By the SYZ construction, a mirror pair of a complex torus X and a mirror partner of the complex torus X is described as the special Lagrangian torus fibrations and on the same base space B. Then, by the SYZ transform, we can construct a simple projectively flat bundle on X from each affine Lagrangian multisection of with a unitary local system along it. However, there are ambiguities of the choices of transition functions of it, and this causes difficulties when we try to construct a functor between the symplectic geometric category and the complex geometric category. In this paper, we prove that there exists a bijection between the set of the isomorphism classes of their objects by solving this problem.
Kyoto J. Math.
62(3):
655-682
(September 2022).
DOI: 10.1215/21562261-2022-0021
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