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

Abstract

AbstractThe 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 $$\Phi _{10}$$ Φ 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 $$1/\Phi _{10}$$ 1 / Φ 10 . The construction uses two distinct $$\textrm{SL}(2, {\mathbb {Z}})$$ SL ( 2 , Z ) subgroups of $$\textrm{Sp}(2, {\mathbb {Z}})$$ Sp ( 2 , 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 $$1/\eta ^{24}$$ 1 / η 24 by means of a continued fraction structure.