Diagonal free field matrix correlators, global symmetries and giant gravitons

TL;DR

This paper develops a systematic basis for diagonal multi-matrix correlators in theories with global symmetries, applying it to  SYM's SL(2) sector, revealing a Fock space structure linked to giant graviton excitations.

Contribution

It introduces a new basis for multi-matrix correlators incorporating global symmetries and connects it to giant graviton physics in  SYM.

Findings

Diagonal correlator basis for multi-matrix operators with global symmetry G.

Reduction of SL(2) Clebsch-Gordan problems to symmetric group computations.

Identification of a Fock space structure related to giant graviton excitations.

Abstract

We obtain a basis of diagonal free field multi-matrix 2-point correlators in a theory with global symmetry group G. The operators fall into irreducible representations of G. This applies for gauge group U(N) at finite N. For composites made of n fundamental fields, this is expressed in terms of Clebsch-Gordan coefficients for the decomposition of the n-fold tensor products of the fundamental field representation in terms of G \times S_n representations. We use this general construction in the case of the SL(2) sector of \cN=4 SYM. In this case, by using oscillator constructions, we reduce the computation of the relevant Clebsch-Gordans coupling infinite dimensional discrete series irreps of SL(2) to a problem in symmetric groups. Applying these constructions we write down gauge invariant operators with a Fock space structure similar to that arising in a large angular momentum limit of worldvolume excitations of giant gravitons. The Fock space structure emerges from Clebsch multiplicities of tensor products of symmetric group representations. We also give the action of the 1-loop dilatation operator of \cN=4 SYM on this basis of multi-matrix operators.