The large N't Hooft limit of Kazama-Suzuki model

We consider N = 2 Kazama-Suzuki model on CP ^N = SU(N+1)/SU(N)×U(1). It is known that the N = 2 current algebra for the supersymmetric WZW model, at level k, is a nonlinear algebra. The N = 2 W ₃ algebra corresponding to N = 2 was recovered from the generalized GKO coset construction previously. For N = 4, we construct one of the higher spin currents, in N = 2 W ₅ algebra, with spins (2, 5/2, 5/2,3). The self-coupling constant in the operator product expansion of this current and itself depends on N as well as k explicitly. We also observe a new higher spin primary current of spins (3,7/2,7/2,4). From the behaviors of N = 2,4 cases, we expect the operator product expansion of the lowest higher spin current and itself in N = 2 W _N+1 algebra. By taking the large (N, k) limit on the various operator product expansions in components, we reproduce, at the linear order, the corresponding operator product expansions in N = 2 classical W ^cl∞ algebra which is the asymptotic symmetry of the higher spin AdS ₃ supergravity found recently.