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arXiv · 2501.13827

Rademacher expansion of modular integrals

Marco Maria Baccianti·Jeevan Chandra·Lorenz Eberhardt·Thomas Hartman·Sebastian Mizera
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Abstract

We develop a method to evaluate integrals of non-holomorphic modular functions over the fundamental domain of the torus with modular parameter $\tau$ 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 $\tau \to i \infty$, $\bar{\tau} \to x \in \mathbb{Q}$. 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.

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Marco Maria Baccianti, Jeevan Chandra, Lorenz Eberhardt, Thomas Hartman, Sebastian Mizera. 2025-01-23. Rademacher expansion of modular integrals. https://doi.org/10.21468/scipostphys.19.4.103

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