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--- abstract: 'I describe recent high-resolution X-ray spectroscopy of surface emission from nearby, thermally emitting neutron stars. I focus on RX J0720.4$-$3125, RX J1308.6+2127, and RX J1605.3+3249, all of which have similar temperature, but differ in the presence and strength of absorption features in their spectra. I discuss possible causes for the absorption we see in two sources, and conclude that it may be proton cyclotron line absorption, but weakened due to the strong-field quantum electrodynamics effect of vacuum resonance mode conversion.' author: - 'M. H. van Kerkwijk' title: 'X-ray Spectroscopy of Thermally Emitting Neutron Stars' --- Introduction ============ [*ROSAT*]{} discovered a number of nearby neutron stars whose emission appears to be entirely thermal, uncontaminated by accretion or magnetospheric processes. At present, six (possibly seven) sources are known (Treves et al. 2000; Haberl 2003). For four sources, optical counterparts have been identified. The high X-ray to optical flux ratios leave no model but an isolated neutron star. As a class, the sources are interesting because of the implied existence of a fair number of neutron stars different from the usual radio pulsars and X-ray binaries. At present, their nature remains unclear. The original idea, of old neutron stars accreting slowly from the interstellar medium, has become unlikely because of the very low accretion rates implied by high proper motions. Instead, they might be radio pulsars beamed away from us, although in this case the long periods are surprising. Perhaps they have very strong magnetic fields, and are descendants of anomalous X-ray pulsars and soft gamma-ray repeaters. As individuals, the sources are of particular interest because of the opportunity to study uncontaminated emission from a neutron-star atmosphere. The hope is that this will allow one to infer precise values of the temperature, surface gravity, gravitational redshift, and magnetic field strength. In turn, these could be used to constrain the interior, and hence learn about the equation of state of cold, ultradense matter, an unexplored region in QCD parameter space. X-ray Spectra ============= Given the interest, long spectroscopic observations were taken with the [*Chandra X-ray Observatory*]{} and [*XMM-Newton*]{}. The first results were discouraging: no lines were found in [*XMM*]{} spectra of RX J0720.4$-$3125 (Paerels et al. 2001), nor in [*Chandra*]{} spectra of the prototype of the class, RX J1856.5$-$3754 (Burwitz et al.2001), not even after 500 ks (Drake et al. 2002; Braje & Romani 2002). Instead, for both sources the spectra were found to be remarkably well described by mildly extincted black bodies. This was unexpected, since for light-element atmospheres one would expect a hard tail (because the opacities decrease towards higher energies), while for anything else one would expect to see lines (for reviews, see Pavlov, Zavlin, & Sanwal 2002; Zavlin & Pavlov 2002).    The situation changed this year. For RX J1308.6+2127, the fourth-brightest object in the class, Haberl et al. (2003) discovered a broad absorption feature in [*XMM*]{} data. It extends from $\sim\!0.5\,$keV to lower energies, and can be described by a Gaussian centred at $\la\!0.3\,$keV. Furthermore, the third-brightest object, RX J1605.3+3249, was found to show a weaker, less wide absorption feature centred at a somewhat higher energy of $0.45\,$keV (Van Kerkwijk et al. 2003), as well as a narrower, marginally significant absorption line at 0.55keV. In Fig. \[fig:rgs\], the RGS spectra obtained with [*XMM*]{} for both sources are shown. In addition, the spectrum of RX J0720.4$-$3125 is shown. This source has a very similar temperature, $kT\simeq90\,$eV, but a featureless spectrum. Proton Cyclotron Absorption and Vacuum Resonance ================================================ For RX J1308.3+2127, Haberl et al. (2003) suggested that the feature was due to proton cyclotron absorption. Given the energy of $\la\!0.3\,$keV, the implied magnetic field strength is $\la\!5(1+z)\times10^{13}\,$G (where $1+z=(1-2GM/Rc^2)^{-1/2}\simeq1.3$ is the gravitational redshift factor). This is not unreasonable, given the observed slow spin period, $P=10.3\,$s. From the temperature, the neutron star should be about half a million years old. Equating this to the characteristic age, one infers a current spin-down rate $\dot{P}\sim P/2t\sim3\times10^{-13}$, and a magnetic field of $3.2\times10^{19}(P\dot{P})^{1/2}\sim6\times10^{13}\,$G, consistent with what is required. The large width of the feature, $\sigma_E/E\ga1/3$, is as expected for proton cyclotron absorption, since the cyclotron energy scales linearly with the magnetic field strength, which will vary over the surface (by a factor two for a centred dipole). Furthermore, as mentioned by Haberl et al. (2003), the equivalent width is consistent with model calculations of Zane et al. (2001). If we assume the absorption in RX J1605.3+3249 is due to proton cyclotron absorption as well, we infer $B\simeq7(1+z)\times10^{13}\,$G. Unfortunately, no pulsations have been found – to a limit of 3% in the frequency range 0.001–800Hz (Van Kerkwijk et al. 2003) – so we cannot verify this. Puzzling in this case, however, is the difference in strength and width of the features in the two sources. As mentioned, the temperatures are very similar, so that cannot be the explanation. The fields strengths inferred above are in excess of the critical quantum electrodynamics field $B_{\rm QED}=4.4\times10^{13}\,$G, at which the electron cyclotron energy equals the electron rest mass. One effect that may become important is that photons propagating down the density gradient in the atmosphere can change polarisation mode at “vacuum resonance,” where the plasma contribution to the dielectric properties is compensated by the QED effect of vacuum polarisation. Vacuum resonance has recently been studied in detail (e.g., Lai & Ho 2003; see also Lai, these proceedings), and it was found to lead to a reduction in the contrast of spectral features when it occurs between the deeper photosphere for the extraordinary mode photons and the shallower one for the ordinary mode photons. For the relevant energies of $\sim\!0.6\,$keV at the surface, this will be the case for magnetic fields in the range 0.7–$50\times10^{14}$G. If this is correct, it will not be very important for RX J1308.6+2127, and it makes sense that that source’s feature could be reproduced by Zane et al.’s 2001 models, which do not take vacuum polarisation into account. It should affect RX J1605.3+3249, however. And indeed, it may explain why the observed feature is so narrow: we might be seeing only absorption from regions with relatively low field, $B\la9\times10^{13}\,$G, the contrast of the absorption in regions with higher field being reduced due to the vacuum resonance. Future Prospects ================ The discovery of absorption features should help further theoretical work on the vacuum resonance (in progress; Ho & Lai 2003b). In particular, in both sources, the absorption extends up to $\sim\!0.5\,$keV. Might it be that the maximum energy out to which absorption is seen is set by vacuum resonance? If so, the observations provide a direct measurement of the critical magnetic field strength: about $7(1+z)\times10^{13}\,$G. Theoretical work is also needed on neutral hydrogen, which is strongly bound in high magnetic fields (for a review, Lai 2001). Hence, it should be present, and could lead to observable features. So far, it has only been studied at lower fields (e.g., Pavlov & Meszaros 1993; Zavlin & Pavlin 2002). It might possibly also help understand the puzzling overall spectral energy distributions. Observationally, the obvious steps are a survey of the remaining sources, and using [*Chandra*]{}/LETGS to obtain spectra over a larger wavelength range and with phase resolution. Timing and astrometric studies could help constrain magnetic field strengths, distances, space velocities, places of origin, and ages. All in all, the future looks bright, even if we may learn more about QED than about QCD. I thank Dong Lai and George Pavlov for the good discussions during the conference, during which the ideas described here arose. , V., [Zavlin]{}, V. E., [Neuh[" a]{}user]{}, R., [Predehl]{}, P., [Tr[ü]{}mper]{}, J., & [Brinkman]{}, A. C. 2001, , 379, L35 , T. M. & [Romani]{}, R. W. 2002, , 580, 1043 , J. J. [et al.]{} 2002, , 572, 996 , F. 2003, in High Energy Studies of Supernova Remnants and Neutron Stars, ed. W. [Becker]{} & W. [Hermsen]{}, in press \[astro-ph/0302540\] , F., [Schwope]{}, A. D., [Hambaryan]{}, V., [Hasinger]{}, G., & [Motch]{}, C. 2003, , 403, L19 , W. C. G. & [Lai]{}, D. 2003, , submitted , D. 2001, Reviews of Modern Physics, 73, 629 , D. & [Ho]{}, W. C. G. 2003, , 588, 962 , F. [et al.]{} 2001, , 365, L298 , G. G. & [Meszaros]{}, P. 1993, , 416, 752 , G. G., [Zavlin]{}, V. E., & [Sanwal]{}, D. 2002, in Neutron Stars, Pulsars, and Supernova Remnants (Garching: MPE), 273 \[astro-ph/0206024\] , A., [Turolla]{}, R., [Zane]{}, S., & [Colpi]{}, M. 2000, , 112, 297 , M. H., [Kaplan]{}, D. L., [Durant]{}, M., [Kulkarni]{}, S. R., [Paerels]{}, F. 2003, , submitted , S., [Turolla]{}, R., [Stella]{}, L., & [Treves]{}, A. 2001, , 560, 384 , V. E. & [Pavlov]{}, G. G. 2002, in Neutron Stars, Pulsars, and Supernova Remnants (Garching: MPE), 263 \[astro-ph/0206025\]

--- author: - Kai Lawonn and Bernhard Preim bibliography: - 'bib.bib' title: 'Feature Lines for Illustrating Medical Surface Models: Mathematical Background and Survey' ---

--- abstract: 'A simple and predictive model is put forward explaining the experimentally observed substantial shift of the glass transition temperature, $T_g$, of sufficiently thin polymer films. It focuses on the limit of small molecular weight, where geometrical ‘finite size’ effects on the chain conformation can be ruled out. The model is based on the idea that the polymer freezes due to memory effects in the viscoelastic eigenmodes of the film, which are affected by the proximity of the boundaries. The elastic modulus of the polymer at the glass transition turns out to be the only fitting parameter. Quantitative agreement is obtained with our experimental results on short chain polystyrene ($M_W$ = 2 kg/mol), as well as with earlier results obtained with larger molecules. Furthermore, the model naturally accounts for the weak dependence of the shift of $T_g$ upon the molecular weight. It furthermore explains why supported films must be thinner than free standing ones to yield the same shift, and why the latter depends upon the chemical properties of the substrate. Generalizations for arbitrary experimental geometries are straightforward.' address: 'Dept. of Applied Physics, University of Ulm, D-89069 Ulm, Germany' author: - 'Stephan Herminghaus, Karin Jacobs, Ralf Seemann' title: 'The Glass Transition of Thin Polymer Films: Some Questions, and a Possible Answer' --- introduction {#introduction .unnumbered} ============ To explain the experimental observation that thin polymer films melt at temperatures well below the glass transition temperature of the bulk polymer, $T_g^{0}$ [@KJ1; @MFB; @DVF; @DVDG], is one of the major current challenges in the theory of homopolymers. At first glance, one might expect such a behavior, due to the impact of the finite size geometry of a thin film upon objects as large as polymer molecules. In fact, for molecular weights larger than about $M_W$ = 300 kg/mol, geometry effects have been recently shown to play a dominant role, resulting in a linear dependence of $T_g$ upon the film thickness, $h$ [@DVDG; @DeGennes; @DeGennes2]. For smaller molecules, however, the problem is conceptually more intricate, since a noticeable reduction of $T_g$ can be observed at film thicknesses orders of magnitude larger than the radius of gyration of the molecules. Furthermore, the shift $T_g^{0}-T_g(h)$ becomes strongly nonlinear and largely independent of molecular weight [@DVF; @Forrest], indicating that this regime must be governed by a different mechanism. A fundamental understanding of this effect would be of great interest not only for polymer physics, but potentially also elucidate the physics of the glass transition in a larger class of systems, since it comes into play at small molecular weight. Many attempts have been undertaken to explain these observations, mostly by considering microscopic models of the inner structure of the films. D. Long and F. Lequeux have envisaged the freezing of the film as a percolation of rigid domains [@Moukarzel], mediated by thermal fluctuations [@LongLequeux]. Other models assume a layered structure of the film, with a particularly mobile region close to the free surface of the film [@KJ1]. Within this framework, J. Forrest and J. Mattsson [@Forrest] have recently been able to achieve quite impressive accordance with the experimental data [@MFB; @Forrest]. Their model makes use of the so-called cooperativity length, $\xi(T)$, which plays a mayor role in a whole class of theoretical concepts of the glass transition. The only drawback is that there is yet no well established theory of $\xi(T)$. Furthermore, as a consequence of the two-layer structure of the film inherent in the model, it is not completely clear why there should not be two glass transitions, rather than a single one shifted in temperature. As a complementary approach, computer simulations of polymer films with free surfaces have recently been carried out, and most of the experimental findings were reproduced qualitatively [@Torres]. However, the polymer chains in these simulations were only 16 monomer units long, very much shorter than those used in the experiments. Before attempting to explain the effect theoretically, it is therefore worthwile to study experimentally the behavior of polymers with short chain length, both to ease comparison with simulation and to explore the range of validity of the apparent independence of the shift of $T_g$ on molecular weight. Before going into the details of our study, let us take a break and summarize the main questions to be answered. 1. [What is the principal mechanism responsible for the reduction of $T_g$ in thin films of low molecular weight ($M_W<$300kg/mol) polymers?]{} 2. [Down to how small molecular weight is this mechanism valid?]{} 3. [Why is the effect stronger in free standing films than in supported ones [@DVF]?]{} 4. [Why is there no significant dependence of this effect on molecular weight [@DVF]?]{} 5. [Why does the effect depend upon the chemical composition of the substrate for supported films [@KJincrease; @vanZanten; @KJCincrease]?]{} 6. [Why is there sometimes an increase of $T_g$ in thin films, instead of a reduction [@KJincrease; @vanZanten; @KJCincrease]?]{} We will try in the present paper to give answers to these questions, or at least show in which direction answers might be found, on the basis of a novel, quite simple model which rests mainly on the viscoelastic eigenmodes of the films. We restrict the discussion to the case of polystyrene (PS), since this is the most thoroughly studied polymer in this context. Furthermore, it is particularly well suited for comparison with theoretical models, since (atactic) polystyrene does not show any propensity to crystallization. In other polymers, which might crystallize at least in part of the film, modelling would be exceedingly difficult, and the main mechanism could well be obscured in experiments. Experiment {#experiment .unnumbered} ========== Let us first explore the range of validity of the aforementioned effect of reduction of $T_g$, as to the molecular weight of the polymer. We have investigated the glass transition in thin supported films of PS with a molecular weight as small as 2 kg/mol ($\approx 20$ monomer units). It was purchased from Polymer Labs (UK) with a polydispersity index $M_W/M_N$ = 1.05, the radius of gyration is 1.3 nm. Effects from the molecular geometry are thus expected only for films of few nanometers thickness. The films were spin cast from toluene solution onto silicon wafers (Silchem GmbH, Freiberg/Germany), which were previously cleaned by ultrasonication in acetone, ethanol, and toluene, subsequently. Residual organics were removed with a 1:1 mixture of $H_2SO_4$ with $H_2O_2$, and the substrates were thoroughly rinsed with hot millipore water afterwards. Films were investigated with thicknesses ranging from 4 to 160 nm. The roughness of the free surface of the films was less than 0.2 nm, as revealed by scanning force microscopy (SFM). The glass transition temperature was determined in two different ways, depending on film thickness. The standard procedure of monitoring the thermal expansion of the film via ellipsometry, as introduced by Keddy and Jones [@KJ1], was used for film thicknesses down to 9.6 nm. Fig. \[TypicalRun\]a shows a typical run. The data were reproducible, irrespective of being taken during heating or cooling. Typical heating or cooling rates were 2 $K/min$. We plotted the changes in refractive index and thickness of the film in fig. \[TypicalRun\]b. The solid line represents the Clausius Mosotti relation. Obviously, there is good agreement, suggesting the absence of any loss or degradation of material. This was found invariably for all samples. In fig. \[ThermalExpansion\] we plotted the thermal expansion coefficients found above and below $T_g(h)$, which coresponds to the kink in fig. \[TypicalRun\]a. At large film thickness, $T_g$ obtained in this way approached $327 \pm 1$ K, which is consistent with the temperature at which macroscopic melting is observed in the bulk ($T_g^{0}$) for PS with this chain length. The shaded areas represent published data of the expansion coefficients [@ExpansionCoefficient]. Good agreement is found both above and below $T_g$. Small deviations at very small film thickness, as might be concluded from our data, have been reported before [@KJ1]. For films thinner than 9 nm, we determined the melting behaviour of the films by observing the buildup of amplified thermal fluctuations (spinodal dewetting) [@Karim; @OurPRL]. These processes were monitored by SFM with [*in-situ*]{} heating. In order to speed up the experiments to a feasible time scale, dewetting was observed at temperatures close to $T_g^{0}$. As an example, we show in fig. \[SpinDewett\] the temporal evolution of the Fourier transform (spatial power spectrum) of the surface topography. The hallmark feature is the clear peak representing the fastest growing mode. The inset shows the peak intensity as a function of time on logarithmic scale. Exponential growth is clearly observed up to a rather well defined time at which coalescence of holes sets in. From the slope of the straight line in the inset, and the known effective interface potential [@OurPRL], the viscosity $\eta$ of the polymer can be determined. Measuring the viscosity as a function of temperature, we found that this obeyed a Vogel-Fulcher law, with the Vogel-Fulcher temperature shifted by a certain amount $\Delta T_{VF}$, which depended on the film thickness. We identified $\Delta T_{VF}(h)$ with the shift in glass temperature. In this way, the apparent glass transition temperature of particularly thin films was inferred from the spinodal dewetting experiments by setting $T_g(h)=T_g^0-\Delta T_{VF}(h)$. Our experimental results are shown in fig. \[Tgresult\] as the full symbols. The circles represent the thermal expansion measurements, the squares were obtained from the spinodal dewetting experiments. As one can clearly see, the glass transition temperature is substantially reduced for all films thinner than about 50 nm. The solid line represents the function $$T_g = T_g^0(1+h_0/h)^{-1} \label{KimFormel}$$ This form has been shown before to account well for the data obtained by others for larger molecular weight films, if $h_0 = 0.68$ nm was assumed for PS [@Kim]. Within experimental scattering, our data exhibit indeed the same dependence of $T_g(h)$ in the full range of film thickness explored. It is remarkable that the data from both the thermal expansion and spinodal dewetting measurments are well fitted by the same curve. This tentatively corroborates the procedure of obtaining $T_g$ for very thin films (squares) as discussed above. Although our polymer chains are roughly by a factor of 50 shorter than those investigated before, we obtain $h_0 = 0.82$ from the fit, which is quite close to the above value. This confirms the weak dependence (if there is a significant one at all) of the reduction of the glass transition temperature on the molecular weight of the polymer, down to a molecular weight as small as 2 kg/mol. We can thus state that the reduction of $T_g$ in thin films, as described phenomenologically by eq. (\[KimFormel\]), is observed in a huge range of molecular weight, from molecules as small as 2 kg/mol up to a few hundred kg/mol. This is in accordance with molecular dynamics simulations [@Torres], and rules out mere finite size effects on the individual coils as the main cause of the reduction of $T_g$ in this regime. Note that the radius of gyration of our polymer is only 1.3 nm, while $T_g$ is significantly reduced at a film thickness of 50 nm already. The fitting parameter used in eq. (\[KimFormel\]), $h_0$, changes only by about 20 % in this range, confirming that the dominant mechanism which is responsible for this effect cannot depend strongly on the molecular weight of the polymer. Eigenmode spectrum of the films {#eigenmode-spectrum-of-the-films .unnumbered} =============================== Since it is clear that the behaviour displayed in fig. \[Tgresult\] can in no way be attributed to the geometrical impact of the finite film thickness upon the microscopic conformation of the individual chains, we present here an approach to the problem which intentionally makes as few reference as possible to the molecular structure of the film. The latter is accounted for merely by the strain in the polymer, i.e., the deviation of the [*local average*]{} gyration ellipsoids of the molecules from a sphere. By ‘[*local*]{}’ we mean a volume much larger than the volume of the backbone of a single molecule, but with a lateral dimension much smaller than the film thickness. Such a deviation from the equilibrium conformation (i.e., from a Gaussian coil, if self-avoiding is neglected), which may be viewed as an entropy fluctuation, can decay either by self diffusion of the individual molecules, or by some center-of-mass rearrangement (i.e., flow) of the melt. It is clear that close to a surface, such rearrangements are much easier to accomplish due to coupling of the capillary waves on the free surface to the bulk flow of the polymer. This coupling is effective down to a depth comparable to the wavelength of the capillary modes, which may be [*much larger than the coil size of the molecules*]{}. Thus we are provided with a mechanism which affects the motion of the polymer molecules, and naturally can act over distances which are large as compared to molecular dimensions. In order to explore the possible relevance of this mechanism for the reduction of $T_g$ in thin films, we have to consider the eigenmode spectrum of the (viscoelastic) polymer film. This can be discussed with all possible boundary conditions at the substrate, such that films with strong slip along the substrate, grafted films, or free standing films, may as well be treated within the same framework. The spectrum of a viscoelastic thin film can be obtained in a straightforward manner by combining standard theory of elasticity [@LL1] and hydrodynamics [@LL2] in the limit of small Reynolds number (Stokes dynamics). The equation of motion reads $$\lbrace \partial_t + \omega_0 + \frac{E}{\eta} \rbrace \nabla^2 {\bf \phi} = \frac{\nabla p}{\eta} \label{EoM}$$ where $E$ is Young’s modulus, $\eta$ is the viscosity, and $\omega_0$ is the Rouse rate of relaxation of the individual chains into their equilibrium configuration [@strobl; @Kimmich]. ${\bf \phi}$ is a vector field related to the strain tensor, ${\bf S}$. For the sake of clarity, we restrict our discussion to a simple model, taking into account only the single intrinsic relaxation rate $\omega_0$, as opposed to more general treatments [@SafranKlein]. Deviations of the real polymer from this simple behaviour will be introduced [*a posteriori*]{} farther below. If we restrict the discussion to one lateral ($x$) and one normal ($z$) coordinate, ${\bf \phi} = (\phi_x ,\phi_z)$ is defined via $${\bf S} = \left(\begin{array}{cc} \partial_x \phi_x & \frac{1}{2} (\partial_x \phi_z + \partial_z \phi_x) \\ \frac{1}{2}(\partial_x\phi_z + \partial_z\phi_x) & \partial_z\phi_z \end{array}\right) \label{strain}$$ Finally, $p$ is the pressure field. For harmonic excursions of the free surface, $\zeta(x) = \zeta_0 \exp\{iqx-\omega t\}$, eq.(\[EoM\]) has solutions $$\begin{aligned} \phi_x = [1+(h+q^{-1})\alpha(q)]\cosh qz - q^{-1}\alpha(q)\sinh qz \nonumber \\ \phi_z = [1+h\alpha(q)]\sinh qz - z\alpha(q)\cosh qz \label{solutionfree}\end{aligned}$$ where for the function $\alpha(q)$, we find $$\alpha(q) = \left(\frac{q}{2}\right) \frac{e^{2qh}-1}{e^{2qh}-1+qh} \label{alphafree}$$ for free standing films (symmetric modes) as well as for supported films with full slippage (zero friction). For supported films with some friction at the substrate, the expressions are of similar form, but considerably more complicated, and will not be discussed here. At the free surface, we used the standard boundary condition of zero tangential stress, and $p = -\sigma\partial_{xx}\zeta$, where $\sigma$ is the surface tension of the polymer. Note that for free standing films, $h$ is defined as [*half*]{} the film thickness. For the relaxation rates of the modes, we get $$\begin{aligned} 2\omega = (\omega_0 + \frac{E}{\eta} + \frac{\sigma q^2}{2 \eta \alpha(q)}) \nonumber \\ \pm \sqrt{\left(\omega_0 + \frac{E}{\eta} + \frac{\sigma q^2}{2 \eta \alpha(q)}\right)^2 + \omega_0\frac{2 \sigma q^2}{\eta \alpha(q)}} \label{dispersion}\end{aligned}$$ Since it is only the ‘fast’ modes which contribute appreciably to the reduction of the glass transition temperature (see below), we consider only the upper branch of eq. (\[dispersion\]). Observing that $\omega_0 << \frac{E}{\eta}$, this is given by $$\omega = \omega_0 + \frac{E}{\eta} + \frac{\sigma q^2}{2 \eta \alpha(q)}, \label{dispersionbranch}$$ as a very good approximation. The dependence of the wave number, $q$, stems solely form the coupling to the capillary waves on the free film surface. The first major assumption of our model is that $T_g(h)$ is determined by the spectrum of the viscoelastic eigenmodes of the film as given by eq. (\[dispersionbranch\]). The freezing mechanism: memory effects {#the-freezing-mechanism-memory-effects .unnumbered} ====================================== The second major assumption is that the physical cause for the melting or freezing of the film, respectively, are [*memory effects*]{} in the polymer material. These are of course not included in the linear theory discussed above, and may be formulated in a generic way by means of a suitable memory kernel, as used in a class of theoretical models of the glass transition, called mode coupling models [@Goetze1; @Goetze2]. Within this framework, memory effects are taken into account by inserting a convolution integral with the memory kernel $m\{\phi(t)\} = a_1 \phi + a_2 \phi^2 + a_3 \phi^3 + ...$ in the otherwise linear differential equation of motion of the modes considered [@footnotestrobl]. In these models, $\phi$ usually describes density fluctuations. In contrast, we consider the material to be essentially incompressible, and $\bf\phi$ denotes here the strain in the polymer material, as defined above. It thus describes the local state of the material, similar to what the density does for simple glass forming liquids [@Goetze1; @Goetze2; @Zaccarelli]. In our case, the equation of motion (2) is of first order in time, hence we have $${\bf\phi}^{\prime} + \omega(q) \phi + \int_0^t m\{{\bf\phi}(\tau)\}{\bf\phi}^{\prime}(t-\tau)d\tau\ = 0 \label{MCTrelax}$$ as the mode coupling equation. This type of equations has been thoroughly analysed [@Goetze1; @Goetze2; @Leutheusser] in relation to the [*microscopic*]{} physics of the glass transition (to which we do not refer here), as well as to large scale degrees of freedom [@Lequeux]. For density fluctuations in glass forming simple liquids, it was found that the coefficients $a_i$ of the memory kernel vary concurrently with temperature, and that upon crossing a certain border in the space spanned by the $a_i$, the system freezes into a nonergodic state [@Goetze2]. The existence of such a freezing transition has been found to be largely independent of the precise form of the memory kernel. In fact, most of the features of a glass transition may be well represented in what has become known as schematic models, which are simple mode coupling equations not referring in their memory kernels to the microscopic physics of the system under study. It is now worthwhile to contemplate on possible memory effects in a polymer melt, as to their scaling with temperature. Let us first consider a polymer molecule in equilibrium, forming a more or less Gaussian coil. If this is elongated by straining the polymer melt, to what extent will it memorize this process after the strain is released? In the strained state, the molecule will relax to some extent. However, this relaxation will not proceed homogeneously along the molecule, since the activation energy, $U$, for local rearrangements will depend upon the local topological environment. In order to discuss the relaxation behaviour of the local molecular geometry, let us define a local geometric exponent of the coil, $\mu$, by the relation $$<\Delta\mid{\bf r}\mid> = (\Delta s)^{\mu} \label{geometricexponent}$$ where $<\Delta\mid{\bf r}\mid>$ is the typical distance travelled in space upon moving alongside the polymer chain by a distance $\Delta s$. In completely stretched parts of the chain, $\mu = 1$, whereas in equilibrated domains, $\mu = \mu_{equilib.} \approx 0.5$. The variation $\Delta U$ of $U(s)$ determines the variation of the relaxation rate of $\mu$ towards $\mu_{equilib.}$. When the strain is released, remnants of this variation will remain, and thus represent a memory of the strain. This holds as long as the strain/release process is fast as compared to the equilibration time, $\omega_0^{-1}$. For the modes to be considered here, this is well fulfilled. When $\Delta U << kT$, as can be safely assumed, these remnant variations in $\mu(s)$ scale as $\Delta U/kT$, thus we can conclude that memory effects in the polymer scale as $1/T$. As a direct consequence, we can replace the memory kernel $m\{ {\bf \phi (t)}\}$ of eq. (\[MCTrelax\]), the coefficients $a_i$ of which are temperature dependent, with $M\{\phi\}/T$, where $M$ is now independent of temperature. By proper normalizaton of time, eq. (\[MCTrelax\]) can now be rewritten as $$\phi^{\prime} + \phi + \frac{1}{T\omega(q)}\int_0^t M\{\phi(\tau)\}\phi^{\prime}(t-\tau)d\tau\ = 0 \label{MCTnormalized}$$ From this equation, we see directly that the modes with the largest relaxation rate freeze at the lowest temperature. Furthermore, the precise form of $M$, which might be derived from a detailed analysis of the non-equilibrium dynamics of the polymer molecules, does not need to be considered here any further. It is only required that it belongs to the class of kernels which yield a freezing transition for ${\bf\phi}$ at all. Let us now turn back to the eigenmodes of the film. The relaxation rates, as given by eq. (\[dispersionbranch\]), are monotonously increasing with $q$. However, modes with $q$ much larger than the inverse film thickness, $h^{-1}$, do not penetrate appreciably into the film, such that only a small fraction of the material takes part in these modes. Hence we are led to considering chiefly the modes with $q \approx h^{-1}$, since these are the highest frequency modes comprising all of the film material. Setting thus $T_g\omega(h^{-1})=$ [*const.*]{} as suggested by eq. (\[MCTnormalized\]), we directly arrive at a simple formula for the glass transition temperature: $$T_g(D) = T^0_g\left(1+\frac{1.16 \ \sigma}{h(E+\eta\omega_0)}\right)^{-1} \label{result}$$ This is precisely the form of eq. (\[KimFormel\]), and fig. \[Tgresult\] shows that it describes our data very well. Discussion {#discussion .unnumbered} ========== The quantity $\eta \omega_0$ is on the order of a few kPa and can in general be safely neglected against $E$. The characteristic length scale which appears here, and which was called $h_0$ in eq. (\[KimFormel\]), is thus the ratio $\sigma/E$, with the surface tension $\sigma = 31$ mN/m for PS. It follows that the elastic modulus determining the dynamics of the relevant modes, $E$, is [*the only physical fitting parameter*]{} in the model. From $h_0 = 0.82$ nm, as obtained from the fit of our model, we find for the elastic modulus $E \approx 44$ MPa, which is, on logarithmic scale, right in between the modulus of the frozen material (a few GPa) and the modulus just above $T_g$ (about 300 kPa). On the basis of the model discussed above, let us now try to formulate answers to the questions asked in the introduction, one after the other. 1\.  What is the main mechanism responsible for the reduction of $T_g$ in thin films of low molecular weight ($M_W<$300kg/mol) polymers? The physical picture which emerges from our model is that as the temperature is increased, melting proceeds as the fastest mode involving all of the film material escapes from its frozen state and fluctuates. This proceeds with the help of the capillary waves on the free film surface, which ease entropy fluctuations in the polymer melt. The concomitantly increased motion of polymer chains reduces the effective viscosity also for modes with smaller $q$, which, as a consequence, are sped up ($\eta$ appears in the denominator in eq. (\[dispersionbranch\])!) and melt in turn. In this way, the film finally melts at all length scales. It is illustrative to note that the softness of the spinodal modes, as used for the determination of $T_g$ at small film thickness, does not contradict our model: these are modes with $q<<1/h$, and are thus not the fastest modes, as considered in the melting mechanism. 2\.  Down to how small molecular weight is this mechanism valid? We found that in agreement with simulation results [@Torres], the effect is present even for molecules as short as about 20 monomer units. This is well below the limit of entanglement. It thus spans a wide range in molecular weight, over more than two orders of magnitude. It is precluded, or obscured, by other effects at molecular weights in excess of about 300 kg/mol [@DVF]. 3\.  Why is there no significant dependence of the effect on molecular weight? The physical property which solely determines the thin film behaviour according to our model is the elastic modulus, $E$. As it is well known, $E$ displays no marked dependence on the molecular weight [@strobl; @stoll], such that on the basis of our model, $T_g(h)$ is expected as well to be largely independent of molecular weight. To be precise, $E$ is slightly less for smaller molecular weight than for larger [@ExpansionCoefficient; @stoll], such that according to eq. (\[result\]), $h_0$ should be larger for smaller molecular weight. In fact, we obtained $h_0=0.82$ nm for our very short molecules, whereas for larger molecules, $h_0=0.68$ nm was found [@Kim]. The observation that the relevant length scale, $h_0$, decreases with increasing molecular weight is another evidence against geometrical effects on individual chains to be relevant in the regime discussed here. The exact physical significance of the somewhat arbitrary ‘choice’ of $E$, which may be viewed as $E$ [*at*]{} $T_g$, is to be investigated in further studies. 4\.  Why is the effect stronger in free standing films than in supported ones? We have mentioned that the eigenmodes of supported films of thickness $h$ with no friction are identical to those (the symmetric ones) of free standing films of thickness $2h$. Thus our model predicts that the effect in a supported film is just as large as in a free standing film of twice the thickness. This is indeed in accordance with experimental observation [@DVF]. 5\.  Why does the effect depend upon the chemical composition of the substrate for supported films? It is clear that the amount of friction of the film material at the substrate changes the eigenmode spectrum of the film, such that a dependence on the chemical composition of the substrate is indeed expected on the basis of our model. For a quantitative comparison with experimental data, however, the friction coefficients would have to be determined for the systems investigated. It should be well noted that this also induces some uncertainty in the numerical value of $E$ derived from our fit (cf. fig. \[Tgresult\]), since we have not characterized the friction of the films on the substrate. 6\.  Why is there sometimes an increase of $T_g$ in thin films [@KJincrease; @vanZanten; @KJCincrease]? We can try to give a rough idea to what extent effects like this may be accounted for within our model, by including the interaction of the free surface with the substrate via long range forces. This is described most conveniently by replacing $\sigma q^2$ by $\sigma q^2 + \frac{d^2V}{dh^2}$, where $V(h)$ is the effective interface potential of the film due to long range forces [@Dietrich; @Schick]. For unretarded van der Waals forces, $V(h)=-A/12\pi h^2$, where $A$ is the Hamaker constant. If $A>0$, $\frac{d^2V}{dh^2}$ is negative, such that the van der Waals forces tend to destabilize film, giving rise to spinodal dewetting for sufficiently thin films. In any case, a positive $A$ will reduce the relaxation rate of all of the modes, including the one at $q=1/h$, and thereby tend to increase the glass transition temperature. Quantitatively, this effect is obtained by accordingly replacing $\sigma$ in eq. (\[result\]) by $\sigma - \frac{A}{2\pi h^2}$. As it turns out, $T_g$ attains a minimum at $h_{min}=3\sqrt{A/2\pi\sigma}$, and increases sharply for smaller thickness. For $A = 2.2 \times 10^{-20}$ J, representing PS on silicon oxide [@GainingControl], this is at $h_{min}\approx$ 1 nm. An increase of $T_g$ above $T_g^0$ at a film thickness significantly larger than 1 nm can be explained only with an unphysically large Hamaker constant. However, one should anyway be cautious with trying to interpret an increase of $T_g$ in thin films in the framework of our model if working with polymers other than PS [@Torres; @KJincrease; @vanZanten; @KJCincrease]. It is not at all clear what impact a possible crystallization (which is not present in atactic PS) can have on $T_g(h)$. Consequently, possible model systems for studying the mechnaism discussed in the present paper have to be carefully chosen to ensure the absence of any pronounced inner texture of the film. Our model can be appropriate only for ‘structureless’ films, and might break down for films with a pronounced layered texture like, e.g., a Langmuir-Blodgett film [@LBpolymer], or for other polymers which may crystallize at least in some part of the film. Outlook {#outlook .unnumbered} ======= It is tempting to apply the view developed here also to the surface of a bulk polymer sample, for which the spectrum is obtained setting $\alpha = q/2$. As it is readily seen, one should expect surface melting at the polymer surface, down to a thickness of $h_{sm} = h_0 T_g^{0}(T_g^{0}-T)^{-1}$. This is very much along the lines of first ideas which had been put forward to explain the observed reduction of $T_g$ in thin films [@KJ1]. To the best of our knowledge, there is yet no conclusive evidence in favor or in disfavor of polymer surface melting. To confirm the theoretical concept put forward here, an [*ab-initio*]{} derivation of the memory kernel, $m\{{\bf \phi}\}$, from the statistics and microscopic transport properties of the individual molecule would be desirable. Furthermore, it is important to elaborate on the significance of Young’s elastic modulus, $E$, at the glass transition. This was determined experimentally using our model to be $E = 44$ MPa, but it is not yet clear what physical principle distinguishes this value. A complete theory of the glass transition on the basis of the proposed model would be rather cumbersome, since it involves two strongly nonlinear mechanisms mutually affecting each other: freezing by the mode coupling mechanism, and non-Newtonian effects on the viscosity. Our model furthermore suggests that the glass transition in a homopolymer may be viewed as a mode-coupling-model freezing of its viscoelastic bulk modes. This needs not to contradict more classical views of the glass transition in homopolymers, but might serve as an interesting, and potentially useful, alternative approach. The authors owe many very helpful hints to J. A. Forrest and K. Dalnoki-Veress. We are furthermore indebted to D. Johannsmann, J. Baschnagel and R. Blossey for stimulating discussions. Funding from the Deutsche Forschungsgemeinschaft within the Priority Program ‘Wetting and Structure Formation at Interfaces’ is gratefully acknowledged. J. L. Keddie, R. A. L. Jones, R. A. Cory, [*Europhys. Lett.*]{} [**27**]{} (1994) 59. J. Mattsson, J. A. Forrest, L. Börgesson, [*Phys. Rev. E*]{} [**62**]{} (2000)5187. J. A. Forrest, K. Dalnoki-Veress, [*Adv. Coll. Int. Sci.*]{}, in print. K. Dalnoki-Veress, J. A. Forrest, P.-G. De Gennes, J. R. Dutcher, [*Journal de Physique IV*]{} (2001), in print. P.-G. de Gennes, [*Europ. Phys. J. E*]{} [**2**]{} (2000) 201. P.-G. de Gennes, [C. R. Acad. Sci. Paris]{} [**1/IV**]{}(2000) 1179. J. A. Forrest, J. Mattsson, [*Phys. Rev. E*]{} [**61**]{} (2000) R53. C. Moukarzel, P. M. Duxbury, [*Phys. Rev. E*]{} [**59**]{} (1999) 2614. D. Long, F. Lequeux, submitted to [*Europhys. Lett.*]{} [****]{} J. A. Torres, P. F. Nealey, J. J. de Pablo, [*Phys. Rev. Lett.*]{} [**85**]{} (2000) 3221. J. L. Keddie and R. A. L. Jones, [*J. Isr. Chem. Soc.*]{} [**35**]{} (1995) 21. J. H. van Zanten, W. E. Wallace, W. Wu, [*Phys. Rev. E*]{} [**53**]{} (1996) R2053. J. L. Keddie, R. A. L. Jones, R. A. Cory, [*Faraday Discuss.*]{} [**98**]{} (1994) 219. J. Brandrup, E. H. Immergut, edts., [*‘Polymer Handbook’*]{} (Wiley, New York 1975). R Xie et al., [*Phys. Rev. Lett.*]{} [**81**]{} (1998) 1251. R. Seemann, S. Herminghaus, K. Jacobs [*Phys. Rev. Lett.*]{} [****]{} (2001) in press. J. H. Kim, J. Jang, W.-Ch. Zin, [*Langmuir*]{} [**16**]{} (2000) 4064, and references therein. L. D. Landau, E. M. Lifshitz, [*Theory of Elasticity*]{} (Vol. [**VII**]{}), Butterworth, London 1995. L. Landau, Lifshitz, [*Hydrodynamics*]{} (Vol. [**VI**]{}), Butterworth, London 1995. G. Strobl, [*‘The Physics of Polymers’*]{} (Springer, Berlin 1996). N. F. Fatkullin, R. Kimmich, M. Kroutieva, [*J. Exp. Theor. Phys.*]{} [**91**]{} (2000) 150. S. A. Safran, J. Klein, [*J. Phys. II France*]{} [**3**]{}(1993) 749. W. Götze and L. Sjögren, [*Rep. Prog. Phys.*]{} [**55**]{} (1992) 241. W. Götze and Th. Vogtmann, [*Phys. Rev. E*]{} [**61**]{} (2000) 4133. The memory kernel discussed here is conceptually different from the memory function of [*stress*]{}, as discussed, e.g., in the context of Lodge liquids [@strobl]. E. Zaccarelli, G. Foffi, F. Sciortino, P. Tartaglia, K.A. Dawson, cond-mat/0101211 (15jan2001). The bifurcation behavior of the particularly simple form emerging here has been discussed, e.g., by E. Leutheusser, [*Phys. Rev. A*]{} [**29**]{} (1984) 2765. P. Hebraud, F. Lequeux, [*Phys. Rev. Lett.*]{} [**81**]{} (1998) 2934. G. Eckert, PhD Thesis (Ulm 1997). S. Dietrich, in [*‘Phase Transitions and Critical Phenomena’*]{} [**12**]{}, C. Domb and J. L. Lebowitz, edts. (Academic Press, London 1988). M. Schick, in [*‘Liquids at Interfaces’*]{}, J. Charvolin et al., edts. (Elsevier, Amsterdam 1990). R. Seemann, S. Herminghaus, K. Jacobs, [*J. Phys. Cond. Mat.*]{} [****]{} (2001) in print. Y.-K. See, J. Cha, T. Chang, M. Ree, [*Langmuir*]{} [**16**]{}(2000) 2351.

--- abstract: 'A new synthesis of the principles of relativity and quantum mechanics is developed by replacing the Poincaré group for the de Sitter one. The new relativistic quantum mechanics is an indefinite mass theory which is reduced to the standard theory on the mass shell. The charge conjugation acquires a geometrical meaning and the Stueckelberg interpretation for antiparticles naturally arises in the formalism. So the idea of the Dirac sea in the second quantized formalism proves to be superfluous. The off-shell theory is free from ultraviolet divergences, which only appear in the process of mass shell reduction.' author: - | Edgardo T. Garcia-Alvarez and Fabián H. Gaioli[^1]\ [*Instituto de Astronomía y Física del Espacio,*]{}\ [*C. C. 67, Suc. 28, 1428 Buenos Aires, Argentina and* ]{}\ [*Departamento de Física, Facultad de Ciencias Exactas y Naturales,*]{}\ [*Universidad de Buenos Aires, 1428 Buenos Aires, Argentina*]{} title: On the quantum electrodynamics of moving bodies --- epsf The advent of quantum theory cherished the hope of reformulating electrodynamics free from anomalies. However, divergences were smoothed but not completely erased by quantization. Such a disappointment was considered as a serious trouble for the physics of that time and the progress in the area was delayed for two decades. After the great advances achieved by the end of the fifties, the new generation of physicists “have learned how to peacefully coexist with the alarming divergences of the old fashioned theory, but these infinities are still with us, even though deeply buried in the formalism” [@Roman]. Due to this fact some workers in the field tried to start again from the beginning formulating the so called axiomatic quantum field theory. Their unsatisfaction was clearly summarized in the statement of Streater and Wightman: “...the quantum theory of fields never reached a stage where one could say with confidence that it was free from internal contradictions –nor the converse” [@Streater]. Unfortunately as Rohrlich [@Rohrlich] has pointed out, this route does not fullfil all aspirations: “We now have a much deeper mathematical understanding of quantum electrodynamics, especially due to the work of axiomatic field theorists; but we have still not solved the basic problem of formulating the theory in a clean mathematical way, not even with all the complicated and highly sophisticated limiting procedures presently used to justify the results of a naive renormalization theory in simpler quantum field theories and in lower dimensionality. The hopes and aspirations indicated in the outlook of twenty years ago remain valid today.” A renovating spirit was present in the more recent movement of string theorists who decided to change some basic principles. As a consequence of it, string models have non-local interactions which provide a way to avoid the ultraviolet divergences from the beginning. However the price payed for this desirable requirement is too high: we have lost the extraordinary power of calculus and predictability of quantum field theory. This is the reason why some theoretical physicists became conservative and, in a radical change to the optic of the problem, tried to justify “the unreasonable effectiveness of quantum field theory” [@Jackiw], arguing that the phenomenologically desirable results are provided by ultraviolet divergences. As in the standard theoretical framework anomalies, as the chiral one, come from the gauge non-invariance of the infinite negative-energy sea. It is argued that “we must assign physical reality to this infinite negative-energy sea” [@Jackiw86]. We see such philosophical position as a new intent of rescuing the theory of the “ether.” Alternatively, Weinberg [@Weinberg] has delayed the present difficulties for quantizing gravity reformulating the problem in this way. He holds the point of view that the standard model and general relativity are the leading terms in effective field theories, and so disregards the problem of renormalizability which is only proper of a fundamental theory still unknown (perhaps a string model). On the contrary, the creators of the quantum field theory, such as Dirac, held a less conservative viewpoint [@Salam]: “Nowadays, most of the theoretical physicists are satisfied with this situation, but I am not. I think that theoretical physicists have taken a wrong way with this new facts and we would not be pleased with this situation. We must understand that we are in front of something wrong radically discarding the infinities from our equations; here we need to respect the basic laws of the logics. Thinking about this point could send us to an important advance. QED is the branch of theoretical physics about we know more, and presumably we have to put it in order until we can make a fundamental progress in other field theories, although this theories continue developing under experimental basis.” In this work we develop the foundations of a new shyntesis of the principles of relativity and quantum mechanics. Following Dirac’s advice we only propose to reformulate QED. As our purpose is humbler than that of the string program (conceived as the theory of everything) the change in the basic principles is also less radical: essentially we propose to substitute once more the standard group of external symmetries, i.e. the Poincaré group for the de Sitter one. It is ironic that, approaching to the end of this century after nine decades from Einstein did the same with the Galilei group, we can motivate the new program rephrasing Einsten’s words [@Einstein]:   It is known that Dirac’s quantum electrodynamics –as usually understood at the present time– leads to asymmetries and inconsistencies which do not appear to be inherent in the phenomena. Take, for example, the description of a pair creation in an external electromagnetic field. The observable phenomenon here always involves finite measurable quantities and does not make any distinction between electron and positron, whereas the customary view draws a sharp distinction between the two particles. While the electron is interpreted as a positive energy state of the Dirac equation, the positron is interpreted as a hole or absence of a negative energy state in the Dirac sea.[^2] This sea of infinite electrons, which fills all the negative energy states of the Dirac equation, is the responsible for ultraviolet divergences in the effective action used for describing such phenomena.[^3] Moreover, from the standpoint of general relativity the zero point energy of the electromagnetic field also seems unsatisfactory since a divergent vacuum stress tensor would imply, via the Einstein field equations, an infinite curvature for the universe corresponding to an infinite cosmological constant, which cannot be removed simply by performing some sort of transfinite shift of the energy scale. Examples of this sort, together with the unsuccessful attempts for quantizing gravity through these methods, suggest that the phenomena of electrodynamics as well as of gravity at a quantum level possess no properties corresponding to the quantum field notion of the vacuum.[^4] They rather suggest that a different route must be taken in order to accommodate the principles of relativity at the quantum level. From our point of view the main difficulty lies in the different role and interpretation of “time” in both theories. In fact, while quantum mechanics privileges an absolute parameter that labels the evolution of the system, the theory of relativity stresses the relative character of the temporal coordinate. Therefore the first concept of time should have the properties of a $c$-number, while the second should be an operator due to the mixing character of the Lorentz transformations. Thus this dual role of time poses a problem in relativistic quantum mechanics at a first quantized level. The standard solution to this dilemma is to give up this vessel and plunge into the sea of quantum field theory, relegating the role of space-time coordinates to be simple parameters of the theory. Unfortunately this mathematical artifact is achieved by means of a choice of vacuum compatible with the idea of the Dirac sea, which actually just swept the problem under the rug. This fact suggests us that such a dual role of time demands the introduction of two different concepts for playing two different roles. In other words we propose that the unification of quantum principles with the theory of relativity requires the introduction of an additional label to describe the events,[^5] increasing in this way the dimension of the space-time manifold [@ap95b; @ga95b; @ga96a]. We will raise this conjecture to the status of a postulate, and also introduce another postulate, namely, laws of physics in our five-dimensional space-time obey the principles of the special theory of relativity. These two postulates suffice for the attainment of a simple and consistent theory of quantum electrodynamics, based on Dirac’s theory in a higher dimension. The introduction of a “Dirac sea” will prove to be superfluous inasmuch as the view here to be developed will not require ordinary time to be the parameter which labels the quantum evolution. Kinematical Part ================ Nowadays, theoretical physicists seem to be more focused on internal symmetries than on external ones, in the search of a grand unified gauge theory. However in the sixties a great effort was made for unifying both symmetries, enlarging the Poincaré group. So for different motivations the simplest extensions of the Poincaré group, such as the five-dimensional Galilei group, the de Sitter group, and conformal group, began to be studied, constituting the antecedents of our program.[^6] However the idea of enlarging the dimension of space-time to take into account particle-antiparticle symmetries is an older fascinating idea. Perhaps the first antecedent can be found in the works of Hinton, who built a model of electricity associating positive and negative charges with right and left handed helixes in higher dimensional spaces. Curiously, this prerelativistic model developed in 1888 has an extraordinary parallelism with the theory of Klein [@gard]. In Sec. 2 we discuss these ideas through a generalization of the Schroedinger [*Zitterbewegung*]{} to four dimensions [@bar84; @ga96a], which is related to the Stueckelberg [@stu], Wheeler and Feynman [@fey48; @fey49; @fey51; @schw; @nam] interpretation of antiparticles. But in this route, the concept of time must be revisited. Time in physics is not an [*a priori*]{} concept in the Newton sense, but enters as a basic concept used to describe the laws of nature. The history of science shows us that physics always adapts and modifies this concept in order to simplify the laws. Then, from this point of view, there is no place to the question why the universe has five dimensions and not four. The important thing is that there is a set of phenomena which can be described in a more simple and symmetrical way if we use two times instead of one. The purpose of this work is to demonstrate that this is the case for QED. We begin considering a five-dimensional manifold as space-time arena in which such phenomena occur. According to the first postulate, each event in our description has associated a point $P$ of the space-time determined by coordinates $x^A=(x^\mu ,x^5)$ ($A=0,1,2,3,5),$ i.e. $P=P(x^A),$ which will be called a super-event. From the second postulate the space-time is endowed with a super-Minkowskian metric $g^{AB}={\rm diag}(+,-,-,-,-),$ so the square of the super-arc element $dS$ reads $$dS^2=g^{AB}dx_Adx_B=g^{\mu \nu }dx_\mu dx_\nu -(dx^5)^2. \label{ds}$$ Any linear transformation of coordinates $x^{A^{\prime }}=L_{.B}^Ax^B+C^A$ which leaves $dS^2$ invariant will be referred to as a coordinate transformation between two super-inertial systems. The super-Poincaré group of such a transformation is the well-known inhomogeneous de Sitter group. The other implicit assumption is that all physical laws adopt the same form in all super-inertial frames, that is to say that they are de Sitter covariant. We do not analyze here all the potentialities of such a description but our intention is to use this new framework to reformulate the physics associated to the Poincaré invariance free from inconsistencies. Keeping this in mind, let us restrict ourselves to the subset of linear transformations $$\label{uno} x^{\mu ^{\prime }}=L_{.\nu }^\mu x^\nu +C^\mu ,$$ $$x^{5^{\prime }}=x^5+C^5, \label{dos}$$ which leaves the square of the standard arc element, $ds^2=g_{\mu \nu }dx^\mu dx^\nu ,$ invariant, maintaining the fifth coordinate $x^5$ as a Poincaré invariant parameter. This means that we are going to describe the super-events posed in a given super-frame, forbidding boosts and rotations between $x^5$ and any of the space-time coordinates. In this case such an evolution parameter works as a Newtonian time in each super-frame and introduces an absolute notion of simultaneity and retarded causality associated to it. The fifth coordinate $x^5$ is arbitrary in principle, however from Eq. (\[ds\]) we see that for the particular case of motions on the super-light cone $(dS=0)$ the coordinate $x^5$ is reduced to $s$. We restrict our analysis of QED to this case. In Fig. 1 we show the super-light cone and its four-dimensional projection. Note that while a super-world line lies on the super-light cone its space-time projection lies inside the standard light cone.   =12truecm   At this point one could ask what we have gained with such a description. The immediate answer is that this description has now an invariant evolution parameter at the classical level, preparing the land for a description at the quantum level that avoids the lack of explicit covariance of the standard canonical formalism. What is not so evident is that it is a natural framework for introducing the notion of antiparticles. Moreover, as we show in Sec. 2, the notion of retarded causality in $x^5$ for super-particles naturally leads to the standard quantum field theoretical boundary conditions for the Green functions on the mass-shell. That is, particles go forward and antiparticles go backward in the coordinate time $x^0.$[^7] Let us consider the world-line of a super-event in a given super-frame. The Poincaré invariance suggests us to parametrize this curve with $x^5,$ i.e. to project the super-world-line in a hyper-plane $x^5={\rm const}$ (the standard space-time)$.$ Thus, at any point of the projected curve (a standard world-line), the four-velocity $\frac{dx^\mu }{dx^5}=\left( \frac{% dx^0}{dx^5},\frac{d\overrightarrow{x}}{dx^5}\right) $ has a new key ingredient with respect to the non-covariant description which takes the coordinate $x^0$ as the evolution parameter, namely the rate $\frac{dx^0}{% dx^5}.$ This new degree of freedom allows us to introduce the concept of antiparticle just at the classical level. Generalizing Stueckelberg’s ideas [@stu; @fey48] we call super-particles and super-antiparticles to those states for which $\frac{dx^0}{dx^5}$ is positive and negative respectively. Therefore for causal propagation ($dx^5>0$), while the super-particles propagate forward in time, the super-antiparticles propagate backward in coordinate time. Notice that for $dx^5=0$ we cannot distinguish the two concepts.[^8] This is the case of the photon in the standard framework, in which we identify the fifth coordinate with the classical proper time. We could expect that the evolution in $x^5$ also interchanges particle and antiparticle states at a first glance. Nevertheless, as we will see below, for the standard electromagnetic interactions this interchange is classically forbidden and only possible at the quantum level as a consequence of the uncertainty principle. Electrodynamical Part ===================== From a dynamical point of view the main difference between the Poincaré and the de Sitter groups is that for the second group the operator $p_\mu p^\mu $ is no longer a Casimir operator. The states of the new theory are off the mass shell $p_\mu p^\mu =m^2$. They are on the super-mass shell hyperboloid $$p_Ap^A=M^2,$$ where $M$ is a super-mass parameter. We are interested in the study of null-super-mass states because in the classical limit they motion is super-luminal and, as we discuss in the kinematical part, we can identify the five coordinate $x^5$ with the proper time $s.$ So, let us begin considering the wave equation satisfied by the non-super-massive ($M=0)$ spin-$\frac 12$ irreducible representation of the de Sitter group $\Psi $ $% \frac {}{}$ $$\Gamma ^Ai\partial _A\Psi =0, \label{irrep}$$ where $\Gamma ^\mu =\gamma ^5\gamma ^\mu ,$ $\Gamma ^5=\gamma ^5=\gamma ^0\gamma ^1\gamma ^2\gamma ^3,$ satisfy the Dirac algebra $$\Gamma ^A\Gamma ^B+\Gamma ^B\Gamma ^A=2g^{AB}.$$ Multiplying on the left by $\gamma ^5,$we can rewrite (\[irrep\]) in the Hamiltonian form $$-i\frac{\partial \Psi }{\partial s}=\gamma ^\mu i\partial _\mu \Psi , \label{Fey}$$ where we have identified $x^5$ with $s$ [@ap95b]. Eq. (\[Fey\]) was originally introduced by Feynman in 1948 in his dissertation at the Pocono Conference.[^9] This is a Schroedinger equation in the invariant parameter $s$ for the evolution of states off the mass-shell. The mass-shell condition is satisfied by stationary states, $\Psi (x^\mu ,s)=\psi _m(x^\mu )e^{ims},$ solutions of the Dirac equation[^10] $$\gamma ^\mu i\partial _\mu \psi _m=m\psi _m. \label{Dir}$$ The Feynman equation minimally coupled to an external electromagnetic field is given by $$-i\frac{\partial \Psi (x,s)}{\partial s}=\gamma ^\mu (i\partial _\mu -eA_\mu )\Psi (x,s), \label{FeyA}$$ where $A_\mu $ is the electromagnetic potential. The key idea of Feynman [@fey51; @schw] was that by Fourier transforming in $s$ any solution $\Psi (x,s)$ of Eq. (\[FeyA\]) a solution $\psi _m(x)$ of the corresponding Dirac equation $$\left[ \gamma ^\mu (i\partial _\mu -eA_\mu )-m\right] \psi _m(x)=0$$ can be obtained$,$ namely $$\label{psi} \psi _m(x)=\int_{-\infty }^{+\infty }\Psi (x,s)e^{-ims}ds.$$ Hence the Fourier transform of the retarded Green function $G(x,x^{\prime },s)$ of Eq. (\[FeyA\]) $$\left[ \gamma ^\mu (i\partial _\mu -eA_\mu )-i\frac \partial {\partial s}\right] G(x,x^{\prime },s)=\delta (x,x^{\prime })\delta (s), \label{green}$$ with $G(x,x^{\prime },s)=0,$ for $s\leq 0,$ enables one to derive the corresponding mass-shell Green function $G_m(x,x^{\prime }),$ i.e. $$\left[ \gamma ^\mu (i\partial _\mu -eA_\mu )-m\right] G_m(x,x^{\prime })=\delta (x,x^{\prime }). \label{shell}$$ From the path integral point of view the retarded condition for the propagator $G(x,x^{\prime },s)$ means that all the classical paths go forward in time ($ds>0$), so the on-shell positive (negative) kinetic energy states must go forward (backward) in coordinate time, since in the classical limit (neglecting spin effects) we have $\frac{dx^0}{ds}=\pm \frac 1{\sqrt{% 1-v^2}}.$ This fact determines the well-known boundary conditions for $% G_m(x,x^{\prime })$ [@fey49]. Moreover if in the Fourier transformation $$G_m(x,x^{\prime })=\int_0^{+\infty }G(x,x^{\prime },s)e^{-ims}ds,$$ for the on-shell retarded Green function $$G(x,x^{\prime },s)=-i\theta (s)\left\langle x\left| e^{i\gamma ^\mu \pi _\mu s}\right| x^{\prime }\right\rangle$$ the Schwinger formal identity $$i/(a+i\epsilon )=\int_0^\infty \exp [is(a+i\epsilon )]ds \label{sch1}$$ is used for $a=\gamma ^\mu \pi _\mu -m$, one immediately sees that such retarded boundary condition for $G(x,x^{\prime },s)$ naturally leads to the Feynman $i\epsilon $ prescription for avoiding the poles in the on-shell Green function $$G_m(x,x^{\prime })=\left\langle x\left| \frac 1{\gamma ^\mu \pi _\mu -m+i\epsilon }\right| x^{\prime }\right\rangle .$$ This formal trick allowed Feynman to discuss external field problems of QED keeping up at a first quantized level. Let us go further these formal tools in order to understand the physical grounds of them. In this formalism the state space is endowed with an indefinite Hermitian form [@ap95a; @ap95b] $$\left\langle \Psi |\Phi \right\rangle =\int d^4x\overline{\Psi }(x)\Phi (x), \label{pe}$$ in which the covariant Hamiltonian or mass operator ${\cal H=}$ $\gamma ^\mu i\partial _\mu $ is self-adjoint and the evolution operator $e^{i{\cal H}s}$ is unitary. It can be proved [@ga96a] that at a semiclassical level $${\rm sign}\left[ \overline{\Psi }(x,s)\Psi (x,s)\right] ={\rm sign}\frac{dx^0% }{ds},$$ that is super-particles and super-antiparticles states have positive and negative norm respectively. This is the root of the indefinite character of the “inner product”. Frequently this fact is considered as an anomaly of the theory, due to it is not possible to straightforward apply the standard probabilistic interpretation. In fact this is one of the reasons why Dirac[^11] originally rejected the Klein-Gordon equation. But as was shown by Feshbach and Villars [@Feshbach] the indefinite metric character of the Klein-Gordon theory can be reinterpreted in the framework of the theory of a charge. This is the interpretation we adopt in this work. We have defined super-particles and super-antiparticles according to the Stueckelberg interpretation in the kinematical part. Let us now show that it is consistent with the more familiar notion based on charge conjugation. For making this let us note that the operation that conjugates the charge in Eq. (\[FeyA\]) is [@han; @ga95b] $$C\Psi (x,s)=c\Psi (x,-s),$$ where $c=\gamma ^5K$ is the standard charge conjugation operator. The remarkable points are that this operation coincides with the $s$-time reversal operation in the Wigner sense [@ga95b] $$C=S, \label{wig}$$ and $PcT$ looks as a “parity” operation in the five-dimensional space-time: $$PcT=\gamma ^5Q,$$ where $$Q\Psi (x)=\Psi (-x),$$ and $\gamma ^5$ plays the role of the “intrinsic parity” operator. The identity (\[wig\]) is the quantum analogous of a celebrated Feynman [@fey48] observation at the classical level, that charge conjugation in the Lorentz force law is equivalent to a proper time reversal. In other words, charge conjugation is equivalent to an inversion of the sign of $\frac{dx^0}{% ds},$ according to the Stueckelberg interpretation for antiparticles. In order to get a more intuitive insight about why this proper time formalism works, let us return to the problem of particle creation in an external electromagnetic field. In this case, the Heisenberg equations of motion are $$\label{gama} \frac{d\gamma ^\mu }{ds}=2i\gamma ^\mu {\cal H}-2i\pi ^\mu ,$$ $$\label{pi} \frac{d\pi ^\mu }{ds}=eF^{\mu \nu }\gamma _\nu ,$$ which form a coupled system of linear differential equations of first order in $\gamma ^\mu $=$\frac{dx^\mu }{ds}$ and $\pi ^\mu =p^\mu -eA^\mu ,$ where the mass operator ${\cal H}=\gamma ^\mu \pi _\mu $ is a constant of motion. Let us restrict to the case of pure electric field, and choose the coordinate system in such a way that $\overrightarrow{E}=E\overrightarrow{e_1% }$, therefore the only non-vanishing components of the electromagnetic field tensor are $F_{10}=-F_{01}=E,$ and the system of differential equations are reduced to $$\frac d{ds}\left[ \begin{array}{c} \gamma ^0 \\ \gamma ^1 \\ \pi ^0 \\ \pi ^1 \end{array} \right] =\left[ \begin{array}{cccc} 2i{\cal H} & 0 & -2i & 0 \\ 0 & 2i{\cal H} & 0 & -2i \\ 0 & -eE & 0 & 0 \\ -eE & 0 & 0 & 0 \end{array} \right] \left[ \begin{array}{c} \gamma ^0 \\ \gamma ^1 \\ \pi ^0 \\ \pi ^1 \end{array} \right] , \label{E}$$ plus uncoupled equations for the components $2$ and $3$ identical to the free case [@bar84; @ga96a] $$\frac{dx^\mu }{ds}=\frac{p^\mu }{{\cal H}}+\left[ \frac{dx^\mu }{ds}(0)-% \frac{p^\mu }{{\cal H}}\right] \cos \left( 2ps\right) -\frac 1{2p}\frac{% d\gamma ^\mu }{ds}(0)\sin \left( 2ps\right) . \label{dd}$$ The system of differential equations could be exactly solved diagonalizing the matrix of Eq. (\[E\]). The eigenvalues are $z_{1,2,3,4}=i{\cal H}\pm \sqrt{-{\cal H}^2\pm 2ieE}.$ In the weak field approximation (${\cal H}^2\gg 2eE)$ the solution of this system adopts a specially simple form [@ga96a] $$\frac{dx^0}{ds}(s)=\left. \frac{dx^0}{ds}(s)\right| _{E=0}\cosh \left( \frac{% eE}{{\cal H}}s\right) -\left. \frac{dx^1}{ds}(s)\right| _{E=0}\sinh \left( \frac{eE}{{\cal H}}s\right) , \label{pri}$$ $$\frac{dx^1}{ds}(s)=\left. \frac{dx^1}{ds}(s)\right| _{E=0}\cosh \left( \frac{% eE}{{\cal H}}s\right) -\left. \frac{dx^0}{ds}(s)\right| _{E=0}\sinh \left( \frac{eE}{{\cal H}}s\right) , \label{seg}$$ where $p=\sqrt{p^\mu p_\mu }$ is the free positive mass operator. The classical picture of Eq. (\[dd\]) together with Eq. (\[seg\]) is a helical motion in the space and the orbital angular momentum of this [*Zitterbewegung*]{} gives rise to the normal magnetic moment of the electron [@bar84; @ga96a]. Eqs. (\[pri\]) and (\[seg\]) describe the classical hyperbolic motion derived from the Lorentz force law modulated by the free [*Zitterbewegung.*]{} This quick oscillatory motion (of a Compton space-time wavelength order) vanishes in the classical limit. Two different $s$-time scales appear, one related to the inverse of the frequency of the [*Zitterbewegung*]{} $\frac 1{2{\cal H}}$ and the other related to the inverse of the electric field strength $\frac{{\cal H}}{eE}.$ Then when $\frac{{\cal H}% }{eE}\gg \frac 1{2{\cal H}},$ the [*Zitterbewegung*]{} does not feel the adiabatic changes in the mean classical motion, so it works as in the free case. The same scales also appear in the space-time trajectories. If the minimal distance $\frac{2{\cal H}}{eE}$ between the two branches of the hyperbola –representing particle and antiparticle solutions at the classical level– is greater than $\frac 1{{\cal H}},$ the particle and antiparticle trajectories are distinguishable. However, when $\frac{2{\cal H}% }{eE}\approx \frac 1{{\cal H}},$ such trajectories overlap, increasing the probability that the particle jumps to the trajectory of the antiparticle and [*vice versa*]{}. These jumps are reinterpreted in the standard viewpoint –which parameterizes the dynamics with the coordinate time $x^0$– as the pair creation and annihilation processes (Dirac picture[^12]). Summarizing, the Schroedinger [*Zitterbewegung* ]{}depicted above gives a very clear semiclassical interpretation of such processes, which dresses the corresponding Feynman diagrams of physical content, disregarding the concept of Dirac’s sea (see Fig. 2). =12truecm   At this point we disagree with some recognized field theorists that regard Feynman’s graphical method as “a convenient pictorial device that enables to keep track of the various terms in the matrix elements which can rigorously derived from quantum field theory” [@Sakurai]. We think that their opinion is due to they do not completely take into account the genesis of Feynman’s ideas originally developed from the proper time method. Unfortunately Feynman due to the misunderstanding of his dissertation at Pocono [@schw] was forced to introduce his space-time visualization of quantum electrodynamical processes in the form written in his 1949 papers [@fey49]. He relegated much of his original physical ideas and motivations to his 1950 and 1951 papers [@fey51]. So there are a generation of field theorists that have learned the derivation of Feynman rules from Dyson’s paper [@Dyson] rather than from Feynman’s ones. In fact when Dyson’s paper appeared most of Feynman’s work was still unpublished. Unfortunately although Dyson himself remarked that “the theory of Feynman differs profoundly from that of Schwinger and Tomonaga,” the announcement of the demonstration of the equivalence (strictly speaking only at the level of the consequences) of both theories had great impact. Moreover the fine Schwinger calculations [@schw51] using a proper time method were considered just as mathematical tools and Nambu’s claims of his deep paper of 1950 [@nam] “The space-time approach to quantum electrodynamics, as has been developed by Feynman, seems to offer a very attractive and useful idea to this domain of physics. His ingenious method is indeed attractive, not only because of its intuitive procedure which enables one to picture to oneself the complicated interactions of elementary particles, its ease and relativistic correctness with which one can calculate the necessary matrix elements or transition probabilities, but also because of its way of thinking which seems somewhat strange at first look and resists our minds that are accustomed to causal laws. According to the new standpoint, one looks upon the world in its four-dimensional entirety. A phenomenon that will come into play in this theatre is now laid out beforehand in full detail from immemorial past to ultimate future and one investigates the whole of it at glance. The time itself loses sense as the indicator of the development of phenomena; there are particles which flow down as well as up the stream of time; the eventual creation and annihilation of pairs that may occur now and then, is no creation nor annihilation, but only a change of directions of moving particles, from past to future, or from future to past; a virtual pair, which, according to the ordinary view, is foredoomed to exist only for a limited interval of time, may also be regarded as a single particle that is circulating round a closed orbit in the four-dimensional theatre; a real particle is then a particle whose orbit is not closed but reaches to infinity ...” received little attention. On the other hand most of quantum field theory treatises which intent to incorporate the Feynman space-time visualization turn out to be contradictory. For example they interpret field operators as operators that create and annihilate particles in space-time points for giving an interpretation to the Green functions. However relativistic and non-relativistic quantum fields exhibit a striking difference concerning the localizability of their respective field quanta [@Lurie]. In fact, while in the non-relativistic case there is in principle no limitation on the accuracy of measuring the position of a particle, the combination of relativity and quantum theory provides an intrinsic limitation on the measurability of the position due to the particle creation mechanism. The understanding of such difficulties have inclined some authors to propose the idea that Minkowsky space-time is not suitable for particle physics and its role was essentially a historical one,[^13] unlike the energy-momentum space which would be fundamental [@Bacry]. On the contrary, in our proposal we prefer to leave Poincaré group and retain the localizability in Minkowsky space-time. Summarizing, those field theories which desire to keep Feynman diagrams interpretative picture, must give up the Poincaré group. There is no space-time localization of particles in this framework. There is only space time localization of charges off the mass-shell. In order to reinforce our pictorial image of the Fig. 2 let us derive the one-loop effective action $W^{(1)},$ which describes the pair creation in an external electromagnetic field, from an argument purely based on the proper time formalism. As $W^{(1)}$ is $i$ times the closed loop amplitude $L$, let us compute $L$ using the proper time formalism. First, let us evaluate the amplitude for a super-particle at $x^\mu $ and polarization $k$ at time $s=0$ remains in the same point and with the same polarization at time $s.$ As a consequence of the indefinite metric ($\ref{pe})$, the spectral resolution of the identity is $$I=\int d^4x\sum_{jk}\gamma _{jk}^0\left| j,x^\mu \right\rangle \left\langle k,x^\mu \right| .$$ Then the expression of such an amplitude per unit of proper time for all the degrees of polarization is $\frac 1s\sum_{jk}\gamma _{jk}^0\left\langle k,x^\mu \left| e^{i(\gamma ^\mu \pi _\mu )s}\right| j,x^\mu \right\rangle .$ The above process is represented through an open diagram in the five-dimensional space-time, but it is a closed loop in four dimensions [@dav55]. Restricting the formalism to the mass-shell by means of a Fourier transformation in proper time with the causal prescription and summing the contributions of each space-time point, we finally have $$W^{(1)}=i\int \int_0^\infty \frac 1s\sum_{jk}\gamma _{jk}^0\left\langle k,x^\mu \left| e^{i(\gamma ^\mu \pi _\mu )s}\right| j,x^\mu \right\rangle e^{-ims}dsd^4x. \label{wef}$$ Schwinger, using quantum field theory, obtained Eq. (\[wef\]), which became the starting point of his 1951 seminal paper [@schw51; @fey51]. The procedure used in the calculation of $W^{(1)}$ also shows that the ultraviolet divergences only appear after the reduction of the off-shell amplitude on the mass shell. Note that this circumstance also suggests a natural regularization method based on a small mass dispersion [@fey51]. Our alternative explanation does not involve the infinite amount of energy and charge of the Dirac sea in order to consider antiparticles, and in this way it avoids the infinities introduced in the standard theory from the very beginning. This is the reason why closed loops do not appear in the off-shell theory. Until now we have only discussed the theory of external fields. In order to concluding, let us briefly discuss the radiative process. Using this formalism and his operator calculus, Feynman presented at Pocono a closed expression for a system of spin half charges interacting via the quantized electromagnetic field for the case in which only virtual photons are present. In the particular case of one charge it reads [@fey51; @schw] $$\begin{aligned} \Psi (x,s) &=&\exp \left\{ -i\left[ \int_0^s\gamma ^\mu (s^{\prime })\pi _\mu (s^{\prime })ds^{\prime }\right. \right. \label{pocono} \\ &&\left. \left. +e^2\int_0^s\int_0^s\gamma ^\mu (s^{\prime })\gamma _\mu (s^{\prime \prime })\delta _{+}\{[x_\mu (s^{\prime })-x_\mu (s^{\prime \prime })]^2\}ds^{\prime }ds^{\prime \prime }\right] \right\} \Psi (x,0), \nonumber\end{aligned}$$ where $\delta _{+}\{[x_\mu (s^{\prime })-x_\mu (s^{\prime \prime })]^2\}$ is the Green function of the d’Alembertian with Feynman’s boundary conditions. From the second term of Eq. (\[pocono\]) Feynman showed that the radiative corrections of QED can be derived. The analogy between the phase of Eq. (\[pocono\]) and the Wheeler-Feynman action [@Wheeler; @fey48] for classical electrodynamics is remarkable. In fact the only substantial difference is the boundary conditions (half-advanced and half-retarded) chosen for the d’Alembertian Green function. The right boundary conditions for QED can be obtained from the retarded condition of the off-shell theory. This fact strongly suggests that Eq. (\[pocono\]) could be derived, from first principles, from a de Sitter invariant formulation of QED. For one super-particle (antiparticle) the de Sitter invariant equations read $$\label{ba} \Gamma ^A(i\partial _A-eA_A)\Psi =0,$$ $$\partial _AF^{AB}=e\overline{\Psi }\Gamma ^B\Psi , \label{bb}$$ where the super-potential $A^A$ =($A^\mu ,A^5)$ arises from a natural extension of the gauge principle [@Horwitz]. The standard four-potential can be obtained from $A^A$ integrating the first four components in the proper time $$A^\mu (x^\nu )=\int_{-\infty }^{+\infty }A^\mu (x^\nu ,s)ds,$$ as in the case of the matter fields. (The exponential factor does not appear in this case because the photon is non-massive. Note also that the transformation $A^\mu (x^\nu ,s)\rightarrow $ $A^\mu (x^\nu ,-s),$ ($% ds\rightarrow -ds),$ leads to the standard notion of charge conjugation for the potentials.) Note added in proof {#note-added-in-proof .unnumbered} =================== After completing this work we discovered a review paper of Fanchi [@nap1] and the closely related works of Herdegen [@nap2] and Kubo [@nap3]. Acknowledgments {#acknowledgments .unnumbered} =============== We are grateful to the organizers of the First International Colloquium on ‘Actual Problems in Quantum Mechanics, Cosmology, and the Primordial Universe’ and the ‘Foyer d’Humanisme’ for thier warm hospitality. We want to express our acknowledgment to Juan León García, Lawrence Horwitz, and Marek Czachor for their encouragement. [99]{} P. Roman, [*Introduction to Quantum Field Theory*]{} (Wiley, New York, 1969), Chap. 6. R.F. Streater and A.S. 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Davidon, Phys. Rev. [**97**]{}[*,* ]{}1131 (1955); [**97**]{}, 1139 (1955). J.A. Wheeler and R.P. Feynman, Rev. Mod. Phys. [**17**]{}, 157 (1945). N. Shnerb and L.P. Horwitz, Phys. Rev. A[** 48**]{}, 4068 (1993). J.R. Fanchi, Found. Phys. [**23**]{}, 487 (1993). A. Herdegen, Acta Phys. Polonica [**B13**]{}, 863 (1982). R. Kubo, Nuovo Cimento A [**85**]{}, 293 (1985). [^1]: E-mail: gaioli@iafe.uba.ar [^2]: The assymetry in the description is more evident from the historical point of view. In fact the holes were originally interpreted by Dirac [@Dirac30] as protons, who thought that he could explain the mass differences by means of the interaction of the electrons of the sea. [^3]: This is analogous to the case of chiral anomaly discused above, and it results specially clear from the Weisskopf derivation of the Heisenberg-Euler Lagrangian [@lif]. In Sec. 2 we discuss the proper time approach to this effective Lagrangian in which becomes clear that divergences appear in the transition from the off-shell theory to the mass shell. [^4]: As we will see we do not discard many “particle” formalisms (we find more appropriate to call them many charge formalisms) nor the notion of field. We only attack the choice of the vacua in standard quantum field theory to implement the charge conjugation symmetry. [^5]: Formulations of relativistic quantum mechanics with an invariant evolution parameter were discused in the past. According to the external group of symmetry they can be classified as five-dimensional Galilean invariant formulations [@Aghassi1; @Horwitz73; @Fanchi] and de Sitter ones. See Refs. [@ap95a; @ap95b] for a critical review about them. [^6]: In connection with this work see Refs. [@Castell; @DeVos; @Aghassi1; @Aghassi2; @Johnson]. [^7]: This formalism allows us to reformulate the “localization problem” [@Kalnay], by following charges “trajectories” instead particles ones. Moreover, the recognition that this strange notion of $x^0-$causality is the only compatible with the requirements of relativistic quantum mechanics enables one to eliminate Hegerfeldt’s paradox [@Heger]. [^8]: Also note that this notion is super-frame dependent, i.e. a state registered as a super-particle from a super-inertial system can be registered as a super-antiparticle from another super-inertial system. The same thing happens with the notion of simultaneity associated to the coordinate $x^5,$ which looses its invariant character under the full de Sitter group transformations. [^9]: Feynman introduced Eq. (\[Fey\]) in a formal way and did not discuss its geometrical meaning. He could not solve Dirac’s doubts about the unitarity of the theory either. For a nice account of these anecdotes, see the review paper of Schweber [@schw]. [^10]: The Dirac equation can be consistently introduced from first principles at a first quantized level interpreting antiparticles as negative energy states going backward in $x^0$-time [@ga95a]. [^11]: Ironically, some years before it was Dirac himself [@Dirac42] who introduced indefinite metric Hilbert spaces in quantum field theory with the hope of removing the true anomaly: the divergences. [^12]: This picture was refined by Sauter by considering the deformation of the energy gap produced by the electric field. Pair creation is interpreted as a tunneling of a negative energy state (not a hole in a sea) to a positive energy state [@lif]. [^13]: Although this hipothesis could work for the Poincaré group in the case of free fields, strong difficulties arise at the time of introducing interactions. Let us bear in mind that localizability and minimal coupling are intimately linked. Moreover, this fact is not compatible with the principle of general covariance. Notice that it would be possible to extend this formulation to develop quantum field theory in curved space-time.

--- abstract: 'Kernel method has been developed as one of the standard approaches for nonlinear learning, which however, does not scale to large data set due to its quadratic complexity in the number of samples. A number of kernel approximation methods have thus been proposed in the recent years, among which the random features method gains much popularity due to its simplicity and direct reduction of nonlinear problem to a linear one. Different random feature functions have since been proposed to approximate a variety of kernel functions. Among them the Random Binning (RB) feature, proposed in the first random-feature paper [@rahimi2007random], has drawn much less attention than the Random Fourier (RF) feature proposed also in [@rahimi2007random]. In this work, we observe that the RB features, with right choice of optimization solver, could be orders-of-magnitude more efficient than other random features and kernel approximation methods under the same requirement of accuracy. We thus propose the first analysis of RB from the perspective of optimization, which by interpreting RB as a Randomized Block Coordinate Descent in the infinite-dimensional space, gives a faster convergence rate compared to that of other random features. In particular, we show that by drawing $R$ random grids with at least $\kappa$ number of non-empty bins per grid in expectation, RB method achieves a convergence rate of $O(1/(\kappa R))$, which not only sharpens its $O(1/\sqrt{R})$ rate from Monte Carlo analysis, but also shows a $\kappa$ times speedup over other random features under the same analysis framework. In addition, we demonstrate another advantage of RB in the L1-regularized setting, where unlike other random features, a RB-based Coordinate Descent solver can be parallelized with guaranteed speedup proportional to $\kappa$. Our extensive experiments demonstrate the superior performance of the RB features over other random features and kernel approximation methods. Our code and data is available at [ <https://github.com/teddylfwu/RB_GEN>]{}.' author: - | Lingfei Wu\ \ \ Ian E.H. Yen [^\*^]{}\ \ \ - | Jie Chen\ \ \ Rui Yan\ \ \ bibliography: - 'randomBinning.bib' title: 'Revisiting Random Binning Features: Fast Convergence and Strong Parallelizability' --- Introduction {#section:introduction} ============ Kernel methods have great promise for learning non-linear model from simple data input representations and have been demonstrated successful for solving various learning problems such as regression, classification, feature extraction, clustering and dimensionality reduction [@LearnKernels2001; @Taskar2003NIPS]. However, they are typically not first choice for large-scale nonlinear learning problems, since large number of samples ($N$) presents significant challenges in terms of computation and memory consumptions to Kernel methods for computing the dense kernel matrix $K \in \mathcal{R}^{N \times N}$ which requires at least a $O(N^2)$ complexity. To scale up the kernel methods, there have been great efforts addressing this challenge from various perspectives such as numerical linear algebra, sampling approximation, optimization and functional analysis [@rahimi2007random; @Dai2014ScalableKernel; @yen2014sparse; @Si2014MEKA; @Drineas2005Nystrom; @Fine2002JMLR]. A line of research [@Seeger2000Nystrom; @Smola2000SparseGreedy; @Fine2002JMLR; @Si2014MEKA] approximates the kernel matrix $K$ using low-rank factorizations, $K \approx Z^TZ$, where $Z \in \mathcal{R}^{N \times R}$ matrix with $R \ll N$. Among them, Nyström method [@Seeger2000Nystrom; @Drineas2005Nystrom; @Kumar2012SamplingNystrom; @Gittens2013RevisitNystrom; @Si2014MEKA] is probably one of the most popular approaches, which reduces the total computational costs to $O(NRd + NR^2 + R^3)$ or $O(NRd + NRm)$ depending whether the algorithm performs on $K$ explicitly or implicitly through $Z$, where $d$ and $m$ are the input data dimension and the number of iterations of an iterative solver respectively. However, the convergence of the low-rank approximation is proportional to $O(1/\sqrt{R} + 1/\sqrt{N})$ [@Drineas2005Nystrom; @yen2014sparse], which implies that the rank $R$ may need to be near-linear to the number of data points in order to achieve comparable generalization error compared to the vanilla kernel method. For large-scale problems, the low-rank approximation could become almost as expensive as the exact kernel method to maintain competitive performance [@Stein2014BlockDiag]. Another popular approach for scaling up kernel method is random features approximation [@rahimi2007random; @rahimi2008RKS]. Unlike previous approach that approximates kernel matrix, Random Features approximate the kernel function directly via sampling from an explicit feature map. Random Fourier (RF) is one of the feature maps that attracted considerable interests due to its easy implementation and fast execution time [@rahimi2008RKS; @Le14Fastfood; @Yang15LearnFastKernel; @Dai2014ScalableKernel; @bengio2009binary], which has total computational cost and storage requirement as $O(NRd + NRm)$ and $O(NR)$ respectively, for computing feature matrix $Z$ and operating the subsequent algorithms on $Z$. A *Fastfood* approach and its extension [@Le14Fastfood; @Yang15LearnFastKernel] was proposed to reduce the time of computing Fourier features from $O(Rd)$ to $O(R\log{d})$ by leveraging Hadamard basis functions, which improves the efficiency for prediction but not necessarily for training if $d\ll m$. Although RF has been successfully applied to speech recognition and vision classifications on very large datasets [@JCLW2016ICCASP; @Huang2014Kernel; @Lu14ScaleKernel], a drawback is that a significant large number of random features are needed to achieve a comparable performance to exact kernel method. This is not surprising since the convergence of approximation error is in the order $O(1/\sqrt{R} + 1/\sqrt{N})$ [@rahimi2008RKS; @Dai2014ScalableKernel], which is the same as that of low-rank kernel approximations. Mercer’s theorem \[19\] guarantees that any positive-definite kernel permits a feature-map decomposition. However, the decomposition is not unique. One may find different feature maps to construct the same kernel function [@rahimi2007random; @yen2014sparse]. Therefore, we ask following question: do some of the feature maps lead to faster convergence than the others in terms of approximation? In this paper, we address this question by reconsidering the Random Binning (RB) feature map, which was proposed in the first Random-Feature paper [@rahimi2007random] but has drawn much less attentions since then compared to the RF feature. Our main contributions are fourfold. First, we propose the first analysis of RB from the perspective of optimization. By interpreting RB as a *Randomized Block Coordinate Descent* (RBCD) in the infinite-dimensional space induced from the kernel, we prove that RB enjoys faster convergence than other random features. Specifically, by drawing $R$ grids with expected number of non-empty bins per grid lower bounded by $\kappa$, RB can achieve a solution comparable to exact kernel method with $O(1/(\kappa R))$ precision in terms of the objective function, which is not only better than the existing $O(1/\sqrt{R})$ rate from Monte Carlo analysis [@rahimi2007random], but also shows a $\kappa$ times speedup over the rate of other random features under the same analysis framework [@yen2014sparse]. Second, we exploit the sparse structure of the feature matrix $Z$, which is the key to rapidly transform the data features into a very high-dimension feature space that is linearly separately by any regressors and classifiers. In addition, we discuss how to efficiently perform the computation for a large, sparse matrix by using state-of-the-art iterative solvers and advanced matrix storage techniques. As a result, the computational complexity and storage requirements in training are still $O(NRd + NRm)$ and $O(NR)$, respectively. Third, we show that Random Binning features is particularly suitable for *Parallel Coordinate Descent* solver. Unlike other random features, RB guarantees a speedup proportional to $\kappa$ due to a sparse feature matrix. This is particularly useful in the *Sparse Random Feature* setting [@yen2014sparse], where L1 regularization is used to induce a compact nonlinear predictor and *Coordinate Descent* is presumably the state-of-the-art solver in such setting. Finally, we provide extensive experiments to demonstrate the faster convergence and better parallelizability of RB in practice. Compared to other popular low-rank approximations, RB shows superior performance on both regression and classification tasks under the same computational budgets, and achieves same performance with one to three orders of magnitude reduction in time and memory consumptions. When combined with Coordinate Descent to solve an L1-regularized objective, RB shows an almost linear speedup, in contrast to RF that has almost no speedup. Random Binning Feature as Kernel Approximation {#section:kernel_approximation} ============================================== In this work, we consider the problem of fitting a nonlinear prediction function $f:{\mathcal{X}}\rightarrow {\mathcal{Y}}$ in Reproducing Kernel Hilbert Space ${\mathcal{H}}$ from training data pairs $\{({\boldsymbol{x}}_n,y_n)\}_{n=1}^N$ via regularized Empirical Risk Minimization (ERM) $$\label{problem} \begin{aligned} &f^* = \underset{f\in{\mathcal{H}}}{argmin} &\frac{\lambda}{2} \|f\|_{{\mathcal{H}}}^2 + \frac{1}{N} \sum_{n=1}^N L(f({\boldsymbol{x}}_n),y_n), \end{aligned}$$ where $L(z,y)$ is a convex loss function with Lipschitz-continuous derivative satisfying $|L'(z_1,y)-L'(z_2,y)|\leq \beta |z_1-z_2|$, which includes several standard loss functions such as the *square-loss* $L(z,y)=\frac{1}{2}(z-y)^2$, *square-hinge loss* $L(z,y)=\max(1-zy,0)^2$ and *logistic loss* $L(z,y)=\log(1+\exp(-yz))$. Learning with RKHS ------------------ The RKHS ${\mathcal{H}}$ can be defined via a positive-definite (PD) kernel function $k({\boldsymbol{x}}_1,{\boldsymbol{x}}_2)$ that measures similarity between samples as $$\label{kernel_H} {\mathcal{H}}= \left\{ f(\cdot)=\sum_{i=1}^K \alpha_i k({\boldsymbol{x}}_i,\cdot) \textit{ }|\textit{ } \alpha_i\in{\mathbb{R}}, {\boldsymbol{x}}_i\in{\mathcal{X}}\right\}.$$ One can also define the RKHS via a possibly infinite-dimensional feature map $\{{\bar{\phi}}_h({\boldsymbol{x}})\}_{h\in H}$ with each $h\in H$ defining a feature function ${\bar{\phi}}_h({\boldsymbol{x}}):{\mathcal{X}}\rightarrow {\mathbb{R}}$. The space can be expressed as $$\label{feature_map_H} {\mathcal{H}}= \left\{ f(\cdot)=\int_{h\in H} w(h){\bar{\phi}}_h(\cdot) dh = \langle {\boldsymbol{w}},{\bar{{\boldsymbol{\phi}}}}(\cdot) \rangle_{{\mathcal{H}}} \textit{ }|\textit{ } \|f\|_{{\mathcal{H}}}^2 < \infty \right\},$$ where $w(h)$ specifies weights over the set of features $\{\phi_h({\boldsymbol{x}})\}_{h\in{\mathcal{H}}}$. The Mercer’s theorem [@mercer1909functions] connects the above two formulations of RKHS by stating that every PD kernel $k(.,.)$ can be expressed as an integration over some basis functions $\{\phi_h(.)\}_{h\in H}$ $$\label{feature_map} k({\boldsymbol{x}}_1,{\boldsymbol{x}}_2)=\int_{h\in H} p(h) \phi_h({\boldsymbol{x}}_1) \phi_h({\boldsymbol{x}}_2) dh = \langle {\bar{{\boldsymbol{\phi}}}}({\boldsymbol{x}}_1), {\bar{{\boldsymbol{\phi}}}}({\boldsymbol{x}}_2)\rangle_{{\mathcal{H}}},$$ However, the decomposition is not unique, so one can find different feature maps $\{{\bar{\phi}}_h(.)\}_{h\in H}$ satisfying for the same kernel $k(.,.)$. In particular, as an example used extensively in this work, the Laplacian Kernel $$\label{laplacian_kernel} k({\boldsymbol{x}}_1,{\boldsymbol{x}}_2)=\exp\left(-\frac{\|{\boldsymbol{x}}_1-{\boldsymbol{x}}_2\|_1}{\sigma}\right),$$ allows decomposition based on (i) Fourier basis map [@rahimi2007random], (ii) RB map [@rahimi2007random], and also (iii) map based on infinite number of decision trees [@lin2008support] to name a few. On the other hand, different kernels can be constructed using the same set of basis function $\{\phi_h(.)\}$ with different distribution $p(h)$. For example, the RB feature map can be used to construct any shift-invariant kernel of the form [@rahimi2007random] $$\label{shift_inv_kernel} K({\boldsymbol{x}}_1,{\boldsymbol{x}}_2) = K({\boldsymbol{x}}_1-{\boldsymbol{x}}_2)= \prod_{j=1}^d k_j(x_{1j}-x_{2j}),$$ by sampling the “width” of bins $\delta_j$ for each feature $j$ from a distribution proportional to $\delta k_j''(\delta)$, where $k_j''(\delta)$ is the second derivative of $k_j(\delta)$, assuming the kernel has a non-negative second derivative. Random Binning Features ----------------------- ![Generating process of RB features.[]{data-label="fig:RB_gen"}](Graphs/RB.png) In this section, we describe the Random Binning (RB) feature map, which has decomposition of the form $$\label{RB_feature_map} K({\boldsymbol{x}}_1,{\boldsymbol{x}}_2)=\int_{{\boldsymbol{\delta}}} p({\boldsymbol{\delta}}) {\boldsymbol{\phi}}_{B_{{\boldsymbol{\delta}}}}({\boldsymbol{x}}_1)^T{\boldsymbol{\phi}}_{B_{{\boldsymbol{\delta}}}}({\boldsymbol{x}}_2) \;d{\boldsymbol{\delta}}$$ where $B_{{\boldsymbol{\delta}}}$ is a grid parameterized by ${\boldsymbol{\delta}}=(\delta_1,u_1,...,\delta_d,u_d)$ that specifies the *width* and *bias* of the grid w.r.t. the $d$ dimensions, and ${\boldsymbol{\phi}}_{B_{{\boldsymbol{\delta}}}}({\boldsymbol{x}})$ is a vector which has $$\phi_{b}({\boldsymbol{x}})=1, \;\textit{if}\; b=(\lfloor \frac{x_{1}-u_1}{\delta_1}\rfloor, ..., \lfloor \frac{x_{d}-u_d}{\delta_d}\rfloor),$$ and $\phi_{b}({\boldsymbol{x}})=0$ otherwise for any $b\in B_{{\boldsymbol{\delta}}}$. Note for each grid $B_{{\boldsymbol{\delta}}}$, the number of bins $|B_{{\boldsymbol{\delta}}}|$ is countably infinite, so ${\boldsymbol{\phi}}_{B_{{\boldsymbol{\delta}}}}({\boldsymbol{x}})$ has infinite dimension but only $1$ non-zero entry (at the bin ${\boldsymbol{x}}$ lies in). Figure \[fig:RB\_gen\] illustrates an example when the raw dimension $d=2$. The kernel $K({\boldsymbol{x}}_1,{\boldsymbol{x}}_2)$ is thus interpreted as the *collision probability* that two data points ${\boldsymbol{x}}_1$, ${\boldsymbol{x}}_2$ fall in the same bin, when the grid is generated from distribution $p({\boldsymbol{\delta}})$. In [@rahimi2007random], it is pointed out for any kernel of form with *nonnegative second derivative* $k_j''(\delta)$, one can derive distribution $p({\boldsymbol{\delta}})=\prod_{j=1}^d p_j(\delta_j)U(u_j;0,\delta_j)$, where $p_j(\delta_j)\propto \delta k_j''(\delta_j)$ and $U(\cdot,a,b)$ is uniform distribution in the range $[a,b]$. To obtain a kernel approximation scheme from the feature map , a simple Monte Carlo method can be used to approximate by averaging over $R$ grids $\{B_{{\boldsymbol{\delta}}_r}\}_{r=1}^R$ with each grid’s parameter ${\boldsymbol{\delta}}_r$ drawn from $p({\boldsymbol{\delta}})$. The procedure for generating $R$ RB features from raw data $\{{\boldsymbol{x}}_n\}_{n=1}^N$ is given in Algorithm \[alg:RB\]. Using a Monte-Carlo analysis, one can show the approximation to yields approximation error of order $O(1/\sqrt{R})$. From the Representer theorem, one can further bound error of the learned predictor $$\left|{\boldsymbol{w}}_{RF}^T{\boldsymbol{z}}({\boldsymbol{x}})-f^*({\boldsymbol{x}})\right|=\left|\sum_{n=1}^N \alpha^{RF}_n{\boldsymbol{z}}({\boldsymbol{x}}_n)^T{\boldsymbol{z}}({\boldsymbol{x}}) - \sum_{n=1}^N \alpha_n^* k({\boldsymbol{x}}_n,{\boldsymbol{x}})\right|$$ as shown in [@rahimi2007random] (appendix C). Unfortunately, the rate of convergence suggests that to achieve small approximation error ${\epsilon}$, one needs significant amount of random features proportional to $\Omega(1/\epsilon^2)$, and furthermore, the Monte-Carlo analysis does not explain why empirically RB feature achieves faster convergence than other random feature map like Fourier basis by orders of magnitude. Given a kernel function $k({\boldsymbol{x}}_1,{\boldsymbol{x}}_2)=\prod_{j=1}^d k_j(|x_{1j}-x_{2j}|)$. Let $p_j(\delta) \propto \delta k_j''(\delta)$ be a distribution over $\delta$. 1. Draw $\delta_{rj} \sim p_j(\delta)$, $\forall j\in [d]$. $u_{rj}\in [0,\delta_{rj}],\forall j\in[d]$. 2. Compute feature ${\boldsymbol{z}}_{r}({\boldsymbol{x}}_n)$ as the the indicator vector of bin index $(\lfloor \frac{x_{n1}-u_1}{\delta_1}\rfloor, ..., \lfloor \frac{x_{nd}-u_d}{\delta_d}\rfloor )$, for $\forall n\in[N]$. . Return ${\boldsymbol{z}}({\boldsymbol{x}}_n)=\frac{1}{\sqrt{D}}[{\boldsymbol{z}}_1({\boldsymbol{x}}_n);...;{\boldsymbol{z}}_D({\boldsymbol{x}}_n)]$ $\forall n\in[N]$ as the data with RB Features. Faster Convergence of Random Binning {#section:faster_convergence} ==================================== In this section, we first illustrate the sparse structure of the feature matrix $Z$ of RB and discuss how to make efficient computation and storage format of $Z$. Then by interpreting RB features as Randomized Block Coordinate Descent in the infinite-dimensional space, we prove that RB has a faster convergence rate than other random features. We illustrate them accordingly in the following sections. Sparse Feature Matrix & Iterative Solvers {#section:iterative} ----------------------------------------- A special characteristic of RB compared to other low-rank approximations is the fact that the feature matrix generated by RB is typically a large, sparse binary matrix $Z \in {\mathbb{R}}^{N \times D}$, where the value of $D$ is determined by both number of grids $R$ and the kernel width parameter (ex. $\sigma$ in the case of Laplacian Kernel). Different from other random features, $D$, rather than $R$, is the actual number of columns of $Z$. A direct connection between $D$ and $R$ is that the matrix has each row $i$ satisfying $nnz(Z(i,:))=R$ and therefore $R \leq D \leq NR$. Intuitively speaking, RB has more expressive power than RF since it generates a large yet sparse feature matrix to rapidly transform the data space to a very high dimension space, where data could become almost linearly separable by the classifiers. Fig. \[fig:RB\_matrix\] gives an example to illustrate the sparse structure of $Z$. In the case of *Kernel Ridge Regression* (L2-regularization with square loss), if using RB feature to approximate the RKHS, one can solve directly in its primal form. The weighting vector is simply the solution of the linear system: $$(Z^TZ + \lambda I) {\boldsymbol{w}}_{RB} = Z^T y.$$ Note since $Z$ is a large sparse matrix, there is no need to explicitly compute the covariance matrix $Z^TZ$, which is much denser than $Z$ itself. One can apply state-of-the-art sparse iterative solvers such as Conjugate Gradient (CG) and GMRES to directly operate on $Z$ [@Saad2003SparseLS]. The main computation in CG or GMRES is the sparse matrix-vector products. Let $m$ be the number of iterations, then the total computational complexity of iterative solver is $O(m\, nnz(Z))=O(m NR)$. In addition, since most elements in $Z$ are zeros, the Compressed Sparse Row type matrix storage format should be employed for economically storing $Z$ [@Golub1996MatrixComp], which gives computational cost and memory requirement as $O(mNR)$ and $O(NR)$ respectively, a similar cost to that of other low-rank approximations despite its much higher dimension. In testing phase, each point produces a sparse feature vector $z(x) \in \mathcal{R}^D$ based on the grids stored during training, yielding a sparse vector $z(x)$ with $nnz(z(x)))=R$ and computing the decision function $z(x)^T{\boldsymbol{w}}_{RB}$ only requires $O(dR + R)$. When the ERM is smooth but not quadratic, a *Newton-CG* method that solves smooth problem via a series of local quadratic approximation gives the same complexity per CG iteration [@lin2008trust], and note that most of state-of-the-art linear classification algorithms have complexity linear to $nnz(Z)$, the number of nonzeros of feature matrix [@fan2008liblinear]. In section \[section:parallel\], we further discuss cases of L1-regularized problem, where a Coordinate Descent algorithm of cost $O(nnz(Z))$ per iteration is discussed. Random Binning Features as Block Coordinate Descent --------------------------------------------------- In [@yen2014sparse], a new approach of analysis was proposed, which interpreted Random Features as Randomized Coordinate Descent in the infinite dimensional space, and gives a better $O(1/R)$ rate in the convergence of objective function. In this section, we extend the approach of [@yen2014sparse] to show that, RB Feature can be interpreted as RBCD in the infinite-dimensional space, which by drawing *a block of features* at a time, produces a number of features $D$ significantly more than the number of blocks $R$, resulting a provably faster convergence rate than other RF. While at the same time, by exploiting state-of-the-art iterative solvers introduced in section \[section:iterative\], the computational complexity of RB does not increase with number of features $D$ but only with the number of blocks $R$. Consequently, to achieve the same accuracy, RB requires significantly less training and prediction time compared to other RF. A key quantity to our analysis is an upper bound on the *collision probability* $\nu_{{\boldsymbol{\delta}}}$ which specifies how unlikely data points will fall into the same bin, and its inverse $\kappa_{{\boldsymbol{\delta}}}:=1/\nu_{{\boldsymbol{\delta}}}$ which lower bounds the number of bins containing at least one data point. We define them as follows. \[def:collision\_prob\] Define collision probability of data ${\mathcal{D}}$ on bin $b\in B_{{\boldsymbol{\delta}}}$ as $$\label{collision_prob} \nu_{b}:=\frac{ |\{n\in[N] \;|\; \phi_{b}({\boldsymbol{x}}_n)=1 \}| }{N}.$$ Let $\nu_{{\boldsymbol{\delta}}} := \max_{b\in B_{{\boldsymbol{\delta}}}} \nu_b$ be an upper bound on , and $\kappa_{{\boldsymbol{\delta}}}:=1/\nu_{{\boldsymbol{\delta}}}$ be a lower bound on the number of nonempty bins of grid ${\boldsymbol{\delta}}$. $$\label{expected_col_prob} \kappa:= E_{{\boldsymbol{\delta}}}[\kappa_{{\boldsymbol{\delta}}}]=E_{{\boldsymbol{\delta}}}[1/\nu_{{\boldsymbol{\delta}}}]$$ is denoted as the lower bound on the expected number of (used) bins w.r.t the distribution $p({\boldsymbol{\delta}})$. In the RB matrix, the empirical collision probability is simply the average number of non-zeros per column, divided by $N$, a number much smaller than $1$ as in the example of Fig. \[fig:RB\_matrix\]. Our analysis assumes a smooth loss function satisfying the following criteria. \[as:smooth\] The loss function $L(z,y)$ is smooth w.r.t. response $z$ so difference between function difference and its linear approximation can be bounded as $$L(z_2,.)-L(z_1,.) \leq \nabla L(z_1,.)(z_2-z_1) + \frac{\beta}{2}(z_2-z_2)^2.$$ for some constant $0\leq\beta\leq \infty$. This assumption is satisfied for a wide range of loss such as square loss $(\beta=1)$, logistic loss $(\beta=1/4)$ and L2-hinge loss ($\beta=1$). We interpret RB as a *Fully Corrective Randomized Block Coordinate Descent (FC-RBCD)* on the objective function $$\label{optimized_obj} \begin{aligned} &\min_{{\bar{\boldsymbol{w}}}} && F({\bar{\boldsymbol{w}}}):={\mathcal{R}}({\bar{\boldsymbol{w}}}) + Loss({\bar{\boldsymbol{w}}};{\boldsymbol{\phi}})\\ \end{aligned}$$ where $Loss({\bar{\boldsymbol{w}}};\phi)=\frac{1}{N}\sum_{n=1}^N L(\langle {\bar{\boldsymbol{w}}}, {\boldsymbol{\phi}}({\boldsymbol{x}}_n) \rangle, y_n)$ and $${\bar{{\boldsymbol{\phi}}}}:=\sqrt{p}\circ {\boldsymbol{\phi}}=(\sqrt{p({\boldsymbol{\delta}})}{\boldsymbol{\phi}}_{B_{{\boldsymbol{\delta}}}}(.))_{{\boldsymbol{\delta}}\in H}$$ with “$\circ$” denoting the component-wise product. The goal is to show that, by performing $R$ steps of FC-RBCD on , one can obtain a ${\bar{\boldsymbol{w}}^{R}}$ with comparable regularized loss to that from optimal solution of . Note one advantage of analysis from this optimization perspective is: it does not rely on Representer theorem, and thus $R({\boldsymbol{w}})$ can be L2 regularizer $\frac{\lambda}{2}\|{\boldsymbol{w}}\|^2$ or L1 regularizer $\lambda\|{\boldsymbol{w}}\|_1$, where the latter has advantage of giving sparse predictor of faster prediction [@yen2014sparse]. The FC-RBCD algorithm maintains an active set of blocks ${\mathcal{A}}^{(r)}$ which is expanded for $R$ iterations. At each iteration $r$, the FC-RBCD does the following: - Draw ${\boldsymbol{\delta}}$ from $p({\boldsymbol{\delta}})$ ( derived from the kernel $k(.,.)$ ). - Expand active set ${\mathcal{A}}^{(r+1)}:= {\mathcal{A}}^{(r)} \cup B_{{\boldsymbol{\delta}}}$. - Minimize subject to a limited support $supp({\bar{\boldsymbol{w}}})\subseteq {\mathcal{A}}^{(r+1)}$. Note this algorithm is only used for analysis. In practice, one can draw $R$ blocks of features at a time, and solve by any optimization algorithm such as those mentioned in section \[section:iterative\] or the CD method we introduce in section \[section:parallel\]. Due to space limit, here we prove the case when ${\mathcal{R}}(.)$ is the non-smooth L1 regularizer $\lambda\|{\bar{\boldsymbol{w}}}\|_1$. The smooth case for ${\mathcal{R}}({\boldsymbol{w}})=\frac{\lambda}{2}\|{\bar{\boldsymbol{w}}}\|^2$ can be shown in a similar way. Note the objective function can be written as $$\label{surrogate} \bar{F}({\boldsymbol{w}}):=F(\sqrt{{\boldsymbol{p}}}\circ{\boldsymbol{w}})={\mathcal{R}}(\sqrt{{\boldsymbol{p}}}\circ {\boldsymbol{w}}) + Loss({\boldsymbol{w}},\bar{{\boldsymbol{\phi}}}).$$ by a scaling of variable ${\bar{\boldsymbol{w}}}=\sqrt{p}\circ{\boldsymbol{w}}$. The below theorem states that, running FC-RBCD for $R$ iterations, it generates a solution ${\bar{\boldsymbol{w}}}^{R}$ close to any reference solution ${\boldsymbol{w}}^*$ in terms of objective with their difference bounded by $O(\frac{1}{\kappa R})$. \[thm:RBconverge\] Let $R$ be the number of blocks (grids) generated by FC-RBCD, and ${\boldsymbol{w}}^*$ be any reference solution, we have $$\label{RBconverge} E[\bar F({\boldsymbol{w}}^{(R)})] - \bar F({\boldsymbol{w}}^*) \leq \frac{\beta\|{\boldsymbol{w}}^*\|^2}{\kappa R'}$$ for $R':=R-c>0$, where $c=\lceil\frac{2\kappa(\bar F({\boldsymbol{0}})-\bar F({\boldsymbol{w}}^*))}{\beta\|{\boldsymbol{w}}^*\|^2}\rceil$. Firstly, we obtain an expression for the progress made by each iteration of FC-RBCD. Let $B:=B_{{\boldsymbol{\delta}}^{(r)}}$ be the block drawn at step 1 of FC-RBCD, and ${\bar{\boldsymbol{w}}}^{(r+1)}$ be the minimizer of subject to support $supp({\bar{\boldsymbol{w}}})\subseteq {\mathcal{A}}^{(r+1)}$ given by the step 3. Since $B \subseteq{\mathcal{A}}^{(r+1)}$, we have $$\label{FC} F({\bar{\boldsymbol{w}}}^{(r+1)})-F({\bar{\boldsymbol{w}}}^{(r)}) \leq F({\bar{\boldsymbol{w}}}^{(r)}+{\boldsymbol{\eta}}_{B})-F({\bar{\boldsymbol{w}}}^{(r)})$$ for any ${\boldsymbol{\eta}}_{B}:supp({\boldsymbol{\eta}})\subseteq B$. Then denote $b_i$ as the bin ${\boldsymbol{x}}_i$ falling in and $L'_i = \nabla L({\bar{\boldsymbol{w}}}^{(r)T}{\boldsymbol{\phi}}({\boldsymbol{x}}_i),y_i)$, by smoothness of the loss (Assumption \[as:smooth\]), we have $$\label{loss_upper_bound} \begin{aligned} &Loss({\bar{\boldsymbol{w}}}^{(r)}+{\boldsymbol{\eta}}_{B}) - Loss({\bar{\boldsymbol{w}}}^{(r)}) \leq \frac{1}{N}\sum_{i=1}^N L'_i \phi_{b_i} \eta +\frac{\beta}{2}(\eta_{b_i}\phi_{b_i})^2 \\ &\leq \langle {\boldsymbol{g}}_{B},{\boldsymbol{\eta}}_{B}\rangle + \frac{\beta\nu_{{\boldsymbol{\delta}}^{(r)}} }{2}\|{\boldsymbol{\eta}}_{B}\|^2 \end{aligned}$$ where the second inequality uses the fact $\phi_{b_i}=1$ and $${\boldsymbol{g}}_{B}:=\nabla_{B} Loss({\bar{\boldsymbol{w}}}^{(r)},{\boldsymbol{\phi}}).$$ Now consider the regularization term, note since block $B$ is drawn from an inifinite-dimensional space, the probability that $B$ is in active set is $0$. Therefore, we have $B\cap {\mathcal{A}}^{(r)}=\emptyset$, ${\bar{\boldsymbol{w}}}^{(r)}_{B}={\boldsymbol{0}}$ and ${\mathcal{R}}_{B}({\bar{\boldsymbol{w}}}^{(r)}_{B})=0$. As a result, $$\label{tmp1} \begin{aligned} &F({\bar{\boldsymbol{w}}}^{(r)}+{\boldsymbol{\eta}}_{B})-F({\bar{\boldsymbol{w}}}^{(r)}) \\ &\leq {\mathcal{R}}_{B}({\boldsymbol{\eta}}_{B})+\langle {\boldsymbol{g}}_{B},{\boldsymbol{\eta}}_{B}\rangle + \frac{\beta\nu_{{\boldsymbol{\delta}}^{(r)}} }{2}\|{\boldsymbol{\eta}}_{B}\|^2 \end{aligned}$$ Let ${\boldsymbol{\eta}}_{B}$ be the minimizer of RHS of . It satisfies ${\boldsymbol{\rho}}_{B}+{\boldsymbol{g}}_{B}+\beta v_{{\boldsymbol{\delta}}^{(r)}} {\boldsymbol{\eta}}_{B}={\boldsymbol{0}}$ for some ${\boldsymbol{\rho}}_{B}\in \partial{\mathcal{R}}({\boldsymbol{\eta}}_B)$, and thus, $$\label{tmp4} \begin{aligned} &F({\bar{\boldsymbol{w}}}^{(r)}+{\boldsymbol{\eta}}_{B})-F({\bar{\boldsymbol{w}}}^{(r)}) \\ &\leq \langle{\boldsymbol{\rho}}_B,{\boldsymbol{\eta}}_B\rangle + \langle {\boldsymbol{g}}_{B},{\boldsymbol{\eta}}_{B}\rangle + \frac{\beta\nu_{{\boldsymbol{\delta}}^{(r)}} }{2}\|{\boldsymbol{\eta}}_{B}\|^2\\ &= -\frac{1}{2\beta\nu_{{\boldsymbol{\delta}}^{(r)}} }\|{\boldsymbol{\rho}}_B+{\boldsymbol{g}}_B\|^2 \end{aligned}$$ Now taking expectation w.r.t. $p({\boldsymbol{\delta}})$ on both sides of , we have $$\label{tmp5} \begin{aligned} E[F({\bar{\boldsymbol{w}}}^{(r)}+{\boldsymbol{\eta}}_{B})]-F({\bar{\boldsymbol{w}}}^{(r)}) &\leq -\frac{1}{2\beta} E\left[\frac{1}{\nu_{{\boldsymbol{\delta}}^{(r)}}}\|{\boldsymbol{\rho}}_B+{\boldsymbol{g}}_B\|^2 \right] \\ &\leq -\frac{1}{2\beta} E\left[\frac{1}{\nu_{{\boldsymbol{\delta}}^{(r)}}}\right]E\left[\|{\boldsymbol{\rho}}_B+{\boldsymbol{g}}_B\|^2 \right]\\ &\leq -\frac{\kappa}{2\beta} \|{\bar{{\boldsymbol{\rho}}}}_B+{\bar{\boldsymbol{g}}}_B\|^2 \end{aligned}$$ where ${\bar{{\boldsymbol{\rho}}}}:=\sqrt{{\boldsymbol{p}}}\circ{\boldsymbol{\rho}}$, $\;{\bar{\boldsymbol{g}}}:=\sqrt{{\boldsymbol{p}}} \circ {\boldsymbol{g}}$, and the second inequality uses the fact that the number of used bins $\kappa_{{\boldsymbol{\delta}}^{(r)}}=1/\nu_{{\boldsymbol{\delta}}^{(r)}}$ has non-negative correlation with the discriminative power of block $B$ measured by the magnitude of gradient with soft-thresholding $\|{\bar{{\boldsymbol{\rho}}}}_B+{\bar{\boldsymbol{g}}}_B\|$ (i.e. fewer collisions on grid $B$ implies $B$ to be a better block of features ). The result of expresses descent amount in terms of the proximal gradient of the reparameterized objective . Note for $B:B\cap{\mathcal{A}}^{(r)}=\emptyset$, we have ${\boldsymbol{w}}_B^{(r)}={\boldsymbol{0}}$, and ${\mathcal{R}}_B({\bar{\boldsymbol{\eta}}})-{\mathcal{R}}_B({\boldsymbol{0}})= \langle{\bar{{\boldsymbol{\rho}}}}, {\bar{\boldsymbol{\eta}}}\rangle$; on the other hand, for $B\subseteq {\mathcal{A}}^{(r)}$, we have $${\boldsymbol{0}}\in arg\min_{{\bar{\boldsymbol{\eta}}}_{B}}\;{\mathcal{R}}_{B}(\sqrt{p_B}{\boldsymbol{w}}_B+{\bar{\boldsymbol{\eta}}}_B)+\langle {\bar{\boldsymbol{g}}}_B, \sqrt{p_B}{\boldsymbol{w}}_{B}+{\bar{\boldsymbol{\eta}}}_B\rangle$$ since they are solved to optimality in the previous iteration. Then $$\label{tmp2} \begin{aligned} &E[F({\bar{\boldsymbol{w}}}^{(r)}+{\boldsymbol{\eta}})]-F({\bar{\boldsymbol{w}}}^{(r)}) \\ &\leq -\frac{\kappa}{2\beta} \|{\bar{{\boldsymbol{\rho}}}}_B+{\bar{\boldsymbol{g}}}_B\|^2 = \langle{\bar{{\boldsymbol{\rho}}}},{\bar{\boldsymbol{\eta}}}\rangle + \langle {\bar{\boldsymbol{g}}},{\bar{\boldsymbol{\eta}}}\rangle + \frac{\beta}{2\kappa}\|{\bar{\boldsymbol{\eta}}}_{\bar{{\mathcal{A}}}^{(r)}}\|^2\\ &= {\mathcal{R}}(\sqrt{{\boldsymbol{p}}} \circ ({\boldsymbol{w}}^{(r)}+{\bar{\boldsymbol{\eta}}}) )-{\mathcal{R}}(\sqrt{{\boldsymbol{p}}}\circ {\boldsymbol{w}}^{(r)}) + \langle {\bar{\boldsymbol{g}}},{\bar{\boldsymbol{\eta}}}\rangle + \frac{\beta}{2\kappa}\|{\bar{\boldsymbol{\eta}}}_{\bar{{\mathcal{A}}}^{(r)}}\|^2 \end{aligned}$$ where ${\bar{\boldsymbol{\eta}}}_{\bar{{\mathcal{A}}}^{(r)}}:=({\bar{\boldsymbol{\eta}}}_B)_{B:B\cap{\mathcal{A}}^{(r)}=\emptyset }$ and ${\bar{\boldsymbol{\eta}}}:=\sqrt{{\boldsymbol{p}}}\circ{\boldsymbol{\eta}}$. Thus the final step is to show the descent amount given by RHS of decreases the suboptimality $\bar{F}({\boldsymbol{w}}^{(r)})-\bar{F}({\boldsymbol{w}}^*)$ significantly. This can be achieved by considering ${\bar{\boldsymbol{\eta}}}$ of the form $\alpha({\boldsymbol{w}}^{*}-{\boldsymbol{w}}^{(r)})$ for some $\alpha\in[0,1]$ as follows: $$\label{tmp3} \begin{aligned} &E[\bar{F}({\boldsymbol{w}}^{(r)}+{\bar{\boldsymbol{\eta}}})]-\bar{F}({\boldsymbol{w}}^{(r)}) \\ &\leq \min_{{\bar{\boldsymbol{\eta}}}} {\mathcal{R}}(\sqrt{{\boldsymbol{p}}} \circ ({\boldsymbol{w}}^{(r)}+{\bar{\boldsymbol{\eta}}}) )-{\mathcal{R}}(\sqrt{{\boldsymbol{p}}}\circ {\boldsymbol{w}}^{(r)})+\langle {\bar{\boldsymbol{g}}},{\bar{\boldsymbol{\eta}}}\rangle + \frac{\beta }{2\kappa}\|{\bar{\boldsymbol{\eta}}}_{\bar{{\mathcal{A}}}^{(r)}}\|^2 \\ &\leq \min_{{\bar{\boldsymbol{\eta}}}} \bar{F}({\boldsymbol{w}}^{(r)}+{\bar{\boldsymbol{\eta}}})-\bar{F}({\boldsymbol{w}}^{(r)}) + \frac{\beta }{2\kappa}\|{\bar{\boldsymbol{\eta}}}_{\bar{{\mathcal{A}}}^{(r)}}\|^2 \\ &\leq \min_{\alpha\in[0,1]} \bar{F}((1-\alpha){\boldsymbol{w}}^{(r)}+\alpha{\boldsymbol{w}}^*)-\bar{F}({\boldsymbol{w}}^{(r)}) + \frac{\beta \alpha^2}{2\kappa}\|{\boldsymbol{w}}^*\|^2\\ &\leq \min_{\alpha\in[0,1]} -\alpha(\bar{F}({\boldsymbol{w}}^{(r)})-\bar{F}({\boldsymbol{w}}^*)) + \frac{\beta \alpha^2}{2\kappa}\|{\boldsymbol{w}}^*\|^2, \end{aligned}$$ where the second and fourth inequalities are from convexity of $\bar{F}(.)$. The $\alpha$ minimizing is $\alpha^*:=\min(\frac{\kappa(\bar{F}({\boldsymbol{w}}^{(r)})-\bar{F}({\boldsymbol{w}}^*))}{\beta\|{\boldsymbol{w}}^*\|^2},1)$, which leads to $$\label{tmp6} \begin{aligned} &E[\bar{F}({\boldsymbol{w}}^{(r)}+{\bar{\boldsymbol{\eta}}})]-\bar{F}({\boldsymbol{w}}^{(r)}) \leq -\frac{\kappa(\bar{F}({\boldsymbol{w}}^{(r)})-\bar{F}({\boldsymbol{w}}^*))^2}{2\beta\|{\boldsymbol{w}}^*\|^2} \end{aligned}$$ if $\bar{F}({\boldsymbol{w}}^{(r)})-\bar{F}({\boldsymbol{w}}^*)\leq \frac{\beta}{\kappa}\|{\boldsymbol{w}}^*\|^2$; otherwise, we have $E[\bar{F}({\boldsymbol{w}}^{(r)}+{\bar{\boldsymbol{\eta}}})]-\bar{F}({\boldsymbol{w}}^{(r)})\leq -\frac{\beta}{2\kappa}\|{\boldsymbol{w}}^*\|^2$. Note the latter case cannot happen more than $c=\lceil\frac{2\kappa(\bar F({\boldsymbol{0}})-\bar F({\boldsymbol{w}}^*))}{\beta\|{\boldsymbol{w}}^*\|^2}\rceil$ times since FC-RBCD is a descent method. Therefore, for $r':=r-c>0$, solving the recursion leads to the conclusion. Note we have $\|\sqrt{{\boldsymbol{p}}}\circ {\boldsymbol{w}}^*\|_1$ $\leq$ $\|\sqrt{{\boldsymbol{p}}}\|\|{\boldsymbol{w}}^*\|$ $=\|{\boldsymbol{w}}^*\|$ in the L1-regularized case, and thus the FC-RBCD guarantees convergence of the L1-norm objective to the (non-square) L2-norm objective. The convergence result of Theorem \[thm:RBconverge\] is of the same form to the rate proved in [@yen2014sparse] for other random features, however, with an additional multiplicative factor $\kappa\geq 1$ that speeds up the rate by $\kappa$ times. Recall that $\kappa$ is the lower bound on the expected number of bins being used by data samples for each block of features $B_{{\boldsymbol{\delta}}}$, which in practice is a factor much larger than $1$, as shown in the Figure \[fig:RB\_matrix\] and also in our experiments. In particular, in case each grid $B_{{\boldsymbol{\delta}}}$ has similar number of bins being used, we have $D\approx \kappa R$, and thus obtain a rate of the form $$\label{D_rate} E[\bar F({\boldsymbol{w}}^{(R)})] - \bar F({\boldsymbol{w}}^*) \lesssim \frac{\beta\|{\boldsymbol{w}}^*\|^2}{D}.$$ Note for a fixed $R$, the total number of features $D$ is increasing with kernel parameter $1/\sigma$ in the case of Laplacian Kernel, which means the less smooth the kernel, the faster convergence of RB. A simple extreme case is when $\sigma\rightarrow 0$, where one achieves $0$ training loss, and the RB, by putting each sample in a separate bin, converges to $0$ loss with $R=1$, $D=N$. On the other hand, other random features, such as Fourier, still require large $R$ for convergence to $0$ loss. In practice, there are many data that require a small kernel bandwidth $\sigma$ to avoid underfitting, for which RB has dramatically faster convergence than other RF. Strong Parallelizability of Random Binning Features {#section:parallel} =================================================== \[section:parallel\] In this section, we study another strength of RB Features in the context of *Sparse Random Feature* [@yen2014sparse], where one aims to train a sparse nonlinear predictor that has faster prediction and more compact representation through an L1-regularized objective. In this case, the CD method is known as state-of-the-art solver [@yuan2010comparison; @richtarik2014iteration], and we aim to show that the structure of RB allows CD to be parallelized with much more speedup than that of other random features. Coordinate Descent Method ------------------------- Given the $N\times D$ data matrix produced by the RB Algorithm \[alg:RB\], a RCD Method solves $$\label{L1_obj} \min_{{\boldsymbol{w}}\in{\mathbb{R}}^D} \;\lambda\|{\boldsymbol{w}}\|_1 + \frac{1}{N}\sum_{n=1}^N L({\boldsymbol{w}}^T{\boldsymbol{z}}_i, y_i)$$ by minimizing w.r.t. a single coordinate $j$ $$\label{cd_obj} \min_{d_j} \; \lambda |w_j+d_j| + g_jd_j + \frac{M_j}{2}d_j^2$$ at a time, where $$\label{cd_grad} g_j:=\frac{1}{N}\sum_{n=1}^N (\nabla_j L({\boldsymbol{w}}^T{\boldsymbol{z}}_i,y_i)) z_{ij}$$ is the gradient of loss term in w.r.t. the $j$-th coordinate, and $M_j:=\beta\frac{1}{N}\sum_{i=1}^N z_{ij}^2$ is an upper bound on $\nabla_{jj}L(.)$. Note, by focusing on single coordinate, has a tighter quadratic upper bound than other algorithms such as Proximal Gradient Method, and allows simple closed-form solution $$\label{cd_sol} d_j^* := {\mathbf{prox}}_{R/M_j}( w_j-\frac{g_j}{M_j} )-w_j$$ where $${\mathbf{prox}}_{R}(v_j):=\left\{\begin{array}{ll} 0 , & |v_j| \leq \lambda \\ v_j-\lambda, & v_j > \lambda \\ v_j+\lambda, & v_j < \lambda \end{array}\right. .$$ To have efficient evaluation of the gradient , a practical implementation maintain the responses $$\label{response} {\hat{y}}_i := {\boldsymbol{w}}^Tz_i$$ after each update ${\boldsymbol{w}}^{t+1}:= {\boldsymbol{w}}^{t} + d^*_j{\boldsymbol{e}}_j$, so the cost for each coordinate-wise minimization takes $O(nnz({\boldsymbol{z}}_j))$ time for both gradient evaluation and maintenance of , where ${\boldsymbol{z}}_j:=(z_{ij})_{i\in[N]}$. The algorithm is summarized in Alg. \[alg:PRCD-RB\], which just like the iterative solver introduced in section \[section:iterative\], has cost $O(nnz(Z))$ for one pass of all variables $j\in[D]$. 0\. Generate RB feature matrix $Z$ by Algorithm \[alg:RB\] 1. ${\boldsymbol{z}}^{1}={\boldsymbol{0}}$, ${\boldsymbol{w}}^{1}={\boldsymbol{0}}$. 2. Draw $j$ from $[D]$ uniformly at random. 3. Compute $d^*_j$ by . 4. ${\boldsymbol{w}}^{t+1}:= {\boldsymbol{w}}^{t} + d^*_j{\boldsymbol{e}}_j$. 5. Maintain ${\hat{y}}_i,\forall i\in[N]$ to satisfy . Parallel Randomized Coordinate Descend on Random Binning Features ----------------------------------------------------------------- The RCD, however, is hard to parallelize [@richtarik2014iteration]. It is known that simultaneous updates of two coordinates $j_1$, $j_2$ could lead to divergence, and although one can enforce convergence by shortening the step size $\frac{1}{M_{p}}\ll \frac{1}{M_j}$, the convergence rate will not be improved with parallelization without additional assumption [@bradley2011parallel; @richtarik2015parallel]. On the other hand, in [@richtarik2015parallel], it is shown that a function with *partially separable* smooth term plus a separable non-smooth term $$\label{partial_separable} \min_{{\boldsymbol{w}}\in {\mathbb{R}}^{D}} \; F({\boldsymbol{w}}):=\Omega({\boldsymbol{w}}) + \sum_{i=1}^N f_{i}({\boldsymbol{w}})$$ can be parallelized with guaranteed speedup in terms of overall complexity, where $\Omega({\boldsymbol{w}})$ is a non-smooth separable function and each function $f_i({\boldsymbol{w}})$ is a smooth depends only on at most $\omega$ number of variables. The form , fortunately, fits our objective with features ${\boldsymbol{z}}_i$ generated by RB. In particular, the generating process of RB guarantees that, for each block of feature $B_{{\boldsymbol{\delta}}}$, the $i$-th sample can fall in exactly one bin $b=(\lfloor \frac{x_{n1}-u_1}{\delta_1}\rfloor, ..., \lfloor \frac{x_{nd}-u_d}{\delta_d}\rfloor)$, therefore each sample inolves at most $R$ features out of $D$. Specifically, let $\Omega({\boldsymbol{w}}):=\lambda\|{\boldsymbol{w}}\|_1$ and $$f_{i}({\boldsymbol{w}}):=\frac{1}{N}L({\boldsymbol{w}}^T{\boldsymbol{z}}_i,y_i),$$ we have $\omega=R$. Then by Theorem 19 of [@richtarik2015parallel], a parallel RCD of $\tau$ threads that selects coordinate $j$ uniformly at random achieves a speed-up (i.e. time-of-sequential/time-of-parallel) of $$\label{parallel_rate_1} \textit{speedup-ratio}=\frac{\tau}{1+\frac{(R-1)(\tau-1)}{D-1}}.$$ When $D, R\gg 1$, and $\tau=a\bar\kappa+1$ where $\bar\kappa:=D/R$, becomes $$\label{parallel_rate_2} \textit{speedup-ratio}=\frac{a\bar\kappa+1}{1+a},$$ which equals $(\bar\kappa+1)/2$ when $a=1$ and approaches $\bar\kappa$ when $a\rightarrow \infty$. Therefore, it is guaranteed in theory that parallelization can speedup RCD significantly as long as $\bar\kappa=D/R\gg 1$. We give our sparse RB Features algorithm based on parallel RCD in Alg. \[alg:PRCD-RB\]. Note for other Random Features, there is no speedup guaranteed and our experiment shows that Parallel RCD performed on Random Fourier features could even have no speedup. Note that the speedup achieved in this section is orthogonal to the faster convergence rate achieved in section \[section:faster\_convergence\], so by increasing $\kappa$, the advantage of RB over other Random Features is super-linearly increasing if a parallel RCD is used. Note also that the results , also apply to algorithms that utilize Coordinate Descent as subproblem solvers such as Proximal (Quasi) Newton Method [@lee2012proximal; @zhong2014proximal]. Those methods are typically employed for computationally expensive loss functions. Experiments {#section:experiments} =========== In this section, we present extensive sets of experiments to demonstrate the efficiency and effectiveness of RB. The datasets are chosen to overlap with those in other papers in the literature, where the details are shown in the table \[tb: info of datasets\]. All sets except census are available at LIBSVM data set [@LIBSVM]. All computations are carried out on a DELL dual socket with Intel Xeon processors at 2.93GHz for a total of 16 cores and 250 GB of memory running the SUSE Linux operating system. We implemented all methods in C++ and all dense matrix operations are performed by using the optimized BLAS and LAPACK routines provided in the OpenBLAS library. Due to the limited space, we only choose subsets of our results to present in each subsection. However, these results are objective and unbiased. Name $C$: Classes $d$: Features $N$: Train $M$: Test ---------- -------------- --------------- ------------ ----------- cadata 1 8 16,512 4,128 census 1 119 18,186 2,273 ijcnn1 2 22 35,000 91,701 cod\_rna 2 8 49,437 271,617 covtype 2 54 464,809 116,203 SUSY 2 18 4,000,000 1,000,000 mnist 10 780 60,000 10,000 acoustic 3 50 78,823 19,705 letter 26 16 10,500 5,000 : Properties of the datasets.[]{data-label="tb: info of datasets"} Effects of $\sigma$ and $R$ on Random Binning --------------------------------------------- We perform experiments to investigate the characteristics of RB by varying the kernel parameter $\lambda$ and the rank $R$, respectively. We use a regularization $\lambda = 0.01$ to make sure the reasonable performance of RB and other low-rank kernels, although we found that RB is not sensitive to this parameter. We increase the $\sigma$ in the large interval from $1e-2$ to $1e2$ so that the optimal $\sigma$ locates within the interval. We apply CG iterative solver to operate on $Z$ directly. In order to make fair runtime comparison in each run, we set the $tol = 1e-15$ to force similar CG iterations with different $\sigma$. We evaluate the training and testing performance of regression and classification, when varying $\sigma$ with fixed $R$. In [@rahimi2007random], it does not consider the effect of $\sigma$ in their analysis, which however has a large impact on the performance since $D$ depends on the number of bins which is controlled by $\sigma$. Fig. \[fig:TrTe\_perf\_time\_lambda\] shows that the training and testing performance coincidentally decrease (increase) before they diverge when $D$ grows by increasing $\sigma$. This confirms with our analysis in Theorem \[thm:RBconverge\] that the larger $\kappa$, the faster convergence of RB Feature (recall that the convergence rate is $O(1/(\kappa R))$). Second, one should not be surprised that the empirical training time increases with $D$. The operations involving the weighting vector $w_{RB}$ could become as expensive as a sparse matrix-vector operation in an iterative solver. However, the total computational costs are still bounded by $O(NR)$ but the constant factor may vary with different datasets. Fortunately, in most of cases, the training time corresponding to the peak performance is just slightly higher than the smallest one. In practice, there are several ways to improve the computation costs by exploiting more advanced sparse matrix techniques such as preconditioning and efficient storage scheme, which is out scope of this paper and left for future study. Finally, we evaluate the training and testing performance when varying $R$ with fixed $\sigma$. Fig.\[fig:TrTe\_perf\_R\] shows that the training and testing performance converge almost linearly with $D$, which again confirms our analysis in Theorem \[thm:RBconverge\]. In addition, we observe that RB has strong overfit ability which turns out to be a strong attribute, especially when the hypothesis space has not yet saturated. Performance Comparisons of All Methods -------------------------------------- We present a large sets of experiments to compare RB with other most popular low-rank kernel approximations, including RF [@rahimi2007random], Nyström [@Seeger2000Nystrom], and recently proposed independent block approximation [@Stein2014BlockDiag]. We also compare all methods with the exact kernel as a benchmark [@LearnKernels2001]. We do not report the results of the vanilla kernel on covtype and SUSY since the programs run out of memory. To make a fair comparison, we also apply CG on RB and Nyström directly on $Z$ to admit similar computational costs. Since the independent block kernel approximation approximates the kernel matrix directly, we employ direct solver of dense matrix for this method. In practice, the CG iterative solver has no need to solve in high precision [@JCLW2016ICCASP], which has also been observed in our experiments. Thus, we set the tolerance to $1e-3$. Fig.\[fig:perf\_time\_R\_mem\_time\_perf\_group1\] clearly demonstrates the superiority of RB compared to other low-rank kernels. For example, in the first column, RB significantly outperforms other methods in testing performance on all of these datasets, especially when $R$ is relatively small. This is because RB enjoys much faster convergence rate to the optimal function than other methods. The advantage generally diminishes when $R$ increases to reasonably large. However, for some large datasets such as covtype and SUSY, increasing number of random features or $R$ boosts the performance extremely slow. This is consistent with our analysis that RB enjoys its fast convergence rate of $O(1/(\kappa R))$ while other methods has slow convergence rates $O(1/\sqrt{R})$. The third and fourth columns further promote the insights about how many number of random features or how large rank $R$ that is needed for achieving similar performance of RB. In particular, RB is often between one and three orders of magnitude faster and less memory consumptions than other methods. In the second column, we also observe that the training time of all low-rank kernels are linear with $R$, which is expected since all these methods has computational complexity of $O(kNR)$. The difference in training time between these low-rank kernels is only within some constant factors. However, we point out that the computations of RF, Nyström and independent block approximation are mainly carried out by the high-optimized BLAS library since they are dense matrices. In contrast, the computations of RB are most involved in sparse matrix operations, which are self-implemented and not yet optimized. In addition, more advanced sparse matrix techniques such as preconditioning can be explored to significantly accelerate the computation, which we leave it as future work. Parallel Performance of Random Binning and Random Fourier --------------------------------------------------------- We perform experiments to compare RB with RF when using RCD to solve L1-regularized Lasso and kernel SVM for both regression and binary classification problems. Since the goal is to demonstrate the strong parallel performance of RB, we implement the basic parallel implementation of RCD based on simple shared memory parallel programming model with OpenMP. We leave the high-performance distributed RCD implementation as one of the future works. We define the speedup of RCD on multicore implementation as follows: $$speedup = \dfrac{\textit{runtime of RCD using single core}}{\textit{runtime using $P$ cores}}$$ As shown in Fig.\[fig:parallel\_rb\_rf\], when the sparsity level of the feature matrix $Z$ is high, the near-linear speedup can be achieved [@marecek2014DBCD; @liu2013asynchronous]. This is because the minimization problem can almost be separated along the coordinate axes, then higher degrees of parallelism are possible. In contrast, if $Z$ is lack of sparsity, then the penalty for data correlations slows the speedup to none. This is confirmed by no gain of parallel speedup of RF since $Z$ is always fully dense. Obviously, in order to empower strong parallel performance of RB, a very large $D$ is expected, which interestingly coincides with power of its faster convergence. Therefore, one can enjoy the double benefits of fast convergence and strong parallelizability of RB, which is especially useful for very large-scale problems. Conclusions {#section:conclusions} =========== In this paper, we revisit RB features, an overlooked yet very powerful random features, which we observe often to be orders of magnitude faster than other random features and kernel approximation methods to achieve the same accuracy. Motivated by these impressive empirical results, we propose the first analysis of RB from the perspective of optimization, to make a solid attempt to quantify its faster convergence, which is not captured by traditional Monte-Carlo analysis. By interpreting RB as a RBCD in the infinite-dimensional space, we show that by drawing $R$ grids with at least $\kappa$ expected number of non-empty bins per grid, RB achieves a convergence rate of $O(1/(\kappa R))$. In addition, in the L1-regularized setting, we demonstrate the sparse structure of RB features allows RCD solver to be parallelized with guaranteed speedup proportional to $\kappa$. Our extensive experiments demonstrate the superior performance of the RB features over other random feature and kernel approximation methods. Acknowledgement =============== This work was done while L. Wu was a research intern at IBM Research. J. Chen is supported in part by the XDATA program of the Advanced Research Projects Agency (DARPA), administered through Air Force Research Laboratory contract FA8750-12-C-0323.

--- abstract: 'Estimation of death counts and associated standard errors is of great importance in armed conflict such as the ongoing violence in Syria, as well as historical conflicts in Guatemala, Perú, Colombia, Timor Leste, and Kosovo. For example, statistical estimates of death counts were cited as important evidence in the trial of General Efraín Ríos Montt for acts of genocide in Guatemala. Estimation relies on both record linkage and multiple systems estimation. A key first step in this process is identifying ways to partition the records such that they are computationally manageable. This step is referred to as blocking and is a major challenge for the Syrian database since it is sparse in the number of duplicate records and feature poor in its attributes. As a consequence, we propose locality sensitive hashing (LSH) methods to overcome these challenges. We demonstrate the computational superiority and error rates of these methods by comparing our proposed approach with others in the literature. We conclude with a discussion of many challenges of merging LSH with record linkage to achieve an estimate of the number of uniquely documented deaths in the Syrian conflict.' address: - | Carnegie Mellon University\ Economics, History, and Statistics Departments\ Undergraduate Student\ 5000 Forbes Avenue, Pittsburgh, PA 15213\ - | Rice University\ Computer Science Department\ Assistant Professor\ Houston, Texas 77005\ - | Human Rights Data Analysis Group (HRDAG)\ Director of Research\ San Francisco, CA 94110, USA\ - | Duke University\ Department of Statistical Science\ Assistant Professor\ Durham, NC 27708, USA\ author: - - - - bibliography: - 'chomp.bib' - 'mybib\_merged.bib' title: Blocking Methods Applied to Casualty Records from the Syrian Conflict --- , ,\ , Introduction ============ The Syrian conflict has been at the center of international news since March 2011. One main question is “how many civilian casualties have occurred as a result of this conflict?" Although this conflict is well documented, answering what might seem to be a simple question is statistically challenging. This stems from the fact that some victims may be reported to multiple sources whereas other victims are not reported at all. We are motivated by finding a death estimate (with associated standard errors) as such information may contribute to future transitional justice and accountability mechanisms. For instance, statistical estimates of death counts have been introduced as evidence in national court cases and international tribunals investigating the responsibility of state leaders for crimes against humanity. Reporting an estimate is a multi-step process, where the first involves data reduction via a process referred to as blocking. Blocking partitions the space of records into similar “blocks" or groups. The second step involves record linkage (within each block), which is the process of merging many noisy databases to remove duplicate entities. On any moderately sized database it is essential to avoid all-to-all record comparisons, thus emphasizing the importance of the initial blocking step and the use of algorithms that are computationally fast. The third step involves multiple systems estimation, where post-blocking and post-record linkage, we attempt to estimate the total number of entries in a closed population. All components of this multi-step process are challenging and complex, and thus, this paper focuses on advancements in the first step, blocking, for the Syrian conflict. We illustrate that traditional blocking methods typically split records referring to the same person across different blocks around 50% of the time, which makes the record linkage step pointless. However, we propose combining the work of [@shrivastava2014densifying; @shrivastava2014improved] and applying these approaches to the Syrian database. In this combined approach, we find that this method only splits records referring to the same person across different blocks less than 1% of the time. Furthermore, this method is linear in the tuning parameters. We illustrate using our `Java` package that for the approximately 300,000 death records, our blocking procedure runs in 10 minutes. See §\[sec:disc\] for a discussion of the remaining challenges of integrating our proposed methods for blocking based on [@shrivastava2014densifying; @shrivastava2014improved] with any subsequent record linkage procedure. Prior Work {#sec:priorwork} ---------- Record linkage, also known as coreference resolution, entity resolution, and de-duplication is a well known but difficult problem, especially for real world applications, which include human rights violations, official statistics, medical applications, and others [@christen_2011; @Herzog_2007; @Herzog:2010]. Such obstacles are due to the noise inherent in the data, which is often hard to accurately model [@pasula_2003; @steorts_2015_jasa]. A more substantial obstacle is the scalability of the approaches [@WYP:2010]. Assuming $d$ databases of $n$ records each, brute-force approaches, using all-to-all comparisons, require $O(n^d)$ comparisons. Such approaches are easily prohibitive for moderate $n$ or $d$. To avoid this computational burden, the number of comparisons made must be drastically reduced, without compromising linkage accuracy. The record linkage literature tries to achieve scalability by blocking, which involves partitioning records into “blocks” and treating records in different blocks as non-co-referent [*a priori*]{} [@christen_2011; @Herzog_2007]. Record linkage methods are only applied [*within*]{} blocks, reducing the comparisons to $O(B n_{\max}^d)$, with $n_{\max}$ being the size of the largest of the $B$ blocks. There are several techniques for constructing a blocking partition. The most basic method picks certain fields (e.g., governorate, or sex and year of death) and places records in the same block if and only if they agree on all such fields. This amounts to an [*a priori*]{} judgment that these fields are error-free. We refer to this as *traditional blocking* (see §\[sec:blocking\]). Other data-dependent blocking methods [@christen_2011; @WYP:2010] are highly application-specific or are based on placing similar records into the same block, using techniques of random projections or locality sensitive hashing (LSH) [@indyk_1998]. LSH is a *probabilistic method* of dimension reduction, which is widely used in computer science and in database engineering as a way of rapidly finding approximate nearest neighbors [@gionis_1999]. Unlike conventional blocking, LSH uses all the fields of a record, and can be adjusted to ensure that blocks are manageably small, but then do not allow for further record linkage within blocks. Such methods are fast and have high recall (true positive rate), but suffer from low precision, rather, too many false positives. [@steorts_2014_hash] proposed clustering-based blocking schemes that are variants on LSH. The first, transitive locality sensitive hashing (TLSH) is based upon the community discovery literature such that *a soft transitivity* (or relaxed form of transitivity) can be imposed across blocks. The second, $k$-means locality sensitive hashing (KLSH) is based upon the information retrieval literature and clusters similar records into blocks using a vector-space representation and projections (KLSH had been used before in information retrieval but never with record linkage [@pauleve_2010]). [@steorts_2014_hash] showed that both KLSH and TLSH gave improvements over popular methods in the literature such as traditional blocking, canopies [@mccallum_2000], and $k$-nearest neighbors clustering. There are many variants of LSH and one popular form is minwise hashing. All LSH methods are defined by a type of similarity and a type of dimension reduction [@broder_1997]. Recently, [@shrivastava2014densifying] showed that minwise hashing based approaches are superior to random projection based approaches when the data is very sparse and feature poor. Furthermore, improvements in computational speed can be obtained by using the recently proposed densification scheme known as densified one permutation hashing (DOPH) [@shrivastava2014densifying; @shrivastava2014improved]. Specifically, the authors proposed an efficient substitute for minwise hashing, which only requires one permutation (or one hash function) for generating many different hash values needed for indexing. In short, the algorithm is linear (or constant) in the tuning parameters, making it very computationally efficient. §\[sec:motivate\] provides a brief background on the Syrian conflict, as well as a description and challenges of the data. §\[sec:blocking\] reviews the blocking literature that has been explored for records from the Syrian conflict. §\[hashing\] introduces an ensemble of hashing methods that we propose as an application for blocking the Syrian database. §\[sec:app\] applies our proposed method for hashing to the Syrian database, illustrating that we only split records across blocks less than 1% of the time, compared to at best 20% of the time for every other blocking method that was considered. We give a thorough discussion of the challenges of integrating LSH with record linkage for the Syrian database in §\[sec:disc\]. Motivation: The Syrian Conflict {#sec:motivate} =============================== Violence broke out in Syria in March 2011 following a series of anti-government protests. In the years since then the conflict has continued to escalate. Restrictions on both local and international media make it difficult to determine the scope of the violence [@FH2012; @CJR2014]. Nonetheless, combinations of conventional and citizen journalists, grassroots non-governmental organizations (NGOs), and international humanitarian organizations continue to do their best to document the conflict. For example, early in the conflict, Human Rights Watch (HRW) documented numerous cases of lethal force used against peaceful protestors, and confirmed from defectors from Syria’s security forces that “…they were given orders to fire on unarmed protesters.” HRW further reported that “…other aspects of the repression - arbitrary and incommunicado detention, rampant torture, and denial of medical care –have continued unabated" [@HRW2011]. More recently, as summarized by the BBC, the United Nations Commission of Inquiry “…has evidence that those on both sides of the conflict have committed war crimes - including murder, torture, rape and enforced disappearances. Government and rebel forces have also been accused by investigators of using civilian suffering, such as blocking access to food, water and health services, as a method (of) war.” Chaos in the region has also been credited with creating an opportunity for the Islamic State to take control of large portions of territory. Fighting now involves a large number of armed groups with shifting allegiances and boundaries. Numerous attempts by members of the international community to broker ceasefires and negotiate dialogues have thus far been unsuccessful [@bbc2015]. Our motivation is to estimate the number of conflict related killings in Syria since March 2011. Such estimation is necessary since many acts of violence are hidden and many victims may not be reported or identified until months or even years after the event. Relying solely on what is observable is inadequate. As described by Maria McFarland, Co-Director of the US Program for Human Rights Watch, in her testimony to the Tom Lantos Human Rights Commission, “…collecting information about what is happening in Syria today is extremely difficult. The situation we have been able to document is extremely disturbing, but we are just as concerned by what we do not know and have been unable to confirm as by what we do know” [@HRW2012]. Machine learning and statistical modeling methods can be applied to information about documented, identifiable victims to estimate a total number of victims, both those currently identified and those not yet identified or documented. These efforts may contribute to future accountability and transitional justice mechanisms in Syria. Additionally, methodological developments for this particular application may support similar efforts following other violent conflicts. A key first step in this analysis is identifying computationally tractable blocking methods to enable record linkage and further statistical modeling. The Data -------- Via collaboration with the Human Rights Data Analysis Group (HRDAG), we have access to four databases. They come from the Violation Documentation Centre (VDC), Syrian Center for Statistics and Research (CSR-SY), Syrian Network for Human Rights (SNHR), and Syria Shuhada website (SS). Each database lists each victim killed in the Syrian conflict, along with identifying information about each person (see [@price_2013] for further details). Data collection by these organizations is carried out in a variety of ways. Three of the groups (VDC, CSR-SY, and SNHR) have trusted networks on the ground in Syria. These networks collect as much information as possible about the victims. For example, information is collected through direct community contacts. Sometimes information comes from a victim’s friends or family members. Other times, information comes from religious leaders, hospital, or morgue records. These networks also verify information collected via social and traditional media sources. The fourth source, SS, aggregates records from multiple other sources, including NGOs as well as social and traditional media sources (see <http://syrianshuhada.com/> for information about specific sources). These lists, despite being products of extremely careful, systematic data collection, are not probabilistic samples [@Sig2015; @IOAS2015; @CJLS2015; @price2014updated]. Thus, these lists cannot be assumed to represent the underlying population of all victims of conflict violence. Records collected by each source are subject to biases, stemming from a number of potential causes, including a group’s relationship within a community, resource availability, and the current security situation. Although it is beyond the scope of this paper, final analyses of these sources must appropriately adjust for such biases before drawing conclusions about patterns of violence. As already mentioned, our ultimate goal is to merge the four databases, each having a different number of recorded victims, so as to remove duplicate entities among them. In this respect, before being able to use record linkage or multiple systems estimation, we first must use blocking to reduce the space from all-to-all record comparisons. To help assess any blocking method, we have a set of training data generated through hand-matching. Four different matchers from HRDAG manually reviewed records in the databases, classifying records that referred to the same individual as matches, and records with no possibility of matching as non-matches. We treat the hand-matched data as a gold standard against which to compare our blocking approaches. Unique Challenges Posed by Data from Syrian Conflict ---------------------------------------------------- With roughly 300,000 total death records, the total possible pairs of record comparisons are of the order of $10^{10}$. The number of comparisons to be made must be drastically reduced, without compromising linkage accuracy. Hence, we turn to blocking, or rather, placing similar records in groups or partitions. There are two main features of the data that make traditional blocking approaches difficult, if not impossible. The first is due to the fact that the data is very sparse in the number of duplicates, and the second is that the data is sparse in the features that can be used to reduce the number of all-to-all record comparisons. In the following section, we introduce traditional blocking methods for record linkage, and then show that such blocking methods perform so poorly that record linkage would never be performed. In particular, note that for the Syrian database, the traditional or simplistic forms of blocking induce errors in the sense that records that refer to the same person are often placed into separate blocks, which then cannot be recovered from any record linkage algorithm. This makes traditional types of blocking inherently impossible on the Syrian database. This calls for special blocking methods to be used on *sparse, feature-poor* data. Blocking {#sec:blocking} ======== Blocking is a set of rules or algorithms that reduces the set of all-to-all record comparisons. Blocking divides records into mutually exclusive and jointly exhaustive blocks or partitions, allowing record linkage to be performed within each block [@winkler_2006; @steorts_2014_hash]. Only records within the same block can be linked and linkage algorithms aggregate information across blocks. The most basic method for constructing a blocking partition picks certain features (e.g. same last name, date of death (DoD), place of death, etc.) and places records in the same block if and only if they agree on all such fields. This amounts to an a priori judgment that these fields are error-free. We call this traditional blocking as was first coined in [@steorts_2014_hash]. In the setting of the Syrian conflict, traditional blocking is not realistic since many blocks are so large that linkage is computationally intractable. Also, since blocks only consider selected features, much time is wasted comparing records that happen to agree on these features but clearly refer to different individuals. Since traditional blocking proves difficult, we next review conjunctions, conjunctions combined with Arabic soundex, and random projections. Conjuctions {#sec:conjuctions} ----------- A conjunction links two sets of variables through the operators union $\cup$ and intersection $\cap$ [@michelson:aaai06]. Suppose we have two variable fields, date of death and governorate. Define a sample conjunction as $DoD \cap Governorate$, which creates a partition for each set of records that agree in these two fields. If two records have a date of death of 2013-06-20 and are from Damascus, then they will be placed in the same partition. A third record, with the same date of death but from Homs, will not meet the rule and will be placed in a second partition. A conjunction specifies the relationship that must exist for two records to be joined together. As a result, each partition contains records that are in some way similar, or agree in a subset of their features. A disjunction of conjunctions is simply the union of two or more conjunction rules. An example of a disjunction of conjunctions is $[(A \cup B) \cap (B \cup C)]$ for a set of record fields $A, B, C, D$. As the number of rules grows, a higher degree of similarity between two records is required to form a candidate pair. A disjunction of conjunctions approach is used on the Syrian dataset as a way of partitioning records by their features. The input to a conjunction rule, in this context, is a feature from the records, and the output is a set of candidate pairs to be compared to ground truth data. The main benefits of using conjunctions for blocking records are the ability to target an approximate number of candidate pairs and ease in isolating certain notable features about the data. For example, if we see a few large clusters of reported locations, then we could subset those records easily. The main disadvantages of using conjunctions is the possibility of over- or under-fitting the data, scalability to moderate/large data, and each conjunction is application specific. Thus, the main challenge is in constructing an optimal set of rules for the conjunction scheme. For $m$ variable fields, there are a minimum of $m^m$ possible combinations of conjunctions, with this number growing as variable fields are subsetted (as in subsetting date of death into its year, month, and day components). Conjunctions and Arabic Soundex ------------------------------- Often, a conjunction scheme by itself is not enough to partition the records effectively. This is due to attributes of the data, including incomplete records, typos, and feature-poor data. In this case, we propose combining a disjunction of conjunctions approach with textual string analysis of the provided Arabic field names. (See [@price_2013] for a description of previous considerations of textual string analysis of Arabic names, in particular comparing pairs in which one name is recorded in English and the other is recorded in Arabic). This approach allows us to make use of the name variable in the data to further segment and partition records by similarity. We create partitions as small and accurate as possible, where candidate pairs are generated only if they match with the ground truth data. Here, we incorporate the strongest conjunction blocks from the §\[sec:conjuctions\] with an Arabic Edit Distance Algorithm (AEDA) [@abdel_2011]. The AEDA algorithm is intended to be an extension of a Levenshtein distance measure for Arabic text. The Levenshtein distance is a metric for comparing the similarity of two strings; it works by finding the minimum number of character edits (either through insertion, deletion, or substitution) needed to convert one string into another. Arabic text does not operate in this way, as [@abdel_2011] suggests, since Arabic words can often be almost identical in characters but have a very low degree of similarity in meaning or intention. Thus, we use AEDA as a way of finding similarity between two Arabic strings, and not just checking for exact matches. Once we can determine how similar two name fields are, we can determine whether two records should remain in the same partition or be separated. Let $a,b$ be any two Arabic characters, and using AEDA we wish to know the cost associated with converting $a$ into $b.$ If the characters are very similar, then the cost to replace them should be very low. As they become more dissimilar, the cost to replace the first character with the second should increase. There are three possible replacement costs to consider in this case: phonetic replacement $\alpha(a, b)$, letter form replacement $\beta(a, b)$, and keyboard distance replacement $\gamma(a, b)$. Each cost is a function of the two input characters. If $a = b$, then the replacement cost for each of the three functions is 0. Let $\omega, \lambda, \sigma$ be the weights associated with each of the costs, respectively. The AEDA formula (Equation \[eqn:fry\]) incorporates these three functions into a single formula, where we must decide how much weight should be given to each feature. $$\begin{aligned} \text{frc}(a, b) = \begin{dcases*} \frac{\alpha(a, b)\cdot\omega + \beta(a, b)\cdot\lambda+\gamma(a, b)\cdot\sigma}{\omega+\lambda+\sigma} & if a $\neq$ b\\ 0 & Otherwise \end{dcases*} \label{eqn:fry} \end{aligned}$$ For determining phonetic similarity, the first cost, the two provided Arabic characters are evaluated for how they sound to a native Arabic speaker, with values for pairwise characters on a scale from 0 to 1. [@abdel_2011] provides a table with these scores. A score of 0 on this scale means that the characters have no similarity, while a 1 means that they are the same letter or sound very similar. For letter form replacement, characters that are often swapped or mistakenly used in place of one another by an Arabic writer define a measure of similarity. This form is especially useful in transcription issues, where two characters could be used for the same spoken sound. The third cost is a keyboard distance, which indicates that characters close to each other on an Arabic keyboard are more similar to one another than characters far away, as they are more likely to have been accidentally transcribed. These three functions attempt to capture the possible transcription issues present in tracking and recording casualty counts for a database. Depending on what methods are used to record and transcribe the information provided, one replacement cost may be more necessary than another. The keyboard relationship is given by $\text{Sim}_{Kb}(a,b) = 1 - \frac{\sqrt{(x_{a}-x_{b})^{2}+(y_{a}-y_{b})^{2}}}{\psi},$ where ${Sim}_{Kb}$ is the keyboard similarity between two characters $a$ and $b$. The similarity of the two characters is found by calculating the x and y distances between the two characters on the keyboard, where $\psi$ is the maximum distance between any two characters on the keyboard (which is 12 units on standard Arabic keyboards). The x and y distances can be found by looking at an Arabic keyboard (or looking at a picture online), and noting the horizontal and vertical distance between the two input characters. Once these three similarity measures are computed individually, the replacement cost $frc$ formula (Equation \[eqn:fry\]) is used to balance the weighting/importance of each. The $\omega$, $\lambda$, and $\sigma$ are the weights given to each cost function, all summing to 1, and must be chosen with care and with respect to the transcription problems identified in the data. For example, if all of the possible error between two strings is in an auditory transcription from a speaker to a typist, then the weighting should all go towards the phonetic component. If no information is known about the data recording process, providing an equal weighting to each term is likely the best option. Once records have been blocked based on conjunction rules, they can be further partitioned based on the results of the replacement cost formula. Pairwise records with low cost values from this function are partitioned and generate final candidate pairs, while records with high cost values are no longer considered as possible pairs. This second partitioning reduces the space of comparisons needed to be made to the ground truth data, which in some cases can dramatically reduce the computational time required for the method to run. Ensemble of Hashing Methods {#hashing} =========================== We consider an ensemble of hashing methods, which we use for comparison on the Syrian database (§\[sec:app\]). They are all based upon LSH, namely, KLSH, minhashing and weighted minhashing. LSH-based blocking schemes “shingle” [@rajaraman_2012] records. That is, each record is treated as a string and is replaced by a “bag” (or “multi-set”) of length-$k$ contiguous sub-strings that it contains. These are known as “$k$-grams”, “shingles”, or “tokens”. The string “TORONTO” yields the bag of length-two shingles “TO”, “OR”, “RO”, “ON”, “NT”, “TO”. (N.B., “TO” appears twice.) As an alternative to shingling, we might use a bag-of-words (BoW) representation, or even to shingle into consecutive pairs (triples, etc.) of words. We first describe some basics of hashing, and then describe KLSH, and then minhashing and weighted minhashing. Minhashing and weighted minhashing are sped up using DOPH (see §\[sec:doph\].) In LSH, a hash function is defined as $y = h(x),$ where $y$ is the *hash code* and $h(\cdot)$ the *hash function*. A *hash table* is a data structure that is composed of buckets (not to be confused with blocks), each of which is indexed by a hash code. Each reference item $x$ is placed into a bucket $h(x).$ For a review of LSH, we refer to [@rajaraman_2012]. KLSH ---- We explore a simple random projection method, $k$-means locality sensitive hashing (KLSH). In KLSH, the number of times each shingle type appears in a record is counted, leading to a bag-of-shingles representation for records. For measuring similarity between records, the inner product of bag-of-shingled vectors (of records) is used, with inverse-document-frequency weighting. Then the bag-of-shingled vectors is reduced first using random projections and second by clustering the low-dimensional projected vectors via the $k$-means algorithm. To put it simply, the mean number of records per cluster is controlled by $n/c,$ where $n$ is the total number of records and $c$ is the number of block-clusters [@steorts_2014_hash]. Minhashing ---------- One of the most popular forms of LSH is known as minhashing, where the similarity between records is Jaccard [@Proc:Broder_STOC98]. Let $\{0,1\}^D$ denote the set of all binary $D$ dimensional vectors, while $\mathbb{R}^D$ refers to the set of all $D$ dimensional vectors (of records). The records can be represented in vector representation via shingling or a BoW method. Given two sets (or equivalently binary vectors) $x,y \in \{0,1\}^D,$ the Jaccard similarity between $x, y \in \{0,1\}^D$ is$$\begin{aligned} \mathcal{J} = \frac{|x \cap y|}{|x \cup y|}, $$ where $|\cdot|$ is the cardinality of the set. Since we use a shingling based approach, our representation of each record is likely to be very sparse. Moreover, [@shrivastava2014defense] showed that minhashing based approaches are superior than the random projection based approaches for very sparse datasets. The minwise hashing family applies a random permutation $\pi$, on the given set $S$, and stores only the minimum value after the permutation mapping, known as the *minhash*. Formally, the minhash is defined as $h_{\pi}^{min}(S) = \min(\pi(S))$, where $h(\cdot)$ is a hash function. Given sets $S_1$ and $S_2$, it can be shown by an elementary probability argument that $$\label{eq:MinHash} Pr_{\pi}({h_{\pi}^{min}(S_1) = h_{\pi}^{min}(S_2)) = \frac{|S_1 \cap S_2|}{| S_1 \cup S_2|}}, $$ where the probability is over uniform sampling of $\pi$. It follows from Equation \[eq:MinHash\] that minhashing is an LSH for Jaccard similarity. Making Minwise Hashing practical: Densified One Permutation Hashing (DOPH) -------------------------------------------------------------------------- Let $K$ be the number of hash functions and let $L$ be the number of hash tables. A $(K,L)$ parameterized blocking scheme requires $K \times L$ hash computations per record. For a single record, this requires storing and processing hundreds (or even thousands) of very large permutations. This in turn requires hundreds or thousands of passes over each record. Thus, traditional minwise hashing is prohibitively expensive for large or moderately sized datasets. In order to cross-validate the optimal $(K,L)$ tuning parameters, we need multiple independent runs of the $(K,L)$ parameterized blocking scheme. This expensive computation is a major computational concern. Instead we utilize the methods of [@shrivastava2014densifying; @shrivastava2014improved], which use only *one permutation* and computes $k = K \times L$ minhashes with the required property (Equation \[eq:MinHash\]) in just one pass over the data. Furthermore, due to sparsity of data vectors (from shingling), empty buckets (in the hash tables) are possible and destroy LSH’s essential property [@rajaraman_2012]. To restore this, we rotate the values of non-empty buckets and assign a number to each of the empty buckets. Our $KL$ hashed values are simply the final assigned values in each of the $KL$ buckets. The final values were shown to satisfy Equation \[eq:MinHash\], for any $S_1, \ S_2$, as in minhash [@shrivastava2014densifying; @shrivastava2014improved]. Weighted Densified One Permutation Hashing {#sec:doph} ------------------------------------------ Minhashing, however, only uses the binary information and ignores the weights (or values) of the components, which as argued before are important for the problem due to the sparsity and feature-poor data (see §\[sec:priorwork\]). This is the reason why we observe slightly better performance for synthetic data of LSH methods used in [@steorts_2014_hash], one of which is based upon random projections. To explore this more broadly, we examine the power of minwise hashing for our sparse representation, while simultaneously utilizing the weighting of various components. Suppose now $\bx, \by$ are non-negative vectors. For our problem, we are only interested in non-negative vectors because shingle based representations are always non-negative. There is a generalization of Jaccard similarity for real valued vectors in $\mathbb{R}^D$, which unlike minhash is sensitive to the weights of the components, defined as $$\begin{aligned} \mathcal{J}_w = \frac{\sum_i \min\{x_i,y_i\}}{ \sum_i \max\{x_i,y_i\}} = 1 - \frac{\| \bx - \by \|_1} { \sum_i \max\{x_i,y_i\}},\end{aligned}$$ where $||\cdot||_1$ represents the $\ell_1$ norm. Consistent weighted sampling [@charikar2002similarity; @gollapudi2006exploiting; @manasse2010consistent; @ioffe2010improved] is used for hashing the weighted Jaccard similarity $J_w$. In our application to the Syrian database, we find minhash and weighted minhash give similar error rates, which can be seen in §\[sec:app\]. With DOPH the traditional minwise hashing scheme is linear or constant in the tuning parameters. For the weighted version of minhashing, we propose a different way of generating hash values for weighted Jaccard similarity, similar to that of [@charikar2002similarity; @gollapudi2006exploiting]. As a result, we obtain the fast and practical one pass hashing scheme for generating many different hash values with weights, analogous to DOPH for the unweighted case. Overall, we require only one scan of the record and only one permutation. Given any two vectors $\bx, \by \in \mathbb{R}^D$ as the shingling representation, we seek hash functions $h(\cdot)$, such that the collision probability between two hash functions is small. That is, $$Pr(h(x) = h(y)) = \frac{\sum_i {\min\{x_i,y_i\}}}{\sum_i\max\{x_i,y_i\}}.$$ Let $\delta$ be a quantity such that all components of any vector $x_i = I_i^x\delta$ for some integer $I_i^x$.[^1] Let the maximum possible component $x_i$ for any record be $x$ and let $M$ be an integer such that $x_i = M \delta$. Thus, $\delta$ and $M$ always exist for finitely bounded datasets over floating points. Consider the transformation $T: \mathbb{R}^D \rightarrow \{0,1\}^{M\times D}$, where for $T(x)$ we expand each component $x_i = I\delta$ to $M$ dimensions and with the first $I$ dimensions have value $1$ and the rest value $0$. Observe that for vectors $x$ and $y$, $T(x)$ and $T(y)$ are binary vectors and $$\begin{aligned} &\frac{|T(x) \cap T(y)|}{|T(x) \cup T(y)|} = \frac{\sum_i \min\{I_i^x,I_i^y\}}{ \sum_i \max\{I_i^x,I_i^y\}} \label{eqn:anshu} \\ &= \frac{\sum_i \min\{I_i^x,I_i^y\} \delta}{ \sum_i \max\{I_i^x,I_i^y\} \delta} = \frac{\sum_i {\min\{x_i,y_i\}}}{\sum_i\max\{x_i,y_i\}} \notag\end{aligned}$$ In other words, the usual resemblance (or Jaccard similarity) between the transformed $T(x)$ and $T(y)$ is precisely the weighted Jaccard similarity between $x$ and $y$ that we are interested in. Thus, we can simply use the DOPH method of [@shrivastava2014densifying; @shrivastava2014improved] on $T(x)$ to get an efficient LSH scheme for weighted Jaccard similarity defined by Equation \[eqn:anshu\]. The complexity here is $O(KL+ \sum_i I_i )$ for generating $k$ hash values, a factor improvement over $O(k\sum_i I_i)$ without the densified scheme. Often $I_i$ is quite large (when shingling) and $\sum_i I_i$ is large as well. When $\sum_i I_i$ is large, [@gollapudi2006exploiting] give simple and accurate approximate hashes for weighted Jaccard similarity. They divide all components $x_i$ by a reasonably big constant so that $x_i \le 1$ for all records $x$. After this normalization, since $x_i \ge 0$, for every $\bx,$ we generate another bag of word $\bx_S$ by sampling each $x_i$ with probability $x_i \le 1$. Then $\bx_S$ is a set (or binary vector) and for any two $\bx$ and $\by$, the resemblance between $\bx_S$ and $\by_S$ sampled in this manner is a very accurate approximation of the weighted Jaccard similarity between $\bx$ and $\by$. After applying the DOPH scheme to the shingled records, we generate $k$ different hash values of each record in time $O(KL+d)$, where $d$ is the number of shingles contained in each record. This is a vast improvement over $O(KL+ \sum_i I_i )$. Algorithm \[alg:hashgeneration\] summarizes our method for generating $k$ different minhashes needed for blocking. $x_S = \phi$ $KL$ densified one permutation hashes (DOPH) of $x_S$ An Application to the Syrian Conflict {#sec:app} ===================================== We apply our methods to a database of 296,245 records of identifiable victims that HRDAG has collected from the four aforementioned sources. We consider four sets of features that include full Arabic name, date of death, high level location (governorate) where the individual died, and sex (M/F). The ultimate blocking method we advocate for (minhashing) is probabilistic and unsupervised. We analyze the Syrian data using the traditional blocking methods and then by the more advanced hashing-based methods from §\[hashing\]. We first review how we evaluate our methods using the training data provided. Recall from §\[sec:motivate\], we have hand matched data which we treat as a gold standard against which to compare our blocking approaches. #### Evaluation Methods We evaluate each of our four hashing methods below using recall and reduction ratio. The recall measures how many of the actual true matching record pairs have been correctly classified as matches. There are four possible classifications. First, record pairs can be linked in both the hand-matched training data (which we refer to as ‘truth’) and under the estimated linked data. We refer to this situation as *true positives* (TP). Second, record pairs can be linked under the truth but *not* linked under the estimate, which are called *false negatives* (FN). Third, record pairs can be *not* linked under the truth but linked under the estimate, which are called *false positives* (FP). Fourth and finally, record pairs can be *not* linked under the truth and also *not* linked under the estimate, which we refer to as *true negatives* (TN). The vast majority of record pairs are classified as true negatives in most practical settings. Then the true number of links is $\text{TP}+\text{FN}$, while the estimated number of links is $\text{TP}+\text{FP}$. The usual definitions of false negative rate and false positive rate are $$\notag \text{FNR}=\frac{\text{FN}}{\text{TP+FN}},\qquad \text{FPR}=\frac{\text{FP}}{\text{TP+FP}},$$ where by convention we take $\text{FPR}=0$ if its numerator and denominator are both zero, i.e., if there are no estimated links. The recall is defined to be $$\text{recall} = 1-FNR.$$ The precision is defined to be $$\text{precision} = 1-FPR.\footnote{Note that the precision for a blocking procedure is not expected to be high since we are only placing similar pair in the same block (not fully running a record linkage procedure or de-duplication procedure, which would try and maximize both the recall and the precision).}$$ The reduction ratio (RR) is defined as $$RR = 1 - \frac{s_M + s_N}{n_M+n_N},$$ where $n_M$ and $n_N$ are the total of matched and non-matched records and the number of true matched and true non-matched candidate record pairs generated by an indexing technique is denoted with $s_M + s_N \leq n_M+n_N.$ The RR measure provides information about how many candidate record pairs were generated by an indexing technique compared to all possible record pairs, without assessing the quality of these candidate record pairs. We also evaluate the methods using the precision, where precision measures the proportion of how many of the classified matches (true positives + false positives) have been correctly classified as true matches (true positives). It thus assesses how precise a classifier is in classifying true matches. Traditional blocking -------------------- Traditional blocking methods, implementing using the feature set considered here, do not scale to the entire dataset well and more importantly they do not perform well either in terms of recall or the reduction ratio. Based on a subset of 20,000 records from the Syrian database, the recall and reduction ratio is never above 0.30. Thus, we find that we split records that refer to the same individual across different blocks around 70% of the time. KLSH ---- We next apply KLSH, illustrating that application to the Syrian database performs poorly in terms of both recall and reduction ratio. This contrasts empirical studies shown in [@steorts_2014_hash]. (The parameters to be set for KLSH are the number of random projections ($p$) and the number of clusters to output ($k$). Through this $k$-means approach to blocking, the mean number of records within a cluster can be fixed. Figure \[fig:klsh-subset\] displays the results of KLSH clustering on the subset of the Syrian database, where we plot the recall versus the total number of blocks. We set the number of random projections to be $p=20$ and allow the shingles to vary from $k=1,2,3,4.$ This figure shows that a 1-shingle always achieves the highest recall. We notice that using a 1-shingle, a block size of 100, the recall is 0.60, meaning that 40% of the time we split records referring to the same individual across different blocks. Of course, performing record linkage would not be useful here. \] ![KLSH on subset of Syria database (20,000 records) using p=20. []{data-label="fig:klsh-subset"}](pics/klsh_20000_p20_8_1_14.pdf){width="60.00000%"} Figure \[fig:klsh\] displays the results of KLSH clustering on the entire Syrian database, using p=20 and a 1-shingle. We see that when we use all of the data available, the recall decreases as the total number of blocks increases, which happens due to the sparsity and feature-poor data. For the entire Syrian database, a reasonable block size corresponds to a recall of 0.40, meaning that 60% of the time we split records that refer to the same individual across different blocks. Of course, performing record linkage would not be useful with a recall this low. \] ![KLSH on entire Syria database using p=20. []{data-label="fig:klsh"}](graphics/recall_klsh_fullname_k1_p20.pdf){width="60.00000%"} #### Conjunctions Next, we investigate the disjunction of conjunctions method on the subset of the Syrian database. In terms of forming conjunctions, we use date of death, governorate, and full Arabic name. Figure \[fig:conj\] displays the results of the formed disjunctions on the subset of the Syrian database. We find that $Year \cup Governorate$ and $Month \cup Year \cup Governorate$ produces the highest recall and reduction ratio. The highest recall performance is around 0.8, which means that around 80% of the training pairs are correctly blocked by the conjunction scheme. The results of the conjunction points to the importance of the governorate and date of death fields in classifying records as matches or non-matches. It is quite easy to calculate the pairs generated by a conjunction blocking scheme. In the conjunction $Year \cup Governorate$, for example, there are 41 blocks created from the total number of records, however these blocks create approximately 1.2 million candidate pairs, which are then compared to the training data. However, there are only 75 training pairs classified as true matches in the Syrian subset of 20,000 records, and so the vast majority of candidate pairs become false positives. Only a few become true positives, and the rest are either false negatives or true negatives. Clearly, the number of candidate pairs only increases in an intractable fashion as the number of records increases. When moving to the entire Syrian database, we find similar results to Figure \[fig:conj\]. ![Highest performing disjunction of conjunctions sets on entire Syria database, in terms of recall and reduction ratio. The $Year \cap Month \cap Governorate$ blocking is the best result from all of the iterated combinations. We achieve very similar results on the entire Syrian database.[]{data-label="fig:conj"}](final_figures_aoas/figure-3){width="75.00000%"} Conjunctions & Arabic Edit Distance ----------------------------------- We next apply the combined approach of conjunctions and AEDA. Due to a lack of information regarding data collection methods, the values for the three weighting parameters, $\omega$, $\lambda$, and $\sigma$, are assumed to be of equal importance, and so each is given the value $1/3$. Once a replacement cost value is obtained for each pairing of names within a block, candidate pairs are created by considering the top 10th percentile of records in string similarity (this value can be adjusted to any size, but we found that this value yields the highest recall). We perform the AEDA within each block formed by the conjunction rules. The benefit of this approach is that it allows us to take into account the similarities between the Arabic string names to form candidate pairs. In contrast, in the previous section only exact matches on full names were considered to form simple conjunctions. First, records are blocked based on a set of features in common. Then, within each block, we compare strings using AEDA. The candidate pair generation provided by the AEDA allows for even more reduction in pairwise comparisons, as not all pairs within a conjunction block have to be compared. The results of the AEDA are displayed in Figure \[fig:arabic\], which points to the relative importance of each metric in determining similarity. In terms of recall, combining the AEDA with conjunctions results in equivalent levels, as the same candidate pairs are formed within the cluster as without the differentiation by string. However, the principal advantage of using the AEDA is that the reduction ratio is higher. ![Distribution of Arabic text replacement cost measures (perfect matches were removed from the graphic). The replacement cost is $1 - \text{Similarity}$, where $\text{Similarity}$ is how much agreement there is between two given text strings. The keyboard measure results in the greatest differentiation between similar and non-similar Arabic names. The other two measures yield quite similar results, and indicate a low degree of similarity among the tested names. The three distributions are evaluated together by Formula \[eqn:fry\], to determine an accurate similarity measure between any two name records. We can also see that the average predicted cost of replacing one string with another is around 0.8, where 1 indicates no agreement at all. This is an indication that the Arabic names present in the databases share very little in common with each other on average.[]{data-label="fig:arabic"}](graphics/arabic-text.png){width="65.00000%"} Minhasing {#sec:minhashing} --------- We apply minhashing using unweighted and weighted DOPH to the full Syrian database using shingles 2—5, where L varies from 100–1000 by steps of 100 and K takes values 15,18,20,23,25,28,30,32,35. We illustrate that regardless of the shingle from 2–5, the recall and RR are close to 1 as illustrated in Figure \[syria-takethatAssad\]. Furthermore, using unweighted DOPH, we see that a shingle of three overall is most stable in having a recall and RR close to 0.99 as illustrated in Figure \[syria-takethatAssadAgain\]. Using weighted DOPH, we see that a shingle of two or three overall is most stable in having a recall and RR close to 0.99. In terms of computational run time, we note that each individual run takes 10 minutes on the full Syrian dataset. We contrast this with the other blocking runs that on 20,000 records from Syria takes many hours or 1-3 days and return a recall and RR that is unacceptable for record linkage purposes. While we can achieve a near perfect recall and reduction ratio, the precision is close to 0, meaning that any minhashing method is not sufficient for the purposes of record linkage. Furthermore, the resulting blocks unfortunately overlap and there is no optimal combination of L and K that leads to blocks that do not overlap and have acceptable recall and reduction ratio measures. For the purposes of record linkage, this means that we cannot simply treat each block as “separate" and run our preferred record linkage procedure in parallel across blocks. Thus, coming up with a reliable estimate of the number of uniquely documented identifiable deaths post-blocking is beyond the scope of the paper and we speak to the potential challenges and solutions to this in §\[sec:disc\] ![For shingles 2–5, we plot the RR versus the recall. Overall, we see the best behavior for a shingle of 3, where the RR and recall can be reached at 0.98 and 1, respectively. We allow L and K to vary on a grid here. L varies from 100–1000 by steps of 100; and K takes values 15, 18, 20, 23, 25, 28, 30, 32, and 35.[]{data-label="syria-takethatAssad"}](plots/syria_full_recall_rr_allShingles){width="\textwidth"} ![For shingles 2–5, we plot the RR versus the recall. Overall, we see the best behavior for a shingle of 2 or 3, where the RR and recall can be reached at 0.98 and 1, respectively. We allow L and K to vary on a grid here. L varies from 100–1000 by steps of 100; and K takes values 15, 18, 20, 23, 25, 28, 30, 32, and 35.[]{data-label="syria-takethatAssadAgain"}](plots/syria_full_recall_rr_Weighted_allShingles){width="\textwidth"} Discussion {#sec:disc} ========== Blocking allows us to reduce the space of all-to-all record comparisons. Specifically, using locality sensitive hashing, we can effectively block without comprising the recall or RR. Specifically, for minhashing approaches, we have shown that we only split the same record across blocks less than 1% of the time, where other blocking methods split records across blocks approximately 20 – 60% of the time. Furthermore, due to recent computational speedups [@shrivastava2014densifying; @shrivastava2014improved] we are able to perform one run in 10 minutes, whereas other blocking methods on the same data (or smaller data) take 1–3 days and the accuracy is unacceptable for record linkage purposes. However, in order to reach such high results on both the recall and RR using minhashing, we sacrifice the precision. Hence, hashing based methods are not enough on their own to be used for this application for simultaneous dimension reduction and record linkage. This implies, that based on blocking alone, we cannot achieve a reliable estimate of the observed death count in Syria. As mentioned in §\[sec:minhashing\], the best blocks we produce are overlapping in the sense that a record can appear in more than one block. When we assess an estimate of the number of uniquely documented identifiable deaths, we seek exact uncertainty quantification from any record linkage procedure, such that the estimate can be as accurate as possible. Currently, exact uncertainty quantification in record linkage is only possible for more than two databases in generative Bayesian methods [@steorts_2015_jasa; @steorts_2014_aistats; @steorts_2015_eb], where the record linkage is done simultaneously across and within all databases. The immediate challenge is that while we have performed a blocking scheme in 10 minutes, the resulting blocks are overlapping, and hence, any Bayesian method will be computationally very slow on moderately sized data, such as the Syrian dataset. For example, suppose that the largest block contains 500 records. Then the fastest Bayesian record linkage procedure would take about 12–24 hours to run in just one block. Given that there are $M$ blocks, this will take M days to run on one processor (which does not account for running different models, assessing model misspecification, etc.). All in all, such a procedure is computationally intractable without additional computational speed ups. Perhaps a promising area of exploration is to assume a generative Bayesian record linkage model. We could then combine the Split and Merge method of [@smmcmc; @steorts_2014_aistats] or the Wormhole method of [@steorts_2015_small_clustering], with proposals from minhashing to traverse the state space more quickly. This involves testing many record linkage models to see which work best. This not only enables record linkage to be performed, but also the computational complexity of the algorithm could be assessed and multiple systems estimation could also be addressed. Such ideas are very promising in the context of human rights but also in a number of other application areas including official statistics, precision medicine, and genetics. ### Acknowledgments {#acknowledgments .unnumbered} We would like to thank Sam Ventura for comments and help with data cleaning that greatly helped our paper. We would also like to thank Patrick Ball, Jerry Reiter, and Abbas Zaidi for suggestions, comments, and discussions that contributed to the paper. RCS was supported by the John Templeton Foundation and all view expressed by this work are of the authors and not of the foundation. [^1]: This assumption is true when dealing with floating point numbers for small enough $\delta$.

--- abstract: 'We have measured the damped motion of a trapped Bose-Einstein condensate, oscillating with respect to a thermal cloud. The cigar-shaped trapping potential provides enough transverse confinement that the dynamics of the system are intermediate between three-dimensional and one-dimensional. We find that oscillations persist for longer than expected for a three-dimensional gas. We attribute this to the suppressed occupation of transverse momentum states, which are essential for damping.' address: 'Centre for Cold Matter, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom' author: - 'B. Yuen, I. J. M. Barr, J. P. Cotter, E. Butler and E. A. Hinds' bibliography: - 'oscillationbibliography.bib' title: 'Enhanced oscillation lifetime of a Bose-Einstein condensate in the 3D/1D crossover' --- February 2014 [*Keywords*]{}: low dimensional quantum gases, Bose-Einstein condensate, dissipation, atom chip, cold atoms Introduction ============ Trapped, ultracold gases offer a versatile way to investigate quantum many-body physics. Well-isolated from their surroundings, they can be controlled to cover a wide parameter space, giving access to regimes beyond the reach of other condensed matter experiments [@Levin12]. Confinement reduces the dimensionality of a gas when the atoms have insufficient energy to reach excited quantum levels. For example, pancake-shaped traps can produce a two-dimensional (2D) gas, while a cigar-shaped trap can confine it to one dimension (1D) [@Bagnato91]. While the static properties of atomic Bose-Einstein condensates (BEC) are generally well understood [@Pethick02] the dynamical behaviour remains an active area of study [@Polkovnikov11]. In the early days of atomic BEC, oscillations of the shape were studied, primarily to establish the superfluidity of the condensate, and it was noticed that these oscillations were damped [@Jin96; @Mewes96] at a rate that depended strongly on the temperature [@Jin97]. An explanation for this was offered by Landau damping [@Liu97; @Pitaevskii1997], in which a low-energy excitation of the condensate is dissipated into the thermal cloud by scattering phonons from lower to higher energy. Fedichev *et al.* [@Fedichev98a; @Fedichev98b] extended this theory to the case of a trapped gas and showed that the damping is determined predominantly by the condensate boundary region, resulting in a different damping rate from that of a spatially homogeneous gas. This theory found reasonable agreement with [@Jin97], and similar agreement was found with the measured damping rate of the scissors mode of oscillation [@Marago01]. Subsequently, Stamper-Kurn *et al.* [@StamperKurn98] excited a cigar-shaped condensate to move rigidly along its length, out of phase with its thermal component. They saw that this second-sound motion [@Zaremba98] was damped, and noted that collisions neglected in the Landau theory might play a role because the hydrodynamicity – the thermal cloud collision rate divided by the oscillation frequency – was not small. The damping of this mode was also noted in [@Ferlaino02] and was studied extensively by Meppelink *et al.* [@Meppelink09]. They found qualitative agreement with [@Fedichev98a] at low values of hydrodynamicity, with a strongly growing discrepancy at higher values, demonstrating the breakdown of the Landau theory at high density. Oscillations of long, thin condensates in the 1D regime [@Moritz03] have very different behaviour, with no damping [@Kinoshita06] unless corrugation is added to the trapping potential [@Fertig05]. This raises the question of how the damping evolves from the 3D rate, through the crossover regime where no analytic theory currently exists, to a complete absence of damping in 1D. Oscillation frequencies have been measured in this crossover regime [@Kottke05; @Fang14], but not the damping rate. In this article, we measure the damping rate for dipole oscillations of a condensate in the crossover regime as a function of temperature, and compare our results with measurements of [@Meppelink09] and the theory of [@Fedichev98a; @Fedichev98b]. We find that the oscillations in our experiment persist for longer than expected for a 3D gas and propose that this is the consequence of suppressed radial excitations due to the tight transverse confinement of the atoms. Condensate oscillations in a thermal background =============================================== We produce highly elongated, finite temperature condensates [@Yuen14] with the apparatus illustrated in figure\[fig:apparatus\]. A magneto-optical trap (MOT) cools and collects $^{87}$Rb atoms a few millimetres away from the surface of an atom chip [@Reichel99]. The MOT is then turned off, and the atoms are transferred to a Ioffe-Pritchard trap approximately from the surface of the chip [@Sewell2010; @Baumgartner2010]. The magnetic trapping field is produced by current in a Z-shaped wire on the chip, with its central section along $z$, together with an external bias field along $x$. The high magnetic field gradient near the centre of the Z-wire gives tight radial ($x$, $y$) confinement with a harmonic oscillation frequency of $\omega_{\rho}/2\pi = 1.4~\mathrm{kHz}$. Axial ($z$) confinement is produced by the currents in the ends of the Z-wire and in the end wires (figure\[fig:apparatus\]), giving an axial frequency of $3~\mathrm{Hz}$. We cool the trapped gas further by forced evaporation, using an rf field to flip the spins of the most energetic atoms so that they are ejected from the trap [@ketterle_evaporative_1996]. By sweeping the escape energy down to a few kilohertz above the bottom of the trap, we produce an almost pure BEC of approximately $10^4$ atoms at a temperature of $\sim150$nK. Minor defects in the chip wire cause the current to meander slightly from side to side, producing small undulations of the trapping potential that make local minima along the $z$ axis up to a microkelvin in depth [@Fortagh02; @Jones03]. We adjust the centre of the axial trap so that the BEC forms in one of these, which is harmonic over a small region, with a characteristic frequency of $\omega_z/2\pi = 10~\mathrm{Hz}$. The condensed atoms are confined to that region, while the higher-energy atoms in the thermal component of the gas explore a larger axial range, and experience an anharmonic potential. When the rf field is turned off, the atoms warm up at approximately $50~\mathrm{nK/s}$, presumably due to noise in the apparatus. To counteract this, we leave the rf field on, so that atoms above some fixed energy are able to leave the trap. Over a few milliseconds the cloud comes to equilibrium at the temperature where the heating is balanced by the evaporative cooling. We select a desired temperature in the range $150-310$ nK by adjusting the rf frequency. The temperature remains fixed over the next 500 ms, while the number of trapped atoms decreases, typically by a few percent. Our aim is to observe the oscillation of condensed atoms moving through the thermal cloud in order to determine the damping rate as the system equilibrates. To resonantly excite axial condensate oscillations, we drive an oscillating current in one of the end wires at $10~\mathrm{Hz}$ for two periods. After this time, the condensate’s centre of mass is left oscillating with an initial amplitude of . The thermal atoms are largely unaffected because they explore the region outside the local potential minimum and are therefore not resonant with this drive. We allow the condensate to oscillate through the thermal background for a time $t$, before switching off the trap and imaging the cloud to determine the condensate’s centre of mass. By increasing $t$ in $12.5\,$ms steps over a total of $400\,$ms, we build up a data set of the damped oscillation. We repeat this process for clouds at different temperatures which we influence by setting the frequency of the rf field as described above. Thus, we observe how the system damps as a function of temperature. Measuring the temperature, condensate centre of mass, and damping rate ====================================================================== We determine the temperature of the gas, and centre of mass of the condensate from an absorption image. To image the atom cloud, we release it from the trap (gravity is up in figure\[fig:apparatus\]), wait for 2 ms, illuminate it with resonant laser light and view the absorption along $x$ using a CCD camera. This image is then integrated over $y$ to obtain the one-dimensional axial number density profile of the cloud, $n(z)= \int \mathrm{d}x \,\mathrm{d}y\,n(\mathbf{r})$. The data points in figure\[fig:measurements\](a) show axial density profiles measured at three different temperatures. At the lowest temperature (red dots), the atoms are nearly all in the condensate, with very little signal in the broad thermal background, whereas the profile at the highest temperature (green triangles) has a clearly visible thermal population on either side of the cloud. Our analysis of the cloud profile builds on the method of [@Naraschewski1998]. The trapping potential is well described by $U(\mathbf{r})={1 \over 2} m\omega_{\rho}^2 \rho^2+V(z)$, where $\rho$ is the radial displacement, and $V(z)$ is the potential on axis, including the irregularity caused by the meandering current. We determine $V(z)$ from the axial density distribution of cold, non-condensed clouds as described in [@Jones2004]. Knowing $U(\mathbf{r})$, we estimate the number density profile of the condensate, $n_{c}(\mathbf{r})$, using the Thomas-Fermi approximation. The profile of the thermal component is calculated by integrating the Bose-Einstein distribution over the effective potential $2 g n_{c}(\mathbf{r}) + U(\mathbf{r})$, where the first term is the mean-field energy of thermal atoms inside the condensate. The cloud is then allowed to evolve freely for $2\,$ms to account for the period of free fall (though we find that this makes no significant difference to the axial profile). We fit this theoretical cloud to the measured density profile $n(z)$ in order to determine the temperature, the position of the condensate, and the peak condensate number density $n_{c}(0)$. We note that the Thomas Fermi approximation is not well satisfied in our 3D/1D condensates, but we find from simulations [@FuturePaper] that this method still yields accurate temperatures, while the peak condensate density is underestimated, typically by 10%. These fits, shown in figure\[fig:measurements\](a) as solid lines, are in excellent agreement with the clouds we observe. For the three clouds that are plotted in figure\[fig:measurements\](a), we determined the temperatures $155(3)$, $251(3)$ and $305(3)$nK. The dotted lines show the thermal cloud density within the condensed regions. In our experiments, the temperature fluctuates by less than $10$nK from one realisation to the next – mainly because of fluctuations in the initial number of magnetically trapped atoms – and drifts by less than $\pm20$nK over an hour. The position of the BEC is very stable, fluctuating from shot to shot by less than , which we associate with mechanical instability of the camera and mirror mounts. It does not drift significantly over an hour. Figures\[fig:measurements\](b-d) show plots of the condensate centre of mass oscillations we measure at each of the three temperatures used in figure\[fig:measurements\](a). It takes approximately 30 minutes to collect the data points for one plot. We have analysed 33 such time sequences, covering a range of temperatures from $150$nK up to $310$nK. In each case, the motion is well described by the exponentially damped sinusoid, $$z(t) = A \mathrm{e}^{-\gamma t} \mathrm{sin}\left( \omega t + \phi \right) + C, \label{eq:dampedSin}$$ where $A$, $\gamma$, $\omega$, $\phi$ and $C$ are fit parameters. Parameters $A$, $\phi$ and $C$ are independent of temperature, and $\omega$ increases only slightly (by 10%) over this range of temperatures. By contrast, $\gamma$ depends significantly on temperature, increasing by a factor of three. Results and discussion ====================== In figure\[fig:results\] we plot the damping rate $\gamma$ as a function of the temperature, each point being the result of fitting one oscillation curve. The temperature assigned to one point is the mean of the $\sim30$ temperatures measured in that curve, and this has a standard error smaller than the symbols in the plot. A vertical error bar indicates the $1\sigma$ uncertainty in $\gamma$ for each fit. We note that the dissipated energy of the oscillation has no significant influence on the temperature, because it corresponds to a negligible rise of $\sim2\,$nK. Damping rates measured in 3D BEC oscillation experiments have generally been consistent with the Landau damping theory [@Jin96; @Mewes96; @Jin97; @Fedichev98a; @Fedichev98b; @Marago01; @Meppelink09]. For a 3D trapped cloud making small oscillations at a frequency $\omega$ close to the trap frequency, this theory gives the damping rate as (see equation (18) of Ref. [@Fedichev98a]) $$\label{eq:gammatheory} \Gamma_{\nu} = A_{\nu}\, \omega \frac{k_{\mathrm B} T}{g\sqrt{n_c(0)}} a^{3/2} \,.$$ Here $A_{\nu}$ is a numerical coefficient that depends on which collective mode $\nu$ is excited, $a$ is the s-wave scattering length, $n_c(0)$ is the peak number density of the condensate and $g=4 \pi \hbar^2 a/m$ is the usual nonlinear coupling parameter (the $\mu$ of [@Fedichev98a] is the same as our $g n_c(0)$). At each temperature our measurements give values for the number density and oscillation frequency, from which we construct empirical functions $n_c(0;T)$ and $\omega(T)$. Using these functions, we fit (\[eq:gammatheory\]) to our data with $A_{\nu}$ as the only free parameter. The result is $A_{\nu}=3.53(15)$. The solid line in figure\[fig:results\] shows this best fit, with the shaded region covering the standard deviation. This theory describes our data well, giving a reduced $\chi^2$ of $0.90$. The measurements of Meppelink *et al.* in [@Meppelink09] involve the same mode as our experiment, but the comparison of their damping at low hydrodynamicity with (\[eq:gammatheory\]) yields a coefficient $A_{\nu}=7$ [@Meppelink09; @Straten14]. The essential difference between these two experiments is in the dimensionality of the trapped gas. In [@Meppelink09] the chemical potential was at least 28 times higher than the radial excitation energy, placing their cloud firmly in the 3D regime. By contrast, the chemical potential of our cloud is aproximately twice the quantum of radial excitation, which places it in the crossover regime between 3D and 1D. The temperature dependence of our result indicates that the same Landau damping idea still applies, even in this crossover regime, but the density of states, which enters through the use of Fermi’s golden rule to obtain (\[eq:gammatheory\]), should be modified to account for the quantisation of the radial excitations [@Yuen14]. Physically, the thermal excitations in this case are more likely to be along $z$, in which case they cannot contribute to the damping, and $A_{\nu}$ is correspondingly reduced. The Utrecht experiment [@Meppelink09] measured the damping over a wide range of hydrodynamicity. Following in the spirit of [@Meppelink09], the red squares in figure\[fig:hydro\] plot the ratio of their measured damping rates $\gamma$ to the $\Gamma_{\nu}$ of (\[eq:gammatheory\]), with $A_{\nu}=7$ [@Straten14], plotted versus hydrodynamicity. At low hydrodynamicity, the ratio approaches 1 in their data, and 0.5 in our data (blue circles), as discussed above. Further, the Utrecht data shows an increase in this ratio as the hydrodynamicity increases, indicating that collisional processes, not incorporated in the model of (\[eq:gammatheory\]), play an important role in damping this dipole mode. Figure\[fig:hydro\] shows that such an increase does not occur in our case. We suggest that this too is a consequence of the discrete radial excitation spectrum, which although broadened at higher collision rates, remains discrete far above a hydrodynamicity of 5 and therefore suppresses the ability of thermal-thermal collisions to contribute to the damping. Following [@Meppelink09], we have taken the hydrodynamicity in figure\[fig:hydro\] to be $ n_{\mathrm{th}} \langle v_{\mathrm{rel}}\rangle \sigma / \omega_z$, where $n_{\mathrm{th}}=N_{\mathrm{th}} m^{3/2} \omega_{\rho}^2 \omega_z / (4 \pi k_B T)^{3/2}$ is the average thermal atom number density experienced by thermal atoms in the harmonic trap, according to the Maxwell Boltzmann distribution. The quantity $\langle v_{\mathrm{rel}}\rangle=4 \left[ k_B T/(\pi m) \right]^{1/2}$ is the mean relative speed between thermal atoms, and $\sigma=8 \pi a^2$ is the s-wave scattering cross-section. In future, it would be better to derive the thermal density from the Bose-Einstein distribution in the harmonic trap, which fixes the mean density at $0.55\, \zeta(\frac{3}{2})(2\pi m k_B T)^{3/2}/ h^3$, $\zeta$ being the Reimann zeta function. This makes no difference to our conclusions here, but will be important for any future quantitative study of the corrections to Landau damping. In all the damping experiments, the energy in the initial coherent motion is very large compared with $\hbar \omega$. Indeed, the ratio of these is generally greater than the number of atoms in the cloud. It is therefore interesting that the analytical theory reproduces the measured 3D damping rates, because the theory assumes a Bogoliubov mode of energy $\hbar \omega$ that is weakly excited. The agreement between experiment and theory indicates that the damping rate calculated for weak excitations is still applicable when the excitation is strong. Collective excitations have been simulated numerically using the method of Zaremba, Nikuni and Griffin [@Zaremba99], which makes Hartree-Fock and semi-classical approximations to derive a mean field equation for the condensate coupled to a Boltzmann equation for the thermal cloud. Simulations by Jackson and Zaremba [@Jackson01; @Jackson02; @Jackson02b], have proved to be in good agreement with the 3D experiments [@Marago01; @Jin97; @Chevy02] respectively. However, in the 3D/1D cross-over where $\hbar \omega_{\rho} \sim \mu \sim k_B T$, the quantisation of the radial excitations is not well approximated by a semi-classical treatment, as we have shown here. A fully quantum treatment may be possible using the perturbative approach of [@Pitaevskii1997; @Guilleumas03], but we are not aware of any such treatment in the 3D/1D crossover regime. Our results provide a point of reference for such simulations. In future we hope to vary the transverse width of our trap in order to elucidate further the damping behaviour in this dimensional crossover region. Acknowledgments {#acknowledgments .unnumbered} =============== We are indebted to Peter van der Straten, Eugene Zaremba, Gora Shlyapnikov and Nathan Welch for valuable discussions. We acknowledge the expert technical support of Jon Dyne, Steve Maine, and Valerius Gerulis. This work was supported by the European FP7 project AQUTE, by the UK EPSRC and by the Royal Society. References {#references .unnumbered} ==========

--- abstract: 'The evolution of quantum walk on a finite wire under a small increment of vector potential $\alpha$ can exhibit intrinsic quantum oscillation of the two topologically protected bound states corresponding to the Majorana modes. By tuning an external electric field corresponding to the addition of an $\alpha$ impulse at the end of each intrinsic period, the intrinsic oscillation is enveloped by a beat modulation with a longer period. This beat oscillation is useful in the preparation of decoherence-free qubit in trapped ion chain and may be observed in several experiments.' author: - Yue Yu - Kwok Yip Szeto bibliography: - 'bound.bib' title: Field Induced Oscillation of Two Majorana Modes for a finite Quantum Wire --- Introduction ============ The recent advances in discrete time quantum walk [@A4; @A5; @B1; @B2; @B3] have created many interesting research endeavors in its application, such as in quantum computation [@A6; @A7; @A9; @A10; @A11]. However, finding a qubit that is free from decoherence is a major challenge. With our increased understanding of topologically protected systems, we find some recent solutions for this challenging problem [@Nielsen; @Nayak]. In the literature of discrete quantum walk[@B4; @B5; @B6; @B7; @B8; @B9; @B10], the Kitaev model [@Kitaev] for quantum wire is one of the simplest systems that supports these topological phases, which are identified as Majorana boundary modes [@Kitagawa; @Hotat; @B11; @B12; @B13; @B14; @B15; @B16]. In this paper we use the exact solution for these modes in quantum walk in one dimension [@Hotat] to compute the response of the quantum walker to the stepwise increment of vector potential. We find interesting field induced oscillation, which can be useful for the preparation of the decoherence-free qubit. Our calculations involve numerical solutions of a set of coupled nonlinear equations, whose solution can be found analytically under certain limiting cases of long chain and small increment of vector potential. The limit of validity of our approximate calculation turns out to be easily satisfied for most cases and thus provides a useful guideline for the experimentalists. For quantum walk in one dimension, such as in a finite wire of length N, we can find the low energy boundary modes that is symmetry protected Majorana bound states by setting parameters in the coin matrix properly [@Hotat]. In this paper, we are going to study the interaction of the two bound states, and provide a description of these states under a coin matrix with a new phase $\alpha$ that corresponds to the addition of a vector potential. In one-dimensional discrete quantum walk, the evolution of the wave function is given by the unitary transformation $ U = S \otimes C$, which composes of the coin operator $C$ and a shifting operator $S = |L\rangle{\langle}L|\otimes\sum|n\rangle{\langle}n+1|+|R\rangle{\langle}R|\otimes\sum|n\rangle{\langle}n-1|$. Here, $|L,R\rangle$ are the basis of the coin space, while $|n\rangle$ are the basis of the position space. Here we focus on the coin operator in the following form $$C=\left( \begin{array}{cc} \cos{\theta}e^{-i\alpha} & \sin{\theta}e^{-i\alpha} \\ -\sin{\theta}e^{i\alpha} & \cos{\theta}e^{i\alpha} \end{array} \right).\label{Eq:1}$$ If the coin matrix is the same at every position in the system, the energy eigenstates can be written in the form a set of plane waves due to the translational symmetry. By substituting the eigenstate into the unitary operator, we can find the dispersion relationship of E and k, $$\begin{aligned} \psi_{k}(n,t) = e^{-iEt+ikn}\left[\begin{array}{ccc} a_k\\ b_k \end{array} \right], \ \cos(E)=\cos{\theta}\cos(q),\label{Eq:3} \end{aligned}$$ where $q=k-\alpha$. For a given energy level, there are two corresponded eigen-momentum; the wave function is the linear combination of these two plane waves [@Hotat2]. The eigenvector $[a_k\,b_k]^{T}$ is $$\left[\begin{array}{c} a_k \\ b_k\end{array} \right]= \frac{1}{\sqrt{2-2\cos\theta\cos(E+q)}}\left[\begin{array}{c} \sin\theta \\ (e^{-i(E+q)}-\cos\theta)\end{array} \right].\label{Eq:4}$$ We are interested in the effect of a step-wise increase of the parameter $\alpha$ as a function of time, which corresponds to the step-wise increase of the vector potential applied to the one-dimensional system [@Hotat]. Experimentally this corresponds to the application of an impulse of electric field at specified time. Here k is the canonical momentum and $q$ is the kinetic momentum. For a given system with translational invariance and known eigenstate $\psi_{\alpha}$ for a given $\alpha$, there exists a simple relation between the eigenstate if $\alpha$ is changed to $\alpha'$ uniformly for every points in the system as, $\psi_{\alpha'}(n,t)=\psi_{\alpha}(n,t)e^{i(\alpha'-\alpha)n }$. The relative phase between the two neighbouring points will compensate the effect of $\alpha'-\alpha$ in the coin matrix, so that $$\begin{aligned} &(C_{\alpha'}\psi_{\alpha'}(n,t))_L=\psi_{\alpha'}(n-1,t+1)_L, \\&(C_{\alpha'}\psi_{\alpha'}(n,t))_R=\psi_{\alpha'}(n+1,t+1)_R. \end{aligned}\label{Eq:6}$$ This implies that the eigen-energy and the parameter $q$ are unchanged, but the quasi-momentum $k=q+\alpha$ is replaced by $k'=q+\alpha'$. This result is important when we tune the $\alpha$ in a step-wise manner.   Evolution of the bound states under single impulse ================================================== We first summarize the known results for the bound state that appears at the topological boundary where the rotation parameter $\theta$ changes its signs. For the system with single boundary, the ground state eigen-energy is exactly zero. Now we consider a finite system of length N+2; the coordinates are set to be $n=0...N+1$. Let us consider the example with the following setup of the finite chain: $\theta_1$ is negative at both ends $n=0,N+1$ while $\theta_0$ is a positive constant in the middle. In this way, we have two topological boundaries, and we anticipate the existence of two symmetry protected bound states[@Kitagawa; @Kitaev; @Hotat]. Specifically, if we set $\theta_1=-\pi/2$ so that the diagonal terms in the coin matrix in Eq.\[Eq:1\] is zero, then the motion of the walker will be completely reflected at the boundaries. Physically this describes quantum walk on a finite wire. Now we consider the bound state of this system. Due to the change of the signs of $\theta$, the wave function is delocalized at the two ends for this ground state. Equivalently, two quasi-particles are formed around $n=1$ and $n=N$. The eigen-energy $E_0$ of the ground state can deviate from zero, as there can be an effective interaction between them[@Hotat]. We first review our exact solution of these two bound states for a zero $\alpha$. Physically, if there were boundary modes, their interaction energy should depend on the size of the system as well as the coin parameter $\theta$, which describes the inertia of the walker. We propose an ansatz for the wave function of the ground state as a linear combination of two momentum eigenstates, $$\begin{aligned} \psi_{k}(n,t)= &e^{-iEt}(c_{L}e^{-\kappa{n}}\left[\begin{array}{ccc}a_{i\kappa}\\b_{i\kappa} \end{array} \right]+\\&c_{R}e^{-\kappa{(N+1-n)}}\left[\begin{array}{ccc}a_{-i\kappa}\\b_{-i\kappa} \end{array} \right]) ,\,n=1...N. \end{aligned}\label{Eq:7}$$ For the bound state, the momentum is purely imaginary, so we set $\kappa=-ik$ for simplicity. The eigenvector $[a_{\pm{i}\kappa},b_{{\pm}i\kappa}]^{T}$ is given by Eq.\[Eq:4\]. At the two ends, the wave function needs to satisfy the continuity requirement, $$\begin{split} \psi_{k}(0,t) = e^{-iEt}[c_{L}a_{i\kappa}+c_{R}e^{-\kappa{(N+1)}}a_{-i\kappa}, 0]^T, \\ \psi_{k}(N+1,t) = e^{-iEt}[0, c_{L}e^{-\kappa{(N+1)}}b_{i\kappa}+c_{R}b_{-i\kappa}]^T. \end{split} \label{Eq:8}$$ The boundary conditions at the two ends provide two linear equations about $c_L$ and $c_R$. By solving the equation for the determinant of the coefficients matrix, we can get solutions for $(\kappa,E)$. We first solve this numerically and we find that the bound state energy is very small, thus we assume the small E limit and obtain analytically the following dispersion relation: $$e^{\kappa}=\frac{1+\sin\theta}{\cos\theta}+O(E^2).\label{Eq:11}$$ In this small E approximation, we get the expression of the energy $E_0$, $$E_0={\pm}2\tan\theta_0(\frac{\cos\theta_0}{1+\sin\theta_0})^{N+1}.\label{Eq:13}$$ With the approximate solutions for $\kappa$ and $E_0$ , we can proceed to compute analytically the change of the system under the application of an impulse realized by a stepwise increase of the parameter $\alpha$. The corresponding boundary mode with energy $+E_0$ evolves as $$\begin{split} \psi_{0}^{+}(n,t) = &e^{-iE_{0}t} \bigg( c_{L}e^{-\kappa{n}}\left[\begin{array}{ccc}a_{i\kappa}\\b_{i\kappa} \end{array} \right]+\\&c_{R}e^{-\kappa{(N+1-n)}}\left[\begin{array}{ccc}a_{-i\kappa}\\b_{-i\kappa} \end{array} \right] \bigg) ,\,n=1...N. \end{split}\label{Eq:14}$$ Here the superscript denotes energy of the mode, in this case it is the mode with energy $+E_0$ . (The other mode has energy $-E_0$.) The subscript denotes the vector potential parameter $\alpha$ and in this case it is 0. At time $t=0^+$, we add a non-zero ${\alpha_0}$ impulse to the entire wire, then the original wave function can be expressed as a linear combination of the set of new eigenstates of the system after the impulse with the nonzero ${\alpha_0}$, $\psi(n,0)_{0}^{+}=c_{+}\psi^{+}_{{\alpha_0}}(n,0)+c_{-}\psi^{-}_{{\alpha_0}}(n,0)+R$, where $ \psi^{\pm}_{\alpha_0}$ are the two ground states after adding ${\alpha_0}$ and the term $R=\sum_{i\neq{\pm}}{c_i}\psi^{i}_{\alpha_0}$ describes the projection to all those states that are not the boundary modes. If we assume that $R$ is small, then we can describe the evolution of the system after the addition of the nonzero vector potential $\alpha_0$ by a new state with a two-level system composed only with the two ground states $ \psi^{\pm}_{\alpha_0}$. We will later verify that this two-level approximation is valid for a small step-wise change of $\alpha$. Using Eq.\[Eq:6\], we can write the new eigenstates as, $\psi^{\pm}_{{\alpha_0}}(n,t)=\psi^{\pm}_{0}(n,t)e^{i{\alpha_0}{n}}$ and then compute the coefficients $c_{\pm}$ by the inner products between the old and new eigenstates. $$\begin{split} c_{\pm}&=\sum_{n}\psi_0^{+}(n,0)\psi_{{\alpha_0}}^{\pm}(n,0)^{*} \\&=\sum_{n}\psi_{0}^{+}(n,0)^{*}\psi_{0}^{\pm}(n,0)e^{-i{\alpha_0}{n}}. \label{Eq:17} \end{split}$$ Note that the new wave function will oscillate since it is no longer a stationary state after the impulse. The frequencies of the oscillation modes are equal to the energy gaps between different energy eigenstates of the system with $\alpha_o >0$. In our two-level system approximation, the only energy gap is $2E_0$. The error of this approximation can be estimated by looking at the effect of neglecting R, which is the inner product of $\psi_0^{+}(n,0)$ with other higher energy levels. This error can be estimated by calculating the total probability of $|c_{+}|^2+|c_{-}|^2$, which should be close to 1 if the higher energy levels terms in R can be ignored. We first consider the case of long chain for which our approximate expression for $\kappa$ and $E_0$ are very good [@Hotat]. Physically, for a long wire, the two quasi-particles are effectively isolated. Under this condition, $\psi_0^{+}(n,0)^{*}\psi_{{\alpha_0}}^{\pm}(n,0)$ can be rewritten as four exponential functions, $$\begin{aligned} &\psi_{0}^{+}(n,0)^{*}\psi_{0}^{\pm}(n,0)_L =A(e^{-2\kappa{n}}{\pm}e^{-2\kappa(N-n)}), \\ &\psi_{0}^{+}(n+1,0)^{*}\psi_{0}^{\pm}(n+1,0)_R=A(e^{-2\kappa{n}}{\pm}e^{-2\kappa(N-n)})\\ &n=0...N. \end{aligned}\label{Eq:18}$$ The first two terms are from the left component and the last two terms are from the right component of the wave function. The $\pm$ sign come from the orthogonality of the original two ground states. For a long wire, $N\gg{1}$, it is reasonable to let the upper limit in the summand in Eq.\[Eq:17\] to go to infinity, $n\rightarrow\infty$, thereby obtaining the analytical result, $$c_{\pm}=\frac{1-e^{-2\kappa}}{4}(\frac{1}{1-e^{-2\kappa-i{\alpha_0}}}\pm\frac{e^{-iN{\alpha_0}}}{1-e^{-2\kappa+i{\alpha_0}}})(1+e^{-i{\alpha_0}}).\label{Eq:19}$$ If now we apply a very small impulse so that $N{\alpha_0}\ll{1}$, then we can expand $c_{+}$ and $c_{-}$ by keeping terms up to $O({\alpha_0}^2)$, $$\begin{aligned} & c_{+}=1-(\frac{N+1}{2})i{\alpha_0}-(\dfrac{1}{4}N^2+\frac{N}{4}-\frac{N} {2(e^{2\kappa}-1)}+\frac{1}{4}+\frac{e^{2\kappa}+1}{2(e^{2\kappa}-1)^2}){\alpha_0}^2, \\ & c_{-}=-(\frac{1}{e^{2\kappa}-1}-\frac{N}{2})i{\alpha_0}+(\dfrac{1}{4}N^2+\frac{N}{4}-\frac{N+1}{2(e^{2\kappa}-1)}){\alpha_0}^2, \\ & |c_{+}|^2+|c_{-}|^2=1-(\frac{1}{4}+\frac{e^{2\kappa}}{(e^{2\kappa}-1)^2}){\alpha_0}^2. \end{aligned}\label{Eq:20}$$ When $N{\alpha_0}\ll{1}$, the total probability becomes linear in ${\alpha_0}^2$ only. However, even when N is small, the above expression for $ c_{\pm}$ is still good as long as $N{\alpha_0}\ll{1}$. This can be seen from a comparison of this simple expression in Eq.\[Eq:20\] with the numerical calculation without ignoring R for different $\theta$ at N=4 ( Fig.\[Fig 1\]). We see that the percentage difference for the Hadamard walk for N=4 is only 5$\%$ for a single impulse that change $\alpha$ from 0 to $\alpha_0=0.01\pi$. The condition for the two-level system approximation to be good is thus $N{\alpha_0}\ll{1}$. ![(a) The percentage difference of the slope in Eq.\[Eq:20\]. The error is determined by N and $\theta$. For $\theta=\pi/4$, which corresponds to the Hadamard walk, the error is smaller than 5$\%$ for N=4. (b) The intrinsic oscillation between the two ground states. The size of the system is N=4. An impulse ${\alpha_0}=0.01\pi$ is added at $t=0$. The intrinsic oscillation $T_0$ is around 128.5.[]{data-label="Fig 1"}](ngg1 "fig:"){width="4.2cm"} ![(a) The percentage difference of the slope in Eq.\[Eq:20\]. The error is determined by N and $\theta$. For $\theta=\pi/4$, which corresponds to the Hadamard walk, the error is smaller than 5$\%$ for N=4. (b) The intrinsic oscillation between the two ground states. The size of the system is N=4. An impulse ${\alpha_0}=0.01\pi$ is added at $t=0$. The intrinsic oscillation $T_0$ is around 128.5.[]{data-label="Fig 1"}](intrinsicoscillation "fig:"){width="4.2cm"} To understand the physics of our two-level approximation, let’s consider a small N (=4) system. The ground state energy is $7.78\times{10}^{-3}\pi$, while the first excited state has energy $0.34\pi$. Thus, we expect that the impulse on the system by switching from zero to a small $\alpha_0\ll{1\over N}$ does not excite the system to higher energy levels through the effective E field produced by the sudden change in vector potential, since the energy gap $\delta E =2E_o$ between the two ground states $E_0$ and $-E_0$ is much smaller than any other energy gaps. We can therefore observe the intrinsic oscillation of the two level system (two Majorana modes) in a finite wire as long as our change in vector potential is very small ($\alpha_0\ll{1/N}$) . Evolution of bound states under multiple impulses ================================================= The characteristic period of the intrinsic oscillation of the two-level system with energy difference $\delta E=2E_0$ is $T_0=2\pi/\delta E=\pi/E_o$. This can be observed by taking some time average of an observable over $T_0$. A more interesting oscillation is related to the increase of the vector potential ${\alpha}$ by hitting the system with multiple impulses so that $\alpha(t)=m\alpha_0,\,t\in[(m-1)T_0,mT_0),\,m=0,1,2...$ with a small $\alpha_0$ such that $N\alpha_0\ll{1}$. From the analysis of single impulse, we expect that it is still reasonable to neglect other states R in the projection if $N\alpha_0\ll{1}$, so that the wave function can be written as a linear combination of the two ground states and the two-level approximation is still good. $$\begin{aligned} \psi(n,t)&=c_{+}e^{-iE_0t} \psi^{+}_{\alpha}(n,t)+c_{-} e^{iE_0t} \psi^{-}_{\alpha}(n,t).\label{Eq:23} \end{aligned}$$ At the end of each intrinsic period $t=mT_0$, a new $\alpha$ is introduced; and the wave function is projected to a new basis of eigenfunctions. During the time interval $[(m-1)T_0,mT_0)$, both eigenstates $ \psi^{\pm}_{\alpha}$ go through half of period $\pi/E_0$, and the wave function before the next projection is the same state as $t=(m-1)T_0$ after the previous projection, $$\begin{aligned} &\psi(n,mT^{-}_0)=-(c_{+} \psi^{+}_{\alpha}(n)+c_{-} \psi^{-}_{\alpha}(n)),\\ &\psi(n,(m-1)T^{+}_0)=c_{+}\psi^{+}_{\alpha}(n)+c_{-} \psi^{-}_{\alpha}(n). \end{aligned}\label{Eq:24}$$ As a result, the two ground states do not change between subsequent projections, except for the overall minus sign. As long as $\alpha_0$ is small, the projection planes are close to each other. Although those states in R, which are outside of our two-level system, are changed in the time interval between impulses, they have higher energy and their projection are oscillating with a much higher frequency, so that the contribution of R is still very small as subsequent projections are close, which is true when $\alpha_0$ is small. Thus, we can use our two-level approximation to compute the time evolution of the two coefficient $c_{+}$ and $c_{-}$ analytically as in the case of single impulse. Because we are interested in the oscillations larger than the intrinsic period, we can measure time by the number of intrinsic periods $m$. For each impulse, the transition matrix $P_{\alpha}$ between states in the two level system can be defined as, $[c_{+},c_{-}]^T_{m+1}=P_{\alpha}[c_{+},c_{-}]^T_{m}$. The elements in the $2\times{2}$ transition matrix $P_{\alpha}$ are the inner products between the new and old basis in the two-level system, $$\begin{split} P_{ij}&=\sum_{n}\psi^{i}_{\alpha+\alpha_0}(n)^{*}\psi^{j}_{\alpha}(n) =\sum_{n}\psi^{i}_{\alpha_0}(n)^{*}\psi^{j}_{0}(n),\, i,j=\pm.\label{Eq:27} \end{split}$$ Note that $P_{ij}(\alpha)$ depends on $\alpha_0$, since the phase $e^{i\alpha{n}}$ of the new and old basis cancels. We thus have $$\begin{aligned} P_{++}& =P_{--}=1-(\frac{N+1}{2})i\alpha_0\\&-(\dfrac{1}{4}N^2+\frac{N}{4}-\frac{N}{2(e^{2\kappa}-1)}+\frac{1}{4}+\frac{e^{2\kappa}+1}{2(e^{2\kappa}-1)^2})\alpha_0^2,\\ P_{+-}&=P_{-+}=-(\frac{1}{e^{2\kappa}-1}-\frac{N}{2})i\alpha_0 \\&+(\dfrac{1}{4}N^2+\frac{N}{4}-\frac{N+1}{2(e^{2\kappa}-1)})\alpha_0^2. \end{aligned}\label{Eq:28}$$ and the eigenvalues and eigenvectors of the transition matrix are, $\lambda_{1,2}=P_{++}\pm{P}_{+-},\, v_{1,2}=[\frac{1}{\sqrt{2}},\pm\frac{1}{\sqrt{2}}]^{T}$. The absolute values of these two eigenvalues are less than one, implying a loss of information. Further calculations show that they have equal magnitude, $|\lambda_{1,2}|^2=1-(\frac{1}{4}+\frac{e^{2\kappa}}{(e^{2\kappa}-1)^2})\alpha_0^2$. We see that the total probability decreases with a speed proportional to $\alpha_0^2$ after each intrinsic period, so that after $m$ intrinsic periods, the total probability becomes $$\begin{aligned} |c_{+}|^2+|c_{-}|^2&=(|\lambda_{1,2}|^2)^m\\&= \left( 1-\left( \frac{1}{4}+\frac{e^{2\kappa}}{(e^{2\kappa}-1)^2}\right) \alpha_0^2\right) ^m. \end{aligned}\label{Eq:31}$$ This expression gives the evolution of the total probability using the two-level system approximation when the system is under the application of impulse $m$ times, each time the value of $\alpha$ is increased by $\alpha_0$ and the impulse is applied periodically with an interval $T_0$. For the same value of $N\alpha_0$, the longer the wire, the smaller is the loss of probability, and the better is the two level system approximation. ![The loss of probability under multiple impulses. Three different $\theta$ are used on the N=5,10 system. $\alpha_0$ is set to be $0.01\pi$ and $0.005\pi$ respectively so that in both cases, $N\alpha_0 =0.05\pi$. The loss of probability is less than 5$\%$ for Hadamard coin after m=100 intrinsic oscillations. The prediction is Eq. \[Eq:31\], shown here by the lines.[]{data-label="Fig 2"}](Malpha){width="6.5cm"} Our numerical results (Fig.\[Fig 2\]) show that for Hadamard coin with $N\alpha_0 =0.05\pi$, the loss of probability is less than 5$\%$ for $N=5$ after 100 intrinsic oscillations. In Fig.\[Fig 3\] we show the average position and variance of the Hadamard walk under the application of multiple impulse. We see the intrinsic oscillation is enclosed by an envelope function with a longer period. The phase difference $\Delta\phi$ is exactly the beat frequency of this two level system, $f_{\text{beat}}=\Delta\phi=(N-\frac{2}{e^{2\kappa}-1})\alpha_0$ with the period of the beat being $T_{\text{beat}}/T_0=\frac{2\pi}{\Delta\phi}$. ![The average position and variance of Hadamard walk with periodic impulse. Only the two ground states are considered and the total probability is normalized. The size of the wire is N=4, and the unit impulse is $\alpha_0=0.01\pi$. The system evolves for m=100 intrinsic periods. Note the system returns to original state $E=E_0$ around $m=\frac{2\pi}{\tan^{-1}(\Delta\phi)}=55.8$.[]{data-label="Fig 3"}](averageposition "fig:"){width="4.2cm"} ![The average position and variance of Hadamard walk with periodic impulse. Only the two ground states are considered and the total probability is normalized. The size of the wire is N=4, and the unit impulse is $\alpha_0=0.01\pi$. The system evolves for m=100 intrinsic periods. Note the system returns to original state $E=E_0$ around $m=\frac{2\pi}{\tan^{-1}(\Delta\phi)}=55.8$.[]{data-label="Fig 3"}](variance "fig:"){width="4.35cm"} Observation of Quantum Oscillation ================================== The intrinsic oscillation originated from the energy gap between the two Majorana modes can be observed if we start the quantum walker at the end of the wire ($n=0$). This initial state $\psi(0,0)_{L}=1$ can be approximated by a linear combination of the two Majorana modes in the form of Eq.\[Eq:14\], since all the other scattering states are not localized. In Fig. \[Fig 4\](a), we show the numerical results of the quantum walk starting with this initial state and the result can be compared with a measurement of the probability distribution at $n=0$ as a function of time. Furthermore, when we apply a small increase of the external electric field at the end of each intrinsic period, the numerical results show the beat oscillation in Fig. \[Fig 4\](b), which can be compared with the measurement of the maximum probability of the quantum walker at $n=0$ after each intrinsic period $T_o$. ![(a) The probability at $n=0$ in the system with length $N+2=10$, for Hadamard walk. Here the intrinsic oscillation is clearly observed in the first two intrinsic periods, which is $T_0\approx4376$. (b) The beat oscillation in the quantum wire can be seen from the maximum probability at $n=0$ at each intrinsic period $T_{0}$. Here the length of the wire is $N+2=10$ for Hadamard walk and the vector potential parameter is $\alpha_0=0.02\pi$. []{data-label="Fig 4"}](run1 "fig:"){width="4cm"} ![(a) The probability at $n=0$ in the system with length $N+2=10$, for Hadamard walk. Here the intrinsic oscillation is clearly observed in the first two intrinsic periods, which is $T_0\approx4376$. (b) The beat oscillation in the quantum wire can be seen from the maximum probability at $n=0$ at each intrinsic period $T_{0}$. Here the length of the wire is $N+2=10$ for Hadamard walk and the vector potential parameter is $\alpha_0=0.02\pi$. []{data-label="Fig 4"}](run2 "fig:"){width="4.1cm"} Now, for a real system of quantum wires, their lengths usually cannot be all equal. We therefore generalize our numerical calculation to an ensemble of quantum wires of different lengths arranged in parallel along the $x$ direction, with the left end points of these wires forming a surface in the $yz$ plane. If the initial state of the walkers is a linear combination of the Majorana modes with maximum probability at $x=0$ on this surface, we can observe the beat oscillation on the wire with a particular length $N_0$ by adding an electric field at the end of the intrinsic period $T_0=2\pi/\delta E=\pi/E_o$ where $E_o(N_0)$ is computed for that specific length $N_0$. To illustrate this selection mechanism on the beat oscillation, let us consider an ensemble of wires of length $N+2= 9, 10, 11$ with corresponding concentration of $0.2, 0.6, 0.2$. Numerical calculation for this ensemble is shown in Fig. \[Fig 8\]. The three sets of beat pattern are superimposed and the beat period for the ensemble is $12.5T_0$ from simulation, which is very close to the value $13.2 T_0$ for a pure sample with all wires of length $N+2=10$ ![The evolution of the probability at $n=0$ when we add a small increment of $\alpha_0=0.02\pi$ to the ensemble of wires of length $N+2= 9, 10, 11$ with corresponding concentration of $0.2, 0.6, 0.2$. We observe that only the correct system can provide a peak of the beat oscillation at $13.2T_0$. The ensemble average is rescaled by ten. []{data-label="Fig 8"}](run4){width="8cm"} Discussion ========== In this paper, we first set the two topological boundary conditions at the ends of a finite wire to construct the two quasi-particles which are shown to be Majorana modes recently [@Hotat; @Kitaev; @Kitagawa]. From the exact calculation of these modes[@Hotat], we obtain the interaction between the two quasi-particles and show the deviation of the ground state energy $\pm {E_o}$, computed for a quantum wire of specific length $N_0$. In order to explore the quantum oscillation of these modes, we initialize the system in the ground state ($c_{+}=1, c_{-}=0$) and apply a single electric impulse at $T_{0}$ by an increase of the vector potential parameter $\alpha_0$ in the coin matrix. Before the application of the impulse, the variance of the intrinsic oscillation is at a maximum because the wave function is delocalized, while the average position is in the middle of the wire. After the addition of an impulse, the wave function starts to oscillate with a frequency equal to the energy gap $2E_0$ in our two-level approximation which is valid when $\alpha_0$ is sufficiently less than $1 /N$ . The oscillating probability density $|\psi|^2$, its associated probability current, and the average speed are all physical observables. This field induced oscillation provides a possible way to realize the delocalized ground state, with a characteristic intrinsic period $T_{0}=\pi / E_o$. We also consider the application of multiple impulse through the increase of the potential parameter $\alpha_0$ in the coin matrix, at designated time $T_m=mT_{0}$. Numerical calculation shows that there is an envelope function describable as a beat oscillation. For example, if the walker starts with zero velocity at the end of the wire of length 4, the wave function will evolve to become the wave package at $t=14T_0$. By slowly increasing the coin parameter $\alpha$ at $t=mT_0$ by the small amount $\alpha_0$ (as used in Fig. \[Fig 3\]), then the delocalized ground state is expected to appear around $t=56T_0$, with very limited mixture with the other states. The variance of probability distribution function goes to minimum (see Fig.\[Fig 3\]), which means that the wave function is similar to a narrow wave package at $t=\frac{T_{\text{beat}}}{4}\approx 14T_0$, while its oscillation amplitude is maximum. In principle, this wave function can go to another delocalized ground state ($c_{+}=0, |c{-}|=1$) around $t=\frac{T_{\text{beat}}}{2}\approx28T_0$. However, this does not happen due to interference because for $N=4$, the intrinsic period is $T_0\approx128.5$, which is not an integer. For $N=3$, where $T_0$ is closer to an integer, the system can reach the other delocalized state clearly. This observation may be useful in the preparation of robust qubit by choosing appropriate N and $\theta_0$ so that the intrinsic period $T_0$ is very close to an integer. Note that one may reduce the beat period by increasing $\alpha_0$, since the beat frequency is proportional to $\alpha_0$ in lowest order. However, there is a trade-off between the reduction of the beat period and the goodness of the approximation of the two-level system that requires a small $\alpha_0$. In conclusion, this field induced oscillation provides a possible way to prepare the delocalized ground state. We expect the one-dimensional quantum walk on finite wire can be realized in a trapped-ion chain with an Majorana qubit encoded [@Mezzacapo]. This qubit is topologically protected against major sources of decoherence, thereby providing an efficient quantum memory. The oscillation of these symmetry protected bound states provide a new phenomenon that can be exploited. Experimentally, we can place the finite wire in $xy$ plane, add electric field along $x$ in a stepwise manner with period $T_0$, then our theory predicts the observation of the oscillation on the average position of the particle with a specified beat frequency. This long period beat oscillation is particularly impressive compared with relatively fast intrinsic oscillation due to the nature of discrete quantum walk. For the $Ca^{2+}$ realization of quantum walk, the hopping time is in the order of $2ns$, while the beat period for $N=4$ system is with the order of $20\mu{s}$, with significant improved coherence lifetimes. A second experiment where our results can be tested is through the electrical measurements on indium antimonide nanowires contacted with two electrodes, one normal and one superconducting [@black1]. In these experiments, the electron density can be changed by varying the gate voltage. By properly measuring the probability current, one may be able to observe our predicted beat oscillation when the electric field is increased at a rate defined by the intrinsic frequency $T_0$ which in turn is fixed by the length of the nanowires. A third experiment that may test our prediction is the work by [@black2]. By placing ferromagnetic atomic chains on the surface of superconducting lead, they fabricated a one-dimensional topological superconductor with the appearance Majorana fermions similar to our model of one-dimensional quantum walk on a finite wire. Other recent works on black phosphorus [@wangning] may be another example for the detection of the beat oscillation. These experimental systems may directly provide evidence of the Majorana modes. We acknowledge discussion with Lam Hotat and Wang Ning.

--- author: - | \ Department of Physics, Tokyo Metropolitan University\ Hachioji, Tokyo 192-0397, Japan\ E-mail: title: 'Constraints on non-standard flavor-dependent interactions from Superkamiokande and Hyperkamiokande' --- Introduction ============ The phenomenon of neutrino oscillation is described by the mixing matrix $$\begin{aligned} U=\left( \begin{array}{ccc} c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\delta}\nonumber\\ -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta} & c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta} & s_{23}c_{13}\nonumber\\ s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta} & -c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta} & c_{23}c_{13}\nonumber\\ \end{array}\right), \label{eqn:mns}\end{aligned}$$ where $s_{jk}\equiv\sin\theta_{jk}$ and $c_{jk}\equiv\cos\theta_{jk}$, $\theta_{jk}$ with $(j,k)=(1,2), (1,3), (2,3)$ are the three mixing angles and $\delta$ is the Dirac CP phase. Thanks to recent neutrino experiments [@Agashe:2014kda], all the mixing angles and the mass squared differences have been measured, and the only unknown quantities which can probed by neutrino oscillation are the mass hierarchy pattern and $\delta$. It is believed that these unknown quantities will be determined in the future neutrino experiments, including those with intense accelerator neutrino beams [@Abe:2014oxa; @Adams:2013qkq]. It is also expected that these future experiments with intense accelerator neutrino beams will enable us to probe new physics beyond the standard model with massive neutrinos, by looking for the deviation from the standard scenario. In the standard model with three massive neutrinos, the Dirac equation for the flavor eigenstate $\Psi^T\equiv(\nu_e,\nu_\mu,\nu_\tau)$ of neutrino in matter is given by $$\begin{aligned} i{d\Psi \over dt}= \left[U \mbox{\rm diag}\left(E_1,E_2,E_3\right) U^{-1} +{\cal A} \right]\Psi, \label{sch1}\end{aligned}$$ where the matter potential is given by $$\begin{aligned} {\cal A}=A\left( \begin{array}{ccc} 1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array} \right). \label{matter-std}\end{aligned}$$ Here $A\equiv \sqrt{2} G_F n_e$ stands for the magnitude of the standard matter effect due to the charged current interaction, $n_e$ is the number density of the electron in the matter, and the matter effect due to the neutral current interaction, which is proportional to the unit matrix in the flavor basis, is ignored because it would affect only the phase of the oscillation probability amplitude. Here I would like to consider the flavor-dependent nonstandard four-fermi interactions $$\begin{aligned} {\cal L}_{\mbox{\rm\scriptsize eff}}^{\mbox{\rm\scriptsize NSI}} = -2\sqrt{2}\, \epsilon_{\alpha\beta}^{fP} G_F (\overline{\nu}_\alpha \gamma_\mu P_L \nu_\beta)\, (\overline{f} \gamma^\mu P f), \label{NSIop}\end{aligned}$$ where only the interactions with $f = e, u, d$ are relevant to the flavor transition of neutrino due to the matter effect, $G_F$ denotes the Fermi coupling constant, $P$ stands for a projection operator and is either $P_L\equiv (1-\gamma_5)/2$ or $P_R\equiv (1+\gamma_5)/2$. (\[NSIop\]) is the most general form of the interactions which conserve electric charge, color, and lepton number [@Davidson:2003ha]. In the presence of these interactions (\[NSIop\]), the matter potential is modified to $$\begin{aligned} {\cal A} \to A\left( \begin{array}{ccc} 1+ \epsilon_{ee} & \epsilon_{e\mu} & \epsilon_{e\tau}\\ \epsilon_{e\mu}^\ast & \epsilon_{\mu\mu} & \epsilon_{\mu\tau}\\ \epsilon_{e\tau}^\ast & \epsilon_{\mu\tau}^\ast & \epsilon_{\tau\tau} \end{array} \right), \label{matter-nsi}\end{aligned}$$ where $\epsilon_{\alpha\beta}$ are defined as $\epsilon_{\alpha\beta} \equiv \sum_{f,P}(n_f/n_e) \epsilon_{\alpha\beta}^{fP} \simeq \sum_{P} \left( \epsilon_{\alpha\beta}^{eP} + 3 \epsilon_{\alpha\beta}^{uP} + 3 \epsilon_{\alpha\beta}^{dP} \right)$, $n_f$ is the number density of $f$ in matter, and we have taken into account the fact that the number density of $u$ quarks and $d$ quarks are three times as that of electrons. The constraint on $\epsilon_{\alpha\beta}$ haven been given by a number of works, and can be summarized as [@Biggio:2009nt] $$\begin{aligned} \left( \begin{array}{ccc} |\epsilon_{ee}|< 4\times 10^0 & |\epsilon_{e\mu}| < 3\times 10^{-1} & |\epsilon_{e\tau}| < 3\times 10^0\\ & |\epsilon_{\mu\mu}| < 7\times 10^{-2} & |\epsilon_{\mu\tau}| < 3\times 10^{-1}\\ & & |\epsilon_{\tau\tau}| < 2 \times 10^1 \end{array} \right). \label{b-eps-0}\end{aligned}$$ From Eq. (\[b-eps-0\]) we see that $$\begin{aligned} \epsilon_{e\mu} \simeq \epsilon_{\mu\mu} \simeq \epsilon_{\mu\tau} \simeq 0 \label{eps-mu}\end{aligned}$$ is satisfied. On the other hand, it was shown [@Friedland:2005vy] that the non-standard matter potential (\[matter-nsi\]) is consistent with the high energy atmospheric neutrino data only if $$\begin{aligned} \epsilon_{\tau\tau} \simeq \frac{|\epsilon_{e\tau}|^2} {1 + \epsilon_{ee}} \label{b-eps-3}\end{aligned}$$ is satisfied. In this talk I will discuss the constraints on the flavor-dependent non-standard interactions from the Superkamiokande atmospheric neutrino data. Taking the constraints (\[eps-mu\]) and (\[b-eps-3\]) into consideration, for simplicity I will take the following ansatz for the matter potential[^1]: $$\begin{aligned} A\left( \begin{array}{ccc} 1+ \epsilon_{ee} & 0 & \epsilon_{e\tau}\\ 0 & 0 & 0\\ \epsilon_{e\tau}^\ast &0 & \frac{|\epsilon_{e\tau}|^2} {1 + \epsilon_{ee}} \end{array} \right). \label{matter-nsi1}\end{aligned}$$ I will also give the expected sensitivity to the same parameters from the future atmospheric neutrino data by the Hyperkamiokande experiment[@Abe:2011ts]. The result will be given as the allowed region in the ($\epsilon_{ee}$, $|\epsilon_{e\tau}|$) plane by marginalizing the $\chi^2$ with respect to the standard oscillation parameters as well as arg($\epsilon_{e\tau}$). Analysis ======== The data we analyzed is for 3903 days[@Itow:2013zza]. The analysis[@Itow:2013zza] by the Superkamiokande collaboration uses information which is even more detailed than the one in Ref. [@Ashie:2005ik], and we have been unable to reproduce their results of the Monte Carlo simulation. So we have combined the two sub-GeV $\mu$-like data set in one, the two multi-GeV e-like in one, the two partially contained event data set and the multi-GeV $\mu$-like in one, and the three upward going $\mu$ in one. The analysis was performed with the code which were used in Refs. [@Foot:1998iw; @Yasuda:1998mh; @Yasuda:2000de]. $\chi^2$ is defined as $$\begin{aligned} \chi^2= \min_{\theta_{23},|\Delta m^2_{32}|,\delta, \mbox{\rm arg}(\epsilon_{e\tau})} \left[ \chi_{\rm sub-GeV}^2+\chi_{\rm multi-GeV}^2 +\chi_{\rm upward}^2\right]\,, \label{eqn:chi}\end{aligned}$$ where $$\begin{aligned} %&{\ }& \displaystyle\chi_{\rm sub-GeV}^2 %\nonumber\\ &=& \min_{\alpha,\beta's}\left[ \frac{\beta_{s1}^2}{\sigma_{\beta s1}^2} +\frac{\beta_{s2}^2}{\sigma_{\beta s2}^2}\right. \nonumber\\ &{\ }&+\sum_{j=1}^{10}\left\{ \frac{1}{n_j^{\rm s}(e)} \left[ \alpha (1-{\beta_{s1} \over 2}+{\beta_{s2} \over 2})N_j^{\rm s}(e) + \alpha (1-{\beta_{s1} \over 2}-{\beta_{s2} \over 2})\bar{N}_j^{\rm s}(e) -n_j^{\rm s}(e)\right]^2 \right. \nonumber\\ &{\ }&\left.\left.+\frac{1}{n_j^{\rm s}(\mu)} \left[ \alpha (1+{\beta_{s1} \over 2}+{\beta_{s2} \over 2})N_j^{\rm s}(\mu) +\alpha (1+{\beta_{s1} \over 2}-{\beta_{s2} \over 2})\bar{N}_j^{\rm s}(\mu) -n_j^{\rm s}(\mu)\right]^2 \right\}\right],\nonumber\\ \displaystyle\chi_{\rm multi-GeV}^2&=& \min_{\alpha,\beta's}\left[ \frac{\beta_{m1}^2}{\sigma_{\beta m1}^2} +\frac{\beta_{m2}^2}{\sigma_{\beta m2}^2}\right. \nonumber\\ &{\ }&+\sum_{j=1}^{10}\left\{ \frac{1}{n_j^{\rm m}(e)} \left[ \alpha (1-{\beta_{m1} \over 2}+{\beta_{m2} \over 2})N_j^{\rm m}(e) + \alpha (1-{\beta_{m1} \over 2}-{\beta_{m2} \over 2})\bar{N}_j^{\rm m}(e) -n_j^{\rm m}(e)\right]^2 \right. \nonumber\\ &{\ }&\left.\left.+\frac{1}{n_j^{\rm m}(\mu)} \left[ \alpha (1+{\beta_{m1} \over 2}+{\beta_{m2} \over 2})N_j^{\rm m}(\mu) +\alpha (1+{\beta_{m1} \over 2}-\frac{\beta_{m2}}{2})\bar{N}_j^{\rm m}(\mu) -n_j^{\rm m}(\mu)\right]^2 \right\}\right],\nonumber\\ \displaystyle\chi_{\rm upward}^2&=& \min_{\alpha}\left[ \frac{\alpha^2}{\sigma_{\alpha}^2} +\sum_{j=1}^{10} \frac{1}{n_j^{\rm u}(\mu)} \left[ \alpha N_j^{\rm u}(\mu) -n_j^{\rm u}(\mu)\right]^2\right].\end{aligned}$$ are $\chi^2$ for the sub-GeV, multi-GeV, and upward going $\mu$ events, respectively, the summation on $j$ runs over the ten zenith angle bins for each $\chi^2$, $N_j^a(\alpha)$ and $n_j^a(\alpha)$ ($a$=s, m, u; $\alpha$=e,$\mu$) stand for the theoretical predictions and data for the numbers of the sub-GeV, multi-GeV, and upward going $\mu$ events, and it is understood that $\chi^2$ is minimized with respect to all the normalization factors $\alpha$, $\beta_s$, $\beta_m$. We have put $\sigma_{s1}=\sigma_{m1}$ = 0.03, $\sigma_{s2}=\sigma_{m2}$ = 0.05, $\sigma_{\alpha}$ = 0.2 and we have assumed that the overall flux normalization $\alpha$ in the contained events is a free parameter as in [@Ashie:2005ik], and we have omitted the other uncertainties, such as the $E_\nu$ spectral index, the relative normalization between PC and FC and up-down correlation, etc., for simplicity. In Eq. (\[eqn:chi\]) the sum of each $\chi^2$ is optimized with respect the mixing angle $\theta_{23}$, the mass squared difference $|\Delta m^2_{32}|$, the Dirac CP phase $\delta$ and the phase arg($\epsilon_{e\tau}$) of the parameter $\epsilon_{e\tau}$. The other oscillation parameters give little effect on $\chi^2$, so we have fixed them as $\sin^22\theta_{12}=0.86$, $\sin^22\theta_{13}=0.1$ and $\Delta m^2_{21}=7.6\times 10^{-5}$eV$^2$. The result for the Superkamiokande data for 3903 days is given in Fig. \[fig:fig1\]. The best-fit point is $\epsilon_{ee}=-1$, $|\epsilon_{e\tau}|=0$, and the value of $\chi^2$ at this point is 74.7 (74.5) for 50 degrees of freedom in the case of the normal (inverted) hierarchy, and goodness of fit is 2.5 $\sigma$CL in both cases. The best-fit point is different from the standard case $\epsilon_{ee}=|\epsilon_{e\tau}|=0$, and this may be because we have been unable to reproduce the Monte Carlo simulation by the Superkamiokande group. The difference of the value of $\chi^2$ for the standard case and that for the best-fit point is $\Delta\chi^2=2.6~(2.0)$ for 2 degrees of freedom in the case of the normal (inverted) hierarchy, and its significance is 1.1 (0.9) $\sigma$CL. So the standard case is certainly acceptable in our analysis. From the Fig. \[fig:fig1\] we observe that the allowed region for $|\tan\beta|\equiv|\epsilon_{e\tau}|/|1+\epsilon_{ee}|$ is approximately $|\tan\beta|\equiv|\epsilon_{e\tau}|/|1+\epsilon_{ee}|\lesssim 0.8$ at 2.5$\sigma$CL. ![\[fig:fig1\] The allowed region in the ($\epsilon_{ee}$, $|\epsilon_{e\tau}|$) plane of the Superkamiokande atmospheric neutrino data for 3903 days for the Normal Hierarchy (left panel) and the Inverted Hierarchy (right panel). ](sknhv04.eps "fig:") ![\[fig:fig1\] The allowed region in the ($\epsilon_{ee}$, $|\epsilon_{e\tau}|$) plane of the Superkamiokande atmospheric neutrino data for 3903 days for the Normal Hierarchy (left panel) and the Inverted Hierarchy (right panel). ](skihv04.eps "fig:") We have also performed the analysis for the Hyperkamiokande case with the same period of time, i.e., for 3903 days. In this case we have assumed that the data $n_j^a(\alpha)$ ($a$=s, m, u; $\alpha$=e,$\mu$) are the numbers of events which are expected from the standard oscillation scenario with parameters $\theta_{23}=\pi/4$, $|\Delta m^2_{32}|=2.5\times 10^{-3}$eV$^2$. We have further assumed for simplicity that $\delta=0$. The result for the Hyperkamiokande case is given in Fig. \[fig:fig2\]. In this case the allowed region for $|\tan\beta|\equiv|\epsilon_{e\tau}|/|1+\epsilon_{ee}|$ is approximately $|\tan\beta|\equiv|\epsilon_{e\tau}|/|1+\epsilon_{ee}|\lesssim 0.4$ at 2.5$\sigma$CL. ![\[fig:fig2\] The expected allowed region in the ($\epsilon_{ee}$, $|\epsilon_{e\tau}|$) plane of the Hyperkamiokande atmospheric neutrino data for 3903 days for the Normal Hierarchy (left panel) and the Inverted Hierarchy (right panel). The best-fit point is the standard case with $\epsilon_{ee}=|\epsilon_{e\tau}|=0$. ](hknhv04.eps "fig:") ![\[fig:fig2\] The expected allowed region in the ($\epsilon_{ee}$, $|\epsilon_{e\tau}|$) plane of the Hyperkamiokande atmospheric neutrino data for 3903 days for the Normal Hierarchy (left panel) and the Inverted Hierarchy (right panel). The best-fit point is the standard case with $\epsilon_{ee}=|\epsilon_{e\tau}|=0$. ](hkihv04.eps "fig:") Conclusion ========== In this talk I have shown that the flavor-dependent neutral current Non-Standard Interactions in propagation can be constrained from the atmospheric neutrino data by Superkamiokande and Hyperkamiokande. With the ansatz (\[matter-nsi1\]) for the parameters of the Non-Standard Interactions, I have shown the allowed region in the ($\epsilon_{ee}$, $|\epsilon_{e\tau}|$) plane. In the case of the Superkamiokande data for 3903 days, the allowed region is described by $|\tan\beta|\equiv|\epsilon_{e\tau}|/|1+\epsilon_{ee}|\lesssim 0.8$ at 2.5$\sigma$CL, while it is given by $|\tan\beta|\equiv|\epsilon_{e\tau}|/|1+\epsilon_{ee}|\lesssim 0.4$ at 2.5$\sigma$CL in the Hyperkamiokande case. [99]{} K. A. Olive [*et al.*]{} \[Particle Data Group Collaboration\], Chin. Phys. C [**38**]{} (2014) 090001. K. Abe [*et al.*]{} \[Hyper-Kamiokande Working Group Collaboration\], arXiv:1412.4673 \[physics.ins-det\]. C. Adams [*et al.*]{} \[LBNE Collaboration\], arXiv:1307.7335 \[hep-ex\]. S. Davidson, C. Pena-Garay, N. Rius and A. Santamaria, JHEP [**0303**]{}, 011 (2003) \[arXiv:hep-ph/0302093\]. C. Biggio, M. Blennow and E. Fernandez-Martinez, JHEP [**0908**]{} (2009) 090 \[arXiv:0907.0097 \[hep-ph\]\]. A. Friedland and C. Lunardini, Phys. Rev. D [**72**]{} (2005) 053009 \[hep-ph/0506143\]. M. C. Gonzalez-Garcia, M. Maltoni and J. Salvado, JHEP [**1105**]{} (2011) 075 \[arXiv:1103.4365 \[hep-ph\]\]. G. Mitsuka [*et al.*]{} \[Super-Kamiokande Collaboration\], Phys. Rev. D [**84**]{} (2011) 113008 \[arXiv:1109.1889 \[hep-ex\]\]. M. C. Gonzalez-Garcia and M. Maltoni, JHEP [**1309**]{} (2013) 152 \[arXiv:1307.3092\]. K. Abe, T. Abe, H. Aihara, Y. Fukuda, Y. Hayato, K. Huang, A. K. Ichikawa and M. Ikeda [*et al.*]{}, arXiv:1109.3262 \[hep-ex\]. Y. Itow, Nucl. Phys. Proc. Suppl.  [**235-236**]{} (2013) 79. Y. Ashie [*et al.*]{} \[Super-Kamiokande Collaboration\], Phys. Rev. D [**71**]{} (2005) 112005 \[hep-ex/0501064\]. R. Foot, R. R. Volkas and O. Yasuda, Phys. Rev. D [**58**]{} (1998) 013006 \[hep-ph/9801431\]. O. Yasuda, Phys. Rev. D [**58**]{} (1998) 091301 \[hep-ph/9804400\]. O. Yasuda, hep-ph/0006319. [^1]: The constraints on $\epsilon_{ee}$ and $\epsilon_{e\tau}$ from the atmospheric neutrino have been discussed in Refs [@GonzalezGarcia:2011my; @Mitsuka:2011ty; @Gonzalez-Garcia:2013usa] with the ansatz different from ours

--- author: - | L. M. Sieberer$^{1,2}$, S. D. Huber$^{3,4}$, E. Altman$^{4,5}$, and S. Diehl$^{1,2}$\ [$^1$*Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria*]{}\ [$^2$*Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria*]{}\ [$^3$*Theoretische Physik, Wolfgang-Pauli-Strasse 27, ETH Zurich, CH-8093 Zurich, Switzerland*]{}\ [$^4$*Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel*]{}\ [$^5$*Department of Physics, University of California, Berkeley, CA 94720, USA*]{}\ title: 'Supplementary Information for “Dynamical Critical Phenomena in Driven-Dissipative Systems”' --- =1 Open system dynamics ==================== Open system dynamics with local particle loss and gain can be modeled microscopically by a many-body quantum master equation ($\hbar = 1$) $$\label{eq:meq} \partial_t \hat{\rho} = - i \left[ \hat{H},\hat{\rho} \right] + \mathcal{L} [\hat{\rho}].$$ The dynamics of the system density matrix $\hat{\rho}$ has both a coherent contribution due to the standard Hamiltonian for bosons of mass $m$ ($\int_{\mathbf{x}} = \int d^3 \mathbf{x}$) $$\hat{H} = \int_{\mathbf{x}} \hat{\psi}^{\dagger}(\mathbf{x}) \left( - \frac{\Delta}{2 m} - \mu \right) \hat{\psi}(\mathbf{x}) + \lambda \int_{\mathbf{x}} \hat{\psi}^{\dagger}(\mathbf{x})^2 \hat{\psi}(\mathbf{x})^2,$$ and a dissipative one that is incorporated by the Liouville operator $$\begin{gathered} \mathcal{L}[\hat{\rho}] = \gamma_p \int_{\mathbf{x}} \left( \hat{\psi}^{\dagger}(\mathbf{x}) \hat{\rho} \hat{\psi}(\mathbf{x}) - \frac{1}{2} \left\{ \hat{\psi}(\mathbf{x}) \hat{\psi}^{\dagger}(\mathbf{x}), \hat{\rho} \right\} \right) \\ + \gamma_l \int_{\mathbf{x}} \left( \hat{\psi}(\mathbf{x}) \hat{\rho} \hat{\psi}^{\dagger}(\mathbf{x}) - \frac{1}{2} \left\{ \hat{\psi}^{\dagger}(\mathbf{x}) \hat{\psi}(\mathbf{x}) ,\hat{\rho} \right\} \right) \\ + 2 \kappa \int_{\mathbf{x}} \left( \hat{\psi}(\mathbf{x})^2 \hat{\rho} \hat{\psi}^{\dagger}(\mathbf{x})^2 - \frac{1}{2} \left\{ \hat{\psi}^{\dagger}(\mathbf{x})^2 \hat{\psi}(\mathbf{x})^2, \hat{\rho} \right\} \right).\end{gathered}$$ Local Lindblad operators $\hat{\psi}^{\dagger}(\mathbf{x})$ and $\hat{\psi}(\mathbf{x})$, respectively, correspond to the processes of incoherent pumping and loss of single particles; $\hat{\psi}(\mathbf{x})^2$ describes the simultaneous loss of two particles. Associated rates are $\gamma_p, \gamma_l$, and $2 \kappa$. The investigation of critical phenomena at the stationary state phase transition exhibited by this model is facilitated by a formulation in terms of a Keldysh partition function [@kamenev09:_keldy; @altlandsimons] $\mathcal{Z} = \int \mathcal{D} \psi_{+} \mathcal{D} \psi_{-} \, e^{i \mathcal{S}}$, which can be subject to renormalization group methods. This partition function is fully equivalent to the master equation  and defined in terms of a Keldysh action $\mathcal{S} = \mathcal{S}_H + \mathcal{S}_D$ with two contributions corresponding to the commutator with the Hamiltonian (from now on we will be using units such that $2 m = 1$; $\int_{t,\mathbf{x}} = \int d t \int d \mathbf{x}$), $$\mathcal{S}_H = \sum_{\sigma = \pm} \sigma \int_{t,\mathbf{x}} \left[ \psi_{\sigma}^{*} \left( i \partial_t + \Delta + \mu \right) \psi_{\sigma} - \lambda \left( \psi_{\sigma}^{*} \psi_{\sigma} \right)^2 \right],$$ and the dissipative Liouvillian, $$\begin{gathered} \mathcal{S}_D = - i \gamma_p \int_{t,\mathbf{x}} \left[ \psi_{+}^{*} \psi_{-} - \frac{1}{2} \left( \psi_{+} \psi_{+}^{*} + \psi_{-} \psi_{-}^{*} \right) \right] \\ - i \gamma_l \int_{t,\mathbf{x}} \left[ \psi_{+} \psi_{-}^{*} - \frac{1}{2} \left( \psi_{+}^{*} \psi_{+} + \psi_{-}^{*} \psi_{-} \right) \right] \\ - i 2 \kappa \int_{t,\mathbf{x}} \left\{ \left( \psi_{+} \psi_{-}^{*} \right)^2 - \frac{1}{2} \left[ \left( \psi_{+}^{*} \psi_{+} \right)^2 + \left( \psi_{-}^{*} \psi_{-} \right)^2 \right] \right\}.\end{gathered}$$ Expressing the Keldysh action in terms of classical and quantum fields, which are defined as $$\phi_c = \frac{1}{\sqrt{2}} \left( \psi_{+} + \psi_{-} \right), \quad \phi_q = \frac{1}{\sqrt{2}} \left( \psi_{+} - \psi_{-} \right),$$ we recover Eq. (2) of the main text. Functional Renormalization group equation ========================================= Our approach to studying critical dynamics is based on the Wetterich functional renormalization group [@wetterich93] adapted to the Keldysh framework (see [@berges02; @salmhofer01; @pawlowski07; @delamotte07; @rosten12; @boettcher12] for reviews on the equilibrium formulation). Central to this method is the functional $\Gamma_{\Lambda}[\phi_c,\phi_q]$ defined by [@footnote_effact] $$\label{eq:effact} e^{i \Gamma_{\Lambda}[\phi_c,\phi_q]} = \int \mathcal D \delta\phi_c\mathcal D \delta\phi_q \, e^{i \mathcal{S}[\phi_c + \delta \phi_c,\phi_q +\delta\phi_q]+ i \Delta \mathcal{S}_{\Lambda}[\delta \phi_c,\delta \phi_q]}.$$ Here $\Delta \mathcal{S}_{\Lambda} $ is a regulator function which suppresses contributions to the above path integral from modes with spatial wave-vector below the running cutoff $\Lambda$. Thus $\Gamma_{\Lambda}$ interpolates between the classical action $\mathcal{S}$, when $\Lambda$ equals the UV cutoff $\Lambda_0$, and the effective action functional $\Gamma[\phi_c,\phi_q]$ [@amitbook] when $\Lambda \to 0$. The latter includes the effects of fluctuations on all scales. The equation $$\label{eq:frg} \partial_{\Lambda} \Gamma_{\Lambda} = \frac{i}{2} \operatorname{Tr}\left[ \left( \Gamma^{(2)}_{\Lambda} + R_{\Lambda} \right)^{-1} \partial_{\Lambda} R_{\Lambda} \right]$$ describes the flow of the interpolating functional as a function of the running cutoff $\Lambda$. In the following sections we first discuss the objects appearing in , namely the second functional derivative $\Gamma_{\Lambda}^{(2)}$ and the cutoff function $R_{\Lambda}$. Then we explain how a closed set of flow equations for a finite number of coupling constants can be obtained from the functional flow equation. Finally we detail the linearized equations for the infrared flow to the Wilson-Fisher fixed point from which the critical properties are inferred. In suitable truncation schemes, results from high order epsilon expansion can be reproduced from the exact flow equation . In our practical calculation, we approach the critical point from the ordered phase. This allows us to calculate the anomalous dimensions at one-loop order, due to the presence of a finite condensate during the flow. Results obtained in this way have proven to be competitive with high-order epsilon expansion or Monte Carlo simulations, as referenced in the main text. The second variational derivative ================================= The second variation $\Gamma_{\Lambda}^{(2)}$ with respect to the fields is the full inverse Green’s function at the scale $\Lambda$, which in the case of an interacting theory is field dependent. Practically we work in a basis of real fields, related to the complex fields by $$\begin{pmatrix} \chi_{\nu,1}(Q) \\ \chi_{\nu,2}(Q) \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ -i & i \end{pmatrix} \begin{pmatrix} \phi_{\nu}(Q) \\ \phi^*_{\nu}(-Q) \end{pmatrix},$$ where $\nu = c, q$ is the Keldysh index. We gather the resulting four independent field components in a field vector, $$\chi(Q) = \left( \chi_{c,1}(Q), \chi_{c,2}(Q), \chi_{q,1}(Q), \chi_{q,2}(Q) \right)^T.$$ In this basis, $\Gamma_{\Lambda}^{(2)}$ is defined as $$\label{eq:6} \left( \Gamma_{\Lambda}^{(2)} \right)_{ij}(Q,Q') = \frac{\delta^2 \Gamma_{\Lambda}}{\delta \chi_i(-Q) \delta \chi_j(Q')},$$ which is a matrix in the discrete field index $i = 1,2,3,4$ and in the continuous momentum variable $Q = (\omega,\mathbf{q})$ collecting frequency and spatial momentum. Accordingly, the trace in  involves both an integration over momenta and a sum over internal indices. $\Gamma_{\Lambda}^{(2)}(Q,Q')$ is conveniently decomposed into a constant part and a fluctuation part. The latter is a polynomial in momentum-dependent fields and, therefore, a non-diagonal matrix in momentum space. In contrast, the constant part is obtained by (i) inserting spatially constant field configurations, i.e., $\chi(Q) = \chi \delta(Q)$ in momentum space, and (ii) evaluating them at their stationary state values in the ordered phase. These read $$\label{eq:backgr} \chi(Q)\bigr\rvert_{\mathrm{ss}} = \left( \sqrt{2 \rho_0}, 0,0,0 \right)^T \delta(Q).$$ (Without loss of generality we choose the condensate amplitude to be real.) As a result, the constant part is diagonal in momentum space, $$P_{\Lambda}(Q) \delta(Q - Q') \equiv \Gamma_{\Lambda}^{(2)}(Q,Q') \bigr\rvert_{\mathrm{ss}},$$ and is structured into retarded, advanced, and Keldysh blocks, $$\label{eq:4} P_{\Lambda}(Q) = \begin{pmatrix} 0 & P^A(Q) \\ P^R(Q) & P^K \end{pmatrix}.$$ (For notational simplicity, we suppress the scale index $\Lambda$ for the different blocks and their respective entries.) The retarded and advanced blocks are mutually hermitian conjugate (we decompose $Z$ and $\bar{K}$ into real and imaginary parts, $Z = Z_R + i Z_I$, $\bar{K} = \bar{A} + i \bar{D}$), $$\begin{aligned} \label{eq:13} P^R(Q) &=& \begin{pmatrix} - i Z_I \omega - \bar{A} \mathbf{q}^2 - 2 {\mathop{\mathrm{Re}}}(\bar{u}) \rho_0 & i Z_R \omega - \bar{D} \mathbf{q}^2 \\ - i Z_R \omega + \bar{D} \mathbf{q}^2 + 2 {\mathop{\mathrm{Im}}}(\bar{u}) \rho_0 & - i Z_I \omega - \bar{A} \mathbf{q}^2 \end{pmatrix}, \nonumber \\ P^A(Q) &=& \left( P^R(Q) \right)^{\dagger}.\end{aligned}$$ Note that $\det P^R (Q=0) = \det P^A (Q=0) =0$; the existence of a gapless mode associated to the broken $U(1)$ symmetry is thus ensured in our truncation at all scales $\Lambda$. For the Keldysh block we have $$P^K = i \bar{\gamma} {\mathbbm{1}}.$$ The regulator function ====================== The cutoff contribution $\Delta \mathcal{S}_{\Lambda}$ is used in Eq.  to generate the effective action $\Gamma_{\Lambda}$ from the microscopic action $\mathcal{S}$ by suppressing contributions from momenta below $\Lambda$. Its second functional derivative $R_{\Lambda}=\Delta \mathcal{S}_{\Lambda}^{(2)}$ enters the exact flow equation . We choose an optimized cutoff function [@litim00] of the form $$\label{eq:7} R_{\Lambda}(Q) = \left( \mathbf{q}^2 - \Lambda^2 \right) \theta(\Lambda^2 - \mathbf{q}^2) \begin{pmatrix} 0 & R^R \\ R^A & 0 \end{pmatrix},$$ where $$R^R = \begin{pmatrix} - \bar{A} & - \bar{D} \\ \bar{D} & - \bar{A} \end{pmatrix}, \quad R^A = \left( R^R \right)^T.$$ Due to the $\theta$-function in , in the regularized inverse Green’s function $$G_{\Lambda}^{-1} = P_{\Lambda} + R_{\Lambda},$$ momenta $\mathbf{q}^2$ smaller than the running scale $\Lambda^2$ acquire an effective mass $\propto \Lambda^2$ and we have $\det G_{\Lambda}^{-1}(Q = 0) \neq 0$, which ensures that momentum integrals over Green’s functions are infrared convergent. Note that it is sufficient for $R_{\Lambda}$ to modify only the retarded and advanced blocks (i.e., the spectrum) of the inverse Green’s function. The choice of a frequency-independent cutoff allows us to perform frequency integrals analytically. The interpolation property of $\Gamma_{\Lambda}$ between the classical action $\mathcal{S}$ and the effective action $\Gamma$ is guaranteed by the limiting behavior [@berges09] $$\lim_{\Lambda^2 \to \Lambda_0^2} R_{\Lambda} \sim \Lambda_0^2, \quad \lim_{\Lambda^2 \to 0} R_{\Lambda} = 0.$$ Flow of the effective potential =============================== In equilibrium problems, an important object for practical calculations is the effective potential. It describes the homogeneous part of the effective action and is obtained by evaluating the full effective action at spatially homogeneous field configurations, $\bar U = \Gamma/\Omega \bigr\rvert_{\chi(Q) = \chi\delta(Q)}$ ($\Omega$ is the quantization volume). In the framework of a derivative expansion, a closed flow equation can be derived for this object, which serves as a compact generating functional for the flow of all local couplings to arbitrarily high order. Here we provide the Keldysh analog of this construction, where the key difference roots in the occurrence of two field variables $\phi_c,\phi_q$, in contrast to a single field in equilibrium. However, for a theory which obeys the power counting discussed in the main text, we can parameterize the homogeneous part of the effective action as $$\bar{V} = \frac{\partial \bar{U}}{\partial \phi_c} \phi_q + \frac{\partial \bar{U}^{*}}{\partial \phi_c^{*}} \phi_q^{*} + i \bar{\gamma} \phi_q^{*} \phi_q,$$ with $\bar U = \bar U (\phi^*_c\phi_c)$ dependent on the $U(1)$ invariant combination of classical fields only, this function thus being the direct counterpart of the effective potential. A flow equation can be derived for the auxiliary object $\bar V$, which reads (we introduce a dimensionless scale derivative $ {\partial_{\ell}}\equiv \Lambda\partial_{\Lambda}$) $$\label{eq:2} {\partial_{\ell}}\bar{V} = - \frac{i}{2} \int_Q \operatorname{tr}\left[ \mathcal G_{\Lambda}(Q) {\partial_{\ell}}R_{\Lambda}(Q) \right].$$ Here, the inverse of $\mathcal G_{\Lambda}$ is obtained from the full second functional variation by evaluating it at homogeneous field configurations (step (i) above Eq. ), however without inserting the stationary state values (step (ii)): $\mathcal G^{-1}_{\Lambda} = \Gamma_{\Lambda}^{(2)} \bigr\rvert_{\chi(Q) = \chi\delta(Q)} + R_{\Lambda}$. $\mathcal G_{\Lambda}$ is then diagonal in momentum space, and so the trace in Eq.  reduces to a single momentum integration, giving rise to the above compact form. In contrast to $G^{-1}_{\Lambda}$, $\mathcal G_{\Lambda}^{-1}$ has a non-vanishing upper left block $P^H$. However, it vanishes when the background fields are set to their stationary state values, $P^H \bigr\rvert_{\mathrm{ss}} = 0,$ which is a manifestation of causality in the Keldysh formalism [@kamenev09:_keldy; @altlandsimons]. From this equation we obtain the $\beta$-functions for the momentum-independent couplings by evaluating appropriate derivatives with respect to the $U(1)$ invariants $$\rho_c = \phi_c^{*} \phi_c, \quad \rho_{cq} = \phi_c^{*} \phi_q = \rho_{qc}^{*}, \quad \rho_q = \phi_q^{*} \phi_q.$$ at their stationary state values $\rho_c\rvert_{\mathrm{ss}} = \rho_0$, $\rho_{cq}\rvert_{\mathrm{ss}} = \rho_{qc}\rvert_{\mathrm{ss}} = \rho_q\rvert_{\mathrm{ss}} = 0$. Specifically, we use the projection prescriptions $$\label{eq:5} \begin{split} {\partial_{\ell}}\rho_0 & = \beta_{\rho_0} = - \frac{1}{u} \left[ \partial_{\rho_{cq}} {\partial_{\ell}}\bar{V} \right]_{\mathrm{ss}}, \\ {\partial_{\ell}}\bar{u} & = \beta_{\bar{u}} = \bar{u}' {\partial_{\ell}}\rho_0 + \left[ \partial_{\rho_c \rho_{cq}}^2 {\partial_{\ell}}\bar{V} \right]_{\mathrm{ss}}, \\ {\partial_{\ell}}\bar{u}' & = \beta_{\bar{u}'} = \left[ \partial_{\rho_c}^2 \partial_{\rho_{cq}} {\partial_{\ell}}\bar{V} \right]_{\mathrm{ss}}, \\ {\partial_{\ell}}\bar{\gamma} & = \beta_{\bar{\gamma}} = i \rho_0 \left[ \partial_{\rho_{cq} \rho_{qc}}^2 {\partial_{\ell}}\bar{V} \right]_{\mathrm{ss}}. \end{split}$$ Calculation of the explicit expressions here and below is largely automatized using [<span style="font-variant:small-caps;">Mathematica</span>]{}. Flow of the inverse propagator ============================== While the flow equation for the effective potential  generates $\beta$-functions for all momentum-independent couplings, the flow of the complex dynamic $Z$ and kinetic $\bar{K}$ couplings, which constitute the momentum-dependent part of the effective action (3), is determined by the flow equation for the inverse propagator. We obtain the latter by taking the second variational derivative of the exact flow equation  and setting the background fields to their stationary state values Eq. , $$\begin{gathered} \label{eq:1} {\partial_{\ell}}P_{\Lambda,ij}(Q) = \\ \frac{i}{2} \int_{Q'} \operatorname{tr}\left[ G_{\Lambda}^2(Q' - Q) {\partial_{\ell}}R_{\Lambda}(Q' - Q) \gamma_i G_{\Lambda}(Q') \gamma_j \right. \\ \left. + G_{\Lambda}(Q' - Q) \gamma_i G_{\Lambda}^2(Q') {\partial_{\ell}}R_{\Lambda}(Q') \gamma_j \right],\end{gathered}$$ where $$\gamma_{i,jl} \delta(P - P' + Q) = \frac{\delta \Gamma^{(2)}_{\Lambda,jl}(P,P')}{\delta \chi_i(Q)} \biggr\rvert_{\mathrm{ss}}.$$ In Eq.  we omit tadpole contributions $\propto \Gamma_{\Lambda}^{(4)}$, which do not depend on the external momentum $Q$ and hence do not contribute to the flow of $Z$ or $\bar{K}$. For these we use the projection prescriptions $$\label{eq:3} \begin{split} {\partial_{\ell}}Z & = \beta_Z = - \frac{1}{2} \partial_{\omega} \operatorname{tr}\left[ \left( {\mathbbm{1}}+ \sigma_y \right) {\partial_{\ell}}P^R(Q) \right] \Bigr\rvert_{Q = 0}, \\ {\partial_{\ell}}\bar{K} & = \beta_{\bar{K}} = \partial_{\mathbf{q}^2} \left[ {\partial_{\ell}}P^R_{22}(Q) + i {\partial_{\ell}}P^R_{12}(Q) \right] \Bigr\rvert_{Q = 0}. \end{split}$$ The $\beta$-functions  and  constitute the components of $\beta_{\mathbf{g}} = \left( \beta_Z, \beta_{\bar{K}}, \beta_{\rho_0}, \beta_{\bar{u}}, \beta_{\bar{u}'}, \beta_{\bar{\gamma}} \right)^T$. Rescaled flow equations ======================= We write the flow equation for the complex dynamic coupling $Z$ in the form $$\label{eq:10} {\partial_{\ell}}Z = - \eta_Z Z.$$ The anomalous dimension $\eta_Z$ is an algebraic function of the rescaled couplings (6) and $\rho_0$. The same applies to the $\beta$-functions of the latter, $$\label{eq:8} \begin{split} {\partial_{\ell}}K & = \beta_K = \eta_Z K + \frac{1}{Z} \beta_{\bar{K}}, \\ {\partial_{\ell}}u & = \beta_u = \eta_Z u + \frac{1}{Z} \beta_{\bar{u}}, \\ {\partial_{\ell}}u' & = \beta_{u'} = \eta_Z u' + \frac{1}{Z} \beta_{\bar{u}'}, \\ {\partial_{\ell}}\gamma & = \beta_{\gamma} = \left( \eta_Z + \eta_Z^{*} \right) \gamma + \frac{1}{{\left\lvert Z \right\rvert}^2} \beta_{\bar{\gamma}}. \end{split}$$ In particular, the very right expressions in these equations ($\beta_{\bar{K}}/Z$ etc.) are functions of the rescaled couplings alone. In terms of these variables, therefore, all explicit reference to the running coupling $Z$ is gone, and we have effectively traded the differential flow equation for $Z$ for the algebraic expression for its anomalous dimension $\eta_Z$. All couplings except for $\gamma$ are complex valued. Taking real and imaginary parts of the $\beta$-functions for $K$, $u$, and $u'$ yields the flow equations for $A$, $D$, $\lambda$, $\kappa$, $\lambda'$, and $\kappa'$ respectively, $$\label{eq:15} \begin{aligned} {\partial_{\ell}}A & = \beta_A = {\mathop{\mathrm{Re}}}\beta_K, & {\partial_{\ell}}D & = \beta_D = {\mathop{\mathrm{Im}}}\beta_K, \\ {\partial_{\ell}}\lambda & = \beta_{\lambda} = {\mathop{\mathrm{Re}}}\beta_u, & {\partial_{\ell}}\kappa & = \beta_{\kappa} = {\mathop{\mathrm{Im}}}\beta_u, \\ {\partial_{\ell}}\lambda' & = \beta_{\lambda'} = {\mathop{\mathrm{Re}}}\beta_{u'}, & {\partial_{\ell}}\kappa' & = \beta_{\kappa'} = {\mathop{\mathrm{Im}}}\beta_{u'}. \\ \end{aligned}$$ The $\beta$-functions for the ratios $\mathbf{r} = \left( r_K, r_u, r_{u'} \right)^T$ are then $$\label{eq:12} \begin{split} {\partial_{\ell}}r_K & = \beta_{r_K} = \frac{1}{D} \beta_A - \frac{r_K}{D} \beta_D, \\ {\partial_{\ell}}r_u & = \beta_{r_u} = \frac{1}{\kappa} \beta_{\lambda} - \frac{r_u}{\kappa} \beta_{\kappa}, \\ {\partial_{\ell}}r_{u'} & = \beta_{r_{u'}} = \frac{1}{\kappa'} \beta_{\lambda'} - \frac{r_{u'}}{\kappa'} \beta_{\kappa'}. \end{split}$$ The number of flow equations can be further reduced by introducing anomalous dimensions for $D$ and $\gamma$, $$\label{eq:9} \begin{split} {\partial_{\ell}}D & = - \eta_D D, \\ {\partial_{\ell}}\gamma & = - \eta_{\gamma} \gamma. \end{split}$$ As for the dynamic coupling $Z$ in terms of the rescaled variables $K$, $u$, $u'$, $\gamma$ and $\rho_0$, all explicit reference to $D$ and $\gamma$ drops out, and we obtain for the couplings $\mathbf{s} = \left( w, \tilde{\kappa}, \tilde{\kappa}' \right)^T$ defined in Eq. (5) $$\label{eq:11} \begin{split} {\partial_{\ell}}w & = \beta_w = - \left( 2 - \eta_D \right) w + \frac{w}{\kappa} \beta_{\kappa} + \frac{2 \kappa}{\Lambda^2 D} \beta_{\rho_0}, \\ {\partial_{\ell}}\tilde{\kappa} & = \beta_{\tilde{\kappa}} = - \left( 1 - 2 \eta_D + \eta_{\gamma} \right) \tilde{\kappa} + \frac{\gamma}{2 \Lambda D^2} \beta_{\kappa}, \\ {\partial_{\ell}}\tilde{\kappa}' & = \beta_{\tilde{\kappa}'} = - \left( - 3 \eta_D + 2 \eta_{\gamma} \right) \tilde{\kappa}' + \frac{\gamma^2}{4 D^3} \beta_{\kappa'}. \end{split}$$ In summary, the transformations (4) and (5) result in the closed system (6) for $\mathbf{r}$ and $\mathbf{s}$ with $\beta_{\mathbf{r}} = \left( \beta_{r_K}, \beta_{r_u}, \beta_{r_{u'}} \right)^T$ given by Eq.  and $\beta_{\mathbf{s}} = \left( \beta_w, \beta_{\tilde{\kappa}}, \beta_{\tilde{\kappa}'} \right)^T$ given by Eq. . The flows of $Z$, $D$, and $\gamma$ are decoupled and determined by the anomalous dimensions  and , which are themselves functions of $\mathbf{r}$ and $\mathbf{s}$. Critical properties =================== For the analysis of critical behavior, we need to find a scaling solution to the flow equations for the bare couplings or, equivalently, a fixed point $\mathbf{r}_{*}$, $\mathbf{s}_{*}$ of the flow of dimensionless rescaled couplings, $$\beta_{\mathbf{r}}(\mathbf{r}_{*},\mathbf{s}_{*}) = \beta_{\mathbf{s}}(\mathbf{r}_{*},\mathbf{s}_{*}) = \mathbf{0}.$$ This non-linear algebraic set of equations has a non-trivial solution given by Eq. (7). In order to characterize the infrared flow in the vicinity of the fixed point (encoding the critical exponents we are interested in here), we study the flow of the couplings linearized around the fixed point, cf.Eq. (8). The stability matrices $N$ and $S$ in this equation read explicitly $$\begin{gathered} N = \nabla_{\mathbf{r}}^T \beta_{\mathbf{r}} \bigr\rvert_{\mathbf{r} = \mathbf{r}_{*}, \mathbf{s} = \mathbf{s}_{*}} = \begin{pmatrix} 0.0525 & 0.0586 & 0.0317 \\ -0.0002 & -0.0526 & 0.1956 \\ 0.4976 & -2.3273 & 1.9725 \end{pmatrix}, \\ S = \nabla_{\mathbf{s}}^T \beta_{\mathbf{s}} \bigr\rvert_{\mathbf{r} = \mathbf{r}_{*}, \mathbf{s} = \mathbf{s}_{*}} = \begin{pmatrix} -1.6204 & 0.0881 & 0.0046 \\ -3.1828 & 0.2899 & 0.0363 \\ -15.3743 & -42.2487 & 2.1828 \end{pmatrix},\end{gathered}$$ without coupling between $\mathbf r$ and $\mathbf s$ sectors. At present we cannot rule out that an extended truncation would couple them. However, since we already include all relevant and marginal couplings, we expect the decoupling to be robust or at least approximately valid to a good accuracy. The infrared flow of $Z$, $D$, and $\gamma$ is determined by the values of the respective anomalous dimensions at the fixed point. Equations  and  imply the scaling behavior $$Z \sim \Lambda^{- \eta_Z}, \quad D \sim \Lambda^{- \eta_D}, \quad \gamma \sim \Lambda^{- \eta_{\gamma}}$$ for $\Lambda \to 0$. While $\eta_D$ and $\eta_{\gamma}$ describe the flow of real quantities and are, therefore, themselves real by definition, $\eta_Z$ is in general a complex valued function of $\mathbf{r}$ and $\mathbf{s}$. At the fixed point, however, the imaginary part vanishes, $${\mathop{\mathrm{Im}}}\eta_Z = 0,$$ which ensures scale invariance of the full effective action at the critical point. As is indicated in the main text, the emergence of $O(2)$ model critical properties in the sector $\mathbf{s}$ is due to the scaling relation $\eta_Z = \eta_{\bar{\gamma}}$, which ensures that these anomalous dimensions compensate each other in the $\beta$-functions for the couplings $\mathbf{s}$. (The anomalous dimensions $\eta_{\bar{\gamma}}$ and $\eta_{\gamma}$ associated with the bare and renormalized noise vertices, respectively, are related via $\eta_{\gamma} = \eta_{\bar{\gamma}} - 2 {\mathop{\mathrm{Re}}}\eta_Z$, as follows from Eq. (4) in the main text.) This can be seen most simply by expressing, e.g., $\tilde{\kappa}$ in terms of bare quantities, $$\label{eq:16} \tilde{\kappa} = \frac{\gamma {\mathop{\mathrm{Im}}}(u)}{2 \Lambda {\mathop{\mathrm{Im}}}(K)^2} = \frac{\gamma {\mathop{\mathrm{Im}}}(\bar{u}/Z)}{2 \Lambda {\mathop{\mathrm{Im}}}(\bar{K}/Z)^2}.$$ In this form it is apparent that the scaling $\sim \Lambda^{- \eta_Z}$ which applies to both $Z$ and $1/\gamma$ drops out. Similar arguments hold for $w$ and $\tilde{\kappa}'$. Alternatively, the cancellation of $\eta_Z$ and $\eta_{\gamma}$ in the $\beta$-functions can be seen explicitly by inserting Eqs.  and  in . What remains is a dependence on $\eta \equiv \eta_D + \eta_Z$ which is just the anomalous dimension associated with the bare kinetic coefficient $\bar{K}$. Ginzburg Criterion ================== We estimate the extent of the universal critical domain governed by the linearized regime of the Wilson-Fisher fixed point, which provides us with an estimate of both the extent of the thermalized regime as well as the energy resolution necessary to probe the critical behavior. This is done by calculating the Ginzburg scale, i.e., the distance from the phase transition where fluctuations on top of the quadratic Bogoliubov-type theory become dominant [@amitbook]: We equate the bare distance from the phase transition $\chi$ to the corresponding one-loop correction, yielding $$\chi_G = \frac{1}{D^3} \left( \frac{\gamma \kappa}{4 \pi} \right)^2.$$ Here, the parameters $\gamma, \kappa$, and $D$ are those appearing in the mesoscopic description of the system. In the case of exciton-polariton condensates, $\gamma$ and $\kappa$ can thus be read off from the dGPE and the noise correlator [@carusotto2005]. The parameter $D$ typically does not appear explicitly in this description. However, it is included effectively in a complex prefactor of the time derivative in the dGPE ($m_{\mathit{LP}}$ is the mass of the lower polariton) $$\label{eq:18} i \left( 1 + i \Delta Z \right) \partial_t \psi = \left( - \frac{1}{2 m_{\mathit{LP}}} \nabla^2 + \dotsb \right) \psi.$$ Such a term results from two physical mechanisms. First, it describes the leading frequency dependence of the pumping process [@wouters10_PRL]. To account for this effect, a convenient parameterization is $\Delta Z = P/ \left( 2 \Omega_K \right)$ which is proportional to the pumping strength $P$, and where $\Omega_K$ is the gain cutoff frequency. Second, it results from energy relaxation due to scattering of the lower polaritons with high frequency photons and excitons [@wouters10_PRB]. These processes are captured by the form $\Delta Z = \kappa \bar{n}/2$ scaling linearly with the time averaged density $\bar{n}$, and a phenomenological relaxation constant $\kappa$. Dividing the dGPE  by $1 + i \Delta Z$ leads to an effective kinetic term $ - \tfrac{1 - i \Delta Z}{1 + \Delta Z^2} \tfrac{\nabla^2}{2 m_{\mathit{LP}}} \psi$, resulting in a mesoscopic coherent propagation coefficient $A = \frac{1}{1 + \Delta Z^2} \frac{1}{2 m_{\mathit{LP}}}$ and an effective mesoscopic diffusion constant $D = \frac{\Delta Z}{1 + \Delta Z^2} \frac{1}{2 m_{\mathit{LP}}}$ entering Eq. (1) in the main text, and the above Ginzburg criterion. Finally, we would like to contrast the Ginzburg scale to a scale identified in [@carusotto2005; @wouters10_PRL]. This scale indicates a crossover between a sonic and a purely diffusive excitation spectrum within the symmetry broken phase, and takes the value $\omega_c = \kappa\rho_0$. The Ginzburg scale identifies the frequency scale below which critical fluctuations become more dominant than the “bare” terms, which occur in Bogoliubov theory. It is therefore only meaningful – and makes a statement about – the physics close to the phase transition, where the order parameter goes to zero. Instead, the crossover scale in [@carusotto2005; @wouters10_PRL] is determined within the symmetry broken phase and is obtained within Bogoliubov theory, without the need of a calculation of fluctuation corrections. This is justified, because within the symmetry broken phase there are no critical fluctuations and mean field plus Bogoliubov theory are valid. As implied by this comparison, these two scales are not directly related to each other and address different physical questions. In particular, the crossover scale in [@carusotto2005; @wouters10_PRL] tends to zero when approaching the phase transition by construction, $\rho_0 \to 0$. Therefore, in the vicinity of the critical point, the discussion of diffusive vs. coherent dynamics is a more subtle issue. How it works quantitatively is addressed by the calculation of the critical exponents, with the key finding of universal decoherence: The coherent dynamics fades out faster than the dissipative one, measured by the critical exponent $\eta_r$. A. Kamenev and A. Levchenko, Advances in Physics, [**58(3)**]{}, 197 (2009). A. Altland and B. Simons, *Condensed Matter Field Theory* (Cambridge University Press, 2010). C. Wetterich, Phys. Lett. B [**301**]{}, 90 (1993); Z. Phys. C [**57**]{}, 451 (1993). J. Berges, N. Tetradis, C. Wetterich, Phys. Rept. [**363**]{} 223 (2002). M. Salmhofer and C. Honerkamp, Prog. Theor. Phys. [**105**]{}, 1 (2001). J. M. Pawlowski, Annals Phys. [**322**]{}, 2831 (2007). B. Delamotte, cond-mat/0702365 (2007). O. J. Rosten, Physics Reports [**511**]{}, 177 (2012). I. Boettcher, J. M. Pawlowski, S. Diehl, Nucl. Phys. Proc. Suppl. [**228**]{}, 63 (2012). This representation holds for the stationary states that obey $\frac{\delta \Gamma_{\Lambda}}{\delta \phi_c} = \frac{\delta \Gamma_{\Lambda}}{\delta \phi_q} = 0$. D. J. Amit and V. Martin-Mayor, *Field Theory, The Renormalization Group, And Critical Phenomena* (World Scientific, Singapore, 2005). D. Litim, Phys. Lett. B [**486**]{}, 92 (2000). J. Berges, G. Hoffmeister, Nucl. Phys. B [**813**]{} 383 (2009). I. Carusotto and C. Ciuti, Physical Review B [**72**]{}, 125335 (2005); M. Wouters and I. Carusotto, Phys. Rev. Lett. [**99**]{}, 140402 (2007); J. Keeling and N.G. Berloff, Phys. Rev. Lett [**100**]{}, 250401 (2008). M. Wouters and I. Carusotto, Phys. Rev. Lett. [**105**]{}, 020602 (2010) M. Wouters, T. C. H. Liew, and V. Savona, Phys. Rev. B [**82**]{}, 245315 (2010).

--- abstract: 'We show that given one $O(a)$ improvement constant, $b_m$, all the remaining quantities needed to define the renormalized and $O(a)$ improved dimension-3 quark bilinears can be obtained by studying the matrix elements of these operators between external quark states in a fixed gauge.' address: 'MS B285, Group T-8, Los Alamos National Laboratory, Los Alamos, NM 87545, USA' author: - 'T. Bhattacharya, R. Gupta, W. Lee' title: 'Renormalization constants using quark states in Landau gauge[^1]' --- INTRODUCTION ============ One of the leading uncertainties in lattice calculations involves the connection between the lattice and the continuum renormalized operators. Current estimates [@us] show that one-loop perturbation theory for $O(a)$ improved Wilson fermions underestimates quantities like $Z_P^0/Z_S^0$ by $\sim 10\%$ at lattice scales of $2$–$4$ GeV. Furthermore, perturbative estimates of the $O(a)$ improvement coefficients are significantly different from their non-perturbative estimates. All the scale independent renormalization constants and the improvement coefficients in the quenched theory have been determined by imposing vector and axial ward identities on the lattice. This Ward identity method is computationally intensive and alternate methods are therefore desirable, especially to determine the scale dependent renormalization constants ($Z_T^0$ and $Z_P^0$ or $Z_S^0$). An alternate well known method that gives all the renormalization constants [@oldrenorm] and some of the improvement constants [@martinelli; @mytalk] involves calculating the matrix elements of the quark bilinears between external quark states in a fixed gauge. This method is far more tractable computationally and the generalization to 4-fermion operators is straightforward. Here, we show how this method gives all but one ($b_m$ as discussed in section \[sec:31\]) of the $O(a)$ renormalization constants. -0.5 ------------------------------------------------------------------------ width 0pt METHOD ====== We start by defining the notation. $O(a)$ improvement of the theory is achieved by improving the action and the operators simultaneously. The improved renormalized quark fields, ${{\hat \psi}}$ can be related to the lattice quark field $\psi$ by $$\begin{aligned} {{\hat \psi}}&=& Z_\psi^{-1/2} (1 - b_\psi a {m_I}) \times{} \nonumber\\ && \qquad [ 1 - a c'_\psi ({\mathop{{\mathpalette{\bgroup\let\style=} {\setbox0=\hbox{$\style D$} \setbox2=\hbox to\wd0{\hss$\style/$\hss} \wd2=0pt\dp2=0pt\box2\box0\egroup}}}}+ {m_I}) - a c_{NGI} {\mathop{{\mathpalette{\bgroup\let\style=} {\setbox0=\hbox{$\style \partial$} \setbox2=\hbox to\wd0{\hss$\style/$\hss} \wd2=0pt\dp2=0pt\box2\box0\egroup}}}}] \psi\,,\end{aligned}$$ where $m_I$ is an $O(a)$ improved quark mass. Thus, apart from a mass dependent renormalization constant, one needs (i) an equation of motion correction, $ c'_\psi$, that does not affect position space correlation functions at finite separations, and (ii) mixing with a gauge non-invariant operator, $c_{NGI}$, that appears because the calculation is performed in a fixed gauge. [@martinelli] We also write the renormalized propagator as $$\langle \mathop{\rm T} [{{\hat \psi}}{{\hat {{\overline\psi}}}}] \rangle \equiv \frac{1}{i {{\hat \Sigma}}_1 {{{\mathpalette{\bgroup\let\style=} {\setbox0=\hbox{$\style p$} \setbox2=\hbox to\wd0{\hss$\style/$\hss} \wd2=0pt\dp2=0pt\box2\box0\egroup}}}}+ {{\hat \Sigma}}_2} \equiv -i {{\hat \sigma}}_1 {{{\mathpalette{\bgroup\let\style=} {\setbox0=\hbox{$\style p$} \setbox2=\hbox to\wd0{\hss$\style/$\hss} \wd2=0pt\dp2=0pt\box2\box0\egroup}}}}+ {{\hat \sigma}}_2\,,$$ where $\mathop{\rm T}$ is the time-ordering symbol, and $p_\mu$ is the four momentum of the quark. Neglecting logarithmic and higher order corrections in $p^{-2}$, the improved renormalized propagator at high momenta is given by $$\begin{aligned} {{\hat \Sigma}}_1 &=& 1 + \frac{\alpha_1 {m_I}^2 + \beta_1}{p^2} \label{sigh1}\\ {{\hat \Sigma}}_2 &=& Z_m {m_I} \left[ 1 + \frac{\alpha_2 {m_I}^2 + \beta_2}{p^2} \right] + \frac{\beta'_2}{p^2} \,,\label{sigh2}\end{aligned}$$ where $Z_m$ is the renormalization constant of $m_I$. The important point to note in these expressions is that chiral symmetry prevents terms proportional to ${m_I}$ and ${m_I}^2$ in ${{\hat \Sigma}}_1$ and ${{\hat \Sigma}}_2$ respectively. ![Plot of $\sigma_2$ versus $p^2$ before (diamonds) and after (squares) subtraction of $O(a^2p^2)$ artefact for $\kappa=0.1344$.[]{data-label="oap2"}](display5m){width="0.9\hsize"} Expanding the lattice quark fields in terms of the continuum field, lattice propagator is $$\begin{aligned} \langle \mathop{\rm T} [ \psi {{\overline\psi}}] \rangle &\equiv& - i \sigma_1 {{{\mathpalette{\bgroup\let\style=} {\setbox0=\hbox{$\style p$} \setbox2=\hbox to\wd0{\hss$\style/$\hss} \wd2=0pt\dp2=0pt\box2\box0\egroup}}}}+ \sigma_2 \label{latsigdef}\\ &=& Z_\psi ( 1 + 2 b_\psi a m ) (1 + 2 i a c_{NGI} {{{\mathpalette{\bgroup\let\style=} {\setbox0=\hbox{$\style p$} \setbox2=\hbox to\wd0{\hss$\style/$\hss} \wd2=0pt\dp2=0pt\box2\box0\egroup}}}}) \times {} \nonumber\\ &&\qquad (- i {{\hat \sigma}}_1 {{{\mathpalette{\bgroup\let\style=} {\setbox0=\hbox{$\style p$} \setbox2=\hbox to\wd0{\hss$\style/$\hss} \wd2=0pt\dp2=0pt\box2\box0\egroup}}}}+ {{\hat \sigma}}_2) + 2 a c'_\psi \,. \label{latcontsigrel}\end{aligned}$$ From this the unknown constants $Z_\psi$, $Z_m$, $b_\psi$, $c_{NGI}$, and $c'_\psi$ can be extracted as follows. We first expand $\sigma_1$ and $\sigma_2$ at large $p^2$ as $$\begin{aligned} p^2 \sigma_1 &=& {\sigma_1^{LO}} + \frac{\sigma_1^{NLO}}{p^2} + O(p^{-4}) \label{sig1d}\\ \sigma_2 &=& {\sigma_2^{LO}} + \frac{\sigma_2^{NLO}}{p^2} + O(p^{-4})\,. \label{sig2d} \end{aligned}$$ where the terms dropped are yet higher order in $p^{-2}$. From Eqns. \[sigh1\]–\[sig2d\], we note that these leading and next to leading coefficients, $\sigma^{(N)LO}_{1,2}$, of the expansion of $\sigma_{1,2}$ in $p^{-2}$ have the following dependence on the quark mass: $$\begin{aligned} \sigma^{LO}_1 &=& \sigma^ {LO,0}_1 + \sigma^ {LO,1}_1 {m_I} \\ \sigma^{NLO}_1 &=& \sigma^{NLO,0}_1 + \sigma^{NLO,1}_1 {m_I} + {}\nonumber\\ &&\qquad \sigma^{NLO,2}_1 {m_I}^2 + \sigma^{NLO,3}_1 {m_I}^3 \\ \sigma^{LO}_2 &=& \sigma^ {LO,0}_2 \\ \sigma^{NLO}_2 &=& \sigma^{NLO,0}_2 + \sigma^{NLO,1}_2 {m_I} + {}\nonumber\\ &&\qquad \sigma^{NLO,2}_2 {m_I}^2 \,,\end{aligned}$$ where, omitting terms of $O(a^2)$, $$\begin{aligned} \sigma^{LO,0}_1 &=& Z_\psi \label{expansion0}\\ \sigma^{LO,1}_1 &=& 2 a Z_\psi (b_\psi - c_{NGI}) \label{expansion1}\\ \sigma^{NLO,0}_1 &=& - 2 Z_\psi (\beta_1 - 2 a c_{NGI} \beta'_2) \label{expansion2}\\ \sigma^{NLO,1}_1 &=& - 2 a Z_\psi [\beta_1 b_\psi - c_{NGI} (2\beta_1 - \beta_2)] \label{expansion3}\\ \sigma^{NLO,2}_1 &=& - Z_\psi (1 + \alpha_1) \label{expansion4}\\ \sigma^{NLO,3}_1 &=& - a Z_\psi [b_\psi (1 + c_{NGI}) - {}\nonumber\\ && \qquad 2 c_{NGI} (1 + 2\alpha_1 - \alpha_2)] \label{expansion5}\\ \sigma^{LO,0}_2 &=& 2 a Z_m ( Z_\psi c_{NGI} + c'_\psi ) \label{expansion6}\\ \sigma^{NLO,0}_2 &=& 2 a Z_m Z_\psi c_{NGI} \beta_1 \label{expansion7}\\ \sigma^{NLO,1}_2 &=& Z_m Z_\psi \label{expansion8}\\ \sigma^{NLO,2}_2 &=& 2 a Z_m Z_\psi [ b_\psi + c_{NGI} (1 + \alpha_1) ] \,. \label{expansion9}\end{aligned}$$ The five renormalization and improvement constants can now be obtained once $\alpha_1$ is eliminated using $\sigma^{LO,0}_1$, $\sigma^{LO,1}_1$, $\sigma^{NLO,2}_1$, $\sigma^{LO,0}_2$, $\sigma^{NLO,1}_2$, and $\sigma^{NLO,2}_2$. Once the improved propagator is known, the remaining constants are obtained as discussed in [@mytalk]. ![Plot of $\sigma_2$ versus $p^{-2}$ after subtraction of $O(a^2p^2)$ artefact for $\kappa=0.1344$.[]{data-label="p2fit"}](prop005m){width="0.9\hsize"} EXAMPLE IMPLEMENTATION ====================== Lattice details {#sec:31} --------------- To illustrate the feasibility of this method, we present a preliminary analysis of 50 $32^4$ quenched configurations at $\beta=6.2$. We take $c_{SW} = 1. 614$ from the ALPHA collaboration [@Luscher] to define the $O(a)$ improved action. Simulations are done at seven values of $\kappa = 0.131$, 0.1321, 0.1333, 0.1339, 0.1344, 0.1348, and 0.1350. The critical value of the hopping constant, $\kappa_c = 0.135899$, and the fourth root of the plaquette, $u_0 = 0.88510$ are taken from [@us]. For the $O(a)$ improved definition of quark mass we take-1 $${m_I} = \ln \left[ 1 + \left( \frac{1}{2\kappa u_0} - \frac{1}{2\kappa_c u_0} \right)\right]$$ since the value of $b_m$ is close to its tadpole improved tree level value [@us]. We scale all lattice fermion fields by $\sqrt{2\kappa u_0}$. The lattice momentum components are $p \equiv \sin (2\pi j/32) $ and we average over momentum combinations equivalent under the hypercubic lattice symmetry group. Our fits use the seventy distinct momentum combinations with $j \leq 4$.-1 $O(a^2)$ subtraction and determination of the constants ------------------------------------------------------- According to Eqs. \[sig1d\] and \[sig2d\], $p^2 \sigma_1$ and $\sigma_2$ are supposed to go to constants at large $p^2$, however, the data, exemplified in Figure \[oap2\], show a linear behavior in $a^2 p^2$ at large $p^2$. These $O(a^2)$ terms are removed by fitting to the large momentum behavior. To extract the desired coefficients $\sigma^{(N)LO}_{1,2}$ we now fit these subtracted $p^2 \sigma_1$ and $\sigma_2$ as a function of $p^{-2}$ (see Figure \[p2fit\] for an example). The data show that one needs to keep at least terms up to $p^{-4}$ in Eqs. \[sig1d\],\[sig2d\] to obtain a reasonable fit. The resulting intercepts and slopes obtained are then fit against ${m_I}$ ([[*e.g.*]{}]{}, see Figure \[mfit\]) to obtain the various expressions defined in Eqns. \[expansion0\]–\[expansion9\]. From these fits, we find $$\begin{aligned} Z_\psi &=& 0.925(6) u_0\\ Z_m &=& 1.03(4) u_0^{-1} \\ c_{NGI} &=& - 0.02(5) \\ b_\psi &=& - 0.53(3) u_0^{-1} \\ c'_\psi &=& 0.27(4)\end{aligned}$$ ![Slope of $O(a^2)$-corrected $\sigma_2$ with respect to $p^{-2}$ plotted versus $m_I$.[]{data-label="mfit"}](fit1_1){width="0.9\hsize"} DISCUSSION ========== We have shown that once $b_m$, or equivalently, ${m_I}$ is known, all other $O(a)$ improvement constants needed to define the renormalized propagator can be determined. In previous work [@martinelli], the constant $c_{NGI}$ was left undetermined. Using perturbation theory Sharpe [@sharpe] has shown that the effect of this term is small. We show that this constant can be determined non-perturbatively and its value is indeed small. [9]{} T. Bhattacharya, [[*et al.*]{}]{}, Phys. Rev. [**D63**]{} (2001) 074505 (hep-lat/0009038). G. Martinelli, [[*et al.*]{}]{}, Nucl. Phys. [**B445**]{} (1995) 81 (hep-lat/9411010). G. Martinelli, [[*et al.*]{}]{}, Nucl. Phys. [**B611**]{} (2001) 311 (hep-lat/0106003). T. Bhattacharya, [[*et al.*]{}]{}, [Nucl. Phys.=1000]{} Proc. Suppl. [**94**]{} (2001) 595 (hep-lat/0106007). M. Lüscher, [[*et al.*]{}]{}, Nucl. Phys. [**B491**]{} (1997) 323 (hep-lat/9609035). S. Sharpe, these proceedings (hep-lat/0110021). [^1]: This research is supported by Department of Energy, under contract W-7405-ENG-36 and the Grand Challenges award at ACL at Los Alamos.

