Rademacher expansion of modular integrals
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911223970267136 |
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| author | Baccianti, Marco Maria Chandra, Jeevan Eberhardt, Lorenz Hartman, Thomas Mizera, Sebastian |
| author_facet | Baccianti, Marco Maria Chandra, Jeevan Eberhardt, Lorenz Hartman, Thomas Mizera, Sebastian |
| contents | We develop a method to evaluate integrals of non-holomorphic modular functions over the fundamental domain of the torus with modular parameter $τ$ analytically. It proceeds in two steps: first the integral is transformed to a Lorentzian contour by the same strategy that leads to the Lorentzian inversion formula in CFT, and then we apply a two-dimensional version of the Rademacher expansion. This computes the integral in terms of an expansion sensitive to the singular behaviour of the integrand near all the Lorentzian cusps $τ\to i \infty$, $\barτ \to x \in \mathbb{Q}$. We apply this technique to a variety of examples such as the evaluation of string one-loop partition functions, where it leads to the first analytic formula for the cosmological constants of the bosonic string and the $\mathrm{SO}(16) \times \mathrm{SO}(16)$ string. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13827 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rademacher expansion of modular integrals Baccianti, Marco Maria Chandra, Jeevan Eberhardt, Lorenz Hartman, Thomas Mizera, Sebastian High Energy Physics - Theory Mathematical Physics We develop a method to evaluate integrals of non-holomorphic modular functions over the fundamental domain of the torus with modular parameter $τ$ analytically. It proceeds in two steps: first the integral is transformed to a Lorentzian contour by the same strategy that leads to the Lorentzian inversion formula in CFT, and then we apply a two-dimensional version of the Rademacher expansion. This computes the integral in terms of an expansion sensitive to the singular behaviour of the integrand near all the Lorentzian cusps $τ\to i \infty$, $\barτ \to x \in \mathbb{Q}$. We apply this technique to a variety of examples such as the evaluation of string one-loop partition functions, where it leads to the first analytic formula for the cosmological constants of the bosonic string and the $\mathrm{SO}(16) \times \mathrm{SO}(16)$ string. |
| title | Rademacher expansion of modular integrals |
| topic | High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2501.13827 |