Adding Complex Fermions to the Grassmannian-like Coset Model

In the ${\cal N}=2$ supersymmetric coset model, $\frac{SU(N+M)_k \times SO(2 N M)_1}{ SU(N)_{k+M} \times U(1)_{ N M (N+M)(k+N+M)}}$, we construct the $SU(M)$ nonsinglet ${\cal N}=2$ multiplet of spins $(1, \frac{3}{2}, \frac{3}{2}, 2)$ in terms of coset fields. The next $SU(M)$ singlet and nonsinglet ${\cal N}=2$ multiplets of spins $(2, \frac{5}{2}, \frac{5}{2}, 3)$ are determined by applying the ${\cal N}=2$ supersymmetry currents of spin $\frac{3}{2}$ 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 ${\cal N}=2$ superconformal algebra and above three kinds of ${\cal N}=2$ multiplets. These currents in two dimensions play the role of the asymptotic symmetry, as the generators of ${\cal N}=2$ "rectangular $W$-algebra", of the $M \times M$ matrix generalization of ${\cal N}=2$ $AdS_3$ 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 ${\cal N}=2$ multiplet of spins $(1, \frac{3}{2}, \frac{3}{2}, 2)$ and itself are analyzed in detail. The complete OPE between the lowest component of the $SU(M)$ singlet ${\cal N}=2$ multiplet of spins $(2, \frac{5}{2}, \frac{5}{2}, 3)$ and itself is described. In particular, when $M=2$, it is known that the above ${\cal N}=2$ supersymmetric coset model provides the realization of the extension of the large ${\cal N}=4$ nonlinear superconformal algebra. We determine the currents of the large ${\cal N}=4$ nonlinear superconformal algebra and the higher spin-$\frac{3}{2}, 2$ currents of the lowest ${\cal N}=4$ multiplet for generic $k$ and $N$ in terms of the coset fields.

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