Unravelling Mathieu moonshine

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, Φ10(Z). We show that the Mathieu group, M24, 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 M24 action. This establishes the correspondence ('moonshine') proposed in arXiv:0907.1410 that relates conjugacy classes of M24 to Siegel modular forms. This also, in a sense that we make precise, subsumes existing moonshines for M24 that relates its conjugacy classes to eta-products and Jacobi forms.