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

Victoria Martin,Andrew Svesko

doi:10.21468/scipostphys.9.3.039

Abstract

In this note we compute the 1-loop partition function of spin-ss fields on Euclidean de Sitter space S^{2n+1}S2n+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}}LAdS→iLdS, appearing in the dS/CFT correspondence, and Wick rotate the normal mode frequencies of fields on thermal \text{AdS}_{2n+1}AdS2n+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}AdS2n+1 can be generalized to calculating 1-loop partition functions on the thermal spherical quotients S^{2n+1}/\mathbb{Z}_{p}S2n+1/ℤp. We further show that the quasinormal mode frequencies encode the group theoretic structure of the spherical spacetimes in question, analogous to the analysis made for thermal AdS in [1-3] .

Highlights

In de Sitter quantum gravity the chief object of interest is the Euclidean partition function, Z = D g e−S[g], (1)written here as a path integral over all compact Euclidean metrics g
In this note we aim to compute 1-loop partition functions of spin-s fields on S2n+1 and its associated Lens spaces S2n+1/ p using the quasinormal mode method, and see how it compares to the heat kernel method
Before we move to comment on the quasinormal mode analysis for Lens space quotients, let us briefly point out another method of computing the 1-loop partition function for scalar fields on Sd+1 [24, 25, 31]

Summary

Introduction

In de Sitter quantum gravity the chief object of interest is the Euclidean partition function,. Γ∈Γ where γ are the generators of Γ This method was used to explicitly compute 1-loop partition functions of symmetric, transverse, traceless (STT) tensors on S3 and Lens spaces S3/ p in [10], and the STT heat kernel for higher odd-dimensional spheres S2n+1 and Lens spaces S2n+1/ p in [11]. In this note we aim to compute 1-loop partition functions of spin-s fields on S2n+1 and its associated Lens spaces S2n+1/ p using the quasinormal mode method, and see how it compares to the heat kernel method. The functional determinants of the Lens spaces and the dihedral case have been worked out and regularized using the Barnes zeta function in [24,25,26]

Quasinormal Mode Method

Comparing to the Heat Kernel Method

Discussion