Dynkin Diagrams and Integrable Models Based on Lie Superalgebras

TL;DR

This paper explores the structure of Toda field theories based on Lie superalgebras, revealing how Dynkin diagrams help identify bosonic root systems and constructing new integrable models with positive kinetic energy and supersymmetry.

Contribution

It introduces a method to determine bosonic root systems from Dynkin diagrams for superalgebras, leading to new integrable models with desirable physical properties.

Findings

Identified two natural bosonic root systems for superalgebras.

Constructed new integrable theories with positive kinetic energy.

Developed models combining massless, massive, and supersymmetric sectors.

Abstract

An analysis is given of the structure of a general two-dimensional Toda field theory involving bosons and fermions which is defined in terms of a set of simple roots for a Lie superalgebra. It is shown that a simple root system for a superalgebra has two natural bosonic root systems associated with it which can be found very simply using Dynkin diagrams; the construction is closely related to the question of how to recover the signs of the entries of a Cartan matrix for a superalgebra from its Dynkin diagram. The significance for Toda theories is that the bosonic root systems correspond to the purely bosonic sector of the integrable model, knowledge of which can determine the bosonic part of the extended conformal symmetry in the theory, or its classical mass spectrum, as appropriate. These results are applied to some special kinds of models and their implications are investigated for features such as supersymmetry, positive kinetic energy and generalized reality conditions for the Toda fields. As a result, some new families of integrable theories with positive kinetic energy are constructed, some containing a mixture of massless and massive degrees of freedom, others being purely massive and supersymmetric, involving a number of coupled sine/sinh-Gordon theories.