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ASYMMETRY
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Advanced Series in Mathematical Physics Vol. 22
^-SYMMETRY
P. Bouwknegt Dept. of Physics University of Southern California Los Angeles, CA
K. Schoutens Joseph Henry Laboratories Princeton University Princeton, NJ
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World Scientific Singapore • New Jersey • London • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, London WC2H 9HE
Library of Congress Cataloging-in-Publkation Data Bouwknegt, P. W-symmetry / P. Bouwknegt, K. Schoutens. p. cm. - (Advanced series in mathematical physics ; vol. 22) ISBN 9810217625 1. Conformal invariants. 2. Quantum field theory. 3. Mathematical physics. I. Schoutens, K. II. Title. III. Series. QC174.52.C66B68 1995 530.1'43~dc20 94-23834 CIP
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V
Preface The study of two-dimensional conformal field theories has been a highly successful undertaking in theoretical physics. One reason for this is the intrinsic elegance of these theories, which are among the few interesting quantum field theories for which exact results can be obtained with relative ease. The other reason is the wide range of applications of conformal field theory in string theory, in the theory of critical phenomena and, recently, in a variety of quantum impurity problems such as the Kondo effect and the Callan-Rubakov effect. Since its introduction in 1985, W-symmetry has evolved to become one of the cen tral notions in the study of conformal field theory. Nine years of effort, by theoretical physicists and, increasingly, by mathematicians (in the context of so-called 'vertex opera tor algebras'), have led to a large body of knowledge on W-algebras, their representation theory, and the role they play in rational conformal field theories and other physical the ories. In the process, interesting connections with the theory of affine Lie algebras (affine Kac-Moody algebras) and with hierarchies of integrable differential equations (such as the Korteweg-de Vries (KdV) and Kadomtsev-Petviashvili (KP) hierarchies) have been unraveled. It is the purpose of the present book to offer a collection of reprints of some of the important papers on W-symmetry. We have organized the material in seven chapters, each of which is preceded by a short reading guide. Among other things, these introductions serve the important goal of pointing out results in the literature which, due to lack of space, could not be included in this book. We include an extensive list of references on various aspects of W-symmetry. The regular reference numbers (such as [1]) in the reading guides refer to this list, whereas the bold-face reference numbers (such as [1.1]) refer to the papers reprinted here. The selection of the reprints included in this book was made at the end of 1993 and the list of references was finalized in February, 1994. As is inevitable in such matters, our choice of material reflects our own knowledge and interest within the wide field that we have tried to cover. We apologize to authors who feel that their contributions have not been properly recognized in this book. For readers who need some background information on topics that are closely related to W-symmetry, we recommend the following sources. Two earlier volumes in the 'Advanced
vi
Series in Mathematical Physics' have been devoted to the algebraic structures that underlie conformal field theory: Vol. 2, by V.G. Kac and A.K. Raina, discusses the representation theory of a number of infinite dimensional Lie algebras and Vol. 3, by P. Goddard and D. Olive, reviews a variety of results on affine Lie algebras and on the Virasoro algebra. Pre-1988 conformal field theory has been reviewed in a reprint volume by C. Itzykson, H. Saleur and J.-B. Zuber [210]; another useful reference is [85]. A textbook on 'Conformal Field Theory in Two Dimensions,' by W. Nahm, is about to appear. We also refer to the abovementioned literature for extensive lists of references on background material such as, in particular, conformal field theory. In a recent issue of Physics Reports [79], the present authors have presented an ex tensive overview of W-symmetry. The organization of the present reprint volume has been motivated by the structure of this review paper and we recommend that the interested reader consult both texts in parallel. Los Angeles, Princeton May 1994
vu
CONTENTS Preface
v
Introductory Chapters 1.
History and background 1.1
1.2
2.
I.M. Gel'fand and L.A. Dickij, k family of hamiltonian structures related to non-linear integrable differential equations, Prepr. Inst. Appl. Mat. 136 (Moscow, 1978), reprinted in 'I.M. Gel'fand, Collected Papers', Vol. 1, S.G. Gindikin et al. editors (Springer-Verlag, 1987), pp. 625-646 V. Drinfel'd and V. Sokolov, Lie algebras and equations of Korteweg-de Vries type, J. Sov. Math. 30 (1984) 1975-2036 . . . .
3 25
Classical W-algebras and their connection to Toda field theories 2.1 A. Bilal and J.-L. Gervais, Systematic approach to conformal systems with extended Virasoro symmetries, Phys. Lett. 206B (1988) 412^20 2.2 I. Bakas, Higher spin fields and the Gel'fand-Dickey algebra, Comm. Math. Phys. 123 (1989) 627-639 2.3 J. Balog, L. Feher, L. O'Raifeartaigh, P. Forgacs and A. Wipf, Toda theory and W-algebra from a gauged WZNW point of view, Ann. Phys. 203 (1990) 76-136 2.4 F. A. Bais, T. Tjin and P. van Driel, Covariantly coupled chiral algebras, Nucl. Phys. B357 (1991) 632-654 2.5 F. Delduc, E. Ragoucy and P. Sorba, Super-Toda theories and W-algebras from superspace Wess-Zumino-Witten models, Comm. Phys. 146 (1992) 403-i26
3.
xi
89 98
Ill 172
195
Quantum W-algebras 3.1
3.2
A. B. Zamoldchikov, Infinite additional symmetries in two dimensional conformal quantum field theory, Theor. Math. Phys. 65 (1985) 1205-1213 P. Bouwknegt, Extended conformal algebras, Phys. Lett. 207B (1988) 295-299
221 230
viii
3.3 R. Blumenhagen, M. Flohr, A. Kliem, W. Nahra, A. Recknagel and R. Varnhagen, W-algebras with two and three generators, Nucl. Phys. B361 (1991) 255-289 3.4 J. de Boer, L. Feher and A. Honecker, A class of W-algebras with infinitely generated classical limit, ITP-SB-93-84, BONN-HE-93-49, submitted to Nucl. Phys. B, 38 pages 4.
