Exact results for supersymmetric abelian vortex loops in 2+1 dimensions

TL;DR

The paper defines and analyzes abelian vortex loop defect operators in 3d ${\cal N}=2$ gauge theories, including their supersymmetric realizations, and provides an exact framework for computing their expectation values on $\mathbb{S}^3$ via localization. It develops three complementary methods—$SL(2,\mathbb{Z})$ transformations, smearing regularization, and explicit singular-background spectral analysis—to obtain consistent 1-loop determinants and demonstrate the loop's dependence on flavor holonomies and Chern–Simons data. The work clarifies how these vortex loops participate in IR dualities, notably mirror symmetry, by mapping to Wilson loops and flavor or topological defects across dual descriptions. It also outlines the generalization to non-abelian defects and the broader implications for duality dictionaries and rigorous tests of 3d dualities through defect operators.

Abstract

We define a class of supersymmetric defect loop operators in N = 2 gauge theories in 2+1 dimensions. We give a prescription for computing the expectation value of such operators in a generic N = 2 theory on the three-sphere using localization. We elucidate the role of defect loop operators in IR dualities of supersymmetric gauge theories, and write down their transformation properties under the SL(2, Z) action on conformal theories with abelian global symmetries.