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

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.