14/03/09 06:24:19.10
英文だが
URLﾘﾝｸ(home.mathematik.uni-freiburg.de)
Mathieu moonshine
By classical results due to Nikulin, Mukai, Xiao and Kondo in the 1980's and 90's, the finite symplectic automorphism groups of K3 surfaces are always subgroups of the Mathieu group M24.
This is a simple sporadic group of order 244823040. However, also by results due to Mukai, each such automorphism group has at most 960 elements and thus is by orders of magnitude smaller than M24.
On the other hand, according to a recent observation by Eguchi, Ooguri and Tachikawa, the elliptic genus of K3 surfaces seems to contain a mysterious footprint of an action of the entire group M24:
If one decomposes the elliptic genus into irreducible characters of the N=4 superconformal algebra, which is natural in view of superconformal field theories (SCFTs) associated to K3,
then the coefficients of the so-called non-BPS characters coincide with the dimensions of representations of M24.
In joint work with Dr. Anne Taormina, first results of which are presented in
Anne Taormina, Katrin Wendland, The overarching finite symmetry group of Kummer surfaces in the Mathieu group M24; JHEP 1308:152 (2013); arXiv:1107.3834 [hep-th]
we develop techniques which eventually should overcome the above-mentioned "order of magnitude problem":
For Kummer surfaces which carry the Kahler class that is induced by their underlying complex torus, we find methods that improve the classical techniques due to Mukai and Kondo,
and we give a construction that allows us to combine the finite symplectic symmetry groups of several Kummer surfaces to a larger group.
Thereby, we generate the so-called overarching finite symmetry group of Kummer surfaces, a group of order 40320, thus already mitigating the "order of magnitude problem".
URLﾘﾝｸ(www.maths.dur.ac.uk)
Mathieu Moonshine