W-Symmetry

The following sections are included:

• Preface

• CONTENTS

• Introductory Chapters

The following sections are included:

• Introduction

• The ring of formal operators

• Resolvents

• Hamiltonian structures

• Lax P, L pairs

• References

The survey contains a description of the connection between the infinite-dimensional Lie algebras of Kats–Moody and systems of differential equations generalizing the Korteweg–de Vries and sine-Gordon equations and integrable by the method of the inverse scattering problem. A survey of the theory of Kats–Moody algebras is also given.

The Toda field theories, which exist for every simple Lie group, are shown to give realizations of extended Virasoro algebras that involve generators of spins higher than or equal to two. They are uniquely determined from the canonical lagrangian formalism. The quantization of the Toda field theories gives a systematic treatment of generalized conformal bosonic models. The well-known pattern of conformal field theories with non-extended Virasoro algebra, appears to be repeated for any simple group, leading to a “periodic table”, parallel to the mathematical classification of simple Lie groups.

We show that in 2-dimensional field theory, higher spin algebras are contained in the algebra of formal pseudodifferential operators introduced by Gelfand and Dickey to describe integrable nonlinear differential equations in Lax form. The spin 2 and 3 algebras are discussed in detail and the generalization to all higher spins is outlined. This provides a conformal field theory approach to the representation theory of Gelfand–Dickey algebras.

A new formulation of Toda theories is proposed by showing that they can be regarded as certain gauged Wess-Zumino-Novikov-Witten (WZNW) models. It is argued that the WZNW variables are the proper ones for Toda theory, since all the physically permitted Toda solutions are regular when expressed in these variables. A detailed study of classical Toda theories and their W-algebras is carried out from this unified WZNW point of view. We construct a primary field basis for the W-algebra for any group, we obtain a new method for calculating the W-algebra and its action on the Toda fields by constructing its Kac-Moody implementation, and we analyse the relationship between W-algebras and Casimir algebras. The W-algebra of G₂ and the Casimir algebras for the classical groups are exhibited explicitly.

New extended conformal algebras are constructed by conformal reductions of sl_N WZWN models. These are associated with the inequivalent sl₂ embeddings into sl_N. Among other things, the conformal weights of the generators and the occurrence of Kac–Moody and W_n subalgebras are determined by the branching rules of the adjoint representation for the particular embedding. For some representative classes the algebras are constructed explicitly. In general they are coupled chiral algebras suggesting that they correspond to the symmetries of certain interacting conformal field theories. Moreover we find that a (minimal) covariant coupling is present which is related to a generalized Gelfand–Dickii structure. Some aspects of the quantization are addressed, in particular the c-values are determined. We introduce a new hybrid realization of KM algebras which interpolates between a realization of currents and of free fields, in which the constraints can be imposed in a very natural way.

Using a superspace approach, it is proved that a N = 1 Super-Toda theory can be seen as a constrained WZW model based on a supergroup. The gauge transformations which survive the constraints are then used à la Drinfeld Sokolov to determine explicitly the super-W algebra underlying this theory. The conformal spin content of any such super-W algebra is provided in the general case.

Additional symmetries in two-dimensional conformal field theory generated by spin currents are investigated. For spins s = 5/2 and s = 3, the generators of the symmetry form associative algebras with quadratic determining relations. “Minimal models” of conformal field theory with such additional symmetries are considered.

We investigate extensions or the Virasoro algebra by a single primary field of integer or halfinteger conformal dimension Δ. We argue that for vanishing structure constant , the extended conformal algebra can only be associative for a generic c-value if Δ = 1/2, 1, 3/2, 2 or 3. For the other Δ ≤ 5 we compute the finite set of allowed c-values and identify the rational solutions. The case is also briefly discussed.

We construct all W-algebras of chiral fields which in addition to the energy-momentum density have a single generator of conformal dimension up to 8. Some of them were unexpected, which indicates that present conjectures concerning the classification of conformally invariant quantum field theories in two dimensions are rather incomplete. We also explicitly construct the WA₃-algebra with generators of dimensions 2,3,4.

There is a relatively well understood class of deformable W-algebras, resulting from Drinfeld-Sokolov (DS) type reductions of Kac-Moody algebras, which are Poisson bracket algebras based on finitely, freely generated rings of differential polynomials in the classical limit. The purpose of this paper is to point out the existence of a second class of deformable W-algebras, which in the classical limit are Poisson bracket algebras carried by infinitely, nonfreely generated rings of differential polynomials. We present illustrative examples of coset constructions, orbifold projections, as well as first class Hamiltonian reductions of DS type W-algebras leading to reduced algebras with such infinitely generated classical limit. We also show in examples that the reduced quantum algebras are finitely generated due to quantum corrections arising upon normal ordering the relations obeyed by the classical generators. We apply invariant theory to describe the relations and to argue that classical cosets are infinitely, nonfreely generated in general. As a by-product, we also explain the origin of the previously constructed and so far unexplained deformable quantum W(2,4,6) and W(2,3,4,5) algebras.