--- abstract: | We compute the exact two-loop matching coefficients for the strong coupling constant $\alpha_s$ and the bottom-quark mass $m_b$ within the Minimal Supersymmetric Standard Model (MSSM), taking into account ${\cal O}(\alpha_s^2)$ contributions from Supersymmetric Quantum Chromodynamics (SQCD). We find that the explicit mass pattern of the supersymmetric particles has a significant impact on the predictions of $\alpha_s$ and $m_b$ at high energies. Further on, the three-loop corrections exceed the uncertainty due to the current experimental accuracy. In case of the the running bottom-quark mass, they can reach in the large $\tan \beta$ regime up to $30\%$ of the tree-level value. PACS numbers: 11.30.Pb, 12.38.-t, 12.38.Bx, 12.10.Kt author: - | A. Bauer, L. Mihaila, J. Salomon\ [*Institut f[ü]{}r Theoretische Teilchenphysik, Universit[ä]{}t Karlsruhe (TH),*]{}\ [*Karlsruhe Institute of Technology (KIT)*]{}\ [*76128 Karlsruhe, Germany*]{}\ title: | -3cm SFB/CPP-08-75\ TTP/08-25 1.5cm Matching coefficients for $\alpha_s$ and $m_b$ to ${\cal O}(\alpha_s^2)$\ in the MSSM --- Introduction ============ Supersymmetry (SUSY) is currently believed to play an important role in physics beyond the Standard Model (SM). A compelling argument in favour of SUSY is the particle content of the MSSM, that leads in a natural way to the unification of the three gauge couplings at a high energy scale $\mu\simeq 10^{16}$ GeV, in agreement with predictions of Grand Unification Theories (GUT) [@Dimopoulos:1981yj; @Ibanez:1981yh; @Amaldi:1991cn].\ Apart from the gauge coupling unification, in GUT models based on simple groups such as $SU(5)$[@Georgi:1974sy] or $SO(10)$ [@Fritzsch:1974nn], also the bottom ($y_b$) and tau ($y_{\tau}$) Yukawa couplings unify at the GUT scale. For some models based on $SO(10)$ (or larger groups) even the unification of the bottom, tau and top ($y_t$) Yukawa couplings is predicted. However, the condition of Yukawa coupling unification can be fulfilled within the MSSM only for two regions of $\tan \beta$, the ratio of Higgs field vacuum expectation values: $\tan \beta \approx 1$ and $\tan \beta \approx 50$ [@Langacker:1994bc; @Hall:1993gn; @DiazCruz:2000mn]. A main feature of SUSY models with large $\tan \beta$ is that the supersymmetric radiative corrections to fermion masses and couplings can be as large as the leading order (LO) contributions  [@Hall:1993gn; @Hempfling:1993kv; @Carena:1994bv]. This renders the knowledge of the higher order (HO) corrections in perturbation theory mandatory. On the other hand, the unification condition becomes very sensitive to the low energy parameters [@Tobe:2003bc]. This property can be exploited to greatly constrain the allowed MSSM parameter space. With the advent of the CERN Large Hadron Collider (LHC), we will be able to probe the realization of SUSY in nature to energy scales of ${\cal O}(1)$ TeV. In particular, precision measurements and computations will allow to test the low-energy supersymmetric relations between the particle masses and couplings. It is often argued (for reviews see e.g. Refs. [@Blair:2002pg; @Aguilar-Saavedra:2005pw]) that, from the precise knowledge of the low-energy supersymmetric parameters one can shed light on the origin and mechanism of supersymmetry breaking and even on physics at much higher energies, like the GUT scale. The extrapolation of the supersymmetric parameters measured at the TeV energy scale to the GUT-scale raises inevitably the question of uncertainties involved. Currently, there are four publicly available spectrum generating codes [@Paige:2003mg; @Allanach:2001kg; @Porod:2003um; @Djouadi:2002ze] based on two-loop order MSSM renormalization group equations (RGEs) [@Martin:1993yx; @Martin:1993zk; @Jack:1994kd; @Yamada:1994id] subjected to two types of boundary conditions. One set of constraints accounts for the weak-scale matching between the MSSM and SM parameters to one-loop order [@Pierce:1996zz]. The second one allows for the SUSY breaking at the high scale according to specific models like minimal supergravity, gauge mediation and anomaly mediation. The approximations within the codes differ by higher order corrections and by the treatment of the low-energy threshold corrections. The typical spread of the results is within few percents, which does not always meet the experimental accuracy [@Allanach:2003jw]. Along the same line, recent analyses [@Harlander:2005wm; @Harlander:2007wh] have proven that the three-loop order effects on the running of the strong coupling constant $\alpha_s$ and the bottom quark mass $m_b$ may exceed those induced by the current experimental accuracy [@Bethke:2006ac; @Kuhn:2007vp].\ Very recently, Ref. [@Noth:2008tw] computed the two-loop SQCD and top-induced Supersymmetric Electroweak (SEW) corrections to the effective bottom Yukawa couplings. The knowledge of the two-loop corrections allows predictions for the branching ratios of the MSSM Higgs bosons with per-cent level accuracy. The aim of this paper is to compute the weak-scale matching relations for the strong coupling constant and the bottom quark mass with two-loop accuracy, taking into account the exact dependence on the particle masses. This will extend the study of Ref. [@Harlander:2005wm] allowing phenomenological analyses based on quasi-realistic mass spectra for SUSY particles, with three-loop order accuracy. However, we will consider in this paper only the ${\cal O}(\alpha_s^2)$ corrections from SQCD and postpone the study of the SEW contributions ${\cal O}(\alpha_s y_t^2, \alpha_s y_b^2, y_t^4, y_t^2 y_b^2, y_b^4 )$ to further investigations. Whereas the SEW corrections to the decoupling coefficient of $\alpha_s$ are expected to be negligible, their numerical impact on the running bottom quark mass can become as important as those from SQCD. At the one-loop order [@Pierce:1996zz], the main contributions to the bottom-quark mass decoupling coefficient arise from diagrams containing gluino- and Higgsino-exchange. In the most regions of the MSSM parameter space the gluino contribution is the dominant one and can be as large as the tree-level bottom quark mass. However, there are domains in the parameter space where the corrections due to gluino- and Higgsino-exchange can become of the same order and have opposite sign. These regions contain the special points for which the Yukawa coupling unification occurs  [@Tobe:2003bc; @Blazek:2002ta]. For the MSSM parameters for which the radiative corrections to the bottom quark mass are comparable with the LO ones, Ref. [@Carena:1999py] proposed a method to resum them to all orders in perturbation theory. A numerical comparison with the results of Ref. [@Carena:1999py] can be found in Section \[sec::num\]. The paper is organized as follows: in Section \[sec::frame\] and Section \[sec::ren\] we discuss the theoretical framework and the renormalization scheme we use. In Section \[sec::res\] we present the analytical one- and two-loop results. The latter ones are displayed in analytical form for three simplifying mass hierarchies among the SUSY particles. The numerical effects are studied in Section \[sec::num\]. \[sec::frame\]Framework ======================= As already stated above, the underlying motivation for the running analysis is to relate physical parameters measured at the electroweak scale with the Lagrange parameters at the GUT scale. The running parameters are most conveniently defined in mass-independent renormalization schemes such as [$\overline{\mbox{{\scalefont{.9}}MS}}$]{} [@Bardeen:1978yd] for the SM parameters and [$\overline{\mbox{{\scalefont{.9}}DR}}$]{} [@Siegel:1979wq] for the MSSM parameters. It is well known that in such “unphysical” renormalization schemes the Appelquist-Carazzone decoupling theorem [@Appelquist:1974tg] does not hold in its naive form. Quantum corrections to low-energy processes contain logarithmically enhanced contributions from heavy particles with masses much greater than the energy-scale of the process under consideration. An elegant approach to get rid of this unwanted behaviour in the [$\overline{\mbox{{\scalefont{.9}}MS}}$]{} or [$\overline{\mbox{{\scalefont{.9}}DR}}$]{} scheme is to formulate an effective theory (ET)  [@Chetyrkin:1997un; @Steinhauser:2002rq] integrating out all heavy particles. The coupling constants defined within the ET must be modified in order to account for the effects of the heavy fields. They are related to the parameters of the full theory by the so-called matching or decoupling relations. For moderate mass splittings between SUSY particles, [*i.e.*]{} there are no large logarithms in the theory that have to be resummed by means of RGEs, the decoupling of heavy particles might be performed in one step [@Ferreira:1996ug]. The energy-scale at which the decoupling is performed is not fixed by the theory. It is usually chosen to be $\mu \simeq \tilde M $, where $\tilde M$ is a typical heavy particle mass. The MSSM parameters at energies $E\simeq \tilde M$ can be determined from the knowledge of the corresponding SM parameters and the associated decoupling relations. The decoupling coefficients for the strong coupling constant and for the light quark masses are known in QCD with four- [@Schroder:2005hy; @Chetyrkin:2005ia] and three-loop [@Chetyrkin:1997un] accuracy, respectively. Due to the presence of many mass scales, their computation within SQCD and SEW is quite involved. At one-loop order, they are known exactly  [@Pierce:1996zz; @Bednyakov:2007vm]. At two-loop order, the decoupling coefficient for the strong coupling is known only for specific mass hierarchies [@Harlander:2005wm]. Recently, the two-loop SQCD corrections for the decoupling coefficient of the bottom-quark mass was computed [@Bednyakov:2007vm]. The focus of this paper is the analytical computation of the decoupling relations for the strong coupling constant and the bottom-quark mass within SQCD through two-loops using a different method as Ref. [@Bednyakov:2007vm]. The comparison of the results will be discussed in the next Section.\ We consider SQCD with $n_f$ active quark and $n_s=n_f$ active squark flavours and $n_{\tilde g}=1$ gluinos. Furthermore, we assume that $n_l=5$ quarks are light (among which the bottom quark) and that the top quark and all squarks and the gluino are heavy. Integrating out the heavy fields from the full SQCD Lagrangian, we obtain the Lagrange density corresponding to the effective QCD with $n_l$ light quarks plus non-renormalizable interactions. The latter are suppressed by negative powers of the heavy masses and will be neglected here. The effective Lagrangian can be written as follows: $$\begin{aligned} {\cal L^\prime}(g_s^{0}, m_q^{0}, \xi^{0}; \psi_q^{0},G_{\mu}^{0,a}, c^{0,a}; \zeta_i^{0}) = {\cal L}^{SQCD}(g_s^{0 \prime}, m_q^{0\prime}, \xi^{0\prime}; \psi_q^{0\prime},G_{\mu}^{0\prime,a},c^{0\prime,a})\,, \label{eq::lag}\end{aligned}$$ where $ \psi_q, G_{\mu}^a ,c^a $ denote the light-quark, the gluon and the ghost fields, respectively, $m_q$ stands for the light quark masses, $\xi$ is the gauge parameter and $g_s=\sqrt{4\pi \alpha_s}$ is the strong coupling. The index $0$ marks bare quantities. ${\cal L}^{SQCD}$ is the usual SQCD Lagrangian from which all heavy fields have been discarded. As a result the fields, masses and couplings associated with light particles have to be rescaled. They are labeled by a prime in Eq. (\[eq::lag\]) and are related with the original parameters through decoupling relations: $$\begin{aligned} g_s^{0\prime} =\zeta_g^0 g_s^0 \,,&\quad m_q^{0\prime} =\zeta_m^0m_q^0 \,,&\quad \xi^{0\prime}-1=\zeta_3^0(\xi^0-1)\,, \nonumber\\ \psi_q^{0\prime} =\sqrt{\zeta_2^0}\psi_q^0 \,,&\quad G_\mu^{0\prime,a}=\sqrt{\zeta_3^0}G_\mu^{0,a} \,,&\quad c^{0\prime,a} =\sqrt{\tilde\zeta_3^0}c^{0,a}\,. \label{eq::bare_dec}\end{aligned}$$ Refs. [@Chetyrkin:1997un] showed that the bare decoupling constants $\zeta_m^0,\, \zeta_2^0,\, \zeta_3^0, \tilde\zeta_3^0 $ can be derived from the quark, the gluon and the ghost propagators, all evaluated at vanishing external momenta. As a result, for calculations performed within the framework of Dimensional Regularization/Reduction (DREG/DRED) only the diagrams containing at least one heavy particle inside the loops do not vanish. In Fig. \[fig::diagrams\] are shown sample Feynman diagrams contributing to the decoupling coefficients for the strong coupling (a) and the bottom-quark mass (b). [c]{} =1.bp (430,100)(6,1) figs/diagcc \ (a)\ =1.bp (430,100)(6,1) figs/diaqq \ (b) For the computation of $\zeta_g$ one has to use the well-known Ward identities. A convenient choice is the relation: $$\begin{aligned} \zeta_g^0 &=& \frac{\tilde{\zeta}_1^0}{\tilde{\zeta}_3^0\sqrt{\zeta_3^0}} \,, \label{eq::zetag0}\end{aligned}$$ where $\tilde{\zeta}_1^0$ denotes the decoupling constant for the ghost-gluon vertex.\ The finite decoupling coefficients are obtained upon the renormalization of the bare parameters. They are given by $$\begin{aligned} \zeta_g = \frac{Z_g}{Z_g^\prime} \zeta_g^0 \,,\quad \zeta_m = \frac{Z_m}{Z_m^\prime} \zeta_m^0\,, \label{eq::zetagren}\end{aligned}$$ where $Z_g^\prime$ and $Z_m^\prime$ correspond to the renormalization constants in the effective theory. Since we are interested in the two-loop results for $\zeta_i,\, i=g,m$, the corresponding renormalization constants for SQCD and QCD have to be implemented with the same accuracy. Analytical results for them can be found in Refs. [@Jack:1994kd; @Steinhauser:2002rq; @Bednyakov:2002sf]. As mentioned above, the decoupling coefficients can be related with vacuum integrals. The latter can be recursively reduced to a master-integral [@Davydychev:1992mt] using the method of integration by parts [@Chetyrkin:1981qh]. Given the large number of diagrams and occurrence of many different mass scales, we computed them with the help of an automated setup. The Feynman diagrams were generated with [QGRAF]{} [@Nogueira:1991ex] and further processed with [q2e]{} [@Seidensticker:1999bb]. The reduction of various vacuum integrals to the master integral was performed by a self written [ FORM]{} [@Vermaseren:2000nd] routine. The reduction of topologies with two different massive and one massless lines requires a careful treatment. The related master integral can be easily derived from its general expression valid for massive lines, given in Ref. [@Davydychev:1992mt]. \[sec::ren\] Regularization and renormalization scheme ====================================================== In our setup, we used the squark mass eigenstates and their mixing angles as input parameters. For convenience, we give below the relations between them and the parameters of the MSSM Lagrangian.\ The squark mass eigenstates $\tilde{q}_{1,2}$ and their mass eigenvalues $m_{\tilde{q}_{1,2}}$ are obtained by diagonalizing the mass matrix $$\begin{aligned} {\cal M}_{\tilde q}= \left( \begin{array}{cc} M_L^2& m_q X_q\\ m_q X_q& M_R^2 \end{array} \right)\,, \label{eq::mixing}\end{aligned}$$ where we used the notation $$\begin{aligned} X_q &=& A_q-\mu \left\{\begin{array}{ll} \tan\beta\,, &\mbox{ for down-type quarks}\\ \cot\beta\,, &\mbox{ for up-type quarks} \end{array}\right . \,,\nonumber\\ M_L^2 &=& M_{\tilde Q}^2+m_q^2+M_Z^2 (I_3^q-Q_q s_W^2)\cos 2\beta\,,\nonumber\\ M_R^2 &=& M_{\tilde{D},\tilde{U}}^2+m_q^2+M_Z^2 Q_q s_W^2\cos 2\beta\,.\end{aligned}$$ Here $m_q$, $I_3^q$ and $Q_q$ are the mass, isospin and electric charge of the quark $q$, respectively, and $s_W=\sin\theta_W$. The parameters $M_{\tilde Q}$ and $M_{\tilde{D},\tilde{U}}$ are the soft supersymmetry breaking masses, $A_q$ is a trilinear coupling and $\mu $ is the Higgs-Higgsino bilinear coupling. The squark mass eigenvalues are defined through the unitary transformation $$\begin{aligned} \left( \begin{array}{cc} m_{\tilde{q}_1}^2& 0\\ 0& m_{\tilde{q}_2}^2 \end{array} \right) = {\cal R}_{\tilde q} {\cal M}_{\tilde q} {\cal R}_{\tilde q}^\dag\,,\quad \mbox{with} \quad{\cal R}_{\tilde q} =\left( \begin{array}{cc} \cos \theta_q & \sin \theta_q \\ -\sin \theta_q& \cos \theta_q \end{array} \right)\,,\end{aligned}$$ and the squark mixing angle through $$\begin{aligned} \sin 2\theta_q = \frac{2 m_q X_q}{m_{\tilde{q}_1}^2-m_{\tilde{q}_2}^2} \,. \label{eq::mixb} \end{aligned}$$ Since we consider the two-loop ${\cal O}(\alpha_s^2)$ corrections, only the one-loop ${\cal O}(\alpha_s)$ counterterms for the input parameters are required. We have chosen the [$\overline{\mbox{{\scalefont{.9}}DR}}$]{} scheme to renormalize the strong coupling constant, and the on-shell scheme for the masses of the heavy particles, [*i.e. *]{} the gluino, squarks and top quark. The corresponding one-loop renormalization constants are known analytically (see, [*e.g. *]{}, Ref. [@Pierce:1996zz]). For the computation of the two-loop $\zeta_{m_b}$, also the one-loop counterterm for the sbottom mixing angle $\theta_b$ is required. We adopted the on-shell renormalization prescription as defined in Ref. [@Guasch:1998as] $$\begin{aligned} \delta\theta_{b}&=&\frac{\mbox{Re} \Sigma_{\tilde{b}_{12}}(m^2_{\tilde{b}_1})+\mbox{Re} \Sigma_{\tilde{b}_{12}}(m^2_{\tilde{b}_2})}{2 (m^2_{\tilde{b}_1}-m^2_{\tilde{b}_2})}\,, \end{aligned}$$ where $\Sigma_{\tilde{b}_{12}}$ is the non-diagonal on-shell sbottom self-energy. As we neglect the bottom-quark mass w.r.t. heavy particle masses, an explicit dependence of the radiative corrections on $m_b$ can occur only through bottom Yukawa couplings. In order to avoid the occurrence of large logarithms of the form $\alpha_s^2\log(\mu^2/m_b^2)$ with $\mu\simeq \tilde M$, we have renormalized the bottom quark mass in the [$\overline{\mbox{{\scalefont{.9}}DR}}$]{} scheme. In this way, the large logarithms are absorbed into the running mass and the higher order corrections are maintained small. The renormalization prescription for the trilinear coupling $A_b$ is fixed by the tree-level relation Eq. (\[eq::mixb\]), while the parameters $\mu$ and $\tan\beta$ do not acquire ${\cal O}(\alpha_s)$ corrections to the one-loop level. Generically, the counterterm for $A_b$ can be expressed as $$\begin{aligned} \delta A_b = \left(2 \cos 2\theta_b \delta\theta_{b}+\sin 2\theta_b \frac{ \delta m^2_{\tilde{b}_1}-\delta m^2_{\tilde{b}_2}}{m^2_{\tilde{b}_1}-m^2_{\tilde{b}_2}}-\sin 2\theta_b\frac{\delta m_b}{m_b}\right) \frac{m^2_{\tilde{b}_1}-m^2_{\tilde{b}_2}}{2 m_b} \,, \label{eq::dab}\end{aligned}$$ where $\delta m_b$ and $ \delta m^2_{\tilde{b}_{1,2}} $ are the counterterms corresponding to bottom-quark and squark masses, respectively. Due to the use of different renormalization prescriptions for the bottom/sbottom masses and mixing angle, the parameter $A_b$ is renormalized in a [*mixed*]{} scheme.\ For the regularization of ultra-violet divergencies, we have implemented DRED with the help of the so-called $\epsilon$-scalars [@Jack:1993ws]. In softly broken SUSY theories, as it is the case of MSSM/SQCD, they get a radiatively induced mass. We choose to renormalize their mass in the on-shell scheme, requiring that the renormalized mass is equal to zero.\ There are also other approaches available in the literature. We want to mention the one proposed in Ref. [@Jack:1994rk], where the $\epsilon$-scalars are treated as massive particles. This approach is known in the literature as the [$\overline{\mbox{{\scalefont{.9}}DR}}$]{}$^\prime$ scheme. In this case, the $\epsilon$-scalars have to be decoupled together with the heavy particles of the theory [@Bednyakov:2007vm]. The advantage of this method is that it directly relates SQCD parameters regularized within DRED with those of QCD regularized within DREG, which are known from experiments. The price of this “shortcut” is on one hand, that additional diagrams containing the $\epsilon$-scalars as massive particles occur in the calculation of the decoupling coefficients. On the other hand, the contributions originating from the change of regularization scheme and those from the decoupling of heavy particles are not distinguishable anymore. In our approach the change of regularization scheme from DRED to DREG has to be performed explicitly within QCD [@Martin:1993zk; @Jack:1994kd; @Harlander:2006rj]. For our purposes, the two-loop conversion relations for the strong coupling constant and the bottom-quark mass are required. The complication that arises at this stage is the occurrence of the [*evanescent coupling*]{} $\alpha_e$ associated with the $\epsilon$-scalar-quark-quark vertex. This has to be distinguished from the gauge coupling within non-supersymmetric theories (e.g. QCD). However, in SQCD they are identical with the gauge couplings, as required by SUSY. Using the ET procedure, we can relate them with the strong coupling within the full theory with the help of decoupling relations similar with those introduced in Eq. (\[eq::bare\_dec\]) $$\begin{aligned} \alpha_e^{\prime}=\zeta_e \alpha_e= \zeta_e \alpha_s\,. \label{eq::zetae}\end{aligned}$$ Following the method described above, one can calculate $\zeta_e$ evaluating the $\epsilon$-scalar and quark-propagators and the $\epsilon$-scalar-quark-quark vertex at vanishing external momentum. In Ref. [@Harlander:2007wh], its one-loop expression was computed under the simplifying assumption of a degenerate SUSY-mass spectrum. In principle, for our numerical analyses, that rely on solving a system of coupled differential equations involving also the evanescent coupling $\alpha_e$, even the two-loop order corrections to $\zeta_e $ are needed. However, from the explicit calculation it turned out that the numerical effects induced by the two-loop corrections to $\zeta_e$ are below the per-mille level. For simplicity, we do not display the corresponding two-loop results in the following. The analytical formulae are available upon request from the authors. The method outlined here and the one introduced in Ref. [@Bednyakov:2007vm] for the calculation of the decoupling coefficient of the bottom-quark mass are equivalent. This has to be understood in the usual sense, that the predictions for physical observables made in one scheme at a given order in perturbation theory can be translated to the other scheme through redefinitions of masses and couplings. We have explicitly checked implementing additionally the method of Ref. [@Bednyakov:2007vm] in our setup the equivalence property for the decoupling coefficient of the bottom-quark mass $\zeta_m$ through two-loop order. Apart from the obvious rescaling of the strong coupling and the bottom quark mass, also the sbottom masses have to be modified  [@Jack:1994rk] $$\begin{aligned} m_{\tilde{b}}^2|_{{\overline{\rm{\scalefont{.9}}DR}}{}^\prime}=m_{\tilde{b}}^2|_{{\overline{\rm{\scalefont{.9}}DR}}{}} -\frac{{\alpha_s^{{\overline{\rm{\scalefont{.9}}DR}}{}}}}{2\pi} C_F m_{\epsilon}^2\,.\end{aligned}$$ Here $C_F$ is the Casimir invariant in the fundamental representation and $m_{\epsilon}$ denotes the mass of the $\epsilon$-scalars. The indices [$\overline{\mbox{{\scalefont{.9}}DR}}$]{} and [$\overline{\mbox{{\scalefont{.9}}DR}}$]{}$^\prime$, respectively, specify the regularization scheme. We also compared numerically the results for the two-loop $\zeta_{m_b}$ obtained using our method with the ones depicted in Fig. 3 of Ref. [@Bednyakov:2007vm] and found very good agreement. \[sec::res\] Analytical results =============================== \[sec::oneloop\]One-loop results -------------------------------- The exact one-loop results for the decoupling coefficients of the strong coupling constant $\zeta_s$ and bottom-quark mass $\zeta_m$ can be found in Refs. [@Pierce:1996zz; @Bednyakov:2007vm]. The analytical formula for $\zeta_e$ is new and we give it below up to $\cal{O}(\epsilon)$. [$$\begin{aligned} \zeta _s =& 1+\frac{\alpha_s^{\rm{ (SQCD)}}}{ \pi } \Bigg[-\frac{1}{6} C_A L_{\tilde{g}}-\frac{1}{6}L_t -\sum_{q}\sum_{i=1,2}\frac{1}{24} L_{\tilde{q}_i} \nonumber \\ &-\epsilon \left(\frac{C_A}{12} \left(L_{\tilde{g}}^2+\zeta(2)\right) +\frac{1}{12} \left( L_t^2+\zeta(2)\right)-\frac{1}{48}\sum_{q}\sum_{i=1,2}\left( L_{\tilde{q}_i}^2+ \zeta(2)\right)\right)\Bigg]\,, \label{eq::zetas1l} \\ \zeta_{e, q} =& 1+ \frac{\alpha_s^{\rm{ (SQCD)}}}{ \pi } \Bigg\{-T_F \frac{ L_t}{2} +\frac{C_A}{4} \Bigg( 2+L_{\tilde{g}} + \sum_{i=1,2}\left(L_{\tilde{g}}-L_{\tilde{q}_i}\right) \frac{m_{\tilde{q}_i}^2}{m_{\tilde{g}}^2-m_{\tilde{q}_1}^2}\Bigg) \nonumber\\ &+ \frac{C_F}{4} \Bigg( \sum_{i=1,2} \left(-1-2 L_{\tilde{g}}+2 L_{\tilde{q}_i} +\left(-L_{\tilde{g}}+L_{\tilde{q}_i}\right) \frac{m_{\tilde{q}_i}^2}{m_{\tilde{g}}^2 - m_{\tilde{q}_i}^2} \right) \frac{ m_{\tilde{q}_i}^2}{ m_{\tilde{g}}^2 -m_{\tilde{q}_i}^2} \nonumber\\ &+ \left(-3-2 L_{\tilde{g}}\right) \Bigg) + \epsilon \Bigg[- \frac{T_F}{4} \left(L_t^2+\zeta(2)\right)+\frac{C_A}{8} \Bigg( 4+4 L_{\tilde{g}}+L_{\tilde{g}}^2+\zeta(2) + \nonumber\\ &+ \frac{1}{2}\sum_{i=1,2} \left(L_{\tilde{g}}-L_{\tilde{q}_i}\right) \left(2+L_{\tilde{g}}+L_{\tilde{q}_i}\right)\frac{ m_{\tilde{q}_i}^2}{ m_{\tilde{g}}^2-m_{\tilde{q}_i}^2} \Bigg) + \frac{C_F}{8} \Bigg( -7-6 L_{\tilde{g}} -2 L_{\tilde{g}}^2 \nonumber\\ & -2 \zeta(2) + \frac{1}{2}\sum_{i=1,2} \Bigg( -3-6 L_{\tilde{g}}-2 L_{\tilde{g}}^2+4 L_{\tilde{q}_i}+2 L_{\tilde{q}_i}^2 \nonumber\\ &- \left(L_{\tilde{g}}-L_{\tilde{q}_1}\right) \left(3+L_{\tilde{g}}+L_{\tilde{q}_1}\right)\frac{ m_{\tilde{q}_i}^2}{ m_{\tilde{g}}^2-m_{\tilde{q}_i}^2 }\Bigg)\frac{ m_{\tilde{q}_i}^2}{ m_{\tilde{g}}^2-m_{\tilde{q}_i}^2 }\Bigg)\bigg] \Bigg\}\,, \label{eq::zetae1l} \\ \zeta_{m_b} = & 1 + \frac{\alpha_s^{\rm{ (SQCD)}}}{ \pi } C_F\sum_{i=1,2}\bigg\{-\frac{(1 + {L_{\tilde{b}_i}})}{4}\frac{ {m_{\tilde{b}_i}}^2}{ ({m_{\tilde{b}_i}}^2-{m_{\tilde g}}^2)} + \frac{ (3 + 2 {L_{\tilde{b}_i}}) {m_{\tilde{b}_i}}^4-(3 + 2 {L_{\tilde g}}) {m_{\tilde g}}^4 }{16 ({m_{\tilde{b}_i}}^2-{m_{\tilde g}}^2)^2} \nonumber\\ &- \frac{(-1)^i\, X_b {m_{\tilde g}}}{m_{\tilde{b}_1}^2-m_{\tilde{b}_2}^2}\frac{ {m_{\tilde{b}_i}}^2 {L_{\tilde{b}_i}}- {m_{\tilde g}}^2 {L_{\tilde g}}}{2 ({m_{\tilde{b}_i}}^2-{m_{\tilde g}}^2)} + \epsilon \bigg[-\frac{{m_{\tilde{b}_i}}^2 (2 + {L_{\tilde{b}_i}}(2 + {L_{\tilde{b}_i}}) +\zeta(2) )}{8 ({m_{\tilde{b}_i}}^2-{m_{\tilde g}}^2)} \nonumber\\ &+ \frac{ {m_{\tilde{b}_i}}^4 (7 + 2 {L_{\tilde{b}_i}}(3 + {L_{\tilde{b}_i}}) + 2 \zeta(2))-{m_{\tilde g}}^4 (7 + 2 {L_{\tilde g}}(3 + {L_{\tilde g}}) + 2 \zeta(2)) }{32 ({m_{\tilde{b}_i}}^2-{m_{\tilde g}}^2)^2} \nonumber\\ &+ \frac{(-1)^i\, X_b {m_{\tilde g}}}{m_{\tilde{b}_1}^2-m_{\tilde{b}_2}^2} \frac{ {m_{\tilde g}}^2 {L_{\tilde g}}(2 + {L_{\tilde g}}) - {m_{\tilde{b}_i}}^2 {L_{\tilde{b}_i}}(2 + {L_{\tilde{b}_i}}) }{4 ({m_{\tilde{b}_i}}^2-{m_{\tilde g}}^2)} \bigg] \bigg\}\,. \label{eq::zetam1l}\end{aligned}$$ ]{} In Eqs. (\[eq::zetas1l\]), (\[eq::zetae1l\]), and (\[eq::zetam1l\]), we have adopted the abbreviations $$L_i = \ln\frac{\mu^2}{m_i^2}\,,\quad i \in \{t,\tilde{g},\tilde{q}_{1,2},\tilde{b}_{1,2}\}\,, \label{eq::abbv}$$ where $\tilde{q}_{i} (\tilde{b}_{i})$ denote the super-partners of the quark $q (b)$. For our purposes, the special case $\zeta_{e, q=b}$ is of interest.\ The colour factors are defined in case of a gauge group $SU(N)$ as follows $$C_F = \frac{N^2-1}{2 N}\,\quad C_A=N\,, \quad T_F=\frac{1}{2}\,.$$ Furthermore, we used the notation $\zeta(2)=\pi^2/6$ and introduced the label “SQCD” to specify that the strong coupling has to be evaluated within the full theory, i.e. the SQCD with $n_f=n_s=6$ active flavours. The presence of the terms proportional with the parameter $X_b$ is a manifestation of the supersymmetry breaking. They are generated by the Yukawa interaction between left- and right-handed bottom squarks and the CP-neutral Higgs fields. Their contribution to the decoupling coefficient of the bottom-quark mass can be related through the Low Energy Theorem [@Ellis:1975ap] to the decay rate of the Higgs boson to $b\bar{b}$ pairs. To one-loop order, the $X_b$-term of Eq. (\[eq::zetam1l\]) coincides with the SQCD corrections to the decay rate $\phi\to b\bar{b}$[@Guasch:2003cv]. To higher orders, the relation between the two parameters becomes more involved.\ The Yukawa-coupling induced contributions attracted a lot of attention in the past years, due to the fact that they are the dominant corrections for large values of $\tan \beta$. They may in general become comparable with the tree-level bottom-quark mass. The resummation of the one-loop corrections was performed in Ref. [@Carena:1999py]. \[sec::twoloop\] Two-loop calculation ------------------------------------- The analytical two-loop results for the decoupling coefficients are too lengthy to be displayed here. They are available in [ MATHEMATICA]{} format from http://www-ttp.particle.uni-karlsruhe.de/Progdata/ttp08-25. Instead, we present the two-loop results for three special cases of the hierarchy among the heavy particle masses. Before displaying the analytical results, let us notice the absence of contributions of the form $\alpha_s^2 X_b^2$ to $\zeta_{m_b}$, in accordance with Refs. [@Carena:1999py; @Guasch:2003cv]. They are suppressed by a factor $m_b/\tilde{M}$, that we neglect in the ET formalism. ### Scenario A We consider first the case of all supersymmetric particles having masses of the same order of magnitude and being much heavier than the top-quark $$\begin{split} &m_{\tilde{u}} = \ldots = m_{\tilde{b}} = m_{\tilde{t}} = m_{\tilde{g}}=\tilde{M}\quad \gg\quad m_t\nonumber\\ & \alpha_s^{(5)} = \zeta_{s}^{\tilde{M}}\, \alpha_s^{\rm{ (SQCD)}} \,,\qquad m_b^{(5)} = \zeta_{m_b}^{\tilde{M}}\, m_b^{\rm{ (SQCD)}}\,. \end{split}$$ $\zeta_{s}^{\tilde{M}},\,\zeta_{m_b}^{\tilde{M}} $ are functions of the supersymmetric mass $\tilde{M}$ and the top-quark mass $m_t$, the soft SUSY breaking parameters $X_q$, $q=b,t$, the strong coupling constant in the full theory $ \alpha_s^{\rm{ (SQCD)}}$ and the decoupling scale $\mu$. The superscript $(5)$ indicates that the parameters are defined in QCD with $n_l=5$ light quarks. In addition to the notations introduced in Eq. (\[eq::abbv\]), the following abbreviation will be used $$\begin{aligned} L_{\tilde{M}}=\ln\frac{\mu^2}{\tilde{M}^2}\,.\end{aligned}$$ The two-loop result for the decoupling coefficient of $\alpha_s$ in case of a degenerate SUSY mass spectrum is known [@Harlander:2005wm], however we give it here for completeness [$$\begin{aligned} \zeta_{s}^{\tilde{M}}&= 1+\frac{\alpha _s^{\rm\tiny{ (SQCD)}}}{\pi } \Bigg[ C_A \left(-\frac{1}{6} L_{\tilde M}\right) +\Bigg(-L_{\tilde M}-\frac{L_t}{3}\Bigg) T_F\Bigg] + \Bigg(\frac{\alpha _s^{\rm\tiny{ (SQCD)}}}{\pi }\Bigg)^2 \Bigg\{C_A^2 \Bigg(-\frac{85}{288}\nonumber\\ &-\frac{L_{\tilde M}}{3}+\frac{L_{\tilde M}^2}{36}\Bigg) + C_F T_F \Bigg[-\frac{31}{16}-\frac{3 L_{\tilde M}}{2}-\frac{L_t}{4}-\frac{m_t^2}{8 \tilde{M}^2}+\frac{\pi m_t^3}{12 \tilde{M}^3}+\Bigg(-\frac{17}{150}-\frac{3L_{\tilde M}}{40 }\nonumber\\ &+\frac{3L_t}{40 }\Bigg) \frac{m_t^4}{\tilde{M}^4} \Bigg] + C_A T_F \Bigg[\frac{41}{36}+L_{\tilde M}+\frac{L_{\tilde M}^2}{3}-\frac{5L_t}{12}+\frac{L_{\tilde M} L_t}{9} +\Bigg(\frac{1}{8 }+\frac{L_{\tilde M}}{4 }-\frac{L_t}{4 }\Bigg) \frac{m_t^2}{m_S^2} \nonumber \\ &-\frac{\pi m_t^3}{6 \tilde{M}^3} +\Bigg(\frac{19}{144}+\frac{ L_{\tilde M}}{24}-\frac{ L_t}{24}\Bigg) \frac{m_t^4}{\tilde{M}^4} \Bigg] +T_F^2\Bigg(L_{\tilde M}^2+\frac{2 L_{\tilde M} L_t}{3}+\frac{L_t^2}{9}\Bigg) \nonumber \\ & + {\cal O}\left( \frac{m_t^5}{\tilde{M}^5} \right) \Bigg\}\,, \label{eq::degas} \\ \zeta_{m_b}^{\tilde{M}} &= 1-\frac{\alpha _s^{\rm\tiny{ (SQCD)}}}{\pi } C_F\left[\frac{ L_{\tilde M} }{4} + \frac{X_b}{4 {\tilde M} }\right]+\left(\frac{\alpha _s^{\rm\tiny{ (SQCD)}}}{\pi }\right)^2 C_F\Bigg\{ -C_A\left(\frac{65}{1152} + \frac{43 L_{\tilde M} }{96} + \frac{L_{\tilde M}^2}{32} \right) \nonumber\\ &+ C_F\left(-\frac{99}{128} - \frac{7 L_{\tilde M}}{32} + \frac{L_{\tilde M}^2}{32} \right) +T_F\bigg[\frac{197}{72} - L_{\tilde M} + \frac{3 L_{\tilde M}^2}{4} - \frac{{L_t}}{12} + \frac{{L_t}^2}{8} \nonumber\\ &+\left( \frac{7}{144} + \frac{L_{\tilde M}}{12} - \frac{{L_t}}{12}\right)\frac{m_t^2}{\tilde{M}^2} -\frac{\pi}{12}\frac{m_t^3}{\tilde{M}^3} +\left(\frac{53}{600} - \frac{7 L_{\tilde M}}{160} + \frac{7 {L_t}}{160}\right)\frac{m_t^4}{\tilde{M}^4} \bigg] \nonumber\\ &+ \frac{X_b}{\tilde{M}}\Bigg[ -C_A\left( \frac{1}{16} + \frac{3 L_{\tilde M}}{16} \right)- C_F\left(\frac{1}{4} - \frac{3 L_{\tilde M}}{16} \right) +T_F\bigg[\frac{-3}{4} + \frac{3 L_{\tilde M} }{4} \nonumber\\ &+ \left(\frac{-7}{72} - \frac{L_{\tilde M}}{24} + \frac{{L_t}}{24}\right)\frac{m_t^2}{\tilde{M}^2} +\frac{\pi}{24}\frac{m_t^3}{\tilde{M}^3} +\left(-\frac{17}{450} + \frac{L_{\tilde M}}{240} - \frac{{L_t}}{240}\right)\frac{m_t^4}{\tilde{M}^4} \bigg] \Bigg] \nonumber\\ &+ \frac{X_t}{\tilde{M}}T_F\Bigg[ \left( \frac{-5}{72} + \frac{L_{\tilde M}}{12} - \frac{{L_t}}{12} \right)\frac{m_t^2}{\tilde{M}^2} -\frac{\pi}{24}\frac{m_t^3}{\tilde{M}^3} +\left(\frac{23}{450} - \frac{L_{\tilde M}}{60} + \frac{{L_t}}{60} \right)\frac{m_t^4}{\tilde{M}^4} \Bigg] \nonumber\\ &+ \frac{X_t X_b}{\tilde{M}^2} T_F\Bigg[ \left( \frac{1}{72} - \frac{L_{\tilde M}}{24} + \frac{{L_t}}{24} \right)\frac{m_t^2}{\tilde{M}^2} +\frac{\pi}{48}\frac{m_t^3}{\tilde{M}^3} +\left(\frac{-59}{1800} - \frac{L_{\tilde M}}{120} + \frac{{L_t}}{120} \right)\frac{m_t^4}{\tilde{M}^4} \Bigg] \nonumber\\ &+ {\cal O}\left( \frac{m_t^5}{\tilde{M}^5} \right) \Bigg\}\,.\end{aligned}$$ ]{} Let us point out that, according to Eq. (\[eq::mixb\]) the assumption of degenerate top-squark masses can be materialized only if $X_t\to 0$, due to the heavy top-quark mass. We display, however, for completeness the full result. Further on, the hypothesis of equal top- and bottom-squark masses is inconsistent with the $SU(2)$ invariance of the $\tilde{t}/\tilde{b}$ isodublet imposed in models like mSUGRA. ### Scenario B In the following, we discuss the possibility that the gluino is the heaviest supersymmetric particle and the squarks have equal masses, much heavier than that of the top-quark $$\begin{split} &m_{\tilde{g}} \quad \gg\quad m_{\tilde{u}} = \ldots = m_{\tilde{b}} = m_{\tilde{t}} = m_{\tilde{q}}\quad \gg\quad m_t\nonumber\\ & \alpha_s^{(5)} = \zeta_{s}^{\tilde{g}}\, \alpha_s^{\rm{ (SQCD)}} \,,\qquad m_b^{(5)} = \zeta_{m_b}^{\tilde{g}}\, m_b^{\rm{ (SQCD)}}\,. \end{split}$$ The two-loop results read [$$\begin{aligned} \zeta_{s}^{\tilde{g}}&= 1+\frac{\alpha _s^{\rm\tiny{ (SQCD)}}}{\pi } \Bigg[C_A\left(-\frac{1}{6} {L_{\tilde{g}}}\right)+\Bigg(-\frac{L_t}{3}-{L_{\tilde{q}}}\Bigg) T_F\Bigg]+ \Bigg(\frac{\alpha _s^{\rm\tiny{ (SQCD)}}}{\pi }\Bigg)^2 \Bigg\{C_A^2 \Bigg(-\frac{85}{288}-\frac{{L_{\tilde{g}}}}{3}\nonumber\\ & +\frac{{L_{\tilde{g}}}^2}{36}\Bigg) + T_F^2\Bigg(\frac{L_t^2}{9}+\frac{2}{3} L_t {L_{\tilde{q}}}+{L_{\tilde{q}}}^2\Bigg) +C_F T_F \Bigg[-\frac{19}{16}-\frac{L_t}{4}+\frac{3 {L_{\tilde{g}}}}{2}-3 {L_{\tilde{q}}}+\Bigg(\frac{1}{6}+{L_{\tilde{g}}}\nonumber\\ & -{L_{\tilde{q}}}\Bigg) \frac{{m_{\tilde{q}}}^2}{{m_{\tilde{g}}}^2} + \Bigg(\frac{1}{24}+{L_{\tilde{g}}}-{L_{\tilde{q}}}\Bigg)\frac{ {m_{\tilde{q}}}^4}{{m_{\tilde{g}}}^4} +\Bigg(-\frac{5}{9}+\frac{L_t}{3}-\frac{{L_{\tilde{g}}}}{3}\Bigg) \frac{ m_t^2}{{m_{\tilde{g}}}^2} +\Bigg(-\frac{9}{8}+\frac{5 L_t}{4}-\frac{5 {L_{\tilde{g}}}}{4}\Bigg) \frac{m_t^4}{{m_{\tilde{g}}}^4} \nonumber\\ &+\frac{m_t^2 {m_{\tilde{q}}}^2}{{m_{\tilde{g}}}^4} \Bigg(-\frac{41}{36}+\frac{11 L_t}{6}-\frac{23 {L_{\tilde{g}}}}{6}+L_t {L_{\tilde{g}}}-{L_{\tilde{g}}}^2+2 {L_{\tilde{q}}}-L_t {L_{\tilde{q}}}+{L_{\tilde{g}}}{L_{\tilde{q}}}-2 \zeta _2\Bigg)\Bigg] \nonumber\\ &+C_A T_F \Bigg[\frac{95}{36}-\frac{5 L_t}{12}+\frac{3 {L_{\tilde{g}}}}{2}+\frac{1}{9} L_t {L_{\tilde{g}}}-\frac{{L_{\tilde{q}}}}{2}+\frac{1}{3} {L_{\tilde{g}}}{L_{\tilde{q}}}+\Bigg(-\frac{3 {L_{\tilde{g}}}}{2}+\frac{3 {L_{\tilde{q}}}}{2}\Bigg)\frac{{m_{\tilde{q}}}^2}{{m_{\tilde{g}}}^2} \nonumber\\ &+ \Bigg(-\frac{3}{2}+\frac{3 {L_{\tilde{g}}}}{2}-\frac{3 {L_{\tilde{q}}}}{2}\Bigg) \frac{{m_{\tilde{q}}}^4}{{m_{\tilde{g}}}^4} +\Bigg(\frac{1}{2}-\frac{L_t}{4}+\frac{{L_{\tilde{g}}}}{4}\Bigg)\frac{ m_t^2}{{m_{\tilde{g}}}^2} +\Bigg(\frac{1}{4}-\frac{L_t}{4}+\frac{{L_{\tilde{g}}}}{4}\Bigg) \frac{m_t^4}{{m_{\tilde{g}}}^4} \nonumber\\ &+\frac{m_t^2 {m_{\tilde{q}}}^2}{{m_{\tilde{g}}}^4} \Bigg(\frac{1}{2}-\frac{3 L_t}{4}+\frac{3 {L_{\tilde{g}}}}{2}-\frac{1}{2} L_t {L_{\tilde{g}}}+\frac{{L_{\tilde{g}}}^2}{2}-\frac{3{L_{\tilde{q}}}}{4}+\frac{1}{2} L_t {L_{\tilde{q}}}-\frac{1}{2} {L_{\tilde{g}}}{L_{\tilde{q}}}+\zeta _2\Bigg)\Bigg] \nonumber\\ & + {\cal O}\left( \frac{{m_{\tilde{q}}}^6}{{m_{\tilde{g}}}^6},\frac{{m_{\tilde{q}}}^4 {m_t}^2}{{m_{\tilde{g}}}^6},\frac{{m_{\tilde{q}}}^2 {m_t}^4}{{m_{\tilde{g}}}^6}, \frac{{m_t}^6}{{m_{\tilde{g}}}^6} \right)\bigg\}\,, \end{aligned}$$ ]{} [$$\begin{aligned} \zeta_{m_b}^{\tilde{g}} &= 1+ \frac{\alpha _s^{\rm\tiny{ (SQCD)}}}{\pi } C_F\bigg\{-\frac{3}{8} - \frac{{L_{\tilde g}}}{4} +\left(-\frac{1}{4} - \frac{{L_{\tilde g}}}{2} + \frac{{L_{\tilde{q}}}}{2} \right)\frac{{m_{\tilde{q}}}^2}{{m_{\tilde{g}}}^2} +\left(-\frac{1}{4} - \frac{3 {L_{\tilde g}}}{4} + \frac{3 {L_{\tilde{q}}}}{4}\right)\frac{{m_{\tilde{q}}}^4}{{m_{\tilde{g}}}^4} \nonumber\\ &+\frac{X_b}{{m_{\tilde{g}}}}\left[ \frac{1}{2} + \frac{{L_{\tilde g}}}{2} - \frac{{L_{\tilde{q}}}}{2}+\left(\frac{1}{2} + {L_{\tilde g}}- {L_{\tilde{q}}}\right)\frac{{m_{\tilde{q}}}^2}{{m_{\tilde{g}}}^2} + \left(\frac{1}{2} + \frac{3{L_{\tilde g}}}{2} - \frac{3{L_{\tilde{q}}}}{2} \right)\frac{{m_{\tilde{q}}}^4}{{m_{\tilde{g}}}^4} \right] \bigg\} \nonumber\\ &+\left(\frac{\alpha _s^{\rm\tiny{ (SQCD)}}}{\pi }\right)^2 C_F\bigg\{ C_A\bigg[ -\frac{209}{1152} - \frac{35 {L_{\tilde g}}}{48} - \frac{{L_{\tilde g}}^2}{32} - \frac{\zeta(2)}{8} +\bigg(-\frac{5}{16} - \frac{3{L_{\tilde g}}}{2} - \frac{5{L_{\tilde g}}^2}{8} \nonumber\\ & +\frac{21{L_{\tilde{q}}}}{16} + \frac{7{L_{\tilde g}}{L_{\tilde{q}}}}{8} - \frac{{L_{\tilde{q}}}^2}{4} - \frac{5\zeta(2)}{8} \bigg)\frac{{m_{\tilde{q}}}^2}{{m_{\tilde{g}}}^2} +\bigg( \frac{7}{32} - \frac{11{L_{\tilde g}}}{4} - \frac{37{L_{\tilde g}}^2}{32} + \frac{41{L_{\tilde{q}}}}{16} + \frac{7{L_{\tilde g}}{L_{\tilde{q}}}}{4} - \frac{19{L_{\tilde{q}}}^2}{32} \nonumber\\ &- \frac{5\zeta(2)}{8} \bigg)\frac{{m_{\tilde{q}}}^4}{{m_{\tilde{g}}}^4} \bigg]+C_F\bigg[ -\frac{221}{128} - \frac{{L_{\tilde g}}}{8} + \frac{{L_{\tilde g}}^2}{32} -\frac{\zeta(2)}{16} +\bigg(\frac{9}{16} + \frac{5{L_{\tilde g}}^2}{8} + \frac{{L_{\tilde{q}}}}{16} - \frac{9{L_{\tilde g}}{L_{\tilde{q}}}}{8} \nonumber\\ &+ \frac{{L_{\tilde{q}}}^2}{2} + \frac{\zeta(2)}{2}\bigg)\frac{{m_{\tilde{q}}}^2}{{m_{\tilde{g}}}^2} +\bigg(\frac{31}{192} + \frac{35{L_{\tilde g}}}{32} + \frac{17{L_{\tilde g}}^2}{16} - \frac{33{L_{\tilde{q}}}}{32} - \frac{31{L_{\tilde g}}{L_{\tilde{q}}}}{16} + \frac{7{L_{\tilde{q}}}^2}{8} +\frac{\zeta(2)}{8} \bigg)\frac{{m_{\tilde{q}}}^4}{{m_{\tilde{g}}}^4} \bigg] \nonumber\\ & + T_F\bigg[ \frac{139}{36} + \frac{3{L_{\tilde g}}}{2} + \frac{3{L_{\tilde g}}^2}{8} - \frac{11{L_{\tilde{q}}}}{8} + \frac{3{L_{\tilde{q}}}^2}{8} - \frac{{L_t}}{12} + \frac{{L_t}^2}{8} + \frac{3\zeta(2)}{4} \nonumber\\ &+\bigg( 3 + \frac{9{L_{\tilde g}}}{2} +3 {L_{\tilde g}}^2- \frac{15{L_{\tilde{q}}}}{4} - \frac{9{L_{\tilde g}}{L_{\tilde{q}}}}{2} + \frac{3{L_{\tilde{q}}}^2}{2} \bigg)\frac{{m_{\tilde{q}}}^2}{{m_{\tilde{g}}}^2} +\bigg(-\frac{1}{2} - \frac{{L_{\tilde g}}}{4} + \frac{{L_{\tilde{q}}}}{4} \bigg)\frac{m_t^2}{{m_{\tilde{g}}}^2} \nonumber\\ &-\bigg( 1 + \frac{{L_{\tilde g}}}{2} - \frac{3{L_{\tilde g}}^2}{8} - \frac{{L_{\tilde{q}}}}{2} + \frac{3{L_{\tilde g}}{L_{\tilde{q}}}}{4} - \frac{3{L_{\tilde{q}}}^2}{8} - \frac{3\zeta(2)}{4} \bigg)\frac{{m_{\tilde{q}}}^2 m_t^2}{{m_{\tilde{g}}}^4} \nonumber\\ &+ \bigg(-\frac{5}{72}+\frac{{L_{\tilde{q}}}}{24}-\frac{{L_t}}{24}\bigg)\frac{ m_t^4}{{m_{\tilde{q}}}^2 {m_{\tilde{g}}}^2}+\bigg( -\frac{155}{96} - \frac{3{L_{\tilde g}}}{16} + \frac{3{L_{\tilde g}}^2}{16} + \frac{7{L_{\tilde{q}}}}{8} - \frac{3{L_{\tilde{q}}}^2}{16} - \frac{11{L_t}}{16} \nonumber\\ &- \frac{3{L_{\tilde g}}{L_t}}{8} + \frac{3{L_{\tilde{q}}}{L_t}}{8} + \frac{3\zeta(2)}{8} \bigg)\frac{m_t^4}{{m_{\tilde{g}}}^4} +\bigg(\frac{69}{16} + \frac{63{L_{\tilde g}}}{8} + 3{L_{\tilde g}}^2 - \frac{57{L_{\tilde{q}}}}{8} - \frac{15{L_{\tilde g}}{L_{\tilde{q}}}}{4} + \frac{3{L_{\tilde{q}}}^2}{4} \bigg)\frac{{m_{\tilde{q}}}^4}{{m_{\tilde{g}}}^4}\bigg] \nonumber\\ &+\frac{X_b}{{m_{\tilde{g}}}}\bigg[ C_A\bigg[\frac{7}{8} + {L_{\tilde g}}+ \frac{3{L_{\tilde g}}^2}{8} - \frac{5{L_{\tilde{q}}}}{8} - \frac{3{L_{\tilde g}}{L_{\tilde{q}}}}{8} - \frac{\zeta(2)}{4} +\bigg(\frac{9}{8} + \frac{17{L_{\tilde g}}}{8} + \frac{11{L_{\tilde g}}^2}{8} - \frac{7{L_{\tilde{q}}}}{4} \nonumber\\ &- 2{L_{\tilde g}}{L_{\tilde{q}}}+ \frac{5{L_{\tilde{q}}}^2}{8} - \frac{\zeta(2)}{4} \bigg)\frac{{m_{\tilde{q}}}^2}{{m_{\tilde{g}}}^2} +\bigg( \frac{19}{16} + \frac{27{L_{\tilde g}}}{8} + \frac{23{L_{\tilde g}}^2}{8} - 3{L_{\tilde{q}}}- \frac{37{L_{\tilde g}}{L_{\tilde{q}}}}{8} + \frac{7{L_{\tilde{q}}}^2}{4} \nonumber\\ &- \frac{\zeta(2)}{4} \bigg)\frac{{m_{\tilde{q}}}^4}{{m_{\tilde{g}}}^4} \bigg] +C_F\bigg[\frac{1}{16} - \frac{3{L_{\tilde g}}}{16} - \frac{3{L_{\tilde g}}^2}{8} - \frac{3{L_{\tilde{q}}}}{16} + \frac{3{L_{\tilde g}}{L_{\tilde{q}}}}{8} + \frac{\zeta(2)}{2} +\bigg(-\frac{9}{16} - \frac{5{L_{\tilde g}}}{24} \nonumber\\ &- \frac{3{L_{\tilde g}}^2}{2} - \frac{{L_{\tilde{q}}}}{6} + \frac{9{L_{\tilde g}}{L_{\tilde{q}}}}{4} - \frac{3{L_{\tilde{q}}}^2}{4} + \frac{\zeta(2)}{2} \bigg)\frac{{m_{\tilde{q}}}^2}{{m_{\tilde{g}}}^2} +\bigg(-\frac{13}{16} - \frac{3{L_{\tilde g}}}{4} - \frac{13{L_{\tilde g}}^2}{4} + \frac{3{L_{\tilde{q}}}}{8} \nonumber\\ &+ \frac{43{L_{\tilde g}}{L_{\tilde{q}}}}{8} - \frac{17{L_{\tilde{q}}}^2}{8} + \frac{\zeta(2)}{2} \bigg)\frac{{m_{\tilde{q}}}^4}{{m_{\tilde{g}}}^4} \bigg] +T_F\bigg[ -\frac{3}{2}(2 + {L_{\tilde g}})(1 + {L_{\tilde g}}- {L_{\tilde{q}}}) +\bigg(-9 - \frac{15{L_{\tilde g}}}{2} \nonumber\\ &- 6{L_{\tilde g}}^2 + 6{L_{\tilde{q}}}+ 9{L_{\tilde g}}{L_{\tilde{q}}}- 3{L_{\tilde{q}}}^2 \bigg)\frac{{m_{\tilde{q}}}^2}{{m_{\tilde{g}}}^2} +\bigg(\frac{3}{2}+\frac{{L_{\tilde g}}}{2}-\frac{{L_{\tilde{q}}}}{2} \bigg)\frac{m_t^2}{{m_{\tilde{g}}}^2} +\bigg(-\frac{33}{4} - \frac{63{L_{\tilde g}}}{4} - 6{L_{\tilde g}}^2 \nonumber\\ &+ \frac{57{L_{\tilde{q}}}}{4} + \frac{15{L_{\tilde g}}{L_{\tilde{q}}}}{2} - \frac{3{L_{\tilde{q}}}^2}{2}\bigg)\frac{{m_{\tilde{q}}}^4}{{m_{\tilde{g}}}^4} +\bigg(\frac{5}{18} - \frac{{L_{\tilde{q}}}}{6} + \frac{{L_t}}{6}\bigg)\frac{m_t^4}{{m_{\tilde{q}}}^2{m_{\tilde{g}}}^2} \nonumber\\ &+\bigg(\frac{5}{2} + \frac{3{L_{\tilde g}}}{2} - \frac{{L_{\tilde g}}^2}{2} - \frac{3{L_{\tilde{q}}}}{2} + {L_{\tilde g}}{L_{\tilde{q}}}- \frac{{L_{\tilde{q}}}^2}{2} - 2\zeta(2)\bigg)\frac{m_t^2 {m_{\tilde{q}}}^2}{{m_{\tilde{g}}}^4} \nonumber\\ &+\bigg(\frac{301}{72} + \frac{5{L_{\tilde g}}}{8} - \frac{{L_{\tilde g}}^2}{4} -\frac{53{L_{\tilde{q}}}}{24} - \frac{{L_{\tilde g}}{L_{\tilde{q}}}}{4} + \frac{{L_{\tilde{q}}}^2}{2} + \frac{19{L_t}}{12} + \frac{3{L_{\tilde g}}{L_t}}{4} - \frac{3{L_{\tilde{q}}}{L_t}}{4} - \zeta(2) \bigg)\frac{{m_t}^4}{{m_{\tilde{g}}}^4} \bigg] \bigg] \nonumber\\ &+\frac{X_t}{{m_{\tilde{g}}}}T_F\bigg[\bigg(\frac{1}{2} - {L_{\tilde g}}- \frac{{L_{\tilde g}}^2}{4} + {L_{\tilde{q}}}+ \frac{{L_{\tilde g}}{L_{\tilde{q}}}}{2} - \frac{{L_{\tilde{q}}}^2}{4} - \frac{\zeta(2)}{2} \bigg)\frac{m_t^2}{{m_{\tilde{g}}}^2} +\bigg( 1 - \frac{7{L_{\tilde g}}}{2} - 2{L_{\tilde g}}^2 + \frac{7{L_{\tilde{q}}}}{2} \nonumber\\ &+ 4{L_{\tilde g}}{L_{\tilde{q}}}- 2{L_{\tilde{q}}}^2 - 2\zeta(2) \bigg)\frac{m_t^2 {m_{\tilde{q}}}^2}{{m_{\tilde{g}}}^4}+\bigg(\frac{4}{9} - \frac{{L_{\tilde{q}}}}{6} + \frac{{L_t}}{6}\bigg)\frac{m_t^4}{{m_{\tilde{g}}}^2 {m_{\tilde{q}}}^2} +\bigg(\frac{17}{9} - 3{L_{\tilde g}}- {L_{\tilde g}}^2 + \frac{{L_{\tilde{q}}}}{6} \nonumber\\ &+ {L_{\tilde g}}{L_{\tilde{q}}}+ \frac{17{L_t}}{6} + {L_{\tilde g}}{L_t}- {L_{\tilde{q}}}{L_t}- 2\zeta(2)\bigg)\frac{m_t^4}{{m_{\tilde{g}}}^4 } \bigg] +\frac{X_t X_b}{{m_{\tilde{g}}}^2}T_F\bigg[-\frac{1}{2}\frac{m_t^2}{{m_{\tilde{q}}}^2 } + \bigg(-\frac{3}{2} + \frac{5{L_{\tilde g}}}{2} \nonumber\\ & + \frac{5{L_{\tilde g}}^2}{4} - \frac{5{L_{\tilde{q}}}}{2} - \frac{5{L_{\tilde g}}{L_{\tilde{q}}}}{2} + \frac{5{L_{\tilde{q}}}^2}{4} + \frac{3\zeta(2)}{2}\bigg)\frac{m_t^2}{{m_{\tilde{g}}}^2 } \bigg] +{\cal O}\left( \frac{{m_{\tilde{q}}}^6}{{m_{\tilde{g}}}^6},\frac{{m_{\tilde{q}}}^4 {m_t}^2}{{m_{\tilde{g}}}^6},\frac{{m_{\tilde{q}}}^2 {m_t}^4}{{m_{\tilde{g}}}^6}, \frac{{m_t}^6}{{m_{\tilde{g}}}^6},\frac{X_t X_b{m_t}^4}{{m_{\tilde{g}}}^6} \right)\bigg\}\,. \end{aligned}$$ ]{} ### Scenario C Finally, we make the assumption that all squark masses are degenerate and are much heavier than the gluino and top masses $$\begin{split} & m_{\tilde{u}} = \ldots = m_{\tilde{b}} = m_{\tilde{t}} = m_{\tilde{q}}\quad \gg\quad m_{\tilde{g}} \quad \gg\quad m_t\nonumber\\ & \alpha_s^{(5)} = \zeta_{s}^{\tilde{q}}\, \alpha_s^{\rm{ (SQCD)}} \,,\qquad m_b^{(5)} = \zeta_{m_b}^{\tilde{q}}\, m_b^{\rm{ (SQCD)}}\,. \end{split}$$ The decoupling coefficients are given by [$$\begin{aligned} \zeta _{s}^{\tilde{q}}&=& 1+\frac{\alpha _s^{\rm{ (SQCD)}}}{\pi } \Bigg[-\frac{1}{6} C_A {L_{\tilde{g}}}+\Bigg(-\frac{L_t}{3}-{L_{\tilde{q}}}\Bigg) T_F\Bigg]+\Bigg(\frac{\alpha _s^{\rm{ (SQCD)}}}{\pi }\Bigg)^2 \Bigg\{C_A^2 \Bigg(-\frac{85}{288}-\frac{{L_{\tilde{g}}}}{3} \nonumber \\&+& \frac{{L_{\tilde{g}}}^2}{36}\Bigg) +C_F T_F \Bigg[-\frac{7}{16}-\frac{L_t}{4}-\frac{3 {L_{\tilde{q}}}}{2}+\Bigg(-\frac{1}{9}-\frac{L_t}{12}+\frac{{L_{\tilde{q}}}}{12}\Bigg) \frac{m_t^4}{{m_{\tilde{q}}}^4}+\Bigg(-\frac{1}{9}-\frac{L_t}{3} \nonumber\\ &+& \frac{{L_{\tilde{q}}}}{3}\Bigg) \frac{m_t^2 {m_{\tilde{g}}}^2}{{m_{\tilde{q}}}^4} +\Bigg(-\frac{3}{4}-\frac{3 {L_{\tilde{g}}}}{2}+\frac{3 {L_{\tilde{q}}}}{2}\Bigg) \frac{{m_{\tilde{g}}}^4}{{m_{\tilde{q}}}^4}+\Bigg(\frac{1}{18}+\frac{L_t}{6}-\frac{{L_{\tilde{q}}}}{6}\Bigg) \frac{m_t^2}{{m_{\tilde{q}}}^2}\Bigg] +T_F^2 \Bigg(\frac{L_t^2}{9}\nonumber\\ &+& \frac{2}{3} L_t {L_{\tilde{q}}}+{L_{\tilde{q}}}^2\Bigg) +C_A T_F \Bigg[\frac{41}{36}-\frac{5 L_t}{12}+\frac{1}{9} L_t {L_{\tilde{g}}}+{L_{\tilde{q}}}+\frac{1}{3} {L_{\tilde{g}}}{L_{\tilde{q}}}+\Bigg(-\frac{L_t}{4}+\frac{{L_{\tilde{q}}}}{4}\Bigg) \frac{m_t^2}{{m_{\tilde{q}}}^2} \nonumber\\ &+& \Bigg(\frac{1}{3}-\frac{{L_{\tilde{g}}}}{2}+\frac{{L_{\tilde{q}}}}{2}\Bigg) \frac{{m_{\tilde{g}}}^2}{{m_{\tilde{q}}}^2} +\Bigg(-\frac{L_t}{4}+\frac{{L_{\tilde{q}}}}{4}\Bigg) \frac{m_t^4}{{m_{\tilde{q}}}^4}+\Bigg(\frac{1}{12}-\frac{{L_{\tilde{g}}}}{2}+\frac{{L_{\tilde{q}}}}{2}\Bigg) \frac{{m_{\tilde{g}}}^4}{{m_{\tilde{q}}}^4} \nonumber\\ &+&\frac{m_t^2 {m_{\tilde{g}}}^2}{{m_{\tilde{q}}}^4} \Bigg(\frac{7}{18}-\frac{3 L_t}{4}-\frac{13 {L_{\tilde{g}}}}{12}+\frac{1}{2} L_t {L_{\tilde{g}}}+\frac{11 {L_{\tilde{q}}}}{6}-\frac{1}{2} L_t {L_{\tilde{q}}}-\frac{1}{2} {L_{\tilde{g}}}{L_{\tilde{q}}}+\frac{{L_{\tilde{q}}}^2}{2}+\zeta _2\Bigg)\Bigg] \nonumber\\ & +& {\cal O}\left( \frac{{m_{\tilde{g}}}^6}{{m_{\tilde{q}}}^6},\frac{{m_{\tilde{g}}}^4 {m_t}^2}{{m_{\tilde{q}}}^6},\frac{{m_{\tilde{g}}}^2 {m_t}^4}{{m_{\tilde{q}}}^6}, \frac{{m_t}^6}{{m_{\tilde{q}}}^6} \right)\Bigg\}\,, \end{aligned}$$ ]{} [$$\begin{aligned} \zeta_{m_b}^{\tilde{q}} &=& 1 + \frac{\alpha _s^{\rm{ (SQCD)}}}{\pi } C_F \Bigg\{-\frac{1}{8}-\frac{{L_{\tilde{q}}}}{4}+\frac{{m_{\tilde{g}}}^2}{4{m_{\tilde{q}}}^2} +\Bigg(\frac{1}{4}-\frac{{L_{\tilde{g}}}}{4}+\frac{{L_{\tilde{q}}}}{4}\Bigg)\frac{{m_{\tilde{g}}}^4}{{m_{\tilde{q}}}^4} + \frac{X_b}{{m_{\tilde{q}}}}\Bigg[-\frac{{m_{\tilde{g}}}}{2{m_{\tilde{q}}}}\nonumber\\&+& \Bigg(-\frac{1}{2}+\frac{{L_{\tilde{g}}}}{2}-\frac{{L_{\tilde{q}}}}{2}\Bigg) \frac{{m_{\tilde{g}}}^3}{{m_{\tilde{q}}}^3}\Bigg]\Bigg\} +\left(\frac{\alpha _s^{\rm{ (SQCD)}}}{\pi }\right)^2 C_F \Bigg\{ C_A \Bigg[\frac{295}{1152}-\frac{{L_{\tilde{g}}}}{24}+\frac{{L_{\tilde{g}}}^2}{16}-\frac{{L_{\tilde{q}}}}{2} -\frac{3{L_{\tilde{q}}}^2}{32}\nonumber\\&-&\frac{\zeta(2)}{2} +\Bigg(\frac{15}{16}+\frac{{L_{\tilde{g}}}}{2}-\frac{5{L_{\tilde{q}}}}{16} -\frac{5\zeta(2)}{8}\Bigg)\frac{{m_{\tilde{g}}}^2}{{m_{\tilde{q}}}^2} +\Bigg(\frac{19}{16}+\frac{3{L_{\tilde{g}}}}{16}+\frac{7{L_{\tilde{g}}}^2}{32} +\frac{13{L_{\tilde{q}}}^2}{32}-\frac{5{L_{\tilde{g}}}{L_{\tilde{q}}}}{8} \nonumber\\&-&\frac{5\zeta(2)}{8}\Bigg)\frac{{m_{\tilde{g}}}^4}{{m_{\tilde{q}}}^4}\Bigg] + C_F \Bigg[-\frac{205}{128}-\frac{3{L_{\tilde{q}}}}{16}+\frac{{L_{\tilde{q}}}^2}{32} +\frac{15\zeta(2)}{16} +\Bigg(-\frac{3}{2}-\frac{{L_{\tilde{q}}}}{16} +\frac{5\zeta(2)}{4}\Bigg)\frac{{m_{\tilde{g}}}^2}{{m_{\tilde{q}}}^2} \nonumber\\&+&\Bigg(-\frac{187}{64}-\frac{39{L_{\tilde{g}}}}{32} -\frac{7{L_{\tilde{g}}}^2}{8}+\frac{37{L_{\tilde{q}}}}{32} +\frac{29{L_{\tilde{q}}}{L_{\tilde{g}}}}{16}-\frac{15{L_{\tilde{q}}}^2}{16} +\frac{13\zeta(2)}{8}\Bigg)\frac{{m_{\tilde{g}}}^4}{{m_{\tilde{q}}}^4}\Bigg] \nonumber\\&+& T_F \Bigg[\frac{28}{9}-\frac{{L_{t}}}{12}+\frac{{L_{t}}^2}{8} -\frac{5{L_{\tilde{q}}}}{8}+\frac{3{L_{\tilde{q}}}^2}{4}-\frac{3\zeta(2)}{4} +\Bigg(\frac{3}{4}-\frac{3{L_{\tilde{q}}}}{4}\Bigg)\frac{{m_{\tilde{g}}}^2}{{m_{\tilde{q}}}^2} +\Bigg(-\frac{1}{2}+\frac{\zeta(2)}{4}\Bigg)\frac{{m_t}^2}{{m_{\tilde{q}}}^2} \nonumber\\&+&\Bigg(-\frac{13}{16}+\frac{3{L_{\tilde{g}}}}{8}-\frac{9{L_{\tilde{q}}}}{8} +\frac{3{L_{\tilde{g}}}{L_{\tilde{q}}}}{4}-\frac{3{L_{\tilde{q}}}^2}{4}\Bigg)\frac{{m_{\tilde{g}}}^4}{{m_{\tilde{q}}}^4} +\Bigg(-\frac{5}{2}+\frac{3\zeta(2)}{2}\Bigg)\frac{{m_t}^2{m_{\tilde{g}}}^2}{{m_{\tilde{q}}}^4} \nonumber\\&+&\Bigg(\frac{61}{288}-\frac{\zeta(2)}{8}+\frac{{L_{t}}}{48} -\frac{{L_{\tilde{q}}}}{48}\Bigg)\frac{{m_t}^4}{{m_{\tilde{q}}}^4} \Bigg] + \frac{X_b}{{m_{\tilde{q}}}} \Bigg[ C_A \Bigg[ \Bigg(-\frac{7}{8}+\frac{3{L_{\tilde{g}}}}{8}-\frac{3{L_{\tilde{q}}}}{4} -\frac{\zeta(2)}{4}\Bigg)\frac{{m_{\tilde{g}}}}{{m_{\tilde{q}}}} \nonumber\\&+&\Bigg(-\frac{5}{8}+2{L_{\tilde{g}}}-\frac{19{L_{\tilde{q}}}}{8}+\frac{3{L_{\tilde{g}}}{L_{\tilde{q}}}}{8} -\frac{3{L_{\tilde{q}}}^2}{8}-\frac{\zeta(2)}{4}\Bigg)\frac{{m_{\tilde{g}}}^3}{{m_{\tilde{q}}}^3} \Bigg] + C_F \Bigg[ \Bigg(-\frac{3}{16}+\frac{3{L_{\tilde{q}}}}{8} \nonumber\\&+&\frac{\zeta(2)}{2}\Bigg)\frac{{m_{\tilde{g}}}}{{m_{\tilde{q}}}} +\Bigg(-\frac{13}{16}-\frac{13{L_{\tilde{g}}}}{16}+\frac{19{L_{\tilde{q}}}}{16}-\frac{3{L_{\tilde{g}}}{L_{\tilde{q}}}}{8} +\frac{3{L_{\tilde{q}}}^2}{8}+\frac{\zeta(2)}{2}\Bigg)\frac{{m_{\tilde{g}}}^3}{{m_{\tilde{q}}}^3} \Bigg] \nonumber\\&+& T_F \Bigg[ \Bigg(-\frac{3}{4}+\frac{3{L_{\tilde{q}}}}{2}\Bigg)\frac{{m_{\tilde{g}}}}{{m_{\tilde{q}}}} +\Bigg(\frac{7}{4}-\frac{3{L_{\tilde{g}}}}{4}+\frac{9{L_{\tilde{q}}}}{4}-\frac{3{L_{\tilde{g}}}{L_{\tilde{q}}}}{2} +\frac{3{L_{\tilde{q}}}^2}{2}\Bigg)\frac{{m_{\tilde{g}}}^3}{{m_{\tilde{q}}}^3} \nonumber\\&+&\Bigg(\frac{13}{4}-2\zeta(2)\Bigg)\frac{{m_t}^2{m_{\tilde{g}}}}{{m_{\tilde{q}}}^3} \Bigg] \Bigg] - \frac{X_t}{{m_{\tilde{q}}}} T_F \frac{\zeta(2)}{2} \frac{{m_t}^2{m_{\tilde{g}}}}{{m_{\tilde{q}}}^3} + \frac{X_tX_b}{{m_{\tilde{q}}}^2} T_F \Bigg[ \frac{\zeta(2)}{2} \frac{{m_t}^2}{{m_{\tilde{q}}}^2}\nonumber\\&+& \Bigg(\frac{3}{4}-3{L_{\tilde{g}}}+ 3{L_{\tilde{q}}}+3\zeta(2)\Bigg)\frac{{m_t}^2{m_{\tilde{g}}}^2}{{m_{\tilde{q}}}^4} +\Bigg(\frac{5}{9}-\frac{2{L_{t}}}{3}+\frac{2{L_{\tilde{q}}}}{3}\Bigg)\frac{{m_t}^4}{{m_{\tilde{q}}}^4} \Bigg] \nonumber\\ &+&{\cal O}\left( \frac{{m_{\tilde{g}}}^6}{{m_{\tilde{q}}}^6},\frac{{m_{\tilde{g}}}^4 {m_t}^2}{{m_{\tilde{q}}}^6},\frac{{m_{\tilde{g}}}^2 {m_t}^4}{{m_{\tilde{q}}}^6}, \frac{{m_t}^6}{{m_{\tilde{q}}}^6},\frac{X_b {m_{\tilde{g}}}^5}{{m_{\tilde{q}}}^6}, \frac{X_t {m_{\tilde{g}}}^5}{{m_{\tilde{q}}}^6} \right) \Bigg\} \,. \end{aligned}$$ ]{} We displayed in the previous expressions only the first three terms of the Taylor expansions in the mass ratios. To get an idea about the convergence of the perturbative series we fix the following input parameters: ${m_t}=172.4$ GeV, $\alpha_s^{\rm{ (SQCD)}}=0.120$, ${m_{\tilde{q}}}=500$ GeV, $X_q=-4000$ GeV, $X_t=-400$ GeV and let ${m_{\tilde{g}}}$ vary. Even for ${m_{\tilde{g}}}/{m_{\tilde{q}}}= 0.5,$ and $2$ the approximations given above agree with the exact results with an accuracy better than 1%. For the case of degenerate SUSY masses, [ *i.e *]{} ${m_{\tilde{g}}}/{m_{\tilde{q}}}=1$ the accuracy is even below the per-mille level. \[sec::num\] Numerical results ============================== In this Section we discuss the numerical impact of the two-loop calculations we presented above. A first phenomenological application is the prediction of the strong coupling and the running bottom-quark mass at high-energy scales like $\tilde{M}= 1$ TeV or $\mu_{\rm GUT}=10^{16}$ GeV, starting from their low-energy values determined experimentally. For the energy evolution of the two parameters, we follow the method proposed in Ref. [@Harlander:2007wh]: first, we compute $\alpha_s^{(5)}(\mu_{\rm dec})$ and $m_b^{(5)}(\mu_{\rm dec})$ from $\alpha_s^{(5)}(M_Z)$ and $m_b^{(5)}(m_b)$, respectively, using the corresponding $i$-loop SM RGEs [@Steinhauser:2002rq]. Here $\mu_{\rm dec}$ denotes the energy scale at which the heavy particles are supposed to become active, [ *i.e. *]{} the scale where the matching between the SM and the MSSM is performed. As pointed out in previous works [@Ferreira:1996ug], one can avoid part of the complications related with the occurrence of the [ *evanescent couplings*]{}, performing the change of the regularization scheme from the [$\overline{\mbox{{\scalefont{.9}}MS}}$]{} to the [$\overline{\mbox{{\scalefont{.9}}DR}}$]{} scheme at the same scale. Nevertheless, one cannot avoid the occurrence of the evanescent coupling $\alpha_e$ in the [$\overline{\mbox{{\scalefont{.9}}MS}}$]{}-[$\overline{\mbox{{\scalefont{.9}}DR}}$]{} relation for the bottom-quark mass. It has to be determined iteratively from the knowledge of the strong coupling at the matching scale. For consistency, the $i$-loop running parameters have to be folded with $(i-1)$-loop conversion and decoupling relations. Above the decoupling scale, the energy dependence of the running parameters is governed by the $i$-loop MSSM RGEs [@Ferreira:1996ug]. We solved numerically the system of coupled differential equations arising from the two sets of RGEs, and implemented this procedure for $i =1,2,3$. The decoupling scale is not a physical parameter and cannot be predicted by the theory. It is usually chosen to be of the order of the heavy particle mass in order to circumvent the appearance of large logarithms. At fixed order perturbation theory, it is expected that the relations between the running parameters evaluated at high-energy scales and their low-energy values become less sensitive to the choice of $\mu_{\rm dec}$ once higher order radiative corrections are considered. The dependence on the precise value of the decoupling scale is interpreted as a measure of the unknown higher order corrections. We discuss the scale dependence of $\alpha_s(\mu_{\rm GUT})$ and $m_b(\mu_{\rm GUT})$ in Fig. \[fig::asGUT\] and Fig. \[fig::mbGUT\], respectively. For the SM parameters we used $\alpha_s(M_Z)=0.1189$ [@Bethke:2006ac], where $ M_Z = 91.1876~\mbox{GeV}\,$ [@Yao:2006px], $m_b(\mu_b)=4.164$ GeV [@Kuhn:2007vp], with $\mu_b=m_b(\mu_b)$, and $M_t = 172.4~\mbox{GeV}\,$ [@:2008vn]. For the SQCD parameters, we implemented their values for the SPS1a$^\prime$ scenario [@Aguilar-Saavedra:2005pw]: ${m_{\tilde{g}}}=607.1$ GeV, $m_{\tilde{t}_1}=366.5$ GeV, $m_{\tilde{t}_2}=585.5$ GeV, $m_{\tilde{b}_1}=506.3$ GeV, $m_{\tilde{b}_2}=545.7$ GeV, $A_t^{{\overline{\rm{\scalefont{.9}}DR}}{}} (1~{\mbox{TeV}})=-565.1$ GeV, $A_b^{{\overline{\rm{\scalefont{.9}}DR}}{}} (1~{\mbox{TeV}})=-943.4$ GeV, $\mu = 396.0$ GeV, and $\tan\beta=10.0$ . For the calculation of $\zeta_{m_b}$ to two-loop accuracy, the [$\overline{\mbox{{\scalefont{.9}}DR}}$]{} parameter $A_b$ has to be converted to the renormalization scheme we used here.[^1] For the accuracy level we are considering, the one-loop conversion relation is required $$\begin{aligned} A_b^{\rm mixed} = A_b^{{\overline{\rm{\scalefont{.9}}DR}}{}}+ \Delta A_b\,, \quad \mbox{where}\quad \Delta A_b =\delta A_b^{{\overline{\rm{\scalefont{.9}}DR}}{}} - \delta A_b^{\rm mixed}\,.\end{aligned}$$ The counterterms $\delta A_b^{i}$ were defined in Eq. (\[eq::dab\]) and the superscript $i$ indicates the renormalization scheme. The shift $\Delta A_b$ is a finite quantity as it can be explicitly checked. It depends in turn on the running bottom quark mass in the MSSM. We use an iterative method and choose the running bottom quark mass in the SM as the initial parameter. A stable solution is obtained after few iterations. In addition, the energy evolution of the parameter $A_b$ has to be taken into account. We use here the one-loop RGE, that can be derived from Eq. (\[eq::dab\]). The dependence on the decoupling scale for $\alpha_s(\mu_{\rm GUT})$ is displayed in Fig. \[fig::asGUT\]. The dotted, dashed and solid lines denote the one-, two-, and three-loop running, where the corresponding exact results for the decoupling coefficients have been implemented. One can see the improved stability of the three-loop results w.r.t.  the decoupling-scale variation. The uncertainty induced by the current experimental accuracy on $\alpha_s(M_Z)$, $\delta \alpha_s= 0.001$[@Bethke:2006ac], is indicated by the hatched band.\ In order to get an idea of the effects induced by the SUSY mass parameters on $\alpha_s({\mu_{{\scalefont{.9}}\rm GUT}})$, we show through the dash-dotted line the three-loop results if the SUSY parameters corresponding to the Snowmass Point SPS2 [@Ghodbane:2002kg] are adopted. Their explicit values are: ${m_{\tilde{g}}}=784.4$ GeV, $m_{\tilde{t}_1}=1003.9$ GeV, $m_{\tilde{t}_2}=1307.4$ GeV, $m_{\tilde{b}_1}=1296.6$ GeV, $m_{\tilde{b}_2}=1520.1$ GeV, and $\tan\beta=10.0$. The curves induced by the other benchmark points SPSi, with $i=3,4,\ldots,9$ would lie between the two curves displayed here. One clearly notices the great impact of the SUSY-mass pattern on the predicted value of the strong coupling at high energies. Accordingly, for precision studies the explicit mass pattern of heavy particles must be taken into account. In Fig. \[fig::mbGUT\] the scale dependence for $m_b(\mu_{\rm GUT})$ is shown. The fine-dotted, dashed and solid lines correspond to the exact one-, two-, and three-loop running obtained in the SPS1a$^\prime$ scenario. As explained above, the energy evolution of the running parameters have to be combined with the appropriate matching conditions between the low- and high-energy regimes. More explicitly, in case of $m_b$ we determine its value within SQCD at the energy-scale $\mu_{\rm dec}$ through the relation $$\begin{aligned} m_b^{\rm SQCD}(\mu_{\rm dec})= \frac{m_b^{(5)}(\mu_{\rm dec})}{\zeta_{m_b}(\mu_{\rm dec})}\,,\quad \mbox{where}\quad \frac{1}{\zeta_{m_b}}=\frac{1}{1+\delta\zeta_{m_b}^{\tan\beta}+ \delta\zeta_{m_b}^{\rm rest}}\,. \label{eq::resumm}\end{aligned}$$ Here $\delta\zeta_{m_b}^{\tan\beta}$ denotes the contributions proportional with $\tan\beta$ and $\delta\zeta_{m_b}^{rest}$ the remaining corrections. For simplicity, we do not show in Eq. (\[eq::resumm\]) the explicit dependence on the MSSM parameters. For the $i$-loop running analysis, we take into account the $(i-1)$-loop contribution to the Eq. (\[eq::resumm\]). As can be seen from the Figure \[fig::mbGUT\], the three-loop results stabilize the scale dependence and reduce further the theoretical uncertainty.\ The dotted line displays the two-loop running bottom-mass, where the contributions proportional with $\tan\beta$ to the one-loop $\zeta_{m_b}$ are resummed following the method proposed in Refs. [@Carena:1999py; @Aguilar-Saavedra:2005pw]. Within this approach, the matching condition can be written as $$\begin{aligned} m_b^{\rm SQCD}(\mu_{\rm dec})= \frac{m_b^{(5)}(\mu_{\rm dec})}{\zeta_{m_b}^{\rm 1-loop}(\mu_{\rm dec})}\quad \mbox{and}\quad \frac{1}{\zeta_{m_b}^{\rm 1-loop}} = \frac{1- \delta\zeta_{m_b}^{\rm rest, 1-loop}}{1+\delta\zeta_{m_b}^{\tan\beta,\rm{ 1-loop}}}\,. \label{eq::resummc}\end{aligned}$$ The superscript ${\rm 1-loop}$ indicates the order in perturbation theory at which the individual contributions are evaluated. The authors of Ref. [@Carena:1999py] showed that, for a consistent analysis not only the $\tan\beta$-enhanced contributions have to be resummed, but also the next-to-leading logarithms (NLL) $\alpha_s^{i+1}\ln^i(\mu^2/m_b^2)$. In our approach based on $i$-loop RGEs and $(i-1)$-loop decoupling coefficients the NLL are implicitly resummed. The very good agreement between the two computations can be explained by the fact that at one-loop order $\delta\zeta_{m_b}^{rest}$ is almost an order of magnitude smaller than $\delta\zeta_{m_b}^{\tan\beta}$.\ The experimental uncertainty generated by $\delta \,\alpha_s=0.001$[@Bethke:2006ac] corresponds to the wider hatched band, and the one due to $\delta\, m_b = 25$ MeV[@Kuhn:2007vp] to the narrow band. Let us notice that the three-loop order effects exceed the uncertainty due to current experimental accuracy $\delta\alpha_s$.\ Finally, the dash-dotted line shows the three-loop running if the SPS2 scenario is implemented. The differences between the three-loop order results are mainly due to the change of masses of the SUSY particles. [l lllll]{}\ $\alpha_s(\mu_{\rm ren})$ & 0.0929 & $\pm 0.0006|_{\delta\,\alpha_s(M_Z)}$ & & $- 0.003|_{\rm SPS2}$ & $\pm 0.0001|_{\rm th}$\ $m_b(\mu_{\rm ren})$ & 2.164 & $\pm 0.017|_{\delta\,\alpha_s(M_Z)}$ & $\pm 0.015|_{\delta\, m_b(m_b)}$ & $+ 0.12|_{\rm SPS2}$ & $\pm 0.01|_{\rm th}$\ \ \ $\alpha_s(\mu_{\rm ren})$ &0.0405 & $\pm 0.0001|_{\delta\,\alpha_s(M_Z)}$ & & $\pm 0.0007|_{\rm SPS2}$ & $\pm 0.0001|_{\rm th}$\ $m_b(\mu_{\rm ren})$ & 1.016 & $\pm 0.011|_{\delta\,\alpha_s(M_Z)}$ & $\pm 0.007|_{\delta\, m_b(m_b)}$ & $+ 0.077|_{\rm SPS2}$ & $\pm 0.005|_{\rm th}$\ For quantitative comparison, we give in Table \[tab::res1\] the numerical values for $\alpha_s(\mu_{\rm ren})$ and $m_b(\mu_{\rm ren})$ for $\mu_{\rm ren}=1000$ GeV and $\mu_{\rm ren}=\mu_{\rm GUT}$, evaluated with three-loop accuracy. For the decoupling scale we choose $\mu_{\rm dec}= 600$ GeV as at this scale the difference between the two- and three-loop order corrections reaches a minimum. The different sources of uncertainties are explicitly displayed. The theoretical uncertainties due to unknown higher order corrections are estimated from the variation of the three-loop results modifying the decoupling scale from $100$ GeV to $1$ TeV. The effects of the SUSY-mass parameters are evaluated as the difference between the three-loop results corresponding to the benchmark points SPS1a$^{\prime}$ and SPS2. One can easily see that the impact of the SUSY-mass pattern is at least five times larger than the experimental accuracy.\ As already pointed out in the previous sections the unification of the Yukawa couplings is very sensitive to the MSSM parameters. The dependence on the soft SUSY breaking parameters is induced in our approximation only through the decoupling coefficients. They comprise an explicit dependence through the $X_b$ parameter (in the case of $\zeta_{m_b}$) and an implicit one through the squark masses. The analytical formulae for $\zeta_s$ and $\zeta_m$ given in Section \[sec::twoloop\] are expressed in terms of the physical squark masses. Since they are not known experimentally, for the following numerical analyses, we computed them making the assumption that the soft SUSY breaking mass parameters defined in the on-shell scheme obey the following relation $M_{\tilde Q}(t)=M_{\tilde{D}}(\tilde{M})=M_{\tilde{U}} (\tilde{M})=A_f=\pm\mu=\tilde{M}$, where $M_{\tilde Q}(t)$ is the on-shell input parameter in the stop-mass matrix.[^2] The corresponding input parameter in the sbottom-mass matrix acquires a finite shift of ${\cal O}(\alpha_s)$ [@Bartl:1997yd]. Upon diagonalization of the squark-mass matrices Eq. (\[eq::mixing\]), one obtains the on-shell squark masses $m_{\tilde{q}_{1,2}}$. The parameter $A_b$ entering through $X_b$ the one-loop results have to be converted from the on-shell scheme in the renormalization scheme we introduced in Section \[sec::frame\]. [c]{}\ (a)\ \ (b) [c]{}\ (a)\ (b) In order to estimate the phenomenological impact, we discuss the difference between $m_b(\mu_{\rm GUT})$ evaluated using $i$-loop running and $(i-1)$-loop decoupling and the one-loop result $$\begin{aligned} \frac{\Delta \, m_b^{\mbox{(i)}}}{m_b}=\frac{m_b^{\mbox{(i-loop)}} - m_b^{\mbox{(1-loop)}}}{m_b^{\mbox{(1-loop)}}}\,. \end{aligned}$$ In Fig. \[fig::tbeta\] we fix the soft SUSY breaking mass parameters to $\tilde{M}=1$ TeV and $M_{\tilde g}=800$ GeV, and study the dependence of $\Delta \, m_b^{\mbox{(i)}}$ for $i=2$ (dashed) and $i=3$ (solid line) as a function of $\tan\beta$. One can clearly see the abrupt increase/decrease of the two- and three-loop order radiative corrections with the increase of $\tan\beta$. Whereas the effects of the the one-loop decoupling can be as large as $65$% for $\mu <0$ and $-37$% for $\mu >0$, the two-loop corrections are moderate, reaching at most $10$% and $-3$%, respectively. The numerical effects for large values of $\tan \beta$ are of special interest for the study of Yukawa-coupling unification. In Fig. \[fig::msusy\] we show $\Delta \,m_b^{\mbox{(i)}}$ as a function of the soft SUSY breaking mass scale for $\tan\beta=50$. One can see the increase in size of radiative corrections for lighter SUSY masses. Again, for $\mu >0$ the bulk of the corrections are comprised in the two-loop running mass, while the three-loop order effects sum up to few percent. For $\mu <0$ and light SUSY masses the three-loop contributions can increase the bottom-quark mass with up to $30$%. \[sec::concl\]Conclusions ========================= The knowledge of fundamental parameters at high energies, such as $\tilde{M}$ or $\mu_{\rm GUT}$, are essential for the reconstruction of the theory beyond the SM. In this paper we presented the exact two-loop decoupling coefficients of the strong coupling and the bottom-quark mass within the SQCD. Together with the known three-loop order RGEs they allow predictions of the two parameters at high energies with three-loop accuracy. This level of precision on the theory side is necessary in order to match with the current experimental accuracy. The values of the gauge and Yukawa couplings at the unification scale $\mu_{\rm GUT}$ are essential ingredients for the determination of the GUT threshold corrections, which in turn are used to identify the underlying GUT model.\ In addition, the dependence on the energy scale at which the supersymmetric particles are integrated out, which reflects the size of the unknown higher order corrections, is significantly reduced in the case of the three-loop order predictions. Furthermore, the approach outlined here accounts for the effects induced by the individual mass parameters. They are phenomenologically significant for both parameters and exceed the experimental uncertainty by more than a factor five. The radiative corrections to the running bottom-quark mass are particularly important for SUSY models with large values of $\tan \beta$. It turns out that for negative values of $\mu $ the three-loop order contributions can reach up to $30\%$ from the tree-level bottom-quark mass. 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--- abstract: 'We construct fixed point lattice models for group supercohomology symmetry protected topological (SPT) phases of fermions in $2+1$D. A key feature of our approach is to construct finite depth circuits of local unitaries that explicitly build the ground states from a tensor product state. We then recover the classification of fermionic SPT phases, including the group structure under stacking, from the algebraic composition rules of these circuits. Furthermore, we show that the circuits are symmetric, implying that the group supercohomology phases can be many body localized. Our strategy involves first building an auxiliary bosonic model, and then fermionizing it using the duality of Chen, Kapustin, and Radicevic. One benefit of this approach is that it clearly disentangles the role of the algebraic group supercohomology data, which is used to build the auxiliary bosonic model, from that of the spin structure, which is combinatorially encoded in the lattice and enters only in the fermionization step. In particular this allows us to study our models on $2$d spatial manifolds of any topology, and to define a lattice-level procedure for ungauging fermion parity.' author: - 'Tyler D. Ellison' - Lukasz Fidkowski bibliography: - 'references.bib' title: Disentangling interacting symmetry protected phases of fermions in two dimensions --- Introduction ============ A major goal in understanding symmetry protected topological (SPT) phases is their classification, i.e. the identification and enumeration of the possible phases. Essential to a classification scheme is the construction of microscopic models for each phase, as well as the identification of quantized many-body invariants which discriminate between the different phases. For bosonic SPT phases in 2+1D with unitary onsite symmetries, the classification is well understood in terms of the framework of group cohomology theory. The algebraic data of group cohomology is used both in the construction of exactly solvable lattice models \[\] and in the identification of quantized invariants, where group cohomology classes appear in the universal statistics of the symmetry flux excitations \[\]. In contrast, despite much recent progress \[\], the classification of fermionic SPT phases is not as well understood. A mathematical structure analogous to group cohomology - termed group supercohomology - was introduced in the pioneering work of Ref. \[\] to describe a subset of fermionic SPT phases. However, group supercohomology has yet to be as directly connected to explicit lattice Hamiltonians or to universal quantized observables. While certain lattice fermionic Hamiltonians were, in fact, written down in terms of group supercohomology data in Ref. \[\], these intricate constructions rely on seemingly arbitrary choices and cannot straightforwardly be put on spatial manifolds of general topology. In a space-time path integral formalism, these arbitrary choices have since been interpreted as choices of spin structure \[\] – now understood to be a crucial ingredient in constructing fixed point fermionic SPT models. Progress has been made in incorporating spin structures directly in a Hamiltonian formalism \[\], in particular in Ref. \[\], where ground state wavefunctions incorporating spin structure were defined implicitly in terms of constraints that involve different lattice structures related by local deformations. However, there is still no general prescription for turning group supercohomology data and a choice of spin structure into a fermionic Hamiltonian on a fixed lattice in a general spatial geometry. Group supercohomology classes have also not yet been directly connected to quantized many-body invariants of gapped, lattice Hamiltonians. It has been shown \[\], that the supercohomology data can be interpreted as quantized topological terms in the effective space-time action for a combination of the global symmetry and fermion parity gauge fields. That being the case, these space-time observables should in principle be encoded in the joint braiding statistics of symmetry and fermion parity fluxes, but such statistics have only been studied in the continuum \[\]. For bosonic SPT phases, the underlying group cohomology data can be extracted using a well defined lattice minimal coupling gauging procedure that maps the SPT system to a system with topological order. An analogous lattice Hamiltonian procedure has so far been missing on the fermionic side - making it difficult to argue that group supercohomology classes are quantized invariants of lattice fermionic SPT Hamiltonians. In this paper, we solve both of these problems in the case of 2+1 dimensions and finite unitary on-site symmetry $G \times {\mathbb{Z}}_2^{\text{f}}$, where ${\mathbb{Z}}_2^{\text{f}}$ is fermion parity. Specifically, we accomplish the following: We construct a representative fermionic lattice SPT Hamiltonian for every choice of group supercohomology data, 2d oriented spatial manifold $M$, and spin structure on $M$. Moreover, we write down an explicit finite depth quantum circuit of local unitaries that constructs its ground state from a trivial product state. Using these finite depth circuits, we recover the group structure of our SPT phases under stacking. We also find that two different sets of group supercohomology data can lead to circuits that differ only by a product of symmetric local unitaries, and hence define the same phase. This leads to a natural equivalence relation on group supercohomology data, which matches that of previous works. Conversely, we prove that for inequivalent group supercohomology data, the corresponding Hamiltonians are in distinct phases. A choice of group supercohomology data is encoded in a pair $(n,\nu)$, where $n$ and $\nu$ are certain ${\mathbb{Z}}_2$ and $U(1)$-valued functions of $G$ variables, respectively (defined precisely in section \[gsd\] below). Given the data $(n,\nu)$, the construction of our fermionic lattice SPT Hamiltonian, inspired by the work of Ref. \[\], proceeds in $3$ steps. \(i) We use $n$ and $\nu$ to construct an auxiliary [*bosonic*]{} SPT Hamiltonian with enlarged symmetry group ${\tilde{G}}$, where ${\tilde{G}}$ is the extension of $G$ by ${\mathbb{Z}}_2$ determined by $n$. ${\tilde{G}}$ contains ${\mathbb{Z}}_2$ as a subgroup and $G$ as a quotient: $G={\tilde{G}}/ {\mathbb{Z}}_2$, so the auxiliary bosonic SPT model has a global ${\mathbb{Z}}_2$ symmetry but is not in general $G$-symmetric. Being a group cohomology bosonic SPT, it can be put on any spatial manifold $M$ with a triangulation and branching structure \[\]. \(ii) We gauge the ${\mathbb{Z}}_2 \subset {\tilde{G}}$ by minimally coupling the auxiliary bosonic SPT to a ${\mathbb{Z}}_2$ lattice gauge field and imposing a Gauss’s law constraint. By choosing an appropriate basis of gauge invariant operators, this gauge theory can be interpreted as an unconstrained bosonic model - which we refer to as the ‘shadow’ model following Ref. \[\] - with global symmetry $G={\tilde{G}}/{\mathbb{Z}}_2$ and toric code topological order. Specifically, the shadow model has generalized $G$-spin vertex degrees of freedom, which transform under the $G$ symmetry in the standard way, and spin-$\frac{1}{2}$ link degrees of freedom, which encode a toric code topological order. \(iii) Finally, we obtain our fermionic SPT by applying the fermionization duality of Ref. \[\] (reviewed below) to trade the bosonic spin-$\frac{1}{2}$ link degrees of freedom in the shadow model for spinless complex fermions located on the triangular faces. The underlying idea behind this fermionization is to represent the fermion as the bound state of a toric code charge and flux excitation \[\]. The fermionization procedure is not unique, however, as it requires a choice of spin structure. Spin structure enters our construction only here, encoded combinatorially in a certain subset of links $\cal{E}$. This step can be thought of as effectively ‘un-gauging’ fermion parity symmetry \[\], resulting in a model defined in a fermionic Fock space. This three-step construction highlights one important advantage of our approach: it clearly disentangles the roles of group supercohomology data and spin structure in fermionic SPT models. One needs just the group supercohomology data to construct the bosonic shadow model (steps $(i)$ and $(ii)$), whereas the spin structure enters only in the fermionization duality that maps this shadow model to the desired fermionic SPT (step $(iii)$). A key part of our approach is the construction of finite depth quantum circuits of local unitaries[^1] \[\], which build the fermionic SPT ground states from a trivial product state. Access to these finite depth circuits has several benefits. First, they give us explicit representations of the corresponding ground states in terms of $G$ domain models decorated with fermions (as opposed to ground state wave functions that are only defined implicitly via constraints). Second, we show that composing these circuits is equivalent to stacking the corresponding fermionic SPT phases, allowing us to extract the stacking group law for supercohomology data just by multiplying circuits. Third, we show that equivalent group supercohomology data gives rise to circuits that differ by a product of symmetric local unitaries, and hence correspond to the same phase. Conversely, by bosonizing our models and using well established classification results for bosonic symmetry enriched toric code phases \[\], we show that inequivalent group supercohomology data always lead to inequivalent phases. An intriguing feature of the finite depth circuit that builds our supercohomology fermionic SPT ground state is that, as a unitary operator, it is $G$-symmetric. This is despite the fact that, when the SPT phase in question is nontrivial, the local unitaries that make it up cannot all be individually $G$-symmetric. This is a property that the supercohomology models share with bosonic group cohomology models, but not with the so-called ‘beyond group cohomology’ models (see [*e.g.*]{} appendix C of Ref. \[\]). One consequence of this property is that the supercohomology phases can be many-body localized \[\]. This is done by disordering the couplings in a trivial commuting projector parent Hamiltonian for the trivial product state and then conjugating by the circuit. The rest of this paper is structured as follows. In section \[sec:bosonic\], we focus on the construction of the bosonic shadow model described in steps $(i)$ and $(ii)$ above. In section \[sec:duality\], we review the bosonization duality of Ref. \[\] and complete step $(iii)$ of our construction. In section \[sec:classification\], we study the group structure of fermionic SPT phases using the finite depth circuits that build their ground states. In particular, we derive a notion of equivalence of group supercohomology data (in agreement with Ref. \[\]) such that equivalent data gives rise to models in the same phase and inequivalent data necessarily yields inequivalent phases. We conclude in section \[sec:discussion\] with some comments about many-body localizability for our models, possible future extensions of our work, and comparisons with other work. Throughout the paper we illustrate our results for the simple case of $G={\mathbb{Z}}_2$ (i.e. total symmetry ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2^{\text{f}}$). In Appendices \[shadowgsderivation\]-\[ap:trivialcircuit\], we provide detailed derivations of the results in the main text. As we were completing this work, we learned of a related preprint by N. Tantivasadakarn and A. Vishwanath \[\], which also constructs a many-body localizable model for the ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2^{\text{f}}$ group supercohomology SPT. Bosonic shadow model from group supercohomology data {#sec:bosonic} ==================================================== In this section we will show how to use group supercohomology data associated to a finite group $G$ to construct a purely bosonic Hamiltonian lattice model, which, in agreement with Ref. \[\], we refer to as the shadow model. The model is defined on a triangulation of a 2d manifold - i.e. a planar graph consisting of vertices $p$ and links ${\langle}pq {\rangle}$, all of whose faces are triangular - with branching structure (FIG. \[fig:torustriangulation\]). Recall that a branching structure is an assignment of an orientation to each link with the property that there are no cycles around any triangle. The notation ${\langle}pq {\rangle}$ always denotes a link oriented from $p$ to $q$. The Hilbert space will consist of generalized $G$-spin degrees of freedom $|g_p{\rangle}$ at vertices $p$ and spin-$\frac{1}{2}$ degrees of freedom on links ${\langle}pq {\rangle}$, with Pauli algebra generated by ${\hat X}_{pq}, {\hat Z}_{pq}$ (see FIG. \[fig:shadowdof\]). Before delving into the construction of the shadow model Hamiltonian, let us first provide some intuition for why a bosonic model built on such a Hilbert space can encode the physics of a fermionic SPT. This intuition is based on interpreting the spin-$\frac{1}{2}$ link degrees of freedom as the Hilbert space of the usual commuting projector toric code Hamiltonian: $$\begin{aligned} \label{toriccode0} \hat{H}^{\text{t.c.}}=-\sum_{p}\prod_{{\langle}st {\rangle}\ni p}{{\hat{X}}}_{st}-\sum_{{\langle}pqr {\rangle}}{{\hat{Z}}}_{pq} {{\hat{Z}}}_{qr} {{\hat{Z}}}_{pr},\end{aligned}$$ where the product in the first sum above is over all oriented links ${\langle}st {\rangle}$ that contain the vertex $p$ (i.e. either $s=p$ or $t=p$). A basis for this toric code Hilbert space can be obtained by specifying, for each basis state, the locations of all the vertex (‘$e$’) and triangular plaquette (‘$m$’) excitations, which are violations of the first and second terms in (\[toriccode0\]), respectively. The key idea is that the bound state of an $e$ and an $m$ excitation is a fermion, so a fermionic Hilbert space can effectively be constructed by restricting to the subspace where all of the $e$ excitations have been bound up with $m$ excitations into fermions. Because the $e$ excitations live on vertices and the $m$ excitations live on plaquettes, there is some arbitrariness in defining their fermionic bound state. This arbitrariness can be resolved by using the branching structure. Following Ref. \[\], we define a fermion on triangle ${\langle}pqr {\rangle}$ to be the bound state of an $m$ excitation on ${\langle}pqr {\rangle}$ with an $e$ excitation on its first vertex $p$. Here the ordering $p,q,r$ of the vertices is specified uniquely by the branching structure (see FIG \[fig:orientations\]). The condition that all the $e$ excitations have been bound up with $m$ excitations into fermions in this way can then be stated as follows. At each vertex $p$, the ${\mathbb{Z}}_2$ charge (i.e. number of $e$ excitations modulo $2$) measured at $p$ must be equal to the total ${\mathbb{Z}}_2$ flux (i.e. number of $m$ excitations modulo $2$) on all triangles ${\langle}pqr {\rangle}$ for which $p$ is the first vertex according to the branching structure. Defining $$\begin{aligned} \label{defW} {{\hat{W}}}_{pqr}\equiv {{\hat{Z}}}_{pq} {{\hat{Z}}}_{qr} {{\hat{Z}}}_{pr}\end{aligned}$$ to be the operator that measures the ${\mathbb{Z}}_2$ flux on ${\langle}pqr {\rangle}$, this is then just the condition that the state be in the $+1$ eigenspace of each operator $$\begin{aligned} \label{def:modifiedgausslaw} {\hat G}_p \equiv \prod_{\substack{{\langle}tqr {\rangle}\\ t=p}} {{\hat{W}}}_{tqr} \prod_{{\langle}st {\rangle}\ni p} {\hat X}_{st}.\end{aligned}$$ The first product above is over all triangles whose first vertex is $p$. We thus expect that the shadow model Hamiltonian will commute with all of the ${{\hat{G}}}_p$ and that its ground states will lie in the $+1$ eigenspace of each ${{\hat{G}}}_p$. Because of the second product in (\[def:modifiedgausslaw\]), ${{\hat{G}}}_p$ resembles a Gauss’s law constraint. In accordance with Ref. \[\], we will refer to it as a ‘modified Gauss’s law’. Our construction of the shadow model Hamiltonian proceeds in two steps. First, we use group supercohomology data to construct an auxiliary bosonic SPT, with an enlarged symmetry group ${\tilde{G}}$ equal to a certain ${\mathbb{Z}}_2$ extension of $G$. Second, we gauge the global ${\mathbb{Z}}_2$ subgroup of ${\tilde{G}}$ in this auxiliary bosonic SPT to end up with our desired bosonic shadow model. Again, we emphasize that because all of these constructions are bosonic, the spin structure does not enter into them at all. To begin, we briefly review group supercohomology. Group supercohomology data {#gsd} -------------------------- For a finite group $G$, group supercohomology data consists of a pair $(n,\nu)$, where $n : G \times G \times G \to {\mathbb{Z}}_2$ is a ${\mathbb{Z}}_2$ valued function of $3$ group variables, and $\nu: G \times G \times G \times G \to U(1)$ is a $U(1)$ valued function of $4$ group variables, satisfying the following two properties:\ 1) $n$ is a homogeneous cocycle, where homogeneity means $$\begin{aligned} \label{neq} n(gg_0,gg_1,gg_2)=n(g_0,g_1,g_2)\end{aligned}$$ for all $g$, and the cocycle property is [^2] $$\begin{aligned} \delta n=0.\end{aligned}$$ 2\) $\nu$ is homogeneous, i.e. $$\begin{aligned} \label{nueq} \nu(gg_0,gg_1,gg_2,gg_3)=\nu(g_0,g_1,g_2,g_3),\end{aligned}$$ for all $g$ and satisfies $$\begin{aligned} \label{Steenrod} \delta \nu(g_0,g_1,g_2,g_3,g_4)=(-1)^{n(g_0,g_1,g_2)n(g_2,g_3,g_4)}.\end{aligned}$$ Just as for ordinary group cocycles, there is an equivalence relation on group supercohomology data. Rather than defining it now, we will postpone the discussion of this equivalence relation to section \[sec:superequivalencerelation\], where we identify it through physical arguments. Group supercohomology classes will then be defined as equivalence classes of group supercohomology data modulo this relation. For convenience, in our constructions below, we will always take $n$ to be a normalized cocycle. This is to say, we choose $n$ such that $$\begin{aligned} \label{normalized} n(g,g,h)=n(g,h,h)=0\end{aligned}$$ for all $g,h$. There is no loss of generality in restricting to normalized cocycles, because each equivalence class of group supercohomology data has a representative $(n,\nu)$ with $n$ normalized. Auxiliary bosonic SPT --------------------- The auxiliary bosonic SPT is again defined on a triangulation of an orientable two dimensional spatial manifold $M$ together with a branching structure. The symmetry group ${\tilde{G}}$ of the auxiliary bosonic SPT is the ${\mathbb{Z}}_2$ extension of $G$ determined by $n$. Explicitly, ${\tilde{G}}$ consists of $2|G|$ elements ${g^{(m)}}$, where $g\in G$ and $m \in {\mathbb{Z}}_2 = \{0,1\}$, obeying the group law: $$\begin{aligned} \label{grouplaw} {g^{(m)}} {h^{(\ell)}} = {(gh)^{(m+\ell+n(1,g,gh))}}.\end{aligned}$$ The degrees of freedom in the auxiliary model are generalized ${\tilde{G}}$-spins $\big|g^{(m_p)}_p\big{\rangle}$ living on the vertices of the triangulation, and the standard bosonic SPT construction of Ref. \[\] allows us to write down the following SPT ground state wave function in the $\big\{g^{(m_p)}_p\big\}$ configuration basis: $$\begin{aligned} \label{gsSPT} \Psi_{\text{SPT}}\big(\big\{g^{(m_p)}_p\big\}\big)=& \big {\langle}\{ g^{(m_p)}_p \} \big| \Psi_{\text{SPT}} \big {\rangle}= \\ \nonumber = &\prod_{{\langle}pqr {\rangle}}\alpha\big(g^{(m_p)}_p,g^{(m_q)}_q,g^{(m_r)}_r,1\big)^{o_{pqr}}.\end{aligned}$$ Here, as below, we do not keep track of the irrelevant overall normalization factor of the ground state wave function. The product in (\[gsSPT\]) is over ordered triangles ${\langle}pqr {\rangle}$, with the ordering determined by the branching structure. $o_{pqr}$ is $+1$ if the orientation of the triangle ${\langle}pqr {\rangle}$ is aligned with the orientation of the manifold and $-1$ otherwise (see FIG. \[fig:orientations\]). Finally, $\alpha$ is defined in terms of the group supercohomology data as \[\]: $$\begin{aligned} \label{def_alpha} \alpha\big({g^{(m_0)}_0}&, {g^{(m_1)}_1}, {g^{(m_2)}_2}, {g^{(m_3)}_3}\big) \equiv \\ & \nu(g_0,g_1,g_2,g_3) (-1)^{{\epsilon}\big(\big({g^{(m_0)}_0}\big)^{-1} {g^{(m_1)}_1}\big) n(g_1,g_2,g_3)}, \nonumber \end{aligned}$$ where we have defined the projector $$\begin{aligned} {\epsilon}\big({g^{(m)}}\big) \equiv m.\end{aligned}$$ One can explicitly verify that $\alpha$ is homogenous and a cocycle ($\delta \alpha=0$) by using equations (\[neq\]) and (\[nueq\]) along with the group law (\[grouplaw\]) of ${\tilde{G}}$, as well as the normalization property (\[normalized\]). Thus (\[gsSPT\]) is a bosonic SPT ground state. The seemingly complicated cocycle $\alpha$ is designed to produce a shadow model wave function that lies in the ${{\hat{G}}}_p=+1$ Hilbert space, as we will see in the next subsection. Bosonic shadow model wave function {#tGSPT} ---------------------------------- We now construct the bosonic shadow model by gauging the ${\mathbb{Z}}_2$ subgroup of ${\tilde{G}}$ in the auxiliary bosonic SPT. This is done in the standard way by introducing a lattice ${\mathbb{Z}}_2$ gauge field $\mu^z_{pq}=\pm 1$ and performing the usual minimal coupling procedure \[\], so we relegate the details to Appendix \[shadowgsderivation\]. A complete set of commuting gauge invariant observables in the resulting gauge theory is given by $\{g_p,Z_{pq} \}$, where $g_p$ is the $G$ component of the ${\tilde{G}}$ degree of freedom $g^{(m_p)}_p$ at vertex $p$, and $$\begin{aligned} Z_{pq}&=\mu^z_{pq} (-1)^{{\epsilon}\big(\big(g^{(m_p)}_p\big)^{-1} g^{(m_q)}_q\big)}\end{aligned}$$ can be thought of as the ${\mathbb{Z}}_2$ part of the lattice gauge covariant derivative of the ${\tilde{G}}$ ‘matter’ fields. We explicitly demonstrate in Appendix \[shadowgsderivation\] that this gauge theory Hilbert space is isomorphic, via a duality transformation, to the unconstrained Hilbert space of generalized $G$-spin degrees of freedom $|g_p{\rangle}$ at vertices $p$ and spin-$1/2$ degrees of freedom on links ${\langle}pq {\rangle}$, with Pauli algebra generated by ${\hat X}_{pq}, {\hat Z}_{pq}$. A ground state wave function $\Psi_{\text{b}}$ of the gauged theory can be obtained by setting the amplitude $\Psi_{\text{b}}(\{g_p, Z_{pq}\})$ of any configuration $\{g_p, Z_{pq}\}$ equal to $\Psi_{\text{SPT}}\big(\big\{g_p^{(m_p)}\big\}\big)$ if there exists $\big \{g_p^{(m_p)} \big\}$ for which $$\begin{aligned} \label{zpq} Z_{pq}&=(-1)^{{\epsilon}\big(\big(g^{(m_p)}_p\big)^{-1} g^{(m_q)}_q\big)}=(-1)^{m_p+m_q+n(1,g_p,g_q)}\end{aligned}$$ and zero otherwise (see FIG. \[fig:z2amplitude\] and \[fig:z2amplitude2\] for an example). Such $\big\{g^{(m_p)}_p\big\}$, if it exists, is ambiguous only up to a global ${\mathbb{Z}}_2$ transformation, i.e. a shift $m_p\rightarrow m_p+1$,[^3] and since $\Psi_{\text{SPT}}$ is invariant under this shift, $\Psi_{\text{b}}$ is well defined. Explicitly, $$\begin{aligned} \label{gsHstd} &\Psi_{\text{b}}\left(\{g_p, Z_{pq}\}\right)= \\ \nonumber &\prod_{{\langle}pqr {\rangle}} \nu(g_p,g_q,g_r,1)^{o_{pqr}} Z_{pq}^{n(g_q,g_r,1)} \\ \nonumber \times & \left(\prod_{{\langle}pqr {\rangle}} \delta_{Z_{pq}Z_{qr}Z_{pr},(-1)^{n(g_p,g_q,g_r)}}\right) h(\{Z_{pq}(-1)^{n(1,g_p,g_q)}\}), \end{aligned}$$ as can be verified by observing that we recover the auxiliary bosonic SPT ground state wave function amplitude by inserting (\[zpq\]) in (\[gsHstd\]). Again, we do not keep track of the irrelevant overall normalization of the wave function. The function $h(\{Z_{pq}(-1)^{n(1,g_p,g_q)}\})=0,1$ is a constraint that enforces trivial $\mu^z$-holonomy around each topologically nontrivial cycle in the geometry. Specifically, it is equal to a product of delta functions over all nontrivial cycles, which enforce the constraint that the product of $Z_{pq}(-1)^{n(1,g_p,g_q)}$ along the links of the cycle is equal to $1$. These holonomy constraints, together with the delta functions in (\[gsHstd\]), ensure that the amplitude of a given configuration $\{g_p, Z_{pq}\}$ is nonzero if and only if there exists $\big\{g_p^{(m_p)}\big\}$ satisfying (\[zpq\]). Once we write down a parent Hamiltonian for $\Psi_{\text{b}}$, we will have other ground states, which will all be of the form (\[gsHstd\]) except with nontrivial holonomy constraints. Because it comes from gauging a global ${\mathbb{Z}}_2$ symmetry in a short range entangled state, the shadow model wave function $\Psi_{\text{b}}$ describes a toric code topological order. Furthermore, since $\Psi_{\text{SPT}}$ is ${\tilde{G}}$ symmetric, $\Psi_{\text{b}}$ is $G$ symmetric, and hence the shadow model wave function describes a $G$-symmetry enriched toric code. One can also explicitly check that $$\begin{aligned} \label{eq:Gpverify} {{\hat{G}}}_p |\Psi_{\text{b}}{\rangle}= |\Psi_{\text{b}}{\rangle}\end{aligned}$$ for all $p$, so that $|\Psi_{\text{b}}{\rangle}$ contains only fermion excitations, without any unbound $e$ excitations or $m$ excitations, in the sense defined above. We will also verify (\[eq:Gpverify\]) below by writing down a finite depth circuit of local unitaries which commutes with all of the ${{\hat{G}}}_p$, and constructs $\Psi_{\text{b}}$ from a state which trivially lies in the ${{\hat{G}}}_p=+1$ eigenspace of each ${{\hat{G}}}_p$. Bosonic shadow model Hamiltonian from a finite depth circuit {#shadowhamiltonian} ------------------------------------------------------------ Our ultimate aim is to use the fermionization duality of Ref. \[\] to turn the bosonic shadow model wave function into the ground state of a fermionic SPT. However, as this fermionization duality is defined at the level of local operators, we must first write down a local parent Hamiltonian for $|\Psi_{\text{b}}{\rangle}$ on which we can apply the duality. One way to obtain such a parent Hamiltonian is to simply start with the form of the bosonic ${\tilde{G}}$ SPT parent Hamiltonian written down in Ref. \[\] and directly couple it to a lattice ${\mathbb{Z}}_2$ gauge field. We outline this approach in appendix \[shadowgsderivation\], but for our purposes, we will find it more useful to construct a different parent Hamiltonian for $|\Psi_{\text{b}}{\rangle}$. Our choice of parent Hamiltonian is based on the insight that $|\Psi_{\text{b}}{\rangle}$, as defined by the wavefunction in (\[gsHstd\]), can be obtained by applying an appropriate finite depth circuit of local unitaries to a ground state of the following Hamiltonian, which describes a trivial generalized $G$-spin paramagnet and a decoupled copy of the toric code: $$\begin{aligned} \label{toriccode} \hat{H}^0_{\text{b}}=-\sum_p {{\hat{P}}}_p^{\text{sym}}-\sum_{p}\prod_{{\langle}st {\rangle}\ni p}{{\hat{X}}}_{st}-\sum_{{\langle}pqr {\rangle}}{{\hat{W}}}_{pqr}.\end{aligned}$$ Here, ${{\hat{P}}}_p^{\text{sym}}$ is the projector onto the symmetric state $\frac{1}{\sqrt{|G|}}\sum_{g_p\in G}|g_p{\rangle}$ at vertex $p$ tensored with the identity on the remaining sites, and ${{\hat{W}}}_{pqr}$, which was defined in (\[defW\]), measures the ${\mathbb{Z}}_2$ flux on ${\langle}pqr {\rangle}$. One ground state of (\[toriccode\]) is $$\begin{aligned} \label{tc} \Psi_{\text{t.c.}}(\{g_p, Z_{pq}\})= \left(\prod_{{\langle}pqr {\rangle}} \delta_{Z_{pq}Z_{qr}Z_{pr},1}\right)h(\{Z_{pq}\}),\end{aligned}$$ where the holonomy constraint $h(\{Z_{pq}\})$ was defined below (\[gsHstd\]). We now claim that $$\begin{aligned} \label{eq:tcSET} |\Psi_{\text{b}}{\rangle}={{\hat{U}}}_{\text{b}} |\Psi_{\text{t.c.}}{\rangle},\end{aligned}$$ where ${{\hat{U}}}_{\text{b}}$ is the following finite depth circuit of local unitaries: $$\begin{aligned} \label{ucirc} {{\hat{U}}}_{\text{b}}=\prod_{{\langle}pqr {\rangle}} \left(\hat{\nu}_{pqr}^{o_{pqr}} {{\hat{Z}}}_{pq} ^{\hat{n}_{qr}}\right) \prod_{{\langle}pq {\rangle}}{{\hat{X}}}_{pq}^{\hat{n}_{pq}}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{ pqr }^{\hat{n}_{pr}}.\end{aligned}$$ Here, $\hat{n}_{pq}$ is the operator defined by $$\begin{aligned} \hat{n}_{pq}\left|\{g_t\}\right{\rangle}=n(g_p,g_q,{{1}})|\{g_t\}{\rangle},\end{aligned}$$ and $\hat{\nu}_{pqr}^{o_{pqr}}$ is given by $$\begin{aligned} \hat{\nu}^{o_{pqr}}_{pqr}|\{g_t\}{\rangle}=\nu(g_p,g_q,g_r,{{1}})^{o_{pqr}}|\{g_t\}{\rangle}.\end{aligned}$$ To see that $|\Psi_{\text{b}}{\rangle}={{\hat{U}}}_{\text{b}} |\Psi_{\text{t.c.}}{\rangle}$, first note that the all of the configurations appearing with non-zero amplitude in $|\Psi_{\text{t.c.}}{\rangle}$ have trivial ${\mathbb{Z}}_2$-flux through all triangles, while the states in (\[gsHstd\]) have nontrivial ${\mathbb{Z}}_2$-flux at triangles ${\langle}pqr {\rangle}$ for which $(-1)^{n(g_p,g_q,g_r)}=-1$. This difference is remedied by the term $\prod_{{\langle}pq {\rangle}}{{\hat{X}}}_{pq}^{\hat{n}_{pq}}$ in (\[ucirc\]). The cocycle condition $\delta n=0$ guarantees that the nontrivial ${\mathbb{Z}}_2$-fluxes are put into the correct positions by this term. Second, the term $\prod_{{\langle}pqr {\rangle}} \hat{\nu}_{pqr}^{o_{pqr}} {{\hat{Z}}}_{pq}^{\hat{n}_{qr}}$ is simply to ensure that the phases assigned to configurations match those in $|\Psi_{\text{b}}{\rangle}$. It is proved in Appendix \[ap:symH\] that ${{\hat{U}}}_{\text{b}}$ is nearly $G$-symmetric - conjugating it by any global symmetry generator yields ${{\hat{U}}}_{\text{b}}$ multiplied by a product of some ${{\hat{G}}}_p$ operators. This property of ${{\hat{U}}}_{\text{b}}$ in particular relies on the term $\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{ pqr }^{\hat{n}_{pr}}$ in (\[ucirc\]), which may have seemed unnecessary at first since it acts trivially on the toric code ground states. Together with the manifest $G$ and ${{\hat{G}}}_p$ invariance of $\hat{H}^0_{\text{b}}$, this property of ${{\hat{U}}}_{\text{b}}$ implies that $$\begin{aligned} \label{Hbos} \hat{H}_{\text{b}}={{\hat{U}}}_{\text{b}}\hat{H}^0_{\text{b}}{{\hat{U}}}_{\text{b}}^\dag,\end{aligned}$$ is a $G$-symmetric parent Hamiltonian for $|\Psi_{\text{b}}{\rangle}$. We will see in section \[sec:fSPT\] that $ {{\hat{U}}}_{\text{b}}$ also commutes with all ${{\hat{G}}}_p$, so that the $\hat{H}_{\text{b}}$ does as well. We have thus constructed, using group supercohomology data, a bosonic shadow model Hamiltonian that commutes with all of the ${{\hat{G}}}_p$, and whose ground states all satisfy ${{\hat{G}}}_p=+1$. This bosonic shadow model describes a $G$-symmetry enriched toric code phase. Example: $G={\mathbb{Z}}_2$ {#Z2sec} --------------------------- Let us describe the above constructions for the simplest nontrivial examples of supercohomology phases, which occur for $G={\mathbb{Z}}_2$ (i.e. total symmetry group ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2^{\text{f}}$). In contrast to the case of general $G$, where we used multiplicative notation for the group law, in the case $G={\mathbb{Z}}_2$, we will use additive notation and denote ${\mathbb{Z}}_2$ elements by $s=0,1$. For $G={\mathbb{Z}}_2$, there are four inequivalent supercohomology classes. Two of these have trivial $n$ and correspond to the trivial phase and the purely bosonic ${\mathbb{Z}}_2$ SPT. The other two both have the same nontrivial $n$: $$\begin{aligned} \label{n2z2} n(s_0,s_1,s_2) = \begin{cases} 1 & (s_0,s_1,s_2)=(0,1,0)\\ 1& (s_0,s_1,s_2)=(1,0,1) \\ 0 & \text{otherwise,} \end{cases}\end{aligned}$$ but different $\nu$: $$\begin{aligned} \label{nu3z2} \nu_{\pm}(s_0,s_1,s_2,s_3) = \begin{cases} \pm i & (s_0,s_1,s_2,s_3)=(1,0,1,0) \\ \pm i & (s_0,s_1,s_2,s_3)=(0,1,0,1) \\ 1 & \text{otherwise.} \end{cases}\end{aligned}$$ This data defines two possible phases according to the choice of sign in (\[nu3z2\]), which turn out to be the index $2$ and $6$ members of the ${\mathbb{Z}}_8$ interacting classification in this symmetry class \[\]. The ${\mathbb{Z}}_2$ extension of $G={\mathbb{Z}}_2$ defined by $n$ is ${\tilde{G}}={\mathbb{Z}}_4$, and ${\epsilon}$ is ${\epsilon}(0)={\epsilon}(3)=0$, ${\epsilon}(1)={\epsilon}(2)=1$. Explicitly computing the cocycle $\alpha$ defined in (\[def\_alpha\]), we obtain $$\begin{aligned} \label{alpha3Z2} \alpha_{\pm}({\tilde{s}}_0,{\tilde{s}}_1,{\tilde{s}}_2,{\tilde{s}}_3)= (\pm i)^{({\tilde{s}}_0-{\tilde{s}}_1)(\overline{{\tilde{s}}_1-{\tilde{s}}_2}) (\overline{{\tilde{s}}_2-{\tilde{s}}_3})},\end{aligned}$$ where ${\tilde{s}}\in {\mathbb{Z}}_4$ and the overline denotes reduction modulo $2$. The corresponding auxiliary ${\mathbb{Z}}_4$ SPT wave function is: $$\begin{aligned} \Psi^{\pm}_{\text{SPT}}(\{{\tilde{s}}_p\}) = \prod_{{\langle}pqr {\rangle}} \alpha_{\pm}({\tilde{s}}_p,{\tilde{s}}_q,{\tilde{s}}_r,0)^{o_{ pqr }}.\end{aligned}$$ The bosonic shadow Hilbert space has ${\mathbb{Z}}_2$ degrees of freedom $s_p=0,1$ on vertices $p$ and spin-$\frac{1}{2}$ degrees of freedom $Z_{pq} = \pm 1$ on links ${\langle}pq {\rangle}$. The shadow model ground states are (see FIG. \[fig:z2amplitude\] and \[fig:z2amplitude2\]) $$\begin{aligned} \label{setgs2} &\Psi^{\pm}_{\text{b}}(\{s_p\},\{ Z_{pq}\})= \\ \nonumber &\prod_{{\langle}pqr {\rangle}} \nu_{\pm}\left(s_p,s_q,s_r,0 \right)^{o_{pqr}} Z_{pq}^{n \left(s_q,s_r,0 \right)} \\ \nonumber \times &\left( \prod_{{\langle}pqr {\rangle}} \delta_{Z_{pq}Z_{qr}Z_{pr},(-1)^{n\left(s_p,s_q,s_r\right)}} \right) h(\{ Z_{pq}(-1)^{n(1,s_p,s_q)}\}).\end{aligned}$$ Here, $h(\{Z_{pq}\})=0,1$ is a function that projects onto a choice of holonomy of the ${\mathbb{Z}}_2$ gauge field. Using the explicit form of the supercohomology data $n(s_p,s_q,0) = (1-s_p)s_q$ and $\nu_{\pm}(s_p,s_q,s_r,0)=(\pm i)^{s_p(1-s_q)s_r}$, we see that the circuit (\[ucirc\]) becomes $$\begin{aligned} \label{z2circuit} {{\hat{U}}}^{\pm}_{\text{b}}=&\prod_{{\langle}pqr {\rangle}} (\pm i)^{o_{pqr}{{\hat{s}}}_p(1-{{\hat{s}}}_q){{\hat{s}}}_r} {{\hat{Z}}}_{pq} ^{(1-{{\hat{s}}}_q){{\hat{s}}}_r} \\ \nonumber \times &\prod_{{\langle}pq {\rangle}}{{\hat{X}}}_{pq}^{(1-{{\hat{s}}}_p){{\hat{s}}}_q}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{ pqr }^{(1-{{\hat{s}}}_p){{\hat{s}}}_r}.\end{aligned}$$ From this circuit, we obtain the Hamiltonian $$\begin{aligned} \hat{H}^{\pm}_{\text{b}} = {{\hat{U}}}^{\pm}_{\text{b}}\hat{H}^0_{\text{b}}({{\hat{U}}}^{\pm}_{\text{b}})^\dag\end{aligned}$$ for the gauged model. For completeness, we note that the global ${\mathbb{Z}}_2$ symmetry generator in the gauged model acts by $$\begin{aligned} |\{s_p,Z_{pq} \} {\rangle}\rightarrow |\{1-s_p,Z_{pq} \} {\rangle}.\end{aligned}$$ This is just the descendant of the ${\mathbb{Z}}_4$ generator in the ${\mathbb{Z}}_4$ SPT. Fermionizing the shadow model {#sec:duality} ============================= In the previous section, we used the supercohomology data to construct a bosonic shadow model ${\hat{H}}_{\text{b}}$ on a Hilbert space consisting of generalized $G$-spin degrees of freedom on vertices $p$ and spin-$\frac{1}{2}$ degrees of freedom on links ${\langle}pq {\rangle}$. In this section, we describe how this bosonic model may be fermionized, i.e. rewritten in terms of local fermionic operators. This fermionization is effectively a procedure for ‘un-gauging’ fermion parity symmetry. Equivalently, it can be viewed as a prescription for a lattice level fermion condensation (see Appendix \[ap:fermioncondensation\] for further detail). We emphasize that this is the only point at which a choice of spin structure enters the construction. Focusing just on the spin-$\frac{1}{2}$ link degrees of freedom, we utilize the fermionization prescription developed in Ref. \[\], reviewed in the next three subsections, which provides an exact duality between the local operator algebra of a bosonic model and that of a fermionic model. To define this duality, one must specify some combinatorial data, which we show amounts to a choice of spin structure for the spatial manifold $M$. We will first define the local bosonic and fermionic operator algebras $\cal{A}_\text{bos}$ and $\cal{A}_\text{fer}$, respectively, and then construct the spin-structure dependent duality between them. Finally, we apply this duality to ${\hat{H}}_{\text{b}}$ to produce our fermionic Hamiltonian ${\hat{H}}_{\text{f}}$ and demonstrate that it describes an SPT. Bosonic operator algebra $\cal{A}_{\text{bos}}$ ----------------------------------------------- On the bosonic side, we consider the spin-$\frac{1}{2}$ degrees of freedom living on links, with Pauli algebra generated by ${{\hat{X}}}_{pq}$ and $ {{\hat{Z}}}_{pq}$. $\cal{A}_{\text{bos}}$ is defined as the operator algebra generated by the subset of local operators that commute with all the ${\hat G}_p$ defined in (\[def:modifiedgausslaw\]): $$\begin{aligned} {\hat G}_p = \prod_{\substack{{\langle}tqr {\rangle}\\ t=p}} {\hat{Z}}_{tq} {\hat{Z}}_{qr} {\hat{Z}}_{tr} \prod_{{\langle}st {\rangle}\ni p} {\hat X}_{st}\end{aligned}$$ and modulo the relations ${\hat G}_p = 1$ for all $p$.[^4] Thus we may think of $\cal{A}_{\text{bos}}$ as the algebra of operators generated by the subset of local operators which are gauge invariant with respect to the modified Gauss’s law ${{\hat{G}}}_p=1$. We now identify two sets of local, modified Gauss’s law invariant operators which generate all of $\cal{A}_{\text{bos}}$ \[\]. The first is ${{\hat{W}}}_{pqr} = {{\hat{Z}}}_{pq} {{\hat{Z}}}_{qr} {{\hat{Z}}}_{pr}$. The second is ${{\hat{U}}}_{pq}$, defined as: $$\begin{aligned} \label{def:Ue} {{\hat{U}}}_{pq} \equiv {{\hat{X}}}_{pq}\hat{K}_{L_{pq}}\hat{K}_{R_{pq}},\end{aligned}$$ with $\hat{K}_{R_{pq}}$ and $\hat{K}_{L_{pq}}$ defined as follows. The action of $\hat{K}_{R_{pq}}$ is dependent upon the triangle $R_{pq}$ to the right of ${\langle}pq {\rangle}$. If the triangle to the right of ${\langle}pq {\rangle}$ has vertex ordering ${\langle}rpq {\rangle}$, with $p$ and $q$ being the second and third vertices, respectively, then $\hat{K}_{R_{pq}}$ acts as ${{\hat{Z}}}_{rp}$. Otherwise, $\hat{K}_{R_{pq}}={{1}}$. The action of $\hat{K}_{L_{pq}}$ is defined similarly but with ‘right’ replaced with ‘left’. Some examples of the action of ${{\hat{U}}}_{pq}$ are depicted in FIG. \[fig:Ueaction\]. Intuition for this seemingly contrived definition can be obtained by recalling that the modified Gauss’s law is a constraint that binds a ${\mathbb{Z}}_2$ flux on a triangle to a ${\mathbb{Z}}_2$ charge at the first vertex of that triangle. The operator ${{\hat{U}}}_{pq}$ then hops a ${\mathbb{Z}}_2$ flux across the link ${\langle}pq {\rangle}$, and also rearranges the ${\mathbb{Z}}_2$ charges in such a way that the modified Gauss’s law remains enforced. As shown in Ref. \[\], the only nontrivial relations among the ${{\hat{U}}}_{pq}$ and ${{\hat{W}}}_{pqr}$ operators are captured in the following operator identity. For any vertex $p$, $$\begin{aligned} \label{bosidentity} \prod_{\substack{{\langle}tq {\rangle}\\ t=p}} {{\hat{U}}}_{tq} \prod_{\substack{{\langle}qt {\rangle}\\ t=p}} {{\hat{U}}}_{qt} = {\hat G}_p \prod_{\substack{{\langle}tqr {\rangle}\\ t=p}} {\hat W}_{tqr} \prod_{\substack{{\langle}qrt {\rangle}\\ t=p}} {\hat W}_{qrt}.\end{aligned}$$ Note that in the first product on the left hand side all the links are oriented away from $p$, while in the second product all the links are oriented towards $p$. Fermionic operator algebra $\cal{A}_\text{fer}$ ----------------------------------------------- On the fermionic side, the degrees of freedom are complex fermions - one at the center of each triangle ${\langle}pqr {\rangle}$. We use the pair of Majorana operators $\gamma_{pqr}$ and ${{\overline{\gamma}}}_{pqr}$ to represent the operator algebra for this complex fermion. The fermion parity at triangle ${\langle}pqr {\rangle}$ is measured by $$\begin{aligned} (-1)^{{\hat F}_{pqr}} \equiv -i \gamma_{pqr} {{\overline{\gamma}}}_{pqr},\end{aligned}$$ and an operator is fermion parity even if it commutes with $\prod_{{\langle}pqr {\rangle}}(-1)^{{\hat F}_{pqr}}$. The algebra $\cal{A}_{\text{fer}}$ of fermion parity even operators is generated by the $(-1)^{{\hat F}_{pqr}}$ and a certain set of ‘hopping operators’, which transfer fermion parity across a link ${\langle}pq {\rangle}$. Specifically, we define the hopping operator $$\begin{aligned} {\hat S'}_{pq} \equiv i\gamma_{L_{pq}} {{{\overline{\gamma}}}}_{R_{pq}},\end{aligned}$$ where we have again denoted the triangles to the left and right of ${\langle}pq {\rangle}$ by $L_{pq}$ and $R_{pq}$, respectively. The $(-1)^{{\hat F}_{pqr}}$ and ${\hat S'}_{pq}$ satisfy nearly the same algebraic relations with each other as do the bosonic operators ${{\hat{W}}}_{pqr}$ and ${{\hat{U}}}_{pq}$. The only difference is that $(-1)^{{\hat F}_{pqr}}$ and ${\hat S'}_{pq}$ satisfy an algebraic relation that is similar to but not exactly the same as (\[bosidentity\]) \[\]: $$\begin{aligned} \label{def:cp} \prod_{\substack{{\langle}tq {\rangle}\\ t=p}} {\hat S'}_{tq} \prod_{\substack{{\langle}qt {\rangle}\\ t=p}} {\hat S'}_{qt} = c(p) \prod_{\substack{{\langle}tqr {\rangle}\\ t=p}} (-1)^{{\hat F}_{tqr}} \prod_{\substack{{\langle}qrt {\rangle}\\ t=p}} (-1)^{{\hat F}_{qrt}}.\end{aligned}$$ In (\[def:cp\]), $c(p)$ is a sign factor determined solely by the branching structure near $p$. We prove (\[def:cp\]) in Appendix \[ap:cpproof\], where we also derive the following graphical method for explicitly calculating $c(p)$. First, we interpolate the branching structure to the interiors of the triangles to give a continuous non-vanishing vector field \[\] ${\cal V}$ (see FIG. \[fig:vecfieldorientation\]). Singularities in this vector field can occur only at vertices, and $c(p)=-1$ if the vertex $p$ has a singularity with odd winding number and $c(p)=1$ otherwise. Spin structure dependent duality between $\cal{A}_\text{bos}$ and $\cal{A}_\text{fer}$ {#sec:spinstructure} -------------------------------------------------------------------------------------- The geometric interpretation of the sign $c(p)$ in (\[def:cp\]) as counting the singularities of a vector field ${\cal V}$ immediately points to a possible modification of the operators generating $\cal{A}_{\text{fer}}$ that makes $\cal{A}_{\text{fer}}$ manifestly isomorphic to $\cal{A}_{\text{bos}}$. To make this modification, first note that there are an even number of vertices with $c(p)=-1$. [^5] Thus, we can find a set ${\cal E}$ of links such that the vertices in the boundary of ${\cal E}$ (the boundary being defined as the set of vertices which are endpoints of an odd number of links in ${\cal E}$) are precisely the vertices with $c(p)=-1$. Then, we can modify the vector field ${\cal V}$ by giving it an extra $2 \pi$ winding as it crosses a link in ${\cal E}$ (see FIG. \[fig:singularedge\]). The result is a new vector field with even singularities only. It is known that in 2 dimensions a vector field with only even singularities defines a spin structure \[\]. Hence, a choice of ${\cal E}$ corresponds to a choice of spin structure. Having made a choice of ${\cal E}$, we now define modified hopping operators $$\begin{aligned} {\hat S}_{pq} \equiv (-1)^{{\cal E}_{pq}} {\hat S'}_{pq}\end{aligned}$$ where ${\cal E}_{pq} = 0,1$ is the indicator function for ${\cal E}$, i.e. ${\cal E}_{pq} = 1$ if ${\langle}pq {\rangle}\in {\cal E}$ and ${\cal E}_{pq}=0$ otherwise. These modified operators then satisfy $$\begin{aligned} \label{Antonrelation} \prod_{\substack{{\langle}tq {\rangle}\\ t=p}} {\hat S}_{tq} \prod_{\substack{{\langle}qt {\rangle}\\ t=p}} {\hat S}_{qt} = \prod_{\substack{{\langle}tqr {\rangle}\\ t=p}} (-1)^{\hat{F}_{tqr}} \prod_{\substack{{\langle}qrt {\rangle}\\ t=p}} (-1)^{\hat{F}_{qrt}}.\end{aligned}$$ Now, comparing with (\[bosidentity\]), we see that the correspondence given by $$\begin{aligned} \label{dictionary} {\hat W}_{pqr} &\longleftrightarrow (-1)^{F_{pqr}}\\ \nonumber {\hat U}_{pq} &\longleftrightarrow {\hat S}_{pq} \end{aligned}$$ defines an explicit isomorphism of operator algebras between $\cal{A}_\text{bos}$ and $\cal{A}_\text{fer}$. We emphasize that this correspondence depends on a choice of spin structure, via the choice of ${\cal E}$. The fermionization duality reviewed here admits an intuitive description in terms of a ‘condensation of fermions’. We elaborate on this point in Appendix \[ap:fermioncondensation\]. Fermionic SPT Hamiltonian {#sec:fSPT} ------------------------- Let us now use the dictionary given in (\[dictionary\]) to rewrite each local term in the shadow model Hamiltonian $$\begin{aligned} \label{Hbos2} \hat{H}_{\text{b}}={{\hat{U}}}_{\text{b}}\hat{H}^0_{\text{b}}{{\hat{U}}}_{\text{b}}^\dag,\end{aligned}$$ defined in (\[Hbos\]), in terms of local fermionic operators. This can be carried out by fermionizing $\hat{H}_{\text{b}}^0$, defined in (\[toriccode\]), and ${{\hat{U}}}_b$, defined in (\[ucirc\]), independently. To fermionize $\hat{H}_{\text{b}}^0$, we first use the definition of ${{\hat{G}}}_p$ to rewrite it as $$\begin{aligned} \label{prefermionize} \hat{H}^0_{\text{b}}=-\sum_p {{\hat{P}}}_p^{\text{sym}} -\sum_{p}\Bigg( {{\hat{G}}}_p\prod_{\substack{{\langle}tqr {\rangle}\\ t=p}}{{\hat{W}}}_{tqr} \Bigg)-\sum_{{\langle}pqr {\rangle}}{{\hat{W}}}_{pqr}.\end{aligned}$$ Then, according to the dictionary in (\[dictionary\]), $\hat{H}^0_{\text{b}}$ fermionizes to $$\begin{aligned} \label{atomicinsulator} \hat{H}_{\text{f}}^0=-\sum_p {{\hat{P}}}_p^{\text{sym}}-\sum_{{\langle}pqr {\rangle}}(-1)^{\hat{F}_{pqr}},\end{aligned}$$ after using the gapped and unfrustrated property of the Hamiltonian to remove the fermionization of the second term in (\[prefermionize\]). This Hamiltonian describes a trivial atomic insulator, and the unique ground state $|\Psi_{\text{f}}^{0}{\rangle}$ is a product state of symmetrized states at the vertices and zero fermion occupancy on the triangles. To fermionize ${{\hat{U}}}_{\text{b}}$, we note that the product $$\begin{aligned} \prod_{{\langle}pqr {\rangle}} {{\hat{Z}}}_{pq} ^{\hat{n}_{qr}} \prod_{{\langle}pq {\rangle}}{{\hat{X}}}_{pq}^{\hat{n}_{pq}}\end{aligned}$$ in (\[ucirc\]) can be rearranged into $$\begin{aligned} \label{Uedgesprod} \hat{\kappa}\prod_{{\langle}pq {\rangle}}{{\hat{U}}}_{pq}^{\hat{n}_{pq}}.\end{aligned}$$ where $\hat{\kappa}$ is a certain diagonal operator in the $\{g_p\}$ configuration basis with eigenvalues $\pm 1$. The eigenvalue is locally determined, in that it is a product of signs, each of which is dependent upon only the $G$-configuration within a disk of finite radius around some point. These signs result from commuting ${{\hat{Z}}}_{pq}$ past ${{\hat{X}}}_{pq}$ and hence the eigenvalues are dependent on the choice of ordering of the ${{\hat{U}}}_{pq}$ operators in (\[Uedgesprod\]). Although the operator $\hat{\kappa}$ is complicated to write out for general $G$, we note that the locality property above makes it a finite depth circuit of local unitaries. Furthermore, we will see below that in the example $G={\mathbb{Z}}_2$ the situation simplifies considerably: $\hat{\kappa}$ is trivial in that case, and all of the terms in the product in (\[Uedgesprod\]) commute. Also, in Appendix \[ap:spinH\] we present another way of circumventing the issue posed by the unwieldy form of $\hat{\kappa}$, by introducing ancillary spin-$\frac{1}{2}$ degrees of freedom on the triangles. This allows for a more canonical finite depth circuit that does not require an arbitrary choice of ordering. We now use (\[dictionary\]) to map (\[Uedgesprod\]) to fermionic operators. The result of fermionizing ${{\hat{U}}}_{\text{b}}$ is the finite depth circuit of local unitaries $$\begin{aligned} \label{ufer} {{\hat{U}}}_{\text{f}}= \hat{\kappa} \prod_{{\langle}pqr {\rangle}} \hat{\nu}_{pqr}^{o_{pqr}} \prod_{{\langle}pq {\rangle}}{{\hat{S}}}_{pq}^{\hat{n}_{pq}}\prod_{{\langle}pqr {\rangle}}\left((-1)^{\hat{F}_{ pqr }}\right)^{\hat{n}_{pr}}.\end{aligned}$$ Therefore, fermionization turns $\hat{H}_{\text{b}}$ into $$\begin{aligned} \hat{H}_{\text{f}}={{\hat{U}}}_{\text{f}}\hat{H}_{\text{f}}^0{{\hat{U}}}_{\text{f}}^\dag.\end{aligned}$$ $\hat{H}_{\text{f}}$ is comprised of two types of terms. First, we have the conjugates of the terms in the second sum in (\[atomicinsulator\]), namely: $$\begin{aligned} -{{\hat{U}}}_{\text{f}}(-1)^{\hat{F}_{pqr}}{{\hat{U}}}_{\text{f}}^\dag=-(-1)^{\hat{n}_{pqr}}(-1)^{\hat{F}_{pqr}}. \end{aligned}$$ These energetically enforce fermions to occupy the triangles ${\langle}pqr {\rangle}$ with nontrivial $n(g_p,g_q,g_r)$. Second, we have the conjugates of the terms in the first sum in (\[atomicinsulator\]): $$\begin{aligned} -{{\hat{U}}}_{\text{f}}{{\hat{P}}}^{\text{sym}}_p{{\hat{U}}}_{\text{f}}^\dag.\end{aligned}$$ These fluctuate the $G$-configuration at vertex $p$ and move the neighboring fermions so that the fermion occupancy conforms to the first term. We will see the action of $\hat{H}_{\text{f}}$ more explicitly below when we treat the case $G={\mathbb{Z}}_2$. $\hat{H}_{\text{f}}$ describes a fermionic SPT phase because (1) it is gapped (2) it has a unique, SRE ground state, and (3) it is symmetric. It is gapped because it is an unfrustrated commuting projector Hamiltonian. The unique ground state is ${{\hat{U}}}_{\text{f}}|\Psi_{\text{f}}^0 {\rangle}$, and since ${{\hat{U}}}_{\text{f}}$ is a finite depth circuit of local unitaries, the ground state is SRE. Lastly, it is $G$-symmetric because ${\hat{H}}_b$ is $G$-symmetric, and the fermionization procedure commutes with the global action of $G$. Example: $G={\mathbb{Z}}_2$ {#Z2cont} --------------------------- Recall that in the $G={\mathbb{Z}}_2$ case, (\[z2circuit\]) is $$\begin{aligned} \label{z2circuit_duplicate} {{\hat{U}}}^{\pm}_{\text{b}}=&\prod_{{\langle}pqr {\rangle}} (\pm i)^{o_{pqr}{{\hat{s}}}_p(1-{{\hat{s}}}_q){{\hat{s}}}_r} {{\hat{Z}}}_{pq} ^{(1-{{\hat{s}}}_q){{\hat{s}}}_r} \\ \nonumber \times &\prod_{{\langle}pq {\rangle}}{{\hat{X}}}_{pq}^{(1-{{\hat{s}}}_p){{\hat{s}}}_q}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{ pqr }^{(1-{{\hat{s}}}_p){{\hat{s}}}_r}.\end{aligned}$$ To avoid confusion, we will for the remainder of this section focus on the case ${{\hat{U}}}^{+}_{\text{b}}$ and drop the $+$ superscript; the case ${{\hat{U}}}^{-}_{\text{b}}$ can be treated similarly. To fermionize ${{\hat{U}}}_{\text{b}}$, we first recognize that it may be written in terms of the local operators ${{\hat{U}}}_{pq}$ of section \[sec:duality\]. The product $$\begin{aligned} \prod_{{\langle}pqr {\rangle}} {{\hat{Z}}}_{pq}^{(1-{{\hat{s}}}_q){{\hat{s}}}_r} \prod_{{\langle}pq {\rangle}}{{\hat{X}}}^{(1-{{\hat{s}}}_p){{\hat{s}}}_q}_{pq}\end{aligned}$$ in (\[z2circuit\_duplicate\]) is exactly equal to $$\begin{aligned} \prod_{{\langle}pq {\rangle}}{{\hat{U}}}^{(1-{{\hat{s}}}_p){{\hat{s}}}_q}_{pq}\end{aligned}$$ without any additional factor of ${\hat{\kappa}}$. This is due to the fact that $(1-s_q)s_r$ and $(1-s_p)s_q$ cannot simultaneously be $1$, so that we never have to move anti-commuting operators past each other to go from one expression to the other. Therefore, the fermionization duality applied to ${{\hat{U}}}_{\text{b}}$ yields $$\begin{aligned} \label{z2Uf} {{\hat{U}}}_{\text{f}}=&\prod_{{\langle}pqr {\rangle}} i^{{o}_{pqr}{{\hat{s}}}_p(1-{{\hat{s}}}_q){{\hat{s}}}_r}\\ \nonumber \times&\prod_{{\langle}pq {\rangle}}{{\hat{S}}}^{(1-{{\hat{s}}}_p){{\hat{s}}}_q}_{pq} \prod_{{\langle}pqr {\rangle}}\left((-1)^{\hat{F}_{pqr}}\right)^{(1-{{\hat{s}}}_p){{\hat{s}}}_r}\end{aligned}$$ with ${{\hat{S}}}_{pq}$ and $(-1)^{\hat{F}_{pqr}}$ defined in section \[sec:duality\]. Hence, ${\hat{H}}_{\text{b}}$ explicitly fermionizes to $$\begin{aligned} \hat{H}_{\text{f}}={{\hat{U}}}_{\text{f}} \hat{H}^0_{\text{f}} {{\hat{U}}}_{\text{f}},\end{aligned}$$ where $$\begin{aligned} \label{atomicinsulator2} \hat{H}_{\text{f}}^0=-\sum_p {{\hat{P}}}_p^{\text{sym}}-\sum_{{\langle}pqr {\rangle}}(-1)^{\hat{F}_{pqr}}.\end{aligned}$$ We have thus constructed a ${\mathbb{Z}}_2$-symmetric fermionic SPT Hamiltonian for the supercohomology data specified in (\[n2z2\]) and (\[nu3z2\]). [**[Picture of the ground state: ]{}**]{} The finite depth circuit of local unitaries ${{\hat{U}}}_\text{f}$ in (\[z2Uf\]) allows us to explicitly construct the ground state $|\Psi_\text{f}{\rangle}$ of $\hat{H}_\text{f}$. This is accomplished by applying ${{\hat{U}}}_\text{f}$ to $|\Psi_\text{f}^0 {\rangle}$, the ground state of $\hat{H}_\text{f}^0$. $|\Psi_\text{f}^0{\rangle}$ is a product state with the ${\mathbb{Z}}_2$-symmetric state $\frac{1}{\sqrt{2}}(|0{\rangle}+|1{\rangle})$ at each vertex $p$ and zero fermion occupancy at every triangle ${\langle}pqr {\rangle}$. Expressed in the configuration basis, $|\Psi_\text{f}^0{\rangle}$ is an equal amplitude superposition of domain configurations – domains containing states $|0{\rangle}$ or $|1{\rangle}$ at vertices. Note that the domain walls between the $|0{\rangle}$ and $|1{\rangle}$ domains run along the edges of the dual lattice. The ground state of $\hat{H}_\text{f}$ is $$\begin{aligned} |\Psi_\text{f}{\rangle}={{\hat{U}}}_\text{f}|\Psi_\text{f}^0{\rangle}=\sum_{\substack{\text{configs} \\ \vcenter{\hbox{\includegraphics[scale=.02,trim={7cm 0cm 7cm 0cm},clip]{fSPT_figures2/domains.pdf}}}}}{{\hat{U}}}_\text{f} \left|\vcenter{\hbox{\includegraphics[scale=.08,trim={7cm 0cm 7cm 0cm},clip]{fSPT_figures2/domains.pdf}}} \right{\rangle}.\end{aligned}$$ The above sum is over all ${\mathbb{Z}}_2$-spin domain configurations tensored with the empty fermionic state. The operator ${{\hat{U}}}_\text{f}$ decorates fermions onto each such domain configuration and multiplies by a configuration-dependent phase, but it does not alter the shape of the domains. We can break the action of ${{\hat{U}}}_\text{f}$ on a domain configuration up into three steps. In the first step, we apply $$\begin{aligned} \label{fermionparityterm} \prod_{{\langle}pqr {\rangle}}\left((-1)^{\hat{F}_{pqr}}\right)^{(1-{{\hat{s}}}_p){{\hat{s}}}_r}.\end{aligned}$$ As the domain configurations in $|\Psi_\text{f}^0{\rangle}$ have no fermions, they are $+1$ eigenvectors of the fermion parity operators in (\[fermionparityterm\]). Thus, this term does not affect the state. In the second step, we act on the domain configuration with $$\begin{aligned} \label{fermionterm} \prod_{{\langle}pq {\rangle}}{{\hat{S}}}^{(1-{{\hat{s}}}_p){{\hat{s}}}_q}_{pq}.\end{aligned}$$ The exponent in (\[fermionterm\]) is $1$ precisely when the link ${\langle}pq {\rangle}$ points from a $|0{\rangle}$ domain to a $|1{\rangle}$ domain. As a result, Majorana operators are applied to the two triangles on either side of the link ${\langle}pq {\rangle}$, and in this way, fermions are only created along the domain wall. The result is a pair of fermions at the two endpoints of each portion of the domain wall where the interpolating vector field points from the $|0{\rangle}$ to the $|1{\rangle}$ domain (see FIG. \[fig:z2gs\] and \[fig:z2gsordering\]). The order in which these two fermions are created depends on the spin structure $\cal{E}$ as follows. First, we locally orient the domain wall so that it runs horizontally with the $|0{\rangle}$ domain below and the $|1{\rangle}$ domain above, as illustrated in FIG. \[fig:z2gsordering\]. If there are an even number of edges in $\cal{E}$ crossing the $|0{\rangle}$ to $|1{\rangle}$ pointing portion of the domain wall, then we create the fermion on the left endpoint first, followed by the fermion on the right endpoint. When there are an odd number of edges in $\cal{E}$ crossing the region, the fermions are created in the opposite order (FIG. (\[fig:z2gsordering\])). Since the difference between these two procedures is just a minus sign, we can alternatively always create the fermions from left to right, and at the end multiply by $-1$ for every edge of ${\cal E}$ that points from the $|0{\rangle}$ to the $|1{\rangle}$ domain. Lastly, we act with $$\begin{aligned} \prod_{{\langle}pqr {\rangle}} i^{o_{pqr}{{\hat{s}}}_p(1-{{\hat{s}}}_q){{\hat{s}}}_r}.\end{aligned}$$ This term assigns a phase to each configuration, which can be thought of as a product of contributions associated to points of tangency of the vector field with the domain wall, or, equivalently, associated to the fermions. These contributions can be determined as follows. Moving from left to right along a domain wall with the $|0{\rangle}$ domain below and the $|1{\rangle}$ domain above, we track the interpolating vector field. If the interpolating vector field rotates clockwise, from initially pointing in the direction of the $|0{\rangle}$ domain to finally pointing in the direction of the $|1{\rangle}$ domain, then we accrue a phase of $i$. If the interpolating vector field rotates clockwise from initially pointing towards the $|1{\rangle}$ domain to finally pointing towards the $|0{\rangle}$ domain, then a phase of $-i$ is picked up (see FIG. \[fig:z2gsphase\]). For the two other possible rotations, no phase is picked up. We would like to emphasize that the ground state constructed according to this prescription admits a continuum interpretation. Namely, in the continuum we can think of the spin structure being encoded in a smooth vector field together with a set of smooth segments $\cal{E}$ connecting the odd singularities of this vector field. The ground state is a superposition over smooth domain wall configurations decorated with fermions. The fermions appear precisely at the locations where the vector field is tangent to a domain wall, and the above prescription gives a specific ordering of fermion creation operators used to create this fermionic state from the empty fermionic state. Finally, the amplitude for each decorated domain wall is multiplied by products of $\pm i$ as determined by the rotation of the vector field at the points of tangency, as detailed above. Classification {#sec:classification} ============== Thus far, we have used a choice of supercohomology data $(n,\nu)$ together with a spin structure on a 2d triangulated spatial manifold with branching structure to construct a zero correlation length fermionic SPT Hamiltonian. The strategy was to first construct a bosonic shadow model using the group supercohomology data. This led us to a finite depth circuit ${{\hat{U}}}_\text{b}$ (\[ucirc\]): $$\begin{aligned} {{\hat{U}}}_{\text{b}}=\prod_{{\langle}pqr {\rangle}} \left(\hat{\nu}_{pqr}^{o_{pqr}} {{\hat{Z}}}_{pq} ^{\hat{n}_{qr}}\right) \prod_{{\langle}pq {\rangle}}{{\hat{X}}}_{pq}^{\hat{n}_{pq}}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{ pqr }^{\hat{n}_{pr}},\end{aligned}$$ which, when applied to an ordinary toric code ground state, produced the ground state of the bosonic shadow model. Furthermore, the fermionization of ${{\hat{U}}}_\text{b}$ yielded ${{\hat{U}}}_\text{f}$ (defined in (\[ufer\])) - a fermionic finite depth circuit that builds a fermionic SPT ground state from a trivial product state. In this section, we show that the composition of these circuits gives insight into the group structure of fermionic SPT phases. First, we clarify the physical meaning behind composing finite depth circuits. Then, we give a physically motivated definition of equivalence for sets of supercohomology data. Lastly, we use this notion of equivalence to establish group supercohomology classes as topological invariants for lattice fermionic SPT Hamiltonians. Stacking as composition of circuits ----------------------------------- The additive group structure on the set of SPT phases is given by stacking. To stack two SPT Hamiltonians, let us imagine that they are defined on identical lattices extending in the $x,y$ directions, and let us put one lattice directly over the other, i.e. separated in the $z$ direction. Then, grouping pairs of vertically separated sites with the same $x,y$-coordinates into supersites, the sum of the two decoupled SPT Hamiltonians for the two layers defines another $2$d gapped SPT Hamiltonian. This stacking operation respects the notion of phase equivalence and thus defines an additive structure on the set of SPT phases. We can reinterpret the stacking operation as composition of finite depth circuits of local unitaries that create the corresponding SPT ground states from a product state. To see this, suppose that ${{\hat{U}}}$ and ${{\hat{U}}}'$ are two such circuits that act on identical Hilbert spaces made out of sites which form identical $G$-representations. The ground state of the stacked system is $$\begin{aligned} \left({{\hat{U}}}\otimes {{\hat{U}}}'\right) \left(|0{\rangle}\otimes |0{\rangle}'\right)= \left({{\hat{U}}}|0{\rangle}\right) \otimes \left({{\hat{U}}}' |0{\rangle}'\right).\end{aligned}$$ Now let ${{\hat{V}}}$ be the unitary operator which exchanges the two layers. Note that ${{\hat{V}}}$ can be defined as a tensor product of finite dimensional unitaries acting on the individual supersites, where they just swap the two sites in each supersite. ${{\hat{V}}}$ clearly commutes with the action of the global symmetry, and we have $$\begin{aligned} \label{eq:stacked} {{\hat{V}}}&\left({{\hat{U}}}\otimes 1\right){{\hat{V}}}^\dag \left( 1 \otimes {{\hat{U}}}' \right) \left(|0{\rangle}\otimes |0{\rangle}'\right) \\ \nonumber &= \left(1 \otimes {{\hat{U}}}{{\hat{U}}}'\right) \left(|0{\rangle}\otimes |0{\rangle}'\right) \\ \nonumber &= |0{\rangle}\otimes \left({{\hat{U}}}{{\hat{U}}}' |0{\rangle}'\right).\end{aligned}$$ Hence (\[eq:stacked\]) is equivalent to the state obtained by composing the two circuits. Notice, ${{\hat{V}}}$ can be continuously connected to the identity via a path in the space of symmetric finite depth circuits. To construct such a path, one just needs to find a path connecting the swap unitary to the identity for a single supersite and tensor these over all the supersites. For a single supersite, the problem is straightforward. This is because, in a finite dimensional Hilbert space, any symmetric unitary is connected to the identity through a path in the space of symmetric unitaries, as can be seen by breaking up the Hilbert space into irreducible representations of $G$ and applying Schur’s lemma. We now use this equivalence between stacking and composing circuits to derive the stacking rule for our supercohomology SPT models. In particular, this will show that the supercohomology SPT phases form a closed subgroup under stacking. Computation of stacking rules by composing circuits --------------------------------------------------- Let $(n,\nu)$ and $(n',\nu')$ be two sets of supercohomology data. Further, denote the bosonic finite depth circuits obtained from $(n,\nu)$ and $(n',\nu')$ via our construction by ${{\hat{U}}}_\text{b}^{n\nu}$ and ${{\hat{U}}}_\text{b}^{n'\nu'}$, respectively. The composition of ${{\hat{U}}}_\text{b}^{n\nu}$ with ${{\hat{U}}}_\text{b}^{n'\nu'}$ yields a finite depth circuit corresponding to yet another set of supercohomology data. This can be seen by explicit computation. The product of ${{\hat{U}}}_\text{b}^{n\nu}$ with ${{\hat{U}}}_\text{b}^{n'\nu'}$ is $$\begin{aligned} {{\hat{U}}}_\text{b}^{n\nu}{{\hat{U}}}_\text{b}^{n'\nu'}=&\prod_{{\langle}pqr {\rangle}} \left(\hat{\nu}_{pqr}^{o_{pqr}} {{\hat{Z}}}_{pq} ^{\hat{n}_{qr}}\right) \prod_{{\langle}pq {\rangle}}{{\hat{X}}}_{pq}^{\hat{n}_{pq}}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{ pqr }^{\hat{n}_{pr}}\\ \nonumber \times &\prod_{{\langle}pqr {\rangle}} \left({\hat{\nu}}'^{o_{pqr}}_{pqr} {{\hat{Z}}}_{pq} ^{{\hat{n}}'_{qr}}\right) \prod_{{\langle}pq {\rangle}}{{\hat{X}}}_{pq}^{{\hat{n}}'_{pq}}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{ pqr }^{{\hat{n}}'_{pr}}.\end{aligned}$$ To obtain an expression in the same form as ${{\hat{U}}}_\text{b}$, and thus reveal the group structure of the fermionic circuits, we group similar terms. In doing so, the only non-trivial signs arise when we move $$\begin{aligned} \prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{ pqr }^{\hat{n}_{pr}}\text{ past }\prod_{{\langle}pq {\rangle}}{{\hat{X}}}_{pq}^{{\hat{n}}'_{pq}}\end{aligned}$$ and $$\begin{aligned} \prod_{{\langle}pqr {\rangle}}{{\hat{X}}}_{ pq }^{\hat{n}_{pq}}\text{ past }\prod_{{\langle}pq {\rangle}}{{\hat{Z}}}_{pq}^{{\hat{n}}'_{qr}}.\end{aligned}$$ Using $\delta n = 0$, we can write the resulting sign as: $$\begin{aligned} \label{cuponesign} \prod_{{\langle}pqr {\rangle}}(-1)^{\hat{n}_{pr}\hat{n}'_{pqr}+\hat{n}_{pq}\hat{n}'_{qr}}.\end{aligned}$$ We then have $$\begin{aligned} \label{compositioncircuit} {{\hat{U}}}_\text{b}^{n\nu}{{\hat{U}}}_\text{b}^{n'\nu'}=&\prod_{{\langle}pqr {\rangle}} \hat{\nu}^{o_{pqr}}_{pqr}{\hat{\nu}}'^{o_{pqr}}_{pqr}(-1)^{\hat{n}_{pr}\hat{n}'_{pqr}+\hat{n}_{pq}\hat{n}'_{qr}} \\ \nonumber \times &\prod_{{\langle}pqr {\rangle}} {{\hat{Z}}}_{pq} ^{\hat{n}_{qr}+{\hat{n}}'_{qr}} \prod_{{\langle}pq {\rangle}}{{\hat{X}}}_{pq}^{\hat{n}_{pq}+\hat{n}'_{pq}}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{ pqr }^{\hat{n}_{pr}+\hat{n}'_{pr}}.\end{aligned}$$ This is precisely the circuit ${{\hat{U}}}_\text{b}$ formed from the input supercohomology data $(n+n',\nu \nu' (-1)^{n \cup_1 n'})$, where[^6] $$\begin{aligned} (n \cup_1 n'&) (g_p,g_q,g_r,1) = \\ \nonumber &n(g_p,g_r,1)n'(g_p,g_q,g_r)+n(g_p,g_q,1)n'(g_q,g_r,1)\end{aligned}$$ agrees with the sign in (\[cuponesign\]) and (\[compositioncircuit\]). Therefore, stacking the fermionic SPT phases corresponding to $(n,\nu)$ and $(n',\nu')$ results in the fermionic SPT phase corresponding to $(n+n',\nu \nu' (-1)^{n \cup_1 n'})$, or $$\begin{aligned} \label{stackingrule} (n,\nu)*(n',\nu')=(n+n',\nu \nu' (-1)^{n \cup_1 n'})\end{aligned}$$ with $*$ denoting the stacking operation. This is in accord with the supercohomology data group law found in Ref. \[\] through continuum space-time methods. Equivalence relation on supercohomology data {#sec:superequivalencerelation} -------------------------------------------- The stacking rules allow us to define a physically motivated notion of equivalence between two sets of supercohomology data, which agrees with the mathematical one given in e.g. Ref. \[\]. We will say that two sets of supercohomology data are equivalent if the corresponding fermionic SPT Hamiltonians $\hat{H}_\text{f}$, constructed in section \[sec:fSPT\], are in the same phase. Consider the supercohomology data[@Note401] $$\begin{aligned} \label{eq:trivialdata} (n_0,\nu_0)=(\delta \beta, (-1)^{\beta \cup \delta \beta}\delta \omega),\end{aligned}$$ where $\beta:G\times G \to {\mathbb{Z}}_2$ and $\omega:G \times G\times G \to U(1)$ are both homogeneous. We claim that this set of data gives a finite depth circuit ${{\hat{U}}}_\text{b}^{n_0 \nu_0}$ built from symmetric local unitaries (up to factors of ${{\hat{G}}}_p$), i.e. the fermionic SPT phase corresponding to this set of data is trivial \[\]. In Appendix \[ap:trivialcircuit\], we compute ${{\hat{U}}}_\text{b}^{n_0 \nu_0}$ in detail, and we simply state the result here: $$\begin{aligned} {{\hat{U}}}_\text{b}^{n_0\nu_0}= &\prod_{{\langle}pqr {\rangle}}\hat{\omega}^{- {o}_{pqr}}_{pqr}(-1)^{\hat{\beta}_{pq}\hat{\beta}_{qr}}\\ \nonumber \times &\prod_{{\langle}pqr {\rangle}}{{\hat{Z}}}_{pq}^{\hat{\beta}_{qr}} \prod_{{\langle}pq {\rangle}}{{\hat{X}}}_{pq}^{\hat{\beta}_{pq}}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{pqr}^{\hat{\beta}_{pr}}\prod_p {{\hat{G}}}_p^{\hat{\beta}_p}.\end{aligned}$$ Above, $\hat{\omega}_{pqr}$, $\hat{\beta}_{pq}$, and $\hat{\beta_{p}}$ are defined by $$\begin{aligned} \label{omega} \hat{\omega}_{pqr}|\{g_t\}{\rangle}&= \omega(g_p,g_q,g_r)|\{g_t\}{\rangle}\\ \label{betapq} \hat{\beta}_{pq}|\{ g_t \}{\rangle}&= \beta(g_p,g_q)|\{ g_t \}{\rangle}\\ \label{betap} \hat{\beta}_p |\{ g_t \}{\rangle}&= \beta(g_p,1)|\{ g_t \}{\rangle}.\end{aligned}$$ The local unitary operators in ${{\hat{U}}}_\text{b}^{n_0\nu_0}$ (besides ${{\hat{G}}}_p^{\hat{\beta}_p}$) are then manifestly symmetric due to the homogeneity properties of $\beta$ and $\omega$. Fermionization maps ${{\hat{G}}}_p^{\hat{\beta}_p}$ to the identity, so the finite depth circuit ${{\hat{U}}}_\text{f}^{n_0 \nu_0}$ obtained from fermionization is indeed built from symmetric local unitaries. Hence, ${{\hat{U}}}_\text{f}^{n_0 \nu_0}$ applied to a trivial product state gives us a trivial SPT. Stacking a trivial SPT phase leaves the system in the same phase. Therefore, composition of ${{\hat{U}}}_\text{b}^{n_0 \nu_0}$ with ${{\hat{U}}}_\text{b}^{n \nu}$ should give us a circuit corresponding to some supercohomology data that is equivalent to $(n,\nu)$. According to the composition rules (\[stackingrule\]) in the previous subsection, the product ${{\hat{U}}}_\text{b}^{n_0 \nu_0}{{\hat{U}}}_\text{b}^{n \nu}$ is the circuit corresponding to the supercohomology data[^7] $$\begin{aligned} \label{superrel0} \left(n+\delta \beta, \nu (-1)^{\beta \cup \delta \beta + n \cup \beta + \beta \cup n}\delta \left[\omega(-1)^{n \cup_1 \beta}\right]\right).\end{aligned}$$ $\omega(-1)^{n \cup_1 \beta}$ in (\[superrel0\]) is some homogeneous function, which we will denote as $\eta$, from $G \times G \times G$ to $U(1)$. Therefore, two sets of supercohomology data $(n,\nu)$ and $(n',\nu')$ are equivalent if there exists a homogeneous function $\beta:G \times G\to {\mathbb{Z}}_2$ and homogeneous function $\eta:G \times G \times G\to U(1)$ such that $$\begin{aligned} \label{eq:equiv} n'&=n+\delta \beta \\ \nonumber \nu'&=\nu (-1)^{\beta \cup \delta \beta + n \cup \beta + \beta \cup n}\delta \eta.\end{aligned}$$ It can be checked that this is a symmetric and transitive relation, and hence defines an equivalence relation. In what follows, we will show that two sets of group supercohomology data that are inequivalent with respect to this relation necessarily give rise to distinct SPT phases. Quantized invariants for fermionic SPT phases {#sec:quantizedinvariants} --------------------------------------------- We are now in a position to establish group supercohomology data as quantized invariants for fermionic SPT phases at the level of gapped lattice Hamiltonians. In the previous subsection, two sets of supercohomology data were said to be equivalent if they correspond to the same fermionic SPT phase. Therefore, we need only argue that inequivalent sets of supercohomology data necessarily correspond to distinct fermionic SPT phases. Suppose $(n',\nu')$ and $(n'',\nu'')$ are inequivalent choices of group supercohomology data with respect to the equivalence relation (\[eq:equiv\]). We will show that the corresponding models are in distinct SPT phases. First, we stack the phase corresponding to $(n',\nu')$ with the inverse of the phase corresponding to $(n'',\nu'')$. Then, using the fact that SPT phases form an abelian group under stacking, the two phases will be distinct if and only if $$\begin{aligned} (n,\nu)\equiv(n',\nu')*(n'',\nu'')^{-1}\end{aligned}$$ gives rise to a nontrivial fermionic SPT phase. In other words, we need to demonstrate that $(n,\nu)$ corresponds to a nontrivial phase whenever it is not of the form (\[eq:trivialdata\]). To show that the phase corresponding to $(n,\nu)$ is nontrivial, we bosonize it, i.e. reverse the fermionization procedure described above. This should simply return our bosonic shadow model. However, because the bosonization dictionary is many-to-one, in the sense that all the ${{\hat{G}}}_p$ operators map to the identity on the fermionic side, we have to define our bosonization procedure carefully to avoid ambiguities. We do this by dressing each local term on the bosonic side with a projector onto the ${{\hat{G}}}_p=1$ Hilbert space everywhere in the vicinity of that term and by adding a term $-\sum_p{{\hat{G}}}_p$ to ensure that the ground state is in the ${{\hat{G}}}_p=1$ subspace. It is important to note that this bosonization can be performed for any gapped fermionic Hamiltonian defined on our Hilbert space, not just on our specific fixed point model. Now, having mapped the fermionic SPT Hamiltonian corresponding to $(n,\nu)$ to a bosonic symmetry enriched toric code Hamiltonian, we look for quantized invariants of the symmetry enriched model that can then be pulled back to give fermionic SPT invariants. If $(n,\nu)$ is nontrivial, i.e. not of the form (\[eq:trivialdata\]), then there are two cases. The first is that $n$ cannot be written in the form $n=\delta \beta$ for any choice of $\beta$ ($\beta$ defined below (\[eq:trivialdata\])). The second is that $n$ can be written as $\delta \beta$, but $\nu$ is nontrivial (clarified below). We treat these cases in turn. [**[Case 1:]{}**]{} Assume that $n$ cannot be written as $\delta \beta$. Then, after bosonizing, we will show that the fermion parity flux excitations ($e$ or $m$ excitations of the bosonic shadow model) carry the nontrivial fractionalization class $n \in H^2(G,{\mathbb{Z}}_2)$. Starting with the ground state of the bosonic shadow model $|\Psi_\text{b}{\rangle}$, we can create a pair of $e$ excitations at some well separated vertices $a$ and $b$ by applying a string operator. From this state, a low energy Hilbert space ${\cal{H}}_{\text{L}}$ is obtained by projecting onto fixed values of the $G$-spins $g_a$ and $g_b$ at vertices $a$ and $b$, respectively. ${\cal{H}}_{\text{L}}$ has dimension $|G|^2$, with a natural basis $\{|\Psi^{ee}_\text{b};g_a, g_b{\rangle}\}$. Explicitly, $$\begin{aligned} |\Psi^{ee}_\text{b};g_a, g_b{\rangle}={{\hat{U}}}_\text{b}^{n \nu} |\Psi^{ee}_\text{t.c.};g_a,g_b{\rangle},\end{aligned}$$ where $|\Psi^{ee}_\text{t.c.};g_a,g_b{\rangle}$ is the toric code state consisting of two $e$ excitations at $a$ and $b$ respectively, tensored with a trivial $G$-spin paramagnet on all vertices $p \neq a,b$ and $G$-spins at $a$ and $b$ fixed to $g_a$ and $g_b$, respectively. Letting $\hat{V}(g)$ be the global on-site symmetry operator corresponding to the group element $g$, we now compute $\hat{V}(g)|\Psi^{ee}_\text{b};g_a, g_b{\rangle}$: $$\begin{aligned} \hat{V}(g)|\Psi^{ee}_\text{b};g_a, g_b{\rangle}= \hat{V}(g) {{\hat{U}}}_\text{b}^{n \nu} |\Psi^{ee}_\text{t.c.};g_a,g_b{\rangle}.\end{aligned}$$ Using the fact (proved in Appendix \[ap:symH\]) that ${{\hat{U}}}_\text{b}^{n \nu}$ is symmetric up to factors of ${{\hat{G}}}_p$: $$\begin{aligned} \label{ucircsymmetry} \hat{V}(g){{\hat{U}}}_\text{b}^{n \nu}={{\hat{U}}}_\text{b}^{n \nu} \prod_p {{\hat{G}}}_p^{\hat{n}^g_p} \hat{V}(g),\end{aligned}$$ where $\hat{n}^g_p$ is defined by $$\begin{aligned} \label{neigenvalues} \hat{n}^g_p|\{ g_t \}{\rangle}=n(g_p,1,g)|\{ g_t \}{\rangle},\end{aligned}$$ we have $$\begin{aligned} \label{longeq} \hat{V}(g)|\Psi^{ee}_\text{b};g_a, g_b{\rangle}&= {{\hat{U}}}_\text{b}^{n \nu}\prod_p {{\hat{G}}}_p^{\hat{n}^g_p}|\Psi^{ee}_\text{t.c.};gg_a,gg_b{\rangle}\nonumber \\ &= {{\hat{U}}}_\text{b}^{n \nu} (-1)^{\hat{n}^g_a} (-1)^{\hat{n}^g_b}|\Psi^{ee}_\text{t.c.};gg_a,gg_b{\rangle}\nonumber \\ &=(-1)^{n(gg_a,1,g)+n(gg_b,1,g)} |\Psi^{ee}_\text{b};g g_a,gg_b{\rangle}.\end{aligned}$$ Focusing on just the $a$ vertex, we see from (\[longeq\]) that the local effective action of ${{\hat{V}}}(g)$ near $a$ is given by the operator: $$\begin{aligned} \label{localsymmetry} {{\hat{V}}}_a(g)|\Psi^{ee}_\text{b};g_a,g_b{\rangle}= (-1)^{n(gg_a,1,g)}|\Psi^{ee}_\text{b};gg_a,g_b{\rangle}.\end{aligned}$$ With ${{\hat{V}}}_b(g)$ defined analogously, we recover ${{\hat{V}}}_a(g) {{\hat{V}}}_b(g)={{\hat{V}}}(g)$, as required. Note that there is a $g$ dependent sign ambiguity in the definition of this local effective action. (A possible phase ambiguity is restricted to just an ambiguity in sign by the ${\mathbb{Z}}_2$ fusion rules of the $e$ excitations \[\].) The fractionalization class captures the failure of the symmetry group law to be satisfied by the effective symmetry action on a single anyon. To compute this fractionalization class, we therefore compute the phase difference between ${{\hat{V}}}_a(g) {{\hat{V}}}_a(h)$ and ${{\hat{V}}}_a(gh)$. For ${{\hat{V}}}_a(g) {{\hat{V}}}_a(h)$, we have $$\begin{aligned} \label{VgVh} {{\hat{V}}}_a(g) &{{\hat{V}}}_a(h) |\Psi^{ee}_\text{b};g_a,g_b{\rangle}= \nonumber \\ &= {{\hat{V}}}_a(g) (-1)^{n(hg_a,1,h)}|\Psi^{ee}_\text{b};hg_a,g_b{\rangle}\nonumber \\ &= (-1)^{n(ghg_a,1,g)+n(hg_a,1,h)}|\Psi^{ee}_\text{b};g h g_a,g_b{\rangle},\end{aligned}$$ while for ${{\hat{V}}}_a(gh)$, we have $$\begin{aligned} \label{Vgh} {{\hat{V}}}_a(gh)|\Psi^{ee}_\text{b};g_a,g_b{\rangle}= (-1)^{n(ghg_a,1,gh)} |\Psi^{ee}_\text{b};g h g_a,g_b{\rangle}.\end{aligned}$$ Using $\delta n=0$ and the homogeneity of $n$, we see that the difference in sign between the far right hand side of (\[VgVh\]) and the right hand side of (\[Vgh\]) is precisely $(-1)^{n(1,g,gh)}$. Thus, the fractionalization class of the local symmetry action is indeed given by $n$. Accounting for the $g$ dependent sign ambiguity in the local symmetry action noted just below (\[localsymmetry\]), one can show \[\] that the symmetry fractionalization is well defined with $n \in H^2(G,{\mathbb{Z}}_2)$. The nontrivial symmetry action on the fermion parity fluxes indicates that the bosonic shadow model corresponding to $(n,\nu)$ is in a nontrivial symmetry enriched phase \[\]. Pulling back via bosonization, this implies that the fermionic SPT corresponding to $(n,\nu)$ is nontrivial. Hence, the fermionic SPT phases given by $(n',\nu')$ and $(n'',\nu'')$ are distinct. Alternatively, the nontrivial symmetry fractionalization can be seen more informally by recalling that the shadow model comes from gauging the ${\mathbb{Z}}_2$ subgroup of ${\tilde{G}}$, with ${\tilde{G}}$ the ${\mathbb{Z}}_2$-extension of $G$ determined by $n$. Therefore, the $G$ group law relations close only modulo a ${\mathbb{Z}}_2$ gauge transformation, and the fermion parity flux, being charged under this gauged ${\mathbb{Z}}_2$, acquires minus signs corresponding to the fractionalization class $n$ when acted on by global $G$ symmetry. [**[Case 2:]{}**]{} Now, suppose instead that $n$ is trivial, i.e. $n=\delta \beta$. Then using the equivalence relation (\[eq:equiv\]), we can ‘gauge’ $n$ away entirely, so that the supercohomology data $(n,\nu)$ is equivalent to $(0,\tilde{\nu})$, with $\delta \tilde{\nu}=0$. For $(0,\tilde{\nu})$ to be nontrivial, it must be that there does not exist an $\omega$ (as defined below (\[eq:trivialdata\])) such that $\delta \omega=\tilde{\nu}$. That is to say, $\tilde{\nu}$ must be nontrivial in $H^3(G,U(1))$. The fixed point fermionic circuit ${{\hat{U}}}_\text{f}^{0\tilde{\nu}}$ corresponding to $(0,\tilde{\nu})$ acts trivially on the fermionic degrees of freedom, whereas the portion of it that acts on the bosonic $G$-spin degrees of freedom is precisely the circuit that constructs a group cohomology SPT ground state from a trivial product state \[\]. To see that this system is nontrivial as a fermionic SPT, we bosonize the system. The result is a trivial toric code phase stacked with the bosonic group cohomology phase corresponding to $\tilde{\nu}$. This symmetry enriched toric code is precisely what one obtains from gauging the ${\mathbb{Z}}_2$ subgroup of $G \times {\mathbb{Z}}_2$ in the ordinary bosonic SPT of $G \times {\mathbb{Z}}_2$ with cocycle $\tilde{\nu} \otimes 1$. $\tilde{\nu} \otimes 1$ is nontrivial in $H^3(G\times{\mathbb{Z}}_2,U(1))$ by K[ü]{}nneth’s theorem \[\] and the assumption that $\tilde{\nu}$ is nontrivial. We have thus shown that (\[eq:equiv\]) generates the maximal possible set of equivalence relations on supercohomology data, with inequivalent data necessarily giving rise to distinct phases. A subtle point is that the fermionic phases corresponding to inequivalent sets of supercohomology data $(n',\nu')$ and $(n'',\nu'')$ might still bosonize into the same symmetry enriched toric code phase \[\]. Hence, it was important in the above argument to bosonize the model corresponding to $(n',\nu')*(n'',\nu'')^{-1}$, rather than bosonizing those corresponding to $(n',\nu')$ and $(n'',\nu'')$ individually. This subtlety arises in the the $G={\mathbb{Z}}_2$ example, which we discuss below. Example: $G={\mathbb{Z}}_2$ {#example-gmathbbz_2} --------------------------- For $G={\mathbb{Z}}_2$, we have $$\begin{aligned} {{\hat{U}}}^{\pm}_{\text{b}}=&\prod_{{\langle}pqr {\rangle}} (\pm i)^{o_{pqr}{{\hat{s}}}_p(1-{{\hat{s}}}_q){{\hat{s}}}_r} {{\hat{Z}}}_{pq} ^{(1-{{\hat{s}}}_q){{\hat{s}}}_r} \\ \nonumber \times &\prod_{{\langle}pq {\rangle}}{{\hat{X}}}_{pq}^{(1-{{\hat{s}}}_p){{\hat{s}}}_q}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{ pqr }^{(1-{{\hat{s}}}_p){{\hat{s}}}_r}.\end{aligned}$$ Let us square this circuit. Then the sign in (\[cuponesign\]) is just $+1$, so that, according to (\[compositioncircuit\]), we get $$\begin{aligned} \left({{\hat{U}}}_{\text{b}}^{\pm}\right)^2 = \prod_{{\langle}pqr {\rangle}} \hat{\nu}^{2}_{pqr}.\end{aligned}$$ But $$\begin{aligned} \hat{\nu}^{2}_{pqr}=(-1)^{{{\hat{s}}}_p (1-{{\hat{s}}}_q) {{\hat{s}}}_r}\end{aligned}$$ is just the nontrivial cocycle in $H^3({\mathbb{Z}}_2,U(1))$ evaluated on $(s_p,s_q,s_r,0)$. Therefore the circuit $\left({{\hat{U}}}_{\text{b}}^{\pm}\right)^2$ builds the nontrivial bosonic ${\mathbb{Z}}_2$ SPT \[\]. Thus, stacking two identical copies of either the $+$ or $-$ group supercohomology phase results in the nontrivial bosonic ${\mathbb{Z}}_2$ SPT phase, and in this sense, these group supercohomology phases are ‘square roots’ of the bosonic phase. Note that bosonizing the $+$ and $-$ phases actually results in the same symmetry enriched topological order. Indeed, after gauging the ${\mathbb{Z}}_2$ global symmetry, the resulting twisted ${\mathbb{Z}}_4$ topological orders are the same. This can be seen from the fact that both can be obtained by gauging ${\mathbb{Z}}_4$ in the corresponding auxiliary ${\mathbb{Z}}_4$ bosonic SPTs, and the $3$-cocycles defining these SPTs differ by the generator of $H^3({\mathbb{Z}}_2,U(1))$ pulled back to $H^3({\mathbb{Z}}_4,U(1))$, which is trivial. Thus, the $+$ and $-$ phases cannot be distinguished in this simple way; nevertheless, we know they correspond to distinct fermionic SPT phases by the argument in the previous section. Discussion {#sec:discussion} ========== We have shown how to use group supercohomology data $(n,\nu)$, together with a choice of spin structure on a 2d oriented manifold $M$, to construct a corresponding lattice fermionic SPT Hamiltonian on $M$. Our procedure cleanly disentangles the roles of the supercohomology data and spin structure. The former is used to build a bosonic ‘shadow’ model, and the latter to fermionize this model. Another advantage of our procedure is that it explicitly builds the finite depth circuit of local unitaries ${{\hat{U}}}_{\text{f}}$ that creates the desired fermionic SPT ground state from a product state. Our SPT Hamiltonian is then $$\begin{aligned} \hat{H}_{\text{f}}={{\hat{U}}}_{\text{f}}\hat{H}_{\text{f}}^0{{\hat{U}}}_{\text{f}}^\dag,\end{aligned}$$ where $\hat{H}_{\text{f}}^0$ is the trivial fermionic Hamiltonian - an atomic insulator tensored with a trivial $G$-spin paramagnet whose ground state is a product state with zero fermion occupancy. Key to this approach is the fact that the circuit ${{\hat{U}}}_{\text{f}}$ is $G$-symmetric. This is the case despite the fact that the individual local unitaries that make up the circuit cannot all be $G$-symmetric, for otherwise the fermionic SPT would be trivial. Note that while we have assumed that the global action of $G$ is unitary, we expect our construction to generalize to anti-unitary symmetries with only minor modifications. Our commuting projector Hamiltonians suffice to show that the supercohomology phases protected by abelian groups \[\] can be many-body localized. The couplings in the Hamiltonian[^8] $\hat{H}^0_{\text{f}}$ can be disordered (or made quasi-periodic), leading to a many-body localized Hamiltonian \[\], or at least one that has a long thermalization time scale. Since $\hat{H}_{\text{f}}$ is just the conjugate of $\hat{H}^0_{\text{f}}$ by a finite depth circuit, the same is true of $\hat{H}_{\text{f}}$. The fermionization duality that was used to construct our zero correlation length lattice models can also be reversed and used to bosonize fermionic SPT Hamiltonians. This, together with a reinterpretation of the stacking structure of SPT phases in terms of composition of the corresponding finite depth circuits, allows well established invariants of the bosonic symmetry enriched toric code to be pulled back to these fermionic SPT Hamiltonians. The result is a classification of fermionic supercohomology SPT phases, with inequivalent supercohomology data necessarily defining distinct phases. One may ask whether a similar construction is possible for the so-called beyond supercohomology phases \[\] in $2+1$D. That is, for these phases, can an exactly solvable model be obtained by conjugating a trivial fermionic Hamiltonian by a symmetric finite depth circuit of local unitaries? We argue that, in contrast to the supercohomology phases, the answer is no. In particular, [we claim that]{} the ground states of beyond supercohomology phases cannot be constructed from a trivial product state by applying a globally symmetric finite depth circuit of local unitaries. See Appendix \[beyondsuper\] for further discussion. It is worth discussing the relation of our work to previous work. Supercohomology models were introduced in the pioneering work of Ref. \[\], where wave functions for these models were written from a lattice path integral. However, the wave functions were only explicitly constructed on a specific planar lattice and required seemingly arbitrary choices to account for a spin structure. In Ref. \[\], a related wave function, for the so-called fermionic toric code, was written down; this is the topological order that would result from gauging the global ${\mathbb{Z}}_2$ in our $G={\mathbb{Z}}_2$ models. The ground states were defined by graphical rules, but again, the spin structure was encoded in these rules in a non-manifest way. The roles of the spin structure and group supercohomology data were disentangled in Ref. \[\], but only in a lattice spacetime formalism. Ref. \[\] extended this to beyond-supercohomology models, and also made the connection between the supercohomology data and the algebraic data defining the shadow models. Insofar as lattice Hamiltonians, Refs. \[, \] clarified the role of spin structure in beyond-supercohomology models, and Ref. \[\] extended this to include supercohomology models; however, Ref. \[\] did not write down explicit Hamiltonians, but rather defined the ground states implicitly using certain self-consistent lattice-deforming local rules. The present work builds on these developments by constructing explicit Hamiltonians, as well as building the ground states explicitly using finite depth circuits, on oriented $2d$ manifolds of any topology. It uses in an essential way the 2+1D bosonization duality introduced in Ref. \[\]. There are many possible avenues for future work. One would be to extend this formalism to group supercohomology models in three spatial dimensions. Another avenue is to extend the present formalism to more complicated groups than $G \times {\mathbb{Z}}_2^{\text{f}}$, such as ones where the fermion parity symmetry forms a nontrivial subgroup of the overall symmetry. Yet another possibility is to extend the quantum circuit formalism to beyond-supercohomology models, both in two and three spatial dimensions. It may also be fruitful to understand our work in terms of tensor network states and operators. Indeed, preliminary investigations suggest that the bosonization duality can naturally be interpreted as a tensor network operator. It would then be nice to understand the relation between the present work and the fermionic models written down in Ref. \[\]. [Futher]{}, our finite depth circuits could be used to study the edge theories of these fermionic SPT phases. Finally, it would be interesting to study the classification of symmetry enriched phases using finite depth circuits applied to the ground states of fixed point Hamiltonians. The circuits ${{\hat{U}}}_\text{b}$, introduced in section \[shadowhamiltonian\], construct ground states of symmetry enriched toric code phases from a trivial toric code state, and thus provide a nontrivial realization of such a construction. [*Acknowledgements –* ]{} We are grateful to Sujeet Shukla, Anton Kapustin, Frank Verstraete, Max Metlitski, Dan Freed, Ryan Thorngren, Dave Aasen, and especially Ashvin Vishwanath for useful conversations. LF is supported by NSF DMR-1519579. Derivation of the bosonic shadow theory ground state and the ‘standard’ parent Hamiltonian {#shadowgsderivation} ========================================================================================== In this appendix, we provide a derivation of the bosonic shadow theory ground state $\Psi_{\text{b}}$ introduced in section \[tGSPT\]. Recall that the first step of the construction is to form an auxiliary ${\tilde{G}}$ bosonic SPT from a choice of normalized[@Note1] supercohomology data ($n$,$\nu$), where ${\tilde{G}}$ is a ${\mathbb{Z}}_2$ extension of $G$ by $n$. The next step is to ‘gauge’ the ${\mathbb{Z}}_2$ subgroup of ${\tilde{G}}$ in the standard way by minimally coupling the SPT Hamiltonian to a ${\mathbb{Z}}_2$ gauge field. We will implement this procedure explicitly and argue that the symmetry enriched Hamiltonian obtained from this procedure - which we refer to as the ‘standard’ symmetry enriched Hamiltonian - can in principle be fermionized since it commutes with the modified Gauss’s law ${{\hat{G}}}_p$ for all $p$. Gauging the ${\mathbb{Z}}_2 \subset {\tilde{G}}$ ------------------------------------------------ As stated in (\[gsSPT\]), the auxiliary ${\tilde{G}}$ SPT ground state wave function in the configuration basis is $$\begin{aligned} \label{gsagain} \Psi_{\text{SPT}}\Big(\Big\{g^{(m_p)}_p\Big\}\Big)= \prod_{{\langle}pqr {\rangle}}\alpha\Big(g^{(m_p)}_p,g^{(m_q)}_q,g^{(m_r)}_r,{{1}}\Big)^{o_{pqr}}.\end{aligned}$$ (Recall that $\alpha$ can be expressed in terms of $n$, $\nu$, and ${\epsilon}$ using (\[def\_alpha\]).) A simple Hamiltonian ${{\hat{H}_{SPT}}}$ with this ground state is $$\begin{aligned} \label{HZ4} {{\hat{H}_{SPT}}}= - {{\hat{U}}}_{\text{SPT}} \left(\sum_p \hat{\tilde{P}}^{\text{sym}}_p \right) {\hat U}^{\dag}_{\text{SPT}}\end{aligned}$$ where ${{\hat{U}}}_{\text{SPT}}$ is the finite depth circuit of local unitaries defined by matrix elements $$\begin{aligned} \label{eq:UtG} \big {\langle}& \big\{ h^{(\ell_p)}_p \big\}\big| {{\hat{U}}}_{\text{SPT}} \big|\big\{g^{(m_p)}_p \big\} \big {\rangle}= \\ \nonumber &\delta_{\big\{ h^{(\ell_p)}_p \big\},\big\{g^{(m_p)}_p \big\}} \prod_{{\langle}pqr {\rangle}} \alpha\big(g^{(m_p)}_p, g^{(m_q)}_q, g^{(m_r)}_r, {{1}}\big)^{o_{pqr}},\end{aligned}$$ and $\hat{\tilde{P}}^{\text{sym}}_p$ is the projector onto the symmetric state at vertex $p$ $$\begin{aligned} \frac{1}{\sqrt{|{\tilde{G}}|}}\sum_{g^{(m_p)}_p\in {\tilde{G}}} \big|g^{(m_p)}_p\big{\rangle}\end{aligned}$$ tensored with the identity on the remaining sites. We gauge the ${\mathbb{Z}}_2$ subgroup of ${\tilde{G}}$ using the usual algorithm as described in Ref. \[\] and Ref. \[\]. First, we introduce at each link ${\langle}pq {\rangle}$ a spin-$\frac{1}{2}$ Hilbert space with Pauli operators $\mu^x_{pq}$ and $\mu^z_{pq}$, and at all sites $p$, impose the gauge constraint $$\begin{aligned} \label{eq:gauge1} \prod_{{\langle}st {\rangle}\ni p} \hat{\mu}_{st}^x = \hat{e}_p^{{\mathbb{Z}}_2}.\end{aligned}$$ Here, the product runs over all links starting or ending on $p$, and ${\hat{e}}_p^{{\mathbb{Z}}_2}$ is the operator that on vertex $p$ takes $$\begin{aligned} \label{def_gpz2} \big|g^{(m_p)}_p \big{\rangle}\rightarrow \big|g^{(m_p+1)}_p\big{\rangle}\end{aligned}$$ and acts as the identity on all other sites. In other words, the action of ${\hat{e}}_p^{{\mathbb{Z}}_2}$ in the configuration basis is multiplication by the generator of the ${\mathbb{Z}}_2$ subgroup, $1^{(1)}$, with the assumption that $n$ is normalized. Second, we minimally couple each term in (\[HZ4\]) to the ${\mathbb{Z}}_2$ gauge field degrees of freedom. In order to make this gauging procedure unambiguous, we multiply each term by a projector onto trivial ${\mathbb{Z}}_2$ flux on triangles in the vicinity of that term, and add the term $$\begin{aligned} \label{eq:noflux} -J\sum_{{\langle}pqr {\rangle}} \hat{\mu}^z_{pq}\hat{\mu}^z_{qr}\hat{\mu}^z_{pr}\end{aligned}$$ with $J$ large enough to ensure that the ground state is in the trivial ${\mathbb{Z}}_2$ flux sector. The result is a Hamiltonian $\hat{H}_\text{gauged}$ which is invariant under the gauge constraints in (\[eq:gauge1\]). A ground state of $\hat{H}_\text{gauged}$ can be written as $$\begin{aligned} \label{gsgauge} &\Psi_{\text{gauged}}\big(\big\{g^{(m_p)}_p\big\},\{ \mu^z_{pq}\}\big)= \\ \nonumber &\prod_{{\langle}pqr {\rangle}} \nu(g_p,g_q,g_r,{{1}})^{o_{pqr}} \left(\mu^z_{pq}(-1)^{{\epsilon}\big(\big(g^{(m_p)}_p\big)^{-1} g^{(m_q)}_q\big)} \right)^{n(g_q,g_r,{{1}})} \\ \nonumber &\times \left( \prod_{{\langle}pqr {\rangle}} \delta_{W'_{pqr},1} \right ) h(\{\mu^z_{pq}\}).\end{aligned}$$ The function $h$ determines the holonomy of the particular ground state. The ground state with trivial holonomy, for example, is obtained with the choice of $h$: $$\begin{aligned} h(\{\mu^z_{pq}\})= \begin{cases} 1 & \text{if $\{\mu^z_{pq}\}\sim \{\mu^z_{pq}=+1\}$} \\ 0 & \text{otherwise} \end{cases}\end{aligned}$$ where $\sim$ means ‘gauge equivalent to’. For the ground states with nontrivial holonomy, $h$ is defined similarly. The $\delta$ function in (\[gsgauge\]) is a consequence of the flux penalizing term in the gauging procedure. The $\mu^z$-flux is $$\begin{aligned} W'_{pqr}=\mu^z_{pq}\mu^z_{qr}\mu^z_{pr},\end{aligned}$$ so the delta function $$\begin{aligned} \delta_{W'_{pqr},1}= \begin{cases} 1 & \text{if $W'_{pqr}=1$}\\ 0 & \text{otherwise} \end{cases}\end{aligned}$$ ensures that all configurations in the ground states have trivial $\mu^z$-flux. In going from (\[gsagain\]) to (\[gsgauge\]), we have also multiplied $(-1)^{{\epsilon}\big(\big(g^{(m_p)}_p\big)^{-1} g^{(m_q)}_q\big)}$ by $\mu_{pq}^z$. This guarantees that $\Psi_{\text{gauged}}$ is gauge invariant and reduces to $\Psi_\text{SPT}$ when all $\mu^z_{pq}=+1$. Mapping to unconstrained variables ---------------------------------- To obtain $\Psi_{\text{b}}$ as expressed in section \[tGSPT\], we must rewrite the system in terms of unconstrained variables. To this end, we define an isomorphism of operator algebras below. This isomorphism will allow us to convert $\hat{H}_\text{gauged}$ into $\hat{H}'_\text{b}$, a Hamiltonian acting on an unconstrained Hilbert space with ground state $\Psi_{\text{b}}$. On one side of the isomorphism, we have the algebra $\cal{A}_\text{constrained}$ appearing in the previous subsection and consisting of gauge invariant operators, modulo the Gauss’s law relation. On the other side of the isomorphism, we have $\cal{A}_\text{unconstrained}$, an operator algebra naturally represented on a tensor product Hilbert space with degrees of freedom matching those of the bosonic shadow theory. We now define $\cal{A}_\text{constrained}$ and $\cal{A}_\text{unconstrained}$ more carefully and write an explicit isomorphism between the two algebras. ### Algebra of constrained operators $\cal{A}_\text{constrained}$ $\cal{A}_\text{constrained}$ admits a representation on the Hilbert space discussed in the previous subsection, i.e. ${\tilde{G}}$ degrees of freedom on vertices and spin-$\frac{1}{2}$ degrees of freedom on links. It can generated by $\hat{e}_p^{{\mathbb{Z}}_2}$, ${{\hat{g}}}^{(0)}_p$, $\hat{\tilde{P}}_p^{g}$, $\hat{\mu}^x_{pq}$, and $(-1)^{\hat{{\epsilon}}_{pq}}\hat{\mu}^z_{pq}$ obeying the relation $$\begin{aligned} \label{gaugerelations} \prod_{{\langle}st {\rangle}\ni p} \hat{\mu}_{st}^x = \hat{e}_p^{{\mathbb{Z}}_2}\end{aligned}$$ for all $p$. Here, ${{\hat{g}}}^{(0)}_p$ and $\hat{\tilde{P}}_p^{g}$ are the defined by their action on a configuration state: $$\begin{aligned} {{\hat{g}}}^{(0)}_p\big|h^{(\ell_p)}_p\big{\rangle}=\big|(gh)^{(\ell_p)}_p\big{\rangle}\end{aligned}$$ and $$\begin{aligned} \hat{\tilde{P}}_p^{g}\big|\big\{h^{(\ell_q)}_q\big\}\big{\rangle}=\delta_{g_p,h_p}\big|\big\{h^{(\ell_q)}_q\big\}\big{\rangle}.\end{aligned}$$ In words, ${{\hat{g}}}^{(0)}_p$ is the operator that multiplies by ${g^{(0)}}$ at vertex $p$ and acts as the identity elsewhere, while $\hat{\tilde{P}}_p^{g}$ is the projector onto the subspace spanned by states with configuration $g^{(0)}$ or $g^{(1)}$ at vertex $p$. Finally, $\hat{{\epsilon}}_{pq}$, appearing in the generator $(-1)^{\hat{{\epsilon}}_{pq}}\hat{\mu}^z_{pq}$, is given by $$\begin{aligned} \hat{{\epsilon}}_{pq}\big|\big\{g^{(m_r)}_r\big\}\big{\rangle}={\epsilon}\big(\big(g^{(m_p)}_p\big)^{-1}g^{(m_q)}_q\big)\big|\big\{g^{(m_r)}_r\big\}\big{\rangle}.\end{aligned}$$ It can be checked that products of ${{\hat{g}}}^{(0)}_p$, $\hat{\tilde{P}}_p^{g}$ $\hat{\mu}^x_{pq}$, and $(-1)^{\hat{{\epsilon}}_{pq}}\hat{\mu}^z_{pq}$ span all gauge invariant operators. ### Algebra of unconstrained operators $\cal{A}_\text{unconstrained}$ We will represent $\cal{A}_\text{unconstrained}$ on a tensor product Hilbert space comprised of $G$ degrees of freedom $|g_p{\rangle}$ on vertices and spin-$\frac{1}{2}$ degrees of freedom on links. The generators of this operator algebra acting on vertex Hilbert spaces are ${{\hat{g}}}_p$ and ${{\hat{P}}}_p^g$ defined by $$\begin{aligned} {{\hat{g}}}_p|h_p{\rangle}=|(gh)_p{\rangle}\end{aligned}$$ and $$\begin{aligned} {{\hat{P}}}_p^g|\{ h_q \}{\rangle}=\delta_{g_p,h_p}|\{ h_q \}{\rangle}.\end{aligned}$$ We take generators acting on the link Hilbert spaces to be the Pauli operators ${{\hat{X}}}_{pq}$ and ${{\hat{Z}}}_{pq}$. ### Isomorphism of $\cal{A}_\text{constrained}$ with $\cal{A}_\text{unconstrained}$ An isomorphism of $\cal{A}_\text{constrained}$ with $\cal{A}_\text{unconstrained}$ is given by the map of generators: $$\begin{aligned} \label{constrainedisomorphism} \begin{split} \hat{e}^{{\mathbb{Z}}_2}_p &\longleftrightarrow \prod_{{\langle}st {\rangle}\ni p}{{\hat{X}}}_{st}\\ {{\hat{g}}}^{(0)}_p &\longleftrightarrow {{\hat{g}}}_p \prod_{\substack{{\langle}tq {\rangle}\\ t=p}}{{\hat{X}}}_{tq}^{\hat{\xi}^{g_p}_{tq}}\prod_{\substack{{\langle}qt {\rangle}\\ t=p}}{{\hat{X}}}_{qt}^{\hat{\xi}^{g_p}_{qt}} \\ \hat{\tilde{P}}_p^{g}&\longleftrightarrow {{\hat{P}}}_p^g \\ \hat{\mu}^x_{pq} &\longleftrightarrow {{\hat{X}}}_{pq} \\ (-1)^{\hat{{\epsilon}}_{pq}}\hat{\mu}_{pq}^z&\longleftrightarrow {{\hat{Z}}}_{pq} \end{split}\end{aligned}$$ where $\hat{\xi}^{g_p}_{pq}$ and $\hat{\xi}^{g_p}_{qp}$ are defined by $$\begin{aligned} {{\hat{g}}}^{(0)}_p(-1)^{\hat{{\epsilon}}_{pq}}=(-1)^{\hat{\xi}^{g_p}_{pq}}(-1)^{\hat{{\epsilon}}_{pq}}{{\hat{g}}}^{(0)}_p\end{aligned}$$ and $$\begin{aligned} {{\hat{g}}}^{(0)}_p(-1)^{\hat{{\epsilon}}_{qp}}=(-1)^{\hat{\xi}^{g_p}_{qp}}(-1)^{\hat{{\epsilon}}_{qp}}{{\hat{g}}}^{(0)}_p.\end{aligned}$$ Explicitly, $\hat{\xi}^{g_p}_{pq}$ and $\hat{\xi}^{g_p}_{qp}$ act on configuration states as $$\begin{aligned} &\hat{\xi}^{g_p}_{pq}|\{ h_t \}{\rangle}=\\ \nonumber &(n(1,h_p,h_q)+n(1,g_p,g_ph_p)+n(1,g_ph_p,h_q))|\{ h_t \} {\rangle}\end{aligned}$$ and $$\begin{aligned} &\hat{\xi}^{g_p}_{qp}|\{ h_t \}{\rangle}=\\ \nonumber &(n(1,h_q,h_p)+n(1,g_p,g_ph_p)+n(1,h_q,g_ph_p))|\{ h_t \} {\rangle}.\end{aligned}$$ $\hat{\xi}^{g_p}_{pq}$ and $\hat{\xi}^{g_p}_{qp}$ defined in this way ensure that the commutation relations exhibited by ${{\hat{g}}}^{(0)}_p$ and $(-1)^{\hat{{\epsilon}}_{pq}}\hat{\mu}_{pq}^z$ are mirrored on the right hand side of the mapping (\[constrainedisomorphism\]). Note that the isomorphism is well defined since, for all $p$, the relation $$\begin{aligned} \prod_{{\langle}st {\rangle}\ni p} \hat{\mu}_{st}^x = \hat{e}_p^{{\mathbb{Z}}_2}\end{aligned}$$ is mapped to the identity. Now, given a system described in terms of the operators in $\cal{A}_\text{constrained}$, one can rewrite it as a system in terms of the operators belonging to $\cal{A}_\text{unconstrained}$. In particular, we can apply this isomorphism to $\hat{H}_\text{gauged}$ to obtain $\hat{H}'_\text{b}$ acting on an unconstrained Hilbert space. $\hat{H}'_\text{b}$ has the ground state $$\begin{aligned} &\Psi_{\text{b}}(\{g_p\},\{ Z_{pq}\})= \\ \nonumber &\prod_{{\langle}pqr {\rangle}} \nu^{o_{pqr}}(g_p,g_q,g_r,{{1}}) Z_{pq}^{n(g_q,g_r,{{1}})} \\ \nonumber \times &\left( \prod_{{\langle}pqr {\rangle}} \delta_{W_{pqr},(-1)^{n(g_p,g_q,g_r)}} \right) h(\{ Z_{pq}(-1)^{n(1,g_p,g_q)}\}),\end{aligned}$$ which is precisely the ground state of the bosonic shadow theory identified in section \[tGSPT\]. Fermionizability of ‘standard’ Hamiltonian ------------------------------------------ To conclude this appendix, we prove that ${{\hat{H}'_{\text{b}}}}$ is fermionizable, i.e. ${{\hat{H}'_{\text{b}}}}$ commutes with the modified Gauss’s law ${{\hat{G}}}_p$ for all sites $p$. To do this, we first note that $\hat{H}_\text{SPT}$ in (\[HZ4\]) commutes with $$\begin{aligned} \label{eq:Z2flip} {{\hat{U}}}_{\text{SPT}} {\hat{e}}_p^{{\mathbb{Z}}_2} {\hat U}^{\dag}_{\text{SPT}}\end{aligned}$$ which follows simply from the fact that ${\hat e}_p^{{\mathbb{Z}}_2}$ commutes with ${\hat P}^{\text{sym}}_p$. After gauging the ${\mathbb{Z}}_2$ subgroup of ${\tilde{G}}$, ${{\hat{H}'_{\text{b}}}}$ must commute with the gauged version of (\[eq:Z2flip\]), which, using the definition of $\alpha$ along with $\delta n=0$ we find to be equal to $$\begin{aligned} \label{eq:term2} \prod_{\substack{{\langle}tqr {\rangle}\\ t=p}} (-1)^{\hat{n}_{pqr}} \prod_{{\langle}st {\rangle}\ni p} {\hat X}_{st}.\end{aligned}$$ Next, we see that (\[eq:noflux\]), the term penalizing $\mu^z$-flux in triangle ${\langle}pqr {\rangle}$, turns into $$\begin{aligned} \label{eq:term1} J(-1)^{\hat{n}_{pqr}} {\hat{Z}}_{pq} {\hat{Z}}_{qr} {\hat{Z}}_{pr}. \end{aligned}$$ when written in terms of the unconstrained variables (here we used the definition of the group law of ${\tilde{G}}$ to simplify the expression). Then, multiplying the operator in (\[eq:term2\]) by the product of the terms in (\[eq:term1\]) over all triangles whose first vertex is $p$ yields an operator proportional to $$\begin{aligned} \prod_{\substack{{\langle}tqr {\rangle}\\ t=p}} {{\hat{W}}}_{tqr} \prod_{{\langle}st {\rangle}\ni p} {\hat X}_{st},\end{aligned}$$ which is just ${{\hat{G}}}_p$, as defined in (\[def:modifiedgausslaw\]). Since the Hamiltonian ${{\hat{H}'_{\text{b}}}}$ commutes with both (\[eq:term1\]) and (\[eq:term2\]), it must commute with ${\hat G}_p$ as well. Thus ${{\hat{H}'_{\text{b}}}}$ is fermionizable. However, explicitly fermionizing it is unwieldy in general, as we do not have an explicit expression for it in terms of the modified Gauss’s law invariant operators ${{\hat{W}}}_{pqr}$ and ${{\hat{U}}}_{pq}$. For this reason, we constructed and worked with the parent Hamiltonian $\hat{H}_\text{b}$ in section \[shadowhamiltonian\] instead. Symmetry of the shadow model Hamiltonian {#ap:symH} ======================================== In this appendix we prove that $\hat{H}_{\text{b}}$ is $G$-symmetric, as claimed in section \[shadowhamiltonian\]. It follows that the fermionic SPT Hamiltonian $\hat{H}_{\text{f}}$ constructed in section \[sec:fSPT\] is also $G$-symmetric, since the fermionization procedure commutes with the global $G$-symmetry action. Concretely, letting $\hat{V}(g)$ be the global symmetry operator representing $g \in G$ and acting as $$\begin{aligned} |g_p{\rangle}\rightarrow |gg_p{\rangle}\end{aligned}$$ on every vertex degree of freedom, we will show that $\hat{V}(g)$ commutes with $\hat{H}_{\text{b}}$. Recall that $$\begin{aligned} \hat{H}_{\text{b}} = {{\hat{U}}}_{\text{b}}\hat{H}^0_{\text{b}} {{\hat{U}}}^\dag_{\text{b}}\end{aligned}$$ so that $$\begin{aligned} \label{conjH} \hat{V}(g) \hat{H}_{\text{b}}\hat{V}^\dag(g)=\left(\hat{V}(g) {{\hat{U}}}_{\text{b}}\hat{V}^\dag(g)\right) \hat{H}^0_{\text{b}} \left(\hat{V}(g) {{\hat{U}}}^\dag_{\text{b}}\hat{V}^\dag(g)\right)\end{aligned}$$ where we have used that $\hat{H}^0_{\text{b}}$ is symmetric. Let us now compute $\hat{V}(g) {{\hat{U}}}_{\text{b}}\hat{V}^\dag(g)$. In (\[ucirc\]), ${{\hat{U}}}_{\text{b}}$ was defined as $$\begin{aligned} {{\hat{U}}}_{\text{b}}=\prod_{{\langle}pqr {\rangle}} \left(\hat{\nu}_{pqr}^{o_{pqr}} {{\hat{Z}}}_{pq} ^{\hat{n}_{qr}}\right) \prod_{{\langle}pq {\rangle}}{{\hat{X}}}_{pq}^{\hat{n}_{pq}}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{ pqr }^{\hat{n}_{pr}},\end{aligned}$$ with the operators $\hat{\nu}_{pqr}^{o_{pqr}}$ and $\hat{n}_{pr}$ defined just below (\[ucirc\]). To proceed, it is useful to first re-express ${{\hat{U}}}_{\text{b}}$ in terms of the generators of the bosonic algebra ${\cal A}_{\text{bos}}$. Following (\[Uedgesprod\]), the result is $$\begin{aligned} {{\hat{U}}}_{\text{b}}= {\hat{\kappa}} \prod_{{\langle}pq {\rangle}}{{\hat{U}}}_{pq}^{\hat{n}_{pq}}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{pqr}^{\hat{n}_{pr}}.\end{aligned}$$ Here, $\kappa$ is a unitary operator that acts as multiplication by a $\{|g_p{\rangle}\}$-dependent eigenvalue and whose explicit form will not be required. Conjugating by $\hat{V}(g)$ gives $$\begin{aligned} \label{conjugated} \hat{V}(g) {{\hat{U}}}_{\text{b}}\hat{V}^\dag(g)=\hat{\phi}\hat{\kappa}\prod_{{\langle}pq {\rangle}}{{\hat{U}}}_{pq}^{\hat{n}^g_{pq}}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{pqr}^{\hat{n}^g_{pr}}.\end{aligned}$$ Here $\hat{\phi}$ and $\hat{n}^g_{pq}$ are operators that act as multiplication by a $\{|g_p{\rangle}\}$-dependent eigenvalue. $\hat{\phi}$ is unitary and its explicit form will again not be required, whereas the eigenvalue of $\hat{n}^g_{pq}$ is $n(g^{-1}g_p,g^{-1}g_q,{{1}})=n(g_p,g_q,g)$. The cocycle condition $\delta n=0$ gives $$\begin{aligned} \label{somens} n(g_p,g_q,g) = n(g_p,{{1}},g)+n(g_q,{{1}},g)+n(g_p,g_q,{{1}}).\end{aligned}$$ Thus, $\hat{n}^g_{pq}$ decomposes into three diagonal operators corresponding to the terms in (\[somens\]), i.e. $$\begin{aligned} \hat{n}^g_{pq}=\hat{n}^g_p+\hat{n}^g_q+\hat{n}_{pq}.\end{aligned}$$ If we substitute for $\hat{n}^g_{pq}$ and do some rearranging, the right hand side of (\[conjugated\]) becomes $$\begin{aligned} \label{conjugated2} \hat{\phi}' {{\hat{U}}}_{\text{b}} \prod_{{\langle}pq {\rangle}}{{\hat{U}}}_{pq}^{\hat{n}^g_p+\hat{n}^g_q}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{pqr}^{\hat{n}^g_p+\hat{n}^g_r}\end{aligned}$$ where, again, $\hat{\phi}'$ multiplies by a $\{|g_p{\rangle}\}$-dependent eigenvalue whose precise form will not be required. It is a combination of $\hat{\phi}$ and a phase picked up in commuting the ${{\hat{U}}}_{pq}$ operators. Next, the product of ${{\hat{U}}}_{pq}$ in (\[conjugated2\]) can be re-organized so that (\[conjugated2\]) is $$\begin{aligned} \hat{\phi}'' {{\hat{U}}}_{\text{b}} \prod_p\left(\prod_{\substack{{\langle}tq {\rangle}\\ t=p}} {{{\hat{U}}}}_{tq} \prod_{\substack{{\langle}qt {\rangle}\\ t=p}} {{{\hat{U}}}}_{qt}\right)^{\hat{n}^g_p} \prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{pqr}^{\hat{n}^g_p+\hat{n}^g_r}.\end{aligned}$$ $\hat{\phi}''$ is yet another diagonal operator in the configuration basis. Employing the identity (\[bosidentity\]), we thus find that $\hat{V}(g) {{\hat{U}}}_{\text{b}}\hat{V}^\dag(g)$ is equal to $$\begin{aligned} \label{justastep} \hat{\phi}'' {{\hat{U}}}_{\text{b}} \prod_p \left({{\hat{G}}}_p \prod_{\substack{{\langle}tqr {\rangle}\\ t=p}} {{\hat{W}}}_{tqr} \prod_{\substack{{\langle}qrt {\rangle}\\ t=p}} {{\hat{W}}}_{qrt} \right)^{\hat{n}^g_p} \prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{pqr}^{\hat{n}^g_p+\hat{n}^g_r}.\end{aligned}$$ The flux operators ${{\hat{W}}}_{pqr}$ in (\[justastep\]) cancel, so we conclude $$\begin{aligned} \label{almostthere} \hat{V}(g){{\hat{U}}}_{\text{b}}\hat{V}^\dag(g)= \hat{\phi}''(\{g_p\}) {{\hat{U}}}_{\text{b}}\prod_p {{\hat{G}}}_p^{\hat{n}^g_p}.\end{aligned}$$ Next, we show that $\hat{\phi}''$ in (\[almostthere\]) must be ${{1}}$. Let us denote the ground state of $\hat{H}_{\text{b}}^0$ with trivial holonomy by $|\Psi_{\text{b}}^0{\rangle}$. It is a tensor product of trivial symmetric states $$\begin{aligned} \frac{1}{|G|} \sum_{g_p} |g_p{\rangle}\end{aligned}$$ at vertices $p$ with the trivial holonomy toric code ground state for the $Z_{pq}$ degrees of freedom. The latter is just a superposition of all trivial holonomy $Z_{pq}$ configurations with trivial ${\mathbb{Z}}_2$-flux $W_{pqr}$ at every triangle. We then have the following chain of equalities $$\begin{aligned} \label{punchline} |\Psi_{\text{b}}^0{\rangle}&=\hat{V}(g)|\Psi_{\text{b}}^0{\rangle}\\ \nonumber &=\hat{V}(g){{\hat{U}}}_{\text{b}}{}^\dag{{\hat{U}}}_{\text{b}}|\Psi_{\text{b}}^0{\rangle}\\ \nonumber &=\hat{V}(g){{\hat{U}}}_{\text{b}}{}^\dag\hat{V}^\dag(g){{\hat{U}}}_{\text{b}}|\Psi_{\text{b}}^0{\rangle}\\ \nonumber &=\hat{\phi}''^*{{\hat{U}}}_{\text{b}}{}^\dag{{\hat{U}}}_{\text{b}}|\Psi_{\text{b}}^0{\rangle}\\ \nonumber &=\hat{\phi}''^*|\Psi_{\text{b}}^0{\rangle}.\end{aligned}$$ In the first equality we used that $|\Psi_{\text{b}}^0{\rangle}$ is symmetric, and in the third equality we used that ${{\hat{U}}}_{\text{b}}|\Psi_{\text{b}}^0{\rangle}$ is symmetric. The fourth equality uses (\[almostthere\]) and the fact that ${{\hat{U}}}_{\text{b}}|\Psi_{\text{b}}^0{\rangle}$ belongs to the ${{\hat{G}}}_p={{1}}$ eigenspace for all $p$. Now, comparing the far left hand side and the far right hand side of (\[punchline\]), we can see that $\hat{\phi}''$ is trivial as follows. $\hat{\phi}''$ is a diagonal operator in the configuration basis, while $|\Psi_{\text{b}}^0{\rangle}$ contains an equal amplitude superposition over all $G$ configurations at vertices. For the equality to hold, it must be that $\hat{\phi}''$ has eigenvalue $1$ on all configurations. Hence, $\hat{\phi}''={{1}}$. Looking back at (\[almostthere\]), we therefore have $$\begin{aligned} \label{conjU} \hat{V}(g){{\hat{U}}}_{\text{b}}\hat{V}^\dag(g)= {{\hat{U}}}_{\text{b}}\prod_p {{\hat{G}}}_p^{\hat{n}^g_p}.\end{aligned}$$ Substituting (\[conjU\]) into (\[conjH\]) and using the fact that $\hat{H}^0_{\text{b}}$ commutes with ${{\hat{G}}}_p$ to cancel the factors of ${{\hat{G}}}_p$, we find $$\begin{aligned} \hat{V}(g) \hat{H}_{\text{b}}\hat{V}^\dag(g)=\hat{H}_{\text{b}}.\end{aligned}$$ Therefore, $\hat{H}_{\text{b}}$ is symmetric, and $\hat{H}_{\text{f}}$ is symmetric since fermionization commutes with the global $G$ symmetry. Graphical interpretation of spin structure dependent relation {#ap:cpproof} ============================================================= Here we prove that $c(p)$, as defined in (\[def:cp\]) and restated here for convenience: $$\begin{aligned} \prod_{\substack{{\langle}tq {\rangle}\\ t=p}} {\hat S'}_{tq} \prod_{\substack{{\langle}qt {\rangle}\\ t=p}} {\hat S'}_{qt} = c(p) \prod_{\substack{{\langle}tqr {\rangle}\\ t=p}} (-1)^{{\hat F}_{tqr}} \prod_{\substack{{\langle}qrt {\rangle}\\ t=p}} (-1)^{{\hat F}_{qrt}},\end{aligned}$$ is proportional to the identity operator, and equal to $\pm 1$ depending on whether the interpolating vector field ${\cal V}$, illustrated in FIG.\[fig:vecfieldorientation\] has an even or odd winding number about $p$. To see this, we first examine the two types of links around $p$. There are links that are oriented towards $p$ and links that are oriented away from $p$. These two types of links form domains as seen in FIG. \[fig:S\_eordering\]. Domains of outward pointing links are separated from domains of inward pointing links by a triangle ${{{\langle}qpr {\rangle}}}$ where $p$ is the second vertex in the ordering. There are necessarily an even number of triangles around $p$ for which $p$ is the second vertex. We will call these types of triangles ${{{\langle}qpr {\rangle}}}$-triangles, and we will think of them in pairs – the two ${{{\langle}qpr {\rangle}}}$-triangles on either side of an inward pointing domain forming a pair. Moving counter-clockwise around $p$, we see that each pair results in a $2\pi$ clockwise rotation of the vector field ${\cal V}$, relative to the outward normal. Without any ${{{\langle}qpr {\rangle}}}$-triangles, all the links are oriented towards $p$ or they are all oriented away from $p$, and the vector field rotates by $2\pi$. Therefore, the winding number of the interpolating vector field around $p$ is, modulo $2$, equal to $1-\frac{N_{qpr}}{2}$, where $N_{qpr}$ is the (even) number of triangles for which $p$ is the second vertex in the ordering. We will now show that $c(p)$ is $-(-1)^{\frac{N_{qpr}}{2}}$. In terms of Majorana operators, the equation for $c(p)$ is $$\begin{aligned} \label{eq:c1} c(p) = &\left ( \prod_{\substack{{\langle}tq {\rangle}\\ t=p}} i \gamma_{L_{tq}} {{\overline{\gamma}}}_{R_{tq}} \right) \left ( \prod_{\substack{{\langle}qt {\rangle}\\ t=p}} i \gamma_{L_{qt}} {{\overline{\gamma}}}_{R_{qt}} \right) \\ \nonumber \times &\left ( \prod_{\substack{{\langle}tqr {\rangle}\\ t=p}} -i \gamma_{{tqr}} {{\overline{\gamma}}}_{{tqr}} \right) \left ( \prod_{\substack{{\langle}qrt {\rangle}\\ t=p}} -i \gamma_{{qrt}} {{\overline{\gamma}}}_{{qrt}} \right).\end{aligned}$$ Each term in the products over links (the first two products on the right hand side of (\[eq:c1\])) has a factor of $i$. The number of such factors of $i$ is equivalent to the total number of triangles having $p$ as a vertex. We can thus assign each of these factors of $i$ to a different triangle having $p$ as a vertex. Each term in the product over ${{{\langle}pqr {\rangle}}}$-triangles and ${{{\langle}qrp {\rangle}}}$-triangles (the last two products on the right hand side of (\[eq:c1\])) contains a factor of $-i$. After multiplying out all of the factors of $i$ and $-i$ we are thus left only with an $i$ for each ${\langle}{qpr} {\rangle}$-triangle. Since these come in pairs we have $$\begin{aligned} \label{eq:edgeorder} c(p) = &(-1)^{\frac{N_{qpr}}{2}} \\ \nonumber \times&\left ( \prod_{\substack{{\langle}tq {\rangle}\\ t=p}} \gamma_{L_{tq}} {{\overline{\gamma}}}_{R_{tq}} \right) \left ( \prod_{\substack{{\langle}qt {\rangle}\\ t=p}} \gamma_{L_{qt}} {{\overline{\gamma}}}_{R_{qt}} \right) \\ \nonumber \times &\left ( \prod_{\substack{{\langle}tqr {\rangle}\\ t=p}} \gamma_{{tqr}} {{\overline{\gamma}}}_{{tqr}} \right) \left ( \prod_{\substack{{\langle}qrt {\rangle}\\ t=p}} \gamma_{{qrt}} {{\overline{\gamma}}}_{{qrt}} \right).\end{aligned}$$ Next, we notice that the terms in the product over inward pointing links all commute with each other. Likewise, the terms in the product over outward pointing links all commute with each other. Therefore, we may choose any ordering of the terms within each product. We choose to have the inward pointing link terms to be ordered counter-clockwise around $p$ and for the outward pointing link terms to be ordered clockwise as in FIG. \[fig:S\_eordering\]. With this ordering, the two Majorana operators corresponding to each ${{{\langle}pqr {\rangle}}}$ triangle end up being positioned next to each other in the product over outward pointing links. Similarly, the two Majorana operators corresponding to each ${{{\langle}qrp {\rangle}}}$ triangle end up positioned next to each other in the product over inward pointing links. This accounts for all but two of the Majorana operators in each such product. The remaining two Majorana operators appearing at the beginning and end of each product are located respectively on the two ${{{\langle}qpr {\rangle}}}$-triangles bordering each domain. Numbering the inward oriented domains by a domain index $d=1,\ldots,D$, we then (by a slight abuse of notation) re-label these ${{{\langle}qpr {\rangle}}}$-triangle Majorana operators by $\gamma_d$, ${{\overline{\gamma}}}_d$, as illustrated in figure \[fig:S\_eordering\]. Note that this labeling scheme accounts for only one Majorana operator located on each ${\langle}qpr {\rangle}$-triangles; the other one does not appear in the expression for $c(p)$ and hence plays no role in the following. Moving the ${{\overline{\gamma}}}_d$ operators past the (even number of) other terms in each product, we obtain $$\begin{aligned} c(p)=&(-1)^{\frac{N_{qpr}}{2}} \\ \nonumber \times& \left( \prod^D_{d=1} {{\overline{\gamma}}}_{d+1} \gamma_d \right) \left( \prod_{\substack{{\langle}tqr {\rangle}\\ t=p}} {{\overline{\gamma}}}_{tqr} \gamma_{tqr} \right) \\ \nonumber \times& \left( \prod^D_{d=1} {{\overline{\gamma}}}_{d} \gamma_d \right) \left( \prod_{\substack{{\langle}qrt {\rangle}\\ t=p}} {{\overline{\gamma}}}_{qrt} \gamma_{qrt} \right) \\ \nonumber \times&\left( \prod_{\substack{{\langle}tqr {\rangle}\\ t=p}} \gamma_{tqr} {{\overline{\gamma}}}_{tqr} \right) \left( \prod_{\substack{{\langle}qrt {\rangle}\\ t=p}} \gamma_{qrt} {{\overline{\gamma}}}_{qrt} \right)\end{aligned}$$ where the subscript $D+1$ is meant to be read as $1$. After canceling the Majorana bilinears corresponding to ${{{\langle}pqr {\rangle}}}$-triangles and ${{{\langle}qrp {\rangle}}}$-triangles (last line in the product above), we find $$\begin{aligned} c(p)&=(-1)^{\frac{N_{qpr}}{2}} \left( \prod^D_{d=1} {{\overline{\gamma}}}_{d+1} \gamma_d \right) \left( \prod^D_{d=1} {{\overline{\gamma}}}_{d} \gamma_d \right) \\ \nonumber &=-(-1)^{\frac{N_{qpr}}{2}} \gamma_D \left( \prod^{D-1}_{d=1} {{\overline{\gamma}}}_{d+1} \gamma_d \right) {{\overline{\gamma}}}_1 \left( \prod^D_{d=1} {{\overline{\gamma}}}_{d} \gamma_d \right) \\ \nonumber &=-(-1)^{\frac{N_{qpr}}{2}} \left( \prod^{D}_{d=1} \gamma_{d} {{\overline{\gamma}}}_d \right) \left( \prod^D_{d=1} {{\overline{\gamma}}}_{d} \gamma_d \right) \\ \nonumber &=-(-1)^{\frac{N_{qpr}}{2}}. \end{aligned}$$ As we argued at the beginning of this appendix, $-(-1)^{\frac{N_{qpr}}{2}}$ is precisely $-1$ raised to the power of the winding number of the interpolating vector field ${\cal V}$. Thus, we have proved the claim: $c(p)$ is $-1$ when the interpolating vector field has an odd winding number at $p$ and $c(p)$ is $1$ otherwise. Fermion condensation and fermionization {#ap:fermioncondensation} ======================================= In this appendix, we will illustrate that the fermionization duality outlined in section \[sec:duality\] and first described in Ref. \[\], can be interpreted as a fermion condensation procedure for certain lattice Hamiltonians. Fermion condensation has been thouroughly studied using a spacetime formulation \[\] and admits the following intuitive picture. We begin with a bosonic system with emergent fermions and introduce a system with bonafide physical fermions. Next, we pair each emergent fermion with a physical fermion and the composite excitation, having bosonic statistics, is then condensed. This results in a fermionic theory in which all particles braiding with the emergent fermion have been confined. As argued in Ref. \[\], it outputs a supercohomology SPT phase when applied to the corresponding bosonic shadow theory. To see the relation with the fermionization duality described in the main text, we must first develop fermion condensation at the lattice Hamiltonian level. To do so, we consider Hamiltonians defined on a Hilbert space consisting of spin-$\frac{1}{2}$ degrees of freedom on links, the same as that of the bosonic shadow models defined in section \[sec:bosonic\], and we assume that the Hamiltonians commute with ${{\hat{G}}}_p$, for all vertices $p$. The restriction to Hamiltonians that commute with ${{\hat{G}}}_p$ can be motivated by interpreting ${{\hat{G}}}_p$ as a short closed emergent fermion string operator around the vertex $p$. Thus, the Hamiltonians considered here have a particular emergent fermion string operator, which we describe in detail below. We note that this particular string operator creates emergent fermion excitations in the bosonic shadow models constructed in section \[sec:bosonic\], since those models commute with ${{\hat{G}}}_p$. We define the emergent fermion string operator $\hat{\widetilde{U}}_\Gamma$ by $$\begin{aligned} \hat{\widetilde{U}}_\Gamma \equiv \prod_{{\langle}pq {\rangle}\in \Gamma} \hat{\widetilde{U}}_{pq},\end{aligned}$$ where $\Gamma$ is a path in the dual lattice and $\hat{\widetilde{U}}_{pq}$ is $$\begin{aligned} \hat{\widetilde{U}}_{pq} \equiv {{\hat{X}}}_{pq}\hat{\widetilde{K}}_{L_{pq}}\hat{\widetilde{K}}_{R_{pq}}.\end{aligned}$$ The action of $\hat{\widetilde{K}}_{R_{pq}}$ is dependent upon the triangle $R_{pq}$ to the right of ${\langle}pq {\rangle}$. If the triangle to the right of ${\langle}pq {\rangle}$ has vertex ordering ${\langle}pqr {\rangle}$, with $p$ and $q$ being the first and second vertices, respectively, then $\hat{K}_{R_{pq}}$ acts as ${{\hat{Z}}}_{qr}$. Otherwise, $\hat{K}_{R_{pq}}={{1}}$. The action of $\hat{K}_{L_{pq}}$ is defined similarly but with ‘right’ replaced with ‘left’. Intuitively, $\hat{\widetilde{U}}_{pq}$ creates a pair of ${\mathbb{Z}}_2$ fluxes and moves ${\mathbb{Z}}_2$ charges so that they are bound to the fluxes at the third vertex in the vertex ordering. Letting $\Gamma$ be a path in the dual lattice around the vertex $p$, we find that $\hat{\widetilde{U}}_{\Gamma}$ is equal to ${{\hat{G}}}_p$ up to an inconsequential sign. We may now describe the fermion condensation procedure for a Hamiltonian $\hat{H}$ which commutes with ${{\hat{G}}}_p$. First, as ${{\hat{U}}}_{pq}$ and ${{\hat{W}}}_{pqr}$ generate all the local operators that commute with ${{\hat{G}}}_p$, $\hat{H}$ can be expressed in terms of ${{\hat{U}}}_{pq}$ and ${{\hat{W}}}_{pqr}$. To make this explicit, we write $\hat{H}$ as $\hat{H}({{\hat{U}}}_{pq},{{\hat{W}}}_{pqr})$. Next, we introduce fermionic degrees of freedom into our system by adding a complex fermion degree of freedom to each triangle. The fermion parity even operators are generated by fermion parity $(-1)^{\hat{F}_{pqr}}$ and hopping operators ${{\hat{S}}}_{pq}$. Here, ${{\hat{S}}}_{pq}$ includes the spin structure related sign for edges in $\mathcal{E}$, as described in section \[sec:spinstructure\]. The next step in fermion condensation is to bind physical fermions to emergent fermion excitations and to condense the composite particles. The binding of physical fermions to emergent fermion excitations is accomplished by replacing ${{\hat{W}}}_{pqr}$ with ${{\hat{W}}}_{pqr}(-1)^{\hat{F}_{pqr}}$ so that $$\begin{aligned} \hat{H}({{\hat{U}}}_{pq},{{\hat{W}}}_{pqr}) \longrightarrow \hat{H}({{\hat{U}}}_{pq},{{\hat{W}}}_{pqr}(-1)^{\hat{F}_{pqr}}).\end{aligned}$$ In the resulting system, a pair of emergent fermions can be created for free as long as there is a physical fermion attached to each emergent fermion. As a consequence, the Hamiltonian becomes highly degenerate. This degeneracy, however, is eliminated by adding a term that proliferates emergent fermion-physical fermion pairs. A pair of composite excitations is created by the operator $\hat{\widetilde{U}}_{pq}{{\hat{S}}}_{pq}$, so the term $$\begin{aligned} \label{Jcondense} -J\sum_{{\langle}pq {\rangle}} \hat{\widetilde{U}}_{pq}{{\hat{S}}}_{pq}, \quad J>0\end{aligned}$$ energetically prefers states where the composite excitations have been proliferated. Adding the term in \[Jcondense\] to the Hamiltonian, we have $$\begin{aligned} \hat{H}({{\hat{U}}}_{pq},{{\hat{W}}}_{pqr}&(-1)^{\hat{F}_{pqr}}) \nonumber \\ &\longrightarrow \hat{H}({{\hat{U}}}_{pq},{{\hat{W}}}_{pqr}(-1)^{\hat{F}_{pqr}})-J\sum_{{\langle}pq {\rangle}} \hat{\widetilde{U}}_{pq}{{\hat{S}}}_{pq}.\end{aligned}$$ One may be concerned that the $J$-term will not proliferate the composite excitations as promised due to possible competition with $\hat{H}({{\hat{U}}}_{pq},{{\hat{W}}}_{pqr}(-1)^{\hat{F}_{pqr}})$. However, it can be shown that ${{\hat{U}}}_{pq}$ commutes with $\hat{\widetilde{U}}_{st}$ for every ${\langle}st {\rangle}$ and ${{\hat{W}}}_{pqr}(-1)^{\hat{F}_{pqr}}$ commutes with $\hat{\widetilde{U}}_{st}{{\hat{S}}}_{st}$ for every ${\langle}st {\rangle}$. Therefore, $\hat{H}({{\hat{U}}}_{pq},{{\hat{W}}}_{pqr}(-1)^{\hat{F}_{pqr}})$ commutes with $-J\sum_{{\langle}pq {\rangle}} \hat{\widetilde{U}}_{pq}{{\hat{S}}}_{pq}$ and the $J$-term is indeed minimized in the ground state. Finally, we drive the system deep into the fermion condensed regime and consider the limit as $J \to \infty$. In the resulting effective Hilbert space, $\hat{\widetilde{U}}_{pq}{{\hat{S}}}_{pq}=1$ for all ${\langle}pq {\rangle}$. Thus, in this effective Hilbert space, the Hamiltonian acts as $$\begin{aligned} \hat{H}({{\hat{U}}}_{pq}\hat{\widetilde{U}}_{pq}{{\hat{S}}}_{pq}, {{\hat{W}}}_{pqr}(-1)^{\hat{F}_{pqr}}),\end{aligned}$$ where we have inserted $\hat{\widetilde{U}}_{pq}{{\hat{S}}}_{pq}=1$ and removed the $J$-term. Relabeling ${{\hat{W}}}_{pqr}(-1)^{\hat{F}_{pqr}}$ as $(-1)^{\hat{\widetilde{F}}_{pqr}}$ (note the tilde above $F$) and ${{\hat{U}}}_{pq}\hat{\widetilde{U}}_{pq}{{\hat{S}}}_{pq}$ as $\hat{\widetilde{S}}_{pq}$, we have $$\begin{aligned} \hat{H}(\hat{\widetilde{S}}_{pq},(-1)^{\hat{\widetilde{F}}_{pqr}}).\end{aligned}$$ $(-1)^{\hat{\widetilde{F}}_{pqr}}$ and $\hat{\widetilde{S}}_{pq}$ satisfy the same commutation relations as fermion parity operators $(-1)^{\hat{F}_{pqr}}$ and hopping operators $\hat{{S}}_{pq}$, respectively, and it can be checked that they satisfy a relation analogous to (\[Antonrelation\]). Functionally, our prescription for fermion condensation maps a Hamiltonian $\hat{H}({{\hat{U}}}_{pq},{{\hat{W}}}_{pqr})$ to $\hat{H}(\hat{\widetilde{S}}, (-1)^{\hat{\widetilde{F}}_{pqr}})$. In effect, we have replaced ${{\hat{U}}}_{pq}$ with $\hat{\widetilde{S}}_{pq}$ and ${{\hat{W}}}_{pqr}$ with $(-1)^{\hat{\widetilde{F}}_{pqr}}$, which is precisely the result of applying the fermionization duality. Hence, we have shown that, for Hamiltonians that commute with ${{\hat{G}}}_p$, fermionization agrees with fermion condensation. We expect that the steps described above can be generalized to a wider class of emergent fermion string operators. This may yield new fermionization dualities and further extend our understanding of fermion condensation at the lattice level. Ancillary spin-$\frac{1}{2}$ degrees of freedom and evaluation of the fermionic SPT Hamiltonian {#ap:spinH} =============================================================================================== In section \[sec:fSPT\], we presented the construction of a fermionic SPT Hamiltonian obtained by conjugating a trivial fermionic Hamiltonian by ${{\hat{U}}}_{\text{f}}$. However, ${{\hat{U}}}_{\text{f}}$ was written with an unspecified locally determined configuration dependent sign $\hat{\kappa}$. $\hat{\kappa}$ is dependent upon the triangulation of the manifold as well as an ordering of operators. Here we discuss a work around to calculating $\hat{\kappa}$ applicable to arbitrary triangulations. Recall that $\hat{\kappa}$ is a consequence of rearranging terms in $$\begin{aligned} \label{ucirc2} {{\hat{U}}}_{\text{b}}=\prod_{{\langle}pqr {\rangle}} \left(\hat{\nu}_{pqr}^{o_{pqr}} {{\hat{Z}}}_{pq} ^{\hat{n}_{qr}}\right) \prod_{{\langle}pq {\rangle}}{{\hat{X}}}_{pq}^{\hat{n}_{pq}}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{ pqr }^{\hat{n}_{pr}},\end{aligned}$$ to make it manifestly fermionizable. We will show that by adding ancillary spin-$\frac{1}{2}$ degrees of freedom on triangles and composing ${{\hat{U}}}_{\text{b}}$ with a certain trivial circuit, we can reorganize the expression into a fermionizable operator without accruing a sign. To this end, let us add a spin-$\frac{1}{2}$ degree of freedom to each triangle so that in total we have $G$ degrees of freedom at vertices, a spin-$\frac{1}{2}$ degree of freedom at every link, and a spin-$\frac{1}{2}$ at each triangle. The Pauli X and Pauli Z operators acting on the spin-$\frac{1}{2}$ at ${\langle}pqr {\rangle}$ will be denoted as $\hat{\tau}^x_{pqr}$ and $\hat{\tau}^z_{pqr}$, respectively. Next, we compose the operator in (\[ucirc2\]) with a circuit that acts trivially on the triangle spin-$\frac{1}{2}$ degrees of freedom. Namely, we compose with $$\begin{aligned} \label{trivialtrianglecircuit} 1=&\left(\prod_{{\langle}pqr {\rangle}} \left( {{\hat{\tau}}}^z_{pqr} \right)^{\hat{n}_{qr}} \prod_{{\langle}pqr {\rangle}} \left( {{\hat{\tau}}}^x_{pqr} \right)^{\hat{n}_{pq}}\right)\\ \nonumber \times &\left( \prod_{{\langle}pqr {\rangle}} \left( {{\hat{\tau}}}^x_{pqr} \right)^{\hat{n}_{pq}} \prod_{{\langle}pqr {\rangle}} \left( {{\hat{\tau}}}^z_{pqr} \right)^{\hat{n}_{qr}}\right)\end{aligned}$$ to obtain $$\begin{aligned} {{\hat{U}}}^\tau_\text{b}\equiv &\prod_{{\langle}pqr {\rangle}} \left(\hat{\nu}_{pqr}^{o_{pqr}} {{\hat{Z}}}_{pq} ^{\hat{n}_{qr}}\right) \prod_{{\langle}pq {\rangle}}{{\hat{X}}}_{pq}^{\hat{n}_{pq}}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{ pqr }^{\hat{n}_{pr}}\\ \nonumber \times &\left(\prod_{{\langle}pqr {\rangle}} \left( {{\hat{\tau}}}^z_{pqr} \right)^{\hat{n}_{qr}} \prod_{{\langle}pqr {\rangle}} \left( {{\hat{\tau}}}^x_{pqr} \right)^{\hat{n}_{pq}}\right)\\ \nonumber \times &\left( \prod_{{\langle}pqr {\rangle}} \left( {{\hat{\tau}}}^x_{pqr} \right)^{\hat{n}_{pq}} \prod_{{\langle}pqr {\rangle}} \left( {{\hat{\tau}}}^z_{pqr} \right)^{\hat{n}_{qr}}\right).\end{aligned}$$ The circuit in (\[trivialtrianglecircuit\]) is equal to the identity, and as such, stacking it with ${{\hat{U}}}_{\text{b}}$ certainly does not affect the phase of our system. However, we can use the anti-commutativity of ${{\hat{\tau}}}^z_{pqr}$ and ${{\hat{\tau}}}^x_{pqr}$ to make up for the anti-commutivity of ${{\hat{Z}}}_{pq}$ and ${{\hat{X}}}_{pq}$. In particular, we arrange the ${{\hat{\tau}}}^z_{pqr}$ and ${{\hat{\tau}}}^x_{pqr}$ so as to ‘dress’ the ${{\hat{Z}}}_{pq}$ and ${{\hat{X}}}_{pq}$ and avoid incurring the sign $\hat{\kappa}$. First, we move ${{\hat{\tau}}}^z_{pqr}$ operators next to the ${{\hat{Z}}}_{pq}$ operators: $$\begin{aligned} {{\hat{U}}}^\tau_{\text{b}}= &\prod_{{\langle}pqr {\rangle}} \left(\hat{\nu}_{pqr}^{o_{pqr}} \left({{\hat{Z}}}_{pq} {{\hat{\tau}}}^z_{pqr} \right)^{\hat{n}_{qr}}\right) \prod_{{\langle}pq {\rangle}}{{\hat{X}}}_{pq}^{\hat{n}_{pq}}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{ pqr }^{\hat{n}_{pr}}\\ \nonumber \times & \prod_{{\langle}pqr {\rangle}} \left( {{\hat{\tau}}}^x_{pqr} \right)^{\hat{n}_{pq}}\\ \nonumber \times &\left( \prod_{{\langle}pqr {\rangle}} \left( {{\hat{\tau}}}^x_{pqr} \right)^{\hat{n}_{pq}} \prod_{{\langle}pqr {\rangle}} \left( {{\hat{\tau}}}^z_{pqr} \right)^{\hat{n}_{qr}}\right).\end{aligned}$$ Next, we rewrite the product $\prod_{{\langle}pqr {\rangle}} \left( {{\hat{\tau}}}^x_{pqr} \right)^{\hat{n}_{pq}}$ as a product over edges. This will allow us to dress the ${{\hat{X}}}_{pq}$ terms appearing in a product over edges. To the ${\langle}pq {\rangle}$ edge of triangle ${\langle}pqr {\rangle}$ we associate the the operator ${{\hat{\tau}}}^x_{pqr}$. This gives $$\begin{aligned} \prod_{{\langle}pqr {\rangle}} \left( {{\hat{\tau}}}^x_{pqr} \right)^{\hat{n}_{pq}}=\prod_{{\langle}pq {\rangle}} \left( \hat{A}_{L_{pq}}\hat{A}_{R_{pq}} \right)^{\hat{n}_{pq}},\end{aligned}$$ where the action of $\hat{A}_{R_{pq}}$, appearing in the formula above, is determined as follows. If the triangle to the right of ${\langle}pq {\rangle}$ is ${\langle}pqr {\rangle}$, where $p$ and $q$ are the first and second vertices, respectively, then $\hat{A}_{R_{pq}}$ acts as ${{\hat{\tau}}}^x_{pqr}$. Otherwise, $\hat{A}_{R_{pq}}={{1}}$. The action of $\hat{A}_{L_{pq}}$ is defined analogously but we look at the triangle to the left of ${\langle}pq {\rangle}$ instead. Now we write $$\begin{aligned} {{\hat{U}}}^\tau_{\text{b}}= &\prod_{{\langle}pqr {\rangle}} \hat{\nu}_{pqr}^{o_{pqr}} \left({{\hat{Z}}}_{pq} {{\hat{\tau}}}^z_{pqr} \right)^{\hat{n}_{qr}} \\ \nonumber \times &\prod_{{\langle}pq {\rangle}}\left({{\hat{X}}}_{pq}\hat{A}_{L_{pq}}\hat{A}_{R_{pq}}\right)^{\hat{n}_{pq}}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{ pqr }^{\hat{n}_{pr}} \\ \nonumber \times & \prod_{{\langle}pqr {\rangle}} \left( {{\hat{\tau}}}^x_{pqr} \right)^{\hat{n}_{pq}} \prod_{{\langle}pqr {\rangle}} \left( {{\hat{\tau}}}^z_{pqr} \right)^{\hat{n}_{qr}}.\end{aligned}$$ One can check that ${{\hat{Z}}}_{pq} {{\hat{\tau}}}^z_{pqr}$ and ${{\hat{X}}}_{pq}\hat{A}_{L_{pq}}\hat{A}_{R_{pq}}$ commute. Therefore, we are free to rearrange the ${{\hat{Z}}}_{pq} {{\hat{\tau}}}^z_{pqr}$ and ${{\hat{X}}}_{pq}\hat{A}_{L_{pq}}\hat{A}_{R_{pq}}$ operators to form a product of ${{\hat{U}}}_{pq}$ (defined in (\[def:Ue\])) without picking up the sign $\hat{\kappa}$. Explicitly, rearranging yields $$\begin{aligned} {{\hat{U}}}^\tau_{\text{b}}= &\prod_{{\langle}pqr {\rangle}} \hat{\nu}_{pqr}^{o_{pqr}}\\ \nonumber \times &\prod_{{\langle}pq {\rangle}}\left({{\hat{U}}}_{pq}\hat{B}_{L_{pq}}\hat{B}_{R_{pq}}\hat{A}_{L_{pq}}\hat{A}_{R_{pq}}\right)^{\hat{n}_{pq}}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{ pqr }^{\hat{n}_{pr}} \\ \nonumber \times & \prod_{{\langle}pqr {\rangle}} \left( {{\hat{\tau}}}^x_{pqr} \right)^{\hat{n}_{pq}} \prod_{{\langle}pqr {\rangle}} \left( {{\hat{\tau}}}^z_{pqr} \right)^{\hat{n}_{qr}},\end{aligned}$$ where we have introduced $\hat{B}_{L_{pq}}$ and $\hat{B}_{R_{pq}}$. $\hat{B}_{R_{pq}}$ is ${{\hat{\tau}}}^z_{rpq}$ when the triangle to the right is of the form ${\langle}rpq {\rangle}$, i.e. $p$ is the second vertex in the ordering and $q$ is the third vertex in the ordering. $\hat{B}_{R_{pq}}$ is ${{1}}$ otherwise. $\hat{B}_{L_{pq}}$ is defined analogously, but for the triangle to the left of ${\langle}pq {\rangle}$. With a choice of spin structure to define the duality, ${{\hat{U}}}^\tau_{\text{b}}$ can be fermionized straightforwardly. If we let $\hat{S}^\tau_{pq}$ be ${{\hat{S}}}_{pq}\hat{B}_{L_{pq}}\hat{B}_{R_{pq}}\hat{A}_{L_{pq}}\hat{A}_{R_{pq}}$, then $$\begin{aligned} \label{def:Ufprime} {{\hat{U}}}^\tau_{\text{f}}= &\prod_{{\langle}pqr {\rangle}} \hat{\nu}_{pqr}^{o_{pqr}} \\ \nonumber \times &\prod_{{\langle}pq {\rangle}}\left(\hat{S}^\tau_{pq}\right)^{\hat{n}_{pq}}\prod_{{\langle}pqr {\rangle}}\left((-1)^{\hat{F}_{ pqr }}\right)^{\hat{n}_{pr}} \\ \nonumber \times & \prod_{{\langle}pqr {\rangle}} \left( {{\hat{\tau}}}^x_{pqr} \right)^{\hat{n}_{pq}} \prod_{{\langle}pqr {\rangle}} \left( {{\hat{\tau}}}^z_{pqr} \right)^{\hat{n}_{qr}} .\end{aligned}$$ A fermionic SPT Hamiltonian $\hat{H}^\tau_{\text{f}}$ can be formed by conjugating the trivial fermionic Hamiltonian $$\begin{aligned} -\sum_p {{\hat{P}}}_p^{\text{sym}}-\sum_{{\langle}pqr {\rangle}}(-1)^{\hat{F}_{pqr}}-\sum_{{\langle}pqr {\rangle}}{{\hat{\tau}}}^x_{pqr}\end{aligned}$$ by ${{\hat{U}}}^\tau_{\text{f}}$. $\hat{H}^\tau_{\text{f}}$ acts identically to $\hat{H}_{\text{f}}$ on the vertex and complex fermion degrees of freedom. We have simply encoded the sign $\hat{\kappa}$ appearing in $\hat{H}_{\text{f}}$ in the ordering of the Pauli operators of $\hat{H}^\tau_{\text{f}}$. Indeed, if one were to cancel the triangle Pauli operators in (\[def:Ufprime\]), one would obtain the sign $\hat{\kappa}$. In the end, we have arrived at an explicit form for a Hamiltonian in the same phase as $\hat{H}_{\text{f}}$ for an arbitrary triangulation of a 2+1D manifold with spin structure. Supercohomology equivalence relation: trivial fermionic finite depth circuit {#ap:trivialcircuit} ============================================================================ In section \[sec:superequivalencerelation\], we claimed that the supercohomology data $(n_0,\nu_0)=\left(\delta \beta, (-1)^{\beta \cup \delta \beta}\delta \omega \right)$ corresponds to a trivial fermionic SPT phase. We prove this claim here by showing that the finite depth circuit ${{\hat{U}}}^{n_0\nu_0}_\text{b}$ can be written in terms of symmetric local unitaries, up to factors of ${{\hat{G}}}_p$. This implies that the fermionized circuit ${{\hat{U}}}_\text{f}^{n_0\nu_0}$ constructs a trivial SPT ground state from a trivial product state because fermionization respects the $G$-symmetry and maps ${{\hat{G}}}_p$ to the identity. Plugging the data $(n_0,\nu_0)$ into the expression for ${{\hat{U}}}_\text{b}$ (\[ucirc\]) we obtain $$\begin{aligned} {{\hat{U}}}^{n_0\nu_0}_\text{b}=&\prod_{{\langle}pqr {\rangle}}(-1)^{\hat{\beta}_{pq}\delta \hat{\beta}_{qr}} \left( \delta \hat{\omega}_{pqr} \right)^{{o}_{pqr}}\\ \nonumber \times &\prod_{{\langle}pqr {\rangle}}{{\hat{Z}}}_{pq}^{\delta \hat{\beta}_{qr}} \prod_{{\langle}pq {\rangle}} {{\hat{X}}}_{pq}^{\delta \hat{\beta}_{pq}}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{pqr}^{\delta \beta_{pr}},\end{aligned}$$ with $\delta \hat{\beta}_{pq}$ and $\delta \hat{\omega}_{pqr}$ defined by $$\begin{aligned} \delta \hat{\beta}_{pq}|\{ g_t \}{\rangle}&=\delta \beta(g_p,g_q,1)|\{ g_t \}{\rangle}\\ \delta \hat{\omega}_{pqr}|\{ g_t \}{\rangle}&=\delta \omega(g_p,g_q,g_r,1)|\{ g_t \}{\rangle}.\end{aligned}$$ Now we notice $$\begin{aligned} \prod_{{\langle}pqr {\rangle}} (\delta \omega(g_p,&g_q,g_r,1))^{o_{pqr}} = \\ \nonumber &=\prod_{{\langle}pqr {\rangle}} \left(\frac{\omega(g_q,g_r,1)\omega(g_p,g_q,1)}{\omega(g_p,g_r,1)\omega(g_p,g_q,g_r)}\right)^{o_{pqr}} \\ \nonumber &=\prod_{{\langle}pqr {\rangle}}\omega(g_p,g_q,g_r)^{-o_{pqr}}.\end{aligned}$$ The last equality follows from treating $\omega(g_s,g_t,1)$ as corresponding to the edge ${\langle}st {\rangle}$ and canceling factors of $\omega(g_s,g_t,1)$ from neighboring triangles. Therefore, $$\begin{aligned} \label{omegatrianglecancel} \prod_{{\langle}pqr {\rangle}} \left(\delta \hat{\omega}_{pqr}\right)^{{o}_{pqr}}=\prod_{{\langle}pqr {\rangle}}\hat{\omega}_{pqr}^{-o_{pqr}}\end{aligned}$$ for $\hat{\omega}_{pqr}$: $$\begin{aligned} \hat{\omega}_{pqr}|\{g_t\}{\rangle}=\omega(g_p,g_q,g_r)|\{g_t\}{\rangle}.\end{aligned}$$ Using $\delta \hat{\beta}_{pq}=\hat{\beta}_{pq}+\hat{\beta}_p+\hat{\beta}_q$ (with $\hat{\beta}_{pq}$ and $\hat{\beta}_p$ defined in (\[betapq\]) and (\[betap\])) as well as the equality in (\[omegatrianglecancel\]), ${{\hat{U}}}^{n_0\nu_0}_\text{b}$ becomes $$\begin{aligned} {{\hat{U}}}^{n_0\nu_0}_\text{b}=&\prod_{{\langle}pqr {\rangle}}\hat{\omega}_{pqr}^{-o_{pqr}}\prod_{{\langle}pqr {\rangle}}(-1)^{\hat{\beta}_{pq}(\hat{\beta}_{qr}+\hat{\beta}_q+\hat{\beta}_r)} \\ \nonumber \times &\prod_{{\langle}pqr {\rangle}}{{\hat{Z}}}_{pq}^{\hat{\beta}_{qr}+\hat{\beta}_q+\hat{\beta}_r} \prod_{{\langle}pq {\rangle}} {{\hat{X}}}_{pq}^{\hat{\beta}_{pq}+\hat{\beta}_p+\hat{\beta}_q}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{pqr}^{\hat{\beta}_{pr}+\hat{\beta}_p+\hat{\beta}_r},\end{aligned}$$ Rearranging and keeping track of the resulting sign we have $$\begin{aligned} \label{intermediatecircuit1} {{\hat{U}}}^{n_0\nu_0}_\text{b}=&\prod_{{\langle}pqr {\rangle}}\hat{\omega}_{pqr}^{-o_{pqr}} \\ \nonumber \times&\prod_{{\langle}pqr {\rangle}}(-1)^{\hat{\beta}_{pq}(\hat{\beta}_{qr}+\hat{\beta}_q+\hat{\beta}_r)}\prod_{{\langle}pqr {\rangle}}(-1)^{\hat{\beta}_{q}\hat{\beta}_{pq}+\hat{\beta}_{r}\hat{\beta}_{pq}} \\ \nonumber \times &\prod_{{\langle}pqr {\rangle}}{{\hat{Z}}}_{pq}^{\hat{\beta}_{qr}} \prod_{{\langle}pq {\rangle}} {{\hat{X}}}_{pq}^{\hat{\beta}_{pq}}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{pqr}^{\hat{\beta}_{pr}} \\ \nonumber \times &\prod_{{\langle}pqr {\rangle}}{{\hat{Z}}}_{pq}^{\hat{\beta}_q+\hat{\beta}_r} \prod_{{\langle}pq {\rangle}} {{\hat{X}}}_{pq}^{\hat{\beta}_p+\hat{\beta}_q}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{pqr}^{\hat{\beta}_p+\hat{\beta}_r}.\end{aligned}$$ Next we write $\prod_{{\langle}pq {\rangle}} {{\hat{X}}}_{pq}^{\hat{\beta}_p+\hat{\beta}_q}$ in (\[intermediatecircuit1\]) as a product over vertices: $$\begin{aligned} \prod_{{\langle}pq {\rangle}} {{\hat{X}}}_{pq}^{\hat{\beta}_p+\hat{\beta}_q}=\prod_p\left( \prod_{{\langle}st {\rangle}\ni p}{{\hat{X}}}_{st}^{\hat{\beta}_p} \right).\end{aligned}$$ Further, one can check that $$\begin{aligned} \label{Gptrivialcircuit} \left[\prod_p\left( \prod_{{\langle}st {\rangle}\ni p}{{\hat{X}}}_{st}^{\hat{\beta}_p} \right) \right]\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{pqr}^{\hat{\beta}_p}=\prod_p\hat{G}_p^{\hat{\beta}_p}.\end{aligned}$$ Hence, substituting (\[Gptrivialcircuit\]) in (\[intermediatecircuit1\]) and canceling signs we are left with $$\begin{aligned} \label{intermediatecircuit2} {{\hat{U}}}^{n_0\nu_0}_\text{b}=&\prod_{{\langle}pqr {\rangle}}\hat{\omega}_{pqr}^{-o_{pqr}}\prod_{{\langle}pqr {\rangle}}(-1)^{\hat{\beta}_{pq}\hat{\beta}_{qr}} \\ \nonumber \times &\prod_{{\langle}pqr {\rangle}}{{\hat{Z}}}_{pq}^{\hat{\beta}_{qr}} \prod_{{\langle}pq {\rangle}} {{\hat{X}}}_{pq}^{\hat{\beta}_{pq}}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{pqr}^{\hat{\beta}_{pr}} \\ \nonumber \times &\prod_{{\langle}pqr {\rangle}}{{\hat{Z}}}_{pq}^{\hat{\beta}_q+\hat{\beta}_r} \prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{pqr}^{\hat{\beta}_r} \prod_p\hat{G}_p^{\hat{\beta}_p}.\end{aligned}$$ The product $\prod_{{\langle}pqr {\rangle}}{{\hat{Z}}}_{pq}^{\hat{\beta}_q+\hat{\beta}_r}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{pqr}^{\hat{\beta}_r}$ above expressed in terms of ${{\hat{Z}}}_{pq}$ operators is $$\begin{aligned} \label{trivialcircuitZidentity} \prod_{{\langle}pqr {\rangle}}{{\hat{Z}}}_{pq}^{\hat{\beta}_q+\hat{\beta}_r}{{\hat{Z}}}_{pq}^{\hat{\beta}_{r}}{{\hat{Z}}}_{qr}^{\hat{\beta}_{r}}{{\hat{Z}}}_{pr}^{\hat{\beta}_{r}}&=\prod_{{\langle}pqr {\rangle}}{{\hat{Z}}}_{pq}^{\hat{\beta}_q}{{\hat{Z}}}_{qr}^{\hat{\beta}_{r}}{{\hat{Z}}}_{pr}^{\hat{\beta}_{r}}\\ \nonumber &=\prod_{{\langle}pq {\rangle}}{{\hat{Z}}}_{pq}^{\hat{\beta}_{q}}{{\hat{Z}}}_{pq}^{\hat{\beta}_{q}}=1.\end{aligned}$$ With (\[trivialcircuitZidentity\]) we finally have that $$\begin{aligned} \label{trivialcircuit2} {{\hat{U}}}^{n_0\nu_0}_\text{b}=&\prod_{{\langle}pqr {\rangle}}\hat{\omega}_{pqr}^{-o_{pqr}}\prod_{{\langle}pqr {\rangle}}(-1)^{\hat{\beta}_{pq}\hat{\beta}_{qr}} \\ \nonumber \times &\prod_{{\langle}pqr {\rangle}}{{\hat{Z}}}_{pq}^{\hat{\beta}_{qr}} \prod_{{\langle}pq {\rangle}} {{\hat{X}}}_{pq}^{\hat{\beta}_{pq}}\prod_{{\langle}pqr {\rangle}}{{\hat{W}}}_{pqr}^{\hat{\beta}_{pr}} \prod_p\hat{G}_p^{\hat{\beta}_p},\end{aligned}$$ as claimed in section \[sec:superequivalencerelation\]. The homogeneity of $\omega$ and $\beta$ guarantees that the local unitaries (not including $\prod_p\hat{G}_p^{\hat{\beta}_p}$) in (\[trivialcircuit2\]) are symmetric. Fermionization commutes with the global symmetry operator and it takes $\prod_p\hat{G}_p^{\hat{\beta}_p}$ to $1$. Therefore, the resulting circuit ${{\hat{U}}}_\text{f}^{n_0\nu_0}$ is a finite depth circuit built from symmetric local unitaries, and consequently, creates a trivial fermionic SPT from a trivial product state. Nonexistence of symmetric quantum circuits for ‘beyond supercohomology’ phases {#beyondsuper} ============================================================================== Here, we present an argument that the ground states of the beyond supercohomology phases cannot be constructed from a trivial product state using a globally symmetric finite depth quantum circuit. It has been shown, using a spacetime formulation, that the shadow models for beyond supercohomology phases are symmetry enriched toric code phases with the property that certain global symmetries transform $e$ excitations into $m$ excitations and vice-versa \[\]. We begin by giving an independent argument that this property has to hold for the lattice Hamiltonian shadow models associated to a beyond supercohomology SPT. We also show how existing lattice Hamiltonian models for beyond supercohomology phases, namely those of Ref. \[\], can be slightly modified so as to be amenable to our bosonization procedure. This argument can be generalized to show that any lattice Hamiltonian SPT model can be modified so as to be bosonizable via our procedure. We then demonstrate that this property of a global symmetry operator exchanging $e$ and $m$ in the shadow model, together with the assumption of the existence of a globally symmetric finite depth circuit disentangling the ground state of the original fermionic SPT, lead to a contradiction. The beyond supercohomology models constructed in Ref. \[\] are not immediately bosonizable, because the fermionic degrees of freedom live on the links instead of at the centers of triangles. This can be remedied as follows. First, we introduce a second complex fermion degree of freedom at each link and add a term to the Hamiltonian that energetically favors zero fermion occupancy at each of the new complex fermion degrees of freedom. Functionally, we have stacked an atomic insulator onto the original system, so it remains in the same phase. Then, we perform a barycentric subdivision of the lattice, remove the original lattice, and associate the two complex fermions per link to different triangles. This procedure is illustrated in FIG. \[fig:beyondlattices\]. In this process, we have not changed the dynamics of the system, and in the end, we have a beyond supercohomology model which lives in the same Hilbert space as the supercohomology models constructed in this paper, and hence is readily bosonizable. Now we argue that bosonizing these beyond supercohomology models gives symmetry enriched toric code models in which global symmetries convert $e$ excitations into $m$ excitations. Key to this argument is the following property of the symmetry action in beyond supercohomology models with symmetry defects. For beyond supercohomology phases with symmetry $G$, there is an additional piece of data relative to supercohomology phases - a homomorphism . According to Refs. \[\], the effective symmetry action near a fermion parity defect is fermion parity odd when acting with on-site symmetry operator ${{\hat{V}}}(g)$ representing $g \in G$ for which $\sigma(g)$ is non-trivial. We analyze this effect in our beyond supercohomology models by inserting a pair of fermion parity defects at well separated vertices $a$ and $b$. Loosely, we create the pair of fermion parity defects by choosing a path $\Gamma$ connecting $a$ and $b$ and modifying hopping operators in the Hamiltonian corresponding to links in $\Gamma$. Heuristically, the modification of the Hamiltonian makes it so that when a fermion moves around one of the fermion parity defects, it picks up an extra $-1$ sign. To include fermion parity defects, we first write the beyond supercohomology model $\hat{H}^\text{b.s.}$ as a sum of local terms $$\begin{aligned} \label{bshamiltonian} \hat{H}^\text{b.s.}=\sum_j {{\hat{h}}}^\text{b.s.}_j,\end{aligned}$$ with each ${{\hat{h}}}^\text{b.s.}_j$ supported on the spatially bounded region $R_j$. Now, we define $$\begin{aligned} \hat{\mathcal{P}}_p = \frac{1}{2} \left( {{1}}+ \prod_{\substack{{\langle}tq {\rangle}\\ t=p}} {\hat S}_{tq} \prod_{\substack{{\langle}qt {\rangle}\\ t=p}} {\hat S}_{qt} \prod_{\substack{{\langle}tqr {\rangle}\\ t=p}} (-1)^{\hat{F}_{tqr}} \prod_{\substack{{\langle}qrt {\rangle}\\ t=p}} (-1)^{\hat{F}_{qrt}} \right)\end{aligned}$$ and $$\begin{aligned} \hat{\mathcal{P}}_{R_j}=\prod_{p \subset R_j} \hat{\mathcal{P}}_p,\end{aligned}$$ and write $$\begin{aligned} \hat{H'}^\text{b.s.} = \sum_j \hat{\mathcal{P}}_{R_j} {{\hat{h}}}^\text{b.s.}_j \hat{\mathcal{P}}_{R_j} -\sum_p \hat{\mathcal{P}}_p.\end{aligned}$$ $\hat{\mathcal{P}}_p$ is identically equal to $1$, so $\hat{H'}^\text{b.s.}$ is equivalent to $\hat{H}^\text{b.s.}$. Next, let us modify this Hamiltonian to insert a pair of defects at two well separated vertices $a$ and $b$. Let $\Gamma$ be a path of links connecting $a$ and $b$, and let $\Gamma_{pq}$ be the indicator function $$\begin{aligned} \Gamma_{pq}= \begin{cases} 1 & \text{if }{\langle}pq {\rangle}\in \Gamma \\ 0 & \text{otherwise}. \end{cases}\end{aligned}$$ Now we write each local term ${{\hat{h}}}^\text{b.s.}_j$ explicitly as a linear combination of products of ${{\hat{S}}}_{pq}$ and $(-1)^{\hat{F}_{pqr}}$, and we make the replacement: $$\begin{aligned} {{\hat{h}}}^\text{b.s.}_j ({{\hat{S}}}_{pq}, (&-1)^{\hat{F}_{pqr}}) \nonumber \\ &\longrightarrow {{\hat{h}}}^\text{b.s.}_{j,\Gamma}\equiv {{\hat{h}}}^\text{b.s.}_j((-1)^{\Gamma_{pq}}{{\hat{S}}}_{pq}, (-1)^{\hat{F}_{pqr}}).\end{aligned}$$ Making the same replacement in $\hat{\mathcal{P}}_{j}$ yields a new Hamiltonian $\hat{H}^\text{b.s.}_\Gamma$. Notice that replacing ${{\hat{S}}}_{pq}$ with $(-1)^{\Gamma_{pq}}{{\hat{S}}}_{pq}$ in the expression defining $\hat{\mathcal{P}}_R$ yields $0$ if $a$ or $b$ is contained in the region $R$. As a consequence, Hamiltonian terms whose support contains the defects are removed from the Hamiltonian. Now we bosonize $\hat{H}^\text{b.s.}$ and $\hat{H}^\text{b.s.}_\Gamma$. This yields $$\begin{aligned} \hat{H}^\text{b.s.}_\text{b} \equiv \hat{H'}^\text{b.s.}({{\hat{U}}}_{pq},{{\hat{W}}}_{pqr}) \end{aligned}$$ and $$\begin{aligned} \hat{H}^\text{b.s.}_{\text{b},\Gamma} \equiv \hat{H'}^\text{b.s.}((-1)^{\Gamma_{pq}}{{\hat{U}}}_{pq},{{\hat{W}}}_{pqr}).\end{aligned}$$ The operators $\hat{\mathcal{P}}_R$ become projectors onto the ${{\hat{G}}}_p=1$ subspace for all $p \subset R$. As a consequence, the ground states of $\hat{H}^\text{b.s.}_\text{b}$ are in the ${{\hat{G}}}_p=1$ subspace, and away from the defects, the ground states of $\hat{H}^\text{b.s.}_{\text{b},\Gamma}$ are in the ${{\hat{G}}}_p=1$ subspace. We also note that, by construction, the bosonized Hamiltonians commute with ${{\hat{G}}}_p$ for all $p$, and thus, ${{\hat{G}}}_p$ can be interpreted as a small loop of an emergent fermion string operator around the vertex $p$. We can obtain a ground state of $\hat{H}^\text{b.s.}_{\text{b},\Gamma}$ by applying a certain string operator to a ground state of $\hat{H}^\text{b.s.}_\text{b}$. In particular, the string operator ${{\hat{U}}}_\Gamma=\prod_{{\langle}pq {\rangle}\in \Gamma}{{\hat{Z}}}_{pq}$ does the job. Explicitly, $$\begin{aligned} \hat{H}_{\text{b},\Gamma}^\text{b.s.}{{\hat{U}}}_\Gamma|\Psi_\text{b}^\text{b.s.}{\rangle}={{\hat{U}}}_\Gamma \hat{H}_\text{b}^\text{b.s.} |\Psi_\text{b}^\text{b.s.}{\rangle}= E_\text{min}{{\hat{U}}}_\Gamma|\Psi_\text{b}^\text{b.s.}{\rangle},\end{aligned}$$ where in the first equality, we used that ${{\hat{U}}}_\Gamma$ anticommutes with ${{\hat{U}}}_{pq}$ for ${\langle}pq {\rangle}\in \Gamma$. Applying ${{\hat{G}}}_p$ to this ground state at either endpoint of $\Gamma$, we find that moving an emergent fermion around the endpoint produces a minus sign: $$\begin{aligned} {{\hat{G}}}_{a/b} {{\hat{U}}}_\Gamma|\Psi_\text{b}^\text{b.s.}{\rangle}= - {{\hat{U}}}_\Gamma |\Psi_\text{b}^\text{b.s.}{\rangle}.\end{aligned}$$ Hence, ${{\hat{U}}}_\Gamma$ creates either $e$ excitations or $m$ excitations at its endpoints. Since ground states of $\hat{H}_{\text{b},\Gamma}^\text{b.s.}$ have a pair of $e$ or $m$ excitations relative to ground states of $\hat{H}_\text{b}^\text{b.s.}$, we can determine the effective symmetry action on a pair of $e$ or $m$ excitations at $a$ and $b$ by bosonizing the effective symmetry action on the fermionic state with fermion parity defects at $a$ and $b$. For $g$ such that $\sigma(g)$ is non-trivial, the effective symmetry action on the state with fermion parity defects splits into a fermion parity odd operator associated to each defect. Expressing the symmetry action in terms of local fermion parity even operators, so that it may be bosonized, requires a string of hopping operators and fermion parity operators connecting the two fermion parity odd operators. Hence, bosonization yields an effective symmetry action that includes an $em$ string connecting $a$ and $b$. This $em$ string converts an $e$ ($m$) string into an $m$ ($e$) string, and we have recovered the expected effective symmetry action in the bosonic shadow theory of a beyond supercohomology phase at the lattice level. Finally, to demonstrate that beyond supercohomology SPT ground states cannot be constructed by applying a symmetric finite depth quantum circuit to a trivial product state, we assume that this is indeed possible and derive a contradiction. If such a circuit ${{\hat{U}}}^\text{b.s.}$ exists, then in bosonizing the circuit we obtain a circuit ${{\hat{U}}}_\text{b}^\text{b.s.}$ which is globally symmetric up to factors of ${{\hat{G}}}_p$. (The ${{\hat{G}}}_p$ generate the kernel of the fermionization duality.) Explicitly, $$\begin{aligned} \hat{V}(g){{\hat{U}}}_\text{b}^\text{b.s.}={{\hat{U}}}_\text{b}^\text{b.s.} f_g({{\hat{G}}}_p) \hat{V}(g)\end{aligned}$$ for some $g$ dependent function $f_g$ of the ${{\hat{G}}}_p$. In what follows, let us assume that ${{\hat{U}}}_\Gamma$ creates a pair of $e$ excitations. An analogous argument can be made if ${{\hat{U}}}_\Gamma$ instead creates $m$ excitations. Writing a ground state of the toric code with a pair of $e$ excitations as $|\Psi^{ee}_\text{t.c.}{\rangle}$, the ground state of $\hat{H}_\text{b}^\text{b.s.}$ with $e$ excitations is ${{\hat{U}}}_\text{b}^\text{b.s.}|\Psi^{ee}_\text{t.c.}{\rangle}$. We then compute the effective symmetry action on a pair of $e$ excitations: $$\begin{aligned} \hat{V}(g){{\hat{U}}}_\text{b}^\text{b.s.}|\Psi^{ee}_\text{t.c.}{\rangle}&={{\hat{U}}}_\text{b}^\text{b.s.} f_g({{\hat{G}}}_p) \hat{V}(g)|\Psi^{ee}_\text{t.c.}{\rangle}\nonumber \\ &={{\hat{U}}}_\text{b}^\text{b.s.} f_g({{\hat{G}}}_p)|\Psi^{ee}_\text{t.c.}{\rangle}.\end{aligned}$$ As we have argued (at least for the anyons created by ${{\hat{U}}}_\Gamma$), the effective symmetry action should convert the $e$ excitations into $m$ excitations. However, $f_g$ is a function of small emergent fermion loop operators. Loops of $em$ string are unable to transform $e$ excitation into $m$ excitations. This contradicts the expected affect of global symmetry action in the bosonic shadow model for beyond supercohomology phases. We have now shown that the ground state of a specific beyond supercohomology model cannot be constructed by applying a symmetric quantum circuit to a trivial product state. This is sufficient to argue that no ground state of any supercohomology model can by constructed from a trivial product state with a symmetric quantum circuit. This is because, by definition, ground states of two beyond supercohomology phases can be related by a quantum circuit built of symmetric local unitaries. While we have shown that a symmetric quantum circuit is incapable of building the ground state of a beyond supercohomology phase from a trivial product state, it would be interesting to identify a quantum circuit, albeit not symmetric, which is capable of creating the ground state of a beyond supercohomology SPT from a trivial product state. We leave this for future work. [^1]: A finite depth quantum circuit of local unitaries is a unitary operator that can be expressed in the form where the unitaries satisfy the following properties. First, each acts as the identity everywhere except on spins located in a disk of finite radius. In this sense, is a local unitary. Furthermore, for each $j \neq k$, $U_{i,j}$ has non-overlapping support with . The collection of unitaries sharing the first index define a ‘layer’ of the quantum circuit. That is, is the layer of the quantum circuit. ‘Finite depth’ means that the number of layers remains finite in the thermodynamic limit of large system size. [^2]: Let $h$ be a map from $G^k$ to ${\mathbb{Z}}_2$. The coboundary of $h$ is given by $$\begin{aligned} \unexpanded{\delta h(g_0,...,g_k)=\sum_{j=0}^k (-1)^j h(g_0,...,\widehat{g_j},...,g_k)} \end{aligned}$$ where means that $g_j$ is omitted. Let $f$ be a map from $G^k$ to $U(1)$. Then the coboundary of $f$ is given by $$\begin{aligned} \unexpanded{\delta f(g_0,...,g_k)=\prod_{j=0}^k f(g_0,...,\widehat{g_j},...,g_k)^{(-1)^j}.}\end{aligned}$$ [^3]: Here we use the fact that $n$ is a normalized $2$-cocycle. [^4]: Note that on manifolds $M$ with nontrivial $H_1(M)$ global relations need to be specified to ensure that the duality is consistent. These additional relations can be seen as coming from operator identities on the fermionic side of the duality - certain products of fermionic ‘hopping’ operators and parity operators along nontrivial $1$-cycles are equivalent to the identity. [^5]: The fact that the number of vertices $p$ with $c(p)=-1$ is even is just a consequence of the fact that the winding number of singularities is additive: a contour that encloses several singularities has a winding number equal to the sum of the winding numbers of those singularities. On a compact manifold, a small contour enclosing no singularities can equivalently be thought of as a large contour enclosing all the singularities (by exchanging the notion of ‘inside’ and ‘outside’ the contour). [^6]: As a reminder, the cup product between homogeneous functions and (for abelian group $A$) is $$\begin{aligned} (f \cup h)(g_0,...,g_{\ell+k-1}) = f(g_0,...,g_\ell)h(g_\ell,...,g_{\ell+k-1}). \end{aligned}$$ The product of $f$ and $h$ is $$\begin{aligned} \unexpanded{&(f \cup_1 h)(g_0,...,g_{\ell+k-1})= \\ &\sum_{i=0}^{\ell-1}f(g_0,...,g_i,g_{k+i},g_{\ell+k-1})h(g_i,...,g_{k+i})}. \end{aligned}$$ [^7]: Here we use the definition of $\cup_1$ (see Appendix A of \[\]) to write $$\begin{aligned} \unexpanded{n\cup_1 \delta \beta=n \cup \beta + \beta \cup n + \delta(n \cup_1 \beta)}.\end{aligned}$$ [^8]: Strictly speaking, the argument holds for a slightly modified (see appendix A of Ref. \[[23](https://arxiv.org/abs/1506.00592)\]).

--- abstract: 'The process of the two-step conversion of the neutrino helicity, $\nu_L \to \nu_R \to \nu_L$, is analysed in the supernova conditions, where the first stage is realized due to the interaction of the neutrino magnetic moment with the plasma electrons and protons in the supernova core. The second stage is caused by the neutrino resonant spin-flip in a magnetic field of the supernova envelope. Given the neutrino magnetic moment within the interval $10^{-13} \, \mu_{\rm B} < \mu_\nu < 10^{-12} \, \mu_{\rm B}$, and with the existence of the magnetic field at the scale $\sim 10^{13}$ G between the neutrinosphere and the shock-wave stagnation region, it is shown that an additional energy of the order of $10^{51}$ erg can be injected into this region during the typical time of the shock-wave stagnation. This energy could be sufficient for stumulation of the damped shock wave.' author: - | A. V. Kuznetsov[^1], N. V. Mikheev[^2], A. A. Okrugin[^3]\ \ title: '**Dirac neutrino magnetic moment and the shock wave revival in a supernova explosion**' --- Introduction ============ In a modelling of the supernova explosion, two main problems arise [@Imshennik:1988; @Bethe:1990; @Raffelt:1996; @Buras:2005; @Janka:2007]. First, the mechanism of the damped shock wave stimulation has not been developed completely yet. It is believed that the explosion cannot be realized without the shock wave revival. Let us remind, that the main reason of the shock-wave damping is the energy loss by the nuclei dissociation. The second problem is that even in the case of the “successful” theoretical supernova explosion, the energy release turns out to be essentialy less than the observed kinetic energy of the envelope $\sim 10^{51}$ erg. That is known as the FOE problem (ten to the Fifty One Ergs). Thus, it is necessary for the self-consistent description of the explosion dynamics, that the neutrino flux, outgoing from the supernova core, could transfer by some mechanism the energy $\sim 10^{51}$ erg to the supernova envelope. A possible solution of those problems, first proposed by A. Dar [@Dar:1987], was based on the assumption of the existence of the neutrino magnetic moment being not too small. A huge number of left-handed electron neutrinos $\nu_e$ is produced in the collapsing supernova core, and a part of them could convert into right-handed neutrinos due to the interaction of the neutrino magnetic moment with plasma electrons and protons. These right-handed neutrinos, being sterile with respect to the weak interaction, freely escape from the central part of the supernova, if the neutrino magnetic moment is not too large, $\mu_\nu < 10^{-11} \, \mu_{\rm B}$, where $\mu_{\rm B}$ is the Bohr magneton. In the supernova envelope, a part of these neutrinos can flip back to the left-handed ones due to the interaction of the neutrino magnetic moment with a magnetic field. It is now believed that the magnetic field strength in the supernova envelope could achieve the critical value $B_e = m_e^2/e \simeq 4.41 \times 10^{13}$ G [^4] and even exceed it. The produced left-handed neutrinos, being absorbed in beta-processes, $\nu_e n \to e^- p$, can transfer an additional energy to the supernova envelope. In our opinion, a reason arises at the present time to reconsider in more detail the Dar’s mechanism. In the recent paper [@Kuznetsov:2007] we have shown that the evaluations of the right-handed neutrino flux and the luminosity from the supernova core were essentially understated in the previous papers on the subject. In this paper, we perform an analysis of the two-step conversion of the neutrino helicity, $\nu_L \to \nu_R \to \nu_L$, under the supernova conditions, and of the possibility of the damped shock wave stumulation by this process. The right-handed neutrino luminosity {#sec:nu_R_luminosity} ==================================== The process of the neutrino chirality flip $\nu_L \to \nu_R$ under the physical conditions of the supernova core was investigated in the papers [@Barbieri:1988; @Ayala:1999; @Ayala:2000; @Kuznetsov:2007]. The process is possible due to the interaction of the Dirac neutrino magnetic moment with a virtual plasmon, which can be both produced and absorbed: $$\nu_L \to \nu_R + \gamma^*; \quad \nu_L + \gamma^* \to \nu_R.$$ The detailed calculation of the plasma polarization effect on the photon propagator reveals, in particular, that the contribution of the proton component of plasma is dominant. As a result a new astrophysical bound on the electron-type neutrino magnetic moment was established [@Kuznetsov:2007] from the supernova $SN1987A$ data: $$\begin{aligned} \mu_\nu < (0.7 - 1.5) \, \times 10^{-12} \, \mu_{\rm B}\,, \label{eq:mu_fr_Q}\end{aligned}$$ which improved the existed constraint by the factor of 2. Particularly, a function $\Gamma_{\nu_R} (E)$ defining the spectrum of the right-handed neutrino energies, was calculated in Ref. [@Kuznetsov:2007]. In other words, this function defines the number of right-handed neutrinos emitted per 1 MeV of the neutrino energy spectrum, per unit time, from the unit volume of the supernova core: $$\begin{aligned} \frac{\mathrm{d} n_{\nu_R}}{\mathrm{d} E} = \frac{E^2}{2 \, \pi^2} \, \Gamma_{\nu_R} (E) \,. \label{eq:dn/dE}\end{aligned}$$ The function $\Gamma_{\nu_R} (E)$ also determines the spectral density of the right-handed neutrino luminosity of the supernova core: $$\begin{aligned} \frac{\mathrm{d} L_{\nu_R}}{\mathrm{d} E} = V\, \frac{\mathrm{d} n_{\nu_R}}{\mathrm{d} E} \, E = V \, \frac{E^3}{2 \, \pi^2} \, \Gamma_{\nu_R} (E) \,, \label{eq:dL/dE}\end{aligned}$$ where $V$ is the volume of the area emitting neutrinos. The function $\mathrm{d} L_{\nu_R}/\mathrm{d} E$ was calculated in Ref. [@Kuznetsov:2007], and it is shown in the figure \[fig:emissivity\] for the typical supernova core parameter values: the temperature $T \simeq$ 30 MeV, the electron and neutrino chemical potentials $\tilde \mu_e \simeq$ 300 MeV, $\tilde \mu_{\nu_e} \simeq$ 160 MeV, the volume $V \simeq 4 \times 10^{18} \, \mbox{cm}^3$ and for the neutrino magnetic moment $\mu_\nu = 3 \times 10^{-13} \, \mu_{\rm B}$. ![The energy spectrum of the right-handed neutrino luminosity for the plasma temperature $T =$ 30 MeV and for $\mu_\nu = 3 \times 10^{-13} \, \mu_{\rm B}$.[]{data-label="fig:emissivity"}](Mikheev_NV_fig1.eps){width="90.00000%"} The integral luminosity of the right-handed neutrinos appeared to be the following: $$\begin{aligned} L_{\nu_R} = 4 \times 10^{51} \, \frac{\mbox{erg}}{\mbox{s}} \,. \label{eq:L}\end{aligned}$$ Hereafter we use for the definiteness the neutrino magnetic moment value $\mu_\nu = 3 \times 10^{-13} \, \mu_{\rm B}$. On the one hand, this value is sufficiently small to avoid a distortion of the supernova dynamics. On the other hand, it is large enough to provide the required level of the luminosity (\[eq:L\]). If the right-handed neutrino energy was converted into the energy of the left-handed neutrinos, for example due to the well-known mechanism of the spin oscillations, then during the typical stagnation time of the shock wave of the order of some tenths of a second, an additional energy of order $10^{51}$ erg could be injected into the supernova envelope. The resonant transition $\nu_R \to \nu_L$ in the magnetic field\ of the supernova envelope {#sec:resonance} ================================================================ We consider a part of the supernova envelope between the neutrinosphere (of the radius $R_\nu$) and the shock wave stagnation region (of the radius $R_s$). By the present conceptions, typical values of $R_\nu$ and $R_s$ vary rather slightly during the stagnation time. These values could be estimated as $R_\nu \sim$ 20–50 km, $R_s \sim$ 100–200 km. If a sufficiently large magnetic field $\sim 10^{13}$ G exists in this region, then the spin oscillation phenomenon takes place, which can be of the resonant type at certain conditions. It is convenient to illustrate the magnetic field influence on a neutrino with a magnetic moment by means of the equation of the neutrino helicity evolution in an external uniform magnetic field. Taking into account the additional energy $C_L$, which the left-handed electron type neutrino $\nu_e$ acquires in medium, the equation of the helicity evolution can be written in the form [@Voloshin:1986a; @Voloshin:1986b; @Okun:1986; @Voloshin:1986c; @Okun:1988] $${\mathrm i}\,\frac{\partial}{\partial t} \left( \begin{array}{c} \nu_R \\ \nu_L \end{array} \right) = \left[\hat E_0 + \left( \begin{array}{cc} 0 & \mu_\nu B_{\perp} \\ \mu_\nu B_{\perp} & C_L \end{array} \right) \right] % \left( \begin{array}{c} \nu_R \\ \nu_L \end{array} \right) \,, \label{eq:evolution}$$ where $$C_L = \frac{3 \, G_{\mathrm F}}{\sqrt{2}} \, \frac{\rho}{m_N} \left( Y_e + \frac{4}{3} \, Y_{\nu_e} - \frac{1}{3} \right) \,. \label{eq:C_L}$$ Here, the ratio $\rho/m_N = n_B$ is the nucleon density, $Y_e = n_e/n_B = n_p/n_B, \, Y_{\nu_e} = n_{\nu_e}/n_B$, $n_{e,p,\nu_e}$ are the densities of electrons, protons and neutrino respectively. $B_{\perp}$ is the transverse component of the magnetic field with respect to the neutrino movement direction, $\hat E_0$ is proportional to the unit matrix and is inessential for our analysis. The expression (\[eq:C\_L\]) for the additional energy of left-handed neutrinos $C_L$ deserves a special analysis. It is remarkable that the possibility exists for this value to be zero just in the region of the supernova envelope we are interested in. And in turn this is the condition of the resonant transition $\nu_R \to \nu_L$. Taking into account that the neutrino density in the supernova envelope is sufficiently small, one may neglect the value $Y_{\nu_e}$ in the expression (\[eq:C\_L\]), that gives the condition of the resonance in the form $Y_e = 1/3$. It should be noted that the values $Y_e$ which are realized in the supernova envelope, typical for the collapsing matter, are: $Y_e \sim$ 0.4–0.5. However, the shock wave causes the nuclei dissociation and makes the substance to be more transparent for neutrinos. This leads to the so-called “short” neutrino outburst and consequently to the significiant matter deleptonization in this region. According to the existing conceptions, a typical gap arises in the radial distribution of the value $Y_e$, where $Y_e$ may fall down to the value $\sim 0.1$, see, for example [@Bethe:1990; @Buras:2005]. The qualitative behaviour of the dependence $Y_e (r)$ is represented in the figure \[fig:gap\]. Thus, a point necessarily exists where $Y_e$ takes the value of $1/3$. It is remarkable, that only one such point appears, with $\mathrm{d} Y_e / \mathrm{d} r > 0$, see [@Bethe:1990; @Buras:2005]. ![The qualitative behaviour of the dependence $Y_e (r)$ about 0.1 to 0.2 s after the shock formation, with the typical gap caused by the “short” neutrino outburst, see e.g. [*Buras et al., 2005*]{}. The dashed line corresponds to the value $Y_e = 1/3$.[]{data-label="fig:gap"}](Mikheev_NV_fig2.eps){width="70.00000%"} Notice, that the condition $Y_e = 1/3$ is the necessary but still not the sufficient one for the resonant conversion $\nu_R \to \nu_L$. The realization is also necessary of the so-called adiabatic condition. This means that the diagonal element $C_L$ in the equation (\[eq:evolution\]), at least, should not exceed the nondiagonal element $\mu_\nu B_{\perp}$, when the shift is made from the resonance point at the distance of the order of the oscillations length. This leads to the condition [@Voloshin:1988]: $$\begin{aligned} \mu_\nu B_{\perp} \gtrsim \left( \frac{\mathrm{d} C_L}{\mathrm{d} r} \right)^{1/2} \simeq \left( \frac{3 \, G_{\mathrm F}}{\sqrt{2}} \, \frac{\rho}{m_N} \, \frac{\mathrm{d} Y_e}{\mathrm{d} r}\right)^{1/2} . \label{eq:res_cond}\end{aligned}$$ The typical parameter values in the considered area are the following, see [@Bethe:1990; @Buras:2005]: $$\begin{aligned} \frac{\mathrm{d} Y_e}{\mathrm{d} r} \sim 10^{-8} \, \mbox{cm}^{-1} \,, \quad \rho \sim 10^{10} \, \mbox{g} \cdot \mbox{cm}^{-3} \,. \label{eq:param}\end{aligned}$$ For the magnetic field value, providing the realization of the resonance condition, one can find: $$\begin{aligned} B_{\perp} \gtrsim 2.6 \times 10^{13} \mbox{G} \left( \frac{10^{-13} \mu_{\rm B}}{\mu_\nu} \right) \left( \frac{\rho}{10^{10} \mbox{g} \cdot \mbox{cm}^{-3}} \right)^{1/2} \left( \frac{\mathrm{d} Y_e}{\mathrm{d} r} \times 10^8 \, \mbox{cm} \right)^{1/2} . \label{eq:res_cond_B}\end{aligned}$$ Thus, the performed analysis shows that the Dar scenario of the two-step conversion of the neutrino helicity, $\nu_L \to \nu_R \to \nu_L$, can be realized, if the value of the neutrino magnetic moment is in the interval $$\begin{aligned} 10^{-13} \, \mu_{\rm B} < \mu_\nu < 10^{-12} \, \mu_{\rm B} \,, \label{eq:munu_int}\end{aligned}$$ and under the condition that the magnetic field of the scale $10^{13}$ G exists in the region $R_\nu < R < R_s$. During the shock wave stagnation time $\Delta t \sim$ 0.2–0.4 sec the additional energy can be injected into this region, of the order of $$\begin{aligned} \Delta E \simeq L_{\nu_R} \, \Delta t \sim 10^{51} \, \mbox{erg} \,, \label{eq:DeltaE}\end{aligned}$$ which is just enough for the problem solution. Conclusion ========== We have re-analysed the two-step conversion of the neutrino helicity, $\nu_L \to \nu_R \to \nu_L$, under the supernova conditions. As we have shown, this conversion process could provide an additional energy of the order of $10^{51}$ erg which can be injected into the region between the neutrinosphere and the shock-wave stagnation area, $R_\nu < R < R_s$, during the typical stagnation time of the order of some tenths of a second. This energy could be sufficient for stumulation of the damped shock wave. The conditions for the realization of this scenario appear to be not very rigid. The Dirac neutrino magnetic moment should belong to the interval $10^{-13} \, \mu_{\rm B} < \mu_\nu < 10^{-12} \, \mu_{\rm B}$, and the magnetic field $\sim 10^{13}$ G should exist in the region $R_\nu < R < R_s$. Acknowledgments {#acknowledgments .unnumbered} =============== The authors are grateful to M. I. Vysotsky for useful discussion. A. K. and N. M. express their deep gratitude to the organizers of the Seminar “Quarks-2008” for warm hospitality. The work was supported in part by the Russian Foundation for Basic Research under the Grant No. 07-02-00285-a, and by the Council on Grants by the President of the Russian Federation for the Support of Young Russian Scientists and Leading Scientific Schools of Russian Federation under the Grant No. NSh-497.2008.2. [99]{} V.S. Imshennik and D.K. Nadyozhin, Usp. Fiz. Nauk [**156**]{}, 561 (1988) \[Sov. Sci. Rev., Sect. E [**8**]{}, 1 (1989)\]. H. A. Bethe, Rev. Mod. Phys. [**62**]{}, 801 (1990). G. G. 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[**44**]{}, 544 (1986)\]. L. B. Okun, Yad. Fiz. [**44**]{}, 847 (1986) \[Sov. J. Nucl. Phys. [**44**]{}, 546 (1986)\]. M. B. Voloshin, M. I. Vysotsky and L. B. Okun, Zh. Eksp. Teor. Fiz. [**91**]{}, 754 (1986); Erratum: [*ibid.*]{} [**92**]{}, 368 (1987) \[Sov. Phys. JETP [**64**]{}, 446 (1986); Erratum: [*ibid.*]{} [**65**]{}, 209 (1987)\]. L. B. Okun, Yad. Fiz. [**48**]{}, 1519 (1988) \[Sov. J. Nucl. Phys. [**48**]{}, 967 (1988)\]. M. B. Voloshin, Phys. Lett. B [**209**]{}, 360 (1988). [^1]: [**e-mail**]{}: avkuzn@uniyar.ac.ru [^2]: [**e-mail**]{}: mikheev@uniyar.ac.ru [^3]: [**e-mail**]{}: alexander.okrugin@gmail.com [^4]: We use the natural system of units $c = \hbar = 1$. $e > 0$ is an elementary charge.

--- author: - The ATLAS Collaboration bibliography: - 'DG2Tau.bib' title: 'Search for the direct production of charginos, neutralinos and staus in final states with at least two hadronically decaying taus and missing transverse momentum in $pp$ collisions at $\sqrt{s}$ = 8TeV with the ATLAS detector' --- =1

--- abstract: 'Generally the adiabatic quantum pumping phenomenon can be interpreted by the surface integral of the Berry curvature inside the cyclic loop. Spin angular momentum flow without charge current can be pumped out by magnetization precession in ferromagnet-based structures. When an electron is scattered by a helimagnet, spin-dependent diffraction occurs due to the spatial modulation of the spiral. In this work, we consider the charge and spin flow driven by magnetization precession in normal-metal/multiferroic-helimagnet/ferromagnet heterostructures. The pumping behavior is governed by the diffracted states. Gauge dependence in the pumped current was encountered, which does not occur in the static transport properties or pumping behaviors in other systems.' address: 'Department of Physics, South China University of Technology, Guangzhou 510641, People’s Republic of China ' author: - 'Rui Zhu[^1]' title: 'Diffraction induced Spin Pumping in Normal-Metal/Multiferroic-Helimagnet/Ferromagnet Heterostructures' --- Introduction ============ A dc charge and spin current can be generated by cyclic variation of system parameters[@Ref1; @Ref2; @Ref14]. In the adiabatic condition, i.e., the escape rate of the particle is much smaller than the speed of the parameter variation, the transport procedure can be viewed as the accumulated effect of static tunneling with the time-dependent parameter frozen at a certain value[@Ref5]. The pumped current can be evaluated from the surface integral of the scattering Berry curvature[@Ref3; @Ref4] or by Taylor’s expansion of the instant scattering matrix at the equilibrium parameter values[@Ref5; @Ref31]. For large pumping frequencies, Floquet scattering theory[@Ref6; @Ref32; @Ref33] and the Green’s function technique[@Ref9; @Ref11] are developed covering both the adiabatic and nonadiabatic situations. To this date, the pumping behavior in almost all novel quantum states has been studied, from the quantum Hall liquids[@Ref34] to superconductors[@Ref36; @Ref37], from graphene[@Ref17; @Ref18; @Ref19] to topological insulators[@Ref35; @Ref20], and etc. Along with theoretical development, quantum charge and spin pumps were realized in various nanoscale transport systems such as the quantum dot[@Ref14; @Ref15; @Ref25], superconductors[@Ref16; @Ref36; @Ref37], spin pumping driven by precessing ferromagnet[@Ref26], and etc. The quantum pumping process acts as a platform to display quantum novelties of different quantum states and structures. As a bifurcation of quantum pumping, research on spin pumping with its counterpart spin transfer torque prospers in its own direction due to its promising spintronic applications[@Ref22]. Despite its practical significance, one naturally wonders its role in revealing unknown properties of novel magnetic structures such as domain walls[@Ref38], multiferroic helimagnets[@Ref39; @Ref40], and skyrmion lattices[@Ref41] etc. Usually spin pumping was investigated in ferromagnet-based multi-layer structures driven by magnetization precession in which spin polarization is uniform in space. In these structures spin momentum flow can be generated with vanishing net charge current. Tserkovnyak et al. considered[@Ref45] the time-dependent magnetic order parameter driven quantum pumping properties in helimagnets and analyzed the evolution of the magnetic spiral, in which the broken symmetry is different from the precessing ferromagnet driven helimagnet heterostructures. Spatially nonuniform spin structure induces diffracted transmission in spiral helimagnets[@Ref42; @Ref43]. It can be predicted that the skyrmion lattice should display sophisticated transmission spectra due to its rich Fourier components of the spin vortex in space. Little in literature addressed the quantum pumping properties featured by diffraction. The topic interests us a lot. So in this work, we consider the spin pumping properties in normal-metal/multiferroic-helimagnet/ferromagnet heterostructures and investigate the physical properties induced by diffracted transmission or what can be coined as spatial nonadiabaticity and the approach can be extended to the skyrmion-lattice-based heterostructures. Theoretical formulation ======================= We consider the normal-metal/multiferroic-helimagnet/ferromagnet (NM/MF/FM) triple-layer heterostructure depicted in Fig. 1. The Hamiltonians in different layers can be formulated as: $$\begin{array}{c} \begin{array}{*{20}c} {H_{\texttt{NM}} = - \frac{{\hbar ^2 }}{{2m_e }}\nabla ^2 ,} & {z < 0,} \\ \end{array} \\ \begin{array}{*{20}c} {H_{\texttt{MF}} = - \frac{{\hbar ^2 }}{{2m^* }}\nabla ^2 + J{\bf{n}}_{\bf{r}} \cdot {\bf{\sigma }} + V_0 ,} & {0 \le z \le d,} \\ \end{array} \\ \begin{array}{*{20}c} {H_{\texttt{FM}} = - \frac{{\hbar ^2 }}{{2m_e }}\nabla ^2 - \Delta {\bf{m}} \cdot {\bf{\sigma }},} & {z > d,} \\ \end{array} \\ \end{array} \label{eq1}$$ where $d$ is the thickness of the MF layer. $m_{e}$ and $m^{*}$ are the free and multiferroic oxides’ effective electron masses respectively. ${\bf{\sigma}}$ is the Pauli vector. ${\bf{m}} = \left[ {\sin \theta \cos \phi ,\sin\theta \sin \phi ,\cos \theta } \right]$ is the magnetization unit vector in the FM layer with respect to the \[100\] crystallographic direction. $\Delta$ is the half width of the Zeeman splitting in the FM electrode. $J{\bf{n}}_{\bf{r}} $ is the space-dependent exchange field following the helicity of the MF spiral with ${\bf{n}}_{\bf{r}} = \left[ {\sin \theta _r ,0,\cos \theta _r } \right]$, $\theta _r = \bar q_m \cdot {\bf{r}}$, and $\bar q_m = \left[ {\bar q,0,0} \right]$. From Eq. (\[eq1\]) it can be seen that the exchange coupling between the electron and the localized noncollinear magnetic moments within the barrier acts as a nonhomogenous magnetic field. Therefore, spin-dependent diffraction of transmission can be foreseen in the situation. We consider an ultrathin film of MF-helimagnet with thickness $d=2$ nm, which can be approximated by a Dirac-delta function. The MF barrier reduces to a plane barrier. Its Hamiltonian can be rewritten as $$H_{MF} = - \frac{{\hbar ^2 }}{{2m^* }}\nabla ^2 + \left( {\tilde J{\bf{n}}_r \cdot {\bf{\sigma }} + V_0 d} \right)\delta \left( z \right),$$ where we assume a single spiral layer. $\tilde J = \left\langle {J\left( z \right)} \right\rangle d $ refers to space and momentum averages of the exchange coupling strength. It should be noted that the helimagnetic field is sinusoidally space dependent. A multichannel-tunneling picture should be considered and integer numbers of the helical wave vector $\bar q$ could be absorbed or emitted in transmission and reflection. With a plane wave incidence, the general wave function of the incident, transmitted and reflected electrons can be written as $$\psi _{\texttt{NM}}^\sigma \left( x,y,z \right) = e^{ik_x x} e^{ik_y y} e^{ik_z z} \chi _\sigma + e^{ik_y y} \sum\limits_{\sigma ',n} {r_n^{\sigma \sigma '} e^{i {k_{x}^{n} } x} e^{ - ik_z^n z} \chi _{\sigma '} } ,$$ $$\psi _{\texttt{FM}}^\sigma \left( x,y,z \right) = e^{ik_y y} \sum\limits_{\sigma ',n} {t_n^{\sigma \sigma '} e^{i {k_x^n} x} e^{ik_z^{n\sigma '} z} \chi _{\sigma '} } ,$$ with ${k_z} = \sqrt {2mE} \cos {\theta _{\texttt{in}}}/\hbar,$ ${k_x} = \sqrt {2mE} \sin {\theta _{\texttt{in}}}\cos {\phi _{\texttt{in}}}/\hbar,$ ${k_y} = \sqrt {2mE} \sin {\theta _{\texttt{in}}}\sin {\phi _{\texttt{in}}}/\hbar,$ $\theta_{\texttt{in}}$ and $\phi _{\texttt{in}}$ the incident polar and azimuthal angles, respectively. Here, $n$ is an integer ranging from $ - \infty $ to $ \infty $ indexing the diffraction order. And $ k_z^n = \sqrt {2mE - \hbar ^2 k_y^2 - \hbar ^2 (k_x^n) ^2 }/\hbar $, $k_z^{n\sigma } = \sqrt {2mE + \sigma \Delta - \hbar ^2 k_y^2 - \hbar ^2 (k_x^n) ^2 }/ \hbar $, $k_x^n = k_x + n\bar q$. By frame rotation, the FM eigenspinors can be obtained as $${\chi _ + } = \left( {\begin{array}{*{20}{c}} {\cos \frac{\theta }{2}{e^{ - \frac{{i\phi }}{2}}}}\\ {\sin \frac{\theta }{2}{e^{\frac{{i\phi }}{2}}}} \end{array}} \right),\begin{array}{*{20}{c}} {}&{{\chi _ - } = \left( {\begin{array}{*{20}{c}} { - \sin \frac{\theta }{2}{e^{ - \frac{{i\phi }}{2}}}}\\ {\cos \frac{\theta }{2}{e^{\frac{{i\phi }}{2}}}} \end{array}} \right),} \end{array} \label{eq2}$$ corresponding to an electron spin parallel ($\sigma =+$) and antiparallel ($\sigma =-$) to the magnetization direction in the FM electrode, whose gauge is the same with eigenspinors $${\chi _ \uparrow } = \left( {\begin{array}{*{20}{c}} 1\\ 0 \end{array}} \right),\begin{array}{*{20}{c}} {}&{{\chi _ \downarrow } = \left( {\begin{array}{*{20}{c}} 0\\ 1 \end{array}} \right),} \end{array} \label{eq3}$$ of $\sigma _z$. Besides Eq. (\[eq2\]), arbitrary gauge selection generates an arbitrary eigenspinor of ${\bf{m}}$ as $$\chi _ + ^f = \left( {\begin{array}{*{20}{c}} {\cos \frac{\theta }{2}{e^{ - \frac{{i\phi }}{2}}}}\\ {\sin \frac{\theta }{2}{e^{\frac{{i\phi }}{2}}}} \end{array}} \right)f\left( \phi \right),\begin{array}{*{20}{c}} {}&{\chi _ - ^g = \left( {\begin{array}{*{20}{c}} { - \sin \frac{\theta }{2}{e^{ - \frac{{i\phi }}{2}}}}\\ {\cos \frac{\theta }{2}{e^{\frac{{i\phi }}{2}}}} \end{array}} \right)g\left( \phi \right),} \end{array} \label{eq5}$$ with $f\left (\phi \right)$ and $g\left (\phi \right)$ arbitrary unitary complex functions (${\left| {f\left( \phi \right)} \right|^2} = 1$ and ${\left| {g\left( \phi \right)} \right|^2} = 1$). In the plane ($x$-$y$ plane) perpendicular to the transport direction ($z$-axis), free motion of the electron is assumed. Diffraction appears in the $x$-direction and $k_y$ is conserved under translational invariance. The reflection ($r_n^{\sigma \sigma '}$) and transmission ($t_n^{\sigma \sigma '}$) amplitude in the $n$th diffraction order can be numerically obtained from the continuity conditions[@Ref40] for $\Psi (x,y,z)$ at $z=0$. $$\Psi _{NM}^\sigma \left( {x,y,0^ - } \right) = \Psi _{FM}^\sigma \left( {x,y,0^ + } \right),$$ $$\begin{array}{c} \frac{{\hbar ^2 }}{{2m_e }}\left. {\frac{{\partial \Psi _{NM}^\sigma \left( {x,y,z} \right)}}{{\partial z}}} \right|_{z = 0^ - } + \left[ {V_0 d + {\bf{\tilde w}}\left( {\theta _r } \right)} \right]\Psi _{NM}^\sigma \left( {x,y,0^ - } \right) \\ = \frac{{\hbar ^2 }}{{2m_e }}\left. {\frac{{\partial \Psi _{FM}^\sigma \left( {x,y,z} \right)}}{{\partial z}}} \right|_{z = 0^ + } , \\ \end{array}$$ with $${\bf{\tilde w}}\left( {\theta _r } \right) = \tilde J\left[ {\begin{array}{*{20}c} {\cos \theta _r } & {\sin \theta _r } \\ {\sin \theta _r } & { - \cos \theta _r } \\ \end{array}} \right].$$ The continuity equation can be expressed in each diffracted order. Transmissivity of a spin-$\sigma$ electron through the MF tunnel junction with the incident wave vector $[k_x,k_y,k_z]$ to the $n$-th diffracted order and spin-$\sigma '$ channel with the outgoing wave vector $[k_x^n,ky,k_z^{n\sigma '}]$ reads $$T_n^{\sigma \sigma '} \left( {E,k_y ,\theta _{xz} } \right) = \frac{{\texttt{Re}\left( {k_z^{n\sigma '} } \right) }}{{ k_z }}\left| {t_n^{\sigma \sigma '} } \right|^2.$$ The scattering matrix can be expressed as $$\left( {\begin{array}{*{20}{c}} {{b_{L \uparrow }}}\\ {{b_{L \downarrow }}}\\ {{b_{R \uparrow }}}\\ {{b_{R \downarrow }}} \end{array}} \right) = {U^{fg \dag} }\left( {\begin{array}{*{20}{c}} {{r_{ + + }}}&{{r_{ - + }}}&{t{'_{ + + }}}&{t{'_{ - + }}}\\ {{r_{ + - }}}&{{r_{ - - }}}&{t{'_{ + - }}}&{t{'_{ - - }}}\\ {{t_{ + + }}}&{{t_{ - + }}}&{r{'_{ + + }}}&{r{'_{ - + }}}\\ {{t_{ + - }}}&{{t_{ - - }}}&{r{'_{ + - }}}&{r{'_{ - - }}} \end{array}} \right)U^{fg}\left( {\begin{array}{*{20}{c}} {{a_{L \uparrow }}}\\ {{a_{L \downarrow }}}\\ {{a_{R \uparrow }}}\\ {{a_{R \uparrow }}} \end{array}} \right),$$ where the primed terms indicate inversive transport and $U^{fg}$ is the transform matrix from ${\sigma _z}$ to ${\bf{m}}$ representation in arbitrary gauge. $${U^{fg}} = \left( {\begin{array}{*{20}{c}} 1&0\\ 0&1 \end{array}} \right) \otimes \left( {\begin{array}{*{20}{c}} {\cos \frac{\theta }{2}{e^{ - i\frac{\phi }{2}}}f}&{\sin \frac{\theta }{2}{e^{i\frac{\phi }{2}}}f}\\ { - \sin \frac{\theta }{2}{e^{ - i\frac{\phi }{2}}}g}&{\cos \frac{\theta }{2}{e^{i\frac{\phi }{2}}}g} \end{array}} \right).$$ We consider spin pumping driven by ferromagnet magnetization precession (see Fig. 1), so this transformation is necessary. The time-dependent parameter is $\phi$, which is both a physical quantity and also a gauge factor in $f$ and $g$. Transport of different diffraction order would not be correlated by time-dependent variation. Therefore, the adiabatically pumped $2 \times 2$ tensor current in the NM electrode can be calculated by[@Ref3; @Ref22] $${\hat I} = \frac{{e\omega }}{{4{\pi ^2}}}\int_0^{2\pi } {dXk_{\max }^2\sin {\theta _{\texttt{in}}}d{\theta _{\texttt{in}}}d{\phi _{\texttt{in}}}\sum\limits_{m = - \infty }^\infty {{\mathop{\rm Im}\nolimits} {{\left[ {\hat S_m^\dag \frac{{\partial {{\hat S}_m}}}{{\partial X}}} \right]}_{LL}}} } ,$$ where ${k_{\max }} = \sqrt {2mE} /\hbar $ and $m$ is the diffraction order. The pumped charge and spin current follows as $$\begin{array}{l} {I_c} = {{\hat I}_{11}} + {{\hat I}_{22}},\\ I_s^z = {{\hat I}_{22}} - {{\hat I}_{11}},\\ I_s^x = - \left( {{{\hat I}_{12}} + {{\hat I}_{21}}} \right),\\ I_s^y = - i\left( {{{\hat I}_{12}} - {{\hat I}_{21}}} \right), \end{array}$$ with the spin angular momentum flow defined in $\hbar {\bf{\sigma }} \cdot {\bf{I}}/2$. Numerical results and interpretations ===================================== We consider spin pumping driven by magnetization precession in the NM/MF/FM heterostructure (see Fig. 1). In numerical calculations, the NM Fermi energy $E_F$ is chosen to be $5.5$ eV. Different values of $E_F$ would not change the primary pumping properties. The spatial average of the helimagnetic exchange coupling strength $\tilde J = 0.2$ ${\rm{eV}} \cdot {\rm{nm}}$, which is reasonable compared to the Fermi energy. Periods of short-period and long-period helimagnets are $3$-$6$ nm and $18$-$90$ nm, respectively[@Ref44]. In our model we set the period to be $10$ nm and hence $\bar q=2 \pi /10$ $\texttt{nm} ^{-1}$. The magnetization of the FM electrode precesses anticlockwise around the $z$-axis. Zeeman splitting in the FM electrode $\Delta =2$ eV. Barrier height of the MF oxide plane $V_0 =0.5$ eV and width $d=2$ nm. We consider diffraction orders of $0$ and $\pm 1$ and numerically proved that keeping the three orders is sufficient for physical $\bar q$’s as higher orders decrease exponentially. The magnitude of the pumped current is in the order of $10^{-17}$ A. So it is far below the strength to rotate the helimagnet spiral or induce Gilbert damping in the FM electrode. During transmission, the incident electron with wave vector $[k_x,k_z]$ would absorb or emit $n \bar q$ from the helimagnet and be diffracted into tunnels with wave vector $[k_x^n,k_z^{n \sigma}]$. Numerical results of the transmission of a single incident beam with $\theta _{\texttt{in}}$ and $\phi _{\texttt{in}}$ fixed are shown in Figs. 1 and 2. It can be seen that zero-order spectrum governs the spin-conserved transmission whereas first-order diffracted spectrum governs the spin-flipped transmission due to the grating effect in the spin space. The second-order spectrum is four orders smaller, which justifies the $0$, $\pm 1$ order cutoff. The exponential decrease in diffracted orders agrees with general grating properties. The $0$, $\pm 1$ order cutoff was also justified by numerical confirmation of the unitarity of the scattering matrix $\hat S$ including the $0$, $\pm 1$ orders. The transmission of one-way incident light through sinusoidal gratings is delta-function-like strict lines. Analogously, direction of transmission of one-way incident electron through sinusoidal helimagnet is discrete strict lines of different grating orders and the spin is conserved or flipped. For arbitrary $\textbf{m}$ relative to the chirality of the helimagnet, $+ 1$ and $-1$ order transmission may not be symmetric. Physically, an electron with spin polarization along the FM magnetization is transmitted in different direction with its spin rotating an angle. It can be seen from Fig. 2 that the $1$ and $-1$ order diffracted transmission varies with $\phi $ in trigonometric functions. The $0$ order transmission varies with $\phi $ much more slowly than the diffracted $\pm 1$ orders, which can be clearly seen from the scale extension in Fig. 3. Also the $0$ order transmission varies with $\phi $ in trigonometric functions with higher harmonics and the variation range is one to two digits smaller than the diffracted $\pm 1$ orders. These observations give rise to the pumping properties driven by cyclic modulation of $\phi$, which is shown in Fig. 4. Two prominent properties highlight the pumping physics in the NM/MF/FM heterostructure. The first is the non-vanishing pumped charge current. It is well known that in FM based tunnel junctions, nonzero spin current with zero charge current can be generated by cyclic magnetization precession. In the latter structure, spatial uniformity gives rise to exactly the same charge scattering matrix for different magnetization azimuthal angles. And transmission difference in the spin space accumulates during precession. However the spatial symmetry is broken by the helimagnet spiral. Non-zero spin as well as non-zero charge current is driven out by ferromagnet precession. The other prominent property is gauge dependence of the pumped charge and spin currents. The considered situation is special in two points. The first is spatial nonuniformity. As a result, spin-dependent diffraction occurs. And also the diffraction effect cannot be averaged out by the cyclic integral of $\phi$. The second one is that the time-dependent parameter $\phi$ is physically the precession angle and nonphysically the gauge factor simultaneously. The two roles play independently. This point makes distinct demonstration at the two poles of the precession sphere. Fig. 4 shows the pumped charge and spin current using the eigenspinors of Eq. (\[eq2\]) and Fig. 5 shows its order expansion of one incident electron beam. It can be seen that the pumped current does not vanish at the two poles of the precession sphere. We have calculated the transmission spectrum at the two poles and find that all of the three orders of the transmission probabilities do not vary with the magnetization azimuthal angle $\phi$ at the two poles and and the nonzero pumped current is a pure result of the phase of the transmission, the latter of which is not unusual in quantum pumps. Also we replace the MF helimagnet layer by a plane delta barrier and reobtained the magnetization-precession-driven pumped spin current of the FM tunnel junction and it vanishes at the two poles of the precession sphere. From Fig. 5 it can be seen that the nonzero pumped current at the two poles is a pure effect of diffraction with the zero-order pumped vanishing at the two poles. Although the numerical results and analysis given above seem self-consistent, it is nonphysical. Precession at zero precession angle makes no sense and to say the least the area of the cyclic loop vanishes. Eq. (\[eq2\]) is the eigenspinor obtained by frame rotation. It should have the same gauge phase as the eigenspinor (\[eq3\]) of ${\sigma _z}$. However, if we make $\theta =0$ in Eq. (\[eq2\]), it could not return to Eq. (\[eq3\]). The remaining $\phi$ is a pure gauge phase but over the cyclic loop integral its effect accumulates instead of cancelling out giving rise to nonvanishing pumped current at zero $\theta$. The problem could be solved by the gauge transformation Eq. (\[eq5\]) with $$f\left( \phi \right) = {e^{\frac{{i\phi }}{2}}},\begin{array}{*{20}{c}} {}&{g\left( \phi \right) = {e^{ - \frac{{i\phi }}{2}}}.} \end{array} \label{eq4}$$ Numerical results of the pumped current in this gauge are shown in Fig. 6. It can be seen that the pumped current vanishes at the $\theta =0$ pole. It does not vanish at the $\theta = \pi$ pole since the gauge of Eq. (\[eq4\]) also could not return to the eigenspinors of $-\sigma _z$, which are the swap of the two spinors of Eq. (\[eq3\]). Also under gauge transformation Eq. (\[eq5\]) with $$f\left( \phi \right) = {e^{-\frac{{i\phi }}{2}}},\begin{array}{*{20}{c}} {}&{g\left( \phi \right) = {-e^{ \frac{{i\phi }}{2}}},} \end{array} \label{eq6}$$ the pumped current vanishes at the $\theta =\pi$ pole and is nonzero at the $\theta=0$ pole, which is not shown in the manuscript to avoid tediousness. In consideration of the pumping properties, gauges (\[eq5\]) and (\[eq2\]) are equivalent only when $f(\phi)=g(\phi)$. In the two cases the relation is not satisfied. The gauge selection in eigenspinors seldom matters. But it really does sometimes. Therefore the numerical results raised a question in the considered situation: which gauge is appropriate and why? We would like to argue that the question is nontrivial in at least two aspects. The first is that gauge difference in the pumped current only occurs in the contribution from diffracted orders. We show the order expansion of the pumped current of gauge Eq. (\[eq4\]) in Fig. 7. It can be seen that the zero order contribution of the pumped current is both qualitatively and quantitatively identical to that of the rotation-frame gauge in Fig. 5. Obtained numerical results of other gauges show the same properties. This is probably the reason why the gauge difference is not encountered in spatially uniform quantum pumps without diffraction. The second is that physically sound results at the two poles of the precession sphere can only be achieved by two un-equivalent gauges. Usually in situations when the intrinsic phase of eigenspinors matters frame rotation gauge of Eq. (\[eq2\]) is thought to secure correctness. However here it leads to nonphysical results at the two poles. From Figs. 4 to 7, it can be seen that diffraction induced gauge dependence of the pumped current is spectacular, which could not be explained by existed theory to our knowledge. Conclusions =========== In this work, we considered the adiabatically pumped charge and spin current driven by precessing ferromagnet in the NM/MF-helimagnet/FM heterostructure. In this structure, space modulation of the helimagnet spin spiral gives rise to diffraction in the transmission spectrum. It is found that the usually neglected gauge difference in the ferromagnet eigenspinor cannot be overlooked trivially. Usually ubiquitous gauge phase obtained by frame rotation of the $\sigma _z$ eigenspinors gives non-vanishing pumped current at the two poles of the precessing sphere, which makes no sense. We numerically confirmed the unitarity of the scattering matrix of the combined zero and $\pm 1$ order diffracted states, which justifies the three order cutoff. We also show that the gauge dependence is a pure effect of diffraction with the zero order contribution gauge independent. 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Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. Böni, Science **323**, 915 (2009). A. Manchon, N. Ryzhanova, A. Vedyayev, and B. Dieny, J. Appl. Phys. **103**, 07A721 (2008); R. Zhu, arXiv:1204.6095. N. Kanazawa, Y. Onose, T. Arima, D. Okuyama, K. Ohoyama, S. Wakimoto, K. Kakurai, S. Ishiwata, and Y. Tokura, Phys. Rev. Lett. **106**, 156603 (2011). {height="14cm" width="16cm"} {height="10cm" width="14cm"} {height="10cm" width="14cm"} ![Pumped charge and spin currents as a function of the precession angle in the gauge of Eq. (\[eq2\]). Inset is the zoom-in of the pumped spin angular momentum flow in the $x$ and $y$ directions.](Fig4.eps){height="10cm" width="14cm"} ![Order expansion of the pumped charge and spin currents as a function of the precession angle in the gauge of Eq. (\[eq2\]). Shown is results of a single incident beam with $\theta_{\texttt{in}}=0.5$ and $\phi_{\texttt{in}}=0.2$ in radian.](Fig5.eps){height="10cm" width="14cm"} ![Pumped charge and spin currents as a function of the precession angle in the gauge of Eq. (\[eq4\]). $I_{sx}$ and $I_{sy}$ are zero to the accuracy of $10^(-10)$ order.](Fig6.eps){height="10cm" width="14cm"} ![Order expansion of the pumped charge and spin currents as a function of the precession angle in the gauge of Eq. (\[eq4\]). Shown is results of a single incident beam with $\theta_{\texttt{in}}=0.5$ and $\phi_{\texttt{in}}=0.2$ in radian. ](Fig7.eps){height="10cm" width="14cm"} [^1]: Corresponding author. Electronic address: rzhu@scut.edu.cn

