Topological sigma-models with \(H\)-flux and twisted generalized complex manifolds.

*(English)*Zbl 1192.81310Sigma-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.

Reviewer: Helmut Rumpf (Wien)

##### 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 |