The bijectivity of mirror functors on tori

DOI10.1215/21562261-2022-0021zbMATH Open1502.14042arXiv1905.00692OpenAlexW2943734510WikidataQ121850321 ScholiaQ121850321MaRDI QIDQ2093559

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Publication date: 27 October 2022

Published in: (Search for Journal in Brave)

Abstract: By the SYZ construction, a mirror pair $(X,checkX)$ of a complex torus $X$ and a mirror partner $checkX$ of the complex torus $X$ is described as the special Lagrangian torus fibrations $XightarrowB$ and $checkXightarrowB$ 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 multi section of $checkXightarrowB$ 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.

Full work available at URL: https://arxiv.org/abs/1905.00692

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