Uniformizing higher-spin equations

TL;DR

This work unifies higher-spin (HS) theories across dimensions by formulating them as a single, algebraically driven system defined by a dynamical-symmetry algebra $\mathfrak{g}$ and an embedding algebra $\mathfrak{A}$. The core equations, $dW+W\star W=0$, $dT_a+ [W,T_a]_{\star}=0$, and $[T_a,T_b]_{\star}^{\pm}-\mathcal{C}{}^{c}{}_{ab} T_c=0$, are gauge-invariant and understood via a map $\tau: \mathfrak{g} \to \mathfrak{A}$, with the last condition ensuring $\tau$ is a homomorphism; the framework encompasses Vasiliev’s HS theories in various dims, along with partially-massless, massive, conformal, and off-shell massless variants. A twisted star-product realizes the embedding algebra as a nontrivial tensor product of the HS algebra and the dynamical-symmetry algebra, and the resulting unfolded, linearized dynamics are controlled by $\mathbb{H}^0(\mathfrak{g},\mathfrak{A})$ and $\mathbb{H}^1(\mathfrak{g},\mathfrak{A})$, yielding a concise AKSZ formulation via Chevalley–Eilenberg differentials. The paper also analyzes the 3d and 4d Vasiliev systems through Serre-type osp(1|2) relations, the role of the Klein operator, and factorization procedures that produce on-shell HS dynamics from off-shell formalisms, clarifying when local degrees of freedom appear. Overall, the work provides a compact, algebraically transparent platform for HS physics, highlights the significance of the embedding algebra and twisted product, and points to future directions in invariant constructions and integrability of HS observables.

Abstract

Vasiliev's higher-spin theories in various dimensions are uniformly represented as a simple system of equations. These equations and their gauge invariances are based on two superalgebras and have a transparent algebraic meaning. For a given higher-spin theory these algebras can be inferred from the vacuum higher-spin symmetries. The proposed system of equations admits a concise AKSZ formulation. We also discuss novel higher-spin systems including partially-massless and massive fields in AdS, as well as conformal and massless off-shell fields.