The Volume of the Quiver Vortex Moduli Space

TL;DR

This paper derives an exact formula for the volume of vortex moduli spaces in quiver gauge theories on Riemann surfaces using supersymmetric localization and graph theory, revealing bounds and properties of these spaces.

Contribution

It introduces a novel contour integral formula for the moduli space volume in quiver gauge theories, generalizing Jeffrey-Kirwan residues and connecting to Bradlow bounds.

Findings

Exact volume formula derived via localization

Generalization of Jeffrey-Kirwan residue formula

Application to gauged non-linear sigma models

Abstract

We study the moduli space volume of BPS vortices in quiver gauge theories on compact Riemann surfaces. The existence of BPS vortices imposes constraints on the quiver gauge theories. We show that the moduli space volume is given by a vev of a suitable cohomological operator (volume operator) in a supersymmetric quiver gauge theory, where BPS equations of the vortices are embedded. In the supersymmetric gauge theory, the moduli space volume is exactly evaluated as a contour integral by using the localization. Graph theory is useful to construct the supersymmetric quiver gauge theory and to derive the volume formula. The contour integral formula of the volume (generalization of the Jeffrey-Kirwan residue formula) leads to the Bradlow bounds (upper bounds on the vorticity by the area of the Riemann surface divided by the intrinsic size of the vortex). We give some examples of various quiver gauge theories and discuss properties of the moduli space volume in these theories. Our formula are applied to the volume of the vortex moduli space in the gauged non-linear sigma model with $CP^N$ target space, which is obtained by a strong coupling limit of a parent quiver gauge theory. We also discuss a non-Abelian generalization of the quiver gauge theory and "Abelianization" of the volume formula.