Mathieu Moonshine and Orbifold K3s

Abstract

The current status of `Mathieu Moonshine', the idea that the Mathieu group M24 organises the elliptic genus of K3, is reviewed. While there is a consistent decomposition of all Fourier coefficients of the elliptic genus in terms of Mathieu M24 representations, a conceptual understanding of this phenomenon in terms of K3 sigma-models is still missing. In particular, it follows from the recent classification of the automorphism groups of arbitrary K3 sigma-models that (i) there is no single K3 sigma-model that has M24 as an automorphism group; and (ii) there exist `exceptional' K3 sigma-models whose automorphism group is not even a subgroup of M24. Here we show that all cyclic torus orbifolds are exceptional in this sense, and that almost all of the exceptional cases are realised as cyclic torus orbifolds. We also provide an explicit construction of a Z5 torus orbifold that realises one exceptional class of K3 sigma-models.

29 pages. Partially based on talk given by M.R.G. at the conference "Conformal Field Theory, Automorphic Forms and Related Topics", Heidelberg, 19-23 September 2011

References in corpus (11)

• Notes on the K3 Surface and the Mathieu group M_24
• Note on Twisted Elliptic Genus of K3 Surface
• K3 Surfaces, N=4 Dyons, and the Mathieu Group M24
• On the Relationship between the Uniqueness of the Moonshine Module and Monstrous Moonshine
• BKM Lie superalgebras from counting twisted CHL dyons
• The Largest Mathieu Group and (Mock) Automorphic Forms
• Unravelling Mathieu Moonshine
• BKM Lie superalgebras from counting twisted CHL dyons -- II
• Brewing moonshine for Mathieu
• BPS Saturated String Amplitudes: K3 Elliptic Genus and Igusa Cusp Form
• Symmetries of K3 sigma models

Cited by in corpus (8)

• Fricke S-duality in CHL models
• A K3 sigma model with Z_2^8:M_{20} symmetry
• On symmetries of N=(4,4) sigma models on T^4
• Umbral Moonshine and the Niemeier Lattices
• Generalised Mathieu Moonshine
• K3 String Theory, Lattices and Moonshine
• Eta Products, BPS States and K3 Surfaces
• Conway Subgroup Symmetric Compactifications Redux