Topological sigma-models with H-flux and twisted generalized complex manifolds

TL;DR

This work extends topological twisting to $(2,2)$ sigma-models with $H$-flux by recasting target-space geometry in terms of twisted generalized complex structures. It shows that the space of topological observables is governed by the Lie algebroid cohomology of one twisting structure, and that the topological metric endows this space with a Frobenius algebra, with quantum corrections constrained by generalized holomorphic instantons. The results unify A- and B-models in a generalized complex setting and illuminate a path toward twisted generalized Calabi–Yau mirror symmetry, including RR-state interpretations. The framework provides a robust geometric language for non-Kähler flux backgrounds and suggests new directions for TG-brane categories and quantum cohomology.

Abstract

We study the topological sector of N=2 sigma-models with H-flux. It has been known for a long time that the target-space geometry of these theories is not Kahler and can be described in terms of a pair of complex structures, which do not commute, in general, and are parallel with respect to two different connections with torsion. Recently an alternative description of this geometry was found, which involves a pair of commuting twisted generalized complex structures on the target space. In this paper we define and study the analogues of A and B-models for N=2 sigma-models with H-flux and show that the results are naturally expressed in the language of twisted generalized complex geometry. For example, the space of topological observables is given by the cohomology of a Lie algebroid associated to one of the two twisted generalized complex structures. We determine the topological scalar product, which endows the algebra of observables with the structure of a Frobenius algebra. We also discuss mirror symmetry for twisted generalized Calabi-Yau manifolds.