Higher-spin Realisations of the Bosonic String 1
It has been shown that certain W algebras can be linearised by the inclusion of a spin–1 current. This provides a way of obtaining new realisations of the W algebras. Recently such new realisations of W3 were used in order to embed the bosonic string in the critical and non-critical W3 strings. In this paper, we consider similar embeddings in W2,4 and W2,6 strings. The linearisation of W2,4 is already known, and can be achieved for all values of central charge. We use this to embed the bosonic string in critical and non-critical W2,4 strings. We then derive the linearisation of W2,6 using a spin–1 current, which turns out to be possible only at central charge c = 390. We use this to embed the bosonic string in a non-critical W2,6 string. Supported in part by the U.S. Department of Energy, under grant DE-FG05-91-ER40633 With the discovery of the property of the W3 algebra that it can be linearised by the inclusion of a spin–1 current , new realisations were constructed for the purpose of building the corresponding W3 strings [2, 3]. An unusual feature of these realisations is that the spin–3 current contains a term linear in a ghost-like field. The realisations also close when this term is omitted, under which circumstance the corresponding string theory is equivalent to the one that is based on the Romans’ free-scalar realisation [4]. However, when this term is included, the corresponding BRST operator is equivalent to that of the bosonic string, which can be shown by making a local canonical field redefinition . Thus the new realisations provide embeddings of the bosonic string in the W3 string. It is interesting to generalise the above consideration to the embedding of the bosonic string in W2,s strings, where W2,s denotes the conformal algebra generated by a spin–s current together with the energy-momentum tensor. The W2,s strings based on free-scalar realisations were extensively discussed in Ref. , where it was shown that when s ≥ 3 the cohomologies describe Virasoro strings coupled to certain minimal models. The W2,s algebras exist at the classical level for all positive integer values of s. However, at the quantum level, for generic values of s, a W2,s algebra exists only for a finite set of special values of central charge [6, 7], which in particular does not include the critical value. The exceptions are the W2,s algebras for s = 1, 2, 3, 4 and 6, for which the central charge can be arbitrary. Although the W2,s algebra does not close at the critical central charge for generic values of s, it is nevertheless possible to build W2,s strings with free-scalar realisations. It was shown in Ref.  that one can first use the free-scalar realisation to write down the classical BRST operator, and then quantise the theory by renormalising the transformation rules and adding necessary quantum counter-terms. The new realisations that were constructed in Ref. , which provide embeddings of the bosonic string in W3 = W2,3 strings, do not generate the W3 algebra at the classical level. The W3 symmetry arises only as a consequence of quantisation. Thus it seems that if we are to use such new realisations for values of s other than 3, we must restrict our attention to the cases s = 1, 2, 4 and 6, for which the quantum algebras exist. The embeddings of the bosonic string in the W2,s string for s = 1, 2 and 3 were discussed in Ref. . In this paper, we shall focus our attention on the remaining cases s = 4 and 6. It is instructive to begin by studying the form of the linearisation of the W3 and W2,4 algebras, for which the results were obtained in Refs. . The associated linearised W1,2,3 and W1,2,4 algebras take the form: T0(z) T0(0) ∼ c 2z4 + 2T0 z2 + ∂T0 z , T0(z) W0(0) ∼ sW0 z2 + ∂W0 z , T0(z) J0(0) ∼ c1 z3 + J0 z2 + ∂J0 z , J0(z) J0(w) ∼ − 1 z2 , (1) J0(z) W0(w) ∼ hW0 z , W0(z) W0(w) ∼ 0 , where s = 3 and 4 respectively. The coefficients c, c1 and h are given by c = 50 + 24t + 24 t2 , c1 = − √ 6(t + 1 t ) , h = √ 3 2 t , (s = 3) c = 86 + 30t + 60 t2 , c1 = −3t− 4 t , h = t . (s = 4) (2) The currents of the W3 and W2,4 algebras are then given by T = T0 , W = W0 + WR , (3) where WR is the Romans type realisation constructed from T0 and J0. For the cases where s = 3, 4 and 6, WR takes the form

title: SYMMETRY IN TWO-DIMENSIONAL GRAVITY
abstract: We study the symmetry of two-dimensional gravity by choosing a generic gauge. A local action is derived which reduces to either the Liouville action or the Polyakov one by reducing to the conformal or light-cone gauge respectively. The theory is also solved classically. We show that an SL(2, R) covariant gauge can be chosen so that the two-dimensional gravity has a manifest Virasoro and the sl(2, R)-current symmetry discovered by Polyakov. The symmetry algebra of the light-cone gauge is shown to be isomorphic to the Beltrami algebra. By using the contour integration method we construct the BRST charge QB corresponding to this algebra following the Fradkin-Vilkovisky procedure and prove that the nilpotence of QB requires c=28 and α0=1. We give a simple interpretation of these conditions.

title: QUANTIZING HIGHER-SPIN STRING THEORIES
abstract: In this paper, we examine the conditions under which a higher-spin string theory can be quantized. The quantizability is crucially dependent on the way in which the matter currents are realized at the classical level. In particular, we construct classical realizations for the W2,s algebra, which is generated by a primary spin-s current in addition to the energy-momentum tensor, and discuss the quantization for s≤8. From these examples we see that quantum BRST operators can exist even when there is no quantum generalization of the classical W2,s algebra. Moreover, we find that there can be several inequivalent ways of quantizing a given classical theory, leading to different BRST operators with inequivalent cohomologies. We discuss their relation to certain minimal models. We also consider the hierarchical embeddings of string theories proposed recently by Berkovits and Vafa, and show how the already known W strings provide examples of this phenomenon. Attempts to find higher-spin fermionic generalizations lead us to examine whether classical BRST operators for (n odd) algebras can exist. We find that even though such fermionic algebras close up to null fields, one cannot build nilpotent BRST operators, at least of the standard form.

