One-Loop Non-Planar Anomalous Dimensions in Super Yang-Mills Theory

TL;DR

This paper analyzes non-planar one-loop anomalous dimensions in supersymmetric Yang-Mills theory, providing explicit calculations, perturbative corrections, and numerical studies of spectral statistics to understand integrability and chaos.

Contribution

It introduces a method to compute non-planar one-loop anomalous dimensions using Bethe states and hexagon functions, and explores the transition from integrability to chaos at finite N.

Findings

Explicit formulas for matrix elements of the dilatation operator.

Leading 1/N^2 corrections to operator dimensions computed.

Level spacing distribution transitions from Poisson to Wigner-Dyson.

Abstract

We consider non-planar one-loop anomalous dimensions in maximally supersymmetric Yang-Mills theory and its marginally deformed analogues. Using the basis of Bethe states, we compute matrix elements of the dilatation operator and find compact expressions in terms of off-shell scalar products and hexagon-like functions. We then use non-degenerate quantum-mechanical perturbation theory to compute the leading $1/N^2$ corrections to operator dimensions and as an example compute the large $R$-charge limit for two-excitation states through subleading order in the $R$-charge. Finally, we numerically study the distribution of level spacings for these theories and show that they transition from the Poisson distribution for integrable systems at infinite $N$ to the GOE Wigner-Dyson distribution for quantum chaotic systems at finite $N$.