270
Quantum Drinfel'd-Sokolov reduction 4.1 V. A. Fateev and S. L. Lukyanov, The models of two dimensional conformal quantum field theory with Z„ symmetry, Int. Jour. Mod. Phys. A 3 (1988) 507-520 4.2 V. A. Fateev and S. L. Lukyanov, Exactly soluble models of conformal quantum field theory associated with the simple Lie algebra Dn, Sov. J. Nuc. Phys. 49 (1989) 925-932 4.3 M. Bershadsky and H. Ooguri, Hidden SL(n) symmetry in conformal field theories, Comm. Math. Phys. 126 (1989) 49-83 4.4 J. M. Figueroa-O'Farrill, On the homological construction of Casimiar algebras, Nucl. Phys. B343 (1990) 450-466 4.5 E. Frenkel, W-algebras and Langlands-Drinfeld correspondence, in Proc. of the 1991 Cargese Summer School 'New Symmetry Principles in Quantum Field Theory', eds. J. Frohlich et al. (Plenum Press, New York, 1992), 433^47 4.6 E. Frenkel, V. Kac and M. Wakimoto, Characters and fusion rules for W-algberas via quantized Drinfeld-Sokolov reduction, Comm. Math. Phys. 147 (1992) 295-328 4.7 P. Bowcock and G. M. T. Watts, On the classification of quantum W-algebras, Nucl. Phys. B379 (1992) 63-95 4.8 J. de Boer and T. Tjin, The relation between quantum W-algebras and Lie algebras, Comm. Math. Phys. 160 (1994) 317-332 . . . 4.9 A. Sevrin and W. Troost, Extensions of the Virasoro algebra and gauged WZW models, Phys. Lett. 315B (1993) 304-310
5.
235
311
325
333 368
385
400 434 467 483
Coset constructions 5.1
F. A. Bais, P. Bouwknegt, K. Schoutens and M. Surridge, Extensions of the Virasoro algebra constructed from Kac-Moody algebras using higher order Casimir invariants, Nucl. Phys. B304 (1988) 348-370
493
ix
5.2
5.3 5.4
6.
F. A. Bais, P. Bouwknegt, K. Schoutens and M. Surridge, Coset construction for extended Virasoro algebras, Nucl. Phys. B304 (1988) 371-391 P. Bowcock and P. Goddard, Coset constructions and extended conformal algebras, Nucl. Phys. B305[FS23] (1988) 685-709 . . . G. M. T. Watts, W-algebras and coset models, Phys. Lett. 245B (1990) 65-71
516 537 562
Woo t y P e algebras 6.1
I. Bakas, The large-N limit of extended conformal symmetries, Phys. Lett. 228B (1989) 57-63 6.2 C. N. Pope, L. J. Romans and X. Shen, Woo ana> the Racah-Wigner algebra, Nucl. Phys. B339 (1990) 191-221 6.3 C. N. Pope, L. J. Romans and X. Shen, A new higher-spin algebra and the lone-star product, Phys. Lett. 242B (1990) 401^406 6.4 I. Bakas and E. Kiritsis, Beyond the large-N limit: non-linear Woo a3 symmetry of the SL(2,R/U(1) coset model, Int. Jour. Mod. Phys. A7[Suppl. 1A] (1992) 55-81 6.5 J. Figueroa-O'Farrill, J. Mas and E. Ramos, The topography of Woe-type algebras, Phys. Lett. 299B (1993) 41-i8 7.
