Abstract
We construct linearized solutions to Vasiliev’s four-dimensional higher spin gravity on warped AdS3 ×ξS1 which is an Sp(2) × U(1) invariant non-rotating BTZ-like black hole with ℝ2 × T2 topology. The background can be obtained from AdS4 by means of identifications along a Killing boost K in the region where ξ2 ≡ K2 ≥ 0, or, equivalently, by gluing together two Bañados-Gomberoff-Martinez eternal black holes along their past and future space-like singularities (where ξ vanishes) as to create a periodic (non-Killing) time. The fluctuations are constructed from gauge functions and initial data obtained by quantizing inverted harmonic oscillators providing an oscillator realization of K and of a commuting Killing boost tilde{K} . The resulting solution space has two main branches in which K star commutes and anti-commutes, respectively, to Vasiliev’s twisted-central closed two-form J. Each branch decomposes further into two subsectors generated from ground states with zero momentum on S1. We examine the subsector in which K anti-commutes to J and the ground state is mathrm{U}{(1)}_Ktimes mathrm{U}{(1)}_{tilde{K}} -invariant of which U(1)K is broken by momenta on S1 and mathrm{U}{(1)}_{tilde{K}} by quasi-normal modes. We show that a set of mathrm{U}{(1)}_{tilde{K}} -invariant modes (with n units of S1 momenta) are singularity-free as master fields living on a total bundle space, although the individual Fronsdal fields have membrane-like singularities at {tilde{K}}^2=1 . We interpret our findings as an example where Vasiliev’s theory completes singular classical Lorentzian geometries into smooth higher spin geometries.
Highlights
1.1 Higher spin resolution of gravitational singularitiesAn interesting problem in gravity is whether classical spacetime singularities can be resolved by switching on higher spin gauge fields
To concord with basic properties of the holographic correspondence between generally covariant theories with anti-de Sitter vacua and conformal field theories in the context of higher spin theory [1,2,3,4], we shall a) presume a higher spin symmetry breaking mechanism whereby weakly coupled gauge fields with spins greater than two acquire masses so as to leave a spectrum with massless subsector corresponding to matter-coupled gravity; and b) construct exact solutions to unbroken higher spin gravities that describe smooth higher spin geometries containing asymptotically locally anti-de Sitter (ALAdS) regions where the full theory can be approximated by Fronsdal fields
We envisage weakly coupled asymptotic regions described by an effective gravity theory glued to strongly coupled core regions described by an unbroken higher spin gravity; that is, we trust the latter when its curvatures are large, and the former when its curvatures are small
Summary
An interesting problem in gravity is whether classical spacetime singularities can be resolved by switching on higher spin gauge fields. Higher spin gravities contain infinite towers of massless fields at weak coupling that one may argue become massive due to quantum effects, associated to screened charges in weakly coupled asymptotic regions, while supporting moduli spaces of classical solutions interpolating between asymptotic regions and strongly coupled core regions with nontrivial topology. Vasiliev’s theory has been conjectured [2,3,4, 21] to undergo dynamical symmetry breaking due to mixing between (massless) one-particle states and multi-particle Goldstone modes in the presence of special boundary conditions in anti-de Sitter spacetime As this mechanism does not require any coupling to additional fields, the theory, possibly including Yang-Mills-like gauge fields and fermions [22, 23], provides a relatively minimalistic framework for studying singularity resolutions already at the classical level in accordance with (a) and (b).. We shall focus, on the issue of topological extensions of the background and higher spin fluctuation fields in the spinless case, leaving the construction and analysis of more complicated vacuum solutions to future work.
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