An infinite set of conformally invariant solutions of the two-dimensional quantum field theory, possessing a global symmetry Z_n is constructed. These solutions can describe the critical behavior of Z_n symmetric statistical systems.

We construct a class of exactly soluble models of two-dimensional conformal quantum field theory, which describes certain critical points of RSOS statistical systems, associated with the D_n series of simple Lie algebras. The infinite-dimensional symmetry algebras of these models are obtained by quantization of the classical Hamiltonian structures of generalized KdV equations.

We prove that an irreducible representation of the Virasoro algebra can be extracted from an irreducible representation space of the SL(2,R) current algebra by putting a constraint on the latter using the Becchi–Rouet–Stora–Tyutin formalism. Thus there is a SL(2,R) symmetry in the Virasoro algebra, but it is gauged and hidden. This construction of the Virasoro algebra is the quantum analogue of the Hamiltonian reduction. We then are naturally lead to consider a constrained SL(2,R) Wess-Zumino-Witten model. This system is also related to quantum field theory of coadjoint orbit of the Virasoro group. Based on this result, we present a canonical derivation of the SL(2,R) current algebra in Polyakov's theory of two-dimensional gravity; it is a manifestation of the SL(2,R) symmetry in conformal field theory hidden by the quantum Hamiltonian reduction. We also discuss the quantum Hamiltonian reduction of the SL(n,R) current algebra and its relation to the W_n-algebra of Zamolodchikov. This makes it possible to define a natural generalization of the geometric action for the W_n-algebra despite its non-Lie-algebraic nature.

We propose a BRST (homological) construction of the Casimir extended conformal algebras by quantizing a classical observation of Drinfeld and Sokolov. We give the explicit expression for the Virasoro generators and compute the discrete series of the Casimir algebras. The unitary subseries agrees with that of the coset construction for the case of simply laced algebras. With the help of a free-field realization of the affine algebra, we compute the BRST cohomology of this system with coefficients in a Fock module. This allows us to generalize and prove a conjecture of Bershadsky and Ooguri.

The W-algebras, associated to arbitrary simple Lie algebras, are defined as the cohomologies of certain BRST complexes. This allows to prove many important facts about them, such as determinant formulas, duality and free field resolutions for generic values of the central charge. A classical limit of a W-algebra can be identified with the center of the universal enveloping algebra of the corresponding affine Kac-Moody algebra. This gives some information on the geometric Langlands-Drinfeld correspondence for complex algebraic curves.

Using the cohomological approach to W-algebras, we calculate characters and fusion coefficients for their representations obtained from modular invariant representations of affine algebras by the quantized Drinfeld–Sokolov reduction.

In this paper we consider the structure of general quantum W-algebras. We introduce the notions of deformability, positive-definiteness, and reductivity of a W-algebra. We show that one can associate a reductive finite Lie algebra to each reductive W-algebra. The finite Lie algebra is also endowed with a preferred sl(2) subalgebra, which gives the conformal weights of the W-algebra. We extend this to cover W-algebras containing both bosonic and fermionic fields, and illustrate our ideas with the Poisson bracket algebras of generalised Drinfeld-Sokolov hamiltonian systems. We then discuss the possibilities of classifying deformable W-algebras which fall outside this class in the context of automorphisms of Lie algebras. In conclusion we list the cases in which the W-algebra has no weight-one fields, and further, those in which it has only one weight-two field.

By quantizing the generalized Drinfeld-Sokolov reduction scheme for arbitrary sl₂ embeddings we show that a large set W of quantum W algebras can be viewed as (BRST) cohomologies of affine Lie algebras. The set W contains many known W algebras such as W_N and W₃⁽²⁾. Our formalism yields a completely algorithmic method for calculating the W algebra generators and their operator product expansions, replacing the cumbersome construction of W algebras as commutants of screening operators. By generalizing and quantizing the Miura transformation we show that any W algebra in W can be embedded into the universal enveloping algebra of a semisimple affine Lie algebra which is, up to shifts in level, isomorphic to a subalgebra of the original affine algebra. Therefore any realization of this semisimple affine Lie algebra leads to a realization of the W algebra. In particular, one obtains in this way a general and explicit method for constructing the free field realizations and Fock resolutions for all algebras in W. Some examples are explicitly worked out.

