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

TL;DR

This paper explicitly computes the complete ${ m N}=4$ higher spin algebra for generic parameter $$, revealing its structure constants and subalgebra relations, extending previous partial results in higher spin theory.

Contribution

It provides the full ${ m N}=4$ higher spin algebra for generic $$, including structure constants and subalgebra relations, based on explicit (anti)commutator calculations.

Findings

Complete ${ m N}=4$ higher spin algebra determined for generic $$.

Structure constants involve hypergeometric functions and are symmetric under $ o 1-$.

Contains the ${ m N}=2$ higher spin algebra as a subalgebra.

Abstract

The ${\cal N}=4$ higher spin generators for general superspin $s$ in terms of oscillators in the matrix generalization of $AdS_3$ 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 ${\cal N}=4$ higher spin generators for low spins $s_1$ and $s_2$ ($s_1+s_2 \leq 11$) explicitly, we determine the complete ${\cal N}=4$ 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 ${\cal N}=4$ higher spin algebra contains the ${\cal N}=2$ higher spin algebra, as a subalgebra, found by Fradkin and Linetsky some time ago.