Nonplanar integrability

Abstract

In this article we study operators with a dimension ∆ ∼ 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 × 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 diagonalization of 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.

Anomalous dimensions of heavy operators from magnon energies

“…We have a clearer understanding of the non-planar integrability discovered in [29][30][31][32][33][34]. The magnons in these systems remain separated and hence free, so they are actually noninteracting.…”

“…The operators introduced in [24,27] are the restricted Schur polynomials. Further, significant progress was made in understanding the spectrum of anomalous dimensions of these operators in the studies [25,26,[29][30][31][32][33][34]. Extensions which consider orthogonal and symplectic gauge groups and other new ideas, have also been achieved [35][36][37][38][39][40].…”

“…In this case one must consider operators that treat these "open strings" symmetrically, leading to the operators constructed in [27]. In a specific limit, the action of the dilatation operator factors into an action on the Zs and an action on the Y s [31,32]. The action on the Y s can be diagonalized by Fourier transforming to a double coset which describes how the magnons are attached to the giant gravitons [32,33].…”

“…These results are useful since they provide data against which the distant corners approximation could be compared. Further, we have demonstrated that the action of the dilatation operator reduces to a set of decoupled harmonic oscillators in [31][32][33][34]. However, to obtain this result we needed to…”

“…It is worth discussing these details and explaining why we do indeed obtain the correct large N limit. This point is not made explicitly in [31][32][33][34].…”