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

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 gaugeequivalent to the extremal Didenko-Vasiliev solution [1]. The solutions activate all spins and can be characterized either via generalized electric and magnetic charges defined asymptotically in weakfield 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. suggest that higher-spin gravities are sub-sectors of particular tensionless limits of closed string field theories. Higher-spin gravities are thus tractable models for studying large-curvature effects in stringy completions of ordinary gravities.In particular, this opens a new window on holography in regimes where the boundary theories are weakly coupled and the bulk theories contain higher-spin gravities (and that are hence complementary to the more widely studied dual pairs involving strongly-coupled boundary theories and string theories with low-energy effective gravity descriptions on the bulk side). In this regime, the massless higherspin fields correspond to the bilinears in free fields on the boundary, which is a manifestation of the Vasiliev's equations [2,3] (see [4,5,6] for reviews) provide a fully nonlinear framework for higherspin gravities. They encode the classical dynamics of a highly complicated system -in which infinitely many fields of all spins are coupled through higher-derivative interaction vertices -into a combination 4 of zero-curvature constraints, for suitable master fields with simple higher-spin gauge transformations, and algebraic constraints, that actually describe a deformed ⋆-product algebra. This elegant description is achieved within the unfolded formulation of dynamics [14,15,4,5], which is a generalization based on differential algebras of the covariant Hamiltonian formulation of dynamics. The resulting unfolded systems consist of finitely many fundamental differential forms, which are the aforementioned master fields, living on extensions of spacetime referred to as correspondence spaces. Locally, these are products T * X × T , where X contains spacetime, and T is a non-commutative twistor space in the case of four-dimensional spacetime. The unexpected simplicity of the equations resides in that all spacetime component fields required for the unfolded formulation are packed away into the master fields in such a way that contractions of the coordinates of T , controlled by the deformed ⋆-p...

New classes of bi-axially symmetric solutions to four-dimensional Vasiliev higher spin gravity

“…In this paper, we shall construct new perturbatively defined solution spaces to Vasiliev's equations in Weyl order, by taking into account classes of functions that resemble closely those used in [26]. We shall then demonstrate explicitly that they can be mapped to Vasiliev's normal order, at least at the linearized level, thus providing further evidence in favour of the covariant Hamiltonian approach outlined above.…”

“…To this end, one starts from Hamilton's principle applied to a covariant Hamiltonian action formulated using Weyl order on a noncommutative manifold whose boundary is given by the direct product of spacetime and twistor space [24,25]; the Weyl order is required for the noncommutative version of the Stokes' theorem to hold and for the imposition of boundary conditions. The resulting variational principle yields Vasiliev's equations in Weyl order, that can be mapped back to Vasiliev's normal order for special classes of initial data in twistor space following the perturbative scheme set up in [26,27]. The resulting form of the higher spin amplitudes [28][29][30] is closely related to first-quantized topological open…”

“…To this end, we recall that at the linearized level, the fluctuations in the master fields that are asymptotic to anti-de Sitter spacetime form various representation spaces of the anti-de Sitter isometry algebra, including lowest-weight spaces as well as spaces associated to linearized solitons [33] and generalized Petrov type-D solutions [26,33,34]. Nonlinear completions of various Type-D solution spaces were constructed in [26,33,34]; for a review, see [35].…”

“…Nonlinear completions of various Type-D solution spaces were constructed in [26,33,34]; for a review, see [35]. Of direct relevance for the work in this paper is the subspace that contains the the black-hole-like solutions, 3 including spherically symmetric solutions.…”

“…In these solutions, each individual Fronsdal field has a point-like source at the origin, showing up as a divergence in its Weyl tensor. However, upon packing all curvatures into a master zero-form, one obtains the symbol of a quantum-mechanical operator that approaches a delta function distribution at the origin [26], which defines a smooth state as seen via classical observables given by zero-form charges [29,30,37]. In this sense, the black-hole-like Type-D solutions to Vasiliev's theory are source free at the origin.…”