1/2-BPS vortex strings in $\mathcal{N}=2$ supersymmetric ${\rm U}(1)^N$ gauge theories

TL;DR

This paper analyzes BPS vortex solutions in general ${ m U}(1)^N$ supersymmetric gauge theories, revealing linear tension relations, classifying solutions, and deriving simplified vortex equations that appear Abelian despite underlying non-Abelian symmetries.

Contribution

The paper provides a comprehensive classification of BPS vortex solutions, proves the non-existence of certain bounds, and derives simplified vortex equations in ${ m U}(1)^N$ theories with arbitrary parameters.

Findings

All BPS solutions have tension linear in magnetic flux.

Existence of solutions is constrained by theorems on the constraint equations.

Vortex equations reduce to Abelian-like forms after solving constraints.

Abstract

Strings in $\mathcal{N}=2$ supersymmetric ${\rm U}(1)^N$ gauge theories with $N$ hypermultiplets are studied in the generic setting of an arbitrary Fayet-Iliopoulos triplet of parameters for each gauge group and an invertible charge matrix. Although the string tension is generically of a square-root form, it turns out that all existing BPS (Bogomol'nyi-Prasad-Sommerfield) solutions have a tension which is linear in the magnetic fluxes, which in turn are linearly related to the winding numbers. The main result is a series of theorems establishing three different kinds of solutions of the so-called constraint equations, which can be pictured as orthogonal directions to the magnetic flux in ${\rm SU}(2)_R$ space. We further prove for all cases, that a seemingly vanishing Bogomol'nyi bound cannot have solutions. Finally, we write down the most general vortex equations in both master form and Taubes-like form. Remarkably, the final vortex equations essentially look Abelian in the sense that there is no trace of the ${\rm SU}(2)_R$ symmetry in the equations, after the constraint equations have been solved.