Higher spin fluctuations on spinless 4D BTZ black hole

We construct linearized solutions to Vasiliev's four-dimensional higher spin gravity on warped $AdS_3 \times_\xi S^1$ which is an $Sp(2)\times U(1)$ invariant non-rotating BTZ-like black hole with $\mathbb{R}^2\times T^2$ topology. The background can be obtained from $AdS_4$ by means of identifications along a Killing boost $K$ in the region where $\xi^2\equiv K^2\geqslant 0$, or, equivalently, by gluing together two Ba\~nados--Gomberoff--Martinez eternal black holes along their past and future space-like singularities (where $\xi$ 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 $\widetilde K$. 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 $U(1)_K\times U(1)_{\widetilde K}$-invariant of which $U(1)_K$ is broken by momenta on $S^1$ and $U(1)_{\widetilde K}$ by quasi-normal modes. We show that a set of $U(1)_{\widetilde K}$-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 $\widetilde K^2=1$. We interpret our findings as an example where Vasiliev's theory completes singular classical Lorentzian geometries into smooth higher spin geometries.