Abstract
We derive a remainder formula for the coefficients of modular invariant partition functions of extremal conformal field theories of central charge c = 24k, where k is a positive integer. The formula encodes, in particular, asymptotics of these coefficients and it provides for additional corrections to Bekenstein–Hawking black hole entropy. We also relate these partition functions to a Patterson–Selberg zeta function. More generally, when c is divisible by 8 we relate this zeta function to vacuum characters of affine E 8 and \(E_{8} \times E_{8}\).
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Höhn, G., Selbstduale Vertexoperatorsuperalgebren und das Babymonster. Bonner Mathematische Schriften, 286, Universität Bonn Mathematisches Institut, Bonn, 1996, Dissertation, Rheinische Friedrich-Wilhelms Universitat Bonn, 1995.
Witten, E., Three-dimensional gravity revisited, 2007 preprint, arXiv: 0706.3359.
Maloney, A.; Witten, E., Quantum gravity partition function in three dimensions. J. High Energy Phys. 2010, No. 2, 1–58; arXiv: 0712.0155, 2007.
Maloney, A., Physics and the monster-quantum gravity, modular forms, and Faber polynomials. Concordia University lecture, 2007.
Osorio, A.V., Modular invariance and 3d gravity, 2008 thesis supervised by Vandoren, A.; pdf available online.
Avramis, S.; Kehagias, A.; Mattheopoulou, C., Three-dimensional AdS gravity and extremal CFTs at c = 8m. arXiv: 0708.3386, 2007.
Gaiotto, D., Monster symmetry and extremal CFTs, 2008 preprint, arXiv: 0801.0988.
Frenkel, I.; Lepowsky, J.; Meurman. A., A moonshine module for the monster, in Vertex Operators in Mathematics and Physics. Edited by Lepowsky, J., Mandelstam, S., and Singer, I. (Berkeley, CA., 1983). Math. Sci. Research Inst. Publ. Vol. 3, Springer, New York, 1985, 231–273.
Brisebarre, N.; Philibert, G., Effective lower and upper bounds for the Fourier coefficients of powers of the modular invariant j. Journal of the Ramanujan Math. Soc. 2005, 20, 255–282.
Knopp, M., Automorphic forms of non-negative dimension and exponential sums. Michigan Math. J., 1960, 7, 257–281.
Petersson, H., Über die Entwicklungskoeffizienten der automorphen formen. Acta Math. 1932, 58, 501–512.
Rademacher, H., The Fourier coefficients of the modular invariant J(τ). Amer. J. Math. 1938, 60, 501–512.
Dijkgraaf, R.; Maldacena, J.; Moore, G.; Verlinde, E., A black hole Farey tail. arXiv: hepth/0005003, 2007.
Cardy, J., Operator content of two-dimensional conformally invariant theories. Nuclear Phys. B, 1986, 270, 186–204.
Giombi, S.; Maloney, A.; Yin, X., One-loop partition functions of 3d gravity. J. High Energy Phys. 2008, 007, 1–24; arXiv:0804. 1173, 2008.
Giombi, S., One-loop partition functions of 3d gravity. Harvard University lecture, 2008.
Williams, F., Lectures on zeta functions, L-functions and modular forms with some physical applications, and also the lecture The role of the Patterson–Selberg zeta function of a hyperbolic cylinder in three-dimensional gravity with a negative cosmological constant, in A Window into Zeta and Modular Physics. Edited by Kirsten, K. and Williams, F. Math. Sci. Research Inst. Publ. Vol. 57, Cambridge Univ. Press, 2010, 7–100 and 329–351.
Birmingham, D.; Siddhartha, S., Exact black hole entropy bound in conformal field theory. Phys. Rev. D 2001, 63, 047501-1-047501-3, arXiv:hep-th/0008051, 2000.
Williams, F., Conical defect zeta function for the BTZ black hole, from One Hundred Years of Relativity: Proceedings of the Einstein Symposium (Iais, Romania, 2005). Scientific Annals of Alexandru Ioan Univ. 2005, TOM LI-LII, 54–58.
Mann, R.; Solodukhin, S., Quantum scalar field on a three-dimensional (BTZ) black hole instanton; heat kernel, effective action, and thermodynamics. Phys. Rev. D, 1997, 55, 3622–3632.
Patterson, S., The Selberg zeta function of a Kleinian group, in Number Theory, Trace Formulas, and Discrete Groups (Oslo, Sweden, 1987). Academic Press, Boston, MA. 1989, 409–441.
Kac, V. G., An elucidation of “Infinite-dimensional algebras and the very strange formula”, E 8 (1) and the cube root of the modular invariant j, Adv. in Math. 1980, 35, 264–273.
Lepowsky, J., Euclidean Lie algebras and the modular function j, from the Santa Cruz Conf. on Finite Groups, AMS Proc. Sympos. Pure Math, 1980, 37, 567–570.
Patterson, S. J.; Perry, P., The divisor of Selberg’s zeta function for Kleinian groups. Duke Math. J. 2001, 106, 321–390 (with an appendix by Epstein, C.).
Perry, P.; Williams, F., Selberg zeta function and trace formula for the BTZ black hole. Internat. J. Pure and Appl. Math. 2003, 9, 1–21.
Guillopé, L.; Zworski, M., Upper bounds on the number of resonances for non-compact Riemann surfaces. J. Funct. Anal. 1995, 129, 364–389.
Williams, F., Remarks on the Patterson–Selberg zeta function, black hole vacua and extreme CFT partition functions, in a volume dedicated to the 75th birthday of Stuart Dowker, J. Phys. A, Math. Theor. 2012, 45 (19 pp).
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Dedicated to Nolan Wallach. Thank you Nolan for your mentorship and your inspiration that have markedly shaped my mathematical life.
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Williams, F.L. (2014). Remainder formula and zeta expression for extremal CFT partition functions. In: Howe, R., Hunziker, M., Willenbring, J. (eds) Symmetry: Representation Theory and Its Applications. Progress in Mathematics, vol 257. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-1590-3_18
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DOI: https://doi.org/10.1007/978-1-4939-1590-3_18
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