Free geometric equations for higher spins

TL;DR

The paper develops a non-local, gauge-invariant geometric framework for free higher-spin fields, showing that allowing non-local terms exposes a curvature-based structure generalizing Maxwell and Einstein equations. It builds iterative kinetic operators ${\cal F}^{(n)}$ and Einstein-like tensors ${\cal G}^{(n)}$, whose Bianchi identities lead to gauge-invariant field equations expressed in terms of traces of higher-spin curvatures ${\cal R}_{\alpha_1 \cdots \alpha_s;\beta_1 \cdots \beta_s}$ (and their $n$-fold traces). The geometric construction, following de Wit–Freedman, yields spin-$s$ equations ${1}/{\Box^{n-1}} {\cal R}^{[n]}_{;\mu_1 \cdots \mu_{2n}}=0$ (even) and ${1}/{\Box^{n}} \partial \cdot {\cal R}^{[n]}_{;\mu_1 \cdots \mu_{2n+1}}=0$ (odd), which reduce to the local Fang-Fronsdal form via partial gauge fixing, and it extends coherently to fermions, linking bosonic and fermionic higher spins through non-local operator relations. This unifies the treatment of all spins within a single geometric language and hints at deeper connections to string-field theory BRST structures.

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.