Magnetic monopoles on manifolds with boundary

TL;DR

This paper proves the existence and uniqueness of singular monopoles on manifolds with boundary, linking them to Hecke modifications of holomorphic bundles, and confirms a geometric invariant theory principle.

Contribution

It establishes a rigorous proof of existence and uniqueness of singular monopoles with prescribed boundary conditions for any Hecke modification data.

Findings

Proves existence of singular monopoles for given boundary conditions.

Establishes uniqueness of these monopoles.

Confirms the geometric invariant theory principle in this context.

Abstract

Kapustin and Witten associate a Hecke modification of a holomorphic bundle over a Riemann surface to a singular monopole on a Riemannian surface times an interval satisfying prescribed boundary conditions. We prove existence and uniqueness of singular monopoles satisfying prescribed boundary conditions for any given Hecke modification data confirming the underlying geometric invariant theory principle.