Mathieu Moonshine in the elliptic genus of K3

By M. Gaberdiel, S. Hohenegger, R. Volpato. 2010

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It has recently been conjectured that the elliptic genus of K3 can be written in terms of dimensions of Mathieu group $$ {\mathbb{M}_{24}} $$ 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 $$ {\mathbb{M}_{24}} $$-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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bf9f8ebac80e722cf9c7aa7b8b792aba858c075f

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Citations in the corpus

Symmetries of K3 sigma models (2011, background): “18) where φ0,1 and φ−2,1 are the standard Jacobi forms of SL(2,Z) of index one and weight 0 and −2, respectively, and {ψ(3), ψ((9)), E((9)) 2 } form a basis of weight 2 modular forms of Γ0(9) (for more information and explicit definitions see [5]).”
Proof of the umbral moonshine conjecture (2015, background): “ed mock modular forms. (See [45,54,55] for introductory accounts of the theory of mock modular forms.) Following the work of Cheng [10], Eguchi and Hikami [27], and Gaberdiel, Hohenegger, and Volpato [33,34], Gannon established the existence of this inﬁnite-dimensional graded M24-module in [37]. It ”
Umbral Moonshine (2012, background): “This idea has been implemented successfully in [19, 20, 21, 22] and provides strong”