The bijectivity of mirror functors on tori
Given an \(n\)-dimensional complex torus \(X\) and its mirror \(\check{X}\), with trivial SYZ fibrations over base \(B\), the paper studies SYZ transformations between objects in the full subcategory \(\mathrm{Fuk}_{\mathrm{aff}}(\check{X})\) of the Fukaya category \(\mathrm{Fuk}(\check{X})\) consisting of affine Lagrangian multisections of \(\check{X}\to B\) with unitary local systems on them, and objects in the DG category \(DG_X\) of simple projectively flat bundles over \(X\). The main result of the paper is that for a fixed transition function, the SYZ transform \[ \iota:\mathrm{Ob}(\mathrm{Fuk}_{\mathrm{aff}}(\check{X}))\rightarrow\mathrm{Ob}(DG_X) \] admits an inverse up to isomorphism of objects. The inverse map is constructed explicitly using natural parameters. It follows that the induced map \[ \iota^{\mathrm{isom}}: \mathrm{Ob}^{\mathrm{isom}}(\mathrm{Fuk}_{\mathrm{aff}}(\check{X}))\rightarrow \mathrm{Ob}^{\mathrm{isom}}(DG_X), \] which is known to be injective, is indeed a bijection.
scientific article