--- abstract: 'Using the spin-hole coherent state representation and taking a long range antiferromagnetic Nèel order as a background of the localized spin degree part, we have studied the normal state behavior of the t-J model, and shown that a strongly short-range antiferromagnetic correlation of the localized spin degree part is responsible for the anomalous non-Korringa-like relaxation behavior of the planar copper spin, the Korringa-like behavior of the planar oxygen spin may derive from the charge degree part describing a Zhang-Rice spin-singlet; The charge degree part feels a strongly staggered magnetic field induced by this short-range antiferromagnetic correlation as a doping hole hopping, this staggered magnetic field enforces the charge degrees to have different responses to external magnetic and electric fields and to show two relaxation rate behaviors corresponding to the planar resistivity and Hall angle, respectively. We have found that the temperature dependence of magnetoresistance is $T^{-n}$, $n\simeq 3$, near the optimal doping, $n\simeq 4$, in the underdoping region, violating Kohler’s rule, the transport relaxation rate is of the order of $2k_{B}T$, all that are consistent with the normal state of the cuprate superconductors.' address: 'International Center for Theoretical Physics, P. O. Box 586, 34100 Trieste, Italy' author: - 'Yu-Liang Liu' title: 'Normal State Property of the t-J Model' --- Introduction ============ Recently, the significant progress has been made in the understanding of the low energy spin dynamics of the normal state of the cuprate superconducting materials in both theoretical[@1]-[@4b] and experimental[@5]-[@7b] aspects. In the undoping case, the spin dynamics of the cuprates, such as $La_{2}CuO_{4}$, is well described by the quantum Heisenberg model on a square lattice of $Cu$ sites. The authors of Refs.\[1,2\] have extensively studied it by using the scalling and renormalization group theory and /or large-N expansion methods, and have given some valuable results which are in good agreement with the current experimental data. However, in the doped case, up to now there is not a general consensus on choosing a microscopic theory qualitatively to describe the unusually magnetic and transport properties of the normal state over the entire doping range from insulator to high doped compounds, although many models have been proposed to describe them. In all hole-type cuprates near optimal doping, the mostest important properties of the normal state are that: a). The linear dependence of the in-plane resistivity $\rho$ on temperature (T) has been confirmed from $T_{c}$ up to temperature as high as 1000K[@7c]. b). The in-plane Hall resistivity $\rho_{xy}=R_{H}B$ varies strongly over a wide range of T ($B$, external magnetic field; $R_{H}$, the Hall coefficient), it falls as $\sim T^{-1}$ between $T_{c}$ and temperature as high as 500K. Anderson[@7d] prodicted that the cotangent of the Hall angle should vary with impurity content $n_{i}$ as $cot\theta_{H}=\alpha T^{2}+\beta n_{i}$, which was immediately confirmed by Ong[@7e] and others[@7f][@7g]. c). The magnetoresistance $\Delta\rho/\rho\propto B^{2}T^{-n},\; n\simeq 3\sim 4$, strongly violates Kohler’s rule[@7i][@7j]. d). The relaxation behaviors at various sites show sharply contrast. The relaxation behavior of the planar oxygen obeys Korringa-like behavior, while the planar copper sharply shows non-Korringa behavior. e). Dynamical antiferromagnetic correlations persist in all the metallic state and the superconducting state. These properties provide significant constraints on candidate descriptions of their anomalous normal state behavior. The items a),b),c) strongly indicate that there exist two different relaxation times in the system[@7d] $1/\tau_{tr}\sim T$, and $1/\tau_{H}\sim T^{2}$. While the items d),e) strongly show that among the planar copper spins there exists a strongly commensurate antiferromagnetic correlation, the planar oxygen, residing at the middle point between two nearest neighbor coppers, is not affected by this commensurate antiferromagnetic correlation and shows the Korringa-like relaxation behavior, at least it is true for the $YBa_{2}Cu_{3}O_{6+x}$ materials. For the $La_{2-x}Sr_{x}CuO_{4}$ system, although the neutron scattering experiments show at low temperature four incommensurate peaks in the spin fluctuation spectrum, whose position depends on the level of $Sr$ doping[@7l], the nuclear magnetic resonance experiment shows that the property d) is remaining invariance[@7b]. The unusually physical properties of the normal state of the cuprate superconducting materials may originate from their strongly antiferromagnetic correlation. The doping will destroy the long range antiferromagnetic correlation, but the system still maintains a strongly short range antiferromagnetic correlation. In Refs. 4-6, we have given a detail study following this idea, and obtained some results which can qualitatively explain the unusually physical properties of the normal state. In this paper, using the similar method as in Refs. 4-6, we study the normal state behavior of the t-J model. The t-J model carries on the important electronic strongly correlated property of the cuprate superconducting materials, through completing studying its property we hope to get more understanding of the strongly correlated electronic system and the cuprate superconducting materials. It is well-known that the gauge theory of the t-J model[@8]-[@10] gives a better description to the temperature dependence of the in-plane resistivity of the normal state, but up to now one has not known whether it can also give a reasonable description to the temperature dependence of the Hall coefficient and the magnetic behavior of the normal state, i.e., can it show not only the two relaxation time behaviors but also the strongly antifrromagnetic correlation behavior? In this paper, we show that the unusual magnetic behavior of the normal state is induced by a strongly short-range antiferromagnetic correlation among the localized spin degrees on the copper sites, while because of there existing this strongly short-range antiferromagnetic correlation in the localized spin degree part, the charge degree part will feel a strong staggered magnetic field as the doping hole hopping, this staggered magnetic field drastically influences the behavior of the charge degree part, and enforces the charge degree part to have different responses to external magnetic field and electric field and to show two relaxation time behaviors corresponding to the in-plane resistivity and Hall angle, respectively. In the usual slave boson(or fermion) description of the t-J model, the spin degree and the charge degree of electrons are separated, the spin degree part effectively describes the localized spins on the copper sites, while the charge degree part effectively describes the spin-singlet (or Zhang-Rice singlet) of the doping hole spin and the copper spin[@10a]. However, because the strongly dynamical antiferromanetic correlations persist in all the metal state, it is reasonable that we use the long range antiferromagnetic Nèel order of the localized spin degerees as our starting point to study the normal state behavior of the t-J model. Spin-hole coherent state representation and spin-charge separation ================================================================== We adopt an usual method to deal with the single occupation condition by introducing a slave fermion, so the Hamiltonian of the t-J model can be written as in a hole representation $$\begin{array}{rl} H= & t\displaystyle{\sum_{<ij>}}(f^{+}_{j}f_{i}b^{+}_{i\sigma}b_{j\sigma} +h.c)\\ & +J\displaystyle{\sum_{<ij>}}(1-f^{+}_{i}f_{i})\hat{S}_{i}\cdot\hat{S}_{j} (1-f^{+}_{j}f_{j})+\displaystyle{\sum_{i}}\lambda_{i}(1-f^{+}_{i}f_{i} -b^{+}_{i\sigma}b_{i\sigma}) \end{array}$$ where $\hat{S}_{i}=\frac{1}{2}b^{+}_{i\alpha}\hat{\sigma}_{\alpha\beta} b_{i\beta}$, $b_{i\sigma}$ is a hard-core boson operator which describes the spin degree of the electron, and $f_{i}$ is a fermion operator which describes the charge degree of the electron. The electron operator is $c_{i\sigma}=f^{+}_{i}b_{i\sigma}$, $\lambda_{i}$ is a Lagrangian multiplier which ensures the single occupation condition of the electrons. In the spin-hole coherent state representation introduced by Auerbach[@11] $$|\hat{\Omega}, \xi>_{S}\equiv |\hat{\Omega}>_{S}\otimes |0>_{f}+ |\hat{\Omega}>_{S-\frac{1}{2}}\otimes\xi f^{+}|0>_{f}$$ where $|\hat{\Omega}>_{S}$ is a spin coherent state[@12] and $\xi$ is an anticommuting Grassmann variable, the partition functional of the Hamiltonian (1) can be written as $$Z=\int D\hat{\Omega}D\xi^{*}D\xi exp\{-\int^{\beta}_{0}[{\it L}_{\Omega}+ {\it L}_{\xi}]\}$$ $${\it L}_{\Omega}=-i\sum_{i}2S\omega_{i}+JS^{2}\sum_{<ij>}(1-\xi^{*}_{i} \xi_{i})\hat{\Omega}_{i}\cdot\hat{\Omega}_{j}(1-\xi^{*}_{j}\xi_{j})$$ $${\it L}_{\xi}=\sum_{i}\xi^{*}_{i}(\partial_{\tau}+i\omega_{i}+\mu_{i}) \xi_{i}+\sqrt{2}tS\sum_{<ij>}(\xi^{*}_{j}\xi_{i}e^{i\gamma_{ij}}\sqrt{ 1+\hat{\Omega}_{i}\cdot\hat{\Omega}_{j}}+h.c)$$ where the Berry phase $\omega$ is a functional of the spin order parameter $\hat{\Omega}(\tau)$. It is ambiguous modulo $4\pi$, and its functional derivative is quite well-behaved[@12] $$\int d\tau\delta\omega=\int d \tau\hat{\Omega}\cdot(\partial_{\tau} \hat{\Omega}\times\delta\hat{\Omega})$$ The parameter $\mu_{i}$ is a chemical potential of the slave fermion $\xi$, $\gamma_{ij}$ is the phase factor of $_{S}<\hat{\Omega}|b^{+}_{i\sigma}b_{j\sigma}|\hat{\Omega}>_{S}$. The Lagrangian ${\it L}_{\xi}$ is invariant under following gauge transformations $$\xi_{i}\rightarrow\xi_{i}e^{i\theta_{i}},\;\; \gamma_{ij}\rightarrow\gamma_{ij}-\theta_{i}+\theta_{j},\;\; \mu_{i}\rightarrow\mu_{i}+i\partial_{\tau}\theta_{i}$$ which derives from the slave fermion representation of the electron operator $c_{i\sigma}=f^{+}_{i}b_{i\sigma}$. The single occupation condition in (1) disappears in (4) and (5), because in the spin-hole coherent state representation the term $(1-f^{+}_{i}f_{i}-b^{+}_{i\sigma}b_{i\sigma})$ is equal to zero at each site. From the equations (4) and (5), we see that the Lagrangian ${\it L}_{\Omega}$ dominates the antiferromagnetic behavior of the system, then the Lagrangian ${\it L}_{\xi}$ dominates the ferromagnetic behavior (or destroys the antiferromagnetic behavior) of the system because the factor $\sqrt{1+\hat{\Omega}_{i}\cdot\hat{\Omega}_{j}}$ is zero for antiferromagnetic order and is biggest for ferromagnetic order. According to the current experimental data of the cuprate superconducting materials, almost all of them show a strongly short range antiferromagnetic behavior in the normal state, even in the superconducting state, the short range antiferromagnetic behavior also appears. Therefore, according to this fact, we take a long range antiferromagnetic Néel order as a background of the spin order parameter $$\hbar S\hat{\Omega}_{i}\simeq\hbar\eta_{i}\hat{\Omega}(x_{i}) +a^{2}\hat{L}(x_{i})$$ where $a^{2}$ is the unit cell volume, $\hat{\Omega}(x_{i})$ is the slowly varying Néel unit vector order, i.e., spin parameter field $|\hat{\Omega}(x_{i})|=1$, and $\hat{L}(x_{i})$ is the slowly varying magnetization density field, $\hat{\Omega}(x_{i})\cdot\hat{L}(x_{i})=0$. The Berry phase term may be separated into two parts $$S\sum_{i}\omega_{i}\simeq S\sum_{i}\eta_{i}\omega(x_{i})+ \frac{1}{\hbar}\int d^{2}x\hat{\Omega}\cdot(\frac{\partial\hat{\Omega}} {\partial\tau}\times\hat{L})$$ where $\omega(x)$ is the solid angle subtended on the unit sphere by the closed curve $\hat{\Omega}(x,\tau)$ (parametrized by $\tau$). Because of in the long range antiferromagnetic Néel order approximation, the electron hoping must be accompanied with a $\pi-phase$ rotation in spin space to match with the nextest neighbor spin orientations, so the t-term in (1) must be changed as $$\begin{array}{rl} f^{+}_{i}f_{j}b^{+}_{j\sigma}b_{i\sigma}=& \displaystyle{e^{-2i\displaystyle{\sum_{l\neq i,j}\theta_{ij}(l)S^{z}_{l}}}}f^{+}_{i} f_{j}\displaystyle{ e^{2i\displaystyle{\sum_{l\neq i,j}\theta_{ij}(l)S^{z}_{l}}}} b^{+}_{j\sigma}b_{i\sigma}\\ =& \displaystyle{e^{-2i\displaystyle{\sum_{l\neq i,j}\theta_{ij}(l) S^{z}_{l}}}}f^{+}_{i}f_{j}\tilde{b}^{+}_{j\sigma}\tilde{b}_{i\sigma} \end{array}$$ where $\theta_{ij}(l)=\theta_{i}(l)-\theta_{j}(l)$, $\theta_{i}(l)$ is an angle between the direction from site $i$ to site $l$ and some fixed direction, the $x$ axis for example; $S^{z}_{l}=\frac{1}{2}b^{+}_{l\alpha} \sigma^{z}_{\alpha\beta}b_{l\beta}$, the $z$-component of the spin operator; $\tilde{b}_{i\sigma}=e^{2i\sum_{l\neq i}\theta_{i}(l)S^{z}_{l}}b_{i\sigma}$, is a fermion operator. Under the approximations (8) and (9), and eliminated the magnetization density field $\hat{L}(x)$, the Lagrangians in (4) and (5) can be written as, respectively $${\it L}_{\Omega}=\frac{1}{2g_{0}}\int d^{2}x[(\vec{\partial}\hat{\Omega})^{2}+\frac{1}{c^{2}}(\partial_{\tau} \hat{\Omega})^{2}]$$ $${\it L}_{\xi}= \displaystyle{\sum_{i}}\xi^{*}_{i}(\partial_{\tau}-\mu_{i}) \xi_{i} + \sqrt{2}tS\displaystyle{\sum_{<ij>}\{\xi^{*}_{j}\xi_{i} e^{i\gamma_{ij}^{'}}[1+\eta_{i}\eta_{j}\hat{\Omega}(x_{i})\hat{\Omega} (x_{j})]^{\frac{1}{2}}}+h.c\}$$ where $\gamma^{'}_{ij}=\gamma_{ij}+\displaystyle{\sum_{l\neq i,j}} \theta_{ij}(l)_{S}<\hat{\Omega}|(b^{+}_{l\uparrow}b_{l\uparrow}- b^{+}_{l\downarrow}b_{l\downarrow})|\hat{\Omega}>_{S}$, $g_{0}=(J(1-\delta)^{2}S^{2})^{-1}$, $c^{2}=8(aJ(1-\delta)S)^{2}$. For the $J$-term in (4), we have replaced the $f^{+}_{i}f_{i}$ and $f^{+}_{j}f_{j}$ by $\delta=<f^{+}_{i}f_{i}>=<f^{+}_{j}f_{j}>$, the doping density. We have omitted the terms $\sum_{i}\eta_{i}\omega(x_{i})$ and $\sum_{i}\eta_{i}\omega(x_{i})\xi^{*}_{i}\xi_{i}$. If $\omega(x)$ is a slowly varying function of space coordinates $\vec{x}$ and “time” $\tau$ and the occupation number of the quasiparticle $\xi$ is equal at the even and odd sites, these two terms have a little contribution to the system. However, the quantity $\omega(x)$ provides an attractive interaction between the fermions $\xi_{i}$ and $\xi_{i+\hat{\delta}}$, $\hat{\delta}=(\pm a, \pm a)$, at the even and odd sites, respectively, which may induce the pairing between the slave fermions at the even and odd sites. Here we assume this effect is very small, and do not consider it, or we only consider the normal state of the system. For the strongly antiferromagnetic correlation among the spin degrees of the system, taking the Hartree-Fock approximation, the Lagrangian (12) can be written as $$\begin{array}{rl} {\it L}_{\xi}=& \displaystyle{\sum_{i}}\xi^{*}_{i}(\partial_{\tau}- \mu_{i})\xi_{i}+\sqrt{2}tS\chi\displaystyle{\sum_{<ij>}\xi^{*}_{j} \xi_{i}e^{i\gamma^{'}}}\\ +& 2atS\eta\displaystyle{\sum_{i}\xi^{*}_{i}\xi_{i}|\vec{\partial} \hat{\Omega}(x_{i})|} \label{a}\end{array}$$ where $\chi=<[1+\eta_{i}\eta_{j}\hat{\Omega}(x_{i})\cdot\hat{\Omega} (x_{j})]^{1/2}>,\;\; \eta=<e^{i\gamma_{ij}^{'}}>, \;\; |\vec{\partial} \hat{\Omega}|\equiv|\partial_{x}\hat{\Omega}|+|\partial_{y} \hat{\Omega}|$. We have omitted the fluctuation phase of the fields $\chi$ and $\eta$, and taken them as constants. The effective Hamiltonian for the charge part can be written as $$H_{\xi}=\bar{t}\sum_{<ij>}\xi^{+}_{j}\xi_{i}e^{i\gamma_{ij}^{'}}+ V\sum_{i}\xi^{+}_{i}\xi_{i}|\vec{\partial}\hat{\Omega}(x_{i})| \label{b}$$ where $\bar{t}=\sqrt{2}tS\xi, \; V=2atS\eta$. Because of the strongly antiferromagnetic correlation among the spin degrees, the spin parameter field $\hat{\Omega}(x)$ is slowly varying in the coordinate space, so the phase factor $\gamma_{ij}$ is very small and can be omitted, the phase factor $\gamma_{ij}^{'}$ is $$\gamma_{ij}^{'}=\sum_{l\neq i,j}\theta_{ij}(l)\;_{S}\!<\hat{\Omega}| b^{+}_{l\uparrow}b_{l\uparrow}-b^{+}_{l\downarrow}b_{l\downarrow} |\hat{\Omega}>_{S} \label{c}$$ which is a rapid varying quantity of the lattice sites, so generally, we cannot treat it in the continuous limit. Here we omit a gauge field $\vec{A}$ which describes the interaction between the spin and charge degree parts of the electrons. Under the spin-hole coherent state representation, if taking the long range antiferromagnetic Nèel order as a background of the spin degree part, the charge and spin degree parts are only coupled via the rapidly varying phase factor $\gamma^{'}_{ij}=\gamma_{ij}+\sum_{l\neq i,j}\theta_{ij}(l)\;_{S}\!<\hat{\Omega}|2S_{z}|\hat{\Omega}>_{S}$, $\gamma_{ij}=(\vec{x}_{i}-\vec{x}_{j})\cdot\vec{A}(\frac{x_{i}-x_{j}}{2})$ being contributed from the localized spin degree part. If the phase factor $\gamma_{ij}$ is a smooth varying function in coordinate space, this gauge field $\vec{A}$ must be massive, because the current corresponding to $\vec{A}$ must be conserved, there appears a term $<\xi^{+}_{j}\xi_{i}|\vec{\partial}\hat{\Omega}|>\cdot e^{i\gamma^{'}_{ij}}$ in equations (13) and (14), which provides a massive term to $\vec{A}$. On the other hand, it is reasonable to omit this gauge field $\vec{A}$ that for strongly antiferromagnetic correlation among the localized spin degrees, the phace factor $\sum_{l\neq i,j}\theta_{ij}(l)\;_{S}\!<\hat{\Omega}|2S_{z}|\hat{\Omega}>_{S}$ is a rapidly varying function in coordinate space, so the phase factor induced by the gauge field $\vec{A}$ can be omitted. Then for a weakly antiferromagnetic correlation case, we must consider the effect produced by this gauge field. Just done as above, we can also adopt the slave boson method to deal with the t-J model, and obtain the similar Lagrangian as (\[a\]) or effective Hamiltonian as (\[b\]) only if we consider $\xi$ as a hard-core boson field[@4b]. So we consider the Lagrangian (\[a\]) or effective Hamiltonian (\[b\]) is valid for slave fermion and boson descriptions, for slave fermion description, $\xi$ is a fermion field, for the slave boson description, $\xi$ is a hard-core boson field. Transport property of the normal state ====================================== Now we study the effective Hamiltonian (\[b\]). In Ref.[@12a], the authors have studied the effect of a strongly fluctuating gauge field on a degenerate hard-core Bose liquid, shown that the gauge fluctuation causes the boson world lines to retrace themselves, and found a transport relaxation rate of the order of $1/\tau_{tr}\sim 2k_{B}T$, consistent with the normal state of the cuprate superconductors. The results obtained in [@12a] are also valid for the effective Hamiltonian (\[b\]), because the rapidly varying phase factor $\gamma^{'}_{ij}$ provides a strongly staggered magnetic field which enforces the world lines of the slave boson (or fermion) to retrace themselves and induces the charge degrees having the order of $1/\tau_{tr}\sim 2k_{B}T$ transport relaxation rate. However, we can use this result only to explain the linear dependence of the resistivity $\rho$ on temperature. In order to study the temperature dependence of the Hall coeficient (or more important, the Hall angle), we must introduce an external magnetic field to the phase factor $\gamma^{'}_{ij}$, while because the phase factor $\gamma^{'}_{ij}$ is a rapid varying function of the coordinate space, we cannot treat it in the continuous limit. To get more valid informations, we adopt this scenario that we separate the rapidly varying phase factor $\gamma^{'}_{ij}$ into two parts $$\begin{array}{rl} \gamma^{'}_{ij}=& \gamma^{(1)}_{ij}+\gamma^{(2)}_{ij}\\ \gamma^{(1)}_{ij}=& \displaystyle{\sum_{l\neq i,j}}\theta_{ij}(l) \;_{S}\!<\hat{\Omega}|b^{+}_{l\uparrow}b_{l\uparrow}|\hat{\Omega}>_{S}\\ \gamma^{(2)}_{ij}=& \displaystyle{-\sum_{l\neq i,j}}\theta_{ij}(l) \;_{S}\!<\hat{\Omega}|b^{+}_{l\downarrow}b_{l\downarrow}|\hat{ \Omega}>_{S}\label{d} \end{array}$$ and introduce three slave particles $$\xi=\psi\bar{\chi}\chi, \;\;\; \xi^{+}\xi=\psi^{+}\psi =\bar{\chi}^{+}\bar{\chi}=\chi^{+}\chi \label{e}$$ to describe the charge degree part. $\bar{\chi}$ describes a slave fermion moving in a background “magnetic” field produced by the phase factor $\gamma^{(1)}_{ij}$, $\chi$ describes a slave fermion moving in a background “magnetic” field produced by the phase factor $\gamma^{(2)}_{ij}$, $\psi$ describes a slave boson or a slave fermion only responsing to external magnetic and electric fields, or more intuitively, it can be considered as describing the “mass-centre” of the slave fermions $\bar{\chi}$ and $\chi$. However, corresponding to these slave boson and slave fermions, there exist two gauge freedoms $$\begin{array}{rl} \psi & \rightarrow\displaystyle{e^{i\theta}\psi, \;\;\; \bar{\chi}\rightarrow e^{-i\theta}\bar{\chi}, \;\;\; \chi\rightarrow\chi}\\ \psi & \rightarrow\psi, \;\;\; \bar{\chi}\rightarrow\displaystyle{ e^{i\bar{\theta}}\bar{\chi}, \;\;\; \chi\rightarrow e^{-i\bar{\theta}}\chi} \label{f}\end{array}$$ that introduce two gauge fields. While two current conservation equations corresponding to these two gauge fields and the gauge invariances will maitain the freedom of the system being conservative. Substituting equation (\[e\]) into equation (\[b\]), we have $$\bar{H}=\bar{t}\displaystyle{\sum_{<ij>}\psi^{+}_{j}\psi_{i}( \bar{\chi}^{+}_{j}\bar{\chi}_{i}e^{i\gamma^{(1)}_{ij}})(\chi^{+}_{j} \chi_{i}e^{i\gamma^{(2)}_{ij}})} + V\displaystyle{\sum_{i}}\psi^{+}_{i}\psi_{i}|\vec{\partial}\hat{ \Omega}|\label{g}$$ Under the Hartree-Fock approximation, we can have the following Lagrangian correponding to the Hamiltonian (\[g\]) $$\begin{array}{rl} {\it L}=& \displaystyle{\sum_{i}}\{\psi^{*}_{i}(\partial_{\tau}- \lambda_{i})\psi_{i}+\bar{\chi}^{*}_{i}(\partial_{\tau}+\lambda_{i} +\eta_{i})\bar{chi}_{i}\\ +& \chi^{*}_{i}(\partial_{\tau}-\eta_{i})\chi_{i}\} +V\displaystyle{\sum_{i}}\psi^{*}_{i}\psi_{i}|\vec{\partial}\hat{ \Omega}|\\ +& \bar{t}\displaystyle{\sum_{<ij>}\{A_{ij}\psi^{*}_{j}\psi_{i} +B_{ij}e^{i\gamma^{(1)}_{ij}}\bar{\chi}^{*}_{j}\bar{\chi}_{i} +C_{ij}e^{i\gamma^{(2)}_{ij}}\chi^{*}_{j}\chi_{i}\}} \label{h}\end{array}$$ where, $m_{\psi}=(A\bar{t})^{-1},\; m_{\bar{\chi}}=(B\bar{t})^{-1}, \; m_{\chi}=(C\bar{t})^{-1}$, $A_{ij}=<\bar{\chi}^{+}_{j}\bar{\chi}_{i}e^{i\gamma^{(1)}_{ij}}> <\chi^{+}_{j}\chi_{i}e^{i\gamma^{(2)}_{ij}}>=Ae^{i\Theta_{ij}}, \; B_{ij}=<\psi^{+}_{j}\psi_{i}><\chi^{+}_{j}\chi_{i}e^{i\gamma^{(2)}_{ij}}> =Be^{-i\Theta_{ij}+i\bar{\Theta}_{ij}}, \; C_{ij}=<\psi^{+}_{j}\psi_{i}><\bar{\chi}^{+}_{j}\bar{\chi}_{i} e^{i\gamma^{(1)}_{ij}}>=Ce^{-i\bar{\Theta}_{ij}}, \; \Theta_{ij}=(\vec{x}_{i}-\vec{x}_{j})\cdot\vec{a}(\frac{x_{i}-x_{j}}{2}),\; \bar{\Theta}_{ij}=(\vec{x}_{i}-\vec{x}_{j})\cdot\vec{\bar{a}}(\frac {x_{i}-x_{j}}{2})$. We introduce two Lagrangian multipliers $\lambda_{i}$ and $\eta_{i}$ to add the constraints \[e\] to the system. Under the gauge transformations (\[f\]), the Lagrangian (\[h\]) remains invariance. In the continuous limit, the Lagrangian (\[h\]) can be rewritten as $$\begin{array}{rl} {\it L}=& \int d^{2}x\{\psi^{*}(\partial_{\tau}-ia_{0})\psi +\bar{\chi}^{*}(\partial_{\tau}+ia_{0}+i\bar{a}_{0})\bar{\chi}\\ +& \chi^{*}(\partial_{\tau}-i\bar{a}_{0})\chi\} +\int d^{2}x\{\displaystyle{\frac{1}{2m_{\psi}}\psi^{*}(\vec{\partial} -i\vec{a})^{2}\psi}\\ +& \displaystyle{\frac{1}{2m_{\bar{\chi}}}\bar{\chi}^{*}(\vec{\partial} +i\vec{a}+i\vec{\bar{a}}+i\vec{A})^{2}\bar{\chi} +\frac{1}{2m_{\chi}}\chi^{*}(\vec{\partial}-i\vec{\bar{a}}-i\vec{A}^{'}) ^{2}\chi\}}\\ + & V^{'}\int d^{2}x\psi^{*}\psi|\vec{\partial}\hat{\Omega}| \label{k}\end{array}$$ where, $V^{'}=V/a^{2}, \; (\vec{x}_{i}-\vec{x}_{j})\cdot\vec{A}(\frac {x_{i}-x_{j}}{2})=\gamma^{(1)}_{ij}, \; -(\vec{x}_{i}-\vec{x}_{j}) \cdot\vec{A}^{'}(\frac{x_{i}-x_{j}}{2})=\gamma^{(2)}_{ij}$. It is reasonable in the continuous limit to study the property of the Lagrangian \[h\], because the phase factors $\gamma^{(1)}_{ij}$ and $\gamma^{(2)}_{ij}$ are slowly varying functions, so we can introduce gauge fields to describe them. However, in thermodynamic limit, we have $<b^{+}_{l\uparrow}b_{l\uparrow}>=<b^{+}_{l\downarrow}b_{l\downarrow}>$, so the gauge fields $\vec{A}$ and $\vec{A}^{'}$ can be generally written as $\vec{A}=\vec{A}^{'}=\vec{\bar{A}}+\delta\vec{A}, \; \nabla\times\vec{\bar{A}}=\bar{B}=\pi(1-\delta)$, $\delta$ is the doping density, while the fluctuation field can be absorbed into $\vec{\bar{a}}$. We see that the phase factors $\gamma^{(1)}_{ij}$ and $\gamma^{(2)}_{ij}$ only provide uniform “magnetic” fields to the slave fermions $\bar{\chi}$ and $\chi$, respectively. Under these approximations, we can easily treat the Lagrangian (\[k\]). First we show that the gauge field $\vec{a}$ is massive and the gauge field $\vec{\bar{a}}$ enforces the slave fermions $\bar{\chi}$ and $\chi$ to be confined. To do so, we consider the current-current correlations of the slave fermions $\bar{\chi}$ and $\chi$. Because of appearance of the uniform magnetic field $\bar{B}$ in the slave fermions $\bar{\chi}$ and $\chi$ systems, there exists a zero-field Hall conductance dynamically produced by this field $\bar{B}$ in their current-current correlations[@12b][@12c], so their current-current correlations can be generally written as $$\begin{array}{rl} \Pi_{\chi\alpha\beta}=& \Pi_{\chi\bot}\displaystyle{(\delta_{\alpha\beta}- \frac{k_{\alpha}k_{\beta}}{k^{2}})+\Pi_{\chi\|}\frac{k_{\alpha}k_{\beta}}{ k^{2}}+i\epsilon_{\alpha\beta}\omega\sigma_{xy}}\\ \Pi_{\bar{\chi}\alpha\beta}=& \Pi_{\bar{\chi}\bot}\displaystyle{( \delta_{\alpha\beta}-\frac{k_{\alpha}k_{\beta}}{k^{2}})+ \Pi_{\bar{\chi}\|}\frac{k_{\alpha}k_{\beta}}{k^{2}}-i\epsilon_{ \alpha\beta}\omega\sigma_{xy}}\label{l} \end{array}$$ where $\sigma_{xy}$, Hall conductance, is a constant. $\Pi_{\chi\alpha\beta}$ has opposite sign Hall conductance against $\Pi_{\bar{\chi}\alpha\beta}$ because the slave fermion $\chi$ carries negative charge to $\vec{A}$ while the slave fermion $\bar{\chi}$ carries positive charge to $\vec{A}$. In the low energy and long wavelength limit, $\Pi_{a\bot}$ and $\Pi_{a\|}, \;a=\chi, \bar{\chi}$, are the quadratic functions of $\omega$ and $k$[@8]. Generally, they can be written as $$\Pi_{a\bot}=\eta_{a}k^{2}-\varepsilon_{a}\omega^{2},\;\;\; \Pi_{a\|}=\bar{\eta}_{a}k^{2}-\bar{\varepsilon}_{a}\omega^{2}. \label{la}$$ where $\eta_{a}, \bar{\eta}_{a}, \varepsilon_{a}\;and\;\bar{\varepsilon}_{a}$ are constants. For the gauge field $\vec{a}$, after integrating out the slave fermions $\chi, \;\bar{\chi}$, and gauge field $\vec{\bar{a}}$, its propagator is $$D^{-1}=\Pi_{\chi}(\Pi_{\chi}+\Pi_{\bar{\chi}})^{-1}\Pi_{\bar{\chi}} \label{lb}$$ We see that, in the long wavelength limit $k\rightarrow 0$, the Hall conductance terms in (\[lb\]) produce a mass term for the gauge field $\vec{a}$, so the gauge field $\vec{a}$ has a little influence on the system although the slave boson (or fermion) $\psi$ dynamically produces an unusual term $\frac{i\omega}{k}$, we can omit it in equation (\[k\]). However, the propagator of the gauge field $\vec{\bar{a}}$ reads $$\bar{D}^{-1}=\Pi_{\chi}+\Pi_{\bar{\chi}}\label{lc}$$ the Hall conductance terms in (\[lc\]) are cancelled. After integrating out the slave fermions $\chi$ and $\bar{\chi}$, we obtain an effective action of the gauge field $\vec{\bar{a}}$ as taking a suitable scalling for “time” $\tau$ $$S[\vec{\bar{a}}]=\frac{1}{4g^{2}}\int d^{3}xF^{2}_{\mu\nu}, \;\;\; \frac{1}{g^{2}}\sim\frac{1}{\sqrt{\delta}}.\label{ld}$$ Here for simplicity we include the $\bar{a}_{0}$ term. If we consider the topologically nontrivial hedgehog configurations of the gauge field $\vec{\bar{a}}$ with integer topological charge $q=\frac{1}{2\pi}\int ds_{\mu}\epsilon_{\mu\nu\lambda}\partial_{\nu}\bar{A}_{\lambda}$, the confinement length of the slave fermions $\chi$ and $\bar{\chi}$ is[@12c][@12d][@12e] $$\xi=\frac{ag}{2\pi}e^{const./g^{2}}\label{le}$$ where $a$ is an in-plane lattice constant. However, we have two basic length parameters, the confinement length $\xi$ and the Landau length $l_{B}\propto\frac{1}{\sqrt{\bar{B}}}$. In the half filling limit, the confinement length of the slave fermions $\bar{\chi}$ and $\chi$ is determined by the Landau length $l_{B}$. On the other hand, in the overdoping limit, their confinement length is determined by $\xi$. Based upon the above discussions, the Lagrangian (\[k\]) can be rewritten as $$\begin{array}{rl} {\it L}=& \int d^{2}x\{\psi^{*}(\partial_{\tau}-ia_{0}+iA^{ex}_{0})\psi +\bar{\chi}^{*}(\partial_{\tau}+ia_{0}+i\bar{a}_{0})\bar{\chi} +\chi^{*}(\partial_{\tau}-i\bar{a}_{0})\chi\\ +& \displaystyle{\frac{1}{2m_{\psi}}\psi^{*}(\vec{\partial}+i\vec{A}^{ex} )^{2}\psi+\frac{1}{2m_{\bar{\chi}}}\bar{\chi}^{*}(\vec{\partial}+ i\vec{A})^{2}\bar{\chi}}\\ +& \displaystyle{\frac{1}{2m_{\chi}} \chi^{*}(\vec{\partial}-i\vec{A})^{2}\chi+V^{'} \psi^{*}\psi|\vec{\partial}\hat{\Omega}|}\}\label{lf} \end{array}$$ where we add an external gauge fields $\vec{A}^{ex}$ and $A^{ex}_{0}$, and omit the gauge field $\vec{a}$ and $\vec{\bar{a}}$. Although the slave boson (or fermion) $\psi$ dynamically contributes a term $(\chi_{F}k^{2}-\frac{i\omega}{v_{F}k})(\delta_{\alpha\beta}-\frac{ k_{\alpha}k_{\beta}}{k^{2}})a_{\alpha}a_{\beta}$ to the gauge field $\vec{a}$, the mass term derived from the slave fermions $\bar{\chi}$ and $\chi$ for the gauge field $\vec{a}$ will remove the singular behavior of its propagator, and maitains the Fermi liquid behavior of the slave boson (or fermion) $\psi$ invariance. However, the density constraints in (\[e\]) and the current conservation law of the slave particle fields $\psi,\; \chi$ and $\bar{\chi}$ enforce the longitudial currents to satisfy the following equation $$\vec{J}_{\psi\|}=\vec{J}_{\chi\|}=\vec{J}_{\bar{\chi}\|}\label{lg}$$ while for the transversal currents there are not any constraints. We see that the slave particle fields $\psi, \; \chi$ and $\bar{\chi}$ interact on each other only via the scalar gauge fields $a_{0}$ and $\bar{a}_{0}$. If we redefine the scalar gauge field, $a_{0}+\bar{a}_{0}=-a^{'}_{0}$, then we obtain the similar Lagrangian as that in Ref.[@12f], so we can use their results about the calculations of relaxation rates. The slave fermions $\bar{\chi}$ and $\chi$ have the same relaxation rate induced by the quasiparticle-scalar-gauge fluctuation scattering $$\frac{\hbar}{\tau_{\chi}}=\frac{\hbar}{\tau_{\bar{\chi}}} \simeq 2\eta(0)k_{B}T\label{lh}$$ where $\eta(0)\sim 1$ is a constant, while the slave boson (or fermion) $\psi$ has the relaxation rate $$\frac{\hbar}{\tau_{\psi}}=\eta^{'}(0)\frac{(k_{B}T)^{2}}{t} \label{lk}$$ where $\eta^{'}(0)$ is a constant. Because in the real case, we have $k_{B}T/t\ll 1$, so for external electric field we have the transport relaxation rate $\tau_{tr}=\tau_{\chi}\simeq 2k_{B}T$, consistent with that one directly calculates it\[26\] using the effective Hamiltonian (14), it also shows that the separations in (16) and (17) are reasonable. We see that the scalar gauge fields $a_{0}$ and $\bar{a}_{0}$ do not change the Fermi liquid behavior of the slave boson (or fermion) $\psi$, so we find that the charge degree part described by the Lagrangian (\[a\]) or effective Hamiltonain (\[b\]) has two relaxation rates corresponding to different responses to external magnetic and electric fields, respectively. Here we must give a detail explanation about the equations (30) and (31). First we only turn on an external electric field, so we have gauge fields $\vec{A}^{ex}_{\|}$ and $A^{ex}_{0}$. If we take a gauge transformation to the slave boson (or fermion) $\psi$, we can cancel the gauge field $\vec{A}^{ex}_{\|}$, and obtain an effective scalar gauge field $\bar{A}^{ex}_{0}$, so the response of the external electric is only the density-density correlations of the slave particle fields $\psi, \; \bar{\chi}$ and $\chi$. Because of the constraints (\[e\]) and (\[lg\]), there exist strongly interactions among the slave particles $\psi, \; \bar{\chi}$ and $\chi$ via the gauge fields $a_{0}$ and $\bar{a}_{0}$, which will drastically change this response of the external electric field. In the normal state, the resistivity of the system is $$\rho(T)\propto 2k_{B}T(1+O(\frac{k_{B}T}{t}))+\gamma n_{i} \label{Ak}$$ where the last term in bracket is very small $k_{B}T/t\ll 1$, $\gamma$ is a constantn, $n_{i}$ is the density of impurity, the last term derives from the impurity scattering. However, if we only switch on an external magnetic field, we have a gauge field $\vec{A}^{ex}_{\bot}$, the response of the external magnetic field is only the current-current correlation of the slave boson (or fermion) $\psi$. Although there exist strongly interactions among the slave particles $\psi, \; \bar{\chi}$ and $\chi$ via the scalar gauge fields $a_{o}$ and $\bar{a}_{0}$, the Fermi behavior of the slave boson (or fermion) $\psi$ is not destroyed by these scalar gauge interactions, so for the external magnetic field, the charge degree part only show its Fermi liquid behavior because only the slave boson (or fermion) $\psi$ response for the external magnetic field, the Hall angle of the system is $$cot\theta_{H}=\frac{\rho^{\psi}_{xx}}{\rho^{\psi}_{xy}} =\alpha T^{2}+\beta n_{i}\label{Bl}$$ where $\alpha(\propto\frac{1}{B})$ and $\beta$ are constants, the last term derives from the impurity scattering. According to the above discussions, the anomalous transverse magnetoresistance is closely related to the temperature dependence of the Hall angle, they are derived from the same origin, the slave boson (or fermion) $\psi$ system. Because of the slave boson (or fermion) $\psi$ system remains the Fermi liquid behavior, according to the Kohler’s rule we should have a temperature dependence of the magnetoresistance of the $\psi$ system $\Delta\rho^{\psi}/\rho^{\psi}\propto(tan\theta_{H})^{2} \propto B^{2}T^{-4}$. since we have the relation $\Delta\rho^{\psi}=\Delta\rho$, so we can obtain the following expression of the magnetoresistance of the charge degree part $$\frac{\Delta\rho}{\rho}=\frac{\rho^{\psi}}{\rho} \frac{\Delta\rho^{\psi}}{\rho^{\psi}}\propto B^{2}T^{-n} \label{Cl}$$ For the resistivity $\rho\sim T$, we have $n=3$; For the resistivity $\rho\sim T^{2}$, we have $n=4$. Generally, in the underdoping range, the resistivity is $\rho\sim T^{\alpha}, \; 1<\alpha\leq 2$ in the low temperature range, the magnetoresistance has the temperature dependence $\Delta\rho/\rho\propto T^{-n}, \; 3<n\leq 4$, consistent with the experimental data in [@7i][@7j]. In the $YBa_{2}Cu_{3}O_{7-\delta}$ samples between $100$ and $375K$[@7i], $\Delta\rho/\rho$ follows a power law $T^{-n}$, with $n=3.5$ and $3.9$ in the ($T_{c}=$) 90-K and 60-K crystals, respectively. In the $BSCCO\;2:2:1:2$ single-crystal samples[@7j], $\Delta\rho/\rho$ is shown to vary as $\sim T^{-3}$ from $T_{c}$ up to room temperatures. For the $YBCO$ samples, there exists a $Cu-O$ chain which may affect the experimental results, but the changing trend of the exponential $n$, from the optimal doping to the underdoping cases, is consistent with the equation (34). We need more experimental data to testify the temperature dependence of the magnetoresistance given in (34). Magnetic property of the normal state ===================================== Because the slave boson (or fermion) $\psi$ system remains the Fermi liquid behavior, after integrating out the field $\psi$, we can obtain an effective term provded by the interaction between the spin parameter field $\hat{\Omega}$ and the slave boson (or fermion) $\psi$ $${\it L}_{\psi}[\hat{\Omega}]=-\beta\sum_{n}\int\frac{d^{2}q}{ (2\pi)^{2}}\frac{|\omega_{n}|}{\omega_{F}}|\hat{\Omega}|^{2}(q,\omega_{n}) \label{ll}$$ where $\omega_{F}\propto\frac{1}{V^{'2}k_{F}}$, a character energy scale describing the damping of the quasiparticle-hole pairing excitation to the spin wave spectrum. We must carefully pay attention on the term (\[ll\]) which is not directly derived from an usual the quasiparticle-hole pairing excitation of a magnon, because the interaction term between the spin parameter field $\hat{\Omega}$ and the slave boson (or fermion) $\psi$ is $\psi^{*}\psi|\vec{\partial}\hat{\Omega}|$, a complicated interaction. Meanwhile, in the momentum space, we remain in mind that origin point of the momentum for the spin parameter field is at $\vec{q}=\vec{Q}=(\pm\pi/a,\pm\pi/a)$, while origin point of the momentum for the slave field $\psi$ is at $\vec{q}=(0,0)$. From equations (11) and (\[ll\]), we obtain an effective action of the spin parameter field of the t-J model $$S_{eff.}[\hat{\Omega}]=\beta\sum_{n}\int\frac{d^{2}q}{(2\pi)^{2}}\{ \frac{1}{2g_{0}}(q^{2}+\frac{1}{c^{2}}\omega^{2}_{n})-\frac{| \omega_{n}|}{\omega_{F}}\}|\hat{\Omega}|^{2}(q,\omega_{n}) \label{ka}$$ where $|\hat{\Omega}(x,\tau)|=1$, the origin points of $\vec{q}$ are in the corner points $\vec{Q}=(\pm\frac{\pi}{a},\pm\frac{\pi}{a})$. The action (\[ka\]) is the same as that in Refs.[@4][@4a] that we obtained from a p-d model or an effective Hamiltonian derived from a three-band Hubbard model. This action has two critical regions: one is a $z=1$ (where $z$ is a dynamic exponent) region which is consisted of three regimes: a renormalized classical (RC) regime, a quantum critical (QC) regime and a quantum disorder (QD) regime[@1]; another one is a $z=2$ region which maybe is also divided into the same two (QC and QD) regimes as above, but their behavior is completely different from that in the $z=1$ region. In the undoping case, $\omega_{F}\rightarrow\infty$, the system is in the RC regime[@1][@2]. In the underdoping case, $\omega_{c}<\omega_{F}<\infty$, the system is in the $z=1$ QC and/or QD regimes\[3\]\[4\]. In the optimal doping case, $\omega_{F}<\omega_{c}$, the system goes into the $z=2$ region[@3][@4][@13]. $\omega_{c}$ is a characteristic energy scale which indicases a crossover of the system from the $z=1$ region to the $z=2$ region as doping. We see that the $\omega_{F}$ term in (\[ka\]) which derives from the damping of the quasiparticle-hole pairing excitation to the spin wave spectrum is very important for determining the doping influence on the system, especially in the optimal doping case, this term is dominant. Generally, in the $z=1$ region, the $\omega_{F}$ term is very small, and can be treated perturbatively, in the low energy limit we can obtain following spin susceptibility $$\chi(q,\omega)=\frac{\chi_{0}}{\xi^{-2}+q^{2}-\frac{1}{c^{2}}\omega^{2} -\frac{i\omega}{\omega^{R}_{F}}}\label{kb}$$ where $\xi$ is a coherent length, $\omega^{R}_{F}$ is a renormalized characteristic energy scale of the spin fluctuation. In the ($z=1$) QC regime[@4][@4a], $\xi\sim\frac{1}{T}, \omega^{R}_{F}\sim\frac{\omega_{F}(\hat{l})}{T}$; In the ($z=1$) QD regime, $\xi$ and $\omega^{R}_{F}$ take constants. In the $z=2$ region, the $\omega_{F}$ term is dominant, the $\omega^{2}$ term is irrelevant and can be omitted, in the low energy limit we can obtain following spin susceptibility $$\bar{\chi}(q,\omega)=\frac{\bar{\chi}_{0}}{\bar{\xi}^{-2}+q^{2}- \frac{i\omega}{\bar{\omega}_{F}}}\label{kc}$$ where $\bar{\omega}_{F}=\frac{\omega_{F}}{2g_{0}}$ is a renormalization group invariant quantity. In the ($z=2$) QC regime[@4][@4a], $\bar{\xi}^{2}\sim\frac{1}{T}$. Using these spin susceptibilities in (\[kb\]) and (\[kc\]), we can betterly explain the current experimental data[@5]-[@7b] of the nuclear magnetic resonance spin-lattice relaxation rate and the spin echo decay rate about the copper spin. The NMR spin lattice relaxation rate $T_{1}$ and the spin echo decay rate $T_{2G}$ can be written as $$\begin{array}{rl} \displaystyle{\frac{1}{T_{1}T}}& \propto\displaystyle {\lim_{\omega\rightarrow 0}}\int d^{2}q|A(q)|^{2} \frac{\chi^{\prime\prime}(q,\omega)}{\omega} \propto\displaystyle{\frac{\xi^{2}_{i}}{\omega_{i}}}\\ \displaystyle{\frac{1}{T_{2G}}}& \propto[\int d^{2}qf(q)\chi^{\prime 2}(q,0)]^{1/2} \propto\xi_{i} \end{array}$$ where, $\omega_{i}=\omega^{R}_{F}$ (for z=1) or $\bar{\omega}_{F}$ (for z=2), $\xi_{i}=\xi$ (for z=1) or $\bar{\xi}$ (for z=2), $A(q)\sim A$ is the hyperfine coupling constant and $f(q)\sim f$ is the form factor originating from the hyperfine interaction between the nuclear spin and the surrounding electron spins. In the QD regime we have $$\begin{array}{rl} \displaystyle \frac{1}{T_{1}T}& \propto\left\{ \begin{array}{ll}\displaystyle{\frac{\xi}{\omega_{F}(\hat{l})}}, &\mbox{z=1}\\ \displaystyle{\frac{\bar{\xi}^{2}}{\bar{\omega}}}, &\mbox{z=2}\end{array}\right.\\ \displaystyle{\frac{1}{T_{2G}}}& =const. \end{array}$$ Similarly, in the QC regime we have $$\begin{array}{rl} \displaystyle \frac{1}{T_{1}T}& \propto\left\{ \begin{array}{ll}\frac{1}{\omega_{F}(\hat{l})T}, &\mbox{z=1}\\ \frac{1}{T{\bar{\omega}}^{2}_{F}}, &\mbox{z=2}\end{array}\right.\\ \displaystyle{\frac{1}{T_{2}}}& \propto\left\{ \begin{array}{ll}\frac{1}{T}, &\mbox{z=1}\\ \frac{1}{{\bar{\omega}}^{1/2}_{F}T^{1/2}}, &\mbox{z=2}\end{array}\right. \end{array}$$ We see that the spin lattice relaxation rate $T_{1}$ is considerably affected by the doping because of the quantity $\omega_{F}(\hat{l})\sim 1/k_{F}$, while the spin echo decay rate $T_{2G}$ depends upon doping through the correlation length $\xi$. For the spin lattice relaxation rate of the oxygen spin, we need more explanation, because in the t-J model the spin degree of the planar oxygen, composed a Zhang-Rice spin-singlet with the localized planar copper spin, is completely suppressed. The slave fermion (or boson) operator $f_{i}$ in (1) really expresses a Zhang-Rice spin-singlet, if there only exists a commensurate strongly short-range antiferromagnetic correlation for the localized planar copper spins, at least it is true for the $YBCO$ samples, the planar oxygen spin will be not influenced by this commensurate antiferromagnetic correlation because the planar oxygen resides in the middle point of two nearest neighbor copper sites. So only the slave fermion (or boson) $f$ system, describeing the charge degree of electron, can influence the spin lattice relaxation rate of the planar oxygen spins[@13a], just shown as in Section III, which obeys the Korringa-like rule because the response of the charge degree part shows the Fermi liquid behavior to external magnetic field. Discussion and Conclusion ========================= Using the spin-hole coherent state representation, we have studied the normal state property of the t-J model in the usual slave boson and slave fermion treatment of the single occupation constraint, and shown that we can qualitatively explain the unusually magnetic and transport behaviors of the normal state of the cuprate superconducting materials by the t-J model. We think that the short range antiferromagnetic correlation induces the unusual behavior of the normal state of the cuprate materials, so it is a reasonable approximation that we take a long range antiferromagnetic Néel order as a background of the spin degree part of the system. Although the interaction between the charge degree and spin degree will destroy this long range order, but the system still has the short range antiferromagnetic order. In the undoping case, the system can be described by a non-linear $\sigma$-model (the t-J model reduces to the Heisenberg model). In the doping case, the interaction between the charge degree and spin degree provides a decay term to the non-linear $\sigma$-model, which describes the damping of the quasiparticle-hole pairing excitation to the spin wave spectrum, but this decay term is not directly derived from the quasiparticle-hole pairing excitation of a magnon because of the complicated interaction term between the spin parameter field $\hat{\Omega}$ and the slave boson (or fermion) field $\psi$. Using this effective Lagrangian \[ka\], we can betterly explain the unusually magnetic behavior of the planar copper spin of the normal state of the cuprate superconducting materials. For the planar oxygen spin, we think that its normal Korringa-like relaxation behavior is coming from the contribution of the slave particle $f$ in (1), described the Zhang-Rice spin-singlet and charge degree of electron. While because of there existing the strongly short-range antiferromagnetic coreelation in the localized spin degree part, the charge degree part will feel a strongly staggered magnetic field as the doping hole hopping, this staggered magnetic field drastically influences the behavior of the charge degree part, and enforces it to have different responses to external magnetic field and electric field and to show two relaxation rate behaviors corresponding to the planar resistivity and Hall angle, respectively. This character of the charge degree part responsed to external magnetic field is compatible with the Korringa-like relaxation behavior of the planar oxygen spin. 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=cmr10 scaled 1 amssym -1,5cm -1.5cm 0.5cm =msbm6 =msbm7 scaled 1 =msbm7 scaled 1 =msbm9 scaled 0 =msbm7 scaled 2 =msbm9 scaled 1 =msbm10 scaled 1 =msbm10 scaled 2 =msbm10 scaled 3 =msbm10 scaled 4 \#1 \#1 \#1 \#1 \#1 \#1 \#1 \#1 \#1 \#1 $ $ **Azumaya-type noncommutative spaces and morphisms therefrom:** **Polchinski’s D-branes in string theory from Grothendieck’s viewpoint** Chien-Hao Liu and Shing-Tung Yau > **Abstract** > > [ Grothendieck’s equivalence of a commutative function ring and a local geometric space gives rise to the language of schemes and functor of points in 1960s that rewrote commutative algebraic geometry while Polchinski’s identification/recognition in 1995 of D-branes – studied since the second half of 1980s as boundary conditions for open strings – as the source of Ramond-Ramond fields created by closed superstrings in the space-time rewrote string theory. In this work, we explain how a noncommutative version of Grothendieck’s equivalence gives rise to a prototype intrinsic definition of D-branes that can reproduce the key, originally open-string-induced, properties of D-branes described in Polchinski’s works. After the discussion of Azumaya-type noncommutative spaces and morphisms therefrom that form the algebro-geometric foundation of the current work, basic properties of D0-branes on a smooth curve/surface or a quasi-projective variety, the associated Chan-Paton modules, the Higgsing/un-Higgsing behavior – all under the current setting –, and their relation with Hilbert schemes and Chow varieties are given. When applied to the case of D0-branes on a (commutative) projective complex smooth surface, this gives also a picture in the current pure algebro-geometric setting that resembles gas of D0-branes in a work of Vafa. Related supplementary discussions/remarks are given in footnotes. ]{} [MSC number 2000: 14A22, 81T30; 14A10, 16G30, 81T75. ]{} [[**Acknowledgements.**]{} We thank Andrew Strominger and Cumrun Vafa for lectures/discussions that influence the project; Duiliu-Emanuel Diaconescu, Gang Liu, Kefeng Liu, Pan Peng for discussions on open/closed string duality; Dep’t of Mathematics of UCLA for hospitality while the work is in preparation. Strings/branes is a topic that mixes notions/techniques/art from physics/QFT/SUSY with languages/techniques/naturality from mathematics. For that, C.-H.L. thanks in addition Paul Aspinwall, Michael Douglas, Jeffrey Harvey, Joseph Polchinski, Ashoke Sen for discussions at three TASI schools at U. Colorado at Boulder and Orlando Alvarez, Philip Candelas, Hungwen Chang, Chong-Sun Chu, Xenia de la Ossa, Jacques Distler, Daniel Freed, Joe Harris, Pei-Ming Ho, Kentaro Hori, Shinobu Hosono, Albrecht Klemm, Shiraz Minwalla, Gregory Moore, Mircea Mustata, Rafael Nepomechie, Mihnea Popa, Lisa Randall, Margaret Symington, Zheng Yin, Barton Zwiebach for joint influence/education over the years in the background; John Beachy for a discussion on noncommutative localizations; Melanie Becker, Alex Maloney for talks; D.-E.D., G.L. for enlightenment on subtleties of open-string world-sheet instantons; Lubos Motl, Li-Sheng Tseng, Ilia Zharkov for conversations; William Oxbury for preprint/communication \[Ox\]; participants of GSS, organized by Monica Maria Guica, for discussions on themes in string theory; Nima Arkani-Hamed, John Duncan, Dennis Gaitsgory, Katrin Wehrheim for topic courses; Rev. Robert Campbell Willman and Betty for hospitality/dinner and a discussion on a book of Brian Greene; Ling-Miao Chou for the moral support. The project is supported by NSF grants DMS-9803347 and DMS-0074329. ]{} $ $ *Chien-Hao Liu dedicates this work to his teacher Ann L. Willman,* *who is giving him yet another lesson* – [*the grace, courage, will power, and inner peace while in the turmoil of life*]{} – *throughout the treatment of her cancer.* $ $ Azumaya-Type Noncommutative Spaces, Morphisms, and D-Branes [**0. Introduction and outline.**]{} [**Introduction.**]{} A [*D-brane*]{} (in full name: [*Dirichlet brane*]{} or [*Dirichlet membrane*]{})[^1] in string theory is by definition (i.e. by the very word ‘Dirichlet’) a boundary condition for the end-points of open strings. From the viewpoint of the field theory on the open-string world-sheet aspect, it is a boundary state in the $d=2$ conformal field theory with boundary. From the viewpoint of open string target space(-time) $M$, it is a cycle or a union of submanifolds $Z$ in $M$ with a gauge bundle (on $Z$) that carries the Chan-Paton index for the end-points of open strings. For the second viewpoint, Polchinski recognized in 1995 in \[Pol2\] that a D-brane is indeed a source of the Ramond-Ramond fields on $M$ created by the oscillations of closed superstrings in $M$. In particular, in a specific region of the Wilson’s theory-space for D-branes, D-branes can be identified with the solitonic/black branes studied earlier[^2] in supergravity and (target) space-time aspect of superstrings. This recognition is so fundamental that it gave rise to the second revolution of string theory. When $M$ is compactified on a Calabi-Yau space $Y$, the preservation of supersymmetries in either the field theory on the open-string world-sheet or in the effective field theory after the compactification requires the D-brane to be supported on a union of Lagrangian submanifolds/subspaces or holomorphic cycles, (cf. \[B-B-St\], \[H-I-V\], and \[O-O-Y\]). When we focus only on the internal/compactified part of space-time, this gives us a preliminary mathematical definition of supersymmetric D-branes as a union of Lagrangian submanifolds with gauge bundles or a coherent (possibly torsion) sheaf on $Y$. While such definitions of D-branes is already very convenient in the study of superstring theory with branes and of stringy dualities, they are not adequate to serve as the intrinsic definition of D-branes as, among other issues, in general they cannot reproduce by themselves a key property of D-branes – the Higgsing/un-Higgsing behavior of D-branes – in its own mathematical framework in a natural way. This subtlety actually does not seem to bother string theorists, likely for two reasons: - The picture of supersymmetric D-branes as cycles in $Y$ with a gauge bundle is generically correct/enough in the regime where branes are still branes. - Under deformations of D-branes for which the mathematical picture in Item (1) is not complete enough to dictate the details, the very definition of D-branes as where open strings end tells us that we can look at the related open string theory, particularly its induced fields and their effective action on the brane, to determine what happens to the deformed D-branes. Depending on one’s taste/weight on such a subtlety, one is either satisfied with this picture or not. And if not, one is led to the following question: - [**Q. \[D-brane\]**]{}$\;$ [*What is a D-brane intrinsically?*]{} In other words, what is the intrinsic definition of D-branes so that by itself it can produce the properties of D-branes that are consistent with, governed by, or originally produced by open strings as well? This is the guiding question of the current work. The answer to this question is indeed already suggested by string theorists: it is hinted already in the works (e.g. \[Pol3\]) of Polchinski and later put with even more weight by other string theorists[^3] that D-branes have a close tie with noncommutative geometry. One cannot expect to have a good answer to Question \[D-brane\] without bringing appropriate noncommutative geometry into the intrinsic definition of D-branes. Indeed, Polchinski’s [*description of deformations of stacked D-branes*]{} together with Grothendieck’s [*local equivalence of rings and spaces/geometries*]{} and the notion of [*functors of points*]{} (Sec. 2.1) implies immediately (Sec. 2.2): - This brings us to a technical world in mathematics: noncommutative geometry. Due to the different languages used in differential geometry and in algebraic geometry for noncommutative geometry (though the philosophy to equate locally a space and a function ring in each category is in common), we focus now on supersymmetric D-branes of B-type, for which algebro-geometric language is appropriate. From the basic properties of D-branes spelt out explicitly in the work of Polchinski, there are a special class of noncommutative spaces that are particularly related to D-branes, namely the Azumaya-type noncommutative spaces. These are the noncommutative spaces that locally have their function ring the matrix ring $M_n(R)$ over a commutative ring $R$. The ansatz of Grothendieck on the equivalence of a ring and a local geometry, when extended to the noncommutative case as well, enables us to directly look at rings themselves without having to deal with the technical subtle issue of the functorial construction of an associated space (i.e. a set of points with topology and other structures) to a ring as Grothendieck did in 1960s for commutative rings that rewrote commutative algebraic geometry. His ansatz of the contravariant equivalence of morphisms-between-spaces and morphisms-between-rings-locally, and the ansatz of composability, which says that the composition of morphisms $X\rightarrow Y$, $Y\rightarrow Z$ between spaces should be a morphism $X\rightarrow Z$, can then be used to give the notion of morphisms from an Azumaya-type noncommutative space to a (either commutative or noncommutative) space without having the spaces themselves. In this way, an Azumaya-type noncommutative space $X$ can be phrased purely as a gluing system ${\cal R}$ of matrix rings and a morphism from $X$ can be phrased purely as a gluing system of ring-homomorphisms to ${\cal R}$. A quasi-coherent sheaf on $X$ is then a gluing system of modules over rings in ${\cal R}$. (Sec. 1.) Once this language is formulated precisely, the following prototype definition of D-branes (of B-type and when a “brane" is still a brane) (Definition 2.2.3): - alone gives a Higgsing/un-Higgsing property of D-branes in its own right that is consistent and originally deduced via open strings in the work of Polchinski; (Sec. 2.2 for highlights for general D-branes; Sec. 3.2 for the case of D0-branes; and Sec. 4.1 - 4.4 for D0-branes on a commutative quasi-projective space). In particular, except that we have to stay on algebraic groups in the pure algebro-geometric setting, D0-branes in the current setting that move on a (commutative) smooth complex projective surface $Y$ has the same Higgsing/un-Higgsing feature of gas of D0-branes in \[Vafa1\] of Vafa when we choose the morphims of the D0-brane to $Y$ appropriately; (the last theme in Sec. 4.4). The anticipation (Sec. 4.5) that: - is supported in the study of the moduli space of D0-branes; (Sec. 3 and Sec. 4.1- Sec. 4.4). Finally, a word about reading the current work: Noncommutative geometry, in the language of either differential geometry or algebraic geometry, is a demanding topic and there is no way to bypass it. Readers who already know D-branes in the string-theoretic aspect from \[Pol3\] or \[Pol4\] are suggested to read Sec. 4.1 first to see how algebraic geometry in the line of Grothendieck is used to implement Polchinski’s picture in a most elementary case: D0-branes on the complex line ${\Bbb C}$. Various general features of D-branes and their moduli space, following the above prototype definition, reveal themselves already in this example in a simplified form. [*Remark 0.1 $[$diverse D-“branes"$]$.*]{} [ Mathematicians should be aware that there are numerous string theorists whose collective contribution shaped the understanding of D-branes nowadays, cf. the limited “short” list of stringy references of the current work, which have influenced us and became part of the background of the project. Their works led to diverse meanings/roles of D-branes in various physical contents. The current work addresses D-branes when they are “still branes", i.e. in the sense of \[D-L-P\], \[P-C\], \[Pol2\], \[Pol3\], \[Pol4\], and, e.g., \[B-V-S1\], \[B-V-S2\], \[Vafa1\], \[Vafa2\] that they are manifold/variety-type objects. The terms ‘Polchinski’s D-brane’ and ‘D0-brane gas’ occasionally used in this work refer to \[D-L-P\], \[Vafa1\], and Polchinski’s special contribution to this topic. Physicists use the same term ‘D-branes’ in the various different physical contents with good reasons, particularly from the aspect of stringy dualities. However, this is unfortunate/inconvenient for us as these other types of D-“branes" are no longer branes and have/involve very different mathematical contents/language as well. Lacking an official terminology, we use above-mentioned terms and terms like ‘D-branes in the sense of Polchinski’ to single out the particular meaning/type of D-branes studied in the above-quoted stringy works in the earlier years of D-branes for convenience. ]{} [*Remark 0.2 $[$other brane$]$.*]{} [ It should be mentioned that, while D-branes have been a central object in string theory since 1995, there are other types of branes, (e.g.. NS-branes) in string theory as well that serve as the source for other types of fields created by closed strings in space-time; see \[Pol4\], \[Jo\], and \[B-B-Sc\] for a review. It is also worth noting that, since the work of Randall and Sundrum \[R-S\] in 1999, the use of branes has been extended outside of string theory and gives a new insight to the weakness of gravity in comparison with electro-magnetic, weak, and strong interactions in nature. That route hints at a connection of hyperbolic geometry and branes – a topic in its own right. ]{} [**Convention.**]{} Standard notations, terminology, operations, facts in (1) (noncommutative; commutative) ring theory; (2) (commutative) algebraic geometry; (3) quantum field theory, supersymmetry; string theory can be found respectively in (1) \[Jac\]; \[Mat\]; (2) \[Ha\], \[E-H\]; (3) \[I-Z\], \[P-S\], \[W-B\]; \[B-B-Sc\], \[G-S-W\], \[Jo\], \[Pol4\], \[Zw\]. - Except the [*zero-ring*]{} $0$, all [*rings*]{} or [*algebras*]{} (over an algebraically closed field) $R$ in the general discussion of this work are [*associative*]{} with an [*identity*]{} $1$ and are both [*left- and right-Noetherian*]{}. The term “[*$R$-modules*]{}", including “[*ideals*]{}" in $R$, means “[*left*]{} $R$-modules" (cf. [*left*]{} ideals in $R$) unless otherwise noted. $Z(R) :=$ the [*center*]{} of $R$. $M_n(R) :=$ the $n\times n$ [*matrix ring*]{} with entries in $R$. - The term [*field*]{} has two completely different meanings: field in [*quantum field theory*]{} vs. field in the [*theory of rings*]{}. - The [*analytic space*]{} ${\Bbb C}^n$, with the standard topology, of closed points in the affine space ${\Bbb A}^n$ over ${\Bbb C}$ is constantly denoted directly by ${\Bbb A}^n$. Similarly for ${\Bbb P}^n$ and other varieties. (In this work, we use the term ‘varieties/schemes’ mainly only to manifest/emphasize the fact that they arise from gluing of affine charts associated to rings.) In this way, ${\Bbb C}^n$ is kept to mean ${\Bbb C}^n$ as a ${\Bbb C}$- or $M_n({{\Bbb C}})$-[*module*]{} as best possible. ${\Bbb C}^n$ as the $n$-th [*product ring*]{} of ${\Bbb C}$ will be denoted also by $\prod_n{\Bbb C}$. - A representation (resp. commuting) scheme with the reduced scheme structure will be called [*representation*]{} (resp. [*commuting*]{}) [*variety*]{} for simplicity. Irreducibility is [*not*]{} implied here. (In fact, in general they are not irreducible.) - Omitted subscripts (resp. superscripts) are indicated by $_{\bullet}$ (resp. $^{\bullet}$). [**Outline.**]{} - Azumaya-type noncommutative spaces and morphisms therefrom. - Azumaya-type noncommutative spaces and morphisms therefrom. - A noncommutative space as a gluing system of rings. - D-branes from the viewpoint of Grothendieck. - The notion of a space(-time): functor of points vs. probes. - D-branes as Azumaya-type noncommutative spaces. - ${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\smallBbb C}),Y)$ as a coarse moduli space. - Central localizations of Artinian rings and their modules. - ${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\smallBbb C}),Y)$ as a coarse moduli space. - D0-branes on a commutative quasi-projective variety. - D0-branes on the complex affine line ${\smallBbb A}^1$. - D0-branes on the complex projective line ${\smallBbb P}^1$. - D0-branes on the complex affine plane ${\smallBbb A}^2$. - D0-branes on a complex quasi-projective variety. - A remark on D-branes and universal moduli space. Azumaya-type noncommutative spaces and morphisms\ therefrom. ================================================= We introduce in Sec. 1.1 a class of noncommutative spaces that are relevant to D-branes. Its foundation, central localizations of noncommutative rings, is given in Sec. 1.2. The ring-theoretic description of a space in Sec. 1.2 allows us to study as well the space of morphisms between noncommutative spaces without having to construct the noncommutative spaces. Azumaya-type noncommutative spaces and morphisms therefrom. ----------------------------------------------------------- [**Definition 1.1.1 \[Azumaya-type noncommutative space\].**]{} An [*Azumaya-type noncommutative space*]{} is a triple $(X,{\cal O}_X,{\cal O}_X^{{\mbox{\scriptsize\it nc}}})$, where $(X,{\cal O}_X)$ is a (commutative Noetherian) scheme, as defined in \[Ha\], and ${\cal O}_X^{{\mbox{\scriptsize\it nc}}}$ is a coherent sheaf of noncommutative ${\cal O}_X$-algebras[^4] on $X$ that contains ${\cal O}_X$ by $1\cdot {\cal O}_X$ in its center ${\cal Z}({\cal O}_X^{{\mbox{\scriptsize\it nc}}})$. We will call ${\cal O}_X$ (resp. ${\cal O}_X^{{\mbox{\scriptsize\it nc}}}$) the [*commutative*]{} (resp. [*noncommutative*]{}) [*structure sheaf*]{} of $X$. A [*strict morphism*]{} from $(X,{\cal O}_X,{\cal O}_X^{{\mbox{\scriptsize\it nc}}})$ to $(Y,{\cal O}_Y,{\cal O}_Y^{{\mbox{\scriptsize\it nc}}})$ is a triple $(f,f^{\sharp}, f^{\sharp{\mbox{\scriptsize\it nc}}})$, where $(f:X\rightarrow Y\,,\, f^{\sharp}:{\cal O}_Y\rightarrow f_{\ast}{\cal O}_X)$ gives a morphism of schemes from $(X,{\cal O}_X)$ to $(Y,{\cal O}_Y)$ and $f^{\sharp{\mbox{\scriptsize\it nc}}}:{\cal O}_Y^{{\mbox{\scriptsize\it nc}}}\rightarrow f_{\ast}{\cal O}_X^{{\mbox{\scriptsize\it nc}}}$ is a homomorphism of ${\cal O}_Y$-algebras that extends $f^{\sharp}$. A [*general morphism*]{} from $(X,{\cal O}_X,{\cal O}_X^{{\mbox{\scriptsize\it nc}}})$ to $(Y,{\cal O}_Y,{\cal O}_Y^{{\mbox{\scriptsize\it nc}}})$ consists of the following data: - an inclusion pair ${\cal O}_X\subset {\cal A}\subset {\cal A}^{{\mbox{\scriptsize\it nc}}} \subset {\cal O}_X^{{\mbox{\scriptsize\it nc}}}$ of ${\cal O}_X$-subalgebras such that ${\cal A}\subset {\cal Z}({\cal A}^{{\mbox{\scriptsize\it nc}}})\,$; - a strict morphism $(f,f^{\sharp},f^{\sharp{\mbox{\scriptsize\it nc}}})$ from $(X^{\prime},{\cal O}_{X^{\prime}},{\cal O}_{X^{\prime}}^{{\mbox{\scriptsize\it nc}}})$ to $(Y,{\cal O}_Y,{\cal O}_Y^{{\mbox{\scriptsize\it nc}}})\,$, where - $X^{\prime}:={\mbox{\it\bf Spec}\,}{\cal A}$ is equipped with the tautological dominant finite morphism $X^{\prime}\stackrel{\jmath}{\rightarrow} X$ of schemes, - ${\cal O}_{X^{\prime}}^{{\mbox{\scriptsize\it nc}}}$ is the ${\cal O}_{X^{\prime}}$-algebra on $X^{\prime}$ associated to ${\cal A}^{{\mbox{\scriptsize\it nc}}}$ as an ${\cal A}$-algebra. A strict morphism is automatically a general morphism. A general morphism will also be called simply a [*morphism*]{}. Define ${\mbox{\it Mor}\,}(X,Y)$ to be the set of morphisms from $X$ to $Y$. To simplify the notation, we will also denote $(X,{\cal O}_X,{\cal O}_X^{{\mbox{\scriptsize\it nc}}})$ collectively by $X$ and both a strict morphism $(f,f^{\sharp},f^{\sharp{\mbox{\scriptsize\it nc}}})$ and a general morphism $(({\cal A},{\cal A}^{{\mbox{\scriptsize\it nc}}}),(f,f^{\sharp},f^{\sharp{\mbox{\scriptsize\it nc}}}))$ collectively by $f:X\rightarrow Y$. [**Definition/Example 1.1.2 \[tautological morphism/surrogate\].**]{} [ With notations from Definition 1.1.1, the (strict) identity morphism $(X^{\prime},{\cal O}_X,{\cal O}_X^{{\mbox{\scriptsize\it nc}}}) \rightarrow (X^{\prime},{\cal O}_X,{\cal O}_X^{{\mbox{\scriptsize\it nc}}})$ defines a (general) morphism $X=(X,{\cal O}_X,{\cal O}_X^{{\mbox{\scriptsize\it nc}}}) \rightarrow X^{\prime}=(X^{\prime},{\cal O}_X,{\cal O}_X^{{\mbox{\scriptsize\it nc}}})$. Given $X$, we will call an $X\rightarrow X^{\prime}$ arising this way a [*tautological morphism*]{} from $X$ and $X^{\prime}$ an [*surrogate*]{} of $X$. ]{} [**Example 1.1.3 \[noncommutative point\].**]{} Let $k$ be an algebraically closed field and $M_n(k)$ be the $k$-algebra of $n\times n$-matrices with entries in $k$. Then, $X=({\mbox{\it Spec}\,}k, k, M_n(k)) =: {\mbox{\it Space}\,}M_n(k)$ defines an [*Azumaya-type noncommutative point*]{}. See Sec. 3.1 for more details. [**Example 1.1.4 \[morphism of commutative schemes\].**]{} An Azumaya-type noncommutative space $X=(X,{\cal O}_X,{\cal O}_X^{{\mbox{\scriptsize\it nc}}})$ is a commutative scheme if and only if ${\cal O}_X={\cal O}^{{\mbox{\scriptsize\it nc}}}$. In this case, $X$ has no surrogates except $X$ itself and any morphism from $X$ to $Y=(Y, {\cal O}_Y, {\cal O}_Y^{{\mbox{\scriptsize\it nc}}})$ is a strict morphism from $X$ to $Y$. In particular, the natural inclusion $\;{\mbox{\it ${\cal S}$cheme}\,}\; \hookrightarrow\; {\mbox{\it ${\cal A}$zumaya${\cal S}$pace}\,}\;$ of the category of commutative schemes into the category of Azumaya-type noncommutative spaces is fully faithful. The foundation of Definition 1.1.1 (i.e. of the sheaf ${\cal O}_X^{{\mbox{\scriptsize\it nc}}}$) is on central localizations of (noncommutative) rings. This will be discussed in Sec. 1.2. The following lemma follows immediately from the definition: [**Lemma 1.1.5 \[exhaustion\].**]{} [ *Let $X$ and $Y$ be Azumaya-type noncommutative spaces and $X^{\prime}$ be a surrogate of $X$. Then there is a canonical embedding ${\mbox{\it Mor}\,}(X^{\prime},Y) \hookrightarrow {\mbox{\it Mor}\,}(X,Y)$.* ]{} [*Remark 1.1.6 $[$noncommutative geometry$]$.*]{} Noncommutative algebraic geometry was developed with vigor by several schools of mathematicians immediately after Grothendieck’s re-writing of commutative algebraic geometry in the 1960s. There are several classes of noncommutative spaces in existence; each is described in its own appropriate language. While many demanding fundamental issues have prevented it from reaching at the moment the same glory and a unified language as its commutative counterpart from Grothendieck’s school, it is a constant growing subject. Readers are referred to, e.g. (in rough historical order) \[Go\], \[vO-V\], \[A-Z\], \[J-V-V\], \[Ro1\], \[Ro2\], \[K-R1\], \[K-R2\] from the algebraic aspect; \[Co\] from the analytic aspect; and \[Man2\], \[Man3\], \[Kapr\], \[Lau\], \[leB1\] from other aspects for details and more references. [*Remark 1.1.7 $[$Azumaya-type noncommutative space$]$.*]{} The class of noncommutative spaces we define here, namely $(X,{\cal O}_X,{\cal O}_X^{{\mbox{\scriptsize\it nc}}})$, are chosen with D-branes in mind. While they may be thought of as noncommutative “clouds" (i.e. ${\cal O}_X^{{\mbox{\scriptsize\it nc}}}$) over (commutative) schemes (i.e. $(X,{\cal O}_X)$), the way we define a morphism from $X$ to $Y$ says that the main object of focus in the triple $(X,{\cal O}_X,{\cal O}_X^{{\mbox{\scriptsize\it nc}}})$ is ${\cal O}_X^{{\mbox{\scriptsize\it nc}}}$, rather than $(X,{\cal O}_X)$. This particular point is important in the realization of a D-brane of B-type as an Azumaya-type noncommutative space. We suggest readers to think of $$\begin{array}{ccl} (X,{\cal O}_X,{\cal O}_X^{{\mbox{\scriptsize\it nc}}}) & \hspace{.8em}\mbox{as}\hspace{.8em} & \begin{array}{l} {\cal O}_X^{{\mbox{\scriptsize\it nc}}}\,,\hspace{1em} \mbox{{\it together with} the system $L_{{\cal O}_X^{{\mbox{\tiny\it nc}}}}$ of sub-${\cal O}_X$-algebra pairs:} \\[1.6ex] L_{{\cal O}_X^{{\mbox{\tiny\it nc}}}}\; =\;\left\{\, ({\cal A},{\cal A}^{{\mbox{\scriptsize\it nc}}})\; \left|\; \begin{array}{l} {\cal O}_X \subset {\cal A} \subset {\cal A}^{{\mbox{\scriptsize\it nc}}} \subset {\cal O}_X^{{\mbox{\scriptsize\it nc}}}\,; \\[.6ex] \mbox{${\cal A}$, ${\cal A}^{{\mbox{\scriptsize\it nc}}}\,$: sub-${\cal O}_X$-algebras}\,;\, {\cal A} \subset {\cal Z}({\cal A}^{{\mbox{\scriptsize\it nc}}})\, \end{array} \right. \right\}. \end{array} \end{array}$$ I.e. $(X,{\cal O}_X,{\cal O}_X^{{\mbox{\scriptsize\it nc}}})$ together with the system $\{X \rightarrow X^{\prime}\}_{X^{\prime}}$ of surrogates in ${\mbox{\it ${\cal A}$zumaya${\cal S}$pace}\,}$. [**Example 1.1.8 \[noncommutative point revisitd\].**]{} (Continuing Example 1.1.3.) A surrogate of the Azumaya-type noncommutative point ${\mbox{\it Space}\,}M_n({k})$ over $k$ is given by a sub-$k$-algebra pair $k\subset C\subset R\subset M_n(k)$ with $C\subset Z(R)$. In particular, while ${\mbox{\it Space}\,}M_n(k)$ consists geometrically of only one point (i.e. ${\mbox{\it Spec}\,}k$), its surrogate $X^{\prime}=({\mbox{\it Spec}\,}C,C,R)$ can have more than one geometric points in ${\mbox{\it Spec}\,}C$. All these $X^{\prime}$’s should be thought of as part of the “geometry" of noncommutative point ${\mbox{\it Space}\,}M_n(k)$. [**Definition 1.1.9 \[left/right quasi-coherent/coherent sheaf\].**]{} [ A [*left quasi-coherent sheaf*]{} on $(X,{\cal O}_X, {\cal O}_X^{{\mbox{\scriptsize\it nc}}})$ is a sheaf of left ${\cal O}_X^{{\mbox{\scriptsize\it nc}}}$-modules that is quasi-coherent on $(X,{\cal O}_X)$. Similarly for the definitin of a [*right quasi-coherent sheaf*]{}, a [*left coherent sheaf*]{}, and a [*right coherent sheaf*]{} on $(X,{\cal O}_X,{\cal O}_X^{{\mbox{\scriptsize\it nc}}})$. A [*${\cal O}_X^{{\mbox{\scriptsize\it nc}}}$-module*]{} is by convention a left ${\cal O}_X^{{\mbox{\scriptsize\it nc}}}$-module. ]{} Hidden in the notion of ${\cal O}_X^{{\mbox{\scriptsize\it nc}}}$ in the tuple $(X,{\cal O}_X, {\cal O}_X^{{\mbox{\scriptsize\it nc}}})$ is the notion of central localizations, which we will discuss more thoroughly in Sec. 1.2. A noncommutative space as a gluing system of rings. --------------------------------------------------- The purely ring[^5]-theoretic construction in this subsection enables us to talk about a “noncommutative scheme" without having to construct one[^6]. The ring system to be defined is meant to carry the same information as the noncommutative scheme associated to ${\cal O}_X^{{\mbox{\scriptsize\it nc}}}$ in $X=(X,{\cal O}_X,{\cal O}_X^{{\mbox{\scriptsize\it nc}}})$ would. Such a description will later be used to study ${\mbox{\it Mor}\,}(X,Y)$. Behind the messy notations is the notion of Grothendieck-descent-data description of spaces/stacks/sheaves and morphisms between them. [**Noncommutative localizations.**]{} The notion of noncommutative localizations can be traced back to Ore in \[Or1\] and \[Or2\] in 1930s. Here we recall only definitions that will be needed later. See e.g. \[Ga\], \[Goldm\], \[Jat\], \[St\] for more details and thorough discussions. A [*Gabriel filter*]{} on a ring $R$ is a collection ${\frak F}$ of ideals in $R$ that satisfies[^7]: - if $I\in{\frak F}$ and $J$ is an ideal that contains $I$, then $J\in {\frak F}$; - if $I$, $J\in {\frak F}$, then $I\cap J\in {\frak F}$; - if $I\in {\frak F}$, then $(I:r)\in {\frak F}$ for $r\in R$; - if $I\in {\frak F}$ and $J$ is an ideal such that $(J:r)\in{\frak F}$ for all $r\in I$, then $J\in{\frak F}$. Each Gabriel filter ${\frak F}$ on $R$ determines the subcategory ${\cal T}_{\frak F}$ of ${\frak F}$-[*torsion*]{} objects and the subcategory ${\cal F}_{\frak F}$ of ${\frak F}$-[*torsion-free*]{} objects in the category $R$-${\mbox{\it Mod}\,}$ of (left) $R$-modules. An object $M$ in ${\cal T}_{\frak F}$ is characterized by that each element $m$ of $M$ has its annihilator ${\mbox{\it Ann}\,}(m)\in {\frak F}$; and an object $N$ in ${\cal F}_{\frak F}$ is characterized by that $N$ contains no submodule in ${\cal T}_{\frak F}$ except the zero submodule $0$. Each object in $M\in R$-${\mbox{\it Mod}\,}$ fits into an exact sequence $0 \rightarrow M^{\prime} \rightarrow M \rightarrow M^{\prime\prime} \rightarrow 0$, where $t_{\frak F}(M):= M^{\prime}\in {\cal T}_{\frak F}$ and $M^{\prime\prime}\in {\cal F}_{\frak F}$. In particular, $M\in {\cal T}_{\frak F}$ (resp. ${\cal F}_{\frak F}$) if and only if $t_{\frak F}(M)=M$ (resp. $t_{\frak F}(M)=0$). The [*localization*]{} $M_{\frak F}$ of $M\in R$-${\mbox{\it Mod}\,}$ with respect to ${\frak F}$ is defined to be the ${\frak F}$-[*injective envelop*]{} $E_{\frak F}(M/t(M))$ of the ${\frak F}$-torsion-free quotient module $M/t_{\frak F}(M)$ of $M$. When ${\frak F}$ is clear or omitted from the text, $E_{\frak F}$, ${\cal F}_{\frak F}$, $t_{\frak F}$, ${\cal T}_{\frak F}$, “${\frak F}$-torsion", and “${\frak F}$-torsion-free" will be denoted/called simply $E$, ${\cal F}$, $t$, ${\cal T}$, “[*torsion*]{}", and “[*torsion-free*]{}" respectively. The following kind of localizations is closest to the localizations in the case of commutative rings. It is the one used in Definition 1.1.1 for ${\cal O}_X^{{\mbox{\scriptsize\it nc}}}$: [**Definition 1.2.1 \[central localization\][^8].**]{} [ Given a ring $R$, a [*central localization*]{} of $R$ is the localization $R_{{\frak F}_S}$ of $R$ with respect to the Gabriel filter ${\frak F}_S$ associated to a multiplicatively closed subset $S$ in the center $Z(R)$ of $R$. ]{} Explicitly, the Gabriel filter in the above definition is given by ${\frak F}_S=\{I\,:\,\mbox{ideal of $R$}\;,\; I\cap S\ne \emptyset\}$ and the central localization is given by $R_{{\frak F}_S} = [S^{-1}]R = R[S^{-1}] := (R\times S)/\sim$, where $(r_1,s_1)\sim (r_2,s_2)$ if and only if $s(r_1s_2-r_2s_1)=0$ for some $s\in S$. [**Definition 1.2.2 \[push-out, admissibility, and descent\].**]{} [ (1) Let $\varphi: R\rightarrow R^{\prime}$ be a ring-homomorphism, $S\subset Z(R)$ be a multiplicatively closed subset in $R$ such that $\varphi(S)\subset Z(R^{\prime})$, and $\psi:R\rightarrow R_{{\frak F}_{S}}$ be the central localization of $R$ with respect to $S$. Then the central localization $\psi^{\prime}: R^{\prime} \rightarrow R^{\prime}_{{\frak F}_{\varphi(S)}}$ of $R^{\prime}$ is called the [*push-out*]{} of $\psi$ to $R^{\prime}$ via $\varphi$. (2) Given central localizations $\psi:R\rightarrow R_{{\frak F}_{S}}$ and $\psi^{\prime}:R^{\prime} \rightarrow R^{\prime}_{{\frak F}_{S^{\prime}}}$, a ring-homomorphism $\varphi:R\rightarrow R^{\prime}$ is called [*admissible*]{} to $(S,S^{\prime})$ if $\varphi(S)\subset S^{\prime}$. For such $\varphi$, there is a canonical/unique ring-homomorphism $\varphi_{(S,S^{\prime})}:R_{{\frak F}_S} \rightarrow R^{\prime}_{{\frak F}_{S^{\prime}}}$ that makes the following diagram commute: $$\begin{array}{cccl} R & \stackrel{\varphi}{\longrightarrow} & R^{\prime}\hspace{.8em} \\ \mbox{\scriptsize $\psi$}\downarrow\hspace{.8em} & & \downarrow\mbox{\scriptsize $\psi^{\prime}$} \\[-.6ex] \hspace{.8em}R_{{\frak F}_S} & \stackrel{\varphi_{(S,S^{\prime})}}{\longrightarrow} & R^{\prime}_{{\frak F}_{S^{\prime}}} & . \end{array}$$ $\varphi_{(S,S^{\prime})}$ is called the [*descent*]{} of $\varphi$ under the central localizations. ]{} [**Example 1.2.3 \[2-step consecutive central localization\].**]{} Given central localizations $\psi_1: R\rightarrow R_1$ and $\psi_2:R\rightarrow R_2$ of $R$, one has the push-outs $\psi_{12}: R_1\rightarrow R_{12}$ and $\psi_{21}: R_2\rightarrow R_{21}$ of $\psi_2$ via $\psi_1$ and of $\psi_1$ via $\psi_2$ respectively. Then there is a canonical isomorphism $R_{12}\simeq R_{21}$ such that the following diagram $$\begin{array}{lcr} R & \stackrel{\psi_2}{\longrightarrow} & R_2\hspace{1ex} \\ \hspace{-.9em}\mbox{\scriptsize $\psi_1$}\downarrow & & \downarrow\mbox{\scriptsize $\psi_{21}$}\hspace{-.7em} \\[-.6ex] R_1 & \stackrel{\psi_{12}}{\longrightarrow} & R_{12}\; \simeq\; R_{21} \end{array}$$ commutes. Both the compositions $\psi_{12}\circ\psi_1:R\rightarrow R_{12}$ and $\psi_{21}\circ\psi_2:R\rightarrow R_{21}$ give central localizations of $R$. Such 2-step consecutive central localizations will appear in stating the cocycle conditions for the gluing of rings along their central localizations. [**A ring-theoretic description of noncommutative spaces and their morphisms.**]{} We give a description of a class of noncommutative spaces and their morphisms solely in terms of rings, ring-homomorphisms, and central localizations, [*without*]{} employing the notion of “points" and “topology" of a “space". This class contains the class of Azumaya-type noncommutative spaces introduced in Sec. 1.1 as a subclass. [**Definition 1.2.4 \[finite central cover of a ring\].**]{} [ Let $A$ be a finite set and ${\cal U}:= \{\varphi_{\alpha}:R\rightarrow R_{\alpha}\}_{\alpha\in A}$ be a finite collection of central localizations of $R$ with respect to Gabriel filters ${\frak F}_{\alpha}$, $\alpha\in A$, on $R$. We say that ${\cal U}$ is a [*finite central cover*]{} of $R$ if $\sum_{\alpha\in A} I_{\alpha}=R$ for any tuple $(I_{\alpha})_{\alpha}\in \prod_{\alpha\in A}{\frak F}_{\alpha}$. ]{} [**Definition 1.2.5 \[gluing system of rings\][^9].**]{} A (finite) [*gluing system*]{} of rings $${\cal R}\; =\; \left( \{R_{\alpha}\}_{\alpha\in A} {\begin{array}{c} \rightarrow \\[-2ex] \rightarrow \end{array}}\{R_{\alpha_1\alpha_2}\}_{\alpha_1,\alpha_2\in A} \right)$$ from central localizations consists of the following data: - \[[*local ring-charts*]{}\]\ a finite collection $\{R_{\alpha}\}_{\alpha\in A}$ of rings; ($A$: the [*index set*]{} of ${\cal R}$) - \[[*transition ring-homomorphisms*]{}\]\ a finite central cover $\{R_{\alpha_1}\rightarrow R_{\alpha_1\alpha_2}\}_{\alpha_2\in A}$ for each $R_{\alpha_1}$ and a choice of ring-isomorphisms $\varphi_{\alpha_1\alpha_2}: R_{\alpha_1\alpha_2} \stackrel{\sim}{\rightarrow} R_{\alpha_2\alpha_1}$ for each $(\alpha_1,\alpha_2)\in A\times A$ such that $R_{\alpha\alpha}=R_{\alpha}$, $\varphi_{\alpha_1\alpha_2}=\varphi_{\alpha_2\alpha_1}^{\;\;-1}$, and $\varphi_{\alpha\alpha}={\mbox{\it Id}}_{R_{\alpha}}$; - \ the ring-homomorphism $R_{\alpha_1}\rightarrow R_{\alpha_1\alpha_2}$ pushes out the finite central cover $\{R_{\alpha_1}\rightarrow R_{\alpha_1\alpha_3}\}_{\alpha_3}$ of $R_{\alpha_1}$ to a finite central cover $\{R_{\alpha_1\alpha_2}\rightarrow R_{\alpha_1\alpha_2\alpha_3}\}_{\alpha_3}$ of $R_{\alpha_1\alpha_2}$ and one has the canonical isomorphisms $R_{\alpha_1\alpha_2\alpha_3}\simeq R_{\alpha_1\alpha_3\alpha_2}$ from the push-out diagrams; it is then required that the gluing ring-isomorphisms $R_{\alpha_1\alpha_2}\rightleftharpoons R_{\alpha_2\alpha_1}$ descend to ring-isomorphisms $R_{\alpha_1\alpha_2\alpha_2} \rightleftharpoons R_{\alpha_2\alpha_1\alpha_3}$ that make the following diagrams $$\begin{array}{ccc} R_{\alpha_1\alpha_2} & \rightleftharpoons & R_{\alpha_2\alpha_1} \\[.6ex] \downarrow & & \downarrow \\ R_{\alpha_1\alpha_2\alpha_3} & \rightleftharpoons & R_{\alpha_2\alpha_1\alpha_3} \end{array} \hspace{1em}\mbox{and}\hspace{1em} \begin{array}{c} R_{\alpha_1\alpha_2\alpha_3} \simeq R_{\alpha_1\alpha_3\alpha_2} \\ \swarrow\hspace{-1.2ex}\nearrow \hspace{10em} \nwarrow\hspace{-1.2ex}\searrow \\[.6ex] R_{\alpha_2\alpha_1\alpha_3} \simeq R_{\alpha_2\alpha_3\alpha_1} \hspace{.8em}\rightleftharpoons\hspace{.8em} R_{\alpha_3\alpha_2\alpha_1}\simeq R_{\alpha_3\alpha_1\alpha_2} \end{array}$$ commute. Note that under the requirement of the first diagram above the isomorphisms $R_{\alpha_1\alpha_2\alpha_3} \rightleftharpoons R_{\alpha_2\alpha_1\alpha_3}$, when exists, are unique. We will write $R_{\alpha}\in {\cal R}$ to indicate that $R_{\alpha}$ is a ring-chart in the system ${\cal R}$. A ([*finite central*]{}) [*refinement*]{} of ${\cal R}$ is a gluing system ${\cal R}^{\prime} =( \{R^{\prime}_{\alpha^{\prime}}\}_{\alpha^{\prime}\in A^{\prime}} {\begin{array}{c} \rightarrow \\[-2ex] \rightarrow \end{array}}\{R^{\prime}_{\alpha_1^{\prime},\alpha_2^{\prime}}\} _{\alpha_1^{\prime},\alpha_2^{\prime}\in A^{\prime}} )$ of rings together with the following data: - a surjective map $\tau:A^{\prime}\rightarrow A\,$; - a central localization ring-homomorphism $R_{\alpha}\rightarrow R^{\prime}_{\alpha^{\prime}}$ for each $\alpha\in A$ and $\alpha^{\prime}\in \tau^{-1}(\alpha)$ such that - for each $\alpha\in A$, $\{R_{\alpha}\rightarrow R^{\prime}_{\alpha^{\prime}}\} _{\alpha^{\prime}\in \tau^{-1}(\alpha)}$ is a finite central cover of $R_{\alpha}$; - for all $(\alpha_1,\alpha_2)\in A\times A$ and $(\alpha^{\prime}_1,\alpha^{\prime}_2) \in \tau^{-1}(\alpha_1)\times\tau^{-1}(\alpha_2)$, $R_{\alpha_1}\rightarrow R^{\prime}_{\alpha^{\prime}_1}$ descends to $R_{\alpha_1\alpha_2}\rightarrow R^{\prime}_{\alpha^{\prime}_1\alpha^{\prime}_2}$ and all the diagrams $$\begin{array}{ccc} R_{\alpha_1\alpha_2} & \stackrel{\varphi_{\alpha_1\alpha_2}}{\longrightarrow} & R_{\alpha_2\alpha_1} \\[.6ex] \downarrow\hspace{1ex} & & \downarrow\hspace{1ex} \\[-1.6ex] R^{\prime}_{\alpha^{\prime}_1\alpha^{\prime}_2} & \stackrel{\varphi^{\prime} _{\alpha^{\prime}_1\alpha^{\prime}_2}}{\longrightarrow} & R^{\prime}_{\alpha^{\prime}_2\alpha^{\prime}_1} \end{array}$$ commute. We will denote ${\cal R}^{\prime}$, together with this data of arrows from ${\cal R}$ to ${\cal R}^{\prime}$, by ${\cal R}^{\prime}{\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}{\cal R}$. Two gluing systems of rings ${\cal R}_1$ and ${\cal R}_2$ are said to be [*equivalent*]{}, in notation ${\cal R}_1\sim{\cal R}_2$, if there exists a gluing system ${\cal R}_3$ such that both ${\cal R}_3{\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}{\cal R}_1$ and ${\cal R}_3{\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}{\cal R}_2$ exist/hold. The [*equivalence class*]{} of ${\cal R}$ under refinements is denoted by $[{\cal R}]$. [**Definition 1.2.6 \[gluing system of ring-homomorphisms\][^10].**]{} A [*gluing system of ring-homomorphisms*]{} from a gluing system ${\cal R}=( \{R_{\alpha}\}_{\alpha\in A} {\begin{array}{c} \rightarrow \\[-2ex] \rightarrow \end{array}}\{R_{\alpha_1\alpha_2}\}_{\alpha_1,\alpha_2\in A} )$ to another such system ${\cal S}=( \{S_{\beta}\}_{\beta\in B} {\begin{array}{c} \rightarrow \\[-2ex] \rightarrow \end{array}}\{S_{\beta_1\beta_2}\}_{\beta_1,\beta_2\in B} )$ consists of the following data: - a map $\tau: B\rightarrow A$ on the index sets; - \[[*ring-homomorphisms on ring-charts*]{}\]\ a collection $\{ \varphi_{\beta}: R_{\tau(\beta)}\rightarrow S_{\beta} \} _{\beta\in B}$ of ring-homomorphisms such that - \[[*compatibility with localizations*]{}\]\ for all $\beta_1$, $\beta_2\in B$, $\varphi_{\beta_1}:R_{\tau(\beta_1)}\rightarrow S_{\beta_1}$ is admissible and, hence, descends to a unique $\varphi_{\beta_1}|_{\beta_2}:R_{\tau(\beta_1)\tau(\beta_2)} \rightarrow S_{\beta_1\beta_2}$ that makes the diagram $$\begin{array}{ccc} R_{\tau(\beta_1)} & \stackrel{\varphi_{\beta_1}}{\longrightarrow} & S_{\beta_1} \\[.6ex] \downarrow && \downarrow \\[-.6ex] R_{\tau(\beta_1)\tau(\beta_2)} & \stackrel{\varphi_{\beta_1}|_{\beta_2}}{\longrightarrow} & S_{\beta_1\beta_2} \end{array}$$ commute, cf. Definition 1.2.2; - \[[*gluing conditions*]{}\]\ the diagrams $$\begin{array}{ccc} S_{\beta_1\beta_2} & \rightleftharpoons & S_{\beta_2\beta_1} \\[.6ex] \mbox{\scriptsize $\varphi_{\beta_1}|_{\beta_2}$}\uparrow\hspace{3em} && \hspace{2em}\uparrow \mbox{\scriptsize $\varphi_{\beta_2}|_{\beta_1}$} \\ R_{\tau(\beta_1)\tau(\beta_2)} & \rightleftharpoons & R_{\tau(\beta_2)\tau(\beta_1)} \end{array}$$ commute for all $(\beta_1,\beta_2)\in B\times B$. We will call the system $\Phi := (\tau,\{\varphi_{\beta}\}_{\beta})$ also a [*morphism*]{} from ${\cal R}$ to ${\cal S}$. [**Example 1.2.7 \[refinement as a morphism\].**]{} A refinement ${\cal R}^{\prime} {\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}{\cal R}$ contains a system $\Phi:{\cal R}\rightarrow {\cal R}^{\prime}$ of ring-homomorphisms in its data. In particular, a central cover $\{R\rightarrow R_{\alpha}\}_{\alpha}$ of $R$ gives rise to a morphism $\{R\}\rightarrow \{R_{\alpha}\}_{\alpha}$. Ring-homomorphisms have the following affine-gluing property: [**Lemma 1.2.8 \[morphism: affine-gluing\].**]{} [ *Given finitely generated rings $R$ and $S$, let $(\{S_{\alpha}\}_{\alpha\in A} {\begin{array}{c} \rightarrow \\[-2ex] \rightarrow \end{array}}\{S_{\alpha_1\alpha_2}\}_{\alpha_1,\alpha_2\in A})$ be a gluing system of rings associated to a finite central cover $\{S\rightarrow S_{\alpha}\}_{\alpha\in A}$ of $S$ and $\Phi=\{ \varphi_{\alpha}: R \rightarrow S_{\alpha}\}_{\alpha\in A}$ be a gluing system of ring-homomorphisms from $R$. Then, there exists a unique ring-homomorphism $\varphi:R\rightarrow S$ such that $\varphi$ descends to $\Phi$.* ]{} We will call $\varphi$ in the above lemma the [*gluing*]{} of the system $\Phi$. A reverse of this lemma gives rise to the following definition: [**Definition 1.2.9 \[refinement of morphism\].**]{} [ Given a morphism $\Phi=(\tau,\{\varphi_{\beta}\}_{\beta}): {\cal R}\rightarrow {\cal S}$ and a pair $( {\cal R}^{\prime}{\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}{\cal R}\,,\, {\cal S}^{\prime}{\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}{\cal S})$ of refinements, denote the index set of ${\cal R}$, ${\cal R}^{\prime}$, ${\cal S}$, ${\cal S}^{\prime}$ by $A$, $A^{\prime}$, $B$, $B^{\prime}$ respectively. Let $\tau:B\rightarrow A$ and $( \tau_{A^{\prime},A}:A^{\prime}\rightarrow A\,,\, \tau_{B^{\prime},B}:B^{\prime}\rightarrow B )$ be the maps on the index sets corresponding to $\Phi$ and the pair of refinements respectively. Then $( {\cal R}^{\prime}{\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}{\cal R}\,,\, {\cal S}^{\prime}{\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}{\cal S})$ is said to be [*$\Phi$-admissible*]{} if, for all $\beta\in B$, $\varphi_{\beta}$ is admissible with respect to the localizations maps in the system pair $( {\cal R}^{\prime}{\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}{\cal R}\,,\, {\cal S}^{\prime}{\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}{\cal S})$; cf. Definition 1.2.2. When this is the case, fix a $\tau^{\prime}:B^{\prime}\rightarrow A^{\prime}$ so that the diagram $$\begin{array}{ccc} A & \stackrel{\tau}{\longleftarrow} & B \\ \hspace{-2em}\mbox{\scriptsize $\tau_{A^{\prime},A}$}\uparrow & & \uparrow\mbox{\scriptsize $\tau_{B^{\prime},B}$}\hspace{-2.2em} \\ A^{\prime} & \stackrel{\tau^{\prime}}{\longleftarrow} & B^{\prime} \end{array}$$ commute. Then $\Phi$ descends to a unique morphism $\Phi^{\prime} =( \tau^{\prime}, \{\varphi^{\prime}_{\beta^{\prime}}\}_{\beta^{\prime}} ) : {\cal R}^{\prime}\rightarrow {\cal S}^{\prime}$, called a [*refinement*]{} of $\Phi$ with respect to $( {\cal R}^{\prime}{\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}{\cal R}\,,\, {\cal S}^{\prime}{\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}{\cal S})$. ]{} [**Definition 1.2.10 \[equivalence of morphisms\].**]{} [ Given equivalent ring-systems ${\cal R}_1\sim {\cal R}_2$ and ${\cal S}_1\sim {\cal S}_2$ and morphisms $\Phi_1:{\cal R}_1\rightarrow {\cal S}_1$ and $\Phi_2:{\cal R}_2\rightarrow {\cal S}_2$, we say that $\Phi_1$ and $\Phi_2$ are [*equivalent*]{}, in notation $\Phi_1\sim \Phi_2$, if there exist common refinements ${\cal R}_1 {\vspace{-.2ex}\begin{array}{c} \succ \\[-1.9ex] \rightarrow \end{array}}{\cal R}^{\prime} {\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}{\cal R}_2$ and ${\cal S}_1 {\vspace{-.2ex}\begin{array}{c} \succ \\[-1.9ex] \rightarrow \end{array}}{\cal S}^{\prime} {\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}{\cal S}_2$ such that (1) $({\cal R}^{\prime} {\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}{\cal R}_1, {\cal S}^{\prime} {\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}{\cal S}_1)$ and $({\cal R}^{\prime} {\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}{\cal R}_2, {\cal S}^{\prime} {\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}{\cal S}_2)$ are $\Phi_1$- and $\Phi_2$-admissible respectively and (2) $\Phi_1$ and $\Phi_2$ can be descended to identical morphisms $\Phi_1^{\prime}=\Phi_2^{\prime}: {\cal R}^{\prime} \rightarrow {\cal S}^{\prime}$. The equivalence class of $\Phi$ will be denoted by $[\Phi]$. An element in $[\Phi]$ will be called a [*representative*]{} of $[\Phi]$. ]{} [**Definition 1.2.11 \[strict morphism on equivalence classes\].**]{} [ By a [*strict morphism*]{} from $[{\cal R}_0]$ to $[{\cal S}_0]$, we mean an equivalence class $[\Phi:{\cal R}\rightarrow {\cal S}]$, where ${\cal R}\in [{\cal R}_0]$ and ${\cal S}\in [{\cal S}_0]$. ]{} By descending to a refinement ${\cal R}$ of ${\cal R}_0$ and taking the pre-composition with the localizations maps in ${\cal R}{\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}{\cal R}_0$, one has the following lemma: [**Lemma 1.2.12 \[one-side refinement enough\].**]{} [ *A strict morphism from $[{\cal R}_0]$ to $[{\cal S}_0]$ can be represented by a $\Phi:{\cal R}_0\rightarrow {\cal S}$, for some ${\cal S}\in [{\cal S}_0]$.* ]{} Thus, in the discussion below, only the refinements on the $[{\cal S}_0]$-side are required. [**Definition 1.2.13 \[injective strict morphism\].**]{} [ A [*injective strict morphism*]{} $[\Phi_0]:[{\cal R}]\rightarrow [{\cal S}_0]$ is a strict morphism that can be represented by a $\Phi=(\tau,\{\varphi_{\beta}\}_{\beta}) :{\cal R}\rightarrow {\cal S}$, ${\cal S}\in [{\cal S}_0]$, such that (1) $\tau$ is surjective and (2) for each $R_{\alpha}\in {\cal R}$, there exists a $\beta\in \tau^{-1}(\alpha)$ such that $\varphi_{\beta}:R_{\alpha}\rightarrow S_{\beta}\in {\cal S}$ is a ring-monomorphism. ]{} [**Definition 1.2.14 \[(general) morphism\].**]{} A [*general morphism*]{} from $[{\cal R}]$ to $[{\cal S}_0]$ consists of the following data: - an injective strict morphism $[\Phi_0]: [{\cal S}_0^{\prime}]\rightarrow [{\cal S}_0]$, - a strict morphism $[\Phi_0^{\prime}]:[{\cal R}]\rightarrow [{\cal S}_0^{\prime}]$. We will denote the tuple $([{\cal S}_0^{\prime}], [\Phi_0], [\Phi_0^{\prime}])$ collectively by $[\Phi_0^{\prime}]$ and a general morphism also by $[\Phi_0^{\prime}]:[{\cal R}]\rightarrow [{\cal S}_0]$. A [*representative*]{} of $[\Phi_0^{\prime}]:[{\cal R}]\rightarrow [{\cal S}_0]$ is given by a [*$3$-step ring-system-morphism diagram*]{} $${\cal R}\; \stackrel{\Phi^{\prime}}{\longrightarrow}\; {\cal S}^{\prime\prime}\; {\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}\; {\cal S}^{\prime}\; \stackrel{\Phi}{\longrightarrow}\; {\cal S}$$ with ${\cal S}\in [{\cal S}_0]$; ${\cal S}^{\prime}$, ${\cal S}^{\prime\prime}\in [{\cal S}_0^{\prime}]$; $\Phi\in [\Phi_0]$, and $\Phi^{\prime}\in [\Phi_0^{\prime}]$. A strict morphism is automatically a general morphism. A general morphism will also be called simply a [*morphism*]{}[^11]. Define ${\mbox{\it Mor}\,}([{\cal R}],[{\cal S}_0])$ to be the set of morphisms from $[{\cal R}]$ to $[{\cal S}_0]$. [**Example 1.2.15 \[non-strict morphism\].**]{} Let $S$ be a subring of $S_0$ such that $Z(S)\supsetneqq Z(S_0)$ and $\Sigma=\{s_{\beta}\}_{\beta}$ be a finite subset in $Z(S)-Z(S_0)$ such that $S=\sum_{s_{\beta}\in \Sigma}s_{\beta}\cdot S$. Let $S\rightarrow S_{\beta}$ (resp. $S\rightarrow S_{\beta_1\beta_2}$) be the central localization with respect to $s_{\beta}$ (resp. $s_{\beta_1}$ and then $s_{\beta_2}$) , then $\{S\rightarrow S_{\beta}\}_{\beta}$ is a cover of $S$. Then the $3$-step diagram $$\begin{aligned} \lefteqn{ \left( \{S_{\beta}\}_{\beta} {\begin{array}{c} \rightarrow \\[-2ex] \rightarrow \end{array}}\{S_{\beta_1\beta_2}\}_{\beta_1, \beta_2} \right) }\\[.6ex] && \stackrel{{\mbox{\scriptsize\it Id}}}{\longrightarrow}\; \left( \{S_{\beta}\}_{\beta} {\begin{array}{c} \rightarrow \\[-2ex] \rightarrow \end{array}}\{S_{\beta_1\beta_2}\}_{\beta_1, \beta_2} \right)\; {\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}\; \left( \{S\}{\begin{array}{c} \rightarrow \\[-2ex] \rightarrow \end{array}}\{S\} \right)\; \longrightarrow\; \left( \{S_0\}{\begin{array}{c} \rightarrow \\[-2ex] \rightarrow \end{array}}\{S_0\} \right) \end{aligned}$$ represents a morphism $[{\mbox{\it Id}}]:( \{S_{\beta}\}_{\beta} {\begin{array}{c} \rightarrow \\[-2ex] \rightarrow \end{array}}\{S_{\beta_1\beta_2}\}_{\beta_1,\beta_2} ) \rightarrow (\{S_0\}{\begin{array}{c} \rightarrow \\[-2ex] \rightarrow \end{array}}\{S_0\})$ that is not strict. See Sec. 4.2 for such examples with $S_0=M_n({\Bbb C})$. [**Grothendieck Ansatz \[ring vs. space\].**]{} [ *We shall hiddenly think of an equivalence class $[{\cal R}]$ of ring-systems as a “space" ${\mbox{\it Space}\,}[\cal R]$ with an equivalence class of atlases $\{{\mbox{\it Space}\,}R_{\alpha}\}_{\alpha}$ (with the gluing data from the arrows $\{{\mbox{\it Space}\,}R_{\alpha_1\alpha_2}\}_{\alpha_1,\alpha_2} {\begin{array}{c} \rightarrow \\[-2ex] \rightarrow \end{array}}\{{\mbox{\it Space}\,}R_{\alpha}\}_{\alpha}$), and a morphism $[{\cal R}]\rightarrow [{\cal S}]$ as a morphism ${\mbox{\it Space}\,}[{\cal S}]\rightarrow {\mbox{\it Space}\,}[{\cal R}]$. Cf. footnote 11.* ]{} [*Remark 1.2.16 $[$morphism vs. map$]$.*]{} In defining a [*morphism*]{} in a category of noncommutative spaces, we mean to keep both the [*domain*]{} and the [*target*]{} of the morphism [*fixed*]{}. In terms of the ring-system language, this is reflected in the fact that a refinement of a ring-system ${\cal R}$ is another ring system ${\cal R}^{\prime}$ [*together with*]{} a localization morphism ${\cal R}\rightarrow {\cal R}^{\prime}$ and the fact that the trivial localization is the identity map (not just a ring-isomorphism). In contrast, later (Sec. 4) when we discuss the space of [*maps*]{} or of [*D0-brane probes*]{}, we remain to keep the target-space fixed but the [*domain*]{}-space will be taken as [*not fixed*]{}. The issue of automorphisms of the domain will then enter. [*Remark 1.2.17 $[$$k$-algebra$]$.*]{} When all the rings $R_{\alpha}\in {\cal R}$ involved are $k$-algebras for a fixed ground field $k$, we will take as a convention that all the ring-homomorphisms involved are then required to be $k$-algebra-homomorphisms unless otherwise noted. [*Remark 1.2.18 $[$Azumaya-type noncommutative space$]$.*]{} For an Azumaya-type noncommutative space $X=(X,{\cal O}_X,{\cal O}_X^{{\mbox{\scriptsize\it nc}}})$, an affine cover $\{U_{\alpha}\}_{\alpha}$ of $(X,{\cal O}_X)$ gives rise to a ring-system representation ${\cal R}_X$ of $X$ defined by $${\cal R}_X\; =\; (\{R_{\alpha}\}_{\alpha} {\begin{array}{c} \rightarrow \\[-2ex] \rightarrow \end{array}}\{R_{\alpha_1\alpha_2}\}_{\alpha_1,\alpha_2})\; :=\; (\{{\cal O}_X^{{\mbox{\scriptsize\it nc}}}(U_{\alpha})\}_{\alpha} {\begin{array}{c} \rightarrow \\[-2ex] \rightarrow \end{array}}\{{\cal O}_X^{{\mbox{\scriptsize\it nc}}}(U_{\alpha_1}\cap U_{\alpha_2})\} _{\alpha_1,\alpha_2})\,.$$ Morphisms $X\rightarrow Y$ between Azumaya-type noncommutative spaces can be expressed contravariantly as morphisms ${\cal R}_Y\rightarrow {\cal R}_X$ of associated ring-systems. In particular, the notion of surrogates $X\rightarrow X^{\prime}$ of $X$ corresponds to the notion of injective strict morphisms $[{\cal R}^{\prime}]\rightarrow [{\cal R}]$ into $[{\cal R}]$. Before leaving this theme, we note that Lemma 1.2.8 and Lemma 1.2.12 together imply that: [**Lemma 1.2.19 \[local description of morphisms\].**]{} [ *Let $R$ and $S$ be rings. Then $${\mbox{\it Mor}\,}({\mbox{\it Space}\,}[\{S\}], {\mbox{\it Space}\,}[\{R\}])\; \stackrel{\mbox{\tiny Grothendieck Ansatz}}{:=}\; {\mbox{\it Mor}\,}([\{R\}], [\{S\}])\; \simeq\; {\mbox{\it Mor}\,}(R,S)$$ canonically, where ${\mbox{\it Mor}\,}(R,S)$ is the set of ring-homomorphisms from $R$ to $S$.* ]{} D-branes from the viewpoint of Grothendieck. ============================================ The notion of a space(-time): functor of points vs. probes. ----------------------------------------------------------- [**Space from a functor of points in algebraic geometry: a space without a space.**]{} In the commutative case[^12], let ${\mbox{\it ${\cal S}$cheme}\,}/S$ be the category of schemes over a base scheme $S$ with a Grothendieck topology. A [*functor of points*]{} on ${\mbox{\it ${\cal S}$cheme}\,}/S$ is a presheaf ${\cal F}$ of sets on ${\mbox{\it ${\cal S}$cheme}\,}/S$. For example, take $S={\mbox{\it Spec}\,}{\Bbb C}$, then a scheme $Y/{\Bbb C}$ determines an ${\cal F}_Y$ on ${\mbox{\it ${\cal S}$cheme}\,}/{\Bbb C}$ with ${\cal F}_Y(Z) := {\mbox{\it Mor}\,}_{{\scriptsizeBbb C}-{\mbox{\scriptsize\it scheme}}}(Z,Y)$ for $Z\in {\mbox{\it ${\cal S}$cheme}\,}/{\Bbb C}$. In this case, $Y$ can be recovered from ${\cal F}_Y$, cf. Yoneda lemma. One can think of a functor of points ${\cal F}$ as a generalized space ${\frak Y}_{\cal F}$ and ${\cal F}(Z)$ as the set ${\mbox{\it Mor}\,}(Z,{\frak Y}_{\cal F})$ of $Z$-valued points on ${\frak Y}_{\cal F}$. The construction of the moduli space that satisfies the functorial/universal property for a moduli problem leads one in general to such a generalized space. Encoded in the functor of points ${\cal F}$ on ${\mbox{\it ${\cal S}$cheme}\,}/S$ is the data of extension property of morphisms into ${\frak Y}_{\cal F}$. In particular, ${\cal F}$ contains the information of tangent-obstruction structure of ${\frak Y}_{\cal F}$ as well as of local properties like smoothness at a point (i.e. an element in, e.g., ${\cal F}({\mbox{\it Spec}\,}{\Bbb C})=:{\mbox{\it Mor}\,}({\mbox{\it Spec}\,}{\Bbb C},{\frak Y}_{\cal F})$) of ${\frak F}_{\cal F}$. It is in this way that ${\cal F}$ describes the geometry of a “space" [*without*]{} giving the space beforehand, for example, as a point-set with a topology and other structures. Schemes, Deligne-Mumford stack (i.e. orbifolds), Artin stacks, and many moduli functors are all examples of functors of points. There are diverse ways/versions to generalize the above to the noncommutative case. The particular one that is selected from Sec. 1.2 is to consider the category ${\mbox{\it ${\cal R}$ing${\cal S}ystem$}}$ of gluing systems of rings with a Grothendieck topology defined by central covers, étale central covers, or fppf central covers. (The étale or fppf condition of a morphism can be defined purely ring-theoretically.) Note that, as we are dealing directly with rings, all the arrows in the commutative case above are reversed here. (However, if one wishes, one may write a ring system ${\cal R}$ by a formal symbol ${\mbox{\it Space}\,}{\cal R}$, meaning the associated space/geometry to ${\cal R}$, to preserve all the arrow directions.) A [*functor of points*]{} on ${\mbox{\it ${\cal R}$ing${\cal S}ystem$}}$ is then a presheaf ${\cal F}$ of sets on ${\mbox{\it ${\cal R}$ing${\cal S}ystem$}}$. Again, one can directly think of ${\cal F}$ as a generalized noncommutative space ${\frak Y}_{\cal F}$. The data of extension properties of morphisms to ${\frak Y}_{\cal F}$ is encoded in ${\cal F}$. Through this, ${\cal F}$ describes the geometry of a generalized noncommutative space ${\frak Y}_{\cal F}$ without ${\frak Y}_{\cal F}$ being given beforehand. [**Space(-time) from probes in QFT/string theory: space(-time)s emerge from QFT.**]{} There are two particular classes of quantum field theories (QFT’s) that are directly relevant to the notion of target space(-time): - [*Nonlinear sigma models*]{} are, by definition, quantum field theories whose field contents contain, among other fields, bosonic fields corresponding to maps from a domain (cf. world-volume of branes) to a target space(-time). - In string theory, D0-brane physics is described by matrix theory. As the moduli space of a single D0-brane moving in a space(-time) is the space(-time) itself, the [*moduli space of a single D0-brane*]{} can be identified as the target space(-time). These two concretely target-space(-time)-related situations can be hidden implicitly in a general quantum field theory that is seemingly irrelevant to a target space-time. Furthermore, depending on where we look at the theory in the related Wilson’s theory-space[^13], there can be more than one target space(-time)s hidden in one combinatorial class of quantum field theories. Even more, such target spaces can be taken either at the classical level – which usually involve only algebraic manipulations of the Lagrangian of the theory – or at the quantum level – which has to bring in the core techniques (and some arts as well) from quantum field theory. A quantum-corrected target space(-time) can be different from its associate classical target space(-time). The following three examples have been around for a while in the string-theory community: [**Example 2.1.1 \[gauged linear sigma model\].**]{} [*Geometric phases*]{} of a gauged linear sigma model are realized effectively by nonlinear sigma models. Birationally equivalent target spaces emerge. See \[Wi1\] and \[M-P\]. [**Example 2.1.2 \[D0-brane probe of space(-time) and singularities\].**]{} A D0-brane moving in a singular space(-time) recognizes various (partial) resolutions of the singular space(-time) as the [*moduli space of D0-branes*]{} at different phases in the Wilson’s theory-space of the $0+1$ dimensional matrix theory involved. Birationally equivalent smooth or partially resolved target spaces emerge from a single singular target space. See \[D-G-M\], \[Do-M\], and \[G-L-R\]. [**Example 2.1.3 \[conformal field theory with boundary\].**]{} D0-branes are realized in a conformal field theory with boundary as a special class of boundary states. The [*moduli space of*]{} such [*boundary states*]{} gives rise to a target space(-time). See \[M-M-S-S\] and \[S-S\]. These examples suggest that quantum field theories, as probes to a target space(-time), can be more fundamental than the space(-time) itself. The latter may even lose its absolute meaning under dualities of quantum field theories, like what happens in mirror symmetry. [**Functor of points vs. probes.**]{} A comparison of these two notions is given below: ---------------------------------------------------------------------------- -- -------------------------------------------------------------------------------------------------------------------- \[.6ex\] \[-1.6ex\] $\cdot$ ${\mbox{\it ${\cal S}$cheme}\,}/S$ a category ${\mbox{\it ${\cal B}$rane}\,}$ of branes \[1ex\] $\cdot$ a functor of point ${\cal F}$ a compatible system $\{\mbox{QFT}_{\Sigma}\}_{\Sigma\in{\mbox{\scriptsize\it ${\cal B}$rane}\,}}$ of effective QFT on ${\mbox{\it ${\cal S}$cheme}\,}/S$ on branes that have isomorphic target space(-time)s \[1ex\] $\cdot$ ${\cal F}(T)\,$, $\,T\in {\mbox{\it ${\cal S}$cheme}\,}/S$ bosonic fields on a brane that correspond to maps from the brane to a target space(-time) ---------------------------------------------------------------------------- -- -------------------------------------------------------------------------------------------------------------------- Here, a ‘brane’ means the defining domain of a quantum field theory. For example, it can be the world-volume of a string, a D-brane, or an NS-brane. Note also that a functor of points ${\cal F}$ encodes the data of a space while an effective QFT from a QFT as a probe encodes more than just the information of the target space(-time). D-branes as Azumaya-type noncommutative spaces. ----------------------------------------------- [**Question: What is a D-brane intrinsically?**]{} A [*D-brane*]{} (in full name: [*Dirichlet brane*]{} or [*Dirichlet membrane*]{}) in string theory is by definition (i.e. by the very word ‘Dirichlet’) a boundary condition for the end-points of open strings moving in a space-time. In the geometric/target-space-time aspect[^14], one may start by thinking of the world-volume (cf. Remark/Definition 2.2.4) of a D-brane as an embedded submanifold $f: Z\hookrightarrow M$ in an open-string target space-time $M$ such that: - [**\[defining property of D-brane: D = Dirichlet\]**]{}\ [*The boundary of open-string world-sheets are mapped to $f(Z)$ in $M$.*]{} Via this defining property, open strings induce then additional structures on $Z$, including a gauge field (from the vibrations of open-strings with end-points on $f(Z)$) and a Chan-Paton bundle (from the Chan-Paton index on the end-points of such an open string) on $Z$. Basic properties of D-branes under such a setting are given in \[Pol3\] and \[Pol4\]. To bring the relevant part of the work of Polchinski into the discussions and to enable a direct comparison/referral, let us introduce notations the-same-as/as-close-as-possible-to those in \[Pol4: vol. I, Sec. 8.7\]: let $\xi:=(\xi^a)_a$ be local coordinates on $Z$ and $X:=(X^a;X^{\mu})_{a,\mu}$ be local coordinates on $M$ such that the embedding $f:Z\hookrightarrow M$ is locally expressed as $$X\; =\; X(\xi)\; =\; (X^a(\xi); X^{\mu}(\xi))_{a,\mu}\; =\; (\xi^a, X^{\mu}(\xi))_{a,\mu}\,;$$ i.e., $X^a$’s (resp. $X^{\mu}$’s) are local coordinates along (resp. transverse to) $f(Z)$ in $M$. This choice of local coordinates removes redundant degrees of freedom of the map $f$, and $X^{\mu}=X^{\mu}(\xi)$ can be regarded as (scalar) fields on $Z$ that collectively describes the postions/shapes/fluctuations of $Z$ in $M$ locally. Here, both $\xi^a$’s, $X^a$’s, and $X^{\mu}$’s are ${\Bbb R}$-valued. The gauge field on $Z$ is locally given by the connection $1$-form $A=\sum_a A_a(\xi)d\xi^a$ of a $U(1)$-bundle on $Z$. When $n$-many such D-branes $Z$ are coincident, from the associated massless spectrum of (oriented) open strings with both end-points on $f(Z)$ one can draw the conclusion that - The gauge field $A=\sum_a A_a(\xi)d\xi^a$ on $Z$ is enhanced to $u(n)$-valued. - Each scalar field $X^{\mu}(\xi)$ on $Z$ is also enhanced to matrix-valued, cf. footnote 17. Property (1) says that there is now a $U(n)$-bundle on $Z$. But - [**Q.**]{} [*What is the meaning of Property (2)?*]{} For this, Polchinski remarks that: - \[[*quote from*]{} \[Pol4: vol. I, Sec. 8.7, p.272\]\] “[ *For the collective coordinate $X^{\mu}$, however, the meaning is mysterious: the collective coordinates for the embedding of $n$ D-branes in space-time are now enlarged to $n\times n$ matrices. This ‘noncommutative geometry’ has proven to play a key role in the dynamics of D-branes, and there are conjectures that it is an important hint about the nature of space-time.*]{}" Particularly from the mathematical/geometric perspective, Property (2) of D-branes when they are coincident, the above question, and Polchinski’s remark are more appropriately incorporated into the following guiding question: - [**Q. \[D-brane\]**]{}$\;$ [*What is a D-brane intrinsically?*]{} In other words, what is the [*intrinsic*]{} definition of D-branes so that [*by itself*]{} it can produce the properties of D-branes (e.g. Property (1) and Property (2) above) that are consistent with, governed by, or originally produced by open strings as well?[^15] [**The noncommutativity ansatz: from Polchinski to Grothendieck.**]{} To understand Property (2) of D-branes, one has two aspects that are dual to each other: - \[[*coordinate tuple as point*]{}\] A tuple $(\xi^a)_a$ (resp. $(X^a; X^{\mu})_{a,\mu}$) represents a point on the world-volume $Z$ of the D-brane (resp. on the target space-time $M$). - \[[*local coordinates as generating set of local functions*]{}\] Each local coordinate $\xi^a$ of $Z$ (resp. $X^a$, $X^{\mu}$ of $M$) is a local function on $Z$ (resp. on $M$) and the local coordinates $\xi^a$’s (resp. $X^a$’s and $X^{\mu}$’s) together form a generating set of local functions on the world-volume $Z$ of the D-brane (resp. on the target space-time $M$). While Aspect (A1) leads one to the anticipation of a noncommutative space from a noncommutatization of the target space-time $M$ when probed by coincident D-branes, Aspect (A2) of Grothendieck leads one to a different/dual[^16] conclusion: a noncommutative space from a noncommutatization of the world-volume $Z$ of coincident D-branes, as follows. Denote by ${\Bbb R}\langle \xi^a\rangle_{a}$ (resp. ${\Bbb R}\langle X^a; X^{\mu}\rangle_{a, \mu}$) the local function ring on the associated local coordinate chart on $Z$ (resp. on $M$). Then the embedding $f:Z\rightarrow M$, locally expressed as $X=X(\xi)=(X^a(\xi); X^{\mu}(\xi))_{a,\mu}=(\xi^a; X^{\mu}(\xi))$, is locally contravariantly equivalent to a ring-homomorphism $$f^{\sharp}\;:\; {\Bbb R}\langle X^a; X^{\mu}\rangle_{a, \mu}\; \longrightarrow\; {\Bbb R}\langle \xi^a\rangle_{a}\,, \hspace{1em}\mbox{generated by}\hspace{1em} X^a\;\longmapsto\; \xi^a\,,\; X^{\mu}\;\longmapsto\;X^{\mu}(\xi)\,.$$ When $n$-many such D-branes are coincident, $X^{\mu}(\xi)$’s become $M_n({\Bbb C})$-valued.[^17] Thus, $f^{\sharp}$ is promoted to a new local ring-homomorphism: $$\hat{f}^{\sharp}\;:\; {\Bbb R}\langle X^a; X^{\mu}\rangle_{a, \mu}\; \longrightarrow\; M_n({\Bbb C}\langle \xi^a\rangle_{a})\,, \hspace{1em}\mbox{generated by}\hspace{1em} X^a\;\longmapsto\; \xi^a\cdot{\mathbf 1}\,,\; X^{\mu}\;\longmapsto\;X^{\mu}(\xi)\,.$$ Under Grothendieck’s contravariant local equivalence of function rings and spaces, $\hat{f}^{\sharp}$ is equivalent to saying that we have now a map $\hat{f}: Z_{{\mbox{\scriptsize\rm noncommutative}}}\rightarrow M$. Thus, the result of Polchinski re-read from the viewpoint of Grothendieck implies the following ansatz: [**Polchinski-Grothendieck Ansatz \[D-brane: noncommutativity\].**]{} [ *The world-volume of a D-brane carries a noncommutative structure locally associated to a function ring of the form $M_n(R)$ for some $n\in {\Bbb Z}_{\ge 1}$ and ring $R$.[^18]* ]{} This ansatz is further enforced if one recalls that scalar fields on the world-volume of a brane are supposed to come from elements in the function ring of that world-volume and the comparison of a functor of points vs. probes in Sec. 2.1.[^19] [*Remark 2.2.1 $[$D-brane and noncommutative geometry$]$.*]{} [ The observation that D-brane should be related to noncommutative geometry was made soon after the second-revolution year 1995 of string theory; see \[Dou4\] and \[Dou5\] for a survey and, e.g., \[Ho-W\] for an earlier study and \[Laz\] for a more recent study in the differential/symplectic geometry category. Noncommutative structures on a D-brane itself and on a space-time are two related but separate issues, e.g. \[Dou2\], \[C-H1\], and \[C-H2\]. It is worth pointing out that, [*from the viewpoint of Grothendieck, it is the noncommutative structure on the world-volume of a D-brane that comes first*]{}. It is exactly because of such a structure on D-branes that a space-time may reveal its noncommutative nature when probed by a D-brane. Said algebro-geometrically in terms of function rings, since a ring-homomorphism from a noncommutative ring $R$ to a commutative ring $S$ must factor through a ring-homomorphism $R/[R,R]\rightarrow S$ from the commutatization $R/[R,R]$ of $R$, D-branes without a noncommutative structure thereon cannot probe/sense any noncommutativity, if any, of a space-time at all. ]{} [*Remark 2.2.2 $[$B-field and noncommutativity on D-brane$]$.*]{} It is known that when the target space(-time) $M$ has the B-field $B$ turned on, the gauge theory on a D-brane world-volume $Z$ can be expressed as a noncommutative gauge theory; (see \[Ch-K\] and \[S-W2\] for details and more references on this subject.) From the underlying formulation, this implies in particular that, in this case, the commutative product of a local function ring $R$ on $Z$ is deformed to a noncommutative $\ast$-product depending on $B$. When $n$-many D-branes $Z$ coincide, these string-induced property on D-branes compared with our discussion above says that: - If $B=0$, then a local function ring on the world-volume of the coincident D-branes is of the form $M_n(R)$, where $R$ is commutative. - If $B\ne 0$, then a local function ring on the world-volume of the coincident D-branes can become $M_n(R_B)$, where $R_B$ is a noncommutatization of $R$ depending on/induced by $B$. In this work, we ignore the effect of B-field. [*The Polchinski-Grothendieck Ansatz for D-branes applies to both nonsupersymmetric and supersymmetric D-branes, and to both D-branes of A-type and D-branes of B-type*]{} (cf. \[B-B-St\], \[H-I-V\], and \[O-O-Y\]) [*in the latter case.*]{} Due to the different languages used in differential geometry and in algebraic geometry for noncommutative geometry (though the philosophy to equate a space and a function ring in each category is common), we will focus entirely on supersymmetric D-branes of B-type, for which algebro-geometric language is appropriate. The ansatz leads thus to a prototype[^20] intrinsic definition of D-branes of B-type as follows: [**Definition 2.2.3 \[D-brane of B-type and Chan-Paton sheaf\].**]{} [ (1) A [*D-brane of B-type*]{} is an Azumaya-type noncommutative space $(X,{\cal O}_X, {\cal O}_X^{{\mbox{\scriptsize\it nc}}})$ over ${\Bbb C}$, together with a fundamental ${\cal O}_X^{{\mbox{\scriptsize\it nc}}}$-module ${\cal E}_X$. ${\cal E}_X$ is called the [*Chan-Paton sheaf*]{} on the D-brane $X$. We say that ${\cal E}_X$ has [*rank*]{} $r$ if it has rank $r$ as an ${\cal O}_X$-module. Note that ${\cal E}_X|_{\eta} \simeq \kappa_{\eta}^{n_1}\oplus\,\cdots\,\oplus\kappa_{\eta}^{n_s}$ at a generic point $\eta$ of $(X,{\cal O}_X)$ with residue field $\kappa_{\eta}$ if ${\cal O}_X^{{\mbox{\scriptsize\it nc}}}|_{\eta}/J({\cal O}_X^{{\mbox{\scriptsize\it nc}}}|_{\eta}) \simeq M_{n_1}(\kappa_{\eta}) \times\,\cdots\,\times M_{n_s}(\kappa_{\eta})$. Here ${\cal O}_{X}^{{\mbox{\scriptsize\it nc}}}|_{\eta}$ is the fiber of ${\cal O}_X^{{\mbox{\scriptsize\it nc}}}$ at $\eta$ and $J(\,\cdot\,)$ is the Jacobson radical of $(\,\cdot\,)$. (2) A [*D-brane*]{} (of B-type) [*in a target space $Y$*]{} is a morphism $\Phi:X\rightarrow Y$. Here, $Y$ can be a (commutative) scheme, an Azumaya-type noncommutative space, a noncommutative space represented by a ring-system, or whatever noncommutative space to which the notion of morphisms from $X$ can be defined. The image Azumaya-type noncommutative space $\Phi(X)$ is called the [*image D-brane*]{} of $X$ in $Y$. (3) The [*Chan-Paton sheaf of a D-brane $\Phi:X\rightarrow Y$ on $Y$*]{} is the push-forward $\Phi_{\ast}{\cal E}_X$ of ${\cal E}_X$, a coherent sheaf supported on $\Phi(X)$ in $Y$. ]{} [**Remark/Definition 2.2.4 \[D-brane vs. D-brane world-volume\].**]{} [ The world-volume of a D-brane is what a D-brane sweeps out in a space-time and, hence, has the extra time-dimension than the D-brane has. It has a Lorentzian structure by definition. The world-volume after Wick rotation is called a Euclidean D-brane world-volume, which has now a Riemannian structure. We will define a [*Eulcidean D-brane world-volume of B-type*]{} the same as in Definition 2.2.3 with ‘D-brane’ replaced by ‘Euclidean D-brane world-volume’. Similarly, for a [*Euclidean D-brane world-volume*]{} (of B-type) [*in a target space*]{} $Y$ and the [*Chan-Paton sheaf*]{} and [*its push-forward*]{} on $Y$. In general, we keep the word ‘Euclidean’ implicit and call it simply [*D-brane world-volume*]{} (of B-type) (resp. [*D-brane world-volume*]{} (of B-type) [*in $Y$*]{}). Readers should compare these simplified terminologies with the term ‘world-sheet’ in the commonly used statement by physicists: “The world-sheet of a string is a Riemann surface.", which takes the same interpretation implicitly. ]{} How these two definitions fit in string theory and, by themselves, reproduce three key open-string-induced properties of D-branes can be summarized/highlighted as follows: - - The Chan-Paton sheaf ${\cal E}_X$ should be identified with a singular coherent analytic sheaf on $X$ with a (singular) connection $A$ via a Kobayashi-Hitchin correspondence. An end-point of an open string in $Y$ can then be coupled to the D-brane $X$ via a morphism $\Phi:X\rightarrow Y$ and the connection $A$, regarded as on ${\cal E}_X$. <!-- --> - - (Subject to that $X$ here has to be interpreted as a Euclidean D-brane world-volume.) Identify $(X, {\cal O}_X)$ canonically with an analytic space $X_{{\mbox{\scriptsize\it an}}}$ (with the structure sheaf ${\cal O}_{X_{{\mbox{\scriptsize\it an}}}}$ of analytic functions). A Ramond-Ramond field (i.e. a differential form) on $Y$ can be pulled back and integrate over $X_{{\mbox{\scriptsize\it an}}}$ via $\Phi:X\rightarrow Y$. <!-- --> - - The Azumaya-type noncommutative structure ${\cal O}_X^{{\mbox{\scriptsize\it nc}}}$ on $X$ makes the deformations of $\Phi:X\rightarrow Y$ locally matrix-valued, as in \[Pol4\]. It realizes the [*Higgsing/un-Higgsing*]{} behavior of the gauge theory on D-branes on $Y$ via (a continuous family of) deformations of a morphism $\Phi:X\rightarrow Y$, as explained below: - - Associated to the (associative, unital) ${\cal O}_X$-algebra ${\cal O}_X^{{\mbox{\scriptsize\it nc}}}$ is the (non-associative, non-unital) Lie ${\cal O}_X$-algebra ${\cal O}_X^{{\mbox{\scriptsize\it nc}},{\mbox{\scriptsize\it Lie}}}:= ({\cal O}_X^{{\mbox{\scriptsize\it nc}}}, [\,\cdot\,,\,\cdot\,])$ with the commutator product $[s_1, s_2]:= s_1\cdot s_2 - s_2\cdot s_1$ for local sections of ${\cal O}_X^{{\mbox{\scriptsize\it nc}}}$. A gauge theory on the D-brane $X$ corresponds to a choice of a gauge sheaf ${\cal G}_X$ embedded in ${\cal O}_X^{{\mbox{\scriptsize\it nc}},{\mbox{\scriptsize\it Lie}}}$. Here, a [*gauge sheaf*]{} is a sheaf of ${\cal O}_X$-Lie-algebras that generalizes the notion of the Lie-algebra bundle associated to the adjoint representation of the gauge group of a principal bundle.[^21] This renders ${\cal E}_X$ a ${\cal G}_X$-module. Thus, it is enough to consider ${\cal O}_X^{{\mbox{\scriptsize\it nc}}}$ and ${\cal E}_X$ as an ${\cal O}_X^{{\mbox{\scriptsize\it nc}}}$-module. - A D-brane $\Phi: X\rightarrow Y$ on $Y$ determines a sheaf ${\cal O}_X \subset {\cal A}^{{\mbox{\scriptsize\it nc}}} \subset {\cal O}_X^{{\mbox{\scriptsize\it nc}}}$ of subalgebras of ${\cal O}_X^{{\mbox{\scriptsize\it nc}}}$, namely the image of the ring-system homomorphism ${\cal R}_Y\rightarrow {\cal R}_X$ that defines $\Phi$. The associated [*gauge symmetry on the D-brane on*]{} $Y$ is given by the sheaf ${\mbox{\it ${\cal C}$entralizer}\,}_{{\cal O}_X^{{\mbox{\tiny\it nc}}}}({\cal A}^{{\mbox{\scriptsize\it nc}}})$ of [*centralizer subalgebras*]{} of ${\cal A}^{{\mbox{\scriptsize\it nc}}}$ in ${\cal O}_X^{{\mbox{\scriptsize\it nc}}}$. A continuous family $\Phi_t:X_t\rightarrow Y$ of deformations of the morphism $\Phi:X\rightarrow Y$ gives rise to a (not-necessarily flat) family ${\mbox{\it ${\cal C}$entralizer}\,}_{{\cal O}_{X_t}^{{\mbox{\tiny\it nc}}}}({\cal A}^{{\mbox{\scriptsize\it nc}}}_t)$ of sheaves of algebras. This realizes the Higgsing/un-Higgsing behavior of the gauge symmetry on D-branes on $Y$ under deformations of D-branes on $Y$.[^22] These highlights explain why we take Definition 2.2.3 as a prototype intrinsic definition for D-branes (or D-brane world-volumes) in the region of the theory-space where “branes are still branes". Details of the case of D0-branes are given in Sec. 3 and Sec. 4. The general higher-dimensional brane case can be thought of as sheafifying/smearing the discussion for D0-branes along a higher-dimensional cycle, chain, or more generally current in the sense of \[G-H\] or \[Fe\]; cf. \[L-Y4\]. [*Remark 2.2.5 $[$other intrinsic definitions$]$.*]{} [ There have been other working mathematical intrinsic definitions for D-branes by other authors aiming also to understanding D-branes (in the region of Wilson’s theory-space where “branes are still branes"). For example, there were the interpretation of D-branes as [*stable torsion sheaves*]{}, given, e.g., in \[H-S-T\] in the algebro-geometric category from the viewpoint of BPS states and Gopakumar-Vafa invariants, and the notion of ‘[*flat D-branes*]{}’, given in \[B-M-R-S\] in the smooth differential-geometric category from the viewpoint of K-theory. Each of these definitions singles out important key properties/features of D-branes in stringy literatures. In contrast, our prototype intrinsic definition of D-branes follows from the Grothendieck’s viewpoint of Polchinski’s work, phrased as the Polchinski-Grothendieck Ansatz for D-branes. This starting point is lower than these other existing intrinsic definitions and can reach up/be linked, for example, to \[H-S-T\] by considering D-brane images with the push-forward Chan-Paton sheaf on the target space and to \[B-M-R-S\] by considering formal linear combinations of D-branes with Chan-Paton sheaves and their equivalence classes in the K-group of the D-brane. ]{} ${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\LargeBbb C}), Y)$ as a coarse moduli space. ========================================================================================= We realize in this section the space $${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\Bbb C}), Y)\; :=\; {\mbox{\it Mor}\,}([{\cal R}], [\{M_n({\Bbb C})\}])\; =\; {\mbox{\it Mor}\,}({\cal R}, [\{M_n({\Bbb C})\}])$$ of morphisms from ${\mbox{\it Space}\,}M_n({\Bbb C})$ to $Y = {\mbox{\it Space}\,}[{\cal R}] = {\mbox{\it Space}\,}( [\{R_{\gamma}\}_{\gamma\in C} {\begin{array}{c} \rightarrow \\[-2ex] \rightarrow \end{array}}\{R_{\gamma_1\gamma_2}\}_{\gamma_1,\gamma_2\in C}] )$ as a constructible set in a topological space from an adhesion of affine varieties/${\Bbb C}$. Central localizations of Artinian rings and their modules. ---------------------------------------------------------- Recall first the [*Structure Theorem of Artinian Rings*]{}: [**Theorem 3.1.1 \[Artinian ring\].**]{} (\[A-M\], \[A-N-T\], and \[Jat\].) ** - Let $R$ be an Artinian ring. The center $Z(R)$ of $R$ is a commutative Artinian ring and hence has finitely-many maximal ideals. Let $t$ be the number of maximal ideals in $Z(R)$. - There exist a unique collection $\{e_1,\,\cdots\,,\,e_t\}$ of orthogonal primitive idempotents in $Z(R)$ such that $1=e_1+\,\cdots\,+e_t$ and that $R$ is the direct sum of the two-sided ideals $R=Re_1+\,\cdots\,+Re_t$. Up to permutations, the collection $\{Re_1\,,\, \cdots\,,\,Re_t\}$ is unique with respect to the following property: - $R=I_1+\,\cdots\,+I_{t^{\prime}}$, where $I_i$ are two-sided ideals of $R$, $I_i\cdot I_j=0$ for $i\ne j$, and each $I_i$ is indecomposable in the sense that $I_i$ cannot be decomposed as a direct sum $I_{i^{\prime}}+I_{i^{\prime\prime}}$ with $I_{i^{\prime}}$ and $I_{i^{\prime\prime}}$ non-zero two-sided ideals. Under such decomposition of $R$, each $R_i:=Re_i$ is itself an Artinian ring with identity $e_i$ and the decomposition $R=Re_1+\,\cdots\,+Re_t$ can be written as the product of rings $R=R_1\times\,\cdots\,\times R_t$. This decomposition restricts to a decomposition $Z(R)=Z(R_1)\times\,\,\cdots\,\times Z(R_t)$ with each $Z(R_i)$, $i=1,\,\ldots\,, t$, an Artinian local ring. - Let $J(R)$ be the Jacobson radical of $R$. Then there is an orthogonal idempotent decomposition $$1\;=\; \sum_{j_1=1}^{l_1}e_{1j_1}\, +\,\cdots\,+\sum_{j_s=1}^{l_t}e_{tj_t}$$ in $R$ that refines the decomposition $1=e_1+\,\cdots\,+e_t$ in $Z(R)$, with $e_i=\sum_{j_i=1}^{l_i}e_{ij_i}$, such that the image $\bar{e}_{ij_i}$ of $e_{ij_i}$ in $R/J(R)$ lies in $Z(R/J(R))$ and that $$\bar{1}\;=\; \sum_{j_1=1}^{l_1}\bar{e}_{1j_1}\, +\,\cdots\, +\, \sum_{j_t=1}^{l_t}\bar{e}_{tj_t}$$ is an orthogonal primitive idempotent decomposition in $Z(R/J(R))$. Let $${\frak m}_{ij_i}\; :=\; R\, (1-e_{ij_i})\,R\,, \hspace{2em}\mbox{for $1\le i\le t$ and $1\le j_i\le l_i$}\,,$$ and ${\mbox{\it Spec}\,}R$ be the set of all prime ideals in $R$. Then all prime ideals in $R$ are maximal ideals and $${\mbox{\it Spec}\,}R\; =\; \{\, {\frak m}_{ij_i}\;:\; 1\le i\le t\,,\, 1\le j_i\le l_i\, \}\,.$$ - Consider the directed graph $\Gamma_R$ with the set of vertices ${\mbox{\it Spec}\,}R$ and a directed edge ${\frak m}_{i_1j_{i_1}}\rightarrow {\frak m}_{i_2j_{i_2}}$ for each pair $(e_{i_1j_{i_1}}, e_{i_2j_{i_2}})$ with $e_{i_1j_{i_1}} J(R) e_{i_2j_{i_2}}\ne 0$. Then $\Gamma_R$ has exactly $t$-many connected components $\Gamma_R^{(i)}$, $i=1,\,\cdots\,,t$, with the set of vertices of $\Gamma_R^{(i)}$ being $\{{\frak m}_{ij_i}\,:\, 1\le j_i\le l_i\}\,$. The two graphs $\Gamma_R^{(i)}$ and $\Gamma_{R_i}$ are canonically isomorphic. In particular, each $\Gamma_{R_i}$ is connected. - By definition, $J(R)=\cap_{i=1}^t\cap_{j=1}^{l_i}\, {\frak m}_{ij_i}$. The quotient ${\frak m}_{ij_i}/J(R)$, with the induced addition and multiplication from those of $R$, is a simple ring and hence is isomorphic to a matrix ring $M_{n_{ij_i}}(k_{ij_i})$ for some skew-field $k_{ij_i}$. The decomposition $R=Re_1+\,\cdots\,+Re_t$ restricts to a decomposition $J(R)=J(R)_1+\,\cdots\,+J(R)_t$, which can be written canonically as $J(R)=J(R_1)\times\,\cdots\,\times J(R_t)$. With respect to this, one has isomorphisms $$R/J(R)\;\simeq\; \prod_{i=1}^t\,R_i/J(R_i)\; \simeq\; \prod_{i=1}^t\prod_{j_i=1}^{l_i}\, M_{n_{ij_i}}(k_{ij_i})\,.$$ [*Remark 3.1.2 $[$quiver$]$.*]{} [ The graph $\Gamma_R$ associated to an Artinian ring $R$ (as an $R$-module) in Theorem is an example of (various) quivers associated to an $R$-module. See Sec. 4.1 and footnote 36 for a theme in which we bring this in again. ]{} The theorem gives a visualization of an Artinian algebra $R/{\Bbb C}$ (e.g. $M_n({\Bbb C})$ and its subalgebras) as a noncommutative space of the form: > “a finite collection of commutative points (i.e. ${\mbox{\it Spec}\,}Z(R)$), with each point dominated/shadowed by a noncommutative cloud (i.e. $Z(R_i)\subset R_i$, where $R_i:=Re_i$); associated to each noncommutative cloud (i.e. $R_i$) over a commutative point (i.e. ${\mbox{\it Spec}\,}Z(R_i)$) are a refined collection of commutative points (i.e. ${\mbox{\it Spec}\,}(\sum_{j=1}^{l_i}{\Bbb C}\cdot e_{ij})$) split off from and stacked over that point (more precisely, ${\mbox{\it Spec}\,}Z(R_i)_{{\mbox{\scriptsize\rm red}\,}}$) and are dominated/shadowed by that cloud (i.e.  $\sum_{j=1}^{l_i}e_{ij}=e_i$ and ${\Bbb C}\cdot e_i > \subset \sum_{j=1}^{l_i}{\Bbb C}\cdot e_{ij} > \subset R_i$) and bound by directed bonds (i.e. $e_{ij_1} J(R_i) e_{ij_2}$ with the direction from $e_{ij_1}$ to $e_{ij_2}$) created through that cloud (i.e. $R_i$)". The following are immediate consequences of the theorem. [**Lemma 3.1.3 \[central non-zero-divisor invertible\].**]{} [ *Let $R$ be an Artinian ring and $r\in Z(R)$ be a non-zero-divisor in $R$. Then $r$ is invertible in $R$.* ]{} [**Lemma 3.1.4 \[direct-sum decomposition of module\].**]{} (Cf. [*Peirce decomposition*]{}.) [ *Let $R$ be an Artinian ring and $R=Re_1+\,\cdots\,+Re_t =: R_1+\,\cdots\,+R_t$ be a decomposition of $R$ as in Theorem 3.1.1 (2). Let $M$ be an $R$-module. Then, $M=e_1M+\,\cdots\,+e_tM =: M_1+\,\cdots\,+M_t$ is a direct-sum decomposition of $M$ such that $R_iM_i=M_i$ and $R_jM_i=0$ for $j\ne i$. In particular, $M_i$ is a $R_i$-module for $i=1,\,\ldots\,, t$.* ]{} [**Corollary 3.1.5 \[localization = quotient\].**]{} [ *[(With notations from above.)]{} $R_i$ is canonically isomorphic to both the quotient $R/(e_j:j\ne i)=R/(\sum_{j\ne i}e_j)$ of $R$ and the localization $R[S_i^{-1}]$ of $R$, where $S_i$ is the multiplicatively closed subset $\{1, e_i\}$. Similarly, $M_i$ is canonically isomorphic to both the quotient $M/(\sum_{j\ne i} M_j)$ of $M$ and the localization $M[S_i^{-1}]$ of $M$.* ]{} [**Corollary 3.1.6 \[localization: standard form\].**]{} [ *$(1)$ Any nonzero central localization $R\rightarrow R^{\prime}$ of an Artinian ring $R$ is realized by inverting a finite multiplicatively closed subset $S \subset Z(R)$ that consists only of idempotents. I.e. $R^{\prime}=R[S^{-1}]$ and $R^{\prime}\rightarrow R$ is $R\rightarrow R[S^{-1}]$ for an afore-mentioned $S$. $(2)$ Any central localization $f:R\rightarrow R^{\prime}$ of an Artinian ring $R$ is a quotient of $R$ that admits a ring-set-homomorphism[^23] $g:R^{\prime}\rightarrow R$ such that $f\circ g={\mbox{\it Id}}_{R^{\prime}}$. $(3)$ Fix a direct-sum decomposition $R=R_1+\,\cdots\,+R_t$ from Theorem 3.1.1 (2). Then the localization $f: R\rightarrow R^{\prime}$ in $(2)$ is simply the projection of $R$ onto the sum $R_{i_1}+\,\cdots\,+R_{i_{t^{\prime}}}$ of some direct summands and $g:R^{\prime}\rightarrow R$ in $(2)$ can be taken to be the inclusion of $R_{i_1}+\,\cdots\,+R_{i_{t^{\prime}}}$ into $R$.* ]{} [*Proof.*]{} Let $R=R_1+\,\cdots\,+R_t$ be a direct-sum decomposition of $R$ from Theorem 3.1.1 (2). Then $S=S_1+\,\cdots\,+ S_t$, where $S_i:=e_iS \subset Z(R_i)$, is a direct-sum decomposition of $S$ and $R[S^{-1}]=R_1[S_1^{-1}]\times\,\cdots\,\times R_t[S_t^{-1}]$ canonically. This reduces the proof to the case that $t=1$ in the decomposition of $R$ (i.e. the case $Z(R)$ is an Artinian local ring). When $Z(R)$ is an Artinian local ring, $R[S^{-1}]=0$ if $S$ contains an element in the maximal ideal of $Z(R)$, as such an element is nilpotent. Otherwise, all elements of $S$ are not in the maximal ideal of $Z(R)$; then they are all invertible and, hence, $R[S^{-1}]=R$. In the former (resp. latter) case, we may replace $S$ by $\{1,0\}$ (resp. $\{1\}$). The corollary now follows. $\Box$ [**Lemma 3.1.7 \[localization in terms of generators of $S$\].**]{} [ *Let $R$ be an Artinian ring and $S$ be a multiplicatively closed subset in $Z(R)$, generated by[^24] $\{s_1,\,\cdots\,, s_l\}$. Let $n_0$ be a positive integer such that every nilpotent element $r$ of $R$ satisfies $r^{n_0}=0$. Then $R[S^{-1}]=R/\sum_{i=1}^l(s_i^{n_0})^{\perp}$, where $(\,\bullet\,)^{\perp}:=\{r\in R: (\,\bullet\,)\cdot r =0\}$.* ]{} [*Proof.*]{} This follows immediately from Corollary 3.1.6. $\Box$ ${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\largeBbb C}),Y)$ as a coarse moduli space. ---------------------------------------------------------------------------------------- [**Definition 3.2.1 \[ring-subset\].**]{} [ Let $R=(R, 0,1,+,\,\cdot\,)$ be a ring, with the identity $1$. An additive subgroup $R^{\prime}\subset R$ is called a [*ring-subset*]{} of $R$ if, in addition, (1) $R^{\prime}$ is closed under the multiplication $\cdot$ in $R$, and (2) there is an element $e\in R^{\prime}$ such that $(R^{\prime},0,e,+,\,\cdot\,)$ is a ring with the identity $e$. ]{} [**Definition 3.2.2 \[ring-set-homomorphism\].**]{} [ Let $R$ and $S$ be rings with the identity $1_R$ and $1_S$ respectively. A map $\varphi: R\rightarrow S$ is called a [*ring-set-homomorphism*]{} if $\varphi$ satisfies all the requirement for a ring-homomorphism [*except*]{} that it is not required that $\varphi(1_R)=1_S$. ]{} Note that $e$ in Definition 3.2.1 is unique and satisfies $e^2=e$. [**Example 3.2.3 \[ring-subset\].**]{} The image $\varphi(R)$ in Definition 3.2.2 is a ring-subset of $S$ with the identity $\varphi(1_R)$. In particular, $\{0\}\subset R$ is the minimal ring-subset of $R$. We will retain these terminologies for algebras and algebra-homomorphisms over a fixed ground field as well. [**Surrogates of the Azumaya-type noncommutative point ${\mbox{\it Space}\,}M_n({\Bbb C})$.**]{} $M_n({\Bbb C})$ is a simple ring in the sense that it is semi-simple as a left $M_n({\Bbb C})$-module and has the only two-sided ideals the zero-ideal $({\mathbf 0})$ and itself $M_n({\Bbb C})$. In particular, the only prime ideal of $M_n({\Bbb C})$ is $({\mathbf 0})$ and the center $Z(M_n({\Bbb C}))$ of $M_n({\Bbb C})$ is given by ${\Bbb C}\cdot{\mathbf 1}$. There are only two Gabriel filters on $M_n({\Bbb C})$: ${\frak F}_0$ that is generated by $({\mathbf 0})$ and is given by the set of all left ideals of $M_n({\Bbb C})$ and ${\frak F}_1:= \{M_n({\Bbb C})\}$. The localization of $M_n({\Bbb C})$ with respect to ${\frak F}_0$ (resp. ${\frak F}_1$) is the zero-ring $0$ (resp. $M_n({\Bbb C})$ itself). The former (resp. latter) covers the notion of the localization of $M_n({\Bbb C})$ with respect to a non-invertible (resp. invertible) element. Thus, directly on $M_n({\Bbb C})$, we see only a seemingly barren geometry. Things change when we bring in the notion of surrogates introduced in Sec. 1.1. A surrogate of the Azumaya-type noncommutative point ${\mbox{\it Space}\,}M_n({\Bbb C})=({\mbox{\it Spec}\,}{\Bbb C},{\Bbb C}, M_n({\Bbb C}))$ is given ring-theoretically by a subalgebra pair ${\Bbb C}\subset C \subset R \subset M_n({\Bbb C})$ with $C\subset Z(R)$. It follows from Corollary 3.1.6 that a finite central cover of the sub-${\Bbb C}$-algebra $R$ of $M_n({\Bbb C})$ can be described by a finite collection $\{(R_{\alpha}, e_{\alpha})\}_{\alpha\in A}$ of ring-subsets of $R$ (and hence of $M_n({\Bbb C})$) that satisfies the following conditions: - $R=\sum_{\alpha\in A} R_{\alpha}$. - $e_{\alpha_1}$ commutes with elements of $R_{\alpha_2}$ for all $\alpha_1$, $\alpha_2\in A$. - $e_{\alpha_1}R_{\alpha_2}=e_{\alpha_2}R_{\alpha_1}$ for all $\alpha_1$, $\alpha_2\in A$. - $e_{\alpha_1}e_{\alpha_2}\in R_{\alpha_1}$ for all $\alpha_1$, $\alpha_2\in A$. - Fix a well-ordering of the index set $A$; then $$\begin{aligned} \lefteqn{ 1\;=\; \sum_{\alpha} e_{\alpha}\; -\, \sum_{\alpha_1<\alpha_2} e_{\alpha_1}e_{\alpha_2}\; +\, \sum_{\alpha_1<\alpha_2<\alpha_3} e_{\alpha_1}e_{\alpha_2}e_{\alpha_3} }\\[.6ex] && \hspace{6em} \pm\; \cdots\; +\, (-1)^{|A|+1}\, \sum_{\alpha_1<\,\cdots\,<\alpha_{|A|}} e_{\alpha_1}\,\cdots\,e_{\alpha_{|A|}}\,. \end{aligned}$$ Conditions (1), (2), and (3) imply that $e_{\alpha_2}R_{\alpha_1} =R_{\alpha_1}\cap R_{\alpha_2} = e_{\alpha_1}R_{\alpha_2}$, which is itself a ring with the identity $e_{\alpha_1}e_{\alpha_2}$. In particular, $(e_{\alpha_2}R_{\alpha_1}, e_{\alpha_2}e_{\alpha_1}) =((e_{\alpha_1}e_{\alpha_2})R_{\alpha_1}, e_{\alpha_1}e_{\alpha_2})$ is a ring-subset of both rings $(R_{\alpha_1},e_{\alpha_1})$ and $(R_{\alpha_2},e_{\alpha_2})$. Condition (4) simplifies to $1=\sum_{\alpha}e_{\alpha}$ when $R=\sum_{\alpha}R_{\alpha}$ is a direct sum. Conversely, one has the following proposition: [**Proposition 3.2.4 \[subring in terms of a collection of ring-subsets\].**]{} [ *$(1)$ Let $\{(R_{\alpha},e_{\alpha})_{\alpha\in A}\}$ be a finite collection of ring-subsets of $M_n({\Bbb C})$ that satisfies Conditions $(1)$, $(2)$, $(3)$, and $(4)$ above. Then $R:=\sum_{\alpha\in A} R_{\alpha}$ contains the identity ${\mathbf 1}$ of $M_n({\Bbb C})$ and is a sub-${\Bbb C}$-algebra of $M_n({\Bbb C})$. $(2)$ There are tautological ring-homomorphisms $R\rightarrow R_{\alpha}$, $\alpha\in A$, that render the collection $\{R\rightarrow R_{\alpha}\}_{\alpha\in A}$ a finite central cover of $R$.* ]{} [*Proof.*]{} Observe that for a ring-subset $(P,e_P)$ and an idempotent $e^{\prime}$ of a ring $Q$ that commutes with the elements in $P$, $(e^{\prime}P, e^{\prime}e_P)$ is another ring-subset of $Q$. In particular, elements in $e^{\prime}P$ are closed under the multiplication in $Q$. Moreover, if, in addition, $e^{\prime}e_p\in P$, then $P=(e_P-e^{\prime}e_P)P+(e^{\prime}e_P)P$ is an orthogonal direct-sum decomposition for $P$ (when neither summand is zero). Using these observations, one can show that Properties (1), (2), and (3) imply that $R_{\alpha_1}R_{\alpha_2}\subset R_{\alpha_1}+R_{\alpha_2}$ for all $\alpha_1$, $\alpha_2\in A$. This proves that $R:=\sum_{\alpha}R_{\alpha}$ is closed under the multiplication in $M_n({\Bbb C})$ as $R\cdot R \subset \sum _{\alpha_1,\alpha_2\in A} (R_{\alpha_1}+R_{\alpha_2}) = R$. Now let $$\begin{aligned} \lefteqn{ e\; :=\; \sum_{\alpha} e_{\alpha}\; -\, \sum_{\alpha_1<\alpha_2} e_{\alpha_1}e_{\alpha_2}\; +\, \sum_{\alpha_1<\alpha_2<\alpha_3} e_{\alpha_1}e_{\alpha_2}e_{\alpha_3} }\\[.6ex] && \hspace{6em} \pm\; \cdots\; +\, (-1)^{|A|+1}\, \sum_{\alpha_1<\,\cdots\,<\alpha_{|A|}} e_{\alpha_1}\,\cdots\,e_{\alpha_{|A|}}\,. \end{aligned}$$ Then it follows from the above that $e\in R$. For $r\in R_{\alpha_i}$, $\alpha_i\in A$, one can check directly that $re=r$, using the property that $re_{\alpha_i}=r$ and the above defining expression of $e$. This implies that $er=r$ for every $r\in R$. It follows that $(R, e)$ is a ring-subset of $M_n({\Bbb C})$. The additional Condition (4), $e=1$, implies then that $R$ is a subalgebra of $M_n({\Bbb C})$. This proves Statement (1). Condition (1) implies that $\{e_{\alpha}\}_{\alpha\in A}\subset Z(R)$. For each $\alpha\in A$, the commutativity, idempotent property, and that both $e_{\alpha}R_{\alpha}=R_{\alpha}$ and $e_{\alpha}(e_{\alpha}R)=e_{\alpha}R$ hold imply that the orthogonal direct-sum decomposition $R=e_{\alpha}R+(1-e_{\alpha})R$ of $R$ coincides with the decomposition $R=R_{\alpha}+e_{\alpha}^{\perp}$, where $e_{\alpha}^{\perp}:= \{r\in R: e_{\alpha}r=0\}$. This shows that the projection map $R\rightarrow R_{\alpha}$ from the above decomposition is identical with the central localization of $R$ with respect to the multiplicatively closed subset $\{1,e_{\alpha}\}$. Furthermore, $\sum_{\alpha\in A}e_{\alpha}$ is invertible in $Z(R)$. Thus, $\{(R\rightarrow R_{\alpha})\}_{\alpha\in A}$ is a central finite cover of $R$. This proves Statement (2). $\Box$ [**The space ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))$ of ring-set-homomorphisms from $R$ to $M_n({\Bbb C})$.**]{} Let $R$ be a finitely-presentable algebra over ${\Bbb C}$ and ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))$ be the set of ring-set-homomorphisms from $R$ to $M_n({\Bbb C})$. We will construct a topology on ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))$ in this theme. Let - $R\, =\, \langle g_0, g_1,\,\cdots\,, g_l\rangle /(r_1,\,\cdots\,, r_m)\;$ be a presentation of $R$ as a quotient of the free unital associative ${\Bbb C}$-algebra $\langle g_0, g_1,\,\cdots\,, g_l\rangle$ generated by $g_0, g_1,\,\cdots\,, g_l$ by the two-sided ideal $(r_1,\,\cdots\,,r_m)$ generated by $\{r_i=r_i(g_0,\,\cdots\,,g_l): i=1,\,\ldots\,,m\}$. Here, for later use, we have the redundant generator $g_0 =$ the identity $1$ and the redundant relators $g_0g_i=g_ig_0=g_i$, $i=0,1,\,\ldots\,,l$, contained in the relator set $\{r_1,\,\cdots\,, r_m\}$. - ${\mbox{\it Gr}\,}^{(2)}(n; d,n-d)\,\simeq\, {\mbox{\it GL}}_n({\Bbb C})/({\mbox{\it GL}}_d({\Bbb C})\times{\mbox{\it GL}}_{n-d}({\Bbb C}))\;$ be the Grassmannian manifold of ordered pairs $(\Pi_1, \Pi_2)$ of ${\Bbb C}$-linear subspaces of ${\Bbb C}^n$ with ${\mbox{\it dim}\,}\Pi_1=d$, ${\mbox{\it dim}\,}\Pi_2=n-d$, and $\Pi_1+\Pi_2={\Bbb C}^n$; - ${\mathbf 1}_d$, $d=0,\,\ldots\,,n$, be the diagonal matrix ${\mbox{\it Diag}\,}(1,\,\cdots\,,1,0,\,\cdots\,,0)$ in $M_n({\Bbb C})$ whose first $d$ diagonal entries are $1$ and the rest $0$, (here, ${\mathbf 1}_0=$ the zero-matrix $0$ and ${\mathbf 1}_n={\mathbf 1}$ by convention); and - ‘$m_1\sim m_2$’$\;$ means that $m_1$ and $m_2$ are in the same adjoint ${\mbox{\it GL}}_n({\Bbb C})$-orbit in $M_n({\Bbb C})$. Let ${\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))$ be the subvariety of the affine space ${\Bbb A}^{n^2}_{(0)}\times{\Bbb A}^{n^2}_{(1)}\times\,\cdots\, \times {\Bbb A}^{n^2}_{(l)}$ (here ${\Bbb A}^{n^2}_{(i)}$ has the polynomial coordinate ring ${\Bbb C}[m_{i, jk}: 1\le j, k\le n]$, $i=0,\,\ldots\,,l$) determined[^25] by the system of equations $$r_1(M_0, M_1,\,\cdots\,, M_l)\;=\; \cdots\; =\; r_m(M_0, M_1,\,\cdots\,, M_l)\; =\; \mbox{the zero-matrix}\; 0\,\in\, M_n({\Bbb C})\,,$$ where $M_i=(m_{i,jk})_{jk}$. Note that this is like the ordinary representation variety of $R$ in $M_n({\Bbb C})$ [*except*]{} that it is not required that $M_0=$ the identity ${\mathbf 1}\in M_n({\Bbb C})$. For convenience, we will call the reduced affine scheme ${\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))$ the [*representation variety*]{} in our discussion. By construction, we have the Zariski topology on ${\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))$ and the analytic topology on the set ${\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))_{\scriptsizeBbb C}$ of ${\Bbb C}$-points of ${\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))$. Regard ${\Bbb C}^n$ as the unique non-zero irreducible $M_n({\Bbb C})$-module. Then, the correspondence $e\mapsto (e\cdot{\Bbb C}^n, e^{\perp})$, where $e^{\perp}$ here $=\{v\in {\Bbb C}^n: e\cdot v=0\}$, gives rise to a (continuous) map from the set of idempotents $\sim {\mathbf 1}_d$ in $M_n({\Bbb C})$ to ${\mbox{\it Gr}\,}^{(2)}(n; d, n-d)$. It follows that the projection map $\pi_{(0)}: {\Bbb A}^{n^2}_{(0)}\times{\Bbb A}^{n^2}_{(1)}\times\,\cdots\, \times {\Bbb A}^{n^2}_{(l)} \rightarrow {\Bbb A}^{n^2}_{(0)}$ restricts to a map $$\pi_{(0)}\; :\; {\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))\; \longrightarrow\; \amalg_{d=0}^n {\mbox{\it Gr}\,}^{(2)}(n; d, n-d)\,.$$ Let $${\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))_{(d)}\; :=\; \pi_{(0)}^{-1}({\mbox{\it Gr}\,}^{(2)}(n; d, n-d))\,.$$ As a set, ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n({\Bbb C})) ={\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n({\Bbb C}))_{\scriptsizeBbb C}$. This identification defines a preliminary analytic topology ${\cal T}_0$ on ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C)) = \amalg_{d=0}^n {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))_{(d)}$ by bringing over the analytic topology on ${\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n({\Bbb C}))_{\scriptsizeBbb C}$. We then modify this preliminary analytic topology, following an analytic format of a [*valuative criterion*]{}, so that each ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))_{(d^{\prime})}$, $d^{\prime}<d$, adheres to ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))_{(d)}$ appropriately, for $d=1,\,\cdots\,, n$, in the new topology. Let $T$ be a (commutative, Noetherian) integral domain over ${\Bbb C}$ and $(D:=({\mbox{\it Spec}\,}T)_{\scriptsizeBbb C}\,,\,p)$ be the associated analytic space together with a base ${\Bbb C}$-point $p$. Note that the residue field $\kappa_p$ of $T$ at $p$ is canonically isomorphic to ${\Bbb C}$. Let $T_p$ be the localization of $T$ at $p$ and $Q_T$ be the field of fractions of $T$. Then, $T\subset T_p\subset Q_T$, $T_p$ is a valuation ring of $Q_T$ (regarded now as the field of fractions of $T_p$), and $M_n(T)\subset M_n(T_p)\subset M_n(Q_T)$. [**Definition 3.2.5 \[limit of family ring-set-homomorphisms\].**]{} [ Let $\phi:R\rightarrow M_n(Q_T)$ be a ring-set-homomorphism such that there exists a unique idempotent $e\in M_n(T_p)$ such that (1) $e\in Z({\mbox{\it Im}\,}\phi)$; (2) $e\cdot\phi$ is a ring-set-homomorphism from $R$ to $T_p$; in particular, $(e\cdot\phi)|_p: R\rightarrow M_n(\kappa_p)=M_n({\Bbb C})$ makes sense; (3) ${\mbox{\it Im}\,}(e\cdot\phi)|_p$ is the unique maximum (with respect to inclusion) in the set of ring-subsets ${\mbox{\it Im}\,}(e^{\prime}\cdot\phi)$ of $M_n(\kappa_p)$, where $e^{\prime}$ satisfies Condition $(1)$ and Condition $(2)$ above. For such a $\phi$, we call $(e\cdot\phi)|_p$ the [*limit*]{} of $\phi$ over $D$ at $p$. ]{} Such a $\phi$ defines a rational map $\Phi_{\phi}: (D,p) {\;\; \mbox{-}\;\mbox{-}\;\mbox{-}\rightarrow\,}{\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))^{{\cal T}_0}$ that is assigned the value $(e\cdot\phi)|_p$ at $p$. [**Definition 3.2.6 \[${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))$ with analytic topology\].**]{} With the notations from above, let ${\cal T}$ be the weakest topology on ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))_{\scriptsizeBbb C}$ such that - the tautological inclusion ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))^{{\cal T}_0}_{(d)} \hookrightarrow {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))^{\cal T}$ of sets is an embedding of topological spaces, for $d=0,\,\cdots\,,n$, and that - $\Phi_{\phi}$ is continuous at $p$ for all $T$, $(D,p)$, and $\phi$ in Definition 3.2.5.[^26] ${\cal T}$ is called the [*analytic topology*]{} on ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))$. [**Proposition 3.2.7 \[independence of presentation\].**]{} [ *$({\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C)),{\cal T})$ is independent of the choice of presentations of $R$ in the construction.* ]{} [*Proof.*]{} Associated to a new presentation $$R\; =\; \langle g^{\prime}_0, g^{\prime}_1,\,\cdots\,, g^{\prime}_{l^{\prime}}\rangle / (r^{\prime}_1,\,\cdots\,, r^{\prime}_{m^{\prime}})$$ of $R$ is a canonical ring-isomorphism $$f^{\sharp}\;:\; \langle g^{\prime}_0, g^{\prime}_1,\,\cdots\,, g^{\prime}_{l^{\prime}}\rangle / (r^{\prime}_1,\,\cdots\,, r^{\prime}_{m^{\prime}})\; \stackrel{\sim}{\longrightarrow}\; \langle g_0, g_1,\,\cdots\,, g_l\rangle /(r_1,\,\cdots\,, r_m)\,,$$ represented by a noncanonical ring-homomorphism $\tilde{f}^{\sharp}: \langle g^{\prime}_0, g^{\prime}_1,\,\cdots\,, g^{\prime}_{l^{\prime}}\rangle \rightarrow \langle g_0, g_1,\,\cdots\,, g_l\rangle$. $\tilde{f}^{\sharp}$ induces contravariantly a morphism $$\tilde{f}\;:\; {\Bbb A}^{n^2}_{(0)}\times{\Bbb A}^{n^2}_{(1)} \times\,\cdots\,\times {\Bbb A}^{n^2}_{(l)}\; \longrightarrow\; {\Bbb A}^{n^2}_{(0)}\times{\Bbb A}^{n^2}_{(1)}\times\,\cdots\, \times {\Bbb A}^{n^2}_{(l^{\prime})}$$ that restricts to a morphism $$f\;:\; {\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))\; \longrightarrow\; {\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))^{\prime}\,,$$ where ${\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))^{\prime} \subset {\Bbb A}^{n^2}_{(0)}\times{\Bbb A}^{n^2}_{(1)}\times\,\cdots\, \times {\Bbb A}^{n^2}_{(l^{\prime})}$ is the representation variety associated to the new presentation of $R$. Reverse this argument, now from $\langle g_0, g_1,\,\cdots\,, g_l\rangle /(r_1,\,\cdots\,, r_m)$ to $\langle g^{\prime}_0, g^{\prime}_1,\,\cdots\,, g^{\prime}_{l^{\prime}}\rangle / (r^{\prime}_1,\,\cdots\,, r^{\prime}_{m^{\prime}})$, implies that $f$ is indeed an isomorphism. Since a ring-isomorphism sends the identity to the identity, $f$ restricts to isomorphisms $f_{(d)}:{\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))_{(d)} \stackrel{\sim}{\rightarrow} {\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))^{\prime}_{(d)}$, for $d=0,\,\cdots\,,n$. In other words, $f_{(d)}:{\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))^{{\cal T}_0}_{(d)} \stackrel{\sim}{\rightarrow} {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C)) ^{\prime\,{\cal T}^{\prime}_0}_{(d)}$, for $d=0,\,\cdots\,,n$. Furthermore, each valuative criterion setup $\Phi_{\phi}: (D,p){\;\; \mbox{-}\;\mbox{-}\;\mbox{-}\rightarrow\,}{\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))$ gives a valuative criterion setup $\Phi_{\phi^{\prime}}= f\circ \Phi_{\phi}: (D,p){\;\; \mbox{-}\;\mbox{-}\;\mbox{-}\rightarrow\,}{\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))^{\prime}$ and vice versa. As we choose the topology ${\cal T}$ on ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))$ (resp. ${\cal T}^{\prime}$ on ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))^{\prime}$) to be the weakest topology that renders all inclusions ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))^{{\cal T}_0}_{(d)} \hookrightarrow {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))$ (resp. ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))^{\prime\,{\cal T}_0}_{(d)} \hookrightarrow {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))^{\prime}$) embeddings of topological spaces and all $\Phi_{\phi}$’s (resp. $\Phi_{\phi^{\prime}}$’s) continuous, this implies that $$f\; :\; ({\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C)),{\cal T})\; \longrightarrow\; ({\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))^{\prime}, {\cal T}^{\prime})$$ is an isomorphism. This completes the proof. $\Box$ By construction, there is a canonical (continuous) bijective embedding $$\tau_{R,n}\,:\; {\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))_{\scriptsizeBbb C}\; \longrightarrow\; {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))\,.$$ [*Remark 3.2.8 $[$moduli problem$]$.*]{} By construction, ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))$ is a coarse moduli space of ring-set-homomorphisms from $R$ to $M_n({\Bbb C})$. Since a ring-set-homomorphism with a fixed domain and target does not have non-trivial automorphisms, it is instructive to think of ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R,M_n(\Bbb C))$ as representing the functor $$\begin{array}{ccccc} {\cal F} & : & \left(\,\mbox{\parbox{27ex}{(commutative) varieties/${\Bbb C}$\\ with analytic topology}}\,\right)^{\circ} & \longrightarrow & (\,\mbox{sets}\,) \\[2.6ex] && V & \longmapsto & {\mbox{\it Mor}\,}_{{\cal O}_V\mbox{\scriptsize\it -Alg}} ( {\cal O}_V\otimes R\,,\, {\cal O}_V\otimes M_n({\Bbb C}) ) \end{array}$$ similar to a functor of points. Here, $(\,\cdots\,)^{\circ}$ is the category $(\,\cdots\,)$ with the arrows reversed. [**${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\largeBbb C}),Y)$ as a coarse moduli space.**]{} Let $Y$ be a noncommutative space presented as a gluing system of finitely-presentable rings ${\cal R}= (\{R_{\alpha}\}_{\alpha\in A} {\begin{array}{c} \rightarrow \\[-2ex] \rightarrow \end{array}}\{R_{\alpha_1\alpha_2}\}_{\alpha_1,\alpha_2\in A})$. We fix a well-ordering of the index set $A$ for convenience. Denote the identity of $R_{\alpha}$ by $1_{R_{\alpha}}$. Assume that each central localization $\varphi_{\alpha_1\alpha_2}:R_{\alpha_1}\rightarrow R_{\alpha_1\alpha_2}$ is associated to a finitely-generated multiplicatively closed subset $S_{\alpha_1\alpha_2}$ in $Z(R_{\alpha_1})$. [**Definition 3.2.9 \[admissible tuple\].**]{} A tuple $(\varphi_{\alpha}:R_{\alpha}\rightarrow M_n({\Bbb C}))_{\alpha\in A}$ of ring-set-homomorphisms to $M_n({\Bbb C})$ is called [*admissible*]{} if it satisfies the following conditions: - $\varphi_{\alpha_1}(1_{R_{\alpha_1}})$ commutes with elements of $\varphi_{\alpha_2}(R_{\alpha_2})$ for all $\alpha_1$, $\alpha_2\in A$. - $\varphi_{\alpha_1}(1_{R_{\alpha_1}})\, \varphi_{\alpha_2}(R_{\alpha_2}) = \varphi_{\alpha_2}(1_{R_{\alpha_2}})\, \varphi_{\alpha_1} (R_{\alpha_1})$ for all $\alpha_1$, $\alpha_2\in A$. - $\varphi_{\alpha_1}(1_{R_{\alpha_1}})\, \varphi_{\alpha_2}(1_{R_{\alpha_2}}) \in \varphi_{\alpha_1}(R_{\alpha_1})$ for all $\alpha_1$, $\alpha_2\in A$. - Let ${\mathbf 1}$ be the identity matrix in $M_n({\Bbb C})$. Then $$\begin{aligned} \lefteqn{ {\mathbf 1}\;=\; \sum_{\alpha} \varphi_{\alpha}(1_{R_{\alpha}})\; -\, \sum_{\alpha_1<\alpha_2} \varphi_{\alpha_1} (1_{R_{\alpha_1}})\, \varphi_{\alpha_2} (1_{R_{\alpha_2}})\; } \\[.6ex] && \hspace{6.2em} +\, \sum_{\alpha_1<\alpha_2<\alpha_3} \varphi_{\alpha_1} (1_{R_{\alpha_1}})\, \varphi_{\alpha_2} (1_{R_{\alpha_2}})\, \varphi_{\alpha_3} (1_{R_{\alpha_3}}) \\[.6ex] && \hspace{6.2em} \pm\; \cdots\; +\, (-1)^{|A|+1}\, \sum_{\alpha_1<\,\cdots\,<\alpha_{|A|}} \varphi_{\alpha_1} (1_{R_{\alpha_1}})\,\cdots\, \varphi_{\alpha_{|A|}} (1_{R_{\alpha_{|A|}}})\,. \end{aligned}$$ - $\varphi_{\alpha_2}(1_{R_{\alpha_2}}) \cdot \left( \varphi_{\alpha_1}(s)^{\perp} \cap \varphi_{\alpha_1}(R_{\alpha_1}) \right) = 0$, where $\varphi_{\alpha_1}(s)^{\perp} := \{ m\in M_n({\Bbb C}): \varphi_{\alpha_1}(s)\cdot m =0 \}$, for all $\alpha_1$, $\alpha_2\in A$ and $s\in S_{\alpha_1\alpha_2}$. This condition is equivalent to the existence of push-out $\varphi_{\alpha_1}|_{\alpha_2}$ under localizations in the following commutative diagram: $$\begin{array}{lccl} R_{\alpha_1} & \stackrel{\varphi_{\alpha_1}}{\longrightarrow} & \varphi_{\alpha_1}(R_{\alpha_1}) \\ \hspace{1ex}\downarrow & & \downarrow \\[-1ex] R_{\alpha_1\alpha_2} & \stackrel{\varphi_{\alpha_1}|_{\alpha_2}}{\longrightarrow} & \varphi_{\alpha_2}(1_{R_{\alpha_2}}) \cdot \varphi_{\alpha_1}(R_{\alpha_1}) &. \end{array}$$ - $\varphi_{\alpha_1}|_{\alpha_2} = \varphi_{\alpha_2}|_{\alpha_1} \circ \varphi_{\alpha_1\alpha_2}$ for all $\alpha_1$, $\alpha_2\in A$. The meaning of these conditions is given below. - Conditions (1) - (4): The finite collection $\{ ( \varphi_{\alpha}(R_{\alpha}), e_{\alpha}:=\varphi_{\alpha}(1_{R_{\alpha}}) ) \}_{\alpha\in A}$ of ring-subsets of $M_n({\Bbb C})$ glue to $\sum_{\alpha\in A}\varphi_{\alpha}(R_{\alpha})$ that is a subalgebra of $M_n({\Bbb C})$. Cf. Proposition 3.2.4. - Condition (5): Elements in $\varphi_{\alpha_1}(S_{\alpha_1\alpha_2})$ become invertible after being mapped to $e_{\alpha_2}\cdot \varphi_{\alpha_1}(R_{\alpha_1})$ and, hence, $\varphi_{\alpha_1}$ can be pushed out to a ring-homomorphism $\varphi_{\alpha_1}|_{\alpha_2}$ from $R_{\alpha_1\alpha_2}$ to the localization $e_{\alpha_2} \cdot \varphi_{\alpha_1}(R_{\alpha_1})$ of $\varphi_{\alpha_1}(R_{\alpha_1})$. Cf. Lemma 3.1.3. - Condition (6): The gluing conditions on the tuple $\{\varphi_{\alpha}:R_{\alpha}\rightarrow M_n({\Bbb C})\} _{\alpha\in A}$ as a system of ring-homomorphisms from ${\cal R}$ to $( \{\varphi_{\alpha}(R_{\alpha})\}_{\alpha\in A} {\begin{array}{c} \rightarrow \\[-2ex] \rightarrow \end{array}}\{ e_{\alpha_2} \cdot\varphi_{\alpha_1}(R_{\alpha_1}) \} _{\alpha_1,\alpha_2\in A} )$. Cf. Condition (2) above and Definition 1.2.6. Thus, Conditions (1) - (6) are necessary conditions for the tuple $\{\varphi_{\alpha}\}_{\alpha\in A}$ to represent a morphism from $({\mbox{\it Spec}\,}{\Bbb C}, {\Bbb C}, M_n({\Bbb C}))$ to $Y$. It follows from Definition 1.2.14 that they are also sufficient and that such presentations are effective in the sense that different admissible tuples give different morphisms. This proves the following lemma: [**Lemma 3.2.10 \[admissible tuple = morphism\].**]{} [ *A tuple $\Phi = (\varphi_{\alpha}:R_{\alpha}\rightarrow M_n({\Bbb C}))_{\alpha\in A}$ of ring-set-homomorphisms to $M_n({\Bbb C})$ corresponds to a morphism from ${\mbox{\it Space}\,}M_n({\Bbb C})$ to $Y={\mbox{\it Space}\,}{\cal R}$ if and only if $\Phi$ is admissible. As sets, ${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n(\Bbb C), Y) = \{\,\mbox{admissible tuples}\,\}$.* ]{} Fix now the following data of presentations and representatives: - \ a finite presentation for each ring-chart $R_{\alpha}$ in ${\cal R}$ $$R_{\alpha}\; =\; \langle g^{(\alpha)}_0, g^{(\alpha)}_1,\,\cdots\,, g^{(\alpha)}_{l^{(\alpha)}}\rangle /(r^{(\alpha)}_1,\,\cdots\,, r^{(\alpha)}_{m^{(\alpha)}})\,,$$ with the redundant generator $g^{(\alpha)}_0 = 1_{R_{\alpha}}$ and the redundant relators $$g^{(\alpha)}_0g^{(\alpha)}_i\; =\; g^{(\alpha)}_ig^{(\alpha)}_0\; =\; g^{(\alpha)}_i\,, \hspace{1ex} i\,=\,0,\,1,\,\ldots\,,l^{(\alpha)}\,,$$ contained in the relator set $\{r^{\alpha}_1,\,\cdots\,, r^{\alpha}_{m^{\alpha}}\}$, as before; - \ a lifting (as sets) $\tilde{S}_{\alpha_1\alpha_2}$ of $S_{\alpha_1\alpha_2}$ in $\langle g^{(\alpha_1)}_0, g^{(\alpha_1)}_1,\, \cdots\,, g^{(\alpha_1)}_{l^{(\alpha_1)}}\rangle$ for each $(\alpha_1, \alpha_2)\in A\times A$; - \ a representative in the induced presentation of $R_{\alpha_2\alpha_1}$ for each $g^{(\alpha_1)}_i$, $i=0,\,\ldots\,, l^{(\alpha_1)}$, $\tilde{s}\in \tilde{S}_{\alpha_1\alpha_2}$, and $(\alpha_1, \alpha_2)\in A\times A$, $$(g^{(\alpha_1\alpha_2)}_i, s^{(\alpha_1\alpha_2)}_i)\;,\;\; (g^{(\alpha_1\alpha_2)}_{\tilde{s}}, s^{(\alpha_1\alpha_2)}_{\tilde{s}})\; \in\; \langle g^{(\alpha_2)}_0, g^{(\alpha_2)}_1,\,\cdots\,, g^{(\alpha_2)}_{l^{(\alpha_2)}}\rangle \times \tilde{S}_{\alpha_2\alpha_1}$$ so that $\varphi_{\alpha_1\alpha_2}(g^{(\alpha_1)}_i, 1_{R_{\alpha_1}}) =(g^{(\alpha_1\alpha_2)}_i,s^{(\alpha_1\alpha_2)}_i)$ and $\varphi_{\alpha_1\alpha_2}(1_{R_{\alpha_1}},\tilde{s}) =(g^{(\alpha_1\alpha_2)}_{\tilde{s}}, s^{(\alpha_1\alpha_2)}_{\tilde{s}})$. (Here, to simplify notations, we identify elements in a presentation of a ring with the corresponding elements in that ring.) Let - ${\Bbb A}^{n^2}_{(\alpha,\,i)}$, $\alpha\in A$, $i=0,\,\ldots\,,l^{(\alpha)}$, be the affine space with the polynomial coordinate ring ${\Bbb C}[m^{(\alpha)}_{i, jk}: 1\le j, k\le n]$; - ${\mathbf A}_{\alpha}$ be the affine space ${\Bbb A}^{n^2}_{(\alpha, 0)}\times{\Bbb A}^{n^2}_{(\alpha, 1)} \times\,\cdots\, \times {\Bbb A}^{n^2}_{(\alpha,\,l^{(\alpha)})}$ and ${\mathbf A}$ be the affine space $\prod_{\alpha\in A}{\mathbf A}_{\alpha} =\; {\Bbb A}^{\sum_{\alpha\in A}(1+l^{(\alpha)})\,n^2}$; - $R({\mathbf A}_{\alpha}) := {\cal O}_{{\mathbf A}_{\alpha}}({\mathbf A}_{\alpha}) = \otimes_{i=0}^{l^{(\alpha)}} {\Bbb C}[m^{(\alpha)}_{i, jk}: 1\le j, k\le n]$ and $R({\mathbf A}) :={\cal O}_{\mathbf A}({\mathbf A}) = \otimes_{\alpha\in A} R({\mathbf A}_{\alpha})$; - $\Psi_{\alpha}: R({\mathbf A})\otimes_{\scriptsizeBbb C} \langle g^{(\alpha)}_0, g^{(\alpha)}_1,\,\cdots\,, g^{\alpha}_{l^{(\alpha)}}\rangle \rightarrow R({\mathbf A})\otimes_{\scriptsizeBbb C} M_n({\Bbb C}) = M_n(R({\mathbf A}))$ be the tautological $R({\mathbf A})$-algebra-homomorphism defined/generated by[^27] $$1\otimes g^{(\alpha)}_i\; \longmapsto\; 1\otimes \left(m^{(\alpha)}_{i,jk}\right)_{jk}$$ and ${\mbox{\it Im}\,}\Psi_{\alpha}$ be the image $R({\mathbf A})$-submodule of $\Psi_{\alpha}$ in $M_n(R({\mathbf A}))$; - $E_{\mathbf A}={\mathbf A}\times M_n({\Bbb C})$ be the trivialized trivial vector bundle on ${\mathbf A}$ with fiber the ${\Bbb C}$-algebra $M_n({\Bbb C})$; the associated sheaf of local sections of $E_{\mathbf A}$ is ${\cal O}_{\mathbf A}\otimes M_n({\Bbb C})$; elements and sub-$R({\mathbf A})$-modules in ${\cal O}_{\mathbf A}\otimes M_n({\Bbb C})$ are canonically identified respectively with global sections and constructible sets in $E_{\mathbf A}$. Define $${\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\cal R},M_n(\Bbb C))\; \subset\; \prod_{\alpha\in A} {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R_{\alpha},M_n(\Bbb C))$$ to be the locus in the indicated product space determined by the following system of constraints from the defining conditions of admissible tuples, via the canonical bijective embedding $$\prod_{\alpha\in A} {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R_{\alpha},M_n(\Bbb C))\; \stackrel{\prod_{\alpha\in A}\tau_{R_{\alpha},n}} {\longleftarrow\hspace{-1ex}\mbox{------------}}\; \prod_{\alpha\in A} {\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R_{\alpha},M_n(\Bbb C)) _{\scriptsizeBbb C}\; \subset\; {\mathbf A}\,:$$ - $$\begin{aligned} \lefteqn{\hspace{3.6em} r^{(\alpha)}_1(M_{\alpha,0}, M_{\alpha,1},\,\cdots\,, M_{\alpha,\,l^{(\alpha)}})\; =\; \cdots\; =\; r^{(\alpha)}_{m^{(\alpha)}} (M_{\alpha,0}, M_{\alpha, 1},\, \cdots\,, M_{\alpha,\,l^{(\alpha)}})\; } \\[,6ex] && \hspace{4em} =\; \mbox{the zero-matrix $0\,\in\, M_n({\Bbb C})$, where $M_{\alpha,\,i} =\left(m^{(\alpha)}_{i,jk}\right)_{jk}$.} \hspace{6em} \end{aligned}$$ - $M_{\alpha_1,0}\, M_{\alpha_2, i}=M_{\alpha_2,i}\, M_{\alpha_1,0}\;$ for all $\alpha_1$, $\alpha_2\in A$, $i=0,\,\ldots\,, l^{(\alpha_2)}$. - $M_{\alpha_1,0}\, {\mbox{\it Im}\,}\Psi_{\alpha_2} = M_{\alpha_2,0}\, {\mbox{\it Im}\,}\Psi_{\alpha_1}$ for all $\alpha_1$, $\alpha_2\in A$. - $M_{\alpha_1,0}\, M_{\alpha_2,0} \in {\mbox{\it Im}\,}\Psi_{\alpha_1}$ for all $\alpha_1$, $\alpha_2\in A$. - (${\mathbf 1}\in M_n({\Bbb C})$ is the identity) $$\begin{aligned} \lefteqn{\hspace{2em} {\mathbf 1}\;=\; \sum_{\alpha} M_{\alpha,0}\; -\, \sum_{\alpha_1<\alpha_2} M_{\alpha_1,0}\, M_{\alpha_2,0}\, +\, \sum_{\alpha_1<\alpha_2<\alpha_3} M_{\alpha_1,0}\, M_{\alpha_2,0}\, M_{\alpha_3,0} }\\[.6ex] && \hspace{10em} \pm\; \cdots\; +\, (-1)^{|A|+1}\, \sum_{\alpha_1<\,\cdots\,<\alpha_{|A|}} M_{\alpha_1,0}\,\cdots\, M_{\alpha_{|A|,0}}\,. \end{aligned}$$ - $M_{\alpha_2,0} \cdot \left( \Psi_{\alpha_1}(\tilde{s})^{\perp}_{E_{\mathbf A}} \cap {\mbox{\it Im}\,}\Psi_{\alpha_1} \right) = 0$ for all $\alpha_1$, $\alpha_2\in A$ and $\tilde{s}\in \tilde{S}_{\alpha_1\alpha_2}$. Here[^28] $$\Psi_{\alpha_1}(\tilde{s})^{\perp}_{E_{\mathbf A}}\; :=\; \{ m\in E_{\mathbf A}: \Psi_{\alpha_1}(\tilde{s})\cdot m =0 \}\,.$$ - $$\begin{aligned} \lefteqn{ \left( M_{\alpha_2,0}\, M_{\alpha_1, i} \rule{0ex}{2ex}\right)\, \left( M_{\alpha_1,0}\; s^{(\alpha_1\alpha_2)}_i (M_{\alpha_2,0},\,\cdots\,, M_{\alpha_2,\,l^{(\alpha_2)}}) \right) }\\[.6ex] && =\; M_{\alpha_1,0}\: g^{(\alpha_1\alpha_2)}_i (M_{\alpha_2,0},\,\cdots\,, M_{\alpha_2,\,l^{(\alpha_2)}}) \hspace{11.2em} \end{aligned}$$ $\mbox{\hspace{3.2em}}$ and $$\begin{aligned} \lefteqn{ \left( M_{\alpha_2,0}\,M_{\alpha_1,0} \rule{0ex}{2ex}\right)\, \left( M_{\alpha_1,0}\; s^{(\alpha_1\alpha_2)}_{\tilde{s}} (M_{\alpha_2,0},\,\cdots\,, M_{\alpha_2,\,l^{(\alpha_2)}}) \right) }\\[.6ex] && =\; \left( M_{\alpha_1,0}\: g^{(\alpha_1\alpha_2)}_{\tilde{s}} (M_{\alpha_2,0},\,\cdots\,, M_{\alpha_2,\,l^{(\alpha_2)}} \right)\, \left( M_{\alpha_2,0}\; \tilde{s} (M_{\alpha_1,0},\,\cdots\,, M_{\alpha_1,\,l^{(\alpha_1)}}) \right)\, \end{aligned}$$ $\mbox{\hspace{3.2em}}$ for all $\alpha_1$, $\alpha_2\in A$, $i=0,\,\ldots\,,l^{(\alpha_1)}$, and $\tilde{s}\in \tilde{S}_{\alpha_1\alpha_2}$. [**Proposition 3.2.11 \[${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\largeBbb C}),Y)$\].**]{} [ *${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\largeBbb C}),Y)$ is given by a constructible set in the product space $\prod_{\alpha\in A} {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R_{\alpha},M_n(\Bbb C))$, independent of the data of presentation chosen in the construction.* ]{} [*Proof.*]{} Conditions (0.1), (1.1), and (1.4) are closed conditions. Condition (1.6) can be restricted to a finite generating set of $S_{\alpha_1\alpha_2}$ and, hence, gives also a closed condition. Conditions (1.2), (1.3), and (1.5) involve image $R({\mathbf A})$-submodules ${\mbox{\it Im}\,}\Psi_{\bullet}$ in $M_n(R({\mathbf A}))$. Let $S_{\alpha_1\alpha_2}^0$ be a finite generating set of $S_{\alpha_1\alpha_2}$ and $\tilde{S}_{\alpha_1\alpha_2}^0 \subset \tilde{S}_{\alpha_1\alpha_2}$ its corresponding lifting in $\langle g^{(\alpha_1)}_0, g^{(\alpha_1)}_1,\, \cdots\,, g^{(\alpha_1)}_{l^{(\alpha_1)}}\rangle$. Then, it follows from Lemma 3.1.7 and the fact that every nilpotent element $m$ of $M_n({\Bbb C})$ satisfies $m^n=0$ that the seemingly possibly-infinite system of constraints from Condition (1.5) can be replaced by the following finite system: - $M_{\alpha_2,0} \cdot \left( \Psi_{\alpha_1}(\tilde{s}^n)^{\perp}_{E_{\mathbf A}} \cap {\mbox{\it Im}\,}\Psi_{\alpha_1} \right) = 0$ for all $\alpha_1$, $\alpha_2\in A$ and $\tilde{s}\in \tilde{S}_{\alpha_1\alpha_2}^0$. Thus, the solution set to Conditions (1.2), (1.3), and (1.5) is described by a finite intersection of constructible sets on ${\mathbf A}$ described via determinantal varieties. This shows that the solution set to the system of constraints from Condition (0.1) and Conditions (1.1) - (1.5) is a constructible set in $\prod_{\alpha\in A} {\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R_{\alpha},M_n(\Bbb C))$ and, hence, in $\prod_{\alpha\in A} {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}(R_{\alpha},M_n(\Bbb C))$. That different choices of data of presentations give isomorphic solution sets (with the subset topology) follows the same discussion as that in the proof of Proposition 3.2.7. Since ${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\Bbb C}), Y) ={\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\cal R},M_n({\Bbb C}))$ as sets, this concludes the proof. $\Box$ We remark that from the proof above, the constructible set referred to in Proposition 3.2.11 is of algebraic kind. It is the set of ${\Bbb C}$-points (with the analytic topology) of a finite union of constructible sets in varieties/${\Bbb C}$. Finally, note that in discussing the space of morphisms from ${\mbox{\it Space}\,}M_n({\Bbb C})$ to $Y$, both ${\mbox{\it Space}\,}M_n({\Bbb C})$ and $Y$ are thought of as fixed. The automorphism group of $M_n({\Bbb C})$ as a ${\Bbb C}$-algebra is given by ${\mbox{\it GL}}_n({\Bbb C})$ via the adjoint ${\mbox{\it GL}}_n({\Bbb C})$-action on $M_n({\Bbb C})$. This induces a ${\mbox{\it GL}}_n({\Bbb C})$-action on ${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\Bbb C}), Y)$. [**Definition 3.2.12 \[isomorphism between morphisms\].**]{} [ Two morphisms from ${\mbox{\it Space}\,}M_n({\Bbb C})$ to $Y$ are said to be [*isomorphic*]{}, in notation $\Phi_1\sim \Phi_2$, if they are in the same ${\mbox{\it GL}}_n({\Bbb C})$-orbit in ${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\Bbb C}), Y)$. Define the [*space ${\mbox{\it Map}\,}({\mbox{\it Space}\,}M_n({\Bbb C}), Y)$ of maps*]{} from ${\mbox{\it Space}\,}M_n({\Bbb C})$ to $Y$ to be the quotient space ${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\Bbb C}), Y)/\!\sim\,$ (with the quotient topology). It parameterizes isomorphism classes of morphisms from ${\mbox{\it Space}\,}M_n({\Bbb C})$ to $Y$. ]{} D0-branes on a commutative quasi-projective variety. ==================================================== A D0-brane in the sense of Definition 2.2.3 is simply an Azumaya-type noncommutative point ${\mbox{\it Space}\,}M_n({\Bbb C})$ (cf. Example 1.1.3 and Example 1.1.8) together with the irreducible $M_n({\Bbb C})$-module ${\Bbb C}^n$ as the Chan-Paton space/module. A D0-brane on a target space $Y$ is given by an isomorphism class of morphisms from ${\mbox{\it Space}\,}M_n({\Bbb C})$ to $Y$. The moduli space of D0-branes on $Y$ in this sense is given then by ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n),Y) ={\mbox{\it Map}\,}({\mbox{\it Space}\,}M_n({\Bbb C}),Y) = {\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\Bbb C}),Y)/\!\sim$. This moduli space for the case of $Y$ being a (commutative) complex quasi-projective smooth curve/surface, or a variety is given in this section to illustrate Sec. 1 - Sec. 3. These examples already reveal simplified key features of D-branes that are fundamental for beyond. Details involving only linear algebras in, e.g., \[Ho-K\] or straightforward manipulations are omitted. D0-branes on the complex affine line ${\Bbb A}^1$. -------------------------------------------------- Various themes concerning D0-branes on ${\Bbb A}^1$ are given in this subsection to illustrate the far-reaching/power of the Polchinski-Grothendieck Ansatz for D-branes, in particular the reproduction of D-brane properties in the work of Polchinski. Same/Similar phenomena occur also for other targets in later subsections by same/similar reasons, which we then omit but focus mainly on the moduli problem. The general discussions in Sec. 1 - Sec. 3 are intentionally made explicit in this example. For that reason, some important algebro-geometric notions are slightly repeated in this subsection for concreteness. [**The moduli space ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n)\,,\,{\Bbb A}^1)$ of D0-branes on ${\Bbb A}^1$.**]{} Let $Y={\Bbb A}^1={\mbox{\it Spec}\,}{\Bbb C}[y]$ be the affine line over ${\Bbb C}$. Then the Grothendieck Satz or Lemma 1.2.19 says that ${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\Bbb C}), Y)={\mbox{\it Mor}\,}({\Bbb C}[y], M_n({\Bbb C}))$. The corresponding ${\Bbb C}$-algebra representation variety ${\mbox{\it Rep}\,}({\Bbb C}[y], M_n({\Bbb C}))$ is given by ${\Bbb A}^{n^2}$ with a closed point represented by $m=(m_{ij})_{i,j}\in M_n({\Bbb C})$ corresponding to the ${\Bbb C}$-algebra-homomorphism $$\varphi_m: {\Bbb C}[y]\rightarrow M_n({\Bbb C})\,, \hspace{1ex} \mbox{generated by $1\mapsto{\mathbf 1}$ and $y\mapsto m$}\,.$$ We will call the ${\mbox{\it GL}}_n({\Bbb C})$-action on ${\mbox{\it Rep}\,}({\Bbb C}[y], M_n({\Bbb C}))$ by post-compositions with the conjugations on $M_n({\Bbb C})$ still the [*adjoint action*]{}. It follows that $${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n), {\Bbb A}^1)\; =\; {\mbox{\it Map}\,}({\mbox{\it Space}\,}M_n({\Bbb C}), {\Bbb A}^1)\; =\; {\mbox{\it Rep}\,}({\Bbb C}[y],M_n({\Bbb C}))/\!\sim\,,$$ the orbit-space[^29] of the adjoint action with the quotient topology. This space is a connected non-Hausdorff topological space, well-understood in other contents from algebraic geometry and Lie groups and Lie algebras as follows. Each adjoint-orbit $O_{\varphi_m}$ is represented by a Jordan form $J_m$ of $m$, unique up to permutations of diagonal blocks of $J_m$ with distinct characteristic values. An adjoint-orbit on ${\mbox{\it Rep}\,}({\Bbb C}[y], M_n({\Bbb C}))$ is closed if and only of it is represented by $\varphi_m$ associated to a diagonal matrix $m$. Given an orbit $O_{\varphi_m}$, let $\overline{O_{\varphi_m}}$ be the closure of $O_{\varphi_m}$ in ${\Bbb A}^{n_2}$. It has the property that $O_{\varphi_m}$ is an open dense subset in $\overline{O_{\varphi_m}}$ and that $\overline{O_{\varphi_m}}$ is a union of $O_{\varphi_m}$ and finitely many lower-dimensional orbits, (e.g. \[Stei\]). Note that any two orbits $O_{\varphi_{m_1}}$ and $O_{\varphi_{m_2}}$ satisfy either $O_{\varphi_{m_1}}\cap \overline{O_{\varphi_{m_2}}}=\emptyset$ or $O_{\varphi_{m_1}}\subset \overline{O_{\varphi_{m_2}}}$. [**Definition 4.1.1 \[partial order on ${\mbox{\it Rep}\,}({\Bbb C}[y], M_n({\Bbb C}))/\!\sim$\].**]{} [ Define a [*partial order*]{} on the orbit-space ${\mbox{\it Rep}\,}({\Bbb C}[y], M_n({\Bbb C}))/\!\sim$ by setting $O_{\varphi_{m_1}}\prec O_{\varphi_{m_2}}$ if $O_{\varphi_{m_1}}\subset\overline{O_{\varphi_{m_2}}}$. ]{} This partial order can be described in terms of Jordan forms, as follows. Let $J^{(\lambda)}_j\in M_j({\Bbb C})$ be the matrix [ $$\left[ \begin{array}{cccc} \lambda & & & 0 \\ 1 & \lambda & & \\ & \ddots & \ddots & \\ 0 & & 1 & \lambda \end{array} \right]_{j\times j}$$ [A]{}]{} Jordan form $J$ in $M_n({\Bbb C})$ is a matrix of the following form $$\left[ \begin{array}{ccc} A_1 & & 0 \\ & \ddots & \\ 0 & & A_k \end{array} \right] \hspace{1em} \mbox{with each $A_i\in M_{n_i}({\Bbb C})$ of the form}\hspace{1em} \left[ \begin{array}{ccc} J^{(\lambda_i)}_{d_{i1}} & & \\ & \ddots & \\ & & J^{(\lambda_i)}_{d_{ik_i}} \end{array} \right]\,.$$ Here, omitted entries are all zero, $n_1\ge \,\cdots\,\ge n_k> 0$, and $d_{i1}\ge\,\cdots\,\ge d_{ik_i}>0$. We thus have a [*double partition*]{} of $n$ by non-increasing positive integers: $$\pi(n)\;:\; n\;=\; n_1+\,\cdots\,+ n_k\,; \hspace{1em} \pi(n_i)\;:\; n_i\; =\; d_{i1}+\,\cdots\,+ d_{ik_i}\,,\; i=1\,,\,\ldots\,, k\,.$$ We will call this double partition the [*type*]{}, in notation ${\mbox{\it type}}(J)$, of $J$. Denote also the set of all such double partitions of $n$ by ${\mbox{\it P$\!$P}}(n)$. Then the admissible permutations of the blocks $A_i, \, \cdots\,, A_k$ induces a finite group action on ${\mbox{\it P$\!$P}}(n)$. The quotient set is denoted by ${\mbox{\it P$\!$P}}(n)/\!\sim$. For a general $m\in M_n({\Bbb C})$, define its type by ${\mbox{\it type}}(m)={\mbox{\it type}}(J_m)$, which is uniquely defined after passing to ${\mbox{\it P$\!$P}}(n)/\!\sim$. [**Definition 4.1.2 \[partial order between Jordan forms\].**]{} *Given two Jordan forms $J_1$ and $J_2$, we say that $J_1\prec J_2$ if the following two conditions are satisfied:* - $J_1$, $J_2$ have the same characteristic values $\lambda_1,\,\cdots\,,\lambda_k$ of the same multiplicities $n_i$ for $\lambda_i$. - Let $A_{1i},\, A_{2i}\in M_{n_i}({\Bbb C})$ be the diagonal blocks of $J_1$ and $J_2$ respectively that are associated to $\lambda_i$ and ${\mathbf 1}_{n_i}$ be the identity of $M_{n_i}({\Bbb C})$. Then ${\mbox{\it rank}\,}((A_{1i}-\lambda_i{\mathbf 1}_{n_i})^j) \le {\mbox{\it rank}\,}((A_{2i}-\lambda_i{\mathbf 1}_{n_i})^j)$ for all $j\in {\Bbb N}$. This defines a partial order $\prec$ on the set of Jordan matrices in $M_n({\Bbb C})$ that is invariant under admissible permutations of diagonal blocks of distinct characteristic values. [**Proposition 4.1.3 \[partial order of orbits via Jordan forms\].**]{} (\[M-T\], \[Ge\], \[Dj\].) [ *$$\mbox{ $O_{\varphi_{m_1}}\prec O_{\varphi_{m_2}}$ if and only if $J_{m_1}\prec J_{m_2}$. }$$* ]{} The following simplified/coarser partial order helps us to see things more directly. [**Definition 4.1.4 \[isotopic decay\].**]{}[^30] [ The composition of a sequence of operations of the form $J^{(\lambda)}_j \rightarrow {\mbox{\it Diag}\,}(J^{(\lambda)}_{j_1}\,,\, J^{(\lambda)}_{j_2})$ with $j=j_1+j_2$, $j_1\ge j_2$, will be called an [*isotopic decay*]{}. ]{} Given two Jordan forms $J_1$ and $J_2$, define $J_1{\prec\!\prec}J_2$ if $J_1$ is obtained from $J_2$ by a sequence of isotopic decays and an re-arrangement of the sub-blocks in each diagonal block associated to a characteristic value. [**Lemma 4.1.5 \[coarser partial order\].**]{} [ *$(1)$ $O_{m_1}\prec O_{m_2}$ if $J_{m_1}{\prec\!\prec}J_{m_2}$. $(2)$ $\prec$ and ${\prec\!\prec}$ generate the same equivalence relation, in notation $\approx$, on the set of Jordan forms.* ]{} [*Remark 4.1.6 $[$orbit dimension drop under ${\prec\!\prec}$$]$.*]{} (E.g. \[We\]; also \[Ge\] or \[Bas\].) Let [ $$\begin{array}{c} \\[1ex] T^{(b_1,\,\cdots\,b_i)}_{i\times j}\\[1em] (i\le j) \end{array}\; =\; \left[ \begin{array}{cccccc} b_1 & & & & \\[.6ex] b_2 & b_1 & & & \\ & b_2 & \ddots & & & \hspace{1em}0 \\ \vdots & \ddots & \ddots & b_1 & \\[.6ex] b_i & \cdots & & b_2 & b_1 & \end{array} \right]_{i\times j}\,, \hspace{1em} \begin{array}{c} \\[1ex] T^{(b_1,\,\cdots\,b_j)}_{i\times j}\\[1em] (i\ge j) \end{array}\; =\; \left[ \begin{array}{ccccc} & & 0 \\[1em] b_1 & & & & \\[.6ex] b_2 & b_1 & & & \\ & b_2 & \ddots & & \\ \vdots & \ddots & \ddots & b_1 & \\[.6ex] b_j & \cdots & & b_2 & b_1 \end{array} \right]_{i\times j}\,.$$ [Here]{}]{}, all omitted entries are zero. The centralizer of $J$, in the form given previously, consists of all matrices of the form $$\left[ \begin{array}{ccc} B_1 & & 0 \\ & \ddots & \\ 0 & & B_k \end{array} \right]$$ with each $B_i\in M_{n_i}({\Bbb C})$ of the block form $\left[ B_{i,rs}\right]_{k_i\times k_i}$ where $$B_{i,rs}\; =\; T^{(b_{i,rs; 1}, \,\cdots\,b_{i,rs; di_r})} _{di_r\times di_s} \hspace{1ex}\mbox{for $r\ge s$},\hspace{1em} B_{i,rs}\; =\; T^{(b_{i,rs; 1}, \,\cdots\,b_{i,rs; di_s})} _{di_r\times di_s} \hspace{1ex}\mbox{for $r<s$.}$$ (Again, omitted entries are all zero.) The dimension of the stabilizer of $J$, as given, is thus $$\begin{aligned} \lefteqn{ n\; \le \; {\mbox{\it dim}\,}_{\scriptsizeBbb C}{\mbox{\it Stab}\,}(J) }\\[.6ex] && =\;\sum_{i=1}^k\, \left(\rule{0em}{1em} (d_{i1}+\,\cdots\,+ d_{ik_i}) + 2(d_{i2}+\,\cdots\,+ d_{ik_i}) + \, \cdots\; + 2(d_{ik_i}) \right)\; \le\; n^2\,. \end{aligned}$$ Thus, for each $J^{(\lambda)}_j \rightarrow {\mbox{\it Diag}\,}(J^{(\lambda)}_{j_1}\,,\,J^{(\lambda)}_{j_2})$ with $j_1\ge j_2$ the corresponding new adjoint-orbit drops the dimension by an integral amount $\ge j_2$. Some properties of ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n), {\Bbb A}^1)$ are listed below: - The equivalence relation $\approx$ in Lemma 4.1.5 descends to an equivalence relation, still denoted by $\approx$, on the topological space ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n), {\Bbb A}^1)$. The associated quotient space ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n), {\Bbb A}^1)/\!\!\approx$ is the $n$-th symmetric product $S^n{{\Bbb A}^n} :=({\Bbb A}^1)^n/{\mbox{\it Sym}}_n\simeq\footnote{However, caution that under this isomorphism that comes from the ring generated by elementary symmetric polynomials, the diagonal locus in $({\footnotesizeBbb A}^1)^n$ becomes a complicated discriminant locus in ${\footnotesizeBbb A}^n$.}\; {\Bbb A}^n$ of ${\Bbb A}^1$, where ${\mbox{\it Sym}}_n$ is the permutation group of $n$ letters. Each $\approx$-equivalence class of points on ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n), {\Bbb A}^1)$ contains a unique maximal point and a unique minimal point with respect to $\prec$ on ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n), {\Bbb A}^1)$. Any other point in the same class is sandwiched between the two by $\prec$. - The types of Jordan forms give rise to a finite stratification $\{S_t\}_t$ of\ ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n), {\Bbb A}^1)$. The stratum associated to the double partition $$\pi(n)\;:\; n\;=\; n_1+\,\cdots\,+ n_k\,; \hspace{1em} \pi(n_i)\;:\; n_i\; =\; d_{i1}+\,\cdots\,+ d_{ik_i}\,,\; i=1\,,\,\ldots\,, k\,,$$ of $n$ is homeomorphic to $({\Bbb C}^k-\mbox{(diagonal locus)})/{\mbox{\it Sym}}_k$. Here, ‘diagonal locus’ means the set of all points whose coordinates have some identical entries. The stratum $S_{(n=1+\,\cdots\,+1)}$ is open dense in ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n), {\Bbb A}^1)$. [**The Chan-Paton space/module on D0-branes on ${\Bbb A}^1$.**]{} Let $m\in M_n({\Bbb C})$ with the Jordan form as given above and $\langle {\mathbf 1}, m\rangle$ be the sub-algebra of $M_n({\Bbb C})$ generated by ${\mathbf 1}$ and $m$. $\langle {\mathbf 1}, m\rangle$ is commutative. The [*c*]{}haracteristic polynomial and the [*min*]{}imal polynomial of $m$ are then respectively $$f_m^c(\lambda)\; =\; (\lambda-\lambda_1)^{n_1}\,\cdots\,(\lambda-\lambda_k)^{n_k} \hspace{1em}\mbox{and}\hspace{1em} f_m^{{\mbox{\scriptsize\rm min}}}(\lambda)\; =\; (\lambda-\lambda_1)^{d_{11}}\,\cdots\,(\lambda-\lambda_k)^{d_{k1}}\,.$$ [**Lemma 4.1.7 \[interpolation formula\].**]{} [ *Given $g(\lambda) := (\lambda-\lambda_1)^{d_1}\,\cdots\,(\lambda-\lambda_k)^{d_k} \in {\Bbb C}[\lambda]$ with the $\lambda_i$’s distinct from each other, then the inverse of $g(\lambda)/(\lambda-\lambda_i)^{d_i}$ in ${\Bbb C}[\lambda]/((\lambda-\lambda_i)^{d_i})$ exists, for $i=1,\,\ldots\,, k$. Denote this inverse by $(1/g_{(i)})(\lambda)$, which is a polynomials of degree $\le d_i-1$. Let $d=d_1+\,\cdots\,+d_k$ and $f(\lambda)$ be a polynomial of degree $< d$. Then there exist unique polynomials $f_i(\lambda)$ with ${\mbox{\it deg}\,}f_i(\lambda) < d_i$ such that $$f(\lambda)\; =\; \sum_{i=1}^k\: f_i(\lambda) \cdot (1/g_{(i)})(\lambda) \cdot \frac{g(\lambda)}{(\lambda-\lambda_i)^{d_i}}\,.$$ Indeed, $f_i(\lambda)$ is the Taylor expansion of $f(\lambda)$ in $(\lambda-\lambda_i)$ up to (including) degree $d_i-1$.* ]{} It follows that, as a ${\Bbb C}$-algebra, $$\begin{aligned} \lefteqn{ \langle{\mathbf 1},m \rangle\; \simeq\; {\Bbb C}[\lambda]/(f_m^{{\mbox{\scriptsize\rm min}}}(\lambda)) }\\[.6ex] &&\hspace{1.4em} =\; \sum_{i=1}^k \left( (1/{f_{m\; (i)}^{{\mbox{\scriptsize\rm min}}}})(\lambda) \cdot \frac{ f_m^{{\mbox{\scriptsize\rm min}}}(\lambda)} {(\lambda-\lambda_i)^{d_{i1}} } \right)\; \simeq\; \prod_{i=1}^k \left( {\Bbb C}[\lambda]/(\lambda-\lambda_i)^{d_{i1}} \right)\,. \end{aligned}$$ The sum in the above expression is a direct sum of orthogonal indecomposable ideals in\ ${\Bbb C}[\lambda]/(f_m^{{\mbox{\scriptsize\rm min}}}(\lambda))$ associated to the decomposition $$1 \; =\; \sum_{i=1}^k (1/{f_{m\;(i)}^{{\mbox{\scriptsize\rm min}}}})(\lambda) \cdot \frac{ f_m^{{\mbox{\scriptsize\rm min}}}(\lambda)} {(\lambda-\lambda_i)^{d_{i1}} }$$ through the complete set of primitive orthogonal idempotents in ${\Bbb C}[\lambda]/(f_m^{{\mbox{\scriptsize\rm min}}}(\lambda))$. The length $l_{\langle{\mathbf 1},m\rangle}$ of $\langle{\mathbf 1},m\rangle$ is ${\mbox{\it deg}\,}f_m^{{\mbox{\scriptsize\rm min}}}(\lambda)=d_{11}+\,\cdots\,+d_{k1}$. Let ${\Bbb C}^n$ be the unique non-zero irreducible representation of $M_n({\Bbb C})$. Up to the ${\mbox{\it GL}}_n({\Bbb C})$ adjoint action, we may assume that $m$ is already a Jordan form $J={\mbox{\it Diag}\,}(A_1,\,\cdots\,,A_k)$ given earlier. Let ${\mathbf 1}_{(i)} = {\mbox{\it Diag}\,}(0,\,\cdots\,, 0, {\mathbf 1}_{n_i},0, \,\cdots\,,0)$, where ${\mathbf 1}_{n_i}$ in the $i$-th position is the identity matrix $\in M_{n_i}({\Bbb C})$ and the $0$ in the $j$-th position are the zero-matrix $\in M_{n_j}({\Bbb C})$ for $j=1,\,\cdots\,, i-1, i+1, \,\cdots\,, k$. Then $$\left. \left( (1/{f_{J\;(i)}^{{\mbox{\scriptsize\rm min}}}})(\lambda) \cdot \frac{ f_J^{{\mbox{\scriptsize\rm min}}}(\lambda)} {(\lambda-\lambda_i)^{d_{i1}} } \right)\right|_{\,\lambda\,=\,J}\; =\; {\mathbf 1}_{(i)}\,.$$ This implies that ${\mathbf 1}_{(i)}\in \langle{\mathbf 1}, J\rangle$ for $i=1,\,\ldots\,,k$ and that ${\mathbf 1}={\mathbf 1}_{(1)}+\,\cdots\,+{\mathbf 1}_{(k)}$ is an orthogonal primitive idempotent decomposition in $\langle{\mathbf 1},J\rangle$. The corresponding direct-sum decomposition, now as $\langle{\mathbf 1}, J\rangle$-modules, $${\Bbb C}^n\; =\; {\mathbf 1}_{(1)}\cdot {\Bbb C}^n \,+\,\cdots\, +\, {\mathbf 1}_{(k)}\cdot {\Bbb C}^n\; =\; {\Bbb C}^{n_1}\,+\,\cdots\,+\,{\Bbb C}^{n_k}\; =:\; V_1\,+\,\cdots\,+V_k$$ is the same decomposition of ${\Bbb C}^n$ that renders $J$ the given diagonal block form. As a $\langle{\mathbf 1},J\rangle$-module, $V_i$ $(={\Bbb C}^{n_i})$ decomposes into a direct sum $V_i = {\Bbb C}^{d_{i1}}+\,\cdots\,+{\Bbb C}^{d_{ik_i}} =: V_{i1}+\,\cdots\,+V_{ik_i}$ of indecomposable $\langle{\mathbf 1},J\rangle$-modules. ${\mbox{\it Spec}\,}\langle{\mathbf 1},J\rangle$ has $k$-many connected components, associated respectively to ideals $({\mathbf 1}-{\mathbf 1}_{(i)})$ in $\langle{\mathbf 1},J\rangle$, $i=1,\,\ldots\,,k$. One has that $$\langle{\mathbf 1}, J\rangle/({\mathbf 1}-{\mathbf 1}_{(i)})\; =\; \langle{\mathbf 1}, J\rangle\cdot{\mathbf 1}_{(i)}\; \simeq\; \langle{\mathbf 1}_{n_i}, A_i\rangle\; \simeq\; {\Bbb C}[\lambda]/((\lambda-\lambda_i)^{n_i})$$ and that the annihilator ${\mbox{\it Ann}\,}(V_i)$ of $V_i$ $(={\Bbb C}^{n_i})$ as an $\langle{\mathbf 1}, J\rangle$-module is $({\mathbf 1}-{\mathbf 1}_{(i)})$. In terms of $\langle{\mathbf 1}, J\rangle \simeq \prod_{i=1}^k\langle{\mathbf 1}_{n_i}, A_i\rangle$, the $\langle{\mathbf 1},J\rangle$-modules $V_i$, $V_{i1}, \,\cdots\,, V_{ik_i}$ are also $\langle{\mathbf 1}_{n_i}, A_i\rangle$-modules automatically. The above algebraic statements correspond to the following geometric picture of Chan-Paton modules on the associated D0-branes on ${\Bbb A}^1\,$: - Under Grothendieck Ansatz or Lemma 1.2.19, $\varphi_J: {\Bbb C}[y]\rightarrow M_n({\Bbb C})$ gives(/is equivalent to) a morphism $\hat{\varphi}_J:{\mbox{\it Space}\,}M_n({\Bbb C})\rightarrow {\Bbb A}^1$ with the image subscheme ${\mbox{\it Im}\,}\hat{\varphi}_J \simeq {\mbox{\it Spec}\,}\langle{\mathbf 1},J\rangle$ associated to the ideal $${\mbox{\it Ker}\,}(\varphi_J)\; =\; (f_m^{{\mbox{\scriptsize\rm min}}}(y))\; =\; \left( (y-\lambda_1)^{d_{i1}}\,\cdots\,(y-\lambda_k)^{d_{k1}} \right)$$ in ${\Bbb C}[y]$. Thus, on ${\Bbb A}^1$ there are $k$-many (generally non-reduced) points located respectively at $y=\lambda_1,\,\cdots\,,\lambda_k$ (in the underlying complex plane ${\Bbb C}$ of ${\Bbb A}^1$) where D0-branes in Polchinski’s sense may sit upon. These are the [*D0-branes on ${\Bbb A}^1$ associated to $\varphi_J$*]{} in the sense of Definition 2.2.3. From the discussion, for a general $\varphi_m$, they depend only on the minimal polynomial $f_m^{{\mbox{\scriptsize\rm min}}}(\lambda)$ of $m$. - The push-forward[^31] $\hat{\varphi}_{J\ast}{\Bbb C}^n =\sum_{i=1}^k\hat{\varphi}_{J\ast}V_i =\sum_{i=1}^k\sum_{j=1}^{k_i}\hat{\varphi}_{J\ast}V_{ij}$ is now an ${\cal O}_{{\mbox{\scriptsize\it Im}\,}\hat{\varphi}_J}$-module of length $n$. Decompose ${\mbox{\it Im}\,}\hat{\varphi}_J$ into a disjoint union $\amalg_{i=1}^kZ_i$, where $Z_i$ is the subscheme of ${\Bbb A}^1$ associated to the ideal $((y-\lambda_i)^{d_{i1}})$. Then $\hat{\varphi}_{J\ast}V_i$ is supported on $Z_i$ and, hence, is an ${\cal O}_{Z_i}$-module of length $n_i$. The decomposition $\hat{\varphi}_{J\ast}V_i =\sum_{j=1}^{k_i}\hat{\varphi}_{J\ast}V_{ij}$ is automatically a direct-sum decomposition as ${\cal O}_{Z_i}$-modules as well. Let $Z_i^{(l)}$, $l\le n_i$, be the subscheme of $Z_i$ associated to the ideal $((y-\lambda_i)^l)$ in ${\Bbb C}[y]$. Note that $Z_i^{(l)}$ has length $l$ and that $Z_i^{(1)}$ is the ${\Bbb C}$-point in $Z_i$ and $Z_i^{(n_i)}=Z_i$. Then, $\hat{\varphi}_{J\ast}V_{ij}$ is a rank-$1$ ${\cal O}_{Z_i}$-module of length $d_{ij}$ and is supported on $Z_i^{(d_{ij})}$. As ${\cal O}_{Z_i}$-modules, $$\hat{\varphi}_{J\ast}V_{i1}\; \simeq\; {\cal O}_{Z_i}$$ and $$\mbox{$\hat{\varphi}_{J\ast}V_{ij}\; \simeq\;$ the ideal $(y-\lambda_i)^{d_{i1}-d_{ij}}\cdot{\cal O}_{Z_i}$ of ${\cal O}_{Z_i}\; \simeq\;$ the quotient ${\cal O}_{Z_i^{(d_{ij})}}$ of ${\cal O}_{Z_i}$}\,.$$ In our setting[^32], we call $\hat{\varphi}_{J\ast}V_i$ the [*Chan-Paton module on the D0-brane supported on $Z_i\subset {\Bbb A}^1$ associated to $\varphi_J$*]{}. From the discussion, for a general $\varphi_m$, their isomorphism class depends only on both $f_m^{{\mbox{\scriptsize\rm min}}}(\lambda)$ and the type of $m$. [**Comparison with Hilbert schemes and Chow varieties.**]{} The Hilbert scheme ${\mbox{\it Hilb}\,}^n_{{\scriptsizeBbb A}^1}=: ({\Bbb A}^1)^{[n]}$ of $n$ points on ${\Bbb A}^1$ parameterizes $0$-dimensional subschemes of length $n$ on ${\Bbb A}^1$. Such a subscheme of ${\Bbb A}^1$ is given uniquely by an ideal $(f)\subset {\Bbb C}[y]$, where $f$ is a monic polynomial of degree $n$. In terms of matrices, it is thus represented by an $m\in M_n({\Bbb C})$ such that both the characteristic polynomial and the minimal polynomial of $m$ are $f$.[^33] Observe that the Jordan form of (omitted entries are zero; the multiplicity of $\lambda_i=n_i$) [ $$\mbox{\normalsize $J^{(\lambda_1,\,\cdots\,,\lambda_k)}_+\; :=\;$} \left[ \begin{array}{ccccccccccccc} \lambda_1 & \\ 1 & \ddots \\ & \ddots &\ddots \\ & & 1 & \lambda_1 \\[3ex] & & & {\mathbf 1} & \lambda_2 \\ &&& & 1 & \ddots \\ &&&&& \ddots & \ddots \\ &&&&&& 1 & \lambda_2 \\ &&&&&&& \ddots & \ddots \\[1ex] &&&&&&&& {\mathbf 1} & \lambda_k \\ &&&&&&&&& 1 & \ddots \\ &&&&&&&&&& \ddots & \ddots \\ &&&&&&&&&&& 1 & \lambda_k \end{array} \right]\,,$$]{} where $(\lambda_1,\,\cdots\,,\,\lambda_k)$ is the $n$-tuple $(\lambda_1,\,\cdots\,,\,\lambda_1,\, \lambda_2,\,\cdots\,,\,\lambda_2,\, \cdots\,,\, \lambda_k,\,\cdots\,,\,\lambda_k)$ with the specified multiplicity $n_i$ for $\lambda_i$, is ${\mbox{\it Diag}\,}(J^{(\lambda_1)}_{n_1}\,,\,\cdots\,,\,J^{(\lambda_k)}_{n_k})$, up to a permutation of the blocks. Its characteristic polynomial and minimal polynomial are identical: $(y-\lambda_1)^{n_1}\,\cdots\,(y-\lambda_k)^{n_k}$. Let ${\Bbb C}^n$ parameterizes the ordered tuples of roots of monic polynomial of degree $n$, then the embedding $${\Bbb C}^n\; \hookrightarrow\; M_n({\Bbb C})\,, \hspace{1em} (\lambda_1,\,\cdots\,,\,\lambda_n)\; \mapsto\; J^{(\lambda_1,\,\cdots\,,\,\lambda_n)}_+$$ descends to an embedding $$\Phi_{{\mbox{\scriptsize\it Hilb}}}: ({\Bbb A}^1)^{[n]}\; \longrightarrow\; {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n), {\Bbb A}^1)\,, \hspace{1em} \mbox{$\prod_{i=1}^n(y-\lambda_i)$}\; \longmapsto\; \varphi_{J^{(\lambda_1,\,\cdots\,,\,\lambda_n)}_+}\,.$$ On the other hand, the Chow variety ${\mbox{\it Chow}\,}^{(n)}_{{0, \scriptsizeBbb A}^1}$ of $n$ points on ${\Bbb A}^1$ parameterizes $0$-cycles of order $n$ on ${\Bbb A}^1$ and is identical to the $n$-th symmetric product $S^n({\Bbb A}^1)$ of ${\Bbb A}^1$. Such a $0$-cycle on ${\Bbb A}^1$ happens to be represented uniquely by a monic polynomial in $y$ of degree $n$ as well. Thus there is a canonical isomorphism $({\Bbb A}^1)^{[n]}\simeq S^n({\Bbb A}^1)$. However, from the general ground of Chow groups, the support of a cycle is meant to be a reduced subscheme with each of its irreducible components marked with a multiplicity. Thus, in terms of matrices, it is represented by an $m\in M_n({\Bbb C})$ such that the minimal polynomial of $m$ has only simple roots. Such matrices are exactly the diagonalizable matrices. Again, let ${\Bbb C}^n$ parameterizes the ordered tuples of roots of monic polynomial of degree $n$, then it follows that the embedding $${\Bbb C}^n\; \hookrightarrow\; M_n({\Bbb C})\,, \hspace{1em} (\lambda_1,\,\cdots\,,\,\lambda_n)\; \mapsto\; {\mbox{\it Diag}\,}(\lambda_1,\,\cdots\,,\,\lambda_n)$$ descends to an embedding $$\Phi_{{\mbox{\scriptsize\it Chow}}}: S^n({\Bbb A}^1)\; \longrightarrow\; {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n), {\Bbb A}^1)\,, \hspace{1em} \mbox{$\prod_{i=1}^n(y-\lambda_i)$}\; \longmapsto\; \varphi_{{\mbox{\scriptsize\it Diag}\,}(\lambda_1,\,\cdots\,,\,\lambda_n)}\,.$$ In other words, ${\mbox{\it Im}\,}\Phi_{{\mbox{\scriptsize\it Hilb}}}$ parameterizes conjugacy classes of regular representations of ${\Bbb C}[y]$ in $M_n({\Bbb C})$ while ${\mbox{\it Im}\,}\Phi_{{\mbox{\scriptsize\it Chow}}}$ parameterizes conjugacy classes of diagonal representations of ${\Bbb C}[y]$ in $M_n({\Bbb C})$. Note that, under the isomorphism $({\Bbb A}^1)^{[n]}\simeq S^n({\Bbb A}^1)$, $\Phi_{{\mbox{\scriptsize\it Hilb}}}$ and $\Phi_{{\mbox{\scriptsize\it Chow}}}$ coincide only on the open dense subset, points of which correspond to $0$-dimensional reduced subschemes of length $n$ on ${\Bbb A}^1$. For all $p$ in the complement of this subset, $\Phi_{{\mbox{\scriptsize\it Chow}}}(p) \prec \Phi_{{\mbox{\scriptsize\it Hilb}}}(p)$ by an isotopic decay. In particular, ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n), {\Bbb A}^1)$ contains $({\Bbb A}^1)^{[n]}$ and $S^n({\Bbb A}^1)$ distinctly and, for $n\ge 3$, has more points than $\Phi_{{\mbox{\scriptsize\it Hilb}}}(({\Bbb A}^1)^{[n]}) \cup \Phi_{{\mbox{\scriptsize\it Chow}}}(S^n({\Bbb A}^1))$. In the current case, it happens that $\Phi_{{\mbox{\scriptsize\it Hilb}}}$ and $\Phi_{{\mbox{\scriptsize\it Chow}}}$ give rise to $$({\Bbb A}^1)^{[n]}\; \stackrel{\sim}{\longrightarrow}\; {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n), {\Bbb A}^1)/\!\approx \;\; \stackrel{\sim}{\longleftarrow}\; S^n({\Bbb A}^1)\,.$$ This is only accidental and does not generalize to $Y$ of dimension $\ge 2$. Note also that, for all $p$, the Chan-Paton module at $\Psi_{{\mbox{\scriptsize\it Hilb}}}(p)$ gives exactly the structure sheaf ${\cal O}_{Z_p}$ of the subscheme $Z_p$ $p$ represents while the Chan-Paton module at $\Psi_{{\mbox{\scriptsize\it Chow}}}(p)$ gives an association of ${\Bbb C}^{n_i}$ to each $p_i$ (as an ${\cal O}_{p_i} (={\Bbb C})$-module), for $p=\sum_{i=1}^kp_i$ as a $0$-cycle. Thus, Chan-Paton spaces/modules in the sense of Definition 2.2.3 tells the difference of subschemes versus cycles as well.[^34] This is a general feature. Finally, the map that sends $\varphi_m$ to the diagonal of $J_m$ gives rise to a continuous map $\pi_{{\mbox{\scriptsize\it Chow}}}: {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n),{\Bbb A}^1) \rightarrow S^n({\Bbb A}^1)$. It has $\Phi_{{\mbox{\scriptsize\it Chow}}}$ as a section. [**Associated quiver.**]{} Given a finite-dimensional ${\Bbb C}$-algebra $R$, one can associate a [*quiver*]{}[^35] $\Gamma_R$ to $R$ as follows: - Let $\{e_1,\,\cdots\,,e_k\}$ be a complete set of primitive orthogonal idempotents in $R$. Then associate to each $e_i$ a vertex, denoted also by $e_i$. - Let $J(R)$ be the radical of $R$. Then, associate ${\mbox{\it dim}\,}_{\scriptsizeBbb C}\,e_i(J(R)/J(A)^2)e_j$-many arrows from $e_i$ to $e_j$. Applying this to $\varphi_m$, representing a point in ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n)\,,\,{\Bbb A}^1)$, by associating a graph to the Artinian ${\Bbb C}$-algebra ${\Bbb C}[y]/{\mbox{\it Ker}\,}\varphi_m\simeq \langle{\mathbf 1},m\rangle$, following the rules above, we obtain a quiver $\Gamma_{\varphi_m}$ that captures part of the geometry of the D0-brane on ${\Bbb A}^1$ associated to $\varphi_m$: - a [*vertex*]{} $e_i$ for the connected component $Z_i$ of ${\mbox{\it Spec}\,}{\Bbb C}[y]/{\mbox{\it Ker}\,}\varphi_m = {\mbox{\it Im}\,}\hat{\varphi}_m=\amalg_{i=1}^k Z_i$ of the D0-brane on ${\Bbb A}^1$; - an [*arrow*]{} with both ends attached to $e_i$ if $Z_i$ has the embedded dimension $1$ (i.e. if $Z_i$ is a non-reduced point on ${\Bbb A}^1$); there are no other arrows for any pair $(e_i,e_j)$, $1\le i,j\le k$. The Chan-Paton module discussed in an earlier theme is realized now as a representation of $\Gamma_{\varphi_m}$: (without loss of generality, we take $m$ to be the Jordan form $J=J_m$ and adopt earlier notations) - assign the ${\cal O}_{Z_i}$-module $(\hat{\varphi}_{m\ast}{\Bbb C}^n)|_{Z_i}=\hat{\varphi}_{m\ast}V_i$ to vertex $e_i$ for $i=1,\,\ldots\,,k$; - if there is an arrow on $e_i$, then assign to that arrow the nilpotent endomorphism on $\hat{\varphi}_mV_i$ associated to the multiplication by $(y-\lambda_i)\,$ (i.e. the push-forward of the endomorphism $A_i-\lambda_i{\mathbf 1}_{n_i}$ on $V_i$). The quiver $\Gamma_{\varphi_m}$ together with this representation now encodes the full geometry of the connected components of the D0-brane on ${\Bbb A}^1$ except their exact locations $y=\lambda_1,\,\cdots\,,\lambda_k$. [**Higgsing/un-Higgsing of D-branes via deformations of morphisms.**]{}[^36] The important open-string-induced Higgsing (i.e. gauge symmetry-breaking)/un-Higgsing (i.e. gauge symmetry enhancement) behavior on D-branes can be reproduced in the current content as follows. As any associative ${\Bbb C}$-algebra $R$ gives rise to a Lie algebra $(R, [\cdot,\cdot])$ over ${\Bbb C}$ by taking the Lie bracket to be $[m_1,m_2]=m_1m_2-m_2m_1$, we can equivalently make the discussion directly for associative algebras in our problem. Since on ${\mbox{\it Space}\,}M_n({\Bbb C})$, $M_n({\Bbb C})$ acts on the Chan-Paton space ${\Bbb C}^n$ as the endomorphism algebra ${\mbox{\it End}\,}({\Bbb C}^n)$ of the Chan-Paton space, this is the counterpart of (the Lie algebra of) the gauge symmetry on a D-brane in physicists’ picture. Given a $[\varphi_m:{\Bbb C}[y]\rightarrow M_n({\Bbb C})] \in {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n),{\Bbb A}^1)$, the Chan-Paton space ${\Bbb C}^n$ on ${\mbox{\it Space}\,}M_n({\Bbb C})$ is turned into the Chan-Paton module on ${\mbox{\it Im}\,}\hat{\varphi}_m$ by taking ${\Bbb C}^n$ now as a (left) $\langle{\mathbf 1},m\rangle$-module, as discussed earlier. To distinguish them, we will denote the latter by $_{\langle{\mathbf 1},m\rangle}{\Bbb C}^n$. Let $${\mbox{\it Centralizer}\,}\langle{\mathbf 1}, m\rangle\; :=\; \{m^{\prime\prime}\in M_n({\Bbb C})\,:\, m^{\prime\prime}m^{\prime}=m^{\prime}m^{\prime\prime}\;\; \mbox{for all $m^{\prime}\in \langle{\mathbf 1},m\rangle$} \}$$ be the centralizer of $\langle{\mathbf 1},m\rangle$ in $M_n({\Bbb C})$. Then, [**Lemma 4.1.8 \[centralizer vs. pushed-forward endomorphism\].**]{} [ *A ${\Bbb C}$-vector-space endomorphism $m^{\prime\prime}\in M_n({\Bbb C})$ of ${\Bbb C}^n$ can be pushed forward to a $\langle{\mathbf 1}, m \rangle$-module endomorphism on $_{\langle{\mathbf 1},m\rangle}{\Bbb C}^n$ if and only if $m^{\prime\prime}\in{\mbox{\it Centralizer}\,}\langle{\mathbf 1},m\rangle \subset M_n({\Bbb C})$.* ]{} This gives a correspondence: - [ *${\mbox{\it Centralizer}\,}\langle{\mathbf 1}, m\rangle\subset M_n({\Bbb C})$ $\hspace{1em}\Longleftrightarrow\hspace{1em}$ gauge symmetry on the D0-brane ${\mbox{\it Im}\,}\hat{\varphi}_m$ on ${\Bbb A}^1$.* ]{} Recall further from earlier discussions the connected-component-decomposition ${\mbox{\it Im}\,}\hat{\varphi}_m=:Z=\amalg_{i=1}^kZ_i$ and the $\langle{\mathbf 1},m\rangle$-module direct-sum decomposition $_{\langle{\mathbf 1},m\rangle}{\Bbb C}^n=\sum_{i=1}^kV_i$ with $\hat{\varphi}_{m\ast}V_i$ supported on $Z_i$. Then, there is a natural direct-product decomposition as ${\Bbb C}$-algebras: $${\mbox{\it Centralizer}\,}\langle{\mathbf 1},m\rangle\; =\; \prod_{i=1}^k {\mbox{\it Centralizer}\,}\langle{\mathbf 1},m\rangle_{(i)}\; \subset\; \prod_{i=1}^k {\mbox{\it End}\,}(V_i)\; \simeq\; \prod_{i=1}^kM_{n_i}({\Bbb C})\,.$$ Up to conjugation, we may assume that $m=J_m=J$ a Jordan form, then ${\mbox{\it Centralizer}\,}\langle{\mathbf 1},m\rangle_{(i)} \subset M_{n_i}({\Bbb C})$ consists of $n_i\times n_i$-matrices is of the form $B_i$ given in Remark 4.1.6. Thus, each $Z_i$ can be regarded as a D0-brane on ${\Bbb A}^1$ in its own right, associated to $[\varphi_{B_i}] \in {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_{n_i}({\Bbb C});{\Bbb C}^{n_i}),{\Bbb A}^1)$, with the Chan-Paton module $\hat{\varphi}_{B_i\ast}{\Bbb C}^{n_i}$ and the gauge symmetry associated to the endomorphism subalgebra ${\mbox{\it Centralizer}\,}\langle{\mathbf 1}_{n_i},B_i\rangle$ in $M_{n_i}({\Bbb C})$. When $\varphi_m$ varies, this gives rise to Higgsing/un-Higgsing of gauge symmetry of D0-branes on ${\Bbb A}^1$. In particular, if we restrict $\varphi_m$ to vary in $\Phi_{{\mbox{\scriptsize\it Chow}}}(S^n({\Bbb A}^1)) \subset {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C}); {\Bbb C}^n),{\Bbb A}^1)$, then the Higgsing/un-Higgsing pattern of $n$ D0-branes on ${\Bbb A}^1$ is as follows: - For $\varphi_m$ in the stratum associated to the type $(n=d_1+\,\cdots\,+d_k)$: - \ $Z=\amalg_{i=1}^kZ_i\simeq\amalg_{i=1}^k{\mbox{\it Spec}\,}{\Bbb C}$, (i.e. $k$-collection of stacked D0-branes on ${\Bbb A}^1$); - \ ${\Bbb C}^{n_i}$ supported at the D0-brane $Z_i$ on ${\Bbb A}^1$ for $i=1,\,\ldots\,k$; - \ a factor $M_{n_i}({\Bbb C})\simeq {\mbox{\it End}\,}({\Bbb C}^{n_i})$ on $Z_i$ for $i=1,\,\cdots\,,k$; the total gauge symmetry of the $k$-many D0-brane system is the Lie algebra associated to the product $\prod_{i=1}^kM_{n_i}({\Bbb C})$. - As a consequence of Item (1) above, when we vary $[\varphi_m]\in\Phi_{{\mbox{\scriptsize\it Chow}}}(S^n({\Bbb A}^1))$ so that, for example, - \ $Z_1$ splits to $j$-many separated D0-brane collections $Z^{\prime}_1$, $\cdots\,$, $Z^{\prime}_j$ on ${\Bbb A}^1$, governed by the partition $n_1=n^{\prime}_1+\,\cdots\,+n^{\prime}_j$. Then the Chan-Paton space ${\Bbb C}^{n_1}$ splits as well and turns into a Chan-Paton-space ${\Bbb C}^{n^{\prime}_i}$ at $Z^{\prime}_i$ for $i=1,\,\ldots\,,j$. The gauge symmetry associated to $M_{n_1}({\Bbb C})$ is now broken to the one associated to the sub-endomorphism-algebra $\prod_{i=1}^j M_{n^{\prime}_i}({\Bbb C})$ with the factor $M_{n^{\prime}_i}({\Bbb C})$ assigned to $(Z^{\prime}_i,{\Bbb C}^{n^{\prime}}_i)$ for $i=1,\,\ldots\,,j$. - \ $Z_1,\,\cdots\,, Z_j$ collide/merge to a new $Z^{\prime}_j$. Then there is now a D0-brane collection at $Z^{\prime}_j$ with Chan-Paton space ${\Bbb C}^{n_1+\,\cdots\,+n_j}$. The original gauge symmetry for the collection $\{(Z_1,{\Bbb C}^{n_1}) \,\cdots\,,(Z_j,{\Bbb C}^{n_j})\}$, which is the one associated to $M_{n_1}({\Bbb C})\times\,\cdots\,\times M_{n_j}({\Bbb C})$, is now enhanced to the gauge symmetry associated to $M_{n_1+\,\cdots\,+n_j}({\Bbb C})$, acting on $(Z^{\prime}_j,{\Bbb C}^{n_1+\,\cdots\,+n_j})$. Except that we have to use algebraic groups – in particular the ${\mbox{\it GL}}_{\,\bullet}({\Bbb C})$-series in the current content – in the pure algebro-geometric setting, this is exactly the pattern of the oriented-open-string-induced Higgsing/un-Higgsing of unitary gauge symmetry of D-branes that Polchinski concluded in \[Pol3: Sec. 3.3 and Sec. 3.4\][^37]. In summary: [**Proposition 4.1.9 \[Higgsing/un-Higgsing of D0-branes on ${\Bbb A}^1$\].**]{}[^38] [ *The pattern of open-string-induced Higgsing/un-Higgsing behavior of $n$ D0-branes on ${\Bbb A}^1$ can be reproduced in the current content via deformations of morphisms $[\varphi_m:{\Bbb C}[y]\rightarrow M_n({\Bbb C})]$ in $\Phi_{{\mbox{\scriptsize\it Chow}}}(S^n({\Bbb A}^1)) \subset {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C}); {\Bbb C}^n),{\Bbb A}^1)$.* ]{} [**Comparison with the spectral cover construction and the Hitchin system.**]{} Fix a complex line bundle $\pi_L: L\rightarrow {\mbox{\it pt}}\,$ over a point ${\mbox{\it pt}}$. We will identify ${\mbox{\it pt}}$ with the zero-section of $L$ whenever needed. Let $\lambda$ be the tautological section of $\pi_L^{\ast}L$ over $L$. [**Definition 4.1.10 \[semi-simple pair\].**]{}[^39] [ A pair $(E,\phi)$, where $\pi_E:E\rightarrow {\mbox{\it pt}}\,$ is a rank-$n$ complex vector bundle over ${\mbox{\it pt}}$ and $\phi:E\rightarrow E\otimes L$ a complex-vector-bundle-homomorphism over ${\mbox{\it pt}}\,$ is called [*semi-simple*]{} if $\phi$ is semi-simple (i.e. diagonalizable) with respect to a (hence any) trivialization $E\simeq {\Bbb C}^n$ and $L\simeq {\Bbb C}$. ]{} Associated to a semi-simple pair $(E,\phi)$, with the $\phi$ of type $(n=n_1+\,\cdots\,+n_k)$, are the following objects: - the reduced zero-locus $Z_{\phi}=\amalg_{i=1}^kZ_{\phi;i}$ of the section $\det(\pi_L^{\ast}\phi-{\mathbf 1}\otimes\lambda)$ of $\det(\pi_L^{\ast}E)\otimes (\pi_L^{\ast}L)^{\otimes n}$; - a direct-sum decomposition $E=\sum_{i=1}^k V_i$ of bundles over ${\mbox{\it pt}}\,$ so that\ $\hat{V}_i := (\pi_L^{\ast}V_i)|_{Z_{\phi,i}} = ({\mbox{\it Ker}\,}(\pi_L^{\ast}\phi-{\mathbf 1}\otimes \lambda))| _{Z_{\phi;i}}$ for $i=1,\,\ldots\,,k$; - $\prod_{i=1}^k{\mbox{\it End}\,}(V_i)\subset {\mbox{\it End}\,}(E)\simeq M_n({\Bbb C})$ acting on $E$ leaving each $V_i$ invariant for $i=1,\,\ldots\,,k$. This is the $0$-dimensional spectral cover construction in the sense of \[Hi\]; see also \[B-N-R\], \[Don1\], and \[Ox\]. The Hitchin system in this content takes the form of the isomorphism $S^n{\Bbb C}\stackrel{\sim}{\longrightarrow} {\Bbb C}^n$ that sends $[\lambda_1, \,\cdots\,,\lambda_n]$ to the monic polynomial $\prod_{i=1}^n(\lambda-\lambda_i)$ of degree $n$ in $\lambda$. Now identify $L$ with ${\Bbb A}^1$ by $y\mapsto \lambda$ and $E$ with the Chan-Paton space of $n$ D0-branes stacked at the origin $y=0$. Then $\phi$ corresponds to a D0-brane configuration supported at $Z_{\phi}$, with the Chan-Paton space $\hat{V}_i$ and endomorphism algebra ${\mbox{\it End}\,}(\hat{V}_i)={\mbox{\it End}\,}(V_i)$ at $Z_{\phi,i}$. One may regard $Z_{\phi}$ as a deformation of the stacked D0-branes at $y=0$ (which corresponds to $\phi=0$). [*This reproduces also the Higgsing/un-Higgsing behavior of Polchinski’s D-branes.*]{} Note that D0-branes on ${\Bbb A}^1$ described through this construction correponds to the locus ${\mbox{\it Im}\,}\Phi_{{\mbox{\scriptsize\it Chow}}}$ in ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n),{\Bbb A}^1)$. This spectral cover picture of D-branes is particularly fascinating when one recalls the Seiberg-Witten integrable system and the associated gauge-symmetry-breaking pattern revealed there; cf. \[S-W1\] and \[Don2\], \[D-W\], \[Le\].[^40] For the rest of this section, we will focus mainly on the moduli problem. D0-branes on the complex projective line ${\Bbb P}^1$. ------------------------------------------------------ Let $Y$ be the projective line over ${\Bbb C}$: $$Y\; =\; {\Bbb P}^1\; =\; U_0\cup_{U_0\cap U_{\infty}} U_{\infty}\; =\; {\mbox{\it Spec}\,}{\Bbb C}[y_0] \cup_{{\mbox{\scriptsize\it Spec}\,}{\scriptsizeBbb C}[y_0\,,\,1/y_o] \simeq {\mbox{\scriptsize\it Spec}\,}{\scriptsizeBbb C}[1/y_{\infty}\,,\,y_{\infty}]} {\mbox{\it Spec}\,}{\Bbb C}[y_{\infty}]\,,$$ where ${\mbox{\it Spec}\,}{\Bbb C}[y_0\,,\,1/y_o] \stackrel{\sim}{\rightarrow} {\mbox{\it Spec}\,}{\Bbb C}[1/y_{\infty}\,,\,y_{\infty}]$ is given by $y_{\infty}\mapsto 1/y_0$. Having discussed the details of D0-branes on ${\Bbb A}^1$ in Sec. 4.1, we focus now on the issue of gluings for D0-branes on ${\Bbb P}^1$. Recall the Grassmannian-like manifold ${\mbox{\it Gr}\,}^{(2)}(n; d,n-d)$; the idempotents ${\mathbf 1}_d$, $d=0,\,\ldots\,,n$, in $M_n({\Bbb C})$; and the notation ‘$m_1\sim m_2$’ for similar matrices in $M_n({\Bbb C})$ from Sec. 3.2. Then, the ring-set representation variety $$\begin{array}{rcl} {\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y], M_n({\Bbb C})) & = & \{(e,m)\in M_n({\Bbb C})\times M_n({\Bbb C})\,:\, e^2=e\,,\; em=me=m\} \\[.6ex] & \subset & {\Bbb A}^{n^2}\times {\Bbb A}^{n^2} \end{array}$$ has $(n+1)$-many connected components, given by $${\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y], M_n({\Bbb C}))_{(d)} \; :=\; \{ (e,m)\in{\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y], M_n({\Bbb C}))\;:\; e\sim {\mathbf 1}_d \}\,,$$ $d=0,\,\ldots\,,n$. (Here we identify the pair $(e,m)$ with the ring-set-homomorphism $$\varphi_{(e,m)}:{\Bbb C}[y]\rightarrow M_n({\Bbb C}) \hspace{2em}\mbox{with $1\mapsto e$ and $y\mapsto m$.})$$ ${\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y], M_n({\Bbb C}))_{(d)}$ is a ${\mbox{\it GL}}_n({\Bbb C})$-manifold that goes with a natural ${\mbox{\it GL}}_n({\Bbb C})$-equivariant bundle map ${\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y], M_n({\Bbb C}))_{(d)} \rightarrow {\mbox{\it Gr}\,}^{(2)}(n; d, n-d)$ with fiber $\simeq M_d({\Bbb C})$. In particular, ${\mbox{\it dim}\,}_{\scriptsizeBbb C} {\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y], M_n({\Bbb C}))_{(d)} = d^2+2d(n-d) = n^2 -(n-d)^2$, which increases strictly when $d$ goes from $0$ to $n$. The space ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y], M_n({\Bbb C}))$ of ring-set-homomorphisms from ${\Bbb C}[y]$ to $M_n({\Bbb C})$ can be thought of as the ${\mbox{\it GL}}_n({\Bbb C})$-space $\amalg_{d=0}^n {\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y], M_n({\Bbb C}))_{(d)}$, but with the topology ${\cal T}$ in Definition 3.2.6. It has the following properties: - ${\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y], M_n({\Bbb C}))_{(n)} = {\mbox{\it Rep}\,}({\Bbb C}[y], M_n({\Bbb C}))$ is an open dense subset of\ ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y], M_n({\Bbb C}))$. - A neighborhood of $(e,m)$ with $e\sim {\mathbf 1}_d$ consists of all $(e^{\prime},m^{\prime}) \in {\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y], M_n({\Bbb C}))$ such that - $e^{\prime}\sim {\mathbf 1}_{d^{\prime}}$ for some $d^{\prime}\ge d$; - there is an idempotent $e^{\prime\prime}$ in $Z(\langle e^{\prime},m^{\prime}\rangle)$ with the properties: - $e^{\prime\prime}\sim {\mathbf 1}_d$ and is in a neighborhood of $e$, - $e^{\prime\prime}m^{\prime}$ is in a neighborhood of $m$ in $M_n({\Bbb C})$, - besides the characteristic value $0$ of multiplicity $d+(n-d^{\prime})$, the matrix $$(e^{\prime}-e^{\prime\prime})m^{\prime}\; =\; (e^{\prime}-e^{\prime\prime})m^{\prime} (e^{\prime}-e^{\prime\prime}) \in M_n({\Bbb C})$$ has all the remaining $(d^{\prime}-d)$-many characteristic values in a neighborhood of $\infty$ in ${\Bbb C}\cup \{\infty\}$. The space ${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\Bbb C}),{\Bbb P}^1)$ of morphisms from ${\mbox{\it Space}\,}M_n({\Bbb C})$ to ${\Bbb P}^1$ is given by the locus in ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y_0], M_n({\Bbb C})) \times {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y_{\infty}], M_n({\Bbb C}))$ described by the following conditions: - $( \varphi_{(e_{(0)}, m_{(0)})}, \varphi_{(e_{(\infty)}, m_{(\infty)})} ) \in {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y_0], M_n({\Bbb C})) \times {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y_{\infty}], M_n({\Bbb C}))$, - $e_{(0)}e_{(\infty)}=e_{(\infty)}e_{(0)}$, $\,{\mathbf 1}=e_{(0)}+e_{(\infty)}-e_{(0)}e_{(\infty)}$; - $e_{(0)}m_{(\infty)}=m_{(\infty)}e_{(0)}$, $\,e_{(\infty)}m_{(0)}=m_{(0)}e_{(\infty)}$; - $e_{(\infty)}\langle e_{(0)}, m_{(0)}\rangle = e_{(0)}\langle e_{(\infty)}, m_{(\infty)} \rangle$ in $M_n({\Bbb C})$, (note that under Condition (2),\ $e_{(\infty)}\langle e_{(0)}, m_{(0)}\rangle =\langle e_{(\infty)}e_{(0)}, e_{(\infty)}m_{(0)}\rangle$ and $\,e_{(0)}\langle e_{(\infty)}, m_{(\infty)} \rangle =\langle e_{(0)}e_{(\infty)}, e_{(0)}m_{(\infty)}\rangle$); - $e_{(\infty)}m_{(0)}$ is invertible in $\langle e_{(\infty)}e_{(0)}, e_{(\infty)}m_{(0)}\rangle$, $\,e_{(0)}m_{(\infty)}$ is invertible in $\langle e_{(0)}e_{(\infty)}, e_{(0)}m_{(\infty)}\rangle$; - The identity in Condition (3) takes $e_{(\infty)}m_{(0)}$ to the inverse of $e_{(0)}m_{(\infty)}$ and $e_{(0)}m_{(\infty)}$ to the inverse of $e_{(\infty)}m_{(0)}$. Note that Conditions (1) and (2) says that $${\mathbf 1}\;\in\; \langle e_{(0)}, e_{(\infty)}\rangle\; \subset\; Z(\langle e_{(0)}, e_{(\infty)}, m_{(0)}, m_{(\infty)} \rangle)\; \subset\; M_n({\Bbb C})\,.$$ Conditions (3), (4), and (5) are the descendability to localizations and the gluability of pairs of ring-set-morphisms in ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y_0], M_n({\Bbb C})) \times {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y_{\infty}], M_n({\Bbb C}))$. ${\mbox{\it GL}}_n({\Bbb C})$ acts diagonally on ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y_0], M_n({\Bbb C})) \times {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y_{\infty}], M_n({\Bbb C}))$, leaving Conditions (1) - (5) invariant. [**Lemma 4.2.1 \[closed condition\].**]{} *Assuming Conditions $(1)$ and $(2)$, then Conditions $(3)$, $(4)$, and $(5)$ together are equivalent to* - $e_{(0)}e_{(\infty)}m_{(0)}m_{(\infty)}=e_{(0)}e_{(\infty)}$. In particular, the system $\{(1), (2), (3), (4), (5)\}$ realizes ${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\Bbb C}),{\Bbb P}^1)$ as a ${\mbox{\it GL}}_n({\Bbb C})$-invariant closed subset in ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y_0], M_n({\Bbb C})) \times {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y_{\infty}], M_n({\Bbb C}))$. ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n),{\Bbb P}^1) ={\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\Bbb C}),{\Bbb P}^1)/\!\sim\,$ is now given by the orbit-space of the ${\mbox{\it GL}}_n({\Bbb C})$-action on the above locus in ${\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y_0], M_n({\Bbb C})) \times {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[y_{\infty}], M_n({\Bbb C}))$. For ${\cal R}= (\varphi_{(e_{(0)},m_{(0)})},\varphi_{(e_{(\infty)},m_{(\infty)})}) \in {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n),{\Bbb P}^1)$, the Chan-Paton module on each local chart $U$, where $U=U_0$ or $U_{\infty}$, is given by the $(e,m)$-module $e\cdot{\Bbb C}$ but now regarded as a ${\Bbb C}[y]$-module $_{{\scriptsizeBbb C}[y]}(e\cdot{\Bbb C})$ via $\varphi_{(e,m)}$. We will denote this ${\cal O}_U$-module on $U$ by $\hat{\varphi}_{(e,m)\ast}(e\cdot{\Bbb C}^n)$. It is supported on the image scheme ${\mbox{\it Im}\,}\hat{\varphi}$ on $U$ associated to the ideal ${\mbox{\it Ker}\,}\varphi_{(e,m)}$ in ${\Bbb C}[y]$. Here, $(e,m)=(e_{(0)},m_{(0)})$ or $(e_{(\infty)},m_{(\infty)})$ respectively and ${\Bbb C}[y]={\Bbb C}[y_0]$ or ${\Bbb C}[y_{\infty}]$ respectively. Except that $e\cdot{\Bbb C}^n$ now replaces ${\Bbb C}^n$, all the local details of $\hat{\varphi}_{(e,m)\ast}(e\cdot{\Bbb C}^n)$ are the same as those in the case $Y={\Bbb A}^1$. The total length of $\hat{\varphi}_{(e,m)\ast}(e\cdot{\Bbb C}^n)$ is ${\mbox{\it dim}\,}_{\scriptsizeBbb C}(e\cdot{\Bbb C}^n)$, ($=d$ for $e\sim {\mathbf 1}_d$). The pair $\{ {\mbox{\it Im}\,}\hat{\varphi}_{(e_{(0)},m_{(0)})}\,,\, {\mbox{\it Im}\,}\hat{\varphi}_{(e_{(\infty)},m_{(\infty)})} \}$ of local image schemes glue to a $0$-dimensional subscheme, denoted ${\mbox{\it Im}\,}\hat{\varphi}_{\cal R}$ or $\hat{\varphi}_{\cal R}({\mbox{\it Space}\,}M_n({\Bbb C}))$, of length $\le n$ on ${\Bbb P}^1$. Idempotency of $e_{\bullet}$ and Conditions (1) and (2) imply that $\{ \hat{\varphi}_{(e_{(0)},m_{(0)})\ast}(e_{(0)}\cdot{\Bbb C}^n)\,,\, \hat{\varphi}_{(e_{(\infty)},m_{(\infty)})\ast} (e_{(\infty)}\cdot{\Bbb C}^n) \}$ glues to a (torsion) ${\cal O}_{{\scriptsizeBbb P}^1}$-module on ${\Bbb P}^1$. This is the push-forward $\hat{\phi}_{{\cal R}\ast}{\Bbb C}^n$ of ${\Bbb C}^n$ on ${\mbox{\it Space}\,}M_n({\Bbb C})$ to ${\Bbb P}^1$ under $\hat{\phi}_{\cal R}$; cf. footnote 32. It is the Chan-Paton module of the D0-branes $\hat{\varphi}({\mbox{\it Space}\,}M_n({\Bbb C}))$ on ${\Bbb P}^1$ in the current setting. Note that the total length of $\hat{\phi}_{{\cal R}\,\ast}{\Bbb C}^n$ on ${\Bbb P}^1$ remains $n$. The Higgsing/un-Higgsing behavior of Chan-Paton modules of D0-branes on any target $Y$ is a local issue and hence, for $Y={\Bbb P}^1$, is the same as that for $Y={\Bbb A}^1$ in Sec. 4.1. The local discussions in Sec. 4.1 can be glued to global statements. In particular, [**Proposition 4.2.2 \[D0-branes on ${\Bbb P}^1$\].**]{} [ *There is an embedding $\Phi_{{\mbox{\scriptsize\it Hilb}}}: {\mbox{\it Hilb}\,}^n_{{\scriptsizeBbb P}^1}=:({\Bbb P}^1)^{[n]} \rightarrow {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n),{\Bbb P}^1)$, whose image is characterized by $\varphi_{\cal R}$ such that ${\mbox{\it Im}\,}\hat{\varphi}_{\cal R}$ is a subscheme of length $n$ on ${\Bbb P}^1$. There is an embedding $\Phi_{{\mbox{\scriptsize\it Chow}}}: S^n({\Bbb P}^1) \rightarrow {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n),{\Bbb P}^1)$, whose image is characterized by $\varphi_{\cal R}$ such that ${\mbox{\it Im}\,}\hat{\varphi}_{\cal R}$ is a reduced subscheme (of length $\le n$) on ${\Bbb P}^1$. There is a map ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n),{\Bbb P}^1) \rightarrow S^n({\Bbb P}^1)$ that has $\Phi_{{\mbox{\scriptsize\it Chow}}}$ as a section. The pattern of open-string-induced Higgsing/un-Higgsing behavior of $n$ D0-branes on ${\Bbb P}^1$ can be reproduced in the current content via deformations of morphisms $[\varphi_{\cal R}]$ in $\Phi_{{\mbox{\scriptsize\it Chow}}}(S^n({\Bbb P}^1)) \subset {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C}); {\Bbb C}^n),{\Bbb P}^1)$.* ]{} [*Remark 4.2.3 $[$strict morphism$]$.*]{} A strict morphism from ${\mbox{\it Space}\,}M_n({\Bbb C})$ to ${\Bbb P}^1$ is given by a strict morphism (cf. Definition 1.2.11 and Definition 1.1.1) from $[( \{ {\Bbb C}[y_0], {\Bbb C}[y_{\infty}] \} {\begin{array}{c} \rightarrow \\[-2ex] \rightarrow \end{array}}\{ {\Bbb C}[y, 1/y]\} )]$ to $[\{M_n({\Bbb C})\}]$. Since $Z(M_n({\Bbb C}))={\Bbb C}$, such a morphism factors as $$[( \{ {\Bbb C}[y_0], {\Bbb C}[y_{\infty}] \} {\begin{array}{c} \rightarrow \\[-2ex] \rightarrow \end{array}}\{ {\Bbb C}[y, 1/y]\} )]\; \longrightarrow\; [\{{\Bbb C}\}]\; \longrightarrow\; [\{M_n({\Bbb C})\}]$$ and, hence, corresponds to a morphism ${\mbox{\it Spec}\,}{\Bbb C}\rightarrow {\Bbb P}^1$. The corresponding D0-brane on ${\Bbb P}^1$ is supported at a reduced ${\Bbb C}$-point on ${\Bbb P}^1$ with the Chan-Paton module ${\Bbb C}^n$, i.e. $n$-many coincident D0-branes on ${\Bbb P}^1$ in the picture of Polchinski. The moduli space of such morphisms (i.e. coincident D0-branes) is ${\Bbb P}^1$. Thus, we see that the inclusion of general morphisms (cf. Definition 1.2.14 and Definition 1.1.1) in the definition of ${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\Bbb C}), {\Bbb P}^1)$ and, hence, in the definition of ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n),{\Bbb P}^1)$ is also required if one wants to incorporate the Higgsing/un-Higgsing behavior of, in this case, D0-branes on ${\Bbb P}^1$. Similar phenomenon occurs for other projective target spaces as well. This is another incident of the mysterious harmony between stringy requirement and mathematical naturality for a string-theory-related mathematical object. D0-branes on the complex affine plane ${\Bbb A}^2$. --------------------------------------------------- For a commutative $Y$ of dimension $\ge 2$, an additional ingredient than those in Sec. 4.1 and Sec. 4.2 is commuting schemes/varieties[^41]. We discuss in this subsection the case $Y={\Bbb A}^2$, for which the commuting variety that occurs is known slightly better. Let $Y={\Bbb A}^2={\mbox{\it Spec}\,}{\Bbb C}[y_1,y_2]$ be the affine plane over ${\Bbb C}$. Then ${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\Bbb C}), Y) ={\mbox{\it Mor}\,}({\Bbb C}[y_1,y_2], M_n({\Bbb C}))$ is the variety[^42] what parameterizes the elements in the set $$C_2M_n({\Bbb C})\; :=\; \{(m_1,m_2)\in M_n({\Bbb C})\times M_n({\Bbb C})\,:\, m_1m_2=m_2m_1\}$$ of pairs of commuting matrices in $M_n({\Bbb C})$. This variety is identical with ${\Bbb C}$-algebra representation variety ${\mbox{\it Rep}\,}({\Bbb C}[y_1,y_2], M_n({\Bbb C}))$ with a point represented by $(m_1,m_2) \in M_n({\Bbb C})\times M_n({\Bbb C})$ corresponding to the ${\Bbb C}$-algebra-homomorphism $$\varphi_{(m_1,m_2)}: {\Bbb C}[y_1,y_2]\rightarrow M_n({\Bbb C})\,, \hspace{1ex} \mbox{generated by $1\mapsto{\mathbf 1}$, $y_1\mapsto m_1$, and $y_2\mapsto m_2$}\,.$$ [**Proposition 4.3.1 \[irreducibility\].**]{} (\[Ge\], \[Bas\], and \[Vac2\].) [ *${\mbox{\it Rep}\,}({\Bbb C}[y_1,y_2], M_n({\Bbb C}))$ is an irreducible variety of dimension $n^2+n$ in ${\Bbb A}^{n^2}\times {\Bbb A}^{n^2}$. The ${\mbox{\it GL}}_n({\Bbb C})$-action on ${\mbox{\it Rep}\,}({\Bbb C}[y_1,y_2], M_n({\Bbb C}))$ has stabilizer subgroups of minimal dimension $n$. A generic ${\mbox{\it GL}}_n({\Bbb C})$-orbit thus has dimension $n^2-n$, that achieves the maximum orbit-dimension and the subset that consists of $\varphi_{(m_1,m_2)}$, where $(m_1,m_2)$ is a diagonalizable commuting pair with both $m_1$ and $m_2$ having distinct characteristic values, is a smooth open dense subset in ${\mbox{\it Rep}\,}({\Bbb C}[y_1,y_2], M_n({\Bbb C}))$.* ]{} It follows that $${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n), {\Bbb A}^2)\; \simeq\; {\mbox{\it Map}\,}({\mbox{\it Space}\,}M_n({\Bbb C}), {\Bbb A}^2)\; =\; {\mbox{\it Rep}\,}({\Bbb C}[y_1,y_2],M_n({\Bbb C}))/\!\sim\,,$$ the orbit-space of the ${\mbox{\it GL}}_n({\Bbb C})$-action with the quotient topology, is a connected non-Hausdorff topological space that contains a connected smooth open dense Hausdorff subset of dimension $2n$, namely the subset of $S^n({\Bbb A}^2)$ that consists of $[(\lambda_1,\mu_1),\,\cdots\,,(\lambda_n,\mu_n)]$ such that $\lambda_i$, $i=1,\,\ldots\,,n$, are all distinct from each other and so are $\mu_i$, $i=1,\,\ldots\,,n$. Here $S^n({\Bbb A}^2) := ({\Bbb A}^2)^{n}/Sym_n$ is the $n$-th symmetric product of ${\Bbb A}^2$. The complete set of dominance relations of the ${\mbox{\it GL}}_n({\Bbb C})$-orbits in ${\mbox{\it Rep}\,}({\Bbb C}[y_1,y_2], M_n({\Bbb C}))$, which generalizes \[Ge\], are not known. However, there are two distinguished Hausdorff subspace in ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n), {\Bbb A}^2)$ that can be understood through the work of Nakajima \[Na\] and of Vaccarino \[Vac2\]: - the naturally embedded image of the Hilbert scheme $({\Bbb A}^2)^{[n]}:= {\mbox{\it Hilb}\,}^n_{{\scriptsizeBbb A}^2}$ (with the reduced scheme structure) of $0$-dimensional subschemes of length $n$ on ${\Bbb A}^2$; - the naturally embedded image of the Chow variety ${\mbox{\it Chow}\,}_{{0, \scriptsizeBbb A}^2}^{(n)} = S^n({\Bbb A}^2)$ of $0$-cycles of order $n$ on ${\Bbb A}^2$. We now explain the details. [**Proposition 4.3.2 \[regular representation\].**]{} [ *Let $R$ be a commutative Artinian algebra over ${\Bbb C}$ of dimension $n$. Then, the regular representation[^43] of $R$ realizes $R$ as a maximal commutative subalgebra $R^{\prime}$ of $M_n({\Bbb C})$. Furthermore, as an $R^{\prime}$-module, $_{R^{\prime}}{\Bbb C}^n\simeq R^{\prime}$.* ]{} [*Proof.*]{} This is an immediate corollary of \[S-T: Sec.2.7, Theorem 11\]. When $R$ is generated by two commuting elements and the identity, as is in our case, there are two other independent proofs: (1) The first part of the proof of \[Na: Sec. 1.2, Theorem 1.9\] can be adapted directly to give another more analytic proof of the statement, cf. proof of Proposition 4.3.3 below. (2) This is a corollary of \[Ge\], which says that the maximum dimension of a commutative subalgebra in $M_n({\Bbb C})$ generated by two commuting matrices and the identity is $n$. $\Box$ Note that, in the above statement, different choices of $R\simeq {\Bbb C}^n$ as ${\Bbb C}$-vector spaces give rise to $R^{\prime}$’s in the same adjoint ${\mbox{\it GL}}_n({\Bbb C})$-orbit. It follows that there is an embedding of sets $$\Phi_{{\mbox{\scriptsize\it Hilb}}}: ({\Bbb A}^2)^{[n]}\; \longrightarrow\; {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n), {\Bbb A}^2)\,, \hspace{1em} {\Bbb C}[y_1,y_2]/I\; \longmapsto\; \varphi_{(m_1,m_2)}\,.$$ Here, $I$ is an ideal of ${\Bbb C}[y_1,y_2]$ so that ${\mbox{\it dim}\,}_{\scriptsizeBbb C}({\Bbb C}[y_1,y_2]/I)=n$; it gives then the subalgebra $({\Bbb C}[y_1,y_2]/I)^{\prime}\subset M_n({\Bbb C})$ as in Proposition 4.3.2, unique up conjugation; the corresponding matrix $m_i$ for $y_i$, $i=1,\,2$. under the built-in ${\Bbb C}$-algebra-isomorphism ${\Bbb C}[y_1,y_2]/I \stackrel{\sim}{\rightarrow} ({\Bbb C}[y_1,y_2]/I)^{\prime}$ determines then $\varphi_{(m_1,m_2)}$. [**Proposition 4.3.3 \[stable subset\].**]{} (Cf. \[Na: Theorem 1.9\].) [ *Let ${\mbox{\it Rep}\,}({\Bbb C}[y_1,y_2], M_n({\Bbb C}))^{{\mbox{\scriptsize\it st}}}$ be the subset of ${\mbox{\it Rep}\,}({\Bbb C}[y_1,y_2], M_n({\Bbb C}))$ that consists of $\varphi_{(m_1,m_2)}$ such that $_{\langle{\mathbf 1}, m_1, m_2\rangle}{\Bbb C}^n \simeq \langle{\mathbf 1}, m_1, m_2\rangle$ as $\langle{\mathbf 1}, m_1, m_2\rangle$-modules. Then ${\mbox{\it Rep}\,}({\Bbb C}[y_1,y_2], M_n({\Bbb C}))^{{\mbox{\scriptsize\it st}}}$ is smooth and ${\mbox{\it GL}}_n({\Bbb C})$-invariant with stabilizers all of the same dimension $n$.* ]{} [*Proof.*]{} This is actually \[Na: Theorem 1.9\] in disguise. Note that the stability condition in the defining condition of the set $\tilde{H}$ in ibidem is precisely the condition “$_{\langle{\mathbf 1}, m_1, m_2\rangle}{\Bbb C}^n \simeq \langle{\mathbf 1}, m_1, m_2\rangle$ as $\langle{\mathbf 1}, m_1, m_2\rangle$-modules" in the statement here. Having said so, let us give a sketch of the proof in terms of the current setting. Using the trace map $M_n({\Bbb C})\rightarrow {\Bbb C}$ as a complex bilinear inner product on the ${\Bbb C}$-vector space $M_n({\Bbb C})$, one can show that the (analytic quadric) commutator map (on analytic spaces) $$c\;:\; M_n({\Bbb C})\times M_n({\Bbb C})\;\longrightarrow\; M_n({\Bbb C})\,, \hspace{1em} (m_1,m_2)\;\longmapsto\; [m_1,m_2]:= m_1m_2-m_2m_1$$ has cokernel ${\mbox{\it coker}\:}dc_{(m_1,m_2)}$ at $(m_1,m_2)$ being $\{\xi\in M_n({\Bbb C}):[\xi,m_1]=[\xi,m_2]=0\}$, i.e. the centralizer ${\mbox{\it Centralizer}\,}\langle{\mathbf 1}, m_1, m_2\rangle$ of the subalgebra $\langle{\mathbf 1}, m_1, m_2\rangle$ in the algebra $M_n({\Bbb C})$. Note that for $(m_1,m_2)\in C_2M_n({\Bbb C})$, $\langle{\mathbf 1}, m_1, m_2\rangle \subset {\mbox{\it Centralizer}\,}\langle{\mathbf 1}, m_1, m_2\rangle$. If, furthermore, $\varphi_{(m_1,m_2)} \in {\mbox{\it Rep}\,}({\Bbb C}[y_1,y_2], M_n({\Bbb C}))^{{\mbox{\scriptsize\it st}}}$, then $_{\langle{\mathbf 1}, m_1, m_2\rangle}{\Bbb C}^n = \langle{\mathbf 1}, m_1, m_2\rangle\cdot v_0$ for some $v_0\in {\Bbb C}^n$. The ${\Bbb C}$-linear map ${\mbox{\it Centralizer}\,}\langle{\mathbf 1}, m_1, m_2\rangle\rightarrow {\Bbb C}^n$, defined by $\xi\mapsto \xi\cdot v_0$, is then invertible and hence a ${\Bbb C}$-vector-space-isomorphism. It follows that $\langle{\mathbf 1}, m_1, m_2\rangle = {\mbox{\it Centralizer}\,}\langle{\mathbf 1}, m_1, m_2\rangle$ and $\dim_{\scriptsizeBbb C}{\mbox{\it coker}\:}dc_{(m_1,m_2)}=n$ for $\varphi_{(m_1,m_2)} \in {\mbox{\it Rep}\,}({\Bbb C}[y_1,y_2], M_n({\Bbb C}))^{{\mbox{\scriptsize\it st}}}$. This shows that ${\mbox{\it Rep}\,}({\Bbb C}[y_1,y_2], M_n({\Bbb C}))^{{\mbox{\scriptsize\it st}}}$ is smooth. Finally, note that ${\mbox{\it Stab}\,}(\varphi_{(m_1,m_2)}) ={\mbox{\it GL}}_n({\Bbb C})\cap{\mbox{\it Centralizer}\,}\langle{\mathbf 1}, m_1, m_2\rangle$, which has the same dimension as ${\mbox{\it Centralizer}\,}\langle{\mathbf 1}, m_1, m_2\rangle$. The proposition follows. $\Box$ Since the closure of $\overline{O}$ of a $G$-orbit $O$ of an action of a reductive algebraic group $G$ on an affine variety $V$ (both over ${\Bbb C}$) is a union of $O$ with $G$-orbits of strictly smaller dimension, one has: [**Corollary 4.3.4 \[good quotient\].**]{} [ *All the ${\mbox{\it GL}}_n({\Bbb C})$-orbits are closed in ${\mbox{\it Rep}\,}({\Bbb C}[y_1,y_2], M_n({\Bbb C}))^{{\mbox{\scriptsize\it st}}}$ and the map ${\mbox{\it Rep}\,}({\Bbb C}[y_1,y_2], M_n({\Bbb C}))^{{\mbox{\scriptsize\it st}}} \rightarrow {\mbox{\it Im}\,}\Phi_{{\mbox{\scriptsize\it Hilb}}}$ to the orbit-space is a good quotient.* ]{} This realizes the map $\Phi_{{\mbox{\scriptsize\it Hilb}}}: ({\Bbb A}^2)^{[n]} \rightarrow {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n), {\Bbb A}^2)$ as an embedding of the (reduced) Hilbert scheme as a variety/analytic space. Let ${\Bbb C}^n$ parameterizes the diagonal matrices in $M_n({\Bbb C})$. Then, the embedding $$\begin{array}{ccc} {\Bbb C}^n\times {\Bbb C^n}=({\Bbb C}^2)^n & \hookrightarrow & M_n({\Bbb C})\times M_n({\Bbb C}) \\[.6ex] ((\lambda_1,\mu_1),\,\cdots\,,\,(\lambda_n,\mu_n)) & \mapsto & ({\mbox{\it Diag}\,}(\lambda_1,\,\cdots\,,\,\lambda_n)\,,\, {\mbox{\it Diag}\,}(\mu_1,\,\cdots\,,\mu_n)) \end{array}$$ descends to an embedding $$\begin{array}{ccccc} \Phi_{{\mbox{\scriptsize\it Chow}}} & : & S^n({\Bbb A}^2) & \longrightarrow & {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n), {\Bbb A}^2) \\[.6ex] && [(\lambda_1,\mu_1),\,\cdots\,,(\lambda_n,\mu_n)] & \longmapsto\; & \varphi_{( {\mbox{\scriptsize\it Diag}\,}(\lambda_1,\,\cdots\,,\,\lambda_n)\,,\, {\mbox{\scriptsize\it Diag}\,}(\mu_1,\,\cdots\,,\mu_n) )} \end{array}$$ of the Chow variety. $S^n({\Bbb A}^2)$ is the categorical quotient of ${\mbox{\it Rep}\,}({\Bbb C}[y_1,y_2], M_n({\Bbb C}))$ under the adjoint ${\mbox{\it GL}}_n({\Bbb C})$-action. The affine morphism ${\mbox{\it Rep}\,}({\Bbb C}[y_1,y_2], M_n({\Bbb C}))\rightarrow S^n({\Bbb A}^2)$ induced by the ${\mbox{\it GL}}_n({\Bbb C})$-invariant function ring on ${\mbox{\it Rep}\,}({\Bbb C}[y_1,y_2], M_n({\Bbb C}))$ descends to a morphism ${\mbox{\it Im}\,}\Phi_{{\mbox{\scriptsize\it Hilb}}}\rightarrow {\mbox{\it Im}\,}\Phi_{{\mbox{\scriptsize\it Chow}}}$ of varieties that realizes $({\Bbb A}^2)^{[n]}$ as a desingularization of $S^n({\Bbb A}^2)$. ${\mbox{\it Im}\,}\Phi_{{\mbox{\scriptsize\it Chow}}}$ consists of all the closed points in ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n), {\Bbb A}^2)$ and the closure of any point in ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n), {\Bbb A}^2)$ contains a unique point in ${\mbox{\it Im}\,}\Phi_{{\mbox{\scriptsize\it Chow}}}$. Cf. \[Na\], \[Pro\], \[Ri\], and \[Vac2\]. Note that, for $(m_1,m_2)\in C_2M_n({\Bbb C})$, as $m_1$ and $m_2$ commute, they can be simultaneously triangularized. If they have a simultaneous triangularization with the diagonal entries $(\lambda_1,\,\cdots\,,\lambda_n)$ and $(\mu_1,\,\cdots\,,\mu_n)$ respectively, let $I_{\{(\lambda_1,\mu_1),\,\cdots\,,(\lambda_n,\mu_n)\}}\; :=\; (y_1-\lambda_1, y_2-\mu_1) \cap\,\cdots\,\cap (y_1-\lambda_n,y_2-\mu_n)$ be the ideal in ${\Bbb C}[y_1,y_2]$ for the set of closed points $\{(\lambda_1,\mu_1),\,\cdots\,, (\lambda_n, \mu_n)\}$ (as points on the analytic space ${\Bbb C}^2$ with repeated points dropped). Then, $$I_{\{(\lambda_1,\mu_1),\,\cdots\,,(\lambda_n,\mu_n)\}}^{\;n}\; \subset\; {\mbox{\it Ker}\,}\varphi_{(m_1,m_2)}\; \subset\; I_{\{(\lambda_1,\mu_1),\,\cdots\,,(\lambda_n,\mu_n)\}}\,.$$ In particular, the image scheme ${\mbox{\it Im}\,}\hat{\varphi}_{(m_1,m_2)}\simeq {\mbox{\it Spec}\,}({\Bbb C}[y_1,y_2]/{\mbox{\it Ker}\,}\varphi_{(m_1,m_2)})$ on ${\Bbb A}^2$ has the reduced scheme structure exactly the set $\{(\lambda_1,\mu_1),\,\cdots\,, (\lambda_n, \mu_n)\}$ above. For $\varphi_{(m_1,m_2)}\in{\mbox{\it Im}\,}\Phi_{{\mbox{\scriptsize\it Hilb}}}$, the Chan-Paton module $\hat{\varphi}_{(m_1,m_2)\ast}{\Bbb C}^n$, as a ${\cal O}_{{\mbox{\scriptsize\it Im}\,}\hat{\varphi}_{(m_1,m_2)}}$-module, is isomorphic to the structure sheaf ${\cal O}_{{\mbox{\scriptsize\it Im}\,}\hat{\varphi}_{(m_1,m_2)}}$ while for $\varphi_{(m_1,m_2)}\in{\mbox{\it Im}\,}\Phi_{{\mbox{\scriptsize\it Chow}}}$, the Chan-Paton module $\hat{\varphi}_{(m_1,m_2)\ast}{\Bbb C}^n$, as a ${\cal O}_{{\mbox{\scriptsize\it Im}\,}\hat{\varphi}_{(m_1,m_2)}}$-module, is isomorphic to $\oplus_{i=1}^n{\cal O}_{(\lambda_i,\mu_i)}$. Here, $(\lambda_i,\mu_i)$, $i=1,\,\ldots\,,n$, are the image point from earlier notations with repeated $(\lambda_i,\mu_i)$ kept to contribute to the direct sum. Behavior of Higgsing/un-Higgsing follows similar pattern as in Sec. 4.1. D0-branes on a complex quasi-projective variety. ------------------------------------------------ A picture of D0-branes on a (commutative) complex quasi-projective variety that follows from a combination and an immediate generalization of Sec. 4.1 - Sec. 4.3 is given in this subsection. A comparison with gas of D0-branes in \[Vafa1\] of Vafa is given in the end. [**D0-branes on ${\Bbb P}^r$.**]{} Let $Y$ be the projective space over ${\Bbb C}$: $$Y\;=\; {\Bbb P}^r\; =\; {\mbox{\it Proj}\,}{\Bbb C}[y_0,y_1,\,\cdots\,, y_r]\; =\; \cup_{i=0}^r\, U_i\; =\; \cup_{i=0}^r\, {\mbox{\it Spec}\,}{\Bbb C}[\mbox{$\frac{y_0}{y_i}$}\,,\,\cdots\,,\, \mbox{$\frac{y_r}{y_i}$}]\,.$$ Here $y_{\bullet}/y_i$ are treated as formal variables with $y_i/y_i =$ the identity $1$ of the ring ${\Bbb C}[\mbox{$\frac{y_0}{y_i}$}\,,\,\cdots\,,\, \mbox{$\frac{y_r}{y_i}$}]$; the gluings $U_i \supset U_{ij}:= U_i\cap U_j\stackrel{\sim}{\leftarrow} U_{ji}:= U_j\cap U_i \subset U_j$ of local affine charts are given by $$\begin{array}{cccccccl} {\Bbb C}[\frac{y_0}{y_i}\,,\,\cdots\,,\,\frac{y_r}{y_i}] & \hookrightarrow & \frac{{\smallBbb C}[\frac{y_0}{y_i}\,,\,\cdots\,,\, \frac{y_r}{y_i}\,,\,\frac{y_i}{y_j}]} {\left( \frac{y_j}{y_i}\cdot\frac{y_i}{y_j}-1 \right)} & \stackrel{\sim}{\longrightarrow} & \frac{{\smallBbb C}[\frac{y_0}{y_j}\,,\,\cdots\,,\, \frac{y_r}{y_j}\,,\,\frac{y_j}{y_i}]} {\left( \frac{y_i}{y_j}\cdot\frac{y_j}{y_i}-1 \right)} & \hookleftarrow & {\Bbb C}[\frac{y_0}{y_j}\,,\,\cdots\,,\,\frac{y_n}{y_j}] \\[3ex] && \frac{y_{\bullet}}{y_i} & \longmapsto & \frac{y_{\bullet}}{y_j}\cdot\frac{y_j}{y_i} \\[1ex] && \frac{y_i}{y_j} & \longmapsto & \frac{y_i}{y_j} &&&. \end{array}$$ Let $$C_{r+1}M_n({\Bbb C})\; :=\; \{ (m_0,\,\cdots\,,m_r)\in M_n({\Bbb C})^{r+1}\,:\, m_im_j=m_jm_i\,, i,j = 0,\,\ldots\,, r \}\,.$$ The ring-set representation variety $$\begin{aligned} \lefteqn{ {\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}( {\Bbb C}[\mbox{$\frac{y_0}{y_i}$}\,,\,\cdots\,,\, \mbox{$\frac{y_n}{y_i}$}], M_n({\Bbb C})) }\\[.6ex] && =\; \{(m_{(i),0},\,\cdots\,,m_{(i),r}) \in C_{r+1}M_n({\Bbb C})\,:\, m_{(i),i}m_{(i),i^{\prime}}=m_{(i),i^{\prime}}m_{(i),i} =m_{(i),i^{\prime}},\, i^{\prime}=0,\,\ldots\,,r \} \\[.6ex] && \subset\; \prod_{r+1}{\Bbb A}^{n^2}\;=\; {\Bbb A}^{n^2(r+1)}\,, \end{aligned}$$ (in particular, $e_{(i)} := m_{(i),i}$ is an idempotent), is a disjoint union of $$\begin{aligned} \lefteqn{ {\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}( {\Bbb C}[\mbox{$\frac{y_0}{y_i}$}\,,\,\cdots\,,\, \mbox{$\frac{y_n}{y_i}$}],M_n({\Bbb C}) )_{(d)} }\\[.6ex] && :=\; \{ (m_{(i),\bullet})_{\bullet} \in {\mbox{\it Rep}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}( {\Bbb C}[\mbox{$\frac{y_0}{y_i}$}\,,\,\cdots\,,\, \mbox{$\frac{y_n}{y_i}$}], M_n({\Bbb C}) )\;:\; m_{(i),i}\sim {\mathbf 1}_d \}\,, \hspace{1em} d\,=\,0,\,\ldots\,,n\,. \end{aligned}$$ Here, again, we identify the ring-set-homomorphism $\varphi_{(m_{(i),0},\,\cdots\,,m_{(i),r})}: {\Bbb C}[\frac{y_0}{y_i}\,,\,\cdots\,,\,\frac{y_r}{y_i}] \rightarrow M_n({\Bbb C})$ that sends $y_{\bullet}/y_i$ to $m_{(i),\bullet}$ with $(m_{(i),0},\,\cdots\,,m_{(i),r})\in C_{r+1}M_n({\Bbb C})$. Similar to the case $Y={\Bbb P}^1$ in Sec. 4.2, the space ${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\Bbb C}),{\Bbb P}^r)$ of morphisms from ${\mbox{\it Space}\,}M_n({\Bbb C})$ to ${\Bbb P}^r$ is given by the locus on $\prod_{i=0}^r\, {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[\frac{y_0}{y_i}\,,\, \cdots\,,\,\frac{y_r}{y_i}], M_n({\Bbb C}))$ described by the following conditions:[^44] - $( \varphi_{(m_{(0),\bullet})_{\bullet}}\,,\, \cdots\,,\,\varphi_{(m_{(r),\bullet})_{\bullet}} ) \in \prod_{i=0}^r\, {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[\frac{y_0}{y_i}\,,\, \cdots\,,\,\frac{y_r}{y_i}], M_n({\Bbb C}))$, - $m_{(i), i}m_{(j),j}=m_{(j),j}m_{(i),i}\,$,\ ${\mathbf 1} =\sum_i m_{(i),i} -\sum_{i_1<i_2} m_{(i_1),i_1}m_{(i_2),i_2} + \,\cdots\,\\[.6ex] \mbox{\hspace{6em}} + (-1)^{k-1}\sum_{i_1<\,\cdots\,<i_k} m_{(i_1),i_1}\,\cdots\,m_{(i_k),i_k} +\,\cdots\,+(-1)^r m_{(0),0}\,\cdots\,m_{(r),r}$; - $m_{(i),i}m_{(j),\bullet}=m_{(j),\bullet}m_{(i),i}$, $\,i,j, \bullet =0,\,\ldots\,,r$; - $( m_{(j),j} m_{(i),j} )\,( m_{(j),j} m_{(j),i})\, =\, m_{(i),i} m_{(j),j}\,$, $\,i,j = 0,\,\ldots\,,r$; cf. Lemma 4.2.1; - $m_{(j),j} m_{(i),\bullet}\, =\, m_{(j),\bullet} \cdot ( m_{(j),j} m_{(i),j} )\,$ $\,i,j, \bullet =0,\,\ldots\,,r$; cf. the gluing $U_{ij}\stackrel{\sim}{\leftarrow}U_{ji}$. ${\mbox{\it GL}}_n({\Bbb C})$ acts diagonally on $\prod_{i=0}^r\, {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[\frac{y_0}{y_i}\,,\, \cdots\,,\,\frac{y_r}{y_i}], M_n({\Bbb C}))$, via the post-composition with the adjoint ${\mbox{\it GL}}_n({\Bbb C})$-action on $M_n({\Bbb C})$, and the above system of conditions describes a ${\mbox{\it GL}}_n({\Bbb C})$-invariant closed subset therein. The space of D0-branes on ${\Bbb P}^r$ is given by\ ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n),{\Bbb P}^r) ={\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\Bbb C}),{\Bbb P}^r)/\!\sim\,$, described by the orbit-space of the ${\mbox{\it GL}}_n({\Bbb C})$-action on the above subset in $\prod_{i=0}^r\, {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[\frac{y_0}{y_i}\,,\, \cdots\,,\,\frac{y_r}{y_i}], M_n({\Bbb C}))$. The Chan-Paton modules of D0-branes on ${\Bbb P}^r$ and their Higgsing/un-Higgsing behavior follow the reasoning that combines the cases $Y={\Bbb P}^1$ and $Y={\Bbb A}^2$. Together with the simultaneous triangularizability of any family of commuting matrices and the map that takes a tuple of triangularized matrices to the tuple of the respective diagonal, one has: (cf. Proposition 4.2.2) [**Proposition 4.4.1 \[D0-branes on ${\Bbb P}^r$\].**]{} [ *There is an embedding $\Phi_{{\mbox{\scriptsize\it Hilb}}}: {\mbox{\it Hilb}\,}^n_{{\scriptsizeBbb P}^r}=:({\Bbb P}^r)^{[n]} \rightarrow {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n),{\Bbb P}^r)$. $\varphi_{\cal R}\in\Phi_{{\mbox{\scriptsize\it Hilb}}}(({\Bbb P}^r)^{[n]})$ has the property that ${\mbox{\it Im}\,}\hat{\varphi}_{\cal R}$ is a subscheme of length $n$ on ${\Bbb P}^r$. There is an embedding $\Phi_{{\mbox{\scriptsize\it Chow}}}: S^n({\Bbb P}^r) \rightarrow {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n),{\Bbb P}^r)$, whose image is characterized by $\varphi_{\cal R}$ associated to a system of commuting diagonalizable matrices. (In particular, ${\mbox{\it Im}\,}\hat{\varphi}_{\cal R}$ is a reduced subscheme of length $\le n$ on ${\Bbb P}^r$.) There is a map ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n),{\Bbb P}^r) \rightarrow S^n({\Bbb P}^r)$ that has $\Phi_{{\mbox{\scriptsize\it Chow}}}$ as a section. The pattern of open-string-induced Higgsing/un-Higgsing behavior of $n$ D0-branes on ${\Bbb P}^r$ can be reproduced in the current content via deformations of morphisms $[\varphi_{\cal R}]$ in $\Phi_{{\mbox{\scriptsize\it Chow}}}(S^n({\Bbb P}^r)) \subset {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C}); {\Bbb C}^n),{\Bbb P}^r)$.* ]{} [**D0-branes on a quasi-projective variety.**]{} Let $Y$ be a quasi-projective variety and suppose that $Y$ is embedded in ${\Bbb P}^r$ as $Y_1-Y_2$, where both $Y_1$ and $Y_2$ are closed subschemes of ${\Bbb P}^r$. Let $I_1=\langle f_{11},\,\cdots\,,f_{1l_1}\rangle$ (resp. $I_2=\langle f_{21},\,\cdots\,,f_{2l_2}\rangle$) be the homogeneous ideal in ${\Bbb C}[y_0,\,\cdots\,,y_r]$ associated to $Y_1$ (resp. $Y_2$) in ${\Bbb P}^r$. Recall the local affine charts $\cup_{i=0}^rU_i$ of ${\Bbb P}^r$. Consider the (in general only quasi-affine) open cover $\cup_{i=0}^r((Y_1-Y_2)\cap U_i)$ of $Y$. Then, the pair $(I_1, I_2)$ gives rise to a pair $$\left(\, I_{1,(i)}=( f_{11,(i)},\,\cdots\,,f_{1l_1,(i)} )\,,\, I_{2,(i)}=( f_{21,(i)},\,\cdots\,,f_{2l_2,(i)} )\, \right)$$ of ideals in ${\Bbb C}[\frac{y_0}{y_i}\,,\,\cdots\,,\,\frac{y_r}{y_i}]$ via the dehomogenization of $(I_1, I_2)$ on the affine chart $U_i$ of ${\Bbb P}^r$ for $i=0,\,\ldots\,,r$. The space ${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\Bbb C}),Y)$ of morphisms from ${\mbox{\it Space}\,}M_n({\Bbb C})$ to $Y$ is given by further restricting the locus ${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\Bbb C}),{\Bbb P}^r)$ in $\prod_{i=0}^r\, {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[\frac{y_0}{y_i}\,,\, \cdots\,,\,\frac{y_r}{y_i}], M_n({\Bbb C}))$, described by Conditions (1) - (4) in the previous theme, to the following system of incidence relation from $I_1$ and exclusion relations from $I_2$: - \[([*closed*]{}) [*incidence conditions from*]{} $I_1$\]$\,$:\ $\mbox{\hspace{1em}} f_{1\bullet, (i)}(m_{(i),0}\,,\,\cdots\,,\,m_{(i),r})\,=\,0\, \in M_n({\Bbb C})\,$, $\,\bullet=1,\,\ldots\,, l_1\,$, $\,i=0,\,\ldots,\,r\,$; - \[([*open*]{}) [*exclusion conditions from*]{} $I_2$\]$\,$:\ $\mbox{\hspace{1em}} m_{(i),i}\;\in\; \left\langle\, f_{2\bullet, (i)}(m_{(i),0}\,,\,\cdots\,,\,m_{(i),r})\, \right\rangle_{\bullet=1}^{l_2}\; \subset\; M_n({\Bbb C})\,$, $\,i=0,\,\ldots,\,r\,$. The diagonal ${\mbox{\it GL}}_n({\Bbb C})$-action on $\prod_{i=0}^r\, {\mbox{\it Mor}\,}^{{\mbox{\scriptsize\rm ring-set}\,}}({\Bbb C}[\frac{y_0}{y_i}\,,\, \cdots\,,\,\frac{y_r}{y_i}], M_n({\Bbb C}))$ leaves the locally-closed subset that satisfies Conditions (1) - (6) invariant. The space of D0-branes on $Y$ is given then by ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n),Y) ={\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\Bbb C}),Y)/\!\sim\,$, described by the orbit-space of the ${\mbox{\it GL}}_n({\Bbb C})$-action on the above locally-closed subset in ${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\Bbb C}),{\Bbb P}^r)$. [*Remark 4.4.2 $[$Independence of embedding$]$.*]{} [ The open cover $\cup_{i=0}^r((Y_1-Y_2)\cap U_i)$ of $Y$ can be refined to an affine open cover of $Y$, which realizes $Y$ as a gluing system of rings. Different embeddings of $Y$ in projective spaces realizes $Y$ as different gluing systems of rings that have a common refinement. It follows then from Sec. 1.2 that ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n), Y)$ thus constructed is independent of the embedding of $Y$ in a projective space. ]{} Proposition 4.4.1 implies then: [**Theorem 4.4.3 \[D0-branes on quasi-projective variety\].**]{} [ *Let $Y$ be a quasi-projective variety over ${\Bbb C}$. $\;(1)$ There is an embedding $\Phi_{{\mbox{\scriptsize\it Hilb}}}: {\mbox{\it Hilb}\,}^n_Y=:Y^{[n]} \rightarrow {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n),Y)$. $\varphi_{\cal R}\in\Phi_{{\mbox{\scriptsize\it Hilb}}}(Y^{[n]})$ has the property that ${\mbox{\it Im}\,}\hat{\varphi}_{\cal R}$ is a subscheme of length $n$ on $Y$. $\;(2)$ There is an embedding $\Phi_{{\mbox{\scriptsize\it Chow}}}: S^nY\rightarrow {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n),Y)$, whose image is characterized by $\varphi_{\cal R}$ associated to a system of commuting diagonalizable matrices. (In particular, ${\mbox{\it Im}\,}\hat{\varphi}_{\cal R}$ is a reduced subscheme of length $\le n$ on $Y$.) $\;(3)$ There is a map ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n),Y)\rightarrow S^nY$ that has $\Phi_{{\mbox{\scriptsize\it Chow}}}$ as a section. $\;(4)$ The pattern of open-string-induced Higgsing/un-Higgsing behavior of $n$ D0-branes on $Y$ can be reproduced in the current content via deformations of morphisms $[\varphi_{\cal R}]$ in $\Phi_{{\mbox{\scriptsize\it Chow}}}(S^nY) \subset {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C}); {\Bbb C}^n),Y)$.* ]{} [*Remark 4.4.4 $[$toric variety$]$.*]{} [ The discussions for D0-branes on ${\Bbb P}^r$ (resp. a quasi-projective variety) generalize immediately to D0-branes on a toric variety (resp. a subscheme of a toric variety). ]{} [**D0-branes, gauged matrix models, and quantum moduli spaces.**]{} When $Y$ is a closed subvariety of a toric variety/${\Bbb C}$, the space ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C}); {\Bbb C}^n),Y)$ are described by a system of noncommutative-polynomial-like algebraic equations that give only closed conditions. In this case, ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C}); {\Bbb C}^n),Y)$ is realizable as the classical moduli space of vacua (also known as vacuum manifold/variety) of a gauged matrix model. The construction is similar to that of \[Wi1\] but adjusted to $d=0+1$ matrix models. See also the discussions in \[D-G-M\], \[Do-M\], and \[G-L-R\] for related situations and \[L-Y5\] for further discussions. The real issue, particularly from the mathematical/geometric aspect, is whether there is or needs to be also a good/mathematical notion of [*quantum moduli space*]{} in this case to incorporate more physics into the current mathematical setting. In the next theme, we will see an example from string theory in which ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C}); {\Bbb C}^n),Y)$ already contains both a classical and a quantum moduli space of D0-branes on $Y$ in the sense of \[Vafa1\]. [**A comparison with the moduli problem of gas of D0-branes in [\[Vafa1\]]{} of Vafa.**]{} In \[Vafa1\], Vafa studied, among other things, the physics of finitely many D0-branes and D4-branes. In particular, for a gas of $n$-many identical D0-branes on one D4-brane supported on a complex torus ${\Bbb T}^4$, except the additional $U(1)$-factor in the whole gauge group that comes from the simple D4-brane, the Higgsing/un-Higgsing behavior of such D0-D4 systems is the same as that for $n$-many D0-branes alone and the classical moduli/configuration space of the $n$-many D0-branes on the ${\Bbb T}^4$ is given by $S^n({\Bbb T}^4)$, which is a singular complex space. This moduli space is subject to a quantum correction to a quantum moduli space $\widetilde{S^n({\Bbb T}^4)}$, dictated by the requirement that the cohomology $H^{\ast}(\widetilde{S^n({\Bbb T}^4)},{\Bbb C})$ should be the orbifold cohomology (e.g. \[V-W1\] and \[V-W2\]) of $S^n({\Bbb T}^4)$ from string theory. It is also anticipated that $\widetilde{S^n({\Bbb T}^4)}$ should be a hyperkähler resolution of $S^n({\Bbb T}^4)$. See also related discussions in \[B-V-S1\], \[B-V-S2\], and \[Vafa2\]. The related orbifold cohomology was later constructed mathematically by Chen and Ruan in \[C-R1\] and \[C-R2\]. In \[Ru: Conjecture 6.3\], Ruan conjectured in particular that, for $Y$ a smooth projective surface over ${\Bbb C}$ such that $Y^{[n]}$ has a hyperkähler structure, the orbifold cohomology ring $H^{\ast}_{{\mbox{\scriptsize\rm orb\,}}}(S^nY,{\Bbb C})$ of $S^nY$ is isomorphic to the (ordinary) cohomology ring $H^{\ast}(X^{[n]};{\Bbb C})$ of $X^{[n]}$. For the case $Y$ is a smooth projective surface$/{\Bbb C}$ with trivial canonical line bundle, this was proved by Uribe \[Ur: Theorem 3.2.3\] together with previous result of Lehn and Sorger in \[L-S\]. Thus, for $Y$ a smooth projective Calabi-Yau surface, the $\widetilde{S^nY}$ anticipated in \[Vafa1\] is $Y^{[n]}$. In our current setting, a gas of $n$-many D0-branes on a D4-brane[^45], supported on a smooth projective surface $Y$, is regarded as the image of a morphism from $({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n)$ to $Y$. The moduli space ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n),Y)$ of such morphisms contains both $Y^{[n]}\simeq {\mbox{\it Im}\,}\Phi_{{\mbox{\scriptsize\it Hilb}}}$ and $S^nY\simeq {\mbox{\it Im}\,}\Phi_{{\mbox{\scriptsize\it Chow}}}$, and the restriction of $\pi_{{\mbox{\scriptsize\it Hilb}}}: {\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n),Y)\rightarrow S^nY$ to ${\mbox{\it Im}\,}\Phi_{{\mbox{\scriptsize\it Hilb}}}$ realizes the resolution $Y^{[n]}\rightarrow S^nY$. In the special case that $Y$ is in addition Calabi-Yau, we see that ${\mbox{\it Map}\,}(({\mbox{\it Space}\,}M_n({\Bbb C});{\Bbb C}^n),Y)$ contains [*both*]{} the classical and the quantum moduli space of D0-brane configurations on $Y$ in \[Vafa1\]. A remark on D-branes and universal moduli space. ------------------------------------------------ In the previous subsections, we see an interesting feature of the moduli space of D0-branes on a (commutative) quasi-projective variety: namely, it incorporates both the Hilbert scheme and the Chow variety. We also see in the end of Sec. 4.4 that in a special occasion this is interpreted as containing both the classical and the quantum moduli space of D0-branes in physics. While the encompassing of both the classical and the quantum moduli space of a D-brane system on a string target space in general is an issue that will be subject to how we formulate the intrinsic definition of D-brane bound system, the unifying feature of the moduli space of D-branes on a target space (in the sense of Definition 2.2.3 and its extension/generalization to systems that contains NS-branes as well) for different moduli spaces (e.g. Hilbert schemes and Chow varieties in the above example) in commutative geometry should be an anticipated feature when the mathematical definition/formulation of D-branes is “correct". Indeed, since 1995 new stringy dualities have made predictions that relate invariants of different mathematical origins, e.g. from the stable maps, the stable/torsion sheaves, and subschemes respectively (when put in the setting of algebraic geometry). These stringy dualities involve D-branes at work. 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([hep-th/9810188]{}) J. Wess and J. Bagger, [*Supersymmetry and supergravity*]{}, 2nd ed., Princeton Univ. Press, 1992. Z. Yin, [*The principle of least action and the geometric basis of D-branes*]{}, [*J. High Energy Phys.*]{} [**0606**]{} (2006) 002. ([hep-th/0601160]{}) B. Zwiebach, [*A first course in string theory*]{}, Cambridge Univ. Press, 2004 [^1]: D-brane theory and open string theory are in a way counterpart to and interacting with each other. As a consequence, supersymmetric D-brane theory and open Gromov-Witten theory are closely related. In a train of communications with Duiliu-Emanuel Diaconescu \[Dia\] on a vanishing lemma in the last section of \[L-Y3\] and its comparison with \[D-F\], he drew our attention to the important distinction between pure open GW-invariants and open-string world-sheet instantons. The former depends only on the boundary condition set up on the stable maps by supersymmetric D-branes and a decoration on the brane (cf. \[L-Y2: Sec. 7.2\]) while the latter may interact via Wilson loops with the [*general*]{} gauge fields on the D-branes as well (cf. \[Wi2: Sec. 4.2 and Sec. 4.4\] and \[D-F: introduction part of Sec. 3\]). Thus, [*D-brane theory and the field theory thereupon are a part in understanding open-string world-sheet instantons beyond the pure Gromov-Witten sector*]{}. We attribute this footnote to him and thank him for the patient explanations of \[D-F\] to us. [^2]: See \[D-K-L\] for a review and more references. [^3]: See, for example, \[Dou4\] and \[Dou5\] of Douglas and \[S-W2\] of Seiberg and Witten for the development and more references up to 1999. [^4]: The category of noncommutative algebras includes also commutative algebras. We will call a sheaf ${\cal G}$ of ${\cal O}_X$-algebras simply an ${\cal O}_X$-algebra. The [*center*]{} ${\cal Z}({\cal G})$ of ${\cal G}$ is, by definition, the sheaf associated to the presheaf that assigns to each open set $U$ of $X$ the sub-${\cal O}_X(U)$-algebra $Z({\cal G}(U))$ of ${\cal G}(U)$. [^5]: For non-algebraic-geometers: A [*ring*]{} $R$ here is meant to be the [*ring of functions*]{} on a “space" $X_R$ these functions are supposed to take as their defining domain, and a [*ring-homomorphism*]{} $R\rightarrow S$ is meant to be the [*pulling-back*]{} of functions on the underlying spaces when there is a map/morphism $X_S\rightarrow X_R$ between the spaces. Algebraic geometers have turn the picture of “space first, function-ring second" around to make the function-ring first and space – if functorially constructible at all – second. Indeed, physicists have already adopted such “function-ring first" philosophy (without knowing the “space") when studying supersymmetry and superfields on a superspace. [^6]: The general functorial construction of noncommutative schemes that generalizes Grothendieck’s school on commutative geometry is a subtle issue. See, e.g., \[J-V-V: introduction\] and Remark 1.1.6. [^7]: Property (1) and Property (2) together define the notion of a [*filter*]{} of ideals in $R$; Property (3) and Property (4) together actually imply Property (1) and Property (2). [^8]: Central localizations are particularly akin to Azumaya-type noncommutative spaces. It should be noted that most of the definitions, statements, and constructions we give based on central localizations [*cannot*]{} be taken directly for general localizations without additional works or modifications. [^9]: This is a Grothendieck’s descent-data-of-objects description. [^10]: This is a Grothendieck’s descent-data-of-morphisms description. [^11]: For non-algebraic-geometers: A few words follow on why the [*morphisms*]{} in Sec. 1.1 and here are defined as they are. In the case of systems of [*commutative rings*]{}, “general morphism" is a redundant notion as the 3-step ring-system-morphism diagram ${\cal R} \stackrel{\Phi^{\prime}}{\longrightarrow} {\cal S}^{\prime\prime} {\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}{\cal S}^{\prime} \stackrel{\Phi}{\longrightarrow} {\cal S}$ can always be reduced to a $2$-step diagram ${\cal R}\; \stackrel{\Phi}{\longrightarrow}\; {\cal S}^{\prime\prime\prime}\; {\vspace{-.2ex}\begin{array}{c}\prec \\[-1.9ex] \leftarrow \end{array}}{\cal S}$, which represents a strict morphism. In this case, $[{\cal R}]$ and $[{\cal S}]$ (resp. ${\cal R}$ and ${\cal S}$) are contravariantly associated to schemes (resp. atlases of affine charts on schemes). This reducibility from a $3$-step diagram to a $2$-step diagram no longer holds in general in the case of [*noncommutative rings*]{}, as the ring-homomorphisms $\varphi_{\beta}$ on ring-charts are required to be admissible to the central localizations in the construction in order that gluings make sense and work. On the other hand, when we shrink the rings $S_{\beta}$ and take only a system of their subrings $S^{\prime}_{\beta^{\prime}}$, the center can increase: $Z(S^{\prime}_{\beta^{\prime}}) \supset Z(S_{\beta})$. Thus, a ring-homomorphism that is not admissible as a map to $S_{\beta}$ but with the image contained in $S^{\prime}_{\beta^{\prime}}$ may become admissible as a map to $S^{\prime}_{\beta^{\prime}}$. In other words, the notion of general morphism partially takes care of the more subtle issue of a functorial construction of general localizations, allowing us to stay in the much more tractable central localizations. This is not the whole story. In the correspondence of the category of [*commutative rings*]{} with the category of (commutative) [*affine schemes*]{}, one has the canonical identification: ${\mbox{\it Mor}\,}(R,S) ={\mbox{\it Mor}\,}({\mbox{\footnotesize\it Spec}\,}S, {\mbox{\footnotesize\it Spec}\,}R)$ by construction. In the [*noncommutative*]{} case, the functorial construction of the operation “" that associates to a ring a “space" is subtle. Indeed, what Grothendieck’s school accomplished in the decade 1960s for commutative algebraic geometry is only partially realized through the work of several independent schools on noncommutative algebraic geometry in the four decades after then. There are several nonequivalent constructions/realization of the notion of “", with each maintaining part of the equivalent characterizing properties of ${\mbox{\footnotesize\it Spec}\,}$ in the commutative case, cf. sample references in Remark 1.1.6. In the current work, we take rings and ring-homomorphisms as more fundamental for “geometry" than the notion of “points" and “topologies". An injective strict morphism $[\Phi_0]: [{\cal S}_0^{\prime}] \rightarrow [{\cal S}_0]$, is then meant to give a dominant morphism $\phi_0: {\mbox{\footnotesize\it Space}\,}[{\cal S}_0] \rightarrow {\mbox{\footnotesize\it Space}\,}[{\cal S}_0^{\prime}]$, should the latter “spaces" be constructed functorially. Geometrically, a general morphism is then simply an ordinary morphism precomposed with a pinching and, hence, must be still an allowable morphism if the setting is natural. From these hidden words to the main text, one sees that we do want to include general morphisms to ${\mbox{\footnotesize\it Mor}\,}([{\cal R}],[{\cal S}_0])$ in any natural setting/definition. Surprisingly, these independent purely mathematical reasonings that attempt to extend Grothendieck’s language of (commutative) algebraic geometry to the noncommutative case give rise to ${\mbox{\it Mor}\,}([{\cal R}], [{\cal S}])$ that is also required for modeling D-branes in string theory correctly! [^12]: Unfamiliar readers are referred to \[L-L-Y: Sec. 1\] for a brief introduction of and literature guide for the notions of [*Grothendieck topology*]{}, [*site*]{}, and [*stack*]{}. All that is said here is standard from algebraic deformation theory. [^13]: See \[L-Y1: appendix A.1\] for highlights and a literature guide for mathematicians on this very important notion from quantum field theory. In particular, a [*Wilson’s theory-space*]{} goes with [*universal objects*]{} over it that encode QFT contents, and a [*duality*]{} is a local isomorphism on Wilson’s theory-space with these structures. [^14]: It should be noted that there are also algebraic properties of D-branes realized as states or operators in a $2$-dimensional conformal field theory with boundary. These algebraic properties from the open-string world-sheet perspective reflect the geometric properties of D-branes in the target space-time of strings. Our focus in this work is on the geometric aspect as given in \[Pol3\] and \[Pol4\]. [^15]: Since the work of Ramond and of Neveu and Schwarz in 1971 that initiated string theory, there are by now at least three ways to enter superstring theory: Gate (1) the string-world-sheet/CFT way ($d=1+1$ or $d=2$ theory), Gate (2) the target-space-time/supergravity/soliton way ($d=9+1$ or $d=10+1$ theory), and Gate (3) the matrix-theory way ($d=0+1$ theory). In Gate (1), after Wick-rotation, one can have Riemann surfaces, conformal field theories, moduli space of Riemann surfaces, ..., etc. before asking how strings move in a space-time. D-branes entered string theory in the second half of 1980s and took a central role after 1995 mainly from the development of Gate (2) during 1990 - 1995. In asking this question, we mean also to repeat Gate (1) but for D-branes instead of for strings. In other words, we are taking a “D-brane" as a fundamental object and asking, “What is (the definition of) a D-brane?", before addressing how they “move" in – i.e. are mapped into – a space-time. [^16]: In what precise sense the noncommutativity of target space-time and the noncommutativity of world-volume of branes are dual to each other deserves more thoughts. [^17]: Strictly as induced by open-strings, $X^{\mu}(\xi)'s$ are $u(n)$-valued for oriented open strings and either [*so*]{}$(n)$- or [*sp*]{}$(n/2)$-valued for unoriented open strings. Instead of any of these Lie algebras, here we directly think of $X^{\mu}(\xi)$ as $M_n(\footnotesizeBbb C)$-valued, where $M_n(\footnotesizeBbb C)$ is regarded not as a Lie algebra with a bracket (i.e. Lie product) but rather as an associative algebra (from the matrix multiplication) with an identity ${\mathbf 1}$, for two reasons: - These Lie algebras are not associative nor with an identity with respect to the Lie product. This makes the notion of localizations and covers, which are crucial in algebraic geometry for the local-to-global setup, difficult to implement. In view of noncommutative algebraic geometry over ${\footnotesizeBbb C}$, it is more natural to think of $X^{\mu}(\xi)$’s as in a special class of $M_n(\footnotesizeBbb C)$-valued functions with $M_n({\footnotesizeBbb C})$ as an associative algebra with an identity. Any associative algebra defines also a tautological Lie algebra, with the Lie product $[x,y] := x\cdot y - y\cdot x$. One can use this to translate back to Lie algebras whenever needed. - In seeking the intrinsic definition/structure of a D-brane (or D-brane world-volume), it is more natural to select the structures thereon as encompassing/universal as possible so that they contain all what different types of open strings can detect/see. Each specific sector of structures on D-brane world-volume seen by a particular type of open strings is then realized by a reduction from the universal structures on D-brane world-volume, as in the reductions of the structure group of principal [*GL*]{}$_n$ fiber bundles. Cf. footnote 21. [^18]: On purely mathematical ground, the $M_n(R)$ in the ansatz can be generalized in some cases. For example, in the case that $R$ is a Noetherian (commutative) integral domain, $M_n(R)$ can be replaced by the more general notion of an [*$R$-order in a central simple $Q_R$-algebra*]{}, where $Q_R$ is the quotient field of $R$; cf. \[Re\]. [^19]: From C.-H.L: Several teachers and colleagues influenced my painfully slow realization/appreciation of this ansatz and its importance through the personal journey of string theory: [*Orlando Alvarez*]{} brought me to the beauty of string theory and T-duality at the dawn of its second revolution. [*Rafael Nepomechie*]{} shared with me his experience in the early days of higher-dimensional extended objects before they became dominating in “string theory". [*Pei-Ming Ho*]{} communicated the work \[Ho-W\] to me. The group meetings of the school of [*Philip Candelas*]{} and the insightful debates between [*Jacques Distler*]{} and [*Vadim Kaplunovsky*]{} promoted my understandings and kept me aware of subtleties as well. Teaching the late Professor [*Raoul Bott*]{} mirror symmetry, fall 2000, assigned by [*Shing-Tung Yau*]{} gave me a rare chance to slow down and to map out what I had still been ignorant of in the big picture. The heat and enthusiasm [*Shiraz Minwalla*]{} brought in to his various topic courses from field theory to strings, from phase structures in QFT to supersymmetry, $\cdots\,$ over the years helped me to access the mind of physicists at the frontier. [*Shinobu Hosono*]{} explained \[H-S-T\] to me in March 2002, in which the subtle issue of the multiplicity/wrapping of D-branes in the torsion-sheaf picture was brought out among other things. Discussions with [*Mihnea Popa*]{}, spring 2002, and his joint Seminar on Derived Category with [*Mircea Mustata*]{}, fall 2002, influenced my mathematical understanding of D-branes of B-type. The semester-long communications with [*Barton Zwiebach*]{} on the draft of \[Zw\], spring 2003, improved my understanding of the physical fundamentals of string theory. [*Paul Aspinwall*]{} emphasized many subtleties of D-branes in his lectures at TASI 2003. The topic courses and talks of [*Kentaro Hori*]{}, [*Andrew Strominger*]{}, and [*Cumrun Vafa*]{} on string theory over the years printed in my mind various pictures of how, physicists think, D-branes should function. The daily summary of work to each other with [*Ling-Miao Chou*]{} over the years helped to clarify my thoughts. The vanishing lemma derived in \[L-Y3\] and its comparison with \[D-F\] led me to a train of discussions with [*Duiliu-Emanuel Diaconescu*]{}, December 2006, on the meaning of open-string world-sheet instantons in the open/closed string duality. These discussions propelled me to come back to re-think about D-brane theory as a companion theory to topological open strings and their instantons, particularly the virtual ones. Finally, it should be noted that, even with this ansatz, there are still other things missing mathematically to understand D-branes fully in a larger scope, cf. footnote 20. Incidentally, while this work is under writing, [*William Thurston*]{} came to give a talk, May 2007, on the future of $3$-dimensional geometry and topology after the justification of the geometrization conjecture of $3$-manifolds. Hyperbolic geometry has now applications to cosmology and AdS/CFT correspondence. It is surprising how a change of course of life of a teacher can lead to a completely unexpected journey of his student. This detour is very demanding, yet only particularly lucky one is given a chance to it. [^20]: For non-string-theorists: There are two reasons we call this a “prototype" definition. The first one is mild: we focus only on the most essential fields on the brane and ignore the others. The second one is the true reason: the definition we give here reflects only what one should think mathematically about a D-brane [*in a special region of the relevant Wilson’s theory-space of string theory*]{} (cf. \[L-Y1: appendix A.1\]) and, furthermore, we ignore also here the variation to the definition required to incorporate all forms of [*D-brane bound states*]{}. Once we move away from this region, what one should think of D-branes can become more complicated or even not that clear when trying to incorporate both mathematics and physics involved. However, since the mathematical definition given here naturally reproduces the key features of D-branes in its beginning years after Polchinski \[Pol2\], it is our strong belief that those more involved and languagewise more demanding features/descriptions of D-branes by string theorists in its growing years can finally be reached, beginning with the current prototype definition. While the detail of this advanced step remains challenging, there is definitely a related Floer-Gromov-Witten-type theory involved so that the coupling of D-branes and strings is always incorporated, cf. footnote 1. [^21]: Readers may wonder why we do not take ${\cal O}_X^{{\mbox{\tiny\it nc}},{\mbox{\tiny\it Lie}}}$ or ${\cal G}_X$ directly to define the noncommutative structure on $X$. There are two reasons: (1) The “geometry" (in the sense of “points" and “topology") associated to a non-associative, non-unital ring is less clear than that for an associative unital ring at the moment. (2) Since the function ring of local charts of the target space is associative and unital, if we use ${\cal O}_X^{{\mbox{\tiny\it nc}},{\mbox{\tiny\it Lie}}}$ for $X$, we will have to consider ring-homomorphisms from an associative unital ring to a Lie ring. The only such ring-homomorphism is the zero-homomorphism. This renders such setting containing no contents as long as “probing a space(-time) via morphisms into it" is concerned. Cf. footnote 17. [^22]: \(1) A priori, one has a choice of whether or not the Higgsing/un-Higgsing of D-branes should be described as nearby points in the to-be-constructed moduli space of D-branes. For a fixed string target-space $Y$, the Wilson theory-space of “D-branes" in the region where they are still branes resembles the Wilson theory-space of a gauge system. With the type of the gauge system fixed, we have a continuum for the latter theory-space. The gauge group and hence the gauge bundle under Higgsing/un-Higgsing jump discontinuously but the situation is like that on the theory-space in Seiberg-Witten theory: there is a continuum as the theory-space. Another similar situation occurs in the geometric engineering of gauge theories, in which the compactification of a superstring theory on a degeneration family ${\cal X}$ of Calabi-Yau $3$-spaces over a base $B$ gives rise to a family $\{{\mbox{\footnotesize\it QFT}\,}_b\}_{b\in B}$ of $d=4$ effective field theories, parameterized by $B$, whose gauge symmetry is enhanced at special locus of $B$ that corresponds to singular fibers of ${\cal X}/B$. Mathematicians may also recall the moduli space ${\cal M}$ of coherent sheaves of a fixed Hilbert polynomial on a projective variety. Even when ${\cal M}$ is connected, the function on ${\cal M}$ that assigns to an $[{\cal F}]\in {\cal M}$ its sheaf-cohomology dimensions or Betti numbers is in general discontinuous. The upper-semicontinuity of such a function, in particular $h^0$ from the global section functor, on ${\cal M}$ can be taken as a resemblance of the phenomenon of enhancement of gauge symmetry due to additional zero/massless modes. \(2) It can happen that the “good part" of the (coarse) moduli space of objects of different nature admit canonical identifications. For example, the moduli space of maps, the moduli space of subschemes, and the moduli space of cycles canonically coincide when the maps are embeddings of reduced schemes with the trivial automorphism group. Ignoring the issue of automorphisms, it is the behavior under degenerations (i.e. moving away from such “good part" of the moduli space) that the nature of the objects we intend to parameter reveals itself. It is only when the degeneration feature distinct for each moduli problem is captured in the setting may one now hope to have a correct description of the objects and hence their moduli space. Definition 2.2.3 is made with both (1) and (2) in mind. [^23]: See Definition 3.2.2. [^24]: I.e. an element of $S$ is either the identity $1$ or a monomial of $s_1,\,\cdots\,, s_l$. [^25]: I.e. taking the reduced scheme associated to the possibly nonreduced subscheme described by the ideal generated by these equations. [^26]: This is a valuative criterion. The meaning of this topology in terms of analytic geometry is as follows. Under deformations of a morphism from ${\mbox{\footnotesize\it Space}\,}M_n({\footnotesizeBbb C})$ to ${\mbox{\it Space}\,}R$, some connected components of the image points of ${\mbox{\footnotesize\it Space}\,}M_n({\footnotesizeBbb C})$ may move away toward the boundary at infinity of ${\mbox{\footnotesize\it Space}\,}R$ and disappear in the end. This corresponds to a drop from $M_0\sim {\mathbf 1}_d$ to some $M_0\sim {\mathbf 1}_{d^{\prime}}$ with $d^{\prime}<d$. When we consider only ${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\footnotesizeBbb C}), {\mbox{\it Space}\,}R)$ by itself, $M_0\sim {\mathbf 1}$ must always hold. However, when we consider ${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\footnotesizeBbb C}), {\mbox{\it Space}\,}R)$ that occurs as a subset in ${\mbox{\it Mor}\,}({\mbox{\it Space}\,}M_n({\footnotesizeBbb C}), {\mbox{\it Space}\,}{\cal R})$ for a gluing system ${\cal R}$ of rings that contains $R$ as a member, it can happen that some of the connected components of the image of a morphism ${\mbox{\it Space}\,}M_n({\footnotesizeBbb C}) \rightarrow {\mbox{\it Space}\,}{\cal R}$ is not contained in ${\mbox{\it Space}\,}R$. This explains geometrically why, in the equivalent ring-theoretic language, we enlarge here the class of maps from ring-homomorphisms to ring-set-homomorphisms. Furthermore, when the morphism deforms, the number of connected components in ${\mbox{\it Space}\,}R$ of the image of morphisms from ${\mbox{\it Space}\,}M_n({\footnotesizeBbb C})$ to ${\mbox{\it Space}\,}{\cal R}$ can change. The topology on ${\mbox{\it Mor}\,}^{{\mbox{\tiny\rm ring-set}\,}}(R,M_n({\footnotesizeBbb C}))$ defined in Definition 3.2.6 ring-theoretically takes all these issues into account. Such treatment automatically comes up and is rerquired in building a (general) morphism from $[{\cal R}]$ to $[\{M_n({\Bbb C})\}]$, following Definition 1.2.14. [^27]: Recall that the multiplication $\cdot$ in the tensor product ${\footnotesizeBbb C}$-algebra $R\otimes_{\tinyBbb C}S$ of two ${\footnotesizeBbb C}$-algebras $R$ and $S$ is ${\footnotesizeBbb C}$-linearly generated by defining $(r_1\otimes s_1)\cdot(r_2\otimes s_2) = (r_1r_2)\otimes (s_1s_2)$. [^28]: Caution that $\Psi_{\alpha_1}(\tilde{s})^{\perp} _{E_{\mathbf A}}$ here is defined to be the union of fiberwise $\perp$ of $\Psi_{\alpha_1}(\tilde{s})$ in $E_{\mathbf A}$. In general, it is not a sub-$R({\mathbf A})$-module of $M_n(R({\mathbf A}))$. [^29]: We shall always think of such an orbit-space $M/\!\sim$ as an Artin stack with atlas $M$. When $M$ is smooth, it is in this sense that we define a smooth map to $M/\!\sim$. [^30]: For topologists: Here the term “[*isotopic*]{}" comes from the notion of “isotope" in physics/chemitry, not topology. The reason why we choose this term is partially enlightened in footnote 35. [^31]: For non-algebro-geometers: ${\footnotesizeBbb C}^n$ as a $\langle{\mathbf 1},J\rangle$-module is now a ${\footnotesizeBbb C}[y]$-module via $\varphi_J$, with annihilator ${\mbox{\it Ker}\,}(\varphi_J)$. Thus, though $\hat{\varphi}_J$ is not directly defined, $\hat{\varphi}_{J\ast}{\footnotesizeBbb C}^n$ is well-defined. This is the Grothendieck Ansatz on quasi-coherent modules versus quasi-coherent sheaves, similar to that on rings versus spaces. [^32]: See footnote 35 for remarks on the original setting in string theory. [^33]: In other words, $m$ is a regular matrix in $M_n({\footnotesizeBbb C})$. [^34]: Some stringy comments follow. When generalized to higher-dimensional D-branes, these notions produce different notions of “wrappings" of a D-brane around a submanifold/subvariety in the target space(-time) of strings. Such a subtlety, among other things, was recognized seriously only by a smaller group of string theorists, e.g. \[G-S\] and \[H-S-T\]. For most of the stringy literatures, the simpler cycle-picture are more dominating (in the region of the related Wilson’s theory-space where “branes are really branes"). In the hind sight, there might be a reason for this: Recall that an open string interacts with D-branes via its end-points. In most disscussions/literatures, these end-points are only taken to be simple points (i.e. reduced points in the algebro-geometric language) and hence, despite the fact that D-brane warpping can be a more complicated notion than usually thought of, open strings do not see anything beyond the cycle picture with a gauge bundle supported thereon. Should one remember that an end-point is attached to the open string and there are jets (in the sense of differential topology or, in the open-string world-sheet picture, in the sense of real algebraic geometry) at the end-point, then one may expect to draw out some open-string-parameterization-invariant details of such hidden “thickened structure" (e.g. non-reducedness of subschemes, embedded points, torsion-subsheaves within a torsion sheaf, ..., etc.). (However, except in the elementary discussion of momentum conservation of open strings, in which $1$-jet is involved, we are not aware of any other use of jets at the end-point of open string in string theory.) On the other hand, since a D-brane (again in the “brane is really a brane" region) is now taken as an extended dynamical object in its own right and hence has its own definition and deformation-obstruction theory, while it must contain contents induced from open strings, it is completely legitimate that it could also have contents without contradictions with open strings and yet open strings cannot see. In the current example and in Polchinski’s picture, open strings do not “see" the (more complicated) structure sheaf ${\cal O}_{Z_p}$ when it is non-reduced but, rather, only see the (simpler) cycle $\Sigma_{i=1}^kn_ip_i$ with the sub-Chan-Paton space ${\footnotesizeBbb C}^{n_i}$ attached to each $p_i$. Furthermore, what open strings do not see is nevertheless transformable to what open strings do see via an isotopic decay. This is actually a general feature. In our setting, we take both as different yet allowable existences of D-branes on the target space(-time) from deformations of D-branes in the sense of deformations of morphisms from an Azumaya-type noncommutative space to the open-string target space(-time). This explains also the term in Definition 4.1.4, cf. footnote 30. [^35]: A few definitions/remarks for readers’ reference are put here to make precise of the discussion while avoiding distractions. A ‘[*quiver*]{}’ is an oriented graph $\Gamma$ introduced in, e.g., the work of Gabriel in early 1970s to study representations of algebras. A [*representation of a quiver*]{} $\Gamma$ over ${\footnotesizeBbb C}$ is an assignment to each vertex $v_i\in\Gamma$ a ${\footnotesizeBbb C}$-vector space $V_i$ and to each arrow (i.e. oriented edge) $\in \Gamma$ from $v_i$ to $v_j$ a ${\footnotesizeBbb C}$-linear homomorphism $\varphi_{ij}:V_i\rightarrow V_j$. Such representations have now become also a standard tool for string theorists to encode the field contents in a supersymmetric gauge field theory coupled with matters. Such field theories occur particularly on (the world-volume of) D-branes. Due to the rigidity of supersymmetric field theory, a quiver representation pretty much fixes the combinatorial type of the field theory under investigation. There are [*different*]{} quivers that can be associated to a finite-dimensional ${\footnotesizeBbb C}$-algebra $R$, regarded as a (left) $R$-module from the algebra multiplication. The one we choose here encodes the embedded dimension (i.e. the dimension of the tangent space when re-phrased in geometry) of of the Artinian ${\footnotesizeBbb C}$-algebra in our problem. See, e.g., \[A-R-S\], \[G-R\], and \[Jat\] for more discussions. [^36]: Readers who already know the stringy side of Polchinski’s D-branes are suggested to compare it with the mathematical picture described in this theme. The Higgsing/un-Higgsing phenomenon described in this theme following Definition 2.2.3 is a general feature. [^37]: E-print version: [hep-th/9611050]{}: Sec. 2.3 and Sec. 2.4. [^38]: For non-string-theorists: On the physics side, the Higgsing of gauge symmetry on D-branes in the sense of Polchinski is originated from the induced stretching of open strings whose end-points are attached to D-branes that are originally stacked and then are deformed and separated. Such stretching turns part of the massless spectrum of open strings that contribute to the gauge fields on the D-branes into massive spectrum and hence reduces the gauge fields on the D-branes. The fact that this crucial open-string-induced behavior of D-branes can be reproduced by following Definition 2.2.3 alone [*without*]{} resorting to open strings is what convince us that it makes sense to take Definition 2.2.3 as the prototype intrinsic mathematical definition for Polchinski’s D-branes. Unfamiliar readers are encouraged to study \[P-S\] and \[Pol4\] to get a feeling. [^39]: The adjoint action of ${\mbox{\it GL}}_n({\footnotesizeBbb C})$ on $M_n({\footnotesizeBbb C})$ does not have stable points in the sense of Mumford in \[M-F-K\]. With Polchinski’s D-branes in mind, we choose semi-simple pairs for the role of stable pairs in \[Hi\]. [^40]: However, this setting has two drawbacks one should be aware of: (1) it obscures the important noncommutative nature of D-branes for it treats D-branes (of B-type) only as coherent torsion sheaves with a gauge symmetry, which we know now is not a complete picture, (see also \[Di-M\] for subtleties in the case of D-brane bound-state systems), and (2) while this construction is immediately generalizable to D-branes of complex codimension-$1$ in a complex target space, the further extension to describe higher-[*co*]{}dimensional D-branes becomes cumbersome. These indicate that the spectral cover setting might be just accidental for the cases it is applicable and is overall not most natural for D-branes. Cf. \[Liu1\]. [^41]: For pure algebraic geometers: Moduli problems in commutative algebraic geometry tends to boil down to Hilbert schemes, which in projective cases are realized as a locus in an appropriate Grassmannian variety. In that sense, commuting schemes/varieties play the same fundamental role as Grassmannian varieties do for the moduli problem of morphisms from an Azumaya-type noncommutative space to a commutative variety. We hope this gives further motivation to study commuting schemes/varieties. See, e.g., \[Bas\], \[Ge\], \[Ri\], \[S-T\], \[Vac1\], \[Vac2\]. [^42]: Throughout, we only consider the reduced scheme structure on a commuting scheme or a representation scheme that occurs in the problem. [^43]: Recall that a [*regular representation*]{} of an algebra $R$ is the representation of $R$ on $R$ itself by, in our convention, left multiplications; i.e. $R$ as a (left) $R$-module. [^44]: For readers who are familiar with toric geometry: Such system of conditions can be formally associated to the fan (or polytope in the projective case) of a toric variety. [^45]: A complete treatment of this involves an intrinsic mathematical construction/definition of a bound system of D-branes. Here, we only consider the pure D0-brane sector/factor in such a system.

--- abstract: 'We consider a natural new inflationary model in broken supergravity based on an $R$ symmetry. The model predicts a concrete relation between the amplitude of primordial density fluctuations and the scale of supersymmetry breaking. The observed value of the density fluctuations is obtained for the gravitino mass of order the weak scale along with a power-law spectral index considerably less than one, which may be tested in future observations.' author: - | Izawa K.-I. and T. Yanagida\ \ [*Department of Physics, University of Tokyo, Tokyo 113, Japan*]{} date: 'August, 1996' title: | UT-757 Natural New Inflation in Broken Supergravity --- c ł u v Ł ¶ § ø Introduction ============ Low-energy supersymmetry has attracted much attention in particle physics, since it provides a conceivable solution to the hierarchy problem [@A; @Nil]. Supersymmetric theories naturally accommodate gravity in the form of supergravity [@Nil], which may give us a consistent description of physics below the Planck scale. However, supergravity generically allows a constant term in a superpotential. Thus we expect a negative cosmological constant of order the Planck scale, which yields an anti de Sitter universe. This leads us to seek a further symmetry which avoids such a disastrous situation. Supersymmetric field theories admit a peculiar symmetry called $R$ symmetry [@Nil]. It is unique in that it can forbid a constant term in a superpotential and thus restrict a cosmological constant in supergravity. It is also ubiquitous in phenomenological models with supersymmetry. Indeed it is a generic ingredient for causing dynamical supersymmetry breaking [@Nel]. These considerations lead us to impose an $R$ symmetry in the framework of supergravity. In this paper, we consider an $R$-invariant model of an inflaton where spontaneous breakdown of the $R$ symmetry naturally generates an inflationary universe. The vanishing cosmological constant in the present vacuum implies that the contributions of inflaton potential and supersymmetry breaking sector to the vacuum energy cancel out between each other provided the other contributions are negligible. [^1] Then we obtain a concrete relation between the amplitude of primordial density fluctuations and the scale of supersymmetry breaking in our universe. We see that the value of primordial density fluctuations predicted for the gravitino mass of order the weak scale is just around that obtained in the observational analyses. The prediction of the spectral index tends to be considerably less than one for the observed value of the density fluctuations, which may be tested in future observations. The model ========= Let us introduce an inflaton superfield $\f$ with $R$ charge $2/(n+1)$, where $n$ denotes a positive integer of order one. Namely, it transforms as $$\f(\h) \rightarrow e^{i{2 \o n+1}\a} \f(e^{-i\a}\h).$$ This charge assignment allows a tree-level superpotential $$W_0 = - {g \o n+1} \f^{n+1},$$ where $g$ is a coupling constant of order one. Here and henceforth, we set the gravitational scale $M \simeq 2.4 \times 10^{18}$ GeV equal to unity and regard it as a plausible cutoff in supergravity. Note that the superpotential $W_0$ by itself yields a fairly flat potential for $n \geq 3$, which is desirable for a slow-roll inflationary scenario. We further assume the presence of a (composite) superfield [^2] with $R$ charge $2 - 2/(n+1)$ which condenses to give a tiny scale $v^2 \ll 1$. This condensation breaks the $U(1)_R$ symmetry [^3] down to a discrete $R$ symmetry $Z_{2n}$ [@Kum]. Then we expect an effective superpotential $$W = v^2 \f - {g \o n+1} \f^{n+1}. \label{ADDEQ}$$ The $R$-invariant effective Kähler potential is given by $$K = |\f|^2 + {k \o 4}|\f|^4 + \cdots,$$ where $k$ is naturally of order one and assumed to be positive. The ellipsis denotes higher-order terms, which we may ignore in the following analysis. The effective potential for the field $\f$ in supergravity is given by [@Nil] $$V = e^K \left\{ \left( {\q^2 K \o \q \f \q \f^*} \right)^{-1} |DW|^2 - 3|W|^2 \right\}, \label{EPOT}$$ where we have defined $$DW = {\q W \o \q \f} + {\q K \o \q \f}W.$$ This indicates that the vacua [^4] with $|\f| < 1$ satisfy the condition [@Wei] $$DW \simeq v^2 - g \f^n = 0,$$ which yields a vacuum $$\langle \f \rangle \simeq \left({v^2 \o g}\right)^{1 \o n}.$$ The potential at the vacuum is given by $$\langle V \rangle = -3 e^{\langle K \rangle} |\langle W \rangle|^2 \simeq -3\left({n \o n+1}\right)^2v^4|\langle \f \rangle|^2, \label{VAC}$$ whose magnitude is much smaller than the inflation scale $V(0) = v^4$. We propose a scenario that the negative vacuum energy Eq.(\[VAC\]) is canceled out by a supersymmetry-breaking effect which gives a positive contribution $\L^4$ to the cosmological constant: $$-3\left({n \o n+1}\right)^2v^4\left|{v^2 \o g}\right|^{2 \o n} + \L^4 = 0. \label{VCC}$$ This cancellation results in our flat vacuum. [^5] We note that $\L^2 \ll v^2$ for $v^2 \ll 1$. In the hidden sector models of supersymmetry breaking, the scale $\L$ is chosen so as to give a mass of the weak scale to the gravitino: $$m_{3/2} \simeq {\L^2 \o \sqrt{3}} \simeq 10^{-16}-10^{-15}. \label{GRA}$$ The inflaton mass $m_\f$ in the vacuum is given by $$m_\f \simeq n|g|^{1 \o n}v^{2-{2 \o n}}. \label{IM}$$ The inflaton $\f$ may have the following $R$-invariant interactions with the ordinary light fields $\p_i$ in the Kähler potential: $$K(\f, \p_i) = \sum_i \l_i |\f|^2 |\p_i|^2 + \cdots, \label{DI}$$ where $\l_i$ is a coupling constant of order one. The decay width $\G_\f$ of the inflaton is then estimated as $$\G_\f \simeq \sum_i \l_i^2 |\langle \f \rangle|^2 m_\f^3. \label{DW}$$ This decay results in a reheating temperature $$T_R \simeq g_*^{-{1 \o 4}}\sqrt{\G_\f}, \label{RT}$$ where $g_*$ is the relativistic degrees of freedom at the temperature $T_R$. Hence we get $$T_R \simeq n^{3 \o 2} |g|^{2 \o n+1} m_{3/2}^{3n-1 \o 2(n+1)}. \label{RHT}$$ Inflationary Dynamics ===================== Let us investigate the inflationary dynamics of the above model by means of a slow-roll approximation [@Kol]. We may set $g > 0$ and $\langle \f \rangle > 0$ without loss of generality and describe the system approximately in terms of the inflaton field $\v$ $(\geq 0)$ which is $\sqrt{2}$ times the real part of the field $\f$. Then the potential for the inflaton reads $$V(\v) \simeq v^4 - {k \o 2}v^4\v^2 - {g \o 2^{{n \o 2}-1}}v^2\v^n + {g^2 \o 2^n}\v^{2n} \label{POT}$$ for $\v < \langle \v \rangle = \sqrt{2} \langle \f \rangle$. The $k$-independent contribution of $\v^2$ term in $e^K|DW|^2$ is exactly canceled by that in $-3|W|^2$, as was noted in Ref.[@Kum]. The slow-roll inflationary regime is determined by the condition [@Kol] $$\e(\v) = {1 \o 2} \left({V'(\v) \o V(\v)} \right)^2 \leq 1, \quad |\y(\v)| \leq 1, \label{COND}$$ where $$\y(\v) = {V''(\v) \o V(\v)}.$$ For the potential Eq.(\[POT\]), we obtain $$\begin{array}{l} \displaystyle \e(\v) \simeq {1 \o 2} \left({-kv^4\v - {g \o 2^{{n \o 2}-1}}nv^2\v^{n-1} \o v^4}\right)^2 = {\v^2 \o 2}(k + {g^2 \o 2^{n-1}}nv^{-2}\v^{n-2})^2, \\ \noalign{\vskip 2ex} \displaystyle \y(\v) \simeq {-kv^4 - {g \o 2^{{n \o 2}-1}}n(n-1)v^2\v^{n-2} \o v^4} = -k - {g \o 2^{{n \o 2}-1}}n(n-1)v^{-2}\v^{n-2}. \end{array}$$ The slow-roll condition Eq.(\[COND\]) is satisfied for $k \leq 1$ and $\v \leq \v_f$ where $$\v_f \simeq \sqrt{2} \left({(1-k)v^2 \o gn(n-1)}\right)^{1 \o n-2},$$ which provides the value of the inflaton field at the end of inflation. [^6] This value is smaller than the vacuum expectation value $\langle \v \rangle$ due to $v^2 \ll 1$, which is consistent with the approximation Eq.(\[POT\]) of the inflaton potential for discussing the inflationary dynamics. The Hubble parameter during the inflation ($0 < \v \leq \v_f$) is given by $$H \simeq \sqrt{V(0) \o 3} \simeq {v^2 \o \sqrt{3}}. \label{HBL}$$ Let us turn to consideration on the horizon of the present universe. The $e$-fold number $N$ of the present horizon is given by [@Kol] $$N \simeq 67 + {1 \o 3}\ln H + {1 \o 3}\ln T_R \simeq 67 + {1 \o 3}\ln(n^{3 \o 2} m_{3/2}^{5n-1 \o 2(n+1)}). \label{EFLD}$$ Let $\v_N$ be the value of the field $\v$ when the observable universe crossed the horizon during the inflation. Then the $e$-fold number $N$ is also given by $$N = \int_{\v_f}^{\v_N} \! d\v \, {V(\v) \o V'(\v)}.$$ $(i)$ For $1/n \leq k < 1$, we obtain $$N \simeq \int_{\v_f}^{\v_N} \! d\v \, {v^4 \o -kv^4\v} = {1 \o k}\ln({\v_f \o \v_N}).$$ That is, $$\v_N \simeq \v_f e^{-kN}.$$ $(ii)$ For $1/N \leq k < 1/n$, we obtain $$N \simeq \int_{\bar \v}^{\v_N} \! d\v \, {v^4 \o -kv^4\v} + \int_{\v_f}^{\bar \v} \! d\v \, {v^4 \o -{g \o 2^{{n \o 2}-1}}nv^2\v^{n-1}} = {1 \o k}\ln({{\bar \v} \o \v_N}) + {1-nk \o (n-2)k(1-k)},$$ where $\bar \v$ is determined by $$kv^4{\bar \v} = {g \o 2^{{n \o 2}-1}}nv^2{\bar \v}^{n-1}.$$ That is, $$\v_N \simeq {\bar \v} e^{-k{\bar N}},$$ where $${\bar \v} = \sqrt{2} \left({kv^2 \o gn}\right)^{1 \o n-2}, \quad {\bar N} = N - {1-nk \o (n-2)k(1-k)}.$$ $(iii)$ We do not consider the region $k < 1/N$ since the coupling $k$ seems unnaturally small for $N$ of several decades. [^7] The value $\v_N$ should exceed the amplitude of quantum fluctuations of the inflaton field in the de Sitter universe [@Kol]: $$\D \v \simeq {H \o 2\pi} \simeq {v^2 \o 2\pi \sqrt{3}}.$$ For $n = 3$, we obtain $N \simeq 47$ from Eq.(\[GRA\]) and Eq.(\[EFLD\]). The condition $$\v_N \simeq {\sqrt{2} k v^2 \o 3g} \exp(-kN + {1-3k \o 1-k}) > \D \v$$ implies that $k$ seems too small to be natural for $g$ of order one. Hence we discard this possibility and restrict ourselves to $n \geq 4$, where the condition $\v_N > \D \v$ is satisfied for a natural range of the parameter $k$. The Density Fluctuations and Spectral Index =========================================== In the above inflationary model, the amplitude of primordial density fluctuations $\d \r / \r$, which arises from quantum fluctuations $\D \v$ of the inflaton field, is given by [@Kol] $${\d \r \o \r} \simeq {3 \o 5 \pi}{H^3 \o |V'(\v_N)|} \simeq {1 \o 5\sqrt{3}\pi}{V^{3 \o 2}(\v_N) \o |V'(\v_N)|}$$ and the spectral index $n_s$ of the density fluctuations is given by [@Kol] $$\label{X} \begin{array}{l} \displaystyle n_s \simeq 1 - 6\e(\v_N) + 2\y(\v_N) \\ \noalign{\vskip 1ex} \displaystyle \ \quad \simeq 1 - 2k\left\{1+(n-1)\exp\left[-k(n-2)N+{1-nk \o 1-k}\right]\right\}. \end{array}$$ For $1/N \ll k < 1$, we obtain $n_s \simeq 1-2k$. The lower bound of the tilt allowed by observations implies $n_s > 0.6 $ [@COBE; @ST], which is realized for $k < 0.2$. Thus we adopt the range $1/N \leq k < 0.2$ and evaluate the density fluctuations by means of an input Eq.(\[GRA\]). ([*a*]{}) For $n = 4$, we obtain $$N \simeq 45,$$ which gives $$8 \times 10^{-6} g^{3 \o 5} \leq 2 \times 10 g^{3 \o 5} m_{3/2}^{2 \o 5} \leq {\d \r \o \r} < 5 \times 10^3 g^{3 \o 5} m_{3/2}^{2 \o 5} \leq 5 \times 10^{-3} g^{3 \o 5}.$$ The lower and upper bounds correspond to the cases of $k=1/45$, $m_{3/2}=10^{-16}$ and $k=0.2$, $m_{3/2}=10^{-15}$, respectively. ([*b*]{}) For $n \geq 5$, we obtain $$0 < {\d \r \o \r} < 2 \times 10^3 g^{4 \o 9} m_{3/2}^{5 \o 9} \leq 1 \times 10^{-5} g^{4 \o 9}.$$ The observational data yield $\d \r / \r \simeq 2 \times 10^{-5}$ [@COBE; @ST], which implies that a realistic inflationary model is given [^8] in the case of $n=4$. Then the required amplitude of the density fluctuations is obtained for $k \simeq 0.03 - 0.13$ and $g$ of order one, which results in the spectral index $n_s \simeq 0.91 - 0.74$. This tilted power spectrum of the primordial density fluctuations may be adequate for structure formation in our universe [@ST]. Conclusion ========== Let us summarize the model with $n=4$. For the gravitino mass Eq.(\[GRA\]) of the weak scale, we obtain the inflation scale $v \simeq 10^{-6}$ and the Hubble parameter during the inflation $H \simeq 10^{-12}$ from Eq.(\[VCC\]) and Eq.(\[HBL\]). The inflaton mass Eq.(\[IM\]) and the reheating temperature Eq.(\[RHT\]) turn out to be $$m_\f \simeq 10^{-9}, \quad T_R \simeq 10^{-16}.$$ For the coupling $k \simeq 0.1$ in the Kähler potential, we get the amplitude $\d \r/\r \simeq 10^{-5}$. The model with the observed amplitude $\d \r/\r \simeq 2 \times 10^{-5}$ predicts a tilted power spectrum of the primordial density fluctuations with the index $n_s \simeq 0.8$. If the mass of some right-handed neutrino is less than $m_\f/2$, the inflaton can decay to a pair of the neutrinos through the interaction Eq.(\[DI\]). In that case, baryogenesis may be followed by leptogenesis from decay of the right-handed neutrino [@Fuk] since $T_R$ is of order the weak scale in the above model. The right-handed neutrino with the mass of this order induces, through the seesaw mechanism [@Yan], a tiny mass of a left-handed neutrino in an interesting range for the solution to the solar neutrino problem [@F]. We note that the reheating temperature is possibly higher than the weak scale when the inflaton field is involved in a stronger interaction than the one in Eq.(\[DI\]) [@Kum]. Let us comment on the supersymmetry breaking sector. The $\L^4$ contribution to the vacuum energy in Eq.(\[VAC\]) is obtained, for example, by introducing a superfield $Z$ and its superpotential $W(Z) = \L^2 Z$ with an origin of the scale $\L$ presumably dynamical [@Iza]. During inflation $\v \simeq 0$, the field $Z$ acquires a mass of the Hubble scale, which keeps the condition $Z \simeq 0$ at the inflationary epoch [^9] and the contribution of the $Z$ sector is negligible during the inflation for $\L^2 \ll v^2$. Thus the introduction of the field $Z$ scarcely affects the inflationary dynamics. We note that the gravitino mass is possibly as light [^10] as $10^{-24}$ for the lower bound of the spectral index $n_s \simeq 0.6$ though we have regarded $m_{3/2}$ as the weak scale throughout the paper. On the other hand, if we consider the case that the contribution of the inflaton potential to the cosmological constant is canceled by some GUT scale physics instead of the supersymmetry breaking sector, the vacuum expectation value of the inflaton $\langle \v \rangle$ turns out to be of order one. Then the present model realizes supersymmetric topological inflation: The model possesses the discrete $R$ symmetry $Z_{2n}$, which is spontaneously broken to the $R$ parity symmetry by the inflaton condensation $\langle \v \rangle \neq 0$. Thus we have $n$ degenerate vacua in this model, which cause domain wall structures in the whole universe. The initial value of the inflaton field $\v \simeq 0$ may be naturally achieved in a sufficiently large region inside a domain wall for topological reasons and the resultant defects serve as seeds for inflation [@Lin]. Acknowledgement {#acknowledgement .unnumbered} =============== We would like to thank E.D. Stewart for valuable comments. [99]{} M.J.G. Veltman, Acta Phys. Pol. [**12**]{} (1981) 437; L. Maiani, in Gif-sur-Yvette Summer School on Particle Physics, 11th Gif-sur-Yvette, France (Inst. Nat. Phys. Nucl. Phys. Particles, 1979). For a review, H.P. Nilles, Phys. Rep. [**110**]{} (1984) 1. A.E. Nelson and N. Seiberg, (1994) 46. K. Kumekawa, T. Moroi, and T. Yanagida, (1994) 437. S. Weinberg, (1982) 1776. For reviews, E.W. Kolb and M.S. Turner, [*The Early Universe*]{} (Addison-Wesley, 1990); A.R. Liddle and D.H. Lyth, Phys. Rep. [**231**]{} (1993) 1. C.L. Bennett [*et al.*]{}, astro-ph/9601067. A.R. Liddle, D.H. Lyth, R.K. Schaefer, Q. Shafi, and P.T.P. Viana, astro-ph/9511057; A.R. Liddle, D.H. Lyth, P.T.P. Viana, and M. White, astro-ph/9512102; M. White and D. Scott, astro-ph/9601170; M. White, P.T.P. Viana, A.R. Liddle, and D. Scott, astro-ph/9605057; E.F. Bunn and M. White, astro-ph/9607060; J. Primack and A. Klypin, astro-ph/9607061. M. Fukugita and T. Yanagida, (1986) 45. T. Yanagida, in [*Proceedings of the Workshop on the Unified Theory and Baryon Number in the Universe*]{}, ed. O. Sawada and A. Sugamoto (KEK report 79-18, 1979); M. Gell-Mann, P. Ramond and R. Slansky, in [*Supergravity*]{}, ed. F. van Nieuwenhuizen and D.Z. Freedman (North Holland, 1979). For a review, M. Fukugita and T. Yanagida, in [*Physics and Astrophysics of Neutrinos*]{}, ed. M. Fukugita and A. Suzuki (Springer-Verlag, 1994). Izawa K.-I. and T. Yanagida, (1995) 1105; [**95**]{} (1996) 829; T. Hotta, Izawa K.-I., and T. Yanagida, hep-ph/9606203. G.D. Coughlan, W. Fischler, E.W. Kolb, S. Raby, and G.G. Ross, (1983) 59; T. Banks, D.B. Kaplan, and A.E. Nelson, (1994) 779; B. de Carlos, J.A. Casas, F. Quevedo, and E. Roulet, (1993) 447. A. Linde, (1994) 208; A. Linde and D. Linde, (1994) 2456; A. Vilenkin, (1994) 3137; R. Basu and A. Vilenkin, (1994) 7150. [^1]: This cancellation requires a fine tuning of parameters, which we postulate in this paper. [^2]: This $R$ charge is chosen so that the inflaton $\f$ couples to this (composite) superfield. [^3]: This U$(1)_R$ symmetry may be anomalous due to a dynamical origin of the scale $v^2$, which avoids the presence of an $R$ axion. We also note that one may impose a discrete $R$ symmetry from the start instead of the continuous one. [^4]: Although these vacua may only correspond to local minima, possible vacua with $|\f| \geq 1$ do not affect the following analysis. [^5]: This is none other than a fine tuning of the cosmological constant, which is the unique unnatural point in the present model. We do not specify the supersymmetry breaking sector since its details are unnecessary for our purposes in this paper (see the final section). [^6]: A sufficiently large expansion of the universe is achieved under the condition that the initial amplitude of the inflaton $\v$ is localized near the origin. We suspect that such an initial condition is derived from some underlying physics. For example, let us consider that the inflation scale $v^2$ in Eq.(\[ADDEQ\]) arises from a hypercolor quark condensation $\l \langle Q \bar{Q} \rangle = v^2$, where the inflaton couples to the hyperquarks as $\l Q \bar{Q} \v$. If the initial values of $Q$ and $\bar{Q}$ are large as order one, the inflaton field $\v$ is set to be near the origin. [^7]: Roughly speaking, this is the situation analyzed in Ref.[@Kum]. [^8]: In the case of $n = 5$, the required value $\d \r /\r \simeq 2 \times 10^{-5}$ implies that the spectral index $n_s \simeq 0.6$, which may be marginally consistent with the observations. [^9]: This may result in a supersymmetry-breaking vacuum without the so-called Polonyi problem [@Ban] if the vacuum lies near the origin $\langle Z \rangle \simeq 0$. [^10]: Such a light gravitino ($m_{3/2} \simeq 10^{-24}$) is realized in the framework of dynamical supersymmetry breaking at low energies. If $n_s > 0.7$ is confirmed in future observations, we will see that our scenario of the vacuum-energy cancelation may be incompatible with the low-energy supersymmetry breaking.

