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Higher Spin Currents in Wolf Space: Part III

The large N=4 linear superconformal algebra (generated by four spin-1/2 currents, seven spin-1 currents, four spin-3/2 currents and one spin-2 current) found by Sevrin, Troost and Van Proeyen (and other groups) was realized in the N=4 superconformal coset SU(5)/SU(3) theory previously. The lowest 16 higher spin currents of spins (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) are obtained by starting with the operator product expansions (OPEs) between the four spin-3/2 currents from the above large N=4 linear superconformal algebra and the lowest higher spin-1 current which is the same as the one in the Wolf space coset SU(5)/[SU(3) x SU(2) x U(1)] theory. These OPEs determine the four higher spin-3/2 currents and the next six higher spin-2 currents are obtained from the OPEs between the above four spin-3/2 currents associated with the N=4 supersymmetry and these four higher spin-3/2 currents. The four higher spin-5/2 currents can be determined by calculating the OPEs between the above four spin-3/2 currents and the higher spin-2 currents. Similarly, the higher spin-3 current is obtained from the OPEs between the four spin-3/2 currents and the higher spin-5/2 currents. The explicit relations between the above 16 higher spin currents and the corresponding 16 higher spin currents which were found in the extension of large N=4 nonlinear superconformal algebra previously are given. By examining the OPEs between the 16 currents from the large N=4 linear superconformal algebra and the 16 higher spin currents, the match with the findings of Beccaria, Candu and Gaberdiel is also given. The next 16 higher spin currents of spins (2, 5/2, 5/2, 3), (5/2, 3, 3, 7/2), (5/2, 3, 3, 7/2) and (3, 7/2, 7/2, 4) occur from the OPEs between the above lowest 16 higher spin currents.