Abstract
Inspired by the multiplicative nature of the Ramanujan modular discriminant, Δ, we consider physical realizations of certain multiplicative products over the Dedekind eta-function in two parallel directions: the generating function of BPS states in certain heterotic orbifolds and elliptic K3 surfaces associated to congruence subgroups of the modular group. We show that they are, after string duality to type II, the same K3 surfaces admitting Nikulin automorphisms. In due course, we will present identities arising from q-expansions as well as relations to the sporadic Mathieu group M 24.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Dummit, H. Kisilevsky and J. McKay, Multiplicative products of eta functions, in Finite groups — coming of age, Montreal Canada (1982), pg. 89 [Contemp. Math. 45 (1985) 89].
B. Gordon and D. Sinor, Multiplicative properties of eta-products, Lecture Notes in Mathematics. Vol. 1395, Springer, Berlin Germany (1989).
Y. Martin, Multiplicative eta-quotients, Trans. Am. Math. Soc. 348 (1996) 4825.
L.J.P. Kilford, Generating spaces of modular forms with η-quotients, math/0701478.
Y. Martin and K. Ono, Eta-quotients and elliptic curves, Proc. Am. Math. Soc. 125 (1997) 3169.
C.L. Siegel, A simple proof of \( \eta \left( {{1 \left/ {\tau } \right.}} \right)=\eta \left( \tau \right)\sqrt{{{\tau \left/ {i} \right.}}} \), Mathematika 1 (1954) 4.
J.-P. Serre, Graduate Texts in Mathematics. Vol. 7: Cours d’arithmétique, Springer, New York U.S.A. (1973).
N. Koblitz, Graduate Texts in Mathematics. Vol. 97: Introduction to Elliptic Curves and Modular Forms, Springer, New York U.S.A. (1993).
M.B.Green, J.H.Schwarz and E.Witten, Superstring Theory, Cambridge University Press, Cambridge U.K. (1987).
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].
A. Sen, Black Hole Entropy Function, Attractors and Precision Counting of Microstates, Gen. Rel. Grav. 40 (2008) 2249 [arXiv:0708.1270] [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].
E. Witten, Supersymmetric index in four-dimensional gauge theories, Adv. Theor. Math. Phys. 5 (2002) 841 [hep-th/0006010] [INSPIRE].
C. Romelsberger, Counting chiral primaries in N = 1, D = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].
J. Bhattacharya, S. Bhattacharyya, S. Minwalla and S. Raju, Indices for Superconformal Field Theories in 3,5 and 6 Dimensions, JHEP 02 (2008) 064 [arXiv:0801.1435] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge Theories and Macdonald Polynomials, Commun. Math. Phys. 319 (2013) 147 [arXiv:1110.3740] [INSPIRE].
S. Benvenuti, B. Feng, A. Hanany and Y.-H. He, Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics, JHEP 11 (2007) 050 [hep-th/0608050] [INSPIRE].
B. Feng, A. Hanany and Y.-H. He, Counting gauge invariants: The Plethystic program, JHEP 03 (2007) 090 [hep-th/0701063] [INSPIRE].
D. Bayer and M. Stillman, Computation of Hilbert functions, J. Symb. Comput. 14 (1992) 31.
W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symb. Comput. 24 (1997) 235.
R. Miranda and U. Persson, Configurations of I n Fibers on Elliptic K3-Surfaces, Math. Z. 201 (1989) 339.
F. Beukers and H. Montanus, Explicit calculation of elliptic fibrations of K3-surfaces and their Belyi-maps, London Mathematical Society Lecture Note Series. Vol. 352: Number theory and polynomials, Cambridge University Press, Cambridge U.K. (2008), pg. 33.
F. Beukers, Dessins Coming from the Miranda-Persson table (2008), available at http://www.staff.science.uu.nl/ beuke106/mirandapersson/Dessins.html.
Y.-H. He, J. McKay and J. Read, Modular Subgroups, Dessins d’Enfants and Elliptic K3 Surfaces, LMS J. Comp. Math. 16 (2013) 271 [arXiv:1211.1931] [INSPIRE].
Y.-H. He and J. McKay, N = 2 Gauge Theories: Congruence Subgroups, Coset Graphs and Modular Surfaces, J. Math. Phys. 54 (2013) 012301 [arXiv:1201.3633] [INSPIRE].
J. McKay and A. Sebbar, Arithmetic Semistable Elliptic Surfaces, Proceedings on Moonshine and related topics, Montréal Canada (1999), pg. 119 [CRM Proc. Lecture Notes. Vol. 30, AMS, Providence U.S.A. (2001)].
A. Sebbar, Modular subgroups, forms, curves and surfaces, Canad. Math. Bull. 45 (2002) 294.
A. Sebbar, Classification of torsion-free genus zero congruence groups, Proc. Am. Math. Soc. 129 (2001) 2517.
M. Schuett and T. Shioda, Elliptic Surfaces, arXiv:0907.0298.
V.V. Nikulin, Finite groups of automorphisms of Kählerian K3 surfaces (in Russian), Trudy Moskov. Mat. Obshch. 38 (1979) 75 [Trans. Moscow Math. Soc. 38 (1980) 71].
