Exact multi-restricted Schur polynomial correlators

Abstract

We derive a product rule satisfied by restricted Schur polynomials. We focus mostly on the case that the restricted Schur polynomial is built using two matrices, although our analysis easily extends to more than two matrices. This product rule allows us to compute exact multi-point correlation functions of restricted Schur polynomials, in the free field theory limit. As an example of the use of our formulas, we compute two point functions of certain single trace operators built using two matrices and three point functions of certain restricted Schur polynomials, exactly, in the free field theory limit. Our results suggest that gravitons become strongly coupled at sufficiently high energy, while the restricted Schur polynomials for totally antisymmetric representations remain weakly interacting at these energies. This is in perfect accord with the half-BPS (single matrix) results of hep-th/0512312. Finally, by studying the interaction of two restricted Schur polynomials we suggest a physical interpretation for the labels of the restricted Schur polynomial: the composite operator χ R,(rn,rm) (Z, X) is constructed from the half BPS "partons" χ rn (Z) and χ rm (X).

Anomalous dimensions of heavy operators from magnon energies

“…Inspired by the Gauss Law which will arise in the world volume description of the giant graviton branes, the authors of [23] suggested operators in the gauge theory that are dual to excited giant graviton brane states. This inspired idea was pursued both in the case that the open strings are described by an open string word [24][25][26] and in the case of minimal open strings, with each open string represented by a single magnon [27,28]. The operators introduced in [24,27] are the restricted Schur polynomials.…”

“…each open string is a single magnon. In this case one must use the correlators computed in [27,28] as opposed to the correlators computed in [24]. The case with distinguishable open strings is much simpler since when the correlators are computed, only contractions between corresponding open strings contribute; when the open strings are identical, it is possible to contract any two of them.…”

Restricted Schur polynomials and finiteNcounting

“…For a nice recent review see Ramgoolam [24]. Finally a product rule for restricted Schur polynomials has been obtained in [25]. Together with the two-point function derived in [4] this allows us to derive higher point functions.…”

Refined counting of necklaces in one-loop N = 4 $$ \mathcal{N}=4 $$ SYM

“…New bases of gauge-invariant operators have been discovered, which diagonalize the tree-level two-point functions at finite N c [5][6][7][8]; see [9] for a review.…”

“…The convention of [24] is used here for efficient evaluation. 7 This formula is elaborated further as the highest weight generating function [25].…”