Group Theory of Non-Abelian Vortices

TL;DR

This paper explores the structure of the moduli space of multiple non-Abelian vortices in U(N) gauge theory, revealing how the global SU(N) symmetry acts on these configurations and determining the Kähler potential for each irreducible orbit.

Contribution

It provides a detailed analysis of the moduli space of non-Abelian vortices, including the representation theory and explicit Kähler potential calculations for various vortex configurations.

Findings

Moduli space of a single vortex is CP(N-1).

Multi-vortex moduli space decomposes into irreducible SU(N) representations.

Kähler potential is exactly determined for each irreducible orbit.

Abstract

We investigate the structure of the moduli space of multiple BPS non-Abelian vortices in U(N) gauge theory with N fundamental Higgs fields, focusing our attention on the action of the exact global (color-flavor diagonal) SU(N) symmetry on it. The moduli space of a single non-Abelian vortex, CP(N-1), is spanned by a vector in the fundamental representation of the global SU(N) symmetry. The moduli space of winding-number k vortices is instead spanned by vectors in the direct-product representation: they decompose into the sum of irreducible representations each of which is associated with a Young tableau made of k boxes, in a way somewhat similar to the standard group composition rule of SU(N) multiplets. The K\"ahler potential is exactly determined in each moduli subspace, corresponding to an irreducible SU(N) orbit of the highest-weight configuration.