# Computing black hole partition functions from quasinormal modes

## Abstract

We propose a method of computing one-loop determinants in black hole space-times (with emphasis on asymptotically anti-de Sitter black holes) that may be used for numerics when completely-analytic results are unattainable. The method utilizes the expression for one-loop determinants in terms of quasinormal frequencies determined by Denef, Hartnoll and Sachdev in [1]. A numerical evaluation must face the fact that the sum over the quasinormal modes, indexed by momentum and overtone numbers, is divergent. A necessary ingredient is then a regularization scheme to handle the divergent contributions of individual fixed-momentum sectors to the partition function. To this end, we formulate an effective two-dimensional problem in which a natural refinement of standard heat kernel techniques can be used to account for contributions to the partition function at fixed momentum. We test our method in a concrete case by reproducing the scalar one-loop determinant in the BTZ black hole background. Furthermore, we then discuss the application of such techniques to more complicated spacetimes.

- Authors:

- Univ. of Virginia, Charlottesville, VA (United States)
- Utrecht Univ., Utrecht (The Netherlands)
- Univ. of Virginia, Charlottesville, VA (United States); College of William and Mary, Williamsburg, VA (United States)

- Publication Date:

- Research Org.:
- U.S. Dept. of the Navy, Arlington County, VA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1326940

- Grant/Contract Number:
- SC0007894

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2016; Journal Issue: 7; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTRONOMY AND ASTROPHYSICS; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; AdS-CFT Correspondence; Black Holes; Holography and condensed matter physics (AdS/CMT); Holography and quark-gluon plasmas

### Citation Formats

```
Arnold, Peter, Szepietowski, Phillip, and Vaman, Diana.
```*Computing black hole partition functions from quasinormal modes*. United States: N. p., 2016.
Web. doi:10.1007/JHEP07(2016)032.

```
Arnold, Peter, Szepietowski, Phillip, & Vaman, Diana.
```*Computing black hole partition functions from quasinormal modes*. United States. doi:10.1007/JHEP07(2016)032.

```
Arnold, Peter, Szepietowski, Phillip, and Vaman, Diana. Thu .
"Computing black hole partition functions from quasinormal modes". United States.
doi:10.1007/JHEP07(2016)032. https://www.osti.gov/servlets/purl/1326940.
```

```
@article{osti_1326940,
```

title = {Computing black hole partition functions from quasinormal modes},

author = {Arnold, Peter and Szepietowski, Phillip and Vaman, Diana},

abstractNote = {We propose a method of computing one-loop determinants in black hole space-times (with emphasis on asymptotically anti-de Sitter black holes) that may be used for numerics when completely-analytic results are unattainable. The method utilizes the expression for one-loop determinants in terms of quasinormal frequencies determined by Denef, Hartnoll and Sachdev in [1]. A numerical evaluation must face the fact that the sum over the quasinormal modes, indexed by momentum and overtone numbers, is divergent. A necessary ingredient is then a regularization scheme to handle the divergent contributions of individual fixed-momentum sectors to the partition function. To this end, we formulate an effective two-dimensional problem in which a natural refinement of standard heat kernel techniques can be used to account for contributions to the partition function at fixed momentum. We test our method in a concrete case by reproducing the scalar one-loop determinant in the BTZ black hole background. Furthermore, we then discuss the application of such techniques to more complicated spacetimes.},

doi = {10.1007/JHEP07(2016)032},

journal = {Journal of High Energy Physics (Online)},

number = 7,

volume = 2016,

place = {United States},

year = {Thu Jul 07 00:00:00 EDT 2016},

month = {Thu Jul 07 00:00:00 EDT 2016}

}

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