Rademacher expansion of a Siegel modular form for $\mathcal{N}=4$ counting

Abstract

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 $\mathrm{SL}(2,\mathbb{Z})$ subgroups of $\mathrm{Sp}(2,\mathbb{Z})$ 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.

50 pages, 2 figures; v2: various improvements, published version

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Cited by in corpus (1)

• Rademacher expansion of modular integrals