Higher genus Siegel forms and multi-center black holes in N=4 supersymmetric string theory

We conjecture that the Fourier coefficients of a degree three Siegel form, $1\mathrm{/}\sqrt{{\chi }_{18}}$, count the degeneracy of three-center BPS bound states in type II string theory compactified on $K3\times T^2$. We provide evidence for our conjecture in the form of consistency with physical considerations of wall-crossing, holographic bounds, and the appearance of suitable counting functions (involving the inverse of the modular discriminant $\mathrm{\Delta }$ and the inverse of the Igusa cusp form ${\mathrm{\Phi }}_{10}$) in limits where the count degenerates to involve single-center or two-center objects.