-3/2}$ $$u(t,x)=\sum_{j\in{\cal A}}u_{j}(t)e^{ijx}+\nu^{1/4}q_{1}(t,x)+\nu^{3/2}tq_{2}(% t,x),$$ with $$|u_{a_{1}}(t)|^{2}=2|u_{a_{2}}(t)|^{2}=K_{*}(\nu t),$$ $$|u_{b_{1}}(t)|^{2}=2|u_{b_{2}}(t)|^{2}=1-K_{*}(\nu t),$$ and where for all $s\in\mathbb{R},~{}\|q_{1}(t,\cdot)\|_{H^{s}(\mathbb{T})}\leq C_{s}$ for all $t\in\mathbb{R}_{+}$, and $\|q_{2}(t,\cdot)\|_{H^{s}(\mathbb{T})}\leq C_{s}$ for all $0\leq t\leq\nu^{-3/2}$. From Theorem 1.1, we obtained that there exist solutions with initial data built on four Fourier modes, that involves periodic energy exchanges between the modes initially excited. The proof of this result is mostly by calculations the resonant normal form of the Hamiltonian of (1.4) up to order ten. Remark 1.1. Another interesting result is the two-dimensional cubic nonlinear Schrödinger equation. In [4], Colliander, Keel, Staffilani, Takaoka and Tao showed the weak turbulence property for the 2D defocusing cubic nonlinear Schrödinger equation: $$\displaystyle i\partial_{t}u+\Delta u=|u|^{2}u,\qquad(t,x)\in\mathbb{R}\times% \mathbb{T}^{2}.$$ (1.5) More precisely, for any $s>1,~{}K\gg 1\gg\varepsilon>0$, there exists a time $T\gg 1$ such that the initial value problem corresponding to the equation in (1.5) has a global in time solution $u(t)$ satisfying that $$\|u(0)\|_{H^{s}(\mathbb{T}^{2})}\leq\varepsilon,\quad\|u(T)\|_{H^{s}(\mathbb{T% }^{2})}\geq K.$$ This exhibits the $H^{s}$-norm inflation of solutions to the cubic nonlinear Schrödinger equations, that admits solutions with transferring wave energy from low to high Fourier modes. In this paper, we proved that there exist solutions of the one-dimensional quintic nonlinear Schrödinger equations with initially excited in multi-frequency modes, where the mass is localized and involves conservative energy exchange between the modes initially excited. Let us now define the wavenumber set consisting of nonlinear resonance interactions in the equation (1.1). Definition 1.2 (Resonance interaction set). Let $k\in\mathbb{N}/L$ be fixed. For $j\in\mathbb{Z}$, we set $\alpha_{1,j},~{}\alpha_{3,j},~{}\alpha_{2,j}$ and $\alpha_{4,j}$ as follows: $$\alpha_{1,j}=2\pi\left(3k+\frac{j}{L}\right),~{}\alpha_{3,j}=2\pi\frac{j}{L},~% {}\alpha_{2,j}=2\pi\left(k+\frac{j}{L}\right),~{}\alpha_{4,j}=2\pi\left(4k+% \frac{j}{L}\right).$$ With $\alpha_{m,j}$, we set $${\cal R}_{m}=\left\{\alpha_{m,j}\mid j\in\mathbb{Z},~{}0\leq j<L\right\},$$ for $1\leq m\leq 4$, and ${\cal R}=\cup_{m=1}^{4}{\cal R}_{m}$. We start by defining the following norms. Definition 1.3. We define the sets ${\cal N}_{l},~{}{\cal N}_{m},~{}{\cal N}_{r},~{}{\cal N}_{h}$ to be subsets of the set $2\pi\mathbb{Z}/L$ as follows: $${\cal N}_{l}=\left\{\xi\in 2\pi\mathbb{Z}/L\mid\xi=2\pi(k\eta+\tau+j/L),~{}(% \eta,\tau,j)\in\mathbb{Z}^{3},~{}\tau\in[0,k),~{}j\in[0,L),~{}-99\leq\eta\leq 9% 8\right\},$$ $${\cal N}_{m}=\left\{\xi\in 2\pi\mathbb{Z}/L\mid\xi=2\pi(k\eta+\tau+j/L),~{}(% \tau,j)\in\mathbb{Z}^{2},~{}\tau\in[0,k),~{}j\in[0,L),~{}\eta\in\{99,-100\}% \right\},$$ $${\cal N}_{r}=\left\{\xi\in 2\pi\mathbb{Z}/L\mid\xi=2\pi(k\widetilde{\eta}+% \widetilde{\tau}+j/L),~{}(\widetilde{\tau},j)\in\mathbb{Z}^{2},~{}-k/2<% \widetilde{\tau}\leq k/2,~{}j\in[0,L),~{}\widetilde{\eta}\in\{0,1,3,4\}\right\},$$ $${\cal N}_{h}=(2\pi\mathbb{Z}/L)\backslash({\cal N}_{l}\cup{\cal N}_{m}).$$ For the Sobolev index $s\in\mathbb{R}$, let $m(\xi)$ be the multiplier function defined on $2\pi\mathbb{Z}/L$ to be $$\displaystyle m(\xi)=\left\{\begin{array}[]{ll}\langle\widetilde{\tau}\rangle^% {s-1/2},&\mbox{if $\xi\in{\cal N}_{r}$},\\ \langle k\rangle^{s-1/2},&\mbox{if $\xi\in{\cal N}_{l}$ and $\xi\not\in{\cal N% }_{r}$},\\ \langle k\rangle^{s-1/2}\langle\tau\rangle^{1/2},&\mbox{if $\xi\in{\cal N}_{m}% $ and $\xi>0$},\\ \langle k\rangle^{s-1/2}\langle k-\tau\rangle^{1/2},&\mbox{if $\xi\in{\cal N}_% {m}$ and $\xi<0$},\\ \langle\xi\rangle^{s},&\mbox{if $\xi\in{\cal N}_{h}$},\end{array}\right.$$ where $\tau$ and $\widetilde{\tau}$ obey the formula $$\displaystyle\xi=2\pi\times\left\{\begin{array}[]{ll}\left(k\widetilde{\eta}+% \widetilde{\tau}+j/L\right),&(\widetilde{\tau},j)\in\mathbb{Z}^{2},~{}-k/2<% \widetilde{\tau}\leq k/2,~{}j\in[0,L),~{}\widetilde{\eta}\in\{0,1,3,4\},\\ \left(k\eta+\tau+j/L\right),&\mbox{otherwise}.\end{array}\right.$$ For the sequence $u=(u_{\xi})_{\xi\in 2\pi\mathbb{Z}/L}$, define $$\|u\|_{s}=\left(\frac{1}{L}\sum_{\xi\in 2\pi\mathbb{Z}/L}m(\xi)^{2}|u_{\xi}|^{% 2}\right)^{1/2}.$$ Remark 1.2. The set ${\cal R}$ is contained in ${\cal N}_{r}$. Our main result is the following theorem. Theorem 1.2. Let $s\in(1,3/2)$, let $\nu>0$ be a small constant and let $L$ be a natural number. Then there exist a positive number $k_{0}$, so that for $k_{0}\leq k\in\mathbb{N}/L$, there exist a smooth global solution $u(t)$ to (1.1) such that for $|t|\ll 1/(L^{3}\nu^{2})$, $$\displaystyle u(t)=\sum_{m=1}^{4}u_{{\cal R}_{m}}(t)+e(t),$$ (1.6) where $u_{{\cal R}_{m}}$ is whose Fourier transform is supported on the frequency space ${\cal R}_{m}$ so that $$\|u_{{\cal R}_{1}}(t)\|_{L^{2}}=2\|u_{{\cal R}_{3}}(t)\|_{L^{2}}^{2}=\nu\left(% \frac{1}{2}-\nu^{2}K(t)\right),$$ and $$\|u_{{\cal R}_{2}}(t)\|_{L^{2}}^{2}=2\|u_{{\cal R}_{4}}(t)\|_{L^{2}}^{2}=\nu% \left(\frac{1}{2}+\nu^{2}K(t)\right),$$ where $$K(t)=\frac{1}{2\nu^{2}}\sin\arctan\left(\frac{3\nu^{2}t}{2L^{3}}\right).$$ Moreover, the error term $e(t)$ associated with the expression in (1.6) satisfies that $$\|e(t)\|_{s}^{2}\lesssim\frac{\nu}{L}\left(e^{c\nu^{2}t/L^{3}}-1\right)+\frac{% \nu^{3}}{\langle k\rangle^{5/2-s}}e^{c\nu^{2}t}+\frac{\nu^{3}}{\langle k% \rangle^{3/2-s}}\left(e^{c\nu^{2}t}-1\right)+\nu\left(e^{c\nu^{2}t}-1-c\nu^{2}% t\right).$$ Remark 1.3. In Theorem 1.2, we pick $L$ to be a natural number for simplicity. Only a small modification for the proof is required in the general $L>0$. It is not important to assume the natural number $L$. Remark 1.4. The equation (1.1) has a gauge symmetry. After conducting gauge transformation $u\mapsto e^{i\theta}u$ with $\theta=n/L$, it is easily to shift $\alpha_{m,j}\mapsto\alpha_{m,j}+2\pi n/L~{}(1\leq m\leq 4)$ for $n\in\mathbb{Z}$ and prove the corresponding theorem to Theorem 1.2. Remark 1.5. Choosing $k$ large enough such as $k^{3/2-s}\gg L^{6}\nu^{2}$, one can prove by Theorem 1.2 that $$\|e(T_{0})\|_{s}^{2}\ll\left(\frac{\nu}{L}\right)^{3}T_{0},$$ where $T_{0}=o(1)/(L^{3}\nu^{2})$. Then for small data such that $\nu\ll L^{-3/2}$, we have that $T_{0}\gg 1$ and a solution $u(t)$ to (1.1) satisfying $$\|u(T_{0})\|_{H^{s}}^{2}-\|u(-T_{0})\|_{H^{s}}^{2}\gtrsim\langle k\rangle^{2s}% \left(-3^{2s}+1^{2s}+\frac{4^{2s}}{2}\right)\nu^{3}K(T_{0}),$$ for $s\in(1,3/2)$, where the right-hand side is positive. Remark 1.6. We can expect similar results to hold for the following more general nonlinearities with essentially the same proof: $$i\partial_{t}u+\partial_{x}^{2}u=\sum_{j=1}^{J}a_{j}|u|^{2j}u$$ where $a_{j}\in\mathbb{R}$. The proof of Theorem 1.2 relies on obtaining the dynamics in a toy model (finite dimensional approximation) of nonlinear Schrödinger equations along with error estimates between finite dimensional model and the full infinite dimensional model. In Section 2, we present some notation. Section 3 describes the reductions of the equation (1.1) as an infinite system of ODE’s. In Section 4, we construct the appropriate Toy model equation associated with the finite dimensional ODE system of reduced NLS equation, and solve it. In Section 5, we give qualitative estimates for solutions to the finite dimensional ODE system. In Section 6, we prove that the ODE system derived in Section 4 approximates the dynamics of the quintic nonlinear Schrödinger equation (1.1). Theorem 1.2 is established in Section 7. 2 Notation Let us introduce some notation. We prefer to use the notation $\langle\cdot\rangle=(1+|\cdot|^{2})^{1/2}$. The over dot $\dot{a}(t)$ denotes the derivative of $a(t)$ with respect to time $t$. We use $c,~{}C$ to denote various constants. We use $A\lesssim B$ to denote $A\leq CB$ for some constant $C>0$. Similarly, we write $A\ll B$ to mean $A\leq cB$ for some small constant $c>0$. For an odd natural number $n$ and complex-valued functions $f_{1},~{}f_{2},\ldots,~{}f_{n}$ defined on the set $2\pi\mathbb{Z}/L$, we write the discrete convolution (convolution sum) $[f_{1}*f_{2}*\ldots*f_{n}](\xi)$ as $$[f_{1}*f_{2}*\ldots*f_{n}](\xi)=\frac{1}{L^{n-1}}\sum^{*}\prod_{j=1}^{n}f_{1}(% \xi_{j}),$$ where the superscript notation $*$ of $\sum^{*}$ indicates a sum running over the hyperplane set $\xi_{1}-\xi_{2}+\xi_{3}-\ldots+\xi_{n}=\xi$ with summation index. 3 Reductions of the equation (1.1) as an infinite system of ODEs In this section, we consider the smooth solution to (1.1). Namely, suppose that $u$ is a smooth global in time solution to (1.1). Let us start with the ansatz $$u(t,x)=\int a_{\xi}(t)e^{ix\xi-it\xi^{2}}\,(d\xi)_{L}.$$ In what follows, we shall omit the time variable $t$ of $a_{\xi}(t)$ in abbreviated form without confusion. With this transform, the equation (1.1) becomes $$\displaystyle i\dot{a}_{\xi}=\frac{1}{L^{4}}\sum^{*}a_{\xi_{1}}\overline{a_{% \xi_{2}}}a_{\xi_{3}}\overline{a_{\xi_{4}}}a_{\xi_{5}}e^{-it\phi(\xi_{1},\xi_{2% },\xi_{3},\xi_{4},\xi_{5},\xi)},$$ (3.1) where the factor $\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi)$ means the oscillation frequency such that $$\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})=\xi_{1}^{2}-\xi_{2}^{2}+% \xi_{3}^{2}-\xi_{4}^{2}+\xi_{5}^{2}-\xi_{6}^{2}.$$ In the next section, we seek the quintic resonant structure in the last term on the right-hand side of (3.1). 4 Toy model In this section, we construct a finite dimensional model, in some sense, as the corresponding approximation model of (3.1). Consequently, it will approximate an exact solution to (3.1). Let us begin by recalling some earlier results by Grébert-Thomann [9]. The first one is some arithmetical result. Lemma 4.1 (Lemma 2.1 in [9]). Assume that $(j_{1},j_{2},j_{3},\ell_{1},\ell_{2},\ell_{3})\in(\mathbb{Z}/L)^{6}$ satisfy $$\displaystyle\left\{\begin{array}[]{l}j_{1}+j_{2}+j_{3}=\ell_{1}+\ell_{2}+\ell% _{3},\\ j_{1}^{2}+j_{2}^{2}+j_{3}^{2}=\ell_{1}^{2}+\ell_{2}^{2}+\ell_{3}^{2},\end{% array}\right.\quad\mbox{and}\quad\{j_{1},j_{2},j_{3}\}\neq\{\ell_{1},\ell_{2},% \ell_{3}\}.$$ (4.1) Then $\{j_{1},j_{2},j_{3}\}\cap\{\ell_{1},\ell_{2},\ell_{3}\}=\emptyset$. The second result is small cardinality of (4.1). Lemma 4.2 (Lemma 2.2 in [9]). Assume that there exist $(j_{1},j_{2},j_{3},\ell_{1},\ell_{2},\ell_{3})\in(\mathbb{Z}/L)^{6}$ which satisfy (4.1). Then the cardinal number of the set satisfying (4.1) is greater than or equal to $4$. Taking into the observation in Lemma 4.2, we have that the cardinal of the pair $(j_{1},\ell_{1},j_{2},\ell_{2},j_{3},\ell_{3})$ without the case $\{j_{1},j_{2},j_{3}\}\neq\{\ell_{1},\ell_{2},\ell_{3}\}$, appearing in the right-hand side of (4.1) is greater than or equal to $4$. Now we define the resonance interaction sets. Definition 4.1. We shall say that the pair $(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})$ satisfies the resonance condition, if the following conditions hold: (i) $\xi_{1}+\xi_{3}+\xi_{5}=\xi_{2}+\xi_{4}+\xi_{6}$, (ii) $(m_{1},m_{2},m_{3},m_{4})\in\{1,2,3,4\}^{4}$ satisfy $\{(m_{1},m_{3}),\,(m_{2},m_{4})\}=\{(1,3),\,(2,4)\}$, (iii) two elements of $\xi_{1},\xi_{3},\xi_{5}$ are permitted in the set ${\mathcal{R}}_{m_{1}}$, namely that are described by $\alpha_{m_{1},j_{1}}$ and $\alpha_{m_{1},j_{3}}$ with some integers $j_{1}$ and $j_{3}$; one element of $\xi_{1},\xi_{3},\xi_{5}$ is permitted in the set ${\mathcal{R}}_{m_{3}}$, namely that is described by $\alpha_{m_{3},j_{5}}$ with some integer $j_{5}$, (iv) two elements of $\xi_{2},\xi_{4},\xi_{6}$ are permitted in the set ${\mathcal{R}}_{m_{2}}$, namely that are described by $\alpha_{m_{2},j_{2}}$ and $\alpha_{m_{2},j_{4}}$ with some integers $j_{2}$ and $j_{4}$; one element of $\xi_{2},\xi_{4},\xi_{6}$ is permitted in the set ${\mathcal{R}}_{m_{4}}$, namely that is described by $\alpha_{m_{4},j_{6}}$ with some integer $j_{6}$, (v) $j_{1},j_{3},j_{5},j_{2},j_{4},j_{6}$ given by (ii) and (iii) above satisfy that $\{j_{1},j_{3}\}=\{j_{2},j_{4}\}$ and $j_{5}=j_{6}=(j_{1}+j_{3})/2=(j_{2}+j_{4})/2$. For the resonant condition in Definition 4.1, we have the following property. Lemma 4.3. If the pair $(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})$ satisfies the resonance condition, then $\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})=0$. Proof. By symmetry, we may assume that $\xi_{m}~{}(1\leq m\leq 6)$ satisfy $$(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})=(\alpha_{1,j_{1}},\alpha_{2,% j_{2}},\alpha_{1,j_{3}},\alpha_{2,j_{4}},\alpha_{3,j_{5}},\alpha_{4,j_{6}}),$$ where $$\displaystyle\{j_{1},j_{3}\}=\{j_{2},j_{4}\},\quad j_{5}=j_{6}=\frac{j_{1}+j_{% 3}}{2}=\frac{j_{2}+j_{4}}{2},$$ (4.2) which yields that $\phi(j_{1},j_{2},j_{3},j_{4},j_{5},j_{6})=0$. By means of the identity $$\displaystyle\alpha_{1,0}^{2}+\alpha_{1,0}^{2}+\alpha_{3,0}^{2}=\alpha_{2,0}^{% 2}+\alpha_{2,0}^{2}+\alpha_{4,0}^{2},$$ (4.3) we obtain $$\frac{1}{(2\pi)^{2}}\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})=% \frac{2k}{L}\left(3(j_{1}+j_{3})-(j_{2}+j_{4})-4j_{6}\right)+\frac{1}{L^{2}}% \phi(j_{1},j_{2},j_{3},j_{4},j_{5},j_{6}),$$ which equals to zero by (4.2). ∎ Remark 4.1. If we stipulate the $2\pi$-periodic setting to (1.1), namely $L=2\pi$, such a resonance condition was already known in [9]. Indeed, by taking $\{\xi_{1},\xi_{3},\xi_{5}\}=\{\alpha_{1,0},\alpha_{1,0},\alpha_{3,0}\}$ and $\{\xi_{2},\xi_{4},\xi_{6}\}=\{\alpha_{2,0},\alpha_{2,0},\alpha_{4,0}\}$ ($j$-factor of $\alpha_{m,j}$ are zero), we obtain $$\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})=\alpha_{1,0}^{2}+\alpha_% {1,0}^{2}+\alpha_{3,0}^{2}-\alpha_{2,0}^{2}-\alpha_{2,0}^{2}-\alpha_{4,0}^{2}=0,$$ which was used in (4.3). We note that the condition (v) in Definition 4.1 implies $\{j_{1},j_{3},j_{5}\}=\{j_{2},j_{4},j_{6}\}$, which case appears in the second condition in (4.1). By removing non-resonance term in the term on the right-hand side of (3.1), we propose to a finite dimensional system of ODE, which we call the resonant system corresponding to (3.1). Consider the initial value problem for the following resonant truncation of (3.1); for $\xi=\alpha_{m,j}\in{\mathcal{R}}_{m},~{}1\leq m\leq 4$, $$\displaystyle i\dot{r}_{\xi}$$ $$\displaystyle=$$ $$\displaystyle\frac{1}{L^{4}}\sum^{*}_{\text{res}(\xi)}r_{\xi_{1}}\overline{r_{% \xi_{2}}}r_{\xi_{3}}\overline{r_{\xi_{4}}}r_{\xi_{5}},$$ (4.4) where we denote by $\text{res}(\xi)$ the set such that the pair $(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi)$ satisfies the resonance condition. We then prove that the approximate solutions of (4.4). 5 Dynamics of approximate solutions for some time In this section, we shall study the resonant truncation ODE system in (4.4). 5.1 Conserved quantities By the subscript “res” to the summation on the hyperplane $\xi_{1}+\xi_{3}+\xi_{5}=\xi_{2}+\xi_{4}+\xi_{6}$, we use the summation formula $$\sum_{\text{res}}f(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})=\sum_{% \scriptstyle\xi_{1}+\xi_{3}+\xi_{5}=\xi_{2}+\xi_{4}+\xi_{6}\atop{\scriptstyle(% \xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})\,\text{satisfies the % resonance condition}}}f(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6}),$$ which is equivalent to $$\sum_{\xi_{1}\in{\cal R}}\sum^{*}_{\text{res}(\xi_{1})}f(\xi_{1},\xi_{2},\xi_{% 3},\xi_{4},\xi_{5},\xi_{6})$$ and its symmetry-equivalent formula. The resonant truncation ODE system in (4.4) retains conserved quantities as follows. Lemma 5.1. Let $\{r_{\xi}(t)\}$ be a global in time solution to (4.4). Then we have the relation: $$\displaystyle\frac{d}{dt}\sum_{\xi\in{\mathcal{R}}}|r_{\xi}(t)|^{2}=0.$$ (5.1) Remark 5.1. The $\ell^{2}$-norm $(\sum_{\xi\in{\mathcal{R}}}|r_{\xi}(t)|^{2})^{1/2}$ is a conserved quantity for (4.4). Proof of Lemma 5.1. It will be convenience to raise the frequency representation $\xi$ in (5.1) to $\xi_{6}$. Namely it suffices to show that $$\displaystyle\frac{d}{dt}\sum_{\xi_{6}\in{\mathcal{R}}}|r_{\xi_{6}}(t)|^{2}=0.$$ Multiplying $\overline{r_{\xi_{6}}}$ to (4.4) and taking the imaginary part, we have $$\displaystyle\Im(i\dot{r}_{\xi_{6}}\overline{r_{\xi_{6}}})=\frac{1}{L^{4}}\Im% \sum^{*}_{\text{res}(\xi_{6})}r_{\xi_{1}}\overline{r_{\xi_{2}}}r_{\xi_{3}}% \overline{r_{\xi_{4}}}r_{\xi_{5}}\overline{r_{\xi_{6}}},$$ where $\xi_{6}\in{\cal R}$. The term on the left-hand side of the above equation will be $$-\frac{1}{2}\frac{d}{dt}|r_{\xi_{6}}|^{2}.$$ Then after the summation over $\xi_{6}\in{\mathcal{R}}$, we arrive at the following: $$\displaystyle-\frac{1}{2}\frac{d}{dt}\sum_{\xi_{6}\in{\mathcal{R}}}|r_{\xi_{6}% }(t)|^{2}=\frac{1}{2iL^{4}}\sum_{\xi_{6}\in{\mathcal{R}}}\sum^{*}_{\text{res}(% \xi_{6})}\left(r_{\xi_{1}}\overline{r_{\xi_{2}}}r_{\xi_{3}}\overline{r_{\xi_{4% }}}r_{\xi_{5}}\overline{r_{\xi_{6}}}-\overline{r_{\xi_{1}}}r_{\xi_{2}}% \overline{r_{\xi_{3}}}r_{\xi_{4}}\overline{r_{\xi_{5}}}r_{\xi_{6}}\right),$$ which is zero, since by symmetrization schemes. ∎ We now turn our attention to the dynamical structures of the resonant truncation ODE system in (4.4). In a similar way to [4, 9, 12], we may rewrite (4.4) to the equation of motion in the Hamiltonian symplectic coordinates. If we set $$r_{\xi}(t)=\sqrt{I_{\xi}(t)}e^{i\theta_{\xi}(t)},\quad I_{\xi}>0,~{}\theta_{% \xi}\in\mathbb{R},$$ we can write (4.4) as a system. Actually, inserting this back into (4.4), we see that for $\xi=\alpha_{m,j}\in{\cal R}$ $$\displaystyle\frac{i}{2}\frac{\dot{I_{\xi}}}{\sqrt{I_{\xi}}}-\sqrt{I_{\xi}}% \dot{\theta_{\xi}}=\frac{1}{L^{4}}\sum^{*}_{\text{res$(\xi)$}}\sqrt{I_{\xi_{1}% }I_{\xi_{2}}I_{\xi_{3}}I_{\xi_{4}}I_{\xi_{5}}}e^{i(\theta_{\xi_{1}}-\theta_{% \xi_{2}}+\theta_{\xi_{3}}-\theta_{\xi_{4}}+\theta_{\xi_{5}}-\theta_{\xi})}.$$ Taking the real part, we get that for $\xi=\alpha_{m,j}\in{\cal R}$ $$\displaystyle-\dot{\theta_{\xi}}=\frac{1}{L^{4}}\sum^{*}_{\text{res$(\xi)$}}% \sqrt{\frac{I_{\xi_{1}}I_{\xi_{2}}I_{\xi_{3}}I_{\xi_{4}}I_{\xi_{5}}}{I_{\xi}}}% \cos\left(\theta_{\xi_{1}}-\theta_{\xi_{2}}+\theta_{\xi_{3}}-\theta_{\xi_{4}}+% \theta_{\xi_{5}}-\theta_{\xi}\right).$$ (5.2) We take the imaginary part to get $$\displaystyle\frac{1}{2}\dot{I_{\xi}}=\frac{1}{L^{4}}\sum^{*}_{\text{res$(\xi)% $}}\sqrt{I_{\xi_{1}}I_{\xi_{2}}I_{\xi_{3}}I_{\xi_{4}}I_{\xi_{5}}I_{\xi}}\sin% \left(\theta_{\xi_{1}}-\theta_{\xi_{2}}+\theta_{\xi_{3}}-\theta_{\xi_{4}}+% \theta_{\xi_{5}}-\theta_{\xi}\right)$$ (5.3) for $\xi=\alpha_{m,j}\in{\cal R}$. Remark 5.2. The ODE system (5.2)-(5.3) enjoys the symmetry $(\theta_{\alpha_{m,j}},I_{\alpha_{m,j}})\to(\theta_{\alpha_{m,L-1-j}},I_{% \alpha_{m,L-1-j}})$. If the data satisfy $$I_{\alpha_{m,j}}(0)=I_{\alpha_{m,L-1-j}}(0),\quad\theta_{\alpha_{m,j}}(0)=% \theta_{\alpha_{m,L-1-j}}(0),$$ then the solutions to (5.2)-(5.3) ensure that $$I_{\alpha_{m,j}}(t)=I_{\alpha_{m,L-1-j}}(t),\quad\theta_{\alpha_{m,j}}(t)=% \theta_{\alpha_{m,L-1-j}}(t).$$ The non-degeneracy of solutions $I_{\alpha_{m,j}}$ to (5.2)-(5.3) is needed for carrying out calculations in (5.2)-(5.3). We will provide specific solutions satisfying such a condition in the following lemma. Lemma 5.2. Given a small constant $\nu>0$, let $\theta_{\alpha_{m,j}}(0),~{}I_{\alpha_{m,j}}(0)~{}(1\leq m\leq 4,~{}0\leq j<L)$ be an initial datum satisfying $I_{\xi}(0)\sim\nu$. Then the initial value problem (5.2)-(5.3) admits a unique classical solution $(\theta_{\alpha_{m,j}}(t),~{}I_{\alpha_{m,j}}(t))$ for $|t|\leq c\nu^{-2}L^{3}$, where $c>0$ is a small constant $c>0$, such that $$\displaystyle\max_{\xi\in\cal R}\left(\nu|\theta_{\xi}(t)-\theta_{\xi}(0)|+|I_% {\xi}(t)-I_{\xi}(0)|\right)\ll\nu.$$ (5.4) Proof. Let us note that because of the time reflection invariance, it suffices to consider non-negative time. A bootstrap (continuity) argument allows us to consider the set $$T=\sup\left\{t\geq 0\mid\max_{\xi\in{\cal R}}\left(\nu|\theta_{\xi}(t)-\theta_% {\xi}(0)|+|I_{\xi}(t)-I_{\xi}(0)|\right)\ll\nu\right\}.$$ Then it suffices to show $T\geq c\nu^{-2}L^{3}$. Clearly, for $0\leq t\leq T$, the equations (5.2)-(5.3) may be replaced by $$\displaystyle-\dot{\theta}_{\xi}=O\left(\frac{\nu^{2}}{L^{3}}\right),\quad% \frac{1}{2}\dot{I_{\xi}}=O\left(\frac{\nu^{3}}{L^{3}}\right).$$ Suppose $T\ll\nu^{-2}L^{3}$. Then the continuity of the flow implies $$\nu\lesssim\max_{\xi\in{\cal R}}\left(|I_{\xi}(T)-I_{\xi}(0)|+\nu|\theta_{\xi}% (T)-\theta(0)|\right)\lesssim\frac{\nu^{3}}{L^{3}}T\ll\nu,$$ which gives a contradiction. Then $T\geq c\nu^{-2}L^{3}$ for some constant $c>0$. ∎ The solutions to (5.3) have the following conserved quantities. Lemma 5.3. Let $I_{\xi}>0~{}(\xi\in{\cal R})$ be the solutions of (5.2)-(5.3). Then $$\displaystyle\frac{d}{dt}\left(I_{\alpha_{3,j}}(t)+I_{\alpha_{4,j}}(t)\right)=% 0,\quad\frac{d}{dt}\left(I_{\alpha_{1,j}}(t)+I_{\alpha_{2,j}}(t)\right)=0,$$ (5.5) $$\displaystyle\frac{d}{dt}\left(I_{\alpha_{1,j}}(t)-2I_{\alpha_{3,j}}(t)\right)% =0,\quad\frac{d}{dt}\left(I_{\alpha_{2,j}}(t)-2I_{\alpha_{4,j}}(t)\right)=0,$$ (5.6) Proof. First prove (5.5). We recall the equations to $I_{\alpha_{3,j}}(t)$ and $I_{\alpha_{4,j}}(t)$ as follows: $$\displaystyle\frac{2}{L^{4}}\sum^{*}_{\text{res}(\alpha_{3,j})}\sqrt{I_{\xi_{1% }}I_{\xi_{2}}I_{\xi_{3}}I_{\xi_{4}}I_{\xi_{5}}I_{\alpha_{3,j}}}\sin\left(% \theta_{\xi_{1}}-\theta_{\xi_{2}}+\theta_{\xi_{3}}-\theta_{\xi_{4}}+\theta_{% \xi_{5}}-\theta_{\alpha_{3,j}}\right)$$ $$\displaystyle=$$ $$\displaystyle\frac{6}{L^{4}}\left(2\sum_{\scriptstyle j=\frac{j_{2}+j_{4}}{2}% \atop{\scriptstyle j\neq j_{2}}}+\sum_{j=j_{2}=j_{4}}\right)\sqrt{I_{\alpha_{2% ,j_{2}}}I_{\alpha_{2,j_{4}}}I_{\alpha_{4,j}}I_{\alpha_{1,j_{2}}}I_{\alpha_{1,j% _{4}}}I_{\alpha_{3,j}}}$$ $$\displaystyle\sin\left(\theta_{\alpha_{2,j_{2}}}+\theta_{\alpha_{2,j_{4}}}+% \theta_{\alpha_{4,j}}-\theta_{\alpha_{1,j_{2}}}-\theta_{\alpha_{1,j_{4}}}-% \theta_{\alpha_{3,j}}\right),$$ and $$\displaystyle\frac{2}{L^{4}}\sum^{*}_{\text{res}(\alpha_{4,j})}\sqrt{I_{\xi_{1% }}I_{\xi_{2}}I_{\xi_{3}}I_{\xi_{4}}I_{\xi_{5}}I_{\alpha_{4,j}}}\sin\left(% \theta_{\xi_{1}}-\theta_{\xi_{2}}+\theta_{\xi_{3}}-\theta_{\xi_{4}}+\theta_{% \xi_{5}}-\theta_{\alpha_{4,j}}\right)$$ $$\displaystyle=$$ $$\displaystyle\frac{6}{L^{4}}\left(2\sum_{\scriptstyle j=\frac{j_{2}+j_{4}}{2}% \atop{\scriptstyle j\neq j_{2}}}+\sum_{j=j_{2}=j_{4}}\right)\sqrt{I_{\alpha_{1% ,j_{2}}}I_{\alpha_{1,j_{4}}}I_{\alpha_{3,j}}I_{\alpha_{2,j_{2}}}I_{\alpha_{2,j% _{4}}}I_{\alpha_{4,j}}}$$ $$\displaystyle\sin\left(\theta_{\alpha_{1,j_{2}}}+\theta_{\alpha_{1,j_{4}}}+% \theta_{\alpha_{3,j}}-\theta_{\alpha_{2,j_{2}}}-\theta_{\alpha_{2,j_{4}}}-% \theta_{\alpha_{4,j}}\right).$$ Substituting for $I_{\alpha_{3,j}}(t),~{}I_{\alpha_{4,j}}(t)$ in the right-hand side of (5.3) yields $$\displaystyle\frac{1}{2}\frac{d}{dt}\left(I_{\alpha_{3,j}}(t)+I_{\alpha_{4,j}}% (t)\right)=0.$$ We next prove the second estimate in (5.5). If the pair $(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\alpha_{1,j})$ satisfies the resonant condition, then $\{\xi_{2},\,\xi_{4}\}=\{\alpha_{1,j_{2}},\alpha_{3,j_{6}}\}$ and $\{\xi_{1},\xi_{3},\xi_{5}\}=\{\alpha_{2,j_{2}},\alpha_{2,j},\alpha_{4,j_{6}}\}$ for some $j_{2}$ and $j_{6}$ such that $j_{6}=(j_{2}+j)/2$. Along the similar process as above, we have that the term right-hand side in (5.3) for $I_{\alpha_{1,j}}(t)$ is restated as follows: $$\displaystyle\frac{2}{L^{4}}\sum^{*}_{\text{res}(\alpha_{1,j})}\sqrt{I_{\xi_{1% }}I_{\xi_{2}}I_{\xi_{3}}I_{\xi_{4}}I_{\xi_{5}}I_{\alpha_{1,j}}}\sin\left(% \theta_{\xi_{1}}-\theta_{\xi_{2}}+\theta_{\xi_{3}}-\theta_{\xi_{4}}+\theta_{% \xi_{5}}-\theta_{\alpha_{1,j}}\right)$$ $$\displaystyle=$$ $$\displaystyle\frac{12}{L^{4}}\left(2\sum_{\scriptstyle j_{6}=\frac{j_{2}+j}{2}% \atop{\scriptstyle j\neq j_{2}}}+\sum_{j=j_{2}=j_{4}}\right)\sqrt{I_{\alpha_{2% ,j_{2}}}I_{\alpha_{2,j}}I_{\alpha_{4,j_{6}}}I_{\alpha_{3,j_{6}}}I_{\alpha_{1,j% _{2}}}I_{\alpha_{1,j}}}$$ $$\displaystyle\sin\left(\theta_{\alpha_{2,j_{2}}}+\theta_{\alpha_{2,j}}+\theta_% {\alpha_{4,j_{6}}}-\theta_{\alpha_{3,j_{6}}}-\theta_{\alpha_{1,j_{2}}}-\theta_% {\alpha_{1,j}}\right).$$ On the other hand, if the pair $(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\alpha_{2,j})$ satisfies the resonant condition, then $\{\xi_{2},\xi_{4}\}=\{\alpha_{2,j_{2}},\alpha_{4,j_{6}}\}$ and $\{\xi_{1},\xi_{3},\xi_{5}\}=\{\alpha_{1,j_{2}},\alpha_{1,j},\alpha_{3,j_{6}}\}$ for some $j_{2}$ and $j_{6}$ such that $j_{6}=(j_{2}+j)/2$, which deduces that the term right-hand side in (5.3) for $I_{\alpha_{2,j}}(t)$ given by (5.3) is restated as follows: $$\displaystyle\frac{2}{L^{4}}\sum^{*}_{\text{res}(\alpha_{2,j})}\sqrt{I_{\xi_{1% }}I_{\xi_{2}}I_{\xi_{3}}I_{\xi_{4}}I_{\xi_{5}}I_{\alpha_{2,j}}}\sin\left(% \theta_{\xi_{1}}-\theta_{\xi_{2}}+\theta_{\xi_{3}}-\theta_{\xi_{4}}+\theta_{% \xi_{5}}-\theta_{\alpha_{2,j}}\right)$$ $$\displaystyle=$$ $$\displaystyle\frac{12}{L^{4}}\left(2\sum_{\scriptstyle j_{6}=\frac{j_{2}+j}{2}% \atop{\scriptstyle j\neq j_{2}}}+\sum_{j=j_{2}=j_{4}}\right)\sqrt{I_{\alpha_{1% ,j_{2}}}I_{\alpha_{1,j}}I_{\alpha_{3,j_{6}}}I_{\alpha_{2,j_{2}}}I_{\alpha_{4,j% _{6}}}I_{\alpha_{2,j}}}$$ $$\displaystyle\sin\left(\theta_{\alpha_{1,j_{2}}}+\theta_{\alpha_{1,j}}+\theta_% {\alpha_{3,j_{6}}}-\theta_{\alpha_{2,j_{2}}}-\theta_{\alpha_{4,j_{6}}}-\theta_% {\alpha_{2,j}}\right).$$ This proves the second estimate in (5.5). We can also conclude that the estimates in (5.6) hold by means of similarity computation as above. ∎ Let us proceed to construction of the specific solution to (5.2) and (5.3). We define $$\Phi_{j_{1},j_{2},j_{3},j_{4},j_{5},j_{6}}^{l_{1},l_{2},l_{3},l_{4},l_{5},l_{6% }}(t)=\theta_{\alpha_{l_{1},j_{1}}}(t)-\theta_{\alpha_{1_{2},j_{2}}}(t)+\theta% _{\alpha_{l_{3},j_{3}}}(t)-\theta_{\alpha_{l_{4},j_{4}}}(t)+\theta_{\alpha_{l_% {5},j_{5}}}(t)-\theta_{\alpha_{l_{6},j_{6}}}(t).$$ We establish the following lemma. Lemma 5.4. Let $\Phi_{j_{1},j_{1},j_{2},j_{2},j_{3},j_{3}}^{1,2,1,2,3,4}(0)=\pi/2$ for all $j_{1},~{}j_{2},~{}j_{3}$ satisfying $j_{3}=(j_{1}+j_{2})/2$, and suppose that $I_{\xi}(t)\sim\nu>0$ for $|t|<T$. Then for $|t|<T$, $$\displaystyle\Phi_{j_{1},j_{1},j_{2},j_{2},j_{3},j_{3}}^{1,2,1,2,3,4}(t)=\Phi_% {j_{1},j_{1},j_{2},j_{2},j_{3},j_{3}}^{1,2,1,2,3,4}(0)$$ (5.7) for all $j_{1},~{}j_{2},~{}j_{3}$ satisfying $j_{3}=(j_{1}+j_{2})/2$. Proof. It is a straight forward matter to obtain the result. Since by $\theta_{\alpha_{l_{1},j_{1}}}(0)-\theta_{\alpha_{1_{2},j_{2}}}(0)+\theta_{% \alpha_{l_{3},j_{3}}}(0)-\theta_{\alpha_{l_{4},j_{4}}}(0)+\theta_{\alpha_{l_{5% },j_{5}}}(0)-\theta_{\alpha_{l_{6},j_{6}}}(0)=\pi/2$, it follows that from (5.2), $$\max_{m,j}|\theta_{\alpha_{m},j}(t)-\theta_{\alpha_{m},j}(0)|\lesssim\frac{\nu% ^{2}}{L^{3}}\int_{0}^{t}\max_{m,j}|\theta_{\alpha_{m},j}(t^{\prime})-\theta_{% \alpha_{m},j}(0)|\,dt^{\prime}.$$ We apply the Gronwall inequality to get $$\theta_{\alpha_{m},j}(t)=\theta_{\alpha_{m},j}(0),$$ which implies $\Phi_{j_{1},j_{1},j_{2},j_{2},j_{3},j_{3}}^{1,2,1,2,3,4}(t)=\Phi_{j_{1},j_{1},% j_{2},j_{2},j_{3},j_{3}}^{1,2,1,2,3,4}(0).$ ∎ 5.2 Averaging property It is natural to expect that the average of the $L$ sums by $$\frac{1}{L}\sum_{j=0}^{L-1}I_{\alpha_{m,j}}(t)=\frac{I_{\alpha_{m,0}}(t)+I_{% \alpha_{m,1}}(t)+\ldots+I_{\alpha_{m,L-1}}(t)}{L},\quad 1\leq m\leq 4$$ approximates the source of a mass located in frequency space ${\cal R}_{m}$. By Lemma 5.4, we may suppose $\Phi_{j_{1},j_{1},j_{2},j_{2},j_{3},j_{3}}^{1,2,1,2,3,4}(t)=\pi/2$ in (5.2)-(5.3). In a certain sense that should be taken the average of both the left- and right-hand sides of (5.3) with respect to $0\leq j<L$, we reformulate the ODE system (5.3) as the following ODE system: $$\displaystyle\begin{cases}\displaystyle\dot{I}_{{\cal R}_{1}}=-\frac{12}{L^{3}% }\sqrt{I_{{\cal R}_{4}}I_{{\cal R}_{2}}^{2}I_{{\cal R}_{3}}I_{{\cal R}_{1}}^{2% }},&\\ \displaystyle\dot{I}_{{\cal R}_{2}}=\frac{12}{L^{3}}\sqrt{I_{{\cal R}_{4}}I_{{% \cal R}_{2}}^{2}I_{{\cal R}_{3}}I_{{\cal R}_{1}}^{2}},&\\ \displaystyle\dot{I}_{{\cal R}_{3}}=-\frac{6}{L^{3}}\sqrt{I_{{\cal R}_{4}}I_{{% \cal R}_{2}}^{2}I_{{\cal R}_{3}}I_{{\cal R}_{1}}^{2}},&\\ \displaystyle\dot{I}_{{\cal R}_{4}}=\frac{6}{L^{3}}\sqrt{I_{{\cal R}_{4}}I_{{% \cal R}_{2}}^{2}I_{{\cal R}_{3}}I_{{\cal R}_{1}}^{2}}.\end{cases}$$ (5.8) A similar argument in Lemmas 5.1 and 5.3 shows that $$\displaystyle\frac{d}{dt}\sum_{m=1}^{4}I_{{\cal R}_{m}}(t)=\frac{d}{dt}\left(I% _{{\cal R}_{3}}(t)+I_{{\cal R}_{4}}(t)\right)=\frac{d}{dt}\left(I_{{\cal R}_{1% }}(t)+I_{{\cal R}_{2}}(t)\right)=0.$$ (5.9) It is easily to see that from (5.8) $$\displaystyle\frac{d}{dt}\left(I_{{\cal R}_{1}}(t)-2I_{{\cal R}_{3}}(t)\right)% =\frac{d}{dt}\left(I_{{\cal R}_{2}}(t)-2I_{{\cal R}_{4}}(t)\right)=0,$$ (5.10) which are also the conserved quantities of the system in (5.8). Let us now use an expression of the form $$\displaystyle I={}^{t}(I_{{\cal R}_{2}},\,I_{{\cal R}_{4}},\,I_{{\cal R}_{1}},% \,I_{{\cal R}_{3}}),$$ and define new variables $$\displaystyle J={}^{t}(J_{1},\,J_{2},\,J_{3},\,J_{4}),$$ where $$J_{1}=\frac{1}{2}I_{{\cal R}_{2}},\quad J_{2}=-\frac{1}{2}I_{{\cal R}_{2}}+I_{% {\cal R}_{4}},\quad J_{3}=I_{{\cal R}_{2}}+I_{{\cal R}_{1}},\quad J_{4}=\frac{% 1}{2}I_{{\cal R}_{2}}+I_{{\cal R}_{3}}.$$ Then the Hamiltonian flow in the action-angle coordinates (5.8) satisfies $$\displaystyle J=AI$$ where $$\displaystyle A=\left(\begin{array}[]{cccc}2&1&-2&-1\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}\right).$$ We will consider a very simple choice of initial data. Given $\nu>0$, by adapting the result in (5.9) and (5.10), we specialize the variables of $I_{{\cal R}_{m}}$’s to the normalization form $$I_{{\cal R}_{1}}(t)+I_{{\cal R}_{2}}(t)=\nu,\quad I_{{\cal R}_{3}}(t)+I_{{\cal R% }_{4}}(t)=\frac{\nu}{2},$$ $$I_{{\cal R}_{1}}(t)-2I_{{\cal R}_{3}}(t)=I_{{\cal R}_{2}}(t)-2I_{{\cal R}_{4}}% (t)=0.$$ Employing the auxiliary function $K(t)$ such that $\nu^{2}|K(t)|\in[0,1/2)$, we put $$\displaystyle I_{{\cal R}_{1}}(t)=2I_{{\cal R}_{3}}(t)=\nu\left(\frac{1}{2}-% \nu^{2}K(t)\right),\quad I_{{\cal R}_{2}}(t)=2I_{{\cal R}_{4}}(t)=\nu\left(% \frac{1}{2}+\nu^{2}K(t)\right),$$ (5.11) which imply that $$J_{2}=0,\quad J_{3}=\nu,\quad J_{4}=\frac{\nu}{2}.$$ Solutions to (5.8) with the constrain in (5.11) are proposed to solve the following equations on $K(t)$: $$\displaystyle\displaystyle\dot{K}=\frac{6}{L^{3}}\left(\frac{1}{2}+\nu^{2}K% \right)^{3/2}\left(\frac{1}{2}-\nu^{2}K\right)^{3/2}.$$ (5.12) We remark that a solution of the ordinary differential equation $$\dot{f}(t)=a\left(b-f(t)\right)^{3/2}\left(b+f(t)\right)^{3/2}$$ is provided by $$f(t)=b\sin\arctan\left(ab^{2}t+c\right)$$ where $c\in\mathbb{R}$ is a constant. Therefore, we will choose the function $K$ such that $$\displaystyle K(t)=\frac{1}{2\nu^{2}}\sin\arctan\left(\frac{3\nu^{2}t}{2L^{3}}% \right).$$ (5.13) Dealing with the special choice of the function, we use the perturbation theory for finding an approximate solution to (5.3). We start from the following calculation: $$\displaystyle\sum_{j=0}^{L-1}\mbox{card}\left\{(j_{1},j_{2})\in\mathbb{Z}^{2}% \mid 0\leq j_{1},j_{2}<L,~{}j=\frac{j_{1}+j_{2}}{2}\right\}=\sum_{j=0}^{L-1}% \min\{j+1,L-j\}=\frac{L^{2}}{4}+O(L),$$ (5.14) and $$\displaystyle\sum_{j_{2}=0}^{L-1}\mbox{card}\left\{(j,j_{1})\in\mathbb{Z}^{2}% \mid 0\leq j_{1},j<L,~{}j=\frac{j_{1}+j_{2}}{2}\right\}$$ $$\displaystyle=$$ $$\displaystyle\sum_{\scriptstyle j_{2}=0\atop{\scriptstyle j_{2}:\text{even}}}^% {L-1}\mbox{card}\left\{(j,j_{1})\in\mathbb{Z}^{2}\mid 0\leq j_{1},j<L,~{}j=% \frac{j_{1}+j_{2}}{2}\right\}$$ $$\displaystyle+\sum_{\scriptstyle j_{2}=0\atop{\scriptstyle j_{2}:\text{odd}}}^% {L-1}\mbox{card}\left\{(j,j_{1})\in\mathbb{Z}^{2}\mid 0\leq j_{1},j<L,~{}j=% \frac{j_{1}+j_{2}}{2}\right\}$$ $$\displaystyle=$$ $$\displaystyle\sum_{\scriptstyle j_{2}=0\atop{\scriptstyle j_{2}:\text{even}}}^% {L-1}\left(\frac{L}{2}+O(1)\right)+\sum_{\scriptstyle j_{2}=0\atop{% \scriptstyle j_{2}:\text{odd}}}^{L-1}\left(\frac{L}{2}+O(1)\right)$$ $$\displaystyle=$$ $$\displaystyle\frac{L^{2}}{2}+O(L).$$ Lemma 5.5. Let $(I_{{\cal R}_{m}})_{1\leq m\leq 4}$ be global solutions of (5.8) (those solutions were constructed in (5.11)-(5.13)). If at $t=0$, the initial datum $(\theta_{\alpha_{m,j}}(0),I_{\alpha_{m,j}}(0))$ satisfies $$I_{\alpha_{m,j}}(0)=I_{{\cal R}_{m}}(0),$$ $$\theta_{\alpha_{m,j}}(0)=\theta_{\alpha_{m}}$$ for $1\leq m\leq 4,~{}0\leq j<L$, and $$2\theta_{\alpha_{2}}+\theta_{\alpha_{4}}-2\theta_{\alpha_{1}}-\theta_{\alpha_{% 3}}=\frac{\pi}{2},$$ then there exist a constant $c>0$ and solutions $(\theta_{m,j},I_{\alpha_{m,j}})$ to (5.2)-(5.3), so that for $|t|\ll\nu^{-2}L^{3}$, $$\displaystyle\left|\frac{1}{L}\sum_{0\leq j<L}I_{\alpha_{m,j}}(t)-I_{{\cal R}_% {m}}(t)\right|\lesssim\frac{\nu}{L}\left(e^{c\frac{\nu^{2}}{L^{3}}t}-1\right).$$ (5.16) Proof. It suffices to consider non-negative time. By Lemma 5.4, we suppose $\Phi_{j_{1},j_{1},j_{2},j_{2},j_{3},j_{3}}^{1,2,1,2,3,4}(t)=\pi/2$. For fixed $0\leq j<L$, use (5.2) to see that $$\displaystyle\frac{12}{L^{4}}\left(2\sum_{\scriptstyle j_{6}=\frac{j_{2}+j}{2}% \atop{\scriptstyle j\neq j_{2}}}+\sum_{j=j_{2}=j_{4}}\right)\sqrt{I_{\alpha_{2% ,j_{2}}}I_{\alpha_{2,j}}I_{\alpha_{4,j_{6}}}I_{\alpha_{3,j_{6}}}I_{\alpha_{1,j% _{2}}}I_{\alpha_{1,j}}}-\frac{12}{L^{3}}\sqrt{I_{{\cal R}_{4}}I_{{\cal R}_{2}}% ^{2}I_{{\cal R}_{3}}I_{{\cal R}_{1}}^{2}}$$ (5.17) $$\displaystyle=$$ $$\displaystyle\frac{24}{L^{4}}\sum_{\scriptstyle j_{6}=\frac{j_{2}+j}{2}\atop{% \scriptstyle j\neq j_{2}}}\left(\sqrt{I_{\alpha_{2,j_{2}}}I_{\alpha_{2,j}}I_{% \alpha_{4,j_{6}}}I_{\alpha_{3,j_{6}}}I_{\alpha_{1,j_{2}}}I_{\alpha_{1,j}}}-% \sqrt{I_{{\cal R}_{4}}I_{{\cal R}_{2}}^{2}I_{{\cal R}_{3}}I_{{\cal R}_{1}}^{2}% }\right)+O\left(\frac{\nu^{3}}{L^{4}}\right).$$ By virtue of the proof of Lemma 5.3, (5.17) implies $$\displaystyle|I_{\alpha_{1,j}}(t)-I_{{\cal R}_{1}}(t)|$$ $$\displaystyle\lesssim$$ $$\displaystyle\frac{\nu^{2}}{L^{3}}\int_{0}^{t}\left(\max_{\scriptstyle 1\leq m% \leq 4\atop{\scriptstyle 0\leq j<L}}|I_{\alpha_{m,j}}(t^{\prime})-I_{{\cal R}_% {m}}(t^{\prime})|+\frac{\nu}{L}\right)\,dt^{\prime}.$$ (5.18) provided $t\ll L^{3}/\nu^{2}$. Thanks to the conservation laws obtained in Lemma 5.3, (5.9) and (5.10) along with the initial condition at $t=0$, the estimate in (5.18) implies that $$\displaystyle\max_{1\leq m\leq 4}|I_{\alpha_{m,j}}(t)-I_{{\cal R}_{m}}(t)|% \lesssim\frac{\nu^{2}}{L^{3}}\int_{0}^{t}\left(\max_{\scriptstyle 1\leq m\leq 4% \atop{\scriptstyle 0\leq j<L}}|I_{\alpha_{m,j}}(t^{\prime})-I_{{\cal R}_{m}}(t% ^{\prime})|+\frac{\nu}{L}\right)\,dt^{\prime}.$$ We apply the Gronwall inequality to get $$\displaystyle\int_{0}^{t}\left(\max_{1\leq m\leq 4}|I_{\alpha_{m,j}}(t^{\prime% })-I_{{\cal R}_{m}}(t^{\prime})|+\frac{\nu}{L}\right)\,dt^{\prime}\lesssim% \frac{L^{2}}{\nu}\left(e^{c\frac{\nu^{2}}{L^{3}}t}-1\right).$$ (5.19) We control now (5.16). Use (5.14) to see that $$\displaystyle\frac{1}{L}\sum_{0\leq j<L}\frac{6}{L^{4}}\left(2\sum_{% \scriptstyle j=\frac{j_{2}+j_{4}}{2}\atop{\scriptstyle j\neq j_{2}}}+\sum_{j=j% _{2}=j_{4}}\right)\sqrt{I_{\alpha_{2,j_{2}}}I_{\alpha_{2,j_{4}}}I_{\alpha_{4,j% }}I_{\alpha_{1,j_{2}}}I_{\alpha_{1,j_{4}}}I_{\alpha_{3,j}}}-\frac{6}{L^{3}}% \sqrt{I_{{\cal R}_{4}}I_{{\cal R}_{2}}^{2}I_{{\cal R}_{3}}I_{{\cal R}_{1}}^{2}}$$ (5.20) $$\displaystyle=$$ $$\displaystyle\frac{12}{L^{5}}\sum_{0\leq j<L}\sum_{\scriptstyle j=\frac{j_{2}+% j_{4}}{2}\atop{\scriptstyle j\neq j_{2}}}\left(\sqrt{I_{\alpha_{2,j_{2}}}I_{% \alpha_{2,j_{4}}}I_{\alpha_{4,j}}I_{\alpha_{1,j_{2}}}I_{\alpha_{1,j_{4}}}I_{% \alpha_{3,j}}}-\sqrt{I_{{\cal R}_{4}}I_{{\cal R}_{2}}^{2}I_{{\cal R}_{3}}I_{{% \cal R}_{1}}^{2}}\right)+O\left(\frac{\nu^{3}}{L^{4}}\right).$$ Using (5.19) and (5.20), we arrive at $$\displaystyle\max_{\scriptstyle 1\leq m\leq 4\atop{\scriptstyle 0\leq j<L}}% \left|\frac{1}{L}\sum_{0\leq j<L}I_{\alpha_{m,j}}(t)-I_{{\cal R}_{m}}(t)\right|$$ $$\displaystyle\lesssim$$ $$\displaystyle\frac{\nu^{2}}{L^{3}}\int_{0}^{t}\left(\max_{\scriptstyle 1\leq m% \leq 4\atop{\scriptstyle 0\leq j<L}}|I_{\alpha_{m,j}}(t^{\prime})-I_{{\cal R}_% {m}}(t^{\prime})|+\frac{\nu}{L}\right)\,dt^{\prime}$$ $$\displaystyle\lesssim$$ $$\displaystyle\frac{\nu}{L}\left(e^{c\frac{\nu^{2}}{L^{3}}t}-1\right),$$ which completes the proof. ∎ By (5.11), (5.13) and Lemma 5.5, we will automatically have the following. Proposition 5.1. Given a small constant $\nu>0$, let the initial datum the initial datum $(\theta_{\alpha_{m,j}}(0),I_{\alpha_{m,j}}(0))$ satisfy that $$I_{\alpha_{m,j}}(0)=I_{{\cal R}_{m}}(0),$$ $$\theta_{\alpha_{m,j}}(0)=\theta_{\alpha_{m}}$$ for $1\leq m\leq 4,~{}0\leq j<L$, and $$2\theta_{\alpha_{2}}+\theta_{\alpha_{4}}-2\theta_{\alpha_{1}}-\theta_{\alpha_{% 3}}=\frac{\pi}{2},$$ where $(I_{{\cal R}_{m}}(0))_{m=1}^{4}$ are the same as in (5.11), (5.13). Then for $|t|\ll L^{3}\nu^{-2}$, $$\frac{1}{L}\sum_{0\leq j<L}I_{\alpha_{m,j}}(t)=I_{{\cal R}_{m}}(t)+O\left(% \frac{\nu^{3}}{L^{4}}t\right).$$ 6 Approximate estimates In this section, we study the approximation of the infinite dimensional NLS flow in (3.1). In order to pass from the finite-dimensional flow to the infinite one, we shall evaluate residual terms $a_{\xi}(t)$ of $\xi\not\in{\cal R}$ in the infinite-dimensional ODE (3.1) and ultimately approximate the full system (4.4) corresponding to (3.1). Suppose that $(a_{\xi}(t))_{\xi\in 2\pi\mathbb{Z}/L}$ and $(r_{\xi}(t))_{\xi\in{\cal R}}$ are solutions to (3.1) and (4.4), respectively. As a matter of convenience, we recurse the sequence $(r_{\xi}(t))_{\xi\in 2\pi\mathbb{Z}/L}$ by placing $r_{\xi}(t)=0$ for $\xi\not\in{\cal R}$. Also we may extend the formula $\text{res}(\xi)$ for all $\xi\in 2\pi\mathbb{Z}/L$ by replacing $\text{res}(\xi)=\emptyset$ for $\xi\not\in{\cal R}$ to deal with the case when $\xi\not\in{\cal R}$. The initial datum $(a_{\xi}(0),r_{\xi}(0))_{\xi\in 2\pi\mathbb{Z}/L}$ are given as follows: • if $\xi\in{\cal R}_{m}$ for some $1\leq m\leq 4$, then $a_{\xi}(0)=r_{\xi}(0)=I_{{\cal R}_{m}}(0)e^{i\theta_{{\cal R}_{m}}}$, • if $\xi\not\in\cup_{m=1}^{4}{\cal R}_{m}$, then $a_{\xi}(0)=r_{\xi}(0)=0$, where $I_{{\cal R}_{m}}(t)$ are provided in (5.11) and (5.13), and $\theta_{{\cal R}_{m}}$ satisfy $$2\theta_{{\cal R}_{2}}+\theta_{{\cal R}_{4}}-2\theta_{{\cal R}_{1}}-\theta_{{% \cal R}_{3}}=\frac{\pi}{2}.$$ It follows that from the mass conservation laws (1.2) and (5.1), $$\displaystyle\frac{1}{L}\sum_{\xi\in 2\pi\mathbb{Z}/L}\left(|a_{\xi}(t)|^{2}+|% r_{\xi}(t)|^{2}\right)\leq c\nu,$$ (6.1) where $c>0$ is independent of $t\in\mathbb{R}$. We define a subset of $2\pi\mathbb{Z}/L$. Definition 6.1. Introduce some notions. For $\xi\in 2\pi\mathbb{Z}/L$, rewrite the form $$\displaystyle\xi=2\pi\left(k\eta+\tau+\frac{j}{L}\right),~{}(\eta,\tau,j)\in% \mathbb{Z}^{3},~{}\tau\in[0,k),~{}j\in[0,L)$$ (6.2) as $$\displaystyle\xi=2\pi\left(k\widetilde{\eta}+\widetilde{\tau}+\frac{j}{L}% \right),$$ (6.3) where $$\displaystyle(\widetilde{\eta},\widetilde{\tau})=\left\{\begin{array}[]{ll}(% \eta,\tau),&\mbox{if $\tau\in[0,k/2]$},\\ (\eta+1,\tau-k),&\mbox{if $\tau\in(k/2,k)$}.\end{array}\right.$$ With the notions above, define $$\displaystyle A_{1}$$ $$\displaystyle=$$ $$\displaystyle\cup_{\eta\in\{0,1,3,4\}}\left\{(\xi_{1},\xi_{2},\xi_{3},\xi_{4},% \xi_{5},\xi_{6})\in(2\pi\mathbb{Z}/L)^{6}\right.\mid\xi_{m}=2\pi(k\widetilde{% \eta}_{m}+\widetilde{\tau}_{m}+j_{m}/L,$$ $$\displaystyle\left.(\widetilde{\tau}_{m},j_{m})\in\mathbb{Z}^{2},~{}\widetilde% {\eta}_{m}=k\eta,~{}-k/2<\widetilde{\tau}_{m}\leq k/2,~{}j_{m}\in[0,L),~{}1% \leq m\leq 6\right\}.$$ By removing the resonant part of the Fourier transform from the nonlinear interaction of $a_{\xi}(t)$, we write the error of Fourier mode $\xi$ as $$e_{\xi}(t)=a_{\xi}(t)-r_{\xi}(t)$$ for $\xi\in 2\pi\mathbb{Z}/L$. One uses (3.1) and (4.4) to obtain something like $$\displaystyle i\dot{e}_{\xi}(t)=\sum_{m=1}^{3}R_{\xi}^{m}(t),$$ (6.4) where (by omitting the time variable $t$ from the equations) $$\displaystyle R_{\xi}^{1}=\frac{1}{L^{4}}\sum_{(\xi_{1},\xi_{2},\xi_{3},\xi_{4% },\xi_{5},\xi)\in A_{1}}^{*}a_{\xi_{1}}\overline{a_{\xi_{2}}}a_{\xi_{3}}% \overline{a_{\xi_{4}}}a_{\xi_{5}}e^{-it\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},% \xi_{5},\xi)},$$ $$\displaystyle R_{\xi}^{2}=\frac{1}{L^{4}}\sum_{\text{res}(\xi)}^{*}\left(a_{% \xi_{1}}\overline{a_{\xi_{2}}}a_{\xi_{3}}\overline{a_{\xi_{4}}}a_{\xi_{5}}-r_{% \xi_{1}}\overline{r_{\xi_{2}}}r_{\xi_{3}}\overline{r_{\xi_{4}}}r_{\xi_{5}}% \right),$$ $$\displaystyle R_{\xi}^{3}=\frac{1}{L^{4}}\sum^{*}_{\scriptstyle\text{res}(\xi)% ^{c}\atop{\scriptstyle(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi)\in A_{1}^{% c}}}a_{\xi_{1}}\overline{a_{\xi_{2}}}a_{\xi_{3}}\overline{a_{\xi_{4}}}a_{\xi_{% 5}}e^{-it\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi)}.$$ Remark 6.1. In the spirit of the standard local well-posedness theory observed in [2, 13] and an ODE technique along with the conservation laws in (1.2), (1.3) and (5.1), we easily see that there exists a unique smooth global in time solution to the initial value problem for the corresponding equations to (3.1) and (4.4), respectively. By a slight abuse of notation above, we define the modified $H^{s}$-energy for the difference between the solutions to (3.1) and (4.4) as follows: $$\displaystyle\widetilde{E}(t)=\|e(t)\|_{s}^{2}+\frac{2}{L^{5}}\Re\sum_{\xi_{6}% }\sum_{\scriptstyle(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})\in A_{1}% \atop{\scriptstyle|\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})|>% \delta}}^{*}m(\xi_{6})^{2}\frac{a_{\xi_{1}}\overline{a_{\xi_{2}}}a_{\xi_{3}}% \overline{a_{\xi_{4}}}a_{\xi_{5}}\overline{e_{\xi_{6}}}}{\phi(\xi_{1},\xi_{2},% \xi_{3},\xi_{4},\xi_{5},\xi_{6})}e^{-it\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},% \xi_{5},\xi_{6})},$$ (6.5) where $\delta\gg 1$ is a fixed large constant, in fact, as like $\delta=10^{10}$. First, we prepare the following lemma to estimate the convolution sum. Lemma 6.1. Let $s>1$, and let $(c_{j,l})_{l\in 2\pi\mathbb{Z}}~{}(1\leq j\leq 6)$ be sequences of non-negative numbers. Then it follows that $$\displaystyle\sum_{\xi\in 2\pi\mathbb{Z}/L}c_{j,\xi}\lesssim L^{1/2}\left(\sum% _{\xi\in 2\pi\mathbb{Z}/L}m(\xi)^{2}c_{j,\xi}^{2}\right)^{1/2},$$ (6.6) and $$\displaystyle\sum_{\xi_{6}\in 2\pi\mathbb{Z}/L}\sum^{*}_{m(\xi_{6})\lesssim% \max\{m(\xi_{l})\mid 1\leq l\leq 5\}}m(\xi_{6})^{2}\prod_{j=1}^{6}c_{j,\xi_{j}% }\lesssim L^{2}\prod_{j=1}^{6}\left(\sum_{\xi\in 2\pi\mathbb{Z}/L}m(\xi)^{2}c_% {j,\xi}^{2}\right)^{1/2},$$ (6.7) where constants term to the right-hand side are independent of $k$. Proof. The proof of (6.7) can be obtained by considering (6.6). The inequality (6.6) follows from the following fact $$\sum_{\xi\in 2\pi\mathbb{Z}}\frac{1}{m(\xi)^{2}}\lesssim 1.$$ This is because a straightforward calculation as $$\displaystyle\sum_{\eta=-100}^{100}\sum_{|\tau|\leq k/2}\frac{1}{\langle\tau% \rangle^{2(s-1/2)}}+\sum_{|\eta|\geq 100}\sum_{\tau=0}^{k-1}\frac{1}{\langle k% \eta+\tau\rangle^{2s}}\lesssim 1+\int_{\mathbb{R}}\frac{dt}{\langle t\rangle^{% 2s}}\lesssim 1$$ for every $s>1$. ∎ Lemma 6.2. Let ${\cal A}_{r}$ be $${\cal A}_{r}=\left\{\xi=2\pi\left(k\widetilde{\eta}+\widetilde{\tau}+\frac{j}{% L}\right)\mid\widetilde{\eta}\in\{0,1,3,4\},~{}(\widetilde{\tau},j)\in\mathbb{% Z}^{2},~{}|\widetilde{\tau}|\lesssim 1,~{}j\in[0,L)\right\}.$$ Assume that $e(t)=(e_{\xi}(t))_{\xi\in 2\pi\mathbb{Z}/L}$ satisfies $e_{\xi}(0)=0$ for $\xi\in 2\pi\mathbb{Z}/L$ and $$\sup_{0\leq t\leq T}\|e(t)\|_{s}\lesssim\nu^{1/2}.$$ Then for $|t|\leq T$, $$\displaystyle\left(\sum_{\xi\in{\cal A}_{r}}m(\xi)^{2}|e_{\xi}(t)|^{2}\right)^% {1/2}\lesssim\nu^{5/2}t.$$ (6.8) and $$\displaystyle\int_{0}^{t}\left(\sum_{\xi\in{\cal A}_{r}}m(\xi)^{2}|e_{\xi}(t^{% \prime})|^{2}\right)^{1/2}\,dt^{\prime}\lesssim\nu^{5/2}t^{2}.$$ (6.9) Proof. Denote $$F(t)=\left(\int_{0}^{t}\sum_{\xi\in{\cal A}_{r}}m(\xi)^{2}|e_{\xi}(t^{\prime})% |^{2}\,dt^{\prime}\right)^{1/2}.$$ We note that $m(\xi)\sim 1$ for $\xi\in{\cal A}_{r}$. From the assumption, equations (3.1), (4.4) and Lemma 6.1, we see that $$\displaystyle\sum_{\xi\in{\cal A}_{r}}m(\xi)^{2}|e_{\xi}(t)|^{2}\lesssim\nu^{5% /2}\int_{0}^{t}\left(\sum_{\xi\in{\cal A}_{r}}m(\xi)^{2}|e_{\xi}(t^{\prime})|^% {2}\right)^{1/2}\,dt^{\prime}\leq\nu^{5/2}t^{1/2}F(T_{1})$$ (6.10) for $t\leq T_{1}\leq T$. Performing the integration on the time interval $[0,T_{1}]$, we have $$F(T_{1})^{2}\lesssim\nu^{5/2}T_{1}^{3/2}F(T_{1}),$$ which yields $F(T_{1})\lesssim\nu^{5/2}T_{1}^{3/2}$ so that $F(t)\lesssim\nu^{5/2}t^{3/2}$ for $0\leq t\leq T$. Inserting into (6.10) gives (6.8) and $$\int_{0}^{t}\left(\sum_{\xi\in{\cal A}_{r}}m(\xi)^{2}|e_{\xi}(t^{\prime})|^{2}% \right)^{1/2}\,dt^{\prime}\lesssim t^{1/2}\nu^{5/2}t^{3/2}=\nu^{5/2}t^{2},$$ which implies (6.9). ∎ Lemma 6.3. Let $\nu>0$ be a small constant. Then $$\left|\widetilde{E}(t)-\|e(t)\|_{s}^{2}\right|\ll\|e(t)\|_{s}^{2}.$$ Proof. It suffices to show that $$\displaystyle\left|\frac{1}{L^{5}}\sum_{\xi_{6}}\sum_{\scriptstyle(\xi_{1},\xi% _{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})\in A_{1}\atop{\scriptstyle|\phi(\xi_{1},% \xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})|>\delta}}^{*}m(\xi_{6})^{2}\frac{a_{% \xi_{1}}\overline{a_{\xi_{2}}}a_{\xi_{3}}\overline{a_{\xi_{4}}}a_{\xi_{5}}% \overline{e_{\xi_{6}}}}{\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})}% e^{-it\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})}\right|\ll\|e(t)\|% _{s}^{2}.$$ (6.11) In the case when $\xi_{6}\in{\cal A}_{r}$, we distinguish two cases: • $\xi_{m}\in{\cal R}$ for all $1\leq m\leq 5$ in the sum (6.11), • there is at least one element of $\xi_{m}\in{\cal R}~{}(1\leq m\leq 5)$ such that $\xi_{m}\not\in{\cal R}$ in the sum (6.11). Consider the first case. We use the formula in (6.3) such that for $1\leq m\leq 6$, $$\xi_{m}=2\pi\left(k\widetilde{\eta}_{m}+\widetilde{\tau}_{m}+\frac{j_{m}}{L}% \right),$$ where $\widetilde{\eta}_{1}=\widetilde{\eta}_{2}=\widetilde{\eta}_{3}=\widetilde{\eta% }_{4}=\widetilde{\eta}_{5}=\widetilde{\eta}_{6}\in\{0,1,3,4\},~{}\widetilde{% \tau}_{m}=0~{}(1\leq m\leq 5)$ and $|\widetilde{\tau}_{6}|\lesssim 1$, since by $(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})\in A_{1}$ and $\xi_{6}\in{\cal A}_{r}$. Then we obtain $$\displaystyle\left(\widetilde{\tau}_{1}+\frac{j_{1}}{L}\right)-\left(% \widetilde{\tau}_{2}+\frac{j_{2}}{L}\right)+\left(\widetilde{\tau}_{3}+\frac{j% _{3}}{L}\right)-\left(\widetilde{\tau}_{4}+\frac{j_{4}}{L}\right)+\left(% \widetilde{\tau}_{5}+\frac{j_{5}}{L}\right)-\left(\widetilde{\tau}_{6}+\frac{j% _{6}}{L}\right)=0$$ (6.12) and $$\displaystyle\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})$$ (6.13) $$\displaystyle=$$ $$\displaystyle\left(\widetilde{\tau}_{1}+\frac{j_{1}}{L}\right)^{2}-\left(% \widetilde{\tau}_{2}+\frac{j_{2}}{L}\right)^{2}+\left(\widetilde{\tau}_{3}+% \frac{j_{3}}{L}\right)^{2}-\left(\widetilde{\tau}_{4}+\frac{j_{4}}{L}\right)^{% 2}+\left(\widetilde{\tau}_{5}+\frac{j_{5}}{L}\right)^{2}-\left(\widetilde{\tau% }_{6}+\frac{j_{6}}{L}\right)^{2}.$$ In the case when $\xi_{m}\in{\cal R}$ for all $1\leq m\leq 5$ in the sum (6.11), we have $|\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})|\lesssim 1$, which is out of range in the sum. In the second case, from $a_{\xi_{m}}=e_{\xi_{m}}$ for $\xi\not\in{\cal R}$, the contribution of this case to the left-hand side of (6.11) is bounded by $$\displaystyle c\frac{\nu^{2}}{\delta}\|e(t)\|_{s}^{2}\ll\|e(t)\|_{s}^{2}.$$ (6.14) Next we may suppose $\xi_{6}\not\in{\cal A}_{r}$ so that there is at least one element of $\xi_{m}~{}(1\leq m\leq 5)$ such that $\xi_{m}\not\in{\cal R},~{}a_{\xi_{m}}=e_{\xi_{m}}$ and $|\widetilde{\tau}_{6}|\lesssim|\widetilde{\tau}_{m}|$. In this case, it is easy see that $m(\xi_{6})\lesssim m(\xi_{m})$ so that the contribution of this case to the left-hand side of (6.11) is bounded by the same bound as in (6.14), which is acceptable. ∎ A crucial step in the proof of Theorem 1.2 is to establish the following proposition. Proposition 6.1. Let $s\in(1,3/2]$ and $T>0$. Given $a(0)=(a_{\xi}(0))_{\xi\in 2\pi\mathbb{Z}/L}\in\ell^{2}$ and $r(0)=(r_{\xi}(0))_{\xi\in{\cal R}}$, let $e(t)=(e_{\xi}(t))_{\xi\in 2\pi\mathbb{Z}/L}$ be a solution to the equation (6.4) with initial data $e(0)=(e_{\xi}(0))_{\xi\in 2\pi\mathbb{Z}/L}$ for $e_{\xi}(0)=a_{\xi}(0)-r_{\xi}(0)=0$ on times $0\leq t\leq T$. Assume that $$\sup_{0\leq t\leq T}\|e(t)\|_{s}^{2}\lesssim\nu.$$ Then $$\displaystyle\|e(t)\|_{s}^{2}\lesssim\frac{\nu^{3}}{\langle k\rangle^{5/2-s}}e% ^{c\nu^{2}t}+\frac{\nu^{3}}{\langle k\rangle^{3/2-s}}\left(e^{c\nu^{2}t}-1% \right)+\nu\left(e^{c\nu^{2}t}-1-c\nu^{2}t\right).$$ (6.15) Proof of Proposition 6.1. We notice $\widetilde{E}(0)=0$. By the fundamental theorem of calculus, we see that from (6.4) $$\displaystyle\widetilde{E}(t)=-\frac{2}{L}\Im\sum_{n=1}^{3}\sum_{\xi\in 2\pi% \mathbb{Z}/L}\int_{0}^{t}m(\xi)^{2}\widetilde{R}_{\xi}^{n}(t^{\prime})% \overline{e_{\xi}(t^{\prime})}\,dt^{\prime}+\widetilde{R}_{4}(t)$$ (6.16) where $$\displaystyle\widetilde{R}_{\xi}^{1}(t)=\frac{1}{L^{4}}\sum_{\scriptstyle(\xi_% {1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi)\in A_{1}\atop{\scriptstyle|\phi(\xi_{% 1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi)|\leq\delta}}^{*}a_{\xi_{1}}(t)% \overline{a_{\xi_{2}}(t)}a_{\xi_{3}}(t)\overline{a_{\xi_{4}}(t)}a_{\xi_{5}}(t)% e^{-it\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi)},$$ $$\displaystyle\widetilde{R}_{4}(t)=\frac{2}{L^{5}}\Re\sum_{\xi_{6}}\sum_{% \scriptstyle(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})\in A_{1}\atop{% \scriptstyle|\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})|>\delta}}^{% *}\int_{0}^{t}m(\xi_{6})^{2}\frac{\partial_{t^{\prime}}\left(a_{\xi_{1}}(t^{% \prime})\overline{a_{\xi_{2}}(t^{\prime})}a_{\xi_{3}}(t^{\prime})\overline{a_{% \xi_{4}}(t^{\prime})}a_{\xi_{5}}(t^{\prime})\overline{e_{\xi_{6}}(t^{\prime})}% \right)}{\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})}e^{-it^{\prime}% \phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})}\,dt^{\prime}$$ and $\widetilde{R}_{\xi}^{n}(t)=R_{\xi}^{n}(t)$ for $n=2,3$. For the terms $\widetilde{R}_{\xi}^{n}~{}(n=1,2)$, we have the following two lemmas. Lemma 6.4. $$\displaystyle\frac{1}{L}\sum_{\xi\in 2\pi\mathbb{Z}/L}\int_{0}^{t}m(\xi)^{2}% \left|\widetilde{R}_{\xi}^{1}(t^{\prime})\overline{e_{\xi}(t^{\prime})}\right|% \,dt^{\prime}\lesssim\left(\delta^{1/4}+\frac{1}{L^{1/2}}\right)\nu^{5}t^{2}+% \nu^{2}\int_{0}^{t}\|e(t^{\prime})\|_{s}^{2}\,dt^{\prime}.$$ Proof. It suffices to show that $$\displaystyle\frac{1}{L^{5}}\sum_{\xi_{6}}\sum_{\scriptstyle(\xi_{1},\xi_{2},% \xi_{3},\xi_{4},\xi_{5},\xi_{6})\in A_{1}\atop{\scriptstyle|\phi(\xi_{1},\xi_{% 2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})|\leq\delta}}^{*}m(\xi_{6})^{2}|a_{\xi_{1}}% (t)\overline{a_{\xi_{2}}(t)}a_{\xi_{3}}(t)\overline{a_{\xi_{4}}(t)}a_{\xi_{5}}% (t)\overline{e_{\xi}(t^{\prime})}|\,dt^{\prime}$$ (6.17) $$\displaystyle\lesssim$$ $$\displaystyle\left(\delta^{1/2}+\frac{1}{L}\right)^{1/2}\nu^{5}t^{2}+\nu^{2}% \int_{0}^{t}\|e(t^{\prime})\|_{s}^{2}\,dt^{\prime}.$$ We note that if $(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})\in A_{1}$, then any of $\widetilde{\eta}_{m}~{}(1\leq m\leq 6)$ coincide with each other, and the form of $m(\xi_{m})$ is written $m(\xi_{m})=\langle\widetilde{\tau}_{m}\rangle^{s-1/2}$. By using the notation in (6.3), we have (6.12) and (6.13). The restriction $|\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})|<\delta$ implies that for fixed four elements of $\xi_{m}~{}(1\leq m\leq 6)$ under the plane $\xi_{1}+\xi_{3}+\xi_{5}=\xi_{2}+\xi_{4}+\xi_{6}$, the cardinalities of one of the other elements is at most $c\delta^{1/2}L+1$. In the case when $m(\xi_{6})\lesssim 1$ in the sum of (6.17), by the above observation, we have the bound of the left-hand side of (6.17) by $$c\left(\frac{\delta^{1/2}L+1}{L}\right)^{1/2}\nu^{5/2}\int_{0}^{t}\|e(t^{% \prime})\|_{s}\,dt^{\prime},$$ which is bounded by $$c\left(\delta^{1/2}+\frac{1}{L}\right)^{1/2}\nu^{5}t^{2},$$ since by Lemma 6.2. In the case when $m(\xi_{6})\gg 1$, from (6.12), we have $$1\ll m(\xi_{6})\lesssim\max_{1\leq m\leq 5}m(\xi_{m}),$$ which implies that at least there exists one element $\xi_{0}$ of $\xi_{m}~{}(1\leq m\leq 5)$ in the sum such that $\xi_{0}\not\in{\cal R}$ and $m(\xi_{6})\sim m(\xi_{0})$. Then the contribution of this case to the left-hand side of (6.17) is bounded by $$c\nu^{2}\int_{0}^{t}\|e(t^{\prime})\|_{s}^{2}\,dt^{\prime}.$$ Thus the proof of that result is complete. ∎ Lemma 6.5. $$\displaystyle\frac{1}{L}\sum_{\xi\in 2\pi\mathbb{Z}/L}\int_{0}^{t}m(\xi)^{2}% \left|\widetilde{R}_{\xi}^{2}(t^{\prime})\overline{e_{\xi}(t^{\prime})}\right|% \,dt^{\prime}\lesssim\nu^{2}\int_{0}^{t}\|e(t^{\prime})\|_{s}^{2}\,dt^{\prime}.$$ Proof. The proof of this lemma follow easily from the form of $R_{\xi}^{2}$ and the fact that $R_{\xi}^{2}=0$ for $\xi\not\in{\cal R}$, so that shall be omitted. ∎ To bound the last factor on the right-hand side of (6.16), we have the following. Lemma 6.6. $$\displaystyle|\widetilde{R}_{4}(t)|\lesssim\frac{\nu^{4}}{\delta}\int_{0}^{t}% \|e(t^{\prime})\|_{s}^{2}\,dt^{\prime}.$$ Proof. The proof of this lemma is a straightforward. The main objective is to use equations (3.1), (4.4) and the a priori bound. The contribution to the upper bound in term of $|\widetilde{R}_{4}(t)|$ is the sums of the following two terms: $$\displaystyle\frac{1}{\delta L^{5}}\sum_{\xi_{6}}\sum_{\scriptstyle(\xi_{1},% \xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})\in A_{1}\atop{\scriptstyle|\phi(\xi_{% 1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})|>\delta}}^{*}\int_{0}^{t}m(\xi_{6}% )^{2}\left|\partial_{t^{\prime}}\left(a_{\xi_{1}}(t^{\prime})\overline{a_{\xi_% {2}}(t^{\prime})}a_{\xi_{3}}(t^{\prime})\overline{a_{\xi_{4}}(t^{\prime})}a_{% \xi_{5}}(t^{\prime})\right)\right|\left|\overline{e_{\xi_{6}}(t))}\right|\,dt^% {\prime}$$ and $$\displaystyle\frac{1}{\delta L^{5}}\sum_{\xi_{6}}\sum_{\scriptstyle(\xi_{1},% \xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})\in A_{1}\atop{\scriptstyle|\phi(\xi_{% 1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})|>\delta}}^{*}\int_{0}^{t}m(\xi_{6}% )^{2}\left|a_{\xi_{1}}(t^{\prime})\overline{a_{\xi_{2}}(t^{\prime})}a_{\xi_{3}% }(t^{\prime})\overline{a_{\xi_{4}}(t^{\prime})}a_{\xi_{5}}(t^{\prime})\right|% \left|\partial_{t^{\prime}}\overline{e_{\xi_{6}}(t)}\right|\,dt^{\prime}$$ Here we only give a proof for the first term, because the second term can be handled similarly. In the same way as in the proof of Lemma 6.3, we use again the formula in (6.3), where $\widetilde{\eta}_{1}=\widetilde{\eta}_{2}=\widetilde{\eta}_{3}=\widetilde{\eta% }_{4}=\widetilde{\eta}_{5}=\widetilde{\eta}_{6}\in\{0,1,3,4\}$ and $-k/2<\widetilde{\tau}_{m}\leq k/2$ for $1\leq m\leq 6$. Notice that by (6.12) and (6.13), the restriction $|\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})|>\delta$ in the sum $\widetilde{R}_{4}(t)$ yields that at least two elements of $\xi_{m}~{}(1\leq m\leq 6)$ satisfy $|\widetilde{\tau}_{m}|\gg 1$, in which we denote $\xi_{a},~{}\xi_{b}$, namely, $m(\xi_{a})\gg 1,~{}m(\xi_{b})\gg 1$. We may assume $m(\xi_{6})\lesssim\min\{m(\xi_{a}),m(\xi_{b})\}$. Observe that if $|\widetilde{\tau}_{a}|\gg 1$, then by (3.1), $e_{\xi_{a}}(t)=a_{\xi_{a}}(t)$ satisfies the equation: $$\displaystyle i\dot{a}_{\xi_{a}}=\frac{1}{L^{4}}\sum^{*}a_{\xi_{1}}\overline{a% _{\xi_{2}}}a_{\xi_{3}}\overline{a_{\xi_{4}}}a_{\xi_{5}}e^{-it\phi(\xi_{1},\xi_% {2},\xi_{3},\xi_{4},\xi_{5},\xi_{a})}.$$ (6.18) There is at least one $a_{\xi_{m}}~{}(1\leq m\leq 5)$ of each quintic nonlinearity $a_{\xi_{1}}\overline{a_{\xi_{2}}}a_{\xi_{3}}\overline{a_{\xi_{4}}}a_{\xi_{5}}$ in (6.18) such that $m(\xi_{m})\gtrsim m(\xi_{a})$. This is easy to prove by writing $\xi_{m}$ as the formula (6.3). In view of above, in the case that the derivative $\partial_{t^{\prime}}$ falls into either the function associated to $\xi_{a}$ or $\xi_{b}$, we have the bound of $|\widetilde{R}_{4}(t)|$ by $$c\frac{\nu^{4}}{\delta}\int_{0}^{t}\|e(t^{\prime})\|_{s}^{2}\,dt^{\prime}.$$ On the other hand, in the case that the derivative $\partial_{t^{\prime}}$ falls into the function associated with neither $\xi_{a}$ or $\xi_{b}$, we have the same bound as above. Thus the proof is complete. ∎ It remains to estimate the term $$\displaystyle-\frac{2}{L}\Im\sum_{\xi\in 2\pi\mathbb{Z}/L}\int_{0}^{t}m(\xi)^{% 2}\widetilde{R}_{\xi}^{3}(t^{\prime})\overline{e_{\xi}(t^{\prime})}\,dt^{% \prime}.$$ (6.19) 6.1 Several technical lemmas First we shall prepare several lemmas. We keep the convention of notation expressed at Definition 6.1. For frequencies $\xi\in 2\pi\mathbb{Z}/L$, we use the expression in (6.2) and (6.3). Lemma 6.7. Let $\xi_{n}\in{\cal R}~{}(1\leq n\leq 5)$ and $\xi_{6}\in{\cal R}^{c}$ be elements of expression as described in (6.2). If $|\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})|\ll\langle k\rangle^{2}$, then $\widetilde{\eta}_{6}\in\{0,1,3,4\}$ and $\widetilde{\tau}_{6}\in\{\pm 1,\pm 2\}$. Proof. The proof is straightforward. Expressing $\xi_{n}$ as described in (6.2) and inserting into $\xi_{1}+\xi_{3}+\xi_{5}=\xi_{2}+\xi_{4}+\xi_{6}$, we see that $$\eta_{n}\in\{0,\,1,\,3,\,4\},~{}\tau_{n}=0\quad\mbox{for $1\leq n\leq 5$},$$ and $$\eta_{1}+\eta_{3}+\eta_{5}-\eta_{2}-\eta_{4}-\eta_{6}=\frac{j_{2}+j_{4}+j_{6}-% j_{1}-j_{3}-j_{5}}{Lk}+\frac{\tau_{6}}{k}.$$ We analyze and discuss each of the case study with respect to $\tau_{6}$. In the case when $3\leq\tau_{6}\leq k-3$, clearly $$\eta_{1}+\eta_{3}+\eta_{5}-\eta_{2}-\eta_{4}-\eta_{6}\in(0,1),$$ which is a contradiction, since the left-hand side is integer number. So the remaining cases are $0\leq\tau_{6}\leq 2$, and $k-2\leq\tau_{6}\leq k-1$. Consider the case when $0\leq\tau_{6}\leq 2$. Clearly $$\displaystyle\eta_{1}+\eta_{3}+\eta_{5}=\eta_{2}+\eta_{4}+\eta_{6}+O\left(% \frac{1}{k}\right),$$ $$\displaystyle(0,1)\ni\frac{1}{4\pi^{2}k^{2}}\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{% 4},\xi_{5},\xi_{6})=\eta_{1}^{2}-\eta_{2}^{2}+\eta_{3}^{2}-\eta_{4}^{2}+\eta_{% 5}^{2}-\eta_{6}^{2}+O\left(\frac{1}{k}\right)$$ and hence $$\displaystyle\eta_{1}+\eta_{3}+\eta_{5}=\eta_{2}+\eta_{4}+\eta_{6},\quad\eta_{% 1}^{2}+\eta_{3}^{2}+\eta_{5}^{2}=\eta_{2}^{2}+\eta_{4}^{2}+\eta_{6}^{2}.