--- abstract: 'In this paper we address the use of rare event computation techniques to estimate small over-threshold probabilities of observables in deterministic dynamical systems. We demonstrate that genealogical particle analysis algorithms can be successfully applied to a toy model of atmospheric dynamics, the Lorenz ’96 model. We furthermore use the Ornstein-Uhlenbeck system to illustrate a number of implementation issues. We also show how a time-dependent objective function based on the fluctuation path to a high threshold can greatly improve the performance of the estimator compared to a fixed-in-time objective function.' address: - | $\dag$ School of Mathematics and Statistics,\ University of Sydney, Sydney, Australia - | $\ddag$ Meteorological Institute,\ University of Hamburg, Hamburg, Germany - | $\star$ Laboratoire de Physique,\ École Normale Supérieure de Lyon, Lyon, France author: - 'J. Wouters$^{\dag\ddag\star}$ and F. Bouchet$^\star$' bibliography: - 'rare\_event\_simulation2.bib' title: Rare event computation in deterministic chaotic systems using genealogical particle analysis --- Introduction ============ Rare events may have a large impact on the dynamics of geophysical turbulent flows and the climate. In bistability situations, a rare transition can drastically change the structure of the flow, like for instance the bistability of the Kuroshio current [@qiu_kuroshio_2000; @schmeits_bimodal_2001] or a change of polarity of the Earth’s magnetic field due to the turbulent dynamics of the Earth metal core [@glatzmaier_three-dimensional_1995]. Rare events can also be extremely important because of their impact on society, ecosystems or the economy. There are many such examples in climate dynamics, for example extreme droughts, heat waves, rainfalls and storms [@field_managing_2012]. The probability and the impact of these events is likely to change in the future due to a changing climate [@field_managing_2012]. The magnitudes of possible changes are however still uncertain [@barriopedro_hot_2011]. On the one hand, for climate dynamics, there is a lack of sufficient reliable empirical data [@wetter_underestimated_2013]. How could one assess faithfully the probability of events with recurrence times longer than one decade with only one or two century long reliable data? In the last decade, many methods have been developed to extract the most information possible from this too short time-series. For instance extreme value statistics [@de_haan_extreme_2006; @leadbetter_extremes_1983] has interestingly allowed to extrapolate from the information available from empirical observation [@kharin_estimating_2005; @kysely_probability_2002]. Another approach would be to critically study and understand the dynamics of rare events produced by complex climate models. This second approach seems to be the only available one for events with a recurrence time longer than decades or centuries. However, the current scientific state of the art does not yet allow to obtain many results by following this route. The first critical issue is a sampling problem. Indeed, if one wants to study events with century or millennial recurrence time and assess the reliability of the model dynamics to produce those events, a direct numerical simulation would require to have model runs of at least hundreds of thousands of years long in order to get reliable statistics on both the probability of and the dynamics leading to these events. As it is not always reasonable to trade computational length with model complexity, especially for the turbulent part which is responsible for most of the fluctuations, it is clear that we are facing an extremely difficult scientific challenge. Is there a way to produce reliable statistics of specific rare events of a given model, without having to rely on prohibitively long direct numerical simulations? The same issue has been faced in many other scientific fields and has led to the development of some interesting approaches. Indeed, many of the complex systems studied in different branches of science feature events that are very rare but nevertheless very relevant due to their high impact. Take for example buffer overflows in digital communication networks, the insolvency of an insurer or bank, collisions in planetary systems, the dynamics of phase transitions in condensed matter, the long time dynamics of complex molecules in chemistry or biology, to name but a few. In recent years a number of promising algorithms have been developed to tackle these problems [@rubino_rare_2009; @bucklew_introduction_2004; @del_moral_mean_2013; @moral_feynman-kac_2004]. These rare event simulation algorithms can drastically reduce the error made on the estimation of small probabilities. Generally speaking, the objective of the algorithms is to make rare events less rare, either by altering the dynamics (importance sampling) or by targeted killing and cloning of an ensemble of realizations (genealogical particle analysis or interacting particle algorithms). Upon estimation the intervention of the algorithm is then taken into account to obtain an estimate for quantities of the original system. Within the class of genealogical particle analysis algorithms, a number of different strategies exist. A first crucial difference is the type of quantity one aims at estimating. One can be interested in the distribution of first entrance time to a set or to sample transition paths [@cerou_adaptive_2007; @rolland_computing_2015; @rolland_statistical_2015; @heymann2008geometric], the probability of a rare event [@del_moral_genealogical_2005; @garnier_simulations_2006; @hairer_improved_2014] or expectation values of long time averages such as the scaled cumulant generation function [@tailleur_probing_2007; @laffargue_large_2013; @giardina_simulating_2011; @lecomte_numerical_2007; @giardina_direct_2006; @kurchan_large_2015] as it appears in large deviation theory. For these different aims again different algorithms exist, for example geneological algorithms with fixed [@del_moral_genealogical_2005; @garnier_simulations_2006] versus variable [@hairer_improved_2014] particle numbers, minimum action algorithms [@heymann2008geometric], or milestoning [@vanden2009exact]. These algorithms have already been applied to a wide range of systems, for example percolation problems [@adams_harmonic_2008], in complex chemistry [@chandler_interfaces_2005], polymer and biomolecule dynamics [@noe_constructing_2009; @metzner_illustration_2006; @bolhuis_transition_2002; @wolde_enhancement_1997], magnetism [@e_energy_2003; @kohn_magnetic_2005], Burger turbulence [@grafke2013instanton; @grafke2014arclength].\ The aim of this work is to make a first step in the application of those approaches to climate dynamics problems. Climate dynamics has specificities that make past approaches not directly adaptable. First, the climate is clearly out of equilibrium (without time reversal symmetry or detailed balance), therefore only non-equilibrium approaches can be considered. Second, the phenomenology of geophysical turbulent flows is dominated by large scale synoptic scales and is rather different from other complex dynamics, for instance molecular dynamics. And third, most climate models are deterministic models, or sometimes include a stochasticity that does not affect directly the synoptic scales. The aim of this paper is to consider the latter specificity of many climate model. We address the following question: can rare event algorithms based on genealogical particle analysis be used effectively and efficiently for deterministic dynamics? Most algorithms rely on a Markov assumption, which is verified for deterministic models. However at the cloning stage, a new trajectory is branched from another one in order to produce a new ensemble member. For a strictly deterministic system, the offspring trajectory will not be different from its parent. To ensure separation of the two trajectories, one has therefore to add either a very small noise on the overall dynamics or a small change on the initial condition to the offspring trajectory and rely on the dynamics chaoticity. A key issue is then to verify a posteriori that the noise is small enough so as not to distort the measured statistics and probabilities. A test using different decreasing noise strengths and checking for stability should therefore be used. In order to perform the first study of the effectiveness of these approaches for chaotic deterministic dynamics, we have chosen to study a simple chaotic system with many degrees of freedom, and of relevance for climate dynamics: the Lorenz ’96 model [@lorenz_predictability:_1996; @lorenz_designing_2005]. We have also chosen the conceptually simplest and most robust genealogical algorithm that allows to sample invariant measure or transition probabilities: the genealogical particle analysis algorithm. We give a detailed heuristic presentation of the algorithm and a benchmark on the Ornstein-Uhlenbeck process in section \[sec:ornstein-uhlenbeck\]. The genealogical particle analysis algorithms explore the statistics of solutions of the dynamical systems by running an ensemble of realizations, interrupting the ensemble simulation at given times and killing ensemble members that do not perform well as measured by a weight or objective function and cloning the ones with a high weight. This selective procedure explains the terminology genealogical particle analysis. The individual realizations are also sometimes referred to as particles. The design of a good objective function is then arguably the main design issue one faces when using genealogical particle analysis algorithms. Other choices that have to be made are the number and timing of interactions and the number of particles to use. We will address these practical issues in a detailed study of the genealogical particle analysis algorithms on the Ornstein-Uhlenbeck process. This process is easy to simulate numerically and allows for analytic expressions to be derived; it is therefore well suited for the purpose of illustration and testing. Another aim of this paper is to propose a systematic approach and procedure to get reliable results and error estimates. We propose to build the tail of the cumulant distribution funciton of interest by gluing together pieces of results obtained for different cloning parameter by a systematic study of the most reliable one, through an empirical estimate of the algorithm variance. Moreover, we propose a procedure to test empirically this class of algorithms against the real dynamics. Indeed, for a model like the Lorenz ’96 model, we have no theoretical results that can serve as a benchmark.\ The paper is organized as follows. In Section \[sec:Rare-event-computation\] we discuss how the need for rare event simulation techniques arises, what the objective of such algorithms is (making rare events typical) and how this goal can be achieved for stochastic processes by implementing a genealogical particle analysis simulation. In Section \[sec:Most-probable-fluctuation\] we present a brief discussion of the theory of large deviations and what it can say about the way in which rare events are reached by a process. This theory can be used to implement a more efficient rare event sampling method. In Section \[sec:ornstein-uhlenbeck\] we proceed by implementing the genealogical particle analysis simulation to the Ornstein-Uhlenbeck system. We discuss in depth the selection of the parameters in the algorithm. In Section \[sec:Interacting-particle-algorithm\] we then present the implementation of the genealogical particle analysis simulation on a chaotic deterministic dynamical system. Finally, we present our conclusions in Section \[sec:Conclusion\]. Rare event computation for Markov dynamics\[sec:Rare-event-computation\] ======================================================================== In sections \[sub:Motivation\] and \[sub:Importance-sampling\] we present a classical discussion of the inefficiency of brute force Monte Carlo simulation for estimating small probabilities. This motivates the need for rare event computation techniques. We introduce the genealogical particle analysis algorithm and the related theory in Section \[sub:Interacting-particle-algorithm\]. Motivation\[sub:Motivation\] ---------------------------- The goal of rare event computation techniques is to make the numerical estimation of small probabilities more efficient. The necessity of using such techniques is demonstrated by the sampling of the tail of a distribution $P$ using independent samples identically distributed according to the distribution $P$. Say one wants to estimate a small probability $\gamma_{A}=P(X\in A)\ll1$ by means of a brute force Monte Carlo estimate $$\begin{aligned} \hat{\gamma}_{A} & = & \frac{1}{N}\sum_{i=1}^{N}1_{A}(X_{i})\label{eq:naiveMC}\end{aligned}$$ where $1_{A}$ is the indicator function on the set $A$. The estimator $\hat{\gamma}_{A}$ is an unbiased estimator of $\gamma_{A}$ since the expectation value of $\hat{\gamma}_{A}$ is clearly $\gamma_{A}$. When the number of samples $N$ is large enough for $\hat{\gamma}_{A}$ to follow a central limit theorem, the statistical error of the estimator can be quantified by its variance $Var(\hat{\gamma}_{A})=Var(1_{A}(X))/N$. Furthermore $$\begin{aligned} Var(1_{A}(X)) & = & E((1_{A}(X)-\gamma_{A})^{2})\nonumber \\ & = & E(1_{A}(X))-\gamma_{A}^{2}=\gamma_{A}-\gamma_{A}^{2}\nonumber \\ & \approx & \gamma_{A}\label{eq:naiveMCVar}\end{aligned}$$ when $\gamma_{A}$ is small. The relative error of the estimator $RE$ being proportional to the standard deviation divided by the estimated quantity, we have $RE\sim\frac{1}{\sqrt{\gamma_{A}N}}$. The relative error quickly becomes large as $\gamma_{A}$ goes to zero for fixed sample size $N$. Fortunately there exist methods for estimating small probabilities more efficiently. Importance sampling\[sub:Importance-sampling\] ---------------------------------------------- The main ingredient of rare event computation techniques is a sampling from a modified distribution together with an adapted estimator to counteract this change of measure. This method to lower the estimator variance of a rare event probability is termed importance sampling. Again the example of the sampling from independent identically distributed random variables provides valuable insights. Say we want to estimate $$\begin{aligned} \gamma_{A} & = & \int dX\rho(X)1_{A}(X)\ll1\end{aligned}$$ where $\rho$ is the density for our random variable $X$. Instead of doing a straightforward sampling of $X$ as in (\[eq:naiveMC\]), assume we can sample from a modified measure $\tilde{\rho}$ for which $\tilde{\rho}(X)>0$ whenever $X\in A$ and $\rho(X)>0$. In such a case, the probability we want to estimate can be rewritten as $$\begin{aligned} \gamma_{A} & = & \int dX\tilde{\rho}(X)\frac{\rho(X)}{\tilde{\rho}(X)}1_{A}(X)\\ & = & \tilde{E}(L(X)1_{A}(X))\\ L(X) & := & \frac{\rho(X)}{\tilde{\rho}(X)}\mbox{ whenever }1_{A}(X)\rho(X)>0\end{aligned}$$ and we can therefore estimate $\gamma_{A}$ using the estimator $$\begin{aligned} \tilde{\gamma}_{A} & = & \frac{1}{N}\sum_{i=1}^{N}L(\tilde{X}_{i})1_{A}(\tilde{X}_{i})\label{eq:importance}\end{aligned}$$ on samples $\tilde{X}_{i}$ distributed according to $\tilde{\rho}$. The variance for such an estimator is $$\begin{aligned} \tilde{V}ar(L(X)1_{A}(X))=\tilde{E}(L^{2}(X)1_{A}(X))-\gamma_{A}^{2} & = & E(L(X)1_{A}(X))-\gamma_{A}^{2}\end{aligned}$$ If we could take $\tilde{\rho}$ as the conditional measure with $\tilde{\rho}(X)=\rho(X)/\gamma_{A}$ for $X\in A$ and zero elsewhere, such that $L(X)=1_{A}(X)\gamma_{A}$, this would result in a zero variance estimator. This estimator is however not practically implementable, since for this we would need to know the value of $\gamma_{A}$, which is the value we seek to calculate. This calculation demonstrates some important points however. First of all, it shows that a change of measure can indeed reduce the variance of the estimator. Although the ideal change of measure is not feasible in practice, a change of measure that is in a sense close to it should also give a substantial variance reduction. This modified measure should therefore have most of its weight on the set of interest $A$. On the other hand, this also implies that one needs to have some understanding of the shape of the set $A$ and the distribution on it to construct an efficient importance sampling. ### Skewing a normal distribution\[sub:Skewing-a-normal\] To illustrate how a change of measure can provide significant variance reductions, even if the modified measure is not the ideal conditional measure, we discuss an example for normally distributed random variables. This example will also be useful to illustrate and validate our rare event algorithm for dynamical systems. Say we want to estimate the probability of the rare event $A=\{x>a\}$ for a normally distributed random variable $x\sim\mathcal{N}_{0,1}$ with zero average and standard deviation equal to one. Assume that we can skew the distribution with an exponential function $\tilde{\rho}(X)=\rho(X)\exp(CX)/E(\exp(CX))=\frac{1}{\sqrt{2\pi}}\exp\left[-\frac{(X-C)^{2}}{2}\right]$ which constitutes of a shift of the average by $C$. Since $L(X)=E\left(e^{CX}\right)/\exp(CX)=\exp\left(-CX+C^{2}/2\right)$, the variance of the terms in the importance sampling estimator is now $$\begin{aligned} \tilde{V}ar(L(X)1_{A}(X)) & = & P_{-c,1}(x>a)e^{C^{2}}-\gamma_{A}^{2}\label{eq:gaussian-importance}\end{aligned}$$ where $P_{\mu,\sigma}$ denotes probabilities under a normal distribution with mean $\mu$ and variance $\sigma$. The standard deviation, the square root of the variance (\[eq:gaussian-importance\]), is plotted for $a=2$ in Figure (\[fig:gaussian-importance\]). The standard deviation has a single minimum, which is obtained for a value of $C$ which is close to $C=2$, for which the mean of the tilted value coincides with the threshold. This basic example illustrates how importance sampling can lower dramatically the estimator variance. As Fig. (\[fig:gaussian-importance\]) shows, the relative error, which is proportional to the plotted quantity $Std(\tilde{\gamma}_{A})/\gamma_{A}$, can be reduced by a factor of more than $4$ for the case where $A=[2,+\infty)$. Since the error decreases as $1/\sqrt{N}$, this means a at least $16$-fold longer brute force simulation would be necessary to obtain a similarly accurate result. For graphical purposes a relatively low threshold $2$ was chosen here. For higher thresholds, the performance gains increase drastically, with a reduction of computational effort by a factor $6\times10^{5}$ when $A=[5,+\infty)$. ![The ratio of the standard deviation to estimated probability of an exponentially tilted gaussian importance sampling estimator for a threshold $a=2$ with $N=1$\[fig:gaussian-importance\]](skewing_normal){width="50.00000%"} Genealogical particle analysis algorithm\[sub:Interacting-particle-algorithm\] ------------------------------------------------------------------------------ The motivation for rare event simulation and the discussion of importance sampling have shown that it is necessary to make rare events less rare. This concept can be applied to stochastic processes such as the paths followed by either stochastic dynamics or chaotic deterministic dynamics. In those cases the objective is to alter the probability of certain paths that are connected to the rare event one wants to study. Two different strategies are employed to alter the path sampling in stochastic dynamical systems. The first one is to alter the dynamical equations of the system by introducing a forcing term [@vanden-eijnden_rare_2012]. By tuning a parameter of the added forcing term, one can then attempt to decrease the variance of the rare event estimator. A second strategy consists in calculating an ensemble of realizations of the stochastic system in parallel and manipulating the ensemble members by performing selections at a finite number of selection times so as to bias the population. Here we will use the second strategy, by employing a variant of the so called genealogical particle analysis algorithms. The selections applied to the ensemble consist of dynamical trajectories, called particles, being copied and killed depending on weight factor asigned to every ensemble member. This strategy has the advantage of not altering the dynamical trajectories themselves, such that their dynamics can be studied a posteriori. Extensive analysis of the convergence of genealogical particle analysis algorithms can be found for example in [@del_moral_mean_2013; @moral_feynman-kac_2004]. In the following sections, we perform a simpler calculation, assuming a mean field approximation, to demonstrate the evolution of the expected particle distributions in a genealogical particle analysis. The validity of this mean field approximation for large particle number is the subject of the complete proofs given in [@del_moral_mean_2013; @moral_feynman-kac_2004]. Before going to a truly interacting genealogical particle analysis in Section \[sub:Interacting-particles\], we first get some insight by looking at an algorithm where particles are reweighted, but by a factor depending only on the evolution of the particle itself, in Section \[sub:A-non-interacting-weighted\]. ### A non-interacting genealogical particle analysis \[sub:A-non-interacting-weighted\] We calculate rare events of a continuous time Markov chain. $P^{(2)}(y|x,\Delta t)$ denotes the transition probabilities from configuration $x$ to $y$ over a time interval $\Delta t$. We are interested in the probability of being in a set of configurations $A$ at a time $t=t_{n}$, given that the process started in configuration $x_{0}$ at time $t=0$. $N_{t}$ denotes the number of particles at time $t$, whereas $\{\xi_{i,t}\}_{1\leqslant i\leqslant N_{t}}$ denote the particle configuration at time $t$. $E_{0,t}$ denotes expectation values under the original Markov dynamics $P^{(2)}$ at time $t$. The algorithm to generate the particles is described in the box “Algorithm \[alg:non-interacting-ips\]”. [[ The algorithm can be summarized as follows: after initialization (step \[alg1:initial\]) the ensemble members are evolved forward in time (step \[alg1:evolve\]) and weight values are calculated from the previous and current configurations of the particle, $\zeta_{i,t_{k}}$ and $\xi_{i,t_{k-1}}$ respectively (step \[alg1:non-interacting-weights\]). Based on these weight values, ensemble members are killed or cloned (step \[alg1:non-interacting-cloning\]). Repeating this procedure results in a reweighted sample of paths, from which expectation values of the unweighted path distribution can be estimated (step \[enu:non-interacting-final\]). ]{}]{} The rare event probability for a set $A$ can be obtained by taking as observable $F(x)=1_{A}e^{-V(x)}e^{V(x_{0})}$ such that $E_{0,t_{n}}(Fe^{V})e^{-V(x_{0})}=E_{0,t_{n}}(1_{A})$. Note that the random number $N_{i,k}$ generated in step \[alg1:non-interacting-cloning\] can be zero, such that particles can be killed as well as cloned (when $N_{i,k}>1$). A way to generate the random number described in step \[alg1:non-interacting-cloning\] is to take $N_{i,k}=\left\lfloor W_{i,k}+u\right\rfloor $ where $u$ is uniformly distributed on $\left[0,1\right]$ and $\left\lfloor x\right\rfloor $ is the floor of $x$ (the largest integer smaller than $x$). ****** 1. Initiate $M$ particles in configuration $x_{0}$: $\xi_{i,0}=x_{0}\mbox{ for }1\leqslant i\leqslant N_{t_0}=M$ \[alg1:initial\] 2. For every time step $k\in\{1,\ldots,n\}$ 1. Propagate $\xi_{i,t_{k-1}}$ under the dynamics, resulting in $\zeta_{i,t_{k}}$ distributed according to $P^{(2)}(\zeta_{i,t_{k}}|\xi_{i,t_{k-1}},\Delta t_{k})$ with $\Delta t_{k}=t_{k}-t_{k-1}$ \[alg1:evolve\] 2. Calculate weights $W_{i,k}$ for particle $i$:\[alg1:non-interacting-weights\] $$\begin{aligned} W_{i,k} = W(\zeta_{i,t_{k}},\xi_{i,t_{k-1}}) & \mathrel{\mathop:}= & \exp(V(\zeta_{i,t_{k}})-V(\xi_{i,t_{k-1}}))\end{aligned}$$ for a suitably chosen weight function $V$ 3. \[alg1:non-interacting-cloning\]Generate a new particle distribution $\xi_{j,t_{k}}$ consisting of $N_{i,k}$ copies of particle with configuration $\zeta_{i,t_{k}}$where $N_{i,k}$ is chosen at random such that $E(N_{i,k})=W_{i,k}$ (note that $N_{t_{k}}=\sum_{i}N_{i,k}$) 3. Finally, for any $F$, calculate $\breve{F}=\frac{1}{M}\Sigma_{i=1}^{N_{t_{n}}}F(\xi_{i,t_{n}})$ to estimate $E_{0,t_{n}}(Fe^{V})e^{-V(x_{0})}$ (to estimate $\gamma_{A}$ take $F(x)=F_A(x):=1_{A}(x)\exp(V(x_{0})-V(x))$)\[enu:non-interacting-final\] **** ### Unbiased estimator We first show that Algorithm \[alg:non-interacting-ips\] provides an unbiased estimator for the quantity $E_{0,t_n}(Fe^{V})e^{-V(x_{0})}$, i.e. the algorithm results in a random estimate whose expectation value equals the quantity to be estimated: ***$$\begin{aligned} E_{1}\left(\frac{1}{M}\Sigma_{i=1}^{N_{t_{n}}}F(\xi_{i,t_{n}})\right) & = & E_{0,t_{n}}(Fe^{V})e^{-V(x_{0})}\end{aligned}$$*** where $E_{1}$ is the expectation over the random variables in the algorithm. Write $N(x,t)$ the particle number at configuration $x$, i.e. $N(x,t)dx$ is the number of particles with $x \leqslant x_{i,t} < x+dx$: $$\begin{aligned} N(x,t_{k-1}) & = & \Sigma_{i=1}^{N_{t_{k-1}}}\delta(x-\xi_{i,t_{k-1}})\end{aligned}$$ According the algorithm \[alg:non-interacting-ips\], if a particle sits at $\xi_{i,t_{k-1}}$ at time step $k-1$, $N_{i,k}$ copies are created of $\zeta_{i,t_{k}}$ at the next time step. Hence, the particle number at the next time step will be $$\begin{aligned} N(x,t_{k}) & = & \Sigma_{i=1}^{N_{t_{k-1}}}N_{i,k}\delta(x-\zeta_{i,t_{k}})\label{eq:particle-distr-update}\end{aligned}$$ One step in the algorithm involves the generation of two sets of random variables, the updated particle configurations $\zeta_{i,t_{k}}$, which is conditioned on $\xi_{i,t_{k-1}}$, and the number of particle copies $N_{i,t_{k}}$, which depends on both $\zeta_{i,t_{k}}$ and $\xi_{i,t_{k-1}}$. The expectation value of functions depending on the particle configurations $\xi_{i,t_{k}}$ at step $k$ can therefore be expressed as the expectation value $$\begin{aligned} E_{\xi_{i,t_{k}}}(\bullet) & = & E_{\xi_{i,t_{k-1}}}(E_{\zeta_{i,t_{k}}|\xi_{i,t_{k-1}}}(E_{N_{i,k}|\zeta_{i,t_{k}},\xi_{i,t_{k-1}}}(\bullet)))\end{aligned}$$ Applying this expression to Eq. \[eq:particle-distr-update\] and using the probabilities for the updated particle configurations $P(\zeta_{i,t_{k}}|\xi_{i,t_{k-1}})=P^{(2)}(\zeta_{i,t_{k}}|\xi_{i,t_{k-1}},\Delta t_{k})$ and that the number of particle copies $E_{N_{i,{k}}|\zeta_{i,t_{k}},\xi_{i,t_{k-1}}}(N_{i,k})=W_{i,k}(\zeta_{i,t_{k}},\xi_{i,t_{k-1}})$, we have $$\begin{aligned} E_{1}(N(x,t_{k})) & = & E_{\xi_{i,t_{k-1}}}(E_{\zeta_{i,t_{k}}|\xi_{i,t_{k}}}(\Sigma_{i=1}^{N_{t_{k-1}}}W_{i,k}(\zeta_{i,t_{k}},\xi_{i,t_{k-1}})\delta(x-\zeta_{i,t_{k}})))\\ & = & E_{\xi_{i,t_{k-1}}}\left(\Sigma_{i=1}^{N_{t_{k-1}}}\int dy P^{(2)}(y|\xi_{i,t_{k-1}},\Delta t_k)W_{i,k}(y,\xi_{i,t_{k-1}})\delta(x-y)\right)\\ & = & E_{\xi_{i,t_{k-1}}}\left(\Sigma_{i=1}^{N_{t_{k-1}}}P^{(2)}(x|\xi_{i,t_{k-1}},\Delta t_k)W_{i,k}(x,\xi_{i,t_{k-1}})\right)\\ & = & E_{\xi_{i,t_{k-1}}}\left(\int dzP^{(2)}(x|z,\Delta t_k)W(x,z)\Sigma_{i=1}^{N_{t_{k-1}}}\delta(z-\xi_{i,t_{k-1}})\right)\\ & = & \int dzP^{(2)}(x|z,\Delta t_k)W(x,z)E_{1}(N(z,t_{k-1}))\\ & = & \int dzP^{(2)}(x|z,\Delta t_k)e^{V(x)-V(z)}E_{1}(N(z,t_{k-1}))\end{aligned}$$ This equation relates the expected particle density at step $k$ to the density at step $k-1$. By iteration we can relate the density at step $k$ to the density at the start of the algorithm, which is $M\delta(x-x_{0})$: $$\begin{aligned} E_{1}(N(x,t_k)) & = & \int dx_{k-1}\ldots dx_{1} dz P^{(2)}(x|x_{k-1},\Delta t_k)\ldots P^{(2)}(x_{1}|z,\Delta t_k) \\ && \phantom{\int dx_{k-1}\ldots dx_{1} dz P^{(2)}} \times e^{V(x)-V(z)}M\delta(z-x_{0})\\ & = & M P^{(2)}(x|x_{0},t_k - t_0)e^{V(x)-V(x_{0})}\end{aligned}$$ The expectation value of the quantity calculated at the end of the algorithm in step \[enu:non-interacting-final\] is therefore $$\begin{aligned} E_{1}\left(\frac{1}{M}\Sigma_{i=1}^{N_{t_n}}F(\xi_{i,t_n})\right) & = & \frac{1}{M} E_{1}\left(\int dx\Sigma_{i=1}^{N_{t_n}} \delta(x-\xi_{i,t_n}) F(x)\right)\\ & = & \frac{1}{M}\int dxE_{1}(N(x,t_n))F(x)\\ & = & \int dx F(x)e^{V(x)-V(x_{0})}P^{(2)}(x|x_{0},t_n - t_0)\\ & = & E_{0,t_n}(F e^{V})e^{-V(x_{0})}\end{aligned}$$ Note that the expected total particle number $E_{1}(N(t_k))=\int dxE_{1}(N(x,t_{k}))=M\int dxP^{(2)}(x|x_{0},t_k - t_0)e^{V(x)-V(x_{0})}$ is in general not preserved over time. The particle number can strongly increase, which entails a large numerical cost. The solution to this problem is to renormalize the weights calculated in step \[alg1:non-interacting-weights\] of the algorithm, hence introducing an interaction between the ensemble members. We discuss this new algorithm in the next section. As we will see, the interaction complicates the algorithm analysis. ### Interacting particles \[sub:Interacting-particles\] We now add interaction to the weights of the particle system, so as to control the particle number. A similar analysis as in the previous section can still be carried out, if one assumes that the number of particles used in algorithm is large enough, such that averages over particle configurations can be replaced by an expectation value under the law of large numbers (mean field approximation). The corresponding algorithm is described in the box “Algorithm \[alg:Interacting-particle-system\]”. By applying the algorithm to a function $F_A(x)=1_{A}(x)\exp(V(x_{0})-V(x))$ with $1_{A}$ the indicator function of the set $A$, estimates $\breve{\gamma}_{A}$ of the probability $\gamma_{A}$ can be obtained. ****** 1. Initiate $M$ particles in configuration $x_{0}$, $\xi_{0}^{i}=x_{0}\mbox{ for }1\leqslant i\leqslant N_{0}=M$ 2. For every time step $k\in\{1,\ldots,n\}$ 1. Propagate $\xi_{i,t_{k-1}}$ under the dynamics, resulting in $\zeta_{i,t_{k}}$ distributed according to $P^{(2)}(\zeta_{i,t_{k}}|\xi_{i,t_{k-1}},\Delta t_{k})$ with $\Delta t_{k}=t_{k}-t_{k-1}$ 2. Calculate weights for particle $i$: $$\begin{aligned} \bar{W}_{i,k} & = & \frac{W_{i,k}(\zeta_{i,t_{k}},\xi_{i,t_{k-1}})}{Z_{k}}\\ Z_{k} & = & \frac{1}{N_{t_{k}}}\Sigma_{i}W_{i,k}(\zeta_{i,t_{k}},\xi_{i,t_{k-1}})\end{aligned}$$ 3. Store the value of the normalizing factor $Z_{k}$ 4. \[enu:interacting-cloning\]Generate a new particle distribution $\xi_{j,t_{k}}$ consisting of $N_{i,k}$ copies of particle with configuration $\zeta_{i,t_{k}}$where $N_{i,k}$ is chosen at random such that $E(N_{i,k})=\bar{W}_{i,k}$ 3. Finally calculate $\frac{1}{M}\Sigma_{i=1}^{N_{t_{n}}}F(\xi_{i,t_{n}})\prod_{k=1}^{n}Z_{k}$ to estimate $E_{0}(Fe^{V})e^{-V(x_{0})}$ (for $\gamma_{A}$ take $F(x)=1_{A}(x)\exp(V(x_{0})-V(x))$) ****  We again perform an analysis of the evolution of the expected particle distribution in Algorithm \[alg:Interacting-particle-system\]. For simplicity of the derivation we assume that the number of particles $N_{t_k}$ in the algorithm is large, such that by the law of large numbers, $$\begin{aligned} Z_{k}=\frac{1}{N_{t_{k}}}\Sigma_{i}^{N_{t_{k}}} W(\zeta_{i,t_{k}},\xi_{i,t_{k-1}}) & \approx & E_1(W(\zeta_{i,t_k}, \xi_{i,t_{k-1}})) \label{eq:normalization-factor}\\ & = & \int dx \, dy\frac{E_1(N(x,t_{k-1}))}{E_1(N_{t_{k-1}})}P^{(2)}(y|x,\Delta t_k)W(y,x)\end{aligned}$$ Using this estimate and the same reasoning as for the non-interacting particle algorithm, we have for the expected particle distribution that $$\begin{aligned} E_1(N(x,t_{k})) & \approx & \int dz P^{(2)}(x|z,\Delta t_k)\frac{W(x,z)}{E_1(W)}E_1(N(z,t_{k-1}))\label{partdens3}\end{aligned}$$ The expectation value of $W$ in the denominator can be substituted using Eq. \[eq:normalization-factor\]. The particle number is now constant: $$\begin{aligned} E_1(N_{t_{k}})) & = & \int dx E_1(N(x,t_{k}))=E_1(N_{t_{k-1}})=\ldots=E_1(N_{t_0})=M\end{aligned}$$ We therefore have that $$\begin{aligned} E_1(W(\zeta_{i,t_k}, \xi_{i,t_{k-1}})) & \approx & \int dxdyP^{(2)}(x|y, \Delta t_k)W(x,y)\frac{E(N(x,t_{k-1}))}{M}\end{aligned}$$ Inserting this into \[partdens3\], we have $$\begin{aligned} E(N(x,t_{k})) & \approx & M\frac{\int dyP^{(2)}(x|y, \Delta t_k)W(x,y)E(N(y,t_{k-1}))}{\int dxdyP^{(2)}(x|y, \Delta t_k)W(x,y)E(N(y,t_{k-1}))}\end{aligned}$$ Therefore by iteration $$\begin{aligned} \frac{1}{M}\Sigma_{i=1}^{N}F(x_{i,t_{n}}) & \approx & \frac{E_0(F(x_{n})W(x_{n},x_{n-1})\ldots W(x_{1},x_{0}))}{E_0(W(x_{n},x_{n-1})\ldots W(x_{1},x_{0}))} \\ &=&\frac{E_0(F(x_{n})W(x_{n},x_{n-1})\ldots W(x_{1},x_{0}))}{\mathcal{Z}_{n}}\end{aligned}$$ where $\mathcal{Z}_{n}=E_0(W(x_n,x_{n-1}) \ldots W(x_1,x_0))$. From Eqs. \[eq:normalization-factor\] and \[partdens3\] we see that $Z_k \approx \frac{\mathcal{Z}_k}{\mathcal{Z}_{k-1}}$ and therefore $\frac{1}{M}\Sigma_{i=1}^{N}F(x_{i,t_{n}}) \prod_{k=1}^n Z_k \approx E_0(F(x_{n})W(x_{n},x_{n-1})\ldots W(x_{1},x_{0})) = E_{0}(Fe^{V})e^{-V(x_{0})}$ The above reasoning can also be extended to show that path dependent quantities (such as $E[x(\tau)|x(T)>a]$ for $\tau<T$) can be estimated from the ancestral paths of the particle system. ### Time-dependent weighting\[sub:Time-dependent-weighting\] The weighting function $W(x,y)=\exp(V(x)-V(y))$ results in a particle distribution tilted by $\exp(V(x))$ at all selection times $t_k$. More flexibility can be obtained by using time-dependent weighting, for example with a weighting function of the form $$W(t_k,x,y)=\exp(V_{t_k}(x)-V_{t_{k-1}}(y))\label{eq:time-dep-weight}$$ This way the telescoping canceling of $V_{t_k}$ is preserved in products of weights that appear in the calculation of the tilted measure. For example, $$\begin{aligned} W(t_k,x,y)W(t_{k-1},y,z) & = & e^{V_{t_k}(x)-V_{t_{k-1}}(y)}e^{V_{t_{k-1}}(y)-V_{t_{k-2}}(z)}\\ & = & \exp(V_{t_k}(x)-V_{t_{k-2}}(z))\end{aligned}$$ The result is again a particle distribution tilted by $\exp(V_{t_k}(x))$ at time $t_k$, as with the time-independent weight function. However, paths up to the final time will have different weights, which can make a large difference in the algorithm performance, as we will demonstrate in Section \[sub:Selections-along-the\]. Fluctuation paths and the weighting function\[sec:Most-probable-fluctuation\] ============================================================================= The ideal change of measure discussed in Section \[sub:Importance-sampling\] suggests to make the rare event that is the least rare the most probable one under the reweighted dynamics. This rationale extends not only to the distribution of the system at the final time, but also to the entire path up to the final time. This means that variance can be reduced if the least unlikely path leading to a high threshold is made more likely under the particle system dynamics. For stochastic differential equations in the weak noise limit, the least unlikely path from an attractor can be calculated from Freidlin-Wentzell type large deviation theory and is called a fluctuation path (also sometimes an instanton). The particle system dynamics can be made to more closely follow the fluctuation path by using the time-dependent weighting discussed in Section \[sub:Time-dependent-weighting\]. Even if the particle distribution at the final time is the same as the particle distribution obtained with a constant weighting, there is still a variance reduction since less particles are killed, increasing the independence of the particle and thus the effective particle number. Fluctuation paths\[sub:Fluctuation-paths\] ------------------------------------------ The probability of a given path in a stochastic differential equation with small noise, $$dX^{\epsilon}=b(X^{\epsilon})dt+\sqrt{\epsilon}dW,$$ where $W$ is a Brownian motion, can be estimated using the Freidlin-Wentzell large deviation theory. The theory determines the probability of seeing a path that is close to a specified continuous function in the limit of $\epsilon$ going to zero. It roughly states that $$\lim_{\epsilon\rightarrow0}\epsilon\log P[X^{\epsilon}\in F]=-\inf_{\omega\in F}I(\omega)$$ where $F$ is any closed subset of the set of continuous trajectories and the rate functional $I$ is called the action. The action is given by $$\begin{aligned} I(\omega)=\frac{1}{2}\int_{0}^{T}dt(\dot{\omega}(t)-b(\omega(t)))^{2}=\int_{0}^{T}dt\mathcal{L}[\omega,\frac{\partial\omega}{\partial t}]\label{eq:Action} \\ \mathcal{L}[\omega,\frac{\partial\omega}{\partial t}] := \frac{1}{2} (\dot{\omega}(t)-b(\omega(t)))^{2} \end{aligned}$$ The distribution of paths leading to rare fluctuations then concentrates around action minima as $\epsilon$ decreases, with given constraints. If the set of paths $F$ contains the evolution along the deterministic dynamics $\dot{x}=b(x)$, this path will obviously minimize the above action, hence the need for constraints to obtain more interesting results. For example, in the simple case were the deterministic dynamics $\dot{x}=b(x)$ has a single attractor $x_{0}$, the distribution of the paths conditioned on $X(0)=X_0, \, X_0 \neq x_0$ concentrate close to the minima of the action $\int_{-\infty}^{0}dt\mathcal{L}[\omega,\frac{\partial\omega}{\partial t}]$ with the boundary conditions $X(-\infty)=x_{0}$ and $X(0)=X_0$. Such a path is called a fluctuation path leading to $X_0$ (it is also sometimes called an instanton, but instanton usually rather refers to those fluctuation paths that connect attractors to saddle points). Rare event simulation for a stochastic process: the Ornstein-Uhlenbeck process {#sec:ornstein-uhlenbeck} ============================================================================== We now illustrate some of the practical issues arising when implementing a genealogical particle analysis algorithm for rare event estimation. We start off with a stochastic process for which we can calculate explicitly all of the probabilities that we want to estimate, for pedagogical reasons, and so that we can compare the numerical results to the analytic expressions. Description of the Ornstein-Uhlenbeck process --------------------------------------------- We consider the Ornstein-Uhlenbeck process $$dx=-\lambda xdt+\sigma dW\label{eq:ornstein-uhlenbeck}$$ As the transition probabilities $P(x(t)|x(0))$ are Gaussian, the Ornstein-Uhlenbeck process preserves Gaussianity. Using the Itô formula, one can derive that the mean $m(t)=E(x(t))$ and the variance $v(t)=E((x(t)-m(t))^{2})$ evolve according to the equations $$\begin{aligned} \dot{m} & = & -\lambda m\nonumber \\ \dot{v} & = & \sigma^{2}-2\lambda v\label{eq:v}\end{aligned}$$ which can be easily solved explicitly. The probability that $x$ exceeds a certain threshold $a$ at a time $t$, given that the process started at $x(0)=0$ at time zero can be calculated explicitly as $$P(x(t)>a|x(0)=0)=\int_{a}^{\infty}dx \mathcal{N}_{m(t),v(t)}(x),\label{eq:Pa}$$ where $m$ and $v$ solve (\[eq:v\]) and $\mathcal{N}$ is the probability density function of the normal distribution with mean $m$ and variance $v$. We consider in the following the estimation of this probability through a genealogical particle analysis algorithm. [[Below we will use the parameter values $\lambda=1$ and $\sigma=1$.]{}]{} Algorithm implementation ------------------------ Let us assume we seek to estimate a small probability $\gamma_{A}$, for instance $\gamma_{A}=P(x(t)\in A|x(0)=0)$. We denote $M$ the number of particles for each realization of the algorithm. Then each independent realization $i$ of the algorithm, with $M$ particle each, will give an estimate $\breve{\gamma}_{A,i}$. According to a theorem discussed in [@del_moral_genealogical_2005], asymptotically for large $M$, the random number $\breve{\gamma}_{A,i}$ is distributed according to a Gaussian distribution with standard deviation $\sigma_{A}(M)=\sigma_{A}/\sqrt{M}$ and a corresponding relative error $RE(M)=\sigma_{A}(M)/\gamma_{A}$. The value of the estimator relative error $RE(M)$ is essential as it quantifies the relative error one should expect for each realization of the algorithm, and thus the quality of the result. How the estimator relative error $RE(M)$ depends on the number of selections, on their timing, and the type observables are critical questions that we analyze in this section. ### Number of particles The result in [@del_moral_genealogical_2005] proves the existence of the central limit theorem, but does not give a value for the estimator variance $RE(M)$. In order to get an estimate of $RE(M)$, we compute it empirically by performing $K$ independent algorithm realization and using the estimator $$RE(M)\simeq\sqrt{\frac{1}{K}\sum_{i=1}^{K}(\breve{\gamma}_{A,i}-\gamma_{A})^{2}}/\gamma_{A}.\label{eq:Estimated_RE}$$ In this formula the value of $\gamma_{A}$ will be either the theoretical value when it is available, for instance for the Ornstein-Uhlenbeck process, or the estimated value of the probability by averaging $\breve{\gamma}_{A,i}$ over $K$ realizations. In the following, by an abuse of notation, $RE$ denotes either the theoretical estimator variance of the estimator variance evaluated from (\[eq:Estimated\_RE\]), which one should be clear from the context.\ We first study the estimator variance $RE$ for the Ornstein–Uhlenbeck case. We first test whether or not the regime of the central limit theorem has been reached by changing the number of particles $M$, and verifying whether $RE$ (\[eq:Estimated\_RE\]) reduces by the corresponding $\sqrt{M}$ factor. Figure \[fig:numparta\] shows the expected decrease in relative error as the number of particles is increased for a range of thresholds $a$ (see eq. (\[eq:Pa\])). The inverse square root behavior of the error with increasing number of particles is demonstrated for a fixed threshold in Figure \[fig:numpartb\]. The parameters are specified in the figure captions. We study how the estimator variance $RE$ depends on the other numerical parameters in the following sections.   ### Number of selections and their timing Since little theoretical analysis has been performed on the optimal number of selection steps, this is the most heuristic choice to be made. Some numerical analysis of this issue has been performed in [@el_makrini_diffusion_2007]. For the problem they investigate, changing the number of selections, and using equidistant in time selections, the estimator variance clearly shows a minimum for a certain number of selections. This result can be interpreted as follows. Selections shouldn’t be performed too frequently, as cloning increases correlations between the particles and therefore reduces the effective number of independent particles, increasing the estimator variance. If not performing selections frequently enough however, the particle distribution relaxes to the unbiased particle measure, leading to the poor brute force Monte-Carlo variance. This can be seen in Figure \[fig:numselections\]: for low thresholds $a$, importance sampling is useless and estimations with a small number of selections have the lowest estimator relative error $RE$. Due to the large time between selections, the particles have relaxed to the particle measure of brute force Monte Carlo simulations and therefore have a similar estimator variance. For higher thresholds, for instance for $a>1.7$, it becomes advantageous to kill a larger number of particles to obtain a more skewed final particle distribution, in order to lower the variance. For the threshold value $a=2$ the optimal number of interactions among the values in the figure is $N=16$. For higher thresholds there is a small reduction in error by increasing the number of selections, although increasing the number of selections much further beyond $N=64$ results in an overall increase of error. Figure \[fig:numselections\] also illustrates the large estimator variance improvement for the genealogical particle analysis algorithm compared with Monte-Carlo sampling, as soon as $a\geq2$.\ Besides the number of selections, there also seems to be little theoretical understanding of the optimal timing for selections. One strategy to selection timing is to calculate on-the-fly a criterion on the distribution of particle weights (such as the squared coefficient of variation or entropy) and only perform selection if a fixed threshold is exceeded. The convergence of such adaptive selection strategies is discussed in [@del_moral_adaptive_2012]. ### Estimating a range of over-threshold probabilities\[sub:Estimating-the-tail\] In the following the weight function $W(x,y)=\exp(C(x-y))$ is used. From the point of view of the estimator variance $RE$ (\[eq:Estimated\_RE\]), to each value of the threshold $a$ corresponds an optimal value of $C$, denoted $C^{*}(a)$, or equivalently for each value of $C$ the estimator variance has a minimum for a given value of $a$, denoted $a^{*}(C)$. For instance Figure \[fig:numselections\] shows that the value $C=4$ is optimal for $a\simeq2.5=a^{*}$. In simple cases, we expect $C^{*}(a)$ to increase monotonically with $a$. There is an optimal value of $a$ for each value of $C$, however the estimate is good for a range of thresholds around this optimum. When instead of a particular over-threshold probability one is interested in the tail of the complete distribution probability, one can perform a number of genealogical particle analysis simulations each with different value of $C$, and select for each threshold the value corresponding to the lowest estimator variance $RE$. Figure \[fig:tail-estimation\] illustrates how the tail of $P(a)=P(x(t)\geq a|x(0)=0)$ can be estimated this way, for $x$ the Ornstein-Uhlenbeck process. For large values of the threshold (above $a\approx4.6$) all estimates have a high error and the highest value of $C$ is chosen by default. As can be seen on this figure there is very good agreement with the theoretical value up to probabilities as low as $10^{-10}$. Using this strategy, we can accurately estimate the tail of the over-threshold distribution down to probabilities as small as $10^{-10}$, with relative error lower than one. ### Selections along the fluctuation paths\[sub:Selections-along-the\] We have discussed in Section \[sub:Time-dependent-weighting\] that a time-dependent weighting function can be used. In this way the particle distribution can be weighted with different exponential factors $C(t_k)$ all along the path, but still lead to the same exponentially tilted final particle distribution. Furthermore, in Section \[sub:Fluctuation-paths\] we have discussed how for small noises, most of the paths leading to a rare event will concentrate around fluctuation paths that minimize the action functional. The aim of this section is to demonstrate the interest of using fluctuation paths to construct time-dependent weighting functions in order to increase the efficiency of the genealogical particle analysis algorithm. Since the Ornstein-Uhlenbeck process is linear, taking limits of higher thresholds is equivalent to taking a weak-noise limit through a rescaling of the $x$ coordinate. Hence, for fixed noise intensity $\sigma$, paths starting at $x_{0}=0$ conditioned on reaching the final threshold $a$ will concentrate around the fluctuation paths in the limit $a\rightarrow\infty$. The action (\[eq:Action\]) for the Ornstein-Uhlenbeck process (\[eq:ornstein-uhlenbeck\]) is given by $I\left[X\right]=\int_{0}^{T}d\tau(\dot{X}+\lambda X)^{2}$. Taking as boundary conditions $X(0)=0$ and $X(T)=a$ the fluctuation paths are easily computed to be $X_{f}(t)=a\frac{\sinh\lambda t}{\sinh\lambda T}.$ By using the potential function $W(t,x,y)=\exp(C(t_{k})x-C(t_{k-1})y)$ with a weight parameter $C(t_{k})$ dependent on the selection time $t_{k}$, we can control $\tilde{\mu}(t_{k})$, the mean particle position at $t_{k}$, by fixing $C(t_{k})$. The expected particle distribution for the Ornstein-Uhlenbeck process tilted with this weighting function after the selection at $t_{k}$ is $$\exp\left(C(t_{k})x\right)\mathcal{N}_{0,v(t_{k})}(x)/\int\mbox{d}x\,\exp\left(C(t_{k})x\right)\mathcal{N}_{0,v(t_{k})}(x)$$ as discussed in Section \[sub:Time-dependent-weighting\]. The corresponding expected mean particle position is therefore $\tilde{\mu}(t_{k})=C(t_{k})v(t_{k})$ where $v(t)=(1-\exp(-2t))/2$ is the variance of the Ornstein-Uhlenbeck process at time $t$ (the solution of Eq. \[eq:v\] with $v(0)=0$). Choosing $C(t_{k})=X_{f}(t_{k})/v(t_{k})$, $\tilde{\mu}(t)$ follows $X_{f}(t)$ and the algorithm particle distribution closely follows the fluctuation path leading to the threshold $a$. Figure \[fig:ancestral-paths\] shows the effect of using a weight function based on a fluctuation path versus an exponential weight function. The bottom two plots show how using the fluctuation path significantly decreases the fraction of particles that are killed during the selection steps ($N_{k}^{(-)}$) to the number of particles at that time step ($N_{k}$). This is also illustrated in the plots in the top and middle rows. The top plots show the paths from the initial state for all surviving particles at the final time. Paths that have been killed during the process are not shown. We call these paths the ancestral paths. As can be seen on the top left plot, only few trajectories from the initial stage of the algorithm are ancestors of the final positions. This is not the case for the top right plot. The algorithm using a weight based on a fluctuation path has a much larger number of ancestors. This richer ancestral tree results in a decreased estimator error for the over-threshold probabilities, as is demonstrated in Figure \[fig:instanton-variance\]. Note that for both the exponential weighting function and the weighting based on the fluctuation path, the paths reaching the threshold follow the fluctuation path. Other paths reaching the threshold are so rare that few of them are generated, even if they are more likely to survive selection in the case of exponential weighting. Note that the killed paths, partially shown in the middle row of Fig. \[fig:ancestral-paths\], tend to have a negative change in position before being killed. The higher target path in the exponential makes for a higher average dissipative force $-x$ on the particles, leading to a large discrepancy between the actual particle distribution and the target distribution at selection times. ![Ancestral paths (top), the final portion of killed paths, plotted only between $t_{k-1}$ and $t_{k}$ if the path is killed at $t_{k}$ (middle) and the fraction of the number of particles killed $N_{k}^{(-)}$ to the total number of particles $N_{k}$ (bottom) for genealogical particle analysis algorithms with either exponential weighting with $C=6.0$ (left) or weighting based on the fluctuation path (right) for the fluctuation path ending at $a=3.0$ at the final time $T=2$. The dashed black lines in the top plots show the fluctuation path. The dash-dotted line in the top left plot shows the mean of the target particle distributions after selection (equals $Cv(t)$). The average number of particles for both simulations is $M=10^{4}$ and the number of selections steps is $32$. For graphical purposes a randomly selected sample of 2% of the ancestral and killed paths are shown in the first two rows \[fig:ancestral-paths\]](exponential_vs_instanton){width="80.00000%"} ![The over-threshold probability $P(x>a)$ as estimated by the genealogical particle analysis algorithm either with an exponential weight (blue long-dashed line) or a weight based on fluctuation paths (red medium-dashed line). The two short-dashed lines, at equal distance from the estimated averages, correspond to a 2 standard deviation interval of the estimator. The full line is the analytic result. Both implementations use the same number of particles **$N=1e4$** and $32$ selections steps and both have roughly the same computational cost.\[fig:instanton-variance\]](exponential_vs_instanton2){width="75.00000%"} Genealogical particle analysis algorithm for a deterministic dynamical system: the Lorenz ’96 model\[sec:Interacting-particle-algorithm\] ========================================================================================================================================= The Lorenz ’96 model is a deterministic dynamical system that is often used as a toy model in the meteorology community. It was proposed by Lorenz as part of a study on error growth and predictability for chaotic dynamical systems [@lorenz_predictability:_1996; @lorenz_designing_2005]. A crucial difference between the famous Lorenz ’63 model and the less well-known Lorenz ’96 model is that the latter has a large number of degrees of freedom. Indeed macroscopic variables of deterministic systems with a large number of degrees of freedom often behave qualitatively similar to solutions of stochastic differential equations with much less degrees of freedom. Such results can be proven for some specific types of models (feauturing separation of time scale, independence, ...). It is believed however that similar results remain true for a wide range of models and observables even though mathematical proofs are out of reach. If this conjecture is correct, then the sampling of rare events through genealogical particle analysis algorithms should be applicable to macroscopic variables of deterministic systems with a large number of degrees of freedom. In this section, we demonstrate empirically, through numerical simulation, that genealogical particle analysis algorithms can indeed efficiently sample the tail of the energy distribution for the Lorenz ’96 model. Description of the model\[sub:Description-of-the\] -------------------------------------------------- The Lorenz ’96 model consists of $L$ variables $x_{i}$ on a ring $i\in\{0,..,L-1\}$, with dynamics $$\dot{x}_{i}=x_{i-1}(x_{i+1}-x_{i-2})+R-x_{i}$$ where indices $i$ are in $\mathbb{Z}_{L}$, i.e. the index $i$ is identified with $i\bmod L$ if $i\notin\{0,..,L-1\}$. The non-linear part of the dynamics $x_{i-1}(x_{i+1}-x_{i-2})$ conserves the energy $E(x)=\frac{1}{2L}\sum_{i=1}^{L}x_{i}^{2}$, while $R$ is a forcing and $-x_{i}$ a linear dissipation. The dynamics is chaotic for $R\geq8$ [@lorenz_predictability:_1996; @lorenz_designing_2005]. We will estimate the probability of reaching a certain energy threshold after a time $t$, starting from the zero vector **$x_{0,i}=0\,\forall i$**. [[A small perturbation $\epsilon\mathcal{\vec{N}}_{0,1}$ is added to the particle configuration to make the trajectories diverge. For large enough times $T$, the system will therefore relax to its physical invariant measure and it makes sense to determine probabilities of exceedances of macroscopic observables $\gamma_{E_t}:=P(E(x(T))> E_t)$.]{}]{} Throughout the article we will use a number of variables $L=32$ and a forcing $R=2^{8}=256$. Figure \[fig:energy\_hist\] shows a plot of the over-threshold probabilities of the energy of the Lorenz ’96 system, estimated through a brute force Monte-Carlo simulation from randomly perturbed initial conditions. Given that we have finite computer resources at our disposal, assume we can generate at most $M=10^{5}$ independent measurements of the energy. If the maximal relative error that we are willing to tolerate is for example $0.5$ then since $RE=\sqrt{\gamma_{E_t}-\gamma_{E_t}^{2}}/(\sqrt{M}\gamma_{E_t})\approx1/(\sqrt{M\gamma_{E_t}})$ the lowest probability that we can estimate is approximately $\gamma_{E_t}=1/M(RE)^{2}=4.10^{-5}$. From Figure \[fig:energy\_hist\] we can deduce that the corresponding highest energy threshold obtainable lies around an energy threshold $E_{t}=1785$. Beyond this threshold the use of rare event algorithms becomes necessary. ![Over-threshold probability $\gamma_{E_t} = P(E(t)>E_t)$ estimated from a brute force simulation and estimator variance $RE=\sqrt{\gamma_{E_t}-\gamma_{E_t}^{2}}/(\sqrt{M}\gamma_{E_t})$ of the energy $E$ of the Lorenz ’96 system with $R=2^{8}$ and $M=10^{5}$. []{data-label="fig:energy_hist"}](l96_energy_hist){width="50.00000%"} Algorithm implementation ------------------------ We use the following settings for the genealogical particle analysis simulation. The initial condition is set to [[$x_{i}=\epsilon\mathcal{\vec{N}}_{0,1}$]{}]{}. The total integration time per realization is **$T=1,27$**. This time interval corresponds to roughly $5$ times the decorrelation time of the energy observable. The standard deviation of the estimator $\sqrt{\frac{1}{K}\sum_{i=1}^{K}(\breve{\gamma}_{A,i}-\gamma_{A})^{2}}$ is estimated from $K=10$ independent runs of the algorithm and the truth $\gamma_{A}$ is taken from a long brute force Monte Carlo simulation. The number of interaction is set to $64$. ### Weight function\[sub:Weight-function\] For simplicity, we have employed an exponential weight function $W=\exp(C\Delta E)$ where $\Delta E$ is the change in energy between two interactions. This choice doesn’t require any a priori knowledge of the dynamics and is easy to implement. This weight function is not optimal, but, as we will show, it already gives good results. For the value of the forcing parameter $R=2^{8}$ the distribution of the energy values is roughly Gaussian. One can therefore estimate the mean $\mu_{E}$ and the variance $\sigma_{E}^{2}$ from a brute force Monte Carlo simulation and use these values along with the reasoning of Section \[sub:Skewing-a-normal\] to determine an appropriate value of the exponential weighting factor $C$ in the weighting function $W=\exp(C\Delta E)$. One can then choose a value $C=\Delta\mu_{E}/\sigma_{E}^{2}$ where $\Delta\mu_{E}$ is the desired change of the mean energy of the final particle distribution. The values of $C$ in the weighting function $W=\exp(C\Delta E)$ for the calculations presented in this section are taken as $C_{r}=r/(2\sigma_{E})$ with $r\in\{1,2,3,,4\}$ and $\sigma_{E}$ being the standard deviation of the energy so as to increase the mean energy $\Delta\mu_{E}$ by steps of size $\sigma_{E}/2$. ### Noise perturbation For deterministic dynamical systems, in order for two trajectories to have different dynamics after selection, a small perturbation can be added. This can be achieved by adding for example a weak Brownian perturbation at all times, or by adding a small instantaneous perturbation to offspring at the selection times. The former approach provides a simpler mathematical framework. Indeed the study of the noise effect would amount to the study of the stochastic differential equation properties in the weak noise limit, independent of the genealogical particle analysis algorithm. By contrast the latter approach intertwines the random perturbation with the genealogical particle analysis algorithm effects and is therefore more complicated to analyze. The latter approach, however, has the practical advantage of being computationally simpler. In this study, as we will proceed purely empirically, we have opted for the latter approach. The clones are perturbed by $\epsilon\mathcal{\vec{N}}_{0,1}$ where $\mathcal{\vec{N}}_{0,1}$ is a standard $L$-dimensional Gaussian random variable, i.e. the noise acts independently on all of the variables. The small noise perturbation invariably adds an error to the estimates of the tail probabilities. To obtain a rough upper bound on the strength of the perturbation that can be added without significantly perturbing the tail, we first perform a brute force simulation with the added noise for different noise strengths and verify that the tail probabilities do not change significantly compared to the sampling error of the brute force calculation. A set of independent realizations is performed like in the brute force Monte Carlo approach, the only difference being that at the selection times $t_{k}$ the same noise perturbations is added as in the genealogical particle analysis simulation. No selection is performed however in these runs. This way we can estimate the effect of the noise on the final time particle distribution. Figures \[fig:epsilons-1\] and \[fig:epsilons\] show that below a perturbation strength of $\epsilon=0.87$ and for thresholds higher than $a=1600$, the noise does not have a significant effect on the over-threshold probabilities. More complex schemes of noise perturbation could be implemented to assure that the perturbed trajectory remains close to the attractor, for example by storing a configuration at a time point before $t_{k}$, adding a small perturbation to it and evolve it up to $t_{k}$ to have the perturbation relax towards the attractor. Furthermore, after performing the genealogical particle analysis algorithm, we check that the perturbing noise intensity $\epsilon$ is small enough by decreasing $\epsilon$ and checking that the estimates of the over-threshold tail statistics are consistent. Figure \[fig:epsilons\] shows that for $\epsilon=0.1$ the results remain stable upon halving the noise intensity. ### Estimating the tail of the energy distribution We use for the Lorenz ’96 model the procedure described in Section \[sub:Estimating-the-tail\]: we increase the values of the weight parameter $C$ and use for each threshold value the best estimate from the point of view of the empirical estimator variance. The result is shown in Figure \[fig:l96\_tail\_estimation\]. As there is no analytic expression for the energy distribution tail of the Lorenz ’96 system, we use a long brute force Monte Carlo estimation as comparison. The estimator variance markedly decreases when using the genealogical particle analysis algorithm. When constructing the over-threshold probability, we see that the tail can be reliably reproduced when compared to the longer brute force calculation. The improvements in efficiency from using a rare event simulation scheme can be quantitatively estimated from Figure \[fig:l96\_tail\_estimation\]. The plot of the empirical relative error shows how for a threshold around $a=1800$ a brute force Monte Carlo calculation yields a relative error of $0.5$, whereas the genealogical particle analysis simulation yields a relative error of approximately $0.05$. A reduction in relative error by a factor $10$ is achieved. Since the brute force Monte Carlo error scales as $1/\sqrt{M}$, a similar reduction by a raw increase of processing power would require $M$ to increase by a factor of $100$. For higher thresholds and with more fine tuning of the selection process, a much larger reduction is likely to be achievable. Conclusion\[sec:Conclusion\] ============================ In this paper we have addressed the use of rare event computation techniques to estimate small over-threshold probabilities of observables in deterministic dynamical systems. We have demonstrated that the genealogical particle analysis algorithms can be successfully applied to a toy model of atmospheric dynamics, the Lorenz ’96 model as presented in Section \[sub:Description-of-the\]. We have furthermore used the Ornstein-Uhlenbeck system to illustrate a number of implementation issues. The example of the Ornstein-Uhlenbeck has illustrated the importance of the choice of the objective function for the performance of the genealogical particle analysis algorithm estimator. We have shown how a time-dependent objective function based on the fluctuation path to a high threshold can greatly improve the performance of the estimator compared to a fixed-in-time objective function. Furthermore we have discussed how the number of particles and the number of selection steps influence the performance of the estimator. For the deterministic chaotic system a complication arises in that a stochastic perturbation needs to be added to the system to make identical clones of one parent diverge and explore the system’s path space. We have demonstrated in this example how the estimates of the rare event simulation are stable for small perturbations and agree with results from brute force Monte Carlo estimations. We therefore can have confidence in the correctness of these estimates. For the example of a deterministic chaotic system that we have studied we have not yet used the fluctuation path approach, since this would require information on the dynamics to the rare event that we a priori do not possess. This lack of knowledge can be improved by iterating the estimation procedure, where one uses estimates of an initial brute force simulation to estimate the fluctuation path, after which an genealogical particle analysis simulation based on this path can be used to estimate a higher fluctuation path, which can be used for a next iteration of the algorithm. However, the results of the straightforward implementation of the rare event simulation already shows significant improvements compared to brute force estimation. Acknowledgements {#acknowledgements .unnumbered} ---------------- The authors would like to thank Eric Vanden-Eijnden, Eric Simonnet, Joran Rolland, Pascal Yiou, Bérengère Dubrulle, Francesco Ragone and Takahiro Nemoto for fruitful discussions. JW acknowledges the support of the AXA Research Fund. The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement n° PIOF-GA-2013-626210.

--- author: - | Gunnar Bali,\ Institut für Theoretische Physik, Universität Regensburg,\ 93040 Regensburg, Germany\ E-mail: ,\ - '(QCDSF Collaboration)' title: 'Static-light meson-meson potentials' --- Introduction ============ We numerically determine ground and excited states of static-light mesons (${\mathcal B}=Q\bar{q}$) as well as intermeson potentials between pairs of static-light mesons, ${\mathcal B}({\mathbf r}){\mathcal B}({\mathbf 0})$ and ${\mathcal B}({\mathbf r})\overline{\mathcal B}({\mathbf 0})$, with a static quark-quark (or quark-antiquark) separation $r=|{\mathbf r}|=Ra$. $a$ denotes the lattice spacing, $Q$ a static colour source and the positions of the light quarks $q$ are not fixed. For large heavy quark masses the spectra of heavy-light mesons are determined by excitations of the light quark and gluonic degrees of freedom. In particular, the vector-pseudoscalar splitting vanishes and the static-light meson ${\mathcal B}$ can be interpreted as either a $\overline{B}$, a $\overline{B}^*$, a $D$ or a $D^*$ heavy-light meson. Static-light intermeson potentials were first evaluated on the lattice by Michael and Pennanen in the quenched approximation [@Michael:1999nq] and with $N_{\mathrm f}=2$ Sheikholeslami-Wohlert sea quarks [@Pennanen:1999xi]. A more detailed quenched study can be found in ref. [@Detmold:2007wk] and the un-quenched case is revisited with twisted mass fermions by Wagner in these proceedings [@Wagner:2010ad]. Meson-antimeson potentials were computed in ref. [@Pennanen:1999xi] and, with Wilson sea quarks, in ref. [@Sesam]. Graphically, the quark line diagrams that we evaluate can be depicted as, $$\begin{aligned} \parbox{.03\textwidth}{ \begin{fmffile}{diag1} \begin{fmfgraph*}(10,40) \fmfstraight \fmfkeep{static-light} \fmfleft{lu,lo} \fmfright{ru,ro} \fmf{fermion}{lu,lo} \fmf{wiggly,tension=0,left=0.5}{lo,lu} \fmf{phantom_arrow,tension=0,left=0.5}{lo,lu} \end{fmfgraph*} \end{fmffile}} \;\;,\;\;\qquad \parbox{.08\textwidth}{ \begin{fmffile}{diag2} \begin{fmfgraph*}(30,40) \fmfstraight \fmfkeep{static-light} \fmfleft{lu,lo} \fmfright{ru,ro} \fmfbottom{lu,um,ru} \fmf{fermion}{lu,lo} \fmf{fermion}{ru,ro} \fmflabel{$\leftarrow$ $R$ $\rightarrow$}{um} \fmf{wiggly,tension=0,left=0.5}{lo,lu} \fmf{phantom_arrow,tension=0,left=0.5}{lo,lu} \fmf{wiggly,tension=0,right=0.5}{ro,ru} \fmf{phantom_arrow,tension=0,right=0.5}{ro,ru} \end{fmfgraph*} \end{fmffile}} \;\;,\;\;\;\quad \parbox{.08\textwidth}{ \begin{fmffile}{diag3} \begin{fmfgraph*}(30,40) \fmfstraight \fmfkeep{static-light} \fmfleft{lu,lo} \fmfright{ru,ro} \fmfbottom{lu,um,ru} \fmf{fermion}{lu,lo} \fmf{fermion}{ro,ru} \fmflabel{$\leftarrow$ $R$ $\rightarrow$}{um} \fmf{wiggly,tension=0,left=0.5}{lo,lu} \fmf{phantom_arrow,tension=0,left=0.5}{lo,lu} \fmf{wiggly,tension=0,left=0.5}{ru,ro} \fmf{phantom_arrow,tension=0,left=0.5}{ru,ro} \end{fmfgraph*} \end{fmffile}}\;\; , \label{equ:feynman1} \\ \nonumber\end{aligned}$$ where the static-light correlation function is shown on the left, and those for the meson-meson and meson-antimeson cases in the centre and on the right, respectively. Straight lines represent static quark propagators and wiggly lines light quark propagators. In this paper we neglect explicit meson exchange (i.e. “box” and “cross”) diagrams. The analyses of these as well as of a larger lattice volume are in progress. This means that here we only consider the isospin $I=0$ ${\mathcal B}{\mathcal B}$ and the $I=1$ ${\mathcal B}\overline{\mathcal B}$ combinations. The static-light correlation function in Euclidean time $t=Ta$ is given by, $$\begin{aligned} C(t) = \langle 0 | (\overline{Q}_{\alpha} \; \mathcal{O}_{\alpha \beta} \; q_{\beta})_{x+t\hat{4}} \; (\bar{q}_{\gamma} \; \mathcal{O}_{\gamma\delta} \; Q_{\delta})_{x} | 0 \rangle = \left\langle \; \mathrm{Tr} \; \left[ \prod^{T -1}_{k=0} \, U^{\dagger}_{x + ka\hat{4},4} \; \frac{\Eins + \gamma_4}{2} \;\mathcal{O} \; M^{-1}_{x+t\hat{4},x} \; \mathcal{O} \right] \right\rangle_{\!\!\!U} \; . \label{equ:correlator}\end{aligned}$$ The trace is over colour (not displayed) and Dirac indices, $\left\langle \cdot\right\rangle_U$ indicates the expectation value over gauge configurations and $\hat{\mu}$ denotes a unit vector in $\mu$-direction. We average over all possible source points $x/a \in \{1,\ldots,L_{\sigma}\}^3\times \{1,\ldots,L_{\tau}\}$ to reduce statistical errors where $x=({\mathbf x},x_4)$. The correlator is automatically zero-momentum projected since ${\mathbf x}$ is the same at source and sink, due to the static propagator. $M^{-1}_{yx}=\langle q_y\bar{q}_x\rangle$ is the propagator for the light quark $q$ on a given gauge configuration and $U_{x,\mu}$ is the gauge link connecting the lattice site $x$ with $x+a\hat{\mu}$. The absence of the spin in the static propagator necessitates the $(\Eins+\gamma_4)/2$ Dirac projection of the (fermionic) static-light “meson” to fix the parity $P$. This is very similar to baryonic correlation functions where a spin $\frac12$ source is created by three (rather than one) light quarks. Meson-(anti)meson correlation functions can be obtained by combining the above correlator with another one that is spatially shifted by a distance ${\mathbf r}$, before taking the gauge average. Representations and classification of states ============================================ In the continuum limit, the static-light states can be classified according to fermionic representations $J^P$ of the rotation group ${\mathrm O(3)}$. At vanishing distance ${\mathbf r}={\mathbf 0}$ the ${\mathcal B}{\mathcal B}$ and ${\mathcal B}\overline{\mathcal B}$ states can be characterized by integer $J^P$ and $J^{PC}$ quantum numbers, respectively. However at $r=|{\mathbf r}|>0$ the $\mathrm{O(3)}$ (or $\mathrm{O(3)}\otimes{\mathcal C}$) symmetry is broken down to its cylindrical $\mathrm{D_{\infty h}}$ subgroup. The irreducible representations of this are conventionally labelled by the spin along the axis $\Lambda$, where $\Sigma,\Pi,\Delta$ refer to $\Lambda=0,1,2$, respectively, with a subscript $\eta=g$ for gerade (even) $PC=+$ or $\eta=u$ for ungerade (odd) $PC=-$ transformation properties with respect to the midpoint. All $\Lambda\geq 1$ representations are two-dimensional. The one-dimensional $\Sigma$ representations carry an additional $\sigma_v=\pm$ superscript for their reflection symmetry with respect to a plane that includes the two endpoints. To create states of different $J^{P(C)}$ we use operators $\mathcal{O}$ that contain combinations of Dirac $\gamma$-matrices and covariant lattice derivatives $\nabla[U]$ that act on a fermion spinor $q$ as, $$\begin{aligned} \nabla_{\mu} q_x = U_{x,\mu} q_{y+a\hat{\mu}} - U_{x,-\mu} q_{x-a\hat{\mu}}\,, \quad \mbox{where} \quad U_{x,-\mu} = U^{\dagger}_{y - a\hat{\mu},\mu}\,.\end{aligned}$$ On the lattice the continuum rotational symmetry is broken and the groups $\mathrm{O(3)}$ and $\mathrm{D_{\infty h}}$ need to be replaced by their finite dimensional subgroups $\mathrm{O_h}$ and $\mathrm{D_{4h}}$, respectively. We label fermionic representations of the octahedral group $\mathrm{O_h}$ as $\mathrm{O_h}'$. For fermionic representations of $\mathrm{D_{\infty h}}$ that we do not need in the present context, see ref. [@Najjar:2009da]. It is well known, see e.g. ref. [@Bali:2000gf], that the assignment of a continuum spin to a lattice result can be ambiguous, in particular for radial excitations because a given $\mathrm{O_h}$ representation can be subduced from several continuum $J$s. For instance, $$G_1 \leftarrow J = \frac{1}{2}, \frac{7}{2}, \dots\,,\quad H \leftarrow J = \frac{3}{2}, \frac{5}{2}, \dots\,,\quad A_1 \leftarrow J = 0, 4,\dots\,,\quad T_1 \leftarrow J = 1, 3, 4,\dots\,. \label{equ:mapping}$$ For $\Lambda\leq 2$ the mapping of continuum $\mathrm{D_{\infty h}}$ onto discrete $\mathrm{D_{4h}}$ representations is more straight forward. Hence in this case we adopt the continuum notation only. The operators that we used to create the static-light mesons are displayed in table \[tab:operators\_sl\] (see, e.g., ref. [@Michael:1998sg]). The intermeson potentials were obtained by combining two static-light mesons of different (or the same) quantum numbers. This can be projected into an irreducible $\mathrm{D_{\infty h}}$ representation, either by coupling the light quarks together in spinor space [@Wagner:2010ad] or by projecting the static-light meson spins into the direction $\hat{\mathbf r}$ of the static source distance, by applying $\frac12(\Eins\pm i\gamma_5\pmb{\gamma}\cdot\hat{\mathbf{r}})$, and taking appropriate symmetric ($\Lambda_s=1$) or antisymmetric ($\Lambda_s=0$) spin combinations. These two approaches can be related to each other via a Fierz transformation. For the preliminary results presented here we have not yet performed this projection and different representations will mix. The analyzed operators and the corresponding representations are listed in table \[tab:operators\_potential\]. For $J>0$ and $r>0$ the irreducible representations of $\mathrm{O(3)}$ split up into two or more irreducible representations of $\mathrm{D_{\infty h}}$. For instance the angular momentum of the $P_+$ operator within our $1^-,2^-$ $r=0$ state can be perpendicular or parallel to the intermeson axis. For the axis pointing into the $\hat{3}$-direction, we call the $S\otimes P_+$ operator $\gamma_5\otimes(\gamma_1\nabla_1 - \gamma_2\nabla_2)$ “parallel" ($\|$) and the other combinations “perpendicular" ($\bot$). The $\bot$ state has no angular momentum pointing into the direction of the axis and hence only couples with the light quarks to $\Sigma_u^+$ and $\Pi_u$. Vice versa, the $\|$ operator can only create $\Pi_u$ and $\Delta_u$ states but not the $\Sigma_u^+$. Simulation and analysis ======================= We employ $N_{\mathrm f}=2$ Sheikholeslami-Wohlert configurations generated by the QCDSF Collaboration [@AliKhan:2003br]. The parameter values are listed in table \[tab:lattice\], where the scale is set using $r_0(\beta,\kappa)=0.467$ fm. The pseudoscalar mass corresponds to its infinite volume value. We use the Chroma software system [@Edwards:2004sx]. To achieve high statistics in the evaluation of the diagrams eq. (\[equ:feynman1\]), all-to-all propagators need to be computed. This is done using stochastic estimator techniques, see ref. [@Bali:2009hu] and references therein. We generate 300 complex ${\mathbb Z}_2$ noise sources and apply the hopping parameter expansion to reduce the stochastic variance [@Thron; @Sesam; @Bali:2009hu]. Furthermore we enhance the signal over noise ratio by employing a static action with reduced self-energy [@Sesam]. This is done by applying one stout smearing step [@Stout] with the parameter $\rho=1/6$ to the temporal links, used to calculate the static propagators. Wuppertal smearing [@Wuppertalsmear] is applied to the source and sink operators, where we employ spatially smeared parallel transporters [@Sesam] with the parameters $n_{\mathrm{iter}}=15$, $\alpha=2.5$. The Wuppertal smearing hopping parameter value $\kappa_{\mathrm w}=0.3$ is combined with three iteration numbers $N_{\mathrm{iter}}\in\{16,50,100\}$. Masses are then extracted from the resulting correlation matrices, by means of the variational method [@Vari], solving a generalized eigenvalue problem. Errors are calculated using the jackknife method. Results ======= The eigenvalues $\lambda^{(k)}(t,t_0)$ of the generalized eigenvalue problem [@Vari], $$C_{ij}(t)u_j^{(k)}=\lambda^{(k)}(t,t_0)C_{ij}(t_0)u_j^{(k)}\,,$$ are fitted to one- and two-exponential ansätze, to obtain the $k$th mass. The appropriate values of $t_0$ and the fit ranges in $t$ are determined from monitoring the effective masses, $$m_{\mathrm{eff},t_0}^{(k)}(t+a/2)=a^{-1}\ln\left(\frac{\lambda^{(k)}(t,t_0)} {\lambda^{(k)}(t+a,t_0)}\right)\,.$$ On the left hand side of figure \[fig:mp\_para\], we display the effective ground state energy levels $E_{\mathrm{eff}}$ for $t_0=2a$ of the $\gamma_5 \otimes \gamma_5$ ${\mathcal B}{\mathcal B}$ system for different distances $R=r/a$. In the limit $r\rightarrow\infty$ these will approach the sum of two $\frac{1}{2}^+$ static-light meson effective masses (dotted curve). At short distances we see attraction in this channel. The quality of the effective mass plateaus deteriorates with decreasing distance since the wavefunction of such interacting states becomes more than the mere product of our two static-light meson interpolators. We should also keep in mind that so far we did not perform the singlet spin projection and hence there will be additional pollution from the $\Pi_g$ state, see table \[tab:operators\_potential\]. For the example displayed, we perform two-exponential fits to the $t/a\in\{4,\ldots,10\}$ data to obtain the masses. We define intermeson potentials as the difference between the meson-meson energy levels and the $r\rightarrow\infty$ two static-light meson limiting cases: $$V_{\!\mathcal{B}_1\mathcal{B}_2}(r)=E_{\!\mathcal{B}_1\mathcal{B}_2}(r) -\left(m_{\!\mathcal{B}_1}+ m_{\!\mathcal{B}_2}\right)\quad\stackrel{r\rightarrow\infty}{\longrightarrow}\quad 0\,.$$ The results for the groundstate ($\Sigma_g^+$) and the first excited state ($\Sigma_g^{+\prime}$) of the $\gamma_5 \otimes \gamma_5$ operator and the $\Sigma_u^-$ groundstate of $\gamma_5 \otimes \Eins$ are plotted on the right hand side of figure \[fig:mp\_para\]. In all these $I=0$ channels there is attraction of the order of 50 MeV at a distance of $0.2$ fm. In figure \[fig:mp\_split\_antipara\] we display the ground state and first excited state energy levels for the ${\mathcal B}\overline{\mathcal B}$ meson-antimeson case in the $\gamma_5 \otimes \gamma_5$ channel. The two horizontal lines correspond to twice the ground state mass of the $\frac12^+$ static-light meson and to the sum of its ground and its first excited state masses, the expected $r\rightarrow\infty$ limits. At first sight there appear to be very substantial short distance attractive forces in this channel. However, states consisting of a $Q\overline{Q}$ static potential and a scalar $I=1$ particle will have the same quantum numbers. For our lattice parameters the $a_0$ meson is the lowest such state, with masses of two pseudoscalars as well as of a $P$-wave vector lying higher. We include the sum of these two masses in the figure. The ground state ${\mathcal B}\overline{\mathcal B}$ energy still lies below this level but to decide whether we effectively see the sum of $a_0$ and the static potential and to disentangle which $I=1$ $\Sigma_g^+$ energy level is the lowest one, interactions of the $a_0$ with the static potential will have to be taken into account. We hope that simulations on a larger volume will help to clarify this. Conclusions =========== We investigated interactions between pairs of static-light mesons and found attraction in the $I=S=0$ sector. Meson-antimeson potentials are also very interesting with respect to charmonium threshold states [@Brambilla:2010cs] ($D\overline{D}$ molecules or tetraquarks) but difficult to disentangle from mesons that are bound to static-static states (hadro-quarkonium [@Dubynskiy:2008mq]). 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--- abstract: 'In this work, we propose a method for calculating the free energy of anisotropic classical spin systems. We use a Hubbard-Stratonovich transformation to express the partition function of a generic bilinear super-exchange Hamiltonian in terms of a functional integral over classical time-independent fields. As an example, we consider an anisotropic spin-exchange Hamiltonian on the cubic lattice as is found for compounds with strongly correlated electrons in multiorbital bands and subject to strong spin-orbit interaction. We calculate the contribution of Gaussian spin fluctuations to the free energy. While the mean-field solution of ordered states for such systems usually has full rotational symmetry, we show here that the fluctuations lead to a pinning of the spontaneous magnetization along some preferred direction of the lattice.' author: - Yuriy Sizyuk - 'Natalia B. Perkins' - Peter Wölfle title: 'Lifting mean-field degeneracies in anisotropic classical spin systems' --- Introduction ============ Recent research activities on transition metal oxides suggest that the interplay of the strong spin-orbit coupling (SOC), crystal field (CF) interactions, and electron correlations may lead to compasslike anisotropic interactions between magnetic degrees of freedom.[@nussinov15] These anisotropic interactions have a generic form $J_{ij}^{\alpha}S_{i}^{\alpha }S_{j}^{\alpha}$ in which $\alpha$ depends on the direction of the particular link or bond and $S$ denotes spin or pseudospin degrees of freedom describing magnetic or orbital degrees of freedom. The models in which compasslike anisotropies are dominating, or also the pure compass models, have been known for a long time. These models appear naturally in strongly correlated electron systems as minimal models to account for interactions between pseudospins describing orbital degrees of freedom.[@kk82; @nussinov04; @nussinov05; @cincio10; @nasu12; @oles13] The compass-like anisotropies also arise as interactions between magnetic degrees of freedom in systems with strong SOC, which might be realized in 4d and 5d transition metal oxides.[@jackeli09] However, in these systems, due to the extended nature of 4d and 5d orbitals, the compass interactions are always accompanied by the usual SU(2) symmetric Heisenberg-type exchange. These models are especially interesting because while the pure compasslike models are rare, the combined Heisenberg-compass models have been shown to be minimal models describing the magnetic properties of various materials. A review of the different realizations of compass models, [@kk82; @nussinov04; @nussinov05; @cincio10; @nasu12; @oles13; @jackeli09; @batista05; @biskup05; @kitaev06; @jackeli10; @chern10; @oitmaa10; @fabien10; @khali2001; @wenzel10; @perkins14; @sizyuk14; @plakida14; @gerlach15] their physical motivations, symmetries, unconventional orderings and excitations may be found in the recent paper by Nussinov and van den Brink.[@nussinov15] One of the common features induced by compasslike anisotropies is frustration, arising from a competition of interactions along different directions and leading to the macroscopic degeneracy of the classical ground state and in addition to rich quantum behavior. In many cases, the pure compass models do not show conventional magnetic ordering because the degeneracy of the classical ground state is connected to discrete sliding symmetries of the model.[@nussinov05; @batista05] Because these symmetries are intrinsic symmetries of the model, they can not be lifted by the order-by disorder mechanisms. Instead, the direct consequence of the existence of these symmetries is that the natural order parameters for pure compass models are nematic, which are invariant under discrete sliding symmetries. The nematic order present in the compass model is fragile and is easily destroyed by the presence of the isotropic Heisenberg interaction which breaks some of the intrinsic symmetries of the model. In Heisenberg-compass models, some of the degeneracies become accidental. In these models, the true magnetic order might be selected by fluctuations via an order by disorder mechanism, removing accidental degeneracies and determining both the nature and the direction of the order parameter. Despite the simplicity of these models, the interplay of the Heisenberg and compass interaction leads to very rich phase diagrams even in the simplest case of the square lattice.[@fabien10] For classical systems this mechanism requires finite temperatures, where entropic contributions of fluctuations to the free energy become effective. In this work, we will be interested in studying the directional ordering transitions in the Heisenberg-compass model on the cubic lattice.[@khali2001] From a historical perspective, the three-dimensional 90$^{\circ}$-compass model was the first model of this kind proposed by Kugel and Khomskii[@kk82] in the context of the ordering of the $t_{2g}$ orbitals in transition metal oxides with perovskite structure and then studied in more details by Khaliullin[@khali2001] in application to LaTiO$_{3}$. The formal procedure which we will be using here is based on the derivation of the fluctuational part of the free energy by integrating out the Gaussian fluctuations, and determining which orientations of the vector order parameter correspond to the free energy minimum. To do so, we first express the partition function as a functional integral over classical fields. In this first paper, we consider classical spins at finite temperature. Our starting point in evaluating this exact representation of the partition function is the mean-field solution, which usually does not reflect the anisotropic character of the interaction referring to the crystal lattice axes. As a next step, we evaluate the contribution of Gaussian fluctuations to the free energy of the mean field ordered state. The latter carries the information embodied in the anisotropic spin interaction and therefore allows to define preferred directions of the spin order with respect to the lattice. We will not go beyond the simple evaluation of the contribution of fluctuations, e.g., by incorporating the fluctuation contribution self-consistently. For simplicity, we choose the parameters of the model such that the ground state is ferromagnetic, i.e. we consider the Heisenberg interaction to be ferromagnetic and allow the compass interaction to be both ferromagnetic and antiferromagnetic. For any ferromagnetic and weak antiferromagnetic compass interactions, the minima of the fluctuational part of the free energy are attained if the spontaneous magnetization vector points along one of the cubic axes. This paper is organized as follows. In section II we introduce the functional integral representation of the partition function for the spin systems with interactions described by the most general bilinear form of the super-exchange Hamiltonian. The details of the method are outlined in the Appendix. In Sec. III, we apply this framework to compute the angular dependence of the fluctuational part of the free energy for the ferromagnetic Heisenberg-compass model on the cubic lattice. Our results are presented and discussed in Section IV. Representation of the partition function {#sec:partition function} ======================================== We consider a system of identical classical spins $\mathbf{S}$ on a lattice, interacting in an anisotropic fashion as indicated in the introduction, defined by the Hamiltonian $$H=\frac{1}{2}\sum_{j,j^{\prime}}\,\sum_{\alpha\alpha^{\prime}}\,J_{j,j^{\prime }}^{\alpha,\alpha^{\prime}}\,S_{j}^{\alpha}\,S_{j^{\prime}}^{\alpha^{\prime}}, \label{ham}$$ where $j,j^{\prime}$ label the lattice sites, $\alpha,\alpha^{\prime }=x,y,z$ label the three components of the spin and $\mathbf{S}^2=1$. For the models with compasslike anisotropic and Heisenberg isotropic interactions of spins, the interaction is diagonal in spin space, $\alpha=\alpha^{\prime}$. The $J_{j,j^{\prime}}^{\alpha,\alpha}$-matrix elements are different for the $(j,j^{\prime})$-bonds along direction $\gamma$ with $\gamma=\alpha$ and $\gamma\neq\alpha$. However, since our consideration is also valid for the case when $\alpha\neq\alpha^{\prime}$, in the following, we will keep both indices. We will be interested in the long-range ordered phases of the system. The mean field approximation of the order parameter usually leads to a highly degenerate manifold of states, e.g., a ferromagnetic state with spontaneous magnetization pointing in any direction. This degeneracy is lifted by the anisotropic components of the spin interaction, but only at the level of the fluctuation contribution to the free energy (action) $S_{fl}$. In the following, we outline a method allowing to calculate $S_{fl}$, which is based on the Hubbard-Stratonovich transformation of the partition function for spin systems described by the generic Hamiltonian (\[ham\]). We present details and discuss justifications for this method in the Appendix. The partition function of the system is given by the integral over the Boltzmann weights of configurations $$Z=\int[dS_{j}]\exp[-\beta\sum_{j\alpha,j^{\prime}\alpha^{\prime}}J_{jj^{\prime}}^{\alpha\alpha^{\prime}}S_{j}^{\alpha}S_{j^{\prime}}^{\alpha^{\prime}}] \delta(\mathbf{S}_{j}^{2}-1),$$ where $\beta=1/k_BT$ is the inverse temperature, $S_{j}^{\alpha}$ are the components of the spin operator at site $j$. It is useful to represent the Hamiltonian in the basis of the eigenfunctions $\chi_{n;j,\alpha}$ of the spin exchange matrix, defined by$$\sum_{j^{\prime},\alpha^{\prime}}J_{jj^{\prime}}^{\alpha\alpha^{\prime}}\chi_{n;j^{\prime},\alpha^{\prime}}=\kappa_{n}\chi_{n;j,\alpha}\,.$$ For spins on a periodic lattice these eigenstates are labeled by a wavevector $\mathbf{q}$ (inside the first Brillouin zone) and index $\nu$, characterizing three principle axes of the matrix ${\hat{J}}$. Thus $|n\rangle=|\mathbf{q},\nu\rangle$ and the normalized eigenfunctions take the form$$\chi_{\mathbf{q},\nu;j,\alpha}=\frac{1}{\sqrt{N}}e^{i\mathbf{q\cdot R}_{j}}u_{\nu,\alpha}\,,$$ where $N$ is the number of lattice sites, the $u_{\nu,\alpha}$ are orthonormal real-valued eigenvectors, i.e., $\sum_{\alpha}u_{\nu,\alpha}u_{\nu^{\prime },\alpha}=\delta_{\nu\nu^{\prime}}$ and $\kappa_{\mathbf{q},\nu}$ are the eigenvalues of the spin exchange interaction matrix. We now define the normal amplitudes of the spins as$$S_{\mathbf{q},\nu}=\sum_{j,\alpha}\chi_{\mathbf{q},\nu;j,\alpha}S_{j}^{\alpha}$$ and express the Hamiltonian as$$\begin{aligned} H=\sum_{\mathbf{q},\nu}\kappa_{\mathbf{q},\nu}S_{\mathbf{q},\nu}^{\ast }S_{\mathbf{q},\nu}\,,\end{aligned}$$ where $S_{\mathbf{q},\nu}^{\ast}=S_{-\mathbf{q},\nu}$. Commutation of classical spins allows us to employ a Hubbard-Stratonovich transformation in terms of classical fields $\varphi_{\mathbf{q,}\nu}$ in order to represent the interaction operator as a Zeeman energy operator of spins in a spatially varying magnetic field. As a result, one finds the following representation of the partition function: $$\begin{aligned} &&Z=\int[d\varphi]\\\nonumber &&\exp\Bigl(-\beta\bigl[\sum_{\mathbf{q,}\nu}|\kappa_{\mathbf{q,}\nu}|^{-1}\varphi_{\mathbf{q,}\nu}^{\ast}\varphi_{\mathbf{q,}\nu}-{\mathcal{S}}_{loc}(\{\varphi_{\mathbf{q,}\nu}^{\ast},\varphi_{\mathbf{q,}\nu}\})\bigr]\Bigr),\end{aligned}$$ where the integration volume element is given by $$\lbrack d\varphi]=\Pi_{\mathbf{q,}\nu}\frac{i\beta{d\varphi_{\mathbf{q,}\nu}^{\ast}}d\varphi_{\mathbf{q,}\nu}}{2\pi|\kappa_{\mathbf{q,}\nu}|}\,.$$ The contribution to the action in the case of classical spins is given by$$S_{loc}(\{\varphi_{\mathbf{q,}\nu}^{\ast},\varphi_{\mathbf{q,}\nu}\})=\beta^{-1}\sum_{j}\ln\bigl[\sinh(2\beta{\varphi}_{j})/2\beta {\varphi}_{j}\bigr],$$ where ${\varphi}_{j}^{2}=({\varphi}_{j}^{x})^{2}+({\varphi}_{j}^{y})^{2}+({\varphi}_{j}^{z})^{2}$ , with ${\varphi}_{j}^{\alpha}\equiv\sum_{\mathbf{q,}\nu}s(\kappa_{\mathbf{q,}\nu })\varphi_{\mathbf{q,}\nu}\chi_{\mathbf{q,}\nu;j,\alpha}^{\ast}$ and $s(\kappa_{\mathbf{q,}\nu})=1$ for $\kappa_{\mathbf{q,}\nu}<0$ and $s(\kappa_{\mathbf{q,}\nu})=i$ for $\kappa_{\mathbf{q,}\nu}>0$. The Hubbard-Stratonovich identity used to derive the above functional integral is different for eigenmodes $\varphi_{\mathbf{q,}\nu}$ with positive or negative eigenvalue $\kappa_{\mathbf{q,}\nu}$, leading to the appearance of a complex-valued ${\varphi}_{j}$. The details of evaluating ${\mathcal{S}}_{loc}(\{\varphi_{\mathbf{q,}\nu}^{\ast},\varphi_{\mathbf{q,}\nu}\})$ can be found in the Appendix. Application to the cubic lattice ================================ Isotropic Heisenberg interaction -------------------------------- In order to demonstrate how to perform the evaluation of the above representation of the partition function, we consider first the isotropic ferromagnetic Heisenberg model with nearest neighbor interactions on the cubic lattice. In this case, the Hamiltonian (\[ham\]) reads $$\begin{aligned} H=J\sum_{\langle j;j^{\prime}\rangle}\sum_{\alpha}S_{j}^{\alpha}S_{j^{\prime}}^{\alpha},\end{aligned}$$ where the lattice summation is over nearest neighbors $\langle j,j^{\prime }\rangle-$bonds and $J<0$. For the isotropic exchange interaction, the eigenvalues, $\kappa _{\mathbf{q,}\nu}=J\sum_{\alpha}\cos q_{\alpha}$, are independent of $\nu$, $\kappa_{\mathbf{q,}\nu}=\kappa_{\mathbf{q}}$, and hence are degenerate. A uniform ferromagnetic mean-field solution is found by solving the saddle point equation $$\begin{aligned} \label{MF} &\frac{\partial}{\partial\varphi_{{\small MF}}}{\mathcal{S}} =-\frac{\partial}{\partial\varphi_{{\small MF}}}N \Bigl[|\kappa_{\mathbf{q=0}}|^{-1}(\varphi_{MF})^{2}\\\nonumber& -\beta^{-1}\ln[\sinh(2\beta\varphi_{MF})/2\beta\varphi_{MF}]\Bigr]=0 ,\nonumber\end{aligned}$$ where we used $\varphi_{\mathbf{q},\nu}^{{\small MF}}=\sqrt{N}\varphi_{{\small MF}}\delta_{\mathbf{q},0}m_{0,\nu}$, ${\varphi}_{j}=\varphi_{{\small MF}}$, $m_{0,\nu}$ for the components of the unit vector along the magnetization in the reference frame defined by the principal axes of the interaction matrix (which are the cubic axes in this case), and $N$ is the number of lattice sites. The solution of Eq.(\[MF\]) gives us a non-linear equation for the mean-field parameter: $$\label{eq:MF} 2|\kappa_{\mathbf{q=0}} |^{-1}\varphi_{ MF}-2\coth(2\beta\varphi_{MF})+\frac{1}{\beta\varphi_{MF}}=0$$ We solve this equation numerically at each temperature and get $\varphi_{{\small MF}} (T)$. Linearizing Eq. (\[eq:MF\]) near the transition, we find the transition temperature $T_{c}=\beta_{c}^{-1}=2|\kappa_{\mathbf{q=0}}|/3$. We note in passing that a different length of the classical spin vector $|\mathbf{S}|=S_0$ may be simply scaled back to the unit length by changing the temperature as $T'=S_{0}^2 T$. Choosing $S_{0}^2=3/4$ appropriate for quantum spin $S=1/2$, we find the renormalized transition temperature $T'_{c}=|\kappa_{\mathbf{q=0}}|/2$ , which agrees with the quantum mean-field transition temperature. The fluctuation contribution is obtained by expanding the action in the fluctuation field $\delta\varphi_{\mathbf{q},\nu}=\varphi_{\mathbf{q},\nu}-\varphi_{\mathbf{q},\nu}^{{\small MF}}$ about the mean field solution to the lowest order: $$\begin{aligned} {\mathcal{S}} & =&{\mathcal{S}}_{0}+{\mathcal{S}}_{\mathrm{fl}}\\\nonumber {\mathcal{S}}_{0} & =& N|\kappa_{\mathbf{q=0}}|^{-1} \varphi_{\small MF}^2 -NT\ln[\sinh(2\beta{\varphi}_{MF})/2\beta {\varphi}_{MF}].\end{aligned}$$ For Gaussian fluctuations, the fluctuation part of the free energy, or equivalently the action, ${\mathcal{S}}_{\mathrm{fl}}$, is a bilinear function of $\delta\varphi_{\mathbf{q,}\nu}$. It is given by $${\mathcal{S}}_{\mathrm{fl}}\{\delta\varphi_{\mathbf{q,}\nu}\}=\sum_{\mathbf{q};\nu,\nu^{\prime}}A_{\mathbf{q,}\nu\nu^{\prime}}\delta\varphi_{\mathbf{q,}\nu }^{\ast}\delta\varphi_{\mathbf{q,}\nu^{\prime}},\label{fluct-free-energy}$$ where we defined matrix elements of $A_{\mathbf{q},\nu\nu^{\prime}}$ describing the weight of the Gaussian fluctuations of wavevector $\mathbf{q}$ and polarization $\nu$ as $$\begin{aligned} \label{fluct-matrix1} &&A_{\mathbf{q},\nu\nu^{\prime}} =|\kappa_{\mathbf{q},\nu}|^{-1}\delta_{\nu ,\nu^{\prime}} -\frac{2}{3}\Bigl[\beta_{c}(\delta_{\nu,\nu^{\prime}}-m_{0,\nu }m_{0,\nu^{\prime}})+ \nonumber \\&& 3\beta rm_{0,\nu}m_{0,\nu^{\prime}}]s(\kappa _{\mathbf{q},\nu})s(\kappa_{\mathbf{q},\nu^{\prime}})\Bigr]\end{aligned}$$ Here, for shortness we introduced $r=1/(2\beta \varphi_{MF})^{2}-1/\sinh^{2}(2\beta\varphi_{MF})$. \[fig1\] ![(Colors online) The magnitude of the action ${\mathcal{S}}_{\mathrm{fl}}(\theta,\phi)$ defined by Eq. (\[action1\]) is plotted on the surface of the unit sphere. The preferred directions of the magnetization, corresponding to the minima of the free energy, are shown by deep blue color. The energy scale is shown in units of $J$. $J=-1$ and $K=0.75$: the preferred directions of the magnetization are along the cubic axes.](fig1a.pdf "fig:"){width="0.65\columnwidth"} In the limit of small $\mathbf{q}$, it is instructive to separate the fluctuations into longitudinal (along $\mathbf{m}_{0}$) and transverse (perpendicular to $\mathbf{m}_{0}$) components, $\delta\varphi_{\bm q}^{\mathrm{l}}=\mathbf{m}_{0}\cdot\delta {\bm\varphi}_{\bm q}$ and $\delta{\bm\varphi}_{\bm q}^{\mathrm{tr}}=\sum_{\mu =1,2}\mathbf{m}_{\mu}\varphi_{{\bm q},\mu}^{\mathrm{tr}}$, respectively. We defined $\delta\varphi_{{\bm q},\mu}^{\mathrm{tr}}=\mathbf{m}_{\mu}\cdot \delta{\bm\varphi}_{\bm q}$, with $\mathbf{m}_{1}=(\mathbf{m}_{0}\mathbf{\times z)/|\sin}\theta\mathbf{|}$ and $\mathbf{m}_{2}=\mathbf{m}_{1}\times \mathbf{m}_{0}$, where $\cos\theta=\mathbf{m}_0\mathbf{\cdot z}$. Despite the complex nature of fluctuational fields, their separation into transverse and longitudinal modes is possible in the limit of small $\mathbf{q}$, because the interaction eigenvalues $\kappa_{\mathbf{q},\nu}<0$ and thus, $s(\kappa_{\mathbf{q},\nu})=1$ in this region of the BZ for any polarization component $\nu$. Then, the longitudinal fluctuations contribute to the free energy as $${\mathcal{S}}_{\mathrm{fl,l}}=\sum_{\bm q} \left[ |\kappa_{\bm q}|^{-1}-2\beta r\right] ({\bm m}_{0}\mathbf{\cdot}\delta{\bm\varphi}_{\bm q}^{\mathrm{l}})(\mathbf{m}_{0}\mathbf{\cdot}\delta{\bm\varphi}_{-{\bm q}}^{\mathrm{l}}).$$ The transverse fluctuations are gapless in agreement with Goldstone’s theorem:$${\mathcal{S}}_{\mathrm{fl,tr}}=\sum_{\bm q,\kappa_{\mathbf{q}}<0}\left[|\kappa_{\mathbf{q}}|^{-1}-\frac{2}{3}\beta_{c}\right](\delta{\bm\varphi}_{\mathbf{q}}^{\mathrm{tr}}\mathbf{\cdot}\delta{\bm\varphi}_{-\mathbf{q}}^{\mathrm{tr}})$$ since $\lim_{\mathbf{q\rightarrow0}}\left[|\kappa_{\mathbf{q}}|^{-1}-\frac{2}{3}\beta_{c}\right]=0$. Fluctuations due to anisotropic compass interactions ---------------------------------------------------- Next, in addition to the isotropic Heisenberg term, let us take into consideration an anisotropic compass interaction, $K$. The constraint that the ferromagnetic mean field solution remains stable is satisfied for all negative (ferromagnetic) values of $K$ and for positive values $K<|J|$. In the presence of the anisotropic compass interaction, the model (\[ham\]) reads $$\begin{aligned} H=\sum_{j;j^{\prime}}\sum_{\alpha}J_{jj^{\prime}}^{\alpha}S_{j}^{\alpha }S_{j^{\prime}}^{\alpha}, \label{hamaniso}$$ where the exchange interaction is given by $$\begin{aligned} J_{jj^{\prime}}^{\alpha}=\frac{1}{2}\delta_{j^{\prime},j+\tau}[J+K\delta _{\alpha,|\tau|}]\end{aligned}$$ The index $\tau=\pm x,\pm y,\pm z$ labels nearest neighbor sites, where $|\tau|=x,y,z$ specifies a direction in spin space ($x$ for bonds along the $x$-direction, etc.). The eigenvalues of the operator $J_{jj^{\prime}}^{\alpha\alpha^{\prime}}$ defined in the previous section are given by $$\begin{aligned} \kappa_{\mathbf{q},\nu}=\sum_{\alpha}(J+K\delta_{\alpha,\nu})\cos q_{\alpha}.$$ The eigenvectors $\mathbf{u}_{\nu}$ are again along the three cubic axes, such that the components are $u_{\nu,\alpha}=\delta_{\nu,\alpha}$. This time the three eigenvalues for given $\mathbf{q}$ are not degenerate (except in the limit $\mathbf{q}\rightarrow0$) and the fluctuation contribution to the free energy will therefore depend on the orientation of the spontaneous magnetization. We may again use the representation of the partition function $Z$ as a functional integral over the Fourier components $\varphi_{\mathbf{q},\nu}$ of the auxilliary field. Provided $J<0$ and $K<|J|$, the mean-field solution $\varphi_{MF}$ is given as before by solving the transcendental equation (\[eq:MF\]) numerically. The fluctuation contribution to the free energy is obtained by expanding the action in the fluctuation field about the mean field solution to lowest order. We get $$Z=C\exp(-\beta{\mathcal{S}}_{0})\int[d\delta\varphi]\exp(-\beta{\mathcal{S}}_{\mathrm{fl}}\{\delta\varphi_{\mathbf{q,}\nu}\}),\label{sflaniso}$$ where the fluctuation part of the action is given by Eqs.(\[fluct-free-energy\]) and (\[fluct-matrix1\]) In the following, we show that by comparison to the isotropic model, Eq. (\[sflaniso\]) manifestly breaks rotational invariance, which results in a selection of preferred directions of the order parameter, which minimize the free energy. The $3\times3$-matrix $A_{\mathbf{q,}\nu\nu^{\prime}}$ may be diagonalized and has eigenvalues $\lambda_{\gamma,{\bm q}}$ and eigenvectors ${\bm v}_{\gamma,{\bm q}}$, $\gamma=0,1,2$. This allows us to express $\sum_{\nu \nu^{\prime}}A_{\mathbf{q,}\nu\nu^{\prime}}\delta\varphi_{\mathbf{q,}\nu}^{\ast }\delta\varphi_{\mathbf{q,}\nu^{\prime}}=\sum_{\gamma}\lambda_{\gamma,{\bm q}}\delta\varphi_{{\bm q},\gamma}\delta\varphi_{-{\bm q},\gamma}$, where $\delta \varphi_{{\bm q},\gamma}={\bm v}_{\gamma,{\bm q}}\cdot\delta\boldsymbol{\varphi }_{\bm q}$. The integration over the fluctuation amplitudes may now be performed and gives $$S_{\mathrm{fl}}=\beta^{-1}\frac{1}{2}\sum_{\mathbf{q}}\ln |\lambda_{0,{\bm q}}\lambda_{1,{\bm q}}\lambda_{2,{\bm q}}|,\label{action1}$$ where we chose $s(\kappa_{\mathbf{q},\nu})=\pm i$ for $\kappa_{\mathbf{q},\nu}>0$, following the procedure described at the end of the Appendix. Alternatively, we may use that $|\lambda_{0,{\bm q}}\lambda_{1,{\bm q}}\lambda_{2,{\bm q}}|=|\det\{A_{\mathbf{q,}\nu\nu^{\prime}}\}|$, saving the trouble of having to determine the eigenstates of $A_{\mathbf{q,}\nu \nu^{\prime}}$. Let us now derive the explicit expression for the fluctuation contribution for an arbitrary orientation of $\mathbf{m}_{0}=(\sin\theta\cos\phi,\sin\theta \sin\phi,\cos\theta).$ Inserting this into the definition of $A_{\mathbf{q,}\nu\nu^{\prime}}$ given by Eq.(\[fluct-matrix1\]), we find its elements to be $$\begin{array} [c]{l}A_{{\bm q},00}=|\kappa_{{\bm q},x}|^{-1}-\frac{2}{3} s(\kappa_{{\bm q},x})s(\kappa_{{\bm q},x}) (\beta_c(1-s_{\theta}^{2}c_{\phi}^{2})+3\beta r s_{\theta}^{2}c_{\phi}^{2}) \\ A_{{\bm q},01}=-\frac{2}{3} s(\kappa_{{\bm q},x})s(\kappa_{{\bm q},y})(3\beta r-\beta_c) c_{\phi}s_{\phi}s_{\theta}^2\\ A_{{\bm q},10}=A_{{\bm q},01}\\ A_{{\bm q},02}=-\frac{2}{3} s(\kappa_{{\bm q},x})s(\kappa_{{\bm q},z})(3\beta r-\beta_c) c_{\phi}c_{\theta}s_{\theta}\\ A_{{\bm q},20}=A_{{\bm q},02}\\ A_{{\bm q},11}=|\kappa_{{\bm q},y}|^{-1} -\frac{2}{3} s(\kappa_{{\bm q},y})s(\kappa_{{\bm q},y}) (\beta_c(1-s_{\theta}^{2}s_{\phi}^{2})+3\beta r s_{\theta}^{2}s_{\phi}^{2})\\ A_{{\bm q},12}=-\frac{2}{3} s(\kappa_{{\bm q},y})s(\kappa_{{\bm q},z})(3\beta r-\beta_c) s_{\phi}c_{\theta}s_{\theta}\\ A_{{\bm q},21}=A_{{\bm q},12}\\ A_{{\bm q},22}=|\kappa_{{\bm q},z}|^{-1} -\frac{2}{3} s(\kappa_{{\bm q},z})s(\kappa_{{\bm q},z}) (\beta_c s_{\theta}^{2}+3\beta r c_{\theta}^{2}), \end{array} \label{cubicmatrix}$$ where, to shorten notations, we denote $\sin\theta(\phi)\equiv s_{\theta (\phi)}$ and $\cos\theta(\phi)\equiv c_{\theta(\phi)}$. The interactions are defined as $\kappa_{{\bm q},x}^{-1}=1/\left[(J+K)\cos q_{x}+J\cos q_{y}+J\cos q_{z}\right]$, $\kappa_{{\bm q},y}^{-1}=1/\left[(J+K)\cos q_{y}+J\cos q_{x}+J\cos q_{z}\right]$ and $\kappa_{{\bm q},z}^{-1}=1/\left[(J+K)\cos q_{z}+J\cos q_{x}+J\cos q_{y}\right]$. We see that the matrix $A_{{\bm q},\nu\nu^{\prime}}$ has a rather complex structure as a function of ${\bm q}$ and angles $\theta$ and $\phi$. This gives rise to a complex behavior of the eigenvalues $\lambda_{0,{\bm q}}$, $\lambda_{1,{\bm q}}$ and $\lambda_{2,{\bm q}}$. Results and discussions ======================= We now present the results obtained for ${\mathcal{S}}_{\mathrm{fl}}(\theta,\phi)$ by performing numerical integration in Eq.(\[action1\]). The angular dependence of ${\mathcal{S}}_{\mathrm{fl}}(\theta,\phi)$ is presented in Figs.1, where the magnitude of $S_{\mathrm{fl}}(\theta,\phi)$ as a function of orientation of the spontaneous magnetization is shown as a color-coded plot on the unit sphere. The calculations in Figs.1 are performed at temperature $\beta=\beta_{c}+1$ and assuming $J=-1$. We see that ${\mathcal{S}}_{\mathrm{fl}}(\theta,\phi)$ has a non-trivial dependence on the direction of the order parameter defined by angles $\theta$ and $\phi$. This peculiar angular dependence of ${\mathcal{S}}_{\mathrm{fl}}(\theta,\phi)$ is inherited from non-trivial angular dependencies of $\lambda_{0,{\bm q}}$, $\lambda_{1,{\bm q}}$ and $\lambda_{2,{\bm q}}$. In Fig.1, we present the profile of ${\mathcal{S}}_{\mathrm{fl}}(\theta,\phi)$ computed for $K=0.75$. We can see that ${\mathcal{S}}_{\mathrm{fl}} (\theta,\phi)$ is minimized when the magnetization is directed along one of the cubic axes. We note that the cubic directions are also selected for other values of the compass interactions, both antiferromagnetic as well as ferromagnetic, where the ferromagnetic state is the mean field solution ($K<|J|$). Conclusion ========== The magnetic properties of heavy transition metal oxides such as iridates and others are emerging as a new fascinating field offering opportunities to realize strongly frustrated quantum spin systems in the laboratory. In these systems, the combination of multiband electronic structure and strong Coulomb and Hund’s couplings with strong spin-orbit interaction can give rise to extremely anisotropic spin exchange interactions of the compass type. Mean field solutions of these models are often untouched by the anisotropies of the model and show the full isotropy of pure Heisenberg models, in contrast with experimental observations. In this paper, we addressed the question how the system selects special preferred directions of the mean field order parameter vector. We restricted ourselves to the case of a ferromagnetic order parameter, but an analogous question exists for antiferromagnetic or more complicated ordered structures. We find that the high degeneracy of the ferromagnetic mean-field solution is lifted by the free energy contribution from thermal fluctuations. We calculated the fluctuation contribution for a Heisenberg-compass model of classical spins on a three dimensional cubic lattice with nearest neighbor interactions - an isotropic Heisenberg coupling $J<0$ (which we take as the energy unit), and a compass coupling $K$. The ferromagnetic state is found if $K<|J|$. Rather than exploring the full phase diagram, we focused on one typical temperature $T=T_{c}/(1+T_{c})$, where $T_{c}$ is the mean-field transition temperature. For values of $K<1$, the system is found to choose preferred directions of the spontaneous magnetization along one of the cubic axes. In the temperature regime considered here, we expect the classical approximation to be valid. A generalization to quantum spin systems of the approach presented here is in preparation. *Acknowledgements.* We thank Michel Gingras, George Jackeli, Yoshi Kamiya, Alberto Hinojosa-Alvarado and Ioannis Rousochatzakis for useful discussions. N.P. and Y.S. acknowledge the support from NSF Grants DMR-1005932 and DMR-1511768. P.W. thanks the Department of Physics at the University of Wisconsin-Madison for hospitality during several stays as a visiting professor. P.W. also acknowledges partial support by an ICAM senior fellowship and through the DFG research unit “Quantum phase transitions”. N.P. acknowledges the hospitality of KITP and partial support by the National Science Foundation under Grant No. NSF PHY11-25915. Hubbard-Stratonovich transformation of the partition function for spin systems ============================================================================== General formulation ------------------- The Hubbard-Stratonovich (H-S) transformation is based on the mathematical identitiy $$\exp[-ax^{2}]=\frac{1}{\sqrt{\pi|a|}}\int dy\exp\left[-\frac{y^{2}}{|a|}+2s(a)xy)\right],$$ where we defined $$s(a)={\Big \{}\begin{array} [c]{cc}1\,, & \mathrm{if}\,a<0\\ \imath\,, & \mathrm{if}\,a>0. \end{array}$$ For $a>0$ we may as well use $s(a)=-i$. We will later make use of this ambiguity when we evaluate the $y$-integrals approximately, which may lead to imaginary-valued contributions.  In the above H-S-transformation, $x$ may be a number or an operator. In the case it is an operator, we use the eigenfunctions $|n\rangle$ of $\widehat{x}$ defined by $$\widehat{x}|n\rangle=x_{n}|n\rangle$$ to prove that $$\begin{aligned} && \exp \left[-a\widehat{x}^{2}\right]|n\rangle= \exp\left[-ax_{n}^{2}\right] |n\rangle\\ && =\frac{1}{\sqrt{\pi|a|}}\int dy \exp\left[-\frac{y^{2}}{|a|}+2s(a)x_{n}y) \right] |n\rangle \nonumber \\ && =\frac{1}{\sqrt{\pi |a|}}\int dy \exp\left[-\frac{y^{2}}{|a|}+2s(a)\widehat{x}y)\right] |n\rangle . \nonumber\end{aligned}$$ This identity also works for complex (non-Hermitian) $x$ and $y$: $$\exp[-a\widehat{x}^{\dag}\widehat{x}]=\frac{i}{2\pi |a|}\int dy^{\ast}dy\exp\left[-\frac{y^{\ast}y}{|a|}+s(a)(\widehat{x}^{\dag}y+H.c.)\right]\nonumber$$ We now turn to the case of the partition function of a spin system with generic interaction Hamiltonian (1). In order to use the mathematical identities we need to represent the Hamiltonian (1) in terms of normal coordinates. To this end we define the normalized eigenstates of the exchange interaction operator $$\sum_{j^{\prime},\alpha^{\prime}}J_{jj^{\prime}}^{\alpha\alpha^{\prime}}\chi_{n;j^{\prime},\alpha^{\prime}}=\kappa_{n}\chi_{n;j,\alpha},$$ in terms of which we have$$J_{jj^{\prime}}^{\alpha\alpha^{\prime}}=\sum_{n}\kappa_{n}\chi_{n;j,\alpha }^{\ast}\chi_{n;j^{\prime},\alpha^{\prime}},$$ where $\chi_{n;j^{\prime},\alpha^{\prime}}$ form a complete and orthonormal set of eigenfunctions and thus obey $$\begin{aligned} & \sum_{j,\alpha}\chi_{n;j,\alpha}^{\ast}\chi_{n^{\prime};j,\alpha}=\delta_{n,n^{\prime}},\\ & \sum_{n}\chi_{n;j,\alpha}^{\ast}\chi_{n;j^{\prime},\alpha^{\prime}}=\delta_{j,j^{\prime}}\delta_{\alpha,\alpha^{\prime}}.\nonumber\end{aligned}$$ For spins on a periodic lattice, the eigenstates $|n\rangle=|\mathbf{q},\nu\rangle$ are labeled by wavevector $\mathbf{q}$ and spin component $\nu$, and the eigenfunctions take the form$$\chi_{\mathbf{q},\nu;j,\alpha}=\frac{1}{\sqrt{N}}e^{i\mathbf{q\cdot R}_{j}}u_{\mathbf{q}\nu}^{\alpha}$$ where $u_{\mathbf{q}\nu}^{\alpha}$ are normalized real valued eigenvectors, i.e. $\sum_{\alpha}u_{\mathbf{q}\nu}^{\alpha}u_{\mathbf{q},\nu}^{\alpha}=1$, and $\kappa_{\mathbf{q},\nu}$ are the eigenvalues of the spin exchange operator. We now define the normal amplitudes of the spin operators as $$S_{\mathbf{q},\nu}=\sum_{j,\alpha}\chi_{\mathbf{q},\nu;j,\alpha}S_{j}^{\alpha}$$ and express the Hamiltonian (1) as$$H=\sum_{\mathbf{q},\nu}\kappa_{\mathbf{q},\nu}S_{\mathbf{q},\nu}^{\ast }S_{\mathbf{q},\nu},$$ where $S_{\mathbf{q},\nu}^{\ast}=S_{-\mathbf{q},\nu}$. We seek to apply the above mathematical identities (A1)-(A3) to each normal component separately. This requires the normal components of the spin operators to commute with each other, which is certainly true for the classical spins. Then using the Hubbard-Stratonovich transformation one may express the Boltzmann weight operator of each normal mode in terms of normal field amplitudes $\varphi_{\mathbf{q,}\nu}$ as$$\begin{aligned} & \exp[-\beta\kappa_{\mathbf{q,}\nu}S_{\mathbf{q,}\nu}^{\ast}S_{\mathbf{q,}\nu}]= \frac{\imath\beta}{2\pi|\kappa_{\mathbf{q,}\nu}|}\int\int d\varphi _{\mathbf{q,}\nu}^{\ast}d\varphi_{\mathbf{q,}\nu}\\ & \exp\left[-\beta\{|\kappa_{\mathbf{q,}\nu}|^{-1}\varphi_{\mathbf{q,}\nu}^{\ast}\varphi_{\mathbf{q,}\nu}+s(\kappa_{\mathbf{q,}\nu})(S_{\mathbf{q,}\nu}^{\ast}\varphi_{\mathbf{q,}\nu}+H.c.)\}\right]\nonumber\end{aligned}$$ The complete Boltzmann weight operator may be expressed, again using the commutability of the normal mode operators, as $$\begin{aligned} & \exp[-\beta\sum_{\mathbf{q},\nu}\kappa_{\mathbf{q},\nu}S_{\mathbf{q},\nu }^{\ast}S_{\mathbf{q},\nu}]=\int[d\varphi]\\ & \exp[-\beta\sum_{\mathbf{q,}\nu}\{|\kappa_{\mathbf{q,}\nu}|^{-1}\varphi_{\mathbf{q,}\nu}^{\ast}\varphi_{\mathbf{q,}\nu}+s(\kappa_{\mathbf{q,}\nu })(S_{\mathbf{q,}\nu}^{\ast}\varphi_{\mathbf{q,}\nu}+h.c.)\}],\nonumber\end{aligned}$$ where $\varphi_{\mathbf{q,}\nu}^{\ast}=\varphi_{-\mathbf{q,}\nu}$ . The integration volume element is given by $$\lbrack d\varphi]=\Pi_{\mathbf{q,}\nu}\frac{i\beta{d\varphi_{\mathbf{q,}\nu}^{\ast}}d\varphi_{\mathbf{q,}\nu}}{2\pi|\kappa_{\mathbf{q,}\nu}|}$$ Next, we find that the partition function of an interacting classical spin system on an infinite periodic lattice may be expressed as $$\begin{aligned} \label{ZZ} &Z=\\ & =C\int[d\varphi]\exp\left[-\beta\sum_{\mathbf{q,}\nu}|\kappa_{\mathbf{q,}\nu}|^{-1}\varphi_{\mathbf{q,}\nu}^{\ast}\varphi_{\mathbf{q,}\nu}-S_{loc}(\{\varphi _{\mathbf{q,}\nu}\})\right],\nonumber\end{aligned}$$ where $C$ is a constant. The contribution ${\mathcal{S}}_{loc}(\{\varphi _{\mathbf{q,}\nu}\})$ to the action is given by$${\mathcal{S}}_{loc}(\{\varphi_{\mathbf{q,}\nu}\})=\frac{1}{\beta}\sum_{j}\ln W_j$$ and $W_{j} $ is computed by taking into account the constraint of the unit length of classical spins, $\mathbf{S}_{j}^{2}=1$, and integrating over all directions of spin at each lattice site:$$\begin{aligned} W_{j} & =\int\frac{dS_{j}d\Omega_{j}}{2\pi}\exp\left[2\beta\sum_{\alpha }{\varphi}_{j}^{\alpha}S_{j}^{\alpha}\right]\delta(\mathbf{S}_{j}^{2}-1)\nonumber\\ & =\int\frac{d\Omega_{j}}{4\pi}\exp\left[2\beta\sum_{\alpha}{\varphi}_{j}^{\alpha}S_{j}^{\alpha}\right]\\\nonumber & =\frac{\sinh\,2\beta|{\varphi}_{j}|}{2\beta|{\varphi}_{j}|}.$$ This gives $$\begin{aligned} \label{SlocAP} {\mathcal{S}}_{loc}(\{\varphi_{\mathbf{q,}\nu}\})=\frac{1}{\beta} \sum_{j}\ln[\frac{\sinh2\beta|{\varphi}_{j}|}{2\beta|{\varphi}_{j}|}].$$ Here we defined the complex-valued three-component field ${\varphi}_{j}^{\alpha}$ at each lattice site $j$ as $$\begin{aligned} {\varphi}_{j}^{\alpha} & =\sum_{\mathbf{q,}\nu}s(\kappa _{\mathbf{q,}\nu}){\mathcal{R}e}\{\varphi_{\mathbf{q,}\nu}^{\ast}\chi _{\mathbf{q,}\nu;j,\alpha}\}\\\nonumber & =\sum_{\mathbf{q,}\nu}s(\kappa_{\mathbf{q,}\nu})\varphi_{\mathbf{q,}\nu}\chi_{\mathbf{q,}\nu;j,\alpha}^{\ast}\nonumber\\ & ={\varphi}_{R,j}^{\alpha}+i{\varphi}_{I,j}^{\alpha}.\nonumber\end{aligned}$$ Observing that $\kappa_{\mathbf{q,}\nu}=\kappa_{-\mathbf{q,}\nu}$, we get$$\begin{aligned} \label{IR} {\varphi}_{R,j}^{\alpha} & =\operatorname{Re}\{{\varphi}_{j}^{\alpha}\}=\sum_{\mathbf{q,}\nu,\kappa_{\mathbf{q,}\nu}<0}\varphi _{\mathbf{q,}\nu}\chi_{\mathbf{q,}\nu;j,\alpha}^{\ast}\\\nonumber {\varphi}_{I,j}^{\alpha} & =\operatorname{Im}\{{\varphi}_{j}^{\alpha}\}=\sum_{\mathbf{q,}\nu,\kappa_{\mathbf{q,}\nu}>0}\varphi _{\mathbf{q,}\nu}\chi_{\mathbf{q,}\nu;j,\alpha}^{\ast}.$$ The field amplitude is determined by $$\begin{aligned} {\varphi}_{j}=\sqrt{({\boldsymbol\varphi}_{R,j}+i\,{\boldsymbol\varphi}_{I,j})^{2}},\end{aligned}$$ where $\boldsymbol{\varphi}_{R,j} =\left({\varphi}^x_{R,j}, \,{\varphi}^y_{R,j},\,{\varphi}^z_{R,j}\right)$ and $\boldsymbol{\varphi}_{I,j} =\left({\varphi}^x_{I,j}, \,{\varphi}^y_{I,j},\,{\varphi}^z_{I,j}\right)$. We now derive the contribution of Gaussian fluctuations to the free energy for the ferromagnetic mean field state which we denote as $\boldsymbol{\varphi}_{MF}$. To this end, we expand ${\mathcal{S}}_{loc}(\{\varphi_{\mathbf{q,}\nu}\})$ (\[SlocAP\]) in terms of the fluctuation amplitudes and separate the mean-field and fluctuational contributions. First, we expand the field amplitude ${\varphi}_{j}$ to bilinear order in the fluctuation amplitudes:$$\begin{aligned} {\varphi}_{j} & ={\varphi}_{MF}+\delta {\varphi}_{j} ,\\\nonumber \delta {\varphi}_{j} & =\frac{1}{2\varphi_{MF}}[2{\boldsymbol\varphi}_{MF}\cdot(\delta{\boldsymbol\varphi} _{R,j}\mathbf{+}i\delta{\boldsymbol\varphi}_{I,j})+\delta{\boldsymbol\varphi}_{R,j}^{2}\,-\delta{\boldsymbol\varphi}_{I,j}^{2}]\\\nonumber & -\frac{1}{2\varphi_{MF}^3}[{\boldsymbol\varphi}_{MF}\cdot(\delta{\boldsymbol\varphi}_{R,j}+i\,\delta{\boldsymbol\varphi}_{I,j})]^{2}.\end{aligned}$$ Using Eq. (\[IR\]), we now obtain the expressions for $\delta \varphi_{j} $ and $\delta \varphi_{j}^{2}$ in terms of  $\varphi_{\mathbf{q,}\nu}^{\ast}$ and $\varphi_{\mathbf{q,}\nu}$, keeping quadratic (Gaussian) terms only: $$\begin{aligned} && \sum_{j}\delta \varphi_{j} =\frac{1}{2{\varphi}_{MF}}\sum_{\mathbf{q},\nu,\nu^{\prime}}\delta_{\nu,\nu^{\prime}}s(\kappa_{\mathbf{q},\nu})s(\kappa_{\mathbf{q}, \nu^{\prime}})\varphi_{\mathbf{q,}\nu}^{\ast}\varphi_{\mathbf{q,}\nu^{\prime}}\nonumber\\ && -\frac{1}{2{\varphi}_{MF}}\sum_{j}\delta \varphi_{j}^{2}\\\nonumber && \sum_{j}\delta \varphi_{j}^{2} =\sum_{\mathbf{q},\nu,\nu^{\prime}}s(\kappa _{\mathbf{q},\nu})s(\kappa_{\mathbf{q},\nu^{\prime}})m_{0,\nu}\varphi _{\mathbf{q,}\nu}^{\ast}\varphi_{\mathbf{q,}\nu^{\prime}}m_{0,\nu^{\prime}}.$$ Next, we expand Eq. (\[SlocAP\]) step by step as $$\begin{aligned} &&\sinh2\beta|{\varphi}_{j}| =\sinh(2\beta({\varphi}_{MF}+\delta \varphi_j))\nonumber\\\nonumber &&=\sinh(2\beta{\varphi}_{MF})[1+2(\beta\delta \varphi_j)^{2}]+\cosh(2\beta {\varphi}_{MF} )2\beta\delta \varphi_j\end{aligned}$$ and further $$\begin{aligned} &&\ln\left[\sinh(2\beta{\varphi}_{j})/2\beta{\varphi}_{j}\right] \nonumber\\ &&=\ln\left[ \sinh(2\beta({\varphi}_{MF}+\delta\varphi_j ))\right]-\ln\left[2\beta({\varphi}_{MF}+\delta \varphi_j)\right]\nonumber\\ \nonumber && =\ln\left[\sinh(2\beta {\varphi}_{MF})/(2\beta {\varphi}_{MF})\right] \\\nonumber &&+\left[2\beta {\varphi}_{MF}\coth(2\beta {\varphi}_{MF})-1\right]\frac{\delta \varphi_j}{{\varphi}_{MF}}\\\nonumber && +\frac{1}{2}\left[-\frac{(2\beta {\varphi}_{MF})^{2}}{\sinh^{2}(2\beta {\varphi}_{MF})}+1\right](\frac{\delta \varphi_j}{{\varphi}_{MF}})^{2}.$$ The fluctuation part of the local part of the free energy is then given by$$\begin{aligned} &&-\beta^{-1}\delta\sum_{j}\ln[\sinh(2\beta{\varphi}_{j})/2\beta{\varphi}_{j}]=\nonumber\\ && =-\frac{4}{3}\beta_{c} {\varphi}_{MF}\sum_{j}\delta \varphi_{j}\\\nonumber&& -\frac{1}{2\beta {\varphi}_{MF}^{2}}[1-\frac{(2\beta {\varphi}_{MF})^{2}}{\sinh^{2}(2\beta {\varphi}_{MF})}]\sum_{j}\delta \varphi_{j}^{2},$$ where we have used that $2\beta {\varphi}_{MF}\coth(2\beta {\varphi}_{MF})-1=\frac{4}{3}\beta_{c}\beta {\varphi}_{MF}^{2}$ . Substituting the expressions for $\delta \varphi_{j},\delta \varphi_{j}^{2}$ and defining $r=1/(2\beta {\varphi}_{MF})^{2}-1/\sinh^{2}(2\beta{\varphi}_{MF})$, we get the fluctuation contribution to the free energy $${\mathcal{S}}_{\mathrm{fl}}\{\delta\varphi_{\mathbf{q,}\nu}\}=\sum_{\mathbf{q};\nu,\nu^{\prime}}A_{\mathbf{q,}\nu\nu^{\prime}}\delta\varphi_{\mathbf{q,}\nu }^{\ast}\delta\varphi_{\mathbf{q,}\nu^{\prime}},\label{fluct-free-energyAP}$$ where we defined matrices $A_{\mathbf{q,}\nu\nu^{\prime}}$ describing the weight of Gaussian fluctuations of wavevector $\mathbf{q}$ and polarization $\nu$ as $$\begin{aligned} \label{fluct-matrix1AP} &&A_{\mathbf{q},\nu\nu^{\prime}} =|\kappa_{\mathbf{q},\nu}|^{-1}\delta_{\nu ,\nu^{\prime}}\\\nonumber && -\frac{2}{3}[\beta_{c}(\delta_{\nu,\nu^{\prime}}-m_{0,\nu }m_{0,\nu^{\prime}})+3\beta rm_{0,\nu}m_{0,\nu^{\prime}}]s(\kappa _{\mathbf{q},\nu})s(\kappa_{\mathbf{q},\nu^{\prime}})\end{aligned}$$ The fluctuation matrix $A_{\mathbf{q},\nu\nu^{\prime}}$ will in general be non-Hermitian, and its eigenvalues will be complex. We now use that $A_{\mathbf{q},\nu\nu^{\prime}}$ is an even function of $\mathbf{q}$ and divide $\mathbf{q}$-space into $q_{x}>0$ ($M_{>}$) and $q_{x}<0$ ($M_{<}$). Note that the choice of $q_x$ for dividing the BZ in half is arbitrary, and we could also do it with a help of $q_y$ and $q_z$. For modes $\varphi_{\mathbf{q},\nu}$ with $\mathbf{q\in}M_{>}$, we choose $s(\kappa_{\mathbf{q},\nu})=+i$, whereas for modes with $\mathbf{q\in}M_{<}$ we choose $s(\kappa_{\mathbf{q},\nu})=-i$, where $\kappa_{\mathbf{q},\nu}>0$ in both cases. Then we have $A_{-\mathbf{q},\nu\nu^{\prime}}=A_{\mathbf{q},\nu \nu^{\prime}}^{\ast}$ and  as a result of the functional integration we will get $$\begin{aligned} Z & =Z_{MF}\int[d\delta\varphi]\exp\left[-\beta\sum_{\mathbf{q,}\nu,\nu^{\prime}}A_{\mathbf{q},\nu\nu^{\prime}}\delta\varphi_{\mathbf{q,}\nu}^{\ast}\delta\varphi_{\mathbf{q,}\nu^{\prime}}\right]\nonumber\\ & =Z_{MF}\exp\left[-\frac{1}{2}\sum_{\mathbf{q\in}M_{>}}\ln(\det(A_{\mathbf{q},\nu\nu^{\prime}})\det(A_{\mathbf{q},\nu\nu^{\prime}}^{\ast}))\right]\nonumber\\ & =Z_{MF}\exp\left[-\frac{1}{2}\sum_{\mathbf{q}}\ln|\det(A_{\mathbf{q},\nu\nu^{\prime}})|\right],\end{aligned}$$ where $$\begin{aligned} Z_{MF}=\exp\left[ -\beta N |\kappa_{{\bm q}=0}|^{-1}\varphi_{MF}^2 \right]\left[\sinh(2\beta {\varphi}_{MF})/(2\beta {\varphi}_{MF})\right]^N. \nonumber\end{aligned}$$ [99]{} Z. Nussinov, J. van den Brink, Rev. Mod. Phys. **87** 1 (2015). K. I. Kugel and D. I. Khomskii, Sov. Phys. Usp. **25**, 231 (1982). Z. Nussinov, M. Biskup, L. 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--- abstract: 'A variety of real world and experimental systems can display a drastic regime shift, as the evolution in one its paramaters crosses a threshold value. Assimilation of such a transition with a bifurcation has allowed to identify so called “early warning signals”, at the level of the time series generated by the system underscope. The literature in early warning detection methods is currently expanding and their potential for practical applicability is being discussed in different contexts. In this work, we elaborate on the use of the variance of a system variable, which constitutes the simplest early warning indicator, to gain control on the long-term dynamics of the system, while extending an exploitation phase. In particular, we address the cases of the cusp and Hopf normal forms, as prototypical examples of bistability and oscillations. Our results provide insights on the interplay between the time-scale for the system observation, the degree of sensitivity of the control feedback and the intensity of the random perturbations, in shaping the long-term control efficiency.' author: - 'García Cantú Ros, Fluschnik, Kropp' title: 'Variance-based control of regime shifts: bistability and oscillations' --- Introduction ============ Along the last few decades an increasing number of scholars has been contributing to develop a coherent and further integral body of knowledge on complex real world systems. Cross-fertilization of concepts and tools stemming from dynamical systems, statistical physics and information and computation theories has allowed to identify and analyse systems whose observable behaviors result from similar underlying mechanisms [@NicolisI; @Politi]. As a result of these attempts, a key finding has been realization of the ubiquity of nonlinear feedbacks in the dynamics of different classes of systems. In particular, the occurence of drastic, qualitative shifts of the functioning regime of systems –as resulting from strong positive and negative feedbacks– has attracted interest in diverse research fields, as diverse as microscopic physical systems at equilibrium [@Stanley] and their macroscopic non-equilibrium counterpart [@NicolisII; @Cross], ecology [@SchefferIV; @Scheffer], socio-physics [@Castellano] and Earth system science [@Lenton; @Scheffer]. Insights into the phenomenology, related to regime shifts and the development of new methods for their early detection, have relied on minimal models. A frequent idea in this approach is to cast the description of the system evolution as a stochastic differential equation [@Gardiner], of the form: $$\textbf{dx}=\textbf{F}(\textbf{x},\lambda)dt+d\textbf{$\textbf{W}_x$}\label{EqGen1}$$ The key variables of the system are represented as a vector $\textbf{x}=(x_1,x_2,\dots,x_N)$ and their dynamic interrelationships are considered as deterministic laws, described as a vector field **F**. Vector $\textbf{W}_x$ aims at introducing the effect of incessant perturbations on the **x** components, as typically resulting from endogenous meso-scale dynamic complexity and from an infinite number of uncorrelated environmental factors. Consequently, $\textbf{W}_x$ is assimilated to a Gaussian Wiener process [@Gardiner] with standard deviation $\omega_{x}$. The parameter $\lambda$ controls the qualitative behavior of solutions to the deterministic part of system around a stationary state $\textbf{x}_s$. In this framework, a regime shift is represented as a local bifurcation occuring in the deterministic part of the system at a critical point $(\textbf{x}_s,\lambda_c)$, i.e., that the Jacobian matrix of **F**, evaluated at $\textbf{x}_c$, possesses one eigenvalue whose real part tends to zero as $\lambda$ approaches the value $\lambda_c$ [@Guckenheimer]. Thereby, the response of the system to random fluctuations can be studied along the evolution towards the bifurcation point. Given the relevance of applications and implications of regime shifts in real world systems, it is of increasing interest to construct early warning indicators (**EWI**) [@SchefferI; @SchefferII] – with focus on identifying statistical signatures of the loss of stability in the time series generated by the evolution of the system towards criticality. Conjointly, a relevant phenomenon is the slowing down of the fast stabilizing dynamic modes –often referred as critical slowing down. This results from the real part of the leading eigenvalue becoming zero as the system closely approaches the bifurcation point [@SchefferI]. As illustrated in a pioneering work by S. Carpenter [@Carpenter] on **EWI**-based monitoring of shallow lake ecosystems, the onset of the critical slowing down can lead to a drastic increase in the variance of a variable of the system. More recently, the skewness [@Guttal], kurtosis[@Biggs] and the autocorrelation function [@Livina] of a system variable have been shown to capture relevant information on the increased loss of stability close to the bifurcation point. The literature in early warning detection is rapidly expanding and their potential applications within a broad range of real world problems is being discussed [@SchefferI; @SchefferII]. Within this contextual frame, it becomes natural to inquire about the possible long-term complex behavior that would arise if **EWI**s are repeatedly used to control systems potentially undergoing regime shifts. As far as we know, this question remains open, in the light of earlier and further recent advanced methods in the construction of **EWI**’s [@Dakos]. In order to tackle this question, we propose to analyze the dynamics of a system represented by subject to a control process. The basic idea is that such a steering control should: 1) re-establish the stability of the system before it crosses a critical point, or 2) re-establish a lost initial system regime once a critical point has been crossed, to then let the system evolve again towards criticality. Let us explore this idea by considering a control system which reacts upon the behavior of an **EWI**, denoted by $I_{\tau}$, whose construction is based on information about the past states of the system within a sliding observation time window of size $\tau$. In other words, we are interested in analyzing the coupled dynamics of with an evolution equation for $\lambda$, of the form $$d\lambda=\Phi(I_{\tau},\alpha)dt+dW_{\lambda} \label{EqGen2}$$ Here we require the control functional $\Phi$ to operate a change of sign, at the level of $d\lambda$, once $I_{\tau}$ crosses a reference value $V_R$. In order to allow for differences in the degree of sensitivity of the control system, let us consider the control response to changes in the ratio $I_{\tau}/V_R$ as modulated by a parameter $\alpha$. Moreover, let us assume that the control response may exhibit random variability as a result of the influence of many uncorrelated factors. Thus, we denote by $W_{\lambda}$ a Gaussian Wiener process with standard deviation $\omega_{\lambda}$. In the domain of application of **EWI**s, relevant cases are those where the control variable $\lambda$ is driven by human interventions. In many of these instances, the evolution of the system towards criticality is the result of an “exploitation” process, that is sought to be intensified and mantained, while avoiding a system regime shift. Consider for instance catching rates in sustainable fisheries [@Ray], agriculture related leaching rates of phosphorus into a shallow lake, whose oligotrophic regime is to be preserved [@SchefferIII], or the increase in the autocatalytic production of chemical species in a well-stirred reactor [@NicolisIII], where a low entropy production regime may be desirable. Consequently, we characterize the dynamics of system - in terms of two phases: ***exploitation*** if $\Phi>0$ and ***recovery*** if $\Phi<0$. Central questions for us are: - The role of the time-scale $\tau$, considered for the construction of the **EWI**s, in shaping the emergence of control properties in a system -. - The long-term complex dynamics that can result from the temporal non-locality introduced by the **EWI** into the dynamics of the coupled system -. In other words, in the long-term, the control exerts changes on the system on the basis of the past dynamics of the system and its control as a whole. This opens the possibility to observe different complex behaviors, as the effective dimension of the system-control is augmented by its dependency on the past. - The efficiency of the control process in terms of: its capacity to increase and mantain explotation in a selected regime as a function of the observation window $\tau$, of the control response sensitivity $\alpha$ and of the variance of stochastic perturbations $\omega_x$ and $\omega_{\lambda}$. In Sec.II we address these questions, by exploring numerically the performance of control on minimal dynamical systems of generic character, that reproduce the prototypical phenomena of bistability and oscillations. Each of these controlled systems is first studied in absence of noise, in order to reveal the role of the observation time scale in shaping the leading deterministic dynamics. Upon the basis of the insights thereby generated, for each system, we introduce a suitable measure of control efficiency. Subsequently, we address the influence of the variance of noise, for different observation time windows and different degrees of control response, on the efficiency of both systems under consideration. Finally, in Sec.III we summarize our main results and conclusions. ![ a) plot of a fold bifurcation diagram, as provided by the stationary solutions of the cusp normal form \[Cusp\], as a function of parameter $\lambda$ and for $\mu=3/2$. The solid lines correspond to the stable branches and the dashed line to the unstable one. Unstable and stable branches meet and anhihilate each other at the saddle-node bifurcation points $f_1$ and $f_2$; b) plot of the branches formed by the critical values of $\lambda$ corresponding to both $f_1$ and $f_2$ bifurcation points, as a function of a second parameter $\mu$. Both branches converge to the cusp bifurcation point CB at $\mu=0$. Finally, plot $\Phi/k_{\lambda}$ vs $M^2_\tau/V_R$, for $\alpha=1$ (dot), $\alpha=2$ (dot-dot-dash), $\alpha=5$ (dot-dash), $\alpha=10$ (dash) and $\alpha=20$ (solid). []{data-label="Bif"}](Bifi.PDF){width=".65\textwidth" height=".35\textheight"} Normal forms and control ======================== Regardless of the complexity of its dynamics, as a system evolves further into the vicinity of a bifurcation point, there is a decrease in the number of degrees of freedom necessary to effectively describe the dynamics. This remarkable fact can be understood as a direct consequence of the gradual run out of the fast-relaxation dynamic modes that occurs along with the critical slowing down. A corner stone in local bifurcation theory is the application of the centre manifold method [@Guckenheimer] to identifying canonical descriptions of the nonlinear behavior around a bifurcation point, in terms of a reduced number of variables that contain the whole information about the dynamics of the slow leading modes. Such simple representations –so called normal forms– describe the long-term behavior of the normalized amplitude of the stable solutions that emerge at criticality. The importance of normal forms lies on the fact that the dynamics of different types of systems, around the critical point, can be mapped one-to-one continuously onto the one displayed by a normal form [@NicolisIV]. This property –a property known as topological equivalence– allows to define classes of universality, where near the critical point, the dynamics of systems belonging to the same class can be mapped onto each other by a change of variables preserving the direction of trajectories. Naturally, a simple and most representative element in a class of universality is thus the corresponding normal form. With the aim of tackling questions ***i***)-***iii***) above in a general context, we shall focus our analysis of control dynamics on selected normal forms. As an **EWI** we consider the variance $M^2_{\tau}[\delta x_j]=\frac{1}{\tau}\int_{t-\tau}^{t} (\delta x_j(t'))^2 dt'$ of deviations of a system variable around its mean value, within the time window $\tau$, i.e. $\delta x_j=x_j-\frac{1}{\tau}\int_{t-\tau}^{t} x_j(t') dt'$. Consistently with the minimal character of the normal form description, let us consider a simple expression for the control functional in , of the form $$\Phi(\tau,\alpha)=k_{\lambda}\Bigg(1-2\frac{M_{\tau}^2[\delta x_j]^\alpha}{M_{\tau}^2[\delta x_j]^\alpha+V_R^\alpha}\Bigg) \mbox{, $k_{\lambda},\alpha>0$}\label{Control}$$ where $k_{\lambda}$ corresponds to the maximum exploitation and recovery rates. For low values in $\alpha$ both the exploitation and recovery phases develop weakly as $M_{\tau}^2[\delta x_j]$ crosses the threshold value $V_R$. As $\alpha$ is arbitrarily increased, the functional $\Phi$ sharply approaches a Heavyside function (see (Fig.c)) –which emulates a full blown response of the control system to a threshold overshooting $$\Phi(\tau,\alpha\rightarrow\infty)=\left\{ \begin{array}{c l} k_{\lambda} & \mbox{$\frac{V_R}{M_{\tau}^2[\delta x_j]}>1$}\\ \\ 0 & \frac{V_R}{M_{\tau}^2[\delta x_j]}=1 \\ \\ -k_{\lambda} & \mbox{otherwise} \label{ControlI} \end{array}\right.$$ Here, a comment is in order with regard to the choice of the variance of a system variable as an **EWI**. As discussed in [@DakosI], in the more general case where the standard deviation of the random perturbations is a time dependent function, the variance of a system variable may not constitute a reliable **EWI**. Consequently, we shall restrain ourselves to the case where both standard deviations $\omega_x$ and $\omega_{\lambda}$ in - are constants. ![For $\tau=10$ (red), $\tau=11$ (green) and $\tau=35$ (blue), a) plot of a **CCS** deterministic trajectories ($\omega_x=0, \omega_{\lambda}=0$) in the ($\lambda,x$)-plane (in black lines the bifurcation diagram of the cusp normal form); b) plot *x* *vs* time (in orange the critical value $x_c$) and c) Plot of $\Phi/k_{\lambda}$ *vs* $\lambda-\lambda_c$, as related to the system trajectories in panels a) and b), for $\tau=11$ and $\tau=35$. Here, values $\Phi>0$ and $\Phi<0$ are considered as exploitation and recovery, respectively. Parameters and initial condition: $\mu=3/2$, $k_{\lambda}=1/10$, $\alpha=2$, $V_R=1/100$. $x(0)=-3/2$, $\lambda(0)=-5$. Time step $dt=5\times 10^{-3}$, numerical integration over $5\times 10^{5}$ time steps.[]{data-label="Traj"}](Traj.PDF){width=".65\textwidth" height=".35\textheight"} The cusp-control system (CCS) ----------------------------- As pointed out in [@SchefferI; @Hastings], the phenomenology observed in so called catasrophic regime shifts in different types of systems can be suitably assimilated with a saddle-node bifurcation. As a parameter evolves, a saddle-node bifurcation occurs when a stable (node) and an unstable (saddle) fixed point solution, to the deterministic evolution law of the system, approach each other and anhihilate upon meeting. This entails the destabilization of the system, which is thus led to an abrupt runaway –this constitutes a catastrophic regime shift. However, since in real systems unbounded runaways are ruled out, saddle-node bifurcations are typically found in cases where the branch formed by the system fixed points folds upon itself, within a range of values in the control parameter $\lambda$ (see Fig.\[Bif\]a). The presence of such a folding entails the existence of a region where two stable solution branches coalesce along with an unstable one. In this picture, the unstable branch marks the boundary line between the two basins of attraction, associated to both stable branches. In general, the distance between the bifurcation points at the extremes of the s-shape folding ($f_1$ and $f_2$ in Fig.\[Bif\]b) is determined by a second paramater $\mu$. Examples of such a bistability are reported in different types of systems, ranging from laser and cell division, to ecoystems and climate [@SchefferI]. In order to explore the question of bifurcation control in the context of bistability, let us consider the cusp normal form $$F(x,\lambda)=x(\mu \pm x^2) +\lambda \label{Cusp}$$ The solutions to the stationary form $F(x_s,\lambda)=0$ (with the sign minus in ), as a function of $\lambda$, correspond to those plotted in Fig.\[Bif\]a. The position of the saddle-node bifurcation points $f_{1,2}$ are determined by the relation $f_{1,2}=\{(\lambda,\mu)|\lambda=\pm\frac{2}{3\sqrt{3}}\mu^{3/2}\}$. Accordingly, both branches approach each other and merge as $\mu$ tends to zero, at the so-called cusp bifurcation point (Fig.\[Bif\]b). A similar behavior occurs when changing the sign in the second term in , which amounts to inverting the s-shaped folding. We carry out the numerical integration of the variance-based cusp-control system (**CCS**) (\[EqGen1\]), (\[EqGen2\]), (\[Control\] and \[Cusp\]) via a standard Euler-Maruyama approximation [@Gardiner]. ### CCS deterministic dynamics In order to illustrate the dynamics of the **CCS** in absence of noise ($\omega_x=0, \omega_{\lambda}=0$), let us consider exploitation as initially occuring along the negative branch in Fig.\[Bif\]a, a moderate degree of control response $(\alpha=5)$ (see Fig.\[Bif\]c) and a small variance reference threshold $V_R=1/100$. According to the numerical results, we distinguish three basic long-term **CCS** behaviors: - *Exploitation runaway*: If the time window is small as compared to a critical value $\tau_c$, the system crosses the bifurcation point $f_1$ and a regime shift occurs. As the system is expelled onto the upper branch, the control mechanism reacts to the overshooting of the reference value $V_R$. This triggers a recovery stage ($\Phi<0$) along the upper branch. However, as the system evolves along the upper branch, it eventually losses completely the “memory” on the original low branch regime. As a result, the variance decreases gradually below $V_R$, recovering is interrupted and the **CCS** switches to runaway explotation (red lines in Figs.\[Traj\]a,b). - *Hysteretic pathways*: When the observation window is now greater than the critical value $\tau_c$, a transition occurs, from runaway exploitation to long-term control. For specific $\tau$-ranges, the system crosses the bifurcation point $f_1$ and it starts recovering along the upper branch. However, differently from **B1**, the upper branch recovering continues further and the bifurcation point $f_2$ is crossed. Thus, the system undergoes again a regime shift towards the original lower branch. As the system stability increases and information about the upper branch is lost, the **CCS** eventually switches to exploitation, beyond $f_1$ (green lines in Figs.\[Traj\]a,b). This exploitation-recovery process is mantained in the long-term. Processes describing cycles along the zone of bistability are commonly refered as hysteretic. Figure \[Traj\]c depicts the long-term hysteretic exploitation-recovery process, as a function of the proximity to the critical value $\lambda_c$ corresponding to the bifurcation point $f_1$. - *Branch-confined pathways*: For large $\tau$ values, escapes to the upper branch are supressed and the system remains oscillating aperiodically without crossing the bifurcation point $f_1$ (blue line, Figs.\[Traj\]a,b). This aperiodic dynamics is predictable in the sense that the average growth of an initially small deviation around a reference state [@NicolisI] is sub-exponential (not shown). The increase in complexity of the dynamics occuring along with $\tau$ is illustrated by the inset of Fig.\[Traj\]a and by Fig.\[Traj\]c (blue lines). The overall range of exploitation and recovery becomes larger as $\tau$ is increased Fig.\[Traj\]c. In contrast with case **B1**, the evolution towards the bifurcation point is marked by short exploitation-recovery-exploitation cycles Fig.\[Traj\]c, where effective recovery avoids a system regime shift. For certain $\tau$-ranges long-term intermittence between behaviors **B2** and **B3** arises (not shown). It is worth noticing from Fig.\[Traj\]a that the regime shifts occur past and not at the crossing of the critical point $(x_c,\lambda_c)$ –following apparent extensions of the lower stable branch, beyond the critical point. Such an “inertia” effect is a result of the critical slowing down, where the dynamics of *x* becomes ’slaved’ by the slower dynamics of $\lambda$. ![For $\alpha=1$ (yellow), $\alpha=5$ (green) and $\alpha=20$ (magenta) and $\omega_{\lambda}=0$, a) plot of $\epsilon$ vs $\tau$ for deterministic **CCS** trajectories ($\omega_x=0$). Similar plots for stochastic trajectories with b) $\omega_x=5$, c) $\omega_x=10$ and d) $\omega_x=25$. Parameter values, initial condition and numerical integration as in Fig.\[Traj\]. Here, each $\epsilon$ value has been computed for $1\times10^6$ time steps.[]{data-label="Eff"}](Eff.PDF){width=".65\textwidth" height=".35\textheight"} ### CCS control efficiency A meaningful way to characterize further the **CCS** dynamics is to quantify the control capacity to increase and mantain long-term explotation within a selected stable branch, as a function of the model paramaters. Accordingly, we quantify the efficiency as $$\epsilon(\tau,\alpha)=\lim_{T\rightarrow \infty} \frac{\omega^2_R}{T M_{T}^2} \int_{0}^{T} \frac{\Phi(\tau,\alpha) H(\Phi(\tau,\alpha)) H(-x(t))}{k_\lambda} dt\label{Effi}$$ *H* denotes a Heavyside function, whose value is zero for negative argument or one otherwise. The product of the Heavyside functions in the integral plays the role of an AND boolean operator, entailing that the contributions to the time average of $\Phi$ are considered only if $\Phi$ is positive and $x$ negative. The variance of deviations from the initial to the final time *T*, in the denominator of , aims at amplifying differences between hysteretic (**B2**) and branch-confined (**B3**) control regimes. Without loss of generality, hereafter we set the reference value $\omega^2_R=1$ in . In the next subsections we focus on the behavior of the efficiency as a function of the size of the observation window $\tau$, of the degree of control response $\alpha$ and of the variance of random fluctuations $\omega_x$ and $\omega_{\lambda}$. ### Deterministic CCS efficiency Figure \[Eff\]a summarizes the behavior of the efficiency over the deterministic **CCS** trajectories corresponding to different values in $\tau$ and $\alpha$. Efficient control rises sharply as the observation window is increased above an $\alpha$-specific threshold $\tau_c$ –since below this value the **CCS** undergoes exploitation runaway (**B1**). Above $\tau_c$, the behavior of the efficiency becomes highly dependent on the degree of control response. For a low control response $(\alpha=1)$, the efficiency exhibits a quasi-constant plateau for $\tau>\tau_c$. This plateau consists of hysteretic cases (**B2**). In cases of moderate ($\alpha=5$) and high ($\alpha=20$) degree of control response, the behavior of the **CCS** efficiency becomes non-trivial. For values $\tau\gg\tau_c$ the efficiency is remarkably enhanced (see the appearance of a second plateau occurring for $\tau\geq 25$ in Fig\[Eff\]a), as a result of the suppression of the **CCS** hysteretic control pathways, in favour of branch-confined recovery-exploitation cycles (**B3**). For intermediate values $\tau>\tau_c$, a combination of pathways (**B2**) and (**B3**) is observed –where low efficiency values correspond to hysteretic control pathways (**B2**). Such a $\tau$-dependent selection of control pathways is further illustrated by the histogram of the *x* variable in Fig.\[EffHist\]a, for the high control response case. It shows that the dynamics of the **CCS** becomes confined to $\textit{x}<0$ values as $\tau$ is slightly shifted, which explains the alternace of low and high efficiency ranges appearing at intermediate $\tau$ values. Comparing the different control response cases in Fig.\[Eff\]a, it can be observed that the moderate one is the most efficient for $\tau \gg \tau_c$. ![ For $\alpha=20$, a) frequency histograms of **CCS** deterministic trajectories ($\omega_x=0$, $\omega_\lambda=0$) for $\tau=22.5$ (red), $\tau=22$ (green) and $\tau=19.5$ (blue). For $\tau=22.5$ and $\omega_\lambda=0$, b) histograms of stochastic trajectories with $\omega_x=10$ (brown) and $\omega_x=25$ (yellow), as well as histogram of the deterministic case (red), as in panel a). Parameter values, initial condition and numerical integration as in Fig..[]{data-label="EffHist"}](Effi.PDF){width=".65\textwidth" height=".35\textheight"} ### Stochastic CCS efficiency Let us address now the case where the evolution of *x* is marked by continuous random perturbations (i.e., $\omega_x\neq0$, $\omega_{\lambda}=0$ in Eqs. (\[EqGen1\]) and (\[EqGen2\])). Panels b) to d) in Fig. illustrate the influence of noise on the **CCS** efficiency. In the case of low control response, the behavior of the efficiency with $\tau$ appears to be independent of the standard deviation of noise $\omega_x$. In contrast, for moderate and high control responses, the overall efficiency becomes enhanced by an increase in the intensity of noise, in the sense that low efficiency values tend to disappear while high efficiency occurs all the way above the $\tau_c$-value (compare Fig.\[Eff\]a with Figs. \[Eff\]b–d). However, if the variance of noise is marginally large (Fig.\[Eff\]a-c), the high control response case is less efficient as compared with the moderate one. For strong noise (Fig.\[Eff\]d), both moderate and high control response cases converge to a similar efficiency value for large $\tau$. Since, for high and moderate response cases, low efficiency values tend to disappear when increasing intensity noise, it is natural to inquire about the influence of strong noise on hystheretic pathways. This effect is shown by the histogram of x Fig.\[EffHist\]b for a single realization with $\alpha=20$. Notice the similarity between panels (a) and (b), which illustrates the equivalence in the control pathway selection that occurs either by changes in $\tau$ (Fig.\[EffHist\]a) or by an increase in the standard deviation of the random fluctuations $\omega_x$ (Fig.\[EffHist\]b). ![For $\alpha=1$ (yellow), $\alpha=5$ (green) $\alpha=20$ (magenta) and $\omega_x=0$, plot of $\epsilon$ vs $\tau$ for stochastic **CCS** trajectories with a) $\omega_{\lambda}=5$, b) $\omega_{\lambda}=10$ and c) $\omega_{\lambda}=20$. In d), similar plots for $\omega_{\lambda}=25$ and $\omega_x=20$. Parameter values, initial condition and numerical integration as in Fig.\[Traj\]. Here, each $\epsilon$ value has been computed for $1\times10^6$ time steps.[]{data-label="EffLambda"}](EffLambda.PDF){width=".65\textwidth" height=".35\textheight"} Interestingly enough, we observe a similar effect of efficiency enhancement and uniformization in the case of noise at the level of the control system (i.e., $\omega_x=0$, $\omega_{\lambda}\neq0$) Fig.\[EffLambda\]a-c, as compared with the deterministic **CCS** (Fig.\[Eff\]a). However, in this case the emergence of efficient control occurs for larger values in $\tau$, as the intensity of noise increases. Finally, in the case of strong noise at the level of both the state variable *x* and control (i.e., $\omega_x\gg0$, $\omega_{\lambda}\gg0$) Fig.\[EffLambda\]d, the case of high control response becomes the most efficient one, in comparison with the ($\omega_x\neq0$, $\omega_{\lambda}=0$) cases (Fig.\[Eff\]d). ![ Behavior of the deterministic **HCS** ($\omega_x=0, \omega_{\lambda}=0$). Plot of $x_1$ *vs* time for a) $\tau=0.5$, b) $\tau=3$, c) $\tau=10$. Corresponding to these $\tau$ values, panels a’)-c’) show the plot $\Phi/k_{\lambda}$ *vs* $\lambda$.[]{data-label="OscilaPhase"}](OscilPhase.PDF){width=".65\textwidth" height=".35\textheight"} The Hopf-control system (HCS) ----------------------------- A relevant and also frequently observed type of regime shift consists in the onset of time periodic behavior, as a stationary system regime becomes unstable at a threshold value in a system parameter. Examples of processes and systems where such phenomenon is present are for instance, chemical clocks at the mesoscopic [@McEwen] and macroscopic [@Epstein] scales, cellular biochemical cycles [@Goldbeter], prey-predator population dynamics [@Murray], shallow lake ecosystems [@Scheffer], the ocean-atmosphere system [@Vallis], or non-equilibrium economics models [@Hallegatte]. All models accounting for the emergence of oscillatory behavior involve at least two variables and in all cases it has been possible to assimilate the mechanisms underlying the critical onset of oscillations to a Hopf bifurcation [@Guckenheimer]. In what follows we focus on the dynamics of the coupled system of control - and the deterministic evolution law in , given by the Hopf normal form –which we refer hereafter as the Hopf-control system (**HCS**). For illustrative purposes and without loss of generality, we consider the supercritical version of the Hopf normal form $$\begin{array}{c l} F_1(x_1,x_2,\lambda)=\lambda x_1-x_2-x_1(x_1^2+x_2^2), & F_2(x_1,x_2,\lambda)=\lambda x_2-x_1-x_2(x_1^2+x_2^2) \end{array}\label{Hnormalform}$$ The stationary solution $x_1=0$, $x_2=0$ to the Hopf normal form losses stability as $\lambda$ approaches zero along the negative axis. Above the critical value $\lambda_c=0$, the system variables $(x_1,x_2)$ describe stable oscillations whose amplitudes increase as $\sqrt{\lambda}$. Here, the control functional is fed by either of both system variables $(x_1,x_2)$, whose dynamics is subjected to stochastic perturbations with same standard deviation $\omega_x$. ### Deterministic HCS dynamics For an initial condition around the stationary solution $x_1=0$, $x_2=0$ and for an intial value $\lambda<0$, Figs.\[OscilaPhase\]a-c show the deterministic dynamics of the $x_1$ component under the influence of control for different values of the time window $\tau$. In complete absence of control, the linear growth in the value of $\lambda$, beyond the critical point, leads to oscillations of increasing amplitude $\sqrt{\lambda}$. In contrast, for small values $\tau\gtrsim0$ the amplitude of oscillations becomes constant (Fig.\[OscilaPhase\]a). In this situation, both recovery ($\Phi<0$) and exploitation ($\Phi>0$) phases succeed each other rapidly, while the control variable always remains above the critical value $\lambda_c=0$ (Fig. \[OscilaPhase\]a’). A further increase in $\tau$ induces a temporal alternance of small and large oscillation amplitudes (Fig.\[OscilaPhase\]b), as the recovery and exploitation phases lead the system back and forth accross $\lambda_c$ (Fig.\[OscilaPhase\]b’). If $\tau$ is large enough, oscillations are temporally suppressed (Fig.\[OscilaPhase\]c), during a time interval that increases with $\tau$. This situation results from the fact that the $\lambda$-interval covered by the recovery and exploitation phases becomes enlarged (Fig.\[OscilaPhase\]c’). In contrast with the **CCS** Fig.\[Cusp\]c, no substantial increase in the complexity of the dynamics is observed as the time window $\tau$ is arbitrarily increased. ![For $\alpha=1$ (yellow), $\alpha=5$ (green), $\alpha=20$ (magenta) and $\omega_{\lambda}=0$, a) plot of $\epsilon$ vs $\tau$ for deterministic **HCS** trajectories ($\omega_x=0$). Similar plots for stochastic trajectories with b) $\omega_x=5$, c) $\omega_x=10$ and d) $\omega_x=25$. Parameter values, initial condition and numerical integration as in Fig.\[Traj\]. Here, each $\epsilon$ value has been computed for $1\times10^6$ time steps.[]{data-label="EffHopf"}](EffHopf.PDF){width=".65\textwidth" height=".35\textheight"} ### Deterministic and stochastic HCS efficiency In a similar vein as for the **CCS**, we define the efficiency of the **HCS** as the capacity to increase and mantain long-term subcritical explotation, while keeping the oscillation amplitudes to a minimum. Thus we write the efficiency as $$\epsilon(\tau,\alpha)=\lim_{T\rightarrow \infty} \frac{\omega^2_R}{T M_{T}^2} \int_{0}^{T} \frac{\Phi(\tau,\alpha) H(\Phi(\tau,\alpha))H(-\lambda(t))}{k_\lambda} dt\label{EffiH}$$ where H denotes a Heavyside function, such as defined for .\ As shown by Figs.\[OscilaPhase\]a’–c’, subcritical exploitation increases with the size of the observation window $\tau$. However, it is also observed that the amplitude of oscillations along the recovery phase also increases with $\tau$ (Figs.\[OscilaPhase\]a–c). Therefore, an optimum is expected to occur in the efficiency as a function of $\tau$. This situation is actually illustrated in Fig.\[EffHopf\]a for different degrees of control response. Notice that compared to the efficiency of the deterministic **CCS** Fig.\[Effi\]a, the sharp rise towards efficient control occurs at a small value $\tau_c$, above which efficiency gradually decays for larger $\tau$ values. ![For $\alpha=1$ (yellow), $\alpha=5$ (green) $\alpha=20$ (magenta) and ($\omega_x=0$), plot of $\epsilon$ vs $\tau$ for stochastic HCS trajectories with a) $\omega_{\lambda}=5$, b) $\omega_{\lambda}=10$ and c) $\omega_{\lambda}=20$. In d), similar plots for $\omega_{\lambda}=25$ and $\omega_x=20$. Parameter values, initial condition and numerical integration as in Fig.\[Traj\].[]{data-label="EffHLambda"}](EffHLambda.PDF){width=".65\textwidth" height=".35\textheight"} In presence of noise at the level of the system variables ($x_1,x_2$) (i.e. $\omega_x\neq0, \omega_{\lambda}=0$), it is observed that the efficiency increases along with $\tau$, for the moderate and high control response cases, while for low control response it remains at low practically constant value (Fig.\[EffHopf\]b). Similarly as for the **CCS** (Fig.\[Eff\]), the efficiency is in overall enhanced by the intensity of noise Fig.\[EffHopf\]b–d. Moreover, the behavior of the efficiency is practically the same for moderate and high control response cases, with the exception of the strong noise case Fig.\[EffHopf\]d, where high control response is highest for arbitrary large $\tau$ values. When considering the effect of stochastic perturbations on the control dynamics (i.e. $\omega_x=0, \omega_{\lambda}\neq0$) Figs.\[EffHLambda\]a–c, the behavior of the efficiency remains qualitatively the same as in the purely deterministic case Fig.\[EffHopf\]a. Regardless the intensity of control fluctuations, a maximum in efficiency is observed at an approximately same $\tau$ value. The efficiency associated with high and moderate control responses exhibit a similar dependency on $\tau$. In both cases, the maximum in efficiency is higher as compared to the low response one. However, the highest efficiency peaks are slightly lowered when augmenting the intensity of noise at the level of control. For all the control response cases, when increasing $\tau$ the corresponding efficiencies tend to converge to a same low value, independently of the intensity of noise in control. Finally, in presence of strong fluctuations at the level of the dynamics of both the system and the control (i.e. $\omega_x\gg0, \omega_{\lambda}\gg0$), the efficiency rises towards a plateau for larger values of $\tau$. In this case it is observed that the presence of strong fluctuations enhances the control efficiency and to a greater extent in the cases of high and moderate control response. Summary and conclusions ======================= In this work we addressed the variance-based control of systems evolving towards criticality. As testbed models we considered the cusp and Hopf normal forms, displaying the prototypical phenomena of bistability and oscillations, respectively. In relation to the core questions ***i)***–***iii)*** exposed in Sec. I, our main results can be summarized as follows: ***i)*** In absence of noise, effective control in both the cusp and the Hopf systems rises above a threshold value in the size of the observation time window. ***ii)*** In the cusp-control system, changes in the observation time window provide a means to selecting among different control pathways, namely, hysteretic, branch-confined control, as well as an intermittent combination of both. The branch-confined regime becomes dominant when considering a large observation window. This entails an increase in the complexity of the dynamics, in the form of aperiodic control-system trajectories. In the case of the Hopf-control system, the crossing of the bifurcation point could not be avoided, for the range of paramaters explored. Instead, as the observation time window is enlarged, exploitation and recovery phases occur for longer periods. This gives rise to the alternace of time intervals where oscillations are suppressed, followed by amplified oscillations. ***iii)*** In both systems, random fluctuations, at the level of the system variables, enhance the efficiency of control, and especially for large observation time windows. In the case of the cusp-control system, the intensity of noise induces a control pathway selection, similarly as in the deterministic case when varying the observation time window. From the deterministic to the strong fluctuations cases, the efficiency of the cusp-control system tends to increase for large values in the size of the observation window. In the Hopf-control system, an increase in the intensity of noise induces a reduction in the amplitude of oscillations, which entails an increase in the control efficiency. Random fluctuations at the level of the control dynamics exert opposite influences on the control efficiency of the cusp and Hopf control systems. In the former it increases, while for the latter it tends to diminish the peak of efficiency. For both cusp and Hopf control systems, the combination of fluctuations in the control and in the system dynamics amplifies the efficiency in particular for large observation time windows. Regarding the interplay between the efficiency, the degree of control response, the intensity of noise, and the observation time window, we observe: in both systems, the low control response case exhibits a low efficiency, independently of the intensity of noise and observation window. When increasing the variance of fluctuations in the dynamics of the cusp system, a moderate control response leads to a higher efficiency; the opposite situation occurs when considering fluctuations on the control dynamics. In the Hopf system, the moderate and high control response perform very similar, regardless the intensity of noise and size of the time window. In this situation, for both moderate and high response cases, the efficiency converges to the lower value associated with a low control response, as the observation time window is increased. For the systems here addressed, it remains to be studied the influence of other parameters, such as the rate of change in the control system ($k_{\lambda}$), the control reference parameter ($V_R$), as well as alternative approaches towards bifurcation, such as variation of the parameter $\mu$ in the cusp map –which would induce flickering patterns at the vicinity of the critical point. Potential applications of this approach to real world systems include ecosystem models, non-equilibrium mesoscopic physico-chemical systems or non-equilibrium models in economics, among others. Similarly, it is of interest to extend this work to address the control properties associated to different alternative early warning indicators. Moreover, our analisis could be extended to address the control of coupled systems, forming networks whose nodes consist in dynamical systems subject to control. Certainly, the structure of the evolution equations of real world models, involving a large number of parameters, lacks the degree of symmetry that confers to normal forms their characteristic simplicity. Moreover, not every dynamical system exhibiting a bifurcation belongs to a class of universality. It is thus important to address further the relation between the observation time scales leading to effective control and the geometrical properties, such as the curvature, of the system solution branches, while considering different early warning indicators. Acknowledgements ================ We acknowledge the Federal Ministry of Education and Research (BMBF) through the program “Spitzenforschung und Innovation in den Neuen Länden” (contract “Potsdam Research Cluster for Georisk Analysis, Environmental Change and Sustainability” D.1.1) for financial support. A.G.C.R. acknowledges Christian Pape, Prof. Gregoire Nicolis and Doctors Elena Surovyatkina, Vasileos Basios and Flavio Pinto for fruitful discussions and for encouraging this work. [10]{} Nicolis G. and Prigogine I. . W. H. Freeman Press, 1989. R. Badii and Politi A. . Cambridge Press, 1997. H. E. Stanley. . Oxford Press, 1987. Nicolis G. and Prigogine I. . John Wiley and Sons Inc., 1977. Cross M. and Greenside H. . Cambridge Press, 2009. Scheffer M., Carpenter S., Foley J.A., Folke C., and Walker B. Catastrophic shifts in ecosystems. , 413:591–596, 2001. Scheffer M. . Princeton Press, 2009. Castellano C., Fortunato S., and Loret V. 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--- abstract: 'For strongly screened Coulomb interactions, quantum Hall interferometers can operate in a novel regime: the intrinsic energy gap can be larger than the charging energy, and addition of flux quanta can occur without adding quasi-particles. We show that flux superperiods are possible, and reconcile their appearance with the Byers-Yang theorem. We explain that the observation of anyonic statistical phases is possible by tuning to the transition from a regime with constant chemical potential to a regime with constant particle density, where a flux superperiod changes to a periodicity with one flux quantum at a critical magnetic field strength.' author: - Bernd Rosenow - Ady Stern date: July 14 2019 title: Flux superperiods and periodicity transitions in quantum Hall interferometers --- Fractionally charged quasi-particles (qps) are among the most intriguing aspects of the fractional quantum Hall effect. Their existence and their fractional anyonic statistics are believed to be unavoidable consequences of the quantization of the Hall conductivity to a fractional value of $e^2/h$, with $e$ the electron charge, and $h$ Planck’s constant. Much experimental effort has been devoted towards a measurement of the fractional charge and the unconventional statistics. The fractional charge has been measured through noise [@shot1; @shot2; @shot3; @Dolev+08] and electrometry [@Goldmann95; @Martin+04; @Vivek+11]. For the statistics, defined by the geometric phase obtained when one qp encircles another, the natural tool to use is interferometry [@Chamon+97]. Indeed, many attempts to study integer and fractional quantum Hall states through Fabry-Perot [@Zhang+09; @Lin+09; @Ofek+10; @McClure+09; @McClure+12; @Choi+11; @Baer+13; @Camino+05; @Camino+07; @Willett+09; @Willett+13] and Mach-Zehnder interferometry [@Ji+03; @Roulleau+08; @Gurman+16] have been carried out. Interpreting interference experiments in quantum Hall (QH) Fabry-Perot devices was found to be complicated by the strong Coulomb interaction, which can drive the interferometer into a Coulomb-dominated regime [@subperiod; @FPtheory]. This problem is particularly severe in small (micron size) interferometers, which are favored from the point of view of battling decoherence. Attempts to screen the Coulomb interaction using Ohmic contacts situated inside or close to a Fabry-Perot interferometer were successful in the integer regime, resulting in novel halving of flux and gate voltage periodicities [@Choi+15; @Frigeri+17]. More recently, the long-range Coulomb coupling has been addressed by the introduction of screening layers both above and below the transport layer [@manfra19]. In this setup, the phenomenology of the weakly interacting Aharonov-Bohm (AB) regime was observed, and an interference signal for the fractional $1/3$ state was reported. Rather unexpectedly, the oscillating part of the conductance through the interferometer followed a flux “superperiod” of three flux quanta $\Delta\Phi=3\Phi_0= 3 h/e$, as compared to the previously predicted period, $\Phi_0$ [@FPtheory] (for a discussion of superperiods in other geometries see [@ThouGe; @GeThou; @Jain+93]). Since the interior of the interferometer is incompressible and not accessible to the interfering qps, one may wonder whether such a super-period violates the Byers-Yang theorem [@ByYa61], which states that in the presence of a “hole” in a multiply connected system, all observables are periodic when the magnetic flux through the hole is adiabatically varied by $\Phi_0$. ![Left panel: Side view - schematic heterostructure with a transport layer (blue) and two screening layers, separated by insulating barriers (grey). Right panel: Top view - Fabry-Perot interferometer with $N_{qp}$ qps localized in the interference cell. \[fig-setup\]](setup_combined_v2){width="45.00000%"} In this letter, we formulate a theory of QH Fabry-Perot interferometers in the presence of screened Coulomb interaction. Taking into account the competition between the charging energy and the energy gap for qp excitations, we find that around the center of QH plateaus there is a region of magnetic field $B$ in which small variations of $B$ does not lead to qp addition in the bulk. The width of this region sets a lower bound for the extension of QH plateaus. For an interferometer, this implies that there is a region of $B$ in which the number of qps in the interference loop is $B$–independent. When that happens, the interfering qp accumulates an AB phase which is reduced compared to the electronic one by the ratio of qp charge $e^*$ and electron charge $e$, hence leading to a magnetic flux superperiod. Beyond the critical magnetic field limiting this region, every additional flux quantum causes the creation of bulk qp(s), augmenting the AB phase by an anyonic phase and reducing the the interference period to $\Phi_0$. Before discussing interferometry, we consider a bulk transport layer with particle density $n_1$ capacitively coupled to a screening layer with density $n_2$. We assume no tunneling between the layers. Coulomb coupling between the two layers creates image charges in the screening layer, making the interaction in the transport layer short-ranged. Viewing the combination of the two layers as a planar capacitor, the charging of the capacitor can be described by a density variation $\delta n$, such that the density of the transport layer is $n_1 = {\nu\over 2 \pi \ell_{B_0}^2} + \delta n$ and the density of the screening layer is $n_2 = n_2^{(0)} - \delta n$. Here, $B_0$ is the reference magnetic field in the center of the quantum Hall plateau, and $\ell_B = \sqrt{\hbar/e B}$ is the magnetic length. We consider an incompressible QH state, with a possible finite density of qps if the magnetic field is lowered as compared to $B_0$. Then, the energy density is given by $$\begin{aligned} E & = & \left(E_0 - \mu \right) {\nu \over 2 \pi \ell_B^2} + \Delta_{e^*}(B){1 \over e^*} \left({\delta n} + { \nu\over 2 \pi \ell_{B_0}^2} - {\nu \over 2 \pi \ell_B^2}\right)\nonumber \\ & & + {\delta n^2 e^2\over 2} \left({1 \over C_g} + {1 \over C_q}\right) \ \ . \label{energydensity.eq}\end{aligned}$$ Here, $E_0$ is the ground state energy per electron in the pristine QH state with filling factor $\nu$, and $e^*$ is measured in units of the electron charge. Furthermore, $\Delta_{e^*}(B)$ is the magnetic field dependent energy gap for adding a qp, $C_g$ denotes the geometric capacitance between transport and screening layer, and $C_q$ is the quantum capacitance of the screening layer. In the second term on the r.h.s. of Eq. (\[energydensity.eq\]), the overall factor of $1/e^*$ is due to the ratio of the density of qps to the density of electrons, and the factors $\nu$ combined with the area per electron account for the filling of the Landau level. The extra filling above the quantized fraction is $$\delta \nu \ = \ 2 \pi \ell_B^2 \delta n \ + \ \nu {\ell_B^2 \over \ell_{B_{0}}^2 } \ - \ \nu \ \ .$$ When minimizing the energy density Eq.(\[energydensity.eq\]) with respect to $\delta n$, we find $$\delta n \ = \ - {{\Delta_{e^*}(B)} \over e^* e^2 \left({1 \over C_{\rm g}} + {1 \over C_q} \right) } \ \ . \label{density_fractional.eq}$$ According to our assumption $B < B_0$, the density of qps cannot be negative, and we find the constraint $$\delta n + { \nu \over 2 \pi \ell_{B_0}^2} - {\nu \over 2 \pi \ell_B^2} \ \geq \ 0 \ \ . \label{inequality_fractional.eq}$$ Clearly, for small changes of $B$ with $(B_0 - B)/B_0 \ll1$ the density change found in Eq. (\[density\_fractional.eq\]) violates the above inequality. Therefore, the actual density change is determined by viewing Eq. (\[inequality\_fractional.eq\]) as an equality, which implies that $\delta\nu=0$, i.e., the filling factor stays constant while $B$ is varied away from $B_0$. Thus, $$\delta n \ = \ - \nu {B_0 - B \over \Phi_0} \ , \ \ \ B_c < B < B_0 \ \ , \label{deltan_inequality.eq}$$ When $B$ is further lowered away from $B_0$, a critical magnetic field $B_c$ is reached at which the expressions for the density change Eqs. (\[density\_fractional.eq\]), (\[deltan\_inequality.eq\]) are equal. From this condition, we obtain $${ B_c } \ = \ {B_0 } \ - \ {\Delta_{e^*}(B) \Phi_0 \over \nu e^* \left({e^2 \over C_{g}} + {e^2 \over C_q}\right)} \ \ . \label{criticalfield.eq}$$ For $B < B_c$, the filling factor does not remain constant anymore, the density change is determined by Eq. (\[density\_fractional.eq\]), and its $B$–dependence is determined by the $B$–dependence of the energy gap $\Delta_{e^*}$. For integer QH states $e^*=1$, and the energy gap is given by either the exchange enhanced spin splitting, or the cyclotron gap, depending on whether the lowest unoccupied Landau level has the same orbital index as the highest occupied one. For concreteness, we discuss a gap given by the cyclotron energy $\hbar e B/m^*$, where $m^*$ denotes the effective mass of electrons. Then, $$B_{c,\rm int} \left( 1 + {2 \pi \hbar^2 C_{\rm tot} \over e^2 m_{\rm eff}}\right) \ = \ B_0 \ \ ,$$ where ${1 \over C_{\rm tot}} \equiv {1 \over C_g} + {1 \over C_q}$. We note that usually the geometric capacitance dominates over the quantum capacitance, such that $${1 \over C_{\rm tot}} \ \gtrapprox \ {d \over \epsilon } \ \ .$$ With one screening layer, $d$ denotes the spatial distance between transport layer and screening layer. With screening layers on both sides of the transport layer, the capacitances add up and $d$ is half of this distance. In the experiment reported in [@manfra19], there are two such screening layers, each about $50 nm$ from the transport layer, and $d \approx 25 nm$ [@manfra19]. Introducing the Bohr radius $a_0 = {4 \pi \epsilon_0 \hbar^2 \over m_e e^2} $, we express the critical magnetic field as $$B_{c, \rm int} \ = \ B_0 \left(1 \ + \ {\epsilon_r \over 2} {m_e \over m_{\rm eff}} \, {a_0 \over d} \right)^{-1} \ \ . \label{bc_integer.eq}$$ For the experimentally relevant parameters $m_{\rm eff} = 0.07 m_e$, $\epsilon_r = 13$, $d = 25 nm$ we find that $B_{c, \rm int} \approx {5 \over 6} B_0$, implying that the total magnetic field width of the $\nu=2$ plateau should be at least $1/3$ of the magnetic field at its center, in agreement with the plateau width in [@manfra19]. Turning now to a Fabry-Perot interferometer with an interference cell of radius $R$, we note that the screening layer(s) significantly weakens the interaction coupling between the edge and bulk of an interferometer: only an annulus of bulk area $2\pi R d \ll \pi R^2$ interacts with the edge. Since only a small part of charges accumulated in the bulk interacts with the edge, one can expect that the bulk-edge coupling is parametrically smaller than the self-interaction of edge charges, placing the interferometer in the AB regime [@subperiod; @FPtheory]. It is indeed this regime which was observed in Ref. [@manfra19]. Since there are no anyonic phases for integer QH states the flux periodicity is not affected by whether or not electrons enter the interference cell. For filling fractions higher than two however, a sufficiently strong inter-mode Coulomb coupling, together with a weak bulk-edge coupling, halves both flux and gate voltage periods [@Frigeri+17]. Observation of such a halving in the experiment [@manfra19] indicates that the above conditions are indeed satisfied. For the fractional QH effect the energy gap is set by the Coulomb interaction $\Delta_{e^*} = \alpha_{e^*} {e^2 \over 4 \pi \epsilon \ell_B}$, with $0.01 \lesssim \alpha \lesssim 0.05$ parametrizing the specific value of the energy gap for different fractional quantum Hall states [@Morf03]. With this, we obtain $$\begin{aligned} B_{c,\rm frac} & = & B_0 \left( 1 \ + \ {\gamma^2 \over 2} \ - \ \gamma \sqrt{1 + {\gamma^2 \over 4} }\right) \label{bc_fractional.eq} \end{aligned}$$ with $$\gamma \ = \ {\alpha_{e^*} \over 2 \nu e^*} \, {\ell_{B_0} \over d} \ \ .$$ For a gap of $7K$ [@manfra19] for $\nu=1/3$ we obtain $\gamma \approx 0.018$. As a consequence, the relative change in $B$ before qps enter in the bulk is about 2%. ![Change in oscillation periodicity of a Fabry-Perot interferometer as a function of magentic field for the $\nu=1/3$ quantum Hall state. For magnetic fields near the plateau center with $B_c < B < B_0$, no qps enter the interference cell and the magnetic fiel period is $\Delta B = 3 \Phi_0/A$. For $B < B_c $, a quasi-hole enters for each additional flux quantum, and the period changes to $\Delta B = \Phi_0/A$ due to the combination of Aharonov-Bohm phase and anyonic statistical phase. \[fig-oscillations\]](oscillations_labels_v2){width="40.00000%"} Using the expressions we derived for the bulk density variation with $B$, we may estimate the number of qps inside an interferometer of area $A$. As long as that number does not change with the magnetic field, the interference follows a superperiod. We find the interferometer to host $$\begin{aligned} N_{qp} & = & {A(B_0 - B) \over \Phi_0}\ - \ { 1 \over 2 \nu e^*} \, {\Delta_{e^*} \over E_c} \ , \ \ \ B < B_c \end{aligned}$$ qps, where the charging energy is $E_c = {e^2 d \over 2 \epsilon A}$. Using the same parameters as above, the area per flux quantum is $2 \pi \ell_B^2 \approx 6 \cdot 49 nm^2 \approx 300 nm^2$. The area of the interferometer [@manfra19] can be estimated from the magnetic field periodicity $\delta B \approx 7 mT$ in the integer regime according to $A = \phi_0 / \delta B \approx 0.56 \mu m^2$. For this area, there would be approximately $1.87 \cdot 10^3$ flux quanta in the interferometer cell, and the offset in the qp number would be $33$ for a gap of $2K$ and $117$ for a gap of $7K$, i.e. there would be $11-39$ oscillations with periodicity $3 \Phi_0$ before extra qps enter. After that, the periodicity should be reduced to $\Phi_0$, see the discussion below. We get a similar estimate by using the experimental values [@manfra19] for $\Delta_{e^*} \approx 700 \mu eV$ and the charging energy $E_c \approx 17 \mu eV$ to obtain an offset of $700/(2\cdot 17/9)\approx 126$ flux quanta through the interferometer, corresponding to 42 oscillations. Focusing on $\nu=1/3$, we denote the range $B_c < B < B_0$ as regime I, and the range of $B < B_c$ but with the system still staying on the $\nu=1/3$ plateau as regime II. The range of the Coulomb interaction is determined by the distance $d$ between the transport and screening layer. As long as $d\gg l_B$, the short range part of the Coulomb interaction, which is responsible for the formation of fractional quantum Hall states, is unchanged, and one expects to find a stable hierarchy of fractional quantum Hall states. For $\nu=1/3$ Laughlin states in regime I the $B$–dependence of the interference phase is purely due to the Aharonov-Bohm phase $e^* A B/\Phi_0$. As a consequence, the flux periodicity is $\Phi_0/e^*$, giving rise to a flux superperiod. Similarly, for $\nu = 1/(2 m +1)$ superperiods with $e^*=1/(2m+1)$ occur in regime I. In contrast, in regime II a reduction of the magnetic flux by $\Phi_0$ introduces a qp in the interference loop. Then, the phase accumulated due to a flux change of one $\Phi_0$ is the sum of the AB phase $2 \pi e^*$ and a statistical phase $4 \pi e^*$, which in sum yield a phase change of $2 \pi$, such that the flux period is $\Phi_0$. A similar outcome holds for $\nu=1/(2m+1)$. Experimental observation of the phase shift $4 \pi m e^*$ as a jump in the interference pattern would be a highly desirable demonstration of anyonic statistics. At finite temperature however, the phase jump is smeared due to fluctuations in the qp number [@FPtheory] if the characteristic relaxation time is shorter than the measurement time. When Fourier transforming a phase jump, higher harmonics in the magnetic flux dependence arise. The amplitude of the $n$-th harmonic is proportional to $\exp\left[-\pi^2 (n-1+2 e^* m)^2 k_B T/E_c (e^*)^2\right]$ [@FPtheory]. For this reason, the temperature needs to satisfy $k_B T \leq (e^*)^2 E_c/\pi^2$ for phase jumps to be observable. While for the sample used in [@manfra19] this would necessitate a temperature of $2 mK$, a moderate increase of the charging energy could make the observation of phase jumps feasible. As an additional experimental parameter, we consider an external gate which changes the area of the interference cell according to $\delta A = {d A \over d V_G} \delta V_G$, where ${d A \over d V_G}$ is assumed to depend only weakly on magnetic field. Such an area gate does not add qps to the bulk of the interferometer, so the gate voltage periodicity will be the same in regimes I and II. Then, the gate voltage periodicity is given by $\Delta V_G = \Phi_0/(e^* B {d A \over d V_G})$. Due to the fact that the magnetic field $B_{1/3}$ at the filling fraction $\nu =1/3$ is three times larger than the field $B_1$ at $\nu=1$, we find that the gate voltage periodicities are expected to be equal in the two cases, as observed in Ref. [@manfra19]. The Byers-Yang theorem [@ByYa61] states that in a multiply connected geometry with flux through a “hole”, all physical properties are periodic under a change of the magnetic flux by one flux quantum. In the context of QH Fabry-Perot interferometers, there is no actual “hole” through the interferometer, unless an anti-dot is placed inside the interference cell. However, the gapped bulk is inaccessible to interfering qps, and it is interesting to ask under which conditions the Byers-Yang theorem applies. In QH interferometers, the interference phase is determined by the magnetic flux enclosed by the interference path. In a pristine fractional QH state with filling fraction $\nu$, the flux is tied to the number of electrons $N_{el} = \nu B A/\Phi_0$ encircled by the interfering electron [@Arovas], and the interference phase can be interpreted as a statistical phase. In the limit of weak backscattering, the area $A$ encircled by the interfering particle is fixed due to electrostatic constraints, and in regime I with fixed chemical potential, the filling fraction $\nu$ is independent of magnetic field. For these reasons, the number of encircled electrons is directly proportional to magnetic field times filling fraction, and a super-period arises. Such a change in the number of encircled electrons however would not be possible if there was a hole without electrons (an anti-dot) inside the interferometer. In this situation, only electrons in the narrow annulus defined by the interfering edge on the outside and the perimeter of the anti-dot on the inside would contribute a statistical phase. If the area of this annulus is much smaller than the total interferometer area, then the number of electrons inside the annulus is approximately independent of magnetic field. The period of one flux quantum would then arise due to the modulation of the anti-dot energy spectrum with magnetic flux, which indeed is governed by the prediction of the Byers-Yang theorem. For filling factor $\nu=5/2$ interferometers are expected to be able to distinguish between abelian and non-abelian candidate states [@Stern+2010] through the even-odd effect [@evenodd1; @evenodd2; @evenodd3]. The magnetic field period for interfering non-abelian charges $e^* = {1 \over 4}$ qps is $4 \Phi_0$ if there is an even number of qps in the bulk of the interference cell, and $2 \Phi_0$ for an odd number of qps inside the interference cell. In addition, for an even number of qps, the interference phase can be shifted by $\pi$ depending on the internal state of the non-abelian degree of freedom of these qps. In regime I, the number of qps in the interior is independent of mangetic field, and is determined by the state of the pristine interferometer in the center of the plateau. Depending on the parity of the number of trapped qps in the bulk, the flux period could be $4 \Phi_0$ or $2 \Phi_0$. In regime II, a change of magnetic field is accompanied by a change of the number of qps in the bulk. Depending on the dynamics of the non-abelian degree of freedom associated with the bulk qps (the so-called neutral fermion), the flux period in the Pfaffian state (anti-Pfaffian state) for weak bulk-edge coupling can be $1 \Phi_0$ ($1.5 \Phi_0$) for fixed fermion parity, and $2 \Phi_0$ ($3 \Phi_0$) for random fermion parity, see [@keyserlingk+15]. For $\nu=2/3$, the magnetic field periodicity in regime II in the limit of a closed interferometer was discussed in [@Viola+12; @PaGeSi15]. Due to the presence of a neutral mode [@KaFiPo94], pairs of conductance peaks bunch together, giving rise to a doubling of the magnetic field period of the conductance as compared to a situation with a charge mode only. Here we discuss an interferometer with weak backscattering, operating in regime I. Generally, the charge and neutral modes have different velocities, $v_n,v_c$. When $v_n\ll v_c$, the charge $e^* = 2/3$ quasiparticle will have the highest visibility since its tunneling operator does not excite the neutral mode. In regime I, this implies a flux period of $\phi_0/e^* = {3 / 2}$, somewhat larger than the experimentally observed value [@manfra19] of $ \phi_0$. In contrast, if $v_n$ is sufficiently high the interference of charge $e^*=1/3$ qps will be observable and contribute to the interference signal, with a flux period of $3 \phi_0$. In regime II, the statistical phase due to addition of qps in the interferometer cell will again reduce the flux period. In the limit of a slow neutral mode, a subperiod of $\phi_0/2$ is expected [@PaGeSi15], which is doubled to $\phi_0$ for the case of $v_n\approx v_c$ [@Viola+12]. Our analysis of the critical field $B_c$ assumed that the pristine QH state is strictly incompressible and has a finite gap for excitations. In reality however, disorder will give rise to a finite density of states in the transport layer, which we parametrize with the help of a quantum capacitance $C_{q,t}$. Then, the second term Eq. (\[energydensity.eq\]) is replaced by ${e^2 \over 2 C_{q,t}} \left( \delta n + {\nu \over 2 \pi \ell_{B_0}^2} - { \nu \over 2 \pi \ell_B^2} \right)^2$. Minimizing this modified energy, one finds that the change in density is $$\delta n \ = - \ { {\nu \over 2 \pi \ell_{B_0}^2} - { \nu \over 2 \pi \ell_B^2} \over 1 + C_{q,t} \left({1 \over C_g} + {1 \over C_q}\right) } \ \ ,$$ slightly smaller that the density change implied from (\[inequality\_fractional.eq\]). Assuming the small density of states in the transport layer is independent of energy, the behavior described above continues until the energy of excited states is lowered to the level of the chemical potential. This reduces $B_c$ to $$B_{c,\rm dis} \ = \ B_c \left[ 1 + C_{q,t}\left({1 \over C_g} + {1 \over C_q}\right)\right]^{-1} \ \ ,$$ When the states below the gap are localized and do not contribute to transport, disorder affects the bulk plateaus only by slightly increasing their width as compared to the pristine case. Inside the interferometer, excited states occur at discrete energies smaller than the energy gap, with a spectrum that varies with disorder realization. Due to these states, qps will enter the interference cell at discrete values of the magnetic field, below the critical field. For fractional interferometers, this will lead to phase shifts in the interference pattern, which allow to determine the anyonic statistical phase. 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--- abstract: 'To gain a better understanding of the interplay between frustrated long-range interactions and zero-temperature quantum fluctuations, we investigate the ground-state phase diagram of the transverse-field Ising model with algebraically-decaying long-range Ising interactions on quasi one-dimensional infinite-cylinder triangular lattices. Technically, we apply various approaches including low- and high-field series expansions. For the classical long-range Ising model, we investigate cylinders with an arbitrary even circumference. We show the occurrence of gapped stripe-ordered phases emerging out of the infinitely-degenerate nearest-neighbor Ising ground-state space on the two-dimensional triangular lattice. Further, while cylinders with circumferences $6$, $10$, $14$ et cetera are always in the same stripe phase for any decay exponent of the long-range Ising interaction, the family of cylinders with circumferences $4$, $8$, $12$ et cetera displays a phase transition between two different types of stripe structures. For the full long-range transverse-field Ising model, we concentrate on cylinders with circumference four and six. The ground-state phase diagram consists of several quantum phases in both cases including an $x$-polarized phase, stripe-ordered phases, and clock-ordered phases which emerge from an order-by-disorder scenario already present in the nearest-neighbor model. In addition, the generic presence of a potential intermediate gapless phase with algebraic correlations and associated Kosterlitz-Thouless transitions is discussed for both cylinders.' author: - Jan Koziol - Sebastian Fey - 'Sebastian C. Kapfer' - Kai Phillip Schmidt bibliography: - 'bibliography.bib' title: 'Quantum criticality of the transverse-field Ising model with long-range interactions on triangular-lattice cylinders' --- The search for exotic phases of quantum matter and the identification of unconventional quantum-critical behavior is one prominent theme in current research on correlated quantum many-body systems. One important knob to trigger such exotic quantumness is frustration, which can either be present due to the lattice geometry like in antiferromagnetic quantum magnets on the triangular, Kagome or pyrochlore lattice containing odd loops or result from conflicting interactions like, most prominently, in Kitaev’s honeycomb model [@Kitaev2006] realizing a topologically-ordered quantum spin liquid. Typically, all the paradigmatic models studied in this context have short-range interactions. There are, however, also important instances where long-range interactions give rise to non-trivial properties, e.g. in the spin-ice systems where the occurrence of magnetic monopoles is a consequence of the long-range dipole-dipole interaction [@Castelnovo2008], or in ferromagnetic, unfrustrated long-range transverse-field Ising models (LRTFIMs) where critical exponents can vary continuously as a function of the strength of an algebraically decaying Ising interaction [@Nagle1970; @Fisher1972; @Dutta2001; @Knap2013; @Fey2016; @Defenu2017; @Fey2019]. It is therefore natural to investigate the interplay between frustration and long-range interactions, which we expect to result in unconventional quantum behavior. Further, this interplay is of direct relevance for experimental systems, most importantly in quantum simulators with Rydberg atoms displaying an effective van-der-Waals coupling [@Schauss2012] as well as with trapped cold ions allowing to realize a LRTFIM with tunable interactions on the geometrically frustrated triangular lattice [@Britton2012; @Islam2013; @Bohnet2016]. The transverse-field Ising model with algebraically decaying long-range interaction on the triangular lattice represents therefore the paradigmatic model to study the interplay of frustration and long-range interactions. While several studies have focused on the same model on a one-dimensional chain [@Koffel2012; @Knap2013; @Fey2016; @Sun2017; @Horita2017; @Vanderstraeten2018] including the frustrated antiferromagnetic case, there are less works on the two-dimensional problem on the triangular lattice [@Humeniuk2016; @Fey2019], which also reflects the higher complexity from a numerical perspective. Using the recently developed high-order series expansion approach for such systems [@Fey2019], it was found that the system displays the same quantum phase transition as the short-range nearest-neighbor model as long as the decay exponent of the long-range Ising interactions is not too small: There is a 3D-XY transition separating the high-field polarized phase from a clock-ordered phase [@Moessner2001; @Moessner2003; @Powalski2013], which results from an order-by-disorder scenario about the Ising limit possessing an extensive ground-state degeneracy of spin-ice states [@Moessner2001]. However, the situation for more slowly decaying Ising interactions is far less understood. This lead Saadatmand [@Saadatmand2018] to investigate the LRTFIM on a quasi-one-dimensional triangular cylinder lattice with circumference six by infinite-size density matrix renormalization group (iDMRG) calculations. Interestingly, apart from a similar type of clock order as well as a trivial polarized phase, a symmetry-broken columnar stripe phase is present in the ground-state phase diagram. However, the limit of a small magnetic field has not been studied in great detail. The latter findings have motivated the current paper where we investigate the ground-state phase diagram of the LRTFIM in a comprehensive fashion. To this end, we first concentrate on the classical long-range Ising model (LRIM) for a generic circumference of even length. We find that cylinders with circumference $4$, $8$, $12$, et cetera display two distinct striped phases as a function of the long-range interaction, while cylinders with circumference $6$, $10$, $14$, and so on always realize the same stripe structure for all long-range Ising interactions, which is, however, distinct from the one found numerically in Ref. . Next we focus on the cylinder with minimal circumference of both families, namely of length $4$ and $6$, and study the ground-state phase diagram of the full LRTFIM. The paper is structured as follows: We start by introducing the model and discussing several limiting cases as well as different representations in Sect. \[sec:model\]. In Sect. \[sec:methods\] an overview of the implemented methods used to derive the ground-state phase diagrams of the LRTFIM on the YC($4$) and YC($6$) is given. Sect. \[sec:pure\_long\_range\_ising\_model\] contains results for the LRIM without a magnetic field. The ground-state phase diagram for the full LRTFIM is presented and discussed for both lattices in Sect. \[sec:phase\_diagrams\]. Finally, we conclude our work in Sect. \[sec:conclusion\] which includes a discussion on the presence of critical intermediate phases in the ground-state phase diagram of the LRTFIM. Model {#sec:model} ===== The Hamiltonian of the LRTFIM is given by $$\begin{aligned} \mathcal{H} = \frac{J}{2}\sum_{{\bf i} \neq {\bf j}} \frac{1}{|{\bf i}-{\bf j}|^\alpha}\sigma_{{\bf i}}^z\;\sigma_{{\bf j}}^z -h\sum_{{\bf j}}\sigma_{{\bf j}}^x ~ , \label{eq:H_tfim_orig}\end{aligned}$$ with Pauli matrices $\sigma_{{\bf i}}^{x/z}$ describing spins-1/2 located on lattice sites ${\bf i}$, the transverse field $h>0$, and the antiferromagnetic coupling constant $J>0$. Tuning the positive parameter $\alpha$ changes the long-range behavior of the Ising interaction from an all-to-all coupling $\alpha=0$ up to the nearest-neighbor case $\alpha=\infty$. The triangular cylinders consist of rings with circumference $n$ which are coupled in the direction of infinite extension to form a triangular lattice as illustrated in Fig. \[fig:cylinder\_2d\_and\_3d\_illustration\]. According to the number of spins per ring $n$, these lattices are labeled YC($n$), while we focus on $n\in\{4,6\}$ for most parts in this paper. In Fourier space the momentum orthogonal to the infinite cylinder extension becomes discrete while it is continuous in the other direction due to the infinite extension. ![Illustration of the YC6 lattice. Upper panel: YC$(6)$ embedded in 3D to illustrate the periodic boundary conditions. Lower panel: Illustration as a cutout of the 2D triangular lattice with periodic boundary conditions denoted with dashed lines. The vectors ${\boldsymbol{\mathbf{e}}}_1$ and ${\boldsymbol{\mathbf{e}}}_2$ are the unit vectors which span the whole triangular cylinder.[]{data-label="fig:cylinder_2d_and_3d_illustration"}](fig1.pdf) It is convenient to rewrite the summations in Eq.  with respect to the characteristics of the YC$(n)$ cylinders. A lattice site can either be addressed by unit vectors ${\boldsymbol{\mathbf{e}}}_1$ and ${\boldsymbol{\mathbf{e}}}_2$ (see Fig. \[fig:cylinder\_2d\_and\_3d\_illustration\]) like for the 2D triangular lattice with periodic boundary conditions, or by the ring $R\in\mathbb{Z}$ (highlighted in red in Fig. \[fig:cylinder\_2d\_and\_3d\_illustration\]) the site is located on and the position within the ring $\nu\in\{ 0,1,2,...,n-1 \}$. The Hamiltonian Eq.  can then be expressed as $$\mathcal{H}=\frac{J}{2}\sum_{R,R'}\sum_{\nu,\nu'=0}^{n-1}d_{R,\nu}^{R',\nu'}(\alpha) \,\sigma^z_{R,\nu}\sigma^z_{R',\nu'}-h\sum_{R}\sum_{\nu=0}^{n-1}\sigma^x_{R,\nu}$$ with $d_{R,\nu}^{R',\nu'}(\alpha)$ representing the $\alpha$-dependent long-range Ising interaction between the sites $(R,\nu)$ and $(R',\nu')$. Obviously one has $d_{R,\nu}^{R,\nu}(\alpha)=0$, since no self-interactions are present. Note that we measure distances in the planar geometry as illustrated in the lower panel of Fig. \[fig:cylinder\_2d\_and\_3d\_illustration\]. The LRTFIM has several interesting limiting cases. Studying them can lead to a better intuition for the involved physics and they are later used as starting points for perturbative calculations. For $h=\infty$ the system is in an $x$-polarized phase where all spins align in direction of the transverse field independently of the lattice. This is a commonly-used starting point for high-field perturbative approaches due to the clear reference state including the long-range case $\alpha<\infty$ [@Fey2016; @Fey2019]. For $\alpha=\infty$ and $n=\infty$ the system corresponds to the nearest-neighbor transverse-field Ising model with an infinitely-degenerate ground-state space for $h=0$ on the 2D triangular lattice. Here every state is a ground state of the system that obeys the rule that on every triangle there are one ferro- and two antiferromagnetic bonds. For the 2D triangular lattice it is known that this highly-degenerate nearest-neighbor ground-state space is not stable against an infinitesimal transverse field $h$ which leads to an order-by-disorder scenario inducing a clock-ordered phase. This is most easily seen by performing first-order degenerate perturbation theory in $h/J$ yielding an effective quantum dimer model on the dual honeycomb lattice [@Moessner2001] $$\label{eq:qdm} \mathcal{H}_{\text{QDM}} = -h\sum_{{\boldsymbol{\mathbf{\nu}}}}\left(\ket{\dimerlefttriangle}_{{\boldsymbol{\mathbf{\nu}}}}\bra{\dimerrighttriangle}_{{\boldsymbol{\mathbf{\nu}}}} +{\mbox{H.\,c.}\xspace}\right)$$ up to an irrelevant constant. The sum over ${\boldsymbol{\mathbf{\nu}}}$ runs over all hexagonal plaquettes of the dual honeycomb lattice as illustrated in Figs. \[fig:dimer\_covering\_YC4\] and \[fig:dimer\_covering\_YC6\] for the YC($4$) and YC($6$) lattice. In this representation bonds with ferromagnetically-oriented neighbor spins are interpreted as a dimer in contrast to bonds with antiferromagnetically-oriented spins. In general, when acting on a spin with the field term, all dimers of a hexagonal plaquette are flipped to non-dimers and vice versa. If the total number of dimers on a plaquette remains unchanged, a different ground state is produced. Therefore, the effective describes quantum fluctuations in the ground-state space between the two configurations on plaquettes with three ferro- and antiferromagnetic bonds. Plaquettes having such a configuration are called *flippable*. Obviously, states with a maximum number of flippable plaquettes are selected energetically. For the 2D triangular lattice the resulting ground state is the above mentioned clock order which breaks the translational symmetry of the lattice. The same mechanism takes place on the cylindric triangular lattices. For the nearest-neighbor model we find a quasi momentum of the clock-ordered state of $(5\pi/4,\pi/2)$ \[$(2\pi/3,-2\pi/3)$\] for the YC($4$) \[YC($6$)\] lattice (see Figs. \[fig:dimer\_covering\_YC4\] and \[fig:dimer\_covering\_YC6\]). Note that for the YC($6$) cylinder this corresponds to the same order as in 2D while the clock order for YC($4$) is distinct, since the unit cell of the 2D clock order does not fit on this cylinder. For the NNTFIM on the 2D triangular lattice, the quantum phase transition between the clock order and the high-field $x$-polarized phase is known to be second order in the 3D-XY universality class [@Moessner2001; @Moessner2003; @Powalski2013]. For finite $\alpha<\infty$, the nature of this quantum phase transition is unchanged for all $\alpha\gtrapprox 2.5$ [@Fey2019; @Humeniuk2016]. Saadatmand investigated the LRTFIM on the YC$(6)$ cylinder for $\alpha\in(1,5)$ as well as the NNTFIM using iDMRG [@Saadatmand2018]. They find for $\alpha>2.40(5)$ the same quantum phases as for the 2D triangular lattice, with a transition between the clock order and the $x$-polarized phase. The critical point for the NNTFIM is located at $h_c=1.5(1)\,J$ [@Saadatmand2018]. For $\alpha<2.40(5)$, they observe a direct phase transition from the $x$-polarized phase into a different ordered phase, which we will call zigzag-stripe phase (see Fig. \[fig:stripes\] for an illustration). Interestingly, our investigation of the pure LRIM for $h=0$ confirms the appearance of stripe-ordered phases (for any finite $\alpha$), although we find a different stripe order to be realized which we attribute to the chosen unit cell in Ref. . Finally, the LRTFIM Eq.  reduces to a fully connected graph with equal all-to-all coupling in the limit $\alpha=0$. It is then convenient to rewrite the Hamiltonian by introducing the total spin $\sigma_{\text{tot}}^{x/z}=\sum_{{\bf i}}\sigma_{{\bf i}}^{x/z}$ and the total number of spins $N$ with $N\rightarrow \infty$ to obtain $$\begin{aligned} \mathcal{H} = -h\sigma^x_{\text{tot}}+\frac{J}{2}(\sigma^z_{\text{tot}})^2-\frac{J}{2}N\quad , \label{eq:H_tfim_meanfield}\end{aligned}$$ which immediately shows that every state with vanishing magnetization is a ground state for $h=0$ so that a large degeneracy results. For all finite transverse fields $h>0$ the system breaks this degeneracy and is directly located in the $x$-polarized phase [@Humeniuk2016]. One then regains the full LRTFIM by adding $$\frac{J}{2}\sum_{{\bf i} \neq {\bf j}} \left(\frac{1}{|{\bf i}-{\bf j}|^\alpha}-1\right)\,\sigma_{{\bf i}}^z\;\sigma_{{\bf j}}^z\ . \label{eq:smallalphaunperturbed}$$ to the all-to-all limit Eq. . It is therefore also possible to consider the infinitely degenerate limit $h=0$ and $\alpha=0$ as perturbative starting point. For the long-range Ising interactions this demands a Taylor expansion of Eq.  giving in leading order in $\alpha$ $$-\alpha\frac{J}{2}\sum_{{\bf i} \neq {\bf j}}\log(|{\bf i}-{\bf j}|)\,\sigma_{{\bf i}}^z\;\sigma_{{\bf j}}^z \ . \label{eq:smallalphalogperturb}$$ This perturbation describes an extensive ferromagnetic LRIM with logarithmically increasing Ising interaction strength. Our goal is to determine the full ground-state phase diagram of the LRTFIM on the YC$(4)$ and YC$(6)$ cylinder. To this end, we apply several perturbative expansions in the $x$-polarized phase and in the stripe- and clock-ordered phases. The technical aspects are discussed next. ![Illustration of the maximally-flippable state of YC$(4)$ in the quantum-dimer model. Flippable plaquettes are shown as red and green hexagons. The original triangular cylinder is shown in the background together with a spin configuration resulting in the displayed dimer configuration. The periodic boundary is reflected in lighter gray.[]{data-label="fig:dimer_covering_YC4"}](fig2.pdf){width=".9\columnwidth"} ![Illustration of the maximally-flippable state of YC$(6)$ in the quantum-dimer model. Flippable plaquettes are shown as red and green hexagons. The original triangular cylinder is shown in the background together with a spin configuration resulting in the displayed dimer configuration. The periodic boundary is reflected in lighter gray.[]{data-label="fig:dimer_covering_YC6"}](fig3.pdf){width="\columnwidth"} Methods {#sec:methods} ======= To map out the ground-state phase diagram of the LRTFIM on the triangular cylinders we set up several methods for the calculation of the ground-state energy as well as the elementary excitation energy in the different quantum phases just introduced in the last section. First, we describe the perturbative expansion of both quantities about the high-field limit in the $x$-polarized phase and the Padé extrapolation of the corresponding series. Then, we explain the perturbative low-field expansion about the stripe-ordered ground states of the LRIM in $h/J$. Finally, the perturbative evaluation of the clock-ordered ground state in the presence of the transverse field and the long-range Ising interaction is explained. High-field expansion {#ssec:high_field_expansion} -------------------- The high-field expansion for the LRTFIM is most efficiently done by combining a white-graph expansion [@Coester2015] for a perturbative continuous unitary transformation (PCUT) [@Knetter2000; @Knetter2003] with Markov-chain Monte Carlo (MCMC) as shown for two-dimensional LRTFIMs recently in Ref. . For details of this approach we therefore refer to Refs.  where this approach was discussed before. Here, we only concentrate on the essential aspects. Using the Matsubara-Matsuda transformation and [@Matsubara1956], the LRTFIM Eq.  can be written, up to the constant $-N/2$, in a quasi-particle (QP) language as $$\begin{aligned} \frac{\mathcal{H}}{2h} &= \sum_{R}\sum_{\nu=0}^{n-1} \hat{n}_{R,\nu} - \frac{\lambda}{2}\sum_{R,R'}\sum_{\nu,\nu'=0}^{n-1} d_{R,\nu}^{R',\nu'}(\alpha) \notag\\ &\quad\quad\quad\left( \hat b^\dagger_{R,\nu} \hat b^\dagger_{R',\nu'} + \hat b^\dagger_{R,\nu} \hat b^{\phantom{\dagger}}_{R',\nu'} + {\rm H.c.}\right)~, \label{eq:H_tfim_orig_boson}\end{aligned}$$ where $\hat{n}_{R,\nu}=\hat{b}^\dagger_{R,\nu}\hat{b}^{\phantom{\dagger}}_{R,\nu}$ counts the number of hardcore bosons on site $(R,\nu)$ and $\lambda\equiv J/(2h)$ is the expansion parameter. The PCUT using a white-graph expansion scheme [@Coester2015] maps this Hamiltonian, order by order in $\lambda$, to an effective block-diagonal Hamiltonian $\mathcal{H}_{\text{eff}}$, which preserves the total number of QP’s, i.e. , $[\mathcal{H}_{\text{eff}},\mathcal{Q}]=0$ with $\mathcal{Q}\equiv \sum_{R,\nu}\hat{n}_{R,\nu}$. Here we focus on the zero- and one-QP block of $\mathcal{H}_{\text{eff}}$ so that we have access to the ground-state energy per site $e_0^{\mathrm{pol}}\equiv E_0^{\mathrm{pol}}/N$ as well as to the one-QP excitation energies. The effective Hamiltonian in the one-QP sector reads $$\mathcal{H}_{\rm eff}^{\rm 1QP} = E_0^{\mathrm{pol}} + \sum_{R,R'}\sum_{\nu,\nu'=0}^{n-1} a_{{\boldsymbol{\mathbf{\delta}}}} \left( \hat{b}^\dagger_{R,\nu} \hat b^{\phantom{\dagger}}_{R',\nu'}+{\rm H.c.}\right) ~. \label{eq:H_eff_1QP}$$ with the ground-state energy $E_0^{\mathrm{pol}}$ and the one-QP hopping amplitudes $a_{{\boldsymbol{\mathbf{\delta}}}}\equiv a_{R,\nu}^{R',\nu'}$, where ${\boldsymbol{\mathbf{\delta}}}$ denotes the vector between lattice sites $(R,\nu)$ and $(R',\nu')$. Exploiting the translational invariance along the infinite cylinder direction, a Fourier transformation with respect to the unit vector ${\boldsymbol{\mathbf{e}}}_1$ depicted in Fig. \[fig:cylinder\_2d\_and\_3d\_illustration\] transforms Eq.  to $$\mathcal{H}_{\rm eff}^{\rm 1QP} = E^{\rm pol}_0+\sum_{k_1}\sum_{\nu,\nu'=0}^{n-1} \omega_{\nu,\nu'}(k_1)\,\hat{b}^\dagger_{k_1,\nu} \hat b^{\phantom{\dagger}}_{k_1,\nu'}~,$$ where $k_1\in\mathds{R}$ is the one-dimensional quasi-momentum in the cylinder direction. The matrix elements $\omega_{\nu,\nu'}(k_1)$ contain information of the QP hopping from site $\nu$ to site $\nu'$ of the $n$-site ring. The one-QP Hamiltonian $\mathcal{H}_{\rm eff}^{\rm 1QP}$ can be fully diagonalized with the help of the discrete translational invariance in the finite periodic cylinder direction in ${\boldsymbol{\mathbf{e}}}_2$-direction so that $$\mathcal{H}_{\rm eff}^{\rm 1QP} = E^{\rm pol}_0+\sum_{{\boldsymbol{\mathbf{k}}}}\omega ({\boldsymbol{\mathbf{k}}})\,\hat{b}^\dagger_{{\boldsymbol{\mathbf{k}}}} \hat b^{\phantom{\dagger}}_{{\boldsymbol{\mathbf{k}}}}$$ defining the two-dimensional quasi-momentum with $k_2\in\{0,\ldots,2\pi/n\}$. The one-QP hopping amplitudes $a_{{\boldsymbol{\mathbf{\delta}}}}$ are calculated on minimal graphs in the white-graph expansion and analytically exact for any lattice. To get the perturbative series in the bulk limit the amplitudes need to be embedded into the lattice. The embedding consists of a summation of the contributions of the hopping elements from all possible configurations of each graph on the lattice in a given order. As a consequence, each matrix element $\omega_{\nu,\nu'}(k_1)$ as well as each one-QP energy $\omega ({\boldsymbol{\mathbf{k}}})$ is given as a high-dimensional nested sum. These summations are most efficiently evaluated numerically by a Markov-chain Monte Carlo (MCMC) method as we demonstrated recently [@Fey2019]. We indeed implemented two schemes. First, we sampled each element of $\omega_{\nu,\nu'}(k_1)$ separately and then calculate the one-particle dispersion $\omega ({\boldsymbol{\mathbf{k}}})$ in a subsequent step. To this end, we extended the code from Ref.  to arbitrary unit cells. Second, we directly sampled with MCMC the one-particle dispersion $\omega ({\boldsymbol{\mathbf{k}}})$ for a fixed ${\boldsymbol{\mathbf{k}}}$. Numerically, it turned out that the second approach yields smaller error bars and we therefore concentrate on this approach in the following. In both cases, for each perturbative order $r$, a separate MCMC calculation for all graphs with $\mu\in[2,r+1]$ vertices is done. For each MCMC calculation up to 80 runs with different random number generator seeds are computed to obtain an error estimation from the standard deviation of the mean sum value. Using this approach, we calculated PCUT results for the ground-state energy per site $e_0^{\rm pol}$ and the one-QP dispersion $\omega ({\boldsymbol{\mathbf{k}}})$ up to order 10 in the thermodynamic limit with high accuracy. Even for the demanding limit of small values of $\alpha$ we get standard deviations in the highest-order coefficientes of a low single-digit percentage magnitude. In the one-QP sector, we are mostly interested in the one-QP gap $\Delta\equiv {\rm min}_{{\boldsymbol{\mathbf{k}}}}\,\omega ({\boldsymbol{\mathbf{k}}})$. In the numerical evaluation we therefore focus on the momenta $(2\pi/3,-2\pi/3)$ \[$(5\pi/4,\pi/2)$\] for the YC($6$) \[YC($4$)\] lattice, where the gap between the $x$-polarized and the clock-ordered phase closes at the phase transition in the nearest-neighbor limit. Further, we consider the excitation energies at the momenta of the relevant stripe-ordered phases which are introduced in Subsect. \[ssec:low\_field\_expansion\]. As the value of the parameter $\alpha$ needs to be fixed for the numerical MCMC evaluation, we compute perturbative series for a set of fixed values of $\alpha$ in the range $[1.5,10]$. In the end, the resulting series for the one-QP gap are extrapolated using Pad[é]{} extrapolations [@Guttmann1989] in order to enlarge the convergence radius. The extrapolants are then used to obtain the critical values $\lambda_{\text{c}}$ of the phase transition for fixed $\alpha$ where the one-QP gap closes. Padé extrapolations {#ssec:pade} ------------------- Padé and Dlog-Padé approximations are standard methods in the field of series expansions as they allow for an evaluation of the series beyond their original radius of convergence [@Guttmann1989]. Therefore, they are well-suited to study our results from the high-field expansion. While Padé expansions are generally used for extrapolating ground-state energies [@Roechner2016], the criticality and location in parameter space of a second-order phase transition is usually studied using Dlog-Padé extrapolations of the one-QP energy gap, because it is capable of incorporating an algebraic gap-closing. As discussed below, we do not expect a second-order phase transition and consequently estimate the phase-transition point via Padé extrapolations. The perturbation series $$F(\lambda)=\sum_{m\geq 0}^{r} c_m \lambda^m=c_0+c_1\lambda+c_2\lambda^2+\dots c_{r}\lambda^r,$$ with $\lambda\in \mathbb{R}$ and $c_m \in \mathds{R}$ is interpreted as a Taylor expansion of a rational function $$G^{L/M}(\lambda) = \frac{p_0 + p_1 \lambda + p_2 \lambda^2 +\cdots+p_L \lambda^L}{1 + q_1 \lambda + q_2 \lambda^2 +\cdots+q_M \lambda^M}~.$$ Comparing the series expansion of $G^{L/M}(\lambda)$ with the original series $F(\lambda)$ one obtains a linear system of equations that can be solved for a given parameter set $(L,M)$. The parameters $L,M\in\mathds{N}$ fulfill the condition $L+M=r$ for an extrapolation in order $r$. Typically the diagonal extrapolations where $|L-M|$ is small give the best results. Extrapolations with unphysical singularities need to be sorted out, as well as defective Padé extrapolants that have a singularity at the same point in the numerator and denominator which effectively cancel out. The phase transition point $\lambda_\mathrm{c}$ studied in the present paper is found by calculating the zeros of the Padé extrapolation of the 1QP gap. To locate the phase transition, we calculate several values for the critical parameter $\lambda_\mathrm{c}$ for different selected combinations $(L,M)$ out of for non-defective Padé extrapolants. In the phase diagrams presented in Sect. \[sec:phase\_diagrams\] we show the mean and standard deviation of the extrapolations. If the standard deviation is zero only a single extrapolation could be selected. Low-field expansion {#ssec:low_field_expansion} ------------------- The opposite limit of small transverse fields $h/J\ll 1$ can be also be treated by high-order series expansions for $\alpha<\infty$, since the extensive ground-state degeneracy for the nearest-neighbor case $\alpha=\infty$ is lifted by the long-range Ising interaction. We therefore have determined the ground state of the LRIM as a function of $\alpha$ by considering large but finite triangular cylinders YC($n$), with periodic boundary conditions, for general even $n$. These findings are outlined and discussed in Sect. \[sec:pure\_long\_range\_ising\_model\]. As a result of these calculations we find that the pure LRIM realizes different types of ordered stripe structures depending on $n$ and $\alpha$ and we can determine the associated ground-state energy per site $e_0^{\rm stripe}$ in units of $J$ by considering finite cylinders of the order of $N=10^5$ spins. These ordered stripe structures represent gapped phases which allows us to set up a high-order (non-degenerate) series expansion about the zero-field ground state. To this end, we apply Takahashi’s perturbation theory [@Takahashi1997] in real space and we obtain the ground-state energy per site $e_0^{\rm stripe}$ for various stripe structures up to order six in the parameter $h/J$. To do this, we calculate the even-order contributions directly by evaluating the expectation value of the perturbation-operator sequences with respect to the considered classical stripe state. We stress that only even orders are present in the low-field expansion of $e_0^{\rm stripe}$, while odd orders vanish exactly. This originates from a double-touch property, because each excitation created locally by the perturbing magnetic field in virtual states has to be destroyed by acting again with the magnetic field on the same site. As a consequence, every site has to be touched an even number of times by the magnetic field to get a non-vanishing result. The perturbation operator sequences in $r$-th order read as follows [@Takahashi1997] $$\hat{P}\mathcal{V}\hat{S}^{k_1}\mathcal{V}\hat{S}^{k_2}\mathcal{V}...\mathcal{V}\hat{S}^{k_{r-1}}\mathcal{V}\hat{P}\, ,$$ where $\mathcal{V}\equiv\mathcal{H}_{\rm h}$ is the perturbation, $\hat{P}$ the projection operator on the ground-state space, the resolvent $\hat{S}$ is given as $$\begin{aligned} \hat{S}=\frac{(1-\hat{P})}{E_0^{\text{stripe}}-\mathcal{H}_{\rm Ising}} &\quad \mathrm{with}\quad \hat{S}^k= \begin{cases} \hat{P} & k=0\\ \hat{S}^k & k>0 \end{cases} \ , \end{aligned}$$ and the constraint $\sum_{i=1}^{n-1}k_i=r-1$. We evaluate all contributions up to order six in $h/J$ by calculating the expectation values $$\begin{aligned} \bra{{\rm stripe}} \hat{P}\mathcal{V}\hat{S}^{k_1}\mathcal{V}\hat{S}^{k_2}\mathcal{V}\ldots\mathcal{V}\hat{S}^{k_{r-1}}\hat{P} \ket{{\rm stripe}}\end{aligned}$$ for the classical stripe state $\ket{{\rm stripe}}$ on finite clusters with $1000n$ spins for the YC$(n)$ cylinder and by treating the perturbing magnetic field in real space. We have further reduced the summation effort by identifying the non-vanishing processes in advance and exploiting the translational invariance for the first excitation that is created by the perturbation $\mathcal{V}$. Finally, for a fixed $\alpha$ we obtain the following order-six series of the ground-state energy per site for $J=1$ $$\begin{aligned} e_0^{\text{stripe}}(\alpha,h) =& e_0^{\text{stripe}}(\alpha,h=0)+\rho_2^{\text{stripe}}(\alpha)\;h^2\\ &+\rho_4^{\text{stripe}}(\alpha)\; h^4+\rho_6^{\text{stripe}}(\alpha)\;h^6 \quad .\end{aligned}$$ As a representative example, the bare series in order two, four, and six of the ground-state energy per site of orthogonal stripes for $\alpha=6$ are displayed in Fig. \[fig:convergence\_low\_field\]. In general, we observe that the first-order phase transition out of the stripe-ordered phase is well located in the regime where the bare series is still converged. Consequently, we do not extrapolate the series of the low-field expansions. ![Bare series of the ground-state energy per site $e_0^{\rm stripe}$ for orthogonal stripes as a function of $h/J$ for $\alpha=6$ for the YC$(4)$ cylinder. The bare series is converged up to $h/J\approx 0.6$. The behavior of other stripe configurations as well as other $\alpha$-values is similar. The calculated phase transition into the clock order, visualized by the black vertical line, takes place at approximately $0.17\,h/J$.[]{data-label="fig:convergence_low_field"}](fig4.pdf) Clock-ordered states {#ssec:clock_orders} -------------------- As already outlined above, apart from the high-field $x$-polarized phase and the low-field stripe phases for $\alpha<\infty$, one expects also clock-ordered phases in the ground-state phase diagram. These clock-ordered states are stabilized by an order-by-disorder phenomenon, i.e. this order is selected by the quantum fluctuations induced by an infinitely small transverse field on the extensive ground-state manifold for $\alpha=\infty$ and $h=0$. We are therefore determining approximately the ground-state energy per site $e_0^{\rm clock}$ of these clock-ordered states as a function of the long-range interaction $\alpha$ and the transverse field $h$. To this end, we split the Hamiltonian $$\frac{\mathcal{H}}{J}=\mathcal{H}_{\rm Ising}^{\alpha=\infty}+\frac{h}{J}\mathcal{H}_{\rm field}+\xi\Delta\mathcal{H}_{\rm Ising}\quad ,$$ where the last term is defined as the difference between the full long-range Ising interaction and the nearest-neighbor contribution $$\Delta\mathcal{H}_{\rm Ising} \equiv \mathcal{H}_{\rm Ising} - \mathcal{H}_{\rm Ising}^{\alpha=\infty}$$ and one recovers the original LRTFIM for $\xi=1$. In the following, we consider the perturbative limit , i.e. we expand about the nearest-neighbor Ising model using degenerate perturbation theory so that an effective description in terms of a quantum dimer model is appropriate. Indeed, due to the frustration, the ground-state subspace consists of infinitely many degenerate states which have in each elementary triangle exactly one ferromagnetic bond. Interpreting the ferromagnetic bond as the presence of a dimer on the dual honeycomb lattice, each ground state for $\alpha=\infty$ and $h=0$ can therefore be represented by a dimer covering $\ket{c}$ and the (perturbative) action of the two perturbations $\mathcal{H}_{\rm field}$ and $\Delta\mathcal{H}_{\rm Ising}$ can be captured by an effective quantum dimer model of the form $$\begin{aligned} \label{eq:qdm2} \mathcal{H}_{\text{QDM}} = E_0& +\sum_{c}E_{c}(\xi,h)\ket{c}\bra{c}\nonumber\\ & -h\sum_{{\boldsymbol{\mathbf{\nu}}}}\left(\ket{\dimerlefttriangle}_{{\boldsymbol{\mathbf{\nu}}}}\bra{\dimerrighttriangle}_{{\boldsymbol{\mathbf{\nu}}}} +{\mbox{H.\,c.}\xspace}\right)\,,\end{aligned}$$ where the sum runs over all dimer coverings $\ket{c}$ so that $E_{c}(\xi,h)$ is the covering-dependent diagonal energy. We have determined this effective quantum dimer model up to order three in the parameters $\xi$ and $h$. The diagonal elements $E_{c}(\xi,h)$ have several contributions. In first order in $\xi$, it depends on the long-range part $\Delta\mathcal{H}_{\rm Ising}$ of the Ising interaction. Note however that this first-order contribution actually represents the exact pure energy correction, since $[\mathcal{H}_{\rm Ising}^{\alpha=\infty},\Delta\mathcal{H}_{\rm Ising}]=0$. Further, $E_{c}(\xi,h)$ depends on $h^2$ and $h^2\xi$ in second- and third-order degenerate perturbation theory. The only off-diagonal term comes in first order which mediates between two different dimer coverings from the transverse field $h$ as already discussed above. Since we consider the hierarchy , it is the (dressed) maximally-flippable plaquette state (see Figs. \[fig:dimer\_covering\_YC4\] and \[fig:dimer\_covering\_YC6\]), which is selected due to an infinitesimal transverse field and which gives rise to the clock-ordered phase on the YC($n$) cylinders. Calculating the ground-state energy $e_0^{\rm clock}$ as a function of $\alpha$ and $h$ is still a highly non-trivial task due to the fact that the quantum dimer model Eq.  still lives in an infinite-dimensional Hilbert space. We therefore have used the following scheme to calculate $e_0^{\rm clock}$ approximatively. First, we restrict the calculations to finite cylinders with the number of $8$ ($6$) rings for $n=4$ ($n=6$), which obviously yields a finite-dimensional problem. In order to focus on the most relevant dimer coverings on these finite cylinders for the description of the clock-ordered state, we generate the following reduced basis. We start from the maximally-flippable state (see Figs. \[fig:dimer\_covering\_YC4\] and \[fig:dimer\_covering\_YC6\]) and then we act subsequently with the magnetic field on flippable plaquettes. This generates new states having less flippable plaquettes on which we can again act with the magnetic field to generate further dimer coverings and so on. Consequently, we iteratively construct a finite basis of dimer coverings where all states are connected to the maximally-flippable state via the magnetic field. The Hamiltonian can then be represented as a finite matrix in this basis and the lowest eigenvalue corresponds to an approximation of $e_0^{\rm clock}$. For the YC($4$) \[YC($6$)\] cylinder, the maximally-flippable state has an 8-site (3-site) unit cell. Our results are therefore computed on four and twelve unit cells for the respective cylinders. The finite cluster size has two competing effects on the calculated ground-state energy $e_0^{\rm clock}$: First, especially relevant for large $\alpha$, the limitation to a finite cylinder results in an underestimation of the quantum fluctuations introduced by the transverse field. Consequently, we expect that in the bulk-limit $e_0^{\rm clock}$ should be pushed down stronger with the magnetic field compared to our finite-size calculation. Second, the long-range Ising interaction $\Delta\mathcal{H}_{\rm Ising}$, which becomes more and more important for smaller $\alpha$, should lead to an increased energy in the bulk limit. Considering both contributions, we expect the method to underestimate the energy for small $\alpha$ and to overestimate it slightly for large $\alpha$. Pure Long-range Ising Model {#sec:pure_long_range_ising_model} =========================== In the following, we consider the pure LRIM with vanishing transverse field $h=0$. Here it is a priori not clear what classical state is the ground state as a function of $\alpha$. In the following we want to clarify the nature of the ground state for the YC($n$) cylinders and its ground-state energy per site, which is then used to set up the low-field expansion. ![Illustration of orthogonal-, zigzag-, and plain-stripe states in real space on a YC($4$) lattice. Red and blue circles denote spins pointing in opposite directions. The infinite extension of the cylinder is in horizontal direction.[]{data-label="fig:stripes"}](fig5.pdf) By studying the energy of the nearest-neighbor ground states on finite cylinders with $N\approx40$ spins for $\alpha<\infty$, we observe that the relevant states for the LRIM consist only of *non-flippable* plaquettes in the quantum dimer language on the dual lattice. These states are symmetry-broken, therefore gapped and stable against quantum fluctuations introduced by small transverse magnetic fields. We name the three relevant occurring order patterns *orthogonal* \[${\boldsymbol{\mathbf{k}}}=(\pi,0)^T$\], *plain* \[${\boldsymbol{\mathbf{k}}}=(\pi,\pi)^T$\], and *stripes*, which are illustrated in Fig. \[fig:stripes\]. ![Energy per site $e_0^{\rm stripe}$ for the considered stripe patterns evaluated on the YC($4$) lattice for $N=4\cdot1000$ (upper panel) and on the YC($6$) lattice for $N=6\cdot1000$ (lower panel) with periodic boundary conditions. The dotted vertical line in the upper panel indicates the first-order phase transition $\alpha_{\rm c}$ between zigzag ($\alpha<\alpha_{\rm c}$) and orthogonal ($\alpha>\alpha_{\rm c}$) stripes for the YC($4$) cylinder. For the YC($6$) orthogonal stripes are realized for all $\alpha$.[]{data-label="fig:zero_field_46"}](fig6.pdf) In the orthogonal-stripe configuration spins with the same orientation order orthogonally to the direction of the cylinder, which leads to a two-fold degeneracy which results from the $\mathbb{Z}_2$ spin-flip symmetry of the state. In the plain-stripe state spins of the same orientation align in plain chains winding around the cylinder in direction of infinite extension. This pattern has a $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry, which results from a spin-flip symmetry and a decoupling of the state into two sublattices. In the 2D limit orthogonal and plain stripes are degenerate with a $\mathbb{Z}_2\times\mathbb{Z}_3$ symmetry due to spin flips and the threefold rotational symmetry of the lattice. The absence of this rotational symmetry for the YC($n$) cylinders leads to the energetic separation of orthogonal and plain stripes, where orthogonal stripes have lower energies for all decay exponents $\alpha$ on the YC($4$) and YC($6$) lattice. In fact, the preference of orthogonal stripes is seen for all studied YC$(n)$ cylinders with even $n$. The third identified ground-state pattern are the zigzag stripes where spins of the same orientation align in a zigzag shape in cylinder direction. This results in a four-fold degeneracy ($\mathbb{Z}_2\times\mathbb{Z}_2$) due to spin-flip symmetry and a decoupling into two sublattices. On the YC$(n)$ cylinders with $n=4s\ (s \in \mathbb{N})$ it is actually possible to rotate the zigzag stripes by $2\pi/3$ and to remain in the subspace of states with only non-flippable plaquettes. These rotated zigzag stripes are always energetically less beneficial than the zigzag stripes with an alignment in infinite direction and we will not consider them further. At small $\alpha$ zigzag stripes are energetically lower compared to the orthogonal stripes for the YC($4$), YC($8$), and YC($12$) cylinder (see Fig. \[fig:zero\_field\_46\]). Consequently, there must be a first-order phase transition between these two stripe phases for these cylinders and we can determine the associated critical $\alpha_{\text{c}}$ which are listed in Tab. \[tab:krit\_alpha\]. $n$ $4$ $8$ $12$ --------------------- --------- --------- --------- $\alpha_{\text{c}}$ 2.55(1) 1.41(1) 1.13(1) : Critical decay exponents $\alpha_c$ for a phase transition between orthogonal ($\alpha>\alpha_{\rm c}$) and zigzag ($\alpha<\alpha_{\rm c}$) stripes on the YC($n$) lattices with $n\in\{4,8,12\}$. The denoted lattices are the ones where the transition occurs in the considered range of $\alpha>1$ in the framework of our numerical real-space implementation.[]{data-label="tab:krit_alpha"} The favoring of these stripe patterns can be explained by looking at the dominant further-neighbor Ising interactions, which are contributing differently to the energy, depending on the type of stripe under consideration. For zigzag stripes the second-nearest neighbors are contributing less beneficial than for the orthogonal stripes, but the third-nearest neighbors are lowering the energy more than they do for the orthogonal stripes. Together with the periodicity of the YC($n$) cylinder this leads to a favoring of the zigzag stripes at low $\alpha$ for the lattices described above, since then the long-range Ising interactions come more and more into play. Such stripe patterns have also been found by Smerald  for a truncated long-range Ising interaction [@Smerald2016]. For a classical Hamiltonian with arbitrary tunable antiferromagnetic nearest and next-nearest neighbor interaction it is a well established result, that on the triangular 2D lattice $\mathbb{Z}_2\times\mathbb{Z}_3$ plain stripes realize the ground state of the system [@Metcalf1974; @Korshunov2005]. On any YC$(n)$ lattice, this degeneracy of plain and orthogonal stripes is lifted. Compared to the plain-stripe order, the $(n+1)$-nearest neighbors of the orthogonal stripes are missing two ferromagnetic interactions for each site along the cylinder ring due to the finite cylinder extension. Instead two additional antiferromagnetic interactions per site are present which leads in total to a lower energy of orthogonal stripes. A second consequence of the cylinder geometry is the relevance of a zigzag-striped order. Further, we note that these three stripe patterns are part of the $\alpha=0$ and $\alpha=\infty$ ground-state space. Using the $\alpha=0$ limit of the LRIM with the perturbation described in Eq.  confirms the above findings. To this end, we considered all $\alpha=0$ ground states on a cluster of $N \approx 40$ spins which leads to the same ground-state space of non-flippable plaquettes. Evaluating the energy for the three relevant stripe patterns on large clusters $N\approx10000$, we find the same stripe-ordered ground states as calculated directly from the full LRIM. We therefore find that the physical behavior of the LRIM is different for the two families of cylinders with $n=4s\ (s\in \mathbb{N})$ and $n=4s+2 \ (s\in \mathbb{N})$. For $n=4s+2$ orthogonal stripes are realized for all studied $\alpha$, while in the other case orthogonal stripes become unstable towards a zigzag-stripe order for small $\alpha$. This can be clearly seen in the energy evolution of the different stripe structures as a function of $\alpha$, which is shown for the smallest member YC($4$) and YC($6$) of both families in Fig. \[fig:zero\_field\_46\], where we have studied finite cylinders with $N($YC$(n))=n\cdot1000$ spins using periodic boundary conditions. In the following, we focus on these two cylinders and study the ground-state phase diagram of the full LRTFIM. Phase diagrams {#sec:phase_diagrams} ============== As described above, we performed series expansions to extract the ground-state energies of the $x$-polarized, stripe, and clock-ordered phases for the LRTFIM on the YC($4$) and YC($6$) cylinder. Quantum phase transitions between two of these phases can then be located by determining crossing points between these energies for fixed $\alpha$. Note that this approach is not sensitive enough to extract the nature of the phase transition, although strong first-order phase transitions are expected to be located with highest precision. Furthermore, we study the breakdown of the $x$-polarized phase by investigating the behavior of the one-particle excitation energies for specific momenta, which are associated with the translational symmetry of the clock- or stripe-ordered phases. A continuous gap-closing of a certain mode signals a continuous quantum phase transition between the $x$-polarized phase and the associated ordered phase. We start our discussion with the YC($6$) cylinder, where the clock order is identical to the one known for the LRTFIM on the 2D triangular lattice and which has already been studied by Saadatmand et al.[@Saadatmand2018]. Afterwards, we turn our attention to the YC($4$) cylinder on which this clock order is frustrated and the ground-state phase diagram is even richer. YC$(6)$ {#sub:YC6} ------- ![Ground-state phase diagram calculated for the YC($6$) cylinder using series expansions. The red line is determined by the closing of the one-QP gap at momentum with the standard deviation of the Pad[é]{} approximants up to tenth order. The blue line is determined by the energy intersection between the clock-order energy $e_0^{\rm clock}$ and the stripe energies in sixth order. The question mark indicates the region of the phase diagram where the used methods break down.[]{data-label="fig:phasediagramYC6"}](fig7.pdf) ![Exemplary depiction of the phase transition points for $\alpha=6$ between the x-polarized phase and the clock ordered phase for the YC$(6)$ cylinder. The figure displays the one-QP gap at momentum as a function of $\lambda=h/2J$ using the order nine and ten Padé extrapolants with no poles before the gap closing. Depicted Padé extrapolants are $(4,5)$, $(5,4)$, $(5,5)$, $(6,4)$ and $(4,6)$.[]{data-label="fig:a6pcut"}](fig8.pdf) ![Exemplary depiction of the phase-transition points for $\alpha=6$ between the orthogonal-stripe phase and the clock-ordered phase for the YC$(6)$ cylinder. The figure shows the ground-state energies of the stripe and clock-order phases as a function of $h/J$. Red crosses denote calculated points in the parameter space of $h$ and $\alpha=6$ for the clock-order energy, the blue line denotes the orthogonal-stripe energy, the yellow line denotes the plain-stripe energy, and the violet line represents the zigzag-stripe energy. The crossing between the red symbols and the blue line indicates the first-order phase transition.[]{data-label="fig:a6stripe"}](fig9.pdf) The obtained ground-state phase diagram for the on the YC$(6)$ cylinder is shown in Fig. \[fig:phasediagramYC6\]. It displays the $x$-polarized phase, the clock order, and the orthogonal-stripe phase. The quantum phase transition between the $x$-polarized phase and the clock order is located by investigating the one-particle excitation energies of the $x$-polarized phase using the high-field expansion. For a non-first-order phase transition one expects that the one-particle gap of the $x$-polarized phase closes at the quantum-critical point and has a momentum ${\boldsymbol{\mathbf{k}}}=(2\pi/3,-2\pi/3)^T$, which is associated with the clock order. Here we locate such a gap-closing quantum-critical point by applying Pad[é]{} extrapolations on the bare order 10 series (see Fig. \[fig:a6pcut\]) and we quantify the uncertainty of this extrapolation scheme by the standard deviation of different extrapolations shown as error bars in Fig. \[fig:phasediagramYC6\]. This gap-closing we can track up to decay exponents $\alpha=1.5$. The calculated phase-transition points for $\alpha\gtrsim 2.4$ are within error bars in good agreement with the numerical findings by Saadatmand [@Saadatmand2018]. Specifically, for the NNTFIM $\alpha\rightarrow\infty$, the pCUT high-field calculation yields a gap closing at a transverse field $h=1.54(7)\,J$, which has to be compared to $h=1.5(1)\,J$ determined numerically by investigating the order parameter for the clock order [@Saadatmand2018]. The second type of phase transition present in the phase diagram is between the clock order and the orthogonal-stripe phase (see Fig. \[fig:a6stripe\] and blue line in Fig. \[fig:phasediagramYC6\]). Since both phases break a different type of discrete translational lattice symmetry, this transition is first order. It can therefore be located by determining the level crossing $e_0^{\rm clock}=e_0^{\rm stripe}$ for a given value of $\alpha$. As the ground-state energy $e_0^{\rm clock}$ of the clock-ordered state is evaluated on a finite cluster of six rings and the long-range interactions are included perturbatively, one has to be aware that the blue line in Fig. \[fig:phasediagramYC6\] is certainly not quantitative for small values of $\alpha$. In fact, we expect the phase transition to occur at higher transverse fields $h/J$ for small $\alpha$, because a better treatment of the long-range interactions would result in an increased ground-state energy $e_0^{\rm clock}$ so that the orthogonal-stripe phase is enlarged with respect to the clock order. In contrast, we (slightly) overestimate $e_0^{\rm clock}$ for large $\alpha$ due to the finite cluster extension and the approximate treatment of the field-induced quantum fluctuations. It is therefore plausible that for $\alpha \lesssim 2.4$ no clock order is present anymore in the phase diagram as suggested by Saadatmand [@Saadatmand2018] and there is a direct phase transition between the $x$-polarized and the orthogonal-stripe phase. Another scenario is the presence of an intermediate (gapless) phase as we discuss in Sect. \[sec:conclusion\]. We stress again that orthogonal stripes are the true ground states of the zero-field LRIM and no zigzag stripes are realized in the LRTFIM as found by Ref. . If there is a direct continuous phase transition between the orthogonal-stripe and the $x$-polarized phase, one expects that the high-field gap in the $x$-polarized phase closes at the critical point and is located at the associated momentum of the orthogonal-stripe phase ${\boldsymbol{\mathbf{k}}}=(\pi,0)^T$ for $\alpha \lesssim 2.4$. To study the transition to the orthogonal-stripe phase, we therefore evaluated the one-QP energy in the high-field limit at momentum ${\boldsymbol{\mathbf{k}}}=(\pi,0)^T$ using Padé extrapolations. Interestingly, no closing of the gap could be observed so that we cannot confirm a direct phase transition between the $x$-polarized and the orthogonal-stripe phase using series expansion methods. So either the situation is similar to the case of the 2D triangular lattice, where the direct phase transition is known to be generically first order due to the $\mathbb{Z}_2\cross \mathbb{Z}_3$ symmetry of the stripe order, or the phase diagram contains an intermediate phase which we elaborate on further in the conclusion. YC$(4)$ {#sub:YC4} ------- ![Ground-state phase diagram calculated for the YC($4$) cylinder using series expansions. The red line is determined by the closing of the one-QP gap with the standard deviation of the Pad[é]{} approximants up to order ten. The blue line is determined by the energy intersection between the clock-order energy $e_0^{\rm clock}$ and the orthogonal ($\perp$) stripe energy in sixth order. The violet line denotes the intersection between the energy of the quantum dimer model in third oder and the zigzag stripe energy in sixth order. The almost vertical black line represents the first-order phase transition line between the zigzag and orthogonal stripes, which is determined by comparing the ground-state energies of both stripe phases. The question mark indicates the region of the phase diagram where the used methods break down.[]{data-label="fig:phasediagram_YC4"}](fig10.pdf) Let us turn to the ground-state phase diagram for the LRTFIM on the YC$(4)$ cylinder, which is shown in Fig. \[fig:phasediagram\_YC4\]. The phase boundaries were calculated analogously to the YC($6$) lattice, where the $x$-polarized- to clock-order transition was determined by Padé extrapolations of the perturbative expansion from the high-field limit and the other transition lines were derived from the crossing of the respective ground-state energies. Besides the $x$-polarized phase a clock-ordered phase with a different momentum ${\boldsymbol{\mathbf{k}}} = (5\pi/4,\pi/2)^T$ from the YC($6$) lattice arises. Additional to the orthogonal-stripe order with ${\boldsymbol{\mathbf{k}}}=(\pi,0)^T$ at finite large $\alpha$ already discussed for the YC($6$) lattice, we find zigzag stripes with for small fields. The first-order phase transition from the orthogonal to a zigzag order for small fields occurs at $\alpha \approx 2.55(1)$. Interestingly, the phase transition between these two stripe orders is almost independent of $h/J$ so that a nearly vertical phase transition line results (see black line in Fig. \[fig:phasediagram\_YC4\]). Further, for small $\alpha$, one might predict a direct phase transition between the $x$-polarized phase realized at high field strength and the zigzag stripes at low field strength in similarity to the YC($6$) cylinder. Padé extrapolations of the one-QP gap with ${\boldsymbol{\mathbf{k}}}=(\pi/2,\pi)^T$ again do not point towards a continuous phase transition signaled by a gap closing in this $\alpha$-regime. We therefore expect that the physical situation is similar to the YC($6$) cylinder as discussed below. In the calculated phase diagram the intersection between the clock-order energy and the different stripe orders is calculated for all $\alpha$ values for which the clock-order expansion is available. Even though the clock-order is not expected to be the ground state for small $\alpha$, the calculated line gives a reference point for the extension of the stripe-ordered phases. As for the YC$(6)$ cylinder, we have to stress that the calculated energy for the clock order is underestimated, because the long-range interactions is cutted due to the evaluation on finite clusters. This implies that the transition between stripes and clock order occurs at higher transverse fields for small decay exponents ($\alpha \lessapprox 3.5$). Conclusions {#sec:conclusion} =========== We have investigated the ground-state phase diagram of the LRTFIM on triangular-lattice cylinders using various approaches. The physical behavior of the classical LRIM is different for the two families of cylinders with $n=4s\ (s\in \mathbb{N})$ and $n=4s+2 \ (s\in \mathbb{N})$. For $n=4s+2$ orthogonal stripes are realized for all studied $\alpha$, while in the other case orthogonal stripes become unstable towards a zigzag-stripe order for small $\alpha$. For the YC($6$) cylinder, our results are therefore distinct from the zigzag stripes obtained numerically [@Saadatmand2018], which is most likely due to the chosen unit cell in the iDMRG approach [@McCulloch_Private]. The full quantum phase diagram of the LRTFIM on the YC($4$) and YC($6$) cylinder contains at least three different types of gapped quantum phases. An $x$-polarized paramagnetic high-field phase, stripe phases triggered by the long-range Ising interaction, as well as clock-ordered phases being stabilized via an order-by-disorder mechanism about the highly-degenerate classical spin liquid of the nearest-neighbor Ising model. The extension of these phases in the parameter space of the LRFTIM has been located approximately by a variety of different perturbative expansions. In contrast to the high- and low-field high-order expansion, the obtained ground-state energy of the clock-ordered phase is the least accurate due to the low order three of the effective Hamiltonian and due to the finite cluster size. The obtained ground-state phase diagrams are valid as long as one assumes that no other phase is present. However, this is not obvious, which leads to the following two points which deserve further investigations: First, we turn our attention to the nearest-neighbor TFIM on the YC($6$) cylinder, which realizes the same kind of clock order as the TFIM on the two-dimensional triangular lattice. For the 2D triangular lattice the quantum phase transition between the $x$-polarized and the clock-ordered phase is a continuous second-order transition which falls into the $(2+1)$D-XY universality class [@Moessner2001; @Moessner2003; @Powalski2013]. This follows from a mapping of the TFIM onto a classical XY-model in three dimensions [@Blankschtein1984]. By applying the same kind of quantum to classical mapping on our quasi one-dimensional cylinders, one would expect that the transition falls into the $(1+1)$D-XY universality class, which is known to be the archetype of an infinite-order Kosterlitz-Thouless phase transition [@Kosterlitz1973]. However, since the Kosterlitz-Thouless phase transition involves one phase with critical (algebraically-decaying) correlations and both, the $x$-polarized and the clock-ordered phase, are gapped, this implies the existence of a gapless intermediate phase in the ground-state phase diagram of the nearest-neighbor TFIM on the YC(6) cylinder so that there are two Kosterlitz-Thouless transitions out of this intermediate phase. For the corresponding classical phase transitions in the NNTFIM on the 2D triangular lattice as a function of temperature, such an intermediate phase as well as the associated Kosterlitz-Thouless transitions are well established theoretically [@Moessner2000; @Moessner2003; @Wang2017] and confirmed experimentally recently in the Ising-type triangular antiferromagnet TmMgGaO$_4$ [@Li2019a]. Clearly, this intermediate phase in the YC($6$) cylinder will also extend in a finite $\alpha$-window in the phase diagram of the LRFTIM and, by similarity, one would expect a similar phase also for the YC(4) cylinder. Let us note that the detection of a Kosterlitz-Thouless transition is not possible with our high-field expansion due to the non-analytic behavior of the gap close to such a phase transition. Further, we find it interesting that iDMRG calculations based on translational invariant states are in good agreement with our findings with respect to the phase transition line between the $x$-polarized and clock-ordered phase as a function of $\alpha$ [@Saadatmand2018]. It is likely that both approaches are not sensitive enough to pinpoint the intermediate phase on the quasi one-dimensional YC(6) cylinder, but rather yield a good estimation for the transition line of the corresponding two-dimensional system on the triangular lattice. In any case, the existence and nature of the intermediate phase has to be clarified in the future. Second, it is not clear how the ground-state phase diagram looks for smaller values of $\alpha$, when the clock order (and potentially the just-discussed intermediate phase, are not realized anymore. The numerical work of Ref.  on the YC(6) cylinder suggests a direct second-order phase transition between the $x$-polarized and a stripe-ordered phase. We stress again that our investigation yields clearly a different ordering pattern for the stripes, namely orthogonal stripes. Extrapolations of the one-particle high-field gap with the corresponding stripe-momentum give no evidence for a gap-closing (second-order) phase transition. So this phase transition might be either (weakly) first order as for the LRFTIM on the triangular lattice [@Korshunov2005; @Smerald2016; @Fey2019] or, again, an intermediate phase could be present between the $x$-polarized and the stripe phase which prevents a controlled extrapolation of the gap. An indication for the latter scenario might be the presence of an intermediate classical spin liquid as a function of temperature for a deformed classical Ising model with dipolar interactions on the triangular lattice [@Smerald2018]. Overall, the interplay of geometric frustration and long-range interactions in low-dimensional quantum magnets displays a variety of interesting quantum phenomena which certainly need further investigations in the future. Acknowledgments =============== We thank Ian McCulloch and Andrew Smerald for fruitful discussions. We gratefully acknowledge the compute resources and support provided by the HPC group of the Erlangen Regional Computing Center (RRZE).

--- abstract: 'We discuss the Hopf-Wess-Zumino term in the effective action of the $6d$ $(2, 0)$ theory of the type $A_{N-1}$ in a generic Coulomb branch. For such terms, the supergravity calculation could be trusted. We calculate the WZ term on supergravity side and show that it could compensate the anomaly deficit, as is required by the anomaly matching condition. In contrast with the SYM theory, in which each WZ term involves one root $e_{i}-e_{j}$, here, the typical WZ term involves two roots $e_{i}-e_{j}$ and $e_{k}-e_{j}$. Such kind of triple interaction may come from the integrating out of the massive states carrying three indices. A natural candidate is the recently proposed $1/4$ BPS objects in the Coulomb phase of the $6d$ $(2, 0)$ theories. The WZ term could be derived from the field theory by the integration out of massive degrees of freedom. Without the $6d$ $(2, 0)$ theory at hand, we take the supersymmetric equations for the 3-algebra valued (2, 0) tensor multiplet as the prototype to see how far we can go. The $H_{3}\wedge A_{3}$ part of the WZ term is obtained, while the $A_{3}\wedge F_{4}$ part, which is the term accounting for the anomaly matching, cannot be produced by the standard fermion loop integration.' --- =msbm10 at 12pt =msbm7 =msbm5 ===\#1[[\#1]{}]{} = cmss12 /\#1[[ /]{}]{} \#1[.3ex]{} =6.5in [**Hopf-Wess-Zumino term in the effective action of the 6d, (2, 0) field theory revisted\ **]{} [**Shan Hu $^{1,a}$, and Dimitri Nanopoulos $^{1,2,3,b}$**]{} \ [*College Station, TX 77843, USA*]{}\ [*$^{2}$Astroparticle physics Group, Houston Advanced Research Center (HARC),*]{}\ [*Mitchell Campus, Woodlands, TX 77381, USA*]{}\ [*$^{3}$Academy of Athens, Division of Nature Sciences,*]{}\ [*28 panepistimiou Avenue, Athens 10679, Greece*]{}\ [$^{a}$hushan@physics.tamu.edu, $^{b}$dimitri@physics.tamu.edu]{}\ Keywords: M-Theory, Brane Dynamics in Gauge Theories, Anomalies in Field and String Theories, Gauge-gravity correspondence Introduction ============ Low energy effective action of the field theory in the Coulomb branch may contain the Wess-Zumino term arising from the integration out of massive fermions getting masses via the Yukawa coupling with the vacuum expectation value of the scalar fields [@1; @2]. The existence of the Wess-Zumino term is also required by the anomaly matching condition [@3]. At a generic point of the moduli space, the gauge symmetry is broken, and then, the ’t Hooft anomaly produced by massless degrees of freedom is different from the anomaly at the origin. On the other hand, the anomaly matching condition states that the ’t Hooft anomaly should be the same everywhere on the moduli space of vacua. As a result, away from the origin, the integrating out of the massive degrees of freedom should generate the Wess-Zumino term in the low energy effective action compensating the deficit so that the total anomaly remains the same [@3; @A]. The WZ term is a topological term that does not depend on the metric nor the coupling, so it is protected without the need of invoking any supersymmetric non-renormalization theorems. For such terms, we may expect that the 1-loop calculation in field theory and the supergravity calculation would match. On supergravity side, the Wess-Zumino term is associated with the magnetic-electric coupling. For Dp-branes with $p \geq 3$, it is given by $\int_{W_{p+2}} F_{8-p}(\wedge dA)^{p-3} = (-1)^{p}\int_{W_{p+1}} F_{8-p}\wedge A (\wedge dA)^{p-4}$ [@D; @E], where $F_{8-p}$ is the magnetic field strength, while $(\wedge dA)^{p-3}$ offers the electric charge. When $p=3$, $\int_{W_{p+2}} F_{8-p}(\wedge dA)^{p-3} \rightarrow \int_{W_{5}} F_{5}$, because the D3-brane carries magnetic as well as the electric charge [@E]. For M5 branes, the WZ term is composed by $\int_{W_{6}}db_{2}\wedge A_{3}$ and $\int_{W_{7}}A_{3} \wedge F_{4}$, which are discussed in [@J] and [@A] respectively. $\int_{W_{6}}db_{2}\wedge A_{3}$ does not contribute to the anomaly. It is $\int_{W_{7}}A_{3} \wedge F_{4}$ that accounts for the anomaly deficit. [@A] considered the situation when the gauge symmetry is broken from $SU(N+1)$ to $SU(N)\times U(1)$ by the vacuum expectation value $\phi^{a}$. The corresponding WZ term takes the form of $\int_{W_{7}}\sigma_{3}(\hat{\phi}) \wedge d \sigma_{3}(\hat{\phi})$, where $d \sigma_{3}(\hat{\phi})$ is the pullback of the 4-form field strength generated by a single M5 brane while $\sigma_{3}(\hat{\phi}) $ is the corresponding 3-form potential. $\hat{\phi} = \phi/|\phi|$. It was shown that with the coefficient given by $N(N+1)/2$, the WZ term could reproduce the anomaly deficit between $SU(N+1)$ and $SU(N)\times U(1)$. In this note, we will extend the discussion to the generic Coulomb branch $(\phi^{a}_{1},\cdots,\phi^{a}_{N})$. We will show that the supergravity calculation could give the right coefficient, while the WZ term, although takes the form of $\int_{W_{7}}\sigma_{3}(\hat{\phi}_{ij}) \wedge d \sigma_{3}(\hat{\phi}_{kj})$ with $\hat{\phi}_{ij} = (\phi_{i}-\phi_{j})/|\phi_{i}-\phi_{j}|$, could produce the same amount of anomaly as that of $\int_{W_{7}}\sigma_{3}(\hat{\phi}) \wedge d \sigma_{3}(\hat{\phi})$. So the WZ term obtained from the supergravity calculation indeed compensates the anomaly deficit thus should appear in the low energy effective action, as is required by the anomaly matching condition. In the generic Coulomb branch, the WZ term in SYM theory is $(-1)^{p}\int_{W_{p+1}} F_{8-p}(\hat{\phi}_{ij})\wedge A_{ij}[\wedge dA_{ij}]^{p-4}$,[^1] which is the typical pair-wise interaction arising from the the integration out of massive fermions carrying index $(i, j)$, or open strings connecting the $i_{th}$ and the $j_{th}$ D-brane [@D; @E]. The term $\int_{W_{7}}\sigma_{3}(\hat{\phi}_{ij}) \wedge d \sigma_{3}(\hat{\phi}_{kj})$ for M5 branes seems indicate some kind of triple interaction: three M5 branes could interact simultaneously. One may naturally expect that such term comes from the integration out of massive fermions with $(i, j, k)$ index, or open M2 branes connecting the $i_{th}$, the $j_{th}$, and the $k_{th}$ M5 branes. In [@K] and more recently, [@L], the $1/4$ BPS objects in the Coulomb phase of the ADE-type 6d $(2, 0) $ superconformal theories are considered. They are made of waves on selfdual strings and junctions of selfdual strings. In [@L], it was shown that the number of $1/4$ BPS objects matches exactly one third of the anomaly constant $c_{G} = d_{G}h_{G}$ for all ADE types, indicating that the anomaly may be produced by these $1/4$ BPS objects. Moreover, the tension of the string junctions is characterized by $(|\phi_{i}- \phi_{j}|, |\phi_{j}- \phi_{k}|, |\phi_{k}- \phi_{i}|)$, which is just what is needed to produce the WZ term, since the selfdual string with tension $|\phi_{i}- \phi_{j}|$ is not enough to give the coupling like $\sigma_{3}(\hat{\phi}_{ij}) \wedge d \sigma_{3}(\hat{\phi}_{kj})$. For SYM theories, the WZ term could be derived by a 1-loop calculation [@D; @E; @F]. It is expected that the WZ term for $6d$ $(2, 0)$ theories could also be obtained via a proper integration. The $6d$ $(2, 0)$ theory is not constructed yet. Nevertheless, the form of the WZ term may offer some hint on the possible structure the underlying theory. Of course, it is possible that to construct the $6d$ SCFT, some new ingredient must be added making the theory different from the QFT in normal sense, and so the way to calculate the WZ term is also beyond the present knowledge. On the other hand, if the $6d$ $(2, 0)$ theory could be built as an ordinary quantum field theory just as that for M2 branes, we will be able to calculate the WZ term with the standard field theory methods. In [@Z], the supersymmetric equations of motion for the 3-algebra valued $(2,0)$ tensor multiplet were found, which may shed light on our understanding of the mysterious $6d$ SCFT. We will calculate the WZ term for the 3-algebra valued $(2,0)$ tensor multiplet to see how far we can go. The $H_{3}\wedge A_{3}$ part of the WZ term is obtained. Especially, without the constraint equations in [@Z], $H_{3}\wedge A_{3}$ cannot be derived. However, the $A_{3}\wedge F_{4}$ part cannot be obtained by the 1-loop fermion integration, so either a refined calculation method or a refined theory is needed. This paper is organized as follows: In section 2, we get the WZ term for $6d$ $A_{N-1}$ $(2, 0)$ theory in a generic Coulomb branch from the supergravity calculation. Part of the details is given in appendix A. In section 3, we show that the WZ terms obtained from the supergravity calculation could indeed compensate the anomaly deficit thus guarantee the anomaly matching condition. In section 4, we discuss the possible degrees of freedom in M5 branes producing the WZ term. In section 5, we caculate the WZ term for the 3-algebra valued $(2, 0)$ tensor multiplet. The conclusion is in section 6. The Hopf-Wess-Zumino term from the supergravity calculation =========================================================== Consider the $6d$ $(2, 0)$ field theory describing $N$ M5 branes. On a generic Coulomb branch $(\phi^{a}_{1},\cdots,\phi^{a}_{N})$ with $a = 1\cdots5$ and $\phi_{i}\neq \phi_{j}$, for $i\neq j$, the gauge symmetry is broken to $U(1)^{N}$. On supergravity side, we have $N$ M5 branes locating at $(\phi_{1},\cdots,\phi_{N})$. On field theory side, $N$ copies of $(2, 0) $ tensor multiplets remain massless, while the rest fields get masses. Integrating out these massive degrees of freedom, one may obtain the effective action of the $6d$, $(2, 0)$ field theory on Coulomb branch. At least for WZ terms, the calculation on both sides should coincide. $6d$ $(2, 0) $ field theory is still mysterious to us, while the multi-centered supergravity solution of M5 branes is more tractable, so we will try to get the WZ term in the effective action through the supergravity calculation. The action for the coupling of M5 branes with the 11d supergravity could be written as[^2] [@a1; @a11; @a2] $$\begin{aligned} \label{a} S &=& S_{g}+S_{M5} \nonumber \\ &=& \frac{1}{2 \kappa^{2}}\int_{M_{11}} *R-\frac{1}{2} *\hat{F}_{4}\wedge \hat{F}_{4} - \frac{1}{6}F_{4}\wedge F_{4} \wedge A_{3} \nonumber \\ &-& T_{5}\int_{W_{6}} d^{6}\xi \sqrt{-\det(g_{\mu \nu}+ (i_{v_{1}} \tilde{*} h_{3})_{\mu \nu})} + \frac{1}{2} v_{1}\wedge h_{3} \wedge \tilde{*} (v_{1}\wedge \tilde{*} h_{3})\nonumber \\ &+& \frac{T_{5}}{2} \int_{W_{6}} db_{2} \wedge A_{3} + \frac{T_{5}}{2} \int_{W_{7}} A_{3} \wedge F_{4}\end{aligned}$$ where $F_{4} = dA_{3}$, $W_{6} = \partial W_{7}$, $$\label{h} \hat{F}_{4} = F_{4} + 2 \kappa^{2} T_{5} * G_{7},$$ $$h_{3} = db_{2}-A_{3}.$$ $d* G_{7} = *J_{6}$. $*J_{6}$ is the the M5-brane current. The last term in (\[a\]) is just the Hopf-Wess-Zumino term proposed in [@A]. The field equations for $\hat{F}_{4}$ are $$\label{101} d \hat{F}_{4} = 2\kappa^{2} T_{5} *J_{6},$$ $$\label{100} d * \hat{F}_{4} + \frac{1}{2}\hat{F}_{4}\wedge \hat{F}_{4} = - 2\kappa^{2} T_{5} \; h_{3}\wedge *J_{6}.$$ $* G_{7}$, $A_{3}$, and $F_{4}$ have the dependence on gauge, while $*J_{6}$, $h_{3}$, and $\hat{F}_{4}$ are gauge independent. (\[101\]) and (\[100\]) only contain gauge invariant quantities. Suppose the vacuum expectation values of $b_{2}$ are equal to zero, consider $N$ M5 branes locating at $(\phi_{1},\cdots,\phi_{N})$. The WZ term is related with the electric-magnetic coupling, so we only need to calculate the magnetic field generated by M5 branes in the given configuration, which, in present case, is $$\hat{F}_{4} = \sum^{N}_{i = 1} \hat{F}_{4i} = Q_{1}\sum^{N}_{i = 1} \omega_{4i},$$ where $\omega_{4i}$ is the unite volume form of $S^{4}$ surrounding the $i_{th}$ brane. The corresponding 3-form field is $$A_{3} = \sum^{N}_{i = 1} A_{3i}= Q_{1}\sum^{N}_{i = 1} \sigma_{3i}.$$ $d \sigma_{3i} = \omega_{4i}$. To calculate the WZ term on the $j_{th}$ M5 brane, we need the pullback of $A_{3}$ and $F_{4}$ on the corresponding $W_{7}$. The pullback of $* G_{7}$ on $W_{7}$ vanishes, so we simply have $$\int_{W_{7j}} A_{3} \wedge F_{4} = \int_{W_{7j}} A_{3} \wedge \hat{F}_{4}= Q_{1}^{2} \int_{W_{7j}} \sum^{N}_{i = 1} \sigma_{3ij} \wedge \sum^{N}_{k = 1} \omega_{4kj},$$ where $d \sigma_{3ij} = \omega_{4ij}$. $\omega_{4ij}$ is the pullback of $\omega_{4i}$ on $W_{7j}$. Altogether, $$\label{b} \frac{T_{5}}{2} \sum^{N}_{j = 1} \int_{W_{7j}} A_{3} \wedge F_{4} = \frac{Q_{1}^{2} T_{5}}{2} \sum^{N}_{i = 1} \sum^{N}_{j = 1} \sum^{N}_{k = 1} \int_{W_{7j}} \sigma_{3ij} \wedge \omega_{4kj}$$ Aside from the $F_{4}\wedge F_{4}\wedge A_{3}$ term in supergravity, $\int A_{3}\wedge F_{4}$ is the other term which has the $N^{3}$ scaling. However, (\[b\]) is still not exactly the WZ term in the effective action. First, when $i=j=k$, we get a self-interaction term. There are totally $N$ such terms. These self-interaction terms will be produced only after the $N$ massless tensor multiplets are also integrated out. Since we only integrate massive degrees of freedom, these terms will not appear in the effective action. Second, as is shown in the Matrix theory calculation [@C], for the given brane configuration, or equivalently, the Coulomb branch, the more accurate expression for the effective action on supergravity side should be $$\label{c} S_{eff} = S_{g}+S_{M5},$$ where $S_{g}$ is the action of the supergravity fields generated by M5 branes, while $S_{M5}$ is the action of M5 branes on the background generated by themselves. $S_{eff}$ is on-shell with respect to supergravity as it should be. (\[b\]) comes from $S_{M5}$. In appendix A, we will show that $S_{g}$ contains a term which is $-2/3$ of (\[b\]), so altogether, we have $$\Gamma_{WZ} = \frac{Q_{1}^{2} T_{5}}{6} ( \sum^{N}_{i = 1} \sum^{N}_{j = 1} \sum^{N}_{k = 1}\int_{W_{7j}} \sigma_{3ij} \wedge \omega_{4kj}-\sum^{N}_{i=1}\int_{W_{7i}} \sigma_{3ii} \wedge \omega_{4ii})$$ or $$\label{j} 2\kappa^{2} \Gamma_{WZ} = \frac{Q_{1}^{3} }{6} ( \sum^{N}_{i = 1} \sum^{N}_{j = 1} \sum^{N}_{k = 1}\int_{W_{7j}} \sigma_{3ij} \wedge \omega_{4kj}-\sum^{N}_{i=1}\int_{W_{7i}} \sigma_{3ii} \wedge \omega_{4ii}),$$ where $Q_{1} = 2\kappa^{2} T_{5}$. The anomaly matching ==================== At the origin of the moduli space, 6d, $A_{N-1}$ (2, 0) field theory has the following form of anomaly when coupled to a background $SO(5)_{R}$ gauge field 1-form $A$, and in a general gravitational background [@C1a; @C1b]. $$I_{8}(N) = (N-1) I_{8}(1)+\frac{1}{24}(N^{3}-N)p_{2}(F).$$ $I_{8}(1)$ is the anomaly polynomial for a single, free, (2, 0) tensor multiplet [@C2a; @C2b]: $$I_{8}(1) = \frac{1}{48}\left[p_{2}(F)- p_{2}(R) +\frac{1}{4} (p_{1}(F)- p_{1}(R))^{2} \right].$$ $p_{2}(F)$ is the second Pontryagin class for the background $SO(5)_{R}$ field strength $F$: $$p_{2}(F) = \frac{1}{8}(\frac{i}{2\pi})^{4}\left[(trF^{2})\wedge(tr F^{2}) -2 trF^{4} \right].$$ At a generic point of the moduli space, the only massless degrees of freedom are $N-1$ copies of tensor multiplets giving rise to the anomaly of $(N-1) I_{8}(1)$. However, based on ’t Hooft anomaly matching condition, the integration out of the massive degrees of freedom will produce the WZ term in the effective action, which will offer the missing $(N^{3}-N)p_{2}(F)/24$ part so that the total anomaly is still the same as before [@A]. In the following, we will show that the WZ term in (\[j\]) could indeed give the $(N^{3}-N)p_{2}(F)/24$ part of the normal bundle anomaly. Turn on the background $SO(5)_{R}$ gauge field $A$ on $W_{6}$. $\partial W_{7} = W_{6}$, so $A$ could be smoothly extended to $W_{7}$. On $W_{7}$, we have gauge field $A^{ab}_{i} = -A^{ba}_{i}$, with $a, b = 1\cdots5$, $i = 1\cdots7$. In presence of the background field $A$, the pullback of the $S^{4}$ unite volume form on $W_{7}$ becomes $$\begin{aligned} \omega_{4}(\hat{\phi}, A) &=& \frac{1}{2}e_{4}(\hat{\phi}, A) = \frac{1}{64\pi^{2}}\epsilon_{a_{1}\cdots a_{5}}[(D_{i_{1}}\hat{\phi})^{a_{1}} (D_{i_{2}}\hat{\phi})^{a_{2}}(D_{i_{3}}\hat{\phi})^{a_{3}} (D_{i_{4}}\hat{\phi})^{a_{4}} \nonumber \\ && -2F^{a_{1}a_{2}}_{i_{1}i_{2}} (D_{i_{3}}\hat{\phi})^{a_{3}} (D_{i_{4}}\hat{\phi})^{a_{4}} + F^{a_{1}a_{2}}_{i_{1}i_{2}}F^{a_{3}a_{4}}_{i_{3}i_{4}} ] \hat{\phi}^{a_{5}} dx^{i_{1}}\wedge \cdots \wedge dx^{i_{4}},\end{aligned}$$ $(D_{i}\hat{\phi})^{a} = \partial_{i} \hat{\phi}^{a}- A^{ab}_{i} \hat{\phi}^{b}$, $F^{ab}_{ij}$ is the field strength. $\hat{\phi}$ is a unite vector in the transverse space $R^{5}$. If $\omega_{4}(\hat{\phi}, A)$ represents the pullback of the 4-form field strength generated by the $i_{th}$ M5 brane on $W_{7j}$, $\hat{\phi}$ is determined by the relative position of $W_{6i}$ and $W_{7j}$. Especially, at the boundary of $W_{7j}$, $\hat{\phi}$ is simply determined by the relative position of $W_{6i}$ and $W_{6j}$ in the transverse space, i.e. $$\hat{\phi}^{a} = \frac{\phi^{a}_{i}- \phi^{a}_{j}}{|\phi_{i}- \phi_{j}|},$$ where $\phi^{a}_{i}$ is the vacuum expectation value of scalar field for the $i_{th}$ M5 brane. $e_{4}(\hat{\phi}, A)$ is the global angular form defined over the sphere bundle with fiber $S^{4}$ and base space $W_{7}$. $$d e_{4} = 0.$$ Under the $SO(5)$ transformation, $$\begin{aligned} \label{k} \hat{\phi}^{a} & \rightarrow & \hat{\phi}^{a} + \Lambda^{ab} \hat{\phi}^{b} \nonumber \\ A^{ab}& \rightarrow & A^{ab} + d \Lambda^{ab} + [\Lambda, A]^{ab}.\end{aligned}$$ $D \hat{\phi}^{a}$ and $F^{ab}$ transform covariantly under (\[k\]), while $e_{4}(\hat{\phi}, A)$ is $SO(5)$ invariant. For the present problem, we have $N^{2}-N$ global angular forms $e_{4}(\hat{\phi}, A)$ with different $\hat{\phi}$ but the same $A$. Since $e_{4}$ is $SO(5)$ invariant, they can also be equivalently represented by $e_{4}(\hat{\phi}, A)$ with the same $\hat{\phi}$ but different $A$. $p_{2}(F)$ is the second Pontryagin class of a rank 5 real vector bundle, nevertheless, we still have $$p_{2}(F) = \chi (F)^{2},$$ where $\chi (F)$ is the Euler class of a rank 4 subbundle with the orthogonal line bundle trivial. One can always choose particular $\hat{\phi}_{0}$ so that $$e_{4}(\hat{\phi}_{0}, A) = \chi (F)$$ Actually, for such $\hat{\phi}_{0}$, $D \hat{\phi}_{0}= 0$, so $e_{4}(\hat{\phi}_{0}, A)$ reduces to the Euler class. We take this $\hat{\phi}_{0}$ as the standard and transform all of the angular forms into the form of $e_{4}(\hat{\phi}_{0}, \tilde{A})$, where $\tilde{A}$ are different connections defined on the same normal bundle. Just as the invariant polynomials, if $\tilde{A}$ and $\tilde{A}'$ are two different connections, $$e_{4}(\hat{\phi}_{0}, \tilde{A})- e_{4}(\hat{\phi}_{0}, \tilde{A}') = de_{3}(\hat{\phi}_{0}, \tilde{A})- de_{3}(\hat{\phi}_{0}, \tilde{A}')= d R (\hat{\phi}_{0}, \tilde{A}, \tilde{A}'),$$ with $e_{3}$ the corresponding Chern-Simons forms [@C1a]. $$R (\hat{\phi}_{0}, \tilde{A}, \tilde{A}') = - \frac{1}{32\pi^{2}} \int^{1}_{0} dt \;\; \epsilon_{a_{1}\cdots a_{5}} [(D_{t}\hat{\phi}_{0})^{a_{1}}(D_{t}\hat{\phi}_{0})^{a_{2}}- F_{t}^{a_{1}a_{2}} ]\eta^{a_{3}a_{4}}\hat{\phi}_{0}^{a_{5}},$$ where $$\eta = \tilde{A}- \tilde{A}', \;\;\;A_{t} = \tilde{A}' + t \eta, \;\;\;F_{t} = dA_{t} - A_{t}^{2}, \;\;\; D_{t} = (d-A_{t}).$$ $R (\hat{\phi}_{0}, \tilde{A}, \tilde{A}')$ is $SO(5)$ invariant. $$e_{3}(\hat{\phi}_{0}, \tilde{A})- e_{3}(\hat{\phi}_{0}, \tilde{A}')= R (\hat{\phi}_{0}, \tilde{A}, \tilde{A}').$$ For different connections, $e_{3}$ only differ by a $SO(5)$ invariant term. Return to the original global angular form $e_{4}(\hat{\phi}, A)$, we will have $$e_{4}(\hat{\phi}, A) = \chi (F) + d \alpha (\hat{\phi}, A),$$ $$e_{4}(\hat{\phi}, A) \wedge e_{4}(\hat{\phi}', A) = p_{2}(F) + d \beta (\hat{\phi}, \hat{\phi}', A),$$ where both $\alpha$ and $\beta$ are $SO(5)$ invariant. $$p_{2}(F) =d [e_{3}(\hat{\phi}, A) \wedge e_{4}(\hat{\phi}', A) - \beta (\hat{\phi}, \hat{\phi}', A)].$$ By descent equations, $$\delta [e_{3}(\hat{\phi}, A) \wedge e_{4}(\hat{\phi}', A)] = \delta p_{2}^{0}(A) = d p_{2}^{1}(A).$$ Now, consider the $SO(5)$ gauge transformation of (\[j\]). For each term, $$\begin{aligned} \delta[ \frac{1}{6} \int_{W_{7}} \sigma_{3}(\hat{\phi}, A) \wedge \omega_{4}(\hat{\phi}', A) ] &=& \delta[ \frac{1}{24} \int_{W_{7}} e_{3}(\hat{\phi}, A) \wedge e_{4}(\hat{\phi}', A)] \nonumber \\ &=&\frac{1}{24} \int_{W_{7}}d p_{2}^{1}(A) = \frac{1}{24} \int_{W_{6}} p_{2}^{1}(A). \end{aligned}$$ There are totally $N^{3}- N$ such terms, so $\Gamma_{WZ}$ could indeed reproduce the $(N^{3}- N)p_{2}(F)/24$ part of the anomaly. If the $SU(N)$ group is broken to some subgroup like $U(N_{1})\times U(N_{2})\times SU(N_{3})$ with $N_{1}+N_{2}+N_{3} = N$, the deficit of the anomaly produced by massless degrees of freedoms is $(N^{3}- N_{1}^{3}- N_{2}^{3}- N_{3}^{3})p_{2}(F)/24$. Corresponding, $\Gamma_{WZ}$ will contain $N^{3}- N_{1}^{3}- N_{2}^{3}- N_{3}^{3}$ terms exactly compensating the deficit. The degrees of freedom in M5 branes producing the WZ term ========================================================= The supergravity interaction between D-branes is pairwise. This is consistent with the fact that the WZ term for N D-branes in a generic Coulomb branch could be written as the sum of $N(N-1)/2$ terms labeled by $(ij)$ index [@D; @E]. On the other hand, the $\int A_{3}\wedge F_{4}$ term in the action of M5 branes gives a triple interaction. That is, three M5 branes could interact simultaneously. The supergravity interaction for D-branes is produced by open strings connecting two D-branes. Similarly, one may expect that the triple interaction $\int A_{3}\wedge F_{4}$ could be produced by open M2 branes connecting three M5 branes. Another example of $N^{3}$ interaction is given by M theory compactified on a Calabi-Yau threefold with M5-branes wrapping 4-cycles, giving rise to $N = 1$ 5d supergravity along with the chiral strings [@G; @H]. In the bulk, we have Chern-Simons term $C_{1} \wedge dC_{1} \wedge dC_{1}$, while in the worldsheet of chiral strings, $\int C_{1} \wedge dC_{1}$ may exist [@A]. These $N^{3}$ degrees of freedom in entropy are explained as states living at the triple-intersection of M5 branes [@III; @I]. In [@K] and more recently, [@L], the $1/4$ BPS objects in the Coulomb phase of the ADE-type 6d $(2, 0) $ superconformal theories are explored. They are made of waves on selfdual strings and junctions of selfdual strings. Especially, in [@L], it is shown that the number of $1/4$ BPS objects matches exactly one third of the anomaly constant $c_{G} = d_{G}h_{G}$ for all ADE types, which strongly indicates that the anomaly may be produced by these $1/4$ BPS objects. In $A_{N-1}$ case, there are $N(N-1)/2$ $1/2$ BPS selfdual strings with tension $T_{ij}\propto |\phi_{i}- \phi_{j}|$. On each selfdual string, there are left and right $1/4$ BPS waves. Turning on these BPS waves, we get $N(N-1)$ $1/4$ BPS objects. For every three M5 branes $ijk$, $1/4$ BPS junction exists. The tension of the string junctions is characterized by $(|\phi_{i}- \phi_{j}|, |\phi_{j}- \phi_{k}|, |\phi_{k}- \phi_{i}|)$. The junction forms a dual lattice to the triangle $\Delta_{ijk}$, if one indentify the $SO(5)$ in $W_{6}$ with the $SO(5)$ in the transverse space. For such configuration, the tension of selfdual strings is balanced and the junction is $1/4$ BPS. There are totally $N(N-1)(N-2)/3$ such objects because of the junction and anti-junction. Altogether, the $1/4$ BPS objects on $N$ M5 branes in a generic Coulomb branch is $N(N^{2}-1)/3$. Let us rewrite (\[j\]) in a more symmetric way. $$\label{n} 2\kappa^{2} \Gamma_{WZ} = \sum_{i\neq j,j\neq k, k\neq i} \Omega_{ijk} + \sum_{i\neq j}\Omega_{ij},$$ where $$\label{l} \Omega_{ijk} = \frac{Q_{1}^{3}}{6}[\int_{W_{7i}} (\sigma_{3ji} \wedge \omega_{4ki}+ \sigma_{3ki} \wedge \omega_{4ji})+ \int_{W_{7j}} (\sigma_{3ij} \wedge \omega_{4kj}+ \sigma_{3kj} \wedge \omega_{4ij}) + \int_{W_{7k}} (\sigma_{3ik} \wedge \omega_{4jk}+ \sigma_{3jk} \wedge \omega_{4ik}) ]$$ $$\label{m} \Omega_{ij} = \frac{Q_{1}^{3}}{6}[\int_{W_{7i}} (\sigma_{3ji} \wedge \omega_{4ii}+ \sigma_{3ii} \wedge \omega_{4ji}+ \sigma_{3ji} \wedge \omega_{4ji})+ \int_{W_{7j}} (\sigma_{3ij} \wedge \omega_{4jj}+ \sigma_{3jj} \wedge \omega_{4ij}+ \sigma_{3ij} \wedge \omega_{4ij}) ].$$ It seems that junction and anti-junction may produce the term $\Omega_{ijk}$, while left and right waves on selfdual strings could give $\Omega_{ij}$. Recall that in D-brane case, the WZ term arising from the integration out of massive fermions $\psi_{ij}$ is expressed in terms of the vector $\phi_{i}- \phi_{j}$ [@D; @E]; here, the WZ term produced by string junctions ($ijk$) could be calculated from the vectors $(\phi_{i}- \phi_{j}, \phi_{j}- \phi_{k}, \phi_{k}- \phi_{i})$. When $i = k$, the three string junctions degenerate to one selfdual string with tension $T_{ij}\propto |\phi_{i}- \phi_{j}|$ and the other tensionless selfdual string perpendicularly ending on it. So, in some sense, selfdual string with waves is a degeneration of the string junction. $\Omega_{ij} = \frac{1}{2}(\Omega_{iji}+ \Omega_{jij})$. Except for $\int_{W_{7}} A_{3}\wedge F_{4}$, M5 brane action contains another term $\int_{W_{6}} db_{2} \wedge A_{3} =-\int_{W_{6}} b_{2} \wedge F_{4} = \int_{W_{6}} H_{3} \wedge A_{3}$. Now suppose the vacuum expectation value of $b_{2}$ on the $i_{th}$ M5 brane is $b_{2i}$, $\int_{W_{6}} H_{3} \wedge A_{3}$ part of the WZ term should also enter into the low energy effective action, although it does not contribute to the anomaly since $d H_{3} = 0$. In [@J], based on the supergravity calculation, it is shown that $$\label{asdfg} \Gamma_{H} = \int_{W_{6}} H_{3} \wedge A_{3} \propto - \sum_{_{i\neq j}} \int_{W_{6}} b_{2ij} \wedge \omega_{4ij} = \sum_{_{i\neq j}} \int_{W_{6}} H_{3ij} \wedge \sigma_{3ij},$$ where $b_{2ij} = b_{2i} - b_{2j}$, $H_{3ij} = d b_{2ij} = H_{3i}-H_{3j}$. This is the typical pairwise interaction. The reduction of $\int_{W_{6}} H_{3} \wedge A_{3}$ on $S^{1}$ gives $\int_{W_{5}} F_{2} \wedge A_{3}$, the WZ term of the 5d SYM theory. In 5d SYM theory, $\int_{W_{5}} F_{2} \wedge A_{3}$ is generated by the integration out of massive fermions coming from the selfdual strings wrapping $S^{1}$, so it is quite possible that (\[asdfg\]) is produced by $1/2$ BPS selfdual strings. The non-abelian part of the R-symmetry anomaly are all accounted for by $1/4$ BPS objects. R-symmetry anomaly and Weyl anomaly are related by supersymmetry. In [@M], the conformal anomaly of 6d (2, 0) SCFT of $A_{N-1}$ type is calculated as $$A_{2, 0}(N) = (N-1) A_{tens} + (N^{3}-N) A,$$ where $A_{tens}$ is the conformal anomaly of the free $(2,0)$ tensor multiplet. It is expected that the $1/4$ BPS objects could produce $(N^{3}-N) A$, if they could give the corresponding part in R-symmetry anomaly. Note that $N-1$ 1/2 BPS massless particles and $N(N-1)(N-2)/3$ junctions of selfdual strings always contribute to the anomaly and entropy. However, the $N(N-1)/2$ selfdual strings have no contribution to the anomaly nor entropy unless the BPS waves are turned on thus the supersymmetry is reduced to $1/4$. Once the selfdual strings become $1/4$ BPS, the anomaly polynomial of them is the same as that of the string junctions, since the $1/4$ BPS selfdual strings could be taken as the degeneration of the string junctions. Then the question is why the $1/2$ BPS selfdual strings have no contribution to the anomaly nor the entropy. In $N=4$ SYM theory, $1/4$ BPS states arising from string junctions ending on three D3 branes also exist [@N], however, the anomaly and entropy are both give by $1/2$ BPS particles. In some sense, the $1/4$ BPS states could be taken as the bound states of the $1/2$ BPS states, so it is likely that for $N=4$ SYM theory, $1/2$ BPS states are fundamental, while for $6d$ $(2, 0)$ theory, it is the bound states which are dominating. The general form of the anomaly for a 6d $(2, 0)$ SCFT of the ADE type G is $$A_{2, 0} = r_{G} A_{tens} + c_{G} A_{X},$$ where $c_{G} = d_{G}h_{G} = r_{G} h_{G} (h_{G}+1)$. $r_{G}$, $d_{G}$ and $h_{G}$ are the rank, the dimension, and the Coxeter number of the Lie algebra of type G. The theory contains $r_{G}$ $1/2$ BPS massless particles, $r_{G} h_{G}$ $1/2$ BPS selfdual strings, and $c_{G}/3$ $1/4$ BPS objects. The anomaly of the single M5 brane does not have the $A_{X}$ part, so $r_{G}$ $1/2$ BPS massless particles only contribute to $A_{tens}$. Then $A_{X}$ should be generated by $1/2$ BPS selfdual strings or $1/4$ BPS objects. If one wants to interpret it in terms of selfdual strings, each selfdual string should give the anomaly of $(h_{G}+1) A_{X}$, which in $SU(N)$ case, is $(N+1) A_{X}$. It is difficult to explain this $h_{G}+1$ factor. Otherwise, since the total number of the $1/2$ BPS states is $d_{G}$, they can account for the $A_{X}$ part if each one contributes $h_{G} A_{X}$. This looks more reasonable, but the problem is that the $r_{G}$ $1/2$ BPS massless particles will contribute to $A_{tens}$ as well as $A_{X}$. The most natural possibility is that $A_{X}$ is produced by $1/4$ BPS objects, which are intrinsically three selfdual string junctions. Finally, notice that for $N=4$ SYM theory, the anomaly takes the form $A_{4} = (N^{2}-1)A_{vec}$, where $A_{vec}$ is the anomaly of a free vector multiplet. The anomaly is not renormalized from weak to strong coupling, so we can calculate it from the free field value. Besides, $N^{2}-1$ elements in the Lie algebra give the same contribution to the anomaly, indicating that they are allowed to transform into each other. On the other hand, for $6d$ (2, 0) SCFT, the anomaly polynomial is of the form $A_{2, 0} = (N-1) A_{tens} + (N^{3}-N) A_{X}$ other than $(N^{3}-1) A$, which seems indicate that there are something special about the non-abelian part. $6d$ $(2, 0)$ SCFT compactified on $S^{1}$ gives $5d$ SYM theory. Selfdual strings wrapping on $S^{1}$ become $1/2$ BPS particles. The unwrapped selfdual strings and $1/4$ BPS string junctions in $6d$ descend to the corresponding string-like objects in $5d$ [@K; @L]. String junctions may also appear as point-like particles in the compactified theory. Consider the $6d$ SCFT compactified on a Riemann surface $\Sigma_{g}$ with $g>1$ [@P; @Q; @R], the T part of $\Sigma_{g}$ is the natural place for string junctions to wrap. $\Sigma_{g}$ is built from $2(g-1)$ $T_{N}$ blocks and $3(g-1)$ $I_{N}$ blocks. $T_{N}$ and $I_{N}$ are spheres with $3$ and $2$ full punctures respectively. The dimension of the Coulomb branch for $T_{N}$ and $I_{N}$ are $$d_{c}T_{N} = \frac{(N-1)(N-2)}{2}, \;\;\;\;\;\;d_{c}I_{N} = N-1.$$ The effective number of vector multiplets for $T_{N}$ and $I_{N}$ are $$n_{v}T_{N} = \frac{2N^{3}}{3}-\frac{3N^{2}}{2}-\frac{N}{6}+1, \;\;\;\;\;\; n_{v}I_{N} = N^{2}-1.$$ Note that $$2(g-1) d_{c}T_{N} + 3(g-1) d_{c}I_{N} = (g-1)(N^{2}-1)$$ and $$2(g-1) n_{v}T_{N} + 3(g-1) n_{v}I_{N} = (g-1)(\frac{4N^{3}}{3}-\frac{N}{3}-1)$$ are the dimension of the Coulomb branch and the effective number of vector multiplets for the $\Sigma_{g}$ theory. Especially, when $g=1$, $\Sigma_{1}$ is simply constructed from one $I_{N}$. The degrees of freedom arising from strings could be calculated as $$n_{T_{N} } =n_{v}T_{N} -d_{c}T_{N} = 4 C^{3}_{N}, \;\;\;\;\;\; n_{I_{N} }= n_{v}I_{N} - d_{c}I_{N}= 2C^{2}_{N}.$$ $n_{T_{N} }$ could be naturally accounted for by string junctions[^3], while $n_{I_{N} }$ is associated with the selfdual strings. In the generic point of the Coulomb branch of $T_{N}$, the Seiberg-Witten curve is a Riemann surface with $(N-1)(N-2)/2$ genus and $3N$ simple punctures [@Q; @SSB]. In that case, $n_{T_{N} }$ could also be explained as the number of M2 branes with two boundaries. However, at the origin of the moduli space, the only nontrivial configurations are M2 branes with three boundaries. The anomaly polynomial $I_{6}(N)$ in $4d$ is obtained from $I_{8}(N)$ in $6d$ by the integration over $\Sigma_{g}$ [@S; @T]. Both of them have a $N^{3}$ scaling part. WZ term from the integration of massive fermions ================================================ The WZ term could be derived by a 1-loop calculation in field theory. For $SU(N)$ gauge theories, this has been done in [@D; @E; @F]. In Coulomb branch, fermions get mass due to the Yukawa coupling. The integration of the fermion loop gives WZ terms in the low energy effective action. We don’t know the structure of the $6d$ $(2, 0)$ field theory. A recent calculation on scattering amplitudes [@U] indicates that an interacting $6d$ Lagrangian with classical $OSp(8|4)$ symmetry cannot be constructed using only $(2, 0)$ tensor multiplets, even if the Lagrangian is non-local. A Lagrangian description may exist, however, if one includes additional degrees of freedom, for example, the selfdual strings. It is possible that the $6d$ $(2, 0)$ theory may be quite different from the ordinary field theories, and so, the WZ term in the effective action should also be calculated in a peculiar way. Alternatively, if the theory is constructed as a normal QFT containing fermionic degrees of freedom and the corresponding Dirac operators, then at a generic Coulomb branch, the WZ term may be calculated by a standard fermion-loop integration. In both cases, the Hopf-Wess-Zumino term should be generated to compensate the anomaly deficit. In the following, we will discuss the WZ term for the second possibility, especially, for the 3-algebra valued tensor multiplet. For the calculation of the WZ term, the input from the field theory is the Dirac operator involving Yukawa couplings as well as the gauge couplings. For $SU(N)$ gauge theories, the Yukawa coupling is $\Gamma_{I}[X^{I}, \psi]$. In Coulomb branch, it becomes $\Gamma_{I}(\phi^{I}_{i}-\phi^{I}_{j})\psi_{ij}$, giving mass $|\phi_{i}-\phi_{j}|$ to $\psi_{ij}$. $|\phi_{i}-\phi_{j}|$ is the length of the string connecting the $i_{th}$ and the $j_{th}$ D-branes. On the other hand, the Lagrangian of the M2 branes has a sextic potential [@X1; @X2; @X3; @Y], so the fermion mass scales as the area other than the length, reflecting the fact that M2 branes are connected by M2 branes other than strings [@V; @W]. M5 branes are also connected by M2 branes, so we may expect that the fermion mass will give the area of the M2 branes connecting different M5 branes. However, there is a difference. The M2 branes connecting parallel M2 branes are totally located in the transverse space. The endpoint is simply a point. As a result, the Yukawa coupling in BLG model takes the form of $\Gamma_{I}\Gamma_{J}[X^{I},X^{J}, \psi]$ [@X1; @X2; @X3], where $X^{I}$ and $X^{J}$ carry the transverse index. Conversely, the M2 branes connecting parallel M5 branes have one dimension living in the worldvolume of M5. The endpoint is a string. Correspondingly, we may have $\Gamma_{\mu}\Gamma_{I}[C^{\mu},X^{I}, \psi]$, where $\mu = 0 \cdots 5$, $I = 6 \cdots 10$. Actually, in [@Z], an attempt to find the $6d$ $(2, 0)$ theory with the 3-algebra structure has already been made. It was shown that for the closure of the supersymmetry, an additional vector $C^{\mu}$ must be introduced, while the 3-brackets appearing in the equations of motion always take the form $[C^{\mu},A, B]$. Later, in [@a; @b], the equations of motion found in [@Z] get a natural interpretation as the supersymmetric gauge field equations in loop space. $C^{\mu}$ is associated with the vector tangential to the loop in the worldvolume of M5 branes. The field content in [@Z] includes the tensor multiplet composed by $X^{I}$ with $I = 6 \cdots 10$, $\psi$, and $H^{\mu\nu\lambda}$ with $\mu, \nu, \lambda = 0 \cdots 5$, an auxiliary gauge field $A^{\mu}$, and a vector field $C^{\mu}$. $X^{I}$, $\psi$, $H^{\mu\nu\lambda}$, and $C^{\mu}$ take values in a vector space $\Lambda$ with the basis $t^{a}$, i.e. $X^{I} = X^{I}_{a}t^{a}$, etc. As a 3-algebra, $\Lambda$ has an associated Lie algebra $\textit{g}_{\Lambda}$ spanned by the transformations $[t^{a},t^{b}, *]$, where $*$ stands for an arbitrary element of $\Lambda$. $A^{\mu}$ takes values in $\textit{g}_{\Lambda}$. $A^{\mu}X^{I} = A^{\mu}_{ab}[t^{a},t^{b},X^{I}]$, etc. $A^{\mu}$ and $H^{\mu\nu\lambda}$ are related by $F^{\mu\nu}_{ab}[t^{a},t^{b}, *] = [C_{\lambda}, H^{\mu\nu\lambda}, *]$, with $F^{\mu\nu}_{ab}$ the field strength of $A^{\mu}_{ab}$. So, for the given $C_{\lambda}$, $F^{\mu\nu}$ is actually a transgression of $H^{\mu\nu\lambda}$ [@a]. The equations of motion for 3-algebra valued (2, 0) tensor multiplets found in [@Z] are $$\begin{aligned} \label{18} &&[C^{\mu},C^{\nu}, *]=0, \\ && \nabla_{\nu}C^{\mu} = 0, \\ && [C^{\rho},\nabla_{\rho} X^{I}, *] = 0,\;\;\ [C^{\rho},\nabla_{\rho} \psi, *] = 0,\;\;\ [C^{\rho},\nabla_{\rho} H_{\mu\nu\lambda}, *] = 0, \label{26}\\ && \tilde{F}^{\mu\nu} - [C_{\lambda}, H^{\mu\nu\lambda}, *] = 0, \label{19} \\ && \Gamma_{\mu} \nabla^{\mu} \psi + [C^{\mu}, X^{I}, \Gamma_{\mu} \Gamma_{I} \psi] = 0, \label{20} \\ && \nabla^{2}X^{I} -\frac{i}{2}[C^{\mu}, \bar{\psi}, \Gamma_{\mu} \Gamma^{I} \psi]+ [C^{\mu}, X^{J}, [C_{\mu},X_{J}, X^{I} ]] = 0,\\ && \nabla_{[\mu}H_{\nu\kappa\lambda]} + \frac{1}{4} \epsilon_{\mu\nu\kappa\lambda\sigma\tau}[C^{\sigma}, X^{I},\nabla^{\tau}X_{I}]+ \frac{i}{8}\epsilon_{\mu\nu\kappa\lambda\sigma\tau}[C^{\sigma}, \bar{\psi}, \Gamma^{\tau} \psi]=0, \label{21}\end{aligned}$$ where $\nabla^{\mu}=\partial^{\mu}-i A^{\mu}$, $X^{I}, \psi, H_{\mu\nu\lambda} \in \Lambda$, $C^{\mu}\in \bar{\Lambda}$. $\forall \; C_{1}, C_{2} \in \bar{\Lambda}$, $[C_{1},C_{2},*] = 0$. $$\tilde{F}^{\mu\nu} = \partial^{\nu}A^{\mu} - \partial^{\mu}A^{\nu} +[A^{\nu}, A^{\mu}].$$ The supersymmetry transformations are [@Z] $$\begin{aligned} \label{27} && \delta X^{I} = i \bar{\varepsilon} \Gamma^{I} \psi, \nonumber \\ && \delta \psi = \Gamma_{\mu}\Gamma_{I} \nabla^{\mu} X^{I} \varepsilon +\frac{1}{12} \Gamma_{\mu\nu\lambda}H^{\mu\nu\lambda} \varepsilon -\frac{1}{2} \Gamma_{IJ}\Gamma_{\lambda} [X^{I},X^{J},C^{\lambda}]\varepsilon, \nonumber \\ && \delta H_{\mu\nu\lambda} = 3i \bar{\varepsilon} \Gamma_{[\mu\nu}\nabla_{\lambda]}\psi + i \bar{\varepsilon} \Gamma_{I} \Gamma_{\mu\nu\lambda\kappa}[X^{I},\psi,C^{\kappa}], \nonumber \\ && \delta A_{\mu} = i \bar{\varepsilon} \Gamma_{\mu\lambda} [C^{\lambda},\psi,*], \nonumber \\ && \delta C_{\mu} = 0. \end{aligned}$$ When $C^{\mu}=0$, (\[18\])-(\[21\]) reduce to the equations of motion for free tensor multiplets. We may take $C^{\mu}$ as a vector associated with the selfdual strings. The length of $C^{\mu}$ may characterize the length of strings. When the strings shrink to points, which are described by tensor multiplets, the interaction disappears. A particular $C^{\mu}$ corresponds to a particular set of selfdual strings with the given length and the orientation. We want to take $C^{\mu}$ as the new degrees of freedom added, so we will not specify it. The path integral may cover all possible configurations of $C^{\mu}$. Now, consider the Coulomb branch of the theory. The supersymmetry transformation (\[27\]) suggests that the vacuum configuration is given by constant $X^{I}$ satisfying $[C^{\mu}, X^{I}, X^{J}] = 0$. Choose a maximal subspace $\Lambda_{0}$, $\Lambda_{0} \subset \Lambda$, $\forall A, B \in \Lambda_{0}$, $\forall C \in \bar{\Lambda}$, $[C, A, B]=0$. $\Lambda = \Lambda_{0}\oplus \Lambda_{1}$, there is a special set of basis $\left\{t^{1}\cdots t^{M} \right\}$ for $\Lambda_{1}$, $\forall C \in \bar{\Lambda}$, $\forall A \in \Lambda_{0}$, $[C, A, t^{m}] \propto t^{m}$. $\Lambda_{0}$ and $\left\{t^{1}\cdots t^{M} \right\}$ could be taken as the Cartan subalgebra and the roots respectively. Suppose the vacuum expectation value of $X^{I}$ is given by $\bar{X}^{I}$, $\bar{X}^{I} \in \Lambda_{0}$. $[C_{\mu}, \bar{X}^{I}, t^{m}] = \phi^{I}_{m \mu} t^{m}$. Similarly, suppose the vacuum expectation value of $H^{\mu\nu\lambda}$ is $\bar{H}^{\mu\nu\lambda}$, $\bar{H}^{\mu\nu\lambda} \in \Lambda_{0}$, then $[C_{\lambda}, \bar{H}^{\mu\nu\lambda}, t^{m}] = f^{\mu\nu}_{m} t^{m}$. Plug the vacuum expectation values into the Dirac equation for fermions. Let $\psi^{m}$ denote fermions taking values in the root $t^{m}$. From (\[20\]), $$\label{re} \Gamma_{\mu} \nabla^{\mu} \psi^{m} + \phi^{I}_{m \mu} \Gamma^{\mu} \Gamma_{I} \psi^{m} = 0.$$ Besides, $$[C_{\lambda}, \bar{H}^{\mu\nu\lambda},\Gamma_{\mu}\Gamma_{\nu} \psi^{m}] = f^{\mu\nu}_{m} \Gamma_{\mu}\Gamma_{\nu} \psi^{m}.$$ The form of the Dirac operator in (\[re\]) is quite natural, so it may also appear in other models for $6d$ $(2, 0)$ theories, for example, [@cccc]. $\psi^{m}$ are the $6d$ anti-chiral fermions. Written as the $11d$ Majorana spinors, $\Gamma_{7} \psi^{m} = - \psi^{m}$, where $\Gamma_{7} = \Gamma_{012345}$. Just as that in [@D; @E], the WZ term could be written as $$\Gamma_{m}= \texttt{Tr} \left\{\texttt{ln} \: [i \Gamma_{0}\Gamma_{\mu}\partial^{\mu} + \Gamma_{0} \Gamma_{\mu}A_{m}^{\mu} + i \Gamma_{0} \Gamma^{\mu} \Gamma_{I} \phi^{I}_{m \mu} ]\frac{1-\Gamma_{7}}{2} \right\},$$ $$\label{4} \frac{\delta \Gamma_{m}}{\delta \phi^{I}_{m \mu}(x)} = Sp [\left\langle x \right| \frac{ 1}{i \Gamma_{\nu}\partial^{\nu} + \Gamma_{\nu}A_{m}^{\nu} + i \Gamma^{\nu} \Gamma_{I} \phi^{I}_{m \nu}} \left| x\right\rangle i \Gamma^{\mu} \Gamma_{I} (\frac{1-\Gamma_{7}}{2})],$$ where $Sp$ is the trace in spinor indices. WZ term comes from the imaginary part of the effective action. Taking the difference of (\[4\]) with its complex conjugate, $$\begin{aligned} \label{5} \frac{\delta Im \Gamma_{m}}{\delta \phi^{I}_{m \mu}(x)} &=& - \frac{1}{2} Sp [\left\langle x \right| \frac{ 1}{i \Gamma_{\nu}\partial^{\nu} + \Gamma_{\nu}A_{m}^{\nu} + i \Gamma^{\nu} \Gamma_{I} \phi^{I}_{m \nu}} \left| x\right\rangle \Gamma^{\mu} \Gamma_{I} \Gamma_{7}] \nonumber \\ &=& -\frac{1}{2} Sp [\left\langle x \right| \frac{ 1}{\slash{D}} \left| x\right\rangle \Gamma^{\mu} \Gamma_{I} \Gamma_{7}]\nonumber \\ &=& - \frac{1}{2} Sp [\left\langle x \right| \frac{ \slash{D}}{\slash{D}^{2}} \left| x\right\rangle \Gamma^{\mu} \Gamma_{I} \Gamma_{7}].\end{aligned}$$ (\[5\]) could be expanded as the sum of the terms proportional to $1$ or $Tr(\Gamma_{M_{1}}\cdots \Gamma_{M_{k}})$, where $M_{1},\ldots, M_{k}$ are distinct indices. Note that $Tr(\Gamma_{M_{1}}\cdots \Gamma_{M_{k}})=0$ unless $k=11$, so we need to extract the term proportional to $Tr(\Gamma_{0}\cdots \Gamma_{10}) \propto \epsilon_{0\cdots10}$. In the numerator, $i \Gamma^{\nu} \Gamma_{I} \phi^{I}_{m \nu}$ in $\slash{D}$ will be kept, while in the denominator, $$\slash{D}^{2}= - \partial^{2} + \phi^{I}_{m \mu} \phi^{\mu}_{m I} + \frac{i}{2}\Gamma_{\mu}\Gamma_{\nu}f_{m}^{\mu\nu}- \Gamma_{\mu}\Gamma^{\nu}\Gamma_{I}\partial^{\mu}\phi^{I}_{m \nu}+ \Gamma^{\mu\nu}\Gamma_{IJ} \phi^{I}_{m\mu}\phi^{J}_{m\nu}+\cdots,$$ where $f_{m}^{\mu\nu}=\partial^{\mu}A_{m}^{\nu}- \partial^{\nu}A_{m}^{\mu}+[A_{m}^{\mu}, A_{m}^{\nu}]$, $$\frac{1}{\slash{D}^{2}}=-\sum^{\infty}_{n=0}\frac{[\frac{i}{2}\Gamma_{\mu}\Gamma_{\nu}f_{m}^{\mu\nu}-\Gamma_{\mu}\Gamma^{\nu}\Gamma_{I}\partial^{\mu}\phi^{I}_{m \nu}+ \Gamma^{\mu\nu}\Gamma_{IJ} \phi^{I}_{m\mu}\phi^{J}_{m\nu}+ \cdots]^{n}}{(\partial^{2}- \phi^{I}_{m \mu} \phi^{\mu}_{m I})^{n+1}}.$$ The integral that needs to be performed is $$\begin{aligned} \left\langle x \right| \frac{ 1}{(\partial^{2}- \phi^{I}_{m \mu} \phi^{\mu}_{m I})^{n+1}} \left| x\right\rangle &=& (-1)^{n+1} \int \frac{d^{d}p}{(2 \pi)^{d}} \frac{1}{(p^{2}+ \phi^{I}_{m \mu} \phi^{\mu}_{m I})^{n+1}} \nonumber \\ &=& \frac{i \Gamma(n+1-\frac{d}{2})}{(2\sqrt{\pi})^{d}\Gamma(n+1)} \frac{1}{(\sqrt{\phi^{I}_{m \mu} \phi^{\mu}_{m I}})^{2n+2-d}}.\end{aligned}$$ The constraint $[C^{\mu}, \nabla_{\mu}\psi^{m}, *] = 0$ reduces the dynamics from $6d$ to $5d$. If $C^{\mu}$ is taken as the vector tangential to the selfdual string, the constraint means that the physical momentum of the string is along the transverse direction. As a result, $d=5$. For $n=4$, $2n+2-d = 5$. $$\frac{\delta Im \Gamma_{m}}{\delta \phi^{I}_{m \mu}(x)} \propto \frac{ Sp \left\{ [\frac{i}{2}\Gamma_{\rho}\Gamma_{\nu}f_{m}^{\rho\nu}-\Gamma_{\rho}\Gamma^{\nu}\Gamma_{J}\partial^{\rho}\phi^{J}_{m \nu}+ \Gamma^{\rho\nu}\Gamma_{KJ} \phi^{K}_{m \rho}\phi^{J}_{m\nu}+ \cdots]^{4} \phi^{L}_{m \lambda} \Gamma^{\lambda} \Gamma_{L} \Gamma^{\mu} \Gamma_{I} \Gamma_{7}\right\}}{(\sqrt{\phi^{I}_{m \mu} \phi^{\mu}_{m I}})^{5}}$$ The term containing one $f_{m}^{\rho\nu}$ and three $\partial^{\rho}\phi^{J}_{m \nu}$ is $$\begin{aligned} \label{31} \frac{\delta L_{1}}{\delta \phi^{I}_{m \mu}(x)} &\propto & - \frac{ f_{m\rho\nu} \partial_{\rho_{1}}\phi^{J_{1}}_{m \nu_{1}}\partial_{\rho_{2}}\phi^{J_{2}}_{m \nu_{2}}\partial_{\rho_{3}}\phi^{J_{3}}_{m \nu_{3}}\phi^{L}_{m \lambda}}{(\sqrt{\phi^{I}_{m \mu} \phi^{\mu}_{m I}})^{5}} \nonumber \\ && Sp[\Gamma^{\rho}\Gamma^{\nu} \Gamma^{\rho_{1}}\Gamma^{\nu_{1}}\Gamma^{\rho_{2}}\Gamma^{\nu_{2}}\Gamma^{\rho_{3}}\Gamma^{\nu_{3}} \Gamma^{\lambda} \Gamma^{\mu} \Gamma_{7}\Gamma_{J_{1}} \Gamma_{J_{2}}\Gamma_{J_{3}}\Gamma_{L}\Gamma_{I} ] \end{aligned}$$ The counterpart of $L_{1}$ in the low energy effective action is $L'_{1} \propto \int_{W_{6}} H_{3} \wedge A_{3}$, $$\label{33} \frac{\delta L'_{1}}{\delta \phi^{I}_{m }(x)} \propto \frac{\epsilon^{\rho\nu\mu\rho_{1}\rho_{2}\rho_{3}}\epsilon_{J_{1}J_{2}J_{3}LI}h_{m \rho\nu\mu} \partial_{\rho_{1}}\phi^{J_{1}}_{m }\partial_{\rho_{2}}\phi^{J_{2}}_{m }\partial_{\rho_{3}}\phi^{J_{3}}_{m} \phi^{L}_{m}}{(\sqrt{\phi^{I}_{m } \phi_{m I}})^{5}},$$ Here $m$ denotes a particular root. If $m \sim (i, j)$, $\phi^{I}_{m } = \phi^{I}_{i } - \phi^{I}_{j } = \phi^{I}_{ij }$, $h_{m \rho\nu\mu} = h_{i \rho\nu\mu}- h_{j \rho\nu\mu} = h_{ij \rho\nu\mu}$. In general, one may expect that $\phi^{I}_{\mu}$ could be expanded as $\phi^{I}_{\mu} = \sum_{k} c_{k\mu} a_{k}^{I}$, where $c_{k\mu}$ and $a_{k}^{I}$ are vectors along the longitudinal and transverse directions respectively. The simplest situation is $\phi^{J}_{m \nu} = c_{m \nu} \phi_{m}^{J}$. $\partial_{\rho}\phi^{J}_{m \nu} = \partial_{\rho}c_{m \nu} \phi_{m}^{J} + c_{m \nu}\partial_{\rho}\phi^{J}_{m } = c_{m \nu}\partial_{\rho}\phi^{J}_{m }$. Therefore, $\partial_{\rho_{3}}\phi^{J_{3}}_{m \nu_{3}}\Gamma^{\rho_{3}}\Gamma^{\nu_{3}} \Gamma^{\lambda} \phi^{L}_{m \lambda} = |c_{m}|^{2} \Gamma^{\rho_{3}}\partial_{\rho_{3}}\phi^{J_{3}}_{m}\phi^{L}_{m}$. Similarly, $\partial_{\rho_{1}}\phi^{J_{1}}_{m \nu_{1}}\partial_{\rho_{2}}\phi^{J_{2}}_{m \nu_{2}}\Gamma^{\rho_{1}}\Gamma^{\nu_{1}}\Gamma^{\rho_{2}}\Gamma^{\nu_{2}} \sim -|c_{m}|^{2}\partial_{\rho_{1}}\phi^{J_{1}}_{m }\partial_{\rho_{2}}\phi^{J_{2}}_{m }\Gamma^{\rho_{1}}\Gamma^{\rho_{2}}$, where we have neglected the term involving $c^{ \nu}_{m}\partial_{\nu}\phi^{J}_{m } $. (\[31\]) is simplified to $$\label{32} \frac{\delta L_{1}}{\delta \phi^{I}_{m }(x)} \propto \frac{ f_{m\rho\nu} c_{m \mu} \partial_{\rho_{1}}\phi^{J_{1}}_{m}\partial_{\rho_{2}}\phi^{J_{2}}_{m }\partial_{\rho_{3}}\phi^{J_{3}}_{m }\phi^{L}_{m }}{|c_{m}|(\sqrt{\phi^{I}_{m } \phi_{m I}})^{5}} Sp[\Gamma^{\rho}\Gamma^{\nu} \Gamma^{\rho_{1}}\Gamma^{\rho_{2}}\Gamma^{\rho_{3}} \Gamma^{\mu} \Gamma_{7}\Gamma_{J_{1}} \Gamma_{J_{2}}\Gamma_{J_{3}}\Gamma_{L}\Gamma_{I} ]$$ To get the nonzero result, the first six Gamma matrices should multiply to $1$, while the last five Gamma matrices should cover $\Gamma_{6}\cdots \Gamma_{10}$. The trace then becomes $Sp[\Gamma_{0}\cdots \Gamma_{10}] \sim \epsilon_{0\cdots 10}$. $\rho_{1}$, $\rho_{2}$ and $\rho_{3}$ must be different, so $\left\{\rho, \nu, \mu \right\}=\left\{\rho_{1}, \rho_{2}, \rho_{3} \right\}$. One can similarly expand $f_{m\rho\nu}$ as $f_{m\rho\nu} = h_{m \rho\nu \sigma} c^{\sigma}_{m}$. $f_{m\rho\nu} c_{m \mu} = h_{m \rho\nu \sigma} c^{\sigma}_{m} c_{m \mu}$. As is mentioned before, $c_{m}$ is not fixed but should also be integrated in the path integral. The orientation of the selfdual string in $6d$ spacetime is arbitrary. Summing over all possible directions, $\sum c^{\sigma}_{m} c_{m \mu} = |c_{m}|^{2} g^{\sigma}_{\mu}$, so $\sum f_{m\rho\nu} c_{m \mu} = h_{m \rho\nu \sigma} \sum c^{\sigma}_{m} c_{m \mu} = |c_{m}|^{2} h_{m \rho\nu \mu} $. (\[32\]) then becomes $$\label{34} \frac{\delta L_{1}}{\delta \phi^{I}_{m }(x)} \propto \frac{ |c_{m}| (*h_{m})_{\rho\nu\mu} \partial_{\rho_{1}}\phi^{J_{1}}_{m }\partial_{\rho_{2}}\phi^{J_{2}}_{m }\partial_{\rho_{3}}\phi^{J_{3}}_{m} \phi^{L}_{m }}{(\sqrt{\phi^{I}_{m } \phi_{m I}})^{5}}\epsilon^{\rho\nu\mu\rho_{1}\rho_{2}\rho_{3}}\epsilon_{J_{1}J_{2}J_{3}LI}.$$ Compared with (\[33\]), (\[34\]) contains $*$ which is resulted from $\Gamma_{7}$ inside the trace. This is not quite satisfactory, but luckily, since $*h=h$, (\[33\]) and (\[34\]) still coincide up to a $|c_{m}|$ factor. Notice that to get $L_{1}$ which is close to $L'_{1}$, $d=5$ is quite crucial. For an ordinary $6d$ theory without the constraint, the denominator is $(\sqrt{\phi^{I}_{m \mu} \phi^{\mu}_{m I}})^{2n-4}$, so one cannot get $L'_{1}$ no matter which $n$ is taken. In other words, to get the WZ term $H_{3} \wedge A_{3}$, the basic degrees of freedom should be the $1d$ object with $5d$ momentum other than the $0d$ object with $6d$ momentum. For $1/2$ BPS selfdual strings, the momentum is along the transverse direction, since the momentum along the longitudinal direction may reduce the selfdual string to a $1/4$ BPS state [@ccc]. So the constraint in [@Z] may indicate that the selfdual strings involved in equations are $1/2$ BPS states. The WZ term $H_{3} \wedge A_{3}$ is generated by $1/2$ BPS selfdual strings. In [@Ber], the anomaly of the $1/2$ BPS selfdual strings coupling with tensor multiplets was discussed in analogy with the anomaly of M5 branes coupling with supergravity. In the generic Coulomb branch, there are $N(N-1)$ selfdual strings acting as the sources for the relative 3-forms $h_{ij}$. $H_{3} \wedge A_{3}$ is exactly the term in the bulk cancelling the selfdual string normal bundle anomaly [@Ber]. When reduced to $5d$, $H_{3} \wedge A_{3}$ becomes $F_{2} \wedge A_{3}$, which is the term needed to cancel the normal bundle anomaly of the magnetic monostring [@Ber]. On the other hand, $F_{2} \wedge A_{3}$ could be obtained in $5d$ SYM theory by the integration out of the massive fields which are the quantization of the electric dual of the monostring. The dual of the selfdual string is itself, so we may expect that $H_{3} \wedge A_{3}$ could be derived by the integration out of the fields which are the quantization of the selfdual string. The next problem is to get the WZ term corresponding to $A_{3}\wedge F_{4} \sim \sigma_{3}\wedge \omega_{4}$, which is far more difficult. The field theory calculation gives $\delta \Gamma_{WZ}/\delta \phi^{I}$, the counterpart of the Lorentz force on supergravity side. Consider a M5 brane in a background field $\hat{F}_{4}$, the action contains the term $S = - \int_{W_{6}} A_{6}$, $d A_{6} = *\hat{F}_{4}+ A_{3}\wedge \hat{F}_{4}/2$. $$\begin{aligned} \label{2wq} && \frac{\delta S}{\delta \phi^{I}} = - \frac{1}{6!}\epsilon^{\rho_{1}\rho_{2}\rho_{3}\rho_{4}\rho_{5}\rho_{6}} \partial_{\rho_{1}} Y^{n_{1}}\partial_{\rho_{2}} Y^{n_{2}}\partial_{\rho_{3}} Y^{n_{3}}\partial_{\rho_{4}} Y^{n_{4}}\partial_{\rho_{5}} Y^{n_{5}}\partial_{\rho_{6}} Y^{n_{6}}(d A_{6})_{In_{1}n_{2}n_{3}n_{4}n_{5}n_{6}} \nonumber \\ &=& - \frac{1}{6!}\epsilon^{\rho_{1}\rho_{2}\rho_{3}\rho_{4}\rho_{5}\rho_{6}} \partial_{\rho_{1}} Y^{n_{1}}\partial_{\rho_{2}} Y^{n_{2}}\partial_{\rho_{3}} Y^{n_{3}}\partial_{\rho_{4}} Y^{n_{4}}\partial_{\rho_{5}} Y^{n_{5}}\partial_{\rho_{6}} Y^{n_{6}} (*\hat{F}_{4}+ \frac{1}{2}A_{3}\wedge \hat{F}_{4})_{In_{1}n_{2}n_{3}n_{4}n_{5}n_{6}} \nonumber \\ &=& f_{6I}+g_{6I}. \end{aligned}$$ $Y^{n_{i}}$ are embedding coordinates. $n_{i} = 0\cdots 10$, $Y^{I} = \phi^{I}$. $f_{6I}$ and $g_{6I}$ are forces related with the magnetic-magnetic interaction and the electric-magnetic interaction respectively. $g_{6I}$ is the Lorentz force derived from the WZ term. If $db_{2}$ is also taken into account, $$\label{wq2} g_{6I} = \frac{1}{6!}\epsilon^{\rho_{1}\rho_{2}\rho_{3}\rho_{4}\rho_{5}\rho_{6}} \partial_{\rho_{1}} Y^{n_{1}}\partial_{\rho_{2}} Y^{n_{2}}\partial_{\rho_{3}} Y^{n_{3}}\partial_{\rho_{4}} Y^{n_{4}}\partial_{\rho_{5}} Y^{n_{5}}\partial_{\rho_{6}} Y^{n_{6}} ( \frac{1}{2}h_{3}\wedge \hat{F}_{4})_{In_{1}n_{2}n_{3}n_{4}n_{5}n_{6}}.$$ For the WZ term in (\[l\]),[^4] $$\begin{aligned} \label{44} \frac{\delta \int_{W_{7i}}\sigma_{3ji}\wedge \omega_{4ki} }{\delta \phi^{I}_{i}(x)} &=& -\int_{W_{7i}} [ \frac{\delta \sigma_{3ji}}{\delta \phi^{I}_{ji}(x) }\wedge \omega_{4ki}+ \sigma_{3ji}\wedge \frac{\delta \omega_{4ki}}{\delta \phi^{I}_{ki}(x)}] \nonumber \\ &=& (\sigma_{3ji} \wedge F_{3kiI})(x) - (\omega_{4ji} \wedge F_{3kiI} + \omega_{4ki} \wedge F_{3jiI})(x). \end{aligned}$$ $\partial W_{7i} = W_{6i}$, $x \in W_{6i}$. $$F_{3kiI} = \alpha \; \frac{\epsilon_{ILJ_{1}J_{2}J_{3}}\phi^{L}_{ki}d \phi^{J_{1}}_{ki}\wedge d \phi^{J_{2}}_{ki} \wedge d \phi^{J_{3}}_{ki}}{|\phi_{ki}|^{5}}$$ up to a total derivative. $\alpha$ is a constant. Except for the 6-form, (\[44\]) also contains the 7-form because $\sigma_{3}\wedge \omega_{4}$ alone is not closed. $$\label{40} \frac{\delta \int_{W_{7i}}(\sigma_{3ji}\wedge \omega_{4ki} + \sigma_{3ki}\wedge \omega_{4ji})}{\delta \phi^{I}_{i}(x)} = (\sigma_{3ji} \wedge F_{3kiI} + \sigma_{3ki} \wedge F_{3jiI})(x)-2(\omega_{4ji} \wedge F_{3kiI} + \omega_{4ki} \wedge F_{3jiI})(x),$$ $$\begin{aligned} \label{41} \frac{\delta \int_{W_{7}} \Omega_{ijk}}{\delta \phi^{I}_{i}(x)} & \propto & (\sigma_{3ji}\wedge F_{3kiI} + \sigma_{3ki} \wedge F_{3jiI} - \sigma_{3kj} \wedge F_{3ijI}- \sigma_{3jk} \wedge F_{3ikI} )(x) \nonumber \\ && - 2( \omega_{4ji}\wedge F_{3kiI} + \omega_{4ki}\wedge F_{3jiI} - \omega_{4kj} \wedge F_{3ijI} - \omega_{4jk} \wedge F_{3ikI})(x).\end{aligned}$$ We need to get the 6-form of (\[41\]) from the field theory calculation. The first thing is to find an explicit expression for $\sigma_{3}$. Consider the $5d$ transverse space with coordinate $\phi^{a_{i}}$, $a_{i} = 6\cdots 10$, $$\omega_{4} = \epsilon_{a_{1}a_{2}a_{3}a_{4} a_{5}}\frac{1}{|\phi|^{5}}\phi^{a_{1}}d\phi^{a_{2}}\wedge d\phi^{a_{3}}\wedge d\phi^{a_{4}} \wedge d\phi^{a_{5}} = *\hat{\phi}.$$ $\sigma_{3}$ could be constructed in analogy with the gauge field describing the magnetic monopole in $3d$ space. Select an arbitrary vector $v$, $v \cdot \phi= |v||\phi| \cos \theta$, $$\begin{aligned} \label{37} \sigma_{3} &=& \frac{(\cos^{3} \theta -3\cos \theta+2 )*(v\wedge \phi)}{3 \sin^{4} \theta |v||\phi|} \nonumber \\ &=& \frac{(v \cdot \phi)^{3} -3 (v \cdot \phi)|v|^{2}|\phi|^{2}+2|v|^{3}|\phi|^{3} }{3 \sin^{4} \theta |v|^{4}|\phi|^{7}} \epsilon_{a_{1}a_{2}a_{3}a_{4} a_{5}}v^{a_{1}}\phi^{a_{2}} d\phi^{a_{3}}\wedge d\phi^{a_{4}} \wedge d\phi^{a_{5}}. \end{aligned}$$ $\sigma_{3}$ is singular on a ray $OV$ starting from the origin and extending in $-v$ direction. In $11d$ spacetime, $OV \times W_{6} = W_{7}$. $W_{7}$ is the Dirac brane similar to the Dirac string [@a1; @a2]. Replace $v$ by $-v$, we get another $\sigma_{3}$. Take the average of these two $\sigma_{3}$’s, the last term in (\[37\]) could be dropped. Of course, in this case, the singularity exists in a straight line. One may have $$\label{110} \sigma_{3ji} = \frac{(v \cdot \phi_{ji})^{3} -3 (v \cdot \phi_{ji})|v|^{2}|\phi_{ji}|^{2}}{3 \sin^{4} \theta |v|^{4}|\phi_{ji}|^{7}} \epsilon_{a_{1}a_{2}a_{3}a_{4} a_{5}} v^{a_{1}}\phi_{ji}^{a_{2}} \; d\phi_{ji}^{a_{3}}\wedge d\phi_{ji}^{a_{4}} \wedge d\phi_{ji}^{a_{5}} + d\chi_{2ji},$$ $d \sigma_{3ji}= \omega_{4ji}$. $\sigma_{3ji}$ is only determined up to an exact form $d\chi_{2ji}$. One may want to take the first term of (\[110\]) as $\sigma_{3ji}$ for simplicity, but there is a problem. The only input on field theory side is the vacuum expectation value $( \phi_{1}, \cdots, \phi_{N} )$ on $W_{6}$. As a result, we have to construct the 6-form from $(\phi_{ij}, \phi_{jk}, \phi_{ki})$. The first term of $\sigma_{3ji}$ in (\[110\]) contains a constant vector $v$, which has no relevance with the vacuum expectation value. We may try to replace $v$ by vectors constructed from $( \phi_{1}, \cdots, \phi_{N} )$. For example, for the 6-form $\sigma_{3ji} \wedge F_{3kiI}$ living on $W_{6}$, $$\begin{aligned} \label{1011} \sigma_{3ji} \wedge F_{3kiI} &=& \alpha \; \frac{(v \cdot \phi_{ji})^{3} -3 (v \cdot \phi_{ji})|v|^{2}|\phi_{ji}|^{2}}{3 \sin^{4}\theta |v|^{4} |\phi_{ji}|^{7}|\phi_{ki}|^{5}} \epsilon_{I b_{1}b_{2}b_{3}b_{4}} \epsilon_{L a_{1}a_{2}a_{3}a_{4}} v^{L}\phi_{ji}^{a_{1}} \nonumber \\ && \phi^{b_{1}}_{ki} \; d\phi_{ji}^{a_{2}}\wedge d\phi_{ji}^{a_{3}} \wedge d\phi_{ji}^{a_{4}}\wedge d \phi^{b_{2}}_{ki}\wedge d \phi^{b_{3}}_{ki} \wedge d \phi^{b_{4}}_{ki}. \end{aligned}$$ $v$ may be replaced by $\phi_{ki}$, but then the extra terms should be added because $\phi_{ki}$ may not be constant. Notice that the epsilon symbols involved in $H_{3}\wedge F_{3}$ and the Lorenz force in SYM theories are $\epsilon_{M_{1}M_{2}\cdots M_{11}}$ and $\epsilon_{N_{1}N_{2}\cdots N_{10}}$ which are equal to the traces of the Gamma matrices product, thus could be derived via the 1-loop fermion integration. On the other hand, $\sigma_{3} \wedge F_{3}$ contains $\epsilon_{a_{1}\cdots a_{5}}\epsilon_{b_{1}\cdots b_{5}}\epsilon_{\mu_{1} \cdots \mu_{6}}$ with $a_{i}, b_{i} = 6\cdots 10$, $\mu_{i}=0\cdots 5$ that cannot be identified with the single trace. As a result, a 1-loop calculation cannot produce $A_{3}\wedge F_{4}$. Maybe a refined integration is needed. For example, since each fermion loop gives a single epsilon symbol, one may consider two fermion loops, one for $\phi_{ji}$ and the other for $\phi_{ki}$, giving rise to $\epsilon_{a_{1}\cdots a_{5}}\epsilon_{\mu_{1} \cdots \mu_{3}\nu_{1} \cdots \nu_{3}}$ and $\epsilon_{b_{1}\cdots b_{5}}\epsilon_{\mu_{4} \cdots \mu_{6}\nu_{4} \cdots \nu_{6}}$ respectively. With a suitable contraction, $\epsilon_{a_{1}\cdots a_{5}}\epsilon_{b_{1}\cdots b_{5}}\epsilon_{\mu_{1} \cdots \mu_{3}\nu_{1} \cdots \nu_{3}}\epsilon_{\mu_{4} \cdots \mu_{6}}\!^{\nu_{1} \cdots \nu_{3}}$ may be produced, from which, one can at most get $*A_{3}\wedge F_{3}$ other than $A_{3}\wedge F_{3}$. This is acceptable if the self duality condition is imposed. If we take $\psi_{ij}$ as the fundamental fields, the two loop fermion integration seems indicate that the anomaly is produced by the bound state of $\psi_{ji}$ and $\psi_{ki}$, which are again three index objects. Given that in standard field theory, the only way to get the epsilon symbols is through the fermion loop integration, one may also expect that the $6d$ $(2,0)$ theory may not be the ordinary QFT thus cannot be analyzed in this way. No matter in which case, even if we could get the correct epsilon symbol, we still need to ensure that each term takes the form of $\sigma_{3ji}\wedge \omega_{4ki}$ with two roots involved. One possibility is to start from the three indices fundamental fields at the beginning. Or one can take two indices fields as fundamental, but there must be a justification for why $\psi_{ji}$ and $\psi_{ki}$ appear simultaneously in the integration. Conclusion ========== In this paper, we discussed the WZ term in the low energy effective action of the $6d$ $(2, 0)$ field theory in the generic Coulomb branch. As a topological term, WZ term does not depend on the metric nor the coupling, so it is protected. For such terms, the supergravity calculation and the field theory calculation will give the same result. There is no available $6d$ $(2,0)$ field theory at present, so we will first calculate the WZ term on supergravity side. We then show that the obtained WZ term could indeed compensate the anomaly deficit, as is required by the anomaly matching condition, thus should appear in the low energy effective action. For SYM theory in a generic Coulomb branch, each WZ term involves one root $e_{i}-e_{j}$, which is consistent with the fact that the supergravity interaction is produced by the integrating out of massive strings connecting the $i_{th}$ and the $j_{th}$ D branes. On the other hand, for M5 branes, each WZ term involves two roots $e_{i}-e_{j}$ and $e_{k}-e_{j}$. One may expect that such kind of triple interaction may be generated by the integrating out of the massive objects carrying $(i, j, k)$ indices. A natural candidate is the string junction with tension $(|\phi_{ij}|,|\phi_{jk}|,|\phi_{ki}|)$ proposed in [@K; @L]. The $6d$ $(2,0)$ theory may have a mathematical structure which is different from the ordinal QFT, or it could just be constructed as a normal quantum field theory. In the latter case, the WZ term could be calculated by a standard 1-loop integration. The input from the field theory is the Dirac operator on the given background scalar fields and tensor fields. The algebra structure is involved. For $SU (N) $ SYM theories, the 2-algebra gives $(N^{2}-N)/2$ terms characterized by roots $e_{i}-e_{j}$. The Hopf-Wess-Zumino term for $6d$ $(2,0)$ theory is composed by $(N^{3}-N)/3$ terms with roots $(e_{i}-e_{j},e_{j}-e_{k},e_{k}-e_{i})$, so it may indicate a new algebra structure. The 3-algebra may be a natural candidate, but it has no finite dimensional Euclidean representation. We calculate the WZ term for the 3-algebra valued (2, 0) tensor multiplet theory proposed in [@Z]. The $H_{3}\wedge A_{3}$ part of the WZ term could be obtained. However, the $A_{3}\wedge F_{4}$ part, which is responsible for the anomaly compensating, cannot be produced by the 1-loop fermion integration, indicating that some key ingredient is still missing. [**Acknowledgments:**]{} We are grateful to Ken Intriligator for helpful comment. The work is supported in part by the Mitchell-Heep Chair in High Energy Physics and by the DOE grant DE-FG03-95-Er-40917. The on-shell action for brane-gravity coupled system ==================================================== Consider the $(d-1)$-brane couples with the supergravity fields. The action is [@B] $$S = S_{brane} + S_{gravity}$$ $$\begin{aligned} \label{400} S_{brane} & =& T_{d} \int d^{d}\xi [ -\frac{1}{2} \sqrt{-\gamma}\gamma^{ij}\partial_{i} X^{M} \partial_{j} X^{N} g_{MN} e^{\alpha (d)\phi/d} + \frac{d-2}{2}\sqrt{-\gamma} \nonumber \\ & & - \frac{1}{d!}\epsilon^{i_{1}i_{2} \cdots i_{d}} \partial_{i_{1}} X^{M_{1}} \partial_{i_{2}} X^{M_{2}} \cdots \partial_{i_{d}} X^{M_{d}} A_{M_{1}M_{2}\cdots M_{d}}] \nonumber \\ & =& S_{1}+ S_{2} + S_{3}\end{aligned}$$ $$S_{gravity} = \frac{1}{2 \kappa^{2}} \int d^{D} x \sqrt{-g} [ R - \frac{1}{2} (\partial \phi)^{2} - \frac{1}{2(d+1)!} e^{- \alpha (d)\phi} F^{2}_{d+1}]$$ The action of this form is valid for both electric and magnetic branes, while for magnetic branes, just let $\alpha(d)\rightarrow -\alpha(d)$. Variation with respect to $g_{MN}$, $A_{M_{1}M_{2}\cdots M_{d}}$, and $\gamma_{ij}$ gives $$\begin{aligned} T^{MN}_{brane} & =& - T_{d} \int d^{d}\xi \sqrt{-\gamma}\gamma^{ij}\partial_{i} X^{M} \partial_{j} X^{N} e^{\alpha (d)\phi/d} \frac{\delta^{D} (x-X)}{\sqrt{-g}} \nonumber \\ & =& \frac{1}{\kappa^{2}} [ R^{MN} - \frac{1}{2}g^{MN} R - \frac{1}{2} (\partial^{M} \phi \partial^{N} \phi - \frac{1}{2}g^{MN}(\partial \phi)^{2} ) \nonumber \\ & & - \frac{1}{2 d!} ( F^M\,_{M_1 \cdots M_d} F^{N M_{1}\cdots M_{d}} - \frac{1}{2(d+1)} g^{MN} F^{2}) e^{- \alpha (d)\phi}] \end{aligned}$$ $$\begin{aligned} \label{600} J^{M_1 \cdots M_d}_{brane} & =& T_{d} \int d^{d}\xi \, \epsilon^{i_{1}i_{2} \cdots i_{d}} \partial_{i_{1}} X^{M_{1}} \partial_{i_{2}} X^{M_{2}} \cdots \partial_{i_{d}} X^{M_{d}} \frac{\delta^{D} (x-X)}{\sqrt{-g}} \nonumber \\ & =& \frac{1}{2\kappa^{2}\sqrt{-g}} \partial_{M} (\sqrt{-g}e^{- \alpha (d)\phi} F^{M M_{1}\cdots M_{d}} )\end{aligned}$$ and $$\label{300} \gamma_{ij} = \partial_{i} X^{M} \partial_{j} X^{N} g_{MN} e^{\alpha (d)\phi/d}.$$ Then $$\begin{aligned} S_{1} & =& \frac{1}{2} \int d^{D} x \sqrt{-g} \, T^{MN}_{brane} \, g_{MN} \nonumber \\ & =& \frac{1}{2 \kappa^{2}} \int d^{D} x \sqrt{-g} [ (1- \frac{D}{2} )(R- \frac{1}{2}(\partial \phi)^{2}) - \frac{1}{2 d!} ( 1 - \frac{D}{2(d+1)}) F^{2} e^{- \alpha (d)\phi}] \end{aligned}$$ Plug (\[300\]) into (\[400\]), we get $$\label{f} S_{1}+S_{2}= \frac{2}{d} S_{1}$$ From (\[600\]), $$\begin{aligned} S_{3} & =& - \frac{1}{d!} \int d^{D} x \sqrt{-g} \, J^{M_1 \cdots M_d}_{brane} A_{M_{1}M_{2}\cdots M_{d}} \nonumber \\ & =& \frac{1}{2 \kappa^{2}} \int d^{D} x \frac{1}{(d+1)!} \sqrt{-g}e^{- \alpha (d)\phi} F^{2}\nonumber \\ &&- \frac{1}{d!} \partial_{M} (\sqrt{-g}e^{- \alpha (d)\phi} A_{M_{1}M_{2}\cdots M_{d}} F^{M M_{1}\cdots M_{d}}) \end{aligned}$$ As a result, $$\begin{aligned} S_{brane} & =& S_{1}+ S_{2} + S_{3} \nonumber \\ & =& \frac{2-D}{d} \frac{1}{2 \kappa^{2}} \int d^{D} x \sqrt{-g} [ R - \frac{1}{2} (\partial \phi)^{2} - \frac{1}{2(d+1)!} e^{- \alpha (d)\phi} F^{2}_{d+1}] \nonumber \\ && - \frac{1}{2 \kappa^{2}} \int d^{D} x \frac{1}{d!} \partial_{M} (\sqrt{-g}e^{- \alpha (d)\phi} A_{M_{1}M_{2}\cdots M_{d}} F^{M M_{1}\cdots M_{d}}) \nonumber \\ & =& \frac{2-D}{d} S_{gravity}+S_{boundary}\end{aligned}$$ For $d<3$, or equivalently, for purely electric branes, $S_{boundary}=0$. If $M_{D}$ has no boundary, $S_{boundary}$ could also be dropped. Then for the given brane configuration, if $S$ is on-shell with respect to supergravity, we have $$\label{700} S_{brane} : S_{gravity} : S = (D-2) : (-d) : (D-2-d),$$ where $$\begin{aligned} S_{brane} & =& T_{d} \int d^{d}\xi [- \sqrt{-\det (\partial_{i} X^{M} \partial_{j} X^{N} g_{MN} e^{\alpha (d)\phi/d})} \nonumber \\ & & - \frac{1}{d!}\epsilon^{i_{1}i_{2} \cdots i_{d}} \partial_{i_{1}} X^{M_{1}} \partial_{i_{2}} X^{M_{2}} \cdots \partial_{i_{d}} X^{M_{d}} A_{M_{1}M_{2}\cdots M_{d}}] \end{aligned}$$ The extension to multi-brane configurations is straightforward, and (\[700\]) still holds. When the dimensions of the branes are different, the exact proportional relation is not valid anymore. Besides, when $F_{ij}$ does not vanish, i.e. the $(p-1) $-brane carries $p-1-2n$ charge, (\[700\]) does not hold. Naively, when $D = 11$, $d=6$, neglecting the boundary term, $S_{brane} : S_{gravity} : S = 3 : (-2) : 1$[^5]. However, $S$ here is not exactly the action for M5 branes coupling with supergravity. In the following, we will use (\[a\]) as the action to get the same conclusion. Now, consider $$S=S_{g}+S_{M5}$$ with $$S_{g} = \frac{1}{2 \kappa^{2}}\int_{M_{11}} *R-\frac{1}{2} *\hat{F}_{4}\wedge \hat{F}_{4} - \frac{1}{6} F_{4}\wedge F_{4} \wedge A_{3}$$ $$\begin{aligned} S_{M5} &=& - T_{5}\int_{W_{6}} d^{6}\xi \sqrt{-\det(g_{\mu \nu}+ (i_{v_{1}} \tilde{*} h_{3})_{\mu \nu})} + \frac{1}{2} v_{1}\wedge h_{3} \wedge \tilde{*} (v_{1}\wedge \tilde{*} h_{3}) \nonumber \\ &+& \frac{T_{5}}{2} \int_{W_{6}} db_{2} \wedge A_{3} + \frac{T_{5}}{2} \int_{W_{7}} A_{3} \wedge F_{4}\end{aligned}$$ The field equations are [@a1] $$\label{d} T^{MN}_{M5} = \frac{1}{\kappa^{2}}\left[ R^{MN} - \frac{1}{2}g^{MN} R - \frac{1}{12}(\hat{F}^{M}_{4}\!_{PQL}\hat{F}^{NPQL}_{4}-\frac{1}{8}g^{MN}\hat{F}^{2}_{4}) \right]$$ $$\label{e} d* \hat{F}_{4}+ \frac{1}{2}F_{4} \wedge F_{4} = -2\kappa^{2} T_{5}( -A_{3}\wedge *J_{6} +F_{4}\wedge *G_{7} )$$ $$d \hat{F}_{4} = 2\kappa^{2}T_{5} *J_{6},$$ The vacuum expectation value of $b_{2}$ are taken to be zero, otherwise, (\[700\]) does not hold. From (\[d\]), $$\frac{1}{2} \int_{W_{6}} d^{6} \xi \sqrt{-g} \, T^{MN}_{M5} \, g_{MN} = \frac{1}{2\kappa^{2}}\int_{M_{11}} - \frac{9}{2} *R +\frac{3}{4} *\hat{F}_{4} \wedge \hat{F}_{4}$$ From (\[e\]), $$\label{g} \frac{T_{5}}{2} \int_{W_{7}} A_{3} \wedge F_{4} = \frac{1}{2\kappa^{2}}\int_{M_{11}}\frac{1}{2}*\hat{F}_{4} \wedge F_{4} + \frac{1}{4} F_{4}\wedge F_{4}\wedge A_{3}-\frac{1}{4 \kappa^{2}} \int_{\partial M_{11}} A_{3} \wedge * \hat{F}_{4}$$ We will still use the general relation (\[f\]) for M5 branes. The Nambu-Goto action for M5 branes is more involved than that for D branes or M2 branes, so the correction may exist, but that will not bring too many problems, since our main concern is the WZ term. $$S_{M5} = \frac{1}{2\kappa^{2}}\int_{M_{11}} - \frac{3}{2} *R +\frac{3}{4} *\hat{F}_{4} \wedge \hat{F}_{4} + \frac{1}{4} F_{4}\wedge F_{4}\wedge A_{3} + \frac{T_{_{5}}}{2}\int _{W_{7}} * \hat{F}_{4}-\frac{1}{4 \kappa^{2}} \int_{\partial M_{11}} A_{3} \wedge * \hat{F}_{4}$$ $$S = \frac{1}{2\kappa^{2}}\int_{M_{11}} - \frac{1}{2} *R +\frac{1}{4} *\hat{F}_{4} \wedge \hat{F}_{4} + \frac{1}{12} F_{4}\wedge F_{4}\wedge A_{3} + \frac{T_{_{5}}}{2}\int _{W_{7}} * \hat{F}_{4}-\frac{1}{4 \kappa^{2}} \int_{\partial M_{11}} A_{3} \wedge * \hat{F}_{4}$$ Because of the last two terms, the exact proportional relation does not hold. 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[^2]: Just as the (6.15) in [@a2], $*F_{4}$ could be added into the worldvolume of the M5 brane, but then an equal term will appear in the bulk. [^3]: String junctions could give $2 C^{3}_{N}$, but the extra factor $2$ is a little difficult to explain. [^4]: Here, for simplicity, the last line in (\[44\]) is denoted as the differential form, but it should be more accurately written in the form like that in (\[2wq\]) and (\[wq2\]). [^5]: Note that for D3, M2, M5 branes, $S_{brane}/S$ equals to $2$, $3/2$, and $3$, while the degrees of freedom on these branes scale as $N^{2}$, $N^{3/2}$, and $N^{3}$. This is not the coincidence. Suppose the degrees of freedom on N branes scale as $N^{\alpha}$. Also suppose that in $S_{brane}$, there is a term $T$ has the $N^{\alpha}$ scaling. Consider $N+1$ branes with large $N$, when the symmetry is broken from $SU(N+1)$ to $SU(N)\times U(1)$, $(N+1)^{\alpha}-N^{\alpha}\sim \alpha N^{\alpha-1}$ number of $T$ will enter into $S_{brane}$. On the other hand, the effective action of the system could be approximated as the action of a single brane on the background generated by the rest $N$ branes. In $S_{eff}$, one may get $N^{\alpha-1}T$. Obviously, $T$ in $S_{brane}$ and $T$ in $S_{eff}$ differ by a $\alpha$ factor. For $D3$, $T$ is $\int F_{5}$, for $M5$, $T$ is $\int A_{3}\wedge F_{4}$, while for $M2$, $T$ is obscure.

