SYZ transforms for immersed Lagrangian multi-sections

TL;DR

This paper explores the SYZ transform for immersed Lagrangian multi-sections, establishing a new equivalence in the immersed Fukaya category that preserves Floer cohomology and relates to holomorphic vector bundles on the mirror elliptic curve.

Contribution

It introduces a novel notion of equivalence in the immersed Fukaya category and demonstrates its invariance and mirror correspondence under SYZ transform.

Findings

New equivalence in immersed Fukaya category preserving Floer cohomology

Relation between Lagrangian surgery and holomorphic vector bundles

Mirror symmetry correspondence for Lagrangian torus fibrations

Abstract

In this paper, we study the geometry of the SYZ transform on a semi-flat Lagrangian torus fibration. Our starting point is an investigation on the relation between Lagrangian surgery of a pair of straight lines in a symplectic 2-torus and extension of holomorphic vector bundles over the mirror elliptic curve, via the SYZ transform for immersed Lagrangian multi-sections. This study leads us to a new notion of equivalence between objects in the immersed Fukaya category of a general compact symplectic manifold $(M, \omega)$, under which the immersed Floer cohomology is invariant; in particular, this provides an answer to a question of Akaho-Joyce. Furthermore, if $M$ admits a Lagrangian torus fibration over an integral affine manifold, we prove, under some additional assumptions, that this new equivalence is mirror to isomorphism between holomorphic vector bundles over the dual torus fibration via the SYZ transform.