571 578 609
615 642
W-gravity and W-strings 7.1
J. Thierry-Mieg, BRS-analysis of Zamolodchikov's spin 2 and 3 current algebra, Phys. Lett. 197B (1987) 368-372 7.2 C. M. Hull, Gauging the Zamolodchikov W-algebra, Phys. Lett. 240B (1990) 110-116 7.3 K. Schoutens, A. Sevrin and P. van Nieuwenhuizen, Properties of covariant W-gravity, Int. Journ. Mod. Phys. A 6 (1991) 2891-2912 7.4 S. R. Das, A. Dhar and S. K. Rama, Physical properties of W-gravities and W-strings, Mod. Phys. Lett. A6 (1991) 3055-3069 7.5 K. Schoutens, A. Sevrin and P. van Nieuwenhuizen, Induced gauge theories and W-gravity, in Proc. of the conference 'Strings and Symmetries 1991', Stony Brook, May 1991, N. Berkovits et al. (World Scientific, 1992), pages 558-590 . . . . 7.6 J. de Boer and J. Goeree, W-gravity from Chem-Simons theory, Nucl. Phys. B381 (1992) 329-359
659 664
671
693
708 741
X
7.7
H. Lu, C. N. Pope, S. Schrans and X.-J. Wang, On the spectrum and scattering of W3 strings, Nucl. Phys. B408 (1993) 3-42 772 7.8 M. Bershadsky, W. Lerche, D. Nemeschansky and N. P. Warner, A BRST operator for non-critical W-strings, Phys. Lett. 282B (1992) 35-41 812 7.9 P. Bouwknegt, J. McCarthy and K. Pilch, Semi-infinite cohomology of W-algebras, Lett. Math. Phys. 29 (1993) 91-102 . 826 7.10 H. Lu, C. N. Pope, X. J. Wang and K. W. Xu, The complete cohomology of the W3 string, Class. Quant. Grav. 11 (1994) 967-982 838 References
857
Introductory Chapters
xii
1. History and background Soon after the first quantum W-algebras were written down in 1985 (see reprint [3.1]), it became clear that a number of results that had already been developed in the math ematical literature were going to be of great help for understanding these algebras. In this chapter we reprint two mathematical papers that have had a major influence on the development of a systematic description of W-algebras. The papers [1.1] and [1.2] are both devoted to the study of integrable hierarchies of non-linear differential equations. Let us explain why they are at the same time relevant for the understanding of W-symmetry. In the work of I.M. Gel'fand and L. Dickey ([1.1] and references therein) it is shown that it is often possible to write hierarchies of integrable differential equations in hamiltonian form. Among other things, this involves the specification of a generalized Poisson bracket, called the Gel'fand-Dickey bracket. It has been found [243,168,165,230,14] that, for the special case of the so-called Korteweg-de Vries (KdV) hierarchy, the Gel'fand-Dickey bracket for the second hamiltonian structure gives rise to a Virasoro algebra. In a similar way, the W/v algebras are related to the second hamiltonian structure of generalized KdV hierarchies, see [1.1], [2.2] and [1,335,247,15,114]. Multi-component generalizations of the KdV hierarchy have been shown to give rise to non-local analogues of W-algebras [54]. The paper [2.1] makes a connection between affine Lie algebras and, again, integrable hierarchies of differential equations. Combining this with the possibility, described above, to associate a W-algebra structure with such integrable hierarchies, we obtain a direct link between affine Lie algebras on the one hand and W-algebras on the other. Once this association, referred to as the Drinfel'd-Sokolov reduction, has been understood it can be formulated without reference to integrable hierarchies of differential equations and in that form it is a powerful and elegant tool for the analysis of W-algebras. In Chapters 2 and 4 we reprint a number of papers where this Drinfel'd-Sokolov reduction is worked out in detail.
Xlll
2. Classical W-algebras and their connection to Toda field theories The defining relations of a W-algebra express the bracket of any two generators of the algebra in terms of a non-linear expression built from the fundamental generators. If we view the generators as functions on a (classical) phase space, the brackets can be viewed as Poisson brackets. The corresponding algebra is called a classical W-algebra. However, in the context of quantum mechanics the generators should rather be viewed as operators acting in a Hilbert space of states, and in that situation the bracket acquires the interpretation of a commutator bracket. In the quantum mechanical case the non-linear expressions appearing on the right hand side of the defining brackets have to be normal ordered. Because of this, a consistent set of structure constants for a quantum W-algebra (consistent in the sense of the Jacobi identities) will be different from an analogous set in the classical case. Obviously, this distinction does not arise in the case of (linear) Lie algebra symmetries. Once the distinction between classical and quantum W-algebras has been made, it is natural to look for relations. One idea is to obtain a classical W-algebra by taking the '/i —y 0' limit of a quantum W-algebra. On the other hand, one may also try to obtain a quantum W-algebra through the quantization of a classical solution to the Jacobi identities. A classical limit can be defined for most quantum W-algebras, although there are cases where the resulting classical algebra is highly degenerate (see reprint [4.6] for more precise remarks). It was shown in [3.4] that a number of quantum W-algebras for which the naive classical limit is degenerate, can be viewed as quantizations of certain classical Poisson bracket algebras which are, however, infinitely and non-freely generated (see also [107]). In general, the process of 'quantizing' a classical W-algebra has not been developed very well, and it has proven more effective to construct quantum W-algebras directly at the quantum level (see Chapters 3, 4 and 5). Nevertheless, the interplay between classical and quantum W-algebras has been a strong guide in the search for and construction of quantum W-algebras. In particular, the role of the Gel'fand-Dickey bracket and the idea of Lie algebra reductions a la Drinfel'd-Sokolov have been studied first in the classical context. An interesting observation, which was first clearly stated in [2.1], is that a particular class of classical field theories in 1 + 1 dimensions, the so-called Toda field theories, possess
xiv
a symmetry algebra that is precisely a W-algebra. In the simplest case, the sfo Toda theory, this algebra is simply the Virasoro algebra, which expresses the conformal invariance of that theory. We refer to [57,58,59,289,244,245,324,222,223] for detailed results on W-symmetry in classical and quantum Toda field theory. Supersymmetric extensions were discussed in [121,262,261,229,201]. A closely related occurrence of classical W-symmetry is in so-called constrained or gauged Wess-Zumino-Witten (WZW) models. The presence of the constraints leads to a reduction of the affine current algebra of these models, and in this way the models provide concrete realizations of the Drinfel'd-Sokolov reduction scheme. We reprint [2.3] (see also [23,25,211,138]), which discusses these reductions and shows that with a certain choice of gauge (the so-called diagonal gauge) results from Toda field theory are reproduced. A different choice of gauge (Drinfel'd-Sokolov gauge) directly points at the W-algebra generators and through this connection (the so-called Miura transformation) a free field representation of many classical W-algebras can be obtained. The papers [142,143,144,202,203,200,81,209,153] discuss supersymmetric W-algebras and extended superconformal algebras in relation to supersymmetric integrable hierarchies and hamiltonian reduction. Important progress in the study of classical W-algebras was made in [2.4], see also [130,267,163]. In [2.4] it is pointed out that for each embedding of the algebra sfe into a semisimple Lie algebra 0, one may define a reduction of the current alge bra of the (untwisted) affine extension 0 and that this leads to a classical W-algebra. The earlier work on such reductions had mostly dealt with the principal s[ 2 embed ding.
