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

Understand

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.