On the General Structure of Hamiltonian Reductions of the Wznw Theory

TL;DR

This paper analyzes the Hamiltonian reduction of WZNW theory, establishing conditions for integrability and symmetry, and introduces new generalized Toda theories with associated W-algebras, advancing understanding of conformal and W-symmetries.

Contribution

It provides a comprehensive framework for Hamiltonian reductions of WZNW theory, including a gauged Lagrangian approach, and constructs new Toda theories with specific W-algebra symmetries.

Findings

Lie algebraic conditions for integrability and symmetry are identified.

A gauged WZNW formulation for quantum reductions is developed.

New generalized Toda theories with W-algebra symmetries are constructed.

Abstract

The structure of Hamiltonian reductions of the Wess-Zumino-Novikov-Witten (WZNW) theory by first class Kac-Moody constraints is analyzed in detail. Lie algebraic conditions are given for ensuring the presence of exact integrability, conformal invariance and $\cal W$-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 ${\cal W}$-algebra is associated to each embedding of $sl(2)$ into the simple Lie algebras by using purely first class constraints. The importance of these $sl(2)$ systems is demonstrated by showing that they underlie the $W_n^l$-algebras as well. New generalized Toda theories are found whose chiral algebras are the ${\cal W}$-algebras belonging to the half-integral $sl(2)$ embeddings, and the ${\cal W}$-symmetry of the effective action of those generalized Toda theories associated with the integral gradings is exhibited explicitly.