--- abstract: 'Quantum toroidal algebras are obtained from quantum affine algebras by a further affinization, and, like the latter, can be used to construct integrable systems. These algebras also describe the symmetries of instanton partition functions for 5D $\mathcal{N}=1$ supersymmetric quiver gauge theories. We consider here the gauge theories defined on an orbifold $S^1\times\mathbb{C}^2/\mathbb{Z}_p$ where the action of $\mathbb{Z}_p$ is determined by two integer parameters $(\nu_1,\nu_2)$. The corresponding quantum toroidal algebra is introduced as a deformation of the quantum toroidal algebra of $\mathfrak{gl}(p)$. We show that it has the structure of a Hopf algebra, and present two representations, called *vertical* and *horizontal*, obtained by deforming respectively the Fock representation and Saito’s vertex representations of the quantum toroidal algebra of $\glp$. We construct the vertex operator intertwining between these two types of representations. This object is identified with a $(\nu_1,\nu_2)$-deformation of the refined topological vertex, allowing us to reconstruct the Nekrasov partition function and the $qq$-characters of the quiver gauge theories.' --- KIAS - Q19005\ YITP-SB-19-18 [**New quantum toroidal algebras\ from 5D $\CN=1$ instantons on orbifolds**]{} [Jean-Emile Bourgine$^\dagger$, Saebyeok Jeong$^\ast$]{}\ [*$^\dagger$Korea Institute for Advanced Study (KIAS)*]{}\ [*Quantum Universe Center (QUC)*]{}\ [*85 Hoegiro, Dongdaemun-gu, Seoul, South Korea*]{}\ `bourgine@kias.re.kr`\ [*$^\ast$C.N. Yang Institute for Theoretical Physics*]{}\ [*Stony Brook University*]{}\ [*Stony Brook, NY 11794-3840, USA*]{}\ `saebyeok.jeong@gmail.com`\ Introduction ============ Non-perturbative dynamics of supersymmetric gauge theories is a prolific research subject in theoretical physics. Since the innovative work [@Nekrasov2002], direct microscopic studies on four-dimensional gauge theories with $\mathcal{N}=2$ supersymmetry became largely accessible, from exact computations of their partition functions on the (non-commutative) $\mathbb{C}^2$. The divergence in the partition function coming from the non-compactness of $\mathbb{C}^2$ is properly regularized by introducing the $\Omega$-background [@Nekrasov2002; @Nekrasov2003], effectively localizing the four-dimensional theory to the origin. In turn, the path integral reduces to an equivariant integration on the finite dimensional framed moduli space of non-commutative instantons, for which equivariant localization can be applied for its exact evaluation. The Nekrasov partition function has been a powerful tool to investigate the correspondences of four-dimensional $\mathcal{N}=2$ supersymmetric quiver gauge theories with other objects in mathematical physics, i.e., quantum integrable systems [@NS2009; @NS2009'; @NS2009''], two-dimensional CFTs [@Alday2010; @Wyllard2009; @Nekrasov_BPS1; @Nekrasov_BPS2; @Nekrasov_BPS3; @Nekrasov_BPS4; @Nekrasov_BPS5], flat connections on Riemann surfaces [@NRS2011; @Jeong2018], and isomonodromic deformations of Fuchsian systems [@LLNZ2013; @GIL2012; @ILT2014]. Very rich algebraic structures lie at the heart of these correspondences [@NPS]. For instance, the AGT correspondence [@Alday2010; @Wyllard2009] between Nekrasov partition functions and conformal blocks of Liouville/Toda 2D CFTs can be understood algebraically as the action of W-algebras on the cohomology of instantons moduli space [@Nakajima1997; @MO2012; @Schiffmann2012]. In this context, the W-algebra currents are coupled to an infinite Heisenberg algebra, and the total action is formulated in terms of a quantum algebra, namely the Spherical Hecke central algebra [@Schiffmann2012] (isomorphic to the affine Yangian of $\mathfrak{gl}(1)$ [@Tsymbaliuk2014; @Prochazka2015]). The coupling to an Heisenberg algebra is essential for the definition of a coalgebraic structure, thus emphasizing the underlying quantum integrability since the coproduct provides the R-matrix satisfying the celebrated quantum Yang-Baxter equation. A closely related but different connection with quantum algebras arises from the type IIB strings theory realization of the five-dimensional uplifts of 4D $\CN=2$ gauge theories, that is the 5D $\CN=1$ quiver gauge theories compactified on $S^1$. In this construction, $\CN=1$ gauge theories emerge as the low-energy description of the dynamics of 5-branes webs [@Aharony1997; @Aharony1997a]. Here, each brane carries the charges $(p,q)$, generalizing D5-branes (charge $(1,0)$) and NS5-branes (charge $(0,1)$). Their world-volume include the five-dimensional gauge theory spacetime, together with an extra line segment in the 56-plane. Individual branes’ line segments are joined by trivalent vertices, and form the $(p,q)$-branes web. Alternatively, the $(p,q)$-brane web can be seen as the toric diagram of a Calabi-Yau threefold on which topological strings can be compactified [@Leung1998]. The trivalent vertices are then identified with the (refined) topological vertex, thereby leading to a very efficient method of computing 5D Nekrasov partition functions as topological strings amplitudes [@Aganagic2005; @Iqbal2007]. Awata, Feigin and Shiraishi observed in [@Awata2011'] that a specific representation of the quantum toroidal $\mathfrak{gl}(1)$ algebra (or Ding-Iohara-Miki algebra [@Ding1997; @Miki2007]) can be associated to each edge of the $(p,q)$-branes web. The charges $(p,q)$ are identified with the values of the two central charges while the brane position define the weight of the representation. As such, the D5-branes correspond to the so-called *vertical representation* while an *horizontal representation* is associated to NS5-branes (possibly dressed by extra D5-branes).[^1] The refined topological vertex is then identified with an intertwiner between vertical and horizontal representations, that is in fact the toroidal version of the vertex operator introduced in [@Davies1992] for the quantum group $U_q(\widehat{\mathfrak{sl}(2)})$. In this way, the Nekrasov partition function is written as a purely algebraic object using the quantum toroidal algebra, just like conformal blocks with W-algebras [@Awata2016a; @Mironov2016]. This algebraic construction turns out to be useful in probing various properties of the partition function, e.g. in addressing the (q-deformed) AGT correspondence [@Awata2009; @Awata2011], or in studying strings’ S-duality [@Awata2009a; @Bourgine2018a]. In [@Nekrasov_BPS1], an important class of half-BPS observables, called *$qq$-characters*, were defined, whose characteristic property is the regularity of their gauge theory expectation values. This regularity property encodes efficiently an infinite set of constraints on the partition function called *non-perturbative Dyson-Schwinger equations* [@Nekrasov_BPS1]. The algebraic nature of these constraints was observed in [@Bourgine2015c; @Bourgine2016]. Actually, the constraints take an even more elegant form in the algebraic construction described above as they express the invariance of an operator $\CT$ under the adjoint action of the quantum toroidal algebra [@Bourgine2017b]. This operator is obtained by gluing intertwiners along the edges of the $(p,q)$-branes web, and its vacuum expectation value reproduces the 5D Nekrasov instanton partition. A natural question is how to generalize the algebraic construction to gauge theories on more complicated manifolds. Among other manifolds, the $\mathbb{Z}_p$-orbifolded $\mathbb{C}^2$ are of a particular interest, since the partition functions on these spaces can be computed by simply projecting out the contributions which are not invariant under the $\mathbb{Z}_p$-action [@Kanno2011; @Bonelli2012; @Nekrasov_BPS3]. The generalization of the algebraic construction is not entirely straightforward since it is necessary to introduce the information of the *coloring* corresponding to the $\Zp$-action of the orbifolding. In this scope, deformations of the quantum toroidal $\mathfrak{gl}(1)$ algebra must be considered. A special case of the $\mathbb{Z}_p$-orbifolded $\mathbb{C}^2$ is the (un-resolved) $A_p$-type ALE spaces. The ALE instantons were introduced by Kronheimer in [@Kronheimer1989], and the ALE instanton moduli spaces were constructed as quiver varieties in [@Kronheimer1990; @Nakajima1994]. The algebraic construction of the corresponding 5D Nekrasov partition functions has been realized recently using an underlying quantum toroidal $\mathfrak{gl}(p)$ algebra [@Awata2017]. There, the index carried by the Drinfeld currents renders the $\Zp$-coloring due to the orbifolding. Incidentally, the vertical representation of this quantum toroidal algebra should coincide with the q-deformation of the affine Yangian of $\mathfrak{gl}(p)$ acting on the cohomology of the moduli space of ALE instantons, extending by further affinization the algebraic actions discovered in [@Nakajima1994; @Nakajima1999]. In this work, we extend the algebraic construction of 5D Nekrasov partition functions to a more general $\mathbb{Z}_p$-orbifolding depending on two integer parameters $(\nu_1,\nu_2)$. We propose an extended quantum toroidal algebra relevant to the construction, and prove its Hopf algebra structure. We define both horizontal and vertical representations, and derive the vertex operator which intertwines between these representations. Finally, using these ingredients, we give an algebraic construction of Nekrasov partition functions and $qq$-characters. The orbifolds considered in this work incorporate the case of codimension-two defect insertion, whose applications to BPS/CFT correspondence, Bethe/gauge correspondence, and Nekrasov-Rosly-Shatashvili correspondence have been largely investigated [@Nekrasov_BPS4; @Nekrasov_BPS5; @Jeong2017'; @Jeong2017; @Jeong2018]. This paper is written in such a way that mathematicians interested only in the formulation of the extended algebra can focus on the reading of section three, together with the appendices (quantum toroidal $\glp$), (representations) and (automorphisms and gradings) for more details. Instead, the section two provides a brief description of the physical context in which the algebra emerges, i.e. instantons of 5D $\CN=1$ gauge theories on the spacetime $\mC^2/\Zp$. Finally, the section four is dedicated to the algebraic construction of gauge theories observables, giving the expression of the $(\nu_1,\nu_2)$-colored refined topological vertex and a few examples of application. Instantons on orbifolds ======================= We consider the moduli space of solutions to the anti-self-duality equation $F = - \star F$ defined on the orbifold $\mathbb{C}^2/\mathbb{Z}_p$. The gauge field is allowed to be singular along the orbifold singularity, and its singular behavior determines how the gauge group is broken on the singularity and how the gauge coupling fractionalizes. For the purpose of computing the instanton partition function, it is convenient to encode these additional data directly in the ADHM construction, following [@Kanno2011; @Bonelli2012] for example. Action of the abelian group $\mathbb{Z}_p$ on the ADHM data ----------------------------------------------------------- In order to derive the group action on the instanton moduli space, we focus first on the case of a pure $U(m)$ gauge theory. In this case, the ADHM construction of the moduli space [@Atiyah1978] involves only two vector spaces $M$ and $K$ of dimension $m$ and $k$ respectively, where $k$ is the instanton number. Introducing the four matrices $B_1,B_2: K\to K$ $I:M\to K$ and $J: K\to M$, the instanton moduli space is identified with the quiver variety (see, for instance, [@Nakajima2003]) $$\CM_k=\{B_1,B_2,I,J\diagup[B_1,B_2]+IJ=0,\mathbb{C}[B_1;B_2]I(M)=K\}\diagup\text{GL}(K).$$ The complexified global symmetry group $\text{GL}(M)\times\text{SL}(2,\mathbb{C})^2$ acts on the ADHM matrices, preserving the quiver variety $\CM_k$. It contains an $(m+2)$-dimensional torus that acts follows, $$\label{torus_action} (B_1,B_2,I,J)\to (t_1B_1,t_2B_2,It,t^{-1}Jt_1t_2),\quad (t,t_1,t_2)\in (\mathbb{C}^\times)^m\times (\mathbb{C}^\times)^2.$$ The fixed points of this action parameterize the configurations of instantons with total charge $k$. They are in one-to-one correspondence with the $m$-tuple partitions $\bl=(\l^{(1)},\cdots\l^{(m)})$ of the integer $k$, here identified with the $m$-tuple Young diagrams with $|\bl|=k$ boxes. At the fixed point, the vector space $K$ is decomposed into $$\label{basis_K} K=\bigoplus_{\a=1}^m\bigoplus_{(i,j)\in\l^{(\a)}}B_1^{i-1}B_2^{j-1}I(M_\a),$$ where $M_\a$ denotes the one-dimensional vector spaces generated by the basis vectors of $M$. Thus, each box $\Abox=(\a,i,j)$ of the $m$-tuple partition $\bl$ with coordinate $(i,j)\in\bl^{(\a)}$ corresponds to a one-dimensional vector space $B_1^{i-1}B_2^{j-1}I(M_\a)$. We further associate to the box $\Abox$ the complex variable $\phi_{\sAbox}=a_\a+(i-1)\e_1+(j-1)\e_2$ called *instanton position* or, sometimes, the *box content of* $\Abox$. The parameters $a_1,\cdots,a_m$ are the *Coulomb branch vevs* of the gauge theory. We also define the exponentiated quantities $v_\a=e^{R a_\a}$, $(q_1,q_2)=(e^{R\e_1},e^{R\e_2})$ and $\chi_{\sAbox}=e^{R\phi_{\sAbox}}=v_\a q_1^{i-1}q_2^{j-1}$. In this paper, gauge theories are considered on the 5D orbifolded omega-background $S_R^1\times(\mathbb{C}_{\e_1}\times\mathbb{C}_{\e_2})/\mathbb{Z}_p$ where $\Zp=\mZ/p\mZ$ is a subgroup of the torus ${U}(1)^2 \subset {SO}(4)$. The action of the group $\Zp$ on the spacetime is parameterized by two integers $\nu_1,\nu_2$, $$\label{Zp_action} (\th,z_1,z_2)\in S_R^1\times\mathbb{C}_{\e_1}\times\mathbb{C}_{\e_2} \to(\th,e^{2i\pi\nu_1/p}z_1,e^{2i\pi\nu_2/p}z_2),\quad\text{with}\quad(\nu_1,\nu_2)\in\mathbb{Z}_p\times\mathbb{Z}_p.$$ Furthermore, it is possible to combine it with a global gauge transformation in the subgroup $U(1)^m\subset U(m)$. As a result, we obtain an action of $\mZ_p$ on the ADHM data by specialization of the $(m+2)$-torus action \[torus\_action\], taking $$t=\text{diag}(e^{2i\pi c_\a/p})_{\a =1, \cdots, m},\quad t_1=e^{2i\pi\nu_1/p},\quad t_2=e^{2i\pi\nu_2/p}.$$ This action of the abelian group $\mZ_p$ is parameterized by the $m+2$ integers $(c_\a,\nu_1,\nu_2)$ considered modulo $p$. The transformation of the vector spaces in the decomposition \[basis\_K\] of $K$ leads to associate to each box $\Abox=(\a,i,j)\in\bl$, in addition to the complex variables $\phi_{\sAbox}$ and $\chi_{\sAbox}$, the integer $c(\Abox)$ such that $$B_1^{i-1}B_2^{j-1}I(M_\a)\to e^{2i\pi c(\sAbox)/p}B_1^{i-1}B_2^{j-1}I(M_\a),\quad\text{with}\quad c(\Abox)=c_\a+(i-1)\nu_1+(j-1)\nu_2\in\mZ_p.$$ We call *color* any integer parameter defined modulo $p$. For short, we also say that $c_\a$ and $\nu_1,\nu_2$ are respectively color of the Coulomb branch vevs, and of the parameters $q_1,q_2$. The map $c:\bl\to\mZ_p$ defines a coloring of the $m$-tuple partitions $\bl$, and $K$ has a natural decomposition into sectors of a given color $c(\Abox)=\o$, $$K=\bigoplus_{\o\in\mZ_p}K_\o(\bl).$$ #### Notations We denote $C_\o(m)$ the subset of $[\![1,m]\!]$ such that the Coulomb branch vevs $a_\a$ (or $v_\a$) with $\a\in C_\o(m)$ have color $c_\a=\o$ (or, equivalently, that the box $(1,1)\in\l^{(\a)}$ with $\a\in C_\o(m)$ is of color $c(\a,1,1)=c_\a=\o$). Similarly, $K_\o(\bl)$ denotes the set of boxes $\Abox\in\bl$ of the $m$-tuple colored partition $\bl$ that carry the color $c(\Abox)=\o$. Besides, in the generic case $\nu_1+\nu_2\neq0$, the shift of color indices $\o$ by the quantity $\nu_1+\nu_2$ appears in many formulas. To simplify these expressions, we introduce the notation $\bar\o=\o+\nu_1+\nu_2$ for the shifted indices, along with the map $\bar c(\Abox)=c(\Abox)+\nu_1+\nu_2$. Finally, we also introduce the extra variables $q_3$ and $\nu_3$ such that $q_1q_2q_3=1$ and $\nu_1+\nu_2+\nu_3=0$. Due to the fact that the $\Zp$-action coincides with a subgroup of the torus action, in all formulas the shift of color indices $\o+\nu_i$ coincide with a factor $q_i$ multiplying the parameters associated to instanton positions in the moduli space. #### McKay subgroups in $SO(4)$ Although we are considering here a different problem, it is interesting to make a short parallel with the action of $SU(2)_L\times SU(2)_R\subset SO(4)$ on the omega-background (see, for instance, [@Johnson1996]). This action takes a simpler form if we employ the quaternionic coordinates $$Z=\left({\begin{smallmatrix} z_1 & -\bz_2\\ z_2 & \bz_1 \end{smallmatrix}}\right),\quad (z_1,z_2)\in\mathbb{C}_{\e_1}\times\mathbb{C}_{\e_2}.$$ Then the $2\times2$ matrices $(G_L,G_R)\in SU(2)_L\times SU(2)_R$ act on the quaternions as $Z\to G_LZG_R$. The McKay subgroups of $SU(2)$ possess an ADE-classification. For instance, the $A_{p-1}$ series corresponds to the action of $\mathbb{Z}_p$, it is generated (multiplicatively) by the diagonal matrices $$G=\left({\begin{smallmatrix} e^{2i\pi/p} & 0\\ 0 & e^{-2i\pi/p} \end{smallmatrix}}\right).$$ Considering only the action of the $A_{p-1}$ subgroup on the left, the background coordinates transform as $(z_1,z_2)\to(e^{2i\pi/p}z_1,e^{-2i\pi/p}z_2)$. This transformation can be recovered from the action \[Zp\_action\] of $\mathbb{Z}_p$ by choosing $\nu_1=-\nu_2=1$. The orbifold of the spacetime under this action of $\mathbb{Z}_p$ reproduces the ALE space constructed in [@Kronheimer1989]. Instantons of $\CN=1$ gauge theories defined on ALE spacetimes have been extensively studied [@Kronheimer1989; @Kronheimer1990; @Nakajima1994; @Nakajima1999]. In [@Awata2017], their contributions to the gauge theory partition functions have been reproduced using algebraic techniques based on the quantum toroidal algebra of $\glp $. The generalization to DE-type McKay subgroups with only left action is expected to involve quantum toroidal algebras based on either $\mathfrak{so}(p)$ or $\mathfrak{sp}(p)$ Lie algebras [@Nakajima1999]. It is also possible to consider simultaneously the action of two McKay subgroups $A_{p_1-1}$ and $A_{p_2-1}$, with one acting on the left, the other on the right. As a result, coordinates now transform as $$(z_1,z_2)\to (e^{2i\pi(p_1+p_2)/(p_1p_2)}z_1,e^{2i\pi(p_1-p_2)/(p_1p_2)}z_2).$$ We recognize here another particular case of the $\mathbb{Z}_p$-action defined in \[Zp\_action\], albeit more general than before. It is simply obtained by the specialization $\nu_1=p_1+p_2$, $\nu_2=p_1-p_2$ and $p=p_1p_2$. Thus, the action \[Zp\_action\] leads to a particularly rich context. Moreover, taking $\nu_1=0$, the first coordinate $z_1$ is invariant and the orbifolded spacetime can be reinterpreted as the insertion of a codimension-two defect in a 5D omega background with no orbifold [@Nekrasov_BPS4; @Kanno2011]. We build here a general algebraic framework to address this kind of problems. It may be possible to further generalize our approach to the action of DE-type McKay subgroups with both left and right actions, but this is beyond the scope of this paper. Instantons partition function ----------------------------- The computation of the Nekrasov instanton partition function on such $\Zp$-orbifolds has been performed in [@Kanno2011; @Bonelli2012; @Nekrasov_BPS3].[^2] For simplicity, we do not introduce fundamental matter multiplets, those being obtained in the limit $\qf\to0$ of the gauge coupling parameters. Furthermore, we only discuss the case of linear quiver gauge theories $A_r$, with $U(m^{(i)})$ gauge groups at each node $i=1\cdots r$. Thus, the node $i$ carries the following parameters: - a set of colored exponentiated gauge couplings $\qf_{\o,i}$, - a $p$-vector of colored Chern-Simons levels $\boldsymbol{\k}^{(i)}=(\k_\o^{(i)})_{\o\in\Zp}$, - an $m^{(i)}$-vector of Coulomb branch vevs $\boldsymbol{a}^{(i)}=(a_\a^{(i)})_{\a=1}^{m^{(i)}}$ defining the exponentiated parameters $\bv^{(i)}=(v_\a^{(i)})_{\a=1}^{m^{(i)}}$ with $v_\a^{(i)}=e^{R a_\a^{(i)}}$, - an associated vector of colors $\boldsymbol{c}^{(i)}=(c_\a^{(i)})_{\a=1}^{m^{(i)}}$. In addition, each link $i\to j$ between two nodes $i$ and $j$ represent a chiral multiplet of matter fields in the bifundamental representation of the gauge group $U(m^{(i)})\times U(m^{(j)})$, with mass $\mu_{ij}\in\mathbb{C}$. For linear quivers, all bifundamental masses can be set to $q_3^{-1}$ by a rescaling of the Coulomb branch vevs. The instantons contribution to the gauge theories partition function is expressed as a sum over the content of $r$ $m^{(i)}$-tuple Young diagrams $\bl^{(i)}$ describing the configuration of instantons at the $i$th node. Each term can be further decomposed into the contributions of vector (gauge) multiplets, bifundamental chiral (matter) multiplets, and Chern-Simons factors: $$\begin{aligned} \label{def_Zv_Zbf} \begin{split} &\Zinst=\sum_{\bl^{(i)}}\prod_{i=1}^{r}\left(\prod_{\o\in\Zp}\qf_{\o,i}^{|K_\o(\bl^{(i)})|}\Zv(\bv^{(i)},\bl^{(i)})\ZCS(\boldsymbol{\k}^{(i)},\bl^{(i)})\right)\prod_{i\to j}\Zbf(\bv^{(i)},\bl^{(i)},\bv^{(j)},\bl^{(j)}|\mu_{ij}),\\ &\text{with}\quad \Zv(\bv,\bl)=N(\bv,\bl|\bv,\bl)^{-1},\quad \Zbf(\bv,\bl,\bv',\bl'|\mu)=N(\bv,\bl|\mu\bv',\bl'),\quad \ZCS(\boldsymbol{\k},\bl)=\prod_{\mAbox\in\bl}\chi_{\sAbox}^{\k_{c(\sAbox)}}. \end{split}\end{aligned}$$ Vector and bifundamental contributions are written in terms of the Nekrasov factor $N(\bv,\bl|\mu\bv',\bl')$. For a better readability, we drop the node indices in the following, and simply distinguish the two nodes involved in the definition of the Nekrasov factor with a prime. In order to write down the expression of $N(\bv,\bl|\mu\bv',\bl')$ given in [@Nekrasov_BPS3], we need to introduce the equivariant character $M_\bv$ and $K_\bl$ of the vector spaces $M$ and $K$ associated to each node, $$\label{equiv_char} M_{\bv}=\sum_{\a=1}^m e^{Ra_\a},\quad K_\bl=\sum_{\mAbox\in\bl}e^{R\phi_{\sAbox}},$$ A linear involutive operation $\ast$ acts on such characters by flipping the sign of $R$: $(e^{R a_\a})^\ast=e^{-Ra_\a}$, $(q_1^\ast,q_2^\ast)=(q_1^{-1},q_2^{-1})$ and thus $(e^{R\phi_{\sAbox}})^\ast=e^{-R\phi_{\sAbox}}$ (see [@Nekrasov_BPS1; @Nekrasov_BPS2; @Nekrasov_BPS3] for more details on these notations). Introducing $S_\bl=M-P_{12}K_\bl$ with $P_{12}=(1-q_1)(1-q_2)$, the Nekrasov factor writes $$\label{expr_N} N(\bv,\bl|\bv',\bl')=\mathbb{I} \left[\dfrac{M_{\bv}M_{\bv'}^\ast-S_{\bl}S_{\bl'}^\ast}{P_{12}^\ast}\right]^{\mathbb{Z}_p} = \mathbb{I} \left[ M_{\bv} K_{\bl'} ^* + q_{3} ^{-1} M_{\bv'} ^* K_{\bl} -P_{12} K_{\bl} K_{\bl'}^* \right]^{\mathbb{Z}_p},$$ where the $\mathbb{I}$-symbol is the equivariant index functor, $$\mathbb{I}\left[\sum_{i\in I_+}e^{R w_i}-\sum_{i\in I_-}e^{R w_i}\right]=\dfrac{\prod_{i\in I_+} 1-e^{R w_i}}{\prod_{i\in I_-} 1-e^{R w_i}},$$ and $\left[ \cdots \right]^{\mathbb{Z}_p}$ denotes the operation of keeping only the $\mathbb{Z}_p$-invariant parts. In particular, the RHS of involves a coloring function $c:\mZ[a_\a,\e_1,\e_2]\to\Zp$ defined on weights $w_i$ as the linear map taking the values $c(a_\a)=c_\a$, $c(\e_1)=\nu_1$ and $c(\e_2)=\nu_2$ so that $c(\phi_{\sAbox})=c(\Abox)$ (justifying our slight abuse of notations). The $[\cdots]^{\mathbb{Z}_p}$ projects on $\Zp$-invariant factors. Replacing the equivariant characters by their expressions \[equiv\_char\], the Nekrasov factor can be written in a more explicit form, $$\label{Nekrasov} N(\bv,\bl|\bv',\bl')=\prod_{{\genfrac{}{}{0pt}{}{\mAbox\in\bl}{\mAboxF\in\bl'}}}S_{c(\sAbox)c(\sAboxF)}(\chi_{\sAbox}/\chi_{\sAboxF})\times\prod_{\mAbox\in\bl}\prod_{\a\in C_{\bar c(\sAbox)}(m')}\left(1-\dfrac{\chi_{\sAbox}}{q_3v_{\a}'}\right)\times \prod_{\mAbox\in\bl'}\prod_{\a\in C_{c(\sAbox)}(m)}\left(1-\dfrac{v_\a}{\chi_{\sAbox}}\right).$$ The function $S_{\o\o'}(z)$ is sometimes called the *scattering function*, it carries two color indices $\o,\o'$: $$\label{def_S} S_{\o\o'}(z)=\dfrac{(1-q_1z)^{\d_{\o,\o'-\nu_1}}(1-q_2z)^{\d_{\o,\o'-\nu_2}}}{(1-z)^{\d_{\o,\o'}}(1-q_1q_2z)^{\d_{\o,\o'-\nu_1-\nu_2}}}.$$ In this expression, the non-zero matrix elements have been expressed in a compact way using the delta function $\d_{\o,\o'}$ defined modulo $p$ (i.e. $\d_{\o,\o'}=1$ iff $\o=\o'$ modulo $p$, zero otherwise). In fact, $S_{\o\o'}(z)$, and more generally all the matrices of size $p\times p$ with indices $\o,\o'$ appearing in this paper, are circulant matrices: their matrix elements only depend on the difference $\o-\o'$ of row and column indices. In particular, $S_{\o+\nu\ \o'}(z)=S_{\o\ \o'-\nu}(z)$ for all $\nu\in\mathbb{Z}_p$. Finally, the function $S_{\o\o'}(z)$ satisfies a sort of *crossing symmetry*, $$\label{crossing} S_{\o\bar\o'}(q_3/z)=f_{\o\o'}(z)S_{\o'\o}(z),$$ with the function $f_{\o\o'}(z)=F_{\o\o'}z^{\b_{\o\o'}}$ defined by[^3] $$\label{def_beta} \b_{\o\o'}=\d_{\o\o'}+\d_{\o\o'+\nu_1+\nu_2}-\d_{\o\o'+\nu_1}-\d_{\o\o'+\nu_2},\quad F_{\o\o'}=(-1)^{\d_{\o\o'}}(-q_3)^{-\d_{\o,\o'-\nu_3}} (-q_1)^{-\d_{\o\o'+\nu_1}}(-q_2)^{-\d_{\o\o'+\nu_2}}.$$ $\CY$-observables ----------------- A new class of BPS-observables for supersymmetric gauge theories was introduced in [@Nekrasov_BPS1], they are called *$qq$-characters*. As the name suggests, they correspond to a natural deformation of the $q$-characters of Frenkel-Reshetikhin [@Frenkel1998] from the gauge theory point of view [@NPS]. They were defined in [@Nekrasov_BPS1] as particular combinations of chiral ring observables in such a way that their expectation values exhibit an important regularity property [@Nekrasov_BPS1; @Nekrasov_BPS2]. This regularity property encodes an infinite set of constraints called *non-perturbative Dyson-Schwinger equations*. From a different viewpoint, $qq$-characters in 5D gauge theories can also be defined in terms of Wilson loops [@Kim2016] (see also [@Kimura2017] for a string theory perspective).[^4] The $qq$-characters are half-BPS observables written as combinations of *$\CY$-observables*. In the case of a $\Zp$-orbifold, it is natural to introduce two inequivalent $\CY$-observables $\CYY(z)=\mathbb{I}\left[e^{-R\z}S_\bl \right]^{\mathbb{Z}_p}$ and $\CY_\o^{[\bl]\ast}(z)=\mathbb{I}\left[e^{R\z}S_\bl^\ast \right]^{\mathbb{Z}_p}$ where $z=e^{R\z}$ and $c(\z)=\o$. These two observables encode the recursion relations satisfied by Nekrasov factors, $$\label{rec_N} \dfrac{N(\bv,\bl|\bv',\bl'+\Abox)}{N(\bv,\bl|\bv',\bl')}=\CY_{c(\sAbox)}^{[\bl]}(\chi_{\sAbox})\quad\dfrac{N(\bv,\bl+\Abox|\bv',\bl')}{N(\bv,\bl|\bv',\bl')}=\CY_{\bc(\sAbox)}^{[\bl']\ast}(q_3^{-1}\chi_{\sAbox}),$$ Replacing the equivariant characters with the expressions \[equiv\_char\], we find the explicit formulas $$\label{def_Y} \CYY(z)=\prod_{\a\in C_\o(m)}(1-v_\a/z)\times\prod_{\mAbox\in\bl}S_{c(\sAbox)\o}(\chi_{\sAbox}/z),\quad \CY_\o^{[\bl]\ast}(z)=\prod_{\a\in C_\o(m)}(1-z/v_\a)\prod_{\mAbox\in\bl}S_{\o\bc(\sAbox)}(q_3z/\chi_{\sAbox}).$$ Due to the crossing symmetry \[crossing\], these two $\CY$-observables satisfy the relation $$\begin{aligned} \label{eq:yandy} \CY_\o^{[\bl]\ast}(z)=f\oY(z)\CY\oY(z),\end{aligned}$$ with $$\label{eq:f} \fYY(z)=\prod_{\a\in C_\o(m)}(-z/v_\a)\prod_{\mAbox\in\bl}f_{\o c(\sAbox)}(\chi_{\sAbox}/z).$$ It will allow us to express all the equations below in terms of $\CYY(z)$ only.[^5] Furthermore, the $\CY$-observables possess an alternative expression following from the *shell formula* derived in appendix , $$\label{shell_Y} \CYY(z)=\dfrac{\prod_{\mAbox\in A_\o(\bl)}(1-\chi_{\sAbox}/z)}{\prod_{\mAbox\in R_{\o-\nu_1-\nu_2}(\bl)}(1-\chi_{\sAbox}/(q_3z))},\quad \fYY(z)=\dfrac{\prod_{\mAbox\in R_{\o-\nu_1-\nu_2}(\bl)}(-\chi_{\sAbox}/(q_3z))}{\prod_{\mAbox\in A_\o(\bl)}(-\chi_{\sAbox}/z)}.$$ Here, the sets $A_\o(\bl)$ and $R_\o(\bl)$ denote respectively the set of boxes of color $\o$ that can be added to or removed from the $m$-tuple Young diagram $\bl$. This expression arises from the cancellations of contributions by neighboring boxes, it plays an essential role in the definition of the vertical representation of the algebra. New quantum toroidal algebras ============================= In order to reconstruct the instanton partition functions on the general orbifold \[Zp\_action\], the definition of a new quantum toroidal algebra is necessary. In addition to the complex parameters $q_1,q_2$ and the rank $p\in\mZ^{>0}$, this algebra will depend on the integers $(\nu_1,\nu_2)$ modulo $\Zp$. Taking $\nu_1=-\nu_2=1$, the $\Zp$-action \[Zp\_action\] reduces to the standard action defining ALE spaces. Thus, in this limit the $(\nu_1,\nu_2)$-deformed algebra should reduce to the quantum toroidal algebra of $\glp$. In fact, this is true only up to a twist in the definition of the Drinfeld currents (see the appendix \[AppE\]). A brief reminder on the quantum toroidal algebra of $\glp$ is given in appendix , it includes its two main representations called, in the gauge theory context, *vertical* and *horizontal* representations. The key ingredient to define the deformation of the quantum toroidal algebra of $\glp$ is the scattering function $S_{\o\o'}(z)$ defined in \[def\_S\]. Indeed, this function plays an essential role in the two elementary representations involved in the algebraic engineering of partition functions and $qq$-characters. In the vertical representation, it enters through the definition \[def\_Y\] of the $\CY$-observables that describe the recursion relations among Nekrasov factors. Instead, in the horizontal representation, it expresses the normal-ordering relations between vertex operators. Thus, from the physics perspective, the scattering function is the natural object to consider for the deformation of the algebra. Moreover, through the crossing symmetry relation \[crossing\], this function defines the $p\times p$ matrix $\b_{\o\o'}$ that could be identified with the underlying Cartan matrix of the deformed quantum toroidal algebra (see the subsection ). Note that the matrix $\b_{\o\o'}$ naturally reduces to the generalized Cartan matrix of the Kac-Moody algebra $\widehat{\glp}$ when $\nu_1=-\nu_2=1$. In general, it is non-symmetrizable, yet, like in the case of $\widehat{\glp}$, it is a circulant matrix. Its eigenvectors $v_j=(1,\O_j,\O_j^2,\cdots,\O_j^{p-1})$ are written in terms of the $p$th root of unity $\O_j=e^{2i\pi j/p}$, and the corresponding eigenvalues read $$\l_j=-4e^{i\pi\nu_3 j/p}\sin(\pi\nu_1 j/p)\sin(\pi\nu_2 j/p).$$ In particular, the eigenvector $v_0=(1,1,\cdots,1)$ has the eigenvalue zero which relates $\b_{\o\o'}$ to the Cartan matrix of affine Lie algebras, and thus justifies the designation *toroidal* of the deformed algebra. Definition of the algebra ------------------------- Like in the case of $\glp$, the $(\nu_1,\nu_2)$-deformed quantum toroidal algebra is defined in terms of a central element $c$ and $4p$ Drinfeld currents, denoted $x_\o^\pm(z)$ and $\psi_\o^\pm(z)$, with $\o\in\Zp$. The currents $\psi_\o^\pm(z)$ (together with $c$) generate the Cartan subalgebra, while the currents $x_\o^\pm(z)$ deform the notion of Chevalley generators $e_\o,f_\o$. The algebraic relations obeyed by the currents resemble those defining the quantum toroidal algebra of $\glp$ in \[algebra\_glp\], the main difference being the presence of shifts in the indices $\o$ by the product $\nu_3 c$:[^6][^7] $$\begin{aligned} \label{algebra} \begin{split} &x_\o^\pm(z)x_{\o'}^\pm(w)=g_{\o\o'}(z/w)^{\pm1}x_{\o'}^\pm(w)x_\o^\pm(z),\quad \psi^+_\o(z)x^\pm_{\o'}(w)=g_{\o\o'}(z/w)^{\pm1}x^\pm_{\o'}(w)\psi^+_\o(z),\\ &\psi^-_\o(z)x^+_{\o'}(w)=g_{\o-\nu_3c\ \o'}(q_3^{-c}z/w)x^+_{\o'}(w)\psi^-_\o(z),\quad \psi^-_\o(z)x^-_{\o'}(w)=g_{\o\o'}(z/w)^{-1}x^-_{\o'}(w)\psi^-_\o(z),\\ &\psi_\o^+(z)\psi_{\o'}^-(w)=\dfrac{g_{\o\o'-\nu_3c}(q_3^{c} z/w)}{g_{\o\o'}(z/w)}\psi_{\o'}^-(w)\psi_\o^+(z),\quad [\psi_\o^\pm(z),\psi_{\o'}^\pm(w)]=0,\\ &[x_\o^+(z),x_{\o'}^-(w)]=\kO\left[\d_{\o,\o'}\d(z/w)\psi_\o^+(z)-\d_{\o,\o'-\nu_3c}\d(q_3^{c} z/w)\psi^-_{\o+\nu_3c}(q_3^{c}z)\right]. \end{split}\end{aligned}$$ In the last relation $\d(z)=\sum_{k\in\mZ}z^k$ denotes the multiplicative Dirac delta function and we introduced the complex parameter $$\kO=\dfrac{(1-q_1)^{\d_{\nu_1,0}}(1-q_2)^{\d_{\nu_2,0}}}{(1-q_1q_2)^{\d_{\nu_1+\nu_2,0}}}F^{1/2},\quad F=F_{\o\o}=-\prod_i(-q_i)^{-\d_{\nu_i,0}}.$$ The other relations in \[algebra\] involve the *structure function* $g_{\o\o'}(z)$ defined as a ratio of two scattering functions. This function depends on the variables $(q_1,q_2)\in\mathbb{C}^\times\times\mathbb{C}^\times$ and the integers $(\nu_1,\nu_2)\in\Zp\times\Zp$: $$\label{def_g} g_{\o\o'}(z)=\dfrac{S_{\o\o'}(z)}{S_{\o'\o}(z^{-1})}=f_{\o\o'}(z^{-1})\prod_{i=1,2,3}\dfrac{(1-q_iz)^{\d_{\o,\o'-\nu_i}}}{(1-q_i^{-1}z)^{\d_{\o,\o'+\nu_i}}},$$ where the extra variables $q_3$ and $\nu_3$ obey $q_1q_2q_3=1$ and $\nu_1+\nu_2+\nu_3=0$. Note that the invariance under the $S_3$-permutation of indices $(\nu_i,q_i)$ is broken to $S_2$ corresponding to exchange $(\nu_1,q_1)$ and $(\nu_2,q_2)$. The structure function satisfies the property $g_{\o\o'}(z)g_{\o'\o}(z^{-1})=1$ necessary for the definiteness of the algebraic relations. The algebraic relations \[algebra\] are expected to include additional Serre relations. However, the Drinfeld currents employed here are a twisted version of those used in the formulation of the quantum toroidal algebra of $\glp$. This explains why the function $g_{\o\o'}(z)$ defined in \[def\_g\] does not quite reproduce the $\glp$ structure function \[def\_g\_glp\] as we set $\nu_1=-\nu_2=1$. Even in the case of $\glp$, the twist of the currents make the derivation of Serre relations difficult. We hope to come back to this question in the near future. Due to the non-trivial power of $z$ in the asymptotics of the functions $g_{\o\o'}(z)$, namely $$g_{\o\o'}(z){\genfrac{}{}{0pt}{3}{\sim}{0}}f_{\o\o'}(z^{-1}),\quad g_{\o\o'}(z){\genfrac{}{}{0pt}{3}{\sim}{\infty}}f_{\o'\o}(z)^{-1},$$ the Cartan currents $\psi_\o^\pm(z)$ cannot be expanded in powers of $z^{\mp k}$ with $k>0$ as it is usually the case for quantum groups. Instead, it is necessary to introduce a *zero modes* part using extra operators $a_{\o,0}^\pm$: $$x_\o^\pm(z)=\sum_{k\in\mathbb{Z}}z^{-k}x_{\o,k}^\pm,\quad \psi_\o^\pm(z)=\psi_{\o,0}^\pm z^{\mp a_{\o,0}^\pm}\exp\left(\pm\sum_{k>0}z^{\mp k}a_{\o,\pm k}\right).$$ In the appendix \[AppD\], the operators $\psi_{\o,0}z^{\mp a_{\o,0}^\pm}$ are constructed as a specific combination of grading operators. The Cartan zero modes $\psi_{\o,0}^\pm$ are invertible, they can be used to define another central element $\bc$ setting $$\label{def_bc} q_3^{-\bc}=\left(\prod_{\o\in\Zp}\psi_{\o,0}^+\right)\left(\prod_{\o\in\Zp}\psi_{\o,0}^-\right)^{-1}=\prod_{{\genfrac{}{}{0pt}{}{\o,\o'=0}{\o\leq\o'}}}^{p-1}\dfrac{F_{\o'\o}}{F_{\o'\o+\nu_3c}}\prod_{\o\in\Zp}\psi_{\o,0}^+(\psi_{\o,0}^-)^{-1}.$$ Note that the ordering of the zero modes is important since they do not commute. It is chosen here such that the expression of the coproduct defined below simplifies. #### Coalgebraic structure A Hopf algebra $\CA$ over the field $\mC$ is a $\mC$-module equipped with a unit $1_\CA$, a product $\nabla$, a counit $\e$, a coproduct $\D$ and an antipode $S$ satisfying the following properties [@Chari1995]. - $\CA$ is both an algebra and a coalgebra. This implies the property $\nabla(1\otimes\e)\D=\nabla(\e\otimes1)\D=1$ and the coassociativity of the coproduct $(1\otimes\D)\D=(\D\otimes1)\D$. - The counit $\e:\CA\to\mC$ and the coproduct $\D:\CA\to\CA\otimes\CA$ are homomorphisms of algebras. The compatibility with the scalar multiplication and the addition are trivially satisfied. On the other hand, the compatibility with the product requires to verify $\e(ee')=\e(e)\e(e')$ and $\D(e)\D(e')=\D(ee')$ for any two elements $e,e'\in\CA$. - The unit $1_\CA:\mC\to\CA$ and product $\nabla:\CA\otimes\CA\to\CA$ are homomorphisms of coalgebras. This means $\D(1_\CA)=1_\CA\otimes 1_\CA$, $\e(1_\CA)=1$, and, once again, $\D(e)\D(e')=\D(ee')$. - The antipode $S:\CA\to\CA$ is a bijective $\mC$-module map satisfying $\nabla(S\otimes1)\D=\e=\nabla(1\otimes S)\D$. The algebra \[algebra\] is a Hopf algebra with the coproduct, counit and antipode given by $$\begin{aligned} \begin{split}\label{Drinfeld_coproduct} &\D(x_\o^+(z))=x_\o^+(z)\otimes 1+\psi_{\o+\nu_3 c_{(1)}}^-(q_3^{c_{(1)}}z)\otimes x_\o^+(z),\\ &\D(x_\o^-(z))=x_\o^-(z)\otimes \psi_{\o-\nu_3c_{(1)}}^+(q_3^{-c_{(1)}}z)+1\otimes x_{\o-\nu_3c_{(1)}}^-(q_3^{-c_{(1)}}z),\\ &\D(\psi_\o^+(z))=\psi_\o^+(z)\otimes\psi_{\o-\nu_3 c_{(1)}}^+(q_3^{-c_{(1)}}z),\quad \D(\psi_\o^-(z))=\psi_{\o-\nu_3c_{(2)}}^-(q_3^{-c_{(2)}}z)\otimes\psi_{\o-\nu_3c_{(1)}}^-(q_3^{-c_{(1)}}z),\\ &S(x_\o^+(z))=-\psi_{\o+\nu_3c}^-(q_3^cz)^{-1}x_\o^+(z),\quad S(x_\o^-(z))=-x_{\o+\nu_3c}^-(q_3^cz)\psi_{\o+\nu_3c}^+(q_3^cz)^{-1},\quad \e(x_\o^\pm(z))=0,\\ &S(\psi_\o^+(z))=\psi_{\o+\nu_3c}^+(q_3^{c}z)^{-1},\quad S(\psi_\o^-(z))=\psi_{\o+2\nu_3c}^-(q_3^{2c}z)^{-1},\quad \e(\psi_\o^\pm(z))=1, \end{split}\end{aligned}$$ with the standard notation $c_{(1)}=c\otimes1$, $c_{(2)}=1\otimes c$. The central element $c$ obeys $\D(c)=c_{(1)}+c_{(2)}$, $S(c)=-c$ and $\e(c)=0$. The proof is a tedious but straightforward calculation that the axioms defining a Hopf algebra hold for any pair of currents. The antipode is an anti-homomorphism of algebra, it satisfies $S^2=(-1)^{1+\e}\text{Id}$. Using the coproduct of the Cartan zero modes $\psi_{\o,0}^\pm$, it is possible to compute the coproduct of the central charge $\bc$ defined in \[def\_bc\], we find[^8] $$\label{coprod_bc} \D(q_3^{-\bc})=(q_3^{-\bc}\otimes q_3^{-\bc})\left(q_3^{c_{(2)}\sum_{\o\in\Zp} a_{\o,0}^-}\otimes q_3^{c_{(1)}\sum_{\o\in\Zp} a_{\o,0}^+}q_3^{c_{(1)}\sum_{\o\in\Zp} a_{\o,0}^-}\right).$$ In order to reconstruct the instanton partition functions, we need to introduce two types of representations: a vertical representation $\rho^{(V)}$ with level $c=0$ and a horizontal representation $\rho^{(H)}$ with level $c=1$. Such representations are already known in the case of quantum toroidal algebras of $\glp$ (see [@Feigin2012; @Saito1996], or the brief summary presented in appendix ), but also for the quantum toroidal $\mathfrak{gl}(1)$ algebra (or Ding-Iohara-Miki algebra [@Ding1997; @Miki2007]) [@feigin2011quantum; @Feigin2009a]. In fact, there are two different point of view concerning these representations. In the mathematics literature [@feigin2011quantum; @Feigin2009a; @Feigin2012], one often considers a single module, the *Fock module*, and present the action of two subalgebras called horizontal and vertical. Miki’s automorphism $\CS$ [@Miki1999; @Miki2007] exchanges the two subalgebras, allowing us to define (for instance) $\rho^{(H)}=\rho^{(V)}\circ\CS$. On the opposite, physicists usually introduce two different types of modules referred as vertical and horizontal modules, somehow fixing the choice of subalgebra. Of course, the modules are isomorphic thanks to Miki’s automorphism and the two point of views are equivalent [@Bourgine2018a]. However, no analogue of Miki’s automorphism is known yet for the $(\nu_1,\nu_2)$-deformed algebra. Thus, at this stage, we have no choice but to follow the second approach and define two distinct representations. This will be done in the next two subsections. Vertical representation {#sec_vert} ----------------------- The vertical representation presented here is a deformation of the Fock representation for the quantum toroidal algebra of $\glp$ [@Feigin2012] (see appendix ). This representation is similar to the usual finite dimensional representations of quantum groups. Indeed, the Cartan currents $\psi_\o^\pm(z)$ are diagonal on a set of weight vectors. The currents $x_\o^-(z)$ annihilates the highest weight (or vacuum) ${\ket{\vac}\!\rangle}$, and $x_\o^+(z)$ creates excitations. However, the weight vectors are labeled here by the box configurations of an $m$-tuple Young diagrams $\bl$. Thus, this representation is infinite dimensional, yet it is graded by the total number of boxes $|\bl|$. From the gauge theory perspective, the vertical representation of the algebra \[algebra\] describes the relation between sectors of different instanton numbers. Thus, vertical modules are characterized by a basis of states ${\ket{\bl}\!\rangle}$ labeled by instanton configurations. Accordingly, the representation depend on a set of $m$ (highest) weights $\bv=(v_\a)_{\a=1\cdots m}$ and a choice of color $c_\a$ for each weight. This coloring defines the integers $m_\o=|C_\o(m)|$ corresponding to the number of weights $v_\a$ of color $\o$. The integers $m_\o$ provide the levels of the vertical representation: $\rho^{(V)}(c)=0$ and $\rho^{(V)}(\bc)=m$ with $m=\sum_{\o\in\Zp}m_\o$. As mentioned previously, the Cartan currents $\psi_\o^\pm(z)$ are diagonal on the basis ${\ket{\bl}\!\rangle}$. On the other hand, the operators $x_\o^\pm(z)$ relate the sectors of instanton charge $|\bl|$ and $|\bl|\pm1$ by adding/removing a box to the $m$-tuple Young diagram $\bl$. Their action encodes the recursion relation \[rec\_N\] obeyed by Nekrasov factors [@Kanno2013]. The action of the Drinfeld currents on the states ${\ket{\bl}\!\rangle}$ is derived in appendix , it reads[^9] $$\begin{aligned} \begin{split}\label{vert_x_psi_mr} &\rho^{(V)}(x_\omega^+(z)){\ket{\bl}\!\rangle}=F^{1/2}\sum_{\mAbox\in A_\omega(\boldsymbol\l)}\d(z/\chi_{\sAbox}) \res_{z=\chi_{\sAbox}}z^{-1}\CYY(z)^{-1} {\ket{\bl+\Abox}\!\rangle},\\ &\rho^{(V)}(x_\omega^-(z)){\ket{\bl}\!\rangle}= {\mathring{f}}_{\bo}^{[\bl]}(q_3^{-1}z)\sum_{\mAbox\in R_\omega(\boldsymbol\l)}\d(z/\chi_{\sAbox})\res_{z=\chi_{\sAbox}}z^{-1}\CY_{\bo}^{[\bl]}(q_3^{-1}z) {\ket{\bl-\Abox}\!\rangle},\\ &\rho^{(V)}(\psi_\omega^\pm(z)){\ket{\bl}\!\rangle}=\left[\PsiY(z)\right]_\pm{\ket{\bl}\!\rangle}. \end{split}\end{aligned}$$ In the first two lines, $A_\o(\bl)$ and $R_\o(\bl)$ correspond respectively to the set of boxes of color $\o$ that can be added to or removed from $\bl$. In the last line, the subscript $\pm$ denotes the expansion of the function $\PsiY(z)$ for $|z|^{\pm1}\to\infty$. This function is written as a ratio of the $\CY$-observables defined in \[def\_Y\], $$\label{def_Psi} \PsiY(z)={\mathring{f}}_{\bo}^{[\bl]}(q_3^{-1}z)\dfrac{\CY_{\bar\o}^{[\bl]}(q_3^{-1}z)}{\CYY(z)},\quad\text{with}\quad {\mathring{f}}\oY(z)=f_{\o}^{[\bl]}(z)\prod_{\a\in C_\o(m)}(-v_\a/z)=\prod_{\mAbox\in\bl}f_{\o c(\sAbox)}(\chi_{\sAbox}/z).$$ We notice that the highest weights are still encoded in the form of a Drinfeld polynomial $p_\o(z)$: $$\Psi_\o^{[\vac]}(z)=z^{m_\o-m_\bo}\dfrac{\prod_{\a\in C_{\bo}(m)}(-q_3 v_\a)}{\prod_{\a\in C_{\o}(m)}(-v_\a)}\dfrac{p_\bo(q_3^{-1/2}z)}{p_\o(q_3^{1/2}z)},\quad\text{with}\quad p_\o(z)=\prod_{\a\in C_\o(m)}(1-q_3^{-1/2}z/v_\a).$$ When $\nu_3=0$, we have $\bo=\o$ and the prefactor reduces to the usual expression $q_3^{m_\o}$ where $m_\o=\deg p_\o(z)$. The functions $\fYY(z)$ and ${\mathring{f}}\oY(z)$ control the asymptotics of the functions $\CYY(z)$ and $\PsiY(z)$, $$\label{asympt_PsiY_g} \CYY(z){\genfrac{}{}{0pt}{3}{\sim}{\infty}}1,\quad \CYY(z){\genfrac{}{}{0pt}{3}{\sim}{0}}\fYY(z)^{-1}\implies \PsiY(z){\genfrac{}{}{0pt}{3}{\sim}{0}}\fYY(z)\dfrac{{\mathring{f}}_{\bo}^{[\bl]}(q_3^{-1}z)}{f_{\bo}^{[\bl]}(q_3^{-1}z)},\quad \PsiY(z){\genfrac{}{}{0pt}{3}{\sim}{\infty}}{\mathring{f}}_{\bo}^{[\bl]}(q_3^{-1}z).$$ As a result, the action of the zero-modes of the Cartan currents read $$\begin{aligned} \begin{split} &\rho^{(V)}(\psi_{\o,0}^+){\ket{\bl}\!\rangle}={\mathring{f}}\boY(q_3^{-1}){\ket{\bl}\!\rangle},\quad \rho^{(V)}(\psi_{\o,0}^-){\ket{\bl}\!\rangle}={\mathring{f}}\oY(1)\dfrac{\prod_{\a\in C_\bo(m)}(-q_3v_\a)}{\prod_{\a\in C_\o(m)}(-v_\a)}{\ket{\bl}\!\rangle},\\ &\rho^{(V)}(a_{\o,0}^+){\ket{\bl}\!\rangle}=\left(\sum_{\mAbox\in\bl}\b_{\bo c(\sAbox)}\right){\ket{\bl}\!\rangle},\quad \rho^{(V)}(a_{\o,0}^-){\ket{\bl}\!\rangle}=\left(m_\o-m_\bo-\sum_{\mAbox\in\bl}\b_{\o c(\sAbox)}\right){\ket{\bl}\!\rangle} \end{split}\end{aligned}$$ The value of the second central charge is obtained by taking the product over $\o$, we recover $\rho^{(V)}(\bc)=m$. #### Contragredient representation The definition of intertwiners in the next section requires the introduction of the dual basis ${\langle\!\bra{\bl}}$. The algebra \[algebra\] acts on the dual basis with the contragredient representation $\rho^{(V)\ast}$, defined such that $${\langle\!\bra{\bl}}\left(\rho^{(V)}(e){\ket{\bl'}\!\rangle}\right)=\left({\langle\!\bra{\bl}}\rho^{(V)\ast}(e)\right){\ket{\bl'}\!\rangle},$$ for any element $e$ of the algebra. Thus, the action of the contragredient representation depends on the choice of a scalar product for the vertical states. It turns out that the analysis of intertwining relations simplifies for a particular choice of scalar product for which states are orthogonal but not orthonormal, $${\langle\!\bra{\bl}}\bl'\rangle\!\rangle=a_{\bl}(\bv)^{-1}\d_{\bl,\bl'}.$$ The norms $a_{\bl}(\bv)^{-1}$ are chosen so that the contragredient representation of $x^\pm_\o(z)$ acts on ${\langle\!\bra{\bl}}$ in the same way as the original representation $\rho^{(V)}(x^\mp_\o(z))$ acts on ${\ket{\bl}\!\rangle}$ (note that $x^\pm_\o$ becomes $x^\mp_\o$). As a result, the norms have to obey the two following recursion relations for a box $x$ of color $c(\Abox)=\o$: $$\begin{aligned} \begin{split} &\dfrac{a_{\bl-\mAbox}(\bv)}{a_\bl(\bv)}=\kO^{-1}f_{\bo}^{[\bl]}(q_3^{-1}\chi_{\sAbox})\res_{z=\chi_{\sAbox}}z^{-1}\CY_\o^{[\bl]}(z)\CY_\bo^{[\bl]}(q_3^{-1}z),\\ &\dfrac{a_{\bl+\mAbox}(\bv)}{a_\bl(\bv)}=-\kO^{-1}Ff_{\bo}^{[\bl]}(q_3^{-1}\chi_{\sAbox})^{-1}\res_{z=\chi_{\sAbox}}z^{-1}\CY_\o^{[\bl]}(z)^{-1}\CY_\bo^{[\bl]}(q_3^{-1}z)^{-1}. \end{split}\end{aligned}$$ The solution is expressed in terms of the vector contribution $\Zv(\bv,\bl)$ defined in \[def\_Zv\_Zbf\], $$a_{\bl}(\bv)=(-F^{1/2})^{|\bl|}\Zv(\bv,\bl)\prod_{\mAbox\in\bl}\prod_{\a\in C_{\bc(\sAbox)}(m)}(-\chi_{\sAbox}/(q_3v_\a)).$$ Horizontal representation ------------------------- The horizontal representation of the algebra \[algebra\] is the equivalent of the vertex representations constructed by Saito in [@Saito1996] for quantum toroidal algebras of $\glp$. It has level $\rho^{(H)}(c)=1$ and depends on $p$ weights $u_\o\in\mathbb{C}^\times$ and $p$ integers $n_\o\in\mZ$. In this representation, Drinfeld currents are constructed as a direct product of two (commuting) algebras. The first algebra is called here the *zero modes* factor, it is defined in terms the two operators $Q_\o(z),P_\o(z)$ satisfying the exchange relation $$\label{rel_PQ} P_\o(z)Q_{\o'}(w)=f_{\o\o'}(w/z)Q_{\o'}(w)P_\o(z).$$ In appendix , these operators are constructed explicitly in terms of $2p$ Heisenberg algebras. As a result, the operator $P_\o(z)$ acts on the vacuum state $\ket{\vac}$ as $P_\o(z)\ket{\vac}=\ket{\vac}$, and $Q_\o(z)$ acts on the dual vacuum $\bra{\vac}$ as $\bra{\vac}Q_\o(z)=\bra{\vac}$. Accordingly, we define the normal ordering of these operators by writing the $Q_\o(z)$-dependence on the left. The second algebra involved in the horizontal representation is defined upon the modes $\a_{\o,k}$ of $p$ coupled free bosons ($\o\in\mathbb{Z}_p$ and $k\in\mathbb{Z}^\times$) satisfying the commutation relations,[^10] $$\label{com_rel} [\a_{\o,k},\a_{\o',l}]=k\d_{k+l}q_3^{k/2}\left[\d_{\o\o'}+q_3^{-k}\d_{\o\bo'}-q_1^k\d_{\o\ \o'+\nu_1}-q_2^k\d_{\o\ \o'+\nu_2}\right],\quad (k>0).$$ As usual, the vacuum state $\ket{\vac}$ is annihilated by the positive modes ($k>0$), while negative modes create excitations. The dual state $\bra{\vac}$ is annihilated by negative modes. Thus, these modes are normal ordered by moving the positive modes to the right. The representation of the Drinfeld currents $x_\o^\pm$ and $\psi_\o^\pm$ is given in terms of the vertex operators $$\begin{aligned} \begin{split}\label{def_eta_vphi} &\eta_\o^+(z)=\exp\left(\sum_{k>0}\dfrac{z^k}{k}\a_{\o,-k}\right)\exp\left(-\sum_{k>0}\dfrac{z^{-k}}{k}q_3^{-k/2}\a_{\o,k}\right),\quad \eta_\o^-(z)=\exp\left(-\sum_{k>0}\dfrac{z^k}{k}\a_{\o,-k}\right)\exp\left(\sum_{k>0}\dfrac{z^{-k}}{k}q_3^{k/2}\a_{\bo,k}\right),\\ &\vphi_\o^+(z)=\exp\left(-\sum_{k>0}\dfrac{z^{-k}}{k}(q_3^{-k/2}\a_{\o,k}-q_3^{k/2}\a_{\bo,k})\right),\quad \vphi_\o^-(z)=\exp\left(\sum_{k>0}\dfrac{z^{k}}{k}(q_3^{-k}\a_{\bo,-k}-\a_{\o,-k})\right). \end{split}\end{aligned}$$ Combining the zero modes and vertex operators, the horizontal representation writes $$\begin{aligned} \begin{split}\label{rep_H} &\rho^{(H)}(x^+_\o(z))=u_\o z^{-n_\o}Q_\o(z)\eta_\o^+(z),\quad \rho_{u}^{(H)}(x^-_\o(z))=u_\o^{-1}z^{n_\o}Q_\o(z)^{-1}P_{\bo}(q_3^{-1}z)\eta_\o^-(z),\\ &\rho^{(H)}(\psi^+_\o(z))=F^{-1/2}P_{\bo}(q_3^{-1}z)\vphi_\o^+(z),\\ &\rho^{(H)}(\psi_\o^-(z))=F^{1/2}\dfrac{u_{\bo}}{u_\o}q_3^{n_{\bo}}z^{n_\o-n_{\bo}}\dfrac{Q_{\bo}(q_3^{-1}z)}{Q_{\o}(z)}P_{\bo}(q_3^{-1}z)\vphi_\o^-(z). \end{split}\end{aligned}$$ It is shown in appendix that the expressions in the RHS obey the algebraic relations \[algebra\] at the levels $\rho^{(H)}(c)=1$ and $\rho^{(H)}(\bc)=n+p$ if $\nu_1+\nu_2<p$ and $\rho^{(H)}(\bc)=n$ otherwise, where $n=\sum_{\o\in\Zp} n_\o$. Note that even in the ALE case $\nu_1=-\nu_2=1$, the horizontal representation given here is slightly more general than the one proposed in [@Zenkevich2018]. Indeed, in the latter the $\Zp$-symmetry is broken by a choice of color $\o_0$, setting $u_\o=u\d_{\o,\o_0}$ and $n_\o=n\d_{\o,\o_0}$. Instead, in our construction of the gauge theory partition functions, it is necessary to keep $u_\o$ and $n_\o$ arbitrary in order to be able to assign a different gauge coupling $\qf_\o$ and Chern-Simons level $\k_\o$ for each color $\o$. Algebraic engineering ===================== The algebraic engineering of 5D $\CN=1$ quiver gauge theories on $\mC_{\e_1}\times\mC_{\e_2}\times S_R^1$ (without orbifold) follows from their correspondence with topological string theories in which the Nekrasov instanton partition function is obtained as a topological strings amplitude [@Aganagic2005]. Indeed, these amplitudes are computed using the (refined) topological vertex [@Iqbal2007; @Awata2005; @Taki2007] that was identified in [@Awata2011] with an intertwiner between certain modules of the Ding-Iohara-Miki algebra [@Ding1997; @Miki2007], also known as the quantum toroidal algebra of $\mathfrak{gl}(1)$. This intertwiner is in fact the toroidal analogue of the vertex operators introduced in [@Davies1992] to compute the form factors of the XXZ Heisenberg spin chain. As result, the powerful topological strings computational methods for supersymmetric gauge theories can be reformulated in the language of quantum integrability. The correspondence between 5D $\CN=1$ gauge theories and quantum toroidal algebras is better formulated using the $(p,q)$-brane realization of the gauge theories in type IIB string theory [@Aharony1997; @Aharony1997a]. In this realization, quiver gauge theories are reproduced by the low energy dynamics of a network of 5-branes with charges $(p,q)$. These branes generalize both NS5-branes $(0,1)$ and D5-branes $(1,0)$. They wrap the 5-dimensional spacetime, and define a line segment in the 56-plane of the ten dimensional strings spacetime. These segments meet at trivalent vertices and form a web called the *$(p,q)$-branes web*. For instance, in the case of linear quivers, a set of $m$-D5 branes is associated to each node bearing a $U(m)$ gauge group. These D5-branes are suspended between dressed NS5-branes (i.e. branes of charge $(n,1)$). In this context, the relevant quantum toroidal algebra is determined by the spacetime of the gauge theory. Then, each brane of the $(p,q)$-branes web is associated to a representation of the algebra, identifying the levels with the charges $\rho(c)=q$, $\rho(\bc)=p$ and the weights with the (exponentiated) position of the branes [@Awata2017; @Mironov2016; @Awata2016a]. Thus, to a D5-brane corresponds a vertical representation with $m=1$, while horizontal representations are associated to dressed NS-branes of charge $(n,1)$. It was further noticed in [@Bourgine2017b] that the set of $m$ D5-branes of a single node (with a $U(m)$ gauge group) can be directly described by a vertical representation with $\rho^{(V)}(\bc)=m$. Following the identification of the $(p,q)$-branes web with the toric diagram of the Calabi-Yau in topological strings [@Leung1998], the trivalent junctions of branes coincide with the vertex operator of the algebra acting on the modules determined by the branes charge. Finally, the automorphisms of the algebra renders the various geometrical operations (translations, rotations) applied to the branes web [@Bourgine2018a]. For each $(p,q)$-branes web it is possible to write down an operator $\CT$ constructed by ‘gluing’ the vertex operators of nodes connected by an edge. The gluing procedure is done by a product of operators in horizontal representations (NS5), and a scalar product in vertical ones (D5). The $\CT$-operator obtained in this way acts on the tensor product of representations corresponding to the external branes of the web (i.e. the semi-infinite line segments). These representations are in fact horizontal modules, and the vacuum expectation value of the $\CT$-operator reproduces the instantons partition function. The $qq$-characters are further obtained by introducing algebra elements (in the proper representation) within the vacuum expectation value [@Bourgine2017b]. We will give several examples below. This algebraic construction of gauge theories BPS-observables has been generalized in several directions: D-type quivers [@Bourgine2017c], 6D spacetime and elliptic algebras [@Foda2018], 4D $\CN=2$ gauge theories and the affine Yangian of $\mathfrak{gl}(1)$ [@Bourgine2018], 5D $\CN=1$ gauge theories on ALE spaces [@Awata2017], and 3D $\CN=2^\ast$ gauge theories [@Zenkevich2018]. In this section, we present yet another generalization corresponding to deformed ALE spaces with the $\Zp$-action described in section two. However, we do not wish to reproduce the whole construction here as it is a straightforward application of the methods developed earlier [@Awata2017; @Mironov2016; @Awata2016a; @Bourgine2017b]. Instead, we will only provide the main ingredient, namely the expression of the vertex operators, and a few selected examples to illustrate the construction. Vertex operators {#sec_Intw} ---------------- We consider two types of vertex operators, denoted $\Phi$ and $\Phi^\ast$, and obtained, up to a normalization factor, by solving the following equations $$\label{intw} \rho^{(H')}(e)\Phi=\Phi\ \left(\rho^{(V)}\otimes\rho^{(H)}\ \D(e)\right),\quad \left(\rho^{(V)}\otimes\rho^{(H)}\ \D'(e)\right)\ \Phi^\ast=\Phi^\ast\rho^{(H')}(e),$$ where $e$ is any of the currents $x_\o^\pm(z),\psi_\o^\pm(z)$ or the central charge $c$.[^11] Here $\D'$ denotes the opposite coproduct obtained by permutation $\D'=\CP\D\CP$. In order to distinguish the two horizontal representations, we denoted them $\rho^{(H)}$ and $\rho^{(H')}$, they depend on the parameters $u_\o,n_\o$ and $u_\o', n_\o'$ respectively. Thus, the vertex operator $\Phi$ (and also $\Phi^\ast)$ depends on the set of weights $u_\o,u_\o',v_\o$ and integers $n_\o,n'_\o,m_\o$. A solution to the equations \[intw\] is found only if these parameters satisfy the two constraints $$\label{rel_u_n} u'_\o=u_\o\prod_{\a\in C_\bo}(-q_3v_\a),\quad n'_\o=n_\o+m_\bo.$$ The first relation expresses a constraint among the position of the branes in the 56-planes. The second equation is the charge conservation at the vertex. Due to the spacetime orbifold, the branes charges $p$ in $(p,q)$ degenerates into charges $p_\o$ with $\o\in\Zp$ identified with the integers $n_\o$ and $m_\bo$ of horizontal/vertical representations.[^12] Summing over $\o$, these constraints reproduce the conservation of the levels $n'=n+m$ that follows from the application of the intertwining relations \[intw\] to the element $e=\bc$ with the coproduct \[coprod\_bc\]. Due to the presence of an algebra automorphism exchanging $c$ and $\bc$ in the $\glp$-case [@Miki1999], we expect a similar degeneration of the charge $q$ into $q_\o$. It is not observed here because only a single charge $q=1$ flow through the topological vertex. By definition, the vertex operator $\Phi^\ast$ is a vector in the vertical module while $\Phi$ is a dual vector, $$\label{Phi_vert} \Phi=\sum_{\bl}\Phi_{\bl}\ {\langle\!\bra{\bl}},\quad \Phi^\ast=\sum_{\bl}\Phi_{\bl}^{\ast}\ {\ket{\bl}\!\rangle}.$$ Each vertical component $\Phi_\bl$ (or $\Phi_\bl^\ast$) is a Fock vertex operator acting on the horizontal module, $$\begin{aligned} \label{def_Phi} \begin{split} &\Phi_{\bl}=t_{\bl}:\Phi_\vac\prod_{\mAbox\in\bl}\eta_{c(\sAbox)}^+(\chi_{\sAbox}):,\quad \Phi_{\bl}^\ast=t_{\bl}^\ast:\Phi_\vac^\ast\prod_{\mAbox\in\bl}\eta_{c(\sAbox)-\nu_1-\nu_2}^-(q_3\chi_{\sAbox}):,\\ &t_{\bl}=F^{-|\bl|/2}\prod_{\mAbox\in\bl}u_{c(\sAbox)}'\chi_{\sAbox}^{-n_{c(\sAbox)}'}\prod_{\mAbox\in\bl}Q_{c(\sAbox)}(\chi_{\sAbox}),\\ &t_{\bl}^\ast=F^{-|\bl|/2}\prod_{\mAbox\in\bl}(-u_{c(\sAbox)-\nu_1-\nu_2})^{-1}(q_3\chi_{\sAbox})^{n_{c(\sAbox)-\nu_1-\nu_2}}\prod_{\mAbox\in\bl}:Q_{c(\sAbox)-\nu_1-\nu_2}(q_3\chi_{\sAbox})^{-1}P_{c(\sAbox)}(\chi_{\sAbox}):. \end{split}\end{aligned}$$ A sketch of the derivation can be found in the appendix , together with the (rather lengthy) expressions of the vacuum components $\Phi_\vac$ and $\Phi_\vac^\ast$. The vertex operators $\Phi$ and $\Phi^\ast$ given here are a generalization of the colored refined topological vertex derived in [@Awata2017; @Chaimanowong2018] with extra parameters $(\nu_1,\nu_2)$. The relation with the vertex operators of the quantum toroidal $\glp$ algebra is investigated in appendix \[AppE\]. It is seen that an extra zero-modes factor is necessary due to the twist of the coproduct. This factor leads to a different identification between the algebraic parameters (levels, weights) and the physical quantities, i.e. the branes charges (or Chern-Simons levels) and positions (or instanton counting parameters and Coulomb branch vevs). The vertical components \[def\_Phi\] of the vertex operators obey important normal ordering relations, from which we recover the vector and bifundamental contributions to the partition functions [@Bourgine2017b],[^13] $$\begin{aligned} \label{ordering_Phi} \begin{split} &\Phi_{\bl}\Phi_{\bl'}=\CG(\bv'|\bv)^{-1}N(\bv',\bl'|\bv,\bl)^{-1}:\Phi_{\bl}\Phi_{\bl'}:\\ &\Phi_{\bl}\Phi_{\bl'}^\ast=\CG(\bv'|q_3^{-1}\bv)N(\bv',\bl'|q_3^{-1}\bv,\bl):\Phi_{\bl}\Phi_{\bl'}^\ast:\\ &\Phi_{\bl}^\ast\Phi_{\bl'}=\CG(\bv'|\bv)N(\bv',\bl'|\bv,\bl):\Phi_{\bl}^\ast\Phi_{\bl'}:\\ &\Phi_{\bl}^\ast\Phi_{\bl'}^\ast=\CG(\bv'|q_3^{-1}\bv)^{-1}N(\bv',\bl'|q_3^{-1}\bv,\bl)^{-1}:\Phi_{\bl}^\ast\Phi_{\bl'}^\ast:, \end{split}\end{aligned}$$ The expression of the one-loop factors $\CG(\bv|\bv')$ can be found in appendix , formula \[def\_CG\]. Note also that, following the method presented in [@Awata2016a; @Awata2017], it is a priori possible to show that $\Phi_{\bl}$ and $\Phi_{\bl}^\ast$ are solutions of the double deformed Knizhnik - Zamolodchikov (or $(q,t)$-KZ) equations. Partition functions and $qq$-characters --------------------------------------- The simplest example of algebraic engineering is given by the pure $U(m)$ gauge theory with quiver $A_1$. In this case, the $(p,q)$-brane web can be described roughly as a set of $m$ D5-branes suspended between two (dressed) NS5-branes. The corresponding $\CT$-operator is obtained as a product of vertex operators $\Phi$ and $\Phi^\ast$ in the vertical channel [@Bourgine2017b], it acts on the tensor product of two horizontal modules, $$\begin{aligned} \CT[U(m)] = \Phi\cdot \Phi^\ast = \sum_{\boldsymbol\lambda} a_{\bl}(\bv)\ \Phi_\bl\otimes\Phi_\bl^\ast\ :\ H\otimes H'_\ast\to H'\otimes H_\ast.\end{aligned}$$ In order to distinguish the horizontal modules, we added the subscript $\ast$ to the ones on which $\Phi^\ast$ act. Accordingly, we denote the parameters of these representations $(n_\o^\ast,u_\o^\ast)$ and $(n_\o^{\ast\prime},u_\o^{\ast\prime})$. Evaluating the vacuum expectation value of this operator, we recover the instanton partition function of the underlying gauge theory: $$\mathcal{Z}_{\text{inst}}= \bra{\vac}\otimes\bra{\vac}\ \mathcal{T}[U(m)] \ket{\vac}\otimes\ket{\vac} = \sum_{\boldsymbol\lambda} \prod_{\omega \in \mathbb{Z}_p} \mathfrak{q}_\omega ^{K_\omega (\boldsymbol\lambda)} \mathcal{Z}_{\text{vect}} (\bv, \bl) \mathcal{Z}_{\text{CS}} (\boldsymbol{\k},\bl)$$ where we have identified the colored gauge coupling $\qf_\o$ and Chern-Simons level $\k_\o$ with $$\begin{aligned} \label{id_q_k} \qf_\o = F^{-1/2} \dfrac{u_\o}{u_{\o+\nu_3}^\ast}q_3 ^{n _{\o+\nu_3}^\ast} , \quad \k_\o=n_{\o+\nu_3}^\ast-n_\o.\end{aligned}$$ By construction, the operator $\CT[U(m)]$ commutes with the action of the algebra defined by the opposite coproduct $\D'$ [@Bourgine2017b], namely, $$\begin{aligned} \label{eq:commute} \left(\rho^{(H')}\otimes\rho^{(H_\ast)}\right) \D'(e ) \, \mathcal{T}[U(m)] = \mathcal{T}[U(m)] \, \left( \rho^{(H_\ast)} \otimes \rho^{(H')} \right) \D'(e), \quad e \in \mathcal{A}.\end{aligned}$$ For this reason, $\CT[U(m)]$ plays the role of the *screening operator* in [@Awata2016a]. The gauge theory expectation value of the fundamental $qq$-characters is obtained by insertion of $\D' ( x^- _{\o +\nu_3} (q_3 z))$ in the horizontal vacuum expectation value, $$\label{qq_character} \la\mathcal{X}_\bo^{[\bl]\ast} (q_3^{-1}z) \rag=u_\o^{-1}z^{n_\o} \frac{ \bra{\vac}\otimes\bra{\vac}\ \left(\rho^{(H')}\otimes\rho^{(H_\ast)}\ \D'(x_{\o+\nu_3}^-(q_3z))\right)\mathcal{T}[U(m)] \ket{\vac}\otimes\ket{\vac} } {\bra{\vac}\otimes\bra{\vac}\ \mathcal{T}[U(m)] \ket{\vac}\otimes\ket{\vac}}$$ where the gauge averaging of a chiral ring observable $\CO^{[\bl]}$ is performed over the instanton configurations weighted by the vector (and Chern-Simons) contributions to the partition function, $$\la \CO^{[\bl]}\rag=\dfrac1{\Zinst}\sum_{\boldsymbol\lambda} \prod_{\omega \in \mathbb{Z}_p} \mathfrak{q}_\omega ^{K_\omega (\boldsymbol\lambda)} \mathcal{Z}_{\text{vect}} (\bv, \bl) \mathcal{Z}_{\text{CS}} (\boldsymbol{\k},\bl) \CO^{[\bl]},$$ and the $qq$-character writes $$\mathcal{X}_\o^{[\bl]\ast} (z)=\CY_\o^{[\bl]\ast}(z)+\qf_{\o+\nu_3} \dfrac{(q_3z)^{\k_{\o+\nu_3}}}{\CY_{\o+\nu_3}^{[\bl]}(q_3z)}.$$ Note that the first term involves the $\CY$-observable $\CY_\o^{[\bl]\ast}(z)=f\oY(z)\CY\oY(z)$. As shown in [@Bourgine2017b], it follows from the commutation relations that the quantity $\la\mathcal{X}_\bo^{[\bl]} (q_3^{-1}z) \rag$ is a finite Laurent series in $z$ (i.e. a polynomial upon multiplication by a positive power of $z$). This is in fact due to the radial ordering of operators in the horizontal Fock spaces. Indeed, when $x_\o^-(z)$ is inserted on the left of $\CT$, the correlator as a well-defined expansion around $z=\infty$. On the other hand, when $x_\o^-(z)$ in inserted on the right, the expansion around $z=0$ is now well-defined. The non-trivial equality between the two expansions implies that both series are finite, and thus that the correlator is a finite Laurent series in $z$. Asymptotically, the $\CY$-observables behave as $\CYY{\genfrac{}{}{0pt}{3}{\sim}{0}}z^{-\gbY}$, $\CY_\o^{[\bl]\ast}{\genfrac{}{}{0pt}{3}{\sim}{0}}1$ and $\CYY{\genfrac{}{}{0pt}{3}{\sim}{\infty}}1$, $\CY_\o^{[\bl]\ast}{\genfrac{}{}{0pt}{3}{\sim}{\infty}}z^{\gbY}$ with $\gbY=|A_\o(\bl)|-|R_{\o+\nu_3}(\bl)|$. When $\nu_3=0$, the exponent $\gbY$ becomes independent of $\bl$, $\gbY=m_\o$. As a result, the gauge average of the $qq$-character $\mathcal{X}_{\o}(z)$ is a polynomial of degree $m_\o$ when $|\k_\o|<m_\o$. Unfortunately, when $\nu_3\neq0$ not much can be said. Another fundamental $qq$-character can be obtained using the generator $x_\o^+(z)$ instead, $$\begin{aligned} \label{eq:antifund} \begin{split} &\la\mathcal{X}_\bo^{[\bl]} (q_3^{-1}z)\rag=(u_\o^\ast)^{-1}z^{n_\o^\ast} \frac{ \bra{\vac}\otimes\bra{\vac}\ \left(\rho^{(H')}\otimes\rho^{(H_\ast)}\ \D'(x_\o^+(z))\right)\mathcal{T}[U(m)] \ket{\vac}\otimes\ket{\vac} } {\bra{\vac}\otimes\bra{\vac}\ \mathcal{T}[U(m)] \ket{\vac}\otimes\ket{\vac}},\\ &\text{with}\quad \mathcal{X}_\o^{[\bl]} (z)=\CY\oY(z)+\qf_{\o+\nu_3} F \dfrac{(q_3z)^{\k_{\o+\nu_3}}}{f\oY(z)\CY_{\o+\nu_3}^{[\bl]}(q_3z)}. \end{split}\end{aligned}$$ The presence of two different fundamental $qq$-characters is a specificity of 5D $\CN=1$ gauge theories on orbifolds: when $p=1$, the two $qq$-characters are equivalent (they only differ by multiplication of a constant times a power of $z$). Further, as we shall see below, in the 4D limit $R\to0$, the $qq$-characters $\mathcal{X}_\o^{[\bl]} (z)$ and $\mathcal{X}_\o^{[\bl]\ast}(z)$ reduce to the same expression. The gauge averages and for the $qq$-characters have been computed at the first few orders in the gauge couplings $\qf_\o$ for the gauge groups $U(1)$ and $U(2)$ and various orbifold parameters. In all cases, it has been observed that these quantities are indeed finite Laurent series in the argument $z$. Finally, it is worth mentioning that higher $qq$-characters can be obtained by multiple insertions of the coproducts $\D'(x_\o^\pm(z))$. We refer to [@Bourgine2017b] for more details on the computation of $qq$-characters, and to the appendix for explicit formulas at the first few orders in the instanton counting parameters. #### 4D limit When the radius $R$ of the background circle $S_R^1$ is sent to zero, the gauge theory reduces to a 4D $\CN=2$ gauge theory. This limit can be performed directly on the partition functions and $qq$-characters, re-introducing the radius dependences in the parameters $(q_1,q_2)=(e^{R\e_1},e^{R\e_2})$, $v_\a=e^{R a_\a}$, $\chi_{\sAbox}=e^{R\phi_{\sAbox}}$,... Sending $R\to0$ in the expression \[def\_Zv\_Zbf\] of the instanton partition function, we observe that the Chern-Simons contribution is subdominant while, after setting the spectral variable to $z=e^{R\z}$, the scattering function \[def\_S\] becomes $$S_{\o\o'}^{(\text{4D})}(\z)=\dfrac{(\z+\e_1)^{\d_{\o,\o'-\nu_1}}(\z+\e_2)^{\d_{\o,\o'-\nu_2}}}{\z^{\d_{\o,\o'}}(\z+\e_1+\e_2)^{\d_{\o,\o'-\nu_1-\nu_2}}}.$$ This function satisfies a simpler crossing symmetry $S_{\o\bo'}^{(\text{4D})}(-\z-\e_1-\e_2)=f_{\o\o'}^{(\text{4D})}S_{\o'\o}^{(\text{4D})}(\z)$ where $f_{\o\o'}^{(\text{4D})}=(-1)^{\b_{\o\o'}}$ in now independent of the spectral variable $\z$. As a result, the function $f\oY(z)$ reduces to a sign. When $\nu_3=0$, this sign is simply $(-1)^{m_\o}$, it can be absorbed in the definition of $\qf_\o$. In this way, both $\mathcal{X}_\o^{[\bl]} (z)$ and $\mathcal{X}_\o^{[\bl]\ast}(z)$ reproduce the expression of the 4D fundamental $qq$-character given in [@Nekrasov_BPS3; @Nekrasov_BPS4]. #### $A_2$ quiver Linear quiver gauge theories can be treated along the same lines. For instance, the $A_2$ quiver gauge theory with gauge group $U(m_1)\times U(m_2)$ is obtained by considering two sets of $m_1$ and $m_2$ D5-branes suspended between three dressed NS5-branes. The $\CT$-operator is simply the product of the single nodes operators $\CT[U(m_1)]$ and $\CT[U(m_2)]$ in a common horizontal representation, $$\CT[U(m_1)\times U(m_2)] = \Phi_1\cdot \Phi_2\Phi_1^\ast\cdot \Phi_2^\ast = \sum_{\bl^{(1)},\bl^{(2)}} a_{\bl^{(1)}}(\bv^{(1)})a_{\bl^{(2)}}(\bv^{(2)})\ \Phi_{\bl^{(1)}}^{(1)}\otimes\Phi_{\bl^{(2)}}^{(2)}\Phi_{\bl^{(1)}}^{(1)\ast}\otimes\Phi_{\bl^{(2)}}^{(2)\ast}.$$ The vacuum expectation value is computed using the normal ordering relation \[ordering\_Phi\] for the product $\Phi_{\bl^{(2)}}^{(2)}\Phi_{\bl^{(1)}}^{(1)\ast}$, $$\begin{aligned} \begin{split} \Zinst&= \dfrac1{\CG(\bv^{(1)}|q_3^{-1}\bv^{(2)})}\bra{\vac}\otimes\bra{\vac}\otimes\bra{\vac}\ \mathcal{T}[U(m_1)\times U(m_2)]\ket{\vac}\otimes\ket{\vac} \otimes\ket{\vac}.\end{split}\end{aligned}$$ It reproduces the instanton partition function \[def\_Zv\_Zbf\] for the $A_2$ quiver gauge theory, with the identification \[id\_q\_k\] of the parameters at each node $i=1,2$. The $qq$-characters can also be constructed along the lines of [@Bourgine2017b]. Concluding remarks ================== In this paper, we have reconstructed algebraically the instanton partition functions for $\CN=1$ linear quiver gauge theories with unitary gauge groups on the five dimensional background $S_R^1\times(\mathbb{C}_{\e_1}\times\mathbb{C}_{\e_2})/\mathbb{Z}_p$. The action of the abelian group considered here is a generalization by two integers $(\nu_1,\nu_2)$ of the standard action defining ALE spaces. These extra parameters led us to introduce a deformation of the quantum toroidal algebra of $\glp$. Conjecturally, this new algebra should coincide with the quantum toroidal algebra built upon a Kac-Moody algebra with the (non-symmetrizable) generalized Cartan matrix $\b_{\o\o'}$ given in \[def\_beta\]. We have shown that this deformed algebra still possesses the structure of a Hopf algebra with the coproduct given in \[Drinfeld\_coproduct\]. We have also presented two different representations, called *vertical* and *horizontal*, that are respectively the deformation of the Fock module [@Feigin2012] and the vertex representation [@Saito1996] of the quantum toroidal algebra of $\glp$. Other types of representations should exist, like the Macmahon representation obtained for $\glp$ as a tensor products of Fock modules in [@Feigin2012]. Although the definition of this new algebra may appear intricate, the physical context in which it emerges is very natural, and its representations are simple generalizations of the usual ones. Quantum toroidal algebras extend the definition of quantum affine algebras (or quantum groups) by an extra affinization. In fact, the quantum toroidal algebra of $\glp$ is generated by two orthogonal quantum affine subalgebra $U_q(\widehat{\mathfrak{sl}(p)})$ [@Ginzburg1995; @Saito1997]. Then, one may wonder if the $(\nu_1,\nu_2)$-deformed algebra possesses a similar property. Of course, it is assuming that a quantum affine algebra built upon the Cartan matrix $\b_{\o\o'}$ can be defined properly. In fact, we expect that this is indeed the case, and that such quantum affine algebra retains a quasitriangular Hopf algebra structure, making it suitable for the construction of new quantum integrable systems. On the gauge theory side, several generalizations of our approach could be implemented. For instance, the abelian group $\Zp$ could be replaced by a Mckay subgroup of $SU(2)$ of type DE, with either left, right, or both left-right action. As shown by Nakajima in [@Nakajima1994; @Nakajima1999], in the first two cases a quantum affine algebra of type $\mathfrak{so}/\mathfrak{sp}$ acts on the cohomology of the instanton moduli space. This action is expected to be lifted to a quantum toroidal algebra in K-theory. Accordingly, the algebraic engineering should involve the quantum toroidal $\mathfrak{so}/\mathfrak{sp}$ algebras. However, the effective construction requires some new developments in the representation theory of these algebras. When $\nu_2=0$, the orbifold can be interpreted as the presence of a surface defect [@Nekrasov_BPS4]. In this case, the Cartan matrix $\b_{\o\o'}$ appears to vanish but the algebra remains non-trivial, $$\begin{aligned} \begin{split} &S_{\o\o'}(z)=\left(\dfrac{1-q_2z}{1-z}\right)^{\d_{\o,\o'}}\left(\dfrac{1-q_1z}{1-q_1q_2z}\right)^{\d_{\o,\o'-\nu_1}},\\ &g_{\o\o'}(z)=\left(q_2^{-1}\dfrac{1-q_2z}{1-q_2^{-1}z}\right)^{\d_{\o\o'}}\left(\dfrac{1-q_1z}{1-q_3^{-1}z}\right)^{\d_{\o,\o'-\nu_1}}\left(q_2\dfrac{1-q_3z}{1-q_1^{-1}z}\right)^{\d_{\o,\o'+\nu_1}}. \end{split}\end{aligned}$$ When $\nu_1=1$, the structure function $g_{\o\o'}(z)$ reproduces the one that defines the quantum toroidal algebra of $\glp$ with $q_2$ and $q_3$ exchanged (up to a factor $q_3^{m_{\o\o'}/2}$). However, the function $S_{\o\o'}(z)$ is different from the one appearing in \[S\_glp\_II\], and thus horizontal and vertical representations of the $(\nu_1,\nu_2)$-deformed algebra degenerate into new representations for the quantum toroidal algebra of $\glp$. We hope to come back to the study of this problem in a future publication. Finally, an important question was left behind in our study, namely the correspondence with (q-deformed) W-algebras. This type of correspondences is now well-understood in the case of quantum toroidal $\mathfrak{gl}(1)$. There, the q-W-algebras appearing in horizontal or vertical representations play different roles. In the horizontal case, a representation of level $c=m$ can be built by tensoring $m$ level one representations. It is thus expressed in terms of $m$ sets of bosonic modes that are coupled through their commutation relations. Diagonalizing these relations, the Drinfeld currents can be expressed in terms of q-$W_m$ currents coupled to an infinite Heisenberg algebra. This dual q-W-algebra corresponds to the quiver W-algebra of Kimura and Pestun [@Kimura2015]. Using Miki’s automorphism [@Miki1999; @Miki2007], vertical representations of level $\bc=m$ can be mapped on horizontal ones, and thus expressed in terms of q-$W_m$ currents coupled to the Heisenberg algebra. In the vertical case, the dual W-algebra is responsible for the AGT-like correspondence with q-deformed conformal blocks [@Awata2009]. Alternatively, the AGT correspondence can also be seen directly in the degenerate limit $R\to0$ in which the vertical representation of the toroidal algebra reduces to a representation of the affine Yangian of $\mathfrak{gl}(1)$ that is known to contain the action of $W_m$-currents [@Schiffmann2012; @Tsymbaliuk2014; @Prochazka2015]. A similar type of duality is believed to hold between the degenerate limit of the quantum toroidal algebra of $\glp$ and the coset[^14] $\widehat{\mathfrak{gl}(\a)_m}/\widehat{\mathfrak{gl}(\a-p)_m}$, leading to an AGT correspondence between instantons on ALE spaces and parafermionic conformal field theories [@Belavin2011; @Bonelli2011a; @Bonelli2011b; @Belavin2011a; @Pedrini2014]. This conjecture has been verified for small values of $p$ and $m$ by comparing the conformal blocks of the coset theory with the gauge theories instanton partition functions [@Belavin2011; @Belavin2011b; @Wyllard2011; @Nishioka2011; @Ito2011; @Alfimov2011], or the limit $R\to0$ of 5D topological strings amplitudes [@Chaimanowong2018]. There are two main strategies to extend this duality to the $(\nu_1,\nu_2)$-deformed algebra. One possibility is again to compare instanton partition functions with conformal blocks. This approach was taken in [@Bonelli2012] where the gauge theory calculations led to conjectural expressions for these conformal blocks. But, unfortunately, the corresponding conformal field theory appears to be unknown. Another possible approach consists in identifying directly the (q-deformed) coset algebra generators acting on the vertical modules of the quantum toroidal algebra. For this purpose, one could diagonalize the commutation relations for the modes $\a_{\o,k}$ in the horizontal representations, and then define the analogue of Miki’s automorphism to map the horizontal representations to the vertical ones. From the strings theory perspective, the latter is expected to exist since it should describe the fiber-base duality of the topological strings (or, the S-duality in Type IIB string theory) [@Awata2009a; @Bourgine2018a]. This approach appears very promising and we hope to be able to report soon on this problem. Acknowledgments {#acknowledgments .unnumbered} =============== JEB would like to thank Omar Foda for discussions and generous support during his visit of the university of Melbourne. SJ is greatly indebted to Nikita Nekrasov for numerous discussions and supports. SJ is also grateful to Korea Institute for Advanced Study for providing support during his visit. The work of SJ was supported in part by the NSF grant PHY 1404446 and also by the generous support of the Simons Center for Geometry and Physics. Quantum toroidal algebra of $\glp$ {#AppA} ================================== In this appendix, we remind the definition of the quantum toroidal algebra of $\glp$, give its vertical and horizontal representations, and comment on the reduction $\nu_1=-\nu_2=1$ of the algebraic relations \[algebra\]. Definition ---------- Quantum toroidal algebras were introduced by V. Ginzburg and M. Kapranov and E. Vasserot in [@Ginzburg1995]. In general, they can be built over an affine Kac-Moody algebra $\hat{\gf}$, but we will focus in this appendix on the case of an algebra of type $A_{p-1}^{(1)}$, also called quantum toroidal $\gf=\glp $ algebra. This algebra is formulated in terms of the Drinfeld currents $$x_\o^\pm(z)=\sum_{k\in\mathbb{Z}}z^{-k}x_{\o,k}^\pm,\quad \psi_\o^\pm(z)=\sum_{k\geq0}z^{\mp k}\psi_{\o,\pm k}^\pm.$$ Like the Chevalley generators, the operators $x_\o^\pm(z)$ are associated to the simple roots $\a_\o$ of $\hat{\gf}$. On the other hand, the operators $\psi_{\o}^\pm(z)$ describe the Cartan sector of the algebra, they are naturally associated to the coroots $\a_\o^\vee$. We denote the Cartan matrix $\b_{\o\o'}=\la \a_\o^\vee,\a_{\o'}\ra$, in the case of $\glp $, we have $\b_{\o\o'}=2\d_{\o,\o'}-\d_{\o,\o'+1}-\d_{\o,\o'-1}$ (here $\d_{\o,\o'}$ denotes the Kronecker delta with indices taken modulo p). In this case, the original relations can be deformed by an extra central parameter $\k$, using the antisymmetric matrix $m_{\o\o'}=\d_{\o,\o'-1}-\d_{\o,\o'+1}$ [@Saito1996]:[^15] $$\begin{aligned} \label{algebra_glp} \begin{split} &[\psi_\o^\pm(z),\psi_{\o'}^\pm(w)]=0,\quad \psi_\o^+(z)\psi_{\o'}^-(w)=\dfrac{g_{\o\o'}(q^c z/w)}{g_{\o\o'}(q^{-c}z/w)}\psi_{\o'}^-(w)\psi_\o^+(z),\quad x_\o^\pm(z)x_{\o'}^\pm(w)=g_{\o\o'}(z/w)^{\pm1}x_{\o'}^\pm(w)x_\o^\pm(z),\\ &\psi_{\o}^+(z)x_{\o'}^\pm(w)=g_{\o\o'}(q^{\pm c/2}z/w)^{\pm1}x_{\o'}^\pm(w)\psi_{\o}^+(z),\quad \psi_{\o}^-(z)x_{\o'}^\pm(w)=g_{\o\o'}(q^{\mp c/2}z/w)^{\pm1}x_{\o'}^\pm(w)\psi_{\o}^-(z)\\ &[x_\o^+(z),x_{\o'}^-(w)]=\dfrac{\d_{\o,\o'}}{q-q^{-1}}\left[\d(q^{-c}z/w)\psi_\o^+(q^{-c/2}z)-\d(q^c z/w)\psi^-_{\o}(q^{c/2} z)\right]\\ &\sum_{\s\in S_2}\left[x_\o^\pm(z_{\s(1)})x_\o^\pm(z_{\s(2)})x_{\o\pm1}^\pm(w)-(q+q^{-1})x_\o^\pm(z_{\s(1)})x_{\o\pm1}^\pm(w)x_\o^\pm(z_{\s(2)})+x_{\o\pm1}^\pm(w)x_\o^\pm(z_{\s(1)})x_\o^\pm(z_{\s(2)})\right]=0, \end{split}\end{aligned}$$ and $\psi_{\o,0}^+\psi_{\o,0}^-=\psi_{\o,0}^-\psi_{\o,0}^+=1$. In these relations, $q\in\mathbb{C}^\times$, $c$ is a central element, and the matrix $g_{\o\o'}(z)$ writes[^16] $$g_{\o\o'}(z)=q^{-\b_{\o\o'}}\dfrac{1-q^{\b_{\o\o'}}\k^{m_{\o\o'}}z}{1-q^{-\b_{\o\o'}}\k^{m_{\o\o'}}z},\quad g_{\o\o'}(z^{-1})=g_{\o'\o}(z)^{-1}=g_{\o\o'}(\k^{-2m_{\o\o'}}z)^{-1}.$$ In order to compare with the gauge theory quantities, we should set $q=q_3^{1/2}$, $\k=(q_1/q_2)^{1/2}$, then $$\label{def_g_glp} g_{\o\o'}(z)=\left(q_3^{-1}\dfrac{1-q_3z}{1-q_3^{-1}z}\right)^{\d_{\o,\o'}}\left(q_3^{1/2}\dfrac{1-q_1 z}{1-q_2^{-1} z}\right)^{\d_{\o,\o'-1}}\left(q_3^{1/2}\dfrac{1-q_2 z}{1-q_1^{-1} z}\right)^{\d_{\o,\o'+1}}.$$ #### Modes decomposition The algebraic relations \[algebra\_glp\] can also be directly written for the modes of the Drinfeld currents. In particular, introducing $$\psi_\o^\pm(z)=\psi_{\o,0}^\pm\exp\left(\pm\sum_{k\geq1}z^{\mp k}a_{\o,\pm k}\right),$$ we find, $$\psi_{\o,0}^+x_{\o'}^\pm(z)=q^{\pm \b_{\o\o'}}x_{\o'}^\pm(z)\psi_{\o,0}^+,\quad[a_{\o,k},a_{\o',l}]=(q^{kc}-q^{-kc})c_{\o\o'}^{(k)}\d_{k+l},\quad [a_{\o,k},x^\pm_{\o',l}]=\pm q^{\mp|k|c/2}c_{\o\o'}^{(k)}x_{\o',k+l}^\pm,$$ where the coefficients $c_{\o\o'}^{(k)}$ appear in the expansion of $\log g_{\o\o'}(z)$:[^17] $$[g_{\o\o'}(z)]_\pm=q^{\pm \b_{\o\o'}}\exp\left(\pm\sum_{k>0}z^{\mp k}c_{\o\o'}^{(\pm k)}\right),\quad c_{\o\o'}^{(k)}=c_{\o'\o}^{(-k)}=\dfrac1k\k^{-km_{\o\o'}}(q^{k\b_{\o\o'}}-q^{-k\b_{\o\o'}}),$$ where $[\cdots]_\pm$ denotes the expansion in powers of $z^{\mp1}$. In addition to the central charge $c$, it is possible to define a second central charge using the zero modes of the Cartan currents: $$\prod_{\o=0}^{p-1}\psi_{\o,0}^\pm=q^{\mp\bc}.$$ Finally, the algebra can be supplemented with the following grading operators, $$\begin{aligned} \begin{split} &q^d x_{\o}^\pm(z)q^{-d}=x^\pm_\o(q^{-1}z),\quad q^d\psi^\pm_{\o}(z)q^{-d}=\psi_{\o}^\pm(q^{-1}z),\\ &q^{\bd_\o} x_{\o'}^\pm(z)q^{-\bd_\o}=q^{\pm\d_{\o,\o'}}x_{\o'}^\pm(z),\quad q^{\bd_\o}\psi_{\o'}^\pm(z)q^{-\bd_\o}=\psi_{\o'}^\pm(z). \end{split}\end{aligned}$$ #### Coproduct The Drinfeld coproduct for the quantum toroidal $\glp$ algebra takes the following form, $$\begin{aligned} \begin{split}\label{Drinfeld_glp} &\D(x_\o^+(z))=x_\o^+(z)\otimes 1+\psi_\o^-(q^{c_{(1)}/2}z)\otimes x_\o^+(q^{c_{(1)}}z),\\ &\D(x_\o^-(z))=x_\o^-(q^{c_{(2)}}z)\otimes \psi_\o^+(q^{c_{(2)}/2}z)+1\otimes x_\o^-(z),\\ &\D(\psi_\o^\pm(z))=\psi_\o^\pm(q^{\pm c_{(2)}/2}z)\otimes\psi_\o^\pm(q^{\mp c_{(1)}/2}z), \end{split}\end{aligned}$$ with $d$, $\bd_\o$, and $c$ co-commutative. This coproduct, and the algebraic relations \[algebra\_glp\], are the same as those presented in [@Awata2017], upon the change of notations $$x_\o^+(z)\to E_i(z),\quad x_\o^-(z)\to F_i(z),\quad \psi_\o^\pm(z)\to K_\o^\pm(\qf^{c/2}z),\quad q^{c_{(1)}}\to C_1,\quad q^{c_{(2)}}\to C_2.$$ Horizontal representation ------------------------- Representations of this type have central charge $c=1$, they have been constructed by Saito in [@Saito1996] under the name *vertex representations*. We review here this construction. For $c\neq0$, the Cartan modes $a_{\o,k}$ define $p$ coupled Heisenberg subalgebras. For later convenience we introduce the rescaled modes $$\label{com_alpha} \a_{\o,k}=\dfrac{k}{q^k-q^{-k}}\rho^{(H)}(a_{\o,k})\implies [\a_{\o,k},\a_{\o',l}]=k\d_{k+l}\dfrac{q^{k\b_{\o\o'}}-q^{-k\b_{\o\o'}}}{q^k-q^{-k}}\k^{-km_{\o\o'}}.$$ The representation of the currents $x_\o^\pm(z)$ and $\psi_\o^\pm(z)$ can be factorized into two commuting parts: a *zero mode* part ($X_\o^\pm(z)$, $Y_\o^\pm(z)$), and a vertex operator part ($\eta_\o^\pm(z)$, $\vphi_\o^\pm(z)$) built over the Cartan modes $\a_{\o,k}$: $$\rho^{(H)}(x_\o^\pm(z))=X_\o^\pm(z)\eta_\o^\pm(z),\quad \rho^{(H)}(\psi_\o^\pm(z))=Y_\o^\pm\vphi_\o^\pm(z).$$ We focus first on the vertex operators part, it writes $$\eta_\o^\pm(z)=:\exp\left(\mp\sum_{k\in\mathbb{Z}}\dfrac{z^{-k}}{k}q^{\mp|k|/2}\a_{\o,k}\right):,\quad \vphi_\o^\pm(z)=\exp\left(\pm\sum_{k>0}\dfrac{z^{\mp k}}{k}(q^k-q^{-k})\a_{\o,\pm k}\right),$$ note that $\vphi_\o^\pm(z)=:\eta_\o^+(q^{\pm1/2}z)\eta_\o^-(q^{\mp1/2}z):$. The Fock vacuum $\ket{\vac}$ is annihilated by positive modes $\a_{\o,k>0}$, and we define accordingly the normal ordering $:\cdots:$ by writing positive modes on the right. It is a matter of simple algebra to derive the following normal-ordering relations:$$\begin{aligned} \begin{split} &\eta_\o^+(z)\eta_{\o'}^+(w)=S_{\o'\o}(w/z)^{-1}:\eta_\o^+(z)\eta_{\o'}^+(w):,\quad\eta_\o^-(z)\eta_{\o'}^-(w)=S_{\o'\o}(q^2w/z)^{-1}:\eta_\o^-(z)\eta_{\o'}^-(w):,\\ &\eta_\o^\pm(z)\eta_{\o'}^\mp(w)=S_{\o'\o}(qw/z):\eta_\o^\pm(z)\eta_{\o'}^\mp(w):,\\ &\vphi_\o^+(q^{\mp1/2}z)\eta_{\o'}^\pm(w)=\left(\dfrac{S_{\o'\o}(q^2w/z)}{S_{\o'\o}(w/z)}\right)^{\pm1}:\vphi_\o^+(q^{\mp1/2}z)\eta_{\o'}^\pm(w):,\\ &\eta_{\o}^\pm(z)\vphi_{\o'}^-(q^{\pm1/2}w)=\left(\dfrac{S_{\o'\o}(q^2w/z)}{S_{\o'\o}(w/z)}\right)^{\pm1}:\eta_{\o}^\pm(z)\vphi_{\o'}^-(q^{\pm1/2}w):,\\ &\vphi_\o^+(z)\vphi_{\o'}^-(w)=\left(\dfrac{S_{\o'\o}(qw/z)^2}{S_{\o'\o}(q^{-1}w/z)S_{\o'\o}(q^3w/z)}\right):\vphi_\o^+(z)\vphi_{\o'}^-(w):, \end{split}\end{aligned}$$ with the function $$\label{S_glp} S_{\o\o'}(z)=\exp\left(\sum_{k>0}\dfrac1kz^kq^{-k}\k^{km_{\o\o'}}\dfrac{q^{k\b_{\o\o'}}-q^{-k\b_{\o\o'}}}{q^k-q^{-k}}\right).$$ In fact, it is possible to resum the infinite series and write the matrix elements $S_{\o\o'}(z)$ as simple rational functions: $$\label{S_glp_II} S_{\o\o'}(z)=\prod_{r=0}^{|\b_{\o\o'}|-1}(1-\k^{m_{\o\o'}}q^{2r-|\b_{\o\o'}|}z)^{-\sign(\b_{\o\o'})}=\dfrac{(1-q_1z)^{\d_{\o,\o'-1}}(1-q_2z)^{\d_{\o,\o'+1}}}{(1-z)^{\d_{\o,\o'}}(1-q_1q_2z)^{\d_{\o,\o'}}}.$$ We then observe the crossing symmetry, $$\label{rel_S_glp} S_{\o\o'}(q^2/z)=f_{\o\o'}(z)S_{\o'\o}(z),\quad f_{\o\o'}(z)=F_{\o\o'}z^{\b_{\o\o'}},\quad F_{\o\o'}=(-q)^{-\b_{\o\o'}}\k^{-m_{\o\o'}\b_{\o\o'}},$$ and $F_{\o\o'}F_{\o'\o}=q^{-2\b_{\o\o'}}$, $f_{\o\o'}(q^2/z)f_{\o'\o}(z)=1$. The structure function $g_{\o\o'}(z)$ can be written as a ratio of functions $S_{\o\o'}(z)$ with shifted arguments, $$\label{rel_g_S_glp} g_{\o\o'}(z)=q^{-\b_{\o\o'}}\dfrac{S_{\o\o'}(z)}{S_{\o\o'}(q^2z)}=f_{\o'\o}(qz)\dfrac{S_{\o\o'}(z)}{S_{\o'\o}(z^{-1})}.$$ We now turn to the analysis of the zero-modes. In [@Saito1996], Saito introduces the symbols $e^{\a_\o}$ associated to the roots $\a_\o$, and obeying the commutation relations $e^{\a_\o}e^{\a_{\o'}}=(-1)^{\b_{\o\o'}}e^{\a_{\o'}}e^{\a_\o}$ (in particular symbols attached to the same root commute). These symbols, together with the operators $a_{\o,0}$ and $\p_{\a_\o}$ act on states parameterized by a root $\a=\sum_{\o\in\Zp} r_\o\a_\o$ ($r_\o\in\mZ$) and a fundamental weight $\L_{\o_0}$, $$\begin{aligned} \begin{split}\label{zm_glp} &e^{\a_\o}\ket{\a,\L_{\o_0}}=\prod_{\o'<\o}(-1)^{r_{\o'}\b_{\o\o'}}\ket{\a+\a_\o,\L_{\o_0}},\quad \p_{\a_\o}\ket{\a,\L_{\o_0}}=\la \a_\o^\vee,\b+\L_{\o_0}\ra\ \ket{\a,\L_{\o_0}},\\ &z^{a_{\o,0}}\ket{\a,\L_{\o_0}}=z^{\la \a_\o^\vee,\a+\L_{\o_0}\ra}\prod_{\o'}\k^{r_{\o'}\b_{\o\o'}m_{\o\o'}/2}\ket{\a,\L_{\o_0}}. \end{split}\end{aligned}$$ In this representation, $$q^{\p_{\a_\o}}e^{\a_{\o'}}=q^{\b_{\o\o'}}e^{\a_{\o'}}q^{\p_{\a_\o}},\quad z^{a_{\o,0}}e^{\a_{\o'}}=z^{\b_{\o\o'}}\k^{m_{\o\o'}\b_{\o\o'}/2}e^{\a_{\o'}}z^{a_{\o,0}}.$$ Thus, introducing $X_\o^\pm(z)=e^{\pm\a_\o}z^{1\pm a_{\o,0}}$ and $Y_\o^\pm=q^{\pm\p_{\a_\o}}$, we find the algebraic relations$$\begin{aligned} \begin{split} &X_\o^\pm(z)X_{\o'}^\pm(w)=f_{\o'\o}(qz/w)X_{\o'}^\pm(w)X_\o^\pm(z),\quad X_\o^\pm(z)X_{\o'}^\mp(w)=f_{\o'\o}(qz/w)^{-1}X_{\o'}^\mp(w)X_\o^\pm(z),\\ &Y_\o^+X_{\o'}^\pm(w)=q^{\pm \b_{\o\o'}}X_{\o'}^\pm(w)Y_\o^+,\quad Y_\o^-X_{\o'}^\pm(w)=q^{\mp \b_{\o\o'}}X_{\o'}^\pm(w)Y_\o^-,\quad [Y_\o^+,Y_{\o'}^-]=0. \end{split}\end{aligned}$$ It is easy to verify that these are indeed the factors needed to reproduce the algebraic relations \[algebra\_glp\]. The only difficulty appears in the verification of the commutation relation $[x_\o^+,x_{\o'}^-]$ for which we need to use the property $z^{a_{\o,0}}w^{-a_{\o,0}}=z^{\p_{\a_\o}}w^{-\p_{\a_\o}}$ to treat the zero mode dependence. The value of the central charge $\bc$ can be recovered by noticing that $$\sum_{\o\in\Zp}\la \a_\o^\vee,\b+\L_{\o_0}\ra=1\implies \prod_{\o\in\Zp} q^{\p_{\a_\o}}=q.$$ This representation has been extended to higher level $\bc$ in [@Awata2017]. Note however that in the definition of the $(\nu_1,\nu_2)$-deformed horizontal representation, a set of $4p$ Heisenberg algebras will be employed to define to the zero-modes $X_\o^\pm$ and $Y_\o^\pm$ instead of the symbols introduced in \[zm\_glp\]. Vertical representations {#AppA3} ------------------------ The vertical representations have central charge $c=0$ and thus the Cartan currents $\psi_{\omega}^\pm(z)$ commute. They are diagonal in the basis of states ${\ket{\bl}\!\rangle}$ labeled by $m$-tuple Young diagram $\bl=(\l^{(1)},\cdots,\l^{(m)})$. The representation depends on an $m$-vector of weights $\bv=(v_1,\cdots,v_m)$ and a choice of coloring $c_\a$ for each component $v_\a$. We denote $m_\o=|C_\o(m)|$ the number of weights $v_\a$ of color $c_\a=\o$ (obviously, $m=\sum_{\o\in\Zp} m_\o$). The action of the Drinfeld currents on the states ${\ket{\bl}\!\rangle}$ reads $$\begin{aligned} \begin{split}\label{vertical} &\rho^{(V)}(x_\omega^+(z)){\ket{\bl}\!\rangle}=(qz)^{-\gbY}\prod_{\mAbox\in\bl}(-\k)^{-m_{\o c(\sAbox)}/2}\sum_{\mAbox\in A_\omega(\boldsymbol\l)}\d(z/\chi_{\sAbox})\res_{z=\chi_{\sAbox}}\dfrac1{z\tilde{\CY}\oY(z)} {\ket{\bl+\Abox}\!\rangle},\\ &\rho^{(V)}(x_\omega^-(z)){\ket{\bl}\!\rangle}=qz^{\gbY+2}\prod_{\mAbox\in\bl}(-\k)^{m_{\o c(\sAbox)}/2} \sum_{\mAbox\in R_\omega(\boldsymbol\l)}\d(z/\chi_{\sAbox})\res_{z=\chi_{\sAbox}}z^{-1}\tilde{\CY}\oY(q_3^{-1}z){\ket{\bl-\Abox}\!\rangle},\\ &\rho^{(V)}(\psi_\omega^\pm(z)){\ket{\bl}\!\rangle}=\left[\tilde{\Psi}\oY(z)\right]_\pm{\ket{\bl}\!\rangle}. \end{split}\end{aligned}$$ In the first two lines, summations are performed over the set of boxes of color $\o$ that can be added ($A_\o(\bl)$) to or removed from ($R_\o(\bl)$) the $m$-tuple Young diagram $\bl$. The summands are expressed in terms of residues involving the functions $\tilde{\CY}\oY(z)$, and the action of the Cartan is given as an expansion of the functions $\tilde{\Psi}\oY(z)$ in powers of $z^{\mp1}$. These two sets of functions are defined as follows:$$\begin{aligned} \begin{split}\label{def_PsiY} &\tilde{\Psi}\oY(z)=q^{-m_{\o}}\prod_{\a\in C_\o(m)}\dfrac{1-q^2v_\a/z}{1-v_\a/z}\prod_{\mAbox\in\bl}g_{\o c(\sAbox)}(z/\chi_{\sAbox})=q^{-\gbY}\dfrac{\tilde{\CY}\oY(q^{-2}z)}{\tilde{\CY}\oY(z)},\\ &\tilde{\CY}\oY(z)=\prod_{\a\in C_\o(m)}(1-v_\a/z)\prod_{\mAbox\in\bl}S_{c(\sAbox)\o}(\chi_{\sAbox}/z),\quad \gbY=m_\o-\sum_{\mAbox\in\bl}\b_{\o c(\sAbox)}. \end{split}\end{aligned}$$ The zero modes of the Cartan act as $$\rho^{(V)}(\psi_{\o,0}^\pm){\ket{\bl}\!\rangle}=q^{\mp\gbY}{\ket{\bl}\!\rangle}.$$ and, taking the product over the index $\o$, we deduce the level $\rho^{(V)}(\bc)=m$. Deformation of the algebra {#AppA4} -------------------------- In order to define the algebraic engineering for the gauge theory on $(\nu_1,\nu_2)$-deformed $\Zp$-orbifolds, the physical quantity we need to reproduce is the scattering function $S_{\o\o'}(z)$ defined in \[def\_S\]. In this scope, it is easier to deform first the horizontal representations, and reproduce the commutation of the Heisenberg subalgebras \[def\_sigma\] using the $\glp$ formula \[com\_alpha\]. It leads to identify the Cartan matrix with the matrix $\b_{\o\o'}$ defined in \[def\_beta\]. Furthermore, the factor $\k^{m_{\o\o'}}$ has to be replaced with a more general matrix $\k_{\o\o'}$ that reads[^18] $$\k_{\o\o'}=q^{-1}q_3^{\d_{\o,\bo'}}q_1^{-\d_{\o,\o'+\nu_1}}q_2^{-\d_{\o,\o'+\nu_2}}.$$ Using this identification, the crossing symmetry relation \[crossing\] reduces to \[rel\_S\_glp\] for $\nu_3=0$, and the formula \[rel\_S\_glp\] for $F_{\o\o'}$ reproduces the definition \[def\_beta\]. The function $S_{\o\o'}(z)$ defined in \[def\_S\] is recovered by replacing the expression \[S\_glp\] with $$S_{\o\o'}(z)=\exp\left(\sum_{k>0}\dfrac{z^k}{k}q^{-k}\k_{\o'\o}^{-k}\dfrac{q^{k\b_{\o'\o}}-q^{-k\b_{\o'\o}}}{q^k-q^{-k}}\right).$$ The definition of the deformed structure functions $g_{\o\o'}(z)$ is a little more difficult because of the freedom in defining the *zero modes* factors. Comparing with the formula \[rel\_g\_S\_glp\] for the case of $\glp$, the most natural choice would be $$g_{\o\o'}^{(1)}(z)=f_{\o\o'}(qz)\dfrac{S_{\o\o'}(z)}{S_{\o'\o}(z^{-1})}=q^{\b_{\o\o'}}F_{\o\o'}^2\prod_{i=1,2,3}\dfrac{(1-q_iz)^{\d_{\o,\o'-\nu_i}}}{(1-q_i^{-1}z)^{\d_{\o,\o'+\nu_i}}}.$$ Unfortunately, with this definition, the identity $g^{(1)}_{\o\o'}(z)g^{(1)}_{\o'\o}(z^{-1})=1$ is NOT satisfied, yet it is necessary for the consistency of the algebraic relations. It prompted us to propose instead the definition given in \[def\_g\] where the factor $f_{\o\o'}(qz)$ is missing. Unfortunately, this redefinition of the structure functions $g_{\o\o'}(z)$ breaks the natural symmetry between positive and negative currents, and makes the definition of the central charge $\bc$ more difficult. Note also that another possibility could have been to define $$g_{\o\o'}^{(2)}(z)=\prod_{i=1,2,3}\dfrac{(1-q_iz)^{\d_{\o,\o'-\nu_i}}}{(1-q_iz^{-1})^{\d_{\o,\o'+\nu_i}}},$$ but it would require us to modify the definition of the functions $S_{\o\o'}(z)$: $$g_{\o\o'}^{(2)}(z)=-\dfrac{S_{\o\o'}^{(2)}(z)}{S_{\o'\o}^{(2)}(z^{-1})},\quad S_{\o\o'}^{(2)}(z)=\dfrac{(1-q_1z)^{\d_{\o,\o'-\nu_1}}(1-q_2z)^{\d_{\o,\o'-\nu_2}}}{(1-z^{-1})^{\d_{\o,\o'}}(1-q_3z^{-1})^{\d_{\o,\o'-\nu_1-\nu_2}}}.$$ In fact, the proper generalization of the quantum toroidal $\glp$ algebra’s structure functions is found in appendix \[AppE\] (formula \[def\_tg\]). It takes into account the twist of the Drinfeld currents, and involves half-integer powers of the spectral variable $z$. Shell formula {#AppB0} ============= (3,0) – (4,0); (0,3) – (0,4); (3,0) – (0,0) – (0,3); (1,0) – (1,1) – (0,1); (1,0) – (2,0) – (2,1) – (1,1) – (1,2) – (0,2); at (0.5,0.5) [$\Abox$]{}; at (1.5,0.5) [$q_2\Abox$]{}; at (0.5,1.5) [$q_1\Abox$]{}; (-2,1) – (-3,1); (2,0) – (3,0); (-2,1) – (0,1) – (0,0) – (2,0) ; (0,1) – (1,1) – (1,0) ; (1,1) – (2,1) – (2,0); (-1,1) – (-1,0) – (0,0) ; at (0.5,0.5) [$\Abox$]{}; at (1.5,0.5) [$q_1\Abox$]{}; at (-0.5,0.5) [$q_1^{-1}\Abox$]{}; (1,-2) – (1,-3); (0,2) – (0,3); (0,2) – (0,0) – (1,0) – (1,-2) ; (0,1) – (1,1) – (1,0) ; (0,2) – (1,2) – (1,1); (0,0) – (0,-1) – (1,-1); at (0.5,0.5) [$\Abox$]{}; at (0.5,1.5) [$q_2\Abox$]{}; at (0.5,-0.5) [$q_2^{-1}\Abox$]{}; (-2,2) – (-3,2); (0,0) – (0,-1); (-2,2) – (0,2) – (0,0) ; (-1,2) – (-1,1) – (0,1) ; (-2,2) – (-2,1) – (-1,1) – (-1,0) – (0,0) ; at (-0.5,1.5) [$\Abox$]{}; at (-1.5,1.5) [$q_1^{-1}\Abox$]{}; at (-0.5,0.5) [$q_2^{-1}\Abox$]{}; We provide here a short derivation of the shell formula \[shell\_Y\] for the functions $\CYY(z)$. Since $K_\o(\bl)$ is a direct sum of $K_\o(\l^{(\a)})$, the function $\CYY(z)$ factorizes into contributions of the individual Young diagrams $\CY^{[\l^{(\a)}]}_\o(z)$. Thus, it is possible to focus on the case of a single Young diagram $\l^{(\a)}$ corresponding to a weight $v_\a$ of color $c_\a$. The proof will be done by recursion on the number of boxes. We start with an empty Young diagram, for which $R_\o(\vac)=\vac$. The box $\Abox=(1,1)$ is of color $c(\Abox)=c_\a$, thus $A_\o(\vac)=\{(1,1)\}$ if $\o=c_\a$ and $A_\o(\vac)=\vac$ otherwise. Accordingly, we recover $\CY_\o^{[\vac]}(z)=(1-v_\a/z)^{\d_{\o,c_\a}}$. Now, let’s add a box $\Abox$ to $\l^{(\a)}$. From the definition \[def\_Psi\], we have $$\label{var_CYY} \dfrac{\CY_\o^{[\l^{(\a)}+\mAbox]}(z)}{\CY_\o^{[\l^{(\a)}]}(z)}=\dfrac{(1-q_1\chi_{\sAbox}/z)^{\d_{c(\sAbox),\o-\nu_1}}(1-q_2\chi_{\sAbox}/z)^{\d_{c(\sAbox),\o-\nu_2}}}{(1-\chi_{\sAbox}/z)^{\d_{c(\sAbox),\o}}(1-q_1q_2\chi_{\sAbox}/z)^{\d_{c(\sAbox),\o-\nu_1-\nu_2}}}.$$ There are four possible configurations for adding box in $\l^{(\a)}$, all represented in figure \[fig\_Young\]. We start with the generic case, for which $$\begin{aligned} \begin{split} &A_\o(\l^{(\a)}+\Abox)=\left(A_\o(\l^{(\a)})\setminus\{\Abox\mid c(\Abox)=\o\}\right)\cup\{q_1\Abox\mid c(\Abox)=\o-\nu_1\}\cup\{q_2\Abox\mid c(\Abox)=\o-\nu_2\},\\ &R_{\o-\nu_1-\nu_2}(\l^{(\a)}+\Abox)=R_{\o-\nu_1-\nu_2}(\l^{(\a)})\cup\{\Abox\mid c(\Abox)=\o-\nu_1-\nu_2\}. \end{split}\end{aligned}$$ We employed here the shortcut notation $q_1^{\pm1}\Abox$ ($q_2^{\pm1}\Abox$) to designate the box of coordinate $(i\pm1,j)$ (resp. $(i,j\pm1)$) next to $\Abox=(i,j)$. In this generic case, the factors induced by the variation of the content of the sets $A_\o$ and $R_{\o-\nu_1-\nu_2}$ reproduce the extra factor $S_{c(\sAbox)\o}(\chi_{\sAbox}/z)$ in the RHS of \[var\_CYY\]. We now turn to the first case of the degenerate configurations represented on figure . In this case, only one more box can be added to $A_\o(\l^{(\a)}+\Abox)$. On the other hand, the addition of the box $\Abox$ prevents the removal of the box $q_1^{-1}\Abox$. As a result, $$\begin{aligned} \begin{split} &A_\o(\l^{(\a)}+\Abox)=\left(A_\o(\l^{(\a)})\setminus\{\Abox\mid c(\Abox)=\o\}\right)\cup\{q_1\Abox\mid c(\Abox)=\o-\nu_1\},\\ &R_{\o-\nu_1-\nu_2}(\l^{(\a)}+\Abox)=\left(R_{\o-\nu_1-\nu_2}(\l^{(\a)})\cup\{\Abox\mid c(\Abox)=\o-\nu_1-\nu_2\}\right)\setminus\{q_1^{-1}\Abox\mid c(\Abox)=\o-\nu_2\}. \end{split}\end{aligned}$$ Once again, we observe the agreement between the variation of the RHS \[shell\_Y\] and the recursion relation \[var\_CYY\]. The other two cases are treated in the same way. Representations of the extended algebra {#AppC} ======================================= Vertical representation {#AppB1} ----------------------- The vertical representation is of the highest weight type. The highest state ${\ket{\vac}\!\rangle}$, also called *vacuum state*, is annihilated by the currents $x_\o^-(z)$, while $x_\o^+(z)$ create excitations. The excited states ${\ket{\bl}\!\rangle}$ are parameterized by an $m$-tuple Young diagram $\bl$. The weights $\bv=(v_1,\cdots,v_m)$ parameterize the action of the Cartan $\psi_\o^\pm(z)$ on the vacuum state. The two Cartan currents commute, they are diagonal in the basis ${\ket{\bl}\!\rangle}$, with the eigenvalue $[\PsiY(z)]_\pm$ where $\pm$ denotes an expansion in powers of $z^{\mp1}$. The action of $x_\o^\pm(z)$ add/remove a box of color $\o$. In order to produce the Dirac $\d$-function in the commutator $[x^+,x^-]$, it is natural to assume that modes $x_{\o,k}^\pm$ depends on the index $k$ only through a factor of $\chi_{\sAbox}^k$ where $\Abox$ is the box that is added/removed. Taking all these assumptions in consideration, we arrive at the following ansatz: $$\begin{aligned} \begin{split}\label{ansatz_vert} &x_\omega^+(z){\ket{\bl}\!\rangle}=\sum_{\mAbox\in A_\omega(\boldsymbol\l)}\d(z/\chi_{\sAbox}) A^{+[\bl]}_\o(x) {\ket{\bl+\Abox}\!\rangle},\\ &x_\omega^-(z){\ket{\bl}\!\rangle}= \sum_{\mAbox\in R_\omega(\boldsymbol\l)}\d(z/\chi_{\sAbox})A^{-[\bl]}_\o(x) {\ket{\bl-\Abox}\!\rangle},\\ &\psi_\omega^\pm(z){\ket{\bl}\!\rangle}=\left[\PsiY(z)\right]_\pm{\ket{\bl}\!\rangle}, \end{split}\end{aligned}$$ where $A^{\pm[\bl]}_\o(x)$ are the coefficients to be determined. When the central charge $c$ is vanishing, the algebra \[algebra\] simplifies drastically, $$\begin{aligned} \label{algebra_vert} \begin{split} &x_\o^\pm(z)x_{\o'}^\pm(w)=g_{\o\o'}(z/w)^{\pm1}x_{\o'}^\pm(w)x_\o^\pm(z),\quad [\psi_\o^+(z),\psi_{\o'}^-(w)]=[\psi_\o^\pm(z),\psi_{\o'}^\pm(w)]=0,\\ &\psi^+_\o(z)x^\pm_{\o'}(w)=g_{\o\o'}(z/w)^{\pm1}x^\pm_{\o'}(w)\psi^+_\o(z),\quad \psi^-_\o(z)x^\pm_{\o'}(w)=g_{\o\o'}(z/w)^{\pm1} x^\pm_{\o'}(w)\psi^-_\o(z),\\ &[x_\o^+(z),x_{\o'}^-(w)]=\kO\d_{\o,\o'}\d(z/w)\left[\psi_\o^+(z)-\psi^-_{\o}(z)\right]. \end{split}\end{aligned}$$ Plugging in the ansatz \[ansatz\_vert\], and the expression \[def\_Psi\] for $\PsiY(z)$, we find that these relations are satisfied provided that $$\begin{aligned} \label{3constraints} \begin{split} &A^{-[\bl]}_\o(x)A^{+[\bl-\Abox]}_\o(x)=\kO\res_{z=\chi_{\sAbox}}z^{-1}\PsiY(z),\quad \mAbox\in R_\o(\l),\\ &A^{+[\bl]}_\o(x)A^{-[\bl+\Abox]}_\o(x)=-\kO\res_{z=\chi_{\sAbox}}z^{-1}\PsiY(z),\quad \mAbox\in A_\o(\l),\\ &\dfrac{A_\o^{\pm[\bl\pm x]}(y)}{A_\o^{\pm[\bl]}(y)}=g_{\o\o'}(\chi_y/\chi_{\sAbox})^{\pm1}\dfrac{A_{\o'}^{\pm[\bl\pm y]}(x)}{A_{\o'}^{\pm[\bl]}(x)},\quad c(\sAbox)=\o',\quad c(y)=\o. \end{split}\end{aligned}$$ The first two relations come from the projection of the commutator $[x^+,x^-]$ on the basis ${\ket{\bl}\!\rangle}$, decomposing the RHS as $$\label{poles_Psi} \left[\PsiY(z)\right]_+-\left[\PsiY(z)\right]_-=\sum_{\mAbox\in A_\o(\bl)}\d(z/\chi_{\sAbox})\res_{z=\chi_{\sAbox}}z^{-1}\PsiY(z)+\sum_{\mAbox\in R_\o(\bl)}\d(z/\chi_{\sAbox})\res_{z=\chi_{\sAbox}}z^{-1}\PsiY(z).$$ The last equation in \[3constraints\] arises from the exchange relations $x^\pm x^\pm$. Then, it is simply a matter of calculation to check that the following coefficients do indeed satisfy the relations \[3constraints\], $$\begin{aligned} \begin{split} &A^{+[\bl]}_\o(\Abox)=F^{1/2}\res_{z=\chi_{\sAbox}}z^{-1}\CYY(z)^{-1}=\kO\CY_\o^{[\bl+\mAbox]}(\chi_{\sAbox})^{-1},\\ &A^{-[\bl]}_\o(\Abox)={\mathring{f}}_{\bo}^{[\bl]}(q_3^{-1}z)\res_{z=\chi_{\sAbox}}z^{-1}\CY_{\bo}^{[\bl]}(q_3^{-1}z)=-\kO F^{1/2}{\mathring{f}}_{\bo}^{[\bl]}(q_3^{-1}z)\CY_{\bo}^{[\bl-\mAbox]}(q_3^{-1}\chi_{\sAbox}). \end{split}\end{aligned}$$ Horizontal representation {#AppB2} ------------------------- Here the strategy is to start by computing the algebraic relations satisfied by the vertex operators $\eta_\o^\pm$ and $\vphi_\o^\pm$, compare them with \[algebra\], and introduce the zero-modes factors to compensate unwanted factors. Using the definition \[def\_eta\_vphi\], we can compute the normal-ordering relations $$\begin{aligned} \begin{split} &\eta_\o^+(z)\eta_{\o'}^+(w)=S_{\o'\o}(w/z)^{-1}:\eta_\o^+(z)\eta_{\o'}^+(w):,\quad \eta_\o^+(z)\eta_{\o'}^-(w)=S_{\o'\o}(w/z):\eta_\o^+(z)\eta_{\o'}^-(w):,\\ &\eta_\o^-(z)\eta_{\o'}^-(w)=S_{\o'\bo}(q_3w/z)^{-1}:\eta_\o^-(z)\eta_{\o'}^-(w):=f_{\o'\o}(z/w)^{-1}S_{\o\o'}(z/w)^{-1}:\eta_\o^-(z)\eta_{\o'}^-(w):,\\ &\eta_\o^-(z)\eta_{\o'}^+(w)=S_{\o'\ \bo}(q_3w/z):\eta_\o^-(z)\eta_{\o'}^+(w):=f_{\o'\o}(z/w)S_{\o\o'}(z/w):\eta_\o^-(z)\eta_{\o'}^+(w):, \end{split}\end{aligned}$$ and, since $\vphi_\o^+(z)=:\eta_\o^+(z)\eta_\o^-(z):$ and $\vphi_\o^-(z)=:\eta_{\bo}^+(q_3^{-1}z)\eta_\o^-(z):$, $$\begin{aligned} \begin{split} &\vphi_\o^+(z)\eta_{\o'}^\pm(w)=f_{\o'\o}(z/w)^{\pm1}g_{\o\o'}(z/w)^{\pm1}:\vphi_\o^+(z)\eta_{\o'}^\pm(w):,\\ &\eta_{\o}^+(z)\vphi_{\o'}^-(w)=f_{\bar\o'\o}(q_3z/w)g_{\o\bar\o'}(q_3z/w):\eta_{\o}^+(z)\vphi_{\o'}^-(w):\\ &\eta_{\o}^-(z)\vphi_{\o'}^-(w)=f_{\o'\o}(z/w)^{-1}g_{\o\o'}(z/w)^{-1}:\eta_{\o}^-(z)\vphi_{\o'}^-(w):,\\ &\vphi_\o^+(z)\vphi_{\o'}^-(w)=\dfrac{f_{\bo'\o}(q_3z/w)}{f_{\o'\o}(z/w)}\dfrac{g_{\o\bar\o'}(q_3z/w)}{g_{\o\o'}(z/w)}:\vphi_\o^+(z)\vphi_{\o'}^-(w):. \end{split}\end{aligned}$$ We deduce the algebraic relations between vertex operators. Comparing them with the currents algebra \[algebra\] at $c=1$, we observe that the latter are satisfied provided that we set $$\rho^{(H)}_{u}(x_\o^\pm(z))=X_\o^\pm(z)\eta_\o^\pm(z),\quad \rho^{(H)}_{u}(\psi_\o^\pm(z))=Y_\o^\pm(z)\vphi_\o^\pm(z),$$ with[^19] $$\begin{aligned} \label{rel_X_Y} \begin{split} &X_\o^+(z)X^+_{\o'}(w)=X^+_{\o'}(w)X_\o^+(z),\quad X_\o^-(z)X^-_{\o'}(w)=\dfrac{f_{\o'\o}(z/w)}{f_{\o\o'}(w/z)}X^-_{\o'}(w)X_\o^-(z),\\ &X_\o^+(z)X^-_{\o'}(w)=:X_\o^+(z)X^-_{\o'}(w):=f_{\o\o'}(w/z)X^-_{\o'}(w)X_\o^+(z),\\ &Y_\o^+(z)X_{\o'}^\pm(w)=f_{\o'\o}(z/w)^{\mp1}X_{\o}^\pm(z)Y_{\o'}^+(w),\\ &X_{\o}^+(z)Y_{\o'}^-(w)=f_{\bo'\o}(q_3z/w)^{-1}Y_{\o'}^-(w)X_{\o}^+(z),\quad X_{\o}^-(z)Y_{\o'}^-(w)=f_{\o'\o}(z/w)Y_{\o'}^-(w)X_{\o}^-(z),\\ &Y_\o^+(z)=F^{-1/2}:X_\o^+(z)X_\o^-(z):,\quad Y_\o^-(z)=F^{1/2}:X_{\bar\o}^+(q_3^{-1}z)X_\o^-(z):. \end{split}\end{aligned}$$ The last two relations come from the commutator $[x^+,x^-]$, they have been obtained using the pole decomposition of the function $S_{\o'\o}(w/z)$ which brings $$\label{decomp_S_g} \left[S_{\o'\o}(w/z)\right]_+-\left[S_{\o'\o}(w/z)\right]_-=\kO\left[\d_{\o,\o'}\d(z/w)F^{-1/2}-\d_{\o,\bar\o'}\d(q_3 z/w)F^{1/2}\right].$$ The relations \[rel\_X\_Y\] are satisfied if we set $$X_\o^+(z)=Q_\o(z),\quad X^-_\o(z)=Q_\o(z)^{-1}P_{\bo}(q_3^{-1}z),\quad Y^+_\o(z)=F^{-1/2}P_{\bo}(q_3^{-1}z),\quad Y^-_\o(z)=F^{1/2}\dfrac{Q_{\bo}(q_3^{-1}z)}{Q_{\o}(z)}P_{\bo}(q_3^{-1}z),$$ where $Q_\o(z)$ and $P_{\o'}(w)$ obey \[rel\_PQ\]. These operators can be constructed in terms of $2p$ Heisenberg algebras $[p_\o,q_{\o'}]=\d_{\o,\o'}$ and $[\tp_\o,\tq_{\o'}]=\d_{\o,\o'}$ by setting $$\label{def_PQ} Q_\o(z)=e^{q_\o+\tq_\o\log z},\quad P_\o(z)=z^{-\sum_{\o'}\b_{\o\o'}p_{\o'}}e^{\sum_{\o'}\b_{\o\o'}\tp_{\o'}}(-1)^{p_\o}(-q_3)^{-p_{\o+\nu_3}}(-q_1)^{-p_{\o-\nu_1}}(-q_2)^{-p_{\o-\nu_2}}.$$ Combining the operators $X_\o^\pm$, $Y_\o^\pm$ and the vertex operators $\eta_\o^\pm$, $\psi_\o^\pm$, we find the representation \[rep\_H\]. The dependence in the weights $u_\o$ and levels $n_\o$ is recovered using the freedom to shift the operators $q_\o$, $\tq_\o$ as $q_\o\to q_\o+\log(u_\o)$, $\tq_\o\to\tq_\o-n_\o$. It remains to compute the central charge $\bc$. The zero modes of the Cartan currents write $$\rho^{(H)}(\psi^+_{\o,0})=F^{-1/2}P_{\bo}(q_3^{-1})\quad \rho^{(H)}(\psi_{\o,0}^-)=F^{1/2}\dfrac{u_{\bo}}{u_\o}q_3^{n_{\bo}}\dfrac{Q_{\bo}(q_3^{-1})}{Q_{\o}(1)}P_{\bo}(q_3^{-1}).$$ We deduce that $$\rho^{(H)}\left(\prod_{\o\in\Zp}\psi_{\o,0}^+(\psi_{\o,0}^-)^{-1}\right)=q_3^{-n-\tq},\quad \tq=\sum_{\o\in\Zp}\tq_\o.$$ Since $[\tq_\o,P_{\o'}(w)]=\b_{\o'\o}P_{\o'}(w)$, the operator $\tq$ commute with $P_\o(z)$, thus it is central in this representation. Moreover, since $Q_\o(z)$ acts trivially on the dual state $\bra{\vac}$, we have $\tq=0$. Finally, we also have to take into account the non-commutation of the zero modes which brings the extra factor $$\prod_{{\genfrac{}{}{0pt}{}{\o,\o'=0}{\o\leq\o'}}}^{p-1}\dfrac{F_{\o'\o}}{F_{\bo'\o}}=\prod_{{\genfrac{}{}{0pt}{}{\o,\o'=0}{\o\leq\o'}}}^{p-1}q_3^{\b_{\o\o'}}F_{\o\o'}F_{\o'\o}=\prod_{{\genfrac{}{}{0pt}{}{\o,\o'=0}{\o\leq\o'}}}^{p-1}q_3^{\b_{\o\o'}}\times F^{-p}\prod_{\o,\o'=0}^{p-1}F_{\o\o'}=\prod_{{\genfrac{}{}{0pt}{}{\o,\o'=0}{\o\leq\o'}}}^{p-1}q_3^{\b_{\o\o'}}.$$ Since $\b_{\o\o'}$ is circulant, it is easy to compute $$\sum_{{\genfrac{}{}{0pt}{}{\o,\o'=0}{\o\leq\o'}}}^{p-1}\b_{\o\o'}= \left\{ \begin{array}{cc} p & (\nu_1+\nu_2<p),\\ 0 & \text{(else)}, \end{array} \right.$$ assuming $0\leq\nu_1,\nu_2\leq p-1$. This gives us the value of the central charge $\bc$. Automorphisms, gradings and modes expansion {#AppD} =========================================== Automorphisms and gradings {#sec_gradings} -------------------------- The algebraic relations \[algebra\] can be supplemented with the grading operators $d$ and $\bd_\o$ ($\o\in\mZ_p$) acting on the currents as $$\begin{aligned} \begin{split} &e^{\a d} x_{\o}^\pm(z)e^{-\a d}=x^\pm(e^\a z),\quad e^{\a d}\psi^\pm_{\o}(z)e^{-\a d}=\psi_{\o}^\pm(e^{\a}z),\\ &e^{\a\bd_\o} x_{\o'}^\pm(z)e^{-\a\bd_\o}=e^{\pm\a\d_{\o,\o'}}x_{\o'}^\pm(z),\quad e^{\a\bd_\o}\psi_{\o'}^+(z)e^{-\a\bd_\o}=\psi_{\o'}^+(z),\quad e^{\a\bd_\o}\psi_{\o'}^-(z)e^{-\a\bd_\o}=e^{\a(\d_{\o,\o'-\nu_3c}-\d_{\o\o'})}\psi_{\o'}^-(z), \end{split}\end{aligned}$$ for any parameter $\a\in\mC$. The grading operator $d$ reflects the invariance of the algebra under rescaling of the variable $z\to e^{\a}z$, it defines the automorphisms $\t_\a$ acting on an element $x$ of the algebra as $\t_\a(x)=e^{\a d}xe^{-\a d}$. Similarly, the grading operators $\bd_\o$ defines the automorphisms $\bar{\t}_{\o,\a}(x)=e^{\a\bd_\o}xe^{-\a\bd_\o}$ associated to the invariance under the following rescaling of the currents for a fixed $\o$: $$x_\o^\pm(z)\to e^{\pm\a}x_\o^\pm(z),\quad \psi_\o^-(z)\to e^{-\a}\psi^-_\o(z),\quad \psi_{\o+\nu_3c}^-(z)\to e^{\a}\psi_{\o+\nu_3c}^-(z),$$ while the currents $x_{\o'\neq\o}^\pm(z)$, $\psi_{\o'}^+(z)$ and $\psi_{\o'\neq\o,\o+\nu_3c}^-(z)$ remain invariant. In addition to the automorphisms $\t_\a$ and $\bar\t_{\o,\a}$, the algebraic relations are invariant under a third class of automorphisms $\tilde{\t}_{\o,\a}(x)=e^{\a\td_\o}xe^{-\a\td_\o}$ defined as $$\begin{aligned} \begin{split} &e^{\a\td_\o} x_{\o'}^\pm(z)e^{-\a\td_\o}=z^{\pm\a\d_{\o,\o'}}x_{\o'}^\pm(z),\quad e^{\a\td_\o}\psi_{\o'}^+(z)e^{-\a\td_\o}=\psi_{\o'}^+(z),\\ &e^{\a\td_\o}\psi_{\o'}^-(z)e^{-\a\td_\o}=(q_3^{-c}z)^{\a\d_{\o,\o'-\nu_3c}}z^{-\a\d_{\o\o'}}\psi_{\o'}^-(z). \end{split}\end{aligned}$$ This transformation is the generalization of the element $\CT$ of the SL$(2,\mathbb{Z})$ group of automorphisms for the quantum toroidal algebra of $\mathfrak{gl}_1$ (or Ding-Iohara-Miki algebra) [@Bourgine2017b]. With a slight abuse of terminology, we will also call $\td_\o$ a grading operator. Modes expansion --------------- In order to define properly the modes expansion of the currents $x_\o^\pm(z)$ and $\psi_\o^\pm(z)$, we need to remove some part of the zero modes factors. For this purpose, we use a twist by a combination of automorphisms to define the new currents $\tx_\o^\pm(z)$ and $\tpsi_\o^\pm(z)$ with proper modes expansion. First, we introduce the following combinations of grading operators, $$\label{expr_xi} F_\o=(-1)^{\bd_\o}(-q_3)^{-\bd_{\o+\nu_3}}(-q_1)^{-\bd_{\o-\nu_1}}(-q_2)^{-\bd_{\o-\nu_2}},\quad \b_\o=-\sum_{\o'}\b_{\o\o'}\bd_{\o'},\quad D_\o=e^{\sum_{\o'}\b_{\o\o'}\td_{\o'}},$$ such that $$\begin{aligned} \begin{split} &F_\o x_{\o'}^\pm(w)F_\o^{-1}=F_{\o\o'}^{\pm1}\ x_{\o'}^\pm(w),\quad z^{\b_\o}x_{\o'}^\pm(w)z^{-\b_\o}=z^{\mp\b_{\o\o'}}x_{\o'}^\pm(w),\quad D_\o x_{\o'}^\pm(w)D_\o^{-1}=w^{\pm\b_{\o\o'}}\ x_{\o'}^\pm(w),\\ &F_\o \psi_{\o'}^-(w)F_\o^{-1}=\dfrac{F_{\o\o'-\nu_3c}}{F_{\o\o'}}\ \psi_{\o'}^-(w),\quad z^{\b_\o}\psi_{\o'}^-(w)z^{-\b_\o}=z^{\b_{\o\o'}-\b_{\o\o'-\nu_3c}}\psi_{\o'}^-(w),\\ &D_\o \psi_{\o'}^-(w)D_\o^{-1}=q_3^{-c\b_{\o,\o'-\nu_3c}}w^{-\b_{\o\o'}+\b_{\o\o'-\nu_3c}}\ \psi_{\o'}^-(w), \end{split}\end{aligned}$$ and $\psi_{\o'}^+(w)$ remains invariant. Defining $\xi_\o(z)=z^{\b_\o}D_\o F_\o$, we find $$\begin{aligned} \begin{split}\label{rel_xi} &\xi_\o(z)x^\pm_{\o'}(w)=f_{\o\o'}(w/z)^{\pm1}x^\pm_{\o'}(w)\xi_\o(z),\quad [\xi_\o(z),\xi_{\o'}(w)]=0,\\ &\xi_\o(z)\psi_{\o'}^+(w)=\psi_{\o'}^+(w)\xi_\o(z),\quad \xi_\o(z)\psi_{\o'}^-(w)=\dfrac{f_{\o\ \o'-\nu_3c}(q_3^{-c}w/z)}{f_{\o\o'}(w/z)}\psi_{\o'}^-(w)\xi_\o(z). \end{split}\end{aligned}$$ The operator $\xi_\o(z)$ is used to define the twisted currents $$x_\o^+(z)=\tx_\o^+(z),\quad x_\o^-(z)=\tx_\o^-(z)\xi_{\bo}(q_3^{-1}z),\quad \psi_\o^+(z)=\tpsi_\o^+(z)\xi_{\bo}(q_3^{-1}z),\quad\psi_\o^-(z)=\tpsi_\o^-(z)\xi_{\o-\nu_3c}(q_3^{-c}z),$$ that satisfy the following algebraic relations, $$\begin{aligned} \begin{split}\label{algebra_tilde} &\tx_\o^+(z)\tx_{\o'}^+(w)=g_{\o\o'}(z/w)\tx_{\o'}^+(w)\tx_\o^+(z),\quad \tx_\o^-(z)\tx_{\o'}^-(w)=\dfrac{f_{\o\o'}(w/z)}{f_{\o'\o}(z/w)}g_{\o\o'}(z/w)^{-1}\tx_{\o'}^-(w)\tx_\o^-(z),\\ &\tpsi^+_\o(z)\tx^\pm_{\o'}(w)=f_{\o'\o}(z/w)^{\pm1}g_{\o\o'}(z/w)^{\pm1}\tx^\pm_{\o'}(w)\tpsi^+_\o(z),\\ &\tpsi^-_\o(z)\tx^+_{\o'}(w)=f_{\o-\nu_3c\ \o'}(q_3^cw/z)^{-1}g_{\o-\nu_3c\ \o'}(q_3^{-c}z/w)\tx^+_{\o'}(w)\tpsi^-_\o(z),\\ &\tpsi^-_\o(z)\tx^-_{\o'}(w)=f_{\o\o'}(w/z)g_{\o\o'}(z/w)^{-1}\tx^-_{\o'}(w)\tpsi^-_\o(z),\\ &\tpsi_\o^+(z)\tpsi_{\o'}^-(w)=\dfrac{f_{\o'-\nu_3c\ \o}(q_3^{c} z/w)g_{\o\o'-\nu_3c}(q_3^{c} z/w)}{f_{\o'\o}(z/w)g_{\o\o'}(z/ w)}\tpsi_{\o'}^-(w)\tpsi_\o^+(z),\\ &\tx_\o^+(z)\tx_{\o'}^-(w)-f_{\o\o'}(w/z)^{-1}\tx_{\o'}^-(w)\tx_\o^+(z)=\kO\d_{\o,\o'}\d\left(\dfrac{z}{ w}\right)\tpsi_\o^+(z)-\kO\d_{\o,\o'-\nu_3c}\d\left(\dfrac{q_3^{c} z}{w}\right)\tpsi^-_{\o'}(q_3^cz)\dfrac{\xi_{\o}(z)}{\xi_{\bo'}(q_3^{c-1}z)}. \end{split}\end{aligned}$$ The operators $\xi_\o(z)$ do not fully decouple from the twisted algebra as it appears in the commutation relation $[\tx^+,\tx^-]$. The exchange relations $\tpsi^\pm-\tx$ now have the correct behavior as $z^{\pm1}\to\infty$ to define the expansions $$\label{exp_tx_tpsi} \tx_\o^\pm(z)=\sum_{k\in\mathbb{Z}}z^{-k}\tx_{\o,k}^\pm,\quad \tpsi_\o^+(z)=\tpsi_{\o,0}^+\exp\left(\sum_{k>0}z^{-k}a_{\o,k}\right),\quad \tpsi_\o^-(z)=\tpsi_{\o,0}^-z^{\ta_{\o,0}}\exp\left(\sum_{k>0}z^{k}a_{\o,-k}\right).$$ The currents $\tpsi_\o^-(z)$ still conserve a zero mode dependence $\ta_{\o,0}$ that is required to reproduce the exchange relation \[rel\_xi\] with the grading operator $\xi_\o(z)$. From the asymptotic behavior of the algebraic relations, we deduce that this zero mode operator $\ta_{\o,0}$ commutes with all the twisted currents $\tx_\o^\pm$, $\tpsi_\o^\pm$ but not with the gradings $\xi_\o(z)$: $$\label{rel_a_xi} \xi_\o(z)w^{\ta_{\o',0}}=w^{\b_{\o\o'-\nu_3c}-\b_{\o\o'}}w^{\ta_{\o',0}}\xi_\o(z).$$ Note that this operator becomes central if $\nu_3=0$ or $c=0$. Expanding in powers of the spectral parameters, the exchange relations $\tpsi-\tx$ and $\tpsi-\tpsi$ given in \[algebra\_tilde\] provide the commutation relations between the modes, $$\begin{aligned} \begin{split} &[a_{\o,k>0},a_{\o',l}]=\d_{k+l}(q_3^{-kc}c_{\o\o'-\nu_3c}^{(k)}-c_{\o\o'}^{(k)}),\quad [a_{\o,k>0},\tx_{\o',l}^\pm]=\pm c_{\o\o'}^{(k)}\tx_{\o',l+k}^\pm,\\ &[a_{\o,-k<0},\tx_{\o',l}^+]=q_3^{-kc}c_{\o\o'+\nu_3c}^{(-k)}\tx_{\o',l-k}^+,\quad [a_{\o,-k<0},\tx_{\o',l}^-]=-c_{\o\o'}^{(-k)}\tx_{\o',l-k}^+,\\ \end{split}\end{aligned}$$ where[^20] $$c_{\o\o'}^{(k)}=c_{\o'\o}^{(-k)}=\dfrac1k\sum_{i=1,2,3}(q_i^k\d_{\o,\o'+\nu_i}-q_i^{-k}\d_{\o,\o'-\nu_i}).$$ In particular, when $c\neq0$, the modes $a_{\o,k}$ of the Cartan currents define $p$ Heisenberg subalgebras. This property is used to build the horizontal representation in appendix \[AppB2\]. The exchange relations $\tx-\tx$ can also be written in terms of modes by projecting the following relations: $$\begin{aligned} \begin{split} &z^{\b_{\o\o'}}\prod_{i=1,2,3}(w-q_i^{-1}z)^{\d_{\o,\o'+\nu_i}}\tx_\o^+(z)\tx_{\o'}^+(w)=F_{\o\o'}w^{\b_{\o'\o}}\prod_{i=1,2,3}(w-q_iz)^{\d_{\o\o'-\nu_i}}\tx_{\o'}^+(w)\tx_\o^+(z),\\ &z^{\b_{\o'\o}}\prod_{i=1,2,3}(w-q_iz)^{\d_{\o,\o'-\nu_i}}\tx_\o^-(z)\tx_{\o'}^-(w)=F_{\o'\o}^{-1}w^{\b_{\o\o'}}\prod_{i=1,2,3}(w-q_i^{-1}z)^{\d_{\o\o'+\nu_i}}\tx_{\o'}^-(w)\tx_\o^-(z). \end{split}\end{aligned}$$ A priori, the commutator $[\tx^+,\tx^-]$ could also be written in terms of modes, but the expression is rather cumbersome. On the other hand, the grading operators have simple actions on the modes: $$\begin{aligned} \label{def_gradings} \begin{split} &[d,\tx_{\o',k}^\pm]=-k\tx_{\o',k}^\pm,\quad [d,a_{\o,k}]=-ka_{\o,k},\quad [d,\tpsi_{\o',0}^\pm]=0,\quad [d,\ta_{\o',0}]=0,\\ &[\bd_\o,\tx_{\o',k}^\pm]=\pm \d_{\o,\o'}\tx_{\o',k}^\pm,\quad[\bd_\o,a_{\o',k}]=0,\quad [\bd_\o,\psi_{\o',0}^+]=[\bd_\o,\ta_{\o',0}]=0,\quad[\bd_{\o},\tpsi_{\o',0}^-]=(\d_{\o,\o'-\nu_3c}-\d_{\o,\o'})\tpsi_{\o',0}^-,\\ &e^{\td_\o}\tx_{\o',k}^\pm e^{-\td_\o}=\tx_{\o',k\pm\d_{\o,\o'}}^\pm,\quad e^{\td_\o}\tpsi_{\o',0}^+e^{-\td_\o}=\tpsi_{\o',0}^+,\quad e^{\td_\o}\tpsi_{\o',0}^-e^{-\td_\o}=q_3^{-c\d_{\o,\o'-\nu_3c}}\tpsi_{\o',0}^-,\\ &[\td_\o,a_{\o',k}]=0,\quad e^{\td_\o}z^{\ta_{\o',0}}e^{-\td_\o}=z^{\d_{\o,\o'-\nu_3c}-\d_{\o,\o'}}z^{\ta_{\o',0}}. \end{split}\end{aligned}$$ Coproduct --------- The Hopf algebra structure can be extended to include the grading operators, provided we define the coproduct, counit and antipode as $$\begin{aligned} \begin{split} &\D(d)=d\otimes1+1\otimes d,\quad \D(\bd_\o)=\bd_\o\otimes1+1\otimes\bd_{\o-\nu_3c_{(1)}},\\ &\D(\td_\o)=\td_\o\otimes1+1\otimes\td_{\o-\nu_3c_{(1)}}+(\log q_3)\ c\otimes\bd_{\o-\nu_3c_{(1)}},\\ &\e(d)=\e(\bd_\o)=\e(\td_\o)=0,\quad S(d)=-d,\quad S(\bd_\o)=-d_{\o+\nu_3c},\\ &S(\td_\o)=-\td_{\o+\nu_3c}+(\log q_3)\ c\bd_{\o+\nu_3c}. \end{split}\end{aligned}$$ We deduce, for the composite operators, $$\begin{aligned} \begin{split} &\D(\b_\o)=\b_\o\otimes1+1\otimes\b_{\o-\nu_3c_{(1)}},\quad \e(\b_\o)=0,\quad S(\b_\o)=-\b_{\o+\nu_3c},\\ &\D(F_\o)=F_\o\otimes F_{\o-\nu_3c_{(1)}},\quad \e(F_\o)=1,\quad S(F_\o)=F_{\o+\nu_3c}^{-1},\\ &\D(D_\o)=D_\o\otimes q_3^{-c_{(1)}\b_{\o-\nu_3c_{(1)}}}D_{\o-\nu_3c_{(1)}},\quad \e(D_\o)=1,\quad S(F_\o)=q_3^{c\b_{\o+\nu_3c}}D_{\o+\nu_3c}^{-1}, \end{split}\end{aligned}$$ and, finally, $$\D(\xi_\o(z))=\xi_\o(z)\otimes\xi_{\o-\nu_3c_{(1)}}(q_3^{-c_{(1)}}z),\quad \e(\xi_\o(z))=1,\quad S(\xi_\o(z))=\xi_{\o+\nu_3c}(q_3^{c}z)^{-1}.$$ We can also compute the coproduct for the twisted currents, $$\begin{aligned} \begin{split} &\D(\tx_\o^+(z))=\tx_\o^+(z)\otimes1+\tpsi^-_{\o+\nu_3c_{(1)}}(q_3^{c_{(1)}}z)\xi_\o(z)\otimes\tx_\o^+(z)\\ &\D(\tx_\o^-(z))=\xi_{\bo}(q_3^{-1}z)^{-1}\otimes\tx^-_{\o-\nu_3c_{(1)}}(q_3^{-c_{(1)}}z)+\tx_\o^-(z)\otimes\tpsi^+_{\o-\nu_3c_{(1)}}(q_3^{-c_{(1)}}z),\\ &\D(\tpsi_\o^+(z))=\tpsi_\o^+(z)\otimes\tpsi_{\o-\nu_3c_{(1)}}^+(q_3^{-c_{(1)}}z),\\ &\D(\tpsi_\o^-(z))=\tpsi_{\o-\nu_3c_{(2)}}^-(q_3^{-c_{(2)}}z)\otimes\tpsi_{\o-\nu_3c_{(1)}}^-(q_3^{-c_{(1)}}z)\xi_{\o-\nu_3c_{(1)}-\nu_3c_{(2)}}(q_3^{-c_{(1)}-c_{(2)}}z)\xi_{\o-2\nu_3c_{(1)}-\nu_3c_{(2)}}(q_3^{-2c_{(1)}-c_{(2)}}z)^{-1}, \end{split}\end{aligned}$$ and deduce $$\begin{aligned} \begin{split} &\D(a_{\o,k>0})=a_{\o,k}\otimes1+q_3^{ck}\otimes a_{\o-\nu_3c_{(1)},k},\quad \D(a_{\o,-k<0})=a_{\o-\nu_3c_{(2)},-k}\otimes q_3^{-kc}+q_3^{-kc}\otimes a_{\o-\nu_3c_{(1)},-k},\\ &\D(a_{\o,0})=a_{\o-\nu_3c_{(2)},0}\otimes1+1\otimes a_{\o-\nu_3c_{(1)},0}+1\otimes\left(\b_{\o-\n_3c_{(1)}-\nu_3c_{(2)}}-\b_{\o-2\n_3c_{(1)}-\nu_3c_{(2)}}\right),\\ \end{split}\end{aligned}$$ together with the coproduct of the zero modes $\psi_{\o,0}^\pm$. Vertical representation {#vertical-representation} ----------------------- In the vertical representation, the grading operators $\bd_\o$ and $\td_\o$ commute with the currents $\psi_\o^\pm(z)$, therefore they are diagonal in the basis ${\ket{\bl}\!\rangle}$. Their eigenvalues can be determined recursively using the relations with the currents $x^\pm(z)$, $$\rho_{\bv}^{(0,m)}(\bd_\o){\ket{\bl}\!\rangle}=|K_\o(\bl)|{\ket{\bl}\!\rangle},\quad \rho_{\bv}^{(0,m)}(\td_\o){\ket{\bl}\!\rangle}=\left(\sum_{\mAbox\in\bl}\d_{\o,c(\sAbox)}\log\chi_{\sAbox}\right){\ket{\bl}\!\rangle},$$ where the eigenvalues on the vacuum have been chosen to be zero. Then, the representation of $\xi_\o(z)$ takes the simple form $$\label{vert_xi} \rho^{(V)}(\xi_\o(z)){\ket{\bl}\!\rangle}={\mathring{f}}\oY(z){\ket{\bl}\!\rangle},$$ with the function ${\mathring{f}}\oY(z)$ defined in \[def\_Psi\]. We find the representation of the twisted currents to be $$\begin{aligned} \label{vertical_g} \begin{split} &\rho^{(V)}(\tx_\omega^+(z)){\ket{\bl}\!\rangle}=F^{1/2}\sum_{\mAbox\in A_\omega(\boldsymbol\l)}\d(z/\chi_{\sAbox}) \res_{z=\chi_{\sAbox}}z^{-1}\CYY(z)^{-1} {\ket{\bl+\Abox}\!\rangle},\\ &\rho^{(V)}(\tx_\omega^-(z)){\ket{\bl}\!\rangle}=\sum_{\mAbox\in R_\omega(\boldsymbol\l)}\d(z/\chi_{\sAbox})\res_{z=\chi_{\sAbox}}z^{-1}\CY_{\bo}^{[\bl]}(q_3^{-1}z) {\ket{\bl-\Abox}\!\rangle},\\ &\rho^{(V)}(\tpsi_\omega^+(z)){\ket{\bl}\!\rangle}={\mathring{f}}_{\bo}^{[\bl]}(q_3^{-1}z)^{-1}\left[\PsiY(z)\right]_+{\ket{\bl}\!\rangle},\quad \rho^{(V)}(\tpsi_{\omega,0}^+)=1,\\ &\rho^{(V)}(\tpsi_\omega^-(z)){\ket{\bl}\!\rangle}={\mathring{f}}\oY(z)^{-1}\left[\PsiY(z)\right]_-{\ket{\bl}\!\rangle},\quad \rho^{(V)}(\tpsi_{\omega,0}^+)=\dfrac{\prod_{\a\in C_\bo(m)}(-q_3v_\a)}{\prod_{\a\in C_\o(m)}(-v_\a)},\\ &\rho^{(V)}(a_{\o,k}){\ket{\bl}\!\rangle}=\left(\sum_{\mAbox\in\bl}c_{\o c(\sAbox)}^{(k)}\chi_{\sAbox}^k\right){\ket{\bl}\!\rangle},\quad \rho^{(V)}(\ta_{\o,0})=m_\o-m_\bo. \end{split}\end{aligned}$$ Horizontal representation {#horizontal-representation-2} ------------------------- Computing the exchange relations of the operators $e^{\a p_\o}$ and $e^{\a\tp_\o}$ with the Drinfeld currents leads to the identification of the representation for grading operators $$\rho^{(H)}(\bd_\o)=p_\o,\quad\rho^{(H)}(\td_\o)=\tp_\o\implies \rho^{(H)}(\xi_\o(z))=P_\o(z).$$ Thus, we find the representation for the twisted currents, $$\begin{aligned} \begin{split} &\rho^{(H)}(\tx^+_\o(z))=u_\o z^{-n_\o}Q_\o(z)\eta_\o^+(z),\quad \rho_{u}^{(1,n)}(\tx^-_\o(z))=u_\o^{-1}z^{n_\o}Q_\o(z)^{-1}\eta_\o^-(z),\\ &\rho^{(H)}(\tpsi^+_\o(z))=F^{-1/2}\vphi_\o^+(z),\\ &\rho^{(H)}(\tpsi_\o^-(z))=F^{1/2}\dfrac{u_{\bo}}{u_\o}q_3^{n_{\bo}}z^{n_\o-n_{\bo}}\dfrac{Q_{\bo}(q_3^{-1}z)}{Q_{\o}(z)}\vphi_\o^-(z), \end{split}\end{aligned}$$ and the modes $$\begin{aligned} \begin{split} &\rho^{(H)}(a_{\o,k>0})=-\dfrac1k(q_3^{-k/2}\a_{\o,k}-q_3^{k/2}\a_{\bo,k}),\quad \rho^{(H)}(a_{\o,-k<0})=-\dfrac1k(q_3^{-k}\a_{\bo,-k}-\a_{\o,-k}),\\ &\rho^{(H)}(\tpsi_{\o,0}^+)=F^{-1/2},\quad \rho^{(H)}(\tpsi_{\o,0}^-)=F^{1/2}\dfrac{u_{\bo}}{u_\o}q_3^{n_{\bo}-\tq_\bo}e^{q_\bo-q_\o},\quad \rho^{(H)}(\ta_{\o,0})=n_\o-n_{\bo}+\tq_{\bo}-\tq_\o. \end{split}\end{aligned}$$ We can verify that $\ta_{\o,0}$ commutes with the twisted currents, and satisfies the relation \[rel\_a\_xi\] with the grading operator $\xi_\o(z)$. Relation with the quantum toroidal $\glp$ algebra {#AppE} ================================================= In order to reproduce the Drinfeld currents of the quantum toroidal $\glp$ algebra when $\nu_1=-\nu_2\to1$, we need to introduce another twist by grading elements, namely $$\begin{aligned} \label{twist_currents_II} \begin{split} &\tx_\o^+(z)=x_\o^+(z)\xi_\o(z)^{-1/2},\quad \tx_\o^-(z)=x_\o^-(z)\xi_{\bo}(q_3^{-1}z)^{-1/2},\\ &\tpsi_\o^+(z)=\psi_\o^+(z)\xi_{\bo}(q_3^{-1/2}z)^{-1},\quad \tpsi_\o^-(z)=\psi_\o^-(z)\xi_{\o-\nu_3c}(q_3^{-(c+1)/2}z)^{-1}. \end{split}\end{aligned}$$ Note that we keep here the notation $\tx^\pm_\o(z)$, $\tpsi_\o^\pm(z)$ for the twisted currents, but the twist is different from the one studied in appendix . All the grading operators appearing in the definition of $\xi_\o(z)$ commute, and the powers $\xi_\o(z)^\a$ can be defined without ambiguity. The twisted currents \[twist\_currents\_II\] obey the same algebraic relations as in \[algebra\] with the structure functions $g_{\o\o'}(z)$ replaced by new functions $\tg_{\o\o'}(z)$, $$\begin{aligned} \label{algebra_II} \begin{split} &\tx_\o^\pm(z)\tx_{\o'}^\pm(w)=\tg_{\o\o'}(z/w)^{\pm1}\tx_{\o'}^\pm(w)\tx_\o^\pm(z),\quad \tpsi^+_\o(z)\tx^\pm_{\o'}(w)=\tg_{\o\o'}(z/w)^{\pm1}\tx^\pm_{\o'}(w)\tpsi^+_\o(z),\\ &\tpsi^-_\o(z)\tx^+_{\o'}(w)=\tg_{\o-\nu_3c\ \o'}(q_3^{-c}z/w)\tx^+_{\o'}(w)\tpsi^-_\o(z),\quad \tpsi^-_\o(z)\tx^-_{\o'}(w)=\tg_{\o\o'}(z/w)^{-1}\tx^-_{\o'}(w)\tpsi^-_\o(z),\\ &\tpsi_\o^+(z)\tpsi_{\o'}^-(w)=\dfrac{\tg_{\o\o'-\nu_3c}(q_3^{c} z/w)}{\tg_{\o\o'}(z/w)}\tpsi_{\o'}^-(w)\tpsi_\o^+(z),\quad [\tpsi_\o^\pm(z),\tpsi_{\o'}^\pm(w)]=0,\\ &[\tx_\o^+(z),\tx_{\o'}^-(w)]=\kO\left[F^{1/2}\d_{\o,\o'}\d(z/w)\tpsi_\o^+(z)-f_{\o\o'}(q_3^c)^{1/2}\d_{\o,\o'-\nu_3c}\d(q_3^{c} z/w)\tpsi^-_{\o+\nu_3c}(q_3^{c}z)\right]. \end{split}\end{aligned}$$ The new structure functions write $$\label{def_tg} \tg_{\o\o'}(z)=\left(\dfrac{f_{\o'\o}(z)}{f_{\o\o'}(z^{-1})}\right)^{1/2}g_{\o\o'}(z)=(-1)^{\d_{\o,\o'}}\prod_{i=1,2,3}\dfrac{\left((-q_iz)^{1/2}+(-q_iz)^{-1/2}\right)^{\d_{\o,\o'-\nu_i}}}{\left((-q_i^{-1}z)^{1/2}+(-q_i^{-1}z)^{-1/2}\right)^{\d_{\o,\o'+\nu_i}}}.$$ They possess the necessary property $\tg_{\o\o'}(z)\tg_{\o'\o}(z^{-1})=1$, and exhibit the following asymptotics, $$\tg_{\o\o'}(z){\genfrac{}{}{0pt}{3}{\sim}{0}}(F_{\o\o'}F_{\o'\o})^{1/2}z^{-(\b_{\o\o'}-\b_{\o'\o})/2},\quad \tg_{\o\o'}(z){\genfrac{}{}{0pt}{3}{\sim}{\infty}}(F_{\o\o'}F_{\o'\o})^{-1/2}z^{(\b_{\o\o'}-\b_{\o'\o})/2}.$$ When $\nu_3=0$, the matrix $\b_{\o\o'}$ is symmetric, and the $z$-dependence disappears from the asymptotics. In this case, zero modes are no longer needed for the Drinfeld currents $\tpsi_\o^\pm(z)$. In fact, when $\nu_1=-\nu_2=1$, the function $\tg_{\o\o'}(z)$ reduces to the structure function of quantum toroidal $\glp$ given in \[def\_g\_glp\]. On the other hand, when $\nu_3\neq 0$, the algebraic relations involve half-integer powers of $z$, and a choice of branch cut for the square root is necessary. We will ignore these issues in this appendix as we have only in mind the application to the case $\nu_3=0$. The relations \[algebra\_II\] do not reduce to the algebraic relation \[algebra\_glp\] of the quantum toroidal $\glp$ algebra yet. An extra shift involving the central charge $c$ is necessary, $$\label{shift_currents} \tx_\o^\pm(z)\to \tx_\o^\pm(q_3^{\mp c/4}),\quad \tpsi_\o^-(z)\to\tpsi_\o^-(q_3^{c/2}z),$$ the current $\tpsi_\o^+(z)$ remaining invariant. Then, the shifted currents satisfy the relations \[algebra\_glp\] when $\nu_3=0$, with the exception of the Serre relations that still have to be imposed by hand. The coproduct of the twisted currents, without taking into account the previous shift, takes the form $$\begin{aligned} \begin{split} \D(\tx_\o^+(z))&=\tx_\o^+(z)\otimes\xi_{\o-\nu_3c_{(1)}}(q_3^{-c_{(1)}}z)^{-1/2}\\ &+\tpsi^-_{\o+\nu_3c_{(1)}}(q_3^{c_{(1)}}z)\xi_{\bo+\nu_3c_{(1)}}(q_3^{c_{(1)}-1}z)^{1/2}\otimes\tx_\o^+(z)\xi_\o(z)^{1/2}\xi_{\o-\nu_3c_{(1)}}(q_3^{-c_{(1)}}z)^{-1/2},\\ \D(\tx_\o^-(z))&=\tx_\o^-(z)\otimes\tpsi^+_{\o-\nu_3c_{(1)}}(q_3^{-c_{(1)}}z)\xi_{\o-\nu_3c_{(1)}}(q_3^{-c_{(1)}}z)^{1/2}\\ &+\xi_{\bo}(q_3^{-1}z)^{-1/2}\otimes\tx_{\o-\nu_3c_{(1)}}^-(q_3^{-c_{(1)}}z)\\ \D(\tpsi_\o^+(z))&=\tpsi_\o^+(z)\otimes\tpsi^+_{\o-\nu_3c_{(1)}}(q_3^{-c_{(1)}}z)\\ \D(\tpsi_\o^-(z))&=\tpsi_{\o-\nu_3c_{(2)}}^-(q_3^{-c_{(2)}}z)\xi_{\bo-\nu_3c_{(2)}}(q_3^{-c_{(2)}-1}z)^{1/2}\xi_{\bo}(q_3^{-1}z)^{-1/2}\\ &\otimes\tpsi_{\o-\nu_3c_{(1)}}^-(q_3^{-c_{(1)}}z)\xi_{\o-\nu_3c_{(1)}-\n_3c_{(2)}}(q_3^{-c_{(1)}-c_{(2)}}z)^{1/2}\xi_{\o-2\nu_3c_{(1)}-\nu_3c_{(2)}}(q_3^{-2c_{(1)}-c_{(2)}}z)^{-1/2}\\ \end{split}\end{aligned}$$ In order to recover the Drinfeld coproduct of the quantum toroidal $\glp$ algebra, we need to remove the dependence in the grading elements. This is done using a twist by the two-tensor $\Xi$, i.e. we define $\tD=\Xi^{-1}\D\Xi$ with $$\begin{aligned} \begin{split} \Xi=&\exp\left(\hf\sum_{\o,\o'}\b_{\o-\nu_3c_{(1)},\o'}\left(\td_\o\otimes\bd_{\o'}-\bd_\o\otimes\td_{\o'}-c_{(1)}(\log q_3)\bd_\o\otimes\bd_{\o'}\right)\right)\\ &\times (-1)^{-\frac12\bd^\otimes_{\nu_3c_{(1)}}}(-q_3)^{\frac12\bd^\otimes_{\nu_3c_{(1)}-\nu_3}}(-q_1)^{\frac12\bd^\otimes_{\nu_3c_{(1)}+\nu_1}}(-q_2)^{\frac12\bd^\otimes_{\nu_3c_{(1)}+\nu_2}}, \end{split}\end{aligned}$$ and the shortcut notation $\bd^\otimes_\a=\sum_{\o} \bd_{\o+\a}\otimes\bd_\o$. Then, the twisted coproduct $\tD$ of the twisted currents $\tx^\pm_\o(z)$, $\tpsi_\o^\pm(z)$ reproduces precisely the formulas \[Drinfeld\_coproduct\] for the coproduct $\D$ of the untwisted currents. However, this is not quite the usual coproduct of quantum toroidal $\glp$ algebra, and, together with the shifts \[shift\_currents\], an extra twist by the two-tensor $$\G=q_3^{\frac14(d\otimes c+3c\otimes d)}$$ is needed to recover the coproduct \[Drinfeld\_glp\] for the shifted currents. The aim of this appendix is to recover the intertwining operators of the quantum toroidal $\glp$ algebra. Since the intertwining relations \[intw\] are also satisfied by the grading operators, the twist \[twist\_currents\_II\] of the currents has no effect: it is equivalent to solve these relations for the currents $x_\o^\pm(z)$, $\psi_\o^\pm(z)$, or for their twisted version. On the other hand, the twist of the coproduct by the two-tensor $\Xi$ has to be taken into account. It is easily seen that if the intertwiner $\Phi$ and $\Phi^\ast$ satisfy the relations \[intw\] for the coproduct $\D$, then the following intertwiners satisfy the same relations for the twisted coproduct $\tD$, $$\tPhi=\Phi\left(\rho^{(V)}\otimes\rho^{(H)}\ \Xi\right),\quad \tPhi^\ast=\left(\rho^{(V)}\otimes\rho^{(H)}\ \Xi'^{-1}\right)\Phi^\ast,$$ where $\Xi'$ denotes the permutation of the two-tensor $\Xi$. The two-tensor $\Xi$ is diagonal in the vertical basis ${\ket{\bl}\!\rangle}$, and these relations can be projected on this basis. As result, we find the relations $\tPhi_\bl=\Phi_\bl\Xi_\bl$, and $\tPhi_\bl^\ast=\Xi_\bl^\ast\Phi_\bl^\ast$ for the components $\bl$, with $$\begin{aligned} \begin{split} &\Xi_\bl=\left(\rho^{(V)}\otimes\rho^{(H)}\ \Xi\right)\left({\ket{\bl}\!\rangle}\otimes 1\right)=\prod_{\sAbox\in\bl}P_{c(\sAbox)}(\chi_{\sAbox})^{-1/2},\\ &\Xi_\bl^\ast=\left({\langle\!\bra{\bl}}\otimes 1\right)\left(\rho^{(V)}\otimes\rho^{(H)}\ \Xi'^{-1}\right)=\prod_{\sAbox\in\bl}P_{c(\sAbox)}(q_3^{-1}\chi_{\sAbox})^{-1/2}. \end{split}\end{aligned}$$ Finally, we also need to take into account the twist by the two tensor $\G$, and multiply by $$\left(\rho^{(V)}\otimes\rho^{(H)}\ \G\right)=q_3^{\frac14\rho^{(V)}(d)},\quad \left(\rho^{(V)}\otimes\rho^{(H)}\ \G'^{-1}\right)=q^{-\frac34\rho^{(V)}(d)}.$$ The vertical representation of the grading operator $e^{\a d}$ is a shift operator acting on the weights $\bv$, sending $\bv\to e^{\a}\bv$. Derivation of the vertex operators {#AppF} ================================== Definition of the vacuum components ----------------------------------- Before sketching the derivation of the solution for the intertwining relations, we would like to provide a bold argument for the definition of the vacuum components $\Phi_\vac$ and $\Phi_\vac^\ast$ entering in the definition \[def\_Phi\] of the intertwiners. In fact, the full partition function of the gauge theory, including classical, one-loop and instantons contributions, has a nice description in terms of the melting crystal picture [@Nekrasov2003; @Bourgine2017]. Indeed, the one-loop contribution can be written as a double product over the boxes of completely filled (infinite) Young diagrams $\blinf=\{(\a,i,j)\diagup\a=1\cdots m, i=1\cdots\infty, j=1\cdots\infty\}$, assuming a $\z_2$-regularization for the infinite product. Then, the instanton correction of order $O(\qf^k)$ is obtained by removing $k$ boxes to $\blinf$, taking the double product over $\blc=\blinf\setminus\bl=\{(\a,i,j)\diagup\a=1\cdots m,i=\l_j^{(\a)}+1\cdots\infty,j=1\cdots\infty\}$ and summing over the configurations $\bl$ of $k=|\bl|$ boxes. The vacuum component $\Phi_\vac$ of the intertwiner $\Phi$ is associated to this infinite product over boxes in $\blinf$, so that formally $$\Phi_\vac\simeq:\prod_{\mAbox\in\blinf}\eta_{c(\sAbox)}^+(\chi_{\sAbox})^{-1}:,\quad \Phi_\bl\simeq t_\bl:\prod_{\mAbox\in\blc}\eta_{c(\sAbox)}^+(\chi_{\sAbox})^{-1}:,$$ and similarly for $\Phi_\bl^\ast$, replacing $\eta_{c(\sAbox)}^+(\chi_{\sAbox})$ with $\eta_{c(\sAbox)+\nu_3}^-(q_3\chi_{\sAbox})$. In order to develop this idea, we may introduce a very crude cut-off $N$ such that $\blinf$ is obtained as the limit $N\to\infty$ of $m$ Young diagrams consisting of squares of size $(pN)\times(pN)$, i.e. $\bl_N=\{(\a,i,j)\diagup\a=1\cdots m, i=1\cdots pN, j=1\cdots pN\}$. Then, we may consider the product over boxes $(\a,i,j)\in\bl_N$ and decompose the indices $(i,j)$ as $i=\bi+1+k_ip$, $j=\bj+1+k_jp$ with $\bi,\bj=0\cdots p-1$ and $k_i,k_j=0\cdots N-1$. We end up with $$\begin{aligned} \begin{split} :\prod_{\mAbox\in\bl_N}\eta_{c(\sAbox)}^+(\chi_{\sAbox})^{-1}:=:\prod_{\a=1}^{m}\prod_{\bi,\bj=0}^{p-1}&\exp\left(-\sum_{k>0}\dfrac{(v_\a q_1^{\bi}q_2^{\bj})^k}{k}\sum_{k_i,k_j=0}^{N-1}q_1^{pk_ik}q_2^{pk_jk}\a_{c_\a+\bi\nu_1+\bj\nu_2,-k}\right)\\ &\times\exp\left(\sum_{k>0}\dfrac{(v_\a q_1^{\bi}q_2^{\bj}\g)^{-k}}{k}\sum_{k_i,k_j=0}^{N-1}q_1^{-pk_ik}q_2^{-pk_jk}\a_{c_\a+\bi\nu_1+\bj\nu_2,k}\right):, \end{split}\end{aligned}$$ Performing the sum over $k_i$ and $k_j$, we find $$\begin{aligned} \begin{split} :\prod_{\mAbox\in\bl_N}\eta_{c(\sAbox)}^+(\chi_{\sAbox})^{-1}:=:\prod_{\a=1}^{m}\prod_{\bi,\bj=0}^{p-1}&\exp\left(-\sum_{k>0}\dfrac{(v_\a q_1^{\bi}q_2^{\bj})^k}{k}\dfrac{1-q_1^{pkN}}{1-q_1^{pk}}\dfrac{1-q_2^{pkN}}{1-q_2^{pk}}\a_{c_\a+\bi\nu_1+\bj\nu_2,-k}\right)\\ &\times\exp\left(\sum_{k>0}\dfrac{(v_\a q_1^{\bi}q_2^{\bj}\g)^{-k}}{k}\dfrac{1-q_1^{-pkN}}{1-q_1^{-pk}}\dfrac{1-q_2^{-pkN}}{1-q_2^{-pk}}\a_{c_\a+\bi\nu_1+\bj\nu_2,k}\right):, \end{split}\end{aligned}$$ At this stage, the limit $N\to\infty$ is ill-defined because the first exponential converges when $|q_1|,|q_2|<1$ while the second exponential for $|q_1|,|q_2|>1$. However, we notice that each color can be treated independently, and their contribution written in terms of the vacuum component for the intertwiner describing instantons on a omega-background with no orbifold [@Awata2011; @Bourgine2017b], with the replacement $\e_1,\e_2\to p\e_1,p\e_2$. Thus, we can borrow the corresponding operator and simply define $$\begin{aligned} \label{def_Phi_vac} \begin{split} \Phi_\vac=:\prod_{\a=1}^{m}\prod_{\bi,\bj=1}^{p-1}&\exp\left(-\sum_{k>0}\dfrac{(v_\a q_1^{\bi}q_2^{\bj})^k}{k(1-q_1^{pk})(1-q_2^{pk})}\a_{c_\a+\bi\nu_1+\bj\nu_2,-k}\right)\exp\left(\sum_{k>0}\dfrac{(v_\a q_1^{\bi}q_2^{\bj}\g)^{-k}}{k(1-q_1^{-pk})(1-q_2^{-pk})}\a_{c_\a+\bi\nu_1+\bj\nu_2,k}\right):. \end{split}\end{aligned}$$ The appearance of quantities defined on the background $\mC_{p\e_1}\times\mC_{p\e_2}\times S_R^1$ is reminiscent of the surface defect interpretation of the orbifold developed in [@Nekrasov_BPS4; @Jeong2018]. It may also be related to the *abelianization* procedure described in the case of $\glp$ (unrefined, i.e. $q_3=1$) in [@Awata2017]. Using the definition \[def\_Phi\_vac\], we obtain the following normal-ordering relations[^21] $$\begin{aligned} \begin{split} &\eta_\o^+(z)\Phi_{\vac}=\prod_{\a\in C_\o(m)}(1-v_\a/z)^{-1}:\eta_\o^+(z)\Phi_{\vac}:,\quad\Phi_{\vac}\eta_\o^+(z)=\prod_{\a\in C_\bo(m)}(1-z/(q_3v_\a))^{-1}:\eta_\o^+(z)\Phi_{\vac}:,\\ &\eta_\o^-(z)\Phi_{\vac}=\prod_{\a\in C_\bo(m)}(1-q_3v_\a/z):\eta_\o^-(z)\Phi_{\vac}:,\quad\Phi_{\vac}\eta_\o^-(z)=\prod_{\a\in C_\bo(m)}(1-z/(q_3v_\a)):\eta_\o^-(z)\Phi_{\vac}:. \end{split}\end{aligned}$$ Since $\vphi_\o^\pm(z)$ can be expressed in terms of $\eta_\o^\pm(z)$, we easily deduce the normal-ordering relations for these vertex operators as well. This argument can also be applied to $\Phi_\vac^\ast$, it leads to define $$\begin{aligned} \begin{split} \Phi_\vac^\ast=:\prod_{\a=1}^{m}\prod_{\bi,\bj=0}^{p-1}&\exp\left(\sum_{k>0}\dfrac{(v_\a q_1^{\bi}q_2^{\bj}q_3)^k}{k(1-q_1^{pk})(1-q_2^{pk})}\a_{c_\a+(\bi-1)\nu_1+(\bj-1)\nu_2,-k}\right)\\ \times&\exp\left(-\sum_{k>0}\dfrac{(v_\a q_1^{\bi}q_2^{\bj}\g)^{-k}}{k(1-q_1^{-pk})(1-q_2^{-pk})}\a_{c_\a+\bi\nu_1+\bj\nu_2,k}\right):, \end{split}\end{aligned}$$ and we obtain $$\begin{aligned} \begin{split} &\eta_\o^+(z)\Phi_{\vac}^\ast=\prod_{\a\in C_\bo(m)}(1-q_3v_\a/z):\eta_\o^+(z)\Phi_{\vac}^\ast:,\quad\Phi_{\vac}^\ast\eta_\o^+(z)=\prod_{\a\in C_\bo(m)}(1-z/(q_3v_\a)):\eta_\o^+(z)\Phi_{\vac}^\ast:,\\ &\eta_\o^-(z)\Phi_{\vac}^\ast=\prod_{\a\in C_{\o+2\nu_1+2\nu_2}(m)}(1-q_3^2v_\a/z)^{-1}:\eta_\o^-(z)\Phi_{\vac}^\ast:,\quad\Phi_{\vac}^\ast\eta_\o^-(z)=\prod_{\a\in C_\bo(m)}(1-z/(q_3v_\a))^{-1}:\eta_\o^-(z)\Phi_{\vac}^\ast:. \end{split}\end{aligned}$$ We can also compute $$\begin{aligned} \begin{split} &\Phi_\vac\Phi_{\vac}'=\CG(\bv'|\bv)^{-1}:\Phi_\vac\Phi_{\vac}':,\quad \Phi_\vac^\ast\Phi_{\vac}^{\ast\prime}=\CG(\bv'|q_3^{-1}\bv)^{-1}:\Phi_\vac^\ast\Phi_{\vac}^{\ast\prime}:,\\ &\Phi_\vac\Phi_{\vac}^{\ast\prime}=\CG(\bv'|q_3^{-1}\bv):\Phi_\vac\Phi_{\vac}^{\ast\prime}:,\quad \Phi_\vac^\ast\Phi_{\vac}'=\CG(\bv'|\bv):\Phi_\vac^\ast\Phi_{\vac}':, \end{split}\end{aligned}$$ where we $\CG(\bv|\bv')$ denotes the bifundamental contribution at one-loop expressed in terms of the function $\CG_{q_1,q_2}(z)$,[^22] $$\label{def_CG} \CG(\bv|\bv')=\prod_{\a=1}^m\prod_{\a'=1}^{m'}\prod_{\bi,\bj=0}^{p-1}\CG_{q_1^p,q_2^p}(v_\a q_1^{\bi+1}q_2^{\bj+1}/v'_{\a'})^{-\d_{c'_{\a'},c_\a+(\bi+1)\nu_1+(\bj+1)\nu_2}},\quad \CG_{q_1,q_2}(z)=\exp\left(-\sum_{k=1}^\infty\dfrac1k\dfrac{z^k}{(1-q_1^{k})(1-q_2^{k})}\right).$$ Solution of intertwining relations ---------------------------------- Once projected on the vertical states using the decomposition \[Phi\_vert\], the intertwining relations \[intw\] write $$\begin{aligned} \label{intw_I} \begin{split} &x_\o^+(z)\Phi_\bl=\PsiY(z)\Phi_\bl x_\o^+(z)+\rho^{(V)}(x_\o^+(z))\cdot\Phi_\bl,\\ &x_\o^-(z)\Phi_\bl=\Phi_\bl x_\o^-(z)+\left[\rho^{(V)}(x_\o^-(z))\cdot\Phi_\bl\right]\psi_\o^+(z),\\ &\psi_\o^+(z)\Phi_\bl=\PsiY(z)\Phi_\bl\psi_\o^+(z),\quad \psi_\o^-(z)\Phi_\bl(z)=\Psi\boY(q_3^{-1}z)\Phi_\bl\psi_\o^-(z), \end{split}\end{aligned}$$ and $$\begin{aligned} \label{intw_II} \begin{split} &x_\o^+(z)\Phi_\bl^\ast=\Phi_\bl^\ast x_\o^+(z)-\psi_{\o-\nu_1-\nu_2}^-(q_3z)\left[\rho^{(V)\ast}(x_\o^+(z))\cdot\Phi_\bl^\ast\right],\\ &\Psi\boY(q_3^{-1}z)x_\o^-(z)\Phi_\bl^\ast=\Phi_\bl^\ast x_\o^-(z)-\rho^{(V)\ast}(x_\bo^-(q_3^{-1}z))\cdot\Phi_\bl^\ast,\\ &\psi_\o^+(z)\Phi_\bl^\ast=\Psi\boY(q_3^{-1}z)^{-1}\Phi_\bl^\ast\psi_\o^+(z),\quad \psi_\o^-(z)\Phi_\bl^\ast(z)=\Psi\boY(q_3^{-1}z)^{-1}\Phi_\bl^\ast\psi_\o^-(z). \end{split}\end{aligned}$$ To lighten the notations, we have omitted the horizontal representations $\rho^{(H)}$ and $\rho^{(H')}$ and indicated the vertical action with a central dot. In order to show that the operators $\Phi_\bl$ and $\Phi_\bl^\ast$ defined in \[def\_Phi\] satisfy these relations, we need to compute the factors coming from the normal ordering of products with the Drinfeld currents in the horizontal representation. It is easier to treat separately the vertex operators part, $$\begin{aligned} \begin{split} &\eta_\o^+(z)\Phi_{\bl}=\CYY(z)^{-1}:\eta_\o^+(z)\Phi_{\bl}:,\quad\Phi_{\bl}\eta_\o^+(z)=f_{\bo}^{[\bl]}(q_3^{-1}z)^{-1}\CY_{\bo}^{[\bl]}(q_3^{-1}z)^{-1}:\eta_\o^+(z)\Phi_{\bl}:,\\ &\eta_\o^-(z)\Phi_{\bl}=\CY_{\bo}(q_3^{-1}z):\eta_\o^-(z)\Phi_{\bl}:,\quad\Phi_{\bl}\eta_\o^-(z)=f_{\bo}^{[\bl]}(q_3^{-1}z)\CY_{\bo}^{[\bl]}(q_3^{-1}z):\eta_\o^-(z)\Phi_{\bl}:,\\ &\vphi_\o^+(z)\Phi_{\bl}={\mathring{f}}_{\bo}^{[\bl]}(q_3^{-1}z)^{-1}\PsiY(z):\vphi_\o^+(z)\Phi_{\bl}:,\quad\Phi_{\bl}\vphi_\o^+(z)=:\vphi_\o^+(z)\Phi_{\bl}:,\\ &\vphi_\o^-(z)\Phi_{\bl}=:\vphi_\o^-(z)\Phi_{\bl}:,\quad\Phi_{\bl}\vphi_\o^-(z)=f_{\bo}^{[\bl]}(q_3^{-1}z)\dfrac{{\mathring{f}}_{\o+2\nu_1+2\nu_2}(q_3^{-2}z)}{f_{\o+2\nu_1+2\nu_2}(q_3^{-2}z)}\Psi_{\bo}^{[\bl]}(q_3^{-1}z)^{-1}:\vphi_\o^-(z)\Phi_{\bl}:, \end{split}\end{aligned}$$ $$\begin{aligned} \begin{split} &\eta_\o^+(z)\Phi_{\bl}^\ast=\CY_{\bo}^{[\bl]}(q_3^{-1}z):\eta_\o^+(z)\Phi_{\bl}^\ast:,\quad\Phi_{\bl}^\ast\eta_\o^+(z)=f_{\bo}^{[\bl]}(q_3^{-1}z)\CY_{\bo}(q_3^{-1}z):\eta_\o^+(z)\Phi_{\bl}^\ast:,\\ &\eta_\o^-(z)\Phi_{\bl}^\ast=\CY_{\o+2\nu_1+2\nu_2}^{[\bl]}(q_3^{-2}z)^{-1}:\eta_\o^-(z)\Phi_{\bl}^\ast:,\quad\Phi_{\bl}^\ast\eta_\o^-(z)=f_{\bo}^{[\bl]}(q_3^{-1}z)^{-1}\CY_{\bo}(q_3^{-1}z)^{-1}:\eta_\o^-(z)\Phi_{\bl}^\ast:,\\ &\vphi_\o^+(z)\Phi_{\bl}^\ast={\mathring{f}}_{\o+2\nu_1+2\nu_2}^{[\bl]}(q_3^{-2}z)\Psi_{\bo}^{[\bl]}(q_3^{-1}z)^{-1}:\vphi_\o^+(z)\Phi_{\bl}^\ast:,\quad\Phi_{\bl}^\ast\vphi_\o^+(z)=:\vphi_\o^+(z)\Phi_{\bl}^\ast:,\\ &\vphi_\o^-(z)\Phi_{\bl}^\ast=:\vphi_\o^-(z)\Phi_{\bl}^\ast:,\quad\Phi_{\bl}^\ast\vphi_\o^-(z)=f_{\bo}^{[\bl]}(q_3^{-1}z)^{-1}\dfrac{f_{\o+2\nu_1+2\nu_2}^{[\bl]}(q_3^{-2}z)}{{\mathring{f}}_{\o+2\nu_1+2\nu_2}^{[\bl]}(q_3^{-2}z)}\Psi_{\bo}^{[\bl]}(q_3^{-1}z):\vphi_\o^-(z)\Phi_{\bl}^\ast:, \end{split}\end{aligned}$$ and the zero-modes part, $$\begin{aligned} \begin{split} &P_\o(z)t_\bl={\mathring{f}}_\o^{[\bl]}(z):P_\o(z)t_\bl:,\quad [Q_\o(z),t_\bl]=0,\\ &P_\o(z)t_\bl^\ast={\mathring{f}}_\bo^{[\bl]}(q_3^{-1}z)^{-1}:P_\o(z)t_\bl^\ast:,\quad t_\bl^\ast Q_\o(z)={\mathring{f}}_\bo^{[\bl]}(q_3^{-1}z)^{-1}:t_\bl^\ast Q_\o(z):. \end{split}\end{aligned}$$ From these relations, we to deduce the normal ordering relations for the currents $x_\o^\pm$ and $\psi_\o^\pm$. Then, the relation \[intw\_I\] and \[intw\_II\] for the Cartan currents $\psi_\o^\pm$ follow directly, provided that the weights and levels satisfy the relation \[rel\_u\_n\]. This condition is related to the difference between the functions $f\oY$ and ${\mathring{f}}\oY$, $$\dfrac{u'_\o z^{-n'_\o}}{u_\o z^{-n_\o}}=\dfrac{{\mathring{f}}\boY(q_3^{-1}z)}{f\boY(q_3^{-1}z)}.$$ The relations \[intw\_I\] and \[intw\_II\] involving the currents $x_\o^\pm$ are harder to prove. This is done by decomposition of the functions $\CYY(z)$ as sum over poles. We refer the reader to [@Bourgine2017b] for a more detailed explanation. Example of $qq$-characters {#AppG} ========================== We provide here the expansion of the fundamental $qq$-characters for several simple theories at the first few orders in the exponentiated gauge couplings $\qf_\o$. For simplicity, all the theories considered here are pure $U(m)$ gauge theories without Chern-Simons terms. This expansion of $qq$-characters has been done using a short Python script. $U(1)$ gauge group ------------------ In this case, we can set $v_0=1$, and choose, for definiteness, $c_0=0$. #### $\mZ_2$ orbifold (surface defect $\nu_1=1$, $\nu_2=0$) $$\begin{aligned} \begin{split} \Zinst&=1+\dfrac{\qf_0}{1-q_2}+\dfrac{\qf_0^2}{(1-q_2)(1-q_2^2)}+\dfrac{\qf_0^3}{(1-q_2)(1-q_2^2)(1-q_2^3)}+\dfrac{\qf_0\qf_1}{(1-q_2)(1-q_1^2)}+\dfrac{\qf_0^2\qf_1}{(1-q_1^2)(1-q_2)^2}+O(\qf^4)\\ \chi_0(z)&=1-z-\qf_0\dfrac{z}{q_2}-\qf_0^2\dfrac{z}{q_2^2}-\qf_0^3\dfrac{z}{q_2^3}+\qf_1-\qf_0\qf_1+O(\qf^4),\quad \chi_1(z)=1+O(\qf^4)\\ \chi_0^\ast(z)&=1-\dfrac1z+\dfrac{\qf_0}{z}-\dfrac{\qf_0^2}{z}-\dfrac{\qf_1}{q_2z}+\dfrac{\qf_0\qf_1(1+q_2)}{q_2z}+O(\qf^4),\quad \chi_1^\ast(z)=1+\dfrac{\qf_0}{q_2}+\dfrac{\qf_0^2}{q_2^2}+\dfrac{\qf_0^3}{q_2^3}+O(\qf^4). \end{split}\end{aligned}$$ #### $\mZ_2$ orbifold (ALE $\nu_1=-\nu_2=1$) $$\begin{aligned} \begin{split} \Zinst&=1+\qf_0+\dfrac{\qf_0\qf_1(1+q_1q_2)}{(1-q_1^2)(1-q_2^2)}+O(\qf^3)\\ \chi_0(z)&=1-z+\qf_0z-\qf_0^2(z-1-q_3^{-2}z^{-1})+\qf_0\qf_1q_3z+O(\qf^3),\\ \chi_1(z)&=1+\qf_0(1-\qf_0)q_3z(z-q_1-q_2)+\qf_1(1-\qf_0)+O(\qf^3)\\ \chi_0^\ast(z)&=1-z^{-1}+\qf_0z^{-1}-\qf_0^2(z-q_3^{-2}+z^{-1})+\qf_0\qf_1z^{-1}+O(\qf^3),\\ \chi_1^\ast(z)&=1+\qf_0(1-\qf_0)z^{-1}((q_3z)^{-1}-q_1-q_2)+\qf_1(1-\qf_0)q_3^{-1}+O(\qf^3). \end{split}\end{aligned}$$ #### $\mZ_3$ orbifold (surface defect $\nu_1=1$, $\nu_2=0$) $$\begin{aligned} \begin{split} \Zinst&=1+\dfrac{\qf_0}{1-q_2}+\dfrac{\qf_0^2}{(1-q_2)(1-q_2^2)}+\dfrac{\qf_0\qf_1}{(1-q_2)}+O(\qf^3)\\ \chi_0(z)&=1-z-\qf_0q_2^{-1}z+\qf_2-\qf_0\qf_1q_2^{-1}z-\qf_0^2q_2^{-2}z+O(\qf^3),\quad \chi_1(z)=1+O(\qf^3),\quad \chi_2(z)=1-\qf_0+\qf_1+O(\qf^3)\\ \chi_0^\ast(z)&=1-z^{-1}+\dfrac{\qf_0}{z}-\dfrac{\qf_2}{q_2z}+\dfrac{\qf_0\qf_1}{z}+\dfrac{\qf_0\qf_2}{q_2z}+O(\qf^3),\quad \chi_1^\ast(z)=1+\dfrac{\qf_0}{q_2}+\dfrac{\qf_0^2}{q_2^2}+O(\qf^3),\quad \chi_2^\ast(z)=1+\dfrac{\qf_1}{q_2}+O(\qf^3). \end{split}\end{aligned}$$ Taking instead $\nu_1=2$ simply exchanges $\qf_1\leftrightarrow\qf_2$ and $\chi_{1,2}\leftrightarrow\chi_{2,1}$. #### $\mZ_3$ orbifold (ALE $\nu_1=-\nu_2=1$) $$\begin{aligned} \begin{split} \Zinst&=1+\qf_0+\qf_0\qf_1+\qf_0\qf_2+O(\qf^3)\\ \chi_0(z)&=1-z+\qf_0z-\qf_0^2(z-1-q_3^{-2}z^{-1})+\qf_0\qf_1z+\qf_0\qf_2z+O(\qf^3),\\ \chi_1(z)&=1-\qf_0q_1^{-1}z+\qf_1+\qf_0^2q_1^{-1}z-\qf_0\qf_1+\qf_0\qf_2q_1^{-1}q_2^{-2}z(z-q_1-q_2^2)+O(\qf^3),\\ \chi_2(z)&=1-\qf_0q_2^{-1}z+\qf_2+\qf_0^2q_2^{-1}z-\qf_0\qf_2+\qf_0\qf_1q_1^{-2}q_2^{-1}z(z-q_1^2-q_2)+O(\qf^3),\\ \chi_0^\ast(z)&=1-z^{-1}+\qf_0z^{-1}-\qf_0^2(z-1-q_3^{-2}z^{-1})+\qf_0\qf_1z^{-1}+\qf_0\qf_2z^{-1}+O(\qf^3),\\ \chi_1^\ast(z)&=1-\qf_0q_1z^{-1}+\qf_1q_3^{-1}+\qf_0^2q_1z^{-1}-\qf_0\qf_1q_3^{-1}-\qf_0\qf_2(q_1+q_2^2-q_1q_2^2z^{-1})z^{-1}+O(\qf^3),\\ \chi_2^\ast(z)&=1-\qf_0q_2z^{-1}+\qf_2q_3^{-1}+\qf_0^2q_2z^{-1}-\qf_0\qf_2q_3^{-1}-\qf_0\qf_1(q_1^2+q_2-q_1^2q_2z^{-1})z^{-1}+O(\qf^3). \end{split}\end{aligned}$$ #### $\mZ_3$ orbifold (case $\nu_1=\nu_2=1$) $$\begin{aligned} \begin{split} \Zinst&=1+\qf_0+\qf_0\qf_1+O(\qf^3)\\ \chi_0(z)&=1-z+\qf_0z+\qf_1-\qf_0^2z-\qf_0\qf_1(1-z)+O(\qf^3),\\ \chi_1(z)&=1+\qf_0(1-\qf_0)q_3z(z-q_1-q_2)+\qf_2-\qf_0\qf_1q_3(q_1+q_2)z-\qf_0\qf_2q_3^{-2}z^{-1}+O(\qf^3),\\ \chi_2(z)&=1+\qf_0q_3z+\qf_0^2+O(\qf^3),\\ \chi_0^\ast(z)&=1-z^{-1}+(\qf_0+\qf_1-\qf_0^2)z^{-1}-\qf_0\qf_1+O(\qf^3),\\ \chi_1^\ast(z)&=1-\qf_0(1-\qf_0)(q_1+q_2-q_3^{-1}z^{-1})z^{-1}-\qf_2+\qf_0\qf_2(1-q_3^{-1}z^{-2}+q_3^{-3}z^{-3})+O(\qf^3),\\ \chi_2^\ast(z)&=1-\qf_0+\qf_0^2(q_3z+1)-\qf_0\qf_1(q_1^2+q_1q_2+q_2^2)z^{-1}+O(\qf^3). \end{split}\end{aligned}$$ $U(2)$ gauge group ------------------ For a gauge group $U(2)$, and a $\mZ_p$-orbifold, one of the important examples is the case of surface defects. When the surface defect lies on the $z_1$-plane, this is realized by choosing $(\nu_1,\nu_2) = (0,1)$. Then, the qq-characters take the following form $$\begin{aligned} \begin{split} &\mathcal{X}_\o^{[\bl]} (z)=\CY\oY(z)+\qf_{\o-1} \dfrac{(q_3z)^{\k_{\o-1}}}{q_1 f\oY(z)\CY_{\o-1}^{[\bl]}(q_3z)},\\ &\mathcal{X}_\o^{[\bl]\ast} (z)=\CY_\o^{[\bl]\ast}(z)+\qf_{\o-1} \dfrac{(q_3z)^{\k_{\o-1}}}{\CY_{\o-1}^{[\bl]}(q_3z)}, \end{split}\end{aligned}$$ with $\o \in\Zp$. Their expansion at first orders in the instanton counting parameters can be obtained with the same Python sketch. For instance, in the case of $p=3$ with colors $(c_1,c_2)=(0,1)$, taking the Chern-Simons levels $\k_\o=0$, and setting $(v_1,v_2)=(1,v)$ by a suitable rescaling of the weight and spectral parameter $z$, we find that the partition function expands in the instanton counting parameters as $$\begin{aligned} \begin{split} \Zinst&=1+\dfrac{\qf_0v}{(1-q_2)(v-q_1q_2)}+\dfrac{\qf_1}{1-q_1q_2v}+\dfrac{\qf_0^2v^2}{(1-q_2)(1-q_2^2)(v-q_1q_2)(v-q_1q_2^2)}\\ &+\dfrac{\qf_1^2}{(1-q_2)(1-q_2^2)}+\dfrac{\qf_0\qf_1v}{(1-q_2)^2(v-q_1q_2)}+\dfrac{\qf_1\qf_2}{(1-q_2)(1-vq_1^2q_2)}+O(\qf^3), \end{split}\end{aligned}$$ and the qq-characters expansion reads $$\begin{aligned} \begin{split} \chi_0(z)&=1-z-\dfrac{\qf_0vz}{q_2(v-q_1q_2)}+\dfrac{\qf_1(z-1+vq_1^2q_2)}{1-vq_1^2q_2}+\qf_2-\qf_0^2\dfrac{zv^2(v-q_1q_2-q_1q_2^2)}{q_2^2(v-q_1q_2)^2(v-q_1q_2^2)}+O(\qf^3)\\ \chi_1(z)&=1-\dfrac{z}{v}+\dfrac{\qf_0z}{v-q_1q_2}-\dfrac{\qf_1z}{q_2v}-\dfrac{\qf_0^2zvq_1q_2}{(v-q_1q_2)^2(v-q_1q_2^2)}-\dfrac{\qf_1^2z}{q_2^2v}+\dfrac{\qf_0\qf_1z}{q_2(v-q_1q_2)}-\dfrac{\qf_1\qf_2z}{vq_2(1-vq_1^2q_2)}+O(\qf^3)\\ \chi_2(z)&=1+O(\qf^3)\\ \chi_0^\ast(z)&=1-z^{-1}+\dfrac{\qf_0v}{z(v-q_1q_2)}-\dfrac{\qf_1vq_1^2}{z(1-vq_1^2q_2)}-\dfrac{\qf_2}{zq_2}-\dfrac{\qf_0^2v^2q_1q_2}{z(v-q_1q_2)^2(v-q_1q_2^2)}+\dfrac{\qf_0\qf_2v}{zq_2(v-q_1q_2)}+O(\qf^3),\\ \chi_1^\ast(z)&=1-\dfrac{v}{z}-\dfrac{\qf_0v^2}{zq_2(v-q_1q_2)}+\dfrac{\qf_1v}{z}-\dfrac{\qf_0^2v^3(v-q_1q_2-q_1q_2^2)}{zq_2^2(v-q_1q_2)^2(v-q_1q_2^2)}+\dfrac{\qf_0\qf_1v^2}{zq_2(v-q_1q_2)}+\dfrac{\qf_1\qf_2v}{z(1-vq_1^2q_2)}+O(\qf^3),\\ \chi_2^\ast(z)&=1+\dfrac{\qf_1}{q_2}+\dfrac{\qf_1^2}{q_2^2}+O(\qf^3). \end{split}\end{aligned}$$ Similar expansions can be obtained for other values of the parameters. 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The equivalent of the horizontal representation can also be defined for 4D $\CN=2$ theories, thus extending the whole algebraic construction of the Nekrasov partition functions [@Bourgine2018]. However, for this purpose, it is necessary to consider the central extension of the Drinfeld double of the affine Yangian following from the construction given in [@Khoroshkin1996]. [^2]: Strictly speaking, in [@Bonelli2012], the authors consider the instantons on the minimal resolution of the orbifold. Instead, here, following [@Kanno2011; @Nekrasov_BPS3], we simply consider the $\Zp$-invariant part of the instanton moduli space $\CM_k$. Both approaches should provide the same result [@Belavin2011a]. [^3]: The function $f_{\o\o'}(z)$ also controls the asymptotics of the scattering function since $S_{\o\o'}(z){\genfrac{}{}{0pt}{3}{\sim}{0}}1$ and $S_{\o\o'}(z){\genfrac{}{}{0pt}{3}{\sim}{\infty}}f_{\o'\o}(z)^{-1}$. It obeys an important reflection symmetry $f_{\bo\o'}(q_3/z)=f_{\o'\o}(z)^{-1}$ coming from $F_{\o\o'}F_{\bar\o'\o}=q_3^{-\b_{\o\o'}}$ and $\b_{\bar\o'\o}=\b_{\o\o'}$. [^4]: In [@Nekrasov_BPS4], the $qq$-characters of 4D $\CN=2$ gauge theories with the insertion of surface defects were considered. In this case, the non-perturbative Dyson-Schwinger equations produce either Knizhnik - Zamolodchikov equations or BPZ equations that are satisfied by the surface defect partition functions [@Nekrasov_BPS5; @Jeong2017]. These surface defect partition functions were investigated in the context of Bethe/gauge correspondence in [@Jeong2017'], and in their relation to the oper submanifold of the moduli space of flat connections on Riemann surfaces in [@Jeong2018]. [^5]: The presence of the function $\fYY(z)$ can be interpreted as follows. Note that $\mathbb{I}(X^\ast)=(-1)^{\text{rk}X^\ast}\det X^\ast\ \mathbb{I}(X)$, for $X=\sum_{i\in I_+}e^{R w_i}-\sum_{i\in I_- }e^{Rw_i}$, $\text{rk} X=|I_+|-|I_-|$, and $\det X=\prod_{i\in I_+}e^{Rw_i}/\prod_{i\in I_-}e^{Rw_i}$. Applying this reflection relation to $X=\left[e^{-R\z}S_\bl \right]^{\mathbb{Z}_p}$, we recover the relation with $\fYY(z)$ given in identified with $(-1)^{\text{rk}X^\ast}\det X^\ast$. [^6]: Comparing with the standard definition of quantum toroidal algebras, the Drinfeld currents have been redefined as follows: $x^\pm(z)\to x^\pm(q_3^{\pm c/4}z)$, $\psi_\o^+(z)\to\psi_\o^+(z)$ and $\psi_\o^-(z)\to\psi_\o^-(q_3^{-c/2}z)$. This redefinition makes the coincidence between shifts of indices $\o\pm\nu_3c$ and spectral parameters $zq_3^{\pm c}$ manifest. In fact, this asymmetric form of the algebraic relations appears naturally in the construction of a central extension of the Yangian double [@Khoroshkin1996]. [^7]: We note that the algebra is well-defined only for representations with integer level $c$ because of the shifts $-\nu_3c$ in the indices of the matrix $g_{\o\o'}(z)$. More rigorously, one should introduce the structure functions $g(z,\o,\o')$ that reduce to the expression $g_{\o\o'}(z)$ given in \[def\_g\] for $\o,\o'\in\mZ$. [^8]: The extra factor in the RHS comes from the shifts of the currents’ arguments in the coproduct that brings $$\D(\psi_{\o,0}^+)=\psi_{\o,0}^+\otimes q_3^{2c_{(1)}a_{\o-\nu_3c_{(1)}}^+}\psi_{\o-\nu_3c_{(1)},0}^+,\quad \D(\psi_{\o,0}^-)=\psi_{\o-\nu_3c_{(2)},0}^-q_3^{-2c_{(2)}a_{\o-\nu_3c_{(2)}}^-}\otimes \psi_{\o-\nu_3c_{(1)},0}^-q_3^{-2c_{(1)}a_{\o-\nu_3c_{(1)}}^-}.$$ [^9]: The definition of the vertical representation is not unique due, for instance, to the following invariance of Drinfeld currents at $c=0$: $$\psi_\o^\pm(z)\to C_\o z^{\a_\o}\psi_\o^\pm(z),\quad x_\o^+(z)\to x_\o^+(z),\quad x_\o^-(z)\to C_\o z^{\a_\o}x_\o^-(z).$$ Here a particular choice is made to simplify the derivation of intertwiners in section 4 below. [^10]: The RHS of these commutation relations involves the coefficients $\s_{\o\o'}^{(k)}=-\s_{\bo'\o}^{(-k)}=kq_3^{k/2}\left[\d_{\o\o'}+q_3^{-k}\d_{\o\bo'}-q_1^k\d_{\o\ \o'+\nu_1}-q_2^k\d_{\o\ \o'+\nu_2}\right]$ appearing in the expansion of the scattering function \[def\_S\], $$\label{def_sigma} \left[S_{\o\o'}(z)\right]_-=\exp\left(\sum_{k>0}\dfrac{z^k}{k^2}q_3^{-k/2}\s_{\o'\o}^{(k)}\right),\quad \left[S_{\o\o'}(z)\right]_+=f_{\o'\o}(z)^{-1}\exp\left(-\sum_{k>0}\dfrac{z^{-k}}{k^2}q_3^{k/2}\s_{\o'\o}^{(-k)}\right).$$ [^11]: In fact, these relations are also satisfied for the grading operator $\xi_\o(z)$ (see appendix ). [^12]: There is an unfortunate conflict of notations here since the integer $p$ labeling the $\Zp$-orbifold is unrelated to the charge $p=\sum_\o p_\o$ of the branes. [^13]: To simplify the notations, we have omitted the dependence of the Nekrasov factors in the vectors of colors $\boldsymbol{c}=(c_\a)_{\a=1}^m$ and $\boldsymbol{c}'$. The shortcut notation $q_3^{-1}\bv'$ in $N(\bv,\bl|q_3^{-1}\bv',\bl')$ should be understood as a shift of the weights $q_3^{-1}v_\a'$ together with the corresponding shift of indices $c_\a'-\nu_3=\bc_\a'$. Thus, we have the important relation $$\begin{aligned} \begin{split} &N(\bv,\bl|q_3^{-1}\bv',\bl')=N(\bv',\bl'|\bv,\bl)f(\bv,\bl|\bv',\bl'),\quad \text{with:}\\ &f(\bv,\bl|\bv',\bl')=\prod_{{\genfrac{}{}{0pt}{}{\sAbox\in\bl}{\sAboxF\in\bl'}}}f_{c(\sAbox)c(\sAboxF)}(\chi_{\sAboxF}/\chi_{\sAbox})\times\prod_{\sAbox\in\bl}\prod_{\a\in C_{c(\sAbox)}(m')}\left(-\dfrac{\chi_{\sAbox}}{v_{\a}'}\right)\times \prod_{\sAbox\in\bl'}\prod_{\a\in C_{\bc(\sAbox)}(m)}\left(-\dfrac{v_\a}{q_3\chi_{\sAbox}}\right). \end{split}\end{aligned}$$ [^14]: The parameter $\a$ is determined by the omega-background parameters $(\e_1,\e_2)$. [^15]: The generators of this algebra are sometimes denoted $x_\o^+(z)\to E_i(z)$, $x_\o^-(z)\to F_i(z)$, $\psi_\o^\pm(z)\to K_i^\pm(z)$. More rigorously the $x^\pm-x^\pm$ exchange relation should be written $$(\k^{m_{\o\o'}}z-q^{\pm \b_{\o\o'}}w)x_\o^\pm(z)x_{\o'}^\pm(w)=(\k^{m_{\o\o'}}q^{\pm \b_{\o\o'}}z-w)x_{\o'}^\pm(w)x_\o^\pm(z).$$ This subtlety only affects the colliding points $z=q^{\pm \b_{\o\o'}}\k^{-m_{\o\o'}}w$. The parameter $\k$ here bears no connection with the Chern-Simons levels $\k_\o$ of the gauge theory. [^16]: This matrix is sometimes also written $$g_{\o\o'}(z)=\th_{\b_{\o\o'}}(\k^{m_{\o\o'}}z),\quad \th_m(z)=q^{-m}\dfrac{1-q^mz}{1-q^{-m}z},\quad \th_m(z^{-1})=\th_m(z)^{-1}=\th_{-m}(z).$$ [^17]: Alternatively, $$c_{\o\o'}^{(k)}=\dfrac1k\left[(q_3^k-q_3^{-k})\d_{\o,\o'}+(q_2^k-q_1^{-k})\d_{\o,\o'-1}+(q_1^k-q_2^{-k})\d_{\o,\o'+1}\right]=k(\b_{\o\o'}^{[k]}-\b_{\o'\o}^{[-k]}),$$ where $\b_{\o\o'}^{[k]}=(1+q_3^k)\d_{\o\o'}-q_1^{-k}\d_{\o,\o'-1}-q_2^{-k}\d_{\o,\o'+1}$ is the mass-deformed Cartan matrix of Kimura and Pestun [@Kimura2015] with the mass $\mu_e=q_1$ associated to each link $e:\o\to\o+1$ of the necklace quiver. [^18]: We could also express the coefficients $\s_{\o\o'}^{[k]}=k\b_{\o\o'}\k_{\o\o'}^{-k}=kq_3^{k/2}\b_{\o\o'}^{[k]}$ in terms of the mass-deformed Cartan matrix $\b_{\o\o'}^{[k]}=\d_{\o,\o'}+q_3^k\d_{\o,\bo'}-q_1^{-k}\d_{\o,\o'+\nu_1}-q_2^{-k}\d_{\o,\o'+\nu_2}$. [^19]: Note that these relations imply $$Y_\o^+(z)Y_{\o'}^-(w)=\dfrac{f_{\o'\o}(z/w)}{f_{\bo'\o}(q_3z/w)}Y_{\o'}^-(w)Y_\o^+(z).$$ [^20]: These coefficients appear in the expansions $$\label{exp_g} [g_{\o\o'}(z)]_+=f_{\o'\o}(z)^{-1}\exp\left(\sum_{k>0}z^{-k}c_{\o\o'}^{(k)}\right),\quad [g_{\o\o'}(z)]_-=f_{\o\o'}(z^{-1})\exp\left(-\sum_{k>0}z^{k}c_{\o\o'}^{(-k)}\right).$$ [^21]: We have used the following property to perform the sum over indices $\bi,\bj$: $$\dfrac1k\sum_{\bi,\bj=0}^{p-1}(q_1^{\bi}q_2^{\bj})^kq_3^{-k/2}\s_{\o,c_\a+\bi\nu_1+\bj\nu_2}^{(k)}=(1-q_1^{pk})(1-q_2^{pk})\d_{\o,c_\a}.$$ [^22]: Note that when the weights are shifted as $\bv\to q_3\bv$, we have to shift the colors $c_\a\to c_\a+\nu_3$ accordingly. For instance, $$\CG(\bv|q_3^{-1}\bv')=\prod_{\a=1}^m\prod_{\a'=1}^{m'}\prod_{\bi,\bj=0}^{p-1}\CG_{q_1^p,q_2^p}(v'_{\a'} q_1^{\bi+1}q_2^{\bj+1}/v_\a)^{\d_{c_\a,c'_{\a'}+(\bi+1)\nu_1+(\bj+1)\nu_2}}.$$

--- abstract: 'We consider the AGT relation [@AGT], expressing conformal blocks for the Virasoro and W-algebras in terms of Nekrasov’s special functions, in the simplest case of the 4-point functions for the first non-trivial $W_3$ algebra. The standard set of Nekrasov functions is sufficient only if additional null-vector restriction is imposed on a half of the external $W$-primaries and this is just the case when the conformal blocks are fully dictated by $W$-symmetry and do not depend on a particular model. Explicit checks confirm that the AGT relation survives in this restricted case, as expected.' author: - '[**A.Mironov**]{}[^1]  and [**A.Morozov**]{}[^2]' date: title: '[**On AGT relation in the case of $U(3)$** ]{} ' --- =-1.4in = - 1.0in FIAN/TD-16/09\ ITEP/TH-32/09 Introduction ============ A recent paper [@AGT], see also [@Wyl]-[@mmNF], provided a long-anticipated relation between $2d$ conformal theories [@BPZ] and Nekrasov’s functions [@Nek], decomposing the Seiberg-Witten prepotential into sums over $4d$ instantons. An exact formulation of the AGT relation [@AGT] is that the [**universal conformal blocks [@BPZ; @Za; @ZZ] for generic Verma modules of the Virasoro algebra coincide with the Nekrasov partition functions, provided the $\alpha$-parameters of the Virasoro primary fields are linear combinations of the $a$ and $\mu$-parameters of the Nekrasov functions.**]{} The central charge is related to Nekrasov’s peculiar additional parameter $\epsilon=\epsilon_1+\epsilon_2$, $c = 1 + \frac{6\epsilon^2}{\epsilon_1\epsilon_2}$ and dimensions are given by $\Delta_\alpha = \frac{\alpha(\epsilon-\alpha)}{\epsilon_1\epsilon_2}$. For details of explicit check of this relation see [@AGT; @Wyl; @MMMagt]. An additional observation of [@AGT] is that with the same $\{\alpha\} - \{a,\mu\}$ identification of the perturbative part of the Nekrasov partition function (exponentiated perturbative part of the Seiberg-Witten prepotential) coincides with the DOZZ functions [@DOZZ]: the structure constants of concrete, Liouville conformal model, that is, a certain reduction of the $SU(2)$ WZNW model, which has a natural description in terms of a single $2d$ free field [@fref]. With no surprise Nekrasov’s functions, appearing in this original AGT relation are associated with the gauge group $SU(2)$. An obvious generalization [@Wyl] of the AGT relation is to take a $2d$ conformal model made from the $r=N-1$ free fields with the extended $W_N$-symmetry [@ZW]-[@FLit] and look if its conformal blocks are expanded into the $SU(N)$ Nekrasov functions. The first problem here is that the theory of W-symmetries is under-developed. In the Virasoro sector, the crucial idea is to identify Virasoro operators with the $2d$ stress-tensor, which generates shifts in the holomorphic $z$ coordinate. This identification results in the two well-known postulates: L\_[-1]{}V(z) = \_zV(z) \[L-1\] for arbitrary vertex operator $V(z)$ and  \~ z\_[12]{}\^[\_3-\_1-\_2]{} z\_[23]{}\^[\_1-\_2-\_3]{} z\_[31]{}\^[\_2-\_1-\_3]{} \[3pfV\] for an arbitrary spherical 3-point function of the Virasoro [*quasi-primaries*]{} [@BPZ]. The 3-point correlator of secondary fields can be then immediately constructed from this one by exploiting the Virasoro symmetry. After that, arbitrary conformal blocks and correlators can be built from these three point functions by a “gluing-of-pants” procedure, at least in principle [@BPZ; @Sonoda; @MS]. Though it is a difficult procedure, involving infinite summations, it is unambiguously defined. In the case of $W$-symmetries, there is still no clear choice for the $W$-generalization of $z$ (a modulus of the $W$-geometry, see, e.g., [@Wgeom]) and, hence, no formulas like (\[L-1\]) and (\[3pfV\]). In result, an arbitrary theory with $W$-symmetry has many additional free parameters: the entire set of the $3$-point functions in the $W_3$ case is  &lt;V\_[\_1]{}(1) V\_[\_2]{}()  (W\_[-1]{})\^kV\_(0)&gt; \[omegadef\] involving a single insertion of arbitrary power of $W_{-1}$. In a theory with underlying $W$-geometry, like the conformal Toda proposed as an obvious $SU(N)$-counterpart of the Liouville theory in [@Wyl], these infinitely many parameters are of course fixed, but the relevant theory is still unavailable. Another obvious problem concerns the AGT relations themselves. For $N>2$ there is a clear mismatch in the number of parameters. Each $W_N$ primary, external or internal, is labeled by the highest-weight, which is arbitrary $r=N-1$ dimensional vector $\vec\alpha$, thus the 4-point conformal block depends on $(4+1)\cdot(N-1) + 1 = 5N-4$ free parameters ($4+1$ is the number of external + internal states, an extra $1$ is the central charge). At the same time, Nekrasov function depends on $N-1$ “expectation values” $\vec a$, $2N$ “masses” $\mu_f$ and two $\epsilon$-parameters, what gives $(N-1)+2N+2-1 = 3N$ ($1$ is subtracted because conformal invariance forces Nekrasov functions to be dimensionless: to depend only on ratios of parameters). For $N=2$ there was a nice matching [@MMMagt]: $5N-4$ = $3N$, but it breaks down for $N>2$: some $2(N-2)$ parameters of the conformal block do not have immediate counterparts in the Nekrasov functions. These two difficulties could be considered as a strong appeal for developing the theory of $W$-symmetries and looking for a further generalization of Nekrasov functions. However, Niclas Wyllard suggested in [@Wyl] an elegant way out which overcomes both problems at once. One can restrict consideration to the case when the appropriate number of “external” states are the [*special*]{} ones [@FLit], satisfying additional $W$-null-vector constraints at level one[^3]. In the case of $W_3$ just one is enough, (3w L\_[-1]{} - 2 W\_[-1]{})V\_[,w]{} = 0, \[Wzc\] for higher $N$ the number of constraints grows up to $N-2$.[^4] Such “special” primary states form a one-parametric family instead of an $r$-parametric one. “The appropriate number” to solve the matching problem is two out of four external states for the $4$-point conformal block. At the same time, if one of the three primaries in (\[omegadef\]) is special, these $3$-point functions get unambiguously defined without any reference to $W$-geometry. Thus, taking two of external states to be [*special*]{}, one also eliminates the first problem. Moreover, just under this restriction a counterpart of the DOZZ functions for the $SU(3)$ conformal Toda has been evaluated in [@FLit], and [@Wyl] checks that they indeed coincide with the exponential of perturbative $SU(3)$ prepotential in the conformal invariant case $N_f=2N=6$. In particular, the linear relation between $\alpha$ and $a,\mu$ parameters was established in this case. Unfortunately no detailed check of the main part of the AGT relation, between conformal blocks and “instanton sums”, was explicitly made in [@Wyl], despite relevant formulas were extracted from [@ZW]-[@FLit]. It is our task in this paper to partly fill in this gap. We explicitly present this check in the simplest case. The “simplest” involves the following restrictions: only $N=3$ and only contributions at levels one and two are considered. It is not a big problem to lift any of these two restrictions, still calculations are tedious, and it is unclear if they are that necessary to provide even more evidence in support of the AGT relation. In any case, they remain for the future work, maybe they can help to better understand the notion of $W$-symmetry beyond the seemingly-artificial (\[Wzc\]) constraint and to reveal the true meaning of the AGT relations, and the way to prove it in the full generality. This would further promote Nekrasov functions in the role of the crucially important new special functions and stimulate the further study of matrix-model tau-functions (i.e. generalized $\tau$-functions [@gentau], subjected to additional Virasoro-like constraints), already proposed for this role in [@amm]. Presentation in this paper does [*not*]{} follow literally the by-now-standard one, accepted in [@AGT; @Wyl; @MMMagt]. Instead of just writing down the conformal blocks and Nekrasov functions and comparing them (actually reporting the result of computer calculations), we [*also*]{} try to demonstrate that the AGT relation itself can turn into a powerful alternative method of iterative, level-by-level, evaluation of conformal blocks. Therefore, we begin with making anzatze for the elementary constituents of conformal blocks (the $2-$ and $3-$point functions) and then define the remaining coefficients by the requirement that the result matches Nekrasov functions. A part of this comparison is to check that the Kac determinant (that of the Shapovalov matrix made from the scalar products of states, essentially a $2$-point functions at special points $z_1=0$, $z_2=\infty$)) factorizes in an appropriate way: in $\alpha$-parametrization of primary fields its roots are given by linear function $\vec\alpha\vec e_i = m_i \epsilon_1 +n_i \epsilon_2$. The AGT relation strengthens this well known Regge-trajectory-like statement and makes an additional claim about the numerators of conformal blocks, not only denominators: [**the Nekrasov expansion provides a parametrization of the numerators by $\mu$-variables, which are also linear functions of $\alpha$-parameters**]{}. Only after making this kind of presentation at level one and for the simplest case of $c=2$ in s.2, we turn to the standard approach: calculate conformal blocks by the standard CFT methods in s.3, what also justifies the results of s.2. The necessary Ward identities, providing recursive relations for the $3$-point functions, are taken from a separate summary in [@MMMM]. Then in s.4 we extend the check of the $U(3)$ AGT relation to $c\neq 2$ and in s.5 to level two. At last, in s.6 we provide the [**complete**]{} proof of the AGT relation in the very particular case of one of the special states in the conformal block being completely degenerate at the first level. The proof is possible, since there is a complete answer for the conformal block in this case [@FLit]. The simplest example: two free fields with $c=2$ \[c2exa\] ========================================================== Structure of the $W_3$ algebra ------------------------------ In the case of several free fields the set of Virasoro primaries is not exhausted by exponentials $e^{\sqrt{2}\vec\alpha\vec\phi}$: there are many more primaries. If one wants exponentials to remain the only primary fields (in addition to the currents $\partial\vec\phi$), they should be primaries of a larger algebra, extending the Virasoro one. The standard choice in the case of $r$ free fields is the $W_{r+1}$ algebra. Generators $W^{(k)}(z) = \omega^{i_1\ldots i_k}\p\phi_{i_1}\ldots\p\phi_{i_k}(z)$ and tensors $\omega$ are defined so that only minimal singularities are allowed in the algebra. From now on, we switch to the simplest example of two free fields, $r=2$, associated with the $SU(3)$ group and $W_3$ algebra. The $W_3$ algebra has two generators: the stress tensor $T(z) = W^{(2)}(z) = \frac{1}{2}(\partial\vec\phi)^2(z)$ and $W(z) = W^{(3)}(z)$. The stress tensor $T(z)$ is invariant under $SO(2)$ rotations of $\phi$, and this rotation freedom should be fixed in order to define $W(z)$ unambiguously. We require that it is symmetric under $\phi_2 \rightarrow -\phi_2$ and, therefore, antisymmetric under $\phi_1\rightarrow -\phi_1$. This means that $W = (\p\phi_1)^3 + h \p\phi_1(\p\phi_2)^2$ with the single undefined parameter $h$. The operator product expansion is T(z)W(0) = \_1(0) + …and $h$ is fixed by the requirement of absence the most singular term with $z^{-4}$ (which is equivalent to the requiring the $W$-field to be Virasoro primary). This defines $h$ to be $h=-3$ and W(z) = W\^[(3)]{}(z) \~(\_1)\^3 -3\_1(\_2)\^2 = \_1((\_1)\^2 - 3(\_2)\^2) (the normalization coefficient is actually $2^{-3/2}$, see the Appendix). Accordingly, L\_0 e\^[(\_1 + \_2)]{} = \_[,]{}e\^[(\_1 + \_2)]{},\ W\_0 e\^[(\_1 + \_2)]{} = w\_[,]{}e\^[(\_1 + \_2)]{} with \_[,]{} = \^2+\^2,\ w\_[,]{} = (\^2-3\^2) Crucial for the AGT relation in the $SU(3)$ case is the fact, that \[Deltaw\] The r.h.s. has nothing to do with the $W_3$ algebra, but this relation demonstrates that a certain linear combination of $\Delta$ and $w$, that is, the one at the l.h.s. is a full square. $4$-point conformal block for the $W_3$ algebra \[WAL3\] -------------------------------------------------------- As reviewed in [@MMMagt] and (in far more detail) in [@MMMM], the conformal block has the structure \^\_[\_1,\_2,\_3,\_4]{}(x) =\_[|[Y]{}=|[Y]{}’|]{} x\^[|[Y]{}|]{} [B]{}\^\_[\_1,\_2,\_3,\_4]{} (,[Y]{}’) =\_[|[Y]{}|=|[Y]{}’|]{} x\^[|[Y]{}|]{} |\_[\_1,\_2;]{}([Y]{}) Q\_\^[-1]{}([Y]{},[Y]{}’) \_[,\_3,\_4]{}([Y]{}’) \[cbexp\] Here ${\cal Y}$ labels the elements of the Verma module for the $W_3$ algebra, i.e. a generalization to $W_3$ of what was the Young diagram for the Virasoro algebra, $\bar\Gamma({\cal Y})$ and $\Gamma({\cal Y})$ are the two types of the corresponding $3$-point functions (they are essentially the same in the Virasoro case, but interrelations are more sophisticated in the $W$-sector [@MMMM]) and $Q({\cal Y},{\cal Y}')$ is the Shapovalov form: a table made from the scalar products of different elements in the Verma module, which, as usual, does not mix different levels $|{\cal Y}|$. At level one, $|{\cal Y}|=1$ there are exactly two states in the Verma module, $L_{-1}V_{\vec\alpha}$ and $W_{-1}V_{\vec\alpha}$ and the corresponding block of the Shapovalov form is [|c||c|c|]{} &&\ Q\_([Y]{},[Y]{}’) & L\_[-1]{}V\_ & W\_[-1]{}V\_\ &&\ &&\ L\_[-1]{}V\_ & 2\_ & 3w\_\ &&\ &&\ W\_[-1]{}V\_ & 3w\_ & q\^2\_\ &&\ \[SF\] It is straightforwardly evaluated from the commutation relations of the algebra. However, at level one of the $W_3$-algebra the entries of this simple matrix are defined on dimensional grounds up to the coefficients. Coefficients $2$ and $3$ are dictated by the elementary part of $W_3$ commutators, = (n-m)L\_[m+n]{} + n(n\^2-1),\ = (2n-m)W\_[m+n]{} \[LWcom\] while $q$ follows from the sophisticated one for $\left[W_m,W_n\right]$, which also depends on the normalization of $W(z)$. This remaining commutation relation will appear only in (\[WWcom\]) far below. Instead of using it now, one can just [*guess*]{} that $q=9/2$ by looking at (\[Deltaw\]): then the determinant $2q\Delta^3-9w^2$ of the Shapovalov form (Kac determinant), which stands in the denominator of conformal block at level one, factorizes nicely: moreover, it is a full square. For this value of $q=9/2$ \^\_[\_1,\_2,\_3,\_4]{}(x) = 1 + + O(x\^2) or B\_1 =   \[B1c0\] where $v^2$ is the r.h.s. of (\[Deltaw\]) and all dependencies on $\vec\alpha_1,\ldots,\alpha_4$ are contained in the numerator $K_1$. Nekrasov’s formulas ------------------- Nekrasov’s partition function has the form = \_[Y\_1,…,Y\_N]{} x\^[|Y\_1|+…+ |Y\_N|]{} [Z]{}(Y\_1,…,Y\_N) where the sum is over $N$ sets of the [*ordinary*]{} Young diagrams. The level one contribution to ${\cal Z}$ is simply = 1 + x \_[i=1]{}\^N [Z]{}(,…,,…) + O(x\^2) where the only non-empty diagram $Y=\Box$ stands at the $i$-th place. Explicitly, for the conformal invariant case $N_f= 2N$ one has \_1 = \_[i=1]{}\^N [Z]{}(,…,,…) = - \_[i=1]{}\^N \[Z1gen\] where $P(a_i)=\prod_{f=1}^{2N} (a_i+\mu_f)$. In our simple example $\epsilon=0$ and ${Z}_1$ has square of the Van-der-Monde determinant $\Delta(\vec a) = \prod_{i<j}(a_i-a_j)$ in the denominator, Z\_1(=0) = \[Z1c0\] We preserve the standard notation $\Delta$ for both dimensions and the Van-der-Monde determinants, hopefully this will not cause a confusion. In the dimension, the argument $\vec\alpha$ is a subscript, while in the determinant $\vec a$ is an argument in brackets. The vector $\vec a$ is actually $r=N-1$-dimensional, like $\vec\alpha$, since its $N$ components $a_i$ (eigenvalues of the gauge-field vev matrix) are constrained by the zero trace condition \_[i=1]{}\^N a\_i = 0 \[trless\] AGT relation \[agt1\] --------------------- The AGT relation [@AGT; @Wyl; @MMMagt] is the statement that for some linear relation between the whole set of $\vec\alpha$’s and the whole set of $\vec a$ and $\mu_f$, the conformal blocks ${\cal B}$ for the $W_N$ algebra and the Nekrasov partition function ${\cal Z}$ for $U(N)$ group coincide provided $c = (N-1)\left(1+N(N+1)\frac{\epsilon^2} {\epsilon_1\epsilon_2}\right)$. We start from the simplest version of this relation for $N=3$: for the 4-point conformal block and for $c=2$, i.e. $\epsilon=0$. Comparing (\[B1c0\]) and (\[Z1c0\]) one observes that in this case the AGT relation requires that v \~(a) \[vVDMa\] which dictates the relation between $\vec a$ and $\vec\alpha$ of the intermediate state, and K\_1 \~M\_1 \[KM\] which would define the relation between $\mu_f$ and $\vec\alpha_1,\ldots,\vec\alpha_4$ for the four external states. ### Denominator Let us begin with (\[vVDMa\]). Coming back to the parametrization $\vec\alpha = (\alpha,\beta)$, one obtains v   (\^2-3\^2) = 4 \~(a\_1-a\_2)(a\_2-a\_3)(a\_3-a\_1) \[va\] which, together with (\[trless\]), implies a linear relation of the form a\_1 = (-),\ a\_2 = (+),\ a\_3 = -2 \[avsalphabeta\] It can be, of course, written in terms of the root and weight vectors of $SU(3)$. ### Numerator. The case of $\ \vec\alpha_1=\ldots=\vec\alpha_4=0$ Let us now proceed to (\[KM\]). We begin with putting all the four “external” dimensions equal to zero, $\alpha_1=\beta_1=\ldots=\alpha_4=\beta_4=0$. Then (\[KM\]) follows from B\^[(1)]{} = = -( + + ) = Z\^[(1)]{} \[ZB1\] where only the two parameters $p$ and $s$ at the l.h.s. are not defined on dimensional grounds and $P_6(a)=a^6 + \sum_{k=1}^6\sigma_k a^{6-k}$ at the r.h.s. is a polynomial of degree $6$, depending on choice of the $6$ parameters $\mu_f$ through symmetric polynomials \_k = \_[f\_1&lt;…&lt;f\_k]{}\_[f\_1]{}…\_[f\_k]{} \[sigmu\] As usual, $a_{ij} \equiv a_i-a_j$. Like the $SU(2)$ case, [@MMMagt] the parameter $-\epsilon_1\epsilon_2$ should be absorbed into rescaling of $\Delta$ and $w$. Now one substitutes $w^2 = \Delta^3-v^2$ with = -  -(a\_1\^2+a\_2\^2+a\_3\^2) and v\^2  -  - (a\_1-a\_2)\^2(a\_2-a\_3)\^2(a\_3-a\_1)\^2 and obtains for (\[KM\]) (a\_1\^2+a\_2\^2+a\_3\^2)( p(a\_1\^2+a\_2\^2+a\_3\^2)\^3 - s a\_[12]{}\^2a\_[23]{}\^2a\_[31]{}\^2 ) = a\_[23]{}\^2P\_6(a\_1) + a\_[13]{}\^2P\_6(a\_2) + a\_[23]{}\^2P\_6(a\_3) where $\tilde p = \frac{p+s}{12\kappa^2}$, $\tilde s = \frac{s}{6\kappa^2}$, and one should also impose condition (\[trless\]), i.e. substitute $a_3=-a_1-a_2$. When all the external dimensions are vanishing (we still keep $\epsilon=0$), the $\mu$-parameters at the r.h.s. are vanishing as well: \_1=…=\_4==0   \_1=…=\_6=0 Then the difference between the l.h.s. and the r.h.s. is a polynomial of degree $8$ in $a_1$ and $a_2$, equal to 2a\_1\^2a\_2\^2((80p - 13s - 4)(a\_1\^4+a\_2\^4) + (128p + 17s +2)a\_1a\_2(a\_1\^2+a\_2\^2) + (152p + 32s + 5)a\_1\^2a\_2\^2) For the AGT relation to be true this expression should vanish identically in $a_1$ and $a_2$, i.e. the three coefficients should be made vanishing by adjusting just the two parameters $\tilde p$ and $\tilde s$. Surprisingly or not, the three equations are indeed consistent, and their common solution is $\tilde p = 1/72$, $\tilde s = -2/9$. This corresponds to $s/p=-8/9$, i.e. one obtains (\[ZB1\]) in the form of the algebraic identity \[B1c2\] The conformal block at the l.h.s. can be of course calculated from representation theory of the $W_3$ algebra, as we shall see in s.\[CFT\] below. ### Numerator. The case of arbitrary $\vec\alpha_2$ and $\vec\alpha_4$ with no ambiguity in the $W$-conformal block If one switches on dimensions, the l.h.s. of (\[ZB1\]) gets more complicated and additional contributions should be compensated by adjustment of six $\mu$’s at the r.h.s. According to (\[cbexp\]) and (\[SF\]) with $q=9/2$, the conformal block at the l.h.s. is now given by 2(\_\^3-w\_\^2)B\^[(1)]{} = \^2 |\_[\_1,\_2;]{}(L\_[-1]{}) \_[\_3,\_4;]{}(L\_[-1]{})- - ( |\_[\_1,\_2;]{}(L\_[-1]{}) \_[\_3,\_4;]{}(W\_[-1]{}) + |\_[\_1,\_2;]{}(W\_[-1]{}) \_[\_3,\_4;]{}(L\_[-1]{})) + |\_[\_1,\_2;]{}(W\_[-1]{}) \_[\_3,\_4;]{}(W\_[-1]{}) \[cbexp1\] where |\_[\_1,\_2;]{}(L\_[-1]{}) = \_+\_[\_1]{}-\_[\_2]{} and $\Gamma_{\vec\alpha_3,\vec\alpha_4;\vec\alpha}(L_{-1})$ is given by exactly the same formula, while $\Gamma (W_{-1})$ and $\bar\Gamma (W_{-1})$ require additional restrictions to be uniquely determined as was explained in the Introduction. We begin with the case of $\vec\alpha_1=\vec\alpha_3=\vec 0$, when the ambiguity is known not to show up. Then \_[0,\_4;]{}(W\_[-1]{}) = w\_ + w\_[\_2]{},      |\_[0,\_2;]{}(W\_[-1]{}) = |w\_ + |w\_[\_2]{} \[omegaw2\] with some coefficients $\xi$, $\bar\xi$, $\eta$ and $\bar\eta$. In order to find $\xi$, one can also put $\vec\alpha_2=\vec\alpha_4=0$, then (\[cbexp1\]) turns into B\^[(1)]{} = \[B1c2CFT\] Comparison with (\[B1c2\]) implies that $\boxed{3\kappa^2=1}$ and $\boxed{\left(\frac{2\xi}{3}-1\right) \left(\frac{2\bar\xi}{3}-1\right)=1/9}$. Thus, we see that the AGT relation together with the basic definition (\[cbexp\]) can only partly fix the coefficients in formulas for the conformal block at level one. Therefore, we glance at s.3.2 and fix the values $\xi=\bar\xi$. Then, $\xi=1$ or $2$. The formulas of s.3.2 fix $\xi=1$. Then, under switching on $\vec\alpha_2$ and $\vec\alpha_4$, (\[B1c2CFT\]) becomes B\^[(1)]{} = which should be now compared with the r.h.s. of (\[ZB1\]). The resulting equations are identically satisfied for any $a$, provided \_1 = 0,\ \_2 = -(\_2\^2+\_2\^2+\_4\^2+\_4\^2),\ \_3= - (|\_2(\_2\^2-3\_2\^2) +\_4(\_4\^2-3\_4)\^3),\ \_4 = (\_2\^2+\_2\^2)(\_4\^2+\_4\^2),\ \_5 = ((\_2\^2+\_2\^2)\_4(\_4\^2-3\_4\^2) +|(\_4\^2+\_4\^2)\_2(\_2\^2-3\_2\^2)),\ \_6 =  \_2(\_2\^2-3\_2\^2) \_4(\_4\^2-3\_4\^2) \[sigxi1\] However, we still need to decompose these $\sigma$’s in $\mu$ according to (\[sigmu\]). The AGT relation requires $\mu$’s to be linear functions of $\vec\alpha$’s, and it is clear from the simple formula for $\sigma_6$ that this is indeed the case, with \_1=-|\_2,\ \_2=|,\ \_3=|,\ \_4=-\_4,\ \_5=,\ \_6= \[mu1c2\] and $\boxed{\eta,\ \bar\eta=\pm 1}$. This constraint is not a surprise since in (\[sigxi1\]) $\eta$ and $\bar\eta$ appear rather irregularly. This solution also reproduces all other $\sigma$’s and it is unique up to the sign choice for $\eta$, $\bar\eta$ and up to $6!$ permutations of $\mu$’s. ### The case of $\xi=2$ Let us now see what happens if one chooses $\xi=\bar\xi=2$ in (\[omegaw2\]). It turns out that this time the overdefined system of equations for six $\sigma$’s have [*no*]{} solution. Solution appears if one takes into account an extra factor Z\^[U(1)]{} = (1-x)\^[-]{} in the AGT relation, = Z\^[U(1)]{}Z\^[SU(N)]{} At level one introduction of $\nu$ implies that (\[ZB1\]) is deformed into B\^[(1)]{} = Z\^[(1)]{} + \[ZB1nu\] If $\nu\neq 0$ is allowed, then $\xi=2$ also provides a solution to (\[ZB1nu\]), actually with the same parameters $\mu$ as in (\[mu1c2\]) and with \[nuintch\] = \_2\^2+\_4\^2+\_2\^2+\_4\^2=\_2+\_4 The origin for this other solution is that it corresponds to the other choice of the special states, 2 and 4. This is why this value of $\xi$ is not consistent with the table (\[vertices\]) where the special states are chosen to be 1 and 3. ### The $U(1)$ factor $\nu$ and projective transformations Thus we observe the same phenomenon as in the $SU(2)$ case in [@MMMagt]: if one makes the simultaneous interchange of external lines $\vec\alpha_1 \leftrightarrow \vec\alpha_2$ and $\vec\alpha_3 \leftrightarrow \vec\alpha_4$ a simple $\nu$ (zero in the particular case of $\vec\alpha_1=\vec\alpha_3=0$ which we are now considering) turns into a sophisticated one. In principle this is nothing but a result of modular transformation of the conformal block. Indeed, the generic four-point function is [@BPZ; @ZZ] \[4V\] V\_[\_1]{}(z\_1)V\_[\_2]{}(z\_2)V\_[\_3]{}(z\_3)V\_[\_4]{}(z\_4) =\_[i&lt;j]{}\^4 (z\_i-z\_j)\^[d\_[ij]{}]{}(|z\_i-|z\_j)\^[|d\_[ij]{}]{}G(x,|x) where $d_{34}=d_{13}=0$, $d_{14}=-2\Delta_2$, $d_{24}=\Delta_1+\Delta_3- \Delta_2-\Delta_4$, $d_{34}=\Delta_1+\Delta_2-\Delta_3-\Delta_4$, $d_{23}= \Delta_4-\Delta_2-\Delta_1-\Delta_3$, $x={(z_1-z_2)(z_3-z_4)\over (z_3-z_2)(z_1-z_4)}$ and similarly for the complex conjugated part. $G(x,\bar x)$ is a bilinear combination of the conformal blocks $B(\{\Delta_i\},\Delta,,c;x)$ and $B(\{\bar\Delta_i\},\bar\Delta, c;\bar x)$. Due to the projective invariance, one can choose three of these four points arbitrarily. If choosing $z_4=\infty$, $z_3=1$ and $z_2=0$, one obtains $z_1=x$ and the 4-point correlator (\[4V\]) becomes (1-x)\^[d\_[12]{}]{}|x\^[|d\_[13]{}]{}(1-|x)\^[|d\_[12]{}]{}x\^[d\_[13]{}]{}G(x,|x) The simultaneous interchange of external lines $\vec\alpha_1 \leftrightarrow \vec\alpha_2$ and $\vec\alpha_3 \leftrightarrow \vec\alpha_4$ in formula (\[4V\]) does not change $x$ and $\bar x$ and leads to the factor of $(1-x)^{d_{24}}$ instead of $(1-x)^{d_{13}}$ ($d_{13}=0$) which is exactly $\nu=-d_{24}$ in (\[nuintch\]) (since $\Delta_1=\Delta_3=0$ there). ### General case, all $\vec\alpha\neq 0$, $\vec\alpha_1$ and $\vec\alpha_3$ [*special*]{} We now come to the general case of all external $\vec\alpha$ switched on. In this case, one needs some ansatz for the $3$-point functions $\Gamma (W_{-1})$, $\bar\Gamma (W_{-1})$. However, the number of parameters is large enough and the adjusting procedure fails to be very effective to define the coefficients. As we know from the $SU(2)$ case, restrictions get very strong when one includes the AGT relation at higher levels [@MMMagt], see s.5. Therefore, we return to a more straightforward approach and just take the true value of the $3$-point function from the conformal field theory analysis, see [@MMMM] and s.\[3pf\] below. Thus, we take the vertices from s.3.2 and fix the special states to have $\beta_{1,3} = r_{1,3}\alpha_{1,3}$ with some fixed $r_{1,3}$. Then AGT relation is satisfied, provided there is a certain linear relation between the sextuplets of parameters $\alpha_1,\vec\alpha_2,\alpha_3,\vec\alpha_4$ and $\mu_1,\ldots,\mu_6$. One of the possible solutions is \_1=(\_2-\_1),\ \_2=-,\ \_3=-,\ \_4=-(\_4+\_3),\ \_5=,\ \_6= \[mu1c2gen\] In this solution, the $U(1)$-factor shows up in the level one AGT relation (\[ZB1nu\]) with $\nu = -4\alpha_1\alpha_3$, while \[r\] r\_[1]{}=-,      r\_3= (43.108,24.13)(-90,-15) (0,0)[(0,1)[17]{}]{} (0,0)[(0,-1)[17]{}]{} (0,0)[(2,1)[15]{}]{} (0,0)[(2,-1)[15]{}]{} (0,0)[(-2,1)[15]{}]{} (0,0)[(-2,-1)[15]{}]{} (0,0)[(1,0)[8]{}]{} (0,0)[(-1,2)[4]{}]{} (0,0)[(-1,-2)[4]{}]{} (20,10) (-20,10) (5,10) (-5,10) (0,19) (7,7)[(0,0)\[cc\][$\lambda_1$]{}]{} (7,-7)[(0,0)\[cc\][$\lambda_2$]{}]{} (-10,-2)[(0,0)\[cc\][$\lambda_3$]{}]{} (0,0)[(-1,0)[8]{}]{} (0,0)[(1,2)[4]{}]{} (0,0)[(1,-2)[4]{}]{} Let us introduce the three vectors, see Fig.\[picroots\] \[fundvec\] \_1 = (1,),     \_2 = (1,-),     \_3= (-1,0) They are three (out of six) minimal vectors in the weight lattice of $sl(3)$. Then $a_i$ in (\[avsalphabeta\]) with $\kappa=1/\sqrt{3}$ and $\mu_f$ in (\[mu1c2gen\]) can be represented as scalar products, a\_[1,2,3]{} = \_[1,2,3]{}= ()\_[1,2,3]{} and \_[1,2,3]{} = -(\_2)\_[1,2,3]{} - \_1,\ \_[4,5,6]{} = (\_4)\_[1,2,3]{} - \_3 \[mu1c2genvect\] Let us remind that the primary exponentials are $V_{\vec\alpha} = e^{(\sqrt{2}\vec\alpha)\vec\phi}$, this can make these formulas looking more natural. The eigenvalues of the $W^{(3)}$ operator are w\_ = -3 (\_1) (\_2) (\_3) = (\^2-3\^2) Note that one could choose the pair of the [*special*]{} states in a way, different from (\[r\]), that is, $r_{1,3}=0,\pm\sqrt{3}$. We return to complete analysis in s.\[muvecs\] below, and the full final result is summarized in the Conclusion, eqs.(\[concmu\])-(\[concshi\]). Intermediate summary -------------------- This ends the explicit check of the AGT conjecture for the 4-point conformal block of $W_3$-algebra with $c=2$ [*at level one*]{}: it is indeed expressed through Nekrasov’s special functions with $\vec a$ and $\mu_f$ linearly expressed through $\vec\alpha$’s. Technical generalizations can go in four obvious directions. First, one can check the AGT relation between the structure constants of conformal model and “perturbative” Nekrasov’s functions. This relation refers to the 3-point functions of particular conformal model: Liouville in the $SU(2)$ and conformal Toda in the more general $SU(N)$ case, which are not yet known in full generality. It was also discussed in some detail in [@Wyl]. The second and third, one can either switch on $\epsilon\neq 0$ or proceed to higher levels or do the both, and repeat the calculations made for the $SU(2)$ case in [@MMMagt]. Fourth, one can consider the $SU(N)/W_N$ examples with $N>3$. These are straightforward exercises in representation theory. In order proceed to a discussion of the second and third directions in ss.\[cn2\] and \[secl\] respectively, we need some additional information from the theory of $W$-algebras, it is reminded in [@MMMM] and, more briefly, in the following section \[CFT\]. As to the fourth direction, we discuss only one, but very interesting subject in s.\[FLpro\]: it concerns a general role of the Nekrasov functions and, more generally, matrix-model $\tau$-functions as a modern substitute for hypergeometric functions. As a particular application, s.\[FLpro\] comments on the complete proof [@mmNF] of the AGT relation for the special case when the conformal blocks are hypergeometric functions (rather than generic “hypergeometric integrals” [@SheVa; @MV], to which they are supposedly related in the Dotsenko-Fateev approach). Some CFT considerations \[CFT\] ================================ In s.\[c2exa\] we constructed conformal blocks at level one mostly from dimensional considerations and demonstrated that remaining few parameters can be adjusted so that the AGT relations are fulfilled. Moreover, even at level one restrictions are rather strong, they actually become exhaustive already at level two. We did this on purpose: to demonstrate that the AGT relations can in the future become a powerful alternative approach to conformal field theory problems, which are pretty hard when attacked directly. Still, conformal blocks are conformal blocks and they can of course be evaluated by the standard CFT methods. In this section we briefly sketch how to evaluate the relevant quantities: the 3-point functions $\Gamma$ and $\bar\Gamma$ at level one and the Shapovalov matrix. The details can be found in [@MMMM]. For original papers and reviews see [@BPZ; @ZZ; @fref; @FLuk; @BW; @FLit]. Note that we use somewhat non-standard normalizations, adjusted to maximally simplify the $W$-algebra formulas. The price for this is certain deviation from conventions in some other papers. We illustrate our conventions with the example of free field theory in the Appendix. $W$-algebra and Shapovalov form ------------------------------- In s.\[WAL3\] we defined the parameter $q$ in (\[SF\]) from the requirement that the corresponding Kac determinant decomposes nicely with the help of (\[Deltaw\]). In CFT $q$ is calculated from the structure constants of $W_3$-algebra. However, the simple ones (\[LWcom\]) are not sufficient, one needs also the remaining, non-linear commutation relation [@ZW]-[@FLit]: = (n-m){\_[n+m]{} + ( -) L\_[n+m]{}} +\ + n(n\^2-1)(n\^2-4)\_[n+m,0]{} \[WWcom\] Here $\Lambda_n$ is a remnant of the $W^{(4)}$ operator. It does not exist as an independent operator in the $W_3$ algebra for the same reason that $\ {\rm tr}\, J^4 = \frac{1}{2}\left({\rm tr}\, J^2\right)^2$ for a $3\times 3$ traceless matrix $J$. In fact, \_n = \_[k=-]{}\^:L\_kL\_[n-k]{}: +L\_n, with $x_{2l} = 1-l^2$,   $x_{2l+1} = (2+l)(1-l)$. We use the notation from [@FLit], but our normalization of the $W$-operator is different by a factor of $3/\sqrt{2}$. Note for $n=0$ that the term with the normal ordering contains an item $L_0^2$ which [*does*]{} contribute to the vacuum average: = \_\^2 + \_= \_(\_+) Thus one obtains for the “difficult” element of the Shapovalov matrix (\[SF\]), (see [@MMMagt] where a similar calculation for the Virasoro group is reminded in detail):  = = ( (\_+) - )    \_\^2 in accordance with the [*guess*]{} in s.\[WAL3\]. The other three entries of the table (\[SF\]) do [*not*]{} depend on $c$. Likewise, more generally,  =( (\_+) + ) = = (D\_+) \[WnWn\] We introduced here a peculiar parameter D\_ (\_+) - \[Ddef\] which helps to get some formulas at low levels shorter (however, $D$ is nothing like a universal effective variable to absorb, say, the $c$-dependence; at most, it is a convenient abbreviation for limited purposes). Two other expressions of this kind are: \[WnW-n\] V\_ = L\_[-1]{}V\_,\ W\_1W\_[-1]{}V\_ = V\_ = V\_ where $V_{\vec\alpha}$ is arbitrary primary. $3$-point functions at level one \[3pf\] ---------------------------------------- Three point functions of interest for us are evaluated simply by moving integration contours. In (\[cbexp\]) we need the two [*types*]{} of such functions, $\Gamma$ and $\bar\Gamma$, which are different in the $W$-sector, see [@MMMM]. We begin with the $\Gamma (L_{1})$, i.e. with the [*stress-tensor*]{} insertion into a [*correlator*]{} of three primary fields (we write it for the operators $V_3$ and $V_4$, rather than $V_1$ and $V_2$, since this is the form in which we actually need it in (\[cbexp\]), also location of the Virasoro operator is underlined to make the formula readable):  V\_[\_3]{}(1) V\_[\_4]{}() = \_0 x\^[n+1]{}dxT(x) V\_(0) V\_[\_3]{}(1) V\_[\_4]{}()=\ = -(\_1 x\^[n+1]{}dx + \_x\^[n+1]{}dx) T(x) V\_(0) V\_[\_3]{}(1) V\_[\_4]{}()=\ = -\_k \_1 V\_(0) V\_[\_4]{}()+ \_k \_x\^[k-2]{} x\^[n+1]{}dx V\_(0)  V\_[\_3]{}(1)=\ = -(n+1)\_3 V\_(0) V\_[\_3]{}(1) V\_[\_4]{}()- V\_(0)   V\_[\_4]{}()+ V\_(0) V\_[\_3]{}(1)\[Lncont\] where we used the fact that primaries are annihilated by $L_n$ with $n>0$ and the action of $L_0$ produces the dimension: because of this, the first sum in the third line is reduced to two terms. Only one term contributes to the second sum even without these conditions. Now one can substitute $n=-1$, $n=0$ and $n=1$ to get (in an obvious abbreviated notation) L\_[-1]{} V\_(0) = - L\_[-1]{} V\_[\_3]{}(1) ,\ = -\_3 - L\_[-1]{} V\_[\_3]{}(1) + \_4,\ 0 = -2\_3 - L\_[-1]{} V\_[\_3]{}(1) + L\_[-1]{} V\_[\_4]{}() which can be resolved for any of the three positions of the $L_{-1}$ operator:  V\_[\_3]{}(1) V\_[\_4]{}() = (+\_3-\_4) V\_(0) V\_[\_3]{}(1) V\_[\_4]{}(),\ \ V\_(0) V\_[\_3]{}(1) = (-+ \_3 + \_4) V\_(0) V\_[\_3]{}(1) V\_[\_4]{}()\[L-1rels\] It is the first position that we denote through $\Gamma_{\alpha_3,\alpha_4;\alpha}(L_{-1})$ in the main text, and it is the second (boxed) one that we use when constructing $\Gamma_{\alpha_3,\vec\alpha_4;\vec\alpha}(W_{-1})$ for the special state $\vec\alpha_3$. Similarly, the $\Gamma (W_{-1})$-function results from the study of $W$-operator insertions:  V\_[\_3]{}(1) V\_[\_4]{}() = \_0 x\^[n+2]{}dxW(x) V\_(0) V\_[\_3]{}(1) V\_[\_4]{}()=\ = -(\_1 x\^[n+2]{}dx + \_x\^[n+2]{}dx) W(x) V\_(0) V\_[\_3]{}(1) V\_[\_4]{}()=\ = -\_k \_1 V\_(0) V\_[\_4]{}() \_k \_x\^[k-3]{} x\^[n+2]{}dx V\_(0)  V\_[\_3]{}(1)=\ =- V\_(0) V\_[\_3]{}(1) V\_[\_4]{}()- (n+2) V\_(0) V\_[\_4]{}()-\ - V\_(0) V\_[\_4]{}()- V\_(0) V\_[\_3]{}(1)\[Wncont\] Note that the sign in the box is [*different*]{} from the Virasoro case, because the odd-spin $3$-differential $W(z)$ is transformed differently from the even-spin $2$-differential $T(z)$ under the change $z\rightarrow 1/z$, $dz/z\rightarrow -dz/z$. Applying the same trick, i.e. putting $n=-1,0,1$ one gets in abbreviated notation: [ccccc]{} W\_[-1]{} V\_(0) = & & - W\_[-1]{} V\_[\_3]{}(1) & - W\_[-2]{} V\_[\_3]{}(1) ,&\ w =& -w\_3& - 2W\_[-1]{} V\_[\_3]{}(1) & - W\_[-2]{} V\_[\_3]{}(1) &- w\_4,\ 0 =& -3w\_3 &- 3W\_[-1]{} V\_[\_3]{}(1) & - W\_[-2]{} V\_[\_3]{}(1) & - W\_[-1]{} V\_[\_4]{}() \[W-1sys\] This time, however, the four unknowns can not be found from these three equations: we can only exclude terms with $W_{-2}$ obtain pair relations between the terms with $W_{-1}$:\  V\_[\_3]{}(1) V\_[\_4]{}() = (2w-w\_3+2w\_4) V\_(0) V\_[\_3]{}(1) V\_[\_4]{}() -V\_(0) V\_[\_3]{}(1),\ V\_(0) V\_[\_4]{}() = (w-2w\_3+w\_4) V\_(0) V\_[\_3]{}(1) V\_[\_4]{}() -V\_(0) V\_[\_3]{}(1)\[W-1rels\] Only if an additional constraint like (\[Wzc\]) is imposed on any [*one*]{} of the three states, all correlators can be unambiguously defined in a universal way: through $\Delta$’s and $w$’s (or, what is equivalent, through the parameters $\vec\alpha$). It is the first (boxed) of these formulas that was relevant for our consideration in s.\[agt1\]. The triple vertices of another type $\bar\Gamma$ are matrix elements rather than correlators (averages), they also obey a Ward identity, similar to (\[Lncont\]), but not exactly the same. We refer to [@MMMM] for details of the derivation and present here only the answers, which we need the most:\ V\_ | V\_2(0) = (\_ - \_1-\_2) V\_ | V\_1(1) V\_2(0), \[barGL\] $$\boxed{ \bar\Gamma(W_{-1})_{\alpha_1\alpha_2;\alpha} =\ \langle \underline{W_{-1}V_\alpha} | V_{\alpha_1}(1)\ V_{\alpha_2}(0)\rangle\ = \left(w_{\alpha}+ {2}w_1-w_2\right) \langle V_{\alpha} | V_1(1)\ V_2(0) \rangle\ + \langle V_{\alpha} | \underline{(W_{-1}V_1)(1)}\ V_2(0)\rangle }$$ The second formula in (\[barGL\]) is used to handle the situation when $V_{\alpha_1}$ is [*special*]{}. Shapovalov matrix at level two ------------------------------ Similar relations at level two are more sophisticated, but can be straightforwardly derived in the same way. They are listed in a separate paper [@MMMM], where they are also checked with the help of the free field model. This check is of importance to make because the derivation is rather tedious and mistakes are not so easy to exclude. We present and use these partly-validated answers in s.5 below. It remains to evaluate the $W_3$ Shapovalov form. At level one it is very simple: if expressed in terms of $D$ from eq.(\[Ddef\]), it turns from (\[SF\]) into [|c|c|]{} 2& 3w\ 3w & 9D/2\ \[SF1\] for all values of $c$. At level two there are five “$W_3$-Young diagrams” ${\cal Y}$ with $|{\cal Y}|=2$ and one gets a $5\times 5$ matrix: [|c||c|c|c|c|c|]{} &&&&&\ Q\_([Y]{},[Y]{}’) & L\_[-2]{} V\_ & L\_[-1]{}\^2 V\_ & L\_[-1]{}W\_[-1]{} V\_ & W\_[-2]{} V\_ & W\_[-1]{}\^2 V\_\ &&&&&\ &&&&&\ L\_[-2]{} V\_&4+&6&9w&6w&\ &&&&&\ &&&&&\ L\_[-1]{}\^2 V\_&6&4(2+1)&6w(2+1)&12w &9(3D+2w\^2)\ &&&&&\ &&&&&\ L\_[-1]{}W\_[-1]{} V\_&9w&6w(2+1)&9(D\^2 + D+ w\^2) &18D&(2+3)\ &&&&&\ &&&&&\ W\_[-2]{} V\_&6w&12w&18D& 9(D+1)&(3D+1)\ &&&&&\ &&&&&\ W\_[-1]{}\^2 V\_&&9(3D+2w\^2) &(2+3)&(3D+1) & D\^2(2+1)+\ &&&&& (D(+1)+4w\^2 )\ &&&&&\ \[SF2\] where special notation $\varkappa$ is introduced for a peculiar combination =  1,     [then]{}    D=(+) -    \[Ddef1\] We describe here only evaluation of the tricky entries of (\[SF2\]), which involves commutators of $W$-operators. The first such example is = &lt;|L\_2W\_[-1]{}\^2|&gt;    DAt the last stage we could of course directly use (\[WnW-n\]). As in [@MMMagt] we use obvious abbreviated notation, which we now abbreviate even further. Similarly, L\_1\^2W\_[-1]{}\^2  L\_1(W\_[-1]{}L\_[-1]{}+3W\_0)W\_[-1]{} = (L\_1W\_[-1]{})\^2+3(L\_1W\_[-1]{})W\_0 + 3L\_1 \[W\_0,W\_[-1]{}\] \[form1\] The first two terms are straightforward: they give $(3w)^2 + 3(3w)w = 18w^2$ when acting on a primary field. The last term is trickier: according to (\[WWcom\]), the commutator turns into $\frac{9}{2}\Big((\varkappa/2) \Lambda_{-1} - (1/5)L_{-1}\Big)$ and then into $\frac{9}{2}L_{-1}\Big(\varkappa(\Delta + 1/5)-1/5\Big) = \frac{9D}{2}L_{-1}$. Finally, $3L_1L_{-1} \rightarrow 6\Delta$ and one obtains for this entry of the table: $18w^2 + 27D\Delta$. Alternatively, in (\[form1\]) one could push $W_0$ to the left: $3L_{1}W_0W_{-1} = 3(W_0L_1+2W_1)W_{-1} \rightarrow 9w^2 + 27D\Delta$, which reproduces the same result in a simpler way. The other entries of the matrix (\[SF2\]) are calculated in a similar way: W\_1L\_1L\_[-1]{}W\_[-1]{} = W\_1(2L\_0+L\_[-1]{}L\_[1]{})W\_[-1]{} 2W\_1(W\_[-1]{}(L\_0+1)) + 3W\_1L\_[-1]{}W\_0 9D(+1) + 9w\^2; W\_1L\_1W\_[-2]{} 4W\_1W\_[-1]{} 18D; W\_1L\_[1]{}W\_[-1]{}\^2 W\_1(W\_[-1]{}L\_1+3W\_0)W\_[-1]{} wD+ wD+ W\_1L\_[-1]{} (2+3) W\_1\^2W\_[-2]{} W\_1 \[W\_1,W\_[-2]{}\] = W\_1 (\_[-1]{} + L\_[-1]{}) W\_1L\_[-1]{} ((+)+) (D+) W\_1\^2W\_[-1]{}\^2 = W\_1(W\_[-1]{}W\_1 + (\_0 - L\_0))W\_[-1]{} ()\^2 +W\_1\_0W\_[-1]{} - D(+1) The middle term in the last expression is a little tedious: since $\Lambda_0$ acts on a first descendant rather than primary, one needs to include also a term from the normal ordered part: $W_1\Lambda_0W_{-1} \rightarrow W_1\left(L_0^2+\frac{1}{5}L_0+2L_{-1}L_1\right)W_{-1} \rightarrow \frac{9D\Delta}{2}(\Delta+\frac{6}{5})(\Delta+1) +18w^2$. Collecting all terms, one finally gets W\_1\^2W\_[-1]{}\^2 ( D\^2(2+1)+(D(+1)+4w\^2 )) AGT relation to $c\neq 2$ at level one \[cn2\] ============================================== As was already mentioned, going to arbitrary central charges $c$ and to higher levels, i.e. to higher powers of $x$ and bigger sizes of the generalized Young diagrams, is a straightforward, but tedious exercise in group theory. The crucial difficult fact is actually well known: the Kac determinant is [*always*]{} (i.e. for arbitrary level and for arbitrary $W_N$ algebra) nicely factorized in terms of $\vec\alpha$-variables, so that its zeroes are always given by an integer combination $\vec\alpha \vec e_i= m\epsilon_1 + n\epsilon_2$. This fact is remarkable, but well known, its best heuristic “explanation” comes from the theory of free fields [@fref], but algebraically it looks somewhat artificial. No external lines, $\vec\alpha_1,\ldots,\vec\alpha_4=0$ \[lev1noext\] --------------------------------------------------------------------- At level one for arbitrary $c$ it follows from existence of the deformation of (\[va\]): \^3-w\^2 = (\^2+\^2)\^3 - ((\^2-3\^2))\^2 = ((\^2-3\^2))\^2  \ \[defDeltaw\] The l.h.s. of this deformed relation is interpreted in accordance with (\[SF1\]), as determinant of the Shapovalov matrix, D\^2-w\^2 = ((\^2+\^2-Q\^2+) - )(\^2+\^2-Q\^2)\^2 - \^2(\^2-3\^2)\^2 =\ ={ (\^2+\^2-Q\^2)\^2(\^2+\^2-Q\^2 ) - \^2(\^2-3\^2)\^2} i.e. $c = 2(1-12Q^2)$ and $\Delta=\alpha^2+\beta^2-Q^2$. Note that the eigenvalue $w$ and, thus, the operator $W^{(3)}$ itself also acquire the $c$-dependent factor $\sqrt{\frac{32}{22+5c}}$. In the AGT relation, the r.h.s. of (\[defDeltaw\]) should match with a deformation of square of the Van-der-Monde determinant in the Nekrasov formula, given by the denominator in eq.(\[Z1gen\]): instead of (\[vVDMa\]) one now has (\^2-Q\^2/4)((-Q)\^2 - 3\^2) ((+Q)\^2-3\^2) \~\ \~(a\_[12]{}-)a\_[12]{}(a\_[12]{}+) (a\_[23]{}-)a\_[23]{}(a\_[23]{}+) (a\_[31]{}-)a\_[31]{}(a\_[31]{}+) \[dencneq2\] Note that the number of different structures at the r.h.s. (\[dencneq2\]), i.e. in the denominator of the Nekrasov formula, increased from six to nine with switching on a non-zero $\epsilon$. Some three zeroes of the denominator should be canceled by the numerator of the Nekrasov formula, and only the remaining six should match the six zeroes at the l.h.s. of (\[dencneq2\]). From experience with the $SU(2)$ case in [@MMMagt], one can guess that [*irrelevant*]{} three factors at the r.h.s. of (\[dencneq2\]) are nothing but $a_{12}a_{23}a_{31}=\Delta(a)$, and the mapping (\[avsalphabeta\]) should be deformed so that ( [c]{} a\_[12]{} = -2\ a\_[23]{} = +\ a\_[31]{} = - )  ( [c]{} a\_[12]{}+= -2+ Q\ a\_[23]{}+= +Q+\ a\_[31]{}+= +Q- )    [and]{}   ( [c]{} a\_[12]{}-= -2- Q\ a\_[23]{}-= -Q+\ a\_[31]{}-= -Q- ) i.e. $\epsilon=Q$. In fact, after appropriate rescalings, see [@MMMagt], Q\^2= and \_ = ,\ w\_ = (\^2-3\^2) Note that (\[avsalphabeta\]) actually remains intact: a\_1 = (-),\ a\_2 = (+),\ a\_3 = -2\[avsalphabetacneq2\] with $\kappa = 1/\sqrt{3}$. For generic $c$ the AGT relation (\[ZB1\]) turns into =\ = ( + + ) + \[ZB1c\] where the conformal block at the l.h.s. is the deformation of (\[B1c2\]) and the r.h.s. is the level one Nekrasov function for non-vanishing $\epsilon=\epsilon_1+\epsilon_2$. The matching of denominators is guaranteed by the basic relation (\[defDeltaw\]), to which we adjusted all our choices of dimensions and the AGT relation $\vec a\ \stackrel{(\ref{avsalphabetacneq2})}{\leftrightarrow}\ \vec\alpha$. Therefore (\[ZB1c\]) defines the polynomial $P(a)$ and the $U(1)$ parameter $\nu$. Like in the $SU(2)$ case in [@MMMagt], some $\mu$-parameters are non-vanishing when $e\neq 0$, even despite the external lines are neglected, therefore, one should not expect the case under consideration makes any sense. However, it turns out in this case still there is a solution of the AGT relation with $\nu\neq 0$: P(a) = a\^2(a\^4+3e a\^3 + a\^2 +a + ) \[Pfree\] \[84\] = - This polynomial, however, does not correspond to any $\mu$’s. Introducing external states \[muvecs\] -------------------------------------- Now we need to switch on non-vanishing $\vec\alpha_1,\ldots, \vec\alpha_4$, and impose the speciality conditions on $\vec\alpha_1$ and $\vec\alpha_3$, this will modify $B^{(1)}$ at the l.h.s. of (\[ZB1c\]). Then we use this relation to find the modified polynomial $P(a)$, its roots. Surprisingly or not, they will indeed be just linear functions of $\alpha$’s and $\beta$’s, as required by the AGT conjecture, and they will provide us with the deformation of (\[mu1c2gen\]) and (\[mu1c2genvect\]). In what follows we omit the normalization factor $-\epsilon_1\epsilon_2$ from most formulas. It can be always restored on dimensional grounds. The [*speciality*]{} condition changes: it is defined by the zeroes of the r.h.s. of (\[defDeltaw\]), and therefore we have six options: \[ssc11\] [either]{}   \_1 = Q/2     [or]{}  \_1 = \_1 Q for such vectors $\frac{w_1}{\Delta_1} = \sqrt{D_1}$, thus when this ratio appears in our formulas, it does not produce new poles. These six options are of course related by Weyl group transformations. However, since we have two special states in a $4$-point conformal block, there is a freedom to choose their relative orientation. Note that the AGT relation itself is correct for all possible choices in (\[ssc11\]). However, if fact, not all of them correspond to the special states (\[Wzc\]). Indeed, (\[Wzc\]) is more restrictive than just the condition of zero Kac determinant (\[defDeltaw\]). Of 6 states (\[ssc11\]) only 3 actually correspond to (\[Wzc\]), [@MMMM] \[ssc1\] [either]{}   \_1 = - Q/2     [or]{}  \_1 = \_1 - Q Now all 9 possible combinations of choosing special states $\vec\alpha_{1,3}$ should be considered. For instance, instead of (\[mu1c2gen\]), [crcc]{} \_1=&(\_2-\_1) =& \_2 &-,\ \_2=&- =& - &-,\ \_3=&- =&- &-,\ \_4=&(\_4-\_3) =&\_4 &-,\ \_5=&- =&- &-,\ \_6=&- =&- &- \[mu1c2gen0\] and = -4\_1\_3 one has, e.g., for [crcc]{} \_1=&+=& \_2 &+ (-\_1+),\ \_2=&-+=& - &+ (-\_1+),\ \_3=&- +=&- &+ (-\_1+),\ \_4=&+=&\_4 &+ (\_3+),\ \_5=&-+=&- &+ (\_3+),\ \_6=&-+=&- &+(\_3+) \[mu1c2gen1-1\] with = - (-\_1+e) (\_3+e) Putting $\alpha_1=0$ and $\alpha_3=0$ here, one reproduces the solution (\[84\]). Other choices of the special states preserve this structure of answer for $\mu$’s. That is, they are equal to the sum of the same first terms as in (\[mu1c2gen1-1\]) plus some quantities $S(\alpha_{1,3})$ which depends on the choice of the special states: \[genmu\] [ccc]{} \_1=\_2+S(\_1), && \_4=-\_4+S(\_3),\ \_2=-(\_2+\_2)+S(\_1), && \_5=(\_4+\_4)+S(\_3),\ \_3=-(\_2-\_2)+S(\_1), && \_6=(\_4-\_4)+S(\_3) with \[genS\] S(\_1,-/2) = -\_1 + ,   S(\_1,-\_1 -) = \_1+,   S(\_1,\_1 - ) = \_1\ S(\_3,-/2) = \_3 + ,    S(\_3,-\_3 - ) = -\_3,   S(\_3,\_3 - ) = -\_3+and \[gennu\] =-3S(\_1)S(\_3) The manifest formulas for $S(\alpha_{1,3})$ for all possible choices of the special states are collected in the Conclusion. Calculations at level two \[secl\] ================================== Level-two Nekrasov functions ---------------------------- Nekrasov functions for $SU(N)$ with $N_f=2N$ fundamentals – the ones appearing in the AGT representation of the $4$-point conformal block with $2$ special external states – are labeled by $N$-plets of ordinary Young diagrams $\vec Y = \{Y_1,\ldots,Y_N\}$ and are given by [@Nek]: Z\_[Y]{} = where for a box with coordinates $(m,n)$ in the Young diagram (a,Y,\_[m,n]{}) = a + \_1(m-1) + \_2(n-1),\ E(a,Y\_1,Y\_2,\_[m,n]{}) = a + \_1(k\^T\_n(Y\_1)-m+1) - \_2(k\_m(Y\_2)-n) $k^T_n(Y)$ and $k_m(Y)$ are the length of the $n$-th row and the height of the $m$-th column in the diagram $Y$. If $Y_2=Y_1$, then at $\epsilon=0$ the combination $h(\Box_{m,n}) = E(0,Y,Y)=\big(k^T_n(Y)-m+1\big) + (\big(k_m(Y)-n\big)$ is the hook length for the box $(m,n)$, which enters, for example, the celebrated hook formula for d\_Y = \_[Y]{} h\^[-1]{}() for the character $\chi_Y(t_k=\delta_{k,1})=d_Y$, which defines up to various simple factors dimensions of representations of symmetric and linear groups. Thus, for $\epsilon=0$ the Nekrasov formulas are more or less natural objects in the theory of character expansions [@charex; @chardeco] and Hurwitz-Kontsevich partition functions [@HuKo], while for $\epsilon\neq 0$ they belong to representation theory of more sophisticated (quantum affine?) algebras. The AGT relation associates the deformation to $\epsilon\neq 0$ with the deformation to $c\neq r=N-1$ in the theory of $r$ free fields (conformal Toda?). The square $d^2_Y$ is also known as the Plancherel measure for integer partitions, which appears in the theory of KP and Kontsevich-Hurwitz $\tau$-functions, beginning from the famous sum rule \_Y \_Y(t)\_Y(t’) = (\_k kt\_kt’\_k) and its particular case \_Y d\^2\_Y x\^[|Y|]{} = e\^x The Nekrasov formulas with $\epsilon\neq 0$ provide a deformation of this measure. At level one, only one kind of $N$-plets $\vec Y$ contributes, containing just one diagram of the unit size: Z\_[\[1\]]{} = -\_[i=1]{}\^N -\_[i=1]{}\^N R\_i(a\_i) At level two, only three kinds of $N$-plets $\vec Y$ contribute: containing just one diagram of size two, either $[2]$ or $[11]$, or containing a pair of the single-box $[1]$ diagrams. The corresponding Nekrasov functions are Z\_[\[2\]]{} = \_[i=1]{}\^N R\_i(a\_i)R\_i(a\_i+\_2),\ Z\_[\[11\]]{} = - \_[i=1]{}\^N R\_i(a\_i)R\_i(a\_i+\_1),\ Z\_[\[1\]\[1\]]{} = \_[i&lt; j]{}\^N R\_i(a\_i)R\_j(a\_j)[a\_[ij]{}\^2 (a\_[ij]{}\^2-\^2)(a\_[ij]{}\^2-\_1\^2) (a\_[ij]{}\^2-\_2\^2)]{} \[Nek2\] See also eq.(62) of [@MMMagt] for a particular case of $N=2$. AGT relations at level two -------------------------- The AGT relation at level two states that B\^[(2)]{} = Z\_[\[2\]]{}+Z\_[\[11\]]{} + Z\_[\[1\]\[1\]]{} +Z\_[\[1\]]{} + \[ZB2\] where $Z_{[1]} = Z_1$. The denominator at the l.h.s. comes from inverse of the Shapovalov form, i.e. equals to the Kac determinant of matrix (\[SF2\]), while at the r.h.s. it contains the product of $a_{ij}$, shifted by linear combinations of $\epsilon_1$ and $\epsilon_2$. In other words, the AGT relation implies, to begin with, the matching of the denominators, which requires that  = ([Shapovalov matrix]{}) \~\_[ij]{} \_[p,q &gt; 0]{} (a\_[ij]{}+p\_1+q\_2) \[KDfact\] The fact that the Kac determinants factorize in such an elegant way, if expressed through the $\vec\alpha$-variables instead of the eigenvalues $\Delta$ and $w$ of the $W$-operators, is one of the central results of representation theory of the Virasoro and $W$-algebras. In s.\[kdfact\] we illustrate this remarkable general result in the particular example of level two for the $W_3$ algebra. Note, however, that the sets $\{(p,q)\}$ are actually not obviously the same at the l.h.s. and the r.h.s. of (\[KDfact\]). In the Kac determinant they are constrained by the condition $0<pq\leq {\rm level}$, while in Nekrasov’s formulas at the same level one can expect that either $p$, or $q$ can be equal to zero. These extra zeroes of the Nekrasov determinant are actually canceled between different terms in the sum over Young diagrams, however, particular Nekrasov functions have [*more*]{} singularities than there are zeroes of the Kac determinant. An additional claim as compared to the Kac determinant factorization property in the AGT conjecture is that the numerators also match for an appropriate [*linear*]{} relation between $\mu_f$ and $\vec\alpha_{1,2,3,4}$ parameters (with $\vec\alpha{1,3}$ subjected to the [*speciality*]{} constraints). We do not present details of this check here: at the present level of understanding it is more a computer exercise than a conceptual calculation. Some necessary comments are given in s.\[coblo2\]. Factorization of Kac determinant \[kdfact\] ------------------------------------------- For $c=2$ the determinants of the level-one and level-two Shapovalov matrices (\[SF\]) and (\[SF2\]) are equal to 3\^2\^2(\^2-3\^2)\^2 \~(a\_[12]{}a\_[23]{}a\_[31]{})\^2 \[den1c2\] and 3\^8\^4(2-1)\^2(2+1)\^2 (\^2-3\^2)\^4 ((-1)\^2-3\^2)\^2 ((+1)\^2-3\^2)\^2 \~\ \~((a\_[12]{}-1)a\_[12]{}\^2(a\_[12]{}+1)(a\_[23]{}-1)a\_[23]{}\^2(a\_[23]{}+1) (a\_[31]{}+1)a\_[31]{}\^2(a\_[31]{}-1))\^2 \[den2c2\] respectively. This is in perfect agreement with (\[KDfact\]), provided $\epsilon_1=-\epsilon_2=1$ and identification (\[avsalphabeta\]) is made: a\_[12]{}=-2,    a\_[23]{}=+,   a\_[31]{} = -,    [i.e.]{}\ a\_1 = (-),   a\_2 = (+),   a\_3 = - \[avsal\] (the relative normalization of $a$ and $\alpha$ in (\[avsal\]) is defined by the full AGT relation and is not obvious at the level of Kac determinants). For generic $c$ the degeneration that gives rise to full squares in the Nekrasov denominators is resolved, and the same happens to the Kac determinant. At level one, one obtains instead of (\[den1c2\]): K\_1=(4\^2-Q\^2) ((-Q)\^2-3\^2) ((+Q)\^2-3\^2) \~\ \~(2-Q)(2+Q) (+-Q)(++Q) (--Q)(-+Q) \~\ \~\_[ij]{} (a\_[ij]{}+) = (a\_[12]{}\^2-\^2)(a\_[23]{}\^2-\^2)(a\_[31]{}\^2-\^2) \[den1\] Therefore, identification (\[avsal\]) remains intact, one should only add that $Q=\epsilon$. At level two, the l.h.s. of (\[den2c2\]) is deformed into K\_2= {(4\^2-Q\^2) ((-Q)\^2-3\^2) ((+Q)\^2-3\^2)}\^2\ (2Q\^2-1-6Q+4\^2)(2Q\^2-1+6Q+4\^2)\ (1+12Q\^3+6\^3 Q-6Q-18\^2Q -15\^2 Q\^2-6\^2-2\^2-4Q\^2-6\^2\^2 +13\^2Q\^2+\^4+4Q\^4+9\^4)\ (1-12Q\^3-6\^3 Q+6Q+18\^2Q -15\^2 Q\^2-6\^2-2\^2-4Q\^2-6\^2\^2 +13\^2 Q\^2+\^4+4Q\^4+9\^4) so that the level one Kac determinant factors out from that of level two for arbitrary $\Delta$ and $w$, and the rest of the formula (including the second power of (\[den1\])) factorizes nicely in the $\alpha$-parametrization. The dictionary for decoding this quantity is simple: for $\epsilon_1+\epsilon_2=\epsilon=Q$ and $\epsilon_1\epsilon_2=-1$ one has (2+\_1)(2+\_2) = 4\^2+2Q- 1,\ (2+2\_1+\_2)(2+\_1+2\_2) = 4\^2+6Q+ 2Q\^2-1 and (++\_1)(++\_2) (-+\_1)(-+\_2) =\ 1-6\^2Q+2Q\^3+2Q-6\^2\^2-3\^2Q\^2 +6\^2+2\^2+\^2Q\^2+9\^4\^4,\ \ (++2\_1+\_2) (++\_1+2\_2) (-+2\_1+\_2) (-+\_1+2\_2) =\ = (1+12Q\^3+6\^3 Q-6Q-18\^2Q -15\^2 Q\^2-6\^2-2\^2-4Q\^2-6\^2\^2 +13\^2Q\^2+\^4+4Q\^4+9\^4) Thus, one can see that K\_2 \~(a\_[12]{}\^2-\^2)\^2(a\_[23]{}\^2-\^2)\^2(a\_[31]{}\^2-\^2)\^2 (a\_[12]{}\^2-(\_1+2\_2)\^2) (a\_[23]{}\^2-(\_1+2\_2)\^2) (a\_[31]{}\^2-(\_1+2\_2)\^2)\ (a\_[12]{}\^2-(2\_1+\_2)\^2) (a\_[23]{}\^2-(2\_1+\_2)\^2) (a\_[31]{}\^2-(2\_1+\_2)\^2) The factors $(a_{ij}^2-\epsilon_1^2)$ and $(a_{ij}^2-\epsilon_2^2)$, which one could expect to arise looking at the denominators in eq.(\[Nek2\]), do not actually appear in the Kac determinant. Instead, $K_1$ comes squared. This means that these factors should cancel out after summing different terms in the Nekrasov formula with the appropriately chosen polynomial $P(a)$ in the numerator. Conformal block at level two \[coblo2\] --------------------------------------- The conformal block $B^{(2)}$ is obtained from the general expression (\[cbexp\]) by substitution of the inverted $5\times 5$ Shapovalov matrix (\[SF2\]) and five pairs triple vertices from [@MMMM]. To define these vertices unambiguously in un-specified conformal model, one should impose the [*speciality*]{} conditions and make use of (\[W-1rels\]), what is also done in [@MMMM]. Note that there are five generalized Young diagrams ${\cal Y}$ of size $|{\cal Y}|=2$, two “pure Virasoro”, two “pure $W$” and one “mixed”. Starting from level two, there is no way to separate $W$ and Virasoro diagrams. Collecting all the seven pairs of vertices for the two first levels in one place, one gets [|ll|]{} &\ |(L\_[-1]{}) =& +\_1-\_2,\ &\ |(W\_[-1]{}) =& w+2w\_1-w\_2 + (-\_1-\_2)\ &\ &\ |(L\_[-2]{}) =& +2\_1 -\_2,\ &\ |(L\_[-1]{}\^2) =& (+\_1 -\_2) (+\_1 -\_2+1),\ &\ |(L\_[-1]{}W\_[-1]{}) =& (+\_1-\_2+1) (w+2w\_1-w\_2 + (-\_1-\_2)),\ &\ |(W\_[-2]{}) =& w+5w\_2-w\_2 + (-\_1-\_2),\ &\ |(W\_[-1]{}\^2) =& (w+2w\_1-w\_2 +(-\_1-\_2)) (w+2w\_1-w\_2 +(-\_1-\_2+1))+\ &\ &                                                          + (+\_1-\_2)\ &\ &\ (L\_[-1]{}) =& +\_3-\_4,\ &\ (W\_[-1]{}) =& w+w\_3+w\_4 - (+\_3-\_4)\ &\ &\ (L\_[-2]{}) =& +2\_3 -\_4,\ &\ (L\_[-1]{}\^2) =& (+\_3 -\_4) (+\_3 -\_4+1),\ &\ (L\_[-1]{}W\_[-1]{}) =& (+\_3-\_4+1) (w+w\_3+w\_4 - (+\_3-\_4)),\ &\ (W\_[-2]{}) =& w+w\_3+w\_4 - (+\_3-\_4),\ &\ (W\_[-1]{}\^2) =& (w+w\_3+w\_4 -(+\_3-\_4)) (w+w\_3+w\_4 -(+\_3-\_4+1)) +\ &\ &                                                          +(+\_3-\_4)\ &\ \[vertices\] Here we assumed that $\vec\alpha_1$ and $\vec\alpha_3$ are the special state, i.e. that W\_[-1]{}V\_[\_1]{}(1)=L\_[-1]{}V\_[\_1]{}(1)   -(+\_1-\_2)V\_[\_1]{}(1) and similarly for $\alpha_3$: this is why these specific combinations appear in the formulas. We also omit the common structure constant factors $C^\alpha_{\alpha_1\alpha_2} =\langle V_{\alpha}(0)| V_{\alpha_1}(1)\ V_{\alpha_2}(\infty)\rangle$ and $C_{\alpha\alpha_3\alpha_4} =\langle V_{\alpha}(0)\ V_{\alpha_1}(1)\ V_{\alpha_2}(\infty)\rangle$ at the r.h.s. Note the delicate and seemingly irregular sign differences between formulas for $\bar\Gamma$ and $\Gamma$ vertices, all these details being essential, since the AGT relations are very sensitive to details. For making up a conformal block, one also needs to invert the Shapovalov matrices (\[SF1\]) and (\[SF2\]). For (\[SF1\]) it is simple: [|c|c|]{} 9D/2& -3w\ -3w & 2\ \[SF11\] but this paper is too short to explicitly write down the inverse of (\[SF2\]), neither in $\Delta,w$, nor in $\alpha,\beta$ variables. Still, multiplying this inverse by the two $5$-vector $\Gamma$ and $\bar\Gamma$ from (\[vertices\]) accordingly to (\[cbexp\]), one obtains the conformal block $B^{(2)}$ which, indeed, coincides with the combination of $Z^{(2)}$ and $Z^{(1)}$ at the r.h.s. of (\[ZB2\]), provided the six $\mu$’s and $\nu$ are given by relations (\[genmu\])-(\[gennu\]) for various choices of [*special*]{} states $\vec\alpha_1$ and $\vec\alpha_3$, which we already found explicitly in the analysis of the first level. The calculation is tedious, it necessarily includes the check of expressions (\[vertices\]) for the $3$-point functions. As usual, it is worth starting from the simplest case of $c=2$, $\vec\alpha_1=\ldots=\vec\alpha_4=0$ (for $c=2$ this does not contradict [*speciality*]{} conditions). In this case, (\[vertices\]) is simplified to [|ll|l|]{} &&\ (L\_[-1]{}) =& &=,\ &&\ (W\_[-1]{}) =& w & = w\ &&\ &&\ (L\_[-2]{}) =& & = ,\ &&\ (L\_[-1]{}\^2) =& (+1) &= (+1),\ &&\ (L\_[-1]{}W\_[-1]{}) =& (+1)w &= (+1)w,\ &&\ (W\_[-2]{}) =& x\_1w & =w,\ &&\ (W\_[-1]{}\^2) =& x\_2\^3+x\_3w\^2 + x\_4\^2 + x\_5&= w\^2+\^2\ &&\ &&\ (L\_[-1]{}) =& &=,\ &&\ (W\_[-1]{}) =& w & = w\ &&\ &&\ (L\_[-2]{}) =& & = ,\ &&\ (L\_[-1]{}\^2) =& (+1) &= (+1),\ &&\ (L\_[-1]{}W\_[-1]{}) =& (+1)w &= (+1)w,\ &&\ (W\_[-2]{}) =& y\_1w & = w,\ &&\ (W\_[-1]{}\^2) =& y\_2\^3+y\_3w\^2 + y\_4\^2 + y\_5&= w\^2+\^2\ &&\ \[vert\] and the Nekrasov formula can be expressed through $\Delta$ and $w$ as follows: $$Z^{(2)} = \frac{2}{K_2}\left\{81\Delta(\Delta^3-w^2) \left(\Delta^3-\frac{8}{9}w^2\right)\right\}^2 +$$ $$+ \frac{81}{4K_2}(\Delta^2-w^2)\left\{ \Delta \Big(-1863 \Delta^9 + 2160w^2\Delta^6 + 608w^4\Delta^3 -896w^6\Big) + 6\Big(81 \Delta^9 + 480w^2\Delta^6 -448 w^4\Delta^3 \Big)\right. +$$ $$+ \Delta^2 \Big( 3807\Delta^6 -6696 w^2\Delta^3 + 2704w^4\Big) + 4\Delta \Big( -1377\Delta^6 + 1422w^2\Delta^3 -152 w^4\Big) + 3\Big(1053 \Delta^6 -960 w^2\Delta^3 + 16w^4\Big) +$$ . - 81(10-1)( \^3 - w\^2) } = = ,\ G\_2 = -72(-1)\^2 (25\^5+20\^4-25\^3+23\^2-8+1),\ G\_4 = 1664\^5+608\^4-2688\^3+2704\^2-608+ 48,\ G\_6 = -128(4+7), where we grouped terms with the same power of $-\epsilon_1\epsilon_2$ (which is suppressed for sake of space, but can be restored in the formulas) or, alternatively, with the same power of $w$. Eq.(\[ZB2\]) with $P(a)=a^6$, $\nu=0$ and AGT relation (\[avsalphabeta\]) between $\vec a$ and $\vec\alpha$ can be [*used*]{} to find the ten free parameters $x_{1},\ldots,y_5$. It is a strongly over-defined system of equations, still it has a solution: $x_1=1$, $x_2=0$, $x_3=1$, $x_4 = 9/2$, $x_5= 0$, $y_1=1$, $y_2=0$, $y_3=1$, $y_4 = 9/2$, $y_5= 0$, as shown in the right column of the table (\[vert\]). Then, validation of the AGT relation at this level comes from comparison of [*such*]{} $x$’s and $y$’s with their values in the CFT table (\[vertices\]). After that, one switches on the external states $\vec\alpha_1,\ldots,\vec\alpha_4$ and finally deforms to $c\neq 2$. [**This completes the explicit check of the $W_3$ AGT relation at level two.**]{} On complete proof of AGT relation in a [*very special*]{} case \[FLpro\] ======================================================================== In [@mmNF], among other things, we provided a proof of the AGT relation in a very restricted setting. The idea (also mentioned in [@Wyl]) is to make use of exact knowledge of the 4-point conformal block in the $sl(N)$ Toda theory [@FLit] for the [*very special*]{} kinematics: when one of external lines is [*special*]{} and another one is further restricted to belong to the [*fully degenerate*]{} $W$-Verma module. In $W^{(N)}$ case [*special*]{} means that $N-2$ conditions of the type (\[Wzc\]) are imposed on the $(N-1)$-component momentum, let it be $\vec\alpha_3$, while [*fully degenerate*]{} means that the last remaining component of the [*special*]{} $\vec\alpha_1$ is further fixed to a certain value. Selection rules of the Toda theory expresses the internal-state momentum $\vec\alpha$ through $\vec\alpha_1$ and $\vec\alpha_2$, much similarly to the free field model rule $\vec\alpha = \vec\alpha_1+ \vec\alpha_2$. The only difference is that, in the Toda case, there are not single, but $N$ possible values of $\vec\alpha$ (i.e. $N$ different non-zero conformal blocks). Thus, this Fateev-Litvinov conformal block depends on $2(N-1) + 1 + 1$ free parameters ($\vec\alpha_2$, $\alpha_3$, $\vec\alpha_4$,$c$) and on this high-codimension subspace in the total moduli space of the $4$-point conformal blocks it is represented by [*generic*]{} hypergeometric series .\_N F\_[N-1]{}(A\_1,…,A\_N;B\_1,…,B\_[N-1]{},x) = 1 + x + +\ + \_[n=3]{}\^\_[k=0]{}\^[n-1]{} \[hyse\] where $2N-1$ parameters $A_i,B_i$ are linear combinations of $\vec\alpha_2,\alpha_3,\vec\alpha_4$ with coefficients, depending on the central charge (actually, on the screening-charge parameters $\epsilon_1$ and $\epsilon_2$). This remarkable result demonstrates how generic hypergeometric series are embedded, as a linear subspace if the $\alpha$-parametrization of this moduli space is used!, into the space of conformal blocks, thus, generalizing the old description [@MV] of their embedding into the space of Dotsenko-Fateev integrals, or of hypergeometric integrals/correlators in the terminology of [@SheVa] (we remind that arbitrary $\phantom._NF_{N-m}$ are obtained in certain limits from $\phantom._NF_{N-1}$ and lie on the boundary of the moduli space, see, for example, [@MV]). As explained in [@mmNF], in these terms the AGT conjecture states that [**the Nekrasov functions provide the generalization of the $N=2$ hypergeometric series, exactly extending it to entire space of Virasoro conformal blocks**]{}. For $N>2$ they are probably sufficient only to extend hypergeometric series to the moduli space of conformal blocks, restricted to [*special*]{} subspace (i.e. when only two out of $m$ external momenta and all the $m-3$ internal momenta in the $m$-point conformal block are arbitrary, while the other $m-2$ external momenta belong to $1$-dimensional [*special*]{} subspaces). If this is true, there remains a question, [**what provides the further extension beyond Nekrasov functions(?)**]{}, which still remains to be answered. However, if one reverse the question: what are hypergeometric series from the point of view of Nekrasov functions, one immediately arrives to complete proof of the AGT conjecture for this very restricted (still huge!) class of conformal blocks. The point is that the answer is very simple: the hypergeometric series arise when only very specific $N$-plets of Young diagrams are non-vanishing: $\vec Y = \Big\{\emptyset,\ldots,[1^n],\ldots,\emptyset\Big\}$ or $\vec Y = \Big\{\emptyset,\ldots,[n],\ldots,\emptyset\Big\}$, i.e. only one diagram in the $N$-plet is not empty, and it is either a row $[1^n]$ or column $[n]$. We call such Nekrasov functions [*chiral*]{} and [*anti-chiral*]{} respectively, explicitly they are Z(,…,\[1\^n\],…,) = and Z(,…,\[n\],…,) = Z(,…,\[1\^n\],…,) (\_1\_2) where $i$ is the position of the non-empty diagram in the $N$-plet, $Q_i(x) = \prod_{j\neq i} (x-a_j)(x-a_j+\epsilon)$ and these functions immediately reproduce the hypergeometric series = the Fateev-Litvinov conformal block (\[hyse\]): P\_i(a\_i) = \_1\^N A\_1…A\_N,       Q\_i(a\_i) = [\_1\^[N]{} B\_1…B\_[N-1]{}-\_1\_2]{} \[PQvsAB\] for the chiral function. However, there is a question: what guarantees that only the chiral functions are non-vanishing? As explained in [@mmNF], this is a condition on polynomial $P$: for given $i$, P\_i(a\_j) = 0    [for]{}    ji       [and]{}     P\_i(a\_i+\_2) = 0 \[Pcons\] These are, in fact, $N$ conditions on the coefficients (or on the parameters $\mu_f$) of the polynomial $P(a) = \prod_{f=1}^{N_f=2N}(a+\mu_f)$ of degree $2N$, and the remaining $N$ parameters can be used to match the arbitrary set of $A_1,\ldots,A_N$ in (\[PQvsAB\]). After that the $N-1$ parameters $a_j$ ($\sum_{j=1}^N a_j=0$) can be adjusted to match $B_1,\ldots,B_{N-1}$. Note that all these relations are [*linear*]{}, in accordance with the general AGT claim. The parameters $\epsilon_1$ and $\epsilon_2$ are associated with the central charge, and the common scale drops out from the Nekrasov functions in the case of $N_f=2N$, as usual. We now use the detailed description of the $N=3$ case in this paper to show explicitly how these matchings work in this particular case, and illustrate what (\[Pcons\]) has to do with the maximal degeneracy condition imposed in [@FLit]. The simplest is to pick up formulas from the Conclusion, where they are collected in a list. For the sake of definiteness, we choose $i=3$ in (\[Pcons\]) (the choice of $i$ corresponds to the choice of non-zero conformal block, there are exactly $N$ of them). Then, it is obvious from (\[concavsal\]) and (\[concmu\]) that there is a natural choice of solution to $P(a_1)=P(a_2)=0$: 0 = (a\_1+ \_3) = (-\_2) - (-\_2) + S(\_1),\ 0 = (a\_2+\_2) = (-\_2) + (-\_2) + S(\_1) i.e. = \_2- S(\_1),     = \_2 \[seru\] which restricts the intermediate momentum $\vec\alpha$ to be a special combination of the two external momenta $\vec\alpha_1$ and $\vec\alpha_2$. The third condition $P(a_3+\epsilon_1)=0$ is then naturally imposed as a constraint 0 = ((a\_3+\_1) +\_1) = -(-\_2)+\_1 + S(\_1)    3S(\_1) + \_1 \[al1co\] which imposes an additional restriction on the already-[*special*]{} momentum $\vec\alpha_1$, i.e. fixes it completely. With these choices, (\[seru\]) and (\[al1co\]) it is guaranteed that the Nekrasov partition function is an $N=3$ hypergeometric series (\[hyse\]) with P(a\_3) = \_[f=1]{}\^6 (a\_3+\_f)  (-\_1)((+\_2)\^2-\_2\^2) \_[f=4]{}\^6 (a\_3+\_f)=\ =(-\_1)((+\_2)\^2-\_2\^2) A\_1A\_2A\_3 and a\_[31]{}a\_[32]{}(a\_[31]{}+)(a\_[32]{}+)  ((+\_2)\^2-\_2\^2) (a\_[31]{}+)(a\_[32]{}+)=\ = ((+\_2)\^2-\_2\^2)B\_1B\_2 i.e. A\_s = a\_3+\_[s+3]{} = {-[2]{}\_2-[23]{}+\_4, -[2]{}\_2-[23]{}+\_5, -[2]{}\_2-[23]{}+\_6 },   s= 1,2,3,\ B\_s = a\_[3s]{}+= {\_2-\_2, \_2+\_2 },   s= 1,2 with $\mu_{4,5,6}$ from (\[concmu\]). It remains to say that (\[al1co\]) is exactly the condition for $\vec\alpha_1$ to describe the maximally degenerate state, and (\[seru\]) is exactly the relevant selection rule in the $sl(3)$ Toda model [@FLit]. [**This completes the proof of AGT conjecture for the [*generic*]{} $N=3$ hypergeometric conformal blocks. The proof remains the same for arbitrary $N$,**]{} provided we accept the straightforward generalization (\[genagtconc\]) of (\[concavsal\]) and (\[concmu\]). A similarly transparent proof for non-hypergeometric blocks, however, remains to be found. Summary and conclusion ====================== Consideration in this paper leaves [**little room for doubts that the AGT relations are true for the Verma modules of generic groups**]{}, [*provided*]{} appropriate [*speciality*]{} (null-vector) constraints are imposed on the modules. Exact linear expressions for the Nekrasov parameters for the $4$-point conformal block are[^5]: SU(2):           a\_1=-a\_2 = ,    \_[1,2]{} = \_2+S(\_1),    \_[3,4]{}=\_4+S(\_3),   =[2S(\_1)S(\_2)\_1\_2]{} for the $SU(2)$ case. There are two different choices for each of the two shifts $S(\alpha_1)$ and $S(\alpha_3)$ (and thus a total of $4$ choices for the pair), S()=[2]{}Similarly, for $SU(3)$ \[concavsal\] \[concmu\] \[concnu\] There are three different choices for each of the two shifts $S(\vec\alpha_1)$ and $S(\vec\alpha_3)$ (and thus a total of $9$ choices for the pair), associated with the three possibilities to choose a [*special*]{} values[^6] $\vec\alpha_1 = (\alpha_1,\beta_1)$ and $\vec\alpha_3 = (\alpha_3,\beta_3)$:\ \[concshi\] If some other pair of external states is chosen to be [*special*]{}, formulas change according to the rule of projective transformation of the conformal block. The resulting $\nu$ can be easily evaluated: 1,32,4      + \_2+\_4-\_1-\_3 (this rule can look more natural, if one mentions that in the free field model there is an identity $2\vec\alpha_2\vec\alpha_4 + \Delta_4+\Delta_2 - \Delta_1-\Delta_3 = 2\vec\alpha_1\vec\alpha_3$). This same permutation with the corresponding change of $\nu\to {2S(\alpha_2)S(\alpha_4)\over\epsilon_1\epsilon_2}- \Delta_2-\Delta_4+\Delta_1+\Delta_3$ was discussed in the Liouville case in [@MMMagt]. With these values of parameters \[135\] where $Z$ and $B$ are explicitly defined in (\[Nek2\]) and (\[cbexp\]) with substituted (\[SF1\]), (\[SF2\]) and (\[vertices\]). Of many formulas in this paper these five are the only ones used in the actual calculation at levels one and two (certainly, to check (\[135\]) one also needs the identifications of parameters in (\[concavsal\])-(\[concshi\])). All the rest is included in order to present the logic and details of from-the-first-principles calculations very explicitly, so that they can be straightforwardly validated and generalized. Note that if formulas are rewritten in terms of the fundamental weights vectors $\lambda_i$ from (\[fundvec\]), they admit immediate generalization to arbitrary $N$. The $sl(N)$ conformal Toda model has $W^{(N)}$ symmetry, and the eigenvalues of its generators in $\alpha$-parametrization are given by =[\^2-[\^2]{}\_1\_2]{},   …,    w\^[(N)]{}\_ \~ \_[i=1]{}\^N (\_i) where $\displaystyle{\vec\epsilon = \epsilon\vec\rho}$ is directed along $\vec\rho$, the half-sum of all positive roots (or the sum of the fundamental weights), the central charge is $\displaystyle{c=(N-1)\left(1+N(N+1) {\epsilon^2\over\epsilon_1\epsilon_2}\right)}$ and $\{\vec\lambda_i\}$ is a Weyl-symmetric set of $N$ minimal weights, see Fig.\[picroots\]. Then the AGT relation between the $sl(N)$ Toda and $U(N)$ Nekrasov’s functions should look like \[genagtconc\] The $\sqrt{2}$ factors appear because with our normalization conditions the free field primaries are $V_{\vec\alpha} = e^{\sqrt{2}\vec\alpha\vec\phi}$. The notation $S$ refers simultaneously to “shift” and “special”. Less technical is the general proof of the AGT relation. A first step is made in [@mmNF] and s.\[FLpro\] of the present paper. It concerns the case, when one of the two [*special*]{} Verma modules in the 4-point conformal block is [*further*]{} specialized by imposing two extra null-vector conditions which leads to additional selection rules on the intermediate state (enforced by the vanishing of the structure constants $C_{\alpha_1,\vec\alpha_2,\vec\alpha}$, which are not included into the definition of conformal block in the present text). In this particular case, the conformal block was explicitly found in [@FLit] and it represents the generic hypergeometric function $\phantom._NF_{N-1}$, which has a straightforward character expansion. As rightly anticipated in [@Wyl], comparing it with Nekrasov formulas, which in this case get contribution only from a single set of “chiral” $N$-plets of Young diagrams, one obtains a complete [*proof*]{} of the AGT relation in this particular case. See [@mmNF] and s.\[FLpro\] above. The AGT relations can provide a new breath to abandoned areas of conformal theory. The beautiful theory of $W$-algebras, for example, may not attract enough attention, because there was nothing to compare it with. Now [*the equation gets the other side*]{}: whatever one obtains by hard calculations in $W$ theory can be compared with the differently looking formulas on the Nekrasov side. We demonstrated in this paper that this, indeed, opens a possibility of validating and improving $W$-algebra calculations. As emphasized in [@MMMagt], the AGT relation implies that there are two different expansions of the same quantity in characters (in sums over the Young diagrams): the natural expansion of conformal blocks and the Nekrasov sum over integer partitions. Already for $SU(2)$ they look absolutely different, like expansions associated with free boson and free fermion formalisms. The difference becomes even more pronounced for $N>2$: it is enough to say that the conformal block expansion is in terms of generalized Young diagrams ${\cal Y}$, labeling elements of the $W_N$-algebra Verma modules, while the Nekrasov functions are labeled by $N$-ples of the ordinary Young diagrams, and there is no any [*a priori*]{} obvious relation between the two. It can deserve mentioning that the AGT relation does not link contributions of individual diagrams on the l.h.s. and the r.h.s., only the entire sums over all diagrams of a given size (level). Thus, at the same stage it can be useful to consider relations between the conformal blocks of a given level and the original integrals from [@Nek] which were afterwards decomposed into sums of the Nekrasov functions. [**The AGT relation implies that these integrals have two different complementary combinatorial expansions**]{} (we use the word combinatorial, like in [@MMMagt], to emphasize that the character structure is not yet explicitly revealed in these expansions). A real mystery at the moment is what substitutes the AGT relations in the case of generic Verma modules of the $W_N$ algebras. The question can be asked in two opposite directions, that is, about more [*general*]{} and more [*special*]{} cases: what happens to generic [*non-degenerate*]{} modules for $N>2$ and what happens to [*stronger*]{} degenerate modules (say, associated with the Ashkin-Teller model [@ATM] and alike), which have more sophisticated conformal blocks, involving elliptic theta-constants already for $N=2$. Answers to these questions can help to better understand the group theory meaning and possible generalizations of the Nekrasov functions. The last but not least: within the general framework, [**the AGT relation between conformal theories and Nekrasov functions looks very much like a “quantization” of the well-known link [@GKMMM] between Seiberg-Witten theory [@SW] and integrable systems**]{}. It is very important to understand the AGT relation from this perspective and to compare it with the other quantizations of the same link, like those of [@RG]-[@NS]. Appendix: Normalization conventions \[norcft\] {#appendix-normalization-conventions-norcft .unnumbered} ============================================== We use somewhat non-standard normalizations, adjusted to maximally simplify the $W$-algebra formulas. The price for this is certain deviations from choices made in some other papers. We illustrate our conventions with the example of free field theory. The freedom in the choice of normalization prescriptions in CFT can be described as follows. First, one can choose the coefficient in the action, i.e. normalization of the free field, or, equivalently, the coefficient in the propagator, we call it $\kappa$. If the stress tensor is defined as a generator of (local) translations, i.e. has the standard operator product expansion with the term $T(z_1)V(z_2) = \ldots + z_{12}^{-1}\p V(z_2) + \ldots$ with the [*unit*]{} coefficient in front of the $V$-derivative term, then, $\kappa^{-1}$ appears as the coefficient in $T$, thus, $\kappa$ can be alternatively considered as a freedom in normalization of $T=W^{(2)}$. Similarly, to each operator $W^{(k)}$ (there are $N-1$ of them in the theory with $N-1$ free fields) can be ascribed an arbitrary new coefficient $\kappa_k$, i.e. there is no way to tie the normalization of $W^{(k)}$’s with $k>2$ to that of the propagator, and there is no [*canonical*]{} way to fix normalizations of some structure constants of the $W$-algebra. When the central charge $c$ deviates from $N-1$, there is a new arbitrariness in the choice of the normalization parameter in the deformation term $\p^2\phi$ in $T$: in fact, it can be considered as a freedom to normalize $W^{(1)}$. Second, in addition to these $N$ free normalizations, different papers define differently the parameter $\lambda$ in the conformal dimensions: $\Delta_{\vec\alpha} = \Delta\big(e^{\vec\alpha\phi/\lambda}\big)$. So far, fortunately the same $\lambda$ always appeared in the definition of other eigenvalues: $w_{\vec\alpha} = w\big(e^{\vec\alpha\phi/\lambda}\big)$ and so on. More explicitly, if the operator product expansion of the free fields is defined as \_i(z)\_j(0) = \_2 + …then the stress tensor T(z) = \_[i=1]{}\^[N-1]{}(\_i)\^2 + \_1\_[i=1]{}\^[N-1]{} Q\_i\^2\_i = ()\^2 + \_1Q\^2satisfies T(z)T(0) = + + + …with c = N-1 - 12\_2\_1\^2Q\^2, while the primary field V\_ =  :(): satisfies T(z) V\_(0) = V\_(0) + L\_[-1]{}V\_(0) + …with the dimension \_ = The free field theory selection rule for the non-vanishing correlator of exponentials is \_i \_i = 2\_1Q = 2It is inspired by zero-mode integration within the functional integral approach, or by conditions like V\_[\_1]{}(z\_1) V\_[\_2]{}(z\_2)\~(z\_1-z\_2)\^[\_2/\^2]{} =       \_[\_1]{}=\_[\_2]{} = -\_1\_2 within CFT itself. Usually in CFT with $N\neq 2$ [@fref] one chooses $\vec Q$ to be directed along the vector $\vec\rho$, the half-sum of all positive roots of $sl(N)$ or, which is the same, the sum of all fundamental weights, $|\vec\rho\,|^2 = \frac{N(N^2-1)}{12}$ (for non-simply-laced groups $\vec Q \sim \sqrt{-\epsilon_1/\epsilon_2}\,\vec\rho + \sqrt{-\epsilon_2/\epsilon_1}\,\vec\rho\, \check{\phantom.}$). The proportionality coefficient, however, is not canonically fixed and remains an arbitrary normalization parameter. The apparent problem with this convention, uniform and convenient for all groups, is that for $SU(2)$ $\rho = 1/\sqrt{2}$, therefore, one rarely uses it in considerations of the single field case, since it makes $k_1$ to contain $\sqrt{2}$. One can continue this “natural” $N=2$ convention to all $N$, using coordinates where $\vec\rho\,\vec\phi/|\rho| = \phi_{N-1}$, and this choice also is present in the literature. Anyhow, At the same time, the central charge of the $sl(N)$ Toda model, expressed by the AGT relation through the Nekrasov parameters $\epsilon=\epsilon_1+\epsilon_2$, is equal to i.e. in this notation = Since the common rescaling of all parameters in the Nekrasov functions with $N_f=2N$ does not affect the answer, one can easily choose $\epsilon_2=-1/\epsilon_1$, as it was done, for example, in [@AGT] and [@mmNF]. The following table contains the comparison of notations in different papers: $ \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline &&&&{\rm selecton}&&&&&\\ {\rm paper} & $N$ & {\rm central\ charge}&{\rm dimension} &{\rm rule:} & {\rm exponential} &1/\lambda & \varepsilon/Q = \kappa_1/|\rho| & \kappa_1\vec Q &\kappa_2\\ &&&&\sum_i \vec\alpha_i = &{\rm primary}&&& &\\ \hline &&&&&&&&&\\ \cite{Wyl}&2&1+6Q^2&\alpha(Q-\alpha)&Q&:e^{2\alpha\phi}:&2&\sqrt{2}&Q&-1/2\\ &&&&&&&&&\\ \hline &&&&&&&&&\\ \cite{Wyl}&N&(N-1)\Big(1+ &\frac{\vec\alpha(2\vec Q -\vec\alpha)}{2} &2\vec Q&:e^{\vec\alpha\vec\phi}:&1&\frac{1}{|\rho|}= \sqrt{\frac{12}{N(N^2-1)}}&\vec Q&-1\\ &&+ N(N+1)\varepsilon^2\Big)&&&&&&&\\ \hline &&&&&&&&&\\ \cite{MMMagt}&2&1+\frac{6Q^2}{\epsilon_1\epsilon_2} &\frac{\alpha(Q-\alpha)}{\epsilon_1\epsilon_2}&Q &:e^{\alpha\phi}:&1&\frac{1}{2\sqrt{2}} &\frac{1}{2}Q&\frac{2}{\epsilon_1\epsilon_2}\\ &&&&&&&&&\\ \hline &&&&&&&&&\\ \cite{MMMM}&2& 1-12Q^2 &\frac{\alpha(\alpha-2Q)}{2}&2Q &:e^{\alpha\phi}:&1&|\rho_{SU(2)}|^{-1}=\sqrt{2}&Q& 1\\ &&&&&&&&&\\ \hline &&&&&&&&&\\ \cite{MMMM}&3&2-12Q^2 &\frac{\alpha^2+\beta(\beta-2Q)}{2}&(0,2Q) &:e^{\alpha\phi_1+\beta\phi_2}:&1 &|\rho_{SU(3)}|^{-1}=\frac{1}{\sqrt{2}}&(0,Q)&1\\ &&&&&&&&&\\ \hline &&&&&&&&&\\ {\rm present}&3&2-24Q^2&\alpha^2+\tilde\beta^2-Q^2&(0,2Q) &:e^{\sqrt{2}(\alpha\phi_1+\beta\phi_2)}:&\sqrt{2}&1 &(0,\sqrt{2}Q)&1\\ {\rm paper}&&&\tilde\beta = \beta-Q&&&&&&\\ &&&&&&&&&\\ \hline &&&&&&&&&\\ \cite{mmNF}\ {\rm and}&&&&&&&&&\\ {\rm present}&N&(N-1)\Big( 1 + &\frac{\vec\alpha(\vec\alpha-2\vec Q)}{-\epsilon_1\epsilon_2} &2\vec Q &:e^{\sqrt{2}\vec\alpha\vec\phi}: &\sqrt{2}&\frac{\sqrt{2}}{|\vec\rho\,|} &\sqrt{2}\vec Q&\frac{1}{-\epsilon_1\epsilon_2}\\ {\rm paper} &&+\frac{N(N+1)\varepsilon^2}{\epsilon_1\epsilon_2}\Big) &=\frac{\overrightarrow{\tilde\alpha}^2-Q^2}{-\epsilon_1\epsilon_2}&&&&&&\\ &&&&&&&&&\\ \hline \end{array} $ In sample free field considerations with $N=3$ in this paper we imply that $V_{\vec\alpha} = e^{\sqrt{2}(\alpha\phi_1 + \beta\phi_2)}$. Thus, it is natural to normalize the $W^{(3)}$ operator to be W\^[(3)]{} = ((\_1)\^3 - 3\_1(\_2)\^2) so that W\^[(3)]{}(z) V\_(0) = V\_(0) + W\_[-1]{}V\_(0) + W\_[-2]{}V\_(0) …with[^7] w\_ = (\^2-3\^2) and W\_[-1]{}V\_ = : ((\^2-\^2)\_1 - 2\_2)V\_: \[W-1V\] Also, of course, L\_[-1]{}V\_ =  :(\_1 + \_2)V\_:= V\_ \[L-1V\] The [*special*]{} primaries of the $W_3$ algebra generate the Verma modules with a null-vector at the first level, with $W_{-1}V_{\vec\alpha} = \zeta L_{-1}V_{\vec\alpha}$ for some $\lambda$. From (\[L-1V\]) and (\[W-1V\]) it follows that 3(\^2-\^2) = 2,\ -6= 2i.e. $\zeta = -3\alpha$ and $\beta^2=3\alpha^2$, or $\zeta=3/2\alpha$ and $\beta=0$. Note that, under these relations, $\zeta = 3w_{\vec\alpha}/2\Delta_{\vec\alpha}$, so that the null-vector condition is exactly (\[Wzc\]) and the Kac determinant $v^2=\Delta^3-w^2 = \Big(\beta(\beta^2-3\alpha^2)\Big)^2$ vanishes. The main peculiarity of our normalization conventions is that the dimension $\Delta = \vec\alpha^2$ does not contain a minus sign, in variance from [@AGT; @Wyl; @MMMagt]: in dealing with $W$-algebras this eliminates unnecessary imaginary units. Factors of $2$ are eliminated from $\Delta$ and $w$ to make the central relation (\[Deltaw\]) as simple as possible. The price for this is the roots of $2$ in the primary exponentials and $W$ operators. Roots of $3$, however, are unavoidable, they come from the roots and weights of the underlying $sl(3)$ algebra. Acknowledgements {#acknowledgements .unnumbered} ================ We are indebted for hospitality and support to Prof.T.Tomaras and the Institute of Theoretical and Computational Physics of University of Crete, where part of this work was done. The work was partly supported by Russian Federal Nuclear Energy Agency and by RFBR grants 07-02-00878 (A.Mir.), and 07-02-00645 (A.Mor.). 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[**251**]{} (2005) 254, hep-th/0506075\ A.Mironov, Theor.Math.Phys. **146** (2006) 63-72, hep-th/0506158;\ A. Dymarsky and V. Pestun, Phys.Rev. **D67** (2003) 125001, hep-th/0301135 N.Nekrasov and S.Shatashvili, arXiv:0901.4748 [^1]: [ and [*ITEP, Moscow, Russia*]{}]{}; mironov@itep.ru; mironov@lpi.ru [^2]: ; morozov@itep.ru [^3]: In fact, restriction to the special states is rather natural within Davide Gaiotto’s construction [@Gpr] behind the AGT relation, where $4d$ SYM theory is described as a result of 5-brane compactification on a Riemann surface (spectral curve) [*a la*]{} [@Wbr; @C]. For $N\geq 3$ the punctures on the Riemann surface are not of general type, and it was suggested in [@Wyl] to identify them exactly with the [*special*]{} states on the CFT side of the AGT relation. We are grateful to the referee of the paper for his comments on this point. In the present text, we discuss a possibility to express conformal blocks through Nekrasov functions [*without*]{} explicit reference to branes or to instanton expansions in Seiberg-Witten theory, i.e. we are using AGT relations as a tool to find explicit expressions for conformal blocks, and then restriction to the [*special*]{} states should be eliminated one day, perhaps, by further extending the set of Nekrasov functions. [^4]: To avoid possible confusion, note that in [@FLit (2.12)] some three constraints were imposed on one of the special states. This was, however, done to make [*a particular method*]{} working, as to the conformal block, it is well defined (but very hard to evaluate) already with a single constraint for the special state. [^5]: Note that as compared with [@MMMagt], our definition of $\alpha$ is shifted: $\alpha\to\alpha+{\epsilon\over 2}$. [^6]: We define these [*special values*]{} simply as zeroes of the Kac determinant $K_1$, i.e. from the condition $D\Delta^2=w^2$. Actual [*special states*]{}, satisfying (\[Wzc\]) do not obligatory exist for each of these values (see, for example, the free-model example in [@MMMM], where only three out of six possible special [*values*]{} are actually associated with special [*states*]{}). It turns out that for this does not matter for AGT relation: it holds just for all special [*values*]{}. [^7]: In fact, it is $w_{\vec\alpha} = {(-\epsilon_1\epsilon_2)^{-3/2}}\cdot \alpha(\alpha^2-3\beta^2)$, but we systematically suppress powers of $(-\epsilon_1\epsilon_2)$ in most formulas to make them more readable. One should remember they are actually there.