title: The complete spectrum of the WN string
abstract: We obtain the complete physical spectrum of the W N string, for arbitrary N . The W N constraints freeze N −2 coordinates, while the remaining coordinates appear in the currents only via their energy-momentum tensor. The spectrum is then effectively described by a set of ordinary Virasoro-like string theories, but with a non-critical value for the central charge and a discrete set of non-standard values for the spin-2 intercepts. In particular, the physical spectrum of the W N string includes the usual massless states of the Virasoro string. By looking at the norms of low-lying states, we find strong indications that all the W N strings are unitary.

title: Embedding of the bosonic string into the W3 string
abstract: We investigate new realisations of the W 3 algebra with arbitrary central charge, making use of the fact that this algebra can be linearised by the inclusion of a spin-1 current. We use the new realisations with c = 102 and c = 100 to build non-critical and critical W 3 BRST operators. Both of these can be converted by local canonical transformations into a BRST operator for the Virasoro string with c = 28 f - 2, together with a Kugo-Ojima topological term. Consequently, these realisations provide embeddings of the Virasoro string into non-critical and critical W 3 strings.

title: Classification of p-branes, NUTs, Waves and Intersections
abstract: We give a full classification of the multi-charge supersymmetric p-brane solutions in the massless and massive maximal supergravities in dimensions D ≥ 2 obtained from the toroidal reduction of eleven-dimensional supergravity. We derive simple universal rules for determining the fractions of supersymmetry that they preserve. By reversing the steps of dimensional reduction, the p-brane solutions become intersections of p-branes, NUTs and waves in D = 10 or D = 11. Having classified the lower-dimensional p-branes, this provides a classification of all the intersections in D = 10 and D = 11 where the harmonic functions depend on the space transverse to all the individual objects. We also discuss the structure of U-duality multiplets of p-brane solutions, and show how these translate into multiplets of harmonic and non-harmonic intersections.

title: SHOCK-ENHANCED C+ EMISSION AND THE DETECTION OF H2O FROM THE STEPHAN'S QUINTET GROUP-WIDE SHOCK USING HERSCHEL
abstract: We present the first Herschel spectroscopic detections of the [OI]63µm and [CII]158µm fine-structure transitions, and a single para-H_2O line from the 35 x 15 kpc^2 shocked intergalactic filament in Stephan's Quintet. The filament is believed to have been formed when a high-speed intruder to the group collided with clumpy intergroup gas. Observations with the PACS spectrometer provide evidence for broad (> 1000 km s^(-1)) luminous [CII] line profiles, as well as fainter [OI]63µm emission. SPIRE FTS observations reveal water emission from the p-H_2O (1_(11)-0_(00)) transition at several positions in the filament, but no other molecular lines. The H_2O line is narrow, and may be associated with denser intermediate-velocity gas experiencing the strongest shock-heating. The [CII]/PAH_(tot) and [CII]/FIR ratios are too large to be explained by normal photo-electric heating in PDRs.  
region excitation or X-ray/Cosmic Ray heating can also be ruled out. The observations lead to the conclusion that a large fraction the molecular gas is diffuse and warm. We propose that the [CII], [OI] and warm H_2 line emission is powered by a turbulent cascade in which kinetic energy from the galaxy collision with the IGM is dissipated to small scales and low-velocities, via shocks and turbulent eddies. 
Low-velocity magnetic shocks can help explain both the [CII]/[OI] ratio, and the relatively high [CII]/H_2 ratios observed. The discovery that [CII] emission can be enhanced, in large-scale turbulent regions in collisional environments has implications for the interpretation of [CII] emission in high-z galaxies.

title: POWERFUL HIGH-VELOCITY DISPERSION MOLECULAR HYDROGEN ASSOCIATED WITH AN INTERGALACTIC SHOCK WAVE IN STEPHAN'S QUINTET
abstract: We present the discovery of strong mid-infrared emission lines of molecular hydrogen of apparently high-velocity dispersion (~870 km s-1) originating from a group-wide shock wave in Stephan's Quintet. These Spitzer Space Telescope observations reveal emission lines of molecular hydrogen and little else. This is the first time an almost pure H2 line spectrum has been seen in an extragalactic object. Along with the absence of PAH-dust features and very low excitation ionized gas tracers, the spectra resemble shocked gas seen in Galactic supernova remnants, but on a vast scale. The molecular emission extends over 24 kpc along the X-ray-emitting shock front, but it has 10 times the surface luminosity as the soft X-rays and about one-third the surface luminosity of the IR continuum. We suggest that the powerful H2 emission is generated by the shock wave caused when a high-velocity intruder galaxy collides with filaments of gas in the galaxy group. Our observations suggest a close connection between galaxy-scale shock waves and strong broad H2 emission lines, like those seen in the spectra of ultraluminous infrared galaxies where high-speed collisions between galaxy disks are common.