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

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 $({\widehat{M}}_{\text{ep}},E_{\text{ep}})$, \ep \in (0, 1]$,andeachhasstructureofaLagrangiantorusfibration$\pi:\hat M_{\ep} \to B$whosefibersareofdiameter$O(\ep)$,andlet$A_{\ep}$ be a family of hermitian Yang-Mills(HYM) connections on $E_{\text{ep}}$. As $\text{ep}$ goes to zero, $A_{\text{ep}}$ 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.