A Sharp Existence Theorem for Vortices in the Theory of Branes

TL;DR

This paper proves precise existence and uniqueness theorems for vortex solutions in D-brane intersection theories, providing explicit conditions for solutions on different domains.

Contribution

It introduces a direct minimization approach to establish sharp existence and uniqueness results for BPS vortex equations in string theory.

Findings

Explicit necessary and sufficient conditions for vortex solutions on doubly periodic domains.

Proved unique solutions exist over the full plane and doubly periodic domains.

Established a new mathematical framework for analyzing BPS equations in brane theories.

Abstract

We investigate the BPS equations arising from the theory of multi-intersection of D-branes. By using the direct minimization method, we establish sharp existence and uniqueness theorems for multiple vortex solutions of the BPS equations over a doubly periodic domain and over the full plane, respectively. In particular, we obtain an explicit necessary and sufficient condition for the existence of a unique solution for the doubly periodic domain case.