Spin-5 Casimir Operator and Its Three-Point Functions with Two Scalars

TL;DR

This paper constructs the spin-5 Casimir operator in a coset model, computes its three-point functions with scalars across all coupling regimes, and establishes their duality with higher spin gravity in AdS_3.

Contribution

It introduces the explicit form of the spin-5 Casimir operator and derives exact three-point functions for all N and k, connecting CFT and higher spin gravity.

Findings

Explicit spin-5 Casimir operator in coset model.

Exact three-point functions for all N and k.

Matching zero-mode eigenvalues with higher spin gravity.

Abstract

By calculating the second-order pole in the operator product expansion (OPE) between the spin-3 Casimir operator and the spin-4 Casimir operator known previously, the spin-5 Casimir operator is obtained in the coset model based on (A_{N-1}^{(1)} \oplus A_{N-1}^{(1)}, A_{N-1}^{(1)}) at level (k,1). This spin-5 Casimir operator consisted of the quintic, quartic (with one derivative) and cubic (with two derivatives) WZW currents contracted with SU(N) invariant tensors. The three-point functions with two scalars for all values of 't Hooft coupling in the large N limit were obtained by analyzing the zero-mode eigenvalue equations carefully. These three-point functions were dual to those in AdS_3 higher spin gravity theory with matter. Furthermore, the exact three-point functions that hold for any finite N and k are obtained. The zero mode eigenvalue equations for the spin-5 current in CFT coincided with those of the spin-5 field in asymptotic symmetry algebra of the higher spin theory on the AdS_3. This paper also describes the structure constant appearing in the spin-4 Casimir operator from the OPE between the spin-3 Casimir operator and itself for N=4, 5 in the more general coset minimal model with two arbitrary levels (k_1, k_2).