Exact results for vortex loop operators in 3d supersymmetric theories

TL;DR

This paper provides exact results for vortex loop operators in 3d N=2 supersymmetric theories, using localization and index theory, and confirms their transformation properties under mirror symmetry.

Contribution

It constructs and computes BPS vortex loop operators in 3d N=2 theories on various manifolds, and clarifies their behavior under mirror symmetry using localization techniques.

Findings

Exact formulas for vortex loop operators on S^3 and S^3_b.

Validation of mirror symmetry predictions for Wilson and vortex loops.

Application of index theory to localization without fixed points.

Abstract

Three dimensional field theories admit disorder line operators, dubbed vortex loop operators. They are defined by the path integral in the presence of prescribed singularities along the defect line. We study half-BPS vortex loop operators for N=2 supersymmetric theories on S^3, its deformation S^3_b and S^1 x S^2. We construct BPS vortex loops defined by the path integral with a fixed gauge or flavor holonomy for infinitesimal curves linking the loop. It is also possible to include a singular profile for matter fields. For vortex loops defined by holonomy, we perform supersymmetric localization by calculating the fluctuation modes, or alternatively by applying the index theory for transversally elliptic operators. We clarify how the latter method works in situations without fixed points of relevant isometries. Abelian mirror symmetry transforms Wilson and vortex loops in a specific way. In particular an ordinary Wilson loop transforms into a vortex loop for a flavor symmetry. Our localization results confirm the predictions of abelian mirror symmetry.