Abstract
A second-quantized version of Mathieu moonshine leads to product formulae for functions that are potentially genus-two Siegel Modular Forms analogous to the Igusa Cusp Form. The modularity of these functions do not follow in an obvious manner. For some conjugacy classes, but not all, they match known modular forms. In this paper, we express the product formulae for all conjugacy classes of M24 in terms of products of standard modular forms. This provides a new proof of their modularity.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, Counting dyons in \( \mathcal{N} \) = 4 string theory, Nucl. Phys. B 484 (1997) 543 [hep-th/9607026] [INSPIRE].
S. Govindarajan and K. Gopala Krishna, Generalized Kac-Moody Algebras from CHL dyons, JHEP 04 (2009) 032 [arXiv:0807.4451] [INSPIRE].
J.H. Conway and S.P. Norton, Monstrous moonshine, Bull. London Math. Soc. 11 (1979) 308.
D. Dummit, H. Kisilevsky and J. McKay, Multiplicative products of η-functions, Contemp. Math. 45 (1985) 89.
G. Mason, M24 and certain automorphic forms, Contemp. Math. 45 (1985) 223.
T. Eguchi, H. Ooguri and Y. Tachikawa, Notes on the K3 Surface and the Mathieu group M24, Exper. Math. 20 (2011) 91 [arXiv:1004.0956] [INSPIRE].
M.C.N. Cheng, K3 Surfaces, \( \mathcal{N} \) = 4 Dyons, and the Mathieu Group M24, Commun. Num. Theor. Phys. 4 (2010) 623 [arXiv:1005.5415] [INSPIRE].
M.R. Gaberdiel, S. Hohenegger and R. Volpato, Mathieu twining characters for K3, JHEP 09 (2010) 058 [arXiv:1006.0221] [INSPIRE].
T. Eguchi and K. Hikami, Note on twisted elliptic genus of K3 surface, Phys. Lett. B 694 (2011) 446 [arXiv:1008.4924] [INSPIRE].
T. Gannon, Much ado about Mathieu, Adv. Math. 301 (2016) 322 [arXiv:1211.5531] [INSPIRE].
S. Govindarajan, Unravelling Mathieu Moonshine, Nucl. Phys. B 864 (2012) 823 [arXiv:1106.5715] [INSPIRE].
D. Persson and R. Volpato, Second Quantized Mathieu Moonshine, Commun. Num. Theor. Phys. 08 (2014) 403 [arXiv:1312.0622] [INSPIRE].
S. Govindarajan and S. Samanta, Two moonshines for L2 (11) but none for M12, Nucl. Phys. B 939 (2019) 566 [arXiv:1804.06677] [INSPIRE].
S. Govindarajan and K. Gopala Krishna, BKM Lie superalgebras from dyon spectra in ZN 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].
S. Govindarajan and S. Samanta, BKM Lie superalgebras from counting twisted CHL dyons — II, Nucl. Phys. B 948 (2019) 114770 [arXiv:1905.06083] [INSPIRE].
M.C.N. Cheng and J.F.R. Duncan, On Rademacher Sums, the Largest Mathieu Group, and the Holographic Modularity of Moonshine, Commun. Num. Theor. Phys. 6 (2012) 697 [arXiv:1110.3859] [INSPIRE].
V. Gritsenko and F. Clery, The Siegel modular forms of genus 2 with the simplest divisor, arXiv:0812.3962.
T. Eguchi and K. Hikami, Twisted Elliptic Genus for K3 and Borcherds Product, Lett. Math. Phys. 102 (2012) 203 [arXiv:1112.5928] [INSPIRE].
V.A. Gritsenko and V.V. Nikulin, Automorphic Forms and Lorentzian Kac-Moody Algebras. Part I, alg-geom/9610022 [INSPIRE].
V.A. Gritsenko and V.V. Nikulin, Automorphic forms and Lorentzian Kac-Moody algebras. Part 2, alg-geom/9611028 [INSPIRE].
H. Aoki and T. Ibukiyama, Simple graded rings of siegel modular forms, differential operators and borcherds products, Int. J. Math. 16 (2005) 249.
M. Raum, M24 -twisted product expansions are Siegel modular forms, Commun. Num. Theor. Phys. 7 (2013) 469.
A.O.L. Atkin and J. Lehner, Hecke operators on γ0(m), Math. Ann. 185 (1970) 134.
M.R. Gaberdiel, D. Persson, H. Ronellenfitsch and R. Volpato, Generalized Mathieu Moonshine, Commun. Num. Theor Phys. 07 (2013) 145 [arXiv:1211.7074] [INSPIRE].
M.C.N. Cheng, J.F.R. Duncan and J.A. Harvey, Umbral Moonshine, Commun. Num. Theor. Phys. 08 (2014) 101 [arXiv:1204.2779] [INSPIRE].
M.C.N. Cheng, J.F.R. Duncan and J.A. Harvey, Umbral Moonshine and the Niemeier Lattices, arXiv:1307.5793 [INSPIRE].
M.R. Gaberdiel, S. Hohenegger and R. Volpato, Symmetries of K3 sigma models, Commun. Num. Theor. Phys. 6 (2012) 1 [arXiv:1106.4315] [INSPIRE].
M.R. Gaberdiel, A. Taormina, R. Volpato and K. Wendland, A K3 sigma model with \( {\mathrm{\mathbb{Z}}}_2^8:{\mathbbm{M}}_{20} \) symmetry, JHEP 02 (2014) 022 [arXiv:1309.4127] [INSPIRE].
A. Taormina and K. Wendland, The overarching finite symmetry group of Kummer surfaces in the Mathieu group M24, JHEP 08 (2013) 125 [arXiv:1107.3834] [INSPIRE].
F. Cléry and V. Gritsenko, Siegel modular forms of genus 2 with the simplest divisor, Proc. Lond. Math. Soc. 102 (2011) 1024.
P. Niemann, Some Generalized Kac-Moody Algebras With Known Root Multiplicities, math/0001029.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2011.07922
Rights and permissions
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.
About this article
Cite this article
Govindarajan, S., Samanta, S. Mathieu moonshine and Siegel Modular Forms. J. High Energ. Phys. 2021, 50 (2021). https://doi.org/10.1007/JHEP03(2021)050
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2021)050