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. We then discuss the application of such techniques to more complicated spacetimes.
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References
F. Denef, S.A. Hartnoll and S. Sachdev, Black hole determinants and quasinormal modes, Class. Quant. Grav. 27 (2010) 125001 [arXiv:0908.2657] [INSPIRE].
G.W. Gibbons and S.W. Hawking, Euclidean quantum gravity, World Scientific (1993).
S.S. Gubser, I.R. Klebanov and A.A. Tseytlin, Coupling constant dependence in the thermodynamics of N = 4 supersymmetric Yang-Mills theory, Nucl. Phys. B 534 (1998) 202 [hep-th/9805156] [INSPIRE].
R.C. Myers, M.F. Paulos and A. Sinha, Quantum corrections to η/s, Phys. Rev. D 79 (2009) 041901 [arXiv:0806.2156] [INSPIRE].
P. Kovtun and L.G. Yaffe, Hydrodynamic fluctuations, long time tails and supersymmetry, Phys. Rev. D 68 (2003) 025007 [hep-th/0303010] [INSPIRE].
S. Caron-Huot and O. Saremi, Hydrodynamic Long-Time tails From Anti de Sitter Space, JHEP 11 (2010) 013 [arXiv:0909.4525] [INSPIRE].
F. Denef, S.A. Hartnoll and S. Sachdev, Quantum oscillations and black hole ringing, Phys. Rev. D 80 (2009) 126016 [arXiv:0908.1788] [INSPIRE].
S.A. Hartnoll and D.M. Hofman, Generalized Lifshitz-Kosevich scaling at quantum criticality from the holographic correspondence, Phys. Rev. B 81 (2010) 155125 [arXiv:0912.0008] [INSPIRE].
D. Anninos, S.A. Hartnoll and N. Iqbal, Holography and the Coleman-Mermin-Wagner theorem, Phys. Rev. D 82 (2010) 066008 [arXiv:1005.1973] [INSPIRE].
T. Faulkner, N. Iqbal, H. Liu, J. McGreevy and D. Vegh, From Black Holes to Strange Metals, arXiv:1003.1728 [INSPIRE].
T. Faulkner, N. Iqbal, H. Liu, J. McGreevy and D. Vegh, Charge transport by holographic Fermi surfaces, Phys. Rev. D 88 (2013) 045016 [arXiv:1306.6396] [INSPIRE].
S.A. Hartnoll and A. Tavanfar, Electron stars for holographic metallic criticality, Phys. Rev. D 83 (2011) 046003 [arXiv:1008.2828] [INSPIRE].
A. Allais, J. McGreevy and S.J. Suh, A quantum electron star, Phys. Rev. Lett. 108 (2012) 231602 [arXiv:1202.5308] [INSPIRE].
A. Allais and J. McGreevy, How to construct a gravitating quantum electron star, Phys. Rev. D 88 (2013) 066006 [arXiv:1306.6075] [INSPIRE].
S. Datta and J.R. David, Higher Spin Quasinormal Modes and One-Loop Determinants in the BTZ black Hole, JHEP 03 (2012) 079 [arXiv:1112.4619] [INSPIRE].
S. Datta and J.R. David, Higher spin fermions in the BTZ black hole, JHEP 07 (2012) 079 [arXiv:1202.5831] [INSPIRE].
H.-b. Zhang and X. Zhang, One loop partition function from normal modes for \( \mathcal{N}=1 \) supergravity in AdS 3, Class. Quant. Grav. 29 (2012) 145013 [arXiv:1205.3681] [INSPIRE].
T. Zojer, On gravity one-loop partition functions of three-dimensional critical gravities, Class. Quant. Grav. 30 (2013) 075005 [arXiv:1210.6887] [INSPIRE].
C. Keeler and G.S. Ng, Partition Functions in Even Dimensional AdS via Quasinormal Mode Methods, JHEP 06 (2014) 099 [arXiv:1401.7016] [INSPIRE].
C. Keeler, P. Lisbao and G.S. Ng, Partition Functions with spin in AdS 2 via Quasinormal Mode Methods, arXiv:1601.04720 [INSPIRE].
