Higher spin Lagrangians from higher derivative diffeomorphisms

Abstract

The covariant description of massless particles of arbitrary spin typically employs symmetric tensors of rank $s$ and rests on a local symmetry carried by symmetric tensor parameters of rank $s-1$, suitably generalizing the $U(1)$ transformations of Maxwell's theory and the diffeomorphisms underlying general relativity. Here we show that Fronsdal's action is actually uniquely identified by the weaker requirement of invariance under higher-derivative gauge transformations driven by a vector parameter. This observation may hint to a tighter, if unexpected, connection between higher-spin symmetry and diffeomorphisms.

Figures (1)

• Figure 1: For spin $s$ and dimension $D$, the circled numbers indicate the degree to which the scalar reduction fails to motivate the unique Fronsdal solution, expressed as the number $N(s,D)$ of remaining free parameters beyond Fronsdal. This number is determined in \ref{['NCalc']} by the rank of the linear system that encodes scalar-reduced gauge invariance in \ref{['ScalarFunction']}. The rank at any point corresponds to the number of non-zero singular values $\sigma_1$, $\sigma_2$ and $\sigma_3$, represented here using a heatmap in arbitrary units, where $\sigma_2$ emerges only for $s\geq2$, and $\sigma_3$ emerges only for $s\geq 4$. The analytic continuation in $s$ and $D$ reveals how the plane is punctuated by singular value roots passing through $D=1$ at $s=3$, $s=4$ and $s=5$. These anomalous points have even larger $N(s,D)$, and so fail more spectacularly. No noteworthy structure exists for $s>7$ or $D>7$.