Lorentz Covariance of the $4d$ Nonlinear Higher-Spin Equations via BRST

TL;DR

The paper addresses the challenge of making local Lorentz covariance manifest in the nonlinear $4d$ higher-spin equations. It introduces a BRST extension tied to the Lorentz sector, which yields a Stueckelberg gauge that trades the original Lorentz connection for a new covariant $\\hat{\omega}$ and introduces curvature-like fields $U^{AB}$, preserving covariance at all perturbative orders. This framework is formulated to be compatible with both conventional and differential homotopy methods and extends to AdS$_3$/AdS$_4$ HS models with higher forms and Coxeter algebras. The approach provides a unified, efficient route to construct Lorentz-covariant HS dynamics and can facilitate analyses of extended HS systems beyond the original Vasiliev theory.

Abstract

We propose a BRST extension of the higher-spin gauge theory in $AdS_4$ with the BRST operator associated with the local Lorentz symmetry. Our construction supports manifest local Lorentz covariance and is applicable both to any homotopy scheme of the perturbative analysis including the recently proposed differential homotopy and to the variety of further extended higher-spin models in $AdS_3$ and $AdS_4$ with higher differential forms and Coxeter higher-spin algebras.