Lagrangian submanifold associated with a degenerating family of stable bundles on T4
Year: 2003
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Takeo Nishinou
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2240526273