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Topological sigma-models with $$H$$-flux and twisted generalized complex manifolds. (English) Zbl 1192.81310
Sigma-models with (2,2) supersymmetry may be “twisted” to obtain topological field theories, known as the A and B-models. When the B-field is a closed 2-form, the target manifold $$M$$ has to be a Kähler manifold. If $$H=dB\not=0$$, $$M$$ has to be “Kähler with torsion”, i.e., there exist two different complex structures which do not commute, in general, and are parallel with respect to two different connections with torsion. An alternative description of this geometry in terms of a (twisted) generalized complex structure on $$M$$ was introduced by N. Hitchin [Q. J. Math. 54, No. 3, 281–308 (2003; Zbl 1076.32019)] and studied in detail by M. Gualtieri [Generalized complex geometry, arXiv:math/0401221v1]. The paper defines and studies the analogs of the A and B-models for $$N=2$$ sigma-models with H-flux in the language of twisted generalized complex geometry. It is proven that the space of topological observables is given by the cohomology of a Lie algebroid associated to one of the two twisted generalized complex structures. The authors also determine the topological scalar product, which endows the algebra of observables with the structure of a Frobenius algebra, and discuss a possible generalization of mirror symmetry to twisted generalized Calabi-Yau manifolds.

##### MSC:
 81T45 Topological field theories in quantum mechanics 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 81T60 Supersymmetric field theories in quantum mechanics 81R25 Spinor and twistor methods applied to problems in quantum theory 32Q15 Kähler manifolds 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
Zbl 1076.32019
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