Non-abelian vortices on compact Riemann surfaces

TL;DR

This paper characterizes non-abelian vortex solutions on compact Riemann surfaces by their zero locations and internal structures, linking gauge theory solutions to moduli spaces of holomorphic pairs.

Contribution

It provides a detailed description of vortex internal spaces and a local factorization approach, advancing understanding of non-abelian vortex moduli spaces on Riemann surfaces.

Findings

Vortex solutions are determined by zero locations and internal structures.

Internal spaces of vortices are compact and connected.

Explicit descriptions of vortex internal spaces are provided.

Abstract

We consider the vortex equations for a U(n) gauge field coupled to a Higgs field with values on the n times n square matrices. It is known that when these equations are defined on a compact Riemann surface, their moduli space of solutions is closely related to a moduli space of tau-stable holomorphic n-pairs on that surface. Using this fact and a local factorization result for the Higgs matrix, we show that the vortex solutions are entirely characterized by (1) the location in the surface of the zeros of the determinant of the Higgs matrix and (2) by the choice of a vortex internal structure at each of these zeros. We describe explicitly the vortex internal spaces and show that they are compact and connected spaces.