--- abstract: '[Classical programming languages cannot model essential elements of complex systems such as true random number generation. This paper develops a formal programming language called the lambda-q calculus that addresses the fundamental properties of complex systems. This formal language allows the expression of quantumized algorithms, which are extensions of randomized algorithms in that probabilities can be negative, and events can cancel out. An illustration of the power of quantumized algorithms is the ability to efficiently solve the satisfiability problem, something that many believe is beyond the capability of classical computers. This paper proves that the lambda-q calculus is not only capable of solving satisfiability but can also simulate such complex systems as quantum computers. Since satisfiability is believed to be beyond the capabilities of quantum computers, the lambda-q calculus may be strictly stronger.]{}' author: - | Philip Maymin\ Long-Term Capital Management date: 'September 21, 1997' title: Programming complex systems --- Introduction ============ The purpose of this paper is to introduce a formalism for expressing models of complex systems. The end result is that modelling any complex system such as human society, evolution, or particle interactions, may be reduced to a programming problem. In addition to the modelling functionalities it provides, a programmable complex system also allows us to see, in its specification, what the distilled and essential elements of a complex system are. In particular, as we will see, interactions like those in a cellular automaton need not be explicit in the formalism, as they may be simulated. Classical programming languages are not strong enough to model complex systems. They do not allow for randomized events and are completely predictable and deterministic, features rarely found in complex systems. Some problems that may be quickly solved on quantum computers, which is a complex system, have no known quick solutions on classical computers or with classical programming languages. In this paper we extend the $\lambda $-calculus, the logical foundation of classical programming languages. The first extension, the $\lambda ^{p}$-calculus, is a new calculus introduced here for expressing randomized functions. Randomized functions, instead of having a unique output for each input, return a distribution of results from which we sample once. The $% \lambda ^{p}$-calculus then provides a formal method for computing distributions. More useful, however, would be the ability to compute conditional distributions. The second extension, the $\lambda ^{q}$-calculus, is a new calculus introduced here for expressing quantumized functions. Quantumized functions also return a distribution of results, called a *superposition*, from which we sample once, but $\lambda ^{q}$-terms have signs, and identical terms with opposite signs are removed before sampling from the result. Quantumized functions can then compute conditional distributions. The effect is that of applying some filter to a superposition to adjust each of the probabilities according to its fitness. One example is the quick solution of satisfiability: by merely filtering out the logical mappings of variables that do not satisfy the given formula, we are left only with satisfying mappings, if any. The $\lambda ^{q}$-calculus is the most general of the three calculi. One of the results of this paper is that the $\lambda ^{q}$-calculus is at least as powerful as quantum computers. Although much research has been done on the hardware of quantum computation (c.f. [@deutsch; @85], [@deutsch; @89], [@simon]), none has focused on formalizing the software. Quantum Turing machines [@deutsch; @85] have been introduced but there has been no quantum analogue to Church’s $\lambda $-calculus. The $\lambda $-calculus has served as the basis for many programming languages since it was introduced by Alonzo Church [@church] in 1936. It and other classical calculi make the implicit assumption that a term may be innocuously observed at any point. Such an assumption is hard to separate from a system of rewriting rules because to rewrite a term, you must have read it. One of the goals of these calculi is to make observation explicit. The $\lambda ^{p}$- and the $\lambda ^{q}$-calculi allow the expression of algorithms that exist and operate in the Heisenberg world of *potentia* [@heisenberg] but whose results are observed. To this end, collections (distributions and superpositions) should be thought of with the following intuition. A collection is a bunch of terms that co-exist in the same place but are not aware of each other. Thus, a collection of three terms takes up no more space than a collection of two terms. A physical analogy is the ability of a particle to be in a superposition of states. When the collection is observed, at most one term in each collection will be the result of the observation. The key point is that in neither calculus can one write a term that can determine if it is part of a collection, how big the collection is, or even if its argument is part of a collection. Despite this inability, the $\lambda ^{q}$-calculus is powerful enough to efficiently solve problems such as satisfiability that are typically believed to be beyond the scope of classical computers. The Lambda Calculus =================== This section is a review of the $\lambda $-calculus and a reference for later calculi. For more details see e.g. [@barendregt]. The $\lambda $-calculus is a calculus of functions. Any computable single-argument function can be expressed in the $\lambda $-calculus. Any computable multiple-argument function can be expressed in terms of computable single-argument functions. The $\lambda $-calculus is useful for encoding functions of arbitrary arity that return at most one output for each input. In particular, the $\lambda $-calculus can be used to express any (computable) *algorithm*. The definition of algorithm is usually taken to be Turing-computable. Syntax ------ The following grammar specifies the syntax of the $\lambda $-calculus. $$\begin{tabular}{|ll|} \hline $ \begin{array}{ll} x & \in \text{\emph{Variable}} \\ M & \in \text{\emph{LambdaTerm}} \\ w & \in \text{\emph{Wff}} \end{array} $ & $ \begin{array}{l} \text{Variables} \\ \text{Terms} \\ \text{Well-formed formulas} \end{array} $ \\ & \\ $ \begin{array}{lll} M & ::= & x \\ & \,\,\,| & M_{1}M_{2} \\ & \,\,\,| & \lambda x.M \\ & & \\ w & ::= & M_{1}=M_{2} \end{array} $ & $ \begin{array}{l} \text{variable} \\ \text{application} \\ \text{abstraction} \\ \\ \text{well-formed formula} \end{array} $ \\ \hline \end{tabular} \label{lambda syntax}$$ To be strict, the subscripts above should be removed (e.g., the rule for well-formed formulas should read $w::=M=M$) because $M_{1}$ and $M_{2}$ are not defined. However, we will maintain this incorrect notation to emphasize that the terms need not be identical. With this abuse of notation, we can easily read the preceding definition as: a $\lambda $-term is a variable, or an application of two terms, or the abstraction of a term by a variable. A well-formed formula of the $\lambda $-calculus is a $\lambda $-term followed by the equality sign followed by a second $\lambda $-term. We also adopt some syntactic conventions. Most importantly, parentheses group subexpressions. Application is taken to be left associative so that the term $MNP$ is correctly parenthesized as $\left( MN\right) P$ and not as $M\left( NP\right) .$ The scope of an abstraction extends as far to the right as possible, for example up to a closing parenthesis, so that the term $\lambda x.xx$ is correctly parenthesized as $\left( \lambda x.xx\right) $ and not as $\left( \lambda x.x\right) x.$ Substitution ------------ We will want to substitute arbitrary $\lambda $-terms for variables. We define the substitution operator, notated$~M\left[ N/x\right] $ and read “$% M $ with all free occurences of $x$ replaced by $N$.” The definition of the free and bound variables of a term are standard. The set of free variables of a term $M$ is written $FV\left( M\right) $. There are six rules of substitution, which we write for reference. $$\begin{array}{l} 1.\;x\left[ N/x\right] \equiv N \\ 2.\;y\left[ N/x\right] \equiv y\,\,\text{for variables }y\not{\equiv}x \\ 3.\;\left( PQ\right) \left[ N/x\right] \equiv \left( P\left[ N/x\right] \right) \left( Q\left[ N/x\right] \right) \\ 4.\;\left( \lambda x.P\right) \left[ N/x\right] \equiv \lambda x.P \\ 5.\;\left( \lambda y.P\right) \left[ N/x\right] \equiv \lambda y.\left( P\left[ N/x\right] \right) \,\,\text{if } \begin{array}{l} y\not{\equiv}x\text{ and} \\ y\notin FV\left( N\right) \end{array} \\ 6.\;\left( \lambda y.P\right) \left[ N/x\right] \equiv \lambda z.\left( P\left[ z/y\right] \left[ N/x\right] \right) \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{if } \begin{array}{l} y\not{\equiv}x\text{,} \\ y\in FV\left( N\right) \text{, and} \\ z\notin FV(P)\bigcup FV\left( N\right) \end{array} \end{array} \label{substitution}$$ This definition will be extended in both subsequent calculi. Reduction {#notions of reduction} --------- The concept of *reduction* seeks to formalize rewriting rules. Given a relation $R$ between terms, we may define the one-step reduction relation, notated$~\rightarrow _{R},$ that is the contextual closure of $R.$ We may also define the reflexive, transitive closure of the one-step reduction relation, which we call $R$-reduction and notate$~\twoheadrightarrow _{R},$ and the symmetric closure of $R$-reduction, called $R$-interconvertibility and notated$~=_{R}.$ The essential notion of reduction for the $\lambda $-calculus is called $% \beta $-reduction. It is based on the $\beta $-relation, which is the formalization of function invocation. $$\beta \triangleq \left\{ \begin{array}{c} \left( \left( \lambda x.M\right) N,M\left[ N/x\right] \right) \,\,\, \\ \text{s.t.}\,\,M,N\in LambdaTerm,\,x\in Variable \end{array} \right\} \label{beta}$$ There is also the $\alpha $-relation that holds of terms that are identical up to a consistent renaming of variables. $$\alpha \triangleq \left\{ \begin{array}{c} \left( \lambda x.M,\lambda y.M\left[ y/x\right] \right) \,\,\, \\ \text{s.t.}\,\,\,M\in LambdaTerm,\,y\notin FV\left( M\right) \end{array} \right\}$$ We will use this only sparingly. Evaluation Semantics -------------------- By imposing an evaluation order on the reduction system, we are providing meaning to the $\lambda $-terms. The evaluation order of a reduction system is sometimes called an operational semantics or an evaluation semantics for the calculus. The evaluation relation is typically denoted $\rightsquigarrow .$ We use call-by-value evaluation semantics. A *value* is the result produced by the evaluation semantics. Call-by-value semantics means that the body of an abstraction is not reduced but arguments are evaluated before being passed into abstractions. There are two rules for the call-by-value evaluation semantics of the $% \lambda $-calculus. $$\begin{aligned} &&\frac {}{v\rightsquigarrow v}\text{(Refl)\qquad \qquad (for }v\text{ a value)} \\ &&\frac{M\rightsquigarrow \lambda x.P\quad N\rightsquigarrow N^{\prime }\quad P\left[ N^{\prime }/x\right] \rightsquigarrow v}{MN\rightsquigarrow v}% \text{(Eval)}\end{aligned}$$ Reference Terms --------------- The following $\lambda $-terms are standard and are provided as reference for later examples. Numbers are represented as Church numerals. $$\begin{aligned} \underline{0} &\equiv &\lambda x.\lambda y.y \\ \underline{n} &\equiv &\lambda x.\lambda y.x^{n}y\end{aligned}$$ where the notation $x^{n}y$ means $n$ right-associative applications of $x$ onto $y.$ It is abbreviatory for the term $ \begin{array}{l} \underbrace{x(x(\cdots (x}y))) \\ \,n\text{ times} \end{array} .$ When necessary, we can extend Church numerals to represent both positive and negative numbers. For the remainder of the terms, we will not provide definitions. The predecessor of Church numerals is written $\underline{\text{% P}}.$ The successor is written $\underline{\text{S}}.$ The conditional is written $\underline{\text{IF}}.$ If its first argument is truth, written $\underline{\text{T}},$ then it returns its second argument. If its first argument is falsity, written $\underline{\text{F}},$ then it returns its third argument. A typical predicate is $\underline{\text{0?}}$ which returns $\underline{\text{T}}$ if its argument is the Church numeral $% \underline{\text{0}}$ and $\underline{\text{F}}$ if it is some other Church numeral. The fixed-point combinator is written $\underline{\text{Y}}.$ The primitive recursive function-building term is written $\underline{\text{PRIM-REC}}$ and it works as follows. If the value of a function $f$ at input $n$ can be expressed in terms of $n-1$ and $f\left( n-1\right) ,$ then that function $f$ is primitive recursive, and it can be generated by providing $\underline{% \text{PRIM-REC}}$ with the function that takes the inputs $n-1$ and $f\left( n-1\right) $ to produce $f\left( n\right) $ and with the value of $f$ at input $0.$ For example, the predecessor function for Church numerals can be represented as $\underline{\text{P}}\equiv \underline{\text{PRIM-REC}}% \,\left( \lambda x.\lambda y.x\right) \,\underline{\text{0}}.$ The Lambda-P Calculus ===================== The $\lambda ^{p}$-calculus is an extension of the $\lambda $-calculus that permits the expression of *randomized* algorithms. In contrast with a computable algorithm which returns at most one output for each input, a randomized algorithm returns a *distribution* of answers from which we sample. There are several advantages to randomized algorithms. 1. Randomized algorithms can provide truly random number generators instead of relying on pseudo-random number generators that work only because the underlying pattern is difficult to determine. 2. Because they can appear to generate random numbers arbitrarily, randomized algorithms can model random processes. 3. Given a problem of finding a suitable solution from a set of possibilities, a randomized algorithm can exhibit the effect of choosing random elements and testing them. Such algorithms can sometimes have an *expected* running time which is considerably shorter than the running time of the computable algorithm that tries every possibility until it finds a solution. Syntax {#section:lambda-p syntax} ------ The following grammar describes the $\lambda ^{p}$-calculus. $$\begin{tabular}{|ll|} \hline $ \begin{array}{ll} x & \in \text{\emph{Variable}} \\ M & \in \text{\emph{LambdaPTerm}} \\ w & \in \text{\emph{WffP}} \end{array} $ & $ \begin{array}{l} \text{Variables} \\ \text{Terms} \\ \text{Well-formed formulas} \end{array} $ \\ & \\ $ \begin{array}{lll} M & ::= & x \\ & \,\,\,| & M_{1}M_{2} \\ & \,\,\,| & \lambda x.M \\ & \,\,\,| & M_{1},M_{2} \\ & & \\ w & ::= & M_{1}=M_{2} \end{array} $ & $ \begin{array}{l} \text{variable} \\ \text{application} \\ \text{abstraction} \\ \text{collection} \\ \\ \text{well-formed formula} \end{array} $ \\ \hline \end{tabular} \newline \label{lambda-p syntax}$$ Since this grammar differs from the $\lambda $-calculus only in the addition of the fourth rule for terms, all $\lambda $-terms can be viewed as $\lambda ^{p}$-terms. A $\lambda ^{p}$-term may be a collection of a term and another collection, so that a $\lambda ^{p}$-term may actually have many nested collections. We adhere to the same parenthesization and precedence rules as the $\lambda $-calculus with the following addition: collection is of lowest precedence and the comma is right associative. This means that the expression $\lambda x.x,z,y$ is correctly parenthesized as $\left( \lambda x.x\right) ,(z,y)$. We introduce abbreviatory notation for collections. Let us write $\left[ M_{i}^{i\in S}\right] $ for the collection of terms $M_{i}$ for all $i$ in the finite, ordered set $S$ of natural numbers. We will write $a..b$ for the ordered set $\left( a,a+1,\ldots ,b\right) .$ In particular, $\left[ M_{i}^{i\in 1..n}\right] $ represents $M_{1},M_{2},\ldots ,M_{n}$ and $% \left[ M_{i}^{i\in n..1}\right] $ represents $M_{n},M_{n-1},\ldots ,M_{1}$. More generally, let us allow multiple iterators in arbitrary contexts. Then, for instance, $$\left[ \lambda x.M_{i}^{i\in 1..n}\right] \equiv \lambda x.M_{1},\lambda x.M_{2},\ldots ,\lambda x.M_{n}$$ and $$\left[ M_{i}^{i\in 1..m}N_{j}^{j\in 1..n}\right] \equiv \begin{array}{c} M_{1}N_{1},M_{1}N_{2},\ldots ,M_{1}N_{n}, \\ M_{2}N_{1},M_{2}N_{2},\ldots ,M_{2}N_{n}, \\ \vdots \\ M_{m}N_{1},M_{m}N_{2},\ldots ,M_{m}N_{n} \end{array} .$$ Note that $\left[ \lambda x.M_{i}^{i\in 1..n}\right] $ and $\lambda x.\left[ M_{i}^{i\in 1..n}\right] $ are not the same term. The former is a collection of abstractions while the latter is an abstraction with a collection in its body. Finally, we allow this notation to hold of non-collection terms as well by identifying $\left[ M_{i}^{i\in 1..1}\right] $ with $M_{1}$ even if $% M_{1}$ is not a collection. To avoid confusion, it is important to understand that although this “collection” notation can be used for non-collections, we do not extend the definition of the word *collection.* A *collection* is still the syntactic structure defined in grammar (\[lambda-p syntax\]). With these additions, every term can be written in this bracket form. In particular, we can write a collection as $\left[ \left[ M_{i}^{i\in S_{i}}\right] _{j}^{j\in S}\right] ,$ or a collection of collections. Unfortunately, collections can be written in a variety of ways with this notation. The term $M,N,P$ can be written as $\left[ M_{i}^{i\in 1..3}\right] $ if $M_{1}\equiv M$ and $M_{2}\equiv N$ and $M_{3}\equiv P;$ as $\left[ M_{i}^{i\in 1..2}\right] $ if $M_{1}\equiv M$ and $M_{2}\equiv N,P;$ or as $\left[ M_{i}^{i\in 1..1}\right] $ if $M_{1}\equiv M,N,P.$ However, it cannot be written as $\left[ M_{i}^{i\in 1..4}\right] $ for any identification of the $M_{i}.$ This observation inspires the following definition. \[dfn: cardinality\]The *cardinality* of a term $M,$ notated$% ~\left| M\right| ,$ is that number $k$ for which $\left[ M_{i}^{i\in 1..k}\right] \equiv M$ for some identification of the $M_{i}$ but $\left[ M_{i}^{i\in 1..\left( k+1\right) }\right] \not{\equiv}M$ for any identification of the $M_{i}$. Note that the cardinality of a term is always strictly positive. Syntactic Identities -------------------- We define substitution of terms in the $\lambda ^{p}$-calculus as an extension of substitution of terms in the $\lambda $-calculus. In addition to the substitution rules of the $\lambda $-calculus, we introduce one for collections. $$\left( P,Q\right) \left[ N/x\right] \equiv \left( P\left[ N/x\right] ,Q\left[ N/x\right] \right) \label{substitution-p}$$ We identify terms that are collections but with a possibly different ordering. We also identify nested collections with the top-level collection. The motivation for this is the conception that a collection is an unordered set of terms. Therefore we will not draw a distinction between a set of terms and a set of a set of terms. We adopt the following axiomatic judgement rules. $$\begin{aligned} &&\dfrac {}{M,N\equiv N,M}\text{(ClnOrd)} \\ &&\dfrac {}{\left( M,N\right) ,P\equiv M,(N,P)}\text{(ClnNest)}\end{aligned}$$ With these axioms, ordering and nesting become innocuous. As an example here is the proof that $A,(B,C),D\equiv A,C,B,D.$ For clarity, we parenthesize fully and underline the affected term in each step. $$\begin{array}{llll} \underline{A,((B,C),D)} & \equiv & ((\underline{B,C}),D),A & \text{(ClnOrd)} \\ & \equiv & (\underline{(C,B),D}),A & \text{(ClnOrd)} \\ & \equiv & \underline{(C,(B,D)),A} & \text{(ClnNest)} \\ & \equiv & A,(C,(B,D)) & \text{(ClnOrd)} \end{array}$$ It can be shown that ordering and parenthesization are irrelevant in general. Aside, it no longer matters that we took the comma to be right associative since any arbitrary parenthesization of a collection does not change its syntactic structure. Because of this theorem, we can alter the abbreviatory notation and allow arbitrary unordered sets in the exponent. This allows us to write, for instance, $\left[ M_{i}^{i\in 1..n-\{j\}}\right] \equiv M_{1},M_{2},\ldots ,M_{j-1},M_{j+1},\ldots ,M_{n}$ where $a..b$ is henceforth taken to be the unordered set $\left\{ a,a+1,\ldots ,b\right\} $ and the subtraction in the exponent represents set difference. This also subtly alters the definition of *cardinality* (\[dfn: cardinality\]). Whereas before the cardinality of a term like $\left( x,y\right) ,z$ was 2, because of this theorem, it is now 3. We may now also introduce a further abbreviation. We let $\left[ \left( M_{i}:n_{i}\right) \right] $ be a rewriting of the term $\left[ N_{i}^{i\in I}\right] $ such each of the $M_{i}$ are distinct and the integer $n_{i}$ represents the count of each $M_{i}$ in $\left[ N_{i}^{i\in I}\right] .$ Reductions ---------- The relation of collection application is called the $\gamma $-relation. It holds of a term that is an application at least one of whose operator or operand is a collection, and the term that is the collection of all possible pairs of applications. $$\gamma ^{p}\triangleq \left\{ \begin{array}{l} \left( \left[ M_{i}^{i\in 1..m}\right] \left[ N_{j}^{j\in 1..n}\right] ,\left[ M_{i}^{i\in 1..m}N_{j}^{j\in 1..n}\right] \right) \\ \text{s.t. }M_{i},N_{j}\in LambdaPTerm,\,m>1\text{ or }n>1 \end{array} \right\} \label{gamma-p}$$ We will omit the superscript except to disambiguate from the $% \gamma $-relation of the $\lambda ^{q}$-calculus. It can be shown that the $\gamma $-relation is Church-Rosser and that all terms have $\gamma $-normal forms. Therefore, we may write $\gamma \left( M\right) $ for the $\gamma $-normal form of $M.$ We extend the $\beta $-relation to apply to collections. $$\beta ^{p}\triangleq \left\{ \begin{array}{l} \left( \left( \lambda x.M\right) \left[ N_{i}^{i\in S}\right] ,\left[ M\left[ N_{i}^{i\in S}/x\right] \right] \right) \\ \text{s.t. }M\text{, }\left[ N_{i}^{i\in S}\right] \in LambdaPTerm,\,x\in Variable \end{array} \right\} \label{beta-p}$$ where $\left[ M\left[ N_{i}^{i\in S}/x\right] \right] $ is the collection of terms $M$ with $N_{i}$ substituted for free occurrences of $x$ in $M,$ for $% i\in S.$ Evaluation Semantics -------------------- We extend the call-by-value evaluation semantics of the $\lambda $-calculus. We modify the definition of a value $v$ to enforce that $v$ has no $\gamma $-redexes. $$\begin{aligned} &&\dfrac {}{v\rightsquigarrow v}\text{(Refl)\qquad \qquad (for }v\text{ a value)} \\ &&\dfrac{\gamma \left( M\right) \rightsquigarrow \lambda x.P\quad \gamma \left( N\right) \rightsquigarrow N^{\prime }\quad \gamma \left( P\left[ N^{\prime }/x\right] \right) \rightsquigarrow v}{MN\rightsquigarrow v}\text{% (Eval)} \\ &&\dfrac{\gamma \left( M\right) \rightsquigarrow v_{1}\quad \gamma \left( N\right) \rightsquigarrow v_{2}}{\left( M,N\right) \rightsquigarrow \left( v_{1},v_{2}\right) }\text{(Coll)}\end{aligned}$$ Observation {#lambda-p observation} ----------- We define an observation function $\Theta $ from $\lambda ^{p}$-terms to $\lambda $-terms. We employ the random number generator $RAND$, which samples one number from a given set of numbers. $$\begin{aligned} \Theta \left( x\right) &=&x \\ \Theta \left( \lambda x.M\right) &=&\lambda x.\Theta \left( M\right) \\ \Theta \left( M_{1}M_{2}\right) &=&\Theta \left( M_{1}\right) \Theta \left( M_{2}\right) \\ \Theta \left( M\equiv \left[ M_{i}^{i\in 1..\left| M\right| }\right] \right) &=&M_{RAND(1..\left| M\right| )}\end{aligned}$$ The function $\Theta $ is total because every $\lambda ^{p}$-term is mapped to a $\lambda $-term. Note that for an arbitrary term $T$ we may write $\Theta \left( T\right) =T_{RAND(S)}$ for some possibly singleton set of natural numbers $S$ and some collection of terms $\left[ T_{i}^{i\in S}\right] .$ We can show that observing a $\lambda ^{p}$-term is statistically indistinguishable from observing its $\gamma $-normal form. Observational Semantics ----------------------- We provide another type of semantics for the $\lambda ^{p}$-calculus called its *observational semantics.* A formalism’s observational semantics expresses the computation as a whole: preparing the input, waiting for the evaluation, and observing the result. The observational semantics relation between $\lambda ^{p}$-terms and $\lambda $-terms is denoted$~\multimap $. It is given by a single rule for the $\lambda ^{p}$-calculus. $$\frac{M\rightsquigarrow v\quad \Theta \left( v\right) =N}{M\multimap N}\text{% (ObsP)} \label{obs-p}$$ Examples -------- A useful term of the $\lambda ^{p}$-calculus is a random number generator. We would like to define a term that takes as input a numeral and computes a collection of numerals from to . This can be represented by the following primitive recursive $\lambda ^{p}$-term. $$\underline{\text{R}}\equiv \underline{\text{PRIM-REC}}\,\left( \lambda k.\lambda p.\left( k,p\right) \right) \,\underline{\text{0}}$$ Then for instance $\underline{\text{R}}\,\underline{\text{3}}% =\left( \underline{3},\underline{2},\underline{1},\underline{0}\right) .$ The following term represents a random walk. Imagine a man that at each moment can either walk forward one step or backwards one step. If he starts at the point $0$, after $n$ steps, what is the distribution of his position? $$\underline{\text{W}}\equiv \underline{\text{PRIM-REC}}\,\left( \lambda k.\lambda p.\left( \underline{\text{P}}p,\underline{\text{S}}p\right) \right) \,\underline{\text{0}}$$ We assume we have extended Church numerals to negative numbers as well. This can be easily done by encoding it is a pair. We will show some of the highlights of the evaluation of $\underline{\text{W}}\,\underline{\text{3% }}.$ Note that $\underline{\text{W}}\,\underline{\text{1}}=\left( \underline{% -1},\underline{1}\right) .$$$\begin{array}{lll} \underline{\text{W}}\,\underline{\text{3}} & = & \underline{\text{P}}\left( \underline{\text{W}}\,\underline{\text{2}}\right) ,\underline{\text{S}}% \left( \underline{\text{W}}\,\underline{\text{2}}\right) \\ & = & \underline{\text{P}}\left( \underline{\text{P}}\left( \underline{\text{% W}}\,\underline{\text{1}}\right) ,\underline{\text{S}}\left( \underline{% \text{W}}\,\underline{\text{1}}\right) \right) ,\underline{\text{S}}\left( \underline{\text{P}}\left( \underline{\text{W}}\,\underline{\text{1}}\right) ,\underline{\text{S}}\left( \underline{\text{W}}\,\underline{\text{1}}% \right) \right) \\ & = & \underline{\text{P}}\left( \underline{\text{P}}\left( \underline{-1},% \underline{1}\right) ,\underline{\text{S}}\left( \underline{-1},\underline{1}% \right) \right) ,\underline{\text{S}}\left( \underline{\text{P}}\left( \underline{-1},\underline{1}\right) ,\underline{\text{S}}\left( \underline{-1% },\underline{1}\right) \right) \\ & = & \underline{\text{P}}\left( \left( \underline{-2},\underline{0}\right) ,\left( \underline{0},\underline{2}\right) \right) ,\underline{\text{S}}% \left( \left( \underline{-2},\underline{0}\right) ,\left( \underline{0},% \underline{2}\right) \right) \\ & = & \left( \left( \underline{-3},\underline{-1}\right) ,\left( \underline{% -1},\underline{1}\right) \right) ,\left( \left( \underline{-1},\underline{1}% \right) ,\left( \underline{1},\underline{3}\right) \right) \\ & \equiv & \left( \underline{-3},\underline{-1},\underline{-1},\underline{1},% \underline{-1},\underline{1},\underline{1},\underline{3}\right) \end{array}$$ Observing $\underline{\text{W}}\,\underline{\text{3}}$ yields $% \underline{-1}$ with probability $\frac{3}{8},$ $\underline{1}$ with probability $\frac{3}{8},$ $\underline{-3}$ with probability $\frac{1}{8}$, and $\underline{3}$ with probability $\frac{1}{8}.$ The Lambda-Q Calculus ===================== The $\lambda ^{q}$-calculus is an extension of the $\lambda ^{p}$-calculus that allows easy expression of *quantumized* algorithms. A quantumized algorithm differs from a randomized algorithm in allowing negative probabilities and in the way we sample from the resulting distribution. Variables and abstractions in the $\lambda ^{q}$-calculus have *phase*. The phase is nothing more than a plus or minus sign, but since the result of a quantumized algorithm is a distribution of terms with phase, we call such a distribution by the special name *superposition*. The major difference between a superposition and a distribution is the observation procedure. Before randomly picking an element, a superposition is transformed into a distribution by the following two-step process. First, all terms in the superposition that are identical except with opposite phase are cancelled. They are both simply removed from the superposition. Second, the phases are stripped to produce a distribution. Then, an element is chosen from the distribution randomly, as in the $\lambda ^{p}$-calculus. The words *phase* and *superposition* come from quantum physics. An electron is in a superposition if it can be in multiple possible states. Although the phases of the quantum states may be any angle from $0{{}^{\circ }}$ to $360{{}^{\circ }}$, we only consider binary phases. Because we use solely binary phases, we will use the words *sign* and *phase* interchangeably in the sequel. A major disadvantage of the $\lambda ^{p}$-calculus is that it is impossible to compress a collection. Every reduction step at best keeps the collection the same size. Quantumized algorithms expressed in the $\lambda ^{q}$-calculus, on the other hand, can do this as easily as randomized algorithms can generate random numbers. That is, $\lambda ^{q}$-terms can contain subterms with opposite signs which will be removed during the observation process. Syntax {#syntax-1} ------ The following grammar describes the $\lambda ^{q}$-calculus. $$\begin{tabular}{|ll|} \hline $ \begin{array}{ll} S & \in \text{\emph{Sign}} \\ x & \in \text{\emph{Variable}} \\ M & \in \text{\emph{LambdaQTerm}} \\ w & \in \text{\emph{WffQ}} \end{array} $ & $ \begin{array}{l} \text{Sign, or phase} \\ \text{Variables} \\ \text{Terms} \\ \text{Well-formed formulas} \end{array} $ \\ & \\ $ \begin{array}{lll} S & ::= & + \\ & \,\,\,| & - \\ & & \\ M & ::= & Sx \\ & \,\,\,| & M_{1}M_{2} \\ & \,\,\,| & S\lambda x.M \\ & \,\,\,| & M_{1},M_{2} \\ & & \\ w & ::= & M_{1}=M_{2} \end{array} $ & $ \begin{array}{l} \text{positive} \\ \text{negative} \\ \\ \text{signed variable} \\ \text{application} \\ \text{signed abstraction} \\ \text{collection} \\ \\ \text{well-formed formula} \end{array} $ \\ \hline \end{tabular} \newline \label{lambda-q syntax}$$ Terms of the $\lambda ^{q}$-calculus differ from terms of the $\lambda ^{p}$-calculus only in that variables and abstractions are *signed*, that is, they are preceded by either a plus (+) or a minus (-) sign. Just as $% \lambda $-terms could be read as $\lambda ^{p}$-terms, we would like $% \lambda ^{p}$-terms to be readable as $\lambda ^{q}$-terms. However, $% \lambda ^{p}$-terms are unsigned and cannot be recognized by this grammar. Therefore, as is traditionally done with integers, we will omit the positive sign. An unsigned term in the $\lambda ^{q}$-calculus is abbreviatory for the same term with a positive sign. With this convention, $\lambda ^{p}$-terms can be seen as $\lambda ^{q}$-terms all of whose signs are positive. Also, so as not to confuse a negative sign with subtraction, we will write it with a logical negation sign ($\lnot $). With these two conventions, the $\lambda ^{q}$-term $+\lambda x.+x-\!x$ is written simply $\lambda x.x\lnot x.$ Finally, we adhere to the same parenthesization and precedence rules as the $% \lambda ^{p}$-calculus. In particular, we continue the use of the abbreviatory notations $\left[ M_{i}^{i\in S}\right] $ and $\left[ \left( M_{i}:n_{i}\right) \right] $ for collections of terms. In addition, we can also $\left[ \left( M_{i}:n_{i}\right) \right] $ as $\left[ \left( M_{i}:a_{i},b_{i},n_{i}\right) \right] $ such that $M_{i}\not{\equiv}M_{j}$ and $M_{i}\not{\equiv}\overline{M_{j}}$ for $i\neq j,$ all of the $M_{i}$ are of positive sign, the integer $a_{i}$ denotes the count of $M_{i},$ the integer $b_{i}$ denotes the count of $\overline{M_{i}},$ and $% n_{i}=a_{i}-b_{i}.$ Syntactic Identities -------------------- We will call two terms *opposites* if they differ only in sign. We define substitution of terms in the $\lambda ^{q}$-calculus as a modification of substitution of terms in the $\lambda ^{p}$-calculus. We rewrite the seven rules of the $\lambda ^{p}$-calculus to take account of the signs of the terms. First, we introduce the function notated by sign concatenation, defined by the following rule in our abbreviatory conventions. $$\lnot \lnot \mapsto \epsilon$$ We also note that the concatenation of a sign $S$ with $\epsilon $ is just $S$ again. Now we can use this function in the following substitution rules. $$\begin{array}{l} 1.\;\left( Sx\right) \left[ N/x\right] \equiv SN \\ 2.\;\left( Sy\right) \left[ N/x\right] \equiv Sy\,\,\text{for variables }y% \not{\equiv}x \\ 3.\;\left( PQ\right) \left[ N/x\right] \equiv \left( P\left[ N/x\right] \right) \left( Q\left[ N/x\right] \right) \\ 4.\;\left( S\lambda x.P\right) \left[ N/x\right] \equiv S\lambda x.P \\ 5.\;\left( S\lambda y.P\right) \left[ N/x\right] \equiv S\lambda y.\left( P\left[ N/x\right] \right) \,\,\text{if } \begin{array}{l} y\not{\equiv}x\text{, and} \\ y\notin FV\left( N\right) \end{array} \\ 6.\;\left( S\lambda y.P\right) \left[ N/x\right] \equiv S\lambda z.\left( P\left[ z/y\right] \left[ N/x\right] \right) \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{if } \begin{array}{l} y\not{\equiv}x\text{,} \\ y\in FV\left( N\right) \text{, and} \\ z\notin FV(P)\bigcup FV\left( N\right) \end{array} \\ 7.\;\left( P,Q\right) \left[ N/x\right] \equiv \left( P\left[ N/x\right] ,Q\left[ N/x\right] \right) \end{array} \label{substitution-q}$$ The use of the sign concatenation function is hidden in rule (1). Consider $% \left( \lnot x\right) \left[ \lnot \lambda y.y/x\right] \equiv \lnot \lnot \lambda y.y.$ This is not a $\lambda ^{q}$-term by grammar (\[lambda-q syntax\]) but applying the sign concatenation function yields the term $% \lambda y.y$. Reduction {#reduction} --------- The $\gamma $-relation of the $\lambda ^{q}$-calculus is of the same form as that of the $\lambda ^{p}$-calculus. $$\gamma ^{q}\triangleq \left\{ \begin{array}{l} \left( \left[ M_{i}^{i\in 1..m}\right] \left[ N_{j}^{j\in 1..n}\right] ,\left[ M_{i}^{i\in 1..m}N_{j}^{j\in 1..n}\right] \right) \\ \text{s.t. }M_{i},N_{j}\in LambdaQTerm,\,m>1\text{ or }n>1 \end{array} \right\} \label{gamma-q}$$ We omit the superscript when it is clear if the terms under consideration are $\lambda ^{p}$-terms or $\lambda ^{q}$-terms. We still write $\gamma \left( M\right) $ for the $\gamma $-normal form of $M.$ We extend the $\beta $-relation to deal properly with signs. $$\beta ^{q}\triangleq \left\{ \begin{array}{l} \left( \left( S\lambda x.M\right) N,SM\left[ N/x\right] \right) \\ \text{s.t. }S\in \text{\emph{Sign}},\text{and }S\lambda x.M,N\in LambdaQTerm \end{array} \right\}$$ Evaluation Semantics -------------------- We modify the call-by-value evaluation semantics of the $\lambda ^{p}$-calculus. $$\begin{aligned} &&\frac {}{v\rightsquigarrow v}\text{(Refl)\qquad \qquad (for }v\text{ a value)} \\ &&\frac{\gamma \left( M\right) \rightsquigarrow S\lambda x.P\,\,\,\,\gamma \left( N\right) \rightsquigarrow N^{\prime }\,\,\,\,\,\gamma \left( SP\left[ N^{\prime }/x\right] \right) \rightsquigarrow v}{MN\rightsquigarrow v}\text{% (Eval)} \\ &&\frac{\gamma \left( M\right) \rightsquigarrow v_{1}\quad \gamma \left( N\right) \rightsquigarrow v_{2}}{\left( M,N\right) \rightsquigarrow \left( v_{1},v_{2}\right) }\text{(Coll)}\end{aligned}$$ Observation {#lambda-q observation} ----------- We define an observation function $\Xi $ from $% \lambda ^{q}$-terms to $\lambda $-terms as the composition of a function $% \Delta $ from $\lambda ^{q}$-terms to $\lambda ^{p}$-terms with the observation function $\Theta $ from $\lambda ^{p}$-terms to $\lambda $-terms defined in (\[lambda-p observation\]). Thus, $\Xi =\Theta \circ \Delta $ where we define $\Delta $ as follows. $$\begin{aligned} \Delta \left( Sx\right) &=&x \\ \Delta \left( S\lambda x.M\right) &=&\lambda x.\Delta \left( M\right) \\ \Delta \left( M_{1}M_{2}\right) &=&\Delta \left( M_{1}\right) \Delta \left( M_{2}\right) \\ \Delta \left( \left[ M_{i}:a_{i},b_{i},n_{i}\right] \right) &=&\left[ \Delta \left( M_{i}^{i\in \left\{ i\,\,\,|\,\,\,n_{i}\neq 0\right\} }:\left| n_{i}\right| \right) \right] \label{Collection case for delta}\end{aligned}$$ Note that unlike the observation function $\Theta $ of the $\lambda ^{p}$-calculus, the observation function $\Xi $ of the $\lambda ^{q}$-calculus is not total. For example, $\Xi \left( x,\lnot x\right) $ does not yield a $% \lambda $-term because $\Delta \left( x,\lnot x\right) $ is the empty collection, which is not a $\lambda ^{p}$-term. Although observing a $\lambda ^{p}$-term is statistically indistinguishable from observing its $\gamma $-normal form, observing a $\lambda ^{q}$-term is, in general, statistically distinguishable from observing its $\gamma $-normal form. Observational Semantics ----------------------- The observational semantics for the $\lambda ^{q}$-calculus is similar to that of the $\lambda ^{p}$-calculus (\[obs-p\]). It is given by a single rule. $$\frac{M\rightsquigarrow v\quad \Xi \left( v\right) =N}{M\multimap N}\text{% (ObsQ)} \label{obs-q}$$ Examples -------- We provide one example. We show how satisfiability may be solved in the $% \lambda ^{q}$-calculus. We assume possible solutions are encoded some way in the $\lambda ^{q}$-calculus and there is a term $\underline{\text{CHECK}_{f}} $ that checks if the fixed Boolean formula $f$ is satisfied by a particular truth assignment, given as the argument. The output from this is a collection of $\underline{\text{T}}$ (truth) and $\underline{\text{F}}$ (falsity) terms. We now present a term that will effectively remove all of the $\underline{\text{F}}$ terms. It is an instance of a more general method. $$\underline{\text{REMOVE-F}}\equiv \lambda x.\,\underline{\text{IF}}% \,x\,x\,\left( x,\lnot x\right)$$ We give an example evaluation. $$\begin{array}{lll} \underline{\text{REMOVE-F}}\,\left( \underline{\text{F}},\underline{\text{T}}% ,\underline{\text{F}}\right) & \equiv & \left( \lambda x.\,\underline{\text{% IF}}\,x\,x\,\left( x,\lnot x\right) \right) \left( \underline{\text{F}},% \underline{\text{T}},\underline{\text{F}}\right) \\ & \rightarrow _{\gamma } & \left( \begin{array}{l} \left( \lambda x.\,\underline{\text{IF}}\,x\,x\,\left( x,\lnot x\right) \right) \underline{\text{F}}, \\ \left( \lambda x.\,\underline{\text{IF}}\,x\,x\,\left( x,\lnot x\right) \right) \underline{\text{T}}, \\ \left( \lambda x.\,\underline{\text{IF}}\,x\,x\,\left( x,\lnot x\right) \right) \underline{\text{F}} \end{array} \right) \\ & \twoheadrightarrow _{\beta } & \left( \left( \underline{\text{F}},\lnot \underline{\text{F}}\right) ,\underline{\text{T}},\left( \underline{\text{F}}% ,\lnot \underline{\text{F}}\right) \right) \\ & \equiv & \left( \underline{\text{F}},\lnot \underline{\text{F}},\underline{% \text{T}},\underline{\text{F}},\lnot \underline{\text{F}}\right) \end{array}$$ Observing the final term will always yield $\underline{\text{T}}.$ Note that the drawback to this method is that if $f$ is unsatisfiable then the term will be unobservable. Therefore, when we insert a distinguished term into the collection to make it observable, we risk observing that term instead of $\underline{\text{T}}.$ At worst, however, we would have a fifty-fifty chance of error. Specifically, consider what happens when the argument to $\underline{\text{% REMOVE-F}}$ is a collection of $\underline{\text{F}}^{\prime }$s$.$ Then $% \underline{\text{REMOVE-F}}\,\underline{\text{F}}=\left( \underline{\text{F}}% ,\lnot \underline{\text{F}}\right) .$We insert $\underline{\text{I}}\equiv \lambda x.x$ which, if we observe, we take to mean that either $f$ is unsatisfiable or we have bad luck. Thus, we observe the term $\left( \underline{\text{I}},\underline{\text{F}},\lnot \underline{\text{F}}\right) . $ This will always yield $\underline{\text{I}}.$ However, we cannot conclude that $f$ is unsatisfiable because, in the worst case, the term may have been $\left( \underline{\text{I}},\underline{\text{REMOVE-F}}\,% \underline{\text{T}}\right) =\left( \underline{\text{I}},\underline{\text{T}}% \right) $ and we may have observed $\underline{\text{I}}$ even though $f$ was satisfiable. We may recalculate until we are certain to an arbitrary significance that $f$ is not satisfiable. Therefore, applying $\underline{\text{REMOVE-F}}$ to the results of $% \underline{\text{CHECK}_{f}}$ and then observing the result will yield $% \underline{\text{T}}$ only if $f$ is satisfiable. Simulation to quantum computers =============================== We show that the $\lambda ^{q}$-calculus can efficiently simulate the *one-dimensional partitioned quantum cellular automata* (1d-PQCA) defined in [@Watrous; @1995]. By the equivalence of 1d-PQCA and quantum Turing machines (QTM) proved in [@Watrous; @1995], the $\lambda ^{q}$-calculus can efficiently simulate QTM. To show that 1d-PQCA can be efficiently simulated by the $\lambda ^{q}$-calculus, we need to exhibit a $\lambda ^{q}$-term $M$ for a given 1d-PQCA $% A$ such that $A$ after $k$ steps is in the same superposition as $M$ after $% P\left( k\right) $ steps, with $P$ a polynomial. We assume for now that the 1d-PQCA has transition amplitudes not over the complex numbers, but over the positive and negative rationals. It has been shown [@bernstein/vazirani] that this is equivalent to the general model in QTM. To express $A$ in $M$, we need to do the following things. 1. Translate states of $A$ into $\lambda ^{q}$-terms that can be compared (e.g. into Church numerals). 2. Translate the acceptance states and the integer denoting the acceptance cell into $\lambda ^{q}$-terms. 3. Create a $\lambda ^{q}$-term $\mathbf{P}$ to mimic the operation of the permutation $\sigma .$ 4. Translate the local transition function into a transition term. For 1d-PQCA this means translating the matrix $\Lambda $ into a term $\mathbf{L}$ comparing the initial state with each of the possible states and returning the appropriate superposition. 5. Determine an injective mapping of configurations of $A$ and configurations of $M$. Although we will not write down $M$ in full, we note that within $M$ are the mechanisms described above that take a single configuration, apply $\mathbf{P% }$, and return the superposition as described by $\mathbf{L}.$ We recall that the contextual closure of the $\beta ^{q}$-relation is such that $M,N\rightarrow _{\beta }M^{\prime },N^{\prime }$ where $M\rightarrow _{\beta }M^{\prime }$and $N\rightarrow _{\beta }N^{\prime }.$ Thus there is parallel reduction within superpositions. By inspection of the mechanisms above it follows that $k$ steps of $A$ is equivalent to a polynomial of $k$ steps of $M$. Steps 1, 2, and 3 are straightforward. Then for step 5, the $\lambda ^{q}$-superposition $\left[ \left( M_{i}:a_{i},b_{i},n_{i}\right) \right] $ (let $% n=\sum n_{i}$) will be equivalent to the 1d-PQCA-superposition $\sum \frac{% n_{i}}{n}\left| c\left( M_{i}\right) \right\rangle ,$ where $c$ takes $% \lambda ^{q}$-terms and translates them into 1d-PQCA configurations. Essentially this means stripping off everything other than the data, that is to say, the structure containing the contents. Note that $c$ is not itself a $\lambda ^{q}$-term. It merely performs a fixed syntactic operation, removing extraneous information such as $\mathbf{P}$ and $\mathbf{L,}$ and translating the Church numerals that represent states into the 1d-PQCA states. This is injective because the mapping from states of $A$ into numerals is injective. Thus, step 5 is complete. Step 4 requires translating the $\Lambda $ matrix into a matrix of whole numbers, and translating an arbitrary 1d–PQCA superposition into a $\lambda ^{q}$-superposition. The latter is done merely by multiplying each of the amplitudes by the product of the denominators of all of the amplitudes, to get integers. We call the product of the denominators here $d$. We perform a similar act on the $\Lambda $ matrix, multiplying each element by the product of all of the denominators of $\Lambda .$ We call this constant $b.$ Then we have that $T=b\Lambda $ is a matrix over integers. This matrix can be considered notation for the $\lambda ^{q}$-term that checks if a given state is a particular state and returns the appropriate superposition. For instance, if $$\Lambda =\left( \begin{array}{ll} \frac{2}{3} & \frac{1}{3} \\ 0 & 1 \end{array} \right)$$ then $$T=b\Lambda =9\Lambda =\left( \begin{array}{ll} 6 & 3 \\ 0 & 9 \end{array} \right)$$ which we can consider as alternate notation for $$\begin{aligned} \mathbf{Q} &\equiv &\lambda s.\text{\textbf{\ IF }(\textbf{EQUAL }}s\text{% \textbf{1}) (\textbf{1,1,1,1,1,1,2,2,2)}} \\ &&\text{(\textbf{IF} (\textbf{EQUAL\ }}s\text{\textbf{2}) (\textbf{% 2,2,2,2,2,2,2,2,2}))}\end{aligned}$$ Then it follows that if $c$ is a superposition of configuration of $A$, applying $\Lambda $ $k$ times results in the same superposition as applying $% T$ $k$ times to the representation of $c$ in the $\lambda ^{q}$-calculus. Conclusion ========== We have seen two new formalisms. The $\lambda ^{p}$-calculus allows expression of randomized algorithms. The $\lambda ^{q}$-calculus allows expression of quantumized algorithms. In these calculi, observation is made explicit, and the notion of superposition common to quantum physics is formalized for algorithms. This work represents a new direction of research. Just as the $\lambda $-calculus found many uses in classical programming languages, the $\lambda ^{p}$-calculus and the $\lambda ^{q}$-calculus may help discussion of randomized and quantum programming languages. It should not be difficult to see that the $\lambda ^{p}$-calculus can simulate a probabilistic Turing machine and we have shown that the $\lambda ^{q}$-calculus can simulate a quantum Turing machine (QTM). However, as we have shown, the $\lambda ^{q}$-calculus can efficiently solve NP-complete problems such as satisfiability, while there is widespread belief (e.g. [@bennett]) that QTM cannot efficiently solve satisfiability. Thus, the greater the doubt that QTM cannot solve NP-complete problems, the greater the justification in believing that the $\lambda ^{q}$-calculus is strictly stronger than QTM. It should also follow that a probabilistic Turing machine can (inefficiently) simulate the $\lambda ^{p}$-calculus. However, it is not obvious that a quantum Turing machine can simulate the $\lambda ^{q}$-calculus. An answer to this question will be interesting. If quantum computers can simulate the $\lambda ^{q}$-calculus efficiently, then the $% \lambda ^{q}$-calculus can be used as a programming language directly. As a byproduct, satisfiability will be efficiently and physically solvable. If quantum computers cannot simulate the $\lambda ^{q}$-calculus efficiently, knowing what the barrier is may allow the formulation of another type of computer that can simulate it. Acknowledgements ================ Thanks to Stuart Shieber for helpful comments. [99]{} <span style="font-variant:small-caps;">Barendregt</span>, Hendrik Pieter, *The lambda calculus: its syntax and semantics,* North-Holland (1981). <span style="font-variant:small-caps;">Bennett</span>, Charles H., Ethan <span style="font-variant:small-caps;">Bernstein</span>, Gilles <span style="font-variant:small-caps;">Brassard</span>, and Umesh <span style="font-variant:small-caps;">Vazirani</span>, “Strengths and Weaknesses of Quantum Computing,” available online as quant-ph/9701001 at http://xxx.lanl.gov/abs/quant-ph/9701001. <span style="font-variant:small-caps;">Bernstein</span>, E. and U. <span style="font-variant:small-caps;">Vazirani</span>, “Quantum complexity theory,” *Proceedings of the 25th Annual ACMSymposium on Theory of Computing* (1993), 11-20. <span style="font-variant:small-caps;">Church</span>, Alonzo, “An unsolvable problem of elementary number theory”, *American Journal of Mathematics* **58** (1936), 345-363. <span style="font-variant:small-caps;">Deutsch</span>, David, “Quantum theory, the Church-Turing principle and the universal quantum computer”, *Proc. R. Soc. Lond.* **A400** (1985), 97-117. <span style="font-variant:small-caps;">Deutsch</span>, David, “Quantum computational networks”, *Proc. R. Soc. Lond.* **A425** (1989), 73-90. <span style="font-variant:small-caps;">Heisenberg</span>, Werner, *Physics and philosophy,* Harper & Bros. (1958). <span style="font-variant:small-caps;">Maymin</span>, Philip, “Extending the Lambda Calculus to Express Randomized and Quantumized Algorithms,” available online as quant-ph/9612052 at http://xxx.lanl.gov/abs/quant-ph/9612052. Many of the proofs omitted from the current paper because of space considerations can be found here. <span style="font-variant:small-caps;">Maymin</span>, Philip, “The lambda-q calculus can efficiently simulate quantum computers,” available online as quant-ph/9702057 at http://xxx.lanl.gov/abs/quant-ph/9702057. <span style="font-variant:small-caps;">Simon</span>, Daniel, “On the power of quantum computation”, *Proc. 35th Annual Symp. FOCS* (1994). <span style="font-variant:small-caps;">Watrous</span>, John, “On One-Dimensional Quantum Cellular Automata,” *Proceedings of the 36th IEEE Symposium on Foundations of Computer Science* (1995), 528-537.

--- abstract: 'In order to study the magnetic properties of frustrated metallic systems, we present Quantum Monte Carlo data on the magnetic susceptibility of the Hubbard model on triangular and Kagomé lattices. We show that the underlying lattice structure is important, and determines the nature and the doping dependence of the magnetic fluctuations. In particular, in the doped Kagomé case we find strong short-range magnetic correlations, which makes the metallic Kagomé systems a promising field for studies of superconductivity.' address: | $^1$Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\ $^2$CREST, Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332-0012, Japan author: - 'N. Bulut$^{1,2}$, W. Koshibae$^1$ and S. Maekawa$^{1,2}$' date: 'February 15, 2005' title: Magnetic correlations in the Hubbard model on triangular and Kagomé lattices --- Frustrated spin systems have received significant attention because of the possibility of novel magnetic ground-states and excitations [@Misguich]. The triangular spin-$1/2$ Heisenberg model has long-range magnetic order in the ground state, while the spin-$1/2$ Heisenberg model on the Kagomé lattice is considered to be disordered. The discovery of superconductivity at $5 K$ in Na$_x$CoO$_2\cdot y$H$_2$O has generated new interest in frustrated interacting systems[@Takada]. Furthermore, recently superconductivity has been discovered in $\beta$-pyrochlore osmate KOs$_2$O$_6$ with $T_c$ of $10K$ [@Yonezawa]. This is interesting since the pyrochlore lattice structure is a three-dimensional analog of the Kagome lattice. In cobaltates, cobalt and oxygen ions form a two-dimensional triangular network. The hopping matrix element of electrons in the cobalt $3d$ orbitals is not isotropic, and it has been shown that the triangular CoO$_2$ lattice consists of four coupled Kagomé sublattices [@Koshibae]. Hence, it is important to compare the magnetic properties of interacting systems on triangular and Kagomé lattices. The electronic properties of the $t$-$J$ and the Hubbard models on the triangular and Kagomé lattices have been studied using various techniques of many-body physics. The triangular $t$-$J$ model was investigated within the RVB framework [@Baskaran; @Kumar; @Wang] and by using high-temperature expansions [@Koretsune]. The triangular Hubbard model was studied with the path-integral renormalization-group (RG) [@Kashima], the one-loop RG [@Honerkamp] and the fluctuation-exchange (FLEX) [@Renner] approaches. The FLEX method was also used for studying the magnetic properties of the Hubbard model on the Kagomé lattice [@Imai]. In this paper, we compare the nature of the magnetic correlations in the Hubbard model on the triangular and the Kagomé lattices using Quantum Monte Carlo (QMC) simulations. We consider the Hubbard model on the Kagomé lattice to be a simple limiting case to explore for new physics due to the underlying orbital structure in frustrated interacting systems. The orbital degrees of freedom create the possibility for mixing the spin and charge channels, and hence of new electronic states. This is important because of the general ongoing research effort on the transition metal oxides. These are our motivations for performing the QMC calculations on the Hubbard model on the triangular and the Kagomé lattices. We are particularly interested in the doped cases of these models for which there are no exact calculations on the magnetic properties. In the following, we will see that the nature of the magnetic fluctuations on the triangular and the Kagomé lattices are different. For the triangular lattice, the QMC results show that there are strong antiferromagnetic (AF) correlations near half-filling at low temperatures, when the Coulomb repulsion $U$ is of the order of the bandwidth. On the other hand, for weak $U$, the magnetic correlations saturate as $T\rightarrow 0$. In the Kagomé lattice, the unit cell consists of three atoms and the unit cells form a triangular lattice, as seen in Fig. 1(a). Consequently, there are three bands of magnetic excitations. Two of these modes involve enhanced short-range AF correlations. We find that, in the doped Kagomé case, the low-frequency short-range AF correlations are stronger in comparison to the triangular lattice. Hence, we note that it would be useful to investigate the possibility of superconductivity in metallic Kagomé systems. The Hubbard model is defined by $$\begin{aligned} \label{Hubbard} H=-t \sum_{\langle i,j\rangle,\sigma} (c^{\dagger}_{i\sigma}c_{j\sigma} + {\rm h.c.}) + U \sum_{i} n_{i\uparrow} n_{i\downarrow} -\mu\sum_{i\sigma}n_{i\sigma},\end{aligned}$$ where $t$ is the hopping matrix element between the nearest neighbor-sites, $U$ is the on-site Coulomb repulsion, and $\mu$ is the chemical potential. Here, $c_{i\sigma}$ ($c^{\dagger}_{i\sigma}$) annihilates (creates) an electron with spin $\sigma$ at site $i$, and $n_{i\sigma}=c^{\dagger}_{i\sigma} c_{i\sigma}$. In the following, we will take $t<0$ and consider $\langle n\rangle \ge 1.0$, which is the appropriate case for the cobaltates [@Koshibae]. In obtaining the QMC data presented here, the determinantal QMC technique [@White] was used. For the triangular lattice, the longitudinal magnetic susceptibility at frequency $\omega=0$ is defined by $$\chi({\bf q}) = \int_0^{\beta} d\tau \sum_{\ell} e^{-i{\bf q}\cdot{\bf r}_{\ell}} \langle m^z({\bf r}_{i+\ell},\tau) m^z({\bf r}_i) \rangle,$$ where $m^z({\bf r}_i) = c^{\dagger}_{i\uparrow}c_{i\uparrow} - c^{\dagger}_{i\downarrow}c_{i\downarrow}$ and $m^z({\bf r}_i,\tau) = e^{H\tau} m^z({\bf r}_i) e^{-H\tau}$. In this paper, results on $\chi$ will be shown in units of $|t|^{-1}$. The Kagomé lattice is a three-band model, since each unit cell consists of three sites. Hence, each lattice on the Kagomé lattice site can be represented by the indices $(\ell,d)$ where $\ell$ is the unit-cell index and $d$ denotes the atomic site in a particular unit cell. The corresponding Brillouin zone (BZ) is reduced with respect to the triangular case as illustrated in Fig. 1(b). For the Kagomé case, we define the longitudinal magnetic susceptibility at $\omega=0$ as $$\chi_{dd'}({\bf q}) = \int_0^{\beta} d\tau \sum_{\ell} e^{-i{\bf q}\cdot{\bf r}_{\ell}} \langle m^z_d({\bf r}_{i+\ell},\tau) m^z_{d'}({\bf r}_i) \rangle,$$ where $m^z_d({\bf r}_i) = c^{\dagger}_{id\uparrow}c_{id\uparrow} - c^{\dagger}_{id\downarrow}c_{id\downarrow}$, $c_{id\sigma}$ ($c^{\dagger}_{id\sigma}$) is the annihilation (creation) operator of an electron with spin $\sigma$ at lattice site $(i,d)$ and the summation is performed over the unit-cell locations. Diagonalizing the $3\times 3$ matrix $\chi_{dd'}({\bf q})$, we obtain $\chi_{\alpha}({\bf q})$ which describes the three modes of the magnetic excitations on the Kagomé lattice, which we define as the Kagomé magnetic bands. At this point, it is useful to note that already for the noninteracting ($U=0$) case, the triangular and the Kagomé lattices have different properties. For $t< 0$, the one-electron density of states $N(\omega)$ of the triangular lattice has a van Hove singularity at $\langle n\rangle=0.5$. For the Kagomé lattice, there is a $\delta$-function singularity in $N(\omega)$ at the bottom of the band and there are van Hove singularities at $\langle n\rangle=1.16$ and 1.51. In addition, $N(\omega=0)$ vanishes at $\langle n\rangle=1.33$. These features of $N(\omega)$ are also reflected in the magnetic susceptibilities of the noninteracting case. We first present results for the triangular lattice at half-filling. Figure 2(a) shows $\chi({\bf q})$ versus ${\bf q}$ for $U=4|t|$ on various size lattices as the temperature is lowered. Here, it is seen that $\chi({\bf q})$ has a broad peak centered at the $K$ point of the BZ, and hence the system exhibits short-range AF correlations. We also note that $\chi({\bf q})$ does not vary significantly with $T$, in particular for $0.25|t| \le T \le 0.17|t|$. For comparison, $\chi_0({\bf q})$ for the noninteracting system at $T=0.17|t|$ is shown by the dotted curve. Figure 2(b) displays $\chi({\bf q})$ versus ${\bf q}$ for $U=8|t|$ at half-filling, where we observe a large Stoner enhancement of the AF correlations. In contrast with the $U=4|t|$ case, here, $\chi({\bf q})$ at the $K$ point grows rapidly with a Curie-like $T$ dependence, as $T$ decreases from $1|t|$ to $0.33|t|$. However, it is not known whether $\chi({\bf q})$ saturates at lower $T$ for $U=8|t|$. These results show that the $T$ dependence of $\chi({\bf q})$ depends strongly on the value of $U/|t|$ in the triangular Hubbard model, in agreement with the findings of the path-integral RG calculations [@Kashima]. Figure 2(c) shows the filling dependence of $\chi({\bf q})$ for $U=8|t|$ while $T$ is kept fixed at $0.33|t|$. Here, we observe that the AF correlations decay monotonically as the electron filling is varied from 1.0 to 1.3. We have also performed calculations for $\chi({\bf q})$ at higher electron fillings. We find that, when $\langle n\rangle$ is increased to 1.5, the peak in $\chi({\bf q})$ shifts to the $M$ point. Upon further doping to $\langle n\rangle=1.75$, we find that, for $U=8|t|$ and $T=0.2|t|$, the Stoner enhancement is about 20%. Hence, for this dilute hole concentration, the magnetic correlations are weakly affected by the presence of the on-site Coulomb repulsion at this temperature. For $U=4|t|$, $\chi({\bf q})$ exhibits a slow monotonic decrease with the electron doping away from half-filling. Next, we discuss the magnetic properties of the Hubbard model on the Kagomé lattice. Figure 3(a) shows QMC results on $\chi_{\alpha}({\bf q})$ for $U=4|t|$ and $T=0.2|t|$ at half-filling. Here, $\chi_{\alpha}({\bf q})$ for the three magnetic bands are plotted as a function of ${\bf q}$, and the red curves represent results for the noninteracting case. In this figure, we observe that the top band ($\alpha=1$) is flat in ${\bf q}$ space, and the second magnetic band ($\alpha=2$) is degenerate with the first one at the zone center. The third mode is weaker in magnitude and exhibits a smooth ${\bf q}$ dependence. Figure 3(b) shows $\chi_{\alpha}({\bf q})$ versus ${\bf q}$ for $U=8|t|$ and $T=0.33|t|$ at half-filling. The general features are similar to those seen in Fig. 3(a), however here the Stoner enhancement is larger. In Fig. 3(c), the QMC results are shown for $U=4|t|$ and $T=0.14|t|$ at $\langle n\rangle=1.15$. These figures show that the features of $\chi_{\alpha}({\bf q})$ are in correspondence with those of the noninteracting case. In order to gain insight into the origin of the Kagomé magnetic bands, we note that the Kagomé lattice is obtained from the triangular lattice, as illustrated in Fig. 1(a), by removing the sites with the empty circles. This is equivalent to putting an infinitely repulsive one-electron potential at these sites. Bragg scattering from this static charge-density-wave field then folds the BZ of the triangular lattice, and also mixes the different wavevector components of the spin fluctuations. It is this process which creates the Kagomé magnetic bands. =4.0cm =5.0cm =4.0cm =5.0cm We next discuss the $T$ dependence of the magnetic correlations in the Kagomé lattice. Figure 4(a) shows $\chi_1({\bf q}=0)$ versus $T$ for $U=4|t|$ and $8|t|$ at $\langle n\rangle=1.0$ and 1.15. Here, we observe that, at half-filling and for $U=4|t|$, $\chi_1({\bf q}=0)$ saturates as $T$ decreases. On the other hand, for $U=8|t|$, $\chi_1({\bf q}=0)$ has a strong $T$-dependence. For $\langle n\rangle=1.15$ and $U=4|t|$, we observe that $\chi_1({\bf q}=0)$ gets enhanced at low $T$, and becomes larger than at half-filling. We also see that the enhancement of $\chi_1({\bf q}=0)$ depends on the lattice size. However, this type of non-monotonic doping dependence was not observed for the triangular Hubbard model. In addition, we observe that, for $U=8|t|$ and at half-filling, $\chi_1({\bf q}= 0)$ exhibits a Curie-like $T$ dependence for $0.25|t| \le T \le 1.0|t|$. In the ground state of the Heisenberg model on the Kagomé lattice, it is considered that a spin gap $\Delta_S\approx J/20$ exists, where $J$ is the magnetic exchange [@Misguich]. For $U=8|t|$, we have $J\approx 4t^2/U=0.5|t|$, which gives $\Delta_S\approx 0.025|t|$. We expect that $\chi_{\alpha}({\bf q})$ saturates before $T$ becomes comparable to $\Delta_S$. However, already at $T=0.33|t|$ we observe enhanced magnetic correlations. The FLEX calculations for the Hubbard model on the Kagomé lattice find that the leading magnetic mode is nearly ${\bf q}$ independent, and the tendency to electronic instabilities is suppressed [@Imai]. We note that while the $\alpha=1$ mode is nearly ${\bf q}$ independent, the $\alpha=2$ mode has ferromagnetic ${\bf q}$ dependence in the sense that it decreases away from the $\Gamma$ point. In addition, the eigenvectors of $\chi_{dd'}({\bf q})$ show that, at ${\bf q}\approx 0$, the $\alpha=1$ and 2 modes describe excitations involving the AF polarization of the spins within a unit cell [@footnote]. Away from ${\bf q}=0$, these modes have additional structures, however they involve AF polarizations over most of the BZ. Hence, these two modes contain enhanced short-range AF fluctuations. Within the context of superconductivity mediated by magnetic fluctuations, an important quantity is the zero-frequency component of the magnetic fluctuations between the two nearest-neighbor sites $i$ and $j$, $$C=\int_0^{\beta} \, d\tau \, \langle m^z({\bf r}_i,\tau) m^z({\bf r}_j) \rangle.$$ In Fig. 4(b), we compare C versus $T$ for the triangular and the Kagomé lattices for $U=4|t|$ and $8|t|$ at $\langle n\rangle=1.15$. This figure shows that, at these temperatures, the nearest-neighbor AF correlations are stronger for the Kagome lattice, even though the ground-state of the spin-$1/2$ Heisenberg model has long-range order on the triangular lattice and it is disordered in the Kagomé case. Hence, it would be useful to investigate the possibility of superconductivity in metallic Kagomé systems. Finally, we discuss the implications of these data for the magnetic correlations observed in the cobaltates. The cobaltates have a rich phase diagram with superconductivity found for $x\approx 0.35$ in Na$_x$CoO$_2\cdot y$H$_2$O [@Takada] and with magnetic order and large quasi-particle renormalizations observed in Na$_x$CoO$_2$ when $x$ is near 0.75 [@Foo]. For the triangular Hubbard model, we find that the magnetic correlations are strongest at half-filling, and the correlation effects are weak in the overdoped regime $\langle n\rangle \ge 1.5$. In contrast, for the Kagomé lattice we have seen that the magnetic correlations can be stronger in the doped case when $U=4|t|$. However, in both of these models, we find that the correlation effects are most prominent in the vicinity of half-filling at the temperatures where the QMC calculations were performed. We note that it would be useful to determine experimentally the ${\bf q}$ dependence of the magnetic fluctuations in the superconducting Na$_x$CoO$_2\cdot y$H$_2$O. In this paper, we have investigated the nature of the magnetic correlations in the Hubbard model on the triangular and Kagomé lattices. At the temperatures where the QMC calculations were performed, we find, in both of these models, that the magnetic correlations grow rapidly as $T$ decreases at half-filling for $U=8|t|$, while they saturate when $U=4|t|$. In the triangular Hubbard model, the AF correlations decay monotonically with the electron doping. In the Kagomé case, on the other hand, we have seen that the magnetic correlations can be stronger in the doped case when $U=4|t|$. We have also seen that in the Kagomé case the BZ is reduced and there are three modes of magnetic excitations. The two leading modes involve short-range AF correlations. In particular, we find that the low-frequency short-range AF correlations are stronger in the doped Kagomé case than in the triangular case. This makes the interacting metallic systems with Kagomé type of lattice structures a promising field for studies of superconductivity. We conclude that in frustrated interacting systems the underlying lattice and orbital structures are important in determining the magnetic properties. The authors are grateful to Y.Y. Bang, C. Honerkamp, M. Imada, K. Ishida, S. Ishihara, T. Koretsune, P. Lee, Y. Motome, B. Normand, T.M. Rice, T. Tohyama and G.-q. Zheng for helpful discussions. One of us (N.B.) would like to thank the International Frontier Center for Advanced Materials at Tohoku University for its kind hospitality, and gratefully acknowledges partial support from the Turkish Academy of Sciences through the GEBIP program (EA-TUBA-GEBIP/2001-1-1). This work was supported by Priority-Areas Grants from the Ministry of Education, Science, Culture and Sport of Japan, NAREGI Japan and NEDO. [999]{} G. Misguich and C. Lhuillier, [*“Frustrated spin systems”*]{}, Ed. H.T. Diep, World-Scientific (2004), and references therein. K. Takada [*et al.*]{}, Nature (London) [**422**]{}, 53 (2003). S. Yonezawa [*et al.*]{}, J. Phys.: Cond. Mat. [**16**]{}, L9 (2004). W. Koshibae and S. Maekawa, , 257003 (2003). G. Baskaran, , 097003 (2003). B. Kumar and B. S. Shastry, , 104508 (2003). Q.-H. Wang, D.-H. Lee and P.A. Lee, , 092504 (2004). T. Koretsune and M. Ogata, , 116401 (2002). T. Kashima and M. Imada, J. Phys. Soc. Jpn. [**70**]{}, 3052 (2001). C. Honerkamp, , 104510 (2003). M. Renner and W. Brenig, cond-mat/0310244. Y. Imai, N. Kawakami and H. Tsunetsugu, , 195103 (2003). S.R. White [*et al.*]{}, , 506 (1989). We also note that, at ${\bf q}\approx 0$, the $\alpha=3$ mode corresponds to fluctuations involving the uniform polarization of the spins within a unit cell, and hence the Knight shift for the Kagomé lattice is determined by $\chi_3({\bf q}\rightarrow 0)$. M.L. Foo [*et al.*]{}, , 247001 (2004).

--- abstract: 'The influence of the stellar plasma on the production and destruction of $K$-isomers is studied for the examples $^{176}$Lu and $^{180}$Ta. Individual electromagnetic transitions are enhanced predominantly by nuclear excitation by electron capture, whereas the other mechanisms of electron scattering and nuclear excitation by electron transition give only minor contributions. It is found that individual transitions can be enhanced significantly for low transition energies below 100keV. Transitions with higher energies above 200keV are practically not affected. Although one low-energy transition in $^{180}$Ta is enhanced by up to a factor of 10, the stellar transition rates from low-$K$ to high-$K$ states via so-called intermediate states in $^{176}$Lu and $^{180}$Ta do not change significantly under s-process conditions. The s-process nucleosynthesis of $^{176}$Lu and $^{180}$Ta remains essentially unchanged.' author: - 'G. Gosselin' - 'P. Morel' - 'P. Mohr' title: 'Modification of nuclear transitions in stellar plasma by electronic processes: $K$-isomers in $^{176}$Lu and $^{180}$Ta under s-process conditions' --- Introduction {#sec:intro} ============ In general, nuclear properties like decay half-lives and radiation widths do not depend on the electronic environment of the atomic nucleus. However, there are several well-known exceptions. Obviously, electron capture decays are affected by the number of available electrons, in particular $K$-electrons, and thus the $K$-capture half-life depends on the ionization of the atom. A second example are low-energy [$\gamma$]{}-transitions where the decay widths are enhanced by additional conversion electrons. The present study focuses on further effects that may affect nuclear transitions in a hot and dense plasma that is found in the interior of stars: inelastic and superelastic electron scattering and nuclear excitation by electron capture (NEEC) [@Doo78; @Gos07]; NEEC is the inverse process of the above mentioned internal conversion (IC). Furthermore, even nuclear excitation by electron transition (NEET) [@Mor04] may be important if matching conditions can be achieved. As will be shown in this study, [$\gamma$]{}-transitions with relatively low energies far below 1MeV are most affected by the surrounding hot and dense plasma. Typical [$\gamma$]{}-transition energies for [(n,$\gamma$)]{}, [(p,$\gamma$)]{}, and [($\alpha$,$\gamma$)]{} capture reactions are of the order of 1MeV and higher and are thus not significantly affected by the stellar plasma. However, low-energy [$\gamma$]{}-transitions play an important role in the production and destruction of low-lying isomers in the astrophysical [s-process]{}. There are two astrophysically relevant examples for heavy odd-odd nuclei where low-lying isomers exist because of the huge difference of the $K$-quantum number between the ground state and the isomer: [$^{176}$Lu]{} and [$^{180}$Ta]{}. The astrophysical transition rates between the [low-$K$]{} and [high-$K$]{} states in [$^{176}$Lu]{} and [$^{180}$Ta]{} may be affected by the temperature dependence of the individual transitions. The interesting astrophysical properties of [$^{176}$Lu]{} and [$^{180}$Ta]{} will not be repeated here. The [s-process]{} nucleosynthesis of [$^{176}$Lu]{} and $^{176}$Hf and the interpretation of the $^{176}$Hf/[$^{176}$Lu]{} ratio as [s-process]{} thermometer are discussed in several recent papers (see [@Heil08; @Mohr09; @Gin09], and references therein). The open question on the nucleosynthetic origin of [$^{180}$Ta]{} in various processes ([s-process]{}, [r-process]{}, [p-process]{} or $\gamma$-process, $\nu$-process) and the survival probability of the $9^-$ isomer under the corresponding conditions was also studied recently (see [@Mohr07] and references therein). The main subject of the present study is the temperature dependence of individual transitions from an initial state $i$ to a final state $f$. This general temperature dependence should not be mixed up with the temperature dependence of the stellar transition rates between [low-$K$]{} states and [high-$K$]{} states in [$^{176}$Lu]{} and [$^{180}$Ta]{} that are defined by low-lying so-called intermediate states and their decay properties – i.e. all possible transitions from these intermediate states. It is obvious that changes in the individual transitions – as studied in this work – do also affect the stellar transition rates. The paper is organized as follows. In Sect. \[sec:rate\] some introductory remarks on the nuclear structure of isomers are given, and the stellar reaction rate between [low-$K$]{} states and [high-$K$]{} states is defined. In Sect. \[sec:mod\] the temperature dependence of individual transitions is discussed. Results for selected individual transitions in [$^{176}$Lu]{} and [$^{180}$Ta]{} are presented in Sect. \[sec:res\], and their influence on the stellar transition rate is discussed. Finally, conclusions are drawn in Sect. \[sec:summ\]. As usual, we will give the “temperature” in units of keV, i.e. the temperature $T$ is multiplied by the Boltzmann constant $k$ leading to the thermal energy $kT$. Stellar reaction rates {#sec:rate} ====================== Nuclear structure {#sec:struc} ----------------- The approximate conservation of the $K$-quantum number leads to a strong suppression of direct transitions between so-called [low-$K$]{} and [high-$K$]{} states in heavy nuclei. As a consequence, the [low-$K$]{} $J^\pi = 1^-;K = 0$ state in [$^{176}$Lu]{} at $E_x = 123$keV practically cannot decay to the [high-$K$]{} $7^-;7$ ground state. Instead, the [low-$K$]{} $1^-;0$ state forms an isomer that $\beta$-decays with a half-life of $t_{1/2} = 3.66$h to $^{176}$Hf. The $\beta$-decay of the $7^-;7$ ground state is also highly suppressed and has a long half-life of about 38 giga-years, i.e. it is practically stable for the timescale of the astrophysical [s-process]{}. In [$^{180}$Ta]{} the roles of the ground state and the isomer are exchanged: the [low-$K$]{} $1^+;1$ state is the ground state and has a short $\beta$-decay half-life of about $8.15$h whereas the [high-$K$]{} $9^-;9$ isomer at $E_x = 77$keV is quasi-stable with $t_{1/2} > 7.1 \times 10^{15}$yr [@Hul06]. Excitation energies, spins and parities, half-lives, and decay properties are in most cases taken from the online data base ENSDF [@ENSDF] that is based on [@Bas06] for [$^{176}$Lu]{} and [@Wu03] for [$^{180}$Ta]{}; other data sources are stated explicitly. Because of the strong suppression of direct transitions between the [low-$K$]{} and the [high-$K$]{} states, two species (a [low-$K$]{} one and a [high-$K$]{} one) of such nuclei like [$^{176}$Lu]{} and [$^{180}$Ta]{} have to be considered in nucleosynthesis calculations (see e.g. [@Heil08]). Within each species, thermal equilibrium is obtained on timescales of far below one second (e.g. explicitly shown in [@Gin09] for [$^{176}$Lu]{}). However, indirect transitions between the [low-$K$]{} and the [high-$K$]{} states are possible via so-called intermediate states (IMS) that are located at higher excitation energies and have intermediate $K$-quantum numbers. Such IMS have been detected experimentally by high-resolution [$\gamma$]{}-ray spectroscopy for [$^{176}$Lu]{} [@Klay91a; @Klay91b; @Les91; @Pet92; @Dra10], and an indirect proof for the existence of IMS was obtained from various photoactivation studies [@Ver70; @Wat81; @Nor85; @Carr89; @Carr91; @Lak91; @Lak95a; @Lak95b; @Van00; @Kn05]. A review of the results for [$^{176}$Lu]{} is given in [@Mohr09]. For [$^{180}$Ta]{} only indirect evidence for the existence of IMS was derived from photoactivation [@Bel99; @Bel02; @Lak00; @Car89; @Col88; @Col90; @Nem92; @Nor84; @Bik99; @Sch94; @Sch98; @Loe96; @Sch01; @Loe03]. A direct detection of IMS by [$\gamma$]{}-spectroscopy was not possible up to now, see e.g. [@Dra98; @Sai99; @Dra00; @Wen01]. Definition of astrophysical reaction rates {#sec:def} ------------------------------------------ The stellar transition rate $\lambda^\ast$ for transitions from the [low-$K$]{} to the [high-$K$]{} species of heavy nuclei is approximately given by $$\begin{aligned} \lambda^\ast(T) & = & \int c \, n_\gamma(E,T) \, \sigma(E) \, dE \nonumber \\ & \approx & c \sum_i n_\gamma(E_{IMS,i},T) \, I^\ast_\sigma(E_{IMS,i}) \label{eq:lam}\end{aligned}$$ with the thermal photon density $$n_\gamma(E,T) = \left( \frac{1}{\pi} \right)^2 \, \left( \frac{1}{\hbar c} \right)^3 \, \frac{E^2}{\exp{(E/kT)} - 1} \label{eq:planck}$$ and the energy-integrated cross section $I^\ast_\sigma$ under stellar conditions for an IMS at excitation energy $E_{IMS}$ $$\begin{aligned} I^\ast_\sigma & = & \int \sigma(E) \, dE = \frac{2J_{IMS}+1}{2J_0+1} \, \left(\frac{\pi \hbar c}{E_{IMS}}\right)^2 \, \times \nonumber \\ & & \, \, \times \, \frac{\Gamma^\ast_{IMS \rightarrow {\rm{low-}}K}\, \Gamma^\ast_{IMS \rightarrow {\rm{high-}}K}}{\Gamma^\ast} \label{eq:isig}\end{aligned}$$ $\Gamma^\ast_{IMS \rightarrow{\rm{low-}}K}$ and $\Gamma^\ast_{IMS \rightarrow{\rm{high-}}K}$ are the total decay widths from the IMS to [low-$K$]{} and to [high-$K$]{} states under stellar conditions (including all cascades), $\Gamma^\ast = \Gamma^\ast_{IMS \rightarrow{\rm{low-}}K} + \Gamma^\ast_{IMS \rightarrow{\rm{high-}}K}$ is the total decay width, $J_{IMS}$ and $J_0$ are the spins of the IMS and the initial state, and the energy $E_{IMS}$ is given by the difference between the excitation energies of the IMS and the initial state: $E_{IMS} = E_x(IMS) - E_0$. The factor $\Gamma^\ast_{IMS \rightarrow{\rm{low-}}K} \times \Gamma^\ast_{IMS \rightarrow{\rm{high-}}K} / \Gamma^\ast$ in Eq. (\[eq:isig\]) may also be written as $b^\ast_{IMS \rightarrow{\rm{low-}}K} \times b^\ast_{IMS \rightarrow{\rm{high-}}K} \times \Gamma^\ast$ where $b^\ast_{IMS \rightarrow{\rm{low-}}K}$ and $b^\ast_{IMS \rightarrow{\rm{high-}}K}$ are the total decay branchings of the IMS under stellar conditions. It is important to point out that the total decay widths (including all cascades) to [low-$K$]{} and [high-$K$]{} states enter into Eq. (\[eq:isig\]). This is a consequence of the thermal population of excited states under stellar conditions; for details, see [@Mohr07; @Mohr06]. The stellar reaction rate $\lambda^\ast$ in Eq. (\[eq:lam\]) is given by the sum over the integrated cross sections $I^\ast_\sigma$ of all IMS where the contribution of each IMS is weighted by the number of thermal photons at the corresponding excitation energy. Because of the exponential dependence of the thermal photon density in Eq. (\[eq:planck\]), practically only very few low-lying IMS do contribute to the sum in Eq. (\[eq:lam\]). In the present study we restrict ourselves to the experimentally confirmed IMS in [$^{176}$Lu]{} at 839keV and a further candidate at 725keV [@Gin09]; for [$^{180}$Ta]{} we analyze the lowest IMS candidate at 594keV [@Mohr07]. The stellar reaction rate $\lambda^\ast(T)$ is strongly temperature dependent because of the roughly exponential factor $E^2/[\exp{(E/kT)} - 1]$ in Eq. (\[eq:planck\]). In addition to this explicit temperature dependence there is further implicit temperature dependence of $\lambda^\ast(T)$ because the widths $\Gamma^\ast$ in Eq. (\[eq:isig\]) also depend on temperature. This further temperature dependence will be discussed in detail in the next Sect. \[sec:mod\]; see also Eq. (\[eq:gammatemp\]). For the sake of clarity we will use the symbol $\lambda^\ast$ in units of s$^{-1}$ only for the stellar reaction rate between [low-$K$]{} and [high-$K$]{} states in Eq. (\[eq:lam\]); the symbol $\lambda$ will be used for transition rates between levels or groups of levels (in the same $K$ group). Levels will be further characterized by their lifetimes $\tau$ instead of their decay constants $\lambda = 1/\tau$. All energies are given in keV. Transitions in [$^{176}$Lu]{} and [$^{180}$Ta]{} {#sec:trans} ------------------------------------------------ ### [$^{176}$Lu]{} {#sec:trans176} A simplified level scheme of [$^{176}$Lu]{} is shown in Fig. \[fig:lu176level\]. There is an experimentally confirmed IMS at 839keV, and a further candidate for an IMS at 725keV has been suggested from the almost degeneracy of a [low-$K$]{} $7^-$ level and a [high-$K$]{} $7^-$ level [@Gin09]. Very recently, new low-lying IMS have been found by coincidence $\gamma$-spectroscopy [@Dra10]. ![ (Color online) Partial level scheme of [$^{176}$Lu]{} with [low-$K$]{} states on the left and [high-$K$]{} states on the right. IMSs are indicated by blue lines over the full width of the diagram. The IMS at 839keV (full line) decays to [low-$K$]{} and to [high-$K$]{} states. Relative $\gamma$-ray branches $b^{\gamma,{\rm{rel}}}$ normalized to the dominating ground state branching $b^{\gamma,{\rm{rel}}}_{839 \rightarrow 0} = 100$ are given for the IMS at 839keV. [[*[K]{}*]{}-mixing]{} of two neighboring $7^-$ levels at 724.7keV and 725.2keV may lead to a further IMS [@Gin09]. New low-lying IMS have been identified in a $K=4$ band at 709keV, 787keV, and 889keV [@Dra10]. The dashed lines indicate IMS that are not studied in detail in this work. []{data-label="fig:lu176level"}](fig01.eps){width="7.4cm"} Here we analyze the experimentally confirmed IMS at 839keV and its decays to the [low-$K$]{} levels at 723keV, 657keV, 635keV, and 596keV and to the [high-$K$]{} levels at 564keV and 0keV (ground state). Further details of the transitions are listed in Table \[tab:lu176trans\]. There are transitions in a wide range of energies for this IMS at 839keV. Thus, conclusions can also be drawn for transitions from other IMS [@Gin09; @Dra10] without a further detailed analysis. ------------- ----- ------------- ----- ------------ ---------- $J^\pi_i;K$ $J^\pi_f;K$ transition $5^-;4$ 839 $4^-;4$ 723 (M1) 1.3[^1] $5^-;4$ 839 $5^+;4$ 657 (E1)[^2] 1.8 $5^-;4$ 839 $4^+;4$ 635 (E1) 3.3 $5^-;4$ 839 $4^-;1$ 596 (M1,E2) 0.3 $5^-;4$ 839 $6^-;6$ 564 M1 6.5 $5^-;4$ 839 $7^-;7$ 0 E2 50.0[^3] $7^-;0$ 725 $5^-;0$ 437 E2 27.3[^4] $7^-;6$ 725 $6^-;6$ 564 (M1) 15.8 ------------- ----- ------------- ----- ------------ ---------- : Transitions in $^{176}$Lu (from [@ENSDF]). \[tab:lu176trans\] A candidate for an IMS at 725keV has been suggested by [@Gin09]; the suggestion is based on a theoretical study of [[*[K]{}*]{}-mixing]{} of two $7^-$ states at 724.7keV and 725.2keV with $K = 0$ and $K = 6$. The 725keV states decay to the [low-$K$]{} state at 437keV and to the [high-$K$]{} state at 564keV. Members of the $K=4$ band with its $4^+$ band head at 635keV have been identified as IMS recently [@Dra10]. Weak branches to the [high-$K$]{} $7^-;7$ ground state have been found for the $6^+$, $7^+$, and $8^+$ members of this band at 709keV, 787keV, and 889keV. The main decay branch from this band goes to the [low-$K$]{} side. From the estimated transition strengths in [@Dra10] it results that only the lowest IMS at 709keV may have significant influence on the stellar transition rate $\lambda^\ast$. Unfortunately, the lifetimes of the two $7^-$ states at 725keV are unknown, and only lower and upper limits for the lifetime of the $5^-$ state at 839keV are available in literature. For the following discussion we take $\Gamma^\gamma_{839 \rightarrow 0} = 50$$\mu$eV that corresponds to a partial lifetime of $\tau_{839 \rightarrow 0} = 13.2$ps. This value is in the experimental limits $10\,{\rm{ps}} \le \tau \le 433\,{\rm{ps}}$ for the lifetime of the 839keV state because this state predominantly (branching $\gtrsim 80\,\%$) decays by the $839 \rightarrow 0$ transition. In agreement with the theoretical arguments in [@Doll99] and the experimental photoactivation yields [@Van00; @Kn05] (see discussion in [@Mohr09] where $\tau \approx 12$ps is suggested with an uncertainty of about a factor of two) we use a value close to the upper experimental limit of the width (or lower limit of the lifetime). ### [$^{180}$Ta]{} {#sec:trans180} Following [@Mohr07], the lowest IMS in [$^{180}$Ta]{} is located at 594keV. It is the band head of a $K = 5$ rotational band, and also the higher members of this band have been assigned as IMS [@Wal01]. The 594keV level has a half-life of $t_{1/2} = 16.1 \pm 1.9$ns and decays by a 72.2keV transition [@Dra98; @Sai99], probably by a M1 transition to the 520keV level on the [low-$K$]{} side. (Note that there is a surprising 2keV discrepancy in the transition energy and the excitation energies that may be related to the 2keV shift of the $9^-$ isomer from $E_x = 75$keV in earlier compilations to $E_x = 77$keV in the latest data base [@ENSDF].) Based on reasonable assumptions for the transition strength of the E2 transition from the 594keV state to the $7^-$ state at 357keV on the [high-$K$]{} side, it has been concluded in [@Mohr07] that the 594keV state is the lowest IMS in [$^{180}$Ta]{}. A simplified level scheme of [$^{180}$Ta]{} is shown in Fig. \[fig:ta180level\]. ![ (Color online) Partial level scheme of [$^{180}$Ta]{} with [low-$K$]{} states on the left and [high-$K$]{} states on the right. The IMS is indicated by a blue line over the full width of the diagram. []{data-label="fig:ta180level"}](fig02.eps){width="7.4cm"} Modifications of transitions by the stellar plasma {#sec:mod} ================================================== Stellar transition rates and detailed balance theorem {#sec:stran} ----------------------------------------------------- In this chapter, we have changed notations to have indices $I$, $L$, and $H$ to designate IMS, [low-$K$]{}, and [high-$K$]{} states, respectively. The stellar reaction rate expression in Eqs. (\[eq:lam\]) to (\[eq:isig\]) only includes radiative excitation and spontaneous photon emission. In a stellar plasma at thermodynamic equilibrium, induced photon emission has also to be included. This can be easily done by changing Eq. (\[eq:isig\]) for a transition from a [high-$K$]{} state to a [low-$K$]{} state into: $$\begin{aligned} I^\ast_\sigma & = & \frac{2J_{I}+1}{2J_H+1} \, \left(\frac{\pi \hbar c}{E_{I}-E_{H}}\right)^2 \, \times \qquad \nonumber \\ & \times & \frac{ \Gamma^\ast_{IL} \Gamma^\ast_{IH} \frac{\exp{\left(\frac{E_{I}-E_{H}}{kT}\right)}}{\exp{\left(\frac{E_{I}-E_{H}}{kT}\right)}-1}}{\Gamma^\ast_{IL} \frac{\exp{\left(\frac{E_{I}-E_{L}}{kT}\right)}}{\exp{\left(\frac{E_{I}-E_{L}}{kT}\right)}-1}+{\Gamma^\ast_{IH} \frac{\exp{\left(\frac{E_{I}-E_{H}}{kT}\right)}}{\exp{\left(\frac{E_{I}-E_{H}}{kT}\right)}-1}}} \label{eq:isigind}\end{aligned}$$ However, it should be noted that $L$ and $H$ must designate single levels here. When several [high-$K$]{} levels or several [low-$K$]{} levels are involved, each stellar transition rate must be dealt with separately. Adding induced photon emission is only relevant when transition energies are not too much larger than the plasma temperature $kT$. In the worst case that will be presented below, a 72 keV transition in [$^{180}$Ta]{} at a temperature of 25 keV, the correction is only $5\%$. Thus, the approximation for the stellar reaction rate in Eq. (\[eq:lam\]) remains valid for typical astrophysical conditions. In a plasma at Local Thermodynamic Equilibrium (LTE), transition rates are related to their corresponding inverse transition rates by the detailed balance theorem. It can be easily proved that this still stands when dealing with indirect (through the IMS) transition rates, so we can write: $$\begin{aligned} \frac{\lambda^\ast_{HL}}{\lambda^\ast_{LH}} &= & \frac{2J_{L}+1}{2J_{H}+1} \exp{\left(\frac{E_{L}-E_{H}}{kT}\right)} \label{eq:revers}\end{aligned}$$ It is possible to define a global excitation and deexcitation rate when the IMS state is excited from, or decays down to, a group of levels by summing over the contributing levels $j$ [@Gos07]: $$\lambda_{IL} = \sum_j \lambda_{IL_{j}} \label{eq:lbdil}$$ and $$\lambda_{HI} = \frac{\displaystyle{\sum_j} \left(2J_{H_{j}}+1\right) e^{-\frac{E_{H_{j}}}{kT}} \lambda_{H_{j}I}}{\displaystyle{\sum_j} \left(2J_{H_{j}}+1\right) e^{-\frac{E_{H_{j}}}{kT}}} \label{eq:lbdhi}$$ These global rates do not verify the detailed balance theorem, as no single energy and spin can be associated to the ‘global level’. The detailed balance theorem can only be verified for a transition between two individual levels, and not when some are grouped together into a global level. However, in the case where one transition dominates all the other transitions from its group, the detailed balance theorem is approximately verified. In particular, such is the case for [$^{176}$Lu]{} in this work. Modifications of transition rates by electronic environment {#sec:modenv} ----------------------------------------------------------- Electronic environment in stellar plasmas may influence decay or excitation properties of nuclei. Internal conversion is strongly dependent on the number of bound electrons, and nuclear transitions may be excited by its inverse process NEEC [@Dzy07; @Gos04]. The huge number of low energy free electrons may also play a role in decay or excitation by electron scattering [@Gos09] even though the transition rate is usually quite small for high energy nuclear transitions. In the particular cases where an atomic transition matches in energy a nuclear transition, NEET (Nuclear Excitation by Electron Transition) and its reverse process BIC (Bound Internal Conversion) become possible [@Mor04; @Mor04b]. However, this last phenomenon is absent for the nuclear transitions in [$^{180}$Ta]{} or [$^{176}$Lu]{} of this study as no atomic transition matches the high energy nuclear transitions of interest. The net effect of all these processes is a modification of the excitation and de-excitation rates leading to modifications of nuclear level lifetimes [@Gos07]. All these processes have been dealt with under the LTE hypothesis, which means that the detailed balance theorem can be used for each individual process as well as for the total transition rate between two levels. The width $\Gamma^\ast_{i \rightarrow f}(T)$ for a transition from an initial state $i$ to a final state $f$ under stellar conditions depends on temperature and is given be the sum over several contributions: $$\begin{aligned} \Gamma^\ast_{i \rightarrow f}(T) & = & \Gamma^{\gamma}_{i \rightarrow f} + \Gamma^{IC}_{i \rightarrow f}(T) + \Gamma^{(e',e)}_{i \rightarrow f}(T) \nonumber \\ & = & \Gamma^{\gamma}_{i \rightarrow f} [ 1 + \alpha^{IC}_{i \rightarrow f}(T) + \alpha^{(e',e)}_{i \rightarrow f}(T) ] \nonumber \\ \label{eq:gammatemp}\end{aligned}$$ $\Gamma^{\gamma}_{i \rightarrow f}$ is the temperature-independent $\gamma$-radiation width that is enhanced by the temperature-dependent widths of conversion electrons $\Gamma^{IC}_{i \rightarrow f}(T)$ and of electron scattering $\Gamma^{(e',e)}_{i \rightarrow f}(T)$. The $\alpha$ are the corresponding dimensionless enhancement factors normalized to the radiation width $\Gamma^{\gamma}_{i \rightarrow f}$. The $\alpha^{IC}_{i \rightarrow f}$ is the well-known internal conversion coefficient modified to take into account the partial ionization of the atom and the modifications it induces on the electronic wavefunctions. The explanation of Eq. (\[eq:gammatemp\]) uses the standard wording for the decay case. Although the underlying physics is exactly the same, the usual wordings for the excitation case are “nuclear excitation by electron capture” $\Gamma^{NEEC}$ instead of “internal conversion” $\Gamma^{IC}$ and “inelastic electron scattering” $\Gamma^{(e,e')}$ instead of “superelastic electron scattering” $\Gamma^{(e',e)}$. For completeness and clarification of the figures in Sect. \[sec:res\] it must be pointed out that the radiation width $\Gamma^\gamma$ itself is temperature-independent. However, the half-life (or decay rate) of a given state becomes temperature-dependent at high temperatures because of induced photon emission (see also Sect. \[sec:stran\]), even in the absence of the further contributions of IC/NEEC and electron scattering in Eq. (\[eq:gammatemp\]). Results {#sec:res} ======= As already mentioned in the introduction, plasma effects are important mainly for transitions with low energies. Thus, capture reactions with typical energies far above 1MeV are practically not affected in any astrophysical scenario, whereas the production and destruction of isomers in the astrophysical [s-process]{} has to be studied in detail. It is generally accepted that the astrophysical [s-process]{} operates in thermally pulsing AGB stars [@Gal98; @Buss99; @Stra06]. In the so-called interpulse phase neutrons are produced by the $^{13}$C[($\alpha$,n)]{}$^{16}$O reaction at relatively low temperatures around $kT \approx 8$keV for about $10^4 - 10^5$years; this temperature is too low to affect isomer production and destruction [@Mohr07; @Mohr09]. During thermal pulses the $^{22}$Ne[($\alpha$,n)]{}$^{25}$Mg neutron source is activated for a few years at temperatures around 25keV and densities of the order of $10^3$g/cm$^3$ [@Gal98]. For the present analysis we adopt this density, and we study the temperature dependence of various transitions in the chosen examples [$^{176}$Lu]{} and [$^{180}$Ta]{}. The results are presented as temperature-dependent enhancement factors ${\cal{F}}(T)$ that relate the plasma effects (mainly NEEC and electron scattering) to the effective radiative transition width $${\cal{F}}^X(T) = \frac{\Gamma^X(T)}{\Gamma^\gamma_{\rm{eff}}(T)} \label{eq:enh}$$ where the index $X$ stands for IC/NEEC, electron scattering, or NEET. The presentation of the relative enhancement factor ${\cal{F}}$ instead of $\Gamma^X(T)$ avoids complications for transitions with unknown radiation widths $\Gamma^\gamma$. For $T \rightarrow 0$ the enhancement factors ${\cal{F}}$ in Eq. (\[eq:enh\]) are identical to the usual factors $\alpha$ in Eq. (\[eq:gammatemp\]). It has to be kept in mind that the radiative width $\Gamma^\gamma$ in Eq. (\[eq:gammatemp\]) is temperature-independent; but the radiative part is enhanced by induced photon transitions at high temperatures leading to the temperature-dependent effective radiation width $\Gamma^\gamma_{\rm{eff}}(T)$ in the denominator in Eq. (\[eq:enh\]): $$\Gamma^\gamma_{\rm{eff}}(T) = \Gamma^\gamma \left[ 1 + \frac{1}{\exp{(\Delta E/kT)}-1} \right] \label{eq:gamgameff}$$ The second part in the parenthesis is the enhancement due to induced photon emission for a transition with energy $\Delta E$; see also Eq. (\[eq:isigind\]) where the same factor was already used for the definition of the integrated cross section $I_\sigma^\ast$. Obviously this enhancement remains small at low temperatures and high transition energies, i.e. $\Delta E \gg kT$. All following results are presented within a range of temperatures from 1 keV to 1 MeV. However, it should be noted that the results are non-relativistic estimates, which can lead to some errors for temperatures above a few hundred keV. Modification of widths in [$^{176}$Lu]{} and [$^{180}$Ta]{} {#sec:resmod} ----------------------------------------------------------- ### [$^{176}$Lu]{} {#sec:res176} The lowest transition energy between the $5^-;4$ IMS state in [$^{176}$Lu]{} at 839keV and a lower state is 116keV. For such a high energy, one should not expect the electrons to have a large influence on the transition rates. First, we study the excitation of the $5^-;4$ IMS at 839keV from the [high-$K$]{} side, i.e. from the $7^-;7$ ground state and the $6^-;6$ state at 564keV. We plot the plasma enhancement factor as a function of temperature for the chosen density of 1000g/cm$^3$ in Fig. \[fig:lu176rate1\]. Only NEEC is not totally negligible against radiative excitation, but it never amounts to more than a few percents. ![ Transition rate enhancement factor for NEEC from [high-$K$]{} levels to the IMS level of [$^{176}$Lu]{} at 1000g/cm$^3$. []{data-label="fig:lu176rate1"}](fig03.eps){width="7.4cm" height="7.4cm"} Excitations of the $5^-;4$ IMS at 839keV from the [low-$K$]{} side are somewhat stronger affected. This is not surprising because of the lower transition energies from the $4^-;1$, $4^+;4$, $5^+;4$, and $4^-;4$ states located between 596keV and 723keV. We find NEEC rates nearly equal to radiative rates for temperatures lower than 10keV as shown in Fig. \[fig:graf\_ratio\_lut176\_nivL7\]. NEEC accounts for a global excitation rate increase by a factor around 1.6 in this temperature range. ![ Transition rate enhancement factor for NEEC from [low-$K$]{} levels to the IMS level of [$^{176}$Lu]{} at 1000g/cm$^3$. []{data-label="fig:graf_ratio_lut176_nivL7"}](fig04.eps){width="7.4cm" height="6.4cm"} This enhancement translates into the same factor on the stellar transition rate Eq. (\[eq:lam\]) shown on Fig. \[fig:graf\_ratioHL\_lut176\]. However, at temperatures below about 15keV the stellar transition rate from [high-$K$]{} to [low-$K$]{} states in [$^{176}$Lu]{} drops below $10^{-15}$/s or $3 \times 10^{-8}$ per year [@Mohr09; @Heil08], i.e. it becomes negligible on the above mentioned timescale of a thermal pulse. Consequently, the plasma modification of the stellar transition rate does not affect the nucleosynthesis of [$^{176}$Lu]{} in the [s-process]{}. ![ Stellar transition rate enhancement factor due to NEEC for [$^{176}$Lu]{} at 1000g/cm$^3$. The enhancement at temperatures below 15keV does not affect the nucleosynthesis of [$^{176}$Lu]{} in the [s-process]{} because the stellar rate drops below $10^{-15}$/s at 15keV. []{data-label="fig:graf_ratioHL_lut176"}](fig05.eps){width="7.4cm" height="7.4cm"} The enhancement of the stellar transition rate is directly related to the decrease of the partial half-life of the IMS level down to [low-$K$]{} levels as shown on Fig. \[fig:graf\_tvie\_lut176\]. The dominating branch to the [high-$K$]{} side is practically not affected. ![ Partial half-lives of the $5^-;4$ IMS level of [$^{176}$Lu]{} towards [low-$K$]{} and [high-$K$]{} levels at 1000g/cm$^3$. At low temperatures the branch to [low-$K$]{} states is enhanced by NEEC/IC. At high temperatures above 100keV induced photon emission shortens the half-lives. []{data-label="fig:graf_tvie_lut176"}](fig06.eps){width="7.4cm" height="7.4cm"} Two almost degenerate $7^-$ states around 725keV and their [[*[K]{}*]{}-mixing]{} have been suggested as a further candidate for a low-lying IMS in [$^{176}$Lu]{} [@Gin09]. The influence of the plasma environment on these two almost degenerate $7^-$ states is small. The decay energies are 288keV for the [low-$K$]{} branch and 161keV for the [high-$K$]{} branch. These transition energies are higher or at least similar to the transition energies in the [low-$K$]{} branch of the $5^-;4$ IMS at 839keV that are enhanced only at very low temperatures (see Fig. \[fig:graf\_ratio\_lut176\_nivL7\] and discussion above). Thus, it can be concluded that the IMS properties of the two $7^-$ states are not affected by the plasma environment. ### [$^{180}$Ta]{} {#sec:res180} The candidate for the lowest IMS in [$^{180}$Ta]{} is a $5^+$ state at 594keV that decays to the [low-$K$]{} branch by a 72keV (M1) transition; the laboratory half-life is $t_{1/2} = 16.1 \pm 1.9$ns. Thus, at first glance, effects on [$^{180}$Ta]{} appear to be stronger because of the relatively low transition energy of only 72keV. Indeed, the excitation rate from the [low-$K$]{} 520 keV state exhibits a large influence of electrons shown in Fig. \[fig:graf\_ratio\_tan180\_niv12\_decale\]. For temperatures below 10keV, electron inelastic scattering reaches $10\,\%$ of the radiative rate and NEEC is 10 times higher than the radiative rate. This factor can also be observed in Fig. \[fig:graf\_tvie\_tan180\] with a factor of 10 decrease on the half-life of the IMS level. ![ Transition rate enhancement factor for NEEC from the [low-$K$]{} $4^+$ level to the $5^+$ IMS level of [$^{180}$Ta]{} at 1000g/cm$^3$. []{data-label="fig:graf_ratio_tan180_niv12_decale"}](fig07.eps){width="7.4cm" height="7.4cm"} ![ Partial half-life of the $5^+$ IMS level of [$^{180}$Ta]{} towards the $4^+$ [low-$K$]{} level at 1000g/cm$^3$. The reduction of the half-life at low temperatures results from enhanced transitions by NEEC. The reduction at high temperatures is due to induced transitions. []{data-label="fig:graf_tvie_tan180"}](fig08.eps){width="7.4cm" height="7.4cm"} The excitation rate enhancement for the 237keV E2 transition from the $5^+$ IMS to the [high-$K$]{} $7^+$ state at 357keV is very small, even though in this case NEEC is not the only contributor as electron inelastic scattering makes an appearance as can be seen on Fig. \[fig:graf\_ratio\_tan180\_niv02\_decale\]. ![ Transition rate enhancement factor for NEEC from the $7^+$ [high-$K$]{} level to the $5^+$ IMS level of [$^{180}$Ta]{} at 1000g/cm$^3$. []{data-label="fig:graf_ratio_tan180_niv02_decale"}](fig09.eps){width="7.4cm" height="7.4cm"} Contrary to the [$^{176}$Lu]{} case, the rate enhancement of the [low-$K$]{} branch of the IMS does not translate into a similar increase on the stellar transition rate between [low-$K$]{} and [high-$K$]{} states. Fig. \[fig:graf\_ratio\_tan180\_niv01\] shows that a $20\,\%$ increase can at best be expected for the lowest temperatures because the excitation from the [high-$K$]{} level is the relevant term in the stellar transition rate. Similar to [$^{176}$Lu]{}, the small enhancement of the stellar reaction rate at low temperatures below about 15keV does not affect the nucleosynthesis in the [s-process]{} because the absolute rates are too small at such low temperatures. ![ Stellar transition rate enhancement factor due to NEEC for [$^{180}$Ta]{} at 1000g/cm$^3$. []{data-label="fig:graf_ratio_tan180_niv01"}](fig10.eps){width="7.4cm" height="7.4cm"} Discussion of the results {#sec:disc} ------------------------- From the above shown examples it can be concluded that transitions with energies above 200keV are practically not affected by the plasma environment that is present under stellar [s-process]{} conditions. The influence of the stellar plasma increases for lower transition energies and may reach about a factor of two for transition energies above 100keV. Low-energy transitions below 100keV may change dramatically; e.g., a factor of about 10 has been found for the 72keV transition in [$^{180}$Ta]{}. NEEC is the main contributor to this increase with capture onto the $1s$ shell amounting to the larger part. This effect disappears when the temperature increases as free electrons have too much energy to be captured onto an atomic shell. The only other influence of electrons is inelastic scattering. However, it is never greater than 10% of the radiative excitation rate or more than 1% of the total transition rate. NEET remains negligible as long as no matching transitions are present. Changes in the strength of a particular transition do not directly translate into modifications of the stellar reaction rate $\lambda^\ast$ for transitions from the [low-$K$]{} to the [high-$K$]{} levels. The stellar reaction rate $\lambda^\ast$ is proportional to the integrated cross section $I_\sigma^\ast$ in Eq. (\[eq:isig\]) and thus proportional to a width factor $$\lambda^\ast \sim I_\sigma^\ast \sim \frac{\Gamma_1 \Gamma_2}{\Gamma_1 + \Gamma_2} \label{eq:gam1gam2}$$ where the $\Gamma_i$ represent the [low-$K$]{} and [high-$K$]{} branches under stellar conditions. As long as one of the partial widths dominates – e.g. $\Gamma_1 \gg \Gamma_2$ and thus $\Gamma = \Gamma_1 + \Gamma_2 \approx \Gamma_1$ – this dominating width $\Gamma_1$ cancels out in Eq. (\[eq:gam1gam2\]), and the stellar rate is approximately proportional to the smaller width $\Gamma_2$. If the smaller width corresponds to a $K$-forbidden transition with relatively high energies above 200keV, then the stellar reaction rate $\lambda^\ast$ is practically not affected by the plasma environment. This is the case for the decay of the lowest IMS in [$^{180}$Ta]{} [@Mohr06] and also for the recently identified lowest IMS in [$^{176}$Lu]{} [@Dra10]. Although [$^{176}$Lu]{} and [$^{180}$Ta]{} appear to have a very different behavior in terms of modification of individual excitation rates by electrons, the global effects on the stellar transition rates are very similar: a 20% to 60% increase of the stellar rate is found for temperatures lower than 20keV. The major change of the 72keV transition in [$^{180}$Ta]{} does not appear as a major modification of the stellar reaction rate because this 72keV transition is the dominating decay branch of the IMS in [$^{180}$Ta]{}. Summary and conclusions {#sec:summ} ======================= Under stellar conditions the radiative transition width $\Gamma^\gamma$ for an individual transition from an initial state $i$ to a final state $f$ is enhanced by electronic transitions which are induced by the surrounding stellar plasma. The enhancement factor ${\cal{F}} = \Gamma^\ast/\Gamma^\gamma_{\rm{eff}}$ is composed of several effects. Under typical [s-process]{} conditions the dominating effect is NEEC. Electron scattering plays a very minor role, and NEET remains completely negligible for practical purposes. Typical [s-process]{} conditions are temperatures around $kT \approx 23$keV and $\rho \approx 10^3$g/cm$^3$ for the helium shell flashes in thermally pulsing AGB stars. Under these conditions we find negligible enhancement factors ${\cal{F}} \approx 1$ for transitions with energies above $\Delta E = 150$keV. At energies around 100keV, ${\cal{F}}$ increases, but remains below a factor of two. Further lowering of the transition energy down to about 50keV leads to dramatic enhancement factors up to one order of magnitude (${\cal{F}} \approx 10$). Transitions with energies below 50keV are even further enhanced; but nuclear transitions with such low transition energies are very rare. The nucleosynthesis of [$^{176}$Lu]{} and [$^{180}$Ta]{} is affected by low-lying $K$-isomers in these nuclei and the production and destruction of these isomers via transitions to IMS. The stellar transition rates $\lambda^\ast$ for transitions from [high-$K$]{} to [low-$K$]{} states are defined by the decay properties of the IMS, i.e. by a combination of the individual transition strengths. For [$^{176}$Lu]{} the stellar plasma does not lead to a significant modification of the stellar transition rate $\lambda^\ast$ because the lowest transition energy of 116keV is sufficiently high, and thus all individual transitions remain unaffected by the plasma. For [$^{180}$Ta]{} a significant enhancement of more than a factor of two is found for the low-energy $\Delta E = 72$keV transition from the lowest IMS at 594keV. This low-energy transition is the dominating decay branch of the IMS; but the stellar rate $\lambda^\ast$ is essentially defined by the weak decay branch to the 357keV state (as suggested in [@Mohr07]) which remains unaffected because of its larger transition energy. Thus, more or less by accident, the stellar rate $\lambda^\ast$ for [$^{180}$Ta]{} is not modified significantly although one individual decay branch is modified by more than a factor of two. In summary, due to the plasma environment the stellar reaction rate $\lambda^\ast$ for the production or destruction of $K$-isomers in [$^{176}$Lu]{} and [$^{180}$Ta]{} does not change by more than about 20% at [s-process]{} temperatures around 25keV and less than about 60% at very low temperatures below 10keV. However, at these low temperatures the absolute rates are too low to have influence on [s-process]{} nucleosynthesis; under these conditions, corresponding to the long-lasting interpulse phase with $kT \approx 8$keV, the [low-$K$]{} and [high-$K$]{} states have to be treated as two separate species that are practically decoupled because the IMS cannot be reached by thermal excitations. Within the present knowledge of IMS in [$^{176}$Lu]{} and [$^{180}$Ta]{} it may be concluded that electronic effects due to the plasma environment do not play a relevant role in the [s-process]{} nucleosynthesis of [$^{176}$Lu]{} and [$^{180}$Ta]{}. However, it should be kept in mind that three new IMS (or a group of IMS) have been suggested in the last few years: 725keV [@Gin09] and 709keV, 787keV, and 889keV [@Dra10] in [$^{176}$Lu]{} and 594keV in [$^{180}$Ta]{} [@Mohr07]. Each newly suggested IMS has its individual decay pattern which has to be studied. 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--- abstract: | In this paper, we review the current theory of very low mass stars model atmospheres including the coolest known M dwarfs, M subdwarfs, and brown dwarfs, i.e. T$_{\rm eff} \leq 5,000\,$K and $-2.0 \le {\rm [M/H]} \le +0.0$. We discuss ongoing efforts to incorporate molecular and grain opacities in cool stellar spectra, as well as the latest progress in deriving the effective temperature scale of M dwarfs. We also present the latest results of the models related to the search for brown dwarfs. author: - 'F. Allard and P. H. Hauschildt' title: | MODEL ATMOSPHERES:\ Brown Dwarfs from the Stellar Perspective --- Very Low Mass Star models and the T$_{\rm eff}$ Scale {#intro} ===================================================== Very Low Mass stars (VLMs) with masses from about $0.3$ M$_{\odot}$ to the hydrogen burning minimum mass ($0.075$ M$_{\odot}$, Baraffe et al. 1995) and young substellar brown dwarfs share similar atmospheric properties. Most of their photospheric hydrogen is locked in H$_2$ and most of the carbon in CO, with the excess oxygen forming important molecular absorbers such as TiO, VO, FeH and H$_2$O. They are subject to an efficient convective mixing often reaching the uppermost layers of their photosphere. Their energy distribution is governed by the millions of absorption lines of TiO and VO in the optical, and H$_2$O in the infrared, which leave [**no**]{} window of true continuum. But as brown dwarfs cool with age, they begin to differentiate themselves with the formation of methane (CH$_4$) in the infrared (Tsuji et al. 1995) at the expense of CO which bands begin to weaken in their spectra (Allard et al. 1996). Across the stellar-to-substellar boundary, clouds of e.g. corundum (Al$_2$O$_3$), perovskite (CaTiO$_3$), iron, enstatite (MgSiO$_3$), and forsterite (Mg$_2$SiO$_4$) may form, depleting the oxygen compounds and heavy elements and profoundly modifying the thermal structure and opacity of their photosphere (Sharp & Huebner 1990, Burrows et al. 1993, Fegley & Loggers 1996, Tsuji et al. 1996ab). Because these processes also occur in the stellar regime where a greater census of cool dwarfs is currently available for study, a proper quantitative understanding of VLM stars near the hydrogen burning limit is a prerequisite to an understanding of the spectroscopic properties and parameters of brown dwarfs and jovian-type planets. Model atmospheres have been constructed by several investigators over recent years with the primary goals of: 1. Determining the effective temperature scale of M dwarf stars down to the substellar regime. 2. Identifying spectroscopic signatures of substellarity i.e. gravity indicators for young brown dwarfs, and spectral features distinctive of cooler evolved brown dwarfs. 3. Providing non-grey surface boundary to evolution calculations of VLMs and brown dwarfs leading to more consistent stellar models, accurate mass-luminosity relations and cooling tracks for these objects. The computation of VLMs and brown dwarf model atmospheres requires a careful treatment of the convective mixing and the molecular opacities. The convection must currently be handled using the mixing length formalism while a variety of approximations have been used to handle the millions of molecular and atomic transitions that define the spectral distributions of VLMs and brown dwarfs. The most accurate of these methods is the so-called opacity sampling (OS) technique which consists in adding the contribution all transitions absorbing within a selected interval around each point of a pre-determined wavelength grid (typically $\approx 22000$ points from 0.001 to 100 $\mu$m). When the detail of the list of transitions is lacking for a molecule as is the case for the important absorber VO, the Just Overlapping Line Approximation (JOLA) offers an alternative by approximating the band structure based on only a few molecular rotational constants. The straight-mean (SM) and K-coefficients techniques, which consist in averaging the opacities over fixed wavelength intervals chosen smaller than the resolution of typical observations, have also been used in modeling late-type dwarf atmospheres. Their main advantage is to save computing time during the calculation of the models, often at the expense of an accurate spectral resolution. The list of recent model atmospheres and the opacity technique they mostly rely upon is given in table \[grids\]. [lcrc]{} Authors & Grid & T$_{\rm eff}$ range (K) & Main Opacity Treatment\ &&&\ Kurucz 1992& Atlas12 & $3500 - \dots$   & OS\ &&&\ Allard 1990& Base & $2000 - 3750$ & SM$+$JOLA\ Saumon et al. 1994& zero-metallicity & $1000 - 5000$ & OS\ Tsuji et al. 1995& grainless & $1000 - 2800$ & JOLA\ Brett 1995& MARCS & $2400 - 4000$ & OS\ Allard & Hauschildt 1995& Extended Base & $1500 - 4500$ & SM\ Tsuji et al. 1996& dusty & $1000 - 2800$ & JOLA$+$Grains\ Allard et al. 1996& NextGen & $900 - 9000$ & OS\ Allard et al. 1997b& NextGen-dusty & $900 - 3000$ & OS$+$Grains\ &&&\ Marley et al. 1996& &   $\dots - 1000$ & K-coefficients\ Because they mask emergent photospheric fluxes that would otherwise escape between absorption lines, the JOLA and SM approximations generally led to an excessive entrapment of heat in the atmosphere which yields systematically hotter model structures, and higher effective temperature (T$_{\rm eff}$) estimates for individual stars. Allard et al. (1997) have reviewed in detail the results of brown dwarfs and VLM model atmosphere calculations with respect to the effective temperature scale of M dwarfs. We reproduce in Figure \[Teffscale\] the T$_{\rm eff} - (V-I)$ relation of Allard et al. (1997) for the models listed in Table \[grids\]. Two double-line spectroscopic and eclipsing M dwarf binary systems, CM Draconis and YY Geminorum, offer some guidance in the sub-solar mass regime and are reported in Figure \[Teffscale\] according to Habets & Heintze (1981). The use of an OS treatment of the main molecular opacities, in particular for TiO, appears to yield a break-through in the agreement of T$_{\rm eff}$ scales with these two M dwarfs binary system. The NextGen and MARCS models yield effective temperatures that are coincidentally in good agreement with those derived empirically from the H$_2$O opacity profile by Jones et al. (1994) [^1]. Note, however, that the Atlas12 OS models suffer from an inaccurate TiO absorption profile and a complete lack of H$_2$O opacities, and are therefore clearly inadequate in the regime of VLM stars (i.e. below T$_{\rm eff} \approx 4500$ K) where molecular opacities dominate the stellar spectra and atmospheric structures. Some uncertainties on the metallicity of the CM Draconis system may soon disqualify the latter as a member of the disk main sequence (Viti et al. 1997). This stresses the importance of finding other low-mass eclipsing binary systems in the disk. These are hopefully soon to be provided by the 2MASS and DENIS surveys (see D. Kirkpatrick and X. Delfosse elsewhere in this volume). Much uncertainty remains, therefore, at the lowermost portion of the main sequence. The inclusion of grain formation (as discussed below) and more complete opacities of TiO promise a better understanding of the stars and brown dwarfs in the vicinity of the hydrogen burning limit (the location of which is roughly indicated in Figure \[Teffscale\] by the termination point of the Allard et al. 1996 model sequence), but still remain to be ascertained. The Infrared Colors of Brown Dwarfs =================================== The DENIS and 2MASS infrared sky surveys will soon deliver large data bases of red dwarfs, brown dwarfs and perhaps extrasolar planets, which will necessitate the best possible theoretical foundation. A proper understanding of their colors is essential in the search for brown dwarfs. Brown dwarfs and giant planets emit over 65 radiation in the infrared ($>1.0 \mu$m). Yet the main difficulties met by VLMs and brown dwarf modelers in recent years has been to reproduce adequately the infrared (1.4 to $2.5 \mu$m) spectral distribution of dwarfs with spectral types later than about M6. All models listed in the central part of table \[grids\] underestimate the emergent flux, most as much as 0.5 mag at the $K$ bandpass, despite the different opacity sources used by the authors. Allard et al. (1994, and subsequent publications) have explored water vapor opacity data from various sources. Figure \[h2o\] summarizes these results. Clearly, the water vapor opacity profile is quite uncertain and has varied with the degree of completeness and the assumptions used in the construction of the molecular model and its potential surface. The most recent and complete line list of Partridge & Schwenke succeeds for the first time in reproducing the $1.6~\mu$m opacity minimum, in the $H$ bandpass, well enough for the atomic Na$\,$I resonance line to finally emerge in the synthetic spectrum, matching the observed feature. However, it fails to provide an improvement in the $K$ bandpass where the less complete list of Miller & Tennyson still yield the best match of the models to the observed spectra. The NextGen models of Allard et al. (1996) are computed using the Miller & Tennyson line list and are the only models to provide a match to the infrared colors of VLMs. This is shown in Figure \[IJK\] where the complete series of NextGen models — as interpolated on the Baraffe et al. (1997) isochrones for 10 Gyrs and 120 Myrs and ranging from metallicities of \[M/H\]$= -2.0$ to 0.0 — are compared to the photometric field dwarfs’ samples of Leggett (1992), Tinney et al. (1993), and Kirkpatrick et al. (1995). Other models series including those of Brett (1995) and the Extended grid of Allard & Hauschildt (1995, not shown) are distinctively bluer than the observed sequence, while the 10 Gyrs NextGen models of solar metallicity follow closely the empirical sequence[^2] of Kirkpatrick & McCarthy (1994) until spectral types of M6 (i.e. $J-K \approx 0.85$). Beyond this point, all models fail to reproduce the bottom of the main sequence into the brown dwarf regime as defined by Gl406, VB10, BRI0021 and GD165B. The models catch up only at the much lower T$_{\rm eff}$ of the evolved brown dwarf Gliese 229B, i.e. 900-1000 K (Allard et al. 1996, Marley et al. 1996). The cause of the model discrepancies at the stellar-to-brown dwarf boundary can only be one that affects the cooler models for Gliese 229B in a far lesser obvious extent. Since the infrared spectral distribution is sensitive to the mixing length, yet without allowing for an improved fit of VLMs spectra, Brett (1995a) suggested that the problem lie in the inadequacy of the mixing length formalism for treating the convective transport in an optically thin photospheric medium. These concerns may also be augmented by uncertainties about the extent of the overshooting phenomenon in VLMs (see F. D’Antona elsewhere in this volume). The convection zone recedes gradually below the photosphere as the mass (and T$_{\rm eff}$) decreases along the isochrones. This implies that the lithium test of substellarity (Rebolo et al. 1992) — which relies on the assumption that the brown dwarf is still fully convective and mixing lithium from its core to its photospheric layers after 10$^8$ yrs of age — is inapplicable for objects cooler than T$_{\rm eff} \leq 2200$ K. The presence of lithium in the spectra of a late-type ($\geq $M10) field dwarfs, if detected, could only reflect their initial abundances and [**not**]{} their substellar nature. The shrinking of the convection zone also allows a very good agreement between the models of Marley et al. (which includes adiabatic convection only for the optically thick layers of the atmosphere) and the models of Allard et al. (1996) (based on a more careful treatment of convection with the mixing length formalism) for the brown dwarf Gliese 229B (see Figure 5 of Allard et al. 1997). Yet the maximum radial extent of the convection zone occurs at around T$_{\rm eff} = 3000\,$K, while the discrepancy with the infrared observations increases steadily towards the bottom of the main sequence. A more promising answer to the so called “infrared problem” may rather be found in the formation of dust grains in the very cool (typically T$_{\rm layer} \approx$ T$_{\rm eff} - 1000\,$K) upper layers of red and brown dwarf’s atmospheres. Tsuji et al. (1996a) proposed, based on their results of including the effects of the formation and opacities of three grain species (Al$_2$O$_3$, Fe, and MgSiO$_3$) in their new “dusty” models, that the greenhouse heating of grain opacities, the resulting enhanced H$_2$O dissociation, and the infrared flux redistribution can explain the infrared spectra of cool M dwarfs. The formation of perovskite dust grains at the expense of TiO may also explain the observed saturation (and disappearance in GD165B and Gliese 229B) of the TiO bands in the optical spectra of late-type red dwarfs (see also Jones & Tsuji elsewhere in this volume). The implications of this result is far reaching. Field brown dwarf candidates such as BRI0021 and GD165B can be far cooler and less massive than previously suspected (see e.g. the NextGen-dusty model predictions in Figure\[Teffscale\]). If grains also form in the young Pleiades brown dwarfs PPl15, Teide1 and Calar3 (T$_{\rm eff} \approx $ 3000, 2800, and 2700 K respectively), lithium abundances derived from grainless models and synthetic spectra such as those of Pavlenko et al. (1995, see also elsewhere in this volume) may be overestimated, and the masses attributed to these objects possibly underestimated. Evolution models of brown dwarfs, which are sensitive to the treatment of the atmospheres (Baraffe et al. 1995, Chabrier et al. 1996), and their predicted Mass-lithium abundance and Mass-Luminosity relations may also be affected. And indeed, the temperatures and pressure conditions of the outer layers of red dwarfs are propice to the formation of dust grains as demonstrated years ago by Sharp & Huebner (1990) and Burrows et al. (1993). However it was not clear at the time if the inward radiation of an active chromosphere, or the efficient convective mixing from the interior, would heat up these upper photospheric layers and disable grain formation. Another concern is that, under the gravities prevailing in M dwarfs, gravitational settling may occur that would eliminate large grains and their opacities from the photospheres over relatively short time scales. These possibilities still need to be thoroughly investigated, but clearly, grain formation is a process that occurs in M dwarf and brown dwarf model atmosphere and it must included in such calculations. In order to investigate which grains may form in the upper layers of M dwarfs, Allard et al. (1997b, in preparation) have modified the equation of states used in the NextGen models to include the detailed calculation of some 1000 liquids and crystals, using the free Gibbs energies compiled by Sharp & Huebner. Their results showed that, besides the three species considered by Tsuji et al., the M dwarfs atmosphere were rich in condensates with ZrO$_2$, Ca$_2$Al$_2$SiO$_7$, Ca$_2$MgSiO$_7$, MgAl$_2$O$_4$, Ti$_2$O$_3$, Ti$_4$O$_7$, CaTiO$_3$, and CaSiO$_3$ showing up in models as hot as T$_{\rm eff} = 2700-3000\,$K (i.e dM8-dM6)! The preliminary NextGen-dusty models have been computed using a continuous distribution of ellipsoid shapes and interstellar grain sizes (between 0.025 and 0.25 $\mu$m) for the treatment of the opacities of the Al$_2$O$_3$, Fe, MgSiO$_3$, and Mg$_2$SiO$_4$ dust grains (see Allard & Alexander elsewhere in this volume for computational details). This contrast with the assumption of spherical grains with 0.1 $\mu$m diameters in the dusty models Tsuji et al. Both model sets are shown in Figures \[Teffscale\] and \[IJK\]. As can be seen, the dusty models of Tsuji et al. provide the correct tendency of the coolest models to get rapidly very red (as much as $J-K = 1.65$ for GD165B) with decreasing mass for a relatively fixed $I-J$ color. Those models are however systematically too red in $I-J$ by as much as 1 mag and do not reproduce even the most massive M dwarfs while over-predicting the effects of grains in Gliese 229B type brown dwarfs (Tsuji et al., 1996b), a problem which must be related to the use of the JOLA treatment of molecular opacities in these models (see section \[intro\] above). The NextGen-dusty models, on the other hand, show the onset of grain formation effects by a progressive deviation from the grainless NextGen models for $J-K \geq 0.85$, bringing an improved agreement with the observed sequence in the region where the grainless NextGen models deviate. Of course, much remains to be improved in the computation of models with dust grains. The size distribution of various grain species, in particular those of the perovskite CaTiO$_3$ which is responsible for the depletion of TiO from the optical spectra of late-type dwarfs (eg. GD165B, see D. Kirkpatrick elsewhere in this volume) and of corundum (Al$_2$O$_3$) which accounts for most of the grain opacities in current models, is unknown for the conditions prevailing in M dwarfs atmospheres. It is conceivable that grains form more efficiently in M dwarfs atmospheres than in the interstellar medium and therefore their opacities are larger than considered in the NextGen-dusty models. We may as well be missing a number of important contributors (e.g. ZrO$_2$) to the total grain opacities in the models. Further investigations including time dependent grain growth analysis will be required to determine the true contribution of dust grains to the infrared colors of red and brown dwarfs. In the meanwhile, diagrams like that of Figure \[IJK\] may help in distinguishing interesting brown dwarfs candidates from large data banks of detected objects, and in obtaining an appreciation of the spectral sensitivity needed to detect new brown dwarfs. Models (Tsuji et al. 1995, Allard et al. 1996, Marley et al. 1996) and observations of Gliese 229B (see B. Oppenheimer elsewhere in this volume) have shown that methane bands at 1.7, 2.4 and 3.3 $\mu$m appear in the spectra of cool evolved brown dwarfs, and cause their $J-K$ colors to get progressively bluer with decreasing mass and as they cool over time. Yet their $I-J$ colors remain very red which allows to distinguish them from hotter low-mass stars, red shifted galaxies, red giant stars, and even from low metallicity brown dwarfs that are also blue due to pressure-induced H$_2$ opacities in the $H-$to$-K$ bandpasses. Fortunately, grain formation and uncertainties in molecular opacities are far reduced under low metallicity conditions (\[M/H\]$<-0.5$). Therefore, model atmospheres of metal-poor subdwarf stars and halo brown dwarfs are more reliable than their metal-rich counterparts at this point. This has been nicely demonstrated by Baraffe et al. (1997) who reproduced closely the main sequences of globular clusters ranging in metallicities from \[M/H\]$= -2.0$ to $-1.0$, as well as the sequence of the Monet et al. (1992) halo subdwarfs in color-magnitude diagrams (see G. Chabrier elsewhere in this volume). The colors of halo brown dwarfs as predicted by the NextGen models are therefore of quantitative quality await confrontation with the infrared colors of metal-poor subdwarfs from e.g. the Luyten catalog and the US Naval Observatory surveys. The sensitivity of the $I-J$ index to the chemical composition of the atmosphere (clearly illustrated by the NextGen model grid) allows to distinguish brown dwarf populations independently of an accurate knowledge of the parallaxes or distances involved. Even young brown dwarfs of lower gravity appear to form a distinct sequence at bluer $I-J$ (and redder $J-K$) values then that of their older field star counterparts as also evident from a comparison of the 10 Gyrs and 120 Myrs NextGen models. This gravity effect, and perhaps enhanced grain formation, may explain the scatter of spectroscopic properties observed among field dwarfs at the bottom of the main sequence (Kirkpatrick, this volume), as well as the systematic differences between Pleiades brown dwarfs and older field stars of same spectral type (i.e. same VO band strengths) noted by Martìn et al. (1996). Gravity effects have also been found to affect the infrared spectra of cool evolved brown dwarfs such as Gliese 229B: Allard et al. (1996) reported a strong response of the 2.2 $\mu$m opacity minimum to gravity changes which allowed to restrain the mass of the brown dwarf [^3]. The general spectral distributions of cool evolved brown dwarfs are well reproduced by current models despite the difference in their respective modeling techniques, and despite the uncertainties tied to grain formation and incomplete opacity data base for methane and ammonia. The models of Allard et al. (1996) and Marley et al. (1996) are compared in Figure \[detectBD\] which also summarizes the predicted absolute fluxes that free-floating brown dwarfs would have at a distance of 50 pc. As can be seen, there is no clear cut distinction between brown dwarfs and planets; molecular bands most gradually form (dust, H$_2$O, CH$_4$ and NH$_3$) and recede (TiO, VO, FeH, and CO) from the stellar to the planetary regime as the atmospheres get cooler. They remain very bright in the $IJK$ region, and become gradually redder in the near-infrared $I$ to $J$ bandpasses, which allows their detection from ground-based facilities. Layers of dust clouds in their upper atmospheres may increase the albedo of extrasolar planets and cool brown dwarfs sufficiently to reflect the light of a close-by parent star, becoming therefore resolvable in the optical where the clouds are densest but the parent star is however brightest. The peak of their intrinsic spectral energy distribution is located at 4.5 $\mu$m. At 5 $\mu$m, the hotter (younger or more massive) brown dwarfs and stars show strong CO bands which cause their flux to drop by nearly 0.5 dex relative to that at 4.5 $\mu$m. And between 4.5 and 10 $\mu$m, opacities of CH$_4$ (and H$_2$O in the hotter brown dwarfs) cause the flux to drop by 0.5 to 1.0 dex. Searches in the 4.5-5 $\mu$m region should therefore offer excellent possibilities of resolving brown dwarfs and EGPs in close binary systems, and to find free-floating brown dwarfs if space-telescope time allocations allow. The detection limits of current and planned ground-based and space-based telescopes Saumon et al. (1996) are also indicated in Figure \[detectBD\] which show that brown dwarfs within 50 pc would be easily detected by SIRTF in the 4.5-5.0 $\mu$m region. The drop in sensitivity of the various instruments redwards of 10 $\mu$m implies, however, that brown dwarfs and planets cooler than Gliese 299B have little chance to be detected in those redder bandpasses. Conclusions =========== In these exciting times where discoveries of brown dwarfs are finally breaking through, model atmospheres are also rapidly becoming up to the task of interpreting the observations and deriving new search strategies. Uniform grids of dwarf stars and brown dwarfs model atmospheres exist that extend from the tip to the toes of the main sequence – and beyond: 9000K to 900K, logg= 3.0-6.0, and \[M/H\]= 0.0 to $-2.0$ for the NextGen models. These large model grids allowed the construction of consistent interior and evolution models for VLMs that yield unprecedent agreement with globular cluster main sequences observed to 0.1 M$_\odot$ with HST. They led to the derivation of the important mass-luminosity relation for halo brown dwarfs and so to the realization that brown dwarfs cannot make up a significant fraction of the halo missing mass. The effective temperature scale of K to M type dwarfs with spectral types earlier than M6 is now unambiguously established, with only small uncertainties remaining from a possible incompleteness of existing TiO line lists. Grain formation has been identified as an important process in M dwarfs and brown dwarfs atmospheres which could explain the long-standing difficulties of the models to reproduce the spectral distribution of dwarfs later than about M6. The results of the models indicate that it may [**not**]{} longer be assumed that the convection zone extends to the photosphere of late-type red dwarfs and brown dwarfs, and that their photospheric lithium abundance reflect their core temperature and mass. The basic assumption supporting the lithium test of substellarity is only valid for young, hot brown dwarfs such as those found in the Pleiades cluster. Fortunately, if the lithium test cannot identify transition objects and brown dwarfs of the field, the OS molecular opacity treatment and grain formation have introduce new gravity (hence age) effects in the NextGen models that were not seen in the previous Extended models and that will potentially allow to separate younger transitional objects from field stars as readily as from their location in color-color diagrams. For this the colors of late-type red dwarfs need to be known with good accuracy i.e. better than about 0.05 magnitude, which we find is not the case of many known late M dwarfs such as Gl406, VB10, and especially LHS2924. As cooler dwarfs are being discovered, spectral types are stretching far beyond the classical Morgan & Keenan scheme. The lack of TiO bands in the optical, and the emergeance of CH$_4$ opacities in the infrared in GD165B and Gl229B call for an extension of the MK system beyond M9 to another spectral class (see D. Kirkpatrick, this volume). While the spectral class should only reflect the effective temperatures and not necessarily the mass of the objects, perhaps a suitable class for these objects would nevertheless be “T dwarfs” as in reminescence of J.C. Tarter who introduced the term “brown dwarf” now commonly accepted to designate substellar dwarfs, and Takashi Tsuji who led the field of late dwarfs atmospheres since the early 1960’s, first introduced methane as a spectral indicator of substellarity, and who is retiring soon. Another spectral class, perhaps “P”, will then be needed for dwarfs cooler then the condensation point of water vapor including planets. In any case, studies of the optical spectra of Gliese 229B, GD165B, the DENIS and 2MASS objects and other late-type dwarfs will soon allow to determine the stellar surface coverage of dust clouds if such are present, and to verify if intrinsic spectral-type variability afflict cool dusty dwarfs. Models will be the subject of further investigations relative to grain formation and its effect on late-type dwarfs until they can reproduce the lower main sequence and lead the way into the regime of cool brown dwarfs. Finally, if brown dwarfs are not abundant in the halo, they certainly are in the galactic disk and their study remains one that shall flourish as the census of the solar neighborhood continues and the gap between planets and stars fills in. This research is supported by a NASA LTSA NAG5-3435 and a NASA EPSCoR grant to Wichita State University. It was also supported in part by NASA ATP grant NAG 5-3018 and LTSA grant NAG 5-3619 to the University of Georgia. Some of the calculations presented in this paper were performed on the IBM SP2 of the UGA UCNS, at the San Diego Supercomputer Center (SDSC) and the Cornell Theory Center (CTC), with support from the National Science Foundation. We thank all these institutions for a generous allocation of computer time. 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Jr. 1994, , 107, 333 Kurucz, R. L. 1992, Rev. Mex. Astro. y Astrofis., 23, 187 Leggett, S. K. 1992, , 82, 351 Leggett, S. K., Allard, F., Berriman, G., Dahn, C. C. & Hauschildt, P. H. 1996, , 104, 117 Marley, M. S., Saumon, D., Guillot, T.,Freedman, R., Hubbard, W. B., Burrows, A. & Lunine, J. I. 1996, Science, 272, 1919 Mart[ì]{}n, E. L., Rebolo, R. & [Zapatero-Osorio]{}, M. R. 1996, , 469, 706 Monet, D. G., Dahn, C. C., Vrba, F. J., Harris, H. C., Pier, J. R., Luginbuhl, C. B. & Ables, H. D. 1992, , 103, 638 Pavlenko, Y. V., Rebolo, R., Mart[í]{}n, E. L. & Garcia-Lopez R. J. 1995, , 303, 807 Rebolo, R., Martìn, E. L. & Magazzú, A. 1992, , 389, L83 Saumon, D., Bergeron, P., Lunine, J. I., Hubbard, W. B. & Burrows, A. 1994, , 424, 333. Saumon, D., Hubbard, W. B., Burrows, A., Guillot, T., Lunine, J. I. & Chabrier, G. 1996, , 460, 993 Schweitzer, A., Hauschildt, P. H., Allard, F. & Basri, G. 1996, , 283, 821 Sharp, C. M. & Huebner, W. F. 1990, , 72, 417 Tinney, C. G., Mould, J. R. & Reid, I. N. 1993, , 105, 1045 Tsuji, T., Ohnaka, K. & Aoki, W. 1995, in Bottom of the Main Sequence — And Beyond, C. G. Tinney, Springer-Verlag, 45 Tsuji, T., Ohnaka, K. & Aoki, W. 1996a, , 305, L1 Tsuji, T., Ohnaka, K., Aoki, W. & Nakajima, T. 1996b, , 308, L29 Viti, S., Jones, H. R. A., Allard, F., Hauschildt, P. H., Tennyson, J., Miller, S. & Longmore, A. J. 1997, , in press , M. R., Rebolo, R. & Martín, E. L. 1997, , 317, 164 [^1]: Note that a comparison to observed spectra reveals uncertainties of the order of 0.2 to 0.5 mag on the published $I$ magnitudes of the latest-type M dwarfs Gl406, VB10 and LHS2924 analyzed by Jones et al. (1994) and reported on Figure \[Teffscale\]. [^2]: Note that this sequence was defined by stars selected from their optical spectroscopic properties. The somewhat erratic aspect of the sequence in this infrared diagram reflects uncertainties in the photometry and perhaps in the age of the selected stars. [^3]: Only within the error on the flux calibration of the observed spectra which are unfortunately large for this object.

[**STUDY OF MULTIPARTICLE PRODUCTION BY GLUON DOMINANCE MODEL (Part II)**]{} [^1] P.F. Ermolov$^1$, E.S. Kokoulina$^{2,3 \dag}$, E.A. Kuraev$^3$ A.V. Kutov$^3$, V.A. Nikitin$^3$, A.A. Pankov$^2$, I.A. Roufanov$^{3}$, N.K. Zhidkov$^3$ **Abstract** The gluon dominance model presents a description of multiparticle production in proton-proton collisions and proton-antiproton annihilation. The collective behavior of secondary particles in $pp$-interactions at 70 GeV/c and higher is studied in the project [**“Thermalization”**]{}. The obtained neutral and charged multiplicity distribution parameters explain some RHIC-data. The gluon dominance model is modified by the inclusion of intermediate quark topology for the multiplicity distribution description in the pure $p\bar p$-annihilation at few tens GeV/c and explains behavior of the second correlative moment. This article proposes a mechanism of the soft photon production as a sign of hadronization. Excess of soft photons allows one to estimate the emission region size. Introduction ============ A new model of investigating multiparticle production (MP) at high energy is proposed. It is based on multiplicity distribution (MD) description of different interactions on basis of QCD and a phenomenological hadronization scheme. It is shown that the proposed model agrees with experimental data in a wide energy region and, perhaps, can be used for analysis of jet quenching and other phenomena at RHIC [@WhPa]. Application of this model approach to pp-interaction (for the beginning see [@BAL]) is given in Section 2. The additional investigations of MD in the $p \bar p $ annihilation channel at a few tens GeV/c are carried out in Section 3. The emission region size for soft photons and the possible mechanisms of their formation are discussed in Section 4. The main results of these studies are given in Section 5. MD in $pp$-interactions (continuation) ====================================== MD of charged particles in proton interactions by means of the gluon dominance model were studied in [@BAL]. It is interesting to get MD for neutral mesons. For this purpose we take experimental mean multiplicity of $\pi ^0$ in pp-interactions at 69 GeV/c ($\sqrt s\simeq $ 11.6  GeV). It was be found $2.57\pm 0.13$ [@MUR]. So the mean multiplicity in this process is calculated as the product of the mean number of evaporated active gluons ($\overline m=2.48$) and hadron parameter $\overline n^h$. We can determine the hadronization parameter for neutral mesons: $\overline n^h_0$=$1.036\pm 0.041$ [@MGD]. We expect approximate equality of probabilities of different hadron production at the second (hadronization) stage. MD for neutral mesons have a form as for charged particles [@BAL]: $$\label{38} P_n(s)=\sum\limits_{m=0}^{ME}\frac{e^{-\overline m} \overline m^m}{m!} C^{n-2}_{mN}\left(\frac{\overline n^h} {N}\right)^{n-2}\left(1-\frac{\overline n^h} {N}\right)^{mN-(n-2)},$$ and can be easily obtained if they are normalized to mean multiplicity $\pi ^0$’s (Fig. 1). From this distribution we see that the maximal possible number of $\pi ^0$ from TSTM [@MGD] is 16. MD for the total multiplicity are shown in Fig. 2. The maximal total number of particles in this case is equal to 42. The dependence of the mean multiplicity of neutral mesons $\overline n_0$ versus the number of charged particles $n_{ch}$ can be determined by means of MD $P_{n_{tot}}(s)$: $$\label{40} \overline n_0(n_{ch},s)=\frac {\sum \limits _{n_{tot}=n_1}^{n_2} P_{n_{tot}}(s) \cdot (n_{tot}-n_{ch})}{\sum \limits _{n_{tot}=n_1}^{n_2} P_{n_{tot}}(s)},$$ where $n_1$ and $n_2$ are lower and top boundaries for the total multiplicity at the given number of charged particles $n_{ch}$. The MD of charged and neutral secondaries obtained by TSTM give the maximal number for charged $n_{ch}=26$, neutral $n_{0}=16$ and total $n_{tot}=42$. That is why we have the following limits for $n_1$ and $n_2$: $n_1 \geq n_{ch}$, $n_2 \leq 16+ n_{ch}$. These restrictions result in great disagreement with experimental data [@MUR] at small multiplicities. It was shown in [@MGD]. A significant improvement will be reached if we decrease the top limit at low multiplicities ($n_{ch}\leq 10$) to $n_2=2 n_{ch}$. This corresponds to the case when the maximal number of neutrals is equal to the number of charged particles, and a double excess of neutral mesons over positive (negative) pions is possible. Fig. 3 shows that multiplicity of neutrals versus $n_{ch}$ when $n_2$ is taken equal to $2 n_{ch}$ at small $n_{ch}$ and $n_2=16+n_{ch}$ at $n_{ch}>10$. This restriction in (\[40\]) indicates that AntiCentauro events (a large number of neutrals and very few charged particles) must be absent. Centauro events (a large number of charged particles and practically no accompanying neutrals) may be realized only in the region of high multiplicity. It is assumed [@KUR] that at the second stage different kinds of quark pairs from the gluon (maximal possible number is equal to $N_{tot}$) occur with equal probabilities. We will try to consider the formation of neutral and charged mesons as an example of the above assumption. The $u \overline u$ and $d \overline d$ quark pairs may appear at sufficient energy. At the end of hadronization the formation of two charged mesons (the law of charge conservation of quarks) may take place. Production of an additional neutral particle is not necessary while formation of a neutral meson. So we can claim that the number of charged hadrons will be larger than the number of the neutral ones, or the probability of the charged hadron production is higher than of the neutral ones. We can estimate these probabilities in GDM. MD of $\pi ^0$ from one gluon at the second stage may be described by the binomial distribution $P_{n_0}= C_{n_t}^{n_0}p_0^{n_0}p_c^{n_t-n_0}$. Here $n_t$ is the total number of hadrons formed from gluon, $n_0$ - the number of neutral mesons among these secondaries (the number of charged hadrons $n_c=n_t-n_0$), $p_c (p_0)$ -the probability of production of charged pair (one $\pi^0$). The normalized condition is $p_0+p_c=1$. From TSTM we have obtained $\overline n_{ch}=1.63$ and $\overline n_{0}=1.036$. The mean multiplicities for binomial distributions will be equal to: $\overline n_{ch}=p_c \overline n_t$, $\overline n_{0} =p_0 \overline n_t$. The probability of the charge particle production is higher than of the neutral mesons ($\overline n_{ch} > \overline n_{0}$). The ratio of these values is $p_c/p_0\sim 1.46$. The mean multiplicity of newly born hadrons (charged or neutral) in proton interactions in GDM is equal to the product of the mean multiplicity of gluons obtained at the first stage and the mean multiplicity of hadrons ($\overline n^h_{ch}$ or $\overline n^h_{0}$) produced from one gluon at the second stage. In the case of binomial distribution $\overline n_{ch}=\overline n_t\cdot p_c$, $\overline n_{0}=\overline n_t \cdot p_0$. Taking into account two leading protons, the mean multiplicity is $ \overline n_{ch}(s)=2+\overline m_g (s)\cdot \overline n^h_{ch}$ for charged particles in pp-interactions. The mean multiplicity of neutral mesons in this process is $ \overline n_{0}(s)=\overline m_g(s) \cdot \overline n^h_{0}.$ The ratio of the mean charged pairs to the neutral mesons in proton interactions is $$\label{43} \frac {\overline n_{ch}(s)/2}{\overline n_{0}}= \frac {1}{\overline m(s)\cdot\overline n_0^h}+\frac {1}{2}\cdot\frac {\overline n_{ch}^h}{\overline n_0^h}.$$ At 69 GeV/c this ratio (\[43\]) is equal to $1.19 \pm .25$. At the higher energy the mean number of active gluons $\overline m$ increases and becomes much more than 3. In this case (\[43\]) it will be around the ratio of $\overline n_{ch}^h/ 2\overline n_0^h$. The experimental data have shown $1.6$ for Au-Au peripheral interactions ($80-92(\%)$ centrality class) at 200 GeV and for pp interactions at 53 GeV [@PHE]. We can compare these results with GDM at higher energies. The application of GDM to describe MD in the energy region (102, 205, 300, 405 and 800 GeV/c) [@PrD] in both schemes (TSMB and TSTM) [@BAL] leads to good results (Fig. 4-8). Parameters of TSTM in this domain are given in Table 2. Table 1. Parameters of TSTM. $\sqrt s$ GeV $\overline m$ $M_g$ $N$ $\overline n^h$ $\Omega $ $\chi^2/$ndf --------------- ----------------- ------- --------------------- ----------------- ----------------- -------------- -- 102 $2.75\pm 0.08$ 8 $3.13\pm 0.56$ $1.64 \pm 0.04$ $1.92\pm 0.08$ 2.2/5 205 $2.82 \pm 0.20$ 8 $4.50 \pm 0.10$ $2.02 \pm 0.12$ $2.00 \pm 0.07$ 2.0/8 300 $2.94 \pm 0.34$ 10 $4.07 \pm 0.86$ $2.22 \pm 0.23$ $1.97 \pm 0.05$ 9.8/9 405 $2.70 \pm 0.30$ 9 $4.60 \pm 0.24$ $2.66 \pm 0.22$ $1.98 \pm 0.07$ 16.4/12 800 $3.41 \pm 2.55$ 10 $ 20.30 \pm 10.40 $ $2.41 \pm 1.69$ $2.01 \pm 0.08$ 10.8/12 We see that the number of active gluons and their mean multiplicity increase, parameters of hadronization $N$ and $\overline n^h_{ch}$ vary very slowly. At these energies the charged hadron/pion ratio (\[43\]) grows up to 1.6. The parameter of hadronization $\overline n_{ch}^h$ has a trend to increase weakly but $\overline n_{0}^h$ does not almost change. This behavior may be related with the production of other charged particles (not only pions): protons, antiprotons, kaons and so on. We consider that parameter $\overline n^h_{ch}$ goes to the limit value (like saturation). On the other side a small growth $\overline n^h_g$ in proton interactions also points at a possible change mechanism of hadronization of gluons in comparison with the transition gluons to hadrons in $e^+e^-$ annihilation. It is considered that in the last case partons transform to hadrons by the fragmentation mechanism at the absence of the thermal medium. Our MD analysis gives $\overline n_g^h \sim 1$ for this fragmentation [@ALU]. The recombination is specific for the hadron and nucleus processes. In this situation a lot of quark pairs from gluons appear almost simultaneously and recombine to various hadrons [@HWA]. The value $\overline n_g^h$ becomes bigger $\sim 2-3$), that indicates to the transition from the fragmentation mechanism to the recombination one.The recombination mechanism provides justification for applying the statistical model to describe ratios of hadron yields (the ratio $Baryon/Meson\approx 1$) [@HWA]. The collective flow of quarks may be explained by the recombination mechanism, too. The rapid local thermalization may be a consequence of this formation of secondary hadrons [@HWA]. In this way we try to compare two kinds of processes which have different values of hadronization parameters. The first one is $e^+e^-$ - annihilation. It is usually supposed that fragmentation dominates in it and newly formed hadrons fragment with a high moment of parton into the surrounding vacuum (such objects can also appear from the hot surface in peripheral events in nucleus and hadron collisions) [@HWA]. The nuclear modification factor $R_{CP}$ and elliptic flow $v_2$ in Au-Au collisions at RHIC have revealed an apparent quark-number dependence in the $p_T$ region from 1.5 to 5 GeV/c. Moreover, the baryon production increases more rapidly with centrality than the meson production. These observations confirm the picture of hadron formation by quark recombination [@HWA] and point out that the hadronization processes in high energy nucleus interactions are modified to the comparison of $e^++e^-$ and partly $p+p$ collisions. The GDM with a branch gives growth of the part of the evaporated gluons to 0.85-0.98 and a small rise of gluon branch number at higher energies. Besides we have got data about emergence of hard constituent in MP [@GIO]. In GDM it can be explained not only by not only evaporation of a single gluon sources but also of groups with several gluons (formed by branch). A simple MD scheme of this superposition will be analyzed below. Let us compare MD (\[38\]) with the descriptions of experimental data obtained by various approaches. We bring two of them. A fortunate expression for KNO function was obtained by a group from IHEP [@SEM] who combined the elastic and inelastic processes. We can see (Fig. 9) good agreement with data [@PrD] at 800 GeV/c both of MD in MGD (solid line) and KNO-function (dot line). A wide research of MD in pp-interactions was fulfilled by L.Van Hove, A.Giovannini and R.Ugoccioni [@GIO]. They proposed a two-step mechanisms of MP. The independent (Poisson) production of groups of ancestor particles (named “clan ancestors”) were supplemented by their decay, according to a hadron shower process (the logarithmic MD within each clan). Such convolution of two mechanisms gives a negative binomial distribution (NBD) for hadrons $$\label{45} P_n(s)=\frac{k_h(k_h+1)\dots(k_h+ n-1)}{n!}\left(\frac{\overline n(s)} {\overline n(s)+k_h}\right)^{n}\left( \frac{k_h}{k_h+\overline n(s)}\right)^{k_h},$$ where $k_h$ - the NBD parameter and $\overline n(s)$ - the mean multiplicity of hadrons. The comparison of NBD (dot line) and our MD in GDM (solid line) with data at 800 Gev/c is given in Fig. 10. A. Giovannini emphasizes that the nature of this clan is gluon bremsstrahlung [@GIO]. Our investigations by GDM allows to give a concrete gluon content. Binomial distributions (BD) describe the hadronization stage. The clan model of [@GIO] uses the logarithmic distribution of secondaries in a single clan. Both of MD have the similar behavior. At the top energy (especially at 900 GeV) the shoulder structure appears in $P_n$ [@UA5]. The comparison of data with one NBD does not describe data well. But the weighted superposition of two NBD gives a good description of the shoulder structure $P_n(s)$ [@GIO]. At 14 TeV A.Giovannini expects the weighted superposition of the three classes of events. We can modify our GDM considering that the gluon fission may be realized at higher energies. The independent evaporation of gluons sources of hadrons may be realized by single gluons and also groups from two and more fission gluons. Following A.Giovannini we name such groups of gluons - clans. Their independent emergence and following hadronization content of GDM. MD in GDM with two kinds of clans are: $$P_n(s)=\alpha _1\sum\limits_{m_1=0}^{Mg_1}\frac{e^{-\overline m_1} \overline m_1^{m_1}}{m_1!} C^{n-2}_{m_1\cdot N}\left(\frac{\overline n^h} {N}\right)^{n-2}\left(1-\frac{\overline n^h} {N}\right)^{m_1\cdot N-(n-2)}+$$ $$\label{48} + \alpha _2\sum\limits_{m_2=0}^{Mg_2}\frac{e^{-\overline m_2} \overline m_2^{m_2}}{m_2!} C^{n-2}_{2\cdot m_2\cdot N}\left(\frac{\overline n^h} {N}\right)^{n-2}\left(1-\frac{\overline n^h} {N}\right)^{2\cdot m_2\cdot N-(n-2)},$$ where $\alpha _1$ and $\alpha _2$ are the contribution single and double gluon clans ($\alpha _1 + \alpha _2 =1$). The comparison (\[48\]) with experimental data for proton interactions at $\sqrt s= 62.2$ GeV [@ISR] is given in Fig. 8. We have obtained the following values of parameters: $N=7.06\pm 3.48$, $\overline m_1 =3.59 \pm 0.03$, $\overline m_2=1.15\pm 0.25$, $\overline n_h=3.23\pm 0.14$, $Mg_1=8$, $Mg_2=4$, $\alpha _1/\alpha _2 \sim 1.8$ at $\chi ^2/$ndf=$9.12/13$. The mean multiplicities of the two kinds of clans are similar. The specific feature of our GDM approach is the dominance of a lot of active gluons in MP. We can expect the emergence of them in nucleus collisions (experiments at RHIC) and the formation of a new kind of matter (quark-gluon plasma)at high energy. We consider that our gluon system can be a candidate for this. So the mean multiplicity of active gluons approached 10 at RHIC. For Au+Au central collisions their number may be equal to $ 200\cdot \overline m \approx 2000$ before the branch. This gluon medium facilitates the quenching. ![ $\overline n_\pi^o(n_{ch})$.[]{data-label="27kdfig"}](ther0-a.eps){width="\linewidth" height="2in"} ![ $\overline n_\pi^o(n_{ch})$.[]{data-label="27kdfig"}](thert-a.eps){width="\linewidth" height="2in"} ![ $\overline n_\pi^o(n_{ch})$.[]{data-label="27kdfig"}](pi0me6-aa.eps){width="\linewidth" height="2in"} MD in $p\bar p$-annihilation ============================ In the midst of interesting and enough inextricable hadron interactions the $p\bar p$ annihilation shows up especially [@Rush]. Experimental data at tens GeV/c [@Rush] point out on some maxima in differences between $p\bar p$ and $pp$ inelastic topological cross sections what may witness about the contribution of different mechanisms of MP $$\label{50} \Delta \sigma _n(p\overline p -pp)= \sigma _n(p \overline p) - \sigma _n (pp).$$ The important information about the MP mechanism may be picked out from the MD moment analysis of charged particles. The second correlative moment for negative particles $f_2^{--}$ are available to study MP $$\label{49} f_2^{--}=\overline {n_-(n_--1)}-{\overline{n}_-}^2.$$ The negative value of second correlative moments is characteristic for a more narrow MD than Poisson, and they indicate the predominance of the hadronization stage in MP. According to MGD, active gluons are a basic source of secondary hadrons. At the initial stage of annihilation three valent $q\overline q$-pairs ($uud$ and $\overline u \overline u \overline d$) are. They can turn to the “leading” mesons which consist from: a) valent quarks or b) valent and vacuum quarks [@KUR]. In the case a) only three “leading” neutral pions (the “0” topology) or two charged and one neutral “leading” mesons (“2” - topology) may form. In b) case the “4”- and “6”- topology is realized for “leading” mesons. We suggest that the formation neutron and antineutron (exchange) can be realized. A simple scheme of MP for annihilation may give the negative second correlative moments in GDM. We suggest that the active gluon emergence together with the formation of intermediate topology occurs. The GF for a single active gluon $Q_1(z)=[1+\overline n/N(z-1)]^N$ gives [@MGD] $$\label{52} f_2=Q_1^{''}(z)|_{z=1} -[Q_1(z)|_{z=1}]^2 = -(\overline n^h)^2/N < 0.$$ Reciprocally for m gluons GF and $f_2$ will be $$\label{53} Q_m(z)=[1+\overline n^h/N(z-1)]^{mN}, \quad f_2=-m(\overline n^h)^2/N.$$ We consider that $m$ grows while increasing the energy of the colliding particles, and $f_2$ will decrease almost linearly from $m$. Such behavior qualitatively agrees with experimental data [@Rush]. If we take concrete MD $P_m^G$ for gluons, then GF for secondary hadrons and $f_2$ are $$\label{54} Q(z)=\sum\limits_m P_m^G [1+\overline n^h/N(z-1)]^{mN}$$ $$\label{55} f_2=[f_2^G +1-1/N]\cdot \overline m \cdot(\overline n^h)^2,\quad f_2^G=\overline {m(m-1)}- \overline m^2,$$ where $f_2^G$ - the second correlative moment for gluons. In this scheme $f_2$ may be negative or positive. We consider that the negative value $f_2$ in the large energy region in comparison with $p+p$ interactions may be related with the destruction of the initial system on three or more shares and the number of active gluons related with a “leading” pion will be less than in the case of a leading proton in pp-collisions at the same energy. Herewith the total number of such gluons at annihilation may be bigger, their manifestation happens independently but the number of them per one pion grows slowly. The explanation of the negative $f_2$ was given R.Lednicky [@Led] at the assumption of the independent MP of charged particles. The second correlative moment has a zero value only in the small energy domain. And so we should restrict the region to apply this explanation. According to GDM for $p\overline p$ annihilation and taking into account three intermediate charged topology and active gluons, GF $Q(z)$ for final MD may be written as the convolution gluon and hadron components: $$Q(z)=c_0\sum\limits_m P_m^G [1+\frac{\overline n^h}{N}(z-1)]^{mN}+c_2\sum\limits_m z^2 P_m^G [1+\frac{\overline n^h}{N}(z-1)]^{mN} +c_4\sum\limits_m z^4 P_m^G [1+\frac{\overline n^h}{N}(z-1)]^{mN}.$$ The parameters of $c_0$, $c_2$ and $c_4$ are determined as the part of intermediate topology (“0”, “2” or “4”) to the annihilation cross section ($c_0 + c_2 + c_4 =1).$ For the simplicity we are limited by Poisson distribution with the finite number of gluons for $P_m^G$. The comparison of the experimental data (Fig. 12) gives the following values of parameters: $\overline m=3.36 \pm 0.18$, $N=4.01\pm 0.61$, $\overline n^h$=1.74 $\pm $0.26, the ratio $c_0$ : $c_2$ : $c_4$ = 15 : 40 : 0.05 at $\chi ^2/ndf=5.77/4$ and the maximum possible number of gluons $M=4$ at “4”-topology. The sum begins from $m=1$ (inelastic events), at $n \geq 2$ - from $m=0$ and finishes up $m<M$ at small multiplicities ($n\leq 4$). We should to emphasize very complicated events $n_{ch}=0$ and $2$. This research of $p\overline p$ annihilation requares to be continued. We will develop MGD to describe MD at energies 200, 500, 900 GeV [@ALN] and higher. ![MD at 303 GeV/c.[]{data-label="33dfig"}](th102-a.eps){width="\linewidth" height="2in"} ![MD at 303 GeV/c.[]{data-label="33dfig"}](th205-a.eps){width="\linewidth" height="2in"} ![MD at 303 GeV/c.[]{data-label="33dfig"}](th303-a.eps){width="\linewidth" height="2in"} ![MD in GDM and KNO-function [@SEM].[]{data-label="36dfig"}](th405-a.eps){width="\linewidth" height="2in"} ![MD in GDM and KNO-function [@SEM].[]{data-label="36dfig"}](th800-a.eps){width="\linewidth" height="2in"} ![MD in GDM and KNO-function [@SEM].[]{data-label="36dfig"}](pros800.eps){width="\linewidth" height="2in"} Soft photons ============ The production of photons in particle collisions at high energies was studied in many experiments [@CHL]. In project “Thermalization” it is planned to investigate low energetic photons with $p_t \leq 0.1GeV/c$ and $x \leq 0.01$ [@THE]. Usually these photons are named soft photons (SP). Experiments shown that measured cross sections of SP are several times larger than the expected ones from QED inner bremsstrahlung. Phenomenological models were proposed to explain the SP excess: the glob model of Lichard and Van Hove and the modified soft annihilation model of Lichard and Thomson [@LVH] . We consider that at a certain moment QGS or excited new hadrons may set in an almost equilibrium state during a short period or finite time. That is why, to describe massless photons, we will try to use the black body emission spectrum [@HER]. From experimental data [@THE] the inelastic cross section is equal to approximately $40 mb$, the cross section of SP formation is about $4 mb$, and since $ \sigma _{\gamma} \simeq n_{\gamma }(T)\cdot \sigma _{in}$, then the number of SP will be equal to $n_{\gamma }\approx 0.1.$ For convenience, we may use the well-known density of MVB at $T_r=2.275 K$ and get the number of photons by means of MVB $n_{\gamma }(T)=n_{\gamma }(T_r)\cdot \left ( \frac {T}{T_r}\right )^3.$ The density of SP in the region $1 fm^3$ will be equal to $$\rho (T)=n_{\gamma }(T)/V=4.112\cdot 10^8\cdot 10^{-6}\cdot 10^{-39}\cdot \left ( \frac {T}{T_r}\right)^3 fm^{-3}.$$ The estimates of temperature are implemented by transfer moment: $T=p\approx p_T\sqrt 2$ ($1MeV=1.16\cdot 10^{10}K$). If $T(p_T)$ is known, using $n_{\gamma }$ we can estimate the linear size of radiation system ($V\simeq L^3$). Dependencies of the linear size of system (L) from the SP moment ($p_T$) are given in Table  2. Table 2. The size of system L (fm) versus $p_T$ (MeV/c) of SP. --------------------- -------------------- ------------------- ---------------------- ------------------- ------------------- ------------------- $ \quad p_T \quad $ $\quad 10 \quad $ $ $ \quad 25 \quad $ $ \quad 30 \quad$ $ \quad 40 \quad$ $ \quad 50 \quad$ \quad 15 \quad $ \[1ex\] $L$ $11$ $6.9$ $4.1$ $3.5$ $2.6$ $2.0$ \[1ex\] --------------------- -------------------- ------------------- ---------------------- ------------------- ------------------- ------------------- It is well-known that the temperature of second hadrons is higher than the temperature of SP. We presume that objects with soft gluon content may not transform into hadrons but turn into SP. The amount of such soft gluons is estimated by $N_g$ in TSMB. ![GDM MD in $p\overline p$ at 14.75 GeV/c.[]{data-label="39dfig"}](pron800-a.eps){width="\linewidth" height="2in"} ![GDM MD in $p\overline p$ at 14.75 GeV/c.[]{data-label="39dfig"}](Clan62s.eps){width="\linewidth" height="2in"} ![GDM MD in $p\overline p$ at 14.75 GeV/c.[]{data-label="39dfig"}](Dsigan.eps){width="\linewidth" height="2in"} Conclusion ========== In our research we have undertaken an attempt to give MP description in different processes by means of a unified approach based on quark-gluon picture using the phenomenological hadronization model. The implemented model investigation allows us to understand deeper the picture of MP at various stages. We have obtained qualitative and quantitative agreements of our schemes with experimental data in $e^+e^-$, $p\overline p$ annihilation and $pp$ and nucleus collisions in a very wide energy domain. The authors appreciate for the support of physicists from JINR, GSTU who encouraged our investigations. [*Project “Thermalization” is partially supported by RFBR grant 03-02-16869*]{}. [99]{} RHIC collaboration white paper: http//www.phenix.bnl.gov/WWW/infor/comment. E.S.Kokoulina and V.A.Nikitin. ISHEPP, JINR, Dubna, Russia (2004), \[hep-ph/0502224\]. V.S.Murzin and L.I.Sarycheva. Interactions of high energy hadrons. (Russian ed.) 288p. (1983); V.G. Grishin, Phys. El.Part. and At.Yad., [**10**]{}, 608 (1979); K.Jager. [*et al.*]{}, Phys. Rev. [**D11**]{}, 2405 (1975). E.S. Kokoulina and V.A. Nikitin, International School-Seminar “The Actual Problems of Microworld Physics”, Gomel, Belarus, 2003 \[hep-ph/0308139\]; E.S.Kokoulina, ISMDXXXIII, Krakow, Poland, Acta Phys.Polon. [**B35**]{} 295 (2004) \[hep-ph/0401223\]. 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[^1]: Talk given at XVII International Baldin - Seminar “Relativistic Nuclear Physics and Quantum Chromodynamics”. JINR. September 27 - October 2, 2004, Dubna, Russia.

