From Liouville to Chern-Simons, Alternative Realization of Wilson Loop Operators in AGT Duality

TL;DR

This paper presents an SL(2,R) Chern-Simons framework for Liouville field theory, providing new insights into Wilson loop operators and their monodromies, and connecting these to gauge theory partition functions and link invariants.

Contribution

It introduces an alternative Chern-Simons description of Liouville theory, linking Wilson loops to link invariants and monodromy, and applies this to compute t'Hooft loops in super Yang-Mills.

Findings

Consistent with previous results in 0909.0945 and 0909.1105.

Provides a new geometric interpretation of Wilson loops as Hopf links.

Calculates t'Hooft loops in N=4 super Yang-Mills using the new framework.

Abstract

We propose an SL(2,R) Chern-Simons description of Liouville field theory (LFT), whose correlation function duals to partition function of N=2 SU(2) gauge theories. We give the dual expressions for conformal blocks, fusion rules, and Wilson loop operators in Chern-Simons theory. By realizing Wilson loop operator in Liouville as a Hopf link in S^3 on which lives an SL(2,R) Chern-Simons theory, we obtain an alternative description of monodromy of this loop operator in Liouville field theory as the ratio of link invariants. We show how to calculate t'Hooft loops in the simplest example -- the N=4 super Yang-Mills theory. The results we obtained are consistant with those in 0909.0945 and 0909.1105.