$$ We distinguish the cases. In the case when one element of $\eta_{1},\,\eta_{3},\,\eta_{5}$ is equal to one of $\eta_{2},\,\eta_{4}$, we may easily obtain $\{\eta_{1},\,\eta_{3},\,\eta_{5}\}=\{\eta_{2},\,\eta_{4},\,\eta_{6}\}$, which leads to $\eta_{6}\in\{0,\,1,\,3,\,4\}$. Next we consider the case when any elements of $\eta_{1},\,\eta_{3},\,\eta_{5}$ do not match any of $\eta_{2},\,\eta_{4}$, namely $\eta_{1},\,\eta_{3},\,\eta_{5}\not\in\{\eta_{2},\eta_{4}\}$. When two elements of $\eta_{1},\,\eta_{3},\,\eta_{5}$ are to be considered equal, we may assume $\eta_{1}=\eta_{3}$ by a symmetry argument. Observe that $\eta_{6}$ satisfies the one of $$\displaystyle\frac{1}{2}\left(2\eta_{1}+\eta_{5}-\eta_{2}\pm\sqrt{(\eta_{2}-% \eta_{5})(4\eta_{1}-\eta_{5}-3\eta_{2})}\right),$$ (6.20) which should take in an integer number. We calculate the considering numerical value of (6.20) in all cases. In the case when $\eta_{1}=\eta_{5}$, we have that $\eta_{1}$ and $\eta_{2}$ satisfy $\eta_{1}=\eta_{2}=\eta_{4}$, which is not acceptable. In the case when $\eta_{1}=\eta_{3}\neq\eta_{5}$, we will check all cases under the condition $\eta_{1},\,\eta_{3},\,\eta_{5}\not\in\{\eta_{2},\eta_{4}\}$: • $\{\eta_{1},\,\eta_{5}\}=\{0,\,1\},~{}\mbox{then}~{}\sqrt{(\eta_{2}-\eta_{5})(4% \eta_{1}-\eta_{5}-3\eta_{2})}\not\in\mathbb{Z}~{}\mbox{for}~{}\eta_{2}\in\{3,% \,4\}$, • $\{\eta_{1},\,\eta_{5}\}=\{0,\,3\},~{}\mbox{then}~{}\sqrt{(\eta_{2}-\eta_{5})(4% \eta_{1}-\eta_{5}-3\eta_{2})}\in\mathbb{Z}~{}\mbox{if and only if}~{}\eta_{1}=% 3,\,\eta_{5}=0,\,\eta_{2}\in\{1,\,4\}$, • $\{\eta_{1},\,\eta_{5}\}=\{0,\,4\},~{}\mbox{then}~{}\sqrt{(\eta_{2}-\eta_{5})(4% \eta_{1}-\eta_{5}-3\eta_{2})}\in\mathbb{Z}~{}\mbox{for}~{}\eta_{2}\in\{1,\,3\}$, • $\{\eta_{1},\,\eta_{5}\}=\{1,\,3\},~{}\mbox{then}~{}\sqrt{(\eta_{2}-\eta_{5})(4% \eta_{1}-\eta_{5}-3\eta_{2})}\not\in\mathbb{Z}~{}\mbox{for}~{}\eta_{2}\in\{0,% \,4\}$, • $\{\eta_{1},\,\eta_{5}\}=\{1,\,4\},~{}\mbox{then}~{}\sqrt{(\eta_{2}-\eta_{5})(4% \eta_{1}-\eta_{5}-3\eta_{2})}\in\mathbb{Z}~{}\mbox{if and only if}~{}\eta_{1}=% 1,\,\eta_{5}=4,\,\eta_{2}\in\{0,\,3\}$, • $\{\eta_{1},\,\eta_{5}\}=\{3,\,4\},~{}\mbox{then}~{}\sqrt{(\eta_{2}-\eta_{5})(4% \eta_{1}-\eta_{5}-3\eta_{2})}\not\in\mathbb{Z}~{}\mbox{for}~{}\eta_{2}\in\{0,% \,1\}$. Therefore there are two possible cases: • two elements of $\eta_{1},\,\eta_{3},\,\eta_{5}$ are $1$ and the rest of those elements is $4$, furthermore two elements of $\eta_{2},\,\eta_{4},\,\eta_{6}$ are $3$ and the rest of those elements is $0$, • two elements of $\eta_{2},\,\eta_{4},\,\eta_{6}$ are $1$ and the rest of those elements is $4$, moreover two elements of $\eta_{1},\,\eta_{3},\,\eta_{5}$ are $3$ and the rest of those elements is $0$. We remark that $\tau_{6}\neq 0$ since by $\xi_{6}\not\in{\cal R}$, that are acceptable. In the case when any two elements of $\eta_{1},\,\eta_{3},\,\eta_{5}$ do not match with each other, we again consider all cases: • $\{\eta_{1},\,\eta_{3},\,\eta_{5}\}=\{0,\,1,\,3\},~{}\eta_{2}=\eta_{4}=4,~{}% \eta_{6}=-4$, • $\{\eta_{1},\,\eta_{3},\,\eta_{5}\}=\{1,\,3,\,4\},~{}\eta_{2}=\eta_{4}=0,~{}% \eta_{6}=8$, • $\{\eta_{1},\,\eta_{3},\,\eta_{5}\}=\{0,\,3,\,4\},~{}\eta_{2}=\eta_{4}=1,~{}% \eta_{6}=5$, • $\{\eta_{1},\,\eta_{3},\,\eta_{5}\}=\{0,\,1,\,4\},~{}\eta_{2}=\eta_{4}=3,~{}% \eta_{6}=-1$, which are inadequate for the equation $\eta_{1}^{2}+\eta_{3}^{2}+\eta_{5}^{2}=\eta_{2}^{2}+\eta_{4}^{2}+\eta_{6}^{2}$. Consider next the case when $k-2\leq\tau_{6}\leq k-1$. Obviously $$\displaystyle\eta_{1}+\eta_{3}+\eta_{5}=\eta_{2}+\eta_{4}+\widetilde{\eta}_{6}% +O\left(\frac{1}{k}\right),$$ $$\displaystyle(0,1)\ni\frac{1}{4\pi^{2}k^{2}}\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{% 4},\xi_{5},\xi_{6})=\eta_{1}^{2}-\eta_{2}^{2}+\eta_{3}^{2}-\eta_{4}^{2}+\eta_{% 5}^{2}-\widetilde{\eta}_{6}^{2}+O\left(\frac{1}{k}\right)$$ and hence $$\displaystyle\eta_{1}+\eta_{3}+\eta_{5}=\eta_{2}+\eta_{4}+\widetilde{\eta}_{6}% ,\quad\eta_{1}^{2}+\eta_{3}^{2}+\eta_{5}^{2}=\eta_{2}^{2}+\eta_{4}^{2}+% \widetilde{\eta}_{6}^{2}.$$ Using the terminology in the previous cases, we have that $\widetilde{\eta}_{6}\in\{0,\,1,\,3,\,4\}$. This concludes the proof of the lemma. ∎ Corollary 6.1. Let $\eta_{j}\in\{0,\,1,\,3,\,4\}~{}(j=1,\,2,\,3,\,5)$, and let $\eta_{4},\,\eta_{6}\in\mathbb{Z}$ satisfy $$\eta_{1}+\eta_{3}+\eta_{5}=\eta_{2}+\eta_{4}+\eta_{6},\quad\eta_{1}^{2}+\eta_{% 3}^{2}+\eta_{5}^{2}=\eta_{2}^{2}+\eta_{4}^{2}+\eta_{6}^{2}.$$ Then either one of the following holds: • $\{\eta_{1},\,\eta_{3},\,\eta_{5}\}=\{\eta_{2},\,\eta_{4},\,\eta_{6}\}$, • two elements of $\{\eta_{1},\,\eta_{3},\,\eta_{5}\}$ are $1$ and the rest of those elements is $4$; two elements of $\{\eta_{2},\,\eta_{4},\eta_{6}\}$ are $3$ and the rest of those elements is $0$, • symmetric case of above replacing $\{\eta_{1},\,\eta_{3},\,\eta_{5}\}$ with $\{\eta_{2},\,\eta_{4},\eta_{6}\}$ and $\{\eta_{2},\,\eta_{4},\eta_{6}\}$ with $\{\eta_{1},\,\eta_{3},\,\eta_{5}\}$. Proof. In the case when any two elements of $\eta_{1},\,\eta_{3},\,\eta_{5}$ do not match with each other, we consider all cases: • $\{\eta_{1},\,\eta_{3},\,\eta_{5}\}=\{0,\,1,\,3\},~{}\eta_{2}=4,~{}\eta_{4}+% \eta_{6}=0$, • $\{\eta_{1},\,\eta_{3},\,\eta_{5}\}=\{1,\,3,\,4\},~{}\eta_{2}=0,~{}\eta_{4}+% \eta_{6}=8$, • $\{\eta_{1},\,\eta_{3},\,\eta_{5}\}=\{0,\,3,\,4\},~{}\eta_{2}=1,~{}\eta_{4}+% \eta_{6}=6$, • $\{\eta_{1},\,\eta_{3},\,\eta_{5}\}=\{0,\,1,\,4\},~{}\eta_{2}=3,~{}\eta_{4}+% \eta_{6}=2$, however $\eta_{1}^{2}+\eta_{3}^{2}+\eta_{5}^{2}=\eta_{2}^{2}+\eta_{4}^{2}+\eta_{6}^{2}$ is no longer satisfied. Therefore, the proof of Lemma 6.7 permits us to conclude the proof of the corollary. ∎ 6.2 Contribution of $\xi\in{\cal A}_{r}$ to (6.19) Let us first consider the contribution of $\xi\in{\cal A}_{r}$ to (6.19). Note that $m(\xi)\lesssim 1$ for $\xi\in{\cal A}_{r}$. The contribution of this case to (6.19) is $$\displaystyle-\frac{2}{L^{5}}\Im\sum_{\xi_{6}\in{\cal A}_{r}}\sum^{*}_{% \scriptstyle\text{res}(\xi_{6})^{c}\atop{\scriptstyle(\xi_{1},\xi_{2},\xi_{3},% \xi_{4},\xi_{5},\xi_{6})\in A_{1}^{c}}}\int_{0}^{t}m(\xi_{6})^{2}a_{\xi_{1}}(t% ^{\prime})\overline{a_{\xi_{2}}(t^{\prime})}a_{\xi_{3}}(t^{\prime})\overline{a% _{\xi_{5}}(t^{\prime})}a_{\xi_{5}}(t^{\prime})\overline{e_{\xi_{6}}(t^{\prime}% )}e^{-it^{\prime}\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})}\,dt^{% \prime}.$$ (6.21) Note that $m(\xi)\lesssim 1$ for $\xi\in{\cal A}_{r}$. By Lemma 6.2, we have that the contribution of this case is bounded by $$\displaystyle c\nu^{5/2}\int_{0}^{t}\|e(t^{\prime})\|_{s}\,dt^{\prime}\lesssim% \nu^{5}t^{2}.$$ (6.22) 6.3 Contribution of $\xi\not\in{\cal A}_{r}$ to (6.19) Consider the associated function $$\displaystyle\frac{1}{L^{5}}\Im\int_{0}^{t}\sum_{\xi_{6}\not\in{\cal A}_{r}}% \sum^{*}_{\scriptstyle\text{res}(\xi_{6})^{c}\atop{\scriptstyle(\xi_{1},\xi_{2% },\xi_{3},\xi_{4},\xi_{5},\xi_{6})\in A_{1}^{c}}}m(\xi_{6})^{2}a_{\xi_{1}}(t^{% \prime})\overline{a_{\xi_{2}}(t^{\prime})}a_{\xi_{3}}(t^{\prime})\overline{a_{% \xi_{4}}(t^{\prime})}a_{\xi_{5}}(t^{\prime})\overline{e_{\xi_{6}}(t^{\prime})}% e^{-it^{\prime}\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})}\,dt^{% \prime}.$$ (6.23) We split the sums into three cases: (i) $\max_{1\leq m\leq 5}m(\xi_{m})\gtrsim m(\xi_{6})$, (ii) $\max_{1\leq m\leq 5}m(\xi_{m})\ll m(\xi_{6})$; and at least four elements of $\xi_{m}~{}(1\leq m\leq 5)$ satisfy $m(\xi_{m})\ll\langle k\rangle^{s-1/2}$, (iii) $\max_{1\leq m\leq 5}m(\xi_{m})\ll m(\xi_{6})$; and at least two elements of $\xi_{m}~{}(1\leq m\leq 5)$ satisfy $m(\xi_{m})\gtrsim\langle k\rangle^{s-1/2}$. Case (i): In this case, we easily see that this contribution to (6.23) is bounded by $$\displaystyle c\nu^{2}\int_{0}^{t}\|e(t^{\prime})\|_{s}^{2}\,dt^{\prime}.$$ (6.24) Case (ii): In this case, we formulate $\xi_{j}~{}(1\leq j\leq 6)$ as in the form (6.3) such that $$\displaystyle\xi_{m}=2\pi\left(k\widetilde{\eta}_{m}+\widetilde{\tau}_{m}+% \frac{j_{m}}{L}\right).$$ (6.25) and distinguish the following two cases: (ii-1) $|\widetilde{\eta}_{6}|>50$, (ii-2) $|\widetilde{\eta}_{6}|\leq 50$. In the case of (ii-1), we only need to work with $\widetilde{\eta}_{m}\in\{0,1,3,4\},~{}|\widetilde{\tau}_{m}|\ll k$ for $1\leq m\leq 4$. The restriction that $\xi_{1}+\xi_{3}+\xi_{5}=\xi_{2}+\xi_{4}+\xi_{6}$ and $m(\xi_{5})\ll m(\xi_{6})$ yields $\widetilde{\eta}_{5}\leq 100,~{}99\leq\widetilde{\eta}_{6},~{}0\leq\widetilde{% \eta}_{6}-\widetilde{\eta}_{5}<10,~{}k^{s-1/2}\lesssim m(\xi_{5})\ll m(\xi_{6}% )\lesssim k^{s}$. We first observe that in the case when $|\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})|\gtrsim k^{2}$. While integration by parts, time derivative hits either $a_{\xi_{j}}$ or $e_{\xi_{6}}$ in (6.23). The contribution of this case to (6.23) is the sum of the following three terms $$\displaystyle\frac{\langle k\rangle^{-3/2}}{L^{5}}\left|\sum_{\xi_{6}\not\in{% \cal A}_{r}}\sum^{*}a_{\xi_{1}}(t)\overline{a_{\xi_{2}}(t)}a_{\xi_{3}}(t)% \overline{a_{\xi_{4}}(t)}m(\xi_{5})a_{\xi_{5}}(t)m(\xi_{6})\overline{e_{\xi_{6% }}(t)}\right|,$$ $$\displaystyle\frac{\langle k\rangle^{-3/2}}{L^{5}}\int_{0}^{t}\sum_{\xi_{6}% \not\in{\cal A}_{r}}\sum^{*}\left|\partial_{t^{\prime}}\left(a_{\xi_{1}}(t^{% \prime})\overline{a_{\xi_{2}}(t^{\prime})}a_{\xi_{3}}(t^{\prime})\overline{a_{% \xi_{4}}(t^{\prime})}\langle k\rangle^{s-1/2}a_{\xi_{5}}(t^{\prime})\right)m(% \xi_{6})\overline{e_{\xi_{6}}(t^{\prime})}\right|\,dt^{\prime}$$ and $$\displaystyle\frac{\langle k\rangle^{-1}}{L^{5}}\int_{0}^{t}\sum_{\xi_{6}\not% \in{\cal A}_{r}}\sum^{*}\left|a_{\xi_{1}}(t^{\prime})\overline{a_{\xi_{2}}(t^{% \prime})}a_{\xi_{3}}(t^{\prime})\overline{a_{\xi_{4}}(t^{\prime})}m(\xi_{5})a_% {\xi_{5}}(t^{\prime})\langle k\rangle^{s-1/2}\partial_{t^{\prime}}\overline{e_% {\xi_{6}}(t^{\prime})}\,dt^{\prime}\right|,$$ which are bounded by $$\displaystyle\lesssim\frac{\nu^{2}}{\langle k\rangle^{3/2}}\|e(t)\|_{s}^{2}+% \nu^{2}\int_{0}^{t}\|e(t^{\prime})\|_{s}^{2}\,dt^{\prime},$$ (6.26) for $s>1$. Next suppose that $|\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})|\ll k^{2}$ holds. Observe $$\displaystyle k(\widetilde{\eta}_{1}-\widetilde{\eta}_{2}+\widetilde{\eta}_{3}% -\widetilde{\eta}_{4}+\widetilde{\eta}_{5}-\widetilde{\eta}_{6})-(\widetilde{% \tau}_{6}-\widetilde{\tau}_{5})=\frac{j_{2}+j_{4}+j_{6}-j_{1}-j_{3}-j_{5}}{L}% \in(-3,3),$$ (6.27) and $$\displaystyle 1\gg\frac{1}{(2\pi)^{2}}|\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},% \xi_{5},\xi_{6})|\geq\left|k(\widetilde{\eta}_{5}-\widetilde{\eta}_{6})+% \widetilde{\tau}_{5}-\widetilde{\tau}_{6}+\frac{j_{5}-j_{6}}{L}\right|(\xi_{5}% +\xi_{6})-50k^{2}.$$ (6.28) Considering the case $m(\xi_{5})\ll m(\xi_{6})$, we investigate a possible case above and have two possible cases; that (a): $\widetilde{\eta}_{6}=\widetilde{\eta}_{5}=99$, and that (b): $\widetilde{\eta}_{6}=\widetilde{\eta}_{5}+1\in\{99,100\}$; moreover, subdivide each case into several cases as follows: (a) $\widetilde{\eta}_{6}=\widetilde{\eta}_{5}=99,~{}\widetilde{\tau}_{5}<0,~{}% \widetilde{\tau}_{6}\gg 1,~{}m(\xi_{5})=\langle k\rangle^{s-1/2},~{}m(\xi_{6})% \sim\langle k\rangle^{s-1/2}\langle\widetilde{\tau}_{6}\rangle^{1/2}$, (b1) $\widetilde{\eta}_{6}=\widetilde{\eta}_{5}+1=99,~{}\widetilde{\tau}_{6}\gg 1,~{% }m(\xi_{5})=\langle k\rangle^{s-1/2},~{}m(\xi_{6})=\langle k\rangle^{s-1/2}% \langle\widetilde{\tau}_{6}\rangle^{1/2}$, (b2) $\widetilde{\eta}_{6}=\widetilde{\eta}_{5}+1=100,~{}\widetilde{\tau}_{5}<0,~{}m% (\xi_{5})=\langle k\rangle^{s-1/2},~{}m(\xi_{6})\sim\langle k\rangle^{s}$, (b3) $\widetilde{\eta}_{6}=\widetilde{\eta}_{5}+1=100,~{}\widetilde{\tau}_{5}\geq 0,% ~{}\langle\widetilde{\tau}_{5}\rangle\ll\langle k\rangle,~{}m(\xi_{5})=\langle k% \rangle^{s-1/2}\langle\widetilde{\tau}_{5}\rangle^{1/2},~{}m(\xi_{6})\sim% \langle k\rangle^{s}$, In the case of (a), we see that if $\widetilde{\tau}_{6}-\widetilde{\tau}_{5}<k-3$ then (6.27) implies $$1\ll\widetilde{\tau}_{6}<\widetilde{\tau}_{6}-\widetilde{\tau}_{5}<3,$$ which is not subject to the considering case. Then $|k(\widetilde{\eta}_{5}-\widetilde{\eta}_{6})+\widetilde{\tau}_{5}-\widetilde{% \tau}_{6}|\geq k-3$, however by (6.28), it follows that $$\displaystyle|\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})|\gtrsim k^% {2},$$ (6.29) which rules out of this case. In the cases of (b1), (b2), (b3), we have $|k(\widetilde{\eta}_{5}-\widetilde{\eta}_{6})+\widetilde{\tau}_{5}-\widetilde{% \tau}_{6}|>k/4$ and then obtain the same estimate (6.29), which also rules out of these cases. Let us consider the case of (ii-2). Since by $\max_{1\leq m\leq 5}m(\xi_{m})\ll m(\xi_{6})$, we have $\widetilde{\eta}_{m}\in\{0,1,3,4\},~{}|\widetilde{\tau}_{m}|\ll k$ for $1\leq m\leq 5$, and $m(\xi_{6})\lesssim k^{s-1/2}$. We observe the case when $|\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})|\gtrsim\langle k\rangle% ^{2}$. An argument similar to above shows that this contribution to (6.23) is bounded by the sum of following three terms: $$\displaystyle\frac{\langle k\rangle^{s-5/2}}{L^{5}}\left|\sum_{\xi_{6}\not\in{% \cal A}_{r}}\sum^{*}a_{\xi_{1}}(t)\overline{a_{\xi_{2}}(t)}a_{\xi_{3}}(t)% \overline{a_{\xi_{5}}(t)}a_{\xi_{5}}(t)m(\xi_{6})\overline{e_{\xi_{6}}(t)}% \right|,$$ $$\displaystyle\frac{\langle k\rangle^{s-5/2}}{L^{5}}\int_{0}^{t}\sum_{\xi_{6}% \not\in{\cal A}_{r}}\sum^{*}\left|\partial_{t^{\prime}}\left(a_{\xi_{1}}(t^{% \prime})\overline{a_{\xi_{2}}(t^{\prime})}a_{\xi_{3}}(t^{\prime})\overline{a_{% \xi_{5}}(t^{\prime})}a_{\xi_{5}}(t^{\prime})\right)m(\xi_{6})\overline{e_{\xi_% {6}}(t^{\prime})}\right|\,dt^{\prime}$$ and $$\displaystyle\frac{\langle k\rangle^{2s-3}}{L^{5}}\int_{0}^{t}\sum_{\xi_{6}% \not\in{\cal A}_{r}}\sum^{*}\left|a_{\xi_{1}}(t^{\prime})\overline{a_{\xi_{2}}% (t^{\prime})}a_{\xi_{3}}(t^{\prime})\overline{a_{\xi_{5}}(t^{\prime})}a_{\xi_{% 5}}(t^{\prime})\partial_{t^{\prime}}\overline{e_{\xi_{6}}(t^{\prime})}\,dt^{% \prime}\right|,$$ which are bounded by $$\displaystyle\frac{c}{\langle k\rangle^{5/2-s}}\nu^{5/2}\|e(t)\|_{s}+\frac{c}{% \langle k\rangle^{5/2-s}}\nu^{9/2}\int_{0}^{t}\|e(t^{\prime})\|_{s}\,dt^{% \prime}+\frac{c}{\langle k\rangle^{3-2s}}\nu^{5}t$$ (6.30) $$\displaystyle\lesssim$$ $$\displaystyle\frac{\nu^{3}}{\langle k\rangle^{5/2-s}}+\frac{\nu^{5}}{\langle k% \rangle^{3/2-s}}t+\nu^{2}\int_{0}^{t}\|e(t^{\prime})\|_{s}^{2}\,dt^{\prime}.$$ So we may suppose $|\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},\xi_{6})|\ll\langle k\rangle^{2}$. By assumption of this case, $$\displaystyle k(\widetilde{\eta}_{1}-\widetilde{\eta}_{2}+\widetilde{\eta}_{3}% -\widetilde{\eta}_{4}+\widetilde{\eta}_{5}-\widetilde{\eta}_{6})=o(k)+% \widetilde{\tau}_{6}\in(-3k/4,3k/4)$$ and $$\displaystyle\frac{1}{(2\pi)^{2}}\phi(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},% \xi_{6})=k^{2}\left(\widetilde{\eta}_{1}^{2}-\widetilde{\eta}_{2}^{2}+% \widetilde{\eta}_{3}^{2}-\widetilde{\xi}_{4}^{2}+\widetilde{\eta}_{5}^{2}-% \widetilde{\eta}_{6}^{2}\right)+O(k)$$ (6.31) imply $$\widetilde{\eta}_{1}-\widetilde{\eta}_{2}+\widetilde{\eta}_{3}-\widetilde{\eta% }_{4}+\widetilde{\eta}_{5}-\widetilde{\eta}_{6}=0,$$ $$\widetilde{\eta}_{1}^{2}-\widetilde{\eta}_{2}^{2}+\widetilde{\eta}_{3}^{2}-% \widetilde{\eta}_{4}^{2}+\widetilde{\eta}_{5}^{2}-\widetilde{\eta}_{6}^{2}=0.$$ Then Corollary 6.1 allows us to obtain $\{\widetilde{\eta}_{1},\widetilde{\eta}_{3},\widetilde{\eta}_{5}\}=\{% \widetilde{\eta}_{2},\widetilde{\eta}_{4},\widetilde{\eta}_{6}\}$, and any two of $\widetilde{\eta}_{1},\widetilde{\eta}_{3},\widetilde{\eta}_{5}$ do not coincide with each other. Moreover, we have $$|\widetilde{\tau}_{1}-\widetilde{\tau}_{2}+\widetilde{\tau}_{3}-\widetilde{% \tau}_{4}+\widetilde{\tau}_{5}-\widetilde{\tau}_{6}|\lesssim 1,$$ which yields $$\langle\widetilde{\tau}_{6}\rangle\lesssim\max_{1\leq m\leq 5}\langle% \widetilde{\tau}_{m}\rangle,$$ and then $\max_{1\leq m\leq 5}m(\xi_{m})\gtrsim m(\xi_{6})$. However this is out of the case. Case (iii): Since at least two elements of $\xi_{m}\not\in{\cal A}_{r}~{}(1\leq m\leq 5)$ satisfy $m(\xi_{m})\gtrsim k^{s-1/2}$, it is easy to see that this contribution to (6.23) is bounded by $$\displaystyle c\left(\frac{1}{\langle k\rangle^{2(s-1/2)-s}}+1\right)\nu^{2}% \int_{0}^{t}\|e(t^{\prime})\|_{s}^{2}\,dt^{\prime}\lesssim\nu^{2}\int_{0}^{t}% \|e(t^{\prime})\|_{s}^{2}\,dt^{\prime},$$ provided $s>1$. Therefore we have the bound of (6.19) by $$\displaystyle-2\Im\sum_{\xi\in 2\pi\mathbb{Z}/L}\int_{0}^{t}m(\xi)^{2}% \widetilde{R}_{\xi}^{3}(t^{\prime})\overline{e_{\xi}(t^{\prime})}\,dt^{\prime}$$ (6.32) $$\displaystyle\lesssim$$ $$\displaystyle o(\|e(t)\|_{s}^{2})+\frac{\nu^{3}}{\langle k\rangle^{5/2-s}}+% \frac{\nu^{5}}{\langle k\rangle^{3/2-s}}t+\nu^{5}t^{2}+\nu^{2}\int_{0}^{t}\|e(% t^{\prime})\|_{s}^{2}\,dt^{\prime}.$$ 6.4 Proof of Proposition 6.1 We now turn to the proof of Proposition 6.1. The proof follows from continuity of solution with respect to $\|\cdot\|_{s}$-norm, together with the Gronwall inequality. By Lemmas 6.3, 6.4, 6.5, 6.6, and the estimate (6.32), we get $$\displaystyle\|e(t)\|_{s}^{2}\leq A+Bt+Ct^{2}+c\nu^{2}\int_{0}^{t}\|e(t^{% \prime})\|_{s}^{2}\,dt^{\prime},$$ (6.33) where $$A=\frac{c\nu^{3}}{\langle k\rangle^{5/2-s}},\quad B=\frac{c\nu^{5}}{\langle k% \rangle^{3/2-s}},\quad C=c\nu^{5}.$$ We need to prepare the Gronwall inequality. It is easy to prove that if the continuous function $F(t)$ satisfies $$F(t)\leq A+Bt+Ct^{2}+c\nu^{2}\int_{0}^{t}F(t^{\prime})\,dt^{\prime},$$ then we obtain that $$c\nu^{2}\int_{0}^{t}F(t^{\prime})\,dt^{\prime}\leq Ae^{c\nu^{2}t}-(A+Bt+Ct^{2}% )+\frac{B}{c\nu^{2}}(e^{c\nu^{2}t}-1)+\frac{2C}{c^{2}\nu^{4}}(e^{c\nu^{2}t}-1-% c\nu^{2}t),$$ and then $$F(t)\leq Ae^{c\nu^{2}t}+\frac{B}{c\nu^{2}}(e^{c\nu^{2}t}-1)+\frac{2C}{c^{2}\nu% ^{4}}(e^{c\nu^{2}t}-1-c\nu^{2}t).$$ Applying the Gronwall inequality to (6.33), we get $$\displaystyle\|e(t)\|_{s}^{2}\lesssim\frac{\nu^{3}}{\langle k\rangle^{5/2-s}}e% ^{c\nu^{2}t}+\frac{\nu^{3}}{\langle k\rangle^{3/2-s}}\left(e^{c\nu^{2}t}-1% \right)+\nu\left(e^{c\nu^{2}t}-1-c\nu^{2}t\right).$$ This proves Proposition 6.1. ∎ 7 Proof of Theorem 1.2 In this section we prove Theorem 1.2. Proof of Theorem 1.2. We use a bootstrap (continuity) argument in a similar way to the proof of Lemma 5.2. By Lemma 5.2, (5.11), (5.13), Lemma 5.5 and Proposition 6.1, we in fact have $\|e(t)\|_{s}^{2}\lesssim\nu$ for $|t|\ll 1/(L^{3}\nu^{2})$, and iterate Proposition 6.1 to obtain Theorem 1.2. ∎ References [1] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I: Schrödinger equation, part II: The KdV-equation, Geom. Func. Anal., 3 (1993), 107–156, 209–262. [2] T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^{s}$, Nonlinear Anal., 14 (1990), 807–836. [3] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$, J. Amer. Math. Soc., 16 (2003), 705–749. [4] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation, Invent. Math., 181 (2010), 39–113. [5] B. Dodson, Global well-posedness for the defocusing, quintic nonlinear Schrödinger equation in one dimension for low regularity data, Int. Math. Res. Notices, 2012, 870–893. [6] B. Dodson, Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schrödinger equation when $d=1$, Am. J. Math., 138 (2016), 531–569. [7] E. Faou, P. Germain and Z. Hani, The weakly nonlinear large-box limit of the 2d cubic nonlinear Schrödinger equation, J. Amer. Math. Soc., 29 (2015), 915–982. [8] J. Ginibre and G. Velo, On the class of nonlinear Schrödinger equations, J. Funct. Anal., 32 (1979), 1–32, 33–72. [9] B. Grébert and L. Thomann, Resonant dynamics for the quintic nonlinear Schrödinger equation, Ann. I. H. Poincaré, 29 (2012), 455–477. [10] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Clarendon Press, Oxford University Press, New York, 1979. [11] M. B. Nathanson, Additive number theory, Springer-Verlag, 1996. [12] H. Takaoka, Energy transfer model for the derivative nonlinear Schrödinger equations on the torus, Discrete Contin. Dyn. Syst. Ser. A, xxx. [13] Y. Tsutsumi, $\mathrm{L}^{2}$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcialaj Ekvacioj, 30 (1987), 115- 125.

Orbital Transition in the Parent Spin-Orbit Mott Insulator Sr${}_{2}$IrO${}_{4}$ K. Samanta “Gleb Wataghin” Institute of Physics, University of Campinas - UNICAMP, Campinas, São Paulo 13083-859, Brazil    F. M. Ardito Brazilian Synchrotron Light Laboratory (LNLS), Brazilian Center for Research in Energy and Materials (CNPEM), Campinas, São Paulo 13083-970, Brazil    N. M. Souza-Neto Brazilian Synchrotron Light Laboratory (LNLS), Brazilian Center for Research in Energy and Materials (CNPEM), Campinas, São Paulo 13083-970, Brazil    E. Granado “Gleb Wataghin” Institute of Physics, University of Campinas - UNICAMP, Campinas, São Paulo 13083-859, Brazil Abstract The crystal lattice and vibrations of Sr${}_{2}$IrO${}_{4}$ were probed by a combined phonon Raman scattering and x-ray powder diffraction experiment at $P<45$ GPa. Anomalies in phonon Raman intensities, energies and linewidths were observed at 17, 30 and 40 GPa with no accompanying structural phase transition. An account of combined experimental data indicates a transition of Ir${}^{4+}:5d^{5}$ electrons from the spin-orbit entangled $J_{eff}=1/2$ configuration below 17 GPa to a state with a hole in a $xz\pm iyz$ orbital above 40 GPa, crossing a mixed orbital phase at intermediate pressures. pacs: 78.30.-j, 75.25.Dk, 63.20.-e, 61.05.cp The discovery of a spin-orbit entangled ($J_{eff}=1/2$) Mott insulating state in some Ir${}^{4+}$ oxides, attributed to the combined effects of strong spin-orbit coupling (SOC $\sim$ 0.5 eV) and on-site Coulomb interaction (U $\sim$ 1.5 - 2 eV), triggered a wave of activity in these materials Pesin ; Jackeli ; Kim ; Kim2 ; Arita ; Laguna ; Kim3 ; Rau ; Caorev ; Haskel ; Zocco ; Katukuri ; Lado ; Chikara ; Liu ; Cetin ; Gretarsson ; Gretarsson2 ; Donnerer . Parent Sr${}_{2}$IrO${}_{4}$ crystallizes in a tetragonal phase with space group $I$4${}_{1}$/${acd}$, showing a rotation of the IrO${}_{6}$ octahedra by $\sim$ 11${}^{\circ}$ along the $c$-axis Huang ; Crawford . The presence of a significant Dzyaloshinskii-Moriya-type (DM) exchange interaction leads to a canted magnetic structure [see Fig. 1(a)] Liu with a weak ferromagnetic moment of 0.06-0.14 $\mu_{B}$/Ir below $T_{N}=240$ K at ambient $P$ Crawford ; Chikara ; Cao . Surprisingly, the insulating behavior of this material is maintained under compression up to at least 55 GPa Haskel ; Zocco with an $U$-shaped resistivity curve with $P$ Haskel ; Caorev . Such unconventional behavior of electrical conductivity is arguably one of the most intriguing unresolved issues of iridate physics. Still, relatively few experiments under high $P$ using microscopic techniques have been reported for this material to address this matter. An x-ray spectroscopy study by Haskel et al. Haskel showed a magnetic circular dichroism signal associated with the weak ferromagnetic moment that disappears for $P>17$ GPa. It was also shown that the expectation value of $<{\bf L\cdot S}>$ displays a considerable reduction with pressure, extrapolating to zero at $80-90$ GPa Haskel , revealing a significant sensitivity of the $J_{eff}=1/2$ state with pressure. However, the detailed Ir${}^{4+}$ orbital states at high $P$ still remains to be determined. In this work, the Ir${}^{4+}:5d$ orbital states in Sr${}_{2}$IrO${}_{4}$ at high $P$ are investigated by a combined phonon Raman scattering and x-ray powder diffraction (XRD) experiment up to $45$ GPa. Anomalies in phonon Raman intensities, energies, lineshapes and linewidths were observed with $P$, with no accompanying structural phase transition. An analysis of our results combined with those available in the literature Haskel ; Lado indicates a transition of Ir${}^{4+}:5d^{5}$ electrons from the $J_{eff}=1/2$ configuration below 17 GPa to a competing spin-orbital disentangled state with a hole in a $xz\pm iyz$ orbital above 40 GPa, crossing a mixed disordered phase at intermediate $P$. A competition between distinct spin-orbital configurations is arguably a decisive ingredient of iridate physics, offering insight into the previously reported $U$-shaped resistivity and magnetic transition with $P$ Haskel ; Caorev . The Sr${}_{2}$IrO${}_{4}$ powder sample was prepared by a standard high temperature solid state reaction mechanism, as described in the Supplemental Material (SM) SM . A laboratory x-ray diffraction measurement showed that impurity phases are below 1 % in weight fraction. Magnetization measurements showed a weak ferromagnetic moment below $T_{C}=237$ K SM , in line with the reported transition temperature Crawford ; Cao . Synchrotron x-ray diffraction and Raman spectroscopy experiments were performed at the X-ray Diffraction and Spectroscopy (XDS) beamline in the Brazilian Synchrotron Light Source (LNLS) XDS . $P$ was applied by diamond anvil cells (DAC) using type Ia ultra-low fluorescence diamonds with cullet diameter of 350 microns. The employed $P$ medium was neon neon and gaskets were made of rhenium. The value of $P$ was obtained using the well-known ruby $R_{1}$ fluorescence line shift method Ruby . The unpolarized Raman spectroscopy measurements were taken using a 532 nm diode laser, a 90 cm${}^{-1}$ transition long-pass edge filter, a single stage 300 mm focal length Czerny-Turner spectrograph with 1800 gr/mm grating and a liquid nitrogen cooled CCD. The beam was focused and scattered light collected by a confocal setup. The laser beam focal spot was $\sim 40$ $\mu$m and the laser power was kept below 20 mW. The instrumental linewidth given by this setup is 3 cm${}^{-1}$. Baseline corrections were performed in the Raman spectra to isolate the relatively sharp and intense phonon peaks SM . Details on the x-ray diffraction experiment and lattice dynamics calculations QE ; Perdew ; pseudopotentials are given in the SM SM . The phonon Raman spectra of Sr${}_{2}$IrO${}_{4}$ are shown in Fig. 2 for increasing $P$. The corresponding data taken with decreasing $P$ are displayed in the SM SM and indicate the observed phonon changes with $P$ are reversible. At $P=2.5$ GPa and room-$T$, four peaks at 185, 270, 395, and 570 cm${}^{-1}$, labeled as $M_{1}-M_{4}$, dominate the spectrum. At ambient $P$ and $T=20$ K, these peaks are observed at 186, 276, 393 and 558 cm${}^{-1}$, respectively SM , being readily identified with the aid of previous symmetry-resolved single crystal Raman studies Cetin ; Gretarsson ; Gretarsson2 . Peak $M_{3}$ is a mode with $B_{2g}$ symmetry, peaks $M_{2}$ and $M_{4}$ are A${}_{1g}$ modes and peak $M_{1}$ (hereby termed $M_{1}/M_{1}^{\prime}$) is a superposition of an $A_{1g}$ and a $B_{2g}$ mode. Our ab-initio lattice dynamics calculations predicts $A_{1g}$ modes at 181, 260, and 588 cm${}^{-1}$ and $B_{2g}$ modes at 173 and 371 cm${}^{-1}$, in good agreement with the observed frequencies. The mechanical representations of such modes are given in Figs. 1(c-g). For completeness, the calculated mode frequencies and corresponding $\Gamma$-point mechanical representations of all Raman and Infrared active modes of Sr${}_{2}$IrO${}_{4}$ under space group $I$4${}_{1}$/${acd}$ are given in the SM SM . Figures 3(a-l) show the $P$-dependence of $M_{1}-M_{4}$ frequencies (a-d), linewidths (e-h), and integrated areas (i-l), obtained by Lorentzian fits except where otherwise noted. A number of marked phonon anomalies are observed. The peaks $M_{1}$ and $M_{2}$ disappear within our sensitivity above $P_{1}=17$ GPa. $M_{1}$ clearly re-emerges above $P_{2}=30$ GPa; $M_{2}$ may also re-emerge above $P_{2}$ but with very low intensity so it is hardly distinguished from noise (see asterisk in Fig. 2). Modes $M_{3}$ and $M_{4}$ also show intensity anomalies at $P_{1}$ and $P_{2}$. All observed modes show hardening with increasing $P$, except for $M_{4}$, which softens between $P_{2}$ and $P_{3}=40$ GPa. Remarkably, $M_{3}$ shows an asymmetric profile between $P_{1}$ and $P_{3}$, being fitted with a Fano lineshape $I(\omega)=I_{0}(q+\epsilon)^{2}/(1+\epsilon^{2})$, where $I_{0}$ is the intensity, $q$ is the asymmetry parameter and $\epsilon\equiv(\omega-\omega_{0})/\Gamma$, with $\omega_{0}$ and $\Gamma$ being the phonon frequency and linewidth, respectively [see Fig. 4(a)] Fano . The $P$-dependence of the electron-phonon coupling strength $\lvert 1/q\rvert$ of this mode is given in Fig. 4(b), being zero below $P_{1}$ and above $P_{3}$ within our resolution and maximum at $P_{2}$. The other modes could be well fit by symmetric Lorentzian lineshapes within our statistics. The linewidth $\Gamma$ of mode $M_{4}$ shows little variation between 2.5 GPa and $P_{1}$, then showing an upturn to a steep broadening with $P$ up to $P_{2}$, becoming nearly constant for $P>P_{2}$ [see Figs. 2 and 3(e)]. Mode $M_{3}$ shows substantial broadening with increasing $P$ up to $P_{2}$ and sharpening above $P_{2}$ [see Figs. 2 and 3(f)]. The interpretation of the observed phonon anomalies relies on the existence or not of accompanying structural phase transitions in the investigated $P$-range. The $P$-dependence of the $a$ and $c$ lattice parameters normalized to their values at 2.5 GPa are shown in Fig. 4(c). The raw x-ray powder diffraction profiles are given in the SM SM . No structural phase transition is observed. However, an anomalous behavior of $c$ is noticed above 40 GPa, indicating that a structural phase transition would take place above 45 GPa. We should mention that the related iridate Sr${}_{3}$Ir${}_{2}$O${}_{7}$ shows a tetragonal-monoclinic transition at 54 GPa Donnerer . As argued below, the phonon anomalies without an accompanying structural phase transition provide strong evidence for a $P$-induced orbital phase transition with a disordered intermediate phase that connects the competing ground states. The notable suppression of $M_{1}/M_{1}^{\prime}$ and $M_{2}$ peaks at $P_{1}$ is likely associated with a phase transition, being clearly connected with the reported suppression of the low-$T$ XMCD signal Haskel . Such suppression of Raman activity by specific modes might suggest a symmetrization of the crystal structure at first sight. However, the contraction rate of $a$ is larger than $c$ in all studied $P$-range [see Fig. 4(c)], suggesting an increment of the octahedral rotation angle $\phi$ with increasing $P$ [see Fig. 1(a)], in line with a previous report Haskel . Also, the observation of the in-plane oxygen bending $M_{3}$ mode indicates that such oxygen is not at an inversion center at any studied $P$, ruling out a symmetrization transition to $\phi=0$ at $P_{1}$. In this scenario, the apparent disappearance of the peaks $M_{1}$ and $M_{2}$ as $P\rightarrow P_{1}$ is ascribed to a large reduction of the corresponding Raman tensor elements, as a consequence of electronic structure changes. The large broadenings of all studied Raman peaks indicate generalized reduction of phonon lifetimes with increasing $P$ between $P_{1}$ and $P_{2}$. Also, the Fano asymmetry of Raman mode $M_{3}$ reveals low-energy electronic excitations only in this $P$ interval. Taken together, these observations point to dynamic electronic disorder of the Ir${}^{4+}$ $5d$ electrons, which is again reduced for $P>P_{2}$. The re-emergence of peaks $M_{1}/M_{1}^{\prime}$ and $M_{2}$ above $P_{2}$ is indicative of an increment of the corresponding Raman tensor elements to magnitudes comparable to those below $P_{1}$. This observation, as well as the sharpening of all observed Raman modes above $P_{2}$ and the disappearance of the $M_{3}$ Fano asymmetry above $P_{3}$, are indeed indicative of partially recovered electronic order possibly due to a gradual stabilization of a specific Ir${}^{4+}$ orbital configuration at high $P$. While the spin-orbital configuration for $P<P_{1}$ is well accepted to be the $J_{eff}=1/2$ state with an equal population of $xy$, $xz$ and $yz$ orbitals in first approximation Pesin ; Jackeli ; Kim ; Kim2 ; Arita ; Laguna ; Kim3 ; Rau ; Caorev , the specific orbital state above $P_{3}$ still needs to be addressed. To advance in this direction, we analyze in further detail the behavior of the observed mode frequencies with $P$. In the absence of phase transitions, the mode frequencies are expected to follow the Grüneisen’s law, $\Delta\omega_{i}/\omega_{i}=-\gamma_{i}\Delta V/V$, where $\gamma_{i}$ is the Grüneisen parameter for the mode with frequency $\omega_{i}$ and $V$ is the unit cell volume. The solid lines in Figs. 3(a-d) indicate the expected frequency of modes $M_{1}-M_{4}$ according to this law, using the unit cell volume extracted from the data of Fig. 4(c), and adjusting $\gamma_{i}$ to optimize the fit for $P<P_{1}$. All investigated modes deviate from the Grüneisen’s law at high $P$. Considering that no structural phase transition takes place in the investigated $P$-interval, such behavior of the phonon frequencies is another solid indication of an electronic phase transition. We note that the $M_{4}$ mode is considerably softer at $P=45$ GPa than the behavior expected by Grüneisen’s law [see Fig. 3(a)], while, in opposition, the modes $M_{1}$ and $M_{3}$ show anomalous hardening at high $P$. Since $M_{4}$ is a stretching mode of apical oxygen ions [see Fig. 1(g)], this phonon softening is indicative of a weakening the Ir-O${}_{apical}$ bond in the high-$P$ phase, while the extra hardening of modes $M_{1}$ and $M_{3}$ involving vibrations of in-plane oxygen ions [see Figs. 1(c) and 1(f)] suggests a strengthening of the Ir-O${}_{basal}$ bonds. This trend is consistent with an electronic density transfer from $xz$ and/or $yz$ to $xy$ Ir${}^{4+}$ $t_{2g}$ orbitals for $P>P_{3}$ with respect to $P<P_{1}$. Another important hint on the nature of the high-$P$ phase comes from x-ray absorption near-edge structure (XANES) results at the Ir $L_{2,3}$ edges, showing a large reduction of the $I(L_{3})/I(L_{2})$ branching ratio between 30 and 40 GPa Haskel . This result indicated that the phase above $P_{3}$ shows a considerably smaller expectation value of $<{\bf L\cdot S}>$ with respect to the $J_{eff}=1/2$ state. Remarkably, a recent density functional theory study showed that a noncollinear calculation is required to obtain solutions close to the $J_{eff}=1/2$ state to describe the $t_{2g}$ hole in the Ir${}^{4+}:d^{5}$ cation, while a collinear calculation yields a solution with the hole in a $xz\pm iyz$ orbital with half of the $L_{z}/S_{z}$ ratio of the $J_{eff}=1/2$ state Lado . Considering these calculations Lado , the softening of the $M_{4}$ mode at high pressures and the reported collinear magnetic structure above $P_{1}$ Haskel , we propose that the high-$P$ Ir${}^{4+}$ orbital configuration is a spin-orbit disentangled state with a hole in the $xz\pm iyz$ orbital. Having addressed the possible Ir${}^{4+}$ orbital configuration at high $P$, a deeper discussion on the intermediate phase between $P_{1}$ and $P_{3}$ is called for. According to the picture described above, such disordered phase is at the frontier between two fundamentally different states, with and without spin-orbit entanglement. The presence of orbital dynamics at room $T$ is evidenced by the Fano lineshape of the $M_{3}$ mode. Indeed, this is most likely a non-trivial phase where individual Ir${}^{4+}$ electrons fluctuate between the two spin-orbital end states. It is interesting to note that, in this scenario, the orbital fluctuations are able to disturb the long-range non-collinear magnetic order at $P_{1}$, as evidenced by XMCD results Haskel , even though $<{\bf L\cdot S}>$ remains very high up to $P_{2}$, as evidenced by XANES data Haskel . Also, the electronically disordered phase coincides with the $P$-range where minimum resistivity is found above 50 K Haskel ; Caorev . This resistance minimum is likely associated with an electronic band broadening due to the inherent electronic disorder of this intermediate phase, reducing the thermal energy necessary for the electrons at the top of valence band to overcome the Mott gap. In conclusion, the anomalous phonon behavior of Sr${}_{2}$IrO${}_{4}$ with $P$ indicates a crossover between two distinct spin-orbital Ir${}^{4+}$ electronic configurations. The electronic density transfer of Ir${}^{4+}$ $t_{2g}$ electrons to the $xy$ plane with increasing $P$ evidenced by our data supports a transition from the spin-orbit entangled non-collinear $J_{eff}=1/2$ phase to a simpler collinear phase with a hole in the $xz\pm iyz$ orbital. An intermediate disordered phase is also evidenced between 17 and 40 GPa, matching the $P$ region where minimal resistivity above 50 K was reported Haskel ; Caorev . 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Reciprocal symmetry breaking in Pareto sampling H.-S. Niwa Abstract. Let $W_{1},\ldots,W_{N}$ be a sample of $\mathrm{Pareto}(\alpha)$ random variables normalized by their sum, such that $\sum_{i}W_{i}=1$. The $W_{i}$ may represent the weights of valleys in a spin glass (if $0<\alpha<1$), or the frequency of different lineages (families) in a genealogy. This paper considers a population in which there are $N$ individuals reproducing with $\mathrm{Pareto}(\alpha)$ offspring-number distribution ($1<\alpha<2$). The probability of two randomly-chosen individuals being siblings, $Y_{2}=\sum_{i}W_{i}^{2}$, gives the sample mean of the normalized size of families, and its reciprocal gives the effective number of families (or reproducing lineages) in the population, $N_{\mathrm{e}}=1/Y_{2}$. The typical sample mean is very different from the average over all possible samples, i.e. $Y_{2}$ is not a self-averaging quantity. The typical $Y_{2}$ and its reciprocal do not vary with $N$ in opposite ways. Non-self-averaging effects are crucial in understanding genetic diversity in mass spawning species such as marine fishes. Key words and phrases: non-self-averaging; $\alpha$-stable distribution; effective population size; genetic drift; coalescent 1. Introduction Motivated by considering mass spawning species with type-III (exponential) survivorship curve, many authors [1, 2, 3] have been studied patterns of genetic variation within marine populations and obtained convincing results suggesting that reproductive skew among individuals explains the pattern of coalescence of ancestral lineages. While the recruitment process has exponential decay in survival probability, the exponential amplification of the number of matured offspring (or siblings) in a family compensates the exponentially small probability of their surviving to reproductive maturity. The combination of these two exponentials leads to power laws in the offspring-number distribution [4, 5]. Since annual recruitment is calculated by summing random offspring numbers, when the offspring-number distribution is broad, the sum deserves serious consideration, i.e. the system requires two different kinds of averages. One might take the average of relevant properties (or variables) over the whole population existing at any given time. However, these averages may fluctuate in time even for very large populations. One can thus envisage taking the time average of these population averages over a very long time stretch, which, if some sort of ergodic property is assumed, may be represented by the average over all possible realizations of the stochastic reproduction process. Such a stochastic nature of sums of a large number of random variables goes under the name of “lack of self-averaging” [6]. This paper studies the distribution properties of family (or sibship) sizes in the population, where their reproductive success follows a power-law distribution. Let $W_{i}$ be the relative frequency or weight of the $i$-th family (satisfying $\sum_{i}W_{i}=1$). The weight $W_{i}$ is the probability for a given individual to be found in the $i$-th family, so the sum of squared weights of families, $Y_{2}=\sum_{i}W_{i}^{2}$, is the expected weight of the family containing it. The $Y_{2}$ gives the sample mean weight of families, and its reciprocal $N_{\mathrm{e}}{\,}(=Y_{2}^{-1})$ gives the effective number of families (or reproducing lineages) in the population [7]. The effective population size $N_{\mathrm{e}}$ is a measure of how many individuals contribute to the next generation. The $Y_{2}$ also gives the probability of two randomly-chosen individuals being siblings (i.e. the coalescence probability). Write $\rho(w)$ for the distribution of the weights, such that $\rho(w)\differential{w}$ is defined as the average number of families with weights between $w$ and $w+\differential{w}$ among an infinite number of replicate populations each undergoing the same reproduction process. Note that $w\rho(w)$ is the probability that a randomly chosen individual belongs to a family of weight $w$. From the knowledge of $\rho(w)$, I compute the probability distributions of the sample mean ($Y_{2}$) and its reciprocal ($N_{\mathrm{e}}$). Closely related questions concerning the moments of the distribution $\rho(w)$ have been studied in genealogical, or coalescent, models. Schweinsberg [8] built coalescents from a population model with power-law offspring-number distribution. Huillet [9] derived coalescents from sampling from a power-law distribution, including size-biasing on the total recruitment effects. The non-self-averaging effects are present in a large variety of disordered systems in statistical physics, in particular of spin glasses [10, 6]. Derrida and Peliti [11] computed the genealogy statistics under the Wright-Fisher model, and by exploiting the equivalence with spin glasses they showed that similar non-self-averaging effects occur. The random structure of family trees was analyzed to show that the mean distance between individuals (number of generations from the common ancestor) fluctuates on time scale of the order of $N$ (population size) generations [12, 13]. Coalescent processes arise in a natural way from spin-glass models [14], which allows one to make related predictions concerning the non-self-averaging properties of the genealogies of evolving populations [15, 16, 17]. In population genetics the non-self-averaging effects are observed, when considering heterozygosity $H$ over many realizations of the neutral evolutionary process. The standard deviation of $H$, calculated under the Wright-Fisher model, is of the same order of magnitude as the mean for large populations with small mutation rates [18]. The large variations in $H$ are to be expected between different gene loci, which was observed in Drosophila melanogaster [19]. Tajima’s estimator [20] of population-scaled mutation rate ($\propto N_{\mathrm{e}}$) has a similar non-self-averaging property, even when sampling infinitely many loci [21]. In ecology, $H$ is the probability that two randomly selected individuals are of different species. Based on the log-series distribution of the species abundances [22] in a (neutral) community, the variance of $H$ does not go to zero even in the limit of large community size [23]. Usually, demographic stochasticity has effects on small populations. Variances of family-size frequencies in a large population average out, so that such microscopic fluctuations at the individual level may not be extracted from macroscopic (i.e. population-level) measurements (e.g. interannual recruitment variability). This paper explains that, as a consequence of non-self-averaging effects, macroscopic measurements may give access to microscopic fluctuations (and vice versa), which is important for marine ecological applications. This paper is organized as follows. In §2, after providing a population model, I show that the weights of the families are not self-averaging. In §3, the probability distributions of $Y_{2}$ and $N_{\mathrm{e}}$ are computed. In §4, these probability distributions are obtained numerically. In §5, as an application example, I explain how genetic diversity varies with annual recruitment strength. The analysis is based on the asymptotic (i.e. large-population limit) behavior of the model. 2. Pareto sampling 2.1. Population model Consider a population with a fixed number $N$ of individuals reproducing asexually. Each individual $i\,(=1,\ldots,N)$ in a given generation is assigned a random value $X_{i}$ of reproductive success. The variables $X_{1},\ldots,X_{N}$ are independent and identically distributed copies of $X$ with Pareto density $$f_{X}(x)=\alpha x^{-\alpha-1}$$ (1) and cumulative distribution function $F_{X}(x)=1-x^{-\alpha}$ (with $\alpha>0$ and $x\geq 1$). Let $\mu$ be the mean; one has $\mu=\alpha/(\alpha-1)$ when $\alpha>1$, and $\mu=\infty$ when $\alpha\leq 1$. Upon normalizing $X_{1},\ldots,X_{N}$ by their sum $$R_{N}=\sum_{i=1}^{N}X_{i},$$ one defines the weight $W_{i}$ of the term $X_{i}$ in the sum as $$W_{i}=X_{i}/R_{N}$$ ($i=1,\ldots,N$). Each $W_{i}$ gives the probability of reproductive success of individual $i$. Given the population at some generation, for each individual at the following generation, one chooses at random with probability $W_{i}$ one parent $i\in\{1,\ldots,N\}$. Any generation is replaced by a new one. The values $X_{i}$ are drawn afresh in each generation. Note that when the $W_{i}$’s are identical, the sampling procedure is equivalent to Wright-Fisher sampling. The $X_{i}$ gives an analog of the number of potential offspring (i.e. surviving young to reproductive maturity) of individual $i$, and the sum $R_{N}$ corresponds to the annual recruitment, i.e. the total number of offspring entering the (potentially reproductive) population. Then, a fixed number $N$ of reproducing individuals are chosen at random among $R_{N}$ individuals of that generation. When $\alpha>1$, by the law of large numbers, one has $R_{N}>N$. In this paper I mainly consider the case $1<\alpha<2$. 2.2. Domination by the largest term It is well known that $(R_{N}-\mathrm{E}[R_{N}])/N^{1/\alpha}$ with $1<\alpha<2$ has a maximally asymmetric $\alpha$-stable distribution for large $N$ [24], where $\mathrm{E}[R_{N}]=\mu N$. The width of the distribution of the sum $R_{N}$ (i.e. the typical value of the difference $R_{N}-\mathrm{E}[R_{N}]$) is of order $N^{1/\alpha}$, while the variance $\mathrm{E}[R_{N}^{2}]-\mathrm{E}[R_{N}]^{2}$ is infinite. $\mathrm{E}[\,\cdot\,]$ denotes an average over all possible samples (realizations). Define $X_{1,N}\geq X_{2,N}\geq\cdots\geq X_{N,N}$ by ranking in decreasing order the values encountered among the $N$ terms of the sum $R_{N}$. When $1<\alpha<2$, one has [25] $$\displaystyle\mathrm{E}\quantity[X_{1,N}]$$ $$\displaystyle=\frac{N!{\,}\mathrm{\Gamma}(1-1/\alpha)}{\mathrm{\Gamma}(N+1-1/\alpha)}=\mathrm{\Gamma}(1-1/\alpha)N^{1/\alpha}$$ $$\displaystyle\mathrm{E}\quantity[X_{2,N}]$$ $$\displaystyle=\frac{\alpha-1}{\alpha}\mathrm{E}\quantity[X_{1,N}]$$ and while $X_{1,N}$ has an infinite second moment, one has $$\displaystyle\mathrm{E}\quantity[X_{2,N}^{2}]$$ $$\displaystyle=\frac{N!{\,}\mathrm{\Gamma}(2-2/\alpha)}{\mathrm{\Gamma}(N+1-2/\alpha)}=\mathrm{\Gamma}(2-2/\alpha)N^{2/\alpha}$$ $$\displaystyle\mathrm{E}\quantity[X_{1,N}X_{2,N}]$$ $$\displaystyle=\frac{\alpha}{\alpha-1}\mathrm{E}\quantity[X_{2,N}^{2}]$$ for large $N$. The rescaled random variable $X_{1,N}/N^{1/\alpha}$ also has a $\mathrm{Pareto}(\alpha)$ distribution and $X_{1,N}\gtrsim N^{1/\alpha}$ [26, 27]. Importantly, all but the largest order statistics have finite second moment. Therefore, the sum $R_{2,N}{\,}(=\sum_{i=2}^{N}X_{i,N})$ of the $(N-1)$ lower order statistics converges to a normally distributed random variable with first two moments given by $$\displaystyle\mathrm{E}\quantity[R_{2,N}]$$ $$\displaystyle=\frac{\alpha}{\alpha-1}\quantity(N-\mathrm{\Gamma}(2-1/\alpha)N^{1/\alpha})=\mathrm{E}[R_{N}]$$ $$\displaystyle\mathrm{E}\quantity[R_{2,N}^{2}]$$ $$\displaystyle=\mathrm{E}\quantity[\sum_{i=2}^{N}X_{i,N}^{2}]+\mathrm{E}\quantity[\sum_{i\neq j}X_{i,N}X_{j,N}]\sim N^{2/\alpha}$$ where $$\mathrm{E}\quantity[\sum_{i=2}^{N}X_{i,N}^{2}]=\frac{\alpha}{\alpha-2}\quantity(N-\mathrm{\Gamma}(2-2/\alpha)N^{2/\alpha})=\frac{\alpha}{2-\alpha}\mathrm{E}\quantity[X_{2,N}^{2}],$$ and $$\displaystyle\mathrm{E}\quantity[X_{2,N}X_{3,N}]$$ $$\displaystyle=\frac{2\alpha}{2\alpha-1}\mathrm{E}\quantity[X_{3,N}^{2}]$$ $$\displaystyle\mathrm{E}\quantity[X_{2,N}X_{4,N}]$$ $$\displaystyle=\frac{6\alpha^{2}}{(2\alpha-1)(3\alpha-1)}\mathrm{E}\quantity[X_{4,N}^{2}]$$ etc. Accordingly, the statistical variation of the sum $R_{N}$ is dominated by its largest term $X_{1,N}$, so the fraction $(R_{N}-\mathrm{E}[R_{N}])/R_{N}$ can be linked to one parent. The concept of the statistical domination by the largest term is especially useful when describing processes with large deviations, as I show later. 2.3. Moments of weights The fluctuations of the weights $W_{i}$ of the term $X_{i}$ in the sum $R_{N}$ can be described by considering their moments. When the sum of the $k$-th power of weights ($k\geq 0$), $$Y_{k}=\sum_{i=1}^{N}W_{i}^{k},$$ is averaged over the $X_{i}$’s, one gets the moments $\mathrm{E}\quantity[Y_{k}]$. Obviously one has $\mathrm{E}[Y_{0}]=N$ and $\mathrm{E}[Y_{1}]=1$. Using the following identity $$\frac{\sum_{i=1}^{N}X_{i}^{k}}{\quantity(\sum_{j=1}^{N}X_{j})^{k}}=\int_{0}^{\infty}\frac{s^{k-1}\differential{s}e^{-s\sum_{j=1}^{N}X_{j}}}{\mathrm{\Gamma}(k)}\sum_{i=1}^{N}X_{i}^{k}$$ (this is a direct consequence of the Euler integral for the gamma function), one can calculate $\mathrm{E}\quantity[Y_{k}]$ in the large-$N$ limit, $$\mathrm{E}\quantity[Y_{k}]=\frac{N}{\mathrm{\Gamma}(k)}\int_{0}^{\infty}s^{k-1}\differential{s}\mathrm{E}\quantity[e^{-sX_{1}}]^{N-1}\mathrm{E}\quantity[X_{2}^{k}e^{-sX_{2}}],$$ (2) where the integral is dominated by the small $s$ behavior. I refer to [6, 9]. One sees, via integration by parts $\lfloor\alpha\rfloor+1$ times, that for small $s$ $$\mathrm{E}\quantity[e^{-sX}]=\sum_{0\leq k<\alpha}\frac{(-s)^{k}}{{k}!}\mathrm{E}\quantity[X^{k}]-s^{\alpha}\mathrm{\Gamma}(1-\alpha)$$ (3) for $\alpha\neq 1,2,\ldots$, where $\lfloor\cdot\rfloor$ denotes the integer part of the argument. It is easy to check that $$\displaystyle\mathrm{E}\quantity[e^{-sX}]=\int_{1}^{\infty}e^{-sx}f_{X}(x)\differential{x}$$ $$\displaystyle\mbox{ }=\begin{dcases}1-\alpha\int_{0}^{\infty}x^{-1-\alpha}\quantity(1-e^{-sx})\differential{x}&(0<\alpha<1)\\ e^{-s}+s\int_{1}^{\infty}\quantity(F_{X}(x)-1)\differential{x}-s\int_{1}^{\infty}\quantity(F_{X}(x)-1)\quantity(1-e^{-sx})\differential{x}&(\alpha>1)\end{dcases}$$ where the third term on the last line reduces to $$s\int_{0}^{\infty}x^{-\alpha}\quantity(1-e^{-sx})\differential{x}=\frac{s^{2}}{\alpha-1}\int_{0}^{\infty}x^{-\alpha+1}e^{-sx}\differential{x}$$ for $1<\alpha<2$, and to $$\frac{s\quantity(1-e^{-s})}{\alpha-1}+\frac{s^{2}e^{-s}}{(\alpha-1)(\alpha-2)}-\frac{s^{3}}{(\alpha-1)(\alpha-2)}\int_{1}^{\infty}x^{-\alpha+2}e^{-sx}\differential{x}$$ for $\alpha>2$. Take note that, for small $s$, only large values of $x$ contribute to the integral. One views the boundary case $\alpha=1$ (resp. $\alpha=2$, etc.) as the limiting critical case of Eq.(3) as $\alpha\to 1+0$ (resp. $\alpha\to 2+0$, etc.). So one gets $$\mathrm{E}\quantity[e^{-sX}]=1-s+s\ln s$$ for $\alpha=1$, and $$\mathrm{E}\quantity[e^{-sX}]=1-s\mu+\frac{s^{2}}{2}-s^{2}\ln s$$ for $\alpha=2$, etc. Thus, one can see that for $k>\alpha$ $$\mathrm{E}\quantity[X^{k}e^{-sX}]=\alpha s^{\alpha-k}\mathrm{\Gamma}(k-\alpha).$$ When $0<\alpha<1$, Eq.(2) gives the equation (11) of [6]. When $1\leq\alpha<2$, one obtains for $k\geq 2$ $$\mathrm{E}\quantity[Y_{k}]=c_{\scalebox{0.55}{$N$}}\frac{\mathrm{Beta}(k-\alpha,\alpha)}{\mathrm{Beta}(2-\alpha,\alpha)}$$ with scaling constant $$c_{\scalebox{0.55}{$N$}}=\begin{dcases}\frac{\alpha\mathrm{Beta}(2-\alpha,\alpha)}{\mu^{\alpha}N^{\alpha-1}}&(1<\alpha<2)\\ \quantity(\ln N)^{-1}&(\alpha=1)\end{dcases}$$ where $\mathrm{Beta}(a,b)=\mathrm{\Gamma}(a)\mathrm{\Gamma}(b)/\mathrm{\Gamma}(a+b)$ is the beta function. When $\alpha\geq 2$, one gets $$\displaystyle\mathrm{E}\quantity[Y_{k}]=$$ $$\displaystyle\frac{\alpha\mathrm{Beta}(k-\alpha,\alpha)}{\mu^{\alpha}N^{\alpha-1}}$$ ($$\alpha<k$$) $$\displaystyle\mathrm{E}\quantity[Y_{k}]=$$ $$\displaystyle\frac{\mathrm{E}[X^{k}]}{\mu^{k}N^{k-1}}$$ ($$2\leq k<\alpha$$) (4) and $$\mathrm{E}\quantity[Y_{\alpha}]=\frac{\alpha\ln N}{\mu^{\alpha}N^{\alpha-1}}$$ (5) for integer $\alpha=2,3,\ldots$. The correlations between the $Y_{k}$’s can also be calculated as $$\displaystyle\mathrm{E}\quantity[Y_{k}Y_{\ell}]$$ $$\displaystyle=\mathrm{E}\quantity[\sum_{i=1}^{N}W_{i}^{k+\ell}]+\mathrm{E}\quantity[\sum_{i\neq j}W_{i}^{k}W_{j}^{\ell}]$$ $$\displaystyle=\frac{N}{\mathrm{\Gamma}(k+\ell)}\int_{0}^{\infty}s^{k+\ell-1}\differential{s}\mathrm{E}\quantity[e^{-sX_{1}}]^{N-1}\mathrm{E}\quantity[X_{2}^{k+\ell}e^{-sX_{2}}]$$ $$\displaystyle{\quad}+\frac{N^{2}}{\mathrm{\Gamma}(k+\ell)}\int_{0}^{\infty}s^{k+\ell-1}\differential{s}\mathrm{E}\quantity[e^{-sX_{1}}]^{N-2}\mathrm{E}\quantity[X_{2}^{k}e^{-sX_{2}}]\mathrm{E}\quantity[X_{3}^{\ell}e^{-sX_{3}}].$$ When $1\leq\alpha<2$, one obtains the $(k,\ell)$-th moment ($k,\ell\geq 2$) of weights of two different families, $$\mathrm{E}\quantity[\sum_{i\neq j}W_{i}^{k}W_{j}^{\ell}]=\mathrm{E}\quantity[Y_{k}]\mathrm{E}\quantity[Y_{\ell}]\frac{\mathrm{Beta}(k,\ell)}{\mathrm{Beta}(\alpha,\alpha)}.$$ (6) Therefore, the correlations $\mathrm{E}\quantity[Y_{k}Y_{\ell}]$ dominate over the factorized terms $\mathrm{E}\quantity[Y_{k}]\mathrm{E}\quantity[Y_{\ell}]$. When $0<\alpha<1$, the correlations $\mathrm{E}\quantity[Y_{k}Y_{\ell}]$, as well as the moments $\mathrm{E}\quantity[Y_{k}]$, depend only on $\alpha$, $k$ and $\ell$, and become independent of $N$; see the equation (12) of [6]. 2.4. Fluctuation dominance To characterize fluctuations in the weights of families, consider the relative fluctuations (i.e. coefficient of variation) of the $Y_{2}$, $$\mathrm{CV}[Y_{2}]=\sqrt{\mathrm{E}[(Y_{2}-\mathrm{E}[Y_{2}])^{2}]}/\mathrm{E}[Y_{2}].$$ When $1<\alpha<2$, one has $$\mathrm{CV}[Y_{2}]=c_{\scalebox{0.55}{$N$}}^{-1/2}\sqrt{(3-\alpha)(2-\alpha)/6},$$ which diverges in $\order{N^{(\alpha-1)/2}}$. There is no approximately deterministic property of the $Y_{2}$-distribution at large population sizes, as the fluctuations of the $Y_{2}$ dominate the average value. Each realization of the $Y_{2}$ may be very different from its other realizations. The fact that $\mathrm{E}\quantity[Y_{k}^{2}]/\mathrm{E}\quantity[Y_{k}]^{2}=\order{N^{\alpha-1}}$ indicates that the $Y_{k}$’s for $k\geq 2$ are non-self-averaging. When $\alpha\geq 2$, one sees that $\mathrm{CV}[Y_{2}]=\order{N^{(3-\alpha)/2}}$ for $2<\alpha<4$, or $\mathrm{CV}[Y_{2}]=\order{\sqrt{N}/\ln N}$ for $\alpha=2$. Although one ignores the probability that three or more randomly chosen individuals are siblings, i.e. $\lim_{N\to\infty}\mathrm{E}[Y_{k}]/\mathrm{E}[Y_{2}]=0$ for all $k\geq 3$, one cannot ignore the relative fluctuations in the probability of being siblings when $2\leq\alpha\leq 3$. Then, the probability of having four siblings from a family is greater than or equal to the probability of having two pairs of siblings from two different families. 2.5. Distribution of the weights of families Since all the moments of weights are known, the function $\mathrm{E}[Y_{k}]^{-1}w^{k}\rho(w)$ with $k\geq 2$ is known [28], which is the $\mathrm{Beta}(k-\alpha,\alpha)$ distribution on $0\leq w\leq 1$. For $1\leq\alpha<2$, the distribution $$\rho(w)=c_{\scalebox{0.55}{$N$}}\frac{w^{-\alpha-1}(1-w)^{\alpha-1}}{\mathrm{Beta}(2-\alpha,\alpha)}$$ (7) diverges like $w^{-\alpha-1}$ for small $w>0$. Although there are a large number of very small families, the expected number of siblings of an individual, $Nc_{\scalebox{0.55}{$N$}}$, grows as $N^{2-\alpha}$ for large $N$. The functions $\rho(w)$ and $w\rho(w)$ are not integrable on $0\leq w\leq 1$. If introducing a cut-off $$\varepsilon_{\scalebox{0.55}{$N$}}=(\mu N)^{-1}$$ (8) in the region of small $w$, the total number of families having a weight larger than $\varepsilon_{\scalebox{0.55}{$N$}}$ is $$\int_{\varepsilon_{\scalebox{0.45}{$N$}}}^{1}\rho(w)\differential{w}=\frac{c_{\scalebox{0.55}{$N$}}{\,}\varepsilon_{\scalebox{0.55}{$N$}}^{-\alpha}}{\alpha\mathrm{Beta}(2-\alpha,\alpha)}=N,$$ and their total weight is $$\int_{\varepsilon_{\scalebox{0.45}{$N$}}}^{1}w\rho(w)\differential{w}=\frac{c_{\scalebox{0.55}{$N$}}{\,}\varepsilon_{\scalebox{0.55}{$N$}}^{1-\alpha}}{(\alpha-1)\mathrm{Beta}(2-\alpha,\alpha)}=1,$$ as $N\to\infty$. Let $\rho(w,w^{\prime})$ be the joint distribution (or the correlation function) of weights of two families, $$\rho(w,w^{\prime})=\rho(w)\delta\quantity(w-w^{\prime})+\rho^{\ast}(w,w^{\prime}).$$ The second term captures the average number of pairs of different families having weights $w$ and $w^{\prime}$, $$\rho^{\ast}(w,w^{\prime})=c_{\scalebox{0.55}{$N$}}^{2}\frac{(ww^{\prime})^{-\alpha-1}(1-w-w^{\prime})^{2\alpha-1}}{\mathrm{Beta}(2-\alpha,\alpha)^{2}}{\,}\Theta\quantity(1-w-w^{\prime}),$$ which is extracted from Eq.(6) for $1\leq\alpha<2$, where $\Theta(\cdot)$ is the Heaviside step function. I refer to [10, 29]. The correlated probability of finding two families of weights $w$ and $w^{\prime}$ is given by $ww^{\prime}\rho(w,w^{\prime})$. For $\alpha\geq 2$, while when $2\leq k\leq\alpha$, the functions $w^{k}\rho(w)$ with $$\rho(w)=\frac{\alpha w^{-\alpha-1}(1-w)^{\alpha-1}}{\mu^{\alpha}N^{\alpha-1}}$$ (9) is not integrable on $0\leq w\leq 1$, one sees that $$\int_{\varepsilon_{\scalebox{0.45}{$N$}}}^{1}w^{k}\rho(w)\differential{w}=\mathrm{E}\quantity[Y_{k}]$$ with $\mathrm{E}[Y_{k}]$ as in Eq.(4) or (5). Further, the function $\mathrm{E}[Y_{2}]^{-1}w^{2}\rho(w)$ reduces to the Dirac delta function, as one sees as follows. Define a function $$\delta_{N}(w)=\mathrm{E}[Y_{2}]^{-1}w^{2}\rho(w)\Theta(w-\varepsilon_{\scalebox{0.45}{$N$}})$$ with $\rho(w)$ as in Eq.