Families of exact solutions to Vasiliev's 4D equations with spherical, cylindrical and biaxial symmetry

TL;DR

The paper constructs six infinite families of exact four-dimensional bosonic higher-spin gravity solutions of Vasiliev's equations, each admitting two commuting Killing vectors and organized into symmetry-enhanced sectors that include generalized Petrov type-D Weyl tensors. The authors combine the gauge-function method with a twistor-space separation of variables, using Di–Rac supersingleton projectors to parameterize internal moduli, yielding static spherically symmetric and cylindrically symmetric, as well as biaxially symmetric configurations with all spins activated. They show the full Weyl zero-forms follow Kerr–Schild-type linearization in these backgrounds, while the spacetime gauge fields remain regular in large regions; they also compute higher-spin invariant zero-form charges that remain finite and encode the configurations via deformation parameters. The work provides gauge-invariant probes of strong-field regions, clarifies the role of regular presentations, and highlights connections to extremal Didenko–Vasiliev solutions and generalized dualities between Type-A and Type-B models, with potential implications for holography and black-hole-like physics in higher-spin gravity.

Abstract

We provide Vasiliev's four-dimensional bosonic higher-spin gravities with six families of exact solutions admitting two commuting Killing vectors. Each family contains a subset of generalized Petrov Type-D solutions in which one of the two so(2) symmetries enhances to either so(3) or so(2,1). In particular, the spherically symmetric solutions are static and we expect one of them to be gauge-equivalent to the extremal Didenko-Vasiliev solution given in arXiv:0906.3898. The solutions activate all spins and can be characterized either via generalized electric and magnetic charges defined asymptotically in weak-field regions or via the values of fully higher-spin gauge-invariant observables given by on-shell closed zero-forms. The solutions are obtained by combining the gauge-function method with separation of variables in twistor space via expansion of the Weyl zero-form in Di-Rac supersingleton projectors times deformation parameters in a fashion that is suggestive of a generalized electromagnetic duality.