Surprisingly Simple Spectra

TL;DR

This paper investigates the large N limit of anomalous dimensions in ${ m extbf{N}=4}$ super Yang-Mills theory using restricted Schur polynomials, revealing an integrable structure akin to harmonic oscillators that extends beyond the planar approximation.

Contribution

It demonstrates that the one-loop dilatation operator for certain operators is equivalent to a lattice second derivative and is integrable, including non-planar contributions.

Findings

Eigenvalues are integer multiples of 8g_{YM}^2.

Spectrum matches that of harmonic oscillators.

Dilitation operator is integrable, extending known BPS sector results.

Abstract

The large N limit of the anomalous dimensions of operators in ${\cal N}=4$ super Yang-Mills theory described by restricted Schur polynomials, are studied. We focus on operators labeled by Young diagrams that have two columns (both long) so that the classical dimension of these operators is O(N). At large N these two column operators mix with each other but are decoupled from operators with $n\ne 2$ columns. The planar approximation does not capture the large N dynamics. For operators built with 2, 3 or 4 impurities the dilatation operator is explicitly evaluated. In all three cases, in a certain limit, the dilatation operator is a lattice version of a second derivative, with the lattice emerging from the Young diagram itself. The one loop dilatation operator is diagonalized numerically. All eigenvalues are an integer multiple of $8g_{YM}^2$ and there are interesting degeneracies in the spectrum. The spectrum we obtain for the one loop anomalous dimension operator is reproduced by a collection of harmonic oscillators. This equivalence to harmonic oscillators generalizes giant graviton results known for the BPS sector and further implies that the Hamiltonian defined by the one loop large $N$ dilatation operator is integrable. This is an example of an integrable dilatation operator, obtained by summing both planar and non-planar diagrams.