Rademacher expansion of a Siegel modular form for ${\cal N}= 4$ counting
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| Format: | Preprint |
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2021
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| _version_ | 1866909059206086656 |
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| author | Cardoso, Gabriel Lopes Nampuri, Suresh Rosselló, Martí |
| author_facet | Cardoso, Gabriel Lopes Nampuri, Suresh Rosselló, Martí |
| contents | 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 $Φ_{10}$ 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 $1/Φ_{10}$. The construction uses two distinct ${\rm SL}(2, \mathbb{Z})$ subgroups of ${\rm Sp}(2, \mathbb{Z})$ 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 $1/η^{24}$ by means of a continued fraction structure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_10023 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Rademacher expansion of a Siegel modular form for ${\cal N}= 4$ counting Cardoso, Gabriel Lopes Nampuri, Suresh Rosselló, Martí High Energy Physics - Theory Mathematical Physics Number Theory 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 $Φ_{10}$ 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 $1/Φ_{10}$. The construction uses two distinct ${\rm SL}(2, \mathbb{Z})$ subgroups of ${\rm Sp}(2, \mathbb{Z})$ 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 $1/η^{24}$ by means of a continued fraction structure. |
| title | Rademacher expansion of a Siegel modular form for ${\cal N}= 4$ counting |
| topic | High Energy Physics - Theory Mathematical Physics Number Theory |
| url | https://arxiv.org/abs/2112.10023 |