We propose a correspondence between loop operators in a family of four dimensional N = 2 gauge theories on S 4 -including Wilson, 't Hooft and dyonic operators -and Liouville theory loop operators on a Riemann surface. This extends the beautiful relation between the partition function of these N = 2 gauge theories and Liouville correlators found by Alday, Gaiotto and Tachikawa. We show that the computation of these Liouville correlators with the insertion of a Liouville loop operator reproduces Pestun's formula capturing the expectation value of a Wilson loop operator in the corresponding gauge theory. We prove that our definition of Liouville loop operators is invariant under modular transformations, which given our correspondence, implies the conjectured action of S-duality on the gauge theory loop operators. Our computations in Liouville theory make an explicit prediction for the exact expectation value of 't Hooft and dyonic loop operators in these N = 2 gauge theories. The Liouville loop operators are also found to admit a simple geometric interpretation within quantum Teichmüller theory as the quantum operators representing the length of geodesics. We study the algebra of Liouville loop operators and show that it gives evidence for our proposal as well as providing definite predictions for the operator product expansion of loop operators in gauge theory.

The path integral of a general N = 2 supersymmetric gauge theory on S 4 is exactly evaluated in the presence of a supersymmetric 't Hooft loop operator. The result we find -obtained using localization techniques -captures all perturbative quantum corrections as well as non-perturbative effects due to instantons and monopoles, which are supported at the north pole, south pole and equator of S 4 . As a by-product, our gauge theory calculations successfully confirm the predictions made for 't Hooft loops obtained from the calculation of topological defect correlators in Liouville/Toda conformal field theory.

We perform an exact localization calculation for the expectation values of Wilson-'t Hooft line operators in N = 2 gauge theories on S 1 × R 3 . The expectation values are naturally expressed in terms of the complexified Fenchel-Nielsen coordinates, and form a quantum mechanically deformed algebra of functions on the associated Hitchin moduli space by Moyal multiplication. We propose that these expectation values are the Weyl transform of the Verlinde operators, which act on Liouville/Toda conformal blocks as difference operators. We demonstrate our proposal explicitly in SU (N ) examples.1

Loop operators and S-duality from curves on Riemann surfacesTo cite this article: Nadav Drukker et al JHEP09(2009) Abstract: We study Wilson-'t Hooft loop operators in a class of N = 2 superconformal field theories recently introduced by Gaiotto. In the case that the gauge group is a product of SU(2) groups, we classify all possible loop operators in terms of their electric and magnetic charges subject to the Dirac quantization condition. We then show that this precisely matches Dehn's classification of homotopy classes of non-self-intersecting curves on an associated Riemann surface -the same surface which characterizes the gauge theory. Our analysis provides an explicit prediction for the action of S-duality on loop operators in these theories which we check against the known duality transformation in several examples.

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 × 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.