Where could one find a short description oft: "two mathematical extensions of the symmetry - to moduli spaces of sheaves and to derived categories", found here? Happen there interesting
things like with the moduli spaces of pointed curves etc.?

Presumably this is an attempt to work around the fact that the sporadic group $M_{24}$ appears to act on some structure underlying the elliptic genera of K3 surfaces (see Eguchi-Ooguri-Tachikawa), but does not actually act on the surfaces themselves. Rather than explicitly constructing an object with $M_{24}$ symmetry, one may cheat and try to redefine the word "symmetry" (although I don't know a canonical choice of new definition).
– S. Carnahan♦Dec 13 '11 at 18:29

Thanks Scott, but for me that passage sounds as if it wold refer to some new general principle, hmm...
– Thomas RiepeDec 14 '11 at 8:41