### 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 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} .

Original language | English (US) |
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Article number | 99 |

Journal | Journal of High Energy Physics |

Volume | 2014 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2014 |

Externally published | Yes |

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### Keywords

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

### ASJC Scopus subject areas

- Nuclear and High Energy Physics

### Cite this

*Journal of High Energy Physics*,

*2014*(6), [99]. https://doi.org/10.1007/JHEP06(2014)099