L. Fehér

Toda Theory and W-Algebra from a Gauged WZNW Point of View

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.

On Hamiltonian reductions of the Wess-Zumino-Novikov-Witten theories

The structure of Hamiltonian symmetry reductions of the Wess-Zumino-Novikov-Witten (WZNW) theories by first class Kac-Moody (KM) constraints is analysed in detail. Lie algebraic conditions are given for ensuring the presence of exact integrability, conformal invariance and W-symmetry in the reduced theories. A Lagrangean, gauged WZNW implementation of the reduction is established in the general case and thereby the path integral as well as the BRST formalism are set up for studying the quantum version of the reduction. The general results are applied to a number of examples. In particular, a W-algebra is associated to each embedding of sl(2) into the simple Lie algebras by using purely first class constraints. The primary fields of these W-algebras are manifestly given by the sl(2) embeddings, but it is also shown that there is an sl(2) embedding present in every polynomial and primary KM reduction and that the Wn l-algebras have a hidden sl(2) structure too. New generalized Toda theories are found whose chiral algebras are the W-algebras based on the half-integral sl(2) embeddings, and the W-symmetry of the effective action of those generalized Toda theories associated with the integral gradings is exhibited explicitly.

Liouville and Toda theories as conformally reduced WZNW theories

It is shown that Liouville theory can be regarded as an SL(2, o) Wess-Zumino-Novikov-Witten theory with conformal invariant constraints and that Polyakovs SL(2, o) Kac-Moody symmetry of induced two-dimensional gravity is just one side of the WZNW current algebra. Analogously, Toda field theories can be regarded as conformal-invariantly constrained WZNW theories for appropriate (maximally non-compact) groups.

Kac-Moody realization of W-algebras

Abstract By realizing the W -algebras of Toda field-theories as the algebras of gauge-invariant polynomials of constrained Kac-Moody systems we obtain a simple algorithm for constructing W -algebras without computing the W -generators themselves. In particular this realization yields an identification of a primary field basis for all the W -algebras, quadratic bases for the A, B, C-algebras, and the relation of W -algebras to Casimir algebras. At the quantum level it yields the general formula for the Virasoro centre in terms of the KM level.

Classical r-matrix and exchange algebra in WZNW and Toda theories

Abstract It is shown that the fundamental Poisson brackets in the chiral sectors of WZNW theory and its Liouville-Toda reduction are of the r -matrix form. In general, the r -matrix is monodromy dependent, but in the case of A l Lie algebras, and only then, this monodromy dependence can be “gauged away” by choosing appropriate representatives in the conjugacy classes of the monodromy. The resulting non-trivial solution of the classical Yang-Baxter equation is the classical limit of the quantum R -matrix of A l Toda theory found recently by Cremmer and Gervais.

Generalized Drinfeld-Sokolov reductions and KdV type hierarchies

Generalized Drinfeld-Sokolov (DS) hierarchies are constructed through local reductions of Hamiltonian flows generated by monodromy invariants on the dual of a loop algebra. Following earlier work of De Groot et al., reductions based upon graded regular elements of arbitrary Heisenberg subalgebras are considered. We show that, in the case of the nontwisted loop algebra ℓ(gln), graded regular elements exist only in those Heisenberg subalgebras which correspond either to the partitions ofn into the sum of equal numbersn=pr or to equal numbers plus onen=pr+1. We prove that the reduction belonging to the grade 1 regular elements in the casen=pr yields thep×p matrix version of the Gelfand-Dickeyr-KdV hierarchy, generalizing the scalar casep=1 considered by DS. The methods of DS are utilized throughout the analysis, but formulating the reduction entirely within the Hamiltonian framework provided by the classical r-matrix approach leads to some simplifications even forp=1.

Generalized Toda theories and W-algebras associated with integral gradings

A general class of conformal Toda theories associated with integral gradings of Lie algebras is investigated. These generalized Toda theories are obtained by reducing the Wess--Zumino--Novikov--Witten (WZNW) theory by first--class constraints, and thus they inherite extended conformal symmetry algebras, generalized W--algebras, and current dependent Kac--Moody (KM) symmetries from the WZNW theory, which are analysed in detail in a non--degenerate case. We recover an

Chiral extensions of the WZNW phase space, Poisson-Lie symmetries and groupoids

sl(2)

Inequivalent quantizations of the three-particle Calogero model constructed by separation of variables

structure underlying the generalized W--algebras, which allows for identifying the primary fields, and give a simple algorithm for implementing the W--symmetries by current dependent KM transformations, which can be used to compute the action of the W--algebra on any quantity. We establish how the Lax pair of Toda theory arises in the WZNW framework, and show that a recent result of Mansfield and Spence, which interprets the W--symmetry of the Toda theory by means of non--Abelian form preserving gauge transformations of the Lax pair, arises immediately as a consequence of the KM interpretation.

A Class of W algebras with infinitely generated classical limit

The chiral WZNW symplectic form Ωρchir is inverted in the general case. Thereby a precise relationship between the arbitrary monodromy dependent 2-form appearing in Ωρchir and the exchange r-matrix that governs the Poisson brackets of the group valued chiral fields is established. The exchange r-matrices are shown to satisfy a new dynamical generalization of the classical modified Yang–Baxter (YB) equation and Poisson–Lie (PL) groupoids are constructed that encode this equation analogously as PL groups encode the classical YB equation. For an arbitrary simple Lie group G, exchange r-matrices are found that are in one-to-one correspondence with the possible PL structures on G and admit them as PL symmetries.