Lagrangian Floer theory for trivalent graphs and homological mirror symmetry for curves

TL;DR

This paper develops a new Lagrangian Floer theory for trivalent graphs and demonstrates homological mirror symmetry between the Fukaya category of rational curve configurations and derived categories of Tate curves.

Contribution

It introduces a novel approach to mirror symmetry for higher genus curves using trivalent graphs and establishes an equivalence of categories in this setting.

Findings

Fukaya category of trivalent rational curves is equivalent to derived category of Tate curves

Explicit formulas for theta functions emerge naturally from the mirror symmetry

Homological mirror symmetry holds in this new graph-based framework

Abstract

Mirror symmetry for higher genus curves is usually formulated and studied in terms of Landau-Ginzburg models; however the critical locus of the superpotential is arguably of greater intrinsic relevance to mirror symmetry than the whole Landau-Ginzburg model. Accordingly, we propose a new approach to the A-model of the mirror, viewed as a trivalent configuration of rational curves together with some extra data at the nodal points. In this context, we introduce a version of Lagrangian Floer theory and the Fukaya category for trivalent graphs, and show that homological mirror symmetry holds, namely, that the Fukaya category of a trivalent configuration of rational curves is equivalent to the derived category of a non-Archimedean generalized Tate curve. To illustrate the concrete nature of this equivalence, we show how explicit formulas for theta functions and for the canonical map of the curve arise naturally under mirror symmetry.