Unravelling Mathieu Moonshine

TL;DR

This paper demonstrates how the Mathieu group M_{24} acts on a module related to the D1-D5-KK-p system, linking it to Siegel modular forms and establishing a new moonshine correspondence.

Contribution

It uncovers the action of M_{24} on a dyonic charge module and derives Borcherds product formulas, extending the Mathieu moonshine to Siegel modular forms.

Findings

M_{24} acts on the dyonic charge module

Derived Borcherds product formulas for modular forms

Established a new Mathieu moonshine correspondence

Abstract

The D1-D5-KK-p system naturally provides an infinite dimensional module graded by the dyonic charges whose dimensions are counted by the Igusa cusp form, Phi_{10}(Z)$. We show that the Mathieu group, M_{24}, acts on this module by recovering the Siegel modular forms that count twisted dyons as a trace over this module. This is done by recovering Borcherds product formulae for these modular forms using the M_{24} action. This establishes the correspondence (`moonshine') proposed in arXiv:0907.1410 that relates conjugacy classes of M_{24} to Siegel modular forms. This also, in a sense that we make precise, subsumes existing moonshines for M_{24} that relates its conjugacy classes to eta-products and Jacobi forms.