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, a simple projectively flat bundle $E(L,\mathcal{ L})\rightarrow X^n$ is constructed from each affine Lagrangian submanifold $L$ in $\check{ X}^n$ with a unitary local system $\mathcal{ L} \rightarrow L$. In this paper, we first interpret these simple projectively flat bundles $E(L,\mathcal{ L})$ in the language of factors of automorphy. Furthermore, we give a geometric interpretation for exact triangles consisting of three simple projectively flat bundles $E(L,\mathcal{ L})$ and their shifts by focusing on the dimension of intersections of the corresponding affine Lagrangian submanifolds $L$. Finally, as an application of this geometric interpretation, we discuss whether such an exact triangle on $X^n$ ($n \geq 2$) is obtained as the pullback of an exact triangle on $X^1$ by a suitable holomorphic projection $X^n \rightarrow X^1$.
Categories: math.DG hep-th