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Line Operators on S^1 x R^3 and Quantization of the Hitchin Moduli Space

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We perform an exact localization calculation for the expectation value of Wilson-'t Hooft line operators in N=2 gauge theories on S^1 x R^3.
The expectation values form a quantum mechanically deformed algebra of functions on the Hitchin moduli space by Moyal multiplication. We demonstrate that these expectation values are the Weyl transform of the Verlinde operators, which acts on conformal blocks as difference operators. Our results are also in exact match with the predictions from wall-crossing in the IR effective theory.