On the breakdown of perturbative integrability in large N matrix models

TL;DR

This paper investigates the perturbative integrability of a large N matrix model derived from super Yang-Mills theory, showing it holds at low orders but breaks down at fourth order, indicating limits of integrability in such models.

Contribution

It demonstrates the breakdown of perturbative integrability in a large N matrix model at higher orders, challenging assumptions based on lower-order integrability features.

Findings

Integrability holds at first order in the SO(6) model.

Up to third order, the SU(2) subsector exhibits integrability features.

Perturbative integrability breaks down at fourth order.

Abstract

We study the perturbative integrability of the planar sector of a massive SU(N) matrix quantum mechanical theory with global SO(6) invariance and Yang-Mills-like interaction. This model arises as a consistent truncation of maximally supersymmetric Yang-Mills theory on a three-sphere to the lowest modes of the scalar fields. In fact, our studies mimic the current investigations concerning the integrability properties of this gauge theory. Like in the field theory we can prove the planar integrability of the SO(6) model at first perturbative order. At higher orders we restrict ourselves to the widely studied SU(2) subsector spanned by two complexified scalar fields of the theory. We show that our toy model satisfies all commonly studied integrability requirements such as degeneracies in the spectrum, existence of conserved charges and factorized scattering up to third perturbative order. These are the same qualitative features as the ones found in super Yang-Mills theory, which were enough to conjecture the all-loop integrability of that theory. For the SO(6) model, however, we show that these properties are not sufficient to predict higher loop integrability. In fact, we explicitly demonstrate the breakdown of perturbative integrability at fourth order.