Rademacher Expansion of a Siegel Modular Form for \({{\mathcal {N}}}= 4\) Counting

The degeneracies of 1/4 BPS states with unit torsion in heterotic string theory compactified on a six torus are given in terms of the Fourier coefficients of the reciprocal of the Igusa cusp Siegel modular form \(\Phi _{10}\) of weight 10. We use the symplectic symmetries of the latter to construct a fine-grained Rademacher-type expansion which expresses these BPS degeneracies as a regularized sum over residues of the poles of \(1/\Phi _{10}\). The construction uses two distinct \(\textrm{SL}(2, {\mathbb {Z}})\) subgroups of \(\textrm{Sp}(2, {\mathbb {Z}})\) which encode multiplier systems, Kloosterman sums and Eichler integrals appearing therein. Additionally, it shows how the polar data are explicitly built from the Fourier coefficients of \(1/\eta ^{24}\) by means of a continued fraction structure.