On the Classification of Real Forms of Non-Abelian Toda Theories and W-algebras

TL;DR

This paper classifies real forms of non-Abelian Toda theories and associated W-algebras by analyzing reality conditions, Lie algebra embeddings, and their correspondences, providing a comprehensive framework for understanding their structure.

Contribution

It introduces a method to classify all real embeddings of sl(2,C) into complex Lie algebras and relates them to reality conditions in Toda theories and W-algebras.

Findings

Complete classification of real embeddings principal in simple regular subalgebras.

Established correspondence between reality conditions and sl(2,R) embeddings.

Developed a method to find all real embeddings from complex ones.

Abstract

We consider conformal non-Abelian Toda theories obtained by hamiltonian reduction from Wess-Zumino-Witten models based on general real Lie groups. We study in detail the possible choices of reality conditions which can be imposed on the WZW or Toda fields and prove correspondences with sl(2,R) embeddings into real Lie algebras and with the possible real forms of the associated W-algebras. We devise a a method for finding all real embeddings which can be obtained from a given embedding of sl(2,C) into a complex Lie algebra. We then apply this to give a complete classification of real embeddings which are principal in some simple regular subalgebra of a classical Lie algebra.