Normal modes in thermal AdS via the Selberg zeta function

TL;DR

This paper links heat kernel and quasinormal mode methods for computing 1-loop partition functions in thermal AdS spacetimes, using the Selberg zeta function to encode normal mode frequencies and validate results.

Contribution

It extends the relation between heat kernel and Selberg zeta function methods to higher-dimensional thermal AdS backgrounds, providing a new way to compute 1-loop partition functions.

Findings

Zeros of the Selberg zeta function encode normal mode frequencies.

Constructed 1-loop partition functions match heat kernel results.

Extended analysis to higher-dimensional thermal AdS backgrounds.

Abstract

The heat kernel and quasinormal mode methods of computing 1-loop partition functions of spin $s$ fields on hyperbolic quotient spacetimes $\mathbb{H}^{3}/\mathbb{Z}$ are related via the Selberg zeta function. We extend that analysis to thermal $\text{AdS}_{2n+1}$ backgrounds, with quotient structure $\mathbb{H}^{2n+1}/\mathbb{Z}$. Specifically, we demonstrate the zeros of the Selberg function encode the normal mode frequencies of spin fields upon removal of non-square-integrable modes. With this information we construct the 1-loop partition functions for symmetric transverse traceless tensors in terms of the Selberg zeta function and find exact agreement with the heat kernel method.