Adding complex fermions to the Grassmannian-like coset model

In the N= 2 supersymmetric coset model, SU(N+M)k×SO(2NM)1SU(N)k+M×U(1)NM(N+M)(k+N+M), we construct the SU(M) nonsinglet N= 2 multiplet of spins (1,32,32,2) in terms of coset fields. The next SU(M) singlet and nonsinglet N= 2 multiplets of spins (2,52,52,3) are determined by applying the N= 2 supersymmetry currents of spin 32 to the bosonic singlet and nonsinglet currents of spin 3 in the bosonic coset model. We also obtain the operator product expansions (OPEs) between the currents of the N= 2 superconformal algebra and above three kinds of N= 2 multiplets. These currents in two dimensions play the role of the asymptotic symmetry, as the generators of N= 2 “rectangular W-algebra”, of the M× M matrix generalization of N= 2 AdS₃ higher spin theory in the bulk. The structure constants in the right hand sides of these OPEs are dependent on the three parameters k, N and M explicitly. Moreover, the OPEs between SU(M) nonsinglet N= 2 multiplet of spins (1,32,32,2) and itself are analyzed in detail. The complete OPE between the lowest component of the SU(M) singlet N= 2 multiplet of spins (2,52,52,3) and itself is described. In particular, when M= 2 , it is known that the above N= 2 supersymmetric coset model provides the realization of the extension of the large N= 4 nonlinear superconformal algebra. We determine the currents of the large N= 4 nonlinear superconformal algebra and the higher spin-32,2 currents of the lowest N= 4 multiplet for generic k and N in terms of the coset fields. For the remaining higher spin-52,3 currents of the lowest N= 4 multiplet, some of the results are given.