Wilson-'t Hooft lines as transfer matrices

TL;DR

This paper links Wilson-'t Hooft lines in 4D supersymmetric gauge theories to transfer matrices in integrable systems, computing their expectation values and connecting them to Toda theory and Chern-Simons theory.

Contribution

It establishes a novel correspondence between supersymmetric gauge line operators and quantum integrable system transfer matrices, including their relation to Toda and Chern-Simons theories.

Findings

Expectation values match Wigner transforms of transfer matrices.

Transfer matrices correspond to Verlinde operators in Toda theory.

Field theory setup relates to 4D Chern-Simons theory via string dualities.

Abstract

We establish a correspondence between a class of Wilson-'t Hooft lines in four-dimensional $\mathcal{N} = 2$ supersymmetric gauge theories described by circular quivers and transfer matrices constructed from dynamical L-operators for trigonometric quantum integrable systems. We compute the vacuum expectation values of the Wilson-'t Hooft lines in a twisted product space $S^1 \times_\epsilon \mathbb{R}^2 \times \mathbb{R}$ by supersymmetric localization and show that they are equal to the Wigner transforms of the transfer matrices. A variant of the AGT correspondence implies an identification of the transfer matrices with Verlinde operators in Toda theory, which we also verify. We explain how these field theory setups are related to four-dimensional Chern-Simons theory via embedding into string theory and dualities.