An excellent review on the general structure of these reductions can be found
in [131]. Integrable hierarchies associated with the more general Drinfel'd-Sokolov re ductions have been discussed in [86,106,87,315,316,248,127,309]. The paper [2.5] (see also [69,225,136,109,154,286,233,282]) discusses general constructions of W-superalgebras, where the relevant data are a Lie superalgebra g and an embedding of osp(l|2) into 0. More recent work has focussed on the possibility of reductions that are different from those described in [2.4]. An important observation is that, in general, one expects alge bras that cannot be obtained in the standard Drinfel'd-Sokolov scheme to be non-freely generated [129]. Examples for this are the W£ (n > 4) algebras [268,42,110,128], and the algebras discussed in [3.4].
XV
3. Quantum W-algebras The first systematic investigation as well as the first examples of (quantum) Walgebras (with nonlinear defining relations) appeared in a paper by A.B. Zamolodchikov [3.1] (for a definition of W-algebras in the context of meromorphic conformal field theory [169] we refer to [5.4] and [79]). In this paper Zamolodchikov investigates the possibility of adding additional generators (of spins A < 3) to the Virasoro algebra in such a way that the resulting algebra closes (albeit, possibly, nonlinearly) and is associative. Associativity is imposed through the crossing symmetry of the four-point functions. Although the investigation in [3.1] was rather limited already an important feature emerged. Namely, it turned out that there are essentially two types of W-algebras, the 'generic' and the 'exotic' ones. Generic W-algebras are those that are associative for all but a finite number of values for the central charge c (t. e. excluding those c-values for which the structure constants blow up), while exotic W-algebras are associative for at most a finite number of c-values. An example of the generic type is the algebra which contains, besides the Virasoro generator of spin 2, one additional generator of spin 3. This algebra has become known as the W3 algebra, and constitutes the prototype W-algebra. An example of the exotic type is the algebra which contains an additional generator of spin | . This algebra is associative if and only if c = - | | . The analysis of Zamolodchikov was continued in [3.2] where, apart from several ex otic cases, a classification of all generic 'rank-2 W-algebras' (i.e. those W-algebras which contain, besides the Virasoro generator, only one additional generator of spin A) was ob tained. Surprisingly, it turned out that (bosonic) generic rank-2 W-algebras only exist for A = 1,2,3,4 and 6. The spin-4 and spin-6 algebra were explicitly constructed afterwards in [178,339,147]. Subsequent investigations of possible W-algebras through the 'crossing symmetry method' as well as a further systematization of the method have appeared in [80,185,188]. Similar computations for extended superconformal algebras and W-superalgebras can be found in [41,227,204,205,148,149]. Realizations of W-algebras were constructed in e.g. [126,30,288,285,3,232,325] (see Chapters 4 and 5 for additional references). The investiga tion of the representation theory of some of the newly discovered (exotic) W-algebras was undertaken in [318,119,151,152].
xvi
As an alternative for checking crossing symmetry of the four-point functions one can attempt to analyse the Jacobi identities (for the modes) directly. This method was de veloped and applied simultaneously in [66,221] (see also [80,256]). We have chosen to reprint [66] (reprint [3.3]). The analogous approach to W-superalgebras was presented in [62,63,64,65]. Further case-by-case studies of a variety of W-algebras can be found in [150,219,226,27,287,263,184,186,120,189]. A systematic construction of some of the alge bras obtained in [221,120,189] is given in the preprint [3.4]. There are various ways in which different W-algebras can be related, c.q. procedures for constructing new W-algebras out of existing ones. Here we would only like to mention the procedure of 'twisting' (».e. 'orbifolding'), see e.g. [180,181,182,183], and the idea of 'factoring out free fields,' see e.g. [173,104].
