Adding Complex Fermions to the Grassmannian-like Coset Model

Abstract

In the N = 2 supersymmetric coset model, SU (N +M ) k ×SO(2N M ) 1 SU (N ) k+M ×U (1) NM (N+M )(k+N+M ) , we construct the SU(M) nonsinglet N = 2 multiplet of spins (1, 3 2 , 3 2 , 2) in terms of coset fields. The next SU(M) singlet and nonsinglet N = 2 multiplets of spins (2, 5 2 , 5 2 , 3) are determined by applying the N = 2 supersymmetry currents of spin 3 2 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 AdS 3 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, 3 2 , 3 2 , 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, 5 2 , 5 2 , 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-3 2 , 2 currents of the lowest N = 4 multiplet for generic k and N in terms of the coset fields. For the remaining higher spin-5 2 , 3 currents of the lowest N = 4 multiplet, some of the results are given.7 Towards the OPEs between the singlet multiplet of spins (2, 5 2 , 5 2 , 3) and itself 8 The extension of the large N = 4 nonlinear superconformal algebra for M = 2 8.