The Next $16$ Higher Spin Currents and Three-Point Functions in the Large ${\cal N}=4$ 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 ${\cal N}=4$ linear superconformal algebra, one determines the OPEs between the lowest $16$ higher spin currents in an extension of the large ${\cal N}=4$ nonlinear superconformal algebra for generic $N$ and $k$. The Wolf space coset contains the group $G =SU(N+2)$ 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 ${\cal N}=4$ 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.