xvii
4. Quantum Drinfel'd-Sokolov reduction A very powerful way of constructing both examples of W-(super)algebras as well as their representation theory is through a quantization of the Drinfel'd-Sokolov reduction (see [1.2]). At first, this was attempted through a direct quantization of the outcome of the classical reduction, i.e. by replacing the classical free fields in the expressions for the generating currents by their quantum counterparts and normal ordering. See, in particular, the reprints [4.1][4.2] and the paper [242], where the closure of the quantum algebra WA„ was established (for a major review on these works, see [124]). The methods of [124], unfortunately, remained rather 'ad hoc,' and only worked well for the W-algebra WA„ associated to the Lie algebra A„ = s[„+i- It was realized that, rather than quantizing the classical reduction at the end, one may quantize the reduction from the outset. The basic idea is to impose the constraints on the quantum currents di rectly through a Becchi-Rouet-Stora-Tyutin (BRST) procedure and obtain the W-algebra as the cohomology of the corresponding BRST operator. These ideas were first put for ward in [4.3] and [4.4] (see also [26,112]). This cohomological approach to W-algebras has become known as the 'quantum Drinfel'd-Sokolov reduction' and was, as such, clearly formulated and further developed in [134,135,159] and in the paper reprinted as [4.5]. A big advantage of formulating W-algebras in the context of homological algebra is that not only the W-algebra itself but, in principle, all its properties (such as representa tion theory) follow from the corresponding properties of the underlying affine Lie algebra. In particular, free field realizations of the W-algebra can be obtained by reducing the free field realizations of the underlying affine Lie algebras, i.e. the so-called Wakimoto realizations [319,133,72,73]. A study of the representation theory of W-algebras from this perspective was started in [4.6] (see also [75,259]). For other studies on various aspects of the representation theory of quantum W-algebras (e.g. Kac determinants) in this context, see [126,124,252,70,320,321,253,254,306,51,52,53,84,258,260]. Recent developments have evolved around the generalization of these reductions to arbitrary Bh embeddings in g, the quantization of the reductions discussed in Chapter 2 (in particular [2.4]). At first, progress was made in the finite-dimensional case where previously unknown 'finite W-algebras' were discovered [313,101] (see also [111]). Subse quently, the methods of [101] were generalized to the infinite-dimensional case [4.8]. See also [4.9], where the quantization was performed in a somewhat different way.
xvm
Some ideas towards a possible classification of W-algebras based on these concepts are contained in the paper reprinted as [4.7] (see also [132]). For other recent developments, extensions to the supersymmetric case and closely related papers we refer the reader to [206,207,208,103,323,324,46,257,224].
XIX
5. Coset constructions The second powerful method to obtain examples of W-algebras is through the so-called coset construction. It is well known that for every affine Lie algebra g there is an associated conformal algebra for which the Virasoro generator is expressed as a bilinear in the affine Lie algebra currents, the so-called Sugawara construction [310] (for generalizations see e.g. [177,105]). It turns out that this Virasoro generator commutes with the finite-dimensional Lie algebra g underlying g, and can thus be interpreted as belonging to the coset pair (g,g). It was found some time ago [170,171] that this construction could be generalized to arbitrary coset pairs (g,g') where g' is a (finite-dimensional or affine) subalgebra of g. The corresponding Virasoro generator has a central charge given by the differences of central charges associated to g and g' and commutes with g'. It was observed in concrete examples that the spectrum of a coset model very often contains chiral fields of higher integer spin (as an example, the spectrum of the c = | three-state Potts model contains a spin 3 field), and thus the question arose whether coset models generally contain higher spin generators that commute with the subalgebra g'. This question was first studied, and answered affirmatively, in [5.1] and [5.2] (see also [162,312]) for the diagonal coset pair (g ©g,g) (and, worked out in detail for g = 5(3). [For some closely related results in the mathematical literature we refer to [175,179].] It turns out, miraculously, that for simply-laced Lie algebras g the coset W-algebra is isomorphic to the W-algebra obtained by the quantum Drinfel'd-Sokolov reduction based on the principal sl? embedding in g. This can be concluded from the explicit construction of the algebra (in the case of W3, see [5.1] and [5.2]) or by comparing the resulting characters [79]. We refer to [5.4] for a discussion of the independent generators of the W-algebra of a diagonal coset model. One of the outstanding problems to date is to gain an a priori insight into which coset W-algebras are isomorphic to which Drinfel'd-Sokolov W-algebras. Other coset-algebras that have been examined in detail are the ones corresponding to the coset pairs (g © g',$j') where g' is a conformal subalgebra of g (see [290,9] for a definition and classification of conformal subalgebras). We reprint [5.3] (see also [172]). For a discussion of other cosets and supersymmetric extensions of the above construc tions the reader may want to consult [116,117,61,93,47,190,4,292]. Not all coset pairs yield different W-algebras. Coset pairs which do yield isomorphic W-algebras are termed 'dual' [213]. The reprinted paper [5.3] contains a discussion and
XX
classification of so-called T-equivalent coset pairs, which are precursors of the aforemen tioned dual coset pairs (see also [6]). Any coset W-algebra comes naturally equipped with a set of (not necessarily irre ducible) representations, whose characters are given by the branching functions [214,216]. Conversely, a priori insight into e.g. the generators of a certain coset W-algebra can be obtained by careful examination of these branching functions, t. e. by means of the so-called character technique (see e.g. [70] and [5.4]). We refer to [216,218,89,74] for the computa tion of branching functions of several coset pairs (fl,g'). Finally, we would like to mention that one important advantage of the coset construction over the quantum Drinfel'd-Sokolov reduction is that the question of unitarity of VV-algebra representations is easier to address in the context of the coset construction (see e.g. [252,253]).
