Partition functions in even dimensional AdS via quasinormal mode methods

  • Cynthia Keeler
  • Gim Seng NgEmail author
Open Access


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 .


Gauge-gravity correspondence AdS-CFT Correspondence 1/N Expansion 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Michigan Center for Theoretical Physics, Randall Laboratory of PhysicsThe University of MichiganAnn ArborU.S.A.
  2. 2.Center for the Fundamental Laws of NatureHarvard UniversityCambridgeU.S.A.

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