A twist in the M 24 moonshine story

Abstract

Prompted by the Mathieu Moonshine observation, we identify a pair of 45-dimensional vector spaces of states that account for the first order term in the massive sector of the elliptic genus of K3 in every Z 2 -orbifold CFT on K3. These generic states are uniquely characterized by the fact that the action of every geometric symmetry group of a Z 2orbifold CFT yields a well-defined faithful representation on them. Moreover, each such representation is obtained by restriction of the 45-dimensional irreducible representation of the Mathieu group M 24 constructed by Margolin. Thus we provide a piece of evidence for Mathieu Moonshine explicitly from SCFTs on K3.The 45-dimensional irreducible representation of M 24 exhibits a twist, which we prove can be undone in the case of Z 2 -orbifold CFTs on K3 for all geometric symmetry groups. This twist however cannot be undone for the combined symmetry group (Z 2 ) 4 A 8 that emerges from surfing the moduli space of Kummer K3s. We conjecture that in general, the untwisted representations are exclusively those of geometric symmetry groups in some geometric interpretation of a CFT on K3. In that light, the twist appears as a representation theoretic manifestation of the maximality constraints in Mukai's classification of geometric symmetry groups of K3.

Calabi-Yau manifolds and sporadic groups

“…Perhaps they do not even have such a symmetry at any point in their moduli space at all. However, the elliptic genus could see some combined symmetry group that arises, for example, from different points in moduli space via symmetry surfing, again in analogy with K3 [14][15][16]. On the other hand it is not impossible that some CY 5-folds have a genuine M 24 symmetry at some point in the moduli space and that this could then explain the corresponding…”

“…These questions are not easily answered but it has been shown that twined elliptic genera for K3 manifolds give all the McKay-Thompson series of Mathieu moonshine plus several extra twined elliptic genera that one would not have expected based on Mathieu moonshine alone [18]. At the same time the idea of symmetry surfing has been pursued, in which one tries to find an explicit M 24 symmetry by combining the symmetry groups at different points in K3 moduli space [13][14][15][16]. In this paper we followed a different path and studied the elliptic genus for other Calabi-Yau manifolds with an eye for potential connections to sporadic groups.…”

“…This has led to the intriguing idea that in order to get M 24 one could combine symmetries at different points in K3 moduli space. Using this so called 'symmetry surfing' it has already been possible to substantially enlarge the symmetry group but it has not yet been possible to find the full M 24 group [13][14][15][16].…”

A K3 sigma model with $ \mathbb{Z}_2^8 $ : $ {{\mathbb{M}}_{20 }} $ symmetry

“…In [23] 45 that are generic to all standard Z 2 -orbifold CFTs on K3 and which govern the massive leading order of the elliptic genus of K3. On V CFT…”

“…Since the elliptic genus is constant along each connected component of the moduli space of N = (4, 4) superconformal field theories at central charge (c, c) = (6, 6), one may then hope that the symmetries at different points in moduli space may be put together. This led two of us to suggest that an 'overarching symmetry group' based on the classical geometric symmetries of K3 nonlinear sigma models could be defined in this manner [21]; indeed already under restriction to symmetries of Kummer K3 s one obtains the group Z 4 2 : A 8 , which is a maximal subgroup of M 24 not contained in M 23 [22,23].…”