Spin-5 Casimir operator its three-point functions with two scalars

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⁾⊕ 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).