Gauge Theory Loop Operators and Liouville Theory

TL;DR

<3-5 sentence high-level summary>The paper extends the AGT-like correspondence by embedding exact loop-operator observables of 4D ${\cal N}=2$ gauge theories ${\cal T}_{g,n}$ into Liouville theory on the associated Riemann surface ${C_{g,n}}$. Loop operators are realized as Liouville loop operators built from monodromies of degenerate fields, whose modular properties automatically implement S-duality and map Wilson, 't Hooft, and dyonic operators across duality frames. The construction is tied to quantum Teichmüller theory via geodesic-length operators, yielding a geometric and algebraic framework (quantum skein relations) that reproduces Pestun’s Wilson-loop results and makes exact predictions for nonperturbative loop observables in ${\cal T}_{g,n}$. The work thus unifies gauge-theory loop operator algebra with Liouville/Teichmüller structures, offering precise tests and predictions for S-duality and the operator product expansions in ${\cal N}=2$ theories. The approach also points to broad extensions to higher rank and refined theories, linking localization, CFT, and quantum geometry in a concrete, computable way.

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.

Figures (9)

• Figure 1: Examples of trivalent graphs (quiver diagrams) corresponding to Riemann surfaces: (a) a 4-puncture sphere, (b) a once-punctured torus, (c) a genus two surface with no puncture.
• Figure 2: Calculating the Wilson loop in the fundamental representation. The solid lines carry arbitrary states, the wiggly ones are degenerate fields and the dashed line carries the identity state.
• Figure 3: Calculating the 't Hooft loop on the one-punctured torus.
• Figure 4: Calculating the 't Hooft loop on the four-punctured sphere as the monodromy associated to moving a $V_{1,2}$ field.
• Figure 5: Basic arcs on a pair-of-pants