Non-abelian vortices on $\mathbb {CP}^{1}$CP1 and Grassmannians

Abstract

Many properties of the moduli space of abelian vortices on a compact Riemann surface Σ are known. For non-abelian vortices the moduli space is less well understood. Here we consider non-abelian vortices on the Riemann sphere $\mathbb {CP}^{1}$CP1, and we study their moduli spaces near the Bradlow limit. We give an explicit description of the moduli space as a Kähler 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 $\mathbb {CP}^{1}$CP1, 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.