Partition functions in even dimensional AdS via quasinormal mode methods

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 AdS_2n, again finding a series of zero modes which are related to discrete series representations of SO(2n, 1), the motion group of H _2n .