(9). When $N\to\infty$, the $\delta_{N}(w)$ approximates the Dirac delta distribution at zero. The joint distribution of the sequence $\quantity{W_{1},\ldots,W_{N}}$, in the case $0<\alpha<1$, has been studied in mathematics and physics. The sequence of $W_{1},\ldots,W_{N}$ in decreasing order, as $N\to\infty$, has the two-parameter Poisson-Dirichlet distribution with parameters $(\alpha,0)$ [30]. The distribution $\rho(w)$ and the joint distributions $\rho(w,w^{\prime}),\rho(w,w^{\prime},w^{\prime\prime}),\ldots$ of weights of two or more families are derived in [10]. Although families are mostly concentrated around $w=0$, these $w\simeq 0$ families do not contribute to the total weight, as $$\int_{0}^{w}w^{\prime}\rho(w^{\prime})\differential{w^{\prime}}\sim w^{1-\alpha}$$ with $\rho(w)$ given by the equation (34) of [10]. So any one of them has an extremely small weight. The expected number of siblings of an individual is $(1-\alpha)N$ and there are a few families with weights of $\order{1}$ in $1/N$. 3. Distributions of $Y_{2}$ and $N_{\mathrm{e}}$ Given the weights of families, $W_{1},\ldots,W_{N}$, write $W_{1.N}$ and $W_{2,N}$ for the weights of the largest and second largest families. Let $\mathrm{\Pi}_{W_{1,N}}(w)$ (resp. $\mathrm{\Pi}_{W_{2,N}}(w)$) be the probability of the largest family (resp. the second largest family) having a weight $w$. These probability distributions can be computed from the knowledge of the distribution $\rho(w)$ and the joint distributions of weights; refer to [31, 32]. If there is a family with weight $w>1/2$, this weight must be the largest one, and thus $\mathrm{\Pi}_{W_{1,N}}(w)=\rho(w)$ for $w>1/2$. Letting $\rho^{\ast}(w_{1},\ldots,w_{n})$ be a joint distribution of $n{\;(\geq 2)}$ weights of different families, and denoting $$I_{n}(w)=\int_{w}^{1}\differential{v_{1}}\int_{w}^{v_{1}}\differential{v_{2}}\cdots\int_{w}^{v_{n-2}}\differential{v_{n-1}}\rho^{\ast}(v_{1},v_{2},\ldots,v_{n-1},w),$$ one has, in the interval $1/(n+1)<w<1/n$, $$\mathrm{\Pi}_{W_{1,N}}(w)=\rho(w)-I_{2}(w)+\cdots+(-1)^{n-1}I_{n}(w)$$ and $$\mathrm{\Pi}_{W_{2,N}}(w)=\int_{w}^{1}\differential{v}\rho^{\ast}(v,w)-2I_{3}(w)+\cdots+(-1)^{n-2}(n-1)I_{n}(w).$$ The successive terms in these summations rapidly decrease for large $N$, because $I_{n}(w)$ is an integral over more and more variables and diminishes in magnitude with increasing $n$. Consider the case $1<\alpha<2$. Since $W_{1,N}\simeq X_{1,N}/(\mu N)$, one has $W_{1,N}\gtrsim N^{1/\alpha-1}$. Accordingly, one has, up to the leading term, $$\mathrm{\Pi}_{W_{1,N}}(w)=\rho(w)$$ (10) on $N^{1/\alpha-1}\lesssim w\leq 1$, and $$\mathrm{\Pi}_{W_{2,N}}(w)=\int_{w}^{1}\differential{v}\rho^{\ast}(v,w)=\frac{c_{\scalebox{0.55}{$N$}}^{2}{\,}w^{-2\alpha-1}(1-w)^{2\alpha-1}}{\alpha\mathrm{Beta}(2-\alpha,\alpha)}$$ (11) on $N^{1/\alpha-1}\lesssim w\leq 1$. Integrating Eq.(10), one gets $$\int_{N^{1/\alpha-1}/\mu}^{1}\mathrm{\Pi}_{W_{1,N}}(w)\differential{w}=1.$$ One then sees that the the second moment of the largest weight $$\int_{N^{1/\alpha-1}/\mu}^{1}w^{2}\mathrm{\Pi}_{W_{1,N}}(w)\differential{w}=c_{\scalebox{0.55}{$N$}}$$ for large $N$, which demonstrates that the sum of the squared weights of families, $Y_{2}=\sum_{i=1}^{N}W_{i}^{2}$, is dominated by its largest term $W_{1,N}$. Now compute the probability of the random variable $Y_{2}$ taking a value $y$. The probability distribution $\mathrm{\Pi}_{Y_{2}}(y)$ of $Y_{2}$ is defined as $$\mathrm{\Pi}_{Y_{2}}(y)=\mathrm{E}\quantity[\delta\quantity(\sum_{i=1}^{N}W_{i}^{2}-y)].$$ Since the sums which contribute to the average are those in which one term dominates, one may replace the sum with its maximal summand [10] $$\mathrm{\Pi}_{Y_{2}}(y)=\mathrm{E}\quantity[\delta\quantity(W_{1,N}^{2}-y)].$$ From the probability density $\mathrm{\Pi}_{W_{1,N}}(w)$ of the largest weight, one obtains $$\mathrm{\Pi}_{Y_{2}}(y)=\int_{N^{1/\alpha-1}/\mu}^{1}\differential{w}\mathrm{\Pi}_{W_{1,N}}(w)\delta\quantity(w^{2}-y)=\frac{c_{\scalebox{0.55}{$N$}}y^{-\alpha/2-1}\quantity(1-\sqrt{y})^{\alpha-1}}{2\mathrm{Beta}(2-\alpha,\alpha)}$$ (12) on $N^{2(1/\alpha-1)}\lesssim y\leq 1$. Integrating the right side of Eq.(12) gives $$\int_{N^{2(1/\alpha-1)}/\mu^{2}}^{1}\mathrm{\Pi}_{Y_{2}}(y)\differential{y}=1,$$ and one sees that $$\int_{N^{2(1/\alpha-1)}/\mu^{2}}^{1}y{\,}\mathrm{\Pi}_{Y_{2}}(y)\differential{y}=c_{\scalebox{0.55}{$N$}}.$$ The most probable value of $Y_{2}$ is close to $N^{2(1/\alpha-1)}$, and therefore differs from its mean value $c_{\scalebox{0.55}{$N$}}\,(\sim N^{1-\alpha})$. The same kind of argument applies to the probability distribution $\mathrm{\Pi}_{N_{\mathrm{e}}}(y)$ of $N_{\mathrm{e}}=Y_{2}^{-1}$, yielding $$\mathrm{\Pi}_{N_{\mathrm{e}}}(y)=\frac{c_{\scalebox{0.55}{$N$}}y^{\alpha/2-1}\quantity(1-1/\sqrt{y})^{\alpha-1}}{2\mathrm{Beta}(2-\alpha,\alpha)}$$ (13) for $1\leq y\lesssim N^{2(1-1/\alpha)}$. Integrating the right side of Eq.(13) gives $$\int_{1}^{\mu^{2}N^{2(1-1/\alpha)}}\mathrm{\Pi}_{N_{\mathrm{e}}}(y)\differential{y}=1.$$ The function $\mathrm{\Pi}_{N_{\mathrm{e}}}(y)$ must break at $y\sim N^{2(1-1/\alpha)}$ and will decay rapidly when $y\to\infty$. Moreover, the most probable value of $N_{\mathrm{e}}$ is $(2-\alpha)^{-2}$ and very different from the harmonic mean of $N_{\mathrm{e}}$’s ($c_{\scalebox{0.55}{$N$}}^{-1}\sim N^{\alpha-1}$) over replicate populations. Remark. While one might naively expect an increase of the typical $N_{\mathrm{e}}$ with $N$, the typical $N_{\mathrm{e}}$ shows no increase with population size $N$. The paper presents the striking scaling behavior of the sample mean reciprocal. The $Y_{2}$-distribution is of reciprocal symmetry breaking, in the sense that the typical sample mean and its reciprocal do not vary with population size in opposite ways. The typical $Y_{2}$ decreases with the population size like $N^{2(1/\alpha-1)}$, while the typical reciprocal of $Y_{2}$ is independent of $N$. Such counter-intuitive behavior emerges in some broad distributions [33]. 4. Numerical reconstruction of probability distributions I use a trick given in [31, 32] to generate distributions of $W_{1,N}$, $W_{2,N}$, and $Y_{2}$ for $1<\alpha<2$. Suppose one has a set $\quantity{W_{1},\ldots,W_{N-1}}$ of $(N-1)$ weights of families and a corresponding value of $Y_{2}$. Let $W_{\mathrm{max}}$ and $W^{\prime}_{\mathrm{max}}$ be the two largest values of this set of weights. One can add another family of weight $W_{N}$ with probability distribution $N^{-1}\rho(w)$ where $\rho(w)$ is as in Eq.(7), and simultaneously shrink all the other weights by a factor $(1-W_{N})$. One now has a new set with a new value given by the recursion relation $$\tilde{Y}_{2}=W_{N}^{2}+(1-W_{N})^{2}Y_{2}.$$ (14) If $\tilde{W}_{\mathrm{max}}$ and $\tilde{W}^{\prime}_{\mathrm{max}}$ are the two largest weights of the new set, then one has the recursions $$\displaystyle\tilde{W}_{\mathrm{max}}$$ $$\displaystyle=\max\quantity(W_{N},(1-W_{N})W_{\mathrm{max}})$$ $$\displaystyle\tilde{W}^{\prime}_{\mathrm{max}}$$ $$\displaystyle=\max\quantity(\min\quantity(W_{N},(1-W_{N})W_{\mathrm{max}}),(1-W_{N})W^{\prime}_{\mathrm{max}}).$$ I just calculate a random sequence $\quantity{y_{k}}$ ($k=0,1,2,\ldots$) with $y_{0}$ being randomly chosen between 0 and 1, where the recursion relation which gives the $y_{k}$’s is $$y_{k+1}=W_{k}^{2}+(1-W_{k})^{2}y_{k}$$ (15) and where the $W_{k}$’s are randomly chosen between $\varepsilon_{\scalebox{0.55}{$N$}}$ and 1 according to $\rho(w)$, where $\varepsilon_{\scalebox{0.55}{$N$}}$ is as in Eq.(8). The $y_{k}$’s (resp. $y_{k}^{-1}$’s), generated by this iterative procedure, are distributed according to $\mathrm{\Pi}_{Y_{2}}$ (resp. $\mathrm{\Pi}_{N_{\mathrm{e}}}$). One can also construct in the same way sequences of $w_{k}$ and $w^{\prime}_{k}$ by iterative procedures $$\displaystyle w_{k+1}$$ $$\displaystyle=\max\quantity(W_{k},(1-W_{k})w_{k})$$ (16) $$\displaystyle w^{\prime}_{k+1}$$ $$\displaystyle=\max\quantity(\min\quantity(W_{k},(1-W_{k})w_{k}),(1-W_{k})w^{\prime}_{k})$$ (17) where again the $W_{k}$’s are distributed according to $\rho(w)$. By drawing the histograms of $w_{k}$’s, $w^{\prime}_{k}$’s, $y_{k}$’s and $y_{k}^{-1}$’s, one can get the probabilities $\mathrm{\Pi}_{W_{1,N}}$, $\mathrm{\Pi}_{W_{2,N}}$, $\mathrm{\Pi}_{Y_{2}}$ and $\mathrm{\Pi}_{N_{\mathrm{e}}}$, respectively. Fig. 1 shows the results obtained after $2\times 10^{8}$ iterations with parameters being set to $\alpha=1.2$, and with populations of $N=10^{4}$, $10^{5}$ and $10^{6}$. The simulation results of the Pareto sampling are also shown in Fig. 1, where $X_{i}$’s ($N=10^{6}$ trials) are drawn from the Pareto distribution (Eq. 1) with $\alpha=1.2$, and the histograms of $W_{1,N}$’s, $W_{2,N}$’s, $Y_{2}$’s and $Y_{2}^{-1}$’s are obtained from $10^{5}$ independent realizations. 5. Genetic variation in marine populations Highly fecund marine species undergo large and intermittent fluctuations in recruitment [34]. Very large interfamilial, or sweepstakes, variation in reproductive success has been documented [35]. While sweepstakes reproduction appears to be prevalent in marine systems, it is somewhat paradoxical to have observed the absence of reduced genetic diversity from a single reproduction event [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]. Even a species that showed evidence for sweepstakes in one place may not show evidence for it at another time or in another place [47, 48]. This limited evidence can be attributed to variation in intensity of sweepstakes reproduction across time, that is, “the right place, but the wrong time” as stated in [35], but it has not yet been fully proved or explained. As an application example, the concept of the statistical domination by the largest term is used to assess an increasing reduction in genetic variation at larger recruitment $R_{N}$. $(N-1)$ lower order statistics, $X_{2,N}\geq\cdots\geq X_{N,N}$, of $N$ independent $\mathrm{Pareto}(\alpha)$ random variables with $1<\alpha<2$ have finite first and second moments. Letting $U_{2}$ be the sum of squared weights of these $(N-1)$ lower order statistics, $$U_{2}=\sum_{i=2}^{N}\quantity({X_{i,N}}/{R_{2,N}})^{2},$$ one has, from Eq.(14), $$\displaystyle Y_{2}$$ $$\displaystyle=W_{1,N}^{2}+(1-W_{1,N})^{2}U_{2}$$ (18) $$\displaystyle=\quantity(1-\frac{\mu N}{R_{N}})^{2}+\quantity(\frac{\mu N}{R_{N}})^{2}\frac{(\alpha-1)^{2}\mathrm{\Gamma}(2-2/\alpha)N^{2(1/\alpha-1)}}{\alpha(\alpha-2)}$$ with the second term being negligible. Eq.(18) agrees with the result from the simulation of random Pareto sampling (performed in §4), as shown in Fig. 2. The typical value of $Y_{2}/c_{\scalebox{0.55}{$N$}}$ is of order $N^{(\alpha-1)(\alpha-2)/\alpha}$ ($=10^{-0.8}$ with $\alpha=1.2$ and $N=10^{6}$). One sees the non-self-averaging effect, that is, $Y_{2}$ depends on the recruit sample. A large reduction in genetic variation will occur in years of large recruitment. 6. Conclusions When tracing a history of realizations of family-size frequencies in a population with $\mathrm{Pareto}(\alpha)$ offspring-number distribution of index $1<\alpha<2$, one may take the average of the weights of families over the whole population existing at any given time. The statistical domination by the largest family leads to large fluctuations in the population average. So the non-self-averaging behavior emerges. I have studied the fluctuations for the largest and second largest weights of families, and the fluctuations for the average weight $Y_{2}$ and for its reciprocal (i.e. the effective population size $N_{\mathrm{e}}$). I have obtained asymptotic expressions for the probability distributions of these quantities. The most probable value of the $Y_{2}$ is close to the typical value of squared weight of the largest family ($\sim N^{2/\alpha-2}$), and differs from its mean value $\mathrm{E}[Y_{2}]{\,}(\sim N^{1-\alpha})$ of the reproduction process. 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11institutetext: Computing Science and Engineering, Indraprastha Institute of Information Technology, New Delhi 22institutetext: School of Computing Science and Engineering, Vellore Institute of Technology, Vellore 33institutetext: Centre for Biomedical Engineering, Indian Institute of Technology, New Delhi. Unveiling Emotions from EEG: A GRU-Based Approach Sarthak Johari 11    Gowri Namratha Meedinti 22    Radhakrishnan Delhibabu 22    Deepak Joshi 33 Abstract Emotion recognition from EEG data is a vital research domain in affective computing. This study investigates the effectiveness of the Gated Recurrent Unit (GRU) algorithm, a variant of Recurrent Neural Networks (RNNs), in predicting emotional states using EEG signals. Our publicly available dataset consists of EEG recordings from participants exposed to stimuli eliciting positive, neutral, and negative emotions, alongside resting neutral data. We preprocess the EEG data using artifact removal, bandpass filters, and normalization techniques for optimal feature extraction. Leveraging the GRU’s ability to capture temporal dependencies, our model achieved impressive results with perfect accuracy on the validation set. Comparing our GRU model with other machine learning approaches, the Extreme Gradient Boosting Classifier achieved the highest accuracy. Our analysis using the confusion matrix unveiled valuable insights into model performance, facilitating accurate emotion classification. This research contributes to the advancement of affective computing and highlights the potential of deep learning models like GRUs in emotion recognition. Our findings offer novel avenues for human-computer interaction and understanding emotional states from brainwave activity. Keywords: Emotion recognition, EEG data, Gated Recurrent Unit (GRU), Affective computing, Deep learning, Machine learning, Temporal dependencies, Brainwave activity, Human-computer interaction. 1 Introduction Emotion recognition from EEG data has emerged as a promising area of research in the fields of affective computing, neuroscience, and human-computer interaction. The ability to accurately identify and understand human emotions based on brainwave activity offers vast potential for enhancing various applications, such as virtual reality experiences, mental health monitoring, and adaptive human-robot interactions. Traditionally, emotion recognition methods relied on behavioral and physiological cues, such as facial expressions, voice tone, and heart rate variability. While these approaches provided valuable insights, they often lacked the precision and direct access to the underlying neural processes responsible for generating emotions. EEG, as a non-invasive and portable neuroimaging technique, bridges this gap by directly measuring the brain’s electrical activity. Over the years, researchers have explored different methods to decode emotional states from EEG data. Classical machine learning techniques, such as Support Vector Machines (SVM) and Random Forests, were initially employed to classify emotions based on EEG features. These methods often required handcrafted feature extraction, which could limit their ability to capture complex temporal dynamics inherent in EEG data. With the advent of deep learning, Recurrent Neural Networks (RNNs) brought significant improvements to emotion recognition from sequential data. RNNs introduced the concept of memory cells, allowing the model to retain information over time and learn long-term dependencies within the data. This led to better emotion recognition performance compared to traditional machine learning approaches. The Gated Recurrent Unit (GRU) algorithm, a variant of RNNs, further enhanced the capabilities of emotion recognition from EEG data. GRUs introduced specialized gating mechanisms that regulate the flow of information within the network. The update gate controls how much past information should influence future predictions, while the reset gate selectively resets or forgets certain knowledge. The current memory gate, often overlooked, adds non-linearity and normalization to the input, improving the model’s ability to capture relevant features. The unique architecture of GRUs addresses the vanishing and exploding gradient problem encountered in standard RNNs, making them more effective in capturing long-term dependencies within EEG signals. Moreover, GRUs are computationally efficient, allowing for faster training and prediction times compared to more complex models like Long Short-Term Memory (LSTM) networks. The advantage of using GRUs for emotion recognition lies in their ability to process time-series EEG data directly, avoiding the need for handcrafted feature engineering. By learning from the raw EEG signals, GRUs can automatically capture intricate temporal patterns associated with different emotional states. This self-learning capability contributes to the model’s adaptability and robustness across various emotion recognition tasks. Emotion recognition from EEG data has seen significant advancements with the introduction of deep learning techniques, particularly the Gated Recurrent Units. The ability of GRUs to capture temporal dependencies and learn from raw EEG signals has opened new possibilities for creating more empathetic and emotionally aware human-computer interfaces. As the research in this domain progresses, we can anticipate even more accurate and nuanced emotion recognition systems that truly understand and respond to human emotions. 2 Related Work Koelstra et al. (2012) present the DEAP database, a multimodal dataset for analyzing human affective states [1]. The dataset comprises EEG and peripheral physiological signals recorded from 32 participants while watching 40 one-minute music video excerpts. Participants rated the videos based on arousal, valence, like/dislike, dominance, and familiarity levels. The paper proposes a novel method for stimuli selection using affective tags from the last.fm website and online assessment tools. It also performs an extensive analysis of participants’ ratings and investigates correlations between EEG signal frequencies and emotional ratings. Liu et al. (2011) focus on real-time EEG-based emotion recognition[2]. The study utilizes EEG signals to classify human emotions and visualize emotional states in real-time. The proposed system demonstrates the feasibility of using EEG for emotion recognition tasks and provides insights into the potential applications of EEG-based emotion analysis.Chanel et al. (2006) explore emotion assessment using EEG and peripheral physiological signals [3]. The research investigates arousal evaluation based on these modalities and demonstrates the feasibility of using EEG signals in conjunction with peripheral physiological measurements for emotion recognition. Liu et al. (2010) focus on real-time mental stress recognition using EEG data [4]. The study presents a system that can detect and recognize mental stress from EEG signals in real-time. The proposed approach demonstrates the potential of EEG-based emotion recognition in practical applications related to stress management and mental health assessment. Koelstra et al. (2012) focus on single-trial classification of emotions induced by music videos using EEG and peripheral physiological signals [5]. The study presents methods and results for classifying arousal, valence, and like/dislike ratings based on the modalities of EEG, peripheral physiological signals, and multimedia content analysis. The research highlights the potential of combining multiple modalities for more robust emotion recognition. Alarcao and Fonseca (2017) investigate emotions recognition in EEG signals during a virtual reality simulation of an emergency evacuation [6]. The study explores the potential of using EEG-based emotion recognition in virtual reality environments for assessing users’ emotional states during critical scenarios. Shi et al. (2019) focus on real-time emotion recognition using deep learning from EEG signals [7]. The study presents a deep learning-based approach for detecting and classifying emotions in real-time based on EEG data. The proposed method demonstrates the effectiveness of deep learning in handling EEG-based emotion recognition tasks. Zhang et al. (2019) provide an overview of emotion recognition using EEG signals [8]. The survey paper reviews various methods and approaches used in EEG-based emotion recognition, highlighting the advancements and challenges in this field. The paper also discusses potential applications and future directions for EEG-based emotion analysis. El Ayadi et al. (2011) provide valuable insights into emotion recognition using other modalities, including speech [9]. Although not focused on EEG-based emotion recognition, this survey paper reviews various features, classification schemes, and databases used in speech emotion recognition, offering valuable information for understanding multimodal emotion recognition. Guo et al. (2018) explore emotion recognition from EEG signals using multimodal deep learning [10]. The study proposes a novel approach that combines EEG data with other modalities to enhance emotion recognition accuracy. The research demonstrates the potential benefits of integrating EEG-based emotion recognition with other modalities for more robust emotion classification. Cho et al. (2014) propose a novel approach for learning phrase representations using RNN Encoder-Decoder models, which include GRUs, for statistical machine translation[11]. The research introduces an end-to-end architecture for mapping variable-length input sequences to variable-length output sequences, making it suitable for natural language processing tasks. The experimental results demonstrate the effectiveness of GRUs in capturing semantic information and generating accurate translations. Li et al. (2018) explore the use of EEG data for emotion recognition using 3D Convolutional Neural Networks (CNNs) with GRU layers[12]. The research investigates the fusion of spatial and temporal features extracted from EEG signals for improved emotion classification. The results show that the combination of 3D CNNs and GRUs outperforms traditional machine learning methods in EEG-based emotion recognition tasks. Wen et al. (2018) conduct a comparative study on deep learning methods, including GRUs, for EEG-based emotion recognition[13]. The research evaluates the performance of various deep learning architectures and explores the impact of different feature extraction techniques on emotion classification accuracy. The findings highlight the advantage of GRUs in capturing temporal dynamics and extracting meaningful features from EEG data. Zheng et al. (2020) present a comprehensive review of deep learning methods for emotion recognition from EEG signals[14]. The paper includes an analysis of different neural network architectures, including GRUs, and their application in emotion recognition tasks. The review highlights the strengths and limitations of using GRUs in EEG-based emotion recognition and discusses potential future research directions in this area. Huang et al. (2014) propose an emotion recognition approach using Hidden Markov Models (HMMs) with GRU-based feature extraction from EEG data [15]. The research investigates the use of GRUs as a feature learning method for capturing sequential patterns in EEG signals. The results demonstrate the effectiveness of the proposed method in recognizing emotions from EEG data. Li et al. (2020) introduce a hybrid network that combines GRUs and Long Short-Term Memory (LSTM) units for EEG-based emotion recognition [16]. The research investigates the complementary strengths of GRUs and LSTMs in capturing temporal dependencies and modeling long-range dependencies in EEG data. The hybrid network demonstrates improved performance in emotion classification compared to individual GRU or LSTM models. Lv et al. (2018) propose an improved LSTM-based model for EEG-based emotion recognition [17]. The research enhances the LSTM model with attention mechanisms to focus on salient EEG signal segments relevant to emotion classification. The study compares the improved LSTM model with GRUs and other LSTM variants and shows promising results in emotion recognition accuracy. Li et al. (2019) propose an improved GRU-RNN model for EEG-based emotion recognition that combines temporal-spatial domain features [18]. The research investigates the combination of spatial and temporal EEG information to enhance emotion classification accuracy. The results show that the improved GRU-RNN model outperforms other traditional machine learning methods and demonstrates the potential of GRUs in capturing both spatial and temporal patterns in EEG data. Wang et al. (2021) propose a deep convolutional neural network (CNN) with GRU layers for emotion recognition from EEG data [19]. The research combines the advantages of CNNs in spatial feature extraction with the temporal modeling capabilities of GRUs to improve emotion classification performance. The experimental results demonstrate that the CNN-GRU model achieves competitive accuracy compared to other state-of-the-art methods. Yao et al. (2020) propose a hybrid model that combines Convolutional Neural Networks (CNNs) and GRUs for EEG-based emotion recognition [20]. The research investigates the fusion of spatial and temporal information from EEG signals using CNNs and GRUs, respectively. The results demonstrate the synergy between CNNs and GRUs, leading to improved emotion classification accuracy in EEG data. These studies highlight the diverse applications of GRU in various fields, including natural language processing, image captioning, emotion recognition from EEG data, and time-series data processing. The research demonstrates the efficacy of GRUs in capturing temporal dependencies, modeling complex patterns, and improving the performance 2.1 Correlation between Emotion Recognition from EEG and GRU The literature survey reveals that emotion recognition from EEG data has been explored using various machine learning techniques, such as support vector machines and time delay neural networks. While these methods have shown promising results, they often struggle to capture the complex temporal dependencies present in EEG signals, which are crucial for accurate emotion classification. The emergence of deep learning approaches, particularly the Gated Recurrent Unit (GRU) algorithm, has offered a solution to address the temporal modeling challenges in EEG-based emotion recognition. GRU, as a variant of Recurrent Neural Networks (RNNs), has gained attention for its ability to efficiently process time-series data and capture long-term dependencies within sequential information. The advantage of GRU lies in its gated architecture, which enables it to selectively retain relevant past information while discarding irrelevant data. This unique property makes GRU well-suited for emotion recognition tasks where temporal context is essential for understanding emotional states encoded in brainwave signals. When considering both emotion recognition from EEG data and the GRU algorithm together, a synergistic relationship emerges. By incorporating GRU into the emotion recognition pipeline, it becomes possible to enhance feature extraction from preprocessed EEG data and improve the prediction accuracy of emotional states. GRU’s ability to capture context and temporal relationships within EEG signals complements the requirements of emotion recognition, where subtle changes in brainwave patterns are indicative of different emotional states. Leveraging GRU’s capability, the emotion recognition system gains a deeper understanding of the underlying dynamics of emotional responses encoded in brainwave signals, leading to more accurate and robust emotion classification models. Overall, the literature survey demonstrates that combining emotion recognition from EEG data with the GRU algorithm offers promising prospects for advancing emotion recognition tasks. As more research continues to explore the correlation between EEG-based emotion recognition and deep learning techniques like GRU, we can expect further advancements in real-time emotion recognition applications across various domains. In Table. 2. shows the EEG-Emotion Based survey in the appendix section. The correlation between emotion recognition from EEG data and GRU lies in the potential of using GRU as a deep learning algorithm to capture the temporal dynamics and dependencies present in EEG signals. EEG-based emotion recognition requires modeling the complex changes in brainwave patterns over time, and GRU’s gated architecture enables it to effectively learn and represent these temporal relationships. By incorporating GRU into the emotion recognition pipeline, it becomes possible to enhance feature extraction from EEG data and improve the prediction accuracy of emotional states. This combination of EEG-based emotion recognition and GRU provides a promising direction for advancing emotion analysis and understanding affective responses encoded in brainwave signals. The publicly available multimodal dataset presented in the research serves as a valuable resource for other researchers to explore and test their own affective state estimation methods. The papers listed in the table explore various aspects of using GRUs in emotion recognition, spanning EEG data analysis, deep learning methods, and temporal dynamics modeling. These studies contribute to the growing body of research on emotion recognition and demonstrate the versatility and effectiveness of GRUs in handling sequential data and capturing complex patterns, making them valuable tools in emotion recognition from EEG signals. 3 Dataset The data used in this research was collected for a duration of 3 minutes from two participants, consisting of one male and one female, for each emotional state: positive, neutral, and negative. The participants were exposed to specific stimuli designed to elicit the desired emotions. Additionally, six minutes of resting neutral data were also recorded. The EEG data was captured using a Muse EEG headgear equipped with dry electrodes. The following EEG placements were utilized: TP9, AF7, AF8, and TP10. These electrode placements are commonly used for capturing brainwave activity related to emotion recognition. The dataset used in this research is publicly available and can be accessed at the following link: https://www.kaggle.com/datasets/birdy654/eeg-brainwave-dataset-feeling-emotions. It provides access to the collected EEG data, which includes the recordings for each participant in different emotional states, as well as the neutral resting data.The dataset consists of time-series EEG recordings, where each sample represents the electrical activity captured at specific time intervals. The data is labeled according to the emotional states experienced by the participants during the recordings. By utilizing this dataset, the research aims to explore the effectiveness of various noise removal methods on ambulatory EEG data and assess the performance of the GRU algorithm in accurately predicting emotional states based on the recorded brainwave activity. 4 Methodology 4.1 Recurrent Neural Networks (RNNs) Recurrent Neural Networks (RNNs) have emerged as a powerful neural network architecture for modeling sequential data. Unlike traditional feedforward neural networks, RNNs have a unique ability to capture temporal dependencies by introducing recurrent connections that allow information to flow from one time step to another. This characteristic makes RNNs well-suited for tasks involving sequential or time-series data analysis, including natural language processing, speech recognition, and, in our case, emotion recognition from EEG signals. In standard neural networks, each input and output is treated as independent, disregarding any contextual information. However, in many real-world applications, such as sentiment analysis or language generation, understanding the context of previous inputs is crucial for accurate predictions. RNNs address this limitation by incorporating a hidden state, which acts as a memory mechanism that retains information about past inputs and influences future predictions. The hidden state in an RNN serves as a crucial element for capturing long-term dependencies within the sequential data. As the RNN processes each input in a sequence, the hidden state is updated and passed to the next time step, allowing the network to remember past information and utilize it to make informed predictions. This recursive nature of RNNs enables them to model complex temporal relationships and extract relevant features from sequential data. However, traditional RNNs suffer from the vanishing or exploding gradient problem, which poses challenges in learning long-term dependencies. To address this issue, advanced variants of RNNs, such as Gated Recurrent Units (GRUs) and Long Short-Term Memory (LSTM) networks, have been developed. These architectures introduce specialized gating mechanisms that selectively retain or forget information, enabling the network to capture long-term dependencies while mitigating the gradient vanishing or exploding problem. In our research, we focus on the GRU algorithm as a variant of RNNs for emotion recognition using EEG data. GRUs have gained attention due to their simplified architecture, which consists of three gates: the update gate, reset gate, and current memory gate. The update gate determines how much past information should be propagated to future time steps, while the reset gate controls the extent to which previous knowledge should be forgotten. The current memory gate, often overlooked in discussions of GRUs, contributes non-linearity and zero-mean normalization to the input, reducing the impact of past data on future information. By leveraging the GRU algorithm, we aim to predict emotional states by analyzing EEG data collected from individuals exposed to various movie scenes or stimuli. The utilization of GRUs offers an efficient approach to capture temporal dependencies within the EEG signals, allowing us to explore the effectiveness of this architecture in emotion recognition tasks. 4.2 Gated Recurrent Units (GRUs) Gated Recurrent Units (GRUs) have gained significant attention as an alternative architecture within the realm of Recurrent Neural Networks (RNNs). GRUs address some of the limitations of traditional RNNs, such as the vanishing or exploding gradient problem, while providing an efficient and effective solution for capturing temporal dependencies in sequential data. In this section, we will delve into the specifics of GRUs and their relevance to our research on emotion recognition using EEG data. GRUs are a type of RNN architecture that incorporates gating mechanisms to regulate the flow of information within the network. Unlike the more complex Long Short-Term Memory (LSTM) networks, GRUs have a simplified structure consisting of three essential gates: the update gate, reset gate, and current memory gate. This architectural design allows GRUs to strike a balance between modeling long-term dependencies and computational efficiency. The update gate, denoted as z, determines the extent to which the previous hidden state is incorporated into the current state. It controls how much of the past information should be carried forward to future time steps. By selectively updating the hidden state, the GRU can adapt to different patterns in the data and retain relevant context over time. The reset gate, denoted as r, determines the extent to which the previous hidden state influences the current state. It selectively resets or forgets some of the previous knowledge, allowing the model to focus on relevant features and adapt to changing patterns within the sequence. The combination of the reset and update gates enables GRUs to capture and adapt to varying dependencies within the data. Another crucial component of GRUs is the current memory gate, often overlooked in discussions of GRUs. It is a sub-component of the reset gate and plays a vital role in introducing non-linearity and zero-mean normalization to the input. This helps in reducing the impact of previous data on the current data being propagated forward, ensuring that relevant information is preserved while minimizing the interference from irrelevant or noisy signals. Compared to a basic RNN, the workflow of a GRU is similar, with the primary distinction lying in the internal functioning of each recurrent unit. By leveraging the gating mechanisms, GRUs excel at capturing and modeling temporal dependencies in the data, making them well-suited for tasks such as emotion recognition from EEG signals. In our research, we adopt the GRU algorithm as the core architecture for predicting emotional states based on EEG data collected during exposure to various movie scenes or stimuli. The GRU’s simplified yet powerful design allows us to effectively model the temporal dynamics within the EEG signals, contributing to the advancement of emotion recognition using EEG-based approaches. 4.3 Preprocessing and Data Preparation The preprocessing and data preparation stage is essential to ensure the quality and suitability of the EEG data for accurate emotion recognition using the GRU algorithm. We follow standard practices in EEG-based emotion recognition to preprocess the data effectively. The raw EEG data collected from participants wearing the Muse EEG headgear with dry electrodes is subjected to artifact removal techniques such as independent component analysis (ICA) or template matching algorithms. This step eliminates unwanted noise and artifacts, allowing us to focus on genuine EEG signals related to emotional states. Next, bandpass filters are applied to remove unwanted frequency components while retaining the relevant frequency ranges associated with brainwave activity. Normalization techniques, such as z-score normalization or min-max scaling, are then employed to address amplitude variations between participants or electrode placements. Feature extraction techniques are utilized to capture relevant information from the preprocessed EEG data. These features, including power spectral density, signal entropy, or time-domain statistics, serve as input for the GRU algorithm, enabling it to learn meaningful patterns and associations with emotional states. To ensure unbiased evaluation, the dataset is partitioned into training, validation, and testing sets. Stratified or random partitioning techniques maintain representative distributions of emotional states across the subsets. This partitioning facilitates model training, hyperparameter optimization, and unbiased evaluation of the GRU model’s generalization capabilities. By implementing these preprocessing and data preparation steps, we enhance the quality and suitability of the EEG data for subsequent analysis using the GRU algorithm. The GRU model can effectively leverage the preprocessed data to accurately predict emotional states. 4.4 Gated Recurrent Unit Algorithm The Gated Recurrent Unit (GRU) algorithm is a variant of Recurrent Neural Networks (RNNs) that excels at capturing long-term dependencies in sequential data while mitigating the vanishing or exploding gradient problem. In this section, we provide an overview of the GRU algorithm and its relevance to our research on emotion recognition using EEG data. The GRU architecture is designed to have a simplified structure compared to traditional RNNs and LSTM networks. It consists of three fundamental gates: the update gate, reset gate, and current memory gate. These gating mechanisms enable the GRU algorithm to effectively model temporal dependencies within the data while maintaining computational efficiency. The update gate (z) determines the extent to which the previous hidden state influences the current state. It controls the flow of information from past time steps to future time steps, allowing the model to adapt and retain relevant context over time. By selectively updating the hidden state, the GRU algorithm can capture long-term dependencies and learn patterns within the sequential data. The reset gate (r) regulates the influence of previous knowledge on the current state. It selectively resets or forgets certain information, enabling the model to focus on relevant features and adapt to changing patterns within the sequence. The combination of the update and reset gates empowers the GRU algorithm to effectively model and adapt to varying dependencies within the data. The current memory gate, often overlooked in discussions of GRUs, plays a critical role in introducing non-linearity and zero-mean normalization to the input. By serving as a sub-component of the reset gate, it helps reduce the impact of previous data on the current data being propagated forward. This mechanism minimizes the interference of irrelevant or noisy signals and ensures the effective transfer of information to future time steps. Compared to a basic RNN, the GRU algorithm follows a similar workflow but excels in its internal functioning within each recurrent unit. Leveraging the gating mechanisms, GRUs excel at capturing temporal dependencies and facilitating the prediction of emotional states from EEG data. In our research, we employ the GRU algorithm as the core architecture for predicting emotional states based on EEG data collected during exposure to various movie scenes or stimuli. The GRU’s simplified yet powerful design allows us to effectively model the temporal dynamics within the EEG signals and contribute to the advancement of emotion recognition using EEG-based approaches. 4.5 Architecture The architecture in Fig 1., used in our research leverages the Gated Recurrent Unit (GRU) algorithm for emotion recognition based on EEG data. This architecture comprises an InputLayer, GRU layer, Flatten layer, and Dense layer. The configuration of this architecture is as follows: 1 InputLayer: The InputLayer serves as the entry point for the EEG data into the neural network. It defines the shape and format of the input data, aligning with the preprocessing steps and feature extraction performed on the EEG data. The input layer represents the initial stage of information flow in the neural network. 2 GRU Layer: The GRU layer is the core component of the architecture and employs the Gated Recurrent Unit algorithm. It processes the sequential EEG data, capturing temporal dependencies and extracting relevant features for emotion recognition. The GRU layer’s hidden state retains information from previous time steps and influences the predictions made by subsequent layers 3 Flatten Layer: Following the GRU layer, the Flatten layer is applied to transform the multi-dimensional output of the GRU into a one-dimensional vector. This flattening operation enables the subsequent layers to receive a flat input, facilitating compatibility with traditional fully connected layers. 4 Dense Layer: The Dense layer, also known as the fully connected layer, receives the flattened output from the preceding layer. It serves as a powerful learning component, responsible for mapping the extracted features to the emotional states being predicted. The dense layer consists of multiple interconnected neurons, and each neuron contributes to the final emotion classification based on learned weights and biases. This architecture efficiently processes the preprocessed EEG data through the GRU layer, capturing temporal dynamics and learning meaningful patterns related to emotional states. The subsequent flatten and dense layers allow for feature extraction and final classification, respectively. 5 Results and Analysis The deep learning accuracy details of the GRU model on the validation set are as follows: loss - 3.4356e-09 and accuracy - 1.0000. These impressive results indicate that the GRU model achieved perfect accuracy in predicting emotional states based on the EEG data. The model’s ability to achieve such high accuracy suggests its proficiency in capturing the temporal dynamics and extracting meaningful features from the EEG signals. Additionally, we compare the performance of the GRU model with other machine learning models using their respective scores. The following table 1 summarizes the scores obtained for various models: From the results, it is evident that the GRU model (95.46% accuracy) performs competitively when compared to other machine learning models. Notably, the Extreme Gradient Boosting Classifier achieves the highest score of 99.39%, closely followed by the Random Forest Classifier with a score of 98.7%. The Linear Support Vector Machine Classifier also demonstrates excellent performance with an accuracy of 96.57%. However, it is important to note that the GRU model showcases the advantage of deep learning in capturing complex temporal dependencies and extracting meaningful features from EEG data. Its accuracy of 95.46% suggests its efficacy in predicting emotional states based on the EEG signals, showcasing (Fib. 2) its potential for real-world applications. Further analysis is required to delve into the strengths and weaknesses of each model, considering factors such as computational complexity, interpretability, and generalization capabilities. Additionally, a more comprehensive evaluation using additional performance metrics like precision, recall, and F1-score could provide deeper insights into the models’ overall effectiveness. Overall, the results demonstrate the effectiveness of the GRU algorithm in accurately predicting emotional responses from EEG data. Its competitive performance against other machine learning models supports the notion that deep learning approaches, such as the GRU, can significantly contribute to the advancement of emotion recognition tasks using EEG signals. The confusion matrix analysis (Fib 3) was conducted to evaluate the performance of the machine learning models, including the GRU algorithm, in predicting emotional states based on EEG data. The confusion matrix provides insights into the model’s accuracy, precision, recall, and overall performance by comparing predicted labels with actual labels. It helps identify potential biases, misclassifications, and areas for improvement in the models’ predictions. 6 Discussion The results obtained from our research demonstrate the effectiveness of the GRU algorithm and other machine learning models in predicting emotional states based on EEG data. The deep learning accuracy of the GRU model showed remarkable performance, achieving perfect accuracy on the validation set. This indicates the ability of the GRU model to capture temporal dependencies and extract meaningful features from EEG signals, making it a promising approach for emotion recognition tasks. Comparing the performance of various machine learning models, we observed competitive results across different models. The Extreme Gradient Boosting Classifier achieved the highest score, closely followed by the Random Forest Classifier, while the Linear Support Vector Machine Classifier also demonstrated excellent performance. These findings highlight the importance of choosing an appropriate model based on the specific requirements and characteristics of the dataset. The confusion matrix analysis provided valuable insights into the models’ predictions, allowing us to assess their accuracy and identify potential biases or misclassifications. By examining the distribution of predicted labels across different emotional states, we gained a deeper understanding of the models’ strengths and weaknesses in capturing specific emotions. This analysis can guide future improvements and refinements in the models’ performance. 7 Conclusion In conclusion, Fib 4 shows the Mindgraph of the our study. Our research explored the application of machine learning models, including the GRU algorithm, for emotion recognition using EEG data. The GRU model demonstrated exceptional performance, achieving perfect accuracy on the validation set, highlighting its capability to capture temporal dynamics and extract meaningful features from EEG signals. Additionally, the comparison with other machine learning models emphasized the competitive performance achieved by various approaches. The choice of model should consider factors such as interpretability, computational complexity, and generalization capabilities, based on the specific requirements of the task. The incorporation of the confusion matrix analysis provided valuable insights into the models’ performance, aiding in identifying potential biases and areas for improvement. By understanding the models’ strengths and weaknesses, future research can focus on enhancing their accuracy and addressing specific challenges related to emotion recognition from EEG data. 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EEG-Based Emotion Recognition using Convolutional Neural Network and Gated Recurrent Unit. Cognitive Neurodynamics, 14(4), 485-495. Appendix: Survey

Relationships between the decoherence-free algebra and the set of fixed points F. Fagnola${}^{1}$, E. Sasso${}^{2}$, V. Umanità${}^{2}$    F. Fagnola${}^{1}$, E. Sasso${}^{2}$, V. Umanità${}^{2}$ () Abstract We show that, for a Quantum Markov Semigroup (QMS) with a faithful normal invariant state, the atomicity of the decoherence-free subalgebra and environmental decoherence are equivalent. Moreover, we characterize the set of reversible states and explicitly describe the relationship between the decoherence-free subalgebra and the fixed point subalgebra for QMSs with the above equivalent properties. 1 Introduction Starting from the fundamental papers of Gorini-Kossakowski-Sudharshan [19] and Lindblad [23] the structure of uniformly continuous quantum Markov semigroups (QMS) $\mathcal{T}=(\mathcal{T}_{t})_{t\geq 0}$, or, in the physical terminology, quantum dynamical semigroups, and their generators, has been the object of significant attention. The increasing interest in mathematical modelling of decoherence, coherent quantum computing and approach to equilibrium in open quantum systems continues to motivate investigation on special features of QMS. Special attention is paid to subalgebras or subspaces where irreversibility and dissipation disappear (see [2, 3, 7, 13, 14, 22, 29] and the references therein). States leaving in such subspaces are promising candidates for storing and manipulating quantum information. The decoherence-free subalgebra, where completely positive maps $\mathcal{T}_{t}$ of the semigroup act as automorphisms, and the set of fixed points, which is a subalgebra when there exists a faithful invariant state, also allow us to gain insight into the structure of a QMS, its invariant states and environment induced decoherence. Indeed, its structure as a von Neumann algebra, has important consequences on the structure and the action of the whole QMS. Recently, we showed in [13] that, if the decoherence-free subalgebra of a uniformly continuous QMS is atomic, it induces a decomposition of the system into its noiseless and purely dissipative parts, determining the structure of invariant states, as well as decoherence-free subsystems and subspaces [29]. In particular, we provided a full description of invariant states extending known ones in the finite dimensional case [6]. In this paper we push further the analysis of uniformly continuous QMS with atomic decoherence-free subalgebra and a faithful invariant state proving a number new results we briefly list and outline below. 1. Environment induced decoherence ([7, 9, 11]) holds if and only if $\mathcal{N(\mathcal{T})}$ is atomic. In this case the decoherence-free subalgebra is generated by the set of eigenvectors corresponding to modulus one eigenvalues of the completely positive maps $\mathcal{T}_{t}$, namely, in an equivalent way, by the eigenvectors with purely imaginary eigenvalue of the generator (Theorem 12). 2. The decoherence-free subalgebra and the set of so-called reversible states, i.e. the linear space generated by eigevectors corresponding to modulus 1 eigenvalues of predual maps $\mathcal{T}_{*t}$ are in the natural duality of a von Neumann algebra with its predual (Theorem 16). Moreover, Theorems 18 and 20 explicitly describe the structure of reversible states. 3. We find a spectral characterization of the decomposition of the fixed point algebra (Theorem 23). Moreover, the exact relationship between $\mathcal{F(\mathcal{T})}$ and $\mathcal{N(\mathcal{T})}$ (Theorems 23 and 25) is established in an explicit and constructive way allowing one to find the structure of each one from the structure of the other. Loosely speaking one can say that, for QMSs with a faithful invariant state, the same conclusions can be drawn replacing finite dimensionality of the system Hilbert space by atomicity of the decoherence-free subalgebra. Counterexamples (Examples 10 and 17) show that, in general, the above conclusions may fail if for QMSs without faithful normal invariant states. The above results, clarify then the relationships between the atomicity of the decoherence-free subalgebra, environmental decoherence, ergodic decomposition of the trace class operators, and the structure of fixed points. In particular the first result implies that the decomposition induced by decoherence coincides with the Jacobs-de-Leeuw-Glickeberg (JDG) splitting. Such decomposition was originally introduced for weakly almost periodic semigroups and generalized to QMSs on von Neumann algebras in [24, 21] at all. It is among the most useful tools in the study of the asymptotic behavior of operators semigroups on Banach spaces or von Neumann algebras. Indeed, it provides a decomposition of the space into the direct sum of the space generated by eigenvectors of the semigroup associated with modulus $1$ eigenvalues, and the remaining space, called stable, consisting of all vectors whose orbits have $0$ as a weak cluster point. Under suitable conditions, we obtain the convergence to $0$ for each vector belonging to the stable space. On the other hand, the ergodic decomposition of trace class operators (which is a particular case of the JDS splitting applied to the predual of $\mathcal{T}$), allows one, for instance, to determine reversible subsytems by spectral calculus. Determining reversible states, in particular, is an important task in the study of irreversible (Markovian) dynamics because these states retain their quantum features that are exploited in quantum computation (see [3, 29] and the references therein). More precisely, reversible (or rotating) and invariant states of a quantum channel (acting on $M_{n}(\mathbb{C})$ for some $n>1$) allow to classify kinds of information that the process can preserve. When the space is finite-dimensional and there exists a faithful invariant state, the structure of these states can be easily found (see e.g. Lemma $6$ and Section $V$ in [8], and Theorems $6.12$, $6.16$ in [31]) through the decomposition of $\mathcal{N(\mathcal{T})}$ and the algebra of fixed points $\mathcal{F(\mathcal{T})}$ in “block diagonal matrices”, i.e. in their canonical form given by the structure theorem for matrix algebras (see Theorem $11.2$ in [28]). Since the same decomposition holds for atomic von Neumann algebras, in this paper we generalize these results to uniformly continuous QMSs acting on $\mathcal{B}(\mathsf{h})$ with $\mathsf{h}$ infinite-dimensional. The paper is organized as follows. In Section 2 we collect some notation and known results on the structure of norm-continuous QMS with atomic decoherence-free subalgebra and the structure of their invariant states. In Section 3, after recalling some known results from [11] on the relationship between EID and Jacobs-de Leeuw-Glickeberg decomposition, we prove the main result of this paper: the equivalence between EID and atomicity of the decoherence free subalgebra. The predual algebra of the decoherence-free subalgebra is characterized in Section 4 as the set of reversible states. Finally, in Section 5, we study the structure of the set of fixed points of the semigroup and its relationships with the decomposition of $\mathcal{N(\mathcal{T})}$ when this algebra is atomic. 2 The structure of the decoherence-free algebra Let $\mathsf{h}$ be a complex separable Hilbert space and let $\mathcal{B}(\mathsf{h})$ the algebra of all bounded operators on $\mathsf{h}$ with unit 1l. A QMS on $\mathcal{B}(\mathsf{h})$ is a semigroup $\mathcal{T}=(\mathcal{T}_{t})_{t\geq 0}$ of completely positive, identity preserving normal maps which is also weakly${}^{*}$ continuous. In this paper we assume $\mathcal{T}$ uniformly continuous i.e. $$\lim_{t\to 0^{+}}\sup_{\|x\|\leq 1}\left\|\mathcal{T}_{t}(x)-x\right\|=0,$$ so that there exists a linear bounded operator $\mathcal{L}$ on $\mathcal{B}(\mathsf{h})$ such that $\mathcal{T}_{t}=e^{t\mathcal{L}}$. The operator $\mathcal{L}$ is the generator of $\mathcal{T}$, and it can be represented in the well-known (see [26]) Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form as $$\mathcal{L}(x)={\mathrm{i}}[H,x]-\frac{1}{2}\sum_{\ell\geq 1}\left(L_{\ell}^{*% }L_{\ell}x-2L_{\ell}^{*}xL_{\ell}+xL_{\ell}^{*}L_{\ell}\right),$$ (1) where $H=H^{*}$ and $(L_{\ell})_{\ell\geq 1}$ are bounded operators on $\mathsf{h}$ such that the series $\sum_{\ell\geq 1}L^{*}_{\ell}L_{\ell}$ is strongly convergent and $[\cdot,\cdot]$ denotes the commutator $[x,y]=xy-yx$. The choice of operators $H$ and $(L_{\ell})_{\ell\geq 1}$ is not unique, but this will not create any inconvenience in this paper. More precisely, we have the following characterization (see [26], Proposition $30.14$ and the discussion below the proof of Theorem $30.16$). Theorem 1. Let $\mathcal{L}$ be the generator of a uniformly continuous QMS on $\mathcal{B}(\mathsf{h})$. Then there exist a bounded selfadjoint operator $H$ and a sequence $(L_{\ell})_{\ell\geq 1}$ of elements in ${\mathcal{B}}(\mathsf{h})$ such that: 1. $\sum_{\ell\geq 1}L^{*}_{\ell}L_{\ell}$ is strongly convergent, 2. if $\sum_{\ell\geq 0}|c_{\ell}|^{2}<\infty$ and $c_{0}\hbox{\rm 1\kern-2.8ptl}+\sum_{\ell\geq 1}c_{\ell}L_{\ell}=0$ for scalars $(c_{\ell})_{\ell\geq 0}$ then $c_{\ell}=0$ for every $\ell\geq 0$, 3. $\mathcal{L}(x)={\mathrm{i}}[H,x]-\frac{1}{2}\sum_{\ell\geq 1}\left(L^{*}_{\ell% }L_{\ell}x-2L^{*}_{\ell}xL_{\ell}+xL^{*}_{\ell}L_{\ell}\right)$ for all $x\in{\mathcal{B}}(\mathsf{h})$. We recall that, for an arbitrary von Neumann algebra $\mathcal{M}$, its predual space $\mathcal{M}_{*}$ is the space of $w^{*}$-continuous functionals on $\mathcal{M}$ (said normal). It is a well-known fact that for all $\omega\in\mathcal{M}_{*}$ there exists $\rho\in\mathfrak{I}(\mathsf{h})$, the space of trace-class operators, such that $\omega(x)={\rm tr}\left(\rho x\right)$ for all $x\in\mathcal{M}$. In particular, if $\omega$ is a positive and $||\omega||=1$, it is called state, and $\rho$ is positive with ${\rm tr}\left(\rho\right)=1$, i.e. $\rho$ is a density. If $\mathcal{M}=\mathcal{B}(\mathsf{h})$, every normal state $\omega$ has a unique density $\rho$. Therefore, in this case, we can identify them. Finally, $\rho$ is faithful if ${\rm tr}\left(\rho x\right)=0$ for a positive $x\in\mathcal{B}(\mathsf{h})$ implies $x=0$ (see [28], Definition 9.4). Given a $w^{*}$-continuous operator $\mathcal{S}:\mathcal{M}\to\mathcal{M}$, we can define its predual map $\mathcal{S}_{*}:\mathcal{M}_{*}\to\mathcal{M}_{*}$ as $\mathcal{S}_{*}(\omega)=\omega\circ\mathcal{S}$. In particular, for $\mathcal{M}=\mathcal{B}(\mathsf{h})$, by considering the predual map of every $\mathcal{T}_{t}$, we obtain the predual semigroup $\mathcal{T}_{*}=(\mathcal{T}_{*t})_{t}$ satisfying $${\rm tr}\left(\mathcal{T}_{*t}(\rho)x\right)={\rm tr}\left(\rho\mathcal{T}_{*t% }(x)\right)\qquad\forall\,\rho\in\mathfrak{I}(\mathsf{h}),\ x\in\mathcal{B}(% \mathsf{h}).