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 $X\rightarrow B$ and $\check{ X}\rightarrow B$ on the same base space $B$ by SYZ construction. Then, we can associate a holomorphic line bundle $E(s,\mathcal{ L})\rightarrow X$ to a pair $(s,\mathcal{ L})$ of a Lagrangian section $s$ of $\check{ X}\rightarrow B$ 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 $(X_{ \mathcal{ G}},\check{ X}_{ \mathcal{ G}})$.
Categories: math.DG hep-th