Moduli Space of Non-Abelian Vortices

TL;DR

This work determines the full moduli space M_{N,k} of k non-Abelian vortices in U(N) gauge theory with N fundamental Higgs fields. It develops a moduli-matrix framework leading to a master equation and a standard triangular form, revealing the separated-vortex sector as a symmetric product of k copies of C × CP^{N-1} and showing that coincident vortices resolve orbifold singularities to yield a smooth moduli manifold. The paper also establishes a concrete relation to Kahler quotient constructions via Z and Ψ data, and provides a detailed example for N=k=2 to illustrate the correspondence. The results offer a complete geometric picture of non-Abelian vortex moduli with implications for vortex dynamics and potential metric computations.

Abstract

We completely determine the moduli space M_{N,k} of k-vortices in U(N) gauge theory with N Higgs fields in the fundamental representation. Its open subset for separated vortices is found as the symmetric product (C x CP^{N-1})^k / S_k. Orbifold singularities of this space correspond to coincident vortices and are resolved resulting in a smooth moduli manifold. Relation to Kahler quotient construction is discussed.