$$ The decoherence-free (DF) subalgebra of $\mathcal{T}$ is defined by $$\mathcal{N(\mathcal{T})}=\{x\in\mathcal{B}(\mathsf{h})\,:\,\mathcal{T}_{t}(x^{% *}x)=\mathcal{T}_{t}(x)^{*}\mathcal{T}_{t}(x),\ \mathcal{T}_{t}(xx^{*})=% \mathcal{T}_{t}(x)\mathcal{T}_{t}(x)^{*}\ \forall\,t\geq 0\}.$$ (2) It is a well known fact that $\mathcal{N(\mathcal{T})}$ is the biggest von Neumann subalgebra of $\mathcal{B}(\mathsf{h})$ on which every $\mathcal{T}_{t}$ acts as a $*$-homomorphism (see e.g. Evans[16] Theorem 3.1). Moreover, the following facts hold (see [14] Proposition 2.1). Proposition 2. Let $\mathcal{T}$ be a QMS on $\mathcal{B}(\mathsf{h})$ and let ${\mathcal{N}}(\mathcal{T})$ be the set defined by (2). Then 1. $\mathcal{N(\mathcal{T})}$ is invariant with respect to every $\mathcal{T}_{t}$, 2. $\mathcal{N(\mathcal{T})}=\{\delta_{H}^{(n)}(L_{k}),\delta_{H}^{(n)}(L_{k}^{*})% \,:\,n\geq 0\}^{\prime}$, where $\delta_{H}(x):=[H,x]$, 3. $\mathcal{T}_{t}(x)=e^{{\mathrm{i}}tH}xe^{-{\mathrm{i}}tH}$ for all $x\in\mathcal{N(\mathcal{T})}$, $t\geq 0$, 4. if $\mathcal{T}$ possesses a faithful normal invariant state, then $\mathcal{N(\mathcal{T})}$ contains the set of fixed points $\mathcal{F(\mathcal{T})}=\{L_{k},L_{k}^{*},H\,:\,k\geq 1\}^{\prime}$. In addition, if the QMS is uniformly continuous, its action on $\mathcal{N(\mathcal{T})}$ is bijective. Theorem 3. If $\mathcal{T}$ is a uniformly continuous QMS, then $\mathcal{N(\mathcal{T})}$ is the biggest von Neumann subalgebra on which every map $\mathcal{T}_{t}$ acts as a $*$-automorphism. Proof. The restriction of every $\mathcal{T}_{t}$ to $\mathcal{N(\mathcal{T})}$ is clearly injective thanks to item $3$ of Proposition 2. Now, given $x\in\mathcal{N(\mathcal{T})}$ and $t>0$, we have to prove that $x=\mathcal{T}_{t}(y)$ for some $y\in\mathcal{N(\mathcal{T})}$. First of all note that, since the QMS is norm continuous, it can be extended to norm continuous group $(\mathcal{T}_{t})_{-\infty<t<+\infty}$ of normal maps on $\mathcal{B}(\mathsf{h})$, and, by analyticity in $t$, $\mathcal{T}_{-t}(z)\in\mathcal{N(\mathcal{T})}$ for all $t>0$ and $z\in\mathcal{N(\mathcal{T})}$, and formula $\mathcal{T}_{-t}(z)=\hbox{\rm e}^{-{\mathrm{i}}tH}z\,\hbox{\rm e}^{{\mathrm{i}% }tH}$ holds. ∎ As we said in the introduction, we will study the relationships between the structure of $\mathcal{N(\mathcal{T})}$ and other problems in the theory of uniformly continuous QMSs in which the atomicity of $\mathcal{N(\mathcal{T})}$ plays a key role. First of all, as shown in [13], the structure of the decoherence-free subalgebra $\mathcal{N(\mathcal{T})}$ gives information on the whole QMS. Let us recall these results. Assume that $\mathcal{N(\mathcal{T})}$ is an atomic algebra, that is, there exists an (at most countable) family $(p_{i})_{i\in I}$ of mutually orthogonal non-zero projections, which are minimal projections in the center of $\mathcal{N(\mathcal{T})}$, such that $\sum_{i\in I}p_{i}=\hbox{\rm 1\kern-2.8ptl}$ and each von Neumann algebra $p_{i}\mathcal{N(\mathcal{T})}p_{i}$ is a type I factor. In that case, the subalgebra $\mathcal{N(\mathcal{T})}$ can be decomposed as $$\mathcal{N(\mathcal{T})}=\oplus_{i\in I}p_{i}\mathcal{N(\mathcal{T})}p_{i}\,.$$ (3) The properties of the projections $p_{i}$ imply their invariance under the semigroup and more generally the $\mathcal{T}_{t}$-invariance of each factor $p_{i}\mathcal{N(\mathcal{T})}p_{i}$. Moreover, each $p_{i}\mathcal{N(\mathcal{T})}p_{i}$ is a type I factor acting on the Hilbert space $p_{i}\mathsf{h}$; thus, there exist two countable sequences of Hilbert spaces $(\mathsf{k}_{i})_{i\in I}$, $(\mathsf{m}_{i})_{i\in I}$, and unitary operators $U_{i}:p_{i}\mathsf{h}\to\mathsf{k}_{i}\otimes\mathsf{m}_{i}$ such that $$U_{i}p_{i}\mathcal{N(\mathcal{T})}p_{i}U_{i}^{*}=\mathcal{B}(\mathsf{k}_{i})% \otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{m}_{i}},\qquad U_{i}p_{i}\mathcal{B}(% \mathsf{h})p_{i}U_{i}^{*}=\mathcal{B}(\mathsf{k}_{i}\otimes\mathsf{m}_{i}).$$ (4) Therefore, defining $U=\oplus_{i\in I}U_{i}$, we obtain a unitary operator $U:\mathsf{h}\to\oplus_{i\in I}(\mathsf{k}_{i}\otimes\mathsf{m}_{i})$ such that $$U\mathcal{N(\mathcal{T})}U^{*}=\oplus_{i\in I}\left(\mathcal{B}(\mathsf{k}_{i}% )\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{m}_{i}}\right).$$ (5) As a consequence, we find the following result. Theorem 4. $\mathcal{N(\mathcal{T})}$ is an atomic algebra if and only if there exist two countable sequences of separable Hilbert spaces $(\mathsf{k}_{i})_{i\in I}$, $(\mathsf{m}_{i})_{i\in I}$ such that $\mathsf{h}=\oplus_{i\in I}(\mathsf{k}_{i}\otimes\mathsf{m}_{i})$ (up to a unitary operator) and $\mathcal{N(\mathcal{T})}=\oplus_{i\in I}\left(\mathcal{B}(\mathsf{k}_{i})% \otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{m}_{i}}\right)$ (up to an isometric isomorphism). In this case: 1. for every GSKL representation of $\mathcal{L}$ by means of operators $H,(L_{\ell})_{\ell\geq 1}$, we have $$L_{\ell}=\oplus_{i\in I}\left(\hbox{\rm 1\kern-2.8ptl}_{\mathsf{k}_{i}}\otimes M% _{\ell}^{(i)}\right)$$ for a collection $(M_{\ell}^{(i)})_{\ell\geq 1}$ of operators in $\mathcal{B}(\mathsf{m}_{i})$, such that the series $\sum_{\ell\geq 1}M_{\ell}^{(i)*}M_{\ell}^{(i)}$ strongly convergent for all $i\in I$, and $$H=\oplus_{i\in I}\left(K_{i}\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{m}_{i}}+% \hbox{\rm 1\kern-2.8ptl}_{\mathsf{k}_{i}}\otimes M_{0}^{(i)}\right)$$ for self-adjoint operators $K_{i}\in\mathcal{B}(\mathsf{k}_{i})$ and $M_{0}^{(i)}\in\mathcal{B}(\mathsf{m}_{i})$, $i\in I$, 2. defining on $\mathcal{B}(\mathsf{m}_{i})$ the GKSL generator $\mathcal{L}^{\mathsf{m}_{i}}$ associated with operators $\{M_{0}^{(i)},M_{\ell}^{(i)})\,:\,\ell^{(i)}\geq 1\}$, we have $$\mathcal{T}_{t}(x_{i}\otimes y_{i})=\hbox{\rm e}^{{\mathrm{i}}tK_{i}}x_{i}% \hbox{\rm e}^{-{\mathrm{i}}tK_{i}}\otimes\mathcal{T}^{\mathsf{m}_{i}}(y_{i})$$ for all $t\geq 0$, $x_{i}\in\mathcal{B}(\mathsf{k}_{i})$ and $y_{i}\in\mathcal{B}(\mathsf{m}_{i})$, where $\mathcal{T}^{\mathsf{m}_{i}}$ is the QMS generated by $\mathcal{L}^{\mathsf{m}_{i}}$, 3. if there exists a faithful normal invariant state, then the QMS $\mathcal{T}^{\mathsf{m}_{i}}$ is irreducible, possesses a unique invariant state $\tau_{\mathsf{m}_{i}}$ which is also faithful, and we have $\mathcal{F}(\mathcal{T}^{\mathsf{m}_{i}})=\mathcal{N}(\mathcal{T}^{\mathsf{m}_% {i}})=\mathbb{C}\hbox{\rm 1\kern-2.8ptl}_{\mathsf{m}_{i}}$. Moreover, for all $i\in I$, $K_{i}$ has pure point spectrum. Proof. The proof of the necessary condition is given in Theorems 3.2 and 4.1, Proposition 4.3 and Lemma 4.2 in [13]. Conversely, given two countable sequences of Hilbert spaces $(\mathsf{k}_{i})_{i}$, $(\mathsf{m}_{i})_{i}$, such that $\mathsf{h}=\oplus_{i\in I}(\mathsf{k}_{i}\otimes\mathsf{m}_{i})$ up to a unitary operator, and $\mathcal{N(\mathcal{T})}=\oplus_{i\in I}\left(\mathcal{B}(\mathsf{k}_{i})% \otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{m}_{i}}\right)$ (up to the corresponding isometric isomorphism), set $p_{i}$ the orthogonal projection onto $\mathsf{k}_{i}\otimes\mathsf{m}_{i}$. Then we obtain a family $(p_{i})_{i\in I}$ of mutually orthogonal non-zero projections, which are minimal projections in the center of $\mathcal{N(\mathcal{T})}$, such that $\sum_{i\in I}p_{i}=\hbox{\rm 1\kern-2.8ptl}$ and each von Neumann algebra $p_{i}\mathcal{N(\mathcal{T})}p_{i}=\mathcal{B}(\mathsf{k}_{i})\otimes\hbox{\rm 1% \kern-2.8ptl}_{\mathsf{m}_{i}}$ is a type I factor. ∎ Now, we recall a characterization of the atomicity of a von Neumann algebra in terms of the existence of a normal conditional expectation, i.e. a weakly*-continuous norm one projection (see [30] Theorem $5$). To this end we will use that, given $x\in\mathsf{h}$ and $\mathsf{k}$ complex separable Hilbert spaces and $\sigma$ a normal state on $\mathsf{k}$, there always exists (see e.g. Exercise $16.10$ in [26]) a normal completely positive linear map $\mathcal{E}_{\sigma}:\mathcal{B}(\mathsf{h}\otimes\mathsf{k})\to\mathcal{B}(% \mathsf{h})$ satisfying $${\rm tr}\left(\mathcal{E}_{\sigma}(X)\eta\right)={\rm tr}\left(X(\eta\otimes% \sigma)\right)\qquad\forall\,X\in\mathcal{B}(\mathsf{h}\otimes\mathsf{k}),\ % \eta\in\mathfrak{I}(\mathsf{h}).$$ (6) Since $\mathcal{E}_{\sigma}$ is positive and $\mathcal{E}_{\sigma}\circ\mathcal{E}_{\sigma}=\mathcal{E}_{\sigma}$, it is a normal conditional expectation called $\sigma$-conditional expectation. Theorem 5 (Tomiyama). Let $\mathcal{M}$ be a von Neumann algebra acting on the Hilbert space $\mathsf{h}$. Then $\mathcal{M}$ is atomic if and only if $\mathcal{M}$ is the image of a normal conditional expectation $\mathcal{E}:\mathcal{B}(\mathsf{h})\to\mathcal{M}$. Proof. If there exists a normal conditional expectation $\mathcal{E}:\mathcal{B}(\mathsf{h})\to\mathcal{M}$ onto $\mathcal{M}$, then $\mathcal{M}$ is atomic by Proposition $3$ and Lemma $5$ in [20], being $\mathcal{B}(\mathsf{h})$ atomic and semifinite. On the other hand, if $\mathcal{M}$ is atomic, let $(p_{i})_{i}$ be a sequence of orthogonal projections in the center of $\mathcal{M}$ such that $p_{i}\mathcal{M}p_{i}$ is a type $I$ factor; moreover, assume $p_{i}\mathcal{M}p_{i}\simeq\mathcal{B}(\mathsf{k}_{i})\otimes\hbox{\rm 1\kern-% 2.8ptl}_{\mathsf{m}_{i}}$ with $(\mathsf{k}_{i})_{i}$ and $(\mathsf{m}_{i})_{i}$ be two sequences of complex and separable Hilbert spaces such that $\mathsf{h}\simeq\oplus_{i}\left(\mathsf{k}_{i}\otimes\mathsf{m}_{i}\right)$. Set $(\sigma_{i})_{i}$ a sequence of normal states on $(\mathsf{m}_{i})_{i}$, define the normal conditional expectations $\pi_{i}:\mathcal{B}(\mathsf{k}_{i}\otimes\mathsf{m}_{i})\to\mathcal{B}(\mathsf% {k}_{i})\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{m}_{i}}$ as $\pi_{i}(x)=\mathcal{E}_{\sigma_{i}}(x)\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf% {m}_{i}}$ for all $x\in\mathcal{B}(\mathsf{k}_{i}\otimes\mathsf{m}_{i})$. So, $$\pi(x):=\sum_{i}\pi(x_{ii})\quad\mbox{for $x=(x_{ij})_{ij}\in\mathcal{B}(% \oplus_{i}\left(\mathsf{k}_{i}\otimes\mathsf{m}_{i}\right))\simeq\mathcal{B}(% \mathsf{h})$}$$ gives a normal conditional expectation onto $\mathcal{M}$. ∎ Corollary 6. Let $\mathcal{M}$ be an atomic von Neumann algebra acting on $\mathsf{h}$ and let $\mathcal{N}\subseteq\mathcal{M}$ be a von Neumann subalgebra. If there exists a normal conditional expectation $\mathcal{E}:\mathcal{M}\to\mathcal{N}$ onto $\mathcal{N}$, then $\mathcal{N}$ is atomic. Proof. By Tomiyama’s Theorem we know that, since $\mathcal{M}$ is atomic, it is the image of a normal conditional expectation $\mathcal{F}:\mathcal{B}(\mathsf{h})\to\mathcal{M}$ (see also the proof of Theorem $5$ in [30]). Therefore, the map $\mathcal{E}\circ\mathcal{F}:\mathcal{B}(\mathsf{h})\to\mathcal{N}$ is a normal conditional expectation onto $\mathcal{N}$. Indeed, since $\mathcal{N}={\rm{Ran\,}}\mathcal{E}$ is contained in $\mathcal{M}={\rm{Ran\,}}\mathcal{F}$, for $x\in\mathcal{B}(\mathsf{h})$ we have $$(\mathcal{E}\circ\mathcal{F})(\mathcal{E}\circ\mathcal{F})x=\mathcal{E}^{2}(% \mathcal{F}(x))=(\mathcal{E}\circ\mathcal{F})(x),$$ i.e. $\mathcal{E}\circ\mathcal{F}$ is a projection. Therefore, $\|\mathcal{E}\circ\mathcal{F}\|\geq 1$. On the other hand, since $\mathcal{E}$ and $\mathcal{F}$ are norm one operators, we clearly obtain $\|\mathcal{E}\circ\mathcal{F}\|=1$. The normality of $\mathcal{E}\circ\mathcal{F}$ is evident, and so we can conclude that the algebra $\mathcal{N}$ is atomic by Tomiyama Theorem. ∎ Remark 7. Theorem 5 is a simplified version of Theorem $5$ in [30], given in terms of atomicity of the subalgebra $\mathcal{M}$. Moreover, Corollary 6 generalizes to atomic algebras one implication of the same theorem. In particular, we give easier proofs of these results. In the following we assume the existence of a faithful normal invariant state; note that, in general, this condition is not necessary for the decoherence-free subalgebra to be atomic. This is always the case for any QMS acting on a finite dimensional algebra. However, we show the following example that will be useful later. Example 8. Let $\mathsf{h}=\mathbb{C}^{3}$ with the canonical orthonormal basis $(e_{i})_{i=1,2,3}$ and $\mathcal{B}(\mathsf{h})=M_{3}(\mathbb{C})$. We consider the operator $\mathcal{L}$ on $M_{3}(\mathbb{C})$ given by $$\mathcal{L}(x)={\mathrm{i}}\omega\,[|{e_{1}}\rangle\langle{e_{1}}|,x]-\frac{1}% {2}\left(|{e_{3}}\rangle\langle{e_{3}}|x-2|{e_{3}}\rangle\langle{e_{2}}|x|{e_{% 2}}\rangle\langle{e_{3}}|+x|{e_{3}}\rangle\langle{e_{3}}|\right)$$ for all $x\in M_{3}(\mathbb{C})$, with $\omega\in\mathbb{R}$, $\omega\neq 0$. Clearly $\mathcal{L}$ is written in the GKSL form with $H=\omega|{e_{1}}\rangle\langle{e_{1}}|$ and $L=|{e_{2}}\rangle\langle{e_{3}}|$, and so it generates a uniformly continuous QMS $\mathcal{T}=(\mathcal{T}_{t})_{t\geq 0}$ on $M_{3}(\mathbb{C})$. An easy computation shows that any invariant functional has the form $$a|{e_{1}}\rangle\langle{e_{1}}|+b|{e_{2}}\rangle\langle{e_{2}}|$$ for some $a,b\in\mathbb{C}$, and so the semigroup has no faithful invariant states. Since $[H,L]=0$, by item $2$ of Proposition 2 we have $\mathcal{N(\mathcal{T})}=\{L,L^{*}\}^{\prime}$, so that an element $x\in\mathcal{B}(\mathsf{h})$ belongs to $\mathcal{N(\mathcal{T})}$ if and only if $$\left\{\begin{array}[]{ll}|{xe_{2}}\rangle\langle{e_{3}}|=|{e_{2}}\rangle% \langle{x^{*}e_{3}}|\\ |{xe_{3}}\rangle\langle{e_{2}}|=|{e_{3}}\rangle\langle{x^{*}e_{2}}|\end{array}% \right.,\quad\mbox{i.e.}\quad\left\{\begin{array}[]{lll}xe_{2}=x_{33}e_{2}\\ xe_{3}=x_{22}e_{3}\\ x_{31}=x_{21}=0\end{array}\right..$$ Therefore we get $$\mathcal{N(\mathcal{T})}=\left\{\,x_{11}|{e_{1}}\rangle\langle{e_{1}}|+x_{22}% \left(|{e_{2}}\rangle\langle{e_{2}}|+|{e_{3}}\rangle\langle{e_{3}}|\right)\,% \mid\,x_{11},x_{22}\in\mathbb{C}\,\right\},$$ i.e. $\mathcal{N(\mathcal{T})}$ is isometrically isomorphic to the atomic algebra $\mathbb{C}\oplus\mathbb{C}p$, where $p$ denotes the identity matrix in $M_{2}(\mathbb{C})$. 3 Atomicity of $\mathcal{N(\mathcal{T})}$ and decoherence In this section, we explore the relationships between the atomicity of $\mathcal{N(\mathcal{T})}$ and the property of environmental decoherence, under the assumption of the existence of a faithful normal invariant state $\rho$. Following [11], we say that there is environment induced decoherence (EID) on the open system described by $\mathcal{T}$ if there exists a $\mathcal{T}_{t}$-invariant and $*$-invariant weakly${}^{*}$ closed subspace $\mathcal{M}_{2}$ of $\mathcal{B}(\mathsf{h})$ such that: (EID1) $\mathcal{B}(\mathsf{h})=\mathcal{N(\mathcal{T})}\oplus\mathcal{M}_{2}$ with $\mathcal{M}_{2}\not=\{0\}$, (EID2) $w^{*}-\lim_{t\to\infty}\mathcal{T}_{t}(x)=0$ for all $x\in\mathcal{M}_{2}$. Unfortunately, if EID holds and $\mathsf{h}$ is infinite-dimensional, it is not clear if the space $\mathcal{M}_{2}$ is uniquely determined. However, $\mathcal{M}_{2}$ is always contained in the $\mathcal{T}$-invariant and $*$-invariant closed subspace $$\mathcal{M}_{0}=\left\{\,x\in\mathcal{B}(\mathsf{h})\,:\,w^{*}-\lim_{t\to% \infty}\mathcal{T}_{t}(x)=0\,\right\}.$$ In [13] we showed that, if $\mathcal{N(\mathcal{T})}$ is atomic, then EID holds (see Theorem 5.1) and, in particular, $\mathcal{N(\mathcal{T})}$ is the image of a normal conditional expectation $\mathcal{E}:\mathcal{B}(\mathsf{h})\to\mathcal{N(\mathcal{T})}$ compatible with the faithful state $\rho$ (i.e. $\rho\circ\mathcal{E}=\rho$) and such that $${\rm{Ker}}\,\mathcal{E}=\mathcal{M}_{2}=\left\{\,x\in\mathcal{B}(\mathsf{h})\,% :\,{\rm tr}\left(\rho xy\right)=0\ \ \forall\,y\in\mathcal{N(\mathcal{T})}\,% \right\}.$$ (7) (see Theorem $19$ in [11]). In the following we will show that, if $\mathcal{N(\mathcal{T})}$ is atomic, the decomposition unique, i.e. the only way to realize it, is taking $\mathcal{M}_{2}$ given by (7) (see Theorem 12 and Remark 13.$2$). Moreover, in this case, we will study the relationships of such a decomposition with another famous asymptotic splitting of $\mathcal{B}(\mathsf{h})$, called the Jacobs-de Leeuw-Glickberg splitting: this comparison is very natural since the decomposition $\mathcal{B}(\mathsf{h})=\mathcal{N(\mathcal{T})}\oplus\mathcal{M}_{2}$ is clearly related too to the asymptotic properties of the semigroup. We recall that, since there exists $\rho$ faithful invariant, the Jacobs-de Leeuw-Glickberg splitting holds (see e.g. Corollary $3.3$ and Proposition $3.3$ in [21]) and is given by $\mathcal{B}(\mathsf{h})=\mathfrak{M}_{r}\oplus\mathfrak{M}_{s}$ with $$\displaystyle\mathfrak{M}_{r}$$ $$\displaystyle:=\overline{{\rm{span}}}^{w^{*}}\{x\in\mathcal{B}(\mathsf{h})\,:% \,\mathcal{T}_{t}(x)=e^{{\mathrm{i}}t\lambda}x\ \mbox{for some $\lambda\in% \mathbb{R}$},\ \forall\,t\geq 0\}$$ (8) $$\displaystyle\mathfrak{M}_{s}$$ $$\displaystyle:=\{x\in\mathcal{B}(\mathsf{h})\,:\,0\in\overline{\{\mathcal{T}_{% t}(x)\}}^{w^{*}}_{t\geq 0}\}.$$ (9) Moreover, in this case $\mathfrak{M}_{r}$ is a von Neumann algebra. The relationship between the decomposition induced by decoherence and the Jacobs-de Leeuw-Glickberg splitting is given by the following result (see Proposition 31 in [11]). Proposition 9. If there exists a faithful normal invariant state $\rho$, then the following conditions are equivalent: 1. EID holds with $\mathcal{M}_{2}=\mathcal{M}_{0}$ and the induced decomposition coincides with the Jacobs-de Leeuw-Glickberg splitting, 2. $\mathcal{N(\mathcal{T})}\cap\mathfrak{M}_{s}=\{0\}$, 3. $\mathcal{N(\mathcal{T})}=\mathfrak{M}_{r}$. Moreover, if one of the previous conditions holds, then $\mathcal{N(\mathcal{T})}$ is the image of a normal conditional expectation $\mathcal{E}$ compatible with $\rho$ and such that ${\rm{Ker}}\,\mathcal{E}=\mathcal{M}_{0}=\mathfrak{M}_{s}$. Clearly, if $\mathfrak{M}_{r}$ is not an algebra, it does not make sense to pose the problem to understand if it coincides with $\mathcal{N(\mathcal{T})}$. In particular, this could happen when $\mathcal{T}$ has no faithful invariant states, as the following example shows. Example 10. Let us consider the uniformly continuous QMS $\mathcal{T}$ on $M_{3}(\mathbb{C})$ defined in Example 8. We have already seen that $\mathcal{T}$ does not posses faithful invariant states, and $$\mathcal{N(\mathcal{T})}=\left\{\,x_{11}|{e_{1}}\rangle\langle{e_{1}}|+x_{22}% \left(|{e_{2}}\rangle\langle{e_{2}}|+|{e_{3}}\rangle\langle{e_{3}}|\right)\,% \mid\,x_{11},x_{22}\in\mathbb{C}\,\right\}.$$ We want now to find the space $\mathfrak{M}_{r}$, generated by eigenvectors of $\mathcal{L}$ corresponding to purely imaginary eigenvalues. Easy computations show that we have $\mathcal{L}(x)={\mathrm{i}}\lambda x$ for some $\lambda\in\mathbb{R}$ if and only if $$\displaystyle{\mathrm{i}}\lambda\sum_{i,j=1}^{3}x_{ij}|{e_{i}}\rangle\langle{e% _{j}}|$$ $$\displaystyle={\mathrm{i}}\omega\sum_{j=1}^{3}\left(x_{1j}|{e_{1}}\rangle% \langle{e_{j}}|-x_{j1}|{e_{j}}\rangle\langle{e_{1}}|\right)$$ $$\displaystyle-\frac{1}{2}\left(\sum_{j=1}^{3}x_{3j}|{e_{3}}\rangle\langle{e_{j% }}|-2x_{22}|{e_{3}}\rangle\langle{e_{3}}|+\sum_{i=1}^{3}x_{i3}|{e_{i}}\rangle% \langle{e_{3}}|\right),$$ i.e., in the case when $\lambda=0$, $x_{ij}=0$ for $i\not=j$ and $x_{22}=x_{33}$, and, in the case $\lambda\not=0$, if and only if the following identities hold $$\begin{array}[]{ccc}x_{11}=0,&x_{22}=0,&x_{12}(\omega-\lambda)=0,\\ x_{13}\left(-\frac{1}{2}+{\mathrm{i}}(\omega-\lambda)\right)=0&x_{21}(\omega+% \lambda)=0,&x_{23}(\frac{1}{2}+{\mathrm{i}}\lambda)=0,\\ x_{31}\left(\frac{1}{2}+{\mathrm{i}}(\omega+\lambda)\right)=0,&x_{32}(\frac{1}% {2}+{\mathrm{i}}\lambda)=0,&x_{33}(1+{\mathrm{i}}\lambda)=x_{22}.\end{array}$$ Since $\omega$ and $\lambda$ belong to $\mathbb{R}$ this is equivalent to have either $x=x_{12}|{e_{1}}\rangle\langle{e_{2}}|$ and $\lambda=\omega$, or $x=x_{21}|{e_{2}}\rangle\langle{e_{1}}|$ and $\lambda=-\omega$. Therefore, we can conclude that $$\mathfrak{M}_{r}=\left\{\left(\begin{array}[]{ccc}x_{11}&x_{12}&0\\ x_{21}&x_{22}&0\\ 0&0&x_{22}\end{array}\right)\,:\,x_{11},x_{22},x_{12},x_{21}\in\mathbb{C}% \right\}.$$ In particular, $\mathfrak{M}_{r}$ is not an algebra and it is strictly bigger that $\mathcal{N(\mathcal{T})}$. Remarks 11. $1.$ Note that if $\mathfrak{M}_{r}$ is contained in $\mathcal{N(\mathcal{T})}$, then it is a $*$-algebra. Indeed, if $\mathfrak{M}_{r}\subseteq\mathcal{N(\mathcal{T})}$, taken $x,y\in\mathfrak{M}_{r}$ such that $\mathcal{T}_{t}(x)=e^{{\mathrm{i}}\lambda t}x$ and $\mathcal{T}_{t}(y)=e^{{\mathrm{i}}\mu t}y$ for some $\lambda,\mu\in\mathbb{R}$ and any $t$, by property $3$ in Proposition 2 we have $$\mathcal{T}_{t}(x^{*}y)=\mathcal{T}_{t}(x)^{*}\mathcal{T}_{t}(y)=e^{{\mathrm{i% }}t(\mu-\lambda)}x^{*}y\qquad\forall\,t\geq 0.$$ As a consequence $x^{*}y$ belongs to $\mathfrak{M}_{r}$. $2.$ If $\mathsf{h}$ is finite-dimensional, then also the opposite implication is true. Indeed, if $\mathfrak{M}_{r}$ is a $*$-algebra, given $x\in\mathcal{B}(\mathsf{h})$ such that $\mathcal{T}_{t}(x)=e^{it\lambda}x$, $\lambda\in\mathbb{R}$, we have $\mathcal{T}_{t}(x^{*})\mathcal{T}_{t}(x)=x^{*}x$. Then, by the Schwarz inequality, $\mathcal{T}_{t}(x^{*}x)\geq x^{*}x$ for all $t\geq 0$. Set $\mathcal{T}^{r}_{t}:=\mathcal{T}_{t|_{\mathfrak{M}_{r}}}$. Since in this case $\mathcal{T}$ is a strongly continuous semigroup, by definition of $\mathfrak{M}_{r}$ and by Corollary $2.9$, Chapter V, of [15], the strong operator closure of $\{\mathcal{T}^{r}_{t}\,:\,t\geq 0\}$ is a compact topological group of operators in $\mathcal{B}(\mathfrak{M}_{r})$. Hence, $(\mathcal{T}^{r}_{t})^{-1}$ is the limit of some net $(\mathcal{T}^{r}_{t_{\alpha}})_{\alpha}$ and so $(\mathcal{T}^{r}_{t})^{-1}$ is a positive operator. Since $x^{*}x\in\mathfrak{M}_{r}$, for all $\alpha$ we have $\mathcal{T}^{r}_{t_{\alpha}}(x^{*}x)\geq x^{*}x$, and so $(\mathcal{T}^{r}_{t})^{-1}(x^{*}x)\geq x^{*}x$. On the other hand, $$(\mathcal{T}^{r}_{t})^{-1}(\mathcal{T}^{r}_{t}(x^{*}x))=x^{*}x\geq(\mathcal{T}% ^{r}_{t})^{-1}(x^{*}x).$$ Therefore, $(\mathcal{T}^{r}_{t})^{-1}(x^{*}x)=x^{*}x$ and this implies $\mathcal{T}_{t}(x^{*}x)=x^{*}x=\mathcal{T}_{t}(x)^{*}\mathcal{T}_{t}(x)$. Similarly we can prove the equality $\mathcal{T}_{t}(xx^{*})=xx^{*}=\mathcal{T}_{t}(x)\mathcal{T}_{t}(x)^{*}$, and so $x$ belongs to $\mathcal{N(\mathcal{T})}$. Now we are able to prove one of the central results of this paper. Theorem 12. Assume that there exists a faithful normal invariant state $\rho$. Then $\mathcal{N(\mathcal{T})}$ is atomic if and only if EID holds with $\mathcal{N(\mathcal{T})}=\mathfrak{M}_{r}$ and $\mathcal{M}_{2}=\mathcal{M}_{0}$. Proof. If $\mathcal{N(\mathcal{T})}$ is atomic, then EID holds by Theorem $5.1$ in [13]. It remains to prove that $\mathcal{N(\mathcal{T})}=\mathfrak{M}_{r}$ and $\mathcal{M}_{2}=\mathcal{M}_{0}$. The atomicity implies $\mathcal{N(\mathcal{T})}=\oplus_{i\in I}\left(\mathcal{B}(\mathsf{k}_{i})% \otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{m}_{i}}\right)$ up to a unitary isomorphism. Let $x=\sum_{i\in I}(x_{i}\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{m}_{i}})$ be in $\mathcal{N(\mathcal{T})}\cap\mathfrak{M}_{s}$, with $x_{i}\in\mathcal{B}(\mathsf{k}_{i})$ for every $i\in I$, and assume $w^{*}-\lim_{\alpha}\mathcal{T}_{t_{\alpha}}(x)=0$. Given $u_{i},v_{i}\in\mathsf{k}_{i}$ and $\tau_{i}$ an arbitrary state on $\mathsf{m}_{i}$, by Theorem 4 $${\rm tr}\left((|{u_{i}}\rangle\langle{v_{i}}|\otimes\tau_{i})\mathcal{T}_{t_{% \alpha}}(x)\right)=\langle{v_{i}},{e^{{\mathrm{i}}t_{\alpha}K_{i}}x_{i}e^{-{% \mathrm{i}}t_{\alpha}K_{i}}u_{i}}\rangle.$$ (10) Choosing $u_{i}$ and $v_{i}$ such that $K_{i}u_{i}=\lambda_{i}u_{i}$ and $K_{i}v_{i}=\mu_{i}v_{i}$, $\lambda_{i},\mu_{i}\in\mathbb{R}$, equation (10) becomes $${\rm tr}\left((|{u_{i}}\rangle\langle{v_{i}}|\otimes\tau_{i})\mathcal{T}_{t_{% \alpha}}(x)\right)=e^{{\mathrm{i}}t_{\alpha}(\mu_{i}-\lambda_{i})}\langle{v_{i% }},{x_{i}u_{i}}\rangle,$$ so that $\langle{v_{i}},{x_{i}u_{i}}\rangle=0$, i.e. $x_{i}=0$ because the eigenvectors of $K_{i}$ from an orthonormal basis of $\mathsf{k}_{i}$ (see item $3$ in Theorem 4). This proves the equality $\mathcal{N(\mathcal{T})}\cap\mathfrak{M}_{s}=\{0\}$. So we can conclude thanks to item $3$ of Proposition 9. Conversely, if EID holds with $\mathcal{N(\mathcal{T})}=\mathfrak{M}_{r}$ and $\mathcal{M}_{2}=\mathcal{M}_{0}$, by Proposition 9 there exists a normal conditional expectation $\mathcal{E}:\mathcal{B}(\mathsf{h})\to\mathcal{N(\mathcal{T})}$ onto $\mathcal{N(\mathcal{T})}$, compatible to $\rho$. Therefore, $\mathcal{N(\mathcal{T})}$ is atomic thanks to Theorem 5. ∎ Remarks 13. As a consequence of Theorem 12 and Proposition 9 the following facts hold: $1.$  $\mathcal{N(\mathcal{T})}$ is atomic if and only if $\mathcal{N(\mathcal{T})}\cap\,\mathfrak{M}_{s}=\{0\}$, if and only if $\mathcal{N(\mathcal{T})}=\mathfrak{M}_{r}$, i.e $\mathcal{N(\mathcal{T})}$ is generated by eigenvectors of $\mathcal{L}$ corresponding to purely imaginary eigenvalues. Moreover, in this case we also have $\mathcal{N(\mathcal{T})}\cap\mathcal{M}_{0}=\{0\}$, being $\mathcal{M}_{0}\subseteq\mathfrak{M}_{s}$: this means that, assuming $\mathcal{N(\mathcal{T})}$ atomic and the existence of a faithful invariant state, the situation is similar to the finite-dimensional case, i.e $\mathcal{N(\mathcal{T})}$ does not contain operators going to $0$ under the action of the semigroup. $2.$ if $\mathcal{N(\mathcal{T})}$ is atomic and $\mathcal{F(\mathcal{T})}=\mathbb{C}\hbox{\rm 1\kern-2.8ptl}$, the semigroup satisfies the following properties given by non-commutative Perron-Frobenius Theorem (see e.g. Propositions $6.1$ and $6.2$ in [5], Theorem $2.5$ in [4]): - the peripheral point spectrum $\sigma_{p}(\mathcal{T}_{t})\cap\mathbb{T}$ of each $\mathcal{T}_{t}$ is a subgroup of the circle group $\mathbb{T}$, - given $t\geq 0$, each peripheral eigenvalue $\alpha$ of $\mathcal{T}_{t}$ is simple and we have $\sigma_{p}(\mathcal{T}_{t})\cap\mathbb{T}=\alpha(\sigma_{p}(\mathcal{T}_{t})% \cap\mathbb{T})$, - the restriction of $\rho$ to $\mathcal{N(\mathcal{T})}$ is a trace. As a consequence, the peripheral point spectrum of each $\mathcal{T}_{t}$ is the cyclic group of all $h$-roots of unit for some $h\in\mathbb{N}$. $3.$  If $\mathcal{N(\mathcal{T})}$ is atomic, the decomposition induced by decoherence is uniquely determined. This fact follows from Proposition $5$ in [11], since we have $\mathcal{N(\mathcal{T})}\cap\mathfrak{M}_{0}=\{0\}$. $4.$  Note that Theorem 12 does not exclude the possibility to have a QMS $\mathcal{T}$ displaying decoherence with $\mathcal{N(\mathcal{T})}$ a non-atomic type I algebra. Clearly, in this case, we will get $\mathcal{N(\mathcal{T})}\supsetneq\mathfrak{M}_{r}$ or $\mathcal{M}_{2}\subsetneq\mathcal{M}_{0}$. Remark 14. In [25] the authors prove that EID holds when the semigroup commutes with the modular group associated with a faithful normal invariant state. However, our result in Theorem 12 is stronger since we find the equivalence between EID and the atomicity of $\mathcal{N(\mathcal{T})}$, which is a weaker assumption of the commutation with the modular group. In fact, it can be shown (see [27], section $3$) that commutation with the modular group implies atomicity of $\mathcal{N(\mathcal{T})}$. Moreover, it is not difficult to find an example of a QMS on $\mathcal{B}(\mathsf{h})$, with $\mathsf{h}$ finite dimensional which does not commute with the modular group. Its decoherence-free subalgebra, as any finite dimensional von Neumann algebra, will be atomic. 4 Structure of reversible states In this section, assuming $\mathcal{N(\mathcal{T})}$ atomic and the existence of a faithful invariant state $\rho$, we study the structure of reversible states, i.e. states belonging to the vector space $$\displaystyle\mathcal{R}(\mathcal{T}_{*}):$$ $$\displaystyle=\overline{{\rm{span}}}\{\sigma\in\mathfrak{I}(\mathsf{h})\,:\,% \mathcal{T}_{*t}(\sigma)=e^{{\mathrm{i}}t\lambda}\sigma\ \mbox{for some $% \lambda\in\mathbb{R},\ \forall\,t\geq 0$}\}$$ (11) $$\displaystyle=\overline{{\rm{span}}}\{\sigma\in\mathfrak{I}(\mathsf{h})\,:\,% \mathcal{L}_{*}(\sigma)={\mathrm{i}}\lambda\,\sigma\ \mbox{for some $\lambda% \in\mathbb{R}$}\}.$$ (12) In particular we will prove that $\mathcal{R}(\mathcal{T}_{*})$ is the predual of the decoherence-free algebra $\mathcal{N(\mathcal{T})}$. To this end, we recall the following result which is a version of the Jacobi-De Leeuw-Glicksberg theorem for strongly continuous semigroup (see Propositions $3.1,3.2$ in [24] and Theorem $2.8$ in [15]). Theorem 15. If there exists a normal density $\rho\in\mathfrak{I}(\mathsf{h})$ satisfying $${\rm tr}\left(\rho\left(\mathcal{T}_{t}(x)^{*}\mathcal{T}_{t}(x)\right)\right)% \leq{\rm tr}\left(x^{*}x\right)\quad\quad\forall\,x\in\mathcal{B}(\mathsf{h}),% \ t\geq 0,$$ (13) then we can decompose $\mathfrak{I}(\mathsf{h})$ as $$\mathfrak{I}(\mathsf{h})=\mathcal{R}(\mathcal{T}_{*})\oplus\{\sigma\in% \mathfrak{I}(\mathsf{h})\,:\,0\in\overline{\{\mathcal{T}_{*t}(\sigma)\}}^{w}_{% t\geq 0}\}.$$ (14) Since each faithful invariant state clearly fulfills (13), we obtain the splitting given by equation (14). On the other hand, denoting by $\,{}^{\perp}A$ the vector space $\{\sigma\in\mathfrak{I}(\mathsf{h})\,:\,{\rm tr}\left(\sigma x\right)=0\ % \forall\ x\in A\}$ for all subset $A$ of $\mathcal{B}(\mathsf{h})$, the atomicity of $\mathcal{N(\mathcal{T})}$ ensures the following facts: (F1). $\mathcal{B}(\mathsf{h})=\mathcal{N(\mathcal{T})}\oplus\mathcal{M}_{0}$ with $\mathcal{N(\mathcal{T})}=\mathfrak{M}_{r}={\rm{Ran\,}}\mathcal{E}$ and $\mathcal{M}_{0}=\mathfrak{M}_{s}={\rm{Ker}}\mathcal{E}$, where $\mathcal{E}:\mathcal{B}(\mathsf{h})\to\mathcal{N(\mathcal{T})}$ is a conditional expectation compatible with the faithful state $\rho$ (see Theorem 12 and Proposition 9); (F2). $\mathfrak{I}(\mathsf{h})=\,^{\perp}\mathcal{M}_{0}\oplus\,^{\perp}\mathcal{N(% \mathcal{T})}$ with $$\,{}^{\perp}\mathcal{M}_{0}={\rm{Ran\,}}\mathcal{E}_{*}\simeq\mathcal{N(% \mathcal{T})}_{*},\qquad\,^{\perp}\mathcal{N(\mathcal{T})}={\rm{Ker}}\mathcal{% E}_{*}\simeq\mathcal{M}_{2*}.$$ Moreover each $\mathcal{T}_{*t}$ acts as a surjective isometry on $\,{}^{\perp}\mathcal{M}_{0}$, and $\lim_{t}\mathcal{T}_{*t}(\sigma)=0$ for all $\sigma\in\,^{\perp}\mathcal{N(\mathcal{T})}$ (see Theorem $10$ in [11]). As a consequence, every state $\omega\in\mathcal{N(\mathcal{T})}_{*}$ is represented by a unique density $\sigma$ in $\,{}^{\perp}\mathcal{M}_{0}$, and, in this case, we write $\omega=\omega_{\sigma}$ to mean that $\omega(x)={\rm tr}\left(\sigma x\right)$ for all $x\in\mathcal{N(\mathcal{T})}$. Therefore, if we denote by $\mathcal{S}=(\mathcal{S}_{t})_{t\geq 0}$ the restriction of $\mathcal{T}$ to $\mathcal{N(\mathcal{T})}$, we have $$(\mathcal{S}_{*t}\omega_{\sigma})(x)=\omega_{\sigma}(\mathcal{T}_{t}(x))={\rm tr% }\left(\sigma\,e^{{\mathrm{i}}tH}xe^{-{\mathrm{i}}tH}\right)={\rm tr}\left(% \mathcal{E}_{*}(e^{-{\mathrm{i}}tH}\sigma e^{{\mathrm{i}}tH})x\right)$$ for all $x=\mathcal{E}(x)\in\mathcal{N(\mathcal{T})}$, concluding that $\mathcal{S}_{*t}\omega_{\sigma}$ is represented by the density $\mathcal{E}_{*}(e^{-{\mathrm{i}}tH}\sigma e^{{\mathrm{i}}tH})\in\,^{\perp}% \mathcal{M}_{0}$. In a equivalent way, we have $$\mathcal{T}_{*t}(\sigma)=\mathcal{E}_{*}(e^{-{\mathrm{i}}tH}\sigma e^{{\mathrm% {i}}tH})\qquad\forall\,\sigma\in\,^{\perp}\mathcal{M}_{0}.$$ (15) Theorem 16. If $\mathcal{N(\mathcal{T})}$ is atomic and there exists a faithful invariant state, then $$\mathcal{R}(\mathcal{T}_{*})=\,^{\perp}\mathcal{M}_{0}=\{\sigma\in\mathfrak{I}% (\mathsf{h})\,:\,\mathcal{T}_{*t}(\sigma)=\mathcal{E}_{*}(e^{-{\mathrm{i}}tH}% \sigma\,e^{{\mathrm{i}}tH})\ \,\forall\,t\geq 0\}\simeq\mathcal{N(\mathcal{T})% }_{*},$$ for every Hamiltonian $H$ in a GKSL representation of the generator of $\mathcal{T}$. Proof. The inclusion $\,{}^{\perp}\mathcal{M}_{0}\subseteq\{\sigma\in\mathfrak{I}(\mathsf{h})\,:\,% \mathcal{T}_{*t}(\sigma)=\mathcal{E}_{*}(e^{-{\mathrm{i}}tH}\sigma\,e^{{% \mathrm{i}}tH})\ \,\forall\,t\geq 0\}$ follows from the previous discussion. On the other hand, if we have $\mathcal{T}_{*t}(\sigma)=\mathcal{E}_{*}(e^{-{\mathrm{i}}tH}\sigma e^{{\mathrm% {i}}tH})$ for all $t\geq 0$, taking $t=0$ we get $\sigma=\mathcal{E}_{*}(\sigma)$, i.e. $\sigma$ belongs to $\,{}^{\perp}\mathcal{M}_{0}$. Now, given $\sigma\in\mathcal{R}(\mathcal{T}_{*})$ such that $\mathcal{T}_{*t}(\sigma)=e^{{\mathrm{i}}t\lambda}\sigma$ for all $t\geq 0$, $\lambda\in\mathbb{R}$, we have $${\rm tr}\left(\sigma x\right)=\lim_{t\to\infty}{\rm tr}\left(\sigma x\right)=% \lim_{t\to\infty}{\rm tr}\left(\mathcal{T}_{*t}(\sigma)e^{-{\mathrm{i}}t% \lambda}x\right)=\lim_{t}e^{-{\mathrm{i}}t\lambda}{\rm tr}\left(\sigma\mathcal% {T}_{t}(x)\right)=0$$ for all $x\in\mathcal{M}_{0}$, so that $\sigma$ belongs to $\,{}^{\perp}\mathcal{M}_{0}$. This proves that $\mathcal{R}(\mathcal{T}_{*})$ is contained in $\,{}^{\perp}\mathcal{M}_{0}$. In order to prove the opposite inclusion it is enough to show that $\,{}^{\perp}\mathcal{N(\mathcal{T})}$ contains $\{\sigma\in\mathfrak{I}(\mathsf{h})\,:\,0\in\overline{\{\mathcal{T}_{*t}(% \sigma)\}}^{w}_{t\geq 0}\}$, since we have $$\mathfrak{I}(\mathsf{h})=\mathcal{R}(\mathcal{T}_{*})\oplus\{\sigma\in% \mathfrak{I}(\mathsf{h})\,:\,0\in\overline{\{\mathcal{T}_{*t}(\sigma)\}}^{w}_{% t\geq 0}\}=\,^{\perp}\mathcal{M}_{0}\oplus\,^{\perp}\mathcal{N(\mathcal{T})}$$ by equation (14), item (F2) and Theorem 12. So, let $\sigma\in\mathfrak{I}(\mathsf{h})$ such that $0\in\overline{\{\mathcal{T}_{*t}(\sigma)\}}^{w}_{t\geq 0}$; given $(t_{\alpha})$ with $w-\lim_{\alpha}\mathcal{T}_{*t_{\alpha}}(\sigma)=0$ and $x\in\mathfrak{M}_{r}$ such that $\mathcal{T}_{t}(x)=e^{{\mathrm{i}}t\lambda}x$ for some $\lambda\in\mathbb{R}$, we have $${\rm tr}\left(\sigma x\right)=\lim_{\alpha}{\rm tr}\left(\sigma e^{-{\mathrm{i% }}t\lambda}\mathcal{T}_{t_{\alpha}}(x)\right)=\lim_{\alpha}e^{-{\mathrm{i}}t% \lambda}{\rm tr}\left(\mathcal{T}_{*t_{\alpha}}(\sigma)x\right)=0.$$ This means that $\sigma$ belongs to $\,{}^{\perp}\mathcal{N(\mathcal{T})}$ by Theorem 12. ∎ In general, when there does not exist a faithful invariant state $\mathcal{R}(\mathcal{T}_{*})$ could be different from $\mathcal{N(\mathcal{T})}_{*}$, as we can see in Example 17. Example 17. Let us consider a generic Quantum Markov Semigroup with $\mathbb{C}^{3}$, more precisely the uniformly continuous QMS generated by $$\mathcal{L}(x)=G^{*}x+\sum_{j=1,2}L_{3j}^{*}xL_{3j}+xG$$ where $$\displaystyle G=\left(-\frac{\gamma_{33}}{2}+i\kappa_{3}\right)|e_{3}\rangle% \langle e_{3}|,\qquad L_{3j}=\sqrt{\gamma_{3j}}\,|e_{j}\rangle\langle e_{3}|% \qquad\mbox{for }j=1,2,$$ with $\kappa_{3}\in\mathbb{R}$, $\gamma_{3j}>0$ for $j=1,2$, and $\gamma_{33}=-\gamma_{31}-\gamma_{32}$. We know that, the restriction of $\mathcal{L}$ to the diagonal matrices is the generator of a continuous time Markov chain $(X_{t})_{t}$ with values in $\{1,2,3\}$. For more details see [1, 10]. Since $1$ and $2$ are absorbing states for $(X_{t})_{t}$, and $3$ is a transient state, by Proposition $2$ in [12] we know that any invariant state of $\mathcal{T}$ is supported on ${\rm{span}}\{e_{1},e_{2}\}$. In particular, this implies there is no faithful invariant state. Moreover, Theorem $8$ in [12] gives $\mathcal{N(\mathcal{T})}=\mathbb{C}\hbox{\rm 1\kern-2.8ptl}$, since the absorbing states are accessible from $3$. As a consequence, $\mathcal{N(\mathcal{T})}_{*}=\mathbb{C}\hbox{\rm 1\kern-2.8ptl}$. On the other hand, since $1$ is absorbing, the state $|{e_{1}}\rangle\langle{e_{1}}|$ is invariant, and so it belongs in particular to $\mathcal{R}(\mathcal{T}_{*})$. Therefore, we have $\mathcal{R}(\mathcal{T}_{*})\neq\mathcal{N(\mathcal{T})}_{*}$ . We can now give the structure of reversible states when $\mathcal{N(\mathcal{T})}$ is a type $I$ factor. Theorem 18. Let $\mathcal{T}$ be a QMS on $\mathcal{B}(\mathsf{k}\otimes\mathsf{m})$ with a faithful invariant state $\rho$ and $\mathcal{N(\mathcal{T})}=\mathcal{B}(\mathsf{k})\otimes\hbox{\rm 1\kern-2.8ptl% }_{\mathsf{m}}$ and let $\tau_{\mathsf{m}}$ be the unique invariant state of the partially traced semigroup $\mathcal{T}^{\mathsf{m}}$ defined in Theorem 4 item 2. Then a state $\eta$ belongs to $\mathcal{R}(\mathcal{T}_{*})\simeq\mathcal{N(\mathcal{T})}_{*}$ if and only if $$\eta=\sigma\otimes\tau_{\mathsf{m}}$$ (16) for some state $\sigma$ on $\mathcal{B}(\mathsf{k})$. Proof. Let $(e_{j})_{j\geq 1}$ be an orthonormal basis of eigenvectors of $K$ so that $Ke_{j}=\kappa_{j}e_{j}$ for some $\kappa_{j}\in\mathbb{R}$. Given a state $\eta$, we can write $$\eta=\sum_{j,k\geq 1}|e_{j}\rangle\langle e_{k}|\otimes\eta_{jk}$$ with $\eta_{jk}$ trace class operator on $\mathsf{m}$, so that $$\mathcal{T}_{*t}(\eta)=\sum_{j,k\geq 1}\hbox{\rm e}^{{\mathrm{i}}(\kappa_{k}-% \kappa_{j})t}|e_{j}\rangle\langle e_{k}|\otimes\mathcal{T}^{\mathsf{m}}_{*t}(% \eta_{jk})$$ by Theorem 4. Therefore, by the linear independence of operators $|e_{j}\rangle\langle e_{k}|$, we have $\mathcal{T}_{*t}(\eta)=e^{{\mathrm{i}}t\lambda}\eta$ for some $\lambda\in\mathbb{R}$ if and only if $$\mathcal{T}^{\mathsf{m}}_{*t}(\eta_{jk})=e^{{\mathrm{i}}t(\lambda-\kappa_{k}+% \kappa_{j})}\eta_{jk}\qquad\forall\,j,k,$$ i.e. if and only if each $\eta_{jk}$ belongs to $\mathcal{R}(\mathcal{T}^{\mathsf{m}}_{*})$. Now, by item $3$ of Theorem 4 we know that $\mathcal{T}^{\mathsf{m}}$ has a unique (faithful) invariant state $\tau_{\mathsf{m}}$ and $\mathcal{N}(\mathcal{T}^{\mathsf{m}})=\mathcal{F}(\mathcal{T}^{\mathsf{m}})$; hence, Theorem 16 and Proposition 21 give $\mathcal{R}(\mathcal{T}^{\mathsf{m}}_{*})\simeq\mathcal{N}(\mathcal{T}^{% \mathsf{m}})_{*}=\mathcal{F}(\mathcal{T}^{\mathsf{m}})_{*}=\mathcal{F}(% \mathcal{T}^{\mathsf{m}}_{*})={\rm{span}}\{\tau_{\mathsf{m}}\}$. As a consequence, we can conclude that $\eta$ belongs to $\mathcal{N(\mathcal{T})}_{*}$ if and only if $\eta_{jk}={\rm tr}\left(\eta_{jk}\right)\tau_{\mathsf{m}}$ for all $j,k\geq 1$, i.e. if and only if $$\eta=\sum_{j,k}\left({\rm tr}\left(\eta_{jk}\right)|e_{j}\rangle\langle e_{k}|% \right)\otimes\tau_{\mathsf{m}}=\sigma\otimes\tau_{\mathsf{m}}$$ with $\sigma:=\sum_{j,k}{\rm tr}\left(\eta_{jk}\right)|e_{j}\rangle\langle e_{k}|$. This is enough to prove the statement since $\mathcal{R}(\mathcal{T}_{*})$ is the vector space generated by eigenstates of $\mathcal{T}_{*t}$ corresponding to modulo $1$ eigenvalues. ∎ If $\mathcal{N(\mathcal{T})}$ is not a type I factor, but it is atomic, we can obtain a similar result using that reversible states are “block-diagonal”, as the following proposition shows. Proposition 19. Assume $\mathcal{N(\mathcal{T})}$ atomic. Let $\mathcal{T}$ be a QMS with a faithful invariant state and let $\mathcal{N(\mathcal{T})}$ as in (3) with $(p_{i})_{i\in I}$ minimal projections in the center of $\mathcal{N(\mathcal{T})}$. Then $p_{i}\sigma p_{j}=0$ for all $i\not=j$ and for all reversible state $\sigma\in\mathcal{R}(\mathcal{T}_{*})$. Proof. Let $\sigma\in\mathcal{R}(\mathcal{T}_{*})$ such that $\mathcal{T}_{*t}(\sigma)=e^{{\mathrm{i}}t\lambda}\sigma$ for some $\lambda\in\mathbb{R}$. Since $\mathcal{R}(\mathcal{T}_{*})=\,^{\perp}\,\mathcal{M}_{0}$, by (F1) we have ${\rm tr}\left(\sigma x\right)={\rm tr}\left(\sigma x_{1}\right)$ for all $\mathcal{B}(\mathsf{h})\ni x=x_{1}+x_{2}$ with $x_{1}\in\mathcal{N(\mathcal{T})}$ and $x_{2}\in\mathcal{M}_{0}$. Moreover, the property $\mathcal{T}_{t}(p_{k})=p_{k}$ for all $k\in I$ and $t\geq 0$ gives $$\mathcal{T}_{*t}(p_{i}\sigma p_{j})=p_{i}\mathcal{T}_{*t}(\sigma)p_{j}=e^{{% \mathrm{i}}t\lambda}\,p_{i}\sigma p_{j}\qquad\,\forall\ i,j\in I,$$ so that each $p_{i}\sigma p_{j}$ belongs to $\mathcal{R}(\mathcal{T}_{*})\simeq\mathcal{N(\mathcal{T})}_{*}$. Therefore, since ${\rm tr}\left(p_{i}\sigma p_{j}x\right)={\rm tr}\left(\sigma p_{j}xp_{i}\right% )=0$ for all $x\in\mathcal{N(\mathcal{T})}=\oplus_{i\in I}p_{i}\mathcal{N(\mathcal{T})}p_{i}$, $i\neq j$, we obtain that $p_{i}\sigma p_{j}=0$ for all $i\neq j$. ∎ As a consequence, by Theorem 4 we have the following characterization of reversible states Theorem 20. Assume $\mathcal{N(\mathcal{T})}$ atomic and suppose there exists a faithful $\mathcal{T}$- invariant state. Let $(p_{i})_{i\in I}$, $(\mathsf{k}_{i})_{i\in I}$, $(\mathsf{m}_{i})_{i\in I}$ be as in Theorem 4. A state $\eta$ belongs to $\mathcal{R}(\mathcal{T}_{*})$ if and only if it can be written in the form $$\eta=\sum_{i\in I}{\rm tr}\left(\eta p_{i}\right)\sigma_{i}\otimes\tau_{% \mathsf{m}_{i}}$$ where, for every $i\in I$, 1. $\tau_{\mathsf{m}_{i}}$ is the unique $\mathcal{T}^{\mathsf{m}_{i}}$-invariant state which is also faithful, 2. $\sigma_{i}$ is a density on $\mathsf{k}_{i}$. We have thus derived the general form of reversible states starting from the structure of the atomic decoherence-free algebra $\mathcal{N(\mathcal{T})}$. This is a well known fact for a completely positive and unitary map (i.e. a channel) on a finite-dimensional space (see e.g. Theorem $6.16$ in [31] and section $V$ in [8]), but the proof of this result is not generalizable to the infinite dimensional case since it is based on a spectral decomposition of the channel based on eigenvectors. 5 Relationships with the structure of fixed points In this section we investigate the structure of the set $\mathcal{F(\mathcal{T})}$ of fixed points of the semigroup and its relationships with the decomposition of $\mathcal{N(\mathcal{T})}$ given in the previous section. First of all we prove the atomicity of $\mathcal{F(\mathcal{T})}$ and relate this algebra with the space of invariant states. Really, the reader can find the proof of these results in [17]. We report them for sake of completeness. Proposition 21. If there exists a faithful normal invariant state, then $\mathcal{F(\mathcal{T})}$ is an atomic algebra and $\mathcal{F(\mathcal{T})}_{*}$ is isomorphic to the space $\mathcal{F}(\mathcal{T}_{*})$ of normal invariant functionals. Proof. Since there exists a faithful invariant state, by Theorem $2.1$ in [17] and in [18] $\mathcal{F(\mathcal{T})}$ it is the image of a normal conditional expectation $\mathcal{E}:\mathcal{B}(\mathsf{h})\to\mathcal{F(\mathcal{T})}$ given by $$\mathcal{E}(x)=w^{*}-\lim_{\lambda\to 0}\lambda\int_{0}^{\infty}e^{-\lambda t}% \mathcal{T}_{t}(x)\,dt=w^{*}-\lim_{t\to+\infty}\frac{1}{t}\int_{0}^{t}\mathcal% {T}_{s}(x)\,ds.$$ (17) Hence, $\mathcal{F(\mathcal{T})}$ is atomic by Theorem 5, the range of the predual operator $\mathcal{E}_{*}$ coincides with $\mbox{}\,^{\perp}{\rm{Ker}}\mathcal{E}$ and it is isomorphic to $\mathcal{F(\mathcal{T})}_{*}$ through the map $${\rm{Ran\,}}\mathcal{E}_{*}=\,^{\perp}{\rm{Ker}}\mathcal{E}\ni\sigma\mapsto% \sigma\circ\mathcal{E}\in\mathcal{F(\mathcal{T})}_{*}.$$ Moreover, we clearly have ${\rm{Ran\,}}\mathcal{E}_{*}=\mathcal{F}(\mathcal{T}_{*})$, (see also Corollary 2.2 in [17]). ∎ Therefore, assuming the existence of a faithful invariant state, we can find a countable set $J$, and two sequences $(\mathsf{s}_{j})_{j\in J}$, $(\mathsf{f}_{j})_{j\in J}$ of separable Hilbert spaces such that $$\displaystyle\mathsf{h}\simeq\oplus_{j\in J}\left(\mathsf{s}_{j}\otimes\mathsf% {f}_{j}\right)\qquad\qquad\quad\ \mbox{(unitary equivalence)}$$ (18) $$\displaystyle\mathcal{F(\mathcal{T})}\simeq\oplus_{j\in J}\left(\mathcal{B}(% \mathsf{s}_{j})\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{f}_{j}}\right),\quad% \mbox{($*$-isomorphism isometric)}$$ (19) where $\hbox{\rm 1\kern-2.8ptl}_{\mathsf{f}_{j}}$ denote the identity operator on $\mathsf{f}_{j}$. Even if the decomposition (19) is given up to an isometric isomorphism, for sake of simplicity we will identify $\mathsf{h}$ with $\oplus_{j\in J}\left(\mathsf{s}_{j}\otimes\mathsf{f}_{j}\right)$ and $\mathcal{F(\mathcal{T})}$ with $\oplus_{j\in J}\left(\mathcal{B}(\mathsf{s}_{j})\otimes\hbox{\rm 1\kern-2.8ptl% }_{\mathsf{f}_{j}}\right)$. Now we can state for $\mathcal{F(\mathcal{T})}$ a similar result to Theorem 4. Note that, it has already been proved for a quantum channel on a matrix algebra in [8] Lemma $6$, and in [31] Theorems $6.12$ and $6.14$. Here, we extend this in the infinite-dimensional framework. Theorem 22. Assume there exists a faithful normal invariant state. Let $(\mathsf{s}_{j})_{j\in J}$ and $(\mathsf{f}_{j})_{j\in J}$ be two countable sequences of Hilbert spaces such that (18) and (19) hold. Then we have the following facts: 1. for every GKSL representation (1) of the generator $\mathcal{L}$ by means of operators $L_{\ell},H$, we have $$\displaystyle L_{\ell}$$ $$\displaystyle=\oplus_{j\in J}\left(\hbox{\rm 1\kern-2.8ptl}_{\mathsf{s}_{j}}% \otimes N^{(j)}_{\ell}\right)\quad\forall\,\ell\geq 1,$$ $$\displaystyle H$$ $$\displaystyle=\oplus_{j\in J}\left(\lambda_{j}\hbox{\rm 1\kern-2.8ptl}_{% \mathsf{s}_{j}}\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{f}_{j}}+\hbox{\rm 1% \kern-2.8ptl}_{\mathsf{s}_{j}}\otimes N^{(j)}_{0}\right),$$ where $N^{(j)}_{\ell}$ are operators on $\mathsf{f}_{j}$ such that the series $\sum_{\ell}(N^{(j)}_{\ell})^{*}N^{(j)}_{\ell}$ are strongly convergent for all $j\in J$, $(\lambda_{j})_{j\in J}$ is a sequence of real numbers, and every $M^{(j)}_{0}$ is a self-adjoint operator on $\mathsf{f}_{j}$; 2. $\mathcal{T}_{t}(x\otimes y)=x\otimes\mathcal{T}^{\mathsf{f}_{j}}_{t}(y)$ for all $x\in\mathcal{B}(\mathsf{s}_{j})$ and $y\in\mathcal{B}(\mathsf{f}_{j})$, for $j\in J$, where $\mathcal{T}^{\mathsf{f}_{j}}$ is the QMS on $\mathcal{B}(\mathsf{f}_{j})$ generated by $\mathcal{L}^{\mathsf{f}_{j}}$, whose GKSL representation is given by $\{N^{(j)}_{\ell},N^{(j)}_{0}\,:\ell\geq 1\}$; 3. every $\mathcal{T}^{\mathsf{f}_{j}}$ is irreducible and possesses a unique (faithful) normal invariant state $\tau_{\mathsf{f}_{j}}$; 4. every invariant state $\eta$ has the form $\eta=\sum_{j\in J}\sigma_{j}\otimes\tau_{\mathsf{f}_{j}}$ with $\sigma_{j}$ an arbitrary positive trace-class operator on $\mathsf{s}_{j}$ such that $\sum_{j\in J}{\rm tr}\left(\sigma_{j}\right)=1$. Proof. Since (19) holds, like in the proof of Theorems 3.1, 3.2 in [13] there exist operators $(N^{(j)}_{\ell})_{\ell}$ on $\mathcal{B}(\mathsf{f}_{j})$ such that $L_{\ell}=\oplus_{j\in J}\left(\hbox{\rm 1\kern-2.8ptl}_{\mathsf{s}_{j}}\otimes N% ^{(j)}_{\ell}\right)$ for all $\ell\geq 1$ and $j\in J$. Now, if $p_{j}$ is the orthogonal projection onto $\mathsf{s}_{j}\otimes\mathsf{f}_{j}$, we have $H=\sum_{l,m}p_{l}H_{lm}p_{m}$ with $H_{lm}:\mathsf{h}_{\mathsf{s}_{m}}\otimes\mathsf{h}_{\mathsf{f}_{m}}\to\mathsf% {h}_{\mathsf{s}_{l}}\otimes\mathsf{h}_{\mathsf{f}_{l}}$ and $H_{lm}^{*}=H_{ml}$ for all $l,m\in J$. Since every $x=\oplus_{j\in J}(x_{j}\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{f}_{j}})\in% \mathcal{F(\mathcal{T})}$ commutes with $H$ we get $0=[x,H]$, i.e. $$0=\sum_{j}p_{j}[x_{j}\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{f}_{j}},H_{jj}]p% _{j}+\sum_{j\neq m}p_{j}\left((x_{j}\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{f% }_{j}})H_{jm}-H_{jm}(x_{m}\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{f}_{m}})% \right)p_{m},$$ which implies $$[x_{j}\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{f}_{j}},H_{jj}]=0\quad\forall\,% j\in J,\quad(x_{j}\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{f}_{j}})H_{jm}=H_{% jm}(x_{m}\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{f}_{m}})\quad\forall\,j\neq m.$$ The first condition is equivalent to have $H_{jj}=\lambda_{j}\hbox{\rm 1\kern-2.8ptl}_{\mathsf{s}_{j}}\otimes\hbox{\rm 1% \kern-2.8ptl}_{\mathsf{f}_{j}}+\hbox{\rm 1\kern-2.8ptl}_{\mathsf{s}_{j}}% \otimes N^{(j)}_{0}$ for some $N^{(j)}_{0}\in\mathcal{B}(\mathsf{f}_{j})$ and $\lambda_{j}\in\mathbb{R}$; the second one gives $H_{jm}=0$ for all $j\neq m$, and so we obtain item 1. Item 2 trivially follows. The proof of items 3 and 4 are similar to the ones of Theorem 4.1 and 4.3, respectively, in [13]. ∎ We want now to understand the relationships between decompositions (5) and (19) making use of the notations introduced in Theorems 4 and 22. In particular, in Theorem 23, we find a spectral characterization of the decomposition of the fixed point algebra, up to an isometric isomorphism. Indeed, in this representation, the spaces $\mathsf{s}_{j}$ undergoing trivial evolutions are the eigenspaces of suitable Hamiltonians $K_{i}$ corresponding to their different eigenvalues. First of all we introduce the following notation: for every $i\in I$ denote by $$\sigma(K_{i}):=\{\kappa_{j}^{(i)}\,:j\in J_{i}\}$$ (20) with $\kappa_{j}^{(i)}\neq\kappa_{l}^{(i)}$ for $j\neq l$ in $J_{i}$, the (pure point) spectrum of the Hamiltonian $K_{i}\in\mathcal{B}(\mathsf{k}_{i})$ for some at most countable set $J_{i}\subseteq\mathbb{N}$. Note that, if $\mathcal{T}$ has a faithful normal invariant state, then $\sigma(K_{i})$ is exactly the spectrum of $K_{i}$ thanks to Theorem 4. Without of loss of generality we can choose the family $\{J_{i}\,:\,i\in I\}$ such that $J_{h}\cap J_{l}=\emptyset$ whenever $h\neq l$. In this way, set $$J:=\cup_{i\in I}J_{i},$$ (21) for $j\in J$ there exists a unique $i\in I$ such that $j=j_{i}\in J_{i}$. Theorem 23. Assume $\mathcal{N(\mathcal{T})}$ atomic and let $\mathcal{N(\mathcal{T})}=\oplus_{i\in I}\left(\mathcal{B}(\mathsf{k}_{i})% \otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{m}_{i}}\right)$ with $(\mathsf{k}_{i})_{i}$, $(\mathsf{m}_{i})_{i}$ two countable sequences of Hilbert spaces such that $\mathsf{h}=\oplus_{i\in I}(\mathsf{k}_{i}\otimes\mathsf{m}_{i})$. If there exists a faithful normal invariant state, up to an isometric isomorphism we have $$\mathcal{F(\mathcal{T})}=\oplus_{j\in J}\left(\mathcal{B}(\mathsf{s}_{j})% \otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{f}_{j}}\right)$$ (22) with $J$ defined in (21), and $$\mathsf{s}_{j}=\mathsf{s}_{j_{i}}:={\rm{Ker}}\left(K_{i}-\kappa_{j}^{(i)}\hbox% {\rm 1\kern-2.8ptl}_{\mathsf{k}_{i}}\right),\qquad\mathsf{f}_{j}=\mathsf{f}_{j% _{i}}:=\mathsf{m}_{i}\qquad\forall\,j_{i}\in J_{i},\ i\in I.$$ (23) Proof. By considering the spectral decomposition $K_{i}=\sum_{j\in J_{i}}\kappa_{j}^{(i)}q_{ji}$ with $(q_{ji})_{j\in J_{i}}$ mutually orthogonal projections such that $$q_{ji}\mathsf{k}_{i}=\rm{Ker}\left(K_{i}-\kappa_{j}^{(i)}\hbox{\rm 1\kern-2.8% ptl}_{\mathsf{k}_{i}}\right)=:\mathsf{s}_{j_{i}}$$ and $\sum_{j\in J_{i}}q_{ji}=\hbox{\rm 1\kern-2.8ptl}_{\mathsf{m}_{i}}$, we immediately obtain $$\mathsf{k}_{i}\otimes\mathsf{m}_{i}=\left(\oplus_{j\in J_{i}}\mathsf{s}_{j_{i}% }\right)\otimes\mathsf{m}_{i}=\oplus_{j\in J_{i}}\left(\mathsf{s}_{j_{i}}% \otimes\mathsf{f}_{j_{i}}\right)$$ by setting $\mathsf{f}_{j_{i}}:=\mathsf{m}_{i}$ for all $j\in J_{i}$. Therefore, by definition of $J$, since every $j\in J$ belongs to a unique $J_{i}$, we have $$\mathsf{h}=\oplus_{i\in I}(\mathsf{k}_{i}\otimes\mathsf{m}_{i})=\oplus_{i\in I% }\oplus_{j\in J_{i}}\left(\mathsf{s}_{j_{i}}\otimes\mathsf{f}_{j_{i}}\right)=% \oplus_{j\in J}\left(\mathsf{s}_{j}\otimes\mathsf{f}_{j}\right).$$ In order to conclude the proof we have to show equality (22). Given $x\in\mathcal{F(\mathcal{T})}\subseteq\mathcal{N(\mathcal{T})}$ (see item $4$ in Proposition 2), we can write $x=\oplus_{i\in I}(x_{i}\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{m}_{i}})$ with $(x_{i})_{i\in I}\subseteq\mathcal{B}(\mathsf{k}_{i})$, and so, by Theorem 4 we have $$x=\oplus_{i\in I}(x_{i}\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{m}_{i}})=% \mathcal{T}_{t}(\oplus_{i\in I}(x_{i}\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{% m}_{i}}))=\oplus_{i\in I}(e^{{\mathrm{i}}tK_{i}}x_{i}e^{-{\mathrm{i}}tK_{i}}% \otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{m}_{i}}).$$ Consequently, $x_{i}=e^{{\mathrm{i}}tK_{i}}x_{i}e^{-{\mathrm{i}}tK_{i}}$ for all $i\in I$, i.e. every $x_{i}$ commutes with $K_{i}$, and then with each projection $q_{ji}$ with $j\in J_{i}$. This means that each $x_{i}=\oplus_{j\in J_{i}}q_{ji}x_{i}q_{ji}$ belongs to the algebra $\oplus_{j\in J_{i}}q_{ji}\mathcal{B}(\mathsf{k}_{i})q_{ji}=\oplus_{j\in J_{i}}% \mathcal{B}(q_{ji}\mathsf{k}_{i})=\oplus_{j\in J_{i}}\mathcal{B}(\mathsf{s}_{j})$, so that $x$ is in $\oplus_{j\in J}\left(\mathcal{B}(\mathsf{s}_{j})\otimes\hbox{\rm 1\kern-2.8ptl% }_{\mathsf{f}_{j}}\right)$. On the other hand, given $i\in I$ and $j\in J_{i}$, for $u,v\in\mathsf{s}_{j}={\rm{Ker}}(K_{i}-\kappa_{j}^{(i)}\hbox{\rm 1\kern-2.8ptl}% _{\mathsf{k}_{i}})$ we get $$\mathcal{T}_{t}(|{u}\rangle\langle{v}|\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf% {m}_{i}})=|{e^{{\mathrm{i}}tK_{i}}u}\rangle\langle{e^{{\mathrm{i}}tK_{i}}v}|% \otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{m}_{i}}=|{u}\rangle\langle{v}|\otimes% \hbox{\rm 1\kern-2.8ptl}_{\mathsf{m}_{i}}\qquad\forall\,t\geq 0,$$ where $u$ and $v$ are eigenvectors of $K_{i}$ associated with the same eigenvalue $\kappa_{j}^{(i)}$. Since ${\rm{Ker}}(K_{i}-\kappa_{j}^{(i)}\hbox{\rm 1\kern-2.8ptl}_{\mathsf{k}_{i}})$ is generated by elements of the form $|{u}\rangle\langle{v}|$, and the net $(\mathcal{T}_{t}(z))_{t}$ is uniformly bounded for all $z$, we obtain the inclusion $\mathcal{B}(\mathsf{s}_{j})\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{m}_{i}}% \subseteq\mathcal{F(\mathcal{T})}$ for all $j\in J_{i}$, $i\in I$, i.e. $\mathcal{B}(\mathsf{s}_{j})\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{f}_{j}}% \subseteq\mathcal{F(\mathcal{T})}$ for all $j\in J=\cup_{i\in I}J_{i}$. ∎ Theorem below shows how we can derive an “atomic decomposition ”of $\mathcal{F(\mathcal{T})}$ from one of $\mathcal{N(\mathcal{T})}$’s. We want now to analyze the opposite procedure. Assuming $\mathcal{F(\mathcal{T})}=\oplus_{j\in J}\left(\mathcal{B}(\mathsf{s}_{j})% \otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{f}_{j}}\right)$ with $(\mathsf{s}_{j})_{j}$, $(\mathsf{f}_{j})_{j}$ two countable sequences of Hilbert spaces such that $\mathsf{h}=\oplus_{j\in j}(\mathsf{s}_{j}\otimes\mathsf{f}_{j})$, and using notations of Theorem 22, we set an equivalence relation on $J$ in the following way: Definition 24. Given $j,k\in J$, we say that $j$ is in relation with $k$ (and write $j\sim k$) if there exist a complex separable Hilbert space $\mathsf{m}$ and unitary isomorphisms $$V_{j}:\mathsf{f}_{j}\to\mathsf{m},\qquad V_{k}:{\mathsf{f}_{k}}\to\mathsf{m}$$ (24) such that operators $\{V_{j}N_{l}^{(j)}V_{j}^{*},V_{j}N_{0}^{(j)}V_{j}^{*}\,:\,l\geq 1\}$ and $\{V_{k}N_{l}^{(k)}V_{k}^{*},V_{k}N_{0}^{(k)}V_{k}^{*}\,:\,l\geq 1\}$ give the same Lindbladian operator on $\mathcal{B}(\mathsf{m})$. We obtain in this way an equivalence relation which induces a partition of $J$, $J=\cup_{n\in I}I_{n}$, for some finite or countable set $I\subseteq\mathbb{N}$, where each $I_{n}$ is an equivalence class with respect to $\sim$. Theorem 25. Assume that there exists a faithful normal invariant state and $\mathcal{N(\mathcal{T})}$ atomic. Let $\mathcal{F(\mathcal{T})}=\oplus_{j\in J}\left(\mathcal{B}(\mathsf{s}_{j})% \otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{f}_{j}}\right)$ with $(\mathsf{s}_{j})_{j}$, $(\mathsf{f}_{j})_{j}$ two countable sequences of Hilbert spaces such that $\mathsf{h}=\oplus_{j\in j}(\mathsf{s}_{j}\otimes\mathsf{f}_{j})$, and let $\{I_{n}\,:\,n\in I\}$ be the set of equivalence classes of $J$ with respect to the relation $\sim$. Then $\mathcal{N(\mathcal{T})}$ is isometrically isomorphic to the direct sum $\oplus_{n\in I}\left(\mathcal{B}(\mathsf{k}_{n})\otimes\hbox{\rm 1\kern-2.8ptl% }_{\mathsf{m}_{n}}\right)$ with $$\mathsf{k}_{n}:=\oplus_{j\in I_{n}}\mathsf{s}_{j},\qquad\mathsf{m}_{n}:=V_{j}% \mathsf{f}_{j}\quad\forall\,j\in I_{n},$$ (25) where $V_{j}$’s are the unitary isomorphisms given in (24). Proof. Given $n\in I$, by definition of $\mathsf{m}_{n}$ we can define a unitary operator by setting $$\begin{array}[]{rcl}U_{n}:\oplus_{j\in I_{n}}\left(\mathsf{s}_{j}\otimes% \mathsf{f}_{j}\right)&\to&\left(\oplus_{j\in I_{n}}\mathsf{s}_{j}\right)% \otimes\mathsf{m}_{n}=\mathsf{k}_{n}\otimes\mathsf{m}_{n}\\ \oplus_{j\in I_{n}}\left(u_{j}\otimes z_{j}\right)&\mapsto&\sum_{j\in I_{n}}% \left(u_{j}^{(j)}\otimes V_{j}z_{l}\right)\end{array}$$ where $u_{j}^{(j)}$ in $\oplus_{i\in I_{n}}\mathsf{h}_{\mathsf{s}_{i}}$ denotes the vector $$u_{j}^{(j)}:=\oplus_{i\in I_{n}}v_{i},\qquad v_{i}:=\left\{\begin{array}[]{ll}% 0&\ \mbox{if $i\neq j$}\\ u_{j}&\ \mbox{if $i=j$}\end{array}\right..$$ Now, since $$\mathsf{h}=\oplus_{j\in j}(\mathsf{s}_{j}\otimes\mathsf{f}_{j})=\oplus_{n\in I% }\left(\oplus_{j\in I_{n}}(\mathsf{s}_{j}\otimes\mathsf{f}_{j})\right)$$ by the equality $J=\cup_{n\in I}I_{n}$, by setting $U:=\oplus_{n\in I}U_{n}$ we get a unitary operator $U:\mathsf{h}\to\oplus_{n\in I}\left(\mathsf{k}_{n}\otimes\mathsf{m}_{n}\right)$ such that $$U\mathcal{F(\mathcal{T})}U^{*}=\oplus_{n\in I}\left(\left(\oplus_{j\in I_{n}}% \mathcal{B}(\mathsf{s}_{j})\right)\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{m}_% {n}}\right).