To any non-trivial embedding of sl(2) in a (super) Lie algebra, one can associate an extension of the Virasoro algebra. We realize the extended Virasoro algebra in terms of a WZW model in which a chiral, solvable group is gauged, the gauge group being determined by the sl(2) embedding. The resulting BRST cohomology is computed and the field content of the extended Virasoro algebra is determined. The closure of the extended Virasoro algebra is shown. Applications such as the quantum Miura transformation and the effective action of the associated extended gravity theory are discussed.

We consider bosonic extensions of the Virasoro algebra that can be obtained from Kac-Moody algebras ĝ by generalizing the Sugawara construction to the higher order Casimirs of g. In this paper we explicitly construct the algebra of a primary field of dimension 3 constructed from the 3rd order Casimir of A_N−1. For N = 3 we compare our results to the Z₃-extended Virasoro algebra proposed by Fateev and Zamolodchikov.

We discuss extensions of the Virasoro algebra obtained by generalizing the Sugawara construction to the higher order Casimir invariants of a Lie algebra . We generalize the GKO coset construction to the dimension-3 operator for and recover results of Fateev and Zamolodchikov if N = 3. Branching rules and generalizations to all simple, simply-laced are discussed.

The structure of coset theories, and the extensions of conformal symmetry which they realise, is considered, using the properties of two special classes of fields, ĥ scalars and scalars. Series of representations of extended conformal symmetry, associated with cosets of the form ĝ_x ⊕ ĥ_m/ĥ_{m + y}, where the subscripts denote the level of the affine algebra, and m runs over the positive integers, are discussed. Those theories possessing a field extending the conformal algebra of weight η, where 1 < η < 2, are listed. The ability of various coset models to be supersymmetrised is established using these techniques. The concept of a dual pair of cosets possessing related partition functions, but not sharing the same extended conformal algebra, is developed.

We define a W-algebra as a special class of meromorphic conformal field theory. We then show that there are closed W-algebraic structures in coset models of the form based on the simple Lie algebras A_n, B_n, D_n and E_n for sufficiently high values of k.

We study the large-N limit of the operator algebra W_N generated by primary conformal fields with integer spin 1, 2, …, N. It is shown that W_∞. provides a representation of a certain infinite dimensional (sub)algebra of the area-preserving diffeomorphisms of the 2-plane (compactified or not). We also discuss certain applications of this result to quantum field theory.

We examine the structure of a recently constructed W_∞ algebra, an extension of the Virasoro algebra that describes an infinite number of fields with all conformal spins 2,3,…, with central terms for all spins. By examining its underlying SL(2, ℝ) structure, we are able to exhibit its relation to the algebras of SL(2, ℝ) tensor operators. Based upon this relationship, we generalise W_∞ to a one-parameter family of inequivalent Lie algebras W_∞(μ), which for general μ requires the introduction of formally negative spins. Furthermore, we display a realisation of the W_∞(μ) commutation relations in terms of an underlying associative product, which we denote with a lone star. This product structure shares many formal features with the Racah-Wigner algebra in angular-momentum theory. We also discuss the relation between W_∞ and the symplectic algebra on a cone, which can be viewed as a co-adjoint orbit of SL(2, ℝ).

In two recent papers, we constructed a new N → ∞ limit of the W_N algebras, which we denote W_∞ having generators of conformal spins 2, 3, …, with central terms for all spins. In this paper, we construct another new algebra, which we denote W_{1 + ∞}, with generators of conformal spins, 1, 2, 3, …, again with central terms for all spins. The requirement that the algebras be closed requires that one include the spin-1 generators in W_{1 + ∞}, and prohibits their inclusion in W_∞. Paralleling our analogous construction for W_∞, we show that the new algebra can also be realised as the antisymmetric part of an associative “lone-star” product, which also closes on the set of generators with conformal spins ≥ 1.

We show that the symmetry algebra, of the SL(2,R)_k/U(1) coset model is a non-linear deformation of W_∞, characterized by k. This is a universal W-algebra which linearizes in the large k limit and truncates to W_N for k = − N. Using the theory of non-compact parafermions we construct a free field realization of the non-linear W_∞ in terms of two bosons with background charge. The W-characters of all unitary SL(2,R)/U(1) representations are computed. Applications to the physics of 2-d black hole backgrounds are also discussed and connections with the KP approach to c=1 string theory are outlined.

