Some of the operator product expansions (OPEs) between the lowest 16 higher spin currents of spins (1, (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 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,

By computing the operator product expansions between the first two [Formula: see text] higher spin multiplets in the unitary coset model, the (anti-)commutators of higher spin currents are obtained under the large [Formula: see text] ’t Hooft-like limit. The free field realization with complex bosons and fermions is presented. The (anti-)commutators for generic spins [Formula: see text] and [Formula: see text] with manifest [Formula: see text] symmetry at vanishing ’t Hooft-like coupling constant are completely determined. The structure constants can be written in terms of the ones in the [Formula: see text] [Formula: see text] algebra found by Bergshoeff, Pope, Romans, Sezgin and Shen previously, in addition to the spin-dependent fractional coefficients and two [Formula: see text] invariant tensors. We also describe the [Formula: see text] higher spin generators, by using the above coset construction results, for general superspin [Formula: see text] in terms of oscillators in the matrix generalization of [Formula: see text] Vasiliev higher spin theory at nonzero ’t Hooft-like coupling constant. We obtain the [Formula: see text] higher spin algebra for low spins and present how to determine the structure constants, which depend on the higher spin algebra parameter, in general, for fixed spins [Formula: see text] and [Formula: see text].

By using the known operator product expansions (OPEs) between the lowest 16 higher spin currents of spins (1,

Some of the operator product expansions (OPEs) between the lowest SO(4) singlet higher spin-2 multiplet of spins (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 an extension of the large N = 4 (non)linear superconformal algebra were constructed in the N = 4 superconformal coset SO(N +4) SO(N)×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-7 2 multiplets, SO(4) singlet higher spin-4 multiplet and four SO(4) vector higher spin-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 N = 4 super OPE between the above lowest SO(4) singlet higher spin-2 multiplet in the N = 4 superspace.