Brackets, Sigma Models and Integrability of Generalized Complex Structures

TL;DR

This paper explores how derived brackets naturally appear in sigma-models with generalized complex structures, providing explicit formulas and linking them to the generalized Nijenhuis tensor, with applications to specific two-dimensional models.

Contribution

It establishes a precise connection between derived brackets and sigma-models in generalized complex geometry, including explicit coordinate expressions and new insights into the Nijenhuis tensor.

Findings

Derived brackets arise naturally in sigma-models via Poisson and antibrackets.

The generalized Nijenhuis tensor coincides with the derived bracket of the structure with itself.

Explicit coordinate expressions for derived brackets and the Nijenhuis tensor are provided.

Abstract

It is shown how derived brackets naturally arise in sigma-models via Poisson- or antibracket, generalizing a recent observation by Alekseev and Strobl. On the way to a precise formulation of this relation, an explicit coordinate expression for the derived bracket is obtained. The generalized Nijenhuis tensor of generalized complex geometry is shown to coincide up to a de-Rham closed term with the derived bracket of the structure with itself, and a new coordinate expression for this tensor is presented. The insight is applied to two known two-dimensional sigma models in a background with generalized complex structure. Introductions to geometric brackets on the one hand and to generalized complex geometry on the other hand are given in the appendix.