$$ In order to conclude the proof we have to show that $$U\mathcal{N(\mathcal{T})}U^{*}=\oplus_{n\in I}\left(\mathcal{B}(\oplus_{j\in I% _{n}}\mathsf{s}_{j})\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{m}_{n}}\right).$$ To this end recall that, by Theorem 22, the operators $(L_{\ell})_{\ell},\ H$ in a GKSL representation of the generator $\mathcal{L}$ can be written as $$L_{\ell}=\oplus_{j\in J}\left(\hbox{\rm 1\kern-2.8ptl}_{\mathsf{s}_{j}}\otimes N% ^{(j)}_{\ell}\right)\quad\forall\,\ell\geq 1,\qquad H=\oplus_{j\in J}\left(% \lambda_{j}\hbox{\rm 1\kern-2.8ptl}_{\mathsf{s}_{j}}\otimes\hbox{\rm 1\kern-2.% 8ptl}_{\mathsf{f}_{j}}+\hbox{\rm 1\kern-2.8ptl}_{\mathsf{s}_{j}}\otimes N^{(j)% }_{0}\right)$$ with $(N_{\ell}^{(j)})_{\ell\geq 1}\subseteq\mathcal{B}(\mathsf{s}_{j})$, $N^{(j)}_{0}=(N^{(j)}_{0})^{*}\in\mathcal{B}(\mathsf{f}_{j})$ and $\lambda_{j}\in\mathbb{R}$ for all $j\in J$; moreover, by definition of $I_{n}$, we can choose operators $(M_{l}^{(n)})_{l}$ and $M_{0}^{(n)}=(M_{0}^{(n)})^{*}$ in $\mathcal{B}(\mathsf{m}_{n})$ such that $\{M_{l}^{(n)},M_{0}^{(n)}\,:\,l\geq 1\}$ is a GKSL representation of the the generator $\mathcal{L}^{\mathsf{m}_{n}}$ of a QMS $\mathcal{T}^{\mathsf{m}_{n}}$ on $\mathcal{B}(\mathsf{m}_{n})$ equivalent to $\{V_{j}N_{l}^{(j)}V_{j}^{*},V_{j}N_{0}^{(j)}V_{j}^{*}\,:\,l\geq 1\}$ for all $j\in I_{n}$. Therefore, the operators $$\displaystyle L_{\ell}^{\prime}$$ $$\displaystyle:=\oplus_{n\in I}\left(\oplus_{j\in I_{n}}\left(\hbox{\rm 1\kern-% 2.8ptl}_{\mathsf{s}_{j}}\otimes V_{j}^{*}M^{(n)}_{\ell}V_{j}\right)\right)$$ $$\displaystyle H^{\prime}$$ $$\displaystyle:=\oplus_{n\in I}\left(\oplus_{j\in I_{n}}\left(\lambda_{j}\hbox{% \rm 1\kern-2.8ptl}_{\mathsf{s}_{j}}\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{f}% _{j}}\right)+\oplus_{j\in I_{n}}\left(\hbox{\rm 1\kern-2.8ptl}_{\mathsf{s}_{j}% }\otimes V_{j}^{*}M^{(n)}_{0}V_{j}\right)\right),$$ clearly give the same GKSL representation of $\{L_{\ell},\ H\,:\,\ell\geq 1\}$. Moreover we have $$\displaystyle UL_{\ell}^{\prime}U^{*}$$ $$\displaystyle=\oplus_{n\in I}U_{n}\left(\oplus_{j\in I_{n}}\left(\hbox{\rm 1% \kern-2.8ptl}_{\mathsf{s}_{j}}\otimes V_{j}^{*}M^{(n)}_{\ell}V_{j}\right)% \right)U_{n}^{*}=\oplus_{n\in I}\left(\hbox{\rm 1\kern-2.8ptl}_{\mathsf{k}_{n}% }\otimes M^{(n)}_{\ell}\right)$$ $$\displaystyle UH^{\prime}U^{*}$$ $$\displaystyle=\oplus_{n\in I}U_{n}\left(\oplus_{j\in I_{n}}\left(\lambda_{j}% \hbox{\rm 1\kern-2.8ptl}_{\mathsf{s}_{j}}\otimes\hbox{\rm 1\kern-2.8ptl}_{% \mathsf{f}_{j}}\right)+\oplus_{j\in I_{n}}\left(\hbox{\rm 1\kern-2.8ptl}_{% \mathsf{s}_{j}}\otimes V_{j}^{*}M^{(n)}_{0}V_{j}\right)\right)U_{n}^{*}$$ $$\displaystyle=\oplus_{n\in I}\left(K_{n}\otimes\hbox{\rm 1\kern-2.8ptl}_{% \mathsf{m}_{n}}+\hbox{\rm 1\kern-2.8ptl}_{\mathsf{k}_{n}}\otimes M^{(n)}_{0}\right)$$ with $K_{n}:=\left(\oplus_{j\in I_{n}}\lambda_{j}\hbox{\rm 1\kern-2.8ptl}_{\mathsf{s% }_{j}}\right)=K_{n}^{*}\in\mathcal{B}(\mathsf{k}_{n})$, so that $$\displaystyle U\delta_{H\prime}^{m}(L_{\ell}^{\prime})U^{*}=\delta_{UH^{\prime% }U^{*}}^{m}(UL_{\ell}^{\prime}U^{*})$$ $$\displaystyle=\oplus_{n\in I}\left(\hbox{\rm 1\kern-2.8ptl}_{\mathsf{k}_{n}}% \otimes\delta_{M_{0}^{(n)}}^{m}(M_{\ell}^{(n)})\right)$$ $$\displaystyle U\delta_{H\prime}^{m}(L_{\ell}^{\prime*})U^{*}=\delta_{UH^{% \prime}U^{*}}^{m}(UL_{\ell}^{\prime*}U^{*})$$ $$\displaystyle=\oplus_{n\in I}\left(\hbox{\rm 1\kern-2.8ptl}_{\mathsf{k}_{n}}% \otimes\delta_{M_{0}^{(n)}}^{m}(M_{\ell}^{(n)*})\right)$$ for all $m\geq 0$. Therefore, since $U\mathcal{N(\mathcal{T})}U^{*}=\left\{U\delta_{H\prime}^{m}(L_{\ell}^{\prime})% U^{*},U\delta_{H\prime}^{m}(L_{\ell}^{\prime*})U^{*}\,:\,m\geq 0\right\}^{\prime}$ by item $2$ of Proposition 2, we obtain that an operator $x\in\mathcal{B}\left(\oplus_{n\in I}\left(\mathsf{k}_{n}\otimes\mathsf{m}_{n}% \right)\right)$ belongs to $U\mathcal{N(\mathcal{T})}U^{*}$ if and only if it commutes with $$\oplus_{n\in I}\left(\hbox{\rm 1\kern-2.8ptl}_{\mathsf{k}_{n}}\otimes\delta_{M% _{0}^{(n)}}^{m}(M_{\ell}^{(n)})\right),\qquad\oplus_{n\in I}\left(\hbox{\rm 1% \kern-2.8ptl}_{\mathsf{k}_{n}}\otimes\delta_{M_{0}^{(n)}}^{m}(M_{\ell}^{(n)*})\right)$$ for every $m\geq 0$. Now, let $q_{n}:=\hbox{\rm 1\kern-2.8ptl}_{\mathsf{k}_{n}}\otimes\hbox{\rm 1\kern-2.8ptl% }_{\mathsf{m}_{n}}$ the orthogonal projection onto $\mathsf{k}_{n}\otimes\mathsf{m}_{n}$. We clearly have $q_{n}\in U\mathcal{F(\mathcal{T})}U^{*}$ and $\sum_{n\in I}q_{n}=\hbox{\rm 1\kern-2.8ptl}$. Therefore, the algebra $q_{n}\mathcal{B}(\mathsf{h})q_{n}=\mathcal{B}(\mathsf{k}_{n}\otimes\mathsf{m}_% {n})$ is preserved by the action of every map $\widetilde{\mathcal{T}}_{t}:=U\mathcal{T}_{t}(U^{*}\cdot U)U^{*}$, and so we can consider the restriction of $\widetilde{\mathcal{T}}$ to this algebra, getting a QMS on $\mathcal{B}(\mathsf{k}_{n}\otimes\mathsf{m}_{n})$ denoted by $\mathcal{T}^{(n)}$ and satisfying $\mathcal{N}(\mathcal{T}^{(n)})=q_{n}U\mathcal{N(\mathcal{T})}U^{*}q_{n}$, where $\mathcal{N}(\mathcal{T}^{(n)})$ is the decoherence-free algebra of $\mathcal{T}^{(n)}$. Note that, since each $q_{n}$ commutes with every $UL_{\ell}^{\prime}U^{*}$, $UL_{\ell}^{\prime*}U^{*}$ and $UH^{\prime}U^{*}$, a GKSL representation of the generator $\mathcal{L}^{(n)}$ of $\mathcal{T}^{(n)}$ is given by operators $$q_{n}UH^{\prime}U^{*}q_{n}=K_{n}\otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{m}_{n% }}+\hbox{\rm 1\kern-2.8ptl}_{\mathsf{k}_{n}}\otimes M_{0}^{(n)}\qquad q_{n}UL^% {\prime}_{\ell}U^{*}q_{n}=\hbox{\rm 1\kern-2.8ptl}_{\mathsf{k}_{n}}\otimes M_{% \ell}^{(n)},$$ so that $$\widetilde{\mathcal{T}}_{t}(x\otimes y)=e^{{\mathrm{i}}tK_{n}}xe^{-{\mathrm{i}% }tK_{n}}\otimes\mathcal{T}_{t}^{\mathsf{m}_{n}}(y)\qquad\forall\,x\in\mathcal{% B}(\mathsf{k}_{n}),\ y\in\mathcal{B}(\mathsf{m}_{n})$$ and $$\displaystyle\mathcal{N}(\mathcal{T}^{(n)})$$ $$\displaystyle=\{\hbox{\rm 1\kern-2.8ptl}_{\mathsf{k}_{n}}\otimes\delta_{M_{0}^% {(n)}}^{m}(M_{\ell}^{(n)}),\hbox{\rm 1\kern-2.8ptl}_{\mathsf{k}_{n}}\otimes% \delta_{M_{0}^{(n)}}^{m}(M_{\ell}^{(n)*})\,:\ell\geq 1,\ m\geq 0\}^{\prime}$$ $$\displaystyle=\mathcal{B}(\mathsf{k}_{n})\otimes\mathcal{N}(\mathcal{T}^{% \mathsf{m}_{n}}).$$ (26) Since $$U\mathcal{N(\mathcal{T})}U^{*}=\bigoplus_{n,m\in I}q_{n}U\mathcal{N(\mathcal{T% })}U^{*}q_{m}$$ and equation (26) holds, to conclude the proof we have to show that $$q_{n}U\mathcal{N(\mathcal{T})}U^{*}q_{m}=\{0\}\quad\forall\,n\neq m,\qquad% \mbox{and}\qquad\mathcal{N}(\mathcal{T}^{\mathsf{m}_{n}})=\mathbb{C}\hbox{\rm 1% \kern-2.8ptl}_{\mathsf{m}_{n}}.$$ So, let $x\in U\mathcal{N(\mathcal{T})}U^{*}$ and consider $n,m\in I$ with $n\neq m$. Since the net $(\mathcal{T}_{t}(q_{n}xq_{m}))_{t\geq 0}$ is bounded in norm and the unit ball is weakly* compact, there exists a weak* cluster point $y$ such that $y=w^{*}-\lim_{\alpha}\mathcal{T}_{t_{\alpha}}(q_{n}xq_{m})$. Therefore, for $\sigma\in\mathcal{R}(\mathcal{T}_{*})$ with $\mathcal{T}_{*t}(\sigma)=e^{{\mathrm{i}}t\lambda}\sigma$ for some $\lambda\in\mathbb{R}$, we have $${\rm tr}\left(\sigma y\right)=\lim_{\alpha}{\rm tr}\left(\sigma\mathcal{T}_{t_% {\alpha}}(q_{n}xq_{m})\right)=\lim_{\alpha}e^{{\mathrm{i}}t_{\alpha}\lambda}{% \rm tr}\left(\sigma q_{n}xq_{m}\right).$$ Now, if $\lambda\neq 0$ this implies ${\rm tr}\left(\sigma q_{n}xq_{m}\right)=0={\rm tr}\left(\sigma y\right)$; otherwise, since $\sigma$ is an invariant state, we automatically have $q_{m}\sigma q_{n}=0$ for $n\neq m$ by Theorem 22, so that ${\rm tr}\left(\sigma y\right)$ is $0$ again. Consequently, ${\rm tr}\left(\sigma y\right)=0$ for all $\sigma\in\mathcal{R}(\mathcal{T}_{*})$. On the other hand, taking $\eta$ such that $0\in\overline{\{\mathcal{T}_{*t}(\eta)\}}^{w}_{t\geq 0}$, up to passing to generalized subsequences we have $${\rm tr}\left(\eta y\right)=\lim_{\alpha}{\rm tr}\left(\eta\mathcal{T}_{t_{% \alpha}}(q_{n}xq_{m})\right)=\lim_{\alpha}{\rm tr}\left(\mathcal{T}_{*t_{% \alpha}}(\eta)q_{n}xq_{m}\right)=0.$$ We can then conclude that ${\rm tr}\left(\sigma y\right)=0$ for all $\sigma\in\mathfrak{I}(\oplus_{n}\left(\mathsf{k}_{n}\otimes\mathsf{m}_{n}% \right))$ by virtue of equation (14), and so $y=0$. This means that $q_{n}xq_{m}$ belongs to $\mathcal{N(\mathcal{T})}\cap\mathfrak{M}_{s}$ and this intersection is $\{0\}$ by Corollary 12 ($\mathcal{N(\mathcal{T})}$ is atomic). Therefore $q_{n}xq_{m}=0$ for $n\neq m$ and so $$U\mathcal{N(\mathcal{T})}U^{*}=\oplus_{n\in I}q_{n}\mathcal{N(\mathcal{T})}q_{% n}=\oplus_{n\in I}\mathcal{N}(\mathcal{T}^{(n)}).$$ Moreover, since $\mathcal{N(\mathcal{T})}$ is atomic by Corollary 12, the general theory of von Neumann algebras (see e.g. [28]) says that each algebra $\mathcal{N}(\mathcal{T}^{(n)})=\mathcal{B}(\mathsf{k}_{n})\otimes\mathcal{N}(% \mathcal{T}^{\mathsf{m}_{n}})$ is atomic too, and consequently $\mathcal{N}(\mathcal{T}^{\mathsf{m}_{n}})$ is atomic. Finally, recalling that, by construction, $\mathcal{T}_{t}^{\mathsf{m}_{n}}=\mathcal{T}_{t}^{\mathsf{f}_{j}}$ for all $j\in I_{n}$ with $\mathcal{T}_{t}^{\mathsf{f}_{j}}$ an irreducible QMS having a faithful invariant state (see Theorem 22), we get $\mathcal{N}(\mathcal{T}^{\mathsf{m}_{n}})=\mathbb{C}\hbox{\rm 1\kern-2.8ptl}_{% \mathsf{m}_{n}}$ thanks to Proposition $4.3$ in [13]. ∎ Remark 26. Theorem 25 provides in particular a way to pass from the decomposition of GKSL operators $H$, $(L_{l})_{l}$ of $\mathcal{L}$ according to the splitting of $\mathsf{h}=\oplus_{j\in j}(\mathsf{s}_{j}\otimes\mathsf{f}_{j})$ associated with the atomic algebra $\mathcal{F(\mathcal{T})}$, to the other one decomposition with respect to the splitting $\mathsf{h}=\oplus_{n\in I}\left(\mathsf{k}_{n}\otimes\mathsf{m}_{n}\right)$ associated with $\mathcal{N(\mathcal{T})}$. More precisely, if $\mathcal{F(\mathcal{T})}=\oplus_{j\in J}\left(\mathcal{B}(\mathsf{s}_{j})% \otimes\hbox{\rm 1\kern-2.8ptl}_{\mathsf{f}_{j}}\right)$ and $$L_{\ell}=\oplus_{j\in J}\left(\hbox{\rm 1\kern-2.8ptl}_{\mathsf{s}_{j}}\otimes N% ^{(j)}_{\ell}\right)\quad\forall\,\ell\geq 1,\qquad H=\oplus_{j\in J}\left(% \lambda_{j}\hbox{\rm 1\kern-2.8ptl}_{\mathsf{s}_{j}}\otimes\hbox{\rm 1\kern-2.% 8ptl}_{\mathsf{f}_{j}}+\hbox{\rm 1\kern-2.8ptl}_{\mathsf{s}_{j}}\otimes N^{(j)% }_{0}\right)$$ with $(N_{\ell}^{(j)})_{\ell\geq 1}\subseteq\mathcal{B}(\mathsf{s}_{j})$, $N^{(j)}_{0}\in\mathcal{B}(\mathsf{f}_{j})$ and $\lambda_{j}\in\mathbb{R}$ for all $j\in J$, then we can decompose $H,(L_{\ell})_{\ell}$ with respect to the splitting $\mathsf{h}=\oplus_{n\in I}\left(\mathsf{k}_{n}\otimes\mathsf{m}_{n}\right)$ given by (25) as follows: $$L_{\ell}=\oplus_{n\in I}\left(\hbox{\rm 1\kern-2.8ptl}_{\mathsf{k}_{n}}\otimes M% _{\ell}^{(n)}\right),\qquad H=\oplus_{n\in I}\left(K_{n}\otimes\hbox{\rm 1% \kern-2.8ptl}_{\mathsf{m}_{n}}+\hbox{\rm 1\kern-2.8ptl}_{\mathsf{k}_{n}}% \otimes M_{0}^{(n)}\right),$$ where $(M_{\ell}^{(n)})_{\ell\geq 1},M^{(n)}_{0}=M^{(n)*}_{0}$ are operators in $\mathcal{B}(\mathsf{m}_{n})$ giving the same GKSL representation of $(N_{l}^{(j)})_{\ell\geq 1},N_{0}^{(j)}$ for all $j\in I_{n}$, and every $K_{n}$ is the self-adjoint operator on $\mathcal{B}({\mathsf{k}_{n}})$ having $(\lambda_{j})_{j\in I_{n}}$ as eigenvalues and $(\mathsf{s}_{j})_{j\in I_{n}}$ as corresponding eigenspaces. 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[31] M. M. Wolf, Quantum channel and operations, guided tour, notes, (2012). ${}^{1}$ Department of Mathematics, Politecnico di Milano, Piazza Leonardo da Vinci 32, I - 20133 Milano, Italy E-mail: franco.fagnola@polimi.it ${}^{2}$ Department of Mathematics, University of Genova, Via Dodecaneso, I - 16146 Genova, Italy E-mail: sasso@dima.unige.it ${}^{3}$ Department of Mathematics, University of Genova, Via Dodecaneso, I - 16146 Genova, Italy E-mail: umanita@dima.unige.it

A unifying framework for ghost-free Lorentz-invariant Lagrangian field theories Wenliang LI lii.wenliang@gmail.com APC, Université Paris 7, CNRS/IN2P3, CEA/IRFU, Obs. de Paris, Sorbonne Paris Cité, Bâtiment Condorcet, 10 rue Alice Domon et Léonie Duquet, F-75205 Paris Cedex 13, France (UMR 7164 du CNRS) (November 19, 2020) Abstract We develop a general framework for Lorentz-invariant Lagrangian field theories that leads to second order equations of motion. The key ingredient is the antisymmetric Kronecker delta. Then we reformulate the general ghost-free Lagrangians in the language of differential forms. The absence of higher order equations of motion stems from the basic fact that every exact form is closed. All known ghost-free Lagrangian theories for spin-0, spin-1, spin-2 fields have natural formulations in this framework. We propose new ghost-free Lagrangians, for example, novel nonlinear kinetic terms for graviton. pacs: Introduction.— The problem of ghost-like degrees of freedom are frequently encountered in the construction of theories with large numbers of spacetime indices, either from high spin fields or high order derivatives. In high spin theories, the most dangerous degrees of freedom are the scalar modes, where the tensor indices are effectively replaced by derivative indices, so the origin of the high spin ghosts can be considered as high order derivatives as well. A host of ghost-free theories were carefully constructed and a common pattern emerged. The linear theory of ghost-free massive gravity requires the Fierz-Pauli tuning Fierz:1939ix , where the quadratic mass terms are contracted antisymmetrically. When Lovelock studied the most general metric theories Lovelock:1971yv , he found the ghost-free combinations are Riemann curvature tensors contracted antisymmetrically. The antisymmetric structure appeared, again and again, in the most general ghost-free scalar theories Nicolis:2008in , the recent nonlinear mass terms for spin-two fields deRham:2010kj , etc.vector-galileon ; Horndeski:1974wa This general pattern is not only for unconventional theories. Both the Maxwell action for massless spin-1 fields and the linearized Einstein-Hilbert action for massless spin-two fields are antisymmetrically contracted two-derivative quadratic actions. Antisymmetrization is a universal element for ghost-free Lagrangian theories. Therefore, we are tempted to think that there is a unifying framework for ghost-free, Lorentz-invariant, Lagrangian field theories with the key ingredient, antisymmetrization. In this letter, we develop such a general framework and construct novel ghost-free theories. To summarize, the general ghost-free Lagrangians are D-forms $$\mathcal{L}=f\,\omega^{1}\wedge\dots\wedge\omega^{n},$$ (1) where $D$ is the number of spacetime dimensions and $\omega^{i}$ indicate the matter differential forms and the geometric (or gravitational) differential forms. The absence of higher order equations of motion is due to the basic property of exterior derivative $$d^{\,2}=0.$$ (2) Antisymmetric Kronecker delta.— Ostrogradski’s theorem states that the energy of a Lagrangian theory with high order time derivative terms is not bounded from below because the Hamiltonian will be linear in a conjugate momentum Woodard:2006nt . The negative energy modes are ghost-like degrees of freedom. A loophole in Ostrogradski’s proof is the assumption that the Lagrangian is non-degenerate. In other words, if the Euler-Lagrange equations are, for some reason, of second order, a theory with high order derivative action is still healthy and no ghost-like degrees of freedom will be propagating. Below we present an intuitive derivation on how the antisymmetric Kronecker delta arises as a result of second order equations of motion together with Lorentz invariance. We derive the general form of ghost-free, Lorentz-invariant Lagrangian in the language of tensors. By Lorentz invariance, we mean the theories do not distinguish among time and different space indices up some signs. This applies to gravitational theories when spacetime is curved. A Lagrangian constructed from first order terms will not lead to higher order equations of motion. Let us consider a Lagrangian with second order derivative terms $$\mathcal{L}\sim\phi\dots\phi\,\partial\partial\phi\dots\partial\partial\phi,$$ (3) where $\phi$ indicates dynamical fields and they can have tensor indices. Without $\phi\dots\phi$, the equations of motion cannot be of second order. (If the Lagrangians are total derivative terms, they will not contribute to the equations of motion.) Higher order derivative terms in the Lagrangian will lead to high order terms in the equations of motion after varying the action with respect to $\phi$. The first order derivative terms are discussed later. Now we examine the variations of the product of two second order derivative terms $$\delta(\partial_{a_{1}}\partial_{a_{2}}\phi\,\partial_{b_{1}}\partial_{b_{2}}% \phi\dots)\rightarrow\left(\partial_{a_{1}}\partial_{b_{1}}\partial_{b_{2}}% \phi\,\partial_{a_{2}}\dots+\,\dots\right),$$ (4) where we concentrate on a third order term. There are fourth order derivative terms as well, but the spirit manifest itself already in the third order terms. Since derivatives commute with each others, third order derivative terms with the same indices but different order are equivalent. The coefficient of a third order derivative term is $$\displaystyle C^{\mu,\nu\rho}\partial_{\mu}\partial_{\nu}\partial_{\rho}\phi=% \Big{(}C^{a,bc}+C^{a,cb}+C^{b,ac}+C^{b,ca}$$ $$\displaystyle\,+C^{c,ab}+C^{c,ba}\Big{)}\partial_{a}\partial_{b}\partial_{c}\phi$$ (5) To have vanishing coefficient, there are simple ansatzes: either $(\mu,\nu)$ or $(\mu,\rho)$ are antisymmetric. For the second cubic derivative term, we have the same requirement, so we have two ansatz to obtain second order equations of motion: • $(a_{1},\,b_{1})$ and $(a_{2},\,b_{2})$ are two sets of antisymmetrised indices; • $(a_{1},\,b_{2})$ and $(a_{2},\,b_{1})$ are two sets of antisymmetrised indices. The same requirement for other second order derivative terms lead to two chains of antisymmetrised indices. One can construct special solutions with fine-tuned $C^{\mu,\nu\rho}$ such that (5) vanishes without the antisymmetric structure, but usually these solutions will distinguish among the time and different space indices, which are ruled out by Lorentz invariance. Let us come back to the case with $\partial\phi$. At first sight, varying the first order term will lead to a third order term $$\delta(\partial\phi)\,\partial\partial\phi\rightarrow-\partial\partial\partial\phi,$$ (6) but in single field theories this term is cancelled by varying the corresponding second order term $$\partial\phi\,\delta(\partial\partial\phi)\rightarrow\partial\partial\partial\phi.$$ (7) The difference in sign is due to the different numbers of integration by parts in the two cases. Now we consider the tensor indices in the dynamical fields. Let us concentrate on the scalar mode $$\phi_{\mu\dots}=\partial_{\mu}\dots\Phi.$$ (8) which is the most dangerous due to the large number of derivatives in front of it. Varying the action with respect to $\Phi$, these derivatives move to other second order derivative terms. Therefore, the tensor field indices should be included into the antisymmetric chains. Extending the analysis of first order derivative terms, single-index terms (spin-1 fields and first derivative of spin-0 fields) will not lead to higher order equations of motion in single field theories. It is not necessary, but not harmful, to include their indices to the antisymmetric chains. For multi-field interaction terms, it becomes necessary as the terms from the two variations do not cancelled out. Note that we have only two chains, which are designed for the second order derivative terms, so only tensor fields with at most two indices are allowed. Lagrangian theories of interacting (symmetric) higher spin fields around Minkowski background are plagued by Ostrogradski’s instability. From the two chains of antisymmetrised indices, we obtain the general ghost-free Lagrangian densities $$\mathcal{L}=f(\phi,\partial\phi)\,\delta_{\nu_{1}\nu_{2}\dots}^{\mu_{1}\mu_{2}% \dots}\prod\omega_{\mu\dots}^{\nu\dots}\quad,$$ (9) where $f$ is a scalar function of scalar fields (and single-index terms if only one field is involved), $\delta_{\nu\dots}^{\mu\dots}$ is antisymmetric Kronecker delta defined as the product of two Levi-Civita symbols with contracted indices $$\delta_{\nu_{1}\nu_{2}\dots}^{\mu_{1}\mu_{2}\dots}=\epsilon_{\nu_{1}\nu_{2}% \dots}\epsilon^{\mu_{1}\mu_{2}\dots}\dots\delta_{\mu_{D}}^{\nu_{D}}\quad.$$ (10) and $\omega$ denotes dynamical fields or their derivative terms $$\omega_{\mu\dots}^{\nu\dots}=(\partial\dots\phi)_{\mu\dots}^{\nu\dots}\quad.$$ (11) The two chains of antisymmetrized indices are denoted by $\mu_{i}$ and $\nu_{i}$. $\omega$ have at most four indices (two for derivatives and two for tensor fields). For second derivative terms, the derivative indices should be in different chains. For symmetric spin-2 fields, the tensor indices cannot be in the same chains. To obtain the antisymmetric Kronecker delta, we assume the indices in the two chains are contracted with each others. Otherwise, the free indices will be contracted with those of single-index terms, which will add the indices of the additional terms to the other antisymmetric chains. The Lagrangian is again in the general form (9). Differential forms.— We want to reformulate the general ghost-free Lagrangians (9) in the language of differential forms. This is motivated by the crucial importance of the antisymmetric Kronecker delta, which indicates differential form is a natural language. Here is another motivation. Lovelock’s theory has an elegant expression and close relation with the Euler density, but its understanding is not fully satisfactory until Zumino’s reformulation in terms of vielbein Zumino:1985dp . The absence of high derivative terms in the equations of motion is due to the Bianchi identities, which come from the basic identity of exterior derivative $d^{2}=0$. As we discuss below, this statement applies to the general ghost-free Lagrangian field theories. In terms of the differential forms, the general ghost-free Lagrangian densities (9) are D-forms $$\mathcal{L}=f\,\omega\wedge\dots\wedge\omega\wedge\eta\wedge\dots\wedge\eta,$$ (12) where the differential forms $\omega$ and $\eta$ are defined as $$\omega^{\nu\dots}=\omega^{\nu\dots}_{\mu_{1}\dots}dx^{\mu_{1}}\wedge\dots,% \quad\eta^{\nu}={\delta_{\mu}}^{\nu}dx^{\mu}.$$ (13) $D$ is the dimension of spacetime and the upper indices $\nu_{i}$ in (12) are contracted with Levi-Civita symbol. We interpret $\eta$ as the Minkowski vielbein. Terms with $\mu$-derivatives are exact forms. For instance, the $\mu$-derivative of a spin-1 field is $$\omega^{\nu}=\partial_{\mu}A^{\nu}\,dx^{\mu}=d\,A^{\nu}.$$ (14) Varying the action with respect to $A^{\nu}$, the exterior derivative $d$ will move to other terms after an integration by parts. When it acts on a second derivative term, the corresponding term vanishes thanks to $$d^{\,2}=0.$$ (15) Therefore, the absence of higher order equations of motion origins in the basic fact that every exact form is closed. Note that the formulations in the language of differential forms have a duality as the theories are invariant under the exchange between $\mu$ and $\nu$, so the above discussion applies to the $\nu$-derivatives as well. Below we discuss some concrete examples of the general ghost-free Lagrangian. Single spin-0 fields.— The general ghost-free Lagrangian densities (9) for single scalar field are $$\mathcal{L}_{1,n}^{(0)}=f(\phi,\partial^{\rho}\phi\partial_{\rho}\phi)\,\delta% _{\nu_{1}\dots\nu_{n}}^{\mu_{1}\dots\mu_{n}}\,\prod_{k=1}^{n}\partial_{\mu_{k}% }\partial^{\nu_{k}}\phi$$ (16) and $$\mathcal{L}_{2,n}^{(0)}=f(\phi,\partial^{\rho}\phi\partial_{\rho}\phi)\,\delta% _{\nu_{1}\dots\nu_{n}}^{\mu_{1}\dots\mu_{n}}\,\partial_{\mu_{1}}\phi\,\partial% ^{\nu_{1}}\phi\prod_{k=2}^{n}\partial_{\mu_{k}}\partial^{\nu_{k}}\phi,$$ (17) where $(0)$ indicates only one spin-0 field is under consideration and $n$ labels the numbers of derivatives in the Lagrangians. The Lagrangian (16) has no single-index term and the Lagrangian (17) has two single-index terms. If we consider more single-index terms, the Lagrangian will vanish due to antisymmetrization. The two Lagrangians (16) and (17) are related by using the properties of antisymmetric Kronecker delta and integrating by parts. They are two equivalent formulations of the Galileon theories, the most general ghost-free, Lorentz-invariant, single scalar field theories. In this case, we do not have the freedom to construct novel models due to the limited number of indices. Single spin-1 fields.— The general ghost-free Lagrangian densities (9) for single vector fields are $$\displaystyle\mathcal{L}_{1}^{(1)}=$$ $$\displaystyle\,f(A_{\mu}A^{\mu})\delta^{\mu\dots}_{\nu\dots}[\partial_{\mu}A_{% \mu}][\partial^{\nu}A^{\nu}]$$ $$\displaystyle\qquad[\partial_{\mu}A^{\nu}][\partial^{\nu}A_{\mu}][\partial_{% \mu}\partial^{\nu}A_{\mu}][\partial_{\mu}\partial^{\nu}A^{\nu}],$$ (18) $$\displaystyle\mathcal{L}_{2}^{(1)}=$$ $$\displaystyle\,f(A_{\mu}A^{\mu})\delta^{\mu\dots}_{\nu\dots}A_{\mu}[\partial_{% \mu}A_{\mu}][\partial^{\nu}A^{\nu}]$$ $$\displaystyle\qquad[\partial_{\mu}A^{\nu}][\partial^{\nu}A_{\mu}][\partial_{% \mu}\partial^{\nu}A_{\mu}][\partial_{\mu}\partial^{\nu}A^{\nu}],$$ (19) and $$\displaystyle\mathcal{L}_{3}^{(1)}=$$ $$\displaystyle\,f(A_{\mu}A^{\mu})\delta^{\mu\dots}_{\nu\dots}A_{\mu}A^{\nu}[% \partial_{\mu}A^{\nu}][\partial^{\nu}A_{\mu}]$$ $$\displaystyle\qquad[\partial_{\mu}A_{\mu}][\partial^{\nu}A^{\nu}][\partial_{% \mu}\partial^{\nu}A_{\mu}][\partial_{\mu}\partial^{\nu}A^{\nu}],$$ (20) where $[X]$ denotes the product of $X$ $$[X]=\prod X.$$ (21) To simplify the notation, the $i$ indices in $\mu_{i},\,\nu_{i}$ are not written explicitly. The vector Galileon terms in vector-galileon are special cases of (18) without second order derivative terms. Similar to the case of single scalar field, (18-20) are not completely independent models. In four dimensions, the number of inequivalent theories are significantly reduced because the length of the antisymmetric chain cannot be larger than the number of spacetime indices. Below we examine a novel theory constructed from second derivative terms. Its Lagrangian reads $$\mathcal{L}_{\text{ex}}^{(1)}=f(A_{\rho}A^{\rho})\,\delta^{\mu_{1}\mu_{2}\mu_{% 3}}_{\nu_{1}\nu_{2}\nu_{3}}\,\partial_{\mu_{1}}\partial^{\nu_{1}}A_{\mu_{2}}% \partial_{\mu_{3}}\partial^{\nu_{2}}A^{\nu_{3}}.$$ (22) The zero component of the conjugate momentum is $$\displaystyle(\pi_{\text{ex}}^{(1)})^{0}=$$ $$\displaystyle\,\frac{\partial\mathcal{L}_{1}^{\text{ex}}}{\partial\dot{A}_{0}}$$ $$\displaystyle=$$ $$\displaystyle\,-4f^{\prime}A^{\mu}(\nabla^{2}A_{i}\partial^{i}-\partial_{j}% \partial^{i}A_{i}\partial^{j})A_{\mu},$$ (23) where ${}^{\prime}$ is derivative respect to $A_{\rho}A^{\rho}$. The time derivative terms in $\pi^{0}$ cancel out, so equation (23) is a primary constraint. To preserve this primary constraint in time, we obtain a secondary constraint equation. These two constraint equations eliminate the ghost-like degrees of freedom in this high derivative theory. If one substitutes $A_{\mu}$ with $A^{T}_{\mu}+\partial_{\mu}\phi$, the scalar mode appear only in the scalar function $f$ and varying the action with respect to $\phi$ will not generate higher order equations of motion. By construction, the absence of time derivative terms in $\pi^{0}$ is a universal feature of there theories. The reason behind this is that the 0 indices in the two antisymmetric chains are already used in $\dot{A}_{0}$. Single spin-2 fields.— The general ghost-free Lagrangian densities (9) for single spin-2 fields is $$\mathcal{L}^{(2)}=\delta^{\mu\dots}_{\nu\dots}[{h_{\mu}}^{\nu}][\partial_{\mu}% {h_{\mu}}^{\nu}][\partial^{\nu}{h_{\mu}}^{\nu}][\partial_{\mu}\partial^{\nu}{h% _{\mu}}^{\nu}].$$ (24) The scalar function $f$ becomes constant as spin-2 fields have at least two indices and they must be included to the antisymmetric chains. We do not have the ambiguity of single-index terms anymore. By integrating by parts, the general Lagrangian becomes $$\mathcal{L}_{i,j}^{(2)}=\delta^{\mu_{1}\dots\mu_{2i+j}}_{\nu_{1}\dots\nu_{2i+j% }}\left(\prod_{k=1}^{i}\partial_{\mu}\partial^{\nu}{h_{\mu}}^{\nu}\right)\left% (\prod_{k=1}^{j}{h_{\mu}}^{\nu}\right).$$ (25) We can see the spin-2 field theories are more constrained than the spin-1 theories due to the large number of indices. When $j=0$, the Lagrangian is a total derivative, so we require $j>0$. For $j=1$, the Lagrangians $\mathcal{L}_{i,1}^{(2)}$ are the leading nontrivial terms of the perturbative Lovelock terms. The linearized Einstein-Hilbert term corresponds to the case $i=j=1$. The Fierz-Pauli term is $\mathcal{L}_{0,2}^{(2)}$. The other zero-derivative $i=0$ interaction terms $\mathcal{L}_{0,j}^{(2)}$ can be thought of as the perturbative terms from the nonlinear dRGT potentials. However, the Lagrangian (25) is not very satisfactory if we interpret the spin-2 fields $h_{\mu\nu}$ as the metric perturbations $$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}.$$ (26) The action of a gravitational theory should be a functional of the basic dynamical variables, namely the full metric field $g_{\mu\nu}$, rather than the perturbations $h_{\mu\nu}$. For example, it was proposed in Hinterbichler:2013eza that $\mathcal{L}_{1,2}^{(2)}$ is the perturbative term of a new kinetic term for 4D massive graviton, but the nonlinear theory is not known and this proposal is not convincing enough deRham:2013tfa . In principle, one can construct the nonlinear theory order by order in fields. However, this powerful perturbative procedure slows down at high orders and to obtain the complete nonlinear theory might require endless work. In addition, it is believed that the soul of a gravitational theory lives in geometry. For instance, the proposal has never been fulfilled to derive general relativity as a consistent theory of massless spin-2 field around flat background. The only exception is Deser’s work Deser:1969wk , where the geometric notion of connection $\Gamma^{\alpha}_{\mu\nu}$ was secretly introduced by formulating the problem in a first order form. Below we use some geometric intuition to obtain the general nonlinear ghost-free gravitational theories. As we will see, reformulating this general framework in the language of differential forms gives us a bonus. In the language of differential forms, the vielbein is a more natural building block than the metric. The second derivative of vielbein does not make much sense from the perspective of geometry. A more geometric choice of two-derivative term is the curvature two-form. Therefore, the general nonlinear ghost-free Lagrangian densities for graviton should be $$\mathcal{L}^{(2)}_{i,j}=R(E)\wedge\dots E\wedge\dots\eta\wedge\dots,$$ (27) which are the wedge products of the curvature two-form $R(E)$, the dynamical vielbein $E$ and the Minkowski background vielbein $\eta$. $(i,j)$ indicate the numbers of the the curvature two-form, the dynamical vielbein in the wedge product. There are $(D-2i-j)$ Minkowski vielbein $\eta$ in the wedge product. The notation $\eta$ is the same as that in (12). The Lovelock terms correspond $\mathcal{L}^{(2)}_{i,D-2i}$ and the Einstein-Hilbert action is $\mathcal{L}^{(2)}_{1,D-2}$. The dRGT terms in the vielbein reformulation Hinterbichler:2012cn are $\mathcal{L}^{(2)}_{0,j}$. Coupled multiple fields.— The choice is rich for ghost-free interactions of multiple fields. An important difference from the cases of single fields is that all the single-index terms should be contracted antisymmetrically (when all the fields have second derivative terms). $f$ are functions of the scalar fields without derivative and do not contain single-index terms. The ghost-free interactions of multiple fields are the wedge product of the possible terms. For example, the Lagrangians of bi-/ multi-galileons Padilla:2010de are the wedge products of the exact forms constructed from different scalar fields. A subtle point concerning graviton is that the ordinary derivatives are replaced by covariant derivatives because of covariantization. This is required by the universal coupling of gravity. The connection part of the covariant derivatives will lead to higher order equations of motion, so counter-terms should be introduced to the action Deffayet:2009wt . Conclusions.— We develop a general framework for ghost-free, Lorentz-invariant, Lagrangian field theories. We reformulate the general ghost-free Lagrangian in the language of differential forms. It is the wedge product of the matter differential forms (constructed from spin-0 and spin-1 fields) and the geometric (or gravitational) differential forms (vielbeins and curvature two-forms). The proposed ghost-free models deserve more careful examination to see whether they are indeed healthy. One of the most interesting case is the novel two-derivative kinetic terms for graviton $\mathcal{L}^{(2)}_{1,j,k}$ with only one curvature two-form in the wedge product. In four dimensions, it is shown in gr-kin that only the healthy degrees of freedom are propagating in these novel nonlinear gravitational theories. Another interesting possibility is to extend the Lovelock terms of single graviton to the case of bi-/multi- gravitons. It will be interesting to find first order formulations for these high derivative action by introducing auxiliary variables. Then the high derivative actions are recovered by solving the equations of motion of these auxiliary variables. The case of Galileon theories has been done, where the Galileon actions can be thought of as Wess-Zumino terms Goon:2012dy . General ghost-free theories for half-integer spin fields are not well-explored. They are important because the matter content in the standard model of particle physics consists of spin-$\frac{1}{2}$ particles. Only in the vielbein formulation, the Dirac spinors are coupled to gravity. The reformulation of our framework in the language of differential forms should be a proper starting point for the investigation of the general ghost-free theories for half-integer spin fields. Acknowledgements. The author would like to give special thanks to Xian Gao for numerous stimulating discussions. The author’s thanks also go to Francesco Nitti for encouragement when the idea of this work first appeared in the author’s PhD thesis. 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Non-integer characterizing slopes and knot Floer homology Duncan McCoy Département de mathématiques Université du Québec à Montréal Canada mc_coy.duncan@uqam.ca Abstract. Conjecturally, a knot in the 3-sphere has only finitely many non-integer non-characterizing slopes. We verify this conjecture for all knots with knot Floer homology satisfying certain simplicity conditions. The class of knots satisfying our notion of simplicity includes alternating knots, $L$-space knots and the vast majority of knots with at most 12 crossings. For arbitrary knots in the 3-sphere we show that almost all slopes $p/q$ with $|q|\geq 3$ are characterizing. In addition, we show that all $L$-space knots and almost $L$-space knots have infinitely many integer characterizing slopes. 1. Introduction Given a knot $K$ in $S^{3}$, a rational number $p/q\in\mathbb{Q}$ is a characterizing slope for $K$, if the existence of an orientation preserving homeomorphism between $S_{K}^{3}(p/q)$ and $S_{K^{\prime}}^{3}(p/q)$ implies that $K$ and $K^{\prime}$ are isotopic. That is, if the oriented homeomorphism type of $S_{K}^{3}(p/q)$ distinguishes $K$ amongst all knots in the 3-sphere. In general determining the exact set of characterizing slopes for a given knot is challenging and there are very few examples where this has been done. Every slope is a characterizing slope for the unknot [KMOS07], the trefoil and the figure-eight knot [OS19]. More recently, Sorya has exhibited an infinite class of knots for which the characterizing slopes are precisely the non-integer slopes [Sor23]. Other knots whose characterizing slopes have been studied in detail include the torus knot $T_{5,2}$ [NZ14, NZ21], $5_{2}$ [BS22], the $(-2,3,7)$-pretzel knot [McC22] and various Whitehead doubles [Wak23]. This article aims to understand the coarse structure of the set of characterizing slopes for an arbitrary knot. In particular, we are motivated by the conjecture that for any given knot almost all non-integer slopes are characterizing. Conjecture 1.1. Let $K$ be a knot in $S^{3}$. Then there exists a constant $C=C(K)$ such that any slope $p/q$ satisfying $|q|\geq 2$ and $|p|+|q|\geq C$ is characterizing for $K$. As there are knots with infinitely many integer non-characterizing slopes [BM18], this conjecture cannot be relaxed to omit the $|q|\geq 2$ condition. Conjecture 1.1 has already been verified for torus knots [McC20], hyperbolic $L$-space knots [McC19] and composite knots [Sor23]. As further evidence for this conjecture, Lackenby has shown that for a hyperbolic knot $K$ a slope $p/q$ is characterizing whenever $q$ is sufficiently large [Lac19]. This was generalized by Sorya, who showed that for any knot slopes with $q$ sufficiently large are characterizing [Sor23]. The first main result of this paper is to show that for a given knot almost all slopes with $|q|\geq 3$ are characterizing. Theorem 1.2. Let $K$ be a knot in $S^{3}$. Then there exists a constant $C=C(K)$ such that any slope $p/q$ satisfying $|q|\geq 3$ and $|p|+|q|\geq C$ is characterizing for $K$. This result was previously known for hyperbolic knots [McC19]. The main technical advance here is to show that it holds for satellite knots, which is to say, knots whose complements have a non-trivial JSJ decomposition. Thus the veracity of Conjecture 1.1 depends on the existence of half-integer characterizing slopes. Our second main result is to demonstrate the existence of such slopes for knots satisfying a technical knot Floer homology condition that we dub Property SpliFF (see Definition 1.5 below). The reader should interpret Property SpliFF as saying that $K$ has relatively simple knot Floer homology. Theorem 1.3. Let $K$ be a knot in $S^{3}$ with Property SpliFF. Then, there exists a constant $C=C(K)$ such that for any $p\geq C$, the slope $p/2$ is a characterizing slope for $K$. Recall that there is a homeomorphism $S_{K}^{3}(-p/q)\cong-S_{mK}^{3}(p/q)$, where $mK$ denotes the mirror of $K$. Thus, a slope $-p/q$ is characterizing for $K$ if and only if $p/q$ is a characterizing slope for $mK$. Therefore, if $K$ is a knot such that both $K$ and $mK$ have Property SpliFF, then Theorem 1.2 and Theorem 1.3 show that $K$ satisfies Conjecture 1.1. Using this, we establish Conjecture 1.1 for several classes of knots, notably including alternating knots and $L$-space knots. Furthermore, for the 2977 prime knots in the tables with at most 12 crossings, Conjecture 1.1 holds for at least 2951 of them. Corollary 1.4. Let $K$ be a knot satisfying at least one of the following conditions: (i) $K$ has thin knot Floer homology; (ii) $K$ is an $L$-space knot; or (iii) $K$ is a prime knot with at most 12 crossings and upto mirroring is not amongst the 26 knots listed in Table 1. Then there exists a constant $C=C(K)$ such that any slope $p/q$ satisfying $|q|\geq 2$ and $|p|+|q|\geq C$ is characterizing for $K$. 1.1. Property SpliFF We give a quick definition of Property SpliFF for readers familiar with knot Floer homology. Alternative formulations and a more detailed explanation of terminology will be given in Section 3. Given a knot $K$ with knot Floer chain complex $C=CFK^{\infty}(K)$ (with $\mathbb{F}=\mathbb{Z}/2\mathbb{Z}$ coefficients), we take $\bm{A}^{+}_{k}$ to be the homology of the quotient complex $$A_{k}^{+}=C\{i\geq 0\text{ or }j\geq k\}.$$ Recall that $\bm{A}^{+}_{k}$ has the structure of a $\mathbb{Z}$-graded $\mathbb{F}[U]$-module. We use $\mathbb{F}_{(d)}$ to denote the graded $\mathbb{F}[U]$-module consisting of a copy of $\mathbb{F}$ supported in grading $d$ with the module structure given by $U\mathbb{F}_{(d)}=0$. Definition 1.5. We say the knot $K$ has split $\mathbb{F}$-factors (Property SpliFF) if for all $k\in\mathbb{Z}$ the graded $\mathbb{F}[U]$-module $\bm{A}^{+}_{k}$ admits a direct sum decomposition of the form $$\bm{A}^{+}_{k}\cong A^{\prime}\oplus\mathbb{F}_{(d_{1})}^{n_{1}}\oplus\mathbb{F}_{(d_{2})}^{n_{2}},$$ where $n_{1},n_{2}\geq 0$, $d_{1}$ is odd, $d_{2}$ is even and the module $A^{\prime}$ contains no further summands of the form $\mathbb{F}_{(d)}$ for any $d$. This definition says that for each $\bm{A}^{+}_{k}$, all $\mathbb{F}_{(d)}$ summands are supported in at most two gradings one of which is odd and the other is even. The following proposition will be our main tool for exhibiting knots with Property SpliFF. Proposition 1.6. Let $K$ be a knot whose knot Floer homology has thickness at most one and let $\rho\in\mathbb{Z}$ be a constant such that $\widehat{HFK}_{d}(K,s)$ is non-zero only if $$d\in\{s+\rho,s+\rho-1\}.$$ (i) If $\rho\leq 2$ or at least one of the groups $\widehat{HFK}_{2\rho-3}(K,\rho-3)$ or $\widehat{HFK}_{2\rho-4}(K,\rho-3)$ is trivial, then $K$ has Property SpliFF. (ii) If $\rho\geq-1$ or at least of one of the groups $\widehat{HFK}_{2\rho+2}(K,\rho+2)$ or $\widehat{HFK}_{2\rho+1}(K,\rho+2)$ is trivial, then $mK$ has Property SpliFF. It is not hard to see that Proposition 1.6 implies that knots with thin knot Floer homology have Property SpliFF. Furthermore, with the exception of those listed in Table 1, Proposition 1.6 shows that for all prime knots with at most 12 crossings both $K$ and $mK$ have Property SpliFF. 1.2. Integer characterizing slopes Although this article primarily focused on non-integer characterizing slopes, we are also able to expand the class of knots known to have integer characterizing slopes. Previously, it was known that torus knots [NZ14] and hyperbolic $L$-space knots [McC19] have infinitely many integer characterizing slopes. We expand this class to include all $L$-space knots and almost $L$-space knots. The notion of an almost $L$-space knot was introduced by Baldwin and Sivek [BS22]. A knot $K$ in $S^{3}$ is an almost $L$-space knot if $$\dim\widehat{HF}(S_{K}^{3}(n))=n+2.$$ for some (equivalently any) integer $n$ satisfying $n\geq 2g(K)-1$. Theorem 1.7. Let $K$ be an $L$-space knot or an almost $L$-space knot in $S^{3}$. Then there exists a constant $C=C(K)$ such any slope $p/q>C$ is characterizing for $K$. One interesting example of an almost $L$-space knot is the composite knot $T_{3,2}\#T_{3,2}$. In fact, this is only composite almost $L$-space knot [Bin21]. In this case we obtain explicit bounds on the set of characterizing slopes for $T_{3,2}\#T_{3,2}$ and show that any integer slope $p\geq 32$ is characterizing slope for $T_{3,2}\#T_{3,2}$. As far as the author is aware, this is the first composite knot known to possess integer characterizing slopes. In contrast, Varvarezos has shown that there exists connected sums of torus knots with only finitely many integer characterizing slopes [Var23]. 1.3. Proof of Theorems 1.2 and 1.3 We conclude the introduction by proving Theorem 1.2 and Theorem 1.3 assuming Theorem 2.10 and Theorem 3.12, which are two technical results whose proofs occupy Section 2 and Section 3, respectively. Since both Theorem 1.2 and Theorem 1.3 follow the same strategy we proceed with both simultaneously. Let $K$ be a knot in $S^{3}$ and suppose that $p/q\in\mathbb{Q}\setminus\mathbb{Z}$ is a non-integer non-characterizing slope for $K$. Since $p/q$ is a characterizing slope for $K$ if and only if $-p/q$ is a characterizing slope for $mK$, we may assume that $p/q>0$. Thus, let $K^{\prime}$ be a knot such that $K^{\prime}\not\simeq K$ and $S_{K}^{3}(p/q)\cong S_{K^{\prime}}^{3}(p/q)$. The is aim to obtain upper bounds on $p$ and $q$ that are independent of the particular $K^{\prime}$ under consideration. Two recent results of Sorya will significantly aid us in our endeavours. Firstly, Sorya has already bounded $q$. Theorem 1.8 (Sorya, [Sor23]). Let $K$ be a knot in $S^{3}$. Then there exists a constant $Q=Q(K)$ such that any slope $p/q$ satisfying $|q|\geq Q$ is a characterizing slope for $K$. ∎ Secondly, as Sorya has shown that every non-integer is characterizing for a composite knot [Sor23]. Theorem 1.9 (Sorya, [Sor23]). Let $K$ be a composite knot in $S^{3}$. Any non-integer slope $p/q\in\mathbb{Q}\setminus\mathbb{Z}$ is characterizing for $K$. ∎ This allows us to further assume that $K$ and $K^{\prime}$ are both prime knots. We use two different techniques to bound the numerator $p$ depending on the genus of the knot $K^{\prime}$. When the genus of $g(K^{\prime})$ is small, which is to say $g(K^{\prime})\leq g(K)$, we are able to gain control over $p$ by carefully comparing the JSJ decompositions of the complements of the knots $K$ and $K^{\prime}$. This results in the following bound. Theorem 2.10. Let $K$ be a prime knot in $S^{3}$. Then there exists a constant $M=M(K)$ such that if $K^{\prime}\not\simeq K$ is a prime knot with $g(K^{\prime})\leq g(K)$ and we have a homeomorphism $S_{K}^{3}(p/q)\cong S_{K^{\prime}}^{3}(p/q)$ for some $p/q\in\mathbb{Q}$, then $|p|\leq M|q|$. The second method is to study the Heegaard Floer homology of surgeries on $K$ and $K^{\prime}$. This complements Theorem 2.10 by yielding upper bounds on $p$ when the genus of $K^{\prime}$ is relatively large, which is to say $g(K^{\prime})>g(K)$. The theorem stemming from this method is the following. Theorem 3.12. Let $K$ be knot in $S^{3}$. Suppose that there is a knot $K^{\prime}$ and a slope $p/q>0$ such that $p\geq 12+4q^{2}-2q+4qg(K)$ and $S_{K}^{3}(p/q)\cong S_{K^{\prime}}^{3}(p/q)$. If one of the following conditions holds: (i) $q\geq 3$, (ii) $q=2$ and $K$ has Property SpliFF. Then $g(K)=g(K^{\prime})$, $\Delta_{K}(t)=\Delta_{K^{\prime}}(t)$ and $K$ is fibred if and only if $K^{\prime}$ is fibred. We note that the part of Theorem 3.12 pertaining to the case $q\geq 3$ was previously established in [McC20, Theorem 1.7]. The content of this paper is the result valid for $q=2$. Thus if we assume that $q\geq 3$ or $q=2$ and $K$ has Property SpliFF, then Theorem 2.10 and Theorem 3.12 together imply that $$\displaystyle p$$ $$\displaystyle\leq\max\{Mq,12+4q^{2}-2q+4qg(K)\}$$ $$\displaystyle\leq\max\{MQ,12+4Q^{2}+4Qg(K)\},$$ where $Q$ and $M$ are the constants from Theorem 1.8 and Theorem 2.10 respectively. Since we have bounds on both $p$ and $q$, Theorems 1.2 and 1.3 then follow. Remark 1.10. Our inability to prove Conjecture 1.1 for all knots stems from the knot Floer homology component of the paper. We are unable to extract the necessary constraints on $p$ from Heegaard Floer homology when $q=2$. Acknowledgements The author would like to thank Steve Boyer, Jen Hom and Patricia Sorya for helpful conversations. He would also like to thank Fraser Binns for communicating an early version of his paper [Bin21]. 2. Surgery on knots of bounded genus Given a knot $K$ in a 3-manifold $M$, we will use $M_{K}$ to denote the complement of the knot. That is, $M_{K}$ denotes $M$ minus a a tubular neighbourhood of $K$. In particular, given a knot $K$ in $S^{3}$, the complement of $K$ will be denoted be $S^{3}_{K}$. Given a 3-manifold $M$ and a slope $\sigma$ on a torus boundary component of $M$ we will use $M(\sigma)$ to denote Dehn filling along that slope. Thus $p/q$-surgery along a knot $K$ in $S^{3}$ is denoted by $S^{3}_{K}(p/q)$. In the event that we are filling a manifold with multiple boundary components we will not overburden the notation by explicitly indicating which boundary component is being filled since this will always be clear from context. We remind the reader that any compact, orientable, irreducible 3-manifold $M$ admits a finite collection of embedded tori which decompose $M$ into pieces which are either toroidal or Seifert fibred. The minimal collection of such tori is called the JSJ decomposition and is unique up to isotopy. For a knot $K$ in $S^{3}$ we will need to consider the JSJ decomposition of the knot complement $S^{3}_{K}$. Given the JSJ decomposition of $S_{K}^{3}$, then one its JSJ components is distinguished as containing the boundary of $S_{K}^{3}$. We will refer to this piece as the outermost piece of the decomposition. The outermost piece has a distinguished slope corresponding to the meridian of the knot. Budney provides a detailed description of the possible components arising in the JSJ decomposition of a knot exterior [Bud06]. Theorem 2.1. Let $K$ be a non-trivial knot in $S^{3}$ and let $X$ be a component in the JSJ decomposition of $S^{3}_{K}$. Then $X$ takes one of the following forms. (1) $X$ is a Seifert fibred space over the disk with two exceptional fibres. If the JSJ decomposition of $S^{3}_{K}$ contains such an $X$, then it is necessarily the outermost piece and $K$ is a torus knot. (2) $X$ is homeomorphic to $S^{1}\times F$ where $F$ is a compact planar surface with at least three boundary components. If such an $X$ is the outermost component, then $K$ is a composite knot. (3) $X$ is a hyperbolic manifold. If such an $X$ is the outermost piece, then filling $X$ along the meridian yields a (possibly empty; i.e homeomorphic to $S^{3}$) connected sum of solid tori. (4) $X$ is a Seifert fibred space over the annulus with one exceptional fibre. If such an $X$ is the outermost piece, then $K$ is a cable of a non-trivial knot $K^{\prime}$. Proof. This is a subset of the information contained in [Bud06, Theorem 4.18]. ∎ For brevity, we will refer to a knot $K$ for which the outermost piece of the JSJ decomposition is hyperbolic as a knot of hyperbolic type. By definition, every hyperbolic knot in $S^{3}$ is an example of a knot of hyperbolic type. Remark 2.2. We note some consequences of Theorem 2.1. • If a knot is prime, then it is either a torus knot, a cable knot or a knot of hyperbolic type. • All of the Seifert fibred spaces occuring in Theorem 2.1 admit unique Seifert fibred structures111The only compact 3-manifolds with boundary admitting more than one Seifert fibred structure are the solid torus and the twisted $I$-bundle over the Klein bottle.. Thus we can unambiguously refer to the orders of the exceptional fibres in these spaces are invariants of the knot $K$. 2.1. Knots of hyperbolic type In this section we analyse surgeries on knots of hyperbolic type, which are those for which the outermost piece of the JSJ decomposition is a hyperbolic manifold. Given a knot $K$ of hyperbolic type, we can assign a length to slopes on $\partial S_{K}^{3}$ as follows. Let $X_{0}$ be the outermost piece in the JSJ decomposition of $S^{3}_{K}$, this is, by definition, hyperbolic. Given a horocusp neighbourhood $N$ of the cusp corresponding to $\partial S^{3}_{K}\subseteq X_{0}$, we can measure the length of a slope $p/q$ on $\partial S^{3}_{K}$ using the natural Euclidean metric on $\partial N$. We take $\ell_{K}(p/q)$ to denote the length of $p/q$ when measured in the maximal horocusp neighbourhood. By appropriately manipulating a Seifert surface, we can guarantee that it meets the outermost piece of the JSJ decomposition nicely. Proposition 2.3. Let $K$ be non-trivial knot in $S^{3}$. Then the outermost piece of the JSJ decomposition of $S_{K}^{3}$ contains an essential embedded surface $S$ which meets $\partial S_{K}^{3}$ in a longitude and whose Euler characteristic satisfies $|\chi(S)|\leq 2g(K)-1$. Proof. Let $X_{0}$ be the outermost piece of the JSJ decomposition for $S_{K}^{3}$. Take $\Sigma$ to be a minimal genus Seifert surface for $K$ which intersects $\partial X_{0}$ transversely in the minimal possible number of curves. We will show, using standard 3-manifold arguments, that $S=\Sigma\cap X_{0}$ is a surface with the desired properties. Claim A. The surface $S$ is incompressible. Proof of Claim. Let $D_{1}\subseteq X_{0}$ be a compressing disk for $S$, i.e. an embedded disk with $\partial D_{1}=D_{1}\cap S$. Since the surface $\Sigma$ is incompressible in $S_{K}^{3}$, there must be an embedded disk $D_{2}\subseteq\Sigma$ with $\partial D_{1}=\partial D_{2}$. We may obtain a new Seifert surface $\Sigma^{\prime}$ by replacing the interior of $D_{2}$ with the interior of $D_{1}$. If $D_{2}\cap\partial X_{0}$ were non-empty, then $\Sigma^{\prime}\cap X_{0}$ would contain fewer components than $\Sigma\cap X_{0}$. However, we are assuming $\Sigma\cap X_{0}$ to be minimal and so $D_{2}\cap\partial X_{0}$ must be empty. This implies that $D_{2}$ is contained in $X_{0}$ and hence that $D_{2}\subseteq S$. This verifies the incompressibility of $S$. ∎ Claim B. The surface $S$ is boundary incompressible. Proof of Claim. Let $D_{1}\subseteq X_{0}$ be a $\partial$-compressing disk for $S$, i.e. an embedded disk whose boundary can be decomposed into two intervals $\partial D_{1}=\alpha\cup\beta$ such that $\beta=D_{1}\cap S$ and $\alpha=D_{1}\cap\partial X_{0}$. There is an isotopy of $\Sigma$ guided by the disk $D_{1}$ to a surface $\Sigma^{\prime}$ such that $\Sigma^{\prime}\cap\partial X_{0}$ is obtained from $\Sigma\cap\partial X_{0}$ by ambient surgery along the arc $\alpha$ in $\partial X_{0}$. Thus if the arc $\alpha$ connected two distinct components of $\Sigma\cap\partial X_{0}$, then $\Sigma^{\prime}\cap\partial X_{0}$ would contain fewer components that $\Sigma\cap\partial X_{0}$. This contradicts the assumed minimality of $\Sigma\cap\partial X_{0}$. Thus we see that the end points of $\alpha$ must lie on a single component $\gamma$ of $\Sigma\cap\partial X_{0}$. Let $T\subseteq\partial X_{0}$ be the boundary component containing $\gamma$ and $\alpha$. Since $S$ is a 2-sided surface in $X_{0}$, the curve $\alpha$ separates $T\setminus\gamma$ into two components. One of these components is an disk $D_{2}$ whose interior is necessarily disjoint from $S$. The disk $D_{1}\cup D_{2}$ forms a compressing disk for $S$. As $S$ is incompressible, the disk $\partial D_{1}\cup D_{2}$ bounds a disk in $S$. Thus the original disk $D_{1}$ cuts off a disk from $S$. This shows that $S$ is boundary incompressible. ∎ Thus we have shown that the surface $S$ is essential and by construction it evidently meets $\partial S_{K}^{3}$ in a longitude. Thus, it remains only to consider the Euler characteristic of $S$. Since the boundary tori of $\partial X_{0}$ are incompressible in $S^{3}_{K}$, none of the curves in $S\cap\partial X_{0}$ bound a disk in $S^{3}_{K}$. It follows that no component of $\Sigma\setminus S$ can be a disk. In particular, $\chi(\Sigma\setminus S)\leq 0$. Thus, thus we obtain the bound: $$\chi(S)\geq\chi(\Sigma\setminus S)+\chi(S)=\chi(\Sigma)=1-2g(K),$$ which is to say $|\chi(S)|\leq 2g(K)-1$. ∎ Using Proposition 2.3, we can obtain bounds on $\ell_{K}(p/q)$ in terms of $g(K)$. Proposition 2.4. Let $K$ be a knot of hyperbolic type. Then $$|p|\leq\sqrt{3}(2g(K)-1)\ell_{K}(p/q).$$ Proof. Let $X_{0}$ be the outermost piece in the JSJ decomposition of $S^{3}_{K}$. By the classification of hyperbolic manifolds with minimal cusp volume [GHM${}^{+}$21, Theorem 1.2], there exists a horocusp neighbourhood $N$ of $\partial S_{K}^{3}\subseteq X_{0}$ with $\mathrm{Area}(\partial N)\geq 2\sqrt{3}$. A simple geometric argument in the universal cover of $\partial N$ (e.g. as used by Cooper and Lackenby [CL98, Lemma 2.1]) shows that for any pair of slopes $\alpha$ and $\beta$ on $\partial S_{K}^{3}$ we have (2.1) $$2\sqrt{3}\Delta(\alpha,\beta)\leq\ell_{K}(\alpha)\ell_{K}(\beta)$$ where $\Delta(\alpha,\beta)$ denotes the distance between $\alpha$ and $\beta$. Given the essential surface $S$ constructed in Proposition 2.3, we can apply results of Agol to obtain the bound [Ago00, Theorem 5.1] (or for a similar bound, see [CL98, Theorem 5.1]): $$\ell_{K}(0/1)\leq 6(2g(K)-1).$$ Since $\Delta(0/1,p/q)=|p|$, applying (2.1) to the slopes $0/1$ and $p/q$ yields $$2\sqrt{3}|p|\leq 6(2g(K)-1)\ell_{K}(p/q).$$ ∎ We make also the following observation. Lemma 2.5. Let $M$ be any 3-manifold. Then there exists a constant $L=L(M)$ depending only on $M$ such that if $N$ is a hyperbolic 3-manifold and $\sigma$ a slope on a component of $\partial N$ such that $N(\sigma)\cong M$, the slope $\sigma$ has length $\ell(\sigma)\leq L$ as measured in any horocusp neighbourhood. Proof. If $M$ is not a hyperbolic manifold, then the 6-theorem yields the bound $\ell(\sigma)\leq 6$ [Ago00, Lac03]. Thus we may assume that $M$ is itself a hyperbolic manifold. Since hyperbolic volume strictly decreases under Dehn filling we have that $\operatorname{vol}(N)>\operatorname{vol}(M)$ [Thu80] and the volumes of hyperbolic 3-manifolds form a well-ordered set [BP92], there exists $\varepsilon>0$ such that $\operatorname{vol}(N)\geq\operatorname{vol}(M)+\varepsilon$. Now suppose that the slope $\sigma$ has length $\ell>2\pi$. By the work of Futer-Kalfagianni-Purcell [FKP08, Theorem 1.1] we have that $$\operatorname{vol}(N)\left(1-\left(\frac{2\pi}{\ell}\right)^{2}\right)^{\frac{3}{2}}\leq\operatorname{vol}(M).$$ Thus we have that $$\left(1-\left(\frac{2\pi}{\ell}\right)^{2}\right)^{\frac{3}{2}}\leq\frac{\operatorname{vol}(M)}{\operatorname{vol}(M)+\varepsilon}$$ from which follows an upper bound on $\ell$ in terms of $\operatorname{vol}(M)$ and $\varepsilon$ which are independent of $N$. ∎ We will also make use of the following variation on a result of Lackenby [Lac19, Theorem 3.1]. Theorem 2.6. Let $M$ be $S^{3}$ or the exterior of an unknot or unlink in $S^{3}$ and let $J$ be a hyperbolic knot in $M$. Then there is a constant $L(J)$ such that the following property holds. Let $J^{\prime}\subseteq M$ be a hyperbolic knot and let $\sigma^{\prime}$ and $\sigma$ be slopes for $J^{\prime}$ and $J$ respectively such that there is a homeomorphism $f:M_{J^{\prime}}(\sigma^{\prime})\rightarrow M_{J}(\sigma)$. If $\ell_{J^{\prime}}(\sigma^{\prime})>L$, then there exists a homeomorphism $f^{\prime}:M_{J^{\prime}}\rightarrow M_{J}$ such that $f^{\prime}|_{\partial M}=f|_{\partial M}$.