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.
Highlights
There has been a lot of recent progress in the study of N = 2 supersymmetric field theories in four dimensions
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 Mflat
Of particular interest is the following special case of the fundamental skein relation obtained by contracting (N − 2) pairs of edges from the upper and lower parts of (2.38): (2.39) We indicate that the edge between the two junctions carries the label N − 2 by drawing it thicker than the other edges associated with the fundamental representation
Summary
A crucial role is played by the relation of the algebra Aflat to the algebra AVer generated by the Verlinde line operators This relation is generalized to theories of higher rank in our paper, thereby supporting the natural generalization of the AGT correspondence to class S theories of type AN−1. The six-dimensional origin of class S theories of type AN−1 suggests that there should exist a family of line operators that correspond to coordinate functions on the moduli space MNg,n ≡ MSflLat(N,C)(Cg,n) of flat SL(N, C)-connections. We first observe that the braiding matrix in Toda field theory, from which the Verlinde network operators are built, is related via a twist to the standard R-matrix of the quantum group Uq(slN ) This R-matrix is used to construct the quantum skein algebra defining the quantized version of ANg,n. The appendices collect some background about Fock-Goncharov coordinates and about quantum groups
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