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1 ASYMMETRY

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3 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 > World Scientific Singapore New Jersey London Hong Kong

4 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 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 Conformal invariants. 2. Quantum field theory. 3. Mathematical physics. I. Schoutens, K. II. Title. III. Series. QC C66B '43~dc CIP Copyright 1995 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form orbyany means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 27 Congress Street, Salem, MA 01970, USA. Printed in Singapore by Uto-Print

5 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 central 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 operator 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 theories. 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, 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

6 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 extensive 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

7 vu CONTENTS Preface Introductory Chapters v xi 1. History and background 1.1 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 V. Drinfel'd and V. Sokolov, Lie algebras and equations of Korteweg-de Vries type, J. Sov. Math. 30 (1984) 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^ I. Bakas, Higher spin fields and the Gel'fand-Dickey algebra, Comm. Math. Phys. 123 (1989) 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) Ill 2.4 F. A. Bais, T. Tjin and P. van Driel, Covariantly coupled chiral algebras, Nucl. Phys. B357 (1991) F. Delduc, E. Ragoucy and P. Sorba, Super-Toda theories and W-algebras from superspace Wess-Zumino-Witten models, Comm. Phys. 146 (1992) 403-i Quantum W-algebras 3.1 A. B. Zamoldchikov, Infinite additional symmetries in two dimensional conformal quantum field theory, Theor. Math. Phys. 65 (1985) P. Bouwknegt, Extended conformal algebras, Phys. Lett. 207B (1988)

8 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) 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 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. A3 (1988) V. A. Fateev and S. L. Lukyanov, Exactly soluble models of conformal quantum field theory associated with the simple Lie algebra D n, Sov. J. Nuc. Phys. 49 (1989) M. Bershadsky and H. Ooguri, Hidden SL(n) symmetry in conformal field theories, Comm. Math. Phys. 126 (1989) J. M. Figueroa-O'Farrill, On the homological construction of Casimiar algebras, Nucl. Phys. B343 (1990) 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^ E. Frenkel, V. Kac and M. Wakimoto, Characters and fusion rules for W-algberas via quantized Drinfeld-Sokolov reduction, Comm. Math. Phys. 147 (1992) P. Bowcock and G. M. T. Watts, On the classification of quantum W-algebras, Nucl. Phys. B379 (1992) J. de Boer and T. Tjin, The relation between quantum W-algebras and Lie algebras, Comm. Math. Phys. 160 (1994) A. Sevrin and W. Troost, Extensions of the Virasoro algebra and gauged WZW models, Phys. Lett. 315B (1993) 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)

9 ix 5.2 F. A. Bais, P. Bouwknegt, K. Schoutens and M. Surridge, Coset construction for extended Virasoro algebras, Nucl. Phys. B304 (1988) P. Bowcock and P. Goddard, Coset constructions and extended conformal algebras, Nucl. Phys. B305[FS23] (1988) G. M. T. Watts, W-algebras and coset models, Phys. Lett. 245B (1990) Woo typ e algebras 6.1 I. Bakas, The large-n limit of extended conformal symmetries, Phys. Lett. 228B (1989) C. N. Pope, L. J. Romans and X. Shen, Woo ana> the Racah-Wigner algebra, Nucl. Phys. B339 (1990) C. N. Pope, L. J. Romans and X. Shen, A new higher-spin algebra and the lone-star product, Phys. Lett. 242B (1990) 401^ 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) J. Figueroa-O'Farrill, J. Mas and E. Ramos, The topography of Woe-type algebras, Phys. Lett. 299B (1993) 41-i8 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) C. M. Hull, Gauging the Zamolodchikov W-algebra, Phys. Lett. 240B (1990) K. Schoutens, A. Sevrin and P. van Nieuwenhuizen, Properties of covariant W-gravity, Int. Journ. Mod. Phys. A6 (1991) S. R. Das, A. Dhar and S. K. Rama, Physical properties of W-gravities and W-strings, Mod. Phys. Lett. A6 (1991) 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 J. de Boer and J. Goeree, W-gravity from Chem-Simons theory, Nucl. Phys. B381 (1992)

10 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) M. Bershadsky, W. Lerche, D. Nemeschansky and N. P. Warner, A BRST operator for non-critical W-strings, Phys. Lett. 282B (1992) P. Bouwknegt, J. McCarthy and K. Pilch, Semi-infinite cohomology of W-algebras, Lett. Math. Phys. 29 (1993) H. Lu, C. N. Pope, X. J. Wang and K. W. Xu, The complete cohomology of the W3 string, Class. Quant. Grav. 11 (1994) References 857

11 Introductory Chapters

12 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 mathematical 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.

13 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 dimensions, the so-called Toda field theories, possess

14 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 algebra 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 embedding. 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 reductions 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 algebras 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].

15 XV 3. Quantum W-algebras The first systematic investigation as well as the first examples of (quantum) W- algebras (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 exotic 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 obtained. 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 investigation of the representation theory of some of the newly discovered (exotic) W-algebras was undertaken in [318,119,151,152].

16 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 developed 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 algebras 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].

17 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 directly 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 forward 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 representation 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]). Subsequently, 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.

18 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].

19 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 constructions 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

20 XX classification of so-called T-equivalent coset pairs, which are precursors of the aforementioned dual coset pairs (see also [6]). Any coset W-algebra comes naturally equipped with a set of (not necessarily irreducible) 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 computation 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]).