Mirror Symmetry in dimension one and Fourier-Mukai equivalences

TL;DR

This paper explores mirror symmetry for certain 1-dimensional Deligne-Mumford stacks called tcnc curves, proposing a sheaf-theoretic model for their Fukaya categories and examining derived equivalences akin to Mukai's duality results.

Contribution

It introduces a conjectural sheaf-theoretic framework for the Fukaya category of punctured Riemann surfaces and applies it to establish derived equivalences of tcnc curves, extending Mukai's classical results.

Findings

Proposed a sheaf-theoretic model for Fukaya categories of punctured Riemann surfaces.

Established derived equivalences for tcnc curves.

Generalized Mukai's duality results to new classes of stacks.

Abstract

In this paper we will describe an approach to mirror symmetry for appropriate 1-dimensional DM stacks of arithmetic genus $g \leq 1$, called tcnc curves, which was developed by the author with Treumann and Zaslow in arXiv:1103.2462 . This involves introducing a conjectural sheaf-theoretic model for the Fukaya category of punctured Riemann surfaces. As an application, we will investigate derived equivalences of tcnc curves, and generalize classic results of Mukai on dual abelian varieties (Mukai 1981).