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

TL;DR

This work constructs an $\ SL(2,\mathbb{R})$ Chern-Simons description of Liouville theory, the 2d side of the AGT duality to 4d $\mathcal{N}=2$ $SU(2)$ gauge theories, recasting conformal data and loop observables in terms of 3d topological data. By realizing Liouville loop operators as Hopf links in $S^3$, the authors express monodromies as ratios of link invariants governed by the modular S-matrix, and they provide explicit formulas for Wilson and $t$'Hooft loops, including the fundamental holonomy $h_{1,2;\alpha}=\dfrac{\cos(2\pi b a)}{\cos(\pi b Q)}$. The approach hinges on a CS/WZW reduction, enabling a surgery-based computation of loop observables and a unifying geometric framework that extends to higher-rank groups; consistency with known AGT results is demonstrated. The framework promises generalizations to $SL(N,\mathbb{R})$ and Toda theories, with potential implications for M-brane constructions, S-duality, and connections to topological strings and matrix models.

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.

Figures (5)

• Figure 1: a) A Riemann surface $\Sigma$ with four punctures. b) A segment of $\Sigma \times S^1$ with four Wilson lines.
• Figure 2: In a), an imaginary loop $C$ was darwn on $M$. In b), the neighborhood of $C$ was cut out, now the $M$ had been cut into $M_L$ and $M_R$ which is a solid torus. In c), the solid torus had been glued back with $M_L$ after a diffeomorphism $K$, and formed a new 3-manifold $\tilde{M}$. In d), It follows that partition function in $\tilde{M}$ can be obtained from that in $M$ with a Wilson loop $C$ and the knowledge of diffeomorphism $K$.
• Figure 3: a) In $S^2\times S^1$, two loops $C$ and $C'$ rounded the uncontractable circle and braided with each other, the generalized surgery was taken on $C$. b) After surgery, $C$ and $C'$ become a linked Hopf link in $S^3$.
• Figure 4: A torus can be cut into two identical spheres with two punctures which are attached with conjugated representations $R$ and $\bar{R}$.
• Figure 5: a) The boundary of two balls $B_1$ and $B_2$ are identified with spheres with two conjugated punctures, these two punctures are endpoints of a Wilson lines in the bulk. b) By gluing $B_1$ and $B_2$ back into $S^3$, one gets an $S^3$ with a single Wilson loop. c) t'Hooft loop generated by (1,2) now can be seen as the (1,2) Wilson loop surrounding the $a$ loop.