Abstract

We review the role of modular forms in counting the microscopic degeneracies  of 1/4 BPS black holes in a class of  N=4 string theories known as CHL models.  We focus on the following 3 key aspects.  For a given CHL model,  the 1/4 BPS index of these states  are Fourier coefficients of an inverse  Siegel modular form.  These Siegel modular forms are associated with the elliptic genera of K3 twisted by a Z_N automorphism, each labelled by the conjugacy classes of the Mathieu group M_24.  We discuss the expansion of the inverse Siegel modular form in terms  of Mock-modular forms to obtain the  index of single centred black holes, and how spherical symmetry of the black hole horizon dictates the sign of this index.  Finally we  obtain  the modular transform of the Siegel  form which captures the large charge expansion of the index and its agreement with the Hawking-Bekenstein-Wald entropy.