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 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.

Highlights

  • Be the mass parameters of the n bifundamental hypermultiplets

  • We establish a correspondence between a class of Wilson-’t Hooft lines in four-dimensional 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 × R2 × R by supersymmetric localization and show that they are equal to the Wigner transforms of the transfer matrices

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Summary

Transfer matrices for Wilson-’t Hooft lines

We discuss the integrable system side of the correspondence. After reviewing L-operators, transfer matrices and their relation to quantum integrable systems, we introduce an L-operator for the elliptic dynamical R-matrix. We define fundamental trigonometric L-operators as certain limits of the elliptic L-operator. These fundamental L-operators are building blocks of transfer matrices that correspond to Wilson-’t Hooft lines in N = 2 supersymmetric circular quiver theories

L-operators and quantum integrable systems
Elliptic L-operator
Trigonometric L-operators
Transfer matrices from circular quiver theories
Monodromy matrices from linear quiver theories
Other representations
Transfer matrices from Verlinde operators
Verlinde operators and Wilson-’t Hooft lines
Verlinde operators on a punctured torus
Brane realization
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