Sporadic and Exceptional

In dedication to the memory of my dear friend Professor John S. McKay


  • Yang-Hui He London Institute, Royal Institution; City, University of London; & University of Oxford
  • John McKay




Generalised Moonshine, Lie algebra, Monster, Baby, Fischer group, del Pezzo surfaces, McKay-Thompson series, Conway's sporadic group, Horrocks-Mumford bundle


We study the web of correspondences linking the exceptional Lie algebras E8,7,6 and the sporadic simple groups Monster, Baby and the largest Fischer group. This is done via the investigation of classical enumerative problems on del Pezzo surfaces in relation to the cusps of certain subgroups of PSL(2,R) for the relevant McKay-Thompson series in Generalized Moonshine. We also study Conway's sporadic group, as well as its association with the Horrocks-Mumford bundle.


J. Conway and S. Norton, “Monstrous moonshine,” Bulletin of the London Mathematical Society, vol. 11, no. 3,

pp. 308–339, 1979.

I. Frenkel, J. Lepowsky, and A. Meurman, Vertex Operator Algebras and the Monster, vol. 134. Cambridge, Massachusetts:

Academic Press, 1989.

V. Kac, “Infinite-dimensional algebras, Dedekind’s η-function, classical Mobius function and the very strange formula,” ¨

Advances in Mathematics, vol. 30, no. 2, pp. 85–136, 1978.

R. E. Borcherds, “Vertex algebras, Kac–Moody algebras, and the Monster,” The Proceedings of the National Academy of

Sciences, vol. 83, no. 10, pp. 3068–3071, 1986.

R. E. Borcherds, “Monstrous moonshine and monstrous Lie superalgebras,” Inventiones mathematicae, vol. 109, no. 2,

pp. 405–444, 1992.

T. Gannon, “Monstrous moonshine: The first twenty-five years,” Bulletin of the London Mathematical Society, vol. 38,

no. 1, pp. 1–33, 2006.

T. Gannon, Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics. Cambridge

Monographs on Mathematical Physics, Cambridge, UK: Cambridge University Press, 2006.

T. Eguchi, H. Ooguri, and Y. Tachikawa, “Notes on the K3 surface and the Mathieu group M24,” Experimental Mathematics, vol. 20, no. 1, pp. 91–96, 2011.

Y.-H. He and J. McKay, “N=2 gauge theories: Congruence subgroups, coset graphs and modular surfaces,” Journal of

Mathematical Physics, vol. 54, no. 1, p. 012301, 2013.

M. C. Cheng, J. F. Duncan, and J. A. Harvey, “Umbral moonshine and the Niemeier lattices.” Available at arXiv:1307.5793,

J. F. Duncan, M. J. Griffin, and K. Ono, “Proof of the umbral moonshine conjecture,” Research in the Mathematical

Sciences, vol. 2, no. 1, p. 26, 2015.

M. C. Cheng, S. M. Harrison, S. Kachru, and D. Whalen, “Exceptional algebra and sporadic groups at c=12.” Available

at arXiv:1503.07219, 2015.

T. Gannon, “The algebraic meaning of genus-zero.” Available at arXiv:math/0512248, 2005.

S. Carnahan, “Generalized Moonshine I: Genus zero functions,” Algebra and Number Theory, vol. 4, no. 6, pp. 649–679,

A. Sebbar, “Classification of torsion-free genus zero congruence groups,” Proceedings of the American Mathematical

Society, vol. 129, no. 9, pp. 2517–2527, 2001.

Y.-H. He, J. McKay, and J. Read, “Modular subgroups, dessins d’enfants and elliptic K3 surfaces,” LMS Journal of

Computation and Mathematics, vol. 16, no. 1, pp. 271–318, 2013.

A. Atkin and J. Lehner, “Hecke operators on γ0(m),” Mathematische Annalen, vol. 185, no. 1, pp. 134–160, 1970.

N. R. Scheithauer, “Generalized Kac–Moody algebras, automorphic forms and Conway’s group I,” Advances in Mathematics, vol. 183, no. 2, pp. 240–270, 2004.

N. R. Scheithauer, “Generalized Kac–Moody algebras, automorphic forms and Conway’s group II,” Journal fur die reine ¨

und angewandte Mathematik, vol. 625, no. 1, pp. 125–154, 2008.

S. Bose, J. Gundry, and Y.-H. He, “Gauge theories and dessins d’enfants: beyond the torus,” Journal of High Energy

Physics, vol. 2015, p. 135, 2015.

