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arXiv · 1505.05898

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

Ioana Coman·Maxime Gabella·Joerg Teschner
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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 $\mathcal{M}_\text{flat}$ 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 $\mathcal{M}_\text{flat}$. 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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Ioana Coman, Maxime Gabella, Joerg Teschner. 2015-05-21. Line operators in theories of class $\mathcal{S}$, quantized moduli space of flat connections, and Toda field theory. https://arxiv.org/abs/1505.05898

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