A gerby deformation of complex tori and the homological mirror symmetry

Abstract

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