Non-abelian vortices on CP^1 and Grassmannians

TL;DR

This paper studies non-abelian vortices on the Riemann sphere, describing their moduli spaces explicitly, revealing unique configurations, and analyzing their geometric and statistical properties near the Bradlow limit.

Contribution

It provides an explicit description of non-abelian vortex moduli spaces on CP^1 as Kahler quotients and explores their geometric structures and statistical mechanics.

Findings

Existence of non-abelian vortex configurations with point moduli spaces.

Moduli space for certain vortices is a Grassmannian.

The vortex partition function resembles the abelian case.

Abstract

Many properties of the moduli space of abelian vortices on a compact Riemann surface are known. For non-abelian vortices the moduli space is less well understood. Here we consider non-abelian vortices on the Riemann sphere CP^1, and we study their moduli spaces near the Bradlow limit. We give an explicit description of the moduli space as a Kahler quotient of a finite-dimensional linear space. The dimensions of some of these moduli spaces are derived. Strikingly, there exist non-abelian vortex configurations on CP^1, with non-trivial vortex number, for which the moduli space is a point. This is in stark contrast to the moduli space of abelian vortices. For a special class of non-abelian vortices the moduli space is a Grassmannian, and the metric near the Bradlow limit is a natural generalization of the Fubini--Study metric on complex projective space. We use this metric to investigate the statistical mechanics of non-abelian vortices. The partition function is found to be analogous to the one for abelian vortices.