Three Dimensional Topological Field Theory induced from Generalized Complex Structure

We construct a three-dimensional topological sigma model which is induced from a generalized complex structure on a target generalized complex manifold. This model is constructed from maps from a three-dimensional manifold $X$ to an arbitrary generalized complex manifold $M$. The theory is invariant under the diffeomorphism on the world volume and the $b$-transformation on the generalized complex structure. Moreover the model is manifestly invariant under the mirror symmetry. We derive from this model the Zucchini’s two dimensional topological sigma model with a generalized complex structure as a boundary action on $\mathrm{\partial }X$. As a special case, we obtain three dimensional realization of a WZ-Poisson manifold.