The Operator Product Expansions in the ${\cal N}=4$ Orthogonal Wolf Space Coset Model

Some of the operator product expansions (OPEs) between the lowest $SO(4)$ singlet higher spin-$2$ multiplet of spins $(2, \frac{5}{2}, \frac{5}{2}, \frac{5}{2}, \frac{5}{2}, 3, 3, 3, 3, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, 4)$ in an extension of the large ${\cal N}=4$ (non)linear superconformal algebra were constructed in the ${\cal N}=4$ superconformal coset $\frac{SO(N+4)}{SO(N) \times SO(4)}$ theory with $N=4$ previously. In this paper, by rewriting the above OPEs with $N=5$, the remaining undetermined OPEs are completely determined. There exist additional $SO(4)$ singlet higher spin-$2$ multiplet, six $SO(4)$ adjoint higher spin-$3$ multiplets, four $SO(4)$ vector higher spin-$\frac{7}{2}$ multiplets, $SO(4)$ singlet higher spin-$4$ multiplet and four $SO(4)$ vector higher spin-$\frac{9}{2}$ multiplets in the right hand side of these OPEs. Furthermore, by introducing the arbitrary coefficients in front of the composite fields in the right hand sides of the above complete 136 OPEs, the complete structures of the above OPEs are obtained by using various Jacobi identities for generic $N$. Finally, we describe them as one single ${\cal N}=4$ super OPE between the above lowest $SO(4)$ singlet higher spin-$2$ multiplet in the ${\cal N}=4$ superspace.