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

TL;DR

This paper computes exact expectation values of line operators in N=2 gauge theories on S^1xR^3, revealing their algebraic structure as a quantum deformation of functions on the Hitchin moduli space, and links them to conformal blocks via Weyl transform.

Contribution

It introduces a novel localization method to evaluate line operator expectation values and connects these to the quantization of the Hitchin moduli space and conformal field theory.

Findings

Expectation values expressed in Fenchel-Nielsen coordinates

Expectation values form a Moyal-deformed algebra of Hitchin moduli space functions

Explicit demonstration in SU(N) gauge theories

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.