The $\mathcal{N}=4$ Higher Spin Algebra for Generic $\mu$ Parameter

Abstract

The $\mathcal{N}=4$ higher spin generators for general superspin $s$ in terms of oscillators in the matrix generalization of $Ad{S}_3$ Vasiliev higher spin theory at nonzero $\mu$ (which is equivalent to the 't Hooft-like coupling constant $\lambda$) were found previously. In this paper, by computing the (anti)commutators between these $\mathcal{N}=4$ higher spin generators for low spins $s_1$ and $s_2$ ($s_1+s_2\le 11$) explicitly, we determine the complete $\mathcal{N}=4$ higher spin algebra for generic $\mu$. The three kinds of structure constants contain the linear combination of two different generalized hypergeometric functions. These structure constants remain the same under the transformation $\mu \leftrightarrow (1-\mu )$ up to signs. We have checked that the above $\mathcal{N}=4$ higher spin algebra contains the $\mathcal{N}=2$ higher spin algebra, as a subalgebra, found by Fradkin and Linetsky some time ago.

arXiv comment67 pages. The last statement of the subsection 2.1, the footnotes 4 and 6 and the bounds of the summation is are added. The equations (2.5),(2.6) and (4.15) are improved. The original section 5 is moved to Appendix F with the addition of the first paragraph. To appear in JHEP

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• Toward A Supersymmetric $w_{1+\infty }$ Symmetry in the Celestial Conformal Field Theory
• The $\mathcal{N}=2$ Supersymmetric $w_{1+\infty }$ Symmetry in the Two-Dimensional SYK Models
• The Structure of the $\mathcal{N}=4$ Supersymmetric Linear $W_{\infty }[\lambda ]$ Algebra
• Adding Complex Fermions to the Grassmannian-like Coset Model
• The Grassmannian-like Coset Model and the Higher Spin Currents