The operator product expansion between the 16 lowest higher spin currents in the N= 4 superspace

Some of the operator product expansions (OPEs) between the lowest 16 higher spin currents of spins (1,32,32,32,32,2,2,2,2,2,2,52,52,52,52,3) in an extension of the large N= 4 linear superconformal algebra were constructed in N= 4 superconformal coset SU(5)SU(3) theory previously. In this paper, by rewriting these OPEs in the N= 4 superspace developed by Schoutens (and other groups), the remaining undetermined OPEs in which the corresponding singular terms possess the composite fields with spins s=72,4,92,5 are completely determined. Furthermore, by introducing arbitrary coefficients in front of the composite fields on the right-hand sides of the above complete 136 OPEs, reexpressing them in the N= 2 superspace, and using the N= 2 OPEs Mathematica package by Krivonos and Thielemans, the complete structures of the above OPEs with fixed coefficient functions are obtained with the help of various Jacobi identities. We then obtain ten N= 2 super OPEs between the four N= 2 higher spin currents denoted by (1,32,32,2), (32,2,2,52), (32,2,2,52), and (2,52,52,3) (corresponding 136 OPEs in the component approach) in the N= 4 superconformal coset SU(N+2)SU(N) theory. Finally, we describe them as one single N= 4 super OPE between the above 16 higher spin currents in the N= 4 superspace. The fusion rule for this OPE contains the next 16 higher spin currents of spins of (2,52,52,52,52,3,3,3,3,3,3,72,72,72,72,4) in addition to the quadratic N= 4 lowest higher spin multiplet and the large N= 4 linear superconformal family of the identity operator. The various structure constants (fixed coefficient functions) appearing on the right-hand side of this OPE depend on N and the level k of the bosonic spin-1 affine Kac–Moody current. For convenience, the above 136 OPEs in the component approach for generic (N, k) with simplified notation are given.