XXI
6. Woo-type algebras W-algebras with an infinite number of independent generating currents, with spins increasing all the way to infinity, form a very special class. We will generically denote them as Woo-type algebras. It is sometimes possible to write these algebras in a basis where all the defining bracket relations are linear. Due to this, those algebras are more tractable than their finitely generated counterparts, and in some cases have a clear geometrical meaning. Linear Woo-type algebras have found applications in a variety of fields including string theory and the study of the quantum Hall effect. However, some of the most interesting Woo-type algebras are non-linear. In this chapter we reprint some of the earlier papers on Woo-type algebras: [6.1] on the Woo algebra, [6.2] and [6.3] discussing the algebras Woo and Wi+oo, respectively, and [6.4], which is one of the first papers on the so-called Woo(fc) algebra. The paper [6.5] discusses relations between various Woo-type algebras. One of the original motivations for considering Woo-type algebras was to study the 'large-N limit' ([6.1], [48,255]) of the W/v symmetries, which were already known at the time. This limit is by no means unique, and a number of different Woo-type algebras have been proposed. The simplest one, called Woo [6.1], has generators of conformal spins 2 , 3 , . . . . The standard central extension of the Virasoro algebra cannot be extended to the full Woo algebra, and this prevents the algebra from playing a direct role in conformal field theory. The algebra Woo, which was first proposed by C. Pope, L. Romans and S. Shen in [272], [6.2] (see also [18]), is a linear deformation of Woo that is precisely such that a non-trivial central extension can be defined on the full algebra. The algebra t«oo
can
be extended by an additional current of spin 1, giving rise to
u>i +00 . This algebra can be viewed as the algebra of area preserving diffeomorphisms of a cylinder. It admits a universal central extension [6.3], which has been called Wi+oo. [In the mathematical literature, Wi+oo is known as V [215,217].] Matrix extensions of Woo-type algebras were considered in [20,265]. The papers [38,35,36], see also [280], discuss an algebra called super-Woo, which is an N = 2 supersymmetric extension of Woo- Various field theoretical realizations of Woo-type algebras have been discussed in [19,20,22,38,305,336]. A systematic discussion of the representation theory of some of the Woo-type algebras can be found in [264]. The unitary representations
XX11
of Wi+oo have been classified in [217]. Useful reviews on the linear W^-type algebras can be found in [275,304]. Non-linear Woo-type algebras were first studied through their connection with integrable hierarchies of differential equations. In Chapter 1 we mentioned that the (classi cal) Wjv algebra arises from the second hamiltonian structure of a generalized Kortewegde Vries (KdV) hierarchy. This, in combination with the fact that the generalized KdV hierarchies can be viewed as reductions of the Kadomtsev-Petviashvili (KP) hierarchy, sug gests that the (second) hamiltonian structure of the KP hierarchy can lead to a Woo-type algebra that plays the role of 'universal W-algebra' for the WN series [146]. The bi-hamiltonian structure of the KP hierarchy was unraveled in a number of papers: in [337,328] it was found that the first hamiltonian structure corresponds to the algebra Wi+oo, and in [113,139] a (classical) Woo-type algebra associated to the second hamiltonian structure, called WKP was obtained. The algebra WKP has non-linear defining relations and it has zero central charge. The same algebra (with the spin-1 generators omitted) was also obtained in [329], where it was called WooIn an independent study, I. Bakas and E. Kiritsis [6.4] studied the chiral symmetry algebra of the conformal field theory based on the non-compact coset SL(2, R)t/{/(l). They obtained a quantum Woo-type algebra which they called Woo(k), and which has a non-zero central charge given by c* = 2(k + l)/(fc — 2). They suggested that this algebra is the quantum version of a suitable deformation of the algebra WKP- In [141], a oneparameter family of deformations of WKP, called Wj^p, was explicitly constructed. The relations between the KP hierarchy, the SL(2, R)k/U(l)
coset conformal field theory and
the nonlinear Woo-type algebras were further worked out by Y.-S. Wu and F. Yu [330]-[334], who used a two-boson realization to connect the classical and quantum constructions. The quantum algebra Woo(fc) truncates on the WN algebra for k = — N, and this confirms its role as the universal W-algebra for the WN series. At the classical level plays a similar role; we reprint paper [6.5] where the relation of W^p
W^p
with other Woo-type
algebras is reviewed. (Notice that [6.5] uses the notion of the 'classical limit' of a Poisson bracket algebra [145], which is not to be confused with the more standard notion of the classical limit of a quantum W-algebra.)
