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Some of the operator product expansions (OPEs) between the lowest 16 higher spin currents of spins (1, 3 2 , 3 2 , 3 2 , 3 2 , 2, 2, 2, 2, 2, 2, 5 2 , 5 2 , 5 2 , 5 2 , 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 = 7 2 , 4, 9 2 , 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 theN = 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, 3 2 , 3 2 , 2), ( 3 2 , 2, 2, 5 2 ), ( 3 2 , 2, 2, 5 2 ), and (2, 5 2 , 5 2 , 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, 5 2 , 5 2 , 5 2 , 5 2 , 3, 3, 3, 3, 3, 3, 7 2 , 7 2 , 7 2 , 7 2 , 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.
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