Abstract
The microscopic spectrum of half-BPS excitations in toroidally compactified heterotic string theory has been computed exactly through the use of results from analytic number theory. Recently, similar quantities have been understood macroscopically by evaluating the gravitational path integral on the M-theory lift of the AdS2 near-horizon geometry of the corresponding black hole. In this paper, we generalize these results to a subset of the CHL models, which include the standard compactification of IIA on K3 × T2 as a special case. We begin by developing a Rademacher-like expansion for the Fourier coefficients of the partition functions for these theories, which are modular forms for congruence subgroups. We then describe a possible macroscopic interpretation of these results, emphasizing the role of twisted boundary conditions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
A. Dabholkar, Exact counting of black hole microstates, Phys. Rev. Lett. 94 (2005) 241301 [hep-th/0409148] [INSPIRE].
A. Dabholkar, F. Denef, G.W. Moore and B. Pioline, Exact and asymptotic degeneracies of small black holes, JHEP 08 (2005) 021 [hep-th/0502157] [INSPIRE].
A. Dabholkar, F. Denef, G.W. Moore and B. Pioline, Precision counting of small black holes, JHEP 10 (2005) 096 [hep-th/0507014] [INSPIRE].
A. Dabholkar, R. Kallosh and A. Maloney, A Stringy cloak for a classical singularity, JHEP 12 (2004) 059 [hep-th/0410076] [INSPIRE].
K. Ranestad, J. Bruinier, G. van der Geer, G. Harder and D. Zagier, The 1-2-3 of Modular Forms: Lectures at a Summer School in Nordfjordeid, Norway, Universitext, Springer Berlin Heidelberg (2008) [DOI:https://doi.org/10.1007/978-3-540-74119-0].
H. Rademacher and H.S. Zuckerman, On the fourier coefficients of certain modular forms of positive dimension, Annals Math. 39 (1938) 433.
H. Rademacher, On the expansion of the partition function in a series, Annals Math. 44 (1943) 416.
T. Apostol, Modular Functions and Dirichlet Series in Number Theory, Graduate Texts in Mathematics, Springer New York (1976) [DOI:https://doi.org/10.1007/978-1-4612-0999-7].
H. Rademacher, Topics in Analytic Number Theory, Grundlehren der mathematischen Wissenschaften, Springer Berlin Heidelberg (2012) [DOI:https://doi.org/10.1007/978-3-642-80615-5].
R. Dijkgraaf, J.M. Maldacena, G.W. Moore and E.P. Verlinde, A Black hole Farey tail, hep-th/0005003 [INSPIRE].
M.C.N. Cheng and J.F.R. Duncan, Rademacher Sums and Rademacher Series, Contrib. Math. Comput. Sci. 8 (2014) 143 [arXiv:1210.3066] [INSPIRE].
A. Sen, Black Hole Entropy Function, Attractors and Precision Counting of Microstates, Gen. Rel. Grav. 40 (2008) 2249 [arXiv:0708.1270] [INSPIRE].
A. Dabholkar and S. Nampuri, Quantum black holes, Lect. Notes Phys. 851 (2012) 165 [arXiv:1208.4814] [INSPIRE].
A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].
S. Murthy and B. Pioline, A Farey tale for N = 4 dyons, JHEP 09 (2009) 022 [arXiv:0904.4253] [INSPIRE].
C. Beasley, D. Gaiotto, M. Guica, L. Huang, A. Strominger and X. Yin, Why Z BH = |Z top|2, hep-th/0608021 [INSPIRE].
J. de Boer, M.C.N. Cheng, R. Dijkgraaf, J. Manschot and E. Verlinde, A Farey Tail for Attractor Black Holes, JHEP 11 (2006) 024 [hep-th/0608059] [INSPIRE].
J. Manschot and G.W. Moore, A Modern Farey Tail, Commun. Num. Theor. Phys. 4 (2010) 103 [arXiv:0712.0573] [INSPIRE].
A. Dabholkar, J. Gomes and S. Murthy, Quantum black holes, localization and the topological string, JHEP 06 (2011) 019 [arXiv:1012.0265] [INSPIRE].
A. Dabholkar, J. Gomes and S. Murthy, Localization & Exact Holography, JHEP 04 (2013) 062 [arXiv:1111.1161] [INSPIRE].
J. Gomes, Quantum entropy of supersymmetric black holes, arXiv:1111.2025 [INSPIRE].
A. Dabholkar, J. Gomes and S. Murthy, Nonperturbative black hole entropy and Kloosterman sums, JHEP 03 (2015) 074 [arXiv:1404.0033] [INSPIRE].
