Integrable sigma models with complex and generalized complex structures

TL;DR

This paper explores conditions for integrability of sigma models on manifolds with complex and generalized complex structures, extending existing methods to Lie groups and providing specific examples with generalized structures.

Contribution

It extends the integrability conditions of sigma models to manifolds with generalized complex structures, including explicit examples on specific Lie groups.

Findings

Integrability conditions are satisfied by zeros of the Nijenhuis tensor.

On certain Lie groups, perturbed actions resemble WZ terms under specific conditions.

The formalism is extended from complex to generalized complex structures.

Abstract

Using the general method presented by Mohammedi \cite{NM} for the integrability of a sigma model on a manifold, we investigate the conditions for having an integrable deformation of the general sigma model on a manifold with a complex structure. On a Lie group, these conditions are satisfied by using the zeros of the Nijenhuis tensor. We then extend this formalism to a manifold, especially a Lie group, with a generalized complex structure. We demonstrate that, for the examples of integrable sigma models with generalized complex structures on the Lie groups $\mathbf{A_{4,8}}$ and $\mathbf{A_{4,10}}$, under special conditions, the perturbed terms of the actions are identical to the WZ terms.