On a B-field transform of generalized complex structures over complex tori

Abstract

Let $(X^n,{\check{X}}^n)$ be a mirror pair of an n-dimensional complex torus $X^n$ and its mirror partner ${\check{X}}^n$. Then, by SYZ transform, we can construct a holomorphic line bundle with an integrable connection from each pair of a Lagrangian section of ${\check{X}}^n\to R^n/Z^n$ and a unitary local system along it, and those holomorphic line bundles with integrable connections form a dg-category $D{G}_{X^n}$. In this paper, we focus on a certain B-field transform of the generalized complex structure induced from the complex structure on $X^n$, and interpret it as the deformation $X_G^n$ of $X^n$ by a flat gerbe $G$. Moreover, we construct the deformation of $D{G}_{X^n}$ associated to the deformation from $X^n$ to $X_G^n$, and also discuss the homological mirror symmetry between $X_G^n$ and its mirror partner on the object level.