G. W. Smith, “Replicant powers for higher genera,” Contemporary Mathematics, vol. 193, no. 1, pp. 337–352, 1996.

J. Thompson, “A finiteness theorem for subgroups of PSL(2,R) which are commensurable with PSL(2,Z),” in Proceedings

of the SYmposium on Pure Mathematics, vol. 37, (Providence, Rhode Island), pp. 553–555, Americal Mathematical

Society, Americal Mathematical Society, 1980.

C. Cummins, “Congruence subgroups of groups commensurable with PSL(2,Z) of genus 0 and 1,” Experimental

Mathematics, vol. 13, no. 3, pp. 361–382, 2004.

J. Jorgenson, L. Smajlovic, and H. Then, “Kronecker’s limit formula, holomorphic modular functions, and q-expansions

on certain arithmetic groups,” Experimental Mathematics, vol. 25, no. 3, pp. 295–319, 2016.

J. H. Conway, H. Coxeter, and G. Shephard, “The centre of a finitely generated group,” Tensor, vol. 25, pp. 405–418,

S. Smith, “On the head characters of the monster simple group,” in Contemporary Mathematics. Finite Groups–coming

of Age: Proceedings of the Canadian Mathematical Society Conference (J. McKay, ed.), pp. 303–313, American

Mathematical Society, 1982.

A. Ogg, “Modular functions,” in The Santa Cruz Conference on Finite Groups (B. Copperstein and G. Mason, eds.),

(Providence, Rhode Island), pp. 521–532, American Mathematical Society, American Mathematical Society, 1979.

J. F. Duncan and K. Ono, “The Jack Daniels problem,” Journal of Number Theory, vol. 161, pp. 230–239, 2016.

J. Brillhart and P. Morton, “Class numbers of quadratic fields, Hasse invariants of elliptic curves, and the supersingular

polynomial,” Journal of Number Theory, vol. 106, no. 1, pp. 79–111, 2004.

A. Pizer, “A note on the conjecture of Hecke,” Pacific Journal of Mathematics, vol. 79, no. 2, pp. 541–548, 1978.

C. Erdenberger, “The Kodaira dimension of certain moduli spaces of Abelian surfaces,” Mathematische Nachrichten,

vol. 274-275, no. 1, pp. 32–39, 2004.

Y.-H. He, C. Kirchoff-Lukat, J. McKay, and J. Read Unpublished Manuscript.

T. O. F. Inc., “The on-line encyclopedia of integer sequences.” Available at http://oeis.org.

J. McKay and R. Friedrich, “Novel approaches to the finite simple groups.” Available at

https://www.birs.ca/workshops/2012/12frg158/report12frg158.pdf, 2012.

A. Degeratu and K. Wendland, “Friendly giant meets pointlike instantons? On a new conjecture by John McKay,” in

Moonshine - The First Quarter Century and Beyond, A Workshop on the Moonshine Conjectures and Vertex Algebras,

vol. 372, pp. 55–127, LMS Lecture Notes Series, 2010.

G. Glauberman and S. Norton, “On McKay’s connection between the affine E8 diagram and the Monster,” in Proceedings

on Moonshine and related topics (J. McKay and A. Sebbar, eds.), pp. 37–42, American Mathematical Society, American

Mathematical Society, 2001.

C. H. Lam, H. Yamada, and H. Yamauchi, “Vertex operator algebras, extended E8 diagram, and McKay’s observation on

the Monster simple group.” Available at arXiv:0403010, 2004.

J. F. Duncan, “Arithmetic groups and the affine e8 dynkin diagram.” Available at arXiv:0810.1465, 2008.

G. Hoehn, C. H. Lam, and H. Yamauchi, “”McKay’s E7 observation on the baby monster” and ”McKay’s E6 observation

on the largest Fischer group”.” Available at arXiv:1002.1777, 2010.

Y.-H. He and J. McKay, “Moonshine and the meaning of life.” Available at arXiv:1408.2083, 2014.

K. Seng Chua and M. Lung Lang, “Congruence subgroups associated to the Monster,” Experimental Mathematics, vol. 13,

no. 3, pp. 343–360, 2012.

J. Conway, J. McKay, and A. Sebbar, “On the discrete groups of Moonshine,” Proceedings of the American Mathematical

Society, vol. 132, pp. 2233–2240, 2004.

B. Kostant, “The graph of the truncated icosahedron and the last letter of Galois,” Mitteilungen der Deutschen

Mathematiker-Vereinigung, vol. 3, no. 4, pp. 8–16, 1995.

