Supersymmetric Yang-Mills Systems And Integrable Systems

TL;DR

This paper constructs an integrable system linked to N=2 supersymmetric SU(n) gauge theories, revealing its duality symmetries and phase structure, advancing understanding of gauge theory moduli spaces.

Contribution

It proposes a new integrable system for N=2 SU(n) gauge theories with an adjoint hypermultiplet, supported by extensive evidence and analysis of duality and phase behavior.

Findings

The integrable system exhibits an SL(2,Z) S-duality group.

The duality permutes different physical phases as expected.

The model provides insights into exotic phases of gauge theories.

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 $SL(2,{\bf Z})$ $S$-duality group (with the central element $-1$ of $SL(2,{\bf Z})$ acting as charge conjugation); $SL(2,{\bf Z})$ permutes the Higgs, confining, and oblique confining phases in the expected fashion. We also study more exotic phases.