The next 16 higher spin currents and three-point functions in the large mathcal{N}=4 holography

Abstract

By using the known operator product expansions (OPEs) between the lowest 16 higher spin currents of spins (1, frac{3}{2}, frac{3}{2}, frac{3}{2}, frac{3}{2}, 2,2,2,2,2,2, frac{5}{2}, frac{5}{2}, frac{5}{2}, frac{5}{2}, 3) in an extension of the large mathcal{N}=4 linear superconformal algebra, one determines the OPEs between the lowest 16 higher spin currents in an extension of the large mathcal{N}=4 nonlinear superconformal algebra for generic N and k. The Wolf space coset contains the group G =mathrm{SU}(N+2) and the affine Kac–Moody spin 1 current has the level k. The next 16 higher spin currents 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) arise in the above OPEs. The most general lowest higher spin 2 current in this multiplet can be determined in terms of affine Kac–Moody spin frac{1}{2}, 1 currents. By careful analysis of the zero mode (higher spin) eigenvalue equations, the three-point functions of bosonic higher spin 2, 3, 4 currents with two scalars are obtained for finite N and k. Furthermore, we also analyze the three-point functions of bosonic higher spin 2, 3, 4 currents in the extension of the large mathcal{N}=4 linear superconformal algebra. It turns out that the three-point functions of higher spin 2, 3 currents in the two cases are equal to each other at finite N and k. Under the large (N, k) ’t Hooft limit, the two descriptions for the three-point functions of higher spin 4 current coincide with each other. The higher spin extension of SO(4) Knizhnik Bershadsky algebra is described.