Notes on the K3 Surface and the Mathieu Group M 24

We point out that the elliptic genus of the K3 surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group M 24. The reason remains a mystery.

[1]  Miranda C. N. Cheng.  K3 Surfaces, N=4 Dyons, and the Mathieu Group M24 , 2010, 1005.5415.

[2]  S. Mukai.  Finite groups of automorphisms of K3 surfaces and the Mathieur group , 1988 .

[3]  E. Witten.  Elliptic genera and quantum field theory , 1987 .

[4]  T. Eguchi,et al.  Character formulas for the N = 4 superconformal algebra , 1988 .

[5]  S. Hohenegger,et al.  Mathieu twining characters for K3 , 2010, 1006.0221.

[6]  Edward Witten,et al.  Three-Dimensional Gravity Revisited , 2007, 0706.3359.

[7]  J. G. Thompson,et al.  Some Numerology between the Fischer-Griess Monster and the Elliptic Modular Function , 1979 .

[8]  V. Kac.  Vertex algebras for beginners , 1997 .

[9]  Monstrous Moonshine: The First Twenty‐Five Years , 2004, math/0402345.

[10]  K. Ono.  THE f(q) MOCK THETA FUNCTION CONJECTURE AND PARTITION RANKS , 2005 .

[11]  C. Dong,et al.  An orbifold theory of genus zero associated to the sporadic groupM24 , 1994 .

[12]  T. Eguchi,et al.  Unitary representations of the N=4 superconformal algebra , 1987 .

[13]  Ozlem Umdu,et al.  Monstrous moonshine , 2019, 100 Years of Math Milestones.

[14]  J. Lepowsky,et al.  Vertex Operator Algebras and the Monster , 2011 .

[15]  S. Kondō.  Niemeier lattices, Mathieu groups, and finite groups of symplectic automorphisms of $K3$ surfaces , 1998 .

[16]  D. Zagier.  Ramanujan's mock theta functions and their applications (after Zwegers and Ono-Bringmann) , 2009 .

[17]  S. Zwegers,et al.  Mock Theta Functions , 2008 .

[18]  Superconformal algebras and mock theta functions 2: Rademacher expansion for K3 surface , 2009, 0904.0911.

[19]  J. Conway,et al.  ATLAS of Finite Groups , 1985 .

[20]  H. Ooguri,et al.  Superconformal Algebras and String Compactification on Manifolds with SU(N) Holonomy , 1989 .

[21]  T. Eguchi,et al.  $ \mathcal{N} = 4 $ superconformal algebra and the entropy of hyperKähler manifolds , 2009, 0909.0410.

[22]  J. Conway,et al.  Atlas of finite groups : maximal subgroups and ordinary characters for simple groups , 1987 .