Higher spin fluctuations on spinless 4D BTZ black hole

We construct linearized solutions to Vasiliev’s four-dimensional higher spin gravity on warped AdS3 ×ξ S1 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 Ba˜nados-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 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 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 nunits 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 K2 = 1. We interpret our findings as an example where Vasiliev’s theory completes singular classical Lorentzian geometries into smooth higher spin geometries.