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

Abstract

We study the topological sector of N = 2 sigma-models with Hflux. It has been known for a long time that the target-space geometry of these theories is not Kähler 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. CALT-68-25141 More precisely, the B-model makes sense on the quantum level if and only if M is a Calabi-Yau manifold. For the A-model, the Calabi-Yau condition is unnecessary.

The biHermitian topological sigma model

“…Both models are also evidently independent from the complex structure of the world sheet Σ. These findings confirm earlier results [12,13].…”

“…Further, when this is the case, the BRST cohomology is equivalent to the Lie algebroid cohomology of the relevant generalized complex structure [13]. This remarkable result was one of the achievements of Kapustin's and Li's work.…”

“…The (3.9) can be rephrased as conditions on the the generalized Kaehler structure (J 1 , J 2 ) corresponding to the given biHermitian structure (g, H, K ± ) according to (2.19) [13]. If E k denotes the +i eigenbundle of J k in (T M ⊕ T * M) ⊗ , then the conditions read…”

“…[12][13][14][15][16][17][18][19][20][21][22][23][24][25]. All these were somehow unsatisfactory either because they remained confined to the analysis of geometrical aspects of the sigma models or because they yielded field theories, which though interesting in their own, were not directly suitable for quantization, showed no apparent kinship with Witten's A and B models and were of limited relevance for string theory.…”

“…In their seminal paper [13], Kapustin and Li defined and studied the analogues of the A and B models for the general biHermitian (2, 2) supersymmetric sigma model. They tackled several crucial issues.…”