TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.19.4.103
TI  - Rademacher expansion of modular integrals
PY  - 2025/10/20
UR  - https://www.scipost.org/SciPostPhys.19.4.103
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 19
IS  - 4
SP  - 103
A1  - Baccianti, Marco Maria
AU  - Chandra, Jeevan
AU  - Eberhardt, Lorenz
AU  - Hartman, Thomas
AU  - Mizera, Sebastian
AB  - 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 $\tau \to i ∞$, $\bar{\tau} \to x ∈ \mathbb{Q}$. 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 $\mathrm{SO}(16) × \mathrm{SO}(16)$ string.
ER  -

@Article{10.21468/SciPostPhys.19.4.103,
	title={{Rademacher expansion of modular integrals}},
	author={Marco Maria Baccianti and Jeevan Chandra and Lorenz Eberhardt and Thomas Hartman and Sebastian Mizera},
	journal={SciPost Phys.},
	volume={19},
	pages={103},
	year={2025},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.19.4.103},
	url={https://scipost.org/10.21468/SciPostPhys.19.4.103},
}