't Hooft surface operators in five dimensions and elliptic Ruijsenaars operators

TL;DR

This paper introduces magnetic surface operators in 5d supersymmetric gauge theories, computes their vevs via localization, and links their deformation quantization to elliptic Ruijsenaars operators, revealing new algebraic structures.

Contribution

It presents the first construction and evaluation of 5d surface operators, connecting their quantization to elliptic integrable systems and extending known localization results.

Findings

Surface operator vevs computed via localization techniques.

Deformation quantization relates to elliptic Ruijsenaars operators.

Mutual commutativity of difference operators corresponds to surface operator algebra.

Abstract

We introduce codimension three magnetically charged surface operators in five-dimensional (5d) $\mathcal{N}=1$ supersymmetric gauge on $T^2 \times \mathbb{R}^3$. We evaluate the vacuum expectation values (vevs) of surface operators by supersymmetric localization techniques. Contributions of Monopole bubbling effects to the path integral are given by elliptic genera of world volume theories on D-branes. Our result gives an elliptic deformation of the SUSY localization formula \cite{Ito:2011ea} (resp. \cite{Okuda:2019emk, Assel:2019yzd}) of BPS 't Hooft loops (resp. bare monopole operators) in 4d $\mathcal{N}=2$ (resp. 3d $\mathcal{N}=4$) gauge theories. We define deformation quantizations of vevs of surface operators in terms of the Weyl-Wigner transform, where the $\Omega$-background parameter plays the role of the Planck constant. For 5d $\mathcal{N}=1^*$ gauge theory, we find that the deformation quantization of the surface operators in the anti-symmetric representations agrees with the type A elliptic Ruijsenaars operators. The mutual commutativity of these difference operators is related to the commutativity of products of 't Hooft surface operators.