A Selberg zeta function for warped AdS$_3$ black holes

TL;DR

This paper develops a new Selberg zeta function formalism for warped AdS3 black holes, linking its zeros to quasinormal modes and introducing a novel warped hyperbolic space metric, with implications for quantum corrections.

Contribution

It extends the Selberg zeta function and trace formula to non-hyperbolic warped AdS3 black holes, providing a new tool to analyze their spectral properties.

Findings

Constructed a warped hyperbolic half-space metric.

Established a correspondence between zeta function zeros and quasinormal modes.

Proposed future applications including quantum corrections and more general orbifolds.

Abstract

The Selberg zeta function and trace formula are powerful tools used to calculate kinetic operator spectra and quasinormal modes on hyperbolic quotient spacetimes. In this article, we extend this formalism to non-hyperbolic quotients by constructing a Selberg zeta function for warped AdS$_3$ black holes. We also consider the so-called self-dual solutions, which are of interest in connection to near-horizon extremal Kerr. We establish a map between the zeta function zeroes and the quasinormal modes on warped AdS$_3$ black hole backgrounds. In the process, we use a method involving conformal coordinates and the symmetry structure of the scalar Laplacian to construct a warped version of the hyperbolic half-space metric, which to our knowledge is new and may have interesting applications of its own, which we describe. We end by discussing several future directions for this work, such as computing 1-loop determinants (which govern quantum corrections) on the quotient spacetimes we consider, as well as adapting the formalism presented here to more generic orbifolds.