Abstract
Abstract There is a ‘Mathieu moonshine’ relating the elliptic genus of K3 to the sporadic group M 24. Here, we give evidence that this moonshine extends to part of the web of dualities connecting heterotic strings compactified on K3 × T 2 to type IIA strings compactified on Calabi-Yau threefolds. We demonstrate that dimensions of M 24 representations govern the new supersymmetric index of the heterotic compactifications, and appear in the Gromov-Witten invariants of the dual Calabi-Yau threefolds, which are elliptic fibrations over the Hirzebruch surfaces $ {{\mathbb{F}}_n} $ .
Highlights
In 2010, Eguchi, Ooguri and Tachikawa (EOT) observed similar mysterious relations between the sporadic group M24 and the elliptic genus of the (4,4) superconformal field theory with K3 target [6]
We demonstrate that dimensions of M24 representations govern the new supersymmetric index of the heterotic compactifications, and appear in the Gromov-Witten invariants of the dual Calabi-Yau threefolds, which are elliptic fibrations over the Hirzebruch surfaces Fn
We show here that in the heterotic theories, with either arbitrary choices of gauge bundles and no Wilson lines or with all instantons embedded in one E8 and only Wilson lines in the other E8, the one-loop prepotential universally exhibits a structure encoding degeneracies of M24 representations
Summary
A heterotic compactification on a manifold X involves, in an obligatory way, data beyond the sigma-model metric on X. The internal worldsheet theory corresponding to the K3 compactification is generically a conformal field theory with (0, 4) supersymmetry In this case, the first condition in (2.1) implies c2(V1) + c2(V2) = n1 + n2 = 24. The elliptic genus of such a sigma-model will match the results computed in [36], and the observation of [6] about the elliptic genus of such a model will continue to hold force This implies immediately that in a very different physical setting from type II compactification on K3 — heterotic standard embeddings enjoy half the supersymmetry and completely different space-time physics — one can expect M24 to emerge. Similar arguments about the universality of the new supersymmetric index can be found in [28, 29, 31, 33]
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