--- abstract: 'When a high symmetry facet joins the rounded part of a crystal, the step line density vanishes as $\sqrt{r}$ with $r$ denoting the distance from the facet edge. This means that the ledge bordering the facet has a lot of space to meander as caused by thermal activation. We investigate the statistical properties of the border ledge fluctuations. In the scaling regime they turn out to be non-Gaussian and related to the edge statistics of GUE multi-matrix models.' author: - 'Patrik L. Ferrari' - Michael Prähofer - Herbert Spohn date: 'October 17, 2003' title: Fluctuations of an Atomic Ledge Bordering a Crystalline Facet --- Equilibrium crystal shapes typically consist of various flat facets connected by rounded surfaces. For a microscopically flat facet there must be an atomic ledge bordering the facet. This border step could be blurred because of thermal excitations, but is clearly visible at sufficiently low temperatures [@SCV; @NBE; @NBEB]. While in the interior of the rounded piece of the crystal the step line density is of order one on the scale of the lattice constant, it decays to zero as the edge of a high symmetry facet is approached. If $r$ denotes the distance from the facet edge, according to Pokrovsky-Talapov [@PT] the step line density vanishes as $\sqrt{r}$. Thus there is a lot of space for the border ledge to meander, in sharp contrast to steps in the rounded part which are so confined by their neighbors that they fluctuate only logarithmically [@Sp]. The goal of our letter is to explore quantitatively the statistics of border ledge fluctuations. To illustrate our set-up we display in Fig. \[fig1\] a typical configuration from a statistical mechanics model which will be discussed below. One clearly recognizes the three facets as joined through a single rounded piece. Our interest is the statistics of the uppermost ledge. Experimentally ledge fluctuations are an elegant tool to determine step energies [@SVR; @NBEB]: One carefully prepares an island, linear size $L$ and single atom height, on a high symmetry facet. Alternatively, one sputters an undercut island. The ledge bordering the island is well described by a random walk, which implies fluctuations of size $\sqrt{L}$ [@KE]. In contrast, as can be clearly observed from Fig. \[fig1\], the border ledge of a facet interacts with its neighbors and a random walk model is not appropriate. In fact as our main result we will establish that the border ledge has fluctuations of size $L^{1/3}$ with a *non-Gaussian* statistics. ![*Crystal corner viewed from the $[1 1 1]$-direction.*[]{data-label="fig1"}](CubesSmall.ps){height="7.8cm"} To gain some understanding of the origin of such anomalous fluctuations let us consider the terrace-ledge-kink (TLK) model, which serves as an accurate description of a vicinal surface, i.e. a crystal cut at a small angle relative to a high symmetry crystal plane. The surface is made up of an array of ledges which on the average run in parallel and are separated by terraces. The ledges are not perfectly straight and meander through kink excitations, only constrained not to touch a neighboring ledge. One can think of these ledges also as discrete random walks constrained not to cross, i.e. with a purely entropic repulsion. Such a line ensemble is very closely related to Dyson’s Brownian motion, in which the random walks are replaced by continuum Brownian motions. As discussed in [@KM; @D], the location of the steps at fixed random walk time $t$ have the same distribution as the eigenvalues of a GUE $(\beta=2)$ random matrix. On this basis it is expected that the ledge-ledge distance is governed by the GUE level spacing [@EPL]. This prediction is verified experimentally [@ERC], however with deviations from $\beta=2$ which are attributed to long range elastic forces mediated through the bulk of the crystal and not included in the TLK model. If in the TLK model one retains the lattice structure in the transverse direction and makes the continuum approximation in the direction along the ledges, then the ledges can be regarded as the world lines of free fermions in space-time $\mathbb Z \times \mathbb R$ [@BV]. The world lines are piecewise constant and have jumps of only one lattice spacing. Consequently the transfer matrix has a nearest neighbor hopping term and the Pauli exclusion principle guarantees entropic repulsion in the sense that ledges never cross. The TLK model, in the version as just explained, has no facet. The crystalline surface has a constant average slope. Slope variations can be enforced through a *volume constraint*. For this purpose we introduce the “occupation” variables $\eta_j(t)$, $|j| \leq N$, $|t| \leq T$, in the surface patch $[-N,-N+1,...,N]\times[-T,T]$: $\eta_j(t)=1$ if there is some ledge passing through $(j,t)$, and $\eta_j(t)=0$ otherwise. In these variables, up to an overall constant, the crystal volume is given by $$\label{vol} A_{\mathrm{v}} = \int_{-T}^T\textrm{d}t \sum_{j=-N}^N j\, \eta_j(t)$$ and volume constraint means to have an ensemble of ledges where the action $A_{\mathrm{v}}$ is kept fixed. ![*Top lines for a TLK model with volume constraint.*[]{data-label="fig2"}](TLK.eps){width="8cm" height="3.5cm"} Without volume constraint the transfer matrix is generated by a free fermion Hamiltonian with nearest neighbor hopping [@BV]. Imposing the volume constraint grand-canonically adds to the fermionic action the term $\lambda^{-1}A_{\mathrm{v}}$ with a suitable Lagrange multiplier $\lambda^{-1}$. Thereby the nearest neighbor hopping Hamiltonian is modified to $$\label{a} H_{\mathrm{F}}=\sum_j\big(-a^\dagger_j a_{j+1}- a^\dagger_{j+1} a_j+2a^\dagger_j a_j+\frac{j}{\lambda}a^\dagger_ja_j\big).$$ $a_j$, resp. $a_j^\dagger$, is the annihilation, resp. creation, operator at lattice site $j\in \mathbbm{Z}$. They satisfy the anticommutation relations $\{a_i,a^\dagger_j\}=\delta_{ij}\,,\{a_i,a_j\}=0=\{a^\dagger_i,a^\dagger_j\}$. In (\[a\]) we have taken already the limit $N\to\infty$. The transfer matrix is $e^{-t H_F}$, $t\geq 0$, and in the limit $T\to\infty$ one has to compute the ground state expectations for $H_F$. A macroscopic facet emerges as $\lambda \to \infty$. In Fig. \[fig2\] we display a typical ledge configuration for the TLK model with volume constraint. There is no further ledge above the one shown and for $j \to -\infty$ ledges are perfectly flat and densely packed. Since a ledge corresponds to a fermionic world line, the average step density $\langle \eta_j(t)\rangle_\lambda = \rho_\lambda(j)$ is independent of $t$ and given by $\langle \eta_j(t)\rangle_\lambda = \langle a_j^\dagger a_j\rangle_\lambda$ with $\langle \cdot \rangle_\lambda$ on the right denoting the ground state expectation for $H_F$. By the linear potential in (\[a\]) steps are suppressed for large $j$. Hence the average surface height $h^\lambda_j(t)$ at $(j,t)$, relative to the high symmetry plane, equals $${\label{height}} h_j^\lambda(t)=-\sum_{k=j}^\infty \langle \eta_k(t)\rangle_\lambda\, .$$ $\langle a_j^\dagger a_j\rangle_\lambda$ can be computed in terms of the Bessel function $J_j(z)$ of integer order $j$ and its derivative $L_j(z)=(\textrm{d}/\textrm{d}j)J_j(t)$ with the result $$\begin{aligned} \label{b} \rho_\lambda(j)=\langle a^\dagger_ja_j\rangle_\lambda&=&\lambda\big(L_{j-1+ 2[\lambda]}(2\lambda)J_{j+2[\lambda]}(2\lambda) \\ & & -L_{j+2[\lambda]}(2\lambda)J_{j-1+2[\lambda]}(2\lambda)\big) \nonumber\end{aligned}$$ where $[\cdot]$ denotes the integer part. For large $\lambda$ the height $h_j^\lambda(t)$ is of order $\lambda$. Therefore we rescale the lattice spacing by $1/\lambda$. Then $\lim_{\lambda \to \infty}\lambda^{-1}h_{[\lambda r]}(\lambda t) = h_{\textrm{eq}}(r,t)$ with the macroscopic equilibrium crystal shape $$\begin{aligned} \label{c} h_{\mathrm{eq}}(r-2,t)=\left\{ \begin{array}{l} r \hspace{30pt}\mbox{for $r\leq -2$\,,}\\[6pt] \frac{1}{\pi}\big(r\arccos(r/2)-\sqrt{4-r^2}\big)\\ \phantom{0} \hspace{30pt} \mbox{for $-2\leq r \leq 2\,$,}\\[6pt] 0\hspace{30pt} \mbox{for $r\geq 2\,$.} \end{array}\right.\end{aligned}$$ Thus under volume constraint the TLK model has two facets, one with slope $1$, the other one with slope $0$, joined by a rounded piece. The upper facet edge is located at $r=0$. It has zero curvature. Expanding near $r=0$ results in $h_{\mathrm{eq}}(r,t) \cong -\frac2{3\pi} (-r)^{3/2}$, consistent with the Pokrovsky-Talapov law. With the exact result (\[b\]) it becomes possible to refine the resolution. The appropriate step size is $\lambda^{1/3}$ lattice constants. For the step density $\rho_\lambda(j)=\langle a_j^\dagger a_j\rangle_\lambda$ close to $r=0$ one finds $$\label{d} \lim_{\lambda\to\infty}\lambda^{1/3}\rho_\lambda(\lambda^{1/3}x) =-x{\mathrm{Ai}}(x)^2+{\mathrm{Ai}}'(x)^2\,,$$ $\mathrm{Ai}$ the Airy function. (\[d\]) has the asymptotics $$\begin{aligned} \label{e} & & \frac{1}{\pi}\sqrt{|x|} \quad \mathrm{for} \quad x\to - \infty\,, \\ & & \frac{1}{8 \pi x}\exp(-4x^{3/2}/3) \quad \mathrm{for}\quad x\to \infty\,. \nonumber\end{aligned}$$ Our real interest are the border ledge fluctuations. Clearly the border ledge is the top fermionic world line which we denote by $b_\lambda(t)$. $b_\lambda(t)$ takes integer values and is piecewise constant with unit size kinks. Since, at fixed $t$, the steps in the bulk have approximately the same statistics as a GUE random matrix, one would expect that the transverse fluctuations of the border ledge equal those of the largest eigenvalue. Indeed, using the fermionic transfer matrix combined with an asymptotic analysis [@PS], one finds that $$\label{g} \lim_{\lambda\to\infty} {\mathrm{Prob}}(\{b_\lambda(0)\leq\lambda^{1/3}a\})=F_2(a),\quad a\in\mathbbm{R}\, .$$ In the random matrix community $F_2(a)$ is known as the Tracy-Widom distribution [@TW]. The corresponding probability density $\textrm{d}F_2(a)/\textrm{d}a$ has an upper tail as $\exp(-\frac43 a^{3/2})$ and a lower tail as $\exp(-\frac{1}{12}|a|^3)$. In our context an experimentally more accessible quantity is the ledge wandering $\langle[b_\lambda(t)-b_\lambda(0)]^2\rangle$. In the limit of large $\lambda$ it has been computed in [@PS] with the result $$\label{j} \text{Var}\big(b_\lambda(t)-b_\lambda(0)\big)\cong\lambda^{2/3}g(\lambda^{-2/3}t),$$ using the short-hand $\text{Var}(X)=\langle(X-\langle X\rangle)^2\rangle$. Thus the transverse fluctuations are on the scale $\lambda^{2/3}$. For small $s$ the scaling function $g(s)$ is linear in $s$, $g(s)\simeq 2|s|$, indicating that for small, on the scale $\lambda^{2/3}$, separations the border ledge has random walk statistics. On the other hand $g(s)$ saturates for large $s$, $g(s)\simeq g(\infty)-c/s^2$, reflecting that the border ledge fluctuations are stationary (on the scale $\lambda^{2/3}$). For the leading term one finds $g(\infty)=\lim_{\lambda\to\infty}\lambda^{-2/3}2 \langle b_\lambda(0)^2\rangle = 1.6264$. The subleading coefficient $c$ has recently been derived in [@AM; @Wid] with the result $c=2$. Within the volume-constrained TLK model we arrived at an interesting prediction for the border ledge fluctuations. To be convincing we have to check against a more realistic model, for which we take the three-dimensional Ising model at low temperatures. At fixed crystal volume the equilibrium shape is then a cube with rounded corners. Taken literally this model is still too complicated and we simplify through an SOS-type approximation by allowing only atomic configurations which are lattice convex. This means that, when the crystal is cut along any line parallel to the major axes, then the atoms fill a single interval (no holes). We use translation invariance to choose our coordinate system in such a way that the crystal lies in the positive octant of $\mathbb{Z}^3$ with three of the facets coinciding with parts of the planes spanned by the three coordinate axes. If one now restricts attention to the piece of the crystal close to the origin, then the actual crystal shape can be represented by a height function $h(i,j)$, where $i$, resp. $j$, refers to the (100), resp. (010), axis. By construction\ [*(i)*]{}:$\,\,\, h(i,j) \geq 0$,\ [*(ii)*]{}: $\,h(i,j) \geq h(i+1,j)\,, \quad h(i,j)\geq h(i,j+1)$.\ The number of atoms missing, relative to the perfect cube, is $$V(h) = \sum _{i,j \geq 0} h(i,j)\,.$$ Thus the volume constraint translates into\ [*(iii)*]{}: $\,\,V(h) = const\,.$\ Every atomic configuration satisfying $(i) - (iii)$ has the same number of broken bonds and thus the same energy. Therefore our simplified version of the 3D Ising equilibrium droplet is to allow only atomic configurations which have a height function satisfying $(i) - (iii)$ and to give them equal statistical weight. Note that our model is purely entropic. Fig. \[fig1\] shows a typical sample with $V(h) = 3 \times 10^5$. By projecting along the (111) direction the Ising corner model is equivalent to tilings of the plane with rhombi of three distinct orientations. In this version the surface tension is computed in [@Wu]. According to the Andreev construction the Legendre transform of the surface free energy yields the equilibrium crystal shape [@A; @BH]. Convenient formulas are available in [@CK], where it is also established that, for the constraint $V(h) = N$, in the limit $N \to \infty$ with corresponding lattice spacing $N^{-1/3}$ the equilibrium crystal shape is attained with probability one. From the implicit formula for the shape it can be deduced that near facet edges the Pokrovsky-Talapov law holds. The facet edge can be computed explicitly. If we consider the facet which lies in the (001) plane and denote the coordinate along (100) by $\tau$ and the one along (010) by $b_{\infty}$, then the macroscopic facet edge is given by $b_\infty(\tau)=-\ln(1-e^{-\tau})$, $\tau > 0$. As observed in [@OR], the 3D Ising corner can be analyzed through fermionic techniques. In particular, one can study the border ledge fluctuations. The details are rather intricate and given elsewhere [@FS]. Here we only report on those results which allow us to gain some understanding of the universal properties of ledge fluctuations. We introduce the scaling parameter $\ell$ by $N=\frac14\zeta(3)\ell^3$, $\zeta(3)=1.202\dots$ being Apery’s constant. The atomic border ledge position in the (001) plane is given by $b_\ell(x)$, $x = 0,1,2,\ldots\,$. $b_\ell(x)$ takes positive integer values and is decreasing as $b_\ell(x+1) \leq b_\ell(x)$. The ledge has only South and East turns and meanders close to its asymptotic mean $\ell b_\infty(x/\ell)$. We zoom at the fixed macroscopic edge point $(\ell \tau,\ell b_{\infty}(\tau))$, $ \tau >0$. Upon proper rescaling [@FS], one recovers exactly the same statistics as in (\[g\]). More precisely, for large $\ell$, $$\label{l} \text{Var}\big(b_\ell(\ell \tau +x)-b_\ell(\ell \tau)\big)\cong \left(\tfrac12 A\ell\right)^{2/3} g\Big(\frac{A^{1/3}}{2^{1/3}\ell^{2/3}}x\Big),$$ with $A=b_\infty''(\tau)$. (\[l\]) differs from (\[j\]) in two respects. Firstly, to obtain the border ledge fluctuations one has to subtract the systematic mean. Since for our particular model the macroscopic facet edge is explicit, the subtraction is $\langle b_\ell(\ell \tau +x)\rangle - \langle b_\ell(\ell \tau)\rangle \cong b_{\infty}'(\tau) x + \frac12 b_\infty''(\tau) \ell^{-1} x^{2}$ with negligible higher order corrections. Secondly, model-dependent properties enter indirectly through the coefficient $A$. Since $g(s) = 2|s|$ for small $|s|$, $\text{Var}\big(b_\ell(\ell \tau +x) -b_\ell(\ell \tau)\big)=A|x|$, $A$ can be identified with the local wandering, resp. diffusion, coefficient. The border ledge of the TLK model and the 3D Ising corner have the same scaling behavior, which suggests the scaling to hold in greater generality. To obtain the form which properly distinguishes between model-dependent and universal properties we have to rely on a few notions from the thermodynamics of equilibrium crystal shapes [@AA]. Let us denote by $h(x,y)$ the height of a vicinal surface relative to the high symmetry reference plane. We find it convenient to measure $h$ in number of atomic layers, whereas $x,y$ are measured in a suitable macroscopic unit. Thus $h$ is dimensionless and $x,y$ have the dimension $[length]$. Further let $k_{B}T f({\bf u})$ be the surface free energy per unit projected area depending on the local slope ${\bf u}=\nabla h$. Below the roughening transition $f$ has a cone at ${\bf u}=0$ and for small ${\bf u}$ behaves as $$\label{m} f({\bf u})\cong \gamma(\theta) |{\bf u}| + B(\theta) |{\bf u}|^3$$ with $\theta$ the polar angle of ${\bf u}$ [@GM]. The line stiffness $\widetilde \gamma$ is defined through $\widetilde\gamma(\theta) = \gamma(\theta)+\gamma''(\theta)$. As argued in [@AAY], for short range surface models the Gaussian curvature of the equilibrium crystal shape has a universal jump across the facet edge, which implies the relation $\widetilde\gamma(\theta) B(\theta) = \pi^2/6$. Let us denote by $\widehat f$ the Legendre transform of $f$. If $\int dx dy f(\nabla h(x,y))$ is minimized under the constraint of fixed volume, then the resulting equilibrium surface is given by $h(x,y)=\ell \widehat f(\ell^{-1} x,\ell^{-1} y)$, where $\ell$ is the Lagrange multiplier adjusted so to give the correct volume. $h$ is convex downwards and has a convex facet lying in the $x$-$y$ plane. The facet boundary is determined by $\gamma(\theta)$ alone. Close to the facet edge, $h\cong -\frac{2}{3} \gamma_{PT} d^{3/2}$ with $d$ the normal distance to the facet edge, which defines the Pokrovsky-Talapov coefficient $\gamma_{PT}$. Under Legendre transformation the angle $\theta$ becomes the angle between the $x$-axis and the outer normal to the facet and, correspondingly, $\gamma_{PT}$, the local curvature $\kappa$, and the distance $r$ of a point on the edge to the origin are parametrized through this angle $\theta$. The relationship between $\widetilde\gamma$ and $B$ implies $$\label{n} \gamma_{PT}^2 \kappa = 2 \ell^{-2} \pi^{-2}.$$ We return to the border ledge fluctuations close to a given angle $\theta_0$. For this purpose it is convenient to center the $x$-$y$ axis coordinate system at $r(\theta_0)$ with the $x$-axis tangential and the $y$-axis along the inner normal to the facet. In this frame, we denote by $y=b(x)$ the fluctuating border step. Then $\langle b(x) \rangle = \frac{1}{2} \kappa(\theta_0) x^2$, in approximation. For sufficiently small $|x|$, still large on the scale of the lattice, $b(x)$ is like a random walk and $\textrm{Var}\big(b(x)-b(0)\big) \cong \sigma^2 |x|$, which defines the local wandering coefficient $\sigma^2$. Following [@AA] it is natural to equate $\sigma(\theta)^2$ with the inverse stiffness $\widetilde \gamma(\theta)^{-1}$. This implies $$\label{o} \sigma^2=\kappa \ell, \qquad \kappa=\pi^2\gamma_{PT}^2 \sigma^4/2$$ valid for any point on the facet edge. The general scaling form is obtained now by using the TLK model as benchmark. Locally the border ledge performs a random walk with nearest neighbor hopping rate 1, see (\[a\]), thus $\sigma^2 = 2$. From (\[c\]) the PT coefficient is $\gamma_{PT} = 1/\pi\sqrt{\ell}$ in our units. Using these two as model-dependent parameters yields the scaling form $$\label{p} \text{Var}\big(b(x)-b(0)\big)\cong (\pi \gamma_{PT})^{-4/3} g\big((\pi\gamma_{PT})^{4/3}\sigma^{2} x/2\big).$$ Of course, through (\[n\]), (\[o\]), any other pair of model-dependent parameter can be used to reexpress (\[p\]). As a control check, the Ising corner must also satisfy (\[p\]). This is indeed the case with coefficients $\kappa= \ell^{-1} \sigma^2$, $\sigma^2= b''_\infty(\tau)(1+b'_\infty(\tau)^2)^{-3/2}$, and . To summarize, the border ledge of a facet has fluctuations of size $\ell^{1/3}$, thus much reduced in comparison with a simple random walk. We claim that the scaling form (\[p\]) is universal within the class of surface models with short range interactions. The scaling function $g$ can be expressed through determinants of infinite dimensional matrices. Short and long distance behavior is known explicitly. In (\[p\]) there are two material parameters. 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[Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras]{} 1em Osamu Iyama and Idun Reiten 1em[ We say that an algebra $\Lambda$ over a commutative noetherian ring $R$ is Calabi-Yau of dimension $d$ ($d$-CY) if the shift functor $[d]$ gives a Serre functor on the bounded derived category of the finite length $\Lambda$-modules. We show that when $R$ is $d$-dimensional local Gorenstein the $d$-CY algebras are exactly the symmetric $R$-orders of global dimension $d$. We give a complete description of all tilting modules of projective dimension at most one for 2-CY algebras, and show that they are in bijection with elements of affine Weyl groups, preserving various natural partial orders. We show that there is a close connection between tilting theory for 3-CY algebras and the Fomin-Zelevinsky mutation of quivers (or matrices). We prove a conjecture of Van den Bergh on derived equivalence of non-commutative crepant resolutions.]{} 1em 0\. Introduction 1\. Cluster algebras and Calabi-Yau conditions 2\. Preliminaries on module-finite algebras 3\. Calabi-Yau algebras and symmetric orders 4\. Construction of tilting modules 5\. Mutation on tilting modules 6\. 2-Calabi-Yau algebras and affine Weyl groups 7\. 3-Calabi-Yau algebras and cluster algebras 8\. Non-commutative crepant resolutions 1.5em[**. Introduction**]{} Let $\Lambda$ be an algebra over a commutative noetherian ring $R$, which is finitely generated as an $R$-module. Sometimes we assume that $R$ in addition satisfies one or more of the following conditions: local, complete, Gorenstein, normal. This paper deals mainly with algebras $\Lambda$ which are Calabi-Yau of dimension $d$, called $d$-CY algebras for short. This means that the shift functor $[d]$ gives a Serre functor on the bounded derived category of the finite length $\Lambda$-modules. The main aspects of $d$-CY algebras which we investigate are the following. We deduce interesting properties of $d$-CY algebras of a ring theoretic and module theoretic nature. In particular, we show that when $R$ is $d$-dimensional local Gorenstein, the $d$-CY algebras are exactly the symmetric $R$-orders of global dimension $d$. A central part of our investigations deals with tilting modules, mainly those of projective dimension at most one. We give a complete description of all such tilting modules for 2-CY algebras, and show that they are in bijection with elements of affine Weyl groups, preserving various natural partial orders. We also investigate tilting theory for 3-CY algebras, where we show that there is a close connection with the Fomin-Zelevinsky mutation of quivers (or matrices), which they introduced in connection with their definition of cluster algebras. Tilting theory for 3-CY algebras also turns out to have a close connection with the theory of non-commutative crepant resolutions of Van den Bergh, and we prove some of our results in this more general setting (actually for a generalization of the definition of Van den Bergh). In particular, we prove a conjecture of Van den Bergh on derived equivalence of non-commutative crepant resolutions in . Main examples of $d$-CY algebras are skew group algebras $S*G$, where $S$ is the formal power series ring in $d$ variables over a field $K$ of characteristic zero and $G$ is a finite subgroup of $\SL_d(K)$. For the 2-dimensional case, these algebras have been a major object in several branches of mathematics. The invariant ring $S^G$ has only finitely many indecomposable (maximal) CM (=Cohen-Macaulay) modules, and $S*G$ is the endomorphism ring of an additive generator $S$ of the category $\cm S^G$ of (maximal) CM $S^G$-modules satisfying $\gl S*G=2$ \[He\]\[A3\]. Consequently, the Auslander-Reiten quiver of $S^G$ coincides with the quiver of $S*G$, and also with the McKay quiver of $G$, and they are the double of an extended Dynkin diagram \[A3\]. This is closely related to the minimal resolution $X$ of the quotient singularity $S^G$. A one-one correspondence between indecomposable non-free CM $S^G$-modules and irreducible exceptional curves of $X$ was constructed in \[AV\]. More strongly, by the McKay correspondence \[Mc\] explained geometrically by Gonzalez-Sprinberg and Verdier \[GV\], the McKay graph of $G$ coincides with the dual graph of irreducible exceptional curves of $X$. Later, Kapranov and Vasserot \[KV\] reformulated McKay correspondence in terms of derived equivalences between $S*G$ and $X$. The tilting modules we construct for 2-CY algebras, and hence for the algebras $S*G$ in dimension 2, give autoequivalences of $\dd^{\b}(\mod(S*G))$, and correspond to twist functors on $\dd^{\b}(\Coh X)$ constructed by Seidel-Thomas \[ST\] under McKay correspondence. It is natural to ask about the geometric meaning of the tilting modules for arbitrary $d$-CY algebras (including $S*G$) which we construct in section . Also the category $\cm S^G$ for the case $d>2$ is studied in \[I3,4\]\[IY\]. Although our study of Calabi-Yau algebras in this paper is done from a purely algebraic viewpoint, it is closely related to the study of the usual ‘geometric’ Calabi-Yau varieties. By using the method of Fourier-Mukai transformations, Bridgeland \[Bri1\] proved a conjecture of Bondal and Orlov \[BO\], which states that any birational 3-dimensional Calabi-Yau varieties are derived equivalent. It was motivated by Bridgeland’s proof and also by the 3-dimensional McKay correspondence due to Bridgeland-King-Reid \[BKR\] that Van den Bergh \[Va1,2\] introduced the concept of non-commutative crepant resolutions (NCCR for short), and gave a new proof of Bridgeland’s theorem. A typical example of NCCR is given again by skew group rings. It was in this connection that Van den Bergh proposed an analogue of a conjecture of Bondal and Orlov, i.e. all crepant resolutions (including NCCR) of a normal Gorenstein domain are derived equivalent, and proved this conjecture for 3-dimensional terminal singularities. In section , we prove the non-commutative part of this conjecture for the 3-dimensional case in a more general form, i.e. all NCCR of a module-finite algebra over a 3-dimensional normal Gorenstein domain are derived equivalent. Our method using tilting modules is surprisingly simple. There are here also interesting relationships with the maximal 1-orthogonal modules of \[I3,4\] and the maximal rigid modules of \[GLS\]. In the philosophy of non-commutative algebraic geometry, we regard algebras with finite global dimension as an algebraic/non-commutative analogue of smooth varieties. Although there is a classical Cohen’s structure theorem for regular algebras in the commutative situation, finiteness of global dimension seems not to be sufficient in the non-commutative situation. Known successful theory of algebras with low global dimension is obtained by Reiten and Van den Bergh \[RV1,2\], Artin and Schelter \[AS\], and Rump \[Rum1,2\] by giving additional assumptions, e.g. order, Auslander condition, and so on. We hope that our CY algebras will provide a source of non-commutative regular algebras. The theory of tilting modules originated in the representation theory of finite dimensional algebras, where a substantial theory has been developed around this concept. There have also been generalizations to other classes of algebras, but not many concrete examples of tilting modules beyond the finite dimensional case. When $T'$ is an almost complete tilting $\Lambda$-module, then there are up to isomorphism at most two complements \[RS\]. For a finite dimensional algebra it is never the case that there are always two complements. However, this holds for $d$-CY algebras for $d=2,3$, and constitutes an important part of our investigations. In particular, it is an essential property for trying to model the Fomin-Zelevinsky mutation of quivers of 3-CY algebras using tilting $\Lambda$-modules. A similar type of program was carried out for acyclic cluster algebras, leading to the notions of cluster categories and cluster tilted algebras \[BMRRT\]\[BMR1\] (see \[CCS\] for the $A_n$ case), along with related work in \[GLS\]. Actually, possible connections with cluster algebras was one of the motivations for investigating tilting theory for 3-CY algebras. Another interesting property of tilting theory for 3-CY algebras is that an equivalence of derived categories induced by a sequence of equivalences determined by tilting modules is actually induced by a tilting module. This is usually not the case for finite dimensional algebras. The tilting theory for 2-CY algebras is also quite interesting from a very different point of view. We give a description of the tilting modules (of projective dimension at most 1), which all turn out to be ideals. The 2-CY algebras have valued quivers which are given by generalized extended Dynkin diagrams. We establish a bijection between the tilting modules and the elements of the associated affine Weyl groups. There is in general a partial order on the tilting modules \[RS\], and in addition there is in this case the reverse inclusion order of ideals. Under our bijection they correspond to the right order and the Bruhat order on the affine Weyl group. We now describe the content of each section. We start with some background material and motivation in section , with particular emphasis on the importance of 2-dimensional and 3-dimensional Calabi-Yau properties for modelling quiver mutation and other essential ingredients in the definition of cluster algebras. In section  we collect basic results on some central concepts from commutative algebra, like dimension and depth, along with properties of reflexive and derived equivalences. We characterize the $d$-CY algebras as symmetric orders of global dimension $d$ in section , and investigate a related class which we call $d$-CY$^-$ algebras, which coincides with the class of symmetric orders. In section  we start with the tilting $\Lambda$-module $\Lambda =\bigoplus_{i=1}^nP_i$, where the $P_i$ are non-isomorphic indecomposable projective $\Lambda$-modules. We describe the indecomposable objects in the bounded derived category which can replace $P_i$ to give a tilting complex, and in particular a tilting module. Our emphasis is on CY algebras. In section  we give some basic properties of tilting theory for module-finite algebras, most of which are analogs of properties of finite dimensional algebras, and give stronger results for complements of almost complete tilting modules over CY algebras. In section  we investigate the structure of tilting modules for 2-CY algebras and their connection with affine Weyl groups. In section  we specialize to 3-CY algebras, and investigate the connection between Fomin-Zelevinsky mutation and tilting theory. The connection between tilting modules, reflexive modules and the non-commutative crepant resolutions of Van den Bergh is discussed in section . Parts of the results in this paper have been presented at conferences in Beijing, Banff, Tokushima, Hanamako, Trieste, Hannover and Oberwolfach. There is related work on various aspects on CY algebras by for example Berger-Taillefer \[BT\], Bocklandt \[Boc\], Braun \[Bra\], Brown-Gordon-Stroppel \[BGS\], Chuang-Rouquier \[CRo\], Ginzburg \[G\], Rickard, Van den Bergh \[Va3\]. .5em[**Notation** ]{} For a noetherian ring $\Lambda$, we denote by $J_\Lambda$ the Jacobson radical of $\Lambda$. By a module we mean a left module. We denote by $\Mod\Lambda$ the category of $\Lambda$-modules, by $\mod\Lambda$ the category of finitely generated $\Lambda$-modules, and by $\flmod\Lambda$ the category of $\Lambda$-modules of finite length. These categories are abelian. Moreover, we denote by $\Pr\Lambda$ the category of projective $\Lambda$-modules, and by $\pr\Lambda$ the category of finitely generated projective $\Lambda$-modules. For an additive category $\cc$, we denote by $J_{\cc}$ the Jacobson radical of $\cc$. For a commutative noetherian ring $R$ we denote by $\Spec R$ the set of prime ideals of $R$, and by $\Max R$ the set of maximal ideals of $R$. For $\dn{p}\in\Spec R$, we denote by $R_{\dn{p}}$ the localization of $R$ at $\dn{p}$, and by $\widehat{R}_{\dn{p}}$ the completion of $R$ at $\dn{p}$. For $X\in\Mod R$, we put $X_{\dn{p}}:=X\otimes_RR_{\dn{p}}$ and $\widehat{X}_{\dn{p}}:=X\otimes_R\widehat{R}_{\dn{p}}$. We denote by $\Supp{}_RX$ the support of $X$. When $R$ is a local ring the maximal ideal $\dn{p}$, we often write $\widehat{X}:=\widehat{X}_{\dn{p}}$. We denote by $(-)^*$ the functor $\hom_R(-,R):\mod R\to\mod R$. .5em[Acknowledgements ]{} In addition to the work of Iyama \[I1,2\] and Van den Bergh \[Va1,2\], some of this work was inspired by the 2004 Edinburgh meeting on “derived categories, quivers and strings” organized by A. King and M. Douglas. The second author would like to thank the participants of the conference, in particular D. Berenstein, T. Bridgeland, J. Chuang, B. Keller, A. King, J. Rickard, R. Rouquier and M. Van den Bergh for interesting conversations. A major part of this work was done while the first author visited Trondheim in February 2005, with improvements during his visit in March 2006. He would like to thank the people at NTNU for hospitality and stimulating discussions. He also would like to thank R.-O. Buchweitz, A. Ishii, B. Keller, H. Krause, J. Miyachi, R. Takahashi, J. Rickard, R. Rouquier, W. Rump, T, Shoji, K. Yoshida and Y. Yoshino for valuable comments. 1.5em[**. Cluster algebras and Calabi-Yau conditions**]{} In this section we recall some basic facts on cluster algebras, and discuss the relevance of Calabi-Yau type conditions for being able to model some of the concepts from the definition of a cluster algebra in a categorical/module theoretical way. We recall the basic definitions from \[FZ1\] in a generality suitable for our purpose. We give the definitions in terms of quivers rather than matrices. Let $\qq$ be a finite quiver with no loops. If $\qq$ has oriented cycles of length two, we can associate with $\qq$ a quiver $\bar{\qq}$ obtained by removing all pairs $\def\arraystretch{.2}\left.\begin{array}{c}\longrightarrow\\ \longleftarrow\end{array}\right.$. Let $F=\qqq(x_1,\cdots, x_n)$ where $\qqq$ denotes the rational numbers and $x_1, \cdots, x_n$ are indeterminates, and let $\qq$ be a finite quiver with $n$ vertices and no oriented cycles of length at most two. The pair $(\underline{x},\qq)$ where $\underline{x} =\{x_1, \cdots, x_n\}$ is called a seed. Let $b_{ij}$ denote the number of arrows from $i$ to $j$ in $\qq$, interpreted as minus the number of arrows from $j$ to $i$ in $\qq$ if $b_{ij} <0$. For each $k=1, \cdots, n$, the mutated quiver $\qq'=\mu_k(\qq)$ is defined as follows, where now $b_{ij}'$ denotes the number of arrows from $i$ to $j$ in $\qq'$: - $b_{ik}' = -b_{ik}$ and $b_{kj}' = -b_{kj}$. - If $b_{ik},\ b_{kj}>0$ (resp. $b_{ik},\ b_{kj}<0$), then $b_{ij}'=b_{ij}+b_{ik}b_{kj}$ (resp. $b_{ij}'=b_{ij}-b_{ik}b_{kj}$). - $b_{ij}' =b_{ij}$ otherwise. There is also defined a mutation $\mu_k(\underline{x},\qq)=(\underline{x}',\qq')$ of seeds, where\ $\underline{x}'=\{x_1,\cdots,x_k',\cdots,x_n\}$ and $x_kx_k'=\prod_{b_{ik}>0}x_i^{b_{ik}} +\prod_{b_{ik}<0}x_i^{-b_{ik}}$ Starting with a seed $(\underline{x},\qq)$ and applying sequences of mutations in all directions, we obtain a collection of seeds. The *clusters* are by definition the $n$ element subsets $\underline{x},\ \underline{x}',\cdots$, and the *cluster variables* are the union of the elements in the clusters. The associated cluster algebra is the subring of $F$ generated by the cluster variables. Mutation of quivers is defined without reference to any clusters or cluster variables. It is an interesting problem to identify classes of quivers where we can ‘lift’ mutation of quivers to ‘mutation’ of algebras associated with the quivers. Secondly, we would want a more global model, by finding some category $\mathcal{C}$ with a special type of objects being the anolog of clusters, and where the indecomposable summands are the anolog of cluster variables. The relevant objects $T$ in $\mathcal{C}$ should have a direct sum decomposition $T=\bigoplus_{i=1}^nT_i$ into $n$ non-isomorphic indecomposable summands, where $n$ is the number of vertices in the quiver. For each $k=1,\cdots,n$ there should be a unique indecomposable object $\check{T}_k\ {\not\simeq}\ T_k$ such that $T'=T/T_k \oplus\check{T}_k$ is an object of the relevant type. Also there should be some nice way of connecting $T_k$ and $\check{T}_k$, and the endomorphism algebras $\endm_{\mathcal{C}}(T)$ and $\endm_{\mathcal{C}}(T')$ should be related via ‘mutation’ of algebras. Inspired by the connection with quiver representations given in \[MRZ\], such a program was carried out for acyclic cluster algebras in \[BMRRT\], \[BMR1,2\]. A cluster algebra is said to be acyclic if the quiver in some seed has no oriented cycles. Here the global approach was done first, and the crucial category was the cluster category ${\mathcal C}_{\qq}$ associated with a finite quiver $\qq$ without oriented cycles, via the path algebra $K{\mathcal Q}$ for a field $K$. The central objects $T$ in ${\mathcal C}_{\qq}$, called (cluster-)tilting objects, are induced by tilting modules over finite dimensional hereditary algebras in the derived equivalence class of $K{\mathcal Q}$. The associated endomorphism algebras $\endm_{{\mathcal C}_{\qq}}(T)$ are called cluster-tilted algebras \[BMR1\], and it was shown later that the passage from $\endm_{{\mathcal C}_{\qq}}(T)$ to $\endm_{{\mathcal C}_{\qq}}(T')$ provides a lifting of the corresponding quiver mutation \[BMR2\]. In addition to providing interesting connections with cluster algebras, including a framework for obtaining results on cluster algebras, the class of cluster-tilted algebras is also interesting in itself. The cluster category ${\mathcal C}_{\qq}$ associated to $H=K{\mathcal Q}$ is by definition the factor category ${\mathcal C}_{\qq}=\dd^{\b}(H)/ \tau^{-1}[1]$, where $\dd^{\b}(H)$ denotes the bounded derived category of the category of finitely generated $H$-modules, $\tau$ the AR- translation on $\dd^{\b}(H)$ and $[1]$ the shift functor. The category ${\mathcal C}_{\qq}$ is known to be triangulated \[K3\] and it has Serre functor $[2]$, since $\tau_{\cc}=[1]$ is an AR translation on ${\mathcal C}_{\qq}$. Hence it is Calabi-Yau of dimension $2$, that is, there is for $M$ and $N$ in ${\mathcal C}_{\qq}$ a functorial isomorphism $D(\hom(M,N)) \simeq \hom(N,M[2])$, where $D$ is the ordinary duality. That the (cluster-)tilting theory in ${\mathcal C}_{\qq}$ works very nicely, including unique exchange of an indecomposable summand of a tilting object in ${\mathcal C}_{\qq}$ to get another one giving rise to a new tilting object, is intimately related to the 2-CY property. Actually it is a consequence of the 2-CY property \[IY\] (see also \[I5\]). In order to prove lifting of the quiver mutation to the class of cluster-tilted algebras, an essential ingredient was proving reduction to three simple modules \[BMR2\]. To be able to lift quiver mutation in this case has to do with establishing a close relationship between arrows and relation spaces, in other words, between $\Ext^1$- and $\Ext^2$-groups (see \[Bon2\]. Here a 3-CY type property of such cluster-tilted algebras would be useful, since this would imply $D\Ext^2(S,S') \simeq \Ext^1(S',S)$ for simple modules $S$ and $S'$. Actually, it has been shown in \[KR\] that the cluster-tilted algebra $\Gamma=\endm_{\cc_{\qq}}(T)$ is Gorenstein of dimension at most 1, with the stable category $\underline{\cm}\Gamma$ of CM modules being 3-CY. There is a more general result starting with $\cc$ being 2-CY \[KR\]. So we see that having 2-CY and 3-CY type properties around is useful for modelling some central ingredients of cluster algebra theory. The quivers we deal with in this paper are coming from algebras satisfying some 3-CY conditions. They are certain algebras $\Lambda$ which are finitely generated as modules over a commutative noetherian ring of Krull dimension $3$, such that the bounded derived category $\dd^{\b}(\flmod\Lambda)$ is 3-CY. In this setting we are able to lift mutations of quivers to such 3-CY algebras. Also, under some additional assumptions, the relevant category $\cc$ is $\mod \Lambda$, or rather the subcategory $\ref \Lambda$ of reflexive $\Lambda$-modules with the relevant objects being the tilting $\Lambda$-modules of projective dimension at most one. In this case there is some trace of a 2-CY property associated with the projective resolution of simple modules. So in this case we have the ‘opposite’ of the case of the categories associated with acyclic cluster algebras; that we have 3-CY and a trace of 2-CY. When we have the 3-CY algebra $S*G$ where $S=K[[x,y,z]]$ with the field $K$ of characteristic zero and $G$ is a finite subgroup of $\SL_3(K)$ acting freely, there is here a close relationship to the stable category $\underline{\cm}R$ of (maximal) CM modules for $R=K[[x,y,z]]^{G}$. The category $\underline{\cm}R$ is 2-CY, as follows from work of Auslander \[A2;III\]. For if $R$ is a commutative complete local Gorenstein isolated singularity of dimension $d$, then the AR-translation $\tau$ is $\Omega^{2-d}$, which is $[d-2]$ in $\underline{\cm}R$, and hence $\underline{\cm}R$ is $(d-1)$-CY. More generally, the stable category of lattices over symmetric orders have the same property (see \[A2\] for definitions and results). Other classes of cluster algebras are investigated from the modelling point of view in \[GLS\]. Here they deal with the case ‘with coefficients’. But at the same time they obtain results on the ‘no coefficients’ case, which is here natural to compare with, from the point of view of CY-conditions. Let $\Lambda$ be the preprojective algebra of a Dynkin diagram and $\underline{\mod} \Lambda$ the stable category of the finitely generated $\Lambda$-modules. Here the AR-translation $\tau$ is $\Omega^{-1}$ (see \[AR2;3.1,1.2\]\[K3;8.5\]), so that $\underline{\mod}\Lambda$ is 2-CY. When $\Lambda$ is of finite representation type, $\underline{\mod} \Lambda$ is equivalent to a cluster category \[BMRRT\]. The special objects in $\underline{\mod}\Lambda$ are, like in the cluster category, objects $C$ maximal with the property $\Ext^1_{\underline{\mod}\Lambda}(C,C)=0$. Then $C$ can be lifted to $\Lambda\oplus C$ in $\mod \Lambda$, which has a similar property. In this case one also has the algebras $\Gamma = \endm_{\Lambda}(\Lambda\oplus C)$, in addition to the factor algebras $\underline{\endm}(C)$. They have global dimension 3, and have some trace of being 3-CY, as shown in \[GLS\]; see also \[KR\]. There are also other similarities between our work and that of \[GLS\], with respect to the role of tilting modules. 1.5em[**. Preliminaries on module-finite algebras** ]{} The main focus in this paper is on $d$-CY algebras and related algebras, especially for $d=2,3$. In this section we give some useful background material on the module theory for module-finite $R$-algebras, for a noetherian commutative ring $R$. First we discuss dimension, depth, global dimension and their relationships. Second, assuming that $R$ is in addition a normal domain, we discuss reflexive modules and reflexive equivalence of algebras, in particular that being a symmetric algebra is preserved under reflexive equivalence. Finally, we give some results on derived categories needed in section 3. Let $\Lambda$ be a module-finite $R$-algebra and $M\in\mod\Lambda$. Put $$\dim{}_RM:=\dim(R/\ann{}_RM)$$ where $\ann{}_RM$ is an annihilator of the $R$-module $M$. Since the value of $\dim{}_RM$ is independent of the choice of central subring $R$ of $\Lambda$, we denote it by $\dim M$. Now assume that $R$ is local. Put $$\depth{}_RM:=\inf\{i\ge0\ |\ \ext^i_R(R/J_R,M)\neq0\}.$$ Then $\depth{}_RM$ coincides with the maximal length of $M$-regular sequences \[Ma;16.7\]. By a result of Goto-Nishida \[GN2;3.2\], we have an equality $$\depth{}_RM=\inf\{i\ge0\ |\ \ext^i_\Lambda(\Lambda/J_\Lambda,M)\neq0\}.$$ In particular, the value of $\depth{}_RM$ is independent of the choice of central subring $R$ of $\Lambda$. Thus we denote it by $\depth M$. The following results will be quite useful. .5em[**Proposition  **]{}\[GN2;3.5\] *For any $M\in\mod\Lambda$, we have* $\depth M\le\dim M\le\id{}_\Lambda M=\sup\{i\in\zzz\ |\ \ext^i_\Lambda(\Lambda/J_\Lambda,M)\neq0\}$. In particular, we have $\depth\Lambda\le\dim \Lambda\le\gl\Lambda$. .5em[**Propostion  **]{}[*For $\Lambda$ as above, we have $\gl\Lambda=\sup\{\pd{}_\Lambda M\ |\ M\in\flmod\Lambda\}$.*]{} .5em[Proof ]{} We put $n:=\pd{}_\Lambda(\Lambda/J_\Lambda)$, and can clearly assume that $n<\infty$. We will show that $\pd{}_\Lambda M\le n$ holds for any $M\in\mod\Lambda$ by using induction on $\dim M$. If $\dim M=0$, then $M\in\flmod\Lambda$ holds, so we have $\pd{}_\Lambda M\le n$. Now assume that $\pd{}_\Lambda M\le n$ holds for any $M\in\mod\Lambda$ with $\dim M<m$, where $m>0$. Take any $M\in\mod\Lambda$ with $\dim M=m$. There is then an exact sequence $0\to L\to M\to N\to 0$ with $L\in\flmod\Lambda$ and $\depth N\ge1$. Since we have $\pd{}_\Lambda L\le n$ by assumption, we only have to show $\pd{}_\Lambda N\le n$. Take an $N$-regular element $r\in R$ and consider the exact sequence $0\to N\stackrel{r}{\to}N\to N/rN\to0$. Since $\dim (N/rN)<m$ holds, we have $\pd{}_\Lambda(N/rN)\le n$ by the induction assumption. Applying Nakayama’s lemma to the exact sequence $\ext^n_\Lambda(N,-)\stackrel{r}{\to}\ext^n_\Lambda(N,-)\to0$, we have $\pd{}_\Lambda N<n$. We see that $\pd{}_\Lambda M\le n$. ------------------------------------------------------------------------ .5em We call $M\in\mod\Lambda$ a [*Cohen-Macaulay*]{} ([*CM*]{} for short) [*$\Lambda$-module of dimension $n$*]{} if $\depth M=\dim M=n$. We simply call a CM $\Lambda$-module of dimension $d(=\dim R)$ a (maximal) [*CM $\Lambda$-module*]{}. We denote by $\cm\Lambda$ the category of CM $\Lambda$-modules. We call $\Lambda$ an [*$R$-order*]{}, or just an [*order*]{} if $\Lambda\in\cm\Lambda$. If $R$ has a canonical module $\omega_R$ (e.g. $R$ is Gorenstein and $\omega_R=R$), then $\depth M=d-\sup\{i\ge0\ |\ \ext^i_R(M,\omega_R)\neq0\}$ and $\dim M=d-\inf\{i\ge0\ |\ \ext^i_R(M,\omega_R)\neq0\}$ \[BH;3.5.11\]. Thus $M$ is CM of dimension $n$ if and only if $\ext^{d-i}_R(M,\omega_R)=0$ for any $i\neq n$. As  suggests, orders with $\gl\Lambda=d$ are very special. For example, we have the following Auslander-Buchsbaum type equality. .5em[**Proposition  **]{}[*Let $\Lambda$ be an order with $\gl\Lambda=d$. For any $M\in\mod\Lambda$, we have $\pd{}_\Lambda M+\depth M=d$.*]{} .5em[Proof ]{} Put $n:=\pd{}_\Lambda M$ and $t:=\depth M$. We have a projective resolution $0\to P_n\stackrel{f_n}{\to}\cdots\stackrel{f_1}{\to}P_0\stackrel{f_0}{\to}M\to0$. Put $M_i:=\Im f_i$ for $0\le i\le n$, then we have an exact sequence $0\to M_{i+1}\stackrel{}{\to}P_i\to M_i\to0$ with $P_i\in\cm\Lambda$. Applying $\hom_{\Lambda}(\Lambda/J_\Lambda,-)$, we have $\depth M_i\ge\depth M_{i+1}-1$. Thus we have $t=\depth M_0\ge \depth M_n-n=d-n$. On the other hand, take an $M$-regular sequence $(x_1,\cdots,x_t)$. Put $N_i:=M/(x_1,\cdots,x_i)M$ for $0\le i\le t$, then we have an exact sequence $0\to N_i\stackrel{x_{i+1}}{\to} N_i\to N_{i+1}\to0$. Applying $\hom_{\Lambda}(\Lambda/J_\Lambda,-)$, we have an exact sequence $\ext^j_\Lambda(\Lambda/J_\Lambda,N_i)\stackrel{x_{i+1}}{\to} \ext^j_\Lambda(\Lambda/J_\Lambda,N_i)\to\ext^{j+1}_\Lambda(\Lambda/J_\Lambda,N_{i+1}) \to\ext^{j+1}_\Lambda(\Lambda/J_\Lambda,N_{i+1})$ for any $j$. Using Nakayama’s Lemma, we have $\pd{}_\Lambda N_{i+1}=\pd{}_\Lambda N_i+1$. Consequently, we have $n=\pd{}_\Lambda N_0=\pd{}_\Lambda N_t-t\le d-t$. ------------------------------------------------------------------------ .5em Let $R$ be an arbitrary commutative noetherian ring. We call $M\in\mod\Lambda$ a [*CM $\Lambda$-module*]{} if $M_{\dn{p}}\in\cm\Lambda_{\dn{p}}$ for any $\dn{p}\in\Spec R$, and we denote by $\cm\Lambda$ the category of CM $\Lambda$-modules. We call $\Lambda$ an [*$R$-order*]{}, or just an [*order*]{} if $\Lambda\in\cm\Lambda$. We call $\Lambda$ a [*symmetric*]{} $R$-algebra if $\hom_R(\Lambda,R)$ is isomorphic to $\Lambda$ as a $(\Lambda,\Lambda)$-module. .5em Now assume that $R$ is a normal domain. We want to investigate reflexive equivalence, especially in connection with symmetric algebras. Recall that $(-)^*$ is the functor $\hom_R(-,R):\mod R\to\mod R$. We call $M\in\mod\Lambda$ a [*reflexive*]{} $\Lambda$-module if the natural map $M\to M^{**}$ (not $\hom_{\Lambda^{\op}}(\hom_\Lambda(M,\Lambda),\Lambda)$!) is an isomorphism. It is well known that $M\in\mod\Lambda$ is reflexive if and only if $M$ satisfies Serre’s S$_2$ condition $\depth{}_{R_{\dn{p}}}M_{\dn{p}}\ge\min\{2,\height\dn{p}\}$ for any $\dn{p}\in\Spec R$ \[EG;0.B,3.6\]. We denote by $\ref\Lambda$ the category of reflexive $\Lambda$-modules. Using the S$_2$ condition, one can easily check that $\ref\Lambda$ is closed under kernels and extensions. We have a functor $(-)^{**}=\hom_R(\hom_R(-,R),R):\mod\Lambda\to\ref\Lambda$ (e.g. (1) below). This gives a left adjoint of the inclusion functor $\ref\Lambda\to\mod\Lambda$. We say that two $R$-algebras $\Lambda$ and $\Gamma$ are [*reflexive equivalent*]{} if the additive categories $\ref\Lambda$ and $\ref\Gamma$ are equivalent. We call $M\in\ref\Lambda$ a [*height one generator*]{} (resp. [*progenerator*]{}, [*projective*]{}) if $M_{\dn{p}}$ is a generator (resp. a progenerator, projective) over $\Lambda_{\dn{p}}$ for any height one prime ideal $\dn{p}$ of $R$. The proposition below shows that many algebras which are not Morita equivalent may be reflexive equivalent. Part (2) is used in \[RV1\]. .5em[**Proposition  **]{} *(1) $\hom_\Lambda(X,Y)\in\ref R$ for any $X\in\mod\Lambda$ and $Y\in\ref\Lambda$.* \(2) Let $\Lambda$ and $\Gamma$ be $R$-algebras which are reflexive $R$-modules. .5em (i) For any height one progenerator $M\in\ref\Lambda$, we have an equivalence $\fff:=\hom_\Lambda(M,-):\ref\Lambda\to\ref\endm_\Lambda(M)$. .5em (ii) Let $\fff:\ref\Lambda\to\ref\Gamma$ be a categorical equivalence. Then there exists a height one progenerator $M\in\ref\Lambda$ such that $\Gamma\simeq\endm_\Lambda(M)$ and $\fff\simeq\hom_\Lambda(M,-)$. \(3) If $\Lambda$ is a symmetric $R$-algebra, then so is $\endm_\Lambda(M)$ for any height one projective $M\in\ref\Lambda$. Thus symmetric algebras are closed under reflexive equivalences. .5em[Proof ]{} (1) Take an exact sequence $\Lambda^n\to\Lambda^m\to X\to0$ in $\mod\Lambda$. Applying $\hom_\Lambda(-,Y)$, we obtain an exact sequence $0\to\hom_\Lambda(X,Y)\to Y^m\to Y^n$. Since $\ref R$ is closed under kernels, we have $\hom_\Lambda(X,Y)\in\ref R$. (2)(i) Let $\Gamma:=\endm_\Lambda(M)$ and consider the functor $\ggg:=\hom_\Gamma(\fff(\Lambda),-):\ref\Gamma\to\ref\Lambda$. We have homomorphisms $f:(\fff(\Lambda)\otimes_\Lambda M)^{**}=(\hom_\Lambda(M,\Lambda)\otimes_\Lambda M)^{**}\to\endm_\Lambda(M)^{**}=\Gamma^{**}=\Gamma$ and $g:(M\otimes_\Gamma\fff(\Lambda))^{**}=(M\otimes_\Gamma\hom_\Lambda(M,\Lambda))^{**}\to\Lambda^{**}=\Lambda$. Since $M$ is a height one progenerator, $f_{\dn{p}}$ and $g_{\dn{p}}$ are isomorphisms for any height one prime ideal $\dn{p}$ of $R$. Since $f$ and $g$ are homomorphisms between reflexive $R$-modules, they are isomorphisms. Using adjointness properties, we obtain [$$\begin{aligned} &\fff\circ\ggg=\hom_\Lambda(M,\hom_\Gamma(\fff(\Lambda),-))=\hom_\Gamma(\fff(\Lambda)\otimes_\Lambda M,-)=\hom_\Gamma((\fff(\Lambda)\otimes_\Lambda M)^{**},-)=1&\\ &\ggg\circ\fff=\hom_\Gamma(\fff(\Lambda),\hom_\Lambda(M,-))=\hom_\Lambda(M\otimes_\Gamma\fff(\Lambda),-)=\hom_\Lambda((M\otimes_\Gamma\fff(\Lambda))^{**},-)=1.&\end{aligned}$$]{} -1em Hence $\fff:\ref\Lambda\to\ref\Gamma$ is an equivalence. \(ii) Left to the reader. \(3) See \[I4;5.4.3(1)\], for example. For completeness, we give a proof here. By the same argument as in the proof of (2), we have an isomorphism $f:(\hom_\Lambda(M,\Lambda)\otimes_\Lambda M)^{**}\to\Gamma$. Thus we have isomorphisms [$$\Gamma^*\stackrel{f^*}{\simeq}(\hom_\Lambda(M,\Lambda)\otimes_\Lambda M)^*\stackrel{}{\simeq}(\hom_\Lambda(M,\Lambda^*)\otimes_\Lambda M)^*\simeq(M^*\otimes_\Lambda M)^*\simeq\hom_\Lambda(M,M^{**})\simeq\Gamma$$]{} of $(\Gamma,\Gamma)$-modules. ------------------------------------------------------------------------ .5em We want to recall some results on derived categories which will be useful in the next section. We start with basic notation and definitions. For the rest of the section $R$ is a commutative noetherian ring and $\Lambda$ is a module-finite $R$-algebra. For an additive category $\aa$, we denote by $\cc(\aa)$ the category of complexes over $\aa$, by $\kk(\aa)$ the [*homotopy category*]{} of $\aa$, and by $\dd(\aa)$ the [*derived category*]{} of $\aa$ provided $\aa$ is abelian \[Hap\]\[Har1\]. For $*=+,-$ or $\b$, we denote by $\cc^*(\aa)$ (resp. $\kk^*(\aa)$, $\dd^*(\aa)$) the full subcategory of $\cc(\aa)$ (resp. $\kk(\aa)$, $\dd(\aa)$) consisting of bounded below, bounded above or bounded complexes respectively. Moreover, for a full subcategory $\bb$ of $\aa$, we denote by $\cc^*_{\bb}(\aa)$ (resp. $\kk^*_{\bb}(\aa)$, $\dd^*_{\bb}(\aa)$) the full subcategory of $\cc^*(\aa)$ (resp. $\kk^*(\aa)$, $\dd^*(\aa)$) consisting of all objects $X$ such that the $i$-th homology $H^i(X)$ belongs to $\bb$ for any $i$. We have natural equivalences $\kk^-(\Pr\Lambda)\stackrel{\sim}{\to}\dd^-(\Mod\Lambda)$, $\kk^-(\pr\Lambda)\stackrel{\sim}{\to}\dd^-(\mod\Lambda)$ and $\dd^{\b}(\mod\Lambda)\stackrel{\sim}{\to}\dd^{\b}_{\mod\Lambda}(\Mod\Lambda)$. The next two results were pointed out to us by Rickard \[Ri2\]. .5em[**Lemma  **]{}[*For any $X\in\cc^{\b}_{\flmod\Lambda}(\mod\Lambda)$, there exists a quasi-isomorphism $X\to Y$ with $Y\in\cc^{\b}(\flmod\Lambda)$. Thus we have equivalences $\dd^{\b}(\flmod\Lambda)\stackrel{\sim}{\to}\dd^{\b}_{\flmod\Lambda}(\mod\Lambda)\stackrel{\sim}{\to}\dd^{\b}_{\flmod\Lambda}(\Mod\Lambda)$.*]{} .5em[Proof ]{} It follows from \[Ve;III.2\] that we only have to check the condition: ($E_2$)$^{\op}$ Let $X\in\mod\Lambda$ and $Y\in\flmod\Lambda$ a submodule of $X$. Then there exists a submodule $Z$ of $X$ such that $Y\cap Z=0$ and $X/Z\in\flmod\Lambda$. To see this, let $I$ be an arbitrary ideal of $R$. Since $R$ is noetherian, there exists $c>0$ such that $I^nX\cap Y=I^{n-c}(I^cX\cap Y)$ holds for any $n>c$ by the Artin-Rees Lemma \[Ma;8.5\]. Applying this to $I:=\ann{}_RY$, we have $I^{c+1}X\cap Y=I(I^cX\cap Y)=0$. Since $R/I$ is an artin ring, $Z:=I^{c+1}X$ satisfies the desired conditions. ------------------------------------------------------------------------ .5em In  we shall use the results below due to Rickard \[Ri2\]. We consider a descending chain $I_1\supset I_2\supset\cdots$ of ideals of $\Lambda$ such that $\Lambda^{(i)}:=\Lambda/I_i\in\flmod R$ for any $i\ge0$. Put $\widehat{\Lambda}:=\plim_{i\ge0}\ \Lambda^{(i)}$. Denote as usual by $\Lotimes$ (resp. $\Rhom$) the left (resp. right) derived functor of $\otimes$ (resp. $\hom$). .5em[**Proposition  **]{} *With the above notation, put $P^{(i)}:=\Lambda^{(i)}\Lotimes_\Lambda P$ and $\widehat{P}:=\widehat{\Lambda}\Lotimes_\Lambda P$ for $P\in \kk^{\b}(\pr\Lambda)$.* \(1) $\plim_{i\ge0}\hom_{\dd(\Mod\Lambda)}(X,P^{(i)})=\hom_{\dd(\Mod\Lambda)}(X,\widehat{P})$ for any $X\in \dd^-(\mod\Lambda)$. \(2) If $I_i=J_\Lambda^i$, then $\ilim_{i\ge0}\hom_{\dd(\Mod\Lambda)}(P^{(i)},X)=\hom_{\dd(\Mod\Lambda)}(P,X)$ for any $X\in\dd^{\b}(\flmod\Lambda)$. .5em[Proof ]{} Let $P$ be a bounded complex $\cdots\to P^{-1}\to P^0\to P^1\to\cdots$ in $\pr\Lambda$. Then $P^{(i)}$ is given by the bounded complex $\cdots\to(P^{-1})^{(i)}\to(P^{0})^{(i)}\to(P^{1})^{(i)}\to\cdots$ in $\pr\Lambda^{(i)}$. \(1) We can assume that $X$ is given by the bounded above complex $\cdots\to Q^{-1}\to Q^0\to Q^1\to\cdots$ in $\pr\Lambda$. Since $\plim\hom_\Lambda(Q^s,(P^{t})^{(i)})=\hom_\Lambda(Q^s,\widehat{P}^t)$ holds, we have an isomorphism of complexes [$$\begin{diag} \cdots&\to&\plim\prod_{-s+t=-1}\hom_\Lambda(Q^s,(P^{t})^{(i)})&\stackrel{}{\to}&\plim\prod_{-s+t=0}\hom_\Lambda(Q^s,(P^{t})^{(i)})&\to&\plim\prod_{-s+t=1}\hom_\Lambda(Q^s,(P^{t})^{(i)})&\to&\cdots\\ &&|\wr&&|\wr&&|\wr\\ \cdots&\to&\prod_{-s+t=-1}\hom_\Lambda(Q^s,\widehat{P}^t)&\stackrel{}{\to}&\prod_{-s+t=0}\hom_\Lambda(Q^s,\widehat{P}^t)&\to&\prod_{-s+t=1}\hom_\Lambda(Q^s,\widehat{P}^t)&\to&\cdots. \end{diag}$$]{} Since each inverse system $(\prod_{-s+t=n}\hom_\Lambda(Q^s,(P^{t})^{(i)}))_i$ consists of finite length $R$-modules, it satisfies the Mittag-Leffler condition (e.g. \[Har2;II.9\]). Thus the $0$-th homology of the upper sequence in the diagram above is isomorphic to $\plim\hom_{\dd(\Mod\Lambda)}(X,P^{(i)})$, and we obtain the desired isomorphism. \(2) We will show that the natural map $\ilim\hom_{\dd(\Mod\Lambda)}(P^{(i)},X)\to\hom_{\dd(\Mod\Lambda)}(P,X)$ is bijective. Fix any $f\in\hom_{\dd(\Mod\Lambda)}(P,X)$. By , we can write $f=gs^{-1}$ for $g\in\hom_{\kk^{\b}(\mod\Lambda)}(P,X^\prime)$ and a quasi-isomorphism $s\in\hom_{\kk^{\b}(\flmod\Lambda)}(X,X^\prime)$. It follows from $X^\prime\in\kk^{\b}(\flmod\Lambda)$ that $g$ factors through some $P^{(i)}$, and hence $f$ also factors through $P^{(i)}$. Thus the above map is surjective. We now show injectivity. Let $p:P\to P^{(i)}$ be the natural map. Assume that $f\in\hom_{\dd(\Mod\Lambda)}(P^{(i)},X)$ satisfies $pf=0$. Again we write $f=gs^{-1}$ for $g\in\hom_{\kk^{\b}(\mod\Lambda)}(P^{(i)},X^\prime)$ and a quasi-isomorphism $s\in\hom_{\kk^{\b}(\flmod\Lambda)}(X,X^\prime)$. Since $pg$ is null-homotopic, we can take a homotopy $a:P\to X^\prime[-1]$. It follows from $X^\prime\in\kk^{\b}(\flmod\Lambda)$ that $a$ factors through $P^{(j)}$ for some sufficiently large $j$. Then the composition of $P^{(j)}\to P^{(i)}$ and $g$ is null-homotopic. ------------------------------------------------------------------------ .5em The concept of [*derived equivalence*]{} is central for our work. Recall that $T\in\kk^{\b}(\pr\Lambda)$ is a [*tilting complex*]{} if $\hom_{\dd(\Mod\Lambda)}(T,T[i])=0$ for any $i\neq0$ and $T$ generates $\kk^{\b}(\pr\Lambda)$. If a $\Lambda$-module $T$ is a tilting complex, it is called a [*tilting module*]{}. We will mostly deal with tilting modules $T$ of projective dimension at most one. They satisfy the conditions (i) $\ext^1_\Lambda(T,T)=0$ and (ii) there exists an exact sequence $0\to\Lambda\to T_0\to T_1\to0$ with $T_i\in\add T$. Rickard proved in \[Ri1\] that the following conditions (1)–(4) are equivalent. \(1) (resp. (2), (3)) $\kk^{\b}(\pr\Lambda)$ and $\kk^{\b}(\pr\Gamma)$ (resp. $\dd^{\b}(\mod\Lambda)$ and $\dd^{\b}(\mod\Gamma)$, $\kk^-(\Pr\Lambda)$ and $\kk^-(\Pr\Gamma)$) are triangle equivalent. \(4) There exists a tilting complex $T\in\kk^{\b}(\pr\Lambda)$ such that $\Gamma\simeq\endm_{\dd(\Mod\Lambda)}(T)$. If these conditions are satisfied, we call $\Lambda$ and $\Gamma$ [*derived equivalent*]{}. We call $T$ in (4) above a [*two-sided tilting complex*]{} if $T\in\dd^{\b}(\mod\Lambda\otimes_{\zzz}\Gamma^{\op})$. We have the following relationship with localizations. .5em[**Lemma  **]{}[*Let $T\in\kk^{\b}(\pr\Lambda)$ be a tilting complex with $\Gamma:=\endm_{\dd(\Mod\Lambda)}(T)$. For any $\dn{p}\in\Spec R$, we have a tilting complex $T_{\dn{p}}\in\kk^{\b}(\pr\Lambda_{\dn{p}})$ with $\endm_{\dd(\Mod\Lambda_{\dn{p}})}(T_{\dn{p}})=\Gamma_{\dn{p}}$.*]{} .5em Recall that a $\Lambda$-module $T$ is said to be a [*partial tilting module*]{} if $\pd{}_\Lambda T\le1$ and $\ext^1_\Lambda(T,T)=0$. The following proposition is a generalization of Bongartz’s result \[Bon1\] for finite dimensional algebras. We sometimes call $X$ a [*Bongartz complement*]{} of $T$. .5em[**Lemma  **]{}[*For any partial tilting $\Lambda$-module $T$, there exists $X\in\mod\Lambda$ such that $T\oplus X$ is a tilting $\Lambda$-module. Moreover, if $R$ is normal and $T\oplus\Lambda\in\ref\Lambda$, then $X\in\ref\Lambda$.*]{} .5em[Proof ]{} Let $P\stackrel{f}{\to}\ext^1_\Lambda(T,\Lambda)\to0$ be exact, with $P$ projective in $\mod\endm_\Lambda(T)$. We can write $P=\hom_\Lambda(T,T^\prime)$ for $T'\in\add T$. It follows from Yoneda’s lemma on $\add T$ that $f$ is given by $\sigma\in\ext^1_\Lambda(T',\Lambda)$. Take an exact sequence $0\to\Lambda\to X\to T^\prime\to0$ corresponding to $\sigma$. Then $\hom_\Lambda(T,T^\prime)\stackrel{f=(\bullet\sigma)}{\to}\ext^1_\Lambda(T,\Lambda)\to0$ is exact and $\pd{}_\Lambda(T\oplus X)\le1$ holds. Applying $\hom_\Lambda(T,-)$, we see that $\ext^1_\Lambda(T,X)=0$. Applying $\hom_\Lambda(-,T\oplus X)$, we get $\ext^1_\Lambda(T\oplus X,T\oplus X)=0$. Thus $T\oplus X$ is a tilting $\Lambda$-module. If $R$ is normal, then $\ref\Lambda$ is closed under extensions. Thus the second assertion follows. ------------------------------------------------------------------------ .5em The following easy lemma is useful (e.g. \[HU2;1.2\], \[Ye2;2.3\]). .5em[**Lemma  **]{}[*Let $T$ be a tilting $\Lambda$-module with a minimal projective resolution $0\to P_1\to P_0\to T\to0$. Then $\add(P_0\oplus P_1)=\add\Lambda$ and $\add P_0\cap\add P_1=0$. In particular, if $\Lambda$ is Morita equivalent to a local ring, then $T$ is projective.*]{} .5em We will use the following canonical isomorphisms (e.g. \[F;pp.153\]), where we denote by $\Flat\Gamma$ the category of flat $\Gamma$-modules. .5em[**Lemma** ]{} *(1) $\Rhom_\Lambda(Y\Lotimes_\Gamma X,Z)\simeq\Rhom_\Gamma(X,\Rhom_\Lambda(Y,Z))$ for any $X\in\dd^-(\Mod\Gamma)$, $Y\in\dd^-(\Mod\Lambda\otimes_R\Gamma^{\op})$ and $Z\in \dd^+(\Mod\Lambda)$.* \(2) $\Rhom_\Lambda(X,Y)\Lotimes_\Gamma Z\simeq\Rhom_\Lambda(X,Y\Lotimes_\Gamma Z)$ for any $X\in\dd^-(\mod\Lambda)$, $Y\in \dd^{\b}(\Mod\Lambda\otimes_R\Gamma^{\op})$ and $Z\in\dd^{\b}(\Mod\Gamma)$, provided $X\in\kk^{\b}(\pr\Lambda)$ or $Z\in\kk^{\b}(\Flat\Gamma)$. 1.5em[**. Calabi-Yau algebras and symmetric orders**]{} Throughout this section, let $R$ be a commutative noetherian ring with $\dim R=d$ and $\Lambda$ a module-finite $R$-algebra. We denote by $E(X)$ the injective hull of $X\in\mod R$ and put $E:=\bigoplus_{\dn{p}\in\Max R}E(R/\dn{p})$. Then $E$ is an injective $R$-module, and we have a duality [$$D:=\hom_R(-,E):\flmod R\to\flmod R\ \ \ (\mbox{resp. }\flmod\Lambda\to\flmod\Lambda^{\op},\ \dd^{\b}(\flmod R)\to\dd^{\b}(\flmod R),\ \dd^{\b}(\flmod\Lambda)\to\dd^{\b}(\flmod\Lambda^{\op}))$$]{} called [*Matlis duality*]{}, such that $D\circ D$ is isomorphic to the identity functor \[BH;3.1.3,3.2.13\]. For example, if $R$ is a polynomial or power series ring over a field $K$, then $D$ is isomorphic to $\hom_K(-,K)$. Obviously, $X\in\Mod\Lambda$ belongs to $\flmod\Lambda$ if and only if it belongs to $\flmod R$ as an $R$-module. For $X,Y\in\dd^{\b}(\mod\Lambda)$, $\hom_{\dd(\Mod\Lambda)}(X,Y)$ belongs to $\flmod R$ if $X$ or $Y$ is in $\dd^{\b}(\flmod\Lambda)$. For an integer $n$, we call $\Lambda$ [*$n$-Calabi-Yau*]{} ([*$n$-CY*]{} for short) if there exists a functorial isomorphism $$\hom_{\dd(\Mod\Lambda)}(X,Y[n])\simeq D\hom_{\dd(\Mod\Lambda)}(Y,X)\ \ \ \ \ \ \ (*)$$ for any $X,Y\in \dd^{\b}(\flmod\Lambda)$. Similarly, we call $\Lambda$ [*$n$-Calabi-Yau$^-$*]{} ([*$n$-CY$^-$*]{} for short) if there exists a functorial isomorphism $(*)$ for any $X\in \dd^{\b}(\flmod\Lambda)$ and $Y\in\kk^{\b}(\pr\Lambda)$. In this section we first give some basic results for $n$-CY and $n$-CY$^-$ algebras in . The main result is a characterization of these algebras in terms of symmetric orders in . Obviously, $n$-CY (resp. $n$-CY$^-$) algebras are closed under Morita equivalences. A finite product $\prod_i\Lambda_i$ of algebras is $n$-CY (resp. $n$-CY$^-$) if and only if so is each $\Lambda_i$. Part (1) and (7) in the following theorem are due to Rickard \[Ri2\]. .5em[**Theorem  **]{} *(1) $n$-CY (resp. $n$-CY$^-$) algebras are closed under derived equivalences.* \(2) $\Lambda$ is $n$-CY if and only if so is $\Lambda^{\op}$ (cf. (1)). (3)(localization) $\Lambda$ is $n$-CY (resp. $n$-CY$^-$) if and only if so is $\Lambda_{\dn{p}}$ for any $\dn{p}\in\Max R\cap\Supp{}_R\Lambda$. (4)(completion) If $R$ is local, then $\Lambda$ is $n$-CY if and only if so is $\widehat{\Lambda}$ (cf. (2)). \(5) Any $n$-CY algebra $\Lambda$ satisfies $\gl\Lambda=n$. \(6) Any $n$-CY$^-$ algebra $\Lambda$ satisfies $\depth{}_{R_{\dn{p}}}\Lambda_{\dn{p}}=\dim \Lambda=\id{}_\Lambda\Lambda=n$ for any $\dn{p}\in\Max R\cap\Supp{}_R\Lambda$. \(7) $\Lambda$ is $n$-CY if and only if it is $n$-CY$^-$ and $\gl\Lambda<\infty$. \(8) If $\Lambda$ is $n$-CY, then there exists a functorial isomorphism $\hom_{\dd(\Mod\Lambda)}(X,Y[n])\simeq D\hom_{\dd(\Mod\Lambda)}(Y,X)$ for any $X\in \dd^{\b}(\flmod\Lambda)$ and $Y\in \dd^{\b}(\mod\Lambda)$. .5em[Proof ]{} (1) By \[Re;6.3\], $\kk^{\b}(\pr\Lambda)$ consists of the compact objects $X$ of $\kk^-(\Pr\Lambda)$, i.e. the functor $\hom_{\dd(\Mod\Lambda)}(X,-)$ on $\kk^-(\Pr\Lambda)$ commutes with arbitrary direct sums. On the other hand, $\dd^{\b}(\flmod\Lambda)$ consists of all objects $X\in\kk^-(\Pr\Lambda)$ such that $\bigoplus_{i\in\zzz}\hom_{\dd(\Mod\Lambda)}(Y,X[i])$ has finite length for any $Y\in\kk^{\b}(\pr\Lambda)$ (see ). Thus any triangle equivalence $\kk^-(\Pr\Lambda)\to\kk^-(\Pr\Gamma)$ induces triangle equivalences $\kk^{\b}(\pr\Lambda)\to\kk^{\b}(\pr\Gamma)$ and $\dd^{\b}(\flmod\Lambda)\to \dd^{\b}(\flmod\Gamma)$. Thus the $n$-CY and $n$-CY$^-$ properties are preserved by derived equivalence. \(2) Immediate from Matlis duality. (3)(4) Put $S:=R_{\dn{p}}$ for (3) and $S:=\widehat{R}$ for (4). Then $S$ is a flat $R$-module. We have a functor $(-\otimes_RS):\dd^{\b}(\Mod\Lambda)\to\dd^{\b}(\Mod\Lambda\otimes_RS)$ with isomorphisms [$$\Rhom_{\Lambda\otimes_RS}(X\otimes_RS,Y\otimes_RS)\simeq\Rhom_\Lambda(X,Y\otimes_RS)\simeq\Rhom_\Lambda(X,Y)\otimes_RS$$]{} for any $X,Y\in\dd^{\b}(\mod\Lambda)$ by . Applying $H^0$, we have a functorial isomorphism [$$\hom_{\dd(\Mod\Lambda\otimes_RS)}(X\otimes_RS,Y\otimes_RS)\simeq\hom_{\dd(\Mod\Lambda)}(X,Y)\otimes_RS.$$]{} If one of $X,Y\in\dd^{\b}(\mod\Lambda)$ is contained in $\dd^{\b}(\flmod\Lambda)$, we have functorial isomorphisms [$$\begin{aligned} \hom_{\dd(\Mod\Lambda)}(X,Y)\simeq\bigoplus_{\dn{p}}\hom_{\dd(\Mod\Lambda)}(X,Y)_{\dn{p}}\simeq\bigoplus_{\dn{p}}\hom_{\dd(\Mod\Lambda_{\dn{p}})}(X_{\dn{p}},Y_{\dn{p}})&&\mbox{for }(3),\\ \hom_{\dd(\Mod\Lambda)}(X,Y)\simeq\hom_{\dd(\Mod\Lambda)}(X,Y)^{\widehat{\ }}\simeq\hom_{\dd(\Mod\widehat{\Lambda})}(\widehat{X},\widehat{Y})&&\mbox{for }(4),\end{aligned}$$]{} where the direct sum is finite. Thus the ‘if’ part follows. The ‘only if’ part also follows since the induced functors $\dd^{\b}(\flmod\Lambda)\to\dd^{\b}(\flmod\Lambda\otimes_RS)$ and $\kk^{\b}(\pr\Lambda)\to\kk^{\b}(\pr\Lambda_{\dn{p}})$ are dense. \(5) We can assume that $R$ is complete local by (3) and (4). For any $X,Y\in\flmod\Lambda$, we have [$$\Ext^{i}_\Lambda(X,Y)\simeq D\ext^{n-i}_\Lambda(Y,X)=\left\{\begin{array}{cc} 0&(i>n),\\ D\hom_\Lambda(Y,X)&(i=n).\\ \end{array}\right.$$]{} Considering a minimal projective resolution $\cdots\to P_1\to P_0\to X\to0$ of $X\in\flmod\Lambda$, we have $\hom_\Lambda(P_i,\Lambda/J_\Lambda)=\ext^i_\Lambda(X,\Lambda/J_{\Lambda})=0$ for $i>n$. Thus we see that $P_{n+1}=0$ and $\pd{}_\Lambda X\le n$. Putting $Y:=X$, we see that $\pd{}_\Lambda X=n$. By , we have $\gl\Lambda=n$. \(6) We can assume that $R$ is local by (3). For any $X\in\flmod\Lambda$, we have [$$\ext^i_\Lambda(X,\Lambda)\simeq D\ext^{n-i}_\Lambda(\Lambda,X)=\left\{\begin{array}{cc} 0&(i\neq n),\\ DX&(i=n).\\ \end{array}\right.$$]{} This implies $\depth\Lambda=\dim \Lambda=\id{}_\Lambda\Lambda=n $ by . \(7) The ‘if’ part holds since $\gl\Lambda<\infty$ implies $\dd^{\b}(\flmod\Lambda)\subset\dd^{\b}(\mod\Lambda)=\kk^{\b}(\pr\Lambda)$. We now show the ‘only if’ part. We can assume that $R$ is local by (3). Then $\Lambda^{(i)}:=\Lambda/J_\Lambda^i$ satisfies $\widehat{\Lambda}=\plim_{i\ge0}\Lambda^{(i)}$. Take $X\in \dd^{\b}(\flmod\Lambda)$ and $Y\in\kk^{\b}(\pr\Lambda)$. Then we have $Y^{(i)}\in \dd^{\b}(\flmod\Lambda)$. Since $\Lambda$ is $n$-CY, we have a functorial isomorphism $D\hom_{\dd(\Mod\Lambda)}(Y^{(i)},X)\simeq\hom_{\dd(\Mod\Lambda)}(X,Y^{(i)}[n])$. Taking $\plim$ on both sides and applying , we obtain functorial isomorphisms [$$D\hom_{\dd(\Mod\Lambda)}(Y,X)\simeq\hom_{\dd(\Mod\Lambda)}(X,\widehat{Y}[n])=\hom_{\dd(\Mod\Lambda)}(X,Y[n])^{\widehat{\ }}=\hom_{\dd(\Mod\Lambda)}(X,Y[n]).$$]{} -1em (8) $\Lambda$ is $n$-CY$^-$ and $\dd^{\b}(\mod\Lambda)=\kk^{\b}(\pr\Lambda)$ by (7). Thus the assertion follows. ------------------------------------------------------------------------ .5em We now state our main theorems in this section, which give a characterization of $n$-CY and $n$-CY$^-$ algebras. .5em[**Theorem  **]{} *Let $R$ be a local Gorenstein ring with $\dim R=d$ and $\Lambda$ a module-finite $R$-algebra. Assume that the structure morphism $R\to\Lambda$ is injective.* \(1) If $\Lambda$ is $n$-CY or $n$-CY$^-$ for some integer $n$, then $n=d$. \(2) $\Lambda$ is $d$-CY$^-$ if and only if $\Lambda$ is a symmetric $R$-order (in the sense of section ). \(3) $\Lambda$ is $d$-CY if and only if $\Lambda$ is a symmetric $R$-order with $\gl\Lambda=d$. .5em[**Theorem  **]{} *Let $R$ be a Gorenstein ring with $\dim R=d$, $\Lambda$ a module-finite $R$-algebra and $n$ an integer.* \(1) The conditions (i)–(iii) are equivalent. 1em (i) $\Lambda$ is $n$-CY$^-$. 1em (ii) $\Lambda_{\dn{p}}$ is a CM $R_{\dn{p}}$-module of dimension $n$ and $\ext^{\height\dn{p}-n}_{R_{\dn{p}}}(\Lambda_{\dn{p}},R_{\dn{p}})\simeq\Lambda_{\dn{p}}$ as $(\Lambda_{\dn{p}},\Lambda_{\dn{p}})$-modules for any $\dn{p}\in\Max R\cap\Supp{}_R\Lambda$. 1em (iii) $\widehat{\Lambda}_{\dn{p}}$ is a CM $\widehat{R}_{\dn{p}}$-module of dimension $n$ and $\ext^{\height\dn{p}-n}_{\widehat{R}_{\dn{p}}}(\widehat{\Lambda}_{\dn{p}},\widehat{R}_{\dn{p}})\simeq\widehat{\Lambda}_{\dn{p}}$ as $(\widehat{\Lambda}_{\dn{p}},\widehat{\Lambda}_{\dn{p}})$-modules for any $\dn{p}\in\Max R\cap\Supp{}_R\Lambda$. \(2) The conditions (i)–(iii) are equivalent. 1em (i) $\Lambda$ is $n$-CY. 1em (ii) (1)(ii) and $\gl\Lambda_{\dn{p}}=n$ for any $\dn{p}\in\Max R\cap\Supp{}_R\Lambda$. 1em (iii) (1)(iii) and $\gl\widehat{\Lambda}_{\dn{p}}=n$ for any $\dn{p}\in\Max R\cap\Supp{}_R\Lambda$. .5em We note that a $d$-CY$^-$ algebra $\Lambda$ over a non-local Gorenstein ring $R$ is not necessarily a symmetric $R$-algebra even if the structure morphism $R\to\Lambda$ is injective.[^1] Before proving our main theorems, we state some easy consequences. .5em[**Corollary  **]{} *Let $R$ be a Gorenstein ring with $\dim R=d$, $\Lambda$ a module-finite $R$-algebra and $n$ an integer.* \(1) $\Lambda$ is $n$-CY$^-$ if and only if so is $\Lambda^{\op}$ (cf. (2)). \(2) If $R$ is local, then $\Lambda$ is $n$-CY$^-$ if and only if so is $\widehat{\Lambda}$ (cf. (4)). \(3) If $\Lambda$ is $n$-CY (resp. $n$-CY$^-$), then $\Lambda_{\dn{p}}$ and $\widehat{\Lambda}_{\dn{p}}$ are $m$-CY (resp. $m$-CY$^-$) for any $\dn{p}\in\Supp{}_R\Lambda$ and $m:=\dim{}_{R_{\dn{p}}}\Lambda_{\dn{p}}$. \(4) If $R$ is a normal domain and $\Lambda$ is $d$-CY ($d\ge1$), then $\Lambda$ is a reflexive $R$-module and a maximal $R$-order. \(5) If $R$ is local and $\Lambda$ is $d$-CY$^-$, then the following assertions hold for any $i$. 1em (i) There exists a functorial isomorphism $\ext^i_\Lambda(-,\Lambda)\simeq\ext^i_R(-,R)$ on $\mod\Lambda$. 1em (ii) $\depth M=d-\sup\{i\ge0\ |\ \ext^i_\Lambda(M,\Lambda)\neq0\}$ and $\dim M=d-\inf\{i\ge0\ |\ \ext^i_\Lambda(M,\Lambda)\neq0\}$ for any $M\in\mod\Lambda$. 1em (iii) $\ext^{d-i}_\Lambda(-,\Lambda)$ gives a duality between CM $\Lambda$-modules of dimension $i$ and CM $\Lambda^{\op}$-modules of dimension $i$. \(6) If $R$ is local, then symmetric $R$-orders (resp. symmetric $R$-orders of global dimension $d$) are closed under derived equivalences. .5em[Proof ]{} (1) follows from the left-right symmetry of the condition (1)(ii), and (2) follows from the equivalence of (1)(ii) and (iii). (3) By (7), we only have to show the assertion for CY$^-$. For any $\dn{p}\in\Supp{}_R\Lambda$, take $\dn{q}\in\Max R$ with $\dn{p}\subseteq\dn{q}$. Since $\ext^i_{R_{\dn{q}}}(\Lambda_{\dn{q}},R_{\dn{q}})=0$ ($i\neq\height\dn{q}-n$) and $\ext^{\height\dn{q}-n}_{R_{\dn{q}}}(\Lambda_{\dn{q}},R_{\dn{q}})\simeq\Lambda_{\dn{q}}$ as $(\Lambda_{\dn{q}},\Lambda_{\dn{q}})$-modules by (1), we have that $\ext^i_{R_{\dn{p}}}(\Lambda_{\dn{p}},R_{\dn{p}})=0$ ($i\neq\height\dn{q}-n$) and $\ext^{\height\dn{q}-n}_{R_{\dn{p}}}(\Lambda_{\dn{p}},R_{\dn{p}})\simeq\Lambda_{\dn{p}}$ as $(\Lambda_{\dn{p}},\Lambda_{\dn{p}})$-modules. Thus $\Lambda_{\dn{p}}$ and $\widehat{\Lambda}_{\dn{p}}$ are $m$-CY$^-$ by (1). \(4) Since $\Lambda$ is a Cohen-Macaulay $R$-module, it is reflexive. By (3), $\widehat{\Lambda}_{\dn{p}}$ is $1$-CY for any $\dn{p}\in\Spec R$ with $\height\dn{p}=1$. Thus $\widehat{\Lambda}_{\dn{p}}$ is a maximal $\widehat{R}_{\dn{p}}$-order (see ). These conditions imply that $\Lambda$ is a maximal order by a result of Auslander-Goldman \[Re;11.4,11.5\]. \(5) $\Lambda^*\simeq\Lambda$ as $(\Lambda,\Lambda)$-modules by . Let $\cdots\to P_0\to M\to0$ be a projective resolution of a $\Lambda$-module $M$. Since $\Lambda\in\cm R$ by , we have $\ext^i_R(P_i,R)=0$ for any $i>0$. Applying $\hom_\Lambda(-,\Lambda)\simeq\hom_\Lambda(-,\Lambda^*)\simeq\hom_R(-,R)$, we see that $\ext^i_\Lambda(M,\Lambda)=\ext^i_R(M,R)$. The other assertions follow immediately. \(6) Since module-finite $R$-algebras are closed under derived equivalences \[Ri1\], the assertion follows by (1) and . ------------------------------------------------------------------------ .5em For proving our main result in this section, it will be useful to investigate Nakayama functors in the context of derived categories. Now let $R$ be a local Gorenstein ring with $\dim R=d$ and $\Lambda$ a module-finite $R$-algebra. Recall that $R$ is a dualizing complex of $R$, i.e. $$(-)^\dagger:=\Rhom_R(-,R):\dd(\mod R)\to \dd(\mod R)$$ gives a duality such that $(-)^{\dagger\dagger}$ is isomorphic to the identity functor \[Har1;V2.1\]. Obviously $(-)^{\dagger}$ induces a duality $(-)^\dagger:\dd^\pm(\mod\Lambda)\leftrightarrow \dd^\mp(\mod\Lambda^{\op})$. Define the [*Nakayama functor*]{} in the derived category by the composition $$\nu:\dd^-(\mod\Lambda)\stackrel{\Rhom_\Lambda(-,\Lambda)}{\longrightarrow}\dd^+(\mod\Lambda^{\op})\stackrel{(-)^\dagger}{\longrightarrow}\dd^-(\mod\Lambda).$$ .5em[**Proposition  **]{} *(1) With the above notation, we have the following isomorphisms of functors, $(-)^\dagger\simeq\Rhom_\Lambda(-,\Lambda^\dagger):\dd(\mod\Lambda)\to \dd(\mod\Lambda^{\op})$ and $(-)^\dagger\simeq\Rhom_{\Lambda^{\op}}(-,\Lambda^\dagger):\dd(\mod\Lambda^{\op})\to \dd(\mod\Lambda)$. Thus $\Lambda^\dagger\in \dd^{\b}(\mod\Lambda\otimes_R\Lambda^{\op})$ is a dualizing complex of $\Lambda$ in the sense of Yekutieli \[Ye1\].* \(2) There exists an isomorphism $\nu\simeq\Lambda^\dagger\Lotimes_\Lambda(-)$ of functors on $\dd^-(\mod\Lambda)$. \(3) There exists a functorial isomorphism $\Rhom_\Lambda(X,\nu(Y))\simeq\Rhom_\Lambda(Y,X)^\dagger$ for any $X\in \dd^{\b}(\mod\Lambda)$ and $Y\in\kk^{\b}(\pr\Lambda)$. .5em[Proof ]{} (1) $\Rhom_\Lambda(-,\Lambda^\dagger)\stackrel{{\rm\XBJ(1)}}{\simeq}(\Lambda\Lotimes_\Lambda-)^\dagger\simeq(-)^\dagger$. \(2) $\nu=\Rhom_\Lambda(-,\Lambda)^\dagger\stackrel{}{\simeq}\Rhom_\Lambda(-,\Lambda^{\dagger\dagger})^\dagger\stackrel{{\rm\XBJ(1)}}{\simeq}(\Lambda^\dagger\Lotimes_\Lambda-)^{\dagger\dagger}\stackrel{}{\simeq}\Lambda^\dagger\Lotimes_\Lambda-$. \(3) $\Rhom_\Lambda(X,\nu(Y))=\Rhom_\Lambda(X,\Rhom_\Lambda(Y,\Lambda)^\dagger)\stackrel{{\rm\XBJ(1)}}{\simeq}(\Rhom_\Lambda(Y,\Lambda)\Lotimes_\Lambda X)^\dagger$ $\stackrel{{\rm\XBJ(2)}}{\simeq}\Rhom_\Lambda(Y,X)^\dagger$. ------------------------------------------------------------------------ .5em We need the following special case of the local duality theorem \[F\]\[Har1\]. .5em[**Lemma  **]{}[*We have an isomorphism $(-)^\dagger\simeq [-d]\circ D$ of functors on $\dd^{\b}(\flmod R)$.*]{} .5em[Proof ]{} We give a proof for completeness. Fix $X\in \dd^{\b}(\flmod R)$. Consider the following morphism in $\dd^{\b}(\Mod R)$ [$$\begin{diag} I&&\cdots&\RA{}&I^0&\RA{}&I^1&\RA{}&\cdots&\RA{}&I^{d-1}&\RA{}&I^d&\RA{}&0&\cdots\\ \uparrow^a&&&&\uparrow&&\uparrow&&&&\uparrow&&\parallel\\ I^d[-d]&&\cdots&\RA{}&0&\RA{}&0&\RA{}&\cdots&\RA{}&0&\RA{}&I^d&\RA{}&0&\cdots, \end{diag}$$]{} where $I$ is a minimal injective resolution of the $R$-module $R$. Then $I^d=E$. Take a triangle $E[-d]\stackrel{a}{\to}I\to I^\prime\to E[1-d]$. Since $\hom_R(X,I^i)=0$ for any $X\in\flmod R$ and for any $i$ with $0\le i<d$, we have $\Rhom_R(X,I^\prime)=0$ for any $X\in \dd^{\b}(\flmod R)$. Thus we have an isomorphism $(DX)[-d]=\Rhom_R(X,E[-d])\simeq\Rhom_R(X,I)=X^\dagger$. ------------------------------------------------------------------------ .5em We have the following ‘Serre duality theorem’ for arbitrary module-finite $R$-algebras. .5em[**Theorem  **]{}[*For any module-finite $R$-algebra $\Lambda$, we have a functorial isomorphism $$\hom_{\dd(\Mod\Lambda)}(X,\nu(Y)[d])\simeq D\hom_{\dd(\Mod\Lambda)}(Y,X)$$ for any $X\in \dd^{\b}(\flmod\Lambda)$ and $Y\in\kk^{\b}(\pr\Lambda)$,*]{} .5em[Proof ]{} $\hom_{\dd(\Mod\Lambda)}(X,\nu(Y)[d])=H^d(\Rhom_\Lambda(X,\nu(Y)))\stackrel{{\rm\XCE(3)}}{\simeq}H^d(\Rhom_\Lambda(Y,X)^\dagger)$ $\stackrel{{\rm\XCF}}{\simeq}H^0(D\Rhom_\Lambda(Y,X))=DH^0(\Rhom_\Lambda(Y,X))=D\hom_{\dd(\Mod\Lambda)}(Y,X)$. ------------------------------------------------------------------------ .5em We now obtain the following crucial result to prove our main theorems  and . .5em[**Theorem  **]{} *Let $R$ be a local Gorenstein ring with $\dim R=d$, $\Lambda$ a module-finite $R$-algebra and $n$ an integer. Then the following conditions are equivalent.* \(1) $\Lambda$ is $n$-CY$^-$. \(2) There exist isomorphisms $\ext^n_\Lambda(-,\Lambda)\simeq D$ and $\ext^i_\Lambda(-,\Lambda)=0$ ($i\neq n$) of functors on $\flmod\Lambda$ which commute with the right action of $\Lambda$. \(3) $\Lambda^\dagger\simeq\Lambda[n-d]$ in $\dd^{\b}(\mod\Lambda\otimes_R\Lambda^{\op})$. \(4) There exists an isomorphism $\nu\simeq[n-d]$ of functors on $\kk^{\b}(\pr\Lambda)$. \(5) $\Lambda$ is a CM $R$-module of dimension $n$, and $\ext^{d-n}_R(\Lambda,R)\simeq\Lambda$ as $(\Lambda,\Lambda)$-modules. ($\widehat{i}$) The condition ($i$) replacing $(R,\Lambda)$ by $(\widehat{R},\widehat{\Lambda})$ ($1\le i\le 5$). .5em[Proof ]{} (3)$\Leftrightarrow$(4) (resp. ($\widehat{3}$)$\Leftrightarrow$($\widehat{4}$)) follows by (2). (4)$\Rightarrow$(1) (resp. ($\widehat{4}$)$\Rightarrow$($\widehat{1}$)) follows by . (1)$\Rightarrow$(2) (resp. ($\widehat{1}$)$\Rightarrow$($\widehat{2}$)) and (2)$\Rightarrow$($\widehat{2}$) are obvious. ($\widehat{2}$)$\Rightarrow$($\widehat{3}$) We put $(R,\Lambda):=(\widehat{R},\widehat{\Lambda})$ for simplicity. It follows from  that ${\Lambda}$ is a CM ${R}$-module of dimension $n$. Take a ${\Lambda}$-regular sequence $a_1,\cdots,a_n\in {R}$, and put $I_i^\prime:=\sum_{j=1}^n{R}a_j^i$ and $I_i:=I_i^\prime{\Lambda}$. Then ${\Lambda}/I_i\in\flmod{\Lambda}$ and ${\Lambda}=\plim_{i\ge0}\ {\Lambda}/I_i$. Since ${\Lambda}\Lotimes_{{R}}{R}/I_i^\prime={\Lambda}/I_i={R}/I_i^\prime\Lotimes_{{R}}{\Lambda}$ holds, we have ${\Lambda}^\dagger\Lotimes_{{\Lambda}}{\Lambda}/I_i={\Lambda}^\dagger\Lotimes_{{R}}{R}/I_i^\prime={R}/I_i^\prime\Lotimes_{{R}}{\Lambda}^\dagger={\Lambda}/I_i\Lotimes_{{\Lambda}}{\Lambda}^\dagger$. We have isomorphisms ${\Lambda}^\dagger\Lotimes_{{\Lambda}}(-)\stackrel{\XCE(2)}{\simeq}\nu=(-)^\dagger\circ\Rhom_{{\Lambda}}(-,{\Lambda})\stackrel{(\widehat{2})}{\simeq}(-)^\dagger\circ[-n]\circ D\stackrel{\XCF}{\simeq}[n-d]$ of functors on $\flmod{\Lambda}$. Thus we have isomorphisms [$${\Lambda}[n-d]\stackrel{{\rm\XBF(1)}}{=}\plim_{i\ge0}\ {\Lambda}/I_i[n-d]\stackrel{}{\simeq}\plim_{i\ge0}\ {\Lambda}^\dagger\Lotimes_{{\Lambda}}{\Lambda}/I_i\simeq\plim_{i\ge0}\ {\Lambda}/I_i\Lotimes_{{\Lambda}}{\Lambda}^\dagger\stackrel{{\rm\XBF(1)}}{=}{\Lambda}^\dagger.$$]{} These isomorphisms commute with the right multiplication of ${\Lambda}$. Thus ($\widehat{3}$) holds. (4)$\Leftrightarrow$(5) (resp. ($\widehat{4}$)$\Leftrightarrow$($\widehat{5}$)) We have $H^i(\Lambda^\dagger)=\ext^i_R(\Lambda,R)$. Thus $H^i(\Lambda^\dagger)=0$ holds for any $i\neq d-n$ if and only if $\Lambda$ is a CM $R$-module of dimension $n$. In this case, $\Lambda^\dagger\simeq\Lambda$ as $(\Lambda,\Lambda)$-modules if and only if $\ext^{d-n}_R(\Lambda,R)\simeq\Lambda$ as $(\Lambda,\Lambda)$-modules. (5)$\Leftrightarrow$($\widehat{5}$) $\Lambda$ is a CM $R$-module of dimension $n$ if and only if $\widehat{\Lambda}$ is a CM $\widehat{R}$-module of dimension $n$. Since $(\Lambda\otimes_R\Lambda^{\op})^{\widehat{\ }}=\widehat{\Lambda}\otimes_{\widehat{R}}\widehat{\Lambda}^{\op}$ holds, it follows from  below that $\Lambda\simeq\hom_R(\Lambda,R)$ as $(\Lambda\otimes_R\Lambda^{\op})$-modules if and only if $\widehat{\Lambda}\simeq\hom_{\widehat{R}}(\widehat{\Lambda},\widehat{R})$ as $(\widehat{\Lambda}\otimes_{\widehat{R}}\widehat{\Lambda}^{\op})$-modules. Now one can easily check that all conditions are equivalent. ------------------------------------------------------------------------ .5em[**Lemma  **]{}[*Let $R$ be a local ring, $\Lambda$ a module-finite $R$-algebra and $M,N\in\mod\Lambda$. If $\widehat{M}\simeq\widehat{N}$ as $\widehat{\Lambda}$-modules, then $M\simeq N$ as $\Lambda$-modules.*]{} .5em[Proof ]{} We modify the proof of \[CRe;30.17\] where the case $\dim R=1$ is treated. Let $f\in\hom_{\widehat{\Lambda}}(\widehat{M},\widehat{N})$ be an isomorphism with $g:=f^{-1}$. Since $\hom_{\widehat{\Lambda}}(\widehat{M},\widehat{N})=\hom_\Lambda(M,N)^{\widehat{\ }}$, we can take $f'\in\hom_\Lambda(M,N)$ and $g'\in\hom_\Lambda(N,M)$ with $f-f'\in\dn{p}\hom_\Lambda(M,N)$ and $g-g'\in\dn{p}\hom_\Lambda(N,M)$. Then $g'f'-1_N=g'f'-gf\in(\dn{p}\endm_{\widehat{\Lambda}}(\widehat{N}))\cap\endm_\Lambda(N)=\dn{p}\endm_\Lambda(N)$. Thus we have $N=g'f'(N)+\dn{p}N$. Using Nakayama’s lemma, we have $g'f'\in\aut_\Lambda(N)$. Similarly, $f'g'\in\aut_\Lambda(M)$ holds, so $f'$ is an isomorphism. ------------------------------------------------------------------------ .5em Using  together with (7), we have now completed the proof of  and . In the rest of this section, we give some examples of $n$-CY algebras. Let us start with considering commutative CY and CY$^-$ algebras. .5em[**Proposition  **]{} *Let $R$ be a commutative noetherian ring and $n$ an integer.* \(1) $R$ is $n$-CY$^-$ if and only if $R$ is Gorenstein and $\dim R_{\dn{p}}=n$ for any $\dn{p}\in\Max R$. \(2) $R$ is $n$-CY if and only if $R$ is regular and $\dim R_{\dn{p}}=n$ for any $\dn{p}\in\Max R$. .5em[Proof ]{} The ‘only if’ part follows from (3)(5) and (6). The ‘if’ part follows from (3) and . ------------------------------------------------------------------------ .5em Next we consider $0$-CY algebras. .5em[**Proposition  **]{}[*A finite dimensional algebra over a field is 0-CY if and only if it is a semisimple algebra.*]{} .5em[Proof ]{} Since any 0-CY algebra has global dimension zero, it is semisimple. Conversely, it is well-known that any semisimple algebra over a field is symmetric \[CRe;9.8\]. ------------------------------------------------------------------------ .5em Now we consider 1-CY algebras over a complete discrete valuation ring $R$ with quotient field $K$. Recall that an $R$-order $\Lambda$ is called [*hereditary*]{} if $\gl\Lambda=1$. Let us recall briefly the structure theory of maximal and hereditary orders \[Re\]\[CRe\]. For a ring $\Delta$ and $n>0$, we put $\tri_n(\Delta):=\{(x_{ij})_{1\le i,j\le n}\in\ma_n(\Delta)\ |\ x_{ij}\in J_{\Delta}\mbox{ if }i>j\}$. The following results are well-known. .5em[**Proposition  **]{} *(1) Any finite dimensional division $K$-algebra $D$ contains a unique maximal $R$-order $\Delta_D$ \[Re;12.8\].* \(2) An $R$-order is maximal if and only if it is Morita equivalent to $\Delta_{D_1}\times\cdots\times\Delta_{D_k}$ for some finite dimensional division $K$-algebras $D_i$ \[Re;17,3\]. \(3) An $R$-order is hereditary if and only if it is Morita equivalent to $\tri_{n_1}(\Delta_{D_1})\times\cdots\times\tri_{n_k}(\Delta_{D_k})$ for some finite dimensional division $K$-algebras $D_i$ and $n_i>0$ \[Re;39.14\]. .5em We have the following relationship between 1-CY algebras and maximal orders. .5em[**Proposition  **]{}[*Let $R$ be a complete discrete valuation ring and $\Lambda$ a module-finite $R$-algebra. If $\Lambda$ is 1-CY, then it is a maximal $R$-order.*]{} .5em[Proof ]{} By , $\Lambda$ is a symmetric $R$-order with $\gl\Lambda=1$. Thus $\Lambda$ is Morita equivalent to $\tri_{n_1}(\Delta_{D_1})\times\cdots\times\tri_{n_k}(\Delta_{D_k})$ by (3). One can check that if $\tri_n(\Delta)$ is a symmetric $R$-algebra, then $n=1$ (e.g. \[Hae;6.3\]). Thus $\Lambda$ is maximal by (2). ------------------------------------------------------------------------ .5em We note that a maximal order is not necessarily symmetric. Let $D$ be a central division $K$-algebra with $\dim_KD=n^2$. If the residue field of $R$ is finite, then $\hom_R(\Delta_D,R)$ is isomorphic to $J_{\Delta_D}^{1-n}$ as a $(\Delta_D,\Delta_D)$-module \[Re;14.9\]. If $n>1$, then $J_{\Delta_D}^{1-n}$ is never isomorphic to $\Delta_D$ by \[Re;37.27\]. We thank Wolfgang Rump for kindly explaining these results to us. .5em We now give other examples of $d$-CY algebras, where $d\ge2$. Let $K$ be a field of characteristic zero and $G$ a finite subgroup of $\SL_d(K)$ acting on $K^d$ naturally. The action of $G$ naturally extends to $S:=K[[x_1,\cdots,x_d]]$. We denote by $S^G$ the invariant subring, and by $S*G$ the skew group ring, i.e. a free $S$-module with a basis $G$, where the multiplication is given by $(s_1g_1)\cdot(s_2g_2)=(s_1g_1(s_2))(g_1g_2)$ for $s_i\in S$ and $g_i\in G$. We have the following result (c.f. \[CRo\]). .5em[**Theorem  **]{}[ *$S*G$ is $d$-CY and a symmetric $S^G$-order with $\gl S*G=d$.*]{} .5em[Proof ]{} Any finite subgroup $G$ of $\SL_d(K)$ is small in the sense that any $g\in G$ with $g\neq1$ satisfies $\rank(g-1)>1$. This implies $\endm_{S^G}(S)=S*G$ by a result of Auslander. See \[A3\]\[Yo\] for $d=2$, and a similar argument works for arbitrary $d$. Since $S*G$ is a free $S$-module, $S*G$ is an $S^G$-order. Since $S*G=\endm_{S^G}(S)$ holds, $S*G$ is a symmetric $S^G$-order by (3). Moreover, $\Ext^i_{S*G}(X,Y)=\Ext^i_S(X,Y)^G$ holds for any $X,Y\in\mod S*G$ and $i\in\zzz$ \[A3\]\[Yo\]. Thus we obtain $\gl S*G=d$ by $\gl S=d$ and  (see also \[RR\]). ------------------------------------------------------------------------ .5em For a finite subgroup $G$ of $\SL_d(K)$, we can draw the quiver of the algebra $S*G$ as the [*McKay quiver*]{} of $G$ \[Mc\] by using irreducible representations of $G$ and tensor products (see \[A3\]\[Yo\]\[I4\]). Now we give some examples. If $G=\langle{\rm diag}(\zeta,\zeta,\cdots,\zeta)\rangle\subset\SL_d(K)$ with $\zeta^d=1$, then $S*G$ is the completion of the path algebra of the following quiver with commutative relations $x_ix_j=x_jx_i$ for any $i$ and $j$. $$\begin{picture}(180,60) \put(0,-2){\circle*{4}} \put(-2,-17){\scriptsize 1} \put(15,6){\tiny $x_1$} \put(15,1){\tiny $x_2$} \put(15,-9){\tiny $\stackrel{\cdots}{x_d}$} \put(4,5){\vector(1,0){34}} \put(4,0){\vector(1,0){34}} \put(4,-10){\vector(1,0){34}} \put(40,-2){\circle*{4}} \put(38,-17){\scriptsize 2} \put(55,6){\tiny $x_1$} \put(55,1){\tiny $x_2$} \put(55,-9){\tiny $\stackrel{\cdots}{x_d}$} \put(44,5){\vector(1,0){34}} \put(44,0){\vector(1,0){34}} \put(44,-10){\vector(1,0){34}} \put(80,-2){\circle*{4}} \put(78,-17){\scriptsize 3} \put(95,6){\tiny $\cdots$} \put(95,1){\tiny $\cdots$} \put(95,-9){\tiny $\stackrel{}{\cdots}$} %\put(82,5){\vector(1,0){38}} %\put(82,0){\vector(1,0){38}} %\put(82,-10){\vector(1,0){38}} \put(120,-2){\circle*{4}} \put(115,-17){\scriptsize $i-1$} \put(135,6){\tiny $x_1$} \put(135,1){\tiny $x_2$} \put(135,-9){\tiny $\stackrel{\cdots}{x_d}$} \put(124,5){\vector(1,0){34}} \put(124,0){\vector(1,0){34}} \put(124,-10){\vector(1,0){34}} \put(160,-2){\circle*{4}} \put(158,-17){\scriptsize $i$} \put(155,6){\vector(0,1){34}} \put(160,6){\vector(0,1){34}} \put(170,6){\vector(0,1){34}} \put(147,25){\tiny $x_1$} \put(155,25){\tiny $x_2$} \put(163,25){\tiny $\vdots$} \put(170,25){\tiny $x_d$} \put(-16,25){\tiny $x_d$} \put(-8,25){\tiny $\vdots$} \put(-1,25){\tiny $x_2$} \put(7,25){\tiny $x_1$} \put(-10,40){\vector(0,-1){34}} \put(0,40){\vector(0,-1){34}} \put(5,40){\vector(0,-1){34}} \put(0,48){\circle*{4}} \put(-2,58){\scriptsize $d$} \put(15,56){\tiny $x_d$} \put(15,46){\tiny $\stackrel{\cdots}{x_2}$} \put(15,41){\tiny $x_1$} \put(38,55){\vector(-1,0){34}} \put(38,45){\vector(-1,0){34}} \put(38,40){\vector(-1,0){34}} \put(40,48){\circle*{4}} \put(28,58){\scriptsize $d-1$} \put(55,56){\tiny $x_d$} \put(55,46){\tiny $\stackrel{\cdots}{x_2}$} \put(55,41){\tiny $x_1$} \put(78,55){\vector(-1,0){34}} \put(78,45){\vector(-1,0){34}} \put(78,40){\vector(-1,0){34}} \put(80,48){\circle*{4}} \put(68,58){\scriptsize $d-2$} \put(95,56){\tiny $\cdots$} \put(95,51){\tiny $\cdots$} \put(95,41){\tiny $\stackrel{}{\cdots}$} %\put(82,55){\vector(1,0){38}} %\put(82,50){\vector(1,0){38}} %\put(82,40){\vector(1,0){38}} \put(120,48){\circle*{4}} \put(110,58){\scriptsize $i+2$} \put(135,56){\tiny $x_d$} \put(135,46){\tiny $\stackrel{\cdots}{x_2}$} \put(135,41){\tiny $x_1$} \put(158,55){\vector(-1,0){34}} \put(158,45){\vector(-1,0){34}} \put(158,40){\vector(-1,0){34}} \put(160,48){\circle*{4}} \put(150,58){\scriptsize $i+1$} \end{picture}$$ .5em If $d=3$ in the example above, then $S*G$ has the left quiver below. If $G=\langle{\rm diag}(\zeta,\zeta^2,\zeta^2)\rangle\subset\SL_5(K)$ with $\zeta^5=1$, then $S*G$ has the right quiver below. $$\begin{picture}(40,40) \put(0,0){\circle*{4}} \put(40,0){\circle*{4}} \put(20,30){\circle*{4}} %\put(-7,-7){\scriptsize 1} %\put(42,-7){\scriptsize 2} %\put(17,34){\scriptsize 3} \put(0,2){\vector(1,0){38}} \put(0,0){\vector(1,0){38}} \put(0,-2){\vector(1,0){38}} \put(40,4){\vector(-2,3){17}} \put(38,3){\vector(-2,3){17}} \put(36,2){\vector(-2,3){17}} \put(16,28){\vector(-2,-3){17}} \put(18,27){\vector(-2,-3){17}} \put(20,26){\vector(-2,-3){17}} \end{picture} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \begin{picture}(60,47) \put(15,0){\circle*{4}} \put(45,0){\circle*{4}} \put(60,30){\circle*{4}} \put(30,45){\circle*{4}} \put(0,30){\circle*{4}} \put(15,0){\vector(1,0){28}} \put(45,0){\vector(1,2){14}} \put(60,30){\vector(-2,1){28}} \put(30,45){\vector(-2,-1){28}} \put(0,30){\vector(1,-2){14}} \put(15,-1){\vector(3,2){43}} \put(15,1){\vector(3,2){43}} \put(45,2){\vector(-1,3){14}} \put(44,-2){\vector(-1,3){15}} \put(60,29){\vector(-1,0){58}} \put(60,31){\vector(-1,0){58}} \put(30,43){\vector(-1,-3){14}} \put(29,47){\vector(-1,-3){15}} \put(0,29){\vector(3,-2){43}} \put(0,31){\vector(3,-2){43}} \end{picture}$$ 1.5em[**. Construction of tilting modules** ]{} Let $R$ be a complete local ring and $\Lambda$ a ring-indecomposable module-finite $R$-algebra. A central theme in this paper is the study of tilting modules for $d$-CY algebras for $d=2,3$, especially the tilting modules of projective dimension at most one. In particular, we are interested in the number of complements of almost complete tilting modules. A basic partial tilting $\Lambda$-module $T$ is said to be an [*almost complete tilting module*]{} if $T$ has $(n-1)$ non-isomorphic indecomposable direct summands, where $n$ is the number of non-isomorphic simple $\Lambda$-modules. In this case, $X$ is called a [*complement*]{} of $T$ if $T\oplus X$ is a basic tilting $\Lambda$-module. For finite dimensional algebras it is known that there are at most two (and at least one) complements, in the case of projective dimension at most one, and it is never the case that all almost complete tilting modules have two complements. In the context of module-finite $R$-algebras $\Lambda$, we see in section  that the result on at most two complements still holds, but now there are algebras $\Lambda$ where all almost complete tilting modules have two complements, as we shall show in section  for 2-CY and 3-CY algebras. In this section we treat the special case of almost complete tilting modules which are projective. We show that for $d$-CY algebras with no loops in the quiver there are exactly two complements to tilting modules of projective dimension at most one, and give an explicit description of the non-projective one. Even though it will not be used later in this paper, we describe more generally all complements which give tilting complexes, in particular those which are tilting modules of projective dimension greater than one. In the first part of this section we work in the general context of module-finite $R$-algebras, and give here necessary (and sufficient) conditions on what the complements are. Then we use this to obtain a nice description for $d$-CY algebras. Fix an indecomposable object $P\in\pr\Lambda$, and let $Q$ be a direct sum of all indecomposable projective $\Lambda$-modules which are not isomorphic to $P$. We want to find conditions for replacing $P$ with a complement $X$ of $Q$. For $n\ge0$, there exists a unique complex up to isomorphism $${\rm RA}_n:\cdots\to0\to A^0\stackrel{a^0}{\longrightarrow}\cdots\stackrel{a^{n-2}}{\longrightarrow}A^{n-1}\stackrel{a^{n-1}}{\longrightarrow}P\to0\to\cdots$$ which gives the first $n$ terms of the minimal right $(\add Q)$-approximation sequence of $P$ \[AS\], i.e. $A^i\in\add Q$ and $a^i\in J_{\pr\Lambda}$ for any $i$ and $H^i(\Rhom_\Lambda(Q,{\rm RA}_n))=0$ for any $i\neq0$. Similarly, for $n\ge0$, there exists a unique complex up to isomorphism $${\rm LA}_n:\cdots\to0\to P %\stackrel{\def\arraystretch{.3}\begin{array}{c}{\scriptstyle P}\\ {\scriptstyle\parallel}\end{array}}{B^{-n}} \stackrel{b^{-n}}{\longrightarrow}B^{1-n}\stackrel{b^{1-n}}{\longrightarrow}\cdots\stackrel{b^{-1}}{\longrightarrow}B^0\to0\to\cdots$$ which gives the first $n$ terms of the minimal left $(\add Q)$-approximation sequence of $P$, i.e. $B^i\in\add Q$ and $b^i\in J_{\pr\Lambda}$ for any $i$ and $H^i(\Rhom_\Lambda({\rm LA}_n,Q))=0$ for any $i\neq0$. We then have the following necessary and sufficient conditions on complements of $Q$. .5em[**Theorem  **]{} *With the above notation and assumptions, we have the following.* \(1) Let $X$ be an indecomposable object in $\kk^{\b}(\pr\Lambda)$. If $X\oplus Q$ is a tilting complex, then $X$ is isomorphic to ${\rm RA}_n$ or ${\rm LA}_n$ for some $n\ge0$. \(2) ${\rm RA}_n\oplus Q$ is a tilting complex if and only if $H^i(\Rhom_\Lambda({\rm RA}_n,Q))=0$ for any $i\neq0$. (3) ${\rm LA}_n\oplus Q$ is a tilting complex if and only if $H^i(\Rhom_\Lambda(Q,{\rm LA}_n))=0$ for any $i\neq0$. .5em[Proof ]{} (1)(i) Let $X$ be a complex $\cdots\stackrel{c^{i-1}}{\longrightarrow}X^i\stackrel{c^{i}}{\longrightarrow}X^{i+1}\stackrel{c^{i+1}}{\longrightarrow}\cdots$ in $\kk^{\b}(\pr\Lambda)$ with $c^i\in J_{\pr\Lambda}$ for any $i$ such that $X\oplus Q$ is a tilting complex. We have $\hom_{\kk^{\b}(\pr\Lambda)}(Q,X[i])=H^i(\Rhom_\Lambda(Q,X))$ and $\hom_{\kk^{\b}(\pr\Lambda)}(X,Q[i])=H^i(\Rhom_\Lambda(X,Q))$. \(ii) Put $m:=\min\{i\ |\ X^i\neq0\}$. We will show that either $X^m\in\add P$ or $m=0$. If $X^m\notin\add P$, then we can choose $f^m\in\hom_\Lambda(X^m,Q)$ not in $J_{\pr\Lambda}$. We extend $f^m$ to a chain morphism $f\in\hom_{\kk^{\b}(\pr\Lambda)}(X,Q[-m])$. If $m\neq0$, then we have $f=0$. Thus $f^m$ factors through $c^m\in J_{\pr\Lambda}$, a contradiction. \(iii) Put $n:=\max\{i\ |\ X^i\neq0\}$. Then the dual argument to (ii) shows that either $X^n\in\add P$ or $n=0$. \(iv) Since $\Lambda$ is ring-indecomposable, so is $\endm_{\kk^{\b}(\pr\Lambda)}(X\oplus Q)$ since $X\oplus Q$ is a tilting complex. Thus $m\le 0\le n$ holds. If $m=n=0$, then we have $X=P$. Otherwise, (ii) and (iii) imply that either ($m<0$ and $X^m\in\add P$) or ($0<n$ and $X^n\in\add P$). \(v) We will show that, if $0<n$, then $X$ is isomorphic to ${\rm RA}_n$. Inductively, we will show that $X^i\in\add Q$ for any $i\neq n$. This is true for any $i$ with $i<m$. Assume that $X^i\in\add Q$ holds for any $i$ with $i<l$. Since $H^i(\Rhom_\Lambda(Q,X))=0$ for any $i\neq0$ by (i), any $f^l\in\hom_\Lambda(X^l,X^n)$ can be extended to the following chain morphism $f\in\hom_{\kk^{\b}(\pr\Lambda)}(X,X[n-l])$. [$$\begin{diag} 0&\longrightarrow&\cdots&\longrightarrow&0&\longrightarrow&X^m&\longrightarrow&\cdots&\longrightarrow&X^{l-1}&\longrightarrow&X^{l}&\stackrel{c^l}{\longrightarrow}&X^{l+1}&\longrightarrow&\cdots&\longrightarrow&X^n\\ \downarrow&&&&\downarrow&&\downarrow&&&&\downarrow&&\downarrow^{f^l}&&\downarrow&&&&\downarrow\\ X^m&\longrightarrow&\cdots&\longrightarrow&X^{n-l+m-1}&\longrightarrow&X^{n-l+m}&\longrightarrow&\cdots&\longrightarrow&X^{n-1}&\stackrel{c^{n-1}}{\longrightarrow}&X^n&\longrightarrow&0&\longrightarrow&\cdots&\longrightarrow&0 \end{diag}$$]{} Since $\hom_{\kk^{\b}(\pr\Lambda)}(X,X[n-l])=0$, we obtain [$$f^l\in c^l\hom_\Lambda(X^{l+1},X^n)+\hom_\Lambda(X^l,X^{n-1})c^{n-1}\subseteq J_{\pr\Lambda}.$$]{} This implies $X^l\in\add Q$. Thus we have proved $X^i\in\add Q$ for any $i\neq n$. Hence (ii) implies $m=0$, and $X$ gives the first $n$ terms of the minimal right $(\add Q)$-approximation sequence of $X^n\in\add P$. Since $X$ is indecomposable, we have $X^n=P$ and $X={\rm RA}_{n}$. \(vi) A dual argument to (v) implies that, if $m<0$, then $X$ is isomorphic to ${\rm LA}_{-m}$. \(2) The ‘only if’ part follows by (1)(i) above. We will show the ‘if’ part. By (1)(i) again, $\hom_{\kk^{\b}(\pr\Lambda)}(Q,{\rm RA}_n[i])=0=\hom_{\kk^{\b}(\pr\Lambda)}({\rm RA}_n,Q[i])$ holds for any $i\neq0$. Take any $f\in\hom_{\kk^{\b}(\pr\Lambda)}({\rm RA}_n,{\rm RA}_n[i])$. If $i>0$, then the conditions $A^i\in\add Q$ ($i\neq n$) and $H^i(\Rhom_\Lambda(Q,{\rm RA}_n))=0$ ($i\neq0$) imply that there exist $s^{n-i}, s^{n-i-1}, \cdots, s^0$ in the diagram below such that $f^j=a^js^{j+1}+s^ja^{j-1}$ for any $j$. [$$\begin{diag} 0&\longrightarrow&\cdots&\longrightarrow&0&\RA{}&A^0&\RA{a^0}&\cdots\cdots&\RA{a^{n-i-2}}&A^{n-i-1}&\RA{a^{n-i-1}}&A^{n-i}&\longrightarrow&A^{n-i+1}&\longrightarrow&\cdots&\longrightarrow&A^n\\ \DA{}&&&&\DA{}&\DLA{s^0}&\DA{f^0}&\DLA{s^1}&&\DLA{s^{n-i-1}}&\DA{f^{n-i-1}}&\DLA{s^{n-i}}&\DA{f^{n-i}}&&\DA{}&&&&\DA{}\\ A^0&\longrightarrow&\cdots&\longrightarrow&A^{i-1}&\RA{a^{i-1}}&A^i&\RA{a^i}&\cdots\cdots&\RA{a^{n-2}}&A^{n-1}&\RA{a^{n-1}}&A^n&\longrightarrow&0&\longrightarrow&\cdots&\longrightarrow&0 \end{diag}$$]{} Thus we have $f=0$. On the other hand, if $i<0$, then the conditions $A^i\in\add Q$ ($i\neq n$) and $H^i(\Rhom_\Lambda({\rm RA}_n,Q))=0$ ($i\neq0$) imply that there exist $s^{1-i}, s^{2-i}, \cdots, s^n$ in the diagram below such that $f^j=a^js^{j+1}+s^ja^{j-1}$ for any $j$. [$$\begin{diag} A^0&\longrightarrow&\cdots&\longrightarrow&A^{-i-1}&\longrightarrow&A^{-i}&\RA{a^{-i}}&A^{1-i}&\RA{a^{1-i}}&\cdots\cdots&\RA{a^{n-1}}&A^n&\longrightarrow&0&\longrightarrow&\cdots&\longrightarrow&0\\ \DA{}&&&&\DA{}&&\DA{f^{-i}}&\DLA{s^{1-i}}&\DA{f^{1-i}}&\DLA{s^{2-i}}&&\DLA{s^{n}}&\DA{f^{n}}&&\DA{}&&&&\DA{}\\ 0&\longrightarrow&\cdots&\longrightarrow&0&\longrightarrow&A^0&\RA{a^0}&A^{1}&\RA{a^{1}}&\cdots\cdots&\RA{a^{n+i-2}}&A^{n+i}&\longrightarrow&A^{n+i+1}&\longrightarrow&\cdots&\longrightarrow&A^n \end{diag}$$]{} Thus we have $f=0$. Consequently, $\hom_{\kk^{\b}(\pr\Lambda)}({\rm RA}_n\oplus Q,({\rm RA}_n\oplus Q)[i])=0$ holds for any $i\neq0$. Thus ${\rm RA}_n\oplus Q$ is a tilting complex since it clearly generates $\kk^{\b}(\pr\Lambda)$. We can show (3) dually. ------------------------------------------------------------------------ .5em As a consequence, we obtain information on the number of possible complements. .5em[**Corollary  **]{}[*For any $n>0$, $Q$ has at most $2n-1$ complements giving rise to tilting complexes with term length at most $n$ in $\kk^{\b}(\pr\Lambda)$. For any $n>0$, $Q$ has at most $n$ complements giving rise to tilting modules of projective dimension at most $n-1$.*]{} .5em[Proof ]{} By , ${\rm LA}_i,{\rm RA}_i$ ($0\le i\le n$) are the possible complements with term length at most $n$ in $\kk^{\b}(\pr\Lambda)$. Note that ${\rm RA}_0=P={\rm LA}_0$. Thus the first assertion follows. Since $H^i({\rm RA}_i)$ never vanishes for any $i\neq0$, ${\rm RA}_i$ can never be isomorphic to a module. Thus the second assertion follows. ------------------------------------------------------------------------ .5em We now give a basic result on $n$-CY algebras $\Lambda$, which we use to obtain more precise information on the number of complements. Following Seidel-Thomas \[ST\], we say that a simple $\Lambda$-module $S$ is [*$n$-spherical*]{} for $n>0$ if $\ext^i_\Lambda(S,S)=0$ for any $i$ with $i\neq0,n$. (See section  for a more general definition.) .5em[**Proposition  **]{} *Let $\Lambda$ be a basic $d$-CY algebra, $e$ a primitive idempotent of $\Lambda$, $P:=\Lambda e$ and $Q:=\Lambda(1-e)$. Take a minimal projective resolution $0\to P_d\stackrel{f_d}{\to}P_{d-1}\stackrel{f_{d-1}}{\to}\cdots\stackrel{f_2}{\to}P_1\stackrel{f_1}{\to}P_0\to S\to0$ of $S:=P/J_\Lambda P$.* \(1) We have an exact sequence $0\to\hom_\Lambda(P_0,\Lambda)\stackrel{f_1\bullet}{\to}\cdots\stackrel{f_d\bullet}{\to}\hom_\Lambda(P_d,\Lambda)\to DS\to0$. \(2) $P_d\simeq P_0=P$. \(3) $\Omega^iS$ is indecomposable for any $i$ ($0\le i\le d$). \(4) $S$ is $d$-spherical if and only if $\bigoplus_{i=1}^{d-1}P_i\in\add Q$. \(5) If $S$ is $d$-spherical, then the induced morphism $\Omega^iS\to P_{i-1}$ by $f_i$ is a minimal left $(\add Q)$-approximation for any $i$ ($1<i\le d$). .5em[Proof ]{} (1) We have an isomorphism [$$\ext^i_\Lambda(S,\Lambda)\simeq D\ext^{d-i}_\Lambda(\Lambda,S)=\left\{\begin{array}{cc}0&(0\le i<d)\\ DS&(i=d) \end{array}\right.$$]{} Thus we have the desired exact sequence by applying $\hom_\Lambda(-,\Lambda)$. \(2) Since the projective cover of $DS$ is $\hom_\Lambda(P_0,\Lambda)$, the assertion follows by (1). \(3) This follows by using that $P=P_d$ is indecomposable. \(4) This is a direct consequence of the definition of $S$ being $d$-spherical. \(5) Applying $-\otimes_\Lambda Q$ to the exact sequence in (1), we get an exact sequence $0\to\hom_\Lambda(P_0,Q)\stackrel{f_1\bullet}{\to}\cdots\stackrel{f_n\bullet}{\to}\hom_\Lambda(P_d,Q)\to0$ using that $(DS)\otimes_\Lambda Q=0$. Thus the assertion follows. ------------------------------------------------------------------------ .5em We now obtain our sufficient conditions for an almost complete projective module to have exactly two completions giving rise to tilting modules of projective dimension at most one. .5em[**Theorem  **]{} *Let $\Lambda$ be a basic $d$-CY algebra, $e$ a primitive idempotent of $\Lambda$, $P:=\Lambda e$ and $Q:=\Lambda(1-e)$. Assume that $S:=P/J_\Lambda P$ is $d$-spherical. (1) Any ${\rm RA}_n$ and ${\rm LA}_n$ ($n\ge0$) are complements of $Q$.* \(2) $Q$ has exactly $d$ complements $\Omega^{n}S$ ($1\le n\le d$) giving rise to tilting modules of finite projective dimension. They satisfy $\pd{}_\Lambda(\Omega^{n}S)=d-n$. \(3) $Q$ has exactly 2 complements $P$ and $\Omega^{d-1}S$ giving rise to tilting modules of projective dimension at most one. They are reflexive if $d\ge3$. .5em[Proof ]{} (1) We use the notation in . Since $S$ is $d$-spherical, ${\rm RA}_n$ is obtained from a minimal projective resolution of $S$ as follows: [$$\cdots\to P_1\to P_{d-1}\to\cdots\to P_1\to P_{d-1}\to\cdots\to P_1\to P_0\to0\to\cdots$$]{} By , $H^i(\Rhom_\Lambda({\rm RA}_n,Q))=0$ holds for any $i\neq0$. Thus ${\rm RA}_n$ is a complement of $Q$. A similar argument works for ${\rm LA}_n$. \(2) $\Omega^nS$ is quasi-isomorphic to the complex $X=(\cdots\to0\to P_d\to P_{d-1}\to\cdots\to P_n\to0\to\cdots)$, which satisfies $\bigoplus_{i=n}^{d-1}P_i\in\add Q$ since $S$ is $d$-spherical. Since $\hom_\Lambda(P_n,Q)\to\cdots\to\hom_\Lambda(P_{d-1},Q)\to\hom_\Lambda(P_d,Q)\to0$ is exact by (5) and $0\to\hom_\Lambda(Q,P_d)\to\hom_\Lambda(Q,P_{d-1})\to\cdots\to\hom_\Lambda(Q,P_n)$ is also exact, we have that $X={\rm LA}_{d-n}$ is a complement of $Q$ by . \(3) Since $\Lambda$ is reflexive and $\ref\Lambda$ is closed under kernels, $\Omega^{d-1}S$ is reflexive if $d\ge3$. ------------------------------------------------------------------------ Let $R$ be a normal complete local Gorenstein domain and $\Lambda$ a module-finite $R$-algebra. Throughout this section, all (almost complete) tilting modules have projective dimension at most one. We denote by $\tilt_1\Lambda$ the set of isomorphism classes of basic tilting $\Lambda$-modules. We have seen in section  that for $d$-CY algebras with no loops in their quiver, the almost complete projective tilting modules have exactly two complements, and we have given an explicit description of the complements. We improve these results, by dropping the assumption that the almost complete tilting module is projective, and show that a more general class of $d$-CY algebras have the same property, including all 2-CY and 3-CY algebras. We also give a description of the complements. We start with some background material on tilting modules. This is taken from the theory of finite dimensional algebras \[RS\]\[HU1\]\[U\], but is stated in our more general context of module-finite $R$-algebras, where the results remain valid. Since the proofs are usually the same as for finite dimensional algebras, they are mostly omitted. Let us start with the following. .5em[**Proposition  **]{}[*Any almost complete tilting module $T$ has at least one complement (called a Bongartz complement constructed in ) and at most two complements.*]{} .5em[Proof ]{} This follows from  and a similar argument as in \[RS;1.3\]\[U\]. ------------------------------------------------------------------------ .5em To study the relationship between two complements of an almost complete tilting module, let us recall the following result \[RS;1.3\]. .5em[**Proposition  **]{} *Let $T$ be an almost complete tilting $\Lambda$-module and $0\to Y\stackrel{g}{\to}T^\prime\stackrel{f}{\to}X\to0$ an exact sequence with $T'\in\add T$. Then the following conditions are equivalent.* \(1) $X$ is a complement of $T$ and $f$ is a minimal right $(\add T)$-approximation. \(2) $Y$ is a complement of $T$, $g$ is a minimal left $(\add T)$-approximation, and $\pd{}_\Lambda X\le 1$. .5em[Proof ]{} (1)$\Rightarrow$(2) Applying $\hom_\Lambda(T,-)$, we obtain $\ext^1_\Lambda(T,Y)=0$. Applying $\hom_\Lambda(-,T\oplus X)$, we obtain $\ext^1_\Lambda(Y,T\oplus X)=0$. We will show that $\hom_\Lambda(Y,T^\prime)\stackrel{\bullet f}{\to}\hom_\Lambda(Y,X)\to0$ is exact. Then we have $\ext^1_\Lambda(Y,Y)=0$ by applying $\hom_\Lambda(Y,-)$. Fix any $a\in\hom_\Lambda(Y,X)$. Since $\ext^1_\Lambda(X,X)=0$, there exists $b$ such that $a=gb$. Since $f$ is a right $(\add T)$-approximation of $X$, there exists $c$ such that $b=cf$. Thus $a=(gc)f$ holds. Since $T\oplus X$ generates $\kk^{\b}(\pr\Lambda)$, it follows from the exact sequence $0\to Y\stackrel{}{\to}T^\prime\stackrel{}{\to}X\to0$ that $T\oplus Y$ also generates $\kk^{\b}(\pr\Lambda)$. Thus $T\oplus Y$ is a tilting $\Lambda$-module. It follows from $\ext^1_\Lambda(X,T)=0$ that $g$ is a left $(\add T)$-approximation. One can show (2)$\Rightarrow$(1) similarly. ------------------------------------------------------------------------ .5em When the conditions of Proposition  hold, put $$\nu^-_X(T\oplus X):=T\oplus Y\ \ \ \mbox{ and }\ \ \ \nu^+_Y(T\oplus Y):=T\oplus X.$$ We call these operations [*mutations*]{}. For example, in (2), we have $\nu^-_{\Omega^{d-1}S}(Q\oplus\Omega^{d-1}S)=\Lambda$ and $\nu^+_{P}(\Lambda)=Q\oplus\Omega^{d-1}S$. For any basic tilting $\Lambda$-module $T$ and any indecomposable direct summand $X$ of $T$, at most one of $\nu^-_X(T)$ and $\nu^+_X(T)$ exists by , and we sometimes denote it by $$\nu_X(T).$$ We put $T^\perp:=\{C\in\mod\Lambda\ |\ \ext^1_\Lambda(T,C)=0\}$. Following \[RS\] (see also \[HU2\]), we write $$T\le U$$ if $T^\perp\supseteq U^\perp$. Then $\tilt_1\Lambda$ forms a partially ordered set with a unique minimal element $\Lambda$. One can easily check that, if $\nu^-_X(T)$ (resp. $\nu^+_X(T)$) exists, then $\nu^-_X(T)<T$ (resp. $T<\nu^+_X(T)$). Recall that the [*Hasse quiver*]{} of $\tilt_1\Lambda$ is the quiver with the set of vertices $\tilt_1\Lambda$, and we draw an arrow $T\to U$ ($T,U\in\tilt_1\Lambda$) if $T<U$ and there is no $V\in\tilt_1\Lambda$ such that $T<V<U$. The following proposition asserts that the arrows of the Hasse quiver of $\tilt_1\Lambda$ are given by mutation. .5em[**Proposition  **]{} *(1) For $T,U\in\tilt_1\Lambda$, the following conditions are equivalent.* 1em (i) $T<U$. 1em (ii) There exists an indecomposable direct summand $X$ of $U$ such that $T\le\nu^-_X(U)$. 1em (iii) There exists an indecomposable direct summand $Y$ of $T$ such that $\nu^+_Y(T)\le U$. \(2) For $T,U\in\tilt_1\Lambda$, the following conditions are equivalent. 1em (i) There exists an arrow $T\to U$ in the Hasse quiver of $\tilt_1\Lambda$. 1em (ii) There exists an indecomposable direct summand $X$ of $U$ such that $T=\nu^-_X(U)$. 1em (iii) There exists an indecomposable direct summand $Y$ of $T$ such that $U=\nu^+_Y(T)$. 1em (iv) There exists an almost complete tilting $\Lambda$-module which is a common direct summand of $T$ and $U$. .5em The next result generalizes  and  in two directions. For one thing, we treat arbitrary tilting modules which are not necessarily projective. In addition, we drop the assumption in  that $S$ is $d$-spherical, and we replace it by a weaker assumption on the depth and injective dimension of $\Gamma/I$. Notice that we can obtain  and  by putting $T=\Lambda$ in . .5em[**Theorem  **]{} *Let $\Lambda$ be a $d$-CY algebra, $T$ a basic tilting $\Lambda$-module and $\Gamma:=\endm_\Lambda(T)$. For a primitive idempotent $e$ of $\Gamma$, put $P:=\Gamma e$, $Q:=\Gamma(1-e)$, $I:=\Gamma(1-e)\Gamma$ and $S:=\Gamma/(I+J_\Gamma)$. Assume that the equality $n:=\depth(\Gamma/I)=\id{}_{\Gamma/I}(\Gamma/I)$ holds.* \(1) There exists a minimal projective resolution $0\to P_{d-n}\stackrel{f_{d-n}}{\to}\cdots\stackrel{f_1}{\to}P_0\stackrel{f_0}{\to}\Gamma/I\to0$ of the $\Gamma$-module $\Gamma/I$. \(2) We have a minimal projective resolution $0\to\hom_\Gamma(P_0,\Gamma)\stackrel{f_1\bullet}{\longrightarrow}\cdots\stackrel{f_{d-n}\bullet}{\longrightarrow}\hom_\Gamma(P_{d-n},\Gamma)\to\Gamma/I\to0$ of the $\Gamma^{\op}$-module $\Gamma/I$. \(3) $P_{d-n}\simeq P_0=P$ and $P_1,P_{d-n-1}\in\add Q$. \(4) Fix $i\ge0$. If $\tor^\Gamma_i(T,S)=0$, then $\tor^\Gamma_i(T,X)=0$ for any $X\in\mod(\Gamma/I)$. \(5) $\nu^+_{\Gamma e}(\Gamma)=\tr_{\Gamma^{\op}}(\Gamma/I)\oplus\Gamma(1-e)$ and $\nu^+_{e\Gamma}(\Gamma)=\tr_\Gamma(\Gamma/I)\oplus(1-e)\Gamma$. \(6) Precisely one of (i) or (ii) holds. 1em (i) $T\otimes_\Gamma S=0$ and $\nu^-_{Te}(T)=\hom_{\Gamma^{\op}}(\nu^+_{e\Gamma}(\Gamma),T)$. 1em (ii) $\tor^\Gamma_1(T,S)=0$ and $\nu^+_{Te}(T)=T\otimes_\Gamma\nu^+_{\Gamma e}(\Gamma)$. \(7) If $d-n\ge3$ and $T$ is reflexive, then $\nu_{Te}(T)$ is reflexive. .5em[Proof ]{} (1) $\pd{}_\Gamma(\Gamma/I)=d-n$ holds by . (2)(3) Since $\Gamma/I$ is a CM $\Gamma$-module of dimension $n$, it follows from (5) that $\ext^i_\Gamma(\Gamma/I,\Gamma)=0$ for any $i\neq d-n$ and $\ext^{d-n}_\Gamma(\Gamma/I,\Gamma)\simeq\ext^{d-n}_R(\Gamma/I,R)$ is a CM $\Gamma^{\op}$-module of dimension $n$. Since $\id{}_{\Gamma/I}(\Gamma/I)=n$, we have that $\ext^{d-n}_R(\Gamma/I,R)$ is a projective $(\Gamma/I)^{\op}$-module by \[GN1;1.1(3)\]. Since $\Gamma/I$ is local, $\ext^{d-n}_R(\Gamma/I,R)\simeq\Gamma/I$ as a $\Gamma^{\op}$-module. Now we can show (2) and (3) by a similar argument as in the proof of (1)(2). \(4) Use induction on $\dim X$ similarly as in the proof of . (5)(6) Applying  to the tilting $\Gamma^{\op}$-module $T$, precisely one of $T\otimes_\Gamma S=0$ or $\tor_1^\Gamma(T,S)=0$ holds. \(i) Assume $T\otimes_\Gamma S=0$. Since $T\otimes_\Gamma(\Gamma/I)=0$ holds by (4), we have an exact sequence $0\to\hom_\Gamma(\tr_\Gamma(\Gamma/I),T)\to T\otimes_\Gamma P_1\stackrel{T\otimes f_1}{\longrightarrow}Te\to0$ by applying $\hom_\Gamma(-,T)$ to the exact sequence $\hom_\Gamma(P_0,\Gamma)\stackrel{f_1\bullet}{\to}\hom_\Gamma(P_1,\Gamma)\to\tr_\Gamma(\Gamma/I)\to0$. Since $f_1$ is a minimal right $(\add Q)$-approximation, $T\otimes f_1$ is a minimal right $(\add T(1-e))$-approximation. Thus we have $\nu^-_{Te}(T)=\hom_\Gamma(\tr_\Gamma(\Gamma/I),T)\oplus T(1-e)$. \(ii) Put $U:=T\otimes_\Gamma\tr_{\Gamma^{\op}}(\Gamma/I)$. Since $\tor^\Gamma_i(T,\Gamma/I)=0$ holds for any $i>0$ by (4), we have an exact sequence $0\to T\otimes_\Gamma P_{d-n}\stackrel{T\otimes f_{d-n}}{\longrightarrow}\cdots\to T\otimes_\Gamma P_0\stackrel{T\otimes f_0}{\longrightarrow}T\otimes_\Gamma(\Gamma/I)\to0$. In particular, $0\to Te\stackrel{T\otimes f_{d-n}}{\longrightarrow}T\otimes_\Gamma P_{d-n-1}\to U\to0$ is exact by (2). Since $f_{d-n}$ is a minimal left $(\add Q)$-approximation, $T\otimes f_{d-n}$ is a minimal left $(\add T(1-e))$-approximation. Thus we only have to show $\pd{}_\Lambda U\le 1$, or equivalently, $\depth U\ge d-1$ by . Take a $\Gamma/I$-regular sequence $(x_1,\cdots,x_n)$, and put $\overline{\Gamma}_i:=(\Gamma/I)/(x_1,\cdots,x_i)(\Gamma/I)$ for $i=1,\cdots,n$. Then for $i<n$ we have an exact sequence $0\to\overline{\Gamma}_i\stackrel{x_{i+1}}{\longrightarrow}\overline{\Gamma}_i\to\overline{\Gamma}_{i+1}\to0$. Applying $T\otimes_\Gamma-$, we have an exact sequence $0\to T\otimes_\Gamma\overline{\Gamma}_i\stackrel{x_{i+1}}{\longrightarrow}T\otimes_\Gamma\overline{\Gamma}_i\to T\otimes_\Gamma\overline{\Gamma}_{i+1}\to0$ since $\tor^\Gamma_1(T,\overline{\Gamma}_{i+1})=0$ by (4). This means that $(x_1,\cdots,x_n)$ is also a $(T\otimes_\Gamma(\Gamma/I))$-regular sequence. In particular, we have $\depth(T\otimes_\Gamma(\Gamma/I))\ge n$. Since $\depth(T\otimes_\Gamma P_i)\ge d-1$, the exact sequence $0\to U\to T\otimes_\Gamma P_{d-n-2}\to\cdots\to T\otimes_\Gamma P_0\stackrel{}{\to}T\otimes_\Gamma(\Gamma/I)\to0$ implies $\depth U\ge d-1$. Putting $\Lambda=T=\Gamma$ in (ii), we have $\nu^+_{\Gamma e}(\Gamma)=\tr_{\Gamma^{\op}}(\Gamma/I)\oplus\Gamma(1-e)$ and $\nu^+_{e\Gamma}(\Gamma)=\tr_{\Gamma}(\Gamma/I)\oplus(1-e)\Gamma$. Thus the equalities in (5) and (6) follow. \(7) This is obvious for the case (6)(i). For the case (6)(ii), the assertion follows from the exact sequence $0\to U\to T\otimes_\Gamma P_{d-n-2}\to T\otimes_\Gamma P_{d-n-3}$. ------------------------------------------------------------------------ .5em We notice here that we can regard the tilting modules constructed in (5) above as analogs of APR tilting modules \[APR\]. To apply  for the case $d=2$ and $3$, we need the following observation. .5em[**Lemma  **]{} *Let $\Gamma$ be a ring-indecomposable $d$-CY algebra. For an idempotent $e\neq1$ of $\Gamma$, put $I:=\Gamma(1-e)\Gamma$.* \(1) $\Gamma\otimes_RK$ is a simple algebra for the quotient field $K$ of $R$. \(2) $\dim (\Gamma/I)\le\max\{0,d-2\}$. \(3) If $d\le 3$, then $\id_{\Gamma/I}(\Gamma/I)\le\max\{0,d-2\}$. \(4) If $d=3$ and $e$ is primitive, then $\depth(\Gamma/I)=\id_{\Gamma/I}(\Gamma/I)$. .5em[Proof ]{} (1) Since $R$ is normal, $\Gamma=\bigcap_{\dn{p}}\Gamma_{\dn{p}}$ holds where $\dn{p}$ runs over all height one prime ideals of $R$. By  and the structure theorem (2) of maximal orders over complete discrete valuation rings, $\Gamma_{\dn{p}}$ contains all central idempotents of $\Gamma\otimes_RK$. Thus $\Gamma$ contains all central idemptents of $\Gamma\otimes_RK$. Since $\Gamma$ is ring-indecomposable, $\Gamma\otimes_RK$ is simple. \(2) By the structure theorem of maximal orders, we have $(\Gamma/I)_{\dn{p}}=\Gamma_{\dn{p}}/\Gamma_{\dn{p}}e\Gamma_{\dn{p}}=0$ for any height one prime ideal $\dn{p}$ of $R$. Thus $\dim (\Gamma/I)\le d-2$ holds. \(3) If $d\le1$, then $\Gamma/I=0$ by (3), (2) and . Assume $d=2$ or $3$. Since $\mod(\Gamma/I)$ is extension closed in $\mod\Gamma$, we have $\ext^1_\Gamma(\Gamma/I,X)=0$ for any $X\in\mod(\Gamma/I)$. Since $\Gamma$ is $d$-CY, we have $\ext^{d-1}_\Gamma(X,\Gamma/I)=0$ for any $X\in\flmod(\Gamma/I)$. Assume $d=2$. Using again that $\mod(\Gamma/I)$ is extension closed in $\mod\Gamma$, we get $\ext^1_{\Gamma/I}(X,\Gamma/I)=0$. Thus $\id_{\Gamma/I}(\Gamma/I)=0$. In the rest, assume $d=3$. For any $X\in\flmod(\Gamma/I)$, take an exact sequence $0\to Y\to(\Gamma/I)^n\to X\to0$. Applying $\hom_\Gamma(-,\Gamma/I)$, we get an exact sequence $0=\ext^1_\Gamma((\Gamma/I)^n,\Gamma/I)\to\ext^1_\Gamma(Y,\Gamma/I)\to\ext^2_\Gamma(X,\Gamma/I)=0$. Thus we have $\ext^1_\Gamma(Y,\Gamma/I)=0$. Since $\mod(\Gamma/I)$ is extension closed in $\mod\Gamma$, we have $\ext^1_{\Gamma/I}(Y,\Gamma/I)=0$. Thus $\ext^2_{\Gamma/I}(X,\Gamma/I)=0$. It follows from  that $\id_{\Gamma/I}(\Gamma/I)\le1$. \(4) Since $\Gamma/I$ is a local algebra with $\id_{\Gamma/I}(\Gamma/I)<\infty$ by (3), it follows from a result of Ramras \[Ra;2.15\] that $\depth(\Gamma/I)=\id_{\Gamma/I}(\Gamma/I)$ (see also \[GN2;3.9\]). ------------------------------------------------------------------------ .5em We now get our desired result for 2-CY and 3-CY algebras. .5em[**Theorem  **]{} *Let $\Lambda$ be a ring-indecomposable $d$-CY algebra with $d=2$ or $3$, $T$ a basic tilting $\Lambda$-module and $\Gamma:=\endm_\Lambda(T)$. For a primitive idempotent $e\neq1$ of $\Gamma$, put $I:=\Gamma(1-e)\Gamma$ and $S:=\Gamma/(I+J_\Gamma)$.* \(1) $n:=\depth(\Gamma/I)=\id_{\Gamma/I}(\Gamma/I)$, and $n=d-2$ or $d-3$. \(2) $\nu^+_{\Gamma e}(\Gamma)=\tr_{\Gamma^{\op}}(\Gamma/I)\oplus\Gamma(1-e)$ and $\nu^+_{e\Gamma}(\Gamma)=\tr_\Gamma(\Gamma/I)\oplus(1-e)\Gamma$. \(3) $\nu^+_{\Gamma e}(\Gamma)=I=\nu^+_{e\Gamma}(\Gamma)$ if $n=d-2$, and $\nu^+_{\Gamma e}(\Gamma)=\hom_{\Gamma^{\op}}(\nu^+_{e\Gamma}(\Gamma),\Gamma)$ if $n=d-3$. \(4) Precisely one of (i) or (ii) holds. 1em (i) $T\otimes_\Gamma S=0$ and $\nu^-_{Te}(T)=\hom_{\Gamma^{\op}}(\nu^+_{e\Gamma}(\Gamma),T)$. 1em (ii) $\tor^\Gamma_1(T,S)=0$ and $\nu^+_{Te}(T)=T\otimes_\Gamma\nu^+_{\Gamma e}(\Gamma)$. \(5) If $n=d-3$ and $T$ is reflexive, then $\nu_{Te}(T)$ is reflexive and $\nu_{Te}(T)=(T\otimes_\Gamma\nu^+_{\Gamma e}(\Gamma))^{**}$. \(6) We have isomorphisms $\endm_\Gamma(\nu^+_{\Gamma e}(\Gamma))\simeq\endm_\Lambda(\nu_{Te}(T))\simeq\endm_{\Gamma^{\op}}(\nu^+_{e\Gamma}(\Gamma))^{\op}$. This is isomorphic to $\Gamma$ if $n=d-2$. .5em[Proof ]{} (1) is shown in , and (2) and (4) are shown in . One can check (3) easily by using the exact sequences in (1)(2). \(5) $\nu_{Te}(T)$ is reflexive by (7). The assertion for the case (4)(ii) is obvious. For the case (4)(i), the assertion follows from $\nu_{Te}(T)=(T\otimes_\Gamma\hom_{\Gamma^{\op}}(\nu^+_{e\Gamma}(\Gamma),\Gamma))^{**}$ and (3). \(6) We have ring morphisms $a:=\hom_{\Gamma^{\op}}(-,T)_{\nu^+_{e\Gamma}(\Gamma),\nu^+_{e\Gamma}(\Gamma)}:\endm_{\Gamma^{\op}}(\nu^+_{e\Gamma}(\Gamma))^{\op}\to\endm_\Lambda(\nu^-_{Te}(T))$ for (4)(i) and $b:=(T\otimes_\Gamma-)_{\nu^+_{\Gamma e}(\Gamma),\nu^+_{\Gamma e}(\Gamma)}:\endm_\Gamma(\nu^+_{\Gamma e}(\Gamma))\to\endm_\Lambda(\nu^+_{Te}(T))$ for (4)(ii) between $d$-CY algebras. For any $\dn{p}\in\Spec R$ with $\height\dn{p}=1$, then $\nu^+_{\Gamma e}(\Gamma)_{\dn{p}}$, $\nu^+_{e\Gamma}(\Gamma)_{\dn{p}}$ and $T_{\dn{p}}$ are progenerators. Thus $a_{\dn{p}}$ and $b_{\dn{p}}$ are isomorphisms. Since $a$ and $b$ are morphisms between reflexive $R$-modules, they are isomorphisms. If $n=d-2$, then we have ring morphisms $\Gamma\to\endm_\Gamma(I)=\endm_\Gamma(\nu^+_{\Gamma e}(\Gamma))$ and $\Gamma\to\endm_{\Gamma^{\op}}(I)^{\op}=\endm_{\Gamma^{\op}}(\nu^+_{e\Gamma}(\Gamma))^{\op}$ by (3), and $I_{\dn{p}}=\Gamma_{\dn{p}}$ for any $\dn{p}\in\Spec R$ with $\height\dn{p}=1$. If $n=d-3$, then we have a ring morphism $\hom_{\Gamma^{\op}}(-,\Gamma)_{\nu^+_{e\Gamma}(\Gamma),\nu^+_{e\Gamma}(\Gamma)}:\endm_{\Gamma^{\op}}(\nu^+_{e\Gamma}(\Gamma))^{\op}\to\endm_\Gamma(\nu^+_{\Gamma e}(\Gamma))$ by (3). We can show that these are isomorphisms by a similar argument as above. ------------------------------------------------------------------------ .5em The following generalization of (3) follows immediately from (4)(5). .5em[**Corollary  **]{} *Let $\Lambda$ be a basic ring-indecomposable non-local $d$-CY algebra with $d=2$ or $3$, and let $n$ be the number of simple $\Lambda$-modules.* \(1) Any almost complete tilting $\Lambda$-module has exactly two complements. Thus any vertex in the Hasse quiver of $\tilt_1\Lambda$ has precisely $n$ neighbours. \(2) Assume $d=3$. If $T$ is a reflexive tilting $\Lambda$-module and the quiver of $\endm_\Lambda(T)$ has no loops, then all $n$ neighbours of $T$ are again reflexive. .5em We have the following natural questions. .5em[**Questions** ]{} (1) Is the Hasse quiver of $\tilt_1\Lambda$ connected? In other words, can any tilting $\Lambda$-module be obtained by applying successive mutations to $\Lambda$? We will show in the next section that this is the case for $d=2$. \(2) Is  valid for arbitrary $d$? By , it is enough to show $\depth(\Gamma/I)=\id{}_{\Gamma/I}(\Gamma/I)$ for any ring-indecomposable $d$-CY algebra $\Gamma$ and any primitive idempotent $e$ of $\Gamma$ with $I:=\Gamma(1-e)\Gamma$. 1.5em [**. 2-Calabi-Yau algebras and affine Weyl groups**]{} Let $R$ be a normal complete local Gorenstein domain with $\dim R=d$ and $\Lambda$ a basic module-finite $R$-algebra which is $d$-CY. Thus $\Lambda$ is a symmetric $R$-order with $\gl\Lambda=d$. If $d=0$ or $1$, then $\Lambda$ is Morita equivalent to a finite product of local rings by  and , and any tilting $\Lambda$-module is projective by . So the next question is to determine tilting $\Lambda$-modules for the case $d=2$. Assume for the rest of this section that $d=2$, and we only deal with tilting modules of projective dimension at most one. In this section we show that each almost complete tilting $\Lambda$-module has exactly two complements, and also give an explicit description of them. The tilting modules are all ideals, and we describe the set $\tilt_1\Lambda$ of tilting modules as a monoid of ideals generated by a finite set of idempotent ideals. We show that this set $\tilt_1\Lambda$ is in bijection with the elements of the affine Weyl group $W$ associated with the quiver of $\Lambda$, which is given by a generalized extended Dynkin diagram. The group $W$ has two natural partial orders, and we show that they coincide with the two orders on $\tilt_1\Lambda$, where one is recalled in section  and the other one comes from inclusion of ideals. When $\Lambda$ is local, then any tilting module over $\Lambda$ is projective. In the rest of this section, we assume that $\Lambda$ is non-local. Let $e_1,\cdots,e_n$ be a complete set of orthogonal primitive idempotents of $\Lambda$. Put $I_i:=\Lambda(1-e_i)\Lambda$. Then $S_i:=\Lambda/(I_i+J_\Lambda)$ is the simple $\Lambda$-module (resp. simple $\Lambda^{\op}$-module) corresponding to $e_i$. Then $I_i$ is maximal amongst left (resp. right) ideals $I$ of $\Lambda$ such that any composition factor of $\Lambda/I$ is $S_i$. We have shown in (3) that $I_i$ is a tilting $\Lambda$-module, and moreover if $T$ is a tilitng $\Lambda$-module, then either $\hom_\Lambda(I_i,T)$ or $I_iT$ is another tilting $\Lambda$-module. For later applications, we investigate these modules more carefully. .5em[**Proposition  **]{} *Let $T$ be a tilting $\Lambda$-module and $1\le i\le n$.* \(1) We have $\depth T\ge1$ and natural inclusions $I_iT\subseteq T\subseteq\hom_\Lambda(I_i,T)$ such that $\hom_\Lambda(I_i,T)/I_iT\in\flmod\Lambda$. \(2) Presisely one of (i) or (ii) holds. 1em (i) $S_i\otimes_\Lambda T=0$, $I_iT=T$ and $\nu^-_{Te_i}(T)=\Rhom_\Lambda(I_i,T)=\hom_\Lambda(I_i,T)$. 1em (ii) $\tor_1^\Lambda(S_i,T)=0$, $\hom_\Lambda(I_i,T)=T$ and $\nu^+_{Te_i}(T)=I_i\Lotimes_\Lambda T=I_i\otimes_\Lambda T=I_iT$. .5em[Proof ]{} (1) We have $\depth T\ge1$ by . It follows from (2) that $\Lambda/I_i$ is artin. Applying $\hom_\Lambda(-,T)$ to the exact sequence $0\to I_i\to\Lambda\to\Lambda/I_i\to0$, we get an exact sequence $0\to\hom_\Lambda(\Lambda/I_i,T)\to T\to\hom_\Lambda(I_i,T)\to\ext^1_\Lambda(\Lambda/I_i,T)\to0$. Since $\ext^i_\Lambda(\Lambda/I_i,T)$ has finite length for any $i$, we have $\hom_\Lambda(\Lambda/I_i,T)=0$ by $\depth T\ge1$. Thus we have inclusions $I_iT\subseteq T\subseteq\hom_\Lambda(I_i,T)$ with $\hom_\Lambda(I_i,T)/I_iT\in\flmod\Lambda$. \(2) By (3) and (4), we only have to show equalities $I_iT=T$ and $\Rhom_\Lambda(I_i,T)=\hom_\Lambda(I_i,T)$ in (i) and $\hom_\Lambda(I_i,T)=T$ and $I_i\Lotimes_\Lambda T=I_i\otimes_\Lambda T=I_iT$ in (ii). \(i) Since any composition factor of $T/I_iT$ is $S_i$, we have $I_iT=T$ since $S_i\otimes_\Lambda T=0$. Since [$$\ext^l_\Lambda(I_i,T)=\ext^{l+1}_\Lambda(\Lambda/I_i,T)=\left\{\begin{array}{cc}0&(l>1)\\ D\hom_\Lambda(T,\Lambda/I_i)=0&(l=1) \end{array}\right.$$]{} holds, we have $\Rhom_\Lambda(I_i,T)=\hom_\Lambda(I_i,T)$. \(ii) Any composition factor of $\hom_\Lambda(I_i,T)/T$ is $S_i$, so we have $\hom_\Lambda(I_i,T)=T$ since $\tor^\Lambda_1(S_i,T)=0$. Since $\tor^l_\Lambda(I_i,T)=\tor^{l+1}_\Lambda(\Lambda/I_i,T)=0$ holds for any $l\neq0$, we have $I_i\Lotimes_\Lambda T=I_i\otimes_\Lambda T$. Applying $-\otimes_\Lambda T$ to the exact sequence $0\to I_i\stackrel{}{\to}\Lambda\to\Lambda/I_i\to0$, we have an exact sequence $0=\tor^1_\Lambda(\Lambda/I_i,T)\to I_i\otimes_\Lambda T\stackrel{f}{\to}T$. Thus we have $I_i\otimes_\Lambda T\simeq\Im f=I_iT$. ------------------------------------------------------------------------ .5em The set of 2-sided ideals of $\Lambda$ forms a monoid by multiplication of ideals. We denote by $\ii(\Lambda)$ the submonoid generated by the ideals $I_1,\cdots,I_n$. Our first main result in this section is that all tilting $\Lambda$-modules are obtained in this way. .5em[**Theorem  **]{}[*$\ii(\Lambda)=\tilt_1\Lambda$ and $\ii(\Lambda)=\tilt_1\Lambda^{\op}$.*]{} .5em[Proof ]{} If $T$ is a tilting $\Lambda$-module, then so is $I_iT$ by (2). Thus $\ii(\Lambda)$ consists of tilting $\Lambda$-modules. We only have to show that any basic tilting $\Lambda$-module is isomorphic to some element of $\ii(\Lambda)$. We will use the functor $(-)^*\simeq\hom_\Lambda(-,\Lambda)$ ((5)). \(i) For any $T\in\tilt_1\Lambda$, we will show that $T^{**}$ is a projective $\Lambda$-module such that $T$ is a submodule of $T^{**}$ and $T^{**}/T$ has finite length. By \[AB\], we have an exact sequence $$0\to\ext^1_\Lambda(\tr T,\Lambda)\to T\to T^{**}\to\ext^2_\Lambda(\tr T,\Lambda)\to0.$$ For any $\dn{p}\in\Spec R\backslash\Max R$, it follows from  that $\widehat{T}_{\dn{p}}$ is a tilting module over $\widehat{\Lambda}_{\dn{p}}$, which is $0$ or $1$-CY by (3). Thus $\widehat{T}_{\dn{p}}$ (and hence $(\tr T)^{\widehat{\ }}_{\dn{p}}$) is a projective $\widehat{\Lambda}_{\dn{p}}$-module as we remarked previously. Hence $\ext^i_\Lambda(\tr T,\Lambda)^{\widehat{\ }}_{\dn{p}}=0$ holds for $i=1,2$. This implies that $\ext^i_\Lambda(\tr T,\Lambda)$ has finite length for $i=1,2$. Since $\depth T\ge1$ holds by , we have $\ext^1_\Lambda(\tr T,\Lambda)=0$. Since $T^{**}\in\ref\Lambda$, we have $\depth T^{**}\ge2$. Thus $T^{**}$ is a projective $\Lambda$-module by . \(ii) Take a simple submodule $S_i$ of $\soc(T^{**}/T)$. Then $\tor^\Lambda_1(S_i,T)=\tor^\Lambda_1(DS_i,T)=D\ext^1_\Lambda(T,S_i)=\ext^1_\Lambda(S_i,T)\neq0$ holds. By , $T_1:=\hom_\Lambda(I_i,T)$ is again a tilting $\Lambda$-module with $T\subset T_1$ and $T_1/T\in\flmod\Lambda$. Applying $(-)^*$ to the exact sequence $0\to T\to T_1\to T_1/T\to0$, we get $T^*=T_1^*$ since $\depth\Lambda=2$. Consequently, we have inclusions $T\subset T_1\subseteq T^{**}=T_1^{**}$. Repeating this process, we obtain an increasing sequence $$T=T_0\subset T_1\subset\cdots\subset T_m=T^{**}$$ of tilting $\Lambda$-modules with $T_{k-1}=I_{i_k}T_{k}$ for any $k$. Since $T^{**}$ is a projective tilting $\Lambda$-module, we have $\add{}_\Lambda T^{**}=\add{}_\Lambda\Lambda$. Thus $\add{}_\Lambda T=\add{}_\Lambda(I_{i_1}\cdots I_{i_m})$. ------------------------------------------------------------------------ .5em For any $T\in\tilt_1\Lambda$ and $1\le i\le n$, there exists an arrow $T\stackrel{}{\to}\nu^+_{Te_i}(T)$ or $\nu^-_{Te_i}(T)\stackrel{}{\to}T$ in the Hasse quiver of the poset $\tilt_1\Lambda$ by (2). We denote these arrows by $\stackrel{i}{\to}$ in the rest of this section. We have the following consequence. .5em[**Corollary  **]{}[*The Hasse quiver of $\tilt_1\Lambda$ is connected, and any vertex $T$ has precisely $n$ neighbours $\nu_{Te_i}(T)$ ($1\le i\le n$). Moreover, $\endm_\Lambda(T)=\Lambda$ holds for any $T\in\tilt_1\Lambda$.*]{} .5em[Proof ]{} The second assertion holds by (2) and . We will show that $\tilt_1\Lambda$ is connected. Any $T_0\in\tilt_1\Lambda$ can be written as $T_0=I_{a_1}\cdots I_{a_m}$ by . We can assume that $m$ is minimal. Put $T_i:=I_{a_1}\cdots I_{a_{m-i}}$. Then we have a strictly increasing sequence $T_0\subset T_1\subset\cdots\subset T_{m-1}\subset T_m=\Lambda$. By , there exists a path $\Lambda\stackrel{a_1}{\to}T_{m-1}\stackrel{a_2}{\to}\cdots\stackrel{a_m}{\to}T_0$ in $\tilt_1\Lambda$. Now the third assertion follows by (6). ------------------------------------------------------------------------ .5em To give an explicit description of $\ii(\Lambda)$, we determine the quiver of $\Lambda$. Following Happel-Preiser-Ringel \[HPR1,2\], we call the valued graphs below [*generalized extended Dynkin diagrams*]{}. \(i) An extended Dynkin diagram, (ii)   $\begin{picture}(10,10)\put(7,4){\circle{15}}\end{picture}\bullet\mbox{-----}\bullet\mbox{-----}\bullet\mbox{-----}\bullet\cdots\cdots\bullet\mbox{-----}\bullet\mbox{-----}\bullet\mbox{-----}\bullet\begin{picture}(10,10)\put(3,4){\circle{15}}\end{picture}$ (iii)   $\begin{picture}(10,10)\put(7,4){\circle{15}}\end{picture}\bullet\mbox{-----}\bullet\mbox{-----}\bullet\mbox{-----}\bullet\cdots\cdots\bullet\mbox{-----}\bullet\mbox{-----}\bullet\stackrel{(a\ b)}{\mbox{-----}}\bullet$          $(a,b)=(2,1)$ or $(1,2)$ .4em (iv)   $\begin{picture}(10,10)\put(7,4){\circle{15}}\end{picture}\bullet\mbox{-----}\bullet\mbox{-----}\bullet\mbox{-----}\bullet\cdots\cdots\bullet\mbox{-----}\bullet\mbox{-----}\bullet\begin{picture}(10,10)\put(-3,3){\line(3,1){22}}\put(-3,3){\line(3,-1){22}}\put(20,7){$\bullet$}\put(20,-7){$\bullet$}\end{picture}$ .5em For a generalized extended Dynkin diagram $\Delta$, define a valued quiver called the [*double*]{} of $\Delta$ as follows: We replace a valued edge $\bullet\stackrel{(a\ b)}{\mbox{-----}}\bullet$ by two valued arrows $\bullet\def\arraystretch{.3}\begin{array}{cc}\stackrel{(a\ b)}{\longrightarrow}\\ \stackrel{(b\ a)}{\longleftarrow}\end{array}\bullet$ of opposite direction. We replace a loop $\begin{picture}(10,10)\put(7,4){\circle{15}}\end{picture}\bullet$ by an arrow from a vertex to itself. We can describe valued quivers of 2-CY algebras (c.f. \[Boc\]). It is an interesting question whether all double of generalized extended Dynkin diagrams occur in this way. .5em[**Proposition  **]{}[*The valued quiver of any basic ring-indecomposable 2-CY algebra is a double of a generalized extended Dynkin diagram.*]{} .5em[Proof ]{} By , the quiver of $\Lambda$ is a double of some graph $\Delta$. For each vertex $e_i$ of $\Lambda$, put $d_i:=\rank_R\Lambda e_i$. Then $d_i$ gives a positive additive function on $\Delta$. By \[HPR1\], $\Delta$ is a generalized extended Dynkin diagram. ------------------------------------------------------------------------ .5em We will give an explicit description of $\ii(\Lambda)$ in terms of affine Weyl groups. By , the quiver of $\Lambda$ is a double of a generalized extended Dynkin diagram $\Delta$. We denote by $W$ the [*affine Weyl group*]{} associated with $\Delta$ defined as follows \[Hu\]\[BB\]: Put $m(i,i):=1$. For $i\neq j$, put $$m(i,j):=\left\{\begin{array}{ll} 2\ \ \ \ \ &\mbox{no edge between $i$ and $j$,}\\ 3&\stackrel{i}{\bullet}\stackrel{}{\mbox{------}}\stackrel{j}{\bullet},\\ 4&\begin{picture}(10,10)\put(7,4){\circle{15}}\end{picture}\stackrel{i}{\bullet}\stackrel{}{\mbox{------}}\stackrel{j}{\bullet},\ \ \stackrel{i}{\bullet}\stackrel{}{\mbox{------}}\stackrel{j}{\bullet}\begin{picture}(10,10)\put(3,4){\circle{15}}\end{picture}\ \mbox{ or } \stackrel{i}{\bullet}\stackrel{(a\ b)}{\mbox{------}}\stackrel{j}{\bullet}\ ((a\ b)=(1\ 2)\mbox{ or }(2\ 1))\\ 6&\stackrel{i}{\bullet}\stackrel{(a\ b)}{\mbox{------}}\stackrel{j}{\bullet}\ ((a\ b)=(1\ 3)\mbox{ or }(3\ 1)),\\ \infty&\stackrel{i}{\bullet}\stackrel{(2\ 2)}{\mbox{------}}\stackrel{j}{\bullet},\ \ \begin{picture}(10,10)\put(7,4){\circle{15}}\end{picture}\stackrel{i}{\bullet}\stackrel{}{\mbox{------}}\stackrel{j}{\bullet}\begin{picture}(10,10)\put(3,4){\circle{15}}\end{picture},\ \ \begin{picture}(10,10)\put(7,4){\circle{15}}\end{picture}\stackrel{i}{\bullet}\stackrel{(a\ b)}{\mbox{------}}\stackrel{j}{\bullet}\ \mbox{ or } \stackrel{i}{\bullet}\stackrel{(a\ b)}{\mbox{------}}\stackrel{j}{\bullet}\begin{picture}(10,10)\put(3,4){\circle{15}}\end{picture}\ ((a\ b)=(1\ 2)\mbox{ or }(2\ 1)). \end{array}\right.$$ Then $W$ is presented by generators $s_1,\cdots,s_n$ and relations $(s_is_j)^{m(i,j)}=1$. We shall also deal with the affine braid group associated to $\Delta$ to study autoequivalences of the derived category $\dd^{\b}(\mod\Lambda)$. This group $B$ is presented by generators $t_1,\cdots,t_n$ and relations $t_it_jt_i\cdots=t_jt_it_j\cdots$, where both sides are product of $m(i,j)$ generators. We have seen in section  that $\tilt_1\Lambda$ has a natural order. In view of  we have in addition the order given by inclusion of ideals. The affine Weyl group $W$ also has two partial orders. We want to show that there is a bijection between the elements of $\tilt_1\Lambda$ and $W$, respecting partial orders. So let us recall the [*Bruhat order*]{} $\le$, [*right order*]{} $\le_R$ and [*left order*]{} $\le_L$ on $W$ \[BB\] ($\le_R$ is called a [*weak order*]{} in \[Hu\]). The [*length*]{} $l(w)$ of $w\in W$ is the minimal value of $k$ for any expression $w=s_{a_1}\cdots s_{a_k}$ of $w$, and we call an expression with $k=l(w)$ [*reduced*]{}. Fix $w,w^\prime\in W$. We draw an arrow $w^\prime\to w$ if both $w=w^\prime s$ and $l(w^\prime)<l(w)$ hold for $s=xs_ix^{-1}$ for some $i$ and $x\in W$. Similarly, we draw an arrow $w^\prime\to_Rw$ (resp. $w^\prime\to_Lw$) if both $w=w^\prime s_i$ (resp. $w=s_iw^\prime$) and $l(w^\prime)<l(w)$ hold for some $i$. For $w,w^\prime\in W$, we define $w^\prime\le w$ (resp. $w^\prime\le_Rw$, $w^\prime\le_Lw$) if and only if there is a path from $w^\prime$ to $w$ consisting of arrows $\to$ (resp. $\to_R$, $\to_L$). For any reduced expression $w=s_{a_1}\cdots s_{a_k}$, it is well-known that $w^\prime\le w$ holds if and only if $w^\prime=s_{a_{i_1}}\cdots s_{a_{i_q}}$ for some $1\le i_1<\cdots<i_q\le k$ \[Hu;5.10\]. Thus the Bruhat order on $W$ is left-right symmetric (i.e. $w^\prime\le w$ if and only if $w^\prime{}^{-1}\le w^{-1}$), but the right order and the left order are not. We are now in the position to state our main result on the connection between $\tilt_1\Lambda$ and the affine Weyl group $W$. The crucial role is played by the mutation of tilting modules given in . .5em[**Theorem  **]{} *(1) $W$ acts transitively and freely on $\tilt_1\Lambda$ by $$T^{s_i}:=\nu_{Te_i}(T)=\left\{\begin{array}{ll} \nu^-_{Te_i}(T)=\hom_{\Lambda^{\op}}(I_i,T)&\mbox{if $T\otimes_\Lambda S_i=0$}\\ \nu^+_{Te_i}(T)=TI_i&\mbox{if $\tor_1^\Lambda(T,S_i)=0$} \end{array}\right.$$ for any $T\in\tilt_1\Lambda$ and $1\le i\le n$.* \(2) Under the induced bijection $W\ni w\mapsto\Lambda^w\in\ii(\Lambda)$, 1em(i) the Bruhat order on $W$ coincides with the reverse inclusion relation on $\ii(\Lambda)$, 1em(ii) the right order on $W$ coincides with the order on $\tilt_1\Lambda=\ii(\Lambda)$ (section ), 1em(iii) the left order on $W$ coincides with the order on $\tilt_1\Lambda^{\op}=\ii(\Lambda)$, 1em(iv) $\Lambda^w=I_{a_1}\cdots I_{a_k}$ holds for any reduced expression $w=s_{a_1}\cdots s_{a_k}$. .5em In order to prove this result we consider the action of tilting complexes on the derived category. Let $K_0(\Lambda)$ be the Grothendieck group of $\Lambda$ and $K_0(\Lambda)_{\ccc}:=K_0(\Lambda)\otimes_{\zzz}\ccc$, which has the basis $\{[P_i]\ |\ 1\le i\le n\}$ because $\Lambda$ has finite global dimension. For an arbitrary two-sided tilting complex $T$ of $\Lambda$, we have an autoequivalence $T\Lotimes_\Lambda-$ of $\dd^{\b}(\mod\Lambda)$. Thus we get a map $\tilt_1\Lambda\to\GL(K_0(\Lambda)_{\ccc})$ defined by $T\mapsto[T\Lotimes_\Lambda-]$. We will compare with the contragredient of the [*geometric representation*]{} of $W$ \[BB;4.1,4.2\] defined as follows: Put $k_{i,i}:=-2$. For $i\neq j$, put $(k_{i,j},k_{j,i}):=(a,b)$ for $\stackrel{i}{\bullet}\stackrel{(a\ b)}{\mbox{------}}\stackrel{j}{\bullet}$, $(k_{i,j},k_{j,i}):=(a,2b)$ for $\begin{picture}(10,10)\put(7,4){\circle{15}}\end{picture}\stackrel{i}{\bullet}\stackrel{(a,b)}{\mbox{------}}\stackrel{j}{\bullet}$, and $(k_{i,j},k_{j,i}):=(2,2)$ for $\begin{picture}(10,10)\put(7,4){\circle{15}}\end{picture}\stackrel{i}{\bullet}\stackrel{}{\mbox{------}}\stackrel{j}{\bullet}\begin{picture}(10,10)\put(3,4){\circle{15}}\end{picture}$. Let $V^*$ be a vector space with basis $\alpha^*_1,\cdots,\alpha^*_n$. Define $\sigma^*_i\in\GL(V^*)$ by $\sigma^*_i(p):=p+p_i\sum_{j=1}^nk_{i,j}\alpha^*_j$ for $p=\sum_{j=1}^np_j\alpha^*_j$. It is well-known that the map $s_i\mapsto\sigma^*_i$ extends uniquely to an injective homomorphism $\sigma^*:W\to\GL(V^*)$, $w\mapsto\sigma^*_w$ \[BB;4.2.7\]. The following result, which is also interesting itself, shows that the autoequivalence induced by a tilting module has similar properties as $\sigma^*$. As we shall explain later, it is closely related to a result of Seidel-Thomas \[ST\]. .5em[**Theorem  **]{}[*(1) Let $V^*\to K_0(\Lambda)_{\ccc}$ be an isomorphism defined by $\alpha^*_i\mapsto[P_i]$. Then the induced isomorphism $\GL(V^*)\to\GL(K_0(\Lambda)_{\ccc})$ satisfies $\sigma^*_w\mapsto[\Lambda^w\Lotimes_\Lambda-]$ for any $w\in W$. (2) We have an action $t_i\mapsto(I_i\Lotimes_\Lambda-)$ of the braid group $B$ on $\dd^{\b}(\mod\Lambda)$.*]{} .5em We will give a proof of  and  after giving a series of preliminary results. Let us start with the following observation on tilting modules associated with a set of simple modules. .5em[**Lemma  **]{} *Let ${\bf S}$ be a set of simple $\Lambda^{\op}$-modules and $T\in\tilt_1\Lambda=\ii(\Lambda)$. Put $$e:=\sum_{1\le i\le n,\ S_i\in{\bf S}}e_i,\ \ \ I_{\bf S}:=\Lambda(1-e)\Lambda,\ \ \ U:=TI_{\bf S}\ \mbox{ and }\ V:=\hom_{\Lambda^{\op}}(I_{\bf S},T).$$* -.5em (1) $U$ is minimal amongst sub $\Lambda^{\op}$-modules of $T$ such that any composition factor of $T/U$ is in ${\bf S}$, and $V$ is maximal amongst sub $\Lambda^{\op}$-modules of $\Lambda$ such that $T\subset V$ and any composition factor of $V/T$ is in ${\bf S}$. \(2) $\top T_\Lambda$ and $\soc(\Lambda/T)_\Lambda$ do not have any common composition factor. \(3) $U,V\in\tilt_1\Lambda$. \(4) There is a path $V\to\cdots\to T\to\cdots\to U$ in $\tilt_1\Lambda$ with arrows indexed by ${\bf S}$. \(5) $V/U$ is a projective-injective $(\Lambda/I_{\bf S})^{\op}$-module which is a generator-cogenerator. .5em[Proof ]{} (1) is obvious. We obtain (2) by (2). Since we can obtain $U$ (resp. $V$) by applying $\nu^+$ (resp. $\nu^-$) to $T$ repeatedly, we have (3) and (4). We will now show (5). Since $\Lambda/I_{\bf S}$ is selfinjective by (3), we only have to show that $V/U$ is progenerator. Let $0\to P_1\to P_0\stackrel{f}{\to}V\to0$ be a minimal projective resolution of the $\Lambda^{\op}$-module $V$. By the choice of $V$, $\soc(\Lambda/V)_\Lambda$ does not contain any module in ${\bf S}$. We have a minimal projective resolution $P_0^*\to P_1^*\to\ext^2_{\Lambda^{\op}}(\Lambda/V,\Lambda)\to0$. Since $\ext^2_{\Lambda^{\op}}(\Lambda/V,\Lambda)=D(\Lambda/V)$ holds, $\top P_1$ does not contain any module in ${\bf S}$. Thus $P_1\subseteq P_0I_{\bf S}$ holds. Thus we have $P_0I_{\bf S}/P_1=U$ and $f^{-1}(U)=P_0I_{\bf S}$. Then $V/U\simeq P_0/P_0I_{\bf S}=P_0\otimes_\Lambda(\Lambda/I_{\bf S})$ implies that $V/U$ is a projective $(\Lambda/I_{\bf S})^{\op}$-module. Moreover,  implies that any module in ${\bf S}$ appears in $\top P_0=\top V$. Thus $V/U$ is a generator. ------------------------------------------------------------------------ .5em The next result is needed for showing that the action of $W$ described in  is well-defined. .5em[**Proposition  **]{}[*$T^{s_i^2}=T$ holds for any $T\in\tilt_1\Lambda$ and $1\le i\le n$.*]{} .5em[Proof ]{} Put ${\bf S}:=\{S_i\}$ and define $U$ and $V$ by . Since any composition factor of the $(\Lambda/I_{\bf S})^{\op}$-module $V/U$ is $S_i$ by (1), either $U=T$ or $V=T$ holds by (2). If $U=T$, then we have $T^{s_i}=V$ and $V^{s_i}=T$. If $V=T$, then we have $T^{s_i}=U$ and $U^{s_i}=T$. ------------------------------------------------------------------------ .5em Now we prove theorem  for the case when $\Lambda$ has only two simple modules. Then the quiver of $\Lambda$ is the double of one of the following valued graphs: $\stackrel{}{\bullet}\stackrel{(2\ 2)}{\mbox{------}}\stackrel{}{\bullet},\ \ \begin{picture}(10,10)\put(7,4){\circle{15}}\end{picture}\stackrel{}{\bullet}\stackrel{}{\mbox{------}}\stackrel{}{\bullet}\begin{picture}(10,10)\put(3,4){\circle{15}}\end{picture}\ \mbox{ or }\ \begin{picture}(10,10)\put(7,4){\circle{15}}\end{picture}\stackrel{}{\bullet}\stackrel{(a\ b)}{\mbox{------}}\stackrel{}{\bullet}\ ((a\ b)=(1\ 2)\mbox{ or }(2\ 1)).$ The corresponding affine Weyl group $W$ is presented by generators $s_1,s_2$ and relations $s_1^2=s_2^2=1$. Thus we obtain (1) in these cases by . Since any element in $W$ can be written as $s_1s_2s_1s_2\cdots$ or $s_2s_1s_2s_1\cdots$, we obtain (2) in these cases by the following proposition. .5em[**Proposition  **]{} *Assume that $\Lambda$ has only two simple modules.* \(1) The Hasse quiver of $\tilt_1\Lambda$ is as follows:           [$\begin{array}{ccccccccccc} \Lambda&\stackrel{2}{\longrightarrow}&I_2&\stackrel{1}{\longrightarrow}&I_2I_1&\stackrel{2}{\longrightarrow}&I_2I_1I_2&\stackrel{1}{\longrightarrow}&I_2I_1I_2I_1&\stackrel{2}{\longrightarrow}&\cdots\\ &\begin{picture}(20,0)\put(2,-2){$\scriptstyle 1$}\put(2,10){\vector(1,-1){15}}\end{picture}&&&&&&&&&\\ &&I_1&\stackrel{2}{\longrightarrow}&I_1I_2&\stackrel{1}{\longrightarrow}&I_1I_2I_1&\stackrel{2}{\longrightarrow}&I_1I_2I_1I_2&\stackrel{1}{\longrightarrow}&\cdots \end{array}$]{} \(2) The Hasse quiver of $\tilt_1\Lambda^{\op}$ is as follows:           [$\begin{array}{ccccccccccc} \Lambda&\stackrel{2}{\longrightarrow}&I_2&&I_2I_1&&I_2I_1I_2&&I_2I_1I_2I_1&&\cdots\\ &\begin{picture}(20,0)\put(2,-1){$\scriptstyle 1$}\put(2,10){\vector(1,-1){15}}\end{picture}&&\begin{picture}(20,0)\put(2,10){$\scriptstyle 1$}\put(12,10){$\scriptstyle 2$}\put(2,-5){\vector(1,1){15}}\put(2,10){\vector(1,-1){15}}\end{picture}&&\begin{picture}(20,0)\put(2,10){$\scriptstyle 1$}\put(12,10){$\scriptstyle 2$}\put(2,-5){\vector(1,1){15}}\put(2,10){\vector(1,-1){15}}\end{picture}&&\begin{picture}(20,0)\put(2,10){$\scriptstyle 1$}\put(12,10){$\scriptstyle 2$}\put(2,-5){\vector(1,1){15}}\put(2,10){\vector(1,-1){15}}\end{picture}&&\begin{picture}(20,0)\put(2,10){$\scriptstyle 1$}\put(12,10){$\scriptstyle 2$}\put(2,-5){\vector(1,1){15}}\put(2,10){\vector(1,-1){15}}\end{picture}&\\ &&I_1&&I_1I_2&&I_1I_2I_1&&I_1I_2I_1I_2&&\cdots \end{array}$]{} \(3) The Hasse quiver of the reverse inclusion order in $\ii(\Lambda)$ is as follows:           [$\begin{array}{ccccccccccc} \Lambda&\stackrel{}{\longrightarrow}&I_2&\stackrel{}{\longrightarrow}&I_2I_1&\stackrel{}{\longrightarrow}&I_2I_1I_2&\stackrel{}{\longrightarrow}&I_2I_1I_2I_1&\stackrel{}{\longrightarrow}&\cdots\\ &\begin{picture}(20,0)\put(2,10){\vector(1,-1){15}}\end{picture}&&\begin{picture}(20,0)\put(2,-5){\vector(1,1){15}}\put(2,10){\vector(1,-1){15}}\end{picture}&&\begin{picture}(20,0)\put(2,-5){\vector(1,1){15}}\put(2,10){\vector(1,-1){15}}\end{picture}&&\begin{picture}(20,0)\put(2,-5){\vector(1,1){15}}\put(2,10){\vector(1,-1){15}}\end{picture}&&\begin{picture}(20,0)\put(2,-5){\vector(1,1){15}}\put(2,10){\vector(1,-1){15}}\end{picture}&\\ &&I_1&\stackrel{}{\longrightarrow}&I_1I_2&\stackrel{}{\longrightarrow}&I_1I_2I_1&\stackrel{}{\longrightarrow}&I_1I_2I_1I_2&\stackrel{}{\longrightarrow}&\cdots \end{array}$]{} .5em[Proof ]{} Using (2) and , we have (1). Considering $\Lambda^{\op}$, we have (2). To show (3), we only have to care about arrows, and it is easy. ------------------------------------------------------------------------ ,5em In the rest of this section, assume that $\Lambda$ has at least three simple modules. We next investigate the Loewy series for the factor algebra given by the ideal associated with a set of two simple modules. .5em[**Lemma  **]{} *(1) The (Loewy) length of $\Lambda/I_i$ is at most two.* \(2) Let ${\bf S}:=\{S_i,S_j\}$ with $i\neq j$. Then the Loewy series of $\Lambda/I_{\bf S}$ is as follows: [$$\begin{array}{|c|c|c|c|c|c|c|}\hline \stackrel{i}{\bullet}\stackrel{}{\mbox{------}}\stackrel{j}{\bullet}& \stackrel{i}{\bullet}\stackrel{(2\ 1)}{\mbox{------}}\stackrel{j}{\bullet}& \stackrel{i}{\bullet}\stackrel{(1\ 2)}{\mbox{------}}\stackrel{j}{\bullet}& \begin{picture}(10,10)\put(7,4){\circle{15}}\end{picture}\stackrel{i}{\bullet}\stackrel{}{\mbox{------}}\stackrel{j}{\bullet}& \stackrel{i}{\bullet}\stackrel{}{\mbox{------}}\stackrel{j}{\bullet}\begin{picture}(10,10)\put(3,4){\circle{15}}\end{picture}& \stackrel{i}{\bullet}\stackrel{(3\ 1)}{\mbox{------}}\stackrel{j}{\bullet}& \stackrel{i}{\bullet}\stackrel{(1\ 3)}{\mbox{------}}\stackrel{j}{\bullet}\\ \hline\def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle i}&{\scriptstyle j}\\ {\scriptstyle j}&{\scriptstyle i}\end{array}\right]& \def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle i}&{\scriptstyle j}\\ {\scriptstyle j\ j}&{\scriptstyle i}\\ {\scriptstyle i}&{\scriptstyle j}\end{array}\right]& \def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle i}&{\scriptstyle j}\\ {\scriptstyle j}&{\scriptstyle i\ i}\\ {\scriptstyle i}&{\scriptstyle j}\end{array}\right]& \def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle i}&{\scriptstyle j}\\ {\scriptstyle i\ j}&{\scriptstyle i}\\ {\scriptstyle i\ j}&{\scriptstyle i}\\ {\scriptstyle i}&{\scriptstyle j}\end{array}\right]& \def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle i}&{\scriptstyle j}\\ {\scriptstyle j}&{\scriptstyle i\ j}\\ {\scriptstyle j}&{\scriptstyle i\ j}\\ {\scriptstyle i}&{\scriptstyle j}\end{array}\right]& \def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle i}&{\scriptstyle j}\\ {\scriptstyle j\ j\ j}&{\scriptstyle i}\\ {\scriptstyle i\ i}&{\scriptstyle j\ j}\\ {\scriptstyle j\ j\ j}&{\scriptstyle i}\\ {\scriptstyle i}&{\scriptstyle j}\end{array}\right]& \def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle i}&{\scriptstyle j}\\ {\scriptstyle j}&{\scriptstyle i\ i\ i}\\ {\scriptstyle i\ i}&{\scriptstyle j\ j}\\ {\scriptstyle j}&{\scriptstyle i\ i\ i}\\ {\scriptstyle i}&{\scriptstyle j}\end{array}\right]\\ \hline\end{array}$$]{} .5em[Proof ]{} By (1), the category $\pr\Lambda$ of projective $\Lambda$-modules over a 2-CY algebra $\Lambda$ forms a $\tau$-category in the sense of \[I1,2\]. For any idempotent $e$ of $\Lambda$, the category $\pr(\Lambda/I)$ also forms a $\tau$-category by \[I2;1.4\]. We give an indication of the proof, referring to \[I1,2\] for definitions. \(1) We only have to consider the case when the quiver of $\Lambda$ has a loop at the vertex $i$. Put $\overline{\Lambda}:=\Lambda/I_i$. Since $\pr\overline{\Lambda}$ is a $\tau$-category, we have a minimal projective resolution $\overline{\Lambda}\stackrel{f}{\to}\overline{\Lambda}\stackrel{}{\to}\overline{\Lambda}\to S_i\to0$ such that the map $(f\cdot):\overline{\Lambda}\to\overline{\Lambda}$, which is obtained by applying $\hom_{\overline{\Lambda}}(-,\overline{\Lambda})$, has a simple cokernel. Thus $f$ does not belong to $J_{\overline{\Lambda}}^2$, and the cokernel of $f$ is simple. Thus $\overline{\Lambda}$ has (Loewy) length two. \(2) Put $\overline{\Lambda}:=\Lambda/I_{\bf S}$. We explain for the case $\stackrel{i}{\bullet}\stackrel{(3\ 1)}{\mbox{------}}\stackrel{j}{\bullet}$. In this case, the first three terms of the minimal projective resolutions of the simple $\overline{\Lambda}^{\op}$-modules $S_i$ and $S_j$ are given by $\overline{P}_i\to \overline{P}_j^3\to \overline{P}_i\to S_i\to0$ and $\overline{P}_j\to \overline{P}_i\to \overline{P}_j\to S_j\to0$. By \[I1;4.1,7.1\], we have commutative diagrams [$$\begin{diag} \overline{P}_i&\supset&\overline{P}_iJ_{\overline{\Lambda}}&\supset&\overline{P}_iJ_{\overline{\Lambda}}^2&\supset&\overline{P}_iJ_{\overline{\Lambda}}^3&\supset&\overline{P}_iJ_{\overline{\Lambda}}^4&\supset&\overline{P}_iJ_{\overline{\Lambda}}^5\\ \uparrow&&\uparrow&&\uparrow&&\uparrow&&\uparrow&&\uparrow\\ \overline{P}_i&\longleftarrow&\overline{P}_j^3&\longleftarrow&\overline{P}_i^2&\longleftarrow&\overline{P}_j^3&\longleftarrow&\overline{P}_i&\longleftarrow&0\\ \uparrow&*&\uparrow&*&\uparrow&*&\uparrow&*&\uparrow&*&\uparrow\\ 0&\longleftarrow&\overline{P}_i&\longleftarrow&\overline{P}_j^3&\longleftarrow&\overline{P}_i^2&\longleftarrow&\overline{P}_j^3&\longleftarrow&\overline{P}_i \end{diag}\ \ \ \ \ \ \ \ \ \ \begin{diag} \overline{P}_j&\supset&\overline{P}_jJ_{\overline{\Lambda}}&\supset&\overline{P}_jJ_{\overline{\Lambda}}^2&\supset&\overline{P}_jJ_{\overline{\Lambda}}^3&\supset&\overline{P}_jJ_{\overline{\Lambda}}^4&\supset&\overline{P}_jJ_{\overline{\Lambda}}^5\\ \uparrow&&\uparrow&&\uparrow&&\uparrow&&\uparrow&&\uparrow\\ \overline{P}_j&\longleftarrow&\overline{P}_i&\longleftarrow&\overline{P}_j^2&\longleftarrow&\overline{P}_i&\longleftarrow&\overline{P}_j&\longleftarrow&0\\ \uparrow&*&\uparrow&*&\uparrow&*&\uparrow&*&\uparrow&*&\uparrow\\ 0&\longleftarrow&\overline{P}_j&\longleftarrow&\overline{P}_i&\longleftarrow&\overline{P}_j^2&\longleftarrow&\overline{P}_i&\longleftarrow&\overline{P}_j \end{diag}$$]{} called ladders, with the following properties: \(i) Each column gives the first two terms of a projective resolution of $\overline{P}_iJ_{\overline{\Lambda}}^k$ and $\overline{P}_jJ_{\overline{\Lambda}}^k$. \(ii) The mapping cone of each commutative square $*$ gives the first three terms of a minimal projective resolution of semisimple $\overline{\Lambda}^{\op}$-modules. By (i), the Loewy series of $\overline{P}_i$ is $(\overline{P}_iJ_{\overline{\Lambda}}^k/\overline{P}_iJ_{\overline{\Lambda}}^{k+1})_{k\ge0}=(S_i,S_j^3,S_i^2,S_j^3,S_i)$, and that of $\overline{P}_j$ is $(\overline{P}_jJ_{\overline{\Lambda}}^k/\overline{P}_jJ_{\overline{\Lambda}}^{k+1})_{k\ge0}=(S_j,S_i,S_j^2,S_i,S_j)$. ------------------------------------------------------------------------ .5em[**Lemma  **]{}[*For ${\bf S}:=\{S_i,S_j\}$ with $i\neq j$, put $\overline{\Lambda}:=\Lambda/I_{\bf S}$. Let $\overline{P}\in\mod\overline{\Lambda}^{\op}$ be a progenerator. Define $X_k,Y_k\in\mod\overline{\Lambda}^{\op}$ by $X_0=Y_0:=\overline{P}$, $X_{2k+1}:=X_{2k}I_i$, $X_{2k+2}:=X_{2k+1}I_j$, $Y_{2k+1}:=Y_{2k}I_j$ and $Y_{2k+2}:=Y_{2k+1}I_i$ for $k\ge0$. Then we have $X_{m(i,j)}=0=Y_{m(i,j)}$ and $X_k\neq0$, $Y_k\neq0$ for any $k<m(i,j)$.*]{} .5em[Proof ]{} Obviously we only have to consider the case $\overline{P}=\overline{\Lambda}$. \(i) We consider the case when the quiver of $\overline{\Lambda}$ has no loop. In this case, we can easily check the assertion by using (2). For example, the calculation of $X_k$ and $Y_k$ for the case $\stackrel{i}{\bullet}\stackrel{(2\ 1)}{\mbox{------}}\stackrel{j}{\bullet}$ is as follows: [$$\begin{aligned} X_0=\overline{\Lambda}=\def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle i}&{\scriptstyle j}\\ {\scriptstyle j\ j}&{\scriptstyle i}\\ {\scriptstyle i}&{\scriptstyle j}\end{array}\right]\supset X_1=\def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle }&{\scriptstyle j}\\ {\scriptstyle j\ j}&{\scriptstyle i}\\ {\scriptstyle i}&{\scriptstyle j}\end{array}\right]\supset X_2=\def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle }&{\scriptstyle }\\ {\scriptstyle }&{\scriptstyle i}\\ {\scriptstyle i}&{\scriptstyle j}\end{array}\right]\supset X_3=\def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle }&{\scriptstyle }\\ {\scriptstyle }&{\scriptstyle }\\ {\scriptstyle }&{\scriptstyle j}\end{array}\right]\supset X_4=0\\ Y_0=\overline{\Lambda}=\def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle i}&{\scriptstyle j}\\ {\scriptstyle j\ j}&{\scriptstyle i}\\ {\scriptstyle i}&{\scriptstyle j}\end{array}\right]\supset Y_1=\def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle i}&{\scriptstyle }\\ {\scriptstyle j\ j}&{\scriptstyle i}\\ {\scriptstyle i}&{\scriptstyle j}\end{array}\right]\supset Y_2=\def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle }&{\scriptstyle }\\ {\scriptstyle j\ j}&{\scriptstyle }\\ {\scriptstyle i}&{\scriptstyle j}\end{array}\right]\supset Y_3=\def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle }&{\scriptstyle }\\ {\scriptstyle }&{\scriptstyle }\\ {\scriptstyle i}&{\scriptstyle }\end{array}\right]\supset Y_4=0\end{aligned}$$]{} Thus the assertion follows. One can check the other cases similarly. \(ii) We consider the case $\begin{picture}(10,10)\put(7,4){\circle{15}}\end{picture}\stackrel{i}{\bullet}\stackrel{}{\mbox{------}}\stackrel{j}{\bullet}$. It follows from (1) that $\Lambda/I_i$ has length $2$. Thus the composition factors of $\overline{P}_i/\overline{P}_iI_i$ are two copies of $S_i$. Since $\overline{\Lambda}$ is selfinjective, $\overline{P}_i$ contains a submodule $X$ whose composition factors are two copies of $S_i$. Then $X$ is contained in $\overline{P}_iI_i$, and the composition factors of $\overline{P}_iI_i/X$ are two copies of $S_j$. Thus we can calculate $M_i$ as follows: [$$\begin{aligned} X_0=\overline{\Lambda}=\def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle i}&{\scriptstyle j}\\ {\scriptstyle i\ j}&{\scriptstyle i}\\ {\scriptstyle i\ j}&{\scriptstyle i}\\ {\scriptstyle i}&{\scriptstyle j}\end{array}\right]\supset X_1=\def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle }&{\scriptstyle j}\\ {\scriptstyle j}&{\scriptstyle i}\\ {\scriptstyle i\ j}&{\scriptstyle i}\\ {\scriptstyle i}&{\scriptstyle j}\end{array}\right]\supset X_2=\def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle }&{\scriptstyle }\\ {\scriptstyle }&{\scriptstyle i}\\ {\scriptstyle i}&{\scriptstyle i}\\ {\scriptstyle i}&{\scriptstyle j}\end{array}\right]\supset X_3=\def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle }&{\scriptstyle }\\ {\scriptstyle }&{\scriptstyle }\\ {\scriptstyle }&{\scriptstyle }\\ {\scriptstyle }&{\scriptstyle j}\end{array}\right]\supset X_4=0\\ Y_0=\overline{\Lambda}=\def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle i}&{\scriptstyle j}\\ {\scriptstyle i\ j}&{\scriptstyle i}\\ {\scriptstyle i\ j}&{\scriptstyle i}\\ {\scriptstyle i}&{\scriptstyle j}\end{array}\right]\supset Y_1=\def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle i}&{\scriptstyle }\\ {\scriptstyle i\ j}&{\scriptstyle i}\\ {\scriptstyle i\ j}&{\scriptstyle i}\\ {\scriptstyle i}&{\scriptstyle j}\end{array}\right]\supset Y_2=\def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle }&{\scriptstyle }\\ {\scriptstyle j}&{\scriptstyle }\\ {\scriptstyle i\ j}&{\scriptstyle }\\ {\scriptstyle i}&{\scriptstyle j}\end{array}\right]\supset Y_3=\def\arraystretch{.5}\left[\begin{array}{c|c} {\scriptstyle }&{\scriptstyle }\\ {\scriptstyle }&{\scriptstyle }\\ {\scriptstyle i}&{\scriptstyle }\\ {\scriptstyle i}&{\scriptstyle }\end{array}\right]\supset Y_4=0\end{aligned}$$]{} Thus the assertion follows. We can treat the case $\stackrel{i}{\bullet}\stackrel{}{\mbox{------}}\stackrel{j}{\bullet}\begin{picture}(10,10)\put(3,4){\circle{15}}\end{picture}$ similarly. ------------------------------------------------------------------------ .5em The following result is crucial for well-defined action of the affine Weyl and braid groups. .5em[**Proposition  **]{} *Let $i\neq j$ be distinct vertices in $\Delta$,* \(1) $T^{(s_is_j)^{m(i,j)}}=T$ holds for any $T\in\tilt_1\Lambda$. \(2) $\tilt_1\Lambda$ has a subquiver [$$\begin{diag} V&\RA{i}&\bullet&\RA{j}&\bullet&\RA{i}&\cdots\cdots&\RA{i\mbox{\tiny\ or }j}&\bullet&\RA{j\mbox{\tiny\ or }i}&U.\\ &\DRA{j}&\bullet&\RA{i}&\bullet&\RA{j}&\cdots\cdots&\RA{j\mbox{\tiny\ or }i}&\bullet&\URA{i\mbox{\tiny\ or }j} \end{diag}$$]{} with two paths of length $m(i,j)$ such that $T$ is one of these vertices. \(3) We have an equality $I_i\Lotimes_\Lambda I_j\Lotimes_\Lambda I_i\Lotimes_\Lambda I_j\Lotimes_\Lambda\cdots=I_iI_jI_iI_j\cdots=\Lambda(1-e_i-e_j)\Lambda=I_jI_iI_jI_i\cdots=I_j\Lotimes_\Lambda I_i\Lotimes_\Lambda I_j\Lotimes_\Lambda I_i\Lotimes_\Lambda\cdots$, where each derived tensor product and product of ideals contains exactly $m(i,j)$ terms. .5em[Proof ]{} Put ${\bf S}:=\{S_i,S_j\}$, and consider $U$ and $V$ as defined in . Then $P:=V/U$ is a progenerator of $\Lambda/I_{\bf S}$ by (5). Consider the sequences $(X_0,X_1,\cdots,X_{m(i,j)})$ and $(Y_0,Y_1,\cdots,Y_{m(i,j)})$ in . Then the preimage of these sequences under the natural surjection $V\to P=V/U$ coincides with the sequences $(V,V^{s_i},V^{s_is_j},V^{s_is_js_i},\cdots)$ and $(V^{s_j},V^{s_js_i},V^{s_js_is_j},\cdots)$. Since $X_{m(i,j)}=0=Y_{m(i,j)}$ holds, we have $V^{s_is_js_i\cdots}=U=V^{s_js_is_j\cdots}$ where both $s_is_js_i\cdots$ and $s_js_is_j\cdots$ are products of $m(i,j)$ simple reflections. Thus we have proved (2). Since $T$ belongs to one of these sequences by (4), we obtain (1). Applying (2) and (2) to $T:=\Lambda$, we have (3). ------------------------------------------------------------------------ .5em We can now prove most of  and . .5em[**Proof of (1) and  **]{} We first show (1). By  and , $W$ acts on $\tilt_1\Lambda$. By , the action is transitive. For freeness of the action, it is enough to prove (1) since the geometric representation $\sigma^*:W\to\GL(V^*)$ is injective. \(2) follows from (3). We will show (1). First we show that $[I_i\Lotimes_\Lambda-]$ corresponds to $\sigma^*_i$. If $i\neq j$, then $[I_i\Lotimes_\Lambda P_j]=[I_ie_j]=[P_j]$. Let us calculate $[I_i\Lotimes_\Lambda P_i]=[I_ie_i]$. If the quiver of $\Lambda$ has no loop at the vertex $i$, then we have a minimal projective resolution $0\to P_i\to\bigoplus_{j\neq i}P_j^{k_{i,j}}\stackrel{f}{\to}P_i\to S_i\to 0$ with $\Im f_i=I_ie_i$. Thus we have $[I_ie_i]=[P_i]+\sum_{j=1}^nk_{i,j}[P_j]$. Assume that the quiver of $\Lambda$ has a loop at the vertex $i$. In this case, the composition factors of $P_i/I_ie_i$ are two copies of $S_i$ by (1). Since we have a minimal projective resolution $0\to P_i\to P_i\oplus(\bigoplus_{j\neq i}P_j^{k_{i,j}/2})\stackrel{}{\to}P_i\to S_i\to 0$ by our definition of $k_{i,j}$, we have $[I_ie_i]=[P_i]-2[S_i]=[P_i]-2([P_i]-\sum_{j\neq i}k_{i,j}[P_j]/2)=[P_i]+\sum_{j=1}^nk_{i,j}[P_j]$. Thus $[I_i\Lotimes_\Lambda-]$ corresponds to $\sigma^*_i$. Now take any $w=s_{a_1}\cdots s_{a_k}\in W$. Then $\Lambda^w=I_{a_1}^{\pm1}\Lotimes_\Lambda\cdots\Lotimes_\Lambda I_{a_k}^{\pm1}$ for $I_{a_i}^{-1}:=\Rhom_{\Lambda^{\op}}(I_{a_i},\Lambda)$. Since $[I_{a_i}^{-1}\Lotimes_\Lambda-]=[I_{a_i}\Lotimes_\Lambda-]$ by $(\sigma^*_i)^2=1_{V^*}$, we have that $[\Lambda^w\Lotimes_\Lambda-]=[I_{a_1}\Lotimes_\Lambda-]\cdots[I_{a_k}\Lotimes_\Lambda-]$ corresponds to $\sigma^*_{a_1}\cdots\sigma^*_{a_k}=\sigma^*_w$. Thus (1) follows. ------------------------------------------------------------------------ .5em It remains to prove (2). For this, we need the following lemma. For simplicity, we write $T\stackrel{i}{\leftrightarrow}U$ if there is an arrow $T\stackrel{i}{\to}U$ or $T\stackrel{i}{\leftarrow}U$ in $\tilt_1\Lambda$. For $w\in W$, we denote by $k(w)$ the length of the shortest path in $\tilt_1\Lambda$ from $\Lambda$ to $\Lambda^w$. .5em[**Lemma  **]{} *Let $w,w^\prime\in W$.* \(1) There is an arrow $\Lambda^{w^\prime}\stackrel{i}{\leftrightarrow}\Lambda^w$ in $\tilt_1\Lambda$ if and only if $w=w^\prime s_i$. \(2) There is a subquiver $\Lambda=T_0\stackrel{a_1}{\leftrightarrow}T_1\stackrel{a_2}{\leftrightarrow}\cdots\stackrel{a_k}{\leftrightarrow}T_k=\Lambda^w$ in $\tilt_1\Lambda$ if and only if $w=s_{a_1}\cdots s_{a_k}$ holds. \(3) Any path in $\tilt_1\Lambda$ from $\Lambda$ to $\Lambda^w$ has length $k(w)$. \(4) $\Lambda^w\le\Lambda^{w^\prime}$ in $\tilt_1\Lambda$ if and only if there exists a path in $\tilt_1\Lambda$ from $\Lambda^w$ to $\Lambda^{w^\prime}$. \(5) If there is an arrow $\Lambda^w\stackrel{}{\to}\Lambda^{w^\prime}$ in $\tilt_1\Lambda$, then $k(w^\prime)=k(w)+1$. \(6) $k(w)=l(w)$. \(7) There is a subquiver $\Lambda=T_0\stackrel{a_1}{\to}T_1\stackrel{a_2}{\to}\cdots\stackrel{a_k}{\to}T_k=\Lambda^w$ in $\tilt_1\Lambda$ if and only if $w=s_{a_1}\cdots s_{a_k}$ is a reduced expression of $w$. .5em[Proof ]{} (1)(2) Immediate from (1). \(3) We use induction on $k(w)$. If $k(w)=0$, then $w=1$, and there is no non-trivial path in $\tilt_1\Lambda$ from $\Lambda$ to $\Lambda$. Assume that the assertion is true for any $w\in W$ with $k(w)<k$. Fix $w\in W$ with $k(w)=k$. Take a path $\Lambda\stackrel{a_1}{\to}\cdots\stackrel{a_k}{\to}\Lambda^w$ of length $k$, and an arbitrary path $\Lambda\stackrel{b_1}{\to}\cdots\stackrel{b_l}{\to}\Lambda^w$ of length $l$. We will show $l=k$. By , $\tilt_1\Lambda$ has a subquiver [$$\begin{diag} \Lambda^{v}&\RA{a_k\mbox{\tiny\ or }b_l}&\cdots\cdots&\RA{a_k}&\Lambda^{wa_kb_l}&\RA{b_l}&\Lambda^{wa_k}&\RA{a_k}&\Lambda^{w}\\ &\DRA{b_l\mbox{\tiny\ or }a_k}&\cdots\cdots&\RA{b_l}&\Lambda^{wb_la_k}&\RA{a_k}&\Lambda^{wb_l}&\URA{b_l} \end{diag}$$]{} consisting of two paths of length $m:=m(a_k,b_l)$. Since there is a path $\Lambda\stackrel{a_1}{\to}\cdots\stackrel{a_{k-1}}{\to}\Lambda^{wa_k}$ of length $k-1$, any path from $\Lambda$ to $\Lambda^{wa_k}$ has length $k-1$ by the inductive assumption. In particular, we have $k(v)=k-m$. Since we have $k(wb_l)\le k(v)+m-1=k-1$ and $k(wb_l)\ge k(w)-1=k-1$, we have $k(wb_l)=k-1$. Since we have a path $\Lambda\stackrel{b_1}{\to}\cdots\stackrel{b_{l-1}}{\to}\Lambda^{wb_l}$ of length $l-1$, we have $l-1=k-1$ by the inductive assumption again. Thus $l=k$. \(4) The ‘if’ part is obvious. We show the ‘only if’ part. By (1)(ii), we have a path $\cdots\to T_1\to T_0=\Lambda^{w^\prime}$ such that $\Lambda^w\le T_i$ for any $i$. We have $\Lambda^w=T_k$ for some $k$ by (3). \(5) Immediate from (3). \(6) By (2), any path from $\Lambda$ to $\Lambda^w$ gives a presentation of $w$. Thus we have $k(w)\ge l(w)$. Take a presentation $w=s_{a_1}\cdots s_{a_l}$ with $l=l(w)$. By (2) again, there exists a subquiver $\Lambda=T_0\stackrel{a_1}{\leftrightarrow}\cdots\stackrel{a_l}{\leftrightarrow}T_l=\Lambda^w$ in $\tilt_1\Lambda$. This implies $k(w)\le l$ by (5). \(7) Immediate from (2)(5) and (6). ------------------------------------------------------------------------ .5em We are now in the position to finish the proof of . .5em[**Proof of (2)** ]{} We first show (ii). By (1)(5) and (6), there is an arrow $\Lambda^{w^\prime}\stackrel{i}{\to}\Lambda^w$ in $\tilt_1\Lambda$ if and only if $w=w^\prime s_i$ and $l(w^\prime)<l(w)$. This implies that the Hasse quiver of $\tilt_1\Lambda$ and that of the poset $W$ with the right order coincide. It follows from (4) that (ii) holds. One can show (iii) by a dual argument. We now show (i) and (iv). Assume $w^\prime\le w$. Take a reduced expression $w=s_{a_1}\cdots s_{a_k}$. Then $\Lambda^w=I_{a_1}\cdots I_{a_k}$ holds by (7). Since $w^\prime=s_{a_{i_1}}\cdots s_{a_{i_q}}$ holds for some $1\le i_1<\cdots<i_q\le k$, we have $\Lambda^{w^\prime}\supseteq I_{a_{i_1}}\cdots I_{a_{i_q}}$. Thus $\Lambda^{w^\prime}\supseteq\Lambda^w$ holds. Conversely, assume $\Lambda^{w^\prime}\supseteq\Lambda^{w}$. Using induction on $\length(\Lambda/\Lambda^w)_\Lambda$, we will show $w^\prime\le w$. Fix $S_i\in\soc(\Lambda/\Lambda^{w})_\Lambda$, so we have $ws_i<w$. If $\Lambda^{w^\prime}\supseteq\Lambda^{ws_i}$, then we have $w^\prime\le ws_i<w$ inductively. If $\Lambda^{w^\prime}\ {\not\supseteq}\ \Lambda^{ws_i}$, then $S_i\in\soc(\Lambda/\Lambda^{w^\prime})_\Lambda$ holds, and we have [$$\begin{diag} \Lambda^{w^\prime s_i}&\ \ \ \supseteq\ \ \ &\Lambda^{ws_i}\\ \cup&&\cup\\ \Lambda^{w^\prime}&\ \ \ \supseteq\ \ \ &\Lambda^w \end{diag}$$]{} -1em Inductively, we have $w^\prime s_i\le ws_i$. Applying \[Hu;5.9\] to $w^\prime s_i$ and $ws_i$, either $w^\prime\le ws_i$ or $w^\prime\le w$ holds. In any case, we have $w^\prime\le w$. Thus we have shown (i). ------------------------------------------------------------------------ .5em The [*derived Picard group*]{} $\dpic_R(\Lambda)$ of $\Lambda$ was introduced by Yekutieli \[Ye2\] (see also \[RZ\]\[MY\]). The elements of $\dpic_R(\Lambda)$ are isoclasses of two-sided tilting complexes $T\in\dd^{\b}(\mod\Lambda\otimes_R\Lambda^{\op})$, and the multiplication of $T$ and $T'$ is given by $T\Lotimes_\Lambda T'$. Then the inverse of $T$ is given by $\Rhom_\Lambda(T,\Lambda)\simeq\Rhom_{\Lambda^{\op}}(T,\Lambda)$. We have a group homomorphism from $\dpic_R(\Lambda)$ to the group $\auteq_R(\dd^{\b}(\mod\Lambda))$ of autoequivalences of $\dd^{\b}(\mod\Lambda)$ defined by $T\mapsto(T\Lotimes_\Lambda-)$. When $\Lambda$ is 2-CY, we have the elements $I_1,\cdots,I_n$ of $\dpic_R(\Lambda)$ which satisfy the braid relations by (3). Inspired by study in algebraic geometry (e.g. \[BO\]\[Bri4\]\[IU\]), we have the following natural questions. .5em[**Questions** ]{} (1) Do $I_1,\cdots,I_n$ together with the shift $[1]$ and the outer automorphism group $\out_R(\Lambda)$ of $\Lambda$ generate $\dpic_R(\Lambda)$? \(2) Is $\dpic_R(\Lambda)$ isomorphic to $(B\timesl\out_R(\Lambda))\times\zzz$ for the affine braid group $B$ and the group $\zzz$ generated by $[1]$? \(3) Is the homomorphism $\dpic_R(\Lambda)\to\auteq_R(\dd^{\b}(\mod\Lambda))$ an isomorphism? .5em Now we consider the case $\Lambda=S*G$ for $S=K[[x,y]]$ for a field $K$ of characteristic 0 and a finite subgroup $G$ of $\SL_2(K)$. In this case, there is a triangle equivalence ([*McKay correspondence*]{}) between $\dd^{\b}(\mod\Lambda)$ and $\dd^{\b}(\Coh X)$ for a minimal resolution $X$ of the singularity $\Spec S^G$ \[KV\]. When $G$ is cyclic, Ishii-Uehara \[IU\] determined generators of the subgroup $\auteq^{\rm FM}(\dd^{\b}(\Coh X))$ consisting of Fourier-Mukai transformations. Consequently, the first question should have a positive answer for this case. .5em We end this section by showing that the spherical objects introduced by Seidel-Thomas \[ST\] give rise to tilting complexes, and hence to autoequivalences of $\dd^{\b}(\mod\Lambda)$. We assume that $\Lambda$ is a projective $R$-module and $\gl\Lambda<\infty$. We call $S\in\dd^{\b}(\mod\Lambda)$ [*$n$-spherical*]{} ($n>0$) if the following conditions (1) and (2) are satisfied. \(1) $E:=\endm_{\dd(\Mod\Lambda)}(S)$ is a division algebra, \(2) $\dim_E\hom_{\dd(\Mod\Lambda)}(S,S[i])=\left\{\begin{array}{cc} 1&\mbox{if $i=0$ or $n$,}\\ 0&\mbox{otherwise.} \end{array}\right.$ It follows from a result of Keller \[K1,2\] that condition (2) implies that $S$ comes from an object in $\dd^{\b}(\mod\Lambda\otimes_RE^{\op})$. Let us recall the definition due to Seidel and Thomas \[ST\] of autoequivalences of $\dd^{\b}(\mod\Lambda)$ called [*twist functors*]{}. We treat here only dual twist functors. To obtain functoriality, we first construct a functor $\ttt_S:\kk^{\b}(\pr\Lambda)\to\kk^{\b}(\mod\Lambda)$. For $P\in\kk^{\b}(\pr\Lambda)$, put $$\ttt_S(P):=(P\stackrel{e_P}{\longrightarrow}\hom^\bullet_{E^{\op}}(\hom^\bullet_{\Lambda}(P,S),S)).$$ where $e_P$ is the evaluation map, and $\ttt_S(P)$ is defined as mapping cones such that $P$ is in degree 0. Composing $\ttt_S$ with natural functors $\dd^{\b}(\mod\Lambda)\stackrel{\sim}{\to}\kk^{\b}(\pr\Lambda)$ and $\kk^{\b}(\mod\Lambda)\to\dd^{\b}(\mod\Lambda)$, we obtain an autoequivalence $\ttt_S$ on $\dd^{\b}(\mod\Lambda)$ \[ST\]. Note that $\ttt_S$ and $\ttt_{S'}$ are isomorphic if $S$ and $S'$ are quasi-isomorphic objects in $\kk^{\b}(\mod\Lambda\otimes_RE^{\op})$. .5em[**Theorem  **]{}[*For any $n$-spherical object $S\in\dd^{\b}(\mod\Lambda)$, $T:=\ttt_S(\Lambda)$ is a two-sided tilting complex of $\Lambda$ and there is an isomorphism $\ttt_S\simeq(T\Lotimes_\Lambda-)$ of functors on $\dd^{\b}(\mod\Lambda)$.*]{} .5em[Proof ]{} By definition, we have $T=(\Lambda\stackrel{e_\Lambda}{\longrightarrow}\hom^\bullet_{E^{\op}}(S,S))$. Since $S$ is a complex of $(\Lambda,E)$-modules, $e_\Lambda$ is a chain homomorphism of complexes of $(\Lambda,\Lambda)$-modules. Thus $T$ is a complex of $(\Lambda,\Lambda)$-modules. Since $\ttt_S$ is an autoequivalence, $T$ is a two-sided tilting complex of $\Lambda$. Now, applying $(-\otimes^\bullet_\Lambda P)$ for any $P\in\kk^{\b}(\pr\Lambda)$, we have $T\otimes^\bullet_\Lambda P=(P\stackrel{e_\Lambda\otimes P}{\longrightarrow}\hom^\bullet_{E^{\op}}(S,S)\otimes^\bullet_\Lambda P)$. We have natural isomorphisms [$$\begin{aligned} &&\hom^\bullet_{E^{\op}}(S,S)\otimes^\bullet_\Lambda P=S\otimes^\bullet_E\hom^\bullet_{E^{\op}}(S,E)\otimes^\bullet_\Lambda P\\ &=&S\otimes^\bullet_E\hom^\bullet_{E^{\op}}(\hom^\bullet_\Lambda(P,S),E)=\hom^\bullet_{E^{\op}}(\hom^\bullet_\Lambda(P,S),S),\end{aligned}$$]{} and one can easily check that the diagram [$$\begin{diag} T\otimes^\bullet_\Lambda P:\ \ \ &P&\RA{e_\Lambda\otimes P}&\hom^\bullet_{E^{\op}}(S,S)\otimes^\bullet_\Lambda P\\ &\parallel&&\downarrow\wr&&\\ \ttt_S(P):\ \ \ &P&\RA{e_P}&\hom^\bullet_{E^{\op}}(\hom^\bullet_\Lambda(P,S),S) \end{diag}$$]{} of complexes of $\Lambda$-modules commutes. Thus we have a functorial isomorphism $T\otimes^\bullet_\Lambda P\simeq\ttt_S(P)$ on $\kk^{\b}(\pr\Lambda)$. ------------------------------------------------------------------------ 1.5em [**. 3-Calabi-Yau algebras and cluster algebras** ]{} Cluster algebras (with ‘no coefficients’ and in the skew-symmetric case) are completely determined by a finite quiver with no loops or 2-cycles. We show that for quivers of 3-CY algebras $\Lambda$, tilting theory (with tilting modules of projective dimension at most one, which we assume here) is a nice framework for modelling some of the ingredients in the definition of the corresponding cluster algebra. This motivates a closer investigation of tilting modules over 3-CY algebras, which we have already started in previous sections. Let $B=(b_{ij})$ be an $n\times n$ skew-symmetric matix with integer entries. The Fomin-Zelevinsky mutation $\mu_k$ ($1\le k\le n$) is defined by $\mu_k(B)= (b_{ij}')$, where [$$b_{ij}'=\left\{\begin{array}{cc} -b_{ij}&\mbox{if $i=k$ or $j=k$}\\ b_{ij}+\frac{|b_{ik}|b_{kj}+b_{ik}|b_{kj}|}{2}&\mbox{otherwise.} \end{array}\right.$$]{} Then $\mu_k(B)$ is skew-symmetric again and satisfies $\mu_k(\mu_k(B))=B$. We identify $B$ with the quiver $\qq$ with vertices $\{1,2,\cdots,n\}$ and $b_{ij}$ arrows from $i$ to $j$ if $b_{ij}>0$. In this way we have a one-one correspondence between skew-symmetric matix with integer entries and finite quivers with no loops or 2-cycles. Thus for a quiver $\qq$ with no loops or 2-cycles, the Fomin-Zelevinsky mutation $\mu_k(\qq)$, which is again a quiver with no loops or 2-cycles, is defined. Let $R$ be a 3-dimensional complete local Gorenstein ring with an algebraically closed residue field and $\Lambda$ a basic module-finite $R$-algebra which is 3-CY. The valued quiver of $\Lambda$ can be regarded as a (non-valued) quiver. We first show that when the quiver of $\Lambda$ has no loops or 2-cycles, we can interpret the Fomin-Zelevinsky mutation of the quiver via endomorphism rings $\Gamma$ of the non-projective completions of almost complete projective tilting modules. The situation is especially nice if the quiver of $\Gamma$ also has no loops or 2-cycles, so that the procedure can be repeated. We do not know if this is the case in general. But we give examples of where any successive application of our mutation of $\Lambda$ will give algebras with quivers having no loops or 2-cycles. We also show that all algebras obtained from $\Lambda$ via a sequence of mutations can be constructed from a tilting $\Lambda$-module, even in the case when the quiver of $\Lambda$ has loops and/or 2-cycles (see (2)). If there are no loops or 2-cycles, it is given by a reflexive tilting $\Lambda$-module (see (4)). Such a result is of interest for the connection with cluster algebras. Write $\Lambda =\bigoplus_{i=1}^nP^{(i)}$, and let $Q^{(k)}:=\bigoplus_{i\neq k}P^{(i)}$. Denote by $\check{P}^{(k)}\neq P^{(k)}$ the unique indecomposable $\Lambda$-module such that $\nu_k(\Lambda)=Q^{(k)}\oplus \check{P}^{(k)}$ is a tilting module. We put $\mu_k(\Lambda):=\endm_\Lambda(\nu_k(\Lambda))$, which is a 3-CY algebra again by (1). Assume that the quiver $\qq_{\Lambda}$ of $\Lambda$, with vertices $1, \cdots, n$, has no loops or 2-cycles. The aim of this section is to use tilting theory to obtain a module theoretical interpretation of the Fomin-Zelevinsky mutation of $\qq_{\Lambda}$ at the vertex $k$. We shall compare $\mu_k(\qq_\Lambda)$ with $\qq_{\mu_k(\Lambda)}$. Recall that for each $i$ we have a minimal projective resolution $$0\longrightarrow P^{(i)} \stackrel{f^{(i)}}\longrightarrow P_2^{(i)}{\longrightarrow}P_1^{(i)}\stackrel{g^{(i)}}\longrightarrow P^{(i)} \longrightarrow S^{(i)}\longrightarrow 0$$ of the simple $\Lambda$-module $S^{(i)}$ (), and $\check{P}^{(i)}:=\Ker g^{(i)}$. The quiver $\qq_{\Lambda}$ is determined by the maps $g^{(i)}:P^{(i)}_1 \to P^{(i)}$. Since by assumption there are no loops, or equivalently, $S_i$ is 3-spherical, then $P^{(i)}_1$ is in $\add Q^{(i)}$. Further $f^{(i)}:P^{(i)}\to P^{(i)}_2 $ is a minimal left $(\add Q^{(i)})$-approximation, and hence gives rise to the arrows in $\qq_{\Lambda}$ starting at $i$. Since there are no 2-cycles, $P^{(i)}_2$ and $P^{(i)}_1$ have no common indecomposable direct summand. Let $b_{ij}$ be the number of arrows from $i$ to $j$ if there are arrows from $i$ to $j$, and otherwise minus the number of arrows from $j$ to $i$. Then if $b_{ij}>0$, it is the multiplicity of $P^{(j)}$ as a summand of $P_1^{(i)}$, or equivalently, the multiplicity of $P^{(i)}$ in $P_2^{(j)}$. Our first goal is to prove the following, which is one of the main results of this section. .5em[**Theorem  **]{}[*Let $\Lambda$ be a basic 3-CY algebra. Assume the quiver $\qq_\Lambda$ of $\Lambda$ has no loops or 2-cycles. Then $\mu_k(\qq_\Lambda)$ is obtained from $\qq_{\mu_k(\Lambda)}$ by removing all 2-cycles.*]{} .5em[Proof ]{} Let $B=(b_{ij})$ be the skew symmetric matrix given by the quiver $\qq_{\Lambda}$ and $B'=(b'_{ij}):=\mu_k(B)$. Put $T:=\nu_k(\Lambda)$ and $\Gamma:=\mu_k(\Lambda)=\endm_\Lambda(T)$. Let $B''=(b_{ij}'')$ be the matrix corresponding to $\qq_{\Gamma}$, after we have removed all 2-cycles. Then we want to show that $b_{ij}'=b_{ij}''$. To avoid confusion we denote the vertex in $\qq_{\Gamma}$ corresponding to $k$ by $\check{k}$, and the same for the matrix $B'$. In order to compare the quivers $\qq_{\Lambda}$ and $\qq_{\Gamma}$, we need to consider (minimal) projective resolutions of simple $\Gamma$-modules and their relationship to the corresponding resolutions of simple $\Lambda$-modules. We divide up into different cases. [**Case 1** ]{} Assume that $i=k$. We want to show that $b_{j\check{k}}=-b_{jk}$ and $b_{\check{k}j}= -b_{kj}$. We have exact sequences [$$\begin{aligned} &0\to (T,P^{(k)})\to (T,P_2^{(k)})\to (T,\check{P}^{(k)})\to \check{S}^{(k)}\to 0,&\\ &0\to (T,\check{P}^{(k)})\to (T,P_1^{(k)})\to (T,P^{(k)})\to \Ext^1_\Lambda(T, \check{P}^{(k)}).&\end{aligned}$$]{} Since $\Ext^1_\Lambda(T,\check{P}^{(k)})=0$, we have an exact sequence of $\Lambda$-modules [$$0\to (T,\check{P}^{(k)})\to (T,P_1^{(k)})\to (T,P_2^{(k)})\to (T,\check{P}^{(k)})\to \check{S}^{(k)}\to 0.$$]{} Since $P_1^{(k)}$ and $P_2^{(k)}$ are in $\add Q^{(k)}$, and hence in $\add T$, and $(T,P_1^{(k)})$ and $(T,P_2^{(k)})$ have no common indecomposable direct summands, we have a minimal projective resolution of the $\Gamma$-module $\check{S}^{(k)}$. We want to show that $\check{S}^{(k)}$ is a simple $\Gamma$-module, or equivalently, that there is no loop in $\qq_{\Gamma}$ at the vertex $\check{k}$. For this we need to show that any non-isomorphic map $a:\check{P}^{(k)}\to \check{P}^{(k)}$ factors through an object in $\add Q^{(k)}$. So consider the commutative diagram [$$\begin{diag} 0&\RA{}&\check{P}^{(k)}&\RA{h}&P_1^{(k)}&\RA{g^{(k)}}&P^{(k)}\\ && \downarrow^{a}&& \downarrow^{b}&& \downarrow^{c}\\ 0&\RA{}&\check{P}^{(k)}&\RA{h}&P_1^{(k)}&\RA{g^{(k)}}&P^{(k)}. \end{diag}$$]{} Here $b$ and $c$ exist by (5) and (1). If $c$ is an isomorphism, then $b$ is also an isomorphism since $g^{(k)}$ is right minimal. Hence we get the contradiction that $a$ is an isomorphism. Since $c$ is not an isomorphism, $c$ factors through $g^{(k)}$. It is then easy to see that $a$ factors through $h$, where $P_1^{(k)}$ is in $\add Q^{(k)}$. Hence there is no loop at $k$. We now compare the projective resolutions of $S^{(k)}$ and $\check{S}^{(k)}$. Since $P_1^{(k)}$ and $P_2^{(k)}$ have no common indecomposable direct summands, the same is the case for $(T,P_1^{(k)})$ and $(T,P_2^{(k)})$. Hence we have $b_{j\check{k}}''=-b_{jk}=b_{jk}'$ and $b_{\check{k}j}''=-b_{kj}=b_{\check{k}j}'$. [**Case 2** ]{} Assume now that $i\neq k$ and $b_{ik}\ge0$, and consider $b_{ij}$ for $j\neq k$. Then $P_2^{(i)}\in\add Q^{(k)}$ and the multiplicity of $P^{(k)}$ in $P_1^{(i)}$ is $m:=b_{ik}$ because there are no cycles of length 2 in $\qq_{\Lambda}$ by assumption. Decompose $P_1^{(i)}=\overline{P}_1^{(i)}\oplus(P^{(k)})^m$ with $\overline{P}_1^{(i)}\in\add Q^{(k)}$. We can take a commutative diagram [$$\begin{diag} 0&\longrightarrow&P^{(i)}&\RA{g^{(i)}}&P_2^{(i)}&\RA{}&\overline{P}_1^{(i)}\oplus(P^{(k)})^m&\RA{f^{(i)}}&P^{(i)}\\ &&\downarrow&&\downarrow&&\downarrow^{{0\choose1}}\\ 0&\longrightarrow&(\check{P}^{(k)})^m&\RA{}&(P_1^{(k)})^m&\RA{(f^{(k)})^m}&(P^{(k)})^m \end{diag}$$]{} of exact sequences. This gives rise to the commutative diagram [$$\begin{diag} 0&\longrightarrow&(T,P^{(i)})&\RA{}&(T,P_2^{(i)})&\RA{}&(T,\overline{P}_1^{(i)}\oplus(P^{(k)})^m)&\RA{}&(T,P^{(i)})&\RA{}&\check{S}^{(i)}&\longrightarrow&0\\ &&\downarrow&&\downarrow&&\downarrow^{{0\choose1}}\\ 0&\longrightarrow&(T,(\check{P}^{(k)})^m)&\RA{}&(T,(P_1^{(k)})^m)&\RA{}&(T,(P^{(k)})^m)&\RA{}&0 \end{diag}$$]{} of exact sequences. Taking the mapping cone, we get a projective resolution [$$0\to(T,P^{(i)})\to(T,P_2^{(i)})\oplus(T,\check{P}^{(k)})^m\to(T,\overline{P}_1^{(i)})\oplus(T,P_1^{(k)})^m\to(T,P^{(i)})\to\check{S}^{(i)}\to0.$$]{} -1em We now compare the minimal projective resolutions for $S^{(i)}$ and $\check{S}^{(i)}$, to see the change in quivers when passing from $\qq_\Lambda$ to $\qq_\Gamma$, for arrows starting or ending at $i$. Consider now $b_{ij}^{\prime\prime}$ with $j\neq k$. When passing from $b_{ij}$ to $b_{ij}^{\prime\prime}$, we see that we get something extra if and only if $P^{(j)}\in\add P_1^{(k)}$, which is equivalent to $b_{kj}>0$. Then we get $b_{ij}^{\prime\prime}=b_{ij}+b_{kj}m=b_{ij}+b_{ik}b_{kj}=b_{ij}^\prime$. And if $b_{kj}\le0$, we see that $b_{ij}^{\prime\prime}=b_{ij}$. Hence we see that $b_{ij}^{\prime\prime}=b_{ij}^\prime$ for any $j\neq k$. [**Case 3** ]{} Assume now that $i\neq k$ and $b_{ik}\le0$. We can show $b_{ij}^{\prime\prime}=b_{ij}^\prime$ for any $j\neq k$ by a dual argument to Case 2. ------------------------------------------------------------------------ .5em In order to continue the process, it is of interest to know if the new quiver $\qq_\Gamma$ also has no 2-cycles. Note that we have seen that it has no loops. We do not know if this is true in general, but we show that if $\Lambda=S*G$ where $S=K[[x,y,z]]$, $G\subset\SL_3(K)$ is given by $G=\langle{\rm diag}(\omega,\omega,\omega)\rangle$ with a primitive third root $\omega$ of $1$, then all iterations of the process in  give 3-CY algebras whose quivers have no loops. This is a consequence of the following proposition, where we call a ring $\Lambda$ [*completely graded*]{} if $\Lambda$ is a direct product(!) $\Lambda=\prod_{i\in\zzz}\Lambda_i$ satisfying $\Lambda_i\Lambda_j\subseteq\Lambda_{i+j}$, and a $\Lambda$-module $M$ [*completely graded*]{} if $M$ is a direct product $M=\prod_{i\in\zzz}M_i$ satisfying $\Lambda_iM_j\subseteq M_{i+j}$. We denote by $\grmod\Lambda$ the category of completely graded $\Lambda$-modules. .5em[**Proposition  **]{} *Let $\Lambda$ be a 3-CY algebra satisfying the conditions (1) and (2) below.* \(1) $\Lambda$ is a completely graded ring $\Lambda=\prod_{j\ge0}\Lambda_j$ with Jacobson radical $J_\Lambda=\prod_{j>0}\Lambda_j$. \(2) $\Lambda=\bigoplus_{i\in\zzz/3\zzz}P^{(i)}$, and each simple $\Lambda$-module $S^{(i)}:=(P^{(i)})_0$ has a projective resolution $0\to P^{(i)}(-3)\to P^{(i+1)}(-2)^{b_{i+2}}\to P^{(i+2)}(-1)^{b_{i+1}}\to P^{(i)}\to S^{(i)}\to0$ in $\grmod\Lambda$. Fix $k\in\zzz/3\zzz$ and put $\Gamma:=\mu_k(\Lambda)$. Then $\Gamma$ satisfies the same conditions as $\Lambda$. Precisely speaking, we introduce a degree on $\check{P}^{(k)}:=\Omega^2(S^{(k)})$ such that the natural inclusion $\check{P}^{(k)}\to(P^{(k+2)})^{b_{k+1}}$ is a morphism in $\grmod\Lambda$. Put $\check{P}^{(k+1)}:=P^{(k+1)}$, $\check{P}^{(k+2)}:=P^{(k+2)}(-1)$ and $T:=\bigoplus_{i\in\zzz/3\zzz}\check{P}^{(i)}\in\grmod\Lambda$. Then the assertions ($\check{1}$) and ($\check{2}$) below hold. ($\check{1}$) $\Gamma$ is a completely graded ring $\Gamma=\prod_{j\ge0}\hom_{\grmod\Lambda}(T,T(j))$ with Jacobson radical $J_{\Gamma}=\prod_{j>0}\hom_{\grmod\Lambda}(T,T(j))$. ($\check{2}$) $\Gamma=\bigoplus_{i\in\zzz/3\zzz}(T,\check{P}^{(i)})$, and each simple $\Gamma$-module $\check{S}^{(i)}:=\hom_\Lambda(T,\check{P}^{(i)})_0$ has projective resolution $0\to(T,\check{P}^{(i)})(-3)\to(T,\check{P}^{(i+2)})(-2)^{b'_{i+1}}\to(T,\check{P}^{(i+1)})(-1)^{b'_{i+2}}\to(T,\check{P}^{(i)})\to\check{S}^{(i)}\to0$ in $\grmod\Lambda$ for $b'_k:=b_{k+1}b_{k+2}-b_k$, $b'_{k+1}:=b_{k+1}$ and $b'_{k+2}:=b_{k+2}$. .5em[Proof ]{} We can prove the assertion by taking care of degree in the proof of . [**Case 1** ]{} We have exact sequences [$$\begin{aligned} &0\to\check{P}^{(k)}(-1)\to P^{(k+2)}(-1)^{b_{k+1}}\to P^{(k)}\to S^{(k)}\to0&\\ &0\to P^{(k)}(-3)\to P^{(k+1)}(-2)^{b_{k+2}}\to\check{P}^{(k)}(-1)\to0&\end{aligned}$$]{} in $\grmod\Lambda$. Applying $\hom_\Lambda(T,-)$ and the same argument as in Case 1 in the proof of , we obtain a minimal projective resolution [$$0\to(T,\check{P}^{(k)})(-3)\to(T,\check{P}^{(k+2)})(-2)^{b_{k+1}}\to(T,\check{P}^{(k+1)})(-1)^{b_{k+2}}\to(T,\check{P}^{(k)})\to\check{S}^{(k)}\to0$$]{} of $S^{(k)}$ in $\grmod\Gamma$. [**Case 2** ]{} We consider the minimal projective resolution of the simple $\Gamma$-module $S^{(k+1)}$. We have a commutative diagram [$$\begin{diag} 0&\longrightarrow&P^{(k+1)}(-3)&\RA{}&P^{(k+2)}(-2)^{b_k}&\RA{}&P^{(k)}(-1)^{b_{k+2}}&\RA{}&P^{(k+1)}\\ &&\downarrow&&\downarrow^{f(-2)}&&\parallel\\ 0&\longrightarrow&\check{P}^{(k)}(-2)^{b_{k+2}}&\RA{}&P^{(k+2)}(-2)^{b_{k+1}b_{k+2}}&\RA{}&P^{(k)}(-1)^{b_{k+2}} \end{diag}$$]{} of exact sequences. We can choose $f$ to be a morphism in $\grmod\Lambda$. Putting $X:=(P^{(k+2)})^{b_k}$ and $Y:=(P^{(k+2)})^{b_{k+1}b_{k+2}}$ and looking at the degree two part, we have a commutative diagram [$$\begin{diag} P^{(k+1)}_{-1}=0&\RA{}&X_0&\RA{}&(P^{(k)}_1)^{b_{k+2}}\\ &&\downarrow^{f_0}&&\parallel\\ &&Y_0&\RA{}&(P^{(k)}_1)^{b_{k+2}} \end{diag}$$]{} of exact sequences. Thus $f_0:X_0\to Y_0$ is a monomorphism between semisimple $\Lambda$-modules because $J_{\Lambda}=\prod_{i>0}\Lambda_i$ by our assumption. We can take $g\in\hom_{\grmod\Lambda}(Y,X)$ such that $f_0g_0=1_{X_0}$. Then $fg-1\in\endm_{\grmod\Lambda}(X)$ satisfies $X_0\subset\Ker(fg-1)$. Since $X$ is generated by $X_0$, we obtain $fg-1=0$. Thus $f$ is a split monomorphism Consequently, taking a mapping cone as in Case 2 in the proof of  and cancelling a trivial direct summand of the complex, we have a complex $$0\to P^{(k+1)}(-3)\to\check{P}^{(k)}(-2)^{b_{k+2}}\to P^{(k+2)}(-2)^{b_{k+1}b_{k+2}-b_k}\to P^{(k+1)}$$ in $\grmod\Lambda$, which induces a projective resolution $$0\to(T,\check{P}^{(k+1)})(-3)\to(T,\check{P}^{(k)})(-2)^{b'_{k+2}}\to(T,\check{P}^{(k+2)})(-1)^{b'_k}\to(T,\check{P}^{(k+1)})\to\check{S}^{(k+1)}\to0$$ of $\check{S}^{(k+1)}$ in $\grmod\Gamma$. [**Case 3** ]{} A dual argument works for $\check{S}^{(k+2)}$. ------------------------------------------------------------------------ . .5em Now we consider all iterations of $S*G$ for $G=\langle{\rm diag}(\omega,\omega,\omega)\rangle\subset\SL_3(K)$ with $\omega^3=1$. The Fomin-Zelevinsky mutation of the quiver (32,25) (-2,-7)[$k$]{} (0,0) (32,0) (16,24) (14,0)[$b$]{} (26,10)[$c$]{} (0,10)[$a$]{} (0,0)[(1,0)[30]{}]{} (30,3)[(-2,3)[13]{}]{} (14,22)[(-2,-3)[13]{}]{} at the vertex $k$ is (32,25) (-2,-7)[$k$]{} (0,0) (32,0) (16,24) (14,0)[$b$]{} (26,10)[$ab-c$]{} (0,10)[$a$]{} (30,0)[(-1,0)[30]{}]{} (17,22.5)[(2,-3)[13]{}]{} (1,2.5)[(2,3)[13]{}]{}  , where $a$, $b$, $c$ and $ab-c$ show the numbers of arrow. The quiver of $S*G$ is the McKay quiver (32,25) (0,0) (32,0) (16,24) (14,0)[3]{} (26,10)[3]{} (0,10)[3]{} (0,0)[(1,0)[30]{}]{} (30,3)[(-2,3)[13]{}]{} (14,22)[(-2,-3)[13]{}]{}  of $G$ (see section ), and we draw quivers of 3-CY algebras obtained by iterated mutations. It is the picture below, where each quiver has precisely three neighbours. One can check inductively that each triple $(a,b,c)$ satisfies the Markov equation $a^2+b^2+c^2=abc$. All integral solutions of the Markov equation appear in the picture below because it is known that the Fomin-Zelevinky mutation rule $(a,b,c)\mapsto(a,b,ab-c)$ in this case gives all of them. $$\begin{array}{ccccccccc} &&\begin{picture}(32,25) %\put(-7,-7){\scriptsize 1} %\put(32,-7){\scriptsize 2} %\put(14,28){\scriptsize 3} \put(0,0){\circle*{4}} \put(32,0){\circle*{4}} \put(16,24){\circle*{4}} \put(14,0){\scriptsize 6} \put(26,10){\scriptsize 15} \put(0,10){\scriptsize 87} \put(30,0){\vector(-1,0){30}} \put(17,22.5){\vector(2,-3){13}} \put(1,2.5){\vector(2,3){13}} \end{picture} &&&& \begin{picture}(32,25) \put(0,0){\circle*{4}} \put(32,0){\circle*{4}} \put(16,24){\circle*{4}} \put(14,0){\scriptsize 6} \put(26,10){\scriptsize 87} \put(0,10){\scriptsize 15} \put(30,0){\vector(-1,0){30}} \put(17,22.5){\vector(2,-3){13}} \put(1,2.5){\vector(2,3){13}} \end{picture}&\\ &&\begin{picture}(2,15)\put(0,0){\line(0,1){15}}\end{picture}&& &&\begin{picture}(2,15)\put(0,0){\line(0,1){15}}\end{picture}&&\\ \begin{picture}(32,25) \put(0,0){\circle*{4}} \put(32,0){\circle*{4}} \put(16,24){\circle*{4}} \put(14,0){\scriptsize 39} \put(26,10){\scriptsize 15} \put(0,10){\scriptsize 3} \put(30,0){\vector(-1,0){30}} \put(17,22.5){\vector(2,-3){13}} \put(1,2.5){\vector(2,3){13}} \end{picture}& \begin{picture}(15,5)\put(0,10){\line(1,0){15}}\end{picture}& \begin{picture}(32,25) \put(0,0){\circle*{4}} \put(32,0){\circle*{4}} \put(16,24){\circle*{4}} \put(14,0){\scriptsize 6} \put(26,10){\scriptsize 15} \put(0,10){\scriptsize 3} \put(0,0){\vector(1,0){30}} \put(30,3){\vector(-2,3){13}} \put(14,22){\vector(-2,-3){13}} \end{picture}&&&& \begin{picture}(32,25) \put(0,0){\circle*{4}} \put(32,0){\circle*{4}} \put(16,24){\circle*{4}} \put(14,0){\scriptsize 6} \put(26,10){\scriptsize 3} \put(0,10){\scriptsize 15} \put(0,0){\vector(1,0){30}} \put(30,3){\vector(-2,3){13}} \put(14,22){\vector(-2,-3){13}} \end{picture}& \begin{picture}(15,5)\put(0,10){\line(1,0){15}}\end{picture}& \begin{picture}(32,25) \put(0,0){\circle*{4}} \put(32,0){\circle*{4}} \put(16,24){\circle*{4}} \put(14,0){\scriptsize 39} \put(26,10){\scriptsize 3} \put(0,10){\scriptsize 15} \put(30,0){\vector(-1,0){30}} \put(17,22.5){\vector(2,-3){13}} \put(1,2.5){\vector(2,3){13}} \end{picture}\\ &&&\begin{picture}(15,10)\put(15,0){\line(-1,1){15}}\end{picture}& &\begin{picture}(15,10)\put(0,0){\line(1,1){15}}\end{picture}&&&\\ &&&&\begin{picture}(32,25) \put(0,0){\circle*{4}} \put(32,0){\circle*{4}} \put(16,24){\circle*{4}} \put(14,0){\scriptsize 6} \put(26,10){\scriptsize 3} \put(0,10){\scriptsize 3} \put(30,0){\vector(-1,0){30}} \put(17,22.5){\vector(2,-3){13}} \put(1,2.5){\vector(2,3){13}} \end{picture}&&&&\\ &&&&\begin{picture}(2,15)\put(0,0){\line(0,1){15}}\end{picture}&&&&\\ &&&&\begin{picture}(32,25) \put(0,0){\circle*{4}} \put(32,0){\circle*{4}} \put(16,24){\circle*{4}} \put(14,0){\scriptsize 3} \put(26,10){\scriptsize 3} \put(0,10){\scriptsize 3} \put(0,0){\vector(1,0){30}} \put(30,3){\vector(-2,3){13}} \put(14,22){\vector(-2,-3){13}} \end{picture}&&&&\\ &&&\begin{picture}(15,10)\put(0,0){\line(1,1){15}}\end{picture}& &\begin{picture}(15,10)\put(15,0){\line(-1,1){15}}\end{picture}&&&\\ \begin{picture}(32,25) \put(0,0){\circle*{4}} \put(32,0){\circle*{4}} \put(16,24){\circle*{4}} \put(14,0){\scriptsize 15} \put(26,10){\scriptsize 6} \put(0,10){\scriptsize 3} \put(0,0){\vector(1,0){30}} \put(30,3){\vector(-2,3){13}} \put(14,22){\vector(-2,-3){13}} \end{picture}& \begin{picture}(15,5)\put(0,10){\line(1,0){15}}\end{picture}& \begin{picture}(32,25) \put(0,0){\circle*{4}} \put(32,0){\circle*{4}} \put(16,24){\circle*{4}} \put(14,0){\scriptsize 3} \put(26,10){\scriptsize 6} \put(0,10){\scriptsize 3} \put(30,0){\vector(-1,0){30}} \put(17,22.5){\vector(2,-3){13}} \put(1,2.5){\vector(2,3){13}} \end{picture} &&&&\begin{picture}(32,25) \put(0,0){\circle*{4}} \put(32,0){\circle*{4}} \put(16,24){\circle*{4}} \put(14,0){\scriptsize 3} \put(26,10){\scriptsize 3} \put(0,10){\scriptsize 6} \put(30,0){\vector(-1,0){30}} \put(17,22.5){\vector(2,-3){13}} \put(1,2.5){\vector(2,3){13}} \end{picture}& \begin{picture}(15,5)\put(0,10){\line(1,0){15}}\end{picture}& \begin{picture}(32,25) \put(0,0){\circle*{4}} \put(32,0){\circle*{4}} \put(16,24){\circle*{4}} \put(14,0){\scriptsize 15} \put(26,10){\scriptsize 3} \put(0,10){\scriptsize 6} \put(0,0){\vector(1,0){30}} \put(30,3){\vector(-2,3){13}} \put(14,22){\vector(-2,-3){13}} \end{picture}\\ &&\begin{picture}(2,15)\put(0,0){\line(0,1){15}}\end{picture}&& &&\begin{picture}(2,15)\put(0,0){\line(0,1){15}}\end{picture}&&\\ &&\begin{picture}(32,25) \put(0,0){\circle*{4}} \put(32,0){\circle*{4}} \put(16,24){\circle*{4}} \put(14,0){\scriptsize 3} \put(26,10){\scriptsize 6} \put(0,10){\scriptsize 15} \put(0,0){\vector(1,0){30}} \put(30,3){\vector(-2,3){13}} \put(14,22){\vector(-2,-3){13}} \end{picture}&&&& \begin{picture}(32,25) \put(0,0){\circle*{4}} \put(32,0){\circle*{4}} \put(16,24){\circle*{4}} \put(14,0){\scriptsize 3} \put(26,10){\scriptsize 15} \put(0,10){\scriptsize 6} \put(0,0){\vector(1,0){30}} \put(30,3){\vector(-2,3){13}} \put(14,22){\vector(-2,-3){13}} \end{picture}&& \end{array}$$ .5em The above picture gives the Hasse graph of $\tilt_1(S*G)$ (see (1) below). We note that a similar picture appeared in the classification of exceptional vector bundles over $\ppp^2$ due to Gorodentsev-Rudakov \[GR\]\[Rud\]. It also appeared in recent work of Bridgeland \[Bri2,3\] on t-structures on the derived category of the total space of the canonical line bundle $\oo_{\ppp^2}(-3)$ on $\ppp^2$. Now we consider other 3-CY algebras $S*G$ for $G=\langle{\rm diag}(\zeta,\zeta^2,\zeta^2)\rangle\subset\SL_3(K)$ with $\zeta^5=1$. We draw a few quivers of 3-CY algebras obtained by iterated mutations. Unfortunately we do not know whether they coincide with mutations of algebras. $$\begin{array}{ccccc} \begin{picture}(60,47) \put(15,0){\circle*{4}} \put(45,0){\circle*{4}} \put(60,30){\circle*{4}} \put(30,45){\circle*{4}} \put(0,30){\circle*{4}} \put(15,0){\vector(1,0){28}} \put(45,0){\vector(1,2){14}} \put(60,30){\vector(-2,1){28}} \put(30,45){\vector(-2,-1){28}} \put(0,30){\vector(1,-2){14}} \put(15,-1){\vector(3,2){43}} \put(15,1){\vector(3,2){43}} \put(45,2){\vector(-1,3){14}} \put(44,-2){\vector(-1,3){15}} \put(60,29){\vector(-1,0){58}} \put(60,31){\vector(-1,0){58}} \put(30,43){\vector(-1,-3){14}} \put(29,47){\vector(-1,-3){15}} \put(0,29){\vector(3,-2){43}} \put(0,31){\vector(3,-2){43}} \end{picture}& \begin{picture}(15,5)\put(0,10){\line(1,0){15}}\end{picture}& \begin{picture}(60,47) \put(15,0){\circle*{4}} \put(45,0){\circle*{4}} \put(60,30){\circle*{4}} \put(30,45){\circle*{4}} \put(0,30){\circle*{4}} \put(43,-2){\vector(-1,0){28}} \put(43,0){\vector(-1,0){28}} \put(43,2){\vector(-1,0){28}} \put(45,0){\vector(1,2){14}} \put(32,44){\vector(2,-1){27}} \put(2,31){\vector(2,1){27}} \put(0,30){\vector(1,-2){14}} \put(31,44){\vector(1,-3){14}} \put(29,43){\vector(1,-3){14}} \put(60,28){\vector(-1,0){58}} \put(60,30){\vector(-1,0){58}} \put(60,32){\vector(-1,0){58}} \put(16,1){\vector(1,3){14}} \put(14,2){\vector(1,3){14}} \end{picture}& \begin{picture}(15,5)\put(0,10){\line(1,0){15}}\end{picture}& \begin{picture}(60,47) \put(15,0){\circle*{4}} \put(45,0){\circle*{4}} \put(60,30){\circle*{4}} \put(30,45){\circle*{4}} \put(0,30){\circle*{4}} \put(15,-2){\vector(1,0){28}} \put(15,0){\vector(1,0){28}} \put(15,2){\vector(1,0){28}} \put(45,0){\vector(1,2){14}} \put(32,44){\vector(2,-1){27}} \put(1,32){\vector(2,1){27}} \put(1,30){\vector(2,1){27}} \put(1,28){\vector(2,1){27}} \put(14,2){\vector(-1,2){14}} \put(42,-1){\vector(-1,3){14}} \put(44,-1){\vector(-1,3){14}} \put(46,-1){\vector(-1,3){14}} \put(48,-1){\vector(-1,3){14}} \put(60,28){\vector(-1,0){58}} \put(60,30){\vector(-1,0){58}} \put(60,32){\vector(-1,0){58}} \put(30,43){\vector(-1,-3){14}} \put(29,47){\vector(-1,-3){15}} \end{picture}\\ &&\begin{picture}(2,15)\put(0,0){\line(0,1){15}}\end{picture}&&\\ &&\begin{picture}(60,47) \put(15,0){\circle*{4}} \put(45,0){\circle*{4}} \put(60,30){\circle*{4}} \put(30,45){\circle*{4}} \put(0,30){\circle*{4}} \put(43,-2){\vector(-1,0){28}} \put(43,0){\vector(-1,0){28}} \put(43,2){\vector(-1,0){28}} \put(45,0){\vector(1,2){14}} \put(60,29){\vector(-2,1){27}} \put(60,31){\vector(-2,1){27}} \put(30,45){\vector(-2,-1){28}} \put(14,2){\vector(-1,2){13}} \put(60,28){\vector(-3,-2){42}} \put(60,30){\vector(-3,-2){42}} \put(60,32){\vector(-3,-2){42}} \put(0,28){\vector(1,0){58}} \put(0,30){\vector(1,0){58}} \put(0,32){\vector(1,0){58}} \put(31,44){\vector(1,-3){14}} \put(29,43){\vector(1,-3){14}} \put(16,1){\vector(1,3){14}} \put(14,2){\vector(1,3){14}} \end{picture}&& \end{array}$$ Motivated by the connection with cluster algebras, we would like to view the tilting modules as analogs to clusters, and hence we want to show that any 3-CY algebra $\Gamma$ obtained via a sequence of mutations from a 3-CY algebra $\Lambda$ can be obtained directly as the endomorphism ring of a tilting $\Lambda$-module. We already know from section  that if $T=\bigoplus_{i=1}^nT_i$ is a basic tilting $\Lambda$-module, with the $T_i$ indecomposable, then there is for each $k=1,\cdots,n$ a unique indecomposable $\Lambda$-module $\check{T}_k$ with $\check{T}_k\ {\not\simeq}\ T_k$ such that $\nu_k(T):=(\bigoplus_{i\neq k}T_i)\oplus\check{T}_k$ is a tilting module, and we have also given an explicit description of $\nu_k(T)$. We now point out the relationship between mutation of algebras and of tilting modules in part (1) of the next result. Part (3) is analogous to a basic property of the Fomin-Zelevinsky mutation. We know that all algebras obtained from a 3-CY algebra $\Lambda$ by successive applications of mutations are derived equivalent, and hence by \[Ri1\] can be obtained from $\Lambda$ as an endomorphism ring of a tilting complex. But we have a better result (2) saying that if $\Gamma$ is obtained from $\Lambda$ by a sequence of our special tilting modules, then it can be obtained directly with one tilting module. A more general version of part (4) (including an alternative approach) is given in the next section (3). .5em[**Proposition  **]{} *Let $\Lambda$ be basic 3-CY.* \(1) $\mu_k(\endm_\Lambda(T))=\endm_\Lambda(\nu_k(T))$ for any tilting $\Lambda$-module $T$. \(2) $\mu_{k_m}\circ\cdots\circ\mu_{k_1}(\Lambda)=\endm_\Lambda(\nu_{k_m}\circ\cdots\circ\nu_{k_1}(\Lambda))$ for any $k_1,\cdots,k_m$. \(3) $\mu_k(\mu_k(\Lambda)))=\Lambda$. \(4) Assume in (2) that the quivers of $\Lambda_i:=\mu_{k_i}\circ\cdots\circ\mu_{k_1}(\Lambda)$ have no loops for any $i$ ($0\le i<m$). Then $\nu_{k_m}\circ\cdots\circ\nu_{k_1}(\Lambda)$ is reflexive and isomorphic to $(T_1\otimes_{\Lambda_1}T_2\otimes_{\Lambda_2}\cdots\otimes_{\Lambda_{m-1}}T_m)^{**}$ for $T_i:=\nu_{k_i}(\Lambda_{i-1})$. .5em[Proof ]{} (1) follows from (6). Using (1) repeatedly, we have (2). We obtain (3) by $\mu_k(\mu_k(\Lambda))=\endm_\Lambda(\nu_k(\nu_k(\Lambda))=\endm_\Lambda(\Lambda)=\Lambda$. We obtain (4) by using (5) repeatedly. ------------------------------------------------------------------------ .5em In addition to considering, for some fixed 3-CY algebra $\Lambda$, the tilting modules to be the analogs of the clusters and the indecomposable partial tilting modules $M$ as the analogs of the cluster variables, we also have an interpretation of the exchange multiplication rule. Let $T=\bigoplus_{i=1}^nT_i$ be a tilting module, and assume that $T$ is connected with the cluster $\underline{x}=\{x_1,\cdots,x_n\}$ in such a way that the $T_i$ are associated with the cluster variables $x_i$. (If we start by fixing a correspondence $\underline{u}=\{u_1,\cdots,u_n\}\mapsto\Lambda=\bigoplus_{i=1}^nQ_i$, we reach $\Lambda^\prime=\endm_\Lambda(T)$ by a sequence of mutations. We choose $\underline{x}=\{x_1\cdots,x_n\}$ to be the cluster obtained by the same sequence of mutations applied to $\underline{u}=\{u_1,\cdots,u_n\}$. This is the procedure used in \[BMR2\] in the context of cluster categories. For $\overline{T}=T/T_k$, we have the minimal right and left $\add\overline{T}$-approximations $B\to T_k$ and $T_k\to B^\prime$, with $B=\bigoplus_{i\neq k}T_i^{r_i}$ and $B^\prime=\bigoplus_{i\neq k}T_i^{s_i}$, where $r_i\ge0$ and $s_i\ge0$. We then have $T_k\cdot\check{T}_k=\prod T_i^{r_i}+\prod T_i^{s_i}$. As we asked in section , it would be interesting to know if every tilting module $T$ can be obtained from the tilting module $\Lambda$ over a 3-CY algebra $\Lambda$ by a finite sequence of mutations, that is, if the Hasse quiver of $\tilt_1\Lambda$ is connected. Another interesting problem is whether there is a one-one correspondence between the cluster variables and the indecomposable partial tilting $\Lambda$-modules, inducing a one-one correspondence between clusters and tilting modules. In addition to the work on cluster categories serving as a model, there is also a connection with the modelling of some class of cluster algebras ‘with coefficients’ by special modules over preprojective algebras, from \[GLS\]. They consider maximal rigid modules $M$ (see section  for definition) over a preprojective algebra $\Lambda$ of a Dynkin diagram. For the case of $\Lambda$ being of finite representation type, the stable category $\underline{\mod}\Lambda$ is actually equivalent to a cluster category \[BMRRT\], and the maximal rigid modules are closely related to the cluster-tilting objects, as well as coinciding with the maximal $1$-orthogonal modules of \[I3,4\]. Here $\endm_\Lambda(M)$ has global dimension $3$, and the tilting theory over this algebra is relevant. These algebras are sort of a degenerate version of 3-CY algebras. The relationship is similar to the relationship between the invariant ring $S^G$ and the skew group ring $S*G$ where $S=K[[x_1,\cdots,x_n]]$ and $G$ is a finite subgroup of $\SL_3(k)$. 1.5em[**. Non-commutative crepant resolutions** ]{} In this section we improve results on tilting modules for 3-CY algebras from section . We show that if we can pass from $\Lambda$ to $\Lambda'$ by taking the endomorphism algebra of a tilting module, and the same way from $\Lambda'$ to $\Lambda''$, then we can also pass directly from $\Lambda$ to $\Lambda''$ in this way. For this, constructing new tilting modules from old ones via homomorphism spaces and tensor products is crucial. Actually, we work in a more general context, investigating the non-commutative crepant resolutions (NCCR) of Van den Bergh, extending his definition to non-commutative algebras. For 3-CY algebras there turns out to be a close relationship to reflexive tilting modules. A main result is the solution of a conjecture of Van den Bergh on derived equivalence of NCCR for 3-dimensional algebras. Throughout this section, let $R$ be a normal Gorenstein domain with $\dim R=d$ and $\Lambda$ a module-finite $R$-algebra such that the structure morphism $R\to\Lambda$ is injective. Generalizing the following definition of Van den Bergh \[Va1,2\], we say that $M$ gives a [*non-commutative crepant resolution*]{} (NCCR for short) $\Gamma:=\endm_\Lambda(M)$ of $\Lambda$ if \(1) $M\in\ref\Lambda$ is a height one generator () of $\Lambda$, and \(2) $\Gamma_{\dn{p}}$ is an $R_{\dn{p}}$-order (in the sense of section ) with $\gl\Gamma_{\dn{p}}=\height\dn{p}$ for any $\dn{p}\in\Spec R$. Obviously, one can replace $\Spec R$ by $\Max R$ in condition (2). If $\Gamma$ is $d$-CY, then condition (2) is satisfied by . In \[Va1,2\], Van den Bergh gave a non-commutative analogue of a conjecture of Bondal-Orlov \[BO\]: [*All NCCR of a normal Gorenstein domain $\Lambda$ are derived equivalent*]{}. In fact, he proved this conjecture for 3-dimensional terminal singularities $\Lambda$. In this section, using a method from \[I4\], we show that [*his conjecture is true for arbitrary 3-dimensional module-finite algebras*]{}. In the original definition in \[Va1,2\], the height one generator condition is not assumed. If $\Lambda$ is a normal domain, then any non-zero reflexive $\Lambda$-module is a height one generator. Thus our definition coincides with the original one in this case. For the non-commutative situation, if we drop the height one generator condition, then $\Lambda:=\def\arraystretch{.5}\left(\begin{array}{cc}{\scriptstyle R}&{\scriptstyle R}\\ {\scriptstyle xR}&{\scriptstyle R}\end{array}\right)$ with $R:=K[[x,y,z]]$ has NCCR $\Lambda=\endm_\Lambda(\Lambda)$ and $R=\endm_\Lambda(\def\arraystretch{.5}\left(\begin{array}{c}{\scriptstyle R}\\ {\scriptstyle R}\end{array}\right))$, and $\Lambda$ and $R$ are not derived equivalent. Thus the height one generator condition seems to be appropriate for our considerations. For NCCR it would be interesting to know for which $R$-algebras $\Lambda$ they exist, what the relationship is with $\Lambda$ when they exist, and what the connection is between different NCCR of the same algebra $\Lambda$. Also, it would be nice to describe the $\Lambda$-modules giving rise to some NCCR. Some such questions will be investigated along the way. We start with some easy properties of NCCR, which will be useful later. In particular, reflexive equivalences play a crucial role. .5em[**Proposition  **]{} *We have the following for a module-finite $R$-algebra $\Lambda$.* \(1) Assume that $M\in\ref\Lambda$ gives a NCCR of $\Lambda$. 1em (i) $M_{\dn{p}}\in\ref\Lambda_{\dn{p}}$ gives a NCCR of $\Lambda_{\dn{p}}$ for any $\dn{p}\in\Spec R$. 1em (ii) $\Lambda_{\dn{p}}$ is an $R_{\dn{p}}$-order with $\gl\Lambda_{\dn{p}}=1$ for any $\dn{p}\in\Spec R$ with $\height\dn{p}=1$. 1em (iii) $M$ is a height one progenerator of $\Lambda$, and we have a reflexive equivalence $\fff:=\hom_\Lambda(M,-):\ref\Lambda\to\ref\endm_\Lambda(M)$. \(2) Let $\fff:\ref\Lambda\to\ref\Gamma$ be any reflexive equivalence. Then $N\in\ref\Lambda$ gives a NCCR of $\Lambda$ if and only if $\fff(N)\in\ref\Gamma$ gives a NCCR of $\Gamma$. .5em[Proof ]{} (1)(i) Obvious from the definition of NCCR. (ii) Assume that $M$ gives a NCCR $\Gamma=\endm_\Lambda(M)$. Since $M_{\dn{p}}$ is a generator for $\Lambda_{\dn{p}}$, there exists an idempotent $e$ in $\ma_n(\Gamma_{\dn{p}})$ for some $n$ such that $\Lambda_{\dn{p}}$ is Morita equivalent to $e\ma_n(\Gamma_{\dn{p}})e$. Since $\ma_n(\Gamma_{\dn{p}})$ is an $R_{\dn{p}}$-order with $\gl\ma_n(\Gamma_{\dn{p}})=1$, we have that $\Lambda_{\dn{p}}$ is an $R_{\dn{p}}$-order with $\gl\Lambda_{\dn{p}}=1$ by taking completion and using (3). \(iii) Fix any $\dn{p}\in\Spec R$ with $\height\dn{p}=1$. By (ii), we have $\gl\Lambda_{\dn{p}}=1$. Since $M_{\dn{p}}$ is a torsionfree $R_{\dn{p}}$-module, $M_{\dn{p}}$ is a projective $\Lambda_{\dn{p}}$-module. Thus the assertion follows by . \(2) Since $\endm_\Lambda(N)=\endm_\Gamma(\fff(N))$ holds, we only have to show that $N$ is a height one generator of $\Lambda$ if and only if $\fff(N)$ is a height one generator of $\Gamma$. Fix $\dn{p}\in\Spec R$ with $\height\dn{p}=1$. Then $\fff$ induces an equivalence $\fff_{\dn{p}}:\ref\Lambda_{\dn{p}}\to\ref\Gamma_{\dn{p}}$ by (2). Since $\gl\Lambda_{\dn{p}}=\gl\Gamma_{\dn{p}}=1$ by (1)(ii), $N_{\dn{p}}$ is a generator if and only if $\ref\Lambda_{\dn{p}}=\add N_{\dn{p}}$ if and only if $\ref\Gamma_{\dn{p}}=\add\fff_{\dn{p}}(N_{\dn{p}})$ if and only if $\fff_{\dn{p}}(N_{\dn{p}})$ is a generator. Thus the assertion follows. ------------------------------------------------------------------------ .5em We note here the following necessary (and sufficient for $d\le2$) conditions for $\Lambda$ to have some NCCR even though we do not use it in the rest of this paper. .5em[**Proposition  **]{} *If $\Lambda$ has a NCCR, then (1) and (2) hold. The converse holds if $d\le2$.* \(1) $\Lambda_{\dn{p}}$ is an $R_{\dn{p}}$-order with $\gl\Lambda_{\dn{p}}=\height\dn{p}$ for any $\dn{p}\in\Spec R$ with $\height\dn{p}\le1$. \(2) There exists $M\in\ref\Lambda$ such that $\ref\Lambda_{\dn{p}}=\add M_{\dn{p}}$ for any $\dn{p}\in\Spec R$ with $\height\dn{p}=2$. In particular, $\Lambda_{\dn{p}}$ is representation-finite. .5em[Proof ]{} (1) follows from (1)(ii). Since $\Gamma_{\dn{p}}$ is a NCCR of $\Lambda_{\dn{p}}$ by (1)(i), we have a reflexive equivalence $\hom_{\Lambda_{\dn{p}}}(M_{\dn{p}},-):\ref\Lambda_{\dn{p}}\to\ref\Gamma_{\dn{p}}$ by (1)(iii). Since $\gl\Gamma_{\dn{p}}=2$, $\ref\Gamma_{\dn{p}}$ consists of projective $\Gamma_{\dn{p}}$-modules. Thus $\ref\Lambda_{\dn{p}}=\add M_{\dn{p}}$. We now show the converse if $d\le2$. If $d\le 1$, then $\Lambda$ itself gives a NCCR. If $d=2$, then we take $M$ in (2). Since $\ref\Lambda_{\dn{p}}$ is closed under kernels, we have that $\gl\endm_{\Lambda_{\dn{p}}}(M_{\dn{p}})=2$ \[A1\]. Thus $M$ gives a NCCR of $\Lambda$. ------------------------------------------------------------------------ .5em For algebras which are $d$-CY$^-$ or $d$-CY, we have some nice properties. .5em[**Proposition  **]{} *(1) If a $n$-CY$^-$ algebra $\Lambda$ has a NCCR, then $\height\dn{p}=n$ and $\Lambda_{\dn{p}}$ is a symmetric $R_{\dn{p}}$-order for any $\dn{p}\in\Max R$.* \(2) Any NCCR of a $d$-CY$^-$ algebra $\Lambda$ is $d$-CY. \(3) Any reflexive tilting module over a $d$-CY algebra $\Lambda$ gives a NCCR. .5em[Proof ]{} (1) Since $R$ is domain and $\ref\Lambda$ is non-empty, we have $0\in\Supp{}_R\Lambda$. Thus the structure morphism $R\to\Lambda$ is injective. Now the assertion follows from (3) and . \(2) Let $\Gamma$ be a NCCR of $\Lambda$. Fix $\dn{p}\in\Max R$. Since $\Gamma_{\dn{p}}$ is reflexive equivalent to a symmetric $R_{\dn{p}}$-order $\Lambda_{\dn{p}}$ by (1) and (1)(iii), it is a symmetric $R_{\dn{p}}$-order with $\gl\Gamma_{\dn{p}}=\height\dn{p}$ by . Thus $\Gamma$ is $d$-CY. \(3) Let $T$ be a reflexive tilting $\Lambda$-module with $\Gamma:=\endm_\Lambda(T)$. Then $\Gamma$ is also $d$-CY by (1). Thus $\Gamma_{\dn{p}}$ is an $R_{\dn{p}}$-order with $\gl\Gamma_{\dn{p}}=\height\dn{p}$ for any $\dn{p}\in\Max R$ by . We only have to show that $T$ is a height one generator of $\Lambda$. For any $\dn{p}\in\Spec R$ with $\height\dn{p}=1$, we have $\depth{}_{R_{\dn{p}}}T_{\dn{p}}=1$ and $\gl\Lambda_{\dn{p}}=1$. Thus $T_{\dn{p}}$ is a projective tilting $\Lambda_{\dn{p}}$-module by  and . Hence it is a generator. ------------------------------------------------------------------------ .5em In the rest of this section, we assume $d\le3$. First we consider the relationship between reflexive tilting modules and tilting modules of projective dimension at most one over a 3-CY algebra $\Lambda$. .5em[**Proposition  **]{} *(1) Assume that $\dim R\le 3$ and $\Lambda_{\dn{p}}$ is an $R_{\dn{p}}$-order with $\gl\Lambda_{\dn{p}}=\height\dn{p}$ for any $\dn{p}\in\Spec R$.* 1em (i) Any $M\in\ref\Lambda$ satisfies $\pd{}_\Lambda M\le1$. 1em (ii) Any reflexive tilting $\Lambda$-module has projective dimension at most one. Conversely, if $\Lambda_{\dn{p}}$ is Morita equivalent to a local ring for any $\dn{p}\in\Spec R$ with $\height\dn{p}\le2$, then any tilting $\Lambda$-module of projective dimension at most one is reflexive. \(2) We use the notation in . Then $R:=S^G$ is an isolated singularity if and only if $G$ acts freely on $K^d\backslash\{0\}$. In this case, reflexive tilting modules over $\Lambda:=S*G$ are exactly tilting $\Lambda$-module of projective dimension at most one. .5em[Proof ]{} (1)(i) For any $\dn{p}\in\Spec R$, we have $\depth{}_{R_{\dn{p}}}M_{\dn{p}}\ge\min\{2,\height{\dn{p}}\}$ and $\gl\Lambda_{\dn{p}}=\height\dn{p}$. Thus $\pd{}_{\Lambda_{\dn{p}}}M_{\dn{p}}\le1$ by , and we have $\pd{}_\Lambda M\le1$. \(ii) Let $T$ be a tilting $\Lambda$-module with $\pd{}_\Lambda T\le1$. For any $\dn{p}\in\Spec R$, it follows from  that $T_{\dn{p}}$ is a tilting $\Lambda_{\dn{p}}$-module. If $\height\dn{p}=3$, then we have $\depth{}_{R_{\dn{p}}}T_{\dn{p}}\ge2$ by . Now assume $\height\dn{p}\le2$. Since $\Lambda_{\dn{p}}$ is Morita equivalent to a local ring, any tilting $\Lambda_{\dn{p}}$-module is projective. Thus we have $\depth{}_{R_{\dn{p}}}T_{\dn{p}}\ge\height\dn{p}$. Since $T$ satisfies the S$_2$ condition, it is reflexive. \(2) For the first assertion, we refer to \[IY;8.2\]. Now we show the second assertion. Fix any $\dn{p}\in\Spec R\backslash\Max R$. Since $S_{\dn{p}}$ is a CM module over a regular local ring $R_{\dn{p}}$, it is a free $R_{\dn{p}}$-module. Thus the assertion follows from (1)(ii) since $(S*G)_{\dn{p}}=\endm_R(S)_{\dn{p}}=\endm_{R_{\dn{p}}}(S_{\dn{p}})$ (e.g. proof of ) is Morita equivalent to $R_{\dn{p}}$. ------------------------------------------------------------------------ .5em Now we put $\Lambda:=S*G$ for $G=\langle{\rm diag}(1,-1,-1)\rangle$. Then $S^G$ is not an isolated singularity, and $\Lambda$ is isomorphic to a complete tensor product $K[[x_1]]\widehat{\otimes}_K(K[[x_2,x_3]]*H)$ for $H:=\langle{\rm diag}(-1,-1)\rangle$. For any $T\in\tilt_1(K[[x_2,x_3]]*H)$, we have $K[[x_1]]\widehat{\otimes}_KT\in\tilt_1\Lambda$. This is not reflexive if $T$ is not projective. Thus $\Lambda$ has non-reflexive tilting modules of projective dimension at most one. Recall that a module-finite $R$-algebra $\Lambda$ is called an [*isolated singularity*]{} if $\gl\Lambda_{\dn{p}}=\height\dn{p}$ for any $\dn{p}\in\Spec R\backslash\Max R$. To prove derived equivalence of different NCCR, we need an easy lemma on depth, and a relationship between depth and vanishing of $\ext^1$ for isolated singularities. .5em[**Lemma  **]{}[*Let $R$ be local and $0\to X_t\stackrel{f_t}{\to}X_{t-1}\stackrel{f_{t-1}}{\to}\cdots\stackrel{f_3}{\to}X_2\stackrel{f_2}{\to}X_1\stackrel{f_1}{\to}X_0\to0$ an exact sequence with $X_0\in\flmod\Lambda$ for some $t\ge0$. If $\depth X_i\ge i$ for any $i>0$, then $X_0=0$.*]{} .5em[Proof ]{} Put $Y_i:=\Im f_{i}$ for $1\le i\le t$. Inductively, we will show $\depth Y_i\ge i$. This is true for $i=t$. Now we assume $\depth Y_{i+1}\ge i+1$ and consider the exact sequence $0\to Y_{i+1}\to X_i\to Y_i\to0$ with $\depth Y_{i+1}\ge i+1$ and $\depth X_i\ge i$. Applying $\hom_\Lambda(\Lambda/J_\Lambda,-)$, we see that $\depth Y_i\ge i$. In particular, $X_0=Y_1\in\flmod\Lambda$ satisfies $\depth X_0\ge0$, i.e. $\hom_\Lambda(\Lambda/J_\Lambda,X_0)=0$, and hence $X_0=0$. ------------------------------------------------------------------------ .5em[**Lemma  **]{} *Assume that $R$ is local with $\dim R=3$ and $\Lambda$ is an isolated singularity. Let $M,N\in\ref\Lambda$.* \(1) For any $\dn{p}\in\Spec R\backslash\Max R$, we have that $M_{\dn{p}}$ is a projective $\Lambda_{\dn{p}}$-module. \(2) $\ext^i_\Lambda(M,X)$ ($i>0$) has finite length for any $X\in\mod\Lambda$. \(3) If $\depth\hom_\Lambda(M,N)\ge3$, then $\ext^1_\Lambda(M,N)=0$. .5em[Proof ]{} (1) follows from the S$_2$ condition $\depth{}_{R_{\dn{p}}}M_{\dn{p}}\ge\min\{2,\height\dn{p}\}$ and . (2) follows from (1). Now we show (3). Consider the exact sequence $0\to\Omega M\to P\to M\to0$ where $P$ is projective. Then $0\to\hom_\Lambda(M,N)\to\hom_\Lambda(P,N)\to\hom_\Lambda(\Omega M,N)\stackrel{}{\to}\ext^1_\Lambda(M,N)\to0$ is an exact sequence with $\depth\hom_\Lambda(M,N)\ge3$, $\depth\hom_\Lambda(P,N)\ge2$ and $\depth\hom_\Lambda(\Omega M,N)\ge2$ (e.g. (1)). Since $\ext^1_\Lambda(M,N)$ has finite length by (2), it follows that $\ext^1_\Lambda(M,N)=0$ by . ------------------------------------------------------------------------ .5em Now we can prove the main result in this section, where we say that a $\Gamma$-module $N$ is [*rigid*]{} if $\ext^1_\Gamma(N,N)=0$. It generalizes results in \[I4\], where $\Gamma$ is assumed to be an order which is an isolated singularity and $M$ is assumed to be a CM $\Gamma$-module. .5em[**Theorem  **]{} *Let $R$ be a normal Gorenstein domain with $\dim R\le3$ and $\Lambda$ a module-finite algebra. For $M_i\in\ref\Lambda$, put $\Gamma_i:=\endm_\Lambda(M_i)$ and $U:=\hom_\Lambda(M_1,M_2)$.* \(1) If $\Gamma_1$ is a NCCR of $\Lambda$ and $\Gamma_2$ is an order, then $U$ is a reflexive rigid $\Gamma_1$-module with $\endm_{\Gamma_1}(U)=\Gamma_2$. \(2) If $\Gamma_i$ ($i=1,2$) is a NCCR of $\Lambda$, then $U$ is a reflexive tilting $\Gamma_1$-module with $\endm_{\Gamma_1}(U)=\Gamma_2$. .5em[Proof ]{} (1) Since we have a reflexive equivalence $\hom_\Lambda(M_1,-):\ref\Lambda\to\ref\Gamma_1$ by (1)(iii), we have $U\in\ref\Gamma_1$ and $\endm_{\Gamma_1}(U)=\Gamma_2$. Fix $\dn{p}\in\Spec R$. If $\height\dn{p}\le2$, then $U_{\dn{p}}\in\ref(\Gamma_1)_{\dn{p}}$ and $\gl(\Gamma_1)_{\dn{p}}=\height\dn{p}$ imply that $U_{\dn{p}}$ is a projective $(\Gamma_1)_{\dn{p}}$-module. In particular, we have $\ext^1_{(\Gamma_1)_{\dn{p}}}(U_{\dn{p}},U_{\dn{p}})=0$. Now we assume $\height\dn{p}=3$. Since $\depth\endm_{(\Gamma_1)_{\dn{p}}}(U_{\dn{p}})=3$ and $(\Gamma_1)_{\dn{p}}$ is an isolated singularity, we have $\ext^1_{(\Gamma_1)_{\dn{p}}}(U_{\dn{p}},U_{\dn{p}})=0$ by (3). Thus we have $\ext^1_{\Gamma_1}(U,U)=0$. \(2) Since $U=\hom_{\Lambda^{\op}}(M_2^*,M_1^*)$, we have $\endm_{\Gamma_2^{\op}}(U)=\Gamma_1$ and $\ext^1_{\Gamma_2^{\op}}(U,U)=0$ by (1). We have $\pd{}_{\Gamma_2^{\op}}U\le1$ by (1)(i). Let $0\to Q_1\to Q_0\to U\to0$ be a projective resolution of the $\Gamma_2^{\op}$-module $U$. Applying $\hom_{\Gamma_2^{\op}}(-,U)$, we have an exact sequence $0\to\Gamma_1\to\hom_{\Gamma_2^{\op}}(Q_0,U)\to\hom_{\Gamma_2^{\op}}(Q_1,U)\to0$ of $\Gamma_1$-modules. Since $\hom_{\Gamma_2^{\op}}(Q_i,U)\in\add{}_{\Gamma_1}U$ holds, $U$ is a tilting $\Gamma_1$-module. ------------------------------------------------------------------------ .5em As a direct consequence, we have the following desired result. .5em[**Corollary  **]{} *Let $R$ be a normal Gorenstein domain with $\dim R\le3$ and $\Lambda$ a module-finite algebra.* \(1) Then all NCCR of $\Lambda$ are derived equivalent. \(2) Assume that $R$ is complete local. If $M_1$ and $M_2$ are $\Lambda$-modules giving NCCR, then $M_1$ and $M_2$ have the same number of non-isomorphic indecomposable summands. .5em[Proof ]{} (1) follows directly from (2). For (2), we use that the Grothendieck group of $\endm_\Lambda(M_i)$ has a basis consisting of isoclasses of indecomposable projective modules, and that derived equivalences preserve Grothendieck groups. ------------------------------------------------------------------------ .5em Note that  also gives a correspondence between $\Lambda$-modules giving rise to NCCR and a subset of the reflexive tilting $\Gamma_1$-modules. For 3-CY$^-$ algebras we have the following improvement of this. .5em[**Theorem  **]{} *Let $\Lambda$ be 3-CY$^-$ and $M$ a $\Lambda$-module giving a NCCR $\Gamma:=\endm_\Lambda(M)$ and $\fff:=\hom_{\Lambda}(M,-):\ref\Lambda\to\ref\Gamma$ the induced reflexive equivalence.* \(1) $\fff$ gives a one-one correspondence between $\Lambda$-modules giving NCCR and reflexive tilting $\Gamma$-modules. \(2) $\fff$ gives a one-one correspondence between reflexive $\Lambda$-modules whose endomorphism rings are orders and reflexive rigid $\Gamma$-modules. \(3) For $N\in\ref\Lambda$, $\endm_\Lambda(N)$ is an order if and only if $N$ is a direct summand of some $\Lambda$-module giving a NCCR. \(4) $\Lambda$ has a generator giving a NCCR. .5em[Proof ]{} (1) Since $\fff$ is full and faithful, we only have to show that $\fff$ gives a surjective map, in view of (2). Take any reflexive tilting $\Gamma$-module $T$. Choose $N\in\ref\Lambda$ such that $\fff(N)=T$. By (2), $\Gamma$ is 3-CY. By (3), $T$ gives a NCCR of $\Gamma$. By (2), $N$ gives a NCCR of $\Lambda$. \(2) Since any reflexive rigid module is a direct summand of a reflexive tilting module by the Bongartz completion , the assertion follows by (1) and (1), and using that if $\endm_\Lambda(M)$ is an order and $N$ is a direct summand of $M$, then $\endm_\Lambda(N)$ is an order. \(3) The ‘if’ part is obvious, and the ‘only if’ part follows by (1)(2) and the Bongartz completion . \(4) Since $\endm_\Lambda(\Lambda)=\Lambda$ is an order, the assertion follows by (3). ------------------------------------------------------------------------ .5em Choosing $M:=\Lambda$ in  for a 3-CY algebra, we have the following remarkable relationship between tilting modules and NCCR. .5em[**Corollary  **]{} *Let $\Lambda$ be 3-CY.* \(1) $\Lambda$-modules giving NCCR are exactly reflexive tilting $\Lambda$-modules. \(2) Reflexive $\Lambda$-modules whose endomorphism rings are orders are exactly reflexive rigid $\Lambda$-modules. \(3) Let $T$ be a reflexive tilting $\Lambda$-module and $\Gamma:=\endm_\Lambda(T)$. Then the reflexive euivalence $\hom_\Lambda(T,-):\ref\Lambda\to\ref\Gamma$ gives a one-one correspondence between reflexive tilting $\Lambda$-modules and reflexive tilting $\Gamma$-modules. .5em The following corollary shows that reflexive tilting modules over 3-CY algebras are closed under taking tensor products and homomorphisms. This is a quite peculiar property of 3-CY algebras. Especially (3) below gives another explanation of (4). .5em[**Corollary  **]{} *Let $\Lambda$ be 3-CY.* \(1) For any reflexive tilting $\Lambda$-modules $T_1$ and $T_2$ and $\Gamma_i:=\endm_\Lambda(T_i)$, we have that $U:=\hom_\Lambda(T_1,T_2)$ is a reflexive tilting $\Gamma_1$-module with $\endm_{\Gamma_1}(U)=\Gamma_2$. \(2) $(-)^*\simeq\hom_\Lambda(-,\Lambda^*)$ gives a one-one correspondence between right reflexive tilting modules and left reflexive tilting modules. \(3) For any reflexive tilting $\Lambda^{\op}$-module $T_1$ and $\Lambda$-module $T_2$ and $\Gamma_i:=\endm_\Lambda(T_i)$, it follows that $U:=(T_1\otimes_\Lambda T_2)^{**}$ is a reflexive tilting $\Gamma_1$-module with $\endm_{\Gamma_1}(U)=\Gamma_2$. .5em[Proof ]{} (1) Immediate from (3). \(2) $\Lambda^*$ is a reflexive tilting $\Lambda$-module by (1). We only have to put $T_2:=\Lambda^*$ in (1). \(3) Since $(T_1\otimes_\Lambda T_2)^{**}=\hom_\Lambda(T_2,T_1^*)^*$ holds, the assertion follows from (1) and (2). ------------------------------------------------------------------------ .5em Van den Bergh raised the following question in \[Va2;4.4\]: [*If $\Lambda$ has a NCCR, then does there exist a CM $\Lambda$-module giving a NCCR?*]{} We give a positive answer for isolated singularities. .5em[**Proposition  **]{}[*For any 3-CY$^-$ algebra $\Lambda$ which is an isolated singularity, then $\Lambda$ has a NCCR if and only if $\Lambda$ has a CM generator giving a NCCR.*]{} .5em[Proof ]{} We only have to show the ‘only if’ part. By (4), $\Lambda$ has a generator $M$ giving a NCCR. Since $\Lambda$ is an isolated singularity, we have $\ext^1_\Lambda(M,M)=0$ by (3), in particular, $\ext^1_\Lambda(M,\Lambda)=0$ holds. Taking localization and applying (5)(ii), we have $M\in\cm\Lambda$ since $M$ is reflexive. ------------------------------------------------------------------------ .5em It was shown in \[I4\] that modules giving NCCR is closed related to maximal $1$-orthogonal modules introduced in \[I3\]. We call $M\in\cm\Lambda$ a [*maximal $1$-orthogonal $\Lambda$-module*]{} if [$$\add M=\{X\in\cm\Lambda\ |\ \ext^1_\Lambda(M,X)=0\}=\{X\in\cm\Lambda\ |\ \ext^1_\Lambda(X,M)=0\}.$$]{} If $\Lambda$ is an order, then any maximal $1$-orthogonal $\Lambda$-module $M$ satisfies $\Lambda\oplus\Lambda^*\in\add M$. Using \[I4;5.2.1\] and , we immediately obtain the following result for complete regular local $R$. Later we shall show in  that this is valid for arbitrary $R$. .5em[**Corollary  **]{}[*Let $R$ be complete regular local and $\Lambda$ a 3-CY$^-$ algebra which is an isolated singularity. Then $\Lambda$ has a NCCR if and only if $\Lambda$ has a maximal $1$-orthogonal module.*]{} .5em[Proof ]{} Since $\Lambda$ is 3-CY$^-$, we have $\Lambda^*\simeq\Lambda$ as $\Lambda$-modules by . Now assume that $M\in\cm\Lambda$ is a generator. It was shown in \[I4;5.2.1\] that $M$ gives a NCCR of $\Lambda$ if and only if $M$ is maximal $1$-orthogonal. Thus the assertion follows from . ------------------------------------------------------------------------ .5em We now investigate the relationship between rigid modules and NCCR for 3-CY$^-$ algebras, in particular for those which are isolated singularities. The following will be useful. .5em[**Lemma  **]{}[*Let $M\in\ref\Lambda$ be a generator, $\Gamma:=\endm_\Lambda(M)$ and $\fff:=\hom_\Lambda(M,-):\ref\Lambda\to\ref\Gamma$ a reflexive equivalence ((2)). Then we have a functorial monomorphism $\ext^1_\Gamma(\fff(X),\fff(Y))\subseteq\ext^1_\Lambda(X,Y)$ for any $X,Y\in\ref\Lambda$.*]{} .5em[Proof ]{} Since $M$ is a generator, there exists an exact sequence $0\to X_1\to M_0\stackrel{f}{\to}X\to0$ with a right $(\add M)$-approximation $f$. Then $0\to\fff(X_1)\to\fff(M_0)\to\fff(X)\to0$ is an exact sequence in $\mod\Gamma$ where $\fff(M_0)$ is projective. We have the following exact commutative diagram: [$$\begin{diag} \hom_\Gamma(\fff(M_0),\fff(Y))&\longrightarrow&\hom_\Gamma(\fff(X_1),\fff(Y))&\longrightarrow&\ext^1_\Gamma(\fff(X),\fff(Y))&\longrightarrow&0\\ \parallel&&\parallel\\ \hom_\Lambda(M_0,Y)&\longrightarrow&\hom_\Lambda(X_1,Y)&\longrightarrow&\ext^1_\Lambda(X,Y) \end{diag}$$]{} -1em Thus we have a monomorphism $\ext^1_\Gamma(\fff(X),\fff(Y))\subseteq\ext^1_\Lambda(X,Y)$. ------------------------------------------------------------------------ .5em[**Theorem  **]{} *Let $\Lambda$ be a 3-CY$^-$ algebra with a NCCR and $N\in\ref\Lambda$. Then (1)$\Rightarrow$(2)$\Leftrightarrow$(3) holds. If $\Lambda$ is an isolated singularity, then (1)–(3) are equivalent.* \(1) $N$ is rigid. \(2) $\endm_\Lambda(N)$ is an order. \(3) $N$ is a direct summand of some $\Lambda$-module giving a NCCR. .5em[Proof ]{} (2)$\Leftrightarrow$(3) follows by (3). If $\Lambda$ is an isolated singularity, then we can show (2)$\Rightarrow$(1) by taking localization and applying (3). (1)$\Rightarrow$(2) Assume that $N$ is rigid. By (4), $\Lambda$ has a generator $M$ giving a NCCR. Put $\Gamma:=\endm_\Lambda(M)$ and let $\fff:=\hom_\Lambda(M,-):\ref\Lambda\to\ref\Gamma$ be a reflexive equivalence. It follows from  that $\ext^1_\Gamma(\fff(N),\fff(N))=0$. Thus $\fff(N)$ is a reflexive rigid $\Gamma$-module. By (2), $\endm_\Lambda(N)$ is an order. ------------------------------------------------------------------------ .5em We illustrate with the following example. .5em[**Theorem  **]{} *Let $K$ be a field of characteristic zero, $G$ a finite subgroup of $\SL_3(K)$, $S:=K[[x,y,z]]$ and $g$ the number of irreducible representations of $G$.* \(1) For any $S^G$-module $M$ giving a NCCR, the number of non-isomorphic indecomposable direct summands of $M$ is exactly $g$. \(2) Any reflexive rigid $S^G$-module $N$ is a direct summand of an $S^G$-module giving a NCCR. Thus the number of non-isomorphic indecomposable direct summands of $N$ is at most $g$. .5em[Proof ]{} $S$ gives a NCCR $S*G$ of a 3-CY$^-$-algebra $S^G$, which has exactly $g$ non-isomorphic indecomposable direct summands \[I4\]. Thus (1) follows by (2). Now (2) follows by (1) and (1)$\Rightarrow$(3). ------------------------------------------------------------------------ .5em We say that a reflexive rigid $\Lambda$-module $N$ [*maximal rigid*]{} if $L\in\ref\Lambda$ and $\ext^1_\Lambda(N\oplus L,N\oplus L)=0$ imply $L$ is in $\add N$ (c.f. \[GLS\]). .5em[**Corollary  **]{} *Let $\Lambda$ be a 3-CY$^-$ algebra which is an isolated singularity and has a NCCR.* \(1) $M$ in $\ref\Lambda$ gives a NCCR if and only if it is maximal rigid. \(2) Any reflexive equivalence $\ref\Lambda\to\ref\Gamma$ gives a one-one correspondence between rigid $\Lambda$-modules and rigid $\Gamma$-modules. .5em[Proof ]{} (1) This follows directly from the equivalence of (1) and (3). \(2) By (1) and (2), rigidity depends only on the endomorphism ring. ------------------------------------------------------------------------ .5em We have the following generalization of  for arbitrary $R$. .5em[**Theorem  **]{} *Let $\Lambda$ a 3-CY$^-$ algebra which is an isolated singularity.* \(1) CM $\Lambda$-modules giving NCCR are exactly maximal $1$-orthogonal $\Lambda$-modules. \(2) $\Lambda$ has a NCCR if and only if $\Lambda$ has a maximal $1$-orthogonal module. .5em[Proof ]{} By , we only have to show (1). \(i) Assume that $M\in\cm\Lambda$ is maximal $1$-orthogonal. Put $\Gamma:=\endm_\Lambda(M)$ and $\fff:=\hom_\Lambda(M,-)$. Take an exact sequence $0\to\Omega M\to P\to M\to0$ where $P$ is projective. Applying $\hom_\Lambda(-,M)$, we have an exact sequence $0\to\Gamma\to\hom_\Lambda(P,M)\to\hom_\Lambda(\Omega M,M)\to\Ext^1_\Lambda(M,M)=0$. Since $\hom_\Lambda(P,M)\in\cm R$ and $\hom_\Lambda(\Omega M,M)\in\ref R$, we have that $\Gamma$ is an $R$-order. Since any $\dn{p}\in\Max R$ satisfies $\height\dn{p}=3$ by (1), we only have to show $\gl\Gamma\le 3$. For any $X\in\mod\Gamma$, take a projective resolution $Q_1\stackrel{f}{\to}Q_0\to X\to0$ where $Q_i$ is projective. By Yoneda’s Lemma on $\add M$, there exists $g\in\hom_\Lambda(M_1,M_0)$ such that $M_i\in\add M$ and $f=\fff(g)$. Put $Y:=\Ker g$ and take an exact sequence $0\to Z\to M_2\stackrel{h}{\to}Y\to0$ with a right $(\add M)$-approximation $h$. Then $Z\in\cm\Lambda$ holds. Applying $\fff$, we have an exact sequence $\fff(M_2)\stackrel{\fff(h)}{\longrightarrow}\fff(Y)\to\Ext^1_\Lambda(M,Z)\to\Ext^1_\Lambda(M,M_2)$. Since $\fff(h)$ is surjective and $M$ is rigid, we have $\Ext^1_\Lambda(M,Z)=0$. Thus $Z\in\add M$. Consequently, $\pd{}_\Gamma X\le 3$ holds since we have a projective resolution $0\to\fff(Z)\to\fff(M_2)\to Q_1\stackrel{f}{\to}Q_0\to X\to0$. \(ii) Assume that $M\in\cm\Lambda$ gives a NCCR $\Gamma:=\endm_\Lambda(M)$. Put $\fff:=\hom_\Lambda(M,-)$. By (1), $M$ is maximal rigid. Since $\Lambda$ is 3-CY$^-$ and $M\in\cm\Lambda$, we have that $M\oplus\Lambda$ and $M\oplus\Lambda^*$ are rigid. In particular, we have $\Lambda\oplus\Lambda^*\in\add M$. Take $X\in\cm\Lambda$ such that $\Ext^1_\Lambda(X,M)=0$. Take a projective resolution $P_1\to P_0\to X^*\to0$ of a $\Lambda^{\op}$-module $X^*$. Applying $(-)^*$, we have an exact sequence $0\to X\to P_0^*\to P_1^*$ with $P_i^*\in\add\Lambda^*\subset\add M$. Applying $\fff$, we have an exact sequence $0\to\fff(X)\to\fff(P_0^*)\to\fff(P_1^*)$ with projective $\Gamma$-modules $\fff(P_i^*)$. Since $\gl\Gamma=3$, we have $\pd{}_\Gamma\fff(X)\le1$. Thus we can take a projective resolution $0\to Q_1\to Q_0\to\fff(X)\to0$. By Yoneda’s Lemma on $\add M$, there exists a complex $0\to M_1\to M_0\to X\to0$ with $M_i\in\add M$ such that $0\to\fff(M_1)\to\fff(M_0)\to\fff(X)\to0$ is isomorphic to the above projective resolution. Since $M$ is a generator, we have that $0\to M_1\to M_0\to X\to0$ is exact. Since $\Ext^1_\Lambda(X,M)=0$ by our assumption, we have $X\in\add M$. Take $Y\in\cm\Lambda$ such that $\Ext^1_\Lambda(M,Y)=0$. Since $M^*\in\cm\Lambda^{\op}$ gives a NCCR and $\Ext^1_{\Lambda^{\op}}(Y^*,M^*)=0$, we have $Y^*\in\add M^*$ and $Y\in\add M$. Consequently, $M$ is maximal $1$-orthogonal. ------------------------------------------------------------------------ .5em While a maximal $1$-orthogonal module is maximal rigid, the converse does not hold in general even if it is CM. For example, the simple singularity $\Lambda:=K[[x_1,x_2,x_3,x_4]]/(x_1^{2n+1}+x_2^2+x_3^2+x_4^2)$ does not have non-projective rigid CM modules \[Yo\]. Thus $\Lambda$ is a maximal rigid $\Lambda$-module, which is not maximal $1$-orthogonal. It seems to be difficult to know when the converse holds. Now we show that some kind of converse holds for 3-CY$^-$ algebras. .5em[**Proposition  **]{}[*Let $\Lambda$ be a 3-CY$^-$ algebra which is an isolated singularity and has a NCCR. Then $M\in\ref\Lambda$ is maximal rigid if and only if there exists a reflexive equivalence $\fff:\ref\Lambda\to\ref\Gamma$ such that $\fff(M)$ is a maximal $1$-orthogonal $\Gamma$-module.*]{} .5em[Proof ]{} Let $M$ be maximal rigid. 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--- abstract: 'Dynamical Component Analysis (DyCA) is a recently-proposed method to detect projection vectors to reduce the dimensionality of multi-variate deterministic datasets. It is based on the solution of a generalized eigenvalue problem and therefore straight forward to implement. DyCA is introduced and applied to EEG data of epileptic seizures. The obtained eigenvectors are used to project the signal and the corresponding trajectories in phase space are compared with PCA and ICA-projections. The eigenvalues of DyCA are utilized for seizure detection and the obtained results in terms of specificity, false discovery rate and miss rate are compared to other seizure detection algorithms.' address: | Ansbach University of Applied Sciences\ Faculty of Engineering Sciences\ Residenzstr. 8, 91522 Ansbach, Germany\ {katharina.korn,bastian.seifert,christian.uhl}@hs-ansbach.de bibliography: - 'Literatur.bib' title: 'Dynamical Component Analysis (DyCA) and its Application on epileptic EEG' --- dimensionality reduction, principal component analysis, independent component analysis, EEG, seizure detection Introduction {#sec:introduction} ============ Automatic detection of epileptic events in EEG data is a challenging problem. On the one hand the detection of all epileptic events, especially during live-monitoring sessions, is desireable. On the other hand not too many epilepsy alarms should be triggered. This represents a common classification problem aiming at high detection rate as well as a high specificity. Current approaches to solve this problem use very sophisticated techniques, like the combination of wavelet transform with classical machine learning classification approaches [@Ahmadi:2017; @Acharya:2012PCA; @Acharya:2012ICA] or instead with a deep convolutional neural network [@Acharya:2018]. Dimensionality reduction of deterministic multi-variate time-series is another ambitious problem. Most of the currently available tools of dimensionality reduction, like PCA [@Pearson:1901] or ICA [@Hyvaerinen.Oja:2000a] and modifications thereof, make a *stochasticity* assumption on the time-series on which they can be applied. Presumably due to the lack of better matching techniques they are often used for dimensionality reduction of deterministic time-series even if its assumptions are not fulfilled. Recently [@Seifert.Korn.Hartmann.Uhl:2018a] the authors proposed a new method for dimensionality reduction of deterministic time-series: dynamical component analysis (DyCA). This method relys on a determinacy assumption on the time-series. The projection onto a lower-dimensional space is then found by solving a generalized eigenvalue problem. The eigenvalues of the generalized eigenproblem of DyCA measure the quality of the assumption of linear determinism for the investigated data. For certain conditions the corresponding eigenvectors together with some linear transforms yield an optimal projection to represent the signal by a deterministic non-linear differential equation. Since EEG data during an epileptic event is known to be of deterministic structure [@vanVeen.Liley:2006], while the EEG data during normal activity is of stochastic nature, we assume that the detection of epileptic events might be possible by investigating the generalized eigenvalues of DyCA. In this article we examine this approach and demonstrate the power of DyCA with respect to the obtained trajectories in phase space by projecting the original signal onto the DyCA eigenvectors. The DyCA eigenvalues are utilized to implement a novel seizure detection algorithm and its results in terms of specificity, false discovery rate and miss rate are compared to other studies [@Acharya:2012PCA; @Acharya:2012ICA; @Acharya:2018]. The paper is structured as follows. In Section \[sec:DyCA\] we present the basic concepts underlying DyCA and demonstrate the formulation of the dimensionality reduction process. Section \[sec:EpilepsyDeterminism\] deals with the assumption and confirmation of a deterministic model of EEG data of epileptic seizures. In Section \[sec:SeizureDetection\] a DyCA-based method for detection of epileptic seizures is evaluated and compared. Finally the results are discussed (Section \[sec:Discussion\]) and concluded (Section \[sec:Conclusion\]). Dynamical Component Analysis (DyCA) {#sec:DyCA} =================================== Dynamical Component Analysis (DyCA) is a recently-proposed [@Seifert.Korn.Hartmann.Uhl:2018a] method for dimensionality reduction of deterministic time-series and can be derived as follows. Assume, given a high-dimensional deterministic time-series $q(t) \in \mathbb{R}^N$ with dynamics governed by a low-dimensional system of ordinary differential equations, the signal can be decomposed into $$\label{eq:DecompositionHighDimSignal} q(t) = \sum_{i=1}^{n} x_i(t) w_i$$ using time-dependent amplitude $x_i(t)$ and vectors $w_i \in \mathbb{R}^N$ with $n \ll N$. The amplitudes are then assumed to be governed by a set of ordinary differential equations, divided into a set of linear differential equations, $$\label{eq:LinearEquationsDynamics} \begin{split} \dot{x}_1 &= \sum_{k=1}^n a_{1,k} x_k \\ &\vdots \\ \dot{x}_m &= \sum_{k=1}^n a_{m,k} x_k , \end{split}$$ and a set of non-linear differential equations with smooth functions $f_i$: $$\label{eq:NonlinearEquationsDynamics} \begin{split} \dot{x}_{m+1} &= f_{m+1}(x_1,\dots,x_n) \\ &\vdots \\ \dot{x}_n &= f_n(x_1,\dots,x_n). \end{split}$$ Furthermore we assume that $m \geq n/2$, i.e. there exist more linear than non-linear equations, and that every amplitude $x_i$ associated to a non-linear equation appears in the right-hand side of at least one of the linear equations without knowing the coefficients $a_{i,k}$ or the smooth functions $f_i$. Then projection vectors, $u_i, v_j \in \mathbb{R}^N $, can be found containing the dynamics by minimizing the cost function $$\label{eq:DyCACostFunction} D(u,v,a) = \frac{\timeavg{\norm{\dot{q}^\top u - \sum_j a_j q^\top v_j}_2^2}}{\timeavg{\norm{\dot{q}^\top u}_2^2}},$$ where $\timeavg{\argument}$ denotes the average over time. The rational behind this is that at a minimum of $D$ all the information on how to project onto the non-linear parts is contained in $\sum_j a_j q^\top v_j$ and for the linear parts it is contained in $\dot{q}^\top u$. The minima of $D$ for the vectors $u$ can be determined by a generalized eigenvalue problem $$\label{eq:DyCAEigenproblem} C_1 C_0^{-1} C_1^\top u = \lambda C_2 u,$$ with correlation matrices $C_0 = \timeavg{q q^\top}, C_1 = \timeavg{\dot{q} q^\top}$, and $C_2 = \timeavg{\dot{q} \dot{q}^\top}$. Furthermore there exists the connection $u = \lambda C_2^{-1} C_1 \sum_j a_j v_j$. Thus by projecting onto $$\label{eq:DyCAProjectionSpace} \mathsf{span}\{u_1, \dots, u_m, C_1^{-1} C_2 u_1, \dots, C_1^{-1} C_2 u_m\} = \mathbb{R}^n$$ all relevant information is received. On the other hand, the eigenvalues of the generalized eigenproblem  reveal something more, as then the cost function  takes the value $$\label{eq:MinValueDyCACostFunction} D_{\min} = 1 - \lambda.$$ Thus the number of the generalized eigenvalues with a value of approximately $1$ are a measure of the number of linear equations contained in the data. In the subsequent sections this connection will be exploited to detect regions in time-series with highly deterministic parts, like epileptic seizures. Epilepsy - deterministic EEG data {#sec:EpilepsyDeterminism} ================================= Unlike a first thought might suggest, during epileptic events the EEG data is much more regular than during normal phases. Indeed there are even models suggesting Shilnikov chaos to appear during epileptic seizures [@vanVeen.Liley:2006; @Friedrich.Uhl:1996a]. In its easiest form a system showing Shilnikov chaos can be described by a set of three differential equations of the form $$\label{eq:ShilnikovChaos} \begin{split} \dot{x}_1 &= x_2 \\ \dot{x}_2 &= x_3 \\ \dot{x}_3 &= f(x_1,x_2,x_3), \end{split}$$ with a non-linear smooth function $f$. Thus one can assume that the assumptions of DyCA are fulfilled and DyCA can be applied to epileptic EEG data. Fig. \[fig:WindowedDyCAEpilepsy\] presents the three largest eigenvalues of DyCA applied to a moving window of an EEG dataset with an epilectic seizure. Each investigated window has a length of three seconds and 90% overlap. An epileptic seizure occurs in-between window number 600 to approx. 700. ![The three largest eigenvalues of DyCA on moving windows of EEG data. In the background the Fz-electrode is shown in grey (right axis). []{data-label="fig:WindowedDyCAEpilepsy"}](Eigenvalues){width="45.00000%"} {width="\textwidth"} A clear jump of the first three eigenvalues is observed during seizure. This confirms the assumed low-dimensional deterministic behaviour in ictal phases of the signal. This observation in one dataset is investigated on a broader data basis in section \[sec:SeizureDetection\] and a possible application of DyCA as seizure detection algorithm is discussed. Considering the ODE  as a model of the epileptic seizure, two eigenvalues are expected to be close to the value of $1$. Fig. \[fig:WindowedDyCAEpilepsy\] shows this behavior within the seizure: the blue and the red line representing the two largest eigenvalues are clearly closer to the value of 1 than the third eigenvalue. By choosing an appropriate threshold, DyCA allows for an identification of projection vectors leading to amplitudes which obey a linear set of ODEs. Fig. \[fig:ShilnikovChaosHomoclinic\] shows on the right hand side the three-dimensional trajectory of the amplitudes corresponding to the eigenvectors $u_1$ and $u_2$ and vector $v_1$. The structure of the trajectory is clearly observable and the typical homoclinic orbit of Shilnikov chaos is shown. DyCA represents a significant improvement compared to the 3D-trajectories in phase space spanned by the projection onto the first three PCA vectors or the best ICA vectors. Detection of seizure events {#sec:SeizureDetection} =========================== We now investigate the utilization of the DyCA-eigenvalues for the detection of epileptic seizure events. For this a moving window frame runs over the data and the DyCA eigenvalues are calculated on the current window. The largest eigenvalues are compared against a threshold. If the eigenvalues are larger than the threshold, it is assumed that the current window contains an epileptic seizure. To measure the quality of the classification method specificity (SPC), false discovery rate (FDR) and miss rate/false negative rate (FNR) were calculated. The specificity is the number of windows correctly classified as not containing a seizure relative to the number of windows not containing a seizure. The false discovery rate is the ratio of windows falsely classified as containing a seizure over all windows classified as containing a seizure. The miss rate is the number of windows falsely classified as not containing a seizure in relation to the number of all windows containing a seizure. In our test setting DyCA detection was applied on six EEG data sets of patients with absences, which are a special kind of epileptic seizures. The mean length of the data sets is 411 seconds containing absences of length ranging from 4 to 25 seconds. The size of the moving window was taken as three seconds. As step size for the movement of the window 10 % of the window size were taken. This results in windows having 90 % overlap. A window was labeled as seizure if the whole window was contained in the pre-labeled seizure. That is windows contained only partly in the seizure are labeled as not being in the seizure. This results in a mean prevalence of $2.32 \pm 2.18~\%$. No further pre-processing techniques, like filtering, were applied to the data. On each window the DyCA eigenvalues were compared to a threshold. In Fig. \[fig:1lambda\] the SPC, FDR, and FNR are shown for classification if only the largest eigenvalue is compared against the threshold. Analogously Fig. \[fig:2lambda\] shows SPC, FDR, and FNR for classification comparing the two largest eigenvalues against the threshold. ![The specificity (green), false discovery rate (blue), and miss rate (red) plotted against the threshold for the largest eigenvalue. The shaded area shows the standard deviation with respect to different data sets.[]{data-label="fig:1lambda"}](ErrorBarPlot){width="45.00000%"} ![The specificity (green), false discovery rate (blue), and miss rate (red) plotted against the threshold for the two largest eigenvalues. The shaded area shows the standard deviation with respect to different data sets.[]{data-label="fig:2lambda"}](ErrorBarPlot2){width="45.00000%"} Discussion {#sec:Discussion} ========== Considering the small prevalence the results in Fig. \[fig:1lambda\] and Fig. \[fig:2lambda\] indicate that the two largest eigenvalues should be considered, since then the specificity reaches 99 %. This supports from a data-driven point of view the theoretical considerations in [@vanVeen.Liley:2006]. The choice of the threshold can be adapted to the application in mind. If the detection of all seizures as soon as possible is wanted and false alarms are acceptable the threshold should be lowered. If one is only interested in finding examples of appearing seizures a higher threshold should be chosen. The high standard deviation and relative high values of the false discovery rate is due to the labeling process of the windows. Since requiring that the whole window is contained in the seizure, windows containing only small parts outside the seizure are labeled as no seizure even though the deterministic part might be dominant. The detection algorithm can be adjusted to obtain a specificity of nearly 99.7 % and still have a miss rate of 20 %. Conclusion {#sec:Conclusion} ========== In this paper we presented dynamical component analysis (DyCA) as an alternative to PCA and ICA reducing the dimensionality of multi-variate time series based on the assumption of an underlying dynamical system. Comparing the 3D-trajectories in phase space we obtained by DyCA more obvious structures than obtained by PCA and ICA (Fig. \[fig:ShilnikovChaosHomoclinic\]). Applying DyCA to EEG data of epileptic seizures we implemented a novel seizure detection algorithm and obtained good results (specificity of nearly 99.7 %) in comparison with other studies using more complex tools for detection: In [@Acharya:2018] using a deep convolutional neural network for detection, a specificity of only 90 % was reached. Further studies used wavelet transform or wavelet packet decomposition combined with ICA [@Acharya:2012ICA] or PCA [@Acharya:2012PCA] and different classifiers. They reach a specificity of 97 % with ICA and 99 % with PCA as intermediate step. Generally, it might be advantageous for some applications to use DyCA instead of ICA or PCA as intermediate step for more refined classification methods. This is subject of work in progress and will be presented in upcoming papers.

--- abstract: | This work is motivated by a hand-collected data set from one of the largest Internet portals in Korea. This data set records the top 30 most frequently discussed stocks on its on-line message board. The frequencies are considered to measure the attention paid by investors to individual stocks. The empirical goal of the data analysis is to investigate the effect of this attention on trading behavior. For this purpose, we regress the (next day) returns and the (partially) observed ranks of frequencies. In the regression, the ranks are transformed into scores, for which purpose the identity or linear scores are commonly used. In this paper, we propose a new class of scores (a score function) that is based on the moments of order statistics of a pre-decided random variable. The new score function, denoted by D-rank, is shown to be asymptotically optimal to maximize the correlation between the response and score, when the pre-decided random variable and true covariate are in the same location-scale family. In addition, the least-squares estimator using the D-rank consistently estimates the true correlation between the response and the covariate, and asymptotically approaches the normal distribution. We additionally propose a procedure for diagnosing a given score function (equivalently, the pre-decided random variable Z) and selecting one that is better suited to the data. We numerically demonstrate the advantage of using a correctly specified score function over that of the identity scores (or other misspecified scores) in estimating the correlation coefficient. Finally, we apply our proposal to test the effects of investors’ attention on their returns using the motivating data set. 0.5cm [**Keywords:**]{} Concomitant variable; investors’ attention; linear regression; moments of order statistics; optimal scaling; partially observed ranks author: - | Yuneung Kim, Johan Lim, Young-Geun Choi\ and\ Sujung Choi, and Do Hwan Park[^1] date: - - title: | Regression with Partially Observed Ranks on a Covariate:\ Distribution-Guided Scores for Ranks --- Introduction ============ This paper is motivated by a hand-collected data set from `Daum.net`, the 2nd largest Internet portal in Korea. The `Daum.net` portal offers an on-line stock message board where investors can freely discuss specific stocks in which they might be interested. This portal also reports a ranked list of the top 30 stocks that are most frequently discussed by users on a daily basis. The data set was collected by the authors during the 537 trading days from October 4th, 2010, to November 23rd, 2012. Along with the rank data, we also collected financial data regarding individual companies from FnGuide (`http://www.fnguide.com`). These additional data include stock-day trading volumes classified in terms of different types of investors, stock prices, stock returns, and so on. The purpose of analyzing the collected data is to investigate the shifts in stock returns caused by variations in investor attention. In finance, researchers are often interested in determining the motivations that drive buying and selling decisions in stock markets. It is commonly assumed that investors efficiently process relevant information in a timely manner, but in reality, it is nearly impossible to be efficient because of information overload. In particular, individual investors are often less sophisticated than are institutional investors and have a limited ability to process all relevant information. For this reason, individual investors may pay attention only to a limited amount of information, perhaps that which is relatively easy to access. The phenomenon of limited attention is a well-documented cognitive bias in the psychological literature [@Kahneman:1973; @Camerer:2003]. This phenomenon affects the information-processing capacities of investors and thus may affect asset prices on the financial market. To empirically prove the effect of investor attention on stock returns, we regress the observed stock returns with respect to the partially observed ranks. Regression on a (partially observed) rank covariate has not previously been extensively studied in the literature. A procedure that is commonly used in practice to address rank covariates is to (i) regroup the ranks into only a few groups (if the number of ranks is high) and (ii) treat the regrouped ranks as an ordinal categorical variable. Ordered categorical variables frequently arise in various applications and have been studied extensively in the literature. Score-based analysis is most commonly used for this purpose; see [@Hajek:1968], @Hora:1984, @Kimeldorf:1992, @Zheng:2008, @Gertheiss:2014 and the references therein. Thus, this typical two-step procedure for addressing a rank covariate is equivalent to defining a score function for the ranks. However, as in the case of ordinal categorical variables, such a score-based approach suffers from an inherent drawback related to the choice of the score function; different choices of scores may lead to conflicting conclusions in the analysis [@Graubard:1987; @Ivanova:2001; @Senn:2007]. The recommendation for selecting the score function according to the literature is (i) to choose meaningful scores for the ordinal categorical variable based on domain knowledge of the data, (ii) to use equally spaced scores if scientifically plausible scores are not available (see @Graubard:1987), and (iii) to find a optimal scaling transformed scores that maximize the correlation with the responses while preserving the assumed characteristics of the ordinal values[@Linting:2007; @Costantini:2010; @deLeeuw:2009; @Mair:2010; @Jacoby:2016]. In this paper, we seek to provide an efficient tool for approach (i) described above, for the case in which some qualitative knowledge is available regarding the ranks or the ranking variable (the variable that is ranked). More specifically, we propose a new set of score functions, denoted by D-rank, and study their use in linear regression. The proposed score function is based on the moments of order statistics (MOS) of a pre-decided random variable $Z$. This score function has several interesting properties related with the regression model, if the pre-decided random variable is correctly specified as listed below. Here, the correct specification implies it is within the same location-scale family with the true (unobserved) covariate $X$. First, the D-rank is asymptotically optimal in the sense that it maximizes the correlation between the response and score if the distribution of the D-rank is correctly specified. Second, the least-squares estimator using the D-rank consistently estimates the true correlation between the response and the covariate and asymptotically approaches the normal distribution. Finally, the residuals of the fitted regression allow us to diagnose the given score function (equivalently, the pre-decided random variable $X$) and to provide a tool for selecting a score function that is better suited to the data. The remainder of this paper is organized as follows. In Section 2, we study the properties of the proposed D-rank. In this section, we show that the proposed D-rank is asymptotically optimal to maximize the correlation between the response and score. In addition, We also demonstrate the asymptotic equivalence between the proposed score function and the quantile function; the quantile function may provide a better illustration of the qualitative features of the score function. In Section 3, we apply the score function to estimate the regression coefficient of the linear model or, more precisely, to estimate the correlation coefficient between the response and the scoring variable $X$. We prove that the least-squares estimator using the D-rank consistently estimates the correlation coefficient and is asymptotically normally distributed. In addition, we discuss the procedure for selecting an appropriate score function using the residuals. In Section 4, we numerically demonstrate that using the correctly specified score function significantly reduces the mean square error on the estimation of the correlation coefficient. In Section 5, we analyze the motivating data set to investigate the existence of the attention effect. Finally, in Section 6, we briefly summarize the paper and discuss the application of the proposed scores to regression using other auxiliary covariates. Distribution-Guided Scores for Ranks (D-rank) ============================================= We consider a simple regression model in which only partial ranks of a covariate are observed. Specifically, suppose that $\big\{ \big( Y_{i}, X_{i} \big), ~i=1,2,\ldots,n \big\}$ is the complete set of observations, where $Y_{i}$ is the variable of primary interest and $X_{i}$ is the covariate related to $Y_i$. For example, in our rank data from `Daum.net`, for $i=1,2,\ldots,n$, $Y_i$ is a relevant outcome such as earning rate or trading volume, $X_i$ is the “unobserved” investors’ attention on the $i$th company measured by the frequency of on-line discussions, and $R_i$ is the “observed” rank of $X_i$ among $X_1,X_2,\ldots,X_n$. We make certain assumptions regarding the distributions of $X$ and $Y$. We assume that the linear model of the relationship between $X_i$ and $Y_i$ is $$\label{eqn:lm} Y_i = \mu_Y + \rho \sigma_Y \frac{X_i - \mu_X}{\sigma_X} + \epsilon_i,$$ where the $\epsilon_i$s are IID values from a distribution of mean $0$ and variance $\sigma_{\epsilon}^2$. The objective of this paper is the estimation and inference of $\rho={\rm corr} \big(Y, X\big)$ (or the regression coefficient between $Y$ and $X$) based on the observed data $\big\{ (Y_i, R_i), i=1,2,\ldots,n \big\}$. To do it, we aim to define a good score function $S(r)$ for the observed rank $r$, and consider the regression of $Y_{[r:n]}$ on $S(r)$, where $Y_{[r:n]}$ is the response $Y_i$ for $R_i=r$. The D-rank, we propose in this paper, is a set of the MOS of pre-decided random variable $Z$, which we assume is in the same location-scale family of the true covariate $X$. To be specific, suppose that $Z_1,Z_2,\ldots,Z_n$ are independent and identically distributed (IID) copies of the random variable $Z$ and that $Z_{(r:n)}$ is the corresponding $r$th-order statistic for $r=1,2,\ldots,n$. The D-rank defines the score of the rank $r$ as $S_n(r)=\alpha_{(r:n)}:={\rm E} \big(Z_{(r:n)} \big)$ for $r=1,2,\ldots,n$. We first show that the D-rank maximizes the sample correlation between $Y_{[r:n]}$ and $\alpha_{(r:n)}$, $r=1,2,\ldots,n$, in asymptotic, among all increasing functions $S_n(r):\{1,2,\ldots,n\} \rightarrow \mathbb{R}$. Let $\overline{S}_n(r)$ and $\overline{\alpha}_{(r:n)}$ be the standardized scores (of $S_n(r)$ and $\alpha_{(r:n)}$) to make $\sum_{r=1}^n \overline{S}_n(r) = \sum_{r=1}^n \overline{\alpha}_{(r:n)}=0$ and $\sum_{r=1}^n \overline{S}^2_n(r) = \sum_{r=1}^n\overline{\alpha}^2_{(r:n)}=1$. Let ${\bf S}_n$ and $\overline{\bf S}_n$ be the collection of all increasing functions $S_n(r)$ and $\overline{S}_n(r)$, respectively. Under the linear model (\[eqn:lm\]), if $Z$ is in the location-scale family of $X$, the D-rank maximizes the limit of the sample correlation between $Y_{[r:n]}$ and $S_n(r)$ among $S_n(r) \in {\bf S}_n$: $$\label{eqn:acorr} \lim_{n \rightarrow \infty} \frac{1}{n \cdot \widehat{\sigma}_Y} \sum_{r=1}^n \overline{S}_n(r) \big( Y_{[r:n]} - \overline{Y}_n \big),$$ where $\widehat{\sigma}^2_Y = \frac{1}{n}\sum_{r=1}^n \big( Y_{[r:n]} - \overline{Y}_n \big)^2$. The proof of Theorem 1 is followed in Appendix. Theorem 1 shows the asymptotic optimality of the D-rank for the regression in view of optimal scaling in the literature. The optimal scaling finds optimally transformed scores that explain mostly well the assumed statistical model. It arises in various contexts including Gifi classification of non-linear multivariate analysis[@deLeeuw:2009], the aspect (correlational and non-cprrelational aspects) of multivariable[@Mair:2010], and non-linear principal component analysis[@Linting:2007; @Costantini:2010]. Here, we adopt the idea of the optimal scaling in @Jacoby:2016, and find the transformation to maximize the correlation between the response and transformed scores. Theorem 1 above shows that the D-rank maximizes the correlation in asymptotic, if pre-determined distribution for the D-rank is correctly specified. The proposed score is closely related to the quantile of the underlying distribution of $Z$. Let $F_{Z} (z)$ for $z \in \mathbb{R}$ and $Q_Z(q)$ for $q \in [0,1]$ be the cumulative distribution function (CDF) and the quantile function (QF), respectively, of $Z$. In the estimation of $F_Z(z)$ for $\{Z_i, i=1,2,\ldots,n\}$, the $r$th-order statistic $Z_{(r:n)}$ is the $(r/n)\times 100$-th percentile point of the empirical CDF, and thus, its expected value is approximately equal to $Q_Z(r/n)$. More specifically, given $p_r=\frac{r} {n+1}$, $q_r=1-p_r$, and $Q_r=Q_Z(p_r)$, we can write $$\nonumber \alpha_{(r:n)}=Q_r+ \frac{p_r q_r}{2(n+2)} Q_r^{(2)}+O\big(\frac{1}{n^2} \big),$$ where $Q_r^{(2)}=-{f_Z^{\prime} (Q_r)} \big/{\{ f_Z (Q_r) \}^3}$ and $f_Z(z)$ is the probability density function of $Z$, which is differentiable. We refer the reader to @David:2003 [Section 4.6] for the details of the relationship between the MOS and the quantiles. Consideration of the QF may provide a better understanding of the qualitative features of the proposed score function. Suppose we expect the score function $S_n(r)$ is convex in tail (for $r \ge [n c]$ for a constant $c$ close to $1$); in other words, $S_n (r+1) - S_n (r) \ge S_n(r) - S_n (r-1)$ for $r \ge [n c]$. From the equivalence between the MOS and quantiles, it is known that the convexity of the scores $S_n(r)$ is approximately equal to that of the quantile function $Q_Z(p)$. Furthermore, the convexity of $Q_Z(p)$ for $p \ge c$ implies the following equivalent statements: (i) $F(z)$ is concave in $z$, (ii) $f^{\prime} (z) \le 0$ or (iii) $\log f (z)$ is decreasing in $z$, all for $z \ge Q_Z([nc])$. Simple Linear Regression ======================== In this section, we consider a simple regression model in which only partial ranks of a covariate are observed. Specifically, suppose that $\big\{ \big( Y_{i}, X_{i} \big), ~i=1,2,\ldots,n \big\}$ is the complete set of observations from the linear model (\[eqn:lm\]), and $R_i$ is the rank of $X_i$ among $X_1,X_2,\ldots,X_n$. The rank $R_i$ of $X_i$ is indirectly measured by the frequency of on-line discussions of the $i$th company. In this paper, we consider the case in which the ranks $R_i$ are partially observed in the sense that we observe only that $U_{i}=R_i {\rm I}\big(R_{i} \le m \big) + m^{+} {\rm I} \big( R_{i}>m \big)$ rather than $R_i$, where $m^+$ is an arbitrary constant that is greater than $m$. Finally, the observations are $$\nonumber \big\{ (Y_i, U_i), ~i=1,2,\ldots,n \big\}.$$ We let $Y_{[r:n]}=Y_{i} {\rm I} \big(R_{i}=r\big)$ for $r=1,2,\ldots,m$, and denote the above partially observed data by ${\bf Y}_{[m]}$ for notational simplicity. The objective of this section is to identify a good estimator of $\rho={\rm corr} \big(Y, X\big)$ (or the regression coefficient between $Y$ and $X$) and to test $\mathcal{H}_0: \rho=0$ versus $\mathcal{H}_1: \rho\neq 0$ or $\rho >0$ using the observed data ${\bf Y}_{[m]}$. Least-Squares Estimator ----------------------- To estimate $\rho$, we recall assumptions regarding the distributions of $X$ and $Y$. We assume that the linear model of the relationship between $X_i$ and $Y_i$ is $$\nonumber Y_i = \mu_Y + \rho \sigma_Y \frac{X_i - \mu_X}{\sigma_X} + \epsilon_i,$$ where the $\epsilon_i$s are IID values from a distribution of mean $0$ and variance $\sigma_{\epsilon}^2$. By ordering on the $X_i$s, we have for $r=1,\ldots,n$ $$Y_{[r:n]} = \mu_Y + \rho \frac{\sigma_Y}{\sigma_X} \big( X_{(r:n)} -\mu_X \big) + \epsilon_{[r:n]}, \label{eqn:con-linear}$$ where $\rho = {\rm corr} \big( Y, X \big)$ and $$\begin{aligned} {\rm E} \big(Y_{[r:n]} \big) &=& \mu_Y + \rho \sigma_Y \alpha_{(r:n)} \label{eqn:con-mean}\\ {\rm var} \big( Y_{[r:n]} \big) &=& \sigma_Y^2 \big( \rho^2 \beta_{(rr:n)} + 1 - \rho^2 \big) \nonumber \\ {\rm cov} \big(Y_{[r:n]},Y_{[s:n]}\big) &=& \rho^2\sigma_Y^2\beta_{(rs:n)},~r \ne s \nonumber\end{aligned}$$ with $$\nonumber \alpha_{(r:n)} = {\rm E} \bigg\{ \frac{ X_{(r:n)} - \mu_X } { \sigma_X} \bigg\} \quad \mbox{and } ~~ \beta_{(rs:n)} = {\rm Cov} \left( \frac{ X_{(r:n)} - \mu_X } { \sigma_X}, \frac{ X_{(s:n)} - \mu_X } { \sigma_X} \right)$$ for $r,s=1,2,\ldots,n$ [@David:1974; @David:2003]. We are motivated by the identities (\[eqn:con-linear\]) and (\[eqn:con-mean\]) given above and propose the least-squares estimator $$\label{eqn:lse} \widehat{\rho} \big(s \big) \equiv \frac{1}{\widehat{\sigma}_Y} \cdot \frac{\sum_{r=1}^{[ns]} \alpha_{(r:n)}\big\{ Y_{[r:n]} -\widehat{\mu}_Y \big\}} {\sum_{r=1}^{[ns]} \alpha_{(r:n)}^2 }$$ as an estimator of $\rho$ with $s=m/n$, where, $\widehat{\mu}_Y=\sum_{i=1}^nY_i/n$ and $\widehat{\sigma}_Y^2=\sum_{i=1}^n(Y_i-\widehat{\mu}_Y)^2/n$ are the empirical estimators of the mean and variance, respectively, of $Y$. We claim that, if $X$ is drawn from a location-scale family generated by $Z$, then the least-squares estimator $ \widehat{\rho} \big(s \big)$ with $s=m/n$ in (\[eqn:lse\]), that is calculated based on the partial observations ${\bf Y}_{[m]}$, is consistent and asymptotically normally distributed with an appropriate scale, as shown in Theorem 2. Suppose that $$\nonumber \Psi_n^{\rm I}(s) := \frac{1}{n} \sum_{r=1}^{[ns]} \alpha_{(r:n)}^2 \sigma_{(r:n)}^2, \Psi_n^{\rm II}(s):= \frac{1}{n} \sum_{r1=1}^{[ns]} \sum_{r2=1}^{[ns]} \alpha_{(r1:n)} \alpha_{(r2:n)}\beta_{(r1,r2:n)}^2,~ \mbox{and}~ \Phi_n(s):= \frac{1}{n} \sum_{r=1}^{[ns]} \alpha_{(r:n)}^2,$$ where $\sigma_{(r:n)}^2= \sigma^2\big(X_{(r:n)} \big)$, and let $\Psi_{\infty}^{\rm I}(s)$, $\Psi_{\infty}^{\rm II}(s)$ and $\Phi_{\infty}(s)$ be the limits of $\Psi_n^{\rm I}(s)$, $\Psi_n^{\rm II}(s)$ and $\Phi_n(s)$, respectively (under the assumption that they exist). Under the assumption that $X$ is drawn from a distribution of a location-scale family with a finite variance, the distribution of $\sqrt{n} \big( \widehat{\rho} (s) -\rho \big) $ converges to the normal distribution of mean $0$ and variance $\big\{\Psi_{\infty}^{\rm I}(s)/\sigma_Y^2+\rho^2\Psi_{\infty}^{\rm II}(s)\big\}/\Phi_{\infty}^2(s)$. The proof of Theorem 2 is provided in the Appendix. We conclude this section with two remarks regarding Theorem 2. First, in Theorem 2, from the tower property of the conditional expectation, $$\nonumber {\rm var}\big(\sqrt{n} \widehat{\rho} \big) > \frac{1}{ \big(1 \big/ n \big) \sum_{r=1}^{[ns]} \alpha_{(r:n)}^2} \ge \frac{1}{{\rm var} \big(X \big)}=1,$$ and when $\rho=0$, the asymptotic variance of $\sqrt{n} \widehat{\rho}$ is larger than $1$, which is the variance of the least-squares estimator in the case where $X$ is completely observed. Second, it is possible to test the hypothesis $\mathcal{H}_0:\rho=0$ using the statistic $ {\rm T}= \sqrt{n} \widehat{\rho}, $ which has an asymptotically normal distribution of mean $0$ and variance $1\big/\Phi_{\infty}(s)$. Residual Analysis ----------------- As in the classical linear model, the residuals can provide guidance for identifying a better model and score function. The residuals are defined as $e_{[r:n]}^* = \big(Y_{[r:n]}-\mu_Y\big)/\sigma_Y-\widehat{\rho}\alpha_{(r:n)}$ for $r=1,2,\ldots,[ns]$. Statistical properties of the residuals, which are analogous to those in the classical linear model, are summarized as follows. Under the assumptions of Theorem 2, the following statements are true for the residuals: (i) ${\rm E} \big( e_{[r:n]}^* \big)=0$; (ii) $$\begin{aligned} {\rm var} \big( e_{[r:n]}^* \big) &=&\Big\{ \rho^2 \beta_{(rr:n)} + \big( 1 -\rho^2 \big) \Big\} + \alpha_{(r:n)}^2 \frac{1}{n\sigma_Y^2} \frac{\Psi_n^{\rm I}(s)}{\Phi_n^2(s)} -2 \frac{1}{\sum_{r=1}^{[ns]}\alpha_{(r:n)}^2} \nonumber\\ &&\times \left\{ \rho^2\sum_{w=1}^{[ns]}\alpha_{(w:n)}\alpha_{(r:n)}\beta_{(rw:n)}+\alpha_{(r:n)}^2(1-\rho^2)\right\}\nonumber\end{aligned}$$ \(iii) ${\rm E} \big( e_{[r:n]}^* \alpha_{(r:n)} \big)=0$; and (iv) ${\rm E} \big( e_{[r:n]}^* \widehat{Y}^*_{[r:n]} \big)=0$, where $\widehat{Y}^*_{[r:n]}=\mu_Y- \widehat{\rho} \alpha_{(r:n)}$. The proof of Theorem 3 requires only simple algebra and is thus omitted here. The theorem states that the residuals have mean $0$ and finite variance, and also states that they are uncorrelated with the scores $\alpha_{(r:n)}$ and the predicted values $\widehat{Y}_{[r:n]}$. Thus, the residual plots, which are the plots of (i) $r$ versus $e_{[r:n]}^*$, (ii) $\alpha_{(r:n)}$ versus $e_{[r:n]}^*$, and (iii) $\widehat{Y}_{[r:n]}$ versus $e_{[r:n]}^*$, have the same interpretations as those of the classical linear model. We plug in $\mu_Y$ and $\sigma_Y$ with their empirical estimators and use $e_{[r:n]}=\big(Y_{[r:n]}-\widehat{\mu}_Y-\big)/\widehat{\sigma}_Y-\widehat{\rho} \alpha_{(r:n)}$. The residual sum of squares may be another useful tool for measuring the goodness of fit of the proposed model, as in the classical linear model. The residual sum of squares in our model is defined as $$\nonumber {\rm RSS} = \sum_{r=1}^{[ns]} \bigg(\frac{Y_{[r:n]}-\widehat{\mu}_Y}{\widehat{\sigma}_Y} -\widehat{Y}_{[r:n]} \bigg)^2$$ and will be used along with the residual plots as a guide for selecting a better score function. Finally, the proposed least-squares estimator (\[eqn:lse\]) assumes that the regression line between $\alpha_{(r:n)}$ and $\big(Y_{[r:n]} - \widehat{\mu}_Y \big)$ has an intercept (at the $y$ axis) of $0$. Thus, if the model (or the score function) is correctly specified, then the intercept estimated by the regression (with intercept) should be close to $0$, and the estimated intercept therefore serves as a measure for checking the correctness of the score function. Note that the regression (without intercept) performed in this paper is based on observations of the top $[ns]$ ranks and assumes that the function passes through the origin (see Figure 4). An Estimator with Unranked Observations --------------------------------------- The least-squares estimator presented in Section 3.2 does not fully use the information contained in $\big\{ Y_{[r:n]}:=Y_i {\rm I}(R_i=r), r >m \big\}$; it is used only to estimate $\mu_Y$ and $\sigma_Y$, not to estimate $\rho$ itself. In this section, we briefly demonstrate how $\widehat{\rho}$ can be modified to incorporate these unranked observations. We consider the following modified estimator: $$\nonumber \widehat{\rho}_{\rm m} \big(s \big) \equiv \frac{1}{\widehat{\sigma}_Y} \cdot \frac{\sum_{r=1}^{[ns]} \alpha_{(r:n)}\big\{ Y_{[r:n]} -\widehat{\mu}_Y \big\} + \big(n-[ns] \big) \overline{\alpha}_{[ns]+} \big(\overline{Y}_{[ns]+} -\widehat{\mu} \big)} {\sum_{r=1}^{[ns]} \alpha_{(r:n)}^2 + \big(n-[ns]\big) \overline{\alpha}_{[ns]+}^2},$$ where $\overline{\alpha}_{[ns]+}=\sum_{r=[ns]+1}^n \alpha_{(r:n)} \big/ \big(n-[ns]\big)$ and $\overline{Y}_{[ns]+}=\sum_{r=[ns]+1}^n Y_{[r:n]}\big/ \big(n-[ns]\big)$. This modified estimator also asymptotically approaches the normal distribution. Specifically, suppose that $$\nonumber \widetilde{\alpha}_{(r:n)}= \left\{ \begin{array}{ll} \alpha_{(r:n)} & \quad r=1,2,\ldots,[ns],\\ & \\ \overline{\alpha}_{[ns]+} & \quad r=[ns]+1,[ns]+2,\ldots,n. \end{array} \right.$$ We also suppose that\ $\widetilde{\Psi}_n^{\rm I} (s)= \big(1\big/n\big) \left\{ \sum_{r=1}^{n} \widetilde{\alpha}_{(r:n)}^2 \sigma_{(r:n)}^2\right\}$, $\widetilde{\Psi}_n^{\rm II} (s)=\big(1\big/n\big)\left\{ \sum_{r1=1}^{n} \sum_{r2=1}^{n} \widetilde{\alpha}_{(r1:n)} \widetilde{\alpha}_{(r2:n)}\beta_{(r1,r2:n)}^2 \right\}$ and $\widetilde{\Phi}_n (s)= \big(1 \big/n \big) \sum_{r=1}^{n} \widetilde{\alpha}_{(r:n)}^2$. As in the previous section, $(1/n)$-scaled limits of $\widetilde{\Psi}_n^{\rm I} (s)$, $\widetilde{\Psi}_n^{\rm II} (s)$ and $\widetilde{\Phi}_n (s)$ exist; let the limits be $\widetilde{\Psi}_{\infty}^{\rm I}(s)= \lim_{n \rightarrow \infty}\widetilde{\Psi}^{\rm I}_n (s)\big/n$, $\widetilde{\Psi}_{\infty}^{\rm II}(s)= \lim_{n \rightarrow \infty}\tilde{\Psi}^{\rm II}_n (s)\big/n$ and $\tilde{\Phi}_{\infty}(s)= \lim_{n \rightarrow \infty} \tilde{\Phi}_n (s) \big/n$, respectively. Then, we can write the following theorem. Under the same assumptions as those of Theorem 2, the distribution of $\sqrt{n} \big(\widehat{\rho}_{\rm m} (s) - \rho \big)$ converges to the normal distribution with mean $0$ and variance $\big(\widetilde{\Psi}_{\infty}^{\rm I}(s)/\sigma_Y^2+\rho^2\widetilde{\Psi}_{\infty}^{\rm II}(s)\big)/\widetilde{\Phi}_{\infty}^2(s)$ $$\begin{aligned} \sqrt{n} \big( \widehat{\rho}_{\rm m} - \rho \big) &=& \sqrt{n} \bigg\{\frac{1}{\sum_{r=1}^{[ns]} \alpha_{(r:n)}^2 + \big(n-[ns]\big) \overline{\alpha}_{[ns]+}^2}\times \nonumber\\ && \qquad \bigg( \sum_{r=1}^{[ns]} \alpha_{(r:n)} \Big( Y_{[r:n]} - \widehat{\mu}_Y \big)+\big(n-[ns] \big) \overline{\alpha}_{[ns]+} \big(\overline{Y}_{[ns]+}-\widehat{\mu}_Y \big) \bigg)-\rho\bigg\} \nonumber\\ &=& \sqrt{n} \bigg(\frac{1}{\widehat{\sigma}_Y} \frac{\sum_{r=1}^n \tilde{\alpha}_{(r:n)} \big( Y_{[r:n]} - \widehat{\mu}_Y \big)} {\sum_{r=1}^n \widetilde{\alpha}_{(r:n)}^2}-\rho\bigg), \nonumber \end{aligned}$$ the distribution of which converges to the normal distribution with mean $0$ and variance\ $\big(\widetilde{\Psi}_{\infty}^{\rm I}(s)/\sigma_Y^2+\rho^2\widetilde{\Psi}_{\infty}^{\rm II}(s)\big)/\widetilde{\Phi}_{\infty}^2(s)$ following the same arguments presented in the proof of Theorem 2. Numerical Study =============== In this section, we numerically investigate the advantage we can gain by choosing the correct score function to estimate $\rho={\rm corr} (Y,X)$. The performance of an estimator is measured in terms of its bias and its mean square error (MSE), which we numerically estimate based on $1000$ simulated data sets and the estimators obtained therefrom. The data sets are generated from the regression model $Y_i = \beta_0 + \beta_1 X_i + \epsilon_i,$ $i=1,2,\ldots,n,$ where the $\epsilon_i$ are independently drawn from $N(0,1)$. We consider three distributions for $X$: the uniform distribution on $[0,1]$, the standard normal distribution, and the gamma distribution with mean $1$ and variance $1/3$. As stated in Section 2, the score function of the uniform distribution is almost equivalent to the identity score function $S_n(r)=r$. However, the normal distribution and the gamma distribution have heavier tails than does the uniform distribution, and their score functions are convex in the right tail. We set the parameters $\delta$ to ensure that $\rho = 0$, $0.3$, $0.5$ and $0.7$, where $\rho=\delta/\sigma_Y$. Finally, in each considered case, the sample size $n$ and the number of partially observed ranks $m$ are set to all possible combinations of $n = 500$ or $2000$ and $r=20$, $50$, or $100$. When estimating $\rho$, we apply four different scores, including the proposed MOS-based score functions obtained from the three distributions listed above and the identity score function, which is commonly used in practice. The approximated bias and MSE values are reported in Tables 1 and 2. We can observe several interesting findings from these tables. First, the correctly specified score function performs better than do others when there exists a strong correlation between $X$ and $Y$ (when $\rho$ is large). However, when $\rho=0$, there is almost no difference among the four considered scores. Second, as the number of observations increases, in the sense that either $r$ or $n$ increases, the superiority of the correctly specified scores with respect to the others becomes apparent even when $\rho$ is not large. Third, as conjectured in the previous section, the scores based on the uniform distribution perform almost identically to the identity scores. Finally, the differences between the correctly specified scores and the others are significant regardless of $\rho$ or the sample size ($r$ or $n$) when the distribution of $X$ has a heavier right tail (the gamma distribution). Data Examples ============= Data Description ---------------- To investigate how the attention of investors affects stock returns, we merge the hand-collected `Daum.net` rank data set and the financial data from `FnGuide`. We illustrate how the returns of attention-grabbing stocks fluctuate around the event dates when investors pay attention to these stocks. The variables to be used in the analysis are as follows. (1) “R": The rank of an individual stock on day $t$; if the rank value is $1$, then the stock is the most frequently discussed stock on the `Daum` stock message board on that day. This is the key variable that measures the degree of investor attention. (2) “RN": Raw returns on day $t+1$ (the next day) (%), which is of primary interest and is the quantity that we wish to predict. (3)“ R0": Raw returns on day $t$ (%). (4) “R1": Raw returns on day $t-1$ (%). (5) “R2": Raw returns on day $t-2$ (%). (6) “R3": Raw returns on day $t-3$ (%). (7) “R4": Raw returns on day $t-4$ (%). (8) “R5": Raw returns on day $t-5$ (%). (9) “ME": Market capitalization (1 trillion Korean won). (10) “T": Turnover ratio defined as the trading volume divided by the number of outstanding shares. (11) “TA": Turnover ratio defined as the trading volume divided by market capitalization. Attention and Predictive Stock Returns -------------------------------------- As stated previously, the primary goal of our analysis is to determine how the returns of attention-grabbing stocks will fluctuate around the event dates when investors pay attention to these stocks. The next-day return can also be influenced by several other factors in addition to investor attention. To account for the effects of these other factors, we consider the residuals obtained after regressing the next-day return against all other covariates except the rank, “R". These residuals are obtained from the multiple linear regression model, which is defined as follows: $$\label{eqn:reg} {\rm RN}_i= \beta_0 + \sum_{l=0}^5 \beta_{l+1} {\rm R}l_i + \beta_7 {\rm ME}_i + \beta_8 {\rm T}_i +\beta_9 {\rm TA}_i + \epsilon_i, \quad i=1,2,\ldots,n,$$ where $n (=1,771)$ is the total number of companies on the market. Let $Y_{i}^t$ be the absolute (value of the) residual of company $i$ obtained from the regression (\[eqn:reg\]). We then select the absolute residuals whose ranks are reported to be within the top 30 for the primary analysis. Below, $Y_{[r:n]}^t$ is the absolute residual corresponding to rank $r$ on day $t$ for $r=1,2\ldots,30$ and $t=1,2,\ldots,T (=537)$. In Figure 1, we plot the quantiles of $\big\{Y_{[r:n]}^t, t=1,2,\ldots,T\big\}$ for each $r=1,2,\ldots,30$. This figure reveals that $Y_{[r:n]}^t$ is not increasing at $r=1$ and $2$, which we hypothesize reflects the heterogeneity of investor expectations with regard to highly attention-grabbing stocks. In other words, the ranking of the `Daum` board is purely determined by the attention of individual investors, and stocks related to news, that is difficult to characterize as either good or bad, often receive the greatest attention and the highest ranks. We introduce an additional term to explain this apparent local non-monotonicity, and consider the model $$\label{eqn:model-realdata} Y_{[r:n]}^t = \mu_Y^t + \rho^t \sigma_Y^t \alpha_{(r:n)} + \gamma^t {\rm I}(r \leq 2) + \eta_{[r:n]}^t, ~~ r = 1, 2, \ldots, 30,$$ for $t= 1,2, \ldots, T$ with $T=537$ and $n=1,771$. ![Plot of the means and quantiles of $\big\{Y_{[r:n]}^t, t=1,2,\ldots,T\big\}$ for each $r=1,2,\ldots,30$.](AbsResid.eps){width="1.0\linewidth"} Regression with Ranks --------------------- In the regression model, we consider the scores from the standardized distributions of the location-scale families generated by the following three distributions: (i) a uniform distribution on $(0,1)$ (called the uniform score), (ii) a positive normal distribution $X = |Z|$, $Z \sim N(0,1)$ (called the half-normal score), and (iii) a power-law distribution $X$ whose CDF is $F(x) = 1 - x^{-\alpha}$ with $\alpha = 2.3$ (called the power-law score). The scores are illustrated on different scales in Figure 2. {width="1.0\linewidth"} We estimate $\rho^t$ and $\gamma^t$ to minimize the empirical squared-error loss of the model (\[eqn:model-realdata\]) by iterating the following steps: 1. Given the least-squares estimator of $\rho$, denoted by $\widehat{\rho}_{(0)}^t$, update the estimate of $\gamma$ as follows: $$\nonumber \widehat{\gamma}^t = \frac{1}{2} \left[ \left( Y_{[1:n]}^t - \mu_Y^t - \sigma_Y^t \widehat{\rho}_{(0)}^t \alpha_{(1:n)} \right) + \left( Y_{[2:n]}^t - \mu_Y^t - \sigma_Y^t \widehat{\rho}_{(0)}^t \alpha_{(2:n)} \right) \right].$$ 2. Given the estimate of $\gamma$, denoted by $\widehat{\gamma}_{(0)}^t$, update the estimate of $\rho$ using the LSE proposed in the previous section as follows: $$\nonumber \widehat{\rho}^t = \frac{1}{\sigma_Y^t} \left\{ \frac{\sum_{r=1}^{30} \alpha_{(r:n)} \big(Y_{[r:n]}^t - \mu_Y^t - \widehat{\gamma}_{(0)}^t {\rm I} (r \le 2) \big) } {\sum_{r=1}^{30} \alpha_{(r:n)}^2 } \right\}.$$ In the analysis, the initial value $\widehat{\rho}_{(0)}^t$ is obtained from the preliminary linear regression on $\left\{ \big( \alpha_{(r:n)}, (Y_{[r:n]}^t - \mu_Y^t)/\sigma_Y^t \big) \right\}_{r=3, \cdots, 30}$, $t=1,\ldots, T$, in which the data corresponding to $r=1,2$ are excluded. By contrast, $\mu_Y^t$ and $\sigma_Y^t$ are estimated based on their empirical values as follows: $\widehat{\mu}_Y^t = \big(\sum_{r=1}^{n} Y_{[r:n]}^t\big) \big/n$ and $\big(\widehat{\sigma}_Y^t)^2 = \sum_{r=1}^{n} \big( Y_{[r:n]}^t - \widehat{\mu}_Y^t \big)^2 \big/n$. To choose the most appropriate score function among the three considered, we follow the guidelines presented in Section 3.2 and perform a residual analysis. First, we plot $\alpha_{(r:n)}$ and the quantiles of the corresponding residuals to identify any remaining trend not explained by the model (see Figure 3). This figure shows that the uniform score and the half-normal score exhibit additional linear trends not explained by the linear model (\[eqn:model-realdata\]), whereas the power-law score performs well. Second, we plot $$\nonumber \left(\alpha_{(r:n)}, \frac{Y_{[r:n]}^t - \widehat{\mu}_Y}{\widehat{\sigma}_Y} \right), ~~r=3,4,\ldots,30, t=1,2,\ldots,T,$$ and apply the least-squares fits with/without intercept. As we know from the model (\[eqn:con-linear\]), the estimated regression line with intercept should cross the origin if the scores are correctly specified. Figure 4 reveals that the (estimate of) the intercept of the power-law score is closest to zero among the intercepts of the three considered scores. Finally, the residual sums of squares of the three scores are found to be $152186.7$, $150706.3$, and $150288.9$, respectively. This finding also supports the superiority of the power-law score function, and in the following analysis, we focus on the power-law score function. {width="1.0\linewidth"} {width="1.0\linewidth"} Test of the Effect of Investor Attention on the Next-day Returns ---------------------------------------------------------------- The primary goal of the analysis is to investigate whether the attention of investors affects the returns of a stock on the following day. Specifically, we are interested in testing $\mathcal{H}_0: \rho=0$ under the assumption that $\rho^t=\rho$ for every $t$. To test this hypothesis, we consider a combined statistic of $\{ \widehat{\rho}^t, t=1,2,\ldots, T \}$, that is, $$\label{eqn:teststat} {\bf t}_{\rho} = \frac{1}{\sqrt{T}} \sum_{t=1}^T U_t,$$ where $U_t= \sqrt{n} \widehat{\rho}^t$. Here, the estimates of $\rho$ for each day $t$, denoted by $\widehat{\rho}^t$, are serially dependent on each other, as are the $U_t$s. Thus, to obtain the reference distribution of ${\bf t}_{\rho}$, we further assume that $\{U_t, ~t=1,2,\ldots,T\}$ is stationary and that ${\rm E} | U_t |^{2+\kappa} < \infty$ for $\kappa>0$. Under these assumptions, the null distribution of ${\bf t}_{\rho}$ is asymptotically normal with mean $0$ and variance $$\nonumber \lim_{T \rightarrow \infty} \frac{1}{T} \sum_{k=0}^T \big( T-k\big) {\rm cov} (U_t, U_{t+k} ).$$ The variance can be empirically estimated from the observed values of $\{U_t,~t=1,2,\ldots,T \}$ as $$\nonumber \frac{1}{T} \sum_{k=0}^m \big( T-k\big) \widehat{\rm cov} (U_t, U_{t+k} )$$ for sufficiently large $m$, where $\widehat{\rm cov} (U_t, U_{t+k} )$ denotes the empirical covariance of the observed statistics $(U_1, U_{1+k}), (U_2, U_{2+k}), \ldots, (U_{T-k+1} ,U_T)$. An additional interesting feature of the combined procedure is that the test statistic ${\bf t}_{\rho}$ is a rough estimator of $\rho$ for all $T$ trading days (after the scaling). It is calculated as $$\begin{aligned} \frac{1}{\sqrt{T}} \sum_{t=1}^T U_t &=& \frac{1}{\sqrt{T}} \sum_{t=1}^T \sqrt{n} \widehat{\rho}^t = \sqrt{n T} \left( \frac{1}{T} \sum_{t=1}^T \widehat{\rho}^t \right) \nonumber\\ &=& \sqrt{n T} \frac{1}{T} \sum_{t=1}^T \frac{ \sum_{r=1}^{30} \alpha_{(r:n)} \big\{ Y_{[r:n]}^t - \widehat{\mu}_Y^t - \widehat{\gamma}^t {\rm I} (r \le 2) \big\}} {\hat{\sigma}_Y^t \sum_{r=1}^{30} \alpha_{(r:n)}^2} \nonumber \\ &\approx& \sqrt{n T} \frac{1}{\widehat{\sigma}_Y} \frac{ \sum_{t=1}^T \sum_{r=1}^{30} \alpha_{(r:n)} \big\{ Y_{[r:n]}^t - \widehat{\mu}_Y- \widehat{\gamma}^t {\rm I} (r \le 2) \big\}} {T \sum_{r=1}^{30} \alpha_{(r:n)}^2} \approx \sqrt{ n T} \widehat{\rho}^{\rm lse}, \label{eqn:equiv}\end{aligned}$$ where $\widehat{\rho}^{\rm lse}$ is the least-squares estimator under the assumption that $\rho^t=\rho$ for all $t$. The difference between the right- and left-hand sides of (\[eqn:equiv\]) lies in the definition of $\widehat{\gamma}^t$, which is defined using $\widehat{\rho}^t$ rather than $\widehat{\rho}^{\rm lse}$. The results of the test indicate that the average value of $\widehat{\rho}^t$, which is an estimator of $\rho$, is $0.043$. The $p$-value obtained when testing $\mathcal{H}_0: \rho=0$ is less than $10^{-5}$ and statistically supports the association between investor attention and the next-day returns of the stocks. Conclusion ========== In this paper, we study a regression problem based on a partially observed rank covariate. We propose a new set of score functions and study their application in simple linear regression. We demonstrate that the least-squares estimator that is calculated based on the newly proposed score consistently estimates the correlation coefficient between the response and the unobserved true covariate if the score function is correctly specified. We also define procedures based on the obtained residuals to identify the correct score function for the given data. The proposed estimator and procedures are applied to rank data collected from `Daum.net`, and we empirically verify the association between investor attention and next-day stock returns. We finally conclude the paper with two discussions on the proposed score function. First, the application of the proposed score function is not restricted to linear regression but may also be appropriate for other statistical procedures based on rank, including the well-known rank aggregation problem [@Breitling:2004; @Eisinga:2013]. Second, the score function still can be used for the the multiple linear regression model $$\nonumber Y_i = X_i \beta + {\bf Z}_i^{\rm T} \eta+ \epsilon_{i}$$ with an additional covariate vector ${\bf Z}=\big(Z_1,Z_2,\ldots,Z_q\big)^{\rm T}$. Similarly to the case of the simple linear regression, we have the representations $$\nonumber Y_{[r:n]} = \mu_Y + \frac{X_{(r:n)}-\mu_X}{\sigma_X} \delta+ \big( {\bf Z}_{[r:n]} - \mu_{{\mathbf{Z}}} \big)^{\rm T} \eta + \epsilon_{[r:n]},$$ where $\delta=\beta \sigma_X$ and $\epsilon_{[r:n]}$, $r=1,2,\ldots,[ns]$, have mean $0$ and independent to each other. Again, the least-squares estimators of $\delta$ and $\eta$ are defined as the solutions to $$\begin{aligned} && \begin{pmatrix} \begin{array}{ll} \sum_{r=1}^{[ns]} \alpha_{(r:n)}^2 & \sum_{r=1}^{[ns]} \alpha_{(r:n)} \big( {\mathbf{Z}}_{[r:n]}-\overline{{\mathbf{Z}}} \big) \\ \sum_{r=1}^{[ns]} \big( {\mathbf{Z}}_{[r:n]}-\overline{{\mathbf{Z}}} \big)^{\rm T} \alpha_{(r:n)} & \sum_{r=1}^{[ns]} \big( {\mathbf{Z}}_{[r:n]}-\overline{{\mathbf{Z}}} \big)^{\rm T} \big( {\mathbf{Z}}_{[r:n]}-\overline{{\mathbf{Z}}} \big) \end{array} \end{pmatrix} \begin{pmatrix} \begin{array}{l} \widehat{\delta} \\ \widehat{\eta} \end{array} \end{pmatrix} \nonumber\\ && \hskip5cm = \begin{pmatrix} \begin{array}{l} \sum_{r=1}^{[ns]} \alpha_{(r:n)} \big( Y_{[r:n]} - \overline{Y} \big) \\ \sum_{r=1}^{[ns]} \big( {\mathbf{Z}}_{[r:n]}-\overline{{\mathbf{Z}}} \big)^{\rm T} \big( Y_{[r:n]} - \overline{Y} \big). \end{array} \end{pmatrix}, \nonumber\end{aligned}$$ and conjecture that they consistently estimate $\delta$ and $\eta$. Appendix {#appendix .unnumbered} ======== A.1 Proof of Theorem 2 {#a.1-proof-of-theorem-2 .unnumbered} ---------------------- Note that $\widehat{\sigma}_Y^2/\sigma_Y^2$ converges in probability to 1 as $n \rightarrow \infty$ and $$\begin{aligned} \sqrt{n} \big( \widehat{\rho}(s) - \rho \big) &=& \sqrt{n} \left\{ \frac{\sigma_Y}{\widehat{\sigma}_Y}\frac{1}{\sigma_Y} \frac{\sum_{r=1}^{[ns]} \alpha_{(r:n)} \big( Y_{[r:n]} - \widehat{\mu}_Y \big)} {\sum_{r=1}^{[ns]} \alpha_{(r:n)}^2} - \rho \right\} \nonumber\\end{aligned}$$ has the same limiting distribution with $$\begin{aligned} && \sqrt{n} \left\{ \frac{1}{\sigma_Y} \frac{\sum_{r=1}^{[ns]} \alpha_{(r:n)} \big( Y_{[r:n]} - \widehat{\mu}_Y \big)} {\sum_{r=1}^{[ns]} \alpha_{(r:n)}^2} - \rho \right\} \nonumber\\ && \qquad = \sqrt{n} \Bigg\{ \frac{1}{\sigma_Y} \frac{ \sum_{r=1}^{[ns]} \alpha_{(r:n)} \big(Y_{[r:n]} - m (X_{(r:n)}) \big)} { \sum_{r=1}^{[ns]} \alpha_{(r:n)}^2 } + \rho \frac{\sum_{r=1}^{[ns]} \alpha_{(r:n)} \left(\left(\frac{X_{(r:n)}-\mu_X}{\sigma_X}\right) - \alpha_{(r:n)} \right)}{\sum_{r=1}^{[ns]} \alpha_{(r:n)}^2} \nonumber \\ && \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad +\frac{1}{\sigma_Y}\frac{\sum_{r=1}^{[ns]}\alpha_{(r:n)}(\mu_Y-\widehat{\mu}_Y)}{\sum_{r=1}^{[ns]}\alpha_{(r:n)}^2} \Bigg\}. \label{eqn:s_stat}\end{aligned}$$ Then, equation ($\ref{eqn:s_stat}$) can be written as $$\nonumber \frac{1}{\sigma_Y}\frac{\sqrt{n}\sqrt{n\Psi_n^{\rm I}(1)}}{\sum_{r=1}^{[ns]}\alpha_{(r:n)}^2}{\rm U}(s)+ \rho\sqrt{n}{\rm V}(s)+\frac{\sqrt{n}}{\sigma_Y}{\rm R}(s),$$ where $\Psi_n^{\rm I}(1)=\sum_{r=1}^n\alpha_{(r:n)}^2\sigma_{(r:n)}^2/n$ and $$\begin{aligned} {\rm U}(s) &=& \frac{1}{\sqrt{n \Psi_n^{\rm I}(1)}}\sum_{r=1}^{[ns]} \alpha_{(r:n)} \big( Y_{[r:n]} - m (X_{(r:n)}) \big) \nonumber\\ &=& \frac{1}{\sqrt{n \Psi_n^{\rm I}(1) } }\sum_{r=1}^{[ns]} \left( \frac{ {\rm E}( X_{(r:n)} ) -\mu_X} { \sigma_X} \right) \Big( Y_{[r:n]} - m (X_{(r:n)}) \Big), \label{eqn:psum}\end{aligned}$$ with $m(X_{(r:n)})={\rm E} \big(Y \big| X_{(r:n)} \big)=\mu_Y + \rho \sigma_Y (X_{(r:n)}-\mu_X)/\sigma_X$ and $$\nonumber {\rm V}(s)=\frac{\sum_{r=1}^{[ns]} \alpha_{(r:n)} \big( (X_{(r:n)}-\mu_X)/\sigma_X - \alpha_{(r:n)} \big)}{\sum_{r=1}^{[ns]} \alpha_{(r:n)}^2},~~{\rm R}(s)=\frac{\sum_{r=1}^{[ns]}\alpha_{(r:n)}(\mu_Y-\widehat{\mu}_Y)}{\sum_{r=1}^{[ns]}\alpha_{(r:n)}^2}.$$ Since ${\rm R}(s)$ converges in probability to 0, we only consider the ${\rm U}(s)$ and ${\rm V}(s)$. Thus, the proof of the theorem is based on the functional central limit theorem for two partial sums of rank statistics, ${\rm U}(s)$ and ${\rm V}(s)$. We first consider the asymptotic distribution of the process of taking the weighted partial sum of the induced rank statistic, which is $$\begin{aligned} {\rm U}(s) &=& \frac{1}{\sqrt{n \Psi_n^{\rm I}(1)}}\sum_{r=1}^{[ns]} \alpha_{(r:n)} \big( Y_{[r:n]} - m (X_{(r:n)}) \big) \nonumber\\ &=& \frac{1}{\sqrt{n \Psi_n^{\rm I}(1) } }\sum_{r=1}^{[ns]} \left( \frac{ {\rm E}( X_{(r:n)} ) -\mu_X} { \sigma_X} \right) \Big( Y_{[r:n]} - m (X_{(r:n)}) \Big). \label{eqn:psum}\end{aligned}$$ The main finding of @Bhattacharya:1974 is the conditional independence of $Y_{[1:n]},\ldots,Y_{[n:n]}$ given $X_1,X_2,\ldots,X_n$ (or equivalently, $X_{(1:n)}, X_{(2:n)}, \ldots, X_{(n:n)}$).\ Thus, given $\mathcal{A}= \sigma\big( X_1,X_2,\ldots,X_n, \ldots \big)$, (\[eqn:psum\]) can be read as $$\label{eqn:psum2} S_{nk}=\frac{1}{\sqrt{n \Psi_n^{\rm I}(1)}}\sum_{r=1}^{k} \alpha_{(r:n)} \sigma_{(r:n)} u_r, \quad k=1,2,\ldots,n,$$ where the $u_r$ are independent, with mean $0$ and variance $\sigma_{(r:n)}^2$. By applying the basic concept of Skorokhod embedding [@Shorack:2009], we obtain a sequence of stopping times $\tau_{n1},\tau_{n2},\ldots,\tau_{nn}$ such that - these stopping times are conditionally independent given $\mathcal{A}$, - ${\rm E} \big(\tau_{nk} \big| \mathcal{A} \big)= \sum_{r=1}^k \alpha_{(r:n)}^2 \sigma_{(r:n)}^2 \big/ \{ n \Psi_n^{\rm I}(1) \} $, - ${\rm var} \big(\tau_{nk} \big| \mathcal{A} \big)= \sum_{r=1}^k \alpha_{(r:n)}^4 {\rm E} \big\{ \big( Y_{[r:n]} - m(X_{(r:n)}) \big)^4 \big| \mathcal{A} \big\} \big/ \{ n \Psi_n^{\rm I}(1) \} ^2 < \infty$, and - $\big(S_{n1}, S_{n2},\ldots,S_{nn} \big)$ has the same distribution as $\big(W(\tau_{n1}), W(\tau_{n1}+\tau_{n2}),\ldots,W (\tau_{n1}+\tau_{n2}+\cdots+\tau_{nn}) \big)$, where $\big\{W(s), ~s \in [0, \infty) \big\}$ is conventional Brownian motion. We now consider the embedded partial-sum process $\big\{W_n(s): 0 \le s \le 1 \big\}$ that is defined by $W_n(s) =S_{n[ns]}$. As in @Bhattacharya:1974, it suffices to show that $$\label{eqn:donsker1} \sup_{0 \le s \le 1} \left|\frac{1}{n}\sum_{r=1}^{[ns]} \tau_{nr} - \frac{\Psi_n^{\rm I}(s)}{\Psi_n^{\rm I}(1)} \right|$$ converges to $0$ probability. For each $s \in [0,1]$, the strong law of large numbers states that $\big(1 \big/ n \big)\sum_{r=1}^{[ns]} \tau_{nr} $ almost certainly converges to $\Psi_{\infty}^{\rm I}(s)\big/\Psi_{\infty}^{\rm I}(1)$. Both $\big(1 \big/n \big) \sum_{r=1}^{[ns]} \tau_{nr}$ and $\Psi_n^{\rm I}(s)\big/\Psi_n^{\rm I}(1)$ are increasing functions of $s$. Thus, using the same arguments [@Shorack:2009 pp. 62], we find that their sup difference also converges to $0$. Second, $$\begin{aligned} \sqrt{n} {\rm V}(s) &=& \frac{1}{\Phi_n(s)} \frac{1}{\sqrt{n}} \left\{\sum_{r=1}^{[ns]} \alpha_{(r:n)} \bigg( \frac{X_{(r:n)}-\mu_X}{\sigma_X} - \alpha_{(r:n)} \bigg) \right\}\end{aligned}$$ is a linear statistic of order statistics and converges to the normal distribution with mean $0$ and variance $\Psi_{\infty}^{\rm II}(s) \big/ \Phi_{\infty}^2(s)$ [@David:2003 Theorem 11.4]. Here, we remark that both $\Psi_{\infty}^{\rm II}(s)$ and $\Phi_{\infty}(s)$ can also be written as functionals of the distribution of $X$, as shown in [@David:2003]. Finally, summing the asymptotic results of ${\rm U}_n(s)$ and ${\rm V}_n(s)$, we find that $ \sqrt{n} \big( \widehat{\rho}(s) - \rho \big)$ converges to the normal distribution with mean $0$ and variance $$\nonumber \frac{\Psi_{\infty}^{\rm I}(s)/\sigma_Y^2+\rho^2\Psi_{\infty}^{\rm II}(s)}{\Phi_{\infty}^2(s)}$$ This concludes the proof. A.2 Proof of Theorem 1 {#a.2-proof-of-theorem-1 .unnumbered} ---------------------- We first decompose the sample correlation between $\overline{S}_n(r)$ and $Y_{[r:n]}$ as ${\rm A}+{\rm B}+{\rm C}$: $$\begin{aligned} {\rm A} &=& \frac{1}{n} \sum_{r=1}^n \overline{S}_n(r) \big\{ Y_{[r:n]} - m(X_{(r:n)})\big\}, \nonumber\\ {\rm B}&=& \frac{1}{n} \sum_{r=1}^n \overline{S}_n(r) \big\{m(X_{(r:n)}) - \mu_Y \big\}, \nonumber\\ {\rm C} &=& \frac{1}{n} \sum_{r=1}^n \overline{S}_n(r) \big\{ \mu_Y - \overline{Y}_n \big\}, \nonumber\end{aligned}$$ where $m(X_{(r:n)})=\mu_Y+\rho \sigma_Y X_{(r:n)}$. In below, we compute the limit of each ${\rm A}, {\rm B}$, and ${\rm C}$. First, similarly to the convergence of ${\rm U}(s)$ (with $s=1$) in Appendix A, we can show that $\sqrt{n} {\rm A}$ converges in distribution to a normal random variable and, thus, ${\rm A}$ converges to $0$ in probability. Second, similarly to the convergence of ${\rm R}(s)$ (with $s=1$) in Appendix A, we can show that $\sqrt{n} {\rm C}$ converges in distribution to a normal random variable and, thus, ${\rm C}$ converges to $0$ in probability. Lastly, $$\begin{aligned} {\rm B} &=& \frac{1}{n} \sum_{r=1}^n \overline{S}_n(r) \big\{m(X_{(r:n)}) - \mu_Y \big\} \nonumber \\ &=& \rho \sigma_Y \cdot \frac{1}{n} \sum_{r=1}^n \overline{S}_n(r) \frac{X_{(r:n)} - \mu_X}{\sigma_X} \nonumber \\ &=& \rho \sigma_Y \cdot \frac{1}{n} \sum_{r=1}^n \overline{S}_n(r) \left\{ \frac{X_{(r:n)} - \mu_X}{\sigma_X} - \alpha_{(r:n)} \right\} + \rho \sigma_Y \cdot \frac{1}{n} \sum_{r=1}^n \overline{S}_n(r) \alpha_{(r:n)}, \nonumber\end{aligned}$$ whose first term converges to $0$ in probability similarly to the convergence of ${\rm V}(s)$ (with $s=1$) in Appendix A. Hence, ${\rm B}$ converges in probability to the limit of $$\label{eqn:inner} \rho \sigma_Y \cdot \frac{1}{n} \sum_{r=1}^n \overline{S}_n(r) \alpha_{(r:n)}.$$ Since $\sum_{r=1}^n \overline{S}^2_n(r)=1$ and $\sum_{r=1}^n \alpha^2_{(r:n)}$ approaches $1$, (\[eqn:inner\]) is maximized when $\overline{S}_n(r)=\alpha_{(r:n)}$ in asymptotic. [99]{} Bhattacharya, P. K. (1974). Convergence of sample paths of normalized sums of induced order statistics. *The Annals of Statistics*, [**2**]{}, 1034–1039. Breitling, R., Armengaud, P., Amtmann, A. and Herzyk, P. (2004). Rank products: a simple, yet powerful, new method to detect differentially regulated genes in replicated microarray experiments. *FEBS Letters*, [**573**]{}, 83-92. Camerer, C. (2003). The behavioral challenge to economics: Understanding normal people. Conference Series, Proceedings, 48. Paper presented at Federal Bank of Boston 48th Conference on ’How humans behave: Implications for economics and policy’. 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--- abstract: 'We reported the monitoring results on spectrophotometry, photometry and imaging of comet 103P/Hartley 2 obtained at Lulin (1m), Calar Alto (2.2m) and Beijing Astronomical (2.16m) Observatory from April to December 2010. We found that a dust feature at sunward direction was detected starting from the end of September until the beginning of December (our last observation from the Lulin and Calar Alto observatory). Two distinct sunward jet features in the processed images were observed on October 11 and after October 29 until November 2. In parallel, the CN images reveal two asymmetrical jet features which are nearly perpendicular to the Sun-nucleus direction and this asymmetrical features implies that the comet was in a nearly side-on view in late-October and early-November. Additional to the jet features, the average result of the C$_2$-to-CN production rate ratio ranges from 0.7 to 1.5 which places 103P/Hartley 2 as being of typical cometary chemistry. We found that the $r_h$ dependence for the dust production rate, Af$\rho$ (5,000 km), is $-$3.75$\pm$0.45 before perihelion and is $-$3.44$\pm$1.20 during post-perihelion period. We detected the higher dust reddening is around the optocenter and getting bluer outward along the sunward jet feature and concluded that the former one, higher dust reddening, could be associated with strong jet activity and the latter one, the lowering of the reddening, might imply the optical properties changed or could be associated with outburst. The average dust color did not appear to vary significantly as the comet passed through perihelion.' author: - 'Z.-Y. Lin' - 'L.M. Lara' - 'W.-H. Ip' title: 'Long-Term Monitoring of Comet 103P/Hartley 2' --- Introduction ============ Comet 103P/Hartley 2, hereafter referred to as Hartley 2, was first spotted by M. Hartley on March 16, 1986. It has a semi-major axis of $\rm a =3.47 AU$, eccentricity $\rm e=0.695$, and inclination $\rm i = 13.617$ and an orbital period of 6.46 years. Its low eccentricity made it a suitable target for the extended mission of NASA’s Deep Impact spacecraft after the impact experiment at comet 9P/Tempel 1 on July 4, 2005. The mission to Hartley 2 was renamed EPOXI and given two missions, Extrasolar Planet Observation and Characterization(EPOCh), and Deep Impact Extended Investigation (DIXI). The EPOXI flyby observations at a closest distance of 694 km on November 4, 2010, brought a wealth of information on the outgassing activity, shape and surface structure of this small Jupiter family comet (A’Hearn et al., 2011). For example, the strong outflows of the CO$_2$-rich jet from the sun-lit end of the bowling-pin shaped and the H$_2$O-rich jet in the waist region came as a surprise. How would they be connected to the large-scale jet structures observed in the coma? How would the outgassing process be modulated by the rotation of the comet nucleus? In fact, based on the time variability of the CN coma morphology and millimeter/sub-millimeter spectra, the rotation period of Hartley 2 has been found to be increasing from 16.7 hr in August, 2010, to 18.4 hr in the first half of November and then to nearly 19 hr in late November [@sam11; @kni11; @mee11; @wan12]. Such time variations of the nucleus rotation period together with the close-up measurements by the EPOXI mission demonstrate the complex nature of the surface outgassing process. In anticipation of the scientific opportunity to compare the large-scale coma structures and gas production rates of Hartley 2 with the EPOXI results, we have made a long-term monitoring program from April to December, 2010, using imaging with both broadband and narrowband filters, and long-slit spectrophotometry. This cooperative effort involved observations at the Lulin Observatory in Taiwan, the Calar Alto Observatory in Spain, and the Beijing Astronomical Observatory in China. The paper is organized as follows. In Section 2, we will explain the observational procedures, instruments and analysis methods. In Section 3, the derived morphology and gas production of the CN coma and jets will be described. In Section 4, we will describe the dust jets and the structure of the dust coma during this period. A summary of the major characteristics of the large-scaled structures of the gas and dust comas of Harley 2 is given in Section 5. Observations, instruments and data analysis =========================================== [**Imaging:**]{} The bulk of the photometric imaging observations was done by using the Lulin One-meter Telescope (LOT) at Lulin observatory. In our first image of Hartley 2 on April 24, 2010, when the comet was 2.42 AU away from the Sun and 2.36 AU from the Earth, only a diffuse coma of 5" diameter was visible with 10-min exposure time. There was no tail feature. In the monitoring program, an Asahi R broadband filter and the narrowband filters of the Rosetta filter set were used. The specifications of these narrowband filters are given as $\lambda_{c}/\Delta \lambda$ both in nm, $\lambda_c$ being the central wavelength and $\Delta \lambda$ the band width: CN (387/5nm), C$_{2}$ (512.5/12.5nm), blue continuum BC (443/4nm), and red continuum RC (684/9nm). Because of the consideration of the signal-to-noise ratio, the narrowband filters were used only in October and November 2 just before the EPOXI close encounter. The camera used on LOT from April to November was PI 1300B which has a pixel scale of 0.516 arcsec and a field of view of $11.2 \times 11.6$ arcmin. In late-November 2010, there was a cooling problem with PI1300B. We, therefore, switched to U42 CCD which has $\rm 2k \times 2k$ pixels and a field of view of $12.17 \times 11.88$ arcmin. The telescope was always operated with non-sidereal tracking so as not to produce trail in the comet images. Typical integration were 600s $\sim$ 900s for the narrowband filters and 30s $\sim$ 300s for the broadband R filter. Table 1 is the observational log of our program. Standard procedure of data reduction was applied. It began with dark current subtraction and flat-field correction of all image frames. This was then followed by the subtraction of the night sky contribution. For the observations obtained before late-September, the night sky levels were determined directly areas of the CCD frames that do not contain contributions from the cometary emission. However, the sky-background of those images taken from late-September to early-November was all influenced by the cometary coma. Therefore, we took sky background images positioned at about 0.5 degrees away from the comet center. The extinction coefficients of the narrowband and broadband R filters were determined for all nights with photometric sky conditions, using the photometric stars like Feige 110 and GD71, observed at different airmasses during the night. For example, the first order extinction coefficient (in units of magnitudes per air mass) measured by Feige 110 with observing airmass range from 1.1 to 1.7 for R-filter on October 29 is 0.10 and for CN and C$_2$ are 0.39 and 0.14, respectively. These data were used to convert the measured counting rates into physical units and the detail has been described in Lin et al. (2007b). Because the CN images contain $29\%$ contribution from the continuum in the blue range while the C$_2$ images have as much as $93\%$, the net CN and C$_2$ gas coma images need to go through the subtraction procedure according to the following formulas: CN= CN$_{obs}$ - 0.29 BC$_{obs}$ , C$_2$ = C$_2^{obs}$ - 0.93 BC$_{obs}$. In addition to Lulin observations, the coma activity of 103P/Hartley 2 was also monitored continuously in R-band from the Calar Alto Observatory (near Almeria, Spain) from July 14 to December 26, 2010 (see Table 1). We used the CAFOS imaging camera ($\rm 2k \times 2k$ pixels, pixel size: 0”53, FOV $18' \times 18'$) which was mounted on the 2.2 m telescope. In our observations, only the central 1k x 1k pixels were used, providing thus a FOV of $9' \times 9'$. Appropriate bias and flat field frames were taken each night. If photometric conditions prevailed, photometric standard stars were observed at airmass similar to the comet observations. Table \[ObsLog\] contains the observations log for the complete dataset. Notice that a larger number of night mentioned in Table \[ObsLog\] is for the Af$\rho$ estimation and some of them are not used to enhance the structures in the coma because the SNR was too low. [**Spectroscopy:**]{} Spectroscopic measurements were planned once every month using CAFOS with grism B400 (see http://www.caha.es/alises/cafos/cafos22.pdf) which renders an spectral range between 3,200 and 8,800 Å with a wavelength scale of 9.4 Å/pixel. The slit of the spectrograph was oriented in the north-south direction, giving dust and gas profiles at different cross-cuts through the coma, depending on the PA of the sun-comet vector on the sky. For absolute calibration, observations of appropriate spectrophotometric standard stars were acquired. All comet observations were done with telescope tracking on Hartley 2. With the exception of Nov. 5, 2010, all observations were done in service mode of the Calar Alto Observatory. Details on the images and spectra reduction and calibration can be found in Lara et al. (2001, 2011a) and they will not be repeated here. If the gas coma covered the whole slit, the sky level was estimated from the edges of the frame. Otherwise, the background could be measured directly by using regions near the edges of the frame. Besides spectra obtained from the Calar Alto Observatory, spectroscopic observations were also performed on October 9 and October 11 at the Beijing Astronomical Observatory using the 2.16m telescope in the spectral range between 3,600 Å$\smallskip$ to 8,400 Å$\smallskip$ at a dispersion 4.8 Å/pixel. The spectroscopic data were reduced following the standard procedures including bias and flat-field corrections and cosmic ray removal. Wavelength calibration was performed based on helium-argon lamps exposed at both the beginning and the end of the observations every night. Flux calibration of all spectra were conducted based on observations of at least one of the spectral standard stars, i.e., HD19445 and the atmospheric extinction effect was corrected by the mean extinction coefficients measured by the Beijing-Arizona-Taiwan-Connecticut (BATC) multicolor survey. See Lin et al. (2007a) for more detailed information. Gas coma morphology and properties ================================== CN jets ------- In order to study the visibility of faint structures of the gas coma of comet Hartley 2, an image enhancement technique was applied to the present set of images. The method used here is the azimuthally averaged profile division, a detailed description which can be found in Lin et al. (2012). This method was applied to all images taken in the CN filter, in continuum filters and in the R-band filter. To estimate the rotational phase from CN morphology, a lot of observing data have to be acquired in consecutive night. However, the images obtained in our observing nights with less temporal coverage were not enough to estimate and display rotational period due to snapshot observations, poor weather and telescope tracking problem. We, therefore, use the known periodicity to estimate the rotational phase in our images. However, we have to face several problems: a non-principal axis rotation of comet Hartley 2 and a rapid change of the viewing geometry might cause different periodicities between rotational cycles. A specific phase is really only applicable to a short stretch of data if we adopt known periodicity such as 18.15 hr in mid-October and 18.7hr in early-November from Knight and Schleicher (2011) or 18.22 hr around perihelion from Harmon et al. (2011). Notice that those ground-based observations have error bars between 0.01 and 0.3. The most robust rotation period at present is from the EPOXI spacecraft lightcurve given in Belton et al. (2013). This gives a spin period of 18.40 hr at encounter and states that it was increasing by 1.3 minutes/day. As the data acquired with the Rosetta filter set spread around one month, it is appropriate to use the midpoint of the observational time interval for this period of time. We extrapolate the rotation period back to midpoint assuming the rotation period was steadily changing during this time frame. Therefore, the rotation period quoted in this work is 18.11 hr on October 21.5 UT which refers to the midpoint of the Oct 10-Nov 2 data. The zero phase is set at 11:40 UT on October 10, 2010. In Figure  \[CNjet\], we can see that the morphology of the CN coma extended almost perfectly along the east-west direction in early October and the north-south direction around Hartley 2’s perihelion. The CN images all showed clear asymmetries before performing the image enhancement. One of these unprocessed CN images is given by contour plot in Figure 1 (top-left panel). The variations in between early-October and around its perihelion in the CN jet features are related to the spin period of the comet nucleus, the changing viewing geometry and non-principal axis rotation as has already been reported by Samarasinha et al. (2011), Knight and Schleicher (2011), Lara et al. (2011b) and Waniak et al. (2012). The processed CN images from the observations between October 11 and November 2 revealed two jets in the coma of comet Hartley 2. The CN jet features being nearly perpendicular to the Sun-tail direction not only varied smoothly during a night but showed similar morphology near its perihelion even though the rotational state was different. We compared the morphology of the CN jet features with those presented by Knight and Schleicher (2011), and Samarasinha et al. (2011) and found that the CN jet features of Hartley 2 did not show the spiral-like structure in early-October but was compatible with the observations obtained by Knight and Schleicher (2011) and Lara et al. (2011b) in late-October. The reason could be the observing geometry, i.e., whether it is observed from the face-on or side-on. Knight and Schleicher (2011) confirmed this effect from the images that revealed the face-on spiral structures in August and September. Furthermore, we found these two CN jet features to be asymmetrical. One of them is always brighter than the other, possibly because it is facing toward the Earth. For example, the southern jet of the images obtained from October 25 to October 27 is slightly stronger than the northern jet of those images. Such asymmetrical features have also been reported in earliler works by Samarasinha et al. (2011), Lara et al. (2011b) and Waniak et al. (2012). Gas production rates -------------------- In order to determine the gas production rates, the mean radial emission profiles of CN and C$_{2}$ were derived from the images with the continuum subtracted. Regarding the spectra of the comet acquired at Calar Alto Observatory and Beijing Astronomical Observatory, they are also used to investigate the CN, C$_{3}$, C$_{2}$ and NH$_{2}$ profiles in the North-South direction and to derive the production rates of these gaseous species. The spectral regions and the subtraction of the underlying continuum in the gas emission bands were done as described by Lara et al. (2001). The conversion of the emission band fluxes into column densities made use of fluorescence efficiency factors ($g-$factors) for C$_{3}$, C$_{2}$ and NH$_{2}$ [@ahe95], whereas the $g-$factors of the CN molecule was calculated for the heliocentric distance and velocity of 103P/Hartley 2 on every date from the set of values given by Schleicher [@sch10]. The gas production rates are obtained by means of the Haser [@has57] model for isotropic emission of cometary neutral molecules and their daughter molecules and radicals. The parameter used for the parent velocity is v$_{p}$ = 0.85 $r_{h}^{-0.5}$ $ \rm km s^{-1}$ [@fra05] and for the daughter velocity it is 1 $\rm km s^{-1}$. For the corresponding set of parameters in the Haser model, we produced theoretical column density profiles for each species by varying the production rate until the best match between observations and theoretical predictions is achieved. The results of nightly averages for Q(CN), Q(C$_3$), Q(C$_{2}$) and Q(NH$_2$) are summarized in Table \[Qs\]. Table \[Qs\] also contains the average gas production rates obtained from the images acquired in one night together with the aperture size we have considered to derive Q. The variation of production rates seen in multiple measurements during a night were less than 5% that is reflected in the uncertainties in Table  \[Qs\]. Our results on Q(C$_{2}$) are less numerous as there were tracking problems at LOT from October 10$\sim$11 and October 25$\sim$27, whereas the long-slit spectroscopic measurements could provide Q(C$_2$) at other dates thus spanning larger heliocentric distances. Our Lulin, BAO and CA results in Table \[Qs\] show that there is no significant variation of Q(CN) from mid-October to early-November. This result is consistent with the Lara et al. (2011b) and Mumma et al. (2011) results that assumed that HCN is the main parent species of CN and that expected variation of Q(HCN) around the perihelion is not very large. Notice that we used the mean radial profile to estimate the gas production rate from the images obtained from Lulin observatory. However, if we averaged the radial profile in the north-south direction where the CN jet feature exists, the derived Q(CN) would be larger in a factor of two to three when compared with the azimuthally averaged radial profile. Figure  \[CNC2\] shows the logarithm of the production rate for CN and C$_{2}$ as a function of the heliocentric distance (r$_{h}$). The data points include those obtained by Lara et al. (2011b), Knight and Schleicher (2013) and the Lulin and CA results (this work) for pre-perihelion and post-perihelion observations during the 2010 apparition are presented here. We used the linear fitting in the log-log scale to estimate the slope of the r$_{h}$ dependence of the gas production rate, Q$\rm \sim r_{h}^{ -\alpha}$, and the slopes ($\alpha$) of CN and C$_{2}$ are 4.57 and 4.84 before perihelion and 3.21 and 3.42 after perihelion, respectively. The corresponding slopes are significantly steeper than the average value estimated for Jupiter-family comets, i.e. Q(gas)$\rm \sim r _h^{-2.7}$ [@ahe95]. Additionally to this, the average C$_2$-to-CN production rate ratio is $0.7 \sim 1.5$ which places 103P/Hartley 2 as being “typical” in terms of cometary chemistry defined by A’Hearn et al (1995) . Our measurement is consistent with the results from the spectroscopic observations (Lara et al. 2011b) and the narrow-band photometry observations (Knight and Schleicher 2013). Dust coma morphology and properties =================================== Jet feature in dust coma ------------------------ We describe the morphology and evolution of the coma structures that can be treated with routine procedures, i.e. Larson-Sekanina algorithm (Larson and Sekanina 1984). In case of doubt, we used additional techniques, such as azimuthal median profile division and Adaptive Laplace filter (Böhnhardt and Birkle 1994) to clearly separate morphological features from artifacts. Figure \[comparison\] compares the jet structure and dust tail feature on October 11 obtained by using three different image enhancement methods: (a) the Larson-Sekanina filtering, (b) the azimuthal median profile, and (c) the adaptive Laplace filtering. In spite of some differences in their appearances, the presence of two jets in the sunward quarter is common to all numerical treatments. It is therefore clear that the jet features are real and not artifacts associated with the image processing procedures. Figure \[Rjet1\] is a summary of the R-band images enhanced by the Larson-Sekanina filtering method to bring out the inhomogeneous structures in the dust coma of 103P/Hartley 2. It can be seen that from April until July, 2010, no clear sign of dust features could be found. However, beginning in August 1, a dust tail of diffuse structure (labeled *T* in Figure  \[Rjet1\]) began to appear in the anti-sunward direction. On September 29, a short jet (indicated by arrows in Figure \[Rjet1\]) in the sunward direction can be seen. Hereafter, this sunward jet feature can be detected in all our images obtained at Lulin and Calar Alto Observatory. It is interesting to note that two distinct sunward dust structures are visible after October 29 lasting until November 2. Around the same time, from November 2 to November 4, Mueller et al. (2013) also reported seeing two separate continuum features in sunward direction. Afterwards, only a single jet could be seen in the sunward direction that became fainter and fainter as Hartley 2’s heliocentric and geocentric distances increased. The sunward jet features showed relatively little variation during a night but its shape and position angle slightly changed from night to night until October 11 when two distinct jet features apparently emerged from the sunward direction (Figure 5). In order to examine the existence of this extremely faint jet feature and to distinguish it from the trail of a background star, we transformed the enhance image into polar coordinates $\rho$-$\theta$ where $\rho$ is the projected cometocentric distance from the nucleus and $\theta$ is the azimuth (position angle). At several distances $\rho$ from the nucleus, we analyzed the resulting azimuthal profile. In Fig. 5 (right panel) we show the azimuthal profile at $\rho = 5,000$ km. It can be seen that this faint jet (referred as main feature in the figure) appears on Oct. 11.76 and persists until Oct. 11.87, that is $\sim$0.7hr later (bottom panel in Fig. 5). It points towards the Sun and it does not display significant changes. On Oct. 11.84, a new faint feature appears nearly perpendicularly to the Sun-comet line. It is interesting to note that the position angle of the secondary jet is roughly the same as that of the CN jet features shown in Figure \[CNjet\] (pointing to the east-south direction in the top-middle panel). At first, one could think that icy grains mixed with the dust grains of this weaker jet could provide the partial fuel to the CN gas jet. However, the gas jets persist for most of a rotation period (Knight and Schleicher 2011, Samarasinha et al. 2011) and are clearly being released over an extended period of time as the nucleus rotates. Thus, the CN jets cannot mainly come from this faint jet feature if it is only active for a few hours as found here. That switching phenomenon may also be explained as a projection effect due to the comet nucleus rotation.\ For Oct. 28 and 29, we obtained a series of images from Lulin and Calar Alto observatories that provide insight into how the sunward feature evolved throughout $\sim$1.4 rotation cycles. Representative images from these nights are shown in Fig. 6, with each panel enhanced by the Larson-Sekanina filtering method. Notice that the position angle (PA) of the Sun during these two days is near 97$^\circ$. Setting the zero phase at 11:40 UT on October 10 and using a period of 18.11 hr (see the Section 3), the rotational phase can be easily estimated in these three images (see the bottom right corner of Figure  \[LotCa\]). A dust jet (labeled J and marked with an arrow in Figure  \[LotCa\]) can be seen in the sunward direction whose shape slightly changes as the rotational phase change from 0.22 (on Oct. 28.68 UT) to 0.82 (Oct. 29.13 UT). Thirteen hours later, (rotational phase of 0.57, Oct. 29.70 UT) two dust jet features emanating in the sunward direction can be seen. One of them, labeled J1, is close to the position of Sun (PAs $\sim$85$^\circ$) and the other, labeled J2, lays at the PAs $\rm \sim 130 ^\circ$. Owing to the similar PAs, we consider the possibility that J1 feature might have the same source region as seen from the previous two images (Oct. 28.68 and Oct. 29.13). Under this assumption, J2 feature is new. Another possibility is J1 feature might be associated with the cometary rotational effect, i.e. local sunrise accompanied by temperature increase turns that jet on. This localized temperature difference in the regions of waist and the sun-lit end of the nucleus have been addressed by Belton et al (2013). The J2 feature which has a collimated-like shape is the persistent feature we detected on Oct. 28.68 and Oct. 29.13 although PA and shape changed between those two dates. We note that the brightness of J2 feature is higher than that of J1 feature and this higher intensity could be related to the dusty ice, or to an outburst from the surface of the comet nucleus. To understand their interrelationship better, our images need to be interpreted in the context of a larger image series that displays the time evolution of the jet structure over two or more rotational period.\ On the tailward side, only the dust tail was readily visible starting in August, 2010. Dust tail was found to point approximately in the antisolar direction. As expected, it appears to be curved slightly counterclock-wise. The properties of dust coma --------------------------- We used Af$\rho$ (A’Hearn et al. 1984) to characterize the dust activity of the comet, the derived values acquired with broadband R-filter from April to November 2010 are presented in Figure  \[afrho\]. Except for the night on October 29, the average values estimated every photometric night were all measured within a projected distance of 5,000km. Notice that Af$\rho$ shows a weak dependence on the $\rho$, projected distance from nucleus, from 5,000km to 20,000km and the variation was found to be less than approx. 5-8%. The reason why we used 5,000km for uniform radius is to reduce the influence of star trails in field of view. The $A f \rho$ values steadily increased with decreasing heliocentric distance, although there was not a noticeable increase when the second jet appeared on Oct. 11.64 UT or at the perihelion. The $A f \rho$ value on October 29.77 $\sim$ 29.85 UT increased from 155 cm to 174 cm in two hours, and at the same time the dust jet seen in Figure 6 (right panel) was more prominent on this night than on any of the other nights and a relatively weak secondary jet feature was also detected. Possible causes for this deviation might include the changes in the physical properties of the grains as they travel outward (i.e. loss of volatiles or fragmentation), the action of solar radiation pressure modifying the straight trajectories of small particles inside the field of view, or a long-lasting population of large particles (Schleicher et al. 1998). Furthermore, the power law index of the $r_h$ dependence for the dust, Af$\rho$ (5,000 km), is $-3.75 \pm 0.45$ before perihelion and is $-3.44 \pm 1.20$ post-perihelion. This is result is completely consistent with Knight and Schleicher’s (2013) when using A($\theta$)f$\rho$.\ The derived Af$\rho$ values for the narrowband filter can be taken to estimate the color of the cometary dust [@jew87] as the normalized gradient of the Af$\rho$ product between the blue (BC,$\lambda_{0}$= 4,430 Å) and red (RC,$\lambda_{0}$= 6,840 Å) continuum filters. The dust color can be converted to a percentage of reddening per 1,000 [Å]{} and is defined by the following relation: $$\label{Afrho_formula} color = \frac{RC_{Af\rho}-BC_{Af\rho}}{6840-4430}\frac{2000}{RC_{Af\rho}+BC_{Af\rho}}$$ The summarized results in Table \[tcolor\] indicate that the averaging dust color within the innermost 5,000 km of the coma did not appear to vary significantly with heliocentric distance. This behavior of the averaging dust color seems to indicate that the innermost coma do not introduce significant changes on the size distribution and/or overall properties of dust grains. As we found a jet feature that switches on and off from our images in Figure \[Rjet1\] to Figure \[LotCa\], we analyzed the entire flux-calibrated images acquired with BC and RC narrowband filters instead of integrating whole flux in the innermost 5,000 km. The resulting two-dimensional dust color map can be seen in Figure \[fcolor\] (the third column). Figure \[fcolor\] displays the dust coma of comet 103P/Hartley 2 from October 10 to November 2 imaged in BC and RC narrowband filters (first two column), the dust color map (the third column) and azimuthal median profile subtracted RC filter images (the fourth column) that displays the jet activity in the dust coma. The data here presented pertaining to October and November give an extremely reddened dust, with a normalized color $\sim$ 30-45 %, within a radius of $\sim 50-100$ km measured from the optocenter of the images. This red dust could be associated with strong jet activity. The sunward jet feature might give rise to higher dust abundances at closer cometocentric distances (i.e. near the optocenter). These dust grains are initially large with a reddening of $\sim 30-40$%/1,000Å, while travelling out they split up and show bluer at $\sim 500$ km with a dust reddening of $\sim$ 10-15 %. In comparison with tailward direction, the color variation is 5% to 10%. The decrease in the dust reddening means that the optical properties of the dust grains change as the dust grains move outward or this blueing of the dust could be also associated with an outburst (Bonev et al., 2002). A possible explanation for color variation is that the larger dust grains mixed with the icy grains dominate the scattering behavior at close distance around the nucleus. When these larger dust grains move outwards, they break up or sublimate into the small sub-micron particles resulting in a bluer continuum due to their smaller sizes (Lara et al. 2011b). Summary ======= We observed the comet Hartley 2 at the Lulin Observatory in Taiwan, the Calar Alto Observatory in Spain, and the Beijing Astronomical Observatory in China, from April to December, 2010 using both broadband and narrowband filters, and long-slit spectrophotometry. The results are summarize below.\ 1. CN morphology: The processed CN images revealed two asymmetric jet features in the coma of comet Hartley 2. The CN jet features detected in the images here presented did not show the sprial-like structure seen by other authors in earlier date due to different observing geometry. One of these CN jet features always shows a higher intensity than the other, possibly because it is facing towards the Earth.\ 2. Gas production rates: Our Lulin, BAO and CA results show that there is no significant variation of Q(CN) from mid-October to early-November. The power law slopes of the heliocentric distance of the gas production rate of CN and C$_{2}$ are $-4.57$ and $-4.84$ before perihelion and $-3.21$ and $-3.42$ after perihelion. The average C$_2$-to-CN production rate ratio is 0.7$\sim$ 1.5 which places 103P/Hartley 2 as a “typical” in terms of $\rm C_2$ enrichment.\ 3. Dust morphology: The sunward jet feature was first detected in images acquired at the end of September, 2010. This sunward jet seems to be non-permanent. Instead, morphology varies with time and two distinct jet features are found on October 11 and after October 29 until November 2.\ 4. Af$\rho$ and dust color: The power law $r_h$ dependence of the dust production rate, Af$\rho$ (5,000 km), is $-3.75\pm0.45$ before perihelion and $-3.44\pm1.20$ during post-perihelion. The higher dust reddening is found around the optocenter and could be associated with a stronger jet activity. The dust color is getting bluer outwards along the sunward jet which implies that the optical properties of the dust grains change with $\rho$. The average dust color did not appear to vary significantly when the heliocentric distance decreased to perihelion. This work was based on observations obtained at Taiwan’s Lulin Observatory. We thank the staff members and Yu-Chi Cheng for their assistances with the observations. We greatfully acknowledge valuable discussions with the referee. The research was supported by project AyA2009-08011 of the Ministerio de Ciencia e Innovacion. Zhong Yi Lin acknowledges a post-doctoral grant awarded by the Junta de Andalucia through project number P07-TIC-274. This work was also supported by grant number NSC 99-2923-M-008-002-MY3 for the Formosa Program (NSC-CSIC). A’Hearn, M.F., Schleicher, D. G., Millis, R. L., Feldman, P. D., Thompson, D. T. 1984, , 89, 579 A’Hearn, M.F., Millis, R. L., Schleicher, D. G., Osip, D. J., Birch, P. V. 1995, Icarus, 118, 223 A’Hearn, M.F. et al. 2011, Science, 332, 1396 Belton, M.J.S. et al. 2013, Icarus, 222, 595 Böhnhardt, H., and Birkle, K. 1994, A&A, 107, 101 Bonev, T., Jockers, K., Petrova, E., Delva, M., Borisov, G., Ivanova, A. 2002, Icarus, 160, 419 Fray, N., Bénilan, Y., Cottin, H., Gazeau, M.-C., Crovisier, J. 2005, P&SS, 53, 1243 Jewitt, D.C., and Meech, K.J. 1987, , 317, 992 Harmon, J.K., Nolan, M.C., Howell, E. S., Giorgini, J. 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L., Gersch, A. 2011, , 734, L3 Schleicher, D.G. 1998, Icarus, 132, 397 Schleicher, D.G. 2010, , 140, 973 Soderblom, L.A. 2002, Science, 296, 1087 Waniak, W., Borisov, G., Drahus, M., Bonev, T. 2012, A&A, 543, A32 [cccccccccc]{} April 24 &19:56-20:17 &2.424 & 2.363 &256.2 & 24.2 & 884.4 & R & Lulin& Phot.\ May 11 &18:39-18:48 &2.283 &2.026 &252.1 &26.3 &758.2 & R & Lulin & Phot.\ May 15 &18:58-19:10 &2.249 &1.948 &251.0 &26.7 &729.1 & R & Lulin & Part. cloudy\ May 16 &19:07-19:36 &2.240 &1.928 &250.8 &26.8 &721.6 & R & Lulin & Part. cloudy\ May 20 &18:34-18:56 &2.206 &1.851 &249.7 &27.1 &692.7 & R & Lulin & Part. cloudy\ July 14 &17:01-18:40 & 1.721 & 0.924 &226.9 &29.0 &345.8 & R & Lulin & Part. cloudy\ July 14 &23:00-23:49 & 1.719 & 0.921 &226.7 &29.0 & 354.0 & R & CA & Phot.\ July 22 &01:25-02:36 & 1.656 & 0.825 &221.7 &29.0 & 316.1 & R & CA & Part. phot.\ July 30 &02:22-02:55 & 1.585 &0.725 &215.2 &29.2 &278.7 & R & CA & Phot.\ August 1 &17:30-18:42 &1.562 &0.694 &212.9 &29.3 &259.7 & R & Lulin & Phot.\ August 19 &14:00-20:15 &1.409 &0.503 &195.5 &31.1 &188.3 & R & Lulin & Part. cloudy\ August 20 &01:33-01:44 & 1.406 &0.500 &195.1 &31.1 &192.2 & R & CA & Phot.\ August 20 &13:17-20:01 &1.402 &0.494 &194.5 &31.2 &184.9 & R & Lulin & Phot.\ August 21 &19:27-20:43 &1.392 &0.483 &193.3 &31.4 &180.8 & R & Lulin & Phot.\ August 25 &22:46-00:23 &1.359 & 0.445 &189.3 &32.3 &174.5 & R,S & CA & Phot.\ August 29 &18:37-19:09 &1.329 &0.412 &185.6 &33.3 &154.2 & R & Lulin & Part. cloudy\ September 2 &02:23-02:27 & 1.304 & 0.384 &182.7 &34.2 &147.6 & R,S & CA & Part. phot.\ September 14 &00:35-00:57 & 1.219 & 0.293 &175.1 &38.3 &109.2 & R,S & CA & Phot.\ September 29 &14:45-20:51 &1.130 &0.192 &182.1 &44.2 &71.9 &R& Lulin & Phot.\ September 30 &13:55-20:06 &1.125 &0.186 &183.8 &44.6 &69.6 &R& Lulin & Phot.\ October 2 &17:55-18;05 &1.116 &0.175 &188.3 &45.3 &65.5 &R& Lulin & Part. phot.\ October 3 &15:21-15:30 &1.112 &0.170 &191.1 &45.6 &63.6 &R& Lulin & Part. phot.\ October 9 &14:40-17:55 &1.090 &0.143 &211.7 &47.7 & &S & BAO & Part. phot.\ October 10 &12:32-18:45 &1.087 &0.140 &215.9 &48.0 &52.4 &R+N & Lulin & Phot.\ October 11 &15:08-19:00 &1.083 &0.136 &220.3 &48.4 & &S & BAO & Phot.\ October 11 &13:05-21:08 &1.083 &0.136 &220.3 &48.4 &50.9 &R+N & Lulin & Phot.\ October 15 &00:28-02:18 & 1.077 &0.129 &230.9 &49.4 &48.8 & R,S & CA & Part. cloudy\ October 18 &04:38-05:10 &1.068 & 0.122 &248.1 &51.4 &46.9 & R,S & CA & Phot.\ October 19 &22:32-23:15 & 1.065& 0.121 &254.4 &52.5 &46.5 & R,S & CA & Part. phot.\ October 22 &02:39-03:13 &1.062 &0.121 &261.0 &53.8 &46.5 & R,S & CA & Phot.\ October 25 &03:53-04:38 & 1.060 &0.125 &268.6 &55.6 &47.6 & R & CA & Part. cloudy\ October 25 &18:09-18:39 &1.059 &0.126 &269.8 &55.9 &47.2 &R+N & Lulin & Phot.\ October 26 &16:09-21:24 &1.059 &0.128 &271.8 &56.4 &47.9 &N & Lulin & Phot.\ October 27 &16:24-26:56 &1.059 &0.130 &273.7 &56.5 &48.7 &N& Lulin & Phot.\ October 28 &17:50-21:28 &1.059 &0.132 &275.3 &57.4 &49.4 &N & Lulin & Phot.\ October 29 &03:06-03:20 & 1.059& 0.133 &275.9 &57.5 &51.1 & R & CA & Part. cloudy\ October 29 &16:24-21:00 &1.059 &0.135 &276.9 &57.7 &50.5 &R+N & Lulin & Phot.\ November 1 &19:14-21:12 &1.060 &0.145 &280.9 &58.6 &54.3 &R & Lulin & Part. cloudy\ November 2 &17:58-20:31 &1.061 &0.150 &282.0 &58.7 &56.1 &R+N & Lulin & Phot.\ November 3 &16:04-16:28 &1.061 &0.149 &281.9 &58.7 &55.7 &R & Lulin & Part. cloudy\ November 5 &18:07-21:09 &1.064 &0.157 &284.2 &58.8 &58.8 &R & Lulin & Part. cloudy\ November 5 &01:25-05:45 & 1.065 &0.162 &285.4 &58.8 & 62.3 & R,S & CA & Part. phot.\ November 16 &01:18-01:47 & 1.090 &0.210 &294.0 &56.1 &80.3 & R,S & CA & Part. phot.\ November 21 &19:07-19:37 &1.112 &0.239 &298.8 &53.2 &60.3 &R & Lulin & Phot.\ December 2 &16:24-18:48 &1.116 &0.295 &308.9 &46.4 &74.5 &R & Lulin & Part. cloudy\ December 17 &17:48-20:58 &1.262&0.377 &326.2 &36.4 &95.2 &R & Lulin & Phot.\ December 26 &23:08-00:12 & 1.323& 0.427 &338.0 &31.3 &163.7 & R,S & CA & Phot.\ [ccccccccc]{} Sept. 14&00:35-00:57&CA && 5.82$\pm$0.13 & 7.28$\pm$1.03 & 0.37$\pm$0.05 & X &\ Oct. 9&14:40-17:55&BAO &&21.9$\pm$4.78 &23.4$\pm$5.24 &4.39$\pm$0.94 &44.4$\pm$9.70&1.07$\pm$0.11\ Oct. 10&12:32-18:45 &Lulin &4.01&6.57$\pm$0.93 & X & X & X &\ Oct. 11&15:08-19:00&BAO &&21.2$\pm$3.57 &24.3$\pm$4.10 &4.37$\pm$0.74 &44.0$\pm$7.38&1.15$\pm$0.11\ Oct. 11&13:05-21:08&Lulin &4.00&9.32$\pm$1.37 & X & X & X &\ Oct. 18&04:38-05:10 &CA && 21.4$\pm$3.56 & 14.8$\pm$1.57 & 1.34$\pm$0.02 & 35.2$\pm$1.41 & 0.69$\pm$0.04\ Oct. 18&04:38-05:10 &CA && 21.2$\pm$1.39 & 15.1$\pm$3.99 & 1.33$\pm$0.002 & 46.9$\pm$0.97 & 0.71$\pm$0.29\ Oct. 22&02:39-03:13 &CA && 20.2$\pm$2.90 & 24.6$\pm$1.57 & 1.56$\pm$0.05 & X &1.22$\pm$0.05\ Oct. 25&18:09-18:39&Lulin &3.62&18.2$\pm$3.1 & X & X & X &\ Oct. 26&16:09-21:24&Lulin &3.66&20.1$\pm$5.4 & X & X & X &\ Oct. 27&16:24-26:56&Lulin &3.82&12.7$\pm$2.7 & X & X & X &\ Oct. 28&17:50-21:28&Lulin &3.98&18.9$\pm$3.2 & 27.8$\pm$5.3 & X & X &1.47$\pm$0.17\ Oct. 29&16:24-21:00 &Lulin &3.99&21.8$\pm$3.6 & 31.6$\pm$5.4 & X & X &1.45$\pm$0.15\ Nov. 2&17:58-20:31 &Lulin &4.17&20.3$\pm$3.1 & 22.0$\pm$3.2 & X & X &1.07$\pm$0.10\ Dec. 26&23:08-00:12 &CA &&7.54$\pm$0.13 & 9.64$\pm$1.6 & 0.29$\pm$0.05 & X &1.29$\pm$1.23\ [ccc]{} Date & color (% / 100 nm)\ October 10 &9.40$\pm$0.82\ October 11 &11.50$\pm$0.934\ October 25 &5.06$\pm$0.78\ October 26 &6.36$\pm$0.82\ October 27 &6.43$\pm$0.87\ October 28 &11.07$\pm$0.83\ October 29 &10.01$\pm$0.83\ November 2 &5.56$\pm$0.88\ ![ \[CNjet\] The CN images after the dust continuum was removed were enhanced by dividing by an azimuthal median profile. The rotational phase is given in the bottom-right corner of each image (see text for details). The original image (left-top) obtained on November 2 is shown with contours overplotted in green. The Sun symbol and arrow indicate the projected direction towards the Sun. North is up, East is to the left. The field of view is 3.44’ $\times$ 3.44’ and the scale bar is shown at the bottom left corner. The images are centered on the optocenter and the color code stretches for white representing the brightest areas and for black representing the darkest areas. ](f1-eps-converted-to.pdf){width="\textwidth"} ![ \[CNC2\] Log of production rates for CN (left) and C$_{2}$(right) plotted as a function of the heliocentric distance. Different symbols come from different data sets: filled square symbols refer to results here presented; open square symbols are taken from results in Knight and Schleicher (2013) and the open circle symbols come from Lara et al. (2011b). “//”is referred to the break heliocentric distance from -0.1 AU(pre-perihelion) to 0.99 AU(post-perihelion). ](f2-eps-converted-to.pdf){width="\textwidth"} ![ \[comparison\] Image of comet 103P/Hartley 2 obtained on October 11, 2010 with R broadband filter. At the top left corner, a contour plot of the original image is shown. In (a) we display the same image after Larson-Sekanina filtering, in (b) the image is divided by an azimuthal median profile, and in (c) the adaptive-Laplace technique has been applied. In all of them, two jet features are visible. North is up, East is to the left., the field of view is 2.92’ $\times$ 1.94’, corresponding to 9,200 x 6,100 km at the comet distance. The images are centered on the nucleus, the arrows point out the jets, T labels the tail, and the streaks are trailed stars. The negatives of the star trails in panel A are the artifacts of the resulting image subtracted using a combination of a 15$^\circ$ counter-clockwise rotation and a 15$^\circ$ clockwise rotation. As the images are normalized, the brightness scales from 0.95 to 1.05. ](f3-eps-converted-to.pdf){width="\textwidth"} ![ \[Rjet1\] Images of comet 103P/Hartley 2 acquired with the R-band images and enhanced by Larson-Sekanina algorithm. The dust sunward jet feature sometimes represents a straight jet but sometimes it shows the multiple jet features during a night. The jet showed minimal change in shape, position angle and extent from night to night. The Sun symbol and arrow indicate the projected direction towards the Sun. North is up, East is to the left. The field of view is 2.92’ $\times$ 1.94’ and the scale bar is shown in the bottom corner. All images are centered on the nucleus, arrows point out the jets, and T represents the tail. ](f4-eps-converted-to.pdf){width="70.00000%"} ![ \[Rjet2\] Dust jet features and tail enhanced by an azimuthal median profile (left panels) and the corresponding azimuthal profiles obtained at $\rho$ $\sim$ 5,000 km (right panels). A sunward dust jet feature is revealed in broadband the R-filter on Oct. 11.76UT (top panel). Two faint dust jet features are detected using both broadband R-filter (middle panel) and narrowband red-continuum filter (bottom panel) on Oct. 11.84 UT and 11.87 UT, respectively. The straight jet pointing towards the Sun (main feature) and the weaker one pointing nearly perpendicular to the Sun-nucleus direction (secondary feature) are marked in the graphs. Position angle is measured from north (up) in the counterclockwise direction (top-left panel). In the left panels, all images are centered on the nucleus, arrows with the indicating the jets and T for the dust tail, North is up, East is to the left, the field of view is 2.92’ $\times$ 1.94’, corresponding to 9,200 $\times$ 6100 km at the comet distance, the Sun symbol and the corresponding arrow indicate the projected direction towards the Sun.](f5-eps-converted-to.pdf){width="70.00000%"} ![ \[LotCa\] Time sequence of images of the comet 103P/Hartley 2 acquired from Lulin (left and right) and from CA (middle) observatories. Dust jet features are enhanced by Larson-Sekanina filtered. The rotational phase is given at the bottom right corner of each image. North is up, East is to the left. The field of view is 3.8’ $\times$ 3.8’ and all images are centered on the nucleus. J, J$_1$ and J$_2$ refer to the jets and the Sun symbol and arrow indicate the projected direction towards the Sun. ](f6-eps-converted-to.pdf){width="\textwidth"} ![ \[afrho\] Af$\rho$ variation as a function of heliocentric distance both pre- and post-perihelion. Filled squares indicate the results obtained from Lulin observatory (LOT) and opened squares pertain to the data from Calar Alto observatory. “//” is referred to the break heliocentric distance from $\rm -0.1~AU$ (pre-perihelion) to 0.99 AU (post-perihelion). The error bars are not clearly seen in this figure because they are lower than 5%. ](f7-eps-converted-to.pdf){width="\textwidth"}

--- abstract: 'The recent years have seen remarkable success in the use of deep neural networks on text summarization. However, there is no clear understanding of *why* they perform so well, or *how* they might be improved. In this paper, we seek to better understand how neural extractive summarization systems could benefit from different types of model architectures, transferable knowledge and learning schemas. Additionally, we find an effective way to improve current frameworks and achieve the state-of-the-art result on CNN/DailyMail by a large margin based on our observations and analyses. Hopefully, our work could provide more clues for future research on extractive summarization. Source code will be available on Github[^1] and our project homepage[^2].' author: - | Ming Zhong [^3]\ `mzhong18@fudan.edu.cn`\ Pengfei Liu\ `pfliu14@fudan.edu.cn`\ Danqing Wang\ `dqwang18@fudan.edu.cn`\ Xipeng Qiu [^4]\ `xpqiu@fudan.edu.cn`\ Xuanjing Huang\ `xjhuang@fudan.edu.cn` - | Ming Zhong[^5], Pengfei Liu, Danqing Wang, Xipeng Qiu[^6] , Xuanjing Huang\ Shanghai Key Laboratory of Intelligent Information Processing, Fudan University\ School of Computer Science, Fudan University\ 825 Zhangheng Road, Shanghai, China\ `{mzhong18,pfliu14,dqwang18,xpqiu,xjhuang}@fudan.edu.cn` bibliography: - './nlp.bib' title: | Searching for Effective Neural Extractive Summarization:\ What Works and What’s Next --- Introduction ============ Recent years has seen remarkable success in the use of deep neural networks for text summarization [@see2017get; @celikyilmaz2018deep; @jadhav2018extractive]. So far, most research utilizing the neural network for text summarization has revolved around architecture engineering [@zhou2018neural; @chen2018fast; @gehrmann2018bottom]. Despite their success, it remains poorly understood why they perform well and what their shortcomings are, which limits our ability to design better architectures. The rapid development of neural architectures calls for a detailed empirical study of analyzing and understanding existing models. In this paper, we primarily focus on extractive summarization since they are computationally efficient, and can generate grammatically and coherent summaries [@nallapati2017summarunner]. and seek to better understand how neural network-based approaches to this task could benefit from different types of model architectures, transferable knowledge, and learning schemas, and how they might be improved. #### Architectures Architecturally, the better performance usually comes at the cost of our understanding of the system. To date, we know little about the functionality of each neural component and the differences between them [@peters2018dissecting], which raises the following typical questions: 1) How does the choice of different neural architectures (CNN, RNN, Transformer) influence the performance of the summarization system? 2) Which part of components matters for specific dataset? 3) Do current models suffer from the over-engineering problem? Understanding the above questions can not only help us to choose suitable architectures in different application scenarios, but motivate us to move forward to more powerful frameworks. #### External Transferable Knowledge and Learning schemas Clearly, the improvement in accuracy and performance is not merely because of the shift from feature engineering to structure engineering, but the flexible ways to incorporate external knowledge [@mikolov2013efficient; @peters2018deep; @devlin2018bert] and learning schemas to introduce extra instructive constraints [@paulus2017deep; @arumae2018reinforced]. For this part, we make some first steps toward answers to the following questions: 1) Which type of pre-trained models (supervised or unsupervised pre-training) is more friendly to the summarization task? 2) When architectures are explored exhaustively, can we push the state-of-the-art results to a new level by introducing external transferable knowledge or changing another learning schema? **Content** -------------------------- ------------------- -------------------- --------------------------------- Sup. & Reinforce. \[eq:schema\] \[2\][\*]{}[Structure]{} Dec. Pointer & SeqLab. \[exp:decoder\] Enc. LSTM & Transformer \[eq:ecoder\] \[2\][\*]{}[Knowledge]{} Exter. GloVe BERT NEWS. \[2\][\*]{}[\[eq:knowledge\]]{} Inter. Random : Outline of our experimental design. Dec. and Enc. represent decoder and encoder respectively. Sup. denotes supervised learning and NEWS. means supervised pre-training knowledge.[]{data-label="tab:addlabel"} To make a comprehensive study of above analytical perspectives, we first build a testbed for summarization system, in which training and testing environment will be constructed. In the training environment, we design different summarization models to analyze how they influence the performance. Specifically, these models differ in the types of **architectures** (Encoders: CNN, LSTM, Transformer [@vaswani2017attention]; Decoders: auto-regressive[^7], non auto-regressive), **external transferable knowledge** (GloVe [@pennington2014glove], BERT [@devlin2018bert], <span style="font-variant:small-caps;">Newsroom</span> [@grusky2018newsroom]) and different **learning schemas** (supervised learning and reinforcement learning). To peer into the internal working mechanism of above testing cases, we provide sufficient evaluation scenarios in the testing environment. Concretely, we present a multi-domain test, sentence shuffling test, and analyze models by different metrics: repetition, sentence length, and position bias, which we additionally developed to provide a better understanding of the characteristics of different datasets. Empirically, our main observations are summarized as: 1\) Architecturally speaking, models with auto-regressive decoder are prone to achieving better performance against non auto-regressive decoder. Besides, LSTM is more likely to suffer from the architecture overfitting problem while Transformer is more robust. 2\) The success of extractive summarization system on the CNN/DailyMail corpus heavily relies on the ability to learn positional information of the sentence. 3\) Unsupervised transferable knowledge is more useful than supervised transferable knowledge since the latter one is easily influenced by the domain shift problem. 4\) We find an effective way to improve the current system, and achieving the state-of-the-art result on CNN/DailyMail by a large margin with the help of unsupervised transferable knowledge (**42.39** R-1 score). And this result can be further enhanced by introducing reinforcement learning (**42.69** R-1 score). Hopefully, this detailed empirical study can provide more hints for the follow-up researchers to design better architectures and explore new state-of-the-art results along a right direction. Related Work ============ The work is connected to the following threads of work of NLP research. #### Task-oriented Neural Networks Interpreting Without knowing the internal working mechanism of the neural network, it is easy for us to get into a hobble when the performance of a task has reached the bottleneck. More recently, @peters2018dissecting investigate how different learning frameworks influence the properties of learned contextualized representations. Different from this work, in this paper, we focus on dissecting the neural models for text summarization. A similar work to us is @kedzie2018content, which studies how deep learning models perform context selection in terms of several typical summarization architectures, and domains. Compared with this work, we make a more comprehensive study and give more different analytic aspects. For example, we additionally investigate how transferable knowledge influence extractive summarization and a more popular neural architecture, Transformer. Besides, we come to inconsistent conclusions when analyzing the auto-regressive decoder. More importantly, our paper also shows how existing systems can be improved, and we have achieved a state-of-the-art performance on CNN/DailyMail. #### Extractive Summarization Most of recent work attempt to explore different neural components or their combinations to build an end-to-end learning model. Specifically, these work instantiate their encoder-decoder framework by choosing recurrent neural networks [@cheng2016neural; @nallapati2017summarunner; @zhou2018neural] as encoder, auto-regressive decoder [@chen2018fast; @jadhav2018extractive; @zhou2018neural] or non auto-regressive decoder [@isonuma2017extractive; @narayan2018ranking; @arumae2018reinforced] as decoder, based on pre-trained word representations [@mikolov2013efficient; @pennington2014glove]. However, how to use Transformer in extractive summarization is still a missing issue. In addition, some work uses reinforcement learning technique [@narayan2018ranking; @wu2018learning; @chen2018fast], which can provide more direct optimization goals. Although above work improves the performance of summarization system from different perspectives, yet a comprehensive study remains missing. A Testbed for Text Summarization ================================ To analyze neural summarization system, we propose to build a *Training-Testing* environment, in which different text cases (models) are firstly generated under different training settings, and they are further evaluated under different testing settings. Before the introduction of our Train-Testing testbed, we first give a description of text summarization. Task Description ---------------- Existing methods of extractive summarization directly choose and output the salient sentences (or phrases) in the original document. Formally, given a document $D = d_1, \cdots, d_n$ consisting of $n$ sentences, the objective is to extract a subset of sentences $R = r_1, \cdots, r_m$ from $D$, $m$ is deterministic during training while is a hyper-parameter in testing phase. Additionally, each sentence contains $|d_i|$ words $d_i = x_1,\cdots, x_{|d_i|}$. Generally, most of existing extractive summarization systems can be abstracted into the following framework, consisting of three major modules: **sentence encoder**, **document encoder** and **decoder**. At first, a sentence encoder will be utilized to convert each sentence $d_i$ into a sentential representation $\mathbf{d}_i$. Then these sentence representations will be contextualized by a document encoder to $\mathbf{s}_i$. Finally, a decoder will extract a subset of sentences based on these contextualized sentence representations. Setup for Training Environment ------------------------------ The objective of this step is to provide typical and diverse testing cases (models) in terms of model architectures, transferable knowledge and learning schemas. ### Sentence Encoder We instantiate our sentence encoder with CNN layer [@kim2014convolutional]. We don’t explore other options as sentence encoder since strong evidence of previous work [@kedzie2018content] shows that the differences of existing sentence encoder don’t matter too much for final performance. ### Document Encoder Given a sequence of sentential representation $\mathbf{d}_1, \cdots, \mathbf{d}_n$, the duty of document encoder is to contextualize each sentence therefore obtaining the contextualized representations $\bs_1, \cdots, \bs_n$. To achieve this goal, we investigate the LSTM-based structure and the Transformer structure, both of which have proven to be effective and achieved the state-of-the-art results in many other NLP tasks. Notably, to let the model make the best of its structural bias, stacking deep layers is allowed. #### LSTM Layer Long short-term memory network (LSTM) was proposed by [@hochreiter1997long] to specifically address this issue of learning long-term dependencies, which has proven to be effective in a wide range of NLP tasks, such as text classification [@liu2017adversarial; @liu2016recurrent], semantic matching [@rocktaschel2015reasoning; @liu2016deep], text summarization [@rush2015neural] and machine translation [@sutskever2014sequence]. #### Transformer Layer Transformer [@vaswani2017attention] is essentially a feed-forward self-attention architecture, which achieves pairwise interaction by attention mechanism. Recently, Transformer has achieved great success in many other NLP tasks [@vaswani2017attention; @dai2018transformer], and it is appealing to know how this neural module performs on text summarization task. ### Decoder Decoder is used to extract a subset of sentences from the original document based on contextualized representations: $\bs_1,\cdots,\bs_n$. Most existing architecture of decoders can divide into auto-regressive and non auto-regressive versions, both of which are investigated in this paper. #### Sequence Labeling (SeqLab) The models, which formulate extractive summarization task as a sequence labeling problem, are equipped with non auto-regressive decoder. Formally, given a document $D$ consisting of $n$ sentences $d_1, \cdots, d_n$, the summaries are extracted by predicting a sequence of label $y_1, \cdots, y_n$ ($y_i \in \{0,1\}$) for the document, where $y_i = 1$ represents the $i$-th sentence in the document should be included in the summaries. #### Pointer Network (Pointer) As a representative of auto-regressive decoder, pointer network-based decoder has shown superior performance for extractive summarization [@chen2018fast; @jadhav2018extractive]. Pointer network selects the sentence by attention mechanism using *glimpse* operation [@vinyals2015order]. When it extracts a sentence, pointer network is aware of previous predictions. ### External transferable knowledge The success of neural network-based models on NLP tasks cannot only be attributed to the shift from feature engineering to structural engineering, but the flexible ways to incorporate external knowledge [@mikolov2013efficient; @peters2018deep; @devlin2018bert]. The most common form of external transferable knowledge is the parameters pre-trained on other corpora. To investigate how different pre-trained models influence the summarization system, we take the following pre-trained knowledge into consideration. #### Unsupervised transferable knowledge Two typical unsupervised transferable knowledge are explored in this paper: context independent word embeddings [@mikolov2013efficient; @pennington2014glove] and contextualized word embeddings [@peters2018deep; @devlin2018bert], have put the state-of-the-art results to new level on a large number of NLP taks recently. #### Supervised pre-trained knowledge Besides unsupervised pre-trained knowledge, we also can utilize parameters of networks pre-trained on other summarization datasets. The value of this investigation is to know transferability between different dataset. To achieve this, we first pre-train our model on the <span style="font-variant:small-caps;">Newsroom</span> dataset [@grusky2018newsroom], which is one of the largest datasets and contains samples from different domains. Then, we fine-tune our model on target domains that we investigate. ### Learning Schemas Utilizing external knowledge provides a way to seek new state-of-the-art results from the perspective of introducing extra data. Additionally, an alternative way is resorting to change the learning schema of the model. In this paper, we also explore how different learning schemas influence extractive summarization system by comparing supervised learning and reinforcement learning. Setup for Testing Environment ----------------------------- In the testing environment, we provide sufficient evaluation scenarios to get the internal working mechanism of testing models. Next, we will make a detailed deception. #### ROUGE Following previous work in text summarization, we evaluate the performance of different architectures with the standard ROUGE-1, ROUGE-2 and ROUGE-L [$\rm F_1$]{} scores [@lin2004rouge] by using pyrouge package[^8]. #### Cross-domain Evaluation We present a multi-domain evaluation, in which each testing model will be evaluated on multi-domain datasets based on CNN/DailyMail and <span style="font-variant:small-caps;">Newsroom</span>. Detail of the multi-domain datasets is descried in Tab. \[tab:dataset\]. #### Repetition We design repetition score to test how different architectures behave diversely on avoiding generating unnecessary lengthy and repeated information. We use the percentage of repeated n-grams in extracted summary to measure the word-level repetition, which can be calculated as: $$\begin{aligned} \mathrm{REP_n} = \frac{\mathrm{Count} {\mathrm{Uniq}(ngram)}}{\mathrm{Count}(ngram)}\end{aligned}$$ where $\mathrm{Count}$ is used to count the number of n-grams and $\mathrm{Uniq}$ is used to eliminate n-gram duplication. The closer the word-based repetition score is to 1, the lower the repeatability of the words in summary. #### Positional Bias It is meaningful to study whether the ground truth distribution of the datasets is different and how it affects different architectures. To achieve this we design a positional bias to describe the uniformity of ground truth distribution in different datasets, which can be calculated as: $$\begin{aligned} \mathrm{PosBias} = \sum_{i=1}^{k}{-{p(i)}\log(p(i))}\end{aligned}$$ We divide each article into $k$ parts (we choose $k=30$ because articles from CNN/DailyMail and <span style="font-variant:small-caps;">Newsroom</span> have 30 sentences by average) and $p(i)$ denotes the probability that the first golden label is in part $i$ of the articles. #### Sentence Length Sentence length will affect different metrics to some extent. We count the average length of the $k$-th sentence extracted from different decoders to explore whether the decoder could perceive the length information of sentences. #### Sentence Shuffling We attempt to explore the impact of sentence position information on different structures. Therefore, we shuffle the orders of sentences and observe the robustness of different architectures to out-of-order sentences. Experiment ========== Datasets -------- Instead of evaluating model solely on a single dataset, we care more about how our testing models perform on different types of data, which allows us to know if current models suffer from the over-engineering problem. #### CNN/DailyMail The CNN/DailyMail question answering dataset [@hermann2015teaching] modified by [@nallapati2016abstractive] is commonly used for summarization. The dataset consists of online news articles with paired human-generated summaries (3.75 sentences on average). For the data prepossessing, we use the data with non-anonymized version as [@see2017get], which doesn’t replace named entities. #### <span style="font-variant:small-caps;">Newsroom</span> Recently, <span style="font-variant:small-caps;">Newsroom</span> is constructed by [@grusky2018newsroom], which contains 1.3 million articles and summaries extracted from 38 major news publications across 20 years. We regard this diversity of sources as a diversity of summarization styles and select seven publications with the largest number of data as different domains to do the cross-domain evaluation. Due to the large scale data in <span style="font-variant:small-caps;">Newsroom</span>, we also choose this dataset to do transfer experiment. 0.07in **R-1** **R-2** **R-L** **R-1** **R-2** **R-L** **R-1** **R-2** **R-L** **R-1** **R-2** **R-L** ---------------------------------------------------------------- ------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- **Dec.** **Enc.** (lr)[1-2]{} (lr)[3-5]{} (lr)[6-8]{} (lr)[9-11]{} (lr)[12-14]{} 40.11 17.64 36.32 28.75 16.10 25.16 22.21 11.40 19.41 54.20 46.60 51.89 55.24 31.14 50.96 52.17 36.10 47.68 42.91 27.11 39.42 73.54 65.50 71.46 LSTM 41.22 18.72 37.52 30.26 17.18 26.58 21.27 10.78 18.56 59.32 51.82 56.95 Transformer 41.31 **18.85** 37.63 30.03 17.01 26.37 21.74 10.92 18.92 59.35 51.82 56.97 LSTM **41.56** 18.77 **37.83** 31.31 **17.28** **27.23** **24.16** **11.84** **20.67** **59.53** **51.89** **57.08** Transformer 41.36 18.59 37.67 **31.34** 17.25 27.16 23.77 11.63 20.48 59.35 51.68 56.90 **Dec.** **Enc.** (lr)[1-2]{} (lr)[3-5]{} (lr)[6-8]{} (lr)[9-11]{} (lr)[12-14]{} 22.51 7.69 17.78 45.26 35.53 42.70 39.63 27.72 36.10 29.44 18.92 26.65 41.08 21.49 35.80 73.99 64.80 72.09 57.15 43.06 53.27 47.17 33.40 44.02 LSTM 23.02 8.12 18.29 53.13 43.52 50.53 41.94 29.54 38.19 30.30 18.96 27.40 Transformer 23.49 8.43 18.65 53.66 44.19 51.07 42.98 **30.22** 39.02 30.97 19.77 28.03 LSTM 24.71 8.55 19.30 53.31 43.37 50.52 43.29 30.20 **39.12** 31.73 19.89 28.50 Transformer **24.86** **8.66** **19.45** **54.30** **44.70** **51.67** **43.30** 30.17 39.07 **31.95** **20.11** **28.78** Training Settings ----------------- For different learning schemas, we utilize cross entropy loss function and reinforcement learning method close to @chen2018fast with a small difference: we use the precision of ROUGE-1 as a reward for every extracted sentence instead of the $\rm F_1$ value of ROUGE-L. hird columns show the scope and methods of interactions for different words $w_i$ in a sentence. For context-independent word representations (GloVe, Word2vec), we directly utilize them to initialize our words of each sentence, which can be fine-tuned during the training phase. For BERT, we truncate the article to 512 tokens and feed it to a feature-based BERT (without gradient), concatenate the last four layers and get a 128-dimensional token embedding after passing through a MLP. Experimental Observations and Analysis -------------------------------------- Next, we will show our findings and analyses in terms of architectures and external transferable knowledge. ### Analysis of Decoders {#exp:decoder} We understand the differences between decoder *Pointer* and *SeqLab* by probing their behaviours in different testing environments. #### Domains From Tab. \[tab:all-models\], we can observe that models with pointer-based decoder are prone to achieving better performance against SeqLab-based decoder. Specifically, among these eight datasets, models with pointer-based decoder outperform SeqLab on six domains and achieves comparable results on the other two domains. For example, in “`NYTimes`”, “`WashingtonPost`” and “`TheGuardian`” domains, Pointer surpasses SeqLab by at least $1.0$ improvment (R-1). We attempt to explain this difference from the following three perspectives. #### Repetition For domains that need to extract multiple sentences as the summary (first two domains in Tab. \[tab:all-models\]), Pointer is aware of the previous prediction which makes it to reduce the duplication of n-grams compared to SeqLab. As shown in Fig. \[fig:test1\], models with Pointer always get higher repetition scores than models with SeqLab when extracting six sentences, which indicates that Pointer does capture word-level information from previous selected sentences and has positive effects on subsequent decisions. #### Positional Bias For domains that only need to extract one sentence as the summary (last six domains in Tab. \[tab:all-models\]), Pointer still performs better than SeqLab. As shown in Fig. \[fig:test2\], *the performance gap between these two decoders grows as the positional bias of different datasets increases*. For example, from the Tab. \[tab:all-models\], we can see in the domains with low-value positional bias, such as “`FoxNews(1.8)`”, “`NYDailyNews(1.9)`”, SeqLab achieves closed performance against Pointer. By contrast, the performance gap grows when processing these domains with high-value positional bias (“`TheGuardian(2.9)`”, “`WashingtonPost(3.0)`”). Consequently, SeqLab is more sensitive to positional bias, which impairs its performance on some datasets. \[fig:three-testing\] #### Sentence length We find *Pointer shows the ability to capture sentence length information based on previous predictions*, while SeqLab doesn’t. We can see from the Fig. \[fig:test3\] that models with Pointer tend to choose longer sentences as the first sentence and greatly reduce the length of the sentence in the subsequent extractions. In comparison, it seems that models with SeqLab tend to extract sentences with similar length. The ability allows Pointer to adaptively change the length of the extracted sentences, thereby achieving better performance regardless of whether one sentence or multiple sentences are required. ### Analysis of Encoders {#eq:ecoder} In this section, we make the analysis of two encoders LSTM and Transformer in different testing environments. #### Domains From Tab. \[tab:all-models\], we get the following observations: 1\) Transformer can outperform LSTM on some datasets “`NYDailyNews`” by a relatively large margin while LSTM beats Transformer on some domains with closed improvements. Besides, during different training phases of these eight domains, the hyper-parameters of Transformer keep unchanged[^9] while for LSTM, many sets of hyper-parameters are used[^10]. Above phenomena suggest that LSTM easily suffers from the architecture overfitting problem compared with Transformer. Additionally, in our experimental setting, Transformer is more efficient to train since it is two or three times faster than LSTM. 2\) When equipped with SeqLab decoder, Transformer always obtains a better performance compared with LSTM, the reason we think is due to the non-local bias [@wang2018non] of Transformer. ![Results of different document encoders with Pointer on normal and shuffled CNN/DailyMail. $\Delta \rm{R}$ denotes the decrease of performance when the sentences in document are shuffled. []{data-label="fig:shuffle"}](./standalone/fig-shuffle){width="0.8\linewidth"} #### Shuffled Testing In this settings, we shuffle the orders of sentences in training set while test set keeps unchanged. We compare two models with different encoders (LSTM, Transformer) and the results can be seen in Fig. \[fig:shuffle\]. Generally, there is significant drop of performance about these two models. However, Transformer obtains lower decrease against LSTM, suggesting that Transformer are more robust. 0.07in 0.07in **R-1** **R-2** **R-L** **R-1** **R-2** **R-L** **R-1** **R-2** **R-L** **R-1** **R-2** **R-L** ---------------------------------------------------------------- ------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- **Dec.** **Enc.** (lr)[1-2]{} (lr)[3-5]{} (lr)[6-8]{} (lr)[9-11]{} (lr)[12-14]{} LSTM 41.22 18.72 37.52 **41.33** **18.78** **37.64** 42.18 19.64 38.53 41.48 **18.95** 37.78 Transformer 41.31 **18.85** 37.63 40.19 18.67 37.51 42.28 **19.73** 38.59 41.32 18.83 37.63 LSTM **41.56** 18.77 **37.83** 41.15 18.38 37.43 **42.39** 19.51 **38.69** 41.35 18.59 37.61 Transformer 41.36 18.59 37.67 41.10 18.38 37.41 42.09 19.31 38.41 **41.54** 18.73 **37.83** #### Disentangling Testing Transformer provides us an effective way to disentangle position and content information, which enables us to design a specific experiment, investigating what role positional information plays. As shown in Tab. \[tab:disen\], we dynamically regulate the ratio between sentence embedding and positional embedding by two coefficients $\alpha$ and $\beta$. **Surprisingly, we find even only utilizing positional embedding (the model is only told how many sentences the document contains), our model can achieve $\mathbf{40.08}$ on R-1**, which is comparable to many existing models. By contrast, once the positional information is removed, the performance dropped by a large margin. This experiment shows that the success of such extractive summarization heavily relies on the ability of learning the positional information on CNN/DailyMail, which has been a benchmark dataset for most of current work. ### Analysis of Transferable Knowledge {#eq:knowledge} Next, we show how different types of transferable knowledge influences our summarization models. #### Unsupervised Pre-training Here, as a baseline, *word2vec* is used to obtain word representations solely based on the training set of CNN/DailyMail. As shown in Tab. \[tab:pre-trained\], we can find that context-independent word representations can not contribute much to current models. However, when the models are equipped with BERT, we are excited to observe that the performances of all types of architectures are improved by a large margin. Specifically, the model `CNN-LSTM-Pointer` has achieved a new state-of-the-art with **42.11** on R-1, surpassing existing models dramatically. #### Supervised Pre-training In most cases, our models can benefit from the pre-trained parameters learned from the <span style="font-variant:small-caps;">newsroom</span> dataset. However, the model `CNN-LSTM-Pointer` fails and the performance are decreased. We understand this phenomenon by the following explanations: The transferring process from CNN/DailyMail to <span style="font-variant:small-caps;">Newsroom</span> suffers from the domain shift problem, in which the distribution of golden labels’ positions are changed. And the observation from Fig. \[fig:shuffle\] shows that `CNN-LSTM-Pointer` is more sensitive to the ordering change, therefore obtaining a lower performance. #### Why does BERT work? We investigate two different ways of using BERT to figure out from where BERT has brought improvement for extractive summarization system. In the first usage, we feed each individual sentence to BERT to obtain sentence representation, which does not contain contextualized information, and the model gets a high R-1 score of 41.7. However, when we feed the entire article to BERT to obtain token representations and get the sentence representation through mean pooling, model performance soared to 42.3 R-1 score. The experiment indicates that though BERT can provide a powerful sentence embedding, the key factor for extractive summarization is contextualized information and this type of information bears the positional relationship between sentences, which has been proven to be critical to extractive summarization task as above. 0.07in Learning Schema and Complementarity {#eq:schema} ----------------------------------- Besides supervised learning, in text summarization, reinforcement learning has been recently used to introduce more constraints. In this paper, we also explore if several advanced techniques be complementary with each other. We first choose the based model `LSTM-Pointer` and `LSTM-Pointer + BERT`, then the reinforcement learning are introduced aiming to further optimize our models. As shown in Tab. \[table:rl\], we observe that even though the performance of `LSTM+PN` has been largely improved by BERT, when applying reinforcement learning, the performance can be improved further, which indicates that there is indeed a complementarity between architecture, transferable knowledge and reinforcement learning. Conclusion ========== In this paper, we seek to better understand how neural extractive summarization systems could benefit from different types of model architectures, transferable knowledge, and learning schemas. Our detailed observations can provide more hints for the follow-up researchers to design more powerful learning frameworks. Acknowledgment {#acknowledgment .unnumbered} ============== We thank Jackie Chi Kit Cheung, Peng Qian for useful comments and discussions. We would like to thank the anonymous reviewers for their valuable comments. The research work is supported by National Natural Science Foundation of China (No. 61751201 and 61672162), Shanghai Municipal Science and Technology Commission (16JC1420401 and 17JC1404100), Shanghai Municipal Science and Technology Major Project(No.2018SHZDZX01)and ZJLab. [^1]: <https://github.com/fastnlp/fastNLP> [^2]: <http://pfliu.com/InterpretSum/> [^3]: These two authors contributed equally. [^4]: Corresponding author. [^5]: These two authors contributed equally. [^6]:    Corresponding author. [^7]: Auto-regressive indicates that the decoder can make current prediction with knowledge of previous predictions. [^8]: [pypi.python.org/pypi/pyrouge/0.1.3](pypi.python.org/pypi/pyrouge/0.1.3) [^9]: 4 layers 512 dimensions for Pointer and 12 layers 512 dimensions for SeqLab [^10]: the number of layers searches in (2, 4, 6, 8) and dimension searches in (512, 1024, 2048)

--- abstract: 'Recent advances in computer vision have made accurate, fast and robust measurement of animal behavior a reality. In the past years powerful tools specifically designed to aid the measurement of behavior have come to fruition. Here we discuss how capturing the postures of animals - pose estimation - has been rapidly advancing with new deep learning methods. While challenges still remain, we envision that the fast-paced development of new deep learning tools will rapidly change the landscape of realizable real-world neuroscience.' author: - Mackenzie Weygandt Mathis - Alexander Mathis title: Deep learning tools for the measurement of animal behavior in neuroscience --- [**Highlights:**]{} Introduction {#introduction .unnumbered} ============ Behavior is the most important output of the underlying neural computations in the brain. Behavior is complex, often multi-faceted, and highly context dependent both in how con-specifics or other observers understand it, as well as how it is emitted. The study of animal behavior - ethology - has a rich history rooted in the understanding that behavior gives an observer a unique look into an animal’s *umwelt*  [@Uexkull; @Tinbergen1963; @Bernstein1967]; what are the motivations, instincts, and needs of an animal? What survival value do they provide? In order to understand the brain, we need to measure behavior in all its beauty and depth, and distill it down into meaningful metrics. Observing and efficiently describing behavior is a core tenant of modern ethology, neuroscience, medicine, and technology.\ In 1973 Tinbergen, Lorenz, and von Frisch were the first ethologists awarded the Nobel Prize in Physiology or Medicine for their pioneering work on the patterns of individual and social group behavior [@tinbergen73]. The award heralded a coming-of-age for behavior, and how rigorously documenting behavior can influence how we study the brain [@tinbergen73]. Manual methods are powerful, but also highly labor intensive and subject to the limits of our senses. Matching (and extending) the capabilities of biologists with technology is a highly non-trivial problem [@Schaefer2012TheSS; @Anderson2014], yet harbors tremendous potential. How does one compress an animal’s behavior over long time periods into meaningful metrics? How does one use behavioral quantification to build a better understanding of the brain and an animal’s *umwelt* [@Uexkull]?\ In this review we discuss the advances, and challenges, in animal pose estimation and its impact on neuroscience. Pose estimation refers to methods for measuring posture, while posture denotes to the geometrical configuration of body parts. While there are many ways to record behavior [@Dell2014review; @egnor2016computational; @camomilla2018trends], videography is a non-invasive way to observe the posture of animals. Estimated poses across time can then, depending on the application, be transformed into kinematics, dynamics, and actions [@Bernstein1967; @gomez2014big; @Schaefer2012TheSS; @Dell2014review; @Anderson2014]. Due to the low-dimensional nature of posture, these applications are computationally tractable. A very brief history of pose estimation {#a-very-brief-history-of-pose-estimation .unnumbered} --------------------------------------- The postures and actions of animals have been documented as far back as cave paintings, illustrating the human desire to distill the essence of an animal for conveying information. As soon as it was possible to store data on a computer, researchers have built systems for automated analysis. Over time, these systems reflected all flavors of artificial intelligence from rule-based via expert systems, to machine learning [@POPPE20074; @litjens2017survey]. Traditionally posture was measured by placing markers on the subject [@Bernstein1967], or markerlessly by using body models (i.e. cylinder-based models with edge features [@HOGG19835]). Other computer vision techniques, such as using texture or color to segment the person from the background to create silhouettes [@wren1997pfinder; @cremers2007review], or using so-called hand-crafted features with decoders [@POPPE20074; @litjens2017survey; @serre2019deep] were also popular before deep learning flourished. The deep learning revolution for posture {#the-deep-learning-revolution-for-posture .unnumbered} ---------------------------------------- Pose estimation is a challenging problem, but it has been tremendously advanced in the last five years due to advances in deep learning. Deep neural networks (DNNs) are computational algorithms that consist of simple units, which are organized in layers and then serially stacked to form “deep networks". The connections between the units are trained on data and therefore learn to extract information from raw data in order to solve tasks. The current deep learning revolution started with achieving human-level accuracy for object recognition on the ImageNet challenge, a popular benchmark with many categories and millions of images [@alom2018history; @serre2019deep]. A combination of large annotated data sets, sophisticated network architectures, and advances in hardware made this possible and quickly impacted many problems in computer vision (see reviews [@SCHMIDHUBER201585; @litjens2017survey; @serre2019deep]). {width="\textwidth"} 2D and 3D (human) pose estimation {#d-and-3d-human-pose-estimation .unnumbered} --------------------------------- In 2014 “DeepPose" was the first paper to apply deep learning to human 2D pose estimation [@toshev2014deeppose], and immediately new networks were proposed that improved accuracy by introducing a translation invariant model [@JainMODEEP], and convolutional networks plus geometric constraints [@NIPS2014_5573; @Tompson_2015_CVPR]. In the few years since, numerous human pose estimation papers (approx. 4,000 on Google Scholar), and new benchmarks with standardized datasets plus evaluation metrics appeared, which allow better comparisons of “state-of-the-art" performance [@andriluka2018posetrack]. This culture has driven rapid and remarkable increases in performance: from  44% of body parts correctly labeled to nearly 94% - with the top 15 networks being within a few percentage points of each other (an example top network is shown in Figure \[fig:2D3D\]a)  [@newell2016stacked; @insafutdinov2016deepercut; @insafutdinov2017cvpr; @cao2017realtime; @kreiss2019pifpaf; @sun2019deep]. The history and many advances in 2D human pose estimation are comprehensively reviewed in [@POPPE20074; @Dang2019].\ 3D human pose estimation is a more challenging task and 3D labeled data is more difficult to acquire. There has been massive improvements in networks; see review [@Sarafianos2016]. Yet currently, the highest accuracy is achieved by using multiple 2D views to reconstruct a 3D estimate (Figure \[fig:2D3D\]b; [@martinez2017simple; @Mehta2017_3D]), but other ways of “lifting" 2D into 3D are being actively explored [@TomeRA17; @chen20173d; @Mehta2017_3D]. Dense-representations of bodies {#dense-representations-of-bodies .unnumbered} ------------------------------- Other video-based approaches for capturing the posture and soft tissue of humans (and other animals) also exist. Depth-cameras such as the Microsoft Kinect have been used in humans [@Shotton2012EfficientHP; @kinectEval2012] and rodents [@wiltschko2015mapping; @hong2015automated]. Recently dense-pose representations, i.e. 3D point clouds or meshes (Figure \[fig:2D3D\]c), have become a popular and elegant way to capture the soft-tissue and shape of bodies, which are highly important features for person identification, fashion (i.e. clothing sales), and in medicine [@guler2018densepose; @SMPL-X:2019; @kanazawaHMR18]. However, state-of-the-art performance currently requires body-scanning of many subjects to make body models. Typically, large datasets are collected to enable the creation of robust algorithms for inference on diverse humans (or for animals, scanning toy models has been fruitful [@Zuffi20163dMenagerie]). Recently, outstanding improvements have been made to capture shapes of animals from images [@biggs2018creatures; @Zuffi_2018_CVPR; @Zuffi2019ICCV]. However, there are no animal-specific toolboxes geared towards neuroscience applications, although we believe that this will change in the near future, as for many applications having the soft-tissue measured will be highly important, i.e. in obesity or pregnancy research, etc. Animal pose estimation {#animal-pose-estimation .unnumbered} ====================== The remarkable performance when using deep learning for human 2D & 3D pose estimation plus dense-representations made this large body of work ripe for exploring its utility in neuroscience (Figure \[fig:2D3D\]d-f). In the past two years, deep learning tools for laboratory experiments have arrived (Figure \[fig:examples\]a-d).\ Many of the properties of DNNs were extremely appealing: remarkable and robust performance, relatively fast inference due to GPU hardware, and efficient code due to modern packages like TensorFlow and PyTorch (reviewed in [@Nguyen2019]). Furthermore, unlike for many previous algorithms, neither body models nor tedious manual tuning of parameters is required. Given the algorithms, the crucial ingredient for human pose estimation success was large-scale well annotated data sets of humans with the locations of the bodyparts.\ Here, we identify [**5**]{} key areas that were important for making DNN-based pose estimation tools useful for neuroscience laboratories, and review the progress in the last two years: 1. Can DNNs be harnessed with small training datasets? Due to the nature of “small-scale" laboratory experiments, labeling $>20,000$ or more frames is not a feasible approach (the typical human benchmark dataset sizes). 2. The end-result must be as accurate as a human manually-applied labels (i.e. the gold standard), and computationally tractable (fast). 3. The resulting networks should be robust to changes in experimental setups, and for long-term storage and re-analysis of video data, to video compression. 4. Animals move in 3D, thus having efficient solutions for 3D pose estimation would be highly valuable, especially in the context of studying motor learning and control. 5. Tracking multiple subjects and objects is important for many experiments studying social behaviors as well as for animal-object interactions. 1. Small training sets for lab-sized experiments {#small-training-sets-for-lab-sized-experiments .unnumbered} ------------------------------------------------ While the challenges discussed above for human pose estimation also apply for other animals, one important challenge for applying these methods to neuroscience was annotated data sets - could DNNs be harnessed for much smaller datasets, at sizes reasonable for typical labs? Thus, while it was clear that given enough annotated frames the same algorithms will be able to learn to track the body parts of any animal, there were feasibility concerns.\ Human networks are typically trained on thousands of images, and nearly all the current state-of-the-art networks provide tailored solutions that utilize the skeleton structure during inference [@cao2017realtime; @insafutdinov2017cvpr]. Thus, applying these tools to new datasets was not immediately straight-forward, and to create animal-specific networks one would need to potentially curate large datasets of the animal(s) they wanted to track. Additionally, researchers would need tailored DNNs to track their subjects (plus the ability to track unique objects, such as the corners of a box, or an implanted fiber).\ Thus, one of the most important challenges is creating tailored DNNs that are robust and generalize well with little training data. One potential solution for making networks for animal pose estimation that could generalize well, even with little data, was to use transfer learning - the ability to take a network that has been trained on one task to perform another. The advantage is that these networks are pretrained on larger datasets (for different tasks where a lot of data is available like ImageNet), therefore they are effectively imbued with good image representations.\ This is indeed what “DeepLabCut," the first tool to leverage the advances in human pose estimation for application to animals did [@mathis2018deeplabcut]. DeepLabCut was built on a subset of “DeeperCut" [@insafutdinov2016deepercut], which was an attractive option due to its use of ResNets, which are powerful for transfer learning [@kornblith2019better; @mathis2019TRANSFER]. Moreover transfer learning reduces training times [@kornblith2019better; @mathis2019TRANSFER; @arac2019deepbehavior], and there is a significant gain over using randomly-initialized networks in performance, especially for smaller datasets [@mathis2019TRANSFER].\ The major result from DeepLabCut was benchmarking on smaller datasets and finding that only a few hundred annotated images are enough to achieve excellent results for diverse pose estimation tasks like locomotion, reaching and trail-tracking in mice, egg-laying in flies and hunting in cheetahs, due to transfer learning (Figure \[fig:examples\]f,g,h) [@MathisWarren2018speed; @nath2019deeplabcut; @mathis2019TRANSFER]. “DeepBehavior," which utilized different DNN-packages for various pose estimation problems, also illustrated the gain of transfer learning [@arac2019deepbehavior]. {width=".85\textwidth"} 2. Accuracy & speed {#accuracy-speed .unnumbered} ------------------- To be useful, pose estimation tools need to be as good as human annotation of frames (or tracking markers, another proxy for a human-applied label). DeepLabCut was shown to reach human labeling accuracy [@mathis2018deeplabcut], and can replace physical markers [@moore2019]. Moreover, they need to be efficient (fast) for both offline analysis and online analysis. Speed is often related to the depth of the network. Stacked-hourglass networks, which use iterative refinement [@newell2016stacked; @SegNet] and fewer layers, are fast. Two toolboxes, “LEAP"  [@pereira2019fast] and “DeepPoseKit" [@graving2019fast] adopted variants of stacked-hourglass networks. LEAP allows the user to rapidly compute postures, and then perform unsupervised behavioral analysis (Figure \[fig:examples\]d,e) [@Berman2014]. This is an attractive solution for real-time applications, but it is not quite as accurate. For various datasets, DeepPoseKit reports it is about three times as accurate as LEAP, yet similar to DeepLabCut [@graving2019fast]. They also report about twice faster video processing compared to DeepLabCut and LEAP for batch-processing (on small frame sizes).\ Deeper networks are slower, but often have more generalization ability [@kornblith2019better]. DeepLabCut was designed for generalization and therefore utilized deeper networks (ResNets) that are inherently slower than stacked-hourglass networks, yet DeepLabCut can match human accuracy in labeling (Figure \[fig:examples\]f) [@mathis2018deeplabcut]. The speed has been shown to be compatible with online-feedback applications [@forys2018real; @vstih2019stytra; @MathisWarren2018speed]. Other networks recently added to DeepLabCut (with a MobileNetV2 backbone) give slightly lower accuracy, with twice the speed [@mathis2019TRANSFER]. Overall, on GPU hardware all packages are fast, and reach speeds of several hundred frames per second in offline modes. 3. Robustness {#robustness .unnumbered} ------------- Neuroscience experiments based on video recordings produce large quantities of data and are collected over extensive periods of time. Thus, analysis pipelines should be robust to a myriad of perturbations: such as changes in setups (backgrounds, light sources, cameras, etc.), subject appearance (due to different animal strains), and compression algorithms (which allow storage of perceptually good videos with little memory demands [@wiegand2003overview]). How can robustness be increased within the DNN? Both transfer learning (discussed above) and data augmentation strategies are popular and rapidly evolving approaches to increase robustness in DNNs (see review [@Shorten2019]). Moreover, active learning approaches allow an experimenter to continuously build more robust and diverse datasets, for large scale projects by expanding the training set with images, where the network fails [@mathis2018deeplabcut; @nath2019deeplabcut; @pereira2019fast]. So far, the toolboxes have been tested on data from the same distribution (i.e. by splitting frames from videos into test and training data), which is important for assessing the performance [@pereira2019fast; @graving2019fast; @mathis2018deeplabcut], but did not directly tested out-of-domain robustness.\ Over the course of long-term experiments the background or even animal strain can change, which means having robust networks would be highly advantageous. We recently tested the generalization ability of DeepLabCut with different network backbones for pose estimation. We find that pretraining on ImageNet strongly improves out-of-domain performance, and that better ImageNet performing networks are more robust (Figure \[fig:examples\]g) [@mathis2019TRANSFER]. There is still a gap to close in out-of-domain performance, however.\ DeepLabCut is also robust to video compression, as compression by more than 1,000X only mildly affects accuracy (less than 1 pixel average error less) [@MathisWarren2018speed]. The International Brain Lab (IBL) independently and synergistically showed that for tracking multiple body parts in a rodent decision making task, DeepLabCut is robust to video compression [@Meijer2019]. Thus, in practice users can substantially compress videos, while retaining accurate posture information. {width=".97\textwidth"} 4. 3D animal pose estimation {#d-animal-pose-estimation .unnumbered} ---------------------------- Currently, there are several animal pose estimation toolboxes that explicitly support 2D and 3D key-point detection [@nath2019deeplabcut; @pierre2019; @pavan2019; @zhang2019multiview]. DeepLabCut uses 2D pose estimation to train a single camera-invariant 2D network (or multiple 2D networks) that is then used to perform traditional triangulation to extract 3D key points (Figure \[fig:examples\]i, j;  [@nath2019deeplabcut; @bova2019automated]). A pipeline built on DeepLabCut called “Anipose" allows for 3D reconstruction from multiple cameras using a wider variety of methods [@pierre2019]. “DeepFly3D" [@pavan2019] uses the network architecture from Newell et al. [@newell2016stacked] and then adds elegant tools to compute an accurate 3D estimate of *Drosophila melanogaster* by using the fly itself vs. standard calibration boards. Zhang et al. use epipolar geometry to train across views and thereby improve 3D pose estimation for mice, dogs, and monkeys [@zhang2019multiview]. 5. Multi-animal & object tracking {#multi-animal-object-tracking .unnumbered} --------------------------------- Many experiments in neuroscience require measuring interactions of multiple animals or interactions with objects. Having the ability to both flexibly track user-defined objects or multiple animals therefore is highly desirable. There are many pre-deep learning algorithms that allow tracking of objects (one such modern example called “Tracktor" also nicely summarizes this body of work [@Sridhar2019]). Recently researchers have also applied deep learning to this problem. For example, the impressive “idTracker:ai" [@romero2019idtracker] allows for users to track a hundred individual, unmarked animals. Arac et al. used YOLO, a popular and fast object localization network, for tracking two mice during a social behavior [@arac2019deepbehavior]. These, and others, can then be combined with pose estimation packages for estimating the pose of multiple animals. Currently, two paths are possible: one is to apply pose estimation algorithms after tracking individuals (for which any package could be used); or, two, extract multiple detections for each part on each animal (Figure \[fig:examples\]h; [@mathis2018deeplabcut; @jiang2019detection]) and link them using part affinity fields [@cao2017realtime], pairwise predictions [@insafutdinov2016deepercut], or geometrical constraints [@jiang2019detection], plus combinatorics. The impact on experimental neuroscience {#the-impact-on-experimental-neuroscience .unnumbered} ======================================= In the short time period these tools have become available there has been a rather wide adoption by the neuroscience and ethology community. Beyond the original publications, DeepLabCut has already been used for pose estimation and behavioral analysis in many contexts. For instance, knee movement quantification during cycling (Figure \[fig:examples\]a) [@kaplan2019video], postural analysis during underwater running [@cronin2019markerless], social behavior in bats [@zhang2019correlated], for benchmarking thermal constraints with optogenetics [@owen2019thermal], fly leg movement analysis [@bidaye2019two; @azevedo2019size], for 3D rat reaching (Figure \[fig:examples\]i) [@bova2019automated], hydra in a thermal stimulation assay [@tzouanas2019thermal] and pupillometry (Figure \[fig:examples\]k)  [@sriram2019sparse]. Also inanimate objects can be tracked, and it has indeed also been used to track metal beads when subjected to a high voltage [@de2019oscillatory], and magic tricks (i.e. coins and the magician) [@zaghi2019playing]. LEAP [@pereira2019fast] has been used to track ants [@clifton2019rough] and mice [@Yue2019_leapmice].\ Pose estimation is just the beginning; the next steps involve careful analysis of kinematics, building detailed, multi-scale ethograms of behaviors, new modeling techniques to understand large-scale brain activity and behaviors across a multitude of timescales, and beyond. We envision three branches where powerful feature tracking and extensions will be useful: motor control studies (often involving complex motor actions), naturalistic behaviors in the lab and in the wild, and better quantification of robust and seemingly simple “non-motor" tasks (Figure \[fig:toolbox\]).\ Many paradigms in neuroscience can be loosely arranged along three branches as natural (i.e. mouse parenting behavior), simple trail-based tasks (i.e. classical conditioning), and/or complex motor actions like skilled reaching (Figure \[fig:toolbox\]). For example, you can have simple and natural tasks such as licking for water, or complex and natural behaviors such as escaping from a looming stimulus that would rarely produce repeated trajectories. For simplicity, here we discuss how pose estimation can potentially enhance these studies along those three branches, namely complex movements (Motor control & kinematics), natural behaviors (Natural behaviors & ethologically relevant features), and during simple motor-output tasks (Revisiting classic tasks). Motor control & kinematics {#motor-control-kinematics .unnumbered} -------------------------- Often in neuroscience-minded motor control studies end-effector proxies (such as manipulandums or joysticks) are used to measure the motor behavior of subjects or animals. There are relatively few marker-tracking based movement neuroscience studies, in which many degrees of freedom were measured alongside neural activity, with notable exceptions like [@vargas2010decoding; @schaffelhofer2015decoding]. The ease with which kinematics of limbs and digits can now be quantified [@mathis2018deeplabcut; @bova2019automated; @azevedo2019size; @arac2019deepbehavior] should greatly simplify such studies in the future. We expect many more highly detailed kinematic studies will emerge that utilize DNN-based analyses, especially for freely moving animals, for small and aquatic animals that cannot be marked, and for motor control studies that can leverage large-scale recordings and behavioral monitoring. Natural behaviors & ethologically relevant features {#natural-behaviors-ethologically-relevant-features .unnumbered} --------------------------------------------------- There is a trend in motor neuroscience towards natural behaviors; i.e. less constrained tasks, everyday-skills, and even “in the wild" studies [@mathis2019highlights]. For instance, we used DeepLabCut for 3D pose estimation in hunting cheetah’s captured via multiple Go-Pro cameras (Figure \[fig:examples\]j; [@nath2019deeplabcut]). Another “in the wild example“ is given by a recent study by Chambers et al. [@chambers2019pose], who revisited the classic question of how people synchronize their walking, but with a modern twist by using videos from YouTube and analysis with OpenPose [@cao2017realtime]. Consistent with studies performed in the laboratory, they found a tendency for pairs of people to either walk in or exactly out of phase [@chambers2019pose].\ How else can DNNs help? Specialized body parts often play a key role in ethologically relevant behaviors. For instance, ants use their antenna to follow odor trails [@draft2018carpenter], while moles use their snouts for sampling bilateral nasal cues to localize odorants [@catania2013stereo]. To accurately measure such behaviors, highly accurate feature-detectors of often tiny, highly dexterous bodyparts are needed. This is a situation where deep learning algorithms can excel. Pose estimation algorithms can not only be used to detect the complete ”pose", but due to their flexibility they are extremely useful to track ethologically relevant body parts in challenging situations; incidentally DeepLabCut was created, in part, to accurately track the snouts of mice following odor trails that were printed onto a treadmill [@mathis2018deeplabcut]. There are of course many other specialized body parts that are hard to track: like whiskers, bee-stingers, jellyfish tentacles, or octopus arms, and we believe that studying these beautiful systems in more natural and ethologically relevant environments has now gotten easier. Revisiting classic tasks {#revisiting-classic-tasks .unnumbered} ------------------------ Measuring behavior is already impacting “classic" decision-making paradigms. For example, several groups could show broad movement encoding across the brain during decision-making tasks by carefully quantifying behavior [@Stringereaav7893; @Musall_NN]. Moreover, large scale efforts to use these “simple" yet robust trial-based behaviors across labs and brain areas are leveraging deep learning, and comparing their utility compared to classical behavior-monitoring approaches. For example, the IBL has surmised that DeepLabCut could replace traditional methods used for eye, paw and lick detection [@Meijer2019]. We believe that detailed behavioral analysis will impact many paradigms, which were historically not considered “motor" studies, as now it is much easier to measure movement. Remaining challenges in pose estimation {#remaining-challenges-in-pose-estimation .unnumbered} --------------------------------------- Advances in deep learning have changed how easily posture can be measured and has impacted many studies. However, pose estimation remains a hard computer vision problem and challenges remain [@andriluka2018posetrack; @golda2019human; @Dang2019; @serre2019deep; @rhodin2018learning; @kordingLimitations2019]. In the context of multi-animal/human pose estimation, dealing with highly crowded scenes, in which different individuals cover each other, remains highly challenging [@andriluka2018posetrack; @Dang2019; @golda2019human]. In general, dealing with occlusions remains a challenge. In some experiments occlusions are hard to avoid. Thus, networks that can constraint body part detection based on anatomical relationships can be advantageous, but are computationally more complex and slower [@insafutdinov2016deepercut]. As we highlighted in the robustness section, it is hard to train networks to generalize to out-of-domain scenarios [@rhodin2018learning; @mathis2019TRANSFER; @michaelis2019benchmarking]. Even though very large data sets have been amassed to build robust DNNs [@insafutdinov2017cvpr; @cao2017realtime; @kreiss2019pifpaf; @sun2019deep], they can still fail in sufficiently different scenarios [@rhodin2018learning; @mathis2019TRANSFER; @kordingLimitations2019]. Making robust networks will highly useful for creating shareable behavior- or animal-specific networks that can generalize across laboratories. There will also be much work towards even faster, lighter models. Outlook & Conclusions {#outlook-conclusions .unnumbered} ===================== The field of 2D, 3D, and dense pose estimation will continue to evolve. For example, with respect to handling occlusions and robustness to out-of-domain data. Perhaps larger and more balanced datasets will be created to better span the behavioral space, more temporal information will be utilized when training networks or analyzing data, and new algorithmic solutions will be found.\ Will we always need training data? A hot topic in object recognition is training from very few examples (one-shot or zero-shot learning) [@Xian2019_ZERO]. Can this be achieved in pose estimation? Perhaps as new architectures and training regimes come to fruition this could be possible. Alternatively, specialized networks could now be built that leverage large datasets of specific animals. It is hard to envision a universal “animal pose detector" network (for object recognition this is possible) as animals have highly diverse body plans and experimentalists often have extremely different needs. Currently many individual labs create their own specialized networks, but we plan to create shareable networks for specific animals (much like the specific networks, i.e. hand network in OpenPose [@simon2017hand], or the human-network in DeepLabCut [@insafutdinov2016deepercut; @mathis2018deeplabcut]). For example, many open field experiments could benefit from robust and easy-to-use DNNs for video analysis across similar body points of the mouse. Indeed, efforts are underway to create networks where one can simply analyze their data without training, and we hope the community will join these efforts. Nothing improves DNNs more than more training data. These efforts, together with making code open source, will contribute to the reproducibility of science and make these tools broadly accessible.\ [**In summary,**]{} we aimed to review the progress in computer vision for human pose estimation, how it influenced animal pose estimation, and how neuroscience laboratories can leverage these tools for better quantification of behavior. Taken together, the tremendous advance of computer vision has provided tools that are practical for the use in the laboratory, and they will only get better. They can be as accurate as human-labeling (or marker-based tracking), and are fast enough for closed-loop experiments, which is key for understanding the link between neural systems and behavior. We expect that in-light of shared, easy-to-use tools and additional deep learning advances, there will be thrilling and unforeseen advances in real-world neuroscience. **Acknowledgments:** We thank Eldar Insafutdinov, Alex Gomez-Marin, the M. Mathis Lab, and the anonymous reviewers for comments on the manuscript, and Julia Kuhl for illustrations. Funding was provided by the Rowland Institute at Harvard University (M.W.M.) and NIH U19MH114823 (A.M). The authors declare no conflicts of interest. References {#references .unnumbered} ========== Highlighted References: {#highlighted-references .unnumbered} ======================= \[1\*\] [**A survey on deep learning in medical image analysis**]{} [@litjens2017survey] *Comprehensive review of all deep learning algorithms used in medical image analysis as of 2017, as well as a discussion of most successful approaches, together with \[2\*\*\] a fantastic introduction for newcomers to the field.*\ \[2\*\*\] [**Deep learning: The good, the bad, and the ugly**]{} [@serre2019deep] *Excellent review of deep learning progress including a detailed description of recent successes as well as limitations of computer vision algorithms.*\ \[3\*\*\] [**Realtime multi-person 2d pose estimation using part affinity fields**]{} [@cao2017realtime] *OpenPose was the first real-time multi-person system to jointly detect human body parts by using part affinity fields, a great way to link body part proposals across individuals. The toolbox is well maintained and now boasts body, hand, facial, and foot keypoints (in total 135 keypoints) as well as 3D support.* \[4\*\] [**Dense-pose: Dense human pose estimation in the wild.**]{}[@guler2018densepose] *Using a large dataset of humans (50K), they build dense correspondences between RGB images and human bodies. They apply this to human “in the wild," and build tools for efficiently dealing with occlusions. It is highly accurate and runs up to 25 frames per second.*\ \[5\*\*\] [**Three-D Safari: Learning to Estimate Zebra Pose, Shape, and Texture from Images “In the Wild"**]{} [@Zuffi2019ICCV] *Zuffi et al. push dense pose estimations by using a new SMALST model for capturing zebras pose, soft-tissue shape, and even texture “in the wild." This is a difficult challenge as zebras are designed to blend into the background in the safari. This paper makes significant improvements on accuracy and realism, and builds on a line of elegant work from these authors [@Zuffi20163dMenagerie; @Zuffi_2018_CVPR]*\ \[6\*\*\] [**DeeperCut: A deeper, stronger, and faster multi-person pose estimation model**]{} [@insafutdinov2016deepercut] *DeeperCut is a highly accurate algorithm for multi-human pose estimation due to improved deep learning based body part detectors, and image-conditioned pairwise terms to predict the location of body parts based on the location of other body parts. These terms are then used to find accurate poses of individuals via graph cutting. In ArtTrack [@insafutdinov2017cvpr] the work was extended to fast multi-human pose estimation in videos.*\ \[7\*\] [**Recovering the shape and motion of animals from video**]{} [@biggs2018creatures] *The authors combine multiple methods in order to efficiently fit 3D shape to multiple quadrupeds from camels to bears. They also provides a novel dataset of joint annotations and silhouette segmentation for eleven animal videos.*\ \[8\*\*\] [**DeepLabCut: markerless pose estimation of user-defined body parts with deep learning**]{} [@mathis2018deeplabcut] *DeepLabCut was the first deep learning toolbox for animal pose estimation. The key advance was to benchmark a subset of the feature detectors in DeeperCut [@insafutdinov2017cvpr]. This paper showed nearly human-level performance with only 50-200 images. It benchmarked flies moving in a 3D chamber, hand articulations and open-field behavior in mice, and provided open-source tools for creating new datasets and data loaders to train deep neural networks, and post-hoc analysis. Subsequent work has improved accuracy, speed, and introduced more network variants into the Python package [@MathisWarren2018speed; @nath2019deeplabcut; @mathis2019TRANSFER].*\ \[9\*\] [**DeepBehavior: A deep learning toolbox for automated analysis of animal and human behavior imaging data**]{} [@arac2019deepbehavior] *Arac et al. use three different DNN-packages (OpenPose [@cao2017realtime], YOLO and Tensorbox) for analyzing 3D analysis of pellet reaching, three-chamber test, social behavior in mice, and 3D human kinematics analysis for clinical motor function assessment with OpenPose [@cao2017realtime]. They also provide MATLAB scripts for additional analysis (after pose estimation).*\ \[10\*\] [**Using DeepLabCut for 3D markerless pose estimation across species and behaviors**]{} [@nath2019deeplabcut]. *A Nature Protocols user-guide to DeepLabCut2.0, with 3D pose estimation of hunting cheetahs and improved network performance. The toolbox is provided as a Python package with graphical user interfaces for labeling, active-learning-based network refinement, together with Jupyter Notebooks that can be run on cloud resources such as Google Colaboratory (for free).*\ \[11\*\] [**Fast animal pose estimation using deep neural networks**]{} [@pereira2019fast]. *LEAP (LEAP estimates animal pose), a DNN method for predicting the positions of animal body parts. This framework consists of a graphical interface for labeling of body parts and training the network. Training and inference times are fast due to the lightweight architecture. The authors also analyzed insect gait based on unsupervised behavioral methods [@Berman2014], which they directly applied to the posture, rather than to compressed image features.*

