Geometry of W-algebras from the affine Lie algebra point of view

TL;DR

This paper explores the geometric structure of W-algebras via affine Lie algebra reductions, classifying symplectic leaves and constructing explicit representatives, with applications to models like Toda and Zamolodchikov.

Contribution

It provides a constructive classification of symplectic leaves of W-algebras from affine Lie algebra reductions, extending previous results for n=2 and n=3 cases.

Findings

Classified symplectic leaves for W_n-algebras via reduction methods.

Constructed explicit representatives on each symplectic leaf.

Described the global nature of W-transformations and classical highest weight states.

Abstract

To classify the classical field theories with W-symmetry one has to classify the symplectic leaves of the corresponding W-algebra, which are the intersection of the defining constraint and the coadjoint orbit of the affine Lie algebra if the W-algebra in question is obtained by reducing a WZNW model. The fields that survive the reduction will obey non-linear Poisson bracket (or commutator) relations in general. For example the Toda models are well-known theories which possess such a non-linear W-symmetry and many features of these models can only be understood if one investigates the reduction procedure. In this paper we analyze the SL(n,R) case from which the so-called W_n-algebras can be obtained. One advantage of the reduction viewpoint is that it gives a constructive way to classify the symplectic leaves of the W-algebra which we had done in the n=2 case which will correspond to the coadjoint orbits of the Virasoro algebra and for n=3 which case gives rise to the Zamolodchikov algebra. Our method in principle is capable of constructing explicit representatives on each leaf. Another attractive feature of this approach is the fact that the global nature of the W-transformations can be explicitly described. The reduction method also enables one to determine the ``classical highest weight (h. w.) states'' which are the stable minima of the energy on a W-leaf. These are important as only to those leaves can a highest weight representation space of the W-algebra be associated which contains a ``classical h. w. state''.