# Quantum Chaos in Perturbative super-Yang-Mills Theory

### Submission summary

 As Contributors: Tristan McLoughlin Preprint link: scipost_202104_00019v1 Date submitted: 2021-04-16 18:38 Submitted by: McLoughlin, Tristan Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Approach: Theoretical

### Abstract

We provide numerical evidence that the perturbative spectrum of anomalous dimensions in maximally supersymmetric SU(N) Yang-Mills theory is chaotic at finite values of N. We calculate the probability distribution of one-loop level spacings for subsectors of the theory and show that for large N it is given by the Poisson distribution of integrable models, while at finite values it is the Wigner-Dyson distribution of the Gaussian orthogonal ensemble random matrix theory. We extend these results to two-loop order and to a one-parameter family of deformations. We further study the spectral rigidity for these models and show that it is also well described by random matrix theory. Finally we demonstrate that the finite-N eigenvectors possess properties of chaotic states.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission scipost_202104_00019v1 on 16 April 2021

## Reports on this Submission

### Report

This paper considers well chosen subsectors of ${\cal N}=4$ super Yang Miils theory and it's $\beta$ deformed cousins. In these subsectors the spectrum of the dilatation operator is computed at finite $N$. This entails using a basis of operators that have arbitrary multi-trace structure and taking into account finite $N$ trace relations. The large $N$ spectrum shows Poisson statistics (expected for an integrable system), while the finite $N$ spectrum follows Wigner-Dyson statistics, a sure sign of ergodicity and clear link to random matrix theory. The connection to random matrix theory is further strengthened by studies of spectral rigidity and chaotic eigenstates.

While the spectrum of the planar dilatation operator is well studied, this paper points out a facinating new direction, clearly extending the existing literature in a highly non-trivial way. Indeed, some of the most interesting questions, motivated by holography, are naturally posed for the finite $N$ theory. This paper has developed an interesting set of finite $N$ questions that can be addressed numerically. In addition, the discovered links to random marix theory are compeling and interesting, although not entirely unexpected.

I thoroughly enjoyed reading this paper and am convinced of its value. For this reason I have recommended that it is accepted. I would however encourage the authors to state the dimension of the space that the dilatation operator acts on, in Figure captions. This would be an extremely useful piece of information for others wishing to reproduce the published results.

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