Free geometric equations for higher spins

TL;DR

This paper introduces a geometric framework for higher-spin field equations that incorporates non-local terms, generalizing Maxwell and Einstein equations, and relates to generalized Riemann curvatures, offering new insights into higher-spin theories.

Contribution

It uncovers a geometric structure for higher-spin equations using non-local terms, connecting them to generalized Riemann curvatures and extending known formulations.

Findings

Unveiled a geometric structure for higher-spin equations.

Connected higher-spin equations to generalized Riemann curvatures.

Showed how local equations are recovered via gauge fixing.

Abstract

We show how allowing non-local terms in the field equations of symmetric tensors uncovers a neat geometry that naturally generalizes the Maxwell and Einstein cases. The end results can be related to multiple traces of the generalized Riemann curvatures R_{alpha_1 ... alpha_s; beta_1 > ... beta_s} introduced by de Wit and Freedman, divided by suitable powers of the D'Alembertian operator \Box. The conventional local equations can be recovered by a partial gauge fixing involving the trace of the gauge parameters Lambda_{alpha_1 ... alpha_{s-1}}, absent in the Fronsdal formulation. The same geometry underlies the fermionic equations, that, for all spins s+(1/2), can be linked via the operator (not hskip 1pt pr)/(\Box) to those of the spin-s bosons.