Mathieu Moonshine in the elliptic genus of K3

Abstract

It has recently been conjectured that the elliptic genus of K3 can be written in terms of dimensions of Mathieu group M24 representations. Some further evidence for this idea was subsequently found by studying the twining genera that are obtained from the elliptic genus upon replacing dimensions of Mathieu group representations by their characters. In this paper we find explicit formulae for all (remaining) twining genera by making an educated guess for their general modular properties. This allows us to identify the decomposition of all expansion coefficients in terms of dimensions of M24-representations. For the first 500 coefficients we verify that the multiplicities with which these representations appear are indeed all non-negative integers. This represents very compelling evidence in favour of the conjecture.Comment: 23 pages; an error corrected, references update

Calabi-Yau manifolds and sporadic groups

“…Let us look, for example, at the quintic, i.e. the CY 3-fold given by a hypersurface in CP 4 1,1,1,1,1 . Its elliptic genus is given by…”

“…In 2010 Eguchi, Ooguri and Tachikawa (EOT) [1] discovered a new moonshine phenomenon [2][3][4][5][6] which connects the elliptic genus of K3 to the largest Mathieu group M 24 . 1 Since the K3 surface plays a central role in mathematics and physics, this new observation 1 Relatedly, M24 has at the same time been connected to the counting functions of half BPS states in four dimensional N = 4 theories [2,7,8].…”

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

“…Recall that the decoupling of the eight Majorana fermions into two sets of four implies that our Z 2 -action breaks the so(8) 1 -symmetry of the underlying toroidal theory to su(2) 4 1 . Indeed, by (2.25) a basis for the Z 2 -invariant (1, 0)-fields is given by 27) where the : ψ j+4 (z)ψ k+4 (z): with j, k ∈ {1, .…”

“…2 Z e,g (τ, z) + Z e,g 3 (τ, z) + Z g 2 ,g (τ, z) + Z g 2 ,g 3 (τ, z) · −ϑ 1 (τ, z) 2 ϑ 3 (τ, z) 2 + ϑ 3 (τ, z) 2 ϑ 1 (τ, z) 2 + ϑ 4 (τ, z) 2 ϑ 2 (τ, z) 2 − ϑ 2 (τ, z) 2 ϑ 4 (τ, z) 2 |η(τ )| 4 .…”

“…In particular, the twining genera, i.e. the elliptic genera with the insertion of a group element g ∈ M 24 , have been determined combining different viewpoints [2][3][4][5].…”