Topological A-type models with flux

Abstract

We study deformations of the A-model in the presence of fluxes, by which we mean rank-three tensors with antisymmetrized upper/lower indices, using the AKSZ construction. There are two natural deformations of the A-model in the AKSZ language: 1) the Zucchini model, which can be defined on a generalized complex manifold and reduces to the A-model when the generalized complex structure comes from a symplectic structure, and 2) a topological membrane model, which naturally accommodates fluxes, and reduces to the Zucchini model on the boundary of the membrane when the fluxes are turned off. We show that the fluxes are related to deformations of the Courant bracket which generalize the twist by a closed 3-from H, in the sense that satisfying the AKSZ master equation implies precisely the integrability conditions for an almost generalized complex structure with respect to the deformed Courant bracket. In addition, the master equation imposes conditions on the fluxes that generalize dH = 0. The membrane model can be defined on a large class of U(m)- and U(m) × U(m)-structure manifolds relevant for string theory, including geometries inspired by (1, 1) supersymmetric σ-models with additional supersymmetries due to almost complex (but not necessarily complex) structures in the target space. In addition we show that the model can be defined on three particular half-flat manifolds related to the Iwasawa manifold. When only the closed 3-form flux is turned on it is possible to obtain a topological string model, which we do for the case of a Calabi-Yau. We argue that deformations from the standard A-model are due to the choice of gauge fixing fermion, rather than a flux deformation of the AKSZ action. The particularly interesting cases arise when the fermion depends on auxiliary fields, the simplest possibility being due to the (2, 0)+(0, 2) component of a non-trivial b-field. The model is generically no longer evaluated on holomorphic maps and defines new topological invariants. Deformations due to H-flux can be more radical, completely preventing auxiliary fields from being integrated out.