Non-abelian vortices, Hecke modifications and singular monopoles

TL;DR

This paper establishes a deep connection between Hecke modifications, non-abelian vortex equations, and singular monopoles for the group U(N), revealing new geometric and physical insights into these moduli spaces.

Contribution

It demonstrates that for U(N), the space of Hecke modifications matches the vortex moduli space, linking it to singular monopoles via recent theoretical developments.

Findings

Hecke modifications correspond to non-abelian vortex solutions

Moduli space of vortices is isomorphic to singular monopoles on C×I

Provides a geometric framework connecting gauge theory and algebraic geometry

Abstract

In this note we show that for the group G = U(N) the space of Hecke modifications of a rank N vector bundle over a Riemann surface C coincides with the moduli space of solutions of certain non-abelian vortex equations over C . Through the recent work of Kapustin and Witten this then leads to an isomorphism between the moduli space of vortices and the moduli space of singular monopoles on the product of C with a closed interval I .