Exact Multi-Restricted Schur Polynomial Correlators

TL;DR

This paper derives a product rule for restricted Schur polynomials, enabling exact computation of multi-point correlation functions in free field theory, revealing insights into operator interactions and energy-dependent coupling behaviors.

Contribution

It introduces a product rule for restricted Schur polynomials, allowing exact multi-point correlator calculations and providing a physical interpretation of their labels.

Findings

Exact two-point functions for operators with two matrices.

Three-point functions of restricted Schur polynomials computed exactly.

High-energy gravitons become strongly coupled, while certain operators remain weakly interacting.

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 $\chi_{R,(r_n,r_m)}(Z,X)$ is constructed from the half BPS ``partons'' $\chi_{r_n}(Z)$ and $\chi_{r_m}(X)$.