Exact results for vortex loop operators in 3d supersymmetric theories

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.

70 pages, 3 figures; version 2: 73 pages, added references, expanded discussions on index theory, and corrected some formulas and arguments

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• Generalized Superconformal Index for Three Dimensional Field Theories
• SL(2,Z) Action On Three-Dimensional Conformal Field Theories With Abelian Symmetry
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• Exact results for boundaries and domain walls in 2d supersymmetric theories
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• Mirror Symmetry And Loop Operators
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