Non-Abelian Vortices of Higher Winding Numbers

TL;DR

This paper analyzes the moduli space of higher winding number non-Abelian vortices in U(N) gauge theories, revealing its geometric structure and singularities, and generalizing previous results for specific cases.

Contribution

It provides a detailed geometric description of the moduli space for k=2 vortices, including singularities and generalization to U(N) theories with N flavors.

Findings

Moduli space for k=2 vortices is a weighted projective space WCP^2_(2,1,1).

The space contains an A_1-type orbifold singularity.

Generalization to U(N) gauge theories results in a weighted Grassmannian manifold with singularities.

Abstract

We make a detailed study of the moduli space of winding number two (k=2) axially symmetric vortices (or equivalently, of co-axial composite of two fundamental vortices), occurring in U(2) gauge theory with two flavors in the Higgs phase, recently discussed by Hashimoto-Tong (hep-th/0506022) and Auzzi-Shifman-Yung (hep-th/0511150). We find that it is a weighted projective space WCP^2_(2,1,1)=CP^2/Z_2. This manifold contains an A_1-type (Z_2) orbifold singularity even though the full moduli space including the relative position moduli is smooth. The SU(2) transformation properties of such vortices are studied. Our results are then generalized to U(N) gauge theory with N flavors, where the internal moduli space of k=2 axially symmetric vortices is found to be a weighted Grassmannian manifold. It contains singularities along a submanifold.