Mirror Symmetry and Generalized Complex Manifolds

Abstract

In this paper we develop a relative version of T-duality in generalized complex geometry which we propose as a manifestation of mirror symmetry. Let M be an n-dimensional smooth real manifold, V a rank n real vector bundle on M, and nabla a flat connection on V. We define the notion of a nabla-semi-flat generalized complex structure on the total space of V. We show that there is an explicit bijective correspondence between nabla-semi-flat generalized complex structures on the total space of V and nabla(dual)-semi-flat generalized complex structures on the total space of the dual of V. Similarly we define semi-flat generalized complex structures on real n-torus bundles with section over an n-dimensional base and establish a similar bijective correspondence between semi-flat generalized complex structures on pair of dual torus bundles. Along the way, we give methods of constructing generalized complex structures on the total spaces of vector bundles and torus bundles with sections. We also show that semi-flat generalized complex structures give rise to a pair of transverse Dirac structures on the base manifold. We give interpretations of these results in terms of relationships between the cohomology of torus bundles and their duals. We also study the ways in which our results generalize some well established aspects of mirror symmetry as well as some recent proposals relating generalized complex geometry to string theory.