Partition functions in even dimensional AdS via quasinormal mode methods

Abstract

In this note, we calculate the one-loop determinant for a massive scalar (with conformal dimension Δ) in even-dimensional AdS d+1 space, using the quasinormal mode method developed in [1] by Denef, Hartnoll, and Sachdev. Working first in two dimensions on the related Euclidean hyperbolic plane H 2, we find a series of zero modes for negative real values of Δ whose presence indicates a series of poles in the one-loop partition function Z(Δ) in the Δ complex plane; these poles contribute temperature-independent terms to the thermal AdS partition function computed in [1]. Our results match those in a series of papers by Camporesi and Higuchi, as well as Gopakumar et al. [2] and Banerjee et al. [3]. We additionally examine the meaning of these zero modes, finding that they Wick-rotate to quasinormal modes of the AdS2 black hole. They are also interpretable as matrix elements of the discrete series representations of SO(2, 1) in the space of smooth functions on S 1. We generalize our results to general even dimensional AdS2n, again finding a series of zero modes which are related to discrete series representations of SO(2n, 1), the motion group of H 2n . © 2014 The Author(s).