Abstract
Non-perturbative aspects of \( \mathcal{N}=2 \) supersymmetric gauge theories of class \( \mathcal{S} \) are deeply encoded in the algebra of functions on the moduli space \( {\mathrm{\mathcal{M}}}_{\mathrm{flat}} \) of flat SL(N )- connections on Riemann surfaces. Expectation values of Wilson and ’t Hooft line operators are related to holonomies of flat connections, and expectation values of line operators in the low-energy effective theory are related to Fock-Goncharov coordinates on \( {\mathrm{\mathcal{M}}}_{\mathrm{flat}} \). 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 find 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 field 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 class \( \mathcal{S} \) theories.
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Coman, I., Gabella, M. & Teschner, J. Line operators in theories of class \( \mathcal{S} \), quantized moduli space of flat connections, and Toda field theory. J. High Energ. Phys. 2015, 143 (2015). https://doi.org/10.1007/JHEP10(2015)143
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DOI: https://doi.org/10.1007/JHEP10(2015)143