The Coset Spin-4 Casimir Operator and Its Three-Point Functions with Scalars

TL;DR

This paper constructs a specific spin-4 Casimir operator within a coset model, determines its coefficients through algebraic constraints, and computes its three-point functions with scalars across all 't Hooft couplings, linking to higher spin gravity duals.

Contribution

It explicitly constructs the coset spin-4 Casimir operator and calculates its three-point functions with scalars for all 't Hooft couplings, extending previous results.

Findings

Explicit form of the spin-4 Casimir operator in the coset model.

Complete determination of coefficients using OPE constraints.

Three-point functions match dual higher spin gravity results.

Abstract

We find the GKO coset construction of the dimension 4 Casimir operator that contains the quartic WZW currents contracted with completely symmetric SU(N) invariant tensors of ranks 4, 3, and 2. The requirements, that the operator product expansion with the diagonal current is regular and it should be primary under the coset Virasoro generator of dimension 2, fix all the coefficients in spin-4 current, up to two unknown coefficients. The operator product expansion of coset primary spin-3 field with itself fixes them completely. We compute the three-point functions with scalars for all values of the 't Hooft coupling in the large N limit. At fixed 't Hooft coupling, these three-point functions are dual to that found by Chang and Yin recently in the undeformed AdS_3 bulk theory (higher spin gravity with matter).