Gauge Theory Loop Operators and Liouville Theory

TL;DR

This paper establishes a correspondence between loop operators in four-dimensional N=2 gauge theories and Liouville theory loop operators on Riemann surfaces, providing new insights into S-duality and operator algebra.

Contribution

It introduces a novel link between gauge theory loop operators and Liouville theory, extending the AGT correspondence to include 't Hooft and dyonic operators.

Findings

Liouville loop operators reproduce Pestun's Wilson loop expectation values.

The invariance under modular transformations supports S-duality conjectures.

Explicit predictions for 't Hooft and dyonic loop operator expectations.

Abstract

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 Teichmuller 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.