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

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 ${\check{X}}^n\to {\mathbb{R}}^n\mathrm{/}{\mathbb{Z}}^n$ and a unitary local system along it, and those holomorphic line bundles with integrable connections forms 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 $X_{\mathcal{G}}^n$ of $X^n$ by a flat gerbe $\mathcal{G}$. Moreover, we construct the deformation of $D{G}_{X^n}$ associated to the deformation from $X^n$ to $X_{\mathcal{G}}^n$, and also discuss the homological mirror symmetry between $X_{\mathcal{G}}^n$ and its mirror partner on the object level.