Non-Abelian Vortices of Higher Winding Numbers

TL;DR

This work analyzes higher-winding-number non-Abelian vortices in $U(N)$ gauge theories, focusing on the winding number two sector. By combining the moduli-matrix approach with a Kähler-quotient perspective, the authors show that the coincident $k=2$ vortices in $U(2)$ form the weighted projective space $W\mathbf{C}P^2_{(2,1,1)}$ (equivalently ${\mathbf C}P^2/{\mathbf Z}_2$), which contains a ${\mathbf Z}_2$ orbifold singularity, while the full moduli space including relative positions is smooth. They generalize to $U(N)$, where the internal moduli space of coincident $k=2$ vortices is the weighted Grassmannian $WGr_{N+1,2}^{(1,\ldots,1,0)}$ with singularities along a submanifold such as $Gr_{N,2}$; this has consequences for vortex reconnection and dynamics. The results bridge moduli-matrix constructions and D-brane/Kähler quotient pictures, and lay groundwork for extensions to semi-local vortices and other gauge groups.

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.

Figures (1)

• Figure 1: Toric diagram of $W{\bf C}P^2_{(2,1,1)}$ and their three patches $\tilde{\cal U}^{(2,0)}$, $\tilde{\cal U}^{(1,1)}$ and $\tilde{\cal U}^{(0,2)}$. The diagram is drawn under a gauge fixing condition (called the $D$ term constraint) $\sum_{a=0}^2q^a|\phi_a|^2 = 1$ where $U(1)^{\bf C}$ charges are $q_a = (2,1,1)$ for $W{\bf C}P^2_{(2,1,1)}$ while $q_a = (1,1,1)$ for the ordinary ${\bf C}P^2$. The triangle with the broken line and $O$ (without singularity) denotes the ordinary ${\bf C}P^2$.