Takuya Okuda

Gauge theory loop operators and Liouville theory

We propose a correspondence between loop operators in a family of four dimensional

On the instantons and the hypermultiplet mass of

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super Yang-Mills on S4

= 2 gauge theories on S4 — 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

Exact results for vortex loop operators in 3d supersymmetric theories

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Spectral curves, emergent geometry, and bubbling solutions for Wilson loops

= 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