Real Forms of Non-abelian Toda Theories and their W-algebras

TL;DR

This paper explores alternative real forms of non-abelian Toda theories and their W-algebras, showing that relaxing traditional restrictions yields new algebraic structures and extending to superalgebras with supersymmetry.

Contribution

It introduces a framework for non-standard reality conditions in non-abelian Toda theories, expanding the class of possible real forms and associated W-algebras, including superalgebra cases.

Findings

New real forms of W-algebras identified

Relaxing maximal non-compactness condition broadens algebraic structures

Explicit examples include N=4 superconformal algebra realization

Abstract

We consider real forms of Lie algebras and embeddings of sl(2) which are consistent with the construction of integrable models via Hamiltonian reduction. In other words: we examine possible non-standard reality conditions for non-abelian Toda theories. We point out in particular that the usual restriction to the maximally non-compact form of the algebra is unnecessary, and we show how relaxing this condition can lead to new real forms of the resulting W-algebras. Previous results for abelian Toda theories are recovered as special cases. The construction can be extended straightforwardly to deal with osp(1|2) embeddings in Lie superalgebras. Two examples are worked out in detail, one based on a bosonic Lie algebra, the other based on a Lie superalgebra leading to an action which realizes the N=4 superconformal algebra.