J. McKay, “Graphs, singularities and finite groups,” in Proceedings of Symposia in Pure Mathematics (B. Cooperstein

and G. Mason, eds.), vol. 37, p. 183, Santa Cruz: American Mathematical Society, 1981.

J. McKay, “Cartan matrices, finite groups of quaternions, and Kleinian singularities,” Proceedings of the American

Mathematical Society, vol. 81, pp. 153–154, 1981.

J. Smith, “Some properties of the spectrum of a graph,” in Combinatorial Structures and their Applications: Proceedings

of the Calgary International Conference, (Calgary), pp. 403–406, 1969.

A. Hanany and Y.-H. He, “Non-Abelian finite gauge theories,” Journal of High Energy Physics, vol. 1999, no. 02,

pp. 013–013, 1999.

P. Du Val, “On isolated singularities of surfaces which do not affect the conditions of adjunction (part I.),” Mathematical

Proceedings of the Cambridge Philosophical Society, vol. 30, no. 4, p. 453–459, 1934.

P. Du Val, “On isolated singularities of surfaces which do not affect the conditions of adjunction (part II.),” Mathematical

Proceedings of the Cambridge Philosophical Society, vol. 30, no. 4, p. 460–465, 1934.

P. Du Val, “On isolated singularities of surfaces which do not affect the conditions of adjunction (part III.),” Mathematical

Proceedings of the Cambridge Philosophical Society, vol. 30, no. 4, p. 483–491, 1934.

V. I. Arnold, “Mysterious mathematical trinities,” in Surveys in Modern Mathematics (V. Prasolov and Y. Ilyashenko,

eds.), London Mathematical Society Lecture Note Series, p. 1–12, Cambridge University Press, 2005.

U. Meierfrankenfeld and S. Spectorov, “The maximal 2-local subgroups of the Monster and the BabyMonster, I & II.”

Available at https://users.math.msu.edu/users/meierfra, 2002.

N. Hitchin, “E6, E7, E8, Clay Academy Lecture.” Available at http://www.claymath.org/programs/outreach/academy/LectureNotes05/Hitchin.pdf.

R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics, New York, NY: Springer, 1977.

W. L. Edge, “Tritangent planes of Bring’s curve,” Journal of the London Mathematical Society, vol. s2-23, no. 2,

pp. 215–222, 1981.

M. Weber, “Kepler’s small stellated dodecahedron as a Riemann surface,” Pacific Journal of Mathematics, vol. 220, no. 1,

pp. 167–182, 2005.

I. Dolgachev, “Classical algebraic geometry: a modern view.” Available at

http://www.math.lsa.umich.edu/ idolga/CAG.pdf.

L. Manivel, “Configurations of lines and models of lie algebras,” Journal of Algebra, vol. 304, no. 1, pp. 457–486, 2006.

A. B. Coble, “An application of finite geometry to the characteristic theory of the odd and even theta functions,”

Transactions of the American Mathematical Society, vol. 14, pp. 241–276, 1913.

J. Conway, R. Curtis, S. Norton, R. Parker, and R. Wilson, The Atlas of finite groups. Oxford, UK: Oxford University

Press, 2003.

T. G. Group, “Gap – groups, algorithms, and programming, version 4.7.6.” Available at http://www.gap-system.org, 2014.

C. H. Lam, “Dihedral groups and subalgebras of Moonshine VOA (finite groups, vertex operator algebras and combinatorics).” Available at http://hdl.handle.net/2433/140873.

D. Ford and J. McKay, “Monstrous moonshine – two footnotes,” in Third Spring Conference on Modular Forms and

Related Topics, (Hamamatsu, Japan), pp. 56–61, 2004.

W. L. Edge, “Fricke’s octavic curve,” Proceedings of the Edinburgh Mathematical Society, vol. 27, no. 1, p. 91–101,

L. Queen, “Modular functions arising from some finite groups,” Mathematics of Computation, vol. 37, p. 547–580, 1981.

G. Hohn, “Generalized moonshine for the baby monster.” Available at ¨ https://www.math.ksu.edu/ gerald/papers/baby8.ps.

R. Farinha Matias, On Modular Forms, Hecke Operators, Replication and Sporadic Groups. PhD thesis, Department of

Mathematics, Concordia University, 2014.

D. Ford, J. McKay, and S. Norton, “More on replicable functions,” Communications in Algebra, vol. 22, no. 13,

pp. 5175–5193, 1994.

D. Alexander, C. Cummins, J. McKay, and C. Simons, “Completely reliable functions,” in Groups, Combinatorics &

Geometry, vol. 165, pp. 87–98, LMS Lecture Notes Series, 1990.