We chart out the landscape of W_∞-type algebras using —a recently discovered one-parameter deformation of W_KP. We relate all hitherto known W_∞-type algebras to and its reductions, contractions, and/or truncations at special values of the parameter.

We construct the BRS operator of Zamolodchikov's spin 2 and 3 associative current algebra. This algebra is not a Lie algebra, since the commutator of two spin 3 fields involves the square of the spin 2 field. Nevertheless, the associated BRS operator is nilpotent when the central charge of the Virasoro subalgebra is 100 and the energy of the vacuum −4.

A world-sheet action is presented whose constraint algebra is the Zamolodchikov w-algebra. It consists of a matter system, such as a free boson model or an SU(3) WZW model, coupled to world-sheet gauge fields of spin two and three which are Lagrange multipliers imposing the constrains. The classical symmetry group has an algebra with field-dependent structure constants. The possibility of a new kind of string theory based on the w-algebra instead of the Virasoro algebra is considered.

We present the beginnings of a new consistent gauge theory with spin 2 and spin 3 gauge fields in two dimensions. It is based on a nonlinear Lie algebra, whose “gauging” is discussed in detail. (The present treatment of the gauge sector is expanded as compared to [3]). For the coupling to matter, we introduce the concept of a “nested covariant derivative”, and we obtain an invariant action by solving a functional integrability condition. It reads

We investigate some basic physical properties of W gravities and W strings, using a free field realization. We argue that the configuration space of W gravities have global characteristics in addition to the Euler characteristic. We identify one such global quantity to be a “monopole” charge and show how this charge appears in the exponents. The free energy would then involve a “θ” parameter. Using a BRST procedure we find all the physical states of W₃ and W₄ gravities, and show that physical operators are nonsingular composites of the screening charge operators. (The latter are not physical operators for N ≥ 3.) For W strings we show how the W constraints lead to the emergence of a single (and not many) extra dimension coming from the W-gravity sector. By analyzing the resulting dispersion relations we find that both the lower and upper critical dimensions are lowered compared to ordinary two-dimensional gravity. The pure W gravity spectrum reveals an intriguing “numerological” connection with unitary minimal models coupled to ordinary gravity.

We review some aspects of induced gauge theories in two dimensions. We focus on W₃ gravity, paying particular attention to the treatment of the non-linearities inherent to W gravity. We show that the induced action Γ_ind[h,b] for chiral W₃ gravity in the c → ±∞ limit is obtained from the induced action of a gauged Sl(3, R) Wess-Zumino-Witten model by imposing constraints on some of the affine currents. Subsequently we investigate the effective action, which is obtained by integrating the induced action over the gauge fields. We show perturbatively that certain subleading terms which appear in the induced action for finite c (and which are related to non-local terms in the Ward identities) get cancelled by similar terms due to loop corrections, and we propose an all-order result for the effective action.

Starting with three-dimensional Chern–Simons theory with gauge group Sl(N, ℝ), we derive an action S_cov, invariant under both left and right W_N transformations. We give an interpretation of S_cov in terms of anomalies, and discuss its relation with Toda theory.

We present a detailed investigation of scattering processes in W₃-string theory. We discover further physical states with continuous momentum, which involve excitations of the ghosts as well as the matter, and use them to gain a better understanding of the interacting theory. The scattering amplitudes display factorisation properties, with states from the different sectors of the theory being exchanged in the various intermediate channels. We find strong evidence for the unitarity of the theory, despite the unusual ghost structure of some of the physical states. Finally, we show that by performing a transformation of the quantum fields that involves mixing the ghost fields with one of the matter fields, the structure of the physical states is dramatically simplified. The new formalism provides a concise framework within which to study the W₃ string.

We construct the BRST operator for non-critical W₃-strings and discuss the tachyon-like spectrum. For N-punctured spheres with N ≥ 5 we briefly describe a formal definition of the integral over W₃-moduli space.

We generalize some of the standard homological techniques to W-algebras, and compute the semi-infinite cohomology of the W₃ algebra on a variety of modules. These computations provide physical states in W₃ gravity coupled to W₃ minimal models and to two free scalar fields.

We present a simple procedure for constructing the complete cohomology of the BRST operator of the two-scalar and multi-scalar W₃ strings. The method consists of obtaining two level–15 physical operators in the two-scalar W₃ string that are invertible, and that can normal order with all other physical operators. They can be used to map all physical operators into non-trivial physical operators whose momenta lie in a fundamental unit cell. By carrying out an exhaustive analysis of physical operators in this cell, the entire cohomology problem is solved.

The following sections are included:

• References