A gerby deformation of complex tori and the homological mirror symmetry

Let $(X,\check{X})$ be a mirror pair of a complex torus $X$ and its mirror partner $\check{X}$. This mirror pair is described as the trivial special Lagrangian torus fibrations $X\to B$ and $\check{X}\to B$ on the same base space $B$ by SYZ construction. Then, we can associate a holomorphic line bundle $E(s,\mathcal{L})\to X$ to a pair $(s,\mathcal{L})$ of a Lagrangian section $s$ of $\check{X}\to B$ and a unitary local system $\mathcal{L}$ along it. In this paper, we first construct the deformation $X_{\mathcal{G}}$ of $X$ by a certain flat gerbe $\mathcal{G}$ and its mirror partner ${\check{X}}_{\mathcal{G}}$ from the mirror pair $(X,\check{X})$, and discuss deformations of objects $E(s,\mathcal{L})$ and $(s,\mathcal{L})$ over the deformed mirror pair $(X_{\mathcal{G}},{\check{X}}_{\mathcal{G}})$.