--- author: - 'A.V. Lipatov, N.P. Zotov' title: | Heavy quark and quarkonium production\ at CERN LEP2: $k_T$-factorization versus data --- [*D.V. Skobeltsyn Institute of Nuclear Physics,\ M.V. Lomonosov Moscow State University,\ 119992 Moscow, Russia*]{}\ [**Abstract** ]{} We present calculations of heavy quark and quarkonium production at CERN LEP2 in the $k_T$-factorization QCD approach. Both direct and resolved photon contribution are taken into account. The conservative error analisys is performed. The unintegrated gluon distribution in the photon is taken from the full CCFM evolution equation. The traditional color-singlet mechanism to describe non-perturbative transition of $Q\bar Q$-pair into a final quarkonium is used. Our analisys covers polarization properties of heavy quarkonia at moderate and large transverse momenta. We find that the total and differential open charm production cross sections are consistent with the recent experimental data taken by the L3, OPAL and ALEPH collaborations. At the same time the DELPHI data for the inclusive $J/\psi$ production exceed our predictions but experimental uncertainties are too large to claim a significant inconsistency. The bottom production in photon-photon collisions at CERN LEP2 is hard to explain within the $k_T$-factorization formalism. Introduction ============ Heavy quark and quarkonium production in photon-photon collisions at high energies is a subject of the intensive studies from both theoretical and experimental point \[1–6\]. From the theoretical point, heavy quark in $\gamma \gamma$ collisions can be produced via direct and resolved production mechanisms. In direct events, the two photons couple directly to a heavy quark pair. In resolved events, one photon (“single-resolved”) or both photon (“double-resolved”) fluctuate into a hadronic state and a gluon or a quark of this hadronic fluctuation takes part in the hard interaction. At LEP2 conditions the contribution of the double-resolved events ($gg \to Q\bar Q$) is small \[7\], and charm and bottom quarks are produced mainly via direct ($\gamma \gamma \to Q\bar Q$) and single-resolved ($\gamma g \to Q\bar Q$) processes. The direct contribution is not dependent on the quark and gluon content in the photon, whereas the single-resolved processes strongly depend on the photon’s gluon density. Therefore detailed knowledge of gluon distributions in the photon is necessary for the theoretical description of such processes at modern (LEP2) and future (TESLA) colliders. Usually quark and gluon densities are described by the Dokshitzer-Gribov-Lipatov-Altarelli-Parizi (DGLAP) evolution equation \[8\] where large logarithmic terms proportional to $\ln(\mu^2)$ are taken into account. The cross sections can be rewritten in terms of process-dependent hard matrix elements convoluted with quark or gluon density functions. In this way the dominant contributions come from diagrams where parton emissions in initial state are strongly ordered in virtuality. This is called collinear factorization, as the strong ordering means that the virtuality of the parton entering the hard scattering matrix elements can be neglected compared to the large scale $\mu^2$. However, at the high energies this hard scale is large compare to the $\Lambda_{\rm QCD}$ parameter but on the other hand $\mu^2$ is much less than the total energy $\sqrt s$ (around 200 GeV for the LEP2 collider). Therefore in such case it was expected that the DGLAP evolution, which is only valid at large $\mu^2$, should break down. The situation is classified as “semihard” \[9–12\]. It is believed that at assymptotically large energies (or small $x \sim \mu^2/s$) the theoretically correct description is given by the Balitsky-Fadin-Kuraev-Lipatov (BFKL) evolution equation \[13\] because here large terms proportional to $\ln(1/x)$ are taken into account. Just as for DGLAP, in this way it is possible to factorize an observable into a convolution of process-dependent hard matrix elements with universal gluon distributions. But as the virtualities (and transverse momenta) of the propagating gluons are no longer ordered, the matrix elements have to be taken off-shell and the convolution made also over transverse momentum ${\mathbf k}_T$ with the unintegrated ($k_T$-dependent) gluon distribution ${\cal F}(x,{\mathbf k}_T^2)$. The unintegrated gluon distribution ${\cal F}(x,{\mathbf k}_T^2)$ determines the probability to find a gluon carrying the longitudinal momentum fraction $x$ and the transverse momentum ${\mathbf k}_T$. This generalized factorization is called $k_T$-factorization \[10, 11\]. The unintegrated gluon distribution is a subject of intensive studies \[14\]. Various approaches to investigate this quantity have been proposed. One such approach, valid for both small and large $x$, have been developed by Ciafaloni, Catani, Fiorani and Marchesini, and is known as the CCFM model \[15\]. It introduces angular ordering of emissions to correctly treat gluon coherence effects. In the limit of asymptotic energies, it almost equivalent to BFKL \[16–18\], but also similar to the DGLAP evolution for large $x$ and high $\mu^2$. The resulting unintegrated gluon distribution depends on two scales, the additional scale ${\bar q}^2$ is a variable related to the maximum angle allowed in the emission and plays the role of the evolution scale $\mu^2$ in the collinear parton densities. We will use the following classification scheme \[14\]: ${\cal F}(x,{\mathbf k}_T^2)$ is used for pure BFKL-type unintegrated gluon distributions and ${\cal A}(x,{\mathbf k}_T^2,\mu^2)$ stands for any other type having two scale involved. The CCFM evolution equation formulated for the proton has been solved numerically or analitically by different ways \[19–21\]. As it was shown \[22–25\], unintegrated gluon distribution in the photon ${\cal A}_{\gamma}(x,{\mathbf k}_T^2,\mu^2)$ can be constructed by the similar method as in the proton[^1]. But situation is a bit different due to the pointlike component which reflects the splitting of the photon into a quark-antiquark pair. Also in the photon there are no sum rules equivalent to those in the proton case that constrain the quark distributions. However, this difference is not significant because CCFM equation contains only gluon splitting $g \to gg$. For the first time the unintegrated gluon density in the photon was obtained \[22\] using a simplified solution of the CCFM equation in single loop approximation, when small-$x$ effects can be neglected. It means that the CCFM evolution is reduced to the DGLAP one with the difference that the single loop evolution takes the gluon transverse momentum $k_T$ into account. Another simplified solution of the CCFM equation for a photon was proposed \[23\] using the Kimber-Martin-Ryskin (KMR) prescription \[20\]. In this way the $\mu^2$-dependence in the unintegrated gluon distribution enters only in last step of the evolution, and one-scale evolution equations can be used up to the last step. Both these methods give the similar results \[23\]. The phenomenological unintegrated gluon density, based on the Golec-Biernat and Wusthoff (GBW) saturation model \[24\] (extended to the large-$x$ region), was proposed also \[23\]. Very recently the unintegrated gluon distribution in the photon ${\cal A}_{\gamma}(x,{\mathbf k}_T^2,\mu^2)$ was obtained \[25\] using the full CCFM evolution equation for the first time. It was shown that the full CCFM evolved effective (integrated over ${\mathbf k}_T^2$) distribution is much higher than the usual DGLAP-based gluon density at $x < 10^{-1}$ region. In the previous studies \[23, 25\] the unintegrated gluon distributions in a photon (obtained from the single loop as well as full CCFM evolution equation) were applied to the calculation of the open charm and bottom production at LEP2. It was found that total cross section of the charm production is consistent with experimental data. In contrast, the theoretical predictions of the bottom total cross section underestimate data by factor 2 or 3. But we note that all these calculations reveal to the total cross sections of the open charm and bottom production only. In this paper we will study heavy flavored production at LEP2 more detail using the full CCFM-evolved unintegrated gluon density \[25\]. We will investigate the total and differential heavy quark cross section (namely pseudo-rapidity and transverse momenta distributions of the $D^*$-meson) and compare our theoretical results with the recent experimental data taken by the L3, OPAL and ALEPH collaborations at LEP2 \[1–5\]. Also we will study here the very intriguing problem connected with the inclusive $J/\psi$ meson production at high energies. It is traces back to the early 1990s, when the CDF data on the $J/\psi$ and $\Upsilon$ hadroproduction cross section revealed a more than order of magnitude discrepancy with theoretical expectations. This fact has resulted to intensive theoretical investigations of such processes. In the so-called non-relativistic QCD (NRQCD) \[28\] there are additional (octet) transition mechanism from $c\bar c$ pair to the $J/\psi$ mesons, where $c\bar c$ pair is produced in the color octet (CO) state and transforms into final color singlet state (CS) by help soft gluon radiation. The CO model describes well the heavy quarkonium production at Tevatron \[29\], although there are also some indications that it does not work well. For example, contributions from the octet mechanism to the $J/\psi$ photo- and leptoproduction processes at HERA are not well reproduce \[30\] experimental data. Also NRQCD is not predict $J/\psi$ polarization properties at HERA and Tevatron \[30, 31\]. At the same time usual CS model supplemented with $k_T$-factorization formalism gives fully correct description of the inclusive $J/\psi$ production at HERA \[32\] and Tevatron \[33\] including spin alignement of the quarkonium. The CO contributions within the $k_T$-factorization approach also will not contradict experimental data if parameters of the non-perturbative matrix elements will be reduced \[33, 34\]. But this fact changes hierarchy of these matrix elements which are obtained within the NRQCD. Recently DELPHI collaboration has presented experimental data \[35\] on the inclusive $J/\psi$ production in $\gamma \gamma$ collisions at LEP2, which wait to be confronted with different theoretical predictions. The theoretical calculations \[36\] within the NRQCD formalism agree well with the DELPHI data. In contrast, the collinear DGLAP-based leading order perturbative QCD calculations in the CS model significantly (by order of magnitude) underestimate \[36\] the data. The aim of this paper, in particular, is to investigate whether the inclusive $J/\psi$ production at LEP2 can be explained in the traditional CS model by using $k_T$-factorization and CCFM-based unintegrated gluon density in a photon. The outline of this paper is following. In Section 2 we present the basic formalism of the $k_T$-factorization approach with a brief review of calculation steps. In Section 3 we present the numerical results of our calculations. Finally, in Section 4, we give some conclusion. The compact analytic expressions for the off-shell matrix elements of the all subprocesses under consideration are given in Appendix. These formulas may be useful for the subsequent applications. Cross sections for heavy flavour production =========================================== In $\gamma \gamma$ collisions heavy quark and quarkonium can be produced by one of the three mechanisms: a direct production, a single-resolved and a double-resolved production processes. The direct contribution is governed by simple QED amplitudes which is independent on the gluon density in the photon. The double-resolved process gives a much smaller contribution than the direct and single-resolved processes \[7\] and will not taken into account in this analysis. Let $p_{\gamma}^{(1)}$ and $p_{\gamma}^{(2)}$ be the momenta of the incoming photons and $p_1$ and $p_2$ the momenta of the produced quarks. The single-resolved contribution to the $\gamma \gamma \to Q\bar Q$ process in the $k_T$-factorization approach has the following form: $$d\sigma_{\rm 1-res}(\gamma \gamma \to Q\bar Q) = \int {dx\over x} {\cal A}_{\gamma}(x,{\mathbf k}_T^2,\mu^2) d{\mathbf k}_T^2 {d\phi \over 2\pi} d \hat \sigma(\gamma g^* \to Q\bar Q), \eqno (1)$$ where $\hat \sigma(\gamma g^* \to Q\bar Q)$ is the heavy quark production cross section via off-shell gluon having fraction $x$ of a photon longitudinal momentum, non-zero transverse momentum ${\mathbf k}_T$ (${\mathbf k}_T^2 = - k_T^2 \neq 0$) and azimuthal angle $\phi$. The expression (1) can be easily rewritten as $${ d\sigma_{\rm 1-res} (\gamma \gamma \to Q\bar Q) \over dy_2 d{\mathbf p}_{2T}^2 } = \int {1\over 16\pi (x s)^2 (1 - \alpha_2)} {\cal A}_{\gamma}(x,{\mathbf k}_T^2,\mu^2) |\bar {\cal M}|^2(\gamma g^* \to Q\bar Q) d{\mathbf k}_T^2 {d\phi \over 2\pi} {d\phi_2 \over 2\pi}, \eqno (2)$$ where $|\bar {\cal M}|^2(\gamma g^* \to Q\bar Q)$ is the off-shell matrix element, $s = (p_{\gamma}^{(1)} + p_{\gamma}^{(2)})^2$ is the total c.m. frame energy, $y_2$ and $\phi_2$ are the rapidity and azimuthal angle of the produced heavy quark having mass $m_Q$, $\alpha_2 = m_{2T}\exp(y_2)/\sqrt s$ and $m_{2T}^2 = m_Q^2 + {\mathbf p}_{2T}^2$. To calculate the single-resolved contribution to the inclusive $J/\psi$ production the same formula (2) can be used where off-shell maxtrix element $|\bar {\cal M}|^2(\gamma g^* \to Q\bar Q)$ should be replaced by one which corresponds to the $\gamma g^* \to J/\psi + g$ production process. The available experimental data \[1–5, 35\] refer to heavy quark or quarkonium production in the $e^+ e^-$ collisions also. In order to obtain these total cross sections, the $\gamma \gamma$ cross sections need to be weighted with the photon flux in the electron: $$d\sigma(e^+ e^- \to e^+ e^- Q\bar Q + X) = \int f_{\gamma/e}(x_1)dx_1 \int f_{\gamma/e}(x_2)dx_2\,d\sigma(\gamma \gamma \to Q\bar Q), \eqno (3)$$ where we use the Weizacker-Williams approximation for the bremsstrahlung photon distribution from an electron: $$f_{\gamma/e}(x) = {\alpha \over 2\pi}\left({1 + (1 - x)^2\over x}\ln{Q^2_{\rm max}\over Q^2_{\rm min}} + 2m_e^2 x\left({1\over Q^2_{\rm max}} - {1\over Q^2_{\rm min}} \right)\right), \eqno (4)$$ with $Q^2_{\rm min} = m_e^2x^2/(1 - x)^2$ and $Q^2_{\rm max} = Q^2_{\rm min} + (E\theta)^2(1 - x)$. Here $x = E_{\gamma}/E_e$, $E = E_e = {\sqrt s}/2$, $s$ is the total energy in the $e^+ e^-$ collision and $\theta \sim 30$ mrad is the angular cut that ensures the photon is quasi-real. The multidimensional integration in (2) and (3) has been performed by means of the Monte Carlo technique, using the routine VEGAS \[37\]. The full C$++$ code is available from the authors on request[^2]. For reader’s convenience, we collect the analytic expressions for the off-shell matrix elements $|\bar {\cal M}|^2(\gamma g^* \to Q\bar Q)$ and $|\bar {\cal M}|^2(\gamma g^* \to J/\psi g)$ in Appendix, including, in particular, the relevant formulas for helicity zero $J/\psi$ production state. Numerical results ================= First of all, we can perform integration of the unintegrated gluon distribution ${\cal A}_{\gamma}(x,k_T^2,\mu^2)$ \[25\] over gluon transverse momenta ${\mathbf k}_T^2$ to obtain the effective gluon density in the photon: $$xg_{\gamma}(x,\mu^2) \sim \int\limits_0^{\mu^2} {\cal A}_{\gamma}(x,{\mathbf k}_T^2,\mu^2) d{\mathbf k}_T^2. \eqno(5)$$ The effective density $xg_{\gamma}(x,\mu^2)$ can be compared with the experimental data \[38, 39\] taken by H1 collaboration at HERA. As seen in Figures 1 and 2, this gluon distribution agrees well to the existing data extracted from the hard dijet (mean $p_T^2 = 38\,{\rm GeV}^2$ \[38\] and $p_T^2 = 74\,{\rm GeV}^2$ \[39\]). In contrast, KMR construction of unintegrated gluon density in the photon tends to underestimate the HERA data at $x < 10^{-1}$ \[23\]. Being sure that the full CCFM-evolved unintegrated gluon density ${\cal A}_{\gamma}(x,k_T^2,\mu^2)$ reproduces well the experimental data for the $xg_{\gamma}(x,\mu^2)$, we now are in a position to present our numerical results. We describe first our theoretical input and the kinematical conditions. The cross sections for heavy quark and quarkonium production depend on the heavy quark mass and the energy scale $\mu^2$. Since there are no free quarks due to confinement effect, their masses cannot be measured directly and should be defined from hadron properties. In our analysis we have examined the following choice: $m_c = 1.4 \pm 0.1$ GeV for charm and $m_b = 4.75 \pm 0.25$ GeV for bottom quark masses. Such variation of the quark masses gives the largest uncertainties in comparison with scale variation[^3] and therefore can be used as an estimate of the total theoretical uncertainties. Then, we will apply standard expression $\mu^2 = m_Q^2 + {\mathbf p}_{T}^2$ for both renormalization and factorization scales. Here ${\mathbf p}_{T}$ is the transverse momentum of the heavy quark in the center-of-mass frame. We use LO formula for the strong coupling constant $\alpha_s(\mu^2)$ with $n_f = 4$ active quark flavours and $\Lambda_{\rm QCD} = 200$ MeV, such that $\alpha_s(M_Z^2) = 0.1232$. But $\Lambda_{\rm QCD} = 340$ MeV \[7\] was tested also. Figure 3 confronts the total cross section $\sigma(\gamma \gamma \to c\bar c + X)$ calculated as a function of the photon-photon total energy $W_{\gamma \gamma}$ with experimental data \[2\] taken by the L3 collaboration in the interval $5 < W_{\gamma \gamma} < 70$ GeV. Solid line represent the calculations with the charm mass $m_c = 1.4$ GeV, whereas upper and lower dashed lines correspond to the $m_c = 1.3$ GeV and $m_c = 1.5$ GeV respectively. It is clear that $k_T$-factorization reproduces well both the energy dependence and the normalization. One can see that sensitivity of the results to the variations of the charm mass is rather large: shifting the mass down to $m_c = 1.3$ GeV changes the estimated cross section by $15 - 20$% at $W_{\gamma \gamma} \sim 60$ GeV. But in general all three curves lie within the experimental uncertainties. Experimental data for the total cross section $\sigma(e^+ e^- \to e^+ e^- c\bar c + X)$ come from the three LEP collaborations L3 \[1, 3\], OPAL \[4\] and ALEPH \[5\]. In Figure 4 we show our predictions in comparison with data. All curves here are the same as in Figure 3. One can see that our calculations describe the experimental data well again. The variation of the quark masses $1.3 < m_c < 1.5$ GeV gives the theoretical uncertainties approximately $15$% in absolute normalization. The available experimental data were obtained for the $D^{*}$ meson production also. Two differential cross section are determined: the first one as a function of the transverse $D^*$ momentum $p_T$, and the second as a function of pseudo-rapidity $|\eta|$. In our calculation we convert charmed quark into $D^*$ meson using the Peterson fragmentation function \[41\]. The default set for the fragmentation parameter $\epsilon_c$ and the fraction $f(c \to D^*)$ is $\epsilon_c = 0.06$ and $f(c \to D^*) = 0.26$. Other values for the parameter $\epsilon_c$ in the NLO perturbative QCD calculations are often used also, namely $\epsilon_c = 0.116$ \[42\] in the massless scheme and $\epsilon_c = 0.031$ \[43\] in the massive one. In the case of the massless calculation, this parameter was determined via a NLO fit to LEP1 data on $D^*$ production in $e^+ e^-$ annihilation measured by OPAL collaboration \[44\]. To investigate the sensitivity of the our numerical results to $\epsilon_c$ parameter we have repeat our calculations using $\epsilon_c = 0.031$. The recent L3 data \[3\] refer to the kinematic region defined by $1 < p_T < 12$ GeV and $|\eta| < 1.4$ with averaged total $e^+ e^-$ energy $\sqrt s = 193$ GeV ($183 < \sqrt s < 209$ GeV). The OPAL data \[4\] were obtained in the region $2 < p_T < 12$ GeV, $|n| < 1.5$ and averaged over $183 < \sqrt s < 189$ GeV. The more recent ALEPH data \[5\] refer to the same kinematic region but averaged over $183 < \sqrt s < 209$ GeV. We compare these three data sets with our calculation at $\sqrt s = 193$ GeV. The different values of $\sqrt s$ are not expected to change the cross section more than the corresponding experimental errors. We have checked directly that shifting $\sqrt s$ from 189 to 193 GeV increase the calculated cross sections by about one percent only. The transverse momenta distributions of the $D^*$ meson for different pseudo-rapidity region in comparison to experimental data shown in Figure 5 and 6. The solid and both dashed curves here are the same as in Figure 3 (calculated with the default value $\epsilon_c = 0.06$), dash-dotted curves represent results obtained using $\epsilon_c = 0.031$ and $m_c = 1.4$ GeV. The overall agreement between the our predictions and the data is good although the ALEPH data points in medium and large $p_T$ range lie slightly above the theoretical curves. Shifting the $\epsilon_c$ default value down to $\epsilon_c = 0.031$ results to a bit broadening of the $p_T$ spectra, which is insufficient to describe the data. The effects come from changing of the charm mass present only at low $p_T$: the predicted cross section $d\sigma/dp_T$ with $m_c = 1.3$ GeV is $10 - 15$% above the one calculated with $m_c = 1.4$ GeV at $p_T \sim 1$ GeV, whereas solid and both dashed theoretical curves practically coincide at medium and large $p_T$. The similar effect was found in the NLO perturbative calculation \[45\], where the difference between the massive and massless approach arises at low $p_T$ only. The $D^*$ pseudo-rapidity distributions compared with the experimental data in different $p_T$ range are shown in Figure 7 and 8. All curves here are the same as in Figure 5. Our calculations agree well with measured differential cross sections but slightly underestimate the OPAL data. However, setting $\epsilon_c = 0.031$ increases the absolute normalization of the pseudo-rapidity distribution by approximately $10$%, and agreement with OPAL data becomes better. Again, one can see that the significant mass dependence has place at low transverse momenta only: the difference between the theoretical curves calculated at $2 < p_T < 12$ GeV and plotted in Figure 8 is much smaller than difference between the results presented in Figure 7 obtained at $1 < p_T < 12$ GeV. We conclude from Figs. 3 — 8, that our calculations agree well with charm data at LEP2. In contrast with charm case, the open bottom production in $\gamma \gamma$ collisions is clear underestimated by $k_T$-factorization. Figure 9 shows our prediction for the open bottom cross section compared to the L3 \[1\] and OPAL \[4\] experimental data. Using low but still reasonable $b$-quark mass $m_b = 4.5$ GeV, we obtain $\sigma(e^+ e^- \to e^+ e^- b\bar b + X) = 2.94$ pb at $\sqrt s = 200$ GeV. The very similar value $\sigma = 2.7$ pb was obtained \[23\] within the GBW saturation model adopted for the photon. The Monte Carlo generator CASCADE predicts $\sigma = 4.9$ pb \[25\] where normalization factor $n = 1.7$ has been applied for the resolved contributions. The calculation \[23\] based on the KMR prescription for unintegrated gluon density gives a lower cross section $\sigma = 1.9$ pb. At the same time the prediction of the massive NLO perturbative QCD calculation \[43\] is 3.88 pb for $m_b = 4.5$ GeV and 2.34 pb for $m_b = 5.2$ GeV respectively. All these results are significantly (about factor 2 or 3) lower than experimental data. Such disagreement between theory and data for bottom production at LEP2 is surprising and needs an explanation. It is known that the similar difference between theory and data was claimed for inclusive bottom hadroproduction at Tevatron. Recent analysis indicates that the overall description of the these data can be improved \[46\] by adopting the non-perturbative fragmentation function of the $b$-quark into $B$ meson: an appropriate treatment of the $b$-quark fragmentation properties considerably reduces the disagreement between measured bottom cross section and the corresponding NLO calculations. It would be interesting to find out whether the similar explanation is also true for the L3 and OPAL experimental data. After we have studied open heavy quark production at LEP2, we will investigate production of the heavy quarkonum in $\gamma \gamma$ interactions. As it was already mentioned above, non-relativistic QCD gives a good description \[36\] of the recent DELPHI data \[35\] on inclusive $J/\psi$ production at LEP2. We will examine whether the DELPHI data can be expained within the CS model using $k_T$-factorization approach and CCFM-based unintegrated gluon density in a photon. Again, only direct and single-resolved contributions are taken into account. Now we change the default set of parameters which were used in the case of open charm calculations. Since we neglect the relative momentum of the $c$-quarks (which form a $J/\psi$ meson) the charm mass should be taken $m_c = m_{\psi}/2$. Therefore as default choice in following we will use $m_c = 1.55$ GeV. On the other hand there are many examples when smaller value $m_c = 1.4$ GeV in the calculation of $J/\psi$ production is used \[30, 32, 36\]. In our analysis we will apply this value as extremal choice to investigate the theoretical uncertainties of the calculations. The DELPHI data \[35\] refer to the kinematic region defined by $-2 < y_{\psi} < 2$ with total $e^+ e^-$ energy $\sqrt s = 197$ GeV, where $y_{\psi}$ is the $J/\psi$ rapidity. However these data were obtained starting from very low transverse momenta $p_{\psi T}^2 \sim 0.2\,{\rm GeV}^2$. We note that $k_T$-factorization as well as usual collinear factorization theorem does not work well for such $p_{\psi T}$ values, and our calculations should be compared with experimental data at approximately $p_{\psi T} > 1$ GeV only. In Figure 10 we confront our theoretical predictions with the measured differential cross section $d\sigma/dp_{\psi T}^2$. Solid line corresponds the default set of parameters and lies below the data by a factor about 2 or 3. This dicrepancy is not catastrofic, because some reasonable variations in $m_c$ and $\Lambda_{\rm QCD}$, namely $200 < \Lambda_{\rm QCD} < 340$ MeV and $1.4 < m_c < 1.55$ GeV, change the estimated cross section by a factor of 3 (dashed line in Figure 9). Thus the visible disagreement is eliminated. However, we do not interpret this as a strong indication of consistency between data and theory, but rather as a concequence of a wide uncertainty band. Better future experimental studies are crucial to determine whether the results of our calculations contradict DELPHI data points. The main difference between $k_T$-factorization and other approaches connects with polarization properties of the final particles because the initial off-shell gluons do promptly manifest in the $J/\psi$ spin alignement \[32–34\]. Only a very small fraction of $J/\psi$ mesons can be produced in the helicity zero state (longitudinal polarization) by massless bosons. This property is totally determined by the subprocess matrix element structure. The degree of spin alignement can be measured experimentally since the different polarization states of $J/\psi$ result in significantly different angular distributions of the $J/\psi \to \mu^+ \mu^-$ decay leptons: $${d\Gamma(J/\psi \to \mu^+ \mu^-) \over d \cos\theta} \sim 1 + \alpha \cos^2\theta, \eqno (6)$$ where $\alpha = 1$ for transverse and $\alpha = - 1$ for longitudinal polarizations, respectively. Here $\theta$ is the angle between the lepton and $J/\psi$ directions, measured in $J/\psi$ meson rest frame. We calculate the $p_T^2$-dependence of the spin alignement parameter as $$\alpha(p_{\psi T}^2) = {1 - 3 \xi(p_{\psi T}^2) \over 1 + \xi(p_{\psi T}^2)}, \eqno (7)$$ with $\xi(p_{\psi T}^2)$ being the fraction of longitudinally polarized $J/\psi$ mesons. The results of our calculations are shown in Figure 11. Solid line represents the $k_T$-factorization predictions and dashed line corresponds to the collinear leading-order pQCD ones with GRV (LO) gluon density \[47\] in a photon. One can see that fraction of longitudinally polarized mesons increases with $p_{\psi T}^2$ within $k_T$-factorization approach. This fact is in clear contrast with usual collinear parton model result. The $k_T$-factorization calculations made for the inelastic $J/\psi$ production at HERA \[32, 34\] and Tevatron \[33\] posses the same behavior of the $\alpha$ parameter, whereas collinear parton model and NRQCD predict the strong transverse polarization at moderate and large $p_{\psi T}^2$ range. We point out that our predictions for the $J/\psi$ polarization are stable with respect to variation of the model parameters, such as charmed quark mass and factorization scale. In fact, there is no dependence on the strong coupling constant which is cancels out. At the same time the DELPHI fit \[35\] gives $\alpha = 0.7 \pm 1.3$ for $p_{\psi T}^2 > 1\,{\rm GeV^2}$, which has, however, huge experimental uncertainties. Since account of the octet contributions does not change predictions of the $k_T$-factorization approach for the spin alignement parameter $\alpha$ \[33\], the future extensive experimental study of such processes will be direct probe of the gluon virtuality. Conclusions =========== We have investigated a heavy flavour production in photon-photon interactions at high energies within the framework of $k_T$-factorization, using unintegrated gluon distribution obtained from the full CCFM evolution equation for a photon. We calculate total and differential cross sections of the open charm and bottom production including $D^*$ meson transverse momenta and pseudo-rapidity distributions. Also we have studied inclusive $J/\psi$ production at LEP2 using color-singlet model supplemented with $k_T$-factorization. We take into account both the direct and single-resolved contribution and investigate the sensitivity of the our results to the different parameters, such as heavy quark mass, charm fragmentation and $\Lambda_{\rm QCD}$ parameter. There are, of course, also some uncertainties due to the renormalization and factorization scales. However, these effects would not to be large enough to change the conclusions presented here, and were not taken into account in our analisys. The results of calculations with default parameter set agree well with open charm production data taken by the L3, OPAL and ALEPH collaborations at LEP2. In contrast, bottom production cross section is clearly underestimated by a factor about 3. A potential explanation of this fact may be, perhaps, connected with the more accurate treatment of the $b$-quark fragmentation function. Our prediction for the inclusive $J/\psi$ production slightly underestimate the DELPHI data. However, a strong inconsistensy cannot be claimed, because of large experimental errors and theoretical uncertainties. Therefore more precise future experiments, espesially polarized quarkonium production, are necessary to know whether our predictions contradict the experimental data. In conclusion, we point out that at CERN LEP2 collider (as well as at HERA and Tevatron) the difference between predictions of the collinear and $k_T$-factorization approaches is clearly visible in polarized heavy quarkonium production. It comes directly from initial gluon off-shellness. The experimental study of such processes should be additional test of non-collinear parton evolution. Acknowledgements ================ We tnank Hannes Jung for possibility to use the CCFM code for unintegrated gluon distribution in a photon in our calculations. The authors are very grateful also to Sergei Baranov for encouraging interest and helpful discussions. One of us (A.L.) was supported by the INTAS grant YSF’2002 N 399 and by the “Dynasty” fundation. Appendix ======== Here we present compact analytic expressions for the off-shell matrix elements which appear in (2). In the following, $\hat s$, $\hat t$, $\hat u$ are usual Mandelstam variables for $2 \to 2$ process and $e_Q$ is the fractional electric charge of heavy quark $Q$. We start from photon-gluon fusion $\gamma (q) g^* (k)\to Q (p_1) \bar Q (p_2)$ subprocess, where initial off-shell gluon has non-zero virtuality ${\mathbf k}_T^2$. The corresponding squared matrix element summed over final polarization states and averaged over initial ones read $$|\bar {\cal M}|^2(\gamma g^* \to Q\bar Q) = - {\displaystyle (4\pi)^2 e_Q^2 \alpha \alpha_s \over \displaystyle (\hat t - m^2)^2 (\hat u - m^2)^2 } F_{Q\bar Q}({\mathbf k}_T^2), \eqno (A.1)$$ where $m$ is the heavy quark mass, and $$F_{Q\bar Q}({\mathbf k}_T^2) = 6m^8 - (2{\mathbf k}_T^4 + 2(\hat t + \hat u){\mathbf k}_T^2 + 3{\hat t}^2 + 3{\hat u}^2 + 14 \hat t\hat u)m^4 +$$ $$(2(\hat t + \hat u){\mathbf k}_T^4 + 8\hat t \hat u{\mathbf k}_T^2 + {\hat t}^3 + {\hat u}^3 + 7 \hat t {\hat u}^2 + 7{\hat t}^2\hat u)m^2 -$$ $$\hat t\hat u(2{\mathbf k}_T^4 + 2(\hat t + \hat u){\mathbf k}_T^2 + {\hat t}^2 + {\hat u}^2). \eqno (A.2)$$ We note that matrix element of the direct contribution $\gamma \gamma \to Q\bar Q$ may be easily obtained from (A.1) and (A.2) in the limit ${\mathbf k}_T^2 \to 0$, if we replace normalization factor $(4\pi)^2 e_Q^2 \alpha \alpha_s$ by the $(4\pi)^2 \alpha^2 e_Q^4 N_c$ (where $N_c$ is the number of colors) and average (A.2) over transverse momentum vector ${\mathbf k}_T$. Now we are in a position to present our formulas for the $\gamma (q) g^* (k) \to J/\psi (p_\psi) g (p_g) $ subprocess. In the color-singlet model the production of $J/\psi$ meson is considered as production of a quark-antiquark system in the color-singlet state with orbital momentum $L = 1$ and spin momentum $S = 1$. The squared off-mass shell matrix element summed over final polarization states and averaged over initial ones can be written as $$|\bar {\cal M}|^2(\gamma g^* \to J/\psi g) = - {64 e_Q^2 (4\pi)^3 \alpha \alpha_s^2 |\psi(0)|^2 \over 3 m_{\psi} (\hat t - m_\psi^2)^2 (\hat u - m_\psi^2 - {\mathbf k}_T^2)^2 (\hat t + \hat u + {\mathbf k}_T^2)^2 {\mathbf k}_T^2 } F_{\psi}({\mathbf k}_T^2), \eqno (A.3)$$ where $|\psi(0)|^2 = 0.0876\,{\rm GeV}^3$ is the $J/\psi$ wave function at the origin, $m_\psi$ is the $J/\psi$ meson mass, ${\mathbf k}_T^2$ is the virtuality of the initial gluon, and function $F_{\psi}({\mathbf k}_T^2)$ is given by $$F_{\psi}({\mathbf k}_T^2) = {\mathbf k}_T^2({\mathbf k}_T^6(m_\psi^2 - \hat t)(m_\psi^2 - \hat t - \hat u) - m_\psi^2 ({\hat t}^2 + \hat t \hat u + {\hat u}^2 -$$ $$m_\psi^2(\hat t + \hat u))^2 + {\mathbf k}_T^4(3m_\psi^6 + \hat t\hat u(\hat t + \hat u) - 3m_\psi^4(2\hat t + \hat u) + m_\psi^2(3{\hat t}^2 + 2\hat t\hat u - {\hat u}^2)) +$$ $${\mathbf k}_T^2(2m_\psi^8 + m_\psi^4\hat t(\hat t - \hat u) - {\hat t}^2 (\hat t + \hat u)^2 - 2m_\psi^6(2\hat t + \hat u) + m_\psi^2\hat t (2{\hat t}^2 + 5\hat t\hat u + 5{\hat u}^2))) +$$ $$2{\mathbf k}_T^2({\mathbf k}_T^4(m_\psi^2 - \hat t)(m_\psi^2 - \hat t - u) + {\mathbf k}_T^2(3m_\psi^6 + \hat t(\hat t + \hat u)^2 - m_\psi^4(5\hat t + 3\hat u) +$$ $$m_\psi^2({\hat t}^2 + \hat t\hat u - 2{\hat u}^2)) + m_\psi^2(2m_\psi^6 - m_\psi^4(3\hat t + 2\hat u) +$$ $$\hat t({\hat t}^2 + 2\hat t\hat u + 3{\hat u}^2)))( - |{\mathbf p}_{\psi T}||{\mathbf k}_T|\cos\phi_2) + 2m_\psi^2(m_\psi^2(-m_\psi^2 + \hat t + \hat u)^2 +$$ $${\mathbf k}_T^2(m_\psi^4 + {\hat t}^2 + 2\hat t\hat u - {\hat u}^2 - 2m_\psi^2(\hat t + \hat u))){\mathbf p}_{\psi T}^2{\mathbf k}_T^2\cos^2\phi_2. \eqno (A.4)$$ Here ${\mathbf p}_{\psi T}$ is the $J/\psi$ transverse momentum, $\phi_2$ is the azimutal angle of the incoming virtual gluon having virtuality ${\mathbf k}_T^2$. To study polarized $J/\psi$ production we introduce the four-vector of the longitudinal polarization $\epsilon_{\psi, L}^{\mu}$. In the frame where the $z$ axis is oriented along the quarkonium momentum vector, $p_\psi^\mu = (E_\psi,0,0,|{\mathbf p}_\psi|)$, this polarization vector is $\epsilon_{\psi, L}^\mu = (|{\mathbf p}_\psi|,0,0,E_\psi)/m_\psi$. The squared off-shell matrix element read $$|\bar {\cal M}|_L^2(\gamma g^* \to J/\psi g) = {32 e_Q^2 (4\pi)^3 \alpha \alpha_s^2 |\psi(0)|^2 m_{\psi} \over 3 (\hat t - m_\psi^2)^2 (\hat u - m_\psi^2 - {\mathbf k}_T^2)^2 (\hat t + \hat u + {\mathbf k}_T^2)^2 {\mathbf k}_T^2 } F_{\psi, L}({\mathbf k}_T^2), \eqno (A.5)$$ where function $F_{\psi, L}({\mathbf k}_T^2)$ is defined as $$F_{\psi, L}({\mathbf k}_T^2) = -4{\Delta}_2{\Delta}_3{\mathbf k}_T^4 m_\psi^4 + 2 {\Delta}_3^2 {\mathbf k}_T^4 m_\psi^4 - 4 {\Delta}_2 {\Delta}_3 {\mathbf k}_T^2 m_\psi^6 +$$ $$8 {\Delta}_2 {\Delta}_3 {\mathbf k}_T^4 m_\psi^2\hat t - 4 {\Delta}_3^2 {\mathbf k}_T^4 m_\psi^2 \hat t + 8 {\Delta}_2 {\Delta}_3 {\mathbf k}_T^2 m_\psi^4 \hat t + 4 {\Delta}_3^2 {\mathbf k}_T^2 m_\psi^4 \hat t -$$ $$4 {\Delta}_2 {\Delta}_3 {\mathbf k}_T^4 {\hat t}^2 + 2 {\Delta}_3^2 {\mathbf k}_T^4 {\hat t}^2 - 4 {\Delta}_2 {\Delta}_3 {\mathbf k}_T^2 m_\psi^2 {\hat t}^2 - 8 {\Delta}_3^2 {\mathbf k}_T^2 m_\psi^2 {\hat t}^2 + 2 {\Delta}_3^2 m_\psi^4 {\hat t}^2 +$$ $$4 {\Delta}_3^2 {\mathbf k}_T^2 {\hat t}^3 - 4 {\Delta}_3^2 m_\psi^2 {\hat t}^3 + 2 {\Delta}_3^2 {\hat t}^4 + 4 {\Delta}_2^2 {\mathbf k}_T^4 m_\psi^2 \hat u -$$ $$4 {\Delta}_2 {\Delta}_3 {\mathbf k}_T^4 m_\psi^2 \hat u + 2 {\Delta}_3^2 {\mathbf k}_T^4 m_\psi^2 \hat u + 4 {\Delta}_2^2 {\mathbf k}_T^2 m_\psi^4 \hat u + 2 {\Delta}_3^2 {\mathbf k}_T^2 m_\psi^4 \hat u -$$ $$4 {\Delta}_2^2 {\mathbf k}_T^4 \hat t \hat u + 4 {\Delta}_2 {\Delta}_3 {\mathbf k}_T^4 \hat t \hat u - 2 {\Delta}_3^2 {\mathbf k}_T^4 \hat t \hat u - 4 {\Delta}_2^2 {\mathbf k}_T^2 m_\psi^2\hat t\hat u - 4 {\Delta}_2 {\Delta}_3 {\mathbf k}_T^2 m_\psi^2\hat t\hat u -$$ $$4 {\Delta}_3^2 {\mathbf k}_T^2 m_\psi^2\hat t\hat u + 2 {\Delta}_3^2 m_\psi^4\hat t\hat u + 4 {\Delta}_2 {\Delta}_3 {\mathbf k}_T^2 {\hat t}^2\hat u + 2 {\Delta}_3^2 {\mathbf k}_T^2 {\hat t}^2\hat u - 6 {\Delta}_3^2 m_\psi^2 {\hat t}^2\hat u +$$ $$4 {\Delta}_3^2 {\hat t}^3\hat u + {\mathbf k}_T^6 {\hat u}^2 - 4 {\Delta}_2^2 {\mathbf k}_T^2 m_\psi^2 {\hat u}^2 + 4 {\Delta}_2 {\Delta}_3 {\mathbf k}_T^2 m_\psi^2 {\hat u}^2 - 2 {\Delta}_3^2 {\mathbf k}_T^2 m_\psi^2 {\hat u}^2 + {\mathbf k}_T^2 m_\psi^4 {\hat u}^2 +$$ $$4 {\Delta}_2 {\Delta}_3 {\mathbf k}_T^2\hat t {\hat u}^2 - 2 {\Delta}_3^2 {\mathbf k}_T^2\hat t {\hat u}^2 - 2 {\mathbf k}_T^4\hat t {\hat u}^2 - 2 {\Delta}_3^2 m_\psi^2\hat t {\hat u}^2 - 2 {\mathbf k}_T^2 m_\psi^2\hat t {\hat u}^2 +$$ $$2 {\Delta}_3^2 {\hat t}^2 {\hat u}^2 + 2 {\mathbf k}_T^2 {\hat t}^2 {\hat u}^2 - 2 {\mathbf k}_T^2 m_\psi^2 {\hat u}^3 + 2 {\mathbf k}_T^2\hat t {\hat u}^3 + {\mathbf k}_T^2 {\hat u}^4 + 8 {\Delta}_1^2 {\mathbf k}_T^4 (-m_\psi^2 + \hat t + \hat u)^2 +$$ $$4 {\Delta}_1 {\mathbf k}_T^2 ({\Delta}_2 ({\mathbf k}_T^4 (2 m_\psi^2 - 2\hat t - \hat u) + m_\psi^2 (m_\psi^2 - 2\hat t - \hat u)\hat u + {\mathbf k}_T^2 (2 m_\psi^4 + {\hat u}^2 -$$ $$2 m_\psi^2 (\hat t + \hat u))) + {\Delta}_3 (-m_\psi^6 + {\mathbf k}_T^4 (-m_\psi^2 + \hat t) + 2\hat t (\hat t + \hat u)^2 +$$ $$m_\psi^4 (3\hat t + \hat u) - m_\psi^2\hat t (4\hat t + 3\hat u) + {\mathbf k}_T^2 (-(m_\psi^2 (2\hat t + \hat u)) + \hat t (2\hat t + 3\hat u)))) +$$ $$4 (-({\Delta}_2 {\Delta}_3 ({\mathbf k}_T^2 + m_\psi^2) (m_\psi^2 - \hat t) (m_\psi^2 - \hat t - \hat u)) + {\mathbf k}_T^2 ({\mathbf k}_T^2 - \hat t) {\hat u}^2 +$$ $$4 {\Delta}_1^2 {\mathbf k}_T^2 (-m_\psi^2 + \hat t + \hat u)^2 + {\Delta}_1 (2 {\Delta}_2 {\mathbf k}_T^2 (2 m_\psi^4 + {\mathbf k}_T^2 (2 m_\psi^2 - 2\hat t - \hat u) -$$ $$2 m_\psi^2 (\hat t + \hat u) + \hat u (\hat t + \hat u)) - {\Delta}_3 (m_\psi^6 + {\mathbf k}_T^4 (m_\psi^2 - \hat t + \hat u) -$$ $$2\hat t (\hat t + \hat u)^2 - m_\psi^4 (3\hat t + 2\hat u) + m_\psi^2 (4 {\hat t}^2 + 5\hat t\hat u + {\hat u}^2) +$$ $${\mathbf k}_T^2 (-2 {\hat t}^2 - t\hat u + {\hat u}^2 + m_\psi^2 (2\hat t + \hat u))))) ( - |{\mathbf p}_{\psi T}||{\mathbf k}_T|\cos\phi_2) +$$ $$4 (2 {\Delta}_1 {\Delta}_2 ({\mathbf k}_T^2 + m_\psi^2) (m_\psi^2 - \hat t - \hat u) +$$ $${\mathbf k}_T^2 {\hat u}^2 + 2 {\Delta}_1^2 (-m_\psi^2 + \hat t + \hat u)^2) {\mathbf p}_{\psi T}^2{\mathbf k}_T^2\cos^2\phi_2, \eqno (A.6)$$ and the following notation has been used: $$\Delta_1 = (\alpha_1 + \alpha_2){\sqrt s\over 2m_\psi}\left(\sqrt{{\mathbf p}_{\psi T}^2 + s(\alpha_1 - \beta_1)^2/4} - {s(\alpha_1^2 - \beta_1^2)\over 4\sqrt{{\mathbf p}_{\psi T}^2 + s(\alpha_1 - \beta_1)^2/4}}\right),$$ $$\displaystyle \Delta_2 = (\beta_1 + \beta_2){\sqrt s\over 2m_\psi}\left(\sqrt{{\mathbf p}_{\psi T}^2 + s(\alpha_1 - \beta_1)^2/4} + {s(\alpha_1^2 - \beta_1^2)\over 4\sqrt{{\mathbf p}_{\psi T}^2 + s(\alpha_1 - \beta_1)^2/4}}\right) + \Delta_3,$$ $$\Delta_3 = - {\sqrt{s} (\alpha_1 + \beta_1) \over 2m_\psi \sqrt{{\mathbf p}_{\psi T}^2 + s(\alpha_1 - \beta_1)^2/4}} |{\mathbf p}_{\psi T}| |{\mathbf k}_{T}| \cos\phi_2,$$ $$\alpha_1 = \sqrt{ m_\psi^2 + {\mathbf p}_{\psi T}^2\over s} \exp (y_\psi), \quad \beta_1 = \sqrt{ m_\psi^2 + {\mathbf p}_{\psi T}^2\over s} \exp ( - y_\psi),$$ $$\alpha_2 = {|{\mathbf p}_{g T}| \over \sqrt s} \exp (y_g), \quad \beta_2 = {|{\mathbf p}_{g T}| \over \sqrt s} \exp (-y_g). \eqno(A.7)$$ [47]{} M. Acciari [*et al.*]{} (L3 Collaboration), Phys. 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[^2]: lipatov@theory.sinp.msu [^3]: It was shown \[25\] that Monte Carlo generator CASCADE \[19, 40\] predicts the very similar results for charm total cross section with both scales $\mu^2 = m_Q^2 + {\mathbf p}_{T}^2$ and $\mu^2 = 4 m_Q^2$.

--- abstract: | #### Abstract. We present a beautiful interplay between combinatorial topology and homological algebra for a class of monoids that arise naturally in algebraic combinatorics. We explore several applications of this interplay. For instance, we provide a new interpretation of the Leray number of a clique complex in terms of non-commutative algebra. #### Résumé. Nous présentons une magnifique interaction entre la topologie combinatoire et l’algèbre homologique d’une classe de monoïdes qui figurent naturellement dans la combinatoire algébrique. Nous explorons plusieurs applications de cette interaction. Par exemple, nous introduisons une nouvelle interprétation du nombre de Leray d’un complexe de clique en termes de la dimension globale d’une certaine algèbre non commutative. address: | Department of Mathematics, Bar Ilan University, Israel\ Laboratoire de Combinatoire et d’informatique Mathématique (LaCIM), UQAM, Canada\ City College of New York, USA author: - 'Stuart Margolis[^1]' - 'Franco Saliola[^2]' - 'Benjamin Steinberg[^3]' bibliography: - 'lrbfpsac-references.bib' title: Poset topology and homological invariants of algebras arising in algebraic combinatorics --- Introduction {#sec:in} ============ In a highly influential paper [@BHR], Bidigare, Hanlon and Rockmore showed that a number of popular Markov chains are random walks on the faces of a hyperplane arrangement. Their analysis of these Markov chains took advantage of the monoid structure on the set of faces. This theory was later extended by Brown [@Brown1; @Brown2] to a larger class of monoids called left regular bands. In both cases, the representation theory of these monoids play a prominent role. It is used to compute the spectrum of the transition operators of the Markov chains and to prove diagonalizability of the transition operators. It develops that there is a close connection between algebraic and combinatorial invariants of these monoids: certain homological invariants of the monoid algebra coincide with the cohomology of order complexes of posets naturally associated with a monoid. We present here a synopsis of the results together with applications of this beautiful interplay between combinatorial topology and homological algebra. The full details will appear in two longer papers [@oldpaper; @newpaper], which include lengthy discussions of the history, examples and applications of these monoids as well as thorough references to the literature. Left Regular Bands {#s:LRBs} ================== A particularly important class of monoids arising in probability theory and in algebraic combinatorics is the class of left regular bands [@Brown1; @Brown2; @DiaconisBrown1; @Saliola; @Saliolahyperplane; @bjorner2; @GrahamLRB]. \[DefLRB1\] A *left regular band* is a semigroup $B$ satisfying the identities $$\begin{gathered} \label{leftregularity} x^2=x\ \text{for all}\ x\in B,\\ xyx = xy\ \text{for all}\ x,y \in B.\end{gathered}$$ In general we will assume in this paper that $B$ is finite and has an identity (is a monoid). An example arises from the set $\{0, +, -\}$, where the monoid operation $\circ$ satisfies $x \circ y = x$ if $x\neq0$ and $x \circ y = y$ otherwise. Several important examples of left regular bands arising in combinatorics are submonoids $\{0,+,-\}^n$. These include the monoid structure on both real hyperplane arrangements, matroids, oriented matriods and interval greedoids (see Section \[s:LRBExamples\]). It is known that left regular bands are precisely the homomorphic images of submonoids of $\{0, +, -\}^n$, for some $n$. The paper [@qvariety] classifies those that are submonoids of some $\{0, +, -\}^n$. Almost all the left regular bands appearing so far in the algebraic combinatorics literature embed in $\{0,+,-\}^n$; the main exception are the complex hyperplane monoids discussed in § \[ss:complexhyperplanemonoids\] below. The following two posets associated with a left regular band play a central role in what follows. #### ${{\mathrel{\mathscr R}}}$-poset. For a monoid $M$ and $m \in M$, we let $mM = \{mm' : m' \in M\}$ denote the *principal right ideal* generated by $m$. *Green’s ${{\mathrel{\mathscr R}}}$-preorder* is defined on $M$ by $m\leq_{{\mathrel{\mathscr R}}}n$ if $mM\subseteq nM$. The associated equivalence relation is denoted ${{\mathrel{\mathscr R}}}$; it is one of *Green’s relations* on a monoid [@Green]. In a left regular band $B$, Green’s ${{\mathrel{\mathscr R}}}$-preorder admits the following description: $$\begin{gathered} a \leq_{{\mathrel{\mathscr R}}}b \text{~~iff~~} ba = a.\end{gathered}$$ Thus, if $a \leq_{{\mathrel{\mathscr R}}}b$ and $b \leq_{{\mathrel{\mathscr R}}}a$, then $a=aba=ab=b$. It follows that $B$ is a poset with respect to $\leq_{{\mathrel{\mathscr R}}}$. We call this partial order the *${{\mathrel{\mathscr R}}}$-order* on $B$ and denote it simply by $\leq$. Figures \[figure: free lrb\] and \[fig:3Lines\] illustrate the ${{\mathrel{\mathscr R}}}$-order on specific examples. #### Support Lattice. For a left regular band $B$, let $\Lambda(B)$ denote the set of principal left ideals of $B$: $$\begin{gathered} \Lambda(B) = \left\{ Ba : a \in B \right\}\end{gathered}$$ where $Ba = \{ba : b \in B\}$. Then $\Lambda(B)$ is a poset under inclusion. Furthermore, it is a lattice with intersection as the meet operation (denoted $\wedge$). The map $\sigma\colon B \xrightarrow{} \Lambda(B)$ given by $\sigma(a)=Ba$ satisfies the following properties for all $a,b \in B$: $$\begin{gathered} \sigma(a b) = \sigma(a) \wedge \sigma(b) \qquad\qquad\text{and}\qquad\qquad a b = a \text{~~iff~~}\sigma(b) \geq \sigma(a)\end{gathered}$$ The first property says that $\sigma$ is a monoid morphism. Note that $\sigma$ is an order preserving poset morphism: that is, if $a \leq_{{\mathrel{\mathscr R}}}b$, then $\sigma(a) \leq \sigma(b)$. Proofs of these statements can be found in [@Clifford; @Brown1]. Following the standard conventions of lattice theory, we denote by ${\widehat}1$ the top of $\Lambda(B)$ (which is $B$, itself) and by ${\widehat}0$ the bottom (called the *minimal ideal* of $B$). In [@Brown1; @Brown2], Brown calls $\Lambda(B)$ the *support lattice* of $B$ (although he uses the reverse ordering) and $\sigma$ is called the *support map*. Examples of Left Regular Bands {#s:LRBExamples} ============================== This section quickly surveys some examples of left regular bands found in the combinatorics literature. Often, these are special cases of certain semigroup-theoretic constructions. Further examples and detailed expositions can be found in [@oldpaper]. We also describe a new class of left regular bands introduced in [@oldpaper]. #### Free left regular bands. The *free left regular band* on a set $A$ will be denoted $F(A)$ and the free left regular band on $n$-generators will be written $F_n$. One can view $F(A)$ as the set of words over $A$ with no repeated letters; multiplication is given by concatenation followed by removal of repetitions (reading from left to right). For example, in $F_5$ we have $523 \cdot 13245 = 52314$. The support lattice of $F(A)$ can be identified with the power set $P(A)$ with the operation of union; the support map $\sigma$ takes a word to the set of letters appearing in the word; Figure \[figure: free lrb\] illustrates the ${{\mathrel{\mathscr R}}}$-order and the support lattice of $F(\{a,b,c\})$. $$\begin{gathered} \xymatrix@l@C=0pt@R=1em{ cba \ar@{-}[d] && cab \ar@{-}[d] && bca \ar@{-}[d] && bac \ar@{-}[d] & & acb \ar@{-}[d] && abc \ar@{-}[d] \\ cb \ar@{-}[dr] && ca \ar@{-}[dl] && bc \ar@{-}[dr] && ba \ar@{-}[dl] & & ac \ar@{-}[dr] && ab \ar@{-}[dl] \\ &c \ar@{-}[drrrr] & && &b\ar@{-}[d]& & & &a\ar@{-}[dllll]& \\ && && &1& } \hspace{3em} \xymatrix@l@C=1em@R=1em{ & \{a,b,c\} \ar@{-}[dl] \ar@{-}[rd] \ar@{-}[d] \\ \{b,c\} \ar@{-}[d]\ar@{-}[dr] & \{a,c\} \ar@{-}[dl]\ar@{-}[dr] & \{a,b\} \ar@{-}[d]\ar@{-}[dl] \\ \{c\} \ar@{-}[dr] & \{b\} \ar@{-}[d] & \{a\} \ar@{-}[dl] \\ & \emptyset }\end{gathered}$$ #### Hyperplane face monoids and oriented matroids Hyperplane arrangements, and more generally oriented matroids, provide an important source of examples of left regular bands, which turn out to be submonoids of the left regular band $\{0,+,-\}^n$ introduced in Section \[s:LRBs\]. We recall the construction and properties of these left regular bands referring the reader to [@Brown1 Appendix A] for details. A *central hyperplane arrangement* in $V = \mathbb R^n$ is a finite collection $\AAA$ of hyperplanes of $V$ passing through the origin. For each hyperplane $H \in \AAA$, fix a labelling $H^+$ and $H^-$ of the two open half spaces of $V$ determined by $H$; the choice of labels $H^+$ and $H^-$ is arbitrary, but fixed throughout. Let $H^0 = H$. The elements of the monoid are the *faces* of $\AAA$: the non-empty intersections of the form $ x = \bigcap_{H \in \AAA} H^{\epsilon_H} $ with $\epsilon_H \in \{0,+,-\}$. Thus, a face $x$ is completely determined by the sequence $\varepsilon(x) = (\epsilon_H)_{H\in\AAA}$. See Figure \[fig:3Lines\]. The image of $\varepsilon$ identifies the set of faces of $\AAA$ with a submonoid of $\{0,+,-\}^{\AAA}$ and so we obtain a monoid structure by defining the product of $x$ and $y$ to be the face with sign sequence $\varepsilon(x) \circ \varepsilon(y)$. The monoid of faces of $\AAA$ is called the *face monoid* and is denoted $\FFF$. The lattice $\Lambda(\FFF)$ is isomorphic to the *intersection lattice* $\LLL$ of $\AAA$; it consists of the subspaces of $V$ that can be expressed as an intersection of hyperplanes from $\AAA$. Under this isomorphism, the support map $\sigma$ corresponds to the map that sends a face $x$ to the smallest subspace that contains $x$. (-2,-2) – (2,2); (2,-2) – (-2,2); (-3,0) – (3,0); (0,0) node\[fill=white,font=\] [$(000)$]{}; (2,0.75) node\[font=\] [$(+++)$]{}; (2,0) node\[fill=white,font=\] [$(0++)$]{}; (2,-0.75) node\[font=\] [$(-++)$]{}; (1.4,-1.4) node\[fill=white,font=\] [$(-+0)$]{}; (0,-1.5) node\[font=\] [$(-+-)$]{}; (-1.4,-1.4) node\[fill=white,font=\] [$(-0-)$]{}; (-2,-0.75) node\[font=\] [$(---)$]{}; (-2,0) node\[fill=white,font=\] [$(0--)$]{}; (-2,0.75) node\[font=\] [$(+--)$]{}; (-1.4,1.4) node\[fill=white,font=\] [$(+-0)$]{}; (0,1.5) node\[font=\] [$(+-+)$]{}; (1.4,1.4) node\[fill=white,font=\] [$(+0+)$]{}; $ \resizebox{3in}{!}{ \xygraph{ !{0;/r18mm/:,p+/u18mm/::} !~:{@{-}} {(---)}( :[u]{(0--)} :[dl]{(+--)} :[u]{(+-0)} :[dl]{(+-+)} :[u]{(+0+)} :[dl]{(+++)} :[u]{(0++)} :[dl]{(-++)} :[u]{(-+0)} :[dl]{(-+-)} :[u]{(-0-)} :"(---)"), "{(-0-)}":[rrru(.85)]{(000)}(:"(-+0)",:"{(0++)}",:"(+0+)",:"(+-0)",:"(0--)") )} } $ #### Free partially commutative left regular bands. If $\Gamma=(V,E)$ is a simple graph, then the *free partially commutative left regular band* associated with $\Gamma$ is the left regular band $B(\Gamma)$ with presentation $$\begin{aligned} B(\Gamma) = \big\langle V \mid xy=yx \text{~for all edges~} \{x,y\}\in E\big\rangle.\end{aligned}$$ This is the left regular band analogue of free partially commutative monoids (also called *trace monoids* or *graph monoids* [@tracebook; @CartierFoata]) and of free partially commutative groups (also called *right-angled Artin groups* or *graph groups* [@Wise]). For example, if $\Gamma$ has no edges, then $B(\Gamma)$ is the free left regular band on the vertex set, whereas if $\Gamma$ is a complete graph, then $B(\Gamma)$ is the free semilattice on the vertex set of $\Gamma$. The elements of $B(\Gamma)$ correspond to acyclic orientations of induced subgraphs of the complementary graph ${\ensuremath{\overline {\Gamma}}}$ of $\Gamma$. Indeed, any element $w$ of the free left regular band $F(V)$, which is a repetition-free sequence of vertices, gives rise to an acyclic orientation $\mathcal O(w)$ as follows: the vertices of the subgraph are the letters appearing in $w$; there is an edge $x {\longrightarrow}y$ if $x$ comes before $y$ in $w$ and if $\{x,y\}$ is not an edge of $\Gamma$. \[wordproblem\] Elements $v,w\in F(V)$ are equal in $B(\Gamma)$ if and only if $\sigma(v)=\sigma(w)$ and $\mathcal O(v)=\mathcal O(w)$. The Tsetlin library Markov chain can be modelled as a hyperplane random walk, but is most naturally a random walk on the free left regular band, see [@Brown1; @Brown2]. Similarly, the random walk on acyclic orientations of a graph considered by Athanasiadis and Diaconis as a function of a hyperplane walk [@DiaconisAthan] is most naturally a random walk on a free partially commutative left regular band. #### Other examples. There are numerous other examples of left regular bands appearing in disguise in the algebraic combinatorics literature and elsewhere, as well as numerous examples of left regular band *semigroups*. These include the face semigroup of an affine hyperplane arrangement, the set of covectors associated with an affine oriented matroid or a $T$-convex set of topes and the face semigroup of a finite $\mathrm{CAT}(0)$ cube complex. These are presented in detail in [@oldpaper; @newpaper]. Statement of the Main Result {#s:mainresult} ============================ Our main result expresses certain algebraic invariants of $B$ in terms of the cohomology of simplicial complexes associated with its ${{\mathrel{\mathscr R}}}$-poset. This allows us to compute these algebraic invariants using the combinatorics of the poset, which we do for certain classes of left regular bands in the next section. We begin by describing the algebraic invariants. Fix a commutative ring with unit $\Bbbk$. For every element $X$ of the support lattice $\Lambda(B)$, the ring $\Bbbk$ becomes a $\Bbbk B$-module via the action $$\begin{aligned} b \cdot \alpha = \begin{cases} \alpha, & \text{if}\ \sigma(b)\geq X,\\ 0, & \text{otherwise}, \end{cases}\end{aligned}$$ for all $b \in B$ and $\alpha \in \Bbbk$. These modules, denoted by $\Bbbk_X$, are precisely the simple $\Bbbk B$-modules when $\Bbbk$ is a field [@Brown1; @Saliola]. We are interested in computing the algebraic invariants ${\mathop{\mathrm{Ext}}\nolimits}^n_{\Bbbk B}(\Bbbk_X,\Bbbk_Y)$ for two such modules [@assem]. It turns out that these invariants coincide precisely with the cohomology of certain simplicial complexes associated with $B$, so a detailed description of ${\mathop{\mathrm{Ext}}\nolimits}^n$ is not necessary for what follows. Indeed, the power of the main result resides in the fact that we can compute these algebraic invariants by making use of combinatorial properties of $B$. For elements $X$ and $Y$ of the support lattice $\Lambda(B)$, pick an element $y \in B$ with $\sigma(y) = Y$ and define $$\begin{gathered} B[X,Y] = \big\{ z \in B : X \leq \sigma(z) \text{~and~} z \leq_{{\mathrel{\mathscr R}}}y \big\}.\end{gathered}$$ Passing from $B$ to the monoid $B[X,Y]$ corresponds to the restriction–contraction operation for oriented matroids. This is a subposet of $B$ with respect to Green’s ${{\mathrel{\mathscr R}}}$-order. Let $\Delta(X,Y)$ be the *order complex* of the poset $B[X,Y] \setminus \{y\}$: it is the simplicial complex whose vertex set is $B[X,Y] \setminus \{y\}$ and whose simplices are the finite chains in the poset. (Up to isomorphism, this poset does not depend on the choice of $y\in Y$.) \[mainresult\] Let $B$ be a finite left regular band, $\Bbbk$ a commutative ring with unit, and $X,Y\in \Lambda(B)$. Then $$\begin{aligned} {\mathop{\mathrm{Ext}}\nolimits}^n_{\Bbbk B}(\Bbbk_X,\Bbbk_Y) = \begin{cases} {\ensuremath{\widetilde {H}}}^{n-1}(\Delta(X,Y),\Bbbk), & \text{if}\ X<Y,\ n\geq 1,\\ \Bbbk, &\text{if}\ X=Y,\ n=0,\\ 0, & \text{otherwise.} \end{cases}\end{aligned}$$ Applications of the Main Result {#s:applications} =============================== Quiver with relations of a left regular band algebra ---------------------------------------------------- We begin by investigating applications of the degree $0$ and degree $1$ cohomology. They encode information about the structure of the monoid algebra $\Bbbk B$ and allows us to describe a *quiver presentation* of $\Bbbk B$. $\vcenter{ \xymatrix@l@C=0pt@R=1em{ cba \ar@{-}[d] && cab \ar@{-}[d] && bca \ar@{-}[d] && bac \ar@{-}[d] & & acb \ar@{-}[d] && abc \ar@{-}[d] \\ cb \ar@{-}[dr] && ca \ar@{-}[dl] && bc \ar@{-}[dr] && ba \ar@{-}[dl] & & ac \ar@{-}[dr] && ab \ar@{-}[dl] \\ &c \ar@{-}[drrrr] & && &b\ar@{-}[d]& & & &a\ar@{-}[dllll]& \\ && && &1& }}$ $\vcenter{ \xymatrix@l@M=1pt{ & \{a,b,c\}\ar@/^7ex/[ddd] \ar@/_7ex/[ddd] \ar@/_1.5ex/[ddl] \ar@/^1.5ex/[rdd] \ar@<-0.4ex>@/^4ex/[dd] \\ \{a,b\} \ar@/_2ex/[ddr] & \ \{a,c\} \ar@<0.4ex>@/_4ex/[dd] & \{b,c\} \ar@/^2ex/[ddl] \\ \{a\} & \{b\} \ & \{c\} \\ & \emptyset }} $ #### Quiver. Assume for the moment that $\Bbbk$ is a field. Then the dimension of the degree $0$ cohomology of $\Delta(X,Y)$ counts its number of connected components [@Wachs]. Combined with Theorem \[mainresult\], we obtain a combinatorial interpretation of the dimension of ${\mathop{\mathrm{Ext}}\nolimits}^1_{\Bbbk B}(\Bbbk_X, \Bbbk_Y)$. The dimensions of these spaces carry all the information needed to construct the quiver of an algebra [@assem], which yields the following new description for the quiver of $\Bbbk B$. This is more conceptual and sometimes easier to apply than [@Saliola]. \[quiver\] Let $B$ be a finite left regular band and $\Bbbk$ a field. Then the quiver of $\Bbbk B$ has vertex set $\Lambda(B)$. The number of arrows from $X$ to $Y$ is zero unless $X<Y$, in which case it is one less than the number of connected components of $\Delta(X,Y)$. The quiver of $\Bbbk F_3$ is depicted in Figure \[fig:F3quiver\]; for instance, there are two arrows from the bottom to the top since there are three connected components in the Hasse diagram of the ${{\mathrel{\mathscr R}}}$-order on $F_3 \setminus \{1\}$. Corollary \[freelrb\] provides a description of the quiver of $F_n$ for all $n \geq 0$. #### Quiver relations. One can think of a quiver as a combinatorial encoding of a special presentation of the algebra by generators and relations: the quiver is a directed graph whose vertices correspond to primitive orthogonal idempotents and whose arrows correspond to a minimal generating set of the radical. Relations take the form of linear combinations of paths of the quiver [@assem]. A result of Bongartz [@assem; @Bongartz] implies that the number of relations involving paths between two vertices in a minimal quiver presentation for a left regular band algebra is determined by the dimension of ${\mathop{\mathrm{Ext}}\nolimits}^2$. Theorem \[mainresult\] admits the following corollary. \[cor:minrelations\] If $X<Y$ in $\Lambda(B)$, then the number of relations involving paths from $X$ to $Y$ in a minimal quiver presentation of a left regular band algebra $\Bbbk B$ is given by $\dim_{\Bbbk} {\ensuremath{\widetilde {H}}}^1(\Delta(X,Y),\Bbbk)$. Algebras of free left regular bands are hereditary {#ss:righthereditary} -------------------------------------------------- A finite dimensional algebra whose left ideals are projective modules is called *hereditary*. This is equivalent to the property that ${\mathop{\mathrm{Ext}}\nolimits}^2$ vanishes; to the property that each submodule of a projective module is projective; and to the property that the global dimension of the algebra is at most $1$. A result of Gabriel says that a split basic algebra is hereditary precisely when it has a quiver presentation with an acyclic quiver and no relations [@assem]. In section \[ss:gldim\] we will see a characterization of those left regular band algebras $\Bbbk B$ that are hereditary in terms of the order complex of $B$, but we foreshadow it by showing that the algebra of the free left regular band is hereditary. The Hasse diagram of the ${{\mathrel{\mathscr R}}}$-order of the free left regular band $F_n$ is a tree, as are the diagrams of the subposets $F_n[X,Y]$. Consequently, the cohomology of the order complexes $\Delta(X,Y)$ vanishes in positive degrees. Theorem \[quiver\] and Corollary \[cor:minrelations\] combine to show that $\Bbbk F_n$ is hereditary and to describe its quiver. \[freelrb\] The algebra $\Bbbk F_n$ is hereditary over any field $\Bbbk$. Its quiver has vertex set the subsets of $\{1,\ldots,n\}$. If $X\supsetneq Y$, then there are $|X\setminus Y|-1$ arrows from $X$ to $Y$. There are no other arrows. This result was first proved by K. Brown using quivers and a counting argument [@Saliola Theorem 13.1]. Our argument for heredity generalizes to any left regular band whose ${{\mathrel{\mathscr R}}}$-order is a tree; we call such left regular bands *right hereditary*. These include the matroid left regular bands of Brown [@Brown1], the interval greedoid left regular bands of Björner [@bjorner2], and the Rhodes and Karnofsky-Rhodes expansions of a lattice [@oldpaper]. We will see in Theorem \[treethm\] another, more transparent, proof of Corollary \[freelrb\], which also makes it simple to compute the quiver of a right hereditary left regular band. Hyperplane face monoids {#ss:hyperplanemonoids} ----------------------- Recall that $\AAA$ denotes a central hyperplane arrangement in a $d$-dimensional real vector space $V$, $\LLL$ its intersection lattice, and $\FFF$ its left regular band of faces. Without loss of generality, we suppose that the intersection of all the hyperplanes in $\AAA$ is the origin: otherwise quotient $V$ by this intersection. We argue that $\Delta({\widehat}0,{\widehat}1)$ is a $(d-1)$-sphere. The ${{\mathrel{\mathscr R}}}$-order on $\FFF$ can be described geometrically as $y \leq x$ if and only if $x \subseteq \overline y$, where $\overline y$ denotes the set-theoretic closure of $y$. This establishes an order-reversing bijection between the faces $\FFF$ and the cells of the regular cell decomposition $\varSigma$ obtained by intersecting the hyperplane arrangement with a sphere centered at the origin. The dual of $\varSigma$ is the boundary of a polytope $Z$ so that the poset of faces of $Z$ is isomorphic to $\FFF$ [@DiaconisBrown1]. Since the order complex of the poset of faces of a polytope is the barycentric subdivision of the polytope, it follows that $\Delta({\widehat}0,{\widehat}1)$ is a $(d-1)$-sphere. This argument also applies to $\Delta(X,Y)$ with $X \leq Y$ since it corresponds to the hyperplane arrangement in $X$ obtained by intersecting $X$ with the hyperplanes in $\AAA$ containing $Y$ and not $X$. It follows that $\Delta(X,Y)$ is a sphere of dimension $\dim(X)-\dim(Y)-1$. Consequently, we recover Lemma 8.3 of [@Saliolahyperplane]. \[ext-spaces\] For $X, Y \in \LLL$ and $n \geq 0$, $${\mathop{\mathrm{Ext}}\nolimits}^n_{\Bbbk\FFF}(\Bbbk_X, \Bbbk_Y) \cong \begin{cases} \Bbbk, & \text{if } Y \subseteq X \text{ and } \dim(X) - \dim(Y) = n, \\ 0, & \text{otherwise}. \end{cases}$$ We apply Theorem \[mainresult\]. Since $\Delta(X,Y)$ is a sphere of dimension $\dim(X)-\dim(Y)-1$, it follows that ${\ensuremath{\widetilde {H}}}^{n-1}(\Delta(X,Y),\Bbbk)$ is $0$ unless $\dim(X)-\dim(Y)=n$, in which case it is $\Bbbk$. It follows that the quiver of $\Bbbk\FFF$ coincides with the Hasse diagram of $\LLL$ ordered by reverse inclusion. \[hyperplanequiver\] The quiver of $\Bbbk\FFF$ has vertex set $\LLL$. The number of arrows from $X$ to $Y$ is zero unless $Y \subsetneq X$ and $\dim(X)-\dim(Y)=1$, in which case there is exactly one arrow. In [@Saliolahyperplane] a set of quiver relations for $\Bbbk\FFF$ was described: for each interval of length two in $\LLL$ take the sum of all paths of length two in the interval. It was also shown that $\Bbbk\FFF$ is a Koszul algebra and that its Koszul dual algebra is isomorphic to the incidence algebra of the intersection lattice $\LLL$. Complex hyperplane face monoids {#ss:complexhyperplanemonoids} ------------------------------- There is a left regular band of faces that one can associated with a complex arrangement [@complexstrat] and it turns out that the situation for complex hyperplane arrangements is similar to that of real hyperplane arrangements. Things are slightly more complicated in this setting because the complexes $\Delta(X, Y)$ do not arise from complex hyperplane face monoids. However, a careful analysis of the PL structure of the cell complex associated with the arrangement reveals that $\Delta(X,Y)$ is a sphere of dimension $\dim_{\mathbb R} X-\dim _{\mathbb R}Y-1$, from which we obtain the following analogues of the results from the real case, *cf.* § \[ss:hyperplanemonoids\]. For $X, Y \in \LLL$ and $n \geq 0$, $${\mathop{\mathrm{Ext}}\nolimits}^n_{\Bbbk\FFF}(\Bbbk_X, \Bbbk_Y) \cong \begin{cases} \Bbbk, & \text{if } Y \subseteq X \text{ and } \dim_{\mathbb R}(X) - \dim_{\mathbb R}(Y) = n, \\ 0, & \text{otherwise}. \end{cases}$$ An immediate consequence is that the quiver of $\Bbbk\FFF$ coincides with the Hasse diagram of the augmented intersection lattice $\LLL$ ordered by reverse inclusion, as was the case for real hyperplane arrangements. Geometric left regular bands and commutation graphs --------------------------------------------------- We say that a left regular band $B$ is *geometric* if, for each $a \in B$, the left stabilizer $\{b\in B\mid b\geq_{{\mathrel{\mathscr R}}}a\}$ of $a$ is commutative (and hence a lattice under the order $\leq_{{\mathrel{\mathscr R}}}$ with the meet given by the product). The left regular bands associated to hyperplane arrangements, oriented matroids and $\mathrm{CAT}(0)$ cube complexes are geometric, whence the name. Almost all the left regular bands appearing so far in the algebraic combinatorics literature embed in $\{0,+,-\}^n$ and hence are geometric (see the discussion in § \[s:LRBs\]). For a finite geometric left regular band $B$, we will use the following special case of Rota’s cross-cut theorem [@Rota; @bjornersurvey] to provide a simplicial complex homotopy equivalent to the order complex $\Delta({\widehat}0, {\widehat}1)$ of $B \setminus \left\{1\right\}$. This second complex simplifies the computation of the quiver of $\Bbbk B$. \[crosscutstated\] Let $P$ be a finite poset such that any subset of $P$ with a common lower bound has a meet. Define a simplicial complex $K$ with vertices the maximal elements of $P$ and with simplices those subsets with a common lower bound. Then $K$ is homotopy equivalent to the order complex $\Delta(P)$. We also need the well-known notion of the *clique complex* or *flag complex* ${\mathsf{Cliq}}(G)$ of a simple graph $G$: it is the simplicial complex whose vertices are the vertices of $G$ and whose simplices are the subsets of vertices that induce a complete subgraph. Notice that $G$ is the $1$-skeleton of ${\mathsf{Cliq}}(G)$ and that ${\mathsf{Cliq}}(G)$ is obtained by ‘filling in’ the $1$-skeleton of every $q$-simplex found in $G$. Let $\mathscr M(B)$ be the set of maximal elements of $B\setminus \{1\}$ and let $\Gamma(\mathscr M(B))$ be the *commutation graph* of $\mathscr M(B)$: that is, the graph whose vertex set is $\mathscr M(B)$ and whose edges are pairs $(a,b)$ such that $ab=ba$. For oriented matroids, the commutation graph is known as the *cocircuit graph* in the literature [@Orientedmatroids]. \[commutationgraph\] Let $B$ be a finite geometric left regular band. Then $\Delta({\widehat}0,{\widehat}1)$ is homotopy equivalent to the clique complex ${\mathsf{Cliq}}(\Gamma(\mathscr M(B)))$ of the commutation graph $\Gamma(\mathscr M(B))$ of the maximal elements of $B\setminus \{1\}$. Theorem \[commutationgraph\] can be used to give another proof that if a left regular band $B$ is right hereditary, then $\Bbbk B$ is hereditary (*cf.* Section \[ss:righthereditary\]). It also leads to an easy computation of the quiver. \[treethm\] Let $B$ be a finite, right hereditary, left regular band and $\Bbbk$ a field. Then $\Bbbk B$ is hereditary. Its quiver has vertex set $\Lambda(B)$. The number of arrows from $X$ to $Y$ is zero if $X \not< Y$; otherwise, it is one less than the number of children of $y$ with support greater than or equal to $X$ (for any $y \in B$ with $\sigma(y) = Y$). Theorem \[treethm\] covers the algebras of nearly all the left regular bands considered by K. Brown in [@Brown1] (except the hyperplane monoids) and the interval greedoid left regular bands of Björner [@bjorner2]. Global dimension of a left regular band algebra {#ss:gldim} ----------------------------------------------- We now turn to applications of the higher degree cohomology spaces, especially the first degree for which the higher degree cohomology vanishes. This “vanishing degree” relates the *Leray number* of the simplicial complex with the *global dimension* of the monoid algebra. The *global dimension* $\mathrm{gl.}\dim A$ of a finite dimensional algebra $A$ over a field $\Bbbk$ is the smallest integer $n$ such that ${\mathop{\mathrm{Ext}}\nolimits}_A^{m}(S_1,S_2)=0$ for all $m>n$ and all simple $A$-modules $S_1$ and $S_2$ (see [@assem]). An algebra $A$ has global dimension zero if and only if it is semisimple; it has $\mathrm{gl.}\dim A\leq 1$ if and only if it is hereditary. \[globaldimension\] Let $B$ be a finite left regular band and $\Bbbk$ a field. Then $$\begin{aligned} \mathrm{gl.}\dim \Bbbk B = \min \{n : {\ensuremath{\widetilde {H}}}^n(\Delta(X,Y),\Bbbk)=0 \text{~for all~} X, Y\in\Lambda(B) \text{~with~} X < Y\}.\end{aligned}$$ In particular, one has $ \mathrm{gl.}\dim \Bbbk B\leq m $ where $m$ is the length of the longest chain in $\Lambda(B)$. As an immediate corollary, we obtain a characterization of those algebras $\Bbbk B$ that are hereditary in terms of the order complex of $B$. The monoid algebra $\Bbbk B$ of a left regular band is hereditary if and only if each connected component of each simplicial complex $\Delta(X,Y)$, for $X < Y \in \Lambda(B)$, is acyclic. Leray numbers and an improved upper bound on global dimension ------------------------------------------------------------- We can improve on the upper bound from Theorem \[globaldimension\] using the Leray number of a simplicial complex. The *$\Bbbk$-Leray number* of a simplicial complex $K$ with vertex set $V$ is $$\begin{aligned} L_\Bbbk(K) =\min\left\{d : {\ensuremath{\widetilde {H}}}^i(K[W],\Bbbk) = 0 \text{~for all~} i\geq d \text{~and all~} W\subseteq V \right\},\end{aligned}$$ where $K[W]$ denotes the subcomplex of $K$ consisting of all simplices whose vertices belong to $W$. \[Leraybound\] Let $B$ be a finite left regular band and $\Bbbk$ a field. Then $\mathrm{gl.}\dim \Bbbk B$ is bounded above by the Leray number $L_\Bbbk(\Delta(B))$ of the order complex of $B$. Originally, interest in Leray numbers came about because of connections with Helly-type theorems [@Wegner; @BolandLek; @Kalai2]: the Leray number of a simplicial complex provides an obstruction for realizing the complex as the nerve of a collection of compact convex subsets of $\mathbb R^d$. Leray numbers also play a role in combinatorial commutative algebra: $L_\Bbbk(K)$ turns out to be the Castelnuovo-Mumford regularity of the Stanley-Reisner ring of $K$ over $\Bbbk$ [@Kalai2]; or equivalently, $L_{\Bbbk}(K)+1$ is the regularity of the face ideal of $K$ [@Kalai1]. In section \[ss:lerayFPCLRBs\], we give a new interpretation to the Leray number of the clique complex of a graph in terms of non-commutative algebra. Leray numbers and free partially commutative left regular bands {#ss:lerayFPCLRBs} --------------------------------------------------------------- In this section, we prove that the global dimension of a free partially commutative left regular band $B(\Gamma)$ is equal to the Leray number of the clique complex of the graph $\Gamma$. \[partiallycommmain\] Let $\Gamma=(V,E)$ be a finite graph and $\Bbbk$ a commutative ring with unit. Then, for $W\subsetneq U\subseteq V$ and $n\geq 1$, we have $$\begin{aligned} {\mathop{\mathrm{Ext}}\nolimits}^n_{\Bbbk B(\Gamma)}(\Bbbk_U,\Bbbk_W)={\ensuremath{\widetilde {H}}}^{n-1}({\mathsf{Cliq}}(\Gamma[U\setminus W]),\Bbbk).\end{aligned}$$ We present here the proof of the special case $W=\emptyset$ and $U=V$ (it turns out that the general case reduces to this case). The maximal elements of $B(\Gamma)\setminus \{1\}$ are the elements of $V$. The commutation graph for this set is exactly $\Gamma$. Since $B(\Gamma)$ is a geometric left regular band, we conclude $\Delta({\widehat}0,{\widehat}1)$ is homotopy equivalent to ${\mathsf{Cliq}}(\Gamma)$ by Theorem \[commutationgraph\]. The theorem now follows from Theorem \[mainresult\]. Our first corollary characterizes the free partially commutative left regular bands with a hereditary $\Bbbk$-algebra. A graph is said to be *chordal* if it contains no induced cycle of length greater than $3$. It is known that $L_\Bbbk(K)\leq 1$ if and only if $K$ is the clique complex of a chordal graph, cf. [@oldpaper; @Wegner; @BolandLek]. \[FPLBgldim\] If $\Bbbk$ is a field and $\Gamma$ a finite graph, then the global dimension of $\Bbbk B(\Gamma)$ is the $\Bbbk$-Leray number $L_\Bbbk({\mathsf{Cliq}}(\Gamma))$. In particular, $\Bbbk B(\Gamma)$ is hereditary if and only if $\Gamma$ is a chordal graph. Our next corollary computes the quiver of the algebra of a free partially commutative left regular band. Let $\Gamma=(V,E)$ be a finite graph. The quiver of $\Bbbk B(\Gamma)$ has vertex set the power set of $V$. If $U\supsetneq W$, then the number of arrows from $U$ to $W$ is one less than the number of connected components of $\Gamma[U\setminus W]$. There are no other arrows. It is easy to see that if $\Gamma$ is *triangle-free*, that is, has no $3$-element cliques, then ${\mathsf{Cliq}}(\Gamma)=\Gamma$ and so $L_\Bbbk({\mathsf{Cliq}}(\Gamma))=2$ unless $\Gamma$ is a forest (in which case $\Gamma$ is chordal). This provides a natural infinite family of finite dimensional algebras of global dimension $2$. CW left regular bands and minimal projective resolutions -------------------------------------------------------- In this last section, we indicate how one can use topological properties of the ${{\mathrel{\mathscr R}}}$-poset of a left regular band $B$ to construct minimal projective resolutions of the modules $\Bbbk_X$ (recall that these are all the simple $\Bbbk B$-modules when $\Bbbk$ is a field, *cf.* § \[s:mainresult\]). For this section, we assume that $B$ is a finite left regular band, but allow it to be a semigroup. To simplify notation below, we write $B_{\geq X}$ for $B[X, {\widehat}1]$, which we call the *contraction* of $B$ to $X$ (it corresponds to the usual notion of contraction in hyperplane and oriented matroid theory). We say that $B$ is *connected* if each $\Delta(B_{\geq X})$ is connected; this is automatic if $B$ is a monoid since $1$ is a greatest element and hence $\Delta(B_{\geq X})$ is in fact contractible. In [@newpaper] it is proved using Theorem \[mainresult\] that $\Bbbk B$ is unital if and only if $B$ is connected and, moreover, the $\Delta(B_{\geq X})$ are all acyclic in this case. \[t:projectiveres\] Let $B$ be a connected left regular band and $\Bbbk$ a commutative ring with unit. Then the augmented simplicial chain complex $C_\bullet(\Delta(B_{\geq X});\Bbbk)\xrightarrow{\,\,\varepsilon\,\,}\Bbbk_X$ is a $\Bbbk B$-projective resolution. Notice that $C_0(\Delta(B_{\geq X});\Bbbk)=\Bbbk B_{\geq X}$ and hence this resolution is almost never minimal. We can improve upon this to get a minimal resolution by taking advantage of the fact that the ${{\mathrel{\mathscr R}}}$-posets of many of the examples we have been discussing have very nice topology. For instance, for real and complex hyperplane face monoids, the subposets $B[X,Y]$ are the face posets of spheres (see §§ \[ss:hyperplanemonoids\]–\[ss:complexhyperplanemonoids\]). Following [@bjornerCW], we say that a poset $P$ is a *CW poset* if it is the face poset of a finite regular CW complex; we denote the associated CW complex by $\Sigma(P)$. Motivated by the many examples, we formulate the following definition. We say that a left regular band $B$ is a *CW left regular band* if $B_{\geq X}$ is a connected CW poset for all $X \in \Lambda(B)$. For example, the set of faces of a central or affine hyperplane arrangement or the set of covectors of an oriented matroid (possibly affine) is a CW left regular band. The face poset of a $\mathrm{CAT}(0)$ cube complex [@Wise; @Chepoi] also admits a CW left regular band structure. Tree space [@BilleraHolmes], sometimes called the tropical Grassmannian, is an example of a $\mathrm{CAT}(0)$ cube complex of combinatorial interest. The enumerative combinatorics of a CW left regular band is like that of a hyperplane arrangement or an oriented matroid. The support semilattice is Cohen-Macaulay and the $f$-vector is determined by the support semilattice via an analogue of Zaslavsky’s theorem. See [@newpaper] for details. \[t:cwlrb\] Let $B$ be a finite CW left regular band, $X \in \Lambda(B)$ and let $\Bbbk$ be a field. Then the augmented cellular chain complex $C_\bullet({\mathop{\Sigma}\nolimits}(B_{\geq X});\Bbbk){\longrightarrow}\Bbbk_X$ is the minimal projective resolution of $\Bbbk_{\geq X}$. In particular, this applies to the case where $B$ is the face monoid of a (central or affine) real or complex hyperplane arrangement, the set of covectors of an oriented matroid or oriented interval greedoid, or the face monoid of a $\mathrm{CAT}(0)$ cube complex. The key ingredient in the proof of Theorem \[t:cwlrb\] is that if $B$ is a CW left regular band, then $B$ acts on the CW complex $\Sigma(B)$. Since $\Sigma(B)$ is acyclic, its augmented chain complex becomes an acyclic chain complex of $\Bbbk B$-modules, which turn out to be projective. For CW left regular bands, we can also prove that $\Bbbk B$ is a Koszul algebra, identify its Koszul dual algebra and its ${\mathop{\mathrm{Ext}}\nolimits}$-algebra and describe a presentation by quiver with relations. Below, ${\operatorname{Inc}}_{\Bbbk}(P)$ denotes the incidence algebra of a poset $P$ over the ring $\Bbbk$. The full details are presented in [@newpaper]. \[t:Koszulfull\] Let $B$ be a finite CW left regular band and let $\Bbbk$ be a field. Then $\Bbbk B$ is a Koszul algebra, its Koszul dual algebra is $\Bbbk B^!\cong {\operatorname{Inc}}_{\Bbbk}(\Lambda(L)^{op})$ and its ${\mathop{\mathrm{Ext}}\nolimits}$-algebra is ${\mathop{\mathrm{Ext}}\nolimits}(\Bbbk B)\cong {\operatorname{Inc}}_{\Bbbk}(\Lambda(L))$. \[sec:ack\] The first author wishes to warmly thank the Center for Algorithmic and Interactive Scientific Software, CCNY, CUNY for inviting him to be a Visiting Professor during part of the research of this paper. The second author was supported in part by NSERC and FQRNT. The third author was supported in part by NSERC. The first and third authors were supported in part by Binational Science Foundation of Israel and the US (BSF) grant number 2012080. Some of the research of this paper was done while the third author was at the School of Mathematics and Statistics of Carleton University. \[sec:biblio\] [^1]: Email: . [^2]: Email: . [^3]: Email: .

--- abstract: 'The relation between the input and output spaces of is investigated to identify those characteristics of the input space that have a large influence on the output for a given task. For this purpose, the function is decomposed into a Taylor expansion in each element of the input space. The Taylor coefficients contain information about the sensitivity of the response to the inputs. A metric is introduced that allows for the identification of the characteristics that mostly determine the performance of the in solving a given task. Finally, the capability of this metric to analyze the performance of the is evaluated based on a task common to data analyses in high-energy particle physics experiments.' author: - Stefan Wunsch - Raphael Friese - Roger Wolf - Günter Quast bibliography: - 'citations.bib' title: Identifying the relevant dependencies of the neural network response on characteristics of the input space --- Introduction {#sec:introduction} ============ A is a multi-parameter system, which, depending on its architecture, can consist of several thousands of weight and bias parameters, subject to one or more non-linear activation functions. Each of these adjustable parameters obtains its concrete value and meaning by minimization during the training process. Thus the same can be applied to several concrete tasks, which are only defined at the training step. In applications in high-energy particle physics, which are supposed to distinguish a signal from one or more backgrounds, the training sample is obtained either from simulation or from an independent dataset without overlap with the sample of interest, to which the is applied. Usually the output itself is then subject to a detailed likelihood based hypothesis test, to infer the presence and yield of the signal [@junk1999confidence; @read2002presentation; @atlas2011procedure; @belforte2012combined; @cowan2011asymptotic]. The likelihood may include information on the shape of a variable that is supposed to discriminate signal from background. This shape could (while it does not have to) be e.g. the output of an . Apart from one or more parameters of interest the hypothesis test may comprise several hundreds of nuisance parameters, steering the response of the test statistic on a corresponding set of uncertainties. The nuisance parameters can be correlated or uncorrelated with the shape of the discriminating variable and (directly or indirectly) depend on the response of the output on its input variables. These kinds of analyses connect the observation of a measurement to a hypothesized truth. For applications they pose the intrinsic problem that, beyond statistical fluctuations, congruency between the training sample and the sample of interest may not be given. Deviations need to be identified and quantified within the uncertainty model of the hypothesis test. They may occur not only in the description of single input variables to the , but also in correlations across input variables, even if the marginal distributions of the individual input variables are reproduced. An can be sensitive to correlations across input variables; in fact this sensitivity is the main reason for potential performance gains, with respect to other approaches, like e.g. profile likelihoods. To make sure that this performance gain is not feigned, in addition to the marginal distributions, all correlations across input variables need to be carefully checked, and their influence on the test statistic identified and eventually mapped into the uncertainty model of the hypothesis test. The complexity of this methodology motivates the interest, not only in keeping the number of inputs to the at a manageable level, but above all in identifying those characteristics of the input space to the with the largest influence on the output. The definition of the uncertainty model of the hypothesis test can then be concentrated on these most influential characteristics. This approach sets the scope of this study to not more than a few tenth, up to a few hundred, partially highly correlated input variables in the context of particle physics experiments, or comparable applications. It differs from the approaches of weak supervision [@metodiev2017weakly; @dery2017weakly; @komiske2018weakly; @cohen2018weakly] and pivoting with adversaries [@louppe2017learning] that have been discussed in the literature. Weak supervision tries to circumvent the problem that we are describing by replacing an originally ground-truth labeled training by a training based on unlabeled training data. The corresponding samples can be obtained from the data themselves. They do not depend on a simulation and may be chosen to be unbiased. This approach is well justified in classification tasks, that are based just on the characteristics of the predefined training data. In the analyses that we are discussing the classification is tied to the hypothesized truth. Replacing the ground-truth labeled training by unlabeled input data does not solve the problem that we are discussing. Our discussion is also beyond the scope of pivoting with adversaries, for which the mismodellings to address have to be known beforehand. Our discussion sets in at an earlier stage, which is the most complete identification of all uncertainties that can be of relevance for the physics analysis. After the most influential features of the input space have been identified the method of pivoting with adversaries could be applied to mitigate potential mismodellings. A related approach to extract information about the characteristics of the input space is to flatten the distributions of sub-spaces so that possible discriminating features vanish [@deoliveira2017flattening; @chang2018flattening]. From the performance degradation after retraining the on the modified inputs, information about the discriminating power of the respective sub-space can be obtained. However, this approach does not allow to evaluate the dependencies of the response of an unique function on the characteristics of the input space, since each retrained function may have learned different features. So far, the questions we are raising have been addressed by methods that have been proposed to relate the output of with certain regions of input pixels in the context of image classification [@bach2015pixel; @montavon2017explaining]. These methods only use first-order derivatives to the function to back propagate the output layer by layer. What we propose is a Taylor expansion of the full function up to an arbitrary order, which allows to connect the input space directly to the output. While with this study we will demonstrate the application of the Taylor expansion only up to second order, we explicitly propose a generalization towards higher-order derivatives in the Taylor expansion to capture relations across variables, which usually play a more important role in data analyses in high-energy particle physics experiments. Due to the high-performance computation of derivatives in modern software frameworks used for the implementation of  [@abadi2016tensorflow; @paszke2017automatic; @bergstra2010theano], this expansion can be obtained at each point of the input space, even if this space is of high dimension. In this way, the sensitivity of the response to the input space can be analyzed by the gradient of the function. For practical reasons we stop the expansion at second order. To facilitate the following interpretation, we define a feature to be a characteristic of a single element or a pair-wise relation between two elements of the input space. The first class of features relates to the coefficients of the expansion to first order (first-order feature); the second class to the coefficients of the second order expansion (second-order feature). First-order features capture the influence of single input elements on the output throughout the input space; second-order features the influence of pair-wise or auto-correlations among the input elements. It is obvious that depending on the given task a certain feature can have large influence on the output of the in a certain region of the input space, while it is less important in others. We propose the arithmetic mean of the absolute value of the corresponding Taylor coefficient, computed from the input space defined by the task to be solved, $$\label{eq:metric} \langle t_{i} \rangle \equiv \frac{1}{N} \sum_{k=1}^{N}\left|t_{i}(\left.\{x_{j}\}\right|_{k})\right|\qquad i\in\mathcal{P}(\{x_{j}\})$$ as a metric for the influence of a given feature of the input space on the output, where the sum runs over the whole testing sample of size $N$, $t_i$ corresponds to the coefficients of the Taylor expansion, $\{x_{j}\}|_{k}$ to the set of variables spanning the input space, evaluated for element $k$ of the testing sample, and $i$ is an element of the powerset of $\{x_j\}$. It should be noted that the $\langle t_i \rangle$ characterize the input space (as covered by the test data) and the sensitivity of the to it, after training, as a whole. In section \[sec:toy\_scenarios\] we illustrate this choice with the help of four simple tasks emphasizing certain single features of the input space or their combination. In section \[sec:explaining\_the\_learning\_progress\] we point out that, when evaluated at each step of the minimization during the training process, the $\langle t_{i} \rangle$ can be utilized to illustrate and monitor the training process and learning strategies adopted by the . In section \[sec:application\_on\_a\_scenario\_from\_particle\_physics\] we show the application of the $\langle t_{i} \rangle$ to a more realistic task common to data analyses in high-energy particle physics experiments. Such tasks usually have the following attributes, which are of relevance for the following discussion: - they consist of not more than several tens of important input parameters, which leads to a moderate dimensionality of the posed problem; - they may rely on relations between elements more than they rely on single elements of the input space; - they usually pose problems, where a signal and background class cannot be separated based on single or few input variables, but only from the combination of several input variables; - they require a good understanding of the performance to turn the output into a reliable measurement. Analysis of features of the input space for simple tasks {#sec:toy_scenarios} ======================================================== In the following we illustrate the relation of the $\langle t_i\rangle$ to certain features of the input space. The applied corresponds to a fully connected feed-forward model with a single hidden layer consisting of 100 nodes. As activation functions a hyperbolic tangent is chosen for the hidden layer and a sigmoid for the output layer. A preprocessing of the inputs is performed following the $(x-\mu)/\sigma$ rule with the mean $\mu$ and the standard deviation $\sigma$ derived independently for each input variable. The free parameters of the are fitted to the training data using the cross-entropy loss and the Adam optimizer algorithm [@kingma2014adam]. The full training dataset with $\SI{e5}{}$ elements is split into two equal halves. One half is used for the calculation of the gradients used by the optimizer. The other half is used as independent validation dataset. The training is stopped if the loss did not improve on the validation dataset for three times in a row (early stopping). The independent test dataset used to calculate the $\langle t_i\rangle$ consists of $\SI{e5}{}$ elements. We use the software packages Keras [@chollet2015keras] and TensorFlow [@abadi2016tensorflow] for the implementation of the and the calculation of the derivatives. For simplicity we choose binary classification tasks with two inputs, $x_1$ and $x_2$. For the signal and background classes we sample Gaussian distributions with parameters, as summarized in Table \[tab:scenarios\]. From the Taylor series we obtain two metrics $\langle t_{x_1}\rangle$ and $\langle t_{x_2}\rangle$ indicating the influence of the marginal distributions of $x_1$ and $x_2$, and three metrics $\langle t_{x_1,x_1}\rangle$, $\langle t_{x_1,x_2}\rangle$, and $\langle t_{x_2,x_2}\rangle$ indicating the influence of the relation between $x_1$ and $x_2$, and the two auto-correlations. In the upper row of Fig. \[fig:scenarios\] the distribution of the (red) signal and (blue) background classes in the input space are shown, where darker colors indicate a higher sample density. In the lower row of Fig. \[fig:scenarios\] the values obtained for the $\langle t_i \rangle$ after the training are shown for each corresponding task. For the task shown in Fig. \[fig:scenario\_a\] the signal and background classes are shifted against each other. In both classes $x_1$ and $x_2$ are uncorrelated and of equal spread. The classification task becomes most difficult along the off-diagonal axis between the two classes through the origin and simpler if both, $x_1$ and $x_2$, take large or small values at the same time. Correspondingly, $\langle t_{x_1}\rangle$ and $\langle t_{x_2}\rangle$ obtain large values indicating the separation power that is already caused by the marginal distributions of $x_1$ and $x_2$. The orientation of the two classes with respect to each other also results in a non-negligible contribution of $\langle t_{x_1,x_2}\rangle$ to the response. For the task shown in Fig. \[fig:scenario\_b\] the signal and background classes are both centered at the origin of the input space, with equal spread in $x_1$ and $x_2$, but with different correlation coefficients in the covariance matrix. The classification task is most difficult in the origin of the input space and becomes simpler if $x_1$ and $x_2$ take large absolute values. Correspondingly, the relation between $x_1$ and $x_2$ is identified as the most influential feature by the value of $\langle t_{x_1,x_2} \rangle$. The fact that large absolute values of $x_1$ and $x_2$ support the separability of the two classes is expressed by the relatively large values for $\langle t_{x_1} \rangle$ and $\langle t_{x_2}\rangle$. A combination of the examples of Fig. \[fig:scenario\_a\] and \[fig:scenario\_b\] is shown in Fig. \[fig:scenario\_c\]. For the task shown in Fig. \[fig:scenario\_d\] the signal and background classes are both centered in the origin of the input space with different spread. In both classes $x_1$ and $x_2$ are uncorrelated. According to the symmetry of the posed problem the relation between $x_1$ and $x_2$ is expected to not strongly contribute to the separability of the signal and background classes. This is confirmed by the lower value of $\langle t_{x_1,x_2} \rangle$. Instead $\langle t_{x_1} \rangle$, $\langle t_{x_2} \rangle$, $\langle t_{x_1,x_1} \rangle$, and $\langle t_{x_2,x_2} \rangle$ take larger values as expected from the previous discussion. -------------------------- ----- ----- ------------------------------ ------------------------------ -------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------- Task Signal Background Fig. \[fig:scenario\_a\] 0.5 0.5 $-0.5$ $-0.5$ $\vphantom{\Biggl(\Biggr)}\left(\begin{array}{cc} \hphantom{-}1\hphantom{.5} & \hphantom{-}0\hphantom{.5} \\ $\vphantom{\Biggl(\Biggr)}\left(\begin{array}{cc} \hphantom{-}1\hphantom{.5} & \hphantom{-}0\hphantom{.5} \\ \hphantom{-}0\hphantom{.5} & \hphantom{-}1\hphantom{.5}\end{array}\right)$ \hphantom{-}0\hphantom{.5} & \hphantom{-}1\hphantom{.5}\end{array}\right)$ Fig. \[fig:scenario\_b\] 0 0 $\hphantom{-}0\hphantom{.0}$ $\hphantom{-}0\hphantom{.0}$ $\vphantom{\Biggl(\Biggr)}\left(\begin{array}{cc} \hphantom{-}1\hphantom{.5} & \hphantom{-}0.5 \\ $\vphantom{\Biggl(\Biggr)}\left(\begin{array}{cc} \hphantom{-}1\hphantom{.5} & -0.5 \\ \hphantom{-}0.5 & \hphantom{-}1\hphantom{.5}\end{array}\right)$ -0.5 & \hphantom{-}1\hphantom{.5}\end{array}\right)$ Fig. \[fig:scenario\_c\] 0.5 0.5 $-0.5$ $-0.5$ $\vphantom{\Biggl(\Biggr)}\left(\begin{array}{cc} \hphantom{-}1\hphantom{.5} & \hphantom{-}0.5 \\ $\vphantom{\Biggl(\Biggr)}\left(\begin{array}{cc} \hphantom{-}1\hphantom{.5} & -0.5 \\ \hphantom{-}0.5 & \hphantom{-}1\hphantom{.5}\end{array}\right)$ -0.5 & \hphantom{-}1\hphantom{.5}\end{array}\right)$ Fig. \[fig:scenario\_d\] 0 0 $\hphantom{-}0\hphantom{.0}$ $\hphantom{-}0\hphantom{.0}$ $\vphantom{\Biggl(\Biggr)}\left(\begin{array}{cc} \hphantom{-}0.5 & \hphantom{-}0\hphantom{.5} \\ $\vphantom{\Biggl(\Biggr)}\left(\begin{array}{cc} \hphantom{-}3\hphantom{.5} & \hphantom{-}0\hphantom{.5} \\ \hphantom{-}0\hphantom{.5} & \hphantom{-}0.5\end{array}\right)$ \hphantom{-}0\hphantom{.5} & \hphantom{-}3\hphantom{.5}\end{array}\right)$ -------------------------- ----- ----- ------------------------------ ------------------------------ -------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------- Analysis of the learning progress {#sec:explaining_the_learning_progress} ================================= When evaluated at each minimization step during the training, the metrics $\langle t_i \rangle$ may serve as a tool to analyze the learning progress of the . We illustrate this for the task shown in Fig. \[fig:scenario\_c\]. In Fig. \[fig:animated\] the evolving values of each $\langle t_i \rangle$ are shown, as continuous lines of different color, for the first 700 gradient steps. The stopping criterion of the training is reached after 339 gradient steps (indicated by the red vertical line in the figure). We measure the performance of the in separating the signal from the background class by the of the . We have added the at each training step to the figure with a separate axis on the right. A rough distinction of two phases can be stated. Approximately up to minimization step 30 the performance of the shows a steep rise up to a plateau value of $0.84$ for the . This rise coincides with increasing values of $\langle t_{x_1}\rangle$ and $\langle t_{x_2}\rangle$. Both metrics have the same progression, which can be explained by the symmetry of the task. Also the values for $\langle t_{x_1,x_1}\rangle$, $\langle t_{x_1,x_2}\rangle$ and $\langle t_{x_2,x_2}\rangle$ show an increase, though much less pronounced. Roughly 100 minimization steps later, a second, more shallow, rise of the sets in, coinciding with increasing values for $\langle t_{x_1,x_2}\rangle$. We interpret this in the following way. During the first phase the adapts to the first-order features related to $\langle t_{x_1}\rangle$ and $\langle t_{x_2}\rangle$, which is the most obvious choice to separate the signal from the background class. During this phase the learning progress of the is concentrated in the areas of the input space with medium to large values of $x_1$ and $x_2$. In the second phase the relation between $x_1$ and $x_2$, as a second-order feature, gains influence. This is when the learning progress concentrates on the region of the input space where the signal and background classes overlap. It can be seen that the influence of the features related to $\langle t_{x_1}\rangle$ and $\langle t_{x_2}\rangle$ decreases from minimization step 50 on. Apparently this influence has been overestimated at first and is successively replaced giving more importance to the more difficult to identify second-order features. From our knowledge of the truth, this is indeed the “more correct” assessment, which from minimization step 250 on, also leads to another gain in performance. Note that by the end of the training the progression of $\langle t_{x_1,x_2}\rangle$ has not converged, yet. The stopping criterion represents a measure of success and not a measure of truth. It might well have happened that the stopping criterion might have been met already between gradient step 50 and 100. In this case the output would have been based on the assessment that $\langle t_{x_1,x_2}\rangle$ plays a less important role. In this case success rules over truth. In our example the a priori known, more correct assessment leads to another performance gain after a few more gradient steps. Stopping the training before gradient step 100 would have missed this performance gain. We would like to emphasize that Fig. \[fig:animated\] is not more but a monitor to visualize what steps have led to the training result of the . This information can help to interpret both the features of the input space and the sensitivity to it. A different configuration might reveal a different sensitivity to any of the $\langle t_{i}\rangle$. Also there is no claim of proof that the increase in $\langle t_{x_1,x_2}\rangle$ causes the increase in the . ![Values of the metrics $\langle t_i \rangle$, as defined in Eq. \[eq:metric\], evaluated at each gradient step of the training, for the task discussed in section \[sec:toy\_scenarios\] and shown in Fig. \[fig:scenario\_c\]. On the axis to the right the of the curve, as a measure of the performance in solving the task at each training step, is shown. The red vertical line indicates after how many gradient steps the predefined stopping criterion, given in section \[sec:toy\_scenarios\], has been met.[]{data-label="fig:animated"}](figures/animated/animated.pdf){width="0.92\linewidth"} Application to a benchmark task from high-energy particle physics {#sec:application_on_a_scenario_from_particle_physics} ================================================================= In the following we are investigating the behavior of the $\langle t_i \rangle$ when applied to a more complex task, typical for data analyses in high-energy particle physics. For this purpose we are exploiting a dataset that was released in the context of the Higgs boson machine learning challenge [@adam2014learning], in 2014. This challenge was inspired by the discovery of a Higgs particle in collisions of high-energy proton beams at the CERN LHC, in 2012 [@chatrchyan2012observation; @aad2012observation]. The search for Higgs bosons in the final state with two $\tau$ leptons [@Chatrchyan2014nva; @Aad2015vsa; @Sirunyan2017khh] at the LHC has two main characteristics of relevance for this challenge: - a Higgs boson will be produced in only a tiny fraction of the recorded collisions. - there is no unambiguous physical signature to distinguish collisions containing Higgs bosons (defining the signal class) from other collisions (defining the background class). Consequently, for such a search the signal needs to be inferred from a larger number of (potentially related) physical quantities of the recorded collisions, using statistical methods, which makes the task suited also for applications. For the challenge a typical set of proton-proton collisions was simulated, of which only a small subset contained Higgs bosons in the final state with two $\tau$ leptons. Important physical quantities to distinguish the signal and background classes are the momenta of certain collision products in the plane, transverse to the incoming proton beams; the invariant mass of pairs of certain collision products; and their angular position relative to each other and to the beam axis. In the context of the challenge the values of 30 such quantities were released, whose names and exact physical meaning are given in [@adam2014learning]. Seventeen of these variables are basic quantities, characterizing a collision from direct measurements; the rest, like all invariant mass quantities, are called derived variables and computed from the basic quantities. These derived variables have a high power to distinguish the signal and background classes. Other variables like the azimuthal angle $\phi$ of single collision products in the plane transverse to the incoming proton beams have no separating power between the signal and background classes, due to the symmetry of the posed problem. The task is solved by the same model and training approach as described in section \[sec:toy\_scenarios\]. Applied to all $30$ input quantities this results in an AUC of $0.92$ and an approximate median significance, as defined in [@adam2014learning], of $2.61$. In total, the $30$ input quantities result in $495$ first- and second-order features. For further discussion we rank these features according to their extracted influence on the output, based on the values of the corresponding $\langle t_i \rangle$, in decreasing order. In Fig. \[fig:ranking\_scores\] the $\langle t_i \rangle$ for all features are shown, split into (orange) first- and (blue) second-order features. The distribution shows a rapidly falling trend, suggesting that only a small number of the investigated features significantly contributes to the solution of the task. The most important input variable is identified as the invariant mass calculated from the kinematics of two distinguished particles in the collision, the identified hadronic $\tau$ lepton decay and the additional light flavor lepton, associated with a leptonic decay of the $\tau$ lepton, `DER_mass_vis`, as defined in [@adam2014learning]. This variable also belongs to the most important quantities to identify Higgs particles in the published analyses [@Chatrchyan2014nva; @Aad2015vsa; @Sirunyan2017khh], with a strong relation to the invariant mass of the new particle. It is a peaking unimodal distribution in the signal class, with a broader distribution, peaking in a different position, in the background class. Among the $10$ most influential features, it appears as the most influential first-order feature (in position $10$), reflecting the difference in the position of the peak in the signal and background classes, and as part of six further second-order features, including the auto-correlation (in position $6$), characterizing the difference in the width of the peak in the signal and background classes. The is thus able to identify the most important features of `DER_mass_vis`: its peak position and width. The usage of this variable in a analysis requires a good understanding not only of the marginal distribution but also of all relevant relations to other variables, which should be reflected in the uncertainty model. The most influential feature is found to be the relation of `DER_mass_vis` with the ratio of the transverse momenta of the two particles that enter the calculation of this variable, named `DER_pt_ratio_lep_tau`. This feature is shown in Fig. \[fig:feature\], visualizing the gain of the relation over a pure marginal distribution on each individual axis. Features related to $\phi$ on the other hand are consequently ranked to the end of the list, as can be seen from Fig. \[fig:ranking\_phi\], with the first occurrence in position $82$. Apart from `DER_mass_vis` only eight more inputs, which are all well motivated from the physics expectation, contribute to the upper $\SI{5}{\percent}$ of the most influential features. When exposed to only these nine input quantities the solves the task with an and curve identical to the one that we observe, when using all 30 input quantities, within the numerical precision, indicating the potential to reduce the input space from $30$ to $9$ dimensions without significant loss of information. We refrain from a more detailed analysis of the complete list of features, which quickly turns very abstract and cannot be fully appreciated without deeper knowledge of the exact physical meaning of the input quantities. We conclude that the metric of Eq. \[eq:metric\] allows for a detailed understanding of the role of each input quantity - even without knowing their exact meaning - and quantitatively confirms the intuition of the high-energy particle physics analyses that have been performed during the search for the Higgs boson in 2012 and afterwards. We would like to emphasize that the reduction of the dimension of the input space (in the demonstrated case from 30 to 9), which can be achieved also by other methods, like the principal component analysis [@abdi2010principal], is not the main goal of our investigation. The main goal is an improved and more intuitive understanding of the features of the input space and the sensitivity of the output on it. ![Metrics $\langle t_i\rangle$, as defined in Eq. \[eq:metric\], obtained from the $30$ inputs of the task discussed in section \[sec:application\_on\_a\_scenario\_from\_particle\_physics\]. The $\langle t_i\rangle$, have been ranked by value, in descending order. A color coding identifies (orange) first-order and (blue) second-order features.[]{data-label="fig:ranking_scores"}](figures/HIGGS/ranking_scores_split.pdf){width="0.85\linewidth"} ![Relation between the variables `DER_mass_vis` and `DER_pt_ratio_lep_tau`, as defined in [@adam2014learning] and discussed in section \[sec:application\_on\_a\_scenario\_from\_particle\_physics\], shown in a subset of the input space. The red (blue) contours correspond to the signal (background) class. Darker colors indicate a higher sample density. This relation is identified as the most influential feature after the training.[]{data-label="fig:feature"}](figures/HIGGS/higgs_DER_mass_vis_DER_pt_ratio_lep_tau_combined.pdf){width="0.70\linewidth"} ![Occurrence of features containing primitive $\phi$ variables and occurrence of `DER_mass_vis`, as discussed in section \[sec:application\_on\_a\_scenario\_from\_particle\_physics\], in the ranked list of features.[]{data-label="fig:ranking_phi"}](figures/HIGGS/ranking_mass_phi.pdf){width="0.80\linewidth"} Summary {#sec:summary} ======= We have discussed the usage of the coefficients $t_i$ from a Taylor expansion in each element of the input space $\{x_j\}$ to identify the characteristics of the input space with the largest influence on the output. For practical reasons we have restricted the discussion to the expansion up to second order, concentrating on the characteristics of marginal distributions of input elements, $x_j$, or relations between them, referred to as first- and second-order features. We propose the arithmetic mean of the absolute value of a corresponding Taylor coefficient $\langle t_i \rangle$, built from the whole input space, as a metric to quantify the influence of the corresponding feature on the output. We have illustrated the relation between features and corresponding $\langle t_i \rangle$ with the help of simple tasks emphasizing single features or relations between them. Evaluating the $\langle t_i \rangle$ at each step of the training allows for the analysis and monitoring of the learning process of the . Finally we have applied the proposed metrics to a more complex task common to high-energy particle physics and found that the most important features, known from physics analyses are reliably identified, while features known to be irrelevant are also identified as such. We consider this as the first step to identify those characteristics of the input space that have the largest influence on the output, in the context of tasks, typical for high-energy particle physics experiments. As shown for the example in section \[sec:application\_on\_a\_scenario\_from\_particle\_physics\] these most influential characteristics may well correspond to relations between different inputs or auto-correlations, and not just to the marginal distribution of single inputs. In subsequent steps the quantification of systematic uncertainties in the inputs can be concentrated on those most relevant inputs.

--- abstract: 'We consider decay processes of scalar-field condensation in the framework of well-established quantum field theory. We postulate that the quantum state corresponding to the scalar-field condensation is so-called coherent state with discussing the validity of such a treatment. We show that, by using the unitarity relation of the scattering matrix, decay rate of the coherent state is systematically calculated. We apply our procedure to derive explicit formulae of decay rates for two cases: (i) we study the case where the scalar condensation decays into a pair of scalar particles and show that our formalism reproduces the results obtained from the parametric-resonance analysis, and (ii) we also calculate the decay rate when the coherent state decays via anomaly.' --- TU-799\ September, 2007\ .75in **** Decay of Scalar Condensation\ in Quantum Field Theory .75in [Shigeki Matsumoto$^{\rm (a,b)}$ and Takeo Moroi$^{\rm (a)}$ ]{} 0.25in $^{\rm (a)}$[*Department of Physics, Tohoku University, Sendai 980-8578, Japan*]{} 0.25in $^{\rm (b)}$[*Tohoku University International Advanced Research and Education Organization,\ Institute for International Advanced Interdisciplinary Research,\ Sendai 980-8578, Japan*]{} .5in Introduction {#sec:introduction} ============ Scalar-field condensations play very crucial roles in various places in cosmology. Probably the most important example is the inflaton field which is necessary for inflation [@Guth:1980zm; @Sato:1980yn]. In particular, in the slow-roll inflation models [@Linde:1981mu; @Albrecht:1982wi], energy density of the inflaton condensation provides the energy density to realize (quasi-) de-Sitter universe during inflation. After inflation, the inflaton oscillates around the minimum of the potential and decays into standard-model particles to reheat the universe realizing hot big-bang cosmology. This class of inflation model not only solves flatness and horizon problems in cosmology but also provides density fluctuation consistent with the Wilkinson Microwave Anisotropy Probe data [@Spergel:2006hy]. In addition, it has been pointed out that the density fluctuation may arise from late-decaying scalar field other than the inflaton, which is called “curvaton” [@Enqvist:2001zp; @Lyth:2001nq; @Moroi:2001ct]. Other important example is the Affleck-Dine field for baryogenesis [@Affleck:1984fy]. In low-energy supersymmetric models, there exist scalar fields, i.e., scalar partners of quarks, which have baryon number. If some of those fields acquire non-vanishing amplitudes in the early universe, non-vanishing baryon number may be imprinted into the motion of the scalar-quark condensations due to baryon-number violating operators at an ultra-high energy scale. Such a scenario is one of the most attractive scenario to generate large enough baryon asymmetry of the universe. All of these exotic scalar-field condensations (i.e, inflaton, curvaton, Affleck-Dine field, and so on) oscillate around the minimum of the potential at some stage of the evolution of the universe, and eventually decay into standard-model particles for the cosmological history consistent with observations. Thus, it is important to understand how the scalar-field condensation decays from the view point of the quantum field theory. The main concern of this paper is to discuss how the oscillating scalar-field condensation decays into other states. Around the minimum, the potential of the scalar field $\varphi$ is well estimated by a parabolic one $$\begin{aligned} V = \frac{1}{2} m_\varphi^2 \varphi^2.\end{aligned}$$ Neglecting the effect of the cosmic expansion for simplicity, the solution to the classical equation of motion is given by $$\begin{aligned} \varphi = A_\varphi \cos m_\varphi t. \label{cos(mt)}\end{aligned}$$ In this case, the energy density of the condensation is given by $\rho_\varphi=\frac{1}{2}m_\varphi^2A_\varphi^2$. One should understand how the energy density stored in the oscillation of $\varphi$ is converted to the that of radiation. In the simplest approach, the decay rate of the scalar-field condensation is estimated from the decay rate of single scalar field (in the vacuum): the energy density of the scalar field in the condensation is approximated to decrease as $$\begin{aligned} \frac{d \rho_\varphi}{d t} = - \gamma_\varphi \rho_\varphi, \label{dot(rho)}\end{aligned}$$ where $\gamma_\varphi$ is the decay rate of $\varphi$ in the vacuum. However, it has been also pointed out that, when the scalar field $\varphi$ is oscillating like Eq. (\[cos(mt)\]), wave functions of fields which couple to $\varphi$ are modified. Consequently, the “decay rate” of the scalar field in the condensation may be significantly different from the one obtained from the field theory in the vacuum. In particular, in some of the cases, instability band may arise in the wave function of the final-state particles, which results in catastrophic particle production (so-called parametric resonance) [@Kofman:1994rk; @Shtanov:1994ce; @Yoshimura:1995gc; @Kofman:1997yn]. Since the decay of scalar-field condensation is very important, it is desirable to have a deep understanding of the decay processes of scalar-field oscillations in the framework of well-established quantum field theory. In this paper, we consider decay processes of scalar condensations in the framework of the quantum field theory. In our analysis, we neglect the effects of cosmic expansion as a first step to understand the behavior of the scalar condensation. We postulate that the scalar condensation corresponds to the so-called coherent state $|\varphi\rangle$ in the quantum field theory; justification of such a treatment will be also discussed. Then, we will show that the decay rate of the coherent state can be systematically calculated by using the unitarity relation of the scattering matrix ($S$-matrix). By using the fact that the decay rate of the coherent state is proportional to the imaginary part of the so-called $T$-matrix element $\Im[\langle\varphi |\hat{\cal T}|\varphi\rangle]$, we formulate the calculation of the decay rate of the coherent state. We apply our procedure to the case where the scalar-field condensation is coupled to a real scalar field and calculate the decay rate for such a case. We will see that our procedure gives the same decay rate as that obtained by the discussion of parametric-resonance. We discuss when Eq. (\[dot(rho)\]) is justified and how the instability band for parametric resonance arises in our framework. We also calculate the decay rate of scalar condensation which decays into gauge-boson pair via anomaly. Organization of this paper is as follows. In Section\[sec:basic\], we first derive basic formulae which are used in the calculation of the decay rate of the coherent state. In particular, we define the coherent state $|\varphi\rangle$ in the quantum field theory and present important properties of $|\varphi\rangle$. Then, we explain how the decay rate of the state $|\varphi\rangle$ is obtained. In Section \[sec:a-term\], we consider the case where the condensation couples to a real scalar field $\chi$ via three-point interaction. Decay of the coherent state via the interaction induced by the anomaly is discussed in Section \[sec:anomaly\]. In Section \[sec:summary\], we summarize our results. Basic Formulae {#sec:basic} ============== In this section, we introduce basic formulae used in our analysis. We discuss how the condensation of scalar field decays via some interaction. We assume that the interaction of the scalar field is weak enough. Total decay rate of any state can be related to the imaginary part of the scattering-matrix element due to the $S$-matrix unitarity. Let us denote the $S$-matrix as $$\begin{aligned} \hat{\cal S} = \hat{\bf 1} + i \hat{\cal T},\end{aligned}$$ where $\hat{\bf 1}$ and $\hat{\cal T}$ are the unit operator and the so-called $T$-matrix, respectively. (Here and hereafter, the “hat” is used for operators.) Then, from the unitarity of the $S$-matrix, the following relation holds: $$\begin{aligned} \hat{\cal T}^\dagger \hat{\cal T} = i (\hat{\cal T}^\dagger - \hat{\cal T}).\end{aligned}$$ This relation is important for our analysis. We expect that there exists a quantum state $|\varphi\rangle$ which describes the state with scalar condensation. (Details about $|\varphi\rangle$ will be explained below.) The probability of the state $|\varphi\rangle$ decaying into all the possible final states is related to the imaginary part of the $T$-matrix element as $$\begin{aligned} {\rm Prob} ( | \varphi \rangle \rightarrow {\rm all} ) = \sum_{f} \left| \langle f | \hat{\cal T} | \varphi \rangle \right|^2 = 2 \Im \left[ \langle \varphi | \hat{\cal T} | \varphi \rangle \right].\end{aligned}$$ Now, we consider what the state $|\varphi\rangle$ is. We first quantize the field operator using the free part of the Lagrangian, then treat the interaction terms as perturbations. We denote the free part of the Lagrangian of the real scalar field $\varphi$ as $$\begin{aligned} {\cal L} = \frac{1}{2} \partial_\mu \varphi \partial^\mu \varphi - \frac{1}{2} m_\varphi^2 \varphi^2.\end{aligned}$$ In our analysis, we use the box normalization of the wave functions with the volume $L^3$. Then, the field operator is given by $$\begin{aligned} \hat{\varphi} (x) = \sum_{\bf k} \frac{1}{\sqrt{2E_{\bf k} L^3}} \left( \hat{a}_{\bf k} e^{-ikx} + \hat{a}_{\bf k}^\dagger e^{ikx} \right), \label{varphi(real)}\end{aligned}$$ where $E_{\bf k}\equiv\sqrt{{\bf k}^2+m_\varphi^2}$. The annihilation and creation operators satisfy the following commutation relations: $$\begin{aligned} [\hat{a}_{\bf k}, \hat{a}_{\bf k'}^\dagger] = \delta_{{\bf k}, {\bf k'}},\end{aligned}$$ while $\hat{a}_{\bf k}$ and $\hat{a}_{\bf k'}$ (as well as $\hat{a}_{\bf k}^\dagger$ and $\hat{a}_{\bf k'}^\dagger$) commute. We postulate that the quantum state describing the scalar-field condensation is the coherent state, which is given by $$\begin{aligned} | \varphi \rangle \equiv e^{-|C_\varphi|^2/2} e^{C_\varphi \hat{a}_{\bf 0}^\dagger} | 0 \rangle,\end{aligned}$$ where $| 0 \rangle$ is the vacuum state satisfying $\hat{a}_{\bf k}| 0 \rangle=0$. Notice that the state $|\varphi\rangle$ is properly normalized: $\langle\varphi |\varphi\rangle=1$. In addition, importantly, the state $|\varphi\rangle$ is an eigenstate of the annihilation operator $\hat{a}_{\bf 0}$: $$\begin{aligned} \hat{a}_{\bf 0} | \varphi \rangle = C_\varphi | \varphi \rangle. \label{a|phi>}\end{aligned}$$ We can also see that $$\begin{aligned} \varphi (x) \equiv \langle \varphi | \hat{\varphi} (x) | \varphi \rangle = \varphi_- (x) + \varphi_+ (x), \label{<varphi>}\end{aligned}$$ where $$\begin{aligned} \varphi_- (x) \equiv \frac{1}{2} A_\varphi e^{-im_\varphi t}, ~~~ \varphi_+ (x) \equiv \frac{1}{2} A_\varphi^* e^{im_\varphi t},\end{aligned}$$ with $$\begin{aligned} A_\varphi = C_\varphi \sqrt{\frac{2}{m_\varphi L^3}}.\end{aligned}$$ One can easily see that, for the coherent state $|\varphi\rangle$, the expectation value of the field operator follows the trajectory of the solution to the classical wave equation. Thus, we expect that the coherent state $|\varphi\rangle$ corresponds to the quantum state where the scalar field is under oscillation. In calculating $T$-matrix elements, it is necessary to calculate the expectation values of time-ordered products of field operators. By using the Wick’s theorem, such products are calculated as $$\begin{aligned} \langle \varphi | T \prod_i \hat{\varphi} (x_i) | \varphi \rangle &=& \langle \varphi | N \prod_i \hat{\varphi} (x_i) | \varphi \rangle + (\mbox{all the possible contractions}) \nonumber \\ &=& \prod_i \varphi (x_i) + (\mbox{all the possible contractions}),\end{aligned}$$ where the symbol $T$ here denotes the time-ordering while $N$ is for normal-ordering. In addition, in the second equality, we have used Eq. (\[a|phi&gt;\]). Even in more complicated cases, we obtain $$\begin{aligned} \langle \varphi | T \prod_i f_i \left( \hat{\varphi} (x_i) \right) | \varphi \rangle &=& \Big\langle \varphi \Big| T \prod_i \sum_n \frac{1}{n!} \left[ \frac{d^n f_i}{d \varphi^n} \right]_{\varphi (x_i)} \left( \hat{\varphi} (x_i) - \varphi (x_i) \right)^n \Big| \varphi \Big\rangle \nonumber \\ &=& \prod_i f_i \left( \varphi (x_i) \right) + (\mbox{all the possible contractions}). \label{Wick's2}\end{aligned}$$ Here, we expand $f_i (\hat{\varphi} (x_i))$ around $\hat{\varphi} (x_i)=\varphi (x_i)$. Then, $\langle\varphi|(\hat{\varphi}(x)-\varphi(x))|\varphi\rangle=0$, and Eq. (\[Wick’s2\]) is applicable even when the function $f_i(\varphi)$ is singular at $\varphi=0$. In the following, we will not consider the processes in which $\varphi$ is produced due to the decay of the coherent state. (The inclusion of such processes is straightforward.) In such a case, propagator of $\varphi$ does not show up in the calculation and the field operator $\hat{\varphi}(x)$ can be simply replaced by the expectation value $\varphi(x)$, in which the contraction terms are irrelevant. It is also instructive to calculate the expectation values of energy-density operator as $$\begin{aligned} \rho_\varphi = L^{-3} \langle \varphi | \sum_{\bf k} E_{\bf k} \hat{a}_{\bf k}^\dagger \hat{a}_{\bf k} | \varphi \rangle = \frac{1}{2} m_\varphi^2 |A_\varphi|^2,\end{aligned}$$ while the expectation value of the number density is also given by $$\begin{aligned} n_\varphi = L^{-3} \langle \varphi | \sum_{\bf k} \hat{a}_{\bf k}^\dagger \hat{a}_{\bf k} | \varphi \rangle = \frac{1}{2} m_\varphi |A_\varphi|^2.\end{aligned}$$ For the complex scalar field (which we denote as $\phi$), similar argument applies. We define the field operator for the complex scalar field as $$\begin{aligned} \hat{\phi} (x) = \sum_{\bf k} \frac{1}{\sqrt{2E_{\bf k} L^3}} \left( \hat{a}_{\bf k} e^{-ikx} + \hat{b}_{\bf k}^\dagger e^{ikx} \right), \label{varphi(complex)}\end{aligned}$$ where $\hat{a}_{\bf k}$ and $\hat{b}_{\bf k}$ ($\hat{a}_{\bf p}^\dagger$ and $\hat{b}_{\bf k}^\dagger$) are annihilation (creation) operators. The coherent state for the complex field is given by $$\begin{aligned} | \phi \rangle \equiv e^{-(|C_\phi|^2 + |C_{\bar{\phi}}|^2)/2} e^{C_\phi \hat{a}_{\bf 0}^\dagger + C_{\bar{\phi}} \hat{b}_{\bf 0}^\dagger} | 0 \rangle, \label{|complex>}\end{aligned}$$ and $$\begin{aligned} \phi (x) \equiv \langle \phi | \hat{\phi} (x) | \phi \rangle = A_\phi e^{-im_\phi t} + A_{\bar{\phi}}^* e^{im_\phi t}, \label{<complex-phi>}\end{aligned}$$ where $$\begin{aligned} A_\phi = \frac{C_\phi}{\sqrt{2m_\phi L^3}},~~~ A_{\bar{\phi}} = \frac{C_{\bar{\phi}}}{\sqrt{2m_\phi L^3}}.\end{aligned}$$ Energy density of this state is given by $$\begin{aligned} \rho_\phi = 2 m_\phi^2 ( |A_\phi|^2 + |A_{\bar{\phi}}|^2 ),\end{aligned}$$ while we can also calculate the expectation values of the number densities of particle $\phi$ and anti-particle $\bar{\phi}$ as $$\begin{aligned} n_\phi &\equiv& L^{-3} \langle \phi | \sum_{\bf k} a_{\bf k}^\dagger a_{\bf k} | \phi \rangle = 2 m_\phi |A_\phi|^2, \label{n_phi}\\ n_{\bar{\phi}} &\equiv& L^{-3} \langle \phi | \sum_{\bf k} b_{\bf k}^\dagger b_{\bf k} | \phi \rangle = 2 m_\phi |A_{\bar{\phi}}|^2. \label{n_phibar}\end{aligned}$$ We can see that the number densities of $\phi$ and its anti-particle are proportional to $|A_\phi|^2$ and $|A_{\bar{\phi}}|^2$, respectively. Thus, when $|A_\phi|>|A_{\bar{\phi}}|$ ($|A_\phi|<|A_{\bar{\phi}}|$), $\phi$ is more (less) abundant than $\bar{\phi}$ in the condensation. It should be also noted that the function $\phi(x)$ given in Eq. (\[&lt;complex-phi&gt;\]) gives an elliptical trajectory on the complex $\phi$-plane. When $A_\phi=0$ or $A_{\bar{\phi}}=0$, the trajectory becomes a circle and, when $|A_\phi|=|A_{\bar{\phi}}|$, the trajectory becomes a straight line. Decay into Scalar Fields {#sec:a-term} ======================== Setup ----- First, we consider the simplest case where the scalar field $\varphi$ couples only to the real scalar field $\chi$ via the interaction $$\begin{aligned} {\cal L}_{\rm int} = -\frac{1}{2} \mu \varphi \chi^2, \label{A-term}\end{aligned}$$ with $\mu$ being the coupling constant. With this interaction, the decay rate of single particle in the vacuum is given by $$\begin{aligned} \gamma_{\varphi\rightarrow\chi\chi} = \frac{\mu^2}{32\pi m_\varphi} \sqrt{1 - \left( \frac{4m_\chi^2}{m_\varphi^2} \right) }. \label{Gamma(1P)}\end{aligned}$$ In this section, with the interaction given in Eq. (\[A-term\]), we discuss how the coherent state decays. As discussed in the previous section, the decay rate of the coherent state can be related to the imaginary part of the diagonal element of the $T$-matrix $\langle\varphi |\hat{\cal T}|\varphi\rangle$. Importantly, $\langle\varphi |\hat{\cal T}|\varphi\rangle$ is obtained by calculating loop diagrams in the quantum field theory. At the one-loop level, in other words, neglecting the fluctuation of $\varphi$, $\langle\varphi |\hat{\cal T}| \varphi\rangle$ is expressed as $$\begin{aligned} \langle\varphi |\hat{\cal T}| \varphi\rangle \equiv \sum_{p=1}^{\infty} \sum_{{\cal F}^{(2p)}} {\cal T}^{{\cal F}^{(2p)}},\end{aligned}$$ where the summation over ${\cal F}^{(2p)}$ is for all the possible Feynman diagrams with $2p$ external $\varphi$. With $p$ being fixed, one can find $$\begin{aligned} \sum_{{\cal F}^{(2p)}} i {\cal T}^{{\cal F}^{(2p)}} = \frac{1}{(2p)!} \left( \frac{i \mu}{2} \right)^{2p} \Big\langle 0 \Big| T \left[ \int d^4 x \left\{ \varphi_- (x) + \varphi_+ (x) \right\} \hat{\chi} (x) \hat{\chi} (x) \right]^{2p} \Big| 0 \Big\rangle, \label{T^F}\end{aligned}$$ where $\varphi_-$ and $\varphi_+$ show up when the field operator $\hat{\varphi}$ is contracted with the creation operator in $|\varphi\rangle$ and the annihilation operator in $\langle\varphi|$, respectively. In order to obtain non-vanishing results, we need to pick up same number of $\varphi_-$ and $\varphi_+$. =0.4 The right-hand side of Eq. (\[T\^F\]) contains contribution from various Feynman diagrams because $\varphi_-$ and $\varphi_+$ can be ordered in many ways. In Fig. \[fig:loop\_general\], we show a typical diagram (after imposing the cut). In our notation, we represent the insertion of $\varphi_-$ by the white dot $\circ$ while $\varphi_+$ by the black dot $\bullet$; non-vanishing diagrams have $p$ black and $p$ white dots. Internal lines are $\chi$-propagator. Expectation value of $\hat{\cal T}$ for the given diagram ${\cal F}^{(2p)}$ (with $2p$ external $\varphi$ insertion) is given by $$\begin{aligned} i {\cal T}^{{\cal F}^{(2p)}} = L^3 T S_{\cal F} \left| \frac{\mu A_\varphi}{2} \right|^{2p} \int \frac{d^4 \tilde{k}}{(2\pi)^4} \prod_{I=1}^{2p} ( k_I^2 - m^2_\chi + i 0^+ )^{-2}, \label{T(2p)}\end{aligned}$$ where $S_{\cal F}$ is the symmetry factor and $$\begin{aligned} L^3 T = \int d^4 x.\end{aligned}$$ In addition, the momentum flowing on the $I$-th propagator is given by $$\begin{aligned} k_I \equiv \tilde{k} + \sum_{J=1}^{I} \varepsilon_{J} Q_\varphi,\end{aligned}$$ with $$\begin{aligned} Q_\varphi \equiv (m_\varphi, {\bf 0}),\end{aligned}$$ and $\varepsilon_J=1$ and $-1$ if the $J$-th $\varphi$ insertion is $\varphi_-$ and $\varphi_+$, respectively. The imaginary part of ${\cal T}^{{\cal F}^{(2p)}}$ is obtained by cutting two propagators (see Fig. \[fig:loop\_general\]), which corresponds to the replacements of those two propagators by the $\delta$-functions (with a relevant numerical factor): $$\begin{aligned} \Im\left[ {\cal T}^{{\cal F}^{(2p)}} \right] = \lim_{\xi\rightarrow m_\chi^2} \sum_{i=1}^{2p-1} \sum_{j=i+1}^{2p} \Im \left[ {\cal T}^{{\cal F}^{(2p)}}_{i,j} (\xi) \right], \label{Im(T)}\end{aligned}$$ where $$\begin{aligned} \Im \left[ {\cal T}^{{\cal F}^{(2p)}}_{i,j} (\xi) \right] \equiv 2\pi^2 L^3 T S_{\cal F} \left| \frac{\mu A_\varphi}{2} \right|^{2p} \int \frac{d^4 \tilde{k}}{(2\pi)^4} \delta ( k_i^2 - \xi_i ) \delta ( k_j^2 - \xi_j ) \prod_{I\neq i,j} ( k_I^2 - \xi_I )^{-1}.\end{aligned}$$ (For details, see Appendix \[app:im\].) $\Im[ {\cal T}^{{\cal F}^{(2p)}}_{i,j}]$ is the contribution from the diagram in which cut is on $i$- and $j$-th propagators. For $\Im[ {\cal T}^{{\cal F}^{(2p)}}_{i,j}]$, we define the energy flow from one side of the cut to the other, which we denote $E_{\chi\chi}$; with the following non-negative integer: $$\begin{aligned} N_\varphi = \left| \sum_{I=i+1}^{j} \varepsilon_I \right|,\end{aligned}$$ the energy flow is given by $$\begin{aligned} E_{\chi\chi} = N_\varphi m_\varphi.\end{aligned}$$ Notice that $\Im [ {\cal T}^{{\cal F}^{(2p)}}_{i,j} ]$ contributes only to the decay rate of the process in which $N_\varphi$ of $\varphi$ in the condensation simultaneously annihilate into two $\chi$ because we neglect the fluctuations of the $\varphi$ field. At the perturbative level, such a decay process is kinematically allowed when $E_{\chi\chi}>2m_\chi$. Small amplitude limit --------------------- =0.4 In this and the next subsections, we concentrate on the case where the amplitude of $\varphi$ is small. In this case, the leading-order contribution to the $E_\varphi=N_\varphi m_\phi$ mode is from the diagram with $2N_\varphi$ external $\varphi$ with $\varphi_-$ and $\varphi_+$ being completely separated by the cut. (See Fig.\[fig:loop\_leading\].) Concentrating on such a leading-order diagram, the imaginary part of the $T$-matrix element is given by $$\begin{aligned} \Im \left[ {\cal T}^{(N_\varphi)}_{\rm Leading} \right] &=& L^3 T \pi^2 \left| \frac{\mu A_\varphi}{2} \right|^{2N_\varphi} \int \frac{d^4\tilde{k}}{(2\pi)^4} \delta \left( \tilde{k}^2 - m_\chi^2 \right) \delta \left( ( \tilde{k} - N_\varphi Q_\varphi )^2 - m_\chi^2 \right) \nonumber \\ && \prod_{I=1}^{N_\varphi-1} \left[ ( \tilde{k} - I Q_\varphi )^2 - m_\chi^2 \right]^{-2}.\end{aligned}$$ Constraints from the $\delta$-functions give $\tilde{k}Q_\varphi=\frac{1}{2}N_\varphi m_\varphi^2$. Thus, $$\begin{aligned} \prod_{I=1}^{N_\varphi-1} \left[ ( \tilde{k} - I Q_\varphi )^2 - m_\chi^2 \right]^{-1} \rightarrow m_\varphi^{ -4( N_\varphi - 1 ) } \left[ ( N_\varphi - 1 )! \right]^{-4},\end{aligned}$$ and hence $$\begin{aligned} \Im \left[ {\cal T}^{(N_\varphi)}_{\rm Leading} \right] &=& L^3 T \frac{\beta_{N_\varphi}}{32\pi} \frac{m_\varphi^4}{[ (N_\varphi-1)! ]^4} \left| \frac{\mu A_\varphi}{2 m_\varphi^2} \right|^{2N_\varphi},\end{aligned}$$ where, for $N_\varphi m_\varphi >2m_\chi$, the velocity $\beta_{N_\varphi}$ is given by $$\begin{aligned} \beta_{N_\varphi} \equiv \sqrt{1 - \frac{4m_\chi^4}{N_\varphi^2 m_\varphi^2}}, \label{beta_N}\end{aligned}$$ while $\beta_{N_\varphi}=0$ for $N_\varphi m_\varphi \leq 2m_\chi$. The decay rate of the coherent state $|\varphi\rangle$ per unit volume is evaluated as $$\begin{aligned} (\mbox{Decay rate per unit volume}) = \frac{{\rm Prob} (|\varphi\rangle \rightarrow {\rm all})} {L^3 T},\end{aligned}$$ and hence is given by $$\begin{aligned} \Gamma^{(N_\varphi)}_{\rm Leading} = \frac{\beta_{N_\varphi}}{16\pi} \frac{m_\varphi^4}{[ (N_\varphi-1)! ]^4} \left| \frac{\mu A_\varphi}{2 m_\varphi^2} \right|^{2N_\varphi}. \label{Gamma^N}\end{aligned}$$ The above expression is consistent with the result given in the study of the parametric resonance [@Yoshimura:1995gc]. In addition, the decay rate for the $N_\varphi=1$ mode is related to the decay rate of single particle, which is given in Eq. (\[Gamma(1P)\]), as $$\begin{aligned} \Gamma^{(N_\varphi=1)}_{\rm Leading} = n_\varphi \gamma_{\varphi\rightarrow\chi\chi}.\end{aligned}$$ Thus, when the amplitude of $\varphi$ is small, decay of the coherent state can be treated as the decay of individual particles in the condensation, which justifies the conventional treatment of the decay processes of scalar condensations. We also comment here that the decay rate given in Eq. (\[Gamma\^N\]) is also derived from the tree-level calculation of the $\chi\chi$ pair creation rate in the external oscillating $\varphi$ field; $\Gamma^{(N_\varphi)}_{\rm Leading}$ is equal to the production rate of $\chi\chi$ pair per unit volume with total energy of $E_{\chi\chi}=N_\varphi m_\varphi$. In general, at the leading order of $A_\varphi$, the decay rate of the coherent state is also obtained by calculating the tree-level production rate of the final-state particles treating the scalar condensation as an external field. If we consider higher order contributions, however, such a calculation breaks down; treating the scalar condensation as the external field, denominators of some propagators vanish in certain types of diagrams. Notice that, in Eq. (\[Im(T)\]), such a difficulty does not exist. (See also the following discussion.) Small velocity limit -------------------- In the previous subsection, we have calculated leading-order contributions to the decay rates of each mode in the small amplitude limit. Calculations of the contributions which are higher order in the amplitude are straightforward. In this subsection, we discuss when the small-amplitude expansion breaks down, taking the $N_\varphi=1$ mode as an example. If we calculate $O(|A_\varphi|^{2p})$ contributions to the decay rate of such mode, which are from diagrams with $2p$ external $\varphi$ insertions, one finds that the imaginary part of the $T$-matrix element is inversely proportional to the powers of $\beta_1$ in the $m_\varphi\rightarrow 2m_\chi$ limit. First, let us derive such a behavior with explicit calculation. For the $N_\varphi=1$ mode, the most important Feynman diagrams in the $m_\varphi\rightarrow 2m_\chi$ limit are those in which $\varphi_+$ and $\varphi_-$ insertions are next to each other. (See Fig.\[fig:loop\_smallb\].) As we will discuss, other types of diagrams with fixed $p$ have less singular behavior when $\beta_1\rightarrow 0$. We also note here that the diagram in Fig. \[fig:loop\_smallb\] contributes only to the $N_\varphi=1$ mode. (The imaginary part of the $T$-matrix vanishes when $N_\varphi=0$.) =0.4 Taking into account the diagram shown in Fig. \[fig:loop\_smallb\], the $T$-matrix element becomes $$\begin{aligned} i {\cal T}^{(N_\varphi=1)}_{\beta_1\rightarrow 0} = L^3 T \frac{1}{2p} \left| \frac{\mu A_\varphi}{2} \right|^{2p} \int \frac{d^4 \tilde{k}}{(2\pi)^4} \left( \tilde{k}^2 - m^2_\chi + i 0^+ \right)^{-p} \left[ (\tilde{k} - Q_\varphi )^2 - m^2_\chi + i 0^+ \right]^{-p}, \label{T(beta)}\end{aligned}$$ and the imaginary part of ${\cal T}^{(N_\varphi=1)}_{\beta_1\rightarrow 0}$ is given by $$\begin{aligned} \Im \left[ {\cal T}^{(N_\varphi=1)}_{\beta_1\rightarrow 0} \right] = - \frac{L^3 T}{32\pi} m_\varphi^4 \beta_1 \left| \frac{\mu A_\varphi}{2} \right|^{2} \times \frac{(4p-7)!!}{p!(p-1)!} \left| \frac{\mu A_\varphi}{2m_\varphi^2 \beta_1^2} \right|^{2(p-1)} + O(\beta_1^{7-4p}), \label{Im[T(beta)]}\end{aligned}$$ where $(2p-1)!!\equiv\prod_{I=1}^{p}(2I-1)$ for $p\geq 1$, and $(-3)!!\equiv -1$. (For details, see Appendix \[app:smallb\].) Hereafter, we neglect $O(\beta_1^{7-4p})$ contribution in Eq.(\[Im\[T(beta)\]\]), and the decay rate of the coherent state becomes $$\begin{aligned} \Gamma^{(N_\varphi=1)}_{\beta_1\rightarrow 0} = -\frac{1}{16\pi} m_\varphi^4 \beta_1 \left| \frac{\mu A_\varphi}{2} \right|^{2} \sum_{p=1}^\infty \frac{(4p-7)!!}{p!(p-1)!} \left| \frac{\mu A_\varphi}{2m_\varphi^2 \beta_1^2} \right|^{2(p-1)}. \label{Gamma_1(b->0)}\end{aligned}$$ As discussed in Appendix \[app:smallb\], the inverse powers of $\beta_1$ stems from the derivative of the function $B(m_\phi^2; \xi_i, \xi_j)$ given in Eq. (\[fnB\]) with respect to $\xi_i$ or $\xi_j$. The order of the derivatives is equal to the number of the propagators whose denominators vanish in the on-shell limit $\xi_i\rightarrow m_\chi^2$ ($i=1-2p$). Number of such propagators is maximized for the diagram given in Fig. \[fig:loop\_smallb\]. Thus, we safely neglect other types of diagrams in studying the case of $\beta_1\rightarrow 0$. As we mentioned, $\Gamma^{(N_\varphi=1)}_{\beta_1\rightarrow 0}$ becomes singular when $\beta_1\rightarrow 0$. In other words, for the $N_\varphi=1$ mode, the small-amplitude expansion breaks down when $|A_\varphi|$ is comparable to $\mu^{-1}m_\varphi^2\beta_1^2$. These behaviors are related to the fact that instability bands appear in the solution to the classical wave equation of the scalar field (i.e., $\chi$ in our argument) which couples to a oscillating scalar field (i.e., $\varphi$). With the interaction given in Eq. (\[A-term\]), the wave equation of $\chi$ (with the 3-momentum ${\bf k}$) in the oscillating background is given by the Mathieu equation: $$\begin{aligned} \frac{d^2 \chi_{\bf k}}{dt^2} + \left( {\bf k}^2 + m_\chi^2 + \mu |A_\varphi| \cos m_\varphi t \right) \chi_{\bf k} = 0.\end{aligned}$$ Parametrizing the momentum of $\chi$ as $$\begin{aligned} {\bf k}^2 = \frac{1}{4} m_\varphi^2 \left( \beta_1^2 + \epsilon \right),\end{aligned}$$ the lowest instability band in the small-amplitude limit is given by [@Yoshimura:1995gc] $$\begin{aligned} -\theta < \epsilon < \theta, \label{eps_pm}\end{aligned}$$ where $$\begin{aligned} \theta \equiv \frac{2 \mu |A_\varphi|}{m_\varphi^2}.\end{aligned}$$ From the study of the parametric resonance, the momentum of $\chi$ produced by the decay of the scalar condensation is in the range given by Eq. (\[eps\_pm\]). For the consistency of the calculation, the mass of the initial-state particle $\varphi$ should be large enough so that ${\bf k}^2$ is positive even for $\epsilon\sim -\theta$; otherwise, $O(\theta^2)$ contributions may be also important. This argument gives the limitation of the small-amplitude approximation; $\beta_1\gg \theta^{1/2}$ is required, which results in $|A_\varphi|\gg\mu^{-1}m_\varphi^2\beta_1^2$. The above argument is supported by the fact that $\Gamma^{(N_\varphi=1)}_{\beta_1\rightarrow 0}$ given in Eq.(\[Gamma\_1(b-&gt;0)\]) is equal to the decay rate obtained by the parametric-resonance analysis. Indeed, $\Gamma^{(N_\varphi=1)}_{\beta_1\rightarrow 0}$ is also expressed as $$\begin{aligned} \Gamma^{(N_\varphi=1)}_{\beta_1\rightarrow 0} &=& \frac{1}{2} m_\varphi \int_{|{\bf k}|_- <|{\bf k}| <|{\bf k}|_+} \frac{d^3 {\bf k}}{(2\pi)^3} \lambda^{(N_\varphi=1)} ({\bf k}) \nonumber \\ &=& \frac{1}{128 \pi^2} m_\varphi^4 \int_{-\theta}^{\theta} d \epsilon \sqrt{ \left(\beta_1^2 + \epsilon \right) \left(\theta^2 - \epsilon^2 \right)}, \label{Gamma_1(Param)}\end{aligned}$$ where, in the first equality, $|{\bf k}|_\pm\equiv\frac{1}{2}m_\varphi\sqrt{\beta_1^2\pm\theta}$, and $\lambda^{(N_\varphi=1)}=\frac{1}{2}\sqrt{\theta^2-\epsilon^2}$ is the “growth-rate factor” for the $N_\varphi=1$ mode obtained in the study of the Mathieu equation [@Yoshimura:1995gc]. The equivalence of Eq. (\[Gamma\_1(b-&gt;0)\]) and Eq.(\[Gamma\_1(Param)\]) can be seen by expanding $\sqrt{\beta_1^2 + \epsilon}$ in the integrand of Eq. (\[Gamma\_1(Param)\]) around $\epsilon=0$ (assuming $\beta_1>\theta^{1/2}$). Eq.(\[Gamma\_1(Param)\]) is nothing but the decay rate of the scalar condensation in the small-amplitude limit derived from the parametric-resonance analysis. Eq. (\[Gamma\_1(Param)\]) (and Eq. (\[Gamma\^N\])) also shows the fact that, at least at the small-amplitude limit, the results from the parametric-resonance analysis is obtained in our procedure where the quantum state describing the scalar condensation is postulated to be the coherent state. The equivalence of two approaches is also expected from the fact that the basic equations governing the behavior of the parametric resonance is derived in our framework. In particular, we can calculate the density matrix of the final-state particle $\chi$ in the quantum field theory. We can see that the Mathieu equation shows up in the calculation and that the resultant density matrix is the same as the one obtained in the study of parametric resonance. These subjects will be discussed in the next subsection. Calculation of the density matrix --------------------------------- In the classical treatment of the parametric resonant system, it is well known that the Mathieu equation appears as the equation of motion for the $\chi$ field. Thus, the equation is also expected to be obtained in the quantum-field-theory treatment. In this subsection, we show the derivations of the Mathieu equation and the density matrix of $\chi$ explicitly. Since we are interested in the case where the scalar field $\varphi$ initially forms the scalar condensation oscillating around the minimum of its potential, we describe the initial state (which is taken at $t=0$ in this subsection) as $$\begin{aligned} | i \rangle = | \varphi \rangle \otimes | 0 \rangle_{\chi},\end{aligned}$$ where the first and second kets represent the states for $\varphi$ and $\chi$, respectively. In addition, $| 0 \rangle_{\chi}$ is the vacuum of the $\chi$ field. (In the following, we omit the subscript $\chi$.) For our argument, it is convenient to use the density matrix of the total system in the Schroedinger picture. The density matrix at the time $T$ is simply given by $$\begin{aligned} \hat{\rho}_{\rm tot} (T) = e^{-i \hat{H} T} | i \rangle \langle i | e^{i \hat{H} T},\end{aligned}$$ with $\hat{H}$ being the Hamiltonian of the total system. We consider the properties of $\hat{\rho}_{\rm tot}$ in the coordinate basis: $$\begin{aligned} | q \otimes X \rangle \equiv | q \rangle \otimes | X \rangle,\end{aligned}$$ where $| q\rangle$ and $| X\rangle$ are eigenstates of the field operators $\hat{\varphi}$ and $\hat{\chi}$, respectively: $$\begin{aligned} \hat{\varphi}(t, {\bf x}) | q \rangle = q({\bf x}) | q \rangle,~~~ \hat{\chi}(t, {\bf x}) | X \rangle = X({\bf x}) | X \rangle.\end{aligned}$$ Then, the density matrix in the coordinate basis, $\rho_{\rm tot}[ q,X;q',X']\equiv\langle q\otimes X|\hat{\rho}_{\rm tot}(T)|q'\otimes X'\rangle$, is given by $$\begin{aligned} \rho_{\rm tot} [ q,X ; q',X' ] &=& \int {\cal D} q_i \int {\cal D} q_i' \int {\cal D} X_i \int {\cal D} X'_i \langle q_i | \varphi \rangle \langle X_i | 0 \rangle \langle \varphi | q'_i \rangle \langle 0 | X'_i \rangle \nonumber \\ && K [ q,X ; q_i,X_i ] K^* [ q',X' ; q_i',X_i' ].\end{aligned}$$ The kernel is represented in the path integral form as $$\begin{aligned} K [ q,X ; q_i,X_i ] = \int^{\chi(T,{\bf x}) = X({\bf x})}_{\chi(0,{\bf x}) = X_i({\bf x})} {\cal D}\chi \int^{\varphi(T,{\bf x}) = q({\bf x})}_{\varphi(0,{\bf x}) = q_i({\bf x})} {\cal D}\varphi e^{i S_{\rm tot}}, \label{kernel}\end{aligned}$$ where $S_{\rm tot}$ is the total action. For the explicit form of the kernel, see [@Feynman:1963fq; @FeynmanHibbs]. In order to study the behavior of the $\chi$ field, we derive the reduced density matrix of $\chi$ by tracing out $q$ and $q'$ variables: $$\begin{aligned} \rho_{\rm red} [X ; X'] \equiv \int {\cal D} q \rho_{\rm tot} [ q,X ; q,X' ].\end{aligned}$$ For this purpose, we define $$\begin{aligned} \tilde{q}_{\bf k} &\equiv& L^{-3/2} \int d^3 {\bf x} q({\bf x}) e^{-i{\bf kx}}, \\ \tilde{X}_{\bf k} &\equiv& L^{-3/2} \int d^3 {\bf x} X({\bf x}) e^{-i{\bf kx}}.\end{aligned}$$ Then, with the use of the properties of the coherent state, we obtain the following relations: $$\begin{aligned} \langle q | \varphi \rangle &=& \exp \left[ -\frac{1}{2} m_\varphi (\tilde{q}_{\bf 0}-L^{3/2} A_\varphi)^2 \right] \prod_{{\bf k}\neq {\bf 0}} \exp \left[ -\frac{1}{2} E_{\bf k} |\tilde{q}_{\bf k}|^2 \right], \label{wf_varphi} \\ \langle X | 0 \rangle &=& \prod_{\bf k} \exp \left[ -\frac{1}{2} \omega_{\bf k} |\tilde{X}_{\bf k}|^2 \right], \label{wf_chi}\end{aligned}$$ where, in this subsection, we consider the case where $A_\varphi$ is real. ($A_\varphi$ can be taken to be real with a relevant shift of the time variable.) In addition, $\omega_{\bf k}^2 = {\bf k}^2 + m^2_\chi$. In Eqs. (\[wf\_varphi\]) and (\[wf\_chi\]), we omit unimportant numerical constants. Using Eq. (\[wf\_varphi\]), the reduced density matrix becomes $$\begin{aligned} \rho_{\rm red} [X;X'] = \int {\cal D}X_i {\cal D} X'_i \langle X_i | 0 \rangle \langle 0 | X'_i \rangle \int^{\chi(T,{\bf x}) = X}_{\chi(0,{\bf x}) = X_i} {\cal D}\chi \int^{\chi'(T,{\bf x}) = X'}_{\chi'(0,{\bf x}) = X'_i} {\cal D}\chi' e^{i \bar{S} [\chi] - i \bar{S}[\chi'] + {\cal C}[\chi, \chi']}, $$ where $$\begin{aligned} \bar{S} [\chi] = \int_0^T dt \int d^3 {\bf x} \left[ \frac{1}{2} \partial_\mu \chi \partial^\mu \chi - \frac{1}{2} m_\chi^2 \chi^2 - \frac{1}{2} \mu A_\varphi \chi^2 \cos m_\varphi t \right],\end{aligned}$$ while ${\cal C}$ gives the collision terms: $$\begin{aligned} {\cal C}[\chi, \chi'] &=& - \frac{\mu^2}{4} \int_0^T dt \int_0^t dt' \int d^3 {\bf x} \int d^3 {\bf x}' \int \frac{d^3 {\bf k}}{ (2\pi)^3 2 E_{\bf k}} \cos {\bf k} \left( {\bf x} - {\bf x}' \right) \nonumber \\ && \left[ \chi^2 (x) - \chi^{\prime 2}(x) \right] \left[ e^{-i E_{\bf k} (t-t')} \chi^2 (x') - e^{ i E_{\bf k} (t-t')} \chi^{\prime 2} (x') \right].\end{aligned}$$ It can be seen that the periodic perturbation term, $\chi^2 \cos m_\varphi t$, appears in the reduced density matrix. Collision terms, which are proportional to $\chi^4$, $\chi^{\prime 4}$, and $\chi^2 \chi^{\prime 2}$, also appear after the integration. Since no approximation was made to derive the density matrix, the above formula can be used at any amplitude of the $\varphi$ field, $A_\varphi$. Furthermore, we can derive the kinetic equation by calculating the correlation function of the $\chi$ field on the reduced density matrix, which allows us to describe the non-linear dynamics of the $\chi$ system, as pointed out in [@Matsumoto:1999us]. When the amplitude $A_\varphi$ is small enough, we can treat the collision terms as perturbations. Then, the reduced density matrix is, at the leading order calculation, written by using the wave functional of the $\chi$ field: $$\begin{aligned} \rho_{\rm red} \left[ X; X' \right] = \Psi[T, X] \times \Psi^*[T; X'],\end{aligned}$$ where $$\begin{aligned} \Psi[T, X] = \int {\cal D} X_i \langle X_i | 0 \rangle \int^{\chi(T,{\bf x})=X}_{\chi(0,{\bf x})=X_i} {\cal D}\chi e^{i \bar{S}[\chi]}.\end{aligned}$$ The wave functional is written as a product of an infinite set of wave functions of harmonic oscillators as $$\begin{aligned} \Psi[T, X] = \prod_{\bf k} \int^\infty_{- \infty} d \tilde{X}_{i,{\bf k}} \exp \left[ -\frac{1}{2} \omega_{\bf k} |\tilde{X}_{i,{\bf k}}|^2 \right] \int^{ \tilde{\chi}_{\bf k}(T)=\tilde{X}_{\bf k}}_{ \tilde{\chi}_{\bf k}(0)=\tilde{X}_{i,{\bf k}}} {\cal D} \tilde{\chi}_{\bf k} e^{i \bar{S}_{\bf k} \left[ \tilde{\chi}_{\bf k} \right]},\end{aligned}$$ where $$\begin{aligned} \bar{S}_{\bf k} \left[ \tilde{\chi}_{\bf k} \right] = \int^T_0 dt \left[ \frac{1}{2} |\dot{\tilde{\chi}}_{\bf k}|^2 - \frac{1}{2} \left( {\bf k}^2 + m^2_\chi + \mu A_\varphi \cos m_\varphi t \right) |\tilde{\chi}_{\bf k}|^2 \right], \label{S_k}\end{aligned}$$ with the “dot” being the derivative with respect to time. $\Psi[T,X]$ satisfies the boundary condition $\Psi[0,X]=\prod_{\bf k}e^{-\omega_{\bf k}|\tilde{X}_{{\bf k}}|^2/2}$ (up to normalization), and its evolution is governed by the wave equation derived from the action given in Eq. (\[S\_k\]). Thus, $\Psi[T,X]$ is nothing but the wave functional obtained in [@Yoshimura:1995gc]: $$\begin{aligned} \Psi[T, X] = \prod_{\bf k} \frac{1}{\sqrt{u_{\bf k} (T)}} \exp \left[ \frac{i}{2} \frac{\dot{u}_{\bf k} (T)}{u_{\bf k} (T)} |\tilde{X}_{\bf k}|^2 \right],\end{aligned}$$ where $u_{\bf k}(t)$ is the solution to the Mathieu equation: $$\begin{aligned} \frac{d^2 u_{\bf k}}{dt^2} + \left( \omega_{\bf k}^2 + \mu A_\varphi \cos m_\varphi t \right) u_{\bf k} = 0,\end{aligned}$$ with the conditions $u_{\bf k}(0) = \sqrt{\pi/\omega_{\bf k}}$ and $\dot{u}_{\bf k}(0) = i \sqrt{\omega_{\bf k} \pi}$. This fact supports that the scalar condensation is well described by the coherent state in the quantum field theory. Before closing this section, we emphasize that the quantity $\Im [\langle\varphi|{\cal T}|\varphi\rangle]$ is relatively easily calculated with wide variety of interactions and final-states. Thus, for some applications, our procedure is more powerful than the approach using the Mathieu equation. Decay via Anomaly {#sec:anomaly} ================= Next, let us consider the case where a complex scalar field $\phi$ may decay via chiral and conformal anomalies. As the fundamental theory, we expect that there exists chiral fermions which have gauge quantum numbers and that the complex scalar field couples to the chiral fermions through a Yukawa interaction. To make our discussion definite, we consider $SU(N_{\rm c})$ gauge interaction; chiral fermions $Q_L$ and $Q_R^c$ are in fundamental and anti-fundamental representations of $SU(N_{\rm c})$, respectively, while $\phi$ is singlet. The complex scalar field couples to chiral fermions $Q_L$ and $Q_R^c$ via the Yukawa interaction $$\begin{aligned} {\cal L}_{\rm Yukawa} = -y (\phi Q_L Q_R^c + {\rm h.c.}).\end{aligned}$$ It should be noted that, in this model, there exists anomalous $U(1)$ symmetry, which we call $U(1)_A$; charges of $\phi$, $Q_L$ and $Q_R^c$ are $1$, $-\frac{1}{2}$, and $-\frac{1}{2}$, respectively. When the amplitude of $\phi$ is large, fermions $Q_L$ and $Q_R^c$ acquire Dirac mass. Thus, when $m_\phi\ll y|\phi|$, effective mass of the fermions are much larger than the mass of $\phi$. In this case, the decay process $\phi\rightarrow Q_LQ_R^c$ is expected to be kinematically forbidden. When $m_\phi\ll y|\phi|$, it is rather convenient to consider the low-energy effective field theory by integrating out the fermions. The relevant (light) fields in the low-energy effective field theory are $\phi$ and gauge fields as far as the effective mass of the fermions are much larger than $m_\phi$. In the following, we concentrate on such a case; thus, we assume that the inequality $m_\phi\ll y|\phi(x)|$ always holds at any point of the trajectory of $\phi$. We first consider the effects of the operator induced by the chiral anomaly: $$\begin{aligned} {\cal L}_{\rm eff} = - i \lambda_{\rm I} \left( \ln \phi - \ln \phi^\dagger \right) F^{\mu\nu} \tilde{F}_{\mu\nu}, \label{phiFF-dual}\end{aligned}$$ where $F_{\mu\nu}$ is the field-strength tensor and $$\begin{aligned} \tilde{F}^{\mu\nu} \equiv \frac{1}{2} \epsilon^{\mu\nu\rho\sigma} F_{\rho\sigma}.\end{aligned}$$ In Eq. (\[phiFF-dual\]) and hereafter, summation over the adjoint gauge index is implicit. In addition, $$\begin{aligned} \lambda_{\rm I} \equiv \frac{g^2}{32\pi^2} T_R,\end{aligned}$$ with $g$ being the gauge coupling constant of $SU(N_{\rm c})$, and $T_R=\frac{1}{2}$. Now, let us discuss the decay of the coherent state given in Eq.(\[|complex&gt;\]). At the leading order in $\lambda_{\rm I}$, which is of $O(\lambda_{\rm I}^2)$ in the calculation of $\langle\phi |\hat{\cal T}|\phi\rangle$, we obtain $$\begin{aligned} \langle \phi | i \hat{\cal T} | \phi \rangle &=& - \frac{1}{2} \lambda_{\rm I}^2 \int d^4 x d^4 x' \langle 0 | T \hat{F}_{\mu\nu} (x) \hat{\tilde{F}}^{\mu\nu} (x) \hat{F}_{\mu'\nu'} (x') \hat{\tilde{F}}^{\mu'\nu'} (x') | 0 \rangle \nonumber \\ && \left[ \ln \phi (x) - \ln \phi^\dagger (x) \right] \left[ \ln \phi (x') - \ln \phi^\dagger (x') \right]. \label{<phi|iT|phi>}\end{aligned}$$ In the following, we consider the case that $A_\phi\geq A_{\bar{\phi}}$. Then, we expand $\ln \phi (x)$ as $$\begin{aligned} \ln \phi (x) = \ln (A_\phi e^{-iQ_\phi x}) + \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} \left( \frac{A_{\bar{\phi}}^*}{A_\phi} \right)^{n} e^{2niQ_\phi x},\end{aligned}$$ where $Q_\phi=(m_\phi,{\bf 0})$. At the lowest order in $\lambda_{\rm I}$, the decay rate of the coherent state can be obtained by calculating the two-point functions of several types of operators with relevant momentum injection. For the local operator $\hat{\cal O}(x)$, let us define $$\begin{aligned} {\cal I}_{\cal O} (Q) \equiv -i \int d^4 x_1 d^4 x_2 \langle 0 | T \hat{\cal O} (x_1) \hat{\cal O} (x_2) | 0 \rangle e^{iQ(x_1-x_2)}.\end{aligned}$$ Then, by using the fact that $F_{\mu\nu}\tilde{F}^{\mu\nu}$ is expressed as a total derivative: $$\begin{aligned} F_{\mu\nu}\tilde{F}^{\mu\nu} = \partial_\mu K^\mu = \frac{1}{2} \partial_\mu \left[ \epsilon^{\mu\nu\rho\sigma} A_\nu (\partial_\rho A_\sigma) + (\mbox{gauge field})^3 \right],\end{aligned}$$ Eq. (\[&lt;phi|iT|phi&gt;\]) becomes $$\begin{aligned} \langle \phi | \hat{\cal T} | \phi \rangle = - 2 \lambda_{\rm I}^2 {\cal I}_{Q_\phi^\mu K_\mu} (0) - \lambda_{\rm I}^2 \sum_{n=1}^{\infty} \frac{1}{n^2} \left| \frac{A_{\bar{\phi}}}{A_\phi} \right|^{2n} {\cal I}_{F_{\mu\nu}\tilde{F}^{\mu\nu}} (2nQ_\phi).\end{aligned}$$ It is notable that ${\cal I}_{Q_\phi^\mu K_\mu}(0)$ has no imaginary part because there is no momentum injection into the internal gauge-boson lines from the $Q_\phi^\mu K_\mu$-vertex. Thus, the coherent state does not decay if $A_\phi=0$ or $A_{\bar{\phi}}=0$. (This statement holds even after taking into account the higher order terms in $\lambda_{\rm I}$.) This fact can be understood by the conservation of the $U(1)_A$ charge. With a fixed value of the total energy of the system, $U(1)_A$ charge is maximized when $A_\phi=0$ or $A_{\bar{\phi}}=0$. Thus, if $U(1)_A$ charge is conserved, the decay of $\phi$ in the condensation into the gauge bosons is forbidden. Of course, the interaction given in Eq.(\[phiFF-dual\]) breaks $U(1)_A$ symmetry because ${\cal L}_{\rm eff}$ is not invariant under the $U(1)_A$ transformation. This is due to the fact that ${\cal L}_{\rm eff}$ is induced by the chiral anomaly. However, we can add new fermions, which we call $Q'_L$ and $Q'^c_R$, to have conserved $U(1)$ symmetry. Indeed, with $Q'_L$ and $Q'^c_R$, which are in fundamental and anti-fundamental representation of $SU(N_{\rm c})$, respectively, we can define non-anomalous $U(1)_A$ symmetry by assigning charge $+\frac{1}{2}$ to both of $Q'_L$ and $Q'^c_R$. (Notice that $Q'_L$ and $Q'^c_R$ do not have to couple to $\phi$.) In this case, conservation of the $U(1)_A$ charge is obvious and the decay of $\phi$ into the gauge bosons is completely forbidden. Thus, the $U(1)_A$ charge stored in the scalar condensation cannot be released by the interaction given in Eq. (\[phiFF-dual\]). This fact may have some relevance in the study of the decay of scalar condensations in various cosmological scenarios, in particular, in the Affleck-Dine scenario [@Affleck:1984fy]. In the absence of $Q'_L$ and $Q'^c_R$, instanton effects may generate new interactions which explicitly breaks $U(1)_A$ symmetry. In such a case, decay of the coherent state occurs via such new interactions. Using the relation $$\begin{aligned} \Im \left[ {\cal I}_{F_{\mu\nu}\tilde{F}^{\mu\nu}} (Q) \right] = - \frac{N_{\rm c}^2 - 1}{4\pi} \left( Q^2 \right)^2 L^3 T,\end{aligned}$$ the decay rate is given by $$\begin{aligned} \Gamma_{F\tilde{F}} = \frac{{\rm Prob} (|\phi\rangle \rightarrow {\rm all})}{L^3 T} = \frac{8}{\pi} (N_{\rm c}^2 - 1) \lambda_{\rm I}^2 m_\phi^4 \sum_{n=1}^{\infty} n^2 \left| \frac{A_{\bar{\phi}}}{A_\phi} \right|^{2n}. \label{Gamma_FFdual}\end{aligned}$$ Since the decay rate vanishes if $A_\phi=0$ or $A_{\bar{\phi}}=0$, the decay of the coherent state in this case should be understood as an annihilation between $\phi$ and $\bar{\phi}$ in the condensation; same number of $\phi$ and $\bar{\phi}$ annihilate into the gauge boson pair. In the study of the decay of coherent state, the energy-loss rate is also important. Using the fact that the imaginary part of ${\cal I}_{F_{\mu\nu}\tilde{F}^{\mu\nu}}(2nQ_\phi)$ is from the decay process into two gauge bosons with the total energy of $2nm_\phi$, the energy-loss rate can be calculated. So far, we have considered the case where $A_\phi\geq A_{\bar{\phi}}$. However, the decay rate for the case of $A_\phi\leq A_{\bar{\phi}}$ is derived by interchanging $A_\phi\leftrightarrow A_{\bar{\phi}}$ in the result. Thus, we obtain $$\begin{aligned} \left[ \frac{d \rho_\phi}{dt} \right]_{F\tilde{F}} = -\frac{16}{\pi} (N_{\rm c}^2 - 1) \lambda_{\rm I}^2 m_\phi^5 \sum_{n=1}^{\infty} n^3 \left[ \frac{\min(n_\phi, n_{\bar{\phi}})} {\max(n_\phi, n_{\bar{\phi}})} \right]^{n}, \label{drho_FFdual}\end{aligned}$$ where we have used the fact that the number densities of $\phi$ and $\bar{\phi}$ are proportional to $|A_\phi|^2$ and $|A_{\bar{\phi}}|^2$, respectively. (See Eqs. (\[n\_phi\]) and (\[n\_phibar\]).) One may simplify the above energy-loss rate by using $$\begin{aligned} \sum_{n=1}^{\infty} n^3 r^n = \frac{r(1 + 4r + r^2)}{(1-r)^4}.\end{aligned}$$ Here, we emphasize that the results given in Eqs.(\[Gamma\_FFdual\]) and (\[drho\_FFdual\]) can be used for any value of the amplitude (as far as the effective mass of $Q_L$ and $Q_R^c$ are much larger than $m_\phi$). When $n_\phi\gg n_{\bar{\phi}}$ or $n_\phi\ll n_{\bar{\phi}}$, which corresponds to the case where the classical motion of the scalar condensation is almost circular, the energy-loss rate is well approximated by the leading term in Eq.(\[drho\_FFdual\]). On the contrary, in the limit of $n_{\bar{\phi}}\rightarrow n_\phi$, higher order terms become important and the energy-loss rate is enhanced. In this case, however, one should note that, at some point of the classical trajectory, $|\phi|$ approaches to the origin. Then, the effective mass of the fermions $Q_L$ and $Q_R^c$ may become so small that the effective field theory, which is obtained by integrating out these fermions, may break down. We also note that, with the ratio $n_\phi/n_{\bar{\phi}}$ being fixed, the decay and energy-loss rates are independent of the amplitude of the scalar condensation. Before closing this section, we also present the result for the case where the scalar field $\phi$ couples to the gauge field as $$\begin{aligned} {\cal L}_{\rm eff} = \lambda_{\rm R} \left( \ln \phi + \ln \phi^\dagger \right) F^{\mu\nu} F_{\mu\nu}. \label{phiFF}\end{aligned}$$ This type of interaction is also generated by integrating out particles which acquire masses from the condensation of $\phi$ (like $Q_L$ and $Q_R^c$). At the leading order in $\lambda_{\rm R}$, the energy-loss rate is given by $$\begin{aligned} \left[ \frac{d \rho_\phi}{dt} \right]_{FF} = -\frac{16}{\pi} (N_{\rm c}^2 - 1) \lambda_{\rm R}^2 m_\phi^5 \sum_{n=1}^{\infty} n^3 \left[ \frac{\min(n_\phi, n_{\bar{\phi}})} {\max(n_\phi, n_{\bar{\phi}})} \right]^{n}. \label{drho_FF}\end{aligned}$$ Summary {#sec:summary} ======= In this paper, we have discussed the decay processes of the scalar condensation. We postulated that the quantum state corresponding to the scalar oscillation is the so-called coherent state in the quantum field theory. Then, by using the $S$-matrix unitarity, we have developed the method to calculate the decay rate of the coherent state. We believe that our procedure can be applied to a large class of models which may contain various types of interactions. Then, in order to demonstrate how the decay rate is calculated, we considered two examples. First, we studied the case where the scalar field $\varphi$ couples to another scalar field $\chi$ via three-point interaction. Using the small-amplitude approximation, we have calculated the decay rate for the process where $N_\varphi$ ($N_\varphi=1,2,3,\cdots$) of $\varphi$ in the condensation simultaneously annihilate into a pair of $\chi$. For the case of $N_\varphi=1$, we have seen that the result is the same as that in the conventional approach where the decay rate of the scalar condensation is estimated by the product of the decay rate of single $\varphi$ in the vacuum and the number density of $\varphi$. We have also pointed out that the small-amplitude approximation breaks down when the amplitude becomes close to $\mu^{-1}m_\varphi^2\beta_1^2$, where $\mu$ is the coupling constant and $\beta_1$ is the velocity of $\chi$ in the $N_\varphi=1$ mode. Such a behavior is also expected from the discussion based on the parametric-resonance. Indeed, our procedure reproduced the decay rate of the scalar condensation calculated from the parametric-resonance analysis. The second example was the case where the complex scalar field decays into gauge bosons via the interaction induced by the chiral anomaly. We have considered the case where the scalar potential has $U(1)_A$ symmetry to rotate $\phi\rightarrow e^{i\alpha}\phi$ at the classical level, which is broken by the effect of the chiral anomaly. In this case, we could calculate the decay rate without using the small-amplitude approximation. We have seen that the decay process is forbidden unless both the particle $\phi$ and its anti-particle $\bar{\phi}$ exist in the condensation, and that the “decay” of the coherent state is due to the annihilation between them. Thus, $U(1)_A$ charge stored in the condensation cannot be released by the effective interaction induced by the chiral anomaly. In our analysis, the effects of the cosmic expansion were completely neglected. However, we believe that our results are applicable to the cosmological discussion as far as the expansion rate of the universe is smaller than the mass of the scalar condensation. [*Acknowledgement*]{}: This work was supported in part by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports, and Culture of Japan, No. 19540255 (T.M.). Calculation of the Imaginary Part {#app:im} ================================= Although the technique which will be explained here is well-known, in this appendix, we show how Eq. (\[Im(T)\]) is derived for the sake of some of the readers. For this purpose, we calculate the imaginary part of the following quantity: $$\begin{aligned} I_{\cal F} (Q_\varphi) \equiv - i \int \frac{d^4 \tilde{k}}{(2\pi)^4} \prod_{I=1}^{2p} ( k_I^2 - m^2_\chi + i 0^+ )^2.\end{aligned}$$ Here, $$\begin{aligned} k_I \equiv \tilde{k} + \sum_{J=1}^{I} \varepsilon_{J} Q_\varphi,\end{aligned}$$ where $\varepsilon_I=\pm 1$ (with $\varepsilon_1 + \cdots + \varepsilon_{2p}=0$), and $Q_\varphi=(m_\varphi, {\bf 0})$. In order to calculate the imaginary part of $I_{\cal F}$, it is convenient to rewrite $I_{\cal F}$ as $$\begin{aligned} I_{\cal F} = - i \lim_{\xi\rightarrow m_\chi^2} \int \frac{d^4 \tilde{k}}{(2\pi)^4} \prod_{I=1}^{2p} ( k_I^2 - \xi_I + i 0^+ )^{-1}, \label{I_F}\end{aligned}$$ where the limit $\xi\rightarrow m_\chi^2$ indicates that $\xi_I\rightarrow m_\chi^2$ ($I=1-2p$); before taking the limit, $\xi_I$ are all set to be different. Integrand of Eq. (\[I\_F\]) has poles at $\tilde{k}_0=-m_\varphi\sum_{J=1}^I\varepsilon_J\pm\sqrt{{\bf k}_I^2+\xi_I}$ ($I=1-2p$) and, after $\tilde{k}_0$-integration, $I_{\cal F}$ becomes $$\begin{aligned} I_{\cal F} &=& - \pi \lim_{\xi\rightarrow m_\chi^2} \sum_{i=1}^{2p} \int \frac{d^3 \tilde{\bf k}}{(2\pi)^4} \left[ \frac{1}{2\tilde{k}_0} \prod_{I\neq i} ( k_I^2 - \xi_I + i 0^+ )^{-1} \right]_{\tilde{k}_0= -m_\varphi\sum_{J=1}^I\varepsilon_J+\sqrt{{\bf k}_I^2+\xi_I}} \nonumber \\ &=& - \pi \lim_{\xi\rightarrow m_\chi^2} \sum_{i=1}^{2p} \int \frac{d^4 \tilde{k}}{(2\pi)^4} \delta (k_i^2 - \xi_i) \prod_{I\neq i} ( k_I^2 - \xi_I + i 0^+ )^{-1},\end{aligned}$$ where, in the second equality, the $\tilde{k}_0$-integration is performed in the region where $k_{i0}\geq 0$. Using the relation $(x+i0^+)^{-1}=P(x^{-1})-i\pi\delta (x)$ (where “$P$” is for the principal value), we obtain $$\begin{aligned} \Im \left[ I_{\cal F} \right] = 2 \pi^2 \lim_{\xi\rightarrow m_\chi^2} \sum_{i=1}^{2p-1} \sum_{j=i+1}^{2p} \int \frac{d^4 \tilde{k}}{(2\pi)^4} \delta (k_i^2 - \xi_i) \delta (k_j^2 - \xi_j) \prod_{I\neq i,j} ( k_I^2 - \xi_I )^{-1}.\end{aligned}$$ Substituting the above expression into Eq. (\[T(2p)\]), we obtain Eq. (\[Im(T)\]). With the quantity $N_\varphi=\left|\sum_{I=i+1}^{j}\varepsilon_I\right|$, $k_i$ and $k_j$ are related as $k_i=k_j+N_\varphi Q_\varphi$ (or $k_i=k_j-N_\varphi Q_\varphi$). Thus, if $N_\varphi=0$, $k_i=k_j$ and the imaginary part vanishes. Notice also that the constraints from the $\delta$-functions can be solved; by shifting the integration variable $\tilde{k}$, $k_i$ and $k_j$ can be taken to be $\tilde{k}$ and $\tilde{k}-N_\varphi Q_\varphi$, respectively. Then, the constraints from the $\delta$-functions become $\tilde{k}Q_\varphi=\frac{1}{2}N_\varphi m_\varphi^2$. Consequently, the product $\prod_{I\neq i,j} ( k_I^2 - \xi_I )^{-1}$ becomes $\tilde{k}$-independent. The remaining part is proportional to the two-body phase space for the process where the parent particle with mass $N_\varphi m_\varphi$ decays into two daughter particles with masses $\xi_i^{1/2}$ and $\xi_j^{1/2}$: $$\begin{aligned} \int \frac{d^4 \tilde{k}}{(2\pi)^4} \delta (k_i^2 - \xi_i) \delta (k_j^2 - \xi_j) = \frac{1}{32 \pi^3} B ( N_\varphi^2 m_\varphi^2; \xi_i, \xi_j ),\end{aligned}$$ where, for $\sqrt{Q^2}>\xi_i^{1/2}+\xi_j^{1/2}$, $$\begin{aligned} B ( Q^2; \xi_i, \xi_j ) \equiv \frac{1}{Q^2} \sqrt{ \left( Q^2 \right)^2 - 2 (\xi_i + \xi_j) Q^2 + (\xi_i - \xi_j)^2 }, \label{fnB}\end{aligned}$$ while $B ( Q^2; \xi_i, \xi_j )=0$ for $\sqrt{Q^2}\leq\xi_i^{1/2}+\xi_j^{1/2}$. Notice that the function $B$ is related to $\beta_{N_\varphi}$ given in Eq. (\[beta\_N\]) as $$\begin{aligned} \beta_{N_\varphi} = B ( N_\varphi^2 m_\varphi^2; m_\chi^2, m_\chi^2).\end{aligned}$$ Derivation of Eq. (\[Im\[T(beta)\]\]) {#app:smallb} ===================================== In this Appendix, we calculate the imaginary part of the following integral $$\begin{aligned} I_{{\rm Fig.}\ref{fig:loop_smallb}} = -i \int \frac{d^4 \tilde{k}}{(2\pi)^4} \left( \tilde{k}^2 - m^2_\chi + i 0^+ \right)^{-p} \left[ (\tilde{k} - Q_\varphi )^2 - m^2_\chi + i 0^+ \right]^{-p},\end{aligned}$$ to derive Eq. (\[Im\[T(beta)\]\]) from Eq. (\[T(beta)\]). Using the procedure given in Appendix \[app:im\], we express $I_{{\rm Fig.}\ref{fig:loop_smallb}}$ as $$\begin{aligned} I_{{\rm Fig.}\ref{fig:loop_smallb}} = -i \lim_{\xi^{(\prime)}\rightarrow m_\chi^2} \int \frac{d^4 \tilde{k}}{(2\pi)^4} \prod_{I=1}^p \left( \tilde{k}^2 - \xi_I + i 0^+ \right)^{-1} \prod_{J=1}^p \left[ (\tilde{k} - Q_\varphi )^2 - \xi'_J + i 0^+ \right]^{-1}.\end{aligned}$$ The imaginary part of this quantity is obtained with $$\begin{aligned} \prod_{I=1}^p \left( \tilde{k}^2 - \xi_I + i 0^+ \right)^{-1} \rightarrow -i \pi \sum_{i=1}^p \prod_{I\neq i} ( \xi_i - \xi_I )^{-1} \delta \left( \tilde{k}^2 - \xi_i \right),\end{aligned}$$ and with the similar replacement of the second product. Then, the imaginary part of $I_{{\rm Fig.}\ref{fig:loop_smallb}}$ becomes $$\begin{aligned} \Im \left[ I_{{\rm Fig.}\ref{fig:loop_smallb}} \right] = \frac{1}{16\pi} \lim_{\xi^{(\prime)}\rightarrow m_\chi^2} \sum_{i=1}^p \sum_{j=1}^p \prod_{I\neq i} ( \xi_i - \xi_I )^{-1} \prod_{J\neq j} ( \xi'_j - \xi'_J )^{-1} B(m_\varphi^2; \xi_i, \xi'_j).\end{aligned}$$ We can use the relation: $$\begin{aligned} \lim_{x_1\rightarrow x} \cdots \lim_{x_p\rightarrow x} \sum_{i=1}^p f(x_i) \prod_{j\neq i} (x_i-x_j)^{-1} = \frac{1}{(p-1)!} \frac{d^{p-1}}{dx^{p-1}} f(x),\end{aligned}$$ to obtain $$\begin{aligned} \Im \left[ I_{{\rm Fig.}\ref{fig:loop_smallb}} \right] = \frac{1}{16\pi} \frac{1}{[(p-1)!]^2} \left[ \frac{\partial^{(p-1)}}{\partial \xi^{(p-1)}} \frac{\partial^{(p-1)}}{\partial \xi'^{(p-1)}} B(m_\varphi^2; \xi, \xi') \right]_{\xi=\xi'=m_\chi^2}.\end{aligned}$$ Taking the $O(\beta_1^{1-4(p-1)})$ term from the above expression, which is the most singular one when $\beta_1\rightarrow 0$, Eq. (\[Im\[T(beta)\]\]) is derived. 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--- abstract: 'Euler’s Gamma function $\Gamma$ either increases or decreases on intervals between two consequtive critical points. The inverse of $\Gamma$ on intervals of increase is shown to have an extension to a Pick-function and similar results are given on the intervals of decrease, thereby answering a question by Uchiyama. The corresponding integral representations are described. Similar results are obtained for a class of entire functions of genus 2, and in particular integral representations for the double gamma function and the $G$-function of Barnes are found.' author: - 'Henrik L. Pedersen[^1]' title: Inverses of gamma functions --- [*MSC:*]{} primary 30D15; secondary 33B15; 30E20\ [*Keywords:*]{} Entire function, inverse, Pick-function, integral representation, gamma function, double gamma function Introduction ============ Euler’s gamma function $\Gamma$ increases on the interval $(\alpha,\infty)$, where $\alpha$ denotes the unique zero of the logarithmic derivative $\psi$ of the gamma function on the positive half line. In [@u] the inverse function of $\Gamma$ defined on $(\Gamma(\alpha),\infty)$ was shown to have an extension to a Pick-function in the cut plane $\mathbb C\setminus (-\infty,\Gamma(\alpha)]$. In [@p] this result was extended to a certain class of entire functions of genus 1. Furthermore, a version of the result was obtained for a class of entire functions of genus 2, including the double gamma function of Barnes. The construction in these papers relied among other things on the theory of positive and negative definite kernels and in particular on Löwner’s theorem linking positive definite Löwner kernels with Pick functions. This method would not seem to work when extending inverses of the gamma function on the interval $(0,\alpha)$ or for that matter on intervals of the negative line where $\Gamma$ has singularities. The first goal of this paper is to extend the inverses of the gamma function on the remaining intervals of the real line to holomorphic functions in the upper half plane and describe those as Pick-functions. This, in particular, answers a question posed in [@u] about the properties of the analytic extension of the inverse of $\Gamma$ on $(0,\alpha)$. See Corollary \[cor:uchiyama\]. We shall base our construction on results relating functions of Laguerre-Pólya class to conformal mappings of the upper half plane onto certain “comb” domains. In sections \[sec:logGamma\], \[sec:construction\] and \[sec:integral\] the case of the Gamma function is described in detail, including a description of the integral representations. See theorems \[thm:mainodd\] and \[thm:maineven\]. The method can also be used to study inverses of Laguerre-Pólya functions, but we do not aim at a complete description of the properties of inverses of Laguerre-Pólya functions. In Remark \[rem:sin\] the situation for inverses to $\sin z$ is sketched briefly. The second goal of this paper is to investigate inverses for a class of entire functions of genus 2, not in the Laguerre-Pólya class. This class includes the double gamma function and the $G$-function of Barnes. We show that inverses of these functions initially defined on a positive half-line can be extended to Pick-functions. See Section \[sec:genus2\]. A Pick-function is a holomorphic function $p$ defined in the upper half plane $\mathbb C_+$ such that $\Im p(z)\geq 0$ for all $z\in \mathbb C_+$. An entire function $f$ is said to belong to the Laguerre-Pólya class if it is the limit of a sequence of real polynomials with all zeros real. This class can be characterized in terms of the Weierstraß factorization and also in terms of conformal mappings of the upper half plane onto so-called $\mathcal V$-comb domains. Suppose that $m,k\in \mathbb Z\cup \{\pm \infty\}$ and $-\infty\leq m<k\leq \infty$. Furthermore let $\{h_j\}$ be a sequence in $[-\infty,\infty)$. The $\mathcal V$-comb domain corresponding to $m,k,\{h_j\}$ is the domain of the form $$\mathcal V=\{ z\, |\, m\pi <\Im z<k\pi\} \setminus \cup_{m<j<k}\{x+i\pi j\, |\, x\leq h_j\}.$$ The connection to conformal mapping goes back to papers of Grunsky ([@g]), Vinberg ([@v]), and MacLane ([@m]). See also the preprint [@kk]. We shall use the following fact from a recent paper by Eremenko and Yuditskii, see [@ey Theorem 2.3]: If $f$ is an entire function of Laguerre-Pólya class with Weierstraß factorization $$f(z)=z^qe^{-az^2+bz}\prod_{k=1}^{\infty}(1-z/z_k)e^{z/z_k},$$ where $a\geq 0$, $b\in \mathbb R$, $q\in \{0,1,\cdots\}$, and $\sum_{k=1}^{\infty}|z_k|^{-2}<\infty$ then $f(z)=e^{\phi(z)}$, where $\phi:\mathbb C_+\to \mathcal V$ is a conformal maping onto a $\mathcal V$-comb. The conformal mapping $\log \Gamma$ {#sec:logGamma} =================================== It is a classical fact that the reciprocal of the gamma function belongs to the Laguerre-Pólya class and admits the following Weierstraß factorization $$\frac{1}{\Gamma(z)}=ze^{\gamma z}\prod_{k=1}^{\infty}\left(1+z/k\right)e^{-z/k}.$$ The function $$\label{eq:logGamma} \log \Gamma(z)=-\gamma z-\log z-\sum_{k=1}^{\infty}\log \left(1+z/k\right)-z/k$$ is holomorphic in the cut plane $\mathbb C\setminus (-\infty,0]$ and extends $\log \Gamma(x)$ for $x>0$. (The logarithms on the right hand side of the relation are the principal logarithm defined in the cut plane.) The logarithmic derivative $\psi=\Gamma'/\Gamma$ is commonly known as the psi-function and it has the representation $$\psi(z)=-\gamma+\sum_{k=0}^{\infty}\left(\frac{1}{k+1}-\frac{1}{k+z}\right).$$ Clearly, $\psi$ increases strictly from $-\infty$ to $\infty$ on each interval $(-k,-k+1)$, for $k\geq 1$. There is thus a unique point $x_k\in (-k,-k+1)$ such that $\Gamma'(x_k)=0$. When $k$ is even $\Gamma$ attains its (positive) minimum value on $(-k,-k+1)$ at $x_k$ and when $k$ is odd its (negative) maximum at $x_k$. We put $x_0=\alpha$, the minimum point of $\Gamma$ on the positive half line. By the results above $\log \Gamma$ defines a conformal mapping of the upper half plane onto the domain $\mathcal V$ of the form $$\label{eq:V} \mathcal V=\mathbb C\setminus \cup_{k=0}^{\infty} [\log|\Gamma(x_k)|,\infty)\times \{-ik\pi\}.$$ We notice that $\log \Gamma$ extends to a continuous function on the closed upper half plane except the points $\{0,-1,-2,\cdots\}$, indeed $$\lim_{z\to x,\, z\in \mathbb C_+ }\log \Gamma(z)=\log|\Gamma(x)|-ik\pi$$ for $x\in (-k,-k+1)$, $k\geq 1$. We remark that conformal properties of the function $\log \Gamma$ in right half planes was investigated in [@ar]. In a rather old paper by Ginzel (see [@gi]) mapping properties of $\Gamma$ were also described. \[rem:argGamma\] From it follows that $$\partial_x\arg \Gamma (x+iy)=\Im \psi(x+iy)=\sum_{k=0}^{\infty}\frac{y}{(k+x)^2+y^2}$$ so for fixed $y>0$, $x\mapsto \arg \Gamma (x+iy)$ is strictly increasing. Since $$m\partial_x\arg \Gamma (m+iy)\geq m\sum_{k=0}^{m}\frac{y}{(k+m)^2+y^2}\geq \frac{ym^2}{4m^2+y^2}\geq \frac{y}{4+y^2}$$ for $m\geq 1$ it follows that $\lim_{x\to \infty}\partial_x\arg \Gamma (x+iy)=\infty$. Furthermore, using the functional equation we see that $\arg \Gamma(-m+iy)\leq -m\pi/2+\arg \Gamma(iy)$ for $m\geq 0$, so $\lim_{x\to -\infty}\partial_x\arg \Gamma (x+iy)=-\infty$. Hence $x\mapsto \arg \Gamma (x+iy)$ maps $\mathbb R$ onto $\mathbb R$. Finally, we record another consequence of : $\arg \Gamma(iy)\to \infty$ as $y\to \infty$. Construction of inverses {#sec:construction} ======================== For an integer $k$ let $S_k$ denote the strip $$S_k=\{z\in \mathbb C| \Im z\in (-(k+1)\pi,-k\pi)\}.$$ We consider $$g_k(z)=(\log \Gamma)^{-1}\left(\log z-i(k+1)\pi\right), \quad z\in \mathbb C_+.$$ This is a holomorphic function in the upper half plane. Let $\mathcal D_k=g_k(\mathbb C_+ )$ be the conformal image of $\mathbb C_+$. Then, clearly, $$\mathcal D_k=\{z\in \mathbb C_+ | \log \Gamma (z)\in S_k\}=\{z\in \mathbb C_+ | -(k+1)\pi<\arg \Gamma (z)<-k\pi\}.$$ \[prop:pick\] For all $k\in \mathbb Z$, $g_k$ is a Pick-function and $\Gamma(g_k(z))=(-1)^{k+1}z$ for $z\in \mathbb C_+ $. [*Proof.*]{} Let $\mathcal V$ be the domain defined in . Since $\log$ maps the upper half plane onto the strip $S_{-1}$, $\log z-i(k+1)\pi \in \mathcal V$ for $z\in \mathbb C_+ $, and hence $(\log \Gamma)^{-1}\left(\log z-i(k+1)\pi\right)\in \mathbb C_+ $. By construction $$\Gamma(g_k(z))=e^{\log \Gamma(g_k(z))}=e^{\log z-i(k+1)\pi}=(-1)^{k+1}z.$$ This proves the proposition. $\Box$ The restriction of $\Gamma$ to $\mathcal D_k$ is conformal. [*Proof.*]{} If $\Gamma(z_1)=\Gamma(z_2)$ for $z_1,z_2\in \mathcal D_k$ then $\log|\Gamma(z_1)|=\log|\Gamma(z_1)|$ and $\arg \Gamma(z_1)=\arg \Gamma(z_2)+2\pi l$, for some $l\in \mathbb Z$. But $\arg \Gamma(z_1)$ and $\arg \Gamma(z_2)$ can only differ at most $\pi$, since both $z_1$ and $z_2$ belong to $\mathcal D_k$. Hence $l=0$ and thus $\log \Gamma(z_1)=\log \Gamma(z_2)$. Since $\log \Gamma$ is conformal this yields $z_1=z_2$.$\Box$ The function $g_{-1}$ maps the upper half plane onto $\mathcal D_{-1}$ and is by reflection in the real line holomorphic in $\mathbb C\setminus \mathbb R$. It has a holomorphic extension to $\mathbb C\setminus (-\infty, \Gamma(x_{0})]$: this can be seen by noting that $g_{-1}$ has a continuous and real valued extension to the interval $(\Gamma(x_{0}), \infty)$ and then conclude using Morera’s theorem. One can also use an argument based on the fact that $\Gamma$ has a local inverse at each point in $(\Gamma(x_{0}), \infty)$. See the proof of Proposition \[prop:G\_k\]. Uchiyama calls the function $g_{-1}$ the principal inverse of the gamma function. From Proposition \[prop:pick\] the result of Uchiyama follows: The principal inverse of $\Gamma$ is a Pick-function. We shall go through similar constructions on the remaining intervals $(x_{k+1},x_k)$ for $k\geq 0$. We notice that $\Gamma(x_{2l-1})<0<\Gamma(x_{2l})$ for $l\geq 0$ and define the interval $I_k$ as $$I_k=\left\{ \begin{array}{ll} [\Gamma(x_k),\Gamma(x_{k+1})], & k\ \text{odd}\\ { } [-\Gamma(x_{k}),-\Gamma(x_{k+1})], & k\ \text{even.} \end{array} \right.$$ When $k$ is odd, $\Gamma$ increases on each of the intervals $(x_{k+1},-k)$ and $(-k,x_k)$, and $\Gamma((x_{k+1},-k)\cup(-k,x_k))=\mathbb R\setminus I_k$. When $k$ is even, $\Gamma$ decreases on $(x_{k+1},-k)$ and on $(-k,x_k)$, and $\Gamma((x_{k+1},-k)\cup(-k,x_k))=\mathbb R\setminus (-I_k)$. For $k\geq 0$ $g_k$ is extended by reflection in the real line to a holomorphic function in $\mathbb C\setminus \mathbb R$ and mapping $\mathbb C\setminus \mathbb R$ onto $\mathcal D_k\cup \overline{\mathcal D_k}$. We now show that $g_k$ can be extended across $\mathbb R\setminus I_k$ where $I_k$ is given above. \[prop:G\_k\] Let $k\geq 0$. The function $g_k$ has a holomorphic extension $G_k$ to $\mathbb C\setminus I_k$ satisfying $G_k(\overline{z})=\overline{G_k(z)}$. [*Proof.*]{} Assume that $k$ is odd. (The argument for $k$ even is similar.) Let $y$ be any real number less than $\Gamma(x_k)$. There exists a unique $x\in (-k,x_k)$ such that $\Gamma(x)=y$. Since $\Gamma'(x)\neq 0$, $\Gamma$ is locally one-to-one and thus there are neighbourhoods $U$ of $x$ and $V$ of $y$ such that for any $w\in V$ there is a unique $z\in U$ satisfying $\Gamma(z)=w$ and the mapping $\Gamma^{-1}:w\mapsto z$ is holomorphic in $V$. Since $\Gamma^{-1}(V\cap \mathbb R)\subseteq \mathbb R$ it follows that $\Gamma^{-1}(\overline{w})=\overline{\Gamma^{-1}(w)}$ for $w\in V$. A similar argument shows that for any $y>\Gamma(x_{k+1})$ there exist $x\in (x_{k+1},-k)$ and neighbourhoods of $U$ of $x$ and $V$ of $y$ and a holomorphic function $\Gamma^{-1}: V\to U$ such that $\Gamma^{-1}(\overline{w})=\overline{\Gamma^{-1}(w)}$ for $w\in V$. Defining $G_k(w)$ as $g_k(w)$ for $w$ in the upper half plane, as $\overline{g_k(\overline{w})}$ for $w$ in the lower half plane and as $\Gamma^{-1}(w)$ for $w\in \mathbb R\setminus I_k$ it follows that $G_k$ is holomorphic in $\mathbb C\setminus I_k$. $\Box$ \[rem:sin\] The trigonometric functions $\sin$ and $\cos$ are also members of the Laguerre-Pólya class. It follows that $\log \sin z$ (defined via the infinite product and the principal logarithms) maps $\mathbb C_+$ conformally onto the $\mathcal V$-comb $$\mathbb C\setminus \left\{\cup_{k\in\mathbb Z}(-\infty,0]\times \{ik\pi\}\right\}.$$ Hence, an inverse to $\sin$ can be constructed by $g(z)=(\log \sin)^{-1}(\log z)$ and this inverse maps $\mathbb C_+$ onto the half strip $\{ |\Re z|< \pi/2, \Im z>0\}$. The function $g$ can be representated in terms of the inverse to the Joukowski transformation: The Joukowski transformation is defined by $J(w)=(w+1/w)/2$ and it maps $\mathbb C_+\setminus \{|z|\leq 1\}$ conformally onto $\mathbb C_+$. Hence $$z\mapsto i\log(J^{-1}(z))+\pi/2$$ also maps $\mathbb C_+$ conformally onto $\{ |\Re z|< \pi/2, \Im z>0\}$. It is easily seen that the two maps agree on $i\mathbb R_+$ and therefore $g(z)=i\log(J^{-1}(z))+\pi/2$ is an inverse to $\sin$ when extended holomorphically from $(-1,1)$ to $(-\pi/2,\pi/2)$. (This formula is not new, see e.g. [@gr 1.622].) Integral representation {#sec:integral} ======================= It is well known that any Pick function $p$ has an integral representation of the form $$p(z)=az+b+\int_{-\infty}^{\infty}\left(\frac{1}{t-z}-\frac{t}{t^2+1}\right)\, d\mu(t),$$ where $a\geq 0$, $b\in \mathbb R$, and $\mu$ is a positive measure on $\mathbb R$ such that $$\int_{-\infty}^{\infty}\frac{d\mu(t)}{t^2+1}<\infty.$$ Furthermore, $a=\lim_{y\to \infty}p(iy)/y$, $b=\Re p(i)$, and $$\mu =\frac{1}{\pi}\lim_{y\to 0_+}\Im p(x+iy)dx$$ in the vague topology. For the function $p=g_{-1}$, the corresponding measure $\mu_{-1}$ is concentrated on $(-\infty, \Gamma(x_{0})]$ and for $k\geq 0$ the measure $\mu_k$ corresponding to $g_{k}$ is supported on $I_k$. We investigate properties of these measures in more detail. \[lemma:mu\_k\] The measure $\mu_k$ in the representation $$g_k(z)=a_kz+b_k+\int_{-\infty}^{\infty}\left(\frac{1}{t-z}-\frac{t}{t^2+1}\right)\, d\mu_k(t),$$ is supported on $I_k$ and on $I_k\setminus \{0\}$ it has a $C^{\infty}$-density $d_k$ given as $$\label{def:d_k} d_k(t)\equiv \frac{1}{\pi}\Im g_k(t+0i), \quad t\in I_k\setminus\{0\}.$$ [*Proof.*]{} We shall give the proof in the situation where $k\geq 0$ is odd (in which case $\Gamma(x_k)<0<\Gamma(x_{k+1})$). It is convenient to consider two holomorphic extensions of the function $g_k$. The first one is defined by $$h^+_k(z)=\left(\log \Gamma\right)^{-1}(\log z-i\pi (k+1)).$$ This is a holomorphic extension of $g_k$ to $\mathbb C\setminus \left((-\infty, 0]\cup [\Gamma(x_{k+1}),\infty)\right)$. The second one is defined as $$h^-_k(z)=\left(\log \Gamma\right)^{-1}(\log(-z)-i\pi k),$$ and since $\log (-z)=\log (z)-i\pi$ for $z\in \mathbb C_+ $, $h^-_k$ is a holomorphic extension of $g_k$ to $\mathbb C\setminus \left((-\infty, \Gamma(x_{k})]\cup [0,\infty)\right)$. Let $\phi$ be a continuous function of compact support in $I_k\setminus \{0\}$. Then, using dominated convergence, $$\begin{aligned} \lim_{y\to 0_+}\frac{1}{\pi}&\int_{-\infty}^{\infty}\phi(x)\Im g_k(x+iy)dx\\ =&\int_{\Gamma(x_{k})}^0\phi(x) h^-_k(x)dx+\int_0^{\Gamma(x_{k+1})}\phi(x) h^+_k(x)dx. \end{aligned}$$ Therefore $\mu_k$ has a $C^{\infty}$-density $d_k$ wrt. Lebesgue measure given as $d_k(t)=\Im g_k(t+0i)/\pi$ for $t\in I_k\setminus\{0\}$. (The density is of course integrable wrt. Lebesgue measure on $I_k$.) $\square$ \[prop:d\_k-increasing\] For odd $k\geq 1$, the function $d_k$ increases from $0$ to $\infty$ on $I_k\cap (-\infty,0)$ and decreases from $\infty$ to 0 on $I_k\cap (0,\infty)$. [*Proof.*]{} Since $\Gamma(g_k(z))=z$ we infer that $g_k'(z)=1/(\psi(g_k(z))z)$, and since $\psi$ is a Pick-function, $\partial_t\Im g_k(t)$ is negative for $t\in I_k\cap (0,\infty)$. Let $y_0$ be a given positive number. From Remark \[rem:argGamma\] it follows that we may choose $x_0\in \mathbb R$ such that $\arg \Gamma(x_0+iy_0)=-\pi(k+1)$. Hence $\log \Gamma(x_0+iy_0)=\log |\Gamma(x_0+iy_0)|-i\pi(k+1)\in \mathcal V$, where $\mathcal V$ is given in . Since $k+1$ is even it follows that $\Gamma(x_0+iy_0)\in (0,\Gamma(x_{k+1}))$ and also $\Im g_k(\Gamma(x_0+iy_0))=y_0$. Similar arguments show that $\Im g_k$ is increasing from 0 to $\infty$ to the left of the origin.$\Box$ \[thm:mainodd\] For odd $k\geq 1$ the function $g_k$ has the integral representation $$g_k(z)=\int_{\Gamma(x_k)}^{\Gamma(x_{k+1})}\frac{d_k(t)}{t-z}dt-k, \quad z\in \mathbb C\setminus I_k,$$ where $d_k$ is given in . [*Proof.*]{} From Lemma \[lemma:mu\_k\] we have $$g_k(z)=a_kz+b_k+\int_{\Gamma(x_k)}^{\Gamma(x_{k+1})}\frac{d_k(t)}{t-z}dt-\frac{c_k}{z}, \quad z\in \mathbb C\setminus I_k,$$ where $a_k\geq 0$, $b_k\in \mathbb R$, and $c_k\geq 0$. Thus $g_k$ is strictly increasing on $\mathbb R\setminus I_k$. Since $\lim_{x\to \infty}g_k(x)=-k$ we obtain $a_k=\lim_{x\to \infty}g_k(x)/x=0$ and then $b_k=-k$. The possible point mass $c_k$ at the origin can be computed via $$c_k=\lim_{y\to 0+}y\Im g_k(iy).$$ We claim that $c_k=0$. Suppose that $c_k>0$. Then $\Im g_k(iy)\to \infty$ as $y\to 0_+$. From this we obtain that $\Re g_k(iy)\leq 0$ for all $y$ sufficiently close to $0$: if for arbitrary small and positive $y$ we would have $\Re g_k(iy)>0$ then $$\arg \Gamma(\Re g_k(iy)+i\Im g_k(iy))>\arg \Gamma(i\Im g_k(iy))$$ since $\arg \Gamma$ increases on horizontal lines (see Remark \[rem:argGamma\]). By the same remark, $ \arg \Gamma(iY)\to \infty$ as $Y\to \infty$ and this contradicts the fact that $$-(k+1)\pi<\arg \Gamma(g_k(iy))<-k\pi.$$ Hence we are in the situation that $\Im g_k(iy)\to \infty$ and $\Re g_k(iy)\leq 0$. The next step is to use Stirling’s formula $$\log \Gamma(w)=\log \sqrt{2\pi}+(w-1/2)\log w-w+\mu(w),$$ where $$\mu(w)=\frac{1}{2}\int_0^{\infty}\frac{Q(t)}{(w+t)^2}\,dt,$$ and $Q$ is the 1-periodic function defined by $Q(t)=t-t^2$ for $t\in (0,1)$. It is a fact that $|\mu(w)|\leq \pi/8$ for $w\in \mathbb C_+ \setminus \{\Re w\leq 1, 0<\Im w\leq 1\}$. By construction of $g_k$ we have $$\log \Gamma(g_k(z))=\log z-i(k+1)\pi.$$ The idea is now to plug $z=iy$ into this relation, and take real parts on both sides. It gives us $$\begin{aligned} \log y=&\log \sqrt{2\pi}+(\Re g_k(iy)-1/2)\log |g_k(iy)|\\ &-\Im g_k(iy)\arg g_k(iy)-\Re g_k(iy)+\Re \mu(g_k(iy)).\end{aligned}$$ After multiplication by $y$ and some rearrangements we obtain: $$\begin{aligned} y\Im g_k(iy)\arg g_k(iy)=&\, -y\log y+y\log \sqrt{2\pi}-\frac{y}{2}\log |g_k(iy)|+y\Re \mu(g_k(iy))\\ &\, +y\Re g_k(iy)(\log |g_k(iy)|-1).\end{aligned}$$ Since $|g_k(iy)|\to \infty$, $\Re g_k(iy)\leq 0$ as $y\to 0_+$ this yields $$y\Im g_k(iy)\arg g_k(iy)\leq |y\log y|+y\log \sqrt{2\pi}+\pi y/8.$$ Since, furthermore, $\arg (g_k(iy))\in [\pi/2,\pi)$ this gives an upper bound on $ y\Im g_k(iy)$, from which we see that this quantity must tend to zero. This is a contradiction and we have shown that $c_k=0$. $\Box$ From Theorem \[thm:mainodd\] it follows that, by monotone convergence, $$\int_{\Gamma(x_k)}^{\Gamma(x_{k+1})}\frac{d_k(t)}{x-t}dt\uparrow \int_{\Gamma(x_k)}^{\Gamma(x_{k+1})}\frac{d_k(t)}{\Gamma(x_{k+1})-t}dt$$ as $x\downarrow \Gamma(x_{k+1})$ and at the same time $g_k(x)\downarrow x_{k+1}$. Thus $$x_{k+1}=\int_{\Gamma(x_k)}^{\Gamma(x_{k+1})}\frac{d_k(t)}{t-\Gamma(x_{k+1})}dt -k$$ Similarly, $$x_{k}=\int_{\Gamma(x_k)}^{\Gamma(x_{k+1})}\frac{d_k(t)}{t-\Gamma(x_{k})}dt-k.$$ The integral representation of $g_k$ can thus be written as $$\begin{aligned} g_k(z)&=x_{k+1}+(z-\Gamma(x_{k+1}))\int_{\Gamma(x_k)}^{\Gamma(x_{k+1})}\frac{d_k(t)dt}{(t-z)(t-\Gamma(x_{k+1}))}\\ &=x_{k}+(z-\Gamma(x_{k}))\int_{\Gamma(x_k)}^{\Gamma(x_{k+1})}\frac{d_k(t)dt}{(t-z)(t-\Gamma(x_{k}))}.\end{aligned}$$ \[prop:end\_point\] For odd $k\geq 1$ both $d_k(\Gamma(x_k)+s)$ and $d_k(\Gamma(x_{k+1})-s)$ are $O(s^{1/2})$ as $s\to 0_+$. [*Proof.*]{} Since $\Gamma'(x_{k+1})=0$ and $\Gamma''(x_{k+1})>0$ there exists a holomorphic function $m(z)$ in a neighbourhood of $x_{k+1}$ such that $$\Gamma(z)=\Gamma(x_{k+1})+m(z)^2$$ and $m'(x_{k+1})>0$. The function $m$ is locally one-to-one so there exist neighbourhoods $U$ of $x_{k+1}$ and $V$ of $0$ and a holomorphic function $p:V\to U$ such that $p(0)=x_{k+1}$, $p'(0)>0$ and $m(p(w))=w$. This gives $$\Gamma(p(w))=\Gamma(x_{k+1})+w^2.$$ Since $p'(0)>0$ we may assume that $V$ is chosen so that the points in $V$ intersected with the first quardrant are mapped by $p$ into the domain $\mathcal D_k$. By construction of the inverse $g_k$, $\Gamma(g_k(w^2+\Gamma(x_{k+1})))=\Gamma(x_{k+1})+w^2$ for $w\in V$ having positive real part. Since $\Gamma$ is conformal in $\mathcal D_k$ we must have $g_k(w^2+\Gamma(x_{k+1}))=p(w)$ for all $w\in V$ having positive real part. Letting $w\to i\sqrt{t}$ for small $t>0$ we obtain $\pi d_k(-t+\Gamma(x_{k+1}))=p(i\sqrt{t})=O(\sqrt{t})$. The investigation at the other end point $\Gamma(x_k)$ is similar, writing $\Gamma(z)=\Gamma(x_{k})-m(z)^2$, where $m$ is chosen so that $m'(x_{k})<0$.$\Box$ In the rest of this section the situation where $k$ is even is briefly described. \[thm:maineven\] For even $k\geq 0$ the function $g_k$ has the integral representation $$g_k(z)=\int_{-\Gamma(x_{k})}^{-\Gamma(x_{k+1})}\frac{d_k(t)}{t-z}dt-k, \quad z\in \mathbb C\setminus I_k,$$ where $d_k$ is given as $d_k(t)=\frac{1}{\pi}\Im g_k(t+i0)$. By considering the function $e_k(z)=g_k(-z)$ in stead of $g_k$ we find: \[cor:uchiyama\] For even $k\geq 0$ the inverse of $\Gamma: (x_{k+1},-k)\cup (-k,x_k)\to (-\infty, \Gamma(x_{k+1}))\cup (\Gamma(x_k),\infty)$ has an extension $e_k$ to $\mathbb C\setminus [\Gamma(x_{k+1}),\Gamma(x_k)]$ with the representation $$e_k(z)=\int_{\Gamma(x_{k+1})}^{\Gamma(x_{k})}\frac{d_k(-t)}{z-t}dt-k, \quad z\in \mathbb C\setminus [\Gamma(x_{k+1}),\Gamma(x_k)].$$ (The imaginary part of $e_k$ is negative in the upper half plane.) This corollary describes for $k=0$ the extension $e_0$ of the inverse of the restriction of $\Gamma$ to $(0,\Gamma(x_0))$ and thus answers the question posed in [@u]. Entire functions of genus 2 {#sec:genus2} =========================== Let $f$ be an entire function having the representation $$\label{eq:G-class} f(z)=z^re^{az^2+bz}\prod_{k=1}^{\infty}\left(1+\frac{z}{\lambda_k}\right)e^{-z/\lambda_k+z^2/(2\lambda_k^2)},$$ where $r$ is a non negative integer, $a$ and $b$ are real numbers and the sequence $\{\lambda_k\}$ satisfies $0<\lambda_1\leq \lambda_2\leq \cdots$, and $$\sum_{k=1}^{\infty}\frac{1}{\lambda_k^2}=\infty,\quad \sum_{k=1}^{\infty}\frac{1}{\lambda_k^3}<\infty.$$ This is expressed as the sequence $\{\lambda_k\}$ be of rank 2. The holomorphic function $\log f$ is defined in the cut plane as $$\label{eq:logf} \log f(z)=r\log z +az^2+bz+\sum_{k=1}^{\infty}\log \left(1+\frac{z}{\lambda_k}\right)-\frac{z}{\lambda_k}+\frac{z^2}{2\lambda_k^2}.$$ In [@p Lemma 3.1] it was shown that if $r>0$ then there exists $u>0$ with the property that $(\log f)''(u)=0$, $\log f(x)$ is concave for $0<x<u$ and convex for $x>u$. From this a mapping property can be deduced: \[lemma:conformal\] Let $f$ satisfy and suppose that $r>0$. Let $u$ denote the unique point in $(0,\infty)$ such that $(\log f)''(u)=0$. Then: 1. $(\log f)'$ is univalent in $\{\Re z>u\}$ and maps $\{\Re z>u\}\cap \mathbb C_+$ into $\mathbb C_+$; 2. $\log f$ is univalent in $\{\Re z>u\}\cap \mathbb C_+$ and in $\{\Re z>u\}\cap \mathbb C_-$. [*Proof.*]{} From the representation of $\log f$ it follows that $$\begin{aligned} \Im ( (\log f)'(&x+iy))\nonumber\\ =y &\left(-\frac{r}{x^2+y^2}+2a+\sum_{k=1}^{\infty}\left(\frac{1}{\lambda_k^2}-\frac{1}{(x+\lambda_k)^2+y^2}\right)\right), \label{eq:imlogf'}\end{aligned}$$ and also that $$\label{eq:logf''} (\log f)''(x+iy) =-\frac{r}{(x+iy)^2}+2a+\sum_{k=1}^{\infty}\left(\frac{1}{\lambda_k^2}-\frac{1}{(x+iy+\lambda_k)^2}\right).$$ Therefore, assuming $y\geq 0$, $$\begin{aligned} \Im \left( (\log f)'(x+iy)\right) &\geq y\left(-\frac{r}{x^2}+2a+\sum_{k=1}^{\infty}\left(\frac{1}{\lambda_k^2}-\frac{1}{(x+\lambda_k)^2}\right)\right)\\ &= y\, (\log f)''(x).\end{aligned}$$ Thus convexity of $\log f$ on $(u,\infty)$ yields that $(\log f)'$ maps $\{\Re z>u\}\cap \mathbb C_+$ into $\mathbb C_+$ and hence also $\{\Re z>u\}\cap \mathbb C_-$ into $\mathbb C_-$. From it follows that $(\log f)''$ maps $\{\Re z>0\} \cap \mathbb C_+$ into $\mathbb C_+$. For $z_1,z_2\in \{\Re z>0\} \cap \mathbb H_+$, $$(\log f)'(z_2)-(\log f)'(z_1)=(z_2-z_1)\int_0^1(\log f)''(z_1+t(z_2-z_1))\, dt$$ and hence $(\log f)'$ is univalent in $\{\Re z>0\} \cap \mathbb C_+$. A similar argument shows that $(\log f)'$ is univalent in $\{\Re z>0\} \cap \mathbb C_-$. Furthermore, $(\log f)'$ is increasing on $(u,\infty)$ and all together it means that $(\log f)'$ is univalent in $\{\Re z>0\}$. Finally, for $z_1,z_2\in \{\Re z>u\} \cap \mathbb C_+$, $$\label{eq:logf-int} \log f(z_2)-\log f(z_1)=(z_2-z_1)\int_0^1(\log f)'(z_1+t(z_2-z_1))\, dt$$ and as before this yields that $\log f$ is univalent in $\{\Re z>u\} \cap \mathbb C_+$ and hence also in $\{\Re z>u\} \cap \mathbb C_-$. $\square$ A function $f$ with the representation either increases on the positive real line or decreases on some open bounded interval $(\alpha,\beta)$, where $0<\alpha$ and increases on $(0,\infty)\setminus (\alpha,\beta)$. (See again [@p Lemma 3.1].) A function $f$ satisfying which decreases on some interval of the positive real line is said to belong to the class $\mathcal G$. Suppose that $f$ has the representation with $r>0$. Then $f\in \mathcal G$ if and only if $(\log f)'(u)<0$ where $u>0$ is the point at which $(\log f)''(u)=0$. If $(\alpha,\beta)$ is the interval of decrease of $f\in \mathcal G$ (with $r>0$) then $u$ belongs to $(\alpha,\beta)$. It also follows that $\log f$ attains its minimum value on $[\beta,\infty)$ at $\beta$. \[lemma:conformal-onto\] The holomorhic function $\log f$ maps $[\beta,\infty)$ onto $[\log f(\beta),\infty)$, and the vertical line $\beta+iy$, $y\in [0,\infty)$ is mapped to a curve where both the real and imaginary parts decrease to $-\infty$ as $y\to \infty$. Furthermore, $\log f$ maps the domain $\{\Re z>\beta\} \cap \mathbb C_+$ conformally onto the domain $\Omega=\Omega_1\cup \Omega_2$, where $$\Omega_1=\{w\in \mathbb C\,|\, \Re w\geq \log f(\beta), \Im w>0\}$$ and $$\Omega_2=\{w\in \mathbb C\,|\, \exists y>0: \Re w=\Re \log f(\beta+iy), \Im w>\Im \log f(\beta+iy) \}.$$ In the proof of this lemma we need a lower bound on $|\log f|$ for $f$ satisfying on quarter circles: \[lemma:quarter-circle\]Suppose that $f$ satisfies . Then $$\inf\{ |\log f(Re^{i\theta})| \, | \, \theta\in[0,\pi/2]\} \to \infty \quad \text{as}\ R\to \infty.$$ [*Proof.*]{} We denote by $n(t)=\#\{ k| \lambda_k\leq t\}$ the zero counting function associated with the sequence $\{\lambda_k\}$. Since $\sum_k1/\lambda_k^3<\infty$ and $\sum_k1/\lambda_k^2=\infty$ it follows that $$\int_0^{\infty}\frac{n(t)}{t^4}\, dt<\infty, \quad \int_0^{\infty}\frac{n(t)}{t^3}\, dt=\infty \quad \text{and}\quad \frac{n(t)}{t^3}\to 0 \text{ for } t\to \infty.$$ Now consider the canonical product $P$ occuring in . For $z$ in the cut plane we can write $\log P(z)$ as a Stieltjes integral w.r.t. the zero counting function, and use integration by parts in order to obtain $$\log P(z)=z^3\int_0^{\infty}\frac{n(t)}{t^3}\frac{1}{t+z}\, dt.$$ (See e.g. [@levin Lecture 12].) This gives $$\log f(z)=r\log z+bz+z^2\left(\int_0^{\infty}\frac{n(t)}{t^3}\frac{z}{t+z}\, dt+a\right),$$ so that $$|\log f(z)|\geq |z^2|\left|\int_0^{\infty}\frac{n(t)}{t^3}\frac{z}{t+z}\, dt+a\right| -r|\log z|-|b||z|.$$ The idea is to obtain a strictly positive lower bound on the factor containing the integral. A computation shows that (for $z=Re^{i\theta}$) $$\frac{z}{z+t}=\frac{R^2+Rt\cos \theta+iRt\sin \theta}{R^2+t^2+2Rt\cos \theta}.$$ This yields (for $\theta\in [0,\pi/2]$) $$\begin{aligned} \left|\int_0^{\infty}\frac{n(t)}{t^3}\frac{z}{t+z}\, dt+a\right| &\geq \int_0^{\infty}\frac{n(t)}{t^3}\Re \left(\frac{z}{t+z}\right)\, dt+a\\ &=\int_0^{\infty}\frac{n(t)}{t^3}\left(\frac{R^2+Rt\cos \theta}{R^2+t^2+2Rt\cos \theta}\right)\, dt+a\\ &\geq \int_0^{\infty}\frac{n(t)}{t^3}\left(\frac{R^2}{R^2+t^2+2Rt}\right)\, dt+a\\ &= \int_0^{\infty}\frac{n(t)}{t^3}\left(\frac{R}{R+t}\right)^2\, dt+a.\end{aligned}$$ By monotone convergence, $$\int_0^{\infty}\frac{n(t)}{t^3}\left(\frac{R}{R+t}\right)^2\, dt\uparrow \int_0^{\infty}\frac{n(t)}{t^3}\, dt=\infty.$$ Since $|z^2|$ dominates both $|z|$ and $|\log z|$ the proof is completed.$\square$ [*Proof of Lemma \[lemma:conformal-onto\].*]{} Let $\iota$ be the simple closed curve consisting of a straight line from $\beta$ to $R$, a circular part from $R$ to the point of intersection of the circle $|z|=R$ with the vertical line $x=\beta$ in the upper half plane, and finally the vertical line segment from the point of intersection to the point $\beta$. The function $\log f$ is one-to-one in $\{\Re z\geq \beta\}\cap \{\Im z\geq0\}$: This can be seen using Lemma \[lemma:conformal\] in $\{\Re z> u\}\cap \{\Im z>0\}$ and noting that holds for $z_1,z_2\in \{\Re z\geq \beta\}\cap \{\Im z\geq0\}$ and $\log f$ is one-to-one in $[\beta,R]$. The image curve $\kappa$ of $\iota$ under the mapping $\log f$ is thus another simple closed curve. Let $w$ belong to the bounded region bounded by the curve $\kappa$. Then $$\frac{1}{2\pi i}\int_{\iota}\frac{(\log f)'(z)}{\log f(z)-w}dz=\frac{1}{2\pi i}\int_{\kappa}\frac{1}{\zeta-w}d\zeta=1,$$ and therefore the function $\log f-w$ has exactly one zero inside $\iota$. (This is a version of a theorem of Darboux, see e.g. [@osgood].) Thus, the conformal image of the region bounded by $\iota$ under $\log f$ is the region bounded by $\kappa$. The image of the interval $[\beta,\infty)$ under $\log f$ is the interval $[\log f(\beta),\infty)$. Let $\log f=U+iV$. Since $\partial_xU=\partial_yV$ and $U$ is increasing at $x=R$ we must have $V(R+iy)>0$ for $y>0$ close to $0$. Thus the circular arc of $\iota$ close to $R$ is mapped into the upper half plane. Next we consider the image of the vertical line $\beta+i(0,\infty)$. For $x>0$ we have from $$\begin{aligned} \partial^2_{yy}V(x+iy)&=-\partial^2_{xx}V(x+iy)\\ &=-y\left(\frac{2xr}{(x^2+y^2)^2}+\sum_{k=1}^{\infty}\frac{2(x+\lambda_k)}{((x+\lambda_k)^2+y^2)^2}\right)<0.\end{aligned}$$ In particular, $\partial_yV(\beta+iy)$ is decreasing as a function of $y>0$ and thus $$\partial_yV(\beta+iy)<\partial_yV(\beta)=\partial_xU(\beta)=(\log f)'(\beta)=0.$$ This means that $V(\beta+iy)$ decreases from its value $0$ at $y=0$. We conlude that the curve $\log f(\beta+i(0,\infty))$ lies in the lower half plane. The relation yields $$\partial_{y}U(x+iy)=y\left(\frac{r}{x^2+y^2}-2a+\sum_{k=1}^{\infty}\left(\frac{1}{(x+\lambda_k)^2+y^2}-\frac{1}{\lambda_k^2}\right)\right),$$ and so in particular $\partial_{y}U(\beta+iy)/y$ is decreasing as a function of $y>0$. Its value at $y=0$ is equal to $$\frac{r}{\beta^2}-2a+\sum_{k=1}^{\infty}\left(\frac{1}{(\beta+\lambda_k)^2}-\frac{1}{\lambda_k^2}\right),$$ which is exactly $-(\log f)''(\beta)$, a negative quantity because $\beta>u$. Therefore $\partial_{y}U(\beta+iy)<0$ for all $y>0$ and hence $U(\beta+iy)$ decreases for $y>0$. Both the real- and imaginary part of the curve $\log f(\beta+iy)$, $y>0$ are therefore decreasing functions. To finish the proof of the asserted description of the conformal image of $\{\Re z>\beta\}\cap \mathbb C_+$ it suffices to take an arbitrary $w\in \Omega$ and show that $w$ is inside the region bounded by $\kappa$ for sufficiently large $R$. By Lemma \[lemma:quarter-circle\], $|\log f(Re^{i\theta})|>|w|$ uniformly for $0\leq \theta\leq \pi/2$, when $R$ is chosen sufficiently large. The image of the circular part of $\iota$ is thus a curve in $\{|z|>|w|\}$ conneting the point $\log f(R)$ and a point in the lower half plane. As mentioned above the curve $\log f(\beta+iy)$ belongs to the upper half plane for small $y>0$ and by conformality it cannot cross the real half line $(\log f(\beta), \infty)$. This means that the curve must surround $w$. $\square$ \[thm:G-main\] Let $f\in \mathcal G$ with $r>0$ and suppose that it has a local minimum at $\beta$, $\beta>0$. Then $f$ has an inverse function $f^{-1}$ defined on $(f(\beta),\infty)$ which can be extended to a univalent Pick-function defined in the cut plane $\mathbb C\setminus (-\infty,f(\beta)]$. [*Proof.*]{} From Lemma \[lemma:conformal-onto\] we know that $\log f$ is a conformal mapping of $\{\Re z>\beta\}\cap \mathbb C_+$ onto $\Omega$. Since the strip $S=\{0<\Im w<\pi\}$ is a subset of $\Omega$ it follows that $f^{-1}\equiv (\log f)^{-1}(\log w)$ maps the upper half plane into the domain $\{\Re z>\beta\}\cap \mathbb C_+$, and in particular into the upper half plane. Hence $f^{-1}$ is a Pick-function and we also have $f((\log f)^{-1}(\log w))=w$. $\square$ Not only $f^{-1}$ but also the function $w\mapsto (f^{-1}(w)-\beta)^2$ is a Pick function. This is seen noting that $f^{-1}$ maps the upper half plane into the quardrant $\{ \Re z>\beta, \Im z>0\}$. If $(\log f)'(u)=0$ Theorem \[thm:G-main\] still holds (with $\beta=u$). However, if $(\log f)'(u)>0$ the conformal image of the quardrant $\{ \Re z>u\}\cap \mathbb C_+$ does not cover the entire strip $\{ 0<\Im w<\pi\}$ and in this situation it is not possible to extend the inverse to a Pick-function. Let $f\in \mathcal G$ with $r>0$ and let $\beta>0$ be a local minimum point for $f$. Let $f^{-1}$ be the inverse defined on $(f(\beta), \infty)$ and extended to a Pick function. Then, for $w\in \mathbb C\setminus (-\infty, f(\beta)]$, $$f^{-1}(w)=\beta+(w-f(\beta))\int_{0}^{\infty}\frac{d(t)}{t+w-f(\beta)}\, dt-\frac{c}{w}$$ where $c\geq 0$, and $d(s)$ is positive and real analytic on $(0,f(\beta))\cup (f(\beta),\infty)$ satisfying $\int_0^{\infty}d(t)/(t+1)<\infty$. [*Sketch of Proof.*]{} Since $w\mapsto f^{-1}(w+f(\beta))-\beta$ is a Pick-function which can be analytically extended across the positive real line, and which maps the positive real line into itself it has the representation $$f^{-1}(w+f(\beta))-\beta=aw+w\int_{0}^{\infty}\frac{d\sigma(t)}{t+w}$$ where $a=\lim_{x\to \infty}f^{-1}(x+f(\beta))/x\geq 0$, and $\sigma$ is a positive measure on $[0,\infty)$ satisfying $\int_0^{\infty}d\sigma(t)/(t+1)<\infty$. See [@b Theorem 2.4]. It readily follows that $a=0$. It can also be shown that $\sigma$ has an analytic density on $(0,f(\beta))\cup (f(\beta),\infty)$, and that this leaves the possibility of a point mass only at $f(\beta)$. $\square$ As also mentioned in [@p], the motivation for studying the class $\mathcal G$ and the results above comes from the multiple gamma functions introduced by Barnes, see [@b1]. The double gamma function is usually denoted by $\Gamma_2$ and it is defined as $\Gamma_2(z)=(2\pi)^{z/2}/G(z)$, where $$G(z+1)=(2\pi)^{z/2}e^{-((1+\gamma)z^2+z)/2}\prod_{k=1}^{\infty}\left( 1+\frac{z}{k}\right) ^ke^{-z+z^2/2k}.$$ The entire function $G$ is the $G$-function of Barnes and it satisfies the relation $G(z+1)=\Gamma(z)G(z)$. The following hold 1. Barnes $G$-function belongs to $\mathcal G$ and increases on the interval $(\beta_G,\infty)$. Its inverse maps $(G(\beta_G),\infty)$ to $(\beta_G,\infty)$ and it can be extended to a univalent Pick-function. 2. The reciprocal to Barnes double gamma function $\Gamma_2$ belongs to $\mathcal G$ and increases on the interval $(\beta_2,\infty)$. Its inverse maps $(1/\Gamma_2(\beta_2),\infty)$ to $(\beta_2,\infty)$ and it can be extended to a univalent Pick-function. According to [@o] and [@p] both $G$ and $1/\Gamma_2$ belong to the class $\mathcal G$ and therefore the corollary follows. Numerical experiments yield that $\beta_G\approx 2.568$, $G(\beta_G)\approx 0.945$, $\beta_2\approx 3.763$, and $1/\Gamma_2(\beta_2)\approx 0.048$. We end this paper with a result concerning the inverse of the double gamma function itself. The proof is almost the same as the proof in [@p Proposition 2.6], so we leave out the details. Barnes’ double gamma function $\Gamma_2$ decreases on $(\beta_2,\infty)$ and maps it onto $(0,\Gamma_2(\beta_2))$. Its inverse $k$ is of the form $k(w)=h(1/w)$, $h$ being a Pick-function with an extension across $(1/\Gamma_2(\beta_2),\infty)$. 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--- abstract: 'Human-centric applications such as virtual reality and immersive gaming will be central to the future wireless networks. Common features of such services include: a) their dependence on the human user’s behavior and state, and b) their need for more network resources compared to conventional cellular applications. To successfully deploy such applications over wireless and cellular systems, the network must be made cognizant of not only the quality-of-service (QoS) needs of the applications, but also of the perceptions of the *human users* on this QoS. In this paper, by explicitly modeling the limitations of the human brain, a concrete measure for the delay perception of human users in a wireless network is introduced. Then, a novel learning method, called probability distribution identification, is proposed to find a probabilistic model for this delay perception based on the brain features of a human user. The proposed learning method uses both supervised and unsupervised learning techniques to build a Gaussian mixture model of the human brain features. Given a model for the delay perception of the human brain, a novel brain-aware resource management algorithm based on Lyapunov optimization is proposed for allocating radio resources to human users while minimizing the transmit power and taking into account the reliability of both machine type devices and human users. The proposed algorithm is shown to have a low complexity. Moreover, a closed-form relationship between the reliability measure and wireless physical layer metrics of the network is derived. Simulation results using real data from actual human users show that a brain-aware approach can yield savings of up to $78\%$ in power compared to the system that only considers QoS metrics. The results also show that, compared with QoS-aware, brain-unaware systems, the brain-aware approach can save substantially more power in low-latency systems.' author: - '\' bibliography: - 'PaperBib.bib' nocite: '[@*]' title: 'Human-in-the-Loop Wireless Communications: Machine Learning and Brain-Aware Resource Management' --- Introduction ============ The next generation of wireless services is expected to be highly human centric. Examples include virtual reality and interactive/immersive gaming [@gobbetti1998virtual; @chen2017virtual]. In order to cope with the quality-of-service (QoS) needs of such human-centric applications, in terms of data rate and ultra-low latency, wireless networks will have to allocate and exploit substantially more radio resources by leveraging heterogeneous spectrum bands across low and high frequencies [@semiari2015context]. However, even though allocating heterogeneous spectrum resources can potentially increase the raw QoS, given the human-centric nature of such emerging applications, their users may not be able to perceive the improved QoS, due to the cognitive limitations of the human brain [@RN1]. Indeed, many empirical studies (anecdotal and otherwise) have shown that the limitations on the human brain can be translated into a limitation on how wireless users translate QoS into actual quality-of-experience (QoE) [@Laghari2013Neuro; @Wechsung2014Springer; @Zhao2017QoE]. For example, the human brain may not be able to perceive any difference between videos transmitted with different QoS (e.g., rates or delays) [@Zhao2017QoE; @Chen2015QoE]. Hence, in order to deploy these services over wireless networks, such as 5G cellular systems, there is a need to enable the system to be strongly cognizant of the human user in the loop. In particular, to deliver such immersive, human-centric services, the network must tailor the usage and optimization of wireless resources to the intrinsic features of its human users such as their behavior and brain processing limitations. By doing so, the network can potentially save resources, accommodate more users, and provide a more realistic QoE to its users. Developing resource management mechanisms that can cater to intrinsic needs of wireless users and their context (e.g., device features or social metrics) has recently been studied in [@Alam2016; @Lin2017TVT; @Letter2017Context; @semiari2015context; @WCNC2017context; @bennis2014proactive; @makris2013survey; @perera2014context; @proebster2011context]. In [@Alam2016], a context-aware scheduling algorithm for 5G systems is proposed. This algorithm exploits the context information of user equipments (UEs), such as battery level, to save energy in the system while satisfying the QoS requirements of users. The authors in [@Lin2017TVT], proposed a user-centric resource allocation framework for ultra-dense heterogeneous networks. Context-aware resource allocation for heterogeneous cellular networks is also studied in [@Letter2017Context; @semiari2015context], and [@WCNC2017context]. In [@semiari2015context], a novel approach to context-aware resource allocation in small cell networks is introduced. Both wireless physical layer metrics and the social ties of human users are exploited in [@semiari2015context] to allocate wireless resource blocks. Proactive caching using context information from social networks is studied in [@bennis2014proactive]. In this work, it is shown that such a socially-aware caching technique reduces the peak traffic in 5G networks. Other context-aware resource allocation algorithms are also studied in [@makris2013survey; @perera2014context; @proebster2011context]. However, despite this surge in literature on context-aware networking [@Alam2016; @semiari2015context; @bennis2014proactive; @makris2013survey; @perera2014context; @proebster2011context; @Letter2017Context; @WCNC2017context; @Lin2017TVT], this prior art is still reliant on device-level features and is agnostic to the human users and their features (e.g., brain limitation or behavior) and, hence, they can potentially waste network resources as they can still allocate more resources to human users that cannot perceive the associated QoS gains, due to cognitive brain limitations. A general framework for modeling the intelligence of communication systems which serve humans is proposed in [@huang2016system]. The author defines intelligence in terms of predicting and serving human demands in advance. However, the work in [@huang2016system] does not account for the cognitive limitations of a human brain. Moreover, demand prediction, as done in [@huang2016system], will not be sufficient to capture the full spectrum of the human user limitations and behavior. By being aware of brain limitations of each user, the network can provide a unique experience for each user and optimize its performance. For example, an increase in the delay of a wireless system may have different effects on the QoE perceived by different human users. In particular, such different delay perceptions can potentially be exploited by the cellular network to minimize power consumption and reduce the amount of wasted resources. To our best knowledge, no existing work has studied the impact of such disparate brain delay perceptions on wireless resource allocation. The main contribution of the paper is, thus, a *novel brain-aware learning and resource management framework* that explicitly factors in the brain state of human users during resource allocation in a cellular network which has both human and machine type devices. In particular, we formulate the brain-aware resource allocation problem using a joint learning and optimization framework. First, we propose a novel learning algorithm to identify the delay perceptions of a human brain. This learning algorithm employs both supervised and unsupervised learning to identify the brain limitations and also creates a statistical model for these limitations based on Gaussian mixture models. Then, using Lyapunov optimization, we address the resource allocation problem with time varying QoS requirements that captures the learned delay perception. Using this approach, the network can allocate radio resources to human users while considering the reliability of both machine type devices and human users. We then identify a closed-form relationship between system reliability and wireless physical layer metrics and derive a closed-form expression for the reliability as a function of the human brain’s delay perception. Simulation results using real data show that the proposed brain-aware approach can substantially save power in the network while preserving the reliability of the users, particularly in low latency applications. In particular, the results show that the proposed brain-aware approach can yield power savings of up to $78$% compared to a conventional, brain-unaware system. The rest of the paper is organized as follows. Section \[sec:SysModel\] introduces the system model. Sections III and IV present the proposed learning algorithm and resource allocation framework, respectively. Section V presents the simulation results and conclusions are drawn in Section VI. System Model and Problem Formulation {#sec:SysModel} ==================================== Consider the downlink of a cellular network with humans-in-the-loop having a single base station (BS) serving a set $\mathcal{H}$ of $N$ human users with their UEs and a set $\mathcal{M}$ of $M$ machine type devices (MTDs). Each UE or MTD can have a different application with different QoS requirements such as sending a command to an actuator (for an MTD) or playing a 3D interactive game (for a UE). We consider a time-slotted system with each slot duration being equal to the LTE transmission time interval (TTI). We define $\mathcal{K}$ as the set of $K$ resource blocks (RBs). In our model, the packets associated with user $i \in \mathcal{H}\cup \mathcal{M}$ arrive at the BS according to independent Poisson processes with rate $a_i(t)$. The lengths $l_i, \forall i \in \mathcal{H}\cup \mathcal{M}$ of the packets follow an exponential distribution. Hence, each user’s buffer at the BS will follow an M/M/1 queuing model. The total queuing and transmission delay of each user $i$ is $D_i(t)=q_i(t)+\frac{l_i}{r_i(t)}$. The data rate for each user is given by: $$\label{eq:rateShannon} r_i(t)=B \sum_{j=1}^{K} \rho_{ij}(t)\,\log_2\left(1+\frac{p_{ij}(t) h_{ij}(t)}{\sigma^2}\right),$$ where $p_{ij}(t)$ is the transmit power between the BS and user $i$ over RB $j$ at time $t$ and $h_{ij}(t)$ is the time-varying Rayleigh fading channel gain. In (\[eq:rateShannon\]) $\rho_{ij}(t)=1$ if RB $j$ is allocated to user $i$ at time slot $t$, and $\rho_{ij}(t) =0$, otherwise. $B$ is the bandwidth of each RB. The BS seeks to allocate RBs and power to the users according to their delay needs and their channel state. The delay that MTD $i$ can tolerate is $\beta_i^m$, i.e., $D_i^{\max}(\beta_i^m)=\beta_m$. $\beta_i^m$ is known to the system. The delay perception of the brain of a human user who is using a given UE is captured by $\beta_i(t)$. $\beta_i(t)$ essentially represents a delay perception threshold for human user $i$ at time $t$. If we decrease the delay below the threshold $\beta_i(t)$, the human user will not be able to discern the difference. This delay perception is determined by the capabilities of the human brain. *By explicitly accounting for the cognitive limitations of the human brain, the BS can better allocate resources to the users that need it, when they can actually use it*. This is in contrast to conventional brain-agnostic networks [@semiari2015context; @huang2016system] in which resources may be wasted, as they are allocated only based on application QoS without being aware on whether the human user can indeed process the actual application’s QoS target. We pose this resource allocation problem as a power minimization problem that is subject to a brain-aware QoS constraint on the latency: \[eq:OrigOptim\] $$\begin{aligned} \min_{{\boldsymbol{\rho}}(t),{\boldsymbol{P}}(t)} \quad &\sum_{j \in \mathcal{K}}\Big[\sum_{i\in \mathcal{H}} \bar P_i^j+\sum_{i\in \mathcal{M}} \bar P_i^j\Big], \label{eq:CostFcn}\\ \text{s.t.} \qquad &\text{Pr}\big\{D_i(t)\geq D_i^{\max}\big(\beta_i(t)\big)\} \leq \epsilon_i\big(\beta_i(t)\big), &\forall i \in \mathcal{H} \cup \mathcal{M}, \label{eq:constrain1} \\ &p_{ij}(t) \geq 0, \,\, \rho_{ij}(t) \in \{0,1\}\quad &\forall i \in \mathcal{H} \cup \mathcal{M}, j\in \mathcal{K}\label{eq:constrain3},\\ &\sum_{i \in \mathcal{H} \cup \mathcal{M}}\rho_{ij}(t)=1, \quad&\forall j\in \mathcal{K}\label{eq:constrain5}, \end{aligned}$$ where ${\boldsymbol{\rho}}(t)$ is an $(M+N)\times K$ matrix having each element $\rho_{ij}(t)$. ${\boldsymbol{P}}(t)$ is an $(M+N)\times K$ matrix with each element $p_{i,j}(t)$, representing the instantaneous power allocated to user $i$ on RB $j$. The term $\bar{P}_i^j=\lim_{t\to \infty} \frac{1}{t} \sum_{\tau=0}^{t-1}\rho_{ij} p_{ij}(\tau) $ is the time average of the power allocated to user $i$ on RB $j$. $D_i^{\max}\big(\beta_i(t)\big)$ is the maximum tolerable delay, and $1-\epsilon_i\big(\beta_i(t)\big)$ is the reliability of user $i$. We define *reliability* as the proportion of time during which the delay of a given user does not exceed a threshold. For notational convenience, hereinafter, we use the terms $D_i^{\max}\left(\beta_i(t)\right)$ and $D_i^{\max}$ interchangeably. The key difference between our problem formulation and conventional RB allocation problems is seen in the QoS delay requirement in (\[eq:constrain1\]). In (\[eq:constrain1\]), the network explicitly accounts for the human brain’s (and the MTDs’) delay needs. By taking into account the features of the brain of the human UEs, the network can avoid wasting resources. This waste of resources can stem from allocating more power to a UE, solely based on the application QoS, while ignoring how the brain of the human carrying the UE perceives this QoS. Clearly, ignoring this human perception can lead to inefficient resource management. For finding $\beta_i(t)$, we propose a machine learning algorithm to identify the human brain delay perception. Each human user has $p$ features, (e.g., age, occupation, location) assumed to be known to the BS. This time-varying feature vector is denoted by ${\boldsymbol{x}}_i(t) \in \mathds{R}^p$. We develop a learning algorithm to build a model that maps these features to $\beta_i(t)$ for each user. We then show that being aware of $\beta_i(t)$ can help the resource allocation algorithm to save a significant amount of resources for low-latency systems. Here, we assume that BS has the access to the user features ${\boldsymbol{x}}_i(t)$. In practice, the BS can collect such data whenever a given user registers in the network or by using the sensors of a user’s mobile device. Machine Learning for Probability Distribution Identification ============================================================ To find the mapping $\beta_i(t)=f({\boldsymbol{x}}_i(t))$ between human features ${\boldsymbol{x}}_i(t)$ and the delay perception of the brain, we use supervised learning [@bishop2007pattern; @chen2017machine]. Since reliability is a key factor in a communication system, we need a supervised learning algorithm that not only predicts $\beta_i(t)$ as function of ${\boldsymbol{x}}_i(t)$, but also gives a measure of reliability for this prediction. Hence, we cannot rely on conventional supervised learning methods, such as neural networks [@chen2017machine]. Here, reliability of predictions is defined as the probability that the prediction of $\beta_i(t)$ lies within a certain range of the true values for $\beta_i(t)$. To this end, we propose a novel supervised learning mechanism dubbed as *probability distribution identification (PDI) method* that can find the mapping between the features of a human user and the human brain’s perception on delay, as captured by $\beta_i(t)$, while quantifying the reliability of this mapping. We will use this reliability to determine the overall reliability of our system. As discussed in [@petkoski2015effects], the delay perception of a human brain typically follows a multi-modal distribution. As a result, we design the proposed PDI approach to capture such a model and find the different modes of a human brain. Then, using the distribution of the brain delay, the PDI approach can find the *effective delay* of the human brain. This effective delay determines relationship between $\beta_i(t)$ and ${\boldsymbol{x}}_i(t)$ along with its reliability. Consider a dataset $\{{\boldsymbol{x}}_1(t),\cdots,{\boldsymbol{x}}_n(t)\}$, where ${\boldsymbol{x}}_i(t)\in \mathds{R}^p$ is one sample data vector. The elements of ${\boldsymbol{x}}_i(t)$ are features which can be both categorical (such as gender) and numerical (such as age). For each input vector ${\boldsymbol{x}}_i(t)$, we have a corresponding output value of delay perception $\beta_i(t)$. This data can be collected using experiments or surveys such as those in [@yang2015prospect]. Since we can remove the data’s time dependency of the data using time-series techniques, hereinafter, we use ${\boldsymbol{x}}$ instead of ${\boldsymbol{x}}(t)$. This dataset can be represented by a matrix ${\boldsymbol{X}}\in \mathds{R}^{n\times p}$, where ${\boldsymbol{x}}_i^T$ is row $i$ of ${\boldsymbol{X}}$. Using PDI, we first create an $n\times(p+1)$ dataset matrix $\boldsymbol W$: $$\label{eq:define w} \boldsymbol{W}=[\boldsymbol{X} \Vert \boldsymbol{\beta}]= \begin{bmatrix} \boldsymbol{w}_1^T\\ \vdots\\ \boldsymbol{w}_n^T \end{bmatrix}= \begin{bmatrix} {\boldsymbol{x}}_1^T &\beta_1(t)\\ \vdots &\vdots\\ {\boldsymbol{x}}_n^T &\beta_n(t)\\ \end{bmatrix},$$ where $\boldsymbol w_i \in \mathds{R}^{p+1}$ is a vector of the delay perception $\beta_i(t)$ and $p$ other correlated features of the human brain. Then, we fit a Gaussian mixture model (GMM) of $m$ modes to our dataset using the expectation-–maximization (EM) algorithm [@moon1996expectation]. In other words, we adopt an EM method for learning the joint probability distribution using the joint dataset $\boldsymbol{W}$, i.e., $p({\boldsymbol{x}},\beta_i(t))$. After finding the probability distribution $p({\boldsymbol{x}},\beta_i(t))$, we are able to cluster the data samples and find $m$ modes in the data. Then, each data vector ${\boldsymbol{x}}_i$ is labeled based on its cluster so that each ${\boldsymbol{x}}_i, \,i=1,\cdots,n$ has a label in the cluster set $\mathcal{C}=\{1,\cdots,m\}$. Subsequently, our dataset is labeled using its cluster number. In the next step, we train a classifier such that it finds a mapping between the input data ${\boldsymbol{x}}_i$ and its cluster number. *These cluster numbers will correspond to the modes of the human brain that determine its effective delay perception.* When this proposed learning approach is deployed in a wireless network, each user will be classified after connecting to the BS, and its brain mode will be identified. Then, each user’s mode can be used to derive a probabilistic model of its delay perception. Before delving into the PDI method for finding the brain’s effective delay, we describe the Gaussian mixture model that we use for the human brain. A multi-modal stochastic model is assumed for the brain features ${\boldsymbol{w}}_i$ for user $i$. The proposed distribution for $\boldsymbol w_i$ is given by [@bishop2007pattern]: $$\label{eq:multi modal distribution} p(\boldsymbol w_i)=\sum_{z} p(z)p(\boldsymbol w_i|z)=\sum_{k=1}^L \pi_k f(\boldsymbol w_i|\boldsymbol{\mu}_k,\boldsymbol{\boldsymbol{\Sigma}}_k),$$ where $f(\boldsymbol w_i|{\boldsymbol{\mu}}_k,{\boldsymbol{\Sigma}}_k)$ is the probability density function for a multivariate normal distribution with mean vector ${\boldsymbol{\mu}}_k$ and covariance matrix ${\boldsymbol{\Sigma}}_k$. ${\boldsymbol{\Sigma}}_k$ and $\boldsymbol{\mu}_k$ represent the covariance matrix and mean vector for mode $k$ of the human brain, respectively. $\pi_k$ is the mixing weight of mode $z=z_k$. $p(\boldsymbol w)$ is the distribution of data points ${\boldsymbol{w}}_i$. $L$ is total number of modes in the GMM. The human brain will be in mode $k$ with probability $\pi_k$, and its features are generated using a multivariate normal distribution with mean and covariance $\boldsymbol{\mu}_k$ and ${\boldsymbol{\Sigma}}_k$, respectively. The posterior probability, i.e. *responsibility*, for mode $k$ will be: $$\label{eq:responsblty} r(z_k)=\frac{\pi_k f(\boldsymbol{w}|\boldsymbol{\mu}_k,{\boldsymbol{\Sigma}}_k)}{\sum_{j=1}^L f(\boldsymbol{w}|\boldsymbol{\mu}_j,{\boldsymbol{\Sigma}}_j)}.$$ This responsibility can be used for clustering the data as well. After fitting the GMM on the dataset, we can find the mode with highest responsibility for each data point and assign the data to this mode. The EM algorithm is used to find these parameters based on a real-time human brain behavior [@moon1996expectation]. The log likelihood function for our dataset can be written as: $$\label{eq:loglikelihood} \ln \mathcal{L}({\boldsymbol{\Sigma}}, \mu, \pi|{\boldsymbol{w}}) =\ln p({\boldsymbol{w}}| {\boldsymbol{\Sigma}}, \mu, \pi) =\sum_i \ln \sum_{k=1}^L \pi_k f(\boldsymbol w_i|\boldsymbol{\mu}_k,\boldsymbol{\boldsymbol{\Sigma}}_k).$$ The likelihood function in (\[eq:loglikelihood\]) has singularities and, hence, it is infeasible to find parameters $\pi_k$, ${\boldsymbol{\Sigma}}_k$, and ${\boldsymbol{\mu}}_k$. The EM algorithm is proposed in [@EM98main] to maximize the likelihood function for a Gaussian mixture model. We first initialize ${\boldsymbol{\Sigma}}_k$, $\boldsymbol{\mu}_k$, and $\pi_k$ randomly. Next, we find the responsibility for each mode using (\[eq:responsblty\]). Then, we reestimate parameters using current responsibilities. Finally, the likelihood in (\[eq:loglikelihood\]) is maximized with respect to ${\boldsymbol{\Sigma}}_k$, $\boldsymbol{\mu}_k$ ,and $\pi_k$. In the next step, we use the derived joint distribution of human features and its associated delay perception to derive a probabilistic model for the human insensitivity of the delay. In other words, we find the relationship between a certain level of the delay $D_i^{\min}$ and the probability $\text{Pr}(\beta_i(t)>D_i^{\min}).$ In the proposed PDI, along with the supervised learning component previously explained, we also propose to use an unsupervised learning step to measure the reliability of our predictions. In the unsupervised learning step, the data will be labeled based on the GMM as follows. For each feature vector ${\boldsymbol{w}}_i$, the responsibilities $r(z_k)=p(z_k=1|{\boldsymbol{w}}_i)$ are found using the EM algorithm. Then, the most probable mode is assigned as the label of this data, i.e., $$c({\boldsymbol{w}}_i)=\underset{k}{\arg \max}\,\, p(z_k=1|{\boldsymbol{w}}_i).$$ The output of the unsupervised learning step, $c_i$, is used for training the supervised learning model. We will form an output vector ${\boldsymbol{y}}$ which is defined as ${\boldsymbol{y}}=\begin{bmatrix} c({\boldsymbol{w}}_1) & \cdots &c({\boldsymbol{w}}_n) \end{bmatrix}^T.$ Then, during the supervised learning step, we train a classifier so that it can find the mode using the human features ${\boldsymbol{x}}$ as input. Given the data matrix ${\boldsymbol{X}}$ and the output vector $c(w_i)$, this supervised learning builds us a model $f$ such that $c_i=f(\boldsymbol{x_i})$,i.e., $$f=\textrm{arg}\min_{\hat f} \sum_{i=1}^n \xi\left(c(w_i),\hat f (x_i)\right),$$ where $\xi(.)$ is a 0-1 loss function. $f$ is approximated using a set of points $({\boldsymbol{x}}_i,c_i)$ and determines the relationship between the features of a user and its cluster. After approximating $f$, given each human user’s feature vector ${\boldsymbol{x}}_i$, we find the modes $c_i$ using model $f$. Finally, we bounds $D_i^{\max}(\beta_i(t))$ based on its features ${\boldsymbol{x}}_i$. Now that the system can identify the human users’ modes, we need to find a relationship between a human user’s mode and the probabilistic model of its delay perception by defining the concept of effective delay. Given the statistical model for human delay perception, $D_i^{\min}(\epsilon')$ is the *effective delay* for human user $i$ that satisfies: $$\label{eq:dminKia} \text{Pr}\big\{\beta_i(t)<D_i^{\min}(\epsilon')\big\}<\epsilon'.$$ To find the effective delay for human user $i$, we first find the probability that the delay perception of human user $i$ is less than a threshold $D_i^{\min}(\epsilon')$. In other words, we want to find the relation between $\epsilon'$ and $D_i^{\min}(\epsilon')$ in (\[eq:dminKia\]). The concept of effective delay is defined using the fact that delays less than $D_i^{\min}(\epsilon')$ cannot be sensed by a human with ($1-\epsilon'$) certainty. The relation between $\epsilon'$ and $D_i^{\min}(\epsilon')$ in (\[eq:dminKia\]) is found in Theorem \[lemma:confidence region single mode\]. For notational simplicity, hereinafter, we use $D_i^{\min}$ instead of $D_i^{\min}(\epsilon')$. \[lemma:confidence region single mode\] The delay perception of the identified brain mode $k$ user $i$ is bounded such that $$\label{eq:lemma1 main result} \text{Pr}\Big\{|\beta_i-\mu_K^D|<\sqrt{Q_{p+1}(\gamma) \boldsymbol{e}_{p+1}^T{\boldsymbol{\Sigma}}_k \boldsymbol{e}_{p+1}}\Big\}>\gamma,$$ where ${\boldsymbol{\Sigma}}_k$ and $\mu_k$ represent, respectively, the covariance matrix and the mean vector of the identified brain mode $k$. Also, $Q_p(\gamma)$ is the quantile function of chi-square distribution with $p$ degrees of freedom, and is defined as $$Q_{p+1}(\gamma)=\inf\Big\{x\in \mathds{R}| \gamma\leq \int_{0}^{x} \chi_{p+1}^2(u) du\Big\},$$ and $\boldsymbol{e}_j$ is a unit vector in $\mathds{R}^{p+1}$, whose $j$th element is $1$ and all other elements are zero. $p$ is number of features used for learning. $\chi_{p+1}^2(x)$ is the probability density function of a chi-square random variable with $p+1$ degrees of freedom. $\mu_K^D$ is the element $p+1$ in vector ${\boldsymbol{\mu}}_k$. See Appendix A. As seen from Theorem \[lemma:confidence region single mode\], in addition to the delay perception element $\mu_K^D$, the only other parameter that affects the delay is $\boldsymbol{e}_{p+1}^T{\boldsymbol{\Sigma}}_k \boldsymbol{e}_{p+1}$, which is the $(p+1)$th diagonal element of the covariance matrix ${\boldsymbol{\Sigma}}_k$. Note that, we did not assume that matrix ${\boldsymbol{\Sigma}}_k$ is diagonal. Fig.\[fig:DuBa\] shows the relationship between $D_i^{min}$ and GMM. From Fig.\[fig:DuBa\], we can see that, after finding the GMM for the dataset, one can find the predictive coverage of each Gaussian distribution. Using this predictive coverage, we can determine the probability with which $\beta_i(t)$ for a user $i$ will be higher than a threshold $D_i^{\min}$. In order to find the effective delay for human user $i$, we first find the probability with which the delay perception for human user $i$ will be less than a threshold $D_i^{\min}$. In other words, we will find the relationship between $\epsilon$ and $D_i^{\min}$ in (\[eq:dminKia\]) using the following corollary that follows directly from Theorem 1. As a direct result of Theorem \[lemma:confidence region single mode\], we can reduce (\[eq:lemma1 main result\]) to $$\label{eq:Dmin epsilon} \text{Pr}\Big\{\beta_i(t)<\mu_K^D-\sqrt{Q(\gamma) \boldsymbol{e}_{p+1}^T{\boldsymbol{\Sigma}}_k \boldsymbol{e}_{p+1}}\Big\}<\frac{1-\gamma}{2}.$$ Therefore, we find $D_i^{\min}(\epsilon)$ and $\epsilon$ in (\[eq:dminKia\]) as $$D_i^{\min}(\epsilon)=\mu_K^D-\sqrt{Q(1-2\epsilon) \boldsymbol{e}_{p+1}^T{\boldsymbol{\Sigma}}_k \boldsymbol{e}_{p+1}}. $$ Since $Q(\gamma)$ can only be calculated numerically, a closed-form relationship cannot be found between $D_i^{\min}(\epsilon)$ and $\epsilon$. However, we can numerically analyze this relationship, as shown in Fig. \[fig:Delay versus epsilon\]. From Fig. \[fig:Delay versus epsilon\], we can first observe that $D_i^{\min}(\epsilon)$ is an increasing function. *This means that the probability of the human brain noticing QoS differences for low delays will be much smaller than for higher delays*, which is an intuitive fact. Furthermore, it can be inferred that, if the delay perception for a group of human users within a cluster is diverse, then the system’s confidence on the delay perception of this group of humans will decrease, i.e., the estimation of the delay perception of this group of human users will be less reliable. Next, we determine constraint (\[eq:constrain1\]) using $D_i^{\min}(\epsilon)$. As stated before, some delays are not perceptible to human users. To capture this feature, we find $D_i^{\max}(\beta_i(t))$ and $\epsilon(\beta_i(t))$ in problem (\[eq:OrigOptim\]) using $D_i^{\min}(\epsilon)$. Recall that $D_i^{\max}(\beta_i(t))$ is a parameter that will be used by the resource allocation system to represent the maximum tolerable delay for the reliable communication of user $i$ with $1- \epsilon(\beta_i(t))$ being the reliability of user $i$. There are three possible cases for $D_i^{\max}$ based on $D_i^{\min}$ of a human user $i$: ### $D_i^{\max}>D_i^{\min}$ In this case, the system will not be reliable even if we satisfy $\text{Pr}(D>D_i^{\max})<\epsilon$. The reason is that the human user has a delay perception of less than the maximum delay $D_i^{\max}$ and hence, the system is not reliable. ### $D_i^{\max}<D_i^{\min}$ In this case, if the system is able to satisfy $\text{Pr}(D>D_i^{\max})<\epsilon$, then the system will be reliable, because user $i$ cannot sense delays less than $D_i^{\min}$ and its service delay will not exceed $D_i^{\max}$. ### $D_i^{\max}=D_i^{\min}$ If this equality holds, the system will be reliable and it will also have prevented a waste of resources. If any given user cannot perceive delays less than $D_i^{\min}$, then it is not effective to allocate more resources to this user. $S$ represents the event where the system delay meets the cognitive perceptions of the human user, which is the desired result (case 2) and case 3)). In other words, if event $S$ happens, the system is reliable. Also, we assume that the events $E_1$ and $E_2$ are defined as $D<D_i^{\max}$ and $\beta_i(t)>D_i^{\min}$, respectively. We know that for case 1, event $E_1\cap E_2 $ is a subset of event $S$, and in case 2, event $S$ is a subset of event $E_1\cap E_2 $. Similarly, in case 3, event $E_1\cap E_2 $ is same as event $S$. Since the probability of $E_1\cap E_2 $ can be computed, if we set $D_i^{\min}$ to $D_i^{\max}$ (case 3), we will be able to find $S$ based on the system parameters and design the resource allocation system in a way that each user is satisfied. We know that: $$\begin{aligned} &\text{Pr}(E_1\cap E_2 )=1-\text{Pr}\Big((D>D_i^{\max}) \cup (\beta_i(t)<D_i^{\min})\Big)\label{eq:De Morgan}\\ &=1-\Big(\text{Pr}(D>D_i^{\max})+\text{Pr}(\beta_i(t)<D_i^{\min})-\text{Pr}(D>D_i^{\max}) \text{Pr}(\beta_i(t)<D_i^{\min})\Big). \label{eq:independent event}\end{aligned}$$ (\[eq:De Morgan\]) follows from De Morgan’s law, and (\[eq:independent event\]) is true since $D$ and $\beta_i(t)$ are two independent random variables. Therefore, if $D_i^{\min}=D_i^{\max}$ for user $i$ and $\epsilon\epsilon'$ is small, we can see that $$1-\Big(\text{Pr}(D>D_i^{\max})+\text{Pr}(\beta_i(t)<D_i^{\min})\Big)\geq1- (\epsilon +\epsilon'),$$ and, hence, $$\text{Pr}(S)=\text{Pr}(E)> 1- (\epsilon +\epsilon'),$$ where [Pr]{}($S$) is the reliability of the system defined in (\[eq:OrigOptim\]). Subsequently, as we design the system, we consider the reliability as a predetermined target design parameter for the system. Using this parameter, we can set $\epsilon$ and $\epsilon'$. Given $\epsilon'$ and numerical function $D_i^{\min}(\epsilon')$ derived in (\[eq:Dmin epsilon\]), $D_i^{\min}$ can be determined. Now, given $\epsilon(\beta_i(t))$ and $D_i^{\max}(\beta_i(t))$, we can fully characterize problem (\[eq:OrigOptim\]). Brain-Aware Resource Management {#sec:BrainAwRA} =============================== To solve problem (\[eq:OrigOptim\]), we propose a novel brain-aware resource management framework that takes into account the time-varying wireless channel and the time-varying brain-aware delay constraint (\[eq:constrain1\]). In this section, we transform this constraint into a mathematically tractable form. First, in the next lemma, the relation between the packet length distribution and the service time distribution for a packet is shown. \[corol:exponential service time\] If a fixed rate $r_i$ is allocated to a user and the packet lengths follow an exponential distribution with parameter $\chi$, then, the distribution of the service time $s$ will also be exponential with parameter $\chi r_i$. The CDF of the exponential distribution is $F_{\psi}(l)=\text{Pr}(l<\psi)=1-e^{-\chi \psi}$. Hence, $$F_S(s)=\text{Pr}(s<S)=\text{Pr}(\frac{l}{r_i}<S)=\text{Pr}(l<r_i S)=F_{r_i S}(s)=1-e^{-\chi r_i S}.$$ This means that the PDF for the service time is $f_S(s)=e^{-\chi r_i s}$. Therefore, without loss of generality, we assume that the service time of each packet is exponential with parameter $r_i$, which is the same as the rate allocated to the user. Here we assume that for any given user, the packets arrive with the rate $a_i(\tau)$, and the user data rate is $r_i(\tau)$ in slot $\tau=1,\cdots,t$. In the next theorem, We derive the probability with which the delay of a given user $i$ exceeds a threshold $D_i^{\max}$. \[prop:packet length\] Assume that user $i$ has a time varying rate $r_i^{\tau}$ at time slot $\tau$. If the duration of each time slot is long enough for the queue to reach its steady state, i.e., $$\label{eq:time slot condition} \frac{1}{r_i(\tau)-a_i(\tau)}<<\delta \tau,$$ then, the probability that the delay exceeds a threshold is $$\label{eq:time average delay} \text{Pr}(D>D_i^{\max})= \lim_{t\to \infty} \frac{1}{t} \sum_{\tau=1}^{t} e^{-\big(r_i(\tau)-a_i(\tau)\big)D_i^{\max}},$$ under the condition that $r_i(\tau)>a_i(\tau)$ for all $\tau>0$. See Appendix B. Theorem \[prop:packet length\] shows that constraint (\[eq:constrain1\]) is satisfied if the network can satisfy the following condition $$\label{eq:2b transformed} \lim_{t\to \infty} \frac{1}{t} \sum_{\tau=1}^{t} e^{-\big(r_i(\tau)-a_i(\tau)\big)D_i^{\max}}<\epsilon.$$ Fig. \[fig:proposition\] shows the relationship between the theoretical result from Theorem \[prop:packet length\] and the simulation results. It shows that the simulation and analytical results are a near-perfect match. The maximum error between the analytical and simulation results is only $0.0146$. Optimal Resource Allocation with Guaranteed Reliability ------------------------------------------------------- Constraint (\[eq:2b transformed\]) is analogous to the drift-plus-penalty method in the Lyapunov optimization framework [@neely2010stochastic] and, hence, we will use this framework to solve (\[eq:OrigOptim\]). The problem has a time-varying nature since the human brain conditions and needs will change from time to time. The users’ processing state $\beta(t)$ is also a function of time, and accordingly, the latency needs in (\[eq:constrain1\]) will be time-varying. Therefore, we need to solve the optimization problem (\[eq:OrigOptim\]) during each time slot efficiently. Here, we propose an algorithm with a low computational complexity for solving problem (\[eq:OrigOptim\]). The drift-plus-penalty approach is used to stabilize a queue network while minimizing time average of a penalty function. In order for constraint (\[eq:constrain1\]) to be satisfied in all time slots, we need to make sure that (\[eq:time average delay\]) will always be smaller than $\epsilon$. For this reason, we define a virtual queue: $$F_i(t+1)=\max\{F_i(t)+e^{-\big(r_i(t+1)-a_i(t+1)\big)D_i^{\max}}-\epsilon\}.$$ We can see that $e^{-\big(r_i(t+1)-a_i(t+1)\big)D_i^{\max}}-\epsilon<F_i(t+1)-F_i(t)$. Consequently, we obtain: $$\sum_{\tau=1}^{t}e^{-\big(r_i(t+1)-a_i(t+1)\big)D_i^{\max}}-\epsilon t<F_i(t)-F_i(0).$$ If $F_i(0)$ is bounded, we have: $$\lim_{t\to \infty} \frac{1}{t}\sum_{\tau=1}^{t}e^{-\big(r_i(t+1)-a_i(t+1)\big)D_i^{\max}}-\epsilon<\lim_{t\to \infty}\frac{F_i(t)}{t}.$$ If the queue $F_i(t)$ is mean-rate stable, that is, $\lim_{t\to \infty}\frac{F_i(t)}{t}=0$, then we have: $$\lim_{t\to \infty} \frac{1}{t}\sum_{\tau=1}^{t}e^{-\big(r_i(t+1)-a_i(t+1)\big)D_i^{\max}}<\epsilon.$$ The Lyapunov function is defined for all the queues in the base station as $$Y(t)=\frac{1}{2}\sum_{i\in \mathcal{M} \cup H} F_i(t)^2.$$ Then, we can find the drift function $\Delta t=Y(t+1)-Y(t)$ as: $$Y(t+1)=\frac{1}{2}\sum_{i\in \mathcal{M} \cup \mathcal{H}} F_i(t+1)^2 \leq \frac{1}{2}\sum_{i\in \mathcal{M} \cup \mathcal{H}}F_i(t)^2+\frac{1}{2}\sum_{i\in \mathcal{M} \cup \mathcal{H}}y(t)^2+\sum_{i\in \mathcal{M} \cup \mathcal{H}}y(t)F_i(t),$$ where $$\label{eq:definition yt} y_i(t)=e^{-\big(r_i(t+1)-a_i(t+1)\big)D_i^{\max}}-\epsilon.$$ Thus, $$\label{eq:lyapunov opt ineq} \Delta t \leq \frac{1}{2}\sum_{i\in \mathcal{M} \cup \mathcal{H}}y(t)^2+\sum_{i\in \mathcal{M} \cup \mathcal{H}}y(t)F_i(t).$$ We can form the drift-plus-penalty by adding $V\sum_{i,j}p_{ij}(t)$ to both sides of inequality (\[eq:lyapunov opt ineq\]), where $\sum_{i,j}p_{ij}$ is the total power of the BS which we want to minimize, and $V$ is a parameter that determines how important minimizing the objective function (\[eq:CostFcn\]) is in comparison with satisfying (\[eq:constrain1\]). We can balance the tradeoff between power and delay. The drift-plus-penalty inequality is $$\label{eq:drift plus pealty} \Delta t +V\sum_{i,j}p_{ij}\leq \frac{1}{2}\sum_{i\in \mathcal{M} \cup \mathcal{H}}y(t)^2+V\sum_{i,j}p_{ij}(t)+\sum_{i\in \mathcal{M} \cup \mathcal{H}}y(t)F_i(t).$$ Given that we assumed $r_i(t)>a_i(t)$ for all $t$, we know that $ |y_i(t)|<1 \, \forall t, i \in \mathcal{M} \cup \mathcal{H}, $ and hence, we can rewrite (\[eq:drift plus pealty\]) as $$\label{eq:drift plus pealty} \Delta t +V\sum_{i,j}p_{ij}\leq B+V\sum_{i,j}p_{ij}(t)+\sum_{i\in \mathcal{M} \cup \mathcal{H}}y(t)F_i(t),$$ where $B$ is the upper bound of $\frac{1}{2}\sum_{i\in \mathcal{M} \cup \mathcal{H}}y(t)^2$, and is equal to $\frac{|\mathcal{H}|+|\mathcal{M}|}{2}$. $|\mathcal{A}|$ is the cardinality of set $\mathcal{A}$. Using the drift-plus-penalty algorithm [@neely2008fairness], we know that, by minimizing the right hand side of equation (\[eq:drift plus pealty\]), queue $F_i(t)$ will be mean-rate stable, and hence, the condition $y_i(t)<0$ will be satisfied. As a result, constraint (\[eq:constrain1\]) will also be satisfied. Furthermore, we know that by minimizing the right hand side of (\[eq:drift plus pealty\]), cost function (\[eq:CostFcn\]) is also minimized, owing to the fact that (\[eq:CostFcn\]) is defined as a penalty function. By minimizing the right hand side of (\[eq:drift plus pealty\]), our optimization problem can be converted to the following time-varying problem: \[eq:converted optimization problem\] $$\begin{aligned} \min_{{\boldsymbol{\rho}}(t),{\boldsymbol{P}}(t)} \qquad &V\sum_{i,j}p_{ij}(t)+\sum_{i\in \mathcal{M} \cup \mathcal{H}}y_i(t)F_i(t), \label{eq:converted cost func}\\ \text{s.t.} \qquad &r_i(t)>a_i(t), \quad &\forall i \in \mathcal{H} \cup \mathcal{M} \label{eq:converted constrain 1}\\ &p_{ij}(t) \geq 0,\quad \rho_{ij}(t) \in \{0,1\}, \quad &\forall i \in \mathcal{H} \cup \mathcal{M}, j\in \mathcal{K},\label{eq:converted constrain 2}\\ &\sum_{i \in \mathcal{H} \cup \mathcal{M}}\rho_{ij}(t)=1, \quad&\forall j\in \mathcal{K}\label{eq:converted constrain 4}. \end{aligned}$$ As discussed earlier, the cost function in (\[eq:converted cost func\]) is equivalent to (\[eq:CostFcn\]) and (\[eq:constrain1\]) in the original optimization problem. Learning the effective delay of each human user using our proposed PDI method determines the parameters $y_i(t)$ and $F_i(t)$ in the problem (\[eq:converted cost func\]). However, in order to satisfy (\[eq:constrain1\]), we need to also satisfy (\[eq:converted constrain 1\]). The reason for adding (\[eq:converted constrain 1\]) is that if this constraint is not satisfied in any time slot, the queue length will approach infinity. Constraints (\[eq:converted constrain 2\]) and (\[eq:converted constrain 4\]) are feasibility conditions and remain the same as (\[eq:OrigOptim\]). Hence, by solving (\[eq:converted optimization problem\]) in each time slot, the original problem (\[eq:OrigOptim\]) will be solved. Nonetheless, problem (\[eq:CostFcn\]) is not a convex optimization problem, due to the fact that it is a mixed integer problem and its complexity increases exponentially with the number of users. Since (\[eq:CostFcn\]) needs to be solved at each time slot, this exponential order of complexity makes the implementation infeasible. Consequently, we should use a dual decomposition method to break down optimization problem (\[eq:converted optimization problem\]) to smaller subproblems, and find the optimal solution to (\[eq:converted optimization problem\]) using a low complexity method. It is rather challenging to solve (\[eq:converted optimization problem\]) using a dual decomposition method, as the structure of $y_i(t)$ makes it infeasible to decompose the objective function for each resource block. In order to overcome this challenge, we convert (\[eq:converted optimization problem\]) to a decomposable form. Then, we will show that this converted problem is equivalent to (\[eq:converted optimization problem\]). For this purpose, the Lagrangian for problem (\[eq:converted optimization problem\]) is written as $$\label{eq:converted lagrange multipliers} V\sum_{i,j}p_{ij}(t)+\sum_{i\in \mathcal{M} \cup \mathcal{H}}y_i(t)F_i(t)+\sum_{i\in \mathcal{M} \cup \mathcal{H}}\lambda_i \big(a_i(t)-r_i(t)\big),$$ where $\lambda_i$ is the Lagrange multiplier. As we know, $y_i(t)=e^{-\big(r_i(t+1)-a_i(t+1)\big)D_i^{\max}}-\epsilon$. Therefore, the only decision variables are allocation of resource blocks to the users and allocating power to each resource block. Although $F_i(t)$ is a function of $y_i(t)$, it is not a decision variable and is treated as a constant. Hence, (\[eq:converted lagrange multipliers\]) can be rewritten as $$\label{eq:lagrangian function} V\sum_{i,j}p_{ij}(t)+\sum_{i\in \mathcal{M} \cup \mathcal{H}}e^{-\big(r_i(t+1)-a_i(t+1)\big)D_i^{\max}}F_i(t)+\sum_{i\in \mathcal{M} \cup \mathcal{H}}\lambda_i\big(a_i(t)-r_i(t)\big).$$ The main optimization problem consists of two components. First, minimizing the total power of the BS with weight $V$, and second, minimizing the summation $\sum_{i\in \mathcal{M} \cup \mathcal{H}}e^{-r_i(t+1)}$ which has a weight $F_i(t)e^{-a_i(t+1)D_i^{\max}}$ for each user $i$. As we can see, (\[eq:lagrangian function\]) is not decomposable for each resource block. Here we will have an approximation of (\[eq:converted optimization problem\]) and then propose an algorithm to solve this approximation efficiently. In this C-additive approximation, $\sum_{i\in \mathcal{M} \cup \mathcal{H}}e^{-\big(r_i(t+1)-a_i(t+1)\big)D_i^{\max}}F_i(t)$ in (\[eq:lagrangian function\]) is substituted with its linear approximation of exponential term $e^{-x}$ at $x=0$. $$\label{eq:C-additive approximation} \sum_{i\in \mathcal{M} \cup \mathcal{H}} -\big(r_i(t+1)-a_i(t+1)\big) F_i(t).$$ In the original problem, if $y_i(t)$ starts to become greater than zero for user $i$, then $F_i(t)$ will increase and it will give more weight to the term $e^{-\big(r_i(t+1)-a_i(t+1)\big)D_i^{\max}}$. As a result, the algorithm allocates more resources to user $i$ such that it minimizes $e^{-\big(r_i(t+1)-a_i(t+1)\big)D_i^{\max}}$ for user $i$, and accordingly, $y_i(t)$ decreases. Hence, $F_i(t) e^{-\big(r_i(t+1)-a_i(t+1)\big)D_i^{\max}}$ plays the role of feedback in the system. As we can see from (\[eq:C-additive approximation\]), this approximation will not change this feedback mechanism and plays the same role in the system. Therefore, we can write $$\begin{aligned} &\underset{\boldsymbol{P}, \mathbf{\rho}}{\min}\big\{V\sum_{i,j}p_{ij}(t)+\sum_{i\in \mathcal{M} \cup \mathcal{H}} -\big(r_i(t+1)-a_i(t+1)\big)D_i^{\max}F_i(t) \big\}\nonumber\\ &<C+\underset{\boldsymbol{P}, \mathbf{\rho}}{\min}\big\{V\sum_{i,j}p_{ij}(t)+\sum_{i\in \mathcal{M} \cup \mathcal{H}}e^{-\big(r_i(t+1)-a_i(t+1)\big)D_i^{\max}}F_i(t)\big\}.\end{aligned}$$ Using this C-additive approximation, it can be easily proved that all terms are mean-rate stable. Hence, (\[eq:constrain1\]) in the original problem is satisfied [@neely2010stochastic]. Finally, problem (\[eq:OrigOptim\]) can be presented as: \[eq:final optimization problem\] $$\begin{aligned} \min_{{\boldsymbol{\rho}}(t),{\boldsymbol{P}}(t)} \qquad &V\sum_{i,j}p_{ij}(t)-\sum_{i\in \mathcal{M} \cup \mathcal{H}} \big(r_i(t+1)-a_i(t+1)\big)D_i^{\max}F_i(t), \nonumber\\ \text{s.t.} \qquad &r_i(t)>a_i(t), \label{eq:minimum rate lambda}\\ &P_i^n(t) \geq 0, \quad \forall i \in \mathcal{H} \cup \mathcal{M}, n\in \mathcal{N},\\ &\rho_i^n(t) \in \{0,1\}, \quad \forall i \in \mathcal{H} \cup \mathcal{M}, n\in \mathcal{N}, \\ &\sum_{i \in \mathcal{H} \cup \mathcal{M}}\rho_{ij}(t)=1, \quad\forall j\in \mathcal{K}.\label{eq:coupling constraint} \end{aligned}$$ In order to solve this problem, we can decompose it into $K$ subproblems. Since these subproblems are coupled through constraint (\[eq:coupling constraint\]), we use the dual decomposition method for solving (\[eq:final optimization problem\]) [@seong2006optimal]. First, the Lagrangian is written for problem (\[eq:final optimization problem\]), and in the second step, it is decomposed for each resource block. After that, the resource block allocation and the power of each RB are found in terms of the Lagrange multiplier $\lambda$. Finally, $\lambda$ is calculated using an ellipsoid method. The Lagrangian for problem (\[eq:final optimization problem\]) is $$\begin{aligned} \mathcal{L}(\boldsymbol{P}, \mathbf{\rho},\lambda)&=V\sum_{i,j}p_{i,j}(t)+\sum_{i\in \mathcal{M} \cup \mathcal{H}} -\big(r_i(t)-a_i(t)\big)D_i^{\max}F_i(t)-\lambda_i\big(r_i(t)-a_i(t)\big)\nonumber\\ &=V\sum_{i,j}p_{i,j}(t)-\sum_{i\in \mathcal{M} \cup \mathcal{H}} \big(\lambda_i+D_i^{\max}F_i(t)\big) \big(r_i(t)-a_i(t)\big).\label{eq:lagrangian}\end{aligned}$$ One major difference between our problem and conventional power minimization problems is that there is an additional term $D_i^{\max} F_i(t)$ added to the Lagrange multiplier (the shadow price). In this problem, $D_i^{\max} F_i(t)$ plays the role of a bias term. Therefore, a new hypothetical Lagrange multiplier $\lambda_i'$ is assumed and defined as $$\lambda_i'=\lambda_i+D_i^{\max}F_i(t).$$ This means that adding constraint (\[eq:constrain1\]) to the problem instead of constraint (\[eq:minimum rate lambda\]) increases the shadow price by a factor of $D_i^{\max}F_i(t)$. Increasing the shadow price for a constraint makes it looser. As a result, in many time slots, constraint (\[eq:minimum rate lambda\]) will not be a tight constraint and the Lagrange multiplier will be set to $\lambda_i=\big[\lambda_i'-D_i^{\max}F_i(t)\big]^+$. the Lagrange dual function is $$\label{eq:lagrange dual problem} g(\lambda)=\min_{{\boldsymbol{\rho}}(t),{\boldsymbol{P}}(t)} \quad \mathcal{L}(\boldsymbol{P}, \boldsymbol{\rho},\lambda).$$ The minimization problem (\[eq:lagrange dual problem\]) can be decomposed to $K$ subproblems. $g_j'(\lambda)$ can be written as $$g_j'(\lambda)=\min_{{\boldsymbol{P}}(t)} \quad V\sum_i p_{i,j}-\sum_{i\in \mathcal{M} \cup \mathcal{H}} \big(\lambda_i+D_i^{\max}F_i(t)\big) \big(W \log_2(1+K \frac{h_{i,j}p_{i,j}}{\sigma^2})\big),$$ where $\mathcal{D}$ is a set of feasible $p_{ij}$s in which for RB $j$, there is only one $i$ that $p_{ij}\neq0$. Hence, $g(\lambda)$ is $$\label{eq:dual_objective} g(\lambda)=\sum_j g_j'(\lambda)+\sum_{i\in \mathcal{M} \cup \mathcal{H}} \big(\lambda_i+D_i^{\max}F_i(t)\big) \big(a_i(t)\big).$$ If $\lambda$ is fixed, $g_j'(\lambda)$ is a convex function of $\boldsymbol{P}$. Therefore, $\boldsymbol{P}$ is found by taking a derivate with respect to $p_{ij}$ and setting it to zero. This results in $$p_{ij}=\Big[\frac{\big(\lambda_i+D_i^{\max}F_i(t)\big) W}{V \log_2 }-\frac{\sigma^2}{K h_{ij}}\Big]^+.$$ The optimal RB allocation for RB $j$ is $k(j)$, and can be written as $$\begin{aligned} k(j)=\underset{i}{\text{argmin}} V\sum_i p_{i,j}-\sum_{i\in \mathcal{M} \cup \mathcal{H}} \big(\lambda_i+D_i^{\max}F_i(t)\big) \big(W \log_2(1+K \frac{h_{i,j}p_{i,j}}{\sigma^2})\big), \label{eq:find the subchannel assignment}\\ g_j'(\lambda)=\underset{i}{\min} V\sum_i p_{i,j}-\sum_{i\in \mathcal{M} \cup \mathcal{H}} \big(\lambda_i+D_i^{\max}F_i(t)\big) \big(W \log_2(1+K \frac{h_{i,j}p_{i,j}}{\sigma^2})\big).\end{aligned}$$ Thus, $\rho^*_{ij}$ and $p^*_{ij}$ will be given by: $$\begin{aligned} &\rho^*_{ij}=\begin{cases} 1, &i=k(j),\\ 0, &\text{otherwise}. \end{cases} & p^*_{ij}=\begin{cases} p_{ij}, &i=k(j),\\ 0, &\text{otherwise}. \end{cases}\end{aligned}$$ Hence, the optimal rate becomes $r^*_i= \sum_j W \log_2(1+K \frac{h_{i,j}p^*_{i,j}}{\sigma^2})$. The only parameter that affects this joint RB and power allocation is $\lambda$. As the number of RBs increases, the duality gap in this problem approaches zero [@seong2006optimal]. We know that the optimal value is found by maximization of $g(\lambda)$ with respect to $\lambda$. In order to find $\lambda$, we use the ellipsoid method [@yu2006dual], and to do so, we have to find the sub-gradient for the dual objective $g(\lambda)$. The following theorem will show that the subgradient for (\[eq:lagrangian\]) is a vector with elements $d_i=a_i-r_i$. The subgradient of the dual optimization problem with dual objective defined in (\[eq:dual\_objective\]), is the vector $\boldsymbol d$ whose elements $d_i, \, \forall i \in \mathcal{H} \cup \mathcal{M}$ are given by: $$d_i=\begin{cases} a_i-r_i^*, &a_i \geq r_i^*,\\ 0, &a_i<r_i^*. \end{cases}$$ Since $$g({\boldsymbol{\lambda}})=\min_{{\boldsymbol{P}},\rho}\mathcal{L}(\boldsymbol{P}, \boldsymbol{\rho},{\boldsymbol{\lambda}})=\mathcal{L}(\boldsymbol{P}^*, \mathbf{\rho}^*,{\boldsymbol{\lambda}}),$$ we have: $$\begin{aligned} g(\boldsymbol{{\boldsymbol{\delta}}} )\leq& \mathcal{L}(\boldsymbol{P}^*, \mathbf{\rho}^*,{\boldsymbol{\delta}})\nonumber\\ =&V\sum_{i,j}p^*_{i,j}(t)-\sum_{i\in \mathcal{M} \cup \mathcal{H}} \big(\delta_i+D_i^{\max}F_i(t)\big) \big(r^*_i(t)-a_i(t)\big)\nonumber\\ =&V\sum_{i,j}p^*_{i,j}(t)-\sum_{i\in \mathcal{M} \cup \mathcal{H}} \big(\lambda_i+D_i^{\max}F_i(t)\big) \big(r^*_i(t)-a_i(t)\big)\nonumber\\ &+(\lambda_i-\delta_i)\big(r^*_i(t)-a_i(t)\big)=g({\boldsymbol{\lambda}})+({\boldsymbol{\lambda}}-{\boldsymbol{\delta}})^T\boldsymbol d',\end{aligned}$$ where $$\boldsymbol d'=\begin{bmatrix} r_1^*-a_1 &\cdots & r_{N+M}^*-a_{N+M} \end{bmatrix}^T.$$ However, because of the term $D_i^{\max}F_i(t)$, when $\lambda=0$ and $a_i<r_i^*$, then the direction of $\boldsymbol d'$ is infeasible. Using the projected subgradient method [@boyd2006subgradient], we can transform this infeasible direction to a feasible one. The update rule for projected subgradient is: $${\boldsymbol{\lambda}}^{(k+1)}=\Pi ({\boldsymbol{\lambda}}^{(k)}- \alpha_k d_k')$$ where $\alpha_k$ is the step size and $\Pi$ is the Euclidan projection on the feasible set. Since the feasible set is $\lambda_i>0$, we can see that $$\Pi ({\boldsymbol{\lambda}}^{(k)}- \alpha_k d_k')={\boldsymbol{\lambda}}^{(k)}- \alpha_k d_k,$$ where: $$d_i=\begin{cases} d_i', &d_i'\geq0\\ 0, &d_i'<0 \end{cases}=\begin{cases} a_i-r_i^*, &a_i \geq r_i^*,\\ 0, &a_i<r_i^*. \end{cases}$$ Complexity Analysis ------------------- Next, we find the complexity of our algorithm which needs to be run in each iteration. There are $K$ RBs in our problem, for each of which (\[eq:find the subchannel assignment\]) needs to be evaluated for $M+N$ users. It takes $\mathcal{O}\big((M+N)K\big)$ times to solve a primal problem. Subsequently, the dual problem will be solved, which gives us the optimal value of $\lambda$ in an $M+N$ dimensional space and has a complexity of $\mathcal{O}\big((M+N)^2\big)$. Therefore, the overall complexity should be $\mathcal{O}\big((M+N)^3K\big)$. However, as mentioned before, adding $D_i^{\max}F_i(t)$ to the Lagrange multiplier sets a major part of it to zero, and as a result, the order of complexity will decrease to $\mathcal{O}\big((M+N)K\big)$. Simulation Results and Analysis =============================== For our simulations, we consider the dataset in [@yang2015prospect] to model the delay perception of a human user. In [@yang2015prospect], the authors conducted human subject studies using 30 human users, where each subject is asked to rate the quality of 5 movies while the delay and packet loss in the system is being increased. We used the average score of each human user to estimate their delay perception. We also used a variation of the bootstrap method [@friedman2001elements] to increase the number of data points to 1000. We can see the histogram of the delay perception for these 1000 data points in Fig. \[fig:Hist\]. Also, since the dataset has no features for each user, we attributed three random continuous features to each user. Hence, each user is associated with a vector ${\boldsymbol{w}}\in \mathds{R}^4$. We consider a network with a bandwidth of $10$ MHz, $a_i(t)=1$ Mbps, $\sigma^2=-173.9$ dBm, and $\epsilon=0.05$. We use a circular cell with the cell radius of $1.5$ km. We set the path loss exponent to $3$ (urban area) and the carrier frequency to $900$ MHz. The packet length is an exponential random variable with an average size of $10$ kbits. We use 5 MTD and 5 UE in the system and we set $D_i^{\max}$ to $20$ ms for them, unless otherwise mentioned. For the brain aware users, we arbitrarily select 5 UE in the system out of all data points. Fig. \[fig:WPS\] shows the within cluster point scatter for the EM algorithm in our dataset. This *within cluster point scatter* for a clustering $C$ is defined as [@friedman2001elements]: $$W(C)=\frac{1}{2}\sum_{k=1}^n\sum_{c(i)=k}\sum_{c(i')=k}d(x_i,x_{i'}),$$ where $d$ is an arbitrary distance metric. In essence, the within cluster point scatter is a loss function that allows the determination of hyper-parameters in the clustering algorithm. The hyper-parameter that we seek to find here is the number of clusters in the dataset. As we can see from Fig. \[fig:WPS\], after the number of clusters reaches $5$, increasing the number of clusters does not decrease the within cluster point scatter substantially. Hence, the optimal number of clusters with is $5$. Fig. \[fig:power versus delay\] shows the total BS power resulting from the proposed brain-aware case and from a brain-unaware case in which UEs have a fixed constraint (\[eq:constrain1\]) with $D_i^{\max}$ between $10$ ms to $60$ ms. Here, the total power is the objective of main optimization problem (\[eq:OrigOptim\]). Fig. \[fig:power versus delay\] shows that, as the latency increases, the total power decreases, because it is easier to satisfy constraint (\[eq:constrain1\]) at higher latencies. Also, at higher delays, being brain-aware will no longer yield substantial gains, since $\beta_i(t)$ and $D_i^{\max}$ become close to each other and learning $\beta_i(t)$ cannot save resources for the system. In contrast, in Fig. \[fig:power versus delay\], we can see that for stringent low-latency requirements, the proposed brain-aware approach yields significant gains in terms of saving power. In particular, for $10$ ms delay in (\[eq:constrain1\]), Fig. \[fig:power versus delay\] shows that the BS in brain-unaware approach uses $44$ % more power compared to the brain-aware case. These results stem from the fact that a brain-aware approach can minimize waste of resources and provide service to the users more precisely based on their real brain processing power. Fig. \[fig:sweepMTD\] shows average BS power for different number of MTDs. As we can see from Fig. \[fig:sweepMTD\], the brain-aware approach will always outperform the brain-unaware approach as the number of MTD increases. For the case of 30 MTD user, the BS in brain-unaware approach uses $16$ % more power compared to the brain-aware case. This is due to fact that brain-aware approach can allocate resources more efficiently in case of a shortage in resources. In Fig. \[fig:sweepuser\], we show the average power usage of the system when the number of UEs increases from $2$ to $30$ with $D_i^{\max}$ set to $20$ ms. As the number of users increases, the average power consumption of the system will also increase. This is due to the fact that increasing the number of users will decrease the bandwidth per user. Since the delay and rate requirements of each user are still unchanged, the system needs to use more power to compensate for the bandwidth deficiency. From Fig. \[fig:sweepuser\], we can see that, in the case of $30$ users, the brain-aware system is able to save $6.7$ dB ($78$%) on average in the BS power. The brain-aware system can allocate resources based on each user’s actual requirement instead of the predefined metrics and this leads to this significant saving in the power consumption of the BS. In Fig. \[fig:youngOldpower\], Fig. \[fig:YoungOldRate\], and Fig. \[fig:YoungoldRely\], we consider the case of 7 UEs and 5 MTDs. Two UEs are chosen as brain-aware users and their delay perception is learned by the PDI method. One of the brain-aware UEs has a delay perception of $\beta_i=133.73$ ms, and the other one has $\beta_i$ equal to $26.8$ ms. The system does not learn the delay perception of the 5 remaining UEs and, hence, it allocates resources to them by using a predefined delay requirement (brain-unaware users). As we can see in Fig. \[fig:youngOldpower\], the power consumption of the first two brain-aware users will be less than that of the brain-unaware users. Moreover, the power consumption for a user with higher delay perception will be less than that of a user with lower delay perception. This shows that the system can successfully allocate resources according to the delay perception of the users. Furthermore, the power consumption related to each user with predetermined delay requirements is different, due to their different channel gains. However, as we will see later, the system is robust to such differences and can guarantee the reliability and rate requirements for users having different channel gains. In Fig. \[fig:YoungOldRate\], we show the transmission rate for four different users. We can see that the rate for brain-unaware users with predetermined delay will converge to $2.5$ Mbps. This rate will ensure the reliability for these users. However, the rate of the users with learned delay perception will converge to a smaller rate. This is due to the fact that these users’ actual requirements are known to the system, and the system uses this knowledge to avoid unnecessarily wasting resources. However, as we will see next, this rate reduction does not change the reliability for these users. Fig. \[fig:YoungoldRely\] shows the reliability for the four aforementioned users. As we can see, the reliability of all the users will converge to $95$ %, which is the target reliability value for the users. We can see that the system is able to ensure reliability for the users with identified delay perceptions as well as the users with predefined delay requirements. However, the system used $45$% less power for those users for which the delay perception is learned. Finally, Fig. \[fig:V\] investigates the effect of parameter $V$ for the system with 5 MTDs and 5 UEs. We can see that, as $V$ increases from $1$ to $1.9$, the convergence time decreases from $40$ iterations to $15$ iterations. Nevertheless, increasing $V$ will make the algorithm unstable, and as we can see, increasing it to $2.2$ will create an overshoot which is $11$% higher than the final value. Hence, parameter $V$, if adjusted correctly, can create a balance between stability and convergence rate of our algorithm. Conclusion ========== In this paper, we have introduced and formulated the notion of delay perception of a human brain, in wireless networks with humans-in-the-loop. Using this notion, we have defined the concept of effective delay of human brain. To quantify this effective delay, we have proposed a novel learning method, named PDI, which consists of an unsupervised and supervised learning part. We have then shown that PDI can predict the effective delay for the human users and find the reliability of this prediction. Then, we have derived a closed-form relationship between the reliability measure and wireless physical layer metrics. Next, using this relationship and the PDI method, we have proposed a novel approach based on Lyapunov optimization for allocating radio resources to human users while considering the reliability of both machine type devices and human users. Our results have shown that the proposed brain-aware approach can save a significant amount of power in the system, particularly for low-latency applications and congested networks. To our best knowledge, this is the first study on the effect of human brain limitations in wireless network design. Proof of Theorem \[lemma:confidence region single mode\] ======================================================== We assume that a single brain mode is dominant for each user at each time. We index this single mode as $k$. For each user $i$ with this dominant mode, ${\boldsymbol{w}}_i=[w_1,\cdots,w_n]$ has the following probability density function: $$f({\boldsymbol{w}}_i)= |2\pi {\boldsymbol{\Sigma}}_k|^{-\frac{1}{2}} \exp\Big[-\frac{1}{2}({\boldsymbol{w}}_i-{\boldsymbol{\mu}}_k)^T{\boldsymbol{\Sigma}}_k^{-1}({\boldsymbol{w}}_i-{\boldsymbol{\mu}}_k)\Big].$$ We want to find the smallest region $\mathcal{D}$ in $\mathds{R}^{p+1}$, in which the delay perception lies with probability $\gamma$, i.e., $$\label{eq:Region D definition} \int \cdots \int_\mathcal{D}\, f(w_1,w_2,\ldots,w_n) \,dw_1 \!\cdots dw_n=\gamma.$$ ${\mathcal{D}}$ is not a unique region. However, the objective is to find the smallest region. To this end, we need to find the region where $f(w_1,w_2,\ldots,w_n)$ has the greatest value, i.e., if $$\label{eq:appendix:equal to} \int \cdots \int_{{\mathcal{D}}_1}\, f(w_1,w_2,\ldots,w_n) \,dw_1 \!\cdots dw_n=\int \cdots \int_{{\mathcal{D}}_2}\, f(y_1,y_2,\ldots,y_n) \,dy_1 \!\cdots dy_n,$$ and also $$\label{eq:appendix:less than} f(y_1,y_2,\ldots,y_n) \leq f(w_1,w_2,\ldots,w_n) \qquad\forall \boldsymbol{y} \in {\mathcal{D}}_2,\,\, \forall {\boldsymbol{w}}_i \in {\mathcal{D}}_1,$$ then $$\int \cdots \int_{{\mathcal{D}}_1} \,dw_1 \!\cdots dw_n \leq \int \cdots \int_{{\mathcal{D}}_2} \,dy_1 \!\cdots dy_n,$$ which implies that the volume of the region ${\mathcal{D}}_1$ is smaller than the volume of ${\mathcal{D}}_2$. Hence, if we find the region ${\mathcal{D}}$ for which (\[eq:Region D definition\]) holds, and, using (\[eq:appendix:less than\]), show that all other regions for which (\[eq:Region D definition\]) holds have greater volumes, then, we would have found the smallest region ${\mathcal{D}}$, in which the human behavior will stay with the probability $\gamma$. Since ${\boldsymbol{w}}_i$ is distributed according to a multivariate Gaussian, we can find the region where it has the highest probability density, i.e., $\big\{{\boldsymbol{w}}_i|f({\boldsymbol{w}}_i)>C_1\big\}$. This region can be written as: $$\bigg\{{\boldsymbol{w}}_i \bigg| \,|2\pi {\boldsymbol{\Sigma}}_k|^{-\frac{1}{2}} \exp\Big[-\frac{1}{2}({\boldsymbol{w}}_i-{\boldsymbol{\mu}}_k)^T{\boldsymbol{\Sigma}}_k^{-1}({\boldsymbol{w}}_i-{\boldsymbol{\mu}}_k)\Big]>C_1 \bigg\},$$ which is equivalent to $$\label{eq:appendix:ellipsoid} {\mathcal{D}}=\Big\{{\boldsymbol{w}}_i\Big|({\boldsymbol{w}}_i-{\boldsymbol{\mu}}_k)^T{\boldsymbol{\Sigma}}_k^{-1}({\boldsymbol{w}}_i-{\boldsymbol{\mu}}_k)<C_2\Big\},$$ where $C_2$ is a positive constant and equals $-\ln |2 \pi{\boldsymbol{\Sigma}}_k|^{\frac{1}{2}} C_1$. Since ${\boldsymbol{\Sigma}}_k$ is a positive definite matrix, (\[eq:appendix:ellipsoid\]) is the inner volume of an ellipsoid in a $p$ dimensional space. We now conjecture that this ellipsoid ${\mathcal{D}}$ is the smallest region, in which the delay perception lies with probability $\gamma$, i.e., the probability of ${\boldsymbol{w}}_i$ being in this region is $\gamma$. We use a proof by contradiction to show this. Consider that there exists any other space ${\mathcal{E}}$ which is smaller than ${\mathcal{D}}$, and the probability of ${\boldsymbol{w}}_i$ being in this region is $\gamma$. We can partition ${\mathcal{E}}$ into two parts ${\mathcal{A}}={\mathcal{E}}\cap {\mathcal{D}}$ and ${\mathcal{E}}_2={\mathcal{E}}\cap {\mathcal{D}}'$, where ${\mathcal{D}}'$ is the complement of the set ${\mathcal{D}}$. We also define ${\mathcal{D}}_2={\mathcal{D}}\cap {\mathcal{E}}'$. We know that $$\begin{aligned} &\int_{\mathcal{D}}f({\boldsymbol{w}}_i) d{\boldsymbol{w}}_i=\int_{{\mathcal{A}}} f({\boldsymbol{w}}_i) d{\boldsymbol{w}}_i+\int_{{\mathcal{D}}_2} f({\boldsymbol{w}}_i) d{\boldsymbol{w}}_i \\ =&\int_{\mathcal{E}}f({\boldsymbol{w}}_i) d{\boldsymbol{w}}_i=\int_{{\mathcal{A}}} f({\boldsymbol{w}}_i) d{\boldsymbol{w}}_i+\int_{{\mathcal{E}}_2} f({\boldsymbol{w}}_i) d{\boldsymbol{w}}_i \\ =&\gamma. \end{aligned}$$ Hence, $\int_{{\mathcal{D}}_2} f({\boldsymbol{w}}_i) d{\boldsymbol{w}}_i=\int_{{\mathcal{E}}_2} f({\boldsymbol{w}}_i) d{\boldsymbol{w}}_i$. Since $$f({\boldsymbol{w}}_i)<C_1\leq f(\boldsymbol{y}) \qquad \forall {\boldsymbol{w}}_i \in {\mathcal{E}}_2, \boldsymbol{y} \in {\mathcal{D}}_2,$$ using (\[eq:appendix:equal to\]) and (\[eq:appendix:less than\]) we have $$\int_{{\mathcal{E}}_2} d{\boldsymbol{w}}_i<\int_{{\mathcal{D}}_2} d{\boldsymbol{w}}_i.$$ This means that the set ${\mathcal{E}}$ has a bigger volume than ${\mathcal{D}}$, which is a contradiction to our first assumption. This proves that region ${\mathcal{D}}$ is the smallest region in $\mathds{R}^{p+1}$ that has the probability $\gamma$. Next, we find the relation between $C_2$ and $\gamma$. $\gamma$ can be defined as $\int_{\mathcal{D}}f({\boldsymbol{w}}_i) d{\boldsymbol{w}}_i$ and can be calculated using chi-square distribution [@berger1980robust]. The region ${\mathcal{D}}$ can be written as $$D=\big\{{\boldsymbol{w}}_i|({\boldsymbol{w}}_i-{\boldsymbol{\mu}}_k)^T{\boldsymbol{\Sigma}}_k^{-1}({\boldsymbol{w}}_i-{\boldsymbol{\mu}}_k)\leq Q_p(\gamma)\big\},$$ where $Q_p(\gamma)$ is the quantile function of the chi-square distribution with $p$ degrees of freedom. It is defined as $$Q_p(\gamma)=\inf\Big\{x\in \mathds{R}| Pc\leq \int_{0}^{x} \chi_p^2(u) du\Big\}.$$ Having defined the confidence region based on $\gamma$, we now must find the edges of this ellipsoid. We know that the center of this ellipsoid is $\mu_k$. We need to solve the following optimization problem: $$\underset{{\boldsymbol{w}}_i}{\min} \,\,\text{or}\,\, \underset{{\boldsymbol{w}}_i}{\max} \,\,e_i^T{\boldsymbol{w}}_i,\qquad \text{s.t.} \,\, {\boldsymbol{w}}_i \in {\mathcal{D}},$$ where $\boldsymbol{e}_i$ is a unit vector in $\mathds{R}^p$, having $1$ in its $i$th element and zero otherwise. Using KKT conditions for solving the above problem, we have: $$\begin{aligned} \qquad &e_i+\lambda\,{\boldsymbol{\Sigma}}_k^{-1}({\boldsymbol{w}}_i-{\boldsymbol{\mu}}_k)=0,\\ \qquad &({\boldsymbol{w}}_i-{\boldsymbol{\mu}}_k)^T{\boldsymbol{\Sigma}}_k^{-1}({\boldsymbol{w}}_i-{\boldsymbol{\mu}}_k)\leq Q_p(\gamma),\label{eq:appendix:KKT:bound}\\ \qquad &\lambda\Big(({\boldsymbol{w}}_i-{\boldsymbol{\mu}}_k)^T{\boldsymbol{\Sigma}}_k^{-1}({\boldsymbol{w}}_i-{\boldsymbol{\mu}}_k)- Q_p(\gamma)\Big)=0, \quad \lambda\geq 0. \end{aligned}$$ The inequality in (\[eq:appendix:KKT:bound\]) is tight. With some algebraic manipulation, we have ${\boldsymbol{w}}_i-{\boldsymbol{\mu}}_k(i)=\frac{1}{\lambda}{\boldsymbol{\Sigma}}\boldsymbol{e}_i$, and so, $\frac{1}{\lambda^2} {\mathbf{e}}_i^T{\boldsymbol{\Sigma}}_k {\boldsymbol{\Sigma}}_k^{-1}{\boldsymbol{\Sigma}}_k{\mathbf{e}}_i=Q(\gamma)$. Therefore, $$\begin{aligned} &{\boldsymbol{w}}_i=\pm \sqrt{\frac{Q(\gamma)}{\boldsymbol{e}_i^T\,{\boldsymbol{\Sigma}}_k\,\boldsymbol{e}_i}}{\boldsymbol{\Sigma}}_k \boldsymbol{e}_i+{\boldsymbol{\mu}}_k,\\ &\lambda =\pm\sqrt{\frac{{\mathbf{e}}_i^T{\boldsymbol{\Sigma}}_k{\mathbf{e}}_i}{Q(\gamma)}},\\ &{\mathbf{e}}_i^T {\boldsymbol{w}}_i=\pm \sqrt{Q(\gamma) {{\mathbf{e}}_i^T{\boldsymbol{\Sigma}}_k{\mathbf{e}}_i}}+\mu_k(i). \end{aligned}$$ If $\lambda$ is positive, we can find the maximum which is $+ \sqrt{Q(\gamma) {{\mathbf{e}}_i^T{\boldsymbol{\Sigma}}_k{\mathbf{e}}_i}}+{\boldsymbol{\mu}}_k(i)$, and if $\lambda$ is negative, we can find the minimum which is $- \sqrt{Q(\gamma) {{\mathbf{e}}_i^T{\boldsymbol{\Sigma}}_k{\mathbf{e}}_i}}+{\boldsymbol{\mu}}_k(i)$. Here, ${\boldsymbol{\mu}}_k(i)$ is the $i$th element of ${\boldsymbol{\mu}}_k$. If we set $i=p+1$, then the delay perception of user $i$ is in the following range: $$-\sqrt{Q(\gamma) \boldsymbol{e}_{p+1}^T{\boldsymbol{\Sigma}}_k \boldsymbol{e}_{p+1}}< \beta_i-\mu_k^D<\sqrt{Q(\gamma) \boldsymbol{e}_{p+1}^T{\boldsymbol{\Sigma}}_k \boldsymbol{e}_{p+1}},$$ at least with probability $\gamma$. Hence, Theorem \[lemma:confidence region single mode\] is proved. Proof of Theorem \[prop:packet length\] ======================================= Since the queuing delay is much smaller than the duration of each time slot, we can assume that each packet arriving at a specific time slot will be served at the same time slot. For analyzing the packet delay, we consider a packet that just arrives in the system in time slot $\tau_k$, and find $\text{Pr}(D>D_i^{\max})$ for this packet. When this packet arrives, there are $m$ packets in the system. From lemma \[corol:exponential service time\], we know that the serving time will be an exponential random variable. Since the exponential distribution is memoryless, there is no distinction between a packet already in service and the other packets. Therefore, the waiting time for the packet that has just arrived is the summation of $m$ exponential distributions. Also, the transmission delay for this packet will be another exponential random variable. Hence, the delay of a packet which arrives at time slot $\tau_k$ while there are $m$ packets in the system can be written as: $$d(\tau_k,m)=t_s+t_1(\tau_k)+t_2(\tau_k)+\cdots+t_{m-1}(\tau_k)+t_c(\tau_k),$$ where $t_i(\tau_k)$ is the service time for packet $i$ in the queue, and $t_c(\tau_k)$ is the service time for packet already in service. Also, $t_s$ is the service time for the packet that has just arrived. we seek to find $\text{Pr}(d(\tau_k,m)>D_i^\textrm{max})$ which can be written as $$\begin{aligned} \text{Pr}\big(d(\tau_k,m)>D_i^\textrm{max}\big)&=\sum_{m,k} \text{Pr}(D>D_i^{\textrm{max}}|m,\tau_k)\text{Pr}(m,\tau_k)\nonumber\\ &=\sum_{m,k} \text{Pr}(D>D_i^{\textrm{max}}|m,\tau_k)\text{Pr}(m|\tau_k) \text{Pr}(\tau_k). \end{aligned}$$ The probability that there are $m$ users in an M/M/1 queue at time slot $\tau_k$, i.e. $\text{Pr}(m|\tau_k)$, can be written as (see [@papoulis2002probability]): $$\text{Pr}(m|\tau_k)=\left(\frac{a_i(\tau_k)}{r_i(\tau_k)}\right)^m \left(1-\frac{a_i(\tau_k)}{r_i(\tau_k)}\right).$$ Since we assumed the time slots have equal lengths, the packets arrive at each time slot with equal probability of $\text{Pr}(\tau_k)=\frac{1}{t}$, where $t$ is the total number of time slots. The sum of $m+1$ identically independent exponential random variables with the mean $\frac{1}{r_i(\tau_k)}$ is a gamma random variable. Consequently, if the users arrive at time slot $\tau_k$ while there are $m$ users in the system at the time of arrival, the distribution of delay is $$f_D(\phi|m,\tau_k)=\frac{r_i(\tau_k)^{m+1}}{\Gamma(m+1)}\phi^{m}e^{-r_i(\tau_k)\phi}.$$ As a result, we can write the probability of delay exceeding a threshold $D_i^{\max}$ as $$\begin{aligned} \text{Pr}(D>D_i^{\max})= &\int_{D_i^{\max}}^{\infty} \sum_{m,k} f_D(\phi|m,\tau_k)\text{Pr}(m|\tau_k) \text{Pr}(\tau_k) d \phi\\ = &\int_{D_i^{\max}}^{\infty} \frac{1}{t}\sum_{m,k} \frac{r_i(\tau_k)^{m+1}}{m!}\phi^{m}e^{-r_i(\tau_k)\phi}(\frac{a_i(\tau_k)}{r_i(\tau_k)})^m (1-\frac{a_i(\tau_k)}{r_i(\tau_k)}) d \phi\\ =&\frac{1}{t}\sum_{k=1}^{t}\int_{D_i^{\max}}^\infty (r_i(\tau_k)-a_i(\tau_k))e^{-r_i(\tau_k)\phi}\sum_{m=0}^{\infty}\frac{\big(\phi a_i(\tau_k)\big)^m}{m!}d \phi\\ =&\frac{1}{t}\sum_{k=1}^{t}\int_{D_i^{\max}}^\infty (r_i(\tau_k)-a_i(\tau_k))e^{-\big(r_i(\tau_k)-a_i(\tau_k)\big)\phi}d \phi\\ =&\frac{1}{t}\sum_{k=1}^{t}e^{-\big(r_i(\tau_k)-a_i(\tau_k)\big)D_i^{\max}}, \end{aligned}$$ which proves the theorem.