Higher spin fluctuations on spinless 4D BTZ black hole

Abstract

We construct linearized solutions to Vasiliev's four-dimensional higher spin gravity on warped $Ad{S}_3{\times }_{\xi }{S}^1$ 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 $Ad{S}_4$ by means of identifications along a Killing boost $K$ in the region where ${\xi }^2\equiv K^2⩾0$, or, equivalently, by gluing together two Bañados--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 $\tilde{K}$. 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 $S^1$. We examine the subsector in which $K$ anti-commutes to $J$ and the ground state is $U(1{)}_K\times U(1{)}_{\tilde{K}}$-invariant of which $U(1{)}_K$ is broken by momenta on $S^1$ and $U(1{)}_{\tilde{K}}$ by quasi-normal modes. We show that a set of $U(1{)}_{\tilde{K}}$-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 ${\tilde{K}}^2=1$. We interpret our findings as an example where Vasiliev's theory completes singular classical Lorentzian geometries into smooth higher spin geometries.

v3: Revised and published on JHEP, with more references, main text 63 pages plus appendices (v2: Clarifications, table and references added, typos corrected)

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• Fronsdal fields from gauge functions in Vasiliev's higher-spin gravity
• Metaplectic representation and ordering (in)dependence in Vasiliev's higher spin gravity
• Planar solutions of higher-spin theory I: free field level
• Scalar Electrodynamics and Higgs Mechanism in the Unfolded Dynamics Approach
• All-Order Quartic Couplings in Highly Symmetric D-brane-Anti-D-brane Systems
• Higher-spin initial data in twistor space with complex stargenvalues