Nonplanar Integrability: Beyond the SU(2) Sector

TL;DR

This paper demonstrates that the one-loop dilatation operator for certain nonplanar operators in N=4 SYM is integrable, extending known results from the SU(2) sector to more complex Young diagram configurations.

Contribution

It generalizes the integrability of the dilatation operator beyond the SU(2) sector by analyzing operators labeled by Young diagrams with two long rows or columns.

Findings

The dilatation operator reduces to harmonic oscillators.

Explicit solutions involve symmetric Kravchuk and Hahn polynomials.

The results extend integrability to nonplanar, large dimension operators.

Abstract

We compute the one loop anomalous dimensions of restricted Schur polynomials with a classical dimension \Delta\sim O(N). The operators that we consider are labeled by Young diagrams with two long columns or two long rows. Simple analytic expressions for the action of the dilatation operator are found. The projection operators needed to define the restricted Schur polynomials are constructed by translating the problem into a spin chain language, generalizing earlier results obtained in the SU(2) sector of the theory. The diagonalization of the dilatation operator reduces to solving five term recursion relations. The recursion relations can be solved exactly in terms of products of symmetric Kravchuk polynomials with Hahn polynomials. This proves that the dilatation operator reduces to a decoupled set of harmonic oscillators and therefore it is integrable, extending a similar conclusion reached for the SU(2) sector of the theory.