Supersymmetric vortex loops in 3D gauge theories

TL;DR

This paper defines and computes BPS vortex loops in 3D ${ m N}=2$ supersymmetric gauge theories, clarifies their relation to Wilson loops, and explores their descriptions via 1D SUSY quantum mechanics, revealing new insights into their properties.

Contribution

It provides a precise definition of BPS vortex loops, computes their expectation values on an ellipsoid, and introduces an alternative 1D SUSY quantum mechanics description, advancing understanding of vortex loops in 3D SUSY theories.

Findings

Expectation values computed on an ellipsoid.

Resolution of parameter shift via anomaly analysis.

Connection to 1D SUSY quantum mechanics and quiver GLSMs.

Abstract

We give a precise definition of BPS vortex loops in 3D non-abelian gauge theories with ${\cal N}=2$ SUSY by the path integral over fields with a prescribed singular behavior. We compute the expectation value of a BPS vortex loop on an ellipsoid. Using the result we revisit the known equivalence between Wilson and vortex loops in pure Chern-Simons theory. Naive computations of expectation values in ${\cal N}=2$ theory leads to an unwanted shift of parameters in the rule of correspondence. We resolve the problem by relating the shift to the global anomaly of ${\cal N}=2$ SUSY quantum mechanics. For theories with $U(N)$ gauge group we also develop an alternative description of vortex loops in terms of 1D ${\cal N}=2$ SUSY quantum mechanics on their worldline. For vortex loops in ${\cal N}=4$ theories, our construction reproduces some of the quiver GLSMs of Assel and Gomis.