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

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).