Nonplanar Integrability

TL;DR

This paper demonstrates that the dilatation operator acting on certain restricted Schur polynomial operators in N=4 SYM is integrable, reducible to harmonic oscillators, through explicit construction of projectors and solving recursion relations.

Contribution

It provides an explicit construction of projectors for restricted Schur polynomials and proves integrability by solving recursion relations exactly.

Findings

Dilatation operator reduces to harmonic oscillators

Explicit projectors constructed via angular momentum addition

Recursion relations solved with Kravchuk polynomials

Abstract

In this article we study operators with a dimension $\Delta\sim O(N)$ and show that simple analytic expressions for the action of the dilatation operator can be found. The operators we consider are restricted Schur polynomials. There are two distinct classes of operators that we consider: operators labeled by Young diagrams with two long columns or two long rows. The main complication in working with restricted Schur polynomials is in building a projector from a given $S_{n+m}$ irreducible representation to an $S_n\times S_m$ irreducible representation (both specified by the labels of the restricted Schur polynomial). We give an explicit construction of these projectors by reducing it to the simple problem of addition of angular momentum in ordinary non-relativistic quantum mechanics. The diagonalizationof the dilatation operator reduces to solving three term recursion relations. The fact that the recursion relations have only three terms is a direct consequence of the weak mixing at one loop of the restricted Schur polynomials. The recursion relations can be solved exactly in terms of symmetric Kravchuk polynomials or in terms of Clebsch-Gordan coefficients. This proves that the dilatation operator reduces to a decoupled set of harmonic oscillators and therefore it is integrable.