On Nonlinear Higher Spin Curvature

TL;DR

This work constructs the first nonlinear term of the higher spin curvature that remains covariant under gauge transformations deformed linearly in the field, working within a metric-like framework and without Fronsdal constraints. The authors derive a second-order curvature form $R = d\Gamma + \Gamma\star\Gamma$ built from squares of the linear connection $\Gamma^{(s-1)}$, present explicit spin-2 and spin-3 constructions, and generalize to arbitrary spin-$s$ using an auxiliary-vector formalism. The key result is a universal expression for the second-order curvature $R_{(2)}$ as a quadratic in the generalized Christoffel symbols, together with a covariant Noether relation $\delta_{(1)} R_{(1)} + \delta_{(0)} R_{(2)} = \mathcal{L}_{\varepsilon} R_{(1)}$ that ensures gauge covariance. The analysis clarifies the geometric structure of higher-spin self-interactions, highlights the role of internal indices for odd spins, and paves the way for AdS extensions and connections to Vasiliev-type constructions.

Abstract

We present the first nonlinear term of the higher spin curvature which is covariant with respect to deformed gauge transformations that are linear in the field. We consider in detail the case of spin 3 after presenting spin 2 as an example, and then construct the general spin s quadratic term of the deWit-Freedman curvature.