The $$ \mathcal{N} $$ = 4 higher spin algebra for generic μ parameter

The $$ \mathcal{N} $$ N = 4 higher spin generators for general superspin s in terms of oscillators in the matrix generalization of AdS3 Vasiliev higher spin theory at nonzero μ (which is equivalent to the ’t Hooft-like coupling constant λ) were found previously. In this paper, by computing the (anti)commutators between these $$ \mathcal{N} $$ N = 4 higher spin generators for low spins s1 and s2 (s1 + s2 ≤ 11) explicitly, we determine the complete $$ \mathcal{N} $$ N = 4 higher spin algebra for generic μ. 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 μ ↔ (1 − μ) up to signs. We have checked that the above $$ \mathcal{N} $$ N = 4 higher spin algebra contains the $$ \mathcal{N} $$ N = 2 higher spin algebra, as a subalgebra, found by Fradkin and Linetsky some time ago.

Spin-locality of η2 and $$ {\overline{\eta}}^2 $$ quartic higher-spin vertices

Higher-spin theory contains a complex coupling parameter η. Different higher-spin vertices are associated with different powers of η and its complex conjugate $$ \overline{\eta} $$ η ¯ . Using Z-dominance Lemma of [1], that controls spin-locality of the higher-spin equations, we show that the third-order contribution to the zero-form B(Z; Y; K) admits a Z-dominated form that leads to spin-local vertices in the η2 and $$ {\overline{\eta}}^2 $$ η ¯ 2 sectors of the higher-spin equations. These vertices include, in particular, the η2 and $$ {\overline{\eta}}^2 $$ η ¯ 2 parts of the ϕ4 scalar field vertex.