Fermionic Construction in the Supersymmetric Coset Model

It is known previously that the operator product expansion (OPE) between the first ${\cal N}=3$ multiplet and itself contains the second ${\cal N}=3$ multiplet in the supersymmetric coset model. In this paper, by using their realizations in terms of various fermions, we compute the four kinds of OPEs between the first and the second ${\cal N}=3$ multiplets for fixed $N$ and $M$ where the group of the coset contains $SU(N+M)$. By supersymmetrizing the above OPEs in ${\cal N}=3$ superspace and using the various Jacobi identities between the currents, we determine the ${\cal N}=3$ supersymmetric OPE between the first and the second ${\cal N}=3$ multiplets completely. The right hand side of this OPE contains the various ${\cal N}=3$ multiplets: the $SO(3)$ singlet ${\cal N}=3$ multiplets of superspin-$\frac{3}{2},2,3,4$ and the $SO(3)$ triplet ${\cal N}=3$ multiplets of superspin-$\frac{5}{2},3,\frac{7}{2}$. The ${\cal N}=2$ superspace description and the decoupling of the spin-$\frac{1}{2}$ current of the ${\cal N}=3$ superconformal algebra are also described.