BTZ one-loop determinants via the Selberg zeta function for general spin

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Springer

Received: February 4, 2020 Accepted: September 12, 2020 Published: October 22, 2020

Cynthia Keeler, Victoria L. Martin and Andrew Svesko Department of Physics, Arizona State University, Tempe, Arizona 85287, U.S.A.

E-mail: [email protected], [email protected], [email protected] Abstract: We relate the heat kernel and quasinormal mode methods of computing the 1-loop partition function of arbitrary spin fields on a rotating (Euclidean) BTZ background using the Selberg zeta function associated with H3 /Z, extending (arXiv:1811.08433) [1]. Previously, Perry and Williams [2] showed for a scalar field that the zeros of the Selberg zeta function coincide with the poles of the associated scattering operator upon a relabeling of integers. We extend the integer relabeling to the case of general spin, and discuss its relationship to the removal of non-square-integrable Euclidean zero modes. Keywords: Black Holes, Higher Spin Gravity ArXiv ePrint: 1910.07607

c The Authors. Open Access, Article funded by SCOAP3 .

https://doi.org/10.1007/JHEP10(2020)138

JHEP10(2020)138

BTZ one-loop determinants via the Selberg zeta function for general spin

Contents 1

2 Review: heat kernels and quasinormal modes on H3 /Z 2.1 Heat kernel method 2.2 Quasinormal mode method 2.3 Comparing the two methods

2 2 5 7

3 Quasinormal modes from the zeros of the Selberg zeta function

8

4 Discussion

1

12

Introduction

Constructing a complete theory of Euclidean quantum gravity requires knowledge of the full partition function Z Z = DgDφ e−SE (g,φ)/~ , (1.1) where g is the dynamical metric and φ represents all other matter fields. Although the partition function Z is often intractable to compute directly, it can be evaluated perturbatively by expanding the Euclidean action SE about a classical solution using a saddle point approximation. This approximation is an asymptotic expansion in ~, where Z (0) ∼ O(~−1 ) is the classical contribution to the full partition function Z, and Z (1) ∼ O(~0 ) captures leading order 1-loop quantum effects. For a free field φ on a gravitational background M, (1) computing the 1-loop partition function Zφ involves calculating functional determinants of kinetic operators ∇2φ,M . For example, when φ is a complex scalar field, (1)

Zφ = [det(−∇2φ,M )]−1 .

(1.2)

This perturbative approach has proven useful in finding quantum corrections to black hole entropy [3–5] and holographic entanglement entropy [6]. Functional determinants are also of pure mathematical interest as their spectral properties provide a classification of smooth manifolds [7]. There are several methods for computing functional determinants of kinetic operators such as those found in (1.2). In this work we focus on further developing a connection between two particular methods, referred to in the literature as the heat kernel method (cf. [8]) and the quasinormal mode method [9]. We reserve a review of these methods for section 2. Recently we showed [1] how to connect the heat kernel and quasinormal mode methods for computing 1-loop dete