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 ∆) 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 ∆ whose presence indicates a series of poles in the one-loop partition function Z(∆) in the ∆ 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 AdS 2 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 .

Tweaking one-loop determinants in AdS3

“…We will not encounter this subtlety here as we are in odd dimensional AdS; consequently we will use "normalizable" to refer to the analytic continuation. For more details, see [24,25].…”

“…Imposing the relations (B.17) or (B.19), one sees that indeed eEE vanishes where kE = 0, kE = 1 (for ingoing) or kE = −1 (for outgoing). 24 Recall from (A.11) that xL = (r+ − r−)t + .…”

Siegel modular forms and black hole entropy

“…For CHL models both d(Q) and its 4D(5D) supersymmetric black hole counterpart are known explicitly, and the agreement is remarkable (see appendix C for a quick review of this class of black holes). Logarithmic corrections have been computed as well for several other supersymmetric configurations [13,14] and using novel techniques in [15][16][17]. There are also very interesting results for non-extremal black holes [18,19].…”

“…What is physically interesting of this example is its intimate relation to the elliptic genera of K3. More concretely 17) and hence another way to write the Igusa form is as…”

“…Assuming that the saddle is at some small imaginary value of τ and z of order one, 17 we can approximate ϕ k,m − 1 τ , z τ by its most polar term…”