XX11I
7. W-gravity and W-strings Conformal symmetry plays a natural role in string theory, where it arises as a residual symmetry associated with the reparametrization invariance of the coordinate maps from the world-sheet into space-time. The fact that, at the algebraic level, conformal symmetry can be extended by including higher spin generators strongly suggests that string models based on extended conformal symmetry can be constructed. This motivation has led to extensive studies of higher spin extensions of two dimensional gravity (W-gravity) and of higher-spin string theories (W-strings). There are two distinct approaches to the subject of W-strings, which are usually referred to as the critical and the non-critical approach. The latter requires a detailed understanding of a fully interacting W-gravity sector as part of the W-string theory. We shall first focus on these W-gravity theories and after that make further comments on W-strings. The study of W-gravity has been guided by the analogy with ordinary gravity in two dimensions, which has been studied in great detail in recent years. However, some features of W-gravity are essentially different from those of ordinary gravity. Most of these have been addressed in the context of W3 gravity. Those features that can be understood from the point of view of affine Lie algebra reductions (see below) can usually be treated for many other examples as well. One of the first issues addressed in W-gravity was the construction of classical Winvariant matter couplings. We reprint the paper by C. Hull [7.2] where the coupling of chiral gauge W3 gravity to scalar matter fields is described. In [294,295] it was found that the matter couplings in more general gauges contain the fundamental W-gravity fields in non-polynomial form. These couplings can be described in closed form by using so-called nested covariant derivatives. Paper [7.3] reviews the construction of the most general matter couplings in classical W3 gravity (see also [192,251,195,269]). These couplings, which were extended to «) M in [39,296,34], are covariant in the sense that they employ covariant vielbein fields for the W-gravity degrees of freedom. The geometrical structure behind the highly non-linear structure of the classical Wgravities has been investigated in a series of papers by C. Hull [193,196,197,199]. Other approaches to the notion of 'W-geometry' can be found in [49,55,307,308,166,167,250,99].
xxiv
At the quantum level, W-gravity is defined by an induced action, which is obtained by integrating out the matter fields. The induced action for W3 gravity shows a number of interesting features. In the chiral gauge, one finds that the dependence on the central charge c of the matter system is not through an overall constant (as it is for ordinary gravity); the result can be written in a 1/c expansion which can be worked out in pertur bation theory [249,297,194] (see also [37,88]). Upon quantization of the W3 gravity fields, which leads to an effective action, the 1/c corrections of the induced action interfere with contributions from VV3 gravity loop diagrams. Keeping track of all contributions allows one to compute order by order in 1/c physical quantities such as the string susceptibility [299] (see also [176]). These perturbative computations are reviewed in [7.5]. In the conformal gauge both the induced and the effective action for W3 gravity take the form of a Toda action. Correspondingly, the W3 currents take the free field form first given in [126]. We reprint the paper [7.4] (see also [91]), which discusses the physical states for conformal gauge W3 gravity. It makes essential use of the explicit form of the BecchiRouet-Stora-Tyutin (BRST) charge for the c = 100 quantum W3 algebra, which was first obtained by J. Thierry-Mieg in [7.1] (see also [293]). (Recently, analogous BRST charges for various higher rank W-algebras have been constructed [240,187,340]; their cohomology was studied in [239,67]). In a somewhat different approach to quantum W3 gravity, one starts from the covariant induced action (without 1/c corrections), which can be quantized by using a BRST procedure. This covariant induced action (in the large c limit) was first given in [96]; we reprint the paper [7.6], where a systematic derivation is presented. The quantization of this action was discussed in [100]. If one goes to a chiral gauge, the nilpotency of the rele vant BRST charge leads to a so-called Knizhnik-Polyakov-Zamolodchikov (KPZ) formula for the critical exponents of W3, which agrees with the result obtained from direct quan tization in the chiral gauge. In the conformal gauge the action reduces to a Toda action and the BRST charge takes the form that was first proposed in [7.8] (see also [44,40]). It should be stressed that this BRST charge, which is appropriate for the coupling of W3 matter to VV3 gravity, is a non-trivial extension of the BRST charge of [7.1]. Many of the results for W-gravity cited above can be understood and extended by using the representation (of both W-matter and W-gravity) as a constrained Wess-ZuminoWitten model, and using the ideas of Drinfel'd-Sokolov reduction. At the classical level this allows one to construct exact expressions for the large-c limit of the chiral and the covariant
XXV
induced actions [266,96]. Furthermore, using quantum Drinfel'd-Sokolov reduction leads to exact results (such as the KPZ formula) for the effective (i.e. quantized) W-gravities [5,45,98,303,246], [4.9], and explains some of the cancellations that had been observed in perturbation theory. We already mentioned that the construction of W-strings (mostly W3 strings) has pro ceeded along two rather different roads. The first road followed has been that of critical W3 strings [60], which are constructed as follows. One first constructs (using the results of [126,285]) a c = 100 realization of the VV3 algebra in terms of scalar matter fields (such con structions necessarily involve background charges). For such a matter system an anomalyfree coupling to VV3 gravity can be constructed [277] and this then forms the starting point for a critical W3 string theory. In this theory, physical states are selected by the cohomology of the BRST operator of [7.1]. We refer to [7.4], [91,283,271,278,234,238,326,231,279] for results on the spectrum of W-strings. We reprint [7.10], in which the complete cohomol ogy of this W3 string is presented. Interacting W3 strings were discussed in [155,236,156] and in the reprint [7.7]. Two recent reviews [270,327] provide a more complete guide to the literature on this subject. Non-critical W3 strings are constructed by starting from a W3 matter system (which should be such that it allows some kind of target space interpretation) and coupling to W3 gravity by using the BRST operator of [7.8]. (Note that these theories cannot be viewed as critical W3 strings with a specific choice of matter system, as is the case for the bosonic string.) The cohomology problem for non-critical W-strings has been studied in [7.9], [44,31,32,33,78]. Further results for W3 gravity coupled to a c = 2 matter system can be found in [77]. There exist intriguing properties (see, e.g. [33]) of the cohomologies of the non-critical W-strings where the matter system is a W-minimal model. These have first been explored for the critical W-strings where the minimal model is a trivial c = 0 theory, see for example [7.4] and [91,198,157,158,239]. A typical example is the fact that the cohomology of the critical W3 string is similar (but not isomorphic!) to the cohomology of an ordinary (W2) string, where the matter system contains an additional c = 1/2 Ising conformal field theory.
