K3 and Moonshining

The moonshining industry links classification results of simple groups and coefficients in expansions such as the ones that you get as partition functions in some field theories.

http://arxiv.org/PS_cache/arxiv/pdf/1004/1004.0956v2.pdf

For Kummer K3 surfaces, Katrin Wendland has constructed a bijection of lattices that provides a way for the Matheiu group to act. On one side there is the lattice of integral homology lattice and on the other there is the Niemeier lattice.

http://de.arxiv.org/PS_cache/arxiv/pdf/1107/1107.3834v1.pdf

The symmetry group acting on these lattices is a subgroup of the Mathieu with order 40320. This gets a little bit closer to getting the entire Mathieu group acting on this nonlinear sigma model with target space Kummer K3.