There are recent papers suggesting that the elliptic genus of K3 exhibits moonshine for the Mathieu group $M_{24}$ (http://arXiv.org/pdf/1004.0956). Does anyone know of constructions of $M_{24}$ analogous to the FLM construction of the monster as the automorphism group of a holomorphic $c=24$ CFT (aka VOA)? In particular, the monster has $2^{1+24}. \cdot O/Z_2$ as the centralizer of an involution and the Conway group acts as automorphisms of the 24-dimensional Leech lattice. $M_{24}$ has $2^{1+6}:L_3(2)$ as the centralizer of an involution and $L_3(2)$ (with an additional $Z_2$) is the automorphism group of a 6-dimensional lattice with 42 vectors of norm 4 (not unimodular obviously). String theory on K3 gives rise to a $c=6$ CFT (not holomorphic). There are obvious differences between the two situations, but enough parallels to make me suspect a connection, hence the question.
[Math] M24 moonshine for K3
finite-groupsk3-surfacesstring-theory
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I can give you half of the answer, but the other half is wide open. I will use the characterization of supersingular primes as those primes p for which the normalizer of Gamma0(p) in SL(2,R) acts on the complex upper half plane to yield a genus zero quotient.
The monstrous moonshine conjecture asserted the existence of an infinite dimensional graded representation of the monster satisfying some exceptional properties. It was conjectured by Conway and Norton, and proved by Borcherds, using the representation constructed by I. Frenkel, Lepowsky, and Meurman. One can take the graded dimension of this representation to get a power series, and it is the q-expansion of the J-function. Furthermore, the graded trace of any element of order n in the monster is the q-expansion of a genus zero modular function that is invariant under Gamma0(nh) for some h|(12,n). One can conclude somewhat abstractly that the normalizer of Gamma0(p) in SL(2,R) has to be genus zero for any prime p dividing the order of the monster.
The proof I've seen that no other primes satisfy the genus zero condition does not seem to have anything to do with the monster. Instead, it is a delicate construction by Mazur involving the Eisenstein ideal, combined with some computations by H. Lenstra. I may be ignorant of more refined arguments developed in the last 30 years, though. [Edit: FC has pointed out that the proof of the bijection is a reasonably straightforward calculation. Still, I haven't seen any good arguments explaining the universality of the monster with respect to the genus zero property.]
Thanks, YangMills, for the references to my papers. I want to elaborate, because I disagree with the statement that mirror symmetry is given by hyperkahler rotation. It may be the case for certain choices of K3, but I think this happens by accident and that it's not a useful principle. Here is how I view mirror symmetry for K3 surfaces. Choose a rank 2 sublattice of the K3 lattice generated by $E$ and $F$ with $E^2=F^2=0, E.F=1$. Consider a K3 surface $X$ with a holomorphic $2$-form with $E.\Omega\not=0$. We can assume after rescaling $\Omega$ that $E.\Omega=1$, and then write $\Omega=F+\check B+i\check\omega \mod E$ for some classes $\check B,\check\omega$ in $E^{\perp}/E$. The K3 surface will be equipped also with a Kaehler form $\omega$ and a B-field $B$, which we write as $B+i\omega$. We choose this data in $E^{\perp}/E \otimes {\mathbb C}$, although the class of $\omega$ is determined in $E^{\perp}$ by its image in $E^{\perp}/E$ by the fact that $\omega\wedge \Omega$ must be zero. Then the mirror $\check X$ is taken to have holomorphic form $\check\Omega=F+B+i\omega\mod E$ and complexified Kaehler class $\check B+i\check\omega$.
Note that there is no particular reason to expect this new K3 surface to be a hyperkaehler rotation, as the mirror complex structure depends on $B$, which gives far too many parameters worth of choices: there is only a two-dimensional family of hyperkaehler rotation of $X$.
Note that we can hyperkaehler rotate $X$ so that special Lagrangians become holomorphic. The new holomorphic form is $\check\omega + i \omega \mod E$. If we multiply this form by $i$, we get $-\omega +i\check\omega\mod E$. A change of Kaehler form followed by another hyperkaehler rotation will give the mirror for certain choices of $B$-field, but note this involves two hyperkaehler rotations with respect to different metrics.
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This is not an answer, but perhaps someone can build off it. I suppose you want something different from the $A_1^{24}$ lattice CFT construction mentioned in the paper that you cited.
I wouldn't be surprised if one could apply a technique along the lines of John Duncan's constructions of vertex superalgebras with actions of larger sporadic groups. For example, you might try to tensor 12 free fermions together to get a $c=6$ superalgebra, then take an orbifold by an involution (but I have no idea if that would work).
An alternative method of construction is by codes. You can get a $c=12$ VOA with an $M_{24}$ action using Golay code construction on $L(1/2,0)^{\otimes 24}$ (see e.g., Miyamoto's paper), but it sounds like this precise construction might not be what you want.