K. Mahler, “On a class of non-linear functional equations connected with modular functions,” Journal of the Australian

Mathematical Society, vol. 22, no. 1, p. 65–118, 1976.

S. Norton, “Moonshine-type functions and the CRM correspondence,” in Groups and Symmetries: From Neolithic Scots

to John McKay, vol. 47, pp. 327–342, CRM Proceedings Lecture Notes, 2009.

M. Koike, “Modular forms and the automorphism group of Leech lattice,” Nagoya Mathematical Journal, vol. 112, no. 1,

p. 63–79, 1988.

T. Kondo, “The automorphism group of Leech lattice and elliptic modular functions,” Journal of the Mathematical

Society of Japan, vol. 37, no. 2, pp. 337–362, 1985.

J. F. R. Duncan and S. Mack-Crane, “The moonshine module for Conway’s group.” Available at arXiv:1409.3829, 2014.

D. Dummit, H. Kisilevsky, and J. McKay, “Multiplicative eta-products,” Contemporary Mathematics, vol. 45, pp. 89–98,

Y. Martin, “Multiplicative eta-quotients,” Transactions of the American Mathematical Society, vol. 348, no. 12, pp. 4825–

, 1996.

Y.-H. He and J. McKay, “Eta products, BPS states and K3 surfaces,” Journal of High Energy Physics, vol. 2014, no. 1,

G. Horrocks and D. Mumford, “A rank 2 vector bundle on P4 with 15,000 symmetries,” Topology, vol. 12, no. 1,

pp. 63–81, 1973.

K. Hulek, “The Horrocks—Mumford bundle,” in Vector Bundles in Algebraic Geometry (N. Hitchin, P. Newstead, and

W. Oxbury, eds.), pp. 139–178, Cambridge: Cambridge University Press, 1995.

A. Grothendieck, “Sur la classification des fibres holomorphes sur la sph ´ ere de Riemann,” ´ American Journal of Mathematics, 1957.

W.-T. Wu, “Sur les espaces fibres et les vari ´ et´ es feuillet ´ ees,” ´ Act. Sci. Ind., vol. 1183, 1952.

M. Atiyah and E. Rees, “Vector bundles on projective 3-space,” Inventiones mathematicae, vol. 35, no. 1, pp. 131–153,

M. Schneider and H. Grauert, “Komplexe unterraume und holomorphe vektorraumb ¨ undel vom rang zwei,” ¨ Mathematische

Annalen, vol. 230, pp. 75–90, 1977.

R. Hartshorne, “Varieties of small codimension in projective space,” Bulletin of the American Mathematical Society,

vol. 80, no. 6, pp. 1017–1032, 1974.

W. Decker and F.-O. Schreyer, “On the uniqueness of the Horrocks–Mumford-bundle,” Mathematische Annalen, vol. 273,

no. 3, pp. 415–443, 1986.

D. Grayson and M. Stillman, “Macaulay2, a software system for research in algebraic geometry.” Available at


V. Braun, “Three generations on the quintic quotient,” Journal of High Energy Physics, vol. 2010, no. 1, 2010.

L. B. Anderson, J. Gray, Y.-H. He, and A. Lukas, “Exploring positive monad bundles and a new heterotic standard model,”

Journal of High Energy Physics, vol. 2010, no. 2, 2010.

C. Schoen, “On the geometry of a special determinantal hypersurface associated to the Mumford–Horrocks vector bundle,”

Journal fur die reine und angewandte Mathematik ¨ , vol. 1986, no. 364, pp. 85–111, 1986.

R. P. Stanley, “Hilbert functions of graded algebras,” Advances in Mathematics, vol. 28, no. 1, pp. 57–83, 1978.

D. Forcella, A. Hanany, Y.-H. He, and A. Zaffaroni, “The master space of n = 1 gauge theories,” Journal of High Energy

Physics, vol. 2008, no. 08, pp. 012–012, 2008.

E. Lee, “A modular quintic Calabi–Yau threefold of level 55,” Canadian Journal of Mathematics, vol. 63, no. 3,

pp. 616–633, 2011.

W. Bosma, J. Cannon, and C. Playoust, “The Magma algebra system I: The user language,” Journal of Symbolic

Computation, vol. 24, no. 3, pp. 235–265, 1997.




How to Cite

He, Y.-H.; McKay, J. Sporadic and Exceptional: In Dedication to the Memory of My Dear Friend Professor John S. McKay. inSTEMM 2024, 1, 61-85.