Higher spin fluctuations on spinless 4D BTZ black hole

Abstract

Abstract We construct linearized solutions to Vasiliev’s four-dimensional higher spin gravity on warped AdS3 × ξ S 1 which is an Sp(2) × U(1) invariant non-rotating BTZ-like black hole with ℝ2 × T 2 topology. The background can be obtained from AdS4 by means of identifications along a Killing boost K in the region where ξ 2 ≡ K 2 ≥ 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 $$ \tilde{K} $$ 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 S 1. We examine the subsector in which K anti-commutes to J and the ground state is $$ \mathrm{U}{(1)}_K\times \mathrm{U}{(1)}_{\tilde{K}} $$ U 1 K × U 1 K ˜ -invariant of which U(1) K is broken by momenta on S1 and $$ \mathrm{U}{(1)}_{\tilde{K}} $$ U 1 K ˜ by quasi-normal modes. We show that a set of $$ \mathrm{U}{(1)}_{\tilde{K}} $$ U 1 K ˜ -invariant modes (with n units of S 1 momenta) are singularity-free as master fields living on a total bundle space, although the individual Fronsdal fields have membrane-like singularities at $$ {\tilde{K}}^2=1 $$ K ˜ 2 = 1 . We interpret our findings as an example where Vasiliev’s theory completes singular classical Lorentzian geometries into smooth higher spin geometries.