Line operators in theories of class \( \mathcal{S} \), quantized moduli space of flat connections, and Toda field theory

Non-perturbative aspects ofN = 2 supersymmetric gauge theories of classS are deeply encoded in the algebra of functions on the moduli spaceMat of at SL(N)- connections on Riemann surfaces. Expectation values of Wilson and 't Hooft line operators are related to holonomies of at connections, and expectation values of line operators in the low-energy eective theory are related to Fock-Goncharov coordinates on Mat . Via the decomposition of UV line operators into IR line operators, we determine their noncommutative algebra from the quantization of Fock-Goncharov Laurent polynomials, and nd that it coincides with the skein algebra studied in the context of Chern-Simons theory. Another realization of the skein algebra is generated by Verlinde network operators in Toda eld theory. Comparing the spectra of these two realizations provides non-trivial support for their equivalence. Our results can be viewed as evidence for the generalization of the AGT correspondence to higher-rank classS theories.