M24 moonshine for K3 - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T01:58:59Z http://mathoverflow.net/feeds/question/44439 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/44439/m24-moonshine-for-k3 M24 moonshine for K3 Jeff Harvey 2010-11-01T13:17:26Z 2010-11-02T14:00:48Z <p>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.</p> http://mathoverflow.net/questions/44439/m24-moonshine-for-k3/44450#44450 Answer by S. Carnahan for M24 moonshine for K3 S. Carnahan 2010-11-01T15:19:01Z 2010-11-01T15:19:01Z <p>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. </p> <p>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).</p> <p>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., <a href="http://arxiv.org/abs/hep-th/9612032" rel="nofollow">Miyamoto's paper</a>), but it sounds like this precise construction might not be what you want.</p>