N. Banerjee, S. Banerjee, R.K. Gupta, I. Mandal and A. Sen, Supersymmetry, Localization and Quantum Entropy Function, JHEP 02 (2010) 091 [arXiv:0905.2686] [INSPIRE].
J. Gomes, Quantum Black Hole Entropy, Localization and the Stringy Exclusion Principle, JHEP 09 (2018) 132 [arXiv:1705.01953] [INSPIRE].
J. Gomes, Generalized Kloosterman Sums from M2-branes, arXiv:1705.04348 [INSPIRE].
J. Gomes, Exact Holography and Black Hole Entropy in \( \mathcal{N} \) = 8 and \( \mathcal{N} \) = 4 String Theory, JHEP 07 (2017) 022 [arXiv:1511.07061] [INSPIRE].
S. Murthy and V. Reys, Functional determinants, index theorems and exact quantum black hole entropy, JHEP 12 (2015) 028 [arXiv:1504.01400] [INSPIRE].
S. Murthy and V. Reys, Single-centered black hole microstate degeneracies from instantons in supergravity, JHEP 04 (2016) 052 [arXiv:1512.01553] [INSPIRE].
S. Chaudhuri, G. Hockney and J.D. Lykken, Maximally supersymmetric string theories in D < 10, Phys. Rev. Lett. 75 (1995) 2264 [hep-th/9505054] [INSPIRE].
S. Chaudhuri and J. Polchinski, Moduli space of CHL strings, Phys. Rev. D 52 (1995) 7168 [hep-th/9506048] [INSPIRE].
S. Chaudhuri and D.A. Lowe, Type IIA heterotic duals with maximal supersymmetry, Nucl. Phys. B 459 (1996) 113 [hep-th/9508144] [INSPIRE].
D.P. Jatkar and A. Sen, Dyon spectrum in CHL models, JHEP 04 (2006) 018 [hep-th/0510147] [INSPIRE].
J.R. David and A. Sen, CHL Dyons and Statistical Entropy Function from D1-D5 System, JHEP 11 (2006) 072 [hep-th/0605210] [INSPIRE].
S. Govindarajan and K. Gopala Krishna, BKM Lie superalgebras from dyon spectra in Z(N) CHL orbifolds for composite N, JHEP 05 (2010) 014 [arXiv:0907.1410] [INSPIRE].
S. Govindarajan, BKM Lie superalgebras from counting twisted CHL dyons, JHEP 05 (2011) 089 [arXiv:1006.3472] [INSPIRE].
D. Persson and R. Volpato, Fricke S-duality in CHL models, JHEP 12 (2015) 156 [arXiv:1504.07260] [INSPIRE].
N.M. Paquette, R. Volpato and M. Zimet, No More Walls! A Tale of Modularity, Symmetry and Wall Crossing for 1/4 BPS Dyons, JHEP 05 (2017) 047 [arXiv:1702.05095] [INSPIRE].
M.C.N. Cheng, S.M. Harrison, R. Volpato and M. Zimet, K3 String Theory, Lattices and Moonshine, arXiv:1612.04404 [INSPIRE].
M.R. Gaberdiel, S. Hohenegger and R. Volpato, Symmetries of K3 σ-models, Commun. Num. Theor. Phys. 6 (2012) 1 [arXiv:1106.4315] [INSPIRE].
J.F. Duncan and I.B. Frenkel, Rademacher sums, Moonshine and Gravity, Commun. Num. Theor. Phys. 5 (2011) 849 [arXiv:0907.4529] [INSPIRE].
D. Persson and R. Volpato, Dualities in CHL-Models, J. Phys. A 51 (2018) 164002 [arXiv:1704.00501] [INSPIRE].
V.V. Nikulin, K3 surfaces with interesting groups of automorphisms, alg-geom/9701011.
Y.-H. He and J. McKay, Eta Products, BPS States and K3 Surfaces, JHEP 01 (2014) 113 [arXiv:1308.5233] [INSPIRE].
G. Bossard, C. Cosnier-Horeau and B. Pioline, Four-derivative couplings and BPS dyons in heterotic CHL orbifolds, SciPost Phys. 3 (2017) 008 [arXiv:1702.01926] [INSPIRE].
J.F.R. Duncan, M.J. Griffin and K. Ono, Moonshine, arXiv:1411.6571 [INSPIRE].
S. Banerjee, A. Sen and Y.K. Srivastava, Generalities of Quarter BPS Dyon Partition Function and Dyons of Torsion Two, JHEP 05 (2008) 101 [arXiv:0802.0544] [INSPIRE].
