Abstract
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 vacuum expectation value 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 the properties of the moduli space volume in these theories. Our formulae are applied to the volume of the vortex moduli space in the gauged non-linear sigma model with $\mathbb{C} P^N$ 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.
Highlights
Vortices are co-dimension two solitons and play an important role for non-perturbative effects in gauge theories
If the vortex equations are considered on compact Riemann surfaces Σh with the genus h, the number of the vortices is restricted by an upper bound which is given by the finite area of Σh divided by the intrinsic size of the vortex
The purpose of our paper is to obtain a formula for the volume of the moduli space of BPS vortices in quiver gauge theories on compact Riemann surfaces
Summary
Vortices are co-dimension two solitons and play an important role for non-perturbative effects in gauge theories. Once a general formula for the volume of the vortex moduli space for the quiver gauge theory is derived, the BPS vortex equations with various kinds of matters or target space can be obtained. The purpose of our paper is to obtain a formula for the volume of the moduli space of BPS vortices in quiver gauge theories on compact Riemann surfaces. We obtain the contour integral formula of the volume of the vortex moduli space in the quiver gauge theory. As concrete examples to apply the contour integral formula for the volume of the vortex moduli space, we consider various quiver gauge theory with Abelian vertices. We apply the contour integral formula of the volume to a quiver gauge theory corresponding to the parent gauged linear sigma model of Abelian GNLSM with CP N target space with n flavors of charge scalar fields. The Laplacian matrix cannot reproduce the whole structure of the quiver diagram including the orientation of the edges
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