Wilson-’t Hooft lines as transfer matrices

Abstract

We establish a correspondence between a class of Wilson-’t Hooft lines in four-dimensional $$ \mathcal{N} $$ 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 S1 × ϵ ℝ2 × 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.

Superspin chains from superstring theory

“…Many of the phenomena that are expected to constitute this web of dualities are yet to be uncovered, but their specializations to the case of gl(m|0) = gl(m) are known and have been studied in recent years. Besides the Bethe/gauge correspondence already described, the structures of rational gl(m) spin chains (and their trigonometric and elliptic generalizations) have appeared in quantization of the Seiberg-Witten geometries of fourdimensional N = 2 supersymmetric gauge theories [4][5][6], the action of surface and line defects on supersymmetric indices of four-dimensional supersymmetric gauge theories [7][8][9][10][11][12], quantization of the Coulomb branches of three-dimensional N = 4 supersymmetric gauge theories [13,14], and correlation functions of local operators on interfaces in fourdimensional N = 4 super Yang-Mills theory [15], to name a few.…”

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

“…In particular, we will see that the commutativity of elliptic Ruijsenaars operators is rephrased as the absence of wall-crossing phenomena (η-independence) of elliptic genera for the monopole bubbling effects. Recently the author of [33] observed that the localization formula [1] of 't Hooft loop operators with particular charges in 4d N = 2 * gauge theory agrees with a trigonometric limit of type-A elliptic Ruijsenaars operators [34]. Since the KK modes along the T 2 -direction give an elliptic deformation of the localization formula of the 't Hooft loops, we expect the deformation quantization of 't Hooft surface operator itself agrees with elliptic Ruisenaars operators.…”

“…In Section 5, we found that the deformation quantization of 't Hooft surface operators in 5d N = 1 * gauge theory agrees with the type-A elliptic Ruijsenaars operators. Although the integrable structure appears in the 't Hooft surface operators is not manifest in the supersymmetric gauge theory, by using dualities in string theory, the brane configuration for 't Hooft operators without the monopole bubbling effect is interpreted as defects in fourdimensional Chern-Simons theory, where the quantum integrable structure naturally appear [33].…”

ABCD of ’t Hooft operators

“…Before this work, localization for 't Hooft operators, especially with monopole screening, had been done only for SU(N ) and U(N ) gauge groups; consequently applications such as [38,[44][45][46][47][48][49] and section 8.2 of [39] had been limited to these groups. Our results here should be useful when extending these applications to SO(N ) and USp(N ).…”