Line operators on S^1xR^3 and quantization of the Hitchin moduli space

TL;DR

This work computes exact localization results for Wilson–t Hooft line operators in $ ext{N}=2$ gauge theories on $S^1 imes bR^3$, showing that their vevs form a noncommutative algebra under the Moyal product and thereby realize the deformation quantization of the Hitchin moduli space in complexified Fenchel–Nielsen coordinates. The authors provide explicit one-loop and monopole-screening contributions, revealing how the twist parameter $oldsymbol heta$ quantizes the Hitchin system, and demonstrate that Verlinde operators in Liouville/Toda theories act as Weyl transforms of these vevs, providing a concrete bridge via the AGT correspondence. They work out detailed SU($N$) examples and discuss the nontrivial link to Liouville/Toda CFTs, S-duality, and potential extensions to non-Lagrangian theories. The results offer a powerful framework to compute and interpret line operator vevs in a broad class of $ ext{N}=2$ theories and to connect 4d gauge dynamics with 2d CFT and Hitchin-system quantization. The work thus solidifies the deformation-quantization picture of Hitchin moduli space arising naturally from localized line operators and suggests new avenues in the study of AGT, wall-crossing, and IR/UV dualities.

Abstract

We perform an exact localization calculation for the expectation values of Wilson-'t Hooft line operators in N=2 gauge theories on S^1xR^3. The expectation values are naturally expressed in terms of the complexified Fenchel-Nielsen coordinates, and form a quantum mechanically deformed algebra of functions on the associated Hitchin moduli space by Moyal multiplication. We propose that these expectation values are the Weyl transform of the Verlinde operators, which act on Liouville/Toda conformal blocks as difference operators. We demonstrate our proposal explicitly in SU(N) examples.