E. Sussman, Rademacher Series for η-Quotients, arXiv:1710.03415.
N. Prabhakar, On the exact entropy of half-bps black holes in heterotic string theory, MSc Thesis, Institute of Mathematical Sciences (2012).
D. Whalen, Vector-Valued Rademacher Sums and Automorphic Integrals, arXiv:1406.0571.
R.K. Gupta and S. Murthy, All solutions of the localization equations for N = 2 quantum black hole entropy, JHEP 02 (2013) 141 [arXiv:1208.6221] [INSPIRE].
G. Lopes Cardoso, B. de Wit and T. Mohaupt, Macroscopic entropy formulae and nonholomorphic corrections for supersymmetric black holes, Nucl. Phys. B 567 (2000) 87 [hep-th/9906094] [INSPIRE].
A. Sen, Extremal black holes and elementary string states, Mod. Phys. Lett. A 10 (1995) 2081 [hep-th/9504147] [INSPIRE].
S. Ferrara, R. Kallosh and A. Strominger, N = 2 extremal black holes, Phys. Rev. D 52 (1995) R5412 [hep-th/9508072] [INSPIRE].
S. Ferrara and R. Kallosh, Supersymmetry and attractors, Phys. Rev. D 54 (1996) 1514 [hep-th/9602136] [INSPIRE].
A. Sen, Entropy Function and AdS 2 /CFT 1 Correspondence, JHEP 11 (2008) 075 [arXiv:0805.0095] [INSPIRE].
A. Sen, Quantum Entropy Function from AdS 2 /CFT 1 Correspondence, Int. J. Mod. Phys. A 24 (2009) 4225 [arXiv:0809.3304] [INSPIRE].
A. Sen, Arithmetic of Quantum Entropy Function, JHEP 08 (2009) 068 [arXiv:0903.1477] [INSPIRE].
J. Gomes, Quantum entropy and exact 4d/5d connection, JHEP 01 (2015) 109 [arXiv:1305.2849] [INSPIRE].
J.M. Maldacena and A. Strominger, AdS 3 black holes and a stringy exclusion principle, JHEP 12 (1998) 005 [hep-th/9804085] [INSPIRE].
B. de Wit and S. Katmadas, Near-Horizon Analysis of D = 5 BPS Black Holes and Rings, JHEP 02 (2010) 056 [arXiv:0910.4907] [INSPIRE].
N. Banerjee, B. de Wit and S. Katmadas, The Off-Shell 4D/5D Connection, JHEP 03 (2012) 061 [arXiv:1112.5371] [INSPIRE].
J. Gomes, U-duality Invariant Quantum Entropy from Sums of Kloosterman Sums, arXiv:1709.06579 [INSPIRE].
A. Sen, A Twist in the Dyon Partition Function, JHEP 05 (2010) 028 [arXiv:0911.1563] [INSPIRE].
A. Sen, Discrete Information from CHL Black Holes, JHEP 11 (2010) 138 [arXiv:1002.3857] [INSPIRE].
A. Chowdhury, R.K. Gupta, S. Lal, M. Shyani and S. Thakur, Logarithmic Corrections to Twisted Indices from the Quantum Entropy Function, JHEP 11 (2014) 002 [arXiv:1404.6363] [INSPIRE].
F. Ferrari and V. Reys, Mixed Rademacher and BPS Black Holes, JHEP 07 (2017) 094 [arXiv:1702.02755] [INSPIRE].
H. Larson, Coefficients of McKay-Thompson series and distributions of the moonshine module, arXiv:1508.03742.
N.M. Paquette, D. Persson and R. Volpato, Monstrous BPS-Algebras and the Superstring Origin of Moonshine, Commun. Num. Theor. Phys. 10 (2016) 433 [arXiv:1601.05412] [INSPIRE].
N.M. Paquette, D. Persson and R. Volpato, BPS Algebras, Genus Zero and the Heterotic Monster, J. Phys. A 50 (2017) 414001 [arXiv:1701.05169] [INSPIRE].
E. Witten, Three-Dimensional Gravity Revisited, arXiv:0706.3359 [INSPIRE].
M. Zimet, Umbral Moonshine and String Duality, arXiv:1803.07567 [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1803.10775
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Nally, R. Exact half-BPS black hole entropies in CHL models from Rademacher series. J. High Energ. Phys. 2019, 60 (2019). https://doi.org/10.1007/JHEP01(2019)060
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2019)060