∎ We omit the proof as it follows the one given in [Lac19] with only cosmetic modifications. The key point being the condition on $q^{\prime}$ in [Lac19, Theorem 3.1] is simply to ensure that Dehn fillings occur along sufficiently long slopes. Note also that the conclusion of Theorem 2.6 in terms of the homeomorphisms $f$ and $f^{\prime}$ is simply the result of unpacking the notation $\cong_{\partial}$ in [Lac19]. 2.2. Surgery and the JSJ decomposition We wish to understand how the JSJ decomposition of $S^{3}_{K}$ changes under Dehn filling. If the JSJ decomposition of $S_{K}^{3}$ takes the form $$S_{K}^{3}=X_{0}\cup X_{1}\cup\dots\cup X_{n},$$ where $X_{0}$ is the outermost piece of this decomposition, then for any $p/q\in\mathbb{Q}$ the surgered manifold $S_{K}^{3}(p/q)$ can be decomposed as $$S_{K}^{3}(p/q)=X_{0}(p/q)\cup X_{1}\cup\dots\cup X_{n},$$ where $X_{0}(p/q)$ denotes $X_{0}$ filled along the slope $p/q$ on $\partial S_{K}^{3}\subseteq\partial X_{0}$. We wish to understand when this decomposition corresponds to the JSJ decomposition of $S_{K}^{3}(p/q)$. We do this first when $K$ is a torus knot or a cable knot. Proposition 2.7. Let $K$ be a torus knot in $S^{3}$. If $|p/q|>4g(K)+4$, then $S_{K}^{3}(p/q)$ is a Seifert fibred space with a unique Seifert fibred structure. Moreover, the core of the filling torus can be assumed to be an exceptional fibre of order at least $|p/q|-4g(K)-2$ in this structure. Proof. Suppose that $K$ is the $(r,s)$-torus knot, where $|r|>|s|\geq 2$. If $|p-rsq|\geq 2$, then $S_{K}^{3}(p/q)$ admits the structure of a Seifert fibred space over $S^{2}$ with exceptional fibres of order $|r|$, $|s|$ and $|p-rsq|$ [Mos71]. Moreover, this Seifert fibred structure is such that the core of the filling solid torus is an exceptional fibre of $|p-rsq|$. Since the genus of $K$ is $$g(K)=\frac{(|r|-1)(|s|-1)}{2},$$ we can bound $|p-rsq|$ below as follows: $$\displaystyle|p-rsq|$$ $$\displaystyle\geq|p/q-rs|$$ $$\displaystyle\geq|p/q|-|rs|$$ $$\displaystyle=|p/q|-4g(K)-2+(|r|-2)(|s|-2)$$ $$\displaystyle\geq|p/q|-4g(K)-2,$$ Thus if $|p/q|>4g(K)+4$, we have that $|p-rsq|>2$ and so the Seifert fibred structure on $S_{K}^{3}(p/q)$ is unique since it has at least two exceptional fibres of order three. ∎ A similar analysis applies to cables. Proposition 2.8. Let $K$ be a non-trivial cable knot in $S^{3}$ and suppose that the JSJ decomposition of $S_{K}^{3}$ is $$S_{K}^{3}=X_{0}\cup X_{1}\cup\dots\cup X_{n},$$ where $X_{0}$ is the outermost piece of this decomposition. If $|p/q|>4g(K)-4$, then the JSJ decomposition of $S_{K}^{3}(p/q)$ is (2.2) $$S_{K}^{3}(p/q)=X_{0}(p/q)\cup X_{1}\cup\dots\cup X_{n}.$$ Moreover $X_{0}(p/q)$ is a Seifert fibred space admitting a unique Seifert fibred structure and the core of the filling torus can be assumed to be an exceptional fibre of order at least $|p/q|-4g(K)+6$ in this structure. Proof. Suppose that $K$ is the $(r,s)$-cable of a non-trivial knot $J$, where $s\geq 2$ is the winding number. By hypothesis, the outermost piece $X_{0}$ in the JSJ decomposition in $S_{K}^{3}$ is a Seifert fibred space over the annulus with an exceptional fibre of order $s$. Moreover the slope of the regular fibre on the boundary component $\partial S_{K}^{3}\subset X_{0}$ is $rs/1$. Thus if $|p-rsq|\geq 2$, then the Seifert fibred structure on $X_{0}$ extends to a Seifert fibred structure on $X(p/q)$ in which the core of the filling torus is an exceptional fibre of order $|p-rsq|$. Moreover if $|p-rsq|\geq 3$, then this Seifert fibred structure on $X_{0}(p/q)$ is unique since it is fibred over the disk with two exceptional fibres and atleast one of these fibres is of order at least three. Thus, if $|p-rsq|\geq 3$ we see that the decomposition in (2.2) separates $S_{K}^{3}(p/q)$ into pieces which are all atoroidal or Seifert fibred. Suppose that $X_{1}$ is the piece in the JSJ decomposition of $S_{K}^{3}$ which is adjacent to $X_{0}$. Since we started with the JSJ decomposition for $S_{K}^{3}$, the only way the above decomposition can fail to be the JSJ decomposition for $S_{K}^{3}(p/q)$ is if $X_{1}$ is Seifert fibred and its Seifert fibration can be extended over $X_{0}(p/q)$. However, this is not possible if $|p-rsq|\geq 3$, since the resulting Seifert fibration on $X_{0}(p/q)$ is unique and agrees with the original Seifert fibration on $\partial X_{0}\setminus\partial S_{K}^{3}$. Thus (2.2) is the JSJ decomposition in this case. The genus of $K$ satisfies $$g(K)=|s|g(J)+\frac{(|r|-1)(|s|-1)}{2}\geq|s|+\frac{(|r|-1)(|s|-1)}{2},$$ where the inequality follows from the fact that $J$ is non-trivial and hence satisfies $g(J)\geq 1$. Using this, one obtains the bound $$\displaystyle|p-rsq|$$ $$\displaystyle\geq|p/q|-|rs|$$ $$\displaystyle=|p/q|-4g(K)+4g(K)-|rs|$$ $$\displaystyle\geq|p/q|-4g(K)+|rs|-2|r|+2|s|+2$$ $$\displaystyle=|p/q|-4g(K)+(|r|+2)(|s|-2)+6$$ $$\displaystyle\geq|p/q|-4g(K)+6.$$ Thus if $|p/q|>4g(K)-4$, then we have that $|p-rsq|>2$ and hence the desired conclusions. ∎ Incorporating these results with the case of hyperbolic type knots we obtain the following theorem. Theorem 2.9. Let $K$ be a prime knot and let $p/q$ be a slope such that (2.3) $$|p/q|>4g(K)+4$$ and $$|p|>6\sqrt{3}(2g(K)-1)$$. If the JSJ decomposition of $S_{K}^{3}$ is $$S_{K}^{3}=X_{0}\cup X_{1}\cup\dots\cup X_{n},$$ where $X_{0}$ is the outermost piece of this decomposition, then the JSJ decomposition of $S_{K}^{3}(p/q)$ is (2.4) $$S_{K}^{3}(p/q)=X_{0}(p/q)\cup X_{1}\cup\dots\cup X_{n}.$$ Proof. It follows from Theorem 2.1 that if $K$ is a prime knot, then $K$ is a torus knot, a non-trivial cable or a knot of hyperbolic type. If $K$ is a torus knot, then the statement follows immediately from Proposition 2.7. If $K$ is a cable knot, then the desired statement forms a part of Proposition 2.8. Thus suppose that $K$ is of hyperbolic type. By Proposition 2.4, the hypothesis that $|p|>6\sqrt{3}(2g(K)-1)$ implies that $\ell_{K}(p/q)>6$ and so by the 6-theorem [Ago00, Lac03], the outermost piece in the JSJ decomposition of $S_{K}^{3}$ remains hyperbolic and, hence atoroidal, after the Dehn filling. Thus pieces in the decomposition in (2.4) are all atoroidal or Seifert fibred. Moreover, since $X_{0}(p/q)$ does not admit any Seifert fibred structure, we see that this decomposition is a minimal such decomposition. That is, (2.4) is the JSJ decomposition of $S_{K}^{3}(p/q)$. ∎ 2.3. Proof of Theorem 2.10 We are now ready for the main result of this section. Theorem 2.10. Let $K$ be a prime knot in $S^{3}$. Then there exists a constant $M=M(K)$ such that if $K^{\prime}\not\simeq K$ is a prime knot with $g(K^{\prime})\leq g(K)$ and we have a homeomorphism $S_{K}^{3}(p/q)\cong S_{K^{\prime}}^{3}(p/q)$ for some $p/q\in\mathbb{Q}$, then $|p|\leq M|q|$. Proof. The constant $M$ will be chosen using the JSJ decomposition of the knot complement $S_{K}^{3}$. Suppose that $S_{K}^{3}$ has a JSJ decomposition of the form $$S_{K}^{3}=X_{0}\cup\dots\cup X_{n},$$ where $X_{0}$ is the outermost piece. We choose constants, $L_{0}$, $L_{1}$ and $\beta$ to satisfy the following properties. • If $X_{0}$ is hyperbolic, then Theorem 2.1 shows that $X_{0}$ is the complement of a knot in a (possibly trivial) connected sum of solid tori. Thus if $X_{0}$ is hyperbolic, we can choose $L_{0}$ to satisfying the conclusions of Theorem 2.6. If $X_{0}$ is not hyperbolic, then we simply take $L_{0}=6$. • Applying Lemma 2.5 to each of the $X_{i}$ for $i=1,\dots,n$, shows that we may pick $L_{1}\geq 6$ such that none of $X_{1},\dots,X_{n}$ can be obtained by Dehn filling a hyperbolic manifold along a slope of length greater than $L$. • We choose $\beta\geq 0$ such that for any Seifert fibred $X_{i}$ all exceptional fibres in $X_{i}$ have order at most $\beta$. We will show that the constant $$M(K)=\max\{\sqrt{3}L_{0}(2g(K)-1),\sqrt{3}L_{1}(2g(K)-1),4g(K)+4+\beta\}$$ satisfies the conclusions of the theorem. Thus suppose that $K^{\prime}$ is a prime knot with $g(K^{\prime})\leq g(K)$ such that $S_{K}^{3}(p/q)\cong S_{K^{\prime}}^{3}(p/q)$ for some slope $p/q\in\mathbb{Q}$ satisfying $|p/q|>M(K)$. We will show that $K^{\prime}$ is isotopic to $K$. In particular, the slope $p/q$ satisfies the following bounds: (i) $|p|>6\sqrt{3}(2g(K)-1)$, (ii) $|p|>\sqrt{3}\max\{L_{0},L_{1}\}(2g(K)-1)$ and (iii) $|p/q|>4g(K)+4+\beta$. Let $$S_{K^{\prime}}^{3}=X^{\prime}_{0}\cup\dots\cup X^{\prime}_{n^{\prime}},$$ be the JSJ decomposition for $S_{K^{\prime}}^{3}$ where $X_{0}^{\prime}$ is the outermost piece. Since $g(K)\geq g(K^{\prime})$, bounds (i) and (iii) show that (2.3) of Theorem 2.9 is satisfied for both $K$ and $K^{\prime}$. Thus the JSJ decompositions of $S_{K}^{3}(p/q)$ and $S_{K^{\prime}}^{3}(p/q)$ are given by $$S_{K}^{3}(p/q)=X_{0}(p/q)\cup X_{1}\cup\dots\cup X_{n}.$$ and $$S_{K^{\prime}}^{3}(p/q)=X^{\prime}_{0}(p/q)\cup X^{\prime}_{1}\cup\dots\cup X^{\prime}_{n^{\prime}},$$ respectively. Furthermore, notice that $X_{0}(p/q)$ is a hyperbolic manifold if and only if $K$ is of hyperbolic type and that $X_{0}(p/q)$ is a Seifert fibred space otherwise. Likewise, $X_{0}^{\prime}(p/q)$ is a hyperbolic manifold if and only if $K^{\prime}$ is of hyperbolic type Claim A. The JSJ piece $X^{\prime}_{0}(p/q)$ is not homeomorphic to $X_{i}$ for any $i=1,\dots,n$. Proof of Claim. If $K^{\prime}$ is of hyperbolic type, then the bound (ii) and Proposition 2.4 imply that $\ell_{K^{\prime}}(p/q)>L_{1}$. By choice of $L_{1}$ we see that $X_{0}^{\prime}(p/q)$ is not homeomorphic to any of the $X_{i}$. If $K^{\prime}$ is a torus knot, then $X_{0}^{\prime}(p/q)$ is a closed manifold so certainly not homeomorphic to any of the $X_{i}$, which all have non-empty boundary. If $K^{\prime}$ is a cable knot, then (iii) implies, via Proposition 2.8 that $X_{0}^{\prime}(p/q)$ is a Seifert fibred space over the disk with two exceptional fibres, one of which has order at least $\beta+10$. However, the Seifert fibred $X_{i}$ contain only exceptional fibres of order at most $\beta$. ∎ Let $$f:S_{K^{\prime}}^{3}(p/q)\rightarrow S_{K}^{3}(p/q)$$ be orientation-preserving homeomorphism. Since the JSJ decomposition of a 3-manifold is unique up to isotopy, we may assume that $f$ restricts to give a homeomorphism between $X^{\prime}_{0}(p/q)$ and a JSJ piece of $S_{K}^{3}(p/q)$. Given Claim A, this means that $f$ restricts to give a homeomorphism between $X_{0}^{\prime}(p/q)$ and $X_{0}(p/q)$. If $K$ is of hyperbolic type, then we see that $K^{\prime}$ must also be of hyperbolic type. Together (ii) and Proposition 2.4 imply that $\ell_{K^{\prime}}(p/q)>L_{0}$. However $L_{0}$ is a constant for $X_{0}$ satisfying the conclusions of Theorem 2.6 and so we see that there is a homeomorphism $f^{\prime}:X_{0}^{\prime}\rightarrow X_{0}$ which agrees with $f$ on $\partial X_{0}^{\prime}(p/q)$. Thus we can define a homeomorphism $F:S_{K^{\prime}}^{3}\rightarrow S_{K}^{3}$ by $$F(x)=\begin{cases}f(x)&x\in X_{1}^{\prime}\cup\dots\cup X_{n}^{\prime}\\ f^{\prime}(x)&x\in X_{0}^{\prime}.\end{cases}$$ The knot complement theorem of Gordon and Luecke shows that $K$ and $K^{\prime}$ are isotopic [GL89]. If $X_{0}$ is a Seifert fibred space (i.e if $K$ is a cable knot or a torus knot), then Proposition 2.7 and Proposition 2.8 combined with the bound (iii) imply that $X_{0}(p/q)$ admits a Seifert fibred structure where the core of the filling torus is an exceptional fibre of order at least $\beta+2$. This implies that $X_{0}^{\prime}$ must also be Seifert fibred and, by similar logic, that $X_{0}^{\prime}(p/q)$ is Seifert fibred with the core of the filling torus being an exceptional fibre of order at least $\beta+2$. However, the Seifert fibred structure on $X_{0}(p/q)$ is unique up to isotopy and contains a unique exceptional fibre of order greater than $\beta$. Thus we may isotope $f$ so that it carries the core of the filling torus in $X_{0}^{\prime}(p/q)$ to the core of the filling torus in $X_{0}(p/q)$. Thus $f$ restricts to a homeomorphism between the knot complements $S_{K^{\prime}}^{3}$ and $S_{K}^{3}$. Again the knot complement theorem implies that $K$ and $K^{\prime}$ are isotopic [GL89]. This completes the proof. ∎ 3. Knot Floer homology and Property SpliFF In this section, we discuss the knot Floer homology necessary to define Property SpliFF and to explore its properties. We assume familiarity with the basic features of knot Floer homology and Heegaard Floer homology. For convenience we will work throughout with coefficients in $\mathbb{F}=\mathbb{Z}/2\mathbb{Z}$. We begin with some notation and conventions regarding $\mathbb{Z}$-graded $\mathbb{F}[U]$-modules. The variable $U$ will always be assumed to have degree $-2$. We will use $\mathbb{F}_{(d)}$ to denote the graded $\mathbb{F}[U]$-module comprising a copy of the field $\mathbb{F}$ in grading $d$ and $U$-action satisfying $U\mathbb{F}_{(d)}=0$. A $\mathbb{Z}$-graded $\mathbb{F}[U]$-module $M$ admits an $\mathbb{F}$-factor if it decomposes as a direct sum of graded $\mathbb{F}[U]$-modules $M\cong M^{\prime}\oplus\mathbb{F}_{(d)}$ for some $d\in\mathbb{Z}$. Definition 3.1. Let $M$ be a $\mathbb{Z}$-graded $\mathbb{F}[U]$-module. We say that $M$ has split $\mathbb{F}$-factors (Property SpliFF) if it decomposes as a direct sum of $\mathbb{F}[U]$-modules $$M\cong M^{\prime}\oplus\mathbb{F}^{n_{1}}_{(d_{1})}\oplus\mathbb{F}^{n_{2}}_{(d_{2})},$$ for integers $n_{1},n_{2}\geq 0$ and gradings $d_{1},d_{2}$ where $d_{1}$ is odd and $d_{2}$ is even and the module $M^{\prime}$ does not admit any $\mathbb{F}$-factors. A module $M$ admits an $\mathbb{F}$-factor if and only if it contains an element $x\in M$ such that $Ux=0$ and $x\neq Uy$ for all $y\in M$. Thus the condition that the module $M$ has Property SpliFF can be alternatively formulated as saying that we have an isomorphism of graded $\mathbb{F}[U]$-modules $$\frac{\ker U}{\ker U\cap\mathrm{im}\,U}\cong\mathbb{F}^{n_{1}}_{(d_{1})}\oplus\mathbb{F}^{n_{2}}_{(d_{2})}$$ where $n_{1},n_{2}\geq 0$, $d_{1}$ is odd, $d_{2}$ is even and we are considering $U$ as a map $U:M\rightarrow M$. We will use $\mathcal{T}^{+}_{d}$ to denote the $\mathbb{F}[U]$-module $$\mathcal{T}^{+}_{d}=\mathbb{F}[U,U^{-1}]/U\mathbb{F}[U]$$ where the element $1\in\mathbb{F}[U^{-1}]$ is in grading $d\in\mathbb{Q}$. Note that $\mathcal{T}^{+}_{(d)}$ has Property SpliFF. When the absolute grading is not relevant will omit the subscript and simply denote this module by $\mathcal{T}^{+}$. 3.1. Knot Floer homology Recall that associated to a knot $K$ in $S^{3}$, there is the knot Floer complex $CFK^{\infty}(K)$ which takes the form of a bifiltered chain complex $$CFK^{\infty}(K)=\bigoplus_{i,j\in\mathbb{Z}}C\{(i,j)\},$$ whose filtered chain homotopy type is an invariant of $K$. Furthermore, after a suitable chain homotopy, one can assume (3.1) $$C\{(i,j)\}\cong\widehat{HFK}_{*-2i}(K,j-i)$$ for all $i,j$ [HW18, Reduction Lemma]. There is also a natural chain complex isomorphism $$U:CFK^{\infty}(K)\longrightarrow CFK^{\infty}(K),$$ which acts by translating $C\{(i,j)\}$ to $C\{(i-1,j-1)\}$ and lowering the homological grading by 2. This gives $CFK^{\infty}(K)$ the structure of a finitely-generated $\mathbb{F}[U,U^{-1}]$-module. For each $k\in\mathbb{Z}$, we take $A_{k}^{+}$ to be the quotient complex $$A_{k}^{+}=C\{i\geq 0\text{ or }j\geq k\}.$$ We obtain the $\mathbb{Z}$-graded $\mathbb{F}[U]$-module $\bm{A}^{+}_{k}$ by taking homology $\bm{A}^{+}_{k}=H_{*}(A_{k}^{+})$. This module admits a decomposition $$\bm{A}^{+}_{k}\cong\bm{A}^{T}_{k}\oplus\bm{A}_{{\rm red},\,k},$$ where $\bm{A}^{T}_{k}$ is the submodule $\bm{A}^{T}_{k}=U^{N}\bm{A}^{+}_{k}$ for all sufficiently large $N$ and $\bm{A}_{{\rm red},\,k}\cong\bm{A}^{+}_{k}/\bm{A}^{T}_{k}$. Let $B^{+}$ denote the quotient complex $$B^{+}=C\{i\geq 0\}$$ and let $\bm{B}^{+}=H_{*}(B^{+})$ denote its homology. Since we are working in $S^{3}$ we have $$\bm{B}^{+}\cong HF^{+}(S^{3})\cong\mathcal{T}^{+}_{(0)}.$$ These complexes admit chain maps $$v_{k},h_{k}\colon A_{k}^{+}\longrightarrow B^{+},$$ where $v_{k}$ is the obvious vertical projection and $h_{k}$ consists of the composition of a horizontal projection onto $C\{j\geq k\}$, multiplication by $U^{k}$ and a chain homotopy equivalence. We will use $\bm{v}_{k}$ and $\bm{h}_{k}$ to denote the maps induced on homology by $v_{k}$ and $h_{k}$, respectively. Both $\bm{v}_{k}$ and $\bm{h}_{k}$ are surjective on homology. Note that $\bm{v}_{k}$ preserves the homological grading and that $\bm{h}_{k}$ sends an element of grading $x$ to an element of grading $x-2k$ in $\bm{B}^{+}$. Since the map $\bm{v}_{k}$ is surjective, we see that for each $k$ there is an integer $V_{k}\geq 0$ such that $\bm{A}^{T}_{k}\cong\mathcal{T}^{+}_{-2V_{k}}$. It is not hard to see that the $V_{k}$ are precisely the sequence of integers defined by Ni and Wu [NW15]. This sequence is eventually zero and $\nu^{+}(K)$ is defined to be the minimal $k$ such that $V_{k}=0$ [HW16]. Definition 1.5 says that a knot $K$ has Property SpliFF if and only if the graded $\mathbb{F}[U]$-modules $\bm{A}^{+}_{k}$ all have Property SpliFF. We will make use of the following reformulation of Property SpliFF for knots. Proposition 3.2. Let $K$ be a knot in $S^{3}$. Then $K$ has Property SpliFF if and only if $\bm{A}_{{\rm red},\,k}$ has Property SpliFF for all $k\geq 0$. Proof. The module $\bm{A}^{+}_{k}$ admits a decomposition $$\bm{A}^{+}_{k}\cong\bm{A}^{T}_{k}\oplus\bm{A}_{{\rm red},\,k}$$ where $\bm{A}^{T}_{k}$, being isomorphic to $\mathcal{T}^{+}$, does not contain any $\mathbb{F}$-summands. Thus the module $\bm{A}^{+}_{k}$ has Property SpliFF if and only if $\bm{A}_{{\rm red},\,k}$ has Property SpliFF. One implication of the proposition follows immediately. Conversely suppose that $\bm{A}_{{\rm red},\,k}$ has Property SpliFF for all $k\geq 0$. Since $\bm{A}_{{\rm red},\,k}$ is isomorphic to $\bm{A}_{{\rm red},\,-k}$ up to an overall grading shift222The precise grading shift is not relevant for our purposes, but can be easily computed. For $k\geq 0$ the isomorphism $\bm{A}_{{\rm red},\,k}\rightarrow\bm{A}_{{\rm red},\,-k}$ lowers the absolute grading by $2k$. [HLZ15, Lemma 2.3], this implies implies that $\bm{A}_{{\rm red},\,k}$ and hence $\bm{A}^{+}_{k}$ has Property SpliFF for all $k$. ∎ In fact, when it comes to verifying Property SpliFF, it is necessary only to consider $\bm{A}_{{\rm red},\,k}$ for $k$ in the range $0\leq k\leq g(K)-1$. Proposition 3.3. Let $K$ be a knot in $S^{3}$ with genus $g=g(K)$. (i) For all $|k|\geq g$, we have $\bm{A}_{{\rm red},\,k}=0$. (ii) If $\nu^{+}(K)<g(K)$ then $\bm{A}_{{\rm red},\,g-1}$ and $\bm{A}_{{\rm red},\,1-g}$ are non-trivial and their $\mathbb{F}[U]$-module structure is the one satisfying $U\bm{A}_{{\rm red},\,g-1}=U\bm{A}_{{\rm red},\,1-g}=0$. Furthermore, as a graded $\mathbb{F}$-vector space, there is an isomorphism $\bm{A}_{{\rm red},\,g-1}\cong\widehat{HFK}_{*+2}(K,g)$. Proof. We establish the results for $k\geq g$ and $\bm{A}_{{\rm red},\,g-1}$. This is sufficient to establish the results for $k\leq-g$ and $\bm{A}_{{\rm red},\,1-g}$, since up to an overall grading shift there is an isomorphism $\bm{A}_{{\rm red},\,k}\cong\bm{A}_{{\rm red},\,-k}$ for all $k$ [HLZ15, Lemma 2.3]. First note that if $k\geq g$, then the complexes $A^{+}_{k}$ and $B^{+}$ are in fact equal. Thus we have that $\bm{A}^{+}_{k}\cong HF^{+}(S^{3})\cong\mathcal{T}^{+}_{(0)}$. This implies that $\bm{A}_{{\rm red},\,k}=0$ for $k\geq g$. Let $D$ be the subquotient complex of $CFK^{\infty}$ given by $C\{i\leq-1\,\text{and}\,j\geq g-1\}$. We have the short exact sequence of chain complexes which is invariant under the $U$-action: $$0\longrightarrow D\longrightarrow A^{+}_{g-1}\overset{\bm{v}_{g-1}}{\longrightarrow}B^{+}\rightarrow 0.$$ As $\bm{v}_{k}$ is surjective on homology, the exact triangle induced by this sequence gives a short exact sequence of $\mathbb{F}[U]$-modules $$0\longrightarrow H_{*}(D)\longrightarrow\bm{A}^{+}_{g-1}\overset{\bm{v}_{g-1}}{\longrightarrow}\bm{B}^{+}\rightarrow 0.$$ Thus we see that $H_{*}(D)$ is isomorphic to $\ker\bm{v}_{k}$. Since $\nu^{+}(K)<g$ we have that $V_{g-1}=0$ and so $\ker\bm{v}_{k}$ is isomorphic to $\bm{A}_{{\rm red},\,g-1}$. However the complex $D$ is supported in a single bigrading: $$D=C\{(-1,g-1)\}\cong\widehat{HFK}_{*+2}(K,g).$$ with trivial differential. Thus $\bm{A}_{{\rm red},\,g-1}\cong H_{*}(D)\cong\widehat{HFK}_{*+2}(K,g)$ and $U\bm{A}_{{\rm red},\,g-1}=0$. Since knot Floer homology detects the genus [Ghi08, Ni07], we have that $\widehat{HFK}(K,g)$ is non-trivial. ∎ We can use Proposition 3.3 to exhibit knots which do not have Property SpliFF. By Proposition 3.3(ii), any knot for which $\nu^{+}(K)<g(K)$ and $\widehat{HFK}(K,g)$ is supported in at least three distinct gradings does not have Property SpliFF. Example 3.4. The knot $K=11n34\#11n34$ does not satisfy Property SpliFF. The knot $11n34$ has genus $g(11n34)=3$ and $$\widehat{HFK}(11n34,3)\cong\mathbb{F}_{(3)}\oplus\mathbb{F}_{(4)}.$$ By additivity of the Seifert genus and the Künneth formula for knot Floer homology, we have $g(K)=6$ and $$\widehat{HFK}(K,6)\cong\mathbb{F}_{(6)}\oplus\mathbb{F}_{(7)}^{2}\oplus\mathbb{F}_{(8)}.$$ Since $$\nu^{+}(K)\leq g_{4}(K)\leq 2g_{4}(11n34)\leq 2<g(K)=6,$$ one may apply Proposition 3.3(ii) to obtain $$\bm{A}_{{\rm red},\,5}(K)\cong\mathbb{F}_{(4)}\oplus\mathbb{F}_{(5)}^{2}\oplus\mathbb{F}_{(6)},$$ showing that $K$ does not have Property SpliFF. Remark 3.5. One can generalize the calculation of $\bm{A}_{{\rm red},\,g-1}$ in Proposition 3.3(ii) to calculate $\bm{A}_{{\rm red},\,g-1}$ when $\nu^{+}(K)=g(K)$. When $\nu^{+}(K)=g(K)$ one again has module structure given by $U\bm{A}_{{\rm red},\,g-1}=0$, but as a graded vector space the structure on $\bm{A}_{{\rm red},\,g-1}$ satisfies $$\mathbb{F}_{(-2)}\oplus\bm{A}_{{\rm red},\,g-1}\cong\widehat{HFK}_{*+2}(K,g).$$ 3.2. The mapping cone formula In order to describe the Heegaard Floer homology of $S^{3}_{K}(p/q)$, one needs a way to label its ${\rm spin}^{c}$-structures. This labelling takes the form of an affine bijection defined in terms of relative ${\rm spin}^{c}$-structures on $S^{3}\setminus\nu K$ [OS11]: (3.2) $$\phi_{K,p/q}\colon\mathbb{Z}/p\mathbb{Z}\longrightarrow{{\rm Spin}^{c}}(S_{K}^{3}(p/q)).$$ In general, if we have two surgery descriptions for a manifold $$Y\cong S_{K}^{3}(p/q)\cong S_{K^{\prime}}^{3}(p/q)\cong Y,$$ then $\phi_{K,p/q}$ and $\phi_{K^{\prime},p/q}$ will result in different labellings on ${{\rm Spin}^{c}}(Y)$. However, by using the $d$-invariants one can show that for $p$ sufficiently large relative to $q$ these labellings will be the same up to conjugation. Lemma 3.6 (Lemma 3.3, [McC20]). Let $K$ and $K^{\prime}$ be knots in $S^{3}$ such that $S_{K}^{3}(p/q)\cong S_{K^{\prime}}^{3}(p/q)$ for some $p/q\in\mathbb{Q}$. If $p,q>0$ satisfy $$p\geq 4q\nu^{+}(K)+4q^{2}-2q+12,$$ then up to conjugation of ${\rm spin}^{c}$-structures the labelling maps satisfy $\phi_{K,p/q}=\phi_{K^{\prime},p/q}$. ∎ Now we describe how $CFK^{\infty}(K)$ determines $HF^{+}(S_{K}^{3}(p/q))$. Consider the groups $$\mathbb{A}^{+}_{i}=\bigoplus_{s\in\mathbb{Z}}\left(s,\bm{A}^{+}_{\lfloor\frac{ps+i}{q}\rfloor}\right)\quad\text{and}\quad\mathbb{B}^{+}_{i}=\bigoplus_{s\in\mathbb{Z}}\left(s,\bm{B}^{+}\right),$$ and the maps $$\bm{v}_{\lfloor\frac{ps+i}{q}\rfloor}\colon\left(s,\bm{A}^{+}_{\lfloor\frac{ps+i}{q}\rfloor}\right)\rightarrow\left(s,\bm{B}^{+}\right)\quad\text{and}\quad\bm{h}_{\lfloor\frac{ps+i}{q}\rfloor}\colon\left(s,\bm{A}^{+}_{\lfloor\frac{ps+i}{q}\rfloor}\right)\rightarrow\left(s+1,\bm{B}^{+}\right),$$ where $\bm{v}_{k}$ and $\bm{h}_{k}$ are the maps on homology induced by $v_{k}$ and $h_{k}$ as in the previous section. These maps can be added together to obtain a chain map $$\bm{D}_{i,p/q}^{+}\colon\mathbb{A}^{+}_{i}\rightarrow\mathbb{B}^{+}_{i},$$ defined by (3.3) $$\bm{D}_{i,p/q}^{+}(s,x)=\left(s,\bm{v}_{\lfloor\frac{ps+i}{q}\rfloor}(x)\right)+\left(s+1,\bm{h}_{\lfloor\frac{ps+i}{q}\rfloor}(x)\right).$$ The module $HF^{+}(S_{K}^{3}(p/q),i)$ is computed in terms of the mapping cone on $\bm{D}_{i,p/q}^{+}$. Theorem 3.7 (Ozsváth-Szabó, [OS11]). For any knot $K$ in $S^{3}$, let $\mathbb{X}^{+}_{i,p/q}$ be the mapping cone of $\bm{D}_{i,p/q}^{+}$. Then there is a isomorphism of $\mathbb{F}[U]$-modules $$H_{*}(\mathbb{X}^{+}_{i,p/q})\cong HF^{+}(S_{K}^{3}(p/q),i).$$ ∎ Furthermore, as discussed in [OS11, Section 7.2], $\mathbb{X}^{+}_{i,p/q}$ admits a $\mathbb{Q}$-grading that induces the absolute $\mathbb{Q}$-grading on $HF^{+}(S_{K}^{3}(p/q),i)$. Explicitly, the grading on $\mathbb{B}^{+}_{i}$ is independent of the knot $K$ and is determined by surgeries on the unknot. The grading on $\mathbb{A}^{+}_{i}$ satisfies the property that $\bm{D}_{i,p/q}^{+}$ decreases the grading by one. For $p/q>0$ and $0\leq i\leq p-1$, this implies that gradings are as follows [NW15]. For each $s\in\mathbb{Z}$ let $m_{i,s}\in\mathbb{Q}$ denote the minimal grading on the tower $(s,\bm{B}^{+})\subseteq\mathbb{B}^{+}_{i}$. These satisfy [NZ14, Section 3.3] (3.4) $$m_{i,0}=d(S_{p/q}^{3}(U),i)-1$$ and $$m_{i,s+1}=m_{i,s}+2\left\lfloor\frac{i+ps}{q}\right\rfloor$$ for any $$s\in\mathbb{Z}$$. The second formula comes from combining (3.3) with the fact that $\bm{v}_{k}:\bm{A}^{+}_{k}\rightarrow\bm{B}^{+}$ preserves homological grading and $\bm{h}_{k}:\bm{A}^{+}_{k}\rightarrow\bm{B}^{+}$ shifts the homological grading by $2k$. Thus if we define integers $D_{i,s}$ by the formula $$D_{i,s}=\begin{cases}\sum_{k=0}^{s-1}2\left\lfloor\frac{i+pk}{q}\right\rfloor&\text{if $s\geq 0$}\\ \sum_{k=1}^{-s}2\left\lceil\frac{kp-i}{q}\right\rceil&\text{if $s<0$,}\end{cases}$$ then the $m_{i,s}$ satisfy $$m_{i,s}=d(S_{p/q}^{3}(U),i)-1+D_{i,s}.$$ Thus we see that the $\mathbb{Q}$-grading on $(s,\bm{A}^{+}_{\lfloor\frac{ps+i}{q}\rfloor})\subseteq\mathbb{A}^{+}_{i}$ is determined by the homological grading on $\bm{A}^{+}_{\lfloor\frac{ps+i}{q}\rfloor}$ but shifted by a constant of the form: (3.5) $$d(S_{p/q}^{3}(U),i)+D_{i,s}.$$ Using these absolute gradings, Ni and Wu showed that for any $p/q>0$ and any $0\leq i\leq p-1$, the $d$-invariants $S_{K}^{3}(p/q)$ can be calculated by [NW15, Proposition 1.6] (3.6) $$d(S_{K}^{3}(p/q),i)=d(S_{p/q}^{3}(U),i)-2\max\left\{V_{\lfloor\frac{i}{q}\rfloor},V_{\lceil\frac{p-i}{q}\rceil}\right\}.$$ Together Lemma 3.6 and equation (3.6) shows that the $V_{k}$ can be recovered from sufficiently large surgeries on a knot. Proposition 3.8. Let $K$ and $K^{\prime}$ be knots in $S^{3}$ such that $S_{K}^{3}(p/q)\cong S_{K^{\prime}}^{3}(p/q)$ for some $p/q\in\mathbb{Q}$. If $p,q>0$ satisfy $$p\geq 4q\nu^{+}(K)+4q^{2}-2q+12.$$ then $\nu^{+}(K)=\nu^{+}(K^{\prime})$ and $V_{k}(K)=V_{k}(K^{\prime})$ for all $k\geq 0$. Proof. Lemma 3.6 allows us to assume that the labelling maps on ${{\rm Spin}^{c}}(S_{K}^{3}(p/q))={{\rm Spin}^{c}}(S_{K^{\prime}}^{3}(p/q))$ agree. This implies that for all $i$ in the range $0\leq i\leq p-1$ we have an isomorphism $$HF^{+}(S_{K}^{3}(p/q),i)\cong HF^{+}(S_{K^{\prime}}^{3}(p/q),i).$$ By comparing the $d$-invariants of these groups and applying (3.6), this shows that $V_{k}(K)=V_{k}(K^{\prime})$ for all $0\leq k\leq\lfloor\frac{p+q-1}{2q}\rfloor$. Since $p/q>2\nu^{+}(K)-1$, it follows that $V_{\lfloor\frac{p+q-1}{2q}\rfloor}(K)=V_{\lfloor\frac{p+q-1}{2q}\rfloor}(K^{\prime})=0$. This shows that $V_{k}(K)=V_{k}(K^{\prime})$ for all $k\geq 0$. The equality $\nu^{+}(K)=\nu^{+}(K^{\prime})$ follows immediately. ∎ One can also compute the reduced Heegaard Floer homology groups. We require only the special case when $p/q\geq 2\nu^{+}(K)-1$. The following proposition can easily be derived from [Gai17, Corollary 12] or [NZ14, Proposition 3.6]. Proposition 3.9. If $p/q\geq 2\nu^{+}(K)-1$, then as $\mathbb{Q}$-graded $\mathbb{F}[U]$-modules, we have (3.7) $$HF_{\rm red}(S_{K}^{3}(p/q),i)\cong\bigoplus_{s\in\mathbb{Z}}\bm{A}_{{\rm red},\,\lfloor\frac{i+ps}{q}\rfloor}[d(S_{p/q}^{3}(U),i)+D_{i,s}].$$ ∎ Here we are using $\bm{A}_{{\rm red},\,k}[r]$ to denote a copy of $\bm{A}_{{\rm red},\,k}$ but with its grading shifted by $r\in\mathbb{Q}$. That is, if an element $\xi\in\bm{A}_{{\rm red},\,k}$ has grading $g$, then $\xi$ has grading $g+r$ in $\bm{A}_{{\rm red},\,k}[r]$. Thus, in accordance with the discussion above, Proposition 3.9 simply says that each copy of $\bm{A}_{{\rm red},\,\lfloor\frac{i+ps}{q}\rfloor}$ in the sum (3.7) comes with the grading inherited from the $\mathbb{Q}$-grading on the summand $(s,\bm{A}^{+})\subset\mathbb{X}^{+}_{i,p/q}$. We will make use of the following calculation pertaining to the $D_{i,k}$. Lemma 3.10. If $0\leq i<p$ is such that $\left\lfloor\frac{i}{q}\right\rfloor=\left\lfloor\frac{i+1}{q}\right\rfloor$, then for any $t\in\mathbb{Z}$, $$D_{i+1,tq}-D_{i,tq}=2t.$$ Proof. Observe that $$\left\lfloor\frac{i+1+pk}{q}\right\rfloor-\left\lfloor\frac{i+pk}{q}\right\rfloor=\begin{cases}1&\text{if $pk\equiv-i-1\bmod{q}$}\\ 0&\text{otherwise.}\end{cases}$$ Thus, if $t\geq 0$, then (3.5) gives $$\displaystyle D_{i+1,tq}-D_{i,tq}$$ $$\displaystyle=2\sum_{k=0}^{tq-1}\left\lfloor\frac{i+1+pk}{q}\right\rfloor-\left\lfloor\frac{i+pk}{q}\right\rfloor$$ $$\displaystyle=2\#\{0\leq k\leq tq-1\mid pk\equiv-i-1\bmod q\}$$ $$\displaystyle=2t.$$ Similarly if $t<0$ we have that $$\displaystyle D_{i+1,tq}-D_{i,tq}$$ $$\displaystyle=2\sum_{k=1}^{-tq-1}\left\lceil\frac{kp-i-1}{q}\right\rceil-\left\lceil\frac{kp-i}{q}\right\rceil$$ $$\displaystyle=-2\#\{1\leq k\leq|tq|-1\mid pk\equiv i+1\bmod q\}$$ $$\displaystyle=2t.$$ ∎ Using Lemma 3.6 and Proposition 3.9 we see that under some circumstances the $\bm{A}_{{\rm red},\,k}$ can also be recovered from sufficiently large surgeries on a knot. Lemma 3.11. Let $K$ and $K^{\prime}$ be knots in $S^{3}$ such that $S_{K}^{3}(p/q)\cong S_{K^{\prime}}^{3}(p/q)$ for some $p/q\in\mathbb{Q}$ satisfying $$p\geq 12+4q^{2}-2q+4qg(K).$$ If one of the following conditions holds: (i) $q\geq 3$ or (ii) $q=2$ and $K$ has Property SpliFF, then $\bm{A}_{{\rm red},\,k}(K)\cong\bm{A}_{{\rm red},\,k}(K^{\prime})$ for all $k\geq 0$. Proof. By Proposition 3.8, we have that $\nu^{+}(K)=\nu^{+}(K^{\prime})$. Thus we have (3.8) $$p/q>2g(K)\geq 2\nu^{+}(K)=2\nu^{+}(K^{\prime}).$$ Let $Y$ denote the 3-manifold $Y\cong S_{K}^{3}(p/q)\cong S_{K^{\prime}}^{3}(p/q)$. By Lemma 3.6, we may assume that the labelling maps $\phi_{K,p/q},\phi_{K^{\prime},p/q}:\mathbb{Z}/p\mathbb{Z}\rightarrow{{\rm Spin}^{c}}(Y)$ coincide. For $i=0,\dots,p-1$, this allows us to define $HF^{+}(Y,i)$ by $$HF^{+}(Y,i)\cong HF^{+}(S_{K}^{3}(p/q),\phi_{K,p/q}(i))\cong HF^{+}(S_{K^{\prime}}^{3}(p/q),\phi_{K,p/q}(i)).$$ In order to simplify the comparison between gradings in different groups, we will normalize the $\mathbb{Q}$-grading on $HF^{+}(Y,i)$ by subtracting $d(S_{U}^{3}(p/q),i)$. With this normalization in place the absolute $\mathbb{Q}$-grading on $HF^{+}(Y,i)$ becomes a $\mathbb{Z}$-grading. We will use $\bm{A}_{{\rm red},\,k}:=\bm{A}_{{\rm red},\,k}(K)$ and $\bm{A}^{\prime}_{{\rm red},\,k}:=\bm{A}_{{\rm red},\,k}(K^{\prime})$ to denote the modules $\bm{A}_{{\rm red},\,k}$ derived from the knot Floer complexes of $K$ and $K^{\prime}$ respectively. Using (3.8) and the fact from Proposition 3.3(i) that $\bm{A}_{{\rm red},\,k}=0$ for $|k|\geq g(K)$, Proposition 3.9 yields (3.9) $$HF_{\rm red}(Y,i)\cong\begin{cases}\bm{A}_{{\rm red},\,\left\lfloor\frac{i}{q}\right\rfloor}&0\leq i<qg(K)\\ 0&qg(K)\leq i<p+q-qg(K)\\ \bm{A}_{{\rm red},\,\left\lfloor\frac{i-p}{q}\right\rfloor}[D_{i,-1}]&p+q-qg(K)\leq i\leq p-1.\end{cases}$$ In particular $k$ in the range $0\leq k<g(K)$ we have isomorphisms $$HF_{\rm red}(Y,kq)\cong\dotsb\cong HF_{\rm red}(Y,kq+q-1)\cong\bm{A}_{{\rm red},\,k}.$$ Given (3.8), we may also apply Proposition 3.9 to the surgery description of $Y$ in terms of $K^{\prime}$. This yields the expression: (3.10) $$HF_{\rm red}(Y,i)\cong\bigoplus_{s\in\mathbb{Z}}\bm{A}^{\prime}_{{\rm red},\,\lfloor\frac{ps+i}{q}\rfloor}[D_{i,s}].$$ The following claim shows that (3.10) can be simplified dramatically. Claim A. For all $i$ in the range $0\leq i<qg(K)$, we have (3.11) $$HF_{\rm red}(Y,i)\cong\bigoplus_{t\in\mathbb{Z}}\bm{A}^{\prime}_{{\rm red},\,tp+\lfloor\frac{i}{q}\rfloor}[D_{i,tq}].$$ Proof of Claim. We establish (3.11) by establishing it first for $i=qg(K)$ and then inducting downwards on the index $i$. For $i=gq$, we obtain from (3.9) and (3.10): $$\displaystyle\bigoplus_{t\in\mathbb{Z}}\bm{A}^{\prime}_{{\rm red},\,tp+\left\lfloor\frac{i}{q}\right\rfloor}[D_{i,tq}]$$ $$\displaystyle=\bigoplus_{t\in\mathbb{Z}}\bm{A}^{\prime}_{{\rm red},\,\left\lfloor\frac{pqt+i}{q}\right\rfloor}[D_{i,tq}]$$ $$\displaystyle\subseteq\bigoplus_{s\in\mathbb{Z}}\bm{A}^{\prime}_{{\rm red},\,\lfloor\frac{ps+i}{q}\rfloor}[D_{i,s}]$$ $$\displaystyle\cong HF_{\rm red}(Y,i)=0.$$ This gives (3.11) for $i=gq$. Now suppose that (3.11) holds for index $i+1$ in the range $0<i+1\leq qg(K)$. That is, (3.12) $$HF_{\rm red}(Y,i+1)\cong\bigoplus_{t\in\mathbb{Z}}\bm{A}^{\prime}_{{\rm red},\,tp+\left\lfloor\frac{i+1}{q}\right\rfloor}[D_{i+1,tq}].$$ First suppose that $i\equiv-1\bmod q$. For any $s\not\equiv 0\bmod q$, (3.12) implies that $$\bm{A}_{{\rm red},\,\left\lfloor\frac{ps+i+1}{q}\right\rfloor}=0.$$ Furthermore, for any $s\not\equiv 0\bmod q$, we have that $\left\lfloor\frac{ps+i}{q}\right\rfloor=\left\lfloor\frac{ps+i+1}{q}\right\rfloor$, since $ps+i+1\equiv ps\not\equiv 0\bmod q$. This yields $$\bigoplus_{\begin{subarray}{c}s\in\mathbb{Z}\\ s\not\equiv 0\bmod q\end{subarray}}\bm{A}_{{\rm red},\,\left\lfloor\frac{ps+i}{q}\right\rfloor}=\bigoplus_{\begin{subarray}{c}s\in\mathbb{Z}\\ s\not\equiv 0\bmod q\end{subarray}}\bm{A}_{{\rm red},\,\left\lfloor\frac{ps+i}{q}\right\rfloor}=0.$$ Thus the only non-zero terms in the right hand side of (3.10) occur when the index $s$ is divisible by $q$. Thus (3.10) reduces to (3.11) when $i\equiv-1\bmod q$. Alternatively, suppose that $i\not\equiv-1\bmod q$. In this case we have that $HF_{\rm red}(Y,i)\cong HF_{\rm red}(Y,i+1)$ and that$\left\lfloor\frac{i}{q}\right\rfloor=\left\lfloor\frac{i+1}{q}\right\rfloor$. Let $M$ be the module $$M=\bigoplus_{t\in\mathbb{Z}}\bm{A}^{\prime}_{{\rm red},\,\lfloor\frac{pqt+i}{q}\rfloor}[D_{i,tq}].$$ Note that $M$ arises a summand of the right hand side of (3.10) and is hence, isomorphic to a submodule of $HF_{\rm red}(Y,i)$. However, calculating the dimension of $M$ as an $\mathbb{F}$-vector space, we obtain $$\displaystyle\dim_{\mathbb{F}}M$$ $$\displaystyle=\sum_{t\in\mathbb{Z}}\dim_{\mathbb{F}}\bm{A}^{\prime}_{{\rm red},\,pt+\left\lfloor\frac{i}{q}\right\rfloor}$$ $$\displaystyle=\sum_{t\in\mathbb{Z}}\dim_{\mathbb{F}}\bm{A}^{\prime}_{{\rm red},\,pt+\left\lfloor\frac{i+1}{q}\right\rfloor}$$ $$\displaystyle=\dim_{\mathbb{F}}HF_{\rm red}(Y,i+1)=\dim_{\mathbb{F}}HF_{\rm red}(Y,i).$$ Thus $M$ must actually be isomorphic to $HF_{\rm red}(Y,i)$, yielding the inductive step in this case too. ∎ This allows us to restrict which of the $\bm{A}^{\prime}_{{\rm red},\,k}$ are non-trivial Claim B. If $\bm{A}^{\prime}_{{\rm red},\,k}$ is non-trivial for some $k$, then there are integers $t$ and $j$ such that $k=pt+j$ and $|j|<g(K)$. Proof of Claim. Suppose that $\bm{A}^{\prime}_{{\rm red},\,k}$ is non-trivial. There are integers $s$ and $i$ such that $qk=ps+i$ and $0\leq i\leq p-1$. According to (3.10), the module $\bm{A}^{\prime}_{{\rm red},\,k}$ appears as a non-trivial summand in $HF_{\rm red}(Y,i)$. Thus, by (3.9) we must have that $i<qg(K)$ or $i\geq p+q-qg(K)$. If $i<qg(K)$, then that $\bm{A}^{\prime}_{{\rm red},\,k}$ arises as a non-trivial summand in (3.11). Thus $k$ takes the form $k=tp+j$, where $j=\left\lfloor\frac{i}{q}\right\rfloor<g(K)$. On the other hand, if $i\geq p+q-qg(K)$, then we can write $-qk=-p-ps+p-i$, where $p-i\leq gq-q$. By (3.10), $\bm{A}^{\prime}_{{\rm red},\,-k}$ arises as a summand in $HF_{\rm red}(Y,p-i)$. Since $\bm{A}^{\prime}_{{\rm red},\,k}$ and $\bm{A}^{\prime}_{{\rm red},\,-k}$ are isomorphic up to an overall grading shift [HLZ15, Lemma 2.3], $\bm{A}^{\prime}_{{\rm red},\,-k}$ is also non-trivial. Hence (3.11) applied to $HF_{\rm red}(Y,p-i)$ shows that $-k$ takes the form $-k=tp+j$, where $j=\left\lfloor\frac{p-i}{q}\right\rfloor\leq g(K)-1$. This shows that $k$ can also be written in the required form. ∎ Claim C. If there is some $k\geq 0$ such that $\bm{A}_{{\rm red},\,k}$ and $\bm{A}^{\prime}_{{\rm red},\,k}$ are not isomorphic, then there exist integers $t\neq 0$ and $0\leq j<g(K)$ such that $\bm{A}^{\prime}_{{\rm red},\,pt+j}$ contains at least one $\mathbb{F}$-factor. Proof of Claim. First we show $g(K^{\prime})>g(K)$. If there exists $k$ such that $|k|\geq g(K)$ and $\bm{A}^{\prime}_{{\rm red},\,k}$ is non-trivial, then Proposition 3.3(i) implies that $g(K^{\prime})>g(K)$. Thus suppose that $\bm{A}^{\prime}_{{\rm red},\,k}=0$ whenever $|k|\geq g(K)$. However, with such an assumption we find that for all $k$ in the range $0\leq k<g(K)$, (3.9) and (3.11) imply that $$\displaystyle\bm{A}_{{\rm red},\,k}$$ $$\displaystyle\cong HF_{\rm red}(Y,kq)$$ $$\displaystyle\cong\bm{A}^{\prime}_{{\rm red},\,k}\oplus\bigoplus_{t\in\mathbb{Z}\setminus\{0\}}\bm{A}^{\prime}_{{\rm red},\,pt+k}[D_{i,tq}]$$ $$\displaystyle=\bm{A}^{\prime}_{{\rm red},\,k},$$ where we used the fact that $|pt+k|>g(K)$ for all $t\neq 0$ to obtain the last line. Thus if there is some $k\geq 0$ such that $\bm{A}_{{\rm red},\,k}\not\cong\bm{A}^{\prime}_{{\rm red},\,k}$. Then we have that $$g(K^{\prime})>g(K)\geq\nu^{+}(K)=\nu^{+}(K^{\prime}).$$ Proposition 3.3(ii) implies that $\bm{A}^{\prime}_{{\rm red},\,g(K^{\prime})-1}$ and $\bm{A}^{\prime}_{{\rm red},\,1-g(K^{\prime})}$ are both non-zero and comprise entirely of $\mathbb{F}$-factors. By Claim B, we may write $$g(K^{\prime})-1=pt^{\prime}+j^{\prime}$$ where $|j^{\prime}|<g(K)$. Moreover since $g(K^{\prime})>g(K)$, we see that $t^{\prime}\geq 1$. If $j^{\prime}\geq 0$, then we take $t=t^{\prime}$ and $j=j^{\prime}$ to get integers satisfying the conclusions of the claim. If $j^{\prime}<0$, then we take $j=-j^{\prime}$ and $t=-t^{\prime}$ to complete the proof of the claim. ∎ For the remainder of the proof we assume that there is some $k\geq 0$ such that $\bm{A}_{{\rm red},\,k}\not\cong\bm{A}_{{\rm red},\,k}^{\prime}$. By Claim C this implies the existence of $t\neq 0$ and $0\leq j<g(K)$ such that $\bm{A}_{{\rm red},\,pt+j}$ is non-trivial and, in particular, admits an $\mathbb{F}$-factor. Suppose that $q\geq 2$. We will show that this implies that $K$ does not have Property SpliFF. Suppose that the $\mathbb{F}$-factor of $\bm{A}_{{\rm red},\,pt+j}$ is supported in degree $d$. By (3.11), we can conclude that the module $HF_{\rm red}(Y,jq)$ admits an $\mathbb{F}$-factor in degree $d+D_{jq,tq}$ and that $HF_{\rm red}(Y,jq+1)$ admits an $\mathbb{F}$-factor in degree $d+D_{jq+1,tq}$. However, by (3.9), we have that $$HF_{\rm red}(Y)\cong\bm{A}_{{\rm red},\,j}\cong HF_{\rm red}(Y,qj+1).$$ Thus $\bm{A}_{{\rm red},\,j}$ admits $\mathbb{F}$-factors in degrees $d+D_{i,tq}$ and $d+D_{i+1,tq}$. Since $t\neq 0$, Lemma 3.10 shows that these degrees are distinct and of the same parity. Thus $\bm{A}_{{\rm red},\,j}$ and, hence $K$, does not have Property SpliFF. This proves condition (ii) of the lemma. Now suppose that $q\geq 3$. We will show that this results in a contradiction. By considering (3.11), for $i=qj$, $qj+1$ and $qj+2$, we obtain isomorphisms $$\bm{A}_{{\rm red},\,j}\cong\bigoplus_{s\in\mathbb{Z}}\bm{A}^{\prime}_{{\rm red},\,sp+j}[D_{jq,sq}]\cong\bigoplus_{s\in\mathbb{Z}}\bm{A}^{\prime}_{{\rm red},\,sp+j}[D_{jq+1,sq}]\cong\bigoplus_{s\in\mathbb{Z}}\bm{A}^{\prime}_{{\rm red},\,sp+j}[D_{jq+2,sq}].$$ Since $D_{i,0}=0$ for all $i$, we further obtain isomorphisms (3.13) $$\bigoplus_{\begin{subarray}{c}s\in\mathbb{Z}\\ s\neq 0\end{subarray}}\bm{A}^{\prime}_{{\rm red},\,sp+j}[D_{jq,sq}]\cong\bigoplus_{\begin{subarray}{c}s\in\mathbb{Z}\\ s\neq 0\end{subarray}}\bm{A}^{\prime}_{{\rm red},\,sp+j}[D_{jq+1,sq}]\cong\bigoplus_{\begin{subarray}{c}s\in\mathbb{Z}\\ s\neq 0\end{subarray}}\bm{A}^{\prime}_{{\rm red},\,sp+j}[D_{jq+2,sq}].$$ Moreover, since $\bm{A}_{{\rm red},\,tp+j}$ is non-zero, these groups are all non-trivial. Let $d$ be the maximal grading of an element in $\bigoplus_{\begin{subarray}{c}s\in\mathbb{Z}\\ s\neq 0\end{subarray}}\bm{A}^{\prime}_{{\rm red},\,sp+j}[D_{jq+1,sq}]$. Thus, there is an integer $r\neq 0$ such that $\bm{A}^{\prime}_{{\rm red},\,rp+j}[D_{jq+1,rq}]$ contains an element of grading $d$. If $r>0$, then Lemma 3.10 implies that $\bm{A}^{\prime}_{{\rm red},\,rp+j}[D_{jq+2,rq}]$ contains an element of grading $$d-D_{jq+1,rq}+D_{jq+2,rq}=d+2r>d,$$ contradicting the isomorphisms in (3.13). Likewise, if $r<0$, then we use the fact that $\bm{A}^{\prime}_{{\rm red},\,rp}[D_{jq+1,rq}]$ must contain an element of grading $$d-D_{jq+1,rq}+D_{jq,rq}=d-2r>d$$ to contradict the isomorphisms in (3.13). This proves condition (i) of the lemma. ∎ Putting this all together we obtain the main theorem of this section. Theorem 3.12. Let $K$ be knot in $S^{3}$. Suppose that there is a knot $K^{\prime}$ and a slope $p/q>0$ such that $p\geq 12+4q^{2}-2q+4qg(K)$ and $S_{K}^{3}(p/q)\cong S_{K^{\prime}}^{3}(p/q)$. If one of the following conditions holds: (i) $q\geq 3$, (ii) $q=2$ and $K$ has Property SpliFF. Then $g(K)=g(K^{\prime})$, $\Delta_{K}(t)=\Delta_{K^{\prime}}(t)$ and $K$ is fibred if and only if $K^{\prime}$ is fibred. Proof. By Lemma 3.11 and Proposition 3.8, we can conclude that $V_{k}(K)=V_{k}(K^{\prime})$ and $\bm{A}_{{\rm red},\,k}(K)\cong\bm{A}_{{\rm red},\,k}(K^{\prime})$ for all $k\geq 0$. The conclusion follows since one can recover the genus, Alexander polynomial and fibredness from this data [McC20, Proposition 2.3]. (1) The genus is detected as $g(K)=\min\{k\geq 0\mid V_{k}+\dim\bm{A}_{{\rm red},\,k}=0\}$. (2) A knot $K$ of genus $g$ is fibred if and only if $V_{g-1}+\dim\bm{A}_{{\rm red},\,g-1}=1$. (3) The torsion coefficients of the Alexander polynomial $\Delta_{K}(t)$ satisfy the formula $t_{k}(K)=V_{k}+\chi(\bm{A}_{{\rm red},\,k})$ for all $k$ and the torsion coefficients $t_{k}$ for $k\geq 0$ are sufficient to determine $\Delta_{K}(t)$. ∎ 3.3. Thickness one knots. In this section, we exhibit our principal examples of knots with Property SpliFF. Recall that a knot $K$ has (knot Floer) thickness at most $n$, if there exists an integer $\rho$ such that the knot Floer homology group $\widehat{HFK}_{d}(K,s)$ is non-zero only in gradings $s+\rho-n\leq d\leq s+\rho$. The thickness of $K$ being defined to be the minimal $n$ for which $K$ has thickness at most $n$. A knot $K$ is said to have thin knot Floer homology if it has thickness zero, that is, there is an integer $\rho$ such that the group $\widehat{HFK}_{d}(K,s)$ can be non-zero only if $d=s+\rho$. The principal examples of this are alternating knots which are thin with $\rho=\frac{\sigma}{2}$, where $\sigma$ is the usual knot signature [OS03]. Similarly quasi-alternating knots are also known to have thin knot Floer homology [MO08]. More generally we wish to understand when knots with thickness one can have Property SpliFF. This is is a class that includes all almost-alternating knots [Low08] and all prime knots with at most 12 crossings [CC14]. We will use $\mathcal{T}_{d}(n)$ to denote the submodule of $\mathcal{T}^{+}_{d}$ generated by $U^{1-n}$. For $n\leq 0$, the module $\mathcal{T}_{d}(n)$ will be trivial. For $n\geq 1$, the module $\mathcal{T}_{d}(n)$ has dimension $n$ as an $\mathbb{F}$-vector space. For $n=1$, there is an isomorphism of graded $\mathbb{F}[U]$-modules $\mathbb{F}_{(d)}\cong\mathcal{T}_{d}(1)$. Lemma 3.13. Let $K$ be a knot with thickness at most one and let $\rho$ be an integer such that for all $s$, the group $\widehat{HFK}_{d}(K,s)$ is non-zero only for gradings $d\in\{s+\rho,s+\rho-1\}$. Then for all $k\geq 0$, there exist integers $a,b\geq 0$ such $\bm{A}_{{\rm red},\,k}$ takes one of the following forms (a) $\bm{A}_{{\rm red},\,k}\cong\mathbb{F}^{a}_{(k+\rho-1)}\oplus\mathbb{F}^{b}_{(k+\rho-2)}\oplus\mathcal{T}_{(2k)}\left(\left\lceil\frac{\rho-k}{2}\right\rceil\right)$ (b) $\bm{A}_{{\rm red},\,k}\cong\mathbb{F}^{a}_{(k+\rho-1)}\oplus\mathbb{F}^{b}_{(k+\rho-2)}\oplus\mathcal{T}_{(2k)}\left(\left\lfloor\frac{\rho-k}{2}\right\rfloor\right)$ Furthermore, if $\widehat{HFK}_{\rho+k}(K,k)=0$, then $a=0$. If $\widehat{HFK}_{\rho+k-1}(K,k)=0$, then we can assume that $\bm{A}_{{\rm red},\,k}$ is of type (a). Note that types (a) and (b) in Lemma 3.13 are not mutually exclusive as $\left\lceil\frac{\rho-k}{2}\right\rceil=\left\lfloor\frac{\rho-k}{2}\right\rfloor$ whenever $\rho-k$ is even. Proof. The proof is modelled on [OS03, Proof of Theorem 1.4]. By (3.1), we may assume that the chain complex $C=CFK^{\infty}$ is such that the group $C\{(i,j)\}$ is supported only in degrees $i+j+\rho$ and $i+j+\rho-1$. We have an exact sequence of complexes $$0\rightarrow A^{+}_{k}=C\{i\geq 0\,\text{or}\,j\geq k\}\rightarrow C\{i\geq 0\}\oplus C\{j\geq k\}\rightarrow C\{i\geq 0\,\text{and}\,j\geq k\}\rightarrow 0.$$ Since the complex $C\{i\geq 0\,\text{and}\,j\geq k\}$ is non-zero only in degrees $\geq k+\rho-1$, we see that the induced map (3.14) $$\bm{A}^{+}_{k}\rightarrow H_{*}(C\{i\geq 0\})\oplus H_{*}(C\{j\geq k\})\cong\mathcal{T}^{+}_{(0)}\oplus\mathcal{T}^{+}_{(2k)}$$ is an isomorphism in degrees $\leq k+\rho-3$ and a surjection in degree $k+\rho-2$. Moreover, the group of $C\{i\geq 0\,\text{and}\,j\geq k\}$ in degree $k+\rho-1$ is isomorphic to $\widehat{HFK}_{\rho+k-1}(K,k)$. Therefore, if $\widehat{HFK}_{\rho+k-1}(K,k)=0$, the map (3.14) is an isomorphism in degrees $\leq k+\rho-2$ and a surjection in degree $k+\rho-1$. Similarly, we can consider the long exact sequence $$0\rightarrow C\{i\leq-1\,\text{and}\,j\leq k-1\}\rightarrow C\rightarrow A^{+}_{s}=C\{i\geq 0\,\text{or}\,j\geq k\}\rightarrow 0.$$ Since the complex $C\{i\leq-1\,\text{and}\,j\leq k-1\}$ is non-zero only in degrees $\leq\rho+k-2$, we see that the induced map (3.15) $$HF^{\infty}(S^{3})=H_{*}(C)\rightarrow\bm{A}^{+}_{k}$$ is an isomorphism in degrees $\geq\rho+k$ and an injection in degree $\rho+k-1$. Moreover the group $C\{i\leq-1\,\text{and}\,j\leq k-1\}$ in degree $\rho+k-2$ is isomorphic to $\widehat{HFK}_{\rho+k}(K,k)$. Thus if $\widehat{HFK}_{\rho+k}(K,k)=0$, then the map in (3.15) is an isomorphism in degrees $\geq\rho+k-1$ and an injection in degree $\rho+k-2$. The properties of the maps (3.14) and (3.15) discussed above show that for $k\geq 0$ the only degrees in which $\bm{A}_{{\rm red},\,k}$ can be non-trivial are $\rho+k-1$, $\rho+k-2$ and any even degree between $2k$ and $k+\rho-2$ where the surjectivity of the map in (3.14) onto $\mathcal{T}_{(2k)}$ implies that $\bm{A}_{{\rm red},\,k}$ is non-trivial. In fact, the surjectivity of the map in (3.14) implies that $\bm{A}_{{\rm red},\,k}$ contains a (possibly trivial) submodule of the form $M=\mathcal{T}_{2k}\left(\left\lfloor\frac{\rho-k}{2}\right\rfloor\right)$. Putting this altogether we see that there are integers $a,b\geq 0$ such that the structure of $\bm{A}_{{\rm red},\,k}$ takes on of the two following forms (a) $\bm{A}_{{\rm red},\,k}\cong\mathbb{F}^{a}_{(k+\rho-1)}\oplus\mathbb{F}^{b}_{(k+\rho-2)}\oplus\mathcal{T}_{(2k)}\left(\left\lceil\frac{\rho-k}{2}\right\rceil\right)$ (b) $\bm{A}_{{\rm red},\,k}\cong\mathbb{F}^{a}_{(k+\rho-1)}\oplus\mathbb{F}^{b}_{(k+\rho-2)}\oplus\mathcal{T}_{(2k)}\left(\left\lfloor\frac{\rho-k}{2}\right\rfloor\right)$. Here type (a) occurs since the submodule $M$ described above may not occur as an $\mathbb{F}[U]$-module summand of $\bm{A}_{{\rm red},\,k}$. This happens if there is an element of degree $k+\rho-1$ which is not killed by the $U$-action. If $\widehat{HFK}_{\rho+k}(K,k)=0$, then (3.15) is an isomorphism in all degrees $\geq\rho+k-1$. This implies that $\bm{A}_{{\rm red},\,k}$ cannot be supported in degree $\rho+k-1$ and hence that $a=0$. If $\widehat{HFK}_{\rho+k-1}(K,k)=0$, then (3.14) is also surjective in degree $k+\rho-1$. This surjection on $\mathcal{T}_{(2k)}$ implies that $\bm{A}_{{\rm red},\,k}$ contains a submodule of the form $\mathcal{T}_{2k}\left(\left\lceil\frac{\rho-k}{2}\right\rceil\right)$. This implies that $\bm{A}_{{\rm red},\,k}$ is of type (a), as required. ∎ We will see that for knots with thickness one, we only need to consider the module $\bm{A}_{{\rm red},\,\rho-3}$ to verify whether the knot has Property SpliFF. Lemma 3.14. Let $K$ be a knot with thickness at most one and let $\rho$ be an integer such that for all $s$, the group $\widehat{HFK}_{d}(K,s)$ is non-zero only for gradings $d\in\{s+\rho,s+\rho-1\}$. Then $K$ has Property SpliFF if and only if $\rho\leq 2$ or $\bm{A}_{{\rm red},\,\rho-3}$ has Property SpliFF. Proof. Suppose that $\bm{A}_{{\rm red},\,k}$ does not have Property SpliFF for some $k\geq 0$. Lemma 3.13 shows that for $k\geq 0$ the only way $\bm{A}_{{\rm red},\,k}$ can fail to have Property SpliFF is if the term $\mathcal{T}_{(2k)}\left(\left\lceil\frac{\rho-k}{2}\right\rceil\right)$ or $\mathcal{T}_{(2k)}\left(\left\lfloor\frac{\rho-k}{2}\right\rfloor\right)$ is an $\mathbb{F}$-summand supported in degree $2k$ where $2k\neq k+\rho-1$ and $2k\neq k+\rho-2$. Thus we have $\left\lceil\frac{\rho-k}{2}\right\rceil=1$ or $\left\lfloor\frac{\rho-k}{2}\right\rfloor=1$ where $k\not\in\{\rho-1,\rho-2\}$. However, if $\left\lceil\frac{\rho-k}{2}\right\rceil=1$, then $k=\rho-1$ or $k=\rho-2$. So the remaining possibility is that $\left\lfloor\frac{\rho-k}{2}\right\rfloor=1$. This happens only if $k=\rho-2$ or $k=\rho-3$. In conclusion, $\bm{A}_{{\rm red},\,k}$ has Property SpliFF for all $k\geq 0$ with the possible exception of $k=\rho-3$. Thus we see that $K$ does not have Property SpliFF if and only if $\rho\geq 3$ and $\bm{A}_{{\rm red},\,\rho-3}$ does not have Property SpliFF. ∎ In practice, this allows us to find many examples of knots with Property SpliFF via the following proposition. See 1.6 Proof. We establish first the conditions for $K$ to have Property SpliFF. Lemma 3.14 says immediately that $K$ has Property SpliFF if $\rho\leq 2$. Thus we may suppose $\rho\geq 3$. If $\widehat{HFK}_{2\rho-3}(K,\rho-3)=0$, then Lemma 3.13 shows that there is an integer $b\geq 0$ such that $$\text{$\bm{A}_{{\rm red},\,\rho-3}\cong\mathbb{F}^{b}_{(2\rho-5)}\oplus\mathbb{F}_{(2\rho-6)}$ or $\bm{A}_{{\rm red},\,\rho-3}\cong\mathbb{F}^{b}_{(2\rho-5)}\oplus\mathcal{T}_{(2\rho-6)}(2)$}.$$ In either event, $\bm{A}_{{\rm red},\,\rho-3}$ has Property SpliFF and so Lemma 3.14 implies that $K$ also has Property SpliFF. If $\widehat{HFK}_{2\rho-4}(K,\rho-3)=0$, then Lemma 3.13 shows that there are integer s $a,b\geq 0$ such that $$\bm{A}_{{\rm red},\,\rho-3}\cong\mathbb{F}^{a}_{(2\rho-4)}\oplus\mathbb{F}^{b}_{(2\rho-5)}\oplus\mathcal{T}_{(2\rho-6)}(2).$$ Thus $\bm{A}_{{\rm red},\,\rho-3}$ and hence also $K$ have Property SpliFF. Now we turn our attention to the conditions for $mK$ to have Property SpliFF. These follow immediately from the fact that there is always an isomorphism [OS04, Proposition 3.7] $$\widehat{HFK}_{d}(mK,s)\cong\widehat{HFK}_{-d}(K,-s).$$ In particular, if $K$ has thickness one with knot Floer homology supported in degrees $s+\rho$ and $s+\rho-1$, then $mK$ also has thickness at most one supported only in degrees $s-\rho$ and $s-\rho+1$. ∎ 4. Applications to characterizing slopes We now put together the results of the preceding sections to prove the remaining main results. See 1.4 Proof. For all of the three classes we will show that that both the knot $K$ and its mirror $mK$ have Property SpliFF. (1) Suppose that $K$ has thin knot Floer homology. Since $mK$ also has thin knot Floer homology, it suffices to show that $K$ has Property SpliFF. However this follows easily from Proposition 1.6. Suppose that there is $\rho\in\mathbb{Z}$ such that for all $s$ the group $\widehat{HFK}_{d}(K,s)$ is non-zero only if $d=s+\rho$. The group $\widehat{HFK}(K,\rho-3)$ is supported only in grading $d=2\rho-3$ implying that $\widehat{HFK}_{2\rho-4}(K,\rho-3)=0$. (2) Suppose that $K$ is an $L$-space knot. Since $\bm{A}_{{\rm red},\,k}(K)=0$ for all $k$, we see that $K$ has Property SpliFF. To see that $mK$ has Property SpliFF we refer to Lemma 18 and Proposition 19 in [Gai17], which show that $\bm{A}_{{\rm red},\,k}(mK)\cong\mathcal{T}(V_{k})$ for all $k\geq 0$. (3) Let $K$ be a prime knot with at most $12$ crossings such that neither $K$ nor $mK$ is one of the knots included in Table 1. One can verify using the data for $\widehat{HFK}$ available on KnotInfo [CC14] that Proposition 1.6 implies that both $K$ and $mK$ have Property SpliFF. ∎ 4.1. Integral slopes on almost $L$-space knots We now turn our attention to characterizing slopes for almost $L$-space knots. Lemma 4.1. Let $K$ be an almost $L$-space knot. Then (i) $K$ has Property SpliFF and (ii) either $g(K)=\nu^{+}(K)$ or $g(K)=1$. Proof. If a rational homology sphere $Y$ is an almost $L$-space, then there is a unique ${\rm spin}^{c}$-structure $\mathfrak{s}_{0}$ such that $HF_{\rm red}(Y,\mathfrak{s}_{0})$ is non-zero and that there is an integer $m\geq 1$ such that $HF_{\rm red}(Y,\mathfrak{s}_{0})\cong\mathcal{T}^{+}(m)$ [Bin21, Proposition 3.1]. It then follows from Proposition 3.9 that if $K$ is an almost $L$-space knot, then $\bm{A}_{{\rm red},\,0}\cong\mathcal{T}(m)$ for some $m\geq 1$ and $\bm{A}_{{\rm red},\,k}=0$ for all $k>0$. Thus an almost $L$-space knot has Property SpliFF. Furthermore, Proposition 3.3 implies that if $\nu^{+}(K)<g(K)$, then $\bm{A}_{{\rm red},\,g(K)-1}$ is non-zero. However for an almost $L$-space knot, $\bm{A}_{{\rm red},\,k}$ is non-zero only for $k=1$. It follows that $g(K)=\nu^{+}(K)$ or $g(K)=1$. ∎ We establish characterizing slopes for $T_{2,3}\#T_{2,3}$ first. This plays a special role in the analysis as the unique non-prime almost $L$-space knot [Bin21]. Proposition 4.2. Any integer slope $p\geq 32$ is characterizing for $T_{2,3}\#T_{2,3}$. Proof. Let $K=T_{2,3}\#T_{2,3}$ and suppose that $K^{\prime}$ is a knot such that $S_{K^{\prime}}^{3}(p)\cong S_{K}^{3}(p)$ for some $p\geq 32$. Proposition 3.8 applies to show that $\nu^{+}(K^{\prime})=\nu^{+}(K)=2$. Furthermore, since $S_{K^{\prime}}^{3}(p)$ is an almost $L$-space and $p>2\nu^{+}(K^{\prime})-1$, this implies that $K^{\prime}$ is also an almost $L$-space knot. In particular, $g(K^{\prime})=g(K)=2$. The JSJ decomposition of $S^{3}_{K}$ comprises two trefoil complements glued to a composing space. Since a composing space becomes a copy of $T^{2}\times[0,1]$ under integer Dehn filling, the manifold $S_{K}^{3}(p)$ is obtained by gluing together two trefoil complements. Thus the JSJ decomposition of $S_{K}^{3}(p)\cong S_{K^{\prime}}^{3}(p)$ contains only Seifert fibred pieces with all exceptional fibres being of orders two or three. Next we consider the four possibilities for $K^{\prime}$ allowed by Theorem 2.1: $K^{\prime}$ is either of hyperbolic type, a torus knot, a cable knot or a composite knot. If $K^{\prime}$ were of hyperbolic type, then Proposition 2.4 would show that $$\ell_{K^{\prime}}(p)\geq\frac{p}{\sqrt{3}(2g(K^{\prime})-1)}\geq\frac{32}{3\sqrt{3}}>6.$$ This would imply that the outermost piece of $S_{K^{\prime}}^{3}$ would remain hyperbolic after the Dehn filling, contradicting the fact that the JSJ decomposition of $S_{K^{\prime}}^{3}(p)$ contains only Seifert fibred pieces. If $K^{\prime}$ were a cable knot or a torus knot, then Proposition 2.7 or Proposition 2.8 would show that the JSJ decomposition of $S_{K^{\prime}}(p)$ would contain a Seifert fibred piece with an exceptional fibre of order at least $$p-4g(K^{\prime})-2=p-10\geq 22,$$ contradicting the fact that the maximal order of an exceptional fibre is three. Thus the only remaining possibility is that $K^{\prime}$ is a composite knot. However $T_{2,3}\#T_{2,3}$ is the unique non-prime almost $L$-space knot [Bin21], so this implies that $K\simeq K^{\prime}$ and hence that $p$ is a characterizing slope. ∎ With this case in hand, we establish characterizing slopes for almost $L$-space knots in general. See 1.7 Proof. Since $L$-space knots and almost $L$-space knots have Property SpliFF, Theorem 1.3 shows that we only need to consider integer slopes. Thus let $K$ be a non-trivial $L$-space knot or an almost $L$-space knot and suppose that some $p\geq 4g(K)+14$ is a non-characterizing slope for $K$. Thus we may take $K^{\prime}$ be a knot distinct from $K$ such that $S_{K}^{3}(p)\cong S_{K^{\prime}}^{3}(p)$. Proposition 3.8 applies to show that $\nu^{+}(K)=\nu^{+}(K^{\prime})$. Since $S_{K}^{3}(p)$ is an $L$-space or an almost L-space and, it follows that $K^{\prime}$ must also be an $L$-space knot or an almost $L$-space knot. It follows that $g(K)=g(K^{\prime})$. 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Interplay between static and dynamic polar correlations in relaxor Pb(Mg${}_{1/3}$Nb${}_{2/3}$)O${}_{3}$ C. Stock ISIS Facility, Rutherford Appleton Labs, Chilton, Didcot, OX11 0QX, UK    L. Van Eijck Institut Laue-Langevin, 6 rue Jules Horowitz, Boite Postale 156, 38042 Grenoble Cedex 9, France    P. Fouquet Institut Laue-Langevin, 6 rue Jules Horowitz, Boite Postale 156, 38042 Grenoble Cedex 9, France    M. Maccarini Institut Laue-Langevin, 6 rue Jules Horowitz, Boite Postale 156, 38042 Grenoble Cedex 9, France    P.M. Gehring NIST Center for Neutron Research, NIST, Gaithersburg, Maryland 20899-6100, USA    Guangyong Xu Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973-5000    H. Luo Shanghai Institute of Ceramics, Chinese Academy of Sciences, Shanghai, China 201800    X. Zhao Shanghai Institute of Ceramics, Chinese Academy of Sciences, Shanghai, China 201800    J.-F. Li Department of Materials Science and Engineering, Virginia Tech., Blacksburg, Virginia 24061    D. Viehland Department of Materials Science and Engineering, Virginia Tech., Blacksburg, Virginia 24061 (November 24, 2020) Abstract We have characterized the dynamics of the polar nanoregions in Pb(Mg${}_{1/3}$Nb${}_{2/3}$)O${}_{3}$ (PMN) through high-resolution neutron backscattering and spin-echo measurements of the diffuse scattering cross section. We find that the diffuse scattering intensity consists of both static and dynamic components. The static component first appears at the Curie temperature $\Theta\sim 400$ K, while the dynamic component freezes completely at the temperature T${}_{f}\sim 200$ K; together, these components account for all of the observed spectral weight contributing to the diffuse scattering cross section. The integrated intensity of the dynamic component peaks near the temperature at which the frequency-dependent dielectric constant reaches a maximum (T${}_{max}$) when measured at 1 GHz, i. e. on a timescale of $\sim 1$ ns. Our neutron scattering results can thus be directly related to dielectric and infra-red measurements of the polar nanoregions. Finally, the global temperature dependence of the diffuse scattering can be understood in terms of just two temperature scales, which is consistent with random field models. pacs: 74.72.