A. Garbagnati and A. Sarti, Elliptic fibrations and symplectic automorphisms on K3 surfaces, Commun. Algebra 37 (2009) 3601 [arXiv:0801.3992].
A. Garbagnati and A. Sarti, Symplectic automorphisms of prime order on K3 surfaces, math/0603742.
J. Top and N. Yui, Explicit Equations of Some Elliptic Modular Surfaces, Rocky Mount. J. Math. 37 (2007) 663.
D. Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. 33 (1976) 193.
M. Kobayashi, Duality of Weights, Mirror Symmetry and Arnold’s Strange Duality, alg-geom/9502004.
S. Mukai, Finite groups of automorphisms of K3 surfaces and the Mathieu group, Invent. Math. 94 (1988) 183.
D. Ford and J. McKay, Ramifications of Ramanujan’s Work on η-products, Proc. Indian Acad. Sci. 99 (1989) 221.
G. Mason, M 24 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 M 24, Exper. Math. 20 (2011) 91 [arXiv:1004.0956] [INSPIRE].
M.C. Cheng, K3 Surfaces, 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].
M.R. Gaberdiel, S. Hohenegger and R. Volpato, Mathieu Moonshine in the elliptic genus of K3, JHEP 10 (2010) 062 [arXiv:1008.3778] [INSPIRE].
M.R. Gaberdiel, D. Persson, H. Ronellenfitsch and R. Volpato, Generalised Mathieu Moonshine, Commun. Num. Theor. Phys. 7 (2013) 145 [arXiv:1211.7074] [INSPIRE].
T. Eguchi and K. Hikami, Note on Twisted Elliptic Genus of K3 Surface, Phys. Lett. B 694 (2011) 446 [arXiv:1008.4924] [INSPIRE].
S. Govindarajan, Unravelling Mathieu Moonshine, Nucl. Phys. B 864 (2012) 823 [arXiv:1106.5715] [INSPIRE].
M.R. Gaberdiel and R. Volpato, Mathieu Moonshine and Orbifold K3s, arXiv:1206.5143 [INSPIRE].
A. Taormina and K. Wendland, Symmetry-surfing the moduli space of Kummer K3s, arXiv:1303.2931 [INSPIRE].
M.C. Cheng, J.F. Duncan and J.A. Harvey, Umbral Moonshine, arXiv:1204.2779 [INSPIRE].
M.C. Cheng et al., Mathieu Moonshine and N = 2 String Compactifications, JHEP 09 (2013) 030 [arXiv:1306.4981] [INSPIRE].
T. Gannon, Much ado about Mathieu, arXiv:1211.5531 [INSPIRE].
L. le Bryun, in The Bourbaki Code, http://www.neverendingbooks.org/index.php/monsieur-mathieu.html and http://win.ua.ac.be/ lebruyn/LeBruyn2012a.pdf.
A. Zvonkin, How to draw a group?, Discrete Math. 180 (1998) 403.
N. Hanusse and A.K. Zvonkin, Cartographic generation of Mathieu groups, Actes du 11eme Colloque, Séries Formelles et Combinatoire Algébrique, Barcelone Spain (1999), pg. 241.
Y.-H. He, On Fields over Fields, arXiv:1003.2986 [INSPIRE].
Y.-H. He, Graph Zeta Function and Gauge Theories, JHEP 03 (2011) 064 [arXiv:1102.1304] [INSPIRE].
Y.-H. He, Bipartita: Physics, Geometry & Number Theory, arXiv:1210.4388 [INSPIRE].
S.-T. Yau and E. Zaslow, BPS states, string duality and nodal curves on K3, Nucl. Phys. B 471 (1996) 503 [hep-th/9512121] [INSPIRE].
J.C. Lagarias, An Elementary Problem Equivalent to the Riemann Hypothesis, Am. Math. Month. 109 (2002) 534.
Y.-H. He, V. Jejjala and D. Minic, On the Physics of the Riemann Zeros, in Proceedings of The 6th International Symposium on Quantum Theory and Symmetries, Lexington U.S.A. (2009) [arXiv:1004.1172] [INSPIRE].
V.G. Kac, Infinite-Dimensional Algebras, Dedekind’s q-Function, Classical Mobius Function and the Very Strange Formula, Adv. Math. 30 (1978) 85.
V.G. Kac, An Elucidation of ‘Infinite-Dimensional Algebras, Dedekind’s q-Function, Classical Mobius Function and the Very Strange Formula, Adv. Math. 35 (1980) 264.
D. Ford, J. McKay and S.P. Norton, More on replicable functions, Commun. Algebra 22 (1994) 5175.
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: 1308.5233
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
He, YH., McKay, J. Eta products, BPS states and K3 surfaces. J. High Energ. Phys. 2014, 113 (2014). https://doi.org/10.1007/JHEP01(2014)113
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2014)113