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

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.