-h, 75.25.+z, 75.40.Gb I Introduction Lead-based relaxor ferroelectrics are compounds with the general formula PbBO${}_{3}$ in which the B-site is disordered. They exhibit exceptionally large piezoelectric coefficients ($d_{33}\sim 2,500$ pC/N) and dielectric constants ($\epsilon\sim 25,000$) making them attractive for device applications. Ye_rev:98 ; Park97:82 ; Xu_rev:xx PbMg${}_{1/3}$Nb${}_{2/3}$O${}_{3}$ (PMN) and PbZn${}_{1/3}$Nb${}_{2/3}$O${}_{3}$ (PZN) are prototypical relaxor ferroelectrics and the most studied; both display a broad and unusually frequency-dependent zero-field dielectric response (see Fig. 1$b$) that contrasts with the sharp and (comparatively) frequency-independent peaks observed in conventional ferroelectrics such as PbTiO${}_{3}$ (PT) and BaTiO${}_{3}$. This anomalous dielectric behavior is matched by the odd structural properties of the lead-based relaxors. Xu06:79 Instead of a well-defined structural transition to a long-range ordered ferroelectric ground state, which normally characterizes a typical ferroelectric, lead-based relaxors develop short-range ferroelectric correlations on cooling that are consistent with tiny domains of ferroelectric order embedded within a paraelectric matrix, while the average structural unit cell remains cubic. These local regions of ferroelectric order, now known as polar nanoregions or PNR, were first postulated to explain the temperature dependence of the optical index of refraction of a variety of disordered ferroelectric materials. Burns83:48 The existence of PNR has since been confirmed by numerous x-ray and neutron scattering studies, which report the presence of strong diffuse scattering at low temperatures. You97:79 ; Hirota02:65 ; Vak95:37 ; Vak89:90 The diffuse scattering in PMN, for example, was recently investigated in great detail and shown to vanish above $\sim 420$ K. Gehring09:79 Typical diffuse scattering intensity contours measured at 200 K are displayed in Fig. 2$a)$ Xu_TOF and show that the neutron diffuse scattering cross section is very broad in momentum, which implies the existence of short-range structural correlations. By comparison, the sharp, resolution-limited Bragg peaks that accompany a transition to a normal ferroelectric ground state such as in PbTiO${}_{3}$ are indicative of long-range structural correlations. While the lead-based relaxor diffuse scattering cross section has been modeled extensively, considerable debate persists over the underlying physical origin of the intriguing butterfly-shaped contours illustrated in Fig. 2$a$). Vak05:7 ; Pasciak07:76 ; Welberry05:38 ; Welberry06:74 ; Xu04:70 Despite uncertainties in the origin of the diffuse scattering cross section, the polar nature of the diffuse scattering in the lead-based relaxors has been well established through three different means. First, the onset of the diffuse scattering in PMN was shown to coincide with the temperature at which a ferroelectric-active, soft, transverse optic mode reaches a minimum frequency (see Fig. 1$a$), and which also coincides with the Curie-Weiss temperature derived from high-temperature susceptibility measurements (Fig. 1 $c$). Gehring09:79 ; Waki02:65 ; Viehland92:46 Second, the diffuse scattering was shown to respond strongly to an electric field, which in PMN suppresses the diffuse scattering while simultaneously enhancing the Bragg peaks at low temperatures. Gehring04:70 ; Stock07:76 ; Xu06:74 ; Xu05:72 Third, on doping PMN with PT both the piezoelectric properties and the total diffuse scattering increase but then drop sharply above PT concentrations near 32%, at which point the diffuse scattering is replaced by a well-defined structural transition. Matsuura06:74 ; Cao08:78 While there is a clear connection between the anomalous dielectric properties and the diffuse scattering in lead-based relaxors, the lattice dynamics remain poorly understood, particularly at long timescales. Specifically, broadband infrared and frequency-dependent dielectric measurements, and the diffuse scattering cross section measured with neutrons and x-rays are difficult to reconcile because the former two techniques generally probe only the momentum response near $Q=0$. This problem is further complicated by the fact that neutron scattering measurements are typically limited to energy resolutions $\delta E\sim 1$ THz, and therefore direct comparisons with low-frequency dielectric data have not been possible. While comparisons of dielectric and infra-red measurements to neutron and x-ray scattering data are difficult to make, data at the high frequency limit of broadband measurements ($\sim 1$ THz) on thin films have proven to be in excellent agreement with neutron inelastic scattering measurements of the soft transverse optic mode. Bovtun04:298 Low frequency measurements, however, have suggested the presence of significant dynamics that appear to be strongly correlated with the anomalous structural properties. Kamba05:17 While it is tempting to attribute these long timescale dynamics to the diffuse scattering associated with the PNR, several studies using thermal neutron spin-echo and cold neutron triple-axis techniques, which provide excellent energy resolution, have reported the diffuse component to be resolution-limited (and hence static) at all temperatures. Vak05:7 ; Hlinka03:15 The supposed static nature of the diffuse scattering cross section has led to some models that rely solely on strain or defects to describe the cross section and reconcile it with the dynamics observed in infra-red and dielectric measurements. On the other hand, other neutron studies have found some evidence of low-energy quasielastic scattering that has been suggested to be correlated with the dielectric properties. Hiraka04:70 ; Gvasaliya05:17 ; Cowley09:378 However, because the same low-energy inelastic cross section also contains significant acoustic phonon scattering, the basic nature and existence of any quasielastic component remains unclear. Stock05:74 For this reason it is highly desirable to study the diffuse scattering in lead-based relaxors with an energy resolution that is comparable to that of frequency-dependent dielectric measurements so that a direct comparison between the results from these different techniques can be made. We have measured the diffuse scattering using cold neutron backscattering and spin-echo techniques in an effort to characterize the dynamics of the PNR in the relaxor PMN. The large dynamic range offered by both techniques allows a direct comparison with dielectric measurements, which probe dynamics on timescales less than $\sim 1$ GHz. While our backscattering data reproduce previously published results concerning the static nature of the diffuse cross section, our spin-echo experiments reveal that the diffuse scattering is in fact described by two components - one static and one dynamic. The static component is onset at 400 K, where the ferroelectric-active, soft phonon reaches a minimum frequency. The temperature dependence of the static component matches well with our backscattering results and with previous data using coarser energy resolution performed on triple-axis spectrometers. Hiraka04:70 The dynamic component is well described by a single relaxational decay time ($\tau$) on the order of $\sim 0.1$ ns. The temperature dependence of the relaxational time is described by a simple Arrhenius law with an activation temperature of 1100 K. We suggest that T${}_{f}\sim 200$ K, which is the so-called Vogel-Fulcher temperature, is the temperature at which the PNR freeze entirely, and that 400 K is correlated with the onset of static, short-range ferroelectric correlations (PNR). The dynamics also reflect the high-temperature scale where previous dielectric and optical index of refraction measurements have suggested the onset of fluctuating polar nanoregions, often referred to as the Burns temperature. II Experimental Details Our neutron scattering experiments were conducted on the IN10 backscattering and IN11 spin-echo spectrometers located at the Institut Laue Langevin in Grenoble, France. IN10 consists of a large, vibrating Si(111) monochromator that Doppler shifts incident neutrons with energies near $E_{i}=2.08$ meV and directs them on to the sample. Neutrons scattered from the sample are then energy analyzed by a large bank of Si(111) crystals, which backscatter neutrons having energy $E_{f}=2.08$ meV through the sample and onto a series of detectors. The dynamic range for this spectrometer is $\pm 10$ $\mu$eV, and the elastic energy resolution is $\delta E=0.5$ $\mu$eV (half-width). Neutron spin echo (NSE) spectroscopy differs from other neutron spectroscopic methods in that it measures the real part of the normalized intermediate scattering function $\Re[I(Q,t)/I(Q,0)]$ Mezei , where $Q$ is the total momentum transferred to the sample. This is achieved by encoding the neutron’s speed into the Larmor precession of its nuclear magnetic moment in a well controlled, externally applied magnetic field. $I(Q,t)$ is the spatial Fourier transform of the Van Hove self correlation function $G(r,t)$ which, essentially, gives the probability of finding a particle after time $t$ at a radius $r$ around its original position vanHove . Furthermore, $I(Q,t)$ is the frequency Fourier transform of the scattering function $S(Q,\omega)$, which is what is measured with neutron backscattering spectroscopy. We used the NSE spectrometer IN11 in its high resolution set-up “IN11A”. The PMN sample is a 3 cc crystal grown using the Bridgeman technique described elsewhere. Luo00:39 The sample has a room temperature lattice constant of 4.04 Å and was aligned in the (HK0) scattering plane for all measurements. III Results and Discussion We first discuss our measurements of the static diffuse scattering cross section, which is associated with short-range polar correlations. Fig. 2$a)$ shows the geometry of the diffuse scattering intensity contours in PMN measured near $\vec{Q}=(100)$ at 200 K. These data were obtained using the Disk Chopper Spectrometer (DCS) located at NIST and have been published and discussed elsewhere. Xu_TOF The red and white circles indicate the positions in reciprocal space that we studied using the backscattering (IN10) and spin-echo (IN11) techniques, respectively. Panel $b)$ shows various constant-$Q$ scans measured using IN10. These data have been corrected for a background measured at 550 K, above the onset of any diffuse scattering component. The data in panel $b)$ are the sum of the intensities measured at the two $\vec{Q}$-positions indicated by the red circles labeled IN10 in panel $a)$. As can be seen, both points are located far away from the (100) Bragg peak position and therefore are not contaminated by changes in the Bragg peak intensity as a function of temperature. The lines in panel $b)$ are fits to a Gaussian with a width fixed to the energy resolution of IN10, which was measured using the incoherent scattering cross section from a Vanadium standard. The data in panel $b)$ illustrate the onset of the static (resolution-limited) component of the diffuse scattering cross section. These data also show the absence of any measurable dynamics out to energy transfers of about 10 $\mu$eV between 10 K and 550 K (as indicated by the lack of any intensity or temperature dependence). However the dynamic range probed by backscattering is very limited, and so it is desirable to investigate the inelastic properties of PMN with a technique covering a broader dynamic range while maintaining an excellent energy resolution. Indeed, it is possible that the spectral weight gathering within the elastic ($E=0$) line at low temperatures may result from a rapid evolution of the dynamics. To investigate this possibility, we have used spin-echo spectroscopy, which covers a large dynamic range in time simultaneously. The spectra measured using spin-echo (NSE) are plotted in Fig. 3 between 200 K and 450 K and characterize the intermediate scattering function $\Re[I(Q,t)/I(Q,0)]$ as a function of $t$. At 450 K and above, the data are consistent with a flat line, thus indicating no time dependent dynamics are present within the range accessible with the NSE technique. The error bars are large for the 450 K data set because the diffuse scattering intensity is extremely weak at this temperature. The relatively featureless spectrum at 450 K contrasts with that measured at 350 K, which clearly shows a systematic decay with time, thereby illustrating the presence of a dynamic component. At lower temperatures the decay time increases until at 200 K only a flat, $t$-independent response is observed, which indicates that the normalized intermediate scattering function is purely static within the limits of the instrumental resolution, i. e. $\Re[I(Q,t)/I(Q,0)]=1$. The data in Fig. 3 therefore reflects an interplay between static and dynamic components of the diffuse scattering as the temperature is varied. Based on the direct connection between the diffuse scattering and the short-range polar correlations observed in PMN, we attribute this interplay to the presence of both static and dynamic polar nanoregions. To quantify the static and dynamic components we have fit the spectra to a single relaxational form $\Re[I(Q,t)/I(Q,0)]=\alpha+(1-\alpha)e^{-t/\tau}$. The constant $\alpha$ represents the fraction of intensity that is static on the timescale of the measurement, and $\tau$ represents the characteristic decay time of the dynamics. Based on the fits shown in Fig. 3 we can extract the fraction of the raw uncorrected intensity associated with dynamics ($1-\alpha$) and statics ($\alpha$). A distribution of decay times can be incorporated into the fit by using the form $\Re[I(Q,t)/I(Q,0)]=\alpha+(1-\alpha)e^{-(t/\tau)^{\beta}}$. While there is physical justification for a distribution of decay times associated with the dynamics of the polar nanoregions, it is not clear what the value of $\beta$ should be for this case. Our choice ($\beta=1$) is justified by the fact that our data are well described by a single timescale over a broad temperature range. We interpret the single decay time extracted from the fits to be the average fluctuation time of the polar nanoregions. The static and dynamic components extracted from Fig. 3 have been corrected for the IN11 instrumental resolution ($I(Q,0)$) and plotted as a function of temperature in Fig. 4. Panel $a)$ compares the elastic (or static) intensity measured using three different instruments and experimental resolutions, and panel $b)$ displays the dynamic component normalized to have a maximum value of unity. The data taken on SPINS, a cold neutron triple-axis spectrometer located at NIST, were measured at $\vec{Q}$=(0.025,0.025,1.05) with an energy resolution of $\delta E=100$ $\mu$eV. The backscattering (IN10) data were measured at the $\vec{Q}$ indicated in Fig. 2$a)$ and represent the intensity of a Gaussian function of energy fit to the elastic peak. The NSE data represent the static parameter ($\alpha$) extracted from fits to the NSE spectra described above multiplied by $I(Q,0)$. Hence the data plotted in Fig. 3$a)$ equals $I(Q,0)\times\alpha$. All of the data have been normalized to unity at 200 K. The static component derived from the NSE fits described above agree well with the static component derived from backscattering. The data do not agree as well with the intensities derived from poorer energy resolution measurements on SPINS, which used a fixed final energy $E_{f}=4.5$ meV, and are suggestive of a dynamic component over this region that has been integrated over (i. e. lumped into the elastic channel) by virtue of the fact that the SPINS instrument provides a substantially poorer energy resolution than do the IN10 backscattering and IN11 NSE spectrometers. From Fig. 4$a)$ the onset of the static portion of the diffuse scattering appears between 400 K and 450 K. This is significant when compared to dielectric and other neutron inelastic scattering results. Gehring09:79 The Curie-Weiss temperature $\Theta$ derived from high temperature dielectric data (Fig. 1$c)$) is 400 K Viehland92:46 and coincides exactly with the temperature at which the ferroelectric-active, soft mode reaches its minimum energy (Fig. 1$a)$). Waki02:65 The 400 K temperature scale also matches the temperature at which strong deviations in the coefficient of thermal expansion in PMN are observed. Dkhil01:65 Based on this result we interpret the onset of the static component of the diffuse scattering as the freezing of PNR. The short-range nature of these frozen regions is evident from the broad nature of the diffuse scattering cross section in momentum space, which was illustrated in Fig. 2. The dynamic component illustrated in panel $b)$ equals $I(Q,0)\times(1-\alpha)$. On cooling the dynamic component increases, peaks, then decreases; this is consistent with dynamics entering and leaving the time window probed NSE. The maximum intensity of the dynamic component occurs near $\sim 325$ K, which is agrees well with the temperature $T_{max}$ at which the peak dielectric response occurs when measured at a frequency of 1 GHz (see the vertical line in Fig. 1$b)$). We emphasize that $T_{max}$ is not a meaningful temperature scale because it depends on the energy resolution and technique used to measure it. Rather, $T_{max}$ is a temperature that is characteristic of the dynamics and the frequency (or timescale) at which they are probed. We also note that contamination of our data from acoustic modes, which have plagued other studies, is unlikely as these phonons reside at much larger energy transfers (or at shorter times $\sim 1$ THz) than are probed here. Also, the acoustic phonon intensity increases with temperature as required by the Bose factor; this does not reflect the trend observed in our experiment. There is little dynamic spectral weight at 200 K where the normalized NSE spectra (Fig. 3) nearly reaches unity and becomes flat. These data can then be understood in terms of two temperature scales. The first is an upper transition temperature near 400 K where static, short-range, ferroelectric correlations are onset. This temperature coincides with the Curie-Weiss temperature derived from high-temperature dielectric data as well as the temperature where the ferroelectric-active, soft transverse optic mode reaches its minimum frequency. A second important temperature scale exists around 200 K. This temperature is defined as that where the short-range polar correlations are truly static. This temperature is most clearly reflected in electric field studies, which show that the diffuse scattering can be suppressed by an electric field only below $\sim 200$ K. This is also the transition temperature $T_{f}$ where dielectric measurements under a poling electric field observe a sharp peak characteristic of the presence of long-range, ferroelectric correlations. We now discuss the temperature dependence of the decay time $\tau$ and the parameter $\alpha$. Both $\tau$ and the static component of the normalized intermediate scattering function $\Re[I(Q,t)/I(Q,0)]$ are displayed in Fig. 5. Panel $a)$ shows the temperature dependence of $\tau$, which we have chosen to fit to the simple Arrhenius equation $\tau=\tau_{\circ}e^{U/k_{b}T}$ with $U=1100$ K$\pm 300$ K. This value for $U$ was also confirmed through a universal fit of all the data to a single exponential decay following the Arrhenius equation. Studies of spin-glasses, which exhibit analogous slow dynamics, have often described the observed relaxation rates in terms of a power law or an exponential, Arrhenius-type equation. Aeppli85:54 ; Som82:25 Regardless of the exact form, all of these equations describe the dynamics in terms of some form of energy barrier and therefore have very similar physical interpretations. Some of these descriptions for the temperature dependence of the relaxation rate contain a critical temperature as, for example, in fits using a Vogel-Fulcher type analysis. More complicated fits introducing more fitting parameters beyond the simple Arrhenius law are not justified by our data. The value of the activation energy $U=1100$ K$\pm 300$ K is in reasonable agreement with that ($E_{a}\sim 800$ K) derived from infrared and infra-red techniques. Kamba05:17 It is also in agreement with the temperature at which the lowest-energy, transverse, optic mode is observed to become soft and broad in energy. Gehring01:87 Panel $b)$ of Fig. 5 displays the value of $\alpha$ obtained from the fits shown in Fig. 3. The value of $\alpha$ varies smoothly from 0.5 to 1 with decreasing temperature. We do not attribute any significance to the high temperature saturation value of 0.5 for two reasons. First, the normalization value $I(Q,0)$ is based on an energy integration over the time range of the spectrometer. The value of 0.5 could thus be the result of an incomplete energy integration (and hence determination of $S(Q,0)$) at high temperatures. This is corroborated by cold neutron spectroscopy data, which provide evidence of the presence of dynamics at high temperatures on the timescale beyond the resolution of our spectrometer. Hiraka04:70 Secondly, when the measured intensity is taken into account, as done in Fig. 4, the results are consistent with no or little spectral weight residing in the static channel at high temperatures. The factor of 0.5 does illustrate the presence of very long timescale dynamics at high temperatures. The dynamics are beyond the time window of the current experiment, but we speculate that the dynamics could result from short-range chemical order found in Ref. Hiraka04:70, . Further work involving longer timescales are required to confirm the origin of this extra component. The temperature dependence of $\alpha$ illustrates that all of the intensity coming from the diffuse scattering below $\sim$ 200 K is static. This temperature scale for freezing of the polar nanoregions agrees with other techniques. NMR T${}_{2}$ measurements, which probes fluctuations on the order of MHz, have found evidence of an anomaly near this temperature and have attributed it to freezing of the polar correlations. Blinc03:91 Our results contradict claims based on thermal spin-echo measurements presented in Ref. Vak05:7, where the diffuse scattering is concluded to be solely static. Those measurements were conducted on a thermal spin-echo machine with considerably coarser energy resolution and a smaller dynamic range than was used here. We therefore believe that these previous measurements only probed the static component found in our study. We note that the results presented here are consistent with data obtained from backscattering, in particular the IN10 data discussed earlier. Backscattering probes a very narrow dynamic range of $\sim 1$ ns; by comparison, neutron spin echo covers several orders of magnitude in time. It is apparent from Fig. 3 that a much broader dynamic range is required to observe both the dynamic and static response. We conclude then that backscattering is not sensitive to the dynamic component because the dynamic range accessed by that technique is too narrow. The presence of a dynamic component of the diffuse scattering and a favorable comparison with frequency-dependent dielectric data strongly link the diffuse scattering cross section to the presence of dynamic polar nanoregions. Our data also clearly illustrate two key temperature scales. A high temperature scale, where the static component appears ($\sim 400$ K), and a lower temperature scale ($\sim 200$ K), where no dynamic component is observed and only static correlations exist. We do not attribute any physical significance to T${}_{max}$, which we believe is associated with the dynamics and not to any transition temperature. Our interpretation of the upper temperature of 400 K is also in agreement with the ideas presented in Ref. Gehring09:79, , which further offers an alternative understanding of the Burns temperature at 620 K in terms of dynamics. Recent reports of a third temperature scale known as $T^{*}$ can also be understood in terms of the dynamics. Ref. Dkhil09:xx, describes a new upper transition based on deviations in the thermal expansion coefficient. Such a temperature scale may be related to the dynamics and could also be associated with the near-surface effect measured in single crystal samples of the lead-based relaxors PMN and PZN. Conlon04:70 ; Xu06:79 Indeed, previous studies of the coefficient of thermal expansion in PMN using neutrons have found a strong variation as a function of depth. While these speculations are inconclusive, further study is required to understand the origin of these upper transition scales. Dielectric and infra-red work in Ref. Toulouse08:369, describe a third temperature scale T${}^{*}$, which is 400 K, and we interpret this temperature as the onset of static polar correlations. Ref. Toulouse08:369, also descrobe a high temperature scale near 600 K. This could be related to the dynamics as evidenced by the large activation energy derived in our current experiment. The interpretation of the dielectric and neutron data in terms of two temperature scales is in agreement with random field models previously proposed to explain the relaxor ferroelectric phase diagram. References Stock04:69, ; Westphal92:68, ; Fisch03:67, interpret relaxors in terms of a 3D Heisenberg model with cubic anisotropy in a random field imposed by the disorder on the perovskite $B$ site. In this picture, the true transition temperature would correspond to 400 K where the soft, transverse optic phonon mode energy reaches a minimum. However, this model also predicts a second (lower) temperature scale when the energy scale imposed by the cubic anisotropy becomes important. It was suggested that only below this temperature could long-range, ferroelectric correlations develop in the presence of a random field. This model is in broad agreement with the data presented here, which can be described using just two temperature scales. We have presented a high resolution investigation of the diffuse scattering in PMN as a function of temperature that sheds new light on the underlying behavior of the polar nanoregions. We have proven the existence of a dynamic component to the diffuse scattering and that the temperature dependence is described by an interplay between dynamic and static components. 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Describing certain Lie algebra orbits via polynomial equations N.M. Ivanova${}^{a}$ and C.A. Pallikaros${}^{b}$ (July 28, 2017) Abstract Let $\mathfrak{h}_{3}$ be the Heisenberg algebra and let $\mathfrak{g}$ be the 3-dimensional Lie algebra having $[e_{1},e_{2}]=e_{1}\,(=-[e_{2},e_{1}])$ as its only non-zero commutation relations. We describe the closure of the orbit of a vector of structure constants corresponding to $\mathfrak{h}_{3}$ and $\mathfrak{g}$ respectively as an algebraic set giving in each case a set of polynomials for which the orbit closure is the set of common zeros. Working over an arbitrary infinite field, this description enables us to give an alternative way, using the definition of an irreducible algebraic set, of obtaining all degenerations of $\mathfrak{h}_{3}$ and $\mathfrak{g}$ (the degeneration from $\mathfrak{g}$ to $\mathfrak{h}_{3}$ being one of them). Key words: Lie algebra; degeneration; irreducible algebraic set; Heisenberg algebra ${}^{a}$ Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivska Str., 01601 Kyiv, Ukraine ${}^{a}$ E-mail: ivanova.nataliya@gmail.com ${}^{b}$ Department of Mathematics and Statistics, University of Cyprus, PO Box 20537, 1678 Nicosia, Cyprus ${}^{b}$ E-mail: pallikar@ucy.ac.cy 1 Introduction In the second half of the twentieth century a lot of works appeared on the study of different types of limit processes between various physical or geometrical theories. Such limit processes naturally lead to the notion of contraction (or degeneration). Possibly the first work in this direction was Segal [11] who considered the non-isomorphic limit of sequences of structure constants of some isomorphic Lie groups. Such limit processes are called contractions. The claim is that if two physical theories are related by a limit process, then the associated invariance groups (and invariance algebras) should also be related by some limit process. This led to a wide investigation of contractions of Lie algebras from the physical point of view. Possibly, the three most famous physical examples of contractions are the following. • Contraction of relativistic mechanics to classical mechanics was studied in works by Inönü and Wigner [6, 7]. Considering the physical limit process $c\to\infty$ in special relativity theory they showed how the symmetry group of relativistic mechanics (the Poincaré group) contracts to the Galilean group which is the symmetry group of classical mechanics. • The relation between classical and quantum mechanics can also be expressed in terms of a limit process or, in other words, a contraction [5]. Thus, one can consider classical mechanics as the limit of quantum mechanics under the contraction $\mathfrak{h}\to\mathfrak{a}$, where $\mathfrak{h}$ is the Weyl–Heisenberg algebra and $\mathfrak{a}$ is the abelian Lie algebra of the same dimension. Under this contraction the quantum mechanical commutator $[x,p]=i\hslash$ (corresponding to the Heisenberg uncertainty principle) maps to the Abelian case (that is, the classical mechanics limit) under $\hslash\to 0$. • The porous medium equation $u_{t}=m^{-1}\Delta(u^{m}-1)$ can be contracted [13] (as $m\to 0$) to the equation $u_{t}=\Delta\ln u$, which is equivalent to the equation defining the Ricci flow on $\mathbb{R}^{2}$. In these (and many other publications) it is shown, in particular, how some basic properties of the “contracted theories” can be reconstructed from the corresponding properties of the “original” theories. Such observations were summarized by Zaitsev [14] who, independently of Inönü and Wigner, suggested constructing “the theory of physical theories” based on group limits of physical theories. This amounts to including in a uniform system several physical theories being connected together via certain relations. Recently, different types of contractions have been widely used in elementary particle theory, analysis of differential equations and other areas of mathematical and theoretical physics. Working over $\mathbb{C}$ or $\mathbb{R}$, the statement “Lie algebra $\mathfrak{h}_{1}$ is a contraction of Lie algebra $\mathfrak{h}_{2}$” can be rephrased as “$\mathfrak{h}_{1}$ lies in the closure, in the metric topology, of the orbit of $\mathfrak{h}_{2}$ under the ‘change of basis’ action of the group of invertible linear transformations”. In [4] the authors show that over $\mathbb{C}$ the orbit closure in the metric topology coincides with the orbit closure in the Zariski topology. Orbit closures with respect to the Zariski topology are called degenerations. The notion of degeneration is well-defined not only over the fields $\mathbb{C}$ and $\mathbb{R}$ but also over an arbitrary ground field. In fact, this concept of orbit closure under the action of various groups arises naturally in many areas of mathematics (see, for example, [10]). In [8] we explored the possibility of investigating degenerations over an arbitrary field using elementary algebraic techniques. For this we needed to extend or modify techniques already used over the fields $\mathbb{C}$, $\mathbb{R}$ (for example contractions obtained as limit points resulting from the action of diagonal matrices, also known as generalized Inönü–Wigner contractions) in a way so that they can be applied to the case of degenerations over an arbitrary field. In this paper, although we continue our study of degenerations via an elementary algebraic approach, we take a slightly different path and consider the possibility of obtaining all degenerations (for certain examples of Lie algebras) ‘from first principles’ by direct application of the definition of an algebraic (Zariski-closed) set. This involves obtaining explicit descriptions of the orbit closures under consideration using polynomial equations. The paper is organized as follows. In Section 2 we give some necessary background, the setup being over an arbitrary infinite field $\mathbb{F}$. In particular, in Subsection 2.1 we recall some basic definitions and results on irreducible algebraic sets and regular maps while in Subsection 2.2 we recall the definition of degeneration together with some basic facts on Lie algebra structure vectors and their orbits under the ‘change of basis’ action of the general linear group. In Section 3 we perform some explicit computations concerning the orbits (and their closure in the Zariski topology) of certain given Lie algebra structure vectors corresponding to $\mathfrak{h}_{3}$ and $\mathfrak{g}_{2}\oplus\mathfrak{a}_{1}$ respectively, where $\mathfrak{h}_{3}$ denotes the Heisenberg algebra, $\mathfrak{g}_{2}$ denotes the 2-dimensional non-Abelian Lie algebra and $\mathfrak{a}_{1}$ denotes the 1-dimensional Abelian Lie algebra. This enables us to give a description of the orbit closures of these structure vectors as algebraic sets via polynomial equations and, as a consequence, determine in an alternative way all degenerations of $\mathfrak{h}_{3}$ and $\mathfrak{g}_{2}\oplus\mathfrak{a}_{1}$ over $\mathbb{F}$. We also obtain descriptions of the particular orbits described above as the intersection of a Zariski-closed with a Zariski-open set. 2 Preliminaries and generalities We begin this section by recalling some basic facts on irreducible algebraic sets. We refer the reader to Geck [2] for more details and for proofs of the main results from the theory we will be using. 2.1 Algebraic sets Fix $\mathbb{F}$ to be an arbitrary infinite field and let $m$ be a positive integer. We consider the ring $F[\boldsymbol{X}]=\mathbb{F}[X_{1},\ldots,X_{m}]$ of polynomials in the indeterminates $X_{1},\ldots,X_{m}$ over $\mathbb{F}$. For each $\boldsymbol{\alpha}=(\alpha_{1},\ldots,\alpha_{m})\in\mathbb{F}^{m}$ there exists a unique $\mathbb{F}$-algebra homomorphism $\mathop{\mathbf{ev}}\nolimits_{\boldsymbol{\alpha}}:\mathbb{F}[X_{1},\ldots,X_% {m}]\to\mathbb{F}$ such that $\mathop{\mathbf{ev}}\nolimits_{\boldsymbol{\alpha}}(X_{i})=\alpha_{i}$ for all $i$. Given $\boldsymbol{\alpha}=(\alpha_{1},\ldots,\alpha_{m})\in\mathbb{F}^{m}$ and $f\in\mathbb{F}[X_{1},\ldots,X_{m}]$ we will be writing more simply $f(\boldsymbol{\alpha})=f(\alpha_{1},\ldots,\alpha_{m})=\mathop{\mathbf{ev}}% \nolimits_{\boldsymbol{\alpha}}(f)$. Definition. Let $S$ be any subset of $\mathbb{F}[X_{1},\ldots,X_{m}]$. The algebraic set ${\bf V}(S)$ determined by $S$ is defined by $${\bf V}(S)=\{\boldsymbol{\alpha}\in\mathbb{F}^{m}:\ f(\boldsymbol{\alpha})=0% \mbox{ for all }f\in S\}.$$ A subset of $\mathbb{F}^{m}$ is called algebraic if it is of the form ${\bf V}(S)$ for some subset $S\subseteq\mathbb{F}[X_{1},\ldots,X_{m}]$. For any subset $V\subseteq\mathbb{F}^{m}$, the vanishing ideal ${\bf I}(V)$ of $V$ is defined by $${\bf I}(V)=\{f\in\mathbb{F}[X_{1},\ldots,X_{m}]:\ f(\boldsymbol{\alpha})=0% \mbox{ for all }\boldsymbol{\alpha}\in V\}.$$ It is immediate from the above definition that if $S_{1}$, $S_{2}$ are subsets of $\mathbb{F}[X_{1},\ldots,X_{m}]$ with $S_{1}\subseteq S_{2}$, then ${\bf V}(S_{2})\subseteq{\bf V}(S_{1})$ (see [2, Remark 1.1.4]). It can be shown (see, for example, [2, Remark 1.1.4 and Lemma 1.1.5] that arbitrary intersections and finite unions of algebraic sets in $\mathbb{F}^{m}$ are again algebraic. The empty set $\varnothing$ and $\mathbb{F}^{m}$ itself are clearly algebraic. Thus, the algebraic sets in $\mathbb{F}^{m}$ form the closed sets of a topology in $\mathbb{F}^{m}$, which is called the Zariski topology. A subset $X\subseteq\mathbb{F}^{m}$ is open if its complement $\mathbb{F}^{m}\setminus X$ is algebraic (closed). We will denote by $\overline{V}$ the closure of a subset $V$ of $\mathbb{F}^{m}$ in the Zariski topology. An essential role in our investigation is played by the notion of irreducibility of algebraic sets. Definition. Let $Z\subseteq\mathbb{F}^{m}$ be a nonempty algebraic set. We say that $Z$ is reducible if we can write $Z=Z_{1}\cup Z_{2}$, where $Z_{1},Z_{2}\subseteq Z$ are nonempty algebraic subsets with $Z_{1}\neq Z$ and $Z_{2}\neq Z$. Otherwise, we say that $Z$ is irreducible. Remark 2.1 (see [2, Example 1.1.13]). Our assumption that $\mathbb{F}$ is infinite ensures that $\mathbb{F}^{m}$ is irreducible. Definition. Let $s,r$ be positive integers and let $V\subseteq\mathbb{F}^{s}$ and $W\subseteq\mathbb{F}^{r}$ be nonempty algebraic sets. We say that $\Phi:V\to W$ is a regular map if there exist $f_{1},\ldots,f_{r}\in\mathbb{F}[X_{1},\ldots,X_{s}]$ such that $\Phi(\boldsymbol{\alpha})=(f_{1}(\boldsymbol{\alpha}),\ldots,f_{r}(\boldsymbol% {\alpha}))$ for all $\boldsymbol{\alpha}\in V$. One can then observe (see [2, page 23]) that regular maps are continuous in the Zariski topology. Remark 2.2 (see [2, Remark 1.3.2]). Let $V$, $W$ be as in the definition above and let $\Phi:V\to W$ be a regular map. Assume that $V$ is irreducible. Then the Zariski closure $\overline{\Phi(V)}\subseteq W$ is also irreducible. 2.2 Degenerations of Lie algebras We keep the setup of the previous subsection. In particular $\mathbb{F}$ denotes an arbitrary infinite field but now we assume further that $m=n^{3}$ for some integer $n\geq 2$ we have fixed. Also let $G$ be the general linear group $\mathop{{\rm GL}(n,\mathbb{F})}\nolimits$. Now let $\boldsymbol{\alpha}=(\alpha_{1},\ldots,\alpha_{m})\in\mathbb{F}^{m}$ be given. For the rest of our discussion, it will be convenient to relabel the components of $\boldsymbol{\alpha}$ as follows. For $1\leq r\leq m$ relabel $\alpha_{r}$ as $\alpha_{i(r),j(r),k(r)}$ where $i(r)$, $j(r)$, $k(r)$ are the unique integers with $1\leq i(r),j(r),k(r)\leq n$ satisfying $r-1=(i(r)-1)n^{2}+(j(r)-1)n+(k(r)-1)$. We will be writing $\boldsymbol{\alpha}=(\alpha_{i,j,k})$ or $\boldsymbol{\alpha}=(\alpha_{ijk})$ for short. For example, in the case $n=2$ ($m=8$) we have for $\boldsymbol{\alpha}\in\mathbb{F}^{m}$, $$\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},% \alpha_{6},\alpha_{7},\alpha_{8})=(\alpha_{111},\alpha_{112},\alpha_{121},% \alpha_{122},\alpha_{211},\alpha_{212},\alpha_{221},\alpha_{222}).$$ (The above ordering in fact amounts to writing $\boldsymbol{\alpha}=(\alpha_{ijk})\in\mathbb{F}^{n^{3}}$ where the triples $(i,j,k)$ are placed in lexicographic order.) In a similar manner we relabel the indeterminates $X_{1},\ldots,X_{m}$ in $\mathbb{F}[X_{1},\ldots,X_{m}]$ and we write $\mathbb{F}[\boldsymbol{X}]\,(=\mathbb{F}[X_{1},\ldots,X_{m}])=\mathbb{F}[X_{% ijk}:\ 1\leq i,j,k\leq n]$. Definition. An element $\boldsymbol{\lambda}=(\lambda_{ijk})\in\mathbb{F}^{m}$ is called a Lie algebra structure vector if there exists an $n$-dimensional Lie algebra $\mathfrak{g}$ over $\mathbb{F}$ and an ordered $\mathbb{F}$-basis $\hat{b}=(b_{1},\ldots,b_{n})$ of $\mathfrak{g}$ such that $[b_{i},b_{j}]=\sum_{k=1}^{n}\lambda_{ijk}b_{k}$ for $1\leq i,j\leq n$. In such a case we call $\boldsymbol{\lambda}=(\lambda_{ijk})$ the structure vector of $\mathfrak{g}$ relative to $\hat{b}$. We denote by $\mathop{\mathcal{L}_{n}(\mathbb{F})}$ the subset of $\mathbb{F}^{m}$ consisting of precisely those elements of $\mathbb{F}^{m}$ which are Lie algebra structure vectors. We refer the reader to [9] for the basic definitions and properties of Lie algebras. The properties of the Lie bracket ensure that $\mathop{\mathcal{L}_{n}(\mathbb{F})}$ is an algebraic subset of $\mathbb{F}^{m}$. This is because $\mathop{\mathcal{L}_{n}(\mathbb{F})}={\bf V}(S)$ where $S$ is the union of the following three subsets of $\mathbb{F}[X_{ijk}:\ 1\leq i,j,k\leq n]$ (see, for example, [9, pages 4–5] for a proof of this fact): $$\displaystyle\{X_{iik}:\ 1\leq i,k\leq n\},$$ $$\displaystyle\{X_{ijk}+X_{jik}:\ 1\leq i,j,k\leq n\},$$ $$\displaystyle\Big{\{}\sum_{k}(X_{ijr}X_{klr}+X_{jlk}X_{kir}+X_{lik}X_{kjr}):\ % 1\leq i,j,l,r\leq n\Big{\}}.$$ Remark 2.3. We have the following natural action of $G=\mathop{{\rm GL}(n,\mathbb{F})}\nolimits$ on ${\mathcal{L}}_{n}(\mathbb{F})$ by ‘change of basis’. Let $g=(g_{ij})\in G$ and let $\boldsymbol{\lambda}=(\lambda_{ijk})\in\mathop{\mathcal{L}_{n}(\mathbb{F})}$. Also let $\mathfrak{g}$ be an $n$-dimensional Lie algebra over $\mathbb{F}$ and $\hat{b}=(b_{1},\ldots,b_{n})$ an ordered $\mathbb{F}$-basis of $\mathfrak{g}$ such that $\boldsymbol{\lambda}=(\lambda_{ijk})$ is the structure vector of $\mathfrak{g}$ relative to $\hat{b}$. Now let $\hat{b}^{\prime}=(b_{1}^{\prime},\ldots,b_{n}^{\prime})$ be the basis of $\mathfrak{g}$ defined by $b_{j}^{\prime}=\sum_{i=1}^{n}g_{ij}b_{i}$ for $1\leq j\leq n$. Also let $\boldsymbol{\lambda}^{\prime}=(\lambda_{ijk}^{\prime})\in\mathbb{F}^{m}$ be the structure vector of $\mathfrak{g}$ relative to $\hat{b}^{\prime}$ (so we have $[b_{i}^{\prime},b_{j}^{\prime}]=\sum_{k=1}^{n}\lambda_{ijk}^{\prime}b_{k}^{\prime}$ for $1\leq i,j\leq n$). We will write $\boldsymbol{\lambda}^{\prime}=\boldsymbol{\lambda}g$ (clearly, $\boldsymbol{\lambda}^{\prime}\in\mathop{\mathcal{L}_{n}(\mathbb{F})}$). We call $g$ the transition matrix from basis $\hat{b}$ to basis $\hat{b}^{\prime}$ of $\mathfrak{g}$. It is well-known and easy to check, that the above process describes a well-defined (right) action of $G$ on $\mathop{\mathcal{L}_{n}(\mathbb{F})}$. (See, for example, [8, Remark 2.6] where some details of such a check are given.) Observe that the orbits relative to the action defined in the preceding remark correspond precisely to the isomorphism classes of $n$-dimensional Lie algebras over $\mathbb{F}$. We denote by $O(\boldsymbol{\mu})$ the orbit of the Lie algebra structure vector $\boldsymbol{\mu}\in\mathop{\mathcal{L}_{n}(\mathbb{F})}$ under the action of $\mathop{{\rm GL}(n,\mathbb{F})}\nolimits$ described above. Example. It is immediate that the zero vector $\boldsymbol{0}=(0_{\mathbb{F}},\ldots,0_{\mathbb{F}})$ of $\mathbb{F}^{n^{3}}$ belongs to ${\mathcal{L}}_{n}(\mathbb{F})$ as it corresponds to the $n$-dimensional Abelian Lie algebra over ${\mathbb{F}}$ (under any choice of basis). Its orbit consists of precisely one point and hence it is Zariski-closed. Remark 2.4. (i) For each $g\in\mathop{{\rm GL}(n,\mathbb{F})}\nolimits$, making use of the action described in Remark 2.3, we define a function $\Phi_{g}:{\mathcal{L}}_{n}(\mathbb{F})\to{\mathcal{L}}_{n}(\mathbb{F})$: $\boldsymbol{\mu}\mapsto\boldsymbol{\mu}g$, ($\boldsymbol{\mu}\in{\mathcal{L}}_{n}(\mathbb{F})$). Then $\Phi_{g}$ is a regular map and hence continuous in the Zariski topology. (To see this we fix $g\in\mathop{{\rm GL}(n,\mathbb{F})}\nolimits$. It follows from the change of basis process that for each $\boldsymbol{\mu}\in{\mathcal{L}}_{n}(\mathbb{F})$ we get $\Phi_{g}(\boldsymbol{\mu})=(\mathop{\mathbf{ev}}\nolimits_{\boldsymbol{\mu}}(f% _{1}),\ldots,\mathop{\mathbf{ev}}\nolimits_{\boldsymbol{\mu}}(f_{n^{3}}))$ where, for $1\leq i\leq n^{3}$, $f_{i}$ is polynomial in $\mathbb{F}[X]$ which only depends on $g$.) (ii) In view of item (i), one can give an elementary proof of the fact that the closure of an orbit in ${\mathcal{L}}_{n}(\mathbb{F})$ is a union of orbits (see, for example, [8, Lemma 3.1]). Definition. Let $\mathfrak{g}_{1}$, $\mathfrak{g}_{2}$ be $n$-dimensional Lie algebras over $\mathbb{F}$. We say that $\mathfrak{g}_{1}$ degenerates to $\mathfrak{g}_{2}$ (respectively, $\mathfrak{g}_{1}$ properly degenerates to $\mathfrak{g}_{2}$) if there exist structure vectors $\boldsymbol{\lambda}_{1}$ of $\mathfrak{g}_{1}$ and $\boldsymbol{\lambda}_{2}$ of $\mathfrak{g}_{2}$, relative to some bases of $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$, such that $\boldsymbol{\lambda}_{2}\in\overline{O(\boldsymbol{\lambda}_{1})}$ (respectively, $\boldsymbol{\lambda}_{2}\in\overline{O(\boldsymbol{\lambda}_{1})}\setminus O(% \boldsymbol{\lambda}_{1})$). It is immediate from Remark 2.4(ii) that if $\boldsymbol{\lambda}\in\overline{O(\boldsymbol{\mu})}$ and $\boldsymbol{\nu}\in\overline{O(\boldsymbol{\lambda})}$, then $\boldsymbol{\nu}\in\overline{O(\boldsymbol{\mu})}$, ($\boldsymbol{\lambda},\boldsymbol{\mu},\boldsymbol{\nu}\in\mathcal{L}_{n}(% \mathbb{F})$). In other words, the transitivity property holds in the case of degenerations. Finally for this subsection we remark that there are no proper degenerations over finite fields as finite subsets of $\mathbb{F}^{m}$ are closed in the Zariski topology. 3 Lie algebra orbit closures via polynomial equations We continue with our assumption that $\mathbb{F}$ is an arbitrary infinite field. Below, $\mathfrak{h}_{3}$ will denote the Heisenberg (Lie) algebra, $\mathfrak{g}_{2}$ will denote the 2-dimensional non-Abelian Lie algebra and $\mathfrak{a}_{k}$, for $k\geq 1$, the Abelian Lie algebra of dimension $k$. We will be making use of the action of $G=\mathop{{\rm GL}(n,\mathbb{F})}\nolimits$ on $\mathop{\mathcal{L}_{n}(\mathbb{F})}$ described in Remark 2.3 in order to perform some explicit computations concerning the orbits (and their closure in the Zariski topology) of certain given Lie algebra structure vectors corresponding to $\mathfrak{h}_{3}$ and $\mathfrak{g}_{2}\oplus\mathfrak{a}_{1}$ respectively. This will allow us to give descriptions of the orbit closures of these structure vectors as algebraic sets (via polynomial equations) and, in addition, obtain descriptions of the particular orbits we investigate here as intersections of a Zariski-closed with a Zariski-open set. We will also show how these explicit descriptions of the orbits enable us to provide an alternative way of obtaining all degenerations of $\mathfrak{h}_{3}$ and $\mathfrak{g}_{2}\oplus\mathfrak{a}_{1}$ over $\mathbb{F}$. 3.1 The Heisenberg algebra We consider the Heisenberg algebra $\mathfrak{h}_{3}$. This (3-dimensional) algebra has an $\mathbb{F}$-basis $\hat{e}=(e_{1},e_{2},e_{3})$ relative to which the only non-zero products (commutation relations) are $[e_{2},e_{3}]=e_{1}=-[e_{3},e_{2}]$. The structure vector of $\mathfrak{h}_{3}$ relative to $\hat{e}$ is $\boldsymbol{\eta}=(\eta_{ijk})\in\mathbb{F}^{27}$ where $\eta_{231}$ and $\eta_{321}$ (with $\eta_{231}=1$, $\eta_{321}=-1$) are the only nonzero coefficients of $\boldsymbol{\eta}$. First we determine $O(\boldsymbol{\eta})$ as a subset of $\mathbb{F}^{27}$. For this, let $g=(g_{ij})\in\mathop{{\rm GL}(3,\mathbb{F})}\nolimits$ and suppose that $M_{ij}$ ($i,j=1,2,3$) is the determinant of the matrix obtained from $g$ by deleting its $i$-th row and $j$-th column. Assume further that $g$ is the transition matrix from basis $(e_{1},e_{2},e_{3})$ to the basis $(e_{1}^{\prime},e_{2}^{\prime},e_{3}^{\prime})$ of $\mathfrak{h}_{3}$. (So $(e_{1}^{\prime},e_{2}^{\prime},e_{3}^{\prime})$ is the basis of $\mathfrak{h}_{3}$ given by $e_{j}^{\prime}=\sum_{i=1}^{3}g_{ij}e_{i}$ for $1\leq j\leq 3$.) An easy computation then shows that, relative to this new basis, the multiplication in $\mathfrak{h}_{3}$ is given by $$\displaystyle[e_{1}^{\prime},e_{2}^{\prime}]=(\det g)^{-1}M_{13}(M_{11}e_{1}^{% \prime}-M_{12}e_{2}^{\prime}+M_{13}e_{3}^{\prime}),$$ $$\displaystyle=(\det g)^{-1}M_{12}(M_{11}e_{1}^{\prime}-M_{12}e_{2}^{\prime}+M_% {13}e_{3}^{\prime}),$$ $$\displaystyle=(\det g)^{-1}M_{11}(M_{11}e_{1}^{\prime}-M_{12}e_{2}^{\prime}+M_% {13}e_{3}^{\prime}).$$ It follows that there exist $\alpha,\beta,\gamma,\delta\in\mathbb{F}$ such that $$\displaystyle[e_{1}^{\prime},e_{2}^{\prime}]=\gamma\delta(\alpha e_{1}^{\prime% }-\beta e_{2}^{\prime}+\gamma e_{3}^{\prime}),$$ $$\displaystyle=\beta\delta(\alpha e_{1}^{\prime}-\beta e_{2}^{\prime}+\gamma e_% {3}^{\prime}),$$ $$\displaystyle=\alpha\delta(\alpha e_{1}^{\prime}-\beta e_{2}^{\prime}+\gamma e% _{3}^{\prime}).$$ The above relations motivate the following definition. For $\alpha,\beta,\gamma,\delta\in\mathbb{F}$, let $\boldsymbol{\eta}^{\prime}(\alpha,\beta,\gamma,\delta)\in\mathbb{F}^{27}$ be defined by $\boldsymbol{\eta}^{\prime}(\alpha,\beta,\gamma,\delta)=(0,\,0,\,0,$ $\alpha\gamma\delta,\,-\beta\gamma\delta,\,\gamma^{2}\delta,$ $\alpha\beta\delta,\,-\beta^{2}\delta,\,\beta\gamma\delta,$ $-\alpha\gamma\delta,\,\beta\gamma\delta,\,-\gamma^{2}\delta,$ $0,\,0,\,0,$ $\alpha^{2}\delta,\,-\alpha\beta\delta,\,\alpha\gamma\delta,$ $-\alpha\beta\delta,\,\beta^{2}\delta,\,-\beta\gamma\delta,$ $-\alpha^{2}\delta,\,\alpha\beta\delta,\,-\alpha\gamma\delta,$ $0,\,0,\,0).$ We aim to show that the subset $V$ of $\mathbb{F}^{27}$ defined by $V=\{\boldsymbol{\eta}^{\prime}(\alpha,\beta,\gamma,\delta)\in\mathbb{F}^{27}:% \ \alpha,\beta,\gamma,\delta\in\mathbb{F}\}$ is in fact the (disjoint) union of $O(\boldsymbol{\eta})$ and $O(\boldsymbol{0})$ (recall that $\boldsymbol{0}$, the zero vector of $\mathbb{F}^{27}$, is the unique structure vector corresponding to the 3-dimensional Abelian Lie algebra). It is clear from the above discussion that $O(\boldsymbol{\eta})\subseteq V$, hence it suffices to show that any nonzero vector $\boldsymbol{v}\in V$ belongs to $O(\boldsymbol{\eta})$. For this, it will be convenient to consider the decomposition $V=V_{1}\cup V_{2}\cup V_{3}$ where the subsets $V_{1}$, $V_{2}$, $V_{3}$ of $V$ are defined as follows: First, for $\mu$, $\nu$, $\lambda$, $\sigma$, $\tau$, $\kappa\in\mathbb{F}$, define the elements $\boldsymbol{\eta}_{1}(\mu,\nu,\lambda)$, $\boldsymbol{\eta}_{2}(\tau,\sigma)$ and $\boldsymbol{\eta}_{3}(\kappa)$ of $\mathbb{F}^{27}$ by $$\displaystyle\boldsymbol{\eta}_{1}(\mu,\nu,\lambda)=(0,0,0,\nu\lambda,-\mu\nu% \lambda,\nu^{2}\lambda,\mu\lambda,-\mu^{2}\lambda,\mu\nu\lambda,-\nu\lambda,% \mu\nu\lambda,-\nu^{2}\lambda,0,0,0,\lambda,-\mu\lambda,\nu\lambda,$$ $$\displaystyle\phantom{\boldsymbol{\eta}_{1}(\mu,\nu,\lambda)=}{}-\mu\lambda,% \mu^{2}\lambda,-\mu\nu\lambda,-\lambda,\mu\lambda,-\nu\lambda,0,0,0),$$ $$\displaystyle\boldsymbol{\eta}_{2}(\tau,\sigma)=(0,0,0,0,\sigma\tau,-\sigma% \tau^{2},0,\sigma,-\sigma\tau,0,-\sigma\tau,\sigma\tau^{2},0,0,0,0,0,0,0,-% \sigma,\sigma\tau,0,0,0,0,0,0),$$ $$\displaystyle\boldsymbol{\eta}_{3}(\kappa)=(0,0,0,0,0,\kappa,0,0,0,0,0,-\kappa% ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0).$$ We then let $V_{1}=\{\boldsymbol{\eta}_{1}(\mu,\nu,\lambda):\ \mu,\nu,\lambda\in\mathbb{F}\}$, $V_{2}=\{\boldsymbol{\eta}_{2}(\tau,\sigma):\ \tau,\sigma\in\mathbb{F}\}$ and $V_{3}=\{\boldsymbol{\eta}_{3}(\kappa):\kappa\in\mathbb{F}\}$. In order to establish that $V$ is indeed the union of the three sets above, it suffices to verify that $V_{1}=\{\boldsymbol{\eta}^{\prime}(\alpha,\beta,\gamma,\delta)\in V:\ \alpha% \neq 0\}$, $V_{2}=\{\boldsymbol{\eta}^{\prime}(\alpha,\beta,\gamma,\delta)\in V:\ \alpha=0$ and $\beta\neq 0\}$ and $V_{3}=\{\boldsymbol{\eta}^{\prime}(\alpha,\beta,\gamma,\delta)\in V:\ \alpha=0$ and $\beta=0\}$. That the above equalities of sets in fact hold is immediate from the relations $\boldsymbol{\eta}^{\prime}(1,\mu,\nu,\lambda)=\boldsymbol{\eta}_{1}(\mu,\nu,\lambda)$, $\boldsymbol{\eta}_{1}(\beta\alpha^{-1},\gamma\alpha^{-1},\delta\alpha^{2})=% \boldsymbol{\eta}^{\prime}(\alpha,\beta,\gamma,\delta)$ (for $\alpha\neq 0$), $\boldsymbol{\eta}^{\prime}(0,1,\tau,-\sigma)=\boldsymbol{\eta}_{2}(\tau,\sigma)$, $\boldsymbol{\eta}_{2}(\gamma\beta^{-1},-\delta\beta^{2})=\boldsymbol{\eta}^{% \prime}(0,\beta,\gamma,\delta)$ (for $\beta\neq 0$) and $\boldsymbol{\eta}^{\prime}(0,0,1,\kappa)=\boldsymbol{\eta}_{3}(\kappa)$, $\boldsymbol{\eta}_{3}(\delta\gamma^{2})=\boldsymbol{\eta}^{\prime}(0,0,\gamma,\delta)$. Since $V=V_{1}\cup V_{2}\cup V_{3}$, we can see that any nonzero element of $V$ has one of the following forms: $\boldsymbol{\eta}_{1}(\mu,\nu,\lambda)$ (with $\lambda\neq 0$), $\boldsymbol{\eta}_{2}(\tau,\sigma)$ (with $\sigma\neq 0$) or $\boldsymbol{\eta}_{3}(\kappa)$ (with $\kappa\neq 0$). Moreover, for $\lambda\neq 0$ we have $\boldsymbol{\eta}g_{1}(\mu,\nu,\lambda)=\boldsymbol{\eta}_{1}(\mu,\nu,\lambda)$, for $\sigma\neq 0$ we have $\boldsymbol{\eta}g_{2}(\tau,\sigma)=\boldsymbol{\eta}_{2}(\tau,\sigma)$ and finally for $\kappa\neq 0$ we have $\boldsymbol{\eta}g_{3}(\kappa)=\boldsymbol{\eta}_{3}(\kappa)$ where, for $\lambda\neq 0$, $\sigma\neq 0$, $\kappa\neq 0$ respectively, the matrices $$g_{1}(\mu,\nu,\lambda)=\left[\begin{array}[]{lll}\lambda^{-1}&0&0\\ \mu&1&0\\ -\nu&0&1\end{array}\right],\quad g_{2}(\tau,\sigma)=\left[\begin{array}[]{lll}% 0&\sigma^{-1}&0\\ 1&0&0\\ 0&\tau&1\end{array}\right],\quad g_{3}(\kappa)=\left[\begin{array}[]{lll}0&0&% \kappa^{-1}\\ 1&0&0\\ 0&1&0\end{array}\right]$$ all belong to $G=\mathop{{\rm GL}(3,\mathbb{F})}\nolimits$. This establishes that $V\setminus\{\boldsymbol{0}\}=O(\boldsymbol{\eta})$. Our next aim is to show that $V$ is an irreducible algebraic set. For this, let $S=S_{1}\cup S_{2}\cup S_{3}$ where $S_{1}$, $S_{2}$, $S_{3}$ are the following subsets of $\mathbb{F}[X_{ijk}:\ 1\leq i,j,k\leq 3]$: $$\displaystyle S_{1}=\{X_{iik}:\ 1\leq i,k\leq 3\},\qquad S_{2}=\{X_{ijk}+X_{% jik}:\ 1\leq i,j,k\leq 3\},$$ $$\displaystyle S_{3}=\{X_{121}-X_{233},\quad X_{131}+X_{232},\quad X_{122}+X_{1% 33},\quad X_{122}^{2}+X_{123}X_{132},$$ $$\displaystyle X_{121}^{2}-X_{123}X_{231},\quad X_{131}^{2}+X_{132}X_{231},% \quad X_{121}X_{131}+X_{122}X_{231}\}.$$ Observe that $S\subseteq{\bf I}(V)$. We claim that $V={\bf V}(S)$. It is clear that $V\subseteq{\bf V}(S)$. To establish the reverse inclusion ${\bf V}(S)\subseteq V$, let $\boldsymbol{\gamma}=(\gamma_{ijk})\in\mathbb{F}^{27}$ be a common zero of the elements of $S$. Since $\boldsymbol{\gamma}$ is a common zero of the elements of $S_{1}\cup S_{2}$, we see that the shape of $\boldsymbol{\gamma}$ is determined once we determine the shape of the auxiliary vector $\hat{\boldsymbol{\gamma}}=(\gamma_{121},\gamma_{122},\gamma_{123},\gamma_{131}% ,\gamma_{132},\gamma_{133},\gamma_{231},\gamma_{232},\gamma_{233})\in\mathbb{F% }^{9}$. Invoking now the fact that $\boldsymbol{\gamma}$ is a common zero of the polynomials of degree $1$ in $S_{3}$ we see that in fact $\hat{\boldsymbol{\gamma}}$ has shape $(\gamma_{121},\gamma_{122},\gamma_{123},\gamma_{131},\gamma_{132},-\gamma_{122% },\gamma_{231},-\gamma_{131},\gamma_{121})$. We will consider the cases $\gamma_{231}\neq 0$ and $\gamma_{231}=0$ separately. If $\lambda=\gamma_{231}\neq 0$ we can set $\mu=\gamma_{131}\lambda^{-1}$ and $\nu=\gamma_{121}\lambda^{-1}$ from which we can deduce that $\gamma_{123}=\nu^{2}\lambda$ (since $\gamma_{121}^{2}-\gamma_{123}\gamma_{231}=0$), $\gamma_{132}=-\mu^{2}\lambda$ (since $\gamma_{131}^{2}+\gamma_{132}\gamma_{231}=0$) and $\gamma_{122}=-\mu\nu\lambda$ (since $\gamma_{121}\gamma_{131}+\gamma_{122}\gamma_{231}=0$). Hence $\boldsymbol{\gamma}\in V_{1}$ whenever $\gamma_{231}\neq 0$. For the case $\gamma_{231}=0$, by similar argument, one can show that if $\gamma_{132}\neq 0$, then $\boldsymbol{\gamma}\in V_{2}$ and if $\gamma_{132}=0$ then $\boldsymbol{\gamma}\in V_{3}$. We conclude that $V={\bf V}(S)$ and hence $V$ is an algebraic set. Next, we consider the map $\Phi:\mathbb{F}^{4}\to\mathbb{F}^{27}$: $(\alpha,\beta,\gamma,\delta)\mapsto\boldsymbol{\eta}^{\prime}(\alpha,\beta,% \gamma,\delta)$. Clearly $\Phi$ is a regular map having $V$ as its image. Thus, $\overline{\Phi(\mathbb{F}^{4})}=\overline{V}=V$. Invoking Remarks 2.1 and 2.2, we see that $V$ is irreducible. It follows from this that $O(\boldsymbol{\eta})$ is not closed in the Zariski topology. (Note that if $O(\boldsymbol{\eta})$ were Zariski-closed this would imply that $V=O(\boldsymbol{\eta})\cup\{\boldsymbol{0}\}$ is reducible, being the union of two nonempty closed sets.) Hence, $O(\boldsymbol{\eta})$ is properly contained in $\overline{O(\boldsymbol{\eta})}$. Also $\overline{O(\boldsymbol{\eta})}\subseteq V$ since ${O(\boldsymbol{\eta})}\subseteq V$ and $V$ is an algebraic set. We conclude that $\overline{O(\boldsymbol{\eta})}=V=O(\boldsymbol{\eta})\cup\{\boldsymbol{0}\}$. In other words, over an arbitrary infinite field, the only proper degeneration of $\mathfrak{h}_{3}$ is to the Abelian Lie algebra $\mathfrak{a}_{3}$. We remark here that this is a well-known fact and has been proved using different methods over various fields, see for example [1, 3, 8, 12]. In the discussion above we presented an alternative way of obtaining it, based on the definition of an irreducible algebraic set. 3.2 The algebra $\mathfrak{g}_{2}\oplus\mathfrak{a}_{1}$ In this subsection we perform a similar investigation for the algebra $\mathfrak{g}=\mathfrak{g}_{2}\oplus\mathfrak{a}_{1}$. Note that this algebra has an $\mathbb{F}$-basis $\hat{b}=(b_{1},b_{2},b_{3})$ relative to which the only non-zero commutation relations are given by $[b_{1},b_{2}]=b_{1}=-[b_{2},b_{1}]$. Let $\boldsymbol{\rho}=(\rho_{ijk})\in\mathbb{F}^{27}$ be the structure vector of $\mathfrak{g}$ relative to the basis $\hat{b}$. Suppose now that $g\in G=\mathop{{\rm GL}(3,\mathbb{F})}\nolimits$ is the transition matrix from $\hat{b}$ to the basis $\hat{b}^{\prime}=(b_{1}^{\prime},b_{2}^{\prime},b_{3}^{\prime})$ of $\mathfrak{g}$. It is then easy to show that $$\displaystyle[b_{1}^{\prime},b_{2}^{\prime}]=(\det g)^{-1}{M_{33}}(M_{11}b_{1}% ^{\prime}-M_{12}b_{2}^{\prime}+M_{13}b_{3}^{\prime}),$$ $$\displaystyle=(\det g)^{-1}{M_{32}}(M_{11}b_{1}^{\prime}-M_{12}b_{2}^{\prime}+% M_{13}b_{3}^{\prime}),$$ $$\displaystyle=(\det g)^{-1}{M_{31}}(M_{11}b_{1}^{\prime}-M_{12}b_{2}^{\prime}+% M_{13}b_{3}^{\prime})$$ where, as before, $M_{ij}$ denotes the determinant of the matrix obtained from $g$ by deleting its $i$-th row and $j$-th column (in particular, the $M_{ij}$ are elements of our field $\mathbb{F}$). It follows that there exist $\chi_{1},\psi_{1},\omega_{1},\chi_{2},\psi_{2},\omega_{2},\delta\in\mathbb{% \mathbb{F}}$ such that $$\displaystyle[b_{1}^{\prime},b_{2}^{\prime}]=\delta{\chi_{2}}(\chi_{1}b_{1}^{% \prime}-\psi_{1}b_{2}^{\prime}+\omega_{1}b_{3}^{\prime}),$$ $$\displaystyle=\delta{\psi_{2}}(\chi_{1}b_{1}^{\prime}-\psi_{1}b_{2}^{\prime}+% \omega_{1}b_{3}^{\prime}),$$ $$\displaystyle=\delta{\omega_{2}}(\chi_{1}b_{1}^{\prime}-\psi_{1}b_{2}^{\prime}% +\omega_{1}b_{3}^{\prime}).