# Higher spin partition functions via the quasinormal mode method in de Sitter quantum gravity

### Submission summary

 As Contributors: Andrew Svesko Arxiv Link: https://arxiv.org/abs/2004.00128v1 Date submitted: 2020-04-17 Submitted by: Svesko, Andrew Submitted to: SciPost Physics Discipline: Physics Subject area: High-Energy Physics - Theory Approach: Theoretical

### Abstract

In this note we compute the 1-loop partition function of spin-$s$ fields on Euclidean de Sitter space $S^{2n+1}$ using the quasinormal mode method. Instead of computing the quasinormal mode frequencies from scratch, we use the analytic continuation prescription $L_{\text{AdS}}\to iL_{\text{dS}}$, appearing in the dS/CFT correspondence, and Wick rotate the normal mode frequencies of fields on thermal $\text{AdS}_{2n+1}$ into the quasinormal mode frequencies of fields on de Sitter space. We compare the quasinormal mode and heat kernel methods of calculating 1-loop determinants, finding exact agreement, and furthermore explicitly relate these methods via a sum over the conformal dimension. We discuss how the Wick rotation of normal modes on thermal $\text{AdS}_{2n+1}$ can be generalized to calculating 1-loop partition functions on the thermal spherical quotients $S^{2n+1}/\mathbb{Z}_{p}$. We further show that the quasinormal mode frequencies encode the group theoretic structure of the spherical spacetimes in question, analogous to the recent analysis made for thermal AdS in (1910.07607) and (1910.11913).

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission 2004.00128v1 on 17 April 2020

## Reports on this Submission

### Strengths

1) The paper is concise

2) Reproduces a previous calculation while using a streamlined method

### Weaknesses

1) Somewhat confusing use of terminology (see report)

### Report

This paper nicely reproduces the partition function for fields of arbitrary spin on the Euclidean sphere $S^N$ and its Lens-space quotients, using the quasinormal mode method. This has the interpretation of a thermal partition function in the static patch.

This paper should be published, however I have two minor comments/confusions:

1) I am puzzled by the use of the word "Dirichlet boundary condition" in the paper. The Euclidean continuation of de Sitter is compact, so there is nowhere to place a Dirichlet condition. As far as I can tell from the paper, the term is used to mean `periodic in Euclidean time.' If I am correct about this interpretation, I think the authors should change this where appropriate.

2) Right before section 3.2 the authors comment that they do not know the quasinormal modes of the Lorentzian Lens spaces. But as far as I'm aware, the zero-mode solutions they compute in Euclidean respecting thermal boundary conditions can be used to define these Lorentzian quasinormal modes. If possible, I would the authors to clarify these remarks somewhat.

Minor typo:

Missing section reference at the end of the introduction

### Requested changes

1) Clarify or remove the use of Dirichlet boundary conditions

2) Clarify the discussion around Lens-space QNMs

• validity: high
• significance: good
• originality: good
• clarity: ok
• formatting: excellent
• grammar: perfect