_2 Reprinted with permission from Prepr. Inst. Appl. Mat. Vol. 136, pp. 625-«46, Moscow 1978 C 1987 Springer-Vcrlag
I. M. Gelfand and L. A. Dikij A family of Hamiltonian structures related to nonlinear integrable differential equations Prepr. Inst. Appl. Mat. 136(1978). MR 81: 58027
1. Introduction 1.1. In [1-3] a construction of nonlinear diflerential equations which admit of the so-called Lax commutation presentation L,=[P,L] was given. Here P and L are differential operators with respect to the variable x; the coefficients of P are differential polynomials in the coefficients of the operator L. It was also proved that these equations are Hamiltonian in a certain sympletic struc ture. The complicated construction of this structure (the Poisson bracket and the symplectic form) was considerably simplified and clarified by Lebedjev and Manin [4] and Adler [5], who showed that these structures can be obtained by a standard Kirillov-Kostant construction for an inflnite Lie algebra. Earlier this fact was proved for the discrete case by Kostant. It is well-known that for a special case of Lax systems, namely for the KdV equation it is possible to introduce two Hamiltonian structures and even a one-parameter family of Hamiltonian structures. Arbitrary linear combinations of two Poisson brackets turn out to be a Poisson bracket (Le. satisfy the Jacobi identity). It is natural to assume that this situation holds for more general Lax equations. In [5] a possible construction of a new Poisson bracket was sug gested but the necessary property (the Jacobi identity) was not proved. Here we shall prove this assertion. 1.2. It is well-known that the theory of the KdV equation can be con structed on the basis of the third-order equation -R'" + 2uR+4(u + QR' = 0
(1.2.1)
for the diagonal of the kernel of a resolvent which is the inverse of the differen tial operator L + ( = —d2/dx2 + u + £. In the case of a general nth order operator L= £ uk(d/dxf the Volterra integral operator T with kernel T(x,y) which k-0
satisfies the equation (Lx + QT(x,y)=0 with respect to the first argument and the equation (L* + £)T(x, y)=0 with respect to the second one is called the resolvent. Here L* is the adjoint of L. Then A + (T)=(L + 0T and A~(T) = T(L + 0 are differential operators of order 5 j n - l . In [3] these operators are the basis of the whole theory. It is obvious that (L + QA-(T)-A + (T)(L+Q=0.
(1.2.2)
4
I.M. Gelfand and L.A. Dikij
Adler [5] suggested considering more general operators A + (X) = [(L + QX~\ + and A~(X)=[X(L + Qi]+. The subscript + distinguishes the differential part of an operator which, generally, can be a sum of an integral (Volterra) and of a differential operators. The equation (L+QA-(X)-A + (X)(L+Q = 0 (1.2.3) is the determining equation for the resolvents, this will be proved later on (see the theorem in 3.7). Thus Eq. (1.2.3) is an analog of (1.2.1) for general operators L. Another analog was introduced earlier in our papers [2] (Eq. (38)) and [3] (Eq. (3.10)). The new equation (1.2.3) suggested by Adler is more suitable; it contains fewer variables. 1.3. The Hamiltonian structures relate to the third-order equation (in the simplest case of the second-order operator L) and to Eq. (1.2.3) (in the general case). Let f({ukn}) be a polynomial in the coefficients uk of the operator L and their derivatives of all orders; denote it as f[u]. L e t / be a functional of the form f=$f[u]dx. This functional generates a Volterra operator Xf whose kernel has the following property: (d/dxfXf(x,y)\ywtX = Sf/5uk where Sf/5uk is the variational derivative of the functional. Then the Poisson bracket depend ing on a parameter ( is defined as {f,l\=Tt{UL
+ QA-(Xf)-A
+ (Xf)(L + Q]Xt}
(1.3.1)
(The trace of an operator which is a sum of a differential operator and an integral operator is the trace of the integral part). We prove here that the Jacobi identity holds for this bracket. 1.4. It is easy to write the expression for the bracket (1.3.1) in explicit form, in terms of variational derivatives. In [3] the expressions
4 - W - "l (£,*..o)(W (1.4.1) were given; here
a.P-0
X.,,=(d\dxnd/dyYX(x,y)\,.x and t# = " i f ( V t a k+»+T + iW < '*> T . * + PSn-l; T "° V " ' and l*t is the adjoint operator of laf. Then (L + QA-(X)-A
+
(X)(L + {)
= 'i [K,+a,)*,.0(^*)"
f.,-0, z + p}>n (1.4.2)
5
A family of Kamiltonian structures related to nonlinear integrable differential equations
where /., = £,-/";.,
f—\fdx. Thus the elements feA are / to within the deriva tives. Accordingly, ) f'gdx = —$fg'dx where f' = df/dx, g'=dg/dx. 2.3. A({z~1)) will denote the differential ring of the formal Laurent series
where z is a parameter. The operations are natural. The invertible elements of this ring are those for which / r o eC. The operation \dx can be extended to the elements of A((z~1)). 2.4. Let A{(£— x)) be the ring of the formal Taylor series 00
(