The Ising model and planar N=4 Yang-Mills

TL;DR

This paper establishes a novel correspondence between the spectral variables in planar N=4 supersymmetric Yang-Mills theory and the transfer matrices of a two-dimensional Ising model, linking integrability to the Yang-Baxter equation.

Contribution

It introduces a mapping from magnon spectral parameters to Ising model couplings, connecting gauge theory integrability with classical statistical mechanics.

Findings

Integrability corresponds to the Yang-Baxter equation in the Ising model.

Ising correlation length relates to magnon momentum, diverging at criticality.

Kramers-Wannier duality has a meaningful interpretation in gauge theory context.

Abstract

The scattering-matrix for planar Yang-Mills with N=4 supersymmetry relies on the assumption that integrability holds to all orders in perturbation theory. In this note we define a map from the spectral variables x^{\pm}, parameterizing the long-range magnon momenta, to couplings in a two-dimensional Ising model. Under this map integrability of planar N=4 Yang-Mills becomes equivalent to the Yang-Baxter equation for the two-dimensional Ising model, and the long-range variables x^{\pm} translate into the entries of the Ising transfer matrices. We explore the Ising correlation length which equals the inverse magnon momentum in the small momentum limit. The critical regime is thus reached for vanishing magnon momentum. We also discuss the meaning of the Kramers-Wannier duality transformation on the gauge theory, together with that of the Ising model critical points.