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 24dimensional 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 6dimensional 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.

$\begingroup$ That analogy is too good not to be true. I guess at a certain radius, the bosons on that lattice has an N=2 supersymmetry. Then taking the elliptic genus w.r.t. the enhanced susy should basically kill the right movers and result in a holomorphic "CFT". Hopefully $L_3(2)$ and the extra special part $2^{1+6}$ survives this procedure. $\endgroup$– Yuji TachikawaCommented Nov 2, 2010 at 15:28

$\begingroup$ Yuji, I don't think that construction works. I'm actually quite skeptical now that any $c=6,N=2$ SCFT exists with $M_{24}$ symmetry and think there must be some deeper voodoo involved in explaining the connection between the K3 elliptic genus and $M_{24}$. $\endgroup$– Jeff HarveyCommented Nov 15, 2010 at 20:28
1 Answer
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.

$\begingroup$ Thanks for pointing me to Miyamoto's paper. The $A_1^{24}$ construction has central charge $c=24$, and as you mention, $L(1/2,0)^{\oplus 24}$ has $c=12$. I'm looking for a $c=6$ CFT, actually a $c=6$, $N=4$ SCFT with an action of $M_{24}$. I haven't been able to make the 12 free fermion construction work, but perhaps it will. $\endgroup$ Commented Nov 1, 2010 at 16:14

$\begingroup$ I'm just arriving at the Moonshine party. If my understanding of chiral CFT is correct (which it might not be), the obvious Z/2 action on 12 real fermions (switching the signs of all fermions) is anomalous, and so the orbifold theory is not defined. If you take the fixed subVOA of 12 fermions and ask for its braided category of modules, I think you get the center of SU(2)atlevel1, not the center of Rep(Z/2). $\endgroup$ Commented Nov 30, 2016 at 1:56

$\begingroup$ @TheoJohnsonFreyd Yes, that seems to be correct. A lot has happened in the last 6 years, but Jeff Harvey's comment about voodoo seems to be remarkably prescient. $\endgroup$– S. Carnahan ♦Commented Nov 30, 2016 at 21:18

$\begingroup$ @S.Carnahan Right. I am in the process of catching up  I don't know what is the cutting edge or anything. The same anomaly calculations (indeed, Duncan's paper about the Conway group) showed up in something else I was thinking about. But among other things I wanted to check my understanding in this case. $\endgroup$ Commented Dec 1, 2016 at 15:24