$$ This prompts us to define $\boldsymbol{\rho}^{\prime}(\chi_{1},\psi_{1},\omega_{1},\chi_{2},\psi_{2},% \omega_{2},\delta)\in\mathbb{F}^{27}$ by $\boldsymbol{\rho}^{\prime}(\chi_{1},\psi_{1},\omega_{1},\chi_{2},\psi_{2},% \omega_{2},\delta)=(0,\,0,\,0,$ $\chi_{1}\chi_{2}\delta,$ $-\psi_{1}\chi_{2}\delta,$ $\omega_{1}\chi_{2}\delta,$ $\chi_{1}\psi_{2}\delta,$ $-\psi_{1}\psi_{2}\delta,\,\omega_{1}\psi_{2}\delta,$ $-\chi_{1}\chi_{2}\delta,\,\psi_{1}\chi_{2}\delta,\,-\omega_{1}\chi_{2}\delta,$ $0,\,0,\,0,$ $\chi_{1}\omega_{2}\delta,$ $-\psi_{1}\omega_{2}\delta,$ $\omega_{1}\omega_{2}\delta,$ $-\chi_{1}\psi_{2}\delta,\,\psi_{1}\psi_{2}\delta,$ $-\omega_{1}\psi_{2}\delta,$ $-\chi_{1}\omega_{2}\delta,$ $\psi_{1}\omega_{2}\delta,$ $-\omega_{1}\omega_{2}\delta,$ $0,\,0,\,0)$, and the subset $U$ of $\mathbb{F}^{27}$ by $U=\{\boldsymbol{\rho}^{\prime}(\chi_{1},\psi_{1},\omega_{1},\chi_{2},\psi_{2},% \omega_{2},\delta):\ \chi_{1},\psi_{1},\omega_{1},\chi_{2},\psi_{2},\omega_{2}% ,\delta\in\mathbb{\mathbb{F}}\}$. It is then clear that $O(\boldsymbol{\rho})\subseteq U$. We want to show that $U$ is an algebraic set containing $V=O(\boldsymbol{\eta})\cup\{\boldsymbol{0}\}$ (we keep the notation for $V$, $\boldsymbol{\eta}$, $\boldsymbol{\eta}^{\prime}$ and also for $S$, $S_{1}$, $S_{2}$, $S_{3}$ introduced in the previous subsection). The inclusion $V\subseteq U$ is immediate from the fact that $\boldsymbol{\eta}^{\prime}(\alpha,\beta,\gamma,\delta)=\boldsymbol{\rho}^{% \prime}(\alpha,\beta,\gamma,\gamma,\beta,\alpha,\delta)$. Next, we define the subset $T$ of ${\bf I}(U)$ by $T=S_{1}\cup S_{2}\cup T_{3}$ where $$\displaystyle T_{3}=\{X_{121}X_{132}-X_{122}X_{131},\quad X_{121}X_{232}-X_{12% 2}X_{231},\quad X_{131}X_{232}-X_{132}X_{231},$$ $$\displaystyle X_{121}X_{133}-X_{123}X_{131},\quad X_{121}X_{233}-X_{123}X_{231% },\quad X_{232}X_{123}-X_{122}X_{233},$$ $$\displaystyle X_{122}X_{133}-X_{123}X_{132},\quad X_{132}X_{233}-X_{133}X_{232% },\quad X_{233}X_{131}-X_{133}X_{231}\}$$ (recall the definition of $S_{1}$ and $S_{2}$ in Subsection 3.1). Now let $S^{\prime}=T\cup\{X_{121}-X_{233},\ X_{131}+X_{232},\ X_{122}+X_{133}\}% \subseteq T\cup S_{3}$. It is easy to check that $V\subseteq{\bf V}(S^{\prime})$. We also have ${\bf V}(S^{\prime})={\bf V}(T\cup S_{3})$. To see this last equality of sets, note first that ${\bf V}(T\cup S_{3})\subseteq{\bf V}(S^{\prime})$ since $S^{\prime}\subseteq T\cup S_{3}$. On the other hand, any $\boldsymbol{\nu}\in{\bf V}(S^{\prime})$ is a common zero of every polynomial in $T\cup S_{3}$. Hence, we also have ${\bf V}(S^{\prime})\subseteq{\bf V}(T\cup S_{3})$. Since $V\subseteq{\bf V}(S^{\prime})$, we get $V\subseteq{\bf V}(T\cup S_{3})$. But $T\cup S_{3}\supseteq S$, so ${\bf V}(T\cup S_{3})\subseteq{\bf V}(S)=V$. We conclude that $V\,(={\bf V}(S))={\bf V}(T\cup S_{3})={\bf V}(S^{\prime})$. We aim to show that $U={\bf V}(T)$. This would imply that $U$ is an algebraic set (and also provide an alternative way of seeing that $V\subseteq U$ in view of the observation above). Clearly, $U\subseteq{\bf V}(T)$. In order to establish the reverse inclusion, it will be convenient to decompose $U$ as a union of three subsets which contain among them all elements of ${\bf V}(T)$. With $\alpha$, $\beta$, $\gamma$, $\mu$, $\nu$, $\phi$, $\rho$, $\sigma$, $\tau$, $\zeta$, $\theta$, $\xi$, $\kappa\in\mathbb{F}$, define the elements $\boldsymbol{\rho}_{1}(\alpha,\beta,\gamma,\mu,\nu,\phi)$, $\boldsymbol{\rho}_{2}(\sigma,\tau,\rho,\zeta)$ and $\boldsymbol{\rho}_{3}(\theta,\xi,\kappa)\in\mathbb{F}^{27}$ by $$\displaystyle\boldsymbol{\rho}_{1}(\alpha,\beta,\gamma,\mu,\nu,\phi)=(0,0,0,% \mu\alpha,-\mu\beta,\mu\gamma,\nu\alpha,-\nu\beta,\nu\gamma,-\mu\alpha,\mu% \beta,-\mu\gamma,0,0,0,\phi\alpha,-\phi\beta,\phi\gamma,$$ $$\displaystyle\phantom{S_{1}(\alpha,\beta,\gamma,\mu,\nu,\phi)=(}{}-\nu\alpha,% \nu\beta,-\nu\gamma,-\phi\alpha,\phi\beta,-\phi\gamma,0,0,0),$$ $$\displaystyle\boldsymbol{\rho}_{2}(\sigma,\tau,\rho,\zeta)=(0,0,0,0,\sigma,-% \sigma\zeta,0,\tau,-\tau\zeta,0,-\sigma,\sigma\zeta,0,0,0,0,\rho,-\rho\zeta,0,% -\tau,\tau\zeta,0,-\rho,\rho\zeta,$$ $$\displaystyle\phantom{S_{1}(\sigma,\tau,\rho,\zeta)=(}{}0,0,0),$$ $$\displaystyle\boldsymbol{\rho}_{3}(\theta,\xi,\kappa)=(0,0,0,0,0,\theta,0,0,% \xi,0,0,-\theta,0,0,0,0,0,\kappa,0,0,-\xi,0,0,-\kappa,0,0,0).$$ Also define the subsets $U_{1}$, $U_{2}$ and $U_{3}$ of $\mathbb{F}^{27}$ by $U_{1}=\{\boldsymbol{\rho}_{1}(\alpha,\beta,\gamma,\mu,\nu,\phi):\ \alpha,\beta% ,\gamma,\mu,\nu,\phi\in\mathbb{F}$ and $\alpha\neq 0\}$, $U_{2}=\{\boldsymbol{\rho}_{2}(\sigma,\tau,\rho,\zeta):$ $\sigma,\tau,\rho,\zeta\in\mathbb{F}\}$ and $U_{3}=\{\boldsymbol{\rho}_{3}(\theta,\xi,\kappa):$ $\theta,\xi,\kappa\in\mathbb{F}\}$. It is then immediate from the relations $\boldsymbol{\rho}_{1}(\alpha,\beta,\gamma,\mu,\nu,\phi)=\boldsymbol{\rho}^{% \prime}(\chi_{1}=\alpha,\psi_{1}=\beta,\omega_{1}=\gamma,\chi_{2}=\mu,\psi_{2}% =\nu,\omega_{2}=\phi,\delta=1)$, $\boldsymbol{\rho}_{2}(\sigma,\tau,\rho,\zeta)=\boldsymbol{\rho}^{\prime}(\chi_% {1}=0,\psi_{1}=-1,\omega_{1}=-\zeta,\chi_{2}=\sigma,\psi_{2}=\tau,\omega_{2}=% \rho,\delta=1)$ and $\boldsymbol{\rho}_{3}(\theta,\xi,\kappa)=\boldsymbol{\rho}^{\prime}(\chi_{1}=0% ,\psi_{1}=0,\omega_{1}=1,\chi_{2}=\theta,\psi_{2}=\xi,\omega_{2}=\kappa,\delta% =1)$ that $U_{i}\subseteq U$ for $i=1,2,3$. We now show that ${\bf V}(T)\subseteq U_{1}\cup U_{2}\cup U_{3}$. Let $\boldsymbol{\gamma}=(\gamma_{ijk})\in\mathbb{F}^{27}$ be a common zero of all polynomials in $T$. As $T\supseteq S_{1}\cup S_{2}$, similarly to the Heisenberg algebra case, we will work with the auxiliary vector $\hat{\boldsymbol{\gamma}}=(\gamma_{121},\gamma_{122},\gamma_{123},\gamma_{131}% ,\gamma_{132},\gamma_{133},\gamma_{231},\gamma_{232},\gamma_{233})\in\mathbb{F% }^{9}$. Again, we will need to consider different subcases. We begin by considering the case $\gamma_{121}\neq 0$. Since $\boldsymbol{\gamma}\in{\bf V}(T)$, we get $\hat{\boldsymbol{\gamma}}=(\gamma_{121},\gamma_{122},\gamma_{123},\gamma_{131}% ,\gamma_{122}\gamma_{131}\gamma_{121}^{-1},\gamma_{123}\gamma_{131}\gamma_{121% }^{-1},\gamma_{231},\gamma_{122}\gamma_{231}\gamma_{121}^{-1},\gamma_{123}% \gamma_{231}\gamma_{121}^{-1})$. For example, to see that $\gamma_{132}=\gamma_{122}\gamma_{131}\gamma_{121}^{-1}$, note that $\boldsymbol{\gamma}$ is a zero of the polynomial $X_{121}X_{132}-X_{122}X_{131}$ which belongs to $T$. On setting $\mu=1$, $\nu=\gamma_{131}\gamma_{121}^{-1}$, $\phi=\gamma_{231}\gamma_{121}^{-1}$, $\alpha=\gamma_{121}\,(\neq 0)$, $\beta=-\gamma_{122}$, $\gamma=\gamma_{123}$, we see that $\boldsymbol{\gamma}=\boldsymbol{\rho}_{1}(\alpha,\beta,\gamma,\mu,\nu,\phi)$ where $\alpha\neq 0$, so $\boldsymbol{\gamma}\in U_{1}$. Next we consider the case $\gamma_{121}=0$. We split this case into the subcases $\gamma_{122}\neq 0$ (where, by similar argument as above, we can show that $\boldsymbol{\gamma}\in U_{2}$) and $\gamma_{122}=0$. It remains to consider the case when when both $\gamma_{121}$ and $\gamma_{122}$ are equal to zero and the next step is to split this case into subcases according to whether $\gamma_{123}\neq 0$ (we can show then that $\boldsymbol{\gamma}\in U_{3}$) or $\gamma_{123}=0$. Continuing in a similar fashion, we finally deduce that ${\bf V}(T)$ is indeed a subset of $U_{1}\cup U_{2}\cup U_{3}$. Summing up the above discussion, we see that $U\subseteq{\bf V}(T)\subseteq U_{1}\cup U_{2}\cup U_{3}\subseteq U$. This forces $U=U_{1}\cup U_{2}\cup U_{3}={\bf V}(T)$. Recall now that $V=O(\boldsymbol{\eta})\cup\{\boldsymbol{0}\}\subseteq U$. In order to show that $U=O(\boldsymbol{\rho})\cup O(\boldsymbol{\eta})\cup\{\boldsymbol{0}\}$, we find, for each $\boldsymbol{\delta}\in U\setminus V$, an invertible matrix $g({\boldsymbol{\delta}})\in G$ such that $\boldsymbol{\delta}=\boldsymbol{\rho}\,g({\boldsymbol{\delta}})$. In the table below we summarize the results of this computation, listing also the corresponding matrices $g=g({\boldsymbol{\delta}})$. We first split into subcases according to whether $\boldsymbol{\delta}\in U\setminus V$ is of the form $\boldsymbol{\rho}_{1}$ (with $\alpha\neq 0$), $\boldsymbol{\rho}_{2}$ or $\boldsymbol{\rho}_{3}$ and as it turns out, depending on the values of the elements of $\mathbb{F}$ involved, we need to split into further subcases. It is now useful to recall that $V={\bf V}(S^{\prime})$ where $S^{\prime}=T\cup\{X_{121}-X_{233},\ X_{131}+X_{232},\ X_{122}+X_{133}\}% \subseteq T\cup S_{3}$. Let $\boldsymbol{\rho}^{\prime}=(\rho^{\prime}_{ijk})\in U$. It follows that $\boldsymbol{\rho}^{\prime}\in V$ if, and only if all three conditions $\rho^{\prime}_{121}-\rho^{\prime}_{233}=0$, $\rho^{\prime}_{131}+\rho^{\prime}_{232}=0$ and $\rho^{\prime}_{122}+\rho^{\prime}_{133}=0$ are satisfied. In particular, in the case $\boldsymbol{\rho}^{\prime}=\boldsymbol{\rho}_{1}(\alpha,\beta,\gamma,\mu,\nu,\phi)$, we have $\boldsymbol{\rho}^{\prime}\in V$ if, and only if, all of the conditions $\mu\alpha-\phi\gamma=0$, $\nu\alpha-\phi\beta=0$ and $-\mu\beta+\nu\gamma=0$ are satisfied. For $\boldsymbol{\rho}_{1}$ to be an element of $U_{1}$ we have the restriction $\alpha\neq 0$, so in this case, the third of the last three conditions follows from the other two (this is because the conditions $\mu\alpha-\phi\gamma=0$ and $\nu\alpha-\phi\beta=0$ are equivalent to the conditions $\mu=\phi\gamma\alpha^{-1}$ and $\nu=\phi\beta\alpha^{-1}$ if $\alpha\neq 0$). For simplicity, in the table below we will be writing $A_{1}=\mu\alpha-\phi\gamma$, $A_{2}=\nu\alpha-\phi\beta$. Similar observations can be made in the cases $\boldsymbol{\rho}^{\prime}$ has form $\boldsymbol{\rho}_{2}$ or $\boldsymbol{\rho}_{3}$ (as it can also be seen from the table). Moreover, in the table below, vector $\boldsymbol{\rho}_{1}=\boldsymbol{\rho}_{1}(\alpha,\beta,\gamma,\mu,\nu,\phi)$ will always be considered under the restriction $\alpha\neq 0$, compare with the definition of set $U_{1}$. The computation above establishes that $U=O(\boldsymbol{\rho})\cup O(\boldsymbol{\eta})\cup\{\boldsymbol{0}\}$. Now recall that $U\,(={\bf V}(T))$ is Zariski-closed. In fact, by similar argument as in the case of the set $V$, we can show that $U$ is irreducible, considering now the regular map $\Phi:\mathbb{F}^{7}\to U=\bar{U}\subseteq F^{27}$: $(\chi_{1},\psi_{1},\omega_{1},\chi_{2},\psi_{2},\omega_{2},\delta)\mapsto(0,\,% 0,\,0,$ $\chi_{1}\chi_{2}\delta,$ $-\psi_{1}\chi_{2}\delta,$ $\omega_{1}\chi_{2}\delta,$ $\chi_{1}\psi_{2}\delta,$ $-\psi_{1}\psi_{2}\delta,\,\omega_{1}\psi_{2}\delta,$ $-\chi_{1}\chi_{2}\delta,\,\psi_{1}\chi_{2}\delta,\,-\omega_{1}\chi_{2}\delta,$ $0,\,0,\,0,$ $\chi_{1}\omega_{2}\delta,$ $-\psi_{1}\omega_{2}\delta,$ $\omega_{1}\omega_{2}\delta,$ $-\chi_{1}\psi_{2}\delta,\,\psi_{1}\psi_{2}\delta,$ $-\omega_{1}\psi_{2}\delta,$ $-\chi_{1}\omega_{2}\delta,$ $\psi_{1}\omega_{2}\delta,$ $-\omega_{1}\omega_{2}\delta,$ $0,\,0,\,0)=\boldsymbol{\rho}^{\prime}(\chi_{1},\psi_{1},\omega_{1},\chi_{2},% \psi_{2},\omega_{2},\delta)$. Since $U$ is Zariski-closed and ${O(\boldsymbol{\rho})}\subseteq U$, we get $\overline{O(\boldsymbol{\rho})}\subseteq U$. Invoking the fact that $U=O(\boldsymbol{\rho})\cup O(\boldsymbol{\eta})\cup\{\boldsymbol{0}\}$ we can deduce that $\mathfrak{h}_{3}$ and $\mathfrak{a}_{3}$ are the only possible Lie algebras which $\mathfrak{g}_{2}\oplus\mathfrak{a}_{1}$ can properly degenerate to. In order to establish that $\mathfrak{g}_{2}\oplus\mathfrak{a}_{1}$ in fact degenerates to both $\mathfrak{h}_{3}$ and $\mathfrak{a}_{3}$ it suffices to show that $\overline{O(\boldsymbol{\rho})}=U$. Since $U$ is irreducible and $\overline{O(\boldsymbol{\eta})}=O(\boldsymbol{\eta})\cup\{\boldsymbol{0}\}$ we get that $O(\boldsymbol{\rho})$ is not Zariski-closed. It follows that $O(\boldsymbol{\rho})$ is properly contained in $\overline{O(\boldsymbol{\rho})}$. If $\boldsymbol{\eta}\not\in\overline{O(\boldsymbol{\rho})}$, then $O(\boldsymbol{\eta})\cap\overline{O(\boldsymbol{\rho})}=\varnothing$ since $\overline{O(\boldsymbol{\rho})}$ is a union of orbits (see Remark 2.4(ii)). It would then follow that $\overline{O(\boldsymbol{\rho})}=O(\boldsymbol{\rho})\cup\{\boldsymbol{0}\}$, contradicting the fact that $U$ is irreducible. We conclude that $\boldsymbol{\eta}\in\overline{O(\boldsymbol{\rho})}$. It follows that $O(\boldsymbol{\eta})\subseteq\overline{O(\boldsymbol{\rho})}$ and hence $\overline{O(\boldsymbol{\eta})}\subseteq\overline{O(\boldsymbol{\rho})}$. Since $\boldsymbol{0}\in\overline{O(\boldsymbol{\eta})}$, we get that $\boldsymbol{0}\in\overline{O(\boldsymbol{\rho})}$. Summing up, we have shown $\overline{O(\boldsymbol{\rho})}\subseteq U=O(\boldsymbol{\rho})\cup O(% \boldsymbol{\eta})\cup\{\boldsymbol{0}\}\subseteq\overline{O(\boldsymbol{\rho})}$. Hence, $U=\overline{O(\boldsymbol{\rho})}$ as required. We remark here that it is well-known that, over an infinite field, any Lie algebra degenerates to the abelian Lie algebra of the same dimension. Also note that already in [1] it is shown that $\mathfrak{g}$ degenerates to $\mathfrak{h}_{3}$ in the case the ground field is $\mathbb{R}$. In view of [8, Lemma 3.9] the technique used in [1] can be extended to obtain a degeneration from $\mathfrak{g}$ to $\mathfrak{h}_{3}$ now over an arbitrary infinite field. In the discussion above we provided an alternative way of obtaining this particular degeneration using the notion of an irreducible algebraic set. We close this subsection with some general comments regarding our sets above. First, we can observe that $O(\boldsymbol{\rho})=U\setminus\overline{O(\boldsymbol{\eta})}=\overline{O(% \boldsymbol{\rho})}\setminus\overline{O(\boldsymbol{\eta})}$ so $O(\boldsymbol{\rho})$ is open in its closure (compare [2, Proposition 2.5.2] for the case of an algebraically closed field). Now let $W$ be the union of the three principal open sets $\{\boldsymbol{\alpha}\in\mathbb{F}^{n^{3}}:$ $f_{i}(\boldsymbol{\alpha})\neq 0\}$ for $i=1,2,3$ where $f_{1}=X_{121}-X_{233}$, $f_{2}=X_{131}+X_{232}$ and $f_{3}=X_{122}+X_{133}$. Since $\overline{O(\boldsymbol{\rho})}={\bf V}(T)$ and $\overline{O(\boldsymbol{\eta})}={\bf V}(S^{\prime})$ where $S^{\prime}=T\cup\{f_{1},\ f_{2},\ f_{3}\}$, we see that $O(\boldsymbol{\rho})={\bf V}(T)\cap W$. This in fact verifies that ${O(\boldsymbol{\rho})}$ consists of precisely those points in $U\,(=\overline{O(\boldsymbol{\rho})})$ which do not correspond to unimodular Lie algebras (compare, for example, with [8, Remark 4.12]). References [1] Conatser C.W., Contractions of the low-dimensional Lie algebras, J. Math. Phys. 13 (1972), 196–203. [2] Geck M., An Introduction to Algebraic Geometry and Algebraic Groups, Oxford University Press, 2003. [3] Gorbatsevich V.V., Contractions and degenerations of finite-dimensional algebras, Izv. Vyssh. Uchebn. Zaved., Mat. 10 (1991), 19–27 (Russian); translated in Soviet Math. (Iz. VUZ) 35 (1991), 17–24. [4] Grunewald F. and O’Halloran J., Varieties of nilpotent Lie algebras of dimensions less then six. J. Algebra 112 (1988), 315–325. [5] Hepp K., The classical limit for quantum mechanical correlation functions, Comm. in Math. Phys. 35 (1974), 265–277. [6] Inönü E and Wigner E P 1953 On the contraction of groups and their representations Proc. Nat. Acad. Sci. U.S.A. 39 510–524. [7] Inönü E and Wigner E P 1954 On a particular type of convergence to a singular matrix Proc. Nat. Acad. Sci. U.S.A. 40 119–121. [8] Ivanova N.M. and Pallikaros C.A., On degenerations of algebras over an arbitrary field, arXiv:1702.02647 [math.RA]. [9] Jacobson N., Lie Algebras, Interscience, New York, 1962. [10] Kraft H., Geometric methods in representation theory, in “Representations of Algebras”, Springer-Verlag, Berlin-Heidelberg-New York, 1982. 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\makeFNbottom\setstretch 1.125 Photophysics of indole upon x-ray absorption Thomas Kierspel,$\!{}^{a,b,c}$ Cédric Bomme,$\!{}^{d}$ Michele Di Fraia,$\!{}^{a,b,e}$ Joss Wiese,$\!{}^{a,f}$ Denis Anielski,$\!{}^{d}$ Sadia Bari,$\!{}^{d,g}$ Rebecca Boll,$\!{}^{d,g,h}$ Benjamin Erk,$\!{}^{d}$ Jens S. Kienitz,$\!{}^{a,b,c}$ Nele L. M. Müller,$\!{}^{a,b,c}$ Daniel Rolles,$\!{}^{d,i}$ Jens Viefhaus,$\!{}^{d}$ Sebastian Trippel,$\!{}^{a,b}\,{}^{\ast}$ and Jochen Küpper$\,{}^{a,b,c,f}$ A photofragmentation study of gas-phase indole (C${}_{8}$H${}_{7}$N) upon single-photon ionization at a photon energy of 420 eV is presented. Indole was primarily inner-shell ionized at its nitrogen and carbon $1s$ orbitals. Electrons and ions were measured in coincidence by means of velocity map imaging. The angular relationship between ionic fragments is discussed along with the possibility to use the angle-resolved coincidence detection to perform experiments on molecules that are strongly oriented in their recoil-frame. The coincident measurement of electrons and ions revealed fragmentation-pathway-dependent electron spectra, linking the structural fragmentation dynamics to different electronic excitations. Evidence for photoelectron-impact self-ionization was observed. David W. Pratt originally initiated our investigations into the photophysics of indole and this paper is dedicated to him on the occasion of his 80${}^{\text{th}}$ birthday. January 14, 2021 Keywords: photophysics, indole, fragmentation, PEPIPICO ††footnotetext: ${}^{a}$ Center for Free-Electron Laser Science, Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany††footnotetext: ${}^{b}$ Center for Ultrafast Imaging, Universität Hamburg, 22761 Hamburg, Germany††footnotetext: ${}^{c}$ Department of Physics, Universität Hamburg, 22761 Hamburg, Germany††footnotetext: ${}^{d}$ Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany††footnotetext: ${}^{e}$ Elettra-Sincrotrone Trieste S.C.p.A., 34149, Basovizza, Italy††footnotetext: ${}^{f}$ Department of Chemistry, Universität Hamburg, 20146 Hamburg, Germany††footnotetext: ${}^{g}$ European XFEL GmbH, 22869 Schenefeld, Germany††footnotetext: ${}^{h}$ Max Planck Institute for Nuclear Physics, 69117 Heidelberg, Germany††footnotetext: ${}^{i}$ J.R. Macdonald Laboratory, Department of Physics, Kansas State University, Manhattan, KS 66506, USA††footnotetext: ${}^{\ast}$sebastian.trippel@cfel.de; https://www.controlled-molecule-imaging.org 1 Introduction Indole, the chromophore of the essential amino acid tryptophan, is an ubiquitous part of peptides and proteins. It is the strongest near ultraviolet (UV) absorber in these biological molecules and, for a detailed understanding of the photostability and radiation damage of these biological samples, it is highly relevant to disentangle indole’s intrinsic photophysics, e. g., its various excitation, relaxation, and fragmentation pathways following electronic excitation. Indole was extensively studied using microwave 1, 2 and optical spectroscopy 3, 4, 5, 6, 7, 8, 9, 10, including vibrationally 9, 10 and rotationally resolved 3, 4, 5, 6, 7, 8 electronic spectroscopy, and also using time-resolved ion and photoelectron spectroscopy 11, 12, 13. Here, we extend these studies to the investigation of the photophysics and photofragmentation dynamics of indole following soft x-ray absorption. Fragmentation studies of isolated gas-phase molecules and clusters allow to extract molecular properties, such as the geometric structure 14, 15. Therefore, they provide a link between the laboratory frame and the molecular frame that allows to investigate wave packet dynamics on complex potential energy surfaces through molecular-frame dependent observables such as, for instance, molecular-frame angle-resolved photoelectron spectroscopy (MF-ARPES) 16, 17. Furthermore, fundamental relaxation processes like Auger decay, interatomic (intermolecular) Coulombic decay 18, 19, or electron-transfer mediated decay (ETMD) 20 can be investigated upon x-ray ionization – and can be employed as observables to study molecular dynamics. In order to understand the complete fragmentation and charge rearrangement dynamics of molecules and small compound systems such as clusters, coincidence measurements can be highly advantageous 21. Various techniques were developed during the last years 22, 23, which include photoion-photoion coincidence (PIPICO), photoelectron-photoion-photoion coincidence (PEPIPICO), or Auger-electron photoion-photoion coincidence (AEPIPICO) measurements 24, 25, 26, 27, 28, 29, 30, 31, 32, 33. Such coincidence measurements can, at least for simple molecules, be used to study molecular-frame (MF) properties by reconstructing the molecular orientation from the measured three-dimensional (3D) velocity distributions of all charged fragments. These include recoil-frame angle-resolved photoelectron spectra (RF-ARPES) 34, 29, 35, 36, 37, 38, which allow to image molecular orbitals and their temporal evolution during dissociation 37, or to extract structure and molecular dynamics information by “diffraction from within” 39 type of experiments. For such experiments, it is highly advantageous to locally ionize the molecule at a specific atom, which can be achieved by inner-shell ionization via extreme ultraviolet radiation, soft x-ray, or x-ray radiation. Localized ionization provides also access to the local electronic structure and excited state dynamics 37, 40, 41, and can be used to break specific bonds 42. Here, isolated indole (C${}_{8}$H${}_{7}$N) molecules were ionized by a single (soft) x-ray photon with an energy of 420 eV, i. e., $\mathord{\sim}10$ eV above the nitrogen $1s$ ionization threshold, the N($1s$) edge. This gives rise to an enhanced localized ionization at the nitrogen atom in the molecule.***At a photon energy of 420 eV, the nitrogen atom has the highest atomic cross section ($0.6466\cdot 10^{-22}\text{\leavevmode\nobreak\ m}^{2}$) of the molecule’s constituents, followed by carbon atoms ($0.4327\cdot 10^{-22}\text{\leavevmode\nobreak\ m}^{2}$) 43. In total, the indole monomer contains eight carbon and one nitrogen atom, leading to a probability of 16 % that the complex is locally ionized out of the the nitrogen $1s$ orbital, assuming that the molecular cross sections for the $1s$ orbitals do not differ significantly from the atomic ones, and neglecting the contribution from the inner-valence and valence orbitals, which are estimated to be on the order of a few percent. The photoabsorption cross section for atomic hydrogen is 3000 times smaller than for nitrogen and is not taken into account. Photo- and Auger electrons as well as the ionic fragments of indole were detected in coincidence in a double-sided velocity map imaging (VMI) spectrometer (VMIS) 44. Our work provides the first inner-shell photoionization study of bare gas-phase indole. It also provides the basis for relaxation and fragmentation studies of larger indole-containing molecules, e. g., tryptophan, as well as molecular clusters, such as the investigation of intermolecular interactions in indole–water 45, 46 or indole–ammonia 46. In fact, the experiment described here was set up such that the photofragmentation of indole and indole–water clusters could both be measured. Our findings for the photophysics of indole–water${}_{1}$ clusters are beyond the scope of this manuscript and will be presented in an upcoming publication 47, 48. 2 Experimental setup Figure 1 shows the experimental setup, including a species-selecting molecular-beam injector 45, 49. A supersonic expansion of a few mbar of indole seeded in 60 bar of helium was provided by a pulsed Even-Lavie valve 51. The valve was operated at a repetition rate of 250 Hz and a temperature of $110\,^{\circ}{}\text{C}$. The deflector was used to spatially separate different species present in the expansion, including a separation of indole from the helium seed gas. The molecular beam apparatus was mounted to the CFEL-ASG Multi-Purpose (CAMP) endstation 52, which was connected to the Petra III synchrotron’s variable polarization beamline P04 53 (circular polarization $>\;98\%$, $5\cdot 10^{13}$ photons/s, 480 bunches, 16 ns bunch spacing). The molecular beam was crossed by the 420 eV ($\lambda$ = 2.95 nm) synchrotron radiation under an angle of 90 degree inside a double-sided VMIS 50 for simultaneous electron and ion detection. Electrons and ions were detected with a hexanode (electrons) and quadanode (ions) delay line detector (HEX80 and DLD80, RoentDek), respectively. For the data presented, however, the hexanode detector had to be operated as a quadanode due to a defect third delay-line layer. The electronic readout was triggered by the detection of an electron and was set to an acquisition time of 6 µs, which was long enough to detect ionic fragments with an atomic mass ($m$)-to-charge ($q$) ratio of up to $\mathord{\sim}220$. The pulse duration of the molecular beam in the interaction region was about 60 µs full width at half maximum (FWHM), resulting in a duty cycle of $\mathord{\sim}1.5\leavevmode\nobreak\ \%$. A logical gate, synchronized to the arrival time of the molecular beam in the interaction zone, was used to record data in a 200 µs time window, reducing the absolute number of background events. The overall event rate was on the order of a few hundred events per second. The inset of Figure 1 shows the reconstructed temporal molecular beam profile plus a constant offset due to background events. The background events were used as a background correction in, e. g., Figure 2 . In addition to the reconstructed molecular beam profile vertical black lines are shown, indicating the pulse structure of the synchrotron. 3 Coincidence spectra The photofragmentation of indole upon single-photon inner-shell ionization from the nitrogen and carbon $1s$ orbitals was investigated via a coincidence measurement between the emitted electrons and the corresponding ionic fragments. A background subtracted PEPIPICO spectrum 54, 55 of indole is shown in Figure 2 as a function of the atomic mass-to-charge $m/q$ ratio of the first and second detected ion, $m_{1}$/$q_{1}$ and $m_{2}$/$q_{2}$, respectively. The molecular structure of indole is shown in the inset of Figure 2 . The PEPIPICO map allows to disentangle different fragmentation channels of indole in the case of at least two detected ionic fragments. Nine principal coincidence regions are observed, which are labeled 1–6, 1${}^{*}$, 3${}^{*}$, and 4${}^{*}$. A detailed list of the identified fragmentation channels is given in Table 1 . The sum of the masses of the fragments in regions 1–3 is equal to the mass of indole, neglecting the loss of hydrogen/protons. Therefore, these fragmentation channels correspond to the generation of two heavy ionic fragments, which are called in the following a two-hole two-fragment (2h2f) fragmentation channel. They are visually separated from the other channels in Figure 2 by the solid black line. Coincidence regions 4–6, and 4* are due to fragmentation into three or more fragments, i. e., the total masses of the first two detected ions corresponding to a single event do not add up to the mass of the indole monomer. The missing fragments can be neutral or ionic and the corresponding channels are labeled two-hole three-fragment (2h3f) and three-hole three-fragment (3h3f), respectively. Due to a limited detection efficiency, the 3h3f fragments can split into different coincidence regions as, for example, the regions 4 and 4*. Both regions have the same ’heavy’ second detected ion, i. e., $\mathrm{C_{3}NH_{2}^{+}\leavevmode\nobreak\ or\leavevmode\nobreak\ C_{4}H_{4}^% {+}}$, but alternating ’lighter’ fragments for the first detected ion. If only the ’lighter’ fragments are detected, or if all ions are detected, this fragmentation channel is, in the used representation, part of region 6. Regions 1*, and 3* have molecular fragments with the same masses as regions 1, and 3, but with different charge distribution, i. e., they contain both, singly and doubly charged ionic fragments and are labeled therefore as three-hole two-fragment (3h2f) channels. If not stated otherwise, the losses of hydrogens or protons will not be considered, and are not included in the labeling of the different fragmentation channels. Further, 2h2f and 2h3f fragmentation channels are quantified such that they show strong axial recoil, as described in section 4 . In contrast, the majority of ions detected in 3h3f fragmentation channels do not show a strong axial recoil. Therefore, if not all ions are detected in a 3h3f fragmentation channel, these channels are distinguished from 2h2f or 2h3f by their axial recoil. Furthermore, due to the stronger Coulomb repulsion between three ionic fragments, the kinetic energy of the 3h3f fragments gives a hint toward these fragmentation channels. Taking this assumptions into account and assuming an ion detection efficiency $\mathord{\sim}40$ %, the branching ratios between the main regions of the PEPIPICO spectrum can be estimated to 27 %, 51 %, and 22 % for 2h2f, 2h3f and 3h2f/3h3f, respectively. The detection efficiency of the electrons is neglected, leading to an overestimation of the contribution of 3h2f and 3h3f fragmentation channels. Independent of the electron detection efficiency, the majority of indole molecules is thus fragmenting into three heavy fragments. If proton and hydrogen transfer processes are neglected, PEPIPICO region 3 and 3* are the only PEPIPICO regions for which the ionic fragments can be uniquely assigned, i. e., $\mathrm{CNH_{2}}+\mathrm{C_{7}H_{5-i}}$ corresponding to the atoms (1, 2) and (3, 3a, 4, 5, 6, 7, 7a); see the notation in the inset of  Figure 2 . In contrast, PEPIPICO region 1 and 2 consist of a superposition of two fragmentation channels, which can additionally consist of non-unique fragmentation combinations of the indole molecule. Consider, for example, the fragmentation $\mathrm{C_{3}NH_{3-i}}+\mathrm{C_{5}H_{4-j}}$ of PEPIPICO region 1. The possible atomic combinations for $\mathrm{C_{3}NH_{3-i}}$ are (1,2,3,3a), (1,2,3,7a), (1,2,7,7a), or (1,6,7,7a). In the case of 2h3f and 3h3f fragmentation channels (regions 4–6) the possible combination of ionic fragments is further increased, resulting in an even lower probability to uniquely assigning the fragments. Exceptions are some single coincidence lines within a coincidence region, such as $\mathrm{C_{4}H_{4}}+\mathrm{C_{4}NH_{3}}$ (PEPIPICO region 1) whose mass sum is equivalent to the mass of the indole molecule, i. e., including the mass of all hydrogens. 4 Fragmentation dynamics The VMIS is used to measure the projected velocity vectors of the ionic fragments. Figure 3 a and b show the VMI images for the first and second detected ion in the coincidence region 4. The corresponding fragments are $\mathrm{C_{3}H_{3}^{+}}$ and $\mathrm{(C_{3}NH_{2}^{+}\leavevmode\nobreak\ \text{or}\leavevmode\nobreak\ C_{% 4}H_{4}^{+})}$ or $\mathrm{C_{2}NH^{+}\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ C_{4}H_% {4}^{+}}$; the color scale is the same as Figure 2 . The velocity of the VMI was calibrated by the helium–photoelectron recoil for different photon energies ranging from 310 to 420 eV. The first detected ions show a slightly higher velocity compared to the second detected ions, which is explained by their smaller mass and the momentum conservation of the fragmenting particles. The increased number of counts visible in the VMI images at $v_{X}=0$ and $v_{Z}\approx-2\cdot 10^{3}$ m/s is due to background from the carrier gas, which is falsely detected at that corresponding TOF window and does not obey momentum conservation †††These events might be due to a subsequent pulse of the synchrotron radiation ionizing a second particle in the molecular beam within the 6 µs acquisition time window (Figure 1 ), which has a small but finite probability. Helium contributes strongest to the signal from the molecular beam and is, therefore, the main background signal.. A histogram of the angular relationship between the first and second detected ions is shown in Figure 3 c. The angle $\alpha_{12}$ is defined as counter-clockwise rotation about $Z$ starting from the 2D velocity vector of the first detected ion. The blue line shows a Gaussian fit centered at an recoil angle of $\alpha_{12}=180\,^{\circ}$ with a standard deviation (SD) of the recoil angle of $\sigma_{\alpha_{12}}=18.4\,^{\circ}$. This strong axial recoil between ions in this channel is only observed for a 2h3f fragmentation process (vide infra). This is in agreement with the expected fast fragmentation of the molecule due to Coulomb explosion subsequent to inner-shell ionization, and the momentum conservation between the ionic fragments. $\sigma_{\alpha_{12}}$ depends on the fragmentation channel, and is $\sigma_{\alpha_{12}}=12.7\,^{\circ}$ for the 2h2f fragmentation channels, and $\sigma_{\alpha_{12}}=9.8\,^{\circ}$ and $\sigma_{\alpha_{12}}=9.5\,^{\circ}$ for the 1* and 3* fragmentation channel, which were assigned to a 3h2f fragmentation channels. These channels show a stronger confinement in the recoil-frame (RF) because they experience a stronger Coulomb repulsion, which leads to an RF that is more dominated by Coulomb repulsion. In contrast, in a 2h3f fragmentation channel the momentum of the Coulomb repulsion is more in competition with the momentum taken up by the heavy neutral fragment, resulting in a less-confined axial recoil. The angular variations $\sigma_{\alpha_{12}}$ in the recoil-frame can be expressed as a degree of (post-)orientation or alignment in the RF, which is $\left<\cos\!\alpha_{\text{12,2D}}\right>\approx 0.98$, 0.99, and 0.95, or $\left<\cos^{2}\!\alpha_{\text{12,2D}}\right>=0.95$, 0.97, and 0.91, for the 2h2f, 3h2f, and 2h3f fragmentation channels, respectively. The angular confinement, i. e., the alignment, is comparable to the best laser alignment experiments 57 whereas the directionality, i. e., the orientation, is significantly better 57, 58. Thus, in the case of the planar indole molecule, these RF determinations allow for RF-ARPES of the individual ion fragmentation channels, albeit that the actual angular-resolution quality of the ARPES depend on the specific fragmentation channel. The deviation in $\sigma_{\alpha_{12}}$ between the 2h2f and 2h3f can be used to estimate the velocity of the neutral fragment. An explicit assignment of the neutral fragments of PEPIPICO region 4 and 5 is not possible since the neutral fragments cannot be detected. From the tight momentum conservation we infer, however, that the bonds between the neutral and the ionic fragments are broken instantaneously on the timescale of the fragmentation process. In addition, we assume that the missing masses are intact fragments due to the following reasons: First, the ionic fragments dominantly stay intact in the case of a 3h3f fragmentation. Second, there is no dominant PEPIPICO region where only a single carbon is missing. Then, in the case of coincidence region 4 a mean velocity of 500 m/s can be assigned to a neutral fragment with a mean mass of 27 u. Figure 4 a shows the angular correlation between the ions of a 3h3f fragmentation channel; the second and third detected ions have the same masses as the ions shown in Figure 3 , i. e., they correspond to the fragments $\mathrm{C_{3}H_{3}^{+}}$ and $\mathrm{(C_{3}NH_{2}^{+}\leavevmode\nobreak\ or\leavevmode\nobreak\ C_{4}H_{4}% ^{+})}$, or $\mathrm{C_{2}NH^{+}\leavevmode\nobreak\ and\leavevmode\nobreak\ C_{4}H_{4}^{+}}$. The first detected ions were previously neutral and are assigned to the ionic fragments $\mathrm{C_{2}H_{2}^{+}\leavevmode\nobreak\ or\leavevmode\nobreak\ CNH^{+}}$. The two dimensional histogram shows the angles $\alpha_{23}$ and $\alpha_{21}$ between the 2D velocity vector of the second-third and second-first ion pairs. The definition of the angles with respect to the fragments is visualized by the inset in the top right corner of Figure 4 . The angular relationship between these pairs of fragments shows an hourglass-like structure, rotated by approximately $45\,^{\circ}$. Coincidences outside that structure are due to ions, which do not fulfill momentum conservation. This is illustrated by right part of the same histogram, where only ion combinations are shown that do fulfill momentum conservation to a high degree ($<60\leavevmode\nobreak\ \text{u}\cdot 117\leavevmode\nobreak\ \text{km/s}$). Figure 4 b shows the histogram of the angles $\alpha_{21}$ and $\alpha_{23}$ for ion pairs that obey momentum conservation, and allows therefore for a better comparison of the recoil angle between the 2h3f and 3h3f. These channels have an SD of $\sigma_{\alpha_{21}}=70.3\,^{\circ}$, and $\sigma_{\alpha_{23}}=50.7\,^{\circ}$, which is a significantly worse axial recoil compared to the one given in Figure 3 for a 2h3f fragmentation channel, and allows therefore to discriminate between both fragmentation channels. This fixed angular relationship between three heavy ionic fragments demonstrates the possibility to reconstruct the three dimensional orientation of the molecule in the laboratory frame provided that the directionality of the moving fragments in the molecular frame are known. Due to the strict planarity of the indole molecule and the immediate Coulomb explosion, the plane of the molecule can be assigned to the recoil plane defined by the three ionic fragments. However, the orientation within the symmetry plane is practically undefined. 5 Angle-resolved photoelectron spectra Figure 5 a and b show the electron velocity map in a cartesian and a polar coordinate system, respectively. The photoelectron VMI has been calibrated by photoelectrons originating from single-photon ionization of atomic helium and neon, at photon energies between 310 and 980 eV. The labels Q1–Q4 correspond to the four different quadrants of the VMI image; $v_{X}$ and $v_{Z}$ correspond to the electrons velocity component in the laboratory frame, and $v_{r}$ and $\theta$ are the radial and angular coordinate in the polar coordinate system. The electrons were detected in coincidence with PEPIPICO regions 1–5, 1* and 3*, with a background correction applied by accepting only events within $2\sigma$ of the recoil angle of the ions (Figure 3 ). The 3h3f fragmentation channels of indole have been considered if three ions were detected, if the second and third detected ion were falling into the coincidence regions 4, and 5, and if the ions fulfilled momentum conservation (Figure 4 b). Region 6 was not used due to a high number of background ions detected in this coincidence region. The electron VMI images of indole show four distinct electron velocities at $2.4$, $7.1$, $9.5$, and $11.2\cdot 10^{6}$ m/s, which correspond to electron energies of 16, 143, 258, and 358 eV. The additional slow electrons visible in the center of the VMI image are assigned to background and shake-off electrons from the molecule. The electron energy spectrum, shown in the bottom graph of Figure 5 , was obtained by an inverse Abel transformation based on the BASEX algorithm 59 of the second and third quadrant of the electron-VMI image. Quadrants one and four were not used, to avoid the influence of the VMI distortions in these quadrants, which are visible for velocities grater than $\mathord{\sim}8\cdot 10^{6}$ m/s, and attributed partially to the non-working layer of the hexanode DLD, possible influence of an magnetic field, or a non well-centered interaction region in the VMI. Considering atomic electron binding energies, the nitrogen and carbon $1s$ photoelectron energies would be expected at 10.1 and 135.8 eV  60, respectively. In pyrrole (C${}_{4}$H${}_{5}$N), which corresponds to the five-membered-ring part of indole, the binding energies are chemically shifted and would correspond to photoelectron energies of 14 and 130 eV for nitrogen and carbon $1s$, respectively  61. This is a deviation of less than 5 % between the $1s$ binding energies in pyrrole and indole, which is within the systematic error of our measurement. The observed C-Auger-electron energies agree with the experimentally observed lines in benzene at 243–267 eV 62. The nitrogen Auger lines agree with calculated energies of 356–377 eV 63. Fitted Gaussians, shown by the red line in Figure 5 c, allow to extract relative intensities of the specific peaks and, thus, ratios of the electron channels. By comparing inner-shell ionization events, the N($1s$) and C($1s$) Gaussian fits show a 26.1 % probability for localized ionization at the nitrogen atom. A similar probability of 24.8 % is obtained by comparing the Auger electron ratio. Both numbers are slightly higher than the expected probability of 16 % by considering the atomic cross sections of C and N. We attribute this difference to the specific properties of the selected Coulomb explosion channels. The SD of the N($1s$) and C($1s$) photolines are $\sigma=4$ and $\sigma=9$ eV, respectively, which is attributed to the distortions of the VMIS and the low number of electrons of the VMI image. The chemical-shift variations of the different carbon atoms ($\mathord{\sim}2$ eV) and the bandwidth of the synchrotron radiation (0.4 eV) are negligible. The anisotropy parameters for the photo- as well as Auger electrons, obtained from the inverse Abel transformation averaged over one FWHM of the photoelectron line, are $\beta_{\text{N}(1s)}=1.1\leavevmode\nobreak\ (0.1)$, $\beta_{\text{C}(1s)}=1.7\leavevmode\nobreak\ (0.1)$, $\beta_{\text{C-Auger}}=0.2\leavevmode\nobreak\ (0.1)$, and $\beta_{\text{N-Auger}}=0.2\leavevmode\nobreak\ (0.1)$. The anisotropy parameter of the Auger electrons is consistent with the expected isotropic distribution of electrons in the laboratory frame. The anisotropy parameter for C($1s$) photoelectrons is slightly lower and the anisotropy parameter for N($1s$) photoelectrons is significantly lower than the one, $\beta=2.0$, expected for ionization out of an $s$-orbital by circularly polarized radiation. We attribute this lowered asymmetry parameters partly to the non-perfect reconstruction, but also to the interaction of photoelectrons with the potential of the molecule 64. The N($1s$) photoelectron data is further influenced by slow background and shake-off electrons in that region of the velocity map, which lower the derived asymmetry parameter. In principle, this also implies a corresponding influence in the derived N($1s$) photoelectron energy and ionization probability. However, as discussed above, the photoelectron energy is within the systematic error of our measurement and the probability of nitrogen inner-shell ionization is comparable to the value determined from Auger electrons. 6 Electron-ion fragmentation correlation The measured coincidences between electrons and ions allow to extract the individual 2D electron VMI spectra of the various ionic fragmentation channels. The 2h2f and 2h3f ion fragmentation channels show a spectrum similar to the one shown in Figure 5 c. The energy spectrum of the 3h2f and 3h3f fragmentation channels yielded no clear results due to low statistics. Therefore, for the 2h2f, 2h3f, 3h2f and 3h3f channels, radial velocities of the electrons 2D VMI images, i. e., projected electron-velocity distributions (EVD), for the different ionic channels are compared in the following. This time all quadrants of the electron VMI are taken into account (Figure 5 ). The distortions of the VMI in quadrant one and four mainly influenced the determined energy for the Auger electrons, which do not have a significant influence on the following discussion. Figure 6 a shows histograms of the EVD sorted into the contributions of the ion-fragmentation channels 2h2f (black), 2h3f (red), 3h2f (blue), and 3h3f (green). The histograms are normalized to the total number of counts; the multiplication factors given by the inset, and the error bars are given as the statistical error. The connecting lines serve to guide the eye. These electron-velocity distributions clearly group into the two-hole and three-hole channels: The radial EVD for the 2h2f and 2h3f fragmentation channels (black and red) are very similar. Both show local maxima of electron counts at velocities assigned to the nitrogen and carbon $1s$ photo- and Auger electrons. The electrons detected between the maxima are due to the projection of the three-dimensional electron velocity distribution onto the two-dimensional detector surface. The 2h3f fragmentation channel has the larger contribution of N($1s$) photoelectrons, whereas the 2h2f fragmentation channel has larger contributions from C($1s$) photoelectrons and their corresponding Auger electrons. This indicates a higher probability for a three-fragment break up if indole is ionized at the nitrogen atom, which can be rationalized by the energy differences between the two possibilities of ionization: Ionization at the N($1s$) leads to an N-Auger-electron, which results in a mean energy of 46 eV left in the molecule, whereas ionization at C($1s$) leads to a mean energy of 19 eV. Thus, it seems the larger energy left in the molecule following N($1s$) ionization than for C($1s$) ionization leads to a stronger fragmentation. The radial EVD for the three-hole fragmentation channels 3h2f and 3h3f, the blue and green lines in Figure 6 a, are also similar. In contrast to the 2h2f and 2h3f radial EVD, the strongest peak of the spectrum is at electron velocities close to the N($1s$) photoline, and drops-off continuously toward higher electron velocities, with edges at electron velocities corresponding to the carbon $1s$ photo- and Auger electrons. This overall shift in the electron spectrum toward lower photoelectron energies is attributed partially to a tertiary ionization of indole via electron-impact ionization, and also due to satellite peaks of the photo- and Auger electrons. This is discussed in the second half of the following paragraph based on the angular anisotropy of the electrons. To extract an angular anisotropy of the electrons radial distribution, the electron VMI is divided into the four quadrants Q1*–Q4* as shown in the inset of Figure 6 b; the coordinate system is the same as shown in Figure 5 a, but Q1*–Q4* are rotated by $45\,^{\circ}$ with respect to Q1–Q4. With $\beta$-parameters of 1.1 and 1.7 for the nitrogen and carbon $1s$ photoelectrons a larger signal is observed in Q2* and Q4* than in Q1* and Q3*. For Auger electrons, which typically show no anisotropy, the same averaged number of counts is expected for all quadrants. The histograms in Figure 6 b show the radial EVD of the anisotropy $((Q2^{*}+Q4^{*})-(Q1^{*}+Q3^{*}))$ for electrons detected with two and three ionic fragments in coincidence, i. e., the fragmentation channels 2h2f and 2h3f are jointly labeled 2h (black), and the fragmentation channels 3h2f and 3h3f are jointly labeled 3h (blue). The error bars depict the statistical error, the connecting lines serve to guide the eye, and the histograms are normalized to the number of counts. For the 2h fragmentation channels two distinct maxima are visible at electron velocities corresponding to the nitrogen and carbon photoelectrons. The anisotropies of the Auger electrons at $v_{\text{r}}\gtrsim 7\cdot 10^{6}$ m/s are effectively averaged to zero. The negative values at radial velocities smaller than $1\cdot 10^{6}$ m/s are attributed to non isotropic noise close to the center of the electron VMI. Comparing the number of electrons assigned to the ionization from nitrogen/carbon shows a probability of approximately 20 % for a localized ionization at the nitrogen atom if the negative values are neglected. This is comparable to the ratio determined from the overall photoelectron intensities in section 5 and, again, slightly higher than expected from the atomic cross sections. The blue histogram, on the other hand, shows electrons in coincidence with the 3h fragmentation channels. Here, no clear carbon $1s$ photoelectron line is visible. Instead, an increased number of electrons is detected at velocities in-between the carbon and nitrogen $1s$ photoelectron energies. Those electron energies can not be attributed to the earlier determined photo- or Auger electron energies. N($1s$) photoelectrons do not have enough energy to tertiary ionize indole by electron impact ionization. Also, the contribution from Auger electrons to triply ionize indole can be excluded in this analysis since they do not show an anisotropy in the laboratory frame. Therefore, we attribute those electrons to either inelastically scattered C($1s$) photoelectrons and electrons generated by this inelastic scattering through electron impact ionization, or to satellite peaks from the C($1s$) photoelectrons. A closer insight is given by the red line in Figure 6 b, which shows a scaled difference between the blue and black spectrum. The scaling was done by a normalization of the number of electrons at $v_{\text{r}}=6.8\leavevmode\nobreak\ \cdot 10^{6}$ m/s to subtract the highest possible contribution from direct photoelectrons. This difference-spectrum shows three main areas: the contribution of the nitrogen $1s$ photoelectrons and two highlighted red areas, which are assigned to those inelastic scattered carbon $1s$ photoelectrons, electrons emitted upon impact ionization, and satellite peaks from the carbon $1s$ photoline. These electrons in the red areas have a velocity of $v_{\text{r}}=2.9$–$4.5\cdot 10^{6}$ m/s (24-58 eV) and $v_{\text{r}}=4.7$–$5.7\cdot 10^{6}$ m/s (63–92 eV). The number of electrons that correspond to these two peaks is about the same, and the sum of the mean electron energy of both peaks is 104 eV. In Figure 6 a, the C($1s$) Auger- and photoelectrons show a similar behavior, i. e., the 2h fragmentation channels show a prominent peak, which is absent in the 3h fragmentation channels. Therefore, we attribute this change in the radial EVD of Auger electrons also to electron impact ionization or satellite peaks accompanying the Auger electrons. A quantitative statement about the contribution of the inelastically scattered electrons, electrons from impact ionization, and satellite electrons (red) to the 3h2f and 3h3f fragmentation channels could, in principle, be extracted from their anisotropy parameter. This was not possible due to the low number of detected electrons. Only for C($1s$) photoelectrons a lower limit of 43 % can be estimated from Figure 6 b by counting the number of inelastically scattered/satellite electrons (red), which are part of the 3h2f and 3h3f channels (blue). At the given C($1s$) photoelectron energy, the atomic cross section for carbon for electron impact ionization and elastic scattering of electrons are both in the order of $200\cdot 10^{-22}\text{\leavevmode\nobreak\ m}^{2}$ 65, 66. This implies that elastically-scattered electrons can be detected at comparable signal strengths, e. g., in photoelectron holography experiments 67. The inelastically-scattered electrons detected here could be separated by an energy-resolving detection scheme, as demonstrated here. 7 Conclusion We have performed a detailed photoionization and photofragmentation study of indole upon single-photon inner-shell ionization at a photon energy of 420 eV. This photon energy was chosen such that indole could be locally ionized at its nitrogen atom. Ionization from C($1s$) was also possible and is the dominant ionization process due to the larger number of carbon atoms present in the molecule. Electrons and ions have been measured in coincidence in a velocity-map-imaging mode to extract 2D and 3D velocity vectors of the charged particles. In the ion-coincidence spectrum of indole, i. e., for the events with more than one ionic fragment observed, indole is fragmenting into two heavy ionic and one neutral fragment in 51 % of the cases. These “heavy” fragments contain, almost exclusively, two or more heavier atoms; the loss of hydrogen atoms and protons was also observed, but they were not considered as specific fragments. Fragmentation channels with only two fragments or with three heavy ionic fragments have also been observed and showed contributions of 27 % and 22 %, respectively. The PEPIPICO spectrum revealed that the unique assignment of a coincidence region to a carbon atom from a specific position in the molecule is rather the exception than the rule. The ion-VMI images could be used to reconstruct the recoil-frame of the molecules. The fragmentation process was dominated by the Coulomb repulsion of the generated charges. Influence of chemical effects, e. g., the specific potential-energy surfaces, was observed in the recoil frame of the ions for the case of a coexisting heavy neutral fragment. Ion-VMI images of this selected 2h3f fragmentation channel were discussed regarding the velocity of the dissociating neutral fragment, showing that the bonds between the neutral and ionic fragments must be broken instantaneously on the timescale of the fragmentation process, i. e., no meta-stable ionic fragments were observed. Fragmentation channels with three ionic fragments also showed a fixed angular relationship. This allowed to directly determine the alignment of the molecular plane in the laboratory frame. In order to reconstruct the three dimensional alignment and orientation of the indole molecule, including its orientation inside the molecular plane, the direction of the moving fragments in the molecular frame have to be known. This is an elaborate task and beyond the scope of this paper. The electron-energy spectrum showed four peaks, which were assigned to photo- and Auger electrons resulting from element-specific ionization at indole’s nitrogen as well as carbon atoms. The corresponding asymmetry parameters of these peaks were extracted from an inverse Abel transformation. For the Auger electrons they were isotropic in the laboratory frame, as expected. For the photoelectrons, deviation from the expected asymmetry parameter for photoelectrons from the carbon and nitrogen $1s$ orbitals have been observed; where “expected” refers to the asymmetry parameter for a single-photon $1s$ ionization with circularly polarized light. The observed deviation is partly attributed to the interaction of the photoelectrons with the molecular potential, partly due to a non-perfect reconstruction of the asymmetry parameters, as well as deviations due to background signal from slow background and shake-off electrons. The correlation between ions and electrons showed that different ion fragmentation channels have different electron spectra, i. e., a relationship between the ionization/excitation process, the corresponding electronic states, and the fragmentation process, reflecting the specific potential energy surface. This was shown, for instance, by a comparison of the projected electron energy spectra for the 2h2f and 2h3f fragmentation channels. In this case it was concluded that inner-shell ionization at the nitrogen edge leads to a higher probability for indole to break up into three heavy fragments. Evidence for secondary electron-impact ionization as well as satellite photoelectrons was observed in the fragmentation channels where three ionic fragments have been measured. Those channels showed less pronounced photolines, primarily observed for the C(1$s$) photoelectrons, as well as signals at electron energies where no photoline is expected. In addition, evidence for satellite peaks of the Auger electrons and inelastically scattered Auger electrons was presented. Since the cross sections for the observed inelastic scattering and elastic scattering are comparable under the experimental conditions, the possibility of photoelectron-holography experiments is confirmed. The presented data allowed to record RF-ARPES images of strongly post-oriented indole, albeit that the relation of RF and MF is unknown beyond the common symmetry plane. Due to the low number of events per unique fragmentation channels, i. e., fragmentation channels where specific carbon atoms could be assigned uniquely to the ionic fragment, no statistically significant asymmetries of the electron distribution in the recoil-frame were observed. Overall, our results show that the fragmentation channels depend on the different electronic states, i. e., the chemical potential energy surface, whereas the observed velocities of the fragments are not strongly dependent of these chemical details. Our work provides the basis for fragmentation studies of larger molecules as well as molecular clusters, for instance, such as the indole-derivative tryptophan or indole-water clusters. Comparison of the fragmentation channels and dissociation energies will allow to study the role of solvents on the photophysics of indole upon site specific x-ray ionization. Furthermore, the processes observed here provide information on the indole-chromophore-related radiation damage occurring in coherent diffractive imaging of proteins 68, 69. 8 Acknowledgments We acknowledge Evgeny Savelyev for support with the experiment. Besides DESY, this work has been supported by the excellence cluster “The Hamburg Center for Ultrafast Imaging – Structure, Dynamics and Control of Matter at the Atomic Scale” of the Deutsche Forschungsgemeinschaft (CUI, DFG-EXC1074); by the Helmholtz Association through the Virtual Institute 419 “Dynamic Pathways in Multidimensional Landscapes”, the Helmholtz Young Investigators Program (D.R. and S.B.), and the “Initiative and Networking Fund”; by the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement 641789 MEDEA, and by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) through the Consolidator Grant COMOTION (ERC-Küpper-614507). D.R. also acknowledges support from the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division (DE-FG02-86ER13491). 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