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 $G=SU(N+2)$ 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.

94 pages; The 43 and 44 pages added, the mathematica notebook files included and to appear epjc

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Cited by in corpus (9)

• The $\mathcal{N}=4$ Higher Spin Algebra for Generic $\mu$ Parameter
• The $\mathcal{N}=4$ Coset Model and the Higher Spin Algebra
• Three point functions in higher spin AdS_3 holography with 1/N corrections
• Adding Complex Fermions to the Grassmannian-like Coset Model
• The Operator Product Expansions in the $\mathcal{N}=4$ Orthogonal Wolf Space Coset Model
• The Grassmannian-like Coset Model and the Higher Spin Currents
• Higher Spin Currents with Manifest $SO(4)$ Symmetry in the Large $\mathcal{N}=4$ Holography
• Wolf Space Coset Spectrum in the Large $\mathcal{N}=4$ Holography
• The Small $\mathcal{N}=4$ Superconformal ${\mathcal{W}}_{\infty }$ Algebra