Convergence of Hermitian-Yang-Mills Connections on K\"ahler Surfaces and mirror symmetry

TL;DR

This paper explores the connection between Hermitian-Yang-Mills connections on Kähler surfaces and mirror symmetry, showing how families of such connections converge to flat connections on fibers, revealing geometric structures related to the mirror dual.

Contribution

It introduces a natural construction linking complex and symplectic geometry via mirror symmetry, demonstrating convergence of HYM connections to flat connections on fibers of Lagrangian torus fibrations.

Findings

HYM connections converge to flat connections on fibers as epsilon approaches zero.

Limit connections determine points in the dual torus, forming a Lagrangian variety.

The construction provides a geometric bridge between complex and symplectic structures in mirror symmetry.

Abstract

The purpose of this paper is to exhibit a natural construction between complex geometry and symplectic geometry following the idea of mirror symmetry. Suppose we are given a family of pairs of 2-dimensional K\"ahler tori and stable holomorphic vector bundles on them $(\hat M_{\ep}, E_{\ep})$, \ep \in (0, 1]$, and each has structure of a Lagrangian torus fibration $\pi:\hat M_{\ep} \to B$ whose fibers are of diameter $O(\ep)$, and let $A_{\ep}$ be a family of hermitian Yang-Mills(HYM) connections on $E_{\ep}$. As $\ep$ goes to zero, $A_{\ep}$ will, modulo possible bubbles, converge to a connection which is flat on each fiber. Since each fiber is a torus, limit connection will determine elements of the dual torus, which are points of the fiber of the mirror $M_1$. These points gather to make up (special) Lagrangian variety.