Non-abelian cubic vertices for higher-spin fields in anti-de Sitter space

TL;DR

Problem: classify non-abelian cubic vertices for totally symmetric higher-spin fields in $AdS_d$ ($d>4$). Approach: Fradkin-Vasiliev procedure in the frame-like formalism, deforming curvature two-forms to generate cubic couplings. Findings: the number of independent non-abelian vertices for spins $(s,s',s'')$ is given by the tensor-product multiplicity; a flat-space cohomological perspective relates these deformations; the work discusses the uniqueness of Vasiliev's simplest HS algebra via (non)associativity and extends to gravitational interactions for mixed-symmetry and partially-massless fields, with nonabelian interactions also among one-form connections. Significance: clarifies the algebraic structure and counting of higher-spin interactions in $AdS_d$, linking to flat-space classifications and HS algebra uniqueness.

Abstract

We use the Fradkin-Vasiliev procedure to construct the full set of non-abelian cubic vertices for totally symmetric higher spin gauge fields in anti-de Sitter space. The number of such vertices is given by a certain tensor-product multiplicity. We discuss the one-to-one relation between our result and the list of non-abelian gauge deformations in flat space obtained elsewhere via the cohomological approach. We comment about the uniqueness of Vasiliev's simplest higher-spin algebra in relation with the (non)associativity properties of the gauge algebras that we classified. The gravitational interactions for (partially)-massless (mixed)-symmetry fields are also discussed. We also argue that those mixed-symmetry and/or partially-massless fields that are described by one-form connections within the frame-like approach can have nonabelian interactions among themselves and again the number of nonabelian vertices should be given by tensor product multiplicities.