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.
-
$\begingroup$ 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 TachikawaNov 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 HarveyNov 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. $\endgroup$ 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 sub-VOA of 12 fermions and ask for its braided category of modules, I think you get the center of SU(2)-at-level-1, not the center of Rep(Z/2). $\endgroup$ Nov 30, 2016 at 1:56
-
$\begingroup$ @TheoJohnson-Freyd 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 ♦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. $\endgroup$ Dec 1, 2016 at 15:24