Geometry and Observables in Vasiliev's Higher Spin Gravity

TL;DR

Sezgin and Sundell provide a global geometric formulation of the $4$-dimensional minimal bosonic Vasiliev theory by identifying structure groups, soldering one-forms, and intrinsic observables, thereby connecting the local unfolded equations to globally defined moduli. They show that in the unbroken phase decorated Wilson loops collapse to on-shell zero-form charges, while in a geometrical metric phase one obtains generalized metrics, minimal-area observables, and abelian $p$-form charges via a soldering mechanism that introduces tensorial coordinates. They also develop off-shell deformations of the bulk action within a duality-extended framework, yielding topological vertex operators that reproduce the zero-form charges and may generate boundary correlators. Overall, the work links the Vasiliev equations to globally defined geometric data and suggests new avenues for holographic interpretation, with open questions about matching standard CFT correlators and exploring alternative structure algebras.

Abstract

We provide global formulations of Vasiliev's four-dimensional minimal bosonic higher spin gravities by identifying structure groups, soldering one-forms and classical observables. In the unbroken phase, we examine how decorated Wilson loops collapse to zero-form charges and exploit them to enlarge the Vasiliev system with new interactions. We propose a metric phase whose characteristic observables are minimal areas of higher spin metrics and on shell closed abelian forms of positive even degrees. We show that the four-form is an on shell deformation of the generalized Hamiltonian action recently proposed by Boulanger and one of the authors. In the metric phase, we also introduce tensorial coset coordinates and demonstrate how single derivatives with respect to coordinates of higher ranks factorize into multiple derivatives with respect to coordinates of lower ranks.