# Non-planar anomalous dimensions in super Yang-Mills theories.

### Anne Speiring

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## Summary

Conformal supersymmetric Yang–Mills theories play an important role in the gauge-gravity correspondence and, despite being highly non-physical, have been a driving force for many new approaches in more realistic theories like QCD and gravity. An important class of objects in conformal field theories is the spectrum of scaling dimensions of local operators, specifically their non-trivial coupling-dependent parts, the anomalous dimensions. The discovery of integrability in planar maximally supersymmetric Yang–Mills theory led to considerable advances in the computation of its anomalous-dimension spectrum. Less is known at the non-planar level where the theory is assumed to be non-integrable. In this thesis we consider non-planar anomalous dimensions in conformal supersymmetric Yang–Mills theories with gauge group SU(N) and approach them by a number of means.

First, we use an on-shell form-factor approach based on the intimate connection between the dilatation operator and scattering amplitudes. The former gives rise to operator mixing and its diagonalisation gives the operators’ anomalous dimensions. The latter are basic observables in any quantum field theory, describing its interactions and linking theoretical developments to experimental investigations. A lot of progress has been made in recent years in the study of scattering amplitudes due to the advent of on-shell methods which circumvent many difficulties of more traditional approaches, and we use some of these here to extract the dilatation operator in certain sectors of the theories considered. In particular, we study a set of dimension-4 operators in N = 4 supersymmetric Yang–Mills theory that is relevant for the mixing of the theory’s on-shell Lagrangian, and compute the spectrum of non-planar anomalous dimensions in this sector. Furthermore, we extract the general form of the one-loop dilatation operator in the sector of purely scalar operators in the β-deformed version of this theory.

In the planar limit of the theories considered in this thesis, the dilatation operator maps to a spin-chain Hamiltonian that can be diagonalised by integrability techniques, in particular a suitable Bethe ansatz. In this mapping the spectrum of anomalous dimensions becomes the energy spectrum of the corresponding spin chains. When going away from the planar limit, integrability is lost, but we can compute non-planar corrections to the planar spectrum using Rayleigh–Schro ̈dinger perturbation theory. Using the basis of Bethe states, we compute matrix elements of the deformed and undeformed dilatation operator relevant in this approach. We find compact expressions in terms of off-shell scalar products and hexagon-like functions. We then use nondegenerate 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.

Finally, we numerically study statistical properties of large sets of anomalous dimensions which we obtain from a direct diagonalisation of the dilatation operators discussed in this thesis. Specifically, we analyse the distribution of level spacings in these spectra and find universal features: in the planar limit it follows the Poisson distribution characteristic of integrable systems, and at finite values of N it transitions to the Wigner–Dyson distribution of the Gaussian orthogonal ensemble of random matrix theory. This provides numerical evidence that perturbative non-planar anomalous-dimension spectra are quantum-chaotic, which is further supported by similar findings in the spectral rigidity measuring long-range interactions in the spectra. We also demonstrate that the finite-N eigenvectors possess properties of chaotic states.

Funded in part by the European Union’s Horizon 2020 Framework Program under the Marie Skłodowska-Curie grant agreement No. 764850 (SAGEX).