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 boostK˜. 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 U (1) K× U (1) K˜-invariant of which U(1)K is broken by momenta on S1 and U (1) K˜ by quasi-normal modes. We show that a set of U (1) 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 K˜ 2= 1. We interpret our findings as an example where Vasiliev’s theory completes singular classical Lorentzian geometries into smooth higher spin geometries.
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