Supersymmetric Yang-Mills theory and integrable systems

TLDR

The Coulomb branch of N=2 supersymmetric gauge theories in four dimensions is generally described by an integrable Hamiltonian system, naturally constructed from two-dimensional gauge theory and spectral curves. The authors propose an integrable system relevant to the N=2 SU(n) gauge theory with an adjoint hypermultiplet. They provide substantial evidence supporting the correctness of this system. The model exhibits an SL(2,Z) S‑duality group, with the central element −1 acting as charge conjugation, and SL(2,Z) permutes the Higgs, confining, and oblique confining phases as expected.

Abstract

The Coulomb branch of N = 2 supersymmetric gauge theories in four dimensions is described in general by an integrable Hamiltonian system in the holomorphic sense. A natural construction of such systems comes from two-dimensional gauge theory and spectral curves. Starting from this point of view, we propose an integrable system relevant to the N = 2 SU(n) gauge theory with a hypermultiplet in the adjoint representation, and offer much evidence that it is correct. The model has an SL(2,Z) S-duality group (with the central element −1 of SL(2,Z) acting as charge conjugation); SL(2,Z) permutes the Higgs, confining, and oblique confining phases in the expected fashion. We also study more exotic phases.