On The Uniqueness of Minimal Coupling in Higher-Spin Gauge Theory
Nicolas Boulanger, Serge Leclercq, Per Sundell
TL;DR
This work analyzes the uniqueness of minimal (and non-minimal) couplings of higher-spin fields to gravity, using BRST–BV cohomology to classify all consistent cubic vertices in Minkowski and AdS backgrounds. It shows that in flat space the familiar two-derivative minimal couplings are inconsistent for spins higher than two, but there exist non-abelian vertices with higher derivatives; in AdS, Fradkin–Vasiliev’s construction yields a finite tower of higher-derivative terms that cancel anomalies, culminating in a unique top vertex whose flat-space limit reproduces the known FV 2-3-3 vertex. The paper extends the uniqueness analysis to spin 4, provides a full cohomological account for 1-s-s and 2-s-s sectors, and clarifies the non-universal nature of the $\Lambda\to 0$ limit, linking the results to unfolded Vasiliev equations and potential microscopic completions. Collectively, these results yield a comprehensive covariant classification of manifestly covariant cubic couplings in Minkowski space and illuminate the structure of higher-spin interactions in curved backgrounds. The findings advance the understanding of higher-spin gauge theory’s interaction structure and underscore the role of background curvature and unfolded formalisms in achieving consistent, unique couplings.
Abstract
We address the uniqueness of the minimal couplings between higher-spin fields and gravity. These couplings are cubic vertices built from gauge non-invariant connections that induce non-abelian deformations of the gauge algebra. We show that Fradkin-Vasiliev's cubic 2-s-s vertex, which contains up to 2s-2 derivatives dressed by a cosmological constant $Λ$, has a limit where: {(i)} $Λ\to 0$; {(ii)} the spin-2 Weyl tensor scales {\emph{non-uniformly}} with s; and {(iii)} all lower-derivative couplings are scaled away. For s=3 the limit yields the unique non-abelian spin 2-3-3 vertex found recently by two of the authors, thereby proving the \emph{uniqueness} of the corresponding FV vertex. We extend the analysis to s=4 and a class of spin 1-s-s vertices. The non-universality of the flat limit high-lightens not only the problematic aspects of higher-spin interactions with $Λ=0$ but also the strongly coupled nature of the derivative expansion of the fully nonlinear higher-spin field equations with $Ł\neq 0$, wherein the standard minimal couplings mediated via the Lorentz connection are \emph{subleading} at energy scales $\sqrt{|Λ|}<< E<< M_{\rm p}$. Finally, combining our results with those obtained by Metsaev, we give the complete list of \emph{all} the manifestly covariant cubic couplings of the form 1-s-s and 2-s-s, in Minkowski background.