A. Maloney and S.F. Ross, Holography on Non-Orientable Surfaces, arXiv:1603.04426 [INSPIRE].
C.M. Warnick, On quasinormal modes of asymptotically anti-de Sitter black holes, Commun. Math. Phys. 333 (2015) 959 [arXiv:1306.5760] [INSPIRE].
S.R. Coleman, The uses of instantons, in Aspects of symmetry, Cambridge University Press, Cambridge U.K. (1988).
J. Natario and R. Schiappa, On the classification of asymptotic quasinormal frequencies for d-dimensional black holes and quantum gravity, Adv. Theor. Math. Phys. 8 (2004) 1001 [hep-th/0411267] [INSPIRE].
S. Musiri, S. Ness and G. Siopsis, Perturbative calculation of quasi-normal modes of AdS Schwarzschild black holes, Phys. Rev. D 73 (2006) 064001 [hep-th/0511113] [INSPIRE].
P. Arnold and P. Szepietowski, Spin 1/2 quasinormal mode frequencies in Schwarzschild-AdS spacetime, Phys. Rev. D 88 (2013) 086002 [arXiv:1308.0341] [INSPIRE].
P. Arnold, P. Szepietowski and D. Vaman, Gravitino and other spin- \( \frac{3}{2} \) quasinormal modes in Schwarzschild-AdS spacetime, Phys. Rev. D 89 (2014) 046001 [arXiv:1311.6409] [INSPIRE].
G. Siopsis, Analytic calculation of quasi-normal modes, Lect. Notes Phys. 769 (2009) 471 [arXiv:0804.2713] [INSPIRE].
D.V. Vassilevich, Heat kernel expansion: User’s manual, Phys. Rept. 388 (2003) 279 [hep-th/0306138] [INSPIRE].
R.B. Mann and S.N. Solodukhin, Quantum scalar field on three-dimensional (BTZ) black hole instanton: Heat kernel, effective action and thermodynamics, Phys. Rev. D 55 (1997) 3622 [hep-th/9609085] [INSPIRE].
G. Festuccia and H. Liu, A Bohr-Sommerfeld quantization formula for quasinormal frequencies of AdS black holes, Adv. Sci. Lett. 2 (2009) 221 [arXiv:0811.1033] [INSPIRE].
P. Breitenlohner and D.Z. Freedman, Stability in Gauged Extended Supergravity, Annals Phys. 144 (1982) 249 [INSPIRE].
B.S. DeWitt, Dynamical theory of groups and fields, Conf. Proc. C 630701 (1964) 585 [Les Houches Lect. Notes 13 (1964) 585] [INSPIRE].
E. Poisson, A. Pound and I. Vega, The Motion of point particles in curved spacetime, Living Rev. Rel. 14 (2011) 7 [arXiv:1102.0529] [INSPIRE].
A.O. Barvinsky and G.A. Vilkovisky, The Generalized Schwinger-Dewitt Technique in Gauge Theories and Quantum Gravity, Phys. Rept. 119 (1985) 1 [INSPIRE].
A.E.M. van de Ven, Index free heat kernel coefficients, Class. Quant. Grav. 15 (1998) 2311 [hep-th/9708152] [INSPIRE].
R. Camporesi, Harmonic analysis and propagators on homogeneous spaces, Phys. Rept. 196 (1990) 1 [INSPIRE].
S. Giombi, A. Maloney and X. Yin, One-loop Partition Functions of 3D Gravity, JHEP 08 (2008) 007 [arXiv:0804.1773] [INSPIRE].
G.V. Dunne, Functional determinants in quantum field theory, lecture notes given at the 14th WE Heraeus Saalburg summer school, Wolfersdorf, Thuringia, September 2008, http://www.itp.uni-hannover.de/saalburg/Lectures/dunne.pdf.
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Arnold, P., Szepietowski, P. & Vaman, D. Computing black hole partition functions from quasinormal modes. J. High Energ. Phys. 2016, 32 (2016). https://doi.org/10.1007/JHEP07(2016)